Properties

Label 6.18.a.a
Level 66
Weight 1818
Character orbit 6.a
Self dual yes
Analytic conductor 10.99310.993
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] = [6,18,Mod(1,6)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: N N == 6=23 6 = 2 \cdot 3
Weight: k k == 18 18
Character orbit: [χ][\chi] == 6.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: 10.993325240710.9933252407
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) == q256q26561q3+65536q4+645150q5+1679616q6+3974432q716777216q8+43046721q9165158400q10500068668q11429981696q125425661314q131017454592q14+21 ⁣ ⁣28q99+O(q100) q - 256 q^{2} - 6561 q^{3} + 65536 q^{4} + 645150 q^{5} + 1679616 q^{6} + 3974432 q^{7} - 16777216 q^{8} + 43046721 q^{9} - 165158400 q^{10} - 500068668 q^{11} - 429981696 q^{12} - 5425661314 q^{13} - 1017454592 q^{14}+ \cdots - 21\!\cdots\!28 q^{99}+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
−256.000 −6561.00 65536.0 645150. 1.67962e6 3.97443e6 −1.67772e7 4.30467e7 −1.65158e8
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 +1 +1
33 +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 6.18.a.a 1
3.b odd 2 1 18.18.a.c 1
4.b odd 2 1 48.18.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.18.a.a 1 1.a even 1 1 trivial
18.18.a.c 1 3.b odd 2 1
48.18.a.f 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5645150 T_{5} - 645150 acting on S18new(Γ0(6))S_{18}^{\mathrm{new}}(\Gamma_0(6)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T+256 T + 256 Copy content Toggle raw display
33 T+6561 T + 6561 Copy content Toggle raw display
55 T645150 T - 645150 Copy content Toggle raw display
77 T3974432 T - 3974432 Copy content Toggle raw display
1111 T+500068668 T + 500068668 Copy content Toggle raw display
1313 T+5425661314 T + 5425661314 Copy content Toggle raw display
1717 T+5466992958 T + 5466992958 Copy content Toggle raw display
1919 T+53889877060 T + 53889877060 Copy content Toggle raw display
2323 T578906836536 T - 578906836536 Copy content Toggle raw display
2929 T+4619583681690 T + 4619583681690 Copy content Toggle raw display
3131 T+6802815567448 T + 6802815567448 Copy content Toggle raw display
3737 T+19571909422138 T + 19571909422138 Copy content Toggle raw display
4141 T57213620756922 T - 57213620756922 Copy content Toggle raw display
4343 T+24501250225084 T + 24501250225084 Copy content Toggle raw display
4747 T184283998832832 T - 184283998832832 Copy content Toggle raw display
5353 T+206542562280354 T + 206542562280354 Copy content Toggle raw display
5959 T+418648048246140 T + 418648048246140 Copy content Toggle raw display
6161 T2501287878088382 T - 2501287878088382 Copy content Toggle raw display
6767 T+145692866050948 T + 145692866050948 Copy content Toggle raw display
7171 T+5364313152664248 T + 5364313152664248 Copy content Toggle raw display
7373 T3302058927938186 T - 3302058927938186 Copy content Toggle raw display
7979 T22 ⁣ ⁣60 T - 22\!\cdots\!60 Copy content Toggle raw display
8383 T20 ⁣ ⁣76 T - 20\!\cdots\!76 Copy content Toggle raw display
8989 T+56 ⁣ ⁣10 T + 56\!\cdots\!10 Copy content Toggle raw display
9797 T+11 ⁣ ⁣18 T + 11\!\cdots\!18 Copy content Toggle raw display
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