Properties

Label 490.8.a.d
Level 490490
Weight 88
Character orbit 490.a
Self dual yes
Analytic conductor 153.069153.069
Analytic rank 11
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] = [490,8,Mod(1,490)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("490.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(490, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")
 
Level: N N == 490=2572 490 = 2 \cdot 5 \cdot 7^{2}
Weight: k k == 8 8
Character orbit: [χ][\chi] == 490.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,8,30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 153.068662487153.068662487
Analytic rank: 11
Dimension: 11
Coefficient field: Q\mathbb{Q}
Coefficient ring: Z\mathbb{Z}
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 70)
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+8q2+30q3+64q4125q5+240q6+512q81287q91000q10+3q11+1920q12+1745q133750q15+4096q163786q1710296q181945q19+3861q99+O(q100) q + 8 q^{2} + 30 q^{3} + 64 q^{4} - 125 q^{5} + 240 q^{6} + 512 q^{8} - 1287 q^{9} - 1000 q^{10} + 3 q^{11} + 1920 q^{12} + 1745 q^{13} - 3750 q^{15} + 4096 q^{16} - 3786 q^{17} - 10296 q^{18} - 1945 q^{19}+ \cdots - 3861 q^{99}+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
8.00000 30.0000 64.0000 −125.000 240.000 0 512.000 −1287.00 −1000.00
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 1 -1
55 +1 +1
77 +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 490.8.a.d 1
7.b odd 2 1 490.8.a.a 1
7.c even 3 2 70.8.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.8.e.a 2 7.c even 3 2
490.8.a.a 1 7.b odd 2 1
490.8.a.d 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T330 T_{3} - 30 acting on S8new(Γ0(490))S_{8}^{\mathrm{new}}(\Gamma_0(490)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8 T - 8 Copy content Toggle raw display
33 T30 T - 30 Copy content Toggle raw display
55 T+125 T + 125 Copy content Toggle raw display
77 T T Copy content Toggle raw display
1111 T3 T - 3 Copy content Toggle raw display
1313 T1745 T - 1745 Copy content Toggle raw display
1717 T+3786 T + 3786 Copy content Toggle raw display
1919 T+1945 T + 1945 Copy content Toggle raw display
2323 T79551 T - 79551 Copy content Toggle raw display
2929 T+94926 T + 94926 Copy content Toggle raw display
3131 T127628 T - 127628 Copy content Toggle raw display
3737 T+128257 T + 128257 Copy content Toggle raw display
4141 T+298077 T + 298077 Copy content Toggle raw display
4343 T+875626 T + 875626 Copy content Toggle raw display
4747 T611559 T - 611559 Copy content Toggle raw display
5353 T+259137 T + 259137 Copy content Toggle raw display
5959 T+2877336 T + 2877336 Copy content Toggle raw display
6161 T+148564 T + 148564 Copy content Toggle raw display
6767 T+1790884 T + 1790884 Copy content Toggle raw display
7171 T+493236 T + 493236 Copy content Toggle raw display
7373 T+2058052 T + 2058052 Copy content Toggle raw display
7979 T+5867074 T + 5867074 Copy content Toggle raw display
8383 T+921132 T + 921132 Copy content Toggle raw display
8989 T+5123082 T + 5123082 Copy content Toggle raw display
9797 T+5878306 T + 5878306 Copy content Toggle raw display
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