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Properties

Label 2.48.a.b
Level 22
Weight 4848
Character orbit 2.a
Self dual yes
Analytic conductor 27.98227.982
Analytic rank 00
Dimension 22
CM no
Inner twists 11

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2,48,Mod(1,2)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 48, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2.1"); S:= CuspForms(chi, 48); N := Newforms(S);
 
Level: N N == 2 2
Weight: k k == 48 48
Character orbit: [χ][\chi] == 2.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 27.981532531027.9815325310
Analytic rank: 00
Dimension: 22
Coefficient field: Q[x]/(x2−⋯ )\mathbb{Q}[x]/(x^{2} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2−x−5897345978580 x^{2} - x - 5897345978580 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 27⋅33⋅5 2^{7}\cdot 3^{3}\cdot 5
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of β=864023589383914321\beta = 8640\sqrt{23589383914321}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q−8388608q2+(−5β+61144922412)q3+70368744177664q4+(460908β+90 ⁣⋯ ⁣70)q5+(41943040β−51 ⁣⋯ ⁣96)q6+(−675210690β+80 ⁣⋯ ⁣16)q7+⋯+(−20 ⁣⋯ ⁣75β+98 ⁣⋯ ⁣44)q99+O(q100) q - 8388608 q^{2} + ( - 5 \beta + 61144922412) q^{3} + 70368744177664 q^{4} + (460908 \beta + 90\!\cdots\!70) q^{5} + (41943040 \beta - 51\!\cdots\!96) q^{6} + ( - 675210690 \beta + 80\!\cdots\!16) q^{7}+ \cdots + ( - 20\!\cdots\!75 \beta + 98\!\cdots\!44) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 2q−16777216q2+122289844824q3+140737488355328q4+18 ⁣⋯ ⁣40q5−10 ⁣⋯ ⁣92q6+16 ⁣⋯ ⁣32q7−11 ⁣⋯ ⁣24q8+42 ⁣⋯ ⁣14q9−15 ⁣⋯ ⁣20q10+⋯+19 ⁣⋯ ⁣88q99+O(q100) 2 q - 16777216 q^{2} + 122289844824 q^{3} + 140737488355328 q^{4} + 18\!\cdots\!40 q^{5} - 10\!\cdots\!92 q^{6} + 16\!\cdots\!32 q^{7} - 11\!\cdots\!24 q^{8} + 42\!\cdots\!14 q^{9} - 15\!\cdots\!20 q^{10}+ \cdots + 19\!\cdots\!88 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
2.42845e6
−2.42844e6
−8.38861e6 −1.48673e11 7.03687e13 2.84306e16 1.24716e18 5.21750e19 −5.90296e20 −4.48523e21 −2.38493e23
1.2 −8.38861e6 2.70963e11 7.03687e13 −1.02521e16 −2.27300e18 1.08843e20 −5.90296e20 4.68319e22 8.60007e22
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 2.48.a.b ✓ 2
4.b odd 2 1 16.48.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.48.a.b ✓ 2 1.a even 1 1 trivial
16.48.a.b 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T32−122289844824T3−40284750299492923142256 T_{3}^{2} - 122289844824T_{3} - 40284750299492923142256 acting on S48new(Γ0(2))S_{48}^{\mathrm{new}}(\Gamma_0(2)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T+8388608)2 (T + 8388608)^{2} Copy content Toggle raw display
33 T2+⋯−40 ⁣⋯ ⁣56 T^{2} + \cdots - 40\!\cdots\!56 Copy content Toggle raw display
55 T2+⋯−29 ⁣⋯ ⁣00 T^{2} + \cdots - 29\!\cdots\!00 Copy content Toggle raw display
77 T2+⋯+56 ⁣⋯ ⁣56 T^{2} + \cdots + 56\!\cdots\!56 Copy content Toggle raw display
1111 T2+⋯−10 ⁣⋯ ⁣36 T^{2} + \cdots - 10\!\cdots\!36 Copy content Toggle raw display
1313 T2+⋯−33 ⁣⋯ ⁣76 T^{2} + \cdots - 33\!\cdots\!76 Copy content Toggle raw display
1717 T2+⋯−58 ⁣⋯ ⁣64 T^{2} + \cdots - 58\!\cdots\!64 Copy content Toggle raw display
1919 T2+⋯−18 ⁣⋯ ⁣00 T^{2} + \cdots - 18\!\cdots\!00 Copy content Toggle raw display
2323 T2+⋯−81 ⁣⋯ ⁣56 T^{2} + \cdots - 81\!\cdots\!56 Copy content Toggle raw display
2929 T2+⋯+43 ⁣⋯ ⁣00 T^{2} + \cdots + 43\!\cdots\!00 Copy content Toggle raw display
3131 T2+⋯+62 ⁣⋯ ⁣04 T^{2} + \cdots + 62\!\cdots\!04 Copy content Toggle raw display
3737 T2+⋯+49 ⁣⋯ ⁣76 T^{2} + \cdots + 49\!\cdots\!76 Copy content Toggle raw display
4141 T2+⋯+82 ⁣⋯ ⁣64 T^{2} + \cdots + 82\!\cdots\!64 Copy content Toggle raw display
4343 T2+⋯−43 ⁣⋯ ⁣96 T^{2} + \cdots - 43\!\cdots\!96 Copy content Toggle raw display
4747 T2+⋯−22 ⁣⋯ ⁣64 T^{2} + \cdots - 22\!\cdots\!64 Copy content Toggle raw display
5353 T2+⋯+30 ⁣⋯ ⁣24 T^{2} + \cdots + 30\!\cdots\!24 Copy content Toggle raw display
5959 T2+⋯−22 ⁣⋯ ⁣00 T^{2} + \cdots - 22\!\cdots\!00 Copy content Toggle raw display
6161 T2+⋯+13 ⁣⋯ ⁣04 T^{2} + \cdots + 13\!\cdots\!04 Copy content Toggle raw display
6767 T2+⋯−16 ⁣⋯ ⁣84 T^{2} + \cdots - 16\!\cdots\!84 Copy content Toggle raw display
7171 T2+⋯+30 ⁣⋯ ⁣84 T^{2} + \cdots + 30\!\cdots\!84 Copy content Toggle raw display
7373 T2+⋯+23 ⁣⋯ ⁣24 T^{2} + \cdots + 23\!\cdots\!24 Copy content Toggle raw display
7979 T2+⋯−11 ⁣⋯ ⁣00 T^{2} + \cdots - 11\!\cdots\!00 Copy content Toggle raw display
8383 T2+⋯−24 ⁣⋯ ⁣36 T^{2} + \cdots - 24\!\cdots\!36 Copy content Toggle raw display
8989 T2+⋯+37 ⁣⋯ ⁣00 T^{2} + \cdots + 37\!\cdots\!00 Copy content Toggle raw display
9797 T2+⋯−35 ⁣⋯ ⁣84 T^{2} + \cdots - 35\!\cdots\!84 Copy content Toggle raw display
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