Embedded newform 353.2.f.a.36.6

• Label: 353.2.f.a.36.6
• Level: $353$
• Weight: $2$
• Character: 353.36
• Analytic conductor: $2.819$
• Analytic rank: $0$
• Dimension: $232$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 353 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	353.f (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.81871919135\)
Analytic rank:	\(0\)
Dimension:	\(232\)
Relative dimension:	\(29\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			36.6
Character	\(\chi\)	\(=\)	353.36
Dual form			353.2.f.a.304.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31564 + 1.31564i) q^{2} +(-0.357461 + 0.0711034i) q^{3} -1.46182i q^{4} +(-1.61496 + 2.41696i) q^{5} +(0.376744 - 0.563837i) q^{6} +(2.12278 - 1.41840i) q^{7} +(-0.708048 - 0.708048i) q^{8} +(-2.64892 + 1.09722i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 232 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/353\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31564	+	1.31564i	−0.930299	+	0.930299i	−0.997724	−	0.0674256i	\(-0.978521\pi\)
							0.0674256	+	0.997724i	\(0.478521\pi\)
\(3\)	−0.357461	+	0.0711034i	−0.206380	+	0.0410516i	−0.297198	−	0.954816i	\(-0.596052\pi\)
							0.0908174	+	0.995868i	\(0.471052\pi\)
\(5\)	−1.61496	+	2.41696i	−0.722231	+	1.08090i	0.270753	+	0.962649i	\(0.412727\pi\)
							−0.992984	+	0.118246i	\(0.962273\pi\)
\(7\)	2.12278	−	1.41840i	0.802336	−	0.536104i	−0.0854598	−	0.996342i	\(-0.527236\pi\)
							0.887796	+	0.460238i	\(0.152236\pi\)
\(11\)	−1.25519	−	1.25519i	−0.378455	−	0.378455i	0.492090	−	0.870545i	\(-0.336233\pi\)
							−0.870545	+	0.492090i	\(0.836233\pi\)
\(13\)	−2.37341	+	0.472102i	−0.658267	+	0.130937i	−0.512911	−	0.858442i	\(-0.671433\pi\)
							−0.145356	+	0.989379i	\(0.546433\pi\)
\(17\)	−3.84274	+	3.84274i	−0.932001	+	0.932001i	−0.997831	−	0.0658299i	\(-0.979031\pi\)
							0.0658299	+	0.997831i	\(0.479031\pi\)
\(19\)	1.21579	−	2.93518i	0.278922	−	0.673377i	−0.720885	−	0.693055i	\(-0.756264\pi\)
							0.999806	+	0.0196785i	\(0.00626427\pi\)
\(23\)	2.52952	−	6.10680i	0.527442	−	1.27336i	−0.405752	−	0.913983i	\(-0.632990\pi\)
							0.933194	−	0.359373i	\(-0.117010\pi\)
\(29\)	−4.46808	−	4.46808i	−0.829701	−	0.829701i	0.157774	−	0.987475i	\(-0.449568\pi\)
							−0.987475	+	0.157774i	\(0.949568\pi\)
\(31\)	−1.49536	−	0.999165i	−0.268574	−	0.179455i	0.413990	−	0.910282i	\(-0.364135\pi\)
							−0.682564	+	0.730826i	\(0.739135\pi\)
\(37\)	−3.46298	−	5.18272i	−0.569311	−	0.852034i	0.429383	−	0.903122i	\(-0.358731\pi\)
							−0.998694	+	0.0510882i	\(0.983731\pi\)
\(41\)	1.28874	+	0.533812i	0.201267	+	0.0833675i	0.481040	−	0.876699i	\(-0.340259\pi\)
							−0.279773	+	0.960066i	\(0.590259\pi\)
\(43\)	1.32917	−	3.20891i	0.202697	−	0.489354i	−0.789542	−	0.613696i	\(-0.789682\pi\)
							0.992239	+	0.124342i	\(0.0396819\pi\)
\(47\)	0.279852	+	0.675622i	0.0408206	+	0.0985496i	0.942975	−	0.332864i	\(-0.108015\pi\)
							−0.902154	+	0.431414i	\(0.858015\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	353.2.f.a.36.6	✓	232
353.304	even	16	inner	353.2.f.a.304.6	yes	232

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
353.2.f.a.36.6	✓	232	1.1	even	1	trivial
353.2.f.a.304.6	yes	232	353.304	even	16	inner