Embedded newform 37.10.e.a.11.12

• Label: 37.10.e.a.11.12
• 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			11.12
Character	\(\chi\)	\(=\)	37.11
Dual form			37.10.e.a.27.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.2408 + 6.48986i) q^{2} +(62.0101 - 107.405i) q^{3} +(-171.763 + 297.503i) q^{4} +(-199.442 - 115.148i) q^{5} +1609.75i q^{6} +(-1112.71 + 1927.27i) q^{7} -11104.5i q^{8} +(2151.00 + 3725.64i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 54 q^{2} - 75 q^{3} + 7082 q^{4} - 186 q^{5} - 4233 q^{7} - 173645 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.2408	+	6.48986i	−0.496776	+	0.286814i	−0.727381	−	0.686233i	\(-0.759263\pi\)
							0.230605	+	0.973047i	\(0.425929\pi\)
\(3\)	62.0101	−	107.405i	0.441994	−	0.765556i	−0.555843	−	0.831287i	\(-0.687604\pi\)
							0.997837	+	0.0657307i	\(0.0209378\pi\)
\(5\)	−199.442	−	115.148i	−0.142709	−	0.0823933i	0.426945	−	0.904277i	\(-0.359590\pi\)
							−0.569655	+	0.821884i	\(0.692923\pi\)
\(7\)	−1112.71	+	1927.27i	−0.175162	+	0.303390i	−0.940217	−	0.340575i	\(-0.889378\pi\)
							0.765055	+	0.643965i	\(0.222712\pi\)
\(11\)	47996.9			0.988430			0.494215	−	0.869340i	\(-0.335456\pi\)
							0.494215	+	0.869340i	\(0.335456\pi\)
\(13\)	−88861.9	−	51304.4i	−0.862920	−	0.498207i	0.00206916	−	0.999998i	\(-0.499341\pi\)
							−0.864989	+	0.501791i	\(0.832675\pi\)
\(17\)	−155522.	+	89790.9i	−0.451619	+	0.260743i	−0.708514	−	0.705697i	\(-0.750634\pi\)
							0.256894	+	0.966439i	\(0.417301\pi\)
\(19\)	18163.0	+	10486.4i	0.0319740	+	0.0184602i	0.515902	−	0.856648i	\(-0.327457\pi\)
							−0.483928	+	0.875108i	\(0.660790\pi\)
\(23\)		−	2.01053e6i		−	1.49808i	−0.662525	−	0.749039i	\(-0.730515\pi\)
							0.662525	−	0.749039i	\(-0.269485\pi\)
\(29\)		−	4.51070e6i		−	1.18428i	−0.805837	−	0.592138i	\(-0.798284\pi\)
							0.805837	−	0.592138i	\(-0.201716\pi\)
\(31\)		−	9.74626e6i		−	1.89544i	−0.319101	−	0.947721i	\(-0.603381\pi\)
							0.319101	−	0.947721i	\(-0.396619\pi\)
\(37\)	−1.12387e7	+	1.91129e6i	−0.985846	+	0.167656i
							\(41\)	−9.72892e6	+	1.68510e7i	−0.537696	+	0.931317i	0.461331	+	0.887228i	\(0.347372\pi\)
−0.999028	+	0.0440894i	\(0.985961\pi\)
\(43\)		−	2.95506e7i		−	1.31813i	−0.752087	−	0.659064i	\(-0.770953\pi\)
							0.752087	−	0.659064i	\(-0.229047\pi\)
\(47\)	1.05408e7			0.315088			0.157544	−	0.987512i	\(-0.449642\pi\)
							0.157544	+	0.987512i	\(0.449642\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.e.a.11.12	✓	56
37.27	even	6	inner	37.10.e.a.27.12	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.e.a.11.12	✓	56	1.1	even	1	trivial
37.10.e.a.27.12	yes	56	37.27	even	6	inner