Properties

Label 9.20.a.a
Level 99
Weight 2020
Character orbit 9.a
Self dual yes
Analytic conductor 20.59420.594
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] = [9,20,Mod(1,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: N N == 9=32 9 = 3^{2}
Weight: k k == 20 20
Character orbit: [χ][\chi] == 9.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: 20.593502690120.5935026901
Analytic rank: 11
Dimension: 11
Coefficient field: Q\mathbb{Q}
Coefficient ring: Z\mathbb{Z}
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 1)
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) == q456q2316352q4+2377410q516917544q7+383331840q81084098960q10+16212108q11+50421615062q13+7714400064q148939761664q16225070099506q17++50 ⁣ ⁣92q98+O(q100) q - 456 q^{2} - 316352 q^{4} + 2377410 q^{5} - 16917544 q^{7} + 383331840 q^{8} - 1084098960 q^{10} + 16212108 q^{11} + 50421615062 q^{13} + 7714400064 q^{14} - 8939761664 q^{16} - 225070099506 q^{17}+ \cdots + 50\!\cdots\!92 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
−456.000 0 −316352. 2.37741e6 0 −1.69175e7 3.83332e8 0 −1.08410e9
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
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 9.20.a.a 1
3.b odd 2 1 1.20.a.a 1
12.b even 2 1 16.20.a.a 1
15.d odd 2 1 25.20.a.a 1
15.e even 4 2 25.20.b.a 2
21.c even 2 1 49.20.a.b 1
24.f even 2 1 64.20.a.h 1
24.h odd 2 1 64.20.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.20.a.a 1 3.b odd 2 1
9.20.a.a 1 1.a even 1 1 trivial
16.20.a.a 1 12.b even 2 1
25.20.a.a 1 15.d odd 2 1
25.20.b.a 2 15.e even 4 2
49.20.a.b 1 21.c even 2 1
64.20.a.b 1 24.h odd 2 1
64.20.a.h 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2+456 T_{2} + 456 acting on S20new(Γ0(9))S_{20}^{\mathrm{new}}(\Gamma_0(9)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T+456 T + 456 Copy content Toggle raw display
33 T T Copy content Toggle raw display
55 T2377410 T - 2377410 Copy content Toggle raw display
77 T+16917544 T + 16917544 Copy content Toggle raw display
1111 T16212108 T - 16212108 Copy content Toggle raw display
1313 T50421615062 T - 50421615062 Copy content Toggle raw display
1717 T+225070099506 T + 225070099506 Copy content Toggle raw display
1919 T+1710278572660 T + 1710278572660 Copy content Toggle raw display
2323 T+14036534788872 T + 14036534788872 Copy content Toggle raw display
2929 T+1137835269510 T + 1137835269510 Copy content Toggle raw display
3131 T+104626880141728 T + 104626880141728 Copy content Toggle raw display
3737 T+169392327370594 T + 169392327370594 Copy content Toggle raw display
4141 T3309984750560838 T - 3309984750560838 Copy content Toggle raw display
4343 T1127913532193492 T - 1127913532193492 Copy content Toggle raw display
4747 T+3498693987674256 T + 3498693987674256 Copy content Toggle raw display
5353 T+29 ⁣ ⁣02 T + 29\!\cdots\!02 Copy content Toggle raw display
5959 T+58 ⁣ ⁣20 T + 58\!\cdots\!20 Copy content Toggle raw display
6161 T23 ⁣ ⁣42 T - 23\!\cdots\!42 Copy content Toggle raw display
6767 T+20 ⁣ ⁣44 T + 20\!\cdots\!44 Copy content Toggle raw display
7171 T17 ⁣ ⁣68 T - 17\!\cdots\!68 Copy content Toggle raw display
7373 T29 ⁣ ⁣22 T - 29\!\cdots\!22 Copy content Toggle raw display
7979 T+92 ⁣ ⁣40 T + 92\!\cdots\!40 Copy content Toggle raw display
8383 T+12 ⁣ ⁣32 T + 12\!\cdots\!32 Copy content Toggle raw display
8989 T+43 ⁣ ⁣30 T + 43\!\cdots\!30 Copy content Toggle raw display
9797 T+63 ⁣ ⁣94 T + 63\!\cdots\!94 Copy content Toggle raw display
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