Embedded newform 37.7.g.a.8.14

• Label: 37.7.g.a.8.14
• Level: $37$
• Weight: $7$
• Character: 37.8
• Analytic conductor: $8.512$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	37.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.51200109393\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			8.14
Character	\(\chi\)	\(=\)	37.8
Dual form			37.7.g.a.14.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.0194 + 2.68468i) q^{2} +(36.8050 + 21.2494i) q^{3} +(37.7546 + 21.7976i) q^{4} +(48.6688 - 13.0408i) q^{5} +(311.715 + 311.715i) q^{6} +(32.6865 - 56.6146i) q^{7} +(-149.662 - 149.662i) q^{8} +(538.570 + 932.831i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 2 q^{2} - 6 q^{3} + 264 q^{4} + 52 q^{5} - 132 q^{6} - 2 q^{7} - 1968 q^{8} + 8094 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	10.0194	+	2.68468i	1.25242	+	0.335585i	0.823271	−	0.567648i	\(-0.192147\pi\)
							0.429150	+	0.903233i	\(0.358813\pi\)
\(3\)	36.8050	+	21.2494i	1.36315	+	0.787013i	0.990042	−	0.140776i	\(-0.0449597\pi\)
							0.373105	+	0.927789i	\(0.378293\pi\)
\(5\)	48.6688	−	13.0408i	0.389351	−	0.104326i	−0.0588327	−	0.998268i	\(-0.518738\pi\)
							0.448183	+	0.893942i	\(0.352071\pi\)
\(7\)	32.6865	−	56.6146i	0.0952958	−	0.165057i	−0.814436	−	0.580253i	\(-0.802954\pi\)
							0.909732	+	0.415196i	\(0.136287\pi\)
\(11\)		−	552.448i		−	0.415063i	−0.978228	−	0.207531i	\(-0.933457\pi\)
							0.978228	−	0.207531i	\(-0.0665429\pi\)
\(13\)	−3194.42	+	855.943i	−1.45399	+	0.389596i	−0.897411	−	0.441196i	\(-0.854554\pi\)
							−0.556583	+	0.830792i	\(0.687888\pi\)
\(17\)	−210.620	+	786.044i	−0.0428699	+	0.159993i	−0.984043	−	0.177934i	\(-0.943059\pi\)
							0.941173	+	0.337926i	\(0.109725\pi\)
\(19\)	6800.32	−	1822.14i	0.991445	−	0.265657i	0.273588	−	0.961847i	\(-0.411790\pi\)
							0.717858	+	0.696190i	\(0.245123\pi\)
\(23\)	6010.02	+	6010.02i	0.493960	+	0.493960i	0.909552	−	0.415591i	\(-0.136425\pi\)
							−0.415591	+	0.909552i	\(0.636425\pi\)
\(29\)	15585.3	−	15585.3i	0.639029	−	0.639029i	−0.311287	−	0.950316i	\(-0.600760\pi\)
							0.950316	+	0.311287i	\(0.100760\pi\)
\(31\)	7736.30	−	7736.30i	0.259686	−	0.259686i	−0.565240	−	0.824926i	\(-0.691217\pi\)
							0.824926	+	0.565240i	\(0.191217\pi\)
\(37\)	−46881.1	−	19180.3i	−0.925535	−	0.378661i
							\(41\)	−76224.1	−	44008.0i	−1.10596	−	0.638528i	−0.168183	−	0.985756i	\(-0.553790\pi\)
−0.937781	+	0.347227i	\(0.887123\pi\)
\(43\)	−12007.4	−	12007.4i	−0.151023	−	0.151023i	0.627552	−	0.778575i	\(-0.284057\pi\)
							−0.778575	+	0.627552i	\(0.784057\pi\)
\(47\)	99293.2			0.956370			0.478185	−	0.878259i	\(-0.341295\pi\)
							0.478185	+	0.878259i	\(0.341295\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.7.g.a.8.14	✓	72
37.14	odd	12	inner	37.7.g.a.14.14	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.7.g.a.8.14	✓	72	1.1	even	1	trivial
37.7.g.a.14.14	yes	72	37.14	odd	12	inner