Properties

Label 539.2.q.b
Level 539
Weight 2
Character orbit 539.q
Analytic conductor 4.304
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 539.q (of order \(15\), degree \(8\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{15})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{2} + \beta_{4} + \beta_{11} - \beta_{12} ) q^{2} + ( \beta_{5} - \beta_{7} ) q^{3} + ( \beta_{3} - 2 \beta_{5} + \beta_{7} + \beta_{14} ) q^{4} + ( 2 - 2 \beta_{2} + \beta_{3} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{10} - \beta_{13} - \beta_{15} ) q^{5} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{6} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{8} - 3 \beta_{10} ) q^{8} + ( 1 - 2 \beta_{2} - \beta_{5} + \beta_{6} + 2 \beta_{12} + \beta_{14} ) q^{9} +O(q^{10})\) \( q + ( \beta_{2} + \beta_{4} + \beta_{11} - \beta_{12} ) q^{2} + ( \beta_{5} - \beta_{7} ) q^{3} + ( \beta_{3} - 2 \beta_{5} + \beta_{7} + \beta_{14} ) q^{4} + ( 2 - 2 \beta_{2} + \beta_{3} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{10} - \beta_{13} - \beta_{15} ) q^{5} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{6} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{8} - 3 \beta_{10} ) q^{8} + ( 1 - 2 \beta_{2} - \beta_{5} + \beta_{6} + 2 \beta_{12} + \beta_{14} ) q^{9} + ( -\beta_{7} + 2 \beta_{11} - \beta_{12} + 4 \beta_{14} - 2 \beta_{15} ) q^{10} + ( -\beta_{3} + 2 \beta_{5} - \beta_{7} + 2 \beta_{11} - \beta_{12} ) q^{11} + ( -3 - \beta_{3} - \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{12} + ( -\beta_{2} - \beta_{4} + 2 \beta_{8} ) q^{13} + ( 2 \beta_{2} + \beta_{4} - \beta_{9} + 2 \beta_{10} ) q^{15} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{8} + 2 \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{15} ) q^{16} + ( 2 \beta_{5} + \beta_{9} - \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{15} ) q^{17} + ( -2 \beta_{3} + \beta_{5} + \beta_{7} + \beta_{9} + \beta_{11} + \beta_{13} + \beta_{14} ) q^{18} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{19} + ( -3 - 4 \beta_{2} - 3 \beta_{4} - 3 \beta_{6} - \beta_{8} ) q^{20} + ( -2 - \beta_{2} - \beta_{4} - 5 \beta_{6} - \beta_{8} - 2 \beta_{9} + 3 \beta_{10} ) q^{22} + ( 3 + 2 \beta_{1} + 3 \beta_{6} - 3 \beta_{7} - 3 \beta_{10} - 3 \beta_{12} - 2 \beta_{13} ) q^{23} + ( 3 \beta_{5} - \beta_{11} + 2 \beta_{12} - 3 \beta_{14} ) q^{24} + ( 2 \beta_{5} - 2 \beta_{7} - 3 \beta_{9} - \beta_{12} - 3 \beta_{13} - \beta_{14} ) q^{25} + ( -\beta_{1} + 3 \beta_{2} + 2 \beta_{5} - 5 \beta_{6} + 3 \beta_{7} + 3 \beta_{10} + 3 \beta_{14} ) q^{26} + ( 3 + \beta_{2} - \beta_{6} - 3 \beta_{10} ) q^{27} + ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{6} + 3 \beta_{9} - 3 \beta_{10} ) q^{29} + ( -5 + 2 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} - 6 \beta_{5} + 6 \beta_{7} - 2 \beta_{8} + 6 \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{30} + ( -\beta_{3} - 2 \beta_{5} + \beta_{11} + \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{31} + ( -3 \beta_{7} - 3 \beta_{12} - 4 \beta_{14} ) q^{32} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{33} + ( 4 - 2 \beta_{1} + 2 \beta_{3} + \beta_{6} - 2 \beta_{9} - \beta_{10} ) q^{34} + ( -\beta_{1} + \beta_{3} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{36} + ( -2 \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{11} + \beta_{13} ) q^{37} + ( \beta_{3} - 6 \beta_{5} + 8 \beta_{7} + \beta_{9} - 2 \beta_{11} + 6 \beta_{12} + \beta_{13} + \beta_{15} ) q^{38} + ( 2 \beta_{3} - \beta_{5} + \beta_{7} - \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{39} + ( -6 \beta_{2} - 3 \beta_{4} + 7 \beta_{5} - \beta_{6} - 6 \beta_{7} - 6 \beta_{10} - 3 \beta_{11} - 3 \beta_{13} - 6 \beta_{14} ) q^{40} + ( 5 - 2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + 2 \beta_{8} - \beta_{10} ) q^{41} + ( 5 + 6 \beta_{6} - 6 \beta_{10} ) q^{43} + ( -5 - \beta_{1} - \beta_{3} - 2 \beta_{4} + 6 \beta_{5} - 5 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{11} - \beta_{12} - 6 \beta_{14} + \beta_{15} ) q^{44} + ( \beta_{3} - 2 \beta_{7} - \beta_{9} - \beta_{11} - 2 \beta_{12} - \beta_{13} - 4 \beta_{14} + \beta_{15} ) q^{45} + ( 3 \beta_{3} - 7 \beta_{5} + 3 \beta_{11} + 2 \beta_{12} + 7 \beta_{14} - 3 \beta_{15} ) q^{46} + ( 2 - 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 4 \beta_{5} + 4 \beta_{7} + 4 \beta_{8} + 4 \beta_{10} + 6 \beta_{12} + \beta_{13} + 6 \beta_{14} - 4 \beta_{15} ) q^{47} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{9} ) q^{48} + ( -11 + \beta_{1} + 7 \beta_{2} - \beta_{3} - \beta_{4} - 7 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} + 11 \beta_{10} ) q^{50} + ( -\beta_{1} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{11} - \beta_{13} ) q^{51} + ( 3 \beta_{5} - 3 \beta_{7} - 5 \beta_{12} - 5 \beta_{14} - \beta_{15} ) q^{52} + ( -3 \beta_{3} + 2 \beta_{5} - 4 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} + 3 \beta_{15} ) q^{53} + ( 3 + \beta_{1} + 3 \beta_{3} + 3 \beta_{4} + 4 \beta_{6} - 4 \beta_{7} + 3 \beta_{8} - 4 \beta_{10} + 3 \beta_{11} - 4 \beta_{12} - 4 \beta_{13} - \beta_{14} - 3 \beta_{15} ) q^{54} + ( -7 + 3 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} - \beta_{6} - 3 \beta_{8} - \beta_{9} + 4 \beta_{10} ) q^{55} + ( -1 - 2 \beta_{2} - 3 \beta_{4} - \beta_{6} - 2 \beta_{8} ) q^{57} + ( 9 - 3 \beta_{1} - 9 \beta_{2} + 3 \beta_{3} + 3 \beta_{5} - 3 \beta_{7} + 3 \beta_{8} - 3 \beta_{10} + 6 \beta_{12} + 6 \beta_{14} - 3 \beta_{15} ) q^{58} + ( -\beta_{3} - 4 \beta_{5} - 3 \beta_{7} - \beta_{9} - \beta_{11} - \beta_{13} - 3 \beta_{14} ) q^{59} + ( -\beta_{3} - 7 \beta_{5} + 10 \beta_{7} + 4 \beta_{9} - 3 \beta_{11} + 7 \beta_{12} + 4 \beta_{13} + 4 \beta_{15} ) q^{60} + ( 2 + \beta_{1} - 3 \beta_{2} - 5 \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - 5 \beta_{8} - 2 \beta_{10} - \beta_{11} + \beta_{12} + 5 \beta_{13} + 5 \beta_{15} ) q^{61} + ( \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{9} - \beta_{10} ) q^{62} + ( 1 + 3 \beta_{4} + \beta_{6} + \beta_{8} ) q^{64} + ( -3 + 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - 3 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - 2 \beta_{13} - 6 \beta_{14} + 2 \beta_{15} ) q^{65} + ( \beta_{3} - 6 \beta_{5} + 8 \beta_{7} + 2 \beta_{9} + \beta_{11} + 6 \beta_{12} + 2 \beta_{13} + 3 \beta_{14} ) q^{66} + ( \beta_{3} + \beta_{7} - \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} - 4 \beta_{14} + \beta_{15} ) q^{67} + ( 1 + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} + 2 \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{68} + ( -3 + 3 \beta_{2} - 2 \beta_{9} + 6 \beta_{10} ) q^{69} + ( 1 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{71} + ( -1 + 2 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{72} + ( 3 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} - 5 \beta_{9} - 5 \beta_{11} - 5 \beta_{13} - 2 \beta_{14} ) q^{73} + ( \beta_{5} - \beta_{7} + \beta_{9} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{74} + ( 2 + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{8} + 3 \beta_{11} - \beta_{12} + 2 \beta_{14} - 3 \beta_{15} ) q^{75} + ( 11 + 4 \beta_{4} + 7 \beta_{6} + 4 \beta_{8} + 4 \beta_{9} - 7 \beta_{10} ) q^{76} + ( -1 + \beta_{4} + 2 \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{78} + ( 3 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{8} + \beta_{11} - 3 \beta_{12} + 2 \beta_{15} ) q^{79} + ( 5 \beta_{5} - 5 \beta_{7} - \beta_{9} + 4 \beta_{12} - \beta_{13} + 4 \beta_{14} - 4 \beta_{15} ) q^{80} + ( 4 \beta_{5} - 6 \beta_{7} - 6 \beta_{14} ) q^{81} + ( 3 - 5 \beta_{1} + 6 \beta_{2} + \beta_{3} + 5 \beta_{4} + 9 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} + \beta_{8} - 3 \beta_{10} + 5 \beta_{11} - 9 \beta_{12} - \beta_{13} - \beta_{15} ) q^{82} + ( -1 - 5 \beta_{1} - \beta_{2} + 5 \beta_{3} + 5 \beta_{4} + \beta_{6} + 3 \beta_{8} + 3 \beta_{9} + \beta_{10} ) q^{83} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{8} + \beta_{9} - 4 \beta_{10} ) q^{85} + ( 6 - \beta_{2} + 6 \beta_{3} + 5 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} + 6 \beta_{8} + 5 \beta_{11} + \beta_{12} + 6 \beta_{14} - 6 \beta_{15} ) q^{86} + ( -3 \beta_{11} - 3 \beta_{14} + 3 \beta_{15} ) q^{87} + ( \beta_{3} + 9 \beta_{5} - \beta_{7} - 4 \beta_{9} - \beta_{11} - 4 \beta_{13} - 3 \beta_{14} - 3 \beta_{15} ) q^{88} + ( -5 + 6 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} - 5 \beta_{13} - 7 \beta_{14} + \beta_{15} ) q^{89} + ( -3 + 7 \beta_{2} + 2 \beta_{4} - 3 \beta_{6} - 2 \beta_{8} ) q^{90} + ( -2 \beta_{1} - 10 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{6} + 3 \beta_{9} - 10 \beta_{10} ) q^{92} + ( -3 + \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 3 \beta_{10} - \beta_{11} + \beta_{12} ) q^{93} + ( -6 \beta_{3} + 9 \beta_{5} + 3 \beta_{7} + 4 \beta_{9} + 2 \beta_{11} - 9 \beta_{12} + 4 \beta_{13} + 4 \beta_{15} ) q^{94} + ( -9 \beta_{5} + 8 \beta_{7} + 3 \beta_{9} + 3 \beta_{11} + 3 \beta_{13} + 8 \beta_{14} ) q^{95} + ( 1 - \beta_{2} - 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{10} + 3 \beta_{12} + 3 \beta_{14} ) q^{96} + ( -7 + \beta_{2} - 7 \beta_{6} - 5 \beta_{8} ) q^{97} + ( -1 + 5 \beta_{1} + \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{8} + 3 \beta_{9} - 2 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + q^{2} - 4q^{3} - 3q^{4} + 3q^{5} - 6q^{6} + 6q^{8} + 2q^{9} + O(q^{10}) \) \( 16q + q^{2} - 4q^{3} - 3q^{4} + 3q^{5} - 6q^{6} + 6q^{8} + 2q^{9} - 28q^{10} - 5q^{11} - 14q^{12} - 10q^{13} + 12q^{15} + 3q^{16} - 11q^{17} - 4q^{18} - 9q^{19} - 42q^{20} - 2q^{22} + 16q^{23} + 21q^{24} - 5q^{25} + 21q^{26} + 44q^{27} - 18q^{29} - 14q^{30} - 11q^{31} + 20q^{32} + 10q^{33} + 48q^{34} - 4q^{36} - 6q^{37} + 35q^{38} + 5q^{39} - 16q^{40} + 44q^{41} + 32q^{43} - 29q^{44} + 18q^{45} - 29q^{46} + 7q^{47} - 8q^{48} - 68q^{50} - 3q^{51} + 21q^{52} - 2q^{53} + 4q^{54} - 52q^{55} - 6q^{57} + 39q^{58} + 25q^{59} + 38q^{60} + 7q^{61} + 10q^{62} + 2q^{64} - 24q^{65} + 18q^{66} + 30q^{67} + 8q^{68} - 16q^{69} - 28q^{71} - 3q^{72} + 3q^{73} + 9q^{74} + 5q^{75} + 104q^{76} - 36q^{78} + 9q^{79} - 33q^{80} + 28q^{81} + 31q^{82} - 46q^{83} - 20q^{85} + 17q^{86} + 12q^{87} + 7q^{88} - 34q^{89} - 4q^{90} - 68q^{92} - 8q^{93} - 30q^{94} - 24q^{95} + 10q^{96} - 60q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - x^{15} - 5 x^{14} + 16 x^{13} + 4 x^{12} - 29 x^{11} + 10 x^{10} - 156 x^{9} + 251 x^{8} + 240 x^{7} + 390 x^{6} - 1375 x^{5} - 300 x^{4} + 500 x^{3} + 375 x^{2} + 625 x + 625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(10747578565 \nu^{15} - 390646149564 \nu^{14} + 595438889614 \nu^{13} + 2023744375705 \nu^{12} - 6934783544984 \nu^{11} + 1387553663334 \nu^{10} + 13772300331741 \nu^{9} - 4306261565400 \nu^{8} + 62191980906094 \nu^{7} - 128695406473289 \nu^{6} - 30916376621980 \nu^{5} - 161148237103450 \nu^{4} + 716191995079275 \nu^{3} - 162385826946000 \nu^{2} - 105743499205250 \nu - 81680866488250\)\()/ 451236007807750 \)
\(\beta_{3}\)\(=\)\((\)\(-11294463 \nu^{15} - 216782207 \nu^{14} + 202974090 \nu^{13} + 547473112 \nu^{12} - 3522648197 \nu^{11} + 496882252 \nu^{10} + 1405593120 \nu^{9} - 7524269367 \nu^{8} + 37111437582 \nu^{7} - 45437262620 \nu^{6} - 564367165 \nu^{5} - 96741534250 \nu^{4} + 118432915400 \nu^{3} - 107615676875 \nu^{2} + 187325355750 \nu + 153447866875\)\()/ 262194077750 \)
\(\beta_{4}\)\(=\)\((\)\(4442741005 \nu^{15} - 50549404333 \nu^{14} + 57359873198 \nu^{13} + 277539827085 \nu^{12} - 847043836223 \nu^{11} + 137356700948 \nu^{10} + 1507562081227 \nu^{9} - 1350437679285 \nu^{8} + 8449126663418 \nu^{7} - 15280985144283 \nu^{6} - 1437348812935 \nu^{5} - 21982568599150 \nu^{4} + 114842922010965 \nu^{3} - 23716493712625 \nu^{2} - 16183969928000 \nu - 11984339635250\)\()/ 90247201561550 \)
\(\beta_{5}\)\(=\)\((\)\(-45506971144 \nu^{15} + 350661785078 \nu^{14} - 149323399494 \nu^{13} - 1669199906909 \nu^{12} + 4758645678158 \nu^{11} - 1497683300668 \nu^{10} - 2364013856961 \nu^{9} + 18065368943364 \nu^{8} - 76629971948588 \nu^{7} + 79980076478579 \nu^{6} - 33669067543990 \nu^{5} + 230037797816170 \nu^{4} - 228790206715425 \nu^{3} + 269200498832200 \nu^{2} - 381835489799000 \nu - 322941345212125\)\()/ 451236007807750 \)
\(\beta_{6}\)\(=\)\((\)\(-83632252305 \nu^{15} + 367332418783 \nu^{14} - 21593146433 \nu^{13} - 2333144026885 \nu^{12} + 5145355461023 \nu^{11} - 151223147723 \nu^{10} - 6612074340177 \nu^{9} + 25735467004125 \nu^{8} - 69457776599693 \nu^{7} + 89320024758233 \nu^{6} - 41006490690815 \nu^{5} + 182426884437775 \nu^{4} - 283504981985225 \nu^{3} + 226617505715125 \nu^{2} + 167804604156125 \nu - 334386818347500\)\()/ 451236007807750 \)
\(\beta_{7}\)\(=\)\((\)\(95874717082 \nu^{15} - 73661012057 \nu^{14} - 732120607075 \nu^{13} + 1820794839302 \nu^{12} + 1771198003753 \nu^{11} - 7015585976493 \nu^{10} + 1645530675560 \nu^{9} - 7418645458657 \nu^{8} + 17312365591157 \nu^{7} + 65255565416770 \nu^{6} - 39013786059435 \nu^{5} - 139014480052425 \nu^{4} - 138675258120350 \nu^{3} + 622151968595825 \nu^{2} - 82629449657375 \nu - 20998151463750\)\()/ 451236007807750 \)
\(\beta_{8}\)\(=\)\((\)\(-30198383055 \nu^{15} + 119405641263 \nu^{14} + 4417433767 \nu^{13} - 767290363435 \nu^{12} + 1643333019103 \nu^{11} - 24649968003 \nu^{10} - 1843743060247 \nu^{9} + 8826339491805 \nu^{8} - 22811269611173 \nu^{7} + 27509782939463 \nu^{6} - 14818453535215 \nu^{5} + 59956356442775 \nu^{4} - 139348684847645 \nu^{3} + 75235145215125 \nu^{2} + 55999400861125 \nu + 38634849837750\)\()/ 90247201561550 \)
\(\beta_{9}\)\(=\)\((\)\(-154162898659 \nu^{15} - 220169951101 \nu^{14} + 919001199770 \nu^{13} - 1104221745559 \nu^{12} - 5399279631671 \nu^{11} + 2032193985536 \nu^{10} - 1747566143665 \nu^{9} + 21076024732419 \nu^{8} + 23154598502726 \nu^{7} - 121627565899485 \nu^{6} - 93373444817095 \nu^{5} - 57339484155250 \nu^{4} + 404789439834025 \nu^{3} - 11405350455625 \nu^{2} + 14502483941500 \nu - 221616106153750\)\()/ 451236007807750 \)
\(\beta_{10}\)\(=\)\((\)\(169638367088 \nu^{15} + 295656442235 \nu^{14} - 1142785999873 \nu^{13} + 737207740433 \nu^{12} + 7017151182115 \nu^{11} - 2533300411795 \nu^{10} - 4606969356647 \nu^{9} - 24623976535863 \nu^{8} - 32955100495645 \nu^{7} + 137833352978403 \nu^{6} + 113501533227125 \nu^{5} + 82238742955775 \nu^{4} - 510650758411475 \nu^{3} + 27853396132625 \nu^{2} - 7878555726875 \nu + 288196456635125\)\()/ 451236007807750 \)
\(\beta_{11}\)\(=\)\((\)\(176090223129 \nu^{15} - 541590004299 \nu^{14} - 1156604341685 \nu^{13} + 4731361480299 \nu^{12} - 2426784958929 \nu^{11} - 13824006840431 \nu^{10} + 5941055817805 \nu^{9} - 25673900926259 \nu^{8} + 101300812445949 \nu^{7} + 55780375629075 \nu^{6} - 94501899542125 \nu^{5} - 458338835044675 \nu^{4} - 81055574547325 \nu^{3} + 499015951885625 \nu^{2} + 156264822677125 \nu + 224933071900000\)\()/ 451236007807750 \)
\(\beta_{12}\)\(=\)\((\)\(-354585769846 \nu^{15} + 508748668505 \nu^{14} + 1993098800331 \nu^{13} - 6592373517306 \nu^{12} - 314121333825 \nu^{11} + 15682266957205 \nu^{10} - 5578051683996 \nu^{9} + 57062946239641 \nu^{8} - 110077052963765 \nu^{7} - 108255183265766 \nu^{6} - 16660884340455 \nu^{5} + 580928878355345 \nu^{4} + 163715215109050 \nu^{3} - 582082324757025 \nu^{2} - 121564313236625 \nu - 236118590095250\)\()/ 451236007807750 \)
\(\beta_{13}\)\(=\)\((\)\(-379898570999 \nu^{15} + 649176782439 \nu^{14} + 1851783118665 \nu^{13} - 6977773859244 \nu^{12} + 1699233441719 \nu^{11} + 13664824546091 \nu^{10} - 2629639309260 \nu^{9} + 59494338686279 \nu^{8} - 131274825328889 \nu^{7} - 35107932262330 \nu^{6} - 146370316576575 \nu^{5} + 719416268648775 \nu^{4} - 167759616228500 \nu^{3} - 109773841167125 \nu^{2} - 88398103091375 \nu - 6717236603125\)\()/ 451236007807750 \)
\(\beta_{14}\)\(=\)\((\)\(-245516587 \nu^{15} + 234222124 \nu^{14} + 1010800728 \nu^{13} - 3725291302 \nu^{12} - 434593236 \nu^{11} + 3597332826 \nu^{10} - 1958283618 \nu^{9} + 39706180692 \nu^{8} - 69148932704 \nu^{7} - 21812543298 \nu^{6} - 141188731550 \nu^{5} + 337020939960 \nu^{4} - 23086558150 \nu^{3} - 4325378100 \nu^{2} - 199684397000 \nu - 228316588875\)\()/ 262194077750 \)
\(\beta_{15}\)\(=\)\((\)\(493169051379 \nu^{15} - 864985162329 \nu^{14} - 2475405530945 \nu^{13} + 9563407738064 \nu^{12} - 2121604357809 \nu^{11} - 18708691847541 \nu^{10} + 8518785412090 \nu^{9} - 76777990078999 \nu^{8} + 190479026663079 \nu^{7} + 86069405698760 \nu^{6} + 138739310527935 \nu^{5} - 894721324555875 \nu^{4} - 56727128738700 \nu^{3} + 499909686111375 \nu^{2} + 408665383985875 \nu + 516245422374375\)\()/ 451236007807750 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{15} - \beta_{13} + \beta_{12} - \beta_{9} + 3 \beta_{7} - \beta_{5} + \beta_{3}\)
\(\nu^{3}\)\(=\)\(\beta_{8} - \beta_{6} + 4 \beta_{4} - 2 \beta_{2} - 1\)
\(\nu^{4}\)\(=\)\(-2 \beta_{15} + 7 \beta_{14} - 7 \beta_{13} + 7 \beta_{12} + 13 \beta_{10} + 2 \beta_{8} + 13 \beta_{7} - 13 \beta_{5} + 2 \beta_{3} + 6 \beta_{2} - 2 \beta_{1} - 6\)
\(\nu^{5}\)\(=\)\(20 \beta_{15} + 12 \beta_{14} + 8 \beta_{13} - 11 \beta_{12} - 20 \beta_{11} + 8 \beta_{9} - 11 \beta_{7} - 8 \beta_{3}\)
\(\nu^{6}\)\(=\)\(36 \beta_{10} + 19 \beta_{9} + 32 \beta_{6} - 19 \beta_{4} + 24 \beta_{3} + 36 \beta_{2} - 24 \beta_{1}\)
\(\nu^{7}\)\(=\)\(111 \beta_{15} + 111 \beta_{13} - 81 \beta_{12} - 55 \beta_{11} - 148 \beta_{10} - 111 \beta_{8} - 148 \beta_{7} + 67 \beta_{6} + 81 \beta_{5} - 55 \beta_{4} - 111 \beta_{3} - 67 \beta_{2} + 55 \beta_{1} + 148\)
\(\nu^{8}\)\(=\)\(-136 \beta_{15} - 221 \beta_{14} + 167 \beta_{12} + 259 \beta_{11} + 221 \beta_{5} + 136 \beta_{3}\)
\(\nu^{9}\)\(=\)\(-913 \beta_{10} - 647 \beta_{9} - 357 \beta_{8} - 357 \beta_{3} - 531 \beta_{2} + 357 \beta_{1} + 531\)
\(\nu^{10}\)\(=\)\(-1560 \beta_{15} - 937 \beta_{14} - 888 \beta_{13} + 1361 \beta_{12} + 1560 \beta_{11} + 1361 \beta_{10} + 1560 \beta_{8} + 1361 \beta_{7} - 1361 \beta_{6} + 1560 \beta_{4} + 1560 \beta_{3} - 672 \beta_{1} - 2298\)
\(\nu^{11}\)\(=\)\(3336 \beta_{14} - 2249 \beta_{13} - 2249 \beta_{11} - 2249 \beta_{9} + 3336 \beta_{7} - 5568 \beta_{5} - 1609 \beta_{3}\)
\(\nu^{12}\)\(=\)\(13823 \beta_{10} + 9426 \beta_{9} + 9426 \beta_{8} - 5467 \beta_{6} + 5585 \beta_{4} + 5585 \beta_{3} + 5467 \beta_{2} - 5585 \beta_{1} - 13823\)
\(\nu^{13}\)\(=\)\(13941 \beta_{15} + 20596 \beta_{14} - 13267 \beta_{12} - 23249 \beta_{11} - 13941 \beta_{8} + 20596 \beta_{6} - 20596 \beta_{5} - 23249 \beta_{4} - 13941 \beta_{3} + 13267 \beta_{2} + 20596\)
\(\nu^{14}\)\(=\)\(34537 \beta_{15} - 32557 \beta_{14} + 57112 \beta_{13} - 32557 \beta_{12} + 57112 \beta_{9} - 83688 \beta_{7} + 83688 \beta_{5}\)
\(\nu^{15}\)\(=\)\(-126186 \beta_{10} - 85668 \beta_{9} - 140800 \beta_{8} + 126186 \beta_{6} - 140800 \beta_{4} - 55132 \beta_{3} + 55132 \beta_{1} + 205873\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/539\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(442\)
\(\chi(n)\) \(-1 - \beta_{14}\) \(\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
214.1
1.62381 + 0.722968i
−0.710267 0.316231i
−0.981435 + 1.08999i
1.65057 1.83314i
0.0812692 + 0.773225i
−0.185798 1.76775i
0.0812692 0.773225i
−0.185798 + 1.76775i
−2.41283 + 0.512862i
1.43468 0.304951i
−2.41283 0.512862i
1.43468 + 0.304951i
1.62381 0.722968i
−0.710267 + 0.316231i
−0.981435 1.08999i
1.65057 + 1.83314i
−0.520239 + 0.577783i −0.564602 + 0.251377i 0.145871 + 1.38787i −0.217653 + 0.0462636i 0.148486 0.456994i 0 −2.13577 1.55173i −1.75181 + 1.94558i 0.0865012 0.149825i
214.2 1.18937 1.32093i −0.564602 + 0.251377i −0.121196 1.15310i −2.71679 + 0.577471i −0.339469 + 1.04478i 0 1.20872 + 0.878189i −1.75181 + 1.94558i −2.46847 + 4.27551i
312.1 −0.257844 + 2.45322i 1.08268 + 1.20243i −3.99550 0.849271i 3.16702 + 1.41005i −3.22899 + 2.34600i 0 1.58914 4.89086i 0.0399263 0.379874i −4.27575 + 7.40581i
312.2 0.153315 1.45870i 1.08268 + 1.20243i −0.147996 0.0314575i −0.426381 0.189837i 1.91998 1.39494i 0 0.837913 2.57883i 0.0399263 0.379874i −0.342285 + 0.592855i
324.1 −1.73864 0.369560i 0.0646021 0.614648i 1.05921 + 0.471591i 1.85850 + 2.06407i −0.339469 + 1.04478i 0 1.20872 + 0.878189i 2.56082 + 0.544320i −2.46847 4.27551i
324.2 0.760494 + 0.161648i 0.0646021 0.614648i −1.27487 0.567608i 0.148892 + 0.165361i 0.148486 0.456994i 0 −2.13577 1.55173i 2.56082 + 0.544320i 0.0865012 + 0.149825i
361.1 −1.73864 + 0.369560i 0.0646021 + 0.614648i 1.05921 0.471591i 1.85850 2.06407i −0.339469 1.04478i 0 1.20872 0.878189i 2.56082 0.544320i −2.46847 + 4.27551i
361.2 0.760494 0.161648i 0.0646021 + 0.614648i −1.27487 + 0.567608i 0.148892 0.165361i 0.148486 + 0.456994i 0 −2.13577 + 1.55173i 2.56082 0.544320i 0.0865012 0.149825i
410.1 −1.33993 0.596574i −1.58268 0.336408i 0.101241 + 0.112439i 0.0487868 0.464175i 1.91998 + 1.39494i 0 0.837913 + 2.57883i −0.348943 0.155360i −0.342285 + 0.592855i
410.2 2.25347 + 1.00331i −1.58268 0.336408i 2.73324 + 3.03557i −0.362372 + 3.44774i −3.22899 2.34600i 0 1.58914 + 4.89086i −0.348943 0.155360i −4.27575 + 7.40581i
422.1 −1.33993 + 0.596574i −1.58268 + 0.336408i 0.101241 0.112439i 0.0487868 + 0.464175i 1.91998 1.39494i 0 0.837913 2.57883i −0.348943 + 0.155360i −0.342285 0.592855i
422.2 2.25347 1.00331i −1.58268 + 0.336408i 2.73324 3.03557i −0.362372 3.44774i −3.22899 + 2.34600i 0 1.58914 4.89086i −0.348943 + 0.155360i −4.27575 7.40581i
471.1 −0.520239 0.577783i −0.564602 0.251377i 0.145871 1.38787i −0.217653 0.0462636i 0.148486 + 0.456994i 0 −2.13577 + 1.55173i −1.75181 1.94558i 0.0865012 + 0.149825i
471.2 1.18937 + 1.32093i −0.564602 0.251377i −0.121196 + 1.15310i −2.71679 0.577471i −0.339469 1.04478i 0 1.20872 0.878189i −1.75181 1.94558i −2.46847 4.27551i
520.1 −0.257844 2.45322i 1.08268 1.20243i −3.99550 + 0.849271i 3.16702 1.41005i −3.22899 2.34600i 0 1.58914 + 4.89086i 0.0399263 + 0.379874i −4.27575 7.40581i
520.2 0.153315 + 1.45870i 1.08268 1.20243i −0.147996 + 0.0314575i −0.426381 + 0.189837i 1.91998 + 1.39494i 0 0.837913 + 2.57883i 0.0399263 + 0.379874i −0.342285 0.592855i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 520.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.q.b 16
7.b odd 2 1 539.2.q.c 16
7.c even 3 1 539.2.f.d 8
7.c even 3 1 inner 539.2.q.b 16
7.d odd 6 1 77.2.f.a 8
7.d odd 6 1 539.2.q.c 16
11.c even 5 1 inner 539.2.q.b 16
21.g even 6 1 693.2.m.g 8
77.i even 6 1 847.2.f.q 8
77.j odd 10 1 539.2.q.c 16
77.m even 15 1 539.2.f.d 8
77.m even 15 1 inner 539.2.q.b 16
77.m even 15 1 5929.2.a.bi 4
77.n even 30 1 847.2.a.k 4
77.n even 30 1 847.2.f.q 8
77.n even 30 2 847.2.f.s 8
77.o odd 30 1 5929.2.a.bb 4
77.p odd 30 1 77.2.f.a 8
77.p odd 30 1 539.2.q.c 16
77.p odd 30 1 847.2.a.l 4
77.p odd 30 2 847.2.f.p 8
231.bc even 30 1 693.2.m.g 8
231.bc even 30 1 7623.2.a.ch 4
231.bf odd 30 1 7623.2.a.co 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.a 8 7.d odd 6 1
77.2.f.a 8 77.p odd 30 1
539.2.f.d 8 7.c even 3 1
539.2.f.d 8 77.m even 15 1
539.2.q.b 16 1.a even 1 1 trivial
539.2.q.b 16 7.c even 3 1 inner
539.2.q.b 16 11.c even 5 1 inner
539.2.q.b 16 77.m even 15 1 inner
539.2.q.c 16 7.b odd 2 1
539.2.q.c 16 7.d odd 6 1
539.2.q.c 16 77.j odd 10 1
539.2.q.c 16 77.p odd 30 1
693.2.m.g 8 21.g even 6 1
693.2.m.g 8 231.bc even 30 1
847.2.a.k 4 77.n even 30 1
847.2.a.l 4 77.p odd 30 1
847.2.f.p 8 77.p odd 30 2
847.2.f.q 8 77.i even 6 1
847.2.f.q 8 77.n even 30 1
847.2.f.s 8 77.n even 30 2
5929.2.a.bb 4 77.o odd 30 1
5929.2.a.bi 4 77.m even 15 1
7623.2.a.ch 4 231.bc even 30 1
7623.2.a.co 4 231.bf odd 30 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\):

\(T_{2}^{16} - \cdots\)
\( T_{3}^{8} + 2 T_{3}^{7} + 2 T_{3}^{5} + 9 T_{3}^{4} + 8 T_{3}^{3} + 5 T_{3}^{2} + 3 T_{3} + 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 4 T^{2} - 7 T^{3} + 14 T^{4} - 34 T^{5} + 48 T^{6} - 115 T^{7} + 169 T^{8} - 299 T^{9} + 523 T^{10} - 758 T^{11} + 1321 T^{12} - 1850 T^{13} + 2863 T^{14} - 4328 T^{15} + 5781 T^{16} - 8656 T^{17} + 11452 T^{18} - 14800 T^{19} + 21136 T^{20} - 24256 T^{21} + 33472 T^{22} - 38272 T^{23} + 43264 T^{24} - 58880 T^{25} + 49152 T^{26} - 69632 T^{27} + 57344 T^{28} - 57344 T^{29} + 65536 T^{30} - 32768 T^{31} + 65536 T^{32} \)
$3$ \( ( 1 + 2 T + 3 T^{2} - 10 T^{3} - 30 T^{4} - 52 T^{5} - T^{6} + 150 T^{7} + 379 T^{8} + 450 T^{9} - 9 T^{10} - 1404 T^{11} - 2430 T^{12} - 2430 T^{13} + 2187 T^{14} + 4374 T^{15} + 6561 T^{16} )^{2} \)
$5$ \( 1 - 3 T + 12 T^{2} - 51 T^{3} + 138 T^{4} - 474 T^{5} + 1189 T^{6} - 3123 T^{7} + 8088 T^{8} - 16326 T^{9} + 38041 T^{10} - 69267 T^{11} + 129643 T^{12} - 221502 T^{13} + 294093 T^{14} - 572844 T^{15} + 907531 T^{16} - 2864220 T^{17} + 7352325 T^{18} - 27687750 T^{19} + 81026875 T^{20} - 216459375 T^{21} + 594390625 T^{22} - 1275468750 T^{23} + 3159375000 T^{24} - 6099609375 T^{25} + 11611328125 T^{26} - 23144531250 T^{27} + 33691406250 T^{28} - 62255859375 T^{29} + 73242187500 T^{30} - 91552734375 T^{31} + 152587890625 T^{32} \)
$7$ \( \)
$11$ \( 1 + 5 T - T^{2} - 20 T^{3} + 50 T^{4} + 265 T^{5} + 1266 T^{6} - 1690 T^{7} - 28761 T^{8} - 18590 T^{9} + 153186 T^{10} + 352715 T^{11} + 732050 T^{12} - 3221020 T^{13} - 1771561 T^{14} + 97435855 T^{15} + 214358881 T^{16} \)
$13$ \( ( 1 + 5 T + 35 T^{2} + 90 T^{3} + 611 T^{4} + 670 T^{5} + 4380 T^{6} - 8865 T^{7} + 38151 T^{8} - 115245 T^{9} + 740220 T^{10} + 1471990 T^{11} + 17450771 T^{12} + 33416370 T^{13} + 168938315 T^{14} + 313742585 T^{15} + 815730721 T^{16} )^{2} \)
$17$ \( 1 + 11 T + 82 T^{2} + 151 T^{3} - 1257 T^{4} - 14573 T^{5} - 43518 T^{6} + 93859 T^{7} + 1568504 T^{8} + 5547395 T^{9} - 2761766 T^{10} - 97790693 T^{11} - 344680251 T^{12} + 129416362 T^{13} + 3449705386 T^{14} + 9240382008 T^{15} + 336684367 T^{16} + 157086494136 T^{17} + 996964856554 T^{18} + 635822586506 T^{19} - 28788039243771 T^{20} - 138848799990901 T^{21} - 66662317386854 T^{22} + 2276310702906835 T^{23} + 10941503449238264 T^{24} + 11130539500131923 T^{25} - 87732022559739582 T^{26} - 499444344891135709 T^{27} - 732356152197809577 T^{28} + 1495591282968796487 T^{29} + 13806981777870876178 T^{30} + 31486653566607973723 T^{31} + 48661191875666868481 T^{32} \)
$19$ \( 1 + 9 T + 38 T^{2} + 45 T^{3} - 713 T^{4} - 9387 T^{5} - 55694 T^{6} - 172071 T^{7} + 11894 T^{8} + 3422331 T^{9} + 28508642 T^{10} + 136939077 T^{11} + 373421069 T^{12} + 123703272 T^{13} - 4687437236 T^{14} - 41957891316 T^{15} - 224298429341 T^{16} - 797199935004 T^{17} - 1692164842196 T^{18} + 848480742648 T^{19} + 48664607133149 T^{20} + 339074711620623 T^{21} + 1341214179003602 T^{22} + 3059124962403609 T^{23} + 202002498809654 T^{24} - 55525194844530309 T^{25} - 341463604161968894 T^{26} - 1093494060277581753 T^{27} - 1578093537294172793 T^{28} + 1892384255801567655 T^{29} + 30362254059749596598 T^{30} + \)\(13\!\cdots\!91\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \)
$23$ \( ( 1 - 8 T - 19 T^{2} + 140 T^{3} + 1118 T^{4} - 1732 T^{5} - 36615 T^{6} + 59134 T^{7} + 444387 T^{8} + 1360082 T^{9} - 19369335 T^{10} - 21073244 T^{11} + 312862238 T^{12} + 901088020 T^{13} - 2812681891 T^{14} - 27238603576 T^{15} + 78310985281 T^{16} )^{2} \)
$29$ \( ( 1 + 9 T - 22 T^{2} - 429 T^{3} - 762 T^{4} + 11754 T^{5} + 81940 T^{6} - 191700 T^{7} - 3727369 T^{8} - 5559300 T^{9} + 68911540 T^{10} + 286668306 T^{11} - 538948122 T^{12} - 8799282921 T^{13} - 13086113062 T^{14} + 155248886781 T^{15} + 500246412961 T^{16} )^{2} \)
$31$ \( 1 + 11 T + 87 T^{2} + 8 T^{3} - 4067 T^{4} - 51562 T^{5} - 279707 T^{6} - 605448 T^{7} + 7071411 T^{8} + 79682271 T^{9} + 453595654 T^{10} + 966291599 T^{11} - 4382753906 T^{12} - 56567005227 T^{13} - 210031083446 T^{14} - 225648686906 T^{15} + 2581204556074 T^{16} - 6995109294086 T^{17} - 201839871191606 T^{18} - 1685187652717557 T^{19} - 4047565270023026 T^{20} + 27664108097802449 T^{21} + 402567812610602374 T^{22} + 2192267573511126081 T^{23} + 6031143063961699251 T^{24} - 16007816357933935608 T^{25} - \)\(22\!\cdots\!07\)\( T^{26} - \)\(13\!\cdots\!22\)\( T^{27} - \)\(32\!\cdots\!87\)\( T^{28} + \)\(19\!\cdots\!28\)\( T^{29} + \)\(65\!\cdots\!27\)\( T^{30} + \)\(25\!\cdots\!61\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( 1 + 6 T + 99 T^{2} + 662 T^{3} + 5739 T^{4} + 29514 T^{5} + 237308 T^{6} + 716560 T^{7} + 4809854 T^{8} + 17329244 T^{9} + 22995413 T^{10} + 480402038 T^{11} + 2695419596 T^{12} + 38023386470 T^{13} + 235277510708 T^{14} + 3056978579838 T^{15} + 9263636009781 T^{16} + 113108207454006 T^{17} + 322094912159252 T^{18} + 1925998594864910 T^{19} + 5051650285458956 T^{20} + 33312978265784366 T^{21} + 58999938419961917 T^{22} + 1645097662215777452 T^{23} + 16894513351359737534 T^{24} + 93125384267560375120 T^{25} + \)\(11\!\cdots\!92\)\( T^{26} + \)\(52\!\cdots\!82\)\( T^{27} + \)\(37\!\cdots\!59\)\( T^{28} + \)\(16\!\cdots\!14\)\( T^{29} + \)\(89\!\cdots\!11\)\( T^{30} + \)\(20\!\cdots\!58\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \)
$41$ \( ( 1 - 22 T + 160 T^{2} - 414 T^{3} + 2641 T^{4} - 46342 T^{5} + 334288 T^{6} - 1062556 T^{7} + 2274033 T^{8} - 43564796 T^{9} + 561938128 T^{10} - 3193936982 T^{11} + 7462834801 T^{12} - 47964467214 T^{13} + 760016678560 T^{14} - 4284594025382 T^{15} + 7984925229121 T^{16} )^{2} \)
$43$ \( ( 1 - 4 T + 45 T^{2} - 172 T^{3} + 1849 T^{4} )^{8} \)
$47$ \( 1 - 7 T + 149 T^{2} - 2128 T^{3} + 19612 T^{4} - 240919 T^{5} + 2462607 T^{6} - 21813848 T^{7} + 211617499 T^{8} - 1862688919 T^{9} + 15071552909 T^{10} - 126763363055 T^{11} + 1005269592991 T^{12} - 7378260675254 T^{13} + 56266129109578 T^{14} - 404849575321212 T^{15} + 2719197771655735 T^{16} - 19027930040096964 T^{17} + 124291879203057802 T^{18} - 766033158086896042 T^{19} + 4905394932795915871 T^{20} - 29072544387192516385 T^{21} + \)\(16\!\cdots\!61\)\( T^{22} - \)\(94\!\cdots\!97\)\( T^{23} + \)\(50\!\cdots\!39\)\( T^{24} - \)\(24\!\cdots\!16\)\( T^{25} + \)\(12\!\cdots\!43\)\( T^{26} - \)\(59\!\cdots\!57\)\( T^{27} + \)\(22\!\cdots\!92\)\( T^{28} - \)\(11\!\cdots\!56\)\( T^{29} + \)\(38\!\cdots\!81\)\( T^{30} - \)\(84\!\cdots\!01\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 + 2 T + 177 T^{2} + 1582 T^{3} + 21921 T^{4} + 231770 T^{5} + 2684050 T^{6} + 23678026 T^{7} + 239791912 T^{8} + 2114692878 T^{9} + 17637992799 T^{10} + 151508122160 T^{11} + 1184498641232 T^{12} + 9380769650632 T^{13} + 69924170534704 T^{14} + 544239771304768 T^{15} + 3774658249521589 T^{16} + 28844707879152704 T^{17} + 196416995031983536 T^{18} + 1396580843277140264 T^{19} + 9346264023166912592 T^{20} + 63360013840205424880 T^{21} + \)\(39\!\cdots\!71\)\( T^{22} + \)\(24\!\cdots\!86\)\( T^{23} + \)\(14\!\cdots\!32\)\( T^{24} + \)\(78\!\cdots\!58\)\( T^{25} + \)\(46\!\cdots\!50\)\( T^{26} + \)\(21\!\cdots\!90\)\( T^{27} + \)\(10\!\cdots\!61\)\( T^{28} + \)\(41\!\cdots\!86\)\( T^{29} + \)\(24\!\cdots\!13\)\( T^{30} + \)\(14\!\cdots\!14\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( 1 - 25 T + 419 T^{2} - 5200 T^{3} + 58281 T^{4} - 598550 T^{5} + 6209893 T^{6} - 62948850 T^{7} + 627371027 T^{8} - 5955201125 T^{9} + 54113866884 T^{10} - 468835214275 T^{11} + 3967819915452 T^{12} - 33125257031325 T^{13} + 272714366831568 T^{14} - 2199975935884200 T^{15} + 17198372786595766 T^{16} - 129798580217167800 T^{17} + 949318710940688208 T^{18} - 6803232163836497175 T^{19} + 48079506298521362172 T^{20} - \)\(33\!\cdots\!25\)\( T^{21} + \)\(22\!\cdots\!44\)\( T^{22} - \)\(14\!\cdots\!75\)\( T^{23} + \)\(92\!\cdots\!67\)\( T^{24} - \)\(54\!\cdots\!50\)\( T^{25} + \)\(31\!\cdots\!93\)\( T^{26} - \)\(18\!\cdots\!50\)\( T^{27} + \)\(10\!\cdots\!61\)\( T^{28} - \)\(54\!\cdots\!00\)\( T^{29} + \)\(25\!\cdots\!59\)\( T^{30} - \)\(91\!\cdots\!75\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( 1 - 7 T + 157 T^{2} - 2810 T^{3} + 23827 T^{4} - 352454 T^{5} + 4248359 T^{6} - 37300018 T^{7} + 438042179 T^{8} - 4457193753 T^{9} + 37214201488 T^{10} - 377860830849 T^{11} + 3428826064944 T^{12} - 26601329050551 T^{13} + 245019222253856 T^{14} - 2011514545995138 T^{15} + 14453308043412074 T^{16} - 122702387305703418 T^{17} + 911716526006598176 T^{18} - 6037996269223116531 T^{19} + 47474980511870297904 T^{20} - \)\(31\!\cdots\!49\)\( T^{21} + \)\(19\!\cdots\!68\)\( T^{22} - \)\(14\!\cdots\!13\)\( T^{23} + \)\(83\!\cdots\!99\)\( T^{24} - \)\(43\!\cdots\!38\)\( T^{25} + \)\(30\!\cdots\!59\)\( T^{26} - \)\(15\!\cdots\!94\)\( T^{27} + \)\(63\!\cdots\!67\)\( T^{28} - \)\(45\!\cdots\!10\)\( T^{29} + \)\(15\!\cdots\!37\)\( T^{30} - \)\(42\!\cdots\!07\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( ( 1 - 15 T - 110 T^{2} + 1095 T^{3} + 30612 T^{4} - 158730 T^{5} - 2667925 T^{6} + 1012950 T^{7} + 259471373 T^{8} + 67867650 T^{9} - 11976315325 T^{10} - 47740110990 T^{11} + 616866116052 T^{12} + 1478386992165 T^{13} - 9950422038590 T^{14} - 90910674079845 T^{15} + 406067677556641 T^{16} )^{2} \)
$71$ \( ( 1 + 14 T - 9 T^{2} - 147 T^{3} + 6927 T^{4} + 52059 T^{5} + 728307 T^{6} + 4047848 T^{7} - 14238189 T^{8} + 287397208 T^{9} + 3671395587 T^{10} + 18632488749 T^{11} + 176026714287 T^{12} - 265221714597 T^{13} - 1152902555289 T^{14} + 127331682217474 T^{15} + 645753531245761 T^{16} )^{2} \)
$73$ \( 1 - 3 T + 11 T^{2} + 696 T^{3} - 1251 T^{4} - 98282 T^{5} + 1060537 T^{6} - 2326410 T^{7} - 65290511 T^{8} + 308702063 T^{9} + 6492765102 T^{10} - 92451983031 T^{11} + 281336636896 T^{12} + 4375330012525 T^{13} - 27106436766918 T^{14} - 268331027179726 T^{15} + 5635646597034826 T^{16} - 19588164984119998 T^{17} - 144450201530906022 T^{18} + 1702077755482437925 T^{19} + 7989465616702099936 T^{20} - \)\(19\!\cdots\!83\)\( T^{21} + \)\(98\!\cdots\!78\)\( T^{22} + \)\(34\!\cdots\!11\)\( T^{23} - \)\(52\!\cdots\!91\)\( T^{24} - \)\(13\!\cdots\!30\)\( T^{25} + \)\(45\!\cdots\!13\)\( T^{26} - \)\(30\!\cdots\!14\)\( T^{27} - \)\(28\!\cdots\!71\)\( T^{28} + \)\(11\!\cdots\!68\)\( T^{29} + \)\(13\!\cdots\!99\)\( T^{30} - \)\(26\!\cdots\!71\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( 1 - 9 T + 156 T^{2} - 861 T^{3} + 2506 T^{4} + 98394 T^{5} - 1985283 T^{6} + 19471377 T^{7} - 143824296 T^{8} + 713258136 T^{9} + 7748031243 T^{10} - 94344490293 T^{11} + 1366072992785 T^{12} - 5625357609390 T^{13} + 11704956821451 T^{14} + 387894213236796 T^{15} - 6232535914916185 T^{16} + 30643642845706884 T^{17} + 73050635522675691 T^{18} - 2773520690376036210 T^{19} + 53208653720888165585 T^{20} - \)\(29\!\cdots\!07\)\( T^{21} + \)\(18\!\cdots\!03\)\( T^{22} + \)\(13\!\cdots\!24\)\( T^{23} - \)\(21\!\cdots\!56\)\( T^{24} + \)\(23\!\cdots\!63\)\( T^{25} - \)\(18\!\cdots\!83\)\( T^{26} + \)\(73\!\cdots\!26\)\( T^{27} + \)\(14\!\cdots\!46\)\( T^{28} - \)\(40\!\cdots\!79\)\( T^{29} + \)\(57\!\cdots\!36\)\( T^{30} - \)\(26\!\cdots\!91\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$ \( ( 1 + 23 T + 98 T^{2} - 1745 T^{3} - 15470 T^{4} + 98632 T^{5} + 1750974 T^{6} - 1504700 T^{7} - 137136651 T^{8} - 124890100 T^{9} + 12062459886 T^{10} + 56396495384 T^{11} - 734180225870 T^{12} - 6873625922035 T^{13} + 32040156590162 T^{14} + 624129172761421 T^{15} + 2252292232139041 T^{16} )^{2} \)
$89$ \( ( 1 + 17 T - 23 T^{2} - 1534 T^{3} + 282 T^{4} + 47093 T^{5} - 689650 T^{6} - 2747686 T^{7} + 49542809 T^{8} - 244544054 T^{9} - 5462717650 T^{10} + 33199105117 T^{11} + 17693311962 T^{12} - 8565947194766 T^{13} - 11430569692103 T^{14} + 751932693223993 T^{15} + 3936588805702081 T^{16} )^{2} \)
$97$ \( ( 1 + 30 T + 361 T^{2} + 2160 T^{3} + 10062 T^{4} + 83370 T^{5} - 577417 T^{6} - 32754150 T^{7} - 458148745 T^{8} - 3177152550 T^{9} - 5432916553 T^{10} + 76089548010 T^{11} + 890781625422 T^{12} + 18548654955120 T^{13} + 300702893779369 T^{14} + 2423948534343390 T^{15} + 7837433594376961 T^{16} )^{2} \)
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