Properties

Label 81.9.d.a
Level $81$
Weight $9$
Character orbit 81.d
Analytic conductor $32.998$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [81,9,Mod(26,81)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(81, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.26");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 81.d (of order \(6\), 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: \(32.9976674150\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - 256 \zeta_{6} q^{4} + (239 \zeta_{6} - 239) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 256 \zeta_{6} q^{4} + (239 \zeta_{6} - 239) q^{7} - 56447 \zeta_{6} q^{13} + (65536 \zeta_{6} - 65536) q^{16} + 100559 q^{19} + (390625 \zeta_{6} - 390625) q^{25} + 61184 q^{28} + 1809406 \zeta_{6} q^{31} - 3468481 q^{37} + (3492194 \zeta_{6} - 3492194) q^{43} + 5707680 \zeta_{6} q^{49} + (14450432 \zeta_{6} - 14450432) q^{52} + (24133919 \zeta_{6} - 24133919) q^{61} + 16777216 q^{64} + 31874833 \zeta_{6} q^{67} - 55236481 q^{73} - 25743104 \zeta_{6} q^{76} + ( - 56007121 \zeta_{6} + 56007121) q^{79} + 13490833 q^{91} + ( - 94775521 \zeta_{6} + 94775521) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 256 q^{4} - 239 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 256 q^{4} - 239 q^{7} - 56447 q^{13} - 65536 q^{16} + 201118 q^{19} - 390625 q^{25} + 122368 q^{28} + 1809406 q^{31} - 6936962 q^{37} - 3492194 q^{43} + 5707680 q^{49} - 14450432 q^{52} - 24133919 q^{61} + 33554432 q^{64} + 31874833 q^{67} - 110472962 q^{73} - 25743104 q^{76} + 56007121 q^{79} + 26981666 q^{91} + 94775521 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 −128.000 221.703i 0 0 −119.500 + 206.980i 0 0 0
53.1 0 0 −128.000 + 221.703i 0 0 −119.500 206.980i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.9.d.a 2
3.b odd 2 1 CM 81.9.d.a 2
9.c even 3 1 27.9.b.a 1
9.c even 3 1 inner 81.9.d.a 2
9.d odd 6 1 27.9.b.a 1
9.d odd 6 1 inner 81.9.d.a 2
36.f odd 6 1 432.9.e.a 1
36.h even 6 1 432.9.e.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.9.b.a 1 9.c even 3 1
27.9.b.a 1 9.d odd 6 1
81.9.d.a 2 1.a even 1 1 trivial
81.9.d.a 2 3.b odd 2 1 CM
81.9.d.a 2 9.c even 3 1 inner
81.9.d.a 2 9.d odd 6 1 inner
432.9.e.a 1 36.f odd 6 1
432.9.e.a 1 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{9}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 239T + 57121 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 3186263809 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 100559)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 3273950072836 \) Copy content Toggle raw display
$37$ \( (T + 3468481)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 12195418933636 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 582446046298561 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 10\!\cdots\!89 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 55236481)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 31\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 89\!\cdots\!41 \) Copy content Toggle raw display
show more
show less