Embedded newform 804.2.u.a.43.6

• Label: 804.2.u.a.43.6
• 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			43.6
Character	\(\chi\)	\(=\)	804.43
Dual form			804.2.u.a.187.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.21380 - 0.725738i) q^{2} +(0.841254 - 0.540641i) q^{3} +(0.946608 + 1.76180i) q^{4} +(-1.70218 - 0.244737i) q^{5} +(-1.41348 + 0.0456983i) q^{6} +(-1.89274 + 4.14453i) q^{7} +(0.129615 - 2.82546i) q^{8} +(0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 340 q - 34 q^{3} - 2 q^{4} + 11 q^{6} - 4 q^{7} + 39 q^{8} - 34 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.21380	−	0.725738i	−0.858284	−	0.513175i
							\(3\)	0.841254	−	0.540641i	0.485698	−	0.312139i
\(5\)	−1.70218	−	0.244737i	−0.761238	−	0.109450i								−0.249247	−	0.968440i	\(-0.580183\pi\)
							−0.511992	+	0.858990i	\(0.671092\pi\)
\(7\)	−1.89274	+	4.14453i	−0.715390	+	1.56649i	0.104864	+	0.994487i	\(0.466559\pi\)
							−0.820254	+	0.571999i	\(0.806168\pi\)
\(11\)	0.121666	−	0.846203i	0.0366836	−	0.255140i	−0.963225	−	0.268698i	\(-0.913407\pi\)
							0.999908	+	0.0135581i	\(0.00431580\pi\)
\(13\)	0.789845	−	2.68997i	0.219064	−	0.746063i	−0.774479	−	0.632600i	\(-0.781988\pi\)
							0.993542	−	0.113462i	\(-0.0361941\pi\)
\(17\)	−0.288131	+	0.332520i	−0.0698819	+	0.0806480i	−0.789614	−	0.613604i	\(-0.789719\pi\)
							0.719732	+	0.694252i	\(0.244265\pi\)
\(19\)	2.28170	−	1.04202i	0.523457	−	0.239055i	−0.136119	−	0.990692i	\(-0.543463\pi\)
							0.659576	+	0.751638i	\(0.270736\pi\)
\(23\)	−3.59624	−	5.59585i	−0.749867	−	1.16682i	−0.981022	−	0.193899i	\(-0.937887\pi\)
							0.231154	−	0.972917i	\(-0.425750\pi\)
\(29\)	−7.55983			−1.40382			−0.701912	−	0.712263i	\(-0.747670\pi\)
							−0.701912	+	0.712263i	\(0.747670\pi\)
\(31\)	−4.36528	+	1.28176i	−0.784027	+	0.230211i	−0.649159	−	0.760652i	\(-0.724879\pi\)
							−0.134868	+	0.990864i	\(0.543061\pi\)
\(37\)	−5.23184			−0.860109			−0.430054	−	0.902803i	\(-0.641506\pi\)
							−0.430054	+	0.902803i	\(0.641506\pi\)
\(41\)	−5.49960	−	4.76543i	−0.858893	−	0.744235i	0.109414	−	0.993996i	\(-0.465102\pi\)
							−0.968308	+	0.249761i	\(0.919648\pi\)
\(43\)	−2.11680	+	2.44291i	−0.322808	+	0.372541i	−0.893839	−	0.448388i	\(-0.851998\pi\)
							0.571031	+	0.820929i	\(0.306544\pi\)
\(47\)	3.87793	+	6.03417i	0.565654	+	0.880174i	0.999786	−	0.0206747i	\(-0.00658143\pi\)
							−0.434133	+	0.900849i	\(0.642945\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.u.a.43.6	✓	340
4.3	odd	2		804.2.u.b.43.5	yes	340
67.53	odd	22		804.2.u.b.187.5	yes	340
268.187	even	22	inner	804.2.u.a.187.6	yes	340

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.u.a.43.6	✓	340	1.1	even	1	trivial
804.2.u.a.187.6	yes	340	268.187	even	22	inner
804.2.u.b.43.5	yes	340	4.3	odd	2
804.2.u.b.187.5	yes	340	67.53	odd	22