Embedded newform 76.7.g.a.7.10

• Label: 76.7.g.a.7.10
• 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			7.10
Character	\(\chi\)	\(=\)	76.7
Dual form			76.7.g.a.11.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.97428 + 3.91912i) q^{2} +(22.3528 - 12.9054i) q^{3} +(33.2810 - 54.6660i) q^{4} +(17.9822 + 31.1462i) q^{5} +(-105.317 + 177.609i) q^{6} -75.3694i q^{7} +(-17.8687 + 511.688i) q^{8} +(-31.4026 + 54.3909i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - q^{2} - 67 q^{4} - 2 q^{5} + 33 q^{6} + 962 q^{8} + 12660 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.97428	+	3.91912i	−0.871784	+	0.489890i
							\(3\)	22.3528	−	12.9054i	0.827880	−	0.477977i	−0.0252461	−	0.999681i	\(-0.508037\pi\)
0.853126	+	0.521704i	\(0.174704\pi\)
\(5\)	17.9822	+	31.1462i	0.143858	+	0.249169i	0.928946	−	0.370215i	\(-0.120716\pi\)
							−0.785088	+	0.619384i	\(0.787382\pi\)
\(7\)		−	75.3694i		−	0.219736i	−0.993946	−	0.109868i	\(-0.964957\pi\)
							0.993946	−	0.109868i	\(-0.0350428\pi\)
\(11\)			2324.15i			1.74617i	0.487568	+	0.873085i	\(0.337884\pi\)
							−0.487568	+	0.873085i	\(0.662116\pi\)
\(13\)	874.867	−	1515.31i	0.398210	−	0.689720i	−0.595295	−	0.803507i	\(-0.702965\pi\)
							0.993505	+	0.113787i	\(0.0362983\pi\)
\(17\)	−305.122	−	528.486i	−0.0621050	−	0.107569i	0.833301	−	0.552819i	\(-0.186448\pi\)
							−0.895406	+	0.445250i	\(0.853115\pi\)
\(19\)	6816.57	−	761.715i	0.993814	−	0.111053i
							\(23\)	10108.1	+	5835.89i	0.830776	+	0.479649i	0.854118	−	0.520079i	\(-0.174097\pi\)
−0.0233422	+	0.999728i	\(0.507431\pi\)
\(29\)	−6244.04	+	10815.0i	−0.256019	+	0.443437i	−0.965172	−	0.261617i	\(-0.915744\pi\)
							0.709153	+	0.705055i	\(0.249078\pi\)
\(31\)			49598.3i			1.66488i	0.554118	+	0.832438i	\(0.313056\pi\)
							−0.554118	+	0.832438i	\(0.686944\pi\)
\(37\)	9540.38			0.188348			0.0941739	−	0.995556i	\(-0.469979\pi\)
							0.0941739	+	0.995556i	\(0.469979\pi\)
\(41\)	41919.1	+	72606.0i	0.608219	+	1.05347i	0.991534	+	0.129849i	\(0.0414492\pi\)
							−0.383314	+	0.923618i	\(0.625217\pi\)
\(43\)	85470.9	−	49346.6i	1.07501	−	0.620658i	0.145464	−	0.989363i	\(-0.453532\pi\)
							0.929546	+	0.368706i	\(0.120199\pi\)
\(47\)	−87391.5	−	50455.5i	−0.841736	−	0.485976i	0.0161182	−	0.999870i	\(-0.494869\pi\)
							−0.857854	+	0.513894i	\(0.828203\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.7.g.a.7.10	✓	116
4.3	odd	2	inner	76.7.g.a.7.49	yes	116
19.11	even	3	inner	76.7.g.a.11.49	yes	116
76.11	odd	6	inner	76.7.g.a.11.10	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.7.g.a.7.10	✓	116	1.1	even	1	trivial
76.7.g.a.7.49	yes	116	4.3	odd	2	inner
76.7.g.a.11.10	yes	116	76.11	odd	6	inner
76.7.g.a.11.49	yes	116	19.11	even	3	inner