Embedded newform 927.2.bg.a.77.27

• Label: 927.2.bg.a.77.27
• Level: $927$
• Weight: $2$
• Character: 927.77
• 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.bg (of order \(102\), degree \(32\), minimal)

Newform invariants

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

Embedding invariants

Embedding label			77.27
Character	\(\chi\)	\(=\)	927.77
Dual form			927.2.bg.a.614.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39307 + 0.589637i) q^{2} +(1.04039 + 1.38477i) q^{3} +(0.200710 - 0.206989i) q^{4} +(1.33540 - 0.945312i) q^{5} +(-2.26585 - 1.31563i) q^{6} +(-2.69154 + 1.04271i) q^{7} +(0.935360 - 2.41444i) q^{8} +(-0.835178 + 2.88140i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3264 q - 48 q^{2} - 37 q^{3} - 118 q^{4} - 48 q^{5} - 37 q^{6} - 17 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(722\)	\(829\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{65}{102}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39307	+	0.589637i	−0.985051	+	0.416936i	−0.821879	−	0.569662i	\(-0.807074\pi\)
							−0.163171	+	0.986598i	\(0.552172\pi\)
\(3\)	1.04039	+	1.38477i	0.600669	+	0.799498i
							\(5\)	1.33540	−	0.945312i	0.597210	−	0.422757i	−0.239114	−	0.970992i	\(-0.576857\pi\)
0.836324	+	0.548235i	\(0.184700\pi\)
\(7\)	−2.69154	+	1.04271i	−1.01731	+	0.394107i	−0.811416	−	0.584468i	\(-0.801303\pi\)
							−0.205890	+	0.978575i	\(0.566009\pi\)
\(11\)	2.20059	−	2.91405i	0.663502	−	0.878619i	−0.334461	−	0.942410i	\(-0.608554\pi\)
							0.997963	+	0.0637904i	\(0.0203189\pi\)
\(13\)	0.508846	−	3.27804i	0.141128	−	0.909164i	−0.806833	−	0.590779i	\(-0.798820\pi\)
							0.947962	−	0.318385i	\(-0.103140\pi\)
\(17\)	2.84507	+	1.00256i	0.690030	+	0.243157i	0.654334	−	0.756206i	\(-0.272949\pi\)
							0.0356956	+	0.999363i	\(0.488635\pi\)
\(19\)	7.08256	−	0.436838i	1.62485	−	0.100217i	0.776665	−	0.629914i	\(-0.216910\pi\)
							0.848187	+	0.529697i	\(0.177694\pi\)
\(23\)	1.02023	+	8.23917i	0.212733	+	1.71798i	0.603388	+	0.797448i	\(0.293817\pi\)
							−0.390655	+	0.920537i	\(0.627752\pi\)
\(29\)	−2.00615	+	4.36050i	−0.372533	+	0.809725i	0.626996	+	0.779022i	\(0.284284\pi\)
							−0.999529	+	0.0307024i	\(0.990226\pi\)
\(31\)	0.604464	−	0.0186233i	0.108565	−	0.00334485i	0.0235328	−	0.999723i	\(-0.492509\pi\)
							0.0850321	+	0.996378i	\(0.472901\pi\)
\(37\)	3.83729	+	4.20930i	0.630846	+	0.692005i	0.968429	−	0.249291i	\(-0.0801975\pi\)
							−0.337583	+	0.941296i	\(0.609609\pi\)
\(41\)	−1.14622	−	0.106213i	−0.179010	−	0.0165877i	0.00239000	−	0.999997i	\(-0.499239\pi\)
							−0.181400	+	0.983409i	\(0.558063\pi\)
\(43\)	−2.95028	−	9.27404i	−0.449914	−	1.41428i	−0.863747	−	0.503925i	\(-0.831889\pi\)
							0.413833	−	0.910353i	\(-0.364190\pi\)
\(47\)	7.74219			1.12931			0.564657	−	0.825326i	\(-0.309008\pi\)
							0.564657	+	0.825326i	\(0.309008\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	927.2.bg.a.77.27	✓	3264
9.2	odd	6		927.2.bm.a.695.27	yes	3264
103.99	odd	102		927.2.bm.a.923.27	yes	3264
927.614	even	102	inner	927.2.bg.a.614.27	yes	3264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
927.2.bg.a.77.27	✓	3264	1.1	even	1	trivial
927.2.bg.a.614.27	yes	3264	927.614	even	102	inner
927.2.bm.a.695.27	yes	3264	9.2	odd	6
927.2.bm.a.923.27	yes	3264	103.99	odd	102