Properties

Label 3024.2.a.bi
Level $3024$
Weight $2$
Character orbit 3024.a
Self dual yes
Analytic conductor $24.147$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3024,2,Mod(1,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, 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("3024.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.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: \(24.1467615712\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 189)
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 \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{5} + q^{7} - \beta q^{11} - 2 q^{13} - 7 q^{19} - 3 \beta q^{23} + 2 q^{25} - 2 \beta q^{29} - 3 q^{31} + \beta q^{35} - 3 q^{37} - \beta q^{41} - 8 q^{43} + q^{49} - 7 q^{55} - 8 q^{61} - 2 \beta q^{65} + 2 q^{67} + 3 \beta q^{71} - \beta q^{77} + 4 q^{79} + 6 \beta q^{83} + 7 \beta q^{89} - 2 q^{91} - 7 \beta q^{95} - 12 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{7} - 4 q^{13} - 14 q^{19} + 4 q^{25} - 6 q^{31} - 6 q^{37} - 16 q^{43} + 2 q^{49} - 14 q^{55} - 16 q^{61} + 4 q^{67} + 8 q^{79} - 4 q^{91} - 24 q^{97}+O(q^{100}) \) 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
−2.64575
2.64575
0 0 0 −2.64575 0 1.00000 0 0 0
1.2 0 0 0 2.64575 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.a.bi 2
3.b odd 2 1 inner 3024.2.a.bi 2
4.b odd 2 1 189.2.a.f 2
12.b even 2 1 189.2.a.f 2
20.d odd 2 1 4725.2.a.bb 2
28.d even 2 1 1323.2.a.w 2
36.f odd 6 2 567.2.f.i 4
36.h even 6 2 567.2.f.i 4
60.h even 2 1 4725.2.a.bb 2
84.h odd 2 1 1323.2.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.f 2 4.b odd 2 1
189.2.a.f 2 12.b even 2 1
567.2.f.i 4 36.f odd 6 2
567.2.f.i 4 36.h even 6 2
1323.2.a.w 2 28.d even 2 1
1323.2.a.w 2 84.h odd 2 1
3024.2.a.bi 2 1.a even 1 1 trivial
3024.2.a.bi 2 3.b odd 2 1 inner
4725.2.a.bb 2 20.d odd 2 1
4725.2.a.bb 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3024))\):

\( T_{5}^{2} - 7 \) Copy content Toggle raw display
\( T_{11}^{2} - 7 \) Copy content Toggle raw display
\( T_{17} \) 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
$5$ \( T^{2} - 7 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 7 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 63 \) Copy content Toggle raw display
$29$ \( T^{2} - 28 \) Copy content Toggle raw display
$31$ \( (T + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T + 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 7 \) Copy content Toggle raw display
$43$ \( (T + 8)^{2} \) 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
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( (T - 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 63 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 252 \) Copy content Toggle raw display
$89$ \( T^{2} - 343 \) Copy content Toggle raw display
$97$ \( (T + 12)^{2} \) Copy content Toggle raw display
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