Embedded newform 354.2.g.b.101.2

• Label: 354.2.g.b.101.2
• 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			101.2
Character	\(\chi\)	\(=\)	354.101
Dual form			354.2.g.b.347.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.561187 - 0.827689i) q^{2} +(-1.41815 + 0.994410i) q^{3} +(-0.370138 - 0.928977i) q^{4} +(-1.02174 + 2.20846i) q^{5} +(0.0272148 + 1.73184i) q^{6} +(-1.16929 - 4.21140i) q^{7} +(-0.976621 - 0.214970i) q^{8} +(1.02230 - 2.82044i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + 10 q^{2} + q^{3} - 10 q^{4} - q^{6} + 2 q^{7} + 10 q^{8} - 3 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.41815	+	0.994410i	−0.818769	+	0.574123i
\(5\)	−1.02174	+	2.20846i	−0.456938	+	0.987655i								0.532995	+	0.846118i	\(0.321066\pi\)
							−0.989933	+	0.141537i	\(0.954796\pi\)
\(7\)	−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	0.775044	−	0.631908i	\(-0.217728\pi\)
							−0.589021	+	0.808118i	\(0.700486\pi\)
\(13\)	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	0.0198692	−	0.999803i	\(-0.493675\pi\)
							−0.498427	+	0.866932i	\(0.666089\pi\)
\(19\)	−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	\(-0.874482\pi\)
							0.200131	−	0.979769i	\(-0.435863\pi\)
\(29\)	4.18950	−	2.84055i	0.777971	−	0.527477i	−0.106366	−	0.994327i	\(-0.533921\pi\)
							0.884337	+	0.466850i	\(0.154611\pi\)
\(31\)	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	−0.214720	−	0.976676i	\(-0.568884\pi\)
							−0.928882	+	0.370376i	\(0.879229\pi\)
\(43\)	−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	−0.786030	−	0.618188i	\(-0.787867\pi\)
							−0.0377061	+	0.999289i	\(0.512005\pi\)

(See \(a_n\) instead)

Twists

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

    

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