Embedded newform 37.8.f.a.7.17

• Label: 37.8.f.a.7.17
• 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			7.17
Character	\(\chi\)	\(=\)	37.7
Dual form			37.8.f.a.16.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.2566 - 3.73308i) q^{2} +(-19.4495 - 7.07902i) q^{3} +(-6.79263 + 5.69969i) q^{4} +(7.78053 + 44.1256i) q^{5} -225.911 q^{6} +(-27.0245 - 153.264i) q^{7} +(-746.939 + 1293.74i) q^{8} +(-1347.17 - 1130.41i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q + 15 q^{2} + 33 q^{3} - 279 q^{4} + 576 q^{5} - 12 q^{6} - 597 q^{7} - 3750 q^{8} - 3243 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	10.2566	−	3.73308i	0.906560	−	0.329961i	0.153682	−	0.988120i	\(-0.450887\pi\)
							0.752879	+	0.658160i	\(0.228665\pi\)
\(3\)	−19.4495	−	7.07902i	−0.415894	−	0.151373i	0.125593	−	0.992082i	\(-0.459917\pi\)
							−0.541488	+	0.840709i	\(0.682139\pi\)
\(5\)	7.78053	+	44.1256i	0.0278365	+	0.157868i	0.995558	−	0.0941553i	\(-0.0300150\pi\)
							−0.967721	+	0.252024i	\(0.918904\pi\)
\(7\)	−27.0245	−	153.264i	−0.0297793	−	0.168887i	0.966291	−	0.257452i	\(-0.0828829\pi\)
							−0.996070	+	0.0885652i	\(0.971772\pi\)
\(11\)	−2705.34	+	4685.79i	−0.612842	+	1.06147i	0.377918	+	0.925839i	\(0.376640\pi\)
							−0.990759	+	0.135633i	\(0.956693\pi\)
\(13\)	−8543.72	+	7169.03i	−1.07856	+	0.905022i	−0.995801	−	0.0915477i	\(-0.970819\pi\)
							−0.0827621	+	0.996569i	\(0.526374\pi\)
\(17\)	6729.32	+	5646.57i	0.332200	+	0.278749i	0.793596	−	0.608445i	\(-0.208207\pi\)
							−0.461395	+	0.887195i	\(0.652651\pi\)
\(19\)	18120.0	+	6595.15i	0.606068	+	0.220591i	0.626782	−	0.779195i	\(-0.284372\pi\)
							−0.0207140	+	0.999785i	\(0.506594\pi\)
\(23\)	−51284.0	−	88826.4i	−0.878889	−	1.52228i	−0.852561	−	0.522627i	\(-0.824952\pi\)
							−0.0263280	−	0.999653i	\(-0.508381\pi\)
\(29\)	−25745.7	+	44592.9i	−0.196025	+	0.339525i	−0.947236	−	0.320537i	\(-0.896137\pi\)
							0.751211	+	0.660062i	\(0.229470\pi\)
\(31\)	−269079.			−1.62224			−0.811118	−	0.584882i	\(-0.801141\pi\)
							−0.811118	+	0.584882i	\(0.801141\pi\)
\(37\)	−275058.	−	138833.i	−0.892728	−	0.450597i
							\(41\)	340316.	−	285559.i	0.771151	−	0.647072i	−0.169853	−	0.985469i	\(-0.554329\pi\)
0.941003	+	0.338397i	\(0.109885\pi\)
\(43\)	668153.			1.28155			0.640776	−	0.767728i	\(-0.278613\pi\)
							0.640776	+	0.767728i	\(0.278613\pi\)
\(47\)	70323.5	+	121804.i	0.0988002	+	0.171127i	0.911188	−	0.411990i	\(-0.135166\pi\)
							−0.812388	+	0.583117i	\(0.801833\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.f.a.7.17	✓	132
37.16	even	9	inner	37.8.f.a.16.17	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.f.a.7.17	✓	132	1.1	even	1	trivial
37.8.f.a.16.17	yes	132	37.16	even	9	inner