Newform orbit 363.2.e.m

• Label: 363.2.e.m
• Level: $363$
• Weight: $2$
• Character orbit: 363.e
• Analytic conductor: $2.899$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $8$

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:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	8.0.324000000.3
Defining polynomial:	\( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_{7} q^{2} - \beta_{6} q^{3} + \beta_{4} q^{4} + (3 \beta_{6} + 3 \beta_{4} + 3 \beta_{2} + 3) q^{5} - \beta_{3} q^{6} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{7} - \beta_1 q^{8} + \beta_{2} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{3} - 2 q^{4} + 6 q^{5} - 2 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 3 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 3 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{4} ) / 9 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{5} ) / 9 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{6} ) / 27 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{7} ) / 27 \)

Character values

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

\(n\)	\(122\)	\(244\)
\(\chi(n)\)	\(1\)	\(\beta_{4}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

124.1

0.535233	+	1.64728i
−0.535233	−	1.64728i
1.40126	−	1.01807i
−1.40126	+	1.01807i
1.40126	+	1.01807i
−1.40126	−	1.01807i
0.535233	−	1.64728i
−0.535233	+	1.64728i

−1.40126

+

1.01807i

−0.309017

−

0.951057i

0.309017

−

0.951057i

2.42705

+

1.76336i

1.40126

+

1.01807i

−1.07047

+

3.29456i

−0.535233

−

1.64728i

−0.809017

+

0.587785i

−5.19615

124.2

1.40126

−

1.01807i

−0.309017

−

0.951057i

0.309017

−

0.951057i

2.42705

+

1.76336i

−1.40126

−

1.01807i

1.07047

−

3.29456i

0.535233

+

1.64728i

−0.809017

+

0.587785i

5.19615

130.1

−0.535233

+

1.64728i

0.809017

−

0.587785i

−0.809017

−

0.587785i

−0.927051

−

2.85317i

0.535233

+

1.64728i

−2.80252

−

2.03615i

−1.40126

+

1.01807i

0.309017

−

0.951057i

5.19615

130.2

0.535233

−

1.64728i

0.809017

−

0.587785i

−0.809017

−

0.587785i

−0.927051

−

2.85317i

−0.535233

−

1.64728i

2.80252

+

2.03615i

1.40126

−

1.01807i

0.309017

−

0.951057i

−5.19615

148.1

−0.535233

−

1.64728i

0.809017

+

0.587785i

−0.809017

+

0.587785i

−0.927051

+

2.85317i

0.535233

−

1.64728i

−2.80252

+

2.03615i

−1.40126

−

1.01807i

0.309017

+

0.951057i

5.19615

148.2

0.535233

+

1.64728i

0.809017

+

0.587785i

−0.809017

+

0.587785i

−0.927051

+

2.85317i

−0.535233

+

1.64728i

2.80252

−

2.03615i

1.40126

+

1.01807i

0.309017

+

0.951057i

−5.19615

202.1

−1.40126

−

1.01807i

−0.309017

+

0.951057i

0.309017

+

0.951057i

2.42705

−

1.76336i

1.40126

−

1.01807i

−1.07047

−

3.29456i

−0.535233

+

1.64728i

−0.809017

−

0.587785i

−5.19615

202.2

1.40126

+

1.01807i

−0.309017

+

0.951057i

0.309017

+

0.951057i

2.42705

−

1.76336i

−1.40126

+

1.01807i

1.07047

+

3.29456i

0.535233

−

1.64728i

−0.809017

−

0.587785i

5.19615

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	inner
11.c	even	5	3	inner
11.d	odd	10	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	363.2.e.m		8
11.b	odd	2	1	inner	363.2.e.m		8
11.c	even	5	1		363.2.a.f	✓	2
11.c	even	5	3	inner	363.2.e.m		8
11.d	odd	10	1		363.2.a.f	✓	2
11.d	odd	10	3	inner	363.2.e.m		8
33.f	even	10	1		1089.2.a.o		2
33.h	odd	10	1		1089.2.a.o		2
44.g	even	10	1		5808.2.a.ca		2
44.h	odd	10	1		5808.2.a.ca		2
55.h	odd	10	1		9075.2.a.bo		2
55.j	even	10	1		9075.2.a.bo		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
363.2.a.f	✓	2	11.c	even	5	1
363.2.a.f	✓	2	11.d	odd	10	1
363.2.e.m		8	1.a	even	1	1	trivial
363.2.e.m		8	11.b	odd	2	1	inner
363.2.e.m		8	11.c	even	5	3	inner
363.2.e.m		8	11.d	odd	10	3	inner
1089.2.a.o		2	33.f	even	10	1
1089.2.a.o		2	33.h	odd	10	1
5808.2.a.ca		2	44.g	even	10	1
5808.2.a.ca		2	44.h	odd	10	1
9075.2.a.bo		2	55.h	odd	10	1
9075.2.a.bo		2	55.j	even	10	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 3T_{2}^{6} + 9T_{2}^{4} + 27T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^8 + 3*T^6 + 9*T^4 + 27*T^2 + 81
$3$	\( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \)
$5$	\( (T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81)^{2} \)
$7$	T^8 + 12*T^6 + 144*T^4 + 1728*T^2 + 20736
$11$	\( T^{8} \)