Properties

Label 405.4.a.i
Level 405405
Weight 44
Character orbit 405.a
Self dual yes
Analytic conductor 23.89623.896
Analytic rank 00
Dimension 33
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [405,4,Mod(1,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 405=345 405 = 3^{4} \cdot 5
Weight: k k == 4 4
Character orbit: [χ][\chi] == 405.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: 23.895773552323.8957735523
Analytic rank: 00
Dimension: 33
Coefficient field: 3.3.7032.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x3x214x+18 x^{3} - x^{2} - 14x + 18 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
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,β21,\beta_1,\beta_2 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2+(β2β1+2)q45q5+(3β2+5β110)q7+(β23β18)q85β1q10+(5β2+7β1+17)q11+(13β2+13β120)q13++(3β2+326β196)q98+O(q100) q + \beta_1 q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} - 5 q^{5} + (3 \beta_{2} + 5 \beta_1 - 10) q^{7} + (\beta_{2} - 3 \beta_1 - 8) q^{8} - 5 \beta_1 q^{10} + (5 \beta_{2} + 7 \beta_1 + 17) q^{11} + ( - 13 \beta_{2} + 13 \beta_1 - 20) q^{13}+ \cdots + (3 \beta_{2} + 326 \beta_1 - 96) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 3q+q2+5q415q525q727q85q10+58q1147q13+159q14127q16+34q175q1925q20+260q2251q23+75q25+253q26++38q98+O(q100) 3 q + q^{2} + 5 q^{4} - 15 q^{5} - 25 q^{7} - 27 q^{8} - 5 q^{10} + 58 q^{11} - 47 q^{13} + 159 q^{14} - 127 q^{16} + 34 q^{17} - 5 q^{19} - 25 q^{20} + 260 q^{22} - 51 q^{23} + 75 q^{25} + 253 q^{26}+ \cdots + 38 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x3x214x+18 x^{3} - x^{2} - 14x + 18 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2+ν10 \nu^{2} + \nu - 10 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2β1+10 \beta_{2} - \beta _1 + 10 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
−3.85028
1.32681
3.52348
−3.85028 0 6.82469 −5.00000 0 −26.3282 4.52526 0 19.2514
1.2 1.32681 0 −6.23958 −5.00000 0 −24.1043 −18.8932 0 −6.63404
1.3 3.52348 0 4.41489 −5.00000 0 25.4325 −12.6321 0 −17.6174
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 +1 +1
55 +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 405.4.a.i yes 3
3.b odd 2 1 405.4.a.g 3
5.b even 2 1 2025.4.a.p 3
9.c even 3 2 405.4.e.s 6
9.d odd 6 2 405.4.e.u 6
15.d odd 2 1 2025.4.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.g 3 3.b odd 2 1
405.4.a.i yes 3 1.a even 1 1 trivial
405.4.e.s 6 9.c even 3 2
405.4.e.u 6 9.d odd 6 2
2025.4.a.p 3 5.b even 2 1
2025.4.a.r 3 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T23T2214T2+18 T_{2}^{3} - T_{2}^{2} - 14T_{2} + 18 acting on S4new(Γ0(405))S_{4}^{\mathrm{new}}(\Gamma_0(405)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T3T2++18 T^{3} - T^{2} + \cdots + 18 Copy content Toggle raw display
33 T3 T^{3} Copy content Toggle raw display
55 (T+5)3 (T + 5)^{3} Copy content Toggle raw display
77 T3+25T2+16140 T^{3} + 25 T^{2} + \cdots - 16140 Copy content Toggle raw display
1111 T358T2+3000 T^{3} - 58 T^{2} + \cdots - 3000 Copy content Toggle raw display
1313 T3+47T2+370352 T^{3} + 47 T^{2} + \cdots - 370352 Copy content Toggle raw display
1717 T334T2++90984 T^{3} - 34 T^{2} + \cdots + 90984 Copy content Toggle raw display
1919 T3+5T2+299645 T^{3} + 5 T^{2} + \cdots - 299645 Copy content Toggle raw display
2323 T3+51T2+1041156 T^{3} + 51 T^{2} + \cdots - 1041156 Copy content Toggle raw display
2929 T3350T2++11237760 T^{3} - 350 T^{2} + \cdots + 11237760 Copy content Toggle raw display
3131 T3638T2+9539064 T^{3} - 638 T^{2} + \cdots - 9539064 Copy content Toggle raw display
3737 T3+414T2+577760 T^{3} + 414 T^{2} + \cdots - 577760 Copy content Toggle raw display
4141 T3179T2++17799627 T^{3} - 179 T^{2} + \cdots + 17799627 Copy content Toggle raw display
4343 T3+836T2++18692992 T^{3} + 836 T^{2} + \cdots + 18692992 Copy content Toggle raw display
4747 T3235T2++9005376 T^{3} - 235 T^{2} + \cdots + 9005376 Copy content Toggle raw display
5353 T3505T2++1500684 T^{3} - 505 T^{2} + \cdots + 1500684 Copy content Toggle raw display
5959 T3535T2+22317657 T^{3} - 535 T^{2} + \cdots - 22317657 Copy content Toggle raw display
6161 T3+104T2+23542832 T^{3} + 104 T^{2} + \cdots - 23542832 Copy content Toggle raw display
6767 T3+40T2+15716208 T^{3} + 40 T^{2} + \cdots - 15716208 Copy content Toggle raw display
7171 T3452T2++116183454 T^{3} - 452 T^{2} + \cdots + 116183454 Copy content Toggle raw display
7373 T3+710T2++8707528 T^{3} + 710 T^{2} + \cdots + 8707528 Copy content Toggle raw display
7979 T3+634T2++5053056 T^{3} + 634 T^{2} + \cdots + 5053056 Copy content Toggle raw display
8383 T31734T2++222334848 T^{3} - 1734 T^{2} + \cdots + 222334848 Copy content Toggle raw display
8989 T3+852T2++17926434 T^{3} + 852 T^{2} + \cdots + 17926434 Copy content Toggle raw display
9797 T31575168T703275008 T^{3} - 1575168 T - 703275008 Copy content Toggle raw display
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