Embedded newform 403.2.be.c.57.6

• Label: 403.2.be.c.57.6
• 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			57.6
Character	\(\chi\)	\(=\)	403.57
Dual form			403.2.be.c.99.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.60653 + 1.60653i) q^{2} +(1.41629 + 0.817697i) q^{3} -3.16189i q^{4} +(-2.93216 - 0.785669i) q^{5} +(-3.58898 + 0.961663i) q^{6} +(4.32754 - 1.15956i) q^{7} +(1.86662 + 1.86662i) q^{8} +(-0.162744 - 0.281882i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 4 q^{2} - 24 q^{3} + 2 q^{5} - 36 q^{6} - 2 q^{7} - 8 q^{8} + 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.60653	+	1.60653i	−1.13599	+	1.13599i	−0.146828	+	0.989162i	\(0.546906\pi\)
							−0.989162	+	0.146828i	\(0.953094\pi\)
\(3\)	1.41629	+	0.817697i	0.817697	+	0.472097i	0.849622	−	0.527393i	\(-0.176830\pi\)
							−0.0319249	+	0.999490i	\(0.510164\pi\)
\(5\)	−2.93216	−	0.785669i	−1.31130	−	0.351362i	−0.465589	−	0.885001i	\(-0.654157\pi\)
							−0.845712	+	0.533639i	\(0.820824\pi\)
\(7\)	4.32754	−	1.15956i	1.63566	−	0.438273i	0.680109	−	0.733111i	\(-0.261932\pi\)
							0.955547	+	0.294838i	\(0.0952656\pi\)
\(11\)	−1.76461	−	0.472826i	−0.532051	−	0.142563i	−0.0172156	−	0.999852i	\(-0.505480\pi\)
							−0.514835	+	0.857289i	\(0.672147\pi\)
\(13\)	1.58381	−	3.23907i	0.439270	−	0.898355i
							\(17\)	1.63021	−	2.82360i	0.395383	−	0.684824i	−0.597767	−	0.801670i	\(-0.703945\pi\)
0.993150	+	0.116846i	\(0.0372784\pi\)
\(19\)	0.727017	+	2.71327i	0.166789	+	0.622466i	0.997805	+	0.0662174i	\(0.0210931\pi\)
							−0.831016	+	0.556249i	\(0.812240\pi\)
\(23\)	7.08450			1.47722			0.738611	−	0.674132i	\(-0.235482\pi\)
							0.738611	+	0.674132i	\(0.235482\pi\)
\(29\)		−	0.969113i		−	0.179960i	−0.995944	−	0.0899799i	\(-0.971320\pi\)
							0.995944	−	0.0899799i	\(-0.0286803\pi\)
\(31\)	−4.36556	−	3.45570i	−0.784078	−	0.620662i
							\(37\)	0.366804	+	1.36893i	0.0603022	+	0.225051i	0.989500	−	0.144531i	\(-0.0461675\pi\)
−0.929198	+	0.369582i	\(0.879501\pi\)
\(41\)	3.39542	+	0.909801i	0.530276	+	0.142087i	0.514016	−	0.857781i	\(-0.328157\pi\)
							0.0162602	+	0.999868i	\(0.494824\pi\)
\(43\)	−3.88533	+	6.72960i	−0.592508	+	1.02625i	0.401386	+	0.915909i	\(0.368529\pi\)
							−0.993893	+	0.110345i	\(0.964805\pi\)
\(47\)	0.257925	+	0.257925i	0.0376223	+	0.0376223i	0.725668	−	0.688045i	\(-0.241531\pi\)
							−0.688045	+	0.725668i	\(0.741531\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.be.c.57.6	✓	136
13.8	odd	4	inner	403.2.be.c.398.6	yes	136
31.6	odd	6	inner	403.2.be.c.161.6	yes	136
403.99	even	12	inner	403.2.be.c.99.6	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.be.c.57.6	✓	136	1.1	even	1	trivial
403.2.be.c.99.6	yes	136	403.99	even	12	inner
403.2.be.c.161.6	yes	136	31.6	odd	6	inner
403.2.be.c.398.6	yes	136	13.8	odd	4	inner