Properties

Label 252.2.bf
Level 252
Weight 2
Character orbit bf
Rep. character \(\chi_{252}(19,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 36
Newform subspaces 7
Sturm bound 96
Trace bound 7

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Defining parameters

Level: \( N \) = \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 252.bf (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 28 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 7 \)
Sturm bound: \(96\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(252, [\chi])\).

Total New Old
Modular forms 112 44 68
Cusp forms 80 36 44
Eisenstein series 32 8 24

Trace form

\( 36q + 6q^{5} + O(q^{10}) \) \( 36q + 6q^{5} + 6q^{10} + 20q^{14} + 4q^{16} + 6q^{17} - 20q^{22} + 4q^{25} - 18q^{26} + 24q^{28} + 16q^{29} - 20q^{32} - 14q^{37} - 36q^{38} - 48q^{40} - 26q^{46} - 20q^{49} - 12q^{50} - 36q^{52} + 6q^{53} - 56q^{56} - 4q^{58} - 18q^{61} - 24q^{64} - 20q^{65} + 36q^{68} - 30q^{70} - 42q^{73} + 44q^{74} - 6q^{77} + 96q^{80} + 24q^{82} - 20q^{85} + 10q^{86} + 56q^{88} - 54q^{89} + 48q^{92} + 78q^{94} + 94q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(252, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
252.2.bf.a \(4\) \(2.012\) \(\Q(\sqrt{-3}, \sqrt{-7})\) None \(-1\) \(0\) \(6\) \(0\) \(q-\beta _{3}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(2-\beta _{2}+\cdots)q^{5}+\cdots\)
252.2.bf.b \(4\) \(2.012\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-2\) \(q+\beta _{3}q^{2}-2q^{4}-2\beta _{1}q^{5}+(1-3\beta _{2}+\cdots)q^{7}+\cdots\)
252.2.bf.c \(4\) \(2.012\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(2\) \(q+\beta _{1}q^{2}+2\beta _{2}q^{4}+(2\beta _{1}-2\beta _{3})q^{5}+\cdots\)
252.2.bf.d \(4\) \(2.012\) \(\Q(\sqrt{-3}, \sqrt{-7})\) None \(1\) \(0\) \(-6\) \(0\) \(q+\beta _{3}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(-2+\beta _{2}+\cdots)q^{5}+\cdots\)
252.2.bf.e \(4\) \(2.012\) \(\Q(\zeta_{12})\) None \(2\) \(0\) \(6\) \(0\) \(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots\)
252.2.bf.f \(8\) \(2.012\) 8.0.562828176.1 None \(-1\) \(0\) \(0\) \(-2\) \(q+\beta _{5}q^{2}+(-\beta _{4}+\beta _{6}-\beta _{7})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
252.2.bf.g \(8\) \(2.012\) 8.0.562828176.1 None \(-1\) \(0\) \(0\) \(2\) \(q+(-\beta _{1}-\beta _{5})q^{2}+(-\beta _{2}-\beta _{6}+\beta _{7})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(252, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(252, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))(\( ( 1 + 2 T^{2} )^{2} \))(\( 1 - 2 T^{2} + 4 T^{4} \))(\( 1 - T - T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} \))(\( 1 + T + T^{2} + 2 T^{3} + 4 T^{5} + 4 T^{6} + 8 T^{7} + 16 T^{8} \))(\( 1 + T + T^{2} + 2 T^{3} + 6 T^{4} + 4 T^{5} + 4 T^{6} + 8 T^{7} + 16 T^{8} \))
$3$ 1
$5$ (\( ( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} )^{2} \))(\( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} \))(\( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} \))(\( ( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} )^{2} \))(\( 1 + 9 T^{2} + 45 T^{4} - 96 T^{5} + 66 T^{6} - 864 T^{7} - 394 T^{8} - 4320 T^{9} + 1650 T^{10} - 12000 T^{11} + 28125 T^{12} + 140625 T^{14} + 390625 T^{16} \))(\( 1 + 9 T^{2} + 45 T^{4} - 96 T^{5} + 66 T^{6} - 864 T^{7} - 394 T^{8} - 4320 T^{9} + 1650 T^{10} - 12000 T^{11} + 28125 T^{12} + 140625 T^{14} + 390625 T^{16} \))
$7$ (\( 1 - 7 T^{2} + 49 T^{4} \))(\( ( 1 + T + 7 T^{2} )^{2} \))(\( ( 1 - T + 7 T^{2} )^{2} \))(\( 1 - 7 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 2 T + 16 T^{3} + 65 T^{4} + 112 T^{5} + 686 T^{7} + 2401 T^{8} \))(\( 1 - 2 T - 16 T^{3} + 65 T^{4} - 112 T^{5} - 686 T^{7} + 2401 T^{8} \))
$11$ (\( 1 + 15 T^{2} + 104 T^{4} + 1815 T^{6} + 14641 T^{8} \))(\( 1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8} \))(\( 1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8} \))(\( 1 + 15 T^{2} + 104 T^{4} + 1815 T^{6} + 14641 T^{8} \))(\( 1 + 21 T^{2} + 320 T^{4} + 2541 T^{6} + 14641 T^{8} \))(\( ( 1 - 8 T + 34 T^{2} - 112 T^{3} + 339 T^{4} - 1232 T^{5} + 4114 T^{6} - 10648 T^{7} + 14641 T^{8} )( 1 + 2 T + 25 T^{2} + 58 T^{3} + 372 T^{4} + 638 T^{5} + 3025 T^{6} + 2662 T^{7} + 14641 T^{8} ) \))(\( ( 1 - 2 T + 25 T^{2} - 58 T^{3} + 372 T^{4} - 638 T^{5} + 3025 T^{6} - 2662 T^{7} + 14641 T^{8} )( 1 + 8 T + 34 T^{2} + 112 T^{3} + 339 T^{4} + 1232 T^{5} + 4114 T^{6} + 10648 T^{7} + 14641 T^{8} ) \))
$13$ (\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( 1 - 66 T^{2} + 2241 T^{4} - 49362 T^{6} + 759092 T^{8} - 8342178 T^{10} + 64005201 T^{12} - 318569394 T^{14} + 815730721 T^{16} \))(\( 1 - 66 T^{2} + 2241 T^{4} - 49362 T^{6} + 759092 T^{8} - 8342178 T^{10} + 64005201 T^{12} - 318569394 T^{14} + 815730721 T^{16} \))
$17$ (\( ( 1 - 12 T + 65 T^{2} - 204 T^{3} + 289 T^{4} )^{2} \))(\( 1 + 26 T^{2} + 387 T^{4} + 7514 T^{6} + 83521 T^{8} \))(\( 1 + 26 T^{2} + 387 T^{4} + 7514 T^{6} + 83521 T^{8} \))(\( ( 1 + 12 T + 65 T^{2} + 204 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 - 3 T + 20 T^{2} - 51 T^{3} + 289 T^{4} )^{2} \))(\( 1 + 40 T^{2} + 786 T^{4} - 1104 T^{5} + 10208 T^{6} - 39072 T^{7} + 129731 T^{8} - 664224 T^{9} + 2950112 T^{10} - 5423952 T^{11} + 65647506 T^{12} + 965502760 T^{14} + 6975757441 T^{16} \))(\( 1 + 40 T^{2} + 786 T^{4} - 1104 T^{5} + 10208 T^{6} - 39072 T^{7} + 129731 T^{8} - 664224 T^{9} + 2950112 T^{10} - 5423952 T^{11} + 65647506 T^{12} + 965502760 T^{14} + 6975757441 T^{16} \))
$19$ (\( ( 1 - 19 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 19 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} + 361 T^{4} )( 1 + 26 T^{2} + 361 T^{4} ) \))(\( 1 - 6 T - 33 T^{2} + 150 T^{3} + 1165 T^{4} - 1968 T^{5} - 34182 T^{6} + 11868 T^{7} + 759066 T^{8} + 225492 T^{9} - 12339702 T^{10} - 13498512 T^{11} + 151823965 T^{12} + 371414850 T^{13} - 1552514073 T^{14} - 5363230434 T^{15} + 16983563041 T^{16} \))(\( 1 + 6 T - 33 T^{2} - 150 T^{3} + 1165 T^{4} + 1968 T^{5} - 34182 T^{6} - 11868 T^{7} + 759066 T^{8} - 225492 T^{9} - 12339702 T^{10} + 13498512 T^{11} + 151823965 T^{12} - 371414850 T^{13} - 1552514073 T^{14} + 5363230434 T^{15} + 16983563041 T^{16} \))
$23$ (\( ( 1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} )( 1 + 8 T + 41 T^{2} + 184 T^{3} + 529 T^{4} ) \))(\( 1 + 38 T^{2} + 915 T^{4} + 20102 T^{6} + 279841 T^{8} \))(\( 1 + 38 T^{2} + 915 T^{4} + 20102 T^{6} + 279841 T^{8} \))(\( ( 1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} )( 1 + 8 T + 41 T^{2} + 184 T^{3} + 529 T^{4} ) \))(\( 1 + 45 T^{2} + 1496 T^{4} + 23805 T^{6} + 279841 T^{8} \))(\( 1 + 52 T^{2} + 1242 T^{4} + 2784 T^{5} + 24080 T^{6} + 140352 T^{7} + 497843 T^{8} + 3228096 T^{9} + 12738320 T^{10} + 33872928 T^{11} + 347562522 T^{12} + 7697866228 T^{14} + 78310985281 T^{16} \))(\( 1 + 52 T^{2} + 1242 T^{4} - 2784 T^{5} + 24080 T^{6} - 140352 T^{7} + 497843 T^{8} - 3228096 T^{9} + 12738320 T^{10} - 33872928 T^{11} + 347562522 T^{12} + 7697866228 T^{14} + 78310985281 T^{16} \))
$29$ (\( ( 1 - 5 T + 29 T^{2} )^{4} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 + 5 T + 29 T^{2} )^{4} \))(\( ( 1 + 4 T + 29 T^{2} )^{4} \))(\( ( 1 - 8 T + 71 T^{2} - 344 T^{3} + 1924 T^{4} - 9976 T^{5} + 59711 T^{6} - 195112 T^{7} + 707281 T^{8} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} - 344 T^{3} + 1924 T^{4} - 9976 T^{5} + 59711 T^{6} - 195112 T^{7} + 707281 T^{8} )^{2} \))
$31$ (\( 1 - 41 T^{2} + 720 T^{4} - 39401 T^{6} + 923521 T^{8} \))(\( ( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} )^{2} \))(\( 1 - 41 T^{2} + 720 T^{4} - 39401 T^{6} + 923521 T^{8} \))(\( ( 1 - 46 T^{2} + 961 T^{4} )( 1 - 13 T^{2} + 961 T^{4} ) \))(\( 1 + 6 T - 4 T^{2} - 336 T^{3} - 2729 T^{4} - 10764 T^{5} + 2216 T^{6} + 444234 T^{7} + 3877768 T^{8} + 13771254 T^{9} + 2129576 T^{10} - 320670324 T^{11} - 2520288809 T^{12} - 9619394736 T^{13} - 3550014724 T^{14} + 165075684666 T^{15} + 852891037441 T^{16} \))(\( 1 - 6 T - 4 T^{2} + 336 T^{3} - 2729 T^{4} + 10764 T^{5} + 2216 T^{6} - 444234 T^{7} + 3877768 T^{8} - 13771254 T^{9} + 2129576 T^{10} + 320670324 T^{11} - 2520288809 T^{12} + 9619394736 T^{13} - 3550014724 T^{14} - 165075684666 T^{15} + 852891037441 T^{16} \))
$37$ (\( ( 1 - 37 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} )^{2} \))(\( 1 - 6 T - 69 T^{2} - 18 T^{3} + 4753 T^{4} + 9780 T^{5} - 152586 T^{6} - 146184 T^{7} + 2893194 T^{8} - 5408808 T^{9} - 208890234 T^{10} + 495386340 T^{11} + 8907887233 T^{12} - 1248191226 T^{13} - 177035122221 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16} \))(\( 1 - 6 T - 69 T^{2} - 18 T^{3} + 4753 T^{4} + 9780 T^{5} - 152586 T^{6} - 146184 T^{7} + 2893194 T^{8} - 5408808 T^{9} - 208890234 T^{10} + 495386340 T^{11} + 8907887233 T^{12} - 1248191226 T^{13} - 177035122221 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16} \))
$41$ (\( ( 1 - 70 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 1681 T^{4} )^{2} \))(\( 1 - 120 T^{2} + 9948 T^{4} - 607176 T^{6} + 27583238 T^{8} - 1020662856 T^{10} + 28110670428 T^{12} - 570012508920 T^{14} + 7984925229121 T^{16} \))(\( 1 - 120 T^{2} + 9948 T^{4} - 607176 T^{6} + 27583238 T^{8} - 1020662856 T^{10} + 28110670428 T^{12} - 570012508920 T^{14} + 7984925229121 T^{16} \))
$43$ (\( ( 1 + 26 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 59 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 59 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 26 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 82 T^{2} + 1849 T^{4} )^{2} \))(\( 1 - 210 T^{2} + 23793 T^{4} - 1729746 T^{6} + 88400276 T^{8} - 3198300354 T^{10} + 81343532193 T^{12} - 1327486240290 T^{14} + 11688200277601 T^{16} \))(\( 1 - 210 T^{2} + 23793 T^{4} - 1729746 T^{6} + 88400276 T^{8} - 3198300354 T^{10} + 81343532193 T^{12} - 1327486240290 T^{14} + 11688200277601 T^{16} \))
$47$ (\( 1 - 10 T^{2} - 2109 T^{4} - 22090 T^{6} + 4879681 T^{8} \))(\( 1 - 70 T^{2} + 2691 T^{4} - 154630 T^{6} + 4879681 T^{8} \))(\( 1 - 70 T^{2} + 2691 T^{4} - 154630 T^{6} + 4879681 T^{8} \))(\( 1 - 10 T^{2} - 2109 T^{4} - 22090 T^{6} + 4879681 T^{8} \))(\( 1 - 19 T^{2} - 1848 T^{4} - 41971 T^{6} + 4879681 T^{8} \))(\( 1 - 4 T - 144 T^{2} + 456 T^{3} + 12722 T^{4} - 27948 T^{5} - 805600 T^{6} + 556940 T^{7} + 41968563 T^{8} + 26176180 T^{9} - 1779570400 T^{10} - 2901645204 T^{11} + 62079301682 T^{12} + 104581323192 T^{13} - 1552207007376 T^{14} - 2026492481852 T^{15} + 23811286661761 T^{16} \))(\( 1 + 4 T - 144 T^{2} - 456 T^{3} + 12722 T^{4} + 27948 T^{5} - 805600 T^{6} - 556940 T^{7} + 41968563 T^{8} - 26176180 T^{9} - 1779570400 T^{10} + 2901645204 T^{11} + 62079301682 T^{12} - 104581323192 T^{13} - 1552207007376 T^{14} + 2026492481852 T^{15} + 23811286661761 T^{16} \))
$53$ (\( ( 1 + 7 T - 4 T^{2} + 371 T^{3} + 2809 T^{4} )^{2} \))(\( 1 - 82 T^{2} + 3915 T^{4} - 230338 T^{6} + 7890481 T^{8} \))(\( 1 - 82 T^{2} + 3915 T^{4} - 230338 T^{6} + 7890481 T^{8} \))(\( ( 1 - 7 T - 4 T^{2} - 371 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + T - 52 T^{2} + 53 T^{3} + 2809 T^{4} )^{2} \))(\( 1 - 4 T - 135 T^{2} + 900 T^{3} + 9413 T^{4} - 70368 T^{5} - 312982 T^{6} + 1938608 T^{7} + 10598262 T^{8} + 102746224 T^{9} - 879166438 T^{10} - 10476176736 T^{11} + 74273097653 T^{12} + 376375943700 T^{13} - 2992188752415 T^{14} - 4698844559348 T^{15} + 62259690411361 T^{16} \))(\( 1 - 4 T - 135 T^{2} + 900 T^{3} + 9413 T^{4} - 70368 T^{5} - 312982 T^{6} + 1938608 T^{7} + 10598262 T^{8} + 102746224 T^{9} - 879166438 T^{10} - 10476176736 T^{11} + 74273097653 T^{12} + 376375943700 T^{13} - 2992188752415 T^{14} - 4698844559348 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 + 71 T^{2} + 1560 T^{4} + 247151 T^{6} + 12117361 T^{8} \))(\( 1 - 94 T^{2} + 5355 T^{4} - 327214 T^{6} + 12117361 T^{8} \))(\( 1 - 94 T^{2} + 5355 T^{4} - 327214 T^{6} + 12117361 T^{8} \))(\( 1 + 71 T^{2} + 1560 T^{4} + 247151 T^{6} + 12117361 T^{8} \))(\( 1 - 91 T^{2} + 4800 T^{4} - 316771 T^{6} + 12117361 T^{8} \))(\( 1 + 14 T - 13 T^{2} - 1110 T^{3} - 3463 T^{4} + 11848 T^{5} - 87914 T^{6} + 1416852 T^{7} + 32978194 T^{8} + 83594268 T^{9} - 306028634 T^{10} + 2433330392 T^{11} - 41962421143 T^{12} - 793565971890 T^{13} - 548346937333 T^{14} + 34841120787466 T^{15} + 146830437604321 T^{16} \))(\( 1 - 14 T - 13 T^{2} + 1110 T^{3} - 3463 T^{4} - 11848 T^{5} - 87914 T^{6} - 1416852 T^{7} + 32978194 T^{8} - 83594268 T^{9} - 306028634 T^{10} - 2433330392 T^{11} - 41962421143 T^{12} + 793565971890 T^{13} - 548346937333 T^{14} - 34841120787466 T^{15} + 146830437604321 T^{16} \))
$61$ (\( ( 1 + 18 T + 169 T^{2} + 1098 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 13 T + 61 T^{2} )^{2}( 1 + T + 61 T^{2} )^{2} \))(\( ( 1 - 13 T + 61 T^{2} )^{2}( 1 + T + 61 T^{2} )^{2} \))(\( ( 1 + 18 T + 169 T^{2} + 1098 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} )^{2} \))(\( 1 - 12 T + 180 T^{2} - 1584 T^{3} + 12426 T^{4} - 46860 T^{5} + 10032 T^{6} + 3576756 T^{7} - 33274477 T^{8} + 218182116 T^{9} + 37329072 T^{10} - 10636329660 T^{11} + 172048420266 T^{12} - 1337840540784 T^{13} + 9273667384980 T^{14} - 37712914032252 T^{15} + 191707312997281 T^{16} \))(\( 1 - 12 T + 180 T^{2} - 1584 T^{3} + 12426 T^{4} - 46860 T^{5} + 10032 T^{6} + 3576756 T^{7} - 33274477 T^{8} + 218182116 T^{9} + 37329072 T^{10} - 10636329660 T^{11} + 172048420266 T^{12} - 1337840540784 T^{13} + 9273667384980 T^{14} - 37712914032252 T^{15} + 191707312997281 T^{16} \))
$67$ (\( ( 1 + 67 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 + 15 T + 142 T^{2} + 1005 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 15 T + 142 T^{2} - 1005 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 67 T^{2} + 4489 T^{4} )^{2} \))(\( 1 + 125 T^{2} + 11136 T^{4} + 561125 T^{6} + 20151121 T^{8} \))(\( 1 + 42 T + 1023 T^{2} + 18270 T^{3} + 261141 T^{4} + 3133152 T^{5} + 32837970 T^{6} + 307895844 T^{7} + 2631022010 T^{8} + 20629021548 T^{9} + 147409647330 T^{10} + 942336194976 T^{11} + 5262283889061 T^{12} + 24666785704890 T^{13} + 92538924958887 T^{14} + 254549887423566 T^{15} + 406067677556641 T^{16} \))(\( 1 - 42 T + 1023 T^{2} - 18270 T^{3} + 261141 T^{4} - 3133152 T^{5} + 32837970 T^{6} - 307895844 T^{7} + 2631022010 T^{8} - 20629021548 T^{9} + 147409647330 T^{10} - 942336194976 T^{11} + 5262283889061 T^{12} - 24666785704890 T^{13} + 92538924958887 T^{14} - 254549887423566 T^{15} + 406067677556641 T^{16} \))
$71$ (\( ( 1 - 16 T + 71 T^{2} )^{2}( 1 + 16 T + 71 T^{2} )^{2} \))(\( ( 1 - 134 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 134 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 16 T + 71 T^{2} )^{2}( 1 + 16 T + 71 T^{2} )^{2} \))(\( ( 1 + 54 T^{2} + 5041 T^{4} )^{2} \))(\( 1 - 288 T^{2} + 41724 T^{4} - 4336608 T^{6} + 350671046 T^{8} - 21860840928 T^{10} + 1060276978044 T^{12} - 36892881769248 T^{14} + 645753531245761 T^{16} \))(\( 1 - 288 T^{2} + 41724 T^{4} - 4336608 T^{6} + 350671046 T^{8} - 21860840928 T^{10} + 1060276978044 T^{12} - 36892881769248 T^{14} + 645753531245761 T^{16} \))
$73$ (\( ( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2}( 1 + 7 T + 73 T^{2} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2}( 1 + 7 T + 73 T^{2} )^{2} \))(\( ( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 15 T + 148 T^{2} - 1095 T^{3} + 5329 T^{4} )^{2} \))(\( 1 + 18 T + 347 T^{2} + 4302 T^{3} + 50601 T^{4} + 463140 T^{5} + 4384558 T^{6} + 34967736 T^{7} + 313616978 T^{8} + 2552644728 T^{9} + 23365309582 T^{10} + 180169333380 T^{11} + 1436979392841 T^{12} + 8918353993086 T^{13} + 52512976522283 T^{14} + 198853173343746 T^{15} + 806460091894081 T^{16} \))(\( 1 + 18 T + 347 T^{2} + 4302 T^{3} + 50601 T^{4} + 463140 T^{5} + 4384558 T^{6} + 34967736 T^{7} + 313616978 T^{8} + 2552644728 T^{9} + 23365309582 T^{10} + 180169333380 T^{11} + 1436979392841 T^{12} + 8918353993086 T^{13} + 52512976522283 T^{14} + 198853173343746 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 + 95 T^{2} + 2784 T^{4} + 592895 T^{6} + 38950081 T^{8} \))(\( ( 1 - 3 T + 82 T^{2} - 237 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 3 T + 82 T^{2} + 237 T^{3} + 6241 T^{4} )^{2} \))(\( 1 + 95 T^{2} + 2784 T^{4} + 592895 T^{6} + 38950081 T^{8} \))(\( 1 + 77 T^{2} - 312 T^{4} + 480557 T^{6} + 38950081 T^{8} \))(\( 1 - 6 T + 132 T^{2} - 720 T^{3} + 9999 T^{4} - 52644 T^{5} - 314880 T^{6} - 1496442 T^{7} - 48671848 T^{8} - 118218918 T^{9} - 1965166080 T^{10} - 25955545116 T^{11} + 389461859919 T^{12} - 2215480607280 T^{13} + 32087544128772 T^{14} - 115223453916954 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 6 T + 132 T^{2} + 720 T^{3} + 9999 T^{4} + 52644 T^{5} - 314880 T^{6} + 1496442 T^{7} - 48671848 T^{8} + 118218918 T^{9} - 1965166080 T^{10} + 25955545116 T^{11} + 389461859919 T^{12} + 2215480607280 T^{13} + 32087544128772 T^{14} + 115223453916954 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( ( 1 + 145 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 145 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 2 T + 229 T^{2} - 802 T^{3} + 24432 T^{4} - 66566 T^{5} + 1577581 T^{6} - 1143574 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 + 2 T + 229 T^{2} + 802 T^{3} + 24432 T^{4} + 66566 T^{5} + 1577581 T^{6} + 1143574 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 + 18 T + 197 T^{2} + 1602 T^{3} + 7921 T^{4} )^{2} \))(\( 1 + 50 T^{2} - 5421 T^{4} + 396050 T^{6} + 62742241 T^{8} \))(\( 1 + 50 T^{2} - 5421 T^{4} + 396050 T^{6} + 62742241 T^{8} \))(\( ( 1 - 18 T + 197 T^{2} - 1602 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 27 T + 332 T^{2} + 2403 T^{3} + 7921 T^{4} )^{2} \))(\( 1 + 264 T^{2} + 38610 T^{4} - 50400 T^{5} + 4052064 T^{6} - 8952768 T^{7} + 353995811 T^{8} - 796796352 T^{9} + 32096398944 T^{10} - 35530437600 T^{11} + 2422477925010 T^{12} + 131203060813704 T^{14} + 3936588805702081 T^{16} \))(\( 1 + 264 T^{2} + 38610 T^{4} - 50400 T^{5} + 4052064 T^{6} - 8952768 T^{7} + 353995811 T^{8} - 796796352 T^{9} + 32096398944 T^{10} - 35530437600 T^{11} + 2422477925010 T^{12} + 131203060813704 T^{14} + 3936588805702081 T^{16} \))
$97$ (\( ( 1 - 119 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 119 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 106 T^{2} + 9409 T^{4} )^{2} \))(\( 1 - 594 T^{2} + 165777 T^{4} - 28537554 T^{6} + 3324136868 T^{8} - 268509845586 T^{10} + 14676118616337 T^{12} - 494785370927826 T^{14} + 7837433594376961 T^{16} \))(\( 1 - 594 T^{2} + 165777 T^{4} - 28537554 T^{6} + 3324136868 T^{8} - 268509845586 T^{10} + 14676118616337 T^{12} - 494785370927826 T^{14} + 7837433594376961 T^{16} \))
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