Properties

Label 1638.2.a.n
Level 16381638
Weight 22
Character orbit 1638.a
Self dual yes
Analytic conductor 13.07913.079
Analytic rank 11
Dimension 11
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] = [1638,2,Mod(1,1638)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.1");
 
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.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: 13.079495851113.0794958511
Analytic rank: 11
Dimension: 11
Coefficient field: Q\mathbb{Q}
Coefficient ring: Z\mathbb{Z}
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)
 
f(q)f(q) == q+q2+q4q5q7+q8q105q11+q13q14+q16q17q19q205q225q234q25+q26q28q296q31++q98+O(q100) q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - q^{10} - 5 q^{11} + q^{13} - q^{14} + q^{16} - q^{17} - q^{19} - q^{20} - 5 q^{22} - 5 q^{23} - 4 q^{25} + q^{26} - q^{28} - q^{29} - 6 q^{31}+ \cdots + q^{98}+O(q^{100}) 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
0
1.00000 0 1.00000 −1.00000 0 −1.00000 1.00000 0 −1.00000
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 1 -1
33 +1 +1
77 +1 +1
1313 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 1638.2.a.n yes 1
3.b odd 2 1 1638.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1638.2.a.g 1 3.b odd 2 1
1638.2.a.n yes 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(1638))S_{2}^{\mathrm{new}}(\Gamma_0(1638)):

T5+1 T_{5} + 1 Copy content Toggle raw display
T11+5 T_{11} + 5 Copy content Toggle raw display
T17+1 T_{17} + 1 Copy content Toggle raw display
T19+1 T_{19} + 1 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T1 T - 1 Copy content Toggle raw display
33 T T Copy content Toggle raw display
55 T+1 T + 1 Copy content Toggle raw display
77 T+1 T + 1 Copy content Toggle raw display
1111 T+5 T + 5 Copy content Toggle raw display
1313 T1 T - 1 Copy content Toggle raw display
1717 T+1 T + 1 Copy content Toggle raw display
1919 T+1 T + 1 Copy content Toggle raw display
2323 T+5 T + 5 Copy content Toggle raw display
2929 T+1 T + 1 Copy content Toggle raw display
3131 T+6 T + 6 Copy content Toggle raw display
3737 T7 T - 7 Copy content Toggle raw display
4141 T+2 T + 2 Copy content Toggle raw display
4343 T1 T - 1 Copy content Toggle raw display
4747 T T Copy content Toggle raw display
5353 T+6 T + 6 Copy content Toggle raw display
5959 T+6 T + 6 Copy content Toggle raw display
6161 T+7 T + 7 Copy content Toggle raw display
6767 T+8 T + 8 Copy content Toggle raw display
7171 T+10 T + 10 Copy content Toggle raw display
7373 T9 T - 9 Copy content Toggle raw display
7979 T2 T - 2 Copy content Toggle raw display
8383 T+2 T + 2 Copy content Toggle raw display
8989 T6 T - 6 Copy content Toggle raw display
9797 T14 T - 14 Copy content Toggle raw display
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