Embedded newform 37.9.g.a.8.4

• Label: 37.9.g.a.8.4
• 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			8.4
Character	\(\chi\)	\(=\)	37.8
Dual form			37.9.g.a.14.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-23.7708 - 6.36938i) q^{2} +(94.2638 + 54.4232i) q^{3} +(302.782 + 174.811i) q^{4} +(460.453 - 123.378i) q^{5} +(-1894.09 - 1894.09i) q^{6} +(1918.35 - 3322.68i) q^{7} +(-1629.17 - 1629.17i) q^{8} +(2643.27 + 4578.28i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 10 q^{2} - 6 q^{3} + 552 q^{4} - 1642 q^{5} - 516 q^{6} - 2 q^{7} - 7440 q^{8} + 117258 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−23.7708	−	6.36938i	−1.48568	−	0.398086i	−0.577403	−	0.816459i	\(-0.695934\pi\)
							−0.908275	+	0.418373i	\(0.862600\pi\)
\(3\)	94.2638	+	54.4232i	1.16375	+	0.671892i	0.952200	−	0.305476i	\(-0.0988154\pi\)
							0.211550	+	0.977367i	\(0.432149\pi\)
\(5\)	460.453	−	123.378i	0.736725	−	0.197405i	0.129103	−	0.991631i	\(-0.458790\pi\)
							0.607622	+	0.794226i	\(0.292123\pi\)
\(7\)	1918.35	−	3322.68i	0.798980	−	1.38387i	−0.121301	−	0.992616i	\(-0.538707\pi\)
							0.920281	−	0.391258i	\(-0.127960\pi\)
\(11\)			7025.05i			0.479820i	0.970795	+	0.239910i	\(0.0771180\pi\)
							−0.970795	+	0.239910i	\(0.922882\pi\)
\(13\)	−9378.66	+	2513.00i	−0.328373	+	0.0879873i	−0.419239	−	0.907876i	\(-0.637703\pi\)
							0.0908662	+	0.995863i	\(0.471036\pi\)
\(17\)	17936.7	−	66940.5i	0.214756	−	0.801482i	−0.771496	−	0.636234i	\(-0.780491\pi\)
							0.986252	−	0.165247i	\(-0.0528422\pi\)
\(19\)	54137.9	−	14506.2i	0.415420	−	0.111311i	−0.0450544	−	0.998985i	\(-0.514346\pi\)
							0.460474	+	0.887673i	\(0.347679\pi\)
\(23\)	−33739.8	−	33739.8i	−0.120568	−	0.120568i	0.644249	−	0.764816i	\(-0.277170\pi\)
							−0.764816	+	0.644249i	\(0.777170\pi\)
\(29\)	732206.	−	732206.i	1.03524	−	1.03524i	0.0358843	−	0.999356i	\(-0.488575\pi\)
							0.999356	−	0.0358843i	\(-0.0114248\pi\)
\(31\)	496638.	−	496638.i	0.537766	−	0.537766i	−0.385106	−	0.922872i	\(-0.625835\pi\)
							0.922872	+	0.385106i	\(0.125835\pi\)
\(37\)	−697983.	+	1.73934e6i	−0.372424	+	0.928063i
							\(41\)	4.25964e6	+	2.45930e6i	1.50743	+	0.870316i	0.999963	+	0.00864573i	\(0.00275206\pi\)
0.507469	+	0.861670i	\(0.330581\pi\)
\(43\)	107546.	+	107546.i	0.0314574	+	0.0314574i	0.722661	−	0.691203i	\(-0.242919\pi\)
							−0.691203	+	0.722661i	\(0.742919\pi\)
\(47\)	5.53554e6			1.13441			0.567203	−	0.823578i	\(-0.308025\pi\)
							0.567203	+	0.823578i	\(0.308025\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.9.g.a.8.4	✓	100
37.14	odd	12	inner	37.9.g.a.14.4	yes	100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.9.g.a.8.4	✓	100	1.1	even	1	trivial
37.9.g.a.14.4	yes	100	37.14	odd	12	inner