Properties

Label 735.2.q
Level 735
Weight 2
Character orbit q
Rep. character \(\chi_{735}(79,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 80
Newform subspaces 8
Sturm bound 224
Trace bound 5

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Defining parameters

Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.q (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 35 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 8 \)
Sturm bound: \(224\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(19\), \(73\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(735, [\chi])\).

Total New Old
Modular forms 256 80 176
Cusp forms 192 80 112
Eisenstein series 64 0 64

Trace form

\( 80q + 40q^{4} - 2q^{5} + 8q^{6} + 40q^{9} + O(q^{10}) \) \( 80q + 40q^{4} - 2q^{5} + 8q^{6} + 40q^{9} + 4q^{10} + 4q^{15} - 48q^{16} + 24q^{19} + 8q^{20} + 12q^{24} + 4q^{25} + 12q^{26} - 40q^{29} + 16q^{30} - 16q^{31} - 16q^{34} + 80q^{36} + 4q^{39} - 32q^{40} - 16q^{41} - 44q^{44} + 2q^{45} + 48q^{46} - 40q^{50} - 4q^{51} + 4q^{54} - 8q^{55} - 4q^{59} - 28q^{60} - 16q^{61} - 112q^{64} + 14q^{65} - 28q^{66} - 40q^{69} + 72q^{71} - 152q^{74} - 8q^{75} + 64q^{76} + 16q^{79} - 52q^{80} - 40q^{81} - 16q^{85} + 80q^{86} - 16q^{89} + 8q^{90} + 32q^{94} + 42q^{95} - 8q^{96} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(735, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
735.2.q.a \(4\) \(5.869\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(-2\) \(0\) \(q+\zeta_{12}q^{2}+(-\zeta_{12}+\zeta_{12}^{3})q^{3}-\zeta_{12}^{2}q^{4}+\cdots\)
735.2.q.b \(4\) \(5.869\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(2\) \(0\) \(q+\zeta_{12}q^{2}+(\zeta_{12}-\zeta_{12}^{3})q^{3}-\zeta_{12}^{2}q^{4}+\cdots\)
735.2.q.c \(8\) \(5.869\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(-12\) \(0\) \(q+(\zeta_{24}-\zeta_{24}^{5}+\zeta_{24}^{7})q^{2}+(-\zeta_{24}^{2}+\cdots)q^{3}+\cdots\)
735.2.q.d \(8\) \(5.869\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(12\) \(0\) \(q+(\zeta_{24}-\zeta_{24}^{5}+\zeta_{24}^{7})q^{2}+(\zeta_{24}^{2}+\cdots)q^{3}+\cdots\)
735.2.q.e \(12\) \(5.869\) 12.0.\(\cdots\).1 None \(0\) \(0\) \(-2\) \(0\) \(q+(-\beta _{3}-\beta _{4})q^{2}+(-\beta _{5}-\beta _{7})q^{3}+\cdots\)
735.2.q.f \(12\) \(5.869\) 12.0.\(\cdots\).1 None \(0\) \(0\) \(2\) \(0\) \(q+(-\beta _{3}-\beta _{4})q^{2}+(\beta _{5}+\beta _{7})q^{3}+(1+\cdots)q^{4}+\cdots\)
735.2.q.g \(16\) \(5.869\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(-2\) \(0\) \(q+(-\beta _{5}+\beta _{6}-\beta _{15})q^{2}-\beta _{3}q^{3}+(1+\cdots)q^{4}+\cdots\)
735.2.q.h \(16\) \(5.869\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{3}+\beta _{8})q^{2}-\beta _{4}q^{3}+(1-\beta _{6}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(735, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(735, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(245, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8} \))(\( 1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8} \))(\( 1 + 4 T^{2} + 7 T^{4} + 4 T^{6} + T^{8} + 16 T^{10} + 112 T^{12} + 256 T^{14} + 256 T^{16} \))(\( 1 + 4 T^{2} + 7 T^{4} + 4 T^{6} + T^{8} + 16 T^{10} + 112 T^{12} + 256 T^{14} + 256 T^{16} \))(\( 1 + T^{2} + 2 T^{4} - 7 T^{6} + 2 T^{8} + 21 T^{10} + 117 T^{12} + 84 T^{14} + 32 T^{16} - 448 T^{18} + 512 T^{20} + 1024 T^{22} + 4096 T^{24} \))(\( 1 + T^{2} + 2 T^{4} - 7 T^{6} + 2 T^{8} + 21 T^{10} + 117 T^{12} + 84 T^{14} + 32 T^{16} - 448 T^{18} + 512 T^{20} + 1024 T^{22} + 4096 T^{24} \))(\( 1 + 4 T^{2} + 4 T^{4} - 8 T^{6} - 20 T^{8} + 32 T^{10} + 96 T^{12} - 208 T^{14} - 944 T^{16} - 832 T^{18} + 1536 T^{20} + 2048 T^{22} - 5120 T^{24} - 8192 T^{26} + 16384 T^{28} + 65536 T^{30} + 65536 T^{32} \))(\( ( 1 - 3 T^{4} - 7 T^{8} - 48 T^{12} + 256 T^{16} )^{2} \))
$3$ (\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 - T^{2} + T^{4} )^{2} \))(\( ( 1 - T^{2} + T^{4} )^{2} \))(\( ( 1 - T^{2} + T^{4} )^{3} \))(\( ( 1 - T^{2} + T^{4} )^{3} \))(\( ( 1 - T^{2} + T^{4} )^{4} \))(\( ( 1 - T^{2} + T^{4} )^{4} \))
$5$ (\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( ( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )^{2} \))(\( 1 + 2 T + T^{2} - 18 T^{3} - 30 T^{4} + 26 T^{5} + 249 T^{6} + 130 T^{7} - 750 T^{8} - 2250 T^{9} + 625 T^{10} + 6250 T^{11} + 15625 T^{12} \))(\( 1 - 2 T + T^{2} + 18 T^{3} - 30 T^{4} - 26 T^{5} + 249 T^{6} - 130 T^{7} - 750 T^{8} + 2250 T^{9} + 625 T^{10} - 6250 T^{11} + 15625 T^{12} \))(\( 1 + 2 T + 4 T^{2} + 20 T^{3} + 62 T^{4} + 118 T^{5} + 200 T^{6} + 670 T^{7} + 1639 T^{8} + 3350 T^{9} + 5000 T^{10} + 14750 T^{11} + 38750 T^{12} + 62500 T^{13} + 62500 T^{14} + 156250 T^{15} + 390625 T^{16} \))(\( 1 + 8 T^{2} + 18 T^{4} - 32 T^{6} - 109 T^{8} - 800 T^{10} + 11250 T^{12} + 125000 T^{14} + 390625 T^{16} \))
$7$ 1
$11$ (\( ( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} )^{6} \))(\( ( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} )^{6} \))(\( ( 1 - 26 T^{2} - 28 T^{3} + 316 T^{4} + 518 T^{5} - 2872 T^{6} - 2968 T^{7} + 31007 T^{8} - 32648 T^{9} - 347512 T^{10} + 689458 T^{11} + 4626556 T^{12} - 4509428 T^{13} - 46060586 T^{14} + 214358881 T^{16} )^{2} \))(\( ( 1 - 2 T^{2} - 117 T^{4} - 242 T^{6} + 14641 T^{8} )^{4} \))
$13$ (\( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{4}( 1 + 6 T + 13 T^{2} )^{4} \))(\( ( 1 - 6 T + 13 T^{2} )^{4}( 1 + 6 T + 13 T^{2} )^{4} \))(\( ( 1 - 34 T^{2} + 359 T^{4} - 2172 T^{6} + 60671 T^{8} - 971074 T^{10} + 4826809 T^{12} )^{2} \))(\( ( 1 - 34 T^{2} + 359 T^{4} - 2172 T^{6} + 60671 T^{8} - 971074 T^{10} + 4826809 T^{12} )^{2} \))(\( ( 1 - 44 T^{2} + 1234 T^{4} - 23988 T^{6} + 358939 T^{8} - 4053972 T^{10} + 35244274 T^{12} - 212379596 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 - 24 T^{2} + 302 T^{4} - 4056 T^{6} + 28561 T^{8} )^{4} \))
$17$ (\( 1 + 18 T^{2} + 35 T^{4} + 5202 T^{6} + 83521 T^{8} \))(\( 1 + 18 T^{2} + 35 T^{4} + 5202 T^{6} + 83521 T^{8} \))(\( ( 1 + 18 T^{2} + 35 T^{4} + 5202 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 + 18 T^{2} + 35 T^{4} + 5202 T^{6} + 83521 T^{8} )^{2} \))(\( 1 + 70 T^{2} + 2485 T^{4} + 66610 T^{6} + 1548890 T^{8} + 31831870 T^{10} + 576997613 T^{12} + 9199410430 T^{14} + 129364841690 T^{16} + 1607803471090 T^{18} + 17334757240885 T^{20} + 141119573031430 T^{22} + 582622237229761 T^{24} \))(\( 1 + 70 T^{2} + 2485 T^{4} + 66610 T^{6} + 1548890 T^{8} + 31831870 T^{10} + 576997613 T^{12} + 9199410430 T^{14} + 129364841690 T^{16} + 1607803471090 T^{18} + 17334757240885 T^{20} + 141119573031430 T^{22} + 582622237229761 T^{24} \))(\( 1 + 76 T^{2} + 2664 T^{4} + 67392 T^{6} + 1583990 T^{8} + 33141472 T^{10} + 603483136 T^{12} + 11039990476 T^{14} + 198790801859 T^{16} + 3190557247564 T^{18} + 50403515001856 T^{20} + 799954567161568 T^{22} + 11049530028969590 T^{24} + 135861860939059008 T^{26} + 1552105639980083304 T^{28} + 12796714818514470604 T^{30} + 48661191875666868481 T^{32} \))(\( ( 1 + 40 T^{2} + 802 T^{4} + 8800 T^{6} + 90883 T^{8} + 2543200 T^{10} + 66983842 T^{12} + 965502760 T^{14} + 6975757441 T^{16} )^{2} \))
$19$ (\( ( 1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} - 114 T^{3} + 361 T^{4} )^{2} \))(\( 1 - 48 T^{2} + 1198 T^{4} - 18432 T^{6} + 261171 T^{8} - 6653952 T^{10} + 156124558 T^{12} - 2258202288 T^{14} + 16983563041 T^{16} \))(\( 1 - 48 T^{2} + 1198 T^{4} - 18432 T^{6} + 261171 T^{8} - 6653952 T^{10} + 156124558 T^{12} - 2258202288 T^{14} + 16983563041 T^{16} \))(\( ( 1 - 6 T - 17 T^{2} + 58 T^{3} + 674 T^{4} - 46 T^{5} - 17305 T^{6} - 874 T^{7} + 243314 T^{8} + 397822 T^{9} - 2215457 T^{10} - 14856594 T^{11} + 47045881 T^{12} )^{2} \))(\( ( 1 + 6 T - 17 T^{2} - 58 T^{3} + 674 T^{4} + 46 T^{5} - 17305 T^{6} + 874 T^{7} + 243314 T^{8} - 397822 T^{9} - 2215457 T^{10} + 14856594 T^{11} + 47045881 T^{12} )^{2} \))(\( ( 1 - 12 T + 30 T^{2} + 80 T^{3} + 689 T^{4} - 9436 T^{5} + 28846 T^{6} - 59944 T^{7} + 291332 T^{8} - 1138936 T^{9} + 10413406 T^{10} - 64721524 T^{11} + 89791169 T^{12} + 198087920 T^{13} + 1411376430 T^{14} - 10726460868 T^{15} + 16983563041 T^{16} )^{2} \))(\( ( 1 - 8 T^{2} + 306 T^{4} + 7712 T^{6} - 94861 T^{8} + 2784032 T^{10} + 39878226 T^{12} - 376367048 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( ( 1 + 23 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 23 T^{2} + 529 T^{4} )^{2} \))(\( 1 + 40 T^{2} + 334 T^{4} + 8320 T^{6} + 476515 T^{8} + 4401280 T^{10} + 93466894 T^{12} + 5921435560 T^{14} + 78310985281 T^{16} \))(\( 1 + 40 T^{2} + 334 T^{4} + 8320 T^{6} + 476515 T^{8} + 4401280 T^{10} + 93466894 T^{12} + 5921435560 T^{14} + 78310985281 T^{16} \))(\( 1 + 106 T^{2} + 6053 T^{4} + 248702 T^{6} + 8184794 T^{8} + 227653746 T^{10} + 5545513725 T^{12} + 120428831634 T^{14} + 2290440937754 T^{16} + 36816821666078 T^{18} + 474016393905893 T^{20} + 4391210188646794 T^{22} + 21914624432020321 T^{24} \))(\( 1 + 106 T^{2} + 6053 T^{4} + 248702 T^{6} + 8184794 T^{8} + 227653746 T^{10} + 5545513725 T^{12} + 120428831634 T^{14} + 2290440937754 T^{16} + 36816821666078 T^{18} + 474016393905893 T^{20} + 4391210188646794 T^{22} + 21914624432020321 T^{24} \))(\( 1 + 20 T^{2} - 1512 T^{4} - 18448 T^{6} + 1612902 T^{8} + 10149224 T^{10} - 1248424544 T^{12} - 1840459676 T^{14} + 770565238915 T^{16} - 973603168604 T^{18} - 349360372817504 T^{20} + 1502449397500136 T^{22} + 126307944781695462 T^{24} - 764236278869396752 T^{26} - 33134912141214725352 T^{28} + \)\(23\!\cdots\!80\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( ( 1 + 64 T^{2} + 2034 T^{4} + 64256 T^{6} + 1805219 T^{8} + 33991424 T^{10} + 569196594 T^{12} + 9474296896 T^{14} + 78310985281 T^{16} )^{2} \))
$29$ (\( ( 1 - 2 T + 29 T^{2} )^{4} \))(\( ( 1 - 2 T + 29 T^{2} )^{4} \))(\( ( 1 + 4 T + 14 T^{2} + 116 T^{3} + 841 T^{4} )^{4} \))(\( ( 1 + 4 T + 14 T^{2} + 116 T^{3} + 841 T^{4} )^{4} \))(\( ( 1 + 2 T + 35 T^{2} + 76 T^{3} + 1015 T^{4} + 1682 T^{5} + 24389 T^{6} )^{4} \))(\( ( 1 + 2 T + 35 T^{2} + 76 T^{3} + 1015 T^{4} + 1682 T^{5} + 24389 T^{6} )^{4} \))(\( ( 1 - 6 T + 78 T^{2} - 332 T^{3} + 2820 T^{4} - 9628 T^{5} + 65598 T^{6} - 146334 T^{7} + 707281 T^{8} )^{4} \))(\( ( 1 + 2 T + 29 T^{2} )^{16} \))
$31$ (\( ( 1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} - 897 T^{4} - 7688 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 8 T^{2} - 897 T^{4} - 7688 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 + 2 T - 37 T^{2} + 202 T^{3} + 530 T^{4} - 5622 T^{5} + 7227 T^{6} - 174282 T^{7} + 509330 T^{8} + 6017782 T^{9} - 34170277 T^{10} + 57258302 T^{11} + 887503681 T^{12} )^{2} \))(\( ( 1 - 2 T - 37 T^{2} - 202 T^{3} + 530 T^{4} + 5622 T^{5} + 7227 T^{6} + 174282 T^{7} + 509330 T^{8} - 6017782 T^{9} - 34170277 T^{10} - 57258302 T^{11} + 887503681 T^{12} )^{2} \))(\( ( 1 + 8 T - 54 T^{2} - 352 T^{3} + 3285 T^{4} + 9272 T^{5} - 157694 T^{6} - 60416 T^{7} + 6248764 T^{8} - 1872896 T^{9} - 151543934 T^{10} + 276222152 T^{11} + 3033766485 T^{12} - 10077461152 T^{13} - 47925198774 T^{14} + 220100912888 T^{15} + 852891037441 T^{16} )^{2} \))(\( ( 1 - 72 T^{2} + 2466 T^{4} - 57312 T^{6} + 1482179 T^{8} - 55076832 T^{10} + 2277402786 T^{12} - 63900265032 T^{14} + 852891037441 T^{16} )^{2} \))
$37$ (\( 1 + 58 T^{2} + 1995 T^{4} + 79402 T^{6} + 1874161 T^{8} \))(\( 1 + 58 T^{2} + 1995 T^{4} + 79402 T^{6} + 1874161 T^{8} \))(\( 1 + 36 T^{2} + 1306 T^{4} - 98928 T^{6} - 3648285 T^{8} - 135432432 T^{10} + 2447654266 T^{12} + 92366150724 T^{14} + 3512479453921 T^{16} \))(\( 1 + 36 T^{2} + 1306 T^{4} - 98928 T^{6} - 3648285 T^{8} - 135432432 T^{10} + 2447654266 T^{12} + 92366150724 T^{14} + 3512479453921 T^{16} \))(\( 1 + 46 T^{2} + 717 T^{4} - 85222 T^{6} - 3398278 T^{8} + 14638566 T^{10} + 4079564917 T^{12} + 20040196854 T^{14} - 6368920094758 T^{16} - 218656336027798 T^{18} + 2518447768461357 T^{20} + 221194881131221054 T^{22} + 6582952005840035281 T^{24} \))(\( 1 + 46 T^{2} + 717 T^{4} - 85222 T^{6} - 3398278 T^{8} + 14638566 T^{10} + 4079564917 T^{12} + 20040196854 T^{14} - 6368920094758 T^{16} - 218656336027798 T^{18} + 2518447768461357 T^{20} + 221194881131221054 T^{22} + 6582952005840035281 T^{24} \))(\( 1 + 128 T^{2} + 6574 T^{4} + 187656 T^{6} + 5183961 T^{8} + 158877780 T^{10} + 5690568750 T^{12} + 459641012708 T^{14} + 25029766954564 T^{16} + 629248546397252 T^{18} + 10665042019068750 T^{20} + 407636915949292020 T^{22} + 18208556502427761081 T^{24} + \)\(90\!\cdots\!44\)\( T^{26} + \)\(43\!\cdots\!94\)\( T^{28} + \)\(11\!\cdots\!92\)\( T^{30} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( ( 1 + 36 T^{2} - 1446 T^{4} + 144 T^{6} + 3870179 T^{8} + 197136 T^{10} - 2710036806 T^{12} + 92366150724 T^{14} + 3512479453921 T^{16} )^{2} \))
$41$ (\( ( 1 - 2 T + 41 T^{2} )^{4} \))(\( ( 1 + 2 T + 41 T^{2} )^{4} \))(\( ( 1 + 10 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 + 10 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 - 2 T + 63 T^{2} + 36 T^{3} + 2583 T^{4} - 3362 T^{5} + 68921 T^{6} )^{4} \))(\( ( 1 + 2 T + 63 T^{2} - 36 T^{3} + 2583 T^{4} + 3362 T^{5} + 68921 T^{6} )^{4} \))(\( ( 1 + 4 T + 114 T^{2} + 346 T^{3} + 5976 T^{4} + 14186 T^{5} + 191634 T^{6} + 275684 T^{7} + 2825761 T^{8} )^{4} \))(\( ( 1 + 24 T^{2} + 3006 T^{4} + 40344 T^{6} + 2825761 T^{8} )^{4} \))
$43$ (\( ( 1 - 70 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 43 T^{2} )^{8} \))(\( ( 1 - 43 T^{2} )^{8} \))(\( ( 1 + 46 T^{2} + 2839 T^{4} + 118948 T^{6} + 5249311 T^{8} + 157264846 T^{10} + 6321363049 T^{12} )^{2} \))(\( ( 1 + 46 T^{2} + 2839 T^{4} + 118948 T^{6} + 5249311 T^{8} + 157264846 T^{10} + 6321363049 T^{12} )^{2} \))(\( ( 1 - 216 T^{2} + 22138 T^{4} - 1464812 T^{6} + 71619331 T^{8} - 2708437388 T^{10} + 75685416538 T^{12} - 1365414418584 T^{14} + 11688200277601 T^{16} )^{2} \))(\( ( 1 - 44 T^{2} + 2902 T^{4} - 81356 T^{6} + 3418801 T^{8} )^{4} \))
$47$ (\( ( 1 + 47 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8} )^{2} \))(\( 1 + 154 T^{2} + 11573 T^{4} + 565358 T^{6} + 20169434 T^{8} + 410981154 T^{10} + 5646494445 T^{12} + 907857369186 T^{14} + 98420403870554 T^{16} + 6094115619972782 T^{18} + 275568020536560053 T^{20} + 8100266364317827546 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 + 154 T^{2} + 11573 T^{4} + 565358 T^{6} + 20169434 T^{8} + 410981154 T^{10} + 5646494445 T^{12} + 907857369186 T^{14} + 98420403870554 T^{16} + 6094115619972782 T^{18} + 275568020536560053 T^{20} + 8100266364317827546 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 + 308 T^{2} + 50892 T^{4} + 5950136 T^{6} + 547353834 T^{8} + 41768130380 T^{10} + 2726163659344 T^{12} + 155042751818668 T^{14} + 7757667166613347 T^{16} + 342489438767437612 T^{18} + 13302809011391389264 T^{20} + \)\(45\!\cdots\!20\)\( T^{22} + \)\(13\!\cdots\!74\)\( T^{24} + \)\(31\!\cdots\!64\)\( T^{26} + \)\(59\!\cdots\!72\)\( T^{28} + \)\(79\!\cdots\!52\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 4 T^{2} + 714 T^{4} + 20464 T^{6} - 4427821 T^{8} + 45204976 T^{10} + 3484092234 T^{12} - 43116861316 T^{14} + 23811286661761 T^{16} )^{2} \))
$53$ (\( 1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8} \))(\( 1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8} \))(\( ( 1 + 52 T^{2} - 105 T^{4} + 146068 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 + 52 T^{2} - 105 T^{4} + 146068 T^{6} + 7890481 T^{8} )^{2} \))(\( 1 + 146 T^{2} + 7213 T^{4} + 289382 T^{6} + 30094394 T^{8} + 2063064026 T^{10} + 100360862165 T^{12} + 5795146849034 T^{14} + 237459244063514 T^{16} + 6413967152232278 T^{18} + 449079146937146893 T^{20} + 25533570673364905154 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 + 146 T^{2} + 7213 T^{4} + 289382 T^{6} + 30094394 T^{8} + 2063064026 T^{10} + 100360862165 T^{12} + 5795146849034 T^{14} + 237459244063514 T^{16} + 6413967152232278 T^{18} + 449079146937146893 T^{20} + 25533570673364905154 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 + 264 T^{2} + 35652 T^{4} + 3177712 T^{6} + 212585578 T^{8} + 12279652216 T^{10} + 710817442960 T^{12} + 42611108403128 T^{14} + 2398437218467955 T^{16} + 119694603504386552 T^{18} + 5608691528144463760 T^{20} + \)\(27\!\cdots\!64\)\( T^{22} + \)\(13\!\cdots\!58\)\( T^{24} + \)\(55\!\cdots\!88\)\( T^{26} + \)\(17\!\cdots\!32\)\( T^{28} + \)\(36\!\cdots\!16\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 160 T^{2} + 14082 T^{4} + 944000 T^{6} + 52863443 T^{8} + 2651696000 T^{10} + 111113753442 T^{12} + 3546297780640 T^{14} + 62259690411361 T^{16} )^{2} \))
$59$ (\( ( 1 + 8 T + 5 T^{2} + 472 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 8 T + 5 T^{2} - 472 T^{3} + 3481 T^{4} )^{2} \))(\( 1 - 124 T^{2} + 7642 T^{4} - 95728 T^{6} - 5826461 T^{8} - 333229168 T^{10} + 92600872762 T^{12} - 5230386171484 T^{14} + 146830437604321 T^{16} \))(\( 1 - 124 T^{2} + 7642 T^{4} - 95728 T^{6} - 5826461 T^{8} - 333229168 T^{10} + 92600872762 T^{12} - 5230386171484 T^{14} + 146830437604321 T^{16} \))(\( ( 1 - 16 T + 143 T^{2} - 592 T^{3} - 3626 T^{4} + 88944 T^{5} - 864085 T^{6} + 5247696 T^{7} - 12622106 T^{8} - 121584368 T^{9} + 1732782623 T^{10} - 11438788784 T^{11} + 42180533641 T^{12} )^{2} \))(\( ( 1 + 16 T + 143 T^{2} + 592 T^{3} - 3626 T^{4} - 88944 T^{5} - 864085 T^{6} - 5247696 T^{7} - 12622106 T^{8} + 121584368 T^{9} + 1732782623 T^{10} + 11438788784 T^{11} + 42180533641 T^{12} )^{2} \))(\( ( 1 + 2 T - 226 T^{2} - 236 T^{3} + 32016 T^{4} + 19820 T^{5} - 2961940 T^{6} - 407278 T^{7} + 205790979 T^{8} - 24029402 T^{9} - 10310513140 T^{10} + 4070611780 T^{11} + 387949429776 T^{12} - 168722134564 T^{13} - 9532800602866 T^{14} + 4977302969638 T^{15} + 146830437604321 T^{16} )^{2} \))(\( ( 1 - 108 T^{2} + 3066 T^{4} - 176688 T^{6} + 21205859 T^{8} - 615050928 T^{10} + 37151828826 T^{12} - 4555497633228 T^{14} + 146830437604321 T^{16} )^{2} \))
$61$ (\( ( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} )^{2} \))(\( 1 - 160 T^{2} + 13486 T^{4} - 747520 T^{6} + 37645555 T^{8} - 2781521920 T^{10} + 186725011726 T^{12} - 8243259897760 T^{14} + 191707312997281 T^{16} \))(\( 1 - 160 T^{2} + 13486 T^{4} - 747520 T^{6} + 37645555 T^{8} - 2781521920 T^{10} + 186725011726 T^{12} - 8243259897760 T^{14} + 191707312997281 T^{16} \))(\( ( 1 - 6 T - 95 T^{2} + 182 T^{3} + 6266 T^{4} + 10162 T^{5} - 492559 T^{6} + 619882 T^{7} + 23315786 T^{8} + 41310542 T^{9} - 1315354895 T^{10} - 5067577806 T^{11} + 51520374361 T^{12} )^{2} \))(\( ( 1 + 6 T - 95 T^{2} - 182 T^{3} + 6266 T^{4} - 10162 T^{5} - 492559 T^{6} - 619882 T^{7} + 23315786 T^{8} - 41310542 T^{9} - 1315354895 T^{10} + 5067577806 T^{11} + 51520374361 T^{12} )^{2} \))(\( ( 1 + 8 T - 80 T^{2} - 376 T^{3} + 3998 T^{4} - 13396 T^{5} - 247632 T^{6} + 798752 T^{7} + 15355491 T^{8} + 48723872 T^{9} - 921438672 T^{10} - 3040637476 T^{11} + 55355672318 T^{12} - 317568209176 T^{13} - 4121629948880 T^{14} + 25141942688168 T^{15} + 191707312997281 T^{16} )^{2} \))(\( ( 1 - 36 T^{2} + 1530 T^{4} + 276336 T^{6} - 18613501 T^{8} + 1028246256 T^{10} + 21184136730 T^{12} - 1854733476996 T^{14} + 191707312997281 T^{16} )^{2} \))
$67$ (\( ( 1 - 109 T^{2} + 4489 T^{4} )( 1 - 13 T^{2} + 4489 T^{4} ) \))(\( ( 1 - 109 T^{2} + 4489 T^{4} )( 1 - 13 T^{2} + 4489 T^{4} ) \))(\( 1 + 60 T^{2} - 3206 T^{4} - 130320 T^{6} + 10010115 T^{8} - 585006480 T^{10} - 64604493926 T^{12} + 5427502930140 T^{14} + 406067677556641 T^{16} \))(\( 1 + 60 T^{2} - 3206 T^{4} - 130320 T^{6} + 10010115 T^{8} - 585006480 T^{10} - 64604493926 T^{12} + 5427502930140 T^{14} + 406067677556641 T^{16} \))(\( 1 + 274 T^{2} + 38973 T^{4} + 3966278 T^{6} + 329505914 T^{8} + 23681216154 T^{10} + 1596172229125 T^{12} + 106304979315306 T^{14} + 6639913543229594 T^{16} + 358783091112496982 T^{18} + 15825675597414969693 T^{20} + \)\(49\!\cdots\!26\)\( T^{22} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 + 274 T^{2} + 38973 T^{4} + 3966278 T^{6} + 329505914 T^{8} + 23681216154 T^{10} + 1596172229125 T^{12} + 106304979315306 T^{14} + 6639913543229594 T^{16} + 358783091112496982 T^{18} + 15825675597414969693 T^{20} + \)\(49\!\cdots\!26\)\( T^{22} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 + 356 T^{2} + 63502 T^{4} + 8090208 T^{6} + 858425049 T^{8} + 80231005476 T^{10} + 6774300927054 T^{12} + 525083738078264 T^{14} + 37066264561921156 T^{16} + 2357100900233327096 T^{18} + \)\(13\!\cdots\!34\)\( T^{20} + \)\(72\!\cdots\!44\)\( T^{22} + \)\(34\!\cdots\!09\)\( T^{24} + \)\(14\!\cdots\!92\)\( T^{26} + \)\(51\!\cdots\!22\)\( T^{28} + \)\(13\!\cdots\!24\)\( T^{30} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( ( 1 + 156 T^{2} + 9594 T^{4} + 899184 T^{6} + 87931139 T^{8} + 4036436976 T^{10} + 193329854874 T^{12} + 14111507618364 T^{14} + 406067677556641 T^{16} )^{2} \))
$71$ (\( ( 1 - 10 T + 71 T^{2} )^{4} \))(\( ( 1 - 10 T + 71 T^{2} )^{4} \))(\( ( 1 - 8 T + 50 T^{2} - 568 T^{3} + 5041 T^{4} )^{4} \))(\( ( 1 - 8 T + 50 T^{2} - 568 T^{3} + 5041 T^{4} )^{4} \))(\( ( 1 - 2 T + 71 T^{2} )^{12} \))(\( ( 1 - 2 T + 71 T^{2} )^{12} \))(\( ( 1 + 14 T + 194 T^{2} + 1648 T^{3} + 14264 T^{4} + 117008 T^{5} + 977954 T^{6} + 5010754 T^{7} + 25411681 T^{8} )^{4} \))(\( ( 1 + 8 T + 138 T^{2} + 568 T^{3} + 5041 T^{4} )^{8} \))
$73$ (\( ( 1 - 16 T + 183 T^{2} - 1168 T^{3} + 5329 T^{4} )( 1 + 16 T + 183 T^{2} + 1168 T^{3} + 5329 T^{4} ) \))(\( ( 1 - 16 T + 183 T^{2} - 1168 T^{3} + 5329 T^{4} )( 1 + 16 T + 183 T^{2} + 1168 T^{3} + 5329 T^{4} ) \))(\( ( 1 + 24 T + 370 T^{2} + 4272 T^{3} + 40371 T^{4} + 311856 T^{5} + 1971730 T^{6} + 9336408 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 - 24 T + 370 T^{2} - 4272 T^{3} + 40371 T^{4} - 311856 T^{5} + 1971730 T^{6} - 9336408 T^{7} + 28398241 T^{8} )^{2} \))(\( 1 + 298 T^{2} + 45029 T^{4} + 5080382 T^{6} + 496253018 T^{8} + 43754080818 T^{10} + 3427677334845 T^{12} + 233165496679122 T^{14} + 14092712802141338 T^{16} + 768835679222562398 T^{18} + 36314091477898573349 T^{20} + \)\(12\!\cdots\!02\)\( T^{22} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 + 298 T^{2} + 45029 T^{4} + 5080382 T^{6} + 496253018 T^{8} + 43754080818 T^{10} + 3427677334845 T^{12} + 233165496679122 T^{14} + 14092712802141338 T^{16} + 768835679222562398 T^{18} + 36314091477898573349 T^{20} + \)\(12\!\cdots\!02\)\( T^{22} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 + 312 T^{2} + 46894 T^{4} + 4662008 T^{6} + 358067385 T^{8} + 22613492772 T^{10} + 1186535858958 T^{12} + 57781939528716 T^{14} + 3463839236542756 T^{16} + 307919955748527564 T^{18} + 33695531277831292878 T^{20} + \)\(34\!\cdots\!08\)\( T^{22} + \)\(28\!\cdots\!85\)\( T^{24} + \)\(20\!\cdots\!92\)\( T^{26} + \)\(10\!\cdots\!74\)\( T^{28} + \)\(38\!\cdots\!08\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( ( 1 + 232 T^{2} + 30210 T^{4} + 3005792 T^{6} + 242394659 T^{8} + 16017865568 T^{10} + 857910860610 T^{12} + 35109540499048 T^{14} + 806460091894081 T^{16} )^{2} \))
$79$ (\( ( 1 - 17 T + 79 T^{2} )^{2}( 1 + 13 T + 79 T^{2} )^{2} \))(\( ( 1 - 17 T + 79 T^{2} )^{2}( 1 + 13 T + 79 T^{2} )^{2} \))(\( ( 1 + 16 T + 46 T^{2} + 832 T^{3} + 17635 T^{4} + 65728 T^{5} + 287086 T^{6} + 7888624 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 + 16 T + 46 T^{2} + 832 T^{3} + 17635 T^{4} + 65728 T^{5} + 287086 T^{6} + 7888624 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 - 12 T - 77 T^{2} + 500 T^{3} + 12470 T^{4} - 4172 T^{5} - 1287289 T^{6} - 329588 T^{7} + 77825270 T^{8} + 246519500 T^{9} - 2999156237 T^{10} - 36924676788 T^{11} + 243087455521 T^{12} )^{2} \))(\( ( 1 - 12 T - 77 T^{2} + 500 T^{3} + 12470 T^{4} - 4172 T^{5} - 1287289 T^{6} - 329588 T^{7} + 77825270 T^{8} + 246519500 T^{9} - 2999156237 T^{10} - 36924676788 T^{11} + 243087455521 T^{12} )^{2} \))(\( ( 1 + 8 T - 98 T^{2} - 1128 T^{3} - 3647 T^{4} - 31028 T^{5} - 546546 T^{6} + 5486636 T^{7} + 152161556 T^{8} + 433444244 T^{9} - 3410993586 T^{10} - 15298014092 T^{11} - 142050945407 T^{12} - 3470919618072 T^{13} - 23822570641058 T^{14} + 153631271889272 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 - 8 T - 30 T^{2} + 512 T^{3} - 2461 T^{4} + 40448 T^{5} - 187230 T^{6} - 3944312 T^{7} + 38950081 T^{8} )^{4} \))
$83$ (\( ( 1 - 102 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 102 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 130 T^{2} + 6889 T^{4} )^{4} \))(\( ( 1 - 130 T^{2} + 6889 T^{4} )^{4} \))(\( ( 1 - 306 T^{2} + 47783 T^{4} - 4793948 T^{6} + 329177087 T^{8} - 14522246226 T^{10} + 326940373369 T^{12} )^{2} \))(\( ( 1 - 306 T^{2} + 47783 T^{4} - 4793948 T^{6} + 329177087 T^{8} - 14522246226 T^{10} + 326940373369 T^{12} )^{2} \))(\( ( 1 - 516 T^{2} + 125904 T^{4} - 18859016 T^{6} + 1891189930 T^{8} - 129919761224 T^{10} + 5975192447184 T^{12} - 168701232658404 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 - 284 T^{2} + 33622 T^{4} - 1956476 T^{6} + 47458321 T^{8} )^{4} \))
$89$ (\( ( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} )^{2} \))(\( 1 - 100 T^{2} + 3946 T^{4} + 978800 T^{6} - 105301325 T^{8} + 7753074800 T^{10} + 247580882986 T^{12} - 49698129096100 T^{14} + 3936588805702081 T^{16} \))(\( 1 - 100 T^{2} + 3946 T^{4} + 978800 T^{6} - 105301325 T^{8} + 7753074800 T^{10} + 247580882986 T^{12} - 49698129096100 T^{14} + 3936588805702081 T^{16} \))(\( ( 1 - 14 T - 123 T^{2} + 598 T^{3} + 37922 T^{4} - 123654 T^{5} - 2788283 T^{6} - 11005206 T^{7} + 300380162 T^{8} + 421571462 T^{9} - 7717295643 T^{10} - 78176832286 T^{11} + 496981290961 T^{12} )^{2} \))(\( ( 1 + 14 T - 123 T^{2} - 598 T^{3} + 37922 T^{4} + 123654 T^{5} - 2788283 T^{6} + 11005206 T^{7} + 300380162 T^{8} - 421571462 T^{9} - 7717295643 T^{10} + 78176832286 T^{11} + 496981290961 T^{12} )^{2} \))(\( ( 1 + 8 T - 98 T^{2} - 1004 T^{3} + 4584 T^{4} + 89290 T^{5} + 1051584 T^{6} - 2927032 T^{7} - 142835833 T^{8} - 260505848 T^{9} + 8329596864 T^{10} + 62946682010 T^{11} + 287610432744 T^{12} - 5606395686796 T^{13} - 48704166514178 T^{14} + 353850679164232 T^{15} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 - 104 T^{2} - 6110 T^{4} - 112736 T^{6} + 138073699 T^{8} - 892981856 T^{10} - 383355092510 T^{12} - 51686054259944 T^{14} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( ( 1 - 190 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 190 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 4 T^{2} - 8826 T^{4} - 37636 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 - 4 T^{2} - 8826 T^{4} - 37636 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 - 26 T^{2} + 8719 T^{4} + 446932 T^{6} + 82037071 T^{8} - 2301761306 T^{10} + 832972004929 T^{12} )^{2} \))(\( ( 1 - 26 T^{2} + 8719 T^{4} + 446932 T^{6} + 82037071 T^{8} - 2301761306 T^{10} + 832972004929 T^{12} )^{2} \))(\( ( 1 - 756 T^{2} + 251932 T^{4} - 48333356 T^{6} + 5838717718 T^{8} - 454768546604 T^{10} + 22303358820892 T^{12} - 629726835726324 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 - 136 T^{2} + 22942 T^{4} - 1279624 T^{6} + 88529281 T^{8} )^{4} \))
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