Embedded newform 805.2.bp.b.61.17

• Label: 805.2.bp.b.61.17
• Level: $805$
• Weight: $2$
• Character: 805.61
• Analytic conductor: $6.428$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 805 = 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	805.bp (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.42795736271\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(32\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			61.17
Character	\(\chi\)	\(=\)	805.61
Dual form			805.2.bp.b.66.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0534238 + 0.0420129i) q^{2} +(-2.26095 - 1.61002i) q^{3} +(-0.470429 - 1.93913i) q^{4} +(0.981929 - 0.189251i) q^{5} +(-0.0531472 - 0.181003i) q^{6} +(0.794489 + 2.52365i) q^{7} +(0.112804 - 0.247006i) q^{8} +(1.53855 + 4.44535i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 2 q^{2} + 34 q^{4} + 32 q^{5} - 3 q^{7} + 12 q^{8} - 34 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/805\mathbb{Z}\right)^\times\).

\(n\)	\(162\)	\(281\)	\(346\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0534238	+	0.0420129i	0.0377763	+	0.0297076i	0.636869	−	0.770972i	\(-0.280229\pi\)
							−0.599092	+	0.800680i	\(0.704472\pi\)
\(3\)	−2.26095	−	1.61002i	−1.30536	−	0.929544i	−0.305618	−	0.952154i	\(-0.598863\pi\)
							−0.999744	+	0.0226101i	\(0.992802\pi\)
\(5\)	0.981929	−	0.189251i	0.439132	−	0.0846357i
							\(7\)	0.794489	+	2.52365i	0.300289	+	0.953848i
\(11\)	0.459909	+	0.584822i	0.138668	+	0.176331i								0.850427	−	0.526093i	\(-0.176344\pi\)
							−0.711759	+	0.702424i	\(0.752101\pi\)
\(13\)	1.46461	+	2.27898i	0.406211	+	0.632077i	0.982738	−	0.185001i	\(-0.0592289\pi\)
							−0.576527	+	0.817078i	\(0.695593\pi\)
\(17\)	1.76951	+	1.68722i	0.429168	+	0.409211i	0.873633	−	0.486585i	\(-0.161758\pi\)
							−0.444465	+	0.895796i	\(0.646606\pi\)
\(19\)	−5.89539	+	5.62124i	−1.35249	+	1.28960i	−0.430398	+	0.902639i	\(0.641627\pi\)
							−0.922096	+	0.386962i	\(0.873525\pi\)
\(23\)	3.31945	+	3.46139i	0.692154	+	0.721750i
							\(29\)	−3.66594	+	1.07642i	−0.680747	+	0.199885i	−0.603783	−	0.797148i	\(-0.706341\pi\)
−0.0769639	+	0.997034i	\(0.524523\pi\)
\(31\)	0.302641	+	3.16940i	0.0543560	+	0.569241i	0.981094	+	0.193533i	\(0.0619946\pi\)
							−0.926738	+	0.375709i	\(0.877399\pi\)
\(37\)	2.66143	−	0.921130i	0.437537	−	0.151433i	−0.0994147	−	0.995046i	\(-0.531697\pi\)
							0.536951	+	0.843613i	\(0.319576\pi\)
\(41\)	6.87499	−	5.95722i	1.07369	−	0.930361i	0.0759242	−	0.997114i	\(-0.475809\pi\)
							0.997770	+	0.0667526i	\(0.0212638\pi\)
\(43\)	6.77394	−	3.09355i	1.03302	−	0.471762i	0.174558	−	0.984647i	\(-0.444150\pi\)
							0.858458	+	0.512884i	\(0.171423\pi\)
\(47\)	4.24597	+	2.45141i	0.619339	+	0.357575i	0.776611	−	0.629980i	\(-0.216937\pi\)
							−0.157273	+	0.987555i	\(0.550270\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	805.2.bp.b.61.17	yes	640
7.3	odd	6		805.2.bp.a.521.16	yes	640
23.20	odd	22		805.2.bp.a.411.16	✓	640
161.66	even	66	inner	805.2.bp.b.66.17	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
805.2.bp.a.411.16	✓	640	23.20	odd	22
805.2.bp.a.521.16	yes	640	7.3	odd	6
805.2.bp.b.61.17	yes	640	1.1	even	1	trivial
805.2.bp.b.66.17	yes	640	161.66	even	66	inner