Properties

Label 315.2.bj
Level 315
Weight 2
Character orbit bj
Rep. character \(\chi_{315}(26,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 24
Newform subspaces 2
Sturm bound 96
Trace bound 5

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Defining parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.bj (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: \(5\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(315, [\chi])\).

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

Trace form

\( 24q + 16q^{4} - 4q^{7} + O(q^{10}) \) \( 24q + 16q^{4} - 4q^{7} - 32q^{16} + 12q^{19} + 64q^{22} - 12q^{25} - 40q^{28} + 12q^{31} - 20q^{37} - 8q^{43} - 8q^{46} + 12q^{49} + 40q^{58} - 112q^{64} + 12q^{67} - 24q^{70} - 84q^{73} + 36q^{79} - 144q^{82} + 8q^{88} - 36q^{91} + 24q^{94} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(315, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
315.2.bj.a \(12\) \(2.515\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(-6\) \(-2\) \(q+\beta _{3}q^{2}+(-\beta _{2}+\beta _{4}+\beta _{5})q^{4}+(-1+\cdots)q^{5}+\cdots\)
315.2.bj.b \(12\) \(2.515\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(6\) \(-2\) \(q+(\beta _{1}-\beta _{3})q^{2}+(1-\beta _{4}-\beta _{5})q^{4}+\beta _{4}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(315, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(315, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} + 2 T^{4} - 12 T^{5} - 24 T^{7} - 8 T^{8} - 24 T^{9} + 52 T^{10} + 24 T^{11} + 148 T^{12} + 48 T^{13} + 208 T^{14} - 192 T^{15} - 128 T^{16} - 768 T^{17} - 1536 T^{19} + 512 T^{20} + 2048 T^{22} + 4096 T^{24} \))(\( 1 + 2 T^{2} + 2 T^{4} + 12 T^{5} + 24 T^{7} - 8 T^{8} + 24 T^{9} + 52 T^{10} - 24 T^{11} + 148 T^{12} - 48 T^{13} + 208 T^{14} + 192 T^{15} - 128 T^{16} + 768 T^{17} + 1536 T^{19} + 512 T^{20} + 2048 T^{22} + 4096 T^{24} \))
$3$ 1
$5$ (\( ( 1 + T + T^{2} )^{6} \))(\( ( 1 - T + T^{2} )^{6} \))
$7$ (\( 1 + 2 T - T^{2} + 6 T^{3} + 130 T^{4} + 130 T^{5} - 131 T^{6} + 910 T^{7} + 6370 T^{8} + 2058 T^{9} - 2401 T^{10} + 33614 T^{11} + 117649 T^{12} \))(\( 1 + 2 T - T^{2} + 6 T^{3} + 130 T^{4} + 130 T^{5} - 131 T^{6} + 910 T^{7} + 6370 T^{8} + 2058 T^{9} - 2401 T^{10} + 33614 T^{11} + 117649 T^{12} \))
$11$ (\( 1 - 12 T + 104 T^{2} - 672 T^{3} + 3641 T^{4} - 16332 T^{5} + 62340 T^{6} - 195852 T^{7} + 477010 T^{8} - 649368 T^{9} - 1262450 T^{10} + 12891636 T^{11} - 53694803 T^{12} + 141807996 T^{13} - 152756450 T^{14} - 864308808 T^{15} + 6983903410 T^{16} - 31542160452 T^{17} + 110439112740 T^{18} - 318264476772 T^{19} + 780480685721 T^{20} - 1584540848352 T^{21} + 2697492158504 T^{22} - 3423740047332 T^{23} + 3138428376721 T^{24} \))(\( 1 + 12 T + 104 T^{2} + 672 T^{3} + 3641 T^{4} + 16332 T^{5} + 62340 T^{6} + 195852 T^{7} + 477010 T^{8} + 649368 T^{9} - 1262450 T^{10} - 12891636 T^{11} - 53694803 T^{12} - 141807996 T^{13} - 152756450 T^{14} + 864308808 T^{15} + 6983903410 T^{16} + 31542160452 T^{17} + 110439112740 T^{18} + 318264476772 T^{19} + 780480685721 T^{20} + 1584540848352 T^{21} + 2697492158504 T^{22} + 3423740047332 T^{23} + 3138428376721 T^{24} \))
$13$ (\( 1 - 54 T^{2} + 1725 T^{4} - 42910 T^{6} + 855078 T^{8} - 14180646 T^{10} + 200139573 T^{12} - 2396529174 T^{14} + 24421882758 T^{16} - 207118374190 T^{18} + 1407135493725 T^{20} - 7444358559846 T^{22} + 23298085122481 T^{24} \))(\( 1 - 54 T^{2} + 1725 T^{4} - 42910 T^{6} + 855078 T^{8} - 14180646 T^{10} + 200139573 T^{12} - 2396529174 T^{14} + 24421882758 T^{16} - 207118374190 T^{18} + 1407135493725 T^{20} - 7444358559846 T^{22} + 23298085122481 T^{24} \))
$17$ (\( 1 - 42 T^{2} + 120 T^{3} + 831 T^{4} - 5256 T^{5} - 3958 T^{6} + 127332 T^{7} - 227496 T^{8} - 2018952 T^{9} + 9082140 T^{10} + 14965308 T^{11} - 192427863 T^{12} + 254410236 T^{13} + 2624738460 T^{14} - 9919111176 T^{15} - 19000693416 T^{16} + 180793231524 T^{17} - 95536498102 T^{18} - 2156740065288 T^{19} + 5796854433471 T^{20} + 14230545179640 T^{21} - 84671743818858 T^{22} + 582622237229761 T^{24} \))(\( 1 - 42 T^{2} - 120 T^{3} + 831 T^{4} + 5256 T^{5} - 3958 T^{6} - 127332 T^{7} - 227496 T^{8} + 2018952 T^{9} + 9082140 T^{10} - 14965308 T^{11} - 192427863 T^{12} - 254410236 T^{13} + 2624738460 T^{14} + 9919111176 T^{15} - 19000693416 T^{16} - 180793231524 T^{17} - 95536498102 T^{18} + 2156740065288 T^{19} + 5796854433471 T^{20} - 14230545179640 T^{21} - 84671743818858 T^{22} + 582622237229761 T^{24} \))
$19$ (\( 1 - 6 T + 75 T^{2} - 378 T^{3} + 2703 T^{4} - 15096 T^{5} + 85196 T^{6} - 494928 T^{7} + 2381289 T^{8} - 12436578 T^{9} + 56967225 T^{10} - 267312150 T^{11} + 1188448446 T^{12} - 5078930850 T^{13} + 20565168225 T^{14} - 85302488502 T^{15} + 310331963769 T^{16} - 1225490725872 T^{17} + 4008120877676 T^{18} - 13493887771944 T^{19} + 45906570899823 T^{20} - 121975949760462 T^{21} + 459829969335075 T^{22} - 698941553389314 T^{23} + 2213314919066161 T^{24} \))(\( 1 - 6 T + 75 T^{2} - 378 T^{3} + 2703 T^{4} - 15096 T^{5} + 85196 T^{6} - 494928 T^{7} + 2381289 T^{8} - 12436578 T^{9} + 56967225 T^{10} - 267312150 T^{11} + 1188448446 T^{12} - 5078930850 T^{13} + 20565168225 T^{14} - 85302488502 T^{15} + 310331963769 T^{16} - 1225490725872 T^{17} + 4008120877676 T^{18} - 13493887771944 T^{19} + 45906570899823 T^{20} - 121975949760462 T^{21} + 459829969335075 T^{22} - 698941553389314 T^{23} + 2213314919066161 T^{24} \))
$23$ (\( 1 - 12 T + 110 T^{2} - 744 T^{3} + 3947 T^{4} - 16104 T^{5} + 39450 T^{6} + 32328 T^{7} - 1232708 T^{8} + 9040500 T^{9} - 50848868 T^{10} + 256937364 T^{11} - 1217123567 T^{12} + 5909559372 T^{13} - 26899051172 T^{14} + 109995763500 T^{15} - 344962239428 T^{16} + 208074096504 T^{17} + 5840015821050 T^{18} - 54831308998488 T^{19} + 309093458904107 T^{20} - 1340057580128472 T^{21} + 4556916233501390 T^{22} - 11433717094967124 T^{23} + 21914624432020321 T^{24} \))(\( 1 + 12 T + 110 T^{2} + 744 T^{3} + 3947 T^{4} + 16104 T^{5} + 39450 T^{6} - 32328 T^{7} - 1232708 T^{8} - 9040500 T^{9} - 50848868 T^{10} - 256937364 T^{11} - 1217123567 T^{12} - 5909559372 T^{13} - 26899051172 T^{14} - 109995763500 T^{15} - 344962239428 T^{16} - 208074096504 T^{17} + 5840015821050 T^{18} + 54831308998488 T^{19} + 309093458904107 T^{20} + 1340057580128472 T^{21} + 4556916233501390 T^{22} + 11433717094967124 T^{23} + 21914624432020321 T^{24} \))
$29$ (\( 1 - 184 T^{2} + 16934 T^{4} - 1029132 T^{6} + 46624399 T^{8} - 1703051756 T^{10} + 52962115192 T^{12} - 1432266526796 T^{14} + 32976551549119 T^{16} - 612151713987372 T^{18} + 8471172757081574 T^{20} - 77410130927236984 T^{22} + 353814783205469041 T^{24} \))(\( 1 - 184 T^{2} + 16934 T^{4} - 1029132 T^{6} + 46624399 T^{8} - 1703051756 T^{10} + 52962115192 T^{12} - 1432266526796 T^{14} + 32976551549119 T^{16} - 612151713987372 T^{18} + 8471172757081574 T^{20} - 77410130927236984 T^{22} + 353814783205469041 T^{24} \))
$31$ (\( 1 - 6 T + 75 T^{2} - 378 T^{3} + 2451 T^{4} - 14268 T^{5} + 15464 T^{6} - 69696 T^{7} - 2002827 T^{8} + 11519550 T^{9} - 72958179 T^{10} + 685842102 T^{11} - 2463303858 T^{12} + 21261105162 T^{13} - 70112810019 T^{14} + 343178914050 T^{15} - 1849652793867 T^{16} - 1995337308096 T^{17} + 13724356922984 T^{18} - 392549978135748 T^{19} + 2090435932767891 T^{20} - 9994177176733638 T^{21} + 61472121523560075 T^{22} - 152450861378428986 T^{23} + 787662783788549761 T^{24} \))(\( 1 - 6 T + 75 T^{2} - 378 T^{3} + 2451 T^{4} - 14268 T^{5} + 15464 T^{6} - 69696 T^{7} - 2002827 T^{8} + 11519550 T^{9} - 72958179 T^{10} + 685842102 T^{11} - 2463303858 T^{12} + 21261105162 T^{13} - 70112810019 T^{14} + 343178914050 T^{15} - 1849652793867 T^{16} - 1995337308096 T^{17} + 13724356922984 T^{18} - 392549978135748 T^{19} + 2090435932767891 T^{20} - 9994177176733638 T^{21} + 61472121523560075 T^{22} - 152450861378428986 T^{23} + 787662783788549761 T^{24} \))
$37$ (\( 1 + 10 T - T^{2} + 206 T^{3} + 1893 T^{4} - 19356 T^{5} - 25070 T^{6} - 11384 T^{7} - 4236169 T^{8} + 22256290 T^{9} - 6662617 T^{10} - 355484390 T^{11} + 9259385786 T^{12} - 13152922430 T^{13} - 9121122673 T^{14} + 1127347857370 T^{15} - 7939262729209 T^{16} - 789411606488 T^{17} - 64322761073630 T^{18} - 1837501413786348 T^{19} + 6649123606272453 T^{20} + 26772118397785862 T^{21} - 4808584372417849 T^{22} + 1779176217794604130 T^{23} + 6582952005840035281 T^{24} \))(\( 1 + 10 T - T^{2} + 206 T^{3} + 1893 T^{4} - 19356 T^{5} - 25070 T^{6} - 11384 T^{7} - 4236169 T^{8} + 22256290 T^{9} - 6662617 T^{10} - 355484390 T^{11} + 9259385786 T^{12} - 13152922430 T^{13} - 9121122673 T^{14} + 1127347857370 T^{15} - 7939262729209 T^{16} - 789411606488 T^{17} - 64322761073630 T^{18} - 1837501413786348 T^{19} + 6649123606272453 T^{20} + 26772118397785862 T^{21} - 4808584372417849 T^{22} + 1779176217794604130 T^{23} + 6582952005840035281 T^{24} \))
$41$ (\( ( 1 - 12 T + 192 T^{2} - 1704 T^{3} + 18087 T^{4} - 122340 T^{5} + 939022 T^{6} - 5015940 T^{7} + 30404247 T^{8} - 117441384 T^{9} + 542546112 T^{10} - 1390274412 T^{11} + 4750104241 T^{12} )^{2} \))(\( ( 1 + 12 T + 192 T^{2} + 1704 T^{3} + 18087 T^{4} + 122340 T^{5} + 939022 T^{6} + 5015940 T^{7} + 30404247 T^{8} + 117441384 T^{9} + 542546112 T^{10} + 1390274412 T^{11} + 4750104241 T^{12} )^{2} \))
$43$ (\( ( 1 + 2 T + 65 T^{2} + 318 T^{3} + 5080 T^{4} + 14554 T^{5} + 241357 T^{6} + 625822 T^{7} + 9392920 T^{8} + 25283226 T^{9} + 222222065 T^{10} + 294016886 T^{11} + 6321363049 T^{12} )^{2} \))(\( ( 1 + 2 T + 65 T^{2} + 318 T^{3} + 5080 T^{4} + 14554 T^{5} + 241357 T^{6} + 625822 T^{7} + 9392920 T^{8} + 25283226 T^{9} + 222222065 T^{10} + 294016886 T^{11} + 6321363049 T^{12} )^{2} \))
$47$ (\( 1 - 162 T^{2} + 192 T^{3} + 11733 T^{4} - 23328 T^{5} - 672838 T^{6} + 760896 T^{7} + 42600858 T^{8} + 8087904 T^{9} - 2486828586 T^{10} - 573202080 T^{11} + 123583933965 T^{12} - 26940497760 T^{13} - 5493404346474 T^{14} + 839710456992 T^{15} + 207878597366298 T^{16} + 174507698446272 T^{17} - 7252665683533702 T^{18} - 11818504154160864 T^{19} + 279377826402441813 T^{20} + 214873050835731264 T^{21} - 8521059422204467938 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 - 162 T^{2} - 192 T^{3} + 11733 T^{4} + 23328 T^{5} - 672838 T^{6} - 760896 T^{7} + 42600858 T^{8} - 8087904 T^{9} - 2486828586 T^{10} + 573202080 T^{11} + 123583933965 T^{12} + 26940497760 T^{13} - 5493404346474 T^{14} - 839710456992 T^{15} + 207878597366298 T^{16} - 174507698446272 T^{17} - 7252665683533702 T^{18} + 11818504154160864 T^{19} + 279377826402441813 T^{20} - 214873050835731264 T^{21} - 8521059422204467938 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \))
$53$ (\( 1 - 12 T + 290 T^{2} - 2904 T^{3} + 41501 T^{4} - 314064 T^{5} + 3427926 T^{6} - 18869580 T^{7} + 172940890 T^{8} - 534860628 T^{9} + 5470600330 T^{10} + 1694225472 T^{11} + 171503876389 T^{12} + 89793950016 T^{13} + 15366916326970 T^{14} - 79628445714756 T^{15} + 1364586806668090 T^{16} - 7891173310802940 T^{17} + 75977789787488454 T^{18} - 368934479421767568 T^{19} + 2583839411761892861 T^{20} - 9582513470593394232 T^{21} + 50717366405998784210 T^{22} - \)\(11\!\cdots\!64\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 + 12 T + 290 T^{2} + 2904 T^{3} + 41501 T^{4} + 314064 T^{5} + 3427926 T^{6} + 18869580 T^{7} + 172940890 T^{8} + 534860628 T^{9} + 5470600330 T^{10} - 1694225472 T^{11} + 171503876389 T^{12} - 89793950016 T^{13} + 15366916326970 T^{14} + 79628445714756 T^{15} + 1364586806668090 T^{16} + 7891173310802940 T^{17} + 75977789787488454 T^{18} + 368934479421767568 T^{19} + 2583839411761892861 T^{20} + 9582513470593394232 T^{21} + 50717366405998784210 T^{22} + \)\(11\!\cdots\!64\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))
$59$ (\( 1 + 24 T + 84 T^{2} - 1128 T^{3} + 9885 T^{4} + 197484 T^{5} - 900376 T^{6} - 12268224 T^{7} + 92002038 T^{8} + 411544404 T^{9} - 9205294494 T^{10} - 19136016684 T^{11} + 471190351413 T^{12} - 1129024984356 T^{13} - 32043630133614 T^{14} + 84522578149116 T^{15} + 1114821907181718 T^{16} - 8770851443174976 T^{17} - 37978340157549016 T^{18} + 491468849827995396 T^{19} + 1451418875718713085 T^{20} - 9771859283442771192 T^{21} + 42933807277253877684 T^{22} + \)\(72\!\cdots\!16\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 - 24 T + 84 T^{2} + 1128 T^{3} + 9885 T^{4} - 197484 T^{5} - 900376 T^{6} + 12268224 T^{7} + 92002038 T^{8} - 411544404 T^{9} - 9205294494 T^{10} + 19136016684 T^{11} + 471190351413 T^{12} + 1129024984356 T^{13} - 32043630133614 T^{14} - 84522578149116 T^{15} + 1114821907181718 T^{16} + 8770851443174976 T^{17} - 37978340157549016 T^{18} - 491468849827995396 T^{19} + 1451418875718713085 T^{20} + 9771859283442771192 T^{21} + 42933807277253877684 T^{22} - \)\(72\!\cdots\!16\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))
$61$ (\( 1 + 138 T^{2} + 5889 T^{4} - 41004 T^{5} - 102418 T^{6} - 5059224 T^{7} - 8898402 T^{8} - 158765292 T^{9} + 1660743870 T^{10} + 9579987756 T^{11} + 189206615181 T^{12} + 584379253116 T^{13} + 6179627940270 T^{14} - 36036704743452 T^{15} - 123205859246082 T^{16} - 4273001876330424 T^{17} - 5276613701304898 T^{18} - 128865027248205084 T^{19} + 1128964366240987809 T^{20} + 98441321809477798938 T^{22} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( 1 + 138 T^{2} + 5889 T^{4} - 41004 T^{5} - 102418 T^{6} - 5059224 T^{7} - 8898402 T^{8} - 158765292 T^{9} + 1660743870 T^{10} + 9579987756 T^{11} + 189206615181 T^{12} + 584379253116 T^{13} + 6179627940270 T^{14} - 36036704743452 T^{15} - 123205859246082 T^{16} - 4273001876330424 T^{17} - 5276613701304898 T^{18} - 128865027248205084 T^{19} + 1128964366240987809 T^{20} + 98441321809477798938 T^{22} + \)\(26\!\cdots\!21\)\( T^{24} \))
$67$ (\( 1 - 6 T - 219 T^{2} + 1306 T^{3} + 22971 T^{4} - 120540 T^{5} - 1928636 T^{6} + 7225512 T^{7} + 163638939 T^{8} - 309627182 T^{9} - 13816766313 T^{10} + 6348931902 T^{11} + 1027244050774 T^{12} + 425378437434 T^{13} - 62023463979057 T^{14} - 93124400139866 T^{15} + 3297508060100619 T^{16} + 9755345162129784 T^{17} - 174461292352891484 T^{18} - 730558176905634420 T^{19} + 9327780621153600411 T^{20} + 35531733921561200782 T^{21} - \)\(39\!\cdots\!31\)\( T^{22} - \)\(73\!\cdots\!98\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 - 6 T - 219 T^{2} + 1306 T^{3} + 22971 T^{4} - 120540 T^{5} - 1928636 T^{6} + 7225512 T^{7} + 163638939 T^{8} - 309627182 T^{9} - 13816766313 T^{10} + 6348931902 T^{11} + 1027244050774 T^{12} + 425378437434 T^{13} - 62023463979057 T^{14} - 93124400139866 T^{15} + 3297508060100619 T^{16} + 9755345162129784 T^{17} - 174461292352891484 T^{18} - 730558176905634420 T^{19} + 9327780621153600411 T^{20} + 35531733921561200782 T^{21} - \)\(39\!\cdots\!31\)\( T^{22} - \)\(73\!\cdots\!98\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))
$71$ (\( 1 - 460 T^{2} + 108410 T^{4} - 17445312 T^{6} + 2127087511 T^{8} - 205744741412 T^{10} + 16152616390624 T^{12} - 1037159241457892 T^{14} + 54052869288615991 T^{16} - 2234749420290428352 T^{18} + 70006140322352950010 T^{20} - \)\(14\!\cdots\!60\)\( T^{22} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 - 460 T^{2} + 108410 T^{4} - 17445312 T^{6} + 2127087511 T^{8} - 205744741412 T^{10} + 16152616390624 T^{12} - 1037159241457892 T^{14} + 54052869288615991 T^{16} - 2234749420290428352 T^{18} + 70006140322352950010 T^{20} - \)\(14\!\cdots\!60\)\( T^{22} + \)\(16\!\cdots\!41\)\( T^{24} \))
$73$ (\( 1 + 42 T + 1119 T^{2} + 22302 T^{3} + 373665 T^{4} + 5463480 T^{5} + 72046694 T^{6} + 868793688 T^{7} + 9715685451 T^{8} + 101446836642 T^{9} + 996453337299 T^{10} + 9237792089286 T^{11} + 81107402998458 T^{12} + 674358822517878 T^{13} + 5310099834466371 T^{14} + 39464544049960914 T^{15} + 275908376917691691 T^{16} + 1801071514770504984 T^{17} + 10903130693170338566 T^{18} + 60357240861116077560 T^{19} + \)\(30\!\cdots\!65\)\( T^{20} + \)\(13\!\cdots\!26\)\( T^{21} + \)\(48\!\cdots\!31\)\( T^{22} + \)\(13\!\cdots\!34\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 + 42 T + 1119 T^{2} + 22302 T^{3} + 373665 T^{4} + 5463480 T^{5} + 72046694 T^{6} + 868793688 T^{7} + 9715685451 T^{8} + 101446836642 T^{9} + 996453337299 T^{10} + 9237792089286 T^{11} + 81107402998458 T^{12} + 674358822517878 T^{13} + 5310099834466371 T^{14} + 39464544049960914 T^{15} + 275908376917691691 T^{16} + 1801071514770504984 T^{17} + 10903130693170338566 T^{18} + 60357240861116077560 T^{19} + \)\(30\!\cdots\!65\)\( T^{20} + \)\(13\!\cdots\!26\)\( T^{21} + \)\(48\!\cdots\!31\)\( T^{22} + \)\(13\!\cdots\!34\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))
$79$ (\( 1 - 18 T - 177 T^{2} + 3346 T^{3} + 40923 T^{4} - 454344 T^{5} - 6959816 T^{6} + 45880152 T^{7} + 889910193 T^{8} - 3054243566 T^{9} - 93786004551 T^{10} + 82100387598 T^{11} + 8378036930038 T^{12} + 6485930620242 T^{13} - 585318454402791 T^{14} - 1505861193537074 T^{15} + 34662074100075633 T^{16} + 141175815298692648 T^{17} - 1691843962334344136 T^{18} - 8725180824407424696 T^{19} + 62084643827806195803 T^{20} + \)\(40\!\cdots\!74\)\( T^{21} - \)\(16\!\cdots\!77\)\( T^{22} - \)\(13\!\cdots\!22\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( 1 - 18 T - 177 T^{2} + 3346 T^{3} + 40923 T^{4} - 454344 T^{5} - 6959816 T^{6} + 45880152 T^{7} + 889910193 T^{8} - 3054243566 T^{9} - 93786004551 T^{10} + 82100387598 T^{11} + 8378036930038 T^{12} + 6485930620242 T^{13} - 585318454402791 T^{14} - 1505861193537074 T^{15} + 34662074100075633 T^{16} + 141175815298692648 T^{17} - 1691843962334344136 T^{18} - 8725180824407424696 T^{19} + 62084643827806195803 T^{20} + \)\(40\!\cdots\!74\)\( T^{21} - \)\(16\!\cdots\!77\)\( T^{22} - \)\(13\!\cdots\!22\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))
$83$ (\( ( 1 + 12 T + 300 T^{2} + 3204 T^{3} + 50769 T^{4} + 448284 T^{5} + 5149954 T^{6} + 37207572 T^{7} + 349747641 T^{8} + 1832005548 T^{9} + 14237496300 T^{10} + 47268487716 T^{11} + 326940373369 T^{12} )^{2} \))(\( ( 1 - 12 T + 300 T^{2} - 3204 T^{3} + 50769 T^{4} - 448284 T^{5} + 5149954 T^{6} - 37207572 T^{7} + 349747641 T^{8} - 1832005548 T^{9} + 14237496300 T^{10} - 47268487716 T^{11} + 326940373369 T^{12} )^{2} \))
$89$ (\( 1 - 12 T - 282 T^{2} + 648 T^{3} + 81261 T^{4} + 90852 T^{5} - 10133890 T^{6} - 68061132 T^{7} + 906987750 T^{8} + 7493819688 T^{9} - 22404081252 T^{10} - 403242866028 T^{11} + 351385075713 T^{12} - 35888615076492 T^{13} - 177462727597092 T^{14} + 5282910571629672 T^{15} + 56906443994547750 T^{16} - 380057407254236268 T^{17} - 5036353734656768290 T^{18} + 4018505237928600708 T^{19} + \)\(31\!\cdots\!41\)\( T^{20} + \)\(22\!\cdots\!32\)\( T^{21} - \)\(87\!\cdots\!82\)\( T^{22} - \)\(33\!\cdots\!68\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))(\( 1 + 12 T - 282 T^{2} - 648 T^{3} + 81261 T^{4} - 90852 T^{5} - 10133890 T^{6} + 68061132 T^{7} + 906987750 T^{8} - 7493819688 T^{9} - 22404081252 T^{10} + 403242866028 T^{11} + 351385075713 T^{12} + 35888615076492 T^{13} - 177462727597092 T^{14} - 5282910571629672 T^{15} + 56906443994547750 T^{16} + 380057407254236268 T^{17} - 5036353734656768290 T^{18} - 4018505237928600708 T^{19} + \)\(31\!\cdots\!41\)\( T^{20} - \)\(22\!\cdots\!32\)\( T^{21} - \)\(87\!\cdots\!82\)\( T^{22} + \)\(33\!\cdots\!68\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))
$97$ (\( 1 - 432 T^{2} + 85686 T^{4} - 12118408 T^{6} + 1564041327 T^{8} - 184182428136 T^{10} + 19001698536324 T^{12} - 1732972466331624 T^{14} + 138463454133595887 T^{16} - 10094294608307633032 T^{18} + \)\(67\!\cdots\!46\)\( T^{20} - \)\(31\!\cdots\!68\)\( T^{22} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 - 432 T^{2} + 85686 T^{4} - 12118408 T^{6} + 1564041327 T^{8} - 184182428136 T^{10} + 19001698536324 T^{12} - 1732972466331624 T^{14} + 138463454133595887 T^{16} - 10094294608307633032 T^{18} + \)\(67\!\cdots\!46\)\( T^{20} - \)\(31\!\cdots\!68\)\( T^{22} + \)\(69\!\cdots\!41\)\( T^{24} \))
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