Embedded newform 37.9.g.a.8.8

• Label: 37.9.g.a.8.8
• 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			8.8
Character	\(\chi\)	\(=\)	37.8
Dual form			37.9.g.a.14.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-14.5407 - 3.89616i) q^{2} +(-127.969 - 73.8828i) q^{3} +(-25.4520 - 14.6947i) q^{4} +(528.293 - 141.556i) q^{5} +(1572.89 + 1572.89i) q^{6} +(-30.8038 + 53.3538i) q^{7} +(3037.83 + 3037.83i) q^{8} +(7636.83 + 13227.4i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 10 q^{2} - 6 q^{3} + 552 q^{4} - 1642 q^{5} - 516 q^{6} - 2 q^{7} - 7440 q^{8} + 117258 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−14.5407	−	3.89616i	−0.908791	−	0.243510i	−0.226003	−	0.974127i	\(-0.572566\pi\)
							−0.682788	+	0.730617i	\(0.739233\pi\)
\(3\)	−127.969	−	73.8828i	−1.57986	−	0.912133i	−0.994877	−	0.101088i	\(-0.967768\pi\)
							−0.584984	−	0.811045i	\(-0.698899\pi\)
\(5\)	528.293	−	141.556i	0.845269	−	0.226489i	0.189905	−	0.981802i	\(-0.439182\pi\)
							0.655364	+	0.755313i	\(0.272515\pi\)
\(7\)	−30.8038	+	53.3538i	−0.0128296	+	0.0222215i	−0.872369	−	0.488848i	\(-0.837417\pi\)
							0.859539	+	0.511070i	\(0.170751\pi\)
\(11\)		−	7049.67i		−	0.481502i	−0.970587	−	0.240751i	\(-0.922606\pi\)
							0.970587	−	0.240751i	\(-0.0773937\pi\)
\(13\)	28251.0	−	7569.83i	0.989146	−	0.265041i	0.272254	−	0.962225i	\(-0.412231\pi\)
							0.716892	+	0.697184i	\(0.245564\pi\)
\(17\)	−5283.88	+	19719.7i	−0.0632641	+	0.236105i	−0.990317	−	0.138826i	\(-0.955667\pi\)
							0.927053	+	0.374931i	\(0.122334\pi\)
\(19\)	100545.	−	26940.9i	0.771516	−	0.206727i	0.148475	−	0.988916i	\(-0.452564\pi\)
							0.623041	+	0.782189i	\(0.285897\pi\)
\(23\)	188843.	+	188843.i	0.674824	+	0.674824i	0.958824	−	0.284000i	\(-0.0916616\pi\)
							−0.284000	+	0.958824i	\(0.591662\pi\)
\(29\)	−507323.	+	507323.i	−0.717286	+	0.717286i	−0.968049	−	0.250762i	\(-0.919319\pi\)
							0.250762	+	0.968049i	\(0.419319\pi\)
\(31\)	1.05188e6	−	1.05188e6i	1.13899	−	1.13899i	0.150361	−	0.988631i	\(-0.451956\pi\)
							0.988631	−	0.150361i	\(-0.0480437\pi\)
\(37\)	−1.86397e6	+	195201.i	−0.994561	+	0.104154i
							\(41\)	3.78821e6	+	2.18712e6i	1.34060	+	0.773994i	0.986895	−	0.161365i	\(-0.0515895\pi\)
0.353702	+	0.935358i	\(0.384923\pi\)
\(43\)	1.03279e6	+	1.03279e6i	0.302092	+	0.302092i	0.841832	−	0.539740i	\(-0.181477\pi\)
							−0.539740	+	0.841832i	\(0.681477\pi\)
\(47\)	3.40916e6			0.698645			0.349322	−	0.937003i	\(-0.386412\pi\)
							0.349322	+	0.937003i	\(0.386412\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.9.g.a.8.8	✓	100
37.14	odd	12	inner	37.9.g.a.14.8	yes	100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.9.g.a.8.8	✓	100	1.1	even	1	trivial
37.9.g.a.14.8	yes	100	37.14	odd	12	inner