Embedded newform 354.2.g.a.23.5

• Label: 354.2.g.a.23.5
• Level: $354$
• Weight: $2$
• Character: 354.23
• 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			23.5
Character	\(\chi\)	\(=\)	354.23
Dual form			354.2.g.a.77.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.725995 + 0.687699i) q^{2} +(-0.639916 - 1.60951i) q^{3} +(0.0541389 - 0.998533i) q^{4} +(-1.86219 - 0.305290i) q^{5} +(1.57143 + 0.728423i) q^{6} +(1.84960 - 3.48871i) q^{7} +(0.647386 + 0.762162i) q^{8} +(-2.18101 + 2.05990i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 10 q^{2} + q^{3} - 10 q^{4} + q^{6} + 2 q^{7} - 10 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{15}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.725995	+	0.687699i	−0.513356	+	0.486277i
							\(3\)	−0.639916	−	1.60951i	−0.369456	−	0.929248i
\(5\)	−1.86219	−	0.305290i	−0.832796	−	0.136530i								−0.269724	−	0.962938i	\(-0.586932\pi\)
							−0.563072	+	0.826408i	\(0.690381\pi\)
\(7\)	1.84960	−	3.48871i	0.699083	−	1.31861i	−0.238600	−	0.971118i	\(-0.576689\pi\)
							0.937683	−	0.347492i	\(-0.112967\pi\)
\(11\)	−2.27842	+	0.247793i	−0.686970	+	0.0747125i	−0.444945	−	0.895558i	\(-0.646777\pi\)
							−0.242024	+	0.970270i	\(0.577811\pi\)
\(13\)	−1.38212	+	2.98739i	−0.383330	+	0.828554i	0.615842	+	0.787870i	\(0.288816\pi\)
							−0.999172	+	0.0406846i	\(0.987046\pi\)
\(17\)	−1.91716	+	1.01641i	−0.464978	+	0.246516i	−0.684390	−	0.729117i	\(-0.739931\pi\)
							0.219411	+	0.975632i	\(0.429586\pi\)
\(19\)	−1.02619	+	0.617439i	−0.235425	+	0.141650i	−0.628378	−	0.777908i	\(-0.716281\pi\)
							0.392954	+	0.919558i	\(0.371453\pi\)
\(23\)	−2.39083	−	6.00053i	−0.498522	−	1.25120i	−0.936011	−	0.351972i	\(-0.885511\pi\)
							0.437488	−	0.899224i	\(-0.355868\pi\)
\(29\)	−2.20500	+	2.32779i	−0.409458	+	0.432260i	−0.897807	−	0.440389i	\(-0.854841\pi\)
							0.488349	+	0.872648i	\(0.337599\pi\)
\(31\)	−2.75658	+	4.58147i	−0.495097	+	0.822857i	−0.999167	−	0.0408055i	\(-0.987008\pi\)
							0.504070	+	0.863663i	\(0.331835\pi\)
\(37\)	−5.48312	−	4.65741i	−0.901420	−	0.765673i	0.0713489	−	0.997451i	\(-0.477270\pi\)
							−0.972769	+	0.231779i	\(0.925545\pi\)
\(41\)	−0.715468	−	0.285069i	−0.111737	−	0.0445202i	0.313600	−	0.949555i	\(-0.398465\pi\)
							−0.425337	+	0.905035i	\(0.639844\pi\)
\(43\)	0.0847651	−	0.779402i	0.0129266	−	0.118858i	−0.985897	−	0.167355i	\(-0.946478\pi\)
							0.998823	+	0.0484968i	\(0.0154431\pi\)
\(47\)	−1.00046	−	6.10254i	−0.145932	−	0.890147i	−0.953725	−	0.300681i	\(-0.902786\pi\)
							0.807793	−	0.589467i	\(-0.200662\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.g.a.23.5	✓	280
3.2	odd	2		354.2.g.b.23.5	yes	280
59.18	odd	58		354.2.g.b.77.5	yes	280
177.77	even	58	inner	354.2.g.a.77.5	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.g.a.23.5	✓	280	1.1	even	1	trivial
354.2.g.a.77.5	yes	280	177.77	even	58	inner
354.2.g.b.23.5	yes	280	3.2	odd	2
354.2.g.b.77.5	yes	280	59.18	odd	58