Embedded newform 76.7.g.a.7.12

• Label: 76.7.g.a.7.12
• Level: $76$
• Weight: $7$
• Character: 76.7
• Analytic conductor: $17.484$
• Analytic rank: $0$
• Dimension: $116$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	76.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.4841103551\)
Analytic rank:	\(0\)
Dimension:	\(116\)
Relative dimension:	\(58\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.12
Character	\(\chi\)	\(=\)	76.7
Dual form			76.7.g.a.11.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.65177 - 4.44454i) q^{2} +(-5.80264 + 3.35015i) q^{3} +(24.4921 + 59.1281i) q^{4} +(84.8928 + 147.039i) q^{5} +(53.4877 + 3.50559i) q^{6} +460.277i q^{7} +(99.8813 - 502.163i) q^{8} +(-342.053 + 592.453i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - q^{2} - 67 q^{4} - 2 q^{5} + 33 q^{6} + 962 q^{8} + 12660 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.65177	−	4.44454i	−0.831471	−	0.555568i
							\(3\)	−5.80264	+	3.35015i	−0.214912	+	0.124080i	−0.603592	−	0.797293i	\(-0.706265\pi\)
0.388680	+	0.921373i	\(0.372931\pi\)
\(5\)	84.8928	+	147.039i	0.679142	+	1.17631i	0.975240	+	0.221151i	\(0.0709814\pi\)
							−0.296097	+	0.955158i	\(0.595685\pi\)
\(7\)			460.277i			1.34192i	0.741495	+	0.670958i	\(0.234117\pi\)
							−0.741495	+	0.670958i	\(0.765883\pi\)
\(11\)			960.371i			0.721541i	0.932655	+	0.360770i	\(0.117486\pi\)
							−0.932655	+	0.360770i	\(0.882514\pi\)
\(13\)	1527.69	−	2646.04i	0.695353	−	1.20439i	−0.274709	−	0.961527i	\(-0.588582\pi\)
							0.970062	−	0.242859i	\(-0.0780851\pi\)
\(17\)	−3376.86	−	5848.89i	−0.687331	−	1.19049i	−0.972698	−	0.232074i	\(-0.925449\pi\)
							0.285367	−	0.958418i	\(-0.407885\pi\)
\(19\)	−4972.33	+	4724.60i	−0.724934	+	0.688818i
							\(23\)	8858.81	+	5114.64i	0.728102	+	0.420370i	0.817727	−	0.575606i	\(-0.195234\pi\)
−0.0896257	+	0.995976i	\(0.528567\pi\)
\(29\)	−8493.98	+	14712.0i	−0.348271	+	0.603223i	−0.985942	−	0.167085i	\(-0.946564\pi\)
							0.637671	+	0.770309i	\(0.279898\pi\)
\(31\)			5998.64i			0.201357i	0.994919	+	0.100679i	\(0.0321014\pi\)
							−0.994919	+	0.100679i	\(0.967899\pi\)
\(37\)	−12699.1			−0.250707			−0.125354	−	0.992112i	\(-0.540007\pi\)
							−0.125354	+	0.992112i	\(0.540007\pi\)
\(41\)	−22353.3	−	38717.0i	−0.324332	−	0.561760i	0.657045	−	0.753851i	\(-0.271806\pi\)
							−0.981377	+	0.192092i	\(0.938473\pi\)
\(43\)	−57133.9	+	32986.2i	−0.718602	+	0.414885i	−0.814238	−	0.580532i	\(-0.802845\pi\)
							0.0956362	+	0.995416i	\(0.469511\pi\)
\(47\)	69566.4	+	40164.2i	0.670048	+	0.386853i	0.796095	−	0.605172i	\(-0.206896\pi\)
							−0.126047	+	0.992024i	\(0.540229\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.7.g.a.7.12	✓	116
4.3	odd	2	inner	76.7.g.a.7.28	yes	116
19.11	even	3	inner	76.7.g.a.11.28	yes	116
76.11	odd	6	inner	76.7.g.a.11.12	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.7.g.a.7.12	✓	116	1.1	even	1	trivial
76.7.g.a.7.28	yes	116	4.3	odd	2	inner
76.7.g.a.11.12	yes	116	76.11	odd	6	inner
76.7.g.a.11.28	yes	116	19.11	even	3	inner