Embedded newform 927.2.y.a.781.78

• Label: 927.2.y.a.781.78
• Level: $927$
• Weight: $2$
• Character: 927.781
• Analytic conductor: $7.402$
• Analytic rank: $0$
• Dimension: $3264$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 927 = 3^{2} \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	927.y (of order \(51\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.40213226737\)
Analytic rank:	\(0\)
Dimension:	\(3264\)
Relative dimension:	\(102\) over \(\Q(\zeta_{51})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{51}]$

Embedding invariants

Embedding label			781.78
Character	\(\chi\)	\(=\)	927.781
Dual form			927.2.y.a.400.78

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.59470 - 0.617788i) q^{2} +(-1.40782 - 1.00898i) q^{3} +(0.683373 - 0.622977i) q^{4} +(-0.740345 + 2.60204i) q^{5} +(-2.86838 - 0.739281i) q^{6} +(2.46295 - 3.71697i) q^{7} +(-0.819682 + 1.64614i) q^{8} +(0.963918 + 2.84093i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3264 q - 15 q^{2} - 29 q^{3} - 215 q^{4} - 11 q^{5} - 29 q^{6} - 17 q^{7} - 56 q^{8} - 19 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/927\mathbb{Z}\right)^\times\).

\(n\)	\(722\)	\(829\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{13}{51}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.59470	−	0.617788i	1.12762	−	0.436842i	0.276081	−	0.961134i	\(-0.410964\pi\)
							0.851539	+	0.524292i	\(0.175670\pi\)
\(3\)	−1.40782	−	1.00898i	−0.812806	−	0.582535i
							\(5\)	−0.740345	+	2.60204i	−0.331092	+	1.16367i	0.600337	+	0.799747i	\(0.295033\pi\)
−0.931430	+	0.363922i	\(0.881438\pi\)
\(7\)	2.46295	−	3.71697i	0.930908	−	1.40488i	0.0165998	−	0.999862i	\(-0.494716\pi\)
							0.914308	−	0.405019i	\(-0.132735\pi\)
\(11\)	0.620242	+	0.499107i	0.187010	+	0.150486i	0.715897	−	0.698206i	\(-0.246018\pi\)
							−0.528887	+	0.848692i	\(0.677391\pi\)
\(13\)	−3.71386	+	5.60477i	−1.03004	+	1.55448i	−0.213710	+	0.976897i	\(0.568555\pi\)
							−0.816329	+	0.577587i	\(0.803994\pi\)
\(17\)	3.21744	−	1.72773i	0.780345	−	0.419036i	−0.0339971	−	0.999422i	\(-0.510824\pi\)
							0.814342	+	0.580386i	\(0.197098\pi\)
\(19\)	0.858010	+	2.43485i	0.196841	+	0.558594i	0.999272	−	0.0381508i	\(-0.0121467\pi\)
							−0.802431	+	0.596745i	\(0.796460\pi\)
\(23\)	0.662440	+	4.26751i	0.138128	+	0.889837i	0.951401	+	0.307954i	\(0.0996444\pi\)
							−0.813273	+	0.581883i	\(0.802316\pi\)
\(29\)	1.14217	−	4.01430i	0.212095	−	0.745438i	−0.780626	−	0.624999i	\(-0.785099\pi\)
							0.992721	−	0.120439i	\(-0.0384301\pi\)
\(31\)	3.46600	−	1.59461i	0.622511	−	0.286401i	−0.0814077	−	0.996681i	\(-0.525942\pi\)
							0.703919	+	0.710280i	\(0.251432\pi\)
\(37\)	−3.87843	+	5.13587i	−0.637610	+	0.844332i	−0.996011	−	0.0892361i	\(-0.971557\pi\)
							0.358401	+	0.933568i	\(0.383322\pi\)
\(41\)	9.77397	+	2.45825i	1.52644	+	0.383914i	0.913706	−	0.406376i	\(-0.133208\pi\)
							0.612732	+	0.790291i	\(0.290071\pi\)
\(43\)	0.728246	+	0.964354i	0.111057	+	0.147063i	0.850151	−	0.526540i	\(-0.176511\pi\)
							−0.739094	+	0.673602i	\(0.764746\pi\)
\(47\)	3.83389	+	6.64049i	0.559230	+	0.968615i	0.997561	+	0.0698014i	\(0.0222366\pi\)
							−0.438331	+	0.898814i	\(0.644430\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	927.2.y.a.781.78	yes	3264
9.4	even	3		927.2.z.a.472.25	yes	3264
103.91	even	51		927.2.z.a.709.25	yes	3264
927.400	even	51	inner	927.2.y.a.400.78	✓	3264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
927.2.y.a.400.78	✓	3264	927.400	even	51	inner
927.2.y.a.781.78	yes	3264	1.1	even	1	trivial
927.2.z.a.472.25	yes	3264	9.4	even	3
927.2.z.a.709.25	yes	3264	103.91	even	51