Properties

Label 1638.2.p.j
Level $1638$
Weight $2$
Character orbit 1638.p
Analytic conductor $13.079$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.p (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} + 15 x^{8} + 14 x^{7} + 110 x^{6} + 36 x^{5} + 233 x^{4} + 164 x^{3} + 345 x^{2} + 76 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 546)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{5} ) q^{2} + \beta_{5} q^{4} + ( -\beta_{1} - \beta_{2} ) q^{5} + \beta_{8} q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 - \beta_{5} ) q^{2} + \beta_{5} q^{4} + ( -\beta_{1} - \beta_{2} ) q^{5} + \beta_{8} q^{7} + q^{8} + \beta_{2} q^{10} + ( 2 - \beta_{2} - \beta_{3} - \beta_{8} + \beta_{9} ) q^{11} + ( -\beta_{1} - \beta_{3} - \beta_{7} ) q^{13} + \beta_{4} q^{14} + ( -1 - \beta_{5} ) q^{16} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{17} + ( -1 + 2 \beta_{2} - \beta_{4} - \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{19} + \beta_{1} q^{20} + ( -2 - \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{22} + ( \beta_{1} + \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{23} + ( 1 - 2 \beta_{1} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{25} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{26} + ( -\beta_{4} - \beta_{8} ) q^{28} + ( \beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{29} + ( -2 - \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{31} + \beta_{5} q^{32} + ( \beta_{2} + \beta_{3} + \beta_{8} - \beta_{9} ) q^{34} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{35} + ( -1 - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{37} + ( 1 + 2 \beta_{1} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{38} + ( -\beta_{1} - \beta_{2} ) q^{40} + ( -\beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{41} + ( 1 - 3 \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{43} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{44} + ( -\beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{46} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{47} + ( -2 + \beta_{1} + \beta_{3} + 2 \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{49} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{9} ) q^{50} + ( -\beta_{2} + \beta_{7} ) q^{52} + ( 3 - \beta_{1} - 2 \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{53} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} ) q^{55} + \beta_{8} q^{56} + ( -\beta_{3} - \beta_{4} - \beta_{6} - \beta_{9} ) q^{58} + ( -4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{59} + ( -3 - \beta_{2} + \beta_{4} + \beta_{6} + \beta_{8} ) q^{61} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{62} + q^{64} + ( -3 + 2 \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} ) q^{65} + ( -1 + 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{9} ) q^{67} + ( \beta_{1} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{68} + ( -\beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{70} + ( -4 - 3 \beta_{1} - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} ) q^{71} + ( 2 + 3 \beta_{1} + 2 \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{9} ) q^{73} + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{74} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + \beta_{9} ) q^{76} + ( -3 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{9} ) q^{77} + ( \beta_{1} + \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{79} + \beta_{2} q^{80} + ( -2 + \beta_{3} + \beta_{8} - \beta_{9} ) q^{82} + ( -\beta_{3} - \beta_{4} - \beta_{6} - \beta_{9} ) q^{83} + ( -1 + \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{85} + ( 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{86} + ( 2 - \beta_{2} - \beta_{3} - \beta_{8} + \beta_{9} ) q^{88} + ( -3 - 7 \beta_{1} - 5 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{89} + ( -2 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{9} ) q^{91} + ( \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{8} - \beta_{9} ) q^{92} + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{94} + ( 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + 3 \beta_{9} ) q^{95} + ( -2 - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{97} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{9} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 5q^{2} - 5q^{4} + 2q^{5} + 4q^{7} + 10q^{8} + O(q^{10}) \) \( 10q - 5q^{2} - 5q^{4} + 2q^{5} + 4q^{7} + 10q^{8} - 4q^{10} + 12q^{11} - 4q^{13} - 2q^{14} - 5q^{16} - 4q^{17} - 6q^{19} + 2q^{20} - 6q^{22} - 6q^{23} - q^{25} + 2q^{26} - 2q^{28} - 10q^{31} - 5q^{32} + 8q^{34} + 2q^{35} + q^{37} + 3q^{38} + 2q^{40} + 4q^{41} + 3q^{43} - 6q^{44} - 6q^{46} + 15q^{47} - 20q^{49} - q^{50} + 2q^{52} + 17q^{53} + 3q^{55} + 4q^{56} - 2q^{59} - 22q^{61} - 10q^{62} + 10q^{64} - 41q^{65} + 2q^{67} - 4q^{68} - 16q^{70} - 18q^{71} + 12q^{73} + q^{74} + 3q^{76} - 18q^{77} - 4q^{79} - 4q^{80} - 8q^{82} + q^{85} + 3q^{86} + 12q^{88} - 7q^{89} - 4q^{91} + 12q^{92} - 30q^{94} - 24q^{95} - 6q^{97} + 16q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} + 15 x^{8} + 14 x^{7} + 110 x^{6} + 36 x^{5} + 233 x^{4} + 164 x^{3} + 345 x^{2} + 76 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -503 \nu^{9} - 2241 \nu^{8} + 8466 \nu^{7} - 67528 \nu^{6} + 19422 \nu^{5} - 156870 \nu^{4} + 1003571 \nu^{3} - 301041 \nu^{2} - 66732 \nu + 438544 \)\()/2044008\)
\(\beta_{3}\)\(=\)\((\)\( -3064 \nu^{9} + 9207 \nu^{8} - 34782 \nu^{7} - 28346 \nu^{6} - 79794 \nu^{5} + 644490 \nu^{4} + 579550 \nu^{3} + 1236807 \nu^{2} + 274164 \nu + 3620228 \)\()/1022004\)
\(\beta_{4}\)\(=\)\((\)\( -3977 \nu^{9} - 10833 \nu^{8} - 12699 \nu^{7} - 334006 \nu^{6} - 625302 \nu^{5} - 1723536 \nu^{4} - 478621 \nu^{3} - 3196425 \nu^{2} - 3391749 \nu - 3118196 \)\()/1022004\)
\(\beta_{5}\)\(=\)\((\)\( -27409 \nu^{9} + 54315 \nu^{8} - 413376 \nu^{7} - 375260 \nu^{6} - 3082518 \nu^{5} - 967302 \nu^{4} - 6543167 \nu^{3} - 3491505 \nu^{2} - 9757146 \nu - 2149816 \)\()/2044008\)
\(\beta_{6}\)\(=\)\((\)\( -14545 \nu^{9} + 29941 \nu^{8} - 216152 \nu^{7} - 200758 \nu^{6} - 1521222 \nu^{5} - 440214 \nu^{4} - 2887781 \nu^{3} - 2349679 \nu^{2} - 3963994 \nu - 1074620 \)\()/340668\)
\(\beta_{7}\)\(=\)\((\)\( 45928 \nu^{9} - 96426 \nu^{8} + 695481 \nu^{7} + 561428 \nu^{6} + 5037264 \nu^{5} + 744876 \nu^{4} + 10649492 \nu^{3} + 5556402 \nu^{2} + 16896855 \nu + 165664 \)\()/1022004\)
\(\beta_{8}\)\(=\)\((\)\( 97465 \nu^{9} - 178473 \nu^{8} + 1399728 \nu^{7} + 1662752 \nu^{6} + 10555542 \nu^{5} + 4653846 \nu^{4} + 20509619 \nu^{3} + 17738031 \nu^{2} + 29957226 \nu + 5377120 \)\()/2044008\)
\(\beta_{9}\)\(=\)\((\)\( 51344 \nu^{9} - 117222 \nu^{8} + 805587 \nu^{7} + 484042 \nu^{6} + 5520312 \nu^{5} + 367938 \nu^{4} + 11604142 \nu^{3} + 5109630 \nu^{2} + 14145267 \nu + 742892 \)\()/1022004\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + 4 \beta_{5} - \beta_{4} + 2 \beta_{2} + 2 \beta_{1}\)
\(\nu^{3}\)\(=\)\(-3 \beta_{8} - 3 \beta_{6} - 3 \beta_{4} - 2 \beta_{3} + 11 \beta_{2} - 6\)
\(\nu^{4}\)\(=\)\(-16 \beta_{9} - 16 \beta_{8} + 6 \beta_{7} - 18 \beta_{6} - 40 \beta_{5} + 2 \beta_{4} - 37 \beta_{1} - 40\)
\(\nu^{5}\)\(=\)\(-65 \beta_{9} + 4 \beta_{8} + 34 \beta_{7} - 4 \beta_{6} - 126 \beta_{5} + 65 \beta_{4} + 34 \beta_{3} - 168 \beta_{2} - 168 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-30 \beta_{9} + 297 \beta_{8} + 267 \beta_{6} + 267 \beta_{4} + 126 \beta_{3} - 657 \beta_{2} + 592\)
\(\nu^{7}\)\(=\)\(1080 \beta_{9} + 1080 \beta_{8} - 564 \beta_{7} + 1176 \beta_{6} + 2280 \beta_{5} - 96 \beta_{4} + 2797 \beta_{1} + 2280\)
\(\nu^{8}\)\(=\)\(5005 \beta_{9} - 468 \beta_{8} - 2256 \beta_{7} + 468 \beta_{6} + 9784 \beta_{5} - 5005 \beta_{4} - 2256 \beta_{3} + 11402 \beta_{2} + 11402 \beta_{1}\)
\(\nu^{9}\)\(=\)\(1788 \beta_{9} - 20451 \beta_{8} - 18663 \beta_{6} - 18663 \beta_{4} - 9542 \beta_{3} + 47603 \beta_{2} - 39726\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(-1 - \beta_{5}\) \(\beta_{5}\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
919.1
−1.10337 1.91109i
−0.623307 1.07960i
−0.114009 0.197470i
0.769836 + 1.33339i
2.07085 + 3.58682i
−1.10337 + 1.91109i
−0.623307 + 1.07960i
−0.114009 + 0.197470i
0.769836 1.33339i
2.07085 3.58682i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −1.10337 + 1.91109i 0 1.44928 + 2.21350i 1.00000 0 2.20674
919.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.623307 + 1.07960i 0 −2.27938 + 1.34329i 1.00000 0 1.24661
919.3 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.114009 + 0.197470i 0 0.848534 2.50599i 1.00000 0 0.228019
919.4 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0.769836 1.33339i 0 −0.131875 2.64246i 1.00000 0 −1.53967
919.5 −0.500000 0.866025i 0 −0.500000 + 0.866025i 2.07085 3.58682i 0 2.11344 + 1.59166i 1.00000 0 −4.14170
991.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.10337 1.91109i 0 1.44928 2.21350i 1.00000 0 2.20674
991.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.623307 1.07960i 0 −2.27938 1.34329i 1.00000 0 1.24661
991.3 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.114009 0.197470i 0 0.848534 + 2.50599i 1.00000 0 0.228019
991.4 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.769836 + 1.33339i 0 −0.131875 + 2.64246i 1.00000 0 −1.53967
991.5 −0.500000 + 0.866025i 0 −0.500000 0.866025i 2.07085 + 3.58682i 0 2.11344 1.59166i 1.00000 0 −4.14170
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 991.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.p.j 10
3.b odd 2 1 546.2.k.e yes 10
7.c even 3 1 1638.2.m.k 10
13.c even 3 1 1638.2.m.k 10
21.h odd 6 1 546.2.j.e 10
39.i odd 6 1 546.2.j.e 10
91.g even 3 1 inner 1638.2.p.j 10
273.bm odd 6 1 546.2.k.e yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.j.e 10 21.h odd 6 1
546.2.j.e 10 39.i odd 6 1
546.2.k.e yes 10 3.b odd 2 1
546.2.k.e yes 10 273.bm odd 6 1
1638.2.m.k 10 7.c even 3 1
1638.2.m.k 10 13.c even 3 1
1638.2.p.j 10 1.a even 1 1 trivial
1638.2.p.j 10 91.g even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{10} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{5} \)
$3$ \( T^{10} \)
$5$ \( 16 + 76 T + 345 T^{2} + 164 T^{3} + 233 T^{4} + 36 T^{5} + 110 T^{6} + 14 T^{7} + 15 T^{8} - 2 T^{9} + T^{10} \)
$7$ \( 16807 - 9604 T + 6174 T^{2} - 1274 T^{3} + 434 T^{4} + 3 T^{5} + 62 T^{6} - 26 T^{7} + 18 T^{8} - 4 T^{9} + T^{10} \)
$11$ \( ( -30 - 21 T + 65 T^{2} - 12 T^{3} - 6 T^{4} + T^{5} )^{2} \)
$13$ \( 371293 + 114244 T + 52728 T^{2} + 7436 T^{3} + 1118 T^{4} + 15 T^{5} + 86 T^{6} + 44 T^{7} + 24 T^{8} + 4 T^{9} + T^{10} \)
$17$ \( 784 - 476 T + 2277 T^{2} + 2327 T^{3} + 4589 T^{4} + 1584 T^{5} + 701 T^{6} + 62 T^{7} + 36 T^{8} + 4 T^{9} + T^{10} \)
$19$ \( ( -111 + 1101 T - 109 T^{2} - 69 T^{3} + 3 T^{4} + T^{5} )^{2} \)
$23$ \( 129600 - 197640 T + 494721 T^{2} + 346653 T^{3} + 246681 T^{4} + 45612 T^{5} + 8955 T^{6} + 642 T^{7} + 108 T^{8} + 6 T^{9} + T^{10} \)
$29$ \( 236196 + 196830 T + 177147 T^{2} + 47385 T^{3} + 25029 T^{4} + 2106 T^{5} + 3195 T^{6} + 54 T^{7} + 60 T^{8} + T^{10} \)
$31$ \( 1254400 + 1647520 T + 1604961 T^{2} + 675789 T^{3} + 221955 T^{4} + 41274 T^{5} + 7137 T^{6} + 738 T^{7} + 126 T^{8} + 10 T^{9} + T^{10} \)
$37$ \( 319225 + 207920 T + 145029 T^{2} + 36684 T^{3} + 14838 T^{4} + 1947 T^{5} + 1059 T^{6} + 72 T^{7} + 39 T^{8} - T^{9} + T^{10} \)
$41$ \( 770884 - 194038 T + 160347 T^{2} - 28125 T^{3} + 19689 T^{4} - 3174 T^{5} + 1311 T^{6} - 126 T^{7} + 48 T^{8} - 4 T^{9} + T^{10} \)
$43$ \( 308025 - 213120 T + 277881 T^{2} - 22980 T^{3} + 92728 T^{4} - 22221 T^{5} + 10725 T^{6} - 164 T^{7} + 111 T^{8} - 3 T^{9} + T^{10} \)
$47$ \( 271854144 - 17164008 T + 19550241 T^{2} - 614784 T^{3} + 1063294 T^{4} - 45738 T^{5} + 18675 T^{6} - 1430 T^{7} + 279 T^{8} - 15 T^{9} + T^{10} \)
$53$ \( 256 - 66704 T + 17364273 T^{2} - 4244362 T^{3} + 1078286 T^{4} - 131550 T^{5} + 21575 T^{6} - 2206 T^{7} + 279 T^{8} - 17 T^{9} + T^{10} \)
$59$ \( 1607824 + 2367356 T + 2927769 T^{2} + 1244992 T^{3} + 502853 T^{4} - 79680 T^{5} + 25142 T^{6} - 1214 T^{7} + 171 T^{8} + 2 T^{9} + T^{10} \)
$61$ \( ( 80 - 1016 T - 400 T^{2} - 8 T^{3} + 11 T^{4} + T^{5} )^{2} \)
$67$ \( ( -20 + 719 T + 199 T^{2} - 137 T^{3} - T^{4} + T^{5} )^{2} \)
$71$ \( 202500 - 2544750 T + 32555925 T^{2} + 7241610 T^{3} + 1686319 T^{4} + 192492 T^{5} + 28812 T^{6} + 2726 T^{7} + 315 T^{8} + 18 T^{9} + T^{10} \)
$73$ \( 37941975369 - 3240866106 T + 887285502 T^{2} - 50704044 T^{3} + 11876944 T^{4} - 622851 T^{5} + 90666 T^{6} - 3100 T^{7} + 408 T^{8} - 12 T^{9} + T^{10} \)
$79$ \( 817216 - 709640 T + 580065 T^{2} - 159768 T^{3} + 60951 T^{4} - 4344 T^{5} + 4416 T^{6} - 204 T^{7} + 87 T^{8} + 4 T^{9} + T^{10} \)
$83$ \( ( -486 + 405 T + 27 T^{2} - 60 T^{3} + T^{5} )^{2} \)
$89$ \( 205435562500 + 22923118750 T + 4266583125 T^{2} + 229948250 T^{3} + 34506950 T^{4} + 1494480 T^{5} + 191111 T^{6} + 4292 T^{7} + 513 T^{8} + 7 T^{9} + T^{10} \)
$97$ \( 1734489 + 1457919 T + 1797027 T^{2} + 159624 T^{3} + 449455 T^{4} + 93495 T^{5} + 60546 T^{6} - 590 T^{7} + 279 T^{8} + 6 T^{9} + T^{10} \)
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