Properties

Label 400.6.c
Level $400$
Weight $6$
Character orbit 400.c
Rep. character $\chi_{400}(49,\cdot)$
Character field $\Q$
Dimension $44$
Newform subspaces $16$
Sturm bound $360$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(360\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(400, [\chi])\).

Total New Old
Modular forms 318 46 272
Cusp forms 282 44 238
Eisenstein series 36 2 34

Trace form

\( 44q - 3400q^{9} + O(q^{10}) \) \( 44q - 3400q^{9} - 724q^{11} - 196q^{19} - 2128q^{21} - 12240q^{29} + 12928q^{31} - 10376q^{39} - 8404q^{41} - 85604q^{49} - 115444q^{51} + 77136q^{59} + 2288q^{61} + 43648q^{69} + 25016q^{71} + 58888q^{79} + 156724q^{81} - 132588q^{89} + 160104q^{91} + 244272q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(400, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
400.6.c.a \(2\) \(64.154\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+13iq^{3}-11iq^{7}-433q^{9}+768q^{11}+\cdots\)
400.6.c.b \(2\) \(64.154\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+12iq^{3}+86iq^{7}-333q^{9}-132q^{11}+\cdots\)
400.6.c.c \(2\) \(64.154\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+11iq^{3}-109iq^{7}-241q^{9}+480q^{11}+\cdots\)
400.6.c.d \(2\) \(64.154\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+10iq^{3}+12iq^{7}-157q^{9}-124q^{11}+\cdots\)
400.6.c.e \(2\) \(64.154\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+9iq^{3}+11^{2}iq^{7}-3^{4}q^{9}-656q^{11}+\cdots\)
400.6.c.f \(2\) \(64.154\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+6iq^{3}-44iq^{7}+99q^{9}-540q^{11}+\cdots\)
400.6.c.g \(2\) \(64.154\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+11iq^{3}-142iq^{7}+122q^{9}-777q^{11}+\cdots\)
400.6.c.h \(2\) \(64.154\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+4iq^{3}-54iq^{7}+179q^{9}+604q^{11}+\cdots\)
400.6.c.i \(2\) \(64.154\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+3iq^{3}+59iq^{7}+207q^{9}-192q^{11}+\cdots\)
400.6.c.j \(2\) \(64.154\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+2iq^{3}+96iq^{7}+227q^{9}+148q^{11}+\cdots\)
400.6.c.k \(2\) \(64.154\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{3}-31iq^{7}+239q^{9}+12^{2}q^{11}+\cdots\)
400.6.c.l \(4\) \(64.154\) \(\Q(i, \sqrt{129})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-3\beta _{1}+\beta _{2})q^{3}+(-13\beta _{1}-3\beta _{2}+\cdots)q^{7}+\cdots\)
400.6.c.m \(4\) \(64.154\) \(\Q(i, \sqrt{409})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{1}+2\beta _{2})q^{3}+(-6\beta _{1}+4\beta _{2}+\cdots)q^{7}+\cdots\)
400.6.c.n \(4\) \(64.154\) \(\Q(i, \sqrt{241})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{1}-2\beta _{2})q^{3}+(-2\beta _{1}+20\beta _{2})q^{7}+\cdots\)
400.6.c.o \(4\) \(64.154\) \(\Q(i, \sqrt{241})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-4\beta _{1}-\beta _{2})q^{3}+(-4\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
400.6.c.p \(6\) \(64.154\) \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{3}+(-24\beta _{1}-\beta _{3}-\beta _{4})q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(400, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(400, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 190 T^{2} + 59049 T^{4} \))(\( 1 + 90 T^{2} + 59049 T^{4} \))(\( 1 - 2 T^{2} + 59049 T^{4} \))(\( 1 - 86 T^{2} + 59049 T^{4} \))(\( 1 - 162 T^{2} + 59049 T^{4} \))(\( 1 - 342 T^{2} + 59049 T^{4} \))(\( 1 - 365 T^{2} + 59049 T^{4} \))(\( 1 - 422 T^{2} + 59049 T^{4} \))(\( 1 - 450 T^{2} + 59049 T^{4} \))(\( 1 - 470 T^{2} + 59049 T^{4} \))(\( 1 - 482 T^{2} + 59049 T^{4} \))(\( 1 + 132 T^{2} + 48150 T^{4} + 7794468 T^{6} + 3486784401 T^{8} \))(\( 1 + 46 T^{2} - 44973 T^{4} + 2716254 T^{6} + 3486784401 T^{8} \))(\( 1 - 290 T^{2} + 42723 T^{4} - 17124210 T^{6} + 3486784401 T^{8} \))(\( 1 - 458 T^{2} + 155115 T^{4} - 27044442 T^{6} + 3486784401 T^{8} \))(\( 1 - 575 T^{2} + 219750 T^{4} - 66390939 T^{6} + 12976017750 T^{8} - 2004901030575 T^{10} + 205891132094649 T^{12} \))
$5$ 1
$7$ (\( 1 - 33130 T^{2} + 282475249 T^{4} \))(\( 1 - 4030 T^{2} + 282475249 T^{4} \))(\( 1 + 13910 T^{2} + 282475249 T^{4} \))(\( 1 - 33038 T^{2} + 282475249 T^{4} \))(\( 1 + 24950 T^{2} + 282475249 T^{4} \))(\( 1 - 25870 T^{2} + 282475249 T^{4} \))(\( 1 - 13450 T^{2} + 282475249 T^{4} \))(\( 1 - 21950 T^{2} + 282475249 T^{4} \))(\( 1 - 19690 T^{2} + 282475249 T^{4} \))(\( 1 + 3250 T^{2} + 282475249 T^{4} \))(\( 1 - 29770 T^{2} + 282475249 T^{4} \))(\( 1 - 56588 T^{2} + 1352943558 T^{4} - 15984709390412 T^{6} + 79792266297612001 T^{8} \))(\( 1 - 36980 T^{2} + 883272198 T^{4} - 10445934708020 T^{6} + 79792266297612001 T^{8} \))(\( 1 - 45300 T^{2} + 1039412998 T^{4} - 12796128779700 T^{6} + 79792266297612001 T^{8} \))(\( 1 - 65268 T^{2} + 1629866758 T^{4} - 18436594551732 T^{6} + 79792266297612001 T^{8} \))(\( 1 + 16002 T^{2} + 295251327 T^{4} + 2616304552796 T^{6} + 83401192111905423 T^{8} + \)\(12\!\cdots\!02\)\( T^{10} + \)\(22\!\cdots\!49\)\( T^{12} \))
$11$ (\( ( 1 - 768 T + 161051 T^{2} )^{2} \))(\( ( 1 + 132 T + 161051 T^{2} )^{2} \))(\( ( 1 - 480 T + 161051 T^{2} )^{2} \))(\( ( 1 + 124 T + 161051 T^{2} )^{2} \))(\( ( 1 + 656 T + 161051 T^{2} )^{2} \))(\( ( 1 + 540 T + 161051 T^{2} )^{2} \))(\( ( 1 + 777 T + 161051 T^{2} )^{2} \))(\( ( 1 - 604 T + 161051 T^{2} )^{2} \))(\( ( 1 + 192 T + 161051 T^{2} )^{2} \))(\( ( 1 - 148 T + 161051 T^{2} )^{2} \))(\( ( 1 - 144 T + 161051 T^{2} )^{2} \))(\( ( 1 + 560 T + 381926 T^{2} + 90188560 T^{3} + 25937424601 T^{4} )^{2} \))(\( ( 1 - 60 T + 230977 T^{2} - 9663060 T^{3} + 25937424601 T^{4} )^{2} \))(\( ( 1 - 196 T + 181081 T^{2} - 31565996 T^{3} + 25937424601 T^{4} )^{2} \))(\( ( 1 - 200 T + 225821 T^{2} - 32210200 T^{3} + 25937424601 T^{4} )^{2} \))(\( ( 1 - 19 T + 398752 T^{2} - 4577119 T^{3} + 64219408352 T^{4} - 492811067419 T^{5} + 4177248169415651 T^{6} )^{2} \))
$13$ (\( 1 - 740470 T^{2} + 137858491849 T^{4} \))(\( 1 + 152330 T^{2} + 137858491849 T^{4} \))(\( 1 - 355702 T^{2} + 137858491849 T^{4} \))(\( 1 - 514102 T^{2} + 137858491849 T^{4} \))(\( 1 - 700150 T^{2} + 137858491849 T^{4} \))(\( 1 - 567862 T^{2} + 137858491849 T^{4} \))(\( 1 + 38870 T^{2} + 137858491849 T^{4} \))(\( 1 - 648950 T^{2} + 137858491849 T^{4} \))(\( 1 + 480650 T^{2} + 137858491849 T^{4} \))(\( 1 - 660790 T^{2} + 137858491849 T^{4} \))(\( 1 - 314870 T^{2} + 137858491849 T^{4} \))(\( 1 - 373292 T^{2} + 167403787638 T^{4} - 51461472139296908 T^{6} + \)\(19\!\cdots\!01\)\( T^{8} \))(\( 1 - 590804 T^{2} + 163581027702 T^{4} - 81447348418356596 T^{6} + \)\(19\!\cdots\!01\)\( T^{8} \))(\( 1 - 1296980 T^{2} + 688260462198 T^{4} - 178799706758316020 T^{6} + \)\(19\!\cdots\!01\)\( T^{8} \))(\( 1 - 6612 T^{2} + 47343141238 T^{4} - 911520348105588 T^{6} + \)\(19\!\cdots\!01\)\( T^{8} \))(\( 1 - 1246462 T^{2} + 812446351367 T^{4} - 359332324029982276 T^{6} + \)\(11\!\cdots\!83\)\( T^{8} - \)\(23\!\cdots\!62\)\( T^{10} + \)\(26\!\cdots\!49\)\( T^{12} \))
$17$ (\( 1 - 2696830 T^{2} + 2015993900449 T^{4} \))(\( 1 - 2790430 T^{2} + 2015993900449 T^{4} \))(\( 1 - 2805118 T^{2} + 2015993900449 T^{4} \))(\( 1 - 1404510 T^{2} + 2015993900449 T^{4} \))(\( 1 + 16386 T^{2} + 2015993900449 T^{4} \))(\( 1 - 2486878 T^{2} + 2015993900449 T^{4} \))(\( 1 - 2838985 T^{2} + 2015993900449 T^{4} \))(\( 1 - 1974814 T^{2} + 2015993900449 T^{4} \))(\( 1 - 2259070 T^{2} + 2015993900449 T^{4} \))(\( 1 - 24030 T^{2} + 2015993900449 T^{4} \))(\( 1 - 1423614 T^{2} + 2015993900449 T^{4} \))(\( 1 + 1613316 T^{2} + 4602934743878 T^{4} + 3252435215496778884 T^{6} + \)\(40\!\cdots\!01\)\( T^{8} \))(\( 1 - 1327586 T^{2} + 3973871730147 T^{4} - 2676405278321486114 T^{6} + \)\(40\!\cdots\!01\)\( T^{8} \))(\( 1 - 2340610 T^{2} + 2927557675523 T^{4} - 4718655483329933890 T^{6} + \)\(40\!\cdots\!01\)\( T^{8} \))(\( 1 - 5517394 T^{2} + 11637628825043 T^{4} - 11123032650373909906 T^{6} + \)\(40\!\cdots\!01\)\( T^{8} \))(\( 1 - 3096363 T^{2} + 4896853719798 T^{4} - 6718235532470229759 T^{6} + \)\(98\!\cdots\!02\)\( T^{8} - \)\(12\!\cdots\!63\)\( T^{10} + \)\(81\!\cdots\!49\)\( T^{12} \))
$19$ (\( ( 1 - 1100 T + 2476099 T^{2} )^{2} \))(\( ( 1 - 500 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 1204 T + 2476099 T^{2} )^{2} \))(\( ( 1 - 3044 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 1364 T + 2476099 T^{2} )^{2} \))(\( ( 1 - 836 T + 2476099 T^{2} )^{2} \))(\( ( 1 - 1145 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 1324 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 2740 T + 2476099 T^{2} )^{2} \))(\( ( 1 - 1060 T + 2476099 T^{2} )^{2} \))(\( ( 1 - 556 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 1000 T + 4533462 T^{2} + 2476099000 T^{3} + 6131066257801 T^{4} )^{2} \))(\( ( 1 - 2092 T + 5218089 T^{2} - 5179999108 T^{3} + 6131066257801 T^{4} )^{2} \))(\( ( 1 + 3180 T + 7185073 T^{2} + 7873994820 T^{3} + 6131066257801 T^{4} )^{2} \))(\( ( 1 + 840 T + 4289677 T^{2} + 2079923160 T^{3} + 6131066257801 T^{4} )^{2} \))(\( ( 1 - 1221 T + 2478936 T^{2} + 986654423 T^{3} + 6138090950664 T^{4} - 7486031900775021 T^{5} + 15181127029874798299 T^{6} )^{2} \))
$23$ (\( 1 - 8928490 T^{2} + 41426511213649 T^{4} \))(\( 1 - 170590 T^{2} + 41426511213649 T^{4} \))(\( 1 - 2722090 T^{2} + 41426511213649 T^{4} \))(\( 1 - 12838830 T^{2} + 41426511213649 T^{4} \))(\( 1 - 8041482 T^{2} + 41426511213649 T^{4} \))(\( 1 + 3970130 T^{2} + 41426511213649 T^{4} \))(\( 1 - 9435370 T^{2} + 41426511213649 T^{4} \))(\( 1 - 12146782 T^{2} + 41426511213649 T^{4} \))(\( 1 - 10420330 T^{2} + 41426511213649 T^{4} \))(\( 1 - 4016110 T^{2} + 41426511213649 T^{4} \))(\( 1 - 8111562 T^{2} + 41426511213649 T^{4} \))(\( 1 - 7126092 T^{2} + 48612883261958 T^{4} - \)\(29\!\cdots\!08\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} \))(\( 1 - 10160180 T^{2} + 108548175292998 T^{4} - \)\(42\!\cdots\!20\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} \))(\( 1 - 24511220 T^{2} + 233031884985798 T^{4} - \)\(10\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} \))(\( 1 - 19211092 T^{2} + 166300343917958 T^{4} - \)\(79\!\cdots\!08\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} \))(\( 1 + 4048866 T^{2} + 106559597795871 T^{4} + \)\(30\!\cdots\!48\)\( T^{6} + \)\(44\!\cdots\!79\)\( T^{8} + \)\(69\!\cdots\!66\)\( T^{10} + \)\(71\!\cdots\!49\)\( T^{12} \))
$29$ (\( ( 1 - 5610 T + 20511149 T^{2} )^{2} \))(\( ( 1 + 2190 T + 20511149 T^{2} )^{2} \))(\( ( 1 + 5526 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 3282 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 2218 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 594 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 4920 T + 20511149 T^{2} )^{2} \))(\( ( 1 + 5902 T + 20511149 T^{2} )^{2} \))(\( ( 1 + 5910 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 3410 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 1578 T + 20511149 T^{2} )^{2} \))(\( ( 1 + 1340 T - 8758306 T^{2} + 27484939660 T^{3} + 420707233300201 T^{4} )^{2} \))(\( ( 1 + 3552 T + 20618074 T^{2} + 72855601248 T^{3} + 420707233300201 T^{4} )^{2} \))(\( ( 1 - 3920 T + 44478298 T^{2} - 80403704080 T^{3} + 420707233300201 T^{4} )^{2} \))(\( ( 1 - 4680 T + 41219034 T^{2} - 95992177320 T^{3} + 420707233300201 T^{4} )^{2} \))(\( ( 1 + 11912 T + 107337223 T^{2} + 544650638288 T^{3} + 2201609774199227 T^{4} + 5011464563071994312 T^{5} + \)\(86\!\cdots\!49\)\( T^{6} )^{2} \))
$31$ (\( ( 1 - 3988 T + 28629151 T^{2} )^{2} \))(\( ( 1 + 2312 T + 28629151 T^{2} )^{2} \))(\( ( 1 + 9356 T + 28629151 T^{2} )^{2} \))(\( ( 1 - 5728 T + 28629151 T^{2} )^{2} \))(\( ( 1 - 1700 T + 28629151 T^{2} )^{2} \))(\( ( 1 + 4256 T + 28629151 T^{2} )^{2} \))(\( ( 1 + 1802 T + 28629151 T^{2} )^{2} \))(\( ( 1 - 3320 T + 28629151 T^{2} )^{2} \))(\( ( 1 - 6868 T + 28629151 T^{2} )^{2} \))(\( ( 1 - 2448 T + 28629151 T^{2} )^{2} \))(\( ( 1 + 9660 T + 28629151 T^{2} )^{2} \))(\( ( 1 - 2248 T + 57017022 T^{2} - 64358331448 T^{3} + 819628286980801 T^{4} )^{2} \))(\( ( 1 - 8888 T + 67804938 T^{2} - 254455894088 T^{3} + 819628286980801 T^{4} )^{2} \))(\( ( 1 - 1096 T - 15343894 T^{2} - 31377549496 T^{3} + 819628286980801 T^{4} )^{2} \))(\( ( 1 - 5008 T + 33327162 T^{2} - 143374788208 T^{3} + 819628286980801 T^{4} )^{2} \))(\( ( 1 + 7442 T + 82803193 T^{2} + 411681472084 T^{3} + 2370585115679143 T^{4} + 6099673711711121042 T^{5} + \)\(23\!\cdots\!51\)\( T^{6} )^{2} \))
$37$ (\( 1 - 138667750 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 12305350 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 107125990 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 32061638 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 137972198 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 138599110 T^{2} + 4808584372417849 T^{4} \))(\( 1 + 34971770 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 22608838 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 108239590 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 138654790 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 126198758 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 211585036 T^{2} + 19959790722550422 T^{4} - \)\(10\!\cdots\!64\)\( T^{6} + \)\(23\!\cdots\!01\)\( T^{8} \))(\( 1 - 33594380 T^{2} - 2634705186058602 T^{4} - \)\(16\!\cdots\!20\)\( T^{6} + \)\(23\!\cdots\!01\)\( T^{8} \))(\( 1 - 257567180 T^{2} + 26166130610514198 T^{4} - \)\(12\!\cdots\!20\)\( T^{6} + \)\(23\!\cdots\!01\)\( T^{8} \))(\( 1 - 86339436 T^{2} + 2659590797035222 T^{4} - \)\(41\!\cdots\!64\)\( T^{6} + \)\(23\!\cdots\!01\)\( T^{8} \))(\( 1 - 132585234 T^{2} + 7422206884581399 T^{4} - \)\(23\!\cdots\!84\)\( T^{6} + \)\(35\!\cdots\!51\)\( T^{8} - \)\(30\!\cdots\!34\)\( T^{10} + \)\(11\!\cdots\!49\)\( T^{12} \))
$41$ (\( ( 1 - 1542 T + 115856201 T^{2} )^{2} \))(\( ( 1 - 1242 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 14394 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 8886 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 1818 T + 115856201 T^{2} )^{2} \))(\( ( 1 - 17226 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 15123 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 17958 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 378 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 9398 T + 115856201 T^{2} )^{2} \))(\( ( 1 - 7462 T + 115856201 T^{2} )^{2} \))(\( ( 1 - 23076 T + 352280470 T^{2} - 2673497694276 T^{3} + 13422659310152401 T^{4} )^{2} \))(\( ( 1 + 12438 T + 217381963 T^{2} + 1441019428038 T^{3} + 13422659310152401 T^{4} )^{2} \))(\( ( 1 - 27754 T + 414643531 T^{2} - 3215473002554 T^{3} + 13422659310152401 T^{4} )^{2} \))(\( ( 1 + 5334 T + 27686155 T^{2} + 617976976134 T^{3} + 13422659310152401 T^{4} )^{2} \))(\( ( 1 - 3223 T + 204190518 T^{2} - 833781103267 T^{3} + 23656737695702118 T^{4} - 43261230956621188423 T^{5} + \)\(15\!\cdots\!01\)\( T^{6} )^{2} \))
$43$ (\( 1 - 268756210 T^{2} + 21611482313284249 T^{4} \))(\( 1 + 131332490 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 293879986 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 209597542 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 183051730 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 147606886 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 232488550 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 208195190 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 288092530 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 292469350 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 243407890 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 313073372 T^{2} + 49182412950657078 T^{4} - \)\(67\!\cdots\!28\)\( T^{6} + \)\(46\!\cdots\!01\)\( T^{8} \))(\( 1 - 540244172 T^{2} + 116157205788519894 T^{4} - \)\(11\!\cdots\!28\)\( T^{6} + \)\(46\!\cdots\!01\)\( T^{8} \))(\( 1 - 37863500 T^{2} + 41125859560630998 T^{4} - \)\(81\!\cdots\!00\)\( T^{6} + \)\(46\!\cdots\!01\)\( T^{8} \))(\( 1 - 29960652 T^{2} + 43433374484607958 T^{4} - \)\(64\!\cdots\!48\)\( T^{6} + \)\(46\!\cdots\!01\)\( T^{8} \))(\( 1 - 78906162 T^{2} - 7556498485091913 T^{4} - \)\(28\!\cdots\!76\)\( T^{6} - \)\(16\!\cdots\!37\)\( T^{8} - \)\(36\!\cdots\!62\)\( T^{10} + \)\(10\!\cdots\!49\)\( T^{12} \))
$47$ (\( 1 + 153278630 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 415288270 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 197996698 T^{2} + 52599132235830049 T^{4} \))(\( 1 + 101294882 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 312908538 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 457010398 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 413370190 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 362728398 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 286503130 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 312570270 T^{2} + 52599132235830049 T^{4} \))(\( 1 + 341860422 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 104863596 T^{2} + 104529727880710502 T^{4} - \)\(55\!\cdots\!04\)\( T^{6} + \)\(27\!\cdots\!01\)\( T^{8} \))(\( 1 - 902407196 T^{2} + 308772689280765702 T^{4} - \)\(47\!\cdots\!04\)\( T^{6} + \)\(27\!\cdots\!01\)\( T^{8} \))(\( 1 - 298326940 T^{2} + 32136931855726598 T^{4} - \)\(15\!\cdots\!60\)\( T^{6} + \)\(27\!\cdots\!01\)\( T^{8} \))(\( ( 1 - 288752718 T^{2} + 52599132235830049 T^{4} )^{2} \))(\( 1 - 503679594 T^{2} + 7712277691779759 T^{4} + \)\(25\!\cdots\!96\)\( T^{6} + \)\(40\!\cdots\!91\)\( T^{8} - \)\(13\!\cdots\!94\)\( T^{10} + \)\(14\!\cdots\!49\)\( T^{12} \))
$53$ (\( 1 - 635715430 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 392614630 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 817259110 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 699828390 T^{2} + 174887470365513049 T^{4} \))(\( 1 + 225456410 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 456374950 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 824735590 T^{2} + 174887470365513049 T^{4} \))(\( 1 + 151705370 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 752228710 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 267759270 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 666192870 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 1357205900 T^{2} + 805869805574922198 T^{4} - \)\(23\!\cdots\!00\)\( T^{6} + \)\(30\!\cdots\!01\)\( T^{8} \))(\( 1 - 1189716620 T^{2} + 656395125486896598 T^{4} - \)\(20\!\cdots\!80\)\( T^{6} + \)\(30\!\cdots\!01\)\( T^{8} \))(\( 1 - 1240678540 T^{2} + 709794326287792598 T^{4} - \)\(21\!\cdots\!60\)\( T^{6} + \)\(30\!\cdots\!01\)\( T^{8} \))(\( 1 - 514633100 T^{2} + 347542581493770198 T^{4} - \)\(90\!\cdots\!00\)\( T^{6} + \)\(30\!\cdots\!01\)\( T^{8} \))(\( 1 + 129510670 T^{2} + 313108616307187447 T^{4} + \)\(44\!\cdots\!60\)\( T^{6} + \)\(54\!\cdots\!03\)\( T^{8} + \)\(39\!\cdots\!70\)\( T^{10} + \)\(53\!\cdots\!49\)\( T^{12} \))
$59$ (\( ( 1 - 28380 T + 714924299 T^{2} )^{2} \))(\( ( 1 - 7980 T + 714924299 T^{2} )^{2} \))(\( ( 1 + 11748 T + 714924299 T^{2} )^{2} \))(\( ( 1 - 16876 T + 714924299 T^{2} )^{2} \))(\( ( 1 - 8668 T + 714924299 T^{2} )^{2} \))(\( ( 1 + 7668 T + 714924299 T^{2} )^{2} \))(\( ( 1 - 33960 T + 714924299 T^{2} )^{2} \))(\( ( 1 - 33228 T + 714924299 T^{2} )^{2} \))(\( ( 1 + 34980 T + 714924299 T^{2} )^{2} \))(\( ( 1 + 20020 T + 714924299 T^{2} )^{2} \))(\( ( 1 + 37092 T + 714924299 T^{2} )^{2} \))(\( ( 1 - 62584 T + 2277965606 T^{2} - 44742822328616 T^{3} + 511116753300641401 T^{4} )^{2} \))(\( ( 1 + 36696 T + 1234961302 T^{2} + 26234862076104 T^{3} + 511116753300641401 T^{4} )^{2} \))(\( ( 1 - 11960 T + 1234152598 T^{2} - 8550494616040 T^{3} + 511116753300641401 T^{4} )^{2} \))(\( ( 1 + 81776 T + 3059970646 T^{2} + 58463649475024 T^{3} + 511116753300641401 T^{4} )^{2} \))(\( ( 1 - 64912 T + 3469849057 T^{2} - 101432669884000 T^{3} + 2480679404711536043 T^{4} - \)\(33\!\cdots\!12\)\( T^{5} + \)\(36\!\cdots\!99\)\( T^{6} )^{2} \))
$61$ (\( ( 1 - 5522 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 16622 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 13202 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 18482 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 34670 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 34738 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 47402 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 40210 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 9838 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 32302 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 39570 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 14108 T + 1110042462 T^{2} - 11915564614508 T^{3} + 713342911662882601 T^{4} )^{2} \))(\( ( 1 - 19204 T + 627029406 T^{2} - 16219627364404 T^{3} + 713342911662882601 T^{4} )^{2} \))(\( ( 1 + 24396 T + 1596983806 T^{2} + 20604771359196 T^{3} + 713342911662882601 T^{4} )^{2} \))(\( ( 1 + 46932 T + 2239379182 T^{2} + 39638593598532 T^{3} + 713342911662882601 T^{4} )^{2} \))(\( ( 1 - 22478 T + 1331345683 T^{2} - 9784946414356 T^{3} + 1124449639214118583 T^{4} - \)\(16\!\cdots\!78\)\( T^{5} + \)\(60\!\cdots\!01\)\( T^{6} )^{2} \))
$67$ (\( 1 - 2088083650 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 2696981350 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 2567032450 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 2459007190 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 437725858 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 2224486870 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 2526616885 T^{2} + 1822837804551761449 T^{4} \))(\( 1 + 764720282 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 1563076930 T^{2} + 1822837804551761449 T^{4} \))(\( 1 + 1017334570 T^{2} + 1822837804551761449 T^{4} \))(\( 1 + 518496542 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 1448611388 T^{2} + 3060385933795078038 T^{4} - \)\(26\!\cdots\!12\)\( T^{6} + \)\(33\!\cdots\!01\)\( T^{8} \))(\( 1 - 974988050 T^{2} + 2516736182389071123 T^{4} - \)\(17\!\cdots\!50\)\( T^{6} + \)\(33\!\cdots\!01\)\( T^{8} \))(\( 1 - 4294993410 T^{2} + 8014205534576787523 T^{4} - \)\(78\!\cdots\!90\)\( T^{6} + \)\(33\!\cdots\!01\)\( T^{8} \))(\( 1 - 2982943418 T^{2} + 5751121538625041883 T^{4} - \)\(54\!\cdots\!82\)\( T^{6} + \)\(33\!\cdots\!01\)\( T^{8} \))(\( 1 - 6588586311 T^{2} + 19273295040514514166 T^{4} - \)\(32\!\cdots\!63\)\( T^{6} + \)\(35\!\cdots\!34\)\( T^{8} - \)\(21\!\cdots\!11\)\( T^{10} + \)\(60\!\cdots\!49\)\( T^{12} \))
$71$ (\( ( 1 + 42372 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 24528 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 29532 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 31960 T + 1804229351 T^{2} )^{2} \))(\( ( 1 + 948 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 46872 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 7548 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 55312 T + 1804229351 T^{2} )^{2} \))(\( ( 1 + 70212 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 32648 T + 1804229351 T^{2} )^{2} \))(\( ( 1 + 45588 T + 1804229351 T^{2} )^{2} \))(\( ( 1 + 47208 T + 4011779662 T^{2} + 85174059202008 T^{3} + 3255243551009881201 T^{4} )^{2} \))(\( ( 1 + 2736 T + 2006886526 T^{2} + 4936371504336 T^{3} + 3255243551009881201 T^{4} )^{2} \))(\( ( 1 - 87296 T + 5453356606 T^{2} - 157502005424896 T^{3} + 3255243551009881201 T^{4} )^{2} \))(\( ( 1 + 7448 T + 3593807902 T^{2} + 13437900206248 T^{3} + 3255243551009881201 T^{4} )^{2} \))(\( ( 1 + 86676 T + 6149511717 T^{2} + 281168787837912 T^{3} + 11095129534129805667 T^{4} + \)\(28\!\cdots\!76\)\( T^{5} + \)\(58\!\cdots\!51\)\( T^{6} )^{2} \))
$73$ (\( 1 - 1429023310 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 3726958510 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 3010587982 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 4122270190 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 164280782 T^{2} + 4297625829703557649 T^{4} \))(\( 1 + 418480658 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 567591145 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 3403144622 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 3662758990 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 2642720110 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 4005910222 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 4251788572 T^{2} + 9098112686809466598 T^{4} - \)\(18\!\cdots\!28\)\( T^{6} + \)\(18\!\cdots\!01\)\( T^{8} \))(\( 1 - 8204978114 T^{2} + 25425197509477462947 T^{4} - \)\(35\!\cdots\!86\)\( T^{6} + \)\(18\!\cdots\!01\)\( T^{8} \))(\( 1 - 5672940770 T^{2} + 16272726164284201923 T^{4} - \)\(24\!\cdots\!30\)\( T^{6} + \)\(18\!\cdots\!01\)\( T^{8} \))(\( 1 - 2368673842 T^{2} + 9983110585747236243 T^{4} - \)\(10\!\cdots\!58\)\( T^{6} + \)\(18\!\cdots\!01\)\( T^{8} \))(\( 1 + 171223381 T^{2} + 1889571079718722886 T^{4} - \)\(12\!\cdots\!47\)\( T^{6} + \)\(81\!\cdots\!14\)\( T^{8} + \)\(31\!\cdots\!81\)\( T^{10} + \)\(79\!\cdots\!49\)\( T^{12} \))
$79$ (\( ( 1 + 39640 T + 3077056399 T^{2} )^{2} \))(\( ( 1 + 46240 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 31208 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 44560 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 46536 T + 3077056399 T^{2} )^{2} \))(\( ( 1 + 76912 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 75830 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 31456 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 4520 T + 3077056399 T^{2} )^{2} \))(\( ( 1 + 33360 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 94216 T + 3077056399 T^{2} )^{2} \))(\( ( 1 + 65904 T + 3994274078 T^{2} + 202790324919696 T^{3} + 9468276082626847201 T^{4} )^{2} \))(\( ( 1 + 16184 T + 6157384362 T^{2} + 49799080761416 T^{3} + 9468276082626847201 T^{4} )^{2} \))(\( ( 1 - 65480 T + 5707696298 T^{2} - 201485653006520 T^{3} + 9468276082626847201 T^{4} )^{2} \))(\( ( 1 + 108104 T + 6816153098 T^{2} + 332642104957496 T^{3} + 9468276082626847201 T^{4} )^{2} \))(\( ( 1 - 21982 T + 8034849513 T^{2} - 109220050198796 T^{3} + 24723685108978683687 T^{4} - \)\(20\!\cdots\!82\)\( T^{5} + \)\(29\!\cdots\!99\)\( T^{6} )^{2} \))
$83$ (\( 1 - 4298931010 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 5217997510 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 6398448130 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 3340172790 T^{2} + 15516041187205853449 T^{4} \))(\( 1 + 3451998 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 3292624630 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 5734483885 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 7275280582 T^{2} + 15516041187205853449 T^{4} \))(\( 1 + 4019056190 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 7598656630 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 6886964962 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 9324010812 T^{2} + 49682906641812448598 T^{4} - \)\(14\!\cdots\!88\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} \))(\( 1 - 7195965410 T^{2} + 40258768525685533923 T^{4} - \)\(11\!\cdots\!90\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} \))(\( 1 - 11381926610 T^{2} + 63038986097658341523 T^{4} - \)\(17\!\cdots\!90\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} \))(\( 1 - 8274508042 T^{2} + 45513777799710942443 T^{4} - \)\(12\!\cdots\!58\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} \))(\( 1 - 6746420239 T^{2} + 48212503738013039526 T^{4} - \)\(20\!\cdots\!07\)\( T^{6} + \)\(74\!\cdots\!74\)\( T^{8} - \)\(16\!\cdots\!39\)\( T^{10} + \)\(37\!\cdots\!49\)\( T^{12} \))
$89$ (\( ( 1 + 57690 T + 5584059449 T^{2} )^{2} \))(\( ( 1 - 110310 T + 5584059449 T^{2} )^{2} \))(\( ( 1 + 119514 T + 5584059449 T^{2} )^{2} \))(\( ( 1 + 71994 T + 5584059449 T^{2} )^{2} \))(\( ( 1 - 104934 T + 5584059449 T^{2} )^{2} \))(\( ( 1 + 29754 T + 5584059449 T^{2} )^{2} \))(\( ( 1 - 30585 T + 5584059449 T^{2} )^{2} \))(\( ( 1 - 90854 T + 5584059449 T^{2} )^{2} \))(\( ( 1 + 38490 T + 5584059449 T^{2} )^{2} \))(\( ( 1 + 101370 T + 5584059449 T^{2} )^{2} \))(\( ( 1 - 94054 T + 5584059449 T^{2} )^{2} \))(\( ( 1 - 55020 T + 10818978262 T^{2} - 307234950883980 T^{3} + 31181719929966183601 T^{4} )^{2} \))(\( ( 1 - 47322 T - 1006825781 T^{2} - 264248861245578 T^{3} + 31181719929966183601 T^{4} )^{2} \))(\( ( 1 - 72810 T + 5926578523 T^{2} - 406575368481690 T^{3} + 31181719929966183601 T^{4} )^{2} \))(\( ( 1 + 70990 T + 6345035107 T^{2} + 396412380284510 T^{3} + 31181719929966183601 T^{4} )^{2} \))(\( ( 1 + 182381 T + 27173913486 T^{2} + 2227663367824697 T^{3} + \)\(15\!\cdots\!14\)\( T^{4} + \)\(56\!\cdots\!81\)\( T^{5} + \)\(17\!\cdots\!49\)\( T^{6} )^{2} \))
$97$ (\( 1 + 3671481410 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 11030942590 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 8214543550 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 14786794558 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 15860327998 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 2193410110 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 6354936190 T^{2} + 73742412689492826049 T^{4} \))(\( 1 + 6759265922 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 17171001790 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 3004635070 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 16612326718 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 1419021116 T^{2} - 92174956617487206138 T^{4} - \)\(10\!\cdots\!84\)\( T^{6} + \)\(54\!\cdots\!01\)\( T^{8} \))(\( 1 - 28221621500 T^{2} + \)\(34\!\cdots\!98\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{6} + \)\(54\!\cdots\!01\)\( T^{8} \))(\( 1 - 11961289340 T^{2} + 68433641741002238598 T^{4} - \)\(88\!\cdots\!60\)\( T^{6} + \)\(54\!\cdots\!01\)\( T^{8} \))(\( 1 - 22699126076 T^{2} + \)\(24\!\cdots\!42\)\( T^{4} - \)\(16\!\cdots\!24\)\( T^{6} + \)\(54\!\cdots\!01\)\( T^{8} \))(\( 1 - 28007218554 T^{2} + \)\(40\!\cdots\!19\)\( T^{4} - \)\(40\!\cdots\!24\)\( T^{6} + \)\(29\!\cdots\!31\)\( T^{8} - \)\(15\!\cdots\!54\)\( T^{10} + \)\(40\!\cdots\!49\)\( T^{12} \))
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