Embedded newform 354.2.g.a.101.6

• Label: 354.2.g.a.101.6
• 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			101.6
Character	\(\chi\)	\(=\)	354.101
Dual form			354.2.g.a.347.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.561187 + 0.827689i) q^{2} +(0.181617 + 1.72250i) q^{3} +(-0.370138 - 0.928977i) q^{4} +(1.12753 - 2.43713i) q^{5} +(-1.52762 - 0.816324i) q^{6} +(0.437143 + 1.57445i) q^{7} +(0.976621 + 0.214970i) q^{8} +(-2.93403 + 0.625671i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 10 q^{2} + q^{3} - 10 q^{4} + q^{6} + 2 q^{7} - 10 q^{8} - 3 q^{9}+O(q^{10}) \)

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
							\(3\)	0.181617	+	1.72250i	0.104857	+	0.994487i
\(5\)	1.12753	−	2.43713i	0.504249	−	1.08992i								−0.473626	−	0.880726i	\(-0.657055\pi\)
							0.977875	−	0.209190i	\(-0.0670827\pi\)
\(7\)	0.437143	+	1.57445i	0.165224	+	0.595085i	0.999071	+	0.0430888i	\(0.0137198\pi\)
							−0.833847	+	0.551996i	\(0.813866\pi\)
\(11\)	2.20426	+	2.08799i	0.664610	+	0.629552i	0.943735	−	0.330702i	\(-0.107286\pi\)
							−0.279125	+	0.960255i	\(0.590044\pi\)
\(13\)	−0.448386	+	4.12284i	−0.124360	+	1.14347i	0.751103	+	0.660185i	\(0.229522\pi\)
							−0.875463	+	0.483286i	\(0.839443\pi\)
\(17\)	3.18616	+	0.884632i	0.772757	+	0.214555i	0.631456	−	0.775412i	\(-0.282458\pi\)
							0.141301	+	0.989967i	\(0.454871\pi\)
\(19\)	3.51543	+	2.67236i	0.806496	+	0.613082i	0.925041	−	0.379866i	\(-0.124030\pi\)
							−0.118545	+	0.992949i	\(0.537823\pi\)
\(23\)	−1.77408	−	3.34627i	−0.369921	−	0.697746i	0.626754	−	0.779217i	\(-0.284383\pi\)
							−0.996676	+	0.0814713i	\(0.974038\pi\)
\(29\)	−5.28077	+	3.58045i	−0.980614	+	0.664872i	−0.942432	−	0.334398i	\(-0.891467\pi\)
							−0.0381817	+	0.999271i	\(0.512157\pi\)
\(31\)	4.84476	+	6.37317i	0.870144	+	1.14465i	0.988430	+	0.151675i	\(0.0484668\pi\)
							−0.118286	+	0.992980i	\(0.537740\pi\)
\(37\)	−1.32940	−	6.03952i	−0.218552	−	0.992891i	−0.950305	−	0.311321i	\(-0.899229\pi\)
							0.731753	−	0.681570i	\(-0.238702\pi\)
\(41\)	−9.90060	−	5.24897i	−1.54621	−	0.819751i	−0.546373	−	0.837542i	\(-0.683992\pi\)
							−0.999842	+	0.0177913i	\(0.994337\pi\)
\(43\)	−3.53914	−	3.73622i	−0.539714	−	0.569769i	0.397908	−	0.917425i	\(-0.369736\pi\)
							−0.937622	+	0.347656i	\(0.886978\pi\)
\(47\)	6.67252	−	3.08704i	0.973287	−	0.450291i	0.132219	−	0.991220i	\(-0.457790\pi\)
							0.841068	+	0.540930i	\(0.181928\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.g.a.101.6	✓	280
3.2	odd	2		354.2.g.b.101.6	yes	280
59.52	odd	58		354.2.g.b.347.6	yes	280
177.170	even	58	inner	354.2.g.a.347.6	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.g.a.101.6	✓	280	1.1	even	1	trivial
354.2.g.a.347.6	yes	280	177.170	even	58	inner
354.2.g.b.101.6	yes	280	3.2	odd	2
354.2.g.b.347.6	yes	280	59.52	odd	58