Properties

Label 3024.2.q.b
Level $3024$
Weight $2$
Character orbit 3024.q
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(2305,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.2305");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.q (of order \(3\), degree \(2\), 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{5} + (2 \zeta_{6} - 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{5} + (2 \zeta_{6} - 3) q^{7} - 5 \zeta_{6} q^{11} + 5 \zeta_{6} q^{13} + ( - 3 \zeta_{6} + 3) q^{17} + \zeta_{6} q^{19} + (3 \zeta_{6} - 3) q^{23} + 4 \zeta_{6} q^{25} + (\zeta_{6} - 1) q^{29} + ( - 3 \zeta_{6} + 1) q^{35} - 3 \zeta_{6} q^{37} - 5 \zeta_{6} q^{41} + (\zeta_{6} - 1) q^{43} + ( - 8 \zeta_{6} + 5) q^{49} + (9 \zeta_{6} - 9) q^{53} + 5 q^{55} - 14 q^{61} - 5 q^{65} - 4 q^{67} - 12 q^{71} + (3 \zeta_{6} - 3) q^{73} + (5 \zeta_{6} + 10) q^{77} - 8 q^{79} + ( - 9 \zeta_{6} + 9) q^{83} + 3 \zeta_{6} q^{85} - 13 \zeta_{6} q^{89} + ( - 5 \zeta_{6} - 10) q^{91} - q^{95} + ( - 9 \zeta_{6} + 9) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} - 4 q^{7} - 5 q^{11} + 5 q^{13} + 3 q^{17} + q^{19} - 3 q^{23} + 4 q^{25} - q^{29} - q^{35} - 3 q^{37} - 5 q^{41} - q^{43} + 2 q^{49} - 9 q^{53} + 10 q^{55} - 28 q^{61} - 10 q^{65} - 8 q^{67} - 24 q^{71} - 3 q^{73} + 25 q^{77} - 16 q^{79} + 9 q^{83} + 3 q^{85} - 13 q^{89} - 25 q^{91} - 2 q^{95} + 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2305.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 −0.500000 0.866025i 0 −2.00000 1.73205i 0 0 0
2881.1 0 0 0 −0.500000 + 0.866025i 0 −2.00000 + 1.73205i 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
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.q.b 2
3.b odd 2 1 1008.2.q.c 2
4.b odd 2 1 189.2.h.a 2
7.c even 3 1 3024.2.t.d 2
9.c even 3 1 3024.2.t.d 2
9.d odd 6 1 1008.2.t.d 2
12.b even 2 1 63.2.h.a yes 2
21.h odd 6 1 1008.2.t.d 2
28.d even 2 1 1323.2.h.a 2
28.f even 6 1 1323.2.f.b 2
28.f even 6 1 1323.2.g.a 2
28.g odd 6 1 189.2.g.a 2
28.g odd 6 1 1323.2.f.a 2
36.f odd 6 1 189.2.g.a 2
36.f odd 6 1 567.2.e.b 2
36.h even 6 1 63.2.g.a 2
36.h even 6 1 567.2.e.a 2
63.h even 3 1 inner 3024.2.q.b 2
63.j odd 6 1 1008.2.q.c 2
84.h odd 2 1 441.2.h.a 2
84.j odd 6 1 441.2.f.a 2
84.j odd 6 1 441.2.g.a 2
84.n even 6 1 63.2.g.a 2
84.n even 6 1 441.2.f.b 2
252.n even 6 1 1323.2.f.b 2
252.o even 6 1 441.2.f.b 2
252.o even 6 1 567.2.e.a 2
252.r odd 6 1 441.2.h.a 2
252.r odd 6 1 3969.2.a.f 1
252.s odd 6 1 441.2.g.a 2
252.u odd 6 1 189.2.h.a 2
252.u odd 6 1 3969.2.a.c 1
252.bb even 6 1 63.2.h.a yes 2
252.bb even 6 1 3969.2.a.d 1
252.bi even 6 1 1323.2.g.a 2
252.bj even 6 1 1323.2.h.a 2
252.bj even 6 1 3969.2.a.a 1
252.bl odd 6 1 567.2.e.b 2
252.bl odd 6 1 1323.2.f.a 2
252.bn odd 6 1 441.2.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.a 2 36.h even 6 1
63.2.g.a 2 84.n even 6 1
63.2.h.a yes 2 12.b even 2 1
63.2.h.a yes 2 252.bb even 6 1
189.2.g.a 2 28.g odd 6 1
189.2.g.a 2 36.f odd 6 1
189.2.h.a 2 4.b odd 2 1
189.2.h.a 2 252.u odd 6 1
441.2.f.a 2 84.j odd 6 1
441.2.f.a 2 252.bn odd 6 1
441.2.f.b 2 84.n even 6 1
441.2.f.b 2 252.o even 6 1
441.2.g.a 2 84.j odd 6 1
441.2.g.a 2 252.s odd 6 1
441.2.h.a 2 84.h odd 2 1
441.2.h.a 2 252.r odd 6 1
567.2.e.a 2 36.h even 6 1
567.2.e.a 2 252.o even 6 1
567.2.e.b 2 36.f odd 6 1
567.2.e.b 2 252.bl odd 6 1
1008.2.q.c 2 3.b odd 2 1
1008.2.q.c 2 63.j odd 6 1
1008.2.t.d 2 9.d odd 6 1
1008.2.t.d 2 21.h odd 6 1
1323.2.f.a 2 28.g odd 6 1
1323.2.f.a 2 252.bl odd 6 1
1323.2.f.b 2 28.f even 6 1
1323.2.f.b 2 252.n even 6 1
1323.2.g.a 2 28.f even 6 1
1323.2.g.a 2 252.bi even 6 1
1323.2.h.a 2 28.d even 2 1
1323.2.h.a 2 252.bj even 6 1
3024.2.q.b 2 1.a even 1 1 trivial
3024.2.q.b 2 63.h even 3 1 inner
3024.2.t.d 2 7.c even 3 1
3024.2.t.d 2 9.c even 3 1
3969.2.a.a 1 252.bj even 6 1
3969.2.a.c 1 252.u odd 6 1
3969.2.a.d 1 252.bb even 6 1
3969.2.a.f 1 252.r odd 6 1

Hecke kernels

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

\( T_{5}^{2} + T_{5} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 5T_{11} + 25 \) 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
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$41$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$43$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 14)^{2} \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$89$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$97$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
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