Properties

Label 8001.2.a.u
Level 8001
Weight 2
Character orbit 8001.a
Self dual yes
Analytic conductor 63.888
Analytic rank 0
Dimension 18
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: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 6 x^{17} - 11 x^{16} + 123 x^{15} - 35 x^{14} - 982 x^{13} + 988 x^{12} + 3872 x^{11} - 5421 x^{10} - 7882 x^{9} + 13376 x^{8} + 7948 x^{7} - 15795 x^{6} - 3858 x^{5} + 8199 x^{4} + 1453 x^{3} - 1610 x^{2} - 380 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-9056232 \nu^{17} + 56952713 \nu^{16} + 63697503 \nu^{15} - 1083863790 \nu^{14} + 1061485425 \nu^{13} + 7576909534 \nu^{12} - 14966699558 \nu^{11} - 22572883200 \nu^{10} + 73124433062 \nu^{9} + 17899620183 \nu^{8} - 170430820251 \nu^{7} + 41863070067 \nu^{6} + 192141347449 \nu^{5} - 79413530246 \nu^{4} - 95483657380 \nu^{3} + 30790742789 \nu^{2} + 19199072948 \nu - 228715354\)\()/ 365415070 \)
\(\beta_{4}\)\(=\)\((\)\(12516449 \nu^{17} - 35576866 \nu^{16} - 322120881 \nu^{15} + 878461675 \nu^{14} + 3459985365 \nu^{13} - 8859247938 \nu^{12} - 20043398304 \nu^{11} + 46965576220 \nu^{10} + 67169495191 \nu^{9} - 139958996266 \nu^{8} - 129295866318 \nu^{7} + 231201547396 \nu^{6} + 134129812547 \nu^{5} - 191805387798 \nu^{4} - 67208803935 \nu^{3} + 60307885677 \nu^{2} + 15314260994 \nu - 1868332582\)\()/ 365415070 \)
\(\beta_{5}\)\(=\)\((\)\(-25066381 \nu^{17} + 165457944 \nu^{16} + 197605069 \nu^{15} - 3294331535 \nu^{14} + 2503016525 \nu^{13} + 24999769422 \nu^{12} - 37913753504 \nu^{11} - 89306242820 \nu^{10} + 187798500441 \nu^{9} + 143893003544 \nu^{8} - 436801227938 \nu^{7} - 57150890634 \nu^{6} + 479794667297 \nu^{5} - 71979148918 \nu^{4} - 217408254535 \nu^{3} + 44636251777 \nu^{2} + 35680796674 \nu - 2904833932\)\()/ 730830140 \)
\(\beta_{6}\)\(=\)\((\)\(-32625911 \nu^{17} + 257402314 \nu^{16} + 38720489 \nu^{15} - 4899416105 \nu^{14} + 7765546185 \nu^{13} + 34400341342 \nu^{12} - 85621658304 \nu^{11} - 104551063080 \nu^{10} + 388747945111 \nu^{9} + 96698100714 \nu^{8} - 864214618628 \nu^{7} + 143202903956 \nu^{6} + 917584185897 \nu^{5} - 295691214778 \nu^{4} - 402286545165 \nu^{3} + 104139037497 \nu^{2} + 68310107754 \nu - 523550232\)\()/ 730830140 \)
\(\beta_{7}\)\(=\)\((\)\(-40185441 \nu^{17} + 349346684 \nu^{16} - 120164091 \nu^{15} - 6504500675 \nu^{14} + 13028075845 \nu^{13} + 43800913262 \nu^{12} - 133329563104 \nu^{11} - 119795883340 \nu^{10} + 589697389781 \nu^{9} + 49503197884 \nu^{8} - 1291628009318 \nu^{7} + 343556698546 \nu^{6} + 1354642874357 \nu^{5} - 520134110778 \nu^{4} - 579856534395 \nu^{3} + 168757634197 \nu^{2} + 87053646174 \nu - 2527247372\)\()/ 730830140 \)
\(\beta_{8}\)\(=\)\((\)\(-20878693 \nu^{17} + 77409897 \nu^{16} + 451307397 \nu^{15} - 1771179645 \nu^{14} - 3907179050 \nu^{13} + 16305558241 \nu^{12} + 17519717313 \nu^{11} - 77527845145 \nu^{10} - 44085517162 \nu^{9} + 202940243242 \nu^{8} + 63666071141 \nu^{7} - 287556922462 \nu^{6} - 53922064354 \nu^{5} + 199447839821 \nu^{4} + 29523656040 \nu^{3} - 50247230349 \nu^{2} - 11117097738 \nu + 723097034\)\()/ 365415070 \)
\(\beta_{9}\)\(=\)\((\)\(8395741 \nu^{17} - 64482942 \nu^{16} - 11290113 \nu^{15} + 1199594231 \nu^{14} - 1945179675 \nu^{13} - 8038582192 \nu^{12} + 21352465070 \nu^{11} + 21465866534 \nu^{10} - 96183962599 \nu^{9} - 5111788540 \nu^{8} + 212368884076 \nu^{7} - 75583142318 \nu^{6} - 226469839587 \nu^{5} + 112899717876 \nu^{4} + 104809072213 \nu^{3} - 38853986253 \nu^{2} - 20711179258 \nu - 76030588\)\()/ 146166028 \)
\(\beta_{10}\)\(=\)\((\)\(50383667 \nu^{17} - 161191148 \nu^{16} - 1204101853 \nu^{15} + 3787887405 \nu^{14} + 11956797505 \nu^{13} - 35940666424 \nu^{12} - 64188753322 \nu^{11} + 176493100810 \nu^{10} + 201153933563 \nu^{9} - 476359522038 \nu^{8} - 365447236374 \nu^{7} + 688564895308 \nu^{6} + 355164023581 \nu^{5} - 472380557164 \nu^{4} - 157197227985 \nu^{3} + 110627356231 \nu^{2} + 31151563462 \nu - 2009629216\)\()/ 730830140 \)
\(\beta_{11}\)\(=\)\((\)\(80549203 \nu^{17} - 492648012 \nu^{16} - 821028837 \nu^{15} + 9925282045 \nu^{14} - 3917777155 \nu^{13} - 77138965156 \nu^{12} + 85375182182 \nu^{11} + 290745846970 \nu^{10} - 439435499053 \nu^{9} - 542716950162 \nu^{8} + 1006286981314 \nu^{7} + 442575263692 \nu^{6} - 1037561853751 \nu^{5} - 95993813136 \nu^{4} + 391615540795 \nu^{3} - 4720823941 \nu^{2} - 38822492922 \nu + 3906261256\)\()/ 730830140 \)
\(\beta_{12}\)\(=\)\((\)\(100575831 \nu^{17} - 597870654 \nu^{16} - 1070376159 \nu^{15} + 11961026425 \nu^{14} - 3958878145 \nu^{13} - 91593180032 \nu^{12} + 99303778874 \nu^{11} + 333681577930 \nu^{10} - 522661745161 \nu^{9} - 568242407784 \nu^{8} + 1220905295328 \nu^{7} + 318287907994 \nu^{6} - 1310588271257 \nu^{5} + 131621699248 \nu^{4} + 559009857575 \nu^{3} - 98821283147 \nu^{2} - 79152820674 \nu + 4483050272\)\()/ 730830140 \)
\(\beta_{13}\)\(=\)\((\)\(-101911933 \nu^{17} + 591076942 \nu^{16} + 1177422127 \nu^{15} - 11988946255 \nu^{14} + 2088222575 \nu^{13} + 94092714686 \nu^{12} - 85031622852 \nu^{11} - 359907129280 \nu^{10} + 466570180813 \nu^{9} + 687964139822 \nu^{8} - 1102540718984 \nu^{7} - 586410163532 \nu^{6} + 1177689206511 \nu^{5} + 145204756046 \nu^{4} - 478582900535 \nu^{3} - 1332536689 \nu^{2} + 52828018842 \nu + 1025152864\)\()/ 730830140 \)
\(\beta_{14}\)\(=\)\((\)\(-10957014 \nu^{17} + 55541023 \nu^{16} + 166873740 \nu^{15} - 1164950524 \nu^{14} - 620946577 \nu^{13} + 9612588205 \nu^{12} - 2152998819 \nu^{11} - 39807766547 \nu^{10} + 21114688373 \nu^{9} + 87382928254 \nu^{8} - 54458600727 \nu^{7} - 98803608930 \nu^{6} + 54227047741 \nu^{5} + 52720255893 \nu^{4} - 14672012293 \nu^{3} - 12177807586 \nu^{2} - 852708688 \nu + 267274566\)\()/73083014\)
\(\beta_{15}\)\(=\)\((\)\(33634461 \nu^{17} - 212119814 \nu^{16} - 304035654 \nu^{15} + 4222090590 \nu^{14} - 2463442190 \nu^{13} - 32123264007 \nu^{12} + 42663320289 \nu^{11} + 116114548090 \nu^{10} - 213829531481 \nu^{9} - 196165560134 \nu^{8} + 491688834403 \nu^{7} + 110801814894 \nu^{6} - 523818348102 \nu^{5} + 37804322218 \nu^{4} + 220034653890 \nu^{3} - 26980577947 \nu^{2} - 29553701509 \nu + 1153552087\)\()/ 182707535 \)
\(\beta_{16}\)\(=\)\((\)\(100069624 \nu^{17} - 532974031 \nu^{16} - 1376337581 \nu^{15} + 10970946660 \nu^{14} + 2569093715 \nu^{13} - 87933730868 \nu^{12} + 45335398986 \nu^{11} + 347183324810 \nu^{10} - 299845169974 \nu^{9} - 699846001231 \nu^{8} + 735383390107 \nu^{7} + 666522143031 \nu^{6} - 770874664093 \nu^{5} - 239724502658 \nu^{4} + 285055129740 \nu^{3} + 31458483877 \nu^{2} - 25149158196 \nu - 346733612\)\()/ 365415070 \)
\(\beta_{17}\)\(=\)\((\)\(212880147 \nu^{17} - 1116211818 \nu^{16} - 2947927153 \nu^{15} + 22837760465 \nu^{14} + 5940214235 \nu^{13} - 181212148554 \nu^{12} + 92102702768 \nu^{11} + 702455085660 \nu^{10} - 618936385827 \nu^{9} - 1362506463858 \nu^{8} + 1525835050196 \nu^{7} + 1169851753908 \nu^{6} - 1615051615589 \nu^{5} - 258422227194 \nu^{4} + 624723711805 \nu^{3} - 36412665269 \nu^{2} - 71450856478 \nu + 5373162964\)\()/ 730830140 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{16} + \beta_{14} - \beta_{10} + \beta_{7} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{16} + \beta_{15} + \beta_{14} + \beta_{13} - \beta_{10} + \beta_{7} + \beta_{6} + 8 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(9 \beta_{16} - \beta_{15} + 9 \beta_{14} - \beta_{13} - 9 \beta_{10} + 8 \beta_{7} + \beta_{6} - \beta_{5} + 9 \beta_{2} + 21 \beta_{1} + 10\)
\(\nu^{6}\)\(=\)\(12 \beta_{16} + 11 \beta_{15} + 11 \beta_{14} + 10 \beta_{13} - 2 \beta_{11} - 13 \beta_{10} + 12 \beta_{7} + 11 \beta_{6} + 59 \beta_{2} + \beta_{1} + 89\)
\(\nu^{7}\)\(=\)\(70 \beta_{16} - 11 \beta_{15} + 69 \beta_{14} - 12 \beta_{13} - \beta_{12} - \beta_{11} - 72 \beta_{10} + 2 \beta_{9} - \beta_{8} + 59 \beta_{7} + 15 \beta_{6} - 14 \beta_{5} + 70 \beta_{2} + 125 \beta_{1} + 86\)
\(\nu^{8}\)\(=\)\(\beta_{17} + 110 \beta_{16} + 92 \beta_{15} + 96 \beta_{14} + 77 \beta_{13} + 2 \beta_{12} - 29 \beta_{11} - 130 \beta_{10} + \beta_{9} - 2 \beta_{8} + 113 \beta_{7} + 97 \beta_{6} - 2 \beta_{5} - \beta_{4} + 427 \beta_{2} + 15 \beta_{1} + 575\)
\(\nu^{9}\)\(=\)\(3 \beta_{17} + 525 \beta_{16} - 88 \beta_{15} + 510 \beta_{14} - 106 \beta_{13} - 14 \beta_{12} - 21 \beta_{11} - 565 \beta_{10} + 30 \beta_{9} - 18 \beta_{8} + 436 \beta_{7} + 158 \beta_{6} - 139 \beta_{5} + \beta_{4} + 4 \beta_{3} + 537 \beta_{2} + 790 \beta_{1} + 698\)
\(\nu^{10}\)\(=\)\(17 \beta_{17} + 926 \beta_{16} + 703 \beta_{15} + 786 \beta_{14} + 543 \beta_{13} + 32 \beta_{12} - 298 \beta_{11} - 1175 \beta_{10} + 24 \beta_{9} - 43 \beta_{8} + 976 \beta_{7} + 799 \beta_{6} - 36 \beta_{5} - 17 \beta_{4} + 4 \beta_{3} + 3077 \beta_{2} + 159 \beta_{1} + 3893\)
\(\nu^{11}\)\(=\)\(56 \beta_{17} + 3904 \beta_{16} - 606 \beta_{15} + 3746 \beta_{14} - 833 \beta_{13} - 137 \beta_{12} - 280 \beta_{11} - 4416 \beta_{10} + 319 \beta_{9} - 224 \beta_{8} + 3268 \beta_{7} + 1458 \beta_{6} - 1195 \beta_{5} + 11 \beta_{4} + 79 \beta_{3} + 4141 \beta_{2} + 5147 \beta_{1} + 5515\)
\(\nu^{12}\)\(=\)\(201 \beta_{17} + 7513 \beta_{16} + 5177 \beta_{15} + 6290 \beta_{14} + 3679 \beta_{13} + 330 \beta_{12} - 2677 \beta_{11} - 10057 \beta_{10} + 348 \beta_{9} - 586 \beta_{8} + 8073 \beta_{7} + 6386 \beta_{6} - 437 \beta_{5} - 205 \beta_{4} + 99 \beta_{3} + 22189 \beta_{2} + 1480 \beta_{1} + 27107\)
\(\nu^{13}\)\(=\)\(688 \beta_{17} + 29038 \beta_{16} - 3737 \beta_{15} + 27583 \beta_{14} - 6182 \beta_{13} - 1180 \beta_{12} - 3051 \beta_{11} - 34447 \beta_{10} + 2981 \beta_{9} - 2384 \beta_{8} + 24810 \beta_{7} + 12611 \beta_{6} - 9514 \beta_{5} + 26 \beta_{4} + 1019 \beta_{3} + 32081 \beta_{2} + 34132 \beta_{1} + 43005\)
\(\nu^{14}\)\(=\)\(2053 \beta_{17} + 59783 \beta_{16} + 37549 \beta_{15} + 49823 \beta_{14} + 24374 \beta_{13} + 2749 \beta_{12} - 22488 \beta_{11} - 83230 \beta_{10} + 4044 \beta_{9} - 6536 \beta_{8} + 65123 \beta_{7} + 50205 \beta_{6} - 4458 \beta_{5} - 2173 \beta_{4} + 1509 \beta_{3} + 160445 \beta_{2} + 12948 \beta_{1} + 192283\)
\(\nu^{15}\)\(=\)\(7080 \beta_{17} + 216596 \beta_{16} - 20422 \beta_{15} + 204059 \beta_{14} - 44489 \beta_{13} - 9693 \beta_{12} - 29755 \beta_{11} - 268136 \beta_{10} + 26170 \beta_{9} - 23273 \beta_{8} + 190018 \beta_{7} + 105128 \beta_{6} - 72351 \beta_{5} - 933 \beta_{4} + 10914 \beta_{3} + 248954 \beta_{2} + 229108 \beta_{1} + 333140\)
\(\nu^{16}\)\(=\)\(19439 \beta_{17} + 470153 \beta_{16} + 271056 \beta_{15} + 392329 \beta_{14} + 159039 \beta_{13} + 19758 \beta_{12} - 181934 \beta_{11} - 673701 \beta_{10} + 41602 \beta_{9} - 65334 \beta_{8} + 517172 \beta_{7} + 390689 \beta_{6} - 41142 \beta_{5} - 21540 \beta_{4} + 18347 \beta_{3} + 1164251 \beta_{2} + 109440 \beta_{1} + 1382442\)
\(\nu^{17}\)\(=\)\(66426 \beta_{17} + 1620828 \beta_{16} - 91207 \beta_{15} + 1516971 \beta_{14} - 314852 \beta_{13} - 78760 \beta_{12} - 271151 \beta_{11} - 2082370 \beta_{10} + 221821 \beta_{9} - 215137 \beta_{8} + 1462990 \beta_{7} + 856189 \beta_{6} - 534253 \beta_{5} - 20129 \beta_{4} + 105540 \beta_{3} + 1931064 \beta_{2} + 1552681 \beta_{1} + 2572050\)

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
−2.55461
−2.24824
−1.90615
−1.70725
−1.33766
−0.665348
−0.432278
−0.375443
0.0366427
0.803961
0.981962
1.52505
1.60620
1.77085
2.40246
2.59597
2.73355
2.77035
−2.55461 0 4.52602 2.12871 0 1.00000 −6.45298 0 −5.43801
1.2 −2.24824 0 3.05458 3.31434 0 1.00000 −2.37095 0 −7.45143
1.3 −1.90615 0 1.63341 −1.97979 0 1.00000 0.698768 0 3.77379
1.4 −1.70725 0 0.914706 −1.08728 0 1.00000 1.85287 0 1.85625
1.5 −1.33766 0 −0.210663 0.660635 0 1.00000 2.95712 0 −0.883705
1.6 −0.665348 0 −1.55731 3.70415 0 1.00000 2.36685 0 −2.46455
1.7 −0.432278 0 −1.81314 4.10000 0 1.00000 1.64833 0 −1.77234
1.8 −0.375443 0 −1.85904 −2.49678 0 1.00000 1.44885 0 0.937397
1.9 0.0366427 0 −1.99866 −1.02187 0 1.00000 −0.146522 0 −0.0374440
1.10 0.803961 0 −1.35365 −0.343335 0 1.00000 −2.69620 0 −0.276028
1.11 0.981962 0 −1.03575 −2.59441 0 1.00000 −2.98099 0 −2.54761
1.12 1.52505 0 0.325779 1.71268 0 1.00000 −2.55327 0 2.61192
1.13 1.60620 0 0.579871 3.54808 0 1.00000 −2.28101 0 5.69892
1.14 1.77085 0 1.13590 −1.38376 0 1.00000 −1.53019 0 −2.45042
1.15 2.40246 0 3.77179 0.666645 0 1.00000 4.25666 0 1.60159
1.16 2.59597 0 4.73905 1.63234 0 1.00000 7.11048 0 4.23750
1.17 2.73355 0 5.47227 −4.25416 0 1.00000 9.49162 0 −11.6289
1.18 2.77035 0 5.67483 3.69380 0 1.00000 10.1806 0 10.2331
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
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.u 18
3.b odd 2 1 2667.2.a.p 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.p 18 3.b odd 2 1
8001.2.a.u 18 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{18} - \cdots\)
\(T_{5}^{18} - \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 6 T + 25 T^{2} - 81 T^{3} + 225 T^{4} - 556 T^{5} + 1256 T^{6} - 2640 T^{7} + 5231 T^{8} - 9842 T^{9} + 17724 T^{10} - 30680 T^{11} + 51321 T^{12} - 83210 T^{13} + 131171 T^{14} - 201431 T^{15} + 301850 T^{16} - 441694 T^{17} + 631856 T^{18} - 883388 T^{19} + 1207400 T^{20} - 1611448 T^{21} + 2098736 T^{22} - 2662720 T^{23} + 3284544 T^{24} - 3927040 T^{25} + 4537344 T^{26} - 5039104 T^{27} + 5356544 T^{28} - 5406720 T^{29} + 5144576 T^{30} - 4554752 T^{31} + 3686400 T^{32} - 2654208 T^{33} + 1638400 T^{34} - 786432 T^{35} + 262144 T^{36} \)
$3$ \( \)
$5$ \( 1 - 10 T + 81 T^{2} - 476 T^{3} + 2454 T^{4} - 10782 T^{5} + 43074 T^{6} - 154195 T^{7} + 511259 T^{8} - 1553151 T^{9} + 4417176 T^{10} - 11648601 T^{11} + 29006707 T^{12} - 67663733 T^{13} + 151179610 T^{14} - 322803966 T^{15} + 682949075 T^{16} - 1441859622 T^{17} + 3174135542 T^{18} - 7209298110 T^{19} + 17073726875 T^{20} - 40350495750 T^{21} + 94487256250 T^{22} - 211449165625 T^{23} + 453229796875 T^{24} - 910046953125 T^{25} + 1725459375000 T^{26} - 3033498046875 T^{27} + 4992763671875 T^{28} - 7529052734375 T^{29} + 10516113281250 T^{30} - 13161621093750 T^{31} + 14978027343750 T^{32} - 14526367187500 T^{33} + 12359619140625 T^{34} - 7629394531250 T^{35} + 3814697265625 T^{36} \)
$7$ \( ( 1 - T )^{18} \)
$11$ \( 1 - 9 T + 123 T^{2} - 853 T^{3} + 7075 T^{4} - 40936 T^{5} + 262862 T^{6} - 1328997 T^{7} + 7225154 T^{8} - 32841991 T^{9} + 157994657 T^{10} - 657379783 T^{11} + 2869404622 T^{12} - 11054117909 T^{13} + 44448793774 T^{14} - 159662650340 T^{15} + 596925478628 T^{16} - 2006578164628 T^{17} + 7011320772848 T^{18} - 22072359810908 T^{19} + 72227982913988 T^{20} - 212510987602540 T^{21} + 650774789645134 T^{22} - 1780276743362359 T^{23} + 5083325321554942 T^{24} - 12810472243263893 T^{25} + 33867557878498817 T^{26} - 77439696846292781 T^{27} + 187401887105613554 T^{28} - 379178354307007167 T^{29} + 824973559961635502 T^{30} - 1413221744323959416 T^{31} + 2686730072601430075 T^{32} - 3563192688511550303 T^{33} + 5651816773219375803 T^{34} - 4549023256493643939 T^{35} + 5559917313492231481 T^{36} \)
$13$ \( 1 + 25 T + 385 T^{2} + 4370 T^{3} + 40821 T^{4} + 328011 T^{5} + 2348286 T^{6} + 15262673 T^{7} + 91519666 T^{8} + 511265569 T^{9} + 2684450093 T^{10} + 13324020516 T^{11} + 62851127050 T^{12} + 282769818174 T^{13} + 1217464085422 T^{14} + 5026930729070 T^{15} + 19945274556838 T^{16} + 76121573741284 T^{17} + 279708125871724 T^{18} + 989580458636692 T^{19} + 3370751400105622 T^{20} + 11044166811766790 T^{21} + 34771991743737742 T^{22} + 104990454099278982 T^{23} + 303370385705083450 T^{24} + 836062527856574772 T^{25} + 2189788409851407053 T^{26} + 5421715405896988237 T^{27} + 12616763129284202434 T^{28} + 27353158057737880901 T^{29} + 54710567119930417566 T^{30} + 99346366588431498783 T^{31} + \)\(16\!\cdots\!69\)\( T^{32} + \)\(22\!\cdots\!90\)\( T^{33} + \)\(25\!\cdots\!85\)\( T^{34} + \)\(21\!\cdots\!25\)\( T^{35} + \)\(11\!\cdots\!29\)\( T^{36} \)
$17$ \( 1 - 17 T + 271 T^{2} - 2811 T^{3} + 27324 T^{4} - 215198 T^{5} + 1613920 T^{6} - 10643807 T^{7} + 67598476 T^{8} - 393720197 T^{9} + 2219851055 T^{10} - 11763913489 T^{11} + 60437526881 T^{12} - 296473981529 T^{13} + 1409837991942 T^{14} - 6467854453086 T^{15} + 28753111843606 T^{16} - 124072891149114 T^{17} + 518698939431416 T^{18} - 2109239149534938 T^{19} + 8309649322802134 T^{20} - 31776568928011518 T^{21} + 117751078924987782 T^{22} - 420950657991821353 T^{23} + 1458814975279492289 T^{24} - 4827188650363060097 T^{25} + 15485142514827950255 T^{26} - 46690442096210509909 T^{27} + \)\(13\!\cdots\!24\)\( T^{28} - \)\(36\!\cdots\!31\)\( T^{29} + \)\(94\!\cdots\!20\)\( T^{30} - \)\(21\!\cdots\!26\)\( T^{31} + \)\(46\!\cdots\!96\)\( T^{32} - \)\(80\!\cdots\!23\)\( T^{33} + \)\(13\!\cdots\!51\)\( T^{34} - \)\(14\!\cdots\!09\)\( T^{35} + \)\(14\!\cdots\!09\)\( T^{36} \)
$19$ \( 1 + 5 T + 147 T^{2} + 706 T^{3} + 11220 T^{4} + 54388 T^{5} + 598059 T^{6} + 2947928 T^{7} + 24930876 T^{8} + 123855871 T^{9} + 863216513 T^{10} + 4247612067 T^{11} + 25659749354 T^{12} + 122670371696 T^{13} + 666353734551 T^{14} + 3046885890738 T^{15} + 15241373221617 T^{16} + 65964543118195 T^{17} + 307973464968892 T^{18} + 1253326319245705 T^{19} + 5502135733003737 T^{20} + 20898590324571942 T^{21} + 86839885040420871 T^{22} + 303743984686093904 T^{23} + 1207185514598110874 T^{24} + 3796820384926674513 T^{25} + 14660492066567696033 T^{26} + 39966765869402810509 T^{27} + \)\(15\!\cdots\!76\)\( T^{28} + \)\(34\!\cdots\!32\)\( T^{29} + \)\(13\!\cdots\!99\)\( T^{30} + \)\(22\!\cdots\!92\)\( T^{31} + \)\(89\!\cdots\!20\)\( T^{32} + \)\(10\!\cdots\!94\)\( T^{33} + \)\(42\!\cdots\!07\)\( T^{34} + \)\(27\!\cdots\!95\)\( T^{35} + \)\(10\!\cdots\!41\)\( T^{36} \)
$23$ \( 1 - 14 T + 241 T^{2} - 2165 T^{3} + 22937 T^{4} - 159770 T^{5} + 1348211 T^{6} - 7982781 T^{7} + 59782924 T^{8} - 318639178 T^{9} + 2236451198 T^{10} - 11093814756 T^{11} + 74232434192 T^{12} - 346225007891 T^{13} + 2204529109109 T^{14} - 9680284070456 T^{15} + 58671559445850 T^{16} - 244022213085305 T^{17} + 1413668128204946 T^{18} - 5612510900962015 T^{19} + 31037254946854650 T^{20} - 117780016285238152 T^{21} + 616917630422171669 T^{22} - 2228422905964182613 T^{23} + 10989064388246716688 T^{24} - 37772502785532895932 T^{25} + \)\(17\!\cdots\!38\)\( T^{26} - \)\(57\!\cdots\!14\)\( T^{27} + \)\(24\!\cdots\!76\)\( T^{28} - \)\(76\!\cdots\!87\)\( T^{29} + \)\(29\!\cdots\!31\)\( T^{30} - \)\(80\!\cdots\!10\)\( T^{31} + \)\(26\!\cdots\!33\)\( T^{32} - \)\(57\!\cdots\!55\)\( T^{33} + \)\(14\!\cdots\!01\)\( T^{34} - \)\(19\!\cdots\!42\)\( T^{35} + \)\(32\!\cdots\!69\)\( T^{36} \)
$29$ \( 1 - 17 T + 417 T^{2} - 5334 T^{3} + 78106 T^{4} - 813038 T^{5} + 9035294 T^{6} - 80040114 T^{7} + 737367708 T^{8} - 5722271524 T^{9} + 45743439048 T^{10} - 317607890130 T^{11} + 2268138488862 T^{12} - 14333282840010 T^{13} + 93413372409634 T^{14} - 545192523955726 T^{15} + 3295455203397803 T^{16} - 17965463101032459 T^{17} + 101749592417859630 T^{18} - 520998429929941311 T^{19} + 2771477826057552323 T^{20} - 13296700466756201414 T^{21} + 66069503451258345154 T^{22} - \)\(29\!\cdots\!90\)\( T^{23} + \)\(13\!\cdots\!02\)\( T^{24} - \)\(54\!\cdots\!70\)\( T^{25} + \)\(22\!\cdots\!28\)\( T^{26} - \)\(83\!\cdots\!56\)\( T^{27} + \)\(31\!\cdots\!08\)\( T^{28} - \)\(97\!\cdots\!06\)\( T^{29} + \)\(31\!\cdots\!54\)\( T^{30} - \)\(83\!\cdots\!82\)\( T^{31} + \)\(23\!\cdots\!86\)\( T^{32} - \)\(46\!\cdots\!66\)\( T^{33} + \)\(10\!\cdots\!57\)\( T^{34} - \)\(12\!\cdots\!53\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \)
$31$ \( 1 - 5 T + 345 T^{2} - 1457 T^{3} + 56911 T^{4} - 201348 T^{5} + 6039940 T^{6} - 17777850 T^{7} + 469396428 T^{8} - 1148134577 T^{9} + 28833727003 T^{10} - 59197956874 T^{11} + 1470588580318 T^{12} - 2592870701857 T^{13} + 64165914583760 T^{14} - 100158175269391 T^{15} + 2432313210501904 T^{16} - 3462335844380129 T^{17} + 80591864885992316 T^{18} - 107332411175783999 T^{19} + 2337452995292329744 T^{20} - 2983812199450427281 T^{21} + 59258569602308618960 T^{22} - 74231686846940033407 T^{23} + \)\(13\!\cdots\!58\)\( T^{24} - \)\(16\!\cdots\!14\)\( T^{25} + \)\(24\!\cdots\!23\)\( T^{26} - \)\(30\!\cdots\!67\)\( T^{27} + \)\(38\!\cdots\!28\)\( T^{28} - \)\(45\!\cdots\!50\)\( T^{29} + \)\(47\!\cdots\!40\)\( T^{30} - \)\(49\!\cdots\!68\)\( T^{31} + \)\(43\!\cdots\!31\)\( T^{32} - \)\(34\!\cdots\!07\)\( T^{33} + \)\(25\!\cdots\!45\)\( T^{34} - \)\(11\!\cdots\!55\)\( T^{35} + \)\(69\!\cdots\!41\)\( T^{36} \)
$37$ \( 1 + 15 T + 366 T^{2} + 4166 T^{3} + 63112 T^{4} + 600801 T^{5} + 7118236 T^{6} + 59182921 T^{7} + 595812423 T^{8} + 4428544822 T^{9} + 39581642519 T^{10} + 267508054476 T^{11} + 2185412894544 T^{12} + 13627736681152 T^{13} + 104028964880528 T^{14} + 607397885300872 T^{15} + 4406318113508128 T^{16} + 24433425678421227 T^{17} + 170043359575354066 T^{18} + 904036750101585399 T^{19} + 6032249497392627232 T^{20} + 30766525084145069416 T^{21} + \)\(19\!\cdots\!08\)\( T^{22} + \)\(94\!\cdots\!64\)\( T^{23} + \)\(56\!\cdots\!96\)\( T^{24} + \)\(25\!\cdots\!08\)\( T^{25} + \)\(13\!\cdots\!99\)\( T^{26} + \)\(57\!\cdots\!94\)\( T^{27} + \)\(28\!\cdots\!27\)\( T^{28} + \)\(10\!\cdots\!73\)\( T^{29} + \)\(46\!\cdots\!16\)\( T^{30} + \)\(14\!\cdots\!97\)\( T^{31} + \)\(56\!\cdots\!68\)\( T^{32} + \)\(13\!\cdots\!38\)\( T^{33} + \)\(45\!\cdots\!06\)\( T^{34} + \)\(68\!\cdots\!55\)\( T^{35} + \)\(16\!\cdots\!29\)\( T^{36} \)
$41$ \( 1 - 17 T + 587 T^{2} - 7983 T^{3} + 157437 T^{4} - 1804251 T^{5} + 26303394 T^{6} - 262317360 T^{7} + 3118583347 T^{8} - 27662680979 T^{9} + 282270497714 T^{10} - 2263125901586 T^{11} + 20458023064018 T^{12} - 150120482591273 T^{13} + 1228554873931248 T^{14} - 8330642142077100 T^{15} + 62666398844152637 T^{16} - 395131743262021859 T^{17} + 2758290334846420354 T^{18} - 16200401473742896219 T^{19} + \)\(10\!\cdots\!97\)\( T^{20} - \)\(57\!\cdots\!00\)\( T^{21} + \)\(34\!\cdots\!28\)\( T^{22} - \)\(17\!\cdots\!73\)\( T^{23} + \)\(97\!\cdots\!38\)\( T^{24} - \)\(44\!\cdots\!66\)\( T^{25} + \)\(22\!\cdots\!94\)\( T^{26} - \)\(90\!\cdots\!19\)\( T^{27} + \)\(41\!\cdots\!47\)\( T^{28} - \)\(14\!\cdots\!60\)\( T^{29} + \)\(59\!\cdots\!14\)\( T^{30} - \)\(16\!\cdots\!71\)\( T^{31} + \)\(59\!\cdots\!57\)\( T^{32} - \)\(12\!\cdots\!83\)\( T^{33} + \)\(37\!\cdots\!67\)\( T^{34} - \)\(44\!\cdots\!77\)\( T^{35} + \)\(10\!\cdots\!21\)\( T^{36} \)
$43$ \( 1 - T + 388 T^{2} - 878 T^{3} + 73702 T^{4} - 275013 T^{5} + 9240295 T^{6} - 48874735 T^{7} + 876579751 T^{8} - 5864621513 T^{9} + 68610867706 T^{10} - 521103359185 T^{11} + 4672717098109 T^{12} - 36435312822151 T^{13} + 282210282216185 T^{14} - 2100029747349757 T^{15} + 15017594037839229 T^{16} - 103445385887703081 T^{17} + 694702110628981988 T^{18} - 4448151593171232483 T^{19} + 27767531375964734421 T^{20} - \)\(16\!\cdots\!99\)\( T^{21} + \)\(96\!\cdots\!85\)\( T^{22} - \)\(53\!\cdots\!93\)\( T^{23} + \)\(29\!\cdots\!41\)\( T^{24} - \)\(14\!\cdots\!95\)\( T^{25} + \)\(80\!\cdots\!06\)\( T^{26} - \)\(29\!\cdots\!59\)\( T^{27} + \)\(18\!\cdots\!99\)\( T^{28} - \)\(45\!\cdots\!45\)\( T^{29} + \)\(36\!\cdots\!95\)\( T^{30} - \)\(47\!\cdots\!59\)\( T^{31} + \)\(54\!\cdots\!98\)\( T^{32} - \)\(27\!\cdots\!46\)\( T^{33} + \)\(53\!\cdots\!88\)\( T^{34} - \)\(58\!\cdots\!43\)\( T^{35} + \)\(25\!\cdots\!49\)\( T^{36} \)
$47$ \( 1 - 31 T + 753 T^{2} - 13091 T^{3} + 197274 T^{4} - 2525137 T^{5} + 29473827 T^{6} - 311740727 T^{7} + 3111974218 T^{8} - 29220169105 T^{9} + 264679717261 T^{10} - 2297383543981 T^{11} + 19369688968200 T^{12} - 157133862498639 T^{13} + 1237282175874877 T^{14} - 9385007085387867 T^{15} + 69206387033259215 T^{16} - 493641831656169754 T^{17} + 3436277445562061196 T^{18} - 23201166087839978438 T^{19} + \)\(15\!\cdots\!35\)\( T^{20} - \)\(97\!\cdots\!41\)\( T^{21} + \)\(60\!\cdots\!37\)\( T^{22} - \)\(36\!\cdots\!73\)\( T^{23} + \)\(20\!\cdots\!00\)\( T^{24} - \)\(11\!\cdots\!03\)\( T^{25} + \)\(63\!\cdots\!21\)\( T^{26} - \)\(32\!\cdots\!35\)\( T^{27} + \)\(16\!\cdots\!82\)\( T^{28} - \)\(77\!\cdots\!81\)\( T^{29} + \)\(34\!\cdots\!07\)\( T^{30} - \)\(13\!\cdots\!99\)\( T^{31} + \)\(50\!\cdots\!06\)\( T^{32} - \)\(15\!\cdots\!13\)\( T^{33} + \)\(42\!\cdots\!13\)\( T^{34} - \)\(82\!\cdots\!97\)\( T^{35} + \)\(12\!\cdots\!89\)\( T^{36} \)
$53$ \( 1 - 35 T + 1002 T^{2} - 20303 T^{3} + 359179 T^{4} - 5356293 T^{5} + 72450797 T^{6} - 876498168 T^{7} + 9860866190 T^{8} - 102370151648 T^{9} + 1007118215698 T^{10} - 9336961305420 T^{11} + 83269959064210 T^{12} - 710236502930910 T^{13} + 5888264586633139 T^{14} - 47091839467003009 T^{15} + 367936393518839916 T^{16} - 2782388009045035942 T^{17} + 20583240410282150440 T^{18} - \)\(14\!\cdots\!26\)\( T^{19} + \)\(10\!\cdots\!44\)\( T^{20} - \)\(70\!\cdots\!93\)\( T^{21} + \)\(46\!\cdots\!59\)\( T^{22} - \)\(29\!\cdots\!30\)\( T^{23} + \)\(18\!\cdots\!90\)\( T^{24} - \)\(10\!\cdots\!40\)\( T^{25} + \)\(62\!\cdots\!78\)\( T^{26} - \)\(33\!\cdots\!84\)\( T^{27} + \)\(17\!\cdots\!10\)\( T^{28} - \)\(81\!\cdots\!96\)\( T^{29} + \)\(35\!\cdots\!77\)\( T^{30} - \)\(13\!\cdots\!89\)\( T^{31} + \)\(49\!\cdots\!51\)\( T^{32} - \)\(14\!\cdots\!71\)\( T^{33} + \)\(38\!\cdots\!42\)\( T^{34} - \)\(71\!\cdots\!55\)\( T^{35} + \)\(10\!\cdots\!89\)\( T^{36} \)
$59$ \( 1 - 46 T + 1709 T^{2} - 44939 T^{3} + 1028319 T^{4} - 19836014 T^{5} + 345643777 T^{6} - 5379928797 T^{7} + 77278229780 T^{8} - 1019115860932 T^{9} + 12568554567858 T^{10} - 144505310405158 T^{11} + 1567113639682192 T^{12} - 15990154268072311 T^{13} + 154748085150649455 T^{14} - 1416913606523974460 T^{15} + 12346169576116035044 T^{16} - \)\(10\!\cdots\!35\)\( T^{17} + \)\(80\!\cdots\!90\)\( T^{18} - \)\(60\!\cdots\!65\)\( T^{19} + \)\(42\!\cdots\!64\)\( T^{20} - \)\(29\!\cdots\!40\)\( T^{21} + \)\(18\!\cdots\!55\)\( T^{22} - \)\(11\!\cdots\!89\)\( T^{23} + \)\(66\!\cdots\!72\)\( T^{24} - \)\(35\!\cdots\!02\)\( T^{25} + \)\(18\!\cdots\!18\)\( T^{26} - \)\(88\!\cdots\!48\)\( T^{27} + \)\(39\!\cdots\!80\)\( T^{28} - \)\(16\!\cdots\!23\)\( T^{29} + \)\(61\!\cdots\!37\)\( T^{30} - \)\(20\!\cdots\!06\)\( T^{31} + \)\(63\!\cdots\!59\)\( T^{32} - \)\(16\!\cdots\!61\)\( T^{33} + \)\(36\!\cdots\!69\)\( T^{34} - \)\(58\!\cdots\!74\)\( T^{35} + \)\(75\!\cdots\!21\)\( T^{36} \)
$61$ \( 1 + 5 T + 313 T^{2} + 823 T^{3} + 52165 T^{4} + 17522 T^{5} + 6076918 T^{6} - 11076242 T^{7} + 573805076 T^{8} - 2160722153 T^{9} + 47466893999 T^{10} - 243623260628 T^{11} + 3639796890962 T^{12} - 20608000137639 T^{13} + 266055733023234 T^{14} - 1442983180432481 T^{15} + 18359172200402956 T^{16} - 91434935405685095 T^{17} + 1172250317548098592 T^{18} - 5577531059746790795 T^{19} + 68314479757699399276 T^{20} - \)\(32\!\cdots\!61\)\( T^{21} + \)\(36\!\cdots\!94\)\( T^{22} - \)\(17\!\cdots\!39\)\( T^{23} + \)\(18\!\cdots\!82\)\( T^{24} - \)\(76\!\cdots\!88\)\( T^{25} + \)\(90\!\cdots\!19\)\( T^{26} - \)\(25\!\cdots\!73\)\( T^{27} + \)\(40\!\cdots\!76\)\( T^{28} - \)\(48\!\cdots\!62\)\( T^{29} + \)\(16\!\cdots\!78\)\( T^{30} + \)\(28\!\cdots\!82\)\( T^{31} + \)\(51\!\cdots\!65\)\( T^{32} + \)\(49\!\cdots\!23\)\( T^{33} + \)\(11\!\cdots\!93\)\( T^{34} + \)\(11\!\cdots\!05\)\( T^{35} + \)\(13\!\cdots\!81\)\( T^{36} \)
$67$ \( 1 - 6 T + 660 T^{2} - 2744 T^{3} + 206847 T^{4} - 546815 T^{5} + 41865837 T^{6} - 58775543 T^{7} + 6267210834 T^{8} - 2850792910 T^{9} + 751996954474 T^{10} + 126073507818 T^{11} + 76266450675238 T^{12} + 35752074266697 T^{13} + 6762717540256435 T^{14} + 3616840565032217 T^{15} + 533791607456110448 T^{16} + 269073439163604974 T^{17} + 37755976465866968420 T^{18} + 18027920423961533258 T^{19} + \)\(23\!\cdots\!72\)\( T^{20} + \)\(10\!\cdots\!71\)\( T^{21} + \)\(13\!\cdots\!35\)\( T^{22} + \)\(48\!\cdots\!79\)\( T^{23} + \)\(68\!\cdots\!22\)\( T^{24} + \)\(76\!\cdots\!14\)\( T^{25} + \)\(30\!\cdots\!34\)\( T^{26} - \)\(77\!\cdots\!70\)\( T^{27} + \)\(11\!\cdots\!66\)\( T^{28} - \)\(71\!\cdots\!69\)\( T^{29} + \)\(34\!\cdots\!57\)\( T^{30} - \)\(29\!\cdots\!05\)\( T^{31} + \)\(75\!\cdots\!63\)\( T^{32} - \)\(67\!\cdots\!92\)\( T^{33} + \)\(10\!\cdots\!60\)\( T^{34} - \)\(66\!\cdots\!62\)\( T^{35} + \)\(74\!\cdots\!09\)\( T^{36} \)
$71$ \( 1 - 22 T + 816 T^{2} - 13317 T^{3} + 296343 T^{4} - 3949811 T^{5} + 67461988 T^{6} - 771642797 T^{7} + 11118168119 T^{8} - 112487037082 T^{9} + 1438298463717 T^{10} - 13144317642374 T^{11} + 154065969757915 T^{12} - 1293495497977143 T^{13} + 14217749002563904 T^{14} - 111372864315380429 T^{15} + 1165348206806512150 T^{16} - 8636949642215036499 T^{17} + 86585290392974234382 T^{18} - \)\(61\!\cdots\!29\)\( T^{19} + \)\(58\!\cdots\!50\)\( T^{20} - \)\(39\!\cdots\!19\)\( T^{21} + \)\(36\!\cdots\!24\)\( T^{22} - \)\(23\!\cdots\!93\)\( T^{23} + \)\(19\!\cdots\!15\)\( T^{24} - \)\(11\!\cdots\!34\)\( T^{25} + \)\(92\!\cdots\!37\)\( T^{26} - \)\(51\!\cdots\!42\)\( T^{27} + \)\(36\!\cdots\!19\)\( T^{28} - \)\(17\!\cdots\!87\)\( T^{29} + \)\(11\!\cdots\!08\)\( T^{30} - \)\(46\!\cdots\!21\)\( T^{31} + \)\(24\!\cdots\!83\)\( T^{32} - \)\(78\!\cdots\!67\)\( T^{33} + \)\(34\!\cdots\!36\)\( T^{34} - \)\(65\!\cdots\!02\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \)
$73$ \( 1 + 16 T + 774 T^{2} + 11991 T^{3} + 310425 T^{4} + 4494649 T^{5} + 84044592 T^{6} + 1122180182 T^{7} + 17029645739 T^{8} + 209114736738 T^{9} + 2728250649635 T^{10} + 30858797839849 T^{11} + 357339169674251 T^{12} + 3732467329882239 T^{13} + 39092259481718292 T^{14} + 377781559757387694 T^{15} + 3620052404771985744 T^{16} + 32383355156256234226 T^{17} + \)\(28\!\cdots\!94\)\( T^{18} + \)\(23\!\cdots\!98\)\( T^{19} + \)\(19\!\cdots\!76\)\( T^{20} + \)\(14\!\cdots\!98\)\( T^{21} + \)\(11\!\cdots\!72\)\( T^{22} + \)\(77\!\cdots\!27\)\( T^{23} + \)\(54\!\cdots\!39\)\( T^{24} + \)\(34\!\cdots\!53\)\( T^{25} + \)\(22\!\cdots\!35\)\( T^{26} + \)\(12\!\cdots\!94\)\( T^{27} + \)\(73\!\cdots\!11\)\( T^{28} + \)\(35\!\cdots\!14\)\( T^{29} + \)\(19\!\cdots\!32\)\( T^{30} + \)\(75\!\cdots\!17\)\( T^{31} + \)\(37\!\cdots\!25\)\( T^{32} + \)\(10\!\cdots\!87\)\( T^{33} + \)\(50\!\cdots\!14\)\( T^{34} + \)\(75\!\cdots\!48\)\( T^{35} + \)\(34\!\cdots\!69\)\( T^{36} \)
$79$ \( 1 - 46 T + 1652 T^{2} - 41934 T^{3} + 913375 T^{4} - 16648169 T^{5} + 272067184 T^{6} - 3944716283 T^{7} + 52754177710 T^{8} - 647408799368 T^{9} + 7513495891584 T^{10} - 82278295480761 T^{11} + 872957218286027 T^{12} - 8937344998796532 T^{13} + 90009306584051938 T^{14} - 881338931700124702 T^{15} + 8477537143157484023 T^{16} - 78804698976013235469 T^{17} + \)\(71\!\cdots\!12\)\( T^{18} - \)\(62\!\cdots\!51\)\( T^{19} + \)\(52\!\cdots\!43\)\( T^{20} - \)\(43\!\cdots\!78\)\( T^{21} + \)\(35\!\cdots\!78\)\( T^{22} - \)\(27\!\cdots\!68\)\( T^{23} + \)\(21\!\cdots\!67\)\( T^{24} - \)\(15\!\cdots\!99\)\( T^{25} + \)\(11\!\cdots\!24\)\( T^{26} - \)\(77\!\cdots\!92\)\( T^{27} + \)\(49\!\cdots\!10\)\( T^{28} - \)\(29\!\cdots\!57\)\( T^{29} + \)\(16\!\cdots\!44\)\( T^{30} - \)\(77\!\cdots\!91\)\( T^{31} + \)\(33\!\cdots\!75\)\( T^{32} - \)\(12\!\cdots\!66\)\( T^{33} + \)\(38\!\cdots\!92\)\( T^{34} - \)\(83\!\cdots\!14\)\( T^{35} + \)\(14\!\cdots\!61\)\( T^{36} \)
$83$ \( 1 - 46 T + 1803 T^{2} - 49450 T^{3} + 1213637 T^{4} - 25057794 T^{5} + 476001267 T^{6} - 8084141739 T^{7} + 128449477860 T^{8} - 1877750632737 T^{9} + 25963046510186 T^{10} - 335627644220575 T^{11} + 4134348251646044 T^{12} - 48084643546611681 T^{13} + 535631470942771309 T^{14} - 5667781749956354908 T^{15} + 57627441353791009762 T^{16} - \)\(55\!\cdots\!66\)\( T^{17} + \)\(52\!\cdots\!58\)\( T^{18} - \)\(46\!\cdots\!78\)\( T^{19} + \)\(39\!\cdots\!18\)\( T^{20} - \)\(32\!\cdots\!96\)\( T^{21} + \)\(25\!\cdots\!89\)\( T^{22} - \)\(18\!\cdots\!83\)\( T^{23} + \)\(13\!\cdots\!36\)\( T^{24} - \)\(91\!\cdots\!25\)\( T^{25} + \)\(58\!\cdots\!26\)\( T^{26} - \)\(35\!\cdots\!11\)\( T^{27} + \)\(19\!\cdots\!40\)\( T^{28} - \)\(10\!\cdots\!13\)\( T^{29} + \)\(50\!\cdots\!87\)\( T^{30} - \)\(22\!\cdots\!22\)\( T^{31} + \)\(89\!\cdots\!73\)\( T^{32} - \)\(30\!\cdots\!50\)\( T^{33} + \)\(91\!\cdots\!43\)\( T^{34} - \)\(19\!\cdots\!58\)\( T^{35} + \)\(34\!\cdots\!09\)\( T^{36} \)
$89$ \( 1 - 42 T + 1798 T^{2} - 48793 T^{3} + 1279302 T^{4} - 26760790 T^{5} + 539059369 T^{6} - 9388573567 T^{7} + 157864657375 T^{8} - 2384001838238 T^{9} + 34852410451996 T^{10} - 467195821431568 T^{11} + 6075757390898767 T^{12} - 73334134581530775 T^{13} + 860024600762049203 T^{14} - 9427780194222755892 T^{15} + \)\(10\!\cdots\!59\)\( T^{16} - \)\(10\!\cdots\!07\)\( T^{17} + \)\(97\!\cdots\!68\)\( T^{18} - \)\(89\!\cdots\!23\)\( T^{19} + \)\(79\!\cdots\!39\)\( T^{20} - \)\(66\!\cdots\!48\)\( T^{21} + \)\(53\!\cdots\!23\)\( T^{22} - \)\(40\!\cdots\!75\)\( T^{23} + \)\(30\!\cdots\!87\)\( T^{24} - \)\(20\!\cdots\!72\)\( T^{25} + \)\(13\!\cdots\!76\)\( T^{26} - \)\(83\!\cdots\!42\)\( T^{27} + \)\(49\!\cdots\!75\)\( T^{28} - \)\(26\!\cdots\!63\)\( T^{29} + \)\(13\!\cdots\!49\)\( T^{30} - \)\(58\!\cdots\!10\)\( T^{31} + \)\(25\!\cdots\!82\)\( T^{32} - \)\(84\!\cdots\!57\)\( T^{33} + \)\(27\!\cdots\!78\)\( T^{34} - \)\(57\!\cdots\!18\)\( T^{35} + \)\(12\!\cdots\!81\)\( T^{36} \)
$97$ \( 1 + 35 T + 1444 T^{2} + 34718 T^{3} + 871617 T^{4} + 16689897 T^{5} + 324272940 T^{6} + 5304501825 T^{7} + 87089442417 T^{8} + 1261878995611 T^{9} + 18253738121908 T^{10} + 239093914197707 T^{11} + 3119110855161003 T^{12} + 37405451259827487 T^{13} + 446307927225846884 T^{14} + 4938726924768483995 T^{15} + 54347074761442941730 T^{16} + \)\(55\!\cdots\!57\)\( T^{17} + \)\(56\!\cdots\!04\)\( T^{18} + \)\(54\!\cdots\!29\)\( T^{19} + \)\(51\!\cdots\!70\)\( T^{20} + \)\(45\!\cdots\!35\)\( T^{21} + \)\(39\!\cdots\!04\)\( T^{22} + \)\(32\!\cdots\!59\)\( T^{23} + \)\(25\!\cdots\!87\)\( T^{24} + \)\(19\!\cdots\!91\)\( T^{25} + \)\(14\!\cdots\!88\)\( T^{26} + \)\(95\!\cdots\!87\)\( T^{27} + \)\(64\!\cdots\!33\)\( T^{28} + \)\(37\!\cdots\!25\)\( T^{29} + \)\(22\!\cdots\!40\)\( T^{30} + \)\(11\!\cdots\!69\)\( T^{31} + \)\(56\!\cdots\!73\)\( T^{32} + \)\(21\!\cdots\!74\)\( T^{33} + \)\(88\!\cdots\!24\)\( T^{34} + \)\(20\!\cdots\!95\)\( T^{35} + \)\(57\!\cdots\!89\)\( T^{36} \)
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