Embedded newform 158.5.f.a.15.3

• Label: 158.5.f.a.15.3
• Level: $158$
• Weight: $5$
• Character: 158.15
• Analytic conductor: $16.332$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 158 = 2 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	158.f (of order \(26\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.3324541672\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(28\) over \(\Q(\zeta_{26})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{26}]$

Embedding invariants

Embedding label			15.3
Character	\(\chi\)	\(=\)	158.15
Dual form			158.5.f.a.137.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.11711 - 1.87559i) q^{2} +(-8.02312 + 5.53796i) q^{3} +(0.964293 + 7.94167i) q^{4} +(31.9338 + 7.87097i) q^{5} +(27.3728 + 3.32366i) q^{6} +(53.7142 - 37.0763i) q^{7} +(12.8538 - 18.6220i) q^{8} +(4.97848 - 13.1272i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 224 q^{4} + 728 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/158\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{21}{26}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.11711	−	1.87559i	−0.529277	−	0.468899i
							\(3\)	−8.02312	+	5.53796i	−0.891458	+	0.615329i	−0.923264	−	0.384165i	\(-0.874489\pi\)
0.0318065	+	0.999494i	\(0.489874\pi\)
\(5\)	31.9338	+	7.87097i	1.27735	+	0.314839i	0.818928	−	0.573896i	\(-0.194569\pi\)
							0.458422	+	0.888734i	\(0.348415\pi\)
\(7\)	53.7142	−	37.0763i	1.09621	−	0.756658i	0.124354	−	0.992238i	\(-0.460314\pi\)
							0.971855	+	0.235579i	\(0.0756988\pi\)
\(11\)	−97.9609	+	24.1452i	−0.809594	+	0.199547i	−0.622322	−	0.782761i	\(-0.713811\pi\)
							−0.187272	+	0.982308i	\(0.559965\pi\)
\(13\)	−6.04150	−	49.7562i	−0.0357485	−	0.294416i	−0.999608	−	0.0280037i	\(-0.991085\pi\)
							0.963859	−	0.266412i	\(-0.0858381\pi\)
\(17\)	−52.9994	+	6.43530i	−0.183389	+	0.0222675i	−0.211716	−	0.977331i	\(-0.567905\pi\)
							0.0283268	+	0.999599i	\(0.490982\pi\)
\(19\)	456.565	−	239.624i	1.26472	−	0.663778i	0.307066	−	0.951688i	\(-0.400653\pi\)
							0.957657	+	0.287911i	\(0.0929606\pi\)
\(23\)	439.089			0.830035			0.415018	−	0.909813i	\(-0.363775\pi\)
							0.415018	+	0.909813i	\(0.363775\pi\)
\(29\)	231.584	−	87.8282i	0.275367	−	0.104433i	−0.213056	−	0.977040i	\(-0.568342\pi\)
							0.488423	+	0.872607i	\(0.337572\pi\)
\(31\)	1087.83	+	963.733i	1.13198	+	1.00284i	0.999933	+	0.0116000i	\(0.00369246\pi\)
							0.132044	+	0.991244i	\(0.457846\pi\)
\(37\)	−31.5737	−	60.1586i	−0.0230633	−	0.0439435i	0.873652	−	0.486552i	\(-0.161746\pi\)
							−0.896715	+	0.442608i	\(0.854053\pi\)
\(41\)	−478.917	+	1943.04i	−0.284900	+	1.15588i	0.637749	+	0.770244i	\(0.279866\pi\)
							−0.922649	+	0.385641i	\(0.873980\pi\)
\(43\)	−133.032	+	539.734i	−0.0719483	+	0.291906i	−0.995883	−	0.0906435i	\(-0.971108\pi\)
							0.923935	+	0.382549i	\(0.124954\pi\)
\(47\)	422.136	−	804.313i	0.191098	−	0.364107i	−0.770989	−	0.636848i	\(-0.780238\pi\)
							0.962088	+	0.272741i	\(0.0879302\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	158.5.f.a.15.3	✓	336
79.58	odd	26	inner	158.5.f.a.137.3	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
158.5.f.a.15.3	✓	336	1.1	even	1	trivial
158.5.f.a.137.3	yes	336	79.58	odd	26	inner