Embedded newform 804.2.u.b.43.18

• Label: 804.2.u.b.43.18
• 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			43.18
Character	\(\chi\)	\(=\)	804.43
Dual form			804.2.u.b.187.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.243897 - 1.39302i) q^{2} +(-0.841254 + 0.540641i) q^{3} +(-1.88103 - 0.679507i) q^{4} +(-2.48843 - 0.357782i) q^{5} +(0.547946 + 1.30375i) q^{6} +(-1.70447 + 3.73227i) q^{7} +(-1.40535 + 2.45459i) q^{8} +(0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 340 q + 34 q^{3} - 2 q^{4} - 11 q^{6} + 4 q^{7} + 27 q^{8} - 34 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.243897	−	1.39302i	0.172461	−	0.985016i
							\(3\)	−0.841254	+	0.540641i	−0.485698	+	0.312139i
\(5\)	−2.48843	−	0.357782i	−1.11286	−	0.160005i								−0.438729	−	0.898619i	\(-0.644571\pi\)
							−0.674128	+	0.738614i	\(0.735480\pi\)
\(7\)	−1.70447	+	3.73227i	−0.644229	+	1.41067i	0.252286	+	0.967653i	\(0.418818\pi\)
							−0.896515	+	0.443012i	\(0.853910\pi\)
\(11\)	0.146160	−	1.01656i	0.0440688	−	0.306505i	−0.955851	−	0.293851i	\(-0.905063\pi\)
							0.999920	−	0.0126540i	\(-0.00402800\pi\)
\(13\)	1.24580	−	4.24280i	0.345522	−	1.17674i	−0.585162	−	0.810916i	\(-0.698969\pi\)
							0.930684	−	0.365824i	\(-0.119213\pi\)
\(17\)	−1.95095	+	2.25152i	−0.473175	+	0.546073i	−0.941292	−	0.337593i	\(-0.890387\pi\)
							0.468117	+	0.883666i	\(0.344933\pi\)
\(19\)	7.50116	−	3.42566i	1.72088	−	0.785901i	0.725684	−	0.688028i	\(-0.241523\pi\)
							0.995199	−	0.0978734i	\(-0.0312040\pi\)
\(23\)	0.0381842	+	0.0594158i	0.00796195	+	0.0123890i	0.845212	−	0.534432i	\(-0.179474\pi\)
							−0.837250	+	0.546821i	\(0.815838\pi\)
\(29\)	0.549827			0.102100			0.0510502	−	0.998696i	\(-0.483743\pi\)
							0.0510502	+	0.998696i	\(0.483743\pi\)
\(31\)	9.08787	−	2.66844i	1.63223	−	0.479266i	0.667962	−	0.744195i	\(-0.267167\pi\)
							0.964268	+	0.264930i	\(0.0853487\pi\)
\(37\)	7.46486			1.22722			0.613608	−	0.789611i	\(-0.289718\pi\)
							0.613608	+	0.789611i	\(0.289718\pi\)
\(41\)	−4.96072	−	4.29849i	−0.774735	−	0.671312i	0.174929	−	0.984581i	\(-0.444030\pi\)
							−0.949664	+	0.313269i	\(0.898576\pi\)
\(43\)	−6.12434	+	7.06786i	−0.933952	+	1.07784i	0.0628572	+	0.998023i	\(0.479979\pi\)
							−0.996810	+	0.0798161i	\(0.974567\pi\)
\(47\)	3.57562	+	5.56377i	0.521557	+	0.811559i	0.997698	−	0.0678102i	\(-0.0216012\pi\)
							−0.476141	+	0.879369i	\(0.657965\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.u.b.43.18	yes	340
4.3	odd	2		804.2.u.a.43.9	✓	340
67.53	odd	22		804.2.u.a.187.9	yes	340
268.187	even	22	inner	804.2.u.b.187.18	yes	340

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.u.a.43.9	✓	340	4.3	odd	2
804.2.u.a.187.9	yes	340	67.53	odd	22
804.2.u.b.43.18	yes	340	1.1	even	1	trivial
804.2.u.b.187.18	yes	340	268.187	even	22	inner