Properties

Label 160.6.a
Level 160
Weight 6
Character orbit a
Rep. character \(\chi_{160}(1,\cdot)\)
Character field \(\Q\)
Dimension 20
Newform subspaces 8
Sturm bound 144
Trace bound 9

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Defining parameters

Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(144\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(160))\).

Total New Old
Modular forms 128 20 108
Cusp forms 112 20 92
Eisenstein series 16 0 16

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

\(2\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(6\)
\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(4\)
Plus space\(+\)\(9\)
Minus space\(-\)\(11\)

Trace form

\( 20q + 1532q^{9} + O(q^{10}) \) \( 20q + 1532q^{9} - 464q^{13} - 2008q^{17} + 10136q^{21} + 12500q^{25} - 7960q^{29} - 7504q^{33} - 4304q^{37} + 44464q^{41} - 11800q^{45} + 5996q^{49} + 35024q^{53} + 183392q^{57} - 125440q^{61} - 42200q^{65} + 147208q^{69} + 212392q^{73} - 149648q^{77} + 38940q^{81} + 282536q^{89} - 667696q^{93} - 81544q^{97} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(160))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
160.6.a.a \(2\) \(25.661\) \(\Q(\sqrt{70}) \) None \(0\) \(-8\) \(50\) \(-104\) \(-\) \(-\) \(q+(-4+\beta )q^{3}+5^{2}q^{5}+(-52-\beta )q^{7}+\cdots\)
160.6.a.b \(2\) \(25.661\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(-50\) \(0\) \(-\) \(+\) \(q-3\beta q^{3}-5^{2}q^{5}-31\beta q^{7}-63q^{9}+\cdots\)
160.6.a.c \(2\) \(25.661\) \(\Q(\sqrt{85}) \) None \(0\) \(0\) \(-50\) \(0\) \(+\) \(+\) \(q-\beta q^{3}-5^{2}q^{5}+3\beta q^{7}+97q^{9}+26\beta q^{11}+\cdots\)
160.6.a.d \(2\) \(25.661\) \(\Q(\sqrt{10}) \) None \(0\) \(0\) \(50\) \(0\) \(-\) \(-\) \(q+\beta q^{3}+5^{2}q^{5}+7\beta q^{7}-203q^{9}+\cdots\)
160.6.a.e \(2\) \(25.661\) \(\Q(\sqrt{70}) \) None \(0\) \(8\) \(50\) \(104\) \(+\) \(-\) \(q+(4+\beta )q^{3}+5^{2}q^{5}+(52-\beta )q^{7}+(53+\cdots)q^{9}+\cdots\)
160.6.a.f \(3\) \(25.661\) 3.3.39180.1 None \(0\) \(-10\) \(-75\) \(6\) \(+\) \(+\) \(q+(-3-\beta _{1})q^{3}-5^{2}q^{5}+(2+\beta _{1}-\beta _{2})q^{7}+\cdots\)
160.6.a.g \(3\) \(25.661\) 3.3.39180.1 None \(0\) \(10\) \(-75\) \(-6\) \(-\) \(+\) \(q+(3+\beta _{1})q^{3}-5^{2}q^{5}+(-2-\beta _{1}+\beta _{2})q^{7}+\cdots\)
160.6.a.h \(4\) \(25.661\) 4.4.81998080.1 None \(0\) \(0\) \(100\) \(0\) \(+\) \(-\) \(q-\beta _{1}q^{3}+5^{2}q^{5}+(-6\beta _{1}+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(160))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(160)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 8 T + 222 T^{2} + 1944 T^{3} + 59049 T^{4} \))(\( 1 + 306 T^{2} + 59049 T^{4} \))(\( 1 + 146 T^{2} + 59049 T^{4} \))(\( 1 + 446 T^{2} + 59049 T^{4} \))(\( 1 - 8 T + 222 T^{2} - 1944 T^{3} + 59049 T^{4} \))(\( 1 + 10 T + 181 T^{2} + 5268 T^{3} + 43983 T^{4} + 590490 T^{5} + 14348907 T^{6} \))(\( 1 - 10 T + 181 T^{2} - 5268 T^{3} + 43983 T^{4} - 590490 T^{5} + 14348907 T^{6} \))(\( 1 + 124 T^{2} + 25686 T^{4} + 7322076 T^{6} + 3486784401 T^{8} \))
$5$ (\( ( 1 - 25 T )^{2} \))(\( ( 1 + 25 T )^{2} \))(\( ( 1 + 25 T )^{2} \))(\( ( 1 - 25 T )^{2} \))(\( ( 1 - 25 T )^{2} \))(\( ( 1 + 25 T )^{3} \))(\( ( 1 + 25 T )^{3} \))(\( ( 1 - 25 T )^{4} \))
$7$ (\( 1 + 104 T + 36038 T^{2} + 1747928 T^{3} + 282475249 T^{4} \))(\( 1 + 14394 T^{2} + 282475249 T^{4} \))(\( 1 + 30554 T^{2} + 282475249 T^{4} \))(\( 1 + 31654 T^{2} + 282475249 T^{4} \))(\( 1 - 104 T + 36038 T^{2} - 1747928 T^{3} + 282475249 T^{4} \))(\( 1 - 6 T + 27153 T^{2} - 1137532 T^{3} + 456360471 T^{4} - 1694851494 T^{5} + 4747561509943 T^{6} \))(\( 1 + 6 T + 27153 T^{2} + 1137532 T^{3} + 456360471 T^{4} + 1694851494 T^{5} + 4747561509943 T^{6} \))(\( 1 - 27060 T^{2} + 497649542 T^{4} - 7643780237940 T^{6} + 79792266297612001 T^{8} \))
$11$ (\( 1 + 320 T + 319702 T^{2} + 51536320 T^{3} + 25937424601 T^{4} \))(\( 1 + 254822 T^{2} + 25937424601 T^{4} \))(\( 1 + 92262 T^{2} + 25937424601 T^{4} \))(\( 1 - 197738 T^{2} + 25937424601 T^{4} \))(\( 1 - 320 T + 319702 T^{2} - 51536320 T^{3} + 25937424601 T^{4} \))(\( 1 - 396 T + 327873 T^{2} - 67617992 T^{3} + 52804274523 T^{4} - 10271220141996 T^{5} + 4177248169415651 T^{6} \))(\( 1 + 396 T + 327873 T^{2} + 67617992 T^{3} + 52804274523 T^{4} + 10271220141996 T^{5} + 4177248169415651 T^{6} \))(\( 1 + 382252 T^{2} + 87516901782 T^{4} + 9914632428581452 T^{6} + \)\(67\!\cdots\!01\)\( T^{8} \))
$13$ (\( 1 + 100 T + 297086 T^{2} + 37129300 T^{3} + 137858491849 T^{4} \))(\( ( 1 - 154 T + 371293 T^{2} )^{2} \))(\( ( 1 - 506 T + 371293 T^{2} )^{2} \))(\( ( 1 + 146 T + 371293 T^{2} )^{2} \))(\( 1 + 100 T + 297086 T^{2} + 37129300 T^{3} + 137858491849 T^{4} \))(\( 1 + 354 T + 845379 T^{2} + 291738444 T^{3} + 313883305047 T^{4} + 48801906114546 T^{5} + 51185893014090757 T^{6} \))(\( 1 + 354 T + 845379 T^{2} + 291738444 T^{3} + 313883305047 T^{4} + 48801906114546 T^{5} + 51185893014090757 T^{6} \))(\( ( 1 + 292 T - 102402 T^{2} + 108417556 T^{3} + 137858491849 T^{4} )^{2} \))
$17$ (\( 1 - 580 T + 2475814 T^{2} - 823517060 T^{3} + 2015993900449 T^{4} \))(\( ( 1 - 178 T + 1419857 T^{2} )^{2} \))(\( ( 1 + 1838 T + 1419857 T^{2} )^{2} \))(\( ( 1 + 702 T + 1419857 T^{2} )^{2} \))(\( 1 - 580 T + 2475814 T^{2} - 823517060 T^{3} + 2015993900449 T^{4} \))(\( 1 - 1158 T + 1914111 T^{2} - 544550612 T^{3} + 2717763902127 T^{4} - 2334520936719942 T^{5} + 2862423051509815793 T^{6} \))(\( 1 - 1158 T + 1914111 T^{2} - 544550612 T^{3} + 2717763902127 T^{4} - 2334520936719942 T^{5} + 2862423051509815793 T^{6} \))(\( ( 1 + 380 T + 2009510 T^{2} + 539545660 T^{3} + 2015993900449 T^{4} )^{2} \))
$19$ (\( 1 + 720 T + 4633798 T^{2} + 1782791280 T^{3} + 6131066257801 T^{4} \))(\( 1 + 4019078 T^{2} + 6131066257801 T^{4} \))(\( 1 + 687238 T^{2} + 6131066257801 T^{4} \))(\( 1 - 2512762 T^{2} + 6131066257801 T^{4} \))(\( 1 - 720 T + 4633798 T^{2} - 1782791280 T^{3} + 6131066257801 T^{4} \))(\( 1 + 3192 T + 9330057 T^{2} + 15933739216 T^{3} + 23102144807643 T^{4} + 19570363494900792 T^{5} + 15181127029874798299 T^{6} \))(\( 1 - 3192 T + 9330057 T^{2} - 15933739216 T^{3} + 23102144807643 T^{4} - 19570363494900792 T^{5} + 15181127029874798299 T^{6} \))(\( 1 + 1633036 T^{2} + 154661897526 T^{4} + 10012251917374313836 T^{6} + \)\(37\!\cdots\!01\)\( T^{8} \))
$23$ (\( 1 + 1688 T + 10950502 T^{2} + 10864546984 T^{3} + 41426511213649 T^{4} \))(\( 1 + 5934266 T^{2} + 41426511213649 T^{4} \))(\( 1 + 9265626 T^{2} + 41426511213649 T^{4} \))(\( 1 - 3871674 T^{2} + 41426511213649 T^{4} \))(\( 1 - 1688 T + 10950502 T^{2} - 10864546984 T^{3} + 41426511213649 T^{4} \))(\( 1 + 6126 T + 29722593 T^{2} + 84141222540 T^{3} + 191304803397399 T^{4} + 253778807694813774 T^{5} + \)\(26\!\cdots\!07\)\( T^{6} \))(\( 1 - 6126 T + 29722593 T^{2} - 84141222540 T^{3} + 191304803397399 T^{4} - 253778807694813774 T^{5} + \)\(26\!\cdots\!07\)\( T^{6} \))(\( 1 + 25651084 T^{2} + 247347299659206 T^{4} + \)\(10\!\cdots\!16\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} \))
$29$ (\( 1 - 108 T + 39233214 T^{2} - 2215204092 T^{3} + 420707233300201 T^{4} \))(\( ( 1 - 4110 T + 20511149 T^{2} )^{2} \))(\( ( 1 + 4530 T + 20511149 T^{2} )^{2} \))(\( ( 1 + 4010 T + 20511149 T^{2} )^{2} \))(\( 1 - 108 T + 39233214 T^{2} - 2215204092 T^{3} + 420707233300201 T^{4} \))(\( 1 - 426 T - 939693 T^{2} + 141773503364 T^{3} - 19274183137257 T^{4} - 179221281385885626 T^{5} + \)\(86\!\cdots\!49\)\( T^{6} \))(\( 1 - 426 T - 939693 T^{2} + 141773503364 T^{3} - 19274183137257 T^{4} - 179221281385885626 T^{5} + \)\(86\!\cdots\!49\)\( T^{6} \))(\( ( 1 + 84 T + 9837118 T^{2} + 1722936516 T^{3} + 420707233300201 T^{4} )^{2} \))
$31$ (\( 1 + 9840 T + 76732702 T^{2} + 281710845840 T^{3} + 819628286980801 T^{4} \))(\( 1 + 47289582 T^{2} + 819628286980801 T^{4} \))(\( 1 + 43928942 T^{2} + 819628286980801 T^{4} \))(\( 1 + 36406942 T^{2} + 819628286980801 T^{4} \))(\( 1 - 9840 T + 76732702 T^{2} - 281710845840 T^{3} + 819628286980801 T^{4} \))(\( 1 + 3276 T + 2292813 T^{2} - 41639420248 T^{3} + 65641289591763 T^{4} + 2685102268149104076 T^{5} + \)\(23\!\cdots\!51\)\( T^{6} \))(\( 1 - 3276 T + 2292813 T^{2} + 41639420248 T^{3} + 65641289591763 T^{4} - 2685102268149104076 T^{5} + \)\(23\!\cdots\!51\)\( T^{6} \))(\( 1 + 24283452 T^{2} + 1637830022018822 T^{4} + \)\(19\!\cdots\!52\)\( T^{6} + \)\(67\!\cdots\!01\)\( T^{8} \))
$37$ (\( 1 - 6540 T + 61572814 T^{2} - 453509478780 T^{3} + 4808584372417849 T^{4} \))(\( ( 1 - 7442 T + 69343957 T^{2} )^{2} \))(\( ( 1 - 338 T + 69343957 T^{2} )^{2} \))(\( ( 1 + 14778 T + 69343957 T^{2} )^{2} \))(\( 1 - 6540 T + 61572814 T^{2} - 453509478780 T^{3} + 4808584372417849 T^{4} \))(\( 1 + 11562 T + 90623691 T^{2} + 525372493468 T^{3} + 6284205331885287 T^{4} + 55596852513895170138 T^{5} + \)\(33\!\cdots\!93\)\( T^{6} \))(\( 1 + 11562 T + 90623691 T^{2} + 525372493468 T^{3} + 6284205331885287 T^{4} + 55596852513895170138 T^{5} + \)\(33\!\cdots\!93\)\( T^{6} \))(\( ( 1 - 9868 T + 159567054 T^{2} - 684286167676 T^{3} + 4808584372417849 T^{4} )^{2} \))
$41$ (\( 1 + 10620 T + 77106582 T^{2} + 1230392854620 T^{3} + 13422659310152401 T^{4} \))(\( ( 1 - 7270 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 6330 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 4350 T + 115856201 T^{2} )^{2} \))(\( 1 + 10620 T + 77106582 T^{2} + 1230392854620 T^{3} + 13422659310152401 T^{4} \))(\( 1 - 12450 T + 384843783 T^{2} - 2886023186300 T^{3} + 44586538676848383 T^{4} - \)\(16\!\cdots\!50\)\( T^{5} + \)\(15\!\cdots\!01\)\( T^{6} \))(\( 1 - 12450 T + 384843783 T^{2} - 2886023186300 T^{3} + 44586538676848383 T^{4} - \)\(16\!\cdots\!50\)\( T^{5} + \)\(15\!\cdots\!01\)\( T^{6} \))(\( ( 1 - 23812 T + 331016342 T^{2} - 2758767858212 T^{3} + 13422659310152401 T^{4} )^{2} \))
$43$ (\( 1 + 25672 T + 364912302 T^{2} + 3774000748696 T^{3} + 21611482313284249 T^{4} \))(\( 1 - 26783614 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 35859614 T^{2} + 21611482313284249 T^{4} \))(\( 1 + 139567886 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 25672 T + 364912302 T^{2} - 3774000748696 T^{3} + 21611482313284249 T^{4} \))(\( 1 + 26346 T + 640605069 T^{2} + 8098987233524 T^{3} + 94174353771597567 T^{4} + \)\(56\!\cdots\!54\)\( T^{5} + \)\(31\!\cdots\!07\)\( T^{6} \))(\( 1 - 26346 T + 640605069 T^{2} - 8098987233524 T^{3} + 94174353771597567 T^{4} - \)\(56\!\cdots\!54\)\( T^{5} + \)\(31\!\cdots\!07\)\( T^{6} \))(\( 1 + 385757980 T^{2} + 71059469286049782 T^{4} + \)\(83\!\cdots\!20\)\( T^{6} + \)\(46\!\cdots\!01\)\( T^{8} \))
$47$ (\( 1 + 28296 T + 617782998 T^{2} + 6489546318072 T^{3} + 52599132235830049 T^{4} \))(\( 1 + 403777034 T^{2} + 52599132235830049 T^{4} \))(\( 1 + 442383274 T^{2} + 52599132235830049 T^{4} \))(\( 1 + 422513974 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 28296 T + 617782998 T^{2} - 6489546318072 T^{3} + 52599132235830049 T^{4} \))(\( 1 + 36762 T + 1119958377 T^{2} + 18490559326820 T^{3} + 256856861812773639 T^{4} + \)\(19\!\cdots\!38\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \))(\( 1 - 36762 T + 1119958377 T^{2} - 18490559326820 T^{3} + 256856861812773639 T^{4} - \)\(19\!\cdots\!38\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \))(\( 1 + 353765740 T^{2} + 66158288717340582 T^{4} + \)\(18\!\cdots\!60\)\( T^{6} + \)\(27\!\cdots\!01\)\( T^{8} \))
$53$ (\( 1 - 31340 T + 755347886 T^{2} - 13106246750620 T^{3} + 174887470365513049 T^{4} \))(\( ( 1 - 32226 T + 418195493 T^{2} )^{2} \))(\( ( 1 + 15486 T + 418195493 T^{2} )^{2} \))(\( ( 1 + 18154 T + 418195493 T^{2} )^{2} \))(\( 1 - 31340 T + 755347886 T^{2} - 13106246750620 T^{3} + 174887470365513049 T^{4} \))(\( 1 + 21162 T + 395789499 T^{2} - 881498066468 T^{3} + 165517384658528007 T^{4} + \)\(37\!\cdots\!38\)\( T^{5} + \)\(73\!\cdots\!57\)\( T^{6} \))(\( 1 + 21162 T + 395789499 T^{2} - 881498066468 T^{3} + 165517384658528007 T^{4} + \)\(37\!\cdots\!38\)\( T^{5} + \)\(73\!\cdots\!57\)\( T^{6} \))(\( ( 1 - 8748 T + 833865262 T^{2} - 3658374172764 T^{3} + 174887470365513049 T^{4} )^{2} \))
$59$ (\( 1 + 30800 T + 1666896598 T^{2} + 22019668409200 T^{3} + 511116753300641401 T^{4} \))(\( 1 + 270997718 T^{2} + 511116753300641401 T^{4} \))(\( 1 + 1376531158 T^{2} + 511116753300641401 T^{4} \))(\( 1 + 1041470358 T^{2} + 511116753300641401 T^{4} \))(\( 1 - 30800 T + 1666896598 T^{2} - 22019668409200 T^{3} + 511116753300641401 T^{4} \))(\( 1 + 35040 T + 1757220897 T^{2} + 50989349593920 T^{3} + 1256279917975876203 T^{4} + \)\(17\!\cdots\!40\)\( T^{5} + \)\(36\!\cdots\!99\)\( T^{6} \))(\( 1 - 35040 T + 1757220897 T^{2} - 50989349593920 T^{3} + 1256279917975876203 T^{4} - \)\(17\!\cdots\!40\)\( T^{5} + \)\(36\!\cdots\!99\)\( T^{6} \))(\( 1 + 488747308 T^{2} + 896846029819225302 T^{4} + \)\(24\!\cdots\!08\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \))
$61$ (\( 1 - 24540 T + 1447225822 T^{2} - 20726393226540 T^{3} + 713342911662882601 T^{4} \))(\( ( 1 - 26770 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 16750 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 42130 T + 844596301 T^{2} )^{2} \))(\( 1 - 24540 T + 1447225822 T^{2} - 20726393226540 T^{3} + 713342911662882601 T^{4} \))(\( 1 + 24138 T + 393086643 T^{2} - 6904061162564 T^{3} + 331999524650307543 T^{4} + \)\(17\!\cdots\!38\)\( T^{5} + \)\(60\!\cdots\!01\)\( T^{6} \))(\( 1 + 24138 T + 393086643 T^{2} - 6904061162564 T^{3} + 331999524650307543 T^{4} + \)\(17\!\cdots\!38\)\( T^{5} + \)\(60\!\cdots\!01\)\( T^{6} \))(\( ( 1 + 31012 T + 1250446302 T^{2} + 26192620486612 T^{3} + 713342911662882601 T^{4} )^{2} \))
$67$ (\( 1 + 34584 T + 2930656478 T^{2} + 46692726700488 T^{3} + 1822837804551761449 T^{4} \))(\( 1 + 219594834 T^{2} + 1822837804551761449 T^{4} \))(\( 1 + 2513562674 T^{2} + 1822837804551761449 T^{4} \))(\( 1 + 2438310974 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 34584 T + 2930656478 T^{2} - 46692726700488 T^{3} + 1822837804551761449 T^{4} \))(\( 1 - 9570 T + 659335509 T^{2} - 83080838420484 T^{3} + 890185424637524463 T^{4} - \)\(17\!\cdots\!30\)\( T^{5} + \)\(24\!\cdots\!43\)\( T^{6} \))(\( 1 + 9570 T + 659335509 T^{2} + 83080838420484 T^{3} + 890185424637524463 T^{4} + \)\(17\!\cdots\!30\)\( T^{5} + \)\(24\!\cdots\!43\)\( T^{6} \))(\( 1 + 3336863100 T^{2} + 6361754088145172822 T^{4} + \)\(60\!\cdots\!00\)\( T^{6} + \)\(33\!\cdots\!01\)\( T^{8} \))
$71$ (\( 1 - 12400 T + 1143670702 T^{2} - 22372443952400 T^{3} + 3255243551009881201 T^{4} \))(\( 1 + 681226622 T^{2} + 3255243551009881201 T^{4} \))(\( 1 + 1737195262 T^{2} + 3255243551009881201 T^{4} \))(\( 1 + 1542914862 T^{2} + 3255243551009881201 T^{4} \))(\( 1 + 12400 T + 1143670702 T^{2} + 22372443952400 T^{3} + 3255243551009881201 T^{4} \))(\( 1 + 88092 T + 7289446053 T^{2} + 329541325840584 T^{3} + 13151832521353701603 T^{4} + \)\(28\!\cdots\!92\)\( T^{5} + \)\(58\!\cdots\!51\)\( T^{6} \))(\( 1 - 88092 T + 7289446053 T^{2} - 329541325840584 T^{3} + 13151832521353701603 T^{4} - \)\(28\!\cdots\!92\)\( T^{5} + \)\(58\!\cdots\!51\)\( T^{6} \))(\( 1 + 6374750812 T^{2} + 16622748841310481702 T^{4} + \)\(20\!\cdots\!12\)\( T^{6} + \)\(10\!\cdots\!01\)\( T^{8} \))
$73$ (\( 1 + 7180 T + 284279286 T^{2} + 14884654037740 T^{3} + 4297625829703557649 T^{4} \))(\( ( 1 + 18534 T + 2073071593 T^{2} )^{2} \))(\( ( 1 + 20806 T + 2073071593 T^{2} )^{2} \))(\( ( 1 - 26266 T + 2073071593 T^{2} )^{2} \))(\( 1 + 7180 T + 284279286 T^{2} + 14884654037740 T^{3} + 4297625829703557649 T^{4} \))(\( 1 - 66750 T + 3875077479 T^{2} - 111360074258500 T^{3} + 8033313042388954047 T^{4} - \)\(28\!\cdots\!50\)\( T^{5} + \)\(89\!\cdots\!57\)\( T^{6} \))(\( 1 - 66750 T + 3875077479 T^{2} - 111360074258500 T^{3} + 8033313042388954047 T^{4} - \)\(28\!\cdots\!50\)\( T^{5} + \)\(89\!\cdots\!57\)\( T^{6} \))(\( ( 1 - 59700 T + 5029368950 T^{2} - 123762374102100 T^{3} + 4297625829703557649 T^{4} )^{2} \))
$79$ (\( 1 + 71840 T + 7430807198 T^{2} + 221055731704160 T^{3} + 9468276082626847201 T^{4} \))(\( 1 - 1369983522 T^{2} + 9468276082626847201 T^{4} \))(\( 1 + 1275471838 T^{2} + 9468276082626847201 T^{4} \))(\( 1 + 6078817438 T^{2} + 9468276082626847201 T^{4} \))(\( 1 - 71840 T + 7430807198 T^{2} - 221055731704160 T^{3} + 9468276082626847201 T^{4} \))(\( 1 - 92952 T + 11164448877 T^{2} - 573846024396496 T^{3} + 34353638858281213923 T^{4} - \)\(88\!\cdots\!52\)\( T^{5} + \)\(29\!\cdots\!99\)\( T^{6} \))(\( 1 + 92952 T + 11164448877 T^{2} + 573846024396496 T^{3} + 34353638858281213923 T^{4} + \)\(88\!\cdots\!52\)\( T^{5} + \)\(29\!\cdots\!99\)\( T^{6} \))(\( 1 + 10884258108 T^{2} + 48452581383610561862 T^{4} + \)\(10\!\cdots\!08\)\( T^{6} + \)\(89\!\cdots\!01\)\( T^{8} \))
$83$ (\( 1 - 31928 T + 5910993662 T^{2} - 125765689649704 T^{3} + 15516041187205853449 T^{4} \))(\( 1 + 1693436786 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 3421895214 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 1875047714 T^{2} + 15516041187205853449 T^{4} \))(\( 1 + 31928 T + 5910993662 T^{2} + 125765689649704 T^{3} + 15516041187205853449 T^{4} \))(\( 1 + 30258 T + 4293702405 T^{2} + 426012342532708 T^{3} + 16913068282241846415 T^{4} + \)\(46\!\cdots\!42\)\( T^{5} + \)\(61\!\cdots\!07\)\( T^{6} \))(\( 1 - 30258 T + 4293702405 T^{2} - 426012342532708 T^{3} + 16913068282241846415 T^{4} - \)\(46\!\cdots\!42\)\( T^{5} + \)\(61\!\cdots\!07\)\( T^{6} \))(\( 1 + 7906443964 T^{2} + 42876570344126662806 T^{4} + \)\(12\!\cdots\!36\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} \))
$89$ (\( 1 + 40748 T + 8570866774 T^{2} + 227539254427852 T^{3} + 31181719929966183601 T^{4} \))(\( ( 1 + 107590 T + 5584059449 T^{2} )^{2} \))(\( ( 1 + 18310 T + 5584059449 T^{2} )^{2} \))(\( ( 1 - 30570 T + 5584059449 T^{2} )^{2} \))(\( 1 + 40748 T + 8570866774 T^{2} + 227539254427852 T^{3} + 31181719929966183601 T^{4} \))(\( 1 - 172686 T + 26445328791 T^{2} - 2103593815517412 T^{3} + \)\(14\!\cdots\!59\)\( T^{4} - \)\(53\!\cdots\!86\)\( T^{5} + \)\(17\!\cdots\!49\)\( T^{6} \))(\( 1 - 172686 T + 26445328791 T^{2} - 2103593815517412 T^{3} + \)\(14\!\cdots\!59\)\( T^{4} - \)\(53\!\cdots\!86\)\( T^{5} + \)\(17\!\cdots\!49\)\( T^{6} \))(\( ( 1 - 104660 T + 13126874198 T^{2} - 584427661932340 T^{3} + 31181719929966183601 T^{4} )^{2} \))
$97$ (\( 1 + 190140 T + 19653817414 T^{2} + 1632796876465980 T^{3} + 73742412689492826049 T^{4} \))(\( ( 1 + 108838 T + 8587340257 T^{2} )^{2} \))(\( ( 1 - 49978 T + 8587340257 T^{2} )^{2} \))(\( ( 1 - 66882 T + 8587340257 T^{2} )^{2} \))(\( 1 + 190140 T + 19653817414 T^{2} + 1632796876465980 T^{3} + 73742412689492826049 T^{4} \))(\( 1 - 170910 T + 30283966671 T^{2} - 2852314667192740 T^{3} + \)\(26\!\cdots\!47\)\( T^{4} - \)\(12\!\cdots\!90\)\( T^{5} + \)\(63\!\cdots\!93\)\( T^{6} \))(\( 1 - 170910 T + 30283966671 T^{2} - 2852314667192740 T^{3} + \)\(26\!\cdots\!47\)\( T^{4} - \)\(12\!\cdots\!90\)\( T^{5} + \)\(63\!\cdots\!93\)\( T^{6} \))(\( ( 1 + 29564 T + 14772618438 T^{2} + 253876127357948 T^{3} + 73742412689492826049 T^{4} )^{2} \))
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