Embedded newform 76.6.f.a.31.16

• Label: 76.6.f.a.31.16
• 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			31.16
Character	\(\chi\)	\(=\)	76.31
Dual form			76.6.f.a.27.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.77038 - 4.93204i) q^{2} +(-1.48160 - 2.56620i) q^{3} +(-16.6500 + 27.3272i) q^{4} +(-49.5981 - 85.9065i) q^{5} +(-8.55202 + 14.4167i) q^{6} -147.623i q^{7} +(180.906 + 6.41167i) q^{8} +(117.110 - 202.840i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 3 q^{2} - 11 q^{4} - 2 q^{5} - 27 q^{6} - 3588 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.77038	−	4.93204i	−0.489738	−	0.871869i
							\(3\)	−1.48160	−	2.56620i	−0.0950446	−	0.164622i	0.814583	−	0.580048i	\(-0.196966\pi\)
−0.909627	+	0.415426i	\(0.863633\pi\)
\(5\)	−49.5981	−	85.9065i	−0.887238	−	1.53674i	−0.843127	−	0.537715i	\(-0.819288\pi\)
							−0.0441117	−	0.999027i	\(-0.514046\pi\)
\(7\)		−	147.623i		−	1.13870i	−0.822095	−	0.569351i	\(-0.807195\pi\)
							0.822095	−	0.569351i	\(-0.192805\pi\)
\(11\)			472.824i			1.17820i	0.808061	+	0.589098i	\(0.200517\pi\)
							−0.808061	+	0.589098i	\(0.799483\pi\)
\(13\)	−652.215	−	376.557i	−1.07037	−	0.617976i	−0.142085	−	0.989854i	\(-0.545381\pi\)
							−0.928282	+	0.371878i	\(0.878714\pi\)
\(17\)	322.528	+	558.635i	0.270673	+	0.468819i	0.969034	−	0.246926i	\(-0.0794205\pi\)
							−0.698361	+	0.715745i	\(0.746087\pi\)
\(19\)	−305.450	+	1543.63i	−0.194114	+	0.980979i
							\(23\)	−256.967	−	148.360i	−0.101288	−	0.0584787i	0.448500	−	0.893783i	\(-0.351958\pi\)
−0.549788	+	0.835304i	\(0.685292\pi\)
\(29\)	−155.910	−	90.0149i	−0.0344255	−	0.0198756i	0.482688	−	0.875792i	\(-0.339660\pi\)
							−0.517114	+	0.855917i	\(0.672994\pi\)
\(31\)	8514.58			1.59132			0.795662	−	0.605740i	\(-0.207123\pi\)
							0.795662	+	0.605740i	\(0.207123\pi\)
\(37\)		−	8785.72i		−	1.05505i	−0.849540	−	0.527525i	\(-0.823120\pi\)
							0.849540	−	0.527525i	\(-0.176880\pi\)
\(41\)	−7674.20	+	4430.70i	−0.712974	+	0.411636i	−0.812161	−	0.583433i	\(-0.801709\pi\)
							0.0991874	+	0.995069i	\(0.468376\pi\)
\(43\)	8297.47	−	4790.55i	0.684344	−	0.395106i	−0.117146	−	0.993115i	\(-0.537374\pi\)
							0.801490	+	0.598009i	\(0.204041\pi\)
\(47\)	−11289.5	−	6518.00i	−0.745470	−	0.430397i	0.0785848	−	0.996907i	\(-0.474960\pi\)
							−0.824055	+	0.566510i	\(0.808293\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.6.f.a.31.16	yes	96
4.3	odd	2	inner	76.6.f.a.31.33	yes	96
19.8	odd	6	inner	76.6.f.a.27.33	yes	96
76.27	even	6	inner	76.6.f.a.27.16	✓	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.f.a.27.16	✓	96	76.27	even	6	inner
76.6.f.a.27.33	yes	96	19.8	odd	6	inner
76.6.f.a.31.16	yes	96	1.1	even	1	trivial
76.6.f.a.31.33	yes	96	4.3	odd	2	inner