Properties

Label 81.2.c.a
Level $81$
Weight $2$
Character orbit 81.c
Analytic conductor $0.647$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [81,2,Mod(28,81)]
 
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([2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.28");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 81.c (of order \(3\), 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: \(0.646788256372\)
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: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{4} + ( - \zeta_{6} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{4} + ( - \zeta_{6} + 1) q^{7} - 5 \zeta_{6} q^{13} + (4 \zeta_{6} - 4) q^{16} - 7 q^{19} + ( - 5 \zeta_{6} + 5) q^{25} + 2 q^{28} + 4 \zeta_{6} q^{31} + 11 q^{37} + (8 \zeta_{6} - 8) q^{43} + 6 \zeta_{6} q^{49} + ( - 10 \zeta_{6} + 10) q^{52} + ( - \zeta_{6} + 1) q^{61} - 8 q^{64} - 5 \zeta_{6} q^{67} - 7 q^{73} - 14 \zeta_{6} q^{76} + (17 \zeta_{6} - 17) q^{79} - 5 q^{91} + ( - 19 \zeta_{6} + 19) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + q^{7} - 5 q^{13} - 4 q^{16} - 14 q^{19} + 5 q^{25} + 4 q^{28} + 4 q^{31} + 22 q^{37} - 8 q^{43} + 6 q^{49} + 10 q^{52} + q^{61} - 16 q^{64} - 5 q^{67} - 14 q^{73} - 14 q^{76} - 17 q^{79} - 10 q^{91} + 19 q^{97}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 1.00000 1.73205i 0 0 0.500000 + 0.866025i 0 0 0
55.1 0 0 1.00000 + 1.73205i 0 0 0.500000 0.866025i 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 81.2.c.a 2
3.b odd 2 1 CM 81.2.c.a 2
4.b odd 2 1 1296.2.i.i 2
9.c even 3 1 27.2.a.a 1
9.c even 3 1 inner 81.2.c.a 2
9.d odd 6 1 27.2.a.a 1
9.d odd 6 1 inner 81.2.c.a 2
12.b even 2 1 1296.2.i.i 2
27.e even 9 6 729.2.e.f 6
27.f odd 18 6 729.2.e.f 6
36.f odd 6 1 432.2.a.e 1
36.f odd 6 1 1296.2.i.i 2
36.h even 6 1 432.2.a.e 1
36.h even 6 1 1296.2.i.i 2
45.h odd 6 1 675.2.a.e 1
45.j even 6 1 675.2.a.e 1
45.k odd 12 2 675.2.b.f 2
45.l even 12 2 675.2.b.f 2
63.l odd 6 1 1323.2.a.i 1
63.o even 6 1 1323.2.a.i 1
72.j odd 6 1 1728.2.a.n 1
72.l even 6 1 1728.2.a.o 1
72.n even 6 1 1728.2.a.n 1
72.p odd 6 1 1728.2.a.o 1
99.g even 6 1 3267.2.a.f 1
99.h odd 6 1 3267.2.a.f 1
117.n odd 6 1 4563.2.a.e 1
117.t even 6 1 4563.2.a.e 1
153.h even 6 1 7803.2.a.k 1
153.i odd 6 1 7803.2.a.k 1
171.l even 6 1 9747.2.a.f 1
171.o odd 6 1 9747.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.a.a 1 9.c even 3 1
27.2.a.a 1 9.d odd 6 1
81.2.c.a 2 1.a even 1 1 trivial
81.2.c.a 2 3.b odd 2 1 CM
81.2.c.a 2 9.c even 3 1 inner
81.2.c.a 2 9.d odd 6 1 inner
432.2.a.e 1 36.f odd 6 1
432.2.a.e 1 36.h even 6 1
675.2.a.e 1 45.h odd 6 1
675.2.a.e 1 45.j even 6 1
675.2.b.f 2 45.k odd 12 2
675.2.b.f 2 45.l even 12 2
729.2.e.f 6 27.e even 9 6
729.2.e.f 6 27.f odd 18 6
1296.2.i.i 2 4.b odd 2 1
1296.2.i.i 2 12.b even 2 1
1296.2.i.i 2 36.f odd 6 1
1296.2.i.i 2 36.h even 6 1
1323.2.a.i 1 63.l odd 6 1
1323.2.a.i 1 63.o even 6 1
1728.2.a.n 1 72.j odd 6 1
1728.2.a.n 1 72.n even 6 1
1728.2.a.o 1 72.l even 6 1
1728.2.a.o 1 72.p odd 6 1
3267.2.a.f 1 99.g even 6 1
3267.2.a.f 1 99.h odd 6 1
4563.2.a.e 1 117.n odd 6 1
4563.2.a.e 1 117.t even 6 1
7803.2.a.k 1 153.h even 6 1
7803.2.a.k 1 153.i odd 6 1
9747.2.a.f 1 171.l even 6 1
9747.2.a.f 1 171.o odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(81, [\chi])\). 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
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( (T - 11)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 64 \) 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
$61$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 7)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 19T + 361 \) Copy content Toggle raw display
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