Properties

Label 324.6.a
Level 324
Weight 6
Character orbit a
Rep. character \(\chi_{324}(1,\cdot)\)
Character field \(\Q\)
Dimension 20
Newform subspaces 5
Sturm bound 324
Trace bound 5

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Defining parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 324.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(324\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(324))\).

Total New Old
Modular forms 288 20 268
Cusp forms 252 20 232
Eisenstein series 36 0 36

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)FrickeDim.
\(-\)\(+\)\(-\)\(11\)
\(-\)\(-\)\(+\)\(9\)
Plus space\(+\)\(9\)
Minus space\(-\)\(11\)

Trace form

\( 20q + 58q^{7} + O(q^{10}) \) \( 20q + 58q^{7} - 1556q^{13} - 956q^{19} + 15332q^{25} - 3350q^{31} + 790q^{37} - 5102q^{43} + 64386q^{49} - 10458q^{55} - 16124q^{61} - 27458q^{67} + 170350q^{73} + 204106q^{79} + 120186q^{85} + 225974q^{91} + 9634q^{97} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(324))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3
324.6.a.a \(3\) \(51.964\) 3.3.513129.1 None \(0\) \(0\) \(-33\) \(-30\) \(-\) \(+\) \(q+(-11-\beta _{1})q^{5}+(-10-\beta _{2})q^{7}+\cdots\)
324.6.a.b \(3\) \(51.964\) 3.3.513129.1 None \(0\) \(0\) \(33\) \(-30\) \(-\) \(+\) \(q+(11+\beta _{1})q^{5}+(-10-\beta _{2})q^{7}+(-10+\cdots)q^{11}+\cdots\)
324.6.a.c \(4\) \(51.964\) \(\Q(\sqrt{3}, \sqrt{91})\) None \(0\) \(0\) \(0\) \(176\) \(-\) \(-\) \(q+(2\beta _{1}+\beta _{2})q^{5}+(44+\beta _{3})q^{7}+(-11\beta _{1}+\cdots)q^{11}+\cdots\)
324.6.a.d \(5\) \(51.964\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(0\) \(-21\) \(-29\) \(-\) \(-\) \(q+(-4-\beta _{2})q^{5}+(-6-\beta _{1}+\beta _{2})q^{7}+\cdots\)
324.6.a.e \(5\) \(51.964\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(0\) \(21\) \(-29\) \(-\) \(+\) \(q+(4+\beta _{2})q^{5}+(-6-\beta _{1}+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(324))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(324)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 + 33 T + 4614 T^{2} + 23217 T^{3} + 14418750 T^{4} + 322265625 T^{5} + 30517578125 T^{6} \))(\( 1 - 33 T + 4614 T^{2} - 23217 T^{3} + 14418750 T^{4} - 322265625 T^{5} + 30517578125 T^{6} \))(\( 1 + 4598 T^{2} + 15971451 T^{4} + 44902343750 T^{6} + 95367431640625 T^{8} \))(\( 1 + 21 T + 5644 T^{2} + 319227 T^{3} + 11115175 T^{4} + 1519942932 T^{5} + 34734921875 T^{6} + 3117451171875 T^{7} + 172241210937500 T^{8} + 2002716064453125 T^{9} + 298023223876953125 T^{10} \))(\( 1 - 21 T + 5644 T^{2} - 319227 T^{3} + 11115175 T^{4} - 1519942932 T^{5} + 34734921875 T^{6} - 3117451171875 T^{7} + 172241210937500 T^{8} - 2002716064453125 T^{9} + 298023223876953125 T^{10} \))
$7$ (\( 1 + 30 T + 6549 T^{2} + 556036 T^{3} + 110069043 T^{4} + 8474257470 T^{5} + 4747561509943 T^{6} \))(\( 1 + 30 T + 6549 T^{2} + 556036 T^{3} + 110069043 T^{4} + 8474257470 T^{5} + 4747561509943 T^{6} \))(\( ( 1 - 88 T + 25722 T^{2} - 1479016 T^{3} + 282475249 T^{4} )^{2} \))(\( 1 + 29 T + 40410 T^{2} + 2467167 T^{3} + 1121022921 T^{4} + 42673614444 T^{5} + 18841032233247 T^{6} + 696913612649583 T^{7} + 191848960616796630 T^{8} + 2313975722630748029 T^{9} + \)\(13\!\cdots\!07\)\( T^{10} \))(\( 1 + 29 T + 40410 T^{2} + 2467167 T^{3} + 1121022921 T^{4} + 42673614444 T^{5} + 18841032233247 T^{6} + 696913612649583 T^{7} + 191848960616796630 T^{8} + 2313975722630748029 T^{9} + \)\(13\!\cdots\!07\)\( T^{10} \))
$11$ (\( 1 - 30 T + 55761 T^{2} + 99601836 T^{3} + 8980364811 T^{4} - 778122738030 T^{5} + 4177248169415651 T^{6} \))(\( 1 + 30 T + 55761 T^{2} - 99601836 T^{3} + 8980364811 T^{4} + 778122738030 T^{5} + 4177248169415651 T^{6} \))(\( 1 + 438068 T^{2} + 92916105558 T^{4} + 11362355720110868 T^{6} + \)\(67\!\cdots\!01\)\( T^{8} \))(\( 1 - 177 T + 427561 T^{2} - 2122014 T^{3} + 77616186361 T^{4} + 7157256911361 T^{5} + 12500164429625411 T^{6} - 55039578127266414 T^{7} + \)\(17\!\cdots\!11\)\( T^{8} - \)\(11\!\cdots\!77\)\( T^{9} + \)\(10\!\cdots\!51\)\( T^{10} \))(\( 1 + 177 T + 427561 T^{2} + 2122014 T^{3} + 77616186361 T^{4} - 7157256911361 T^{5} + 12500164429625411 T^{6} + 55039578127266414 T^{7} + \)\(17\!\cdots\!11\)\( T^{8} + \)\(11\!\cdots\!77\)\( T^{9} + \)\(10\!\cdots\!51\)\( T^{10} \))
$13$ (\( 1 + 273 T + 469230 T^{2} + 237041953 T^{3} + 174221814390 T^{4} + 37635368274777 T^{5} + 51185893014090757 T^{6} \))(\( 1 + 273 T + 469230 T^{2} + 237041953 T^{3} + 174221814390 T^{4} + 37635368274777 T^{5} + 51185893014090757 T^{6} \))(\( ( 1 + 686 T + 378663 T^{2} + 254706998 T^{3} + 137858491849 T^{4} )^{2} \))(\( 1 - 181 T + 1045092 T^{2} - 87339735 T^{3} + 629954553255 T^{4} - 52722114809928 T^{5} + 233897715941708715 T^{6} - 12040524145591320015 T^{7} + \)\(53\!\cdots\!44\)\( T^{8} - \)\(34\!\cdots\!81\)\( T^{9} + \)\(70\!\cdots\!93\)\( T^{10} \))(\( 1 - 181 T + 1045092 T^{2} - 87339735 T^{3} + 629954553255 T^{4} - 52722114809928 T^{5} + 233897715941708715 T^{6} - 12040524145591320015 T^{7} + \)\(53\!\cdots\!44\)\( T^{8} - \)\(34\!\cdots\!81\)\( T^{9} + \)\(70\!\cdots\!93\)\( T^{10} \))
$17$ (\( 1 + 543 T + 1400478 T^{2} + 2899206507 T^{3} + 1988478491646 T^{4} + 1094684687943807 T^{5} + 2862423051509815793 T^{6} \))(\( 1 - 543 T + 1400478 T^{2} - 2899206507 T^{3} + 1988478491646 T^{4} - 1094684687943807 T^{5} + 2862423051509815793 T^{6} \))(\( 1 + 2188094 T^{2} + 4213626465219 T^{4} + 4411184157609054206 T^{6} + \)\(40\!\cdots\!01\)\( T^{8} \))(\( 1 + 1140 T + 4980550 T^{2} + 3443850354 T^{3} + 10068870522169 T^{4} + 5069379208548852 T^{5} + 14296356292995309833 T^{6} + \)\(69\!\cdots\!46\)\( T^{7} + \)\(14\!\cdots\!50\)\( T^{8} + \)\(46\!\cdots\!40\)\( T^{9} + \)\(57\!\cdots\!57\)\( T^{10} \))(\( 1 - 1140 T + 4980550 T^{2} - 3443850354 T^{3} + 10068870522169 T^{4} - 5069379208548852 T^{5} + 14296356292995309833 T^{6} - \)\(69\!\cdots\!46\)\( T^{7} + \)\(14\!\cdots\!50\)\( T^{8} - \)\(46\!\cdots\!40\)\( T^{9} + \)\(57\!\cdots\!57\)\( T^{10} \))
$19$ (\( 1 - 1410 T + 1123593 T^{2} + 1223470420 T^{3} + 2782127503707 T^{4} - 8644803423499410 T^{5} + 15181127029874798299 T^{6} \))(\( 1 - 1410 T + 1123593 T^{2} + 1223470420 T^{3} + 2782127503707 T^{4} - 8644803423499410 T^{5} + 15181127029874798299 T^{6} \))(\( ( 1 + 1472 T + 3832962 T^{2} + 3644817728 T^{3} + 6131066257801 T^{4} )^{2} \))(\( 1 + 416 T + 5046258 T^{2} + 6215761044 T^{3} + 20272296121125 T^{4} + 15898268281316088 T^{5} + 50196212153221491375 T^{6} + \)\(38\!\cdots\!44\)\( T^{7} + \)\(76\!\cdots\!42\)\( T^{8} + \)\(15\!\cdots\!16\)\( T^{9} + \)\(93\!\cdots\!99\)\( T^{10} \))(\( 1 + 416 T + 5046258 T^{2} + 6215761044 T^{3} + 20272296121125 T^{4} + 15898268281316088 T^{5} + 50196212153221491375 T^{6} + \)\(38\!\cdots\!44\)\( T^{7} + \)\(76\!\cdots\!42\)\( T^{8} + \)\(15\!\cdots\!16\)\( T^{9} + \)\(93\!\cdots\!99\)\( T^{10} \))
$23$ (\( 1 + 2766 T + 19152933 T^{2} + 33948849540 T^{3} + 123274846244019 T^{4} + 114585730016953134 T^{5} + \)\(26\!\cdots\!07\)\( T^{6} \))(\( 1 - 2766 T + 19152933 T^{2} - 33948849540 T^{3} + 123274846244019 T^{4} - 114585730016953134 T^{5} + \)\(26\!\cdots\!07\)\( T^{6} \))(\( 1 + 19977668 T^{2} + 175780104226854 T^{4} + \)\(82\!\cdots\!32\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} \))(\( 1 - 399 T + 16236442 T^{2} + 15815242731 T^{3} + 114281668162057 T^{4} + 214964375471995932 T^{5} + \)\(73\!\cdots\!51\)\( T^{6} + \)\(65\!\cdots\!19\)\( T^{7} + \)\(43\!\cdots\!94\)\( T^{8} - \)\(68\!\cdots\!99\)\( T^{9} + \)\(11\!\cdots\!43\)\( T^{10} \))(\( 1 + 399 T + 16236442 T^{2} - 15815242731 T^{3} + 114281668162057 T^{4} - 214964375471995932 T^{5} + \)\(73\!\cdots\!51\)\( T^{6} - \)\(65\!\cdots\!19\)\( T^{7} + \)\(43\!\cdots\!94\)\( T^{8} + \)\(68\!\cdots\!99\)\( T^{9} + \)\(11\!\cdots\!43\)\( T^{10} \))
$29$ (\( 1 - 3063 T + 45202542 T^{2} - 75644272695 T^{3} + 927156074140758 T^{4} - 1288626255598515663 T^{5} + \)\(86\!\cdots\!49\)\( T^{6} \))(\( 1 + 3063 T + 45202542 T^{2} + 75644272695 T^{3} + 927156074140758 T^{4} + 1288626255598515663 T^{5} + \)\(86\!\cdots\!49\)\( T^{6} \))(\( 1 + 68023478 T^{2} + 1951315900018971 T^{4} + \)\(28\!\cdots\!78\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} \))(\( 1 + 6033 T + 32744932 T^{2} + 82954489011 T^{3} + 288237740873407 T^{4} + 1430006647746298968 T^{5} + \)\(59\!\cdots\!43\)\( T^{6} + \)\(34\!\cdots\!11\)\( T^{7} + \)\(28\!\cdots\!68\)\( T^{8} + \)\(10\!\cdots\!33\)\( T^{9} + \)\(36\!\cdots\!49\)\( T^{10} \))(\( 1 - 6033 T + 32744932 T^{2} - 82954489011 T^{3} + 288237740873407 T^{4} - 1430006647746298968 T^{5} + \)\(59\!\cdots\!43\)\( T^{6} - \)\(34\!\cdots\!11\)\( T^{7} + \)\(28\!\cdots\!68\)\( T^{8} - \)\(10\!\cdots\!33\)\( T^{9} + \)\(36\!\cdots\!49\)\( T^{10} \))
$31$ (\( 1 - 3156 T + 78072333 T^{2} - 162943052312 T^{3} + 2235144610379283 T^{4} - 2586746873711407956 T^{5} + \)\(23\!\cdots\!51\)\( T^{6} \))(\( 1 - 3156 T + 78072333 T^{2} - 162943052312 T^{3} + 2235144610379283 T^{4} - 2586746873711407956 T^{5} + \)\(23\!\cdots\!51\)\( T^{6} \))(\( ( 1 + 2072 T + 57977790 T^{2} + 59319600872 T^{3} + 819628286980801 T^{4} )^{2} \))(\( 1 + 2759 T + 62514558 T^{2} - 8483492775 T^{3} + 1879317626697489 T^{4} - 2373615527060533968 T^{5} + \)\(53\!\cdots\!39\)\( T^{6} - \)\(69\!\cdots\!75\)\( T^{7} + \)\(14\!\cdots\!58\)\( T^{8} + \)\(18\!\cdots\!59\)\( T^{9} + \)\(19\!\cdots\!51\)\( T^{10} \))(\( 1 + 2759 T + 62514558 T^{2} - 8483492775 T^{3} + 1879317626697489 T^{4} - 2373615527060533968 T^{5} + \)\(53\!\cdots\!39\)\( T^{6} - \)\(69\!\cdots\!75\)\( T^{7} + \)\(14\!\cdots\!58\)\( T^{8} + \)\(18\!\cdots\!59\)\( T^{9} + \)\(19\!\cdots\!51\)\( T^{10} \))
$37$ (\( 1 - 7143 T + 108119958 T^{2} - 310835233127 T^{3} + 7497465718393806 T^{4} - 34347718172180695407 T^{5} + \)\(33\!\cdots\!93\)\( T^{6} \))(\( 1 - 7143 T + 108119958 T^{2} - 310835233127 T^{3} + 7497465718393806 T^{4} - 34347718172180695407 T^{5} + \)\(33\!\cdots\!93\)\( T^{6} \))(\( ( 1 - 838 T + 86490063 T^{2} - 58110235966 T^{3} + 4808584372417849 T^{4} )^{2} \))(\( 1 + 7586 T + 201201093 T^{2} + 803146672896 T^{3} + 19241810738464926 T^{4} + 60351714230064941916 T^{5} + \)\(13\!\cdots\!82\)\( T^{6} + \)\(38\!\cdots\!04\)\( T^{7} + \)\(67\!\cdots\!49\)\( T^{8} + \)\(17\!\cdots\!86\)\( T^{9} + \)\(16\!\cdots\!57\)\( T^{10} \))(\( 1 + 7586 T + 201201093 T^{2} + 803146672896 T^{3} + 19241810738464926 T^{4} + 60351714230064941916 T^{5} + \)\(13\!\cdots\!82\)\( T^{6} + \)\(38\!\cdots\!04\)\( T^{7} + \)\(67\!\cdots\!49\)\( T^{8} + \)\(17\!\cdots\!86\)\( T^{9} + \)\(16\!\cdots\!57\)\( T^{10} \))
$41$ (\( 1 + 21318 T + 435973047 T^{2} + 4657370441748 T^{3} + 50510180963814447 T^{4} + \)\(28\!\cdots\!18\)\( T^{5} + \)\(15\!\cdots\!01\)\( T^{6} \))(\( 1 - 21318 T + 435973047 T^{2} - 4657370441748 T^{3} + 50510180963814447 T^{4} - \)\(28\!\cdots\!18\)\( T^{5} + \)\(15\!\cdots\!01\)\( T^{6} \))(\( 1 + 451482308 T^{2} + 77787169313251110 T^{4} + \)\(60\!\cdots\!08\)\( T^{6} + \)\(18\!\cdots\!01\)\( T^{8} \))(\( 1 + 18435 T + 457528267 T^{2} + 6389512835910 T^{3} + 92046695080587061 T^{4} + \)\(99\!\cdots\!97\)\( T^{5} + \)\(10\!\cdots\!61\)\( T^{6} + \)\(85\!\cdots\!10\)\( T^{7} + \)\(71\!\cdots\!67\)\( T^{8} + \)\(33\!\cdots\!35\)\( T^{9} + \)\(20\!\cdots\!01\)\( T^{10} \))(\( 1 - 18435 T + 457528267 T^{2} - 6389512835910 T^{3} + 92046695080587061 T^{4} - \)\(99\!\cdots\!97\)\( T^{5} + \)\(10\!\cdots\!61\)\( T^{6} - \)\(85\!\cdots\!10\)\( T^{7} + \)\(71\!\cdots\!67\)\( T^{8} - \)\(33\!\cdots\!35\)\( T^{9} + \)\(20\!\cdots\!01\)\( T^{10} \))
$43$ (\( 1 - 18078 T + 490818129 T^{2} - 5010503105108 T^{3} + 72154408940463147 T^{4} - \)\(39\!\cdots\!22\)\( T^{5} + \)\(31\!\cdots\!07\)\( T^{6} \))(\( 1 - 18078 T + 490818129 T^{2} - 5010503105108 T^{3} + 72154408940463147 T^{4} - \)\(39\!\cdots\!22\)\( T^{5} + \)\(31\!\cdots\!07\)\( T^{6} \))(\( ( 1 + 19160 T + 289469058 T^{2} + 2816681767880 T^{3} + 21611482313284249 T^{4} )^{2} \))(\( 1 + 1469 T + 274021497 T^{2} + 2209484086710 T^{3} + 59712901022608185 T^{4} + \)\(32\!\cdots\!87\)\( T^{5} + \)\(87\!\cdots\!55\)\( T^{6} + \)\(47\!\cdots\!90\)\( T^{7} + \)\(87\!\cdots\!79\)\( T^{8} + \)\(68\!\cdots\!69\)\( T^{9} + \)\(68\!\cdots\!43\)\( T^{10} \))(\( 1 + 1469 T + 274021497 T^{2} + 2209484086710 T^{3} + 59712901022608185 T^{4} + \)\(32\!\cdots\!87\)\( T^{5} + \)\(87\!\cdots\!55\)\( T^{6} + \)\(47\!\cdots\!90\)\( T^{7} + \)\(87\!\cdots\!79\)\( T^{8} + \)\(68\!\cdots\!69\)\( T^{9} + \)\(68\!\cdots\!43\)\( T^{10} \))
$47$ (\( 1 - 41256 T + 1083114141 T^{2} - 18501711697584 T^{3} + 248406820249443987 T^{4} - \)\(21\!\cdots\!44\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \))(\( 1 + 41256 T + 1083114141 T^{2} + 18501711697584 T^{3} + 248406820249443987 T^{4} + \)\(21\!\cdots\!44\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \))(\( 1 + 367391324 T^{2} + 135388412358208134 T^{4} + \)\(19\!\cdots\!76\)\( T^{6} + \)\(27\!\cdots\!01\)\( T^{8} \))(\( 1 + 25155 T + 1034020258 T^{2} + 20179979725617 T^{3} + 464012894081647969 T^{4} + \)\(66\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!83\)\( T^{6} + \)\(10\!\cdots\!33\)\( T^{7} + \)\(12\!\cdots\!94\)\( T^{8} + \)\(69\!\cdots\!55\)\( T^{9} + \)\(63\!\cdots\!07\)\( T^{10} \))(\( 1 - 25155 T + 1034020258 T^{2} - 20179979725617 T^{3} + 464012894081647969 T^{4} - \)\(66\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!83\)\( T^{6} - \)\(10\!\cdots\!33\)\( T^{7} + \)\(12\!\cdots\!94\)\( T^{8} - \)\(69\!\cdots\!55\)\( T^{9} + \)\(63\!\cdots\!07\)\( T^{10} \))
$53$ (\( 1 - 11874 T + 402605979 T^{2} - 12066273873516 T^{3} + 168368005872652647 T^{4} - \)\(20\!\cdots\!26\)\( T^{5} + \)\(73\!\cdots\!57\)\( T^{6} \))(\( 1 + 11874 T + 402605979 T^{2} + 12066273873516 T^{3} + 168368005872652647 T^{4} + \)\(20\!\cdots\!26\)\( T^{5} + \)\(73\!\cdots\!57\)\( T^{6} \))(\( 1 + 728451284 T^{2} + 478272292253360790 T^{4} + \)\(12\!\cdots\!16\)\( T^{6} + \)\(30\!\cdots\!01\)\( T^{8} \))(\( 1 + 58422 T + 3354568213 T^{2} + 110313236959296 T^{3} + 3390725554692289246 T^{4} + \)\(71\!\cdots\!28\)\( T^{5} + \)\(14\!\cdots\!78\)\( T^{6} + \)\(19\!\cdots\!04\)\( T^{7} + \)\(24\!\cdots\!41\)\( T^{8} + \)\(17\!\cdots\!22\)\( T^{9} + \)\(12\!\cdots\!93\)\( T^{10} \))(\( 1 - 58422 T + 3354568213 T^{2} - 110313236959296 T^{3} + 3390725554692289246 T^{4} - \)\(71\!\cdots\!28\)\( T^{5} + \)\(14\!\cdots\!78\)\( T^{6} - \)\(19\!\cdots\!04\)\( T^{7} + \)\(24\!\cdots\!41\)\( T^{8} - \)\(17\!\cdots\!22\)\( T^{9} + \)\(12\!\cdots\!93\)\( T^{10} \))
$59$ (\( 1 + 92964 T + 4241750433 T^{2} + 131176669224216 T^{3} + 3032530454845471467 T^{4} + \)\(47\!\cdots\!64\)\( T^{5} + \)\(36\!\cdots\!99\)\( T^{6} \))(\( 1 - 92964 T + 4241750433 T^{2} - 131176669224216 T^{3} + 3032530454845471467 T^{4} - \)\(47\!\cdots\!64\)\( T^{5} + \)\(36\!\cdots\!99\)\( T^{6} \))(\( 1 + 1998785708 T^{2} + 1995662264722641366 T^{4} + \)\(10\!\cdots\!08\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \))(\( 1 + 90537 T + 5365830529 T^{2} + 236000523803862 T^{3} + 8392018437003638425 T^{4} + \)\(24\!\cdots\!43\)\( T^{5} + \)\(59\!\cdots\!75\)\( T^{6} + \)\(12\!\cdots\!62\)\( T^{7} + \)\(19\!\cdots\!71\)\( T^{8} + \)\(23\!\cdots\!37\)\( T^{9} + \)\(18\!\cdots\!99\)\( T^{10} \))(\( 1 - 90537 T + 5365830529 T^{2} - 236000523803862 T^{3} + 8392018437003638425 T^{4} - \)\(24\!\cdots\!43\)\( T^{5} + \)\(59\!\cdots\!75\)\( T^{6} - \)\(12\!\cdots\!62\)\( T^{7} + \)\(19\!\cdots\!71\)\( T^{8} - \)\(23\!\cdots\!37\)\( T^{9} + \)\(18\!\cdots\!99\)\( T^{10} \))
$61$ (\( 1 - 53439 T + 3142629438 T^{2} - 92218187357903 T^{3} + 2654253198748508838 T^{4} - \)\(38\!\cdots\!39\)\( T^{5} + \)\(60\!\cdots\!01\)\( T^{6} \))(\( 1 - 53439 T + 3142629438 T^{2} - 92218187357903 T^{3} + 2654253198748508838 T^{4} - \)\(38\!\cdots\!39\)\( T^{5} + \)\(60\!\cdots\!01\)\( T^{6} \))(\( ( 1 + 60098 T + 2094592503 T^{2} + 50758548497498 T^{3} + 713342911662882601 T^{4} )^{2} \))(\( 1 + 1403 T + 3538874292 T^{2} + 2256100311765 T^{3} + 5454602684103950271 T^{4} + \)\(18\!\cdots\!56\)\( T^{5} + \)\(46\!\cdots\!71\)\( T^{6} + \)\(16\!\cdots\!65\)\( T^{7} + \)\(21\!\cdots\!92\)\( T^{8} + \)\(71\!\cdots\!03\)\( T^{9} + \)\(42\!\cdots\!01\)\( T^{10} \))(\( 1 + 1403 T + 3538874292 T^{2} + 2256100311765 T^{3} + 5454602684103950271 T^{4} + \)\(18\!\cdots\!56\)\( T^{5} + \)\(46\!\cdots\!71\)\( T^{6} + \)\(16\!\cdots\!65\)\( T^{7} + \)\(21\!\cdots\!92\)\( T^{8} + \)\(71\!\cdots\!03\)\( T^{9} + \)\(42\!\cdots\!01\)\( T^{10} \))
$67$ (\( 1 - 53826 T + 1934468313 T^{2} - 45752080329452 T^{3} + 2611774238077234491 T^{4} - \)\(98\!\cdots\!74\)\( T^{5} + \)\(24\!\cdots\!43\)\( T^{6} \))(\( 1 - 53826 T + 1934468313 T^{2} - 45752080329452 T^{3} + 2611774238077234491 T^{4} - \)\(98\!\cdots\!74\)\( T^{5} + \)\(24\!\cdots\!43\)\( T^{6} \))(\( ( 1 + 53648 T + 3394214562 T^{2} + 72431511740336 T^{3} + 1822837804551761449 T^{4} )^{2} \))(\( 1 + 13907 T + 4070090193 T^{2} - 10411373671926 T^{3} + 6695736735425021001 T^{4} - \)\(80\!\cdots\!07\)\( T^{5} + \)\(90\!\cdots\!07\)\( T^{6} - \)\(18\!\cdots\!74\)\( T^{7} + \)\(10\!\cdots\!99\)\( T^{8} + \)\(46\!\cdots\!07\)\( T^{9} + \)\(44\!\cdots\!07\)\( T^{10} \))(\( 1 + 13907 T + 4070090193 T^{2} - 10411373671926 T^{3} + 6695736735425021001 T^{4} - \)\(80\!\cdots\!07\)\( T^{5} + \)\(90\!\cdots\!07\)\( T^{6} - \)\(18\!\cdots\!74\)\( T^{7} + \)\(10\!\cdots\!99\)\( T^{8} + \)\(46\!\cdots\!07\)\( T^{9} + \)\(44\!\cdots\!07\)\( T^{10} \))
$71$ (\( 1 - 31866 T + 5363863701 T^{2} - 115003148710092 T^{3} + 9677640324107688051 T^{4} - \)\(10\!\cdots\!66\)\( T^{5} + \)\(58\!\cdots\!51\)\( T^{6} \))(\( 1 + 31866 T + 5363863701 T^{2} + 115003148710092 T^{3} + 9677640324107688051 T^{4} + \)\(10\!\cdots\!66\)\( T^{5} + \)\(58\!\cdots\!51\)\( T^{6} \))(\( 1 - 65314012 T^{2} + 4689386407931107110 T^{4} - \)\(21\!\cdots\!12\)\( T^{6} + \)\(10\!\cdots\!01\)\( T^{8} \))(\( 1 + 114684 T + 7758380659 T^{2} + 426246123888336 T^{3} + 19260501229393543450 T^{4} + \)\(77\!\cdots\!40\)\( T^{5} + \)\(34\!\cdots\!50\)\( T^{6} + \)\(13\!\cdots\!36\)\( T^{7} + \)\(45\!\cdots\!09\)\( T^{8} + \)\(12\!\cdots\!84\)\( T^{9} + \)\(19\!\cdots\!51\)\( T^{10} \))(\( 1 - 114684 T + 7758380659 T^{2} - 426246123888336 T^{3} + 19260501229393543450 T^{4} - \)\(77\!\cdots\!40\)\( T^{5} + \)\(34\!\cdots\!50\)\( T^{6} - \)\(13\!\cdots\!36\)\( T^{7} + \)\(45\!\cdots\!09\)\( T^{8} - \)\(12\!\cdots\!84\)\( T^{9} + \)\(19\!\cdots\!51\)\( T^{10} \))
$73$ (\( 1 - 123765 T + 10300430238 T^{2} - 540333156371105 T^{3} + 21353529322076029134 T^{4} - \)\(53\!\cdots\!85\)\( T^{5} + \)\(89\!\cdots\!57\)\( T^{6} \))(\( 1 - 123765 T + 10300430238 T^{2} - 540333156371105 T^{3} + 21353529322076029134 T^{4} - \)\(53\!\cdots\!85\)\( T^{5} + \)\(89\!\cdots\!57\)\( T^{6} \))(\( ( 1 + 46190 T + 3864348579 T^{2} + 95755176880670 T^{3} + 4297625829703557649 T^{4} )^{2} \))(\( 1 - 7600 T + 3606834246 T^{2} - 31056473559714 T^{3} + 12288417972789256281 T^{4} - \)\(80\!\cdots\!84\)\( T^{5} + \)\(25\!\cdots\!33\)\( T^{6} - \)\(13\!\cdots\!86\)\( T^{7} + \)\(32\!\cdots\!22\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(38\!\cdots\!93\)\( T^{10} \))(\( 1 - 7600 T + 3606834246 T^{2} - 31056473559714 T^{3} + 12288417972789256281 T^{4} - \)\(80\!\cdots\!84\)\( T^{5} + \)\(25\!\cdots\!33\)\( T^{6} - \)\(13\!\cdots\!86\)\( T^{7} + \)\(32\!\cdots\!22\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(38\!\cdots\!93\)\( T^{10} \))
$79$ (\( 1 - 129534 T + 12729649965 T^{2} - 811215903905732 T^{3} + 39169850881833376035 T^{4} - \)\(12\!\cdots\!34\)\( T^{5} + \)\(29\!\cdots\!99\)\( T^{6} \))(\( 1 - 129534 T + 12729649965 T^{2} - 811215903905732 T^{3} + 39169850881833376035 T^{4} - \)\(12\!\cdots\!34\)\( T^{5} + \)\(29\!\cdots\!99\)\( T^{6} \))(\( ( 1 - 2512 T - 2000773254 T^{2} - 7729565674288 T^{3} + 9468276082626847201 T^{4} )^{2} \))(\( 1 + 29993 T + 6251931678 T^{2} - 85699438105257 T^{3} + 15422207141619743649 T^{4} - \)\(73\!\cdots\!92\)\( T^{5} + \)\(47\!\cdots\!51\)\( T^{6} - \)\(81\!\cdots\!57\)\( T^{7} + \)\(18\!\cdots\!22\)\( T^{8} + \)\(26\!\cdots\!93\)\( T^{9} + \)\(27\!\cdots\!99\)\( T^{10} \))(\( 1 + 29993 T + 6251931678 T^{2} - 85699438105257 T^{3} + 15422207141619743649 T^{4} - \)\(73\!\cdots\!92\)\( T^{5} + \)\(47\!\cdots\!51\)\( T^{6} - \)\(81\!\cdots\!57\)\( T^{7} + \)\(18\!\cdots\!22\)\( T^{8} + \)\(26\!\cdots\!93\)\( T^{9} + \)\(27\!\cdots\!99\)\( T^{10} \))
$83$ (\( 1 - 226584 T + 25210795929 T^{2} - 1851024586107024 T^{3} + 99306349806709942347 T^{4} - \)\(35\!\cdots\!16\)\( T^{5} + \)\(61\!\cdots\!07\)\( T^{6} \))(\( 1 + 226584 T + 25210795929 T^{2} + 1851024586107024 T^{3} + 99306349806709942347 T^{4} + \)\(35\!\cdots\!16\)\( T^{5} + \)\(61\!\cdots\!07\)\( T^{6} \))(\( 1 + 9492423500 T^{2} + 44942056330712675766 T^{4} + \)\(14\!\cdots\!00\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} \))(\( 1 + 228951 T + 31015128418 T^{2} + 2949331181889681 T^{3} + \)\(22\!\cdots\!25\)\( T^{4} + \)\(14\!\cdots\!60\)\( T^{5} + \)\(88\!\cdots\!75\)\( T^{6} + \)\(45\!\cdots\!69\)\( T^{7} + \)\(18\!\cdots\!26\)\( T^{8} + \)\(55\!\cdots\!51\)\( T^{9} + \)\(94\!\cdots\!43\)\( T^{10} \))(\( 1 - 228951 T + 31015128418 T^{2} - 2949331181889681 T^{3} + \)\(22\!\cdots\!25\)\( T^{4} - \)\(14\!\cdots\!60\)\( T^{5} + \)\(88\!\cdots\!75\)\( T^{6} - \)\(45\!\cdots\!69\)\( T^{7} + \)\(18\!\cdots\!26\)\( T^{8} - \)\(55\!\cdots\!51\)\( T^{9} + \)\(94\!\cdots\!43\)\( T^{10} \))
$89$ (\( 1 - 45897 T + 3780940614 T^{2} - 499185304837341 T^{3} + 21112997161714561686 T^{4} - \)\(14\!\cdots\!97\)\( T^{5} + \)\(17\!\cdots\!49\)\( T^{6} \))(\( 1 + 45897 T + 3780940614 T^{2} + 499185304837341 T^{3} + 21112997161714561686 T^{4} + \)\(14\!\cdots\!97\)\( T^{5} + \)\(17\!\cdots\!49\)\( T^{6} \))(\( 1 - 680992978 T^{2} + 62238193382785937523 T^{4} - \)\(21\!\cdots\!78\)\( T^{6} + \)\(97\!\cdots\!01\)\( T^{8} \))(\( 1 + 299166 T + 52616244181 T^{2} + 6660261403977288 T^{3} + \)\(67\!\cdots\!10\)\( T^{4} + \)\(55\!\cdots\!64\)\( T^{5} + \)\(37\!\cdots\!90\)\( T^{6} + \)\(20\!\cdots\!88\)\( T^{7} + \)\(91\!\cdots\!69\)\( T^{8} + \)\(29\!\cdots\!66\)\( T^{9} + \)\(54\!\cdots\!49\)\( T^{10} \))(\( 1 - 299166 T + 52616244181 T^{2} - 6660261403977288 T^{3} + \)\(67\!\cdots\!10\)\( T^{4} - \)\(55\!\cdots\!64\)\( T^{5} + \)\(37\!\cdots\!90\)\( T^{6} - \)\(20\!\cdots\!88\)\( T^{7} + \)\(91\!\cdots\!69\)\( T^{8} - \)\(29\!\cdots\!66\)\( T^{9} + \)\(54\!\cdots\!49\)\( T^{10} \))
$97$ (\( 1 - 211290 T + 31141328127 T^{2} - 2965626780069260 T^{3} + \)\(26\!\cdots\!39\)\( T^{4} - \)\(15\!\cdots\!10\)\( T^{5} + \)\(63\!\cdots\!93\)\( T^{6} \))(\( 1 - 211290 T + 31141328127 T^{2} - 2965626780069260 T^{3} + \)\(26\!\cdots\!39\)\( T^{4} - \)\(15\!\cdots\!10\)\( T^{5} + \)\(63\!\cdots\!93\)\( T^{6} \))(\( ( 1 + 165932 T + 24001271142 T^{2} + 1424914543524524 T^{3} + 73742412689492826049 T^{4} )^{2} \))(\( 1 + 40541 T + 19537068819 T^{2} + 1527692999826186 T^{3} + \)\(22\!\cdots\!25\)\( T^{4} + \)\(18\!\cdots\!11\)\( T^{5} + \)\(18\!\cdots\!25\)\( T^{6} + \)\(11\!\cdots\!14\)\( T^{7} + \)\(12\!\cdots\!67\)\( T^{8} + \)\(22\!\cdots\!41\)\( T^{9} + \)\(46\!\cdots\!57\)\( T^{10} \))(\( 1 + 40541 T + 19537068819 T^{2} + 1527692999826186 T^{3} + \)\(22\!\cdots\!25\)\( T^{4} + \)\(18\!\cdots\!11\)\( T^{5} + \)\(18\!\cdots\!25\)\( T^{6} + \)\(11\!\cdots\!14\)\( T^{7} + \)\(12\!\cdots\!67\)\( T^{8} + \)\(22\!\cdots\!41\)\( T^{9} + \)\(46\!\cdots\!57\)\( T^{10} \))
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