Properties

Label 336.4.a
Level 336
Weight 4
Character orbit a
Rep. character \(\chi_{336}(1,\cdot)\)
Character field \(\Q\)
Dimension 18
Newform subspaces 15
Sturm bound 256
Trace bound 5

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Defining parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(256\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(336))\).

Total New Old
Modular forms 204 18 186
Cusp forms 180 18 162
Eisenstein series 24 0 24

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

\(2\)\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(1\)
Plus space\(+\)\(11\)
Minus space\(-\)\(7\)

Trace form

\( 18q - 4q^{5} - 14q^{7} + 162q^{9} + O(q^{10}) \) \( 18q - 4q^{5} - 14q^{7} + 162q^{9} + 20q^{11} + 92q^{13} - 60q^{15} + 52q^{17} - 24q^{19} + 164q^{23} + 406q^{25} + 484q^{29} + 264q^{31} - 124q^{37} - 312q^{39} - 236q^{41} + 488q^{43} - 36q^{45} + 744q^{47} + 882q^{49} + 1356q^{51} + 196q^{53} + 1080q^{55} - 120q^{59} + 1068q^{61} - 126q^{63} + 696q^{65} + 672q^{67} + 528q^{69} + 2060q^{71} + 740q^{73} + 1104q^{75} - 952q^{77} - 1784q^{79} + 1458q^{81} - 2224q^{83} + 512q^{85} - 1368q^{87} - 1580q^{89} + 1092q^{91} - 456q^{93} + 656q^{95} - 1884q^{97} + 180q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(336))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 7
336.4.a.a \(1\) \(19.825\) \(\Q\) None \(0\) \(-3\) \(-16\) \(-7\) \(+\) \(+\) \(+\) \(q-3q^{3}-2^{4}q^{5}-7q^{7}+9q^{9}+18q^{11}+\cdots\)
336.4.a.b \(1\) \(19.825\) \(\Q\) None \(0\) \(-3\) \(-10\) \(7\) \(+\) \(+\) \(-\) \(q-3q^{3}-10q^{5}+7q^{7}+9q^{9}+52q^{11}+\cdots\)
336.4.a.c \(1\) \(19.825\) \(\Q\) None \(0\) \(-3\) \(-2\) \(7\) \(+\) \(+\) \(-\) \(q-3q^{3}-2q^{5}+7q^{7}+9q^{9}-52q^{11}+\cdots\)
336.4.a.d \(1\) \(19.825\) \(\Q\) None \(0\) \(-3\) \(2\) \(7\) \(-\) \(+\) \(-\) \(q-3q^{3}+2q^{5}+7q^{7}+9q^{9}+8q^{11}+\cdots\)
336.4.a.e \(1\) \(19.825\) \(\Q\) None \(0\) \(-3\) \(14\) \(7\) \(-\) \(+\) \(-\) \(q-3q^{3}+14q^{5}+7q^{7}+9q^{9}-4q^{11}+\cdots\)
336.4.a.f \(1\) \(19.825\) \(\Q\) None \(0\) \(3\) \(-18\) \(-7\) \(-\) \(-\) \(+\) \(q+3q^{3}-18q^{5}-7q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)
336.4.a.g \(1\) \(19.825\) \(\Q\) None \(0\) \(3\) \(-10\) \(-7\) \(+\) \(-\) \(+\) \(q+3q^{3}-10q^{5}-7q^{7}+9q^{9}+12q^{11}+\cdots\)
336.4.a.h \(1\) \(19.825\) \(\Q\) None \(0\) \(3\) \(-4\) \(7\) \(-\) \(-\) \(-\) \(q+3q^{3}-4q^{5}+7q^{7}+9q^{9}-62q^{11}+\cdots\)
336.4.a.i \(1\) \(19.825\) \(\Q\) None \(0\) \(3\) \(-2\) \(-7\) \(+\) \(-\) \(+\) \(q+3q^{3}-2q^{5}-7q^{7}+9q^{9}-12q^{11}+\cdots\)
336.4.a.j \(1\) \(19.825\) \(\Q\) None \(0\) \(3\) \(4\) \(7\) \(+\) \(-\) \(-\) \(q+3q^{3}+4q^{5}+7q^{7}+9q^{9}+26q^{11}+\cdots\)
336.4.a.k \(1\) \(19.825\) \(\Q\) None \(0\) \(3\) \(6\) \(-7\) \(-\) \(-\) \(+\) \(q+3q^{3}+6q^{5}-7q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
336.4.a.l \(1\) \(19.825\) \(\Q\) None \(0\) \(3\) \(18\) \(-7\) \(-\) \(-\) \(+\) \(q+3q^{3}+18q^{5}-7q^{7}+9q^{9}+72q^{11}+\cdots\)
336.4.a.m \(2\) \(19.825\) \(\Q(\sqrt{57}) \) None \(0\) \(-6\) \(6\) \(-14\) \(-\) \(+\) \(+\) \(q-3q^{3}+(3+\beta )q^{5}-7q^{7}+9q^{9}+(3+\cdots)q^{11}+\cdots\)
336.4.a.n \(2\) \(19.825\) \(\Q(\sqrt{177}) \) None \(0\) \(-6\) \(14\) \(-14\) \(+\) \(+\) \(+\) \(q-3q^{3}+(7-\beta )q^{5}-7q^{7}+9q^{9}+(-9+\cdots)q^{11}+\cdots\)
336.4.a.o \(2\) \(19.825\) \(\Q(\sqrt{337}) \) None \(0\) \(6\) \(-6\) \(14\) \(+\) \(-\) \(-\) \(q+3q^{3}+(-3-\beta )q^{5}+7q^{7}+9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(336))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(336)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T \))(\( 1 + 3 T \))(\( 1 + 3 T \))(\( 1 + 3 T \))(\( 1 + 3 T \))(\( 1 - 3 T \))(\( 1 - 3 T \))(\( 1 - 3 T \))(\( 1 - 3 T \))(\( 1 - 3 T \))(\( 1 - 3 T \))(\( 1 - 3 T \))(\( ( 1 + 3 T )^{2} \))(\( ( 1 + 3 T )^{2} \))(\( ( 1 - 3 T )^{2} \))
$5$ (\( 1 + 16 T + 125 T^{2} \))(\( 1 + 10 T + 125 T^{2} \))(\( 1 + 2 T + 125 T^{2} \))(\( 1 - 2 T + 125 T^{2} \))(\( 1 - 14 T + 125 T^{2} \))(\( 1 + 18 T + 125 T^{2} \))(\( 1 + 10 T + 125 T^{2} \))(\( 1 + 4 T + 125 T^{2} \))(\( 1 + 2 T + 125 T^{2} \))(\( 1 - 4 T + 125 T^{2} \))(\( 1 - 6 T + 125 T^{2} \))(\( 1 - 18 T + 125 T^{2} \))(\( 1 - 6 T + 202 T^{2} - 750 T^{3} + 15625 T^{4} \))(\( 1 - 14 T + 122 T^{2} - 1750 T^{3} + 15625 T^{4} \))(\( 1 + 6 T - 78 T^{2} + 750 T^{3} + 15625 T^{4} \))
$7$ (\( 1 + 7 T \))(\( 1 - 7 T \))(\( 1 - 7 T \))(\( 1 - 7 T \))(\( 1 - 7 T \))(\( 1 + 7 T \))(\( 1 + 7 T \))(\( 1 - 7 T \))(\( 1 + 7 T \))(\( 1 - 7 T \))(\( 1 + 7 T \))(\( 1 + 7 T \))(\( ( 1 + 7 T )^{2} \))(\( ( 1 + 7 T )^{2} \))(\( ( 1 - 7 T )^{2} \))
$11$ (\( 1 - 18 T + 1331 T^{2} \))(\( 1 - 52 T + 1331 T^{2} \))(\( 1 + 52 T + 1331 T^{2} \))(\( 1 - 8 T + 1331 T^{2} \))(\( 1 + 4 T + 1331 T^{2} \))(\( 1 - 36 T + 1331 T^{2} \))(\( 1 - 12 T + 1331 T^{2} \))(\( 1 + 62 T + 1331 T^{2} \))(\( 1 + 12 T + 1331 T^{2} \))(\( 1 - 26 T + 1331 T^{2} \))(\( 1 + 36 T + 1331 T^{2} \))(\( 1 - 72 T + 1331 T^{2} \))(\( 1 - 6 T + 1246 T^{2} - 7986 T^{3} + 1771561 T^{4} \))(\( 1 + 18 T + 1150 T^{2} + 23958 T^{3} + 1771561 T^{4} \))(\( 1 + 26 T + 2494 T^{2} + 34606 T^{3} + 1771561 T^{4} \))
$13$ (\( 1 + 54 T + 2197 T^{2} \))(\( 1 + 10 T + 2197 T^{2} \))(\( 1 - 86 T + 2197 T^{2} \))(\( 1 + 42 T + 2197 T^{2} \))(\( 1 - 54 T + 2197 T^{2} \))(\( 1 + 34 T + 2197 T^{2} \))(\( 1 - 30 T + 2197 T^{2} \))(\( 1 + 62 T + 2197 T^{2} \))(\( 1 + 66 T + 2197 T^{2} \))(\( 1 - 2 T + 2197 T^{2} \))(\( 1 - 62 T + 2197 T^{2} \))(\( 1 + 34 T + 2197 T^{2} \))(\( 1 - 16 T + 2406 T^{2} - 35152 T^{3} + 4826809 T^{4} \))(\( 1 - 48 T + 4262 T^{2} - 105456 T^{3} + 4826809 T^{4} \))(\( 1 - 96 T + 5350 T^{2} - 210912 T^{3} + 4826809 T^{4} \))
$17$ (\( 1 + 128 T + 4913 T^{2} \))(\( 1 + 54 T + 4913 T^{2} \))(\( 1 + 30 T + 4913 T^{2} \))(\( 1 + 2 T + 4913 T^{2} \))(\( 1 + 14 T + 4913 T^{2} \))(\( 1 - 42 T + 4913 T^{2} \))(\( 1 - 34 T + 4913 T^{2} \))(\( 1 - 84 T + 4913 T^{2} \))(\( 1 + 70 T + 4913 T^{2} \))(\( 1 + 36 T + 4913 T^{2} \))(\( 1 - 114 T + 4913 T^{2} \))(\( 1 - 6 T + 4913 T^{2} \))(\( 1 + 6 T + 9778 T^{2} + 29478 T^{3} + 24137569 T^{4} \))(\( 1 - 34 T - 4222 T^{2} - 167042 T^{3} + 24137569 T^{4} \))(\( 1 - 78 T + 8314 T^{2} - 383214 T^{3} + 24137569 T^{4} \))
$19$ (\( 1 + 52 T + 6859 T^{2} \))(\( 1 - 52 T + 6859 T^{2} \))(\( 1 - 4 T + 6859 T^{2} \))(\( 1 - 124 T + 6859 T^{2} \))(\( 1 + 92 T + 6859 T^{2} \))(\( 1 - 124 T + 6859 T^{2} \))(\( 1 + 148 T + 6859 T^{2} \))(\( 1 + 100 T + 6859 T^{2} \))(\( 1 - 92 T + 6859 T^{2} \))(\( 1 - 76 T + 6859 T^{2} \))(\( 1 - 76 T + 6859 T^{2} \))(\( 1 + 92 T + 6859 T^{2} \))(\( 1 + 64 T + 6534 T^{2} + 438976 T^{3} + 47045881 T^{4} \))(\( 1 - 16 T + 10950 T^{2} - 109744 T^{3} + 47045881 T^{4} \))(\( ( 1 + 20 T + 6859 T^{2} )^{2} \))
$23$ (\( 1 - 202 T + 12167 T^{2} \))(\( 1 + 48 T + 12167 T^{2} \))(\( 1 + 120 T + 12167 T^{2} \))(\( 1 + 76 T + 12167 T^{2} \))(\( 1 - 152 T + 12167 T^{2} \))(\( 1 + 12167 T^{2} \))(\( 1 + 152 T + 12167 T^{2} \))(\( 1 - 42 T + 12167 T^{2} \))(\( 1 + 16 T + 12167 T^{2} \))(\( 1 - 114 T + 12167 T^{2} \))(\( 1 - 24 T + 12167 T^{2} \))(\( 1 - 180 T + 12167 T^{2} \))(\( 1 + 6 T + 7870 T^{2} + 73002 T^{3} + 148035889 T^{4} \))(\( 1 + 110 T + 18686 T^{2} + 1338370 T^{3} + 148035889 T^{4} \))(\( 1 + 22 T + 16030 T^{2} + 267674 T^{3} + 148035889 T^{4} \))
$29$ (\( 1 - 302 T + 24389 T^{2} \))(\( 1 + 186 T + 24389 T^{2} \))(\( 1 - 246 T + 24389 T^{2} \))(\( 1 - 254 T + 24389 T^{2} \))(\( 1 + 106 T + 24389 T^{2} \))(\( 1 - 102 T + 24389 T^{2} \))(\( 1 + 106 T + 24389 T^{2} \))(\( 1 + 10 T + 24389 T^{2} \))(\( 1 + 122 T + 24389 T^{2} \))(\( 1 - 6 T + 24389 T^{2} \))(\( 1 - 54 T + 24389 T^{2} \))(\( 1 + 114 T + 24389 T^{2} \))(\( 1 + 252 T + 56446 T^{2} + 6146028 T^{3} + 594823321 T^{4} \))(\( 1 - 212 T + 57182 T^{2} - 5170468 T^{3} + 594823321 T^{4} \))(\( ( 1 - 102 T + 24389 T^{2} )^{2} \))
$31$ (\( 1 - 200 T + 29791 T^{2} \))(\( 1 + 224 T + 29791 T^{2} \))(\( 1 + 80 T + 29791 T^{2} \))(\( 1 - 72 T + 29791 T^{2} \))(\( 1 - 144 T + 29791 T^{2} \))(\( 1 - 160 T + 29791 T^{2} \))(\( 1 + 304 T + 29791 T^{2} \))(\( 1 - 48 T + 29791 T^{2} \))(\( 1 + 64 T + 29791 T^{2} \))(\( 1 - 256 T + 29791 T^{2} \))(\( 1 - 112 T + 29791 T^{2} \))(\( 1 + 56 T + 29791 T^{2} \))(\( 1 + 40 T - 13890 T^{2} + 1191640 T^{3} + 887503681 T^{4} \))(\( 1 - 136 T + 61374 T^{2} - 4051576 T^{3} + 887503681 T^{4} \))(\( 1 + 96 T - 24386 T^{2} + 2859936 T^{3} + 887503681 T^{4} \))
$37$ (\( 1 + 150 T + 50653 T^{2} \))(\( 1 - 94 T + 50653 T^{2} \))(\( 1 + 290 T + 50653 T^{2} \))(\( 1 - 398 T + 50653 T^{2} \))(\( 1 - 158 T + 50653 T^{2} \))(\( 1 - 398 T + 50653 T^{2} \))(\( 1 + 114 T + 50653 T^{2} \))(\( 1 + 246 T + 50653 T^{2} \))(\( 1 + 306 T + 50653 T^{2} \))(\( 1 + 86 T + 50653 T^{2} \))(\( 1 + 178 T + 50653 T^{2} \))(\( 1 + 34 T + 50653 T^{2} \))(\( 1 + 248 T + 98214 T^{2} + 12561944 T^{3} + 2565726409 T^{4} \))(\( 1 + 24 T - 18202 T^{2} + 1215672 T^{3} + 2565726409 T^{4} \))(\( 1 - 504 T + 163462 T^{2} - 25529112 T^{3} + 2565726409 T^{4} \))
$41$ (\( 1 - 172 T + 68921 T^{2} \))(\( 1 + 478 T + 68921 T^{2} \))(\( 1 + 374 T + 68921 T^{2} \))(\( 1 - 462 T + 68921 T^{2} \))(\( 1 + 390 T + 68921 T^{2} \))(\( 1 + 318 T + 68921 T^{2} \))(\( 1 - 202 T + 68921 T^{2} \))(\( 1 + 248 T + 68921 T^{2} \))(\( 1 - 50 T + 68921 T^{2} \))(\( 1 - 160 T + 68921 T^{2} \))(\( 1 - 378 T + 68921 T^{2} \))(\( 1 - 6 T + 68921 T^{2} \))(\( 1 + 450 T + 175642 T^{2} + 31014450 T^{3} + 4750104241 T^{4} \))(\( 1 - 694 T + 243914 T^{2} - 47831174 T^{3} + 4750104241 T^{4} \))(\( 1 + 102 T + 99666 T^{2} + 7029942 T^{3} + 4750104241 T^{4} \))
$43$ (\( 1 + 164 T + 79507 T^{2} \))(\( 1 - 316 T + 79507 T^{2} \))(\( 1 + 164 T + 79507 T^{2} \))(\( 1 + 212 T + 79507 T^{2} \))(\( 1 - 508 T + 79507 T^{2} \))(\( 1 - 268 T + 79507 T^{2} \))(\( 1 + 116 T + 79507 T^{2} \))(\( 1 + 68 T + 79507 T^{2} \))(\( 1 + 20 T + 79507 T^{2} \))(\( 1 - 220 T + 79507 T^{2} \))(\( 1 - 172 T + 79507 T^{2} \))(\( 1 + 164 T + 79507 T^{2} \))(\( 1 + 376 T + 161526 T^{2} + 29894632 T^{3} + 6321363049 T^{4} \))(\( 1 - 584 T + 232950 T^{2} - 46432088 T^{3} + 6321363049 T^{4} \))(\( 1 + 296 T + 94646 T^{2} + 23534072 T^{3} + 6321363049 T^{4} \))
$47$ (\( 1 - 460 T + 103823 T^{2} \))(\( 1 + 256 T + 103823 T^{2} \))(\( 1 + 464 T + 103823 T^{2} \))(\( 1 - 264 T + 103823 T^{2} \))(\( 1 - 528 T + 103823 T^{2} \))(\( 1 + 240 T + 103823 T^{2} \))(\( 1 + 224 T + 103823 T^{2} \))(\( 1 + 324 T + 103823 T^{2} \))(\( 1 - 176 T + 103823 T^{2} \))(\( 1 + 308 T + 103823 T^{2} \))(\( 1 - 192 T + 103823 T^{2} \))(\( 1 + 168 T + 103823 T^{2} \))(\( 1 - 12 T + 141790 T^{2} - 1245876 T^{3} + 10779215329 T^{4} \))(\( 1 - 316 T + 231902 T^{2} - 32808068 T^{3} + 10779215329 T^{4} \))(\( 1 - 780 T + 326046 T^{2} - 80981940 T^{3} + 10779215329 T^{4} \))
$53$ (\( 1 + 190 T + 148877 T^{2} \))(\( 1 + 66 T + 148877 T^{2} \))(\( 1 + 162 T + 148877 T^{2} \))(\( 1 + 162 T + 148877 T^{2} \))(\( 1 - 606 T + 148877 T^{2} \))(\( 1 + 498 T + 148877 T^{2} \))(\( 1 + 274 T + 148877 T^{2} \))(\( 1 - 258 T + 148877 T^{2} \))(\( 1 - 526 T + 148877 T^{2} \))(\( 1 - 258 T + 148877 T^{2} \))(\( 1 + 402 T + 148877 T^{2} \))(\( 1 - 654 T + 148877 T^{2} \))(\( 1 + 1104 T + 602230 T^{2} + 164360208 T^{3} + 22164361129 T^{4} \))(\( 1 - 560 T + 369782 T^{2} - 83371120 T^{3} + 22164361129 T^{4} \))(\( 1 - 192 T + 197782 T^{2} - 28584384 T^{3} + 22164361129 T^{4} \))
$59$ (\( 1 + 96 T + 205379 T^{2} \))(\( 1 + 420 T + 205379 T^{2} \))(\( 1 + 180 T + 205379 T^{2} \))(\( 1 - 772 T + 205379 T^{2} \))(\( 1 - 364 T + 205379 T^{2} \))(\( 1 - 132 T + 205379 T^{2} \))(\( 1 - 660 T + 205379 T^{2} \))(\( 1 + 120 T + 205379 T^{2} \))(\( 1 + 540 T + 205379 T^{2} \))(\( 1 + 264 T + 205379 T^{2} \))(\( 1 + 396 T + 205379 T^{2} \))(\( 1 - 492 T + 205379 T^{2} \))(\( 1 + 804 T + 380614 T^{2} + 165124716 T^{3} + 42180533641 T^{4} \))(\( 1 - 492 T + 351622 T^{2} - 101046468 T^{3} + 42180533641 T^{4} \))(\( 1 + 212 T + 355942 T^{2} + 43540348 T^{3} + 42180533641 T^{4} \))
$61$ (\( 1 - 622 T + 226981 T^{2} \))(\( 1 - 342 T + 226981 T^{2} \))(\( 1 + 666 T + 226981 T^{2} \))(\( 1 - 30 T + 226981 T^{2} \))(\( 1 - 678 T + 226981 T^{2} \))(\( 1 - 398 T + 226981 T^{2} \))(\( 1 - 382 T + 226981 T^{2} \))(\( 1 - 622 T + 226981 T^{2} \))(\( 1 + 818 T + 226981 T^{2} \))(\( 1 - 606 T + 226981 T^{2} \))(\( 1 - 254 T + 226981 T^{2} \))(\( 1 + 250 T + 226981 T^{2} \))(\( 1 + 428 T + 425886 T^{2} + 97147868 T^{3} + 51520374361 T^{4} \))(\( 1 + 604 T + 406398 T^{2} + 137096524 T^{3} + 51520374361 T^{4} \))(\( 1 + 100 T + 370190 T^{2} + 22698100 T^{3} + 51520374361 T^{4} \))
$67$ (\( 1 + 744 T + 300763 T^{2} \))(\( 1 + 668 T + 300763 T^{2} \))(\( 1 - 628 T + 300763 T^{2} \))(\( 1 - 764 T + 300763 T^{2} \))(\( 1 + 844 T + 300763 T^{2} \))(\( 1 + 92 T + 300763 T^{2} \))(\( 1 + 12 T + 300763 T^{2} \))(\( 1 + 904 T + 300763 T^{2} \))(\( 1 - 228 T + 300763 T^{2} \))(\( 1 - 520 T + 300763 T^{2} \))(\( 1 - 1012 T + 300763 T^{2} \))(\( 1 - 124 T + 300763 T^{2} \))(\( 1 + 148 T + 440790 T^{2} + 44512924 T^{3} + 90458382169 T^{4} \))(\( 1 - 1020 T + 843926 T^{2} - 306778260 T^{3} + 90458382169 T^{4} \))(\( 1 + 212 T + 611414 T^{2} + 63761756 T^{3} + 90458382169 T^{4} \))
$71$ (\( 1 - 54 T + 357911 T^{2} \))(\( 1 - 272 T + 357911 T^{2} \))(\( 1 + 296 T + 357911 T^{2} \))(\( 1 - 236 T + 357911 T^{2} \))(\( 1 - 8 T + 357911 T^{2} \))(\( 1 - 720 T + 357911 T^{2} \))(\( 1 - 552 T + 357911 T^{2} \))(\( 1 - 678 T + 357911 T^{2} \))(\( 1 + 864 T + 357911 T^{2} \))(\( 1 - 286 T + 357911 T^{2} \))(\( 1 + 840 T + 357911 T^{2} \))(\( 1 + 36 T + 357911 T^{2} \))(\( 1 + 954 T + 930526 T^{2} + 341447094 T^{3} + 128100283921 T^{4} \))(\( 1 - 1710 T + 1336222 T^{2} - 612027810 T^{3} + 128100283921 T^{4} \))(\( 1 - 534 T + 746334 T^{2} - 191124474 T^{3} + 128100283921 T^{4} \))
$73$ (\( 1 - 742 T + 389017 T^{2} \))(\( 1 + 86 T + 389017 T^{2} \))(\( 1 + 518 T + 389017 T^{2} \))(\( 1 - 418 T + 389017 T^{2} \))(\( 1 + 422 T + 389017 T^{2} \))(\( 1 + 502 T + 389017 T^{2} \))(\( 1 + 614 T + 389017 T^{2} \))(\( 1 + 642 T + 389017 T^{2} \))(\( 1 - 106 T + 389017 T^{2} \))(\( 1 + 530 T + 389017 T^{2} \))(\( 1 - 890 T + 389017 T^{2} \))(\( 1 - 1010 T + 389017 T^{2} \))(\( 1 - 1072 T + 1063278 T^{2} - 417026224 T^{3} + 151334226289 T^{4} \))(\( 1 + 1312 T + 1201998 T^{2} + 510390304 T^{3} + 151334226289 T^{4} \))(\( 1 - 1128 T + 1062430 T^{2} - 438811176 T^{3} + 151334226289 T^{4} \))
$79$ (\( 1 - 92 T + 493039 T^{2} \))(\( 1 + 1360 T + 493039 T^{2} \))(\( 1 - 1184 T + 493039 T^{2} \))(\( 1 + 552 T + 493039 T^{2} \))(\( 1 + 384 T + 493039 T^{2} \))(\( 1 - 1024 T + 493039 T^{2} \))(\( 1 + 880 T + 493039 T^{2} \))(\( 1 + 740 T + 493039 T^{2} \))(\( 1 + 736 T + 493039 T^{2} \))(\( 1 - 44 T + 493039 T^{2} \))(\( 1 + 80 T + 493039 T^{2} \))(\( 1 + 56 T + 493039 T^{2} \))(\( 1 - 572 T + 901662 T^{2} - 282018308 T^{3} + 243087455521 T^{4} \))(\( 1 - 556 T + 751134 T^{2} - 274129684 T^{3} + 243087455521 T^{4} \))(\( 1 + 468 T - 92834 T^{2} + 230742252 T^{3} + 243087455521 T^{4} \))
$83$ (\( 1 - 228 T + 571787 T^{2} \))(\( 1 + 188 T + 571787 T^{2} \))(\( 1 + 220 T + 571787 T^{2} \))(\( 1 + 1036 T + 571787 T^{2} \))(\( 1 - 548 T + 571787 T^{2} \))(\( 1 - 204 T + 571787 T^{2} \))(\( 1 - 108 T + 571787 T^{2} \))(\( 1 + 468 T + 571787 T^{2} \))(\( 1 - 588 T + 571787 T^{2} \))(\( 1 + 1012 T + 571787 T^{2} \))(\( 1 - 108 T + 571787 T^{2} \))(\( 1 + 228 T + 571787 T^{2} \))(\( 1 + 1944 T + 1957030 T^{2} + 1111553928 T^{3} + 326940373369 T^{4} \))(\( 1 - 264 T + 979750 T^{2} - 150951768 T^{3} + 326940373369 T^{4} \))(\( 1 - 824 T + 536870 T^{2} - 471152488 T^{3} + 326940373369 T^{4} \))
$89$ (\( 1 + 116 T + 704969 T^{2} \))(\( 1 + 366 T + 704969 T^{2} \))(\( 1 + 774 T + 704969 T^{2} \))(\( 1 - 30 T + 704969 T^{2} \))(\( 1 - 1194 T + 704969 T^{2} \))(\( 1 - 354 T + 704969 T^{2} \))(\( 1 + 86 T + 704969 T^{2} \))(\( 1 - 200 T + 704969 T^{2} \))(\( 1 - 146 T + 704969 T^{2} \))(\( 1 - 768 T + 704969 T^{2} \))(\( 1 + 1638 T + 704969 T^{2} \))(\( 1 - 390 T + 704969 T^{2} \))(\( 1 - 366 T + 1156090 T^{2} - 258018654 T^{3} + 496981290961 T^{4} \))(\( 1 - 70 T - 186262 T^{2} - 49347830 T^{3} + 496981290961 T^{4} \))(\( 1 + 2118 T + 2285746 T^{2} + 1493124342 T^{3} + 496981290961 T^{4} \))
$97$ (\( 1 + 554 T + 912673 T^{2} \))(\( 1 - 1554 T + 912673 T^{2} \))(\( 1 + 1086 T + 912673 T^{2} \))(\( 1 + 1190 T + 912673 T^{2} \))(\( 1 + 1502 T + 912673 T^{2} \))(\( 1 + 286 T + 912673 T^{2} \))(\( 1 - 1426 T + 912673 T^{2} \))(\( 1 + 1266 T + 912673 T^{2} \))(\( 1 + 1214 T + 912673 T^{2} \))(\( 1 - 222 T + 912673 T^{2} \))(\( 1 - 1010 T + 912673 T^{2} \))(\( 1 + 70 T + 912673 T^{2} \))(\( 1 - 808 T + 903054 T^{2} - 737439784 T^{3} + 832972004929 T^{4} \))(\( 1 + 136 T + 1812270 T^{2} + 124123528 T^{3} + 832972004929 T^{4} \))(\( 1 - 400 T + 214046 T^{2} - 365069200 T^{3} + 832972004929 T^{4} \))
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