Embedded newform 37.8.c.a.10.8

• Label: 37.8.c.a.10.8
• Level: $37$
• Weight: $8$
• Character: 37.10
• Analytic conductor: $11.558$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5582459429\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(21\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			10.8
Character	\(\chi\)	\(=\)	37.10
Dual form			37.8.c.a.26.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.42160 + 5.92638i) q^{2} +(28.0949 + 48.6617i) q^{3} +(40.5853 + 70.2959i) q^{4} +(-242.165 - 419.442i) q^{5} -384.517 q^{6} +(-642.579 - 1112.98i) q^{7} -1431.40 q^{8} +(-485.141 + 840.289i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 16 q^{2} - 68 q^{3} - 1126 q^{4} - 337 q^{5} + 252 q^{6} - 158 q^{7} + 6264 q^{8} - 12773 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.42160	+	5.92638i	−0.302429	+	0.523823i	−0.976686	−	0.214674i	\(-0.931131\pi\)
							0.674256	+	0.738497i	\(0.264464\pi\)
\(3\)	28.0949	+	48.6617i	0.600762	+	1.04055i	0.992706	+	0.120561i	\(0.0384693\pi\)
							−0.391944	+	0.919989i	\(0.628197\pi\)
\(5\)	−242.165	−	419.442i	−0.866395	−	1.50064i	−0.865655	−	0.500641i	\(-0.833098\pi\)
							−0.000739881	−	1.00000i	\(-0.500236\pi\)
\(7\)	−642.579	−	1112.98i	−0.708082	−	1.22643i	−0.965568	−	0.260152i	\(-0.916227\pi\)
							0.257486	−	0.966282i	\(-0.417106\pi\)
\(11\)	763.881			0.173042			0.0865209	−	0.996250i	\(-0.472425\pi\)
							0.0865209	+	0.996250i	\(0.472425\pi\)
\(13\)	−1834.10	−	3176.75i	−0.231537	−	0.401034i	0.726724	−	0.686930i	\(-0.241042\pi\)
							−0.958261	+	0.285896i	\(0.907709\pi\)
\(17\)	−14076.7	+	24381.5i	−0.694911	+	1.20362i	0.275299	+	0.961359i	\(0.411223\pi\)
							−0.970211	+	0.242263i	\(0.922110\pi\)
\(19\)	−25140.5	−	43544.7i	−0.840886	−	1.45646i	−0.889147	−	0.457622i	\(-0.848701\pi\)
							0.0482610	−	0.998835i	\(-0.484632\pi\)
\(23\)	15563.0			0.266714			0.133357	−	0.991068i	\(-0.457424\pi\)
							0.133357	+	0.991068i	\(0.457424\pi\)
\(29\)	109671.			0.835024			0.417512	−	0.908671i	\(-0.362902\pi\)
							0.417512	+	0.908671i	\(0.362902\pi\)
\(31\)	−120814.			−0.728368			−0.364184	−	0.931327i	\(-0.618652\pi\)
							−0.364184	+	0.931327i	\(0.618652\pi\)
\(37\)	−269063.	+	150123.i	−0.873270	+	0.487237i
							\(41\)	−241952.	−	419073.i	−0.548259	−	0.949613i	−0.998394	−	0.0566521i	\(-0.981957\pi\)
0.450135	−	0.892961i	\(-0.351376\pi\)
\(43\)	−571228.			−1.09564			−0.547822	−	0.836595i	\(-0.684543\pi\)
							−0.547822	+	0.836595i	\(0.684543\pi\)
\(47\)	1.19239e6			1.67524			0.837618	−	0.546257i	\(-0.183948\pi\)
							0.837618	+	0.546257i	\(0.183948\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.c.a.10.8	✓	42
37.26	even	3	inner	37.8.c.a.26.8	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.c.a.10.8	✓	42	1.1	even	1	trivial
37.8.c.a.26.8	yes	42	37.26	even	3	inner