gp: [N,k,chi] = [9248,2,Mod(1,9248)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9248, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("9248.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [9,0,-9,0,-6,0,-3,0,12,0,-6,0,-12,0,21,0,0,0,0,0,18,0,3,0,9,0,
-24,0,-21,0,-24,0,27,0,12,0,-12,0,-3,0,15,0,3,0,-33,0,12,0,-6,0,0,0,9,
0,-21,0,-9,0,-45,0,-3,0,-27,0,51,0,-6,0,18,0,9,0,30,0,-39,0,9,0,18,0,9,
0,9,0,0,0,48,0,0,0,-3,0,33,0,-33,0,33,0,-90,0,-15,0,51,0,-9,0,3,0,-6,0,
21,0,27,0,9,0,24,0,0,0,27,0,-48,0,-27,0,-12,0,6,0,-9,0,27,0,45,0,3,0,0,
0,-51,0,-78,0,57]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(145)] == 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 8 1,\beta_1,\ldots,\beta_{8} 1 , β 1 , … , β 8 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 9 − 15 x 7 − 7 x 6 + 69 x 5 + 48 x 4 − 110 x 3 − 87 x 2 + 45 x + 37 x^{9} - 15x^{7} - 7x^{6} + 69x^{5} + 48x^{4} - 110x^{3} - 87x^{2} + 45x + 37 x 9 − 1 5 x 7 − 7 x 6 + 6 9 x 5 + 4 8 x 4 − 1 1 0 x 3 − 8 7 x 2 + 4 5 x + 3 7
x^9 - 15*x^7 - 7*x^6 + 69*x^5 + 48*x^4 - 110*x^3 - 87*x^2 + 45*x + 37
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
ν 2 − ν − 3 \nu^{2} - \nu - 3 ν 2 − ν − 3
v^2 - v - 3
β 3 \beta_{3} β 3 = = =
( ν 7 − 3 ν 6 − 8 ν 5 + 22 ν 4 + 19 ν 3 − 43 ν 2 − 13 ν + 19 ) / 2 ( \nu^{7} - 3\nu^{6} - 8\nu^{5} + 22\nu^{4} + 19\nu^{3} - 43\nu^{2} - 13\nu + 19 ) / 2 ( ν 7 − 3 ν 6 − 8 ν 5 + 2 2 ν 4 + 1 9 ν 3 − 4 3 ν 2 − 1 3 ν + 1 9 ) / 2
(v^7 - 3*v^6 - 8*v^5 + 22*v^4 + 19*v^3 - 43*v^2 - 13*v + 19) / 2
β 4 \beta_{4} β 4 = = =
( ν 8 − 3 ν 7 − 8 ν 6 + 22 ν 5 + 19 ν 4 − 43 ν 3 − 13 ν 2 + 19 ν ) / 2 ( \nu^{8} - 3\nu^{7} - 8\nu^{6} + 22\nu^{5} + 19\nu^{4} - 43\nu^{3} - 13\nu^{2} + 19\nu ) / 2 ( ν 8 − 3 ν 7 − 8 ν 6 + 2 2 ν 5 + 1 9 ν 4 − 4 3 ν 3 − 1 3 ν 2 + 1 9 ν ) / 2
(v^8 - 3*v^7 - 8*v^6 + 22*v^5 + 19*v^4 - 43*v^3 - 13*v^2 + 19*v) / 2
β 5 \beta_{5} β 5 = = =
( 3 ν 8 − 8 ν 7 − 27 ν 6 + 60 ν 5 + 75 ν 4 − 122 ν 3 − 68 ν 2 + 54 ν + 17 ) / 2 ( 3\nu^{8} - 8\nu^{7} - 27\nu^{6} + 60\nu^{5} + 75\nu^{4} - 122\nu^{3} - 68\nu^{2} + 54\nu + 17 ) / 2 ( 3 ν 8 − 8 ν 7 − 2 7 ν 6 + 6 0 ν 5 + 7 5 ν 4 − 1 2 2 ν 3 − 6 8 ν 2 + 5 4 ν + 1 7 ) / 2
(3*v^8 - 8*v^7 - 27*v^6 + 60*v^5 + 75*v^4 - 122*v^3 - 68*v^2 + 54*v + 17) / 2
β 6 \beta_{6} β 6 = = =
( 3 ν 8 − 6 ν 7 − 33 ν 6 + 44 ν 5 + 121 ν 4 − 88 ν 3 − 166 ν 2 + 42 ν + 65 ) / 2 ( 3\nu^{8} - 6\nu^{7} - 33\nu^{6} + 44\nu^{5} + 121\nu^{4} - 88\nu^{3} - 166\nu^{2} + 42\nu + 65 ) / 2 ( 3 ν 8 − 6 ν 7 − 3 3 ν 6 + 4 4 ν 5 + 1 2 1 ν 4 − 8 8 ν 3 − 1 6 6 ν 2 + 4 2 ν + 6 5 ) / 2
(3*v^8 - 6*v^7 - 33*v^6 + 44*v^5 + 121*v^4 - 88*v^3 - 166*v^2 + 42*v + 65) / 2
β 7 \beta_{7} β 7 = = =
( 4 ν 8 − 7 ν 7 − 46 ν 6 + 48 ν 5 + 176 ν 4 − 83 ν 3 − 250 ν 2 + 27 ν + 98 ) / 2 ( 4\nu^{8} - 7\nu^{7} - 46\nu^{6} + 48\nu^{5} + 176\nu^{4} - 83\nu^{3} - 250\nu^{2} + 27\nu + 98 ) / 2 ( 4 ν 8 − 7 ν 7 − 4 6 ν 6 + 4 8 ν 5 + 1 7 6 ν 4 − 8 3 ν 3 − 2 5 0 ν 2 + 2 7 ν + 9 8 ) / 2
(4*v^8 - 7*v^7 - 46*v^6 + 48*v^5 + 176*v^4 - 83*v^3 - 250*v^2 + 27*v + 98) / 2
β 8 \beta_{8} β 8 = = =
( 5 ν 8 − 9 ν 7 − 57 ν 6 + 64 ν 5 + 213 ν 4 − 121 ν 3 − 290 ν 2 + 51 ν + 111 ) / 2 ( 5\nu^{8} - 9\nu^{7} - 57\nu^{6} + 64\nu^{5} + 213\nu^{4} - 121\nu^{3} - 290\nu^{2} + 51\nu + 111 ) / 2 ( 5 ν 8 − 9 ν 7 − 5 7 ν 6 + 6 4 ν 5 + 2 1 3 ν 4 − 1 2 1 ν 3 − 2 9 0 ν 2 + 5 1 ν + 1 1 1 ) / 2
(5*v^8 - 9*v^7 - 57*v^6 + 64*v^5 + 213*v^4 - 121*v^3 - 290*v^2 + 51*v + 111) / 2
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 2 + β 1 + 3 \beta_{2} + \beta _1 + 3 β 2 + β 1 + 3
b2 + b1 + 3
ν 3 \nu^{3} ν 3 = = =
− β 8 + β 7 + β 5 − 2 β 4 + β 2 + 5 β 1 + 1 -\beta_{8} + \beta_{7} + \beta_{5} - 2\beta_{4} + \beta_{2} + 5\beta _1 + 1 − β 8 + β 7 + β 5 − 2 β 4 + β 2 + 5 β 1 + 1
-b8 + b7 + b5 - 2*b4 + b2 + 5*b1 + 1
ν 4 \nu^{4} ν 4 = = =
− 2 β 8 + 2 β 7 + β 6 + β 5 − 4 β 4 − 2 β 3 + 8 β 2 + 9 β 1 + 15 -2\beta_{8} + 2\beta_{7} + \beta_{6} + \beta_{5} - 4\beta_{4} - 2\beta_{3} + 8\beta_{2} + 9\beta _1 + 15 − 2 β 8 + 2 β 7 + β 6 + β 5 − 4 β 4 − 2 β 3 + 8 β 2 + 9 β 1 + 1 5
-2*b8 + 2*b7 + b6 + b5 - 4*b4 - 2*b3 + 8*b2 + 9*b1 + 15
ν 5 \nu^{5} ν 5 = = =
− 10 β 8 + 10 β 7 + 2 β 6 + 9 β 5 − 23 β 4 − 5 β 3 + 15 β 2 + 36 β 1 + 16 -10\beta_{8} + 10\beta_{7} + 2\beta_{6} + 9\beta_{5} - 23\beta_{4} - 5\beta_{3} + 15\beta_{2} + 36\beta _1 + 16 − 1 0 β 8 + 1 0 β 7 + 2 β 6 + 9 β 5 − 2 3 β 4 − 5 β 3 + 1 5 β 2 + 3 6 β 1 + 1 6
-10*b8 + 10*b7 + 2*b6 + 9*b5 - 23*b4 - 5*b3 + 15*b2 + 36*b1 + 16
ν 6 \nu^{6} ν 6 = = =
− 26 β 8 + 28 β 7 + 10 β 6 + 16 β 5 − 60 β 4 − 30 β 3 + 69 β 2 + 87 β 1 + 102 -26\beta_{8} + 28\beta_{7} + 10\beta_{6} + 16\beta_{5} - 60\beta_{4} - 30\beta_{3} + 69\beta_{2} + 87\beta _1 + 102 − 2 6 β 8 + 2 8 β 7 + 1 0 β 6 + 1 6 β 5 − 6 0 β 4 − 3 0 β 3 + 6 9 β 2 + 8 7 β 1 + 1 0 2
-26*b8 + 28*b7 + 10*b6 + 16*b5 - 60*b4 - 30*b3 + 69*b2 + 87*b1 + 102
ν 7 \nu^{7} ν 7 = = =
− 95 β 8 + 101 β 7 + 24 β 6 + 79 β 5 − 238 β 4 − 84 β 3 + ⋯ + 195 - 95 \beta_{8} + 101 \beta_{7} + 24 \beta_{6} + 79 \beta_{5} - 238 \beta_{4} - 84 \beta_{3} + \cdots + 195 − 9 5 β 8 + 1 0 1 β 7 + 2 4 β 6 + 7 9 β 5 − 2 3 8 β 4 − 8 4 β 3 + ⋯ + 1 9 5
-95*b8 + 101*b7 + 24*b6 + 79*b5 - 238*b4 - 84*b3 + 175*b2 + 312*b1 + 195
ν 8 \nu^{8} ν 8 = = =
− 278 β 8 + 312 β 7 + 89 β 6 + 191 β 5 − 696 β 4 − 344 β 3 + ⋯ + 846 - 278 \beta_{8} + 312 \beta_{7} + 89 \beta_{6} + 191 \beta_{5} - 696 \beta_{4} - 344 \beta_{3} + \cdots + 846 − 2 7 8 β 8 + 3 1 2 β 7 + 8 9 β 6 + 1 9 1 β 5 − 6 9 6 β 4 − 3 4 4 β 3 + ⋯ + 8 4 6
-278*b8 + 312*b7 + 89*b6 + 191*b5 - 696*b4 - 344*b3 + 651*b2 + 878*b1 + 846
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
17 17 1 7
− 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 ( 9248 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(9248)) S 2 n e w ( Γ 0 ( 9 2 4 8 ) ) :
T 3 9 + 9 T 3 8 + 21 T 3 7 − 28 T 3 6 − 162 T 3 5 − 111 T 3 4 + 191 T 3 3 + 177 T 3 2 − 60 T 3 − 19 T_{3}^{9} + 9T_{3}^{8} + 21T_{3}^{7} - 28T_{3}^{6} - 162T_{3}^{5} - 111T_{3}^{4} + 191T_{3}^{3} + 177T_{3}^{2} - 60T_{3} - 19 T 3 9 + 9 T 3 8 + 2 1 T 3 7 − 2 8 T 3 6 − 1 6 2 T 3 5 − 1 1 1 T 3 4 + 1 9 1 T 3 3 + 1 7 7 T 3 2 − 6 0 T 3 − 1 9
T3^9 + 9*T3^8 + 21*T3^7 - 28*T3^6 - 162*T3^5 - 111*T3^4 + 191*T3^3 + 177*T3^2 - 60*T3 - 19
T 5 9 + 6 T 5 8 − 9 T 5 7 − 97 T 5 6 − 33 T 5 5 + 432 T 5 4 + 324 T 5 3 − 315 T 5 2 + 51 T 5 + 1 T_{5}^{9} + 6T_{5}^{8} - 9T_{5}^{7} - 97T_{5}^{6} - 33T_{5}^{5} + 432T_{5}^{4} + 324T_{5}^{3} - 315T_{5}^{2} + 51T_{5} + 1 T 5 9 + 6 T 5 8 − 9 T 5 7 − 9 7 T 5 6 − 3 3 T 5 5 + 4 3 2 T 5 4 + 3 2 4 T 5 3 − 3 1 5 T 5 2 + 5 1 T 5 + 1
T5^9 + 6*T5^8 - 9*T5^7 - 97*T5^6 - 33*T5^5 + 432*T5^4 + 324*T5^3 - 315*T5^2 + 51*T5 + 1
T 7 9 + 3 T 7 8 − 24 T 7 7 − 49 T 7 6 + 177 T 7 5 + 99 T 7 4 − 521 T 7 3 + 378 T 7 2 − 48 T 7 − 17 T_{7}^{9} + 3T_{7}^{8} - 24T_{7}^{7} - 49T_{7}^{6} + 177T_{7}^{5} + 99T_{7}^{4} - 521T_{7}^{3} + 378T_{7}^{2} - 48T_{7} - 17 T 7 9 + 3 T 7 8 − 2 4 T 7 7 − 4 9 T 7 6 + 1 7 7 T 7 5 + 9 9 T 7 4 − 5 2 1 T 7 3 + 3 7 8 T 7 2 − 4 8 T 7 − 1 7
T7^9 + 3*T7^8 - 24*T7^7 - 49*T7^6 + 177*T7^5 + 99*T7^4 - 521*T7^3 + 378*T7^2 - 48*T7 - 17
T 19 9 − 57 T 19 7 − 41 T 19 6 + 873 T 19 5 + 870 T 19 4 − 3862 T 19 3 − 1731 T 19 2 + 6663 T 19 − 2447 T_{19}^{9} - 57T_{19}^{7} - 41T_{19}^{6} + 873T_{19}^{5} + 870T_{19}^{4} - 3862T_{19}^{3} - 1731T_{19}^{2} + 6663T_{19} - 2447 T 1 9 9 − 5 7 T 1 9 7 − 4 1 T 1 9 6 + 8 7 3 T 1 9 5 + 8 7 0 T 1 9 4 − 3 8 6 2 T 1 9 3 − 1 7 3 1 T 1 9 2 + 6 6 6 3 T 1 9 − 2 4 4 7
T19^9 - 57*T19^7 - 41*T19^6 + 873*T19^5 + 870*T19^4 - 3862*T19^3 - 1731*T19^2 + 6663*T19 - 2447
T 43 9 − 3 T 43 8 − 339 T 43 7 + 1222 T 43 6 + 38238 T 43 5 − 144075 T 43 4 + ⋯ + 8890633 T_{43}^{9} - 3 T_{43}^{8} - 339 T_{43}^{7} + 1222 T_{43}^{6} + 38238 T_{43}^{5} - 144075 T_{43}^{4} + \cdots + 8890633 T 4 3 9 − 3 T 4 3 8 − 3 3 9 T 4 3 7 + 1 2 2 2 T 4 3 6 + 3 8 2 3 8 T 4 3 5 − 1 4 4 0 7 5 T 4 3 4 + ⋯ + 8 8 9 0 6 3 3
T43^9 - 3*T43^8 - 339*T43^7 + 1222*T43^6 + 38238*T43^5 - 144075*T43^4 - 1553569*T43^3 + 4688835*T43^2 + 16074114*T43 + 8890633
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 9 T^{9} T 9
T^9
3 3 3
T 9 + 9 T 8 + ⋯ − 19 T^{9} + 9 T^{8} + \cdots - 19 T 9 + 9 T 8 + ⋯ − 1 9
T^9 + 9*T^8 + 21*T^7 - 28*T^6 - 162*T^5 - 111*T^4 + 191*T^3 + 177*T^2 - 60*T - 19
5 5 5
T 9 + 6 T 8 + ⋯ + 1 T^{9} + 6 T^{8} + \cdots + 1 T 9 + 6 T 8 + ⋯ + 1
T^9 + 6*T^8 - 9*T^7 - 97*T^6 - 33*T^5 + 432*T^4 + 324*T^3 - 315*T^2 + 51*T + 1
7 7 7
T 9 + 3 T 8 + ⋯ − 17 T^{9} + 3 T^{8} + \cdots - 17 T 9 + 3 T 8 + ⋯ − 1 7
T^9 + 3*T^8 - 24*T^7 - 49*T^6 + 177*T^5 + 99*T^4 - 521*T^3 + 378*T^2 - 48*T - 17
11 11 1 1
T 9 + 6 T 8 + ⋯ − 4672 T^{9} + 6 T^{8} + \cdots - 4672 T 9 + 6 T 8 + ⋯ − 4 6 7 2
T^9 + 6*T^8 - 45*T^7 - 329*T^6 + 144*T^5 + 4104*T^4 + 7216*T^3 - 720*T^2 - 9216*T - 4672
13 13 1 3
T 9 + 12 T 8 + ⋯ − 29179 T^{9} + 12 T^{8} + \cdots - 29179 T 9 + 1 2 T 8 + ⋯ − 2 9 1 7 9
T^9 + 12*T^8 - 15*T^7 - 609*T^6 - 1257*T^5 + 7302*T^4 + 25826*T^3 + 7119*T^2 - 39207*T - 29179
17 17 1 7
T 9 T^{9} T 9
T^9
19 19 1 9
T 9 − 57 T 7 + ⋯ − 2447 T^{9} - 57 T^{7} + \cdots - 2447 T 9 − 5 7 T 7 + ⋯ − 2 4 4 7
T^9 - 57*T^7 - 41*T^6 + 873*T^5 + 870*T^4 - 3862*T^3 - 1731*T^2 + 6663*T - 2447
23 23 2 3
T 9 − 3 T 8 + ⋯ − 71 T^{9} - 3 T^{8} + \cdots - 71 T 9 − 3 T 8 + ⋯ − 7 1
T^9 - 3*T^8 - 42*T^7 + 147*T^6 + 345*T^5 - 1317*T^4 - 735*T^3 + 2724*T^2 + 912*T - 71
29 29 2 9
T 9 + 21 T 8 + ⋯ − 198917 T^{9} + 21 T^{8} + \cdots - 198917 T 9 + 2 1 T 8 + ⋯ − 1 9 8 9 1 7
T^9 + 21*T^8 + 84*T^7 - 905*T^6 - 8493*T^5 - 16407*T^4 + 49079*T^3 + 197316*T^2 + 86400*T - 198917
31 31 3 1
T 9 + 24 T 8 + ⋯ − 239581 T^{9} + 24 T^{8} + \cdots - 239581 T 9 + 2 4 T 8 + ⋯ − 2 3 9 5 8 1
T^9 + 24*T^8 + 120*T^7 - 1157*T^6 - 11871*T^5 - 10725*T^4 + 169203*T^3 + 467916*T^2 + 200073*T - 239581
37 37 3 7
T 9 + 12 T 8 + ⋯ − 8390231 T^{9} + 12 T^{8} + \cdots - 8390231 T 9 + 1 2 T 8 + ⋯ − 8 3 9 0 2 3 1
T^9 + 12*T^8 - 168*T^7 - 2417*T^6 + 4431*T^5 + 138627*T^4 + 293897*T^3 - 1837152*T^2 - 8052921*T - 8390231
41 41 4 1
T 9 − 15 T 8 + ⋯ − 1549 T^{9} - 15 T^{8} + \cdots - 1549 T 9 − 1 5 T 8 + ⋯ − 1 5 4 9
T^9 - 15*T^8 - 42*T^7 + 1167*T^6 - 177*T^5 - 25635*T^4 + 10853*T^3 + 93396*T^2 - 59550*T - 1549
43 43 4 3
T 9 − 3 T 8 + ⋯ + 8890633 T^{9} - 3 T^{8} + \cdots + 8890633 T 9 − 3 T 8 + ⋯ + 8 8 9 0 6 3 3
T^9 - 3*T^8 - 339*T^7 + 1222*T^6 + 38238*T^5 - 144075*T^4 - 1553569*T^3 + 4688835*T^2 + 16074114*T + 8890633
47 47 4 7
T 9 − 12 T 8 + ⋯ + 384067 T^{9} - 12 T^{8} + \cdots + 384067 T 9 − 1 2 T 8 + ⋯ + 3 8 4 0 6 7
T^9 - 12*T^8 - 150*T^7 + 1721*T^6 + 8337*T^5 - 75237*T^4 - 228435*T^3 + 995244*T^2 + 2958597*T + 384067
53 53 5 3
T 9 − 9 T 8 + ⋯ − 503488 T^{9} - 9 T^{8} + \cdots - 503488 T 9 − 9 T 8 + ⋯ − 5 0 3 4 8 8
T^9 - 9*T^8 - 84*T^7 + 639*T^6 + 2496*T^5 - 15228*T^4 - 28912*T^3 + 149760*T^2 + 106752*T - 503488
59 59 5 9
T 9 + 45 T 8 + ⋯ − 1677943 T^{9} + 45 T^{8} + \cdots - 1677943 T 9 + 4 5 T 8 + ⋯ − 1 6 7 7 9 4 3
T^9 + 45*T^8 + 699*T^7 + 3036*T^6 - 31848*T^5 - 444975*T^4 - 2177749*T^3 - 5079039*T^2 - 5313546*T - 1677943
61 61 6 1
T 9 + 3 T 8 + ⋯ + 358085807 T^{9} + 3 T^{8} + \cdots + 358085807 T 9 + 3 T 8 + ⋯ + 3 5 8 0 8 5 8 0 7
T^9 + 3*T^8 - 411*T^7 - 1794*T^6 + 58440*T^5 + 312261*T^4 - 3279241*T^3 - 19757301*T^2 + 55596126*T + 358085807
67 67 6 7
T 9 + 6 T 8 + ⋯ + 4191337 T^{9} + 6 T^{8} + \cdots + 4191337 T 9 + 6 T 8 + ⋯ + 4 1 9 1 3 3 7
T^9 + 6*T^8 - 324*T^7 - 1739*T^6 + 32727*T^5 + 170037*T^4 - 957955*T^3 - 5875638*T^2 - 4025349*T + 4191337
71 71 7 1
T 9 − 9 T 8 + ⋯ + 41761781 T^{9} - 9 T^{8} + \cdots + 41761781 T 9 − 9 T 8 + ⋯ + 4 1 7 6 1 7 8 1
T^9 - 9*T^8 - 405*T^7 + 3150*T^6 + 55446*T^5 - 302757*T^4 - 3413713*T^3 + 8239821*T^2 + 81588318*T + 41761781
73 73 7 3
T 9 − 30 T 8 + ⋯ − 9721 T^{9} - 30 T^{8} + \cdots - 9721 T 9 − 3 0 T 8 + ⋯ − 9 7 2 1
T^9 - 30*T^8 + 234*T^7 + 511*T^6 - 13971*T^5 + 63117*T^4 - 118559*T^3 + 90582*T^2 - 12147*T - 9721
79 79 7 9
T 9 − 18 T 8 + ⋯ − 6839288 T^{9} - 18 T^{8} + \cdots - 6839288 T 9 − 1 8 T 8 + ⋯ − 6 8 3 9 2 8 8
T^9 - 18*T^8 - 147*T^7 + 3986*T^6 - 5265*T^5 - 205134*T^4 + 746227*T^3 + 1964814*T^2 - 7097040*T - 6839288
83 83 8 3
T 9 − 9 T 8 + ⋯ + 9609011 T^{9} - 9 T^{8} + \cdots + 9609011 T 9 − 9 T 8 + ⋯ + 9 6 0 9 0 1 1
T^9 - 9*T^8 - 228*T^7 + 3092*T^6 - 774*T^5 - 140946*T^4 + 575468*T^3 + 556500*T^2 - 6775107*T + 9609011
89 89 8 9
T 9 − 504 T 7 + ⋯ + 10434311 T^{9} - 504 T^{7} + \cdots + 10434311 T 9 − 5 0 4 T 7 + ⋯ + 1 0 4 3 4 3 1 1
T^9 - 504*T^7 + 917*T^6 + 74679*T^5 - 204279*T^4 - 2785783*T^3 + 2107584*T^2 + 25454643*T + 10434311
97 97 9 7
T 9 − 33 T 8 + ⋯ − 19741184 T^{9} - 33 T^{8} + \cdots - 19741184 T 9 − 3 3 T 8 + ⋯ − 1 9 7 4 1 1 8 4
T^9 - 33*T^8 + 15*T^7 + 8221*T^6 - 47820*T^5 - 470712*T^4 + 2333312*T^3 + 10008768*T^2 - 2908416*T - 19741184
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