Embedded newform 37.8.f.a.7.7

• Label: 37.8.f.a.7.7
• Level: $37$
• Weight: $8$
• Character: 37.7
• Analytic conductor: $11.558$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	37.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5582459429\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			7.7
Character	\(\chi\)	\(=\)	37.7
Dual form			37.8.f.a.16.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.0215 + 4.37547i) q^{2} +(-68.2052 - 24.8247i) q^{3} +(27.3180 - 22.9225i) q^{4} +(75.7434 + 429.562i) q^{5} +928.548 q^{6} +(226.548 + 1284.82i) q^{7} +(590.646 - 1023.03i) q^{8} +(2360.34 + 1980.56i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q + 15 q^{2} + 33 q^{3} - 279 q^{4} + 576 q^{5} - 12 q^{6} - 597 q^{7} - 3750 q^{8} - 3243 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.0215	+	4.37547i	−1.06256	+	0.386740i	−0.813390	−	0.581719i	\(-0.802380\pi\)
							−0.249171	+	0.968460i	\(0.580158\pi\)
\(3\)	−68.2052	−	24.8247i	−1.45845	−	0.530834i	−0.513515	−	0.858080i	\(-0.671657\pi\)
							−0.944939	+	0.327246i	\(0.893879\pi\)
\(5\)	75.7434	+	429.562i	0.270988	+	1.53685i	0.751424	+	0.659819i	\(0.229367\pi\)
							−0.480436	+	0.877030i	\(0.659522\pi\)
\(7\)	226.548	+	1284.82i	0.249641	+	1.41579i	0.809462	+	0.587172i	\(0.199759\pi\)
							−0.559821	+	0.828614i	\(0.689130\pi\)
\(11\)	−3990.89	+	6912.42i	−0.904055	+	1.56587i	−0.0818752	+	0.996643i	\(0.526091\pi\)
							−0.822180	+	0.569227i	\(0.807242\pi\)
\(13\)	−4135.39	+	3470.00i	−0.522053	+	0.438055i	−0.865347	−	0.501174i	\(-0.832902\pi\)
							0.343294	+	0.939228i	\(0.388457\pi\)
\(17\)	902.800	+	757.539i	0.0445677	+	0.0373967i	0.664799	−	0.747022i	\(-0.268517\pi\)
							−0.620232	+	0.784419i	\(0.712961\pi\)
\(19\)	4840.41	+	1761.76i	0.161899	+	0.0589264i	0.421698	−	0.906736i	\(-0.361434\pi\)
							−0.259799	+	0.965663i	\(0.583656\pi\)
\(23\)	33977.6	+	58851.0i	0.582299	+	1.00857i	0.995206	+	0.0977980i	\(0.0311799\pi\)
							−0.412908	+	0.910773i	\(0.635487\pi\)
\(29\)	11624.6	−	20134.3i	0.0885082	−	0.153301i	−0.818373	−	0.574688i	\(-0.805123\pi\)
							0.906881	+	0.421387i	\(0.138457\pi\)
\(31\)	−170615.			−1.02861			−0.514304	−	0.857608i	\(-0.671950\pi\)
							−0.514304	+	0.857608i	\(0.671950\pi\)
\(37\)	−79930.3	+	297562.i	−0.259421	+	0.965764i
							\(41\)	402111.	−	337411.i	0.911175	−	0.764567i	−0.0611668	−	0.998128i	\(-0.519482\pi\)
0.972342	+	0.233561i	\(0.0750377\pi\)
\(43\)	303369.			0.581878			0.290939	−	0.956742i	\(-0.406032\pi\)
							0.290939	+	0.956742i	\(0.406032\pi\)
\(47\)	2730.33	+	4729.07i	0.00383595	+	0.00664405i	0.867937	−	0.496674i	\(-0.165446\pi\)
							−0.864101	+	0.503318i	\(0.832112\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.f.a.7.7	✓	132
37.16	even	9	inner	37.8.f.a.16.7	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.f.a.7.7	✓	132	1.1	even	1	trivial
37.8.f.a.16.7	yes	132	37.16	even	9	inner