Embedded newform 37.10.e.a.11.7

• Label: 37.10.e.a.11.7
• Level: $37$
• Weight: $10$
• Character: 37.11
• Analytic conductor: $19.056$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	37.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0563259381\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.7
Character	\(\chi\)	\(=\)	37.11
Dual form			37.10.e.a.27.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-23.8055 + 13.7441i) q^{2} +(131.157 - 227.170i) q^{3} +(121.802 - 210.967i) q^{4} +(-1058.66 - 611.220i) q^{5} +7210.54i q^{6} +(4250.53 - 7362.13i) q^{7} -7377.74i q^{8} +(-24562.7 - 42543.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 54 q^{2} - 75 q^{3} + 7082 q^{4} - 186 q^{5} - 4233 q^{7} - 173645 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−23.8055	+	13.7441i	−1.05207	+	0.607410i	−0.923227	−	0.384255i	\(-0.874458\pi\)
							−0.128839	+	0.991666i	\(0.541125\pi\)
\(3\)	131.157	−	227.170i	0.934857	−	1.61922i	0.159969	−	0.987122i	\(-0.448860\pi\)
							0.774888	−	0.632098i	\(-0.217806\pi\)
\(5\)	−1058.66	−	611.220i	−0.757518	−	0.437353i	0.0708859	−	0.997484i	\(-0.477417\pi\)
							−0.828404	+	0.560131i	\(0.810751\pi\)
\(7\)	4250.53	−	7362.13i	0.669117	−	1.15894i	−0.309035	−	0.951051i	\(-0.600006\pi\)
							0.978151	−	0.207894i	\(-0.0666608\pi\)
\(11\)	12565.8			0.258775			0.129388	−	0.991594i	\(-0.458699\pi\)
							0.129388	+	0.991594i	\(0.458699\pi\)
\(13\)	34114.0	+	19695.7i	0.331274	+	0.191261i	0.656406	−	0.754407i	\(-0.272076\pi\)
							−0.325133	+	0.945668i	\(0.605409\pi\)
\(17\)	282120.	−	162882.i	0.819244	−	0.472991i	−0.0309117	−	0.999522i	\(-0.509841\pi\)
							0.850156	+	0.526531i	\(0.176508\pi\)
\(19\)	−430620.	−	248619.i	−0.758059	−	0.437665i	0.0705395	−	0.997509i	\(-0.477528\pi\)
							−0.828598	+	0.559843i	\(0.810861\pi\)
\(23\)			1.84297e6i			1.37323i	0.727022	+	0.686614i	\(0.240904\pi\)
							−0.727022	+	0.686614i	\(0.759096\pi\)
\(29\)			2.57034e6i			0.674837i	0.941355	+	0.337419i	\(0.109554\pi\)
							−0.941355	+	0.337419i	\(0.890446\pi\)
\(31\)		−	7.41841e6i		−	1.44272i	−0.692558	−	0.721362i	\(-0.743516\pi\)
							0.692558	−	0.721362i	\(-0.256484\pi\)
\(37\)	−4.62101e6	−	1.04215e7i	−0.405349	−	0.914162i
							\(41\)	−8.59488e6	+	1.48868e7i	−0.475021	+	0.822760i	−0.999591	−	0.0286075i	\(-0.990893\pi\)
0.524570	+	0.851367i	\(0.324226\pi\)
\(43\)			3.11304e7i			1.38860i	0.719687	+	0.694298i	\(0.244285\pi\)
							−0.719687	+	0.694298i	\(0.755715\pi\)
\(47\)	−4.47991e7			−1.33915			−0.669575	−	0.742745i	\(-0.733524\pi\)
							−0.669575	+	0.742745i	\(0.733524\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.e.a.11.7	✓	56
37.27	even	6	inner	37.10.e.a.27.7	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.e.a.11.7	✓	56	1.1	even	1	trivial
37.10.e.a.27.7	yes	56	37.27	even	6	inner