↑

Properties

Label 135.2.f.a
Level $135$
Weight $2$
Character orbit 135.f
Analytic conductor $1.078$
Analytic rank $0$
Dimension $8$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [135,2,Mod(53,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("135.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 135.f (of order \(4\), degree \(2\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.12745506816.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 23x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{4}) q^{4} + ( - \beta_{5} - \beta_{3}) q^{5} + (\beta_{6} + \beta_{4} + \beta_{2}) q^{7} + (\beta_{7} + \beta_{5}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{4}) q^{4} + ( - \beta_{5} - \beta_{3}) q^{5} + (\beta_{6} + \beta_{4} + \beta_{2}) q^{7} + (\beta_{7} + \beta_{5}) q^{8} + ( - \beta_{4} - \beta_{2} + 2) q^{10} + ( - 2 \beta_{7} - \beta_{5} + \cdots - \beta_1) q^{11}+ \cdots + ( - 2 \beta_{7} - 9 \beta_{5}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{7} + 20 q^{10} - 4 q^{13} - 4 q^{16} + 4 q^{22} - 16 q^{25} - 44 q^{28} - 8 q^{31} - 40 q^{37} - 12 q^{40} + 44 q^{43} + 4 q^{46} + 44 q^{52} + 28 q^{55} + 48 q^{58} - 8 q^{61} + 32 q^{67} - 48 q^{70} - 28 q^{73} - 12 q^{76} + 4 q^{82} - 28 q^{85} + 36 q^{88} + 88 q^{91} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 23x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 9 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 24\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 24\nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 24\nu^{3} ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{6} - 67\nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{7} - 23\nu^{3} \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 5\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{2} - 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{3} - 24\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -24\beta_{6} - 67\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -24\beta_{7} - 115\beta_{5} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(56\) \(82\)
\(\chi(n)\) \(-1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−1.54779 − 1.54779i
−0.323042 − 0.323042i
0.323042 + 0.323042i
1.54779 + 1.54779i
−1.54779 + 1.54779i
−0.323042 + 0.323042i
0.323042 − 0.323042i
1.54779 − 1.54779i
−1.54779 − 1.54779i 0 2.79129i −1.22474 + 1.87083i 0 −2.79129 + 2.79129i 1.22474 − 1.22474i 0 4.79129 − 1.00000i
53.2 −0.323042 − 0.323042i 0 − 1.79129i 1.22474 + 1.87083i 0 1.79129 − 1.79129i −1.22474 + 1.22474i 0 0.208712 − 1.00000i
53.3 0.323042 + 0.323042i 0 − 1.79129i −1.22474 − 1.87083i 0 1.79129 − 1.79129i 1.22474 − 1.22474i 0 0.208712 − 1.00000i
53.4 1.54779 + 1.54779i 0 2.79129i 1.22474 − 1.87083i 0 −2.79129 + 2.79129i −1.22474 + 1.22474i 0 4.79129 − 1.00000i
107.1 −1.54779 + 1.54779i 0 − 2.79129i −1.22474 − 1.87083i 0 −2.79129 − 2.79129i 1.22474 + 1.22474i 0 4.79129 + 1.00000i
107.2 −0.323042 + 0.323042i 0 1.79129i 1.22474 − 1.87083i 0 1.79129 + 1.79129i −1.22474 − 1.22474i 0 0.208712 + 1.00000i
107.3 0.323042 − 0.323042i 0 1.79129i −1.22474 + 1.87083i 0 1.79129 + 1.79129i 1.22474 + 1.22474i 0 0.208712 + 1.00000i
107.4 1.54779 − 1.54779i 0 − 2.79129i 1.22474 + 1.87083i 0 −2.79129 − 2.79129i −1.22474 − 1.22474i 0 4.79129 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.2.f.a ✓ 8
3.b odd 2 1 inner 135.2.f.a ✓ 8
4.b odd 2 1 2160.2.w.d 8
5.b even 2 1 675.2.f.i 8
5.c odd 4 1 inner 135.2.f.a ✓ 8
5.c odd 4 1 675.2.f.i 8
9.c even 3 2 405.2.m.c 16
9.d odd 6 2 405.2.m.c 16
12.b even 2 1 2160.2.w.d 8
15.d odd 2 1 675.2.f.i 8
15.e even 4 1 inner 135.2.f.a ✓ 8
15.e even 4 1 675.2.f.i 8
20.e even 4 1 2160.2.w.d 8
45.k odd 12 2 405.2.m.c 16
45.l even 12 2 405.2.m.c 16
60.l odd 4 1 2160.2.w.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.f.a ✓ 8 1.a even 1 1 trivial
135.2.f.a ✓ 8 3.b odd 2 1 inner
135.2.f.a ✓ 8 5.c odd 4 1 inner
135.2.f.a ✓ 8 15.e even 4 1 inner
405.2.m.c 16 9.c even 3 2
405.2.m.c 16 9.d odd 6 2
405.2.m.c 16 45.k odd 12 2
405.2.m.c 16 45.l even 12 2
675.2.f.i 8 5.b even 2 1
675.2.f.i 8 5.c odd 4 1
675.2.f.i 8 15.d odd 2 1
675.2.f.i 8 15.e even 4 1
2160.2.w.d 8 4.b odd 2 1
2160.2.w.d 8 12.b even 2 1
2160.2.w.d 8 20.e even 4 1
2160.2.w.d 8 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 23T_{2}^{4} + 1 \) acting on \(S_{2}^{\mathrm{new}}(135, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 23T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 4 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 34 T^{2} + 100)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 49)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 1394T^{4} + 625 \) Copy content Toggle raw display
$29$ \( (T^{4} - 66 T^{2} + 900)^{2} \) Copy content Toggle raw display
$31$ \( (T + 1)^{8} \) Copy content Toggle raw display
$37$ \( (T^{2} + 10 T + 50)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 34 T^{2} + 100)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 22 T^{3} + \cdots + 2500)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 368T^{4} + 256 \) Copy content Toggle raw display
$53$ \( T^{8} + 12098T^{4} + 1 \) Copy content Toggle raw display
$59$ \( (T^{4} - 66 T^{2} + 900)^{2} \) Copy content Toggle raw display
$61$ \( (T + 1)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 16 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 34 T^{2} + 100)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 14 T^{3} + \cdots + 4900)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 114 T^{2} + 225)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 29138 T^{4} + 141158161 \) Copy content Toggle raw display
$89$ \( (T^{4} - 306 T^{2} + 8100)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 4 T^{3} + \cdots + 1600)^{2} \) Copy content Toggle raw display
show more
show less