Properties

Label 693.2.m.j
Level 693
Weight 2
Character orbit 693.m
Analytic conductor 5.534
Analytic rank 0
Dimension 20
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 693.m (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.53363286007\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 231)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{7} q^{2} + ( \beta_{3} - \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{19} ) q^{4} -\beta_{13} q^{5} + \beta_{9} q^{7} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{14} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{8} +O(q^{10})\) \( q -\beta_{7} q^{2} + ( \beta_{3} - \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{19} ) q^{4} -\beta_{13} q^{5} + \beta_{9} q^{7} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{14} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{8} + ( \beta_{3} - \beta_{7} - \beta_{8} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{18} ) q^{10} + ( \beta_{1} - \beta_{2} + \beta_{8} + \beta_{9} - \beta_{12} - \beta_{15} ) q^{11} + ( 1 + \beta_{2} + \beta_{5} - \beta_{9} + \beta_{19} ) q^{13} + ( \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} ) q^{14} + ( -4 - 3 \beta_{1} + \beta_{2} + 5 \beta_{4} + \beta_{5} + 4 \beta_{6} + 5 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{16} + ( -1 + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{18} ) q^{17} + ( \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} + \beta_{14} + \beta_{17} ) q^{19} + ( 3 + \beta_{2} + \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - 5 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} + 2 \beta_{15} + \beta_{19} ) q^{20} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{10} + \beta_{11} - \beta_{13} - \beta_{15} + \beta_{16} + \beta_{18} - \beta_{19} ) q^{22} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} + 3 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{23} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{14} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{25} + ( -1 - 2 \beta_{1} + \beta_{2} + 5 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{13} - \beta_{18} + \beta_{19} ) q^{26} + ( 2 - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{9} + \beta_{18} ) q^{28} + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{8} + 3 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{16} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{29} + ( -1 + \beta_{2} + \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} + \beta_{15} + \beta_{16} - \beta_{17} - \beta_{19} ) q^{31} + ( -3 - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{11} - \beta_{13} - 2 \beta_{14} - \beta_{15} + 2 \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{32} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{4} - 4 \beta_{5} - \beta_{6} + \beta_{8} + 3 \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{16} + \beta_{17} - \beta_{18} ) q^{34} + \beta_{10} q^{35} + ( -1 - \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - 3 \beta_{9} - \beta_{11} + 2 \beta_{12} + 2 \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{18} + \beta_{19} ) q^{37} + ( 1 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{10} - 4 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} - 4 \beta_{17} - \beta_{18} - \beta_{19} ) q^{38} + ( -6 \beta_{1} + 6 \beta_{2} - \beta_{3} + 3 \beta_{4} + 6 \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} + 3 \beta_{15} - \beta_{16} - \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{40} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{8} ) q^{41} + ( \beta_{1} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{18} ) q^{43} + ( -1 + 2 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{18} ) q^{44} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{12} + \beta_{13} + \beta_{14} + 3 \beta_{15} - \beta_{16} - \beta_{17} - \beta_{19} ) q^{46} + ( -2 \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} + 2 \beta_{8} + 3 \beta_{10} - 2 \beta_{11} + \beta_{13} - \beta_{15} + \beta_{18} - \beta_{19} ) q^{47} + ( -1 + \beta_{4} + \beta_{6} + \beta_{9} ) q^{49} + ( 4 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 4 \beta_{4} + 4 \beta_{6} + \beta_{7} + 3 \beta_{9} + 2 \beta_{10} - \beta_{15} - \beta_{19} ) q^{50} + ( -6 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - \beta_{10} + 2 \beta_{12} + \beta_{13} - \beta_{15} + 2 \beta_{16} - \beta_{18} + \beta_{19} ) q^{52} + ( -1 - 2 \beta_{2} - 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{53} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{19} ) q^{55} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} - 2 \beta_{16} - \beta_{17} ) q^{56} + ( 3 + 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{10} - \beta_{11} + \beta_{13} - 2 \beta_{14} - \beta_{15} - 2 \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{58} + ( -1 - \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} + 2 \beta_{17} + \beta_{18} + \beta_{19} ) q^{59} + ( -2 + 3 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{7} - \beta_{9} - \beta_{10} - \beta_{13} + 2 \beta_{15} - 2 \beta_{18} - \beta_{19} ) q^{61} + ( 2 + \beta_{1} - 3 \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{9} - 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{15} - \beta_{18} ) q^{62} + ( -3 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{17} + \beta_{19} ) q^{64} + ( -1 - 5 \beta_{1} + 4 \beta_{2} - \beta_{3} + 2 \beta_{4} + 5 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} - 4 \beta_{11} + 2 \beta_{12} + \beta_{13} + 2 \beta_{14} + 2 \beta_{15} - 4 \beta_{16} - 2 \beta_{17} + 2 \beta_{18} ) q^{65} + ( 5 - \beta_{3} - \beta_{5} + \beta_{7} + \beta_{10} - \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{67} + ( -2 - 3 \beta_{2} - 3 \beta_{5} - 8 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{11} + \beta_{15} - \beta_{19} ) q^{68} + ( -1 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{5} + 3 \beta_{11} - \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{70} + ( -1 + 5 \beta_{1} - 4 \beta_{2} + \beta_{3} - \beta_{5} - 5 \beta_{6} - 3 \beta_{7} - 5 \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} + 2 \beta_{17} - \beta_{18} + \beta_{19} ) q^{71} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} - 2 \beta_{15} + \beta_{18} - \beta_{19} ) q^{73} + ( 1 - 3 \beta_{1} + \beta_{2} + \beta_{3} + 5 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{14} - 2 \beta_{17} - \beta_{18} + \beta_{19} ) q^{74} + ( 5 - 2 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 6 \beta_{9} - \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} + 2 \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{76} + ( -1 + \beta_{4} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{14} - \beta_{18} + \beta_{19} ) q^{77} + ( 1 + \beta_{6} - 3 \beta_{7} + \beta_{10} - \beta_{11} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{79} + ( -11 - \beta_{1} + \beta_{2} + 3 \beta_{3} + 7 \beta_{4} - \beta_{5} + 10 \beta_{6} - \beta_{7} - 3 \beta_{8} - \beta_{10} + \beta_{13} - \beta_{15} - \beta_{18} + \beta_{19} ) q^{80} + ( -4 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} + 2 \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{82} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{15} - 2 \beta_{18} ) q^{83} + ( -3 - \beta_{1} + 2 \beta_{2} + 4 \beta_{4} + \beta_{5} + 4 \beta_{6} - 3 \beta_{11} - \beta_{12} + \beta_{14} - \beta_{16} + \beta_{17} - 3 \beta_{18} + 3 \beta_{19} ) q^{85} + ( 2 - \beta_{2} - \beta_{5} - 4 \beta_{6} + 2 \beta_{8} + 3 \beta_{10} - 2 \beta_{11} - 2 \beta_{15} ) q^{86} + ( 13 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 7 \beta_{6} + 2 \beta_{7} - \beta_{8} - 8 \beta_{9} - 3 \beta_{10} - \beta_{11} + \beta_{13} + \beta_{15} - 2 \beta_{16} - \beta_{17} + 2 \beta_{19} ) q^{88} + ( 1 + 3 \beta_{1} - \beta_{3} + \beta_{4} - 3 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{18} ) q^{89} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{8} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{15} ) q^{91} + ( 3 + 3 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} - 6 \beta_{9} - 3 \beta_{11} - \beta_{12} + 3 \beta_{13} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{92} + ( -\beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{7} - \beta_{9} + 3 \beta_{10} - \beta_{13} - 2 \beta_{14} - 2 \beta_{15} - 2 \beta_{16} + 2 \beta_{18} + \beta_{19} ) q^{94} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + 5 \beta_{4} - \beta_{5} + 7 \beta_{6} + 2 \beta_{8} + 5 \beta_{9} + 2 \beta_{10} + 5 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - 4 \beta_{14} - \beta_{15} + 2 \beta_{16} + 2 \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{95} + ( -1 + \beta_{2} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{9} + \beta_{19} ) q^{97} -\beta_{5} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 14q^{4} + 5q^{5} + 5q^{7} + 9q^{8} + O(q^{10}) \) \( 20q - 14q^{4} + 5q^{5} + 5q^{7} + 9q^{8} + 12q^{10} + q^{11} + 13q^{13} - 24q^{16} + q^{17} + 10q^{19} + 46q^{20} + 26q^{22} - 8q^{25} + 53q^{26} + 4q^{28} - 3q^{29} - 13q^{31} - 82q^{32} + 42q^{34} - 5q^{35} - 32q^{37} - 16q^{38} + 20q^{40} + 3q^{41} + 12q^{43} - 25q^{44} - 13q^{46} - 20q^{47} - 5q^{49} + 83q^{50} - 80q^{52} - 3q^{53} - 28q^{55} + 6q^{56} + 2q^{58} + 9q^{59} - 15q^{61} + 37q^{62} - 49q^{64} - 58q^{65} + 76q^{67} - 51q^{68} + 3q^{70} - 37q^{71} + 27q^{73} + 32q^{74} + 4q^{76} - 6q^{77} + 5q^{79} - 137q^{80} - 55q^{82} + 42q^{83} - 48q^{85} - 3q^{86} + 151q^{88} + 18q^{89} + 7q^{91} - 39q^{92} - 35q^{94} + 96q^{95} - 27q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 12 x^{18} - 3 x^{17} + 94 x^{16} - 10 x^{15} + 662 x^{14} - 153 x^{13} + 4638 x^{12} - 1174 x^{11} + 15808 x^{10} - 3393 x^{9} + 26062 x^{8} - 15494 x^{7} + 11660 x^{6} - 33295 x^{5} + 67756 x^{4} - 43696 x^{3} + 13084 x^{2} - 1155 x + 121\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(46\!\cdots\!50\)\( \nu^{19} - \)\(13\!\cdots\!96\)\( \nu^{18} - \)\(60\!\cdots\!30\)\( \nu^{17} - \)\(15\!\cdots\!69\)\( \nu^{16} - \)\(45\!\cdots\!19\)\( \nu^{15} - \)\(12\!\cdots\!18\)\( \nu^{14} - \)\(34\!\cdots\!39\)\( \nu^{13} - \)\(86\!\cdots\!29\)\( \nu^{12} - \)\(23\!\cdots\!29\)\( \nu^{11} - \)\(59\!\cdots\!21\)\( \nu^{10} - \)\(83\!\cdots\!87\)\( \nu^{9} - \)\(20\!\cdots\!82\)\( \nu^{8} - \)\(16\!\cdots\!59\)\( \nu^{7} - \)\(30\!\cdots\!49\)\( \nu^{6} - \)\(15\!\cdots\!62\)\( \nu^{5} + \)\(14\!\cdots\!80\)\( \nu^{4} - \)\(30\!\cdots\!18\)\( \nu^{3} - \)\(60\!\cdots\!57\)\( \nu^{2} + \)\(15\!\cdots\!72\)\( \nu + \)\(53\!\cdots\!27\)\(\)\()/ \)\(18\!\cdots\!60\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(15\!\cdots\!62\)\( \nu^{19} + \)\(30\!\cdots\!22\)\( \nu^{18} + \)\(18\!\cdots\!69\)\( \nu^{17} - \)\(20\!\cdots\!56\)\( \nu^{16} + \)\(14\!\cdots\!57\)\( \nu^{15} + \)\(20\!\cdots\!07\)\( \nu^{14} + \)\(10\!\cdots\!38\)\( \nu^{13} + \)\(27\!\cdots\!56\)\( \nu^{12} + \)\(70\!\cdots\!24\)\( \nu^{11} + \)\(25\!\cdots\!52\)\( \nu^{10} + \)\(24\!\cdots\!77\)\( \nu^{9} + \)\(23\!\cdots\!21\)\( \nu^{8} + \)\(40\!\cdots\!30\)\( \nu^{7} - \)\(71\!\cdots\!17\)\( \nu^{6} + \)\(17\!\cdots\!86\)\( \nu^{5} - \)\(35\!\cdots\!36\)\( \nu^{4} + \)\(91\!\cdots\!89\)\( \nu^{3} - \)\(42\!\cdots\!65\)\( \nu^{2} + \)\(35\!\cdots\!19\)\( \nu + \)\(73\!\cdots\!95\)\(\)\()/ \)\(18\!\cdots\!60\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(48\!\cdots\!57\)\( \nu^{19} - \)\(50\!\cdots\!50\)\( \nu^{18} - \)\(59\!\cdots\!40\)\( \nu^{17} + \)\(77\!\cdots\!41\)\( \nu^{16} - \)\(47\!\cdots\!17\)\( \nu^{15} - \)\(22\!\cdots\!39\)\( \nu^{14} - \)\(33\!\cdots\!32\)\( \nu^{13} + \)\(35\!\cdots\!92\)\( \nu^{12} - \)\(23\!\cdots\!85\)\( \nu^{11} + \)\(31\!\cdots\!99\)\( \nu^{10} - \)\(82\!\cdots\!87\)\( \nu^{9} + \)\(72\!\cdots\!44\)\( \nu^{8} - \)\(14\!\cdots\!36\)\( \nu^{7} + \)\(56\!\cdots\!09\)\( \nu^{6} - \)\(89\!\cdots\!59\)\( \nu^{5} + \)\(15\!\cdots\!33\)\( \nu^{4} - \)\(32\!\cdots\!12\)\( \nu^{3} + \)\(20\!\cdots\!74\)\( \nu^{2} - \)\(12\!\cdots\!15\)\( \nu + \)\(22\!\cdots\!27\)\(\)\()/ \)\(20\!\cdots\!60\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(49\!\cdots\!48\)\( \nu^{19} - \)\(12\!\cdots\!79\)\( \nu^{18} - \)\(59\!\cdots\!87\)\( \nu^{17} - \)\(80\!\cdots\!48\)\( \nu^{16} - \)\(46\!\cdots\!65\)\( \nu^{15} - \)\(72\!\cdots\!49\)\( \nu^{14} - \)\(32\!\cdots\!53\)\( \nu^{13} - \)\(10\!\cdots\!03\)\( \nu^{12} - \)\(22\!\cdots\!49\)\( \nu^{11} - \)\(24\!\cdots\!00\)\( \nu^{10} - \)\(77\!\cdots\!99\)\( \nu^{9} - \)\(39\!\cdots\!85\)\( \nu^{8} - \)\(12\!\cdots\!08\)\( \nu^{7} + \)\(41\!\cdots\!48\)\( \nu^{6} - \)\(47\!\cdots\!98\)\( \nu^{5} + \)\(14\!\cdots\!81\)\( \nu^{4} - \)\(29\!\cdots\!98\)\( \nu^{3} + \)\(13\!\cdots\!43\)\( \nu^{2} - \)\(11\!\cdots\!86\)\( \nu - \)\(68\!\cdots\!52\)\(\)\()/ \)\(18\!\cdots\!60\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(62\!\cdots\!32\)\( \nu^{19} - \)\(54\!\cdots\!28\)\( \nu^{18} + \)\(73\!\cdots\!15\)\( \nu^{17} - \)\(83\!\cdots\!53\)\( \nu^{16} + \)\(58\!\cdots\!80\)\( \nu^{15} - \)\(56\!\cdots\!35\)\( \nu^{14} + \)\(40\!\cdots\!45\)\( \nu^{13} - \)\(45\!\cdots\!79\)\( \nu^{12} + \)\(28\!\cdots\!83\)\( \nu^{11} - \)\(32\!\cdots\!07\)\( \nu^{10} + \)\(97\!\cdots\!56\)\( \nu^{9} - \)\(10\!\cdots\!65\)\( \nu^{8} + \)\(15\!\cdots\!49\)\( \nu^{7} - \)\(23\!\cdots\!96\)\( \nu^{6} + \)\(11\!\cdots\!48\)\( \nu^{5} - \)\(25\!\cdots\!18\)\( \nu^{4} + \)\(58\!\cdots\!83\)\( \nu^{3} - \)\(59\!\cdots\!50\)\( \nu^{2} + \)\(22\!\cdots\!61\)\( \nu - \)\(19\!\cdots\!06\)\(\)\()/ \)\(20\!\cdots\!60\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(56\!\cdots\!45\)\( \nu^{19} + \)\(15\!\cdots\!62\)\( \nu^{18} + \)\(68\!\cdots\!62\)\( \nu^{17} + \)\(13\!\cdots\!34\)\( \nu^{16} + \)\(53\!\cdots\!74\)\( \nu^{15} + \)\(86\!\cdots\!07\)\( \nu^{14} + \)\(37\!\cdots\!97\)\( \nu^{13} + \)\(14\!\cdots\!53\)\( \nu^{12} + \)\(26\!\cdots\!66\)\( \nu^{11} + \)\(43\!\cdots\!94\)\( \nu^{10} + \)\(89\!\cdots\!12\)\( \nu^{9} + \)\(50\!\cdots\!92\)\( \nu^{8} + \)\(14\!\cdots\!11\)\( \nu^{7} - \)\(46\!\cdots\!00\)\( \nu^{6} + \)\(58\!\cdots\!83\)\( \nu^{5} - \)\(17\!\cdots\!89\)\( \nu^{4} + \)\(34\!\cdots\!84\)\( \nu^{3} - \)\(15\!\cdots\!31\)\( \nu^{2} + \)\(31\!\cdots\!15\)\( \nu - \)\(29\!\cdots\!56\)\(\)\()/ \)\(18\!\cdots\!60\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(26\!\cdots\!84\)\( \nu^{19} - \)\(21\!\cdots\!82\)\( \nu^{18} + \)\(31\!\cdots\!69\)\( \nu^{17} - \)\(33\!\cdots\!03\)\( \nu^{16} + \)\(24\!\cdots\!18\)\( \nu^{15} - \)\(22\!\cdots\!11\)\( \nu^{14} + \)\(17\!\cdots\!49\)\( \nu^{13} - \)\(17\!\cdots\!97\)\( \nu^{12} + \)\(12\!\cdots\!63\)\( \nu^{11} - \)\(12\!\cdots\!71\)\( \nu^{10} + \)\(41\!\cdots\!76\)\( \nu^{9} - \)\(41\!\cdots\!11\)\( \nu^{8} + \)\(67\!\cdots\!35\)\( \nu^{7} - \)\(95\!\cdots\!02\)\( \nu^{6} + \)\(48\!\cdots\!62\)\( \nu^{5} - \)\(10\!\cdots\!74\)\( \nu^{4} + \)\(24\!\cdots\!59\)\( \nu^{3} - \)\(24\!\cdots\!92\)\( \nu^{2} + \)\(91\!\cdots\!97\)\( \nu - \)\(78\!\cdots\!74\)\(\)\()/ \)\(20\!\cdots\!60\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(26\!\cdots\!96\)\( \nu^{19} + \)\(62\!\cdots\!95\)\( \nu^{18} + \)\(32\!\cdots\!34\)\( \nu^{17} - \)\(52\!\cdots\!06\)\( \nu^{16} + \)\(25\!\cdots\!98\)\( \nu^{15} + \)\(31\!\cdots\!54\)\( \nu^{14} + \)\(17\!\cdots\!29\)\( \nu^{13} + \)\(48\!\cdots\!79\)\( \nu^{12} + \)\(12\!\cdots\!31\)\( \nu^{11} - \)\(25\!\cdots\!78\)\( \nu^{10} + \)\(42\!\cdots\!02\)\( \nu^{9} + \)\(76\!\cdots\!04\)\( \nu^{8} + \)\(70\!\cdots\!64\)\( \nu^{7} - \)\(24\!\cdots\!03\)\( \nu^{6} + \)\(26\!\cdots\!60\)\( \nu^{5} - \)\(82\!\cdots\!07\)\( \nu^{4} + \)\(16\!\cdots\!97\)\( \nu^{3} - \)\(78\!\cdots\!92\)\( \nu^{2} + \)\(17\!\cdots\!23\)\( \nu + \)\(41\!\cdots\!85\)\(\)\()/ \)\(20\!\cdots\!60\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(29\!\cdots\!75\)\( \nu^{19} + \)\(23\!\cdots\!79\)\( \nu^{18} + \)\(36\!\cdots\!54\)\( \nu^{17} + \)\(18\!\cdots\!93\)\( \nu^{16} + \)\(27\!\cdots\!13\)\( \nu^{15} + \)\(18\!\cdots\!09\)\( \nu^{14} + \)\(19\!\cdots\!39\)\( \nu^{13} + \)\(10\!\cdots\!31\)\( \nu^{12} + \)\(13\!\cdots\!92\)\( \nu^{11} + \)\(70\!\cdots\!93\)\( \nu^{10} + \)\(46\!\cdots\!19\)\( \nu^{9} + \)\(24\!\cdots\!74\)\( \nu^{8} + \)\(77\!\cdots\!92\)\( \nu^{7} + \)\(84\!\cdots\!80\)\( \nu^{6} + \)\(87\!\cdots\!81\)\( \nu^{5} - \)\(89\!\cdots\!48\)\( \nu^{4} + \)\(12\!\cdots\!03\)\( \nu^{3} + \)\(22\!\cdots\!58\)\( \nu^{2} - \)\(18\!\cdots\!70\)\( \nu + \)\(14\!\cdots\!88\)\(\)\()/ \)\(20\!\cdots\!60\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(30\!\cdots\!21\)\( \nu^{19} + \)\(77\!\cdots\!28\)\( \nu^{18} + \)\(35\!\cdots\!64\)\( \nu^{17} - \)\(14\!\cdots\!09\)\( \nu^{16} + \)\(27\!\cdots\!25\)\( \nu^{15} + \)\(37\!\cdots\!73\)\( \nu^{14} + \)\(19\!\cdots\!06\)\( \nu^{13} + \)\(10\!\cdots\!66\)\( \nu^{12} + \)\(13\!\cdots\!73\)\( \nu^{11} - \)\(22\!\cdots\!05\)\( \nu^{10} + \)\(45\!\cdots\!83\)\( \nu^{9} - \)\(14\!\cdots\!80\)\( \nu^{8} + \)\(70\!\cdots\!36\)\( \nu^{7} - \)\(34\!\cdots\!71\)\( \nu^{6} + \)\(12\!\cdots\!31\)\( \nu^{5} - \)\(10\!\cdots\!97\)\( \nu^{4} + \)\(17\!\cdots\!06\)\( \nu^{3} - \)\(74\!\cdots\!46\)\( \nu^{2} + \)\(63\!\cdots\!67\)\( \nu - \)\(89\!\cdots\!91\)\(\)\()/ \)\(18\!\cdots\!60\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(39\!\cdots\!52\)\( \nu^{19} - \)\(25\!\cdots\!75\)\( \nu^{18} - \)\(47\!\cdots\!43\)\( \nu^{17} - \)\(18\!\cdots\!25\)\( \nu^{16} - \)\(37\!\cdots\!34\)\( \nu^{15} - \)\(19\!\cdots\!87\)\( \nu^{14} - \)\(26\!\cdots\!48\)\( \nu^{13} - \)\(10\!\cdots\!02\)\( \nu^{12} - \)\(18\!\cdots\!16\)\( \nu^{11} - \)\(70\!\cdots\!91\)\( \nu^{10} - \)\(62\!\cdots\!08\)\( \nu^{9} - \)\(25\!\cdots\!95\)\( \nu^{8} - \)\(10\!\cdots\!13\)\( \nu^{7} - \)\(26\!\cdots\!17\)\( \nu^{6} - \)\(20\!\cdots\!30\)\( \nu^{5} + \)\(11\!\cdots\!45\)\( \nu^{4} - \)\(18\!\cdots\!44\)\( \nu^{3} + \)\(24\!\cdots\!36\)\( \nu^{2} + \)\(18\!\cdots\!96\)\( \nu - \)\(74\!\cdots\!47\)\(\)\()/ \)\(20\!\cdots\!60\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(94\!\cdots\!92\)\( \nu^{19} - \)\(62\!\cdots\!32\)\( \nu^{18} + \)\(11\!\cdots\!08\)\( \nu^{17} - \)\(37\!\cdots\!85\)\( \nu^{16} + \)\(86\!\cdots\!69\)\( \nu^{15} - \)\(16\!\cdots\!18\)\( \nu^{14} + \)\(60\!\cdots\!27\)\( \nu^{13} - \)\(19\!\cdots\!45\)\( \nu^{12} + \)\(42\!\cdots\!45\)\( \nu^{11} - \)\(14\!\cdots\!45\)\( \nu^{10} + \)\(14\!\cdots\!63\)\( \nu^{9} - \)\(46\!\cdots\!68\)\( \nu^{8} + \)\(21\!\cdots\!11\)\( \nu^{7} - \)\(17\!\cdots\!67\)\( \nu^{6} + \)\(62\!\cdots\!74\)\( \nu^{5} - \)\(32\!\cdots\!38\)\( \nu^{4} + \)\(64\!\cdots\!56\)\( \nu^{3} - \)\(39\!\cdots\!87\)\( \nu^{2} + \)\(70\!\cdots\!34\)\( \nu - \)\(43\!\cdots\!83\)\(\)\()/ \)\(40\!\cdots\!12\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(50\!\cdots\!16\)\( \nu^{19} + \)\(13\!\cdots\!36\)\( \nu^{18} + \)\(60\!\cdots\!71\)\( \nu^{17} + \)\(55\!\cdots\!22\)\( \nu^{16} + \)\(46\!\cdots\!75\)\( \nu^{15} + \)\(70\!\cdots\!79\)\( \nu^{14} + \)\(32\!\cdots\!42\)\( \nu^{13} + \)\(81\!\cdots\!14\)\( \nu^{12} + \)\(22\!\cdots\!24\)\( \nu^{11} + \)\(61\!\cdots\!98\)\( \nu^{10} + \)\(77\!\cdots\!23\)\( \nu^{9} + \)\(26\!\cdots\!17\)\( \nu^{8} + \)\(12\!\cdots\!36\)\( \nu^{7} - \)\(48\!\cdots\!25\)\( \nu^{6} + \)\(32\!\cdots\!16\)\( \nu^{5} - \)\(15\!\cdots\!50\)\( \nu^{4} + \)\(29\!\cdots\!53\)\( \nu^{3} - \)\(12\!\cdots\!53\)\( \nu^{2} + \)\(11\!\cdots\!87\)\( \nu + \)\(45\!\cdots\!99\)\(\)\()/ \)\(20\!\cdots\!60\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(54\!\cdots\!50\)\( \nu^{19} + \)\(26\!\cdots\!47\)\( \nu^{18} + \)\(66\!\cdots\!95\)\( \nu^{17} + \)\(15\!\cdots\!28\)\( \nu^{16} + \)\(51\!\cdots\!43\)\( \nu^{15} + \)\(19\!\cdots\!71\)\( \nu^{14} + \)\(36\!\cdots\!53\)\( \nu^{13} + \)\(91\!\cdots\!83\)\( \nu^{12} + \)\(25\!\cdots\!93\)\( \nu^{11} + \)\(57\!\cdots\!42\)\( \nu^{10} + \)\(87\!\cdots\!39\)\( \nu^{9} + \)\(22\!\cdots\!19\)\( \nu^{8} + \)\(14\!\cdots\!98\)\( \nu^{7} - \)\(16\!\cdots\!82\)\( \nu^{6} + \)\(48\!\cdots\!94\)\( \nu^{5} - \)\(16\!\cdots\!45\)\( \nu^{4} + \)\(28\!\cdots\!96\)\( \nu^{3} - \)\(10\!\cdots\!01\)\( \nu^{2} + \)\(20\!\cdots\!16\)\( \nu + \)\(33\!\cdots\!56\)\(\)\()/ \)\(20\!\cdots\!60\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(64\!\cdots\!20\)\( \nu^{19} + \)\(50\!\cdots\!70\)\( \nu^{18} + \)\(77\!\cdots\!03\)\( \nu^{17} - \)\(13\!\cdots\!59\)\( \nu^{16} + \)\(60\!\cdots\!62\)\( \nu^{15} - \)\(19\!\cdots\!17\)\( \nu^{14} + \)\(42\!\cdots\!81\)\( \nu^{13} - \)\(66\!\cdots\!21\)\( \nu^{12} + \)\(29\!\cdots\!21\)\( \nu^{11} - \)\(53\!\cdots\!45\)\( \nu^{10} + \)\(10\!\cdots\!96\)\( \nu^{9} - \)\(14\!\cdots\!21\)\( \nu^{8} + \)\(16\!\cdots\!17\)\( \nu^{7} - \)\(89\!\cdots\!24\)\( \nu^{6} + \)\(66\!\cdots\!26\)\( \nu^{5} - \)\(21\!\cdots\!82\)\( \nu^{4} + \)\(41\!\cdots\!67\)\( \nu^{3} - \)\(25\!\cdots\!42\)\( \nu^{2} + \)\(72\!\cdots\!99\)\( \nu - \)\(66\!\cdots\!86\)\(\)\()/ \)\(20\!\cdots\!60\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(64\!\cdots\!30\)\( \nu^{19} - \)\(57\!\cdots\!12\)\( \nu^{18} - \)\(78\!\cdots\!29\)\( \nu^{17} + \)\(12\!\cdots\!27\)\( \nu^{16} - \)\(61\!\cdots\!32\)\( \nu^{15} + \)\(13\!\cdots\!41\)\( \nu^{14} - \)\(43\!\cdots\!01\)\( \nu^{13} + \)\(62\!\cdots\!01\)\( \nu^{12} - \)\(30\!\cdots\!75\)\( \nu^{11} + \)\(50\!\cdots\!01\)\( \nu^{10} - \)\(10\!\cdots\!86\)\( \nu^{9} + \)\(13\!\cdots\!47\)\( \nu^{8} - \)\(16\!\cdots\!79\)\( \nu^{7} + \)\(89\!\cdots\!56\)\( \nu^{6} - \)\(66\!\cdots\!92\)\( \nu^{5} + \)\(21\!\cdots\!02\)\( \nu^{4} - \)\(41\!\cdots\!47\)\( \nu^{3} + \)\(25\!\cdots\!54\)\( \nu^{2} - \)\(73\!\cdots\!41\)\( \nu + \)\(62\!\cdots\!90\)\(\)\()/ \)\(20\!\cdots\!60\)\( \)
\(\beta_{18}\)\(=\)\((\)\(\)\(11\!\cdots\!15\)\( \nu^{19} + \)\(10\!\cdots\!47\)\( \nu^{18} + \)\(13\!\cdots\!24\)\( \nu^{17} - \)\(32\!\cdots\!55\)\( \nu^{16} + \)\(10\!\cdots\!25\)\( \nu^{15} - \)\(10\!\cdots\!27\)\( \nu^{14} + \)\(73\!\cdots\!51\)\( \nu^{13} - \)\(16\!\cdots\!07\)\( \nu^{12} + \)\(51\!\cdots\!44\)\( \nu^{11} - \)\(12\!\cdots\!09\)\( \nu^{10} + \)\(17\!\cdots\!45\)\( \nu^{9} - \)\(36\!\cdots\!20\)\( \nu^{8} + \)\(28\!\cdots\!04\)\( \nu^{7} - \)\(17\!\cdots\!58\)\( \nu^{6} + \)\(11\!\cdots\!27\)\( \nu^{5} - \)\(36\!\cdots\!78\)\( \nu^{4} + \)\(74\!\cdots\!57\)\( \nu^{3} - \)\(46\!\cdots\!66\)\( \nu^{2} + \)\(11\!\cdots\!64\)\( \nu - \)\(52\!\cdots\!28\)\(\)\()/ \)\(20\!\cdots\!60\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(16\!\cdots\!50\)\( \nu^{19} + \)\(20\!\cdots\!98\)\( \nu^{18} + \)\(20\!\cdots\!53\)\( \nu^{17} - \)\(24\!\cdots\!79\)\( \nu^{16} + \)\(15\!\cdots\!86\)\( \nu^{15} + \)\(26\!\cdots\!03\)\( \nu^{14} + \)\(11\!\cdots\!63\)\( \nu^{13} - \)\(11\!\cdots\!23\)\( \nu^{12} + \)\(77\!\cdots\!59\)\( \nu^{11} - \)\(10\!\cdots\!19\)\( \nu^{10} + \)\(26\!\cdots\!68\)\( \nu^{9} - \)\(24\!\cdots\!87\)\( \nu^{8} + \)\(43\!\cdots\!89\)\( \nu^{7} - \)\(20\!\cdots\!10\)\( \nu^{6} + \)\(17\!\cdots\!92\)\( \nu^{5} - \)\(54\!\cdots\!76\)\( \nu^{4} + \)\(10\!\cdots\!91\)\( \nu^{3} - \)\(60\!\cdots\!14\)\( \nu^{2} + \)\(16\!\cdots\!15\)\( \nu - \)\(14\!\cdots\!54\)\(\)\()/ \)\(20\!\cdots\!60\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{19} + \beta_{18} + \beta_{10} + \beta_{6} - 4 \beta_{4}\)
\(\nu^{3}\)\(=\)\(-\beta_{19} - \beta_{17} - \beta_{12} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - 6 \beta_{2}\)
\(\nu^{4}\)\(=\)\(\beta_{17} - \beta_{16} + \beta_{15} + \beta_{11} + 7 \beta_{8} + \beta_{7} - 23 \beta_{6} + \beta_{5} + \beta_{2} - 1\)
\(\nu^{5}\)\(=\)\(2 \beta_{18} + 9 \beta_{17} + 18 \beta_{16} - 9 \beta_{15} - 10 \beta_{14} - 9 \beta_{13} - 8 \beta_{12} + 18 \beta_{11} - 8 \beta_{10} - 11 \beta_{9} - 2 \beta_{8} + \beta_{7} - 9 \beta_{6} - 48 \beta_{5} - 11 \beta_{4} + 11 \beta_{3} - 18 \beta_{2} + 16 \beta_{1} - 11\)
\(\nu^{6}\)\(=\)\(-\beta_{19} - 49 \beta_{18} - 22 \beta_{17} - 10 \beta_{16} + \beta_{15} + 12 \beta_{14} + 10 \beta_{13} + 11 \beta_{12} - 22 \beta_{11} + 10 \beta_{10} + 168 \beta_{9} - 3 \beta_{7} + 157 \beta_{6} + 11 \beta_{5} + 170 \beta_{4} - \beta_{3} + 8 \beta_{2} - 33 \beta_{1} - 160\)
\(\nu^{7}\)\(=\)\(71 \beta_{19} - 71 \beta_{18} + 69 \beta_{17} - 13 \beta_{16} + \beta_{15} + 69 \beta_{14} - \beta_{13} - 13 \beta_{12} + 2 \beta_{11} - 73 \beta_{10} + 26 \beta_{8} + 197 \beta_{7} - 94 \beta_{6} + 266 \beta_{5} + 39 \beta_{4} - 26 \beta_{3} + 276 \beta_{2} - 207 \beta_{1} + 34\)
\(\nu^{8}\)\(=\)\(348 \beta_{19} + 18 \beta_{18} + 113 \beta_{17} + 97 \beta_{16} - 85 \beta_{15} - 194 \beta_{14} - 17 \beta_{13} - 81 \beta_{12} + 114 \beta_{11} - 332 \beta_{10} - 1174 \beta_{9} - 348 \beta_{8} - 16 \beta_{7} - 320 \beta_{6} - 242 \beta_{5} + 12 \beta_{4} + 366 \beta_{3} - 82 \beta_{2} + 242 \beta_{1} + 80\)
\(\nu^{9}\)\(=\)\(-255 \beta_{19} - 635 \beta_{17} - 383 \beta_{16} + 488 \beta_{15} + 509 \beta_{14} + 509 \beta_{13} + 1018 \beta_{12} - 1039 \beta_{11} + 676 \beta_{10} + 930 \beta_{9} - 540 \beta_{8} - 1747 \beta_{7} + 1318 \beta_{6} + 570 \beta_{5} + 1079 \beta_{2} - 1018 \beta_{1} - 400\)
\(\nu^{10}\)\(=\)\(-232 \beta_{18} - 178 \beta_{17} - 356 \beta_{16} + 178 \beta_{15} + 980 \beta_{14} - 419 \beta_{13} - 624 \beta_{12} + 346 \beta_{11} - 1221 \beta_{10} + 42 \beta_{9} + 232 \beta_{8} + 1331 \beta_{7} + 178 \beta_{6} + 2469 \beta_{5} + 42 \beta_{4} - 2727 \beta_{3} + 356 \beta_{2} - 885 \beta_{1} + 7048\)
\(\nu^{11}\)\(=\)\(2265 \beta_{19} + 4052 \beta_{18} + 2224 \beta_{17} - 2604 \beta_{16} - 834 \beta_{15} - 4828 \beta_{14} - 907 \beta_{13} - 1112 \beta_{12} + 2224 \beta_{11} + 1173 \beta_{10} - 11321 \beta_{9} - 1303 \beta_{7} - 8758 \beta_{6} - 1112 \beta_{5} - 3448 \beta_{4} + 2265 \beta_{3} - 2415 \beta_{2} + 12186 \beta_{1} + 6052\)
\(\nu^{12}\)\(=\)\(-18008 \beta_{19} + 18008 \beta_{18} - 1726 \beta_{17} + 6464 \beta_{16} - 834 \beta_{15} - 1726 \beta_{14} + 834 \beta_{13} + 6464 \beta_{12} - 5179 \beta_{11} + 23187 \beta_{10} - 2557 \beta_{8} - 17679 \beta_{7} + 11655 \beta_{6} - 19405 \beta_{5} - 59581 \beta_{4} + 2557 \beta_{3} - 7909 \beta_{2} + 6183 \beta_{1} + 5068\)
\(\nu^{13}\)\(=\)\(-30262 \beta_{19} - 19232 \beta_{18} - 36553 \beta_{17} - 9462 \beta_{16} + 7343 \beta_{15} + 18924 \beta_{14} + 6451 \beta_{13} - 17629 \beta_{12} - 15913 \beta_{11} + 3171 \beta_{10} + 94278 \beta_{9} + 30262 \beta_{8} + 27091 \beta_{7} + 67994 \beta_{6} + 10783 \beta_{5} + 64823 \beta_{4} - 49494 \beta_{3} - 78458 \beta_{2} - 10783 \beta_{1} - 3011\)
\(\nu^{14}\)\(=\)\(25683 \beta_{19} + 67097 \beta_{17} - 35809 \beta_{16} + 15779 \beta_{15} - 15644 \beta_{14} - 15644 \beta_{13} - 31288 \beta_{12} + 62711 \beta_{11} - 27748 \beta_{10} - 27687 \beta_{9} + 130655 \beta_{8} + 108271 \beta_{7} - 376853 \beta_{6} + 80300 \beta_{5} + 64656 \beta_{2} + 31288 \beta_{1} - 19380\)
\(\nu^{15}\)\(=\)\(159501 \beta_{18} + 197887 \beta_{17} + 395774 \beta_{16} - 197887 \beta_{15} - 277088 \beta_{14} - 168664 \beta_{13} - 118686 \beta_{12} + 346521 \beta_{11} - 89463 \beta_{10} - 524470 \beta_{9} - 159501 \beta_{8} - 34051 \beta_{7} - 197887 \beta_{6} - 928965 \beta_{5} - 524470 \beta_{4} + 385359 \beta_{3} - 395774 \beta_{2} + 350624 \beta_{1} - 314427\)
\(\nu^{16}\)\(=\)\(-242805 \beta_{19} - 952108 \beta_{18} - 813282 \beta_{17} - 270246 \beta_{16} + 178806 \beta_{15} + 543036 \beta_{14} + 356663 \beta_{13} + 406641 \beta_{12} - 813282 \beta_{11} + 334245 \beta_{10} + 3389971 \beta_{9} - 377660 \beta_{7} + 3089740 \beta_{6} + 406641 \beta_{5} + 3066295 \beta_{4} - 242805 \beta_{3} + 28981 \beta_{2} - 1184903 \beta_{1} - 2796049\)
\(\nu^{17}\)\(=\)\(1687959 \beta_{19} - 1687959 \beta_{18} + 1450189 \beta_{17} - 657013 \beta_{16} + 178806 \beta_{15} + 1450189 \beta_{14} - 178806 \beta_{13} - 657013 \beta_{12} + 444029 \beta_{11} - 2131988 \beta_{10} + 1304919 \beta_{8} + 4014407 \beta_{7} - 3776479 \beta_{6} + 5464596 \beta_{5} + 2004623 \beta_{4} - 1304919 \beta_{3} + 5011973 \beta_{2} - 3561784 \beta_{1} + 2301504\)
\(\nu^{18}\)\(=\)\(6965155 \beta_{19} + 2201571 \beta_{18} + 4359143 \beta_{17} + 3199137 \beta_{16} - 2807347 \beta_{15} - 6398274 \beta_{14} - 1535964 \beta_{13} - 2039131 \beta_{12} + 4735101 \beta_{11} - 5805149 \beta_{10} - 25499168 \beta_{9} - 6965155 \beta_{8} - 1160006 \beta_{7} - 9158330 \beta_{6} - 9526490 \beta_{5} - 3353181 \beta_{4} + 9166726 \beta_{3} - 2381505 \beta_{2} + 9526490 \beta_{1} + 1663173\)
\(\nu^{19}\)\(=\)\(-10583887 \beta_{19} - 16083999 \beta_{17} - 5250919 \beta_{16} + 8363279 \beta_{15} + 10667459 \beta_{14} + 10667459 \beta_{13} + 21334918 \beta_{12} - 23639098 \beta_{11} + 19674950 \beta_{10} + 33316132 \beta_{9} - 12645097 \beta_{8} - 35720110 \beta_{7} + 49296455 \beta_{6} + 5718483 \beta_{5} + 16385942 \beta_{2} - 21334918 \beta_{1} - 20344493\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/693\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(442\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{9}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
−0.864833 2.66168i
−0.557915 1.71709i
0.0347586 + 0.106976i
0.648215 + 1.99500i
0.739775 + 2.27679i
−2.14548 1.55878i
−1.07866 0.783695i
0.383242 + 0.278442i
0.705143 + 0.512316i
2.13576 + 1.55172i
−0.864833 + 2.66168i
−0.557915 + 1.71709i
0.0347586 0.106976i
0.648215 1.99500i
0.739775 2.27679i
−2.14548 + 1.55878i
−1.07866 + 0.783695i
0.383242 0.278442i
0.705143 0.512316i
2.13576 1.55172i
−0.864833 + 2.66168i 0 −4.71859 3.42825i 0.518429 + 1.59556i 0 0.809017 + 0.587785i 8.67739 6.30449i 0 −4.69523
64.2 −0.557915 + 1.71709i 0 −1.01908 0.740407i −0.858186 2.64122i 0 0.809017 + 0.587785i −1.08138 + 0.785667i 0 5.01400
64.3 0.0347586 0.106976i 0 1.60780 + 1.16813i −0.327964 1.00937i 0 0.809017 + 0.587785i 0.362845 0.263622i 0 −0.119378
64.4 0.648215 1.99500i 0 −1.94181 1.41081i −0.976330 3.00483i 0 0.809017 + 0.587785i −0.679172 + 0.493447i 0 −6.62751
64.5 0.739775 2.27679i 0 −3.01849 2.19306i 1.21700 + 3.74554i 0 0.809017 + 0.587785i −3.35264 + 2.43583i 0 9.42812
190.1 −2.14548 + 1.55878i 0 1.55524 4.78653i 1.38593 + 1.00694i 0 −0.309017 + 0.951057i 2.48543 + 7.64935i 0 −4.54308
190.2 −1.07866 + 0.783695i 0 −0.0686960 + 0.211425i −0.706442 0.513260i 0 −0.309017 + 0.951057i −0.915619 2.81798i 0 1.16425
190.3 0.383242 0.278442i 0 −0.548689 + 1.68869i 3.04094 + 2.20937i 0 −0.309017 + 0.951057i 0.552692 + 1.70101i 0 1.78060
190.4 0.705143 0.512316i 0 −0.383276 + 1.17960i −3.28814 2.38897i 0 −0.309017 + 0.951057i 0.872746 + 2.68604i 0 −3.54252
190.5 2.13576 1.55172i 0 1.53559 4.72606i 2.49476 + 1.81255i 0 −0.309017 + 0.951057i −2.42230 7.45506i 0 8.14075
379.1 −0.864833 2.66168i 0 −4.71859 + 3.42825i 0.518429 1.59556i 0 0.809017 0.587785i 8.67739 + 6.30449i 0 −4.69523
379.2 −0.557915 1.71709i 0 −1.01908 + 0.740407i −0.858186 + 2.64122i 0 0.809017 0.587785i −1.08138 0.785667i 0 5.01400
379.3 0.0347586 + 0.106976i 0 1.60780 1.16813i −0.327964 + 1.00937i 0 0.809017 0.587785i 0.362845 + 0.263622i 0 −0.119378
379.4 0.648215 + 1.99500i 0 −1.94181 + 1.41081i −0.976330 + 3.00483i 0 0.809017 0.587785i −0.679172 0.493447i 0 −6.62751
379.5 0.739775 + 2.27679i 0 −3.01849 + 2.19306i 1.21700 3.74554i 0 0.809017 0.587785i −3.35264 2.43583i 0 9.42812
631.1 −2.14548 1.55878i 0 1.55524 + 4.78653i 1.38593 1.00694i 0 −0.309017 0.951057i 2.48543 7.64935i 0 −4.54308
631.2 −1.07866 0.783695i 0 −0.0686960 0.211425i −0.706442 + 0.513260i 0 −0.309017 0.951057i −0.915619 + 2.81798i 0 1.16425
631.3 0.383242 + 0.278442i 0 −0.548689 1.68869i 3.04094 2.20937i 0 −0.309017 0.951057i 0.552692 1.70101i 0 1.78060
631.4 0.705143 + 0.512316i 0 −0.383276 1.17960i −3.28814 + 2.38897i 0 −0.309017 0.951057i 0.872746 2.68604i 0 −3.54252
631.5 2.13576 + 1.55172i 0 1.53559 + 4.72606i 2.49476 1.81255i 0 −0.309017 0.951057i −2.42230 + 7.45506i 0 8.14075
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 631.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.m.j 20
3.b odd 2 1 231.2.j.g 20
11.c even 5 1 inner 693.2.m.j 20
11.c even 5 1 7623.2.a.cx 10
11.d odd 10 1 7623.2.a.cy 10
33.f even 10 1 2541.2.a.br 10
33.h odd 10 1 231.2.j.g 20
33.h odd 10 1 2541.2.a.bq 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.g 20 3.b odd 2 1
231.2.j.g 20 33.h odd 10 1
693.2.m.j 20 1.a even 1 1 trivial
693.2.m.j 20 11.c even 5 1 inner
2541.2.a.bq 10 33.h odd 10 1
2541.2.a.br 10 33.f even 10 1
7623.2.a.cx 10 11.c even 5 1
7623.2.a.cy 10 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{20} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} - 3 T^{3} + 6 T^{4} + 8 T^{5} + 14 T^{6} - 5 T^{7} + 10 T^{8} - 2 T^{9} + 176 T^{10} - 67 T^{11} + 170 T^{12} - 190 T^{13} + 510 T^{14} + 495 T^{15} + 710 T^{16} - 492 T^{17} + 1850 T^{18} - 927 T^{19} + 8347 T^{20} - 1854 T^{21} + 7400 T^{22} - 3936 T^{23} + 11360 T^{24} + 15840 T^{25} + 32640 T^{26} - 24320 T^{27} + 43520 T^{28} - 34304 T^{29} + 180224 T^{30} - 4096 T^{31} + 40960 T^{32} - 40960 T^{33} + 229376 T^{34} + 262144 T^{35} + 393216 T^{36} - 393216 T^{37} + 524288 T^{38} + 1048576 T^{40} \)
$3$ \( \)
$5$ \( 1 - 5 T + 4 T^{2} + 32 T^{3} - 105 T^{4} + 10 T^{5} + 535 T^{6} - 956 T^{7} - 452 T^{8} + 4251 T^{9} - 8164 T^{10} + 5041 T^{11} + 36064 T^{12} - 140288 T^{13} + 128201 T^{14} + 579422 T^{15} - 1696447 T^{16} + 345724 T^{17} + 4805888 T^{18} - 5640991 T^{19} - 3186474 T^{20} - 28204955 T^{21} + 120147200 T^{22} + 43215500 T^{23} - 1060279375 T^{24} + 1810693750 T^{25} + 2003140625 T^{26} - 10960000000 T^{27} + 14087500000 T^{28} + 9845703125 T^{29} - 79726562500 T^{30} + 207568359375 T^{31} - 110351562500 T^{32} - 1166992187500 T^{33} + 3265380859375 T^{34} + 305175781250 T^{35} - 16021728515625 T^{36} + 24414062500000 T^{37} + 15258789062500 T^{38} - 95367431640625 T^{39} + 95367431640625 T^{40} \)
$7$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5} \)
$11$ \( 1 - T - T^{2} - 25 T^{3} + 246 T^{4} - 17 T^{5} - 807 T^{6} - 1153 T^{7} + 25424 T^{8} + 22607 T^{9} - 174843 T^{10} + 248677 T^{11} + 3076304 T^{12} - 1534643 T^{13} - 11815287 T^{14} - 2737867 T^{15} + 435804006 T^{16} - 487179275 T^{17} - 214358881 T^{18} - 2357947691 T^{19} + 25937424601 T^{20} \)
$13$ \( 1 - 13 T + 66 T^{2} - 139 T^{3} + 51 T^{4} - 2859 T^{5} + 33752 T^{6} - 143455 T^{7} + 202546 T^{8} + 225355 T^{9} + 2636810 T^{10} - 30800939 T^{11} + 94014139 T^{12} + 15525383 T^{13} - 616206028 T^{14} - 489756397 T^{15} + 8014753706 T^{16} + 3731069987 T^{17} - 135009335494 T^{18} + 334766284429 T^{19} - 343304150125 T^{20} + 4351961697577 T^{21} - 22816577698486 T^{22} + 8197160761439 T^{23} + 228909380597066 T^{24} - 181843121911321 T^{25} - 2974308801804652 T^{26} + 974194759107011 T^{27} + 76690221390664219 T^{28} - 326628538313311247 T^{29} + 363506649892361690 T^{30} + 403872305598208135 T^{31} + 4718933949218036626 T^{32} - 43448948416191654115 T^{33} + \)\(13\!\cdots\!28\)\( T^{34} - \)\(14\!\cdots\!63\)\( T^{35} + 33936247068342171891 T^{36} - \)\(12\!\cdots\!87\)\( T^{37} + \)\(74\!\cdots\!14\)\( T^{38} - \)\(19\!\cdots\!01\)\( T^{39} + \)\(19\!\cdots\!01\)\( T^{40} \)
$17$ \( 1 - T - 34 T^{2} + 41 T^{3} + 811 T^{4} + 3837 T^{5} - 18522 T^{6} - 139403 T^{7} + 389000 T^{8} + 3251903 T^{9} + 2685136 T^{10} - 59936211 T^{11} - 176272043 T^{12} + 1070308079 T^{13} + 4384181682 T^{14} - 6978997953 T^{15} - 68502447326 T^{16} + 33742794795 T^{17} + 1218398812778 T^{18} + 458231992717 T^{19} - 15300717915965 T^{20} + 7789943876189 T^{21} + 352117256892842 T^{22} + 165778350827835 T^{23} - 5721392903114846 T^{24} - 9909179096552721 T^{25} + 105823487857811058 T^{26} + 439188796838039167 T^{27} - 1229631015597521963 T^{28} - 7107707987766132867 T^{29} + 5413217797876026064 T^{30} + \)\(11\!\cdots\!99\)\( T^{31} + \)\(22\!\cdots\!00\)\( T^{32} - \)\(13\!\cdots\!11\)\( T^{33} - \)\(31\!\cdots\!38\)\( T^{34} + \)\(10\!\cdots\!41\)\( T^{35} + \)\(39\!\cdots\!91\)\( T^{36} + \)\(33\!\cdots\!57\)\( T^{37} - \)\(47\!\cdots\!06\)\( T^{38} - \)\(23\!\cdots\!53\)\( T^{39} + \)\(40\!\cdots\!01\)\( T^{40} \)
$19$ \( 1 - 10 T - 2 T^{2} + 248 T^{3} + 82 T^{4} - 2032 T^{5} - 23994 T^{6} + 41732 T^{7} + 518146 T^{8} - 926242 T^{9} + 540084 T^{10} - 30787074 T^{11} + 50342382 T^{12} + 703180340 T^{13} - 2471883182 T^{14} - 3864155488 T^{15} - 21312045690 T^{16} + 241124991192 T^{17} + 917942443426 T^{18} - 4413260805162 T^{19} - 5762405945050 T^{20} - 83851955298078 T^{21} + 331377222076786 T^{22} + 1653876314585928 T^{23} - 2777407106366490 T^{24} - 9568031539681312 T^{25} - 116291922026273342 T^{26} + 628553033346411260 T^{27} + 854993018331103662 T^{28} - 9934610030411708646 T^{29} + 3311290788778195284 T^{30} - \)\(10\!\cdots\!98\)\( T^{31} + \)\(11\!\cdots\!06\)\( T^{32} + \)\(17\!\cdots\!88\)\( T^{33} - \)\(19\!\cdots\!74\)\( T^{34} - \)\(30\!\cdots\!68\)\( T^{35} + \)\(23\!\cdots\!42\)\( T^{36} + \)\(13\!\cdots\!72\)\( T^{37} - \)\(20\!\cdots\!82\)\( T^{38} - \)\(19\!\cdots\!90\)\( T^{39} + \)\(37\!\cdots\!01\)\( T^{40} \)
$23$ \( ( 1 + 124 T^{2} - 38 T^{3} + 8330 T^{4} - 3634 T^{5} + 381506 T^{6} - 180342 T^{7} + 12965000 T^{8} - 5936040 T^{9} + 338330610 T^{10} - 136528920 T^{11} + 6858485000 T^{12} - 2194221114 T^{13} + 106761020546 T^{14} - 23389670462 T^{15} + 1233138955370 T^{16} - 129383366986 T^{17} + 9710562174844 T^{18} + 41426511213649 T^{20} )^{2} \)
$29$ \( 1 + 3 T - 4 T^{2} - 59 T^{3} + 609 T^{4} - 7469 T^{5} - 58340 T^{6} - 18675 T^{7} + 564354 T^{8} - 1565053 T^{9} - 1348916 T^{10} + 228651093 T^{11} + 603011151 T^{12} + 2322397715 T^{13} - 7665582012 T^{14} - 36331461243 T^{15} + 746249961816 T^{16} + 576665139899 T^{17} - 30471253369428 T^{18} - 122834837258411 T^{19} + 929104458856239 T^{20} - 3562210280493919 T^{21} - 25626324083688948 T^{22} + 14064286096996711 T^{23} + 527808419243182296 T^{24} - 745200014942898207 T^{25} - 4559666949775701852 T^{26} + 40061073324054233935 T^{27} + \)\(30\!\cdots\!11\)\( T^{28} + \)\(33\!\cdots\!17\)\( T^{29} - \)\(56\!\cdots\!16\)\( T^{30} - \)\(19\!\cdots\!37\)\( T^{31} + \)\(19\!\cdots\!14\)\( T^{32} - \)\(19\!\cdots\!75\)\( T^{33} - \)\(17\!\cdots\!40\)\( T^{34} - \)\(64\!\cdots\!81\)\( T^{35} + \)\(15\!\cdots\!89\)\( T^{36} - \)\(42\!\cdots\!31\)\( T^{37} - \)\(84\!\cdots\!44\)\( T^{38} + \)\(18\!\cdots\!07\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$31$ \( 1 + 13 T + 65 T^{2} + 146 T^{3} + 295 T^{4} + 2802 T^{5} + 48827 T^{6} + 252198 T^{7} + 448847 T^{8} + 5903225 T^{9} + 65860972 T^{10} + 114046885 T^{11} - 1198096047 T^{12} - 10361131234 T^{13} - 7489384555 T^{14} + 470405899010 T^{15} + 3347900754057 T^{16} + 6862384130018 T^{17} + 25638402034775 T^{18} + 239884734794937 T^{19} + 1493178408224822 T^{20} + 7436426778643047 T^{21} + 24638504355418775 T^{22} + 204437285617366238 T^{23} + 3091856652287474697 T^{24} + 13467321514048040510 T^{25} - 6646856360987046955 T^{26} - \)\(28\!\cdots\!74\)\( T^{27} - \)\(10\!\cdots\!27\)\( T^{28} + \)\(30\!\cdots\!35\)\( T^{29} + \)\(53\!\cdots\!72\)\( T^{30} + \)\(14\!\cdots\!75\)\( T^{31} + \)\(35\!\cdots\!67\)\( T^{32} + \)\(61\!\cdots\!18\)\( T^{33} + \)\(36\!\cdots\!67\)\( T^{34} + \)\(65\!\cdots\!02\)\( T^{35} + \)\(21\!\cdots\!95\)\( T^{36} + \)\(32\!\cdots\!06\)\( T^{37} + \)\(45\!\cdots\!65\)\( T^{38} + \)\(28\!\cdots\!23\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \)
$37$ \( 1 + 32 T + 496 T^{2} + 5340 T^{3} + 50712 T^{4} + 444144 T^{5} + 3184776 T^{6} + 17020156 T^{7} + 60430936 T^{8} - 66824016 T^{9} - 4566815524 T^{10} - 57414614544 T^{11} - 482483421256 T^{12} - 3283683956236 T^{13} - 19092317847912 T^{14} - 82953054964976 T^{15} - 130679609958264 T^{16} + 1790702451469140 T^{17} + 24993747942875904 T^{18} + 219578098429763712 T^{19} + 1506082500954951158 T^{20} + 8124389641901257344 T^{21} + 34216440933797112576 T^{22} + 90704451274266348420 T^{23} - \)\(24\!\cdots\!04\)\( T^{24} - \)\(57\!\cdots\!32\)\( T^{25} - \)\(48\!\cdots\!08\)\( T^{26} - \)\(31\!\cdots\!88\)\( T^{27} - \)\(16\!\cdots\!76\)\( T^{28} - \)\(74\!\cdots\!88\)\( T^{29} - \)\(21\!\cdots\!76\)\( T^{30} - \)\(11\!\cdots\!08\)\( T^{31} + \)\(39\!\cdots\!16\)\( T^{32} + \)\(41\!\cdots\!32\)\( T^{33} + \)\(28\!\cdots\!64\)\( T^{34} + \)\(14\!\cdots\!92\)\( T^{35} + \)\(62\!\cdots\!92\)\( T^{36} + \)\(24\!\cdots\!80\)\( T^{37} + \)\(83\!\cdots\!84\)\( T^{38} + \)\(19\!\cdots\!36\)\( T^{39} + \)\(23\!\cdots\!01\)\( T^{40} \)
$41$ \( 1 - 3 T - 151 T^{2} - 283 T^{3} + 15260 T^{4} + 53807 T^{5} - 908059 T^{6} - 5105899 T^{7} + 41696524 T^{8} + 332390297 T^{9} - 1424132099 T^{10} - 18190782425 T^{11} + 13662607978 T^{12} + 883993384739 T^{13} + 2395186168445 T^{14} - 33386146780263 T^{15} - 238454984830174 T^{16} + 934346443615571 T^{17} + 13814720376087777 T^{18} - 13438189208809197 T^{19} - 619586511510882716 T^{20} - 550965757561177077 T^{21} + 23222544952203553137 T^{22} + 64396091240428768891 T^{23} - \)\(67\!\cdots\!14\)\( T^{24} - \)\(38\!\cdots\!63\)\( T^{25} + \)\(11\!\cdots\!45\)\( T^{26} + \)\(17\!\cdots\!59\)\( T^{27} + \)\(10\!\cdots\!38\)\( T^{28} - \)\(59\!\cdots\!25\)\( T^{29} - \)\(19\!\cdots\!99\)\( T^{30} + \)\(18\!\cdots\!77\)\( T^{31} + \)\(94\!\cdots\!44\)\( T^{32} - \)\(47\!\cdots\!79\)\( T^{33} - \)\(34\!\cdots\!99\)\( T^{34} + \)\(83\!\cdots\!07\)\( T^{35} + \)\(97\!\cdots\!60\)\( T^{36} - \)\(73\!\cdots\!23\)\( T^{37} - \)\(16\!\cdots\!71\)\( T^{38} - \)\(13\!\cdots\!83\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$43$ \( ( 1 - 6 T + 312 T^{2} - 1828 T^{3} + 47319 T^{4} - 259736 T^{5} + 4576717 T^{6} - 22838020 T^{7} + 311402464 T^{8} - 1377461130 T^{9} + 15540959670 T^{10} - 59230828590 T^{11} + 575783155936 T^{12} - 1815782456140 T^{13} + 15646884656317 T^{14} - 38183384951048 T^{15} + 299120578115631 T^{16} - 496884421103596 T^{17} + 3646718486611512 T^{18} - 3015555671621058 T^{19} + 21611482313284249 T^{20} )^{2} \)
$47$ \( 1 + 20 T + 141 T^{2} + 502 T^{3} + 4037 T^{4} - 6218 T^{5} - 801001 T^{6} - 7197116 T^{7} - 33989974 T^{8} - 317744608 T^{9} - 1914149671 T^{10} + 9965865086 T^{11} + 157004029439 T^{12} + 867584843046 T^{13} + 9623990203403 T^{14} + 81333165828100 T^{15} + 137676757256356 T^{16} - 1184018410587624 T^{17} - 6791369921638187 T^{18} - 141773649130018030 T^{19} - 1649383233650553737 T^{20} - 6663361509110847410 T^{21} - 15002136156898755083 T^{22} - \)\(12\!\cdots\!52\)\( T^{23} + \)\(67\!\cdots\!36\)\( T^{24} + \)\(18\!\cdots\!00\)\( T^{25} + \)\(10\!\cdots\!87\)\( T^{26} + \)\(43\!\cdots\!98\)\( T^{27} + \)\(37\!\cdots\!79\)\( T^{28} + \)\(11\!\cdots\!62\)\( T^{29} - \)\(10\!\cdots\!79\)\( T^{30} - \)\(78\!\cdots\!24\)\( T^{31} - \)\(39\!\cdots\!34\)\( T^{32} - \)\(39\!\cdots\!32\)\( T^{33} - \)\(20\!\cdots\!69\)\( T^{34} - \)\(75\!\cdots\!74\)\( T^{35} + \)\(22\!\cdots\!77\)\( T^{36} + \)\(13\!\cdots\!74\)\( T^{37} + \)\(17\!\cdots\!49\)\( T^{38} + \)\(11\!\cdots\!60\)\( T^{39} + \)\(27\!\cdots\!01\)\( T^{40} \)
$53$ \( 1 + 3 T - 126 T^{2} - 1842 T^{3} - 51 T^{4} + 176024 T^{5} + 1687499 T^{6} - 693786 T^{7} - 129427770 T^{8} - 1030515651 T^{9} + 22821416 T^{10} + 58177807579 T^{11} + 439171427598 T^{12} + 484180552566 T^{13} - 15615841075935 T^{14} - 125664083280776 T^{15} - 350599946515409 T^{16} + 1684175373629766 T^{17} + 16682975693935818 T^{18} + 76739832358344117 T^{19} + 165299081934488990 T^{20} + 4067211114992238201 T^{21} + 46862478724265712762 T^{22} + \)\(25\!\cdots\!82\)\( T^{23} - \)\(27\!\cdots\!29\)\( T^{24} - \)\(52\!\cdots\!68\)\( T^{25} - \)\(34\!\cdots\!15\)\( T^{26} + \)\(56\!\cdots\!42\)\( T^{27} + \)\(27\!\cdots\!78\)\( T^{28} + \)\(19\!\cdots\!07\)\( T^{29} + \)\(39\!\cdots\!84\)\( T^{30} - \)\(95\!\cdots\!47\)\( T^{31} - \)\(63\!\cdots\!70\)\( T^{32} - \)\(18\!\cdots\!78\)\( T^{33} + \)\(23\!\cdots\!31\)\( T^{34} + \)\(12\!\cdots\!68\)\( T^{35} - \)\(19\!\cdots\!71\)\( T^{36} - \)\(37\!\cdots\!46\)\( T^{37} - \)\(13\!\cdots\!14\)\( T^{38} + \)\(17\!\cdots\!51\)\( T^{39} + \)\(30\!\cdots\!01\)\( T^{40} \)
$59$ \( 1 - 9 T - 11 T^{2} + 2009 T^{3} - 16566 T^{4} + 39875 T^{5} + 1481949 T^{6} - 16040319 T^{7} + 91259286 T^{8} + 482457503 T^{9} - 9939821477 T^{10} + 75582121635 T^{11} - 12007856396 T^{12} - 4148742606411 T^{13} + 36852203890145 T^{14} - 106193218742209 T^{15} - 1378906768809232 T^{16} + 14879013797272333 T^{17} - 71002723235317991 T^{18} - 551840607836296325 T^{19} + 6375429074335094320 T^{20} - 32558595862341483175 T^{21} - \)\(24\!\cdots\!71\)\( T^{22} + \)\(30\!\cdots\!07\)\( T^{23} - \)\(16\!\cdots\!52\)\( T^{24} - \)\(75\!\cdots\!91\)\( T^{25} + \)\(15\!\cdots\!45\)\( T^{26} - \)\(10\!\cdots\!09\)\( T^{27} - \)\(17\!\cdots\!16\)\( T^{28} + \)\(65\!\cdots\!65\)\( T^{29} - \)\(50\!\cdots\!77\)\( T^{30} + \)\(14\!\cdots\!77\)\( T^{31} + \)\(16\!\cdots\!66\)\( T^{32} - \)\(16\!\cdots\!01\)\( T^{33} + \)\(91\!\cdots\!89\)\( T^{34} + \)\(14\!\cdots\!25\)\( T^{35} - \)\(35\!\cdots\!06\)\( T^{36} + \)\(25\!\cdots\!71\)\( T^{37} - \)\(82\!\cdots\!31\)\( T^{38} - \)\(39\!\cdots\!51\)\( T^{39} + \)\(26\!\cdots\!01\)\( T^{40} \)
$61$ \( 1 + 15 T - 90 T^{2} - 3165 T^{3} - 19585 T^{4} + 92683 T^{5} + 2977530 T^{6} + 30435495 T^{7} + 58060720 T^{8} - 2575904645 T^{9} - 32871179972 T^{10} - 119504505635 T^{11} + 1285823633985 T^{12} + 22572191661785 T^{13} + 150060548339790 T^{14} - 142916662288619 T^{15} - 12943544265704350 T^{16} - 115634960295586275 T^{17} - 240942316112795150 T^{18} + 4864605484258100025 T^{19} + 61052179162441039695 T^{20} + \)\(29\!\cdots\!25\)\( T^{21} - \)\(89\!\cdots\!50\)\( T^{22} - \)\(26\!\cdots\!75\)\( T^{23} - \)\(17\!\cdots\!50\)\( T^{24} - \)\(12\!\cdots\!19\)\( T^{25} + \)\(77\!\cdots\!90\)\( T^{26} + \)\(70\!\cdots\!85\)\( T^{27} + \)\(24\!\cdots\!85\)\( T^{28} - \)\(13\!\cdots\!35\)\( T^{29} - \)\(23\!\cdots\!72\)\( T^{30} - \)\(11\!\cdots\!45\)\( T^{31} + \)\(15\!\cdots\!20\)\( T^{32} + \)\(49\!\cdots\!95\)\( T^{33} + \)\(29\!\cdots\!30\)\( T^{34} + \)\(55\!\cdots\!83\)\( T^{35} - \)\(71\!\cdots\!85\)\( T^{36} - \)\(70\!\cdots\!65\)\( T^{37} - \)\(12\!\cdots\!90\)\( T^{38} + \)\(12\!\cdots\!15\)\( T^{39} + \)\(50\!\cdots\!01\)\( T^{40} \)
$67$ \( ( 1 - 38 T + 939 T^{2} - 16920 T^{3} + 253022 T^{4} - 3231648 T^{5} + 37034650 T^{6} - 384823180 T^{7} + 3720320353 T^{8} - 33446868814 T^{9} + 283134556406 T^{10} - 2240940210538 T^{11} + 16700518064617 T^{12} - 115740574086340 T^{13} + 746289713342650 T^{14} - 4363129101786336 T^{15} + 22887960773164718 T^{16} - 102547240362065160 T^{17} + 381297549225685899 T^{18} - 1033848307059207986 T^{19} + 1822837804551761449 T^{20} )^{2} \)
$71$ \( 1 + 37 T + 515 T^{2} + 3274 T^{3} + 13215 T^{4} + 211688 T^{5} + 4684917 T^{6} + 60912592 T^{7} + 467488647 T^{8} + 2980461565 T^{9} + 33268020472 T^{10} + 428318804705 T^{11} + 4339324633273 T^{12} + 35873672109364 T^{13} + 288523918504695 T^{14} + 2785853372108440 T^{15} + 26650571131092837 T^{16} + 220452230480408382 T^{17} + 1885088405364310445 T^{18} + 17288624106824488873 T^{19} + \)\(15\!\cdots\!82\)\( T^{20} + \)\(12\!\cdots\!83\)\( T^{21} + \)\(95\!\cdots\!45\)\( T^{22} + \)\(78\!\cdots\!02\)\( T^{23} + \)\(67\!\cdots\!97\)\( T^{24} + \)\(50\!\cdots\!40\)\( T^{25} + \)\(36\!\cdots\!95\)\( T^{26} + \)\(32\!\cdots\!24\)\( T^{27} + \)\(28\!\cdots\!53\)\( T^{28} + \)\(19\!\cdots\!55\)\( T^{29} + \)\(10\!\cdots\!72\)\( T^{30} + \)\(68\!\cdots\!15\)\( T^{31} + \)\(76\!\cdots\!27\)\( T^{32} + \)\(70\!\cdots\!12\)\( T^{33} + \)\(38\!\cdots\!77\)\( T^{34} + \)\(12\!\cdots\!88\)\( T^{35} + \)\(55\!\cdots\!15\)\( T^{36} + \)\(96\!\cdots\!34\)\( T^{37} + \)\(10\!\cdots\!15\)\( T^{38} + \)\(55\!\cdots\!47\)\( T^{39} + \)\(10\!\cdots\!01\)\( T^{40} \)
$73$ \( 1 - 27 T + 117 T^{2} + 3825 T^{3} - 60386 T^{4} + 206645 T^{5} + 5076633 T^{6} - 88912135 T^{7} + 412271510 T^{8} + 5307403509 T^{9} - 94198639093 T^{10} + 498629925791 T^{11} + 2922251864844 T^{12} - 76647599821181 T^{13} + 554955524573989 T^{14} + 735710433225271 T^{15} - 49117193322293732 T^{16} + 443373919368750715 T^{17} - 1123922652740353551 T^{18} - 21964116604269227841 T^{19} + \)\(31\!\cdots\!04\)\( T^{20} - \)\(16\!\cdots\!93\)\( T^{21} - \)\(59\!\cdots\!79\)\( T^{22} + \)\(17\!\cdots\!55\)\( T^{23} - \)\(13\!\cdots\!12\)\( T^{24} + \)\(15\!\cdots\!03\)\( T^{25} + \)\(83\!\cdots\!21\)\( T^{26} - \)\(84\!\cdots\!57\)\( T^{27} + \)\(23\!\cdots\!64\)\( T^{28} + \)\(29\!\cdots\!83\)\( T^{29} - \)\(40\!\cdots\!57\)\( T^{30} + \)\(16\!\cdots\!93\)\( T^{31} + \)\(94\!\cdots\!10\)\( T^{32} - \)\(14\!\cdots\!55\)\( T^{33} + \)\(61\!\cdots\!97\)\( T^{34} + \)\(18\!\cdots\!65\)\( T^{35} - \)\(39\!\cdots\!46\)\( T^{36} + \)\(18\!\cdots\!25\)\( T^{37} + \)\(40\!\cdots\!73\)\( T^{38} - \)\(68\!\cdots\!99\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$79$ \( 1 - 5 T - 226 T^{2} + 1188 T^{3} + 28293 T^{4} - 182558 T^{5} - 2094253 T^{6} + 20887148 T^{7} + 66066202 T^{8} - 1685198093 T^{9} + 5203173736 T^{10} + 81171391019 T^{11} - 899452563978 T^{12} + 259180666124 T^{13} + 50685690960293 T^{14} - 453111325450158 T^{15} + 2532348314805923 T^{16} + 41547309775010276 T^{17} - 818375975431692878 T^{18} - 1321261704586397645 T^{19} + 85108949992387444270 T^{20} - \)\(10\!\cdots\!55\)\( T^{21} - \)\(51\!\cdots\!98\)\( T^{22} + \)\(20\!\cdots\!64\)\( T^{23} + \)\(98\!\cdots\!63\)\( T^{24} - \)\(13\!\cdots\!42\)\( T^{25} + \)\(12\!\cdots\!53\)\( T^{26} + \)\(49\!\cdots\!16\)\( T^{27} - \)\(13\!\cdots\!58\)\( T^{28} + \)\(97\!\cdots\!61\)\( T^{29} + \)\(49\!\cdots\!36\)\( T^{30} - \)\(12\!\cdots\!47\)\( T^{31} + \)\(39\!\cdots\!82\)\( T^{32} + \)\(97\!\cdots\!72\)\( T^{33} - \)\(77\!\cdots\!93\)\( T^{34} - \)\(53\!\cdots\!42\)\( T^{35} + \)\(65\!\cdots\!53\)\( T^{36} + \)\(21\!\cdots\!92\)\( T^{37} - \)\(32\!\cdots\!86\)\( T^{38} - \)\(56\!\cdots\!95\)\( T^{39} + \)\(89\!\cdots\!01\)\( T^{40} \)
$83$ \( 1 - 42 T + 661 T^{2} - 3786 T^{3} - 19549 T^{4} + 561934 T^{5} - 5968953 T^{6} + 20177190 T^{7} + 432561126 T^{8} - 6143013970 T^{9} + 28916165545 T^{10} - 66576463426 T^{11} + 144701828969 T^{12} + 5596879899702 T^{13} - 23882904399413 T^{14} + 248862974645262 T^{15} - 21903508692713044 T^{16} + 235757216137931078 T^{17} + 483849794965596541 T^{18} - 23802972835548797594 T^{19} + \)\(24\!\cdots\!45\)\( T^{20} - \)\(19\!\cdots\!02\)\( T^{21} + \)\(33\!\cdots\!49\)\( T^{22} + \)\(13\!\cdots\!86\)\( T^{23} - \)\(10\!\cdots\!24\)\( T^{24} + \)\(98\!\cdots\!66\)\( T^{25} - \)\(78\!\cdots\!97\)\( T^{26} + \)\(15\!\cdots\!54\)\( T^{27} + \)\(32\!\cdots\!29\)\( T^{28} - \)\(12\!\cdots\!78\)\( T^{29} + \)\(44\!\cdots\!05\)\( T^{30} - \)\(79\!\cdots\!90\)\( T^{31} + \)\(46\!\cdots\!86\)\( T^{32} + \)\(17\!\cdots\!70\)\( T^{33} - \)\(43\!\cdots\!37\)\( T^{34} + \)\(34\!\cdots\!38\)\( T^{35} - \)\(99\!\cdots\!69\)\( T^{36} - \)\(15\!\cdots\!78\)\( T^{37} + \)\(23\!\cdots\!49\)\( T^{38} - \)\(12\!\cdots\!74\)\( T^{39} + \)\(24\!\cdots\!01\)\( T^{40} \)
$89$ \( ( 1 - 9 T + 640 T^{2} - 4050 T^{3} + 178355 T^{4} - 698876 T^{5} + 29313265 T^{6} - 51732294 T^{7} + 3369419368 T^{8} - 1036172651 T^{9} + 317703552166 T^{10} - 92219365939 T^{11} + 26689170813928 T^{12} - 36469663568886 T^{13} + 1839179937126865 T^{14} - 3902565131479324 T^{15} + 88639098149349155 T^{16} - 179136906326892450 T^{17} + 2519416835649331840 T^{18} - 3153207633367366881 T^{19} + 31181719929966183601 T^{20} )^{2} \)
$97$ \( 1 + 27 T - 146 T^{2} - 10565 T^{3} - 42323 T^{4} + 1720741 T^{5} + 15722390 T^{6} - 118678851 T^{7} - 2079209414 T^{8} - 3980410181 T^{9} + 86169204810 T^{10} + 1330688850015 T^{11} + 13895090921091 T^{12} - 18497220903723 T^{13} - 2497628718197930 T^{14} - 21619068182342995 T^{15} + 116376235825961792 T^{16} + 3142408121720289719 T^{17} + 13315354121489765710 T^{18} - \)\(14\!\cdots\!49\)\( T^{19} - \)\(24\!\cdots\!21\)\( T^{20} - \)\(14\!\cdots\!53\)\( T^{21} + \)\(12\!\cdots\!90\)\( T^{22} + \)\(28\!\cdots\!87\)\( T^{23} + \)\(10\!\cdots\!52\)\( T^{24} - \)\(18\!\cdots\!15\)\( T^{25} - \)\(20\!\cdots\!70\)\( T^{26} - \)\(14\!\cdots\!99\)\( T^{27} + \)\(10\!\cdots\!51\)\( T^{28} + \)\(10\!\cdots\!55\)\( T^{29} + \)\(63\!\cdots\!90\)\( T^{30} - \)\(28\!\cdots\!93\)\( T^{31} - \)\(14\!\cdots\!74\)\( T^{32} - \)\(79\!\cdots\!27\)\( T^{33} + \)\(10\!\cdots\!10\)\( T^{34} + \)\(10\!\cdots\!13\)\( T^{35} - \)\(25\!\cdots\!83\)\( T^{36} - \)\(62\!\cdots\!05\)\( T^{37} - \)\(84\!\cdots\!94\)\( T^{38} + \)\(15\!\cdots\!91\)\( T^{39} + \)\(54\!\cdots\!01\)\( T^{40} \)
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