# Properties

 Label 210.2.r Level 210 Weight 2 Character orbit r Rep. character $$\chi_{210}(101,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 24 Newform subspaces 2 Sturm bound 96 Trace bound 3

# Related objects

## Defining parameters

 Level: $$N$$ = $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 210.r (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$21$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$2$$ Sturm bound: $$96$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(210, [\chi])$$.

Total New Old
Modular forms 112 24 88
Cusp forms 80 24 56
Eisenstein series 32 0 32

## Trace form

 $$24q + 12q^{4} + 16q^{7} + 6q^{9} + O(q^{10})$$ $$24q + 12q^{4} + 16q^{7} + 6q^{9} - 8q^{15} - 12q^{16} - 8q^{18} - 14q^{21} - 6q^{24} - 12q^{25} + 8q^{28} - 2q^{30} + 24q^{31} - 24q^{33} + 12q^{36} - 16q^{37} - 12q^{39} + 4q^{42} + 6q^{45} + 4q^{46} - 28q^{49} - 4q^{51} - 24q^{52} - 36q^{54} - 72q^{57} + 16q^{58} - 4q^{60} - 60q^{61} + 16q^{63} - 24q^{64} - 48q^{66} - 8q^{67} + 12q^{70} + 8q^{72} + 64q^{78} - 8q^{79} + 10q^{81} + 8q^{84} + 48q^{85} + 72q^{87} + 56q^{91} - 28q^{93} + 48q^{94} - 6q^{96} + 96q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(210, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
210.2.r.a $$12$$ $$1.677$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$-2$$ $$6$$ $$8$$ $$q+\beta _{4}q^{2}+(-\beta _{3}+\beta _{9})q^{3}-\beta _{6}q^{4}+\cdots$$
210.2.r.b $$12$$ $$1.677$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$2$$ $$-6$$ $$8$$ $$q-\beta _{4}q^{2}-\beta _{7}q^{3}-\beta _{6}q^{4}+(-1-\beta _{6}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(210, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(210, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - T^{2} + T^{4} )^{3}$$)($$( 1 - T^{2} + T^{4} )^{3}$$)
$3$ ($$1 + 2 T - T^{2} - 2 T^{3} - 8 T^{4} - 6 T^{5} + 33 T^{6} - 18 T^{7} - 72 T^{8} - 54 T^{9} - 81 T^{10} + 486 T^{11} + 729 T^{12}$$)($$1 - 2 T + 2 T^{2} - 4 T^{3} + 16 T^{4} - 18 T^{5} + 6 T^{6} - 54 T^{7} + 144 T^{8} - 108 T^{9} + 162 T^{10} - 486 T^{11} + 729 T^{12}$$)
$5$ ($$( 1 - T + T^{2} )^{6}$$)($$( 1 + T + T^{2} )^{6}$$)
$7$ ($$1 - 8 T + 39 T^{2} - 152 T^{3} + 483 T^{4} - 1360 T^{5} + 3642 T^{6} - 9520 T^{7} + 23667 T^{8} - 52136 T^{9} + 93639 T^{10} - 134456 T^{11} + 117649 T^{12}$$)($$1 - 8 T + 39 T^{2} - 152 T^{3} + 483 T^{4} - 1360 T^{5} + 3642 T^{6} - 9520 T^{7} + 23667 T^{8} - 52136 T^{9} + 93639 T^{10} - 134456 T^{11} + 117649 T^{12}$$)
$11$ ($$1 + 12 T + 104 T^{2} + 672 T^{3} + 3656 T^{4} + 16524 T^{5} + 63276 T^{6} + 198252 T^{7} + 468664 T^{8} + 516768 T^{9} - 2129576 T^{10} - 16995156 T^{11} - 68897930 T^{12} - 186946716 T^{13} - 257678696 T^{14} + 687818208 T^{15} + 6861709624 T^{16} + 31928682852 T^{17} + 112097293836 T^{18} + 322006013604 T^{19} + 783696068936 T^{20} + 1584540848352 T^{21} + 2697492158504 T^{22} + 3423740047332 T^{23} + 3138428376721 T^{24}$$)($$1 - 12 T + 104 T^{2} - 672 T^{3} + 3656 T^{4} - 16524 T^{5} + 63276 T^{6} - 198252 T^{7} + 468664 T^{8} - 516768 T^{9} - 2129576 T^{10} + 16995156 T^{11} - 68897930 T^{12} + 186946716 T^{13} - 257678696 T^{14} - 687818208 T^{15} + 6861709624 T^{16} - 31928682852 T^{17} + 112097293836 T^{18} - 322006013604 T^{19} + 783696068936 T^{20} - 1584540848352 T^{21} + 2697492158504 T^{22} - 3423740047332 T^{23} + 3138428376721 T^{24}$$)
$13$ ($$1 - 24 T^{2} + 632 T^{4} - 13172 T^{6} + 225768 T^{8} - 3379208 T^{10} + 49928150 T^{12} - 571086152 T^{14} + 6448159848 T^{16} - 63578728148 T^{18} + 515541815672 T^{20} - 3308603804376 T^{22} + 23298085122481 T^{24}$$)($$1 - 24 T^{2} + 632 T^{4} - 13172 T^{6} + 225768 T^{8} - 3379208 T^{10} + 49928150 T^{12} - 571086152 T^{14} + 6448159848 T^{16} - 63578728148 T^{18} + 515541815672 T^{20} - 3308603804376 T^{22} + 23298085122481 T^{24}$$)
$17$ ($$1 - 12 T + 10 T^{2} + 240 T^{3} + 929 T^{4} - 10248 T^{5} - 21642 T^{6} + 175188 T^{7} + 650254 T^{8} - 1893108 T^{9} - 17461318 T^{10} + 24663192 T^{11} + 233538997 T^{12} + 419274264 T^{13} - 5046320902 T^{14} - 9300839604 T^{15} + 54309864334 T^{16} + 248741908116 T^{17} - 522385268298 T^{18} - 4205150720904 T^{19} + 6480478662689 T^{20} + 28461090359280 T^{21} + 20159939004490 T^{22} - 411262755691596 T^{23} + 582622237229761 T^{24}$$)($$1 + 12 T + 10 T^{2} - 240 T^{3} + 929 T^{4} + 10248 T^{5} - 21642 T^{6} - 175188 T^{7} + 650254 T^{8} + 1893108 T^{9} - 17461318 T^{10} - 24663192 T^{11} + 233538997 T^{12} - 419274264 T^{13} - 5046320902 T^{14} + 9300839604 T^{15} + 54309864334 T^{16} - 248741908116 T^{17} - 522385268298 T^{18} + 4205150720904 T^{19} + 6480478662689 T^{20} - 28461090359280 T^{21} + 20159939004490 T^{22} + 411262755691596 T^{23} + 582622237229761 T^{24}$$)
$19$ ($$1 + 24 T^{2} - 316 T^{4} + 144 T^{5} - 5636 T^{6} - 33888 T^{7} + 181380 T^{8} - 941760 T^{9} + 1187824 T^{10} + 9129072 T^{11} - 54892570 T^{12} + 173452368 T^{13} + 428804464 T^{14} - 6459531840 T^{15} + 23637622980 T^{16} - 83910042912 T^{17} - 265150585316 T^{18} + 128717530416 T^{19} - 5366805920956 T^{20} + 147145590187224 T^{22} + 2213314919066161 T^{24}$$)($$1 + 24 T^{2} - 316 T^{4} + 144 T^{5} - 5636 T^{6} - 33888 T^{7} + 181380 T^{8} - 941760 T^{9} + 1187824 T^{10} + 9129072 T^{11} - 54892570 T^{12} + 173452368 T^{13} + 428804464 T^{14} - 6459531840 T^{15} + 23637622980 T^{16} - 83910042912 T^{17} - 265150585316 T^{18} + 128717530416 T^{19} - 5366805920956 T^{20} + 147145590187224 T^{22} + 2213314919066161 T^{24}$$)
$23$ ($$1 + 24 T + 367 T^{2} + 4200 T^{3} + 39535 T^{4} + 321816 T^{5} + 2347952 T^{6} + 15699792 T^{7} + 97859137 T^{8} + 573159912 T^{9} + 3169289665 T^{10} + 16552915872 T^{11} + 81697151198 T^{12} + 380717065056 T^{13} + 1676554232785 T^{14} + 6973636649304 T^{15} + 27384998757217 T^{16} + 101049246340656 T^{17} + 347581161649328 T^{18} + 1095727306051752 T^{19} + 3096024803084335 T^{20} + 7564841178144600 T^{21} + 15203529615409183 T^{22} + 22867434189934248 T^{23} + 21914624432020321 T^{24}$$)($$1 - 24 T + 367 T^{2} - 4200 T^{3} + 39535 T^{4} - 321816 T^{5} + 2347952 T^{6} - 15699792 T^{7} + 97859137 T^{8} - 573159912 T^{9} + 3169289665 T^{10} - 16552915872 T^{11} + 81697151198 T^{12} - 380717065056 T^{13} + 1676554232785 T^{14} - 6973636649304 T^{15} + 27384998757217 T^{16} - 101049246340656 T^{17} + 347581161649328 T^{18} - 1095727306051752 T^{19} + 3096024803084335 T^{20} - 7564841178144600 T^{21} + 15203529615409183 T^{22} - 22867434189934248 T^{23} + 21914624432020321 T^{24}$$)
$29$ ($$1 - 158 T^{2} + 12415 T^{4} - 625726 T^{6} + 22774147 T^{8} - 673287380 T^{10} + 19008498890 T^{12} - 566234686580 T^{14} + 16107721464307 T^{16} - 372196417356046 T^{18} + 6210559216910815 T^{20} - 66471742861431758 T^{22} + 353814783205469041 T^{24}$$)($$1 - 158 T^{2} + 12415 T^{4} - 625726 T^{6} + 22774147 T^{8} - 673287380 T^{10} + 19008498890 T^{12} - 566234686580 T^{14} + 16107721464307 T^{16} - 372196417356046 T^{18} + 6210559216910815 T^{20} - 66471742861431758 T^{22} + 353814783205469041 T^{24}$$)
$31$ ($$1 - 12 T + 162 T^{2} - 1368 T^{3} + 10769 T^{4} - 77664 T^{5} + 554878 T^{6} - 3968772 T^{7} + 27631230 T^{8} - 178188132 T^{9} + 1059815698 T^{10} - 6109630032 T^{11} + 33269854325 T^{12} - 189398530992 T^{13} + 1018482885778 T^{14} - 5308402640412 T^{15} + 25518021160830 T^{16} - 113622572872572 T^{17} + 492456267505918 T^{18} - 2136739662316704 T^{19} + 9184783582202129 T^{20} - 36169403115797928 T^{21} + 132779782490889762 T^{22} - 304901722756857972 T^{23} + 787662783788549761 T^{24}$$)($$1 - 12 T + 162 T^{2} - 1368 T^{3} + 10769 T^{4} - 77664 T^{5} + 554878 T^{6} - 3968772 T^{7} + 27631230 T^{8} - 178188132 T^{9} + 1059815698 T^{10} - 6109630032 T^{11} + 33269854325 T^{12} - 189398530992 T^{13} + 1018482885778 T^{14} - 5308402640412 T^{15} + 25518021160830 T^{16} - 113622572872572 T^{17} + 492456267505918 T^{18} - 2136739662316704 T^{19} + 9184783582202129 T^{20} - 36169403115797928 T^{21} + 132779782490889762 T^{22} - 304901722756857972 T^{23} + 787662783788549761 T^{24}$$)
$37$ ($$1 + 8 T - 104 T^{2} - 832 T^{3} + 6788 T^{4} + 46312 T^{5} - 326044 T^{6} - 1583912 T^{7} + 13285604 T^{8} + 33469952 T^{9} - 514631840 T^{10} - 373683016 T^{11} + 19047573830 T^{12} - 13826271592 T^{13} - 704530988960 T^{14} + 1695353478656 T^{15} + 24899360878244 T^{16} - 109834725619784 T^{17} - 836539701295996 T^{18} + 4396485093783496 T^{19} + 23842710533215748 T^{20} - 108128167509504064 T^{21} - 500092774731456296 T^{22} + 1423340974235683304 T^{23} + 6582952005840035281 T^{24}$$)($$1 + 8 T - 104 T^{2} - 832 T^{3} + 6788 T^{4} + 46312 T^{5} - 326044 T^{6} - 1583912 T^{7} + 13285604 T^{8} + 33469952 T^{9} - 514631840 T^{10} - 373683016 T^{11} + 19047573830 T^{12} - 13826271592 T^{13} - 704530988960 T^{14} + 1695353478656 T^{15} + 24899360878244 T^{16} - 109834725619784 T^{17} - 836539701295996 T^{18} + 4396485093783496 T^{19} + 23842710533215748 T^{20} - 108128167509504064 T^{21} - 500092774731456296 T^{22} + 1423340974235683304 T^{23} + 6582952005840035281 T^{24}$$)
$41$ ($$( 1 + 2 T + 101 T^{2} + 150 T^{3} + 6062 T^{4} + 7442 T^{5} + 295913 T^{6} + 305122 T^{7} + 10190222 T^{8} + 10338150 T^{9} + 285401861 T^{10} + 231712402 T^{11} + 4750104241 T^{12} )^{2}$$)($$( 1 - 2 T + 101 T^{2} - 150 T^{3} + 6062 T^{4} - 7442 T^{5} + 295913 T^{6} - 305122 T^{7} + 10190222 T^{8} - 10338150 T^{9} + 285401861 T^{10} - 231712402 T^{11} + 4750104241 T^{12} )^{2}$$)
$43$ ($$( 1 + 131 T^{2} - 196 T^{3} + 7479 T^{4} - 24892 T^{5} + 314798 T^{6} - 1070356 T^{7} + 13828671 T^{8} - 15583372 T^{9} + 447862931 T^{10} + 6321363049 T^{12} )^{2}$$)($$( 1 + 131 T^{2} - 196 T^{3} + 7479 T^{4} - 24892 T^{5} + 314798 T^{6} - 1070356 T^{7} + 13828671 T^{8} - 15583372 T^{9} + 447862931 T^{10} + 6321363049 T^{12} )^{2}$$)
$47$ ($$1 - 16 T + 32 T^{2} + 544 T^{3} + 428 T^{4} - 39000 T^{5} + 312596 T^{6} - 2377608 T^{7} + 2268284 T^{8} + 67449856 T^{9} + 407948184 T^{10} - 5931851872 T^{11} + 27380961750 T^{12} - 278797037984 T^{13} + 901157538456 T^{14} + 7002846399488 T^{15} + 11068502337404 T^{16} - 545292523403256 T^{17} + 3369539594984084 T^{18} - 19758301698057000 T^{19} + 10191230691233708 T^{20} + 608806977367905248 T^{21} + 1683172231546561568 T^{22} - 39554547441344196848 T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$)($$1 + 16 T + 32 T^{2} - 544 T^{3} + 428 T^{4} + 39000 T^{5} + 312596 T^{6} + 2377608 T^{7} + 2268284 T^{8} - 67449856 T^{9} + 407948184 T^{10} + 5931851872 T^{11} + 27380961750 T^{12} + 278797037984 T^{13} + 901157538456 T^{14} - 7002846399488 T^{15} + 11068502337404 T^{16} + 545292523403256 T^{17} + 3369539594984084 T^{18} + 19758301698057000 T^{19} + 10191230691233708 T^{20} - 608806977367905248 T^{21} + 1683172231546561568 T^{22} + 39554547441344196848 T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$)
$53$ ($$1 + 48 T + 1272 T^{2} + 24192 T^{3} + 372104 T^{4} + 4941024 T^{5} + 58639988 T^{6} + 632702016 T^{7} + 6267491832 T^{8} + 57492176256 T^{9} + 491936676296 T^{10} + 3943130126256 T^{11} + 29642342340470 T^{12} + 208985896691568 T^{13} + 1381850123715464 T^{14} + 8559262724464512 T^{15} + 49453525218051192 T^{16} + 264593131503213888 T^{17} + 1299717870632226452 T^{18} + 5804275935001973088 T^{19} + 23167079840829073544 T^{20} + 79827880812877201536 T^{21} +$$$$22\!\cdots\!28$$$$T^{22} +$$$$44\!\cdots\!56$$$$T^{23} +$$$$49\!\cdots\!41$$$$T^{24}$$)($$1 - 48 T + 1272 T^{2} - 24192 T^{3} + 372104 T^{4} - 4941024 T^{5} + 58639988 T^{6} - 632702016 T^{7} + 6267491832 T^{8} - 57492176256 T^{9} + 491936676296 T^{10} - 3943130126256 T^{11} + 29642342340470 T^{12} - 208985896691568 T^{13} + 1381850123715464 T^{14} - 8559262724464512 T^{15} + 49453525218051192 T^{16} - 264593131503213888 T^{17} + 1299717870632226452 T^{18} - 5804275935001973088 T^{19} + 23167079840829073544 T^{20} - 79827880812877201536 T^{21} +$$$$22\!\cdots\!28$$$$T^{22} -$$$$44\!\cdots\!56$$$$T^{23} +$$$$49\!\cdots\!41$$$$T^{24}$$)
$59$ ($$1 - 12 T - 62 T^{2} + 192 T^{3} + 13661 T^{4} - 4488 T^{5} - 291210 T^{6} - 10035204 T^{7} + 17708314 T^{8} + 439140108 T^{9} + 4538367434 T^{10} - 27175198584 T^{11} - 197381258843 T^{12} - 1603336716456 T^{13} + 15798057037754 T^{14} + 90190156240932 T^{15} + 214578033439354 T^{16} - 7174411185021996 T^{17} - 12283393201595610 T^{18} - 11169067863867672 T^{19} + 2005850608112629181 T^{20} + 1663295197181748288 T^{21} - 31689238704639766862 T^{22} -$$$$36\!\cdots\!08$$$$T^{23} +$$$$17\!\cdots\!81$$$$T^{24}$$)($$1 + 12 T - 62 T^{2} - 192 T^{3} + 13661 T^{4} + 4488 T^{5} - 291210 T^{6} + 10035204 T^{7} + 17708314 T^{8} - 439140108 T^{9} + 4538367434 T^{10} + 27175198584 T^{11} - 197381258843 T^{12} + 1603336716456 T^{13} + 15798057037754 T^{14} - 90190156240932 T^{15} + 214578033439354 T^{16} + 7174411185021996 T^{17} - 12283393201595610 T^{18} + 11169067863867672 T^{19} + 2005850608112629181 T^{20} - 1663295197181748288 T^{21} - 31689238704639766862 T^{22} +$$$$36\!\cdots\!08$$$$T^{23} +$$$$17\!\cdots\!81$$$$T^{24}$$)
$61$ ($$1 + 30 T + 633 T^{2} + 9990 T^{3} + 134066 T^{4} + 1612650 T^{5} + 18045535 T^{6} + 190552410 T^{7} + 1899056448 T^{8} + 17814782910 T^{9} + 157346567941 T^{10} + 1317498445782 T^{11} + 10515537066752 T^{12} + 80367405192702 T^{13} + 585486579308461 T^{14} + 4043617239694710 T^{15} + 26294033629032768 T^{16} + 160939860632635410 T^{17} + 929712718744528135 T^{18} + 5068144234509265650 T^{19} + 25701432624293474546 T^{20} +$$$$11\!\cdots\!90$$$$T^{21} +$$$$45\!\cdots\!33$$$$T^{22} +$$$$13\!\cdots\!30$$$$T^{23} +$$$$26\!\cdots\!21$$$$T^{24}$$)($$1 + 30 T + 633 T^{2} + 9990 T^{3} + 134066 T^{4} + 1612650 T^{5} + 18045535 T^{6} + 190552410 T^{7} + 1899056448 T^{8} + 17814782910 T^{9} + 157346567941 T^{10} + 1317498445782 T^{11} + 10515537066752 T^{12} + 80367405192702 T^{13} + 585486579308461 T^{14} + 4043617239694710 T^{15} + 26294033629032768 T^{16} + 160939860632635410 T^{17} + 929712718744528135 T^{18} + 5068144234509265650 T^{19} + 25701432624293474546 T^{20} +$$$$11\!\cdots\!90$$$$T^{21} +$$$$45\!\cdots\!33$$$$T^{22} +$$$$13\!\cdots\!30$$$$T^{23} +$$$$26\!\cdots\!21$$$$T^{24}$$)
$67$ ($$1 + 4 T - 339 T^{2} - 1212 T^{3} + 65398 T^{4} + 197860 T^{5} - 8949237 T^{6} - 19422588 T^{7} + 965230072 T^{8} + 1219402116 T^{9} - 85058357911 T^{10} - 32989889052 T^{11} + 6246904001980 T^{12} - 2210322566484 T^{13} - 381826968662479 T^{14} + 366751038614508 T^{15} + 19450467973710712 T^{16} - 26222923701716916 T^{17} - 809533500666955053 T^{18} + 1199172398229208780 T^{19} + 26556013976849208118 T^{20} - 32974319688309475764 T^{21} -$$$$61\!\cdots\!11$$$$T^{22} +$$$$48\!\cdots\!32$$$$T^{23} +$$$$81\!\cdots\!61$$$$T^{24}$$)($$1 + 4 T - 339 T^{2} - 1212 T^{3} + 65398 T^{4} + 197860 T^{5} - 8949237 T^{6} - 19422588 T^{7} + 965230072 T^{8} + 1219402116 T^{9} - 85058357911 T^{10} - 32989889052 T^{11} + 6246904001980 T^{12} - 2210322566484 T^{13} - 381826968662479 T^{14} + 366751038614508 T^{15} + 19450467973710712 T^{16} - 26222923701716916 T^{17} - 809533500666955053 T^{18} + 1199172398229208780 T^{19} + 26556013976849208118 T^{20} - 32974319688309475764 T^{21} -$$$$61\!\cdots\!11$$$$T^{22} +$$$$48\!\cdots\!32$$$$T^{23} +$$$$81\!\cdots\!61$$$$T^{24}$$)
$71$ ($$1 - 300 T^{2} + 49610 T^{4} - 6300412 T^{6} + 667955583 T^{8} - 59469293656 T^{10} + 4533955360172 T^{12} - 299784709319896 T^{14} + 16973874197365023 T^{16} - 807084566019275452 T^{18} + 32035832685102203210 T^{20} -$$$$97\!\cdots\!00$$$$T^{22} +$$$$16\!\cdots\!41$$$$T^{24}$$)($$1 - 300 T^{2} + 49610 T^{4} - 6300412 T^{6} + 667955583 T^{8} - 59469293656 T^{10} + 4533955360172 T^{12} - 299784709319896 T^{14} + 16973874197365023 T^{16} - 807084566019275452 T^{18} + 32035832685102203210 T^{20} -$$$$97\!\cdots\!00$$$$T^{22} +$$$$16\!\cdots\!41$$$$T^{24}$$)
$73$ ($$1 + 158 T^{2} + 8161 T^{4} - 137736 T^{5} + 413890 T^{6} - 21056400 T^{7} + 34702222 T^{8} - 1012533192 T^{9} + 9774206798 T^{10} - 39652148664 T^{11} + 1268896234901 T^{12} - 2894606852472 T^{13} + 52086748026542 T^{14} - 393892624752264 T^{15} + 985482063591502 T^{16} - 43651424690845200 T^{17} + 62635722918754210 T^{18} - 1521624482426344392 T^{19} + 6581520809947595041 T^{20} +$$$$67\!\cdots\!42$$$$T^{22} +$$$$22\!\cdots\!21$$$$T^{24}$$)($$1 + 158 T^{2} + 8161 T^{4} - 137736 T^{5} + 413890 T^{6} - 21056400 T^{7} + 34702222 T^{8} - 1012533192 T^{9} + 9774206798 T^{10} - 39652148664 T^{11} + 1268896234901 T^{12} - 2894606852472 T^{13} + 52086748026542 T^{14} - 393892624752264 T^{15} + 985482063591502 T^{16} - 43651424690845200 T^{17} + 62635722918754210 T^{18} - 1521624482426344392 T^{19} + 6581520809947595041 T^{20} +$$$$67\!\cdots\!42$$$$T^{22} +$$$$22\!\cdots\!21$$$$T^{24}$$)
$79$ ($$1 + 4 T - 254 T^{2} + 1984 T^{3} + 51521 T^{4} - 543784 T^{5} - 2546434 T^{6} + 103559324 T^{7} - 209408530 T^{8} - 8333233172 T^{9} + 86160957682 T^{10} + 365142325960 T^{11} - 8321189358475 T^{12} + 28846243750840 T^{13} + 537730536893362 T^{14} - 4108608949889708 T^{15} - 8156479205590930 T^{16} + 318657880590314276 T^{17} - 619006161712162114 T^{18} - 10442778444129485656 T^{19} + 78162962995195929281 T^{20} +$$$$23\!\cdots\!96$$$$T^{21} -$$$$24\!\cdots\!54$$$$T^{22} +$$$$29\!\cdots\!16$$$$T^{23} +$$$$59\!\cdots\!41$$$$T^{24}$$)($$1 + 4 T - 254 T^{2} + 1984 T^{3} + 51521 T^{4} - 543784 T^{5} - 2546434 T^{6} + 103559324 T^{7} - 209408530 T^{8} - 8333233172 T^{9} + 86160957682 T^{10} + 365142325960 T^{11} - 8321189358475 T^{12} + 28846243750840 T^{13} + 537730536893362 T^{14} - 4108608949889708 T^{15} - 8156479205590930 T^{16} + 318657880590314276 T^{17} - 619006161712162114 T^{18} - 10442778444129485656 T^{19} + 78162962995195929281 T^{20} +$$$$23\!\cdots\!96$$$$T^{21} -$$$$24\!\cdots\!54$$$$T^{22} +$$$$29\!\cdots\!16$$$$T^{23} +$$$$59\!\cdots\!41$$$$T^{24}$$)
$83$ ($$( 1 + 20 T + 635 T^{2} + 8628 T^{3} + 148691 T^{4} + 1458128 T^{5} + 17076098 T^{6} + 121024624 T^{7} + 1024332299 T^{8} + 4933378236 T^{9} + 30136033835 T^{10} + 78780812860 T^{11} + 326940373369 T^{12} )^{2}$$)($$( 1 - 20 T + 635 T^{2} - 8628 T^{3} + 148691 T^{4} - 1458128 T^{5} + 17076098 T^{6} - 121024624 T^{7} + 1024332299 T^{8} - 4933378236 T^{9} + 30136033835 T^{10} - 78780812860 T^{11} + 326940373369 T^{12} )^{2}$$)
$89$ ($$1 - 26 T + 101 T^{2} + 1838 T^{3} + 1058 T^{4} - 234822 T^{5} - 191269 T^{6} + 17352042 T^{7} - 36283792 T^{8} - 94067578 T^{9} - 6641487 T^{10} - 7507089026 T^{11} - 13384168368 T^{12} - 668130923314 T^{13} - 52607218527 T^{14} - 66314726395082 T^{15} - 2276526422057872 T^{16} + 96894834089544858 T^{17} - 95057114540819509 T^{18} - 10386490522837910838 T^{19} + 4164910956432801698 T^{20} +$$$$64\!\cdots\!42$$$$T^{21} +$$$$31\!\cdots\!01$$$$T^{22} -$$$$72\!\cdots\!14$$$$T^{23} +$$$$24\!\cdots\!21$$$$T^{24}$$)($$1 + 26 T + 101 T^{2} - 1838 T^{3} + 1058 T^{4} + 234822 T^{5} - 191269 T^{6} - 17352042 T^{7} - 36283792 T^{8} + 94067578 T^{9} - 6641487 T^{10} + 7507089026 T^{11} - 13384168368 T^{12} + 668130923314 T^{13} - 52607218527 T^{14} + 66314726395082 T^{15} - 2276526422057872 T^{16} - 96894834089544858 T^{17} - 95057114540819509 T^{18} + 10386490522837910838 T^{19} + 4164910956432801698 T^{20} -$$$$64\!\cdots\!42$$$$T^{21} +$$$$31\!\cdots\!01$$$$T^{22} +$$$$72\!\cdots\!14$$$$T^{23} +$$$$24\!\cdots\!21$$$$T^{24}$$)
$97$ ($$1 - 484 T^{2} + 116170 T^{4} - 18891668 T^{6} + 2452131295 T^{8} - 279695423560 T^{10} + 28680693351980 T^{12} - 2631654240276040 T^{14} + 217085420463948895 T^{16} - 15736230570413031572 T^{18} +$$$$91\!\cdots\!70$$$$T^{20} -$$$$35\!\cdots\!16$$$$T^{22} +$$$$69\!\cdots\!41$$$$T^{24}$$)($$1 - 484 T^{2} + 116170 T^{4} - 18891668 T^{6} + 2452131295 T^{8} - 279695423560 T^{10} + 28680693351980 T^{12} - 2631654240276040 T^{14} + 217085420463948895 T^{16} - 15736230570413031572 T^{18} +$$$$91\!\cdots\!70$$$$T^{20} -$$$$35\!\cdots\!16$$$$T^{22} +$$$$69\!\cdots\!41$$$$T^{24}$$)