Embedded newform 76.7.l.a.23.17

• Label: 76.7.l.a.23.17
• Level: $76$
• Weight: $7$
• Character: 76.23
• Analytic conductor: $17.484$
• Analytic rank: $0$
• Dimension: $348$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	76.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.4841103551\)
Analytic rank:	\(0\)
Dimension:	\(348\)
Relative dimension:	\(58\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			23.17
Character	\(\chi\)	\(=\)	76.23
Dual form			76.7.l.a.43.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.61729 + 6.53304i) q^{2} +(1.04918 - 0.184999i) q^{3} +(-21.3612 - 60.3299i) q^{4} +(59.4109 - 49.8517i) q^{5} +(-3.63577 + 7.70853i) q^{6} +(-459.233 - 265.139i) q^{7} +(492.769 + 139.007i) q^{8} +(-683.969 + 248.944i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q - 6 q^{2} + 192 q^{4} - 12 q^{5} - 492 q^{6} - 3 q^{8} + 1368 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.61729	+	6.53304i	−0.577162	+	0.816630i
							\(3\)	1.04918	−	0.184999i	0.0388585	−	0.00685181i	−0.154185	−	0.988042i	\(-0.549275\pi\)
0.193044	+	0.981190i	\(0.438164\pi\)
\(5\)	59.4109	−	49.8517i	0.475287	−	0.398813i	−0.373432	−	0.927658i	\(-0.621819\pi\)
							0.848719	+	0.528844i	\(0.177375\pi\)
\(7\)	−459.233	−	265.139i	−1.33887	−	0.772999i	−0.352233	−	0.935913i	\(-0.614577\pi\)
							−0.986640	+	0.162914i	\(0.947911\pi\)
\(11\)	1483.76	−	856.650i	1.11477	−	0.643614i	0.174711	−	0.984620i	\(-0.444101\pi\)
							0.940061	+	0.341006i	\(0.110768\pi\)
\(13\)	−416.926	+	2364.50i	−0.189771	+	1.07624i	0.729901	+	0.683553i	\(0.239566\pi\)
							−0.919671	+	0.392689i	\(0.871545\pi\)
\(17\)	7634.02	+	2778.56i	1.55384	+	0.565552i	0.969315	−	0.245823i	\(-0.0790582\pi\)
							0.584526	+	0.811375i	\(0.301280\pi\)
\(19\)	−2450.63	+	6406.27i	−0.357287	+	0.933994i
							\(23\)	−2668.57	+	3180.28i	−0.219328	+	0.261385i	−0.864478	−	0.502671i	\(-0.832351\pi\)
0.645149	+	0.764056i	\(0.276795\pi\)
\(29\)	−3464.56	+	1261.00i	−0.142054	+	0.0517035i	−0.412069	−	0.911153i	\(-0.635194\pi\)
							0.270015	+	0.962856i	\(0.412971\pi\)
\(31\)	−25085.3	−	14483.0i	−0.842045	−	0.486155i	0.0159141	−	0.999873i	\(-0.494934\pi\)
							−0.857959	+	0.513719i	\(0.828268\pi\)
\(37\)	−8623.54			−0.170247			−0.0851237	−	0.996370i	\(-0.527129\pi\)
							−0.0851237	+	0.996370i	\(0.527129\pi\)
\(41\)	20930.4	+	118702.i	0.303687	+	1.72230i	0.629621	+	0.776903i	\(0.283210\pi\)
							−0.325934	+	0.945393i	\(0.605679\pi\)
\(43\)	21179.6	+	25240.9i	0.266387	+	0.317467i	0.882612	−	0.470103i	\(-0.155783\pi\)
							−0.616225	+	0.787570i	\(0.711339\pi\)
\(47\)	48623.5	+	133592.i	0.468331	+	1.28673i	0.919077	+	0.394077i	\(0.128936\pi\)
							−0.450746	+	0.892652i	\(0.648842\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.7.l.a.23.17	✓	348
4.3	odd	2	inner	76.7.l.a.23.31	yes	348
19.5	even	9	inner	76.7.l.a.43.31	yes	348
76.43	odd	18	inner	76.7.l.a.43.17	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.7.l.a.23.17	✓	348	1.1	even	1	trivial
76.7.l.a.23.31	yes	348	4.3	odd	2	inner
76.7.l.a.43.17	yes	348	76.43	odd	18	inner
76.7.l.a.43.31	yes	348	19.5	even	9	inner