Embedded newform 804.2.bf.b.7.10

• Label: 804.2.bf.b.7.10
• Level: $804$
• Weight: $2$
• Character: 804.7
• Analytic conductor: $6.420$
• Analytic rank: $0$
• Dimension: $680$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.bf (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.41997232251\)
Analytic rank:	\(0\)
Dimension:	\(680\)
Relative dimension:	\(34\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			7.10
Character	\(\chi\)	\(=\)	804.7
Dual form			804.2.bf.b.115.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.944404 + 1.05266i) q^{2} +(0.959493 + 0.281733i) q^{3} +(-0.216203 - 1.98828i) q^{4} +(-1.97935 - 1.71512i) q^{5} +(-1.20272 + 0.743954i) q^{6} +(-0.0520043 + 1.09170i) q^{7} +(2.29717 + 1.65015i) q^{8} +(0.841254 + 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q + 68 q^{3} + 2 q^{4} + 11 q^{6} - 4 q^{7} - 27 q^{8} - 68 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\)	\(269\)	\(337\)	\(403\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{23}{66}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.944404	+	1.05266i	−0.667794	+	0.744346i
							\(3\)	0.959493	+	0.281733i	0.553964	+	0.162658i
\(5\)	−1.97935	−	1.71512i	−0.885192	−	0.767023i								0.0882162	−	0.996101i	\(-0.471883\pi\)
							−0.973408	+	0.229078i	\(0.926429\pi\)
\(7\)	−0.0520043	+	1.09170i	−0.0196558	+	0.412626i	0.967501	+	0.252869i	\(0.0813741\pi\)
							−0.987156	+	0.159757i	\(0.948929\pi\)
\(11\)	0.905551	+	0.174531i	0.273034	+	0.0526229i	0.323930	−	0.946081i	\(-0.394996\pi\)
							−0.0508964	+	0.998704i	\(0.516208\pi\)
\(13\)	0.185501	−	0.463359i	0.0514487	−	0.128513i	−0.900417	−	0.435027i	\(-0.856739\pi\)
							0.951866	+	0.306515i	\(0.0991629\pi\)
\(17\)	−3.69119	−	5.18355i	−0.895244	−	1.25719i	−0.965695	−	0.259679i	\(-0.916383\pi\)
							0.0704506	−	0.997515i	\(-0.477556\pi\)
\(19\)	4.57470	−	0.217920i	1.04951	−	0.0499943i	0.484267	−	0.874920i	\(-0.339086\pi\)
							0.565241	+	0.824926i	\(0.308783\pi\)
\(23\)	−1.11022	−	1.16436i	−0.231496	−	0.242787i	0.597732	−	0.801696i	\(-0.296069\pi\)
							−0.829229	+	0.558909i	\(0.811220\pi\)
\(29\)	−1.12277	−	1.94470i	−0.208494	−	0.361122i	0.742746	−	0.669573i	\(-0.233523\pi\)
							−0.951240	+	0.308451i	\(0.900189\pi\)
\(31\)	−0.926711	+	0.370999i	−0.166442	+	0.0666334i	−0.453387	−	0.891314i	\(-0.649785\pi\)
							0.286945	+	0.957947i	\(0.407360\pi\)
\(37\)	2.54715	−	4.41180i	0.418749	−	0.725295i	−0.577065	−	0.816698i	\(-0.695802\pi\)
							0.995814	+	0.0914034i	\(0.0291353\pi\)
\(41\)	−1.14078	−	11.9468i	−0.178160	−	1.86577i	−0.436478	−	0.899715i	\(-0.643774\pi\)
							0.258318	−	0.966060i	\(-0.416832\pi\)
\(43\)	3.88869	−	8.51503i	0.593019	−	1.29853i	−0.340581	−	0.940215i	\(-0.610624\pi\)
							0.933601	−	0.358316i	\(-0.116649\pi\)
\(47\)	2.01309	−	0.488371i	0.293640	−	0.0712362i	−0.0862330	−	0.996275i	\(-0.527483\pi\)
							0.379873	+	0.925039i	\(0.375968\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.bf.b.7.10	yes	680
4.3	odd	2		804.2.bf.a.7.33	✓	680
67.48	odd	66		804.2.bf.a.115.33	yes	680
268.115	even	66	inner	804.2.bf.b.115.10	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.bf.a.7.33	✓	680	4.3	odd	2
804.2.bf.a.115.33	yes	680	67.48	odd	66
804.2.bf.b.7.10	yes	680	1.1	even	1	trivial
804.2.bf.b.115.10	yes	680	268.115	even	66	inner