Embedded newform 76.3.g.c.7.14

• Label: 76.3.g.c.7.14
• Level: $76$
• Weight: $3$
• Character: 76.7
• Analytic conductor: $2.071$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	76.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.07085000914\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.14
Character	\(\chi\)	\(=\)	76.7
Dual form			76.3.g.c.11.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.72876 + 1.00569i) q^{2} +(-0.443154 + 0.255855i) q^{3} +(1.97719 + 3.47717i) q^{4} +(1.99023 + 3.44717i) q^{5} +(-1.02341 - 0.00336270i) q^{6} -9.66827i q^{7} +(-0.0788568 + 7.99961i) q^{8} +(-4.36908 + 7.56746i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 5 q^{2} - 11 q^{4} + 6 q^{5} - 3 q^{6} - 62 q^{8} + 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.72876	+	1.00569i	0.864378	+	0.502843i
							\(3\)	−0.443154	+	0.255855i	−0.147718	+	0.0852850i	−0.572037	−	0.820228i	\(-0.693847\pi\)
0.424319	+	0.905513i	\(0.360513\pi\)
\(5\)	1.99023	+	3.44717i	0.398045	+	0.689435i	0.993485	−	0.113965i	\(-0.0363553\pi\)
							−0.595439	+	0.803400i	\(0.703022\pi\)
\(7\)		−	9.66827i		−	1.38118i	−0.723246	−	0.690591i	\(-0.757351\pi\)
							0.723246	−	0.690591i	\(-0.242649\pi\)
\(11\)			2.62030i			0.238209i	0.992882	+	0.119104i	\(0.0380023\pi\)
							−0.992882	+	0.119104i	\(0.961998\pi\)
\(13\)	11.8012	−	20.4403i	0.907787	−	1.57233i	0.0906551	−	0.995882i	\(-0.471104\pi\)
							0.817132	−	0.576451i	\(-0.195563\pi\)
\(17\)	−3.40777	−	5.90242i	−0.200457	−	0.347201i	0.748219	−	0.663452i	\(-0.230909\pi\)
							−0.948676	+	0.316251i	\(0.897576\pi\)
\(19\)	−12.0371	−	14.7006i	−0.633530	−	0.773718i
							\(23\)	−17.2580	−	9.96389i	−0.750346	−	0.433213i	0.0754728	−	0.997148i	\(-0.475953\pi\)
−0.825819	+	0.563935i	\(0.809287\pi\)
\(29\)	0.445777	−	0.772108i	0.0153716	−	0.0266244i	−0.858237	−	0.513253i	\(-0.828440\pi\)
							0.873609	+	0.486629i	\(0.161774\pi\)
\(31\)			59.6042i			1.92272i	0.275303	+	0.961358i	\(0.411222\pi\)
							−0.275303	+	0.961358i	\(0.588778\pi\)
\(37\)	−7.80746			−0.211012			−0.105506	−	0.994419i	\(-0.533646\pi\)
							−0.105506	+	0.994419i	\(0.533646\pi\)
\(41\)	15.9560	+	27.6366i	0.389171	+	0.674063i	0.992338	−	0.123551i	\(-0.0394283\pi\)
							−0.603168	+	0.797614i	\(0.706095\pi\)
\(43\)	6.09346	−	3.51806i	0.141708	−	0.0818154i	−0.427470	−	0.904030i	\(-0.640595\pi\)
							0.569178	+	0.822214i	\(0.307262\pi\)
\(47\)	47.7806	+	27.5861i	1.01661	+	0.586939i	0.913120	−	0.407692i	\(-0.133666\pi\)
							0.103489	+	0.994631i	\(0.466999\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.3.g.c.7.14	yes	28
4.3	odd	2	inner	76.3.g.c.7.8	✓	28
19.11	even	3	inner	76.3.g.c.11.8	yes	28
76.11	odd	6	inner	76.3.g.c.11.14	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.g.c.7.8	✓	28	4.3	odd	2	inner
76.3.g.c.7.14	yes	28	1.1	even	1	trivial
76.3.g.c.11.8	yes	28	19.11	even	3	inner
76.3.g.c.11.14	yes	28	76.11	odd	6	inner