Properties

Label 170.2.n.b
Level $170$
Weight $2$
Character orbit 170.n
Analytic conductor $1.357$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.n (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.35745683436\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 16 x^{15} + 52 x^{14} + 992 x^{13} + 6181 x^{12} + 8952 x^{11} + 6244 x^{10} - 11448 x^{9} - 14520 x^{8} + 27936 x^{7} + 27880 x^{6} - 121104 x^{5} + 187460 x^{4} - 142208 x^{3} + 73856 x^{2} - 19456 x + 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 16 x^{15} + 52 x^{14} + 992 x^{13} + 6181 x^{12} + 8952 x^{11} + 6244 x^{10} - 11448 x^{9} - 14520 x^{8} + 27936 x^{7} + 27880 x^{6} - 121104 x^{5} + 187460 x^{4} - 142208 x^{3} + 73856 x^{2} - 19456 x + 2048\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(24\!\cdots\!67\)\( \nu^{19} + \)\(64\!\cdots\!72\)\( \nu^{18} + \)\(11\!\cdots\!32\)\( \nu^{17} + \)\(10\!\cdots\!52\)\( \nu^{16} + \)\(12\!\cdots\!32\)\( \nu^{15} - \)\(35\!\cdots\!00\)\( \nu^{14} + \)\(25\!\cdots\!04\)\( \nu^{13} + \)\(25\!\cdots\!68\)\( \nu^{12} + \)\(22\!\cdots\!75\)\( \nu^{11} + \)\(73\!\cdots\!24\)\( \nu^{10} + \)\(15\!\cdots\!92\)\( \nu^{9} + \)\(19\!\cdots\!76\)\( \nu^{8} + \)\(14\!\cdots\!76\)\( \nu^{7} + \)\(45\!\cdots\!88\)\( \nu^{6} + \)\(72\!\cdots\!88\)\( \nu^{5} - \)\(61\!\cdots\!20\)\( \nu^{4} + \)\(73\!\cdots\!40\)\( \nu^{3} - \)\(44\!\cdots\!36\)\( \nu^{2} + \)\(50\!\cdots\!36\)\( \nu - \)\(24\!\cdots\!36\)\(\)\()/ \)\(51\!\cdots\!40\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(47\!\cdots\!67\)\( \nu^{19} - \)\(23\!\cdots\!52\)\( \nu^{18} - \)\(34\!\cdots\!72\)\( \nu^{17} - \)\(39\!\cdots\!12\)\( \nu^{16} - \)\(14\!\cdots\!92\)\( \nu^{15} + \)\(82\!\cdots\!80\)\( \nu^{14} + \)\(14\!\cdots\!76\)\( \nu^{13} - \)\(53\!\cdots\!08\)\( \nu^{12} - \)\(53\!\cdots\!75\)\( \nu^{11} - \)\(22\!\cdots\!44\)\( \nu^{10} - \)\(48\!\cdots\!72\)\( \nu^{9} - \)\(65\!\cdots\!76\)\( \nu^{8} - \)\(31\!\cdots\!76\)\( \nu^{7} + \)\(28\!\cdots\!72\)\( \nu^{6} + \)\(25\!\cdots\!72\)\( \nu^{5} - \)\(10\!\cdots\!20\)\( \nu^{4} - \)\(30\!\cdots\!40\)\( \nu^{3} - \)\(13\!\cdots\!44\)\( \nu^{2} + \)\(63\!\cdots\!84\)\( \nu - \)\(14\!\cdots\!04\)\(\)\()/ \)\(27\!\cdots\!60\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(42\!\cdots\!91\)\( \nu^{19} - \)\(28\!\cdots\!16\)\( \nu^{18} - \)\(10\!\cdots\!56\)\( \nu^{17} + \)\(34\!\cdots\!24\)\( \nu^{16} + \)\(36\!\cdots\!24\)\( \nu^{15} + \)\(64\!\cdots\!80\)\( \nu^{14} + \)\(22\!\cdots\!88\)\( \nu^{13} - \)\(55\!\cdots\!64\)\( \nu^{12} - \)\(54\!\cdots\!55\)\( \nu^{11} - \)\(22\!\cdots\!32\)\( \nu^{10} - \)\(33\!\cdots\!16\)\( \nu^{9} - \)\(19\!\cdots\!88\)\( \nu^{8} + \)\(37\!\cdots\!72\)\( \nu^{7} + \)\(44\!\cdots\!16\)\( \nu^{6} - \)\(84\!\cdots\!44\)\( \nu^{5} - \)\(76\!\cdots\!80\)\( \nu^{4} + \)\(23\!\cdots\!60\)\( \nu^{3} - \)\(31\!\cdots\!72\)\( \nu^{2} + \)\(16\!\cdots\!92\)\( \nu - \)\(49\!\cdots\!32\)\(\)\()/ \)\(10\!\cdots\!80\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(15\!\cdots\!03\)\( \nu^{19} + \)\(15\!\cdots\!36\)\( \nu^{18} + \)\(17\!\cdots\!92\)\( \nu^{17} - \)\(19\!\cdots\!44\)\( \nu^{16} - \)\(18\!\cdots\!72\)\( \nu^{15} + \)\(24\!\cdots\!36\)\( \nu^{14} - \)\(82\!\cdots\!88\)\( \nu^{13} - \)\(15\!\cdots\!76\)\( \nu^{12} - \)\(92\!\cdots\!75\)\( \nu^{11} - \)\(12\!\cdots\!08\)\( \nu^{10} - \)\(70\!\cdots\!80\)\( \nu^{9} + \)\(19\!\cdots\!68\)\( \nu^{8} + \)\(19\!\cdots\!44\)\( \nu^{7} - \)\(51\!\cdots\!24\)\( \nu^{6} - \)\(48\!\cdots\!92\)\( \nu^{5} + \)\(18\!\cdots\!48\)\( \nu^{4} - \)\(29\!\cdots\!60\)\( \nu^{3} + \)\(22\!\cdots\!12\)\( \nu^{2} - \)\(10\!\cdots\!40\)\( \nu + \)\(17\!\cdots\!44\)\(\)\()/ \)\(15\!\cdots\!76\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(38\!\cdots\!43\)\( \nu^{19} + \)\(12\!\cdots\!98\)\( \nu^{18} + \)\(10\!\cdots\!08\)\( \nu^{17} - \)\(16\!\cdots\!52\)\( \nu^{16} - \)\(13\!\cdots\!32\)\( \nu^{15} - \)\(62\!\cdots\!80\)\( \nu^{14} + \)\(18\!\cdots\!76\)\( \nu^{13} + \)\(39\!\cdots\!92\)\( \nu^{12} + \)\(25\!\cdots\!15\)\( \nu^{11} + \)\(42\!\cdots\!26\)\( \nu^{10} + \)\(36\!\cdots\!88\)\( \nu^{9} - \)\(36\!\cdots\!36\)\( \nu^{8} - \)\(72\!\cdots\!56\)\( \nu^{7} + \)\(86\!\cdots\!32\)\( \nu^{6} + \)\(14\!\cdots\!72\)\( \nu^{5} - \)\(42\!\cdots\!40\)\( \nu^{4} + \)\(59\!\cdots\!20\)\( \nu^{3} - \)\(32\!\cdots\!24\)\( \nu^{2} + \)\(14\!\cdots\!64\)\( \nu - \)\(19\!\cdots\!44\)\(\)\()/ \)\(25\!\cdots\!20\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(11\!\cdots\!32\)\( \nu^{19} + \)\(24\!\cdots\!67\)\( \nu^{18} + \)\(64\!\cdots\!72\)\( \nu^{17} + \)\(11\!\cdots\!32\)\( \nu^{16} + \)\(10\!\cdots\!52\)\( \nu^{15} - \)\(18\!\cdots\!80\)\( \nu^{14} + \)\(60\!\cdots\!64\)\( \nu^{13} + \)\(11\!\cdots\!48\)\( \nu^{12} + \)\(73\!\cdots\!60\)\( \nu^{11} + \)\(10\!\cdots\!39\)\( \nu^{10} + \)\(81\!\cdots\!32\)\( \nu^{9} - \)\(11\!\cdots\!44\)\( \nu^{8} - \)\(15\!\cdots\!64\)\( \nu^{7} + \)\(34\!\cdots\!28\)\( \nu^{6} + \)\(33\!\cdots\!48\)\( \nu^{5} - \)\(14\!\cdots\!40\)\( \nu^{4} + \)\(22\!\cdots\!00\)\( \nu^{3} - \)\(16\!\cdots\!16\)\( \nu^{2} + \)\(87\!\cdots\!56\)\( \nu - \)\(17\!\cdots\!56\)\(\)\()/ \)\(51\!\cdots\!40\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(74\!\cdots\!99\)\( \nu^{19} + \)\(31\!\cdots\!44\)\( \nu^{18} + \)\(10\!\cdots\!84\)\( \nu^{17} + \)\(86\!\cdots\!64\)\( \nu^{16} - \)\(12\!\cdots\!16\)\( \nu^{15} - \)\(11\!\cdots\!40\)\( \nu^{14} + \)\(33\!\cdots\!08\)\( \nu^{13} + \)\(75\!\cdots\!16\)\( \nu^{12} + \)\(49\!\cdots\!55\)\( \nu^{11} + \)\(87\!\cdots\!68\)\( \nu^{10} + \)\(81\!\cdots\!64\)\( \nu^{9} - \)\(56\!\cdots\!48\)\( \nu^{8} - \)\(13\!\cdots\!68\)\( \nu^{7} + \)\(15\!\cdots\!16\)\( \nu^{6} + \)\(27\!\cdots\!76\)\( \nu^{5} - \)\(79\!\cdots\!20\)\( \nu^{4} + \)\(10\!\cdots\!20\)\( \nu^{3} - \)\(59\!\cdots\!32\)\( \nu^{2} + \)\(29\!\cdots\!52\)\( \nu - \)\(33\!\cdots\!32\)\(\)\()/ \)\(20\!\cdots\!60\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(11\!\cdots\!25\)\( \nu^{19} - \)\(70\!\cdots\!88\)\( \nu^{18} - \)\(14\!\cdots\!52\)\( \nu^{17} + \)\(33\!\cdots\!72\)\( \nu^{16} - \)\(61\!\cdots\!28\)\( \nu^{15} + \)\(17\!\cdots\!60\)\( \nu^{14} - \)\(47\!\cdots\!36\)\( \nu^{13} - \)\(11\!\cdots\!00\)\( \nu^{12} - \)\(76\!\cdots\!77\)\( \nu^{11} - \)\(14\!\cdots\!84\)\( \nu^{10} - \)\(14\!\cdots\!04\)\( \nu^{9} + \)\(73\!\cdots\!68\)\( \nu^{8} + \)\(23\!\cdots\!68\)\( \nu^{7} - \)\(20\!\cdots\!20\)\( \nu^{6} - \)\(50\!\cdots\!28\)\( \nu^{5} + \)\(11\!\cdots\!56\)\( \nu^{4} - \)\(12\!\cdots\!96\)\( \nu^{3} + \)\(48\!\cdots\!84\)\( \nu^{2} - \)\(17\!\cdots\!00\)\( \nu - \)\(17\!\cdots\!00\)\(\)\()/ \)\(20\!\cdots\!16\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(17\!\cdots\!31\)\( \nu^{19} + \)\(30\!\cdots\!06\)\( \nu^{18} - \)\(31\!\cdots\!72\)\( \nu^{17} - \)\(35\!\cdots\!84\)\( \nu^{16} + \)\(38\!\cdots\!88\)\( \nu^{15} - \)\(27\!\cdots\!52\)\( \nu^{14} + \)\(84\!\cdots\!40\)\( \nu^{13} + \)\(17\!\cdots\!28\)\( \nu^{12} + \)\(10\!\cdots\!63\)\( \nu^{11} + \)\(17\!\cdots\!62\)\( \nu^{10} + \)\(13\!\cdots\!80\)\( \nu^{9} - \)\(18\!\cdots\!28\)\( \nu^{8} - \)\(28\!\cdots\!56\)\( \nu^{7} + \)\(44\!\cdots\!28\)\( \nu^{6} + \)\(58\!\cdots\!28\)\( \nu^{5} - \)\(19\!\cdots\!40\)\( \nu^{4} + \)\(28\!\cdots\!64\)\( \nu^{3} - \)\(18\!\cdots\!28\)\( \nu^{2} + \)\(81\!\cdots\!12\)\( \nu - \)\(12\!\cdots\!56\)\(\)\()/ \)\(30\!\cdots\!52\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(48\!\cdots\!23\)\( \nu^{19} - \)\(24\!\cdots\!14\)\( \nu^{18} + \)\(87\!\cdots\!88\)\( \nu^{17} + \)\(54\!\cdots\!72\)\( \nu^{16} + \)\(27\!\cdots\!68\)\( \nu^{15} + \)\(77\!\cdots\!64\)\( \nu^{14} - \)\(24\!\cdots\!20\)\( \nu^{13} - \)\(48\!\cdots\!72\)\( \nu^{12} - \)\(30\!\cdots\!63\)\( \nu^{11} - \)\(44\!\cdots\!90\)\( \nu^{10} - \)\(26\!\cdots\!48\)\( \nu^{9} + \)\(65\!\cdots\!04\)\( \nu^{8} + \)\(83\!\cdots\!56\)\( \nu^{7} - \)\(13\!\cdots\!80\)\( \nu^{6} - \)\(15\!\cdots\!24\)\( \nu^{5} + \)\(59\!\cdots\!16\)\( \nu^{4} - \)\(83\!\cdots\!68\)\( \nu^{3} + \)\(54\!\cdots\!00\)\( \nu^{2} - \)\(23\!\cdots\!80\)\( \nu + \)\(48\!\cdots\!24\)\(\)\()/ \)\(68\!\cdots\!72\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(10\!\cdots\!29\)\( \nu^{19} - \)\(15\!\cdots\!04\)\( \nu^{18} + \)\(19\!\cdots\!12\)\( \nu^{17} - \)\(73\!\cdots\!52\)\( \nu^{16} - \)\(12\!\cdots\!52\)\( \nu^{15} + \)\(16\!\cdots\!32\)\( \nu^{14} - \)\(50\!\cdots\!76\)\( \nu^{13} - \)\(10\!\cdots\!68\)\( \nu^{12} - \)\(64\!\cdots\!97\)\( \nu^{11} - \)\(10\!\cdots\!84\)\( \nu^{10} - \)\(77\!\cdots\!28\)\( \nu^{9} + \)\(10\!\cdots\!84\)\( \nu^{8} + \)\(15\!\cdots\!24\)\( \nu^{7} - \)\(28\!\cdots\!20\)\( \nu^{6} - \)\(33\!\cdots\!80\)\( \nu^{5} + \)\(12\!\cdots\!44\)\( \nu^{4} - \)\(16\!\cdots\!72\)\( \nu^{3} + \)\(10\!\cdots\!16\)\( \nu^{2} - \)\(46\!\cdots\!60\)\( \nu + \)\(44\!\cdots\!56\)\(\)\()/ \)\(13\!\cdots\!44\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(96\!\cdots\!58\)\( \nu^{19} - \)\(29\!\cdots\!53\)\( \nu^{18} - \)\(11\!\cdots\!16\)\( \nu^{17} + \)\(27\!\cdots\!04\)\( \nu^{16} + \)\(71\!\cdots\!44\)\( \nu^{15} + \)\(15\!\cdots\!68\)\( \nu^{14} - \)\(45\!\cdots\!32\)\( \nu^{13} - \)\(96\!\cdots\!72\)\( \nu^{12} - \)\(62\!\cdots\!46\)\( \nu^{11} - \)\(10\!\cdots\!05\)\( \nu^{10} - \)\(93\!\cdots\!16\)\( \nu^{9} + \)\(84\!\cdots\!96\)\( \nu^{8} + \)\(17\!\cdots\!56\)\( \nu^{7} - \)\(20\!\cdots\!96\)\( \nu^{6} - \)\(33\!\cdots\!48\)\( \nu^{5} + \)\(10\!\cdots\!24\)\( \nu^{4} - \)\(14\!\cdots\!16\)\( \nu^{3} + \)\(97\!\cdots\!32\)\( \nu^{2} - \)\(49\!\cdots\!52\)\( \nu + \)\(87\!\cdots\!56\)\(\)\()/ \)\(10\!\cdots\!08\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(48\!\cdots\!91\)\( \nu^{19} - \)\(18\!\cdots\!21\)\( \nu^{18} - \)\(11\!\cdots\!96\)\( \nu^{17} - \)\(48\!\cdots\!76\)\( \nu^{16} + \)\(11\!\cdots\!44\)\( \nu^{15} + \)\(77\!\cdots\!80\)\( \nu^{14} - \)\(22\!\cdots\!12\)\( \nu^{13} - \)\(48\!\cdots\!64\)\( \nu^{12} - \)\(31\!\cdots\!15\)\( \nu^{11} - \)\(55\!\cdots\!17\)\( \nu^{10} - \)\(53\!\cdots\!56\)\( \nu^{9} + \)\(31\!\cdots\!52\)\( \nu^{8} + \)\(80\!\cdots\!12\)\( \nu^{7} - \)\(95\!\cdots\!84\)\( \nu^{6} - \)\(15\!\cdots\!24\)\( \nu^{5} + \)\(51\!\cdots\!60\)\( \nu^{4} - \)\(73\!\cdots\!40\)\( \nu^{3} + \)\(46\!\cdots\!08\)\( \nu^{2} - \)\(23\!\cdots\!48\)\( \nu + \)\(37\!\cdots\!48\)\(\)\()/ \)\(51\!\cdots\!40\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(42\!\cdots\!55\)\( \nu^{19} + \)\(73\!\cdots\!04\)\( \nu^{18} + \)\(18\!\cdots\!24\)\( \nu^{17} + \)\(26\!\cdots\!36\)\( \nu^{16} + \)\(90\!\cdots\!60\)\( \nu^{15} - \)\(68\!\cdots\!84\)\( \nu^{14} + \)\(21\!\cdots\!84\)\( \nu^{13} + \)\(42\!\cdots\!36\)\( \nu^{12} + \)\(27\!\cdots\!23\)\( \nu^{11} + \)\(43\!\cdots\!44\)\( \nu^{10} + \)\(34\!\cdots\!48\)\( \nu^{9} - \)\(41\!\cdots\!64\)\( \nu^{8} - \)\(66\!\cdots\!80\)\( \nu^{7} + \)\(10\!\cdots\!28\)\( \nu^{6} + \)\(13\!\cdots\!80\)\( \nu^{5} - \)\(50\!\cdots\!12\)\( \nu^{4} + \)\(72\!\cdots\!08\)\( \nu^{3} - \)\(47\!\cdots\!20\)\( \nu^{2} + \)\(21\!\cdots\!84\)\( \nu - \)\(27\!\cdots\!84\)\(\)\()/ \)\(41\!\cdots\!32\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(24\!\cdots\!51\)\( \nu^{19} - \)\(52\!\cdots\!70\)\( \nu^{18} + \)\(12\!\cdots\!84\)\( \nu^{17} + \)\(23\!\cdots\!72\)\( \nu^{16} + \)\(13\!\cdots\!44\)\( \nu^{15} + \)\(39\!\cdots\!04\)\( \nu^{14} - \)\(11\!\cdots\!44\)\( \nu^{13} - \)\(24\!\cdots\!24\)\( \nu^{12} - \)\(15\!\cdots\!31\)\( \nu^{11} - \)\(25\!\cdots\!98\)\( \nu^{10} - \)\(18\!\cdots\!40\)\( \nu^{9} + \)\(27\!\cdots\!40\)\( \nu^{8} + \)\(45\!\cdots\!56\)\( \nu^{7} - \)\(59\!\cdots\!88\)\( \nu^{6} - \)\(86\!\cdots\!92\)\( \nu^{5} + \)\(27\!\cdots\!84\)\( \nu^{4} - \)\(38\!\cdots\!36\)\( \nu^{3} + \)\(24\!\cdots\!08\)\( \nu^{2} - \)\(10\!\cdots\!76\)\( \nu + \)\(18\!\cdots\!16\)\(\)\()/ \)\(20\!\cdots\!16\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(10\!\cdots\!13\)\( \nu^{19} + \)\(82\!\cdots\!48\)\( \nu^{18} + \)\(93\!\cdots\!38\)\( \nu^{17} + \)\(74\!\cdots\!08\)\( \nu^{16} - \)\(55\!\cdots\!32\)\( \nu^{15} - \)\(16\!\cdots\!60\)\( \nu^{14} + \)\(53\!\cdots\!96\)\( \nu^{13} + \)\(10\!\cdots\!92\)\( \nu^{12} + \)\(65\!\cdots\!25\)\( \nu^{11} + \)\(10\!\cdots\!96\)\( \nu^{10} + \)\(79\!\cdots\!78\)\( \nu^{9} - \)\(10\!\cdots\!76\)\( \nu^{8} - \)\(15\!\cdots\!76\)\( \nu^{7} + \)\(27\!\cdots\!32\)\( \nu^{6} + \)\(28\!\cdots\!72\)\( \nu^{5} - \)\(12\!\cdots\!60\)\( \nu^{4} + \)\(19\!\cdots\!60\)\( \nu^{3} - \)\(14\!\cdots\!64\)\( \nu^{2} + \)\(72\!\cdots\!24\)\( \nu - \)\(14\!\cdots\!64\)\(\)\()/ \)\(85\!\cdots\!40\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(49\!\cdots\!77\)\( \nu^{19} - \)\(21\!\cdots\!32\)\( \nu^{18} - \)\(68\!\cdots\!42\)\( \nu^{17} + \)\(48\!\cdots\!88\)\( \nu^{16} + \)\(21\!\cdots\!08\)\( \nu^{15} + \)\(79\!\cdots\!60\)\( \nu^{14} - \)\(22\!\cdots\!84\)\( \nu^{13} - \)\(50\!\cdots\!68\)\( \nu^{12} - \)\(32\!\cdots\!05\)\( \nu^{11} - \)\(58\!\cdots\!44\)\( \nu^{10} - \)\(54\!\cdots\!02\)\( \nu^{9} + \)\(38\!\cdots\!84\)\( \nu^{8} + \)\(94\!\cdots\!44\)\( \nu^{7} - \)\(98\!\cdots\!68\)\( \nu^{6} - \)\(18\!\cdots\!88\)\( \nu^{5} + \)\(52\!\cdots\!80\)\( \nu^{4} - \)\(69\!\cdots\!00\)\( \nu^{3} + \)\(40\!\cdots\!96\)\( \nu^{2} - \)\(19\!\cdots\!96\)\( \nu + \)\(22\!\cdots\!96\)\(\)\()/ \)\(25\!\cdots\!20\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(13\!\cdots\!01\)\( \nu^{19} - \)\(23\!\cdots\!76\)\( \nu^{18} + \)\(33\!\cdots\!04\)\( \nu^{17} + \)\(12\!\cdots\!24\)\( \nu^{16} - \)\(37\!\cdots\!56\)\( \nu^{15} + \)\(22\!\cdots\!00\)\( \nu^{14} - \)\(68\!\cdots\!12\)\( \nu^{13} - \)\(13\!\cdots\!44\)\( \nu^{12} - \)\(88\!\cdots\!25\)\( \nu^{11} - \)\(13\!\cdots\!12\)\( \nu^{10} - \)\(10\!\cdots\!96\)\( \nu^{9} + \)\(14\!\cdots\!12\)\( \nu^{8} + \)\(23\!\cdots\!32\)\( \nu^{7} - \)\(36\!\cdots\!84\)\( \nu^{6} - \)\(46\!\cdots\!24\)\( \nu^{5} + \)\(16\!\cdots\!60\)\( \nu^{4} - \)\(23\!\cdots\!00\)\( \nu^{3} + \)\(15\!\cdots\!48\)\( \nu^{2} - \)\(66\!\cdots\!28\)\( \nu + \)\(10\!\cdots\!68\)\(\)\()/ \)\(51\!\cdots\!40\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{18} - \beta_{16} + \beta_{11} + 4 \beta_{8}\)
\(\nu^{3}\)\(=\)\(\beta_{19} + \beta_{18} - \beta_{16} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 7 \beta_{6} + \beta_{4}\)
\(\nu^{4}\)\(=\)\(9 \beta_{19} - \beta_{16} + \beta_{15} + 28 \beta_{10} - 10 \beta_{9} + \beta_{8} + \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 10 \beta_{4} - \beta_{2} - \beta_{1}\)
\(\nu^{5}\)\(=\)\(13 \beta_{19} - 13 \beta_{17} + 13 \beta_{15} - 11 \beta_{14} + 13 \beta_{12} - 11 \beta_{11} + 23 \beta_{10} - 13 \beta_{9} - 4 \beta_{8} + 23 \beta_{7} + \beta_{6} - 57 \beta_{5} + 13 \beta_{4} + \beta_{2} + 4\)
\(\nu^{6}\)\(=\)\(3 \beta_{19} - 83 \beta_{17} - 2 \beta_{16} + 94 \beta_{15} + \beta_{14} - 14 \beta_{13} + 94 \beta_{12} - 2 \beta_{11} + 17 \beta_{10} - \beta_{9} + 232 \beta_{7} - 15 \beta_{6} - 51 \beta_{5} + 14 \beta_{4} - 3 \beta_{3} + 51 \beta_{2} + 15 \beta_{1} - 17\)
\(\nu^{7}\)\(=\)\(-3 \beta_{19} + 3 \beta_{18} - 146 \beta_{17} - 104 \beta_{16} + 147 \beta_{15} + 147 \beta_{14} - 150 \beta_{13} + 150 \beta_{12} - 57 \beta_{10} + 104 \beta_{9} + 57 \beta_{8} + 308 \beta_{7} - 22 \beta_{5} - 146 \beta_{3} + 499 \beta_{2} - 22 \beta_{1} - 308\)
\(\nu^{8}\)\(=\)\(-68 \beta_{18} - 68 \beta_{17} + 21 \beta_{16} + 21 \beta_{15} + 899 \beta_{14} - 899 \beta_{13} + 166 \beta_{12} - 166 \beta_{11} + 49 \beta_{9} - 233 \beta_{8} + 233 \beta_{7} + 167 \beta_{6} + 167 \beta_{5} + 49 \beta_{4} - 793 \beta_{3} + 647 \beta_{2} - 647 \beta_{1} - 2094\)
\(\nu^{9}\)\(=\)\(50 \beta_{19} - 1567 \beta_{18} + 50 \beta_{17} + 1585 \beta_{16} - 949 \beta_{15} + 1663 \beta_{14} - 1585 \beta_{13} + 20 \beta_{12} - 1663 \beta_{11} + 590 \beta_{10} + 20 \beta_{9} - 3557 \beta_{8} - 590 \beta_{7} - 343 \beta_{6} + 949 \beta_{4} - 1567 \beta_{3} + 343 \beta_{2} - 4587 \beta_{1} - 3557\)
\(\nu^{10}\)\(=\)\(-1057 \beta_{19} - 7757 \beta_{18} + 8766 \beta_{16} - 764 \beta_{15} + 1898 \beta_{14} - 345 \beta_{13} + 764 \beta_{12} - 8766 \beta_{11} - 2925 \beta_{10} + 1898 \beta_{9} - 19876 \beta_{8} - 7403 \beta_{6} - 1647 \beta_{5} - 345 \beta_{4} - 1057 \beta_{3} - 1647 \beta_{2} - 7403 \beta_{1} - 2925\)
\(\nu^{11}\)\(=\)\(-16514 \beta_{19} - 16514 \beta_{18} - 513 \beta_{17} + 17990 \beta_{16} - 596 \beta_{15} + 596 \beta_{14} + 8592 \beta_{13} + 8592 \beta_{12} - 16713 \beta_{11} - 38800 \beta_{10} + 17990 \beta_{9} - 38800 \beta_{8} + 5311 \beta_{7} - 43637 \beta_{6} + 4672 \beta_{5} - 16713 \beta_{4} + 513 \beta_{3} - 4672 \beta_{1} + 5311\)
\(\nu^{12}\)\(=\)\(-77063 \beta_{19} - 14070 \beta_{18} + 14070 \beta_{17} + 21468 \beta_{16} - 21468 \beta_{15} + 9911 \beta_{14} + 9911 \beta_{13} - 5069 \beta_{12} - 5069 \beta_{11} - 194370 \beta_{10} + 86733 \beta_{9} - 35087 \beta_{8} - 35087 \beta_{7} - 80829 \beta_{6} + 80829 \beta_{5} - 86733 \beta_{4} + 15097 \beta_{2} + 15097 \beta_{1}\)
\(\nu^{13}\)\(=\)\(-172631 \beta_{19} + 3540 \beta_{18} + 172631 \beta_{17} + 11440 \beta_{16} - 191543 \beta_{15} + 77849 \beta_{14} + 11440 \beta_{13} - 174291 \beta_{12} + 77849 \beta_{11} - 412605 \beta_{10} + 174291 \beta_{9} + 43136 \beta_{8} - 412605 \beta_{7} - 59295 \beta_{6} + 425077 \beta_{5} - 191543 \beta_{4} - 3540 \beta_{3} - 59295 \beta_{2} - 43136\)
\(\nu^{14}\)\(=\)\(-172989 \beta_{19} + 773659 \beta_{17} + 117234 \beta_{16} - 867100 \beta_{15} - 69075 \beta_{14} + 241486 \beta_{13} - 867100 \beta_{12} + 117234 \beta_{11} - 409591 \beta_{10} + 69075 \beta_{9} - 1934776 \beta_{7} + 130337 \beta_{6} + 861781 \beta_{5} - 241486 \beta_{4} + 172989 \beta_{3} - 861781 \beta_{2} - 130337 \beta_{1} + 409591\)
\(\nu^{15}\)\(=\)\(6085 \beta_{19} - 6085 \beta_{18} + 1797956 \beta_{17} + 707214 \beta_{16} - 1807851 \beta_{15} - 1807851 \beta_{14} + 2019104 \beta_{13} - 2019104 \beta_{12} + 180224 \beta_{11} + 310095 \beta_{10} - 707214 \beta_{9} - 310095 \beta_{8} - 4334562 \beta_{7} + 720730 \beta_{5} - 180224 \beta_{4} + 1797956 \beta_{3} - 4208167 \beta_{2} + 720730 \beta_{1} + 4334562\)
\(\nu^{16}\)\(=\)\(2032620 \beta_{18} + 2032620 \beta_{17} - 891059 \beta_{16} - 891059 \beta_{15} - 8732441 \beta_{14} + 8732441 \beta_{13} - 2702720 \beta_{12} + 2702720 \beta_{11} - 1317975 \beta_{9} + 4694581 \beta_{8} - 4694581 \beta_{7} - 1054423 \beta_{6} - 1054423 \beta_{5} - 1317975 \beta_{4} + 7823869 \beta_{3} - 9072035 \beta_{2} + 9072035 \beta_{1} + 19468618\)
\(\nu^{17}\)\(=\)\(330322 \beta_{19} + 18695535 \beta_{18} + 330322 \beta_{17} - 18707565 \beta_{16} + 6436269 \beta_{15} - 21155071 \beta_{14} + 18707565 \beta_{13} - 2539356 \beta_{12} + 21155071 \beta_{11} - 1770186 \beta_{10} - 2539356 \beta_{9} + 45263765 \beta_{8} + 1770186 \beta_{7} + 8509319 \beta_{6} - 6436269 \beta_{4} + 18695535 \beta_{3} - 8509319 \beta_{2} + 42121419 \beta_{1} + 45263765\)
\(\nu^{18}\)\(=\)\(23228121 \beta_{19} + 79536549 \beta_{18} - 88408294 \beta_{16} + 14388480 \beta_{15} - 30086562 \beta_{14} + 11036645 \beta_{13} - 14388480 \beta_{12} + 88408294 \beta_{11} + 53087193 \beta_{10} - 30086562 \beta_{9} + 197262636 \beta_{8} + 94846311 \beta_{6} + 7784283 \beta_{5} + 11036645 \beta_{4} + 23228121 \beta_{3} + 7784283 \beta_{2} + 94846311 \beta_{1} + 53087193\)
\(\nu^{19}\)\(=\)\(194291250 \beta_{19} + 194291250 \beta_{18} - 7913799 \beta_{17} - 220871206 \beta_{16} + 33338924 \beta_{15} - 33338924 \beta_{14} - 58575976 \beta_{13} - 58575976 \beta_{12} + 193432777 \beta_{11} + 471300144 \beta_{10} - 220871206 \beta_{9} + 471300144 \beta_{8} - 3698703 \beta_{7} + 424838813 \beta_{6} - 98388604 \beta_{5} + 193432777 \beta_{4} + 7913799 \beta_{3} + 98388604 \beta_{1} - 3698703\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/170\mathbb{Z}\right)^\times\).

\(n\) \(71\) \(137\)
\(\chi(n)\) \(\beta_{8}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−2.32088 0.961341i
−1.86202 0.771273i
0.236338 + 0.0978946i
0.953222 + 0.394838i
2.99334 + 1.23988i
−2.32088 + 0.961341i
−1.86202 + 0.771273i
0.236338 0.0978946i
0.953222 0.394838i
2.99334 1.23988i
1.18678 + 2.86514i
0.355063 + 0.857197i
0.254075 + 0.613391i
−0.826884 1.99627i
−0.969032 2.33945i
1.18678 2.86514i
0.355063 0.857197i
0.254075 0.613391i
−0.826884 + 1.99627i
−0.969032 + 2.33945i
0.707107 + 0.707107i −0.961341 + 2.32088i 1.00000i 2.19336 0.434925i −2.32088 + 0.961341i 0.124542 0.0515871i −0.707107 + 0.707107i −2.34100 2.34100i 1.85848 + 1.24340i
9.2 0.707107 + 0.707107i −0.771273 + 1.86202i 1.00000i −2.22682 0.203147i −1.86202 + 0.771273i −2.47378 + 1.02468i −0.707107 + 0.707107i −0.750924 0.750924i −1.43095 1.71825i
9.3 0.707107 + 0.707107i 0.0978946 0.236338i 1.00000i −1.19048 + 1.89281i 0.236338 0.0978946i 3.48449 1.44332i −0.707107 + 0.707107i 2.07505 + 2.07505i −2.18022 + 0.496623i
9.4 0.707107 + 0.707107i 0.394838 0.953222i 1.00000i 0.473610 2.18534i 0.953222 0.394838i 0.363965 0.150759i −0.707107 + 0.707107i 1.36858 + 1.36858i 1.88016 1.21037i
9.5 0.707107 + 0.707107i 1.23988 2.99334i 1.00000i 1.87165 + 1.22349i 2.99334 1.23988i −1.49921 + 0.620992i −0.707107 + 0.707107i −5.30145 5.30145i 0.458323 + 2.18859i
19.1 0.707107 0.707107i −0.961341 2.32088i 1.00000i 2.19336 + 0.434925i −2.32088 0.961341i 0.124542 + 0.0515871i −0.707107 0.707107i −2.34100 + 2.34100i 1.85848 1.24340i
19.2 0.707107 0.707107i −0.771273 1.86202i 1.00000i −2.22682 + 0.203147i −1.86202 0.771273i −2.47378 1.02468i −0.707107 0.707107i −0.750924 + 0.750924i −1.43095 + 1.71825i
19.3 0.707107 0.707107i 0.0978946 + 0.236338i 1.00000i −1.19048 1.89281i 0.236338 + 0.0978946i 3.48449 + 1.44332i −0.707107 0.707107i 2.07505 2.07505i −2.18022 0.496623i
19.4 0.707107 0.707107i 0.394838 + 0.953222i 1.00000i 0.473610 + 2.18534i 0.953222 + 0.394838i 0.363965 + 0.150759i −0.707107 0.707107i 1.36858 1.36858i 1.88016 + 1.21037i
19.5 0.707107 0.707107i 1.23988 + 2.99334i 1.00000i 1.87165 1.22349i 2.99334 + 1.23988i −1.49921 0.620992i −0.707107 0.707107i −5.30145 + 5.30145i 0.458323 2.18859i
49.1 −0.707107 + 0.707107i −2.86514 + 1.18678i 1.00000i −2.09387 + 0.784666i 1.18678 2.86514i 1.09360 2.64018i 0.707107 + 0.707107i 4.67924 4.67924i 0.925748 2.03543i
49.2 −0.707107 + 0.707107i −0.857197 + 0.355063i 1.00000i 1.64404 1.51563i 0.355063 0.857197i 0.939960 2.26926i 0.707107 + 0.707107i −1.51260 + 1.51260i −0.0908004 + 2.23422i
49.3 −0.707107 + 0.707107i −0.613391 + 0.254075i 1.00000i −1.28583 1.82938i 0.254075 0.613391i −1.97319 + 4.76369i 0.707107 + 0.707107i −1.80963 + 1.80963i 2.20279 + 0.384345i
49.4 −0.707107 + 0.707107i 1.99627 0.826884i 1.00000i 0.399242 + 2.20014i −0.826884 + 1.99627i −1.32795 + 3.20595i 0.707107 + 0.707107i 1.18005 1.18005i −1.83804 1.27343i
49.5 −0.707107 + 0.707107i 2.33945 0.969032i 1.00000i −1.78490 1.34690i −0.969032 + 2.33945i 1.26758 3.06021i 0.707107 + 0.707107i 2.41268 2.41268i 2.21452 0.309710i
59.1 −0.707107 0.707107i −2.86514 1.18678i 1.00000i −2.09387 0.784666i 1.18678 + 2.86514i 1.09360 + 2.64018i 0.707107 0.707107i 4.67924 + 4.67924i 0.925748 + 2.03543i
59.2 −0.707107 0.707107i −0.857197 0.355063i 1.00000i 1.64404 + 1.51563i 0.355063 + 0.857197i 0.939960 + 2.26926i 0.707107 0.707107i −1.51260 1.51260i −0.0908004 2.23422i
59.3 −0.707107 0.707107i −0.613391 0.254075i 1.00000i −1.28583 + 1.82938i 0.254075 + 0.613391i −1.97319 4.76369i 0.707107 0.707107i −1.80963 1.80963i 2.20279 0.384345i
59.4 −0.707107 0.707107i 1.99627 + 0.826884i 1.00000i 0.399242 2.20014i −0.826884 1.99627i −1.32795 3.20595i 0.707107 0.707107i 1.18005 + 1.18005i −1.83804 + 1.27343i
59.5 −0.707107 0.707107i 2.33945 + 0.969032i 1.00000i −1.78490 + 1.34690i −0.969032 2.33945i 1.26758 + 3.06021i 0.707107 0.707107i 2.41268 + 2.41268i 2.21452 + 0.309710i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.m even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.2.n.b yes 20
5.b even 2 1 170.2.n.a 20
5.c odd 4 1 850.2.l.h 20
5.c odd 4 1 850.2.l.i 20
17.d even 8 1 170.2.n.a 20
85.k odd 8 1 850.2.l.h 20
85.m even 8 1 inner 170.2.n.b yes 20
85.n odd 8 1 850.2.l.i 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.n.a 20 5.b even 2 1
170.2.n.a 20 17.d even 8 1
170.2.n.b yes 20 1.a even 1 1 trivial
170.2.n.b yes 20 85.m even 8 1 inner
850.2.l.h 20 5.c odd 4 1
850.2.l.h 20 85.k odd 8 1
850.2.l.i 20 5.c odd 4 1
850.2.l.i 20 85.n odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{20} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(170, [\chi])\).