Properties

Label 2058.2.a.m
Level 20582058
Weight 22
Character orbit 2058.a
Self dual yes
Analytic conductor 16.43316.433
Analytic rank 00
Dimension 66
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] = [2058,2,Mod(1,2058)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2058, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2058.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 2058=2373 2058 = 2 \cdot 3 \cdot 7^{3}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 2058.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: 16.433212736016.4332127360
Analytic rank: 00
Dimension: 66
Coefficient field: 6.6.72070817.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x6x525x4+21x3+187x2107x377 x^{6} - x^{5} - 25x^{4} + 21x^{3} + 187x^{2} - 107x - 377 Copy content Toggle raw display
Coefficient ring: Z[a1,,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 1 1
Twist minimal: yes
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,,β51,\beta_1,\ldots,\beta_{5} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+q2q3+q4+(β2β1)q5q6+q8+q9+(β2β1)q10+(β5+β3+β2+1)q11q12+(β4+β3)q13++(β5+β3+β2+1)q99+O(q100) q + q^{2} - q^{3} + q^{4} + (\beta_{2} - \beta_1) q^{5} - q^{6} + q^{8} + q^{9} + (\beta_{2} - \beta_1) q^{10} + (\beta_{5} + \beta_{3} + \beta_{2} + 1) q^{11} - q^{12} + ( - \beta_{4} + \beta_{3}) q^{13}+ \cdots + (\beta_{5} + \beta_{3} + \beta_{2} + 1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q+6q26q3+6q43q56q6+6q8+6q93q10+3q116q123q13+3q15+6q166q17+6q18+5q193q20+3q22+12q23++3q99+O(q100) 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} - 3 q^{5} - 6 q^{6} + 6 q^{8} + 6 q^{9} - 3 q^{10} + 3 q^{11} - 6 q^{12} - 3 q^{13} + 3 q^{15} + 6 q^{16} - 6 q^{17} + 6 q^{18} + 5 q^{19} - 3 q^{20} + 3 q^{22} + 12 q^{23}+ \cdots + 3 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x6x525x4+21x3+187x2107x377 x^{6} - x^{5} - 25x^{4} + 21x^{3} + 187x^{2} - 107x - 377 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (8ν57ν491ν3+120ν2+46ν484)/293 ( 8\nu^{5} - 7\nu^{4} - 91\nu^{3} + 120\nu^{2} + 46\nu - 484 ) / 293 Copy content Toggle raw display
β3\beta_{3}== (11ν5+27ν4235ν3421ν2+1162ν+1239)/293 ( 11\nu^{5} + 27\nu^{4} - 235\nu^{3} - 421\nu^{2} + 1162\nu + 1239 ) / 293 Copy content Toggle raw display
β4\beta_{4}== (19ν520ν4+326ν3+594ν21208ν3392)/293 ( -19\nu^{5} - 20\nu^{4} + 326\nu^{3} + 594\nu^{2} - 1208\nu - 3392 ) / 293 Copy content Toggle raw display
β5\beta_{5}== (20ν5+129ν4374ν32044ν2+1580ν+6408)/293 ( 20\nu^{5} + 129\nu^{4} - 374\nu^{3} - 2044\nu^{2} + 1580\nu + 6408 ) / 293 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β4+β3+β2+9 \beta_{4} + \beta_{3} + \beta_{2} + 9 Copy content Toggle raw display
ν3\nu^{3}== β54β3+3β2+10β1 \beta_{5} - 4\beta_{3} + 3\beta_{2} + 10\beta_1 Copy content Toggle raw display
ν4\nu^{4}== 3β5+16β4+12β3+14β2+92 3\beta_{5} + 16\beta_{4} + 12\beta_{3} + 14\beta_{2} + 92 Copy content Toggle raw display
ν5\nu^{5}== 14β5β450β3+68β2+108β1+6 14\beta_{5} - \beta_{4} - 50\beta_{3} + 68\beta_{2} + 108\beta _1 + 6 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
2.25380
3.20700
3.48647
−1.40506
−3.50078
−3.04143
1.00000 −1.00000 1.00000 −4.05574 −1.00000 0 1.00000 1.00000 −4.05574
1.2 1.00000 −1.00000 1.00000 −3.65204 −1.00000 0 1.00000 1.00000 −3.65204
1.3 1.00000 −1.00000 1.00000 −2.23949 −1.00000 0 1.00000 1.00000 −2.23949
1.4 1.00000 −1.00000 1.00000 0.960022 −1.00000 0 1.00000 1.00000 0.960022
1.5 1.00000 −1.00000 1.00000 1.69885 −1.00000 0 1.00000 1.00000 1.69885
1.6 1.00000 −1.00000 1.00000 4.28841 −1.00000 0 1.00000 1.00000 4.28841
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 1 -1
33 +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 2058.2.a.m 6
3.b odd 2 1 6174.2.a.v 6
7.b odd 2 1 2058.2.a.n yes 6
7.c even 3 2 2058.2.e.n 12
7.d odd 6 2 2058.2.e.m 12
21.c even 2 1 6174.2.a.r 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2058.2.a.m 6 1.a even 1 1 trivial
2058.2.a.n yes 6 7.b odd 2 1
2058.2.e.m 12 7.d odd 6 2
2058.2.e.n 12 7.c even 3 2
6174.2.a.r 6 21.c even 2 1
6174.2.a.v 6 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(2058))S_{2}^{\mathrm{new}}(\Gamma_0(2058)):

T56+3T5524T5467T53+118T52+208T5232 T_{5}^{6} + 3T_{5}^{5} - 24T_{5}^{4} - 67T_{5}^{3} + 118T_{5}^{2} + 208T_{5} - 232 Copy content Toggle raw display
T1163T11536T114+147T113+34T112416T11104 T_{11}^{6} - 3T_{11}^{5} - 36T_{11}^{4} + 147T_{11}^{3} + 34T_{11}^{2} - 416T_{11} - 104 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T1)6 (T - 1)^{6} Copy content Toggle raw display
33 (T+1)6 (T + 1)^{6} Copy content Toggle raw display
55 T6+3T5+232 T^{6} + 3 T^{5} + \cdots - 232 Copy content Toggle raw display
77 T6 T^{6} Copy content Toggle raw display
1111 T63T5+104 T^{6} - 3 T^{5} + \cdots - 104 Copy content Toggle raw display
1313 T6+3T5+56 T^{6} + 3 T^{5} + \cdots - 56 Copy content Toggle raw display
1717 T6+6T5+281 T^{6} + 6 T^{5} + \cdots - 281 Copy content Toggle raw display
1919 T65T5+568 T^{6} - 5 T^{5} + \cdots - 568 Copy content Toggle raw display
2323 T612T5+4523 T^{6} - 12 T^{5} + \cdots - 4523 Copy content Toggle raw display
2929 T621T5+16472 T^{6} - 21 T^{5} + \cdots - 16472 Copy content Toggle raw display
3131 T62T5++9793 T^{6} - 2 T^{5} + \cdots + 9793 Copy content Toggle raw display
3737 T621T5++4696 T^{6} - 21 T^{5} + \cdots + 4696 Copy content Toggle raw display
4141 T6+4T5+301 T^{6} + 4 T^{5} + \cdots - 301 Copy content Toggle raw display
4343 T615T5++104 T^{6} - 15 T^{5} + \cdots + 104 Copy content Toggle raw display
4747 T610T5+19264 T^{6} - 10 T^{5} + \cdots - 19264 Copy content Toggle raw display
5353 T619T5++16136 T^{6} - 19 T^{5} + \cdots + 16136 Copy content Toggle raw display
5959 T67T5++4472 T^{6} - 7 T^{5} + \cdots + 4472 Copy content Toggle raw display
6161 T621T5++9736 T^{6} - 21 T^{5} + \cdots + 9736 Copy content Toggle raw display
6767 T6+T5++16792 T^{6} + T^{5} + \cdots + 16792 Copy content Toggle raw display
7171 T62T5++20693 T^{6} - 2 T^{5} + \cdots + 20693 Copy content Toggle raw display
7373 (T316T2+104)2 (T^{3} - 16 T^{2} + \cdots - 104)^{2} Copy content Toggle raw display
7979 T62T5+99904 T^{6} - 2 T^{5} + \cdots - 99904 Copy content Toggle raw display
8383 T623T5++98728 T^{6} - 23 T^{5} + \cdots + 98728 Copy content Toggle raw display
8989 (T33T2+701)2 (T^{3} - 3 T^{2} + \cdots - 701)^{2} Copy content Toggle raw display
9797 (T328T2+448)2 (T^{3} - 28 T^{2} + \cdots - 448)^{2} Copy content Toggle raw display
show more
show less