Embedded newform 804.2.u.b.43.13

• Label: 804.2.u.b.43.13
• 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			43.13
Character	\(\chi\)	\(=\)	804.43
Dual form			804.2.u.b.187.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.413712 + 1.35235i) q^{2} +(-0.841254 + 0.540641i) q^{3} +(-1.65768 - 1.11896i) q^{4} +(0.112828 + 0.0162222i) q^{5} +(-0.383097 - 1.36134i) q^{6} +(-0.733664 + 1.60650i) q^{7} +(2.19903 - 1.77884i) q^{8} +(0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 340 q + 34 q^{3} - 2 q^{4} - 11 q^{6} + 4 q^{7} + 27 q^{8} - 34 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.413712	+	1.35235i	−0.292539	+	0.956254i
							\(3\)	−0.841254	+	0.540641i	−0.485698	+	0.312139i
\(5\)	0.112828	+	0.0162222i	0.0504581	+	0.00725479i								0.167498	−	0.985872i	\(-0.446431\pi\)
							−0.117040	+	0.993127i	\(0.537340\pi\)
\(7\)	−0.733664	+	1.60650i	−0.277299	+	0.607200i	−0.996121	−	0.0879944i	\(-0.971954\pi\)
							0.718822	+	0.695194i	\(0.244682\pi\)
\(11\)	−0.601336	+	4.18238i	−0.181310	+	1.26104i	0.672361	+	0.740223i	\(0.265280\pi\)
							−0.853671	+	0.520813i	\(0.825629\pi\)
\(13\)	−0.516847	+	1.76022i	−0.143348	+	0.488197i	−0.999598	−	0.0283646i	\(-0.990970\pi\)
							0.856250	+	0.516561i	\(0.172788\pi\)
\(17\)	0.609842	−	0.703795i	0.147908	−	0.170695i	−0.676961	−	0.736019i	\(-0.736703\pi\)
							0.824869	+	0.565324i	\(0.191249\pi\)
\(19\)	0.163309	−	0.0745808i	0.0374657	−	0.0171100i	−0.396594	−	0.917994i	\(-0.629808\pi\)
							0.434060	+	0.900884i	\(0.357081\pi\)
\(23\)	−1.58461	−	2.46570i	−0.330414	−	0.514135i	0.635807	−	0.771848i	\(-0.280667\pi\)
							−0.966221	+	0.257713i	\(0.917031\pi\)
\(29\)	1.23982			0.230229			0.115114	−	0.993352i	\(-0.463277\pi\)
							0.115114	+	0.993352i	\(0.463277\pi\)
\(31\)	−4.30876	+	1.26517i	−0.773877	+	0.227231i	−0.644746	−	0.764397i	\(-0.723037\pi\)
							−0.129131	+	0.991628i	\(0.541219\pi\)
\(37\)	−3.50332			−0.575942			−0.287971	−	0.957639i	\(-0.592981\pi\)
							−0.287971	+	0.957639i	\(0.592981\pi\)
\(41\)	−9.22363	−	7.99232i	−1.44049	−	1.24819i	−0.918607	−	0.395172i	\(-0.870685\pi\)
							−0.521882	−	0.853018i	\(-0.674770\pi\)
\(43\)	−3.68594	+	4.25380i	−0.562101	+	0.648699i	−0.963659	−	0.267134i	\(-0.913923\pi\)
							0.401558	+	0.915833i	\(0.368469\pi\)
\(47\)	−3.53141	−	5.49498i	−0.515109	−	0.801525i	0.482105	−	0.876113i	\(-0.339872\pi\)
							−0.997214	+	0.0745878i	\(0.976236\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.u.b.43.13	yes	340
4.3	odd	2		804.2.u.a.43.23	✓	340
67.53	odd	22		804.2.u.a.187.23	yes	340
268.187	even	22	inner	804.2.u.b.187.13	yes	340

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.u.a.43.23	✓	340	4.3	odd	2
804.2.u.a.187.23	yes	340	67.53	odd	22
804.2.u.b.43.13	yes	340	1.1	even	1	trivial
804.2.u.b.187.13	yes	340	268.187	even	22	inner