Properties

Label 600.8.f.d
Level 600600
Weight 88
Character orbit 600.f
Analytic conductor 187.431187.431
Analytic rank 00
Dimension 22
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [600,8,Mod(49,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.49");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: N N == 600=23352 600 = 2^{3} \cdot 3 \cdot 5^{2}
Weight: k k == 8 8
Character orbit: [χ][\chi] == 600.f (of order 22, degree 11, not minimal)

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: no
Analytic conductor: 187.431015290187.431015290
Analytic rank: 00
Dimension: 22
Coefficient field: Q(1)\Q(\sqrt{-1})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2+1 x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{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 i=1i = \sqrt{-1}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q27iq3+504iq7729q9+3812q119574iq13+26098iq17+38308q19+13608q21+71128iq23+19683iq2774262q29275680q31102924iq33+2778948q99+O(q100) q - 27 i q^{3} + 504 i q^{7} - 729 q^{9} + 3812 q^{11} - 9574 i q^{13} + 26098 i q^{17} + 38308 q^{19} + 13608 q^{21} + 71128 i q^{23} + 19683 i q^{27} - 74262 q^{29} - 275680 q^{31} - 102924 i q^{33} + \cdots - 2778948 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q1458q9+7624q11+76616q19+27216q21148524q29551360q31516996q39+1369524q41+1139054q49+1409292q51+3050648q595280916q61+3840912q69+5557896q99+O(q100) 2 q - 1458 q^{9} + 7624 q^{11} + 76616 q^{19} + 27216 q^{21} - 148524 q^{29} - 551360 q^{31} - 516996 q^{39} + 1369524 q^{41} + 1139054 q^{49} + 1409292 q^{51} + 3050648 q^{59} - 5280916 q^{61} + 3840912 q^{69}+ \cdots - 5557896 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/600Z)×\left(\mathbb{Z}/600\mathbb{Z}\right)^\times.

nn 151151 301301 401401 577577
χ(n)\chi(n) 11 11 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.

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}
49.1
1.00000i
1.00000i
0 27.0000i 0 0 0 504.000i 0 −729.000 0
49.2 0 27.0000i 0 0 0 504.000i 0 −729.000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.8.f.d 2
5.b even 2 1 inner 600.8.f.d 2
5.c odd 4 1 24.8.a.c 1
5.c odd 4 1 600.8.a.a 1
15.e even 4 1 72.8.a.b 1
20.e even 4 1 48.8.a.c 1
40.i odd 4 1 192.8.a.c 1
40.k even 4 1 192.8.a.k 1
60.l odd 4 1 144.8.a.d 1
120.q odd 4 1 576.8.a.s 1
120.w even 4 1 576.8.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.a.c 1 5.c odd 4 1
48.8.a.c 1 20.e even 4 1
72.8.a.b 1 15.e even 4 1
144.8.a.d 1 60.l odd 4 1
192.8.a.c 1 40.i odd 4 1
192.8.a.k 1 40.k even 4 1
576.8.a.s 1 120.q odd 4 1
576.8.a.t 1 120.w even 4 1
600.8.a.a 1 5.c odd 4 1
600.8.f.d 2 1.a even 1 1 trivial
600.8.f.d 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T72+254016 T_{7}^{2} + 254016 acting on S8new(600,[χ])S_{8}^{\mathrm{new}}(600, [\chi]). 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+729 T^{2} + 729 Copy content Toggle raw display
55 T2 T^{2} Copy content Toggle raw display
77 T2+254016 T^{2} + 254016 Copy content Toggle raw display
1111 (T3812)2 (T - 3812)^{2} Copy content Toggle raw display
1313 T2+91661476 T^{2} + 91661476 Copy content Toggle raw display
1717 T2+681105604 T^{2} + 681105604 Copy content Toggle raw display
1919 (T38308)2 (T - 38308)^{2} Copy content Toggle raw display
2323 T2+5059192384 T^{2} + 5059192384 Copy content Toggle raw display
2929 (T+74262)2 (T + 74262)^{2} Copy content Toggle raw display
3131 (T+275680)2 (T + 275680)^{2} Copy content Toggle raw display
3737 T2+71080892100 T^{2} + 71080892100 Copy content Toggle raw display
4141 (T684762)2 (T - 684762)^{2} Copy content Toggle raw display
4343 T2+60494353936 T^{2} + 60494353936 Copy content Toggle raw display
4747 T2+229249440000 T^{2} + 229249440000 Copy content Toggle raw display
5353 T2+324227748100 T^{2} + 324227748100 Copy content Toggle raw display
5959 (T1525324)2 (T - 1525324)^{2} Copy content Toggle raw display
6161 (T+2640458)2 (T + 2640458)^{2} Copy content Toggle raw display
6767 T2+2005724407696 T^{2} + 2005724407696 Copy content Toggle raw display
7171 (T+3511304)2 (T + 3511304)^{2} Copy content Toggle raw display
7373 T2+22454500549924 T^{2} + 22454500549924 Copy content Toggle raw display
7979 (T+4661488)2 (T + 4661488)^{2} Copy content Toggle raw display
8383 T2+32824328479504 T^{2} + 32824328479504 Copy content Toggle raw display
8989 (T+11993514)2 (T + 11993514)^{2} Copy content Toggle raw display
9797 T2+51133282768516 T^{2} + 51133282768516 Copy content Toggle raw display
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