Embedded newform 37.10.e.a.11.3

• Label: 37.10.e.a.11.3
• 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			11.3
Character	\(\chi\)	\(=\)	37.11
Dual form			37.10.e.a.27.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-31.1498 + 17.9843i) q^{2} +(57.1110 - 98.9191i) q^{3} +(390.874 - 677.013i) q^{4} +(-1648.13 - 951.547i) q^{5} +4108.41i q^{6} +(-5484.32 + 9499.11i) q^{7} +9702.46i q^{8} +(3318.18 + 5747.25i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 54 q^{2} - 75 q^{3} + 7082 q^{4} - 186 q^{5} - 4233 q^{7} - 173645 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−31.1498	+	17.9843i	−1.37664	+	0.794803i	−0.991753	−	0.128160i	\(-0.959093\pi\)
							−0.384887	+	0.922964i	\(0.625759\pi\)
\(3\)	57.1110	−	98.9191i	0.407074	−	0.705074i	−0.587486	−	0.809234i	\(-0.699882\pi\)
							0.994560	+	0.104161i	\(0.0332156\pi\)
\(5\)	−1648.13	−	951.547i	−1.17930	−	0.680872i	−0.223451	−	0.974715i	\(-0.571732\pi\)
							−0.955854	+	0.293843i	\(0.905066\pi\)
\(7\)	−5484.32	+	9499.11i	−0.863339	+	1.49535i	0.00534817	+	0.999986i	\(0.498298\pi\)
							−0.868687	+	0.495361i	\(0.835036\pi\)
\(11\)	−69306.4			−1.42727			−0.713635	−	0.700518i	\(-0.752952\pi\)
							−0.713635	+	0.700518i	\(0.752952\pi\)
\(13\)	−57988.0	−	33479.4i	−0.563110	−	0.325112i	0.191283	−	0.981535i	\(-0.438735\pi\)
							−0.754393	+	0.656423i	\(0.772069\pi\)
\(17\)	331810.	−	191570.i	0.963538	−	0.556299i	0.0662778	−	0.997801i	\(-0.478888\pi\)
							0.897260	+	0.441502i	\(0.145554\pi\)
\(19\)	−122291.	−	70604.5i	−0.215279	−	0.124292i	0.388483	−	0.921456i	\(-0.372999\pi\)
							−0.603762	+	0.797164i	\(0.706332\pi\)
\(23\)		−	837729.i		−	0.624207i	−0.950048	−	0.312103i	\(-0.898966\pi\)
							0.950048	−	0.312103i	\(-0.101034\pi\)
\(29\)			3.32427e6i			0.872781i	0.899757	+	0.436391i	\(0.143743\pi\)
							−0.899757	+	0.436391i	\(0.856257\pi\)
\(31\)		−	1.60611e6i		−	0.312354i	−0.987729	−	0.156177i	\(-0.950083\pi\)
							0.987729	−	0.156177i	\(-0.0499171\pi\)
\(37\)	5.08722e6	+	1.02021e7i	0.446244	+	0.894911i
							\(41\)	1.05604e7	−	1.82912e7i	0.583653	−	1.01092i	−0.411389	−	0.911460i	\(-0.634956\pi\)
0.995042	−	0.0994571i	\(-0.0317106\pi\)
\(43\)		−	2.07782e7i		−	0.926831i	−0.886141	−	0.463415i	\(-0.846624\pi\)
							0.886141	−	0.463415i	\(-0.153376\pi\)
\(47\)	7.75710e6			0.231878			0.115939	−	0.993256i	\(-0.463012\pi\)
							0.115939	+	0.993256i	\(0.463012\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.e.a.11.3	✓	56
37.27	even	6	inner	37.10.e.a.27.3	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.e.a.11.3	✓	56	1.1	even	1	trivial
37.10.e.a.27.3	yes	56	37.27	even	6	inner