Properties

Label 8512.2.a.ce
Level 85128512
Weight 22
Character orbit 8512.a
Self dual yes
Analytic conductor 67.96967.969
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] = [8512,2,Mod(1,8512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 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("8512.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 8512=26719 8512 = 2^{6} \cdot 7 \cdot 19
Weight: k k == 2 2
Character orbit: [χ][\chi] == 8512.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,1,0,5,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 67.968662200567.9686622005
Analytic rank: 00
Dimension: 66
Coefficient field: Q[x]/(x6)\mathbb{Q}[x]/(x^{6} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x6x514x4+9x3+52x219x27 x^{6} - x^{5} - 14x^{4} + 9x^{3} + 52x^{2} - 19x - 27 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 4256)
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+β1q3+(β4+1)q5q7+(β2+2)q9+(β5+1)q11+(β5+β21)q13+(β5+β4+β3+1)q15+(β5+β4+1)q17++(2β5+3β4++7)q99+O(q100) q + \beta_1 q^{3} + (\beta_{4} + 1) q^{5} - q^{7} + (\beta_{2} + 2) q^{9} + ( - \beta_{5} + 1) q^{11} + (\beta_{5} + \beta_{2} - 1) q^{13} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots - 1) q^{15} + ( - \beta_{5} + \beta_{4} + 1) q^{17}+ \cdots + ( - 2 \beta_{5} + 3 \beta_{4} + \cdots + 7) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q+q3+5q56q7+11q9+7q118q138q15+6q17+6q19q21+2q23+9q25+10q27+17q2918q31+9q335q35+3q37+2q39++43q99+O(q100) 6 q + q^{3} + 5 q^{5} - 6 q^{7} + 11 q^{9} + 7 q^{11} - 8 q^{13} - 8 q^{15} + 6 q^{17} + 6 q^{19} - q^{21} + 2 q^{23} + 9 q^{25} + 10 q^{27} + 17 q^{29} - 18 q^{31} + 9 q^{33} - 5 q^{35} + 3 q^{37} + 2 q^{39}+ \cdots + 43 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x6x514x4+9x3+52x219x27 x^{6} - x^{5} - 14x^{4} + 9x^{3} + 52x^{2} - 19x - 27 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν25 \nu^{2} - 5 Copy content Toggle raw display
β3\beta_{3}== ν3ν27ν+3 \nu^{3} - \nu^{2} - 7\nu + 3 Copy content Toggle raw display
β4\beta_{4}== (ν42ν37ν2+10ν+3)/2 ( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 10\nu + 3 ) / 2 Copy content Toggle raw display
β5\beta_{5}== (ν53ν47ν3+19ν2+5ν7)/2 ( \nu^{5} - 3\nu^{4} - 7\nu^{3} + 19\nu^{2} + 5\nu - 7 ) / 2 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2+5 \beta_{2} + 5 Copy content Toggle raw display
ν3\nu^{3}== β3+β2+7β1+2 \beta_{3} + \beta_{2} + 7\beta _1 + 2 Copy content Toggle raw display
ν4\nu^{4}== 2β4+2β3+9β2+4β1+36 2\beta_{4} + 2\beta_{3} + 9\beta_{2} + 4\beta _1 + 36 Copy content Toggle raw display
ν5\nu^{5}== 2β5+6β4+13β3+15β2+56β1+34 2\beta_{5} + 6\beta_{4} + 13\beta_{3} + 15\beta_{2} + 56\beta _1 + 34 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
−2.51085
−2.39702
−0.608457
0.982453
2.37796
3.15591
0 −2.51085 0 3.58240 0 −1.00000 0 3.30438 0
1.2 0 −2.39702 0 0.684008 0 −1.00000 0 2.74569 0
1.3 0 −0.608457 0 −1.54426 0 −1.00000 0 −2.62978 0
1.4 0 0.982453 0 3.55156 0 −1.00000 0 −2.03479 0
1.5 0 2.37796 0 −2.86049 0 −1.00000 0 2.65470 0
1.6 0 3.15591 0 1.58678 0 −1.00000 0 6.95979 0
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
22 +1 +1
77 +1 +1
1919 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 8512.2.a.ce 6
4.b odd 2 1 8512.2.a.cc 6
8.b even 2 1 4256.2.a.j 6
8.d odd 2 1 4256.2.a.l yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4256.2.a.j 6 8.b even 2 1
4256.2.a.l yes 6 8.d odd 2 1
8512.2.a.cc 6 4.b odd 2 1
8512.2.a.ce 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(8512))S_{2}^{\mathrm{new}}(\Gamma_0(8512)):

T36T3514T34+9T33+52T3219T327 T_{3}^{6} - T_{3}^{5} - 14T_{3}^{4} + 9T_{3}^{3} + 52T_{3}^{2} - 19T_{3} - 27 Copy content Toggle raw display
T565T557T54+54T5315T52101T5+61 T_{5}^{6} - 5T_{5}^{5} - 7T_{5}^{4} + 54T_{5}^{3} - 15T_{5}^{2} - 101T_{5} + 61 Copy content Toggle raw display
T1167T11510T114+131T11366T112587T11+711 T_{11}^{6} - 7T_{11}^{5} - 10T_{11}^{4} + 131T_{11}^{3} - 66T_{11}^{2} - 587T_{11} + 711 Copy content Toggle raw display
T2362T23558T23432T233+657T232+1442T23+796 T_{23}^{6} - 2T_{23}^{5} - 58T_{23}^{4} - 32T_{23}^{3} + 657T_{23}^{2} + 1442T_{23} + 796 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T6 T^{6} Copy content Toggle raw display
33 T6T5+27 T^{6} - T^{5} + \cdots - 27 Copy content Toggle raw display
55 T65T5++61 T^{6} - 5 T^{5} + \cdots + 61 Copy content Toggle raw display
77 (T+1)6 (T + 1)^{6} Copy content Toggle raw display
1111 T67T5++711 T^{6} - 7 T^{5} + \cdots + 711 Copy content Toggle raw display
1313 T6+8T5++92 T^{6} + 8 T^{5} + \cdots + 92 Copy content Toggle raw display
1717 T66T5++48 T^{6} - 6 T^{5} + \cdots + 48 Copy content Toggle raw display
1919 (T1)6 (T - 1)^{6} Copy content Toggle raw display
2323 T62T5++796 T^{6} - 2 T^{5} + \cdots + 796 Copy content Toggle raw display
2929 T617T5+21 T^{6} - 17 T^{5} + \cdots - 21 Copy content Toggle raw display
3131 T6+18T5++2828 T^{6} + 18 T^{5} + \cdots + 2828 Copy content Toggle raw display
3737 T63T5+3589 T^{6} - 3 T^{5} + \cdots - 3589 Copy content Toggle raw display
4141 T6T5++1701 T^{6} - T^{5} + \cdots + 1701 Copy content Toggle raw display
4343 T611T5++2576 T^{6} - 11 T^{5} + \cdots + 2576 Copy content Toggle raw display
4747 T6+T5+7291 T^{6} + T^{5} + \cdots - 7291 Copy content Toggle raw display
5353 T67T5++6383 T^{6} - 7 T^{5} + \cdots + 6383 Copy content Toggle raw display
5959 T6+15T5++1587 T^{6} + 15 T^{5} + \cdots + 1587 Copy content Toggle raw display
6161 T67T5+57779 T^{6} - 7 T^{5} + \cdots - 57779 Copy content Toggle raw display
6767 T618T5++45148 T^{6} - 18 T^{5} + \cdots + 45148 Copy content Toggle raw display
7171 T63T5+49 T^{6} - 3 T^{5} + \cdots - 49 Copy content Toggle raw display
7373 T618T5+9212 T^{6} - 18 T^{5} + \cdots - 9212 Copy content Toggle raw display
7979 T6+15T5++128848 T^{6} + 15 T^{5} + \cdots + 128848 Copy content Toggle raw display
8383 T6+2T5++144384 T^{6} + 2 T^{5} + \cdots + 144384 Copy content Toggle raw display
8989 T613T5+576 T^{6} - 13 T^{5} + \cdots - 576 Copy content Toggle raw display
9797 T621T5++597249 T^{6} - 21 T^{5} + \cdots + 597249 Copy content Toggle raw display
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