Properties

Label 6480.2.a.bw
Level $6480$
Weight $2$
Character orbit 6480.a
Self dual yes
Analytic conductor $51.743$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6480,2,Mod(1,6480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6480, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6480.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6480.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: \(51.7430605098\)
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.

\(f(q)\) \(=\) \( q + q^{5} + (\beta_1 - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + (\beta_1 - 1) q^{7} - \beta_{2} q^{11} + (\beta_{2} + 2) q^{13} - \beta_{2} q^{17} + (\beta_{2} - 2) q^{19} + ( - \beta_1 - 1) q^{23} + q^{25} + ( - 2 \beta_1 + 1) q^{29} + (\beta_{2} - 2) q^{31} + (\beta_1 - 1) q^{35} + ( - \beta_{2} + 2 \beta_1 + 4) q^{37} + ( - \beta_{2} + 2 \beta_1 - 1) q^{41} + ( - \beta_{2} - 2) q^{43} + (\beta_{2} - \beta_1 + 5) q^{47} + (\beta_{2} + 6) q^{49} + (2 \beta_1 + 2) q^{53} - \beta_{2} q^{55} + (2 \beta_1 + 2) q^{59} + ( - \beta_{2} + 2 \beta_1 + 7) q^{61} + (\beta_{2} + 2) q^{65} + ( - \beta_{2} - \beta_1 - 3) q^{67} + ( - \beta_{2} + 2 \beta_1 + 8) q^{71} + 8 q^{73} + (3 \beta_{2} - 2 \beta_1 - 2) q^{77} - 2 q^{79} + ( - \beta_1 - 7) q^{83} - \beta_{2} q^{85} + 3 q^{89} + ( - 3 \beta_{2} + 4 \beta_1) q^{91} + (\beta_{2} - 2) q^{95} + ( - 2 \beta_1 + 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{5} - 3 q^{7} + 6 q^{13} - 6 q^{19} - 3 q^{23} + 3 q^{25} + 3 q^{29} - 6 q^{31} - 3 q^{35} + 12 q^{37} - 3 q^{41} - 6 q^{43} + 15 q^{47} + 18 q^{49} + 6 q^{53} + 6 q^{59} + 21 q^{61} + 6 q^{65} - 9 q^{67} + 24 q^{71} + 24 q^{73} - 6 q^{77} - 6 q^{79} - 21 q^{83} + 9 q^{89} - 6 q^{95} + 18 q^{97}+O(q^{100}) \) 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 1.00000 0 −4.10083 0 0 0
1.2 0 0 0 1.00000 0 −2.73419 0 0 0
1.3 0 0 0 1.00000 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 6480.2.a.bw 3
3.b odd 2 1 6480.2.a.bt 3
4.b odd 2 1 1620.2.a.j 3
9.c even 3 2 2160.2.q.i 6
9.d odd 6 2 720.2.q.k 6
12.b even 2 1 1620.2.a.i 3
20.d odd 2 1 8100.2.a.u 3
20.e even 4 2 8100.2.d.o 6
36.f odd 6 2 540.2.i.b 6
36.h even 6 2 180.2.i.b 6
60.h even 2 1 8100.2.a.v 3
60.l odd 4 2 8100.2.d.p 6
180.n even 6 2 900.2.i.c 6
180.p odd 6 2 2700.2.i.c 6
180.v odd 12 4 900.2.s.c 12
180.x even 12 4 2700.2.s.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.i.b 6 36.h even 6 2
540.2.i.b 6 36.f odd 6 2
720.2.q.k 6 9.d odd 6 2
900.2.i.c 6 180.n even 6 2
900.2.s.c 12 180.v odd 12 4
1620.2.a.i 3 12.b even 2 1
1620.2.a.j 3 4.b odd 2 1
2160.2.q.i 6 9.c even 3 2
2700.2.i.c 6 180.p odd 6 2
2700.2.s.c 12 180.x even 12 4
6480.2.a.bt 3 3.b odd 2 1
6480.2.a.bw 3 1.a even 1 1 trivial
8100.2.a.u 3 20.d odd 2 1
8100.2.a.v 3 60.h even 2 1
8100.2.d.o 6 20.e even 4 2
8100.2.d.p 6 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6480))\):

\( T_{7}^{3} + 3T_{7}^{2} - 15T_{7} - 43 \) Copy content Toggle raw display
\( T_{11}^{3} - 24T_{11} - 36 \) Copy content Toggle raw display
\( T_{13}^{3} - 6T_{13}^{2} - 12T_{13} + 76 \) Copy content Toggle raw display
\( T_{17}^{3} - 24T_{17} - 36 \) Copy content Toggle raw display
\( T_{19}^{3} + 6T_{19}^{2} - 12T_{19} - 4 \) Copy content Toggle raw display
\( T_{23}^{3} + 3T_{23}^{2} - 15T_{23} + 9 \) 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
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 3 T^{2} + \cdots - 43 \) Copy content Toggle raw display
$11$ \( T^{3} - 24T - 36 \) Copy content Toggle raw display
$13$ \( T^{3} - 6 T^{2} + \cdots + 76 \) Copy content Toggle raw display
$17$ \( T^{3} - 24T - 36 \) Copy content Toggle raw display
$19$ \( T^{3} + 6 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$23$ \( T^{3} + 3 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{3} - 3 T^{2} + \cdots + 279 \) Copy content Toggle raw display
$31$ \( T^{3} + 6 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$37$ \( T^{3} - 12 T^{2} + \cdots + 436 \) Copy content Toggle raw display
$41$ \( T^{3} + 3 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{3} + 6 T^{2} + \cdots - 76 \) Copy content Toggle raw display
$47$ \( T^{3} - 15 T^{2} + \cdots - 27 \) Copy content Toggle raw display
$53$ \( T^{3} - 6 T^{2} + \cdots - 72 \) Copy content Toggle raw display
$59$ \( T^{3} - 6 T^{2} + \cdots - 72 \) Copy content Toggle raw display
$61$ \( T^{3} - 21 T^{2} + \cdots + 409 \) Copy content Toggle raw display
$67$ \( T^{3} + 9 T^{2} + \cdots - 151 \) Copy content Toggle raw display
$71$ \( T^{3} - 24 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$73$ \( (T - 8)^{3} \) Copy content Toggle raw display
$79$ \( (T + 2)^{3} \) Copy content Toggle raw display
$83$ \( T^{3} + 21 T^{2} + \cdots + 243 \) Copy content Toggle raw display
$89$ \( (T - 3)^{3} \) Copy content Toggle raw display
$97$ \( T^{3} - 18 T^{2} + \cdots + 424 \) Copy content Toggle raw display
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