Properties

Label 3024.2.k.j
Level $3024$
Weight $2$
Character orbit 3024.k
Analytic conductor $24.147$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3024,2,Mod(1889,3024)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.1889");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.k (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 378)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5} + ( - \beta_1 + 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{5} + ( - \beta_1 + 3) q^{7} + ( - \beta_{3} + 2 \beta_{2}) q^{11} + (8 \beta_1 - 4) q^{13} - 4 \beta_{3} q^{17} + ( - 4 \beta_1 + 2) q^{19} + (2 \beta_{3} - 4 \beta_{2}) q^{23} - 2 q^{25} + (2 \beta_{3} - 4 \beta_{2}) q^{29} + (6 \beta_1 - 3) q^{31} + ( - 3 \beta_{3} + \beta_{2}) q^{35} - 2 q^{37} - 2 \beta_{3} q^{41} + 2 q^{43} - 2 \beta_{3} q^{47} + ( - 5 \beta_1 + 8) q^{49} + ( - \beta_{3} + 2 \beta_{2}) q^{53} + ( - 6 \beta_1 + 3) q^{55} + 2 \beta_{3} q^{59} + ( - 8 \beta_1 + 4) q^{61} + (4 \beta_{3} - 8 \beta_{2}) q^{65} - 2 q^{67} + (4 \beta_{3} - 8 \beta_{2}) q^{71} + ( - 14 \beta_1 + 7) q^{73} + ( - \beta_{3} + 5 \beta_{2}) q^{77} - 8 q^{79} - \beta_{3} q^{83} + 12 q^{85} - 6 \beta_{3} q^{89} + (20 \beta_1 - 4) q^{91} + ( - 2 \beta_{3} + 4 \beta_{2}) q^{95} + (14 \beta_1 - 7) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{7} - 8 q^{25} - 8 q^{37} + 8 q^{43} + 22 q^{49} - 8 q^{67} - 32 q^{79} + 48 q^{85} + 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1889.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0 0 0 −1.73205 0 2.50000 0.866025i 0 0 0
1889.2 0 0 0 −1.73205 0 2.50000 + 0.866025i 0 0 0
1889.3 0 0 0 1.73205 0 2.50000 0.866025i 0 0 0
1889.4 0 0 0 1.73205 0 2.50000 + 0.866025i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.k.j 4
3.b odd 2 1 inner 3024.2.k.j 4
4.b odd 2 1 378.2.d.a 4
7.b odd 2 1 inner 3024.2.k.j 4
12.b even 2 1 378.2.d.a 4
21.c even 2 1 inner 3024.2.k.j 4
28.d even 2 1 378.2.d.a 4
36.f odd 6 1 1134.2.m.e 4
36.f odd 6 1 1134.2.m.f 4
36.h even 6 1 1134.2.m.e 4
36.h even 6 1 1134.2.m.f 4
84.h odd 2 1 378.2.d.a 4
252.s odd 6 1 1134.2.m.e 4
252.s odd 6 1 1134.2.m.f 4
252.bi even 6 1 1134.2.m.e 4
252.bi even 6 1 1134.2.m.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.d.a 4 4.b odd 2 1
378.2.d.a 4 12.b even 2 1
378.2.d.a 4 28.d even 2 1
378.2.d.a 4 84.h odd 2 1
1134.2.m.e 4 36.f odd 6 1
1134.2.m.e 4 36.h even 6 1
1134.2.m.e 4 252.s odd 6 1
1134.2.m.e 4 252.bi even 6 1
1134.2.m.f 4 36.f odd 6 1
1134.2.m.f 4 36.h even 6 1
1134.2.m.f 4 252.s odd 6 1
1134.2.m.f 4 252.bi even 6 1
3024.2.k.j 4 1.a even 1 1 trivial
3024.2.k.j 4 3.b odd 2 1 inner
3024.2.k.j 4 7.b odd 2 1 inner
3024.2.k.j 4 21.c even 2 1 inner

Hecke kernels

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

\( T_{5}^{2} - 3 \) Copy content Toggle raw display
\( T_{11}^{2} + 9 \) Copy content Toggle raw display
\( T_{13}^{2} + 48 \) 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
$5$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 5 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$43$ \( (T - 2)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$67$ \( (T + 2)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 147)^{2} \) Copy content Toggle raw display
$79$ \( (T + 8)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 147)^{2} \) Copy content Toggle raw display
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