Properties

Label 3600.2.f.k
Level 36003600
Weight 22
Character orbit 3600.f
Analytic conductor 28.74628.746
Analytic rank 00
Dimension 22
CM discriminant -3
Inner twists 44

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3600,2,Mod(2449,3600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3600.2449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 3600=243252 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 3600.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: 28.746144727728.7461447277
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,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 5 5
Twist minimal: no (minimal twist has level 225)
Sato-Tate group: U(1)[D2]\mathrm{U}(1)[D_{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 β=5i\beta = 5i. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+βq7+βq13q19+7q31+2βq37βq4318q4913q61+βq672βq734q7925q91βq97+O(q100) q + \beta q^{7} + \beta q^{13} - q^{19} + 7 q^{31} + 2 \beta q^{37} - \beta q^{43} - 18 q^{49} - 13 q^{61} + \beta q^{67} - 2 \beta q^{73} - 4 q^{79} - 25 q^{91} - \beta q^{97} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q2q19+14q3136q4926q618q7950q91+O(q100) 2 q - 2 q^{19} + 14 q^{31} - 36 q^{49} - 26 q^{61} - 8 q^{79} - 50 q^{91}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/3600Z)×\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times.

nn 577577 901901 28012801 31513151
χ(n)\chi(n) 1-1 11 11 11

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}
2449.1
1.00000i
1.00000i
0 0 0 0 0 5.00000i 0 0 0
2449.2 0 0 0 0 0 5.00000i 0 0 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})
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.2.f.k 2
3.b odd 2 1 CM 3600.2.f.k 2
4.b odd 2 1 225.2.b.c 2
5.b even 2 1 inner 3600.2.f.k 2
5.c odd 4 1 3600.2.a.b 1
5.c odd 4 1 3600.2.a.br 1
12.b even 2 1 225.2.b.c 2
15.d odd 2 1 inner 3600.2.f.k 2
15.e even 4 1 3600.2.a.b 1
15.e even 4 1 3600.2.a.br 1
20.d odd 2 1 225.2.b.c 2
20.e even 4 1 225.2.a.c 1
20.e even 4 1 225.2.a.d yes 1
60.h even 2 1 225.2.b.c 2
60.l odd 4 1 225.2.a.c 1
60.l odd 4 1 225.2.a.d yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.a.c 1 20.e even 4 1
225.2.a.c 1 60.l odd 4 1
225.2.a.d yes 1 20.e even 4 1
225.2.a.d yes 1 60.l odd 4 1
225.2.b.c 2 4.b odd 2 1
225.2.b.c 2 12.b even 2 1
225.2.b.c 2 20.d odd 2 1
225.2.b.c 2 60.h even 2 1
3600.2.a.b 1 5.c odd 4 1
3600.2.a.b 1 15.e even 4 1
3600.2.a.br 1 5.c odd 4 1
3600.2.a.br 1 15.e even 4 1
3600.2.f.k 2 1.a even 1 1 trivial
3600.2.f.k 2 3.b odd 2 1 CM
3600.2.f.k 2 5.b even 2 1 inner
3600.2.f.k 2 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(3600,[χ])S_{2}^{\mathrm{new}}(3600, [\chi]):

T72+25 T_{7}^{2} + 25 Copy content Toggle raw display
T11 T_{11} Copy content Toggle raw display
T132+25 T_{13}^{2} + 25 Copy content Toggle raw display
T17 T_{17} 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 T^{2} Copy content Toggle raw display
55 T2 T^{2} Copy content Toggle raw display
77 T2+25 T^{2} + 25 Copy content Toggle raw display
1111 T2 T^{2} Copy content Toggle raw display
1313 T2+25 T^{2} + 25 Copy content Toggle raw display
1717 T2 T^{2} Copy content Toggle raw display
1919 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
2323 T2 T^{2} Copy content Toggle raw display
2929 T2 T^{2} Copy content Toggle raw display
3131 (T7)2 (T - 7)^{2} Copy content Toggle raw display
3737 T2+100 T^{2} + 100 Copy content Toggle raw display
4141 T2 T^{2} Copy content Toggle raw display
4343 T2+25 T^{2} + 25 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 (T+13)2 (T + 13)^{2} Copy content Toggle raw display
6767 T2+25 T^{2} + 25 Copy content Toggle raw display
7171 T2 T^{2} Copy content Toggle raw display
7373 T2+100 T^{2} + 100 Copy content Toggle raw display
7979 (T+4)2 (T + 4)^{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+25 T^{2} + 25 Copy content Toggle raw display
show more
show less