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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [135,2,Mod(46,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("135.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 135.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.07798042729\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} - \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 3) q^{7} + 3 q^{8} - q^{10} + (2 \zeta_{6} - 2) q^{11} + 2 \zeta_{6} q^{13} - 3 \zeta_{6} q^{14} + ( - \zeta_{6} + 1) q^{16} + \cdots - 2 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{4} - q^{5} + 3 q^{7} + 6 q^{8} - 2 q^{10} - 2 q^{11} + 2 q^{13} - 3 q^{14} + q^{16} - 8 q^{17} - 16 q^{19} + q^{20} + 2 q^{22} + 3 q^{23} - q^{25} + 4 q^{26} + 6 q^{28} - q^{29}+ \cdots - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(135, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$17$ \( (T + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$47$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$71$ \( (T + 2)^{2} \) Copy content Toggle raw display
$73$ \( (T - 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$83$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$89$ \( (T - 15)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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