<figure><img src="image_48.png" /><figcaption>Fig. B.32. Corner diagram of the MCMC fitting result of KIC 8619145.</figcaption></figure>
<figure><img src="image_48.png" /><figcaption>Fig. B.32. Corner diagram of the MCMC fitting result of KIC 8619145.</figcaption></figure>
<figure><img src="image_49.png" /><figcaption>**Fig. B.33.** Normal échelle diagram of KIC 8684542. See Fig. B.1 for the explanations of the symbols.</figcaption></figure>
<figure><img src="image_49.png" /><figcaption>**Fig. B.33.** Normal échelle diagram of KIC 8684542. See Fig. B.1 for the explanations of the symbols.</figcaption></figure>
<figure><img src="image_50.png" /><figcaption>**Fig. B.34.** Stretched échelle diagram of KIC 8684542. Symbols are explained in Fig. 1.</figcaption></figure>

<figure><img src="image_51.png" /><figcaption>**Fig. B.35.** Splitting asymmetries as a function of frequency of KIC 8684542. Symbols are explained in Fig. 4.</figcaption></figure>
<figure><img src="image_50.png" /><figcaption>**Fig. B.34.** Stretched échelle diagram of KIC 8684542. Symbols are explained in Fig. 1.</figcaption></figure>

<figure><img src="image_51.png" /><figcaption>**Fig. B.35.** Splitting asymmetries as a function of frequency of KIC 8684542. Symbols are explained in Fig. 4.</figcaption></figure>
<figure><img src="image_52.png" /><figcaption>Fig. B.36. Corner diagram of the MCMC fitting result of KIC 8684542.</figcaption></figure>
<figure><img src="image_52.png" /><figcaption>**Fig. B.36.** Corner diagram of the MCMC fitting result of KIC 8684542.</figcaption></figure>
<figure><img src="image_53.png" /><figcaption>**Fig. B.37.** Normal échelle diagram of KIC 9202471. See Fig. B.1 for the explanations of the symbols.</figcaption></figure>
<figure><img src="image_53.png" /><figcaption>**Fig. B.37.** Normal échelle diagram of KIC 9202471. See Fig. B.1 for the explanations of the symbols.</figcaption></figure>
<figure><img src="image_54.png" /><figcaption>**Fig. B.38.** Stretched échelle diagram of KIC 9202471. Symbols are explained in Fig. 1.</figcaption></figure>

<figure><img src="image_55.png" /><figcaption>**Fig. B.39.** Splitting asymmetries as a function of frequency of KIC 9202471. Symbols are explained in Fig. 4.</figcaption></figure>
<figure><img src="image_54.png" /><figcaption>Fig. B.38. Stretched échelle diagram of KIC 9202471. Symbols are explained in Fig. 1.</figcaption></figure>

<figure><img src="image_55.png" /><figcaption>Fig. B.39. Splitting asymmetries as a function of frequency of KIC 9202471. Symbols are explained in Fig. 4.</figcaption></figure>
<figure><img src="image_56.png" /><figcaption>Fig. B.40. Corner diagram of the MCMC fitting result of KIC 9202471.</figcaption></figure>
<figure><img src="image_56.png" /><figcaption>Fig. B.40. Corner diagram of the MCMC fitting result of KIC 9202471.</figcaption></figure>
<figure><img src="image_57.png" /><figcaption>**Fig. B.41.** Normal échelle diagram of KIC 9589420. See Fig. B.1 for the explanations of the symbols.</figcaption></figure>
<figure><img src="image_57.png" /><figcaption>**Fig. B.41.** Normal échelle diagram of KIC 9589420. See Fig. B.1 for the explanations of the symbols.</figcaption></figure>
<figure><img src="image_58.png" /><figcaption>**Fig. B.42.** Stretched échelle diagram of KIC 9589420. Symbols are explained in Fig. 1.</figcaption></figure>

<figure><img src="image_59.png" /><figcaption>**Fig. B.43.** Splitting asymmetries as a function of frequency of KIC 9589420. Symbols are explained in Fig. 4.</figcaption></figure>
<figure><img src="image_58.png" /><figcaption>Fig. B.42. Stretched échelle diagram of KIC 9589420. Symbols are explained in Fig. 1.</figcaption></figure>

<figure><img src="image_59.png" /><figcaption>Fig. B.43. Splitting asymmetries as a function of frequency of KIC 9589420. Symbols are explained in Fig. 4.</figcaption></figure>
<figure><img src="image_60.png" /><figcaption>Fig. B.44. Corner diagram of the MCMC fitting result of KIC 9589420.</figcaption></figure>
<figure><img src="image_60.png" /><figcaption>**Fig. B.44.** Corner diagram of the MCMC fitting result of KIC 9589420.</figcaption></figure>
<figure><img src="image_61.png" /><figcaption>**Fig. B.45.** Normal échelle diagram of KIC 10801792. See Fig. B.1 for the explanations of the symbols.</figcaption></figure>
<figure><img src="image_61.png" /><figcaption>**Fig. B.45.** Normal échelle diagram of KIC 10801792. See Fig. B.1 for the explanations of the symbols.</figcaption></figure>
<figure><img src="image_62.png" /><figcaption>**Fig. B.46.** Stretched échelle diagram of KIC 10801792. Symbols are explained in Fig. 1.</figcaption></figure>

<figure><img src="image_63.png" /><figcaption>**Fig. B.47.** Splitting asymmetries as a function of frequency of KIC 10801792. Symbols are explained in Fig. 4.</figcaption></figure>
<figure><img src="image_62.png" /><figcaption>**Fig. B.46.** Stretched échelle diagram of KIC 10801792. Symbols are explained in Fig. 1.</figcaption></figure>

<figure><img src="image_63.png" /><figcaption>**Fig. B.47.** Splitting asymmetries as a function of frequency of KIC 10801792. Symbols are explained in Fig. 4.</figcaption></figure>
<figure><img src="image_64.png" /><figcaption>Fig. B.48. Corner diagram of the MCMC fitting result of KIC 10801792.</figcaption></figure>
<figure><img src="image_64.png" /><figcaption>Fig. B.48. Corner diagram of the MCMC fitting result of KIC 10801792.</figcaption></figure>
<figure><img src="image_65.png" /><figcaption>**Fig. B.49.** Normal échelle diagram of KIC 11515377. See Fig. B.1 for the explanations of the symbols.</figcaption></figure>
<figure><img src="image_65.png" /><figcaption>**Fig. B.49.** Normal échelle diagram of KIC 11515377. See Fig. B.1 for the explanations of the symbols.</figcaption></figure>
<figure><img src="image_66.png" /><figcaption>**Fig. B.50.** Stretched échelle diagram of KIC 11515377. Symbols are explained in Fig. 1.</figcaption></figure>

<figure><img src="image_67.png" /><figcaption>**Fig. B.51.** Splitting asymmetries as a function of frequency of KIC 11515377. Symbols are explained in Fig. 4.</figcaption></figure>
<figure><img src="image_66.png" /><figcaption>**Fig. B.50.** Stretched échelle diagram of KIC 11515377. Symbols are explained in Fig. 1.</figcaption></figure>

<figure><img src="image_67.png" /><figcaption>**Fig. B.51.** Splitting asymmetries as a function of frequency of KIC 11515377. Symbols are explained in Fig. 4.</figcaption></figure>
<figure><img src="image_68.png" /><figcaption>Fig. B.52. Corner diagram of the MCMC fitting result of KIC 11515377.</figcaption></figure>
<figure><img src="image_68.png" /><figcaption>Fig. B.52. Corner diagram of the MCMC fitting result of KIC 11515377.</figcaption></figure>
# Appendix C: Other parameters

## C.1. $\Delta\nu$ vs $\Delta\Pi_1$

<figure><img src="image_69.png" /><figcaption>**Fig. C.1.** The correlation between $\Delta\nu$ and $\Delta\Pi_1$. The grey circles are given by Mosser et al. (2018).</figcaption></figure>

Figure C.1 displays the correlation between the measured period spacing $\Delta\Pi_1$ and the frequency separation $\Delta\nu$. The frequency separations are distributed between ~8 to ~18 µHz and the period spacings range from ~75 to ~90 s. All the stars are located at the well-defined $\Delta\Pi_1-\Delta\nu$ degenerate ridge, as shown by the grey dots by Mosser et al. (2018), which implies that these stars did not undergo any mass transfer or binary merger (e.g. Deheuvels et al. 2022; Rui & Fuller 2021). We find that our stars tend to locate close to the lower boundary of the $\Delta\Pi_1-\Delta\nu$ ridge, implying that they have higher masses (Deheuvels et al. 2022).

## C.2. $q$, $\epsilon_g$, and $f_{\text{shift}}$

We show the values of the coupling factor $q$ in Fig. C.2. The $q$ values of our stars are in the typical range of red giant stars, and we also reproduce the increasing trend between $q$ and $\Delta\Pi_1$ (e.g. Mosser et al. 2018).

The g-mode phases $\epsilon_g$ are displayed in Fig. C.3. We only applied a uniform prior for $\epsilon_g$, but we still find that most of the stars have the typical $\epsilon_g$ values of red giant stars, which is 0.28 ± 0.08 reported by both observations (Mosser et al. 2018) and theoretical predictions that do not include magnetic perturbations (Takata 2016). Therefore, the magnetism-induced perturbation does not affect the distribution of g-mode phase. KIC 8619145 has a large $\epsilon_g$ with large uncertainty, and we cannot provide strong constraint on $\epsilon_g$ of KIC 6936091 either.

The frequency correction of $l = 1$ pure p modes $f_{\text{shift}}$ in Eq. 4 is a new parameter that has not been studied in previous work. We plot $f_{\text{shift}}$ as function of $\Delta\nu$ in Fig. C.4. The best values of $f_{\text{shift}}$ lie between ~0.5 to ~0.75 µHz. It seems that there is a rapid drop when $\Delta\nu \le 10$ µHz, but this feature needs more stars to confirm.

<figure><img src="image_70.png" /><figcaption>**Fig. C.2.** The coupling factor $q$ vs $\Delta\Pi_1$. The grey circles are reported by Mosser et al. (2018).</figcaption></figure>

<figure><img src="image_71.png" /><figcaption>**Fig. C.3.** The g-mode phase $\epsilon_g$ as a function of $\Delta\Pi_1$.</figcaption></figure>

<figure><img src="image_72.png" /><figcaption>**Fig. C.4.** $f_{\text{shift}}$ as a function of $\Delta\nu$.</figcaption></figure>
## Appendix C: Other parameters

### C.1. $\Delta\nu$ vs $\Delta\Pi_1$

<figure><img src="image_69.png" /><figcaption>**Fig. C.1.** The correlation between $\Delta\nu$ and $\Delta\Pi_1$. The grey circles are given by Mosser et al. (2018).</figcaption></figure>

Figure C.1 displays the correlation between the measured period spacing $\Delta\Pi_1$ and the frequency separation $\Delta\nu$. The frequency separations are distributed between ~ 8 to ~ 18 µHz and the period spacings range from ~ 75 to ~ 90 s. All the stars are located at the well-defined $\Delta\Pi_1-\Delta\nu$ degenerate ridge, as shown by the grey dots by Mosser et al. (2018), which implies that these stars did not undergo any mass transfer or binary merger (e.g. Deheuvels et al. 2022; Rui & Fuller 2021). We find that our stars tend to locate close to the lower boundary of the $\Delta\Pi_1-\Delta\nu$ ridge, implying that they have higher masses (Deheuvels et al. 2022).

### C.2. $q$, $\epsilon_g$, and $f_{\text{shift}}$

We show the values of the coupling factor $q$ in Fig. C.2. The $q$ values of our stars are in the typical range of red giant stars, and we also reproduce the increasing trend between $q$ and $\Delta\Pi_1$ (e.g. Mosser et al. 2018).

The g-mode phases $\epsilon_g$ are displayed in Fig. C.3. We only applied a uniform prior for $\epsilon_g$, but we still find that most of the stars have the typical $\epsilon_g$ values of red giant stars, which is 0.28 ± 0.08 reported by both observations (Mosser et al. 2018) and theoretical predictions that do not include magnetic perturbations (Takata 2016). Therefore, the magnetism-induced perturbation does not affect the distribution of g-mode phase. KIC 8619145 has a large $\epsilon_g$ with large uncertainty, and we cannot provide strong constraint on $\epsilon_g$ of KIC 6936091 either.

The frequency correction of $l = 1$ pure p modes $f_{\text{shift}}$ in Eq. 4 is a new parameter that has not been studied in previous work. We plot $f_{\text{shift}}$ as function of $\Delta\nu$ in Fig. C.4. The best values of $f_{\text{shift}}$ lie between ~ 0.5 to ~ 0.75 µHz. It seems that there is a rapid drop when $\Delta\nu \le 10$ µHz, but this feature needs more stars to confirm.

<figure><img src="image_70.png" /><figcaption>**Fig. C.2.** The coupling factor $q$ vs $\Delta\Pi_1$. The grey circles are reported by Mosser et al. (2018).</figcaption></figure>

<figure><img src="image_71.png" /><figcaption>**Fig. C.3.** The g-mode phase $\epsilon_g$ as a function of $\Delta\Pi_1$.</figcaption></figure>

<figure><img src="image_72.png" /><figcaption>**Fig. C.4.** $f_{\text{shift}}$ as a function of $\Delta\nu$.</figcaption></figure>
## Appendix D: Linear relation between $I$ and $\Delta\Pi_1$

Using the best-fitting models, we calculated the core factors $I$ and derived the field strengths, listed in Table 5. The core factor $I$ is vital to calculate the field strength in Eq. 19, but it is time-consuming to run a stellar evolution code to derive it. Hence, we report an empirical relation between $I$ and the period spacing $\Delta\Pi_1$. As shown in Fig. D.1, there is a linear relation between $I$ and $\Delta\Pi_1$, which is characterised as

$$ \frac{I\left(\frac{m}{s^2kg}\right)}{10^{-23}} = (-0.0864\Delta\Pi_1 + 8.914) \pm 0.11. \quad (D.1) $$

<figure><img src="image_73.png" /><figcaption>Fig. D.1. The core factor $I$ as a function of $\Delta\Pi_1$, colour coded by the model-inferred masses. The grey dashed line shows the linear fit and the shaded area gives the $1\sigma$ range.</figcaption></figure>
## Appendix D: Linear relation between $I$ and $\Delta\Pi_1$

Using the best-fitting models, we calculated the core factors $I$ and derived the field strengths, listed in Table 5. The core factor $I$ is vital to calculate the field strength in Eq. 19, but it is time-consuming to run a stellar evolution code to derive it. Hence, we report an empirical relation between $I$ and the period spacing $\Delta\Pi_1$. As shown in Fig. D.1, there is a linear relation between $I$ and $\Delta\Pi_1$, which is characterised as

$$ \frac{I\left(\frac{m}{s^2kg}\right)}{10^{-23}} = (-0.0864\Delta\Pi_1 + 8.914) \pm 0.11. \quad (D.1) $$

<figure><img src="image_73.png" /><figcaption>Fig. D.1. The core factor $I$ as a function of $\Delta\Pi_1$, colour coded by the model-inferred masses. The grey dashed line shows the linear fit and the shaded area gives the $1\sigma$ range.</figcaption></figure>