Properties

Label 1800.4.a.z
Level 18001800
Weight 44
Character orbit 1800.a
Self dual yes
Analytic conductor 106.203106.203
Analytic rank 00
Dimension 11
CM no
Inner twists 11

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1800,4,Mod(1,1800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1800.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 1800=233252 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 1800.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0,0,12,0,0,0,-64,0,-58,0,0,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 106.203438010106.203438010
Analytic rank: 00
Dimension: 11
Coefficient field: Q\mathbb{Q}
Coefficient ring: Z\mathbb{Z}
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 72)
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
f(q)f(q) == q+12q764q1158q13+32q17136q19128q23+144q29+20q31+18q37+288q41+200q43+384q47199q49+496q53+128q59458q61+206q97+O(q100) q + 12 q^{7} - 64 q^{11} - 58 q^{13} + 32 q^{17} - 136 q^{19} - 128 q^{23} + 144 q^{29} + 20 q^{31} + 18 q^{37} + 288 q^{41} + 200 q^{43} + 384 q^{47} - 199 q^{49} + 496 q^{53} + 128 q^{59} - 458 q^{61}+ \cdots - 206 q^{97}+O(q^{100}) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
0
0 0 0 0 0 12.0000 0 0 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 1800.4.a.z 1
3.b odd 2 1 1800.4.a.ba 1
5.b even 2 1 72.4.a.a 1
5.c odd 4 2 1800.4.f.b 2
15.d odd 2 1 72.4.a.d yes 1
15.e even 4 2 1800.4.f.x 2
20.d odd 2 1 144.4.a.a 1
40.e odd 2 1 576.4.a.x 1
40.f even 2 1 576.4.a.w 1
45.h odd 6 2 648.4.i.a 2
45.j even 6 2 648.4.i.l 2
60.h even 2 1 144.4.a.f 1
120.i odd 2 1 576.4.a.c 1
120.m even 2 1 576.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.a.a 1 5.b even 2 1
72.4.a.d yes 1 15.d odd 2 1
144.4.a.a 1 20.d odd 2 1
144.4.a.f 1 60.h even 2 1
576.4.a.c 1 120.i odd 2 1
576.4.a.d 1 120.m even 2 1
576.4.a.w 1 40.f even 2 1
576.4.a.x 1 40.e odd 2 1
648.4.i.a 2 45.h odd 6 2
648.4.i.l 2 45.j even 6 2
1800.4.a.z 1 1.a even 1 1 trivial
1800.4.a.ba 1 3.b odd 2 1
1800.4.f.b 2 5.c odd 4 2
1800.4.f.x 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(Γ0(1800))S_{4}^{\mathrm{new}}(\Gamma_0(1800)):

T712 T_{7} - 12 Copy content Toggle raw display
T11+64 T_{11} + 64 Copy content Toggle raw display
T1732 T_{17} - 32 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T T Copy content Toggle raw display
33 T T Copy content Toggle raw display
55 T T Copy content Toggle raw display
77 T12 T - 12 Copy content Toggle raw display
1111 T+64 T + 64 Copy content Toggle raw display
1313 T+58 T + 58 Copy content Toggle raw display
1717 T32 T - 32 Copy content Toggle raw display
1919 T+136 T + 136 Copy content Toggle raw display
2323 T+128 T + 128 Copy content Toggle raw display
2929 T144 T - 144 Copy content Toggle raw display
3131 T20 T - 20 Copy content Toggle raw display
3737 T18 T - 18 Copy content Toggle raw display
4141 T288 T - 288 Copy content Toggle raw display
4343 T200 T - 200 Copy content Toggle raw display
4747 T384 T - 384 Copy content Toggle raw display
5353 T496 T - 496 Copy content Toggle raw display
5959 T128 T - 128 Copy content Toggle raw display
6161 T+458 T + 458 Copy content Toggle raw display
6767 T496 T - 496 Copy content Toggle raw display
7171 T+512 T + 512 Copy content Toggle raw display
7373 T602 T - 602 Copy content Toggle raw display
7979 T1108 T - 1108 Copy content Toggle raw display
8383 T704 T - 704 Copy content Toggle raw display
8989 T960 T - 960 Copy content Toggle raw display
9797 T+206 T + 206 Copy content Toggle raw display
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