Embedded newform 76.5.g.a.7.2

• Label: 76.5.g.a.7.2
• 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			7.2
Character	\(\chi\)	\(=\)	76.7
Dual form			76.5.g.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.98849 - 0.303244i) q^{2} +(0.697952 - 0.402963i) q^{3} +(15.8161 + 2.41897i) q^{4} +(22.3482 + 38.7083i) q^{5} +(-2.90597 + 1.39556i) q^{6} -79.0280i q^{7} +(-62.3487 - 14.4442i) q^{8} +(-40.1752 + 69.5856i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - q^{2} + 13 q^{4} - 2 q^{5} - 15 q^{6} - 94 q^{8} + 972 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.98849	−	0.303244i	−0.997122	−	0.0758110i
							\(3\)	0.697952	−	0.402963i	0.0775502	−	0.0447736i	−0.460723	−	0.887544i	\(-0.652410\pi\)
0.538274	+	0.842770i	\(0.319077\pi\)
\(5\)	22.3482	+	38.7083i	0.893929	+	1.54833i	0.835124	+	0.550061i	\(0.185396\pi\)
							0.0588050	+	0.998269i	\(0.481271\pi\)
\(7\)		−	79.0280i		−	1.61282i	−0.591359	−	0.806409i	\(-0.701408\pi\)
							0.591359	−	0.806409i	\(-0.298592\pi\)
\(11\)			108.934i			0.900277i	0.892959	+	0.450138i	\(0.148625\pi\)
							−0.892959	+	0.450138i	\(0.851375\pi\)
\(13\)	60.1770	−	104.230i	0.356077	−	0.616743i	−0.631225	−	0.775600i	\(-0.717448\pi\)
							0.987302	+	0.158857i	\(0.0507809\pi\)
\(17\)	195.508	+	338.629i	0.676497	+	1.17173i	0.976029	+	0.217640i	\(0.0698360\pi\)
							−0.299532	+	0.954086i	\(0.596831\pi\)
\(19\)	20.7500	+	360.403i	0.0574793	+	0.998347i
							\(23\)	−47.1661	−	27.2313i	−0.0891608	−	0.0514770i	0.454757	−	0.890616i	\(-0.349726\pi\)
−0.543917	+	0.839139i	\(0.683060\pi\)
\(29\)	−58.0920	+	100.618i	−0.0690749	+	0.119641i	−0.898494	−	0.438985i	\(-0.855338\pi\)
							0.829419	+	0.558626i	\(0.188671\pi\)
\(31\)			792.737i			0.824908i	0.910978	+	0.412454i	\(0.135328\pi\)
							−0.910978	+	0.412454i	\(0.864672\pi\)
\(37\)	1585.14			1.15788			0.578939	−	0.815371i	\(-0.303467\pi\)
							0.578939	+	0.815371i	\(0.303467\pi\)
\(41\)	77.7753	+	134.711i	0.0462673	+	0.0801373i	0.888232	−	0.459396i	\(-0.151934\pi\)
							−0.841964	+	0.539533i	\(0.818601\pi\)
\(43\)	−330.250	+	190.670i	−0.178610	+	0.103121i	−0.586639	−	0.809848i	\(-0.699549\pi\)
							0.408029	+	0.912969i	\(0.366216\pi\)
\(47\)	−918.378	−	530.226i	−0.415744	−	0.240030i	0.277511	−	0.960722i	\(-0.410491\pi\)
							−0.693255	+	0.720693i	\(0.743824\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.g.a.7.2	✓	76
4.3	odd	2	inner	76.5.g.a.7.25	yes	76
19.11	even	3	inner	76.5.g.a.11.25	yes	76
76.11	odd	6	inner	76.5.g.a.11.2	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.g.a.7.2	✓	76	1.1	even	1	trivial
76.5.g.a.7.25	yes	76	4.3	odd	2	inner
76.5.g.a.11.2	yes	76	76.11	odd	6	inner
76.5.g.a.11.25	yes	76	19.11	even	3	inner