Properties

Label 81.10.c.c
Level $81$
Weight $10$
Character orbit 81.c
Analytic conductor $41.718$
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] = [81,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.28");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 10 \)
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: \(41.7179027293\)
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
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.

\(f(q)\) \(=\) \( q + 512 \zeta_{6} q^{4} + ( - 12580 \zeta_{6} + 12580) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 512 \zeta_{6} q^{4} + ( - 12580 \zeta_{6} + 12580) q^{7} - 118370 \zeta_{6} q^{13} + (262144 \zeta_{6} - 262144) q^{16} - 976696 q^{19} + ( - 1953125 \zeta_{6} + 1953125) q^{25} + 6440960 q^{28} - 1691228 \zeta_{6} q^{31} - 15384490 q^{37} + ( - 16577080 \zeta_{6} + 16577080) q^{43} - 117902793 \zeta_{6} q^{49} + ( - 60605440 \zeta_{6} + 60605440) q^{52} + ( - 117903058 \zeta_{6} + 117903058) q^{61} - 134217728 q^{64} - 112542320 \zeta_{6} q^{67} + 296368310 q^{73} - 500068352 \zeta_{6} q^{76} + ( - 616732324 \zeta_{6} + 616732324) q^{79} - 1489094600 q^{91} + (1288928270 \zeta_{6} - 1288928270) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 512 q^{4} + 12580 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 512 q^{4} + 12580 q^{7} - 118370 q^{13} - 262144 q^{16} - 1953392 q^{19} + 1953125 q^{25} + 12881920 q^{28} - 1691228 q^{31} - 30768980 q^{37} + 16577080 q^{43} - 117902793 q^{49} + 60605440 q^{52} + 117903058 q^{61} - 268435456 q^{64} - 112542320 q^{67} + 592736620 q^{73} - 500068352 q^{76} + 616732324 q^{79} - 2978189200 q^{91} - 1288928270 q^{97}+O(q^{100}) \) 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 0 256.000 443.405i 0 0 6290.00 + 10894.6i 0 0 0
55.1 0 0 256.000 + 443.405i 0 0 6290.00 10894.6i 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 81.10.c.c 2
3.b odd 2 1 CM 81.10.c.c 2
9.c even 3 1 9.10.a.b 1
9.c even 3 1 inner 81.10.c.c 2
9.d odd 6 1 9.10.a.b 1
9.d odd 6 1 inner 81.10.c.c 2
36.f odd 6 1 144.10.a.h 1
36.h even 6 1 144.10.a.h 1
45.h odd 6 1 225.10.a.d 1
45.j even 6 1 225.10.a.d 1
45.k odd 12 2 225.10.b.f 2
45.l even 12 2 225.10.b.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.10.a.b 1 9.c even 3 1
9.10.a.b 1 9.d odd 6 1
81.10.c.c 2 1.a even 1 1 trivial
81.10.c.c 2 3.b odd 2 1 CM
81.10.c.c 2 9.c even 3 1 inner
81.10.c.c 2 9.d odd 6 1 inner
144.10.a.h 1 36.f odd 6 1
144.10.a.h 1 36.h even 6 1
225.10.a.d 1 45.h odd 6 1
225.10.a.d 1 45.j even 6 1
225.10.b.f 2 45.k odd 12 2
225.10.b.f 2 45.l even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{10}^{\mathrm{new}}(81, [\chi])\). 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
$7$ \( T^{2} - 12580 T + 158256400 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 118370 T + 14011456900 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 976696)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1691228 T + 2860252147984 \) Copy content Toggle raw display
$37$ \( (T + 15384490)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 274799581326400 \) 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
$61$ \( T^{2} - 117903058 T + 13\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{2} + 112542320 T + 12\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 296368310)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 616732324 T + 38\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1288928270 T + 16\!\cdots\!00 \) Copy content Toggle raw display
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