Properties

Label 2535.2.a.bj
Level 25352535
Weight 22
Character orbit 2535.a
Self dual yes
Analytic conductor 20.24220.242
Analytic rank 00
Dimension 44
CM no
Inner twists 11

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2535,2,Mod(1,2535)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2535, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2535.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 2535=35132 2535 = 3 \cdot 5 \cdot 13^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 2535.a (trivial)

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: yes
Analytic conductor: 20.242076912420.2420769124
Analytic rank: 00
Dimension: 44
Coefficient field: 4.4.13824.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x46x2+6 x^{4} - 6x^{2} + 6 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 195)
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(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 a basis 1,β1,β2,β31,\beta_1,\beta_2,\beta_3 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2q3+(β2+1)q4q5β1q6+(β2β1)q7+β3q8+q9β1q10+β3q11+(β21)q12+(β3β2+β13)q14++β3q99+O(q100) q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - q^{5} - \beta_1 q^{6} + (\beta_{2} - \beta_1) q^{7} + \beta_{3} q^{8} + q^{9} - \beta_1 q^{10} + \beta_{3} q^{11} + ( - \beta_{2} - 1) q^{12} + (\beta_{3} - \beta_{2} + \beta_1 - 3) q^{14}+ \cdots + \beta_{3} q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q4q3+4q44q5+4q94q1212q14+4q158q16+12q194q20+4q254q27+12q2812q29+12q31+12q34+4q36+24q37+24q98+O(q100) 4 q - 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{9} - 4 q^{12} - 12 q^{14} + 4 q^{15} - 8 q^{16} + 12 q^{19} - 4 q^{20} + 4 q^{25} - 4 q^{27} + 12 q^{28} - 12 q^{29} + 12 q^{31} + 12 q^{34} + 4 q^{36} + 24 q^{37}+ \cdots - 24 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x46x2+6 x^{4} - 6x^{2} + 6 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν23 \nu^{2} - 3 Copy content Toggle raw display
β3\beta_{3}== ν34ν \nu^{3} - 4\nu Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2+3 \beta_{2} + 3 Copy content Toggle raw display
ν3\nu^{3}== β3+4β1 \beta_{3} + 4\beta_1 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.

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}
1.1
−2.17533
−1.12603
1.12603
2.17533
−2.17533 −1.00000 2.73205 −1.00000 2.17533 3.90738 −1.59245 1.00000 2.17533
1.2 −1.12603 −1.00000 −0.732051 −1.00000 1.12603 −0.606018 3.07638 1.00000 1.12603
1.3 1.12603 −1.00000 −0.732051 −1.00000 −1.12603 −2.85808 −3.07638 1.00000 −1.12603
1.4 2.17533 −1.00000 2.73205 −1.00000 −2.17533 −0.443277 1.59245 1.00000 −2.17533
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 +1 +1
55 +1 +1
1313 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 2535.2.a.bj 4
3.b odd 2 1 7605.2.a.ci 4
13.b even 2 1 2535.2.a.bk 4
13.f odd 12 2 195.2.bb.b 8
39.d odd 2 1 7605.2.a.ch 4
39.k even 12 2 585.2.bu.d 8
65.o even 12 2 975.2.w.h 8
65.s odd 12 2 975.2.bc.j 8
65.t even 12 2 975.2.w.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bb.b 8 13.f odd 12 2
585.2.bu.d 8 39.k even 12 2
975.2.w.h 8 65.o even 12 2
975.2.w.i 8 65.t even 12 2
975.2.bc.j 8 65.s odd 12 2
2535.2.a.bj 4 1.a even 1 1 trivial
2535.2.a.bk 4 13.b even 2 1
7605.2.a.ch 4 39.d odd 2 1
7605.2.a.ci 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(2535))S_{2}^{\mathrm{new}}(\Gamma_0(2535)):

T246T22+6 T_{2}^{4} - 6T_{2}^{2} + 6 Copy content Toggle raw display
T7412T7212T73 T_{7}^{4} - 12T_{7}^{2} - 12T_{7} - 3 Copy content Toggle raw display
T11412T112+24 T_{11}^{4} - 12T_{11}^{2} + 24 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T46T2+6 T^{4} - 6T^{2} + 6 Copy content Toggle raw display
33 (T+1)4 (T + 1)^{4} Copy content Toggle raw display
55 (T+1)4 (T + 1)^{4} Copy content Toggle raw display
77 T412T2+3 T^{4} - 12 T^{2} + \cdots - 3 Copy content Toggle raw display
1111 T412T2+24 T^{4} - 12T^{2} + 24 Copy content Toggle raw display
1313 T4 T^{4} Copy content Toggle raw display
1717 T418T2+54 T^{4} - 18T^{2} + 54 Copy content Toggle raw display
1919 T412T3+444 T^{4} - 12 T^{3} + \cdots - 444 Copy content Toggle raw display
2323 T460T2+72 T^{4} - 60 T^{2} + \cdots - 72 Copy content Toggle raw display
2929 T4+12T3+234 T^{4} + 12 T^{3} + \cdots - 234 Copy content Toggle raw display
3131 T412T3++33 T^{4} - 12 T^{3} + \cdots + 33 Copy content Toggle raw display
3737 T424T3++528 T^{4} - 24 T^{3} + \cdots + 528 Copy content Toggle raw display
4141 T412T3++222 T^{4} - 12 T^{3} + \cdots + 222 Copy content Toggle raw display
4343 T416T3+1523 T^{4} - 16 T^{3} + \cdots - 1523 Copy content Toggle raw display
4747 T424T3++654 T^{4} - 24 T^{3} + \cdots + 654 Copy content Toggle raw display
5353 T4144T2+3456 T^{4} - 144T^{2} + 3456 Copy content Toggle raw display
5959 T4+12T3+2106 T^{4} + 12 T^{3} + \cdots - 2106 Copy content Toggle raw display
6161 (T2+16T+61)2 (T^{2} + 16 T + 61)^{2} Copy content Toggle raw display
6767 T4+12T3++981 T^{4} + 12 T^{3} + \cdots + 981 Copy content Toggle raw display
7171 T4+12T3+1146 T^{4} + 12 T^{3} + \cdots - 1146 Copy content Toggle raw display
7373 T424T3+1179 T^{4} - 24 T^{3} + \cdots - 1179 Copy content Toggle raw display
7979 T4+8T3+1703 T^{4} + 8 T^{3} + \cdots - 1703 Copy content Toggle raw display
8383 T4132T2+4056 T^{4} - 132T^{2} + 4056 Copy content Toggle raw display
8989 T412T3+66 T^{4} - 12 T^{3} + \cdots - 66 Copy content Toggle raw display
9797 T436T3++5109 T^{4} - 36 T^{3} + \cdots + 5109 Copy content Toggle raw display
show more
show less