Embedded newform 76.5.g.a.7.5

• Label: 76.5.g.a.7.5
• Level: $76$
• Weight: $5$
• Character: 76.7
• Analytic conductor: $7.856$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	76.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85611719437\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.5
Character	\(\chi\)	\(=\)	76.7
Dual form			76.5.g.a.11.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.71892 - 1.47296i) q^{2} +(4.05898 - 2.34345i) q^{3} +(11.6608 + 10.9556i) q^{4} +(-1.00987 - 1.74915i) q^{5} +(-18.5469 + 2.73642i) q^{6} +54.3250i q^{7} +(-27.2284 - 57.9190i) q^{8} +(-29.5164 + 51.1240i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - q^{2} + 13 q^{4} - 2 q^{5} - 15 q^{6} - 94 q^{8} + 972 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.71892	−	1.47296i	−0.929731	−	0.368240i
							\(3\)	4.05898	−	2.34345i	0.450998	−	0.260384i	−0.257254	−	0.966344i	\(-0.582818\pi\)
0.708252	+	0.705960i	\(0.249484\pi\)
\(5\)	−1.00987	−	1.74915i	−0.0403949	−	0.0699660i	0.845121	−	0.534575i	\(-0.179528\pi\)
							−0.885516	+	0.464609i	\(0.846195\pi\)
\(7\)			54.3250i			1.10867i	0.832292	+	0.554337i	\(0.187028\pi\)
							−0.832292	+	0.554337i	\(0.812972\pi\)
\(11\)		−	137.122i		−	1.13324i	−0.823979	−	0.566620i	\(-0.808251\pi\)
							0.823979	−	0.566620i	\(-0.191749\pi\)
\(13\)	−118.197	+	204.723i	−0.699391	+	1.21138i	0.269286	+	0.963060i	\(0.413212\pi\)
							−0.968678	+	0.248321i	\(0.920121\pi\)
\(17\)	147.393	+	255.293i	0.510012	+	0.883366i	0.999933	+	0.0115993i	\(0.00369226\pi\)
							−0.489921	+	0.871767i	\(0.662974\pi\)
\(19\)	170.891	+	317.989i	0.473382	+	0.880857i
							\(23\)	669.385	+	386.470i	1.26538	+	0.730567i	0.974110	−	0.226074i	\(-0.0725892\pi\)
0.291269	+	0.956641i	\(0.405923\pi\)
\(29\)	−677.650	+	1173.72i	−0.805767	+	1.39563i	0.110004	+	0.993931i	\(0.464914\pi\)
							−0.915772	+	0.401699i	\(0.868420\pi\)
\(31\)		−	677.759i		−	0.705265i	−0.935762	−	0.352632i	\(-0.885287\pi\)
							0.935762	−	0.352632i	\(-0.114713\pi\)
\(37\)	−1907.95			−1.39368			−0.696839	−	0.717228i	\(-0.745411\pi\)
							−0.696839	+	0.717228i	\(0.745411\pi\)
\(41\)	−470.042	−	814.137i	−0.279620	−	0.484317i	0.691670	−	0.722214i	\(-0.256875\pi\)
							−0.971290	+	0.237897i	\(0.923542\pi\)
\(43\)	937.371	−	541.191i	0.506961	−	0.292694i	−0.224623	−	0.974446i	\(-0.572115\pi\)
							0.731584	+	0.681752i	\(0.238782\pi\)
\(47\)	729.107	+	420.950i	0.330062	+	0.190561i	0.655869	−	0.754875i	\(-0.272303\pi\)
							−0.325807	+	0.945436i	\(0.605636\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.g.a.7.5	✓	76
4.3	odd	2	inner	76.5.g.a.7.22	yes	76
19.11	even	3	inner	76.5.g.a.11.22	yes	76
76.11	odd	6	inner	76.5.g.a.11.5	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.g.a.7.5	✓	76	1.1	even	1	trivial
76.5.g.a.7.22	yes	76	4.3	odd	2	inner
76.5.g.a.11.5	yes	76	76.11	odd	6	inner
76.5.g.a.11.22	yes	76	19.11	even	3	inner