Embedded newform 37.8.f.a.7.10

• Label: 37.8.f.a.7.10
• 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			7.10
Character	\(\chi\)	\(=\)	37.7
Dual form			37.8.f.a.16.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.25018 + 1.54694i) q^{2} +(-41.1038 - 14.9605i) q^{3} +(-82.3827 + 69.1273i) q^{4} +(-92.7474 - 525.997i) q^{5} +197.841 q^{6} +(9.66574 + 54.8172i) q^{7} +(532.674 - 922.618i) q^{8} +(-209.638 - 175.907i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q + 15 q^{2} + 33 q^{3} - 279 q^{4} + 576 q^{5} - 12 q^{6} - 597 q^{7} - 3750 q^{8} - 3243 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.25018	+	1.54694i	−0.375666	+	0.136731i	−0.522951	−	0.852363i	\(-0.675169\pi\)
							0.147285	+	0.989094i	\(0.452947\pi\)
\(3\)	−41.1038	−	14.9605i	−0.878936	−	0.319907i	−0.137156	−	0.990550i	\(-0.543796\pi\)
							−0.741780	+	0.670643i	\(0.766018\pi\)
\(5\)	−92.7474	−	525.997i	−0.331823	−	1.88186i	−0.456590	−	0.889677i	\(-0.650929\pi\)
							0.124767	−	0.992186i	\(-0.460182\pi\)
\(7\)	9.66574	+	54.8172i	0.0106510	+	0.0604051i	0.989670	−	0.143364i	\(-0.0457920\pi\)
							−0.979019	+	0.203769i	\(0.934681\pi\)
\(11\)	−2796.10	+	4842.99i	−0.633401	+	1.09708i	0.353451	+	0.935453i	\(0.385008\pi\)
							−0.986852	+	0.161629i	\(0.948325\pi\)
\(13\)	−1376.96	+	1155.41i	−0.173828	+	0.145859i	−0.725551	−	0.688168i	\(-0.758415\pi\)
							0.551723	+	0.834027i	\(0.313971\pi\)
\(17\)	11629.7	+	9758.45i	0.574111	+	0.481737i	0.883007	−	0.469359i	\(-0.155515\pi\)
							−0.308896	+	0.951096i	\(0.599959\pi\)
\(19\)	4466.97	+	1625.84i	0.149408	+	0.0543802i	0.415642	−	0.909528i	\(-0.363557\pi\)
							−0.266233	+	0.963909i	\(0.585779\pi\)
\(23\)	20502.0	+	35510.5i	0.351357	+	0.608569i	0.986488	−	0.163836i	\(-0.0523869\pi\)
							−0.635130	+	0.772405i	\(0.719054\pi\)
\(29\)	95164.8	−	164830.i	0.724575	−	1.25500i	−0.234574	−	0.972098i	\(-0.575370\pi\)
							0.959149	−	0.282902i	\(-0.0912971\pi\)
\(31\)	93640.6			0.564544			0.282272	−	0.959334i	\(-0.408912\pi\)
							0.282272	+	0.959334i	\(0.408912\pi\)
\(37\)	−33264.6	−	306309.i	−0.107963	−	0.994155i
							\(41\)	40983.2	−	34389.0i	0.0928672	−	0.0779248i	−0.595172	−	0.803598i	\(-0.702916\pi\)
0.688039	+	0.725673i	\(0.258472\pi\)
\(43\)	−828283.			−1.58869			−0.794345	−	0.607467i	\(-0.792186\pi\)
							−0.794345	+	0.607467i	\(0.792186\pi\)
\(47\)	171333.	+	296757.i	0.240712	+	0.416926i	0.960917	−	0.276835i	\(-0.0892857\pi\)
							−0.720205	+	0.693761i	\(0.755952\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.f.a.7.10	✓	132
37.16	even	9	inner	37.8.f.a.16.10	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.f.a.7.10	✓	132	1.1	even	1	trivial
37.8.f.a.16.10	yes	132	37.16	even	9	inner