Properties

Label 38.6.c.a
Level 38
Weight 6
Character orbit 38.c
Analytic conductor 6.095
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 38.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.09458515289\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Defining polynomial: \(x^{6} + 133 x^{4} - 60 x^{3} + 17689 x^{2} - 3990 x + 900\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 4 - 4 \beta_{3} ) q^{2} + ( -5 + \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{3} -16 \beta_{3} q^{4} + ( 5 - \beta_{1} - \beta_{2} - 5 \beta_{3} - \beta_{5} ) q^{5} + ( 4 \beta_{1} + 20 \beta_{3} ) q^{6} + ( -104 + 4 \beta_{2} ) q^{7} -64 q^{8} + ( -10 \beta_{1} - 138 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( 4 - 4 \beta_{3} ) q^{2} + ( -5 + \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{3} -16 \beta_{3} q^{4} + ( 5 - \beta_{1} - \beta_{2} - 5 \beta_{3} - \beta_{5} ) q^{5} + ( 4 \beta_{1} + 20 \beta_{3} ) q^{6} + ( -104 + 4 \beta_{2} ) q^{7} -64 q^{8} + ( -10 \beta_{1} - 138 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{9} + ( -4 \beta_{1} - 20 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{10} + ( -156 + \beta_{2} - \beta_{4} ) q^{11} + ( 80 - 16 \beta_{2} ) q^{12} + ( -15 \beta_{1} - 247 \beta_{3} + 5 \beta_{4} - 5 \beta_{5} ) q^{13} + ( -416 + 16 \beta_{1} + 16 \beta_{2} + 416 \beta_{3} ) q^{14} + ( 54 \beta_{1} + 321 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} ) q^{15} + ( -256 + 256 \beta_{3} ) q^{16} + ( -335 + 45 \beta_{1} + 45 \beta_{2} + 335 \beta_{3} + 5 \beta_{5} ) q^{17} + ( -552 + 40 \beta_{2} + 16 \beta_{4} ) q^{18} + ( 893 + 19 \beta_{1} + 114 \beta_{3} - 19 \beta_{5} ) q^{19} + ( -80 + 16 \beta_{2} + 16 \beta_{4} ) q^{20} + ( 1944 - 124 \beta_{1} - 124 \beta_{2} - 1944 \beta_{3} - 16 \beta_{5} ) q^{21} + ( -624 + 4 \beta_{1} + 4 \beta_{2} + 624 \beta_{3} - 4 \beta_{5} ) q^{22} + ( -14 \beta_{1} - 711 \beta_{3} - 35 \beta_{4} + 35 \beta_{5} ) q^{23} + ( 320 - 64 \beta_{1} - 64 \beta_{2} - 320 \beta_{3} ) q^{24} + ( -83 \beta_{1} - 1052 \beta_{3} - 31 \beta_{4} + 31 \beta_{5} ) q^{25} + ( -988 + 60 \beta_{2} + 20 \beta_{4} ) q^{26} + ( 2795 - 121 \beta_{2} - 60 \beta_{4} ) q^{27} + ( 64 \beta_{1} + 1664 \beta_{3} ) q^{28} + ( -61 \beta_{1} + 505 \beta_{3} - 29 \beta_{4} + 29 \beta_{5} ) q^{29} + ( 1284 - 216 \beta_{2} - 36 \beta_{4} ) q^{30} + ( 48 + 356 \beta_{2} + 54 \beta_{4} ) q^{31} + 1024 \beta_{3} q^{32} + ( 1196 - 117 \beta_{1} - 117 \beta_{2} - 1196 \beta_{3} + \beta_{5} ) q^{33} + ( 180 \beta_{1} + 1340 \beta_{3} - 20 \beta_{4} + 20 \beta_{5} ) q^{34} + ( -1704 + 300 \beta_{1} + 300 \beta_{2} + 1704 \beta_{3} + 120 \beta_{5} ) q^{35} + ( -2208 + 160 \beta_{1} + 160 \beta_{2} + 2208 \beta_{3} + 64 \beta_{5} ) q^{36} + ( 3326 - 94 \beta_{2} + 52 \beta_{4} ) q^{37} + ( 4028 - 76 \beta_{2} - 3572 \beta_{3} + 76 \beta_{4} - 76 \beta_{5} ) q^{38} + ( 6275 - 542 \beta_{2} - 85 \beta_{4} ) q^{39} + ( -320 + 64 \beta_{1} + 64 \beta_{2} + 320 \beta_{3} + 64 \beta_{5} ) q^{40} + ( -6463 - 212 \beta_{1} - 212 \beta_{2} + 6463 \beta_{3} - 162 \beta_{5} ) q^{41} + ( -496 \beta_{1} - 7776 \beta_{3} + 64 \beta_{4} - 64 \beta_{5} ) q^{42} + ( -1701 + 202 \beta_{1} + 202 \beta_{2} + 1701 \beta_{3} + 45 \beta_{5} ) q^{43} + ( 16 \beta_{1} + 2496 \beta_{3} + 16 \beta_{4} - 16 \beta_{5} ) q^{44} + ( -19074 + 744 \beta_{2} + 18 \beta_{4} ) q^{45} + ( -2844 + 56 \beta_{2} - 140 \beta_{4} ) q^{46} + ( 166 \beta_{1} + 2409 \beta_{3} + 199 \beta_{4} - 199 \beta_{5} ) q^{47} + ( -256 \beta_{1} - 1280 \beta_{3} ) q^{48} + ( -295 - 832 \beta_{2} - 64 \beta_{4} ) q^{49} + ( -4208 + 332 \beta_{2} - 124 \beta_{4} ) q^{50} + ( -780 \beta_{1} - 17395 \beta_{3} + 205 \beta_{4} - 205 \beta_{5} ) q^{51} + ( -3952 + 240 \beta_{1} + 240 \beta_{2} + 3952 \beta_{3} + 80 \beta_{5} ) q^{52} + ( 319 \beta_{1} - 1199 \beta_{3} + 441 \beta_{4} - 441 \beta_{5} ) q^{53} + ( 11180 - 484 \beta_{1} - 484 \beta_{2} - 11180 \beta_{3} - 240 \beta_{5} ) q^{54} + ( 2780 + 176 \beta_{1} + 176 \beta_{2} - 2780 \beta_{3} + 200 \beta_{5} ) q^{55} + ( 6656 - 256 \beta_{2} ) q^{56} + ( -11799 + 1729 \beta_{1} + 1102 \beta_{2} + 3325 \beta_{3} - 19 \beta_{4} + 95 \beta_{5} ) q^{57} + ( 2020 + 244 \beta_{2} - 116 \beta_{4} ) q^{58} + ( -125 - 303 \beta_{1} - 303 \beta_{2} + 125 \beta_{3} + 50 \beta_{5} ) q^{59} + ( 5136 - 864 \beta_{1} - 864 \beta_{2} - 5136 \beta_{3} - 144 \beta_{5} ) q^{60} + ( -243 \beta_{1} + 6481 \beta_{3} + 231 \beta_{4} - 231 \beta_{5} ) q^{61} + ( 192 + 1424 \beta_{1} + 1424 \beta_{2} - 192 \beta_{3} + 216 \beta_{5} ) q^{62} + ( 2296 \beta_{1} + 27632 \beta_{3} - 576 \beta_{4} + 576 \beta_{5} ) q^{63} + 4096 q^{64} + ( -24955 + 1127 \beta_{2} + 107 \beta_{4} ) q^{65} + ( -468 \beta_{1} - 4784 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{66} + ( -1781 \beta_{1} + 20665 \beta_{3} - 158 \beta_{4} + 158 \beta_{5} ) q^{67} + ( 5360 - 720 \beta_{2} - 80 \beta_{4} ) q^{68} + ( 10639 + 759 \beta_{2} + 119 \beta_{4} ) q^{69} + ( 1200 \beta_{1} + 6816 \beta_{3} - 480 \beta_{4} + 480 \beta_{5} ) q^{70} + ( 1951 - 2818 \beta_{1} - 2818 \beta_{2} - 1951 \beta_{3} - 93 \beta_{5} ) q^{71} + ( 640 \beta_{1} + 8832 \beta_{3} - 256 \beta_{4} + 256 \beta_{5} ) q^{72} + ( 6137 - 1132 \beta_{1} - 1132 \beta_{2} - 6137 \beta_{3} + 336 \beta_{5} ) q^{73} + ( 13304 - 376 \beta_{1} - 376 \beta_{2} - 13304 \beta_{3} + 208 \beta_{5} ) q^{74} + ( 36668 - 103 \beta_{2} - 177 \beta_{4} ) q^{75} + ( 1824 - 304 \beta_{1} - 304 \beta_{2} - 16112 \beta_{3} + 304 \beta_{4} ) q^{76} + ( 17888 - 552 \beta_{2} + 88 \beta_{4} ) q^{77} + ( 25100 - 2168 \beta_{1} - 2168 \beta_{2} - 25100 \beta_{3} - 340 \beta_{5} ) q^{78} + ( 26487 - 3960 \beta_{1} - 3960 \beta_{2} - 26487 \beta_{3} + 697 \beta_{5} ) q^{79} + ( 256 \beta_{1} + 1280 \beta_{3} - 256 \beta_{4} + 256 \beta_{5} ) q^{80} + ( -19917 + 3610 \beta_{1} + 3610 \beta_{2} + 19917 \beta_{3} - 188 \beta_{5} ) q^{81} + ( -848 \beta_{1} + 25852 \beta_{3} + 648 \beta_{4} - 648 \beta_{5} ) q^{82} + ( -55748 + 3447 \beta_{2} - 457 \beta_{4} ) q^{83} + ( -31104 + 1984 \beta_{2} + 256 \beta_{4} ) q^{84} + ( 2685 \beta_{1} + 34275 \beta_{3} - 315 \beta_{4} + 315 \beta_{5} ) q^{85} + ( 808 \beta_{1} + 6804 \beta_{3} - 180 \beta_{4} + 180 \beta_{5} ) q^{86} + ( 20931 + 1476 \beta_{2} - 99 \beta_{4} ) q^{87} + ( 9984 - 64 \beta_{2} + 64 \beta_{4} ) q^{88} + ( 1455 \beta_{1} - 55439 \beta_{3} + 89 \beta_{4} - 89 \beta_{5} ) q^{89} + ( -76296 + 2976 \beta_{1} + 2976 \beta_{2} + 76296 \beta_{3} + 72 \beta_{5} ) q^{90} + ( 3428 \beta_{1} + 45848 \beta_{3} - 760 \beta_{4} + 760 \beta_{5} ) q^{91} + ( -11376 + 224 \beta_{1} + 224 \beta_{2} + 11376 \beta_{3} - 560 \beta_{5} ) q^{92} + ( 123256 - 4108 \beta_{1} - 4108 \beta_{2} - 123256 \beta_{3} - 1694 \beta_{5} ) q^{93} + ( 9636 - 664 \beta_{2} + 796 \beta_{4} ) q^{94} + ( 10659 - 1444 \beta_{1} - 1938 \beta_{2} - 77729 \beta_{3} - 950 \beta_{4} - 133 \beta_{5} ) q^{95} + ( -5120 + 1024 \beta_{2} ) q^{96} + ( -29047 + 3472 \beta_{1} + 3472 \beta_{2} + 29047 \beta_{3} + 526 \beta_{5} ) q^{97} + ( -1180 - 3328 \beta_{1} - 3328 \beta_{2} + 1180 \beta_{3} - 256 \beta_{5} ) q^{98} + ( 1494 \beta_{1} + 9784 \beta_{3} - 706 \beta_{4} + 706 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 12q^{2} - 15q^{3} - 48q^{4} + 14q^{5} + 60q^{6} - 624q^{7} - 384q^{8} - 410q^{9} + O(q^{10}) \) \( 6q + 12q^{2} - 15q^{3} - 48q^{4} + 14q^{5} + 60q^{6} - 624q^{7} - 384q^{8} - 410q^{9} - 56q^{10} - 938q^{11} + 480q^{12} - 736q^{13} - 1248q^{14} + 954q^{15} - 768q^{16} - 1000q^{17} - 3280q^{18} + 5681q^{19} - 448q^{20} + 5816q^{21} - 1876q^{22} - 2168q^{23} + 960q^{24} - 3187q^{25} - 5888q^{26} + 16650q^{27} + 4992q^{28} + 1486q^{29} + 7632q^{30} + 396q^{31} + 3072q^{32} + 3589q^{33} + 4000q^{34} - 4992q^{35} - 6560q^{36} + 20060q^{37} + 13528q^{38} + 37480q^{39} - 896q^{40} - 19551q^{41} - 23264q^{42} - 5058q^{43} + 7504q^{44} - 114408q^{45} - 17344q^{46} + 7426q^{47} - 3840q^{48} - 1898q^{49} - 25496q^{50} - 51980q^{51} - 11776q^{52} - 3156q^{53} + 33300q^{54} + 8540q^{55} + 39936q^{56} - 60762q^{57} + 11888q^{58} - 325q^{59} + 15264q^{60} + 19674q^{61} + 792q^{62} + 82320q^{63} + 24576q^{64} - 149516q^{65} - 14356q^{66} + 61837q^{67} + 32000q^{68} + 64072q^{69} + 19968q^{70} + 5760q^{71} + 26240q^{72} + 18747q^{73} + 40120q^{74} + 219654q^{75} - 36784q^{76} + 107504q^{77} + 74960q^{78} + 80158q^{79} + 3584q^{80} - 59939q^{81} + 78204q^{82} - 335402q^{83} - 186112q^{84} + 102510q^{85} + 20232q^{86} + 125388q^{87} + 60032q^{88} - 166228q^{89} - 228816q^{90} + 136784q^{91} - 34688q^{92} + 368074q^{93} + 59408q^{94} - 171266q^{95} - 30720q^{96} - 86615q^{97} - 3796q^{98} + 28646q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 133 x^{4} - 60 x^{3} + 17689 x^{2} - 3990 x + 900\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{3} - 60 \)\()/133\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 133 \nu^{3} - 30 \nu^{2} + 17689 \nu \)\()/3990\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} + 133 \nu^{2} - 30 \nu + 11837 \)\()/133\)
\(\beta_{5}\)\(=\)\((\)\( -89 \nu^{5} - 11837 \nu^{3} + 6660 \nu^{2} - 1574321 \nu + 355110 \)\()/3990\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(\beta_{5} + 89 \beta_{3} - 89\)
\(\nu^{3}\)\(=\)\((\)\(133 \beta_{2} + 60\)\()/2\)
\(\nu^{4}\)\(=\)\(-133 \beta_{5} + 133 \beta_{4} - 11837 \beta_{3} + 15 \beta_{1}\)
\(\nu^{5}\)\(=\)\((\)\(60 \beta_{5} + 13320 \beta_{3} - 17689 \beta_{2} - 17689 \beta_{1} - 13320\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\) \(21\)
\(\chi(n)\) \(-\beta_{3}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
5.70904 9.88835i
0.112825 0.195419i
−5.82187 + 10.0838i
5.70904 + 9.88835i
0.112825 + 0.195419i
−5.82187 10.0838i
2.00000 + 3.46410i −13.9181 24.1068i −8.00000 + 13.8564i 34.6044 + 59.9365i 55.6723 96.4273i −195.345 −64.0000 −265.926 + 460.597i −138.418 + 239.746i
7.2 2.00000 + 3.46410i −2.72565 4.72096i −8.00000 + 13.8564i −41.7489 72.3112i 10.9026 18.8839i −105.805 −64.0000 106.642 184.709i 166.996 289.245i
7.3 2.00000 + 3.46410i 9.14373 + 15.8374i −8.00000 + 13.8564i 14.1445 + 24.4990i −36.5749 + 63.3496i −10.8501 −64.0000 −45.7157 + 79.1819i −56.5781 + 97.9961i
11.1 2.00000 3.46410i −13.9181 + 24.1068i −8.00000 13.8564i 34.6044 59.9365i 55.6723 + 96.4273i −195.345 −64.0000 −265.926 460.597i −138.418 239.746i
11.2 2.00000 3.46410i −2.72565 + 4.72096i −8.00000 13.8564i −41.7489 + 72.3112i 10.9026 + 18.8839i −105.805 −64.0000 106.642 + 184.709i 166.996 + 289.245i
11.3 2.00000 3.46410i 9.14373 15.8374i −8.00000 13.8564i 14.1445 24.4990i −36.5749 63.3496i −10.8501 −64.0000 −45.7157 79.1819i −56.5781 97.9961i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.6.c.a 6
3.b odd 2 1 342.6.g.a 6
4.b odd 2 1 304.6.i.a 6
19.c even 3 1 inner 38.6.c.a 6
19.c even 3 1 722.6.a.e 3
19.d odd 6 1 722.6.a.f 3
57.h odd 6 1 342.6.g.a 6
76.g odd 6 1 304.6.i.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.c.a 6 1.a even 1 1 trivial
38.6.c.a 6 19.c even 3 1 inner
304.6.i.a 6 4.b odd 2 1
304.6.i.a 6 76.g odd 6 1
342.6.g.a 6 3.b odd 2 1
342.6.g.a 6 57.h odd 6 1
722.6.a.e 3 19.c even 3 1
722.6.a.f 3 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 15 T_{3}^{5} + 682 T_{3}^{4} - 1305 T_{3}^{3} + 250474 T_{3}^{2} + 1268175 T_{3} + 7700625 \) acting on \(S_{6}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 4 T + 16 T^{2} )^{3} \)
$3$ \( 1 + 15 T - 47 T^{2} - 4950 T^{3} - 59837 T^{4} + 130935 T^{5} + 17818902 T^{6} + 31817205 T^{7} - 3533315013 T^{8} - 71027089650 T^{9} - 163878866847 T^{10} + 12709329141645 T^{11} + 205891132094649 T^{12} \)
$5$ \( 1 - 14 T - 2996 T^{2} - 196640 T^{3} + 1277528 T^{4} + 385096642 T^{5} + 33053246326 T^{6} + 1203427006250 T^{7} + 12475859375000 T^{8} - 6000976562500000 T^{9} - 285720825195312500 T^{10} - 4172325134277343750 T^{11} + \)\(93\!\cdots\!25\)\( T^{12} \)
$7$ \( ( 1 + 312 T + 74357 T^{2} + 10711824 T^{3} + 1249718099 T^{4} + 88132277688 T^{5} + 4747561509943 T^{6} )^{2} \)
$11$ \( ( 1 + 469 T + 549865 T^{2} + 153971518 T^{3} + 88556308115 T^{4} + 12164652137869 T^{5} + 4177248169415651 T^{6} )^{2} \)
$13$ \( 1 + 736 T - 499140 T^{2} - 287313252 T^{3} + 309244832284 T^{4} + 80793864978472 T^{5} - 97145685360579842 T^{6} + 29998196509451804296 T^{7} + \)\(42\!\cdots\!16\)\( T^{8} - \)\(14\!\cdots\!64\)\( T^{9} - \)\(94\!\cdots\!40\)\( T^{10} + \)\(51\!\cdots\!48\)\( T^{11} + \)\(26\!\cdots\!49\)\( T^{12} \)
$17$ \( 1 + 1000 T - 2408696 T^{2} - 2405502500 T^{3} + 3872248293944 T^{4} + 2245689732583000 T^{5} - 4337536158265406426 T^{6} + \)\(31\!\cdots\!00\)\( T^{7} + \)\(78\!\cdots\!56\)\( T^{8} - \)\(68\!\cdots\!00\)\( T^{9} - \)\(97\!\cdots\!96\)\( T^{10} + \)\(57\!\cdots\!00\)\( T^{11} + \)\(81\!\cdots\!49\)\( T^{12} \)
$19$ \( 1 - 5681 T + 15930569 T^{2} - 29730651014 T^{3} + 39445665970331 T^{4} - 34830587410567481 T^{5} + 15181127029874798299 T^{6} \)
$23$ \( 1 + 2168 T - 8760066 T^{2} - 30615252044 T^{3} + 29724241296454 T^{4} + 116606691007999892 T^{5} + 66348890302406753462 T^{6} + \)\(75\!\cdots\!56\)\( T^{7} + \)\(12\!\cdots\!46\)\( T^{8} - \)\(81\!\cdots\!08\)\( T^{9} - \)\(15\!\cdots\!66\)\( T^{10} + \)\(23\!\cdots\!24\)\( T^{11} + \)\(71\!\cdots\!49\)\( T^{12} \)
$29$ \( 1 - 1486 T - 52804508 T^{2} + 35874759344 T^{3} + 1810629470101136 T^{4} - 507041284560567550 T^{5} - \)\(42\!\cdots\!46\)\( T^{6} - \)\(10\!\cdots\!50\)\( T^{7} + \)\(76\!\cdots\!36\)\( T^{8} + \)\(30\!\cdots\!56\)\( T^{9} - \)\(93\!\cdots\!08\)\( T^{10} - \)\(53\!\cdots\!14\)\( T^{11} + \)\(74\!\cdots\!01\)\( T^{12} \)
$31$ \( ( 1 - 198 T + 4743581 T^{2} + 270431123436 T^{3} + 135804696729731 T^{4} - 162286400822198598 T^{5} + \)\(23\!\cdots\!51\)\( T^{6} )^{2} \)
$37$ \( ( 1 - 10030 T + 220041227 T^{2} - 1356924175348 T^{3} + 15258529383315239 T^{4} - 48230101255351025470 T^{5} + \)\(33\!\cdots\!93\)\( T^{6} )^{2} \)
$41$ \( 1 + 19551 T + 79730791 T^{2} - 2332252585848 T^{3} - 24150006361237239 T^{4} + \)\(13\!\cdots\!21\)\( T^{5} + \)\(43\!\cdots\!42\)\( T^{6} + \)\(16\!\cdots\!21\)\( T^{7} - \)\(32\!\cdots\!39\)\( T^{8} - \)\(36\!\cdots\!48\)\( T^{9} + \)\(14\!\cdots\!91\)\( T^{10} + \)\(40\!\cdots\!51\)\( T^{11} + \)\(24\!\cdots\!01\)\( T^{12} \)
$43$ \( 1 + 5058 T - 391958150 T^{2} - 895138250460 T^{3} + 105353925148109606 T^{4} + \)\(11\!\cdots\!14\)\( T^{5} - \)\(17\!\cdots\!54\)\( T^{6} + \)\(17\!\cdots\!02\)\( T^{7} + \)\(22\!\cdots\!94\)\( T^{8} - \)\(28\!\cdots\!20\)\( T^{9} - \)\(18\!\cdots\!50\)\( T^{10} + \)\(34\!\cdots\!94\)\( T^{11} + \)\(10\!\cdots\!49\)\( T^{12} \)
$47$ \( 1 - 7426 T - 397164762 T^{2} + 4334007544900 T^{3} + 72285854998029634 T^{4} - \)\(58\!\cdots\!14\)\( T^{5} - \)\(10\!\cdots\!10\)\( T^{6} - \)\(13\!\cdots\!98\)\( T^{7} + \)\(38\!\cdots\!66\)\( T^{8} + \)\(52\!\cdots\!00\)\( T^{9} - \)\(10\!\cdots\!62\)\( T^{10} - \)\(47\!\cdots\!82\)\( T^{11} + \)\(14\!\cdots\!49\)\( T^{12} \)
$53$ \( 1 + 3156 T - 21762380 T^{2} + 21897373038084 T^{3} + 22136104052497932 T^{4} + \)\(12\!\cdots\!84\)\( T^{5} + \)\(26\!\cdots\!78\)\( T^{6} + \)\(51\!\cdots\!12\)\( T^{7} + \)\(38\!\cdots\!68\)\( T^{8} + \)\(16\!\cdots\!88\)\( T^{9} - \)\(66\!\cdots\!80\)\( T^{10} + \)\(40\!\cdots\!08\)\( T^{11} + \)\(53\!\cdots\!49\)\( T^{12} \)
$59$ \( 1 + 325 T - 2078392259 T^{2} - 654694327310 T^{3} + 2833968652603107515 T^{4} + \)\(58\!\cdots\!45\)\( T^{5} - \)\(23\!\cdots\!46\)\( T^{6} + \)\(41\!\cdots\!55\)\( T^{7} + \)\(14\!\cdots\!15\)\( T^{8} - \)\(23\!\cdots\!90\)\( T^{9} - \)\(54\!\cdots\!59\)\( T^{10} + \)\(60\!\cdots\!75\)\( T^{11} + \)\(13\!\cdots\!01\)\( T^{12} \)
$61$ \( 1 - 19674 T - 1939800348 T^{2} + 20046443458280 T^{3} + 2801522678585113080 T^{4} - \)\(13\!\cdots\!94\)\( T^{5} - \)\(25\!\cdots\!66\)\( T^{6} - \)\(11\!\cdots\!94\)\( T^{7} + \)\(19\!\cdots\!80\)\( T^{8} + \)\(12\!\cdots\!80\)\( T^{9} - \)\(98\!\cdots\!48\)\( T^{10} - \)\(84\!\cdots\!74\)\( T^{11} + \)\(36\!\cdots\!01\)\( T^{12} \)
$67$ \( 1 - 61837 T + 384165349 T^{2} + 110380695724918 T^{3} - 2801865955161978413 T^{4} - \)\(94\!\cdots\!33\)\( T^{5} + \)\(80\!\cdots\!46\)\( T^{6} - \)\(12\!\cdots\!31\)\( T^{7} - \)\(51\!\cdots\!37\)\( T^{8} + \)\(27\!\cdots\!74\)\( T^{9} + \)\(12\!\cdots\!49\)\( T^{10} - \)\(27\!\cdots\!59\)\( T^{11} + \)\(60\!\cdots\!49\)\( T^{12} \)
$71$ \( 1 - 5760 T - 1162067618 T^{2} + 76725553844268 T^{3} - 968682651749976234 T^{4} - \)\(43\!\cdots\!32\)\( T^{5} + \)\(10\!\cdots\!54\)\( T^{6} - \)\(79\!\cdots\!32\)\( T^{7} - \)\(31\!\cdots\!34\)\( T^{8} + \)\(45\!\cdots\!68\)\( T^{9} - \)\(12\!\cdots\!18\)\( T^{10} - \)\(11\!\cdots\!60\)\( T^{11} + \)\(34\!\cdots\!01\)\( T^{12} \)
$73$ \( 1 - 18747 T - 4569061397 T^{2} + 15800845049124 T^{3} + 12998093967735391937 T^{4} + \)\(36\!\cdots\!31\)\( T^{5} - \)\(30\!\cdots\!66\)\( T^{6} + \)\(74\!\cdots\!83\)\( T^{7} + \)\(55\!\cdots\!13\)\( T^{8} + \)\(14\!\cdots\!68\)\( T^{9} - \)\(84\!\cdots\!97\)\( T^{10} - \)\(71\!\cdots\!71\)\( T^{11} + \)\(79\!\cdots\!49\)\( T^{12} \)
$79$ \( 1 - 80158 T + 6756292046 T^{2} - 548417554298436 T^{3} + 24171391996329566410 T^{4} - \)\(10\!\cdots\!18\)\( T^{5} + \)\(79\!\cdots\!74\)\( T^{6} - \)\(31\!\cdots\!82\)\( T^{7} + \)\(22\!\cdots\!10\)\( T^{8} - \)\(15\!\cdots\!64\)\( T^{9} + \)\(60\!\cdots\!46\)\( T^{10} - \)\(22\!\cdots\!42\)\( T^{11} + \)\(84\!\cdots\!01\)\( T^{12} \)
$83$ \( ( 1 + 167701 T + 13355547801 T^{2} + 802637191479838 T^{3} + 52608045597668276043 T^{4} + \)\(26\!\cdots\!49\)\( T^{5} + \)\(61\!\cdots\!07\)\( T^{6} )^{2} \)
$89$ \( 1 + 166228 T + 2865258232 T^{2} + 177116904254068 T^{3} + \)\(14\!\cdots\!24\)\( T^{4} + \)\(79\!\cdots\!28\)\( T^{5} - \)\(58\!\cdots\!46\)\( T^{6} + \)\(44\!\cdots\!72\)\( T^{7} + \)\(46\!\cdots\!24\)\( T^{8} + \)\(30\!\cdots\!32\)\( T^{9} + \)\(27\!\cdots\!32\)\( T^{10} + \)\(90\!\cdots\!72\)\( T^{11} + \)\(30\!\cdots\!01\)\( T^{12} \)
$97$ \( 1 + 86615 T - 13044791905 T^{2} - 1310411799333648 T^{3} + \)\(11\!\cdots\!05\)\( T^{4} + \)\(75\!\cdots\!05\)\( T^{5} - \)\(52\!\cdots\!38\)\( T^{6} + \)\(64\!\cdots\!85\)\( T^{7} + \)\(82\!\cdots\!45\)\( T^{8} - \)\(82\!\cdots\!64\)\( T^{9} - \)\(70\!\cdots\!05\)\( T^{10} + \)\(40\!\cdots\!55\)\( T^{11} + \)\(40\!\cdots\!49\)\( T^{12} \)
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