Embedded newform 927.2.y.a.781.20

• Label: 927.2.y.a.781.20
• 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			781.20
Character	\(\chi\)	\(=\)	927.781
Dual form			927.2.y.a.400.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.80667 + 0.699907i) q^{2} +(-1.51941 + 0.831492i) q^{3} +(1.29616 - 1.18161i) q^{4} +(-0.182470 + 0.641315i) q^{5} +(2.16311 - 2.56568i) q^{6} +(-0.489827 + 0.739224i) q^{7} +(0.212518 - 0.426793i) q^{8} +(1.61724 - 2.52676i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3264 q - 15 q^{2} - 29 q^{3} - 215 q^{4} - 11 q^{5} - 29 q^{6} - 17 q^{7} - 56 q^{8} - 19 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.80667	+	0.699907i	−1.27751	+	0.494909i	−0.901930	−	0.431882i	\(-0.857850\pi\)
							−0.375578	+	0.926791i	\(0.622556\pi\)
\(3\)	−1.51941	+	0.831492i	−0.877234	+	0.480062i
							\(5\)	−0.182470	+	0.641315i	−0.0816030	+	0.286805i	−0.991877	−	0.127202i	\(-0.959400\pi\)
0.910274	+	0.414007i	\(0.135871\pi\)
\(7\)	−0.489827	+	0.739224i	−0.185137	+	0.279400i	−0.914382	−	0.404853i	\(-0.867323\pi\)
							0.729244	+	0.684253i	\(0.239872\pi\)
\(11\)	1.25243	+	1.00782i	0.377621	+	0.303870i	0.797045	−	0.603920i	\(-0.206395\pi\)
							−0.419424	+	0.907790i	\(0.637768\pi\)
\(13\)	0.960705	−	1.44985i	0.266452	−	0.402116i	−0.675767	−	0.737116i	\(-0.736187\pi\)
							0.942218	+	0.335000i	\(0.108736\pi\)
\(17\)	−4.01138	+	2.15407i	−0.972903	+	0.522438i	−0.880891	−	0.473320i	\(-0.843056\pi\)
							−0.0920126	+	0.995758i	\(0.529330\pi\)
\(19\)	−1.20080	−	3.40762i	−0.275482	−	0.781762i	−0.996084	−	0.0884125i	\(-0.971821\pi\)
							0.720602	−	0.693349i	\(-0.243866\pi\)
\(23\)	0.906859	+	5.84208i	0.189093	+	1.21816i	0.872681	+	0.488290i	\(0.162379\pi\)
							−0.683588	+	0.729868i	\(0.739581\pi\)
\(29\)	−1.80545	+	6.34551i	−0.335264	+	1.17833i	0.592626	+	0.805478i	\(0.298091\pi\)
							−0.927890	+	0.372854i	\(0.878379\pi\)
\(31\)	5.50468	−	2.53256i	0.988670	−	0.454860i	0.143669	−	0.989626i	\(-0.454110\pi\)
							0.845001	+	0.534765i	\(0.179600\pi\)
\(37\)	−1.60241	+	2.12194i	−0.263435	+	0.348844i	−0.910440	−	0.413642i	\(-0.864257\pi\)
							0.647005	+	0.762486i	\(0.276021\pi\)
\(41\)	−12.1232	−	3.04912i	−1.89333	−	0.476192i	−0.999620	−	0.0275479i	\(-0.991230\pi\)
							−0.893711	−	0.448644i	\(-0.851907\pi\)
\(43\)	−5.24439	−	6.94469i	−0.799762	−	1.05906i	−0.996900	−	0.0786798i	\(-0.974930\pi\)
							0.197138	−	0.980376i	\(-0.436835\pi\)
\(47\)	4.78856	+	8.29402i	0.698483	+	1.20981i	0.968992	+	0.247090i	\(0.0794744\pi\)
							−0.270510	+	0.962717i	\(0.587192\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	927.2.y.a.781.20	yes	3264
9.4	even	3		927.2.z.a.472.83	yes	3264
103.91	even	51		927.2.z.a.709.83	yes	3264
927.400	even	51	inner	927.2.y.a.400.20	✓	3264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
927.2.y.a.400.20	✓	3264	927.400	even	51	inner
927.2.y.a.781.20	yes	3264	1.1	even	1	trivial
927.2.z.a.472.83	yes	3264	9.4	even	3
927.2.z.a.709.83	yes	3264	103.91	even	51