Embedded newform 37.9.g.a.8.9

• Label: 37.9.g.a.8.9
• 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			8.9
Character	\(\chi\)	\(=\)	37.8
Dual form			37.9.g.a.14.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.75824 - 2.61471i) q^{2} +(46.3249 + 26.7457i) q^{3} +(-133.316 - 76.9700i) q^{4} +(-1105.12 + 296.117i) q^{5} +(-382.117 - 382.117i) q^{6} +(938.117 - 1624.87i) q^{7} +(2928.42 + 2928.42i) q^{8} +(-1849.84 - 3204.01i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 10 q^{2} - 6 q^{3} + 552 q^{4} - 1642 q^{5} - 516 q^{6} - 2 q^{7} - 7440 q^{8} + 117258 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.75824	−	2.61471i	−0.609890	−	0.163420i	−0.0593625	−	0.998236i	\(-0.518907\pi\)
							−0.550528	+	0.834817i	\(0.685573\pi\)
\(3\)	46.3249	+	26.7457i	0.571912	+	0.330194i	0.757913	−	0.652356i	\(-0.226219\pi\)
							−0.186001	+	0.982550i	\(0.559553\pi\)
\(5\)	−1105.12	+	296.117i	−1.76820	+	0.473786i	−0.988352	−	0.152188i	\(-0.951368\pi\)
							−0.779844	+	0.625975i	\(0.784701\pi\)
\(7\)	938.117	−	1624.87i	0.390719	−	0.676745i	−0.601825	−	0.798628i	\(-0.705560\pi\)
							0.992545	+	0.121882i	\(0.0388930\pi\)
\(11\)			11552.4i			0.789046i	0.918886	+	0.394523i	\(0.129090\pi\)
							−0.918886	+	0.394523i	\(0.870910\pi\)
\(13\)	10060.4	−	2695.68i	0.352243	−	0.0943833i	−0.0783588	−	0.996925i	\(-0.524968\pi\)
							0.430602	+	0.902542i	\(0.358301\pi\)
\(17\)	−911.045	+	3400.07i	−0.0109080	+	0.0407091i	−0.971165	−	0.238407i	\(-0.923375\pi\)
							0.960257	+	0.279116i	\(0.0900414\pi\)
\(19\)	101132.	−	27098.2i	0.776020	−	0.207934i	0.150991	−	0.988535i	\(-0.451754\pi\)
							0.625029	+	0.780601i	\(0.285087\pi\)
\(23\)	336894.	+	336894.i	1.20388	+	1.20388i	0.972978	+	0.230898i	\(0.0741664\pi\)
							0.230898	+	0.972978i	\(0.425834\pi\)
\(29\)	−168849.	+	168849.i	−0.238729	+	0.238729i	−0.816324	−	0.577595i	\(-0.803991\pi\)
							0.577595	+	0.816324i	\(0.303991\pi\)
\(31\)	707497.	−	707497.i	0.766086	−	0.766086i	−0.211329	−	0.977415i	\(-0.567779\pi\)
							0.977415	+	0.211329i	\(0.0677791\pi\)
\(37\)	631993.	+	1.76439e6i	0.337214	+	0.941428i
							\(41\)	−407965.	−	235539.i	−0.144374	−	0.0833541i	0.426073	−	0.904689i	\(-0.359897\pi\)
−0.570447	+	0.821335i	\(0.693230\pi\)
\(43\)	−1.96584e6	−	1.96584e6i	−0.575007	−	0.575007i	0.358516	−	0.933524i	\(-0.383283\pi\)
							−0.933524	+	0.358516i	\(0.883283\pi\)
\(47\)	806026.			0.165180			0.0825900	−	0.996584i	\(-0.473681\pi\)
							0.0825900	+	0.996584i	\(0.473681\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.9.g.a.8.9	✓	100
37.14	odd	12	inner	37.9.g.a.14.9	yes	100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.9.g.a.8.9	✓	100	1.1	even	1	trivial
37.9.g.a.14.9	yes	100	37.14	odd	12	inner