Properties

Label 45.18.a.f
Level $45$
Weight $18$
Character orbit 45.a
Self dual yes
Analytic conductor $82.450$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,18,Mod(1,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 45.a (trivial)

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: \(82.4499393051\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 481686x^{2} + 26523040x + 36023696000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_1 + 8) q^{2} + (\beta_{2} - 65 \beta_1 + 109856) q^{4} - 390625 q^{5} + (23 \beta_{3} + 55 \beta_{2} + \cdots + 4394104) q^{7} + (74 \beta_{3} - 165 \beta_{2} + \cdots - 15929984) q^{8} + ( - 390625 \beta_1 - 3125000) q^{10}+ \cdots + (73134402560 \beta_{3} + \cdots + 13\!\cdots\!88) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 33 q^{2} + 439357 q^{4} - 1562500 q^{5} + 17583104 q^{7} - 63651621 q^{8} - 12890625 q^{10} + 575495184 q^{11} - 5049645832 q^{13} + 6699316032 q^{14} + 7512683905 q^{16} - 45757603848 q^{17} + 198913764368 q^{19}+ \cdots + 55\!\cdots\!13 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 481686x^{2} + 26523040x + 36023696000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 81\nu - 240864 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 189\nu^{2} - 316646\nu - 25909216 ) / 74 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 81\beta _1 + 240864 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 74\beta_{3} - 189\beta_{2} + 331955\beta _1 - 19614080 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−662.567
−265.574
359.218
569.922
−654.567 0 297386. −390625. 0 359681. −1.08863e8 0 2.55690e8
1.2 −257.574 0 −64727.8 −390625. 0 8.43659e6 5.04329e7 0 1.00615e8
1.3 367.218 0 3777.06 −390625. 0 −1.91248e7 −4.67450e7 0 −1.43445e8
1.4 577.922 0 202922. −390625. 0 2.79117e7 4.15238e7 0 −2.25751e8
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.18.a.f 4
3.b odd 2 1 15.18.a.d 4
15.d odd 2 1 75.18.a.f 4
15.e even 4 2 75.18.b.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.18.a.d 4 3.b odd 2 1
45.18.a.f 4 1.a even 1 1 trivial
75.18.a.f 4 15.d odd 2 1
75.18.b.f 8 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 33T_{2}^{3} - 481278T_{2}^{2} + 34227776T_{2} + 35780688384 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(45))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + \cdots + 35780688384 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T + 390625)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 81\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots - 60\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 79\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 45\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 43\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 10\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 72\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 54\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 25\!\cdots\!04 \) Copy content Toggle raw display
show more
show less