Embedded newform 354.2.g.b.11.9

• Label: 354.2.g.b.11.9
• Level: $354$
• Weight: $2$
• Character: 354.11
• Analytic conductor: $2.827$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	354.g (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.82670423155\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(10\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			11.9
Character	\(\chi\)	\(=\)	354.11
Dual form			354.2.g.b.161.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.976621 + 0.214970i) q^{2} +(1.46208 + 0.928615i) q^{3} +(0.907575 - 0.419889i) q^{4} +(2.23908 + 0.621678i) q^{5} +(-1.62752 - 0.592600i) q^{6} +(1.22747 - 1.16272i) q^{7} +(-0.796093 + 0.605174i) q^{8} +(1.27535 + 2.71542i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + 10 q^{2} + q^{3} - 10 q^{4} - q^{6} + 2 q^{7} + 10 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/354\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{25}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.976621	+	0.214970i	−0.690575	+	0.152007i
							\(3\)	1.46208	+	0.928615i	0.844132	+	0.536136i
\(5\)	2.23908	+	0.621678i	1.00135	+	0.278023i								0.729250	−	0.684247i	\(-0.239869\pi\)
							0.272097	+	0.962270i	\(0.412283\pi\)
\(7\)	1.22747	−	1.16272i	0.463941	−	0.439469i	−0.419679	−	0.907673i	\(-0.637857\pi\)
							0.883620	+	0.468204i	\(0.155099\pi\)
\(11\)	1.45348	+	1.71117i	0.438241	+	0.515937i	0.936298	−	0.351207i	\(-0.114229\pi\)
							−0.498057	+	0.867144i	\(0.665953\pi\)
\(13\)	−1.80572	−	5.35918i	−0.500816	−	1.48637i	−0.838390	−	0.545071i	\(-0.816503\pi\)
							0.337573	−	0.941299i	\(-0.390394\pi\)
\(17\)	−0.859268	+	0.907118i	−0.208403	+	0.220008i	−0.821815	−	0.569755i	\(-0.807038\pi\)
							0.613412	+	0.789763i	\(0.289797\pi\)
\(19\)	−0.793976	−	1.99273i	−0.182151	−	0.457164i	0.809340	−	0.587340i	\(-0.199825\pi\)
							−0.991491	+	0.130176i	\(0.958446\pi\)
\(23\)	−2.51643	+	0.273678i	−0.524712	+	0.0570659i	−0.366642	−	0.930362i	\(-0.619493\pi\)
							−0.158070	+	0.987428i	\(0.550527\pi\)
\(29\)	−0.733297	+	3.33140i	−0.136170	+	0.618626i	0.858259	+	0.513217i	\(0.171546\pi\)
							−0.994429	+	0.105409i	\(0.966385\pi\)
\(31\)	0.269115	+	0.107225i	0.0483344	+	0.0192582i	0.394180	−	0.919033i	\(-0.371029\pi\)
							−0.345846	+	0.938291i	\(0.612408\pi\)
\(37\)	−2.89714	+	3.81113i	−0.476287	+	0.626545i	−0.970007	−	0.243076i	\(-0.921844\pi\)
							0.493720	+	0.869621i	\(0.335637\pi\)
\(41\)	0.00968128	−	0.0890179i	0.00151196	−	0.0139023i	−0.993358	−	0.115068i	\(-0.963292\pi\)
							0.994870	+	0.101165i	\(0.0322571\pi\)
\(43\)	7.80229	+	6.62733i	1.18984	+	1.01066i	0.999582	+	0.0289235i	\(0.00920792\pi\)
							0.190257	+	0.981734i	\(0.439068\pi\)
\(47\)	−2.55701	−	9.20953i	−0.372979	−	1.34335i	−0.877346	−	0.479859i	\(-0.840688\pi\)
							0.504367	−	0.863489i	\(-0.331726\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.g.b.11.9	yes	280
3.2	odd	2		354.2.g.a.11.1	✓	280
59.43	odd	58		354.2.g.a.161.1	yes	280
177.161	even	58	inner	354.2.g.b.161.9	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.g.a.11.1	✓	280	3.2	odd	2
354.2.g.a.161.1	yes	280	59.43	odd	58
354.2.g.b.11.9	yes	280	1.1	even	1	trivial
354.2.g.b.161.9	yes	280	177.161	even	58	inner