Newform orbit 135.2.e.a

• Label: 135.2.e.a
• Level: $135$
• Weight: $2$
• Character orbit: 135.e
• Analytic conductor: $1.078$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	135.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.07798042729\)
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:	no (minimal twist has level 45)
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.

\(f(q)\)	\(=\)	\( q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} - \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 3) q^{7} + 3 q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + q^{4} - q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\)	\(56\)	\(82\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

46.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0.500000

−

0.866025i

0

0.500000

+

0.866025i

−0.500000

−

0.866025i

0

1.50000

−

2.59808i

3.00000

0

−1.00000

91.1

0.500000

+

0.866025i

0

0.500000

−

0.866025i

−0.500000

+

0.866025i

0

1.50000

+

2.59808i

3.00000

0

−1.00000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	135.2.e.a		2
3.b	odd	2	1		45.2.e.a	✓	2
4.b	odd	2	1		2160.2.q.a		2
5.b	even	2	1		675.2.e.a		2
5.c	odd	4	2		675.2.k.a		4
9.c	even	3	1	inner	135.2.e.a		2
9.c	even	3	1		405.2.a.b		1
9.d	odd	6	1		45.2.e.a	✓	2
9.d	odd	6	1		405.2.a.e		1
12.b	even	2	1		720.2.q.d		2
15.d	odd	2	1		225.2.e.a		2
15.e	even	4	2		225.2.k.a		4
36.f	odd	6	1		2160.2.q.a		2
36.f	odd	6	1		6480.2.a.x		1
36.h	even	6	1		720.2.q.d		2
36.h	even	6	1		6480.2.a.k		1
45.h	odd	6	1		225.2.e.a		2
45.h	odd	6	1		2025.2.a.b		1
45.j	even	6	1		675.2.e.a		2
45.j	even	6	1		2025.2.a.e		1
45.k	odd	12	2		675.2.k.a		4
45.k	odd	12	2		2025.2.b.d		2
45.l	even	12	2		225.2.k.a		4
45.l	even	12	2		2025.2.b.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
45.2.e.a	✓	2	3.b	odd	2	1
45.2.e.a	✓	2	9.d	odd	6	1
135.2.e.a		2	1.a	even	1	1	trivial
135.2.e.a		2	9.c	even	3	1	inner
225.2.e.a		2	15.d	odd	2	1
225.2.e.a		2	45.h	odd	6	1
225.2.k.a		4	15.e	even	4	2
225.2.k.a		4	45.l	even	12	2
405.2.a.b		1	9.c	even	3	1
405.2.a.e		1	9.d	odd	6	1
675.2.e.a		2	5.b	even	2	1
675.2.e.a		2	45.j	even	6	1
675.2.k.a		4	5.c	odd	4	2
675.2.k.a		4	45.k	odd	12	2
720.2.q.d		2	12.b	even	2	1
720.2.q.d		2	36.h	even	6	1
2025.2.a.b		1	45.h	odd	6	1
2025.2.a.e		1	45.j	even	6	1
2025.2.b.c		2	45.l	even	12	2
2025.2.b.d		2	45.k	odd	12	2
2160.2.q.a		2	4.b	odd	2	1
2160.2.q.a		2	36.f	odd	6	1
6480.2.a.k		1	36.h	even	6	1
6480.2.a.x		1	36.f	odd	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 \(S_{2}^{\mathrm{new}}(135, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T + 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + T + 1 \)
$7$	\( T^{2} - 3T + 9 \)
$11$	\( T^{2} + 2T + 4 \)