Embedded newform 927.2.y.a.781.34

• Label: 927.2.y.a.781.34
• 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			781.34
Character	\(\chi\)	\(=\)	927.781
Dual form			927.2.y.a.400.34

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08374 + 0.419841i) q^{2} +(0.547809 + 1.64314i) q^{3} +(-0.479803 + 0.437398i) q^{4} +(-0.585173 + 2.05667i) q^{5} +(-1.28354 - 1.55073i) q^{6} +(-2.43016 + 3.66747i) q^{7} +(1.37243 - 2.75621i) q^{8} +(-2.39981 + 1.80025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3264 q - 15 q^{2} - 29 q^{3} - 215 q^{4} - 11 q^{5} - 29 q^{6} - 17 q^{7} - 56 q^{8} - 19 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08374	+	0.419841i	−0.766317	+	0.296873i	−0.712494	−	0.701678i	\(-0.752435\pi\)
							−0.0538222	+	0.998551i	\(0.517140\pi\)
\(3\)	0.547809	+	1.64314i	0.316278	+	0.948667i
							\(5\)	−0.585173	+	2.05667i	−0.261697	+	0.919771i	0.713908	+	0.700239i	\(0.246923\pi\)
−0.975605	+	0.219531i	\(0.929547\pi\)
\(7\)	−2.43016	+	3.66747i	−0.918513	+	1.38617i	0.00390988	+	0.999992i	\(0.498755\pi\)
							−0.922422	+	0.386182i	\(0.873794\pi\)
\(11\)	2.49854	+	2.01057i	0.753339	+	0.606209i	0.925451	−	0.378868i	\(-0.123687\pi\)
							−0.172112	+	0.985077i	\(0.555059\pi\)
\(13\)	0.268588	−	0.405341i	0.0744930	−	0.112421i	−0.794459	−	0.607318i	\(-0.792245\pi\)
							0.868952	+	0.494897i	\(0.164794\pi\)
\(17\)	−4.44030	+	2.38439i	−1.07693	+	0.578299i	−0.912911	−	0.408158i	\(-0.866171\pi\)
							−0.164019	+	0.986457i	\(0.552446\pi\)
\(19\)	0.994854	+	2.82319i	0.228235	+	0.647684i	0.999932	+	0.0116934i	\(0.00372222\pi\)
							−0.771696	+	0.635991i	\(0.780592\pi\)
\(23\)	1.35854	+	8.75186i	0.283275	+	1.82489i	0.521000	+	0.853557i	\(0.325559\pi\)
							−0.237724	+	0.971333i	\(0.576401\pi\)
\(29\)	2.00393	−	7.04308i	0.372120	−	1.30787i	−0.519262	−	0.854615i	\(-0.673793\pi\)
							0.891382	−	0.453252i	\(-0.149736\pi\)
\(31\)	1.91966	−	0.883183i	0.344781	−	0.158624i	−0.237976	−	0.971271i	\(-0.576484\pi\)
							0.582757	+	0.812647i	\(0.301974\pi\)
\(37\)	1.12800	−	1.49371i	0.185442	−	0.245565i	−0.695794	−	0.718241i	\(-0.744947\pi\)
							0.881236	+	0.472676i	\(0.156712\pi\)
\(41\)	8.61235	+	2.16609i	1.34502	+	0.338287i	0.848251	−	0.529594i	\(-0.177656\pi\)
							0.496772	+	0.867881i	\(0.334518\pi\)
\(43\)	−7.37670	−	9.76833i	−1.12494	−	1.48966i	−0.848933	−	0.528500i	\(-0.822755\pi\)
							−0.276003	−	0.961157i	\(-0.589010\pi\)
\(47\)	3.89300	+	6.74288i	0.567853	+	0.983550i	0.996778	+	0.0802096i	\(0.0255590\pi\)
							−0.428925	+	0.903340i	\(0.641108\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	927.2.y.a.781.34	yes	3264
9.4	even	3		927.2.z.a.472.69	yes	3264
103.91	even	51		927.2.z.a.709.69	yes	3264
927.400	even	51	inner	927.2.y.a.400.34	✓	3264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
927.2.y.a.400.34	✓	3264	927.400	even	51	inner
927.2.y.a.781.34	yes	3264	1.1	even	1	trivial
927.2.z.a.472.69	yes	3264	9.4	even	3
927.2.z.a.709.69	yes	3264	103.91	even	51