Embedded newform 76.3.g.c.7.8

• Label: 76.3.g.c.7.8
• 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			7.8
Character	\(\chi\)	\(=\)	76.7
Dual form			76.3.g.c.11.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.00657150 + 1.99999i) q^{2} +(0.443154 - 0.255855i) q^{3} +(-3.99991 + 0.0262858i) q^{4} +(1.99023 + 3.44717i) q^{5} +(0.514619 + 0.884621i) q^{6} +9.66827i q^{7} +(-0.0788568 - 7.99961i) q^{8} +(-4.36908 + 7.56746i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 5 q^{2} - 11 q^{4} + 6 q^{5} - 3 q^{6} - 62 q^{8} + 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.00657150	+	1.99999i	0.00328575	+	0.999995i
							\(3\)	0.443154	−	0.255855i	0.147718	−	0.0852850i	−0.424319	−	0.905513i	\(-0.639487\pi\)
0.572037	+	0.820228i	\(0.306153\pi\)
\(5\)	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	\(0.242649\pi\)
							−0.723246	+	0.690591i	\(0.757351\pi\)
\(11\)		−	2.62030i		−	0.238209i	−0.992882	−	0.119104i	\(-0.961998\pi\)
							0.992882	−	0.119104i	\(-0.0380023\pi\)
\(13\)	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	0.825819	−	0.563935i	\(-0.190713\pi\)
−0.0754728	+	0.997148i	\(0.524047\pi\)
\(29\)	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	\(-0.588778\pi\)
							0.275303	−	0.961358i	\(-0.411222\pi\)
\(37\)	−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	−0.569178	−	0.822214i	\(-0.692738\pi\)
							0.427470	+	0.904030i	\(0.359405\pi\)
\(47\)	−47.7806	−	27.5861i	−1.01661	−	0.586939i	−0.103489	−	0.994631i	\(-0.533001\pi\)
							−0.913120	+	0.407692i	\(0.866334\pi\)

(See \(a_n\) instead)

Twists

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

    

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