Embedded newform 37.9.g.a.8.12

• Label: 37.9.g.a.8.12
• Level: $37$
• Weight: $9$
• Character: 37.8
• Analytic conductor: $15.073$
• Analytic rank: $0$
• Dimension: $100$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	37.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.0730085723\)
Analytic rank:	\(0\)
Dimension:	\(100\)
Relative dimension:	\(25\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			8.12
Character	\(\chi\)	\(=\)	37.8
Dual form			37.9.g.a.14.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.47162 - 1.46612i) q^{2} +(-38.0479 - 21.9670i) q^{3} +(-193.913 - 111.956i) q^{4} +(326.464 - 87.4758i) q^{5} +(175.978 + 175.978i) q^{6} +(1762.78 - 3053.23i) q^{7} +(1922.29 + 1922.29i) q^{8} +(-2315.40 - 4010.40i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 10 q^{2} - 6 q^{3} + 552 q^{4} - 1642 q^{5} - 516 q^{6} - 2 q^{7} - 7440 q^{8} + 117258 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.47162	−	1.46612i	−0.341976	−	0.0916323i	0.0837434	−	0.996487i	\(-0.473312\pi\)
							−0.425720	+	0.904855i	\(0.639979\pi\)
\(3\)	−38.0479	−	21.9670i	−0.469727	−	0.271197i	0.246398	−	0.969169i	\(-0.420753\pi\)
							−0.716126	+	0.697972i	\(0.754086\pi\)
\(5\)	326.464	−	87.4758i	0.522342	−	0.139961i	0.0119925	−	0.999928i	\(-0.496183\pi\)
							0.510350	+	0.859967i	\(0.329516\pi\)
\(7\)	1762.78	−	3053.23i	0.734186	−	1.27165i	−0.220893	−	0.975298i	\(-0.570897\pi\)
							0.955079	−	0.296350i	\(-0.0957694\pi\)
\(11\)		−	258.175i		−	0.0176337i	−0.999961	−	0.00881686i	\(-0.997193\pi\)
							0.999961	−	0.00881686i	\(-0.00280653\pi\)
\(13\)	−20861.3	+	5589.77i	−0.730413	+	0.195713i	−0.604813	−	0.796368i	\(-0.706752\pi\)
							−0.125600	+	0.992081i	\(0.540085\pi\)
\(17\)	6233.18	−	23262.6i	0.0746301	−	0.278523i	−0.918519	−	0.395377i	\(-0.870614\pi\)
							0.993149	+	0.116853i	\(0.0372807\pi\)
\(19\)	−131675.	+	35282.3i	−1.01039	+	0.270734i	−0.725794	−	0.687912i	\(-0.758527\pi\)
							−0.284600	+	0.958647i	\(0.591861\pi\)
\(23\)	20391.1	+	20391.1i	0.0728667	+	0.0728667i	0.742601	−	0.669734i	\(-0.233592\pi\)
							−0.669734	+	0.742601i	\(0.733592\pi\)
\(29\)	−179911.	+	179911.i	−0.254370	+	0.254370i	−0.822760	−	0.568389i	\(-0.807567\pi\)
							0.568389	+	0.822760i	\(0.307567\pi\)
\(31\)	−475011.	+	475011.i	−0.514348	+	0.514348i	−0.915856	−	0.401508i	\(-0.868486\pi\)
							0.401508	+	0.915856i	\(0.368486\pi\)
\(37\)	423893.	−	1.82559e6i	0.226178	−	0.974086i
							\(41\)	908934.	+	524774.i	0.321660	+	0.185711i	0.652132	−	0.758105i	\(-0.273875\pi\)
−0.330472	+	0.943816i	\(0.607208\pi\)
\(43\)	413802.	+	413802.i	0.121037	+	0.121037i	0.765031	−	0.643994i	\(-0.222724\pi\)
							−0.643994	+	0.765031i	\(0.722724\pi\)
\(47\)	3.55832e6			0.729212			0.364606	−	0.931162i	\(-0.381204\pi\)
							0.364606	+	0.931162i	\(0.381204\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.9.g.a.8.12	✓	100
37.14	odd	12	inner	37.9.g.a.14.12	yes	100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.9.g.a.8.12	✓	100	1.1	even	1	trivial
37.9.g.a.14.12	yes	100	37.14	odd	12	inner