Embedded newform 37.9.g.a.8.17

• Label: 37.9.g.a.8.17
• 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			8.17
Character	\(\chi\)	\(=\)	37.8
Dual form			37.9.g.a.14.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.5756 + 3.10167i) q^{2} +(46.5342 + 26.8665i) q^{3} +(-97.3286 - 56.1927i) q^{4} +(-129.992 + 34.8313i) q^{5} +(455.330 + 455.330i) q^{6} +(-1422.98 + 2464.68i) q^{7} +(-3121.67 - 3121.67i) q^{8} +(-1836.88 - 3181.57i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 10 q^{2} - 6 q^{3} + 552 q^{4} - 1642 q^{5} - 516 q^{6} - 2 q^{7} - 7440 q^{8} + 117258 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.5756	+	3.10167i	0.723474	+	0.193854i	0.601721	−	0.798706i	\(-0.294482\pi\)
							0.121753	+	0.992560i	\(0.461148\pi\)
\(3\)	46.5342	+	26.8665i	0.574496	+	0.331686i	0.758943	−	0.651157i	\(-0.225716\pi\)
							−0.184447	+	0.982842i	\(0.559049\pi\)
\(5\)	−129.992	+	34.8313i	−0.207987	+	0.0557301i	−0.361308	−	0.932446i	\(-0.617670\pi\)
							0.153321	+	0.988176i	\(0.451003\pi\)
\(7\)	−1422.98	+	2464.68i	−0.592662	+	1.02652i	0.401210	+	0.915986i	\(0.368590\pi\)
							−0.993872	+	0.110535i	\(0.964743\pi\)
\(11\)			27056.2i			1.84798i	0.382422	+	0.923988i	\(0.375090\pi\)
							−0.382422	+	0.923988i	\(0.624910\pi\)
\(13\)	−30787.7	+	8249.54i	−1.07796	+	0.288839i	−0.753761	−	0.657149i	\(-0.771762\pi\)
							−0.324202	+	0.945988i	\(0.605096\pi\)
\(17\)	18951.6	−	70728.5i	0.226909	−	0.846835i	−0.754722	−	0.656045i	\(-0.772228\pi\)
							0.981631	−	0.190790i	\(-0.0611050\pi\)
\(19\)	−4769.42	+	1277.96i	−0.0365975	+	0.00980627i	−0.277071	−	0.960849i	\(-0.589364\pi\)
							0.240474	+	0.970656i	\(0.422697\pi\)
\(23\)	64966.5	+	64966.5i	0.232155	+	0.232155i	0.813592	−	0.581437i	\(-0.197509\pi\)
							−0.581437	+	0.813592i	\(0.697509\pi\)
\(29\)	−793857.	+	793857.i	−1.12241	+	1.12241i	−0.131029	+	0.991379i	\(0.541828\pi\)
							−0.991379	+	0.131029i	\(0.958172\pi\)
\(31\)	585183.	−	585183.i	0.633643	−	0.633643i	−0.315337	−	0.948980i	\(-0.602117\pi\)
							0.948980	+	0.315337i	\(0.102117\pi\)
\(37\)	−67248.2	−	1.87295e6i	−0.0358818	−	0.999356i
							\(41\)	460646.	+	265954.i	0.163017	+	0.0941178i	0.579289	−	0.815122i	\(-0.303330\pi\)
−0.416272	+	0.909240i	\(0.636664\pi\)
\(43\)	2.74167e6	+	2.74167e6i	0.801939	+	0.801939i	0.983398	−	0.181460i	\(-0.0580822\pi\)
							−0.181460	+	0.983398i	\(0.558082\pi\)
\(47\)	6.21952e6			1.27457			0.637287	−	0.770626i	\(-0.280057\pi\)
							0.637287	+	0.770626i	\(0.280057\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.9.g.a.8.17	✓	100
37.14	odd	12	inner	37.9.g.a.14.17	yes	100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.9.g.a.8.17	✓	100	1.1	even	1	trivial
37.9.g.a.14.17	yes	100	37.14	odd	12	inner