Embedded newform 76.3.l.a.23.5

• Label: 76.3.l.a.23.5
• Level: $76$
• Weight: $3$
• Character: 76.23
• Analytic conductor: $2.071$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	76.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.07085000914\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(18\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			23.5
Character	\(\chi\)	\(=\)	76.23
Dual form			76.3.l.a.43.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.57322 + 1.23490i) q^{2} +(-4.66468 + 0.822509i) q^{3} +(0.950039 - 3.88554i) q^{4} +(2.08183 - 1.74687i) q^{5} +(6.32285 - 7.05441i) q^{6} +(-0.551331 - 0.318311i) q^{7} +(3.30364 + 7.28601i) q^{8} +(12.6255 - 4.59531i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 6 q^{2} - 12 q^{5} + 12 q^{6} - 3 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.57322	+	1.23490i	−0.786610	+	0.617450i
							\(3\)	−4.66468	+	0.822509i	−1.55489	+	0.274170i	−0.884037	−	0.467417i	\(-0.845185\pi\)
−0.670857	+	0.741587i	\(0.734074\pi\)
\(5\)	2.08183	−	1.74687i	0.416367	−	0.349373i	−0.410412	−	0.911900i	\(-0.634615\pi\)
							0.826779	+	0.562527i	\(0.190171\pi\)
\(7\)	−0.551331	−	0.318311i	−0.0787616	−	0.0454730i	0.460102	−	0.887866i	\(-0.347813\pi\)
							−0.538864	+	0.842393i	\(0.681146\pi\)
\(11\)	17.4118	−	10.0527i	1.58289	−	0.913883i	0.588457	−	0.808529i	\(-0.299736\pi\)
							0.994435	−	0.105354i	\(-0.0335977\pi\)
\(13\)	1.73716	−	9.85191i	0.133628	−	0.757840i	−0.842178	−	0.539200i	\(-0.818727\pi\)
							0.975806	−	0.218640i	\(-0.0701620\pi\)
\(17\)	−10.4131	−	3.79007i	−0.612537	−	0.222945i	0.0170758	−	0.999854i	\(-0.494564\pi\)
							−0.629613	+	0.776909i	\(0.716787\pi\)
\(19\)	−12.2950	−	14.4856i	−0.647107	−	0.762399i
							\(23\)	12.7612	−	15.2083i	0.554837	−	0.661229i	−0.413609	−	0.910455i	\(-0.635732\pi\)
0.968445	+	0.249226i	\(0.0801763\pi\)
\(29\)	43.3800	−	15.7890i	1.49586	−	0.544450i	0.540878	−	0.841101i	\(-0.318092\pi\)
							0.954986	+	0.296651i	\(0.0958699\pi\)
\(31\)	−19.8612	−	11.4669i	−0.640685	−	0.369900i	0.144193	−	0.989550i	\(-0.453941\pi\)
							−0.784878	+	0.619650i	\(0.787275\pi\)
\(37\)	1.23385			0.0333473			0.0166737	−	0.999861i	\(-0.494692\pi\)
							0.0166737	+	0.999861i	\(0.494692\pi\)
\(41\)	3.29674	+	18.6967i	0.0804083	+	0.456018i	0.998253	+	0.0590791i	\(0.0188164\pi\)
							−0.917845	+	0.396939i	\(0.870072\pi\)
\(43\)	15.3813	+	18.3307i	0.357705	+	0.426296i	0.914646	−	0.404256i	\(-0.132470\pi\)
							−0.556941	+	0.830552i	\(0.688025\pi\)
\(47\)	17.8474	+	49.0354i	0.379732	+	1.04331i	0.971467	+	0.237173i	\(0.0762210\pi\)
							−0.591735	+	0.806133i	\(0.701557\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.3.l.a.23.5	✓	108
4.3	odd	2	inner	76.3.l.a.23.9	yes	108
19.5	even	9	inner	76.3.l.a.43.9	yes	108
76.43	odd	18	inner	76.3.l.a.43.5	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.l.a.23.5	✓	108	1.1	even	1	trivial
76.3.l.a.23.9	yes	108	4.3	odd	2	inner
76.3.l.a.43.5	yes	108	76.43	odd	18	inner
76.3.l.a.43.9	yes	108	19.5	even	9	inner