gp: [N,k,chi] = [8512,2,Mod(1,8512)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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")
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);
Newform invariants
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)
gp: f = lf[1] \\ Warning: the index may be different
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the q q q -expansion are expressed in terms of a basis 1 , β 1 , … , β 5 1,\beta_1,\ldots,\beta_{5} 1 , β 1 , … , β 5 for the coefficient ring described below.
We also show the integral q q q -expansion of the trace form .
Basis of coefficient ring in terms of a root ν \nu ν of
x 6 − x 5 − 14 x 4 + 9 x 3 + 52 x 2 − 19 x − 27 x^{6} - x^{5} - 14x^{4} + 9x^{3} + 52x^{2} - 19x - 27 x 6 − x 5 − 1 4 x 4 + 9 x 3 + 5 2 x 2 − 1 9 x − 2 7
x^6 - x^5 - 14*x^4 + 9*x^3 + 52*x^2 - 19*x - 27
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
ν 2 − 5 \nu^{2} - 5 ν 2 − 5
v^2 - 5
β 3 \beta_{3} β 3 = = =
ν 3 − ν 2 − 7 ν + 3 \nu^{3} - \nu^{2} - 7\nu + 3 ν 3 − ν 2 − 7 ν + 3
v^3 - v^2 - 7*v + 3
β 4 \beta_{4} β 4 = = =
( ν 4 − 2 ν 3 − 7 ν 2 + 10 ν + 3 ) / 2 ( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 10\nu + 3 ) / 2 ( ν 4 − 2 ν 3 − 7 ν 2 + 1 0 ν + 3 ) / 2
(v^4 - 2*v^3 - 7*v^2 + 10*v + 3) / 2
β 5 \beta_{5} β 5 = = =
( ν 5 − 3 ν 4 − 7 ν 3 + 19 ν 2 + 5 ν − 7 ) / 2 ( \nu^{5} - 3\nu^{4} - 7\nu^{3} + 19\nu^{2} + 5\nu - 7 ) / 2 ( ν 5 − 3 ν 4 − 7 ν 3 + 1 9 ν 2 + 5 ν − 7 ) / 2
(v^5 - 3*v^4 - 7*v^3 + 19*v^2 + 5*v - 7) / 2
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 2 + 5 \beta_{2} + 5 β 2 + 5
b2 + 5
ν 3 \nu^{3} ν 3 = = =
β 3 + β 2 + 7 β 1 + 2 \beta_{3} + \beta_{2} + 7\beta _1 + 2 β 3 + β 2 + 7 β 1 + 2
b3 + b2 + 7*b1 + 2
ν 4 \nu^{4} ν 4 = = =
2 β 4 + 2 β 3 + 9 β 2 + 4 β 1 + 36 2\beta_{4} + 2\beta_{3} + 9\beta_{2} + 4\beta _1 + 36 2 β 4 + 2 β 3 + 9 β 2 + 4 β 1 + 3 6
2*b4 + 2*b3 + 9*b2 + 4*b1 + 36
ν 5 \nu^{5} ν 5 = = =
2 β 5 + 6 β 4 + 13 β 3 + 15 β 2 + 56 β 1 + 34 2\beta_{5} + 6\beta_{4} + 13\beta_{3} + 15\beta_{2} + 56\beta _1 + 34 2 β 5 + 6 β 4 + 1 3 β 3 + 1 5 β 2 + 5 6 β 1 + 3 4
2*b5 + 6*b4 + 13*b3 + 15*b2 + 56*b1 + 34
For each embedding ι m \iota_m ι m of the coefficient field, the values ι m ( a n ) \iota_m(a_n) ι m ( a n ) are shown below.
For more information on an embedded modular form you can click on its label.
gp: mfembed(f)
Refresh table
p p p
Sign
2 2 2
+ 1 +1 + 1
7 7 7
+ 1 +1 + 1
19 19 1 9
− 1 -1 − 1
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S 2 n e w ( Γ 0 ( 8512 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(8512)) S 2 n e w ( Γ 0 ( 8 5 1 2 ) ) :
T 3 6 − T 3 5 − 14 T 3 4 + 9 T 3 3 + 52 T 3 2 − 19 T 3 − 27 T_{3}^{6} - T_{3}^{5} - 14T_{3}^{4} + 9T_{3}^{3} + 52T_{3}^{2} - 19T_{3} - 27 T 3 6 − T 3 5 − 1 4 T 3 4 + 9 T 3 3 + 5 2 T 3 2 − 1 9 T 3 − 2 7
T3^6 - T3^5 - 14*T3^4 + 9*T3^3 + 52*T3^2 - 19*T3 - 27
T 5 6 − 5 T 5 5 − 7 T 5 4 + 54 T 5 3 − 15 T 5 2 − 101 T 5 + 61 T_{5}^{6} - 5T_{5}^{5} - 7T_{5}^{4} + 54T_{5}^{3} - 15T_{5}^{2} - 101T_{5} + 61 T 5 6 − 5 T 5 5 − 7 T 5 4 + 5 4 T 5 3 − 1 5 T 5 2 − 1 0 1 T 5 + 6 1
T5^6 - 5*T5^5 - 7*T5^4 + 54*T5^3 - 15*T5^2 - 101*T5 + 61
T 11 6 − 7 T 11 5 − 10 T 11 4 + 131 T 11 3 − 66 T 11 2 − 587 T 11 + 711 T_{11}^{6} - 7T_{11}^{5} - 10T_{11}^{4} + 131T_{11}^{3} - 66T_{11}^{2} - 587T_{11} + 711 T 1 1 6 − 7 T 1 1 5 − 1 0 T 1 1 4 + 1 3 1 T 1 1 3 − 6 6 T 1 1 2 − 5 8 7 T 1 1 + 7 1 1
T11^6 - 7*T11^5 - 10*T11^4 + 131*T11^3 - 66*T11^2 - 587*T11 + 711
T 23 6 − 2 T 23 5 − 58 T 23 4 − 32 T 23 3 + 657 T 23 2 + 1442 T 23 + 796 T_{23}^{6} - 2T_{23}^{5} - 58T_{23}^{4} - 32T_{23}^{3} + 657T_{23}^{2} + 1442T_{23} + 796 T 2 3 6 − 2 T 2 3 5 − 5 8 T 2 3 4 − 3 2 T 2 3 3 + 6 5 7 T 2 3 2 + 1 4 4 2 T 2 3 + 7 9 6
T23^6 - 2*T23^5 - 58*T23^4 - 32*T23^3 + 657*T23^2 + 1442*T23 + 796
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 6 T^{6} T 6
T^6
3 3 3
T 6 − T 5 + ⋯ − 27 T^{6} - T^{5} + \cdots - 27 T 6 − T 5 + ⋯ − 2 7
T^6 - T^5 - 14*T^4 + 9*T^3 + 52*T^2 - 19*T - 27
5 5 5
T 6 − 5 T 5 + ⋯ + 61 T^{6} - 5 T^{5} + \cdots + 61 T 6 − 5 T 5 + ⋯ + 6 1
T^6 - 5*T^5 - 7*T^4 + 54*T^3 - 15*T^2 - 101*T + 61
7 7 7
( T + 1 ) 6 (T + 1)^{6} ( T + 1 ) 6
(T + 1)^6
11 11 1 1
T 6 − 7 T 5 + ⋯ + 711 T^{6} - 7 T^{5} + \cdots + 711 T 6 − 7 T 5 + ⋯ + 7 1 1
T^6 - 7*T^5 - 10*T^4 + 131*T^3 - 66*T^2 - 587*T + 711
13 13 1 3
T 6 + 8 T 5 + ⋯ + 92 T^{6} + 8 T^{5} + \cdots + 92 T 6 + 8 T 5 + ⋯ + 9 2
T^6 + 8*T^5 - 6*T^4 - 126*T^3 - 163*T^2 + 38*T + 92
17 17 1 7
T 6 − 6 T 5 + ⋯ + 48 T^{6} - 6 T^{5} + \cdots + 48 T 6 − 6 T 5 + ⋯ + 4 8
T^6 - 6*T^5 - 31*T^4 + 92*T^3 + 361*T^2 + 256*T + 48
19 19 1 9
( T − 1 ) 6 (T - 1)^{6} ( T − 1 ) 6
(T - 1)^6
23 23 2 3
T 6 − 2 T 5 + ⋯ + 796 T^{6} - 2 T^{5} + \cdots + 796 T 6 − 2 T 5 + ⋯ + 7 9 6
T^6 - 2*T^5 - 58*T^4 - 32*T^3 + 657*T^2 + 1442*T + 796
29 29 2 9
T 6 − 17 T 5 + ⋯ − 21 T^{6} - 17 T^{5} + \cdots - 21 T 6 − 1 7 T 5 + ⋯ − 2 1
T^6 - 17*T^5 + 106*T^4 - 291*T^3 + 322*T^2 - 77*T - 21
31 31 3 1
T 6 + 18 T 5 + ⋯ + 2828 T^{6} + 18 T^{5} + \cdots + 2828 T 6 + 1 8 T 5 + ⋯ + 2 8 2 8
T^6 + 18*T^5 + 61*T^4 - 436*T^3 - 2667*T^2 - 2688*T + 2828
37 37 3 7
T 6 − 3 T 5 + ⋯ − 3589 T^{6} - 3 T^{5} + \cdots - 3589 T 6 − 3 T 5 + ⋯ − 3 5 8 9
T^6 - 3*T^5 - 63*T^4 + 122*T^3 + 1019*T^2 - 1395*T - 3589
41 41 4 1
T 6 − T 5 + ⋯ + 1701 T^{6} - T^{5} + \cdots + 1701 T 6 − T 5 + ⋯ + 1 7 0 1
T^6 - T^5 - 142*T^4 + 425*T^3 + 1372*T^2 - 4123*T + 1701
43 43 4 3
T 6 − 11 T 5 + ⋯ + 2576 T^{6} - 11 T^{5} + \cdots + 2576 T 6 − 1 1 T 5 + ⋯ + 2 5 7 6
T^6 - 11*T^5 - 23*T^4 + 284*T^3 - 84*T^2 - 1904*T + 2576
47 47 4 7
T 6 + T 5 + ⋯ − 7291 T^{6} + T^{5} + \cdots - 7291 T 6 + T 5 + ⋯ − 7 2 9 1
T^6 + T^5 - 123*T^4 - 26*T^3 + 2573*T^2 + 2381*T - 7291
53 53 5 3
T 6 − 7 T 5 + ⋯ + 6383 T^{6} - 7 T^{5} + \cdots + 6383 T 6 − 7 T 5 + ⋯ + 6 3 8 3
T^6 - 7*T^5 - 34*T^4 + 323*T^3 - 34*T^2 - 3683*T + 6383
59 59 5 9
T 6 + 15 T 5 + ⋯ + 1587 T^{6} + 15 T^{5} + \cdots + 1587 T 6 + 1 5 T 5 + ⋯ + 1 5 8 7
T^6 + 15*T^5 - 109*T^4 - 1214*T^3 + 6185*T^2 - 7429*T + 1587
61 61 6 1
T 6 − 7 T 5 + ⋯ − 57779 T^{6} - 7 T^{5} + \cdots - 57779 T 6 − 7 T 5 + ⋯ − 5 7 7 7 9
T^6 - 7*T^5 - 123*T^4 + 718*T^3 + 4581*T^2 - 16739*T - 57779
67 67 6 7
T 6 − 18 T 5 + ⋯ + 45148 T^{6} - 18 T^{5} + \cdots + 45148 T 6 − 1 8 T 5 + ⋯ + 4 5 1 4 8
T^6 - 18*T^5 - T^4 + 1326*T^3 - 4083*T^2 - 16352*T + 45148
71 71 7 1
T 6 − 3 T 5 + ⋯ − 49 T^{6} - 3 T^{5} + \cdots - 49 T 6 − 3 T 5 + ⋯ − 4 9
T^6 - 3*T^5 - 95*T^4 + 582*T^3 - 665*T^2 - 539*T - 49
73 73 7 3
T 6 − 18 T 5 + ⋯ − 9212 T^{6} - 18 T^{5} + \cdots - 9212 T 6 − 1 8 T 5 + ⋯ − 9 2 1 2
T^6 - 18*T^5 + 41*T^4 + 552*T^3 - 819*T^2 - 7742*T - 9212
79 79 7 9
T 6 + 15 T 5 + ⋯ + 128848 T^{6} + 15 T^{5} + \cdots + 128848 T 6 + 1 5 T 5 + ⋯ + 1 2 8 8 4 8
T^6 + 15*T^5 - 131*T^4 - 3188*T^3 - 14044*T^2 + 12160*T + 128848
83 83 8 3
T 6 + 2 T 5 + ⋯ + 144384 T^{6} + 2 T^{5} + \cdots + 144384 T 6 + 2 T 5 + ⋯ + 1 4 4 3 8 4
T^6 + 2*T^5 - 243*T^4 - 1320*T^3 + 11277*T^2 + 92896*T + 144384
89 89 8 9
T 6 − 13 T 5 + ⋯ − 576 T^{6} - 13 T^{5} + \cdots - 576 T 6 − 1 3 T 5 + ⋯ − 5 7 6
T^6 - 13*T^5 - 39*T^4 + 804*T^3 + 216*T^2 - 12896*T - 576
97 97 9 7
T 6 − 21 T 5 + ⋯ + 597249 T^{6} - 21 T^{5} + \cdots + 597249 T 6 − 2 1 T 5 + ⋯ + 5 9 7 2 4 9
T^6 - 21*T^5 - 215*T^4 + 7102*T^3 - 35235*T^2 - 79885*T + 597249
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