Newform orbit 135.2.e.b

• Label: 135.2.e.b
• Level: $135$
• Weight: $2$
• Character orbit: 135.e
• Analytic conductor: $1.078$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• 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:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	6.0.954288.1
Defining polynomial:	\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 3 \)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta_{5} - \beta_{4}) q^{2} + (2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{4} + ( - \beta_{3} + 1) q^{5} + (\beta_{5} - \beta_{4} - 2 \beta_{3}) q^{7} + (\beta_{4} - \beta_{2} - 1) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} - 5 q^{4} + 3 q^{5} - 5 q^{7} - 6 q^{8}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{5} + 4\nu^{4} - \nu^{3} + 9\nu^{2} - 21\nu - 9 ) / 27 \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{5} + 4\nu^{4} - \nu^{3} - 18\nu^{2} + 33\nu - 9 ) / 27 \)
\(\beta_{3}\)	\(=\)	\( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27 \)
\(\beta_{4}\)	\(=\)	\( ( -2\nu^{5} - \nu^{4} + 7\nu^{3} + 9\nu^{2} + 12\nu + 9 ) / 27 \)
\(\beta_{5}\)	\(=\)	\( ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} + 3\nu - 72 ) / 27 \)

Character values

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

\(n\)	\(56\)	\(82\)
\(\chi(n)\)	\(-1 + \beta_{3}\)	\(1\)

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

1.71903	+	0.211943i
−1.62241	+	0.606458i
0.403374	−	1.68443i
1.71903	−	0.211943i
−1.62241	−	0.606458i
0.403374	+	1.68443i

−1.04307

+

1.80664i

0

−1.17597

−

2.03684i

0.500000

+

0.866025i

0

−2.04307

+

3.53869i

0.734191

0

−2.08613

46.2

0.285997

−

0.495361i

0

0.836412

+

1.44871i

0.500000

+

0.866025i

0

−0.714003

+

1.23669i

2.10083

0

0.571993

46.3

1.25707

−

2.17731i

0

−2.16044

−

3.74200i

0.500000

+

0.866025i

0

0.257068

−

0.445256i

−5.83502

0

2.51414

91.1

−1.04307

−

1.80664i

0

−1.17597

+

2.03684i

0.500000

−

0.866025i

0

−2.04307

−

3.53869i

0.734191

0

−2.08613

91.2

0.285997

+

0.495361i

0

0.836412

−

1.44871i

0.500000

−

0.866025i

0

−0.714003

−

1.23669i

2.10083

0

0.571993

91.3

1.25707

+

2.17731i

0

−2.16044

+

3.74200i

0.500000

−

0.866025i

0

0.257068

+

0.445256i

−5.83502

0

2.51414

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.b		6
3.b	odd	2	1		45.2.e.b	✓	6
4.b	odd	2	1		2160.2.q.k		6
5.b	even	2	1		675.2.e.b		6
5.c	odd	4	2		675.2.k.b		12
9.c	even	3	1	inner	135.2.e.b		6
9.c	even	3	1		405.2.a.i		3
9.d	odd	6	1		45.2.e.b	✓	6
9.d	odd	6	1		405.2.a.j		3
12.b	even	2	1		720.2.q.i		6
15.d	odd	2	1		225.2.e.b		6
15.e	even	4	2		225.2.k.b		12
36.f	odd	6	1		2160.2.q.k		6
36.f	odd	6	1		6480.2.a.bs		3
36.h	even	6	1		720.2.q.i		6
36.h	even	6	1		6480.2.a.bv		3
45.h	odd	6	1		225.2.e.b		6
45.h	odd	6	1		2025.2.a.n		3
45.j	even	6	1		675.2.e.b		6
45.j	even	6	1		2025.2.a.o		3
45.k	odd	12	2		675.2.k.b		12
45.k	odd	12	2		2025.2.b.m		6
45.l	even	12	2		225.2.k.b		12
45.l	even	12	2		2025.2.b.l		6

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} + 6T_{2}^{4} - T_{2}^{3} + 28T_{2}^{2} - 15T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(135, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^6 - T^5 + 6*T^4 - T^3 + 28*T^2 - 15*T + 9
$3$	\( T^{6} \)
$5$	\( (T^{2} - T + 1)^{3} \)
$7$	T^6 + 5*T^5 + 22*T^4 + 21*T^3 + 24*T^2 - 9*T + 9
$11$	T^6 + 2*T^5 + 12*T^4 + 8*T^3 + 88*T^2 + 96*T + 144