Embedded newform 804.2.u.b.43.1

• Label: 804.2.u.b.43.1
• 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			43.1
Character	\(\chi\)	\(=\)	804.43
Dual form			804.2.u.b.187.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41407 + 0.0203147i) q^{2} +(-0.841254 + 0.540641i) q^{3} +(1.99917 - 0.0574526i) q^{4} +(-3.35521 - 0.482407i) q^{5} +(1.17861 - 0.781592i) q^{6} +(-1.56720 + 3.43169i) q^{7} +(-2.82580 + 0.121854i) q^{8} +(0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 340 q + 34 q^{3} - 2 q^{4} - 11 q^{6} + 4 q^{7} + 27 q^{8} - 34 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41407	+	0.0203147i	−0.999897	+	0.0143646i
							\(3\)	−0.841254	+	0.540641i	−0.485698	+	0.312139i
\(5\)	−3.35521	−	0.482407i	−1.50050	−	0.215739i								−0.657416	−	0.753527i	\(-0.728351\pi\)
							−0.843080	+	0.537789i	\(0.819260\pi\)
\(7\)	−1.56720	+	3.43169i	−0.592346	+	1.29706i	0.341668	+	0.939821i	\(0.389008\pi\)
							−0.934014	+	0.357236i	\(0.883719\pi\)
\(11\)	−0.610801	+	4.24822i	−0.184163	+	1.28089i	0.662623	+	0.748953i	\(0.269443\pi\)
							−0.846787	+	0.531932i	\(0.821466\pi\)
\(13\)	−1.32456	+	4.51105i	−0.367368	+	1.25114i	0.543838	+	0.839190i	\(0.316971\pi\)
							−0.911206	+	0.411951i	\(0.864848\pi\)
\(17\)	1.85048	−	2.13557i	0.448808	−	0.517952i	−0.485588	−	0.874188i	\(-0.661395\pi\)
							0.934396	+	0.356236i	\(0.115940\pi\)
\(19\)	1.21523	−	0.554975i	0.278792	−	0.127320i	−0.271112	−	0.962548i	\(-0.587391\pi\)
							0.549904	+	0.835228i	\(0.314664\pi\)
\(23\)	−0.450650	−	0.701225i	−0.0939670	−	0.146215i	0.791087	−	0.611703i	\(-0.209515\pi\)
							−0.885054	+	0.465488i	\(0.845879\pi\)
\(29\)	−2.48342			−0.461160			−0.230580	−	0.973053i	\(-0.574062\pi\)
							−0.230580	+	0.973053i	\(0.574062\pi\)
\(31\)	0.303330	−	0.0890656i	0.0544796	−	0.0159967i	−0.254379	−	0.967105i	\(-0.581871\pi\)
							0.308859	+	0.951108i	\(0.400053\pi\)
\(37\)	−9.50235			−1.56218			−0.781088	−	0.624421i	\(-0.785335\pi\)
							−0.781088	+	0.624421i	\(0.785335\pi\)
\(41\)	−8.88261	−	7.69683i	−1.38723	−	1.20204i	−0.953646	−	0.300930i	\(-0.902703\pi\)
							−0.433585	−	0.901113i	\(-0.642752\pi\)
\(43\)	5.20737	−	6.00963i	0.794117	−	0.916460i	−0.203926	−	0.978986i	\(-0.565370\pi\)
							0.998043	+	0.0625265i	\(0.0199158\pi\)
\(47\)	4.76858	+	7.42005i	0.695569	+	1.08233i	0.991873	+	0.127232i	\(0.0406094\pi\)
							−0.296304	+	0.955094i	\(0.595754\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.u.b.43.1	yes	340
4.3	odd	2		804.2.u.a.43.8	✓	340
67.53	odd	22		804.2.u.a.187.8	yes	340
268.187	even	22	inner	804.2.u.b.187.1	yes	340

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.u.a.43.8	✓	340	4.3	odd	2
804.2.u.a.187.8	yes	340	67.53	odd	22
804.2.u.b.43.1	yes	340	1.1	even	1	trivial
804.2.u.b.187.1	yes	340	268.187	even	22	inner