Embedded newform 37.8.f.a.7.16

• Label: 37.8.f.a.7.16
• 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			7.16
Character	\(\chi\)	\(=\)	37.7
Dual form			37.8.f.a.16.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.1961 - 3.71107i) q^{2} +(11.1844 + 4.07079i) q^{3} +(-7.86580 + 6.60019i) q^{4} +(-64.6668 - 366.744i) q^{5} +129.144 q^{6} +(-94.8119 - 537.705i) q^{7} +(-750.134 + 1299.27i) q^{8} +(-1566.82 - 1314.72i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q + 15 q^{2} + 33 q^{3} - 279 q^{4} + 576 q^{5} - 12 q^{6} - 597 q^{7} - 3750 q^{8} - 3243 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	10.1961	−	3.71107i	0.901214	−	0.328015i	0.150475	−	0.988614i	\(-0.451920\pi\)
							0.750739	+	0.660599i	\(0.229698\pi\)
\(3\)	11.1844	+	4.07079i	0.239160	+	0.0870470i	0.458819	−	0.888530i	\(-0.348272\pi\)
							−0.219660	+	0.975577i	\(0.570495\pi\)
\(5\)	−64.6668	−	366.744i	−0.231359	−	1.31210i	−0.850148	−	0.526544i	\(-0.823487\pi\)
							0.618789	−	0.785557i	\(-0.287624\pi\)
\(7\)	−94.8119	−	537.705i	−0.104477	−	0.592517i	−0.991428	−	0.130654i	\(-0.958292\pi\)
							0.886951	−	0.461863i	\(-0.152819\pi\)
\(11\)	2403.45	−	4162.90i	0.544454	−	0.943022i	−0.454187	−	0.890906i	\(-0.650070\pi\)
							0.998641	−	0.0521159i	\(-0.0165965\pi\)
\(13\)	4334.59	−	3637.16i	0.547201	−	0.459156i	−0.326791	−	0.945097i	\(-0.605967\pi\)
							0.873992	+	0.485941i	\(0.161523\pi\)
\(17\)	−4518.62	−	3791.57i	−0.223067	−	0.187175i	0.524405	−	0.851469i	\(-0.324288\pi\)
							−0.747471	+	0.664294i	\(0.768732\pi\)
\(19\)	−21103.8	−	7681.14i	−0.705866	−	0.256914i	−0.0359526	−	0.999353i	\(-0.511447\pi\)
							−0.669913	+	0.742439i	\(0.733669\pi\)
\(23\)	21018.4	+	36405.0i	0.360208	+	0.623898i	0.987995	−	0.154487i	\(-0.0493725\pi\)
							−0.627787	+	0.778385i	\(0.716039\pi\)
\(29\)	−6832.07	+	11833.5i	−0.0520187	+	0.0900990i	−0.890862	−	0.454274i	\(-0.849899\pi\)
							0.838844	+	0.544373i	\(0.183232\pi\)
\(31\)	167088.			1.00735			0.503673	−	0.863894i	\(-0.331982\pi\)
							0.503673	+	0.863894i	\(0.331982\pi\)
\(37\)	−41961.6	+	305239.i	−0.136190	+	0.990683i
							\(41\)	74980.2	−	62915.8i	0.169904	−	0.142566i	−0.553871	−	0.832602i	\(-0.686850\pi\)
0.723775	+	0.690036i	\(0.242405\pi\)
\(43\)	708332.			1.35862			0.679309	−	0.733852i	\(-0.262280\pi\)
							0.679309	+	0.733852i	\(0.262280\pi\)
\(47\)	51231.9	+	88736.3i	0.0719777	+	0.124669i	0.899768	−	0.436369i	\(-0.143736\pi\)
							−0.827790	+	0.561038i	\(0.810402\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.f.a.7.16	✓	132
37.16	even	9	inner	37.8.f.a.16.16	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.f.a.7.16	✓	132	1.1	even	1	trivial
37.8.f.a.16.16	yes	132	37.16	even	9	inner