Embedded newform 403.2.ba.a.6.15

• Label: 403.2.ba.a.6.15
• Level: $403$
• Weight: $2$
• Character: 403.6
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(35\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			6.15
Character	\(\chi\)	\(=\)	403.6
Dual form			403.2.ba.a.336.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.137083 - 0.511599i) q^{2} -1.84118i q^{3} +(1.48911 - 0.859737i) q^{4} +(0.868165 + 3.24004i) q^{5} +(-0.941949 + 0.252395i) q^{6} +(1.36093 - 0.364660i) q^{7} +(-1.39301 - 1.39301i) q^{8} -0.389962 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 12 q^{4} - 2 q^{5} + 6 q^{6} + 12 q^{7} - 10 q^{8} - 124 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.137083	−	0.511599i	−0.0969321	−	0.361755i	0.900373	−	0.435118i	\(-0.143293\pi\)
							−0.997305	+	0.0733630i	\(0.976627\pi\)
\(3\)		−	1.84118i		−	1.06301i	−0.847056	−	0.531504i	\(-0.821627\pi\)
							0.847056	−	0.531504i	\(-0.178373\pi\)
\(5\)	0.868165	+	3.24004i	0.388255	+	1.44899i	0.832971	+	0.553317i	\(0.186638\pi\)
							−0.444716	+	0.895672i	\(0.646695\pi\)
\(7\)	1.36093	−	0.364660i	0.514384	−	0.137829i	0.00771614	−	0.999970i	\(-0.497544\pi\)
							0.506667	+	0.862142i	\(0.330877\pi\)
\(11\)	−0.338638	−	0.0907378i	−0.102103	−	0.0273585i	0.207406	−	0.978255i	\(-0.433498\pi\)
							−0.309509	+	0.950897i	\(0.600165\pi\)
\(13\)	3.54795	−	0.641924i	0.984024	−	0.178038i
							\(17\)	−1.31987	+	2.28608i	−0.320116	+	0.554457i	−0.980512	−	0.196461i	\(-0.937055\pi\)
0.660396	+	0.750918i	\(0.270388\pi\)
\(19\)	0.538757	+	2.01067i	0.123599	+	0.461279i	0.999786	−	0.0206933i	\(-0.00658734\pi\)
							−0.876187	+	0.481972i	\(0.839921\pi\)
\(23\)	−1.65057	+	2.85887i	−0.344167	+	0.596115i	−0.985202	−	0.171397i	\(-0.945172\pi\)
							0.641035	+	0.767512i	\(0.278505\pi\)
\(29\)	−7.43967	−	4.29529i	−1.38151	−	0.797616i	−0.389173	−	0.921165i	\(-0.627239\pi\)
							−0.992339	+	0.123548i	\(0.960573\pi\)
\(31\)	−0.292881	−	5.56006i	−0.0526029	−	0.998616i
							\(37\)	0.981331	−	0.981331i	0.161330	−	0.161330i	−0.621826	−	0.783156i	\(-0.713609\pi\)
0.783156	+	0.621826i	\(0.213609\pi\)
\(41\)	−1.18672	−	0.317980i	−0.185334	−	0.0496601i	0.164958	−	0.986301i	\(-0.447251\pi\)
							−0.350292	+	0.936640i	\(0.613918\pi\)
\(43\)	−2.01356	+	3.48760i	−0.307066	+	0.531854i	−0.977719	−	0.209917i	\(-0.932680\pi\)
							0.670653	+	0.741771i	\(0.266014\pi\)
\(47\)	3.48537	+	3.48537i	0.508393	+	0.508393i	0.914033	−	0.405640i	\(-0.132951\pi\)
							−0.405640	+	0.914033i	\(0.632951\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.ba.a.6.15	✓	140
13.11	odd	12		403.2.bf.a.37.15	yes	140
31.26	odd	6		403.2.bf.a.305.15	yes	140
403.336	even	12	inner	403.2.ba.a.336.15	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.15	✓	140	1.1	even	1	trivial
403.2.ba.a.336.15	yes	140	403.336	even	12	inner
403.2.bf.a.37.15	yes	140	13.11	odd	12
403.2.bf.a.305.15	yes	140	31.26	odd	6