Properties

Label 225.2.e
Level 225
Weight 2
Character orbit e
Rep. character \(\chi_{225}(76,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 32
Newform subspaces 5
Sturm bound 60
Trace bound 2

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Defining parameters

Level: \( N \) = \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 225.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 5 \)
Sturm bound: \(60\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(225, [\chi])\).

Total New Old
Modular forms 72 44 28
Cusp forms 48 32 16
Eisenstein series 24 12 12

Trace form

\( 32q + 2q^{2} + 2q^{3} - 12q^{4} + 2q^{7} + 6q^{9} + O(q^{10}) \) \( 32q + 2q^{2} + 2q^{3} - 12q^{4} + 2q^{7} + 6q^{9} - 6q^{11} - 14q^{12} + 2q^{13} + 6q^{14} - 8q^{16} - 4q^{17} + 20q^{18} + 8q^{19} - 18q^{21} - 6q^{22} + 6q^{23} - 54q^{24} - 24q^{26} + 2q^{27} - 16q^{28} - 6q^{29} + 4q^{31} + 22q^{32} - 20q^{33} - 14q^{34} - 12q^{36} - 4q^{37} - 10q^{38} + 42q^{39} - 6q^{41} + 42q^{42} + 2q^{43} + 72q^{44} - 16q^{46} + 20q^{47} + 10q^{48} + 10q^{49} - 30q^{51} - 10q^{52} + 8q^{53} + 48q^{54} + 48q^{56} - 22q^{57} - 18q^{58} - 42q^{59} + 10q^{61} - 84q^{62} - 24q^{63} + 28q^{64} + 36q^{66} + 8q^{67} - 26q^{68} - 18q^{69} + 24q^{71} - 6q^{72} + 8q^{73} - 48q^{74} - 30q^{76} + 6q^{77} - 2q^{78} + 2q^{79} + 42q^{81} + 48q^{82} - 6q^{83} + 54q^{84} + 12q^{86} + 26q^{87} - 18q^{88} - 60q^{89} - 60q^{91} + 36q^{92} + 42q^{93} - 2q^{94} + 36q^{96} - 16q^{97} + 76q^{98} + 18q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(225, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
225.2.e.a \(2\) \(1.797\) \(\Q(\sqrt{-3}) \) None \(1\) \(3\) \(0\) \(-3\) \(q+(1-\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
225.2.e.b \(6\) \(1.797\) 6.0.954288.1 None \(1\) \(-1\) \(0\) \(5\) \(q+(\beta _{1}-\beta _{2}+\beta _{4}+\beta _{5})q^{2}-\beta _{4}q^{3}+\cdots\)
225.2.e.c \(8\) \(1.797\) 8.0.1223810289.2 None \(-2\) \(1\) \(0\) \(1\) \(q-\beta _{1}q^{2}+(-\beta _{4}+\beta _{7})q^{3}+(-1+\beta _{1}+\cdots)q^{4}+\cdots\)
225.2.e.d \(8\) \(1.797\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{7}q^{2}+(\zeta_{24}^{6}-\zeta_{24}^{7})q^{3}-\zeta_{24}^{2}q^{4}+\cdots\)
225.2.e.e \(8\) \(1.797\) 8.0.1223810289.2 None \(2\) \(-1\) \(0\) \(-1\) \(q+\beta _{1}q^{2}+(\beta _{4}-\beta _{7})q^{3}+(-1+\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(225, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(225, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T - T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 - T + T^{3} - 2 T^{4} - T^{5} + 11 T^{6} - 2 T^{7} - 8 T^{8} + 8 T^{9} - 32 T^{11} + 64 T^{12} \))(\( 1 + 2 T - 6 T^{3} - 9 T^{4} - 2 T^{5} + 9 T^{6} + 13 T^{7} + 7 T^{8} + 26 T^{9} + 36 T^{10} - 16 T^{11} - 144 T^{12} - 192 T^{13} + 256 T^{15} + 256 T^{16} \))(\( 1 - 4 T^{2} + 7 T^{4} - 4 T^{6} + T^{8} - 16 T^{10} + 112 T^{12} - 256 T^{14} + 256 T^{16} \))(\( 1 - 2 T + 6 T^{3} - 9 T^{4} + 2 T^{5} + 9 T^{6} - 13 T^{7} + 7 T^{8} - 26 T^{9} + 36 T^{10} + 16 T^{11} - 144 T^{12} + 192 T^{13} - 256 T^{15} + 256 T^{16} \))
$3$ (\( 1 - 3 T + 3 T^{2} \))(\( 1 + T + 4 T^{2} + 3 T^{3} + 12 T^{4} + 9 T^{5} + 27 T^{6} \))(\( 1 - T - 2 T^{2} + 3 T^{3} - 3 T^{4} + 9 T^{5} - 18 T^{6} - 27 T^{7} + 81 T^{8} \))(\( 1 - 9 T^{4} + 81 T^{8} \))(\( 1 + T - 2 T^{2} - 3 T^{3} - 3 T^{4} - 9 T^{5} - 18 T^{6} + 27 T^{7} + 81 T^{8} \))
$5$ 1
$7$ (\( 1 + 3 T + 2 T^{2} + 21 T^{3} + 49 T^{4} \))(\( 1 - 5 T + T^{2} + 14 T^{3} + 73 T^{4} - 173 T^{5} - 82 T^{6} - 1211 T^{7} + 3577 T^{8} + 4802 T^{9} + 2401 T^{10} - 84035 T^{11} + 117649 T^{12} \))(\( 1 - T - 15 T^{2} - 22 T^{3} + 142 T^{4} + 303 T^{5} - 274 T^{6} - 1448 T^{7} - 633 T^{8} - 10136 T^{9} - 13426 T^{10} + 103929 T^{11} + 340942 T^{12} - 369754 T^{13} - 1764735 T^{14} - 823543 T^{15} + 5764801 T^{16} \))(\( 1 - 16 T^{2} + 121 T^{4} - 592 T^{6} + 3280 T^{8} - 29008 T^{10} + 290521 T^{12} - 1882384 T^{14} + 5764801 T^{16} \))(\( 1 + T - 15 T^{2} + 22 T^{3} + 142 T^{4} - 303 T^{5} - 274 T^{6} + 1448 T^{7} - 633 T^{8} + 10136 T^{9} - 13426 T^{10} - 103929 T^{11} + 340942 T^{12} + 369754 T^{13} - 1764735 T^{14} + 823543 T^{15} + 5764801 T^{16} \))
$11$ (\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 - 2 T - 21 T^{2} + 14 T^{3} + 286 T^{4} + 58 T^{5} - 3673 T^{6} + 638 T^{7} + 34606 T^{8} + 18634 T^{9} - 307461 T^{10} - 322102 T^{11} + 1771561 T^{12} \))(\( 1 - T - 18 T^{2} + 129 T^{3} + 120 T^{4} - 1886 T^{5} + 6087 T^{6} + 14944 T^{7} - 88007 T^{8} + 164384 T^{9} + 736527 T^{10} - 2510266 T^{11} + 1756920 T^{12} + 20775579 T^{13} - 31888098 T^{14} - 19487171 T^{15} + 214358881 T^{16} \))(\( ( 1 + 6 T + 8 T^{2} + 36 T^{3} + 267 T^{4} + 396 T^{5} + 968 T^{6} + 7986 T^{7} + 14641 T^{8} )^{2} \))(\( 1 - T - 18 T^{2} + 129 T^{3} + 120 T^{4} - 1886 T^{5} + 6087 T^{6} + 14944 T^{7} - 88007 T^{8} + 164384 T^{9} + 736527 T^{10} - 2510266 T^{11} + 1756920 T^{12} + 20775579 T^{13} - 31888098 T^{14} - 19487171 T^{15} + 214358881 T^{16} \))
$13$ (\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - 4 T - 19 T^{2} + 60 T^{3} + 370 T^{4} - 536 T^{5} - 4235 T^{6} - 6968 T^{7} + 62530 T^{8} + 131820 T^{9} - 542659 T^{10} - 1485172 T^{11} + 4826809 T^{12} \))(\( 1 + 2 T - 18 T^{2} - 102 T^{3} - 3 T^{4} + 1191 T^{5} + 3365 T^{6} - 5033 T^{7} - 45777 T^{8} - 65429 T^{9} + 568685 T^{10} + 2616627 T^{11} - 85683 T^{12} - 37871886 T^{13} - 86882562 T^{14} + 125497034 T^{15} + 815730721 T^{16} \))(\( ( 1 - 20 T^{2} + 231 T^{4} - 3380 T^{6} + 28561 T^{8} )^{2} \))(\( 1 - 2 T - 18 T^{2} + 102 T^{3} - 3 T^{4} - 1191 T^{5} + 3365 T^{6} + 5033 T^{7} - 45777 T^{8} + 65429 T^{9} + 568685 T^{10} - 2616627 T^{11} - 85683 T^{12} + 37871886 T^{13} - 86882562 T^{14} - 125497034 T^{15} + 815730721 T^{16} \))
$17$ (\( ( 1 + 4 T + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 43 T^{2} - 56 T^{3} + 731 T^{4} - 578 T^{5} + 4913 T^{6} )^{2} \))(\( ( 1 - 11 T + 88 T^{2} - 451 T^{3} + 2111 T^{4} - 7667 T^{5} + 25432 T^{6} - 54043 T^{7} + 83521 T^{8} )^{2} \))(\( ( 1 + 40 T^{2} + 786 T^{4} + 11560 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 + 11 T + 88 T^{2} + 451 T^{3} + 2111 T^{4} + 7667 T^{5} + 25432 T^{6} + 54043 T^{7} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 4 T + 53 T^{2} - 148 T^{3} + 1007 T^{4} - 1444 T^{5} + 6859 T^{6} )^{2} \))(\( ( 1 - 2 T + 49 T^{2} - 34 T^{3} + 1115 T^{4} - 646 T^{5} + 17689 T^{6} - 13718 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 - 2 T + 36 T^{2} - 38 T^{3} + 361 T^{4} )^{4} \))(\( ( 1 - 2 T + 49 T^{2} - 34 T^{3} + 1115 T^{4} - 646 T^{5} + 17689 T^{6} - 13718 T^{7} + 130321 T^{8} )^{2} \))
$23$ (\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 27 T^{2} - 66 T^{3} + 405 T^{4} + 2625 T^{5} - 9794 T^{6} + 60375 T^{7} + 214245 T^{8} - 803022 T^{9} - 7555707 T^{10} - 19309029 T^{11} + 148035889 T^{12} \))(\( 1 + 15 T + 76 T^{2} + 183 T^{3} + 1258 T^{4} + 9882 T^{5} + 28951 T^{6} + 29994 T^{7} + 75241 T^{8} + 689862 T^{9} + 15315079 T^{10} + 120234294 T^{11} + 352039978 T^{12} + 1177850769 T^{13} + 11250727564 T^{14} + 51072381705 T^{15} + 78310985281 T^{16} \))(\( 1 - 88 T^{2} + 4753 T^{4} - 170104 T^{6} + 4569664 T^{8} - 89985016 T^{10} + 1330084273 T^{12} - 13027158232 T^{14} + 78310985281 T^{16} \))(\( 1 - 15 T + 76 T^{2} - 183 T^{3} + 1258 T^{4} - 9882 T^{5} + 28951 T^{6} - 29994 T^{7} + 75241 T^{8} - 689862 T^{9} + 15315079 T^{10} - 120234294 T^{11} + 352039978 T^{12} - 1177850769 T^{13} + 11250727564 T^{14} - 51072381705 T^{15} + 78310985281 T^{16} \))
$29$ (\( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \))(\( 1 - 7 T - 9 T^{2} + 304 T^{3} - 803 T^{4} - 2929 T^{5} + 36038 T^{6} - 84941 T^{7} - 675323 T^{8} + 7414256 T^{9} - 6365529 T^{10} - 143578043 T^{11} + 594823321 T^{12} \))(\( 1 + T - 75 T^{2} + 186 T^{3} + 3234 T^{4} - 10969 T^{5} - 64920 T^{6} + 187910 T^{7} + 1140565 T^{8} + 5449390 T^{9} - 54597720 T^{10} - 267522941 T^{11} + 2287346754 T^{12} + 3815073714 T^{13} - 44611749075 T^{14} + 17249876309 T^{15} + 500246412961 T^{16} \))(\( ( 1 + 6 T - 19 T^{2} - 18 T^{3} + 1140 T^{4} - 522 T^{5} - 15979 T^{6} + 146334 T^{7} + 707281 T^{8} )^{2} \))(\( 1 + T - 75 T^{2} + 186 T^{3} + 3234 T^{4} - 10969 T^{5} - 64920 T^{6} + 187910 T^{7} + 1140565 T^{8} + 5449390 T^{9} - 54597720 T^{10} - 267522941 T^{11} + 2287346754 T^{12} + 3815073714 T^{13} - 44611749075 T^{14} + 17249876309 T^{15} + 500246412961 T^{16} \))
$31$ (\( 1 - 31 T^{2} + 961 T^{4} \))(\( 1 + 8 T + 31 T^{2} + 208 T^{3} - 158 T^{4} - 7756 T^{5} - 34897 T^{6} - 240436 T^{7} - 151838 T^{8} + 6196528 T^{9} + 28629151 T^{10} + 229033208 T^{11} + 887503681 T^{12} \))(\( 1 - 4 T - 66 T^{2} + 362 T^{3} + 1939 T^{4} - 11745 T^{5} - 45361 T^{6} + 144883 T^{7} + 1472277 T^{8} + 4491373 T^{9} - 43591921 T^{10} - 349895295 T^{11} + 1790707219 T^{12} + 10363752662 T^{13} - 58575242946 T^{14} - 110050456444 T^{15} + 852891037441 T^{16} \))(\( ( 1 - 2 T - 56 T^{2} + 4 T^{3} + 2515 T^{4} + 124 T^{5} - 53816 T^{6} - 59582 T^{7} + 923521 T^{8} )^{2} \))(\( 1 - 4 T - 66 T^{2} + 362 T^{3} + 1939 T^{4} - 11745 T^{5} - 45361 T^{6} + 144883 T^{7} + 1472277 T^{8} + 4491373 T^{9} - 43591921 T^{10} - 349895295 T^{11} + 1790707219 T^{12} + 10363752662 T^{13} - 58575242946 T^{14} - 110050456444 T^{15} + 852891037441 T^{16} \))
$37$ (\( ( 1 - 4 T + 37 T^{2} )^{2} \))(\( ( 1 + 6 T + 99 T^{2} + 440 T^{3} + 3663 T^{4} + 8214 T^{5} + 50653 T^{6} )^{2} \))(\( ( 1 + T + 49 T^{2} - 392 T^{3} + 241 T^{4} - 14504 T^{5} + 67081 T^{6} + 50653 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 + 56 T^{2} + 1369 T^{4} )^{4} \))(\( ( 1 - T + 49 T^{2} + 392 T^{3} + 241 T^{4} + 14504 T^{5} + 67081 T^{6} - 50653 T^{7} + 1874161 T^{8} )^{2} \))
$41$ (\( 1 + 5 T - 16 T^{2} + 205 T^{3} + 1681 T^{4} \))(\( 1 - 13 T + 27 T^{2} + 292 T^{3} + 445 T^{4} - 22279 T^{5} + 169790 T^{6} - 913439 T^{7} + 748045 T^{8} + 20124932 T^{9} + 76295547 T^{10} - 1506130613 T^{11} + 4750104241 T^{12} \))(\( 1 - 5 T - 114 T^{2} + 213 T^{3} + 9222 T^{4} - 3430 T^{5} - 507867 T^{6} + 107426 T^{7} + 20984173 T^{8} + 4404466 T^{9} - 853724427 T^{10} - 236399030 T^{11} + 26059167942 T^{12} + 24677370813 T^{13} - 541511883474 T^{14} - 973771369405 T^{15} + 7984925229121 T^{16} \))(\( ( 1 + 12 T + 29 T^{2} + 396 T^{3} + 5640 T^{4} + 16236 T^{5} + 48749 T^{6} + 827052 T^{7} + 2825761 T^{8} )^{2} \))(\( 1 - 5 T - 114 T^{2} + 213 T^{3} + 9222 T^{4} - 3430 T^{5} - 507867 T^{6} + 107426 T^{7} + 20984173 T^{8} + 4404466 T^{9} - 853724427 T^{10} - 236399030 T^{11} + 26059167942 T^{12} + 24677370813 T^{13} - 541511883474 T^{14} - 973771369405 T^{15} + 7984925229121 T^{16} \))
$43$ (\( ( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - 10 T - 25 T^{2} + 462 T^{3} + 1690 T^{4} - 17990 T^{5} + 34975 T^{6} - 773570 T^{7} + 3124810 T^{8} + 36732234 T^{9} - 85470025 T^{10} - 1470084430 T^{11} + 6321363049 T^{12} \))(\( 1 - 10 T - 24 T^{2} + 168 T^{3} + 2043 T^{4} + 8292 T^{5} - 128884 T^{6} - 10646 T^{7} + 2342736 T^{8} - 457778 T^{9} - 238306516 T^{10} + 659272044 T^{11} + 6984610443 T^{12} + 24697418424 T^{13} - 151712713176 T^{14} - 2718186111070 T^{15} + 11688200277601 T^{16} \))(\( 1 - 88 T^{2} + 3838 T^{4} - 18304 T^{6} - 2480621 T^{8} - 33844096 T^{10} + 13121358238 T^{12} - 556279948312 T^{14} + 11688200277601 T^{16} \))(\( 1 + 10 T - 24 T^{2} - 168 T^{3} + 2043 T^{4} - 8292 T^{5} - 128884 T^{6} + 10646 T^{7} + 2342736 T^{8} + 457778 T^{9} - 238306516 T^{10} - 659272044 T^{11} + 6984610443 T^{12} - 24697418424 T^{13} - 151712713176 T^{14} + 2718186111070 T^{15} + 11688200277601 T^{16} \))
$47$ (\( 1 - 7 T + 2 T^{2} - 329 T^{3} + 2209 T^{4} \))(\( 1 - 13 T + 39 T^{2} + 16 T^{3} - 299 T^{4} + 19253 T^{5} - 232366 T^{6} + 904891 T^{7} - 660491 T^{8} + 1661168 T^{9} + 190307559 T^{10} - 2981485091 T^{11} + 10779215329 T^{12} \))(\( 1 + 20 T + 105 T^{2} + 120 T^{3} + 6294 T^{4} + 71140 T^{5} + 174165 T^{6} + 1240450 T^{7} + 19222555 T^{8} + 58301150 T^{9} + 384730485 T^{10} + 7385968220 T^{11} + 30712712214 T^{12} + 27521400840 T^{13} + 1131817609545 T^{14} + 10132462409260 T^{15} + 23811286661761 T^{16} \))(\( 1 - 160 T^{2} + 14809 T^{4} - 1019680 T^{6} + 55015600 T^{8} - 2252473120 T^{10} + 72263195929 T^{12} - 1724674452640 T^{14} + 23811286661761 T^{16} \))(\( 1 - 20 T + 105 T^{2} - 120 T^{3} + 6294 T^{4} - 71140 T^{5} + 174165 T^{6} - 1240450 T^{7} + 19222555 T^{8} - 58301150 T^{9} + 384730485 T^{10} - 7385968220 T^{11} + 30712712214 T^{12} - 27521400840 T^{13} + 1131817609545 T^{14} - 10132462409260 T^{15} + 23811286661761 T^{16} \))
$53$ (\( ( 1 - 2 T + 53 T^{2} )^{2} \))(\( ( 1 - 2 T + 139 T^{2} - 188 T^{3} + 7367 T^{4} - 5618 T^{5} + 148877 T^{6} )^{2} \))(\( ( 1 - 20 T + 298 T^{2} - 3001 T^{3} + 25499 T^{4} - 159053 T^{5} + 837082 T^{6} - 2977540 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 + 196 T^{2} + 15174 T^{4} + 550564 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 + 20 T + 298 T^{2} + 3001 T^{3} + 25499 T^{4} + 159053 T^{5} + 837082 T^{6} + 2977540 T^{7} + 7890481 T^{8} )^{2} \))
$59$ (\( 1 - 14 T + 137 T^{2} - 826 T^{3} + 3481 T^{4} \))(\( 1 - 2 T - 153 T^{2} + 110 T^{3} + 14962 T^{4} - 3194 T^{5} - 1012513 T^{6} - 188446 T^{7} + 52082722 T^{8} + 22591690 T^{9} - 1853956233 T^{10} - 1429848598 T^{11} + 42180533641 T^{12} \))(\( 1 + 17 T + 51 T^{2} - 144 T^{3} + 2220 T^{4} - 17291 T^{5} - 531846 T^{6} - 1855556 T^{7} + 4239943 T^{8} - 109477804 T^{9} - 1851355926 T^{10} - 3551208289 T^{11} + 26900541420 T^{12} - 102949099056 T^{13} + 2151207215691 T^{14} + 42307075241923 T^{15} + 146830437604321 T^{16} \))(\( ( 1 + 12 T + 2 T^{2} + 288 T^{3} + 8187 T^{4} + 16992 T^{5} + 6962 T^{6} + 2464548 T^{7} + 12117361 T^{8} )^{2} \))(\( 1 + 17 T + 51 T^{2} - 144 T^{3} + 2220 T^{4} - 17291 T^{5} - 531846 T^{6} - 1855556 T^{7} + 4239943 T^{8} - 109477804 T^{9} - 1851355926 T^{10} - 3551208289 T^{11} + 26900541420 T^{12} - 102949099056 T^{13} + 2151207215691 T^{14} + 42307075241923 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \))(\( 1 + T - 145 T^{2} - 240 T^{3} + 12217 T^{4} + 14087 T^{5} - 812786 T^{6} + 859307 T^{7} + 45459457 T^{8} - 54475440 T^{9} - 2007646945 T^{10} + 844596301 T^{11} + 51520374361 T^{12} \))(\( 1 - 13 T - 72 T^{2} + 1443 T^{3} + 6378 T^{4} - 120042 T^{5} - 78037 T^{6} + 2420938 T^{7} + 9620433 T^{8} + 147677218 T^{9} - 290375677 T^{10} - 27247253202 T^{11} + 88308773898 T^{12} + 1218752462343 T^{13} - 3709466953992 T^{14} - 40855656868273 T^{15} + 191707312997281 T^{16} \))(\( ( 1 + 4 T - 35 T^{2} - 284 T^{3} - 2096 T^{4} - 17324 T^{5} - 130235 T^{6} + 907924 T^{7} + 13845841 T^{8} )^{2} \))(\( 1 - 13 T - 72 T^{2} + 1443 T^{3} + 6378 T^{4} - 120042 T^{5} - 78037 T^{6} + 2420938 T^{7} + 9620433 T^{8} + 147677218 T^{9} - 290375677 T^{10} - 27247253202 T^{11} + 88308773898 T^{12} + 1218752462343 T^{13} - 3709466953992 T^{14} - 40855656868273 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 + 3 T - 58 T^{2} + 201 T^{3} + 4489 T^{4} \))(\( 1 - 11 T - 41 T^{2} + 152 T^{3} + 4753 T^{4} + 32755 T^{5} - 764902 T^{6} + 2194585 T^{7} + 21336217 T^{8} + 45715976 T^{9} - 826195961 T^{10} - 14851376177 T^{11} + 90458382169 T^{12} \))(\( 1 + 17 T - 15 T^{2} - 1066 T^{3} + 7426 T^{4} + 68373 T^{5} - 833566 T^{6} + 1085128 T^{7} + 111947307 T^{8} + 72703576 T^{9} - 3741877774 T^{10} + 20564068599 T^{11} + 149642224546 T^{12} - 1439233364062 T^{13} - 1356875732535 T^{14} + 103032097290491 T^{15} + 406067677556641 T^{16} \))(\( 1 - 184 T^{2} + 16657 T^{4} - 1512664 T^{6} + 123674896 T^{8} - 6790348696 T^{10} + 335657222497 T^{12} - 16644342319096 T^{14} + 406067677556641 T^{16} \))(\( 1 - 17 T - 15 T^{2} + 1066 T^{3} + 7426 T^{4} - 68373 T^{5} - 833566 T^{6} - 1085128 T^{7} + 111947307 T^{8} - 72703576 T^{9} - 3741877774 T^{10} - 20564068599 T^{11} + 149642224546 T^{12} + 1439233364062 T^{13} - 1356875732535 T^{14} - 103032097290491 T^{15} + 406067677556641 T^{16} \))
$71$ (\( ( 1 - 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 10 T + 121 T^{2} + 712 T^{3} + 8591 T^{4} + 50410 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 + 8 T + 244 T^{2} + 1441 T^{3} + 24947 T^{4} + 102311 T^{5} + 1230004 T^{6} + 2863288 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 - 18 T + 196 T^{2} - 1278 T^{3} + 5041 T^{4} )^{4} \))(\( ( 1 + 8 T + 244 T^{2} + 1441 T^{3} + 24947 T^{4} + 102311 T^{5} + 1230004 T^{6} + 2863288 T^{7} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 + 4 T + 73 T^{2} )^{2} \))(\( ( 1 - 8 T + 155 T^{2} - 1040 T^{3} + 11315 T^{4} - 42632 T^{5} + 389017 T^{6} )^{2} \))(\( ( 1 - 2 T + 196 T^{2} - 679 T^{3} + 18071 T^{4} - 49567 T^{5} + 1044484 T^{6} - 778034 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 + 74 T^{2} + 5329 T^{4} )^{4} \))(\( ( 1 + 2 T + 196 T^{2} + 679 T^{3} + 18071 T^{4} + 49567 T^{5} + 1044484 T^{6} + 778034 T^{7} + 28398241 T^{8} )^{2} \))
$79$ (\( 1 - 6 T - 43 T^{2} - 474 T^{3} + 6241 T^{4} \))(\( 1 + 2 T - 149 T^{2} - 374 T^{3} + 10642 T^{4} + 16154 T^{5} - 772597 T^{6} + 1276166 T^{7} + 66416722 T^{8} - 184396586 T^{9} - 5803562069 T^{10} + 6154112798 T^{11} + 243087455521 T^{12} \))(\( 1 - 7 T - 234 T^{2} + 1199 T^{3} + 36520 T^{4} - 121434 T^{5} - 3999001 T^{6} + 3702556 T^{7} + 362101113 T^{8} + 292501924 T^{9} - 24957765241 T^{10} - 59871697926 T^{11} + 1422456958120 T^{12} + 3689390622401 T^{13} - 56882464591914 T^{14} - 134427362903113 T^{15} + 1517108809906561 T^{16} \))(\( ( 1 + 8 T - 98 T^{2} + 32 T^{3} + 14947 T^{4} + 2528 T^{5} - 611618 T^{6} + 3944312 T^{7} + 38950081 T^{8} )^{2} \))(\( 1 - 7 T - 234 T^{2} + 1199 T^{3} + 36520 T^{4} - 121434 T^{5} - 3999001 T^{6} + 3702556 T^{7} + 362101113 T^{8} + 292501924 T^{9} - 24957765241 T^{10} - 59871697926 T^{11} + 1422456958120 T^{12} + 3689390622401 T^{13} - 56882464591914 T^{14} - 134427362903113 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \))(\( 1 + 15 T - 51 T^{2} - 678 T^{3} + 17133 T^{4} + 80979 T^{5} - 925778 T^{6} + 6721257 T^{7} + 118029237 T^{8} - 387671586 T^{9} - 2420374371 T^{10} + 59085609645 T^{11} + 326940373369 T^{12} \))(\( 1 + 30 T + 280 T^{2} + 1770 T^{3} + 42079 T^{4} + 523485 T^{5} + 2681215 T^{6} + 35481675 T^{7} + 533008645 T^{8} + 2944979025 T^{9} + 18470890135 T^{10} + 299321917695 T^{11} + 1996998689359 T^{12} + 6972101938110 T^{13} + 91543304543320 T^{14} + 814081529688810 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 160 T^{2} + 5929 T^{4} - 942880 T^{6} + 163131520 T^{8} - 6495500320 T^{10} + 281380385209 T^{12} - 52310459739040 T^{14} + 2252292232139041 T^{16} \))(\( 1 - 30 T + 280 T^{2} - 1770 T^{3} + 42079 T^{4} - 523485 T^{5} + 2681215 T^{6} - 35481675 T^{7} + 533008645 T^{8} - 2944979025 T^{9} + 18470890135 T^{10} - 299321917695 T^{11} + 1996998689359 T^{12} - 6972101938110 T^{13} + 91543304543320 T^{14} - 814081529688810 T^{15} + 2252292232139041 T^{16} \))
$89$ (\( ( 1 + 15 T + 89 T^{2} )^{2} \))(\( ( 1 + 3 T + 89 T^{2} )^{6} \))(\( ( 1 + 9 T + 257 T^{2} + 1998 T^{3} + 31929 T^{4} + 177822 T^{5} + 2035697 T^{6} + 6344721 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 6 T + 79 T^{2} - 534 T^{3} + 7921 T^{4} )^{4} \))(\( ( 1 + 9 T + 257 T^{2} + 1998 T^{3} + 31929 T^{4} + 177822 T^{5} + 2035697 T^{6} + 6344721 T^{7} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} \))(\( 1 + 18 T + 69 T^{2} + 214 T^{3} + 906 T^{4} - 163158 T^{5} - 2743251 T^{6} - 15826326 T^{7} + 8524554 T^{8} + 195312022 T^{9} + 6108520389 T^{10} + 154572124626 T^{11} + 832972004929 T^{12} \))(\( 1 - 19 T - 108 T^{2} + 1825 T^{3} + 46754 T^{4} - 332118 T^{5} - 5651993 T^{6} + 4946342 T^{7} + 777287673 T^{8} + 479795174 T^{9} - 53179602137 T^{10} - 303115131414 T^{11} + 4139098003874 T^{12} + 15671895969025 T^{13} - 89960976532332 T^{14} - 1535167405084147 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 52 T^{2} - 12902 T^{4} + 167024 T^{6} + 129576019 T^{8} + 1571528816 T^{10} - 1142204783462 T^{12} - 43314544256308 T^{14} + 7837433594376961 T^{16} \))(\( 1 + 19 T - 108 T^{2} - 1825 T^{3} + 46754 T^{4} + 332118 T^{5} - 5651993 T^{6} - 4946342 T^{7} + 777287673 T^{8} - 479795174 T^{9} - 53179602137 T^{10} + 303115131414 T^{11} + 4139098003874 T^{12} - 15671895969025 T^{13} - 89960976532332 T^{14} + 1535167405084147 T^{15} + 7837433594376961 T^{16} \))
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