Embedded newform 363.2.e.g.202.1

• Label: 363.2.e.g.202.1
• Level: $363$
• Weight: $2$
• Character: 363.202
• Analytic conductor: $2.899$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	363.e (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.89856959337\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{10})\)
Defining polynomial:	\( x^{4} - x^{3} + x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			202.1
Root			\(0.809017 - 0.587785i\) of defining polynomial
Character	\(\chi\)	\(=\)	363.202
Dual form			363.2.e.g.124.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.809017 + 0.587785i) q^{2} +(-0.309017 + 0.951057i) q^{3} +(-0.309017 - 0.951057i) q^{4} +(1.61803 - 1.17557i) q^{5} +(-0.809017 + 0.587785i) q^{6} +(-1.23607 - 3.80423i) q^{7} +(0.927051 - 2.85317i) q^{8} +(-0.809017 - 0.587785i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + q^{3} + q^{4} + 2 q^{5} - q^{6} + 4 q^{7} - 3 q^{8} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(244\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.809017	+	0.587785i	0.572061	+	0.415627i	0.835853	−	0.548953i	\(-0.184973\pi\)
							−0.263792	+	0.964580i	\(0.584973\pi\)
\(3\)	−0.309017	+	0.951057i	−0.178411	+	0.549093i
							\(5\)	1.61803	−	1.17557i	0.723607	−	0.525731i	−0.163928	−	0.986472i	\(-0.552416\pi\)
0.887535	+	0.460741i	\(0.152416\pi\)
\(7\)	−1.23607	−	3.80423i	−0.467190	−	1.43786i	−0.856208	−	0.516632i	\(-0.827186\pi\)
							0.389018	−	0.921230i	\(-0.372814\pi\)
\(11\)		0			0
							\(13\)	−1.61803	−	1.17557i	−0.448762	−	0.326045i	0.340345	−	0.940301i	\(-0.389456\pi\)
−0.789107	+	0.614256i	\(0.789456\pi\)
\(17\)	−1.61803	+	1.17557i	−0.392431	+	0.285118i	−0.766451	−	0.642303i	\(-0.777979\pi\)
							0.374020	+	0.927421i	\(0.377979\pi\)
\(19\)		0			0		−0.951057	−	0.309017i	\(-0.900000\pi\)
							0.951057	+	0.309017i	\(0.100000\pi\)
\(23\)	8.00000			1.66812			0.834058	−	0.551677i	\(-0.186012\pi\)
							0.834058	+	0.551677i	\(0.186012\pi\)
\(29\)	1.85410	+	5.70634i	0.344298	+	1.05964i	0.961958	+	0.273196i	\(0.0880806\pi\)
							−0.617660	+	0.786445i	\(0.711919\pi\)
\(31\)	6.47214	+	4.70228i	1.16243	+	0.844555i	0.990083	−	0.140482i	\(-0.0448651\pi\)
							0.172347	+	0.985036i	\(0.444865\pi\)
\(37\)	1.85410	+	5.70634i	0.304812	+	0.938116i	0.979747	+	0.200239i	\(0.0641718\pi\)
							−0.674935	+	0.737878i	\(0.735828\pi\)
\(41\)	0.618034	−	1.90211i	0.0965207	−	0.297060i	−0.891126	−	0.453755i	\(-0.850084\pi\)
							0.987647	+	0.156695i	\(0.0500840\pi\)
\(43\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(47\)	2.47214	−	7.60845i	0.360598	−	1.10981i	−0.592094	−	0.805869i	\(-0.701699\pi\)
							0.952692	−	0.303938i	\(-0.0983015\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.2.e.g.202.1		4
11.2	odd	10		363.2.e.e.148.1		4
11.3	even	5	inner	363.2.e.g.124.1		4
11.4	even	5	inner	363.2.e.g.130.1		4
11.5	even	5		363.2.a.b.1.1		1
11.6	odd	10		33.2.a.a.1.1	✓	1
11.7	odd	10		363.2.e.e.130.1		4
11.8	odd	10		363.2.e.e.124.1		4
11.9	even	5	inner	363.2.e.g.148.1		4
11.10	odd	2		363.2.e.e.202.1		4
33.5	odd	10		1089.2.a.j.1.1		1
33.17	even	10		99.2.a.b.1.1		1
44.27	odd	10		5808.2.a.t.1.1		1
44.39	even	10		528.2.a.g.1.1		1
55.17	even	20		825.2.c.a.199.2		2
55.28	even	20		825.2.c.a.199.1		2
55.39	odd	10		825.2.a.a.1.1		1
55.49	even	10		9075.2.a.q.1.1		1
77.6	even	10		1617.2.a.j.1.1		1
88.61	odd	10		2112.2.a.bb.1.1		1
88.83	even	10		2112.2.a.j.1.1		1
99.50	even	30		891.2.e.g.298.1		2
99.61	odd	30		891.2.e.e.595.1		2
99.83	even	30		891.2.e.g.595.1		2
99.94	odd	30		891.2.e.e.298.1		2
132.83	odd	10		1584.2.a.o.1.1		1
143.116	odd	10		5577.2.a.a.1.1		1
165.17	odd	20		2475.2.c.d.199.1		2
165.83	odd	20		2475.2.c.d.199.2		2
165.149	even	10		2475.2.a.g.1.1		1
187.50	odd	10		9537.2.a.m.1.1		1
231.83	odd	10		4851.2.a.b.1.1		1
264.83	odd	10		6336.2.a.n.1.1		1
264.149	even	10		6336.2.a.x.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.2.a.a.1.1	✓	1	11.6	odd	10
99.2.a.b.1.1		1	33.17	even	10
363.2.a.b.1.1		1	11.5	even	5
363.2.e.e.124.1		4	11.8	odd	10
363.2.e.e.130.1		4	11.7	odd	10
363.2.e.e.148.1		4	11.2	odd	10
363.2.e.e.202.1		4	11.10	odd	2
363.2.e.g.124.1		4	11.3	even	5	inner
363.2.e.g.130.1		4	11.4	even	5	inner
363.2.e.g.148.1		4	11.9	even	5	inner
363.2.e.g.202.1		4	1.1	even	1	trivial
528.2.a.g.1.1		1	44.39	even	10
825.2.a.a.1.1		1	55.39	odd	10
825.2.c.a.199.1		2	55.28	even	20
825.2.c.a.199.2		2	55.17	even	20
891.2.e.e.298.1		2	99.94	odd	30
891.2.e.e.595.1		2	99.61	odd	30
891.2.e.g.298.1		2	99.50	even	30
891.2.e.g.595.1		2	99.83	even	30
1089.2.a.j.1.1		1	33.5	odd	10
1584.2.a.o.1.1		1	132.83	odd	10
1617.2.a.j.1.1		1	77.6	even	10
2112.2.a.j.1.1		1	88.83	even	10
2112.2.a.bb.1.1		1	88.61	odd	10
2475.2.a.g.1.1		1	165.149	even	10
2475.2.c.d.199.1		2	165.17	odd	20
2475.2.c.d.199.2		2	165.83	odd	20
4851.2.a.b.1.1		1	231.83	odd	10
5577.2.a.a.1.1		1	143.116	odd	10
5808.2.a.t.1.1		1	44.27	odd	10
6336.2.a.n.1.1		1	264.83	odd	10
6336.2.a.x.1.1		1	264.149	even	10
9075.2.a.q.1.1		1	55.49	even	10
9537.2.a.m.1.1		1	187.50	odd	10