Embedded newform 76.5.g.a.7.16

• Label: 76.5.g.a.7.16
• Level: $76$
• Weight: $5$
• Character: 76.7
• Analytic conductor: $7.856$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	76.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85611719437\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.16
Character	\(\chi\)	\(=\)	76.7
Dual form			76.5.g.a.11.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707709 + 3.93690i) q^{2} +(-10.7755 + 6.22123i) q^{3} +(-14.9983 - 5.57236i) q^{4} +(-13.6139 - 23.5800i) q^{5} +(-16.8664 - 46.8248i) q^{6} +71.9337i q^{7} +(32.5522 - 55.1031i) q^{8} +(36.9073 - 63.9254i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - q^{2} + 13 q^{4} - 2 q^{5} - 15 q^{6} - 94 q^{8} + 972 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.707709	+	3.93690i	−0.176927	+	0.984224i
							\(3\)	−10.7755	+	6.22123i	−1.19728	+	0.691247i	−0.959947	−	0.280182i	\(-0.909605\pi\)
−0.237329	+	0.971429i	\(0.576272\pi\)
\(5\)	−13.6139	−	23.5800i	−0.544557	−	0.943200i	−0.998635	−	0.0522384i	\(-0.983364\pi\)
							0.454078	−	0.890962i	\(-0.349969\pi\)
\(7\)			71.9337i			1.46803i	0.679131	+	0.734017i	\(0.262357\pi\)
							−0.679131	+	0.734017i	\(0.737643\pi\)
\(11\)		−	136.755i		−	1.13021i	−0.825020	−	0.565103i	\(-0.808836\pi\)
							0.825020	−	0.565103i	\(-0.191164\pi\)
\(13\)	−34.4173	+	59.6125i	−0.203653	+	0.352737i	−0.949703	−	0.313153i	\(-0.898615\pi\)
							0.746050	+	0.665890i	\(0.231948\pi\)
\(17\)	163.162	+	282.604i	0.564573	+	0.977870i	0.997089	+	0.0762434i	\(0.0242926\pi\)
							−0.432516	+	0.901626i	\(0.642374\pi\)
\(19\)	334.758	−	135.122i	0.927307	−	0.374300i
							\(23\)	−640.548	−	369.820i	−1.21086	−	0.699093i	−0.247917	−	0.968781i	\(-0.579746\pi\)
−0.962948	+	0.269688i	\(0.913079\pi\)
\(29\)	614.468	−	1064.29i	0.730639	−	1.26550i	−0.225971	−	0.974134i	\(-0.572555\pi\)
							0.956610	−	0.291371i	\(-0.0941113\pi\)
\(31\)		−	664.220i		−	0.691176i	−0.938386	−	0.345588i	\(-0.887679\pi\)
							0.938386	−	0.345588i	\(-0.112321\pi\)
\(37\)	−729.503			−0.532873			−0.266436	−	0.963852i	\(-0.585846\pi\)
							−0.266436	+	0.963852i	\(0.585846\pi\)
\(41\)	−1531.21	−	2652.14i	−0.910894	−	1.57771i	−0.812804	−	0.582537i	\(-0.802060\pi\)
							−0.0980893	−	0.995178i	\(-0.531273\pi\)
\(43\)	−256.875	+	148.307i	−0.138926	+	0.0802092i	−0.567852	−	0.823131i	\(-0.692226\pi\)
							0.428926	+	0.903340i	\(0.358892\pi\)
\(47\)	3121.12	+	1801.98i	1.41291	+	0.815744i	0.995662	−	0.0930492i	\(-0.0296614\pi\)
							0.417248	+	0.908793i	\(0.362995\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.g.a.7.16	✓	76
4.3	odd	2	inner	76.5.g.a.7.34	yes	76
19.11	even	3	inner	76.5.g.a.11.34	yes	76
76.11	odd	6	inner	76.5.g.a.11.16	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.g.a.7.16	✓	76	1.1	even	1	trivial
76.5.g.a.7.34	yes	76	4.3	odd	2	inner
76.5.g.a.11.16	yes	76	76.11	odd	6	inner
76.5.g.a.11.34	yes	76	19.11	even	3	inner