Embedded newform 804.2.bf.b.7.5

• Label: 804.2.bf.b.7.5
• 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			7.5
Character	\(\chi\)	\(=\)	804.7
Dual form			804.2.bf.b.115.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32902 + 0.483424i) q^{2} +(0.959493 + 0.281733i) q^{3} +(1.53260 - 1.28496i) q^{4} +(0.0351858 + 0.0304887i) q^{5} +(-1.41138 + 0.0894134i) q^{6} +(0.111602 - 2.34282i) q^{7} +(-1.41568 + 2.44864i) q^{8} +(0.841254 + 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q + 68 q^{3} + 2 q^{4} + 11 q^{6} - 4 q^{7} - 27 q^{8} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32902	+	0.483424i	−0.939761	+	0.341833i
							\(3\)	0.959493	+	0.281733i	0.553964	+	0.162658i
\(5\)	0.0351858	+	0.0304887i	0.0157356	+	0.0136350i								0.662693	−	0.748891i	\(-0.269413\pi\)
							−0.646957	+	0.762526i	\(0.723959\pi\)
\(7\)	0.111602	−	2.34282i	0.0421817	−	0.885504i	−0.874290	−	0.485404i	\(-0.838673\pi\)
							0.916472	−	0.400100i	\(-0.131024\pi\)
\(11\)	−5.54098	−	1.06794i	−1.67067	−	0.321995i	−0.735856	−	0.677138i	\(-0.763220\pi\)
							−0.934812	+	0.355143i	\(0.884432\pi\)
\(13\)	1.21747	−	3.04110i	0.337667	−	0.843450i	−0.658219	−	0.752827i	\(-0.728690\pi\)
							0.995885	−	0.0906236i	\(-0.0288860\pi\)
\(17\)	−0.975434	−	1.36980i	−0.236577	−	0.332226i	0.679170	−	0.733981i	\(-0.262340\pi\)
							−0.915747	+	0.401755i	\(0.868401\pi\)
\(19\)	−5.20541	+	0.247964i	−1.19420	+	0.0568868i	−0.635246	−	0.772310i	\(-0.719101\pi\)
							−0.558956	+	0.829197i	\(0.688798\pi\)
\(23\)	1.10329	+	1.15710i	0.230052	+	0.241272i	0.828635	−	0.559789i	\(-0.189118\pi\)
							−0.598583	+	0.801061i	\(0.704269\pi\)
\(29\)	−2.57381	−	4.45796i	−0.477944	−	0.827823i	0.521737	−	0.853107i	\(-0.325284\pi\)
							−0.999680	+	0.0252840i	\(0.991951\pi\)
\(31\)	−1.90073	+	0.760937i	−0.341381	+	0.136668i	−0.536017	−	0.844207i	\(-0.680072\pi\)
							0.194636	+	0.980876i	\(0.437647\pi\)
\(37\)	−3.92600	+	6.80004i	−0.645431	+	1.11792i	0.338771	+	0.940869i	\(0.389989\pi\)
							−0.984202	+	0.177050i	\(0.943345\pi\)
\(41\)	−0.0743101	−	0.778211i	−0.0116053	−	0.121536i	0.987813	−	0.155647i	\(-0.0497462\pi\)
							−0.999418	+	0.0341107i	\(0.989140\pi\)
\(43\)	1.11881	−	2.44985i	0.170617	−	0.373599i	−0.804937	−	0.593361i	\(-0.797801\pi\)
							0.975554	+	0.219762i	\(0.0705280\pi\)
\(47\)	−2.15933	+	0.523848i	−0.314971	+	0.0764111i	−0.390128	−	0.920761i	\(-0.627569\pi\)
							0.0751572	+	0.997172i	\(0.476054\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.bf.b.7.5	yes	680
4.3	odd	2		804.2.bf.a.7.29	✓	680
67.48	odd	66		804.2.bf.a.115.29	yes	680
268.115	even	66	inner	804.2.bf.b.115.5	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.bf.a.7.29	✓	680	4.3	odd	2
804.2.bf.a.115.29	yes	680	67.48	odd	66
804.2.bf.b.7.5	yes	680	1.1	even	1	trivial
804.2.bf.b.115.5	yes	680	268.115	even	66	inner