Embedded newform 804.2.u.a.43.16

• Label: 804.2.u.a.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.a.187.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.145969 + 1.40666i) q^{2} +(0.841254 - 0.540641i) q^{3} +(-1.95739 - 0.410658i) q^{4} +(3.06720 + 0.440996i) q^{5} +(0.637701 + 1.26227i) q^{6} +(0.888803 - 1.94621i) q^{7} +(0.863375 - 2.69343i) q^{8} +(0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 340 q - 34 q^{3} - 2 q^{4} + 11 q^{6} - 4 q^{7} + 39 q^{8} - 34 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.145969	+	1.40666i	−0.103216	+	0.994659i
							\(3\)	0.841254	−	0.540641i	0.485698	−	0.312139i
\(5\)	3.06720	+	0.440996i	1.37169	+	0.197219i								0.788460	−	0.615086i	\(-0.210879\pi\)
							0.583232	+	0.812306i	\(0.301788\pi\)
\(7\)	0.888803	−	1.94621i	0.335936	−	0.735597i	−0.663990	−	0.747741i	\(-0.731138\pi\)
							0.999926	+	0.0121438i	\(0.00386560\pi\)
\(11\)	−0.0129460	+	0.0900411i	−0.00390335	+	0.0271484i	−0.991680	−	0.128727i	\(-0.958911\pi\)
							0.987777	+	0.155876i	\(0.0498199\pi\)
\(13\)	0.874731	−	2.97906i	0.242607	−	0.826243i	−0.744697	−	0.667402i	\(-0.767406\pi\)
							0.987304	−	0.158841i	\(-0.0507756\pi\)
\(17\)	2.58625	−	2.98469i	0.627257	−	0.723893i	−0.349811	−	0.936820i	\(-0.613754\pi\)
							0.977068	+	0.212927i	\(0.0682997\pi\)
\(19\)	−5.87783	+	2.68431i	−1.34847	+	0.615824i	−0.953088	−	0.302695i	\(-0.902114\pi\)
							−0.395378	+	0.918518i	\(0.629386\pi\)
\(23\)	−3.44404	−	5.35903i	−0.718132	−	1.11744i	−0.987987	−	0.154535i	\(-0.950612\pi\)
							0.269855	−	0.962901i	\(-0.413024\pi\)
\(29\)	6.92950			1.28678			0.643388	−	0.765540i	\(-0.277528\pi\)
							0.643388	+	0.765540i	\(0.277528\pi\)
\(31\)	4.14651	−	1.21753i	0.744736	−	0.218674i	0.112717	−	0.993627i	\(-0.464045\pi\)
							0.632019	+	0.774953i	\(0.282227\pi\)
\(37\)	−2.44251			−0.401546			−0.200773	−	0.979638i	\(-0.564345\pi\)
							−0.200773	+	0.979638i	\(0.564345\pi\)
\(41\)	−1.92122	−	1.66475i	−0.300045	−	0.259990i	0.491806	−	0.870705i	\(-0.336337\pi\)
							−0.791851	+	0.610715i	\(0.790882\pi\)
\(43\)	−5.10868	+	5.89573i	−0.779067	+	0.899091i	−0.997042	−	0.0768636i	\(-0.975509\pi\)
							0.217975	+	0.975954i	\(0.430055\pi\)
\(47\)	3.03093	+	4.71622i	0.442107	+	0.687931i	0.988774	−	0.149419i	\(-0.0477402\pi\)
							−0.546667	+	0.837350i	\(0.684104\pi\)

(See \(a_n\) instead)

Twists

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

    

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