Embedded newform 804.2.bf.b.7.4

• Label: 804.2.bf.b.7.4
• 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			7.4
Character	\(\chi\)	\(=\)	804.7
Dual form			804.2.bf.b.115.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35999 - 0.387841i) q^{2} +(0.959493 + 0.281733i) q^{3} +(1.69916 + 1.05492i) q^{4} +(1.14805 + 0.994791i) q^{5} +(-1.19564 - 0.755285i) q^{6} +(0.0104266 - 0.218882i) q^{7} +(-1.90170 - 2.09369i) q^{8} +(0.841254 + 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q + 68 q^{3} + 2 q^{4} + 11 q^{6} - 4 q^{7} - 27 q^{8} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35999	−	0.387841i	−0.961660	−	0.274245i
							\(3\)	0.959493	+	0.281733i	0.553964	+	0.162658i
\(5\)	1.14805	+	0.994791i	0.513424	+	0.444884i								0.872636	−	0.488372i	\(-0.162409\pi\)
							−0.359212	+	0.933256i	\(0.616954\pi\)
\(7\)	0.0104266	−	0.218882i	0.00394089	−	0.0827295i	−0.996040	−	0.0889059i	\(-0.971663\pi\)
							0.999981	+	0.00617634i	\(0.00196600\pi\)
\(11\)	1.75319	+	0.337899i	0.528606	+	0.101880i	0.446570	−	0.894749i	\(-0.352645\pi\)
							0.0820364	+	0.996629i	\(0.473858\pi\)
\(13\)	1.57343	−	3.93025i	0.436392	−	1.09006i	−0.533343	−	0.845899i	\(-0.679065\pi\)
							0.969736	−	0.244157i	\(-0.0785111\pi\)
\(17\)	−4.35638	−	6.11768i	−1.05658	−	1.48376i	−0.864668	−	0.502343i	\(-0.832471\pi\)
							−0.191909	−	0.981413i	\(-0.561468\pi\)
\(19\)	5.31540	−	0.253204i	1.21944	−	0.0580890i	0.572030	−	0.820233i	\(-0.306156\pi\)
							0.647408	+	0.762144i	\(0.275853\pi\)
\(23\)	2.75858	+	2.89312i	0.575204	+	0.603257i	0.945486	−	0.325663i	\(-0.105587\pi\)
							−0.370282	+	0.928919i	\(0.620739\pi\)
\(29\)	1.87924	+	3.25495i	0.348967	+	0.604429i	0.986066	−	0.166353i	\(-0.0531992\pi\)
							−0.637099	+	0.770782i	\(0.719866\pi\)
\(31\)	7.52089	−	3.01091i	1.35079	−	0.540775i	0.420398	−	0.907340i	\(-0.361890\pi\)
							0.930393	+	0.366565i	\(0.119466\pi\)
\(37\)	−1.05579	+	1.82868i	−0.173570	+	0.300633i	−0.939666	−	0.342094i	\(-0.888864\pi\)
							0.766095	+	0.642727i	\(0.222197\pi\)
\(41\)	0.269125	+	2.81840i	0.0420302	+	0.440160i	0.991854	+	0.127376i	\(0.0406556\pi\)
							−0.949824	+	0.312784i	\(0.898738\pi\)
\(43\)	−4.69236	+	10.2748i	−0.715578	+	1.56690i	0.104422	+	0.994533i	\(0.466701\pi\)
							−0.820000	+	0.572364i	\(0.806026\pi\)
\(47\)	−1.68052	+	0.407690i	−0.245129	+	0.0594677i	−0.356441	−	0.934318i	\(-0.616010\pi\)
							0.111312	+	0.993786i	\(0.464495\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.bf.b.7.4	yes	680
4.3	odd	2		804.2.bf.a.7.21	✓	680
67.48	odd	66		804.2.bf.a.115.21	yes	680
268.115	even	66	inner	804.2.bf.b.115.4	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.bf.a.7.21	✓	680	4.3	odd	2
804.2.bf.a.115.21	yes	680	67.48	odd	66
804.2.bf.b.7.4	yes	680	1.1	even	1	trivial
804.2.bf.b.115.4	yes	680	268.115	even	66	inner