Properties

Label 6480.2.a.ca
Level $6480$
Weight $2$
Character orbit 6480.a
Self dual yes
Analytic conductor $51.743$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.62352.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 11x^{2} + 12x + 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3240)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} + (\beta_{2} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + (\beta_{2} - \beta_1) q^{7} + ( - \beta_{3} + \beta_1 - 2) q^{11} + (\beta_{3} - \beta_{2} + \beta_1) q^{13} + (\beta_{3} - 2 \beta_{2}) q^{17} + ( - 2 \beta_1 + 1) q^{19} + (\beta_1 - 3) q^{23} + q^{25} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{29} + (\beta_1 - 4) q^{31} + (\beta_{2} - \beta_1) q^{35} + ( - \beta_{3} - 2 \beta_1 + 4) q^{37} + ( - 2 \beta_{3} + 3 \beta_{2}) q^{41} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 - 6) q^{43} + (\beta_{3} - 4 \beta_{2} - \beta_1 + 1) q^{47} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{49} + ( - \beta_{3} - 4 \beta_{2} + \beta_1 - 3) q^{53} + ( - \beta_{3} + \beta_1 - 2) q^{55} - \beta_{2} q^{59} + (2 \beta_{3} + 2 \beta_{2} + 2) q^{61} + (\beta_{3} - \beta_{2} + \beta_1) q^{65} + (2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 4) q^{67} + (\beta_{3} + 2 \beta_{2} + 3 \beta_1 - 8) q^{71} + (\beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{73} + (3 \beta_{3} - 6) q^{77} + (2 \beta_{2} - 4) q^{79} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{83} + (\beta_{3} - 2 \beta_{2}) q^{85} + ( - \beta_{3} - 6 \beta_{2} - \beta_1 - 2) q^{89} + ( - 6 \beta_{2} + 3 \beta_1 - 9) q^{91} + ( - 2 \beta_1 + 1) q^{95} + ( - \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} - 2 q^{7} - 4 q^{11} - 2 q^{17} - 10 q^{23} + 4 q^{25} - 4 q^{29} - 14 q^{31} - 2 q^{35} + 14 q^{37} + 4 q^{41} - 22 q^{43} + 10 q^{49} - 8 q^{53} - 4 q^{55} + 4 q^{61} - 24 q^{67} - 28 q^{71} + 2 q^{73} - 30 q^{77} - 16 q^{79} - 6 q^{83} - 2 q^{85} - 8 q^{89} - 30 q^{91} - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 11x^{2} + 12x + 24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - \nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - \nu^{2} - 8\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 2\beta_{2} + 9\beta _1 + 6 \) 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
2.16910
3.61675
−1.16910
−2.61675
0 0 0 1.00000 0 −3.90115 0 0 0
1.2 0 0 0 1.00000 0 −1.88469 0 0 0
1.3 0 0 0 1.00000 0 −0.562950 0 0 0
1.4 0 0 0 1.00000 0 4.34880 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.ca 4
3.b odd 2 1 6480.2.a.by 4
4.b odd 2 1 3240.2.a.v yes 4
12.b even 2 1 3240.2.a.t 4
36.f odd 6 2 3240.2.q.bg 8
36.h even 6 2 3240.2.q.bh 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3240.2.a.t 4 12.b even 2 1
3240.2.a.v yes 4 4.b odd 2 1
3240.2.q.bg 8 36.f odd 6 2
3240.2.q.bh 8 36.h even 6 2
6480.2.a.by 4 3.b odd 2 1
6480.2.a.ca 4 1.a even 1 1 trivial

Hecke kernels

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

\( T_{7}^{4} + 2T_{7}^{3} - 17T_{7}^{2} - 42T_{7} - 18 \) Copy content Toggle raw display
\( T_{11}^{4} + 4T_{11}^{3} - 35T_{11}^{2} - 144T_{11} - 108 \) Copy content Toggle raw display
\( T_{13}^{4} - 39T_{13}^{2} + 36T_{13} + 216 \) Copy content Toggle raw display
\( T_{17}^{4} + 2T_{17}^{3} - 38T_{17}^{2} - 120T_{17} - 72 \) Copy content Toggle raw display
\( T_{19}^{4} - 50T_{19}^{2} + 433 \) Copy content Toggle raw display
\( T_{23}^{4} + 10T_{23}^{3} + 25T_{23}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} - 17 T^{2} - 42 T - 18 \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} - 35 T^{2} - 144 T - 108 \) Copy content Toggle raw display
$13$ \( T^{4} - 39 T^{2} + 36 T + 216 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} - 38 T^{2} - 120 T - 72 \) Copy content Toggle raw display
$19$ \( T^{4} - 50T^{2} + 433 \) Copy content Toggle raw display
$23$ \( T^{4} + 10 T^{3} + 25 T^{2} - 12 \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} - 47 T^{2} + 96 T - 48 \) Copy content Toggle raw display
$31$ \( T^{4} + 14 T^{3} + 61 T^{2} + 84 T + 24 \) Copy content Toggle raw display
$37$ \( T^{4} - 14 T^{3} - 2 T^{2} + \cdots - 1152 \) Copy content Toggle raw display
$41$ \( T^{4} - 4 T^{3} - 122 T^{2} + 684 T + 69 \) Copy content Toggle raw display
$43$ \( T^{4} + 22 T^{3} + 70 T^{2} + \cdots - 6768 \) Copy content Toggle raw display
$47$ \( T^{4} - 113 T^{2} - 462 T - 74 \) Copy content Toggle raw display
$53$ \( T^{4} + 8 T^{3} - 137 T^{2} + \cdots - 498 \) Copy content Toggle raw display
$59$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - 4 T^{3} - 152 T^{2} + \cdots - 1152 \) Copy content Toggle raw display
$67$ \( T^{4} + 24 T^{3} + 4 T^{2} + \cdots - 18944 \) Copy content Toggle raw display
$71$ \( T^{4} + 28 T^{3} + 121 T^{2} + \cdots - 16428 \) Copy content Toggle raw display
$73$ \( T^{4} - 2 T^{3} - 86 T^{2} + 456 T - 552 \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} - 78 T^{2} - 504 T - 72 \) Copy content Toggle raw display
$89$ \( T^{4} + 8 T^{3} - 267 T^{2} + \cdots + 11344 \) Copy content Toggle raw display
$97$ \( T^{4} + 10 T^{3} - 114 T^{2} + \cdots - 32 \) Copy content Toggle raw display
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