Properties

Label 588.4.f.a
Level 588588
Weight 44
Character orbit 588.f
Analytic conductor 34.69334.693
Analytic rank 00
Dimension 22
CM discriminant -3
Inner twists 44

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,4,Mod(293,588)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) 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("588.293"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 588=22372 588 = 2^{2} \cdot 3 \cdot 7^{2}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 588.f (of order 22, degree 11, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-54,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-250,0,0,0, 0,0,0,0,0,0,0,0,-646] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 34.693123083434.6931230834
Analytic rank: 00
Dimension: 22
Coefficient field: Q(3)\Q(\sqrt{-3})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2x+1 x^{2} - x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 2 2
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: U(1)[D2]\mathrm{U}(1)[D_{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)
 

Coefficients of the qq-expansion are expressed in terms of β=3\beta = \sqrt{-3}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q3βq327q953βq13+73βq19125q25+81βq27109βq31323q37477q3971q43+657q57+540βq61127q67+703βq73++792βq97+O(q100) q - 3 \beta q^{3} - 27 q^{9} - 53 \beta q^{13} + 73 \beta q^{19} - 125 q^{25} + 81 \beta q^{27} - 109 \beta q^{31} - 323 q^{37} - 477 q^{39} - 71 q^{43} + 657 q^{57} + 540 \beta q^{61} - 127 q^{67} + 703 \beta q^{73} + \cdots + 792 \beta q^{97} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q54q9250q25646q37954q39142q43+1314q57254q672774q79+1458q811962q93+O(q100) 2 q - 54 q^{9} - 250 q^{25} - 646 q^{37} - 954 q^{39} - 142 q^{43} + 1314 q^{57} - 254 q^{67} - 2774 q^{79} + 1458 q^{81} - 1962 q^{93}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/588Z)×\left(\mathbb{Z}/588\mathbb{Z}\right)^\times.

nn 197197 295295 493493
χ(n)\chi(n) 1-1 11 1-1

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}
293.1
0.500000 + 0.866025i
0.500000 0.866025i
0 5.19615i 0 0 0 0 0 −27.0000 0
293.2 0 5.19615i 0 0 0 0 0 −27.0000 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by Q(3)\Q(\sqrt{-3})
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.4.f.a 2
3.b odd 2 1 CM 588.4.f.a 2
7.b odd 2 1 inner 588.4.f.a 2
7.c even 3 1 84.4.k.a 2
7.c even 3 1 588.4.k.b 2
7.d odd 6 1 84.4.k.a 2
7.d odd 6 1 588.4.k.b 2
21.c even 2 1 inner 588.4.f.a 2
21.g even 6 1 84.4.k.a 2
21.g even 6 1 588.4.k.b 2
21.h odd 6 1 84.4.k.a 2
21.h odd 6 1 588.4.k.b 2
28.f even 6 1 336.4.bc.b 2
28.g odd 6 1 336.4.bc.b 2
84.j odd 6 1 336.4.bc.b 2
84.n even 6 1 336.4.bc.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.k.a 2 7.c even 3 1
84.4.k.a 2 7.d odd 6 1
84.4.k.a 2 21.g even 6 1
84.4.k.a 2 21.h odd 6 1
336.4.bc.b 2 28.f even 6 1
336.4.bc.b 2 28.g odd 6 1
336.4.bc.b 2 84.j odd 6 1
336.4.bc.b 2 84.n even 6 1
588.4.f.a 2 1.a even 1 1 trivial
588.4.f.a 2 3.b odd 2 1 CM
588.4.f.a 2 7.b odd 2 1 inner
588.4.f.a 2 21.c even 2 1 inner
588.4.k.b 2 7.c even 3 1
588.4.k.b 2 7.d odd 6 1
588.4.k.b 2 21.g even 6 1
588.4.k.b 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(588,[χ])S_{4}^{\mathrm{new}}(588, [\chi]):

T5 T_{5} Copy content Toggle raw display
T132+8427 T_{13}^{2} + 8427 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 T2+27 T^{2} + 27 Copy content Toggle raw display
55 T2 T^{2} Copy content Toggle raw display
77 T2 T^{2} Copy content Toggle raw display
1111 T2 T^{2} Copy content Toggle raw display
1313 T2+8427 T^{2} + 8427 Copy content Toggle raw display
1717 T2 T^{2} Copy content Toggle raw display
1919 T2+15987 T^{2} + 15987 Copy content Toggle raw display
2323 T2 T^{2} Copy content Toggle raw display
2929 T2 T^{2} Copy content Toggle raw display
3131 T2+35643 T^{2} + 35643 Copy content Toggle raw display
3737 (T+323)2 (T + 323)^{2} Copy content Toggle raw display
4141 T2 T^{2} Copy content Toggle raw display
4343 (T+71)2 (T + 71)^{2} Copy content Toggle raw display
4747 T2 T^{2} Copy content Toggle raw display
5353 T2 T^{2} Copy content Toggle raw display
5959 T2 T^{2} Copy content Toggle raw display
6161 T2+874800 T^{2} + 874800 Copy content Toggle raw display
6767 (T+127)2 (T + 127)^{2} Copy content Toggle raw display
7171 T2 T^{2} Copy content Toggle raw display
7373 T2+1482627 T^{2} + 1482627 Copy content Toggle raw display
7979 (T+1387)2 (T + 1387)^{2} Copy content Toggle raw display
8383 T2 T^{2} Copy content Toggle raw display
8989 T2 T^{2} Copy content Toggle raw display
9797 T2+1881792 T^{2} + 1881792 Copy content Toggle raw display
show more
show less