Embedded newform 403.2.bg.a.123.4

• Label: 403.2.bg.a.123.4
• Level: $403$
• Weight: $2$
• Character: 403.123
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(36\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			123.4
Character	\(\chi\)	\(=\)	403.123
Dual form			403.2.bg.a.154.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.623506 - 2.32695i) q^{2} +(1.53051 + 0.883643i) q^{3} +(-3.29391 + 1.90174i) q^{4} +(2.78278 - 2.78278i) q^{5} +(1.10191 - 4.11240i) q^{6} +(0.171705 - 0.640810i) q^{7} +(3.07213 + 3.07213i) q^{8} +(0.0616504 + 0.106782i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 8 q^{2} - 12 q^{4} - 8 q^{5} - 16 q^{7} - 28 q^{8} + 64 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.623506	−	2.32695i	−0.440885	−	1.64541i	−0.726578	−	0.687084i	\(-0.758890\pi\)
							0.285693	−	0.958321i	\(-0.407776\pi\)
\(3\)	1.53051	+	0.883643i	0.883643	+	0.510172i	0.871858	−	0.489759i	\(-0.162915\pi\)
							0.0117853	+	0.999931i	\(0.496249\pi\)
\(5\)	2.78278	−	2.78278i	1.24449	−	1.24449i	0.286378	−	0.958117i	\(-0.407549\pi\)
							0.958117	−	0.286378i	\(-0.0924514\pi\)
\(7\)	0.171705	−	0.640810i	0.0648982	−	0.242203i	−0.925855	−	0.377879i	\(-0.876654\pi\)
							0.990753	+	0.135675i	\(0.0433203\pi\)
\(11\)	1.24557	+	4.64854i	0.375554	+	1.40159i	0.852533	+	0.522673i	\(0.175065\pi\)
							−0.476979	+	0.878915i	\(0.658268\pi\)
\(13\)	2.93317	−	2.09678i	0.813516	−	0.581543i
							\(17\)	−1.07392	−	1.86009i	−0.260464	−	0.451137i	0.705901	−	0.708310i	\(-0.250542\pi\)
−0.966365	+	0.257173i	\(0.917209\pi\)
\(19\)	1.85348	+	0.496639i	0.425218	+	0.113937i	0.465080	−	0.885268i	\(-0.346025\pi\)
							−0.0398628	+	0.999205i	\(0.512692\pi\)
\(23\)	−3.42429	+	5.93104i	−0.714013	+	1.23671i	0.249326	+	0.968420i	\(0.419791\pi\)
							−0.963339	+	0.268287i	\(0.913542\pi\)
\(29\)	−4.82190	−	2.78392i	−0.895404	−	0.516961i	−0.0196974	−	0.999806i	\(-0.506270\pi\)
							−0.875706	+	0.482845i	\(0.839604\pi\)
\(31\)	−5.32312	+	1.63229i	−0.956061	+	0.293168i
							\(37\)	−1.44464	−	5.39148i	−0.237498	−	0.886354i	−0.977007	−	0.213208i	\(-0.931609\pi\)
0.739509	−	0.673147i	\(-0.235058\pi\)
\(41\)	0.838659	+	3.12992i	0.130977	+	0.488811i	0.999982	−	0.00599124i	\(-0.00190708\pi\)
							−0.869005	+	0.494802i	\(0.835240\pi\)
\(43\)	4.83361	+	8.37206i	0.737119	+	1.27673i	0.953788	+	0.300482i	\(0.0971474\pi\)
							−0.216669	+	0.976245i	\(0.569519\pi\)
\(47\)	−5.51247	−	5.51247i	−0.804077	−	0.804077i	0.179653	−	0.983730i	\(-0.442502\pi\)
							−0.983730	+	0.179653i	\(0.942502\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bg.a.123.4	yes	144
13.11	odd	12	inner	403.2.bg.a.154.3	yes	144
31.30	odd	2	inner	403.2.bg.a.123.3	✓	144
403.154	even	12	inner	403.2.bg.a.154.4	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bg.a.123.3	✓	144	31.30	odd	2	inner
403.2.bg.a.123.4	yes	144	1.1	even	1	trivial
403.2.bg.a.154.3	yes	144	13.11	odd	12	inner
403.2.bg.a.154.4	yes	144	403.154	even	12	inner