Properties

Label 6018.2.a.ba
Level 6018
Weight 2
Character orbit 6018.a
Self dual yes
Analytic conductor 48.054
Analytic rank 0
Dimension 12
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{11} - 31 x^{10} + 111 x^{9} + 381 x^{8} - 1101 x^{7} - 2301 x^{6} + 4690 x^{5} + 6915 x^{4} - 8325 x^{3} - 10082 x^{2} + 5149 x + 5653\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-37967 \nu^{11} + 239615 \nu^{10} + 833951 \nu^{9} - 6728760 \nu^{8} - 5503599 \nu^{7} + 69490919 \nu^{6} + 4825579 \nu^{5} - 319841161 \nu^{4} + 53318216 \nu^{3} + 631646387 \nu^{2} - 84801718 \nu - 435582988\)\()/1323382\)
\(\beta_{3}\)\(=\)\((\)\(-118047 \nu^{11} + 575314 \nu^{10} + 3132406 \nu^{9} - 15664289 \nu^{8} - 30426407 \nu^{7} + 151503348 \nu^{6} + 126683562 \nu^{5} - 617251320 \nu^{4} - 192373155 \nu^{3} + 993409765 \nu^{2} + 94398553 \nu - 541927834\)\()/2646764\)
\(\beta_{4}\)\(=\)\((\)\(127547 \nu^{11} - 600100 \nu^{10} - 3402596 \nu^{9} + 16221775 \nu^{8} + 33453103 \nu^{7} - 156149770 \nu^{6} - 143333204 \nu^{5} + 634988602 \nu^{4} + 237183129 \nu^{3} - 1020839309 \nu^{2} - 139596031 \nu + 546225598\)\()/2646764\)
\(\beta_{5}\)\(=\)\((\)\(154847 \nu^{11} - 789735 \nu^{10} - 3900430 \nu^{9} + 21598935 \nu^{8} + 34189684 \nu^{7} - 211704075 \nu^{6} - 113127340 \nu^{5} + 886236402 \nu^{4} + 66506323 \nu^{3} - 1492728506 \nu^{2} + 64214944 \nu + 870706107\)\()/2646764\)
\(\beta_{6}\)\(=\)\((\)\(-107852 \nu^{11} + 581451 \nu^{10} + 2636977 \nu^{9} - 15934575 \nu^{8} - 21914010 \nu^{7} + 156316299 \nu^{6} + 63583367 \nu^{5} - 652711197 \nu^{4} - 1932955 \nu^{3} + 1085960100 \nu^{2} - 73257917 \nu - 622275386\)\()/1323382\)
\(\beta_{7}\)\(=\)\((\)\(237037 \nu^{11} - 1126760 \nu^{10} - 6439990 \nu^{9} + 30758583 \nu^{8} + 65032295 \nu^{7} - 298396854 \nu^{6} - 289297782 \nu^{5} + 1220063292 \nu^{4} + 503450773 \nu^{3} - 1967030013 \nu^{2} - 295295825 \nu + 1047731024\)\()/2646764\)
\(\beta_{8}\)\(=\)\((\)\(-128019 \nu^{11} + 656217 \nu^{10} + 3243284 \nu^{9} - 17843661 \nu^{8} - 29046033 \nu^{7} + 172914262 \nu^{6} + 102692323 \nu^{5} - 707942363 \nu^{4} - 93300582 \nu^{3} + 1141470325 \nu^{2} - 3011304 \nu - 625188159\)\()/661691\)
\(\beta_{9}\)\(=\)\((\)\(310088 \nu^{11} - 1606687 \nu^{10} - 7780026 \nu^{9} + 43783498 \nu^{8} + 68395961 \nu^{7} - 426411439 \nu^{6} - 231916382 \nu^{5} + 1762942612 \nu^{4} + 171457996 \nu^{3} - 2893219847 \nu^{2} + 65188905 \nu + 1621505661\)\()/1323382\)
\(\beta_{10}\)\(=\)\((\)\(-324090 \nu^{11} + 1636811 \nu^{10} + 8235372 \nu^{9} - 44361950 \nu^{8} - 74242225 \nu^{7} + 428531541 \nu^{6} + 267410740 \nu^{5} - 1749467342 \nu^{4} - 266922102 \nu^{3} + 2814323071 \nu^{2} + 22537731 \nu - 1533113265\)\()/1323382\)
\(\beta_{11}\)\(=\)\((\)\(1137721 \nu^{11} - 5777705 \nu^{10} - 28952712 \nu^{9} + 157286279 \nu^{8} + 261108042 \nu^{7} - 1528293261 \nu^{6} - 937729774 \nu^{5} + 6293405452 \nu^{4} + 915294055 \nu^{3} - 10269561860 \nu^{2} - 32595192 \nu + 5703167183\)\()/2646764\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{11} - \beta_{10} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + \beta_{1} + 6\)
\(\nu^{3}\)\(=\)\(-2 \beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 10 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(-17 \beta_{11} - 15 \beta_{10} + \beta_{9} - \beta_{8} + 14 \beta_{7} - 16 \beta_{6} + 17 \beta_{5} + 3 \beta_{4} + 11 \beta_{3} + 22 \beta_{1} + 59\)
\(\nu^{5}\)\(=\)\(-48 \beta_{11} - 27 \beta_{10} + 7 \beta_{9} - 16 \beta_{8} + 26 \beta_{7} - 41 \beta_{6} + 48 \beta_{5} + 34 \beta_{4} + 18 \beta_{3} + \beta_{2} + 133 \beta_{1} + 92\)
\(\nu^{6}\)\(=\)\(-268 \beta_{11} - 212 \beta_{10} + 35 \beta_{9} - 23 \beta_{8} + 191 \beta_{7} - 240 \beta_{6} + 264 \beta_{5} + 80 \beta_{4} + 117 \beta_{3} + 4 \beta_{2} + 403 \beta_{1} + 719\)
\(\nu^{7}\)\(=\)\(-902 \beta_{11} - 536 \beta_{10} + 191 \beta_{9} - 218 \beta_{8} + 501 \beta_{7} - 734 \beta_{6} + 886 \beta_{5} + 539 \beta_{4} + 272 \beta_{3} + 29 \beta_{2} + 1973 \beta_{1} + 1728\)
\(\nu^{8}\)\(=\)\(-4271 \beta_{11} - 3111 \beta_{10} + 801 \beta_{9} - 422 \beta_{8} + 2767 \beta_{7} - 3690 \beta_{6} + 4153 \beta_{5} + 1618 \beta_{4} + 1373 \beta_{3} + 123 \beta_{2} + 6942 \beta_{1} + 9879\)
\(\nu^{9}\)\(=\)\(-15745 \beta_{11} - 9639 \beta_{10} + 3856 \beta_{9} - 2964 \beta_{8} + 8818 \beta_{7} - 12646 \beta_{6} + 15255 \beta_{5} + 8628 \beta_{4} + 4044 \beta_{3} + 617 \beta_{2} + 30847 \beta_{1} + 30171\)
\(\nu^{10}\)\(=\)\(-69069 \beta_{11} - 47589 \beta_{10} + 15600 \beta_{9} - 7283 \beta_{8} + 42177 \beta_{7} - 58223 \beta_{6} + 66633 \beta_{5} + 29649 \beta_{4} + 17988 \beta_{3} + 2657 \beta_{2} + 116710 \beta_{1} + 146554\)
\(\nu^{11}\)\(=\)\(-266914 \beta_{11} - 166147 \beta_{10} + 70032 \beta_{9} - 41789 \beta_{8} + 149858 \beta_{7} - 214001 \beta_{6} + 256688 \beta_{5} + 140031 \beta_{4} + 61399 \beta_{3} + 11732 \beta_{2} + 495453 \beta_{1} + 510941\)

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
4.07242
4.06325
3.06946
2.01103
1.37577
1.26686
−1.07790
−1.11808
−1.17036
−2.56401
−2.88446
−3.04399
1.00000 1.00000 1.00000 −3.07242 1.00000 −1.48607 1.00000 1.00000 −3.07242
1.2 1.00000 1.00000 1.00000 −3.06325 1.00000 1.90874 1.00000 1.00000 −3.06325
1.3 1.00000 1.00000 1.00000 −2.06946 1.00000 −4.76338 1.00000 1.00000 −2.06946
1.4 1.00000 1.00000 1.00000 −1.01103 1.00000 −2.46588 1.00000 1.00000 −1.01103
1.5 1.00000 1.00000 1.00000 −0.375767 1.00000 4.48414 1.00000 1.00000 −0.375767
1.6 1.00000 1.00000 1.00000 −0.266858 1.00000 0.661205 1.00000 1.00000 −0.266858
1.7 1.00000 1.00000 1.00000 2.07790 1.00000 3.88310 1.00000 1.00000 2.07790
1.8 1.00000 1.00000 1.00000 2.11808 1.00000 1.85275 1.00000 1.00000 2.11808
1.9 1.00000 1.00000 1.00000 2.17036 1.00000 1.24708 1.00000 1.00000 2.17036
1.10 1.00000 1.00000 1.00000 3.56401 1.00000 −2.46585 1.00000 1.00000 3.56401
1.11 1.00000 1.00000 1.00000 3.88446 1.00000 −1.31412 1.00000 1.00000 3.88446
1.12 1.00000 1.00000 1.00000 4.04399 1.00000 3.45827 1.00000 1.00000 4.04399
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.ba 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.ba 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\(T_{5}^{12} - \cdots\)
\(T_{7}^{12} - \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{12} \)
$3$ \( ( 1 - T )^{12} \)
$5$ \( 1 - 8 T + 51 T^{2} - 241 T^{3} + 1020 T^{4} - 3740 T^{5} + 12725 T^{6} - 39442 T^{7} + 115511 T^{8} - 313980 T^{9} + 811532 T^{10} - 1964637 T^{11} + 4534576 T^{12} - 9823185 T^{13} + 20288300 T^{14} - 39247500 T^{15} + 72194375 T^{16} - 123256250 T^{17} + 198828125 T^{18} - 292187500 T^{19} + 398437500 T^{20} - 470703125 T^{21} + 498046875 T^{22} - 390625000 T^{23} + 244140625 T^{24} \)
$7$ \( 1 - 5 T + 49 T^{2} - 189 T^{3} + 1098 T^{4} - 3451 T^{5} + 15214 T^{6} - 39726 T^{7} + 148435 T^{8} - 331017 T^{9} + 1147558 T^{10} - 2327810 T^{11} + 8085246 T^{12} - 16294670 T^{13} + 56230342 T^{14} - 113538831 T^{15} + 356392435 T^{16} - 667674882 T^{17} + 1789911886 T^{18} - 2842046893 T^{19} + 6329751498 T^{20} - 7626831723 T^{21} + 13841287201 T^{22} - 9886633715 T^{23} + 13841287201 T^{24} \)
$11$ \( 1 - 11 T + 136 T^{2} - 1029 T^{3} + 7610 T^{4} - 44406 T^{5} + 247245 T^{6} - 1183845 T^{7} + 5408719 T^{8} - 22119640 T^{9} + 86817715 T^{10} - 311347963 T^{11} + 1076672844 T^{12} - 3424827593 T^{13} + 10504943515 T^{14} - 29441240840 T^{15} + 79189054879 T^{16} - 190659421095 T^{17} + 438009599445 T^{18} - 865347315426 T^{19} + 1631271084410 T^{20} - 2426328174039 T^{21} + 3527489745736 T^{22} - 3138428376721 T^{23} + 3138428376721 T^{24} \)
$13$ \( 1 - 6 T + 87 T^{2} - 500 T^{3} + 4049 T^{4} - 21192 T^{5} + 126815 T^{6} - 601590 T^{7} + 2945467 T^{8} - 12658772 T^{9} + 53317026 T^{10} - 207298060 T^{11} + 771807910 T^{12} - 2694874780 T^{13} + 9010577394 T^{14} - 27811322084 T^{15} + 84125482987 T^{16} - 223366155870 T^{17} + 612111783335 T^{18} - 1329766572264 T^{19} + 3302893689329 T^{20} - 5302249686500 T^{21} + 11993688790863 T^{22} - 10752962364222 T^{23} + 23298085122481 T^{24} \)
$17$ \( ( 1 + T )^{12} \)
$19$ \( 1 - 3 T + 99 T^{2} - 114 T^{3} + 4257 T^{4} + 2409 T^{5} + 119748 T^{6} + 248425 T^{7} + 2894779 T^{8} + 7766318 T^{9} + 68362009 T^{10} + 159458621 T^{11} + 1437560102 T^{12} + 3029713799 T^{13} + 24678685249 T^{14} + 53269175162 T^{15} + 377250494059 T^{16} + 615124894075 T^{17} + 5633650157988 T^{18} + 2153337019251 T^{19} + 72299027865537 T^{20} - 36786397546806 T^{21} + 606975559522299 T^{22} - 349470776694657 T^{23} + 2213314919066161 T^{24} \)
$23$ \( 1 - 22 T + 366 T^{2} - 4422 T^{3} + 45491 T^{4} - 398585 T^{5} + 3123504 T^{6} - 21985462 T^{7} + 142319212 T^{8} - 849851556 T^{9} + 4750481617 T^{10} - 24858529091 T^{11} + 122908912502 T^{12} - 571746169093 T^{13} + 2513004775393 T^{14} - 10340143881852 T^{15} + 39826750605292 T^{16} - 141505974445466 T^{17} + 462390691435056 T^{18} - 1357112350792495 T^{19} + 3562445031417971 T^{20} - 7964697068989386 T^{21} + 15162103104195534 T^{22} - 20961814674106394 T^{23} + 21914624432020321 T^{24} \)
$29$ \( 1 - 26 T + 561 T^{2} - 8371 T^{3} + 109292 T^{4} - 1182768 T^{5} + 11552999 T^{6} - 99231528 T^{7} + 783090761 T^{8} - 5577822816 T^{9} + 36861839364 T^{10} - 222507792339 T^{11} + 1251583504156 T^{12} - 6452725977831 T^{13} + 31000806905124 T^{14} - 136037520659424 T^{15} + 553865216530841 T^{16} - 2035352656305672 T^{17} + 6871993232689679 T^{18} - 20402601702243312 T^{19} + 54672930965333612 T^{20} - 121439318963999399 T^{21} + 236016757881412761 T^{22} - 317213253908351554 T^{23} + 353814783205469041 T^{24} \)
$31$ \( 1 - T + 154 T^{2} - 5 T^{3} + 12533 T^{4} + 2875 T^{5} + 730314 T^{6} + 116231 T^{7} + 33454419 T^{8} - 392016 T^{9} + 1281066780 T^{10} - 190706776 T^{11} + 42476071278 T^{12} - 5911910056 T^{13} + 1231105175580 T^{14} - 11678548656 T^{15} + 30895858489299 T^{16} + 3327594849881 T^{17} + 648156363285834 T^{18} + 79098765569125 T^{19} + 10689283372248053 T^{20} - 132198110803355 T^{21} + 126222756195043354 T^{22} - 25408476896404831 T^{23} + 787662783788549761 T^{24} \)
$37$ \( 1 - 10 T + 253 T^{2} - 1959 T^{3} + 31588 T^{4} - 214354 T^{5} + 2711359 T^{6} - 16447028 T^{7} + 174724441 T^{8} - 956315644 T^{9} + 8886357896 T^{10} - 44047315133 T^{11} + 364945806300 T^{12} - 1629750659921 T^{13} + 12165423959624 T^{14} - 48440256315532 T^{15} + 327461733069001 T^{16} - 1140502002409796 T^{17} + 6956605390579831 T^{18} - 20349027590967082 T^{19} + 110952200990456548 T^{20} - 254595048258555843 T^{21} + 1216571846221715797 T^{22} - 1779176217794604130 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 - 16 T + 328 T^{2} - 3890 T^{3} + 50483 T^{4} - 488847 T^{5} + 4980234 T^{6} - 41569976 T^{7} + 359619120 T^{8} - 2659016966 T^{9} + 20274652749 T^{10} - 134713999933 T^{11} + 921246452002 T^{12} - 5523273997253 T^{13} + 34081691271069 T^{14} - 183262108313686 T^{15} + 1016197684150320 T^{16} - 4816139495021176 T^{17} + 23656630644572394 T^{18} - 95205042523905207 T^{19} + 403102980341715443 T^{20} - 1273515724792508290 T^{21} + 4402632253729987528 T^{22} - 8805264507459975056 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( 1 - 23 T + 494 T^{2} - 6945 T^{3} + 93452 T^{4} - 1025738 T^{5} + 10856849 T^{6} - 100414985 T^{7} + 894019619 T^{8} - 7179083176 T^{9} + 55580772905 T^{10} - 394278746763 T^{11} + 2696050439680 T^{12} - 16953986110809 T^{13} + 102768849101345 T^{14} - 570787366074232 T^{15} + 3056475167456819 T^{16} - 14761850598718355 T^{17} + 68630084097172601 T^{18} - 278814678519671966 T^{19} + 1092285692342368652 T^{20} - 3490505689901374635 T^{21} + 10676072262762419006 T^{22} - 21373756007838122261 T^{23} + 39959630797262576401 T^{24} \)
$47$ \( 1 - 6 T + 278 T^{2} - 639 T^{3} + 33521 T^{4} + 27941 T^{5} + 2761510 T^{6} + 8534956 T^{7} + 199123987 T^{8} + 742030442 T^{9} + 12685955780 T^{10} + 43652207820 T^{11} + 664921509750 T^{12} + 2051653767540 T^{13} + 28023276318020 T^{14} + 77039826579766 T^{15} + 971661536008147 T^{16} + 1957449543564692 T^{17} + 29766910923186790 T^{18} + 14155556608856683 T^{19} + 798178140188890481 T^{20} - 715124372312668113 T^{21} + 14622558761560753622 T^{22} - 14832955290504073818 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 - 10 T + 329 T^{2} - 2120 T^{3} + 46936 T^{4} - 206497 T^{5} + 4523843 T^{6} - 14954354 T^{7} + 357531419 T^{8} - 939806598 T^{9} + 23562464052 T^{10} - 50164644249 T^{11} + 1326160748168 T^{12} - 2658726145197 T^{13} + 66186961522068 T^{14} - 139915586890446 T^{15} + 2821094868522539 T^{16} - 6253843443526522 T^{17} + 100268089942898747 T^{18} - 242574326242920989 T^{19} + 2922220829147639896 T^{20} - 6995498814620521960 T^{21} + 57537977750253793121 T^{22} - 92690359293721915970 T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( ( 1 - T )^{12} \)
$61$ \( 1 - 15 T + 444 T^{2} - 6249 T^{3} + 108672 T^{4} - 1331380 T^{5} + 17730175 T^{6} - 188692169 T^{7} + 2096224607 T^{8} - 19561240074 T^{9} + 187215880333 T^{10} - 1543732697357 T^{11} + 12938468416416 T^{12} - 94167694538777 T^{13} + 696630290719093 T^{14} - 4440029833236594 T^{15} + 29023992608809487 T^{16} - 159368707965066869 T^{17} + 913465253486043175 T^{18} - 4184184957021638980 T^{19} + 20833217118040520832 T^{20} - 73076718934120547109 T^{21} + \)\(31\!\cdots\!44\)\( T^{22} - \)\(65\!\cdots\!15\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 - 4 T + 376 T^{2} - 791 T^{3} + 72851 T^{4} - 52483 T^{5} + 9803691 T^{6} + 3388737 T^{7} + 1022308687 T^{8} + 1096681215 T^{9} + 87425467265 T^{10} + 123277363470 T^{11} + 6325605398946 T^{12} + 8259583352490 T^{13} + 392452922552585 T^{14} + 329841132267045 T^{15} + 20600666051088127 T^{16} + 4575218904719859 T^{17} + 886826027144785779 T^{18} - 318084327182167009 T^{19} + 29582436377678853491 T^{20} - 21520368707469303077 T^{21} + \)\(68\!\cdots\!24\)\( T^{22} - \)\(48\!\cdots\!32\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 - 10 T + 446 T^{2} - 4173 T^{3} + 94859 T^{4} - 759207 T^{5} + 12478030 T^{6} - 79668674 T^{7} + 1128037419 T^{8} - 5439162632 T^{9} + 79041506420 T^{10} - 296710912754 T^{11} + 5315201586386 T^{12} - 21066474805534 T^{13} + 398448233863220 T^{14} - 1946736136781752 T^{15} + 28665327047691339 T^{16} - 143740559986050574 T^{17} + 1598439185774755630 T^{18} - 6905078890091555937 T^{19} + 61255534220441642699 T^{20} - \)\(19\!\cdots\!63\)\( T^{21} + \)\(14\!\cdots\!46\)\( T^{22} - \)\(23\!\cdots\!10\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 - 24 T + 738 T^{2} - 13420 T^{3} + 252035 T^{4} - 3720767 T^{5} + 53632468 T^{6} - 667254760 T^{7} + 7982391378 T^{8} - 85498289908 T^{9} + 878056045481 T^{10} - 8191908374589 T^{11} + 73293039390982 T^{12} - 598009311344997 T^{13} + 4679160666368249 T^{14} - 33260288245140436 T^{15} + 226685874108766098 T^{16} - 1383266888250032680 T^{17} + 8116428048749551252 T^{18} - 41104795845704987399 T^{19} + \)\(20\!\cdots\!35\)\( T^{20} - \)\(79\!\cdots\!60\)\( T^{21} + \)\(31\!\cdots\!62\)\( T^{22} - \)\(75\!\cdots\!48\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 - 23 T + 838 T^{2} - 14682 T^{3} + 308243 T^{4} - 4383261 T^{5} + 68301864 T^{6} - 821272379 T^{7} + 10455345593 T^{8} - 109310267326 T^{9} + 1193837241830 T^{10} - 11038263616301 T^{11} + 106132418360590 T^{12} - 872022825687779 T^{13} + 7450738226261030 T^{14} - 53894224892143714 T^{15} + 407236557730343033 T^{16} - 2527101429123903221 T^{17} + 16603326327101391144 T^{18} - 84175745306580284499 T^{19} + \)\(46\!\cdots\!23\)\( T^{20} - \)\(17\!\cdots\!58\)\( T^{21} + \)\(79\!\cdots\!38\)\( T^{22} - \)\(17\!\cdots\!17\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 + 394 T^{2} + 838 T^{3} + 85658 T^{4} + 242147 T^{5} + 13279713 T^{6} + 41583946 T^{7} + 1591614515 T^{8} + 5134778456 T^{9} + 159560820573 T^{10} + 506534429703 T^{11} + 13989731333412 T^{12} + 42042357665349 T^{13} + 1099214492927397 T^{14} + 2935999569020872 T^{15} + 75535352561129315 T^{16} + 163800853390317278 T^{17} + 4341674326453163097 T^{18} + 6570913338985209169 T^{19} + \)\(19\!\cdots\!78\)\( T^{20} + \)\(15\!\cdots\!14\)\( T^{21} + \)\(61\!\cdots\!06\)\( T^{22} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 - 13 T + 640 T^{2} - 5864 T^{3} + 186798 T^{4} - 1284097 T^{5} + 35315134 T^{6} - 184266914 T^{7} + 4949887363 T^{8} - 19681831200 T^{9} + 557257097299 T^{10} - 1799959084980 T^{11} + 53294527225442 T^{12} - 160196358563220 T^{13} + 4414033467705379 T^{14} - 13875080859232800 T^{15} + 310567025852200483 T^{16} - 1028957402259770386 T^{17} + 17550960885780703774 T^{18} - 56797324445344102313 T^{19} + \)\(73\!\cdots\!38\)\( T^{20} - \)\(20\!\cdots\!76\)\( T^{21} + \)\(19\!\cdots\!40\)\( T^{22} - \)\(36\!\cdots\!57\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 - 13 T + 624 T^{2} - 6199 T^{3} + 182188 T^{4} - 1381153 T^{5} + 33152915 T^{6} - 182225652 T^{7} + 4296773011 T^{8} - 15422891933 T^{9} + 445946914693 T^{10} - 1004584770090 T^{11} + 43003903870192 T^{12} - 97444722698730 T^{13} + 4195914520346437 T^{14} - 14076057049166909 T^{15} + 380390225284035091 T^{16} - 1564833677277672564 T^{17} + 27615450076790718035 T^{18} - \)\(11\!\cdots\!89\)\( T^{19} + \)\(14\!\cdots\!68\)\( T^{20} - \)\(47\!\cdots\!83\)\( T^{21} + \)\(46\!\cdots\!76\)\( T^{22} - \)\(92\!\cdots\!89\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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