Embedded newform 76.6.f.a.31.17

• Label: 76.6.f.a.31.17
• Level: $76$
• Weight: $6$
• Character: 76.31
• Analytic conductor: $12.189$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(12.1891703058\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(48\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.17
Character	\(\chi\)	\(=\)	76.31
Dual form			76.6.f.a.27.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.64697 - 4.99936i) q^{2} +(-15.2473 - 26.4090i) q^{3} +(-17.9871 + 26.4663i) q^{4} +(-23.0916 - 39.9958i) q^{5} +(-91.6692 + 146.130i) q^{6} +131.131i q^{7} +(179.926 + 19.8689i) q^{8} +(-343.458 + 594.887i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 3 q^{2} - 11 q^{4} - 2 q^{5} - 27 q^{6} - 3588 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.64697	−	4.99936i	−0.467922	−	0.883770i
							\(3\)	−15.2473	−	26.4090i	−0.978112	−	1.69414i	−0.669256	−	0.743032i	\(-0.733387\pi\)
−0.308856	−	0.951109i	\(-0.599946\pi\)
\(5\)	−23.0916	−	39.9958i	−0.413075	−	0.715467i	0.582149	−	0.813082i	\(-0.302212\pi\)
							−0.995224	+	0.0976153i	\(0.968879\pi\)
\(7\)			131.131i			1.01149i	0.862684	+	0.505743i	\(0.168782\pi\)
							−0.862684	+	0.505743i	\(0.831218\pi\)
\(11\)		−	582.590i		−	1.45171i	−0.687845	−	0.725857i	\(-0.741443\pi\)
							0.687845	−	0.725857i	\(-0.258557\pi\)
\(13\)	−337.943	−	195.112i	−0.554607	−	0.320203i	0.196371	−	0.980530i	\(-0.437084\pi\)
							−0.750978	+	0.660327i	\(0.770418\pi\)
\(17\)	359.750	+	623.106i	0.301911	+	0.522925i	0.976569	−	0.215205i	\(-0.0690421\pi\)
							−0.674658	+	0.738131i	\(0.735709\pi\)
\(19\)	−1504.51	−	461.024i	−0.956118	−	0.292981i
							\(23\)	−1699.95	−	981.465i	−0.670063	−	0.386861i	0.126038	−	0.992025i	\(-0.459774\pi\)
−0.796101	+	0.605164i	\(0.793107\pi\)
\(29\)	6022.02	+	3476.82i	1.32968	+	0.767692i	0.985250	−	0.171120i	\(-0.0547387\pi\)
							0.344431	+	0.938812i	\(0.388072\pi\)
\(31\)	−832.313			−0.155554			−0.0777772	−	0.996971i	\(-0.524782\pi\)
							−0.0777772	+	0.996971i	\(0.524782\pi\)
\(37\)			8818.65i			1.05900i	0.848308	+	0.529502i	\(0.177621\pi\)
							−0.848308	+	0.529502i	\(0.822379\pi\)
\(41\)	−382.746	+	220.979i	−0.0355591	+	0.0205301i	−0.517674	−	0.855578i	\(-0.673202\pi\)
							0.482115	+	0.876108i	\(0.339869\pi\)
\(43\)	−3215.97	+	1856.74i	−0.265241	+	0.153137i	−0.626723	−	0.779242i	\(-0.715604\pi\)
							0.361482	+	0.932379i	\(0.382271\pi\)
\(47\)	2413.36	+	1393.36i	0.159360	+	0.0920063i	0.577559	−	0.816349i	\(-0.304005\pi\)
							−0.418199	+	0.908355i	\(0.637339\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.6.f.a.31.17	yes	96
4.3	odd	2	inner	76.6.f.a.31.34	yes	96
19.8	odd	6	inner	76.6.f.a.27.34	yes	96
76.27	even	6	inner	76.6.f.a.27.17	✓	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.f.a.27.17	✓	96	76.27	even	6	inner
76.6.f.a.27.34	yes	96	19.8	odd	6	inner
76.6.f.a.31.17	yes	96	1.1	even	1	trivial
76.6.f.a.31.34	yes	96	4.3	odd	2	inner