Properties

Label 2400.4.a.bt
Level 24002400
Weight 44
Character orbit 2400.a
Self dual yes
Analytic conductor 141.605141.605
Analytic rank 00
Dimension 33
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2400,4,Mod(1,2400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2400.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 2400=25352 2400 = 2^{5} \cdot 3 \cdot 5^{2}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 2400.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: 141.604584014141.604584014
Analytic rank: 00
Dimension: 33
Coefficient field: 3.3.115636.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x3x269x81 x^{3} - x^{2} - 69x - 81 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 23 2^{3}
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,β21,\beta_1,\beta_2 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+3q3+(β21)q7+9q9+(β2β1+15)q11+(2β2β14)q13+(β2+3β1+11)q17+(β2β1+24)q19+(3β23)q21++(9β29β1+135)q99+O(q100) q + 3 q^{3} + (\beta_{2} - 1) q^{7} + 9 q^{9} + ( - \beta_{2} - \beta_1 + 15) q^{11} + (2 \beta_{2} - \beta_1 - 4) q^{13} + (\beta_{2} + 3 \beta_1 + 11) q^{17} + (\beta_{2} - \beta_1 + 24) q^{19} + (3 \beta_{2} - 3) q^{21}+ \cdots + ( - 9 \beta_{2} - 9 \beta_1 + 135) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 3q+9q33q7+27q9+44q1113q13+36q17+71q199q21+160q23+81q27184q2959q31+132q33+350q3739q39+166q41+341q43++396q99+O(q100) 3 q + 9 q^{3} - 3 q^{7} + 27 q^{9} + 44 q^{11} - 13 q^{13} + 36 q^{17} + 71 q^{19} - 9 q^{21} + 160 q^{23} + 81 q^{27} - 184 q^{29} - 59 q^{31} + 132 q^{33} + 350 q^{37} - 39 q^{39} + 166 q^{41} + 341 q^{43}+ \cdots + 396 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x3x269x81 x^{3} - x^{2} - 69x - 81 : Copy content Toggle raw display

β1\beta_{1}== 4ν1 4\nu - 1 Copy content Toggle raw display
β2\beta_{2}== (2ν28ν90)/3 ( 2\nu^{2} - 8\nu - 90 ) / 3 Copy content Toggle raw display
ν\nu== (β1+1)/4 ( \beta _1 + 1 ) / 4 Copy content Toggle raw display
ν2\nu^{2}== (3β2+2β1+92)/2 ( 3\beta_{2} + 2\beta _1 + 92 ) / 2 Copy content Toggle raw display

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
1.1
−1.22200
9.32800
−7.10600
0 3.00000 0 0 0 −26.7458 0 9.00000 0
1.2 0 3.00000 0 0 0 2.13303 0 9.00000 0
1.3 0 3.00000 0 0 0 21.6128 0 9.00000 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 +1 +1
33 1 -1
55 1 -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 2400.4.a.bt yes 3
4.b odd 2 1 2400.4.a.bi 3
5.b even 2 1 2400.4.a.bj yes 3
20.d odd 2 1 2400.4.a.bs yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2400.4.a.bi 3 4.b odd 2 1
2400.4.a.bj yes 3 5.b even 2 1
2400.4.a.bs yes 3 20.d odd 2 1
2400.4.a.bt yes 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 S4new(Γ0(2400))S_{4}^{\mathrm{new}}(\Gamma_0(2400)):

T73+3T72589T7+1233 T_{7}^{3} + 3T_{7}^{2} - 589T_{7} + 1233 Copy content Toggle raw display
T11344T112656T11+24864 T_{11}^{3} - 44T_{11}^{2} - 656T_{11} + 24864 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T3 T^{3} Copy content Toggle raw display
33 (T3)3 (T - 3)^{3} Copy content Toggle raw display
55 T3 T^{3} Copy content Toggle raw display
77 T3+3T2++1233 T^{3} + 3 T^{2} + \cdots + 1233 Copy content Toggle raw display
1111 T344T2++24864 T^{3} - 44 T^{2} + \cdots + 24864 Copy content Toggle raw display
1313 T3+13T2+119313 T^{3} + 13 T^{2} + \cdots - 119313 Copy content Toggle raw display
1717 T336T2+218016 T^{3} - 36 T^{2} + \cdots - 218016 Copy content Toggle raw display
1919 T371T2++2891 T^{3} - 71 T^{2} + \cdots + 2891 Copy content Toggle raw display
2323 T3160T2++2700000 T^{3} - 160 T^{2} + \cdots + 2700000 Copy content Toggle raw display
2929 T3+184T2+1985440 T^{3} + 184 T^{2} + \cdots - 1985440 Copy content Toggle raw display
3131 T3+59T2++5724361 T^{3} + 59 T^{2} + \cdots + 5724361 Copy content Toggle raw display
3737 T3350T2++223640 T^{3} - 350 T^{2} + \cdots + 223640 Copy content Toggle raw display
4141 T3166T2+1460648 T^{3} - 166 T^{2} + \cdots - 1460648 Copy content Toggle raw display
4343 T3341T2++43261577 T^{3} - 341 T^{2} + \cdots + 43261577 Copy content Toggle raw display
4747 T3+394T2+44505544 T^{3} + 394 T^{2} + \cdots - 44505544 Copy content Toggle raw display
5353 T3314T2++7884200 T^{3} - 314 T^{2} + \cdots + 7884200 Copy content Toggle raw display
5959 T3538T2++161430408 T^{3} - 538 T^{2} + \cdots + 161430408 Copy content Toggle raw display
6161 T3+523T2+6303415 T^{3} + 523 T^{2} + \cdots - 6303415 Copy content Toggle raw display
6767 T3+543T2+72402579 T^{3} + 543 T^{2} + \cdots - 72402579 Copy content Toggle raw display
7171 T3910T2++28003320 T^{3} - 910 T^{2} + \cdots + 28003320 Copy content Toggle raw display
7373 T3870T2++462224120 T^{3} - 870 T^{2} + \cdots + 462224120 Copy content Toggle raw display
7979 T3+96T2+4064256 T^{3} + 96 T^{2} + \cdots - 4064256 Copy content Toggle raw display
8383 T3210T2++8776232 T^{3} - 210 T^{2} + \cdots + 8776232 Copy content Toggle raw display
8989 T3+64T2+20736000 T^{3} + 64 T^{2} + \cdots - 20736000 Copy content Toggle raw display
9797 T3943T2+15001373 T^{3} - 943 T^{2} + \cdots - 15001373 Copy content Toggle raw display
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