Properties

Label 162.4.c
Level 162
Weight 4
Character orbit c
Rep. character \(\chi_{162}(55,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 24
Newform subspaces 10
Sturm bound 108
Trace bound 5

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Defining parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 162.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 10 \)
Sturm bound: \(108\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(162, [\chi])\).

Total New Old
Modular forms 186 24 162
Cusp forms 138 24 114
Eisenstein series 48 0 48

Trace form

\( 24q - 48q^{4} - 60q^{7} + O(q^{10}) \) \( 24q - 48q^{4} - 60q^{7} + 120q^{13} - 192q^{16} - 780q^{19} + 36q^{22} - 12q^{25} + 480q^{28} - 78q^{31} - 180q^{34} - 1392q^{37} - 132q^{43} - 1008q^{46} + 1008q^{49} + 480q^{52} + 4644q^{55} - 1260q^{58} + 84q^{61} + 1536q^{64} - 1914q^{67} - 540q^{70} - 924q^{73} + 1560q^{76} + 4602q^{79} + 3024q^{82} + 1026q^{85} + 144q^{88} - 5196q^{91} - 1224q^{94} + 120q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(162, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
162.4.c.a \(2\) \(9.558\) \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-12\) \(7\) \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-12\zeta_{6}q^{5}+\cdots\)
162.4.c.b \(2\) \(9.558\) \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-3\) \(-29\) \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-3\zeta_{6}q^{5}+\cdots\)
162.4.c.c \(2\) \(9.558\) \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(6\) \(16\) \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+6\zeta_{6}q^{5}+\cdots\)
162.4.c.d \(2\) \(9.558\) \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(21\) \(-8\) \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+21\zeta_{6}q^{5}+\cdots\)
162.4.c.e \(2\) \(9.558\) \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-21\) \(-8\) \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-21\zeta_{6}q^{5}+\cdots\)
162.4.c.f \(2\) \(9.558\) \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-6\) \(16\) \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-6\zeta_{6}q^{5}+\cdots\)
162.4.c.g \(2\) \(9.558\) \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(3\) \(-29\) \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+3\zeta_{6}q^{5}+\cdots\)
162.4.c.h \(2\) \(9.558\) \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(12\) \(7\) \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+12\zeta_{6}q^{5}+\cdots\)
162.4.c.i \(4\) \(9.558\) \(\Q(\zeta_{12})\) None \(-4\) \(0\) \(-12\) \(-16\) \(q+(-2+2\zeta_{12})q^{2}-4\zeta_{12}q^{4}+(-6\zeta_{12}+\cdots)q^{5}+\cdots\)
162.4.c.j \(4\) \(9.558\) \(\Q(\zeta_{12})\) None \(4\) \(0\) \(12\) \(-16\) \(q+(2-2\zeta_{12})q^{2}-4\zeta_{12}q^{4}+(6\zeta_{12}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(162, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(162, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 4 T^{2} \))(\( 1 + 2 T + 4 T^{2} \))(\( 1 + 2 T + 4 T^{2} \))(\( 1 + 2 T + 4 T^{2} \))(\( 1 - 2 T + 4 T^{2} \))(\( 1 - 2 T + 4 T^{2} \))(\( 1 - 2 T + 4 T^{2} \))(\( 1 - 2 T + 4 T^{2} \))(\( ( 1 + 2 T + 4 T^{2} )^{2} \))(\( ( 1 - 2 T + 4 T^{2} )^{2} \))
$3$ 1
$5$ (\( 1 + 12 T + 19 T^{2} + 1500 T^{3} + 15625 T^{4} \))(\( 1 + 3 T - 116 T^{2} + 375 T^{3} + 15625 T^{4} \))(\( 1 - 6 T - 89 T^{2} - 750 T^{3} + 15625 T^{4} \))(\( 1 - 21 T + 316 T^{2} - 2625 T^{3} + 15625 T^{4} \))(\( 1 + 21 T + 316 T^{2} + 2625 T^{3} + 15625 T^{4} \))(\( 1 + 6 T - 89 T^{2} + 750 T^{3} + 15625 T^{4} \))(\( 1 - 3 T - 116 T^{2} - 375 T^{3} + 15625 T^{4} \))(\( 1 - 12 T + 19 T^{2} - 1500 T^{3} + 15625 T^{4} \))(\( 1 + 12 T - 115 T^{2} + 108 T^{3} + 33456 T^{4} + 13500 T^{5} - 1796875 T^{6} + 23437500 T^{7} + 244140625 T^{8} \))(\( 1 - 12 T - 115 T^{2} - 108 T^{3} + 33456 T^{4} - 13500 T^{5} - 1796875 T^{6} - 23437500 T^{7} + 244140625 T^{8} \))
$7$ (\( ( 1 - 35 T + 343 T^{2} )( 1 + 28 T + 343 T^{2} ) \))(\( 1 + 29 T + 498 T^{2} + 9947 T^{3} + 117649 T^{4} \))(\( 1 - 16 T - 87 T^{2} - 5488 T^{3} + 117649 T^{4} \))(\( 1 + 8 T - 279 T^{2} + 2744 T^{3} + 117649 T^{4} \))(\( 1 + 8 T - 279 T^{2} + 2744 T^{3} + 117649 T^{4} \))(\( 1 - 16 T - 87 T^{2} - 5488 T^{3} + 117649 T^{4} \))(\( 1 + 29 T + 498 T^{2} + 9947 T^{3} + 117649 T^{4} \))(\( ( 1 - 35 T + 343 T^{2} )( 1 + 28 T + 343 T^{2} ) \))(\( 1 + 16 T - 386 T^{2} - 704 T^{3} + 206707 T^{4} - 241472 T^{5} - 45412514 T^{6} + 645657712 T^{7} + 13841287201 T^{8} \))(\( 1 + 16 T - 386 T^{2} - 704 T^{3} + 206707 T^{4} - 241472 T^{5} - 45412514 T^{6} + 645657712 T^{7} + 13841287201 T^{8} \))
$11$ (\( 1 + 60 T + 2269 T^{2} + 79860 T^{3} + 1771561 T^{4} \))(\( 1 - 57 T + 1918 T^{2} - 75867 T^{3} + 1771561 T^{4} \))(\( 1 - 12 T - 1187 T^{2} - 15972 T^{3} + 1771561 T^{4} \))(\( 1 - 36 T - 35 T^{2} - 47916 T^{3} + 1771561 T^{4} \))(\( 1 + 36 T - 35 T^{2} + 47916 T^{3} + 1771561 T^{4} \))(\( 1 + 12 T - 1187 T^{2} + 15972 T^{3} + 1771561 T^{4} \))(\( 1 + 57 T + 1918 T^{2} + 75867 T^{3} + 1771561 T^{4} \))(\( 1 - 60 T + 2269 T^{2} - 79860 T^{3} + 1771561 T^{4} \))(\( 1 + 36 T + 38 T^{2} - 50544 T^{3} - 1913973 T^{4} - 67274064 T^{5} + 67319318 T^{6} + 84886116876 T^{7} + 3138428376721 T^{8} \))(\( 1 - 36 T + 38 T^{2} + 50544 T^{3} - 1913973 T^{4} + 67274064 T^{5} + 67319318 T^{6} - 84886116876 T^{7} + 3138428376721 T^{8} \))
$13$ (\( 1 - 79 T + 4044 T^{2} - 173563 T^{3} + 4826809 T^{4} \))(\( 1 + 20 T - 1797 T^{2} + 43940 T^{3} + 4826809 T^{4} \))(\( 1 + 38 T - 753 T^{2} + 83486 T^{3} + 4826809 T^{4} \))(\( 1 - 49 T + 204 T^{2} - 107653 T^{3} + 4826809 T^{4} \))(\( 1 - 49 T + 204 T^{2} - 107653 T^{3} + 4826809 T^{4} \))(\( 1 + 38 T - 753 T^{2} + 83486 T^{3} + 4826809 T^{4} \))(\( 1 + 20 T - 1797 T^{2} + 43940 T^{3} + 4826809 T^{4} \))(\( 1 - 79 T + 4044 T^{2} - 173563 T^{3} + 4826809 T^{4} \))(\( 1 + 10 T + 973 T^{2} - 52670 T^{3} - 4284380 T^{4} - 115715990 T^{5} + 4696485157 T^{6} + 106044993730 T^{7} + 23298085122481 T^{8} \))(\( 1 + 10 T + 973 T^{2} - 52670 T^{3} - 4284380 T^{4} - 115715990 T^{5} + 4696485157 T^{6} + 106044993730 T^{7} + 23298085122481 T^{8} \))
$17$ (\( ( 1 + 108 T + 4913 T^{2} )^{2} \))(\( ( 1 + 72 T + 4913 T^{2} )^{2} \))(\( ( 1 - 126 T + 4913 T^{2} )^{2} \))(\( ( 1 + 21 T + 4913 T^{2} )^{2} \))(\( ( 1 - 21 T + 4913 T^{2} )^{2} \))(\( ( 1 + 126 T + 4913 T^{2} )^{2} \))(\( ( 1 - 72 T + 4913 T^{2} )^{2} \))(\( ( 1 - 108 T + 4913 T^{2} )^{2} \))(\( ( 1 - 120 T + 10159 T^{2} - 589560 T^{3} + 24137569 T^{4} )^{2} \))(\( ( 1 + 120 T + 10159 T^{2} + 589560 T^{3} + 24137569 T^{4} )^{2} \))
$19$ (\( ( 1 - 11 T + 6859 T^{2} )^{2} \))(\( ( 1 + 106 T + 6859 T^{2} )^{2} \))(\( ( 1 - 20 T + 6859 T^{2} )^{2} \))(\( ( 1 + 112 T + 6859 T^{2} )^{2} \))(\( ( 1 + 112 T + 6859 T^{2} )^{2} \))(\( ( 1 - 20 T + 6859 T^{2} )^{2} \))(\( ( 1 + 106 T + 6859 T^{2} )^{2} \))(\( ( 1 - 11 T + 6859 T^{2} )^{2} \))(\( ( 1 + 8 T + 666 T^{2} + 54872 T^{3} + 47045881 T^{4} )^{2} \))(\( ( 1 + 8 T + 666 T^{2} + 54872 T^{3} + 47045881 T^{4} )^{2} \))
$23$ (\( 1 - 132 T + 5257 T^{2} - 1606044 T^{3} + 148035889 T^{4} \))(\( 1 + 174 T + 18109 T^{2} + 2117058 T^{3} + 148035889 T^{4} \))(\( 1 - 168 T + 16057 T^{2} - 2044056 T^{3} + 148035889 T^{4} \))(\( 1 - 180 T + 20233 T^{2} - 2190060 T^{3} + 148035889 T^{4} \))(\( 1 + 180 T + 20233 T^{2} + 2190060 T^{3} + 148035889 T^{4} \))(\( 1 + 168 T + 16057 T^{2} + 2044056 T^{3} + 148035889 T^{4} \))(\( 1 - 174 T + 18109 T^{2} - 2117058 T^{3} + 148035889 T^{4} \))(\( 1 + 132 T + 5257 T^{2} + 1606044 T^{3} + 148035889 T^{4} \))(\( 1 + 180 T + 398 T^{2} + 1380240 T^{3} + 481881315 T^{4} + 16793380080 T^{5} + 58918283822 T^{6} + 324207479063340 T^{7} + 21914624432020321 T^{8} \))(\( 1 - 180 T + 398 T^{2} - 1380240 T^{3} + 481881315 T^{4} - 16793380080 T^{5} + 58918283822 T^{6} - 324207479063340 T^{7} + 21914624432020321 T^{8} \))
$29$ (\( 1 + 96 T - 15173 T^{2} + 2341344 T^{3} + 594823321 T^{4} \))(\( 1 - 210 T + 19711 T^{2} - 5121690 T^{3} + 594823321 T^{4} \))(\( 1 - 30 T - 23489 T^{2} - 731670 T^{3} + 594823321 T^{4} \))(\( 1 + 135 T - 6164 T^{2} + 3292515 T^{3} + 594823321 T^{4} \))(\( 1 - 135 T - 6164 T^{2} - 3292515 T^{3} + 594823321 T^{4} \))(\( 1 + 30 T - 23489 T^{2} + 731670 T^{3} + 594823321 T^{4} \))(\( 1 + 210 T + 19711 T^{2} + 5121690 T^{3} + 594823321 T^{4} \))(\( 1 - 96 T - 15173 T^{2} - 2341344 T^{3} + 594823321 T^{4} \))(\( 1 + 324 T + 31277 T^{2} + 8074404 T^{3} + 2276459616 T^{4} + 196926639156 T^{5} + 18604289010917 T^{6} + 4700315296181556 T^{7} + 353814783205469041 T^{8} \))(\( 1 - 324 T + 31277 T^{2} - 8074404 T^{3} + 2276459616 T^{4} - 196926639156 T^{5} + 18604289010917 T^{6} - 4700315296181556 T^{7} + 353814783205469041 T^{8} \))
$31$ (\( 1 + 20 T - 29391 T^{2} + 595820 T^{3} + 887503681 T^{4} \))(\( 1 + 47 T - 27582 T^{2} + 1400177 T^{3} + 887503681 T^{4} \))(\( 1 - 88 T - 22047 T^{2} - 2621608 T^{3} + 887503681 T^{4} \))(\( ( 1 + 19 T + 29791 T^{2} )( 1 + 289 T + 29791 T^{2} ) \))(\( ( 1 + 19 T + 29791 T^{2} )( 1 + 289 T + 29791 T^{2} ) \))(\( 1 - 88 T - 22047 T^{2} - 2621608 T^{3} + 887503681 T^{4} \))(\( 1 + 47 T - 27582 T^{2} + 1400177 T^{3} + 887503681 T^{4} \))(\( 1 + 20 T - 29391 T^{2} + 595820 T^{3} + 887503681 T^{4} \))(\( 1 - 248 T + 21538 T^{2} + 4864768 T^{3} - 1122488189 T^{4} + 144926303488 T^{5} + 19115054281378 T^{6} - 6557026295846408 T^{7} + 787662783788549761 T^{8} \))(\( 1 - 248 T + 21538 T^{2} + 4864768 T^{3} - 1122488189 T^{4} + 144926303488 T^{5} + 19115054281378 T^{6} - 6557026295846408 T^{7} + 787662783788549761 T^{8} \))
$37$ (\( ( 1 + 169 T + 50653 T^{2} )^{2} \))(\( ( 1 - 2 T + 50653 T^{2} )^{2} \))(\( ( 1 - 254 T + 50653 T^{2} )^{2} \))(\( ( 1 + T + 50653 T^{2} )^{2} \))(\( ( 1 + T + 50653 T^{2} )^{2} \))(\( ( 1 - 254 T + 50653 T^{2} )^{2} \))(\( ( 1 - 2 T + 50653 T^{2} )^{2} \))(\( ( 1 + 169 T + 50653 T^{2} )^{2} \))(\( ( 1 + 434 T + 148287 T^{2} + 21983402 T^{3} + 2565726409 T^{4} )^{2} \))(\( ( 1 + 434 T + 148287 T^{2} + 21983402 T^{3} + 2565726409 T^{4} )^{2} \))
$41$ (\( 1 + 192 T - 32057 T^{2} + 13232832 T^{3} + 4750104241 T^{4} \))(\( 1 - 6 T - 68885 T^{2} - 413526 T^{3} + 4750104241 T^{4} \))(\( 1 - 42 T - 67157 T^{2} - 2894682 T^{3} + 4750104241 T^{4} \))(\( 1 + 42 T - 67157 T^{2} + 2894682 T^{3} + 4750104241 T^{4} \))(\( 1 - 42 T - 67157 T^{2} - 2894682 T^{3} + 4750104241 T^{4} \))(\( 1 + 42 T - 67157 T^{2} + 2894682 T^{3} + 4750104241 T^{4} \))(\( 1 + 6 T - 68885 T^{2} + 413526 T^{3} + 4750104241 T^{4} \))(\( 1 - 192 T - 32057 T^{2} - 13232832 T^{3} + 4750104241 T^{4} \))(\( 1 + 192 T - 57922 T^{2} - 8266752 T^{3} + 1693577811 T^{4} - 569752814592 T^{5} - 275135537847202 T^{6} + 62857331403640512 T^{7} + 22563490300366186081 T^{8} \))(\( 1 - 192 T - 57922 T^{2} + 8266752 T^{3} + 1693577811 T^{4} + 569752814592 T^{5} - 275135537847202 T^{6} - 62857331403640512 T^{7} + 22563490300366186081 T^{8} \))
$43$ (\( 1 + 488 T + 158637 T^{2} + 38799416 T^{3} + 6321363049 T^{4} \))(\( 1 + 218 T - 31983 T^{2} + 17332526 T^{3} + 6321363049 T^{4} \))(\( 1 - 52 T - 76803 T^{2} - 4134364 T^{3} + 6321363049 T^{4} \))(\( 1 + 20 T - 79107 T^{2} + 1590140 T^{3} + 6321363049 T^{4} \))(\( 1 + 20 T - 79107 T^{2} + 1590140 T^{3} + 6321363049 T^{4} \))(\( 1 - 52 T - 76803 T^{2} - 4134364 T^{3} + 6321363049 T^{4} \))(\( 1 + 218 T - 31983 T^{2} + 17332526 T^{3} + 6321363049 T^{4} \))(\( 1 + 488 T + 158637 T^{2} + 38799416 T^{3} + 6321363049 T^{4} \))(\( 1 - 608 T + 119206 T^{2} - 55597952 T^{3} + 27016971067 T^{4} - 4420426369664 T^{5} + 753544403619094 T^{6} - 305576308057600544 T^{7} + 39959630797262576401 T^{8} \))(\( 1 - 608 T + 119206 T^{2} - 55597952 T^{3} + 27016971067 T^{4} - 4420426369664 T^{5} + 753544403619094 T^{6} - 305576308057600544 T^{7} + 39959630797262576401 T^{8} \))
$47$ (\( 1 + 204 T - 62207 T^{2} + 21179892 T^{3} + 10779215329 T^{4} \))(\( 1 + 474 T + 120853 T^{2} + 49212102 T^{3} + 10779215329 T^{4} \))(\( 1 + 96 T - 94607 T^{2} + 9967008 T^{3} + 10779215329 T^{4} \))(\( 1 - 84 T - 96767 T^{2} - 8721132 T^{3} + 10779215329 T^{4} \))(\( 1 + 84 T - 96767 T^{2} + 8721132 T^{3} + 10779215329 T^{4} \))(\( 1 - 96 T - 94607 T^{2} - 9967008 T^{3} + 10779215329 T^{4} \))(\( 1 - 474 T + 120853 T^{2} - 49212102 T^{3} + 10779215329 T^{4} \))(\( 1 - 204 T - 62207 T^{2} - 21179892 T^{3} + 10779215329 T^{4} \))(\( 1 - 384 T - 62062 T^{2} - 718848 T^{3} + 17809252707 T^{4} - 74632955904 T^{5} - 668979661748398 T^{6} - 429746101671462528 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 + 384 T - 62062 T^{2} + 718848 T^{3} + 17809252707 T^{4} + 74632955904 T^{5} - 668979661748398 T^{6} + 429746101671462528 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} \))
$53$ (\( ( 1 - 360 T + 148877 T^{2} )^{2} \))(\( ( 1 - 81 T + 148877 T^{2} )^{2} \))(\( ( 1 + 198 T + 148877 T^{2} )^{2} \))(\( ( 1 - 174 T + 148877 T^{2} )^{2} \))(\( ( 1 + 174 T + 148877 T^{2} )^{2} \))(\( ( 1 - 198 T + 148877 T^{2} )^{2} \))(\( ( 1 + 81 T + 148877 T^{2} )^{2} \))(\( ( 1 + 360 T + 148877 T^{2} )^{2} \))(\( ( 1 + 408 T + 183418 T^{2} + 60741816 T^{3} + 22164361129 T^{4} )^{2} \))(\( ( 1 - 408 T + 183418 T^{2} - 60741816 T^{3} + 22164361129 T^{4} )^{2} \))
$59$ (\( 1 + 156 T - 181043 T^{2} + 32039124 T^{3} + 42180533641 T^{4} \))(\( 1 + 84 T - 198323 T^{2} + 17251836 T^{3} + 42180533641 T^{4} \))(\( 1 + 660 T + 230221 T^{2} + 135550140 T^{3} + 42180533641 T^{4} \))(\( 1 - 504 T + 48637 T^{2} - 103511016 T^{3} + 42180533641 T^{4} \))(\( 1 + 504 T + 48637 T^{2} + 103511016 T^{3} + 42180533641 T^{4} \))(\( 1 - 660 T + 230221 T^{2} - 135550140 T^{3} + 42180533641 T^{4} \))(\( 1 - 84 T - 198323 T^{2} - 17251836 T^{3} + 42180533641 T^{4} \))(\( 1 - 156 T - 181043 T^{2} - 32039124 T^{3} + 42180533641 T^{4} \))(\( 1 - 1008 T + 378938 T^{2} - 228178944 T^{3} + 155070797979 T^{4} - 46863163339776 T^{5} + 15983807056853258 T^{6} - 8732299785204178512 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \))(\( 1 + 1008 T + 378938 T^{2} + 228178944 T^{3} + 155070797979 T^{4} + 46863163339776 T^{5} + 15983807056853258 T^{6} + 8732299785204178512 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \))
$61$ (\( 1 + 83 T - 220092 T^{2} + 18839423 T^{3} + 51520374361 T^{4} \))(\( 1 + 56 T - 223845 T^{2} + 12710936 T^{3} + 51520374361 T^{4} \))(\( 1 - 538 T + 62463 T^{2} - 122115778 T^{3} + 51520374361 T^{4} \))(\( 1 - 385 T - 78756 T^{2} - 87387685 T^{3} + 51520374361 T^{4} \))(\( 1 - 385 T - 78756 T^{2} - 87387685 T^{3} + 51520374361 T^{4} \))(\( 1 - 538 T + 62463 T^{2} - 122115778 T^{3} + 51520374361 T^{4} \))(\( 1 + 56 T - 223845 T^{2} + 12710936 T^{3} + 51520374361 T^{4} \))(\( 1 + 83 T - 220092 T^{2} + 18839423 T^{3} + 51520374361 T^{4} \))(\( 1 + 742 T - 32291 T^{2} + 95638606 T^{3} + 163232009380 T^{4} + 21708146428486 T^{5} - 1663644408491051 T^{6} + 8677056400882932622 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \))(\( 1 + 742 T - 32291 T^{2} + 95638606 T^{3} + 163232009380 T^{4} + 21708146428486 T^{5} - 1663644408491051 T^{6} + 8677056400882932622 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \))
$67$ (\( 1 + 47 T - 298554 T^{2} + 14135861 T^{3} + 90458382169 T^{4} \))(\( 1 - 142 T - 280599 T^{2} - 42708346 T^{3} + 90458382169 T^{4} \))(\( 1 + 884 T + 480693 T^{2} + 265874492 T^{3} + 90458382169 T^{4} \))(\( 1 + 272 T - 226779 T^{2} + 81807536 T^{3} + 90458382169 T^{4} \))(\( 1 + 272 T - 226779 T^{2} + 81807536 T^{3} + 90458382169 T^{4} \))(\( 1 + 884 T + 480693 T^{2} + 265874492 T^{3} + 90458382169 T^{4} \))(\( 1 - 142 T - 280599 T^{2} - 42708346 T^{3} + 90458382169 T^{4} \))(\( 1 + 47 T - 298554 T^{2} + 14135861 T^{3} + 90458382169 T^{4} \))(\( 1 - 104 T - 79226 T^{2} + 53194336 T^{3} - 85603873013 T^{4} + 15998888078368 T^{5} - 7166655785721194 T^{6} - 2829479577214674488 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \))(\( 1 - 104 T - 79226 T^{2} + 53194336 T^{3} - 85603873013 T^{4} + 15998888078368 T^{5} - 7166655785721194 T^{6} - 2829479577214674488 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \))
$71$ (\( ( 1 - 216 T + 357911 T^{2} )^{2} \))(\( ( 1 - 360 T + 357911 T^{2} )^{2} \))(\( ( 1 + 792 T + 357911 T^{2} )^{2} \))(\( ( 1 - 888 T + 357911 T^{2} )^{2} \))(\( ( 1 + 888 T + 357911 T^{2} )^{2} \))(\( ( 1 - 792 T + 357911 T^{2} )^{2} \))(\( ( 1 + 360 T + 357911 T^{2} )^{2} \))(\( ( 1 + 216 T + 357911 T^{2} )^{2} \))(\( ( 1 + 1140 T + 1038994 T^{2} + 408018540 T^{3} + 128100283921 T^{4} )^{2} \))(\( ( 1 - 1140 T + 1038994 T^{2} - 408018540 T^{3} + 128100283921 T^{4} )^{2} \))
$73$ (\( ( 1 + 511 T + 389017 T^{2} )^{2} \))(\( ( 1 + 1159 T + 389017 T^{2} )^{2} \))(\( ( 1 - 218 T + 389017 T^{2} )^{2} \))(\( ( 1 - 371 T + 389017 T^{2} )^{2} \))(\( ( 1 - 371 T + 389017 T^{2} )^{2} \))(\( ( 1 - 218 T + 389017 T^{2} )^{2} \))(\( ( 1 + 1159 T + 389017 T^{2} )^{2} \))(\( ( 1 + 511 T + 389017 T^{2} )^{2} \))(\( ( 1 - 850 T + 709827 T^{2} - 330664450 T^{3} + 151334226289 T^{4} )^{2} \))(\( ( 1 - 850 T + 709827 T^{2} - 330664450 T^{3} + 151334226289 T^{4} )^{2} \))
$79$ (\( 1 - 529 T - 213198 T^{2} - 260817631 T^{3} + 243087455521 T^{4} \))(\( 1 - 160 T - 467439 T^{2} - 78886240 T^{3} + 243087455521 T^{4} \))(\( 1 - 520 T - 222639 T^{2} - 256380280 T^{3} + 243087455521 T^{4} \))(\( 1 - 652 T - 67935 T^{2} - 321461428 T^{3} + 243087455521 T^{4} \))(\( 1 - 652 T - 67935 T^{2} - 321461428 T^{3} + 243087455521 T^{4} \))(\( 1 - 520 T - 222639 T^{2} - 256380280 T^{3} + 243087455521 T^{4} \))(\( 1 - 160 T - 467439 T^{2} - 78886240 T^{3} + 243087455521 T^{4} \))(\( 1 - 529 T - 213198 T^{2} - 260817631 T^{3} + 243087455521 T^{4} \))(\( 1 - 440 T - 773378 T^{2} + 8404000 T^{3} + 596506646563 T^{4} + 4143499756000 T^{5} - 187998490175919938 T^{6} - 52734702232352060360 T^{7} + \)\(59\!\cdots\!41\)\( T^{8} \))(\( 1 - 440 T - 773378 T^{2} + 8404000 T^{3} + 596506646563 T^{4} + 4143499756000 T^{5} - 187998490175919938 T^{6} - 52734702232352060360 T^{7} + \)\(59\!\cdots\!41\)\( T^{8} \))
$83$ (\( 1 - 1128 T + 700597 T^{2} - 644975736 T^{3} + 326940373369 T^{4} \))(\( 1 + 735 T - 31562 T^{2} + 420263445 T^{3} + 326940373369 T^{4} \))(\( 1 + 492 T - 329723 T^{2} + 281319204 T^{3} + 326940373369 T^{4} \))(\( 1 - 84 T - 564731 T^{2} - 48030108 T^{3} + 326940373369 T^{4} \))(\( 1 + 84 T - 564731 T^{2} + 48030108 T^{3} + 326940373369 T^{4} \))(\( 1 - 492 T - 329723 T^{2} - 281319204 T^{3} + 326940373369 T^{4} \))(\( 1 - 735 T - 31562 T^{2} - 420263445 T^{3} + 326940373369 T^{4} \))(\( 1 + 1128 T + 700597 T^{2} + 644975736 T^{3} + 326940373369 T^{4} \))(\( 1 + 264 T - 994102 T^{2} - 21060864 T^{3} + 764874544683 T^{4} - 12042328243968 T^{5} - 325012079046869638 T^{6} + 49352227390630666392 T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 - 264 T - 994102 T^{2} + 21060864 T^{3} + 764874544683 T^{4} + 12042328243968 T^{5} - 325012079046869638 T^{6} - 49352227390630666392 T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \))
$89$ (\( ( 1 - 36 T + 704969 T^{2} )^{2} \))(\( ( 1 + 954 T + 704969 T^{2} )^{2} \))(\( ( 1 + 810 T + 704969 T^{2} )^{2} \))(\( ( 1 + 21 T + 704969 T^{2} )^{2} \))(\( ( 1 - 21 T + 704969 T^{2} )^{2} \))(\( ( 1 - 810 T + 704969 T^{2} )^{2} \))(\( ( 1 - 954 T + 704969 T^{2} )^{2} \))(\( ( 1 + 36 T + 704969 T^{2} )^{2} \))(\( ( 1 - 768 T + 1343527 T^{2} - 541416192 T^{3} + 496981290961 T^{4} )^{2} \))(\( ( 1 + 768 T + 1343527 T^{2} + 541416192 T^{3} + 496981290961 T^{4} )^{2} \))
$97$ (\( 1 + 605 T - 546648 T^{2} + 552167165 T^{3} + 832972004929 T^{4} \))(\( 1 + 191 T - 876192 T^{2} + 174320543 T^{3} + 832972004929 T^{4} \))(\( 1 + 1154 T + 419043 T^{2} + 1053224642 T^{3} + 832972004929 T^{4} \))(\( 1 - 1246 T + 639843 T^{2} - 1137190558 T^{3} + 832972004929 T^{4} \))(\( 1 - 1246 T + 639843 T^{2} - 1137190558 T^{3} + 832972004929 T^{4} \))(\( 1 + 1154 T + 419043 T^{2} + 1053224642 T^{3} + 832972004929 T^{4} \))(\( 1 + 191 T - 876192 T^{2} + 174320543 T^{3} + 832972004929 T^{4} \))(\( 1 + 605 T - 546648 T^{2} + 552167165 T^{3} + 832972004929 T^{4} \))(\( 1 - 764 T - 1302902 T^{2} - 46796528 T^{3} + 2193556429267 T^{4} - 42709927599344 T^{5} - 1085280891166003958 T^{6} - \)\(58\!\cdots\!88\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \))(\( 1 - 764 T - 1302902 T^{2} - 46796528 T^{3} + 2193556429267 T^{4} - 42709927599344 T^{5} - 1085280891166003958 T^{6} - \)\(58\!\cdots\!88\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \))
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