Embedded newform 76.5.g.a.7.19

• Label: 76.5.g.a.7.19
• 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			7.19
Character	\(\chi\)	\(=\)	76.7
Dual form			76.5.g.a.11.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.359633 + 3.98380i) q^{2} +(4.35470 - 2.51418i) q^{3} +(-15.7413 - 2.86542i) q^{4} +(9.73967 + 16.8696i) q^{5} +(8.44992 + 18.2524i) q^{6} +18.1632i q^{7} +(17.0764 - 61.6798i) q^{8} +(-27.8578 + 48.2510i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - q^{2} + 13 q^{4} - 2 q^{5} - 15 q^{6} - 94 q^{8} + 972 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.359633	+	3.98380i	−0.0899084	+	0.995950i
							\(3\)	4.35470	−	2.51418i	0.483855	−	0.279354i	−0.238167	−	0.971224i	\(-0.576546\pi\)
0.722022	+	0.691870i	\(0.243213\pi\)
\(5\)	9.73967	+	16.8696i	0.389587	+	0.674784i	0.992394	−	0.123103i	\(-0.0392845\pi\)
							−0.602807	+	0.797887i	\(0.705951\pi\)
\(7\)			18.1632i			0.370678i	0.982675	+	0.185339i	\(0.0593383\pi\)
							−0.982675	+	0.185339i	\(0.940662\pi\)
\(11\)			51.6905i			0.427194i	0.976922	+	0.213597i	\(0.0685179\pi\)
							−0.976922	+	0.213597i	\(0.931482\pi\)
\(13\)	−108.032	+	187.117i	−0.639244	+	1.10720i	0.346355	+	0.938104i	\(0.387419\pi\)
							−0.985599	+	0.169100i	\(0.945914\pi\)
\(17\)	28.9245	+	50.0988i	0.100085	+	0.173352i	0.911719	−	0.410813i	\(-0.134755\pi\)
							−0.811635	+	0.584166i	\(0.801422\pi\)
\(19\)	−358.039	−	46.1407i	−0.991798	−	0.127814i
							\(23\)	−148.501	−	85.7368i	−0.280719	−	0.162073i	0.353030	−	0.935612i	\(-0.385151\pi\)
−0.633749	+	0.773539i	\(0.718485\pi\)
\(29\)	333.195	−	577.110i	0.396189	−	0.686219i	−0.597063	−	0.802194i	\(-0.703666\pi\)
							0.993252	+	0.115975i	\(0.0369992\pi\)
\(31\)			1081.95i			1.12586i	0.826506	+	0.562928i	\(0.190325\pi\)
							−0.826506	+	0.562928i	\(0.809675\pi\)
\(37\)	2364.60			1.72724			0.863622	−	0.504140i	\(-0.168191\pi\)
							0.863622	+	0.504140i	\(0.168191\pi\)
\(41\)	333.631	+	577.867i	0.198472	+	0.343764i	0.948033	−	0.318172i	\(-0.103069\pi\)
							−0.749561	+	0.661935i	\(0.769735\pi\)
\(43\)	1315.03	−	759.231i	0.711210	−	0.410617i	−0.100299	−	0.994957i	\(-0.531980\pi\)
							0.811509	+	0.584340i	\(0.198647\pi\)
\(47\)	805.439	+	465.020i	0.364617	+	0.210512i	0.671104	−	0.741363i	\(-0.265820\pi\)
							−0.306487	+	0.951875i	\(0.599154\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.g.a.7.19	✓	76
4.3	odd	2	inner	76.5.g.a.7.33	yes	76
19.11	even	3	inner	76.5.g.a.11.33	yes	76
76.11	odd	6	inner	76.5.g.a.11.19	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.g.a.7.19	✓	76	1.1	even	1	trivial
76.5.g.a.7.33	yes	76	4.3	odd	2	inner
76.5.g.a.11.19	yes	76	76.11	odd	6	inner
76.5.g.a.11.33	yes	76	19.11	even	3	inner