Properties

Label 8024.2.a.u
Level $8024$
Weight $2$
Character orbit 8024.a
Self dual yes
Analytic conductor $64.072$
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] = [8024,2,Mod(1,8024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8024, 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("8024.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8024.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: \(64.0719625819\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta_{2} + 1) q^{5} + ( - \beta_1 + 1) q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta_{2} + 1) q^{5} + ( - \beta_1 + 1) q^{7} - 2 q^{9} + 2 \beta_{2} q^{11} + ( - \beta_{2} - 4) q^{13} + ( - \beta_{2} + 1) q^{15} + q^{17} + ( - \beta_{2} - 1) q^{19} + ( - \beta_1 + 1) q^{21} - \beta_{2} q^{23} + ( - 2 \beta_{2} - \beta_1 + 2) q^{25} - 5 q^{27} + (\beta_{2} - \beta_1 + 5) q^{29} + ( - \beta_{2} + 6) q^{31} + 2 \beta_{2} q^{33} + ( - 3 \beta_{2} - 2 \beta_1 + 3) q^{35} + 2 q^{37} + ( - \beta_{2} - 4) q^{39} + (3 \beta_{2} - 3) q^{41} + (\beta_{2} + \beta_1 + 4) q^{43} + (2 \beta_{2} - 2) q^{45} + (\beta_{2} - \beta_1 + 2) q^{47} + ( - 4 \beta_{2} + \beta_1 + 8) q^{49} + q^{51} + ( - 2 \beta_{2} + \beta_1 + 5) q^{53} + (2 \beta_{2} + 2 \beta_1 - 12) q^{55} + ( - \beta_{2} - 1) q^{57} - q^{59} + (3 \beta_{2} - \beta_1 - 4) q^{61} + (2 \beta_1 - 2) q^{63} + (3 \beta_{2} - \beta_1 + 2) q^{65} + ( - 2 \beta_{2} + \beta_1 - 6) q^{67} - \beta_{2} q^{69} + ( - 2 \beta_{2} + 4) q^{71} - 2 q^{73} + ( - 2 \beta_{2} - \beta_1 + 2) q^{75} + (6 \beta_{2} + 2 \beta_1 - 4) q^{77} + ( - 2 \beta_{2} + \beta_1 + 3) q^{79} + q^{81} + ( - \beta_1 + 8) q^{83} + ( - \beta_{2} + 1) q^{85} + (\beta_{2} - \beta_1 + 5) q^{87} + (3 \beta_{2} - \beta_1 - 4) q^{89} + ( - 3 \beta_{2} + 3 \beta_1 - 2) q^{91} + ( - \beta_{2} + 6) q^{93} + ( - \beta_1 + 5) q^{95} + (\beta_{2} + 12) q^{97} - 4 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 4 q^{5} + 2 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 4 q^{5} + 2 q^{7} - 6 q^{9} - 2 q^{11} - 11 q^{13} + 4 q^{15} + 3 q^{17} - 2 q^{19} + 2 q^{21} + q^{23} + 7 q^{25} - 15 q^{27} + 13 q^{29} + 19 q^{31} - 2 q^{33} + 10 q^{35} + 6 q^{37} - 11 q^{39} - 12 q^{41} + 12 q^{43} - 8 q^{45} + 4 q^{47} + 29 q^{49} + 3 q^{51} + 18 q^{53} - 36 q^{55} - 2 q^{57} - 3 q^{59} - 16 q^{61} - 4 q^{63} + 2 q^{65} - 15 q^{67} + q^{69} + 14 q^{71} - 6 q^{73} + 7 q^{75} - 16 q^{77} + 12 q^{79} + 3 q^{81} + 23 q^{83} + 4 q^{85} + 13 q^{87} - 16 q^{89} + 19 q^{93} + 14 q^{95} + 35 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 4 \) 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
−1.76156
3.12489
−0.363328
0 1.00000 0 −1.62620 0 1.89692 0 −2.00000 0
1.2 0 1.00000 0 1.48486 0 −4.76491 0 −2.00000 0
1.3 0 1.00000 0 4.14134 0 4.86799 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(17\) \(-1\)
\(59\) \(1\)

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 8024.2.a.u 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8024.2.a.u 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8024))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{3} - 4T_{5}^{2} - 3T_{5} + 10 \) Copy content Toggle raw display
\( T_{7}^{3} - 2T_{7}^{2} - 23T_{7} + 44 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 4 T^{2} - 3 T + 10 \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} - 23 T + 44 \) Copy content Toggle raw display
$11$ \( T^{3} + 2 T^{2} - 32 T - 32 \) Copy content Toggle raw display
$13$ \( T^{3} + 11 T^{2} + 32 T + 20 \) Copy content Toggle raw display
$17$ \( (T - 1)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} + 2 T^{2} - 7 T - 4 \) Copy content Toggle raw display
$23$ \( T^{3} - T^{2} - 8T + 4 \) Copy content Toggle raw display
$29$ \( T^{3} - 13 T^{2} + 31 T + 61 \) Copy content Toggle raw display
$31$ \( T^{3} - 19 T^{2} + 112 T - 200 \) Copy content Toggle raw display
$37$ \( (T - 2)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} + 12 T^{2} - 27 T - 270 \) Copy content Toggle raw display
$43$ \( T^{3} - 12 T^{2} + 8 T + 160 \) Copy content Toggle raw display
$47$ \( T^{3} - 4 T^{2} - 20 T + 64 \) Copy content Toggle raw display
$53$ \( T^{3} - 18 T^{2} + 65 T + 100 \) Copy content Toggle raw display
$59$ \( (T + 1)^{3} \) Copy content Toggle raw display
$61$ \( T^{3} + 16 T^{2} + 8 T - 512 \) Copy content Toggle raw display
$67$ \( T^{3} + 15 T^{2} + 32 T - 32 \) Copy content Toggle raw display
$71$ \( T^{3} - 14 T^{2} + 32 T + 64 \) Copy content Toggle raw display
$73$ \( (T + 2)^{3} \) Copy content Toggle raw display
$79$ \( T^{3} - 12 T^{2} + 5 T + 166 \) Copy content Toggle raw display
$83$ \( T^{3} - 23 T^{2} + 152 T - 236 \) Copy content Toggle raw display
$89$ \( T^{3} + 16 T^{2} + 8 T - 512 \) Copy content Toggle raw display
$97$ \( T^{3} - 35 T^{2} + 400 T - 1492 \) Copy content Toggle raw display
show more
show less