Properties

Label 8001.2.a.w
Level 8001
Weight 2
Character orbit 8001.a
Self dual yes
Analytic conductor 63.888
Analytic rank 1
Dimension 20
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} - 56 x^{11} + 39579 x^{10} - 17664 x^{9} - 52271 x^{8} + 35701 x^{7} + 32493 x^{6} - 25504 x^{5} - 8607 x^{4} + 6812 x^{3} + 609 x^{2} - 425 x + 31\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 889)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} - 56 x^{11} + 39579 x^{10} - 17664 x^{9} - 52271 x^{8} + 35701 x^{7} + 32493 x^{6} - 25504 x^{5} - 8607 x^{4} + 6812 x^{3} + 609 x^{2} - 425 x + 31\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 2 \)
\(\beta_{4}\)\(=\)\((\)\(-80574 \nu^{19} + 88265 \nu^{18} + 4448116 \nu^{17} - 11956345 \nu^{16} - 63248616 \nu^{15} + 232480171 \nu^{14} + 345773311 \nu^{13} - 1910206802 \nu^{12} - 405883602 \nu^{11} + 7970416630 \nu^{10} - 3293131310 \nu^{9} - 17070805117 \nu^{8} + 13762066276 \nu^{7} + 16264479010 \nu^{6} - 19092142347 \nu^{5} - 3272911391 \nu^{4} + 8444727149 \nu^{3} - 1113016448 \nu^{2} - 888775901 \nu + 123446515\)\()/4334246\)
\(\beta_{5}\)\(=\)\((\)\(112041 \nu^{19} - 1197675 \nu^{18} + 2264887 \nu^{17} + 17783216 \nu^{16} - 73262421 \nu^{15} - 57671117 \nu^{14} + 647447518 \nu^{13} - 364390775 \nu^{12} - 2623374358 \nu^{11} + 3210092650 \nu^{10} + 5054834351 \nu^{9} - 9103752501 \nu^{8} - 3527560932 \nu^{7} + 10907649962 \nu^{6} - 969053095 \nu^{5} - 4410403111 \nu^{4} + 1105027913 \nu^{3} + 377557653 \nu^{2} - 99590538 \nu + 2196193\)\()/2167123\)
\(\beta_{6}\)\(=\)\((\)\(16028 \nu^{19} - 143864 \nu^{18} + 122280 \nu^{17} + 2470224 \nu^{16} - 6859382 \nu^{15} - 12972651 \nu^{14} + 67984425 \nu^{13} - 1910609 \nu^{12} - 295889912 \nu^{11} + 245045373 \nu^{10} + 599709512 \nu^{9} - 851828022 \nu^{8} - 418277501 \nu^{7} + 1092553461 \nu^{6} - 169615722 \nu^{5} - 415699503 \nu^{4} + 165471717 \nu^{3} + 18340672 \nu^{2} - 13862243 \nu + 1383690\)\()/309589\)
\(\beta_{7}\)\(=\)\((\)\(-248879 \nu^{19} + 3978132 \nu^{18} - 13988931 \nu^{17} - 48139158 \nu^{16} + 338768723 \nu^{15} - 38956001 \nu^{14} - 2759046818 \nu^{13} + 3143549108 \nu^{12} + 10402831226 \nu^{11} - 18846899174 \nu^{10} - 17448853443 \nu^{9} + 48415589078 \nu^{8} + 5425063462 \nu^{7} - 55791961163 \nu^{6} + 14568997447 \nu^{5} + 22437701529 \nu^{4} - 8633213124 \nu^{3} - 1967839611 \nu^{2} + 676572227 \nu - 44644594\)\()/4334246\)
\(\beta_{8}\)\(=\)\((\)\(130444 \nu^{19} - 1062848 \nu^{18} + 203009 \nu^{17} + 19282840 \nu^{16} - 38105845 \nu^{15} - 122444455 \nu^{14} + 401582731 \nu^{13} + 251551762 \nu^{12} - 1805118745 \nu^{11} + 509262882 \nu^{10} + 3897308205 \nu^{9} - 3070652687 \nu^{8} - 3512303592 \nu^{7} + 4434806592 \nu^{6} + 364613036 \nu^{5} - 1821783462 \nu^{4} + 433924998 \nu^{3} + 117630811 \nu^{2} - 52984391 \nu + 4552939\)\()/2167123\)
\(\beta_{9}\)\(=\)\((\)\(-146869 \nu^{19} + 1305396 \nu^{18} - 1062848 \nu^{17} - 22121079 \nu^{16} + 59524946 \nu^{15} + 117722164 \nu^{14} - 581410080 \nu^{13} - 35646282 \nu^{12} + 2524202668 \nu^{11} - 1796894081 \nu^{10} - 5303665269 \nu^{9} + 6491602221 \nu^{8} + 4606336812 \nu^{7} - 8755673761 \nu^{6} - 337407825 \nu^{5} + 4110360012 \nu^{4} - 557681979 \nu^{3} - 566546630 \nu^{2} + 28187590 \nu + 9434934\)\()/2167123\)
\(\beta_{10}\)\(=\)\((\)\(452378 \nu^{19} - 3636757 \nu^{18} + 479452 \nu^{17} + 64942855 \nu^{16} - 120306036 \nu^{15} - 413198975 \nu^{14} + 1229305933 \nu^{13} + 962389260 \nu^{12} - 5289287134 \nu^{11} + 625710296 \nu^{10} + 10741270610 \nu^{9} - 6399002549 \nu^{8} - 8503960458 \nu^{7} + 8810093974 \nu^{6} - 662651141 \nu^{5} - 2565501807 \nu^{4} + 1987510173 \nu^{3} - 333016752 \nu^{2} - 247405745 \nu + 51386023\)\()/4334246\)
\(\beta_{11}\)\(=\)\((\)\(239299 \nu^{19} - 839918 \nu^{18} - 6727183 \nu^{17} + 25057160 \nu^{16} + 75577807 \nu^{15} - 306448870 \nu^{14} - 430391229 \nu^{13} + 1985694746 \nu^{12} + 1274010673 \nu^{11} - 7326712947 \nu^{10} - 1640919567 \nu^{9} + 15286073938 \nu^{8} - 166042331 \nu^{7} - 16629421700 \nu^{6} + 1992882190 \nu^{5} + 7760740247 \nu^{4} - 741296302 \nu^{3} - 1279308585 \nu^{2} - 50188832 \nu + 33626793\)\()/2167123\)
\(\beta_{12}\)\(=\)\((\)\(513169 \nu^{19} - 3359961 \nu^{18} - 4253023 \nu^{17} + 67902593 \nu^{16} - 49716311 \nu^{15} - 524918352 \nu^{14} + 801626921 \nu^{13} + 1873963420 \nu^{12} - 4115132892 \nu^{11} - 2626187800 \nu^{10} + 9547027119 \nu^{9} - 992559901 \nu^{8} - 8895240438 \nu^{7} + 5258752701 \nu^{6} + 511422678 \nu^{5} - 2384056922 \nu^{4} + 1615698465 \nu^{3} + 53574147 \nu^{2} - 261345936 \nu + 14104789\)\()/4334246\)
\(\beta_{13}\)\(=\)\((\)\(600491 \nu^{19} - 5014091 \nu^{18} + 1433657 \nu^{17} + 92851141 \nu^{16} - 196388497 \nu^{15} - 595632982 \nu^{14} + 2117597099 \nu^{13} + 1139674102 \nu^{12} - 9839915902 \nu^{11} + 3476162040 \nu^{10} + 22105769119 \nu^{9} - 18817836025 \nu^{8} - 21121831192 \nu^{7} + 27608913433 \nu^{6} + 3331895996 \nu^{5} - 12057424380 \nu^{4} + 2030109247 \nu^{3} + 1034875291 \nu^{2} - 246014754 \nu + 34267467\)\()/4334246\)
\(\beta_{14}\)\(=\)\((\)\(-410993 \nu^{19} + 2213625 \nu^{18} + 6917810 \nu^{17} - 52563575 \nu^{16} - 29025299 \nu^{15} + 513862951 \nu^{14} - 139289961 \nu^{13} - 2668384327 \nu^{12} + 1733798795 \nu^{11} + 7888472785 \nu^{10} - 6649332298 \nu^{9} - 13110138889 \nu^{8} + 12281615752 \nu^{7} + 11099392631 \nu^{6} - 10831511067 \nu^{5} - 3600113430 \nu^{4} + 3719573150 \nu^{3} + 165170286 \nu^{2} - 303545763 \nu + 31062379\)\()/2167123\)
\(\beta_{15}\)\(=\)\((\)\(-413388 \nu^{19} + 3462562 \nu^{18} - 1511903 \nu^{17} - 60346403 \nu^{16} + 134466805 \nu^{15} + 354990510 \nu^{14} - 1345384350 \nu^{13} - 525261149 \nu^{12} + 5839741304 \nu^{11} - 2589729038 \nu^{10} - 12179494628 \nu^{9} + 11432686680 \nu^{8} + 10435235153 \nu^{7} - 15517251270 \nu^{6} - 583856352 \nu^{5} + 6479767911 \nu^{4} - 1490693552 \nu^{3} - 545381160 \nu^{2} + 147237033 \nu - 3502341\)\()/2167123\)
\(\beta_{16}\)\(=\)\((\)\(-1008515 \nu^{19} + 6530918 \nu^{18} + 8850087 \nu^{17} - 132965184 \nu^{16} + 86306181 \nu^{15} + 1050649075 \nu^{14} - 1488281912 \nu^{13} - 3982849598 \nu^{12} + 7914662926 \nu^{11} + 6906341856 \nu^{10} - 19645694201 \nu^{9} - 2787853840 \nu^{8} + 22331737484 \nu^{7} - 4642008039 \nu^{6} - 8848072497 \nu^{5} + 2556273551 \nu^{4} + 817071694 \nu^{3} - 21377165 \nu^{2} - 16996363 \nu - 10476190\)\()/4334246\)
\(\beta_{17}\)\(=\)\((\)\(-579954 \nu^{19} + 3614942 \nu^{18} + 6341910 \nu^{17} - 77052369 \nu^{16} + 25536710 \nu^{15} + 659355662 \nu^{14} - 695262384 \nu^{13} - 2896884557 \nu^{12} + 4231996085 \nu^{11} + 6882461020 \nu^{10} - 12292561414 \nu^{9} - 8381823921 \nu^{8} + 18384933518 \nu^{7} + 4109230177 \nu^{6} - 13340039557 \nu^{5} + 136389714 \nu^{4} + 4022677293 \nu^{3} - 483654127 \nu^{2} - 315155203 \nu + 42358963\)\()/2167123\)
\(\beta_{18}\)\(=\)\((\)\(170912 \nu^{19} - 1102231 \nu^{18} - 1613038 \nu^{17} + 22982443 \nu^{16} - 12939568 \nu^{15} - 189215421 \nu^{14} + 248577527 \nu^{13} + 774985956 \nu^{12} - 1408256152 \nu^{11} - 1603313970 \nu^{10} + 3845197606 \nu^{9} + 1393415281 \nu^{8} - 5275677336 \nu^{7} + 3660472 \nu^{6} + 3356197997 \nu^{5} - 455638987 \nu^{4} - 921371945 \nu^{3} + 151871254 \nu^{2} + 82480027 \nu - 10261733\)\()/619178\)
\(\beta_{19}\)\(=\)\((\)\(1492257 \nu^{19} - 9628117 \nu^{18} - 13332635 \nu^{17} + 194372481 \nu^{16} - 107507193 \nu^{15} - 1560069588 \nu^{14} + 1932311183 \nu^{13} + 6369622840 \nu^{12} - 10456617166 \nu^{11} - 13947586220 \nu^{10} + 27726528907 \nu^{9} + 15505936931 \nu^{8} - 38183605866 \nu^{7} - 6618531811 \nu^{6} + 25857674064 \nu^{5} - 907539212 \nu^{4} - 7654017971 \nu^{3} + 1064221961 \nu^{2} + 698367060 \nu - 78687283\)\()/4334246\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{18} + \beta_{15} + \beta_{14} + \beta_{10} + \beta_{3} + 8 \beta_{2} + \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(\beta_{19} + 2 \beta_{18} + \beta_{17} + 2 \beta_{16} + \beta_{15} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + 9 \beta_{3} + 11 \beta_{2} + 27 \beta_{1} + 9\)
\(\nu^{6}\)\(=\)\(12 \beta_{18} + \beta_{17} + \beta_{16} + 11 \beta_{15} + 9 \beta_{14} - \beta_{13} + \beta_{12} - 2 \beta_{11} + 12 \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{4} + 12 \beta_{3} + 58 \beta_{2} + 13 \beta_{1} + 74\)
\(\nu^{7}\)\(=\)\(11 \beta_{19} + 25 \beta_{18} + 12 \beta_{17} + 23 \beta_{16} + 14 \beta_{15} - \beta_{14} - \beta_{13} - 14 \beta_{11} + 15 \beta_{10} - 15 \beta_{9} + 14 \beta_{8} + 13 \beta_{7} + 8 \beta_{6} - 11 \beta_{5} + 26 \beta_{4} + 70 \beta_{3} + 95 \beta_{2} + 159 \beta_{1} + 69\)
\(\nu^{8}\)\(=\)\(3 \beta_{19} + 107 \beta_{18} + 16 \beta_{17} + 17 \beta_{16} + 95 \beta_{15} + 61 \beta_{14} - 12 \beta_{13} + 12 \beta_{12} - 31 \beta_{11} + 108 \beta_{10} - 18 \beta_{9} + 16 \beta_{8} + 16 \beta_{7} - 15 \beta_{6} - 2 \beta_{5} + 34 \beta_{4} + 110 \beta_{3} + 415 \beta_{2} + 122 \beta_{1} + 427\)
\(\nu^{9}\)\(=\)\(94 \beta_{19} + 233 \beta_{18} + 112 \beta_{17} + 200 \beta_{16} + 143 \beta_{15} - 18 \beta_{14} - 14 \beta_{13} + \beta_{12} - 142 \beta_{11} + 159 \beta_{10} - 157 \beta_{9} + 137 \beta_{8} + 125 \beta_{7} + 42 \beta_{6} - 90 \beta_{5} + 249 \beta_{4} + 522 \beta_{3} + 760 \beta_{2} + 998 \beta_{1} + 508\)
\(\nu^{10}\)\(=\)\(55 \beta_{19} + 861 \beta_{18} + 181 \beta_{17} + 195 \beta_{16} + 758 \beta_{15} + 368 \beta_{14} - 104 \beta_{13} + 107 \beta_{12} - 335 \beta_{11} + 881 \beta_{10} - 223 \beta_{9} + 178 \beta_{8} + 183 \beta_{7} - 159 \beta_{6} - 29 \beta_{5} + 388 \beta_{4} + 915 \beta_{3} + 2969 \beta_{2} + 1029 \beta_{1} + 2627\)
\(\nu^{11}\)\(=\)\(749 \beta_{19} + 1951 \beta_{18} + 964 \beta_{17} + 1587 \beta_{16} + 1289 \beta_{15} - 215 \beta_{14} - 130 \beta_{13} + 28 \beta_{12} - 1272 \beta_{11} + 1465 \beta_{10} - 1424 \beta_{9} + 1164 \beta_{8} + 1082 \beta_{7} + 139 \beta_{6} - 655 \beta_{5} + 2129 \beta_{4} + 3829 \beta_{3} + 5886 \beta_{2} + 6543 \beta_{1} + 3700\)
\(\nu^{12}\)\(=\)\(682 \beta_{19} + 6626 \beta_{18} + 1783 \beta_{17} + 1903 \beta_{16} + 5847 \beta_{15} + 2054 \beta_{14} - 784 \beta_{13} + 866 \beta_{12} - 3132 \beta_{11} + 6881 \beta_{10} - 2321 \beta_{9} + 1695 \beta_{8} + 1823 \beta_{7} - 1465 \beta_{6} - 274 \beta_{5} + 3757 \beta_{4} + 7272 \beta_{3} + 21318 \beta_{2} + 8274 \beta_{1} + 16903\)
\(\nu^{13}\)\(=\)\(5849 \beta_{19} + 15528 \beta_{18} + 8007 \beta_{17} + 12128 \beta_{16} + 10902 \beta_{15} - 2161 \beta_{14} - 997 \beta_{13} + 443 \beta_{12} - 10725 \beta_{11} + 12567 \beta_{10} - 12026 \beta_{9} + 9231 \beta_{8} + 8945 \beta_{7} - 237 \beta_{6} - 4470 \beta_{5} + 17254 \beta_{4} + 27907 \beta_{3} + 44880 \beta_{2} + 44173 \beta_{1} + 26897\)
\(\nu^{14}\)\(=\)\(7165 \beta_{19} + 49927 \beta_{18} + 16332 \beta_{17} + 17066 \beta_{16} + 44367 \beta_{15} + 10564 \beta_{14} - 5421 \beta_{13} + 6771 \beta_{12} - 27216 \beta_{11} + 52601 \beta_{10} - 21850 \beta_{9} + 14829 \beta_{8} + 16853 \beta_{7} - 12569 \beta_{6} - 2112 \beta_{5} + 33413 \beta_{4} + 56412 \beta_{3} + 153805 \beta_{2} + 64837 \beta_{1} + 112151\)
\(\nu^{15}\)\(=\)\(45504 \beta_{19} + 120296 \beta_{18} + 65244 \beta_{17} + 91153 \beta_{16} + 88859 \beta_{15} - 19819 \beta_{14} - 6721 \beta_{13} + 5434 \beta_{12} - 87508 \beta_{11} + 103472 \beta_{10} - 97615 \beta_{9} + 70564 \beta_{8} + 72322 \beta_{7} - 10263 \beta_{6} - 29212 \beta_{5} + 135979 \beta_{4} + 203042 \beta_{3} + 339458 \beta_{2} + 304274 \beta_{1} + 195625\)
\(\nu^{16}\)\(=\)\(68741 \beta_{19} + 372420 \beta_{18} + 143070 \beta_{17} + 145540 \beta_{16} + 333767 \beta_{15} + 47771 \beta_{14} - 34977 \beta_{13} + 52411 \beta_{12} - 226950 \beta_{11} + 397689 \beta_{10} - 193095 \beta_{9} + 123205 \beta_{8} + 148769 \beta_{7} - 103573 \beta_{6} - 14172 \beta_{5} + 282868 \beta_{4} + 431591 \beta_{3} + 1115180 \beta_{2} + 500189 \beta_{1} + 760028\)
\(\nu^{17}\)\(=\)\(354430 \beta_{19} + 917916 \beta_{18} + 525239 \beta_{17} + 680149 \beta_{16} + 707465 \beta_{15} - 172044 \beta_{14} - 40089 \beta_{13} + 57847 \beta_{12} - 700514 \beta_{11} + 830610 \beta_{10} - 773876 \beta_{9} + 528454 \beta_{8} + 578015 \beta_{7} - 131379 \beta_{6} - 184387 \beta_{5} + 1055444 \beta_{4} + 1478128 \beta_{3} + 2556417 \beta_{2} + 2126070 \beta_{1} + 1424146\)
\(\nu^{18}\)\(=\)\(623089 \beta_{19} + 2765411 \beta_{18} + 1215932 \beta_{17} + 1201403 \beta_{16} + 2498983 \beta_{15} + 158548 \beta_{14} - 210305 \beta_{13} + 405455 \beta_{12} - 1846013 \beta_{11} + 2990011 \beta_{10} - 1636228 \beta_{9} + 989690 \beta_{8} + 1273224 \beta_{7} - 832969 \beta_{6} - 83371 \beta_{5} + 2322373 \beta_{4} + 3274565 \beta_{3} + 8123538 \beta_{2} + 3819421 \beta_{1} + 5228188\)
\(\nu^{19}\)\(=\)\(2765954 \beta_{19} + 6943503 \beta_{18} + 4191957 \beta_{17} + 5061055 \beta_{16} + 5543561 \beta_{15} - 1440901 \beta_{14} - 201460 \beta_{13} + 563630 \beta_{12} - 5542441 \beta_{11} + 6557790 \beta_{10} - 6045266 \beta_{9} + 3911178 \beta_{8} + 4589665 \beta_{7} - 1330431 \beta_{6} - 1126086 \beta_{5} + 8123077 \beta_{4} + 10779432 \beta_{3} + 19206088 \beta_{2} + 15014219 \beta_{1} + 10376694\)

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
2.75954
2.70334
2.58654
2.24329
2.11976
2.02020
1.78791
1.72132
0.831296
0.697877
0.170762
0.102309
−0.343534
−0.723268
−0.731838
−1.60447
−1.86440
−1.86718
−2.12365
−2.48579
−2.75954 0 5.61505 −0.308409 0 1.00000 −9.97588 0 0.851066
1.2 −2.70334 0 5.30805 3.37987 0 1.00000 −8.94278 0 −9.13693
1.3 −2.58654 0 4.69021 −1.18795 0 1.00000 −6.95835 0 3.07267
1.4 −2.24329 0 3.03233 1.67114 0 1.00000 −2.31581 0 −3.74884
1.5 −2.11976 0 2.49337 −3.20666 0 1.00000 −1.04583 0 6.79735
1.6 −2.02020 0 2.08119 3.61502 0 1.00000 −0.164016 0 −7.30304
1.7 −1.78791 0 1.19664 −2.61034 0 1.00000 1.43634 0 4.66707
1.8 −1.72132 0 0.962958 −2.01516 0 1.00000 1.78509 0 3.46875
1.9 −0.831296 0 −1.30895 1.71517 0 1.00000 2.75071 0 −1.42582
1.10 −0.697877 0 −1.51297 0.682552 0 1.00000 2.45162 0 −0.476337
1.11 −0.170762 0 −1.97084 −2.09248 0 1.00000 0.678071 0 0.357317
1.12 −0.102309 0 −1.98953 −2.62692 0 1.00000 0.408164 0 0.268756
1.13 0.343534 0 −1.88198 4.21457 0 1.00000 −1.33360 0 1.44785
1.14 0.723268 0 −1.47688 1.71672 0 1.00000 −2.51472 0 1.24165
1.15 0.731838 0 −1.46441 −3.93264 0 1.00000 −2.53539 0 −2.87806
1.16 1.60447 0 0.574339 −0.248535 0 1.00000 −2.28744 0 −0.398768
1.17 1.86440 0 1.47599 1.88221 0 1.00000 −0.976959 0 3.50920
1.18 1.86718 0 1.48637 −2.52504 0 1.00000 −0.959047 0 −4.71472
1.19 2.12365 0 2.50989 2.22935 0 1.00000 1.08284 0 4.73436
1.20 2.48579 0 4.17918 −3.35246 0 1.00000 5.41698 0 −8.33354
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
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 8001.2.a.w 20
3.b odd 2 1 889.2.a.d 20
21.c even 2 1 6223.2.a.l 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
889.2.a.d 20 3.b odd 2 1
6223.2.a.l 20 21.c even 2 1
8001.2.a.w 20 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(127\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{20} + \cdots\)
\(T_{5}^{20} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 8 T + 40 T^{2} + 152 T^{3} + 486 T^{4} + 1365 T^{5} + 3477 T^{6} + 8181 T^{7} + 18026 T^{8} + 37522 T^{9} + 74311 T^{10} + 140720 T^{11} + 255933 T^{12} + 448587 T^{13} + 760073 T^{14} + 1247786 T^{15} + 1988749 T^{16} + 3081632 T^{17} + 4648149 T^{18} + 6829961 T^{19} + 9782715 T^{20} + 13659922 T^{21} + 18592596 T^{22} + 24653056 T^{23} + 31819984 T^{24} + 39929152 T^{25} + 48644672 T^{26} + 57419136 T^{27} + 65518848 T^{28} + 72048640 T^{29} + 76094464 T^{30} + 76845056 T^{31} + 73834496 T^{32} + 67018752 T^{33} + 56967168 T^{34} + 44728320 T^{35} + 31850496 T^{36} + 19922944 T^{37} + 10485760 T^{38} + 4194304 T^{39} + 1048576 T^{40} \)
$3$ 1
$5$ \( 1 + 3 T + 41 T^{2} + 102 T^{3} + 852 T^{4} + 1789 T^{5} + 11896 T^{6} + 21175 T^{7} + 126095 T^{8} + 190114 T^{9} + 1091285 T^{10} + 1389004 T^{11} + 8094983 T^{12} + 8669778 T^{13} + 53198371 T^{14} + 48017284 T^{15} + 316996324 T^{16} + 245397320 T^{17} + 1744322639 T^{18} + 1214931511 T^{19} + 8987882722 T^{20} + 6074657555 T^{21} + 43608065975 T^{22} + 30674665000 T^{23} + 198122702500 T^{24} + 150054012500 T^{25} + 831224546875 T^{26} + 677326406250 T^{27} + 3162102734375 T^{28} + 2712898437500 T^{29} + 10657080078125 T^{30} + 9282910156250 T^{31} + 30784912109375 T^{32} + 25848388671875 T^{33} + 72607421875000 T^{34} + 54595947265625 T^{35} + 130004882812500 T^{36} + 77819824218750 T^{37} + 156402587890625 T^{38} + 57220458984375 T^{39} + 95367431640625 T^{40} \)
$7$ \( ( 1 - T )^{20} \)
$11$ \( 1 + 26 T + 451 T^{2} + 5745 T^{3} + 60390 T^{4} + 539974 T^{5} + 4256816 T^{6} + 30041049 T^{7} + 193010981 T^{8} + 1139446445 T^{9} + 6242249652 T^{10} + 31935452615 T^{11} + 153602642308 T^{12} + 697892207262 T^{13} + 3010361369953 T^{14} + 12374514757616 T^{15} + 48663255845401 T^{16} + 183604003037714 T^{17} + 666451122560284 T^{18} + 2331322386267096 T^{19} + 7870077151801638 T^{20} + 25644546248938056 T^{21} + 80640585829794364 T^{22} + 244376928043197334 T^{23} + 712478728832516041 T^{24} + 1992927976228814416 T^{25} + 5333038798915306633 T^{26} + 13599944782482035802 T^{27} + 32926090523786137348 T^{28} + 75302126754579161965 T^{29} + \)\(16\!\cdots\!52\)\( T^{30} + \)\(32\!\cdots\!95\)\( T^{31} + \)\(60\!\cdots\!01\)\( T^{32} + \)\(10\!\cdots\!19\)\( T^{33} + \)\(16\!\cdots\!56\)\( T^{34} + \)\(22\!\cdots\!74\)\( T^{35} + \)\(27\!\cdots\!90\)\( T^{36} + \)\(29\!\cdots\!95\)\( T^{37} + \)\(25\!\cdots\!31\)\( T^{38} + \)\(15\!\cdots\!66\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \)
$13$ \( 1 + 4 T + 147 T^{2} + 612 T^{3} + 10965 T^{4} + 46930 T^{5} + 551382 T^{6} + 2393971 T^{7} + 20938477 T^{8} + 90968199 T^{9} + 637211719 T^{10} + 2733708791 T^{11} + 16093496584 T^{12} + 67342306759 T^{13} + 344740815492 T^{14} + 1391161764554 T^{15} + 6350216567005 T^{16} + 24452183743506 T^{17} + 101441444392152 T^{18} + 368871001513274 T^{19} + 1411805054266120 T^{20} + 4795323019672562 T^{21} + 17143604102273688 T^{22} + 53721447684482682 T^{23} + 181368535370229805 T^{24} + 516528625046548322 T^{25} + 1663998070884125028 T^{26} + 4225629880486326403 T^{27} + 13127959571877357064 T^{28} + 28989613160124088043 T^{29} + 87845046569848778431 T^{30} + \)\(16\!\cdots\!63\)\( T^{31} + \)\(48\!\cdots\!37\)\( T^{32} + \)\(72\!\cdots\!63\)\( T^{33} + \)\(21\!\cdots\!98\)\( T^{34} + \)\(24\!\cdots\!10\)\( T^{35} + \)\(72\!\cdots\!65\)\( T^{36} + \)\(52\!\cdots\!96\)\( T^{37} + \)\(16\!\cdots\!63\)\( T^{38} + \)\(58\!\cdots\!08\)\( T^{39} + \)\(19\!\cdots\!01\)\( T^{40} \)
$17$ \( 1 + 4 T + 183 T^{2} + 657 T^{3} + 16646 T^{4} + 53973 T^{5} + 1003885 T^{6} + 2955994 T^{7} + 45271848 T^{8} + 121833860 T^{9} + 1633898241 T^{10} + 4051003521 T^{11} + 49301704657 T^{12} + 113655742290 T^{13} + 1280577022539 T^{14} + 2768680397715 T^{15} + 29171755167306 T^{16} + 59503142298315 T^{17} + 589114026053452 T^{18} + 1136153545521719 T^{19} + 10601403457616084 T^{20} + 19314610273869223 T^{21} + 170253953529447628 T^{22} + 292338938111621595 T^{23} + 2436454163328564426 T^{24} + 3931130243458426755 T^{25} + 30910016241349667691 T^{26} + 46637346470108581170 T^{27} + \)\(34\!\cdots\!37\)\( T^{28} + \)\(48\!\cdots\!37\)\( T^{29} + \)\(32\!\cdots\!09\)\( T^{30} + \)\(41\!\cdots\!80\)\( T^{31} + \)\(26\!\cdots\!28\)\( T^{32} + \)\(29\!\cdots\!78\)\( T^{33} + \)\(16\!\cdots\!65\)\( T^{34} + \)\(15\!\cdots\!89\)\( T^{35} + \)\(81\!\cdots\!26\)\( T^{36} + \)\(54\!\cdots\!89\)\( T^{37} + \)\(25\!\cdots\!47\)\( T^{38} + \)\(95\!\cdots\!12\)\( T^{39} + \)\(40\!\cdots\!01\)\( T^{40} \)
$19$ \( 1 - T + 168 T^{2} - 22 T^{3} + 14607 T^{4} + 7772 T^{5} + 881922 T^{6} + 903038 T^{7} + 41475084 T^{8} + 56620098 T^{9} + 1610059343 T^{10} + 2542393063 T^{11} + 53307795046 T^{12} + 90285835997 T^{13} + 1535187711513 T^{14} + 2663902287560 T^{15} + 38939941596646 T^{16} + 67089130901567 T^{17} + 877089383861008 T^{18} + 1463016008143316 T^{19} + 17627732755396580 T^{20} + 27797304154723004 T^{21} + 316629267573823888 T^{22} + 460164348853848053 T^{23} + 5074692128816503366 T^{24} + 6596085790325028440 T^{25} + 72224258388502927953 T^{26} + 80703957229707188783 T^{27} + \)\(90\!\cdots\!86\)\( T^{28} + \)\(82\!\cdots\!77\)\( T^{29} + \)\(98\!\cdots\!43\)\( T^{30} + \)\(65\!\cdots\!62\)\( T^{31} + \)\(91\!\cdots\!24\)\( T^{32} + \)\(37\!\cdots\!42\)\( T^{33} + \)\(70\!\cdots\!62\)\( T^{34} + \)\(11\!\cdots\!28\)\( T^{35} + \)\(42\!\cdots\!67\)\( T^{36} - \)\(12\!\cdots\!58\)\( T^{37} + \)\(17\!\cdots\!88\)\( T^{38} - \)\(19\!\cdots\!79\)\( T^{39} + \)\(37\!\cdots\!01\)\( T^{40} \)
$23$ \( 1 + 31 T + 692 T^{2} + 11153 T^{3} + 151248 T^{4} + 1734815 T^{5} + 17720485 T^{6} + 161884250 T^{7} + 1358181700 T^{8} + 10498603348 T^{9} + 76056972053 T^{10} + 517541354301 T^{11} + 3348917400973 T^{12} + 20629077972721 T^{13} + 122097347658439 T^{14} + 694208163015300 T^{15} + 3818664081897215 T^{16} + 20293567559402971 T^{17} + 104766417002832779 T^{18} + 524141981778219074 T^{19} + 2552226400117829118 T^{20} + 12055265580899038702 T^{21} + 55421434594498540091 T^{22} + \)\(24\!\cdots\!57\)\( T^{23} + \)\(10\!\cdots\!15\)\( T^{24} + \)\(44\!\cdots\!00\)\( T^{25} + \)\(18\!\cdots\!71\)\( T^{26} + \)\(70\!\cdots\!87\)\( T^{27} + \)\(26\!\cdots\!13\)\( T^{28} + \)\(93\!\cdots\!63\)\( T^{29} + \)\(31\!\cdots\!97\)\( T^{30} + \)\(10\!\cdots\!96\)\( T^{31} + \)\(29\!\cdots\!00\)\( T^{32} + \)\(81\!\cdots\!50\)\( T^{33} + \)\(20\!\cdots\!65\)\( T^{34} + \)\(46\!\cdots\!05\)\( T^{35} + \)\(92\!\cdots\!28\)\( T^{36} + \)\(15\!\cdots\!59\)\( T^{37} + \)\(22\!\cdots\!48\)\( T^{38} + \)\(23\!\cdots\!97\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$29$ \( 1 + 16 T + 356 T^{2} + 4090 T^{3} + 54774 T^{4} + 503499 T^{5} + 5185947 T^{6} + 40664010 T^{7} + 355565661 T^{8} + 2486474985 T^{9} + 19496647316 T^{10} + 125387959304 T^{11} + 909855586995 T^{12} + 5484428987080 T^{13} + 37452135569274 T^{14} + 213835811773071 T^{15} + 1385703445491510 T^{16} + 7532573989645114 T^{17} + 46469947093723675 T^{18} + 240781898786988183 T^{19} + 1414696368543280646 T^{20} + 6982675064822657307 T^{21} + 39081225505821610675 T^{22} + \)\(18\!\cdots\!46\)\( T^{23} + \)\(98\!\cdots\!10\)\( T^{24} + \)\(43\!\cdots\!79\)\( T^{25} + \)\(22\!\cdots\!54\)\( T^{26} + \)\(94\!\cdots\!20\)\( T^{27} + \)\(45\!\cdots\!95\)\( T^{28} + \)\(18\!\cdots\!76\)\( T^{29} + \)\(82\!\cdots\!16\)\( T^{30} + \)\(30\!\cdots\!65\)\( T^{31} + \)\(12\!\cdots\!01\)\( T^{32} + \)\(41\!\cdots\!90\)\( T^{33} + \)\(15\!\cdots\!07\)\( T^{34} + \)\(43\!\cdots\!51\)\( T^{35} + \)\(13\!\cdots\!54\)\( T^{36} + \)\(29\!\cdots\!10\)\( T^{37} + \)\(74\!\cdots\!16\)\( T^{38} + \)\(97\!\cdots\!04\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$31$ \( 1 - 6 T + 318 T^{2} - 1518 T^{3} + 50004 T^{4} - 189505 T^{5} + 5236681 T^{6} - 15524147 T^{7} + 413848915 T^{8} - 934471947 T^{9} + 26425681433 T^{10} - 43790363774 T^{11} + 1421493925351 T^{12} - 1653825118500 T^{13} + 66176781521440 T^{14} - 51939856393273 T^{15} + 2713695257514807 T^{16} - 1433424696071581 T^{17} + 99105886692289330 T^{18} - 39032877693138405 T^{19} + 3242764231950792144 T^{20} - 1210019208487290555 T^{21} + 95240757111290046130 T^{22} - 42703155120668469571 T^{23} + \)\(25\!\cdots\!47\)\( T^{24} - \)\(14\!\cdots\!23\)\( T^{25} + \)\(58\!\cdots\!40\)\( T^{26} - \)\(45\!\cdots\!00\)\( T^{27} + \)\(12\!\cdots\!91\)\( T^{28} - \)\(11\!\cdots\!54\)\( T^{29} + \)\(21\!\cdots\!33\)\( T^{30} - \)\(23\!\cdots\!57\)\( T^{31} + \)\(32\!\cdots\!15\)\( T^{32} - \)\(37\!\cdots\!77\)\( T^{33} + \)\(39\!\cdots\!01\)\( T^{34} - \)\(44\!\cdots\!55\)\( T^{35} + \)\(36\!\cdots\!24\)\( T^{36} - \)\(34\!\cdots\!98\)\( T^{37} + \)\(22\!\cdots\!38\)\( T^{38} - \)\(13\!\cdots\!26\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \)
$37$ \( 1 - 2 T + 436 T^{2} - 636 T^{3} + 94484 T^{4} - 96806 T^{5} + 13573717 T^{6} - 9079400 T^{7} + 1452889835 T^{8} - 545246494 T^{9} + 123414777016 T^{10} - 16711275309 T^{11} + 8652333784042 T^{12} + 463394612292 T^{13} + 514000404908958 T^{14} + 98194267690708 T^{15} + 26349138021430845 T^{16} + 7256885305064918 T^{17} + 1179935948825639157 T^{18} + 361679909566421909 T^{19} + 46488733595387500290 T^{20} + 13382156653957610633 T^{21} + \)\(16\!\cdots\!33\)\( T^{22} + \)\(36\!\cdots\!54\)\( T^{23} + \)\(49\!\cdots\!45\)\( T^{24} + \)\(68\!\cdots\!56\)\( T^{25} + \)\(13\!\cdots\!22\)\( T^{26} + \)\(43\!\cdots\!36\)\( T^{27} + \)\(30\!\cdots\!82\)\( T^{28} - \)\(21\!\cdots\!93\)\( T^{29} + \)\(59\!\cdots\!84\)\( T^{30} - \)\(97\!\cdots\!22\)\( T^{31} + \)\(95\!\cdots\!35\)\( T^{32} - \)\(22\!\cdots\!00\)\( T^{33} + \)\(12\!\cdots\!13\)\( T^{34} - \)\(32\!\cdots\!58\)\( T^{35} + \)\(11\!\cdots\!44\)\( T^{36} - \)\(29\!\cdots\!12\)\( T^{37} + \)\(73\!\cdots\!44\)\( T^{38} - \)\(12\!\cdots\!46\)\( T^{39} + \)\(23\!\cdots\!01\)\( T^{40} \)
$41$ \( 1 + 25 T + 696 T^{2} + 12098 T^{3} + 210607 T^{4} + 2933554 T^{5} + 39929227 T^{6} + 474261550 T^{7} + 5464594445 T^{8} + 57229450190 T^{9} + 580264984256 T^{10} + 5467190547761 T^{11} + 49861714090633 T^{12} + 428109710343371 T^{13} + 3559450256532479 T^{14} + 28082973862518910 T^{15} + 214665825107524323 T^{16} + 1564541885634587045 T^{17} + 11052155258273894274 T^{18} + 74631837884937612416 T^{19} + \)\(48\!\cdots\!02\)\( T^{20} + \)\(30\!\cdots\!56\)\( T^{21} + \)\(18\!\cdots\!94\)\( T^{22} + \)\(10\!\cdots\!45\)\( T^{23} + \)\(60\!\cdots\!03\)\( T^{24} + \)\(32\!\cdots\!10\)\( T^{25} + \)\(16\!\cdots\!39\)\( T^{26} + \)\(83\!\cdots\!51\)\( T^{27} + \)\(39\!\cdots\!93\)\( T^{28} + \)\(17\!\cdots\!21\)\( T^{29} + \)\(77\!\cdots\!56\)\( T^{30} + \)\(31\!\cdots\!90\)\( T^{31} + \)\(12\!\cdots\!45\)\( T^{32} + \)\(43\!\cdots\!50\)\( T^{33} + \)\(15\!\cdots\!47\)\( T^{34} + \)\(45\!\cdots\!54\)\( T^{35} + \)\(13\!\cdots\!87\)\( T^{36} + \)\(31\!\cdots\!38\)\( T^{37} + \)\(74\!\cdots\!16\)\( T^{38} + \)\(10\!\cdots\!25\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$43$ \( 1 - 13 T + 471 T^{2} - 6283 T^{3} + 120422 T^{4} - 1515822 T^{5} + 21422518 T^{6} - 245586530 T^{7} + 2891592693 T^{8} - 30027599711 T^{9} + 310306876222 T^{10} - 2936062274715 T^{11} + 27307558269017 T^{12} - 237138824626569 T^{13} + 2013092147719312 T^{14} - 16135263794711645 T^{15} + 126104976690593190 T^{16} - 936396434766212447 T^{17} + 6773410299108949813 T^{18} - 46695524115296682981 T^{19} + \)\(31\!\cdots\!58\)\( T^{20} - \)\(20\!\cdots\!83\)\( T^{21} + \)\(12\!\cdots\!37\)\( T^{22} - \)\(74\!\cdots\!29\)\( T^{23} + \)\(43\!\cdots\!90\)\( T^{24} - \)\(23\!\cdots\!35\)\( T^{25} + \)\(12\!\cdots\!88\)\( T^{26} - \)\(64\!\cdots\!83\)\( T^{27} + \)\(31\!\cdots\!17\)\( T^{28} - \)\(14\!\cdots\!45\)\( T^{29} + \)\(67\!\cdots\!78\)\( T^{30} - \)\(27\!\cdots\!77\)\( T^{31} + \)\(11\!\cdots\!93\)\( T^{32} - \)\(42\!\cdots\!90\)\( T^{33} + \)\(15\!\cdots\!82\)\( T^{34} - \)\(48\!\cdots\!54\)\( T^{35} + \)\(16\!\cdots\!22\)\( T^{36} - \)\(36\!\cdots\!69\)\( T^{37} + \)\(11\!\cdots\!79\)\( T^{38} - \)\(14\!\cdots\!91\)\( T^{39} + \)\(46\!\cdots\!01\)\( T^{40} \)
$47$ \( 1 + 19 T + 774 T^{2} + 11872 T^{3} + 273276 T^{4} + 3551724 T^{5} + 60026983 T^{6} + 681902290 T^{7} + 9358794360 T^{8} + 94919407161 T^{9} + 1115094426107 T^{10} + 10251596671240 T^{11} + 106457690519673 T^{12} + 897170926294968 T^{13} + 8415041365773569 T^{14} + 65562559430606980 T^{15} + 563889426336727830 T^{16} + 4086881137017435528 T^{17} + 32568007441163889159 T^{18} + \)\(22\!\cdots\!48\)\( T^{19} + \)\(16\!\cdots\!12\)\( T^{20} + \)\(10\!\cdots\!56\)\( T^{21} + \)\(71\!\cdots\!31\)\( T^{22} + \)\(42\!\cdots\!44\)\( T^{23} + \)\(27\!\cdots\!30\)\( T^{24} + \)\(15\!\cdots\!60\)\( T^{25} + \)\(90\!\cdots\!01\)\( T^{26} + \)\(45\!\cdots\!84\)\( T^{27} + \)\(25\!\cdots\!53\)\( T^{28} + \)\(11\!\cdots\!80\)\( T^{29} + \)\(58\!\cdots\!43\)\( T^{30} + \)\(23\!\cdots\!83\)\( T^{31} + \)\(10\!\cdots\!60\)\( T^{32} + \)\(37\!\cdots\!30\)\( T^{33} + \)\(15\!\cdots\!27\)\( T^{34} + \)\(42\!\cdots\!32\)\( T^{35} + \)\(15\!\cdots\!96\)\( T^{36} + \)\(31\!\cdots\!64\)\( T^{37} + \)\(96\!\cdots\!86\)\( T^{38} + \)\(11\!\cdots\!77\)\( T^{39} + \)\(27\!\cdots\!01\)\( T^{40} \)
$53$ \( 1 + 24 T + 752 T^{2} + 12714 T^{3} + 241627 T^{4} + 3275489 T^{5} + 47980334 T^{6} + 556894589 T^{7} + 6886482945 T^{8} + 71132417785 T^{9} + 778084195202 T^{10} + 7330484721748 T^{11} + 72925359990441 T^{12} + 637044558919314 T^{13} + 5866379765742918 T^{14} + 48032253020185635 T^{15} + 413966785474461514 T^{16} + 3196625429282140999 T^{17} + 25947703842724984442 T^{18} + \)\(18\!\cdots\!27\)\( T^{19} + \)\(14\!\cdots\!16\)\( T^{20} + \)\(10\!\cdots\!31\)\( T^{21} + \)\(72\!\cdots\!78\)\( T^{22} + \)\(47\!\cdots\!23\)\( T^{23} + \)\(32\!\cdots\!34\)\( T^{24} + \)\(20\!\cdots\!55\)\( T^{25} + \)\(13\!\cdots\!22\)\( T^{26} + \)\(74\!\cdots\!18\)\( T^{27} + \)\(45\!\cdots\!01\)\( T^{28} + \)\(24\!\cdots\!84\)\( T^{29} + \)\(13\!\cdots\!98\)\( T^{30} + \)\(65\!\cdots\!45\)\( T^{31} + \)\(33\!\cdots\!45\)\( T^{32} + \)\(14\!\cdots\!97\)\( T^{33} + \)\(66\!\cdots\!46\)\( T^{34} + \)\(23\!\cdots\!73\)\( T^{35} + \)\(93\!\cdots\!67\)\( T^{36} + \)\(26\!\cdots\!82\)\( T^{37} + \)\(81\!\cdots\!28\)\( T^{38} + \)\(13\!\cdots\!08\)\( T^{39} + \)\(30\!\cdots\!01\)\( T^{40} \)
$59$ \( 1 + 23 T + 835 T^{2} + 14786 T^{3} + 314559 T^{4} + 4632805 T^{5} + 74385661 T^{6} + 955001616 T^{7} + 12748655041 T^{8} + 147128015000 T^{9} + 1716324613351 T^{10} + 18163752339784 T^{11} + 190575665724983 T^{12} + 1872219326295796 T^{13} + 17979789322374275 T^{14} + 165115335732762556 T^{15} + 1467551308043410298 T^{16} + 12644429046237961845 T^{17} + \)\(10\!\cdots\!46\)\( T^{18} + \)\(84\!\cdots\!39\)\( T^{19} + \)\(65\!\cdots\!12\)\( T^{20} + \)\(50\!\cdots\!01\)\( T^{21} + \)\(36\!\cdots\!26\)\( T^{22} + \)\(25\!\cdots\!55\)\( T^{23} + \)\(17\!\cdots\!78\)\( T^{24} + \)\(11\!\cdots\!44\)\( T^{25} + \)\(75\!\cdots\!75\)\( T^{26} + \)\(46\!\cdots\!24\)\( T^{27} + \)\(27\!\cdots\!43\)\( T^{28} + \)\(15\!\cdots\!76\)\( T^{29} + \)\(87\!\cdots\!51\)\( T^{30} + \)\(44\!\cdots\!00\)\( T^{31} + \)\(22\!\cdots\!21\)\( T^{32} + \)\(10\!\cdots\!64\)\( T^{33} + \)\(46\!\cdots\!21\)\( T^{34} + \)\(16\!\cdots\!95\)\( T^{35} + \)\(67\!\cdots\!19\)\( T^{36} + \)\(18\!\cdots\!34\)\( T^{37} + \)\(62\!\cdots\!35\)\( T^{38} + \)\(10\!\cdots\!97\)\( T^{39} + \)\(26\!\cdots\!01\)\( T^{40} \)
$61$ \( 1 + 27 T + 875 T^{2} + 17033 T^{3} + 331994 T^{4} + 5116610 T^{5} + 76175542 T^{6} + 982572262 T^{7} + 12203408370 T^{8} + 136820170557 T^{9} + 1484538070309 T^{10} + 14880081183760 T^{11} + 145384403908907 T^{12} + 1332681712720025 T^{13} + 12007469649060554 T^{14} + 102707680608618470 T^{15} + 871687045460596794 T^{16} + 7092973046188373144 T^{17} + 57801758671098466844 T^{18} + \)\(45\!\cdots\!44\)\( T^{19} + \)\(36\!\cdots\!60\)\( T^{20} + \)\(27\!\cdots\!84\)\( T^{21} + \)\(21\!\cdots\!24\)\( T^{22} + \)\(16\!\cdots\!64\)\( T^{23} + \)\(12\!\cdots\!54\)\( T^{24} + \)\(86\!\cdots\!70\)\( T^{25} + \)\(61\!\cdots\!94\)\( T^{26} + \)\(41\!\cdots\!25\)\( T^{27} + \)\(27\!\cdots\!67\)\( T^{28} + \)\(17\!\cdots\!60\)\( T^{29} + \)\(10\!\cdots\!09\)\( T^{30} + \)\(59\!\cdots\!77\)\( T^{31} + \)\(32\!\cdots\!70\)\( T^{32} + \)\(15\!\cdots\!22\)\( T^{33} + \)\(75\!\cdots\!22\)\( T^{34} + \)\(30\!\cdots\!10\)\( T^{35} + \)\(12\!\cdots\!34\)\( T^{36} + \)\(38\!\cdots\!93\)\( T^{37} + \)\(11\!\cdots\!75\)\( T^{38} + \)\(22\!\cdots\!07\)\( T^{39} + \)\(50\!\cdots\!01\)\( T^{40} \)
$67$ \( 1 - 9 T + 591 T^{2} - 4752 T^{3} + 170755 T^{4} - 1206847 T^{5} + 32101040 T^{6} - 197318136 T^{7} + 4462893877 T^{8} - 23737930247 T^{9} + 500089616732 T^{10} - 2319275613331 T^{11} + 48393933613827 T^{12} - 200872169616621 T^{13} + 4252058182022998 T^{14} - 16383764790112677 T^{15} + 347117079633002046 T^{16} - 1274967067532347537 T^{17} + 26328418982796754431 T^{18} - 92887292534777448481 T^{19} + \)\(18\!\cdots\!00\)\( T^{20} - \)\(62\!\cdots\!27\)\( T^{21} + \)\(11\!\cdots\!59\)\( T^{22} - \)\(38\!\cdots\!31\)\( T^{23} + \)\(69\!\cdots\!66\)\( T^{24} - \)\(22\!\cdots\!39\)\( T^{25} + \)\(38\!\cdots\!62\)\( T^{26} - \)\(12\!\cdots\!83\)\( T^{27} + \)\(19\!\cdots\!07\)\( T^{28} - \)\(63\!\cdots\!57\)\( T^{29} + \)\(91\!\cdots\!68\)\( T^{30} - \)\(28\!\cdots\!01\)\( T^{31} + \)\(36\!\cdots\!97\)\( T^{32} - \)\(10\!\cdots\!32\)\( T^{33} + \)\(11\!\cdots\!60\)\( T^{34} - \)\(29\!\cdots\!21\)\( T^{35} + \)\(28\!\cdots\!55\)\( T^{36} - \)\(52\!\cdots\!04\)\( T^{37} + \)\(43\!\cdots\!19\)\( T^{38} - \)\(44\!\cdots\!27\)\( T^{39} + \)\(33\!\cdots\!01\)\( T^{40} \)
$71$ \( 1 + 63 T + 2596 T^{2} + 78560 T^{3} + 1968706 T^{4} + 42263598 T^{5} + 807471444 T^{6} + 13947310273 T^{7} + 221613295558 T^{8} + 3266810090497 T^{9} + 45096447088326 T^{10} + 585913588994251 T^{11} + 7204919382169096 T^{12} + 84116808873980238 T^{13} + 935676611913596316 T^{14} + 9935597569031351192 T^{15} + \)\(10\!\cdots\!58\)\( T^{16} + \)\(98\!\cdots\!37\)\( T^{17} + \)\(91\!\cdots\!02\)\( T^{18} + \)\(82\!\cdots\!29\)\( T^{19} + \)\(70\!\cdots\!14\)\( T^{20} + \)\(58\!\cdots\!59\)\( T^{21} + \)\(46\!\cdots\!82\)\( T^{22} + \)\(35\!\cdots\!07\)\( T^{23} + \)\(25\!\cdots\!98\)\( T^{24} + \)\(17\!\cdots\!92\)\( T^{25} + \)\(11\!\cdots\!36\)\( T^{26} + \)\(76\!\cdots\!58\)\( T^{27} + \)\(46\!\cdots\!56\)\( T^{28} + \)\(26\!\cdots\!81\)\( T^{29} + \)\(14\!\cdots\!26\)\( T^{30} + \)\(75\!\cdots\!87\)\( T^{31} + \)\(36\!\cdots\!78\)\( T^{32} + \)\(16\!\cdots\!03\)\( T^{33} + \)\(66\!\cdots\!64\)\( T^{34} + \)\(24\!\cdots\!98\)\( T^{35} + \)\(82\!\cdots\!26\)\( T^{36} + \)\(23\!\cdots\!60\)\( T^{37} + \)\(54\!\cdots\!56\)\( T^{38} + \)\(94\!\cdots\!53\)\( T^{39} + \)\(10\!\cdots\!01\)\( T^{40} \)
$73$ \( 1 + 21 T + 916 T^{2} + 15097 T^{3} + 380738 T^{4} + 5228211 T^{5} + 98720492 T^{6} + 1169044476 T^{7} + 18261023340 T^{8} + 190285062363 T^{9} + 2593445804090 T^{10} + 24049645680336 T^{11} + 296353479113844 T^{12} + 2462421964424588 T^{13} + 28282525841124194 T^{14} + 212323680749559685 T^{15} + 2347288002490873928 T^{16} + 16218471867962234439 T^{17} + \)\(17\!\cdots\!16\)\( T^{18} + \)\(11\!\cdots\!68\)\( T^{19} + \)\(13\!\cdots\!66\)\( T^{20} + \)\(85\!\cdots\!64\)\( T^{21} + \)\(95\!\cdots\!64\)\( T^{22} + \)\(63\!\cdots\!63\)\( T^{23} + \)\(66\!\cdots\!48\)\( T^{24} + \)\(44\!\cdots\!05\)\( T^{25} + \)\(42\!\cdots\!66\)\( T^{26} + \)\(27\!\cdots\!36\)\( T^{27} + \)\(23\!\cdots\!64\)\( T^{28} + \)\(14\!\cdots\!68\)\( T^{29} + \)\(11\!\cdots\!10\)\( T^{30} + \)\(59\!\cdots\!51\)\( T^{31} + \)\(41\!\cdots\!40\)\( T^{32} + \)\(19\!\cdots\!08\)\( T^{33} + \)\(12\!\cdots\!28\)\( T^{34} + \)\(46\!\cdots\!27\)\( T^{35} + \)\(24\!\cdots\!18\)\( T^{36} + \)\(71\!\cdots\!41\)\( T^{37} + \)\(31\!\cdots\!04\)\( T^{38} + \)\(53\!\cdots\!77\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$79$ \( 1 - 18 T + 1184 T^{2} - 19119 T^{3} + 677618 T^{4} - 9886627 T^{5} + 249669964 T^{6} - 3315237533 T^{7} + 66604748933 T^{8} - 810231025074 T^{9} + 13722285935097 T^{10} - 153803296119898 T^{11} + 2274301424731523 T^{12} - 23596328615581358 T^{13} + 311767076012223811 T^{14} - 3004511485146333437 T^{15} + 36040317878578482083 T^{16} - \)\(32\!\cdots\!52\)\( T^{17} + \)\(35\!\cdots\!28\)\( T^{18} - \)\(29\!\cdots\!26\)\( T^{19} + \)\(30\!\cdots\!92\)\( T^{20} - \)\(23\!\cdots\!54\)\( T^{21} + \)\(22\!\cdots\!48\)\( T^{22} - \)\(15\!\cdots\!28\)\( T^{23} + \)\(14\!\cdots\!23\)\( T^{24} - \)\(92\!\cdots\!63\)\( T^{25} + \)\(75\!\cdots\!31\)\( T^{26} - \)\(45\!\cdots\!22\)\( T^{27} + \)\(34\!\cdots\!03\)\( T^{28} - \)\(18\!\cdots\!62\)\( T^{29} + \)\(12\!\cdots\!97\)\( T^{30} - \)\(60\!\cdots\!46\)\( T^{31} + \)\(39\!\cdots\!53\)\( T^{32} - \)\(15\!\cdots\!87\)\( T^{33} + \)\(92\!\cdots\!84\)\( T^{34} - \)\(28\!\cdots\!73\)\( T^{35} + \)\(15\!\cdots\!78\)\( T^{36} - \)\(34\!\cdots\!21\)\( T^{37} + \)\(17\!\cdots\!24\)\( T^{38} - \)\(20\!\cdots\!42\)\( T^{39} + \)\(89\!\cdots\!01\)\( T^{40} \)
$83$ \( 1 - T + 887 T^{2} - 696 T^{3} + 387443 T^{4} - 295714 T^{5} + 111591936 T^{6} - 99249840 T^{7} + 23963641195 T^{8} - 26427369351 T^{9} + 4117436531106 T^{10} - 5497886185864 T^{11} + 593163051161905 T^{12} - 904000829396146 T^{13} + 73957076079070724 T^{14} - 120559785802295129 T^{15} + 8138620605331576032 T^{16} - 13425709742136785765 T^{17} + \)\(79\!\cdots\!35\)\( T^{18} - \)\(12\!\cdots\!00\)\( T^{19} + \)\(70\!\cdots\!44\)\( T^{20} - \)\(10\!\cdots\!00\)\( T^{21} + \)\(54\!\cdots\!15\)\( T^{22} - \)\(76\!\cdots\!55\)\( T^{23} + \)\(38\!\cdots\!72\)\( T^{24} - \)\(47\!\cdots\!47\)\( T^{25} + \)\(24\!\cdots\!56\)\( T^{26} - \)\(24\!\cdots\!42\)\( T^{27} + \)\(13\!\cdots\!05\)\( T^{28} - \)\(10\!\cdots\!92\)\( T^{29} + \)\(63\!\cdots\!94\)\( T^{30} - \)\(34\!\cdots\!17\)\( T^{31} + \)\(25\!\cdots\!95\)\( T^{32} - \)\(88\!\cdots\!20\)\( T^{33} + \)\(82\!\cdots\!44\)\( T^{34} - \)\(18\!\cdots\!98\)\( T^{35} + \)\(19\!\cdots\!83\)\( T^{36} - \)\(29\!\cdots\!08\)\( T^{37} + \)\(30\!\cdots\!83\)\( T^{38} - \)\(29\!\cdots\!47\)\( T^{39} + \)\(24\!\cdots\!01\)\( T^{40} \)
$89$ \( 1 - 16 T + 1054 T^{2} - 12313 T^{3} + 495452 T^{4} - 4181350 T^{5} + 143352715 T^{6} - 815178841 T^{7} + 29862289153 T^{8} - 95871706672 T^{9} + 4986555809317 T^{10} - 5188938821125 T^{11} + 718166918489319 T^{12} + 484338156975993 T^{13} + 92377826244411923 T^{14} + 171552993935159192 T^{15} + 10685149175823232350 T^{16} + 27269038279907589269 T^{17} + \)\(11\!\cdots\!99\)\( T^{18} + \)\(31\!\cdots\!79\)\( T^{19} + \)\(10\!\cdots\!30\)\( T^{20} + \)\(27\!\cdots\!31\)\( T^{21} + \)\(87\!\cdots\!79\)\( T^{22} + \)\(19\!\cdots\!61\)\( T^{23} + \)\(67\!\cdots\!50\)\( T^{24} + \)\(95\!\cdots\!08\)\( T^{25} + \)\(45\!\cdots\!03\)\( T^{26} + \)\(21\!\cdots\!97\)\( T^{27} + \)\(28\!\cdots\!39\)\( T^{28} - \)\(18\!\cdots\!25\)\( T^{29} + \)\(15\!\cdots\!17\)\( T^{30} - \)\(26\!\cdots\!08\)\( T^{31} + \)\(73\!\cdots\!13\)\( T^{32} - \)\(17\!\cdots\!29\)\( T^{33} + \)\(28\!\cdots\!15\)\( T^{34} - \)\(72\!\cdots\!50\)\( T^{35} + \)\(76\!\cdots\!72\)\( T^{36} - \)\(16\!\cdots\!77\)\( T^{37} + \)\(12\!\cdots\!74\)\( T^{38} - \)\(17\!\cdots\!44\)\( T^{39} + \)\(97\!\cdots\!01\)\( T^{40} \)
$97$ \( 1 + 32 T + 1491 T^{2} + 35175 T^{3} + 973478 T^{4} + 18675375 T^{5} + 393370752 T^{6} + 6490900374 T^{7} + 114399025400 T^{8} + 1681555576436 T^{9} + 26020622897892 T^{10} + 348211972917488 T^{11} + 4860356046062111 T^{12} + 59995895481335366 T^{13} + 767559931026210100 T^{14} + 8808788952687065688 T^{15} + \)\(10\!\cdots\!79\)\( T^{16} + \)\(11\!\cdots\!30\)\( T^{17} + \)\(12\!\cdots\!97\)\( T^{18} + \)\(12\!\cdots\!32\)\( T^{19} + \)\(12\!\cdots\!86\)\( T^{20} + \)\(12\!\cdots\!04\)\( T^{21} + \)\(11\!\cdots\!73\)\( T^{22} + \)\(10\!\cdots\!90\)\( T^{23} + \)\(92\!\cdots\!99\)\( T^{24} + \)\(75\!\cdots\!16\)\( T^{25} + \)\(63\!\cdots\!00\)\( T^{26} + \)\(48\!\cdots\!58\)\( T^{27} + \)\(38\!\cdots\!71\)\( T^{28} + \)\(26\!\cdots\!96\)\( T^{29} + \)\(19\!\cdots\!08\)\( T^{30} + \)\(12\!\cdots\!08\)\( T^{31} + \)\(79\!\cdots\!00\)\( T^{32} + \)\(43\!\cdots\!98\)\( T^{33} + \)\(25\!\cdots\!88\)\( T^{34} + \)\(11\!\cdots\!75\)\( T^{35} + \)\(59\!\cdots\!38\)\( T^{36} + \)\(20\!\cdots\!75\)\( T^{37} + \)\(86\!\cdots\!99\)\( T^{38} + \)\(17\!\cdots\!56\)\( T^{39} + \)\(54\!\cdots\!01\)\( T^{40} \)
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