Embedded newform 134.2.g.a.21.1

• Label: 134.2.g.a.21.1
• Level: $134$
• Weight: $2$
• Character: 134.21
• Analytic conductor: $1.070$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 134 = 2 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	134.g (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.06999538709\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(2\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			21.1
Character	\(\chi\)	\(=\)	134.21
Dual form			134.2.g.a.83.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.981929 + 0.189251i) q^{2} +(-1.08837 + 2.38320i) q^{3} +(0.928368 - 0.371662i) q^{4} +(-2.17083 - 0.637412i) q^{5} +(0.617678 - 2.54610i) q^{6} +(-0.589167 + 1.70229i) q^{7} +(-0.841254 + 0.540641i) q^{8} +(-2.53050 - 2.92035i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 2 q^{2} - 6 q^{3} + 2 q^{4} - 3 q^{6} + q^{7} + 4 q^{8} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/134\mathbb{Z}\right)^\times\).

\(n\)	\(69\)
\(\chi(n)\)	\(e\left(\frac{31}{33}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.981929	+	0.189251i	−0.694328	+	0.133821i
							\(3\)	−1.08837	+	2.38320i	−0.628370	+	1.37594i	0.280901	+	0.959737i	\(0.409367\pi\)
−0.909272	+	0.416203i	\(0.863361\pi\)
\(5\)	−2.17083	−	0.637412i	−0.970823	−	0.285059i	−0.242392	−	0.970178i	\(-0.577932\pi\)
							−0.728431	+	0.685119i	\(0.759750\pi\)
\(7\)	−0.589167	+	1.70229i	−0.222684	+	0.643404i	0.777215	+	0.629235i	\(0.216632\pi\)
							−0.999900	+	0.0141693i	\(0.995490\pi\)
\(11\)	−1.09409	−	4.50989i	−0.329880	−	1.35978i	−0.859321	−	0.511437i	\(-0.829113\pi\)
							0.529441	−	0.848347i	\(-0.322402\pi\)
\(13\)	−5.15774	+	2.65900i	−1.43050	+	0.737474i	−0.987288	−	0.158942i	\(-0.949192\pi\)
							−0.443213	+	0.896416i	\(0.646161\pi\)
\(17\)	−1.84760	−	0.739666i	−0.448108	−	0.179395i	0.136622	−	0.990623i	\(-0.456375\pi\)
							−0.584730	+	0.811228i	\(0.698800\pi\)
\(19\)	2.40640	+	6.95284i	0.552067	+	1.59509i	0.782933	+	0.622106i	\(0.213723\pi\)
							−0.230866	+	0.972986i	\(0.574156\pi\)
\(23\)	3.96753	+	5.57162i	0.827288	+	1.16176i	0.984705	+	0.174231i	\(0.0557440\pi\)
							−0.157417	+	0.987532i	\(0.550317\pi\)
\(29\)	0.844207	+	1.46221i	0.156765	+	0.271525i	0.933700	−	0.358055i	\(-0.116560\pi\)
							−0.776935	+	0.629581i	\(0.783227\pi\)
\(31\)	3.86152	+	1.99075i	0.693549	+	0.357549i	0.768688	−	0.639624i	\(-0.220910\pi\)
							−0.0751392	+	0.997173i	\(0.523940\pi\)
\(37\)	−0.411453	+	0.712657i	−0.0676424	+	0.117160i	−0.897863	−	0.440275i	\(-0.854881\pi\)
							0.830221	+	0.557435i	\(0.188214\pi\)
\(41\)	−4.15951	+	3.27108i	−0.649607	+	0.510856i	−0.887722	−	0.460381i	\(-0.847713\pi\)
							0.238115	+	0.971237i	\(0.423471\pi\)
\(43\)	−0.942389	−	6.55446i	−0.143713	−	0.999546i	−0.926241	−	0.376931i	\(-0.876979\pi\)
							0.782528	−	0.622615i	\(-0.213930\pi\)
\(47\)	0.829207	−	0.0791797i	0.120952	−	0.0115495i	−0.0344043	−	0.999408i	\(-0.510953\pi\)
							0.155357	+	0.987858i	\(0.450347\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	134.2.g.a.21.1	✓	40
67.4	even	33		8978.2.a.o.1.1		20
67.16	even	33	inner	134.2.g.a.83.1	yes	40
67.63	odd	66		8978.2.a.r.1.20		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
134.2.g.a.21.1	✓	40	1.1	even	1	trivial
134.2.g.a.83.1	yes	40	67.16	even	33	inner
8978.2.a.o.1.1		20	67.4	even	33
8978.2.a.r.1.20		20	67.63	odd	66