Properties

Label 360.6.a
Level 360
Weight 6
Character orbit a
Rep. character \(\chi_{360}(1,\cdot)\)
Character field \(\Q\)
Dimension 25
Newform subspaces 16
Sturm bound 432
Trace bound 7

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Defining parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 360.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(432\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(360))\).

Total New Old
Modular forms 376 25 351
Cusp forms 344 25 319
Eisenstein series 32 0 32

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

\(2\)\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(4\)
Plus space\(+\)\(11\)
Minus space\(-\)\(14\)

Trace form

\( 25q - 25q^{5} - 124q^{7} + O(q^{10}) \) \( 25q - 25q^{5} - 124q^{7} - 448q^{11} + 454q^{13} - 3746q^{17} + 3340q^{19} + 1076q^{23} + 15625q^{25} - 8494q^{29} - 4616q^{31} + 8800q^{35} + 5022q^{37} - 25470q^{41} - 9704q^{43} + 8780q^{47} + 109905q^{49} - 61398q^{53} + 13900q^{55} + 73768q^{59} + 12166q^{61} + 14750q^{65} + 107480q^{67} - 36456q^{71} - 70566q^{73} + 81664q^{77} + 57360q^{79} + 136264q^{83} + 14450q^{85} + 202050q^{89} + 64688q^{91} - 37900q^{95} + 128818q^{97} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(360))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5
360.6.a.a \(1\) \(57.738\) \(\Q\) None \(0\) \(0\) \(-25\) \(-160\) \(-\) \(-\) \(+\) \(q-5^{2}q^{5}-160q^{7}+596q^{11}-122q^{13}+\cdots\)
360.6.a.b \(1\) \(57.738\) \(\Q\) None \(0\) \(0\) \(-25\) \(-108\) \(+\) \(-\) \(+\) \(q-5^{2}q^{5}-108q^{7}+604q^{11}-306q^{13}+\cdots\)
360.6.a.c \(1\) \(57.738\) \(\Q\) None \(0\) \(0\) \(-25\) \(-80\) \(+\) \(-\) \(+\) \(q-5^{2}q^{5}-80q^{7}-684q^{11}-978q^{13}+\cdots\)
360.6.a.d \(1\) \(57.738\) \(\Q\) None \(0\) \(0\) \(-25\) \(128\) \(-\) \(-\) \(+\) \(q-5^{2}q^{5}+2^{7}q^{7}+308q^{11}-1058q^{13}+\cdots\)
360.6.a.e \(1\) \(57.738\) \(\Q\) None \(0\) \(0\) \(25\) \(-100\) \(+\) \(-\) \(-\) \(q+5^{2}q^{5}-10^{2}q^{7}+136q^{11}+82q^{13}+\cdots\)
360.6.a.f \(1\) \(57.738\) \(\Q\) None \(0\) \(0\) \(25\) \(-62\) \(-\) \(-\) \(-\) \(q+5^{2}q^{5}-62q^{7}+12^{2}q^{11}-654q^{13}+\cdots\)
360.6.a.g \(1\) \(57.738\) \(\Q\) None \(0\) \(0\) \(25\) \(-28\) \(+\) \(-\) \(-\) \(q+5^{2}q^{5}-28q^{7}+208q^{11}-422q^{13}+\cdots\)
360.6.a.h \(1\) \(57.738\) \(\Q\) None \(0\) \(0\) \(25\) \(108\) \(-\) \(-\) \(-\) \(q+5^{2}q^{5}+108q^{7}+8q^{11}+162q^{13}+\cdots\)
360.6.a.i \(1\) \(57.738\) \(\Q\) None \(0\) \(0\) \(25\) \(242\) \(+\) \(-\) \(-\) \(q+5^{2}q^{5}+242q^{7}-656q^{11}-206q^{13}+\cdots\)
360.6.a.j \(2\) \(57.738\) \(\Q(\sqrt{3289}) \) None \(0\) \(0\) \(-50\) \(-80\) \(+\) \(+\) \(+\) \(q-5^{2}q^{5}+(-40-\beta )q^{7}+(140+\beta )q^{11}+\cdots\)
360.6.a.k \(2\) \(57.738\) \(\Q(\sqrt{2161}) \) None \(0\) \(0\) \(-50\) \(-8\) \(+\) \(-\) \(+\) \(q-5^{2}q^{5}+(-4-\beta )q^{7}+(-244-2\beta )q^{11}+\cdots\)
360.6.a.l \(2\) \(57.738\) \(\Q(\sqrt{129}) \) None \(0\) \(0\) \(-50\) \(52\) \(-\) \(-\) \(+\) \(q-5^{2}q^{5}+(26-\beta )q^{7}+(-280-2\beta )q^{11}+\cdots\)
360.6.a.m \(2\) \(57.738\) \(\Q(\sqrt{3289}) \) None \(0\) \(0\) \(50\) \(-80\) \(-\) \(+\) \(-\) \(q+5^{2}q^{5}+(-40-\beta )q^{7}+(-140-\beta )q^{11}+\cdots\)
360.6.a.n \(2\) \(57.738\) \(\Q(\sqrt{1489}) \) None \(0\) \(0\) \(50\) \(16\) \(-\) \(-\) \(-\) \(q+5^{2}q^{5}+(8+\beta )q^{7}+(-2^{5}+4\beta )q^{11}+\cdots\)
360.6.a.o \(3\) \(57.738\) 3.3.2521041.1 None \(0\) \(0\) \(-75\) \(18\) \(-\) \(+\) \(+\) \(q-5^{2}q^{5}+(6+\beta _{1})q^{7}+(-186-\beta _{1}+\cdots)q^{11}+\cdots\)
360.6.a.p \(3\) \(57.738\) 3.3.2521041.1 None \(0\) \(0\) \(75\) \(18\) \(+\) \(+\) \(-\) \(q+5^{2}q^{5}+(6+\beta _{1})q^{7}+(186+\beta _{1}+\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(360))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(360)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 16}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 + 25 T \))(\( 1 + 25 T \))(\( 1 + 25 T \))(\( 1 + 25 T \))(\( 1 - 25 T \))(\( 1 - 25 T \))(\( 1 - 25 T \))(\( 1 - 25 T \))(\( 1 - 25 T \))(\( ( 1 + 25 T )^{2} \))(\( ( 1 + 25 T )^{2} \))(\( ( 1 + 25 T )^{2} \))(\( ( 1 - 25 T )^{2} \))(\( ( 1 - 25 T )^{2} \))(\( ( 1 + 25 T )^{3} \))(\( ( 1 - 25 T )^{3} \))
$7$ (\( 1 + 160 T + 16807 T^{2} \))(\( 1 + 108 T + 16807 T^{2} \))(\( 1 + 80 T + 16807 T^{2} \))(\( 1 - 128 T + 16807 T^{2} \))(\( 1 + 100 T + 16807 T^{2} \))(\( 1 + 62 T + 16807 T^{2} \))(\( 1 + 28 T + 16807 T^{2} \))(\( 1 - 108 T + 16807 T^{2} \))(\( 1 - 242 T + 16807 T^{2} \))(\( 1 + 80 T + 22058 T^{2} + 1344560 T^{3} + 282475249 T^{4} \))(\( 1 + 8 T - 946 T^{2} + 134456 T^{3} + 282475249 T^{4} \))(\( 1 - 52 T + 29646 T^{2} - 873964 T^{3} + 282475249 T^{4} \))(\( 1 + 80 T + 22058 T^{2} + 1344560 T^{3} + 282475249 T^{4} \))(\( 1 - 16 T + 9854 T^{2} - 268912 T^{3} + 282475249 T^{4} \))(\( 1 - 18 T + 945 T^{2} + 157516 T^{3} + 15882615 T^{4} - 5084554482 T^{5} + 4747561509943 T^{6} \))(\( 1 - 18 T + 945 T^{2} + 157516 T^{3} + 15882615 T^{4} - 5084554482 T^{5} + 4747561509943 T^{6} \))
$11$ (\( 1 - 596 T + 161051 T^{2} \))(\( 1 - 604 T + 161051 T^{2} \))(\( 1 + 684 T + 161051 T^{2} \))(\( 1 - 308 T + 161051 T^{2} \))(\( 1 - 136 T + 161051 T^{2} \))(\( 1 - 144 T + 161051 T^{2} \))(\( 1 - 208 T + 161051 T^{2} \))(\( 1 - 8 T + 161051 T^{2} \))(\( 1 + 656 T + 161051 T^{2} \))(\( 1 - 280 T + 328546 T^{2} - 45094280 T^{3} + 25937424601 T^{4} \))(\( 1 + 488 T + 243334 T^{2} + 78592888 T^{3} + 25937424601 T^{4} \))(\( 1 + 560 T + 381926 T^{2} + 90188560 T^{3} + 25937424601 T^{4} \))(\( 1 + 280 T + 328546 T^{2} + 45094280 T^{3} + 25937424601 T^{4} \))(\( 1 + 64 T - 58058 T^{2} + 10307264 T^{3} + 25937424601 T^{4} \))(\( 1 + 558 T + 216237 T^{2} + 56677948 T^{3} + 34825185087 T^{4} + 14473082927358 T^{5} + 4177248169415651 T^{6} \))(\( 1 - 558 T + 216237 T^{2} - 56677948 T^{3} + 34825185087 T^{4} - 14473082927358 T^{5} + 4177248169415651 T^{6} \))
$13$ (\( 1 + 122 T + 371293 T^{2} \))(\( 1 + 306 T + 371293 T^{2} \))(\( 1 + 978 T + 371293 T^{2} \))(\( 1 + 1058 T + 371293 T^{2} \))(\( 1 - 82 T + 371293 T^{2} \))(\( 1 + 654 T + 371293 T^{2} \))(\( 1 + 422 T + 371293 T^{2} \))(\( 1 - 162 T + 371293 T^{2} \))(\( 1 + 206 T + 371293 T^{2} \))(\( 1 - 480 T + 326570 T^{2} - 178220640 T^{3} + 137858491849 T^{4} \))(\( 1 - 612 T + 525038 T^{2} - 227231316 T^{3} + 137858491849 T^{4} \))(\( 1 - 1388 T + 1149918 T^{2} - 515354684 T^{3} + 137858491849 T^{4} \))(\( 1 - 480 T + 326570 T^{2} - 178220640 T^{3} + 137858491849 T^{4} \))(\( 1 - 1164 T + 866894 T^{2} - 432185052 T^{3} + 137858491849 T^{4} \))(\( 1 + 84 T + 484743 T^{2} + 138297800 T^{3} + 179981682699 T^{4} + 11580113315316 T^{5} + 51185893014090757 T^{6} \))(\( 1 + 84 T + 484743 T^{2} + 138297800 T^{3} + 179981682699 T^{4} + 11580113315316 T^{5} + 51185893014090757 T^{6} \))
$17$ (\( 1 - 1078 T + 1419857 T^{2} \))(\( 1 + 930 T + 1419857 T^{2} \))(\( 1 - 862 T + 1419857 T^{2} \))(\( 1 + 1586 T + 1419857 T^{2} \))(\( 1 + 358 T + 1419857 T^{2} \))(\( 1 - 1190 T + 1419857 T^{2} \))(\( 1 - 146 T + 1419857 T^{2} \))(\( 1 - 714 T + 1419857 T^{2} \))(\( 1 + 1690 T + 1419857 T^{2} \))(\( 1 - 148 T + 1529590 T^{2} - 210138836 T^{3} + 2015993900449 T^{4} \))(\( 1 + 860 T + 2990038 T^{2} + 1221077020 T^{3} + 2015993900449 T^{4} \))(\( 1 + 148 T - 795706 T^{2} + 210138836 T^{3} + 2015993900449 T^{4} \))(\( 1 + 148 T + 1529590 T^{2} + 210138836 T^{3} + 2015993900449 T^{4} \))(\( 1 + 2164 T + 3414838 T^{2} + 3072570548 T^{3} + 2015993900449 T^{4} \))(\( 1 + 726 T + 3953151 T^{2} + 1935854132 T^{3} + 5612909119407 T^{4} + 1463611571725974 T^{5} + 2862423051509815793 T^{6} \))(\( 1 - 726 T + 3953151 T^{2} - 1935854132 T^{3} + 5612909119407 T^{4} - 1463611571725974 T^{5} + 2862423051509815793 T^{6} \))
$19$ (\( 1 - 796 T + 2476099 T^{2} \))(\( 1 + 1324 T + 2476099 T^{2} \))(\( 1 - 916 T + 2476099 T^{2} \))(\( 1 - 2308 T + 2476099 T^{2} \))(\( 1 - 796 T + 2476099 T^{2} \))(\( 1 - 556 T + 2476099 T^{2} \))(\( 1 + 2012 T + 2476099 T^{2} \))(\( 1 + 532 T + 2476099 T^{2} \))(\( 1 + 1364 T + 2476099 T^{2} \))(\( 1 + 48 T + 3637174 T^{2} + 118852752 T^{3} + 6131066257801 T^{4} \))(\( 1 + 96 T + 770806 T^{2} + 237705504 T^{3} + 6131066257801 T^{4} \))(\( 1 + 1000 T + 4533462 T^{2} + 2476099000 T^{3} + 6131066257801 T^{4} \))(\( 1 + 48 T + 3637174 T^{2} + 118852752 T^{3} + 6131066257801 T^{4} \))(\( 1 - 2640 T + 5527222 T^{2} - 6536901360 T^{3} + 6131066257801 T^{4} \))(\( 1 - 876 T + 3345081 T^{2} - 3724581832 T^{3} + 8282751719019 T^{4} - 5370814041833676 T^{5} + 15181127029874798299 T^{6} \))(\( 1 - 876 T + 3345081 T^{2} - 3724581832 T^{3} + 8282751719019 T^{4} - 5370814041833676 T^{5} + 15181127029874798299 T^{6} \))
$23$ (\( 1 - 1088 T + 6436343 T^{2} \))(\( 1 - 852 T + 6436343 T^{2} \))(\( 1 - 1552 T + 6436343 T^{2} \))(\( 1 + 2656 T + 6436343 T^{2} \))(\( 1 + 488 T + 6436343 T^{2} \))(\( 1 + 2182 T + 6436343 T^{2} \))(\( 1 - 1096 T + 6436343 T^{2} \))(\( 1 - 4584 T + 6436343 T^{2} \))(\( 1 + 2198 T + 6436343 T^{2} \))(\( 1 - 3232 T + 10221742 T^{2} - 20802260576 T^{3} + 41426511213649 T^{4} \))(\( 1 + 2984 T + 9255406 T^{2} + 19206047512 T^{3} + 41426511213649 T^{4} \))(\( 1 - 2452 T + 6569198 T^{2} - 15781913036 T^{3} + 41426511213649 T^{4} \))(\( 1 + 3232 T + 10221742 T^{2} + 20802260576 T^{3} + 41426511213649 T^{4} \))(\( 1 + 40 T - 10021778 T^{2} + 257453720 T^{3} + 41426511213649 T^{4} \))(\( 1 + 960 T + 15148581 T^{2} + 10843596928 T^{3} + 97501463279283 T^{4} + 39769450765103040 T^{5} + \)\(26\!\cdots\!07\)\( T^{6} \))(\( 1 - 960 T + 15148581 T^{2} - 10843596928 T^{3} + 97501463279283 T^{4} - 39769450765103040 T^{5} + \)\(26\!\cdots\!07\)\( T^{6} \))
$29$ (\( 1 + 46 T + 20511149 T^{2} \))(\( 1 + 5902 T + 20511149 T^{2} \))(\( 1 - 7314 T + 20511149 T^{2} \))(\( 1 + 1198 T + 20511149 T^{2} \))(\( 1 + 7466 T + 20511149 T^{2} \))(\( 1 - 1578 T + 20511149 T^{2} \))(\( 1 - 1462 T + 20511149 T^{2} \))(\( 1 + 938 T + 20511149 T^{2} \))(\( 1 - 2218 T + 20511149 T^{2} \))(\( 1 - 1340 T + 41260702 T^{2} - 27484939660 T^{3} + 420707233300201 T^{4} \))(\( 1 + 2236 T + 28441822 T^{2} + 45862929164 T^{3} + 420707233300201 T^{4} \))(\( 1 + 1340 T - 8758306 T^{2} + 27484939660 T^{3} + 420707233300201 T^{4} \))(\( 1 + 1340 T + 41260702 T^{2} + 27484939660 T^{3} + 420707233300201 T^{4} \))(\( 1 + 1940 T + 17567422 T^{2} + 39791629060 T^{3} + 420707233300201 T^{4} \))(\( 1 + 1710 T + 42734451 T^{2} + 37260127060 T^{3} + 876532691894199 T^{4} + 719409368943343710 T^{5} + \)\(86\!\cdots\!49\)\( T^{6} \))(\( 1 - 1710 T + 42734451 T^{2} - 37260127060 T^{3} + 876532691894199 T^{4} - 719409368943343710 T^{5} + \)\(86\!\cdots\!49\)\( T^{6} \))
$31$ (\( 1 + 4952 T + 28629151 T^{2} \))(\( 1 + 3320 T + 28629151 T^{2} \))(\( 1 + 9312 T + 28629151 T^{2} \))(\( 1 - 9520 T + 28629151 T^{2} \))(\( 1 - 2728 T + 28629151 T^{2} \))(\( 1 - 9660 T + 28629151 T^{2} \))(\( 1 + 80 T + 28629151 T^{2} \))(\( 1 + 8360 T + 28629151 T^{2} \))(\( 1 + 1700 T + 28629151 T^{2} \))(\( 1 + 6440 T + 62364302 T^{2} + 184371732440 T^{3} + 819628286980801 T^{4} \))(\( 1 - 9352 T + 45895742 T^{2} - 267739820152 T^{3} + 819628286980801 T^{4} \))(\( 1 + 2248 T + 57017022 T^{2} + 64358331448 T^{3} + 819628286980801 T^{4} \))(\( 1 + 6440 T + 62364302 T^{2} + 184371732440 T^{3} + 819628286980801 T^{4} \))(\( 1 - 6328 T + 63242942 T^{2} - 181165267528 T^{3} + 819628286980801 T^{4} \))(\( 1 - 324 T + 78872013 T^{2} - 13928801848 T^{3} + 2258038769850963 T^{4} - 265559564981779524 T^{5} + \)\(23\!\cdots\!51\)\( T^{6} \))(\( 1 - 324 T + 78872013 T^{2} - 13928801848 T^{3} + 2258038769850963 T^{4} - 265559564981779524 T^{5} + \)\(23\!\cdots\!51\)\( T^{6} \))
$37$ (\( 1 + 6114 T + 69343957 T^{2} \))(\( 1 - 10774 T + 69343957 T^{2} \))(\( 1 + 8826 T + 69343957 T^{2} \))(\( 1 - 4470 T + 69343957 T^{2} \))(\( 1 - 7794 T + 69343957 T^{2} \))(\( 1 + 3534 T + 69343957 T^{2} \))(\( 1 + 15750 T + 69343957 T^{2} \))(\( 1 - 1090 T + 69343957 T^{2} \))(\( 1 + 846 T + 69343957 T^{2} \))(\( 1 + 6440 T + 77014058 T^{2} + 446575083080 T^{3} + 4808584372417849 T^{4} \))(\( 1 - 10612 T + 133614014 T^{2} - 735878071684 T^{3} + 4808584372417849 T^{4} \))(\( 1 + 5940 T + 123434318 T^{2} + 411903104580 T^{3} + 4808584372417849 T^{4} \))(\( 1 + 6440 T + 77014058 T^{2} + 446575083080 T^{3} + 4808584372417849 T^{4} \))(\( 1 - 3532 T + 112622270 T^{2} - 244922856124 T^{3} + 4808584372417849 T^{4} \))(\( 1 - 10320 T + 189184575 T^{2} - 1131781251360 T^{3} + 13118807033863275 T^{4} - 49624590723352201680 T^{5} + \)\(33\!\cdots\!93\)\( T^{6} \))(\( 1 - 10320 T + 189184575 T^{2} - 1131781251360 T^{3} + 13118807033863275 T^{4} - 49624590723352201680 T^{5} + \)\(33\!\cdots\!93\)\( T^{6} \))
$41$ (\( 1 - 6 T + 115856201 T^{2} \))(\( 1 - 17958 T + 115856201 T^{2} \))(\( 1 - 3286 T + 115856201 T^{2} \))(\( 1 - 6198 T + 115856201 T^{2} \))(\( 1 + 18234 T + 115856201 T^{2} \))(\( 1 + 7462 T + 115856201 T^{2} \))(\( 1 - 2358 T + 115856201 T^{2} \))(\( 1 - 11238 T + 115856201 T^{2} \))(\( 1 - 1818 T + 115856201 T^{2} \))(\( 1 + 9680 T + 219564178 T^{2} + 1121488025680 T^{3} + 13422659310152401 T^{4} \))(\( 1 + 17156 T + 298517590 T^{2} + 1987628984356 T^{3} + 13422659310152401 T^{4} \))(\( 1 + 23076 T + 352280470 T^{2} + 2673497694276 T^{3} + 13422659310152401 T^{4} \))(\( 1 - 9680 T + 219564178 T^{2} - 1121488025680 T^{3} + 13422659310152401 T^{4} \))(\( 1 + 2404 T - 34529258 T^{2} + 278518307204 T^{3} + 13422659310152401 T^{4} \))(\( 1 + 4080 T + 108883419 T^{2} + 42537842400 T^{3} + 12614819277231219 T^{4} + 54764449985421796080 T^{5} + \)\(15\!\cdots\!01\)\( T^{6} \))(\( 1 - 4080 T + 108883419 T^{2} - 42537842400 T^{3} + 12614819277231219 T^{4} - 54764449985421796080 T^{5} + \)\(15\!\cdots\!01\)\( T^{6} \))
$43$ (\( 1 + 24116 T + 147008443 T^{2} \))(\( 1 - 9264 T + 147008443 T^{2} \))(\( 1 - 7556 T + 147008443 T^{2} \))(\( 1 + 6332 T + 147008443 T^{2} \))(\( 1 + 2444 T + 147008443 T^{2} \))(\( 1 + 7114 T + 147008443 T^{2} \))(\( 1 - 2812 T + 147008443 T^{2} \))(\( 1 + 7692 T + 147008443 T^{2} \))(\( 1 - 10534 T + 147008443 T^{2} \))(\( 1 + 19360 T + 382456886 T^{2} + 2846083456480 T^{3} + 21611482313284249 T^{4} \))(\( 1 - 440 T + 1414022 T^{2} - 64683714920 T^{3} + 21611482313284249 T^{4} \))(\( 1 - 17684 T + 312898614 T^{2} - 2599697306012 T^{3} + 21611482313284249 T^{4} \))(\( 1 + 19360 T + 382456886 T^{2} + 2846083456480 T^{3} + 21611482313284249 T^{4} \))(\( 1 - 31928 T + 510747782 T^{2} - 4693685568104 T^{3} + 21611482313284249 T^{4} \))(\( 1 + 1752 T + 208706289 T^{2} + 626926262672 T^{3} + 30681586590198027 T^{4} + 37863317012874004248 T^{5} + \)\(31\!\cdots\!07\)\( T^{6} \))(\( 1 + 1752 T + 208706289 T^{2} + 626926262672 T^{3} + 30681586590198027 T^{4} + 37863317012874004248 T^{5} + \)\(31\!\cdots\!07\)\( T^{6} \))
$47$ (\( 1 + 13480 T + 229345007 T^{2} \))(\( 1 - 9796 T + 229345007 T^{2} \))(\( 1 - 5960 T + 229345007 T^{2} \))(\( 1 + 14920 T + 229345007 T^{2} \))(\( 1 - 2200 T + 229345007 T^{2} \))(\( 1 - 28294 T + 229345007 T^{2} \))(\( 1 - 7960 T + 229345007 T^{2} \))(\( 1 - 13640 T + 229345007 T^{2} \))(\( 1 + 12074 T + 229345007 T^{2} \))(\( 1 + 15024 T + 178326558 T^{2} + 3445679385168 T^{3} + 52599132235830049 T^{4} \))(\( 1 - 16728 T + 499568094 T^{2} - 3836483277096 T^{3} + 52599132235830049 T^{4} \))(\( 1 - 2908 T + 56660030 T^{2} - 666935280356 T^{3} + 52599132235830049 T^{4} \))(\( 1 - 15024 T + 178326558 T^{2} - 3445679385168 T^{3} + 52599132235830049 T^{4} \))(\( 1 + 38232 T + 811508574 T^{2} + 8768318307624 T^{3} + 52599132235830049 T^{4} \))(\( 1 - 8976 T + 403529613 T^{2} - 2395450153952 T^{3} + 92547501918192291 T^{4} - \)\(47\!\cdots\!24\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \))(\( 1 + 8976 T + 403529613 T^{2} + 2395450153952 T^{3} + 92547501918192291 T^{4} + \)\(47\!\cdots\!24\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \))
$53$ (\( 1 + 20598 T + 418195493 T^{2} \))(\( 1 - 31434 T + 418195493 T^{2} \))(\( 1 - 8698 T + 418195493 T^{2} \))(\( 1 + 38310 T + 418195493 T^{2} \))(\( 1 + 10122 T + 418195493 T^{2} \))(\( 1 - 13046 T + 418195493 T^{2} \))(\( 1 - 7590 T + 418195493 T^{2} \))(\( 1 + 19050 T + 418195493 T^{2} \))(\( 1 + 32586 T + 418195493 T^{2} \))(\( 1 + 1844 T + 311001070 T^{2} + 771152489092 T^{3} + 174887470365513049 T^{4} \))(\( 1 + 31484 T + 851712526 T^{2} + 13166466901612 T^{3} + 174887470365513049 T^{4} \))(\( 1 - 5412 T + 693247822 T^{2} - 2263274008116 T^{3} + 174887470365513049 T^{4} \))(\( 1 - 1844 T + 311001070 T^{2} - 771152489092 T^{3} + 174887470365513049 T^{4} \))(\( 1 - 24572 T + 952934926 T^{2} - 10275899653996 T^{3} + 174887470365513049 T^{4} \))(\( 1 - 16098 T + 974021019 T^{2} - 9312300047724 T^{3} + 407331200233067367 T^{4} - \)\(28\!\cdots\!02\)\( T^{5} + \)\(73\!\cdots\!57\)\( T^{6} \))(\( 1 + 16098 T + 974021019 T^{2} + 9312300047724 T^{3} + 407331200233067367 T^{4} + \)\(28\!\cdots\!02\)\( T^{5} + \)\(73\!\cdots\!57\)\( T^{6} \))
$59$ (\( 1 - 46756 T + 714924299 T^{2} \))(\( 1 + 33228 T + 714924299 T^{2} \))(\( 1 - 42036 T + 714924299 T^{2} \))(\( 1 + 11564 T + 714924299 T^{2} \))(\( 1 - 6776 T + 714924299 T^{2} \))(\( 1 - 37092 T + 714924299 T^{2} \))(\( 1 + 18064 T + 714924299 T^{2} \))(\( 1 - 18936 T + 714924299 T^{2} \))(\( 1 + 8668 T + 714924299 T^{2} \))(\( 1 - 12840 T + 732105634 T^{2} - 9179627999160 T^{3} + 511116753300641401 T^{4} \))(\( 1 - 61464 T + 2273480806 T^{2} - 43942107113736 T^{3} + 511116753300641401 T^{4} \))(\( 1 + 62584 T + 2277965606 T^{2} + 44742822328616 T^{3} + 511116753300641401 T^{4} \))(\( 1 + 12840 T + 732105634 T^{2} + 9179627999160 T^{3} + 511116753300641401 T^{4} \))(\( 1 + 5184 T + 1405691158 T^{2} + 3706167566016 T^{3} + 511116753300641401 T^{4} \))(\( 1 - 23106 T + 1088276733 T^{2} - 11148915546724 T^{3} + 778035480458035167 T^{4} - \)\(11\!\cdots\!06\)\( T^{5} + \)\(36\!\cdots\!99\)\( T^{6} \))(\( 1 + 23106 T + 1088276733 T^{2} + 11148915546724 T^{3} + 778035480458035167 T^{4} + \)\(11\!\cdots\!06\)\( T^{5} + \)\(36\!\cdots\!99\)\( T^{6} \))
$61$ (\( 1 + 9602 T + 844596301 T^{2} \))(\( 1 + 40210 T + 844596301 T^{2} \))(\( 1 - 37518 T + 844596301 T^{2} \))(\( 1 + 48338 T + 844596301 T^{2} \))(\( 1 - 23398 T + 844596301 T^{2} \))(\( 1 - 39570 T + 844596301 T^{2} \))(\( 1 + 19658 T + 844596301 T^{2} \))(\( 1 + 1978 T + 844596301 T^{2} \))(\( 1 + 34670 T + 844596301 T^{2} \))(\( 1 + 24076 T + 1576248446 T^{2} + 20334500542876 T^{3} + 713342911662882601 T^{4} \))(\( 1 - 51596 T + 2049215870 T^{2} - 43577790746396 T^{3} + 713342911662882601 T^{4} \))(\( 1 - 14108 T + 1110042462 T^{2} - 11915564614508 T^{3} + 713342911662882601 T^{4} \))(\( 1 + 24076 T + 1576248446 T^{2} + 20334500542876 T^{3} + 713342911662882601 T^{4} \))(\( 1 + 53956 T + 1627858910 T^{2} + 45571038016756 T^{3} + 713342911662882601 T^{4} \))(\( 1 - 51270 T + 2305394931 T^{2} - 70676310393124 T^{3} + 1947128031066750231 T^{4} - \)\(36\!\cdots\!70\)\( T^{5} + \)\(60\!\cdots\!01\)\( T^{6} \))(\( 1 - 51270 T + 2305394931 T^{2} - 70676310393124 T^{3} + 1947128031066750231 T^{4} - \)\(36\!\cdots\!70\)\( T^{5} + \)\(60\!\cdots\!01\)\( T^{6} \))
$67$ (\( 1 + 17404 T + 1350125107 T^{2} \))(\( 1 - 58864 T + 1350125107 T^{2} \))(\( 1 - 29324 T + 1350125107 T^{2} \))(\( 1 - 56972 T + 1350125107 T^{2} \))(\( 1 + 9676 T + 1350125107 T^{2} \))(\( 1 + 56734 T + 1350125107 T^{2} \))(\( 1 - 31868 T + 1350125107 T^{2} \))(\( 1 - 44212 T + 1350125107 T^{2} \))(\( 1 + 47566 T + 1350125107 T^{2} \))(\( 1 + 42240 T + 1422289750 T^{2} + 57029284519680 T^{3} + 1822837804551761449 T^{4} \))(\( 1 + 45816 T + 1490141302 T^{2} + 61857331902312 T^{3} + 1822837804551761449 T^{4} \))(\( 1 + 85412 T + 4371910566 T^{2} + 115316885639084 T^{3} + 1822837804551761449 T^{4} \))(\( 1 + 42240 T + 1422289750 T^{2} + 57029284519680 T^{3} + 1822837804551761449 T^{4} \))(\( 1 - 95160 T + 4779613558 T^{2} - 128477905182120 T^{3} + 1822837804551761449 T^{4} \))(\( 1 - 69084 T + 4950834057 T^{2} - 182092399391720 T^{3} + 6684245360946369099 T^{4} - \)\(12\!\cdots\!16\)\( T^{5} + \)\(24\!\cdots\!43\)\( T^{6} \))(\( 1 - 69084 T + 4950834057 T^{2} - 182092399391720 T^{3} + 6684245360946369099 T^{4} - \)\(12\!\cdots\!16\)\( T^{5} + \)\(24\!\cdots\!43\)\( T^{6} \))
$71$ (\( 1 + 26568 T + 1804229351 T^{2} \))(\( 1 - 55312 T + 1804229351 T^{2} \))(\( 1 + 84408 T + 1804229351 T^{2} \))(\( 1 + 44856 T + 1804229351 T^{2} \))(\( 1 + 13728 T + 1804229351 T^{2} \))(\( 1 + 45588 T + 1804229351 T^{2} \))(\( 1 + 57216 T + 1804229351 T^{2} \))(\( 1 - 59744 T + 1804229351 T^{2} \))(\( 1 + 948 T + 1804229351 T^{2} \))(\( 1 - 12880 T + 3454118398 T^{2} - 23238474040880 T^{3} + 3255243551009881201 T^{4} \))(\( 1 - 97456 T + 5874446350 T^{2} - 175832975631056 T^{3} + 3255243551009881201 T^{4} \))(\( 1 + 47208 T + 4011779662 T^{2} + 85174059202008 T^{3} + 3255243551009881201 T^{4} \))(\( 1 + 12880 T + 3454118398 T^{2} + 23238474040880 T^{3} + 3255243551009881201 T^{4} \))(\( 1 - 71552 T + 3857659342 T^{2} - 129096218522752 T^{3} + 3255243551009881201 T^{4} \))(\( 1 - 55932 T + 3009690789 T^{2} - 141554371038216 T^{3} + 5430172458948147939 T^{4} - \)\(18\!\cdots\!32\)\( T^{5} + \)\(58\!\cdots\!51\)\( T^{6} \))(\( 1 + 55932 T + 3009690789 T^{2} + 141554371038216 T^{3} + 5430172458948147939 T^{4} + \)\(18\!\cdots\!32\)\( T^{5} + \)\(58\!\cdots\!51\)\( T^{6} \))
$73$ (\( 1 - 75450 T + 2073071593 T^{2} \))(\( 1 - 27258 T + 2073071593 T^{2} \))(\( 1 + 46550 T + 2073071593 T^{2} \))(\( 1 + 19446 T + 2073071593 T^{2} \))(\( 1 + 27390 T + 2073071593 T^{2} \))(\( 1 - 11842 T + 2073071593 T^{2} \))(\( 1 - 9906 T + 2073071593 T^{2} \))(\( 1 - 56994 T + 2073071593 T^{2} \))(\( 1 + 63102 T + 2073071593 T^{2} \))(\( 1 + 96260 T + 5679384470 T^{2} + 199553871542180 T^{3} + 4297625829703557649 T^{4} \))(\( 1 - 58852 T + 4427698262 T^{2} - 122004409391236 T^{3} + 4297625829703557649 T^{4} \))(\( 1 + 67452 T + 4400780438 T^{2} + 139832825091036 T^{3} + 4297625829703557649 T^{4} \))(\( 1 + 96260 T + 5679384470 T^{2} + 199553871542180 T^{3} + 4297625829703557649 T^{4} \))(\( 1 + 96236 T + 6445380086 T^{2} + 199504117823948 T^{3} + 4297625829703557649 T^{4} \))(\( 1 - 100914 T + 9222632103 T^{2} - 440423264194140 T^{3} + 19119176625419150079 T^{4} - \)\(43\!\cdots\!86\)\( T^{5} + \)\(89\!\cdots\!57\)\( T^{6} \))(\( 1 - 100914 T + 9222632103 T^{2} - 440423264194140 T^{3} + 19119176625419150079 T^{4} - \)\(43\!\cdots\!86\)\( T^{5} + \)\(89\!\cdots\!57\)\( T^{6} \))
$79$ (\( 1 - 50472 T + 3077056399 T^{2} \))(\( 1 - 31456 T + 3077056399 T^{2} \))(\( 1 - 26752 T + 3077056399 T^{2} \))(\( 1 + 77328 T + 3077056399 T^{2} \))(\( 1 + 93288 T + 3077056399 T^{2} \))(\( 1 - 94216 T + 3077056399 T^{2} \))(\( 1 - 7872 T + 3077056399 T^{2} \))(\( 1 + 15128 T + 3077056399 T^{2} \))(\( 1 - 46536 T + 3077056399 T^{2} \))(\( 1 + 89896 T + 6990395502 T^{2} + 276615062044504 T^{3} + 9468276082626847201 T^{4} \))(\( 1 + 116776 T + 8933123742 T^{2} + 359326338049624 T^{3} + 9468276082626847201 T^{4} \))(\( 1 + 65904 T + 3994274078 T^{2} + 202790324919696 T^{3} + 9468276082626847201 T^{4} \))(\( 1 + 89896 T + 6990395502 T^{2} + 276615062044504 T^{3} + 9468276082626847201 T^{4} \))(\( 1 - 29288 T + 5671921950 T^{2} - 90120827813912 T^{3} + 9468276082626847201 T^{4} \))(\( 1 - 159492 T + 15447781533 T^{2} - 970158382454328 T^{3} + 47533695016471679667 T^{4} - \)\(15\!\cdots\!92\)\( T^{5} + \)\(29\!\cdots\!99\)\( T^{6} \))(\( 1 - 159492 T + 15447781533 T^{2} - 970158382454328 T^{3} + 47533695016471679667 T^{4} - \)\(15\!\cdots\!92\)\( T^{5} + \)\(29\!\cdots\!99\)\( T^{6} \))
$83$ (\( 1 + 33236 T + 3939040643 T^{2} \))(\( 1 + 24552 T + 3939040643 T^{2} \))(\( 1 - 7956 T + 3939040643 T^{2} \))(\( 1 + 40364 T + 3939040643 T^{2} \))(\( 1 - 23276 T + 3939040643 T^{2} \))(\( 1 - 31482 T + 3939040643 T^{2} \))(\( 1 + 109996 T + 3939040643 T^{2} \))(\( 1 - 21996 T + 3939040643 T^{2} \))(\( 1 - 88778 T + 3939040643 T^{2} \))(\( 1 - 57024 T + 8584451830 T^{2} - 224619853626432 T^{3} + 15516041187205853449 T^{4} \))(\( 1 - 136632 T + 12540178198 T^{2} - 538199001134376 T^{3} + 15516041187205853449 T^{4} \))(\( 1 + 108724 T + 10572459494 T^{2} + 428268254869532 T^{3} + 15516041187205853449 T^{4} \))(\( 1 + 57024 T + 8584451830 T^{2} + 224619853626432 T^{3} + 15516041187205853449 T^{4} \))(\( 1 - 143016 T + 12113227414 T^{2} - 563345836599288 T^{3} + 15516041187205853449 T^{4} \))(\( 1 + 16428 T + 4008885369 T^{2} + 134680015872584 T^{3} + 15791162401619052267 T^{4} + \)\(25\!\cdots\!72\)\( T^{5} + \)\(61\!\cdots\!07\)\( T^{6} \))(\( 1 - 16428 T + 4008885369 T^{2} - 134680015872584 T^{3} + 15791162401619052267 T^{4} - \)\(25\!\cdots\!72\)\( T^{5} + \)\(61\!\cdots\!07\)\( T^{6} \))
$89$ (\( 1 + 133194 T + 5584059449 T^{2} \))(\( 1 - 90854 T + 5584059449 T^{2} \))(\( 1 + 59674 T + 5584059449 T^{2} \))(\( 1 + 35706 T + 5584059449 T^{2} \))(\( 1 + 102354 T + 5584059449 T^{2} \))(\( 1 - 94054 T + 5584059449 T^{2} \))(\( 1 + 62466 T + 5584059449 T^{2} \))(\( 1 + 14066 T + 5584059449 T^{2} \))(\( 1 - 104934 T + 5584059449 T^{2} \))(\( 1 - 25240 T + 10143343298 T^{2} - 140941660492760 T^{3} + 31181719929966183601 T^{4} \))(\( 1 - 124924 T + 13365576758 T^{2} - 697583042606876 T^{3} + 31181719929966183601 T^{4} \))(\( 1 - 55020 T + 10818978262 T^{2} - 307234950883980 T^{3} + 31181719929966183601 T^{4} \))(\( 1 + 25240 T + 10143343298 T^{2} + 140941660492760 T^{3} + 31181719929966183601 T^{4} \))(\( 1 - 139724 T + 16006792406 T^{2} - 780227122452076 T^{3} + 31181719929966183601 T^{4} \))(\( 1 + 120852 T + 12302174427 T^{2} + 1026615424009896 T^{3} + 68696073352335510723 T^{4} + \)\(37\!\cdots\!52\)\( T^{5} + \)\(17\!\cdots\!49\)\( T^{6} \))(\( 1 - 120852 T + 12302174427 T^{2} - 1026615424009896 T^{3} + 68696073352335510723 T^{4} - \)\(37\!\cdots\!52\)\( T^{5} + \)\(17\!\cdots\!49\)\( T^{6} \))
$97$ (\( 1 + 42878 T + 8587340257 T^{2} \))(\( 1 - 154706 T + 8587340257 T^{2} \))(\( 1 - 136898 T + 8587340257 T^{2} \))(\( 1 + 97022 T + 8587340257 T^{2} \))(\( 1 + 49502 T + 8587340257 T^{2} \))(\( 1 - 23714 T + 8587340257 T^{2} \))(\( 1 + 97598 T + 8587340257 T^{2} \))(\( 1 - 75938 T + 8587340257 T^{2} \))(\( 1 + 36254 T + 8587340257 T^{2} \))(\( 1 + 229100 T + 28475803110 T^{2} + 1967359652878700 T^{3} + 73742412689492826049 T^{4} \))(\( 1 + 39260 T + 9987043590 T^{2} + 337138978489820 T^{3} + 73742412689492826049 T^{4} \))(\( 1 - 147668 T + 11612429670 T^{2} - 1268075361070676 T^{3} + 73742412689492826049 T^{4} \))(\( 1 + 229100 T + 28475803110 T^{2} + 1967359652878700 T^{3} + 73742412689492826049 T^{4} \))(\( 1 + 36860 T + 14824711110 T^{2} + 316529361873020 T^{3} + 73742412689492826049 T^{4} \))(\( 1 - 223734 T + 38248594287 T^{2} - 3912864163619380 T^{3} + \)\(32\!\cdots\!59\)\( T^{4} - \)\(16\!\cdots\!66\)\( T^{5} + \)\(63\!\cdots\!93\)\( T^{6} \))(\( 1 - 223734 T + 38248594287 T^{2} - 3912864163619380 T^{3} + \)\(32\!\cdots\!59\)\( T^{4} - \)\(16\!\cdots\!66\)\( T^{5} + \)\(63\!\cdots\!93\)\( T^{6} \))
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