Properties

Label 8100.2.a.v
Level $8100$
Weight $2$
Character orbit 8100.a
Self dual yes
Analytic conductor $64.679$
Analytic rank $0$
Dimension $3$
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] = [8100,2,Mod(1,8100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8100.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8100.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: \(64.6788256372\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 180)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{7} - \beta_{2} q^{11} + ( - \beta_{2} - 2) q^{13} - \beta_{2} q^{17} + ( - \beta_{2} + 2) q^{19} + (\beta_1 + 1) q^{23} + (2 \beta_1 - 1) q^{29} + ( - \beta_{2} + 2) q^{31} + (\beta_{2} - 2 \beta_1 - 4) q^{37} + (\beta_{2} - 2 \beta_1 + 1) q^{41} + ( - \beta_{2} - 2) q^{43} + ( - \beta_{2} + \beta_1 - 5) q^{47} + (\beta_{2} + 6) q^{49} + (2 \beta_1 + 2) q^{53} + (2 \beta_1 + 2) q^{59} + ( - \beta_{2} + 2 \beta_1 + 7) q^{61} + ( - \beta_{2} - \beta_1 - 3) q^{67} + ( - \beta_{2} + 2 \beta_1 + 8) q^{71} - 8 q^{73} + (3 \beta_{2} - 2 \beta_1 - 2) q^{77} + 2 q^{79} + (\beta_1 + 7) q^{83} - 3 q^{89} + (3 \beta_{2} - 4 \beta_1) q^{91} + (2 \beta_1 - 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{7} - 6 q^{13} + 6 q^{19} + 3 q^{23} - 3 q^{29} + 6 q^{31} - 12 q^{37} + 3 q^{41} - 6 q^{43} - 15 q^{47} + 18 q^{49} + 6 q^{53} + 6 q^{59} + 21 q^{61} - 9 q^{67} + 24 q^{71} - 24 q^{73} - 6 q^{77} + 6 q^{79} + 21 q^{83} - 9 q^{89} - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 5x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{2} + 2\beta _1 + 11 ) / 3 \) 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
0.571993
−2.08613
2.51414
0 0 0 0 0 −4.10083 0 0 0
1.2 0 0 0 0 0 −2.73419 0 0 0
1.3 0 0 0 0 0 3.83502 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)

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 8100.2.a.v 3
3.b odd 2 1 8100.2.a.u 3
5.b even 2 1 1620.2.a.i 3
5.c odd 4 2 8100.2.d.p 6
9.c even 3 2 900.2.i.c 6
9.d odd 6 2 2700.2.i.c 6
15.d odd 2 1 1620.2.a.j 3
15.e even 4 2 8100.2.d.o 6
20.d odd 2 1 6480.2.a.bt 3
45.h odd 6 2 540.2.i.b 6
45.j even 6 2 180.2.i.b 6
45.k odd 12 4 900.2.s.c 12
45.l even 12 4 2700.2.s.c 12
60.h even 2 1 6480.2.a.bw 3
180.n even 6 2 2160.2.q.i 6
180.p odd 6 2 720.2.q.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.i.b 6 45.j even 6 2
540.2.i.b 6 45.h odd 6 2
720.2.q.k 6 180.p odd 6 2
900.2.i.c 6 9.c even 3 2
900.2.s.c 12 45.k odd 12 4
1620.2.a.i 3 5.b even 2 1
1620.2.a.j 3 15.d odd 2 1
2160.2.q.i 6 180.n even 6 2
2700.2.i.c 6 9.d odd 6 2
2700.2.s.c 12 45.l even 12 4
6480.2.a.bt 3 20.d odd 2 1
6480.2.a.bw 3 60.h even 2 1
8100.2.a.u 3 3.b odd 2 1
8100.2.a.v 3 1.a even 1 1 trivial
8100.2.d.o 6 15.e even 4 2
8100.2.d.p 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8100))\):

\( T_{7}^{3} + 3T_{7}^{2} - 15T_{7} - 43 \) Copy content Toggle raw display
\( T_{11}^{3} - 24T_{11} - 36 \) Copy content Toggle raw display
\( T_{17}^{3} - 24T_{17} - 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 3 T^{2} - 15 T - 43 \) Copy content Toggle raw display
$11$ \( T^{3} - 24T - 36 \) Copy content Toggle raw display
$13$ \( T^{3} + 6 T^{2} - 12 T - 76 \) Copy content Toggle raw display
$17$ \( T^{3} - 24T - 36 \) Copy content Toggle raw display
$19$ \( T^{3} - 6 T^{2} - 12 T + 4 \) Copy content Toggle raw display
$23$ \( T^{3} - 3 T^{2} - 15 T - 9 \) Copy content Toggle raw display
$29$ \( T^{3} + 3 T^{2} - 69 T - 279 \) Copy content Toggle raw display
$31$ \( T^{3} - 6 T^{2} - 12 T + 4 \) Copy content Toggle raw display
$37$ \( T^{3} + 12 T^{2} - 36 T - 436 \) Copy content Toggle raw display
$41$ \( T^{3} - 3 T^{2} - 81 T - 81 \) Copy content Toggle raw display
$43$ \( T^{3} + 6 T^{2} - 12 T - 76 \) Copy content Toggle raw display
$47$ \( T^{3} + 15 T^{2} + 39 T + 27 \) Copy content Toggle raw display
$53$ \( T^{3} - 6 T^{2} - 60 T - 72 \) Copy content Toggle raw display
$59$ \( T^{3} - 6 T^{2} - 60 T - 72 \) Copy content Toggle raw display
$61$ \( T^{3} - 21 T^{2} + 63 T + 409 \) Copy content Toggle raw display
$67$ \( T^{3} + 9 T^{2} - 21 T - 151 \) Copy content Toggle raw display
$71$ \( T^{3} - 24 T^{2} + 108 T + 324 \) Copy content Toggle raw display
$73$ \( (T + 8)^{3} \) Copy content Toggle raw display
$79$ \( (T - 2)^{3} \) Copy content Toggle raw display
$83$ \( T^{3} - 21 T^{2} + 129 T - 243 \) Copy content Toggle raw display
$89$ \( (T + 3)^{3} \) Copy content Toggle raw display
$97$ \( T^{3} + 18 T^{2} + 36 T - 424 \) Copy content Toggle raw display
show more
show less