Embedded newform 76.4.k.a.3.4

• Label: 76.4.k.a.3.4
• 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			3.4
Character	\(\chi\)	\(=\)	76.3
Dual form			76.4.k.a.51.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.66917 - 0.935695i) q^{2} +(2.28426 - 0.831401i) q^{3} +(6.24895 + 4.99506i) q^{4} +(-1.38264 + 7.84134i) q^{5} +(-6.87501 + 0.0817851i) q^{6} +(-8.80596 + 5.08413i) q^{7} +(-12.0057 - 19.1798i) q^{8} +(-16.1566 + 13.5570i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 6 q^{2} - 24 q^{4} - 12 q^{5} - 24 q^{6} - 9 q^{8} + 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.66917	−	0.935695i	−0.943695	−	0.330818i
							\(3\)	2.28426	−	0.831401i	0.439605	−	0.160003i	−0.112729	−	0.993626i	\(-0.535959\pi\)
0.552335	+	0.833623i	\(0.313737\pi\)
\(5\)	−1.38264	+	7.84134i	−0.123667	+	0.701350i	0.858424	+	0.512941i	\(0.171444\pi\)
							−0.982091	+	0.188409i	\(0.939667\pi\)
\(7\)	−8.80596	+	5.08413i	−0.475477	+	0.274517i	−0.718530	−	0.695496i	\(-0.755185\pi\)
							0.243052	+	0.970013i	\(0.421851\pi\)
\(11\)	36.8082	+	21.2512i	1.00892	+	0.582499i	0.910874	−	0.412684i	\(-0.135408\pi\)
							0.0980427	+	0.995182i	\(0.468742\pi\)
\(13\)	−12.2465	+	33.6471i	−0.261275	+	0.717848i	0.737807	+	0.675012i	\(0.235862\pi\)
							−0.999082	+	0.0428362i	\(0.986361\pi\)
\(17\)	99.7828	+	83.7277i	1.42358	+	1.19453i	0.949388	+	0.314105i	\(0.101704\pi\)
							0.474193	+	0.880421i	\(0.342740\pi\)
\(19\)	−80.8816	−	17.8090i	−0.976606	−	0.215035i
							\(23\)	38.6251	−	6.81065i	0.350169	−	0.0617443i	0.00420284	−	0.999991i	\(-0.498662\pi\)
0.345967	+	0.938247i	\(0.387551\pi\)
\(29\)	−37.3435	−	44.5042i	−0.239121	−	0.284973i	0.633116	−	0.774057i	\(-0.281776\pi\)
							−0.872237	+	0.489084i	\(0.837331\pi\)
\(31\)	−124.321	−	215.330i	−0.720278	−	1.24756i	−0.960888	−	0.276937i	\(-0.910681\pi\)
							0.240610	−	0.970622i	\(-0.422653\pi\)
\(37\)			118.853i			0.528090i	0.964510	+	0.264045i	\(0.0850567\pi\)
							−0.964510	+	0.264045i	\(0.914943\pi\)
\(41\)	−65.9833	−	181.288i	−0.251338	−	0.690545i	−0.999631	−	0.0271767i	\(-0.991348\pi\)
							0.748293	−	0.663369i	\(-0.230874\pi\)
\(43\)	−74.8431	−	13.1969i	−0.265429	−	0.0468024i	0.0393494	−	0.999226i	\(-0.487471\pi\)
							−0.304779	+	0.952423i	\(0.598583\pi\)
\(47\)	−29.3598	−	34.9897i	−0.0911185	−	0.108591i	0.718559	−	0.695466i	\(-0.244802\pi\)
							−0.809677	+	0.586876i	\(0.800358\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.4.k.a.3.4	✓	168
4.3	odd	2	inner	76.4.k.a.3.19	yes	168
19.13	odd	18	inner	76.4.k.a.51.19	yes	168
76.51	even	18	inner	76.4.k.a.51.4	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.k.a.3.4	✓	168	1.1	even	1	trivial
76.4.k.a.3.19	yes	168	4.3	odd	2	inner
76.4.k.a.51.4	yes	168	76.51	even	18	inner
76.4.k.a.51.19	yes	168	19.13	odd	18	inner