Embedded newform 37.11.g.a.8.2

• Label: 37.11.g.a.8.2
• 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			8.2
Character	\(\chi\)	\(=\)	37.8
Dual form			37.11.g.a.14.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-54.2077 - 14.5249i) q^{2} +(-356.538 - 205.848i) q^{3} +(1840.69 + 1062.72i) q^{4} +(1975.98 - 529.461i) q^{5} +(16337.2 + 16337.2i) q^{6} +(14659.3 - 25390.7i) q^{7} +(-43708.4 - 43708.4i) q^{8} +(55221.9 + 95647.1i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} - 6 q^{3} - 3768 q^{4} + 2922 q^{5} - 2052 q^{6} - 2 q^{7} + 20880 q^{8} + 1052964 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−54.2077	−	14.5249i	−1.69399	−	0.453903i	−0.722575	−	0.691292i	\(-0.757042\pi\)
							−0.971415	+	0.237389i	\(0.923708\pi\)
\(3\)	−356.538	−	205.848i	−1.46724	−	0.847109i	−0.467908	−	0.883777i	\(-0.654992\pi\)
							−0.999328	+	0.0366678i	\(0.988326\pi\)
\(5\)	1975.98	−	529.461i	0.632312	−	0.169428i	0.0715935	−	0.997434i	\(-0.477192\pi\)
							0.560719	+	0.828006i	\(0.310525\pi\)
\(7\)	14659.3	−	25390.7i	0.872215	−	1.51072i	0.0125152	−	0.999922i	\(-0.496016\pi\)
							0.859700	−	0.510799i	\(-0.170650\pi\)
\(11\)			60513.5i			0.375741i	0.982194	+	0.187871i	\(0.0601586\pi\)
							−0.982194	+	0.187871i	\(0.939841\pi\)
\(13\)	−505382.	+	135417.i	−1.36114	+	0.364717i	−0.864235	−	0.503088i	\(-0.832197\pi\)
							−0.496906	+	0.867804i	\(0.665531\pi\)
\(17\)	130618.	−	487475.i	0.0919941	−	0.343326i	−0.904553	−	0.426362i	\(-0.859795\pi\)
							0.996547	+	0.0830357i	\(0.0264616\pi\)
\(19\)	3.42145e6	−	916776.i	1.38179	−	0.370250i	0.510021	−	0.860162i	\(-0.329638\pi\)
							0.871772	+	0.489912i	\(0.162971\pi\)
\(23\)	−2.18334e6	−	2.18334e6i	−0.339220	−	0.339220i	0.516853	−	0.856074i	\(-0.327103\pi\)
							−0.856074	+	0.516853i	\(0.827103\pi\)
\(29\)	−1.96942e7	+	1.96942e7i	−0.960169	+	0.960169i	−0.999237	−	0.0390680i	\(-0.987561\pi\)
							0.0390680	+	0.999237i	\(0.487561\pi\)
\(31\)	−2.68094e7	+	2.68094e7i	−0.936435	+	0.936435i	−0.998097	−	0.0616616i	\(-0.980360\pi\)
							0.0616616	+	0.998097i	\(0.480360\pi\)
\(37\)	−5.29928e7	−	4.47253e7i	−0.764201	−	0.644978i
							\(41\)	1.14363e8	+	6.60275e7i	0.987112	+	0.569909i	0.904410	−	0.426665i	\(-0.140312\pi\)
0.0827021	+	0.996574i	\(0.473645\pi\)
\(43\)	−2.29415e7	−	2.29415e7i	−0.156056	−	0.156056i	0.624761	−	0.780816i	\(-0.285197\pi\)
							−0.780816	+	0.624761i	\(0.785197\pi\)
\(47\)	−1.18599e8			−0.517122			−0.258561	−	0.965995i	\(-0.583248\pi\)
							−0.258561	+	0.965995i	\(0.583248\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.11.g.a.8.2	✓	120
37.14	odd	12	inner	37.11.g.a.14.2	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.11.g.a.8.2	✓	120	1.1	even	1	trivial
37.11.g.a.14.2	yes	120	37.14	odd	12	inner