Embedded newform 37.10.f.a.7.3

• Label: 37.10.f.a.7.3
• Level: $37$
• Weight: $10$
• Character: 37.7
• Analytic conductor: $19.056$
• Analytic rank: $0$
• Dimension: $162$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	37.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0563259381\)
Analytic rank:	\(0\)
Dimension:	\(162\)
Relative dimension:	\(27\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			7.3
Character	\(\chi\)	\(=\)	37.7
Dual form			37.10.f.a.16.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-34.4384 + 12.5346i) q^{2} +(-33.2455 - 12.1004i) q^{3} +(636.676 - 534.235i) q^{4} +(217.968 + 1236.16i) q^{5} +1296.60 q^{6} +(860.947 + 4882.67i) q^{7} +(-5847.68 + 10128.5i) q^{8} +(-14119.2 - 11847.4i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 162 q - 57 q^{2} - 228 q^{3} + 249 q^{4} - 3420 q^{5} - 12 q^{6} + 5808 q^{7} + 61698 q^{8} - 30198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−34.4384	+	12.5346i	−1.52198	+	0.553955i	−0.961641	−	0.274310i	\(-0.911551\pi\)
							−0.560337	+	0.828265i	\(0.689328\pi\)
\(3\)	−33.2455	−	12.1004i	−0.236967	−	0.0862488i	0.220807	−	0.975317i	\(-0.429131\pi\)
							−0.457774	+	0.889069i	\(0.651353\pi\)
\(5\)	217.968	+	1236.16i	0.155965	+	0.884523i	0.957898	+	0.287109i	\(0.0926941\pi\)
							−0.801933	+	0.597414i	\(0.796195\pi\)
\(7\)	860.947	+	4882.67i	0.135530	+	0.768628i	0.974489	+	0.224434i	\(0.0720534\pi\)
							−0.838959	+	0.544194i	\(0.816835\pi\)
\(11\)	6025.18	−	10435.9i	0.124080	−	0.214913i	−0.797293	−	0.603593i	\(-0.793735\pi\)
							0.921373	+	0.388679i	\(0.127069\pi\)
\(13\)	142605.	−	119660.i	1.38481	−	1.16199i	0.417411	−	0.908718i	\(-0.362938\pi\)
							0.967395	−	0.253272i	\(-0.0815069\pi\)
\(17\)	−464948.	−	390137.i	−1.35016	−	1.13291i	−0.978890	−	0.204388i	\(-0.934479\pi\)
							−0.371266	−	0.928527i	\(-0.621076\pi\)
\(19\)	−365332.	−	132970.i	−0.643127	−	0.234079i	−0.000192399	−	1.00000i	\(-0.500061\pi\)
							−0.642935	+	0.765921i	\(0.722283\pi\)
\(23\)	1.22504e6	+	2.12183e6i	0.912797	+	1.58101i	0.810095	+	0.586298i	\(0.199415\pi\)
							0.102701	+	0.994712i	\(0.467251\pi\)
\(29\)	1.33753e6	−	2.31666e6i	0.351165	−	0.608236i	−0.635289	−	0.772275i	\(-0.719119\pi\)
							0.986454	+	0.164039i	\(0.0524522\pi\)
\(31\)	5.05892e6			0.983852			0.491926	−	0.870637i	\(-0.336293\pi\)
							0.491926	+	0.870637i	\(0.336293\pi\)
\(37\)	−2.53933e6	+	1.11137e7i	−0.222746	+	0.974876i
							\(41\)	−5.42386e6	+	4.55116e6i	−0.299765	+	0.251533i	−0.780246	−	0.625472i	\(-0.784906\pi\)
0.480481	+	0.877005i	\(0.340462\pi\)
\(43\)	2.87590e7			1.28282			0.641411	−	0.767198i	\(-0.278350\pi\)
							0.641411	+	0.767198i	\(0.278350\pi\)
\(47\)	2.33730e6	+	4.04832e6i	0.0698673	+	0.121014i	0.898843	−	0.438271i	\(-0.144409\pi\)
							−0.828975	+	0.559285i	\(0.811076\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.f.a.7.3	✓	162
37.16	even	9	inner	37.10.f.a.16.3	yes	162

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.f.a.7.3	✓	162	1.1	even	1	trivial
37.10.f.a.16.3	yes	162	37.16	even	9	inner