Embedded newform 805.2.bs.a.37.9

• Label: 805.2.bs.a.37.9
• Level: $805$
• Weight: $2$
• Character: 805.37
• Analytic conductor: $6.428$
• Analytic rank: $0$
• Dimension: $3680$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 805 = 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	805.bs (of order \(132\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.42795736271\)
Analytic rank:	\(0\)
Dimension:	\(3680\)
Relative dimension:	\(92\) over \(\Q(\zeta_{132})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{132}]$

Embedding invariants

Embedding label			37.9
Character	\(\chi\)	\(=\)	805.37
Dual form			805.2.bs.a.718.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.03883 + 1.24273i) q^{2} +(0.879859 + 1.81127i) q^{3} +(1.69598 - 3.28975i) q^{4} +(1.72696 - 1.42043i) q^{5} +(-4.04479 - 2.59943i) q^{6} +(1.66548 - 2.05577i) q^{7} +(0.289777 + 4.05161i) q^{8} +(-0.652056 + 0.829157i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3680 q - 18 q^{2} - 18 q^{3} - 22 q^{5} - 160 q^{6} - 44 q^{7} - 56 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(162\)	\(281\)	\(346\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.03883	+	1.24273i	−1.44167	+	0.878743i	−0.999832	−	0.0183530i	\(-0.994158\pi\)
							−0.441837	+	0.897096i	\(0.645673\pi\)
\(3\)	0.879859	+	1.81127i	0.507987	+	1.04573i	0.985901	+	0.167329i	\(0.0535141\pi\)
							−0.477915	+	0.878406i	\(0.658607\pi\)
\(5\)	1.72696	−	1.42043i	0.772319	−	0.635234i
							\(7\)	1.66548	−	2.05577i	0.629492	−	0.777007i
\(11\)	2.08700	−	0.506300i	0.629253	−	0.152655i								0.0915650	−	0.995799i	\(-0.470813\pi\)
							0.537688	+	0.843144i	\(0.319298\pi\)
\(13\)	−5.81340	−	2.16829i	−1.61235	−	0.601374i	−0.628094	−	0.778138i	\(-0.716165\pi\)
							−0.984253	+	0.176763i	\(0.943437\pi\)
\(17\)	−3.58993	−	3.94905i	−0.870685	−	0.957786i	0.128740	−	0.991678i	\(-0.458907\pi\)
							−0.999425	+	0.0338923i	\(0.989210\pi\)
\(19\)	0.235071	+	4.93475i	0.0539290	+	1.13211i	0.849958	+	0.526851i	\(0.176627\pi\)
							−0.796029	+	0.605259i	\(0.793070\pi\)
\(23\)	−3.60583	−	3.16196i	−0.751868	−	0.659314i
							\(29\)	2.90015	−	4.51271i	0.538544	−	0.837990i	−0.460215	−	0.887808i	\(-0.652228\pi\)
0.998758	+	0.0498175i	\(0.0158640\pi\)
\(31\)	5.11771	+	0.986357i	0.919167	+	0.177155i	0.626826	−	0.779159i	\(-0.284354\pi\)
							0.292341	+	0.956314i	\(0.405566\pi\)
\(37\)	10.0300	−	1.19923i	1.64892	−	0.197153i	0.758302	−	0.651903i	\(-0.226029\pi\)
							0.890619	+	0.454750i	\(0.150272\pi\)
\(41\)	−0.330615	−	2.29948i	−0.0516333	−	0.359118i	−0.999215	−	0.0396052i	\(-0.987390\pi\)
							0.947582	−	0.319512i	\(-0.103519\pi\)
\(43\)	0.162625	−	2.27380i	0.0248001	−	0.346751i	−0.969869	−	0.243629i	\(-0.921662\pi\)
							0.994669	−	0.103122i	\(-0.0328833\pi\)
\(47\)	−1.57215	+	0.421255i	−0.229321	+	0.0614464i	−0.371650	−	0.928373i	\(-0.621208\pi\)
							0.142329	+	0.989819i	\(0.454541\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	805.2.bs.a.37.9	✓	3680
5.3	odd	4	inner	805.2.bs.a.198.84	yes	3680
7.4	even	3	inner	805.2.bs.a.382.9	yes	3680
23.5	odd	22	inner	805.2.bs.a.212.9	yes	3680
35.18	odd	12	inner	805.2.bs.a.543.9	yes	3680
115.28	even	44	inner	805.2.bs.a.373.9	yes	3680
161.74	odd	66	inner	805.2.bs.a.557.84	yes	3680
805.718	even	132	inner	805.2.bs.a.718.9	yes	3680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
805.2.bs.a.37.9	✓	3680	1.1	even	1	trivial
805.2.bs.a.198.84	yes	3680	5.3	odd	4	inner
805.2.bs.a.212.9	yes	3680	23.5	odd	22	inner
805.2.bs.a.373.9	yes	3680	115.28	even	44	inner
805.2.bs.a.382.9	yes	3680	7.4	even	3	inner
805.2.bs.a.543.9	yes	3680	35.18	odd	12	inner
805.2.bs.a.557.84	yes	3680	161.74	odd	66	inner
805.2.bs.a.718.9	yes	3680	805.718	even	132	inner