Properties

Label 135.1.d.a
Level $135$
Weight $1$
Character orbit 135.d
Self dual yes
Analytic conductor $0.067$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -15
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [135,1,Mod(134,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("135.134");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 135.d (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(0.0673737767055\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.135.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.135.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{5} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{5} + q^{8} - q^{10} - q^{16} - q^{17} - q^{19} - q^{23} + q^{25} - q^{31} + q^{34} + q^{38} + q^{40} + q^{46} + 2 q^{47} + q^{49} - q^{50} - q^{53} - q^{61} + q^{62} + q^{64} - q^{79} - q^{80} - q^{83} - q^{85} - 2 q^{94} - q^{95} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(56\) \(82\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
134.1
0
−1.00000 0 0 1.00000 0 0 1.00000 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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.1.d.a 1
3.b odd 2 1 135.1.d.b yes 1
4.b odd 2 1 2160.1.c.b 1
5.b even 2 1 135.1.d.b yes 1
5.c odd 4 2 675.1.c.c 2
9.c even 3 2 405.1.h.b 2
9.d odd 6 2 405.1.h.a 2
12.b even 2 1 2160.1.c.a 1
15.d odd 2 1 CM 135.1.d.a 1
15.e even 4 2 675.1.c.c 2
20.d odd 2 1 2160.1.c.a 1
27.e even 9 6 3645.1.n.d 6
27.f odd 18 6 3645.1.n.e 6
45.h odd 6 2 405.1.h.b 2
45.j even 6 2 405.1.h.a 2
45.k odd 12 4 2025.1.j.c 4
45.l even 12 4 2025.1.j.c 4
60.h even 2 1 2160.1.c.b 1
135.n odd 18 6 3645.1.n.d 6
135.p even 18 6 3645.1.n.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.1.d.a 1 1.a even 1 1 trivial
135.1.d.a 1 15.d odd 2 1 CM
135.1.d.b yes 1 3.b odd 2 1
135.1.d.b yes 1 5.b even 2 1
405.1.h.a 2 9.d odd 6 2
405.1.h.a 2 45.j even 6 2
405.1.h.b 2 9.c even 3 2
405.1.h.b 2 45.h odd 6 2
675.1.c.c 2 5.c odd 4 2
675.1.c.c 2 15.e even 4 2
2025.1.j.c 4 45.k odd 12 4
2025.1.j.c 4 45.l even 12 4
2160.1.c.a 1 12.b even 2 1
2160.1.c.a 1 20.d odd 2 1
2160.1.c.b 1 4.b odd 2 1
2160.1.c.b 1 60.h even 2 1
3645.1.n.d 6 27.e even 9 6
3645.1.n.d 6 135.n odd 18 6
3645.1.n.e 6 27.f odd 18 6
3645.1.n.e 6 135.p even 18 6

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(135, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 1 \) Copy content Toggle raw display
$19$ \( T + 1 \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 1 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 2 \) Copy content Toggle raw display
$53$ \( T + 1 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 1 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T + 1 \) Copy content Toggle raw display
$83$ \( T + 1 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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