Embedded newform 927.2.y.a.781.40

• Label: 927.2.y.a.781.40
• 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			781.40
Character	\(\chi\)	\(=\)	927.781
Dual form			927.2.y.a.400.40

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.647745 + 0.250938i) q^{2} +(-0.216341 - 1.71849i) q^{3} +(-1.12141 + 1.02230i) q^{4} +(0.349776 - 1.22933i) q^{5} +(0.571367 + 1.05885i) q^{6} +(-1.94056 + 2.92860i) q^{7} +(1.08912 - 2.18725i) q^{8} +(-2.90639 + 0.743557i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3264 q - 15 q^{2} - 29 q^{3} - 215 q^{4} - 11 q^{5} - 29 q^{6} - 17 q^{7} - 56 q^{8} - 19 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.647745	+	0.250938i	−0.458025	+	0.177440i	−0.579190	−	0.815193i	\(-0.696631\pi\)
							0.121165	+	0.992632i	\(0.461337\pi\)
\(3\)	−0.216341	−	1.71849i	−0.124904	−	0.992169i
							\(5\)	0.349776	−	1.22933i	0.156424	−	0.549775i	−0.843496	−	0.537136i	\(-0.819506\pi\)
0.999920	−	0.0126388i	\(-0.00402316\pi\)
\(7\)	−1.94056	+	2.92860i	−0.733463	+	1.10691i	0.256632	+	0.966509i	\(0.417387\pi\)
							−0.990095	+	0.140398i	\(0.955162\pi\)
\(11\)	−3.24449	−	2.61083i	−0.978250	−	0.787195i	−0.00115485	−	0.999999i	\(-0.500368\pi\)
							−0.977095	+	0.212805i	\(0.931740\pi\)
\(13\)	−0.757334	+	1.14293i	−0.210047	+	0.316992i	−0.923419	−	0.383793i	\(-0.874618\pi\)
							0.713373	+	0.700785i	\(0.247167\pi\)
\(17\)	5.28129	−	2.83599i	1.28090	−	0.687829i	0.315581	−	0.948899i	\(-0.397801\pi\)
							0.965320	+	0.261070i	\(0.0840752\pi\)
\(19\)	1.78493	+	5.06527i	0.409491	+	1.16205i	0.947678	+	0.319227i	\(0.103423\pi\)
							−0.538187	+	0.842825i	\(0.680891\pi\)
\(23\)	0.386917	+	2.49256i	0.0806779	+	0.519735i	0.993722	+	0.111875i	\(0.0356857\pi\)
							−0.913044	+	0.407860i	\(0.866275\pi\)
\(29\)	0.601417	−	2.11376i	0.111680	−	0.392516i	−0.885750	−	0.464163i	\(-0.846355\pi\)
							0.997430	+	0.0716476i	\(0.0228257\pi\)
\(31\)	−3.23358	+	1.48768i	−0.580767	+	0.267195i	−0.686417	−	0.727208i	\(-0.740818\pi\)
							0.105650	+	0.994403i	\(0.466308\pi\)
\(37\)	4.68739	−	6.20711i	0.770603	−	1.02044i	−0.228299	−	0.973591i	\(-0.573316\pi\)
							0.998902	−	0.0468521i	\(-0.0149190\pi\)
\(41\)	−2.13440	−	0.536822i	−0.333337	−	0.0838376i	0.0736162	−	0.997287i	\(-0.476546\pi\)
							−0.406953	+	0.913449i	\(0.633409\pi\)
\(43\)	3.45973	+	4.58142i	0.527603	+	0.698660i	0.981297	−	0.192498i	\(-0.0616588\pi\)
							−0.453694	+	0.891158i	\(0.649894\pi\)
\(47\)	1.90039	+	3.29158i	0.277201	+	0.480126i	0.970688	−	0.240343i	\(-0.0772599\pi\)
							−0.693487	+	0.720469i	\(0.743927\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	927.2.y.a.781.40	yes	3264
9.4	even	3		927.2.z.a.472.63	yes	3264
103.91	even	51		927.2.z.a.709.63	yes	3264
927.400	even	51	inner	927.2.y.a.400.40	✓	3264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
927.2.y.a.400.40	✓	3264	927.400	even	51	inner
927.2.y.a.781.40	yes	3264	1.1	even	1	trivial
927.2.z.a.472.63	yes	3264	9.4	even	3
927.2.z.a.709.63	yes	3264	103.91	even	51