Properties

Label 720.6.a.l
Level 720720
Weight 66
Character orbit 720.a
Self dual yes
Analytic conductor 115.476115.476
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] = [720,6,Mod(1,720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("720.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: N N == 720=24325 720 = 2^{4} \cdot 3^{2} \cdot 5
Weight: k k == 6 6
Character orbit: [χ][\chi] == 720.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,25,0,-218,0,0,0,-480] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 115.476350265115.476350265
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 20)
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+25q5218q7480q11622q13186q17+1204q193186q23+625q255526q299356q315450q35+5618q37+14394q41+370q43+16146q47++94658q97+O(q100) q + 25 q^{5} - 218 q^{7} - 480 q^{11} - 622 q^{13} - 186 q^{17} + 1204 q^{19} - 3186 q^{23} + 625 q^{25} - 5526 q^{29} - 9356 q^{31} - 5450 q^{35} + 5618 q^{37} + 14394 q^{41} + 370 q^{43} + 16146 q^{47}+ \cdots + 94658 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 25.0000 0 −218.000 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 720.6.a.l 1
3.b odd 2 1 80.6.a.b 1
4.b odd 2 1 180.6.a.e 1
12.b even 2 1 20.6.a.a 1
15.d odd 2 1 400.6.a.m 1
15.e even 4 2 400.6.c.c 2
20.d odd 2 1 900.6.a.b 1
20.e even 4 2 900.6.d.h 2
24.f even 2 1 320.6.a.c 1
24.h odd 2 1 320.6.a.n 1
60.h even 2 1 100.6.a.a 1
60.l odd 4 2 100.6.c.a 2
84.h odd 2 1 980.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.6.a.a 1 12.b even 2 1
80.6.a.b 1 3.b odd 2 1
100.6.a.a 1 60.h even 2 1
100.6.c.a 2 60.l odd 4 2
180.6.a.e 1 4.b odd 2 1
320.6.a.c 1 24.f even 2 1
320.6.a.n 1 24.h odd 2 1
400.6.a.m 1 15.d odd 2 1
400.6.c.c 2 15.e even 4 2
720.6.a.l 1 1.a even 1 1 trivial
900.6.a.b 1 20.d odd 2 1
900.6.d.h 2 20.e even 4 2
980.6.a.b 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S6new(Γ0(720))S_{6}^{\mathrm{new}}(\Gamma_0(720)):

T7+218 T_{7} + 218 Copy content Toggle raw display
T11+480 T_{11} + 480 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 T25 T - 25 Copy content Toggle raw display
77 T+218 T + 218 Copy content Toggle raw display
1111 T+480 T + 480 Copy content Toggle raw display
1313 T+622 T + 622 Copy content Toggle raw display
1717 T+186 T + 186 Copy content Toggle raw display
1919 T1204 T - 1204 Copy content Toggle raw display
2323 T+3186 T + 3186 Copy content Toggle raw display
2929 T+5526 T + 5526 Copy content Toggle raw display
3131 T+9356 T + 9356 Copy content Toggle raw display
3737 T5618 T - 5618 Copy content Toggle raw display
4141 T14394 T - 14394 Copy content Toggle raw display
4343 T370 T - 370 Copy content Toggle raw display
4747 T16146 T - 16146 Copy content Toggle raw display
5353 T4374 T - 4374 Copy content Toggle raw display
5959 T+11748 T + 11748 Copy content Toggle raw display
6161 T13202 T - 13202 Copy content Toggle raw display
6767 T11542 T - 11542 Copy content Toggle raw display
7171 T+29532 T + 29532 Copy content Toggle raw display
7373 T33698 T - 33698 Copy content Toggle raw display
7979 T+31208 T + 31208 Copy content Toggle raw display
8383 T+38466 T + 38466 Copy content Toggle raw display
8989 T+119514 T + 119514 Copy content Toggle raw display
9797 T94658 T - 94658 Copy content Toggle raw display
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