Embedded newform 37.9.g.a.8.6

• Label: 37.9.g.a.8.6
• Level: $37$
• Weight: $9$
• Character: 37.8
• Analytic conductor: $15.073$
• Analytic rank: $0$
• Dimension: $100$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	37.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.0730085723\)
Analytic rank:	\(0\)
Dimension:	\(100\)
Relative dimension:	\(25\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			8.6
Character	\(\chi\)	\(=\)	37.8
Dual form			37.9.g.a.14.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-19.4436 - 5.20990i) q^{2} +(-78.9822 - 45.6004i) q^{3} +(129.209 + 74.5989i) q^{4} +(-557.634 + 149.418i) q^{5} +(1298.13 + 1298.13i) q^{6} +(-564.669 + 978.035i) q^{7} +(1520.19 + 1520.19i) q^{8} +(878.292 + 1521.25i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 10 q^{2} - 6 q^{3} + 552 q^{4} - 1642 q^{5} - 516 q^{6} - 2 q^{7} - 7440 q^{8} + 117258 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−19.4436	−	5.20990i	−1.21523	−	0.325619i	−0.406416	−	0.913688i	\(-0.633222\pi\)
							−0.808811	+	0.588069i	\(0.799888\pi\)
\(3\)	−78.9822	−	45.6004i	−0.975089	−	0.562968i	−0.0743050	−	0.997236i	\(-0.523674\pi\)
							−0.900784	+	0.434268i	\(0.857007\pi\)
\(5\)	−557.634	+	149.418i	−0.892214	+	0.239068i	−0.675670	−	0.737205i	\(-0.736145\pi\)
							−0.216545	+	0.976273i	\(0.569479\pi\)
\(7\)	−564.669	+	978.035i	−0.235181	+	0.407345i	−0.959325	−	0.282303i	\(-0.908901\pi\)
							0.724145	+	0.689648i	\(0.242235\pi\)
\(11\)			26730.4i			1.82572i	0.408273	+	0.912860i	\(0.366131\pi\)
							−0.408273	+	0.912860i	\(0.633869\pi\)
\(13\)	−33735.2	+	9039.32i	−1.18116	+	0.316492i	−0.795388	−	0.606101i	\(-0.792733\pi\)
							−0.385775	+	0.922593i	\(0.626066\pi\)
\(17\)	1510.68	−	5637.93i	0.0180874	−	0.0675031i	−0.956292	−	0.292413i	\(-0.905542\pi\)
							0.974380	+	0.224909i	\(0.0722086\pi\)
\(19\)	26208.4	−	7022.51i	0.201106	−	0.0538862i	−0.156860	−	0.987621i	\(-0.550137\pi\)
							0.357966	+	0.933735i	\(0.383470\pi\)
\(23\)	−351700.	−	351700.i	−1.25678	−	1.25678i	−0.952619	−	0.304165i	\(-0.901623\pi\)
							−0.304165	−	0.952619i	\(-0.598377\pi\)
\(29\)	686043.	−	686043.i	0.969973	−	0.969973i	−0.0295895	−	0.999562i	\(-0.509420\pi\)
							0.999562	+	0.0295895i	\(0.00942002\pi\)
\(31\)	−923871.	+	923871.i	−1.00038	+	1.00038i	−0.000378974		1.00000i	\(0.500121\pi\)
							−1.00000		0.000378974i	\(0.999879\pi\)
\(37\)	−1.72308e6	+	737205.i	−0.919388	+	0.393352i
							\(41\)	−1.05781e6	−	610729.i	−0.374347	−	0.216129i	0.301009	−	0.953621i	\(-0.402677\pi\)
−0.675356	+	0.737492i	\(0.736010\pi\)
\(43\)	1.72654e6	+	1.72654e6i	0.505014	+	0.505014i	0.912992	−	0.407978i	\(-0.133766\pi\)
							−0.407978	+	0.912992i	\(0.633766\pi\)
\(47\)	1.86380e6			0.381951			0.190975	−	0.981595i	\(-0.438835\pi\)
							0.190975	+	0.981595i	\(0.438835\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.9.g.a.8.6	✓	100
37.14	odd	12	inner	37.9.g.a.14.6	yes	100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.9.g.a.8.6	✓	100	1.1	even	1	trivial
37.9.g.a.14.6	yes	100	37.14	odd	12	inner