Embedded newform 37.8.f.a.16.11

• Label: 37.8.f.a.16.11
• Level: $37$
• Weight: $8$
• Character: 37.16
• 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			16.11
Character	\(\chi\)	\(=\)	37.16
Dual form			37.8.f.a.7.11

$q$-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 \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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	0.301474	−	0.953474i	\(-0.402521\pi\)
							−0.381939	+	0.924187i	\(0.624744\pi\)
\(3\)	19.9506	−	7.26143i	0.426611	−	0.155274i	−0.119787	−	0.992800i	\(-0.538221\pi\)
							0.546397	+	0.837526i	\(0.315999\pi\)
\(5\)	96.6240	−	547.982i	0.345693	−	1.96052i	0.0783149	−	0.996929i	\(-0.475046\pi\)
							0.267378	−	0.963592i	\(-0.413843\pi\)
\(7\)	−202.895	+	1150.68i	−0.223578	+	1.26797i	0.641807	+	0.766866i	\(0.278185\pi\)
							−0.865385	+	0.501107i	\(0.832926\pi\)
\(11\)	1003.96	+	1738.91i	0.227427	+	0.393916i	0.957045	−	0.289940i	\(-0.0936352\pi\)
							−0.729617	+	0.683855i	\(0.760302\pi\)
\(13\)	−9437.04	−	7918.62i	−1.19134	−	0.999649i	−0.999835	−	0.0181565i	\(-0.994220\pi\)
							−0.191500	−	0.981493i	\(-0.561335\pi\)
\(17\)	7564.88	−	6347.69i	0.373449	−	0.313361i	−0.436675	−	0.899619i	\(-0.643844\pi\)
							0.810124	+	0.586259i	\(0.199400\pi\)
\(19\)	−35056.7	+	12759.6i	−1.17256	+	0.426775i	−0.853567	−	0.520983i	\(-0.825565\pi\)
							−0.318989	+	0.947759i	\(0.603343\pi\)
\(23\)	40606.0	−	70331.6i	0.695893	−	1.20532i	−0.273985	−	0.961734i	\(-0.588342\pi\)
							0.969879	−	0.243589i	\(-0.0783248\pi\)
\(29\)	56618.7	+	98066.5i	0.431089	+	0.746668i	0.996967	−	0.0778207i	\(-0.0247962\pi\)
							−0.565878	+	0.824489i	\(0.691463\pi\)
\(31\)	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	0.133584	−	0.991037i	\(-0.457351\pi\)
−0.952785	+	0.303647i	\(0.901796\pi\)
\(43\)	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	−0.477273	−	0.878755i	\(-0.658375\pi\)
							0.999661	−	0.0260467i	\(-0.00829187\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.f.a.7.11	✓	132	37.7	even	9	inner
37.8.f.a.16.11	yes	132	1.1	even	1	trivial