Properties

Label 1638.2.c.e
Level 16381638
Weight 22
Character orbit 1638.c
Analytic conductor 13.07913.079
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] = [1638,2,Mod(883,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.883");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1638=232713 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1638.c (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: 13.079495851113.0794958511
Analytic rank: 00
Dimension: 22
Coefficient field: Q(1)\Q(\sqrt{-1})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2+1 x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 546)
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 i=1i = \sqrt{-1}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == qiq2q4+iq5iq7+iq8+q103iq11+(2i+3)q13q14+q167q17+3iq19iq203q22q23+4q25+(3i2)q26++iq98+O(q100) q - i q^{2} - q^{4} + i q^{5} - i q^{7} + i q^{8} + q^{10} - 3 i q^{11} + ( - 2 i + 3) q^{13} - q^{14} + q^{16} - 7 q^{17} + 3 i q^{19} - i q^{20} - 3 q^{22} - q^{23} + 4 q^{25} + ( - 3 i - 2) q^{26} + \cdots + i q^{98} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q2q4+2q10+6q132q14+2q1614q176q222q23+8q254q26+2q29+2q35+6q382q4010q432q496q52+12q53+6q95+O(q100) 2 q - 2 q^{4} + 2 q^{10} + 6 q^{13} - 2 q^{14} + 2 q^{16} - 14 q^{17} - 6 q^{22} - 2 q^{23} + 8 q^{25} - 4 q^{26} + 2 q^{29} + 2 q^{35} + 6 q^{38} - 2 q^{40} - 10 q^{43} - 2 q^{49} - 6 q^{52} + 12 q^{53}+ \cdots - 6 q^{95}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1638Z)×\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times.

nn 379379 703703 911911
χ(n)\chi(n) 1-1 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}
883.1
1.00000i
1.00000i
1.00000i 0 −1.00000 1.00000i 0 1.00000i 1.00000i 0 1.00000
883.2 1.00000i 0 −1.00000 1.00000i 0 1.00000i 1.00000i 0 1.00000
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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.c.e 2
3.b odd 2 1 546.2.c.c 2
12.b even 2 1 4368.2.h.h 2
13.b even 2 1 inner 1638.2.c.e 2
21.c even 2 1 3822.2.c.b 2
39.d odd 2 1 546.2.c.c 2
39.f even 4 1 7098.2.a.o 1
39.f even 4 1 7098.2.a.y 1
156.h even 2 1 4368.2.h.h 2
273.g even 2 1 3822.2.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.c 2 3.b odd 2 1
546.2.c.c 2 39.d odd 2 1
1638.2.c.e 2 1.a even 1 1 trivial
1638.2.c.e 2 13.b even 2 1 inner
3822.2.c.b 2 21.c even 2 1
3822.2.c.b 2 273.g even 2 1
4368.2.h.h 2 12.b even 2 1
4368.2.h.h 2 156.h even 2 1
7098.2.a.o 1 39.f even 4 1
7098.2.a.y 1 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(1638,[χ])S_{2}^{\mathrm{new}}(1638, [\chi]):

T52+1 T_{5}^{2} + 1 Copy content Toggle raw display
T112+9 T_{11}^{2} + 9 Copy content Toggle raw display
T17+7 T_{17} + 7 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+1 T^{2} + 1 Copy content Toggle raw display
33 T2 T^{2} Copy content Toggle raw display
55 T2+1 T^{2} + 1 Copy content Toggle raw display
77 T2+1 T^{2} + 1 Copy content Toggle raw display
1111 T2+9 T^{2} + 9 Copy content Toggle raw display
1313 T26T+13 T^{2} - 6T + 13 Copy content Toggle raw display
1717 (T+7)2 (T + 7)^{2} Copy content Toggle raw display
1919 T2+9 T^{2} + 9 Copy content Toggle raw display
2323 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
2929 (T1)2 (T - 1)^{2} Copy content Toggle raw display
3131 T2+64 T^{2} + 64 Copy content Toggle raw display
3737 T2+1 T^{2} + 1 Copy content Toggle raw display
4141 T2+16 T^{2} + 16 Copy content Toggle raw display
4343 (T+5)2 (T + 5)^{2} Copy content Toggle raw display
4747 T2 T^{2} Copy content Toggle raw display
5353 (T6)2 (T - 6)^{2} Copy content Toggle raw display
5959 T2+100 T^{2} + 100 Copy content Toggle raw display
6161 (T+13)2 (T + 13)^{2} Copy content Toggle raw display
6767 T2+64 T^{2} + 64 Copy content Toggle raw display
7171 T2+36 T^{2} + 36 Copy content Toggle raw display
7373 T2+169 T^{2} + 169 Copy content Toggle raw display
7979 (T+12)2 (T + 12)^{2} Copy content Toggle raw display
8383 T2+4 T^{2} + 4 Copy content Toggle raw display
8989 T2+144 T^{2} + 144 Copy content Toggle raw display
9797 T2+36 T^{2} + 36 Copy content Toggle raw display
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