Embedded newform 37.11.g.a.8.8

• Label: 37.11.g.a.8.8
• 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			8.8
Character	\(\chi\)	\(=\)	37.8
Dual form			37.11.g.a.14.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-33.6907 - 9.02740i) q^{2} +(-34.0392 - 19.6525i) q^{3} +(166.761 + 96.2797i) q^{4} +(3757.77 - 1006.89i) q^{5} +(969.394 + 969.394i) q^{6} +(8651.45 - 14984.7i) q^{7} +(20506.1 + 20506.1i) q^{8} +(-28752.1 - 49800.0i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} - 6 q^{3} - 3768 q^{4} + 2922 q^{5} - 2052 q^{6} - 2 q^{7} + 20880 q^{8} + 1052964 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−33.6907	−	9.02740i	−1.05284	−	0.282106i	−0.309412	−	0.950928i	\(-0.600132\pi\)
							−0.743423	+	0.668822i	\(0.766799\pi\)
\(3\)	−34.0392	−	19.6525i	−0.140079	−	0.0808747i	0.428323	−	0.903626i	\(-0.359105\pi\)
							−0.568402	+	0.822751i	\(0.692438\pi\)
\(5\)	3757.77	−	1006.89i	1.20249	−	0.322205i	0.398677	−	0.917091i	\(-0.369469\pi\)
							0.803810	+	0.594886i	\(0.202803\pi\)
\(7\)	8651.45	−	14984.7i	0.514753	−	0.891578i	−0.485101	−	0.874458i	\(-0.661217\pi\)
							0.999853	−	0.0171196i	\(-0.00544960\pi\)
\(11\)			311513.i			1.93425i	0.254296	+	0.967127i	\(0.418156\pi\)
							−0.254296	+	0.967127i	\(0.581844\pi\)
\(13\)	404837.	−	108476.i	1.09034	−	0.292157i	0.331517	−	0.943449i	\(-0.392439\pi\)
							0.758827	+	0.651292i	\(0.225773\pi\)
\(17\)	−510916.	+	1.90676e6i	−0.359836	+	1.34293i	0.514452	+	0.857519i	\(0.327995\pi\)
							−0.874288	+	0.485408i	\(0.838671\pi\)
\(19\)	125914.	−	33738.5i	0.0508517	−	0.0136257i	−0.233304	−	0.972404i	\(-0.574954\pi\)
							0.284155	+	0.958778i	\(0.408287\pi\)
\(23\)	4.27304e6	+	4.27304e6i	0.663892	+	0.663892i	0.956295	−	0.292403i	\(-0.0944550\pi\)
							−0.292403	+	0.956295i	\(0.594455\pi\)
\(29\)	1.81592e7	−	1.81592e7i	0.885332	−	0.885332i	−0.108738	−	0.994070i	\(-0.534681\pi\)
							0.994070	+	0.108738i	\(0.0346810\pi\)
\(31\)	2.33009e7	−	2.33009e7i	0.813886	−	0.813886i	−0.171328	−	0.985214i	\(-0.554806\pi\)
							0.985214	+	0.171328i	\(0.0548058\pi\)
\(37\)	3.16463e7	−	6.17017e7i	0.456367	−	0.889791i
							\(41\)	2.83231e7	+	1.63523e7i	0.244467	+	0.141143i	0.617228	−	0.786784i	\(-0.288256\pi\)
−0.372761	+	0.927927i	\(0.621589\pi\)
\(43\)	1.61400e8	+	1.61400e8i	1.09790	+	1.09790i	0.994656	+	0.103241i	\(0.0329213\pi\)
							0.103241	+	0.994656i	\(0.467079\pi\)
\(47\)	−2.68728e8			−1.17172			−0.585859	−	0.810413i	\(-0.699243\pi\)
							−0.585859	+	0.810413i	\(0.699243\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.11.g.a.8.8	✓	120
37.14	odd	12	inner	37.11.g.a.14.8	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.11.g.a.8.8	✓	120	1.1	even	1	trivial
37.11.g.a.14.8	yes	120	37.14	odd	12	inner