Embedded newform 37.8.f.a.7.13

• Label: 37.8.f.a.7.13
• Level: $37$
• Weight: $8$
• Character: 37.7
• Analytic conductor: $11.558$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	37.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5582459429\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			7.13
Character	\(\chi\)	\(=\)	37.7
Dual form			37.8.f.a.16.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.24172 - 0.815920i) q^{2} +(67.7355 + 24.6537i) q^{3} +(-93.6941 + 78.6187i) q^{4} +(-36.3337 - 206.059i) q^{5} +171.960 q^{6} +(278.285 + 1578.23i) q^{7} +(-298.567 + 517.134i) q^{8} +(2304.96 + 1934.09i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q + 15 q^{2} + 33 q^{3} - 279 q^{4} + 576 q^{5} - 12 q^{6} - 597 q^{7} - 3750 q^{8} - 3243 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.24172	−	0.815920i	0.198142	−	0.0721179i	−0.241043	−	0.970514i	\(-0.577489\pi\)
							0.439185	+	0.898397i	\(0.355267\pi\)
\(3\)	67.7355	+	24.6537i	1.44841	+	0.527179i	0.942147	−	0.335199i	\(-0.108804\pi\)
							0.506265	+	0.862378i	\(0.331026\pi\)
\(5\)	−36.3337	−	206.059i	−0.129991	−	0.737218i	−0.978217	−	0.207584i	\(-0.933440\pi\)
							0.848226	−	0.529635i	\(-0.177671\pi\)
\(7\)	278.285	+	1578.23i	0.306652	+	1.73911i	0.615624	+	0.788040i	\(0.288904\pi\)
							−0.308972	+	0.951071i	\(0.599985\pi\)
\(11\)	2056.02	−	3561.12i	0.465749	−	0.806701i	−0.533486	−	0.845809i	\(-0.679118\pi\)
							0.999235	+	0.0391079i	\(0.0124516\pi\)
\(13\)	−10957.6	+	9194.49i	−1.38329	+	1.16072i	−0.415310	+	0.909680i	\(0.636327\pi\)
							−0.967978	+	0.251036i	\(0.919229\pi\)
\(17\)	26139.1	+	21933.3i	1.29039	+	1.08276i	0.991722	+	0.128405i	\(0.0409858\pi\)
							0.298665	+	0.954358i	\(0.403459\pi\)
\(19\)	−13050.3	−	4749.93i	−0.436499	−	0.158873i	0.114418	−	0.993433i	\(-0.463500\pi\)
							−0.550917	+	0.834560i	\(0.685722\pi\)
\(23\)	18231.9	+	31578.7i	0.312454	+	0.541186i	0.978893	−	0.204374i	\(-0.0655158\pi\)
							−0.666439	+	0.745559i	\(0.732182\pi\)
\(29\)	−6446.71	+	11166.0i	−0.0490846	+	0.0850170i	−0.889524	−	0.456889i	\(-0.848964\pi\)
							0.840439	+	0.541906i	\(0.182297\pi\)
\(31\)	55442.1			0.334252			0.167126	−	0.985936i	\(-0.446551\pi\)
							0.167126	+	0.985936i	\(0.446551\pi\)
\(37\)	−108513.	−	288369.i	−0.352188	−	0.935929i
							\(41\)	580000.	−	486678.i	1.31427	−	1.10280i	0.326784	−	0.945099i	\(-0.394035\pi\)
0.987486	−	0.157704i	\(-0.0504093\pi\)
\(43\)	197307.			0.378446			0.189223	−	0.981934i	\(-0.439403\pi\)
							0.189223	+	0.981934i	\(0.439403\pi\)
\(47\)	−135317.	−	234377.i	−0.190113	−	0.329285i	0.755175	−	0.655524i	\(-0.227552\pi\)
							−0.945287	+	0.326239i	\(0.894219\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.f.a.7.13	✓	132
37.16	even	9	inner	37.8.f.a.16.13	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.f.a.7.13	✓	132	1.1	even	1	trivial
37.8.f.a.16.13	yes	132	37.16	even	9	inner