Properties

Label 6045.2.a.bh
Level 6045
Weight 2
Character orbit 6045.a
Self dual Yes
Analytic conductor 48.270
Analytic rank 0
Dimension 17
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6045.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2695680219\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{17}\mathstrut -\mathstrut \) \(2\) \(x^{16}\mathstrut -\mathstrut \) \(25\) \(x^{15}\mathstrut +\mathstrut \) \(47\) \(x^{14}\mathstrut +\mathstrut \) \(252\) \(x^{13}\mathstrut -\mathstrut \) \(437\) \(x^{12}\mathstrut -\mathstrut \) \(1319\) \(x^{11}\mathstrut +\mathstrut \) \(2056\) \(x^{10}\mathstrut +\mathstrut \) \(3854\) \(x^{9}\mathstrut -\mathstrut \) \(5201\) \(x^{8}\mathstrut -\mathstrut \) \(6304\) \(x^{7}\mathstrut +\mathstrut \) \(6915\) \(x^{6}\mathstrut +\mathstrut \) \(5469\) \(x^{5}\mathstrut -\mathstrut \) \(4238\) \(x^{4}\mathstrut -\mathstrut \) \(2198\) \(x^{3}\mathstrut +\mathstrut \) \(760\) \(x^{2}\mathstrut +\mathstrut \) \(306\) \(x\mathstrut +\mathstrut \) \(8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(113557\) \(\nu^{16}\mathstrut -\mathstrut \) \(530768\) \(\nu^{15}\mathstrut +\mathstrut \) \(5270067\) \(\nu^{14}\mathstrut +\mathstrut \) \(10382037\) \(\nu^{13}\mathstrut -\mathstrut \) \(83048126\) \(\nu^{12}\mathstrut -\mathstrut \) \(70627697\) \(\nu^{11}\mathstrut +\mathstrut \) \(620969061\) \(\nu^{10}\mathstrut +\mathstrut \) \(171782910\) \(\nu^{9}\mathstrut -\mathstrut \) \(2421255639\) \(\nu^{8}\mathstrut +\mathstrut \) \(78954606\) \(\nu^{7}\mathstrut +\mathstrut \) \(4898277165\) \(\nu^{6}\mathstrut -\mathstrut \) \(898215335\) \(\nu^{5}\mathstrut -\mathstrut \) \(4695220286\) \(\nu^{4}\mathstrut +\mathstrut \) \(1089799149\) \(\nu^{3}\mathstrut +\mathstrut \) \(1669490717\) \(\nu^{2}\mathstrut -\mathstrut \) \(324483746\) \(\nu\mathstrut -\mathstrut \) \(133114498\)\()/14023831\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(312195\) \(\nu^{16}\mathstrut -\mathstrut \) \(3509363\) \(\nu^{15}\mathstrut +\mathstrut \) \(13349902\) \(\nu^{14}\mathstrut +\mathstrut \) \(87408009\) \(\nu^{13}\mathstrut -\mathstrut \) \(196134497\) \(\nu^{12}\mathstrut -\mathstrut \) \(869010094\) \(\nu^{11}\mathstrut +\mathstrut \) \(1349656637\) \(\nu^{10}\mathstrut +\mathstrut \) \(4400726869\) \(\nu^{9}\mathstrut -\mathstrut \) \(4694027971\) \(\nu^{8}\mathstrut -\mathstrut \) \(11993289690\) \(\nu^{7}\mathstrut +\mathstrut \) \(7965510540\) \(\nu^{6}\mathstrut +\mathstrut \) \(17031291127\) \(\nu^{5}\mathstrut -\mathstrut \) \(5497485944\) \(\nu^{4}\mathstrut -\mathstrut \) \(10936962026\) \(\nu^{3}\mathstrut +\mathstrut \) \(511972718\) \(\nu^{2}\mathstrut +\mathstrut \) \(1966684368\) \(\nu\mathstrut +\mathstrut \) \(160614960\)\()/28047662\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(1500797\) \(\nu^{16}\mathstrut -\mathstrut \) \(2833680\) \(\nu^{15}\mathstrut +\mathstrut \) \(44873866\) \(\nu^{14}\mathstrut +\mathstrut \) \(73542695\) \(\nu^{13}\mathstrut -\mathstrut \) \(532269412\) \(\nu^{12}\mathstrut -\mathstrut \) \(761838312\) \(\nu^{11}\mathstrut +\mathstrut \) \(3189061955\) \(\nu^{10}\mathstrut +\mathstrut \) \(4005715774\) \(\nu^{9}\mathstrut -\mathstrut \) \(10164598845\) \(\nu^{8}\mathstrut -\mathstrut \) \(11241989735\) \(\nu^{7}\mathstrut +\mathstrut \) \(16639475285\) \(\nu^{6}\mathstrut +\mathstrut \) \(16241045978\) \(\nu^{5}\mathstrut -\mathstrut \) \(12298440348\) \(\nu^{4}\mathstrut -\mathstrut \) \(10518791456\) \(\nu^{3}\mathstrut +\mathstrut \) \(2699073926\) \(\nu^{2}\mathstrut +\mathstrut \) \(1899638378\) \(\nu\mathstrut +\mathstrut \) \(64950898\)\()/28047662\)
\(\beta_{7}\)\(=\)\((\)\(1604091\) \(\nu^{16}\mathstrut -\mathstrut \) \(384325\) \(\nu^{15}\mathstrut -\mathstrut \) \(41605574\) \(\nu^{14}\mathstrut +\mathstrut \) \(2341871\) \(\nu^{13}\mathstrut +\mathstrut \) \(429886899\) \(\nu^{12}\mathstrut +\mathstrut \) \(53714652\) \(\nu^{11}\mathstrut -\mathstrut \) \(2247822761\) \(\nu^{10}\mathstrut -\mathstrut \) \(657336163\) \(\nu^{9}\mathstrut +\mathstrut \) \(6243488261\) \(\nu^{8}\mathstrut +\mathstrut \) \(2678609842\) \(\nu^{7}\mathstrut -\mathstrut \) \(8866269156\) \(\nu^{6}\mathstrut -\mathstrut \) \(4549058777\) \(\nu^{5}\mathstrut +\mathstrut \) \(5667993268\) \(\nu^{4}\mathstrut +\mathstrut \) \(3045081072\) \(\nu^{3}\mathstrut -\mathstrut \) \(1097688346\) \(\nu^{2}\mathstrut -\mathstrut \) \(548614148\) \(\nu\mathstrut -\mathstrut \) \(18135454\)\()/28047662\)
\(\beta_{8}\)\(=\)\((\)\(823629\) \(\nu^{16}\mathstrut -\mathstrut \) \(1818804\) \(\nu^{15}\mathstrut -\mathstrut \) \(21236683\) \(\nu^{14}\mathstrut +\mathstrut \) \(44900786\) \(\nu^{13}\mathstrut +\mathstrut \) \(221201292\) \(\nu^{12}\mathstrut -\mathstrut \) \(442474917\) \(\nu^{11}\mathstrut -\mathstrut \) \(1194843988\) \(\nu^{10}\mathstrut +\mathstrut \) \(2223167706\) \(\nu^{9}\mathstrut +\mathstrut \) \(3574397427\) \(\nu^{8}\mathstrut -\mathstrut \) \(6016496297\) \(\nu^{7}\mathstrut -\mathstrut \) \(5865410604\) \(\nu^{6}\mathstrut +\mathstrut \) \(8451012595\) \(\nu^{5}\mathstrut +\mathstrut \) \(4905261150\) \(\nu^{4}\mathstrut -\mathstrut \) \(5292910107\) \(\nu^{3}\mathstrut -\mathstrut \) \(1746027224\) \(\nu^{2}\mathstrut +\mathstrut \) \(919914264\) \(\nu\mathstrut +\mathstrut \) \(154119101\)\()/14023831\)
\(\beta_{9}\)\(=\)\((\)\(1839987\) \(\nu^{16}\mathstrut -\mathstrut \) \(267610\) \(\nu^{15}\mathstrut -\mathstrut \) \(47761999\) \(\nu^{14}\mathstrut -\mathstrut \) \(450507\) \(\nu^{13}\mathstrut +\mathstrut \) \(494622034\) \(\nu^{12}\mathstrut +\mathstrut \) \(81684169\) \(\nu^{11}\mathstrut -\mathstrut \) \(2598840645\) \(\nu^{10}\mathstrut -\mathstrut \) \(820692828\) \(\nu^{9}\mathstrut +\mathstrut \) \(7282350710\) \(\nu^{8}\mathstrut +\mathstrut \) \(3310199705\) \(\nu^{7}\mathstrut -\mathstrut \) \(10492827800\) \(\nu^{6}\mathstrut -\mathstrut \) \(6045124051\) \(\nu^{5}\mathstrut +\mathstrut \) \(6902239911\) \(\nu^{4}\mathstrut +\mathstrut \) \(4729675172\) \(\nu^{3}\mathstrut -\mathstrut \) \(1580860484\) \(\nu^{2}\mathstrut -\mathstrut \) \(1170041948\) \(\nu\mathstrut +\mathstrut \) \(88911066\)\()/28047662\)
\(\beta_{10}\)\(=\)\((\)\(2263647\) \(\nu^{16}\mathstrut -\mathstrut \) \(1882134\) \(\nu^{15}\mathstrut -\mathstrut \) \(57777861\) \(\nu^{14}\mathstrut +\mathstrut \) \(38112665\) \(\nu^{13}\mathstrut +\mathstrut \) \(589221704\) \(\nu^{12}\mathstrut -\mathstrut \) \(284971261\) \(\nu^{11}\mathstrut -\mathstrut \) \(3057841575\) \(\nu^{10}\mathstrut +\mathstrut \) \(961805100\) \(\nu^{9}\mathstrut +\mathstrut \) \(8505862278\) \(\nu^{8}\mathstrut -\mathstrut \) \(1437669135\) \(\nu^{7}\mathstrut -\mathstrut \) \(12239294860\) \(\nu^{6}\mathstrut +\mathstrut \) \(850560935\) \(\nu^{5}\mathstrut +\mathstrut \) \(7967091845\) \(\nu^{4}\mathstrut -\mathstrut \) \(321519062\) \(\nu^{3}\mathstrut -\mathstrut \) \(1568781388\) \(\nu^{2}\mathstrut +\mathstrut \) \(286418488\) \(\nu\mathstrut +\mathstrut \) \(58213626\)\()/28047662\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(1322580\) \(\nu^{16}\mathstrut +\mathstrut \) \(593343\) \(\nu^{15}\mathstrut +\mathstrut \) \(34139372\) \(\nu^{14}\mathstrut -\mathstrut \) \(9391330\) \(\nu^{13}\mathstrut -\mathstrut \) \(352121239\) \(\nu^{12}\mathstrut +\mathstrut \) \(36045591\) \(\nu^{11}\mathstrut +\mathstrut \) \(1846126566\) \(\nu^{10}\mathstrut +\mathstrut \) \(109116630\) \(\nu^{9}\mathstrut -\mathstrut \) \(5167779456\) \(\nu^{8}\mathstrut -\mathstrut \) \(1015367914\) \(\nu^{7}\mathstrut +\mathstrut \) \(7401279035\) \(\nu^{6}\mathstrut +\mathstrut \) \(2220420630\) \(\nu^{5}\mathstrut -\mathstrut \) \(4635908462\) \(\nu^{4}\mathstrut -\mathstrut \) \(1815548007\) \(\nu^{3}\mathstrut +\mathstrut \) \(702952785\) \(\nu^{2}\mathstrut +\mathstrut \) \(471493319\) \(\nu\mathstrut +\mathstrut \) \(37102250\)\()/14023831\)
\(\beta_{12}\)\(=\)\((\)\(3448071\) \(\nu^{16}\mathstrut -\mathstrut \) \(591283\) \(\nu^{15}\mathstrut -\mathstrut \) \(91652439\) \(\nu^{14}\mathstrut +\mathstrut \) \(1155319\) \(\nu^{13}\mathstrut +\mathstrut \) \(973882157\) \(\nu^{12}\mathstrut +\mathstrut \) \(133423725\) \(\nu^{11}\mathstrut -\mathstrut \) \(5254617039\) \(\nu^{10}\mathstrut -\mathstrut \) \(1412534597\) \(\nu^{9}\mathstrut +\mathstrut \) \(15091413240\) \(\nu^{8}\mathstrut +\mathstrut \) \(5644078754\) \(\nu^{7}\mathstrut -\mathstrut \) \(22073709939\) \(\nu^{6}\mathstrut -\mathstrut \) \(9857570968\) \(\nu^{5}\mathstrut +\mathstrut \) \(14149820915\) \(\nu^{4}\mathstrut +\mathstrut \) \(6958870154\) \(\nu^{3}\mathstrut -\mathstrut \) \(2435890434\) \(\nu^{2}\mathstrut -\mathstrut \) \(1255967342\) \(\nu\mathstrut -\mathstrut \) \(30204858\)\()/28047662\)
\(\beta_{13}\)\(=\)\((\)\(3616735\) \(\nu^{16}\mathstrut -\mathstrut \) \(5281842\) \(\nu^{15}\mathstrut -\mathstrut \) \(89703162\) \(\nu^{14}\mathstrut +\mathstrut \) \(117460815\) \(\nu^{13}\mathstrut +\mathstrut \) \(886622524\) \(\nu^{12}\mathstrut -\mathstrut \) \(1014336732\) \(\nu^{11}\mathstrut -\mathstrut \) \(4451885089\) \(\nu^{10}\mathstrut +\mathstrut \) \(4327778728\) \(\nu^{9}\mathstrut +\mathstrut \) \(11973831065\) \(\nu^{8}\mathstrut -\mathstrut \) \(9680674593\) \(\nu^{7}\mathstrut -\mathstrut \) \(16642435131\) \(\nu^{6}\mathstrut +\mathstrut \) \(11237279022\) \(\nu^{5}\mathstrut +\mathstrut \) \(10351984080\) \(\nu^{4}\mathstrut -\mathstrut \) \(6141699664\) \(\nu^{3}\mathstrut -\mathstrut \) \(1864907048\) \(\nu^{2}\mathstrut +\mathstrut \) \(1167845648\) \(\nu\mathstrut +\mathstrut \) \(117339748\)\()/28047662\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(2051817\) \(\nu^{16}\mathstrut +\mathstrut \) \(1074872\) \(\nu^{15}\mathstrut +\mathstrut \) \(52769930\) \(\nu^{14}\mathstrut -\mathstrut \) \(18831079\) \(\nu^{13}\mathstrut -\mathstrut \) \(541921869\) \(\nu^{12}\mathstrut +\mathstrut \) \(101643546\) \(\nu^{11}\mathstrut +\mathstrut \) \(2828341110\) \(\nu^{10}\mathstrut -\mathstrut \) \(70556136\) \(\nu^{9}\mathstrut -\mathstrut \) \(7894106494\) \(\nu^{8}\mathstrut -\mathstrut \) \(936265285\) \(\nu^{7}\mathstrut +\mathstrut \) \(11366061330\) \(\nu^{6}\mathstrut +\mathstrut \) \(2597281558\) \(\nu^{5}\mathstrut -\mathstrut \) \(7420642047\) \(\nu^{4}\mathstrut -\mathstrut \) \(2204078055\) \(\nu^{3}\mathstrut +\mathstrut \) \(1476654119\) \(\nu^{2}\mathstrut +\mathstrut \) \(441811730\) \(\nu\mathstrut +\mathstrut \) \(10580640\)\()/14023831\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(2558766\) \(\nu^{16}\mathstrut +\mathstrut \) \(1224081\) \(\nu^{15}\mathstrut +\mathstrut \) \(67288514\) \(\nu^{14}\mathstrut -\mathstrut \) \(21959060\) \(\nu^{13}\mathstrut -\mathstrut \) \(709445968\) \(\nu^{12}\mathstrut +\mathstrut \) \(126249210\) \(\nu^{11}\mathstrut +\mathstrut \) \(3820473522\) \(\nu^{10}\mathstrut -\mathstrut \) \(161031477\) \(\nu^{9}\mathstrut -\mathstrut \) \(11069214955\) \(\nu^{8}\mathstrut -\mathstrut \) \(780657571\) \(\nu^{7}\mathstrut +\mathstrut \) \(16640955208\) \(\nu^{6}\mathstrut +\mathstrut \) \(2515907309\) \(\nu^{5}\mathstrut -\mathstrut \) \(11325212214\) \(\nu^{4}\mathstrut -\mathstrut \) \(2314760375\) \(\nu^{3}\mathstrut +\mathstrut \) \(2267966656\) \(\nu^{2}\mathstrut +\mathstrut \) \(604301492\) \(\nu\mathstrut +\mathstrut \) \(1853237\)\()/14023831\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(3116457\) \(\nu^{16}\mathstrut +\mathstrut \) \(2351568\) \(\nu^{15}\mathstrut +\mathstrut \) \(79728849\) \(\nu^{14}\mathstrut -\mathstrut \) \(45045593\) \(\nu^{13}\mathstrut -\mathstrut \) \(816697554\) \(\nu^{12}\mathstrut +\mathstrut \) \(300324502\) \(\nu^{11}\mathstrut +\mathstrut \) \(4272901584\) \(\nu^{10}\mathstrut -\mathstrut \) \(740835504\) \(\nu^{9}\mathstrut -\mathstrut \) \(12062740125\) \(\nu^{8}\mathstrut -\mathstrut \) \(28301554\) \(\nu^{7}\mathstrut +\mathstrut \) \(17850119443\) \(\nu^{6}\mathstrut +\mathstrut \) \(2491934088\) \(\nu^{5}\mathstrut -\mathstrut \) \(12321599225\) \(\nu^{4}\mathstrut -\mathstrut \) \(2753575910\) \(\nu^{3}\mathstrut +\mathstrut \) \(2845878583\) \(\nu^{2}\mathstrut +\mathstrut \) \(602412121\) \(\nu\mathstrut -\mathstrut \) \(88198496\)\()/14023831\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(28\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{6}\)\(=\)\(10\) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(12\) \(\beta_{10}\mathstrut +\mathstrut \) \(9\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(46\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(87\)
\(\nu^{7}\)\(=\)\(13\) \(\beta_{15}\mathstrut +\mathstrut \) \(2\) \(\beta_{14}\mathstrut +\mathstrut \) \(13\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(14\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(68\) \(\beta_{3}\mathstrut +\mathstrut \) \(13\) \(\beta_{2}\mathstrut +\mathstrut \) \(165\) \(\beta_{1}\mathstrut +\mathstrut \) \(14\)
\(\nu^{8}\)\(=\)\(-\)\(\beta_{15}\mathstrut +\mathstrut \) \(81\) \(\beta_{14}\mathstrut -\mathstrut \) \(16\) \(\beta_{13}\mathstrut -\mathstrut \) \(14\) \(\beta_{12}\mathstrut -\mathstrut \) \(13\) \(\beta_{11}\mathstrut +\mathstrut \) \(111\) \(\beta_{10}\mathstrut +\mathstrut \) \(67\) \(\beta_{9}\mathstrut +\mathstrut \) \(14\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(11\) \(\beta_{5}\mathstrut +\mathstrut \) \(15\) \(\beta_{4}\mathstrut +\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(301\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\) \(\beta_{1}\mathstrut +\mathstrut \) \(541\)
\(\nu^{9}\)\(=\)\(\beta_{16}\mathstrut +\mathstrut \) \(118\) \(\beta_{15}\mathstrut +\mathstrut \) \(33\) \(\beta_{14}\mathstrut +\mathstrut \) \(121\) \(\beta_{13}\mathstrut -\mathstrut \) \(17\) \(\beta_{12}\mathstrut -\mathstrut \) \(27\) \(\beta_{11}\mathstrut +\mathstrut \) \(6\) \(\beta_{10}\mathstrut +\mathstrut \) \(15\) \(\beta_{9}\mathstrut +\mathstrut \) \(13\) \(\beta_{8}\mathstrut +\mathstrut \) \(17\) \(\beta_{7}\mathstrut -\mathstrut \) \(140\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(16\) \(\beta_{4}\mathstrut +\mathstrut \) \(490\) \(\beta_{3}\mathstrut +\mathstrut \) \(121\) \(\beta_{2}\mathstrut +\mathstrut \) \(1007\) \(\beta_{1}\mathstrut +\mathstrut \) \(137\)
\(\nu^{10}\)\(=\)\(2\) \(\beta_{16}\mathstrut -\mathstrut \) \(23\) \(\beta_{15}\mathstrut +\mathstrut \) \(620\) \(\beta_{14}\mathstrut -\mathstrut \) \(174\) \(\beta_{13}\mathstrut -\mathstrut \) \(140\) \(\beta_{12}\mathstrut -\mathstrut \) \(122\) \(\beta_{11}\mathstrut +\mathstrut \) \(933\) \(\beta_{10}\mathstrut +\mathstrut \) \(477\) \(\beta_{9}\mathstrut +\mathstrut \) \(136\) \(\beta_{8}\mathstrut -\mathstrut \) \(20\) \(\beta_{7}\mathstrut -\mathstrut \) \(38\) \(\beta_{6}\mathstrut -\mathstrut \) \(82\) \(\beta_{5}\mathstrut +\mathstrut \) \(158\) \(\beta_{4}\mathstrut +\mathstrut \) \(134\) \(\beta_{3}\mathstrut +\mathstrut \) \(1982\) \(\beta_{2}\mathstrut +\mathstrut \) \(158\) \(\beta_{1}\mathstrut +\mathstrut \) \(3495\)
\(\nu^{11}\)\(=\)\(19\) \(\beta_{16}\mathstrut +\mathstrut \) \(932\) \(\beta_{15}\mathstrut +\mathstrut \) \(371\) \(\beta_{14}\mathstrut +\mathstrut \) \(991\) \(\beta_{13}\mathstrut -\mathstrut \) \(198\) \(\beta_{12}\mathstrut -\mathstrut \) \(250\) \(\beta_{11}\mathstrut +\mathstrut \) \(120\) \(\beta_{10}\mathstrut +\mathstrut \) \(158\) \(\beta_{9}\mathstrut +\mathstrut \) \(120\) \(\beta_{8}\mathstrut +\mathstrut \) \(192\) \(\beta_{7}\mathstrut -\mathstrut \) \(1229\) \(\beta_{6}\mathstrut +\mathstrut \) \(76\) \(\beta_{5}\mathstrut +\mathstrut \) \(181\) \(\beta_{4}\mathstrut +\mathstrut \) \(3470\) \(\beta_{3}\mathstrut +\mathstrut \) \(988\) \(\beta_{2}\mathstrut +\mathstrut \) \(6308\) \(\beta_{1}\mathstrut +\mathstrut \) \(1167\)
\(\nu^{12}\)\(=\)\(38\) \(\beta_{16}\mathstrut -\mathstrut \) \(314\) \(\beta_{15}\mathstrut +\mathstrut \) \(4651\) \(\beta_{14}\mathstrut -\mathstrut \) \(1613\) \(\beta_{13}\mathstrut -\mathstrut \) \(1235\) \(\beta_{12}\mathstrut -\mathstrut \) \(1018\) \(\beta_{11}\mathstrut +\mathstrut \) \(7471\) \(\beta_{10}\mathstrut +\mathstrut \) \(3356\) \(\beta_{9}\mathstrut +\mathstrut \) \(1145\) \(\beta_{8}\mathstrut -\mathstrut \) \(253\) \(\beta_{7}\mathstrut -\mathstrut \) \(480\) \(\beta_{6}\mathstrut -\mathstrut \) \(498\) \(\beta_{5}\mathstrut +\mathstrut \) \(1442\) \(\beta_{4}\mathstrut +\mathstrut \) \(1104\) \(\beta_{3}\mathstrut +\mathstrut \) \(13159\) \(\beta_{2}\mathstrut +\mathstrut \) \(1421\) \(\beta_{1}\mathstrut +\mathstrut \) \(23119\)
\(\nu^{13}\)\(=\)\(240\) \(\beta_{16}\mathstrut +\mathstrut \) \(6878\) \(\beta_{15}\mathstrut +\mathstrut \) \(3556\) \(\beta_{14}\mathstrut +\mathstrut \) \(7618\) \(\beta_{13}\mathstrut -\mathstrut \) \(1968\) \(\beta_{12}\mathstrut -\mathstrut \) \(2001\) \(\beta_{11}\mathstrut +\mathstrut \) \(1571\) \(\beta_{10}\mathstrut +\mathstrut \) \(1447\) \(\beta_{9}\mathstrut +\mathstrut \) \(981\) \(\beta_{8}\mathstrut +\mathstrut \) \(1819\) \(\beta_{7}\mathstrut -\mathstrut \) \(10110\) \(\beta_{6}\mathstrut +\mathstrut \) \(945\) \(\beta_{5}\mathstrut +\mathstrut \) \(1775\) \(\beta_{4}\mathstrut +\mathstrut \) \(24397\) \(\beta_{3}\mathstrut +\mathstrut \) \(7548\) \(\beta_{2}\mathstrut +\mathstrut \) \(40314\) \(\beta_{1}\mathstrut +\mathstrut \) \(9262\)
\(\nu^{14}\)\(=\)\(475\) \(\beta_{16}\mathstrut -\mathstrut \) \(3400\) \(\beta_{15}\mathstrut +\mathstrut \) \(34571\) \(\beta_{14}\mathstrut -\mathstrut \) \(13739\) \(\beta_{13}\mathstrut -\mathstrut \) \(10259\) \(\beta_{12}\mathstrut -\mathstrut \) \(8052\) \(\beta_{11}\mathstrut +\mathstrut \) \(58138\) \(\beta_{10}\mathstrut +\mathstrut \) \(23565\) \(\beta_{9}\mathstrut +\mathstrut \) \(8987\) \(\beta_{8}\mathstrut -\mathstrut \) \(2611\) \(\beta_{7}\mathstrut -\mathstrut \) \(5074\) \(\beta_{6}\mathstrut -\mathstrut \) \(2480\) \(\beta_{5}\mathstrut +\mathstrut \) \(12219\) \(\beta_{4}\mathstrut +\mathstrut \) \(8456\) \(\beta_{3}\mathstrut +\mathstrut \) \(88063\) \(\beta_{2}\mathstrut +\mathstrut \) \(11666\) \(\beta_{1}\mathstrut +\mathstrut \) \(155389\)
\(\nu^{15}\)\(=\)\(2537\) \(\beta_{16}\mathstrut +\mathstrut \) \(48912\) \(\beta_{15}\mathstrut +\mathstrut \) \(31314\) \(\beta_{14}\mathstrut +\mathstrut \) \(56495\) \(\beta_{13}\mathstrut -\mathstrut \) \(17944\) \(\beta_{12}\mathstrut -\mathstrut \) \(14982\) \(\beta_{11}\mathstrut +\mathstrut \) \(17061\) \(\beta_{10}\mathstrut +\mathstrut \) \(12339\) \(\beta_{9}\mathstrut +\mathstrut \) \(7606\) \(\beta_{8}\mathstrut +\mathstrut \) \(15657\) \(\beta_{7}\mathstrut -\mathstrut \) \(80141\) \(\beta_{6}\mathstrut +\mathstrut \) \(9781\) \(\beta_{5}\mathstrut +\mathstrut \) \(16080\) \(\beta_{4}\mathstrut +\mathstrut \) \(171026\) \(\beta_{3}\mathstrut +\mathstrut \) \(55494\) \(\beta_{2}\mathstrut +\mathstrut \) \(261724\) \(\beta_{1}\mathstrut +\mathstrut \) \(70526\)
\(\nu^{16}\)\(=\)\(4970\) \(\beta_{16}\mathstrut -\mathstrut \) \(32423\) \(\beta_{15}\mathstrut +\mathstrut \) \(255583\) \(\beta_{14}\mathstrut -\mathstrut \) \(111214\) \(\beta_{13}\mathstrut -\mathstrut \) \(82428\) \(\beta_{12}\mathstrut -\mathstrut \) \(61937\) \(\beta_{11}\mathstrut +\mathstrut \) \(444161\) \(\beta_{10}\mathstrut +\mathstrut \) \(165707\) \(\beta_{9}\mathstrut +\mathstrut \) \(67887\) \(\beta_{8}\mathstrut -\mathstrut \) \(24073\) \(\beta_{7}\mathstrut -\mathstrut \) \(48548\) \(\beta_{6}\mathstrut -\mathstrut \) \(8425\) \(\beta_{5}\mathstrut +\mathstrut \) \(99116\) \(\beta_{4}\mathstrut +\mathstrut \) \(62302\) \(\beta_{3}\mathstrut +\mathstrut \) \(593508\) \(\beta_{2}\mathstrut +\mathstrut \) \(90319\) \(\beta_{1}\mathstrut +\mathstrut \) \(1056297\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.65328
−2.37579
−2.18223
−1.58644
−1.45848
−1.21506
−0.565863
−0.287376
−0.0282840
0.551239
1.15560
1.39113
1.42963
2.00742
2.54626
2.59255
2.67896
−2.65328 −1.00000 5.03989 1.00000 2.65328 1.22871 −8.06567 1.00000 −2.65328
1.2 −2.37579 −1.00000 3.64437 1.00000 2.37579 2.23288 −3.90666 1.00000 −2.37579
1.3 −2.18223 −1.00000 2.76211 1.00000 2.18223 −1.44741 −1.66310 1.00000 −2.18223
1.4 −1.58644 −1.00000 0.516801 1.00000 1.58644 5.23184 2.35301 1.00000 −1.58644
1.5 −1.45848 −1.00000 0.127166 1.00000 1.45848 3.69177 2.73149 1.00000 −1.45848
1.6 −1.21506 −1.00000 −0.523635 1.00000 1.21506 −0.863273 3.06636 1.00000 −1.21506
1.7 −0.565863 −1.00000 −1.67980 1.00000 0.565863 −3.78718 2.08226 1.00000 −0.565863
1.8 −0.287376 −1.00000 −1.91742 1.00000 0.287376 −1.07674 1.12577 1.00000 −0.287376
1.9 −0.0282840 −1.00000 −1.99920 1.00000 0.0282840 2.73754 0.113114 1.00000 −0.0282840
1.10 0.551239 −1.00000 −1.69614 1.00000 −0.551239 4.16860 −2.03745 1.00000 0.551239
1.11 1.15560 −1.00000 −0.664588 1.00000 −1.15560 3.45180 −3.07920 1.00000 1.15560
1.12 1.39113 −1.00000 −0.0647466 1.00000 −1.39113 1.38286 −2.87234 1.00000 1.39113
1.13 1.42963 −1.00000 0.0438374 1.00000 −1.42963 −4.47441 −2.79659 1.00000 1.42963
1.14 2.00742 −1.00000 2.02973 1.00000 −2.00742 −0.373125 0.0596763 1.00000 2.00742
1.15 2.54626 −1.00000 4.48346 1.00000 −2.54626 −1.02990 6.32355 1.00000 2.54626
1.16 2.59255 −1.00000 4.72133 1.00000 −2.59255 3.82140 7.05518 1.00000 2.59255
1.17 2.67896 −1.00000 5.17683 1.00000 −2.67896 3.10465 8.51061 1.00000 2.67896
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(13\) \(-1\)
\(31\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\(T_{2}^{17} - \cdots\)
\(T_{7}^{17} - \cdots\)
\(T_{11}^{17} - \cdots\)