Embedded newform 76.4.k.a.3.7

• Label: 76.4.k.a.3.7
• 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			3.7
Character	\(\chi\)	\(=\)	76.3
Dual form			76.4.k.a.51.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.90584 + 2.08992i) q^{2} +(1.16156 - 0.422774i) q^{3} +(-0.735541 - 7.96611i) q^{4} +(0.336107 - 1.90616i) q^{5} +(-1.33019 + 3.23331i) q^{6} +(-25.4625 + 14.7008i) q^{7} +(18.0504 + 13.6449i) q^{8} +(-19.5127 + 16.3731i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 6 q^{2} - 24 q^{4} - 12 q^{5} - 24 q^{6} - 9 q^{8} + 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.90584	+	2.08992i	−0.673817	+	0.738899i
							\(3\)	1.16156	−	0.422774i	0.223543	−	0.0813629i	−0.227821	−	0.973703i	\(-0.573160\pi\)
0.451363	+	0.892340i	\(0.350938\pi\)
\(5\)	0.336107	−	1.90616i	0.0300623	−	0.170492i	−0.966080	−	0.258242i	\(-0.916857\pi\)
							0.996142	+	0.0877504i	\(0.0279678\pi\)
\(7\)	−25.4625	+	14.7008i	−1.37485	+	0.793769i	−0.991534	−	0.129849i	\(-0.958551\pi\)
							−0.383314	+	0.923618i	\(0.625217\pi\)
\(11\)	−41.2989	−	23.8439i	−1.13201	−	0.653565i	−0.187568	−	0.982252i	\(-0.560061\pi\)
							−0.944439	+	0.328687i	\(0.893394\pi\)
\(13\)	11.0458	−	30.3482i	0.235659	−	0.647468i	−0.764337	−	0.644816i	\(-0.776934\pi\)
							0.999996	−	0.00265139i	\(-0.000843963\pi\)
\(17\)	24.4160	+	20.4875i	0.348339	+	0.292291i	0.800122	−	0.599837i	\(-0.204768\pi\)
							−0.451784	+	0.892127i	\(0.649212\pi\)
\(19\)	−81.1848	−	16.3714i	−0.980267	−	0.197677i
							\(23\)	−97.1521	+	17.1305i	−0.880766	+	0.155303i	−0.595702	−	0.803206i	\(-0.703126\pi\)
−0.285064	+	0.958508i	\(0.592015\pi\)
\(29\)	−124.128	−	147.930i	−0.794826	−	0.947237i	0.204675	−	0.978830i	\(-0.434386\pi\)
							−0.999501	+	0.0315931i	\(0.989942\pi\)
\(31\)	137.601	+	238.331i	0.797219	+	1.38082i	0.921420	+	0.388567i	\(0.127030\pi\)
							−0.124201	+	0.992257i	\(0.539637\pi\)
\(37\)			6.26769i			0.0278487i	0.999903	+	0.0139244i	\(0.00443240\pi\)
							−0.999903	+	0.0139244i	\(0.995568\pi\)
\(41\)	101.814	+	279.732i	0.387822	+	1.06553i	0.967980	+	0.251027i	\(0.0807684\pi\)
							−0.580158	+	0.814504i	\(0.697009\pi\)
\(43\)	396.570	+	69.9260i	1.40643	+	0.247991i	0.824782	−	0.565452i	\(-0.191298\pi\)
							0.581645	+	0.813443i	\(0.302409\pi\)
\(47\)	−324.223	−	386.394i	−1.00623	−	1.19918i	−0.979893	−	0.199524i	\(-0.936061\pi\)
							−0.0263357	−	0.999653i	\(-0.508384\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.4.k.a.3.7	✓	168
4.3	odd	2	inner	76.4.k.a.3.8	yes	168
19.13	odd	18	inner	76.4.k.a.51.8	yes	168
76.51	even	18	inner	76.4.k.a.51.7	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.k.a.3.7	✓	168	1.1	even	1	trivial
76.4.k.a.3.8	yes	168	4.3	odd	2	inner
76.4.k.a.51.7	yes	168	76.51	even	18	inner
76.4.k.a.51.8	yes	168	19.13	odd	18	inner