Embedded newform 363.2.e.e.130.1

• Label: 363.2.e.e.130.1
• Level: $363$
• Weight: $2$
• Character: 363.130
• 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			130.1
Root			\(-0.309017 - 0.951057i\) of defining polynomial
Character	\(\chi\)	\(=\)	363.130
Dual form			363.2.e.e.148.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.309017 - 0.951057i) q^{2} +(0.809017 - 0.587785i) q^{3} +(0.809017 + 0.587785i) q^{4} +(-0.618034 - 1.90211i) q^{5} +(-0.309017 - 0.951057i) q^{6} +(-3.23607 - 2.35114i) q^{7} +(2.42705 - 1.76336i) q^{8} +(0.309017 - 0.951057i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + q^{3} + q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 3 q^{8} - q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(244\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.309017	−	0.951057i	0.218508	−	0.672499i	−0.780378	−	0.625308i	\(-0.784973\pi\)
							0.998886	−	0.0471903i	\(-0.0150267\pi\)
\(3\)	0.809017	−	0.587785i	0.467086	−	0.339358i
							\(5\)	−0.618034	−	1.90211i	−0.276393	−	0.850651i	−0.988847	−	0.148932i	\(-0.952416\pi\)
0.712454	−	0.701719i	\(-0.247584\pi\)
\(7\)	−3.23607	−	2.35114i	−1.22312	−	0.888648i	−0.226764	−	0.973950i	\(-0.572814\pi\)
							−0.996355	+	0.0853021i	\(0.972814\pi\)
\(11\)		0			0
							\(13\)	−0.618034	+	1.90211i	−0.171412	+	0.527551i	−0.999451	−	0.0331183i	\(-0.989456\pi\)
0.828040	+	0.560670i	\(0.189456\pi\)
\(17\)	−0.618034	−	1.90211i	−0.149895	−	0.461330i	0.847713	−	0.530456i	\(-0.177979\pi\)
							−0.997608	+	0.0691254i	\(0.977979\pi\)
\(19\)		0			0		−0.587785	−	0.809017i	\(-0.700000\pi\)
							0.587785	+	0.809017i	\(0.300000\pi\)
\(23\)	8.00000			1.66812			0.834058	−	0.551677i	\(-0.186012\pi\)
							0.834058	+	0.551677i	\(0.186012\pi\)
\(29\)	4.85410	+	3.52671i	0.901384	+	0.654894i	0.938821	−	0.344405i	\(-0.111919\pi\)
							−0.0374370	+	0.999299i	\(0.511919\pi\)
\(31\)	−2.47214	+	7.60845i	−0.444009	+	1.36652i	0.439558	+	0.898214i	\(0.355135\pi\)
							−0.883567	+	0.468304i	\(0.844865\pi\)
\(37\)	−4.85410	−	3.52671i	−0.798009	−	0.579788i	0.112320	−	0.993672i	\(-0.464172\pi\)
							−0.910330	+	0.413884i	\(0.864172\pi\)
\(41\)	1.61803	−	1.17557i	0.252694	−	0.183593i	−0.454226	−	0.890887i	\(-0.650084\pi\)
							0.706920	+	0.707293i	\(0.250084\pi\)
\(43\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(47\)	−6.47214	+	4.70228i	−0.944058	+	0.685898i	−0.949394	−	0.314087i	\(-0.898301\pi\)
							0.00533600	+	0.999986i	\(0.498301\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.2.e.e.130.1		4
11.2	odd	10		363.2.e.g.124.1		4
11.3	even	5	inner	363.2.e.e.202.1		4
11.4	even	5		33.2.a.a.1.1	✓	1
11.5	even	5	inner	363.2.e.e.148.1		4
11.6	odd	10		363.2.e.g.148.1		4
11.7	odd	10		363.2.a.b.1.1		1
11.8	odd	10		363.2.e.g.202.1		4
11.9	even	5	inner	363.2.e.e.124.1		4
11.10	odd	2		363.2.e.g.130.1		4
33.26	odd	10		99.2.a.b.1.1		1
33.29	even	10		1089.2.a.j.1.1		1
44.7	even	10		5808.2.a.t.1.1		1
44.15	odd	10		528.2.a.g.1.1		1
55.4	even	10		825.2.a.a.1.1		1
55.29	odd	10		9075.2.a.q.1.1		1
55.37	odd	20		825.2.c.a.199.2		2
55.48	odd	20		825.2.c.a.199.1		2
77.48	odd	10		1617.2.a.j.1.1		1
88.37	even	10		2112.2.a.bb.1.1		1
88.59	odd	10		2112.2.a.j.1.1		1
99.4	even	15		891.2.e.e.298.1		2
99.59	odd	30		891.2.e.g.298.1		2
99.70	even	15		891.2.e.e.595.1		2
99.92	odd	30		891.2.e.g.595.1		2
132.59	even	10		1584.2.a.o.1.1		1
143.103	even	10		5577.2.a.a.1.1		1
165.59	odd	10		2475.2.a.g.1.1		1
165.92	even	20		2475.2.c.d.199.1		2
165.158	even	20		2475.2.c.d.199.2		2
187.169	even	10		9537.2.a.m.1.1		1
231.125	even	10		4851.2.a.b.1.1		1
264.59	even	10		6336.2.a.n.1.1		1
264.125	odd	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	11.4	even	5
99.2.a.b.1.1		1	33.26	odd	10
363.2.a.b.1.1		1	11.7	odd	10
363.2.e.e.124.1		4	11.9	even	5	inner
363.2.e.e.130.1		4	1.1	even	1	trivial
363.2.e.e.148.1		4	11.5	even	5	inner
363.2.e.e.202.1		4	11.3	even	5	inner
363.2.e.g.124.1		4	11.2	odd	10
363.2.e.g.130.1		4	11.10	odd	2
363.2.e.g.148.1		4	11.6	odd	10
363.2.e.g.202.1		4	11.8	odd	10
528.2.a.g.1.1		1	44.15	odd	10
825.2.a.a.1.1		1	55.4	even	10
825.2.c.a.199.1		2	55.48	odd	20
825.2.c.a.199.2		2	55.37	odd	20
891.2.e.e.298.1		2	99.4	even	15
891.2.e.e.595.1		2	99.70	even	15
891.2.e.g.298.1		2	99.59	odd	30
891.2.e.g.595.1		2	99.92	odd	30
1089.2.a.j.1.1		1	33.29	even	10
1584.2.a.o.1.1		1	132.59	even	10
1617.2.a.j.1.1		1	77.48	odd	10
2112.2.a.j.1.1		1	88.59	odd	10
2112.2.a.bb.1.1		1	88.37	even	10
2475.2.a.g.1.1		1	165.59	odd	10
2475.2.c.d.199.1		2	165.92	even	20
2475.2.c.d.199.2		2	165.158	even	20
4851.2.a.b.1.1		1	231.125	even	10
5577.2.a.a.1.1		1	143.103	even	10
5808.2.a.t.1.1		1	44.7	even	10
6336.2.a.n.1.1		1	264.59	even	10
6336.2.a.x.1.1		1	264.125	odd	10
9075.2.a.q.1.1		1	55.29	odd	10
9537.2.a.m.1.1		1	187.169	even	10