Embedded newform 403.2.g.a.87.14

• Label: 403.2.g.a.87.14
• Level: $403$
• Weight: $2$
• Character: 403.87
• 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.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(35\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			87.14
Character	\(\chi\)	\(=\)	403.87
Dual form			403.2.g.a.315.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.356919 + 0.618202i) q^{2} +1.24248 q^{3} +(0.745217 + 1.29075i) q^{4} +(-0.797665 + 1.38160i) q^{5} +(-0.443464 + 0.768103i) q^{6} +(0.782978 - 1.35616i) q^{7} -2.49161 q^{8} -1.45625 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 2 q^{2} - 8 q^{3} - 34 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{8} + 58 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.356919	+	0.618202i	−0.252380	+	0.437135i	−0.964181	−	0.265247i	\(-0.914547\pi\)
							0.711801	+	0.702382i	\(0.247880\pi\)
\(3\)	1.24248			0.717345			0.358672	−	0.933464i	\(-0.383230\pi\)
							0.358672	+	0.933464i	\(0.383230\pi\)
\(5\)	−0.797665	+	1.38160i	−0.356727	+	0.617869i	−0.987412	−	0.158170i	\(-0.949441\pi\)
							0.630685	+	0.776039i	\(0.282774\pi\)
\(7\)	0.782978	−	1.35616i	0.295938	−	0.512579i	−0.679265	−	0.733893i	\(-0.737701\pi\)
							0.975203	+	0.221314i	\(0.0710346\pi\)
\(11\)	1.82743	+	3.16520i	0.550991	+	0.954345i	0.998203	+	0.0599171i	\(0.0190836\pi\)
							−0.447212	+	0.894428i	\(0.647583\pi\)
\(13\)	1.70039	+	3.17941i	0.471604	+	0.881810i
							\(17\)	−2.50402	+	4.33709i	−0.607314	+	1.05190i	0.384367	+	0.923180i	\(0.374420\pi\)
−0.991681	+	0.128718i	\(0.958914\pi\)
\(19\)	3.83486	−	6.64216i	0.879776	−	1.52382i	0.0281899	−	0.999603i	\(-0.491026\pi\)
							0.851586	−	0.524214i	\(-0.175641\pi\)
\(23\)	1.58610	−	2.74721i	0.330725	−	0.572833i	−0.651929	−	0.758280i	\(-0.726040\pi\)
							0.982654	+	0.185447i	\(0.0593733\pi\)
\(29\)	−0.219667	+	0.380475i	−0.0407912	+	0.0706524i	−0.885700	−	0.464258i	\(-0.846321\pi\)
							0.844909	+	0.534910i	\(0.179654\pi\)
\(31\)	−5.20665	−	1.97251i	−0.935142	−	0.354273i
							\(37\)	5.44083			0.894468			0.447234	−	0.894417i	\(-0.352409\pi\)
0.447234	+	0.894417i	\(0.352409\pi\)
\(41\)	−0.121319	−	0.210131i	−0.0189468	−	0.0328169i	0.856397	−	0.516319i	\(-0.172698\pi\)
							−0.875343	+	0.483502i	\(0.839365\pi\)
\(43\)	2.02003	−	3.49880i	0.308052	−	0.533562i	−0.669884	−	0.742466i	\(-0.733656\pi\)
							0.977936	+	0.208904i	\(0.0669895\pi\)
\(47\)	11.9925			1.74929			0.874646	−	0.484762i	\(-0.161094\pi\)
							0.874646	+	0.484762i	\(0.161094\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.g.a.87.14	yes	70
13.3	even	3		403.2.e.a.211.14	yes	70
31.5	even	3		403.2.e.a.191.14	✓	70
403.315	even	3	inner	403.2.g.a.315.14	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.e.a.191.14	✓	70	31.5	even	3
403.2.e.a.211.14	yes	70	13.3	even	3
403.2.g.a.87.14	yes	70	1.1	even	1	trivial
403.2.g.a.315.14	yes	70	403.315	even	3	inner