Embedded newform 804.2.u.b.43.16

• Label: 804.2.u.b.43.16
• Level: $804$
• Weight: $2$
• Character: 804.43
• Analytic conductor: $6.420$
• Analytic rank: $0$
• Dimension: $340$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.u (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.41997232251\)
Analytic rank:	\(0\)
Dimension:	\(340\)
Relative dimension:	\(34\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			43.16
Character	\(\chi\)	\(=\)	804.43
Dual form			804.2.u.b.187.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0430767 + 1.41356i) q^{2} +(-0.841254 + 0.540641i) q^{3} +(-1.99629 + 0.121783i) q^{4} +(1.24980 + 0.179694i) q^{5} +(-0.800465 - 1.16587i) q^{6} +(-0.832575 + 1.82308i) q^{7} +(-0.258141 - 2.81662i) q^{8} +(0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 340 q + 34 q^{3} - 2 q^{4} - 11 q^{6} + 4 q^{7} + 27 q^{8} - 34 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(269\)	\(337\)	\(403\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{22}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0430767	+	1.41356i	0.0304598	+	0.999536i
							\(3\)	−0.841254	+	0.540641i	−0.485698	+	0.312139i
\(5\)	1.24980	+	0.179694i	0.558927	+	0.0803616i								0.415988	−	0.909370i	\(-0.363436\pi\)
							0.142939	+	0.989732i	\(0.454345\pi\)
\(7\)	−0.832575	+	1.82308i	−0.314684	+	0.689061i	−0.999202	−	0.0399343i	\(-0.987285\pi\)
							0.684519	+	0.728995i	\(0.260012\pi\)
\(11\)	0.824026	−	5.73123i	0.248453	−	1.72803i	−0.358706	−	0.933451i	\(-0.616782\pi\)
							0.607159	−	0.794580i	\(-0.292309\pi\)
\(13\)	1.93985	−	6.60651i	0.538017	−	1.83232i	−0.0160887	−	0.999871i	\(-0.505121\pi\)
							0.554105	−	0.832446i	\(-0.313060\pi\)
\(17\)	−1.55900	+	1.79919i	−0.378114	+	0.436367i	−0.912627	−	0.408794i	\(-0.865950\pi\)
							0.534513	+	0.845160i	\(0.320495\pi\)
\(19\)	0.0500072	−	0.0228375i	0.0114724	−	0.00523928i	−0.409671	−	0.912234i	\(-0.634356\pi\)
							0.421143	+	0.906994i	\(0.361629\pi\)
\(23\)	−2.81093	−	4.37389i	−0.586119	−	0.912018i	−0.999999	−	0.00168694i	\(-0.999463\pi\)
							0.413880	−	0.910331i	\(-0.364173\pi\)
\(29\)	3.10105			0.575851			0.287926	−	0.957653i	\(-0.407034\pi\)
							0.287926	+	0.957653i	\(0.407034\pi\)
\(31\)	−2.10422	+	0.617853i	−0.377928	+	0.110970i	−0.465179	−	0.885217i	\(-0.654010\pi\)
							0.0872507	+	0.996186i	\(0.472192\pi\)
\(37\)	0.654254			0.107559			0.0537793	−	0.998553i	\(-0.482873\pi\)
							0.0537793	+	0.998553i	\(0.482873\pi\)
\(41\)	1.67718	+	1.45329i	0.261932	+	0.226965i	0.775919	−	0.630833i	\(-0.217287\pi\)
							−0.513987	+	0.857798i	\(0.671832\pi\)
\(43\)	7.39236	−	8.53124i	1.12732	−	1.30100i	0.178946	−	0.983859i	\(-0.442731\pi\)
							0.948378	−	0.317143i	\(-0.102723\pi\)
\(47\)	3.40596	+	5.29978i	0.496811	+	0.773053i	0.995603	−	0.0936688i	\(-0.0298595\pi\)
							−0.498793	+	0.866721i	\(0.666223\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.u.b.43.16	yes	340
4.3	odd	2		804.2.u.a.43.25	✓	340
67.53	odd	22		804.2.u.a.187.25	yes	340
268.187	even	22	inner	804.2.u.b.187.16	yes	340

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.u.a.43.25	✓	340	4.3	odd	2
804.2.u.a.187.25	yes	340	67.53	odd	22
804.2.u.b.43.16	yes	340	1.1	even	1	trivial
804.2.u.b.187.16	yes	340	268.187	even	22	inner