Embedded newform 76.7.g.a.7.2

• Label: 76.7.g.a.7.2
• 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			7.2
Character	\(\chi\)	\(=\)	76.7
Dual form			76.7.g.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.98630 + 0.467925i) q^{2} +(-26.4172 + 15.2520i) q^{3} +(63.5621 - 7.47398i) q^{4} +(-21.3081 - 36.9067i) q^{5} +(203.839 - 134.168i) q^{6} +91.3832i q^{7} +(-504.129 + 89.4318i) q^{8} +(100.747 - 174.500i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - q^{2} - 67 q^{4} - 2 q^{5} + 33 q^{6} + 962 q^{8} + 12660 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.98630	+	0.467925i	−0.998288	+	0.0584906i
							\(3\)	−26.4172	+	15.2520i	−0.978417	+	0.564889i	−0.901792	−	0.432171i	\(-0.857748\pi\)
−0.0766249	+	0.997060i	\(0.524414\pi\)
\(5\)	−21.3081	−	36.9067i	−0.170465	−	0.295253i	0.768118	−	0.640309i	\(-0.221193\pi\)
							−0.938582	+	0.345055i	\(0.887860\pi\)
\(7\)			91.3832i			0.266423i	0.991088	+	0.133212i	\(0.0425290\pi\)
							−0.991088	+	0.133212i	\(0.957471\pi\)
\(11\)			1734.65i			1.30327i	0.758535	+	0.651633i	\(0.225916\pi\)
							−0.758535	+	0.651633i	\(0.774084\pi\)
\(13\)	−885.415	+	1533.58i	−0.403011	+	0.698035i	−0.994088	−	0.108580i	\(-0.965370\pi\)
							0.591077	+	0.806615i	\(0.298703\pi\)
\(17\)	−145.752	−	252.450i	−0.0296667	−	0.0513842i	0.850811	−	0.525472i	\(-0.176111\pi\)
							−0.880478	+	0.474088i	\(0.842778\pi\)
\(19\)	−625.681	+	6830.40i	−0.0912205	+	0.995831i
							\(23\)	−9216.75	−	5321.30i	−0.757521	−	0.437355i	0.0708842	−	0.997485i	\(-0.477418\pi\)
−0.828405	+	0.560130i	\(0.810751\pi\)
\(29\)	2501.77	−	4333.19i	0.102578	−	0.177670i	−0.810168	−	0.586197i	\(-0.800624\pi\)
							0.912746	+	0.408527i	\(0.133958\pi\)
\(31\)		−	1912.99i		−	0.0642136i	−0.999484	−	0.0321068i	\(-0.989778\pi\)
							0.999484	−	0.0321068i	\(-0.0102217\pi\)
\(37\)	−21719.7			−0.428794			−0.214397	−	0.976747i	\(-0.568779\pi\)
							−0.214397	+	0.976747i	\(0.568779\pi\)
\(41\)	−36881.8	−	63881.2i	−0.535132	−	0.926875i	−0.999157	−	0.0410534i	\(-0.986929\pi\)
							0.464025	−	0.885822i	\(-0.346405\pi\)
\(43\)	37669.5	−	21748.5i	0.473789	−	0.273542i	−0.244036	−	0.969766i	\(-0.578471\pi\)
							0.717824	+	0.696224i	\(0.245138\pi\)
\(47\)	−38152.8	−	22027.5i	−0.367479	−	0.212164i	0.304878	−	0.952392i	\(-0.401384\pi\)
							−0.672357	+	0.740227i	\(0.734718\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.7.g.a.7.2	✓	116
4.3	odd	2	inner	76.7.g.a.7.40	yes	116
19.11	even	3	inner	76.7.g.a.11.40	yes	116
76.11	odd	6	inner	76.7.g.a.11.2	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.7.g.a.7.2	✓	116	1.1	even	1	trivial
76.7.g.a.7.40	yes	116	4.3	odd	2	inner
76.7.g.a.11.2	yes	116	76.11	odd	6	inner
76.7.g.a.11.40	yes	116	19.11	even	3	inner