Embedded newform 76.5.l.a.23.20

• Label: 76.5.l.a.23.20
• Level: $76$
• Weight: $5$
• Character: 76.23
• Analytic conductor: $7.856$
• Analytic rank: $0$
• Dimension: $228$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	76.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85611719437\)
Analytic rank:	\(0\)
Dimension:	\(228\)
Relative dimension:	\(38\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			23.20
Character	\(\chi\)	\(=\)	76.23
Dual form			76.5.l.a.43.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.324370 + 3.98683i) q^{2} +(10.8623 - 1.91531i) q^{3} +(-15.7896 - 2.58641i) q^{4} +(23.2093 - 19.4749i) q^{5} +(4.11262 + 43.9272i) q^{6} +(10.4065 + 6.00818i) q^{7} +(15.4332 - 62.1113i) q^{8} +(38.2053 - 13.9056i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 48 q^{4} - 12 q^{5} - 60 q^{6} - 3 q^{8} - 180 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.324370	+	3.98683i	−0.0810924	+	0.996707i
							\(3\)	10.8623	−	1.91531i	1.20692	−	0.212812i	0.466231	−	0.884663i	\(-0.345611\pi\)
0.740687	+	0.671851i	\(0.234500\pi\)
\(5\)	23.2093	−	19.4749i	0.928372	−	0.778996i	−0.0471528	−	0.998888i	\(-0.515015\pi\)
							0.975524	+	0.219891i	\(0.0705703\pi\)
\(7\)	10.4065	+	6.00818i	0.212377	+	0.122616i	0.602416	−	0.798183i	\(-0.294205\pi\)
							−0.390039	+	0.920798i	\(0.627538\pi\)
\(11\)	192.168	−	110.948i	1.58816	−	0.916926i	0.594552	−	0.804057i	\(-0.297330\pi\)
							0.993610	−	0.112868i	\(-0.0360037\pi\)
\(13\)	−14.6509	+	83.0894i	−0.0866918	+	0.491654i	0.910287	+	0.413978i	\(0.135861\pi\)
							−0.996979	+	0.0776756i	\(0.975250\pi\)
\(17\)	−205.158	−	74.6714i	−0.709889	−	0.258378i	−0.0382618	−	0.999268i	\(-0.512182\pi\)
							−0.671627	+	0.740889i	\(0.734404\pi\)
\(19\)	20.4289	+	360.422i	0.0565898	+	0.998398i
							\(23\)	−249.894	+	297.812i	−0.472390	+	0.562972i	−0.948648	−	0.316333i	\(-0.897548\pi\)
0.476258	+	0.879305i	\(0.341993\pi\)
\(29\)	737.002	−	268.247i	0.876340	−	0.318962i	0.135608	−	0.990763i	\(-0.456701\pi\)
							0.740732	+	0.671801i	\(0.234479\pi\)
\(31\)	493.671	+	285.021i	0.513705	+	0.296588i	0.734355	−	0.678765i	\(-0.237485\pi\)
							−0.220650	+	0.975353i	\(0.570818\pi\)
\(37\)	−2110.41			−1.54157			−0.770784	−	0.637096i	\(-0.780135\pi\)
							−0.770784	+	0.637096i	\(0.780135\pi\)
\(41\)	−347.497	−	1970.75i	−0.206721	−	1.17237i	−0.894709	−	0.446649i	\(-0.852617\pi\)
							0.687989	−	0.725721i	\(-0.258494\pi\)
\(43\)	−413.160	−	492.385i	−0.223450	−	0.266298i	0.642659	−	0.766152i	\(-0.277831\pi\)
							−0.866109	+	0.499855i	\(0.833387\pi\)
\(47\)	−267.419	−	734.727i	−0.121059	−	0.332606i	0.864330	−	0.502924i	\(-0.167743\pi\)
							−0.985389	+	0.170318i	\(0.945520\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.l.a.23.20	✓	228
4.3	odd	2	inner	76.5.l.a.23.27	yes	228
19.5	even	9	inner	76.5.l.a.43.27	yes	228
76.43	odd	18	inner	76.5.l.a.43.20	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.l.a.23.20	✓	228	1.1	even	1	trivial
76.5.l.a.23.27	yes	228	4.3	odd	2	inner
76.5.l.a.43.20	yes	228	76.43	odd	18	inner
76.5.l.a.43.27	yes	228	19.5	even	9	inner