Embedded newform 76.5.l.a.23.13

• Label: 76.5.l.a.23.13
• 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			23.13
Character	\(\chi\)	\(=\)	76.23
Dual form			76.5.l.a.43.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.54434 - 3.68985i) q^{2} +(12.3954 - 2.18564i) q^{3} +(-11.2300 + 11.3968i) q^{4} +(-26.3005 + 22.0687i) q^{5} +(-27.2074 - 42.3618i) q^{6} +(82.9404 + 47.8857i) q^{7} +(59.3954 + 23.8368i) q^{8} +(72.7534 - 26.4801i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 48 q^{4} - 12 q^{5} - 60 q^{6} - 3 q^{8} - 180 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.54434	−	3.68985i	−0.386084	−	0.922463i
							\(3\)	12.3954	−	2.18564i	1.37726	−	0.242849i	0.564496	−	0.825436i	\(-0.309070\pi\)
0.812769	+	0.582587i	\(0.197959\pi\)
\(5\)	−26.3005	+	22.0687i	−1.05202	+	0.882749i	−0.993305	−	0.115526i	\(-0.963145\pi\)
							−0.0587151	+	0.998275i	\(0.518700\pi\)
\(7\)	82.9404	+	47.8857i	1.69266	+	0.977259i	0.952356	+	0.304989i	\(0.0986528\pi\)
							0.740306	+	0.672270i	\(0.234681\pi\)
\(11\)	45.2460	−	26.1228i	0.373934	−	0.215891i	−0.301242	−	0.953548i	\(-0.597401\pi\)
							0.675176	+	0.737657i	\(0.264068\pi\)
\(13\)	−18.4064	+	104.388i	−0.108914	+	0.617680i	0.880671	+	0.473728i	\(0.157092\pi\)
							−0.989585	+	0.143952i	\(0.954019\pi\)
\(17\)	172.470	+	62.7741i	0.596784	+	0.217211i	0.622710	−	0.782452i	\(-0.286031\pi\)
							−0.0259268	+	0.999664i	\(0.508254\pi\)
\(19\)	−360.978	+	3.95319i	−0.999940	+	0.0109507i
							\(23\)	430.190	−	512.681i	0.813214	−	0.969151i	−0.186698	−	0.982417i	\(-0.559778\pi\)
0.999912	+	0.0132663i	\(0.00422292\pi\)
\(29\)	159.392	−	58.0139i	0.189527	−	0.0689821i	−0.245513	−	0.969393i	\(-0.578956\pi\)
							0.435040	+	0.900411i	\(0.356734\pi\)
\(31\)	1324.70	+	764.813i	1.37846	+	0.795852i	0.991974	−	0.126445i	\(-0.0403568\pi\)
							0.386482	+	0.922297i	\(0.373690\pi\)
\(37\)	−1216.54			−0.888634			−0.444317	−	0.895870i	\(-0.646554\pi\)
							−0.444317	+	0.895870i	\(0.646554\pi\)
\(41\)	−426.483	−	2418.70i	−0.253708	−	1.43885i	−0.799368	−	0.600842i	\(-0.794832\pi\)
							0.545660	−	0.838007i	\(-0.316279\pi\)
\(43\)	−32.3779	−	38.5864i	−0.0175110	−	0.0208688i	0.757218	−	0.653162i	\(-0.226558\pi\)
							−0.774729	+	0.632294i	\(0.782114\pi\)
\(47\)	−2.66221	−	7.31435i	−0.00120516	−	0.00331116i	0.939089	−	0.343675i	\(-0.111672\pi\)
							−0.940294	+	0.340364i	\(0.889450\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.l.a.23.13	yes	228
4.3	odd	2	inner	76.5.l.a.23.6	✓	228
19.5	even	9	inner	76.5.l.a.43.6	yes	228
76.43	odd	18	inner	76.5.l.a.43.13	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.l.a.23.6	✓	228	4.3	odd	2	inner
76.5.l.a.23.13	yes	228	1.1	even	1	trivial
76.5.l.a.43.6	yes	228	19.5	even	9	inner
76.5.l.a.43.13	yes	228	76.43	odd	18	inner