Embedded newform 13.12.e.a.10.3

• Label: 13.12.e.a.10.3
• Level: $13$
• Weight: $12$
• Character: 13.10
• Analytic conductor: $9.988$
• Analytic rank: $0$
• Dimension: $22$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	13.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.98846134727\)
Analytic rank:	\(0\)
Dimension:	\(22\)
Relative dimension:	\(11\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			10.3
Character	\(\chi\)	\(=\)	13.10
Dual form			13.12.e.a.4.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-47.1587 + 27.2271i) q^{2} +(-219.942 - 380.951i) q^{3} +(458.627 - 794.365i) q^{4} -8334.97i q^{5} +(20744.3 + 11976.8i) q^{6} +(19275.1 + 11128.5i) q^{7} -61573.8i q^{8} +(-8175.47 + 14160.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 242 q^{3} + 8191 q^{4} + 45528 q^{6} + 128496 q^{7} - 322217 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−47.1587	+	27.2271i	−1.04207	+	0.601639i	−0.920419	−	0.390934i	\(-0.872152\pi\)
							−0.121651	+	0.992573i	\(0.538819\pi\)
\(3\)	−219.942	−	380.951i	−0.522566	−	0.905111i	−0.999655	−	0.0262562i	\(-0.991641\pi\)
							0.477089	−	0.878855i	\(-0.341692\pi\)
\(5\)		−	8334.97i		−	1.19280i	−0.802686	−	0.596402i	\(-0.796596\pi\)
							0.802686	−	0.596402i	\(-0.203404\pi\)
\(7\)	19275.1	+	11128.5i	0.433468	+	0.250263i	0.700823	−	0.713335i	\(-0.252816\pi\)
							−0.267355	+	0.963598i	\(0.586150\pi\)
\(11\)	−671148.	+	387487.i	−1.25649	+	0.725434i	−0.972390	−	0.233361i	\(-0.925027\pi\)
							−0.284098	+	0.958795i	\(0.591694\pi\)
\(13\)	−1.27745e6	−	400337.i	−0.954239	−	0.299046i
							\(17\)	−1.46291e6	+	2.53384e6i	−0.249890	+	0.432823i	−0.963495	−	0.267726i	\(-0.913728\pi\)
0.713605	+	0.700548i	\(0.247061\pi\)
\(19\)	1.12656e7	+	6.50422e6i	1.04379	+	0.602630i	0.920903	−	0.389791i	\(-0.127453\pi\)
							0.122882	+	0.992421i	\(0.460786\pi\)
\(23\)	5.03434e6	+	8.71972e6i	0.163094	+	0.282488i	0.935977	−	0.352061i	\(-0.114519\pi\)
							−0.772883	+	0.634549i	\(0.781186\pi\)
\(29\)	−1.60169e6	−	2.77421e6i	−0.0145007	−	0.0251160i	0.858684	−	0.512505i	\(-0.171283\pi\)
							−0.873185	+	0.487389i	\(0.837949\pi\)
\(31\)			2.03119e8i			1.27427i	0.770752	+	0.637135i	\(0.219881\pi\)
							−0.770752	+	0.637135i	\(0.780119\pi\)
\(37\)	−1.47224e8	+	8.50001e7i	−0.349036	+	0.201516i	−0.664261	−	0.747501i	\(-0.731254\pi\)
							0.315224	+	0.949017i	\(0.397920\pi\)
\(41\)	8.99423e7	−	5.19282e7i	0.121242	−	0.0699991i	−0.438153	−	0.898901i	\(-0.644367\pi\)
							0.559394	+	0.828902i	\(0.311034\pi\)
\(43\)	−6.68514e8	+	1.15790e9i	−0.693480	+	1.20114i	0.277211	+	0.960809i	\(0.410590\pi\)
							−0.970690	+	0.240333i	\(0.922743\pi\)
\(47\)		−	1.98484e9i		−	1.26237i	−0.775631	−	0.631187i	\(-0.782568\pi\)
							0.775631	−	0.631187i	\(-0.217432\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.12.e.a.10.3	yes	22
13.2	odd	12		169.12.a.g.1.5		22
13.4	even	6	inner	13.12.e.a.4.3	✓	22
13.11	odd	12		169.12.a.g.1.18		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.e.a.4.3	✓	22	13.4	even	6	inner
13.12.e.a.10.3	yes	22	1.1	even	1	trivial
169.12.a.g.1.5		22	13.2	odd	12
169.12.a.g.1.18		22	13.11	odd	12