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Properties

Label 2.22.a.a
Level 22
Weight 2222
Character orbit 2.a
Self dual yes
Analytic conductor 5.5905.590
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] = [2,22,Mod(1,2)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: N N == 2 2
Weight: k k == 22 22
Character orbit: [χ][\chi] == 2.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: 5.589546885745.58954688574
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−1024q2+71604q3+1048576q4−28693770q5−73322496q6−853202392q7−1073741824q8−5333220387q9+29382420480q10+86731179612q11+75082235904q12−895323442786q13+⋯−46 ⁣⋯ ⁣44q99+O(q100) q - 1024 q^{2} + 71604 q^{3} + 1048576 q^{4} - 28693770 q^{5} - 73322496 q^{6} - 853202392 q^{7} - 1073741824 q^{8} - 5333220387 q^{9} + 29382420480 q^{10} + 86731179612 q^{11} + 75082235904 q^{12} - 895323442786 q^{13}+ \cdots - 46\!\cdots\!44 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
−1024.00 71604.0 1.04858e6 −2.86938e7 −7.33225e7 −8.53202e8 −1.07374e9 −5.33322e9 2.93824e10
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 +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 2.22.a.a ✓ 1
3.b odd 2 1 18.22.a.e 1
4.b odd 2 1 16.22.a.a 1
5.b even 2 1 50.22.a.c 1
5.c odd 4 2 50.22.b.a 2
8.b even 2 1 64.22.a.b 1
8.d odd 2 1 64.22.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.22.a.a ✓ 1 1.a even 1 1 trivial
16.22.a.a 1 4.b odd 2 1
18.22.a.e 1 3.b odd 2 1
50.22.a.c 1 5.b even 2 1
50.22.b.a 2 5.c odd 4 2
64.22.a.b 1 8.b even 2 1
64.22.a.f 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3−71604 T_{3} - 71604 acting on S22new(Γ0(2))S_{22}^{\mathrm{new}}(\Gamma_0(2)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T+1024 T + 1024 Copy content Toggle raw display
33 T−71604 T - 71604 Copy content Toggle raw display
55 T+28693770 T + 28693770 Copy content Toggle raw display
77 T+853202392 T + 853202392 Copy content Toggle raw display
1111 T−86731179612 T - 86731179612 Copy content Toggle raw display
1313 T+895323442786 T + 895323442786 Copy content Toggle raw display
1717 T−3257566804818 T - 3257566804818 Copy content Toggle raw display
1919 T−23032467644420 T - 23032467644420 Copy content Toggle raw display
2323 T−146495714575224 T - 146495714575224 Copy content Toggle raw display
2929 T+734051633521170 T + 734051633521170 Copy content Toggle raw display
3131 T+3146664162057568 T + 3146664162057568 Copy content Toggle raw display
3737 T+12 ⁣⋯ ⁣62 T + 12\!\cdots\!62 Copy content Toggle raw display
4141 T−45 ⁣⋯ ⁣42 T - 45\!\cdots\!42 Copy content Toggle raw display
4343 T+24 ⁣⋯ ⁣56 T + 24\!\cdots\!56 Copy content Toggle raw display
4747 T+44 ⁣⋯ ⁣52 T + 44\!\cdots\!52 Copy content Toggle raw display
5353 T−20 ⁣⋯ ⁣54 T - 20\!\cdots\!54 Copy content Toggle raw display
5959 T+37 ⁣⋯ ⁣40 T + 37\!\cdots\!40 Copy content Toggle raw display
6161 T+76 ⁣⋯ ⁣38 T + 76\!\cdots\!38 Copy content Toggle raw display
6767 T+18 ⁣⋯ ⁣32 T + 18\!\cdots\!32 Copy content Toggle raw display
7171 T+45 ⁣⋯ ⁣28 T + 45\!\cdots\!28 Copy content Toggle raw display
7373 T+25 ⁣⋯ ⁣26 T + 25\!\cdots\!26 Copy content Toggle raw display
7979 T−99 ⁣⋯ ⁣80 T - 99\!\cdots\!80 Copy content Toggle raw display
8383 T−29 ⁣⋯ ⁣84 T - 29\!\cdots\!84 Copy content Toggle raw display
8989 T−11 ⁣⋯ ⁣90 T - 11\!\cdots\!90 Copy content Toggle raw display
9797 T+56 ⁣⋯ ⁣02 T + 56\!\cdots\!02 Copy content Toggle raw display
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