Embedded newform 76.3.g.c.7.3

• Label: 76.3.g.c.7.3
• 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			7.3
Character	\(\chi\)	\(=\)	76.7
Dual form			76.3.g.c.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.82726 + 0.813087i) q^{2} +(-0.851777 + 0.491774i) q^{3} +(2.67778 - 2.97145i) q^{4} +(1.71651 + 2.97309i) q^{5} +(1.15657 - 1.59117i) q^{6} +2.43870i q^{7} +(-2.47697 + 7.60688i) q^{8} +(-4.01632 + 6.95647i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 5 q^{2} - 11 q^{4} + 6 q^{5} - 3 q^{6} - 62 q^{8} + 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.82726	+	0.813087i	−0.913632	+	0.406543i
							\(3\)	−0.851777	+	0.491774i	−0.283926	+	0.163925i	−0.635199	−	0.772348i	\(-0.719082\pi\)
0.351274	+	0.936273i	\(0.385749\pi\)
\(5\)	1.71651	+	2.97309i	0.343302	+	0.594617i	0.985044	−	0.172304i	\(-0.0551211\pi\)
							−0.641741	+	0.766921i	\(0.721788\pi\)
\(7\)			2.43870i			0.348385i	0.984712	+	0.174193i	\(0.0557315\pi\)
							−0.984712	+	0.174193i	\(0.944268\pi\)
\(11\)			7.76499i			0.705908i	0.935641	+	0.352954i	\(0.114823\pi\)
							−0.935641	+	0.352954i	\(0.885177\pi\)
\(13\)	−4.63516	+	8.02833i	−0.356551	+	0.617564i	−0.987382	−	0.158356i	\(-0.949381\pi\)
							0.630832	+	0.775920i	\(0.282714\pi\)
\(17\)	8.87198	+	15.3667i	0.521881	+	0.903924i	0.999676	+	0.0254531i	\(0.00810285\pi\)
							−0.477795	+	0.878471i	\(0.658564\pi\)
\(19\)	6.84864	−	17.7228i	0.360455	−	0.932777i
							\(23\)	10.0507	+	5.80275i	0.436985	+	0.252293i	0.702318	−	0.711863i	\(-0.252148\pi\)
−0.265333	+	0.964157i	\(0.585482\pi\)
\(29\)	2.64340	−	4.57851i	0.0911519	−	0.157880i	−0.816844	−	0.576858i	\(-0.804278\pi\)
							0.907996	+	0.418979i	\(0.137612\pi\)
\(31\)		−	10.2531i		−	0.330744i	−0.986231	−	0.165372i	\(-0.947117\pi\)
							0.986231	−	0.165372i	\(-0.0528825\pi\)
\(37\)	−7.71952			−0.208636			−0.104318	−	0.994544i	\(-0.533266\pi\)
							−0.104318	+	0.994544i	\(0.533266\pi\)
\(41\)	−25.2287	−	43.6973i	−0.615333	−	1.06579i	−0.990326	−	0.138761i	\(-0.955688\pi\)
							0.374992	−	0.927028i	\(-0.377645\pi\)
\(43\)	−43.2661	+	24.9797i	−1.00619	+	0.580923i	−0.910073	−	0.414447i	\(-0.863975\pi\)
							−0.0961149	+	0.995370i	\(0.530642\pi\)
\(47\)	74.9827	+	43.2913i	1.59538	+	0.921091i	0.992362	+	0.123361i	\(0.0393674\pi\)
							0.603015	+	0.797730i	\(0.293966\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.3.g.c.7.3	✓	28
4.3	odd	2	inner	76.3.g.c.7.13	yes	28
19.11	even	3	inner	76.3.g.c.11.13	yes	28
76.11	odd	6	inner	76.3.g.c.11.3	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.g.c.7.3	✓	28	1.1	even	1	trivial
76.3.g.c.7.13	yes	28	4.3	odd	2	inner
76.3.g.c.11.3	yes	28	76.11	odd	6	inner
76.3.g.c.11.13	yes	28	19.11	even	3	inner