Properties

Label 1728.2.i.m
Level 17281728
Weight 22
Character orbit 1728.i
Analytic conductor 13.79813.798
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,2,Mod(577,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1728=2633 1728 = 2^{6} \cdot 3^{3}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1728.i (of order 33, degree 22, 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: 13.798149469313.7981494693
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q(2,3)\Q(\sqrt{-2}, \sqrt{-3})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x42x2+4 x^{4} - 2x^{2} + 4 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 3 3
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

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+(β1+1)q5+(β2+β1)q7+(β2β1)q11+(2β3+2β2++1)q13+2β3q17+4q19+(β3+β23β1+3)q23++(2β2β1)q97+O(q100) q + ( - \beta_1 + 1) q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{2} - \beta_1) q^{11} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 1) q^{13} + 2 \beta_{3} q^{17} + 4 q^{19} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{23}+ \cdots + ( - 2 \beta_{2} - \beta_1) q^{97}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+2q5+2q72q11+2q13+16q19+6q23+8q2510q29+10q31+4q3516q3714q4110q432q47+16q534q55+14q596q61+2q97+O(q100) 4 q + 2 q^{5} + 2 q^{7} - 2 q^{11} + 2 q^{13} + 16 q^{19} + 6 q^{23} + 8 q^{25} - 10 q^{29} + 10 q^{31} + 4 q^{35} - 16 q^{37} - 14 q^{41} - 10 q^{43} - 2 q^{47} + 16 q^{53} - 4 q^{55} + 14 q^{59} - 6 q^{61}+ \cdots - 2 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x42x2+4 x^{4} - 2x^{2} + 4 : Copy content Toggle raw display

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

Character values

We give the values of χ\chi on generators for (Z/1728Z)×\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times.

nn 325325 703703 12171217
χ(n)\chi(n) 11 11 1+β1-1 + \beta_{1}

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}
577.1
1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
0 0 0 0.500000 0.866025i 0 −0.724745 1.25529i 0 0 0
577.2 0 0 0 0.500000 0.866025i 0 1.72474 + 2.98735i 0 0 0
1153.1 0 0 0 0.500000 + 0.866025i 0 −0.724745 + 1.25529i 0 0 0
1153.2 0 0 0 0.500000 + 0.866025i 0 1.72474 2.98735i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.i.m 4
3.b odd 2 1 576.2.i.m 4
4.b odd 2 1 1728.2.i.k 4
8.b even 2 1 864.2.i.e 4
8.d odd 2 1 864.2.i.c 4
9.c even 3 1 inner 1728.2.i.m 4
9.c even 3 1 5184.2.a.bj 2
9.d odd 6 1 576.2.i.m 4
9.d odd 6 1 5184.2.a.bu 2
12.b even 2 1 576.2.i.i 4
24.f even 2 1 288.2.i.e yes 4
24.h odd 2 1 288.2.i.c 4
36.f odd 6 1 1728.2.i.k 4
36.f odd 6 1 5184.2.a.bn 2
36.h even 6 1 576.2.i.i 4
36.h even 6 1 5184.2.a.by 2
72.j odd 6 1 288.2.i.c 4
72.j odd 6 1 2592.2.a.j 2
72.l even 6 1 288.2.i.e yes 4
72.l even 6 1 2592.2.a.n 2
72.n even 6 1 864.2.i.e 4
72.n even 6 1 2592.2.a.o 2
72.p odd 6 1 864.2.i.c 4
72.p odd 6 1 2592.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.c 4 24.h odd 2 1
288.2.i.c 4 72.j odd 6 1
288.2.i.e yes 4 24.f even 2 1
288.2.i.e yes 4 72.l even 6 1
576.2.i.i 4 12.b even 2 1
576.2.i.i 4 36.h even 6 1
576.2.i.m 4 3.b odd 2 1
576.2.i.m 4 9.d odd 6 1
864.2.i.c 4 8.d odd 2 1
864.2.i.c 4 72.p odd 6 1
864.2.i.e 4 8.b even 2 1
864.2.i.e 4 72.n even 6 1
1728.2.i.k 4 4.b odd 2 1
1728.2.i.k 4 36.f odd 6 1
1728.2.i.m 4 1.a even 1 1 trivial
1728.2.i.m 4 9.c even 3 1 inner
2592.2.a.j 2 72.j odd 6 1
2592.2.a.n 2 72.l even 6 1
2592.2.a.o 2 72.n even 6 1
2592.2.a.s 2 72.p odd 6 1
5184.2.a.bj 2 9.c even 3 1
5184.2.a.bn 2 36.f odd 6 1
5184.2.a.bu 2 9.d odd 6 1
5184.2.a.by 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(1728,[χ])S_{2}^{\mathrm{new}}(1728, [\chi]):

T52T5+1 T_{5}^{2} - T_{5} + 1 Copy content Toggle raw display
T742T73+9T72+10T7+25 T_{7}^{4} - 2T_{7}^{3} + 9T_{7}^{2} + 10T_{7} + 25 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4 T^{4} Copy content Toggle raw display
33 T4 T^{4} Copy content Toggle raw display
55 (T2T+1)2 (T^{2} - T + 1)^{2} Copy content Toggle raw display
77 T42T3++25 T^{4} - 2 T^{3} + \cdots + 25 Copy content Toggle raw display
1111 T4+2T3++25 T^{4} + 2 T^{3} + \cdots + 25 Copy content Toggle raw display
1313 T42T3++529 T^{4} - 2 T^{3} + \cdots + 529 Copy content Toggle raw display
1717 (T224)2 (T^{2} - 24)^{2} Copy content Toggle raw display
1919 (T4)4 (T - 4)^{4} Copy content Toggle raw display
2323 T46T3++9 T^{4} - 6 T^{3} + \cdots + 9 Copy content Toggle raw display
2929 T4+10T3++1 T^{4} + 10 T^{3} + \cdots + 1 Copy content Toggle raw display
3131 T410T3++361 T^{4} - 10 T^{3} + \cdots + 361 Copy content Toggle raw display
3737 (T2+8T8)2 (T^{2} + 8 T - 8)^{2} Copy content Toggle raw display
4141 T4+14T3++625 T^{4} + 14 T^{3} + \cdots + 625 Copy content Toggle raw display
4343 T4+10T3++841 T^{4} + 10 T^{3} + \cdots + 841 Copy content Toggle raw display
4747 T4+2T3++2809 T^{4} + 2 T^{3} + \cdots + 2809 Copy content Toggle raw display
5353 (T28T8)2 (T^{2} - 8 T - 8)^{2} Copy content Toggle raw display
5959 T414T3++25 T^{4} - 14 T^{3} + \cdots + 25 Copy content Toggle raw display
6161 T4+6T3++225 T^{4} + 6 T^{3} + \cdots + 225 Copy content Toggle raw display
6767 T4+10T3++841 T^{4} + 10 T^{3} + \cdots + 841 Copy content Toggle raw display
7171 (T2+4T92)2 (T^{2} + 4 T - 92)^{2} Copy content Toggle raw display
7373 (T224)2 (T^{2} - 24)^{2} Copy content Toggle raw display
7979 T422T3++13225 T^{4} - 22 T^{3} + \cdots + 13225 Copy content Toggle raw display
8383 T4+6T3++9 T^{4} + 6 T^{3} + \cdots + 9 Copy content Toggle raw display
8989 (T216T+40)2 (T^{2} - 16 T + 40)^{2} Copy content Toggle raw display
9797 T4+2T3++529 T^{4} + 2 T^{3} + \cdots + 529 Copy content Toggle raw display
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