Embedded newform 37.11.g.a.8.17

• Label: 37.11.g.a.8.17
• Level: $37$
• Weight: $11$
• Character: 37.8
• Analytic conductor: $23.508$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	37.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(23.5082183489\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(30\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			8.17
Character	\(\chi\)	\(=\)	37.8
Dual form			37.11.g.a.14.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.41925 + 2.25593i) q^{2} +(-149.333 - 86.2176i) q^{3} +(-821.016 - 474.014i) q^{4} +(-1530.72 + 410.155i) q^{5} +(-1062.77 - 1062.77i) q^{6} +(8137.72 - 14094.9i) q^{7} +(-12154.2 - 12154.2i) q^{8} +(-14657.6 - 25387.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} - 6 q^{3} - 3768 q^{4} + 2922 q^{5} - 2052 q^{6} - 2 q^{7} + 20880 q^{8} + 1052964 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.41925	+	2.25593i	0.263101	+	0.0704978i	0.387958	−	0.921677i	\(-0.373180\pi\)
							−0.124857	+	0.992175i	\(0.539847\pi\)
\(3\)	−149.333	−	86.2176i	−0.614540	−	0.354805i	0.160200	−	0.987085i	\(-0.448786\pi\)
							−0.774740	+	0.632280i	\(0.782119\pi\)
\(5\)	−1530.72	+	410.155i	−0.489830	+	0.131250i	−0.495276	−	0.868736i	\(-0.664933\pi\)
							0.00544605	+	0.999985i	\(0.498266\pi\)
\(7\)	8137.72	−	14094.9i	0.484186	−	0.838635i	−0.515649	−	0.856800i	\(-0.672449\pi\)
							0.999835	+	0.0181650i	\(0.00578241\pi\)
\(11\)		−	160205.i		−	0.994746i	−0.867537	−	0.497373i	\(-0.834298\pi\)
							0.867537	−	0.497373i	\(-0.165702\pi\)
\(13\)	−462590.	+	123951.i	−1.24589	+	0.333835i	−0.820747	−	0.571292i	\(-0.806443\pi\)
							−0.425142	+	0.905127i	\(0.639776\pi\)
\(17\)	−406103.	+	1.51560e6i	−0.286017	+	1.06743i	0.662076	+	0.749437i	\(0.269676\pi\)
							−0.948093	+	0.317993i	\(0.896991\pi\)
\(19\)	1.77802e6	−	476420.i	0.718074	−	0.192407i	0.118762	−	0.992923i	\(-0.462107\pi\)
							0.599312	+	0.800515i	\(0.295441\pi\)
\(23\)	3.92652e6	+	3.92652e6i	0.610055	+	0.610055i	0.942960	−	0.332905i	\(-0.108029\pi\)
							−0.332905	+	0.942960i	\(0.608029\pi\)
\(29\)	7.33269e6	−	7.33269e6i	0.357498	−	0.357498i	−0.505392	−	0.862890i	\(-0.668652\pi\)
							0.862890	+	0.505392i	\(0.168652\pi\)
\(31\)	2.90255e7	−	2.90255e7i	1.01384	−	1.01384i	0.0139420	−	0.999903i	\(-0.495562\pi\)
							0.999903	−	0.0139420i	\(-0.00443801\pi\)
\(37\)	−6.63644e7	+	2.01084e7i	−0.957033	+	0.289980i
							\(41\)	−8.06437e7	−	4.65597e7i	−0.696067	−	0.401875i	0.109814	−	0.993952i	\(-0.464975\pi\)
−0.805881	+	0.592078i	\(0.798308\pi\)
\(43\)	1.58319e8	+	1.58319e8i	1.07694	+	1.07694i	0.996783	+	0.0801526i	\(0.0255408\pi\)
							0.0801526	+	0.996783i	\(0.474459\pi\)
\(47\)	−2.11094e8			−0.920421			−0.460210	−	0.887810i	\(-0.652226\pi\)
							−0.460210	+	0.887810i	\(0.652226\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.11.g.a.8.17	✓	120
37.14	odd	12	inner	37.11.g.a.14.17	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.11.g.a.8.17	✓	120	1.1	even	1	trivial
37.11.g.a.14.17	yes	120	37.14	odd	12	inner