Embedded newform 55.6.g.b.16.1

• Label: 55.6.g.b.16.1
• Level: $55$
• Weight: $6$
• Character: 55.16
• Analytic conductor: $8.821$
• Analytic rank: $0$
• Dimension: $40$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 55 = 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	55.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.82111008971\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			16.1
Character	\(\chi\)	\(=\)	55.16
Dual form			55.6.g.b.31.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.96862 - 9.13648i) q^{2} +(-17.4748 - 12.6962i) q^{3} +(-48.7740 + 35.4364i) q^{4} +(-7.72542 + 23.7764i) q^{5} +(-64.1223 + 197.348i) q^{6} +(41.3933 - 30.0740i) q^{7} +(219.853 + 159.733i) q^{8} +(69.0839 + 212.618i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 2 q^{2} + 55 q^{3} - 176 q^{4} + 250 q^{5} + 432 q^{6} + 11 q^{7} - 274 q^{8} - 579 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\)	\(12\)	\(46\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.96862	−	9.13648i	−0.524783	−	1.61512i	−0.764744	−	0.644334i	\(-0.777134\pi\)
							0.239961	−	0.970783i	\(-0.422866\pi\)
\(3\)	−17.4748	−	12.6962i	−1.12101	−	0.814459i	−0.136646	−	0.990620i	\(-0.543632\pi\)
							−0.984361	+	0.176161i	\(0.943632\pi\)
\(5\)	−7.72542	+	23.7764i	−0.138197	+	0.425325i
							\(7\)	41.3933	−	30.0740i	0.319289	−	0.231977i	−0.416583	−	0.909098i	\(-0.636772\pi\)
0.735872	+	0.677120i	\(0.236772\pi\)
\(11\)	−394.719	+	72.4449i	−0.983571	+	0.180520i
							\(13\)	−152.975	−	470.809i	−0.251051	−	0.772656i	−0.994582	−	0.103954i	\(-0.966850\pi\)
0.743531	−	0.668702i	\(-0.233150\pi\)
\(17\)	−48.1460	+	148.178i	−0.0404053	+	0.124355i	−0.969224	−	0.246179i	\(-0.920825\pi\)
							0.928819	+	0.370533i	\(0.120825\pi\)
\(19\)	−146.788	−	106.647i	−0.0932836	−	0.0677745i	0.540166	−	0.841559i	\(-0.318362\pi\)
							−0.633449	+	0.773784i	\(0.718362\pi\)
\(23\)	3721.18			1.46677			0.733384	−	0.679814i	\(-0.237940\pi\)
							0.733384	+	0.679814i	\(0.237940\pi\)
\(29\)	4216.65	−	3063.58i	0.931049	−	0.676447i	−0.0152002	−	0.999884i	\(-0.504839\pi\)
							0.946250	+	0.323438i	\(0.104839\pi\)
\(31\)	2794.79	+	8601.47i	0.522329	+	1.60756i	0.769537	+	0.638602i	\(0.220487\pi\)
							−0.247208	+	0.968963i	\(0.579513\pi\)
\(37\)	−11504.0	+	8358.17i	−1.38148	+	1.00371i	−0.384744	+	0.923023i	\(0.625710\pi\)
							−0.996740	+	0.0806830i	\(0.974290\pi\)
\(41\)	−13448.8	−	9771.15i	−1.24947	−	0.907791i	−0.251276	−	0.967915i	\(-0.580850\pi\)
							−0.998191	+	0.0601242i	\(0.980850\pi\)
\(43\)	−8281.23			−0.683005			−0.341502	−	0.939881i	\(-0.610936\pi\)
							−0.341502	+	0.939881i	\(0.610936\pi\)
\(47\)	12821.3	+	9315.21i	0.846617	+	0.615103i	0.924211	−	0.381882i	\(-0.124724\pi\)
							−0.0775941	+	0.996985i	\(0.524724\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	55.6.g.b.16.1	✓	40
11.3	even	5		605.6.a.p.1.3		20
11.8	odd	10		605.6.a.o.1.18		20
11.9	even	5	inner	55.6.g.b.31.1	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.6.g.b.16.1	✓	40	1.1	even	1	trivial
55.6.g.b.31.1	yes	40	11.9	even	5	inner
605.6.a.o.1.18		20	11.8	odd	10
605.6.a.p.1.3		20	11.3	even	5