Embedded newform 804.2.u.a.43.4

• Label: 804.2.u.a.43.4
• 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			43.4
Character	\(\chi\)	\(=\)	804.43
Dual form			804.2.u.a.187.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28564 + 0.589182i) q^{2} +(0.841254 - 0.540641i) q^{3} +(1.30573 - 1.51495i) q^{4} +(-0.935143 - 0.134453i) q^{5} +(-0.763012 + 1.19072i) q^{6} +(-0.374614 + 0.820291i) q^{7} +(-0.786116 + 2.71699i) q^{8} +(0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 340 q - 34 q^{3} - 2 q^{4} + 11 q^{6} - 4 q^{7} + 39 q^{8} - 34 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28564	+	0.589182i	−0.909083	+	0.416614i
							\(3\)	0.841254	−	0.540641i	0.485698	−	0.312139i
\(5\)	−0.935143	−	0.134453i	−0.418209	−	0.0601293i								−0.0700015	−	0.997547i	\(-0.522300\pi\)
							−0.348207	+	0.937418i	\(0.613209\pi\)
\(7\)	−0.374614	+	0.820291i	−0.141591	+	0.310041i	−0.967121	−	0.254318i	\(-0.918149\pi\)
							0.825530	+	0.564359i	\(0.190876\pi\)
\(11\)	0.0334271	−	0.232491i	0.0100787	−	0.0700986i	−0.984161	−	0.177275i	\(-0.943272\pi\)
							0.994240	+	0.107176i	\(0.0341809\pi\)
\(13\)	0.977844	−	3.33023i	0.271205	−	0.923640i	−0.705438	−	0.708772i	\(-0.749250\pi\)
							0.976643	−	0.214868i	\(-0.0689322\pi\)
\(17\)	−1.12451	+	1.29775i	−0.272733	+	0.314751i	−0.875549	−	0.483130i	\(-0.839500\pi\)
							0.602816	+	0.797880i	\(0.294045\pi\)
\(19\)	4.47074	−	2.04172i	1.02566	−	0.468403i	0.169727	−	0.985491i	\(-0.445711\pi\)
							0.855932	+	0.517088i	\(0.172984\pi\)
\(23\)	−0.815205	−	1.26848i	−0.169982	−	0.264497i	0.745802	−	0.666167i	\(-0.232066\pi\)
							−0.915785	+	0.401670i	\(0.868430\pi\)
\(29\)	9.10823			1.69136			0.845678	−	0.533693i	\(-0.179196\pi\)
							0.845678	+	0.533693i	\(0.179196\pi\)
\(31\)	9.03141	−	2.65186i	1.62209	−	0.476288i	0.660510	−	0.750817i	\(-0.270340\pi\)
							0.961579	+	0.274529i	\(0.0885218\pi\)
\(37\)	−9.52689			−1.56621			−0.783106	−	0.621889i	\(-0.786366\pi\)
							−0.783106	+	0.621889i	\(0.786366\pi\)
\(41\)	−5.48376	−	4.75171i	−0.856420	−	0.742092i	0.111390	−	0.993777i	\(-0.464470\pi\)
							−0.967809	+	0.251685i	\(0.919015\pi\)
\(43\)	7.89162	−	9.10741i	1.20346	−	1.38887i	0.303537	−	0.952820i	\(-0.401832\pi\)
							0.899923	−	0.436048i	\(-0.143622\pi\)
\(47\)	−5.33413	−	8.30008i	−0.778064	−	1.21069i	−0.973210	−	0.229916i	\(-0.926155\pi\)
							0.195147	−	0.980774i	\(-0.437482\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.u.a.43.4	✓	340
4.3	odd	2		804.2.u.b.43.14	yes	340
67.53	odd	22		804.2.u.b.187.14	yes	340
268.187	even	22	inner	804.2.u.a.187.4	yes	340

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.u.a.43.4	✓	340	1.1	even	1	trivial
804.2.u.a.187.4	yes	340	268.187	even	22	inner
804.2.u.b.43.14	yes	340	4.3	odd	2
804.2.u.b.187.14	yes	340	67.53	odd	22