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Properties

Label 135.2.b.a
Level $135$
Weight $2$
Character orbit 135.b
Analytic conductor $1.078$
Analytic rank $0$
Dimension $4$
CM discriminant -15
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [135,2,Mod(109,135)]
 
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([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("135.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 135.b (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: no
Analytic conductor: \(1.07798042729\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{3} - 2) q^{4} + (\beta_{2} - \beta_1) q^{5} + ( - \beta_{2} + 3 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{3} - 2) q^{4} + (\beta_{2} - \beta_1) q^{5} + ( - \beta_{2} + 3 \beta_1) q^{8} + (\beta_{3} - 3) q^{10} + ( - \beta_{3} + 7) q^{16} + ( - 3 \beta_{2} - \beta_1) q^{17} + ( - 2 \beta_{3} + 3) q^{19} + (\beta_{2} + 4 \beta_1) q^{20} + (3 \beta_{2} - \beta_1) q^{23} - 5 q^{25} + ( - 2 \beta_{3} - 3) q^{31} + ( - \beta_{2} - 4 \beta_1) q^{32} + (\beta_{3} - 7) q^{34} + (2 \beta_{2} - 9 \beta_1) q^{38} + ( - 2 \beta_{3} + 11) q^{40} + (\beta_{3} - 1) q^{46} + ( - 4 \beta_{2} + 4 \beta_1) q^{47} + 7 q^{49} + 5 \beta_1 q^{50} + (3 \beta_{2} + 5 \beta_1) q^{53} + (4 \beta_{3} - 3) q^{61} + (2 \beta_{2} - 3 \beta_1) q^{62} + (2 \beta_{3} - 3) q^{64} + ( - 7 \beta_{2} + 8 \beta_1) q^{68} + (5 \beta_{3} - 28) q^{76} + ( - 2 \beta_{3} + 9) q^{79} + (4 \beta_{2} - 9 \beta_1) q^{80} + ( - 3 \beta_{2} + 5 \beta_1) q^{83} + (4 \beta_{3} + 3) q^{85} + (5 \beta_{2} + 2 \beta_1) q^{92} + ( - 4 \beta_{3} + 12) q^{94} + ( - 3 \beta_{2} - 7 \beta_1) q^{95} - 7 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{4} - 10 q^{10} + 26 q^{16} + 8 q^{19} - 20 q^{25} - 16 q^{31} - 26 q^{34} + 40 q^{40} - 2 q^{46} + 28 q^{49} - 4 q^{61} - 8 q^{64} - 102 q^{76} + 32 q^{79} + 20 q^{85} + 40 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} + 5 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{2} + 5\beta_1 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
− 1.61803i
0.618034i
− 0.618034i
1.61803i
− 2.61803i 0 −4.85410 − 2.23607i 0 0 7.47214i 0 −5.85410
109.2 − 0.381966i 0 1.85410 2.23607i 0 0 − 1.47214i 0 0.854102
109.3 0.381966i 0 1.85410 − 2.23607i 0 0 1.47214i 0 0.854102
109.4 2.61803i 0 −4.85410 2.23607i 0 0 − 7.47214i 0 −5.85410
\(n\): 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 \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.2.b.a ✓ 4
3.b odd 2 1 inner 135.2.b.a ✓ 4
4.b odd 2 1 2160.2.f.j 4
5.b even 2 1 inner 135.2.b.a ✓ 4
5.c odd 4 1 675.2.a.j 2
5.c odd 4 1 675.2.a.q 2
9.c even 3 2 405.2.j.h 8
9.d odd 6 2 405.2.j.h 8
12.b even 2 1 2160.2.f.j 4
15.d odd 2 1 CM 135.2.b.a ✓ 4
15.e even 4 1 675.2.a.j 2
15.e even 4 1 675.2.a.q 2
20.d odd 2 1 2160.2.f.j 4
45.h odd 6 2 405.2.j.h 8
45.j even 6 2 405.2.j.h 8
60.h even 2 1 2160.2.f.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.b.a ✓ 4 1.a even 1 1 trivial
135.2.b.a ✓ 4 3.b odd 2 1 inner
135.2.b.a ✓ 4 5.b even 2 1 inner
135.2.b.a ✓ 4 15.d odd 2 1 CM
405.2.j.h 8 9.c even 3 2
405.2.j.h 8 9.d odd 6 2
405.2.j.h 8 45.h odd 6 2
405.2.j.h 8 45.j even 6 2
675.2.a.j 2 5.c odd 4 1
675.2.a.j 2 15.e even 4 1
675.2.a.q 2 5.c odd 4 1
675.2.a.q 2 15.e even 4 1
2160.2.f.j 4 4.b odd 2 1
2160.2.f.j 4 12.b even 2 1
2160.2.f.j 4 20.d odd 2 1
2160.2.f.j 4 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 7T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(135, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 7T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 82T^{2} + 961 \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T - 41)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 58T^{2} + 121 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T - 29)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 298 T^{2} + 19321 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T - 179)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 16 T + 19)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 178T^{2} + 5041 \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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