Embedded newform 37.9.g.a.8.15

• Label: 37.9.g.a.8.15
• 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			8.15
Character	\(\chi\)	\(=\)	37.8
Dual form			37.9.g.a.14.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.07517 + 1.89579i) q^{2} +(65.9616 + 38.0829i) q^{3} +(-175.238 - 101.174i) q^{4} +(-317.418 + 85.0520i) q^{5} +(394.492 + 394.492i) q^{6} +(582.766 - 1009.38i) q^{7} +(-2373.96 - 2373.96i) q^{8} +(-379.880 - 657.971i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 10 q^{2} - 6 q^{3} + 552 q^{4} - 1642 q^{5} - 516 q^{6} - 2 q^{7} - 7440 q^{8} + 117258 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.07517	+	1.89579i	0.442198	+	0.118487i	0.473047	−	0.881037i	\(-0.343154\pi\)
							−0.0308483	+	0.999524i	\(0.509821\pi\)
\(3\)	65.9616	+	38.0829i	0.814341	+	0.470160i	0.848461	−	0.529258i	\(-0.177530\pi\)
							−0.0341204	+	0.999418i	\(0.510863\pi\)
\(5\)	−317.418	+	85.0520i	−0.507869	+	0.136083i	−0.503650	−	0.863908i	\(-0.668010\pi\)
							−0.00421959	+	0.999991i	\(0.501343\pi\)
\(7\)	582.766	−	1009.38i	0.242718	−	0.420400i	−0.718770	−	0.695248i	\(-0.755294\pi\)
							0.961488	+	0.274849i	\(0.0886277\pi\)
\(11\)		−	20328.0i		−	1.38843i	−0.719767	−	0.694216i	\(-0.755751\pi\)
							0.719767	−	0.694216i	\(-0.244249\pi\)
\(13\)	−959.056	+	256.978i	−0.0335792	+	0.00899752i	−0.275570	−	0.961281i	\(-0.588866\pi\)
							0.241990	+	0.970279i	\(0.422200\pi\)
\(17\)	1982.38	−	7398.33i	0.0237351	−	0.0885804i	−0.953042	−	0.302837i	\(-0.902066\pi\)
							0.976777	+	0.214257i	\(0.0687329\pi\)
\(19\)	78743.0	−	21099.1i	0.604223	−	0.161901i	0.0562777	−	0.998415i	\(-0.482077\pi\)
							0.547946	+	0.836514i	\(0.315410\pi\)
\(23\)	−335905.	−	335905.i	−1.20034	−	1.20034i	−0.974062	−	0.226281i	\(-0.927343\pi\)
							−0.226281	−	0.974062i	\(-0.572657\pi\)
\(29\)	351414.	−	351414.i	0.496853	−	0.496853i	−0.413604	−	0.910457i	\(-0.635730\pi\)
							0.910457	+	0.413604i	\(0.135730\pi\)
\(31\)	−32963.0	+	32963.0i	−0.0356927	+	0.0356927i	−0.724728	−	0.689035i	\(-0.758035\pi\)
							0.689035	+	0.724728i	\(0.258035\pi\)
\(37\)	−1.83927e6	+	359969.i	−0.981381	+	0.192070i
							\(41\)	2.61270e6	+	1.50844e6i	0.924600	+	0.533818i	0.885100	−	0.465401i	\(-0.154090\pi\)
0.0395004	+	0.999220i	\(0.487423\pi\)
\(43\)	−970443.	−	970443.i	−0.283855	−	0.283855i	0.550789	−	0.834644i	\(-0.314327\pi\)
							−0.834644	+	0.550789i	\(0.814327\pi\)
\(47\)	−2.62268e6			−0.537469			−0.268735	−	0.963214i	\(-0.586605\pi\)
							−0.268735	+	0.963214i	\(0.586605\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.9.g.a.8.15	✓	100
37.14	odd	12	inner	37.9.g.a.14.15	yes	100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.9.g.a.8.15	✓	100	1.1	even	1	trivial
37.9.g.a.14.15	yes	100	37.14	odd	12	inner