gp: [N,k,chi] = [325,2,Mod(126,325)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(325, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("325.126");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [10,0,-3]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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 9 1,\beta_1,\ldots,\beta_{9} 1 , β 1 , … , β 9 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 10 + 8 x 8 − 2 x 7 + 52 x 6 − 5 x 5 + 97 x 4 + 60 x 3 + 141 x 2 + 36 x + 9 x^{10} + 8x^{8} - 2x^{7} + 52x^{6} - 5x^{5} + 97x^{4} + 60x^{3} + 141x^{2} + 36x + 9 x 1 0 + 8 x 8 − 2 x 7 + 5 2 x 6 − 5 x 5 + 9 7 x 4 + 6 0 x 3 + 1 4 1 x 2 + 3 6 x + 9
x^10 + 8*x^8 - 2*x^7 + 52*x^6 - 5*x^5 + 97*x^4 + 60*x^3 + 141*x^2 + 36*x + 9
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( 1304 ν 9 + 1776 ν 8 − 1628 ν 7 + 8447 ν 6 − 14800 ν 5 + 93980 ν 4 + ⋯ − 242496 ) / 983355 ( 1304 \nu^{9} + 1776 \nu^{8} - 1628 \nu^{7} + 8447 \nu^{6} - 14800 \nu^{5} + 93980 \nu^{4} + \cdots - 242496 ) / 983355 ( 1 3 0 4 ν 9 + 1 7 7 6 ν 8 − 1 6 2 8 ν 7 + 8 4 4 7 ν 6 − 1 4 8 0 0 ν 5 + 9 3 9 8 0 ν 4 + ⋯ − 2 4 2 4 9 6 ) / 9 8 3 3 5 5
(1304*v^9 + 1776*v^8 - 1628*v^7 + 8447*v^6 - 14800*v^5 + 93980*v^4 - 511687*v^3 + 165612*v^2 + 42624*v - 242496) / 983355
β 3 \beta_{3} β 3 = = =
( 592 ν 9 − 4020 ν 8 + 3685 ν 7 − 27536 ν 6 + 33500 ν 5 − 212725 ν 4 + ⋯ − 987267 ) / 327785 ( 592 \nu^{9} - 4020 \nu^{8} + 3685 \nu^{7} - 27536 \nu^{6} + 33500 \nu^{5} - 212725 \nu^{4} + \cdots - 987267 ) / 327785 ( 5 9 2 ν 9 − 4 0 2 0 ν 8 + 3 6 8 5 ν 7 − 2 7 5 3 6 ν 6 + 3 3 5 0 0 ν 5 − 2 1 2 7 2 5 ν 4 + ⋯ − 9 8 7 2 6 7 ) / 3 2 7 7 8 5
(592*v^9 - 4020*v^8 + 3685*v^7 - 27536*v^6 + 33500*v^5 - 212725*v^4 + 29124*v^3 - 374865*v^2 - 96480*v - 987267) / 327785
β 4 \beta_{4} β 4 = = =
( − 4744 ν 9 − 20940 ν 8 + 19195 ν 7 − 124843 ν 6 + 174500 ν 5 − 1108075 ν 4 + ⋯ − 1749321 ) / 983355 ( - 4744 \nu^{9} - 20940 \nu^{8} + 19195 \nu^{7} - 124843 \nu^{6} + 174500 \nu^{5} - 1108075 \nu^{4} + \cdots - 1749321 ) / 983355 ( − 4 7 4 4 ν 9 − 2 0 9 4 0 ν 8 + 1 9 1 9 5 ν 7 − 1 2 4 8 4 3 ν 6 + 1 7 4 5 0 0 ν 5 − 1 1 0 8 0 7 5 ν 4 + ⋯ − 1 7 4 9 3 2 1 ) / 9 8 3 3 5 5
(-4744*v^9 - 20940*v^8 + 19195*v^7 - 124843*v^6 + 174500*v^5 - 1108075*v^4 + 1662452*v^3 - 1952655*v^2 - 502560*v - 1749321) / 983355
β 5 \beta_{5} β 5 = = =
( 3213 ν 9 − 12516 ν 8 + 11473 ν 7 − 129958 ν 6 + 104300 ν 5 − 662305 ν 4 + ⋯ − 553661 ) / 327785 ( 3213 \nu^{9} - 12516 \nu^{8} + 11473 \nu^{7} - 129958 \nu^{6} + 104300 \nu^{5} - 662305 \nu^{4} + \cdots - 553661 ) / 327785 ( 3 2 1 3 ν 9 − 1 2 5 1 6 ν 8 + 1 1 4 7 3 ν 7 − 1 2 9 9 5 8 ν 6 + 1 0 4 3 0 0 ν 5 − 6 6 2 3 0 5 ν 4 + ⋯ − 5 5 3 6 6 1 ) / 3 2 7 7 8 5
(3213*v^9 - 12516*v^8 + 11473*v^7 - 129958*v^6 + 104300*v^5 - 662305*v^4 + 22966*v^3 - 1167117*v^2 - 300384*v - 553661) / 327785
β 6 \beta_{6} β 6 = = =
( 26944 ν 9 + 1304 ν 8 + 217328 ν 7 − 55516 ν 6 + 1409535 ν 5 − 149520 ν 4 + ⋯ + 1012608 ) / 983355 ( 26944 \nu^{9} + 1304 \nu^{8} + 217328 \nu^{7} - 55516 \nu^{6} + 1409535 \nu^{5} - 149520 \nu^{4} + \cdots + 1012608 ) / 983355 ( 2 6 9 4 4 ν 9 + 1 3 0 4 ν 8 + 2 1 7 3 2 8 ν 7 − 5 5 5 1 6 ν 6 + 1 4 0 9 5 3 5 ν 5 − 1 4 9 5 2 0 ν 4 + ⋯ + 1 0 1 2 6 0 8 ) / 9 8 3 3 5 5
(26944*v^9 + 1304*v^8 + 217328*v^7 - 55516*v^6 + 1409535*v^5 - 149520*v^4 + 2707548*v^3 + 1104953*v^2 + 3964716*v + 1012608) / 983355
β 7 \beta_{7} β 7 = = =
( − 47884 ν 9 + 55843 ν 8 − 351659 ν 7 + 476704 ν 6 − 2541330 ν 5 + 2272140 ν 4 + ⋯ + 13443 ) / 983355 ( - 47884 \nu^{9} + 55843 \nu^{8} - 351659 \nu^{7} + 476704 \nu^{6} - 2541330 \nu^{5} + 2272140 \nu^{4} + \cdots + 13443 ) / 983355 ( − 4 7 8 8 4 ν 9 + 5 5 8 4 3 ν 8 − 3 5 1 6 5 9 ν 7 + 4 7 6 7 0 4 ν 6 − 2 5 4 1 3 3 0 ν 5 + 2 2 7 2 1 4 0 ν 4 + ⋯ + 1 3 4 4 3 ) / 9 8 3 3 5 5
(-47884*v^9 + 55843*v^8 - 351659*v^7 + 476704*v^6 - 2541330*v^5 + 2272140*v^4 - 4375563*v^3 - 938609*v^2 - 4559898*v + 13443) / 983355
β 8 \beta_{8} β 8 = = =
( − 75292 ν 9 − 9639 ν 8 − 564788 ν 7 + 116165 ν 6 − 3525310 ν 5 + ⋯ − 1809360 ) / 983355 ( - 75292 \nu^{9} - 9639 \nu^{8} - 564788 \nu^{7} + 116165 \nu^{6} - 3525310 \nu^{5} + \cdots - 1809360 ) / 983355 ( − 7 5 2 9 2 ν 9 − 9 6 3 9 ν 8 − 5 6 4 7 8 8 ν 7 + 1 1 6 1 6 5 ν 6 − 3 5 2 5 3 1 0 ν 5 + ⋯ − 1 8 0 9 3 6 0 ) / 9 8 3 3 5 5
(-75292*v^9 - 9639*v^8 - 564788*v^7 + 116165*v^6 - 3525310*v^5 + 63560*v^4 - 5316409*v^3 - 4586418*v^2 - 7114821*v - 1809360) / 983355
β 9 \beta_{9} β 9 = = =
( 26944 ν 9 + 1304 ν 8 + 217328 ν 7 − 55516 ν 6 + 1409535 ν 5 − 149520 ν 4 + ⋯ + 1012608 ) / 327785 ( 26944 \nu^{9} + 1304 \nu^{8} + 217328 \nu^{7} - 55516 \nu^{6} + 1409535 \nu^{5} - 149520 \nu^{4} + \cdots + 1012608 ) / 327785 ( 2 6 9 4 4 ν 9 + 1 3 0 4 ν 8 + 2 1 7 3 2 8 ν 7 − 5 5 5 1 6 ν 6 + 1 4 0 9 5 3 5 ν 5 − 1 4 9 5 2 0 ν 4 + ⋯ + 1 0 1 2 6 0 8 ) / 3 2 7 7 8 5
(26944*v^9 + 1304*v^8 + 217328*v^7 - 55516*v^6 + 1409535*v^5 - 149520*v^4 + 2707548*v^3 + 1432738*v^2 + 3964716*v + 1012608) / 327785
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 9 − 3 β 6 \beta_{9} - 3\beta_{6} β 9 − 3 β 6
b9 - 3*b6
ν 3 \nu^{3} ν 3 = = =
− β 4 + β 3 − 5 β 2 -\beta_{4} + \beta_{3} - 5\beta_{2} − β 4 + β 3 − 5 β 2
-b4 + b3 - 5*b2
ν 4 \nu^{4} ν 4 = = =
− 6 β 9 − β 8 − β 7 + 14 β 6 + β 4 − 6 β 3 + 3 β 1 − 14 -6\beta_{9} - \beta_{8} - \beta_{7} + 14\beta_{6} + \beta_{4} - 6\beta_{3} + 3\beta _1 - 14 − 6 β 9 − β 8 − β 7 + 1 4 β 6 + β 4 − 6 β 3 + 3 β 1 − 1 4
-6*b9 - b8 - b7 + 14*b6 + b4 - 6*b3 + 3*b1 - 14
ν 5 \nu^{5} ν 5 = = =
9 β 9 + 8 β 8 − 6 β 6 + 28 β 2 − 28 β 1 9\beta_{9} + 8\beta_{8} - 6\beta_{6} + 28\beta_{2} - 28\beta_1 9 β 9 + 8 β 8 − 6 β 6 + 2 8 β 2 − 2 8 β 1
9*b9 + 8*b8 - 6*b6 + 28*b2 - 28*b1
ν 6 \nu^{6} ν 6 = = =
− 8 β 5 − 9 β 4 + 37 β 3 − 24 β 2 + 76 -8\beta_{5} - 9\beta_{4} + 37\beta_{3} - 24\beta_{2} + 76 − 8 β 5 − 9 β 4 + 3 7 β 3 − 2 4 β 2 + 7 6
-8*b5 - 9*b4 + 37*b3 - 24*b2 + 76
ν 7 \nu^{7} ν 7 = = =
− 69 β 9 − 53 β 8 − β 7 + 71 β 6 + 53 β 4 − 69 β 3 + 167 β 1 − 71 -69\beta_{9} - 53\beta_{8} - \beta_{7} + 71\beta_{6} + 53\beta_{4} - 69\beta_{3} + 167\beta _1 - 71 − 6 9 β 9 − 5 3 β 8 − β 7 + 7 1 β 6 + 5 3 β 4 − 6 9 β 3 + 1 6 7 β 1 − 7 1
-69*b9 - 53*b8 - b7 + 71*b6 + 53*b4 - 69*b3 + 167*b1 - 71
ν 8 \nu^{8} ν 8 = = =
236 β 9 + 71 β 8 + 52 β 7 − 446 β 6 + 52 β 5 + 211 β 2 − 263 β 1 236\beta_{9} + 71\beta_{8} + 52\beta_{7} - 446\beta_{6} + 52\beta_{5} + 211\beta_{2} - 263\beta_1 2 3 6 β 9 + 7 1 β 8 + 5 2 β 7 − 4 4 6 β 6 + 5 2 β 5 + 2 1 1 β 2 − 2 6 3 β 1
236*b9 + 71*b8 + 52*b7 - 446*b6 + 52*b5 + 211*b2 - 263*b1
ν 9 \nu^{9} ν 9 = = =
− 19 β 5 − 340 β 4 + 499 β 3 − 1022 β 2 + 614 -19\beta_{5} - 340\beta_{4} + 499\beta_{3} - 1022\beta_{2} + 614 − 1 9 β 5 − 3 4 0 β 4 + 4 9 9 β 3 − 1 0 2 2 β 2 + 6 1 4
-19*b5 - 340*b4 + 499*b3 - 1022*b2 + 614
Character values
We give the values of χ \chi χ on generators for ( Z / 325 Z ) × \left(\mathbb{Z}/325\mathbb{Z}\right)^\times ( Z / 3 2 5 Z ) × .
n n n
27 27 2 7
301 301 3 0 1
χ ( n ) \chi(n) χ ( n )
1 1 1
− β 6 -\beta_{6} − β 6
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
This newform subspace can be constructed as the kernel of the linear operator
T 2 10 + 8 T 2 8 − 2 T 2 7 + 52 T 2 6 − 5 T 2 5 + 97 T 2 4 + 60 T 2 3 + 141 T 2 2 + 36 T 2 + 9 T_{2}^{10} + 8T_{2}^{8} - 2T_{2}^{7} + 52T_{2}^{6} - 5T_{2}^{5} + 97T_{2}^{4} + 60T_{2}^{3} + 141T_{2}^{2} + 36T_{2} + 9 T 2 1 0 + 8 T 2 8 − 2 T 2 7 + 5 2 T 2 6 − 5 T 2 5 + 9 7 T 2 4 + 6 0 T 2 3 + 1 4 1 T 2 2 + 3 6 T 2 + 9
T2^10 + 8*T2^8 - 2*T2^7 + 52*T2^6 - 5*T2^5 + 97*T2^4 + 60*T2^3 + 141*T2^2 + 36*T2 + 9
acting on S 2 n e w ( 325 , [ χ ] ) S_{2}^{\mathrm{new}}(325, [\chi]) S 2 n e w ( 3 2 5 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 10 + 8 T 8 + ⋯ + 9 T^{10} + 8 T^{8} + \cdots + 9 T 1 0 + 8 T 8 + ⋯ + 9
T^10 + 8*T^8 - 2*T^7 + 52*T^6 - 5*T^5 + 97*T^4 + 60*T^3 + 141*T^2 + 36*T + 9
3 3 3
T 10 + 3 T 9 + ⋯ + 1 T^{10} + 3 T^{9} + \cdots + 1 T 1 0 + 3 T 9 + ⋯ + 1
T^10 + 3*T^9 + 14*T^8 + 7*T^7 + 52*T^6 + 20*T^5 + 148*T^4 - 56*T^3 + 47*T^2 + 6*T + 1
5 5 5
T 10 T^{10} T 1 0
T^10
7 7 7
T 10 − 2 T 9 + ⋯ + 9025 T^{10} - 2 T^{9} + \cdots + 9025 T 1 0 − 2 T 9 + ⋯ + 9 0 2 5
T^10 - 2*T^9 + 23*T^8 - 50*T^7 + 405*T^6 - 755*T^5 + 2582*T^4 - 1674*T^3 + 6116*T^2 - 4180*T + 9025
11 11 1 1
T 10 + 3 T 9 + ⋯ + 9 T^{10} + 3 T^{9} + \cdots + 9 T 1 0 + 3 T 9 + ⋯ + 9
T^10 + 3*T^9 + 26*T^8 + 61*T^7 + 466*T^6 + 1009*T^5 + 2974*T^4 + 606*T^3 + 249*T^2 - 27*T + 9
13 13 1 3
T 10 − 10 T 9 + ⋯ + 371293 T^{10} - 10 T^{9} + \cdots + 371293 T 1 0 − 1 0 T 9 + ⋯ + 3 7 1 2 9 3
T^10 - 10*T^9 + 60*T^8 - 263*T^7 + 1076*T^6 - 3957*T^5 + 13988*T^4 - 44447*T^3 + 131820*T^2 - 285610*T + 371293
17 17 1 7
T 10 + 4 T 9 + ⋯ + 239121 T^{10} + 4 T^{9} + \cdots + 239121 T 1 0 + 4 T 9 + ⋯ + 2 3 9 1 2 1
T^10 + 4*T^9 + 52*T^8 + 46*T^7 + 1448*T^6 + 2085*T^5 + 15277*T^4 + 13548*T^3 + 98439*T^2 + 111492*T + 239121
19 19 1 9
T 10 − 4 T 9 + ⋯ + 225 T^{10} - 4 T^{9} + \cdots + 225 T 1 0 − 4 T 9 + ⋯ + 2 2 5
T^10 - 4*T^9 + 24*T^8 - 34*T^7 + 194*T^6 - 263*T^5 + 1045*T^4 - 174*T^3 + 499*T^2 - 30*T + 225
23 23 2 3
T 10 + 15 T 9 + ⋯ + 281961 T^{10} + 15 T^{9} + \cdots + 281961 T 1 0 + 1 5 T 9 + ⋯ + 2 8 1 9 6 1
T^10 + 15*T^9 + 197*T^8 + 1086*T^7 + 6802*T^6 + 20835*T^5 + 147498*T^4 + 370395*T^3 + 869706*T^2 + 543213*T + 281961
29 29 2 9
T 10 − T 9 + ⋯ + 881721 T^{10} - T^{9} + \cdots + 881721 T 1 0 − T 9 + ⋯ + 8 8 1 7 2 1
T^10 - T^9 + 66*T^8 - 233*T^7 + 3888*T^6 - 9652*T^5 + 52852*T^4 - 49656*T^3 + 376107*T^2 - 456354*T + 881721
31 31 3 1
( T 5 − 57 T 3 + ⋯ + 225 ) 2 (T^{5} - 57 T^{3} + \cdots + 225)^{2} ( T 5 − 5 7 T 3 + ⋯ + 2 2 5 ) 2
(T^5 - 57*T^3 - 29*T^2 + 495*T + 225)^2
37 37 3 7
T 10 + 17 T 9 + ⋯ + 826281 T^{10} + 17 T^{9} + \cdots + 826281 T 1 0 + 1 7 T 9 + ⋯ + 8 2 6 2 8 1
T^10 + 17*T^9 + 236*T^8 + 1385*T^7 + 8138*T^6 + 27575*T^5 + 138412*T^4 + 390384*T^3 + 1256247*T^2 + 1104435*T + 826281
41 41 4 1
T 10 + 6 T 9 + ⋯ + 50625 T^{10} + 6 T^{9} + \cdots + 50625 T 1 0 + 6 T 9 + ⋯ + 5 0 6 2 5
T^10 + 6*T^9 + 110*T^8 + 354*T^7 + 8254*T^6 + 34359*T^5 + 129435*T^4 + 186516*T^3 + 237231*T^2 - 86400*T + 50625
43 43 4 3
T 10 + 12 T 9 + ⋯ + 14190289 T^{10} + 12 T^{9} + \cdots + 14190289 T 1 0 + 1 2 T 9 + ⋯ + 1 4 1 9 0 2 8 9
T^10 + 12*T^9 + 198*T^8 + 502*T^7 + 8458*T^6 + 2225*T^5 + 358753*T^4 - 374014*T^3 + 4010189*T^2 + 5115586*T + 14190289
47 47 4 7
( T 5 + 12 T 4 + ⋯ + 2025 ) 2 (T^{5} + 12 T^{4} + \cdots + 2025)^{2} ( T 5 + 1 2 T 4 + ⋯ + 2 0 2 5 ) 2
(T^5 + 12*T^4 - 20*T^3 - 311*T^2 + 90*T + 2025)^2
53 53 5 3
( T 5 − 8 T 4 + ⋯ + 6075 ) 2 (T^{5} - 8 T^{4} + \cdots + 6075)^{2} ( T 5 − 8 T 4 + ⋯ + 6 0 7 5 ) 2
(T^5 - 8*T^4 - 117*T^3 + 279*T^2 + 3645*T + 6075)^2
59 59 5 9
T 10 − 12 T 9 + ⋯ + 4782969 T^{10} - 12 T^{9} + \cdots + 4782969 T 1 0 − 1 2 T 9 + ⋯ + 4 7 8 2 9 6 9
T^10 - 12*T^9 + 231*T^8 - 1386*T^7 + 23364*T^6 - 137052*T^5 + 1344276*T^4 - 1856763*T^3 + 4133430*T^2 + 2657205*T + 4782969
61 61 6 1
T 10 + 5 T 9 + ⋯ + 403225 T^{10} + 5 T^{9} + \cdots + 403225 T 1 0 + 5 T 9 + ⋯ + 4 0 3 2 2 5
T^10 + 5*T^9 + 72*T^8 + 145*T^7 + 2888*T^6 + 6855*T^5 + 45662*T^4 + 8200*T^3 + 194091*T^2 + 172085*T + 403225
67 67 6 7
T 10 + ⋯ + 2567550241 T^{10} + \cdots + 2567550241 T 1 0 + ⋯ + 2 5 6 7 5 5 0 2 4 1
T^10 - 16*T^9 + 387*T^8 - 2924*T^7 + 57533*T^6 - 386265*T^5 + 5461592*T^4 - 13807922*T^3 + 127229154*T^2 + 10742252*T + 2567550241
71 71 7 1
T 10 + ⋯ + 101787921 T^{10} + \cdots + 101787921 T 1 0 + ⋯ + 1 0 1 7 8 7 9 2 1
T^10 + 19*T^9 + 368*T^8 + 2633*T^7 + 29920*T^6 + 156342*T^5 + 1695840*T^4 + 5111748*T^3 + 26869923*T^2 - 36259866*T + 101787921
73 73 7 3
( T 5 + 8 T 4 + ⋯ + 10125 ) 2 (T^{5} + 8 T^{4} + \cdots + 10125)^{2} ( T 5 + 8 T 4 + ⋯ + 1 0 1 2 5 ) 2
(T^5 + 8*T^4 - 119*T^3 - 990*T^2 + 900*T + 10125)^2
79 79 7 9
( T 5 + 14 T 4 + ⋯ + 16875 ) 2 (T^{5} + 14 T^{4} + \cdots + 16875)^{2} ( T 5 + 1 4 T 4 + ⋯ + 1 6 8 7 5 ) 2
(T^5 + 14*T^4 - 95*T^3 - 1350*T^2 + 2250*T + 16875)^2
83 83 8 3
( T 5 + 7 T 4 + ⋯ − 53775 ) 2 (T^{5} + 7 T^{4} + \cdots - 53775)^{2} ( T 5 + 7 T 4 + ⋯ − 5 3 7 7 5 ) 2
(T^5 + 7*T^4 - 257*T^3 - 461*T^2 + 19110*T - 53775)^2
89 89 8 9
T 10 − 10 T 9 + ⋯ + 2537649 T^{10} - 10 T^{9} + \cdots + 2537649 T 1 0 − 1 0 T 9 + ⋯ + 2 5 3 7 6 4 9
T^10 - 10*T^9 + 139*T^8 - 558*T^7 + 6387*T^6 - 22599*T^5 + 203832*T^4 - 183978*T^3 + 770958*T^2 + 200718*T + 2537649
97 97 9 7
T 10 + ⋯ + 1683378841 T^{10} + \cdots + 1683378841 T 1 0 + ⋯ + 1 6 8 3 3 7 8 8 4 1
T^10 + 37*T^9 + 1001*T^8 + 14930*T^7 + 180868*T^6 + 1281185*T^5 + 9727402*T^4 + 44083039*T^3 + 419732172*T^2 + 867147915*T + 1683378841
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