Embedded newform 354.2.g.b.11.7

• Label: 354.2.g.b.11.7
• 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			11.7
Character	\(\chi\)	\(=\)	354.11
Dual form			354.2.g.b.161.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.976621 + 0.214970i) q^{2} +(0.745416 - 1.56344i) q^{3} +(0.907575 - 0.419889i) q^{4} +(-1.05087 - 0.291772i) q^{5} +(-0.391895 + 1.68713i) q^{6} +(1.62439 - 1.53870i) q^{7} +(-0.796093 + 0.605174i) q^{8} +(-1.88871 - 2.33083i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + 10 q^{2} + q^{3} - 10 q^{4} - q^{6} + 2 q^{7} + 10 q^{8} - 3 q^{9}+O(q^{10}) \)

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
							\(3\)	0.745416	−	1.56344i	0.430366	−	0.902654i
\(5\)	−1.05087	−	0.291772i	−0.469963	−	0.130485i								0.0244697	−	0.999701i	\(-0.492210\pi\)
							−0.494432	+	0.869216i	\(0.664624\pi\)
\(7\)	1.62439	−	1.53870i	0.613961	−	0.581575i	−0.316216	−	0.948687i	\(-0.602412\pi\)
							0.930176	+	0.367113i	\(0.119654\pi\)
\(11\)	−0.613585	−	0.722368i	−0.185003	−	0.217802i	0.661840	−	0.749645i	\(-0.269776\pi\)
							−0.846843	+	0.531843i	\(0.821500\pi\)
\(13\)	−0.373460	−	1.10839i	−0.103579	−	0.307412i	0.883315	−	0.468780i	\(-0.155306\pi\)
							−0.986894	+	0.161367i	\(0.948410\pi\)
\(17\)	1.44829	−	1.52894i	0.351261	−	0.370822i	−0.526290	−	0.850305i	\(-0.676417\pi\)
							0.877551	+	0.479484i	\(0.159176\pi\)
\(19\)	0.0477440	+	0.119828i	0.0109532	+	0.0274905i	0.934352	−	0.356352i	\(-0.115980\pi\)
							−0.923398	+	0.383843i	\(0.874600\pi\)
\(23\)	−1.83383	+	0.199441i	−0.382381	+	0.0415864i	−0.297291	−	0.954787i	\(-0.596083\pi\)
							−0.0850895	+	0.996373i	\(0.527118\pi\)
\(29\)	0.659075	−	2.99421i	0.122387	−	0.556010i	−0.875023	−	0.484082i	\(-0.839154\pi\)
							0.997410	−	0.0719283i	\(-0.0229153\pi\)
\(31\)	−0.531352	−	0.211710i	−0.0954336	−	0.0380242i	0.321933	−	0.946763i	\(-0.395668\pi\)
							−0.417366	+	0.908738i	\(0.637047\pi\)
\(37\)	2.73291	−	3.59509i	0.449288	−	0.591029i	−0.514569	−	0.857449i	\(-0.672048\pi\)
							0.963857	+	0.266421i	\(0.0858410\pi\)
\(41\)	0.194781	−	1.79098i	0.0304197	−	0.279704i	−0.969042	−	0.246896i	\(-0.920589\pi\)
							0.999462	−	0.0328083i	\(-0.0104451\pi\)
\(43\)	5.30132	+	4.50298i	0.808444	+	0.686698i	0.953057	−	0.302790i	\(-0.0979180\pi\)
							−0.144613	+	0.989488i	\(0.546194\pi\)
\(47\)	−1.66320	−	5.99032i	−0.242603	−	0.873778i	−0.980113	−	0.198441i	\(-0.936412\pi\)
							0.737510	−	0.675337i	\(-0.236002\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.g.b.11.7	yes	280
3.2	odd	2		354.2.g.a.11.5	✓	280
59.43	odd	58		354.2.g.a.161.5	yes	280
177.161	even	58	inner	354.2.g.b.161.7	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.g.a.11.5	✓	280	3.2	odd	2
354.2.g.a.161.5	yes	280	59.43	odd	58
354.2.g.b.11.7	yes	280	1.1	even	1	trivial
354.2.g.b.161.7	yes	280	177.161	even	58	inner