Embedded newform 804.2.l.a.431.8

• Label: 804.2.l.a.431.8
• Level: $804$
• Weight: $2$
• Character: 804.431
• Analytic conductor: $6.420$
• Analytic rank: $0$
• Dimension: $264$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.41997232251\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(132\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			431.8
Character	\(\chi\)	\(=\)	804.431
Dual form			804.2.l.a.707.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39364 + 0.240373i) q^{2} +(-1.35771 + 1.07547i) q^{3} +(1.88444 - 0.669985i) q^{4} -4.20267i q^{5} +(1.63364 - 1.82517i) q^{6} +(3.17693 - 1.83420i) q^{7} +(-2.46518 + 1.38668i) q^{8} +(0.686739 - 2.92034i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 2 q^{4} - 3 q^{6} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(269\)	\(337\)	\(403\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39364	+	0.240373i	−0.985449	+	0.169970i
							\(3\)	−1.35771	+	1.07547i	−0.783873	+	0.620922i
\(5\)		−	4.20267i		−	1.87949i								−0.341877	−	0.939745i	\(-0.611063\pi\)
							0.341877	−	0.939745i	\(-0.388937\pi\)
\(7\)	3.17693	−	1.83420i	1.20077	−	0.693263i	0.240041	−	0.970763i	\(-0.422839\pi\)
							0.960726	+	0.277500i	\(0.0895058\pi\)
\(11\)	2.51242	+	4.35165i	0.757524	+	1.31207i	0.944110	+	0.329632i	\(0.106925\pi\)
							−0.186585	+	0.982439i	\(0.559742\pi\)
\(13\)	1.41203	−	2.44570i	0.391626	−	0.678315i	−0.601039	−	0.799220i	\(-0.705246\pi\)
							0.992664	+	0.120905i	\(0.0385795\pi\)
\(17\)	4.47174	+	2.58176i	1.08456	+	0.626169i	0.932122	−	0.362144i	\(-0.117955\pi\)
							0.152435	+	0.988313i	\(0.451288\pi\)
\(19\)	−3.36608	−	1.94341i	−0.772233	−	0.445849i	0.0614378	−	0.998111i	\(-0.480431\pi\)
							−0.833671	+	0.552262i	\(0.813765\pi\)
\(23\)	2.82525	−	4.89348i	0.589106	−	1.02036i	−0.405244	−	0.914209i	\(-0.632813\pi\)
							0.994350	−	0.106153i	\(-0.0338534\pi\)
\(29\)	−0.445628	+	0.257284i	−0.0827511	+	0.0477764i	−0.540805	−	0.841148i	\(-0.681880\pi\)
							0.458053	+	0.888925i	\(0.348547\pi\)
\(31\)	1.70782	−	0.986008i	0.306733	−	0.177092i	−0.338731	−	0.940883i	\(-0.609998\pi\)
							0.645464	+	0.763791i	\(0.276664\pi\)
\(37\)	−2.46047	+	4.26166i	−0.404499	+	0.700612i	−0.994263	−	0.106963i	\(-0.965887\pi\)
							0.589764	+	0.807575i	\(0.299221\pi\)
\(41\)	3.70968	−	2.14179i	0.579355	−	0.334491i	−0.181522	−	0.983387i	\(-0.558102\pi\)
							0.760877	+	0.648896i	\(0.224769\pi\)
\(43\)		−	0.912766i		−	0.139196i	−0.997575	−	0.0695978i	\(-0.977828\pi\)
							0.997575	−	0.0695978i	\(-0.0221716\pi\)
\(47\)	1.60701	+	2.78341i	0.234406	+	0.406003i	0.959100	−	0.283068i	\(-0.0913522\pi\)
							−0.724694	+	0.689071i	\(0.758019\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.l.a.431.8	✓	264
3.2	odd	2	inner	804.2.l.a.431.125	yes	264
4.3	odd	2	inner	804.2.l.a.431.52	yes	264
12.11	even	2	inner	804.2.l.a.431.81	yes	264
67.37	even	3	inner	804.2.l.a.707.81	yes	264
201.104	odd	6	inner	804.2.l.a.707.52	yes	264
268.171	odd	6	inner	804.2.l.a.707.125	yes	264
804.707	even	6	inner	804.2.l.a.707.8	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.l.a.431.8	✓	264	1.1	even	1	trivial
804.2.l.a.431.52	yes	264	4.3	odd	2	inner
804.2.l.a.431.81	yes	264	12.11	even	2	inner
804.2.l.a.431.125	yes	264	3.2	odd	2	inner
804.2.l.a.707.8	yes	264	804.707	even	6	inner
804.2.l.a.707.52	yes	264	201.104	odd	6	inner
804.2.l.a.707.81	yes	264	67.37	even	3	inner
804.2.l.a.707.125	yes	264	268.171	odd	6	inner