Embedded newform 76.4.k.a.3.10

• Label: 76.4.k.a.3.10
• Level: $76$
• Weight: $4$
• Character: 76.3
• Analytic conductor: $4.484$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	76.k (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.48414516044\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(28\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.10
Character	\(\chi\)	\(=\)	76.3
Dual form			76.4.k.a.51.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.58740 - 2.34097i) q^{2} +(-1.55643 + 0.566493i) q^{3} +(-2.96030 + 7.43213i) q^{4} +(2.88655 - 16.3705i) q^{5} +(3.79682 + 2.74430i) q^{6} +(-17.3301 + 10.0055i) q^{7} +(22.0976 - 4.86782i) q^{8} +(-18.5816 + 15.5919i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 6 q^{2} - 24 q^{4} - 12 q^{5} - 24 q^{6} - 9 q^{8} + 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.58740	−	2.34097i	−0.561232	−	0.827659i
							\(3\)	−1.55643	+	0.566493i	−0.299535	+	0.109022i	−0.487416	−	0.873170i	\(-0.662060\pi\)
0.187881	+	0.982192i	\(0.439838\pi\)
\(5\)	2.88655	−	16.3705i	0.258181	−	1.46422i	−0.529591	−	0.848253i	\(-0.677655\pi\)
							0.787773	−	0.615966i	\(-0.211234\pi\)
\(7\)	−17.3301	+	10.0055i	−0.935737	+	0.540248i	−0.888621	−	0.458641i	\(-0.848336\pi\)
							−0.0471156	+	0.998889i	\(0.515003\pi\)
\(11\)	−10.9189	−	6.30402i	−0.299288	−	0.172794i	0.342835	−	0.939396i	\(-0.388613\pi\)
							−0.642123	+	0.766602i	\(0.721946\pi\)
\(13\)	−24.9383	+	68.5173i	−0.532049	+	1.46179i	0.324580	+	0.945858i	\(0.394777\pi\)
							−0.856629	+	0.515933i	\(0.827445\pi\)
\(17\)	−43.1147	−	36.1775i	−0.615109	−	0.516137i	0.281153	−	0.959663i	\(-0.409283\pi\)
							−0.896262	+	0.443526i	\(0.853728\pi\)
\(19\)	34.9610	+	75.0781i	0.422137	+	0.906532i
							\(23\)	−39.6502	+	6.99140i	−0.359463	+	0.0633830i	−0.350463	−	0.936577i	\(-0.613976\pi\)
−0.00899957	+	0.999960i	\(0.502865\pi\)
\(29\)	−141.341	−	168.444i	−0.905048	−	1.07859i	−0.996567	−	0.0827911i	\(-0.973617\pi\)
							0.0915188	−	0.995803i	\(-0.470828\pi\)
\(31\)	−157.660	−	273.075i	−0.913438	−	1.58212i	−0.809173	−	0.587571i	\(-0.800084\pi\)
							−0.104265	−	0.994550i	\(-0.533249\pi\)
\(37\)		−	112.995i		−	0.502059i	−0.967979	−	0.251030i	\(-0.919231\pi\)
							0.967979	−	0.251030i	\(-0.0807692\pi\)
\(41\)	31.5199	+	86.6002i	0.120063	+	0.329870i	0.985136	−	0.171776i	\(-0.0549504\pi\)
							−0.865073	+	0.501646i	\(0.832728\pi\)
\(43\)	359.089	+	63.3171i	1.27350	+	0.224553i	0.769218	−	0.638986i	\(-0.220646\pi\)
							0.504283	+	0.863539i	\(0.331757\pi\)
\(47\)	91.8819	+	109.501i	0.285156	+	0.339836i	0.889540	−	0.456857i	\(-0.151025\pi\)
							−0.604384	+	0.796693i	\(0.706581\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.4.k.a.3.10	✓	168
4.3	odd	2	inner	76.4.k.a.3.25	yes	168
19.13	odd	18	inner	76.4.k.a.51.25	yes	168
76.51	even	18	inner	76.4.k.a.51.10	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.k.a.3.10	✓	168	1.1	even	1	trivial
76.4.k.a.3.25	yes	168	4.3	odd	2	inner
76.4.k.a.51.10	yes	168	76.51	even	18	inner
76.4.k.a.51.25	yes	168	19.13	odd	18	inner