Properties

Label 735.2.s
Level 735
Weight 2
Character orbit s
Rep. character \(\chi_{735}(521,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 108
Newform subspaces 14
Sturm bound 224
Trace bound 6

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Defining parameters

Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.s (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 14 \)
Sturm bound: \(224\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\), \(13\), \(17\)

Dimensions

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

Total New Old
Modular forms 256 108 148
Cusp forms 192 108 84
Eisenstein series 64 0 64

Trace form

\( 108q + 56q^{4} + 6q^{9} + O(q^{10}) \) \( 108q + 56q^{4} + 6q^{9} + 30q^{12} - 8q^{15} - 92q^{16} + 2q^{18} + 6q^{19} + 88q^{22} - 18q^{24} - 54q^{25} - 2q^{30} + 54q^{31} - 24q^{33} + 4q^{36} + 22q^{37} - 108q^{43} + 18q^{45} + 28q^{46} + 10q^{51} - 48q^{52} - 24q^{54} - 52q^{57} + 56q^{58} - 2q^{60} - 12q^{61} - 264q^{64} + 12q^{66} - 86q^{67} + 38q^{72} - 42q^{73} + 24q^{78} + 18q^{79} + 14q^{81} - 72q^{82} + 48q^{85} + 6q^{87} + 32q^{88} - 66q^{93} + 48q^{94} + 18q^{96} - 68q^{99} + O(q^{100}) \)

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.s.a \(2\) \(5.869\) \(\Q(\sqrt{-3}) \) None \(-3\) \(-3\) \(-1\) \(0\) \(q+(-2+\zeta_{6})q^{2}+(-1-\zeta_{6})q^{3}+(1+\cdots)q^{4}+\cdots\)
735.2.s.b \(2\) \(5.869\) \(\Q(\sqrt{-3}) \) None \(-3\) \(3\) \(1\) \(0\) \(q+(-2+\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.s.c \(2\) \(5.869\) \(\Q(\sqrt{-3}) \) None \(-3\) \(3\) \(1\) \(0\) \(q+(-2+\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.s.d \(2\) \(5.869\) \(\Q(\sqrt{-3}) \) None \(3\) \(-3\) \(1\) \(0\) \(q+(2-\zeta_{6})q^{2}+(-1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.s.e \(2\) \(5.869\) \(\Q(\sqrt{-3}) \) None \(3\) \(0\) \(-1\) \(0\) \(q+(2-\zeta_{6})q^{2}+(-1+2\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.s.f \(2\) \(5.869\) \(\Q(\sqrt{-3}) \) None \(3\) \(3\) \(-1\) \(0\) \(q+(2-\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.s.g \(4\) \(5.869\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(-3\) \(-2\) \(-2\) \(0\) \(q+(-1-\beta _{3})q^{2}+(-1+\beta _{1}-\beta _{3})q^{3}+\cdots\)
735.2.s.h \(4\) \(5.869\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(-3\) \(2\) \(2\) \(0\) \(q+(-1-\beta _{3})q^{2}+(1-\beta _{1}+\beta _{3})q^{3}+\cdots\)
735.2.s.i \(4\) \(5.869\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(3\) \(-1\) \(2\) \(0\) \(q+(1-\beta _{1})q^{2}-\beta _{1}q^{3}+(1-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
735.2.s.j \(4\) \(5.869\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(3\) \(1\) \(-2\) \(0\) \(q+(1-\beta _{1})q^{2}+\beta _{1}q^{3}+(1-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
735.2.s.k \(8\) \(5.869\) 8.0.856615824.2 None \(-3\) \(-1\) \(-4\) \(0\) \(q+(\beta _{1}+\beta _{3})q^{2}+(-1+\beta _{1}-\beta _{4}-\beta _{6}+\cdots)q^{3}+\cdots\)
735.2.s.l \(8\) \(5.869\) 8.0.856615824.2 None \(3\) \(-2\) \(4\) \(0\) \(q-\beta _{3}q^{2}+(-1-\beta _{3}-\beta _{6})q^{3}+(-1+\cdots)q^{4}+\cdots\)
735.2.s.m \(32\) \(5.869\) None \(0\) \(-4\) \(16\) \(0\)
735.2.s.n \(32\) \(5.869\) None \(0\) \(4\) \(-16\) \(0\)

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

\( S_{2}^{\mathrm{old}}(735, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \))(\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \))(\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \))(\( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} \))(\( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} \))(\( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} \))(\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 6 T^{4} + 12 T^{5} + 20 T^{6} + 24 T^{7} + 16 T^{8} \))(\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 6 T^{4} + 12 T^{5} + 20 T^{6} + 24 T^{7} + 16 T^{8} \))(\( 1 - 3 T + 5 T^{2} - 6 T^{3} + 6 T^{4} - 12 T^{5} + 20 T^{6} - 24 T^{7} + 16 T^{8} \))(\( 1 - 3 T + 5 T^{2} - 6 T^{3} + 6 T^{4} - 12 T^{5} + 20 T^{6} - 24 T^{7} + 16 T^{8} \))(\( 1 + 3 T + 7 T^{2} + 12 T^{3} + 16 T^{4} + 12 T^{5} - 8 T^{6} - 36 T^{7} - 68 T^{8} - 72 T^{9} - 32 T^{10} + 96 T^{11} + 256 T^{12} + 384 T^{13} + 448 T^{14} + 384 T^{15} + 256 T^{16} \))(\( 1 - 3 T + 7 T^{2} - 12 T^{3} + 16 T^{4} - 12 T^{5} - 8 T^{6} + 36 T^{7} - 68 T^{8} + 72 T^{9} - 32 T^{10} - 96 T^{11} + 256 T^{12} - 384 T^{13} + 448 T^{14} - 384 T^{15} + 256 T^{16} \))
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( ( 1 + T + 3 T^{2} )^{2} \))(\( ( 1 - T + 3 T^{2} )^{2} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( 1 + T + 3 T^{2} + 4 T^{3} + 16 T^{4} + 12 T^{5} + 27 T^{6} + 27 T^{7} + 81 T^{8} \))(\( 1 + 2 T - 4 T^{3} - 11 T^{4} - 12 T^{5} + 54 T^{7} + 81 T^{8} \))
$5$ (\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{4} \))
$7$ 1
$11$ (\( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 + 3 T + 23 T^{2} + 60 T^{3} + 312 T^{4} + 660 T^{5} + 2783 T^{6} + 3993 T^{7} + 14641 T^{8} \))(\( 1 + 3 T + 23 T^{2} + 60 T^{3} + 312 T^{4} + 660 T^{5} + 2783 T^{6} + 3993 T^{7} + 14641 T^{8} \))(\( 1 - 3 T + 23 T^{2} - 60 T^{3} + 312 T^{4} - 660 T^{5} + 2783 T^{6} - 3993 T^{7} + 14641 T^{8} \))(\( 1 - 3 T + 23 T^{2} - 60 T^{3} + 312 T^{4} - 660 T^{5} + 2783 T^{6} - 3993 T^{7} + 14641 T^{8} \))(\( 1 + 16 T^{2} - 2 T^{4} + 30 T^{5} + 268 T^{6} + 1548 T^{7} + 21079 T^{8} + 17028 T^{9} + 32428 T^{10} + 39930 T^{11} - 29282 T^{12} + 28344976 T^{14} + 214358881 T^{16} \))(\( 1 + 16 T^{2} - 2 T^{4} - 30 T^{5} + 268 T^{6} - 1548 T^{7} + 21079 T^{8} - 17028 T^{9} + 32428 T^{10} - 39930 T^{11} - 29282 T^{12} + 28344976 T^{14} + 214358881 T^{16} \))
$13$ (\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( 1 - T^{2} + 264 T^{4} - 169 T^{6} + 28561 T^{8} \))(\( 1 - T^{2} + 264 T^{4} - 169 T^{6} + 28561 T^{8} \))(\( 1 - T^{2} + 264 T^{4} - 169 T^{6} + 28561 T^{8} \))(\( 1 - T^{2} + 264 T^{4} - 169 T^{6} + 28561 T^{8} \))(\( 1 - 83 T^{2} + 3217 T^{4} - 76058 T^{6} + 1197778 T^{8} - 12853802 T^{10} + 91880737 T^{12} - 400625147 T^{14} + 815730721 T^{16} \))(\( 1 - 83 T^{2} + 3217 T^{4} - 76058 T^{6} + 1197778 T^{8} - 12853802 T^{10} + 91880737 T^{12} - 400625147 T^{14} + 815730721 T^{16} \))
$17$ (\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 19 T^{2} - 18 T^{3} + 342 T^{4} - 306 T^{5} - 5491 T^{6} + 14739 T^{7} + 83521 T^{8} \))(\( 1 - 3 T - 19 T^{2} + 18 T^{3} + 342 T^{4} + 306 T^{5} - 5491 T^{6} - 14739 T^{7} + 83521 T^{8} \))(\( 1 - 3 T - 19 T^{2} + 18 T^{3} + 342 T^{4} + 306 T^{5} - 5491 T^{6} - 14739 T^{7} + 83521 T^{8} \))(\( 1 + 3 T - 19 T^{2} - 18 T^{3} + 342 T^{4} - 306 T^{5} - 5491 T^{6} + 14739 T^{7} + 83521 T^{8} \))(\( 1 - 12 T + 34 T^{2} - 12 T^{3} + 1078 T^{4} - 6882 T^{5} + 8740 T^{6} - 70272 T^{7} + 637627 T^{8} - 1194624 T^{9} + 2525860 T^{10} - 33811266 T^{11} + 90035638 T^{12} - 17038284 T^{13} + 820677346 T^{14} - 4924064076 T^{15} + 6975757441 T^{16} \))(\( 1 + 12 T + 34 T^{2} + 12 T^{3} + 1078 T^{4} + 6882 T^{5} + 8740 T^{6} + 70272 T^{7} + 637627 T^{8} + 1194624 T^{9} + 2525860 T^{10} + 33811266 T^{11} + 90035638 T^{12} + 17038284 T^{13} + 820677346 T^{14} + 4924064076 T^{15} + 6975757441 T^{16} \))
$19$ (\( ( 1 - 7 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( ( 1 - T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 - 12 T + 67 T^{2} - 228 T^{3} + 361 T^{4} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( 1 - 12 T + 67 T^{2} - 228 T^{3} + 361 T^{4} \))(\( ( 1 - T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2} \))(\( ( 1 - T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2} \))(\( 1 + 9 T + 100 T^{2} + 657 T^{3} + 4723 T^{4} + 26244 T^{5} + 148996 T^{6} + 704196 T^{7} + 3331528 T^{8} + 13379724 T^{9} + 53787556 T^{10} + 180007596 T^{11} + 615506083 T^{12} + 1626797043 T^{13} + 4704588100 T^{14} + 8044845651 T^{15} + 16983563041 T^{16} \))(\( 1 + 9 T + 100 T^{2} + 657 T^{3} + 4723 T^{4} + 26244 T^{5} + 148996 T^{6} + 704196 T^{7} + 3331528 T^{8} + 13379724 T^{9} + 53787556 T^{10} + 180007596 T^{11} + 615506083 T^{12} + 1626797043 T^{13} + 4704588100 T^{14} + 8044845651 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 + 6 T + 35 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 35 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 3 T + 26 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 35 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 3 T + 26 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 35 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 18 T + 170 T^{2} + 1116 T^{3} + 5799 T^{4} + 25668 T^{5} + 89930 T^{6} + 219006 T^{7} + 279841 T^{8} \))(\( 1 + 18 T + 170 T^{2} + 1116 T^{3} + 5799 T^{4} + 25668 T^{5} + 89930 T^{6} + 219006 T^{7} + 279841 T^{8} \))(\( 1 - 18 T + 170 T^{2} - 1116 T^{3} + 5799 T^{4} - 25668 T^{5} + 89930 T^{6} - 219006 T^{7} + 279841 T^{8} \))(\( 1 - 18 T + 170 T^{2} - 1116 T^{3} + 5799 T^{4} - 25668 T^{5} + 89930 T^{6} - 219006 T^{7} + 279841 T^{8} \))(\( 1 - 27 T + 391 T^{2} - 3996 T^{3} + 31651 T^{4} - 205875 T^{5} + 1157938 T^{6} - 5917779 T^{7} + 28782226 T^{8} - 136108917 T^{9} + 612549202 T^{10} - 2504881125 T^{11} + 8857247491 T^{12} - 25719626628 T^{13} + 57882032599 T^{14} - 91930287069 T^{15} + 78310985281 T^{16} \))(\( 1 + 27 T + 391 T^{2} + 3996 T^{3} + 31651 T^{4} + 205875 T^{5} + 1157938 T^{6} + 5917779 T^{7} + 28782226 T^{8} + 136108917 T^{9} + 612549202 T^{10} + 2504881125 T^{11} + 8857247491 T^{12} + 25719626628 T^{13} + 57882032599 T^{14} + 91930287069 T^{15} + 78310985281 T^{16} \))
$29$ (\( 1 - 10 T^{2} + 841 T^{4} \))(\( 1 - 10 T^{2} + 841 T^{4} \))(\( 1 - 55 T^{2} + 841 T^{4} \))(\( 1 - 10 T^{2} + 841 T^{4} \))(\( 1 - 55 T^{2} + 841 T^{4} \))(\( 1 - 10 T^{2} + 841 T^{4} \))(\( 1 - 97 T^{2} + 3960 T^{4} - 81577 T^{6} + 707281 T^{8} \))(\( 1 - 97 T^{2} + 3960 T^{4} - 81577 T^{6} + 707281 T^{8} \))(\( 1 - 97 T^{2} + 3960 T^{4} - 81577 T^{6} + 707281 T^{8} \))(\( 1 - 97 T^{2} + 3960 T^{4} - 81577 T^{6} + 707281 T^{8} \))(\( 1 - 53 T^{2} + 3250 T^{4} - 128951 T^{6} + 4063174 T^{8} - 108447791 T^{10} + 2298663250 T^{12} - 31525636013 T^{14} + 500246412961 T^{16} \))(\( 1 - 53 T^{2} + 3250 T^{4} - 128951 T^{6} + 4063174 T^{8} - 108447791 T^{10} + 2298663250 T^{12} - 31525636013 T^{14} + 500246412961 T^{16} \))
$31$ (\( 1 - 6 T + 43 T^{2} - 186 T^{3} + 961 T^{4} \))(\( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} \))(\( 1 - 6 T + 43 T^{2} - 186 T^{3} + 961 T^{4} \))(\( 1 - 6 T + 43 T^{2} - 186 T^{3} + 961 T^{4} \))(\( 1 - 6 T + 43 T^{2} - 186 T^{3} + 961 T^{4} \))(\( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} \))(\( ( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 6 T + 43 T^{2} - 186 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 6 T + 43 T^{2} - 186 T^{3} + 961 T^{4} )^{2} \))(\( 1 - 21 T + 262 T^{2} - 2415 T^{3} + 17293 T^{4} - 101304 T^{5} + 505090 T^{6} - 2328618 T^{7} + 11769748 T^{8} - 72187158 T^{9} + 485391490 T^{10} - 3017947464 T^{11} + 15970448653 T^{12} - 69139399665 T^{13} + 232525964422 T^{14} - 577764896331 T^{15} + 852891037441 T^{16} \))(\( 1 - 21 T + 262 T^{2} - 2415 T^{3} + 17293 T^{4} - 101304 T^{5} + 505090 T^{6} - 2328618 T^{7} + 11769748 T^{8} - 72187158 T^{9} + 485391490 T^{10} - 3017947464 T^{11} + 15970448653 T^{12} - 69139399665 T^{13} + 232525964422 T^{14} - 577764896331 T^{15} + 852891037441 T^{16} \))
$37$ (\( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( 1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( 1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( 1 - 2 T - 38 T^{2} + 64 T^{3} + 247 T^{4} + 2368 T^{5} - 52022 T^{6} - 101306 T^{7} + 1874161 T^{8} \))(\( 1 - 2 T - 38 T^{2} + 64 T^{3} + 247 T^{4} + 2368 T^{5} - 52022 T^{6} - 101306 T^{7} + 1874161 T^{8} \))(\( 1 - 2 T - 38 T^{2} + 64 T^{3} + 247 T^{4} + 2368 T^{5} - 52022 T^{6} - 101306 T^{7} + 1874161 T^{8} \))(\( 1 - 2 T - 38 T^{2} + 64 T^{3} + 247 T^{4} + 2368 T^{5} - 52022 T^{6} - 101306 T^{7} + 1874161 T^{8} \))(\( 1 - 7 T - 24 T^{2} + 493 T^{3} - 973 T^{4} - 16188 T^{5} + 118336 T^{6} + 258098 T^{7} - 5756772 T^{8} + 9549626 T^{9} + 162001984 T^{10} - 819970764 T^{11} - 1823558653 T^{12} + 34186570801 T^{13} - 61577433816 T^{14} - 664523139931 T^{15} + 3512479453921 T^{16} \))(\( 1 - 7 T - 24 T^{2} + 493 T^{3} - 973 T^{4} - 16188 T^{5} + 118336 T^{6} + 258098 T^{7} - 5756772 T^{8} + 9549626 T^{9} + 162001984 T^{10} - 819970764 T^{11} - 1823558653 T^{12} + 34186570801 T^{13} - 61577433816 T^{14} - 664523139931 T^{15} + 3512479453921 T^{16} \))
$41$ (\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 3 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{4} \))(\( ( 1 - 6 T + 41 T^{2} )^{4} \))(\( ( 1 - 6 T + 41 T^{2} )^{4} \))(\( ( 1 + 6 T + 41 T^{2} )^{4} \))(\( ( 1 - 15 T + 218 T^{2} - 1791 T^{3} + 14136 T^{4} - 73431 T^{5} + 366458 T^{6} - 1033815 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + 15 T + 218 T^{2} + 1791 T^{3} + 14136 T^{4} + 73431 T^{5} + 366458 T^{6} + 1033815 T^{7} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 - T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 - T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 2 T + 54 T^{2} + 86 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 2 T + 54 T^{2} + 86 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 2 T + 54 T^{2} + 86 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 2 T + 54 T^{2} + 86 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 8 T + 184 T^{2} - 1022 T^{3} + 12127 T^{4} - 43946 T^{5} + 340216 T^{6} - 636056 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 - 8 T + 184 T^{2} - 1022 T^{3} + 12127 T^{4} - 43946 T^{5} + 340216 T^{6} - 636056 T^{7} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 9 T - 25 T^{2} - 108 T^{3} + 5220 T^{4} - 5076 T^{5} - 55225 T^{6} - 934407 T^{7} + 4879681 T^{8} \))(\( 1 + 9 T - 25 T^{2} + 108 T^{3} + 5220 T^{4} + 5076 T^{5} - 55225 T^{6} + 934407 T^{7} + 4879681 T^{8} \))(\( 1 + 9 T - 25 T^{2} + 108 T^{3} + 5220 T^{4} + 5076 T^{5} - 55225 T^{6} + 934407 T^{7} + 4879681 T^{8} \))(\( 1 - 9 T - 25 T^{2} - 108 T^{3} + 5220 T^{4} - 5076 T^{5} - 55225 T^{6} - 934407 T^{7} + 4879681 T^{8} \))(\( 1 - 6 T - 116 T^{2} + 252 T^{3} + 10126 T^{4} - 1986 T^{5} - 595736 T^{6} + 157218 T^{7} + 25623007 T^{8} + 7389246 T^{9} - 1315980824 T^{10} - 206192478 T^{11} + 49411649806 T^{12} + 57794941764 T^{13} - 1250388978164 T^{14} - 3039738722778 T^{15} + 23811286661761 T^{16} \))(\( 1 + 6 T - 116 T^{2} - 252 T^{3} + 10126 T^{4} + 1986 T^{5} - 595736 T^{6} - 157218 T^{7} + 25623007 T^{8} - 7389246 T^{9} - 1315980824 T^{10} + 206192478 T^{11} + 49411649806 T^{12} - 57794941764 T^{13} - 1250388978164 T^{14} + 3039738722778 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 18 T + 230 T^{2} + 2196 T^{3} + 17799 T^{4} + 116388 T^{5} + 646070 T^{6} + 2679786 T^{7} + 7890481 T^{8} \))(\( 1 + 18 T + 230 T^{2} + 2196 T^{3} + 17799 T^{4} + 116388 T^{5} + 646070 T^{6} + 2679786 T^{7} + 7890481 T^{8} \))(\( 1 - 18 T + 230 T^{2} - 2196 T^{3} + 17799 T^{4} - 116388 T^{5} + 646070 T^{6} - 2679786 T^{7} + 7890481 T^{8} \))(\( 1 - 18 T + 230 T^{2} - 2196 T^{3} + 17799 T^{4} - 116388 T^{5} + 646070 T^{6} - 2679786 T^{7} + 7890481 T^{8} \))(\( 1 - 24 T + 340 T^{2} - 3552 T^{3} + 29050 T^{4} - 180120 T^{5} + 750160 T^{6} - 1659096 T^{7} + 1273315 T^{8} - 87932088 T^{9} + 2107199440 T^{10} - 26815725240 T^{11} + 229218473050 T^{12} - 1485430391136 T^{13} + 7535882783860 T^{14} - 28193067356088 T^{15} + 62259690411361 T^{16} \))(\( 1 + 24 T + 340 T^{2} + 3552 T^{3} + 29050 T^{4} + 180120 T^{5} + 750160 T^{6} + 1659096 T^{7} + 1273315 T^{8} + 87932088 T^{9} + 2107199440 T^{10} + 26815725240 T^{11} + 229218473050 T^{12} + 1485430391136 T^{13} + 7535882783860 T^{14} + 28193067356088 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 58 T^{2} + 144 T^{3} + 3231 T^{4} + 8496 T^{5} - 201898 T^{6} - 1232274 T^{7} + 12117361 T^{8} \))(\( 1 + 6 T - 58 T^{2} - 144 T^{3} + 3231 T^{4} - 8496 T^{5} - 201898 T^{6} + 1232274 T^{7} + 12117361 T^{8} \))(\( 1 + 6 T - 58 T^{2} - 144 T^{3} + 3231 T^{4} - 8496 T^{5} - 201898 T^{6} + 1232274 T^{7} + 12117361 T^{8} \))(\( 1 - 6 T - 58 T^{2} + 144 T^{3} + 3231 T^{4} + 8496 T^{5} - 201898 T^{6} - 1232274 T^{7} + 12117361 T^{8} \))(\( 1 - 12 T - 80 T^{2} + 1164 T^{3} + 7690 T^{4} - 80082 T^{5} - 434420 T^{6} + 1772232 T^{7} + 28861927 T^{8} + 104561688 T^{9} - 1512216020 T^{10} - 16447161078 T^{11} + 93182506090 T^{12} + 832171884036 T^{13} - 3374442691280 T^{14} - 29863817817828 T^{15} + 146830437604321 T^{16} \))(\( 1 + 12 T - 80 T^{2} - 1164 T^{3} + 7690 T^{4} + 80082 T^{5} - 434420 T^{6} - 1772232 T^{7} + 28861927 T^{8} - 104561688 T^{9} - 1512216020 T^{10} + 16447161078 T^{11} + 93182506090 T^{12} - 832171884036 T^{13} - 3374442691280 T^{14} + 29863817817828 T^{15} + 146830437604321 T^{16} \))
$61$ (\( ( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( 1 - 9 T + 88 T^{2} - 549 T^{3} + 3721 T^{4} \))(\( ( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 - 9 T + 88 T^{2} - 549 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( ( 1 - 13 T + 61 T^{2} )^{2}( 1 + T + 61 T^{2} )^{2} \))(\( ( 1 - T + 61 T^{2} )^{2}( 1 + 13 T + 61 T^{2} )^{2} \))(\( ( 1 - 13 T + 61 T^{2} )^{2}( 1 + T + 61 T^{2} )^{2} \))(\( ( 1 - T + 61 T^{2} )^{2}( 1 + 13 T + 61 T^{2} )^{2} \))(\( 1 + 15 T + 223 T^{2} + 2220 T^{3} + 19711 T^{4} + 141723 T^{5} + 816310 T^{6} + 5175267 T^{7} + 31433836 T^{8} + 315691287 T^{9} + 3037489510 T^{10} + 32168428263 T^{11} + 272915371951 T^{12} + 1875003788220 T^{13} + 11489043482503 T^{14} + 47141142540315 T^{15} + 191707312997281 T^{16} \))(\( 1 + 15 T + 223 T^{2} + 2220 T^{3} + 19711 T^{4} + 141723 T^{5} + 816310 T^{6} + 5175267 T^{7} + 31433836 T^{8} + 315691287 T^{9} + 3037489510 T^{10} + 32168428263 T^{11} + 272915371951 T^{12} + 1875003788220 T^{13} + 11489043482503 T^{14} + 47141142540315 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} \))(\( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} \))(\( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 - 2 T - 98 T^{2} + 64 T^{3} + 5647 T^{4} + 4288 T^{5} - 439922 T^{6} - 601526 T^{7} + 20151121 T^{8} \))(\( 1 - 2 T - 98 T^{2} + 64 T^{3} + 5647 T^{4} + 4288 T^{5} - 439922 T^{6} - 601526 T^{7} + 20151121 T^{8} \))(\( 1 - 2 T - 98 T^{2} + 64 T^{3} + 5647 T^{4} + 4288 T^{5} - 439922 T^{6} - 601526 T^{7} + 20151121 T^{8} \))(\( 1 - 2 T - 98 T^{2} + 64 T^{3} + 5647 T^{4} + 4288 T^{5} - 439922 T^{6} - 601526 T^{7} + 20151121 T^{8} \))(\( 1 - 4 T - 234 T^{2} + 412 T^{3} + 35255 T^{4} - 28434 T^{5} - 3551522 T^{6} + 717722 T^{7} + 271900824 T^{8} + 48087374 T^{9} - 15942782258 T^{10} - 8551895142 T^{11} + 710427770855 T^{12} + 556251544084 T^{13} - 21167261427546 T^{14} - 24242846421292 T^{15} + 406067677556641 T^{16} \))(\( 1 - 4 T - 234 T^{2} + 412 T^{3} + 35255 T^{4} - 28434 T^{5} - 3551522 T^{6} + 717722 T^{7} + 271900824 T^{8} + 48087374 T^{9} - 15942782258 T^{10} - 8551895142 T^{11} + 710427770855 T^{12} + 556251544084 T^{13} - 21167261427546 T^{14} - 24242846421292 T^{15} + 406067677556641 T^{16} \))
$71$ (\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 94 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 94 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 100 T^{2} + 4134 T^{4} - 504100 T^{6} + 25411681 T^{8} \))(\( 1 - 100 T^{2} + 4134 T^{4} - 504100 T^{6} + 25411681 T^{8} \))(\( 1 - 100 T^{2} + 4134 T^{4} - 504100 T^{6} + 25411681 T^{8} \))(\( 1 - 100 T^{2} + 4134 T^{4} - 504100 T^{6} + 25411681 T^{8} \))(\( 1 - 464 T^{2} + 99532 T^{4} - 12936548 T^{6} + 1114829374 T^{8} - 65213138468 T^{10} + 2529275433292 T^{12} - 59438531739344 T^{14} + 645753531245761 T^{16} \))(\( 1 - 464 T^{2} + 99532 T^{4} - 12936548 T^{6} + 1114829374 T^{8} - 65213138468 T^{10} + 2529275433292 T^{12} - 59438531739344 T^{14} + 645753531245761 T^{16} \))
$73$ (\( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} \))(\( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} \))(\( 1 + 6 T + 85 T^{2} + 438 T^{3} + 5329 T^{4} \))(\( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} \))(\( 1 + 6 T + 85 T^{2} + 438 T^{3} + 5329 T^{4} \))(\( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} \))(\( ( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} )^{2} \))(\( 1 + 15 T + 280 T^{2} + 3075 T^{3} + 38779 T^{4} + 422928 T^{5} + 4017052 T^{6} + 39506334 T^{7} + 310273396 T^{8} + 2883962382 T^{9} + 21406870108 T^{10} + 164526181776 T^{11} + 1101255387739 T^{12} + 6374695148475 T^{13} + 42373583360920 T^{14} + 165710977786455 T^{15} + 806460091894081 T^{16} \))(\( 1 + 15 T + 280 T^{2} + 3075 T^{3} + 38779 T^{4} + 422928 T^{5} + 4017052 T^{6} + 39506334 T^{7} + 310273396 T^{8} + 2883962382 T^{9} + 21406870108 T^{10} + 164526181776 T^{11} + 1101255387739 T^{12} + 6374695148475 T^{13} + 42373583360920 T^{14} + 165710977786455 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + T - 149 T^{2} - 8 T^{3} + 16180 T^{4} - 632 T^{5} - 929909 T^{6} + 493039 T^{7} + 38950081 T^{8} \))(\( 1 + T - 149 T^{2} - 8 T^{3} + 16180 T^{4} - 632 T^{5} - 929909 T^{6} + 493039 T^{7} + 38950081 T^{8} \))(\( 1 + T - 149 T^{2} - 8 T^{3} + 16180 T^{4} - 632 T^{5} - 929909 T^{6} + 493039 T^{7} + 38950081 T^{8} \))(\( 1 + T - 149 T^{2} - 8 T^{3} + 16180 T^{4} - 632 T^{5} - 929909 T^{6} + 493039 T^{7} + 38950081 T^{8} \))(\( 1 + 29 T + 294 T^{2} + 1975 T^{3} + 27377 T^{4} + 260496 T^{5} + 598654 T^{6} + 2403434 T^{7} + 77714340 T^{8} + 189871286 T^{9} + 3736199614 T^{10} + 128434687344 T^{11} + 1066336367537 T^{12} + 6077186388025 T^{13} + 71467711923174 T^{14} + 556913360598611 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 29 T + 294 T^{2} + 1975 T^{3} + 27377 T^{4} + 260496 T^{5} + 598654 T^{6} + 2403434 T^{7} + 77714340 T^{8} + 189871286 T^{9} + 3736199614 T^{10} + 128434687344 T^{11} + 1066336367537 T^{12} + 6077186388025 T^{13} + 71467711923174 T^{14} + 556913360598611 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 - 9 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 9 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 70 T^{2} + 996 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 - 12 T + 70 T^{2} - 996 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 - 12 T + 70 T^{2} - 996 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 12 T + 70 T^{2} + 996 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 15 T + 380 T^{2} + 3759 T^{3} + 49260 T^{4} + 311997 T^{5} + 2617820 T^{6} + 8576805 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 - 15 T + 380 T^{2} - 3759 T^{3} + 49260 T^{4} - 311997 T^{5} + 2617820 T^{6} - 8576805 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 3 T - 80 T^{2} - 267 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 18 T + 98 T^{2} + 864 T^{3} + 14319 T^{4} + 76896 T^{5} + 776258 T^{6} + 12689442 T^{7} + 62742241 T^{8} \))(\( 1 - 18 T + 98 T^{2} - 864 T^{3} + 14319 T^{4} - 76896 T^{5} + 776258 T^{6} - 12689442 T^{7} + 62742241 T^{8} \))(\( 1 - 18 T + 98 T^{2} - 864 T^{3} + 14319 T^{4} - 76896 T^{5} + 776258 T^{6} - 12689442 T^{7} + 62742241 T^{8} \))(\( 1 + 18 T + 98 T^{2} + 864 T^{3} + 14319 T^{4} + 76896 T^{5} + 776258 T^{6} + 12689442 T^{7} + 62742241 T^{8} \))(\( 1 - 3 T - 53 T^{2} - 2820 T^{3} + 14227 T^{4} + 160275 T^{5} + 3467116 T^{6} - 26593569 T^{7} - 193500020 T^{8} - 2366827641 T^{9} + 27463025836 T^{10} + 112988906475 T^{11} + 892633862707 T^{12} - 15747047646180 T^{13} - 26340008420933 T^{14} - 132694004686587 T^{15} + 3936588805702081 T^{16} \))(\( 1 + 3 T - 53 T^{2} + 2820 T^{3} + 14227 T^{4} - 160275 T^{5} + 3467116 T^{6} + 26593569 T^{7} - 193500020 T^{8} + 2366827641 T^{9} + 27463025836 T^{10} - 112988906475 T^{11} + 892633862707 T^{12} + 15747047646180 T^{13} - 26340008420933 T^{14} + 132694004686587 T^{15} + 3936588805702081 T^{16} \))
$97$ (\( 1 - 146 T^{2} + 9409 T^{4} \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( 1 - 86 T^{2} + 9409 T^{4} \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( 1 - 86 T^{2} + 9409 T^{4} \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( 1 - 265 T^{2} + 32736 T^{4} - 2493385 T^{6} + 88529281 T^{8} \))(\( 1 - 265 T^{2} + 32736 T^{4} - 2493385 T^{6} + 88529281 T^{8} \))(\( 1 - 265 T^{2} + 32736 T^{4} - 2493385 T^{6} + 88529281 T^{8} \))(\( 1 - 265 T^{2} + 32736 T^{4} - 2493385 T^{6} + 88529281 T^{8} \))(\( 1 - 368 T^{2} + 81676 T^{4} - 12257504 T^{6} + 1385094598 T^{8} - 115330855136 T^{10} + 7230717554956 T^{12} - 306533697813872 T^{14} + 7837433594376961 T^{16} \))(\( 1 - 368 T^{2} + 81676 T^{4} - 12257504 T^{6} + 1385094598 T^{8} - 115330855136 T^{10} + 7230717554956 T^{12} - 306533697813872 T^{14} + 7837433594376961 T^{16} \))
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