Embedded newform 37.8.f.a.7.11

• Label: 37.8.f.a.7.11
• 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			7.11
Character	\(\chi\)	\(=\)	37.7
Dual form			37.8.f.a.16.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.910359 + 0.331344i) q^{2} +(19.9506 + 7.26143i) q^{3} +(-97.3347 + 81.6735i) q^{4} +(96.6240 + 547.982i) q^{5} -20.5683 q^{6} +(-202.895 - 1150.68i) q^{7} +(123.550 - 213.994i) q^{8} +(-1330.04 - 1116.04i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q + 15 q^{2} + 33 q^{3} - 279 q^{4} + 576 q^{5} - 12 q^{6} - 597 q^{7} - 3750 q^{8} - 3243 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.910359	+	0.331344i	−0.0804652	+	0.0292869i	−0.381939	−	0.924187i	\(-0.624744\pi\)
							0.301474	+	0.953474i	\(0.402521\pi\)
\(3\)	19.9506	+	7.26143i	0.426611	+	0.155274i	0.546397	−	0.837526i	\(-0.315999\pi\)
							−0.119787	+	0.992800i	\(0.538221\pi\)
\(5\)	96.6240	+	547.982i	0.345693	+	1.96052i	0.267378	+	0.963592i	\(0.413843\pi\)
							0.0783149	+	0.996929i	\(0.475046\pi\)
\(7\)	−202.895	−	1150.68i	−0.223578	−	1.26797i	−0.865385	−	0.501107i	\(-0.832926\pi\)
							0.641807	−	0.766866i	\(-0.278185\pi\)
\(11\)	1003.96	−	1738.91i	0.227427	−	0.393916i	−0.729617	−	0.683855i	\(-0.760302\pi\)
							0.957045	+	0.289940i	\(0.0936352\pi\)
\(13\)	−9437.04	+	7918.62i	−1.19134	+	0.999649i	−0.191500	+	0.981493i	\(0.561335\pi\)
							−0.999835	+	0.0181565i	\(0.994220\pi\)
\(17\)	7564.88	+	6347.69i	0.373449	+	0.313361i	0.810124	−	0.586259i	\(-0.199400\pi\)
							−0.436675	+	0.899619i	\(0.643844\pi\)
\(19\)	−35056.7	−	12759.6i	−1.17256	−	0.426775i	−0.318989	−	0.947759i	\(-0.603343\pi\)
							−0.853567	+	0.520983i	\(0.825565\pi\)
\(23\)	40606.0	+	70331.6i	0.695893	+	1.20532i	0.969879	+	0.243589i	\(0.0783248\pi\)
							−0.273985	+	0.961734i	\(0.588342\pi\)
\(29\)	56618.7	−	98066.5i	0.431089	−	0.746668i	−0.565878	−	0.824489i	\(-0.691463\pi\)
							0.996967	+	0.0778207i	\(0.0247962\pi\)
\(31\)	30741.1			0.185333			0.0926666	−	0.995697i	\(-0.470461\pi\)
							0.0926666	+	0.995697i	\(0.470461\pi\)
\(37\)	−270931.	+	146725.i	−0.879331	+	0.476211i
							\(41\)	−361521.	+	303352.i	−0.819200	+	0.687391i	−0.952785	−	0.303647i	\(-0.901796\pi\)
0.133584	+	0.991037i	\(0.457351\pi\)
\(43\)	226219.			0.433900			0.216950	−	0.976183i	\(-0.430389\pi\)
							0.216950	+	0.976183i	\(0.430389\pi\)
\(47\)	371822.	+	644015.i	0.522387	+	0.904802i	0.999661	+	0.0260467i	\(0.00829187\pi\)
							−0.477273	+	0.878755i	\(0.658375\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.f.a.7.11	✓	132
37.16	even	9	inner	37.8.f.a.16.11	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.f.a.7.11	✓	132	1.1	even	1	trivial
37.8.f.a.16.11	yes	132	37.16	even	9	inner