Properties

Label 549.2.a
Level 549
Weight 2
Character orbit a
Rep. character \(\chi_{549}(1,\cdot)\)
Character field \(\Q\)
Dimension 25
Newform subspaces 9
Sturm bound 124
Trace bound 5

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Defining parameters

Level: \( N \) \(=\) \( 549 = 3^{2} \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 549.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(124\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(549))\).

Total New Old
Modular forms 66 25 41
Cusp forms 59 25 34
Eisenstein series 7 0 7

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

\(3\)\(61\)FrickeDim.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(9\)
\(-\)\(-\)\(+\)\(6\)
Plus space\(+\)\(8\)
Minus space\(-\)\(17\)

Trace form

\( 25q + q^{2} + 27q^{4} - 2q^{5} - 2q^{7} + 3q^{8} + O(q^{10}) \) \( 25q + q^{2} + 27q^{4} - 2q^{5} - 2q^{7} + 3q^{8} + 4q^{10} - 2q^{13} + 20q^{14} + 39q^{16} - 12q^{17} + 8q^{19} + 2q^{20} - 4q^{22} + 4q^{23} + 11q^{25} + 2q^{26} - 28q^{28} + 8q^{29} + 6q^{31} + 17q^{32} - 30q^{34} + 4q^{35} + 8q^{37} - 30q^{38} - 6q^{40} + 6q^{41} + 14q^{43} + 34q^{44} - 20q^{46} + 8q^{47} + 21q^{49} + 11q^{50} - 30q^{52} - 10q^{53} + 2q^{55} + 32q^{56} + 26q^{58} - 18q^{59} + 3q^{61} + 4q^{62} + 55q^{64} + 4q^{65} + 14q^{67} - 32q^{68} - 60q^{70} - 14q^{71} - 2q^{73} - 46q^{74} + 16q^{76} - 24q^{77} + 32q^{79} - 2q^{80} - 22q^{82} + 32q^{83} - 26q^{85} + 44q^{86} - 24q^{88} + 2q^{89} + 42q^{91} - 38q^{92} - 8q^{94} + 40q^{95} - 10q^{97} - 53q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(549))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 61
549.2.a.a \(1\) \(4.384\) \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(q-q^{2}-q^{4}-2q^{7}+3q^{8}+4q^{11}+\cdots\)
549.2.a.b \(1\) \(4.384\) \(\Q\) None \(1\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(q+q^{2}-q^{4}-2q^{7}-3q^{8}-4q^{11}+\cdots\)
549.2.a.c \(1\) \(4.384\) \(\Q\) None \(1\) \(0\) \(3\) \(1\) \(-\) \(+\) \(q+q^{2}-q^{4}+3q^{5}+q^{7}-3q^{8}+3q^{10}+\cdots\)
549.2.a.d \(2\) \(4.384\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(-2\) \(+\) \(-\) \(q+\beta q^{2}+q^{4}+\beta q^{5}-q^{7}-\beta q^{8}+3q^{10}+\cdots\)
549.2.a.e \(2\) \(4.384\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(2\) \(-2\) \(-\) \(+\) \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+q^{5}+(-1+\cdots)q^{7}+\cdots\)
549.2.a.f \(3\) \(4.384\) 3.3.148.1 None \(-1\) \(0\) \(-6\) \(0\) \(-\) \(-\) \(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}-2q^{5}-2\beta _{2}q^{7}+\cdots\)
549.2.a.g \(3\) \(4.384\) 3.3.148.1 None \(-1\) \(0\) \(1\) \(-3\) \(-\) \(-\) \(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(\beta _{1}-\beta _{2})q^{5}+\cdots\)
549.2.a.h \(6\) \(4.384\) 6.6.91407488.1 None \(0\) \(0\) \(-2\) \(2\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-1-\beta _{3}+\beta _{4}+\cdots)q^{5}+\cdots\)
549.2.a.i \(6\) \(4.384\) 6.6.337383424.1 None \(0\) \(0\) \(0\) \(6\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+\beta _{5}q^{5}+(1-\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(549))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(549)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(61))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(183))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 + T^{2} + 4 T^{4} \))(\( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} \))(\( 1 + T + 3 T^{2} + 3 T^{3} + 6 T^{4} + 4 T^{5} + 8 T^{6} \))(\( 1 + T + 3 T^{2} + 3 T^{3} + 6 T^{4} + 4 T^{5} + 8 T^{6} \))(\( 1 + T^{2} - 2 T^{3} + 3 T^{4} - 2 T^{5} + 3 T^{6} - 4 T^{7} + 12 T^{8} - 16 T^{9} + 16 T^{10} + 64 T^{12} \))(\( 1 - T^{2} - 3 T^{4} + 11 T^{6} - 12 T^{8} - 16 T^{10} + 64 T^{12} \))
$3$ 1
$5$ (\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 + 7 T^{2} + 25 T^{4} \))(\( ( 1 - T + 5 T^{2} )^{2} \))(\( ( 1 + 2 T + 5 T^{2} )^{3} \))(\( 1 - T + 6 T^{2} + 3 T^{3} + 30 T^{4} - 25 T^{5} + 125 T^{6} \))(\( 1 + 2 T + 7 T^{2} + 22 T^{3} + 59 T^{4} + 160 T^{5} + 346 T^{6} + 800 T^{7} + 1475 T^{8} + 2750 T^{9} + 4375 T^{10} + 6250 T^{11} + 15625 T^{12} \))(\( 1 + 11 T^{2} + 75 T^{4} + 386 T^{6} + 1875 T^{8} + 6875 T^{10} + 15625 T^{12} \))
$7$ (\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( ( 1 + T + 7 T^{2} )^{2} \))(\( 1 + 2 T + 13 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 + 5 T^{2} - 16 T^{3} + 35 T^{4} + 343 T^{6} \))(\( 1 + 3 T + 20 T^{2} + 41 T^{3} + 140 T^{4} + 147 T^{5} + 343 T^{6} \))(\( 1 - 2 T + 17 T^{2} - 10 T^{3} + 163 T^{4} - 152 T^{5} + 1590 T^{6} - 1064 T^{7} + 7987 T^{8} - 3430 T^{9} + 40817 T^{10} - 33614 T^{11} + 117649 T^{12} \))(\( ( 1 - 3 T + 5 T^{2} + 2 T^{3} + 35 T^{4} - 147 T^{5} + 343 T^{6} )^{2} \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 5 T + 11 T^{2} \))(\( 1 - 5 T^{2} + 121 T^{4} \))(\( 1 - 2 T + 21 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + 2 T + 29 T^{2} + 40 T^{3} + 319 T^{4} + 242 T^{5} + 1331 T^{6} \))(\( 1 + 13 T + 86 T^{2} + 353 T^{3} + 946 T^{4} + 1573 T^{5} + 1331 T^{6} \))(\( 1 - 8 T + 61 T^{2} - 330 T^{3} + 1527 T^{4} - 6058 T^{5} + 21498 T^{6} - 66638 T^{7} + 184767 T^{8} - 439230 T^{9} + 893101 T^{10} - 1288408 T^{11} + 1771561 T^{12} \))(\( 1 + 51 T^{2} + 1211 T^{4} + 16958 T^{6} + 146531 T^{8} + 746691 T^{10} + 1771561 T^{12} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 + 3 T + 13 T^{2} )^{2} \))(\( 1 - 6 T + 35 T^{2} - 116 T^{3} + 455 T^{4} - 1014 T^{5} + 2197 T^{6} \))(\( 1 + 9 T + 50 T^{2} + 197 T^{3} + 650 T^{4} + 1521 T^{5} + 2197 T^{6} \))(\( 1 - 6 T + 55 T^{2} - 274 T^{3} + 1507 T^{4} - 6080 T^{5} + 24378 T^{6} - 79040 T^{7} + 254683 T^{8} - 601978 T^{9} + 1570855 T^{10} - 2227758 T^{11} + 4826809 T^{12} \))(\( ( 1 + 3 T + 23 T^{2} + 86 T^{3} + 299 T^{4} + 507 T^{5} + 2197 T^{6} )^{2} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 22 T^{2} + 289 T^{4} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( 1 + 12 T + 71 T^{2} + 308 T^{3} + 1207 T^{4} + 3468 T^{5} + 4913 T^{6} \))(\( 1 - 2 T + 43 T^{2} - 72 T^{3} + 731 T^{4} - 578 T^{5} + 4913 T^{6} \))(\( 1 + 10 T + 90 T^{2} + 482 T^{3} + 2835 T^{4} + 12484 T^{5} + 60164 T^{6} + 212228 T^{7} + 819315 T^{8} + 2368066 T^{9} + 7516890 T^{10} + 14198570 T^{11} + 24137569 T^{12} \))(\( 1 + 46 T^{2} + 587 T^{4} + 3180 T^{6} + 169643 T^{8} + 3841966 T^{10} + 24137569 T^{12} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( 1 - 4 T + 10 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 + 8 T + 65 T^{2} + 288 T^{3} + 1235 T^{4} + 2888 T^{5} + 6859 T^{6} \))(\( 1 + 9 T^{2} - 20 T^{3} + 171 T^{4} + 6859 T^{6} \))(\( 1 - 8 T + 54 T^{2} - 104 T^{3} + 263 T^{4} + 1280 T^{5} + 532 T^{6} + 24320 T^{7} + 94943 T^{8} - 713336 T^{9} + 7037334 T^{10} - 19808792 T^{11} + 47045881 T^{12} \))(\( ( 1 - 2 T + 19 T^{2} )^{6} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 9 T + 23 T^{2} \))(\( 1 + 19 T^{2} + 529 T^{4} \))(\( 1 - 2 T + 29 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 + 2 T + 25 T^{2} + 72 T^{3} + 575 T^{4} + 1058 T^{5} + 12167 T^{6} \))(\( 1 + 5 T + 74 T^{2} + 229 T^{3} + 1702 T^{4} + 2645 T^{5} + 12167 T^{6} \))(\( 1 + 93 T^{2} - 2 T^{3} + 4215 T^{4} - 346 T^{5} + 119626 T^{6} - 7958 T^{7} + 2229735 T^{8} - 24334 T^{9} + 26025213 T^{10} + 148035889 T^{12} \))(\( 1 + 123 T^{2} + 6611 T^{4} + 198302 T^{6} + 3497219 T^{8} + 34420443 T^{10} + 148035889 T^{12} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 10 T^{2} + 841 T^{4} \))(\( 1 + 26 T^{2} + 841 T^{4} \))(\( 1 + 4 T + 83 T^{2} + 212 T^{3} + 2407 T^{4} + 3364 T^{5} + 24389 T^{6} \))(\( 1 + 4 T + 83 T^{2} + 212 T^{3} + 2407 T^{4} + 3364 T^{5} + 24389 T^{6} \))(\( 1 - 10 T + 118 T^{2} - 618 T^{3} + 4379 T^{4} - 15748 T^{5} + 114652 T^{6} - 456692 T^{7} + 3682739 T^{8} - 15072402 T^{9} + 83459158 T^{10} - 205111490 T^{11} + 594823321 T^{12} \))(\( 1 + 114 T^{2} + 6467 T^{4} + 228980 T^{6} + 5438747 T^{8} + 80630034 T^{10} + 594823321 T^{12} \))
$31$ (\( 1 + 6 T + 31 T^{2} \))(\( 1 + 6 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( 1 - 4 T + 34 T^{2} - 124 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 61 T^{2} + 224 T^{3} + 1891 T^{4} + 7688 T^{5} + 29791 T^{6} \))(\( 1 + 2 T + 17 T^{2} + 240 T^{3} + 527 T^{4} + 1922 T^{5} + 29791 T^{6} \))(\( 1 + 78 T^{2} + 256 T^{3} + 3039 T^{4} + 15680 T^{5} + 105380 T^{6} + 486080 T^{7} + 2920479 T^{8} + 7626496 T^{9} + 72034638 T^{10} + 887503681 T^{12} \))(\( ( 1 - 4 T + 41 T^{2} - 24 T^{3} + 1271 T^{4} - 3844 T^{5} + 29791 T^{6} )^{2} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( ( 1 + 10 T + 37 T^{2} )^{2} \))(\( 1 + 4 T + 70 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( 1 + 6 T + 59 T^{2} + 452 T^{3} + 2183 T^{4} + 8214 T^{5} + 50653 T^{6} \))(\( 1 + 6 T + 75 T^{2} + 336 T^{3} + 2775 T^{4} + 8214 T^{5} + 50653 T^{6} \))(\( 1 + 4 T + 98 T^{2} + 324 T^{3} + 4279 T^{4} + 13768 T^{5} + 142140 T^{6} + 509416 T^{7} + 5857951 T^{8} + 16411572 T^{9} + 183667778 T^{10} + 277375828 T^{11} + 2565726409 T^{12} \))(\( ( 1 - 6 T + 37 T^{2} )^{6} \))
$41$ (\( 1 + 4 T + 41 T^{2} \))(\( 1 - 4 T + 41 T^{2} \))(\( 1 + 5 T + 41 T^{2} \))(\( 1 + 55 T^{2} + 1681 T^{4} \))(\( 1 - 6 T + 83 T^{2} - 246 T^{3} + 1681 T^{4} \))(\( 1 + 2 T + 87 T^{2} + 60 T^{3} + 3567 T^{4} + 3362 T^{5} + 68921 T^{6} \))(\( 1 + 3 T + 62 T^{2} + 55 T^{3} + 2542 T^{4} + 5043 T^{5} + 68921 T^{6} \))(\( 1 - 10 T + 191 T^{2} - 1350 T^{3} + 16051 T^{4} - 89568 T^{5} + 814746 T^{6} - 3672288 T^{7} + 26981731 T^{8} - 93043350 T^{9} + 539720351 T^{10} - 1158562010 T^{11} + 4750104241 T^{12} \))(\( 1 + 43 T^{2} - 957 T^{4} - 92942 T^{6} - 1608717 T^{8} + 121507723 T^{10} + 4750104241 T^{12} \))
$43$ (\( 1 + 2 T + 43 T^{2} \))(\( 1 + 2 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( 1 - 12 T + 90 T^{2} - 516 T^{3} + 1849 T^{4} \))(\( 1 + 9 T^{2} + 16 T^{3} + 387 T^{4} + 79507 T^{6} \))(\( 1 + 14 T + 185 T^{2} + 1272 T^{3} + 7955 T^{4} + 25886 T^{5} + 79507 T^{6} \))(\( 1 - 4 T + 206 T^{2} - 684 T^{3} + 19303 T^{4} - 52984 T^{5} + 1058436 T^{6} - 2278312 T^{7} + 35691247 T^{8} - 54382788 T^{9} + 704273006 T^{10} - 588033772 T^{11} + 6321363049 T^{12} \))(\( ( 1 - 4 T + 77 T^{2} - 120 T^{3} + 3311 T^{4} - 7396 T^{5} + 79507 T^{6} )^{2} \))
$47$ (\( 1 - 4 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 46 T^{2} + 2209 T^{4} \))(\( 1 - 8 T + 38 T^{2} - 376 T^{3} + 2209 T^{4} \))(\( 1 + 4 T + 93 T^{2} + 312 T^{3} + 4371 T^{4} + 8836 T^{5} + 103823 T^{6} \))(\( 1 - 4 T + 53 T^{2} - 392 T^{3} + 2491 T^{4} - 8836 T^{5} + 103823 T^{6} \))(\( 1 - 4 T + 178 T^{2} - 908 T^{3} + 15631 T^{4} - 83848 T^{5} + 881340 T^{6} - 3940856 T^{7} + 34528879 T^{8} - 94271284 T^{9} + 868583218 T^{10} - 917380028 T^{11} + 10779215329 T^{12} \))(\( 1 + 90 T^{2} + 3119 T^{4} + 53036 T^{6} + 6889871 T^{8} + 439171290 T^{10} + 10779215329 T^{12} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 2 T^{2} + 2809 T^{4} \))(\( 1 - 4 T + 38 T^{2} - 212 T^{3} + 2809 T^{4} \))(\( 1 + 12 T + 123 T^{2} + 732 T^{3} + 6519 T^{4} + 33708 T^{5} + 148877 T^{6} \))(\( 1 - 2 T + 147 T^{2} - 204 T^{3} + 7791 T^{4} - 5618 T^{5} + 148877 T^{6} \))(\( 1 - 2 T + 146 T^{2} - 322 T^{3} + 10283 T^{4} - 28788 T^{5} + 545108 T^{6} - 1525764 T^{7} + 28884947 T^{8} - 47938394 T^{9} + 1152010226 T^{10} - 836390986 T^{11} + 22164361129 T^{12} \))(\( 1 + 134 T^{2} + 10995 T^{4} + 654716 T^{6} + 30884955 T^{8} + 1057324454 T^{10} + 22164361129 T^{12} \))
$59$ (\( 1 + 12 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 9 T + 59 T^{2} \))(\( 1 + 43 T^{2} + 3481 T^{4} \))(\( 1 - 10 T + 125 T^{2} - 590 T^{3} + 3481 T^{4} \))(\( 1 + 6 T + 5 T^{2} - 560 T^{3} + 295 T^{4} + 20886 T^{5} + 205379 T^{6} \))(\( 1 + 29 T + 408 T^{2} + 3747 T^{3} + 24072 T^{4} + 100949 T^{5} + 205379 T^{6} \))(\( 1 - 16 T + 389 T^{2} - 4106 T^{3} + 56583 T^{4} - 440906 T^{5} + 4375002 T^{6} - 26013454 T^{7} + 196965423 T^{8} - 843286174 T^{9} + 4713653429 T^{10} - 11438788784 T^{11} + 42180533641 T^{12} \))(\( 1 + 131 T^{2} + 15003 T^{4} + 939614 T^{6} + 52225443 T^{8} + 1587374291 T^{10} + 42180533641 T^{12} \))
$61$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{6} \))(\( ( 1 - T )^{6} \))
$67$ (\( 1 + 10 T + 67 T^{2} \))(\( 1 + 10 T + 67 T^{2} \))(\( 1 + 7 T + 67 T^{2} \))(\( ( 1 - 5 T + 67 T^{2} )^{2} \))(\( 1 - 6 T + 93 T^{2} - 402 T^{3} + 4489 T^{4} \))(\( 1 + 65 T^{2} - 496 T^{3} + 4355 T^{4} + 300763 T^{6} \))(\( 1 - 9 T + 116 T^{2} - 647 T^{3} + 7772 T^{4} - 40401 T^{5} + 300763 T^{6} \))(\( 1 + 2 T + 161 T^{2} + 606 T^{3} + 16899 T^{4} + 67908 T^{5} + 1369430 T^{6} + 4549836 T^{7} + 75859611 T^{8} + 182262378 T^{9} + 3244330481 T^{10} + 2700250214 T^{11} + 90458382169 T^{12} \))(\( ( 1 - 9 T + 209 T^{2} - 1202 T^{3} + 14003 T^{4} - 40401 T^{5} + 300763 T^{6} )^{2} \))
$71$ (\( 1 + 8 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 130 T^{2} + 5041 T^{4} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( 1 + 14 T + 233 T^{2} + 1888 T^{3} + 16543 T^{4} + 70574 T^{5} + 357911 T^{6} \))(\( 1 + 14 T + 201 T^{2} + 1896 T^{3} + 14271 T^{4} + 70574 T^{5} + 357911 T^{6} \))(\( 1 - 18 T + 438 T^{2} - 5406 T^{3} + 76731 T^{4} - 709420 T^{5} + 7223660 T^{6} - 50368820 T^{7} + 386800971 T^{8} - 1934866866 T^{9} + 11130316278 T^{10} - 32476128318 T^{11} + 128100283921 T^{12} \))(\( 1 + 314 T^{2} + 45531 T^{4} + 4012772 T^{6} + 229521771 T^{8} + 7979267834 T^{10} + 128100283921 T^{12} \))
$73$ (\( 1 - 10 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 11 T + 73 T^{2} \))(\( ( 1 - 11 T + 73 T^{2} )^{2} \))(\( 1 + 10 T + 99 T^{2} + 730 T^{3} + 5329 T^{4} \))(\( 1 + 2 T + 71 T^{2} + 828 T^{3} + 5183 T^{4} + 10658 T^{5} + 389017 T^{6} \))(\( 1 + T + 174 T^{2} + 121 T^{3} + 12702 T^{4} + 5329 T^{5} + 389017 T^{6} \))(\( 1 - 30 T + 719 T^{2} - 11614 T^{3} + 160203 T^{4} - 1739636 T^{5} + 16506762 T^{6} - 126993428 T^{7} + 853721787 T^{8} - 4518043438 T^{9} + 20418335279 T^{10} - 62192147790 T^{11} + 151334226289 T^{12} \))(\( ( 1 + 25 T + 387 T^{2} + 3958 T^{3} + 28251 T^{4} + 133225 T^{5} + 389017 T^{6} )^{2} \))
$79$ (\( 1 + 6 T + 79 T^{2} \))(\( 1 + 6 T + 79 T^{2} \))(\( 1 - 3 T + 79 T^{2} \))(\( ( 1 - 11 T + 79 T^{2} )^{2} \))(\( 1 + 2 T + 157 T^{2} + 158 T^{3} + 6241 T^{4} \))(\( 1 + 12 T + 149 T^{2} + 1912 T^{3} + 11771 T^{4} + 74892 T^{5} + 493039 T^{6} \))(\( 1 - 13 T + 186 T^{2} - 1429 T^{3} + 14694 T^{4} - 81133 T^{5} + 493039 T^{6} \))(\( 1 - 6 T + 401 T^{2} - 1650 T^{3} + 68699 T^{4} - 203164 T^{5} + 6836870 T^{6} - 16049956 T^{7} + 428750459 T^{8} - 813514350 T^{9} + 15618982481 T^{10} - 18462338394 T^{11} + 243087455521 T^{12} \))(\( ( 1 - 7 T + 69 T^{2} + 266 T^{3} + 5451 T^{4} - 43687 T^{5} + 493039 T^{6} )^{2} \))
$83$ (\( 1 - 4 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 118 T^{2} + 6889 T^{4} \))(\( 1 - 8 T + 174 T^{2} - 664 T^{3} + 6889 T^{4} \))(\( 1 - 8 T + 185 T^{2} - 1072 T^{3} + 15355 T^{4} - 55112 T^{5} + 571787 T^{6} \))(\( 1 - 8 T + 185 T^{2} - 1072 T^{3} + 15355 T^{4} - 55112 T^{5} + 571787 T^{6} \))(\( 1 - 12 T + 234 T^{2} - 2388 T^{3} + 34631 T^{4} - 291352 T^{5} + 3439916 T^{6} - 24182216 T^{7} + 238572959 T^{8} - 1365427356 T^{9} + 11105247114 T^{10} - 47268487716 T^{11} + 326940373369 T^{12} \))(\( 1 + 306 T^{2} + 45671 T^{4} + 4458716 T^{6} + 314627519 T^{8} + 14522246226 T^{10} + 326940373369 T^{12} \))
$89$ (\( 1 - 2 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 - 4 T + 89 T^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( 1 - 16 T + 210 T^{2} - 1424 T^{3} + 7921 T^{4} \))(\( 1 - 4 T + 159 T^{2} - 660 T^{3} + 14151 T^{4} - 31684 T^{5} + 704969 T^{6} \))(\( 1 - 4 T + 211 T^{2} - 792 T^{3} + 18779 T^{4} - 31684 T^{5} + 704969 T^{6} \))(\( 1 + 26 T + 550 T^{2} + 7986 T^{3} + 104323 T^{4} + 1107204 T^{5} + 11288476 T^{6} + 98541156 T^{7} + 826342483 T^{8} + 5629882434 T^{9} + 34508232550 T^{10} + 145185545674 T^{11} + 496981290961 T^{12} \))(\( 1 + 10 T^{2} + 7835 T^{4} + 663684 T^{6} + 62061035 T^{8} + 627422410 T^{10} + 496981290961 T^{12} \))
$97$ (\( 1 + 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( 1 + 12 T + 198 T^{2} + 1164 T^{3} + 9409 T^{4} \))(\( 1 - 18 T + 383 T^{2} - 3596 T^{3} + 37151 T^{4} - 169362 T^{5} + 912673 T^{6} \))(\( 1 - 10 T + 175 T^{2} - 844 T^{3} + 16975 T^{4} - 94090 T^{5} + 912673 T^{6} \))(\( 1 - 16 T + 402 T^{2} - 5552 T^{3} + 76159 T^{4} - 885888 T^{5} + 9052124 T^{6} - 85931136 T^{7} + 716580031 T^{8} - 5067160496 T^{9} + 35588770962 T^{10} - 137397444112 T^{11} + 832972004929 T^{12} \))(\( ( 1 + 2 T + 235 T^{2} + 500 T^{3} + 22795 T^{4} + 18818 T^{5} + 912673 T^{6} )^{2} \))
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