Properties

Label 6018.2.a.z
Level 6018
Weight 2
Character orbit 6018.a
Self dual yes
Analytic conductor 48.054
Analytic rank 0
Dimension 11
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6018.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.0539719364\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + \beta_{1} q^{5} - q^{6} -\beta_{2} q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} + \beta_{1} q^{5} - q^{6} -\beta_{2} q^{7} + q^{8} + q^{9} + \beta_{1} q^{10} + ( 1 + \beta_{4} ) q^{11} - q^{12} + ( \beta_{1} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{13} -\beta_{2} q^{14} -\beta_{1} q^{15} + q^{16} - q^{17} + q^{18} + \beta_{5} q^{19} + \beta_{1} q^{20} + \beta_{2} q^{21} + ( 1 + \beta_{4} ) q^{22} + ( 1 + \beta_{2} + \beta_{6} ) q^{23} - q^{24} + ( 1 + \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{9} ) q^{25} + ( \beta_{1} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{26} - q^{27} -\beta_{2} q^{28} + ( 2 + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{10} ) q^{29} -\beta_{1} q^{30} + ( 1 - \beta_{4} + \beta_{5} + \beta_{10} ) q^{31} + q^{32} + ( -1 - \beta_{4} ) q^{33} - q^{34} + ( -\beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{35} + q^{36} + ( 2 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{9} ) q^{37} + \beta_{5} q^{38} + ( -\beta_{1} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{39} + \beta_{1} q^{40} + ( 1 + \beta_{1} + \beta_{4} + \beta_{9} ) q^{41} + \beta_{2} q^{42} + ( 1 - \beta_{2} - \beta_{3} + \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{43} + ( 1 + \beta_{4} ) q^{44} + \beta_{1} q^{45} + ( 1 + \beta_{2} + \beta_{6} ) q^{46} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{47} - q^{48} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{49} + ( 1 + \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{9} ) q^{50} + q^{51} + ( \beta_{1} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{52} + ( 1 - \beta_{3} - \beta_{8} + \beta_{9} ) q^{53} - q^{54} + ( -1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{8} + \beta_{9} ) q^{55} -\beta_{2} q^{56} -\beta_{5} q^{57} + ( 2 + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{10} ) q^{58} - q^{59} -\beta_{1} q^{60} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{61} + ( 1 - \beta_{4} + \beta_{5} + \beta_{10} ) q^{62} -\beta_{2} q^{63} + q^{64} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{10} ) q^{65} + ( -1 - \beta_{4} ) q^{66} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{67} - q^{68} + ( -1 - \beta_{2} - \beta_{6} ) q^{69} + ( -\beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{70} + ( 2 - \beta_{1} - \beta_{5} - \beta_{10} ) q^{71} + q^{72} + ( 3 + \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{73} + ( 2 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{9} ) q^{74} + ( -1 - \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{9} ) q^{75} + \beta_{5} q^{76} + ( 2 + 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{77} + ( -\beta_{1} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{78} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{79} + \beta_{1} q^{80} + q^{81} + ( 1 + \beta_{1} + \beta_{4} + \beta_{9} ) q^{82} + ( 1 + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} + 2 \beta_{9} ) q^{83} + \beta_{2} q^{84} -\beta_{1} q^{85} + ( 1 - \beta_{2} - \beta_{3} + \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{86} + ( -2 - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{87} + ( 1 + \beta_{4} ) q^{88} + ( 2 \beta_{1} - \beta_{5} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{89} + \beta_{1} q^{90} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{8} + \beta_{10} ) q^{91} + ( 1 + \beta_{2} + \beta_{6} ) q^{92} + ( -1 + \beta_{4} - \beta_{5} - \beta_{10} ) q^{93} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{94} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{7} - \beta_{9} ) q^{95} - q^{96} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{97} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{98} + ( 1 + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q + 11q^{2} - 11q^{3} + 11q^{4} + 4q^{5} - 11q^{6} + 3q^{7} + 11q^{8} + 11q^{9} + O(q^{10}) \) \( 11q + 11q^{2} - 11q^{3} + 11q^{4} + 4q^{5} - 11q^{6} + 3q^{7} + 11q^{8} + 11q^{9} + 4q^{10} + 9q^{11} - 11q^{12} + 6q^{13} + 3q^{14} - 4q^{15} + 11q^{16} - 11q^{17} + 11q^{18} - q^{19} + 4q^{20} - 3q^{21} + 9q^{22} + 10q^{23} - 11q^{24} + 15q^{25} + 6q^{26} - 11q^{27} + 3q^{28} + 14q^{29} - 4q^{30} + 17q^{31} + 11q^{32} - 9q^{33} - 11q^{34} + 8q^{35} + 11q^{36} + 30q^{37} - q^{38} - 6q^{39} + 4q^{40} + 10q^{41} - 3q^{42} + 11q^{43} + 9q^{44} + 4q^{45} + 10q^{46} - 6q^{47} - 11q^{48} + 18q^{49} + 15q^{50} + 11q^{51} + 6q^{52} + 10q^{53} - 11q^{54} - 11q^{55} + 3q^{56} + q^{57} + 14q^{58} - 11q^{59} - 4q^{60} + 13q^{61} + 17q^{62} + 3q^{63} + 11q^{64} + 32q^{65} - 9q^{66} + 26q^{67} - 11q^{68} - 10q^{69} + 8q^{70} + 14q^{71} + 11q^{72} + 20q^{73} + 30q^{74} - 15q^{75} - q^{76} + 26q^{77} - 6q^{78} + 15q^{79} + 4q^{80} + 11q^{81} + 10q^{82} + 2q^{83} - 3q^{84} - 4q^{85} + 11q^{86} - 14q^{87} + 9q^{88} + q^{89} + 4q^{90} + 17q^{91} + 10q^{92} - 17q^{93} - 6q^{94} + 3q^{95} - 11q^{96} + 33q^{97} + 18q^{98} + 9q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 4 x^{10} - 27 x^{9} + 117 x^{8} + 200 x^{7} - 1023 x^{6} - 484 x^{5} + 3403 x^{4} + 562 x^{3} - 4372 x^{2} - 692 x + 1200\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-68377 \nu^{10} + 100902 \nu^{9} + 1972995 \nu^{8} - 2438247 \nu^{7} - 16451050 \nu^{6} + 12213703 \nu^{5} + 41682174 \nu^{4} - 1530071 \nu^{3} - 16641720 \nu^{2} - 30704680 \nu - 6530016\)\()/9881348\)
\(\beta_{3}\)\(=\)\((\)\(478220 \nu^{10} - 806963 \nu^{9} - 14837488 \nu^{8} + 21958815 \nu^{7} + 147883587 \nu^{6} - 155628260 \nu^{5} - 599066657 \nu^{4} + 301247970 \nu^{3} + 966550127 \nu^{2} + 17946790 \nu - 277474058\)\()/4940674\)
\(\beta_{4}\)\(=\)\((\)\(-690399 \nu^{10} + 1061578 \nu^{9} + 21837033 \nu^{8} - 28769331 \nu^{7} - 224589340 \nu^{6} + 200728579 \nu^{5} + 944194472 \nu^{4} - 362823799 \nu^{3} - 1555875710 \nu^{2} - 115397832 \nu + 437533206\)\()/4940674\)
\(\beta_{5}\)\(=\)\((\)\(-3156869 \nu^{10} + 5380560 \nu^{9} + 98175115 \nu^{8} - 145767653 \nu^{7} - 981973340 \nu^{6} + 1024667347 \nu^{5} + 3999281264 \nu^{4} - 1934315723 \nu^{3} - 6482502258 \nu^{2} - 230680552 \nu + 1824506128\)\()/9881348\)
\(\beta_{6}\)\(=\)\((\)\(1183567 \nu^{10} - 2051153 \nu^{9} - 36518360 \nu^{8} + 55439219 \nu^{7} + 359948269 \nu^{6} - 387860873 \nu^{5} - 1432388663 \nu^{4} + 728576639 \nu^{3} + 2258150935 \nu^{2} + 65727972 \nu - 599619469\)\()/2470337\)
\(\beta_{7}\)\(=\)\((\)\(4973375 \nu^{10} - 8328764 \nu^{9} - 154431069 \nu^{8} + 224826723 \nu^{7} + 1539701372 \nu^{6} - 1563255225 \nu^{5} - 6227037308 \nu^{4} + 2853612557 \nu^{3} + 9978785870 \nu^{2} + 558006780 \nu - 2734225624\)\()/9881348\)
\(\beta_{8}\)\(=\)\((\)\(1707617 \nu^{10} - 2986333 \nu^{9} - 52746178 \nu^{8} + 80817374 \nu^{7} + 521178581 \nu^{6} - 567120079 \nu^{5} - 2085306979 \nu^{4} + 1067360641 \nu^{3} + 3317249635 \nu^{2} + 108639160 \nu - 894863368\)\()/2470337\)
\(\beta_{9}\)\(=\)\((\)\(10702739 \nu^{10} - 18943456 \nu^{9} - 330397033 \nu^{8} + 513011107 \nu^{7} + 3262430232 \nu^{6} - 3608727329 \nu^{5} - 13063793500 \nu^{4} + 6833246697 \nu^{3} + 20866262830 \nu^{2} + 615967628 \nu - 5650580696\)\()/9881348\)
\(\beta_{10}\)\(=\)\((\)\(-20411677 \nu^{10} + 35419276 \nu^{9} + 632398343 \nu^{8} - 959470865 \nu^{7} - 6282554076 \nu^{6} + 6746297163 \nu^{5} + 25332890104 \nu^{4} - 12726418695 \nu^{3} - 40564710178 \nu^{2} - 1489912936 \nu + 10921660816\)\()/9881348\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{9} + 2 \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{1} + 6\)
\(\nu^{3}\)\(=\)\(\beta_{10} + 2 \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{5} - 4 \beta_{4} - \beta_{3} + \beta_{2} + 10 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-\beta_{10} - 20 \beta_{9} + 36 \beta_{8} + \beta_{7} - 16 \beta_{6} - 16 \beta_{5} + 20 \beta_{4} - 4 \beta_{3} + 8 \beta_{2} + 15 \beta_{1} + 66\)
\(\nu^{5}\)\(=\)\(19 \beta_{10} + 44 \beta_{9} - 26 \beta_{8} - 18 \beta_{7} + 37 \beta_{6} + 29 \beta_{5} - 83 \beta_{4} - 14 \beta_{3} + 12 \beta_{2} + 124 \beta_{1} - 40\)
\(\nu^{6}\)\(=\)\(-25 \beta_{10} - 337 \beta_{9} + 572 \beta_{8} + 32 \beta_{7} - 251 \beta_{6} - 253 \beta_{5} + 355 \beta_{4} - 81 \beta_{3} + 181 \beta_{2} + 196 \beta_{1} + 901\)
\(\nu^{7}\)\(=\)\(312 \beta_{10} + 824 \beta_{9} - 587 \beta_{8} - 300 \beta_{7} + 643 \beta_{6} + 596 \beta_{5} - 1472 \beta_{4} - 144 \beta_{3} + 96 \beta_{2} + 1680 \beta_{1} - 937\)
\(\nu^{8}\)\(=\)\(-512 \beta_{10} - 5508 \beta_{9} + 8945 \beta_{8} + 708 \beta_{7} - 4004 \beta_{6} - 4024 \beta_{5} + 6100 \beta_{4} - 1282 \beta_{3} + 3188 \beta_{2} + 2422 \beta_{1} + 13390\)
\(\nu^{9}\)\(=\)\(4996 \beta_{10} + 14746 \beta_{9} - 12143 \beta_{8} - 4849 \beta_{7} + 11135 \beta_{6} + 11092 \beta_{5} - 25076 \beta_{4} - 1149 \beta_{3} + 199 \beta_{2} + 23706 \beta_{1} - 19052\)
\(\nu^{10}\)\(=\)\(-9750 \beta_{10} - 89608 \beta_{9} + 140335 \beta_{8} + 13689 \beta_{7} - 64588 \beta_{6} - 64478 \beta_{5} + 103301 \beta_{4} - 18981 \beta_{3} + 52165 \beta_{2} + 28182 \beta_{1} + 206442\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.08681
−2.64826
−1.55818
−1.33314
−0.760310
0.504544
2.03178
2.25177
2.25950
3.59322
3.74589
1.00000 −1.00000 1.00000 −4.08681 −1.00000 −0.153752 1.00000 1.00000 −4.08681
1.2 1.00000 −1.00000 1.00000 −2.64826 −1.00000 3.55508 1.00000 1.00000 −2.64826
1.3 1.00000 −1.00000 1.00000 −1.55818 −1.00000 −1.66703 1.00000 1.00000 −1.55818
1.4 1.00000 −1.00000 1.00000 −1.33314 −1.00000 −3.20872 1.00000 1.00000 −1.33314
1.5 1.00000 −1.00000 1.00000 −0.760310 −1.00000 −1.62737 1.00000 1.00000 −0.760310
1.6 1.00000 −1.00000 1.00000 0.504544 −1.00000 2.39214 1.00000 1.00000 0.504544
1.7 1.00000 −1.00000 1.00000 2.03178 −1.00000 −2.78135 1.00000 1.00000 2.03178
1.8 1.00000 −1.00000 1.00000 2.25177 −1.00000 3.43625 1.00000 1.00000 2.25177
1.9 1.00000 −1.00000 1.00000 2.25950 −1.00000 3.73494 1.00000 1.00000 2.25950
1.10 1.00000 −1.00000 1.00000 3.59322 −1.00000 3.34551 1.00000 1.00000 3.59322
1.11 1.00000 −1.00000 1.00000 3.74589 −1.00000 −4.02570 1.00000 1.00000 3.74589
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6018.2.a.z 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.z 11 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(17\) \(1\)
\(59\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\(T_{5}^{11} - \cdots\)
\(T_{7}^{11} - \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{11} \)
$3$ \( ( 1 + T )^{11} \)
$5$ \( 1 - 4 T + 28 T^{2} - 83 T^{3} + 360 T^{4} - 843 T^{5} + 2841 T^{6} - 5387 T^{7} + 16212 T^{8} - 25937 T^{9} + 79238 T^{10} - 120820 T^{11} + 396190 T^{12} - 648425 T^{13} + 2026500 T^{14} - 3366875 T^{15} + 8878125 T^{16} - 13171875 T^{17} + 28125000 T^{18} - 32421875 T^{19} + 54687500 T^{20} - 39062500 T^{21} + 48828125 T^{22} \)
$7$ \( 1 - 3 T + 34 T^{2} - 92 T^{3} + 690 T^{4} - 1635 T^{5} + 9612 T^{6} - 20703 T^{7} + 103651 T^{8} - 200742 T^{9} + 889845 T^{10} - 1567634 T^{11} + 6228915 T^{12} - 9836358 T^{13} + 35552293 T^{14} - 49707903 T^{15} + 161548884 T^{16} - 192356115 T^{17} + 568244670 T^{18} - 530361692 T^{19} + 1372022638 T^{20} - 847425747 T^{21} + 1977326743 T^{22} \)
$11$ \( 1 - 9 T + 95 T^{2} - 548 T^{3} + 3577 T^{4} - 16218 T^{5} + 83482 T^{6} - 322251 T^{7} + 1422693 T^{8} - 4864829 T^{9} + 19170922 T^{10} - 59179954 T^{11} + 210880142 T^{12} - 588644309 T^{13} + 1893604383 T^{14} - 4718076891 T^{15} + 13444859582 T^{16} - 28731176298 T^{17} + 69705610667 T^{18} - 117468666788 T^{19} + 224005030645 T^{20} - 233436821409 T^{21} + 285311670611 T^{22} \)
$13$ \( 1 - 6 T + 74 T^{2} - 290 T^{3} + 2111 T^{4} - 6170 T^{5} + 36064 T^{6} - 88300 T^{7} + 446947 T^{8} - 925648 T^{9} + 4430087 T^{10} - 9224868 T^{11} + 57591131 T^{12} - 156434512 T^{13} + 981942559 T^{14} - 2521936300 T^{15} + 13390310752 T^{16} - 29781411530 T^{17} + 132462119387 T^{18} - 236561909090 T^{19} + 784732953602 T^{20} - 827150951094 T^{21} + 1792160394037 T^{22} \)
$17$ \( ( 1 + T )^{11} \)
$19$ \( 1 + T + 128 T^{2} - 29 T^{3} + 7321 T^{4} - 12492 T^{5} + 249745 T^{6} - 894519 T^{7} + 5895112 T^{8} - 33805701 T^{9} + 113851209 T^{10} - 797637736 T^{11} + 2163172971 T^{12} - 12203858061 T^{13} + 40434573208 T^{14} - 116574610599 T^{15} + 618393344755 T^{16} - 587697145452 T^{17} + 6544035001219 T^{18} - 492523328189 T^{19} + 41304025315712 T^{20} + 6131066257801 T^{21} + 116490258898219 T^{22} \)
$23$ \( 1 - 10 T + 195 T^{2} - 1516 T^{3} + 17310 T^{4} - 113347 T^{5} + 972036 T^{6} - 5545469 T^{7} + 38924230 T^{8} - 195399715 T^{9} + 1168720699 T^{10} - 5159745986 T^{11} + 26880576077 T^{12} - 103366449235 T^{13} + 473591106410 T^{14} - 1551849590429 T^{15} + 6256357104348 T^{16} - 16779423910483 T^{17} + 58937528487570 T^{18} - 118719453685996 T^{19} + 351224768985285 T^{20} - 414265112136490 T^{21} + 952809757913927 T^{22} \)
$29$ \( 1 - 14 T + 238 T^{2} - 2399 T^{3} + 25358 T^{4} - 206327 T^{5} + 1689265 T^{6} - 11698249 T^{7} + 80416878 T^{8} - 488919195 T^{9} + 2938404708 T^{10} - 15920222560 T^{11} + 85213736532 T^{12} - 411181042995 T^{13} + 1961287237542 T^{14} - 8273949250969 T^{15} + 34648766115485 T^{16} - 122728111351967 T^{17} + 437422363443622 T^{18} - 1200091144693439 T^{19} + 3452700742256822 T^{20} - 5889901266202814 T^{21} + 12200509765705829 T^{22} \)
$31$ \( 1 - 17 T + 247 T^{2} - 2812 T^{3} + 29042 T^{4} - 260389 T^{5} + 2160344 T^{6} - 16316368 T^{7} + 115440467 T^{8} - 753625354 T^{9} + 4643318967 T^{10} - 26611855720 T^{11} + 143942887977 T^{12} - 724233965194 T^{13} + 3439086952397 T^{14} - 15068508491728 T^{15} + 61848814587944 T^{16} - 231096195991909 T^{17} + 799021339011662 T^{18} - 2398329597284092 T^{19} + 6530586673685737 T^{20} - 13933680878673617 T^{21} + 25408476896404831 T^{22} \)
$37$ \( 1 - 30 T + 516 T^{2} - 6239 T^{3} + 58186 T^{4} - 436507 T^{5} + 2715113 T^{6} - 14343215 T^{7} + 67275402 T^{8} - 301517259 T^{9} + 1443858134 T^{10} - 8033227004 T^{11} + 53422750958 T^{12} - 412777127571 T^{13} + 3407700937506 T^{14} - 26881494167615 T^{15} + 188276679122141 T^{16} - 1119957537613363 T^{17} + 5523706202860738 T^{18} - 21914359313013119 T^{19} + 67060257734259732 T^{20} - 144257531172535470 T^{21} + 177917621779460413 T^{22} \)
$41$ \( 1 - 10 T + 365 T^{2} - 2954 T^{3} + 60470 T^{4} - 408733 T^{5} + 6148614 T^{6} - 35486467 T^{7} + 436134586 T^{8} - 2187405683 T^{9} + 23131300421 T^{10} - 102024602272 T^{11} + 948383317261 T^{12} - 3677028953123 T^{13} + 30058831801706 T^{14} - 100276274476387 T^{15} + 712355059455414 T^{16} - 1941524356736653 T^{17} + 11776790941584070 T^{18} - 23587469126823434 T^{19} + 119494406053795765 T^{20} - 134226593101524010 T^{21} + 550329031716248441 T^{22} \)
$43$ \( 1 - 11 T + 281 T^{2} - 3140 T^{3} + 43317 T^{4} - 432094 T^{5} + 4527972 T^{6} - 38909167 T^{7} + 340428471 T^{8} - 2551415567 T^{9} + 19127035900 T^{10} - 126007952442 T^{11} + 822462543700 T^{12} - 4717567383383 T^{13} + 27066446443797 T^{14} - 133022699048767 T^{15} + 665650113667596 T^{16} - 2731423045294606 T^{17} + 11774366777321919 T^{18} - 36700948871667140 T^{19} + 141228523954252883 T^{20} - 237726305446126739 T^{21} + 929293739471222707 T^{22} \)
$47$ \( 1 + 6 T + 235 T^{2} + 1237 T^{3} + 27328 T^{4} + 144510 T^{5} + 2253074 T^{6} + 12507376 T^{7} + 149276661 T^{8} + 836868396 T^{9} + 8262758745 T^{10} + 44092833302 T^{11} + 388349661015 T^{12} + 1848642286764 T^{13} + 15498350775003 T^{14} + 61032005027056 T^{15} + 516731272301518 T^{16} + 1557704407193790 T^{17} + 13844996636012864 T^{18} + 29454561600598357 T^{19} + 262995661179150245 T^{20} + 315594793414980294 T^{21} + 2472159215084012303 T^{22} \)
$53$ \( 1 - 10 T + 500 T^{2} - 4484 T^{3} + 117486 T^{4} - 937195 T^{5} + 17086977 T^{6} - 120430875 T^{7} + 1707151194 T^{8} - 10546849619 T^{9} + 122980029886 T^{10} - 658040847186 T^{11} + 6517941583958 T^{12} - 29626100579771 T^{13} + 254155548309138 T^{14} - 950257531000875 T^{15} + 7145696770394661 T^{16} - 20772328428293155 T^{17} + 138012112974889782 T^{18} - 279172451804542724 T^{19} + 1649881795901066500 T^{20} - 1748874703655130490 T^{21} + 9269035929372191597 T^{22} \)
$59$ \( ( 1 + T )^{11} \)
$61$ \( 1 - 13 T + 563 T^{2} - 6710 T^{3} + 150081 T^{4} - 1616270 T^{5} + 24905612 T^{6} - 239441415 T^{7} + 2850655351 T^{8} - 24166137901 T^{9} + 235892367236 T^{10} - 1736218384774 T^{11} + 14389434401396 T^{12} - 89922199129621 T^{13} + 647044602225331 T^{14} - 3315267760905015 T^{15} + 21035187769341212 T^{16} - 83270835468453470 T^{17} + 471665987572867701 T^{18} - 1286356070211755510 T^{19} + 6583804250265621383 T^{20} - 9273457851617473813 T^{21} + 43513917611435838661 T^{22} \)
$67$ \( 1 - 26 T + 589 T^{2} - 9799 T^{3} + 148746 T^{4} - 1931036 T^{5} + 23257443 T^{6} - 253616403 T^{7} + 2593504252 T^{8} - 24594268782 T^{9} + 220624587087 T^{10} - 1855391455220 T^{11} + 14781847334829 T^{12} - 110403672562398 T^{13} + 780030119344276 T^{14} - 5110654824437763 T^{15} + 31400457718921401 T^{16} - 174678392470097084 T^{17} + 901506608445374958 T^{18} - 3979057172377525159 T^{19} + 16024648759417723783 T^{20} - 47393782918345797674 T^{21} + \)\(12\!\cdots\!83\)\( T^{22} \)
$71$ \( 1 - 14 T + 589 T^{2} - 6467 T^{3} + 154714 T^{4} - 1395290 T^{5} + 24883062 T^{6} - 190493176 T^{7} + 2830896977 T^{8} - 18945968312 T^{9} + 248604974441 T^{10} - 1490384924298 T^{11} + 17650953185311 T^{12} - 95506626260792 T^{13} + 1013209167935047 T^{14} - 4840751821188856 T^{15} + 44894750803152762 T^{16} - 178737045152132090 T^{17} + 1407142420185305174 T^{18} - 4176088086566336387 T^{19} + 27004766923166479259 T^{20} - 45573409714138336814 T^{21} + \)\(23\!\cdots\!71\)\( T^{22} \)
$73$ \( 1 - 20 T + 281 T^{2} - 4322 T^{3} + 51068 T^{4} - 531865 T^{5} + 5992894 T^{6} - 60219675 T^{7} + 565373456 T^{8} - 5604724289 T^{9} + 51028787635 T^{10} - 428234401720 T^{11} + 3725101497355 T^{12} - 29867575736081 T^{13} + 219939885732752 T^{14} - 1710132843591675 T^{15} + 12423698311260142 T^{16} - 80489378265198985 T^{17} + 564168547573245596 T^{18} - 3485520517166218082 T^{19} + 16542915865023283553 T^{20} - 85952516594071152980 T^{21} + \)\(31\!\cdots\!77\)\( T^{22} \)
$79$ \( 1 - 15 T + 571 T^{2} - 6507 T^{3} + 138192 T^{4} - 1162572 T^{5} + 18355298 T^{6} - 102770505 T^{7} + 1474103091 T^{8} - 3917430053 T^{9} + 87211762843 T^{10} - 67971873592 T^{11} + 6889729264597 T^{12} - 24448680960773 T^{13} + 726790313883549 T^{14} - 4002919494160905 T^{15} + 56480287166451902 T^{16} - 282606669339960012 T^{17} + 2653826590615284528 T^{18} - 9871827026061992427 T^{19} + 68435261306075060149 T^{20} - \)\(14\!\cdots\!15\)\( T^{21} + \)\(74\!\cdots\!79\)\( T^{22} \)
$83$ \( 1 - 2 T + 305 T^{2} + 404 T^{3} + 47807 T^{4} + 172945 T^{5} + 6235712 T^{6} + 25348229 T^{7} + 713062419 T^{8} + 2727590341 T^{9} + 68597783678 T^{10} + 246823772998 T^{11} + 5693616045274 T^{12} + 18790369859149 T^{13} + 407719821372753 T^{14} + 1202984388663509 T^{15} + 24562723006042816 T^{16} + 56542702872301705 T^{17} + 1297293189661097989 T^{18} + 909926061784172564 T^{19} + 57016777856599822915 T^{20} - 31032082374411706898 T^{21} + \)\(12\!\cdots\!67\)\( T^{22} \)
$89$ \( 1 - T + 467 T^{2} + 587 T^{3} + 104971 T^{4} + 426358 T^{5} + 15306383 T^{6} + 114752722 T^{7} + 1669796932 T^{8} + 18343921590 T^{9} + 155470497383 T^{10} + 1963819049122 T^{11} + 13836874267087 T^{12} + 145302202914390 T^{13} + 1177155073355108 T^{14} + 7199842939130002 T^{15} + 85471752621162967 T^{16} + 211891949251550038 T^{17} + 4643007455318574659 T^{18} + 2310777628947121547 T^{19} + \)\(16\!\cdots\!03\)\( T^{20} - 31181719929966183601 T^{21} + \)\(27\!\cdots\!89\)\( T^{22} \)
$97$ \( 1 - 33 T + 1061 T^{2} - 23268 T^{3} + 469045 T^{4} - 7907015 T^{5} + 123492206 T^{6} - 1709966267 T^{7} + 22157957367 T^{8} - 260674072892 T^{9} + 2884883742560 T^{10} - 29273429105914 T^{11} + 279833723028320 T^{12} - 2452682351840828 T^{13} + 20222969424011991 T^{14} - 151382084151764027 T^{15} + 1060469592009536942 T^{16} - 6586322137553676935 T^{17} + 37898031343036512085 T^{18} - \)\(18\!\cdots\!48\)\( T^{19} + \)\(80\!\cdots\!37\)\( T^{20} - \)\(24\!\cdots\!17\)\( T^{21} + \)\(71\!\cdots\!53\)\( T^{22} \)
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