Properties

Label 729.2.a.d
Level 729729
Weight 22
Character orbit 729.a
Self dual yes
Analytic conductor 5.8215.821
Analytic rank 00
Dimension 66
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] = [729,2,Mod(1,729)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("729.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 729=36 729 = 3^{6}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 729.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,3,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 5.821094307355.82109430735
Analytic rank: 00
Dimension: 66
Coefficient field: 6.6.1397493.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x63x53x4+10x3+3x26x+1 x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 27)
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) == q+(β5+β1)q2+(β3β2+β1)q4+(β4+1)q5+(β5+β4+β2)q7+(β52β3β2+1)q8+(2β5+β4++β1)q10++(β5+β4+2β3+6)q98+O(q100) q + (\beta_{5} + \beta_1) q^{2} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{4} + (\beta_{4} + 1) q^{5} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{7} + ( - \beta_{5} - 2 \beta_{3} - \beta_{2} + 1) q^{8} + (2 \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{10}+ \cdots + (\beta_{5} + \beta_{4} + 2 \beta_{3} + \cdots - 6) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q+3q2+3q4+6q5+6q8+3q10+12q11+6q143q16+9q17+3q19+6q20+6q22+15q236q25+15q266q28+12q29+12q35+45q98+O(q100) 6 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{8} + 3 q^{10} + 12 q^{11} + 6 q^{14} - 3 q^{16} + 9 q^{17} + 3 q^{19} + 6 q^{20} + 6 q^{22} + 15 q^{23} - 6 q^{25} + 15 q^{26} - 6 q^{28} + 12 q^{29} + 12 q^{35}+ \cdots - 45 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x63x53x4+10x3+3x26x+1 x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2ν2 \nu^{2} - \nu - 2 Copy content Toggle raw display
β3\beta_{3}== ν32ν22ν+2 \nu^{3} - 2\nu^{2} - 2\nu + 2 Copy content Toggle raw display
β4\beta_{4}== ν43ν3ν2+6ν1 \nu^{4} - 3\nu^{3} - \nu^{2} + 6\nu - 1 Copy content Toggle raw display
β5\beta_{5}== ν53ν43ν3+9ν2+4ν3 \nu^{5} - 3\nu^{4} - 3\nu^{3} + 9\nu^{2} + 4\nu - 3 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2+β1+2 \beta_{2} + \beta _1 + 2 Copy content Toggle raw display
ν3\nu^{3}== β3+2β2+4β1+2 \beta_{3} + 2\beta_{2} + 4\beta _1 + 2 Copy content Toggle raw display
ν4\nu^{4}== β4+3β3+7β2+7β1+9 \beta_{4} + 3\beta_{3} + 7\beta_{2} + 7\beta _1 + 9 Copy content Toggle raw display
ν5\nu^{5}== β5+3β4+12β3+18β2+20β1+18 \beta_{5} + 3\beta_{4} + 12\beta_{3} + 18\beta_{2} + 20\beta _1 + 18 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
0.198473
−1.40162
−1.11662
2.68091
0.584534
2.05432
−1.68091 0 0.825466 1.12954 0 −3.90892 1.97429 0 −1.89866
1.2 −1.05432 0 −0.888399 1.74579 0 2.45925 3.04531 0 −1.84063
1.3 0.415466 0 −1.82739 −2.21519 0 −1.31963 −1.59015 0 −0.920335
1.4 0.801527 0 −1.35755 2.74984 0 2.37683 −2.69117 0 2.20407
1.5 2.11662 0 2.48009 2.68310 0 0.972333 1.01617 0 5.67911
1.6 2.40162 0 3.76778 −0.0930834 0 −0.579861 4.24555 0 −0.223551
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

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 729.2.a.d 6
3.b odd 2 1 729.2.a.a 6
9.c even 3 2 729.2.c.b 12
9.d odd 6 2 729.2.c.e 12
27.e even 9 2 81.2.e.a 12
27.e even 9 2 243.2.e.a 12
27.e even 9 2 243.2.e.b 12
27.f odd 18 2 27.2.e.a 12
27.f odd 18 2 243.2.e.c 12
27.f odd 18 2 243.2.e.d 12
108.l even 18 2 432.2.u.c 12
135.n odd 18 2 675.2.l.c 12
135.q even 36 4 675.2.u.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.e.a 12 27.f odd 18 2
81.2.e.a 12 27.e even 9 2
243.2.e.a 12 27.e even 9 2
243.2.e.b 12 27.e even 9 2
243.2.e.c 12 27.f odd 18 2
243.2.e.d 12 27.f odd 18 2
432.2.u.c 12 108.l even 18 2
675.2.l.c 12 135.n odd 18 2
675.2.u.b 24 135.q even 36 4
729.2.a.a 6 3.b odd 2 1
729.2.a.d 6 1.a even 1 1 trivial
729.2.c.b 12 9.c even 3 2
729.2.c.e 12 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T263T253T24+12T239T2+3 T_{2}^{6} - 3T_{2}^{5} - 3T_{2}^{4} + 12T_{2}^{3} - 9T_{2} + 3 acting on S2new(Γ0(729))S_{2}^{\mathrm{new}}(\Gamma_0(729)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T63T5++3 T^{6} - 3 T^{5} + \cdots + 3 Copy content Toggle raw display
33 T6 T^{6} Copy content Toggle raw display
55 T66T5++3 T^{6} - 6 T^{5} + \cdots + 3 Copy content Toggle raw display
77 T615T4+17 T^{6} - 15 T^{4} + \cdots - 17 Copy content Toggle raw display
1111 T612T5++3 T^{6} - 12 T^{5} + \cdots + 3 Copy content Toggle raw display
1313 T624T4++1 T^{6} - 24 T^{4} + \cdots + 1 Copy content Toggle raw display
1717 T69T5++27 T^{6} - 9 T^{5} + \cdots + 27 Copy content Toggle raw display
1919 T63T5++19 T^{6} - 3 T^{5} + \cdots + 19 Copy content Toggle raw display
2323 T615T5++327 T^{6} - 15 T^{5} + \cdots + 327 Copy content Toggle raw display
2929 T612T5+213 T^{6} - 12 T^{5} + \cdots - 213 Copy content Toggle raw display
3131 T651T4++163 T^{6} - 51 T^{4} + \cdots + 163 Copy content Toggle raw display
3737 T63T5++4933 T^{6} - 3 T^{5} + \cdots + 4933 Copy content Toggle raw display
4141 T615T5++3351 T^{6} - 15 T^{5} + \cdots + 3351 Copy content Toggle raw display
4343 T696T4++1819 T^{6} - 96 T^{4} + \cdots + 1819 Copy content Toggle raw display
4747 T621T5++6537 T^{6} - 21 T^{5} + \cdots + 6537 Copy content Toggle raw display
5353 T69T5+12393 T^{6} - 9 T^{5} + \cdots - 12393 Copy content Toggle raw display
5959 T624T5+13281 T^{6} - 24 T^{5} + \cdots - 13281 Copy content Toggle raw display
6161 T6+9T5++16543 T^{6} + 9 T^{5} + \cdots + 16543 Copy content Toggle raw display
6767 T6+9T5+2879 T^{6} + 9 T^{5} + \cdots - 2879 Copy content Toggle raw display
7171 T627T5++27 T^{6} - 27 T^{5} + \cdots + 27 Copy content Toggle raw display
7373 T6+6T5+431 T^{6} + 6 T^{5} + \cdots - 431 Copy content Toggle raw display
7979 T6177T4++1873 T^{6} - 177 T^{4} + \cdots + 1873 Copy content Toggle raw display
8383 T612T5+83373 T^{6} - 12 T^{5} + \cdots - 83373 Copy content Toggle raw display
8989 T69T5++32589 T^{6} - 9 T^{5} + \cdots + 32589 Copy content Toggle raw display
9797 T6204T4+8171 T^{6} - 204 T^{4} + \cdots - 8171 Copy content Toggle raw display
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