Properties

Label 1840.2.m.a
Level 18401840
Weight 22
Character orbit 1840.m
Analytic conductor 14.69214.692
Analytic rank 00
Dimension 44
Inner twists 44

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,2,Mod(1839,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1840.1839");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1840=24523 1840 = 2^{4} \cdot 5 \cdot 23
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1840.m (of order 22, degree 11, 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: 14.692473971914.6924739719
Analytic rank: 00
Dimension: 44
Coefficient field: Q(6,14)\Q(\sqrt{6}, \sqrt{-14})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4+4x2+25 x^{4} + 4x^{2} + 25 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 1 1
Twist minimal: yes
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 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) == qq3β2q52q9+(2β2+2β1)q11+β3q13+β2q15+(β2β1)q17+(β2+β1)q19+(β2+β1+3)q23++(4β24β1)q99+O(q100) q - q^{3} - \beta_{2} q^{5} - 2 q^{9} + ( - 2 \beta_{2} + 2 \beta_1) q^{11} + \beta_{3} q^{13} + \beta_{2} q^{15} + (\beta_{2} - \beta_1) q^{17} + ( - \beta_{2} + \beta_1) q^{19} + (\beta_{2} + \beta_1 + 3) q^{23}+ \cdots + (4 \beta_{2} - 4 \beta_1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q4q38q9+12q238q25+20q2712q2912q41+36q47+28q49+24q5512q69+8q75+4q8112q85+12q87+12q95+O(q100) 4 q - 4 q^{3} - 8 q^{9} + 12 q^{23} - 8 q^{25} + 20 q^{27} - 12 q^{29} - 12 q^{41} + 36 q^{47} + 28 q^{49} + 24 q^{55} - 12 q^{69} + 8 q^{75} + 4 q^{81} - 12 q^{85} + 12 q^{87} + 12 q^{95}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4+4x2+25 x^{4} + 4x^{2} + 25 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν3+4ν)/5 ( \nu^{3} + 4\nu ) / 5 Copy content Toggle raw display
β3\beta_{3}== ν2+2 \nu^{2} + 2 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β32 \beta_{3} - 2 Copy content Toggle raw display
ν3\nu^{3}== 5β24β1 5\beta_{2} - 4\beta_1 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1840Z)×\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times.

nn 737737 11511151 12011201 13811381
χ(n)\chi(n) 1-1 1-1 1-1 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}
1839.1
−1.22474 + 1.87083i
−1.22474 1.87083i
1.22474 + 1.87083i
1.22474 1.87083i
0 −1.00000 0 −1.22474 1.87083i 0 0 0 −2.00000 0
1839.2 0 −1.00000 0 −1.22474 + 1.87083i 0 0 0 −2.00000 0
1839.3 0 −1.00000 0 1.22474 1.87083i 0 0 0 −2.00000 0
1839.4 0 −1.00000 0 1.22474 + 1.87083i 0 0 0 −2.00000 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
20.d odd 2 1 inner
23.b odd 2 1 inner
460.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.2.m.a 4
4.b odd 2 1 1840.2.m.d yes 4
5.b even 2 1 1840.2.m.d yes 4
20.d odd 2 1 inner 1840.2.m.a 4
23.b odd 2 1 inner 1840.2.m.a 4
92.b even 2 1 1840.2.m.d yes 4
115.c odd 2 1 1840.2.m.d yes 4
460.g even 2 1 inner 1840.2.m.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.m.a 4 1.a even 1 1 trivial
1840.2.m.a 4 20.d odd 2 1 inner
1840.2.m.a 4 23.b odd 2 1 inner
1840.2.m.a 4 460.g even 2 1 inner
1840.2.m.d yes 4 4.b odd 2 1
1840.2.m.d yes 4 5.b even 2 1
1840.2.m.d yes 4 92.b even 2 1
1840.2.m.d yes 4 115.c 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(1840,[χ])S_{2}^{\mathrm{new}}(1840, [\chi]):

T3+1 T_{3} + 1 Copy content Toggle raw display
T7 T_{7} Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4 T^{4} Copy content Toggle raw display
33 (T+1)4 (T + 1)^{4} Copy content Toggle raw display
55 T4+4T2+25 T^{4} + 4T^{2} + 25 Copy content Toggle raw display
77 T4 T^{4} Copy content Toggle raw display
1111 (T224)2 (T^{2} - 24)^{2} Copy content Toggle raw display
1313 (T2+21)2 (T^{2} + 21)^{2} Copy content Toggle raw display
1717 (T26)2 (T^{2} - 6)^{2} Copy content Toggle raw display
1919 (T26)2 (T^{2} - 6)^{2} Copy content Toggle raw display
2323 (T26T+23)2 (T^{2} - 6 T + 23)^{2} Copy content Toggle raw display
2929 (T+3)4 (T + 3)^{4} Copy content Toggle raw display
3131 (T2+21)2 (T^{2} + 21)^{2} Copy content Toggle raw display
3737 (T224)2 (T^{2} - 24)^{2} Copy content Toggle raw display
4141 (T+3)4 (T + 3)^{4} Copy content Toggle raw display
4343 (T2+126)2 (T^{2} + 126)^{2} Copy content Toggle raw display
4747 (T9)4 (T - 9)^{4} Copy content Toggle raw display
5353 (T224)2 (T^{2} - 24)^{2} Copy content Toggle raw display
5959 (T2+84)2 (T^{2} + 84)^{2} Copy content Toggle raw display
6161 (T2+126)2 (T^{2} + 126)^{2} Copy content Toggle raw display
6767 T4 T^{4} Copy content Toggle raw display
7171 (T2+189)2 (T^{2} + 189)^{2} Copy content Toggle raw display
7373 (T2+21)2 (T^{2} + 21)^{2} Copy content Toggle raw display
7979 (T254)2 (T^{2} - 54)^{2} Copy content Toggle raw display
8383 (T2+14)2 (T^{2} + 14)^{2} Copy content Toggle raw display
8989 (T2+14)2 (T^{2} + 14)^{2} Copy content Toggle raw display
9797 (T296)2 (T^{2} - 96)^{2} Copy content Toggle raw display
show more
show less