Embedded newform 403.2.ch.a.110.19

• Label: 403.2.ch.a.110.19
• Level: $403$
• Weight: $2$
• Character: 403.110
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $560$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.ch (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(560\)
Relative dimension:	\(35\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			110.19
Character	\(\chi\)	\(=\)	403.110
Dual form			403.2.ch.a.11.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.124958 - 0.00654878i) q^{2} +(0.895096 - 0.290834i) q^{3} +(-1.97347 - 0.207420i) q^{4} +(0.216048 + 0.806301i) q^{5} +(-0.113754 + 0.0304803i) q^{6} +(1.43092 + 0.549278i) q^{7} +(0.492421 + 0.0779919i) q^{8} +(-1.71044 + 1.24271i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q - 12 q^{2} - 10 q^{3} - 18 q^{4} - 8 q^{5} - 36 q^{6} - 32 q^{7} + 50 q^{8} + 114 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{7}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.124958	−	0.00654878i	−0.0883588	−	0.00463069i	0.00810533	−	0.999967i	\(-0.497420\pi\)
							−0.0964641	+	0.995336i	\(0.530753\pi\)
\(3\)	0.895096	−	0.290834i	0.516784	−	0.167913i	−0.0390013	−	0.999239i	\(-0.512418\pi\)
							0.555785	+	0.831326i	\(0.312418\pi\)
\(5\)	0.216048	+	0.806301i	0.0966195	+	0.360589i	0.997260	−	0.0739761i	\(-0.0235688\pi\)
							−0.900641	+	0.434565i	\(0.856902\pi\)
\(7\)	1.43092	+	0.549278i	0.540837	+	0.207608i	0.613423	−	0.789754i	\(-0.289792\pi\)
							−0.0725866	+	0.997362i	\(0.523125\pi\)
\(11\)	3.78676	+	4.67626i	1.14175	+	1.40995i	0.897937	+	0.440125i	\(0.145066\pi\)
							0.243816	+	0.969822i	\(0.421601\pi\)
\(13\)	−3.02429	+	1.96307i	−0.838788	+	0.544458i
							\(17\)	−0.578672	−	5.50570i	−0.140349	−	1.33533i	−0.807262	−	0.590193i	\(-0.799052\pi\)
0.666913	−	0.745135i	\(-0.267615\pi\)
\(19\)	6.24086	+	0.327070i	1.43175	+	0.0750349i	0.752365	−	0.658746i	\(-0.228913\pi\)
							0.679386	+	0.733781i	\(0.262246\pi\)
\(23\)	0.640816	+	6.09696i	0.133619	+	1.27130i	0.831676	+	0.555261i	\(0.187382\pi\)
							−0.698056	+	0.716043i	\(0.745952\pi\)
\(29\)	−4.11645	+	3.70647i	−0.764405	+	0.688273i	−0.956054	−	0.293190i	\(-0.905283\pi\)
							0.191649	+	0.981463i	\(0.438616\pi\)
\(31\)	5.56558	−	0.155875i	0.999608	−	0.0279959i
							\(37\)	−5.54106	+	5.54106i	−0.910945	+	0.910945i	−0.996347	−	0.0854016i	\(-0.972783\pi\)
0.0854016	+	0.996347i	\(0.472783\pi\)
\(41\)	3.06921	−	4.72617i	0.479330	−	0.738104i	−0.513441	−	0.858125i	\(-0.671630\pi\)
							0.992771	+	0.120021i	\(0.0382962\pi\)
\(43\)	−4.18102	−	4.64349i	−0.637600	−	0.708126i	0.334579	−	0.942368i	\(-0.391406\pi\)
							−0.972178	+	0.234242i	\(0.924739\pi\)
\(47\)	−0.398223	+	0.781558i	−0.0580869	+	0.114002i	−0.918225	−	0.396060i	\(-0.870377\pi\)
							0.860138	+	0.510061i	\(0.170377\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.ch.a.110.19	yes	560
13.11	odd	12		403.2.cc.a.141.17	✓	560
31.11	odd	30		403.2.cc.a.383.17	yes	560
403.11	even	60	inner	403.2.ch.a.11.19	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.cc.a.141.17	✓	560	13.11	odd	12
403.2.cc.a.383.17	yes	560	31.11	odd	30
403.2.ch.a.11.19	yes	560	403.11	even	60	inner
403.2.ch.a.110.19	yes	560	1.1	even	1	trivial