gp: [N,k,chi] = [296,2,Mod(121,296)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(296, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 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("296.121");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [10]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == 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 − x 9 + 9 x 8 + 4 x 7 + 54 x 6 + 6 x 5 + 98 x 4 − 8 x 3 + 148 x 2 − 24 x + 4 x^{10} - x^{9} + 9x^{8} + 4x^{7} + 54x^{6} + 6x^{5} + 98x^{4} - 8x^{3} + 148x^{2} - 24x + 4 x 1 0 − x 9 + 9 x 8 + 4 x 7 + 5 4 x 6 + 6 x 5 + 9 8 x 4 − 8 x 3 + 1 4 8 x 2 − 2 4 x + 4
x^10 - x^9 + 9*x^8 + 4*x^7 + 54*x^6 + 6*x^5 + 98*x^4 - 8*x^3 + 148*x^2 - 24*x + 4
:
β 1 \beta_{1} β 1 = = =
2 ν 2\nu 2 ν
2*v
β 2 \beta_{2} β 2 = = =
( − 13 ν 9 − 605 ν 8 + 1210 ν 7 − 7028 ν 6 + 1815 ν 5 − 27830 ν 4 + ⋯ + 5505 ) / 15127 ( - 13 \nu^{9} - 605 \nu^{8} + 1210 \nu^{7} - 7028 \nu^{6} + 1815 \nu^{5} - 27830 \nu^{4} + \cdots + 5505 ) / 15127 ( − 1 3 ν 9 − 6 0 5 ν 8 + 1 2 1 0 ν 7 − 7 0 2 8 ν 6 + 1 8 1 5 ν 5 − 2 7 8 3 0 ν 4 + ⋯ + 5 5 0 5 ) / 1 5 1 2 7
(-13*v^9 - 605*v^8 + 1210*v^7 - 7028*v^6 + 1815*v^5 - 27830*v^4 + 3582*v^3 - 52030*v^2 + 8470*v + 5505) / 15127
β 3 \beta_{3} β 3 = = =
( 40 ν 9 + 33 ν 8 − 66 ν 7 + 1012 ν 6 − 99 ν 5 + 1518 ν 4 − 15676 ν 3 + ⋯ + 5004 ) / 15127 ( 40 \nu^{9} + 33 \nu^{8} - 66 \nu^{7} + 1012 \nu^{6} - 99 \nu^{5} + 1518 \nu^{4} - 15676 \nu^{3} + \cdots + 5004 ) / 15127 ( 4 0 ν 9 + 3 3 ν 8 − 6 6 ν 7 + 1 0 1 2 ν 6 − 9 9 ν 5 + 1 5 1 8 ν 4 − 1 5 6 7 6 ν 3 + ⋯ + 5 0 0 4 ) / 1 5 1 2 7
(40*v^9 + 33*v^8 - 66*v^7 + 1012*v^6 - 99*v^5 + 1518*v^4 - 15676*v^3 + 2838*v^2 - 462*v + 5004) / 15127
β 4 \beta_{4} β 4 = = =
( 146 ν 9 + 1309 ν 8 − 2618 ν 7 + 17092 ν 6 − 3927 ν 5 + 60214 ν 4 + ⋯ + 49383 ) / 15127 ( 146 \nu^{9} + 1309 \nu^{8} - 2618 \nu^{7} + 17092 \nu^{6} - 3927 \nu^{5} + 60214 \nu^{4} + \cdots + 49383 ) / 15127 ( 1 4 6 ν 9 + 1 3 0 9 ν 8 − 2 6 1 8 ν 7 + 1 7 0 9 2 ν 6 − 3 9 2 7 ν 5 + 6 0 2 1 4 ν 4 + ⋯ + 4 9 3 8 3 ) / 1 5 1 2 7
(146*v^9 + 1309*v^8 - 2618*v^7 + 17092*v^6 - 3927*v^5 + 60214*v^4 - 39065*v^3 + 112574*v^2 - 18326*v + 49383) / 15127
β 5 \beta_{5} β 5 = = =
( − 153 ν 9 + 360 ν 8 − 720 ν 7 + 235 ν 6 − 1080 ν 5 + 16560 ν 4 + ⋯ + 50660 ) / 15127 ( - 153 \nu^{9} + 360 \nu^{8} - 720 \nu^{7} + 235 \nu^{6} - 1080 \nu^{5} + 16560 \nu^{4} + \cdots + 50660 ) / 15127 ( − 1 5 3 ν 9 + 3 6 0 ν 8 − 7 2 0 ν 7 + 2 3 5 ν 6 − 1 0 8 0 ν 5 + 1 6 5 6 0 ν 4 + ⋯ + 5 0 6 6 0 ) / 1 5 1 2 7
(-153*v^9 + 360*v^8 - 720*v^7 + 235*v^6 - 1080*v^5 + 16560*v^4 + 13067*v^3 + 30960*v^2 - 5040*v + 50660) / 15127
β 6 \beta_{6} β 6 = = =
( 863 ν 9 − 1395 ν 8 + 7112 ν 7 + 4762 ν 6 + 27956 ν 5 + 7143 ν 4 + ⋯ − 11542 ) / 30254 ( 863 \nu^{9} - 1395 \nu^{8} + 7112 \nu^{7} + 4762 \nu^{6} + 27956 \nu^{5} + 7143 \nu^{4} + \cdots - 11542 ) / 30254 ( 8 6 3 ν 9 − 1 3 9 5 ν 8 + 7 1 1 2 ν 7 + 4 7 6 2 ν 6 + 2 7 9 5 6 ν 5 + 7 1 4 3 ν 4 + ⋯ − 1 1 5 4 2 ) / 3 0 2 5 4
(863*v^9 - 1395*v^8 + 7112*v^7 + 4762*v^6 + 27956*v^5 + 7143*v^4 + 54444*v^3 + 74520*v^2 + 71394*v - 11542) / 30254
β 7 \beta_{7} β 7 = = =
( − 1251 ν 9 + 1291 ν 8 − 11226 ν 7 − 5070 ν 6 − 66542 ν 5 − 7605 ν 4 + ⋯ + 29562 ) / 30254 ( - 1251 \nu^{9} + 1291 \nu^{8} - 11226 \nu^{7} - 5070 \nu^{6} - 66542 \nu^{5} - 7605 \nu^{4} + \cdots + 29562 ) / 30254 ( − 1 2 5 1 ν 9 + 1 2 9 1 ν 8 − 1 1 2 2 6 ν 7 − 5 0 7 0 ν 6 − 6 6 5 4 2 ν 5 − 7 6 0 5 ν 4 + ⋯ + 2 9 5 6 2 ) / 3 0 2 5 4
(-1251*v^9 + 1291*v^8 - 11226*v^7 - 5070*v^6 - 66542*v^5 - 7605*v^4 - 121080*v^3 - 5668*v^2 - 182310*v + 29562) / 30254
β 8 \beta_{8} β 8 = = =
( − 2432 ν 9 − 1142 ν 8 − 17165 ν 7 − 36462 ν 6 − 156488 ν 5 − 123845 ν 4 + ⋯ − 3432 ) / 30254 ( - 2432 \nu^{9} - 1142 \nu^{8} - 17165 \nu^{7} - 36462 \nu^{6} - 156488 \nu^{5} - 123845 \nu^{4} + \cdots - 3432 ) / 30254 ( − 2 4 3 2 ν 9 − 1 1 4 2 ν 8 − 1 7 1 6 5 ν 7 − 3 6 4 6 2 ν 6 − 1 5 6 4 8 8 ν 5 − 1 2 3 8 4 5 ν 4 + ⋯ − 3 4 3 2 ) / 3 0 2 5 4
(-2432*v^9 - 1142*v^8 - 17165*v^7 - 36462*v^6 - 156488*v^5 - 123845*v^4 - 276076*v^3 - 80924*v^2 - 550194*v - 3432) / 30254
β 9 \beta_{9} β 9 = = =
( 4505 ν 9 − 4117 ν 8 + 40649 ν 7 + 22134 ν 6 + 243578 ν 5 + 65616 ν 4 + ⋯ + 2796 ) / 30254 ( 4505 \nu^{9} - 4117 \nu^{8} + 40649 \nu^{7} + 22134 \nu^{6} + 243578 \nu^{5} + 65616 \nu^{4} + \cdots + 2796 ) / 30254 ( 4 5 0 5 ν 9 − 4 1 1 7 ν 8 + 4 0 6 4 9 ν 7 + 2 2 1 3 4 ν 6 + 2 4 3 5 7 8 ν 5 + 6 5 6 1 6 ν 4 + ⋯ + 2 7 9 6 ) / 3 0 2 5 4
(4505*v^9 - 4117*v^8 + 40649*v^7 + 22134*v^6 + 243578*v^5 + 65616*v^4 + 441952*v^3 + 30596*v^2 + 597888*v + 2796) / 30254
ν \nu ν = = =
( β 1 ) / 2 ( \beta_1 ) / 2 ( β 1 ) / 2
(b1) / 2
ν 2 \nu^{2} ν 2 = = =
( − β 9 + β 8 − 7 β 7 − 2 β 6 + β 5 + β 4 − β 3 + β 1 ) / 2 ( -\beta_{9} + \beta_{8} - 7\beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 ) / 2 ( − β 9 + β 8 − 7 β 7 − 2 β 6 + β 5 + β 4 − β 3 + β 1 ) / 2
(-b9 + b8 - 7*b7 - 2*b6 + b5 + b4 - b3 + b1) / 2
ν 3 \nu^{3} ν 3 = = =
β 4 − 3 β 3 + 2 β 2 − 3 \beta_{4} - 3\beta_{3} + 2\beta_{2} - 3 β 4 − 3 β 3 + 2 β 2 − 3
b4 - 3*b3 + 2*b2 - 3
ν 4 \nu^{4} ν 4 = = =
2 β 9 − 4 β 8 + 21 β 7 + 9 β 6 + 9 β 2 − 6 β 1 − 21 2\beta_{9} - 4\beta_{8} + 21\beta_{7} + 9\beta_{6} + 9\beta_{2} - 6\beta _1 - 21 2 β 9 − 4 β 8 + 2 1 β 7 + 9 β 6 + 9 β 2 − 6 β 1 − 2 1
2*b9 - 4*b8 + 21*b7 + 9*b6 + 9*b2 - 6*b1 - 21
ν 5 \nu^{5} ν 5 = = =
β 9 − 11 β 8 + 40 β 7 + 23 β 6 − β 5 − 11 β 4 + 23 β 3 − 23 β 1 \beta_{9} - 11\beta_{8} + 40\beta_{7} + 23\beta_{6} - \beta_{5} - 11\beta_{4} + 23\beta_{3} - 23\beta_1 β 9 − 1 1 β 8 + 4 0 β 7 + 2 3 β 6 − β 5 − 1 1 β 4 + 2 3 β 3 − 2 3 β 1
b9 - 11*b8 + 40*b7 + 23*b6 - b5 - 11*b4 + 23*b3 - 23*b1
ν 6 \nu^{6} ν 6 = = =
− 11 β 5 − 35 β 4 + 60 β 3 − 79 β 2 + 160 -11\beta_{5} - 35\beta_{4} + 60\beta_{3} - 79\beta_{2} + 160 − 1 1 β 5 − 3 5 β 4 + 6 0 β 3 − 7 9 β 2 + 1 6 0
-11*b5 - 35*b4 + 60*b3 - 79*b2 + 160
ν 7 \nu^{7} ν 7 = = =
− 16 β 9 + 104 β 8 − 409 β 7 − 223 β 6 − 223 β 2 + 197 β 1 + 409 -16\beta_{9} + 104\beta_{8} - 409\beta_{7} - 223\beta_{6} - 223\beta_{2} + 197\beta _1 + 409 − 1 6 β 9 + 1 0 4 β 8 − 4 0 9 β 7 − 2 2 3 β 6 − 2 2 3 β 2 + 1 9 7 β 1 + 4 0 9
-16*b9 + 104*b8 - 409*b7 - 223*b6 - 223*b2 + 197*b1 + 409
ν 8 \nu^{8} ν 8 = = =
− 78 β 9 + 316 β 8 − 1363 β 7 − 705 β 6 + 78 β 5 + 316 β 4 − 565 β 3 + 565 β 1 -78\beta_{9} + 316\beta_{8} - 1363\beta_{7} - 705\beta_{6} + 78\beta_{5} + 316\beta_{4} - 565\beta_{3} + 565\beta_1 − 7 8 β 9 + 3 1 6 β 8 − 1 3 6 3 β 7 − 7 0 5 β 6 + 7 8 β 5 + 3 1 6 β 4 − 5 6 5 β 3 + 5 6 5 β 1
-78*b9 + 316*b8 - 1363*b7 - 705*b6 + 78*b5 + 316*b4 - 565*b3 + 565*b1
ν 9 \nu^{9} ν 9 = = =
176 β 5 + 954 β 4 − 1757 β 3 + 2073 β 2 − 3877 176\beta_{5} + 954\beta_{4} - 1757\beta_{3} + 2073\beta_{2} - 3877 1 7 6 β 5 + 9 5 4 β 4 − 1 7 5 7 β 3 + 2 0 7 3 β 2 − 3 8 7 7
176*b5 + 954*b4 - 1757*b3 + 2073*b2 - 3877
Character values
We give the values of χ \chi χ on generators for ( Z / 296 Z ) × \left(\mathbb{Z}/296\mathbb{Z}\right)^\times ( Z / 2 9 6 Z ) × .
n n n
113 113 1 1 3
149 149 1 4 9
223 223 2 2 3
χ ( n ) \chi(n) χ ( n )
− 1 + β 7 -1 + \beta_{7} − 1 + β 7
1 1 1
1 1 1
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 3 10 + T 3 9 + 11 T 3 8 − 2 T 3 7 + 88 T 3 6 + 16 T 3 5 + 168 T 3 4 + 96 T 3 3 + 288 T 3 2 + 128 T 3 + 64 T_{3}^{10} + T_{3}^{9} + 11T_{3}^{8} - 2T_{3}^{7} + 88T_{3}^{6} + 16T_{3}^{5} + 168T_{3}^{4} + 96T_{3}^{3} + 288T_{3}^{2} + 128T_{3} + 64 T 3 1 0 + T 3 9 + 1 1 T 3 8 − 2 T 3 7 + 8 8 T 3 6 + 1 6 T 3 5 + 1 6 8 T 3 4 + 9 6 T 3 3 + 2 8 8 T 3 2 + 1 2 8 T 3 + 6 4
T3^10 + T3^9 + 11*T3^8 - 2*T3^7 + 88*T3^6 + 16*T3^5 + 168*T3^4 + 96*T3^3 + 288*T3^2 + 128*T3 + 64
acting on S 2 n e w ( 296 , [ χ ] ) S_{2}^{\mathrm{new}}(296, [\chi]) S 2 n e w ( 2 9 6 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 10 T^{10} T 1 0
T^10
3 3 3
T 10 + T 9 + ⋯ + 64 T^{10} + T^{9} + \cdots + 64 T 1 0 + T 9 + ⋯ + 6 4
T^10 + T^9 + 11*T^8 - 2*T^7 + 88*T^6 + 16*T^5 + 168*T^4 + 96*T^3 + 288*T^2 + 128*T + 64
5 5 5
T 10 − 3 T 9 + ⋯ + 6241 T^{10} - 3 T^{9} + \cdots + 6241 T 1 0 − 3 T 9 + ⋯ + 6 2 4 1
T^10 - 3*T^9 + 23*T^8 - 42*T^7 + 301*T^6 - 541*T^5 + 1821*T^4 - 1330*T^3 + 3759*T^2 - 1659*T + 6241
7 7 7
T 10 − 3 T 9 + ⋯ + 64 T^{10} - 3 T^{9} + \cdots + 64 T 1 0 − 3 T 9 + ⋯ + 6 4
T^10 - 3*T^9 + 31*T^8 - 110*T^7 + 812*T^6 - 2328*T^5 + 6312*T^4 - 5984*T^3 + 4800*T^2 + 512*T + 64
11 11 1 1
( T 5 − 8 T 4 + ⋯ − 128 ) 2 (T^{5} - 8 T^{4} + \cdots - 128)^{2} ( T 5 − 8 T 4 + ⋯ − 1 2 8 ) 2
(T^5 - 8*T^4 - 8*T^3 + 144*T^2 - 112*T - 128)^2
13 13 1 3
T 10 − T 9 + ⋯ + 2304 T^{10} - T^{9} + \cdots + 2304 T 1 0 − T 9 + ⋯ + 2 3 0 4
T^10 - T^9 + 45*T^8 + 12*T^7 + 1728*T^6 - 304*T^5 + 10064*T^4 - 640*T^3 + 50944*T^2 - 10752*T + 2304
17 17 1 7
( T 2 − T + 1 ) 5 (T^{2} - T + 1)^{5} ( T 2 − T + 1 ) 5
(T^2 - T + 1)^5
19 19 1 9
T 10 + 3 T 9 + ⋯ + 256 T^{10} + 3 T^{9} + \cdots + 256 T 1 0 + 3 T 9 + ⋯ + 2 5 6
T^10 + 3*T^9 + 21*T^8 + 28*T^7 + 224*T^6 + 304*T^5 + 1168*T^4 - 128*T^3 + 768*T^2 + 256*T + 256
23 23 2 3
( T 5 + 6 T 4 + ⋯ + 1152 ) 2 (T^{5} + 6 T^{4} + \cdots + 1152)^{2} ( T 5 + 6 T 4 + ⋯ + 1 1 5 2 ) 2
(T^5 + 6*T^4 - 36*T^3 - 248*T^2 + 8*T + 1152)^2
29 29 2 9
( T 5 − 72 T 3 + ⋯ + 358 ) 2 (T^{5} - 72 T^{3} + \cdots + 358)^{2} ( T 5 − 7 2 T 3 + ⋯ + 3 5 8 ) 2
(T^5 - 72*T^3 - 222*T^2 + 135*T + 358)^2
31 31 3 1
( T 5 + 8 T 4 + ⋯ + 128 ) 2 (T^{5} + 8 T^{4} + \cdots + 128)^{2} ( T 5 + 8 T 4 + ⋯ + 1 2 8 ) 2
(T^5 + 8*T^4 - 8*T^3 - 144*T^2 - 112*T + 128)^2
37 37 3 7
T 10 + 12 T 9 + ⋯ + 69343957 T^{10} + 12 T^{9} + \cdots + 69343957 T 1 0 + 1 2 T 9 + ⋯ + 6 9 3 4 3 9 5 7
T^10 + 12*T^9 + 61*T^8 - 26*T^7 - 1967*T^6 - 14918*T^5 - 72779*T^4 - 35594*T^3 + 3089833*T^2 + 22489932*T + 69343957
41 41 4 1
T 10 − 9 T 9 + ⋯ + 8543929 T^{10} - 9 T^{9} + \cdots + 8543929 T 1 0 − 9 T 9 + ⋯ + 8 5 4 3 9 2 9
T^10 - 9*T^9 + 167*T^8 - 1494*T^7 + 21053*T^6 - 156719*T^5 + 1015477*T^4 - 3410678*T^3 + 8594719*T^2 - 10087273*T + 8543929
43 43 4 3
( T 5 − 2 T 4 − 16 T 3 + ⋯ + 32 ) 2 (T^{5} - 2 T^{4} - 16 T^{3} + \cdots + 32)^{2} ( T 5 − 2 T 4 − 1 6 T 3 + ⋯ + 3 2 ) 2
(T^5 - 2*T^4 - 16*T^3 + 8*T^2 + 56*T + 32)^2
47 47 4 7
( T 5 − 16 T 4 + ⋯ + 23424 ) 2 (T^{5} - 16 T^{4} + \cdots + 23424)^{2} ( T 5 − 1 6 T 4 + ⋯ + 2 3 4 2 4 ) 2
(T^5 - 16*T^4 - 72*T^3 + 2416*T^2 - 13616*T + 23424)^2
53 53 5 3
T 10 − 5 T 9 + ⋯ + 2304 T^{10} - 5 T^{9} + \cdots + 2304 T 1 0 − 5 T 9 + ⋯ + 2 3 0 4
T^10 - 5*T^9 + 141*T^8 - 28*T^7 + 11648*T^6 - 2032*T^5 + 478224*T^4 + 1000576*T^3 + 11090176*T^2 - 159744*T + 2304
59 59 5 9
T 10 + 5 T 9 + ⋯ + 147456 T^{10} + 5 T^{9} + \cdots + 147456 T 1 0 + 5 T 9 + ⋯ + 1 4 7 4 5 6
T^10 + 5*T^9 + 69*T^8 + 420*T^7 + 4176*T^6 + 20096*T^5 + 76160*T^4 + 171008*T^3 + 286720*T^2 + 245760*T + 147456
61 61 6 1
T 10 − 15 T 9 + ⋯ + 40921609 T^{10} - 15 T^{9} + \cdots + 40921609 T 1 0 − 1 5 T 9 + ⋯ + 4 0 9 2 1 6 0 9
T^10 - 15*T^9 + 247*T^8 - 1834*T^7 + 21565*T^6 - 162937*T^5 + 1159957*T^4 - 4967314*T^3 + 16610647*T^2 - 31031847*T + 40921609
67 67 6 7
T 10 − T 9 + ⋯ + 20736 T^{10} - T^{9} + \cdots + 20736 T 1 0 − T 9 + ⋯ + 2 0 7 3 6
T^10 - T^9 + 41*T^8 - 40*T^7 + 1512*T^6 - 1488*T^5 + 6576*T^4 - 6400*T^3 + 22144*T^2 - 18432*T + 20736
71 71 7 1
T 10 + 17 T 9 + ⋯ + 40000 T^{10} + 17 T^{9} + \cdots + 40000 T 1 0 + 1 7 T 9 + ⋯ + 4 0 0 0 0
T^10 + 17*T^9 + 243*T^8 + 1246*T^7 + 6540*T^6 + 5448*T^5 + 79304*T^4 + 129760*T^3 + 184000*T^2 + 96000*T + 40000
73 73 7 3
( T 5 + 18 T 4 + ⋯ + 1184 ) 2 (T^{5} + 18 T^{4} + \cdots + 1184)^{2} ( T 5 + 1 8 T 4 + ⋯ + 1 1 8 4 ) 2
(T^5 + 18*T^4 - 24*T^3 - 912*T^2 + 1296*T + 1184)^2
79 79 7 9
T 10 + ⋯ + 12908595456 T^{10} + \cdots + 12908595456 T 1 0 + ⋯ + 1 2 9 0 8 5 9 5 4 5 6
T^10 - 21*T^9 + 545*T^8 - 5032*T^7 + 89688*T^6 - 619216*T^5 + 10308912*T^4 - 34831360*T^3 + 419561344*T^2 + 352664064*T + 12908595456
83 83 8 3
T 10 + 15 T 9 + ⋯ + 45481536 T^{10} + 15 T^{9} + \cdots + 45481536 T 1 0 + 1 5 T 9 + ⋯ + 4 5 4 8 1 5 3 6
T^10 + 15*T^9 + 395*T^8 + 2130*T^7 + 62000*T^6 + 344544*T^5 + 5714440*T^4 - 2387040*T^3 + 19780960*T^2 + 13488000*T + 45481536
89 89 8 9
T 10 + ⋯ + 1372480209 T^{10} + \cdots + 1372480209 T 1 0 + ⋯ + 1 3 7 2 4 8 0 2 0 9
T^10 - 29*T^9 + 623*T^8 - 7646*T^7 + 79789*T^6 - 576523*T^5 + 4361213*T^4 - 24802846*T^3 + 146221375*T^2 - 484093149*T + 1372480209
97 97 9 7
( T 5 + 18 T 4 + ⋯ + 88282 ) 2 (T^{5} + 18 T^{4} + \cdots + 88282)^{2} ( T 5 + 1 8 T 4 + ⋯ + 8 8 2 8 2 ) 2
(T^5 + 18*T^4 - 102*T^3 - 2660*T^2 + 429*T + 88282)^2
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