Embedded newform 76.5.g.a.7.10

• Label: 76.5.g.a.7.10
• 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			7.10
Character	\(\chi\)	\(=\)	76.7
Dual form			76.5.g.a.11.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.88768 + 2.76791i) q^{2} +(-6.22437 + 3.59364i) q^{3} +(0.677370 - 15.9857i) q^{4} +(17.9743 + 31.1323i) q^{5} +(8.02710 - 27.6057i) q^{6} +37.3202i q^{7} +(42.2908 + 48.0363i) q^{8} +(-14.6715 + 25.4118i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - q^{2} + 13 q^{4} - 2 q^{5} - 15 q^{6} - 94 q^{8} + 972 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.88768	+	2.76791i	−0.721920	+	0.691977i
							\(3\)	−6.22437	+	3.59364i	−0.691596	+	0.399293i	−0.804210	−	0.594346i	\(-0.797411\pi\)
0.112613	+	0.993639i	\(0.464078\pi\)
\(5\)	17.9743	+	31.1323i	0.718970	+	1.24529i	0.961408	+	0.275126i	\(0.0887196\pi\)
							−0.242438	+	0.970167i	\(0.577947\pi\)
\(7\)			37.3202i			0.761637i	0.924650	+	0.380819i	\(0.124358\pi\)
							−0.924650	+	0.380819i	\(0.875642\pi\)
\(11\)		−	51.9444i		−	0.429293i	−0.976692	−	0.214646i	\(-0.931140\pi\)
							0.976692	−	0.214646i	\(-0.0688599\pi\)
\(13\)	−32.4288	+	56.1683i	−0.191886	+	0.332357i	−0.945875	−	0.324530i	\(-0.894794\pi\)
							0.753989	+	0.656887i	\(0.228127\pi\)
\(17\)	−258.741	−	448.152i	−0.895296	−	1.55070i	−0.833437	−	0.552614i	\(-0.813630\pi\)
							−0.0618589	−	0.998085i	\(-0.519703\pi\)
\(19\)	251.717	+	258.766i	0.697277	+	0.716802i
							\(23\)	−559.127	−	322.812i	−1.05695	−	0.610230i	−0.132363	−	0.991201i	\(-0.542256\pi\)
−0.924587	+	0.380971i	\(0.875590\pi\)
\(29\)	−483.356	+	837.198i	−0.574740	+	0.995479i	0.421330	+	0.906908i	\(0.361564\pi\)
							−0.996070	+	0.0885716i	\(0.971770\pi\)
\(31\)			709.328i			0.738115i	0.929407	+	0.369057i	\(0.120319\pi\)
							−0.929407	+	0.369057i	\(0.879681\pi\)
\(37\)	−2180.08			−1.59246			−0.796231	−	0.604993i	\(-0.793176\pi\)
							−0.796231	+	0.604993i	\(0.793176\pi\)
\(41\)	1068.04	+	1849.91i	0.635363	+	1.10048i	0.986438	+	0.164133i	\(0.0524827\pi\)
							−0.351075	+	0.936347i	\(0.614184\pi\)
\(43\)	2189.08	−	1263.86i	1.18392	−	0.683539i	0.227005	−	0.973894i	\(-0.427107\pi\)
							0.956919	+	0.290355i	\(0.0937733\pi\)
\(47\)	−112.975	−	65.2263i	−0.0511432	−	0.0295275i	0.474210	−	0.880412i	\(-0.342734\pi\)
							−0.525354	+	0.850884i	\(0.676067\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.g.a.7.10	✓	76
4.3	odd	2	inner	76.5.g.a.7.35	yes	76
19.11	even	3	inner	76.5.g.a.11.35	yes	76
76.11	odd	6	inner	76.5.g.a.11.10	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.g.a.7.10	✓	76	1.1	even	1	trivial
76.5.g.a.7.35	yes	76	4.3	odd	2	inner
76.5.g.a.11.10	yes	76	76.11	odd	6	inner
76.5.g.a.11.35	yes	76	19.11	even	3	inner