Properties

Label 630.2.k
Level 630
Weight 2
Character orbit k
Rep. character \(\chi_{630}(361,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 24
Newform subspaces 10
Sturm bound 288
Trace bound 7

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Defining parameters

Level: \( N \) = \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 630.k (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 10 \)
Sturm bound: \(288\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(630, [\chi])\).

Total New Old
Modular forms 320 24 296
Cusp forms 256 24 232
Eisenstein series 64 0 64

Trace form

\( 24q - 12q^{4} - 2q^{5} - 8q^{7} + O(q^{10}) \) \( 24q - 12q^{4} - 2q^{5} - 8q^{7} + 2q^{10} - 6q^{11} + 10q^{14} - 12q^{16} - 12q^{17} + 6q^{19} + 4q^{20} + 8q^{22} + 12q^{23} - 12q^{25} + 6q^{26} + 4q^{28} + 20q^{29} - 8q^{31} - 16q^{34} + 6q^{35} - 12q^{37} - 16q^{38} + 2q^{40} - 40q^{41} + 56q^{43} - 6q^{44} - 16q^{46} + 4q^{47} + 42q^{49} - 20q^{53} - 16q^{55} - 2q^{56} + 36q^{59} - 42q^{61} + 16q^{62} + 24q^{64} - 6q^{65} - 48q^{67} - 12q^{68} + 6q^{70} + 32q^{71} + 28q^{73} + 6q^{74} - 12q^{76} + 32q^{77} - 8q^{79} - 2q^{80} - 16q^{82} + 16q^{83} - 16q^{85} - 18q^{86} - 4q^{88} + 6q^{89} + 76q^{91} - 24q^{92} + 6q^{94} - 4q^{95} - 32q^{97} - 24q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(630, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
630.2.k.a \(2\) \(5.031\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-1\) \(-4\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-\zeta_{6}q^{5}+(-3+\cdots)q^{7}+\cdots\)
630.2.k.b \(2\) \(5.031\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-1\) \(-1\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots\)
630.2.k.c \(2\) \(5.031\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-1\) \(4\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots\)
630.2.k.d \(2\) \(5.031\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(1\) \(-4\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+\zeta_{6}q^{5}+(-3+\cdots)q^{7}+\cdots\)
630.2.k.e \(2\) \(5.031\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(-1\) \(-4\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-\zeta_{6}q^{5}+(-1+\cdots)q^{7}+\cdots\)
630.2.k.f \(2\) \(5.031\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(-1\) \(1\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-\zeta_{6}q^{5}+(-1+\cdots)q^{7}+\cdots\)
630.2.k.g \(2\) \(5.031\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(1\) \(-4\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+\zeta_{6}q^{5}+(-1+\cdots)q^{7}+\cdots\)
630.2.k.h \(2\) \(5.031\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(1\) \(4\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+\zeta_{6}q^{5}+(3+\cdots)q^{7}+\cdots\)
630.2.k.i \(4\) \(5.031\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(-2\) \(0\) \(2\) \(0\) \(q+(-1-\beta _{2})q^{2}+\beta _{2}q^{4}+(1+\beta _{2})q^{5}+\cdots\)
630.2.k.j \(4\) \(5.031\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(2\) \(0\) \(-2\) \(0\) \(q+(1+\beta _{2})q^{2}+\beta _{2}q^{4}+(-1-\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(630, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(630, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(315, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))
$3$ 1
$5$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))
$11$ (\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} \))(\( 1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 + 4 T - 3 T^{2} - 12 T^{3} + 64 T^{4} - 132 T^{5} - 363 T^{6} + 5324 T^{7} + 14641 T^{8} \))(\( 1 - 4 T - 3 T^{2} + 12 T^{3} + 64 T^{4} + 132 T^{5} - 363 T^{6} - 5324 T^{7} + 14641 T^{8} \))
$13$ (\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 - T + 13 T^{2} )^{2} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 7 T + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} + 52 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} + 52 T^{3} + 169 T^{4} )^{2} \))
$17$ (\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 6 T^{2} + 96 T^{3} - 461 T^{4} + 1632 T^{5} + 1734 T^{6} - 19652 T^{7} + 83521 T^{8} \))(\( 1 + 4 T + 6 T^{2} - 96 T^{3} - 461 T^{4} - 1632 T^{5} + 1734 T^{6} + 19652 T^{7} + 83521 T^{8} \))
$19$ (\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 - 3 T - 10 T^{2} - 57 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 - 6 T + 17 T^{2} - 114 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 + 2 T - 7 T^{2} - 54 T^{3} - 316 T^{4} - 1026 T^{5} - 2527 T^{6} + 13718 T^{7} + 130321 T^{8} \))(\( 1 + 2 T - 7 T^{2} - 54 T^{3} - 316 T^{4} - 1026 T^{5} - 2527 T^{6} + 13718 T^{7} + 130321 T^{8} \))
$23$ (\( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - 7 T + 26 T^{2} - 161 T^{3} + 529 T^{4} \))(\( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \))(\( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \))(\( ( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} )^{2} \))
$29$ (\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 - 3 T + 29 T^{2} )^{2} \))(\( ( 1 - 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 9 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 - 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 + 29 T^{2} )^{4} \))
$31$ (\( 1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} \))(\( 1 + 6 T + 5 T^{2} + 186 T^{3} + 961 T^{4} \))(\( ( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} )^{2} \))
$37$ (\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))(\( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( 1 - 3 T - 28 T^{2} - 111 T^{3} + 1369 T^{4} \))(\( 1 + 4 T + T^{2} - 236 T^{3} - 1736 T^{4} - 8732 T^{5} + 1369 T^{6} + 202612 T^{7} + 1874161 T^{8} \))(\( 1 + 4 T + T^{2} - 236 T^{3} - 1736 T^{4} - 8732 T^{5} + 1369 T^{6} + 202612 T^{7} + 1874161 T^{8} \))
$41$ (\( ( 1 + 9 T + 41 T^{2} )^{2} \))(\( ( 1 + 9 T + 41 T^{2} )^{2} \))(\( ( 1 - 11 T + 41 T^{2} )^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( ( 1 - 7 T + 41 T^{2} )^{2} \))(\( ( 1 + 5 T + 41 T^{2} )^{2} \))(\( ( 1 + 9 T + 41 T^{2} )^{2} \))(\( ( 1 - 8 T + 91 T^{2} - 328 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 8 T + 91 T^{2} + 328 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 7 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 2 T + 43 T^{2} )^{2} \))(\( ( 1 + 10 T + 43 T^{2} )^{2} \))(\( ( 1 + 5 T + 43 T^{2} )^{2} \))(\( ( 1 - 12 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 12 T + 94 T^{2} - 516 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 12 T + 94 T^{2} - 516 T^{3} + 1849 T^{4} )^{2} \))
$47$ (\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 5 T - 22 T^{2} + 235 T^{3} + 2209 T^{4} \))(\( 1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4} \))(\( 1 + 11 T + 74 T^{2} + 517 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 + 2 T + 21 T^{2} - 222 T^{3} - 2108 T^{4} - 10434 T^{5} + 46389 T^{6} + 207646 T^{7} + 4879681 T^{8} \))(\( 1 - 2 T + 21 T^{2} + 222 T^{3} - 2108 T^{4} + 10434 T^{5} + 46389 T^{6} - 207646 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 + 11 T + 68 T^{2} + 583 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \))(\( 1 + 2 T - 49 T^{2} + 106 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 + T - 52 T^{2} + 53 T^{3} + 2809 T^{4} \))(\( 1 + 10 T - 3 T^{2} - 30 T^{3} + 2500 T^{4} - 1590 T^{5} - 8427 T^{6} + 1488770 T^{7} + 7890481 T^{8} \))(\( 1 - 10 T - 3 T^{2} + 30 T^{3} + 2500 T^{4} + 1590 T^{5} - 8427 T^{6} - 1488770 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 10 T + 41 T^{2} - 590 T^{3} + 3481 T^{4} \))(\( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} \))(\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 - 8 T - 42 T^{2} + 96 T^{3} + 3979 T^{4} + 5664 T^{5} - 146202 T^{6} - 1643032 T^{7} + 12117361 T^{8} \))(\( 1 + 8 T - 42 T^{2} - 96 T^{3} + 3979 T^{4} - 5664 T^{5} - 146202 T^{6} + 1643032 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} \))(\( 1 - 61 T^{2} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} \))(\( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( 1 + 12 T + 14 T^{2} + 96 T^{3} + 4395 T^{4} + 5856 T^{5} + 52094 T^{6} + 2723772 T^{7} + 13845841 T^{8} \))(\( 1 + 12 T + 14 T^{2} + 96 T^{3} + 4395 T^{4} + 5856 T^{5} + 52094 T^{6} + 2723772 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 67 T^{2} + 4489 T^{4} \))(\( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 - 9 T + 14 T^{2} - 603 T^{3} + 4489 T^{4} \))(\( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} \))(\( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} \))(\( 1 + 20 T + 194 T^{2} + 1440 T^{3} + 11147 T^{4} + 96480 T^{5} + 870866 T^{6} + 6015260 T^{7} + 20151121 T^{8} \))(\( 1 + 20 T + 194 T^{2} + 1440 T^{3} + 11147 T^{4} + 96480 T^{5} + 870866 T^{6} + 6015260 T^{7} + 20151121 T^{8} \))
$71$ (\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 2 T + 71 T^{2} )^{2} \))(\( ( 1 - 14 T + 71 T^{2} )^{2} \))(\( ( 1 + 16 T + 178 T^{2} + 1136 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 16 T + 178 T^{2} - 1136 T^{3} + 5041 T^{4} )^{2} \))
$73$ (\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( 1 - 16 T + 183 T^{2} - 1168 T^{3} + 5329 T^{4} \))(\( 1 - 6 T - 37 T^{2} - 438 T^{3} + 5329 T^{4} \))(\( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} \))(\( ( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} )^{2} \))
$79$ (\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} \))(\( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} \))(\( 1 - 12 T + 65 T^{2} - 948 T^{3} + 6241 T^{4} \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( 1 + 12 T - 22 T^{2} + 96 T^{3} + 9939 T^{4} + 7584 T^{5} - 137302 T^{6} + 5916468 T^{7} + 38950081 T^{8} \))(\( 1 + 12 T - 22 T^{2} + 96 T^{3} + 9939 T^{4} + 7584 T^{5} - 137302 T^{6} + 5916468 T^{7} + 38950081 T^{8} \))
$83$ (\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 3 T + 83 T^{2} )^{2} \))(\( ( 1 + 8 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 7 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 20 T + 238 T^{2} - 1660 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 20 T + 238 T^{2} + 1660 T^{3} + 6889 T^{4} )^{2} \))
$89$ (\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \))(\( 1 + 10 T + 11 T^{2} + 890 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - T - 88 T^{2} - 89 T^{3} + 7921 T^{4} \))(\( 1 - 14 T + 107 T^{2} - 1246 T^{3} + 7921 T^{4} \))(\( 1 + 2 T - 85 T^{2} + 178 T^{3} + 7921 T^{4} \))(\( 1 - 8 T - 18 T^{2} + 768 T^{3} - 6893 T^{4} + 68352 T^{5} - 142578 T^{6} - 5639752 T^{7} + 62742241 T^{8} \))(\( 1 + 8 T - 18 T^{2} - 768 T^{3} - 6893 T^{4} - 68352 T^{5} - 142578 T^{6} + 5639752 T^{7} + 62742241 T^{8} \))
$97$ (\( ( 1 - 8 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( ( 1 + 16 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( ( 1 + 8 T + 97 T^{2} )^{2} \))(\( ( 1 + 16 T + 97 T^{2} )^{2} \))(\( ( 1 + 4 T + 97 T^{2} )^{4} \))(\( ( 1 + 4 T + 97 T^{2} )^{4} \))
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