Embedded newform 37.10.e.a.11.10

• Label: 37.10.e.a.11.10
• 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			11.10
Character	\(\chi\)	\(=\)	37.11
Dual form			37.10.e.a.27.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-16.0906 + 9.28991i) q^{2} +(-49.1825 + 85.1865i) q^{3} +(-83.3953 + 144.445i) q^{4} +(-2260.08 - 1304.86i) q^{5} -1827.60i q^{6} +(2409.21 - 4172.87i) q^{7} -12611.8i q^{8} +(5003.67 + 8666.61i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 54 q^{2} - 75 q^{3} + 7082 q^{4} - 186 q^{5} - 4233 q^{7} - 173645 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−16.0906	+	9.28991i	−0.711110	+	0.410560i	−0.811472	−	0.584391i	\(-0.801333\pi\)
							0.100362	+	0.994951i	\(0.468000\pi\)
\(3\)	−49.1825	+	85.1865i	−0.350562	+	0.607191i	−0.986348	−	0.164674i	\(-0.947343\pi\)
							0.635786	+	0.771865i	\(0.280676\pi\)
\(5\)	−2260.08	−	1304.86i	−1.61718	−	0.933679i	−0.987645	−	0.156705i	\(-0.949913\pi\)
							−0.629534	−	0.776973i	\(-0.716754\pi\)
\(7\)	2409.21	−	4172.87i	0.379257	−	0.656892i	−0.611697	−	0.791092i	\(-0.709513\pi\)
							0.990954	+	0.134200i	\(0.0428464\pi\)
\(11\)	17234.5			0.354920			0.177460	−	0.984128i	\(-0.443212\pi\)
							0.177460	+	0.984128i	\(0.443212\pi\)
\(13\)	−125236.	−	72305.2i	−1.21614	−	0.702141i	−0.252053	−	0.967713i	\(-0.581106\pi\)
							−0.964091	+	0.265572i	\(0.914439\pi\)
\(17\)	−153124.	+	88406.0i	−0.444654	+	0.256721i	−0.705570	−	0.708641i	\(-0.749309\pi\)
							0.260916	+	0.965362i	\(0.415975\pi\)
\(19\)	−524199.	−	302647.i	−0.922795	−	0.532776i	−0.0382693	−	0.999267i	\(-0.512184\pi\)
							−0.884526	+	0.466492i	\(0.845518\pi\)
\(23\)			1.81465e6i			1.35213i	0.736843	+	0.676064i	\(0.236316\pi\)
							−0.736843	+	0.676064i	\(0.763684\pi\)
\(29\)		−	6.52618e6i		−	1.71344i	−0.515784	−	0.856719i	\(-0.672499\pi\)
							0.515784	−	0.856719i	\(-0.327501\pi\)
\(31\)			6.73170e6i			1.30917i	0.755987	+	0.654587i	\(0.227157\pi\)
							−0.755987	+	0.654587i	\(0.772843\pi\)
\(37\)	1.12224e7	+	2.00509e6i	0.984411	+	0.175884i
							\(41\)	1.19563e7	−	2.07090e7i	0.660801	−	1.14454i	−0.319605	−	0.947551i	\(-0.603550\pi\)
0.980406	−	0.196990i	\(-0.0631166\pi\)
\(43\)			1.79395e7i			0.800205i	0.916470	+	0.400103i	\(0.131026\pi\)
							−0.916470	+	0.400103i	\(0.868974\pi\)
\(47\)	2.48693e7			0.743402			0.371701	−	0.928353i	\(-0.378775\pi\)
							0.371701	+	0.928353i	\(0.378775\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.e.a.11.10	✓	56
37.27	even	6	inner	37.10.e.a.27.10	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.e.a.11.10	✓	56	1.1	even	1	trivial
37.10.e.a.27.10	yes	56	37.27	even	6	inner