Embedded newform 37.11.g.a.8.1

• Label: 37.11.g.a.8.1
• 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			8.1
Character	\(\chi\)	\(=\)	37.8
Dual form			37.11.g.a.14.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-57.5818 - 15.4290i) q^{2} +(155.272 + 89.6465i) q^{3} +(2190.80 + 1264.86i) q^{4} +(1994.02 - 534.297i) q^{5} +(-7557.70 - 7557.70i) q^{6} +(1378.57 - 2387.76i) q^{7} +(-63470.2 - 63470.2i) q^{8} +(-13451.5 - 23298.7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} - 6 q^{3} - 3768 q^{4} + 2922 q^{5} - 2052 q^{6} - 2 q^{7} + 20880 q^{8} + 1052964 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−57.5818	−	15.4290i	−1.79943	−	0.482156i	−0.805542	−	0.592539i	\(-0.798126\pi\)
							−0.993889	+	0.110383i	\(0.964792\pi\)
\(3\)	155.272	+	89.6465i	0.638981	+	0.368916i	0.784222	−	0.620481i	\(-0.213062\pi\)
							−0.145241	+	0.989396i	\(0.546396\pi\)
\(5\)	1994.02	−	534.297i	0.638087	−	0.170975i	0.0747505	−	0.997202i	\(-0.476184\pi\)
							0.563337	+	0.826227i	\(0.309517\pi\)
\(7\)	1378.57	−	2387.76i	0.0820238	−	0.142069i	−0.822095	−	0.569350i	\(-0.807195\pi\)
							0.904119	+	0.427280i	\(0.140528\pi\)
\(11\)		−	77616.8i		−	0.481939i	−0.970533	−	0.240970i	\(-0.922535\pi\)
							0.970533	−	0.240970i	\(-0.0774654\pi\)
\(13\)	169598.	−	45443.5i	0.456775	−	0.122393i	−0.0230925	−	0.999733i	\(-0.507351\pi\)
							0.479868	+	0.877341i	\(0.340685\pi\)
\(17\)	−116603.	+	435170.i	−0.0821233	+	0.306488i	−0.994754	−	0.102297i	\(-0.967381\pi\)
							0.912631	+	0.408785i	\(0.134047\pi\)
\(19\)	912318.	−	244455.i	0.368450	−	0.0987258i	−0.0698432	−	0.997558i	\(-0.522250\pi\)
							0.438293	+	0.898832i	\(0.355583\pi\)
\(23\)	−4.51624e6	−	4.51624e6i	−0.701678	−	0.701678i	0.263092	−	0.964771i	\(-0.415258\pi\)
							−0.964771	+	0.263092i	\(0.915258\pi\)
\(29\)	1.08054e7	−	1.08054e7i	0.526806	−	0.526806i	−0.392813	−	0.919619i	\(-0.628498\pi\)
							0.919619	+	0.392813i	\(0.128498\pi\)
\(31\)	−1.63345e7	+	1.63345e7i	−0.570554	+	0.570554i	−0.932283	−	0.361729i	\(-0.882187\pi\)
							0.361729	+	0.932283i	\(0.382187\pi\)
\(37\)	6.91322e7	−	5.41480e6i	0.996947	−	0.0780861i
							\(41\)	−1.24942e8	−	7.21355e7i	−1.07843	−	0.622629i	−0.147954	−	0.988994i	\(-0.547269\pi\)
−0.930471	+	0.366365i	\(0.880602\pi\)
\(43\)	−6.53297e7	−	6.53297e7i	−0.444394	−	0.444394i	0.449092	−	0.893486i	\(-0.351748\pi\)
							−0.893486	+	0.449092i	\(0.851748\pi\)
\(47\)	2.14215e8			0.934028			0.467014	−	0.884250i	\(-0.345330\pi\)
							0.467014	+	0.884250i	\(0.345330\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.11.g.a.8.1	✓	120
37.14	odd	12	inner	37.11.g.a.14.1	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.11.g.a.8.1	✓	120	1.1	even	1	trivial
37.11.g.a.14.1	yes	120	37.14	odd	12	inner