Embedded newform 37.10.c.a.10.27

• Label: 37.10.c.a.10.27
• Level: $37$
• Weight: $10$
• Character: 37.10
• Analytic conductor: $19.056$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0563259381\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(27\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			10.27
Character	\(\chi\)	\(=\)	37.10
Dual form			37.10.c.a.26.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(21.4851 - 37.2132i) q^{2} +(45.7321 + 79.2104i) q^{3} +(-667.215 - 1155.65i) q^{4} +(931.933 + 1614.15i) q^{5} +3930.23 q^{6} +(-4988.17 - 8639.76i) q^{7} -35339.9 q^{8} +(5658.65 - 9801.06i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 32 q^{2} + 73 q^{3} - 6486 q^{4} + 2729 q^{5} + 1020 q^{6} + 4231 q^{7} - 7848 q^{8} - 140840 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	21.4851	−	37.2132i	0.949514	−	1.64461i	0.203064	−	0.979165i	\(-0.434910\pi\)
							0.746450	−	0.665441i	\(-0.231757\pi\)
\(3\)	45.7321	+	79.2104i	0.325969	+	0.564594i	0.981708	−	0.190393i	\(-0.0609763\pi\)
							−0.655739	+	0.754987i	\(0.727643\pi\)
\(5\)	931.933	+	1614.15i	0.666837	+	1.15500i	0.978784	+	0.204896i	\(0.0656855\pi\)
							−0.311947	+	0.950099i	\(0.600981\pi\)
\(7\)	−4988.17	−	8639.76i	−0.785236	−	1.36007i	−0.928858	−	0.370435i	\(-0.879208\pi\)
							0.143623	−	0.989633i	\(-0.454125\pi\)
\(11\)	−62821.3			−1.29372			−0.646860	−	0.762609i	\(-0.723918\pi\)
							−0.646860	+	0.762609i	\(0.723918\pi\)
\(13\)	−57683.9	−	99911.4i	−0.560157	−	0.970220i	−0.997482	−	0.0709164i	\(-0.977408\pi\)
							0.437326	−	0.899303i	\(-0.355926\pi\)
\(17\)	−35862.8	+	62116.3i	−0.104142	+	0.180379i	−0.913387	−	0.407092i	\(-0.866543\pi\)
							0.809246	+	0.587471i	\(0.199876\pi\)
\(19\)	−244978.	−	424315.i	−0.431257	−	0.746959i	0.565725	−	0.824594i	\(-0.308596\pi\)
							−0.996982	+	0.0776351i	\(0.975263\pi\)
\(23\)	2.54480e6			1.89617			0.948087	−	0.318010i	\(-0.103014\pi\)
							0.948087	+	0.318010i	\(0.103014\pi\)
\(29\)	3.14605e6			0.825990			0.412995	−	0.910733i	\(-0.364483\pi\)
							0.412995	+	0.910733i	\(0.364483\pi\)
\(31\)	−4.15160e6			−0.807398			−0.403699	−	0.914892i	\(-0.632276\pi\)
							−0.403699	+	0.914892i	\(0.632276\pi\)
\(37\)	959953.	+	1.13596e7i	0.0842059	+	0.996448i
							\(41\)	−2.61474e6	−	4.52887e6i	−0.144511	−	0.250301i	0.784679	−	0.619902i	\(-0.212828\pi\)
−0.929190	+	0.369601i	\(0.879494\pi\)
\(43\)	9.29076e6			0.414423			0.207211	−	0.978296i	\(-0.433561\pi\)
							0.207211	+	0.978296i	\(0.433561\pi\)
\(47\)	−1.38249e7			−0.413259			−0.206630	−	0.978419i	\(-0.566249\pi\)
							−0.206630	+	0.978419i	\(0.566249\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.c.a.10.27	✓	54
37.26	even	3	inner	37.10.c.a.26.27	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.c.a.10.27	✓	54	1.1	even	1	trivial
37.10.c.a.26.27	yes	54	37.26	even	3	inner