Newform orbit 57.2.e.a

• Label: 57.2.e.a
• Level: $57$
• Weight: $2$
• Character orbit: 57.e
• Analytic conductor: $0.455$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	57.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.455147291521\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} + \zeta_{6} q^{4} + \zeta_{6} q^{6} + q^{7} - 3 q^{8} - \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + q^{3} + q^{4} + q^{6} + 2 q^{7} - 6 q^{8} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\)	\(20\)	\(40\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

7.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−0.500000

−

0.866025i

0.500000

+

0.866025i

0.500000

−

0.866025i

0

0.500000

−

0.866025i

1.00000

−3.00000

−0.500000

+

0.866025i

0

49.1

−0.500000

+

0.866025i

0.500000

−

0.866025i

0.500000

+

0.866025i

0

0.500000

+

0.866025i

1.00000

−3.00000

−0.500000

−

0.866025i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	57.2.e.a	✓	2
3.b	odd	2	1		171.2.f.a		2
4.b	odd	2	1		912.2.q.a		2
12.b	even	2	1		2736.2.s.j		2
19.c	even	3	1	inner	57.2.e.a	✓	2
19.c	even	3	1		1083.2.a.c		1
19.d	odd	6	1		1083.2.a.b		1
57.f	even	6	1		3249.2.a.f		1
57.h	odd	6	1		171.2.f.a		2
57.h	odd	6	1		3249.2.a.c		1
76.g	odd	6	1		912.2.q.a		2
228.m	even	6	1		2736.2.s.j		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
57.2.e.a	✓	2	1.a	even	1	1	trivial
57.2.e.a	✓	2	19.c	even	3	1	inner
171.2.f.a		2	3.b	odd	2	1
171.2.f.a		2	57.h	odd	6	1
912.2.q.a		2	4.b	odd	2	1
912.2.q.a		2	76.g	odd	6	1
1083.2.a.b		1	19.d	odd	6	1
1083.2.a.c		1	19.c	even	3	1
2736.2.s.j		2	12.b	even	2	1
2736.2.s.j		2	228.m	even	6	1
3249.2.a.c		1	57.h	odd	6	1
3249.2.a.f		1	57.f	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(57, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T + 1 \)
$3$	\( T^{2} - T + 1 \)
$5$	\( T^{2} \)
$7$	\( (T - 1)^{2} \)
$11$	\( (T + 2)^{2} \)