Embedded newform 804.2.u.b.43.7

• Label: 804.2.u.b.43.7
• 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			43.7
Character	\(\chi\)	\(=\)	804.43
Dual form			804.2.u.b.187.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13018 - 0.850112i) q^{2} +(-0.841254 + 0.540641i) q^{3} +(0.554618 + 1.92156i) q^{4} +(-2.84397 - 0.408902i) q^{5} +(1.41037 + 0.104138i) q^{6} +(-0.447300 + 0.979450i) q^{7} +(1.00673 - 2.64320i) q^{8} +(0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 340 q + 34 q^{3} - 2 q^{4} - 11 q^{6} + 4 q^{7} + 27 q^{8} - 34 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13018	−	0.850112i	−0.799159	−	0.601120i
							\(3\)	−0.841254	+	0.540641i	−0.485698	+	0.312139i
\(5\)	−2.84397	−	0.408902i	−1.27186	−	0.182866i								−0.526875	−	0.849943i	\(-0.676636\pi\)
							−0.744989	+	0.667077i	\(0.767545\pi\)
\(7\)	−0.447300	+	0.979450i	−0.169063	+	0.370197i	−0.975132	−	0.221625i	\(-0.928864\pi\)
							0.806069	+	0.591822i	\(0.201591\pi\)
\(11\)	0.565094	−	3.93031i	0.170382	−	1.18503i	−0.707696	−	0.706517i	\(-0.750265\pi\)
							0.878078	−	0.478517i	\(-0.158825\pi\)
\(13\)	−0.369419	+	1.25812i	−0.102458	+	0.348941i	−0.994726	−	0.102565i	\(-0.967295\pi\)
							0.892268	+	0.451506i	\(0.149113\pi\)
\(17\)	−2.97900	+	3.43795i	−0.722513	+	0.833825i	−0.991607	−	0.129289i	\(-0.958730\pi\)
							0.269094	+	0.963114i	\(0.413276\pi\)
\(19\)	−3.25991	+	1.48875i	−0.747874	+	0.341543i	−0.752628	−	0.658446i	\(-0.771214\pi\)
							0.00475376	+	0.999989i	\(0.498487\pi\)
\(23\)	−2.21304	−	3.44355i	−0.461450	−	0.718031i	0.530074	−	0.847951i	\(-0.322164\pi\)
							−0.991524	+	0.129921i	\(0.958528\pi\)
\(29\)	7.93353			1.47322			0.736610	−	0.676318i	\(-0.236425\pi\)
							0.736610	+	0.676318i	\(0.236425\pi\)
\(31\)	1.37356	−	0.403315i	0.246699	−	0.0724375i	−0.156045	−	0.987750i	\(-0.549875\pi\)
							0.402744	+	0.915312i	\(0.368056\pi\)
\(37\)	5.00214			0.822347			0.411173	−	0.911557i	\(-0.365119\pi\)
							0.411173	+	0.911557i	\(0.365119\pi\)
\(41\)	4.58146	+	3.96986i	0.715504	+	0.619988i	0.934594	−	0.355717i	\(-0.115763\pi\)
							−0.219090	+	0.975705i	\(0.570309\pi\)
\(43\)	5.15466	−	5.94879i	0.786078	−	0.907182i	−0.211455	−	0.977388i	\(-0.567820\pi\)
							0.997533	+	0.0702055i	\(0.0223655\pi\)
\(47\)	0.617896	+	0.961465i	0.0901294	+	0.140244i	0.883391	−	0.468637i	\(-0.155255\pi\)
							−0.793261	+	0.608881i	\(0.791619\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.u.b.43.7	yes	340
4.3	odd	2		804.2.u.a.43.2	✓	340
67.53	odd	22		804.2.u.a.187.2	yes	340
268.187	even	22	inner	804.2.u.b.187.7	yes	340

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.u.a.43.2	✓	340	4.3	odd	2
804.2.u.a.187.2	yes	340	67.53	odd	22
804.2.u.b.43.7	yes	340	1.1	even	1	trivial
804.2.u.b.187.7	yes	340	268.187	even	22	inner