Embedded newform 804.2.bf.a.7.13

• Label: 804.2.bf.a.7.13
• 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			7.13
Character	\(\chi\)	\(=\)	804.7
Dual form			804.2.bf.a.115.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.546520 - 1.30434i) q^{2} +(-0.959493 - 0.281733i) q^{3} +(-1.40263 + 1.42570i) q^{4} +(3.24307 + 2.81014i) q^{5} +(0.156906 + 1.40548i) q^{6} +(0.0543857 - 1.14170i) q^{7} +(2.62617 + 1.05034i) q^{8} +(0.841254 + 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q - 68 q^{3} + 2 q^{4} - 11 q^{6} + 4 q^{7} - 39 q^{8} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.546520	−	1.30434i	−0.386448	−	0.922311i
							\(3\)	−0.959493	−	0.281733i	−0.553964	−	0.162658i
\(5\)	3.24307	+	2.81014i	1.45035	+	1.25673i								0.909560	+	0.415573i	\(0.136419\pi\)
							0.540787	+	0.841160i	\(0.318127\pi\)
\(7\)	0.0543857	−	1.14170i	0.0205559	−	0.431521i	−0.965008	−	0.262219i	\(-0.915546\pi\)
							0.985564	−	0.169302i	\(-0.0541512\pi\)
\(11\)	−3.42041	−	0.659230i	−1.03129	−	0.198765i	−0.354590	−	0.935022i	\(-0.615379\pi\)
							−0.676702	+	0.736257i	\(0.736592\pi\)
\(13\)	−0.0730411	+	0.182448i	−0.0202580	+	0.0506019i	−0.938152	−	0.346223i	\(-0.887464\pi\)
							0.917894	+	0.396825i	\(0.129888\pi\)
\(17\)	−0.694936	−	0.975902i	−0.168547	−	0.236691i	0.721660	−	0.692248i	\(-0.243380\pi\)
							−0.890207	+	0.455557i	\(0.849440\pi\)
\(19\)	4.64926	−	0.221471i	1.06661	−	0.0508090i	0.493074	−	0.869987i	\(-0.335873\pi\)
							0.573539	+	0.819178i	\(0.305570\pi\)
\(23\)	4.92953	+	5.16995i	1.02788	+	1.07801i	0.996915	+	0.0784877i	\(0.0250091\pi\)
							0.0309637	+	0.999521i	\(0.490142\pi\)
\(29\)	2.10724	+	3.64985i	0.391305	+	0.677760i	0.992622	−	0.121251i	\(-0.0386905\pi\)
							−0.601317	+	0.799011i	\(0.705357\pi\)
\(31\)	−5.41646	+	2.16843i	−0.972826	+	0.389461i	−0.802977	−	0.596010i	\(-0.796752\pi\)
							−0.169848	+	0.985470i	\(0.554328\pi\)
\(37\)	−0.329911	+	0.571422i	−0.0542370	+	0.0939412i	−0.891869	−	0.452293i	\(-0.850606\pi\)
							0.837632	+	0.546235i	\(0.183939\pi\)
\(41\)	−0.0292975	−	0.306817i	−0.00457549	−	0.0479167i	0.992896	−	0.118987i	\(-0.0379646\pi\)
							−0.997471	+	0.0710701i	\(0.977359\pi\)
\(43\)	−3.61522	+	7.91623i	−0.551316	+	1.20721i	0.404849	+	0.914384i	\(0.367324\pi\)
							−0.956165	+	0.292830i	\(0.905403\pi\)
\(47\)	12.2683	−	2.97625i	1.78951	−	0.434130i	0.802750	−	0.596316i	\(-0.203369\pi\)
							0.986760	+	0.162186i	\(0.0518543\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.bf.a.7.13	✓	680
4.3	odd	2		804.2.bf.b.7.12	yes	680
67.48	odd	66		804.2.bf.b.115.12	yes	680
268.115	even	66	inner	804.2.bf.a.115.13	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.bf.a.7.13	✓	680	1.1	even	1	trivial
804.2.bf.a.115.13	yes	680	268.115	even	66	inner
804.2.bf.b.7.12	yes	680	4.3	odd	2
804.2.bf.b.115.12	yes	680	67.48	odd	66