Properties

Label 624.4.q.c
Level 624624
Weight 44
Character orbit 624.q
Analytic conductor 36.81736.817
Analytic rank 00
Dimension 22
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,4,Mod(289,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.289");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 624=24313 624 = 2^{4} \cdot 3 \cdot 13
Weight: k k == 4 4
Character orbit: [χ][\chi] == 624.q (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: 36.817191843636.8171918436
Analytic rank: 00
Dimension: 22
Coefficient field: Q(3)\Q(\sqrt{-3})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2x+1 x^{2} - x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 39)
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 primitive root of unity ζ6\zeta_{6}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(3ζ6+3)q3+7q510ζ6q79ζ6q9+(22ζ622)q11+(13ζ639)q13+(21ζ6+21)q1537ζ6q17+30ζ6q19++198q99+O(q100) q + ( - 3 \zeta_{6} + 3) q^{3} + 7 q^{5} - 10 \zeta_{6} q^{7} - 9 \zeta_{6} q^{9} + (22 \zeta_{6} - 22) q^{11} + ( - 13 \zeta_{6} - 39) q^{13} + ( - 21 \zeta_{6} + 21) q^{15} - 37 \zeta_{6} q^{17} + 30 \zeta_{6} q^{19} + \cdots + 198 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+3q3+14q510q79q922q1191q13+21q1537q17+30q1960q21162q23152q2554q27+113q29392q31+66q3370q35++396q99+O(q100) 2 q + 3 q^{3} + 14 q^{5} - 10 q^{7} - 9 q^{9} - 22 q^{11} - 91 q^{13} + 21 q^{15} - 37 q^{17} + 30 q^{19} - 60 q^{21} - 162 q^{23} - 152 q^{25} - 54 q^{27} + 113 q^{29} - 392 q^{31} + 66 q^{33} - 70 q^{35}+ \cdots + 396 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/624Z)×\left(\mathbb{Z}/624\mathbb{Z}\right)^\times.

nn 7979 145145 209209 469469
χ(n)\chi(n) 11 ζ6-\zeta_{6} 11 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}
289.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.50000 + 2.59808i 0 7.00000 0 −5.00000 + 8.66025i 0 −4.50000 + 7.79423i 0
529.1 0 1.50000 2.59808i 0 7.00000 0 −5.00000 8.66025i 0 −4.50000 7.79423i 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.q.c 2
4.b odd 2 1 39.4.e.b 2
12.b even 2 1 117.4.g.a 2
13.c even 3 1 inner 624.4.q.c 2
52.i odd 6 1 507.4.a.d 1
52.j odd 6 1 39.4.e.b 2
52.j odd 6 1 507.4.a.b 1
52.l even 12 2 507.4.b.d 2
156.p even 6 1 117.4.g.a 2
156.p even 6 1 1521.4.a.h 1
156.r even 6 1 1521.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.b 2 4.b odd 2 1
39.4.e.b 2 52.j odd 6 1
117.4.g.a 2 12.b even 2 1
117.4.g.a 2 156.p even 6 1
507.4.a.b 1 52.j odd 6 1
507.4.a.d 1 52.i odd 6 1
507.4.b.d 2 52.l even 12 2
624.4.q.c 2 1.a even 1 1 trivial
624.4.q.c 2 13.c even 3 1 inner
1521.4.a.e 1 156.r even 6 1
1521.4.a.h 1 156.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(624,[χ])S_{4}^{\mathrm{new}}(624, [\chi]):

T57 T_{5} - 7 Copy content Toggle raw display
T72+10T7+100 T_{7}^{2} + 10T_{7} + 100 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 T23T+9 T^{2} - 3T + 9 Copy content Toggle raw display
55 (T7)2 (T - 7)^{2} Copy content Toggle raw display
77 T2+10T+100 T^{2} + 10T + 100 Copy content Toggle raw display
1111 T2+22T+484 T^{2} + 22T + 484 Copy content Toggle raw display
1313 T2+91T+2197 T^{2} + 91T + 2197 Copy content Toggle raw display
1717 T2+37T+1369 T^{2} + 37T + 1369 Copy content Toggle raw display
1919 T230T+900 T^{2} - 30T + 900 Copy content Toggle raw display
2323 T2+162T+26244 T^{2} + 162T + 26244 Copy content Toggle raw display
2929 T2113T+12769 T^{2} - 113T + 12769 Copy content Toggle raw display
3131 (T+196)2 (T + 196)^{2} Copy content Toggle raw display
3737 T2+13T+169 T^{2} + 13T + 169 Copy content Toggle raw display
4141 T2+285T+81225 T^{2} + 285T + 81225 Copy content Toggle raw display
4343 T2+246T+60516 T^{2} + 246T + 60516 Copy content Toggle raw display
4747 (T462)2 (T - 462)^{2} Copy content Toggle raw display
5353 (T+537)2 (T + 537)^{2} Copy content Toggle raw display
5959 T2576T+331776 T^{2} - 576T + 331776 Copy content Toggle raw display
6161 T2635T+403225 T^{2} - 635T + 403225 Copy content Toggle raw display
6767 T2202T+40804 T^{2} - 202T + 40804 Copy content Toggle raw display
7171 T2+1086T+1179396 T^{2} + 1086 T + 1179396 Copy content Toggle raw display
7373 (T+805)2 (T + 805)^{2} Copy content Toggle raw display
7979 (T+884)2 (T + 884)^{2} Copy content Toggle raw display
8383 (T+518)2 (T + 518)^{2} Copy content Toggle raw display
8989 T2+194T+37636 T^{2} + 194T + 37636 Copy content Toggle raw display
9797 T21202T+1444804 T^{2} - 1202 T + 1444804 Copy content Toggle raw display
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