Embedded newform 804.2.bf.b.7.7

• Label: 804.2.bf.b.7.7
• 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			7.7
Character	\(\chi\)	\(=\)	804.7
Dual form			804.2.bf.b.115.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22292 - 0.710261i) q^{2} +(0.959493 + 0.281733i) q^{3} +(0.991060 + 1.73718i) q^{4} +(-2.24944 - 1.94915i) q^{5} +(-0.973278 - 1.02603i) q^{6} +(0.0649402 - 1.36326i) q^{7} +(0.0218664 - 2.82834i) q^{8} +(0.841254 + 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q + 68 q^{3} + 2 q^{4} + 11 q^{6} - 4 q^{7} - 27 q^{8} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22292	−	0.710261i	−0.864734	−	0.502230i
							\(3\)	0.959493	+	0.281733i	0.553964	+	0.162658i
\(5\)	−2.24944	−	1.94915i	−1.00598	−	0.871686i								−0.0142324	−	0.999899i	\(-0.504530\pi\)
							−0.991747	+	0.128213i	\(0.959076\pi\)
\(7\)	0.0649402	−	1.36326i	0.0245451	−	0.515265i	−0.952798	−	0.303604i	\(-0.901810\pi\)
							0.977343	−	0.211660i	\(-0.0678871\pi\)
\(11\)	−1.00618	−	0.193926i	−0.303375	−	0.0584708i	0.0352918	−	0.999377i	\(-0.488764\pi\)
							−0.338667	+	0.940906i	\(0.609976\pi\)
\(13\)	−2.43017	+	6.07028i	−0.674009	+	1.68359i	0.0544626	+	0.998516i	\(0.482655\pi\)
							−0.728471	+	0.685076i	\(0.759769\pi\)
\(17\)	−2.02161	−	2.83895i	−0.490311	−	0.688546i	0.493092	−	0.869977i	\(-0.335866\pi\)
							−0.983404	+	0.181431i	\(0.941927\pi\)
\(19\)	−6.60554	+	0.314661i	−1.51541	+	0.0721881i	−0.788711	−	0.614764i	\(-0.789251\pi\)
							−0.726703	+	0.686952i	\(0.758948\pi\)
\(23\)	2.50709	+	2.62937i	0.522765	+	0.548261i	0.931191	−	0.364531i	\(-0.118771\pi\)
							−0.408426	+	0.912791i	\(0.633922\pi\)
\(29\)	2.07843	+	3.59994i	0.385954	+	0.668492i	0.991901	−	0.127013i	\(-0.0405390\pi\)
							−0.605947	+	0.795505i	\(0.707206\pi\)
\(31\)	0.724026	−	0.289856i	0.130039	−	0.0520597i	−0.305730	−	0.952118i	\(-0.598901\pi\)
							0.435769	+	0.900058i	\(0.356476\pi\)
\(37\)	−0.208885	+	0.361800i	−0.0343406	+	0.0594796i	−0.882685	−	0.469965i	\(-0.844266\pi\)
							0.848344	+	0.529445i	\(0.177600\pi\)
\(41\)	0.376844	+	3.94649i	0.0588532	+	0.616338i	0.975997	+	0.217783i	\(0.0698824\pi\)
							−0.917144	+	0.398556i	\(0.869512\pi\)
\(43\)	−0.838505	+	1.83607i	−0.127871	+	0.279998i	−0.962729	−	0.270467i	\(-0.912822\pi\)
							0.834859	+	0.550464i	\(0.185549\pi\)
\(47\)	−4.48981	+	1.08922i	−0.654906	+	0.158878i	−0.549425	−	0.835543i	\(-0.685153\pi\)
							−0.105481	+	0.994421i	\(0.533638\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.bf.b.7.7	yes	680
4.3	odd	2		804.2.bf.a.7.18	✓	680
67.48	odd	66		804.2.bf.a.115.18	yes	680
268.115	even	66	inner	804.2.bf.b.115.7	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.bf.a.7.18	✓	680	4.3	odd	2
804.2.bf.a.115.18	yes	680	67.48	odd	66
804.2.bf.b.7.7	yes	680	1.1	even	1	trivial
804.2.bf.b.115.7	yes	680	268.115	even	66	inner