Embedded newform 403.2.be.c.57.2

• Label: 403.2.be.c.57.2
• Level: $403$
• Weight: $2$
• Character: 403.57
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.be (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(34\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			57.2
Character	\(\chi\)	\(=\)	403.57
Dual form			403.2.be.c.99.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.87740 + 1.87740i) q^{2} +(-2.43895 - 1.40813i) q^{3} -5.04929i q^{4} +(1.13969 + 0.305379i) q^{5} +(7.22253 - 1.93527i) q^{6} +(0.691082 - 0.185175i) q^{7} +(5.72474 + 5.72474i) q^{8} +(2.46566 + 4.27065i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 4 q^{2} - 24 q^{3} + 2 q^{5} - 36 q^{6} - 2 q^{7} - 8 q^{8} + 60 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.87740	+	1.87740i	−1.32752	+	1.32752i	−0.420001	+	0.907524i	\(0.637970\pi\)
							−0.907524	+	0.420001i	\(0.862030\pi\)
\(3\)	−2.43895	−	1.40813i	−1.40813	−	0.812984i	−0.412922	−	0.910766i	\(-0.635492\pi\)
							−0.995208	+	0.0977822i	\(0.968825\pi\)
\(5\)	1.13969	+	0.305379i	0.509684	+	0.136569i	0.504491	−	0.863417i	\(-0.331680\pi\)
							0.00519327	+	0.999987i	\(0.498347\pi\)
\(7\)	0.691082	−	0.185175i	0.261204	−	0.0699895i	−0.125840	−	0.992051i	\(-0.540163\pi\)
							0.387045	+	0.922061i	\(0.373496\pi\)
\(11\)	−3.93156	−	1.05346i	−1.18541	−	0.317629i	−0.388339	−	0.921517i	\(-0.626951\pi\)
							−0.797070	+	0.603887i	\(0.793618\pi\)
\(13\)	3.59674	−	0.251857i	0.997557	−	0.0698525i
							\(17\)	−1.67301	+	2.89774i	−0.405764	+	0.702804i	−0.994410	−	0.105587i	\(-0.966328\pi\)
0.588646	+	0.808391i	\(0.299661\pi\)
\(19\)	−0.728131	−	2.71742i	−0.167045	−	0.623419i	−0.997771	−	0.0667376i	\(-0.978741\pi\)
							0.830726	−	0.556682i	\(-0.187926\pi\)
\(23\)	6.91932			1.44278			0.721389	−	0.692530i	\(-0.243504\pi\)
							0.721389	+	0.692530i	\(0.243504\pi\)
\(29\)			1.35049i			0.250779i	0.992108	+	0.125390i	\(0.0400181\pi\)
							−0.992108	+	0.125390i	\(0.959982\pi\)
\(31\)	−0.714773	−	5.52169i	−0.128377	−	0.991725i
							\(37\)	−0.932763	−	3.48112i	−0.153345	−	0.572293i	−0.999241	−	0.0389447i	\(-0.987600\pi\)
0.845896	−	0.533348i	\(-0.179066\pi\)
\(41\)	−9.36185	−	2.50850i	−1.46208	−	0.391762i	−0.561870	−	0.827226i	\(-0.689918\pi\)
							−0.900206	+	0.435464i	\(0.856584\pi\)
\(43\)	3.00163	−	5.19897i	0.457744	−	0.792836i	−0.541097	−	0.840960i	\(-0.681991\pi\)
							0.998841	+	0.0481241i	\(0.0153243\pi\)
\(47\)	−6.36690	−	6.36690i	−0.928708	−	0.928708i	0.0689142	−	0.997623i	\(-0.478047\pi\)
							−0.997623	+	0.0689142i	\(0.978047\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.be.c.57.2	✓	136
13.8	odd	4	inner	403.2.be.c.398.2	yes	136
31.6	odd	6	inner	403.2.be.c.161.2	yes	136
403.99	even	12	inner	403.2.be.c.99.2	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.be.c.57.2	✓	136	1.1	even	1	trivial
403.2.be.c.99.2	yes	136	403.99	even	12	inner
403.2.be.c.161.2	yes	136	31.6	odd	6	inner
403.2.be.c.398.2	yes	136	13.8	odd	4	inner