Embedded newform 354.2.g.a.101.9

• Label: 354.2.g.a.101.9
• Level: $354$
• Weight: $2$
• Character: 354.101
• 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			101.9
Character	\(\chi\)	\(=\)	354.101
Dual form			354.2.g.a.347.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.561187 + 0.827689i) q^{2} +(1.51735 + 0.835252i) q^{3} +(-0.370138 - 0.928977i) q^{4} +(1.02174 - 2.20846i) q^{5} +(-1.54285 + 0.787162i) q^{6} +(-1.16929 - 4.21140i) q^{7} +(0.976621 + 0.214970i) q^{8} +(1.60471 + 2.53474i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 10 q^{2} + q^{3} - 10 q^{4} + q^{6} + 2 q^{7} - 10 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{11}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.561187	+	0.827689i	−0.396819	+	0.585265i
							\(3\)	1.51735	+	0.835252i	0.876043	+	0.482233i
\(5\)	1.02174	−	2.20846i	0.456938	−	0.987655i								−0.532995	−	0.846118i	\(-0.678934\pi\)
							0.989933	−	0.141537i	\(-0.0452044\pi\)
\(7\)	−1.16929	−	4.21140i	−0.441950	−	1.59176i	−0.764117	−	0.645077i	\(-0.776825\pi\)
							0.322167	−	0.946683i	\(-0.395589\pi\)
\(11\)	−0.616967	−	0.584423i	−0.186023	−	0.176210i	0.589021	−	0.808118i	\(-0.299514\pi\)
							−0.775044	+	0.631908i	\(0.782272\pi\)
\(13\)	0.461622	−	4.24455i	0.128031	−	1.17722i	−0.737137	−	0.675743i	\(-0.763823\pi\)
							0.865168	−	0.501482i	\(-0.167212\pi\)
\(17\)	1.97314	+	0.547841i	0.478558	+	0.132871i	0.498427	−	0.866932i	\(-0.333911\pi\)
							−0.0198692	+	0.999803i	\(0.506325\pi\)
\(19\)	−5.20577	−	3.95732i	−1.19429	−	0.907872i	−0.196921	−	0.980419i	\(-0.563094\pi\)
							−0.997365	+	0.0725471i	\(0.976887\pi\)
\(23\)	3.46798	+	6.54131i	0.723124	+	1.36396i	0.923255	+	0.384187i	\(0.125518\pi\)
							−0.200131	+	0.979769i	\(0.564137\pi\)
\(29\)	−4.18950	+	2.84055i	−0.777971	+	0.527477i	−0.884337	−	0.466850i	\(-0.845389\pi\)
							0.106366	+	0.994327i	\(0.466079\pi\)
\(31\)	3.65152	+	4.80350i	0.655833	+	0.862733i	0.997011	−	0.0772656i	\(-0.0246189\pi\)
							−0.341178	+	0.939999i	\(0.610826\pi\)
\(37\)	1.74962	+	7.94861i	0.287636	+	1.30674i	0.869692	+	0.493594i	\(0.164317\pi\)
							−0.582056	+	0.813148i	\(0.697752\pi\)
\(41\)	7.32263	+	3.88221i	1.14360	+	0.606300i	0.928882	−	0.370376i	\(-0.120771\pi\)
							0.214720	+	0.976676i	\(0.431116\pi\)
\(43\)	−3.07553	−	3.24680i	−0.469014	−	0.495132i	0.447974	−	0.894046i	\(-0.352146\pi\)
							−0.916988	+	0.398915i	\(0.869387\pi\)
\(47\)	5.64725	−	2.61270i	0.823737	−	0.381101i	0.0377061	−	0.999289i	\(-0.487995\pi\)
							0.786030	+	0.618188i	\(0.212133\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.g.a.101.9	✓	280
3.2	odd	2		354.2.g.b.101.2	yes	280
59.52	odd	58		354.2.g.b.347.2	yes	280
177.170	even	58	inner	354.2.g.a.347.9	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.g.a.101.9	✓	280	1.1	even	1	trivial
354.2.g.a.347.9	yes	280	177.170	even	58	inner
354.2.g.b.101.2	yes	280	3.2	odd	2
354.2.g.b.347.2	yes	280	59.52	odd	58