Embedded newform 76.7.g.a.7.9

• Label: 76.7.g.a.7.9
• 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			7.9
Character	\(\chi\)	\(=\)	76.7
Dual form			76.7.g.a.11.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.00051 - 3.87205i) q^{2} +(-41.5950 + 24.0149i) q^{3} +(34.0144 + 54.2127i) q^{4} +(-83.5383 - 144.693i) q^{5} +(384.173 - 7.05848i) q^{6} +406.670i q^{7} +(-28.2037 - 511.223i) q^{8} +(788.930 - 1366.47i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - q^{2} - 67 q^{4} - 2 q^{5} + 33 q^{6} + 962 q^{8} + 12660 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.00051	−	3.87205i	−0.875064	−	0.484007i
							\(3\)	−41.5950	+	24.0149i	−1.54056	+	0.889441i	−0.541753	+	0.840538i	\(0.682239\pi\)
−0.998804	+	0.0489027i	\(0.984428\pi\)
\(5\)	−83.5383	−	144.693i	−0.668307	−	1.15754i	−0.978377	−	0.206828i	\(-0.933686\pi\)
							0.310071	−	0.950713i	\(-0.399647\pi\)
\(7\)			406.670i			1.18563i	0.805340	+	0.592813i	\(0.201983\pi\)
							−0.805340	+	0.592813i	\(0.798017\pi\)
\(11\)		−	2402.91i		−	1.80534i	−0.430332	−	0.902671i	\(-0.641603\pi\)
							0.430332	−	0.902671i	\(-0.358397\pi\)
\(13\)	652.775	−	1130.64i	0.297121	−	0.514628i	−0.678355	−	0.734734i	\(-0.737307\pi\)
							0.975476	+	0.220106i	\(0.0706402\pi\)
\(17\)	770.972	+	1335.36i	0.156925	+	0.271802i	0.933758	−	0.357904i	\(-0.116509\pi\)
							−0.776833	+	0.629706i	\(0.783175\pi\)
\(19\)	−6716.92	−	1388.84i	−0.979285	−	0.202485i
							\(23\)	6218.56	+	3590.29i	0.511101	+	0.295084i	0.733286	−	0.679920i	\(-0.237986\pi\)
−0.222185	+	0.975004i	\(0.571319\pi\)
\(29\)	1749.53	−	3030.28i	0.0717345	−	0.124248i	−0.827927	−	0.560836i	\(-0.810480\pi\)
							0.899662	+	0.436588i	\(0.143813\pi\)
\(31\)			20395.8i			0.684630i	0.939585	+	0.342315i	\(0.111211\pi\)
							−0.939585	+	0.342315i	\(0.888789\pi\)
\(37\)	−62333.0			−1.23059			−0.615294	−	0.788298i	\(-0.710963\pi\)
							−0.615294	+	0.788298i	\(0.710963\pi\)
\(41\)	−9681.19	−	16768.3i	−0.140468	−	0.243298i	0.787205	−	0.616691i	\(-0.211527\pi\)
							−0.927673	+	0.373394i	\(0.878194\pi\)
\(43\)	41901.3	−	24191.7i	0.527014	−	0.304271i	−0.212786	−	0.977099i	\(-0.568254\pi\)
							0.739799	+	0.672827i	\(0.234920\pi\)
\(47\)	−139388.	−	80475.8i	−1.34256	−	0.775125i	−0.355374	−	0.934724i	\(-0.615646\pi\)
							−0.987182	+	0.159599i	\(0.948980\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.7.g.a.7.9	✓	116
4.3	odd	2	inner	76.7.g.a.7.30	yes	116
19.11	even	3	inner	76.7.g.a.11.30	yes	116
76.11	odd	6	inner	76.7.g.a.11.9	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.7.g.a.7.9	✓	116	1.1	even	1	trivial
76.7.g.a.7.30	yes	116	4.3	odd	2	inner
76.7.g.a.11.9	yes	116	76.11	odd	6	inner
76.7.g.a.11.30	yes	116	19.11	even	3	inner