Embedded newform 363.2.e.e.148.1

• Label: 363.2.e.e.148.1
• Level: $363$
• Weight: $2$
• Character: 363.148
• 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			148.1
Root			\(-0.309017 + 0.951057i\) of defining polynomial
Character	\(\chi\)	\(=\)	363.148
Dual form			363.2.e.e.130.1

$q$-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 \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(244\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{5}\right)\)

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

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.2.e.e.148.1		4
11.2	odd	10		363.2.e.g.130.1		4
11.3	even	5		33.2.a.a.1.1	✓	1
11.4	even	5	inner	363.2.e.e.124.1		4
11.5	even	5	inner	363.2.e.e.202.1		4
11.6	odd	10		363.2.e.g.202.1		4
11.7	odd	10		363.2.e.g.124.1		4
11.8	odd	10		363.2.a.b.1.1		1
11.9	even	5	inner	363.2.e.e.130.1		4
11.10	odd	2		363.2.e.g.148.1		4
33.8	even	10		1089.2.a.j.1.1		1
33.14	odd	10		99.2.a.b.1.1		1
44.3	odd	10		528.2.a.g.1.1		1
44.19	even	10		5808.2.a.t.1.1		1
55.3	odd	20		825.2.c.a.199.1		2
55.14	even	10		825.2.a.a.1.1		1
55.19	odd	10		9075.2.a.q.1.1		1
55.47	odd	20		825.2.c.a.199.2		2
77.69	odd	10		1617.2.a.j.1.1		1
88.3	odd	10		2112.2.a.j.1.1		1
88.69	even	10		2112.2.a.bb.1.1		1
99.14	odd	30		891.2.e.g.298.1		2
99.25	even	15		891.2.e.e.595.1		2
99.47	odd	30		891.2.e.g.595.1		2
99.58	even	15		891.2.e.e.298.1		2
132.47	even	10		1584.2.a.o.1.1		1
143.25	even	10		5577.2.a.a.1.1		1
165.14	odd	10		2475.2.a.g.1.1		1
165.47	even	20		2475.2.c.d.199.1		2
165.113	even	20		2475.2.c.d.199.2		2
187.135	even	10		9537.2.a.m.1.1		1
231.146	even	10		4851.2.a.b.1.1		1
264.179	even	10		6336.2.a.n.1.1		1
264.245	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	11.3	even	5
99.2.a.b.1.1		1	33.14	odd	10
363.2.a.b.1.1		1	11.8	odd	10
363.2.e.e.124.1		4	11.4	even	5	inner
363.2.e.e.130.1		4	11.9	even	5	inner
363.2.e.e.148.1		4	1.1	even	1	trivial
363.2.e.e.202.1		4	11.5	even	5	inner
363.2.e.g.124.1		4	11.7	odd	10
363.2.e.g.130.1		4	11.2	odd	10
363.2.e.g.148.1		4	11.10	odd	2
363.2.e.g.202.1		4	11.6	odd	10
528.2.a.g.1.1		1	44.3	odd	10
825.2.a.a.1.1		1	55.14	even	10
825.2.c.a.199.1		2	55.3	odd	20
825.2.c.a.199.2		2	55.47	odd	20
891.2.e.e.298.1		2	99.58	even	15
891.2.e.e.595.1		2	99.25	even	15
891.2.e.g.298.1		2	99.14	odd	30
891.2.e.g.595.1		2	99.47	odd	30
1089.2.a.j.1.1		1	33.8	even	10
1584.2.a.o.1.1		1	132.47	even	10
1617.2.a.j.1.1		1	77.69	odd	10
2112.2.a.j.1.1		1	88.3	odd	10
2112.2.a.bb.1.1		1	88.69	even	10
2475.2.a.g.1.1		1	165.14	odd	10
2475.2.c.d.199.1		2	165.47	even	20
2475.2.c.d.199.2		2	165.113	even	20
4851.2.a.b.1.1		1	231.146	even	10
5577.2.a.a.1.1		1	143.25	even	10
5808.2.a.t.1.1		1	44.19	even	10
6336.2.a.n.1.1		1	264.179	even	10
6336.2.a.x.1.1		1	264.245	odd	10
9075.2.a.q.1.1		1	55.19	odd	10
9537.2.a.m.1.1		1	187.135	even	10