Properties

Label 430.2.k.c
Level 430
Weight 2
Character orbit 430.k
Analytic conductor 3.434
Analytic rank 0
Dimension 18
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 430.k (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 5 x^{17} + 20 x^{16} - 61 x^{15} + 142 x^{14} - 195 x^{13} + 244 x^{12} + 320 x^{11} + 64 x^{10} - 562 x^{9} + 4114 x^{8} - 1933 x^{7} + 2941 x^{6} - 2555 x^{5} + 2807 x^{4} - 2366 x^{3} + 1225 x^{2} - 343 x + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(82\!\cdots\!53\)\( \nu^{17} + \)\(15\!\cdots\!30\)\( \nu^{16} - \)\(35\!\cdots\!88\)\( \nu^{15} + \)\(28\!\cdots\!38\)\( \nu^{14} - \)\(11\!\cdots\!13\)\( \nu^{13} + \)\(36\!\cdots\!73\)\( \nu^{12} - \)\(44\!\cdots\!58\)\( \nu^{11} + \)\(10\!\cdots\!06\)\( \nu^{10} + \)\(17\!\cdots\!21\)\( \nu^{9} + \)\(28\!\cdots\!32\)\( \nu^{8} + \)\(83\!\cdots\!25\)\( \nu^{7} + \)\(15\!\cdots\!74\)\( \nu^{6} - \)\(74\!\cdots\!57\)\( \nu^{5} + \)\(71\!\cdots\!03\)\( \nu^{4} - \)\(73\!\cdots\!93\)\( \nu^{3} + \)\(77\!\cdots\!07\)\( \nu^{2} - \)\(67\!\cdots\!23\)\( \nu + \)\(17\!\cdots\!38\)\(\)\()/ \)\(14\!\cdots\!08\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(19\!\cdots\!05\)\( \nu^{17} - \)\(59\!\cdots\!74\)\( \nu^{16} + \)\(21\!\cdots\!66\)\( \nu^{15} - \)\(49\!\cdots\!92\)\( \nu^{14} + \)\(68\!\cdots\!85\)\( \nu^{13} + \)\(84\!\cdots\!95\)\( \nu^{12} - \)\(10\!\cdots\!18\)\( \nu^{11} + \)\(12\!\cdots\!28\)\( \nu^{10} + \)\(15\!\cdots\!47\)\( \nu^{9} - \)\(69\!\cdots\!58\)\( \nu^{8} + \)\(55\!\cdots\!33\)\( \nu^{7} + \)\(10\!\cdots\!42\)\( \nu^{6} + \)\(16\!\cdots\!85\)\( \nu^{5} + \)\(18\!\cdots\!49\)\( \nu^{4} - \)\(36\!\cdots\!73\)\( \nu^{3} - \)\(42\!\cdots\!11\)\( \nu^{2} - \)\(46\!\cdots\!31\)\( \nu + \)\(12\!\cdots\!36\)\(\)\()/ \)\(14\!\cdots\!08\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(27\!\cdots\!98\)\( \nu^{17} - \)\(17\!\cdots\!64\)\( \nu^{16} + \)\(71\!\cdots\!01\)\( \nu^{15} - \)\(23\!\cdots\!63\)\( \nu^{14} + \)\(57\!\cdots\!83\)\( \nu^{13} - \)\(93\!\cdots\!11\)\( \nu^{12} + \)\(11\!\cdots\!86\)\( \nu^{11} + \)\(33\!\cdots\!21\)\( \nu^{10} - \)\(13\!\cdots\!69\)\( \nu^{9} - \)\(26\!\cdots\!69\)\( \nu^{8} + \)\(13\!\cdots\!82\)\( \nu^{7} - \)\(19\!\cdots\!94\)\( \nu^{6} + \)\(68\!\cdots\!81\)\( \nu^{5} - \)\(16\!\cdots\!11\)\( \nu^{4} + \)\(11\!\cdots\!12\)\( \nu^{3} - \)\(14\!\cdots\!63\)\( \nu^{2} + \)\(71\!\cdots\!66\)\( \nu - \)\(18\!\cdots\!83\)\(\)\()/ \)\(14\!\cdots\!08\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(10\!\cdots\!44\)\( \nu^{17} + \)\(16\!\cdots\!29\)\( \nu^{16} - \)\(47\!\cdots\!05\)\( \nu^{15} + \)\(49\!\cdots\!87\)\( \nu^{14} + \)\(36\!\cdots\!01\)\( \nu^{13} - \)\(20\!\cdots\!24\)\( \nu^{12} + \)\(21\!\cdots\!65\)\( \nu^{11} - \)\(92\!\cdots\!41\)\( \nu^{10} - \)\(15\!\cdots\!59\)\( \nu^{9} - \)\(39\!\cdots\!82\)\( \nu^{8} - \)\(26\!\cdots\!06\)\( \nu^{7} - \)\(11\!\cdots\!13\)\( \nu^{6} - \)\(36\!\cdots\!87\)\( \nu^{5} - \)\(66\!\cdots\!76\)\( \nu^{4} + \)\(12\!\cdots\!91\)\( \nu^{3} - \)\(44\!\cdots\!22\)\( \nu^{2} + \)\(33\!\cdots\!49\)\( \nu - \)\(78\!\cdots\!18\)\(\)\()/ \)\(14\!\cdots\!08\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(58\!\cdots\!99\)\( \nu^{17} + \)\(27\!\cdots\!64\)\( \nu^{16} - \)\(10\!\cdots\!31\)\( \nu^{15} + \)\(32\!\cdots\!27\)\( \nu^{14} - \)\(71\!\cdots\!56\)\( \nu^{13} + \)\(86\!\cdots\!12\)\( \nu^{12} - \)\(10\!\cdots\!92\)\( \nu^{11} - \)\(23\!\cdots\!89\)\( \nu^{10} - \)\(90\!\cdots\!64\)\( \nu^{9} + \)\(33\!\cdots\!95\)\( \nu^{8} - \)\(23\!\cdots\!93\)\( \nu^{7} + \)\(35\!\cdots\!68\)\( \nu^{6} - \)\(12\!\cdots\!46\)\( \nu^{5} + \)\(95\!\cdots\!68\)\( \nu^{4} - \)\(10\!\cdots\!95\)\( \nu^{3} + \)\(81\!\cdots\!36\)\( \nu^{2} - \)\(28\!\cdots\!53\)\( \nu - \)\(30\!\cdots\!09\)\(\)\()/ \)\(73\!\cdots\!54\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(11\!\cdots\!40\)\( \nu^{17} + \)\(62\!\cdots\!36\)\( \nu^{16} - \)\(24\!\cdots\!61\)\( \nu^{15} + \)\(77\!\cdots\!55\)\( \nu^{14} - \)\(18\!\cdots\!37\)\( \nu^{13} + \)\(25\!\cdots\!01\)\( \nu^{12} - \)\(29\!\cdots\!22\)\( \nu^{11} - \)\(36\!\cdots\!05\)\( \nu^{10} + \)\(11\!\cdots\!11\)\( \nu^{9} + \)\(93\!\cdots\!33\)\( \nu^{8} - \)\(49\!\cdots\!96\)\( \nu^{7} + \)\(34\!\cdots\!02\)\( \nu^{6} - \)\(20\!\cdots\!75\)\( \nu^{5} + \)\(44\!\cdots\!13\)\( \nu^{4} - \)\(27\!\cdots\!22\)\( \nu^{3} + \)\(31\!\cdots\!93\)\( \nu^{2} - \)\(13\!\cdots\!04\)\( \nu + \)\(19\!\cdots\!91\)\(\)\()/ \)\(14\!\cdots\!08\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(15\!\cdots\!34\)\( \nu^{17} + \)\(66\!\cdots\!89\)\( \nu^{16} - \)\(26\!\cdots\!77\)\( \nu^{15} + \)\(75\!\cdots\!11\)\( \nu^{14} - \)\(16\!\cdots\!13\)\( \nu^{13} + \)\(18\!\cdots\!86\)\( \nu^{12} - \)\(22\!\cdots\!79\)\( \nu^{11} - \)\(67\!\cdots\!49\)\( \nu^{10} - \)\(44\!\cdots\!81\)\( \nu^{9} + \)\(67\!\cdots\!26\)\( \nu^{8} - \)\(58\!\cdots\!68\)\( \nu^{7} - \)\(72\!\cdots\!29\)\( \nu^{6} - \)\(38\!\cdots\!05\)\( \nu^{5} + \)\(13\!\cdots\!78\)\( \nu^{4} - \)\(24\!\cdots\!43\)\( \nu^{3} + \)\(15\!\cdots\!76\)\( \nu^{2} - \)\(35\!\cdots\!77\)\( \nu - \)\(64\!\cdots\!78\)\(\)\()/ \)\(14\!\cdots\!08\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(19\!\cdots\!61\)\( \nu^{17} - \)\(91\!\cdots\!15\)\( \nu^{16} + \)\(35\!\cdots\!43\)\( \nu^{15} - \)\(10\!\cdots\!87\)\( \nu^{14} + \)\(23\!\cdots\!60\)\( \nu^{13} - \)\(29\!\cdots\!09\)\( \nu^{12} + \)\(35\!\cdots\!89\)\( \nu^{11} + \)\(76\!\cdots\!69\)\( \nu^{10} + \)\(33\!\cdots\!14\)\( \nu^{9} - \)\(10\!\cdots\!84\)\( \nu^{8} + \)\(76\!\cdots\!87\)\( \nu^{7} - \)\(12\!\cdots\!41\)\( \nu^{6} + \)\(45\!\cdots\!40\)\( \nu^{5} - \)\(34\!\cdots\!35\)\( \nu^{4} + \)\(39\!\cdots\!60\)\( \nu^{3} - \)\(30\!\cdots\!21\)\( \nu^{2} + \)\(10\!\cdots\!20\)\( \nu - \)\(15\!\cdots\!50\)\(\)\()/ \)\(14\!\cdots\!08\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(22\!\cdots\!40\)\( \nu^{17} - \)\(99\!\cdots\!19\)\( \nu^{16} + \)\(38\!\cdots\!81\)\( \nu^{15} - \)\(11\!\cdots\!75\)\( \nu^{14} + \)\(24\!\cdots\!31\)\( \nu^{13} - \)\(26\!\cdots\!74\)\( \nu^{12} + \)\(33\!\cdots\!01\)\( \nu^{11} + \)\(99\!\cdots\!37\)\( \nu^{10} + \)\(71\!\cdots\!03\)\( \nu^{9} - \)\(99\!\cdots\!20\)\( \nu^{8} + \)\(86\!\cdots\!02\)\( \nu^{7} + \)\(13\!\cdots\!23\)\( \nu^{6} + \)\(59\!\cdots\!95\)\( \nu^{5} - \)\(19\!\cdots\!38\)\( \nu^{4} + \)\(43\!\cdots\!99\)\( \nu^{3} - \)\(21\!\cdots\!04\)\( \nu^{2} + \)\(50\!\cdots\!17\)\( \nu + \)\(10\!\cdots\!72\)\(\)\()/ \)\(14\!\cdots\!08\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(29\!\cdots\!71\)\( \nu^{17} + \)\(17\!\cdots\!34\)\( \nu^{16} - \)\(68\!\cdots\!08\)\( \nu^{15} + \)\(21\!\cdots\!78\)\( \nu^{14} - \)\(52\!\cdots\!17\)\( \nu^{13} + \)\(80\!\cdots\!17\)\( \nu^{12} - \)\(95\!\cdots\!62\)\( \nu^{11} - \)\(62\!\cdots\!38\)\( \nu^{10} + \)\(87\!\cdots\!93\)\( \nu^{9} + \)\(25\!\cdots\!48\)\( \nu^{8} - \)\(12\!\cdots\!19\)\( \nu^{7} + \)\(14\!\cdots\!62\)\( \nu^{6} - \)\(59\!\cdots\!65\)\( \nu^{5} + \)\(14\!\cdots\!55\)\( \nu^{4} - \)\(96\!\cdots\!73\)\( \nu^{3} + \)\(10\!\cdots\!15\)\( \nu^{2} - \)\(55\!\cdots\!43\)\( \nu + \)\(14\!\cdots\!42\)\(\)\()/ \)\(14\!\cdots\!08\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(31\!\cdots\!50\)\( \nu^{17} - \)\(13\!\cdots\!89\)\( \nu^{16} + \)\(54\!\cdots\!85\)\( \nu^{15} - \)\(15\!\cdots\!07\)\( \nu^{14} + \)\(34\!\cdots\!13\)\( \nu^{13} - \)\(38\!\cdots\!90\)\( \nu^{12} + \)\(48\!\cdots\!91\)\( \nu^{11} + \)\(13\!\cdots\!89\)\( \nu^{10} + \)\(96\!\cdots\!69\)\( \nu^{9} - \)\(14\!\cdots\!86\)\( \nu^{8} + \)\(11\!\cdots\!16\)\( \nu^{7} + \)\(15\!\cdots\!37\)\( \nu^{6} + \)\(81\!\cdots\!09\)\( \nu^{5} - \)\(35\!\cdots\!10\)\( \nu^{4} + \)\(54\!\cdots\!15\)\( \nu^{3} - \)\(36\!\cdots\!40\)\( \nu^{2} + \)\(86\!\cdots\!29\)\( \nu - \)\(30\!\cdots\!30\)\(\)\()/ \)\(14\!\cdots\!08\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(32\!\cdots\!24\)\( \nu^{17} + \)\(16\!\cdots\!97\)\( \nu^{16} - \)\(66\!\cdots\!89\)\( \nu^{15} + \)\(20\!\cdots\!03\)\( \nu^{14} - \)\(48\!\cdots\!95\)\( \nu^{13} + \)\(69\!\cdots\!36\)\( \nu^{12} - \)\(86\!\cdots\!07\)\( \nu^{11} - \)\(93\!\cdots\!81\)\( \nu^{10} + \)\(44\!\cdots\!25\)\( \nu^{9} + \)\(18\!\cdots\!78\)\( \nu^{8} - \)\(13\!\cdots\!38\)\( \nu^{7} + \)\(85\!\cdots\!67\)\( \nu^{6} - \)\(97\!\cdots\!79\)\( \nu^{5} + \)\(89\!\cdots\!92\)\( \nu^{4} - \)\(10\!\cdots\!09\)\( \nu^{3} + \)\(87\!\cdots\!18\)\( \nu^{2} - \)\(44\!\cdots\!11\)\( \nu + \)\(12\!\cdots\!70\)\(\)\()/ \)\(14\!\cdots\!08\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(34\!\cdots\!35\)\( \nu^{17} + \)\(17\!\cdots\!53\)\( \nu^{16} - \)\(70\!\cdots\!32\)\( \nu^{15} + \)\(21\!\cdots\!50\)\( \nu^{14} - \)\(50\!\cdots\!23\)\( \nu^{13} + \)\(70\!\cdots\!30\)\( \nu^{12} - \)\(84\!\cdots\!89\)\( \nu^{11} - \)\(10\!\cdots\!62\)\( \nu^{10} + \)\(11\!\cdots\!83\)\( \nu^{9} + \)\(22\!\cdots\!35\)\( \nu^{8} - \)\(14\!\cdots\!85\)\( \nu^{7} + \)\(83\!\cdots\!69\)\( \nu^{6} - \)\(81\!\cdots\!49\)\( \nu^{5} + \)\(90\!\cdots\!24\)\( \nu^{4} - \)\(99\!\cdots\!02\)\( \nu^{3} + \)\(78\!\cdots\!98\)\( \nu^{2} - \)\(40\!\cdots\!28\)\( \nu + \)\(53\!\cdots\!59\)\(\)\()/ \)\(14\!\cdots\!08\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(37\!\cdots\!67\)\( \nu^{17} - \)\(18\!\cdots\!37\)\( \nu^{16} + \)\(73\!\cdots\!76\)\( \nu^{15} - \)\(22\!\cdots\!86\)\( \nu^{14} + \)\(51\!\cdots\!51\)\( \nu^{13} - \)\(67\!\cdots\!82\)\( \nu^{12} + \)\(82\!\cdots\!37\)\( \nu^{11} + \)\(13\!\cdots\!26\)\( \nu^{10} + \)\(27\!\cdots\!09\)\( \nu^{9} - \)\(22\!\cdots\!23\)\( \nu^{8} + \)\(15\!\cdots\!69\)\( \nu^{7} - \)\(59\!\cdots\!29\)\( \nu^{6} + \)\(92\!\cdots\!53\)\( \nu^{5} - \)\(89\!\cdots\!04\)\( \nu^{4} + \)\(89\!\cdots\!58\)\( \nu^{3} - \)\(78\!\cdots\!10\)\( \nu^{2} + \)\(31\!\cdots\!12\)\( \nu - \)\(57\!\cdots\!15\)\(\)\()/ \)\(14\!\cdots\!08\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(52\!\cdots\!74\)\( \nu^{17} + \)\(24\!\cdots\!83\)\( \nu^{16} - \)\(97\!\cdots\!35\)\( \nu^{15} + \)\(29\!\cdots\!77\)\( \nu^{14} - \)\(65\!\cdots\!67\)\( \nu^{13} + \)\(80\!\cdots\!42\)\( \nu^{12} - \)\(96\!\cdots\!85\)\( \nu^{11} - \)\(20\!\cdots\!27\)\( \nu^{10} - \)\(66\!\cdots\!71\)\( \nu^{9} + \)\(32\!\cdots\!46\)\( \nu^{8} - \)\(20\!\cdots\!16\)\( \nu^{7} + \)\(45\!\cdots\!37\)\( \nu^{6} - \)\(10\!\cdots\!99\)\( \nu^{5} + \)\(10\!\cdots\!74\)\( \nu^{4} - \)\(95\!\cdots\!85\)\( \nu^{3} + \)\(86\!\cdots\!56\)\( \nu^{2} - \)\(22\!\cdots\!83\)\( \nu + \)\(53\!\cdots\!74\)\(\)\()/ \)\(14\!\cdots\!08\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(12\!\cdots\!18\)\( \nu^{17} + \)\(56\!\cdots\!98\)\( \nu^{16} - \)\(22\!\cdots\!59\)\( \nu^{15} + \)\(66\!\cdots\!37\)\( \nu^{14} - \)\(14\!\cdots\!49\)\( \nu^{13} + \)\(18\!\cdots\!03\)\( \nu^{12} - \)\(22\!\cdots\!12\)\( \nu^{11} - \)\(47\!\cdots\!55\)\( \nu^{10} - \)\(23\!\cdots\!45\)\( \nu^{9} + \)\(63\!\cdots\!13\)\( \nu^{8} - \)\(47\!\cdots\!14\)\( \nu^{7} + \)\(64\!\cdots\!36\)\( \nu^{6} - \)\(31\!\cdots\!11\)\( \nu^{5} + \)\(20\!\cdots\!43\)\( \nu^{4} - \)\(25\!\cdots\!62\)\( \nu^{3} + \)\(18\!\cdots\!99\)\( \nu^{2} - \)\(73\!\cdots\!12\)\( \nu + \)\(92\!\cdots\!83\)\(\)\()/ \)\(14\!\cdots\!08\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{16} - \beta_{15} + \beta_{14} - \beta_{13} + \beta_{11} + \beta_{10} - \beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(-\beta_{16} + \beta_{14} + 2 \beta_{12} - 2 \beta_{10} - \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} - 7 \beta_{4} - 2 \beta_{3} - 7 \beta_{2} - 8 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{17} + \beta_{16} + 9 \beta_{15} + 8 \beta_{13} - 8 \beta_{11} + 3 \beta_{9} - 10 \beta_{8} + 19 \beta_{7} + 9 \beta_{6} - \beta_{5} - 9 \beta_{4} - 10 \beta_{2} - 7 \beta_{1} + 9\)
\(\nu^{5}\)\(=\)\(10 \beta_{17} + 5 \beta_{15} - 7 \beta_{14} + 10 \beta_{13} + 4 \beta_{12} - 7 \beta_{11} + 18 \beta_{10} + 13 \beta_{9} - 53 \beta_{8} + 4 \beta_{7} + 53 \beta_{6} + 2 \beta_{5} + 8 \beta_{4} - 5 \beta_{3} - 13 \beta_{2} + 8 \beta_{1} - 18\)
\(\nu^{6}\)\(=\)\(53 \beta_{17} - 8 \beta_{16} + 3 \beta_{15} - 53 \beta_{14} + 5 \beta_{13} + 143 \beta_{12} + 8 \beta_{11} - 74 \beta_{10} + 87 \beta_{9} - 79 \beta_{8} + 3 \beta_{7} + 87 \beta_{6} - 74 \beta_{3} - 41 \beta_{2} - 33 \beta_{1}\)
\(\nu^{7}\)\(=\)\(79 \beta_{17} + 79 \beta_{16} + 202 \beta_{15} - 120 \beta_{14} + 120 \beta_{13} + 202 \beta_{12} - 33 \beta_{11} - 219 \beta_{10} + 423 \beta_{9} + 48 \beta_{8} + 219 \beta_{7} + 136 \beta_{6} - 33 \beta_{5} - 48 \beta_{4} + 157\)
\(\nu^{8}\)\(=\)\(-48 \beta_{17} + 423 \beta_{16} + 537 \beta_{15} - 88 \beta_{14} + 423 \beta_{13} - 538 \beta_{12} + 538 \beta_{10} + 697 \beta_{9} + 429 \beta_{6} + 48 \beta_{5} + 697 \beta_{4} + 601 \beta_{3} + 429 \beta_{2} + 389 \beta_{1} - 601\)
\(\nu^{9}\)\(=\)\(737 \beta_{16} - 182 \beta_{15} - 737 \beta_{14} + 308 \beta_{13} + 389 \beta_{11} + 182 \beta_{10} - 224 \beta_{9} - 224 \beta_{8} - 1965 \beta_{7} - 308 \beta_{5} + 3466 \beta_{4} + 599 \beta_{3} + 1317 \beta_{2} - 224 \beta_{1} - 1965\)
\(\nu^{10}\)\(=\)\(224 \beta_{17} + 1317 \beta_{16} - 3690 \beta_{14} + 4028 \beta_{12} - 224 \beta_{11} - 8414 \beta_{10} + 6151 \beta_{8} - 855 \beta_{7} - 4100 \beta_{6} - 3690 \beta_{5} + 6245 \beta_{4} + 855 \beta_{3} + 6245 \beta_{2} + 2145 \beta_{1} + 4028\)
\(\nu^{11}\)\(=\)\(-6151 \beta_{17} + 4100 \beta_{16} + 1643 \beta_{15} - 2145 \beta_{13} - 15533 \beta_{12} + 2145 \beta_{11} + 474 \beta_{9} + 28821 \beta_{8} - 3789 \beta_{7} - 12293 \beta_{6} - 4100 \beta_{5} + 12293 \beta_{4} + 15533 \beta_{3} + 28821 \beta_{2} + 29295 \beta_{1} + 1643\)
\(\nu^{12}\)\(=\)\(-28821 \beta_{17} - 36579 \beta_{15} + 29295 \beta_{14} - 28821 \beta_{13} - 76402 \beta_{12} + 29295 \beta_{11} + 66641 \beta_{10} - 54131 \beta_{9} + 53293 \beta_{8} - 76402 \beta_{7} - 53293 \beta_{6} + 17002 \beta_{5} + 37575 \beta_{4} + 36579 \beta_{3} + 54131 \beta_{2} + 37575 \beta_{1} - 66641\)
\(\nu^{13}\)\(=\)\(-53293 \beta_{17} - 37575 \beta_{16} - 134492 \beta_{15} + 53293 \beta_{14} - 91706 \beta_{13} - 34893 \beta_{12} + 37575 \beta_{11} - 13549 \beta_{10} - 241942 \beta_{9} + 112448 \beta_{8} - 134492 \beta_{7} - 241942 \beta_{6} - 13549 \beta_{3} + 6593 \beta_{2} - 122901 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-112448 \beta_{17} - 112448 \beta_{16} - 102208 \beta_{15} + 119041 \beta_{14} - 119041 \beta_{13} - 102208 \beta_{12} - 122901 \beta_{11} - 224116 \beta_{10} - 458294 \beta_{9} + 336477 \beta_{8} + 224116 \beta_{7} - 474733 \beta_{6} - 122901 \beta_{5} - 336477 \beta_{4} + 532296\)
\(\nu^{15}\)\(=\)\(-336477 \beta_{17} - 458294 \beta_{16} - 430535 \beta_{15} + 811210 \beta_{14} - 458294 \beta_{13} - 1265079 \beta_{12} + 1265079 \beta_{10} - 1016020 \beta_{9} - 134309 \beta_{6} + 336477 \beta_{5} - 1016020 \beta_{4} + 104770 \beta_{3} - 134309 \beta_{2} + 894493 \beta_{1} - 104770\)
\(\nu^{16}\)\(=\)\(-2044822 \beta_{16} - 2686846 \beta_{15} + 2044822 \beta_{14} - 1910513 \beta_{13} + 894493 \beta_{11} + 2686846 \beta_{10} - 2970492 \beta_{9} - 2970492 \beta_{8} - 1668184 \beta_{7} + 1910513 \beta_{5} - 3966718 \beta_{4} - 2611383 \beta_{3} - 4150650 \beta_{2} - 2970492 \beta_{1} - 1668184\)
\(\nu^{17}\)\(=\)\(2970492 \beta_{17} - 4150650 \beta_{16} + 996226 \beta_{14} + 10750085 \beta_{12} - 2970492 \beta_{11} - 6923340 \beta_{10} - 9106201 \beta_{8} + 9987154 \beta_{7} + 1722282 \beta_{6} + 996226 \beta_{5} - 17370932 \beta_{4} - 9987154 \beta_{3} - 17370932 \beta_{2} - 15648650 \beta_{1} + 10750085\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/430\mathbb{Z}\right)^\times\).

\(n\) \(87\) \(261\)
\(\chi(n)\) \(1\) \(-\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.583263 + 0.731388i
0.219976 0.275841i
1.14077 1.43048i
−1.43931 + 0.693135i
0.403095 0.194120i
2.65970 1.28085i
−1.43931 0.693135i
0.403095 + 0.194120i
2.65970 + 1.28085i
−0.635189 2.78294i
0.168441 + 0.737990i
0.565778 + 2.47884i
−0.635189 + 2.78294i
0.168441 0.737990i
0.565778 2.47884i
−0.583263 0.731388i
0.219976 + 0.275841i
1.14077 + 1.43048i
−0.623490 0.781831i −0.583263 + 0.731388i −0.222521 + 0.974928i −0.900969 + 0.433884i 0.935481 2.01178 0.900969 0.433884i 0.472829 + 2.07160i 0.900969 + 0.433884i
11.2 −0.623490 0.781831i 0.219976 0.275841i −0.222521 + 0.974928i −0.900969 + 0.433884i −0.352814 −0.963199 0.900969 0.433884i 0.639864 + 2.80343i 0.900969 + 0.433884i
11.3 −0.623490 0.781831i 1.14077 1.43048i −0.222521 + 0.974928i −0.900969 + 0.433884i −1.82965 4.80227 0.900969 0.433884i −0.0773496 0.338891i 0.900969 + 0.433884i
21.1 0.900969 + 0.433884i −1.43931 + 0.693135i 0.623490 + 0.781831i −0.222521 0.974928i −1.59751 −1.15014 0.222521 + 0.974928i −0.279294 + 0.350223i 0.222521 0.974928i
21.2 0.900969 + 0.433884i 0.403095 0.194120i 0.623490 + 0.781831i −0.222521 0.974928i 0.447401 2.78084 0.222521 + 0.974928i −1.74567 + 2.18900i 0.222521 0.974928i
21.3 0.900969 + 0.433884i 2.65970 1.28085i 0.623490 + 0.781831i −0.222521 0.974928i 2.95205 −1.54255 0.222521 + 0.974928i 3.56299 4.46785i 0.222521 0.974928i
41.1 0.900969 0.433884i −1.43931 0.693135i 0.623490 0.781831i −0.222521 + 0.974928i −1.59751 −1.15014 0.222521 0.974928i −0.279294 0.350223i 0.222521 + 0.974928i
41.2 0.900969 0.433884i 0.403095 + 0.194120i 0.623490 0.781831i −0.222521 + 0.974928i 0.447401 2.78084 0.222521 0.974928i −1.74567 2.18900i 0.222521 + 0.974928i
41.3 0.900969 0.433884i 2.65970 + 1.28085i 0.623490 0.781831i −0.222521 + 0.974928i 2.95205 −1.54255 0.222521 0.974928i 3.56299 + 4.46785i 0.222521 + 0.974928i
121.1 0.222521 0.974928i −0.635189 2.78294i −0.900969 0.433884i 0.623490 0.781831i −2.85451 −1.93646 −0.623490 + 0.781831i −4.63840 + 2.23374i −0.623490 0.781831i
121.2 0.222521 0.974928i 0.168441 + 0.737990i −0.900969 0.433884i 0.623490 0.781831i 0.756969 −2.52891 −0.623490 + 0.781831i 2.18665 1.05304i −0.623490 0.781831i
121.3 0.222521 0.974928i 0.565778 + 2.47884i −0.900969 0.433884i 0.623490 0.781831i 2.54259 2.52637 −0.623490 + 0.781831i −3.12162 + 1.50329i −0.623490 0.781831i
231.1 0.222521 + 0.974928i −0.635189 + 2.78294i −0.900969 + 0.433884i 0.623490 + 0.781831i −2.85451 −1.93646 −0.623490 0.781831i −4.63840 2.23374i −0.623490 + 0.781831i
231.2 0.222521 + 0.974928i 0.168441 0.737990i −0.900969 + 0.433884i 0.623490 + 0.781831i 0.756969 −2.52891 −0.623490 0.781831i 2.18665 + 1.05304i −0.623490 + 0.781831i
231.3 0.222521 + 0.974928i 0.565778 2.47884i −0.900969 + 0.433884i 0.623490 + 0.781831i 2.54259 2.52637 −0.623490 0.781831i −3.12162 1.50329i −0.623490 + 0.781831i
391.1 −0.623490 + 0.781831i −0.583263 0.731388i −0.222521 0.974928i −0.900969 0.433884i 0.935481 2.01178 0.900969 + 0.433884i 0.472829 2.07160i 0.900969 0.433884i
391.2 −0.623490 + 0.781831i 0.219976 + 0.275841i −0.222521 0.974928i −0.900969 0.433884i −0.352814 −0.963199 0.900969 + 0.433884i 0.639864 2.80343i 0.900969 0.433884i
391.3 −0.623490 + 0.781831i 1.14077 + 1.43048i −0.222521 0.974928i −0.900969 0.433884i −1.82965 4.80227 0.900969 + 0.433884i −0.0773496 + 0.338891i 0.900969 0.433884i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.k.c 18
43.e even 7 1 inner 430.2.k.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.k.c 18 1.a even 1 1 trivial
430.2.k.c 18 43.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{18} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{3} \)
$3$ \( 1 - 5 T + 11 T^{2} - 22 T^{3} + 64 T^{4} - 177 T^{5} + 364 T^{6} - 721 T^{7} + 1705 T^{8} - 3727 T^{9} + 6910 T^{10} - 13168 T^{11} + 27619 T^{12} - 53717 T^{13} + 95903 T^{14} - 176450 T^{15} + 331870 T^{16} - 581677 T^{17} + 983731 T^{18} - 1745031 T^{19} + 2986830 T^{20} - 4764150 T^{21} + 7768143 T^{22} - 13053231 T^{23} + 20134251 T^{24} - 28798416 T^{25} + 45336510 T^{26} - 73358541 T^{27} + 100678545 T^{28} - 127722987 T^{29} + 193444524 T^{30} - 282195171 T^{31} + 306110016 T^{32} - 315675954 T^{33} + 473513931 T^{34} - 645700815 T^{35} + 387420489 T^{36} \)
$5$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{3} \)
$7$ \( ( 1 - 4 T + 43 T^{2} - 165 T^{3} + 954 T^{4} - 3272 T^{5} + 13473 T^{6} - 40583 T^{7} + 131977 T^{8} - 340920 T^{9} + 923839 T^{10} - 1988567 T^{11} + 4621239 T^{12} - 7856072 T^{13} + 16033878 T^{14} - 19412085 T^{15} + 35412349 T^{16} - 23059204 T^{17} + 40353607 T^{18} )^{2} \)
$11$ \( 1 - 5 T + 29 T^{2} - 2 T^{3} - 64 T^{4} + 1743 T^{5} - 258 T^{6} + 563 T^{7} + 93155 T^{8} - 93657 T^{9} + 443272 T^{10} + 1889454 T^{11} - 1463949 T^{12} + 6857689 T^{13} + 40881021 T^{14} - 138737432 T^{15} + 197538356 T^{16} - 688470581 T^{17} - 1743464851 T^{18} - 7573176391 T^{19} + 23902141076 T^{20} - 184659521992 T^{21} + 598539028461 T^{22} + 1104437671139 T^{23} - 2593474954389 T^{24} + 36820113194634 T^{25} + 95019289898632 T^{26} - 220838306895987 T^{27} + 2416200788706155 T^{28} + 160630470553993 T^{29} - 809714521194018 T^{30} + 60173087266871733 T^{31} - 24303989349327424 T^{32} - 8354496338831302 T^{33} + 1332542166043592669 T^{34} - 2527235142496468855 T^{35} + 5559917313492231481 T^{36} \)
$13$ \( 1 - 22 T + 189 T^{2} - 776 T^{3} + 2016 T^{4} - 11560 T^{5} + 76130 T^{6} - 262662 T^{7} + 880343 T^{8} - 5919980 T^{9} + 28592867 T^{10} - 84734756 T^{11} + 312372760 T^{12} - 1531317100 T^{13} + 5382091707 T^{14} - 17490109252 T^{15} + 77239495484 T^{16} - 309727210712 T^{17} + 1082670643211 T^{18} - 4026453739256 T^{19} + 13053474736796 T^{20} - 38425770026644 T^{21} + 153717921243627 T^{22} - 568567320010300 T^{23} + 1507763649322840 T^{24} - 5316980277356852 T^{25} + 23324080013367107 T^{26} - 62778424198172540 T^{27} + 121362758289824207 T^{28} - 470732433418546494 T^{29} + 1773683220374478530 T^{30} - 3501236232206444680 T^{31} + 7937750793569766624 T^{32} - 39720252978934427432 T^{33} + \)\(12\!\cdots\!49\)\( T^{34} - \)\(19\!\cdots\!26\)\( T^{35} + \)\(11\!\cdots\!29\)\( T^{36} \)
$17$ \( 1 - T + 17 T^{2} - 71 T^{3} + 163 T^{4} - 1386 T^{5} + 5859 T^{6} - 25092 T^{7} + 79275 T^{8} - 574216 T^{9} + 380351 T^{10} - 8959160 T^{11} + 10289590 T^{12} - 149888072 T^{13} + 943088259 T^{14} - 2036970771 T^{15} + 18923873831 T^{16} - 49782589785 T^{17} + 210671203633 T^{18} - 846304026345 T^{19} + 5468999537159 T^{20} - 10007637397923 T^{21} + 78767674479939 T^{22} - 212819628245704 T^{23} + 248365688606710 T^{24} - 3676289825594680 T^{25} + 2653236318441791 T^{26} - 68095056090601352 T^{27} + 159817916458094475 T^{28} - 859950422151127236 T^{29} + 3413583687929169699 T^{30} - 13727745153607628682 T^{31} + 27445585729182351427 T^{32} - \)\(20\!\cdots\!03\)\( T^{33} + \)\(82\!\cdots\!77\)\( T^{34} - \)\(82\!\cdots\!77\)\( T^{35} + \)\(14\!\cdots\!09\)\( T^{36} \)
$19$ \( 1 + 7 T + 80 T^{2} + 528 T^{3} + 3734 T^{4} + 20920 T^{5} + 120996 T^{6} + 605461 T^{7} + 2904657 T^{8} + 13467676 T^{9} + 54757744 T^{10} + 216429928 T^{11} + 724355904 T^{12} + 2023912444 T^{13} + 3282662694 T^{14} - 10380996494 T^{15} - 115691998692 T^{16} - 856017011464 T^{17} - 3500283086684 T^{18} - 16264323217816 T^{19} - 41764811527812 T^{20} - 71203254952346 T^{21} + 427799884944774 T^{22} + 5011407578675956 T^{23} + 34077961661231424 T^{24} + 193460596113004792 T^{25} + 929981597206939504 T^{26} + 4345853362873491604 T^{27} + 17808644523185479257 T^{28} + 70530308642774573959 T^{29} + \)\(26\!\cdots\!56\)\( T^{30} + \)\(87\!\cdots\!80\)\( T^{31} + \)\(29\!\cdots\!14\)\( T^{32} + \)\(80\!\cdots\!72\)\( T^{33} + \)\(23\!\cdots\!80\)\( T^{34} + \)\(38\!\cdots\!73\)\( T^{35} + \)\(10\!\cdots\!41\)\( T^{36} \)
$23$ \( 1 - 8 T + 13 T^{2} + 20 T^{3} + 794 T^{4} - 8894 T^{5} + 44620 T^{6} - 76030 T^{7} - 348697 T^{8} - 785090 T^{9} + 27523733 T^{10} - 101765222 T^{11} - 302476000 T^{12} + 4014732254 T^{13} - 12572137619 T^{14} - 14995597814 T^{15} + 64743711756 T^{16} + 2079662840830 T^{17} - 17083611139087 T^{18} + 47832245339090 T^{19} + 34249423518924 T^{20} - 182451438602938 T^{21} - 3518199563438579 T^{22} + 25840193839907122 T^{23} - 44777303561164000 T^{24} - 346492817485204234 T^{25} + 2155410649841173973 T^{26} - 1414066942987986670 T^{27} - 14445300180665765353 T^{28} - 72442125894195869810 T^{29} + \)\(97\!\cdots\!20\)\( T^{30} - \)\(44\!\cdots\!02\)\( T^{31} + \)\(92\!\cdots\!46\)\( T^{32} + \)\(53\!\cdots\!40\)\( T^{33} + \)\(79\!\cdots\!93\)\( T^{34} - \)\(11\!\cdots\!24\)\( T^{35} + \)\(32\!\cdots\!69\)\( T^{36} \)
$29$ \( 1 + 6 T - 71 T^{2} - 679 T^{3} + 1326 T^{4} + 34252 T^{5} + 88405 T^{6} - 867289 T^{7} - 8027938 T^{8} - 681278 T^{9} + 325349830 T^{10} + 1194610606 T^{11} - 6989192779 T^{12} - 61491095320 T^{13} - 35442154336 T^{14} + 1740460758039 T^{15} + 9086251390251 T^{16} - 21170258850237 T^{17} - 364928839121169 T^{18} - 613937506656873 T^{19} + 7641537419201091 T^{20} + 42448097427813171 T^{21} - 25067562360920416 T^{22} - 1261253018281722680 T^{23} - 4157334859913999059 T^{24} + 20606885190919533254 T^{25} + \)\(16\!\cdots\!30\)\( T^{26} - 9883399396148080582 T^{27} - \)\(33\!\cdots\!38\)\( T^{28} - \)\(10\!\cdots\!81\)\( T^{29} + \)\(31\!\cdots\!05\)\( T^{30} + \)\(35\!\cdots\!28\)\( T^{31} + \)\(39\!\cdots\!06\)\( T^{32} - \)\(58\!\cdots\!71\)\( T^{33} - \)\(17\!\cdots\!91\)\( T^{34} + \)\(43\!\cdots\!54\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \)
$31$ \( 1 + 5 T - 66 T^{2} - 503 T^{3} + 337 T^{4} + 12028 T^{5} + 33978 T^{6} + 406763 T^{7} + 2995682 T^{8} - 18538475 T^{9} - 206059741 T^{10} - 275557932 T^{11} + 1910953762 T^{12} + 11736742065 T^{13} + 105697996694 T^{14} + 712002834894 T^{15} - 316523157013 T^{16} - 20711586044207 T^{17} - 106673227770053 T^{18} - 642059167370417 T^{19} - 304178753889493 T^{20} + 21211276454327154 T^{21} + 97614319604839574 T^{22} + 336012960826936815 T^{23} + 1695978497995797922 T^{24} - 7581319048341178452 T^{25} - \)\(17\!\cdots\!81\)\( T^{26} - \)\(49\!\cdots\!25\)\( T^{27} + \)\(24\!\cdots\!82\)\( T^{28} + \)\(10\!\cdots\!53\)\( T^{29} + \)\(26\!\cdots\!58\)\( T^{30} + \)\(29\!\cdots\!48\)\( T^{31} + \)\(25\!\cdots\!77\)\( T^{32} - \)\(11\!\cdots\!53\)\( T^{33} - \)\(48\!\cdots\!46\)\( T^{34} + \)\(11\!\cdots\!55\)\( T^{35} + \)\(69\!\cdots\!41\)\( T^{36} \)
$37$ \( ( 1 + 9 T + 218 T^{2} + 1614 T^{3} + 23401 T^{4} + 151257 T^{5} + 1644298 T^{6} + 9291632 T^{7} + 82504230 T^{8} + 404531792 T^{9} + 3052656510 T^{10} + 12720244208 T^{11} + 83288626594 T^{12} + 283479970377 T^{13} + 1622717937757 T^{14} + 4141082424126 T^{15} + 20695149214994 T^{16} + 31612315085289 T^{17} + 129961739795077 T^{18} )^{2} \)
$41$ \( 1 - 16 T - 52 T^{2} + 2313 T^{3} - 7721 T^{4} - 117110 T^{5} + 860282 T^{6} + 1872521 T^{7} - 38916852 T^{8} + 82707819 T^{9} + 1144968899 T^{10} - 8680504064 T^{11} - 18775033826 T^{12} + 451896528557 T^{13} - 387529241846 T^{14} - 14990466569601 T^{15} + 45356087778625 T^{16} + 236151025913079 T^{17} - 2271300748262055 T^{18} + 9682192062436239 T^{19} + 76243583555868625 T^{20} - 1033157946443470521 T^{21} - 1095065017967994806 T^{22} + 52355015043702031957 T^{23} - 89183367801801056066 T^{24} - \)\(16\!\cdots\!84\)\( T^{25} + \)\(91\!\cdots\!79\)\( T^{26} + \)\(27\!\cdots\!59\)\( T^{27} - \)\(52\!\cdots\!52\)\( T^{28} + \)\(10\!\cdots\!61\)\( T^{29} + \)\(19\!\cdots\!42\)\( T^{30} - \)\(10\!\cdots\!10\)\( T^{31} - \)\(29\!\cdots\!81\)\( T^{32} + \)\(35\!\cdots\!13\)\( T^{33} - \)\(33\!\cdots\!32\)\( T^{34} - \)\(41\!\cdots\!96\)\( T^{35} + \)\(10\!\cdots\!21\)\( T^{36} \)
$43$ \( 1 + 13 T + 135 T^{2} + 173 T^{3} - 5485 T^{4} - 91815 T^{5} - 487924 T^{6} - 685681 T^{7} + 23032937 T^{8} + 203701481 T^{9} + 990416291 T^{10} - 1267824169 T^{11} - 38793373468 T^{12} - 313897213815 T^{13} - 806341309855 T^{14} + 1093595807477 T^{15} + 36695512499445 T^{16} + 151946603608813 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 + 36 T + 612 T^{2} + 6578 T^{3} + 56686 T^{4} + 524814 T^{5} + 5385180 T^{6} + 49774124 T^{7} + 394711081 T^{8} + 3091549542 T^{9} + 26602209040 T^{10} + 226026429292 T^{11} + 1707793138576 T^{12} + 12049342225310 T^{13} + 89717535794078 T^{14} + 702257335176408 T^{15} + 5175913711157672 T^{16} + 34826239488864972 T^{17} + 232179354339637252 T^{18} + 1636833255976653684 T^{19} + 11433593387947297448 T^{20} + 72910463310020207784 T^{21} + \)\(43\!\cdots\!18\)\( T^{22} + \)\(27\!\cdots\!70\)\( T^{23} + \)\(18\!\cdots\!04\)\( T^{24} + \)\(11\!\cdots\!96\)\( T^{25} + \)\(63\!\cdots\!40\)\( T^{26} + \)\(34\!\cdots\!14\)\( T^{27} + \)\(20\!\cdots\!69\)\( T^{28} + \)\(12\!\cdots\!72\)\( T^{29} + \)\(62\!\cdots\!80\)\( T^{30} + \)\(28\!\cdots\!78\)\( T^{31} + \)\(14\!\cdots\!34\)\( T^{32} + \)\(79\!\cdots\!54\)\( T^{33} + \)\(34\!\cdots\!52\)\( T^{34} + \)\(95\!\cdots\!32\)\( T^{35} + \)\(12\!\cdots\!89\)\( T^{36} \)
$53$ \( 1 + 18 T + 219 T^{2} + 1873 T^{3} + 21028 T^{4} + 191940 T^{5} + 1927031 T^{6} + 16691841 T^{7} + 157628752 T^{8} + 1143000974 T^{9} + 9987462820 T^{10} + 79427444768 T^{11} + 685958917601 T^{12} + 4783166371000 T^{13} + 40313855954708 T^{14} + 280316920456111 T^{15} + 2208536845954869 T^{16} + 15550871535373675 T^{17} + 128962261045140095 T^{18} + 824196191374804775 T^{19} + 6203780000287227021 T^{20} + 41732742166744437347 T^{21} + \)\(31\!\cdots\!48\)\( T^{22} + \)\(20\!\cdots\!00\)\( T^{23} + \)\(15\!\cdots\!29\)\( T^{24} + \)\(93\!\cdots\!16\)\( T^{25} + \)\(62\!\cdots\!20\)\( T^{26} + \)\(37\!\cdots\!42\)\( T^{27} + \)\(27\!\cdots\!48\)\( T^{28} + \)\(15\!\cdots\!77\)\( T^{29} + \)\(94\!\cdots\!71\)\( T^{30} + \)\(49\!\cdots\!20\)\( T^{31} + \)\(29\!\cdots\!32\)\( T^{32} + \)\(13\!\cdots\!61\)\( T^{33} + \)\(84\!\cdots\!99\)\( T^{34} + \)\(36\!\cdots\!34\)\( T^{35} + \)\(10\!\cdots\!89\)\( T^{36} \)
$59$ \( 1 + 4 T + 182 T^{2} + 889 T^{3} + 19535 T^{4} + 92775 T^{5} + 1403700 T^{6} + 3337697 T^{7} + 60900249 T^{8} - 100866924 T^{9} + 279379214 T^{10} - 22396990778 T^{11} - 136590509711 T^{12} - 1169980179938 T^{13} - 4940983425545 T^{14} + 7131058308663 T^{15} + 145810789722975 T^{16} + 5887152771171894 T^{17} + 24165780228834877 T^{18} + 347342013499141746 T^{19} + 507567359025675975 T^{20} + 1464569624374898277 T^{21} - 59871679862345386745 T^{22} - \)\(83\!\cdots\!62\)\( T^{23} - \)\(57\!\cdots\!51\)\( T^{24} - \)\(55\!\cdots\!82\)\( T^{25} + \)\(41\!\cdots\!94\)\( T^{26} - \)\(87\!\cdots\!36\)\( T^{27} + \)\(31\!\cdots\!49\)\( T^{28} + \)\(10\!\cdots\!23\)\( T^{29} + \)\(24\!\cdots\!00\)\( T^{30} + \)\(97\!\cdots\!25\)\( T^{31} + \)\(12\!\cdots\!35\)\( T^{32} + \)\(32\!\cdots\!11\)\( T^{33} + \)\(39\!\cdots\!62\)\( T^{34} + \)\(50\!\cdots\!76\)\( T^{35} + \)\(75\!\cdots\!21\)\( T^{36} \)
$61$ \( 1 - 42 T + 933 T^{2} - 14861 T^{3} + 192423 T^{4} - 2141538 T^{5} + 21336371 T^{6} - 196037772 T^{7} + 1665767448 T^{8} - 12759457459 T^{9} + 83761862657 T^{10} - 415805126156 T^{11} + 730146937632 T^{12} + 16028186575331 T^{13} - 292933998347776 T^{14} + 3505813369714012 T^{15} - 35535339534952817 T^{16} + 322709127419838593 T^{17} - 2650994802911766219 T^{18} + 19685256772610154173 T^{19} - \)\(13\!\cdots\!57\)\( T^{20} + \)\(79\!\cdots\!72\)\( T^{21} - \)\(40\!\cdots\!16\)\( T^{22} + \)\(13\!\cdots\!31\)\( T^{23} + \)\(37\!\cdots\!52\)\( T^{24} - \)\(13\!\cdots\!76\)\( T^{25} + \)\(16\!\cdots\!17\)\( T^{26} - \)\(14\!\cdots\!19\)\( T^{27} + \)\(11\!\cdots\!48\)\( T^{28} - \)\(85\!\cdots\!92\)\( T^{29} + \)\(56\!\cdots\!91\)\( T^{30} - \)\(34\!\cdots\!78\)\( T^{31} + \)\(19\!\cdots\!43\)\( T^{32} - \)\(89\!\cdots\!61\)\( T^{33} + \)\(34\!\cdots\!13\)\( T^{34} - \)\(94\!\cdots\!82\)\( T^{35} + \)\(13\!\cdots\!81\)\( T^{36} \)
$67$ \( 1 + 19 T - 121 T^{2} - 3509 T^{3} + 17241 T^{4} + 454616 T^{5} - 1635832 T^{6} - 44808601 T^{7} + 41507781 T^{8} + 3267974565 T^{9} + 6835163934 T^{10} - 240159436956 T^{11} - 1558153168075 T^{12} + 15530120367369 T^{13} + 158968204106237 T^{14} - 817261224750396 T^{15} - 11907700881595164 T^{16} + 24442230601190901 T^{17} + 850574669249442709 T^{18} + 1637629450279790367 T^{19} - 53453669257480691196 T^{20} - \)\(24\!\cdots\!48\)\( T^{21} + \)\(32\!\cdots\!77\)\( T^{22} + \)\(20\!\cdots\!83\)\( T^{23} - \)\(14\!\cdots\!75\)\( T^{24} - \)\(14\!\cdots\!88\)\( T^{25} + \)\(27\!\cdots\!94\)\( T^{26} + \)\(88\!\cdots\!55\)\( T^{27} + \)\(75\!\cdots\!69\)\( T^{28} - \)\(54\!\cdots\!83\)\( T^{29} - \)\(13\!\cdots\!52\)\( T^{30} + \)\(24\!\cdots\!92\)\( T^{31} + \)\(63\!\cdots\!89\)\( T^{32} - \)\(86\!\cdots\!87\)\( T^{33} - \)\(19\!\cdots\!01\)\( T^{34} + \)\(20\!\cdots\!13\)\( T^{35} + \)\(74\!\cdots\!09\)\( T^{36} \)
$71$ \( 1 + 27 T + 192 T^{2} - 3064 T^{3} - 71457 T^{4} - 431170 T^{5} + 3006306 T^{6} + 66658940 T^{7} + 407727641 T^{8} - 36960119 T^{9} - 18346711662 T^{10} - 203539733210 T^{11} - 1941959274995 T^{12} - 10817050171315 T^{13} + 59547414301229 T^{14} + 1594397690229843 T^{15} + 9439982388216595 T^{16} - 39777894586305512 T^{17} - 893741411732820099 T^{18} - 2824230515627691352 T^{19} + 47586951218999855395 T^{20} + \)\(57\!\cdots\!73\)\( T^{21} + \)\(15\!\cdots\!49\)\( T^{22} - \)\(19\!\cdots\!65\)\( T^{23} - \)\(24\!\cdots\!95\)\( T^{24} - \)\(18\!\cdots\!10\)\( T^{25} - \)\(11\!\cdots\!82\)\( T^{26} - \)\(16\!\cdots\!89\)\( T^{27} + \)\(13\!\cdots\!41\)\( T^{28} + \)\(15\!\cdots\!40\)\( T^{29} + \)\(49\!\cdots\!46\)\( T^{30} - \)\(50\!\cdots\!70\)\( T^{31} - \)\(59\!\cdots\!17\)\( T^{32} - \)\(17\!\cdots\!64\)\( T^{33} + \)\(80\!\cdots\!32\)\( T^{34} + \)\(79\!\cdots\!57\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \)
$73$ \( 1 - 28 T + 206 T^{2} + 1439 T^{3} - 30467 T^{4} + 115902 T^{5} + 1193696 T^{6} - 12456075 T^{7} - 52758690 T^{8} + 1181272439 T^{9} + 858716819 T^{10} - 81754299384 T^{11} - 22479646170 T^{12} + 7830326968241 T^{13} - 27470831061776 T^{14} - 692066597072037 T^{15} + 6983730927869749 T^{16} + 20546869285875433 T^{17} - 650284685443406899 T^{18} + 1499921457868906609 T^{19} + 37216302114617892421 T^{20} - \)\(26\!\cdots\!29\)\( T^{21} - \)\(78\!\cdots\!16\)\( T^{22} + \)\(16\!\cdots\!13\)\( T^{23} - \)\(34\!\cdots\!30\)\( T^{24} - \)\(90\!\cdots\!48\)\( T^{25} + \)\(69\!\cdots\!39\)\( T^{26} + \)\(69\!\cdots\!07\)\( T^{27} - \)\(22\!\cdots\!10\)\( T^{28} - \)\(39\!\cdots\!75\)\( T^{29} + \)\(27\!\cdots\!16\)\( T^{30} + \)\(19\!\cdots\!66\)\( T^{31} - \)\(37\!\cdots\!03\)\( T^{32} + \)\(12\!\cdots\!23\)\( T^{33} + \)\(13\!\cdots\!66\)\( T^{34} - \)\(13\!\cdots\!84\)\( T^{35} + \)\(34\!\cdots\!69\)\( T^{36} \)
$79$ \( ( 1 + 13 T + 473 T^{2} + 3826 T^{3} + 90460 T^{4} + 498762 T^{5} + 11312861 T^{6} + 50578492 T^{7} + 1144441533 T^{8} + 4524834638 T^{9} + 90410881107 T^{10} + 315660368572 T^{11} + 5577681674579 T^{12} + 19426820299722 T^{13} + 278350521853540 T^{14} + 930052604823346 T^{15} + 9083448950453207 T^{16} + 19722414528785293 T^{17} + 119851595982618319 T^{18} )^{2} \)
$83$ \( 1 - 62 T + 1955 T^{2} - 43474 T^{3} + 791403 T^{4} - 12660991 T^{5} + 182312666 T^{6} - 2380818115 T^{7} + 28454173700 T^{8} - 313999851042 T^{9} + 3203343358059 T^{10} - 30029118019905 T^{11} + 256415447656186 T^{12} - 1966785367099865 T^{13} + 13121718252161262 T^{14} - 70231361214705904 T^{15} + 228113352519056074 T^{16} + 528279145154473206 T^{17} - 13256029529997156100 T^{18} + 43847169047821276098 T^{19} + \)\(15\!\cdots\!86\)\( T^{20} - \)\(40\!\cdots\!48\)\( T^{21} + \)\(62\!\cdots\!02\)\( T^{22} - \)\(77\!\cdots\!95\)\( T^{23} + \)\(83\!\cdots\!34\)\( T^{24} - \)\(81\!\cdots\!35\)\( T^{25} + \)\(72\!\cdots\!19\)\( T^{26} - \)\(58\!\cdots\!26\)\( T^{27} + \)\(44\!\cdots\!00\)\( T^{28} - \)\(30\!\cdots\!05\)\( T^{29} + \)\(19\!\cdots\!26\)\( T^{30} - \)\(11\!\cdots\!33\)\( T^{31} + \)\(58\!\cdots\!87\)\( T^{32} - \)\(26\!\cdots\!18\)\( T^{33} + \)\(99\!\cdots\!55\)\( T^{34} - \)\(26\!\cdots\!26\)\( T^{35} + \)\(34\!\cdots\!09\)\( T^{36} \)
$89$ \( 1 + 12 T + 116 T^{2} + 1499 T^{3} + 3935 T^{4} - 42611 T^{5} - 1895934 T^{6} - 26145279 T^{7} - 209446975 T^{8} - 2196428168 T^{9} - 7262368198 T^{10} + 44027191578 T^{11} + 1289869174743 T^{12} + 22210109524986 T^{13} + 120472292193179 T^{14} + 920900716991509 T^{15} + 1767591597163055 T^{16} - 97723090524153866 T^{17} - 862701798538908603 T^{18} - 8697355056649694074 T^{19} + 14001093041128558655 T^{20} + \)\(64\!\cdots\!21\)\( T^{21} + \)\(75\!\cdots\!39\)\( T^{22} + \)\(12\!\cdots\!14\)\( T^{23} + \)\(64\!\cdots\!23\)\( T^{24} + \)\(19\!\cdots\!62\)\( T^{25} - \)\(28\!\cdots\!38\)\( T^{26} - \)\(76\!\cdots\!12\)\( T^{27} - \)\(65\!\cdots\!75\)\( T^{28} - \)\(72\!\cdots\!31\)\( T^{29} - \)\(46\!\cdots\!14\)\( T^{30} - \)\(93\!\cdots\!59\)\( T^{31} + \)\(76\!\cdots\!35\)\( T^{32} + \)\(26\!\cdots\!51\)\( T^{33} + \)\(17\!\cdots\!76\)\( T^{34} + \)\(16\!\cdots\!48\)\( T^{35} + \)\(12\!\cdots\!81\)\( T^{36} \)
$97$ \( 1 - 71 T + 2368 T^{2} - 50969 T^{3} + 836880 T^{4} - 11828308 T^{5} + 155370763 T^{6} - 1975485248 T^{7} + 25025905391 T^{8} - 318238582892 T^{9} + 3957848635408 T^{10} - 46785223241176 T^{11} + 523567041637077 T^{12} - 5637158848736682 T^{13} + 59668959182508043 T^{14} - 630662658429013549 T^{15} + 6659652238475527590 T^{16} - 69228635653545298441 T^{17} + \)\(69\!\cdots\!39\)\( T^{18} - \)\(67\!\cdots\!77\)\( T^{19} + \)\(62\!\cdots\!10\)\( T^{20} - \)\(57\!\cdots\!77\)\( T^{21} + \)\(52\!\cdots\!83\)\( T^{22} - \)\(48\!\cdots\!74\)\( T^{23} + \)\(43\!\cdots\!33\)\( T^{24} - \)\(37\!\cdots\!88\)\( T^{25} + \)\(31\!\cdots\!88\)\( T^{26} - \)\(24\!\cdots\!64\)\( T^{27} + \)\(18\!\cdots\!59\)\( T^{28} - \)\(14\!\cdots\!44\)\( T^{29} + \)\(10\!\cdots\!83\)\( T^{30} - \)\(79\!\cdots\!16\)\( T^{31} + \)\(54\!\cdots\!20\)\( T^{32} - \)\(32\!\cdots\!17\)\( T^{33} + \)\(14\!\cdots\!28\)\( T^{34} - \)\(42\!\cdots\!27\)\( T^{35} + \)\(57\!\cdots\!89\)\( T^{36} \)
show more
show less