Embedded newform 403.2.v.a.56.13

• Label: 403.2.v.a.56.13
• Level: $403$
• Weight: $2$
• Character: 403.56
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $70$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.v (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(35\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			56.13
Character	\(\chi\)	\(=\)	403.56
Dual form			403.2.v.a.36.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.918845 - 0.530496i) q^{2} -1.70695 q^{3} +(-0.437149 - 0.757164i) q^{4} +(-0.822265 - 0.474735i) q^{5} +(1.56843 + 0.905531i) q^{6} +(3.56148 + 2.05622i) q^{7} +3.04960i q^{8} -0.0863111 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 6 q^{2} + 4 q^{3} + 30 q^{4} + 6 q^{6} - 12 q^{7} + 58 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.918845	−	0.530496i	−0.649722	−	0.375117i	0.138628	−	0.990345i	\(-0.455731\pi\)
							−0.788350	+	0.615227i	\(0.789064\pi\)
\(3\)	−1.70695			−0.985510			−0.492755	−	0.870168i	\(-0.664010\pi\)
							−0.492755	+	0.870168i	\(0.664010\pi\)
\(5\)	−0.822265	−	0.474735i	−0.367728	−	0.212308i	0.304737	−	0.952436i	\(-0.401431\pi\)
							−0.672465	+	0.740129i	\(0.734765\pi\)
\(7\)	3.56148	+	2.05622i	1.34611	+	0.777179i	0.987697	−	0.156383i	\(-0.0499833\pi\)
							0.358417	+	0.933562i	\(0.383317\pi\)
\(11\)	1.60150	−	0.924628i	0.482871	−	0.278786i	−0.238741	−	0.971083i	\(-0.576735\pi\)
							0.721612	+	0.692297i	\(0.243401\pi\)
\(13\)	−2.49752	+	2.60046i	−0.692688	+	0.721238i
							\(17\)	2.85629	−	4.94724i	0.692752	−	1.19988i	−0.278181	−	0.960529i	\(-0.589732\pi\)
0.970933	−	0.239352i	\(-0.0769351\pi\)
\(19\)	3.03229	+	1.75069i	0.695655	+	0.401637i	0.805727	−	0.592287i	\(-0.201775\pi\)
							−0.110072	+	0.993924i	\(0.535108\pi\)
\(23\)	1.09762	−	1.90114i	0.228870	−	0.396415i	−0.728603	−	0.684936i	\(-0.759830\pi\)
							0.957474	+	0.288521i	\(0.0931636\pi\)
\(29\)	3.93714	−	6.81933i	0.731109	−	1.26632i	−0.225300	−	0.974289i	\(-0.572336\pi\)
							0.956409	−	0.292029i	\(-0.0943303\pi\)
\(31\)	−4.26446	−	3.57972i	−0.765919	−	0.642937i
							\(37\)			11.8622i			1.95013i	0.221916	+	0.975066i	\(0.428769\pi\)
−0.221916	+	0.975066i	\(0.571231\pi\)
\(41\)	7.53785	−	4.35198i	1.17722	−	0.679665i	0.221846	−	0.975082i	\(-0.428792\pi\)
							0.955369	+	0.295416i	\(0.0954583\pi\)
\(43\)	3.47334	−	6.01601i	0.529680	−	0.917432i	−0.469721	−	0.882815i	\(-0.655645\pi\)
							0.999401	−	0.0346173i	\(-0.0110212\pi\)
\(47\)			1.27967i			0.186659i	0.995635	+	0.0933294i	\(0.0297510\pi\)
							−0.995635	+	0.0933294i	\(0.970249\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.v.a.56.13	yes	70
13.10	even	6		403.2.s.a.335.13	yes	70
31.5	even	3		403.2.s.a.160.13	✓	70
403.36	even	6	inner	403.2.v.a.36.13	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.s.a.160.13	✓	70	31.5	even	3
403.2.s.a.335.13	yes	70	13.10	even	6
403.2.v.a.36.13	yes	70	403.36	even	6	inner
403.2.v.a.56.13	yes	70	1.1	even	1	trivial