Embedded newform 76.3.l.a.23.13

• Label: 76.3.l.a.23.13
• Level: $76$
• Weight: $3$
• Character: 76.23
• Analytic conductor: $2.071$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	76.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.07085000914\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(18\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			23.13
Character	\(\chi\)	\(=\)	76.23
Dual form			76.3.l.a.43.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.32019 + 1.50237i) q^{2} +(2.09930 - 0.370163i) q^{3} +(-0.514209 + 3.96681i) q^{4} +(3.93098 - 3.29849i) q^{5} +(3.32759 + 2.66523i) q^{6} +(-3.79547 - 2.19131i) q^{7} +(-6.63846 + 4.46440i) q^{8} +(-4.18719 + 1.52401i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 6 q^{2} - 12 q^{5} + 12 q^{6} - 3 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.32019	+	1.50237i	0.660094	+	0.751183i
							\(3\)	2.09930	−	0.370163i	0.699767	−	0.123388i	0.187564	−	0.982252i	\(-0.439941\pi\)
0.512203	+	0.858865i	\(0.328830\pi\)
\(5\)	3.93098	−	3.29849i	0.786197	−	0.659697i	−0.158604	−	0.987342i	\(-0.550699\pi\)
							0.944801	+	0.327645i	\(0.106255\pi\)
\(7\)	−3.79547	−	2.19131i	−0.542209	−	0.313045i	0.203765	−	0.979020i	\(-0.434682\pi\)
							−0.745974	+	0.665975i	\(0.768016\pi\)
\(11\)	2.81191	−	1.62346i	0.255628	−	0.147587i	−0.366711	−	0.930335i	\(-0.619516\pi\)
							0.622339	+	0.782748i	\(0.286183\pi\)
\(13\)	−1.04780	+	5.94238i	−0.0806001	+	0.457106i	0.917619	+	0.397460i	\(0.130108\pi\)
							−0.998220	+	0.0596459i	\(0.981003\pi\)
\(17\)	−22.4046	−	8.15461i	−1.31792	−	0.479683i	−0.415128	−	0.909763i	\(-0.636263\pi\)
							−0.902791	+	0.430080i	\(0.858485\pi\)
\(19\)	18.2241	−	5.37422i	0.959163	−	0.282854i
							\(23\)	14.5136	−	17.2967i	0.631028	−	0.752030i	−0.351897	−	0.936039i	\(-0.614463\pi\)
0.982925	+	0.184009i	\(0.0589076\pi\)
\(29\)	27.0450	−	9.84357i	0.932586	−	0.339433i	0.169352	−	0.985556i	\(-0.445833\pi\)
							0.763234	+	0.646122i	\(0.223610\pi\)
\(31\)	−17.3768	−	10.0325i	−0.560543	−	0.323630i	0.192820	−	0.981234i	\(-0.438237\pi\)
							−0.753364	+	0.657604i	\(0.771570\pi\)
\(37\)	1.95625			0.0528718			0.0264359	−	0.999651i	\(-0.491584\pi\)
							0.0264359	+	0.999651i	\(0.491584\pi\)
\(41\)	5.59322	+	31.7207i	0.136420	+	0.773676i	0.973860	+	0.227148i	\(0.0729400\pi\)
							−0.837440	+	0.546529i	\(0.815949\pi\)
\(43\)	6.25388	+	7.45308i	0.145439	+	0.173327i	0.833846	−	0.551997i	\(-0.186134\pi\)
							−0.688407	+	0.725325i	\(0.741690\pi\)
\(47\)	30.7984	+	84.6180i	0.655286	+	1.80038i	0.597229	+	0.802071i	\(0.296268\pi\)
							0.0580569	+	0.998313i	\(0.481510\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.3.l.a.23.13	✓	108
4.3	odd	2	inner	76.3.l.a.23.17	yes	108
19.5	even	9	inner	76.3.l.a.43.17	yes	108
76.43	odd	18	inner	76.3.l.a.43.13	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.l.a.23.13	✓	108	1.1	even	1	trivial
76.3.l.a.23.17	yes	108	4.3	odd	2	inner
76.3.l.a.43.13	yes	108	76.43	odd	18	inner
76.3.l.a.43.17	yes	108	19.5	even	9	inner