Properties

Label 3969.2.a.bd
Level 39693969
Weight 22
Character orbit 3969.a
Self dual yes
Analytic conductor 31.69331.693
Analytic rank 11
Dimension 66
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3969,2,Mod(1,3969)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3969, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3969.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 3969=3472 3969 = 3^{4} \cdot 7^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 3969.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-2,0,6,0,0,0,-12,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 31.692624562231.6926245622
Analytic rank: 11
Dimension: 66
Coefficient field: 6.6.59351616.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x612x4+21x29 x^{6} - 12x^{4} + 21x^{2} - 9 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 3 3
Twist minimal: no (minimal twist has level 441)
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 a basis 1,β1,,β51,\beta_1,\ldots,\beta_{5} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == qβ2q2+(β4+β2+1)q4β1q5+(β4β22)q8+(β5+2β1)q10+(β41)q11+β3q13+(β4+2β2)q16++(β5+β3+2β1)q97+O(q100) q - \beta_{2} q^{2} + (\beta_{4} + \beta_{2} + 1) q^{4} - \beta_1 q^{5} + ( - \beta_{4} - \beta_{2} - 2) q^{8} + (\beta_{5} + 2 \beta_1) q^{10} + (\beta_{4} - 1) q^{11} + \beta_{3} q^{13} + ( - \beta_{4} + 2 \beta_{2}) q^{16}+ \cdots + (\beta_{5} + \beta_{3} + 2 \beta_1) q^{97}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q2q2+6q412q88q11+6q16+6q224q23+12q2522q2916q326q37+6q43+14q44+12q4656q5028q53+18q5824q64+60q95+O(q100) 6 q - 2 q^{2} + 6 q^{4} - 12 q^{8} - 8 q^{11} + 6 q^{16} + 6 q^{22} - 4 q^{23} + 12 q^{25} - 22 q^{29} - 16 q^{32} - 6 q^{37} + 6 q^{43} + 14 q^{44} + 12 q^{46} - 56 q^{50} - 28 q^{53} + 18 q^{58} - 24 q^{64}+ \cdots - 60 q^{95}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x612x4+21x29 x^{6} - 12x^{4} + 21x^{2} - 9 : Copy content Toggle raw display

β1\beta_{1}== (ν512ν3+21ν)/3 ( \nu^{5} - 12\nu^{3} + 21\nu ) / 3 Copy content Toggle raw display
β2\beta_{2}== (ν412ν2+15)/3 ( \nu^{4} - 12\nu^{2} + 15 ) / 3 Copy content Toggle raw display
β3\beta_{3}== (2ν521ν3+15ν)/3 ( 2\nu^{5} - 21\nu^{3} + 15\nu ) / 3 Copy content Toggle raw display
β4\beta_{4}== (2ν421ν2+15)/3 ( 2\nu^{4} - 21\nu^{2} + 15 ) / 3 Copy content Toggle raw display
β5\beta_{5}== ν511ν3+9ν \nu^{5} - 11\nu^{3} + 9\nu Copy content Toggle raw display
ν\nu== (β5+β3+β1)/3 ( -\beta_{5} + \beta_{3} + \beta_1 ) / 3 Copy content Toggle raw display
ν2\nu^{2}== β42β2+5 \beta_{4} - 2\beta_{2} + 5 Copy content Toggle raw display
ν3\nu^{3}== 3β5+4β3+β1 -3\beta_{5} + 4\beta_{3} + \beta_1 Copy content Toggle raw display
ν4\nu^{4}== 12β421β2+45 12\beta_{4} - 21\beta_{2} + 45 Copy content Toggle raw display
ν5\nu^{5}== 29β5+41β3+8β1 -29\beta_{5} + 41\beta_{3} + 8\beta_1 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
0.820103
−0.820103
1.15750
−1.15750
3.16032
−3.16032
−2.46050 0 4.05408 −3.65808 0 0 −5.05408 0 9.00071
1.2 −2.46050 0 4.05408 3.65808 0 0 −5.05408 0 −9.00071
1.3 −0.239123 0 −1.94282 −2.59179 0 0 0.942820 0 0.619757
1.4 −0.239123 0 −1.94282 2.59179 0 0 0.942820 0 −0.619757
1.5 1.69963 0 0.888736 −0.949271 0 0 −1.88874 0 −1.61341
1.6 1.69963 0 0.888736 0.949271 0 0 −1.88874 0 1.61341
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 1 -1
77 1 -1

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3969.2.a.bd 6
3.b odd 2 1 3969.2.a.be 6
7.b odd 2 1 inner 3969.2.a.bd 6
9.c even 3 2 1323.2.f.g 12
9.d odd 6 2 441.2.f.g 12
21.c even 2 1 3969.2.a.be 6
63.g even 3 2 1323.2.g.g 12
63.h even 3 2 1323.2.h.g 12
63.i even 6 2 441.2.h.g 12
63.j odd 6 2 441.2.h.g 12
63.k odd 6 2 1323.2.g.g 12
63.l odd 6 2 1323.2.f.g 12
63.n odd 6 2 441.2.g.g 12
63.o even 6 2 441.2.f.g 12
63.s even 6 2 441.2.g.g 12
63.t odd 6 2 1323.2.h.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.f.g 12 9.d odd 6 2
441.2.f.g 12 63.o even 6 2
441.2.g.g 12 63.n odd 6 2
441.2.g.g 12 63.s even 6 2
441.2.h.g 12 63.i even 6 2
441.2.h.g 12 63.j odd 6 2
1323.2.f.g 12 9.c even 3 2
1323.2.f.g 12 63.l odd 6 2
1323.2.g.g 12 63.g even 3 2
1323.2.g.g 12 63.k odd 6 2
1323.2.h.g 12 63.h even 3 2
1323.2.h.g 12 63.t odd 6 2
3969.2.a.bd 6 1.a even 1 1 trivial
3969.2.a.bd 6 7.b odd 2 1 inner
3969.2.a.be 6 3.b odd 2 1
3969.2.a.be 6 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(3969))S_{2}^{\mathrm{new}}(\Gamma_0(3969)):

T23+T224T21 T_{2}^{3} + T_{2}^{2} - 4T_{2} - 1 Copy content Toggle raw display
T5621T54+108T5281 T_{5}^{6} - 21T_{5}^{4} + 108T_{5}^{2} - 81 Copy content Toggle raw display
T113+4T112T111 T_{11}^{3} + 4T_{11}^{2} - T_{11} - 1 Copy content Toggle raw display
T13639T134+351T13281 T_{13}^{6} - 39T_{13}^{4} + 351T_{13}^{2} - 81 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T3+T24T1)2 (T^{3} + T^{2} - 4 T - 1)^{2} Copy content Toggle raw display
33 T6 T^{6} Copy content Toggle raw display
55 T621T4+81 T^{6} - 21 T^{4} + \cdots - 81 Copy content Toggle raw display
77 T6 T^{6} Copy content Toggle raw display
1111 (T3+4T2T1)2 (T^{3} + 4 T^{2} - T - 1)^{2} Copy content Toggle raw display
1313 T639T4+81 T^{6} - 39 T^{4} + \cdots - 81 Copy content Toggle raw display
1717 T684T4+3969 T^{6} - 84 T^{4} + \cdots - 3969 Copy content Toggle raw display
1919 T675T4+3969 T^{6} - 75 T^{4} + \cdots - 3969 Copy content Toggle raw display
2323 (T3+2T225T59)2 (T^{3} + 2 T^{2} - 25 T - 59)^{2} Copy content Toggle raw display
2929 (T3+11T2+89)2 (T^{3} + 11 T^{2} + \cdots - 89)^{2} Copy content Toggle raw display
3131 T6129T4+77841 T^{6} - 129 T^{4} + \cdots - 77841 Copy content Toggle raw display
3737 (T3+3T224T+27)2 (T^{3} + 3 T^{2} - 24 T + 27)^{2} Copy content Toggle raw display
4141 T6162T4+6561 T^{6} - 162 T^{4} + \cdots - 6561 Copy content Toggle raw display
4343 (T33T224T27)2 (T^{3} - 3 T^{2} - 24 T - 27)^{2} Copy content Toggle raw display
4747 T6183T4+194481 T^{6} - 183 T^{4} + \cdots - 194481 Copy content Toggle raw display
5353 (T3+14T2+263)2 (T^{3} + 14 T^{2} + \cdots - 263)^{2} Copy content Toggle raw display
5959 T6183T4+149769 T^{6} - 183 T^{4} + \cdots - 149769 Copy content Toggle raw display
6161 T6264T4+558009 T^{6} - 264 T^{4} + \cdots - 558009 Copy content Toggle raw display
6767 (T3111T+353)2 (T^{3} - 111 T + 353)^{2} Copy content Toggle raw display
7171 (T3+19T2++227)2 (T^{3} + 19 T^{2} + \cdots + 227)^{2} Copy content Toggle raw display
7373 T675T4+3969 T^{6} - 75 T^{4} + \cdots - 3969 Copy content Toggle raw display
7979 (T3+3T2+107)2 (T^{3} + 3 T^{2} + \cdots - 107)^{2} Copy content Toggle raw display
8383 T6228T4+227529 T^{6} - 228 T^{4} + \cdots - 227529 Copy content Toggle raw display
8989 T6246T4+3969 T^{6} - 246 T^{4} + \cdots - 3969 Copy content Toggle raw display
9797 T6111T4+9801 T^{6} - 111 T^{4} + \cdots - 9801 Copy content Toggle raw display
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