Properties

Label 572.2.n.a
Level $572$
Weight $2$
Character orbit 572.n
Analytic conductor $4.567$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.56744299562\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} + 22 x^{18} - 72 x^{17} + 236 x^{16} - 556 x^{15} + 1232 x^{14} - 1981 x^{13} + 3407 x^{12} - 4130 x^{11} + 5358 x^{10} - 6093 x^{9} + 10112 x^{8} - 14136 x^{7} + 17259 x^{6} - 10035 x^{5} + 2921 x^{4} + 440 x^{3} + 560 x^{2} - 200 x + 400\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + ( -\beta_{9} - \beta_{10} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} - \beta_{7} + \beta_{11} - \beta_{15} ) q^{7} + ( 1 - \beta_{1} + \beta_{4} - \beta_{8} - \beta_{13} - \beta_{14} ) q^{9} +O(q^{10})\) \( q + \beta_{5} q^{3} + ( -\beta_{9} - \beta_{10} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} - \beta_{7} + \beta_{11} - \beta_{15} ) q^{7} + ( 1 - \beta_{1} + \beta_{4} - \beta_{8} - \beta_{13} - \beta_{14} ) q^{9} + ( -\beta_{1} + \beta_{5} + \beta_{7} + \beta_{9} + \beta_{16} ) q^{11} + ( -1 - \beta_{4} + \beta_{7} - \beta_{10} ) q^{13} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{11} - \beta_{15} ) q^{15} + ( -\beta_{2} - \beta_{3} - \beta_{7} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} - \beta_{16} - \beta_{17} - \beta_{18} ) q^{17} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{10} + \beta_{15} + \beta_{16} - \beta_{18} ) q^{19} + ( -2 + 2 \beta_{6} + \beta_{9} + \beta_{12} - 2 \beta_{17} + \beta_{19} ) q^{21} + ( \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} ) q^{23} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{25} + ( 2 - \beta_{2} + 2 \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{27} + ( -2 + \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{8} - 2 \beta_{10} + \beta_{12} - \beta_{14} - 2 \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{29} + ( -2 + 2 \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{7} + 3 \beta_{8} - 2 \beta_{10} + 2 \beta_{12} + \beta_{13} - 2 \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{31} + ( -2 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{7} - 2 \beta_{8} + \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{33} + ( 1 - 3 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - 4 \beta_{7} - 3 \beta_{8} + 3 \beta_{10} - \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{35} + ( 6 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + 4 \beta_{10} + 2 \beta_{11} + \beta_{14} + \beta_{17} - \beta_{19} ) q^{37} + \beta_{11} q^{39} + ( 3 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 4 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} + 3 \beta_{14} + \beta_{16} + \beta_{17} ) q^{41} + ( \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{9} - 2 \beta_{12} + \beta_{19} ) q^{43} + ( -2 + \beta_{1} + 3 \beta_{2} + \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{14} + \beta_{15} + \beta_{16} - \beta_{19} ) q^{45} + ( -3 + 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{4} + \beta_{5} - \beta_{6} + 7 \beta_{7} + 3 \beta_{8} - \beta_{9} - 4 \beta_{10} + 3 \beta_{13} + \beta_{15} - \beta_{18} ) q^{47} + ( 1 + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} - 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{13} - 2 \beta_{14} - 2 \beta_{18} - \beta_{19} ) q^{49} + ( -3 + \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{14} + 2 \beta_{15} - \beta_{18} ) q^{51} + ( 5 + 2 \beta_{3} + 5 \beta_{4} + \beta_{5} - \beta_{7} + 3 \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} + 4 \beta_{14} + \beta_{15} + 2 \beta_{16} + 2 \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{53} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{15} - \beta_{16} + 2 \beta_{17} + \beta_{18} - \beta_{19} ) q^{55} + ( -\beta_{1} - \beta_{3} + \beta_{5} - 3 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{13} - 2 \beta_{14} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{57} + ( -2 - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} + \beta_{18} ) q^{59} + ( 2 - \beta_{2} + 4 \beta_{3} + \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{16} + 2 \beta_{17} + \beta_{18} + \beta_{19} ) q^{61} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{17} + \beta_{18} ) q^{63} + ( -1 + \beta_{6} + \beta_{9} - \beta_{17} + \beta_{19} ) q^{65} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - \beta_{9} - 2 \beta_{11} + \beta_{12} + 3 \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} - \beta_{19} ) q^{67} + ( \beta_{3} - 3 \beta_{4} - 4 \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{9} - 3 \beta_{10} - 3 \beta_{13} + \beta_{14} - 2 \beta_{15} + \beta_{16} + 2 \beta_{18} ) q^{69} + ( 3 - 2 \beta_{3} - 2 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} + 2 \beta_{13} + \beta_{14} + \beta_{18} - 2 \beta_{19} ) q^{71} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} - 4 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + 4 \beta_{14} + \beta_{15} - 2 \beta_{18} ) q^{73} + ( 1 - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} - 3 \beta_{7} - 4 \beta_{8} - \beta_{9} - 3 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 5 \beta_{14} - 2 \beta_{15} - 3 \beta_{16} - 2 \beta_{17} + 3 \beta_{18} ) q^{75} + ( -7 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + \beta_{6} + 4 \beta_{7} + 3 \beta_{8} - 3 \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} + 2 \beta_{19} ) q^{77} + ( 4 - \beta_{1} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{9} + \beta_{10} - 2 \beta_{11} - 4 \beta_{12} + 2 \beta_{13} + 4 \beta_{14} + 4 \beta_{15} + 2 \beta_{16} + \beta_{17} - 2 \beta_{18} - 2 \beta_{19} ) q^{79} + ( 1 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + \beta_{10} + 2 \beta_{12} - \beta_{14} - \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{81} + ( -1 - 2 \beta_{2} - 3 \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{13} - 2 \beta_{14} - 2 \beta_{16} - 2 \beta_{17} + \beta_{18} + \beta_{19} ) q^{83} + ( -4 \beta_{3} - 5 \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{13} - 4 \beta_{14} + \beta_{15} - 2 \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{85} + ( 4 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} + 3 \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} - 2 \beta_{15} - 2 \beta_{16} ) q^{87} + ( -4 + 5 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} - 5 \beta_{11} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} + \beta_{19} ) q^{89} + ( -\beta_{3} - \beta_{5} - \beta_{7} - \beta_{14} - \beta_{16} - \beta_{17} ) q^{91} + ( -1 - 4 \beta_{3} + 3 \beta_{6} - \beta_{7} + 3 \beta_{8} - 4 \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - \beta_{14} - 2 \beta_{16} - 2 \beta_{17} + \beta_{18} + 3 \beta_{19} ) q^{93} + ( -3 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} - 3 \beta_{10} + \beta_{14} + \beta_{15} + \beta_{17} - \beta_{19} ) q^{95} + ( 2 + 4 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - 3 \beta_{12} + \beta_{13} + 4 \beta_{14} + 3 \beta_{15} + 3 \beta_{16} + 2 \beta_{17} - 3 \beta_{18} + 2 \beta_{19} ) q^{97} + ( 1 - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{9} + 2 \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} - \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + q^{3} + q^{5} + 3q^{7} + 12q^{9} + O(q^{10}) \) \( 20q + q^{3} + q^{5} + 3q^{7} + 12q^{9} - q^{11} - 5q^{13} + 14q^{15} - 3q^{17} - 7q^{19} - 32q^{21} - 30q^{23} - 12q^{25} + 13q^{27} - 14q^{29} + 11q^{31} - 10q^{33} - 10q^{35} + 45q^{37} - 4q^{39} + 9q^{41} - 8q^{43} - 34q^{45} + 39q^{47} + 30q^{49} - 55q^{51} + 36q^{53} + 11q^{55} - 31q^{57} - 31q^{59} - 7q^{61} + 14q^{63} - 14q^{65} + 26q^{67} + 32q^{69} + 33q^{71} - 44q^{73} + 37q^{75} - 73q^{77} + 21q^{79} + 16q^{81} - 25q^{83} + 15q^{85} + 32q^{87} - 2q^{89} + 3q^{91} + 33q^{93} - 37q^{95} + 52q^{97} - 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 4 x^{19} + 22 x^{18} - 72 x^{17} + 236 x^{16} - 556 x^{15} + 1232 x^{14} - 1981 x^{13} + 3407 x^{12} - 4130 x^{11} + 5358 x^{10} - 6093 x^{9} + 10112 x^{8} - 14136 x^{7} + 17259 x^{6} - 10035 x^{5} + 2921 x^{4} + 440 x^{3} + 560 x^{2} - 200 x + 400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(72\!\cdots\!07\)\( \nu^{19} - \)\(73\!\cdots\!48\)\( \nu^{18} + \)\(30\!\cdots\!58\)\( \nu^{17} - \)\(19\!\cdots\!40\)\( \nu^{16} + \)\(67\!\cdots\!56\)\( \nu^{15} - \)\(23\!\cdots\!88\)\( \nu^{14} + \)\(59\!\cdots\!16\)\( \nu^{13} - \)\(13\!\cdots\!13\)\( \nu^{12} + \)\(23\!\cdots\!39\)\( \nu^{11} - \)\(40\!\cdots\!02\)\( \nu^{10} + \)\(52\!\cdots\!86\)\( \nu^{9} - \)\(66\!\cdots\!21\)\( \nu^{8} + \)\(75\!\cdots\!72\)\( \nu^{7} - \)\(11\!\cdots\!76\)\( \nu^{6} + \)\(17\!\cdots\!27\)\( \nu^{5} - \)\(22\!\cdots\!83\)\( \nu^{4} + \)\(17\!\cdots\!45\)\( \nu^{3} - \)\(58\!\cdots\!12\)\( \nu^{2} - \)\(19\!\cdots\!80\)\( \nu + \)\(18\!\cdots\!20\)\(\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(92\!\cdots\!14\)\( \nu^{19} + \)\(10\!\cdots\!57\)\( \nu^{18} - \)\(47\!\cdots\!02\)\( \nu^{17} + \)\(21\!\cdots\!02\)\( \nu^{16} - \)\(69\!\cdots\!36\)\( \nu^{15} + \)\(20\!\cdots\!12\)\( \nu^{14} - \)\(47\!\cdots\!28\)\( \nu^{13} + \)\(95\!\cdots\!06\)\( \nu^{12} - \)\(14\!\cdots\!07\)\( \nu^{11} + \)\(23\!\cdots\!05\)\( \nu^{10} - \)\(27\!\cdots\!96\)\( \nu^{9} + \)\(34\!\cdots\!96\)\( \nu^{8} - \)\(41\!\cdots\!57\)\( \nu^{7} + \)\(72\!\cdots\!06\)\( \nu^{6} - \)\(10\!\cdots\!34\)\( \nu^{5} + \)\(10\!\cdots\!13\)\( \nu^{4} - \)\(35\!\cdots\!99\)\( \nu^{3} - \)\(12\!\cdots\!05\)\( \nu^{2} + \)\(12\!\cdots\!90\)\( \nu + \)\(21\!\cdots\!00\)\(\)\()/ \)\(10\!\cdots\!20\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(31\!\cdots\!65\)\( \nu^{19} - \)\(16\!\cdots\!78\)\( \nu^{18} + \)\(84\!\cdots\!90\)\( \nu^{17} - \)\(30\!\cdots\!52\)\( \nu^{16} + \)\(98\!\cdots\!96\)\( \nu^{15} - \)\(24\!\cdots\!24\)\( \nu^{14} + \)\(54\!\cdots\!20\)\( \nu^{13} - \)\(92\!\cdots\!97\)\( \nu^{12} + \)\(14\!\cdots\!77\)\( \nu^{11} - \)\(18\!\cdots\!20\)\( \nu^{10} + \)\(20\!\cdots\!62\)\( \nu^{9} - \)\(20\!\cdots\!57\)\( \nu^{8} + \)\(34\!\cdots\!10\)\( \nu^{7} - \)\(57\!\cdots\!16\)\( \nu^{6} + \)\(75\!\cdots\!19\)\( \nu^{5} - \)\(35\!\cdots\!69\)\( \nu^{4} - \)\(31\!\cdots\!05\)\( \nu^{3} + \)\(71\!\cdots\!62\)\( \nu^{2} - \)\(95\!\cdots\!00\)\( \nu - \)\(47\!\cdots\!00\)\(\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(21\!\cdots\!59\)\( \nu^{19} - \)\(75\!\cdots\!30\)\( \nu^{18} + \)\(40\!\cdots\!86\)\( \nu^{17} - \)\(12\!\cdots\!28\)\( \nu^{16} + \)\(37\!\cdots\!92\)\( \nu^{15} - \)\(78\!\cdots\!20\)\( \nu^{14} + \)\(15\!\cdots\!16\)\( \nu^{13} - \)\(18\!\cdots\!11\)\( \nu^{12} + \)\(27\!\cdots\!35\)\( \nu^{11} - \)\(17\!\cdots\!96\)\( \nu^{10} + \)\(72\!\cdots\!06\)\( \nu^{9} - \)\(14\!\cdots\!15\)\( \nu^{8} + \)\(65\!\cdots\!38\)\( \nu^{7} - \)\(10\!\cdots\!92\)\( \nu^{6} + \)\(22\!\cdots\!97\)\( \nu^{5} + \)\(20\!\cdots\!85\)\( \nu^{4} - \)\(35\!\cdots\!31\)\( \nu^{3} + \)\(56\!\cdots\!50\)\( \nu^{2} + \)\(20\!\cdots\!00\)\( \nu + \)\(62\!\cdots\!00\)\(\)\()/ \)\(10\!\cdots\!20\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(50\!\cdots\!27\)\( \nu^{19} + \)\(51\!\cdots\!90\)\( \nu^{18} - \)\(23\!\cdots\!38\)\( \nu^{17} + \)\(10\!\cdots\!20\)\( \nu^{16} - \)\(34\!\cdots\!96\)\( \nu^{15} + \)\(10\!\cdots\!56\)\( \nu^{14} - \)\(23\!\cdots\!40\)\( \nu^{13} + \)\(46\!\cdots\!83\)\( \nu^{12} - \)\(76\!\cdots\!79\)\( \nu^{11} + \)\(11\!\cdots\!12\)\( \nu^{10} - \)\(15\!\cdots\!50\)\( \nu^{9} + \)\(17\!\cdots\!83\)\( \nu^{8} - \)\(23\!\cdots\!62\)\( \nu^{7} + \)\(36\!\cdots\!16\)\( \nu^{6} - \)\(52\!\cdots\!17\)\( \nu^{5} + \)\(51\!\cdots\!27\)\( \nu^{4} - \)\(26\!\cdots\!97\)\( \nu^{3} + \)\(65\!\cdots\!82\)\( \nu^{2} - \)\(60\!\cdots\!40\)\( \nu + \)\(41\!\cdots\!00\)\(\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(10\!\cdots\!29\)\( \nu^{19} - \)\(22\!\cdots\!08\)\( \nu^{18} + \)\(15\!\cdots\!54\)\( \nu^{17} - \)\(33\!\cdots\!84\)\( \nu^{16} + \)\(11\!\cdots\!84\)\( \nu^{15} - \)\(15\!\cdots\!80\)\( \nu^{14} + \)\(30\!\cdots\!44\)\( \nu^{13} + \)\(91\!\cdots\!79\)\( \nu^{12} + \)\(21\!\cdots\!67\)\( \nu^{11} + \)\(15\!\cdots\!66\)\( \nu^{10} - \)\(10\!\cdots\!86\)\( \nu^{9} + \)\(25\!\cdots\!63\)\( \nu^{8} + \)\(61\!\cdots\!72\)\( \nu^{7} + \)\(27\!\cdots\!36\)\( \nu^{6} - \)\(53\!\cdots\!33\)\( \nu^{5} + \)\(17\!\cdots\!93\)\( \nu^{4} - \)\(10\!\cdots\!99\)\( \nu^{3} + \)\(45\!\cdots\!72\)\( \nu^{2} + \)\(17\!\cdots\!40\)\( \nu + \)\(92\!\cdots\!20\)\(\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(12\!\cdots\!45\)\( \nu^{19} - \)\(52\!\cdots\!70\)\( \nu^{18} + \)\(27\!\cdots\!38\)\( \nu^{17} - \)\(93\!\cdots\!36\)\( \nu^{16} + \)\(30\!\cdots\!40\)\( \nu^{15} - \)\(72\!\cdots\!56\)\( \nu^{14} + \)\(15\!\cdots\!52\)\( \nu^{13} - \)\(25\!\cdots\!21\)\( \nu^{12} + \)\(41\!\cdots\!69\)\( \nu^{11} - \)\(49\!\cdots\!44\)\( \nu^{10} + \)\(58\!\cdots\!82\)\( \nu^{9} - \)\(62\!\cdots\!13\)\( \nu^{8} + \)\(11\!\cdots\!06\)\( \nu^{7} - \)\(16\!\cdots\!80\)\( \nu^{6} + \)\(19\!\cdots\!91\)\( \nu^{5} - \)\(77\!\cdots\!97\)\( \nu^{4} - \)\(43\!\cdots\!45\)\( \nu^{3} + \)\(80\!\cdots\!10\)\( \nu^{2} + \)\(25\!\cdots\!40\)\( \nu + \)\(15\!\cdots\!00\)\(\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(13\!\cdots\!39\)\( \nu^{19} - \)\(80\!\cdots\!56\)\( \nu^{18} + \)\(35\!\cdots\!54\)\( \nu^{17} - \)\(13\!\cdots\!92\)\( \nu^{16} + \)\(41\!\cdots\!92\)\( \nu^{15} - \)\(10\!\cdots\!44\)\( \nu^{14} + \)\(21\!\cdots\!96\)\( \nu^{13} - \)\(37\!\cdots\!67\)\( \nu^{12} + \)\(50\!\cdots\!61\)\( \nu^{11} - \)\(74\!\cdots\!62\)\( \nu^{10} + \)\(55\!\cdots\!34\)\( \nu^{9} - \)\(87\!\cdots\!23\)\( \nu^{8} + \)\(11\!\cdots\!88\)\( \nu^{7} - \)\(26\!\cdots\!52\)\( \nu^{6} + \)\(24\!\cdots\!01\)\( \nu^{5} - \)\(91\!\cdots\!93\)\( \nu^{4} - \)\(24\!\cdots\!45\)\( \nu^{3} + \)\(10\!\cdots\!24\)\( \nu^{2} + \)\(44\!\cdots\!60\)\( \nu - \)\(42\!\cdots\!40\)\(\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(17\!\cdots\!29\)\( \nu^{19} + \)\(82\!\cdots\!56\)\( \nu^{18} - \)\(42\!\cdots\!54\)\( \nu^{17} + \)\(15\!\cdots\!72\)\( \nu^{16} - \)\(48\!\cdots\!24\)\( \nu^{15} + \)\(12\!\cdots\!48\)\( \nu^{14} - \)\(26\!\cdots\!36\)\( \nu^{13} + \)\(46\!\cdots\!41\)\( \nu^{12} - \)\(75\!\cdots\!59\)\( \nu^{11} + \)\(10\!\cdots\!46\)\( \nu^{10} - \)\(12\!\cdots\!10\)\( \nu^{9} + \)\(14\!\cdots\!09\)\( \nu^{8} - \)\(21\!\cdots\!36\)\( \nu^{7} + \)\(33\!\cdots\!64\)\( \nu^{6} - \)\(40\!\cdots\!35\)\( \nu^{5} + \)\(30\!\cdots\!03\)\( \nu^{4} - \)\(61\!\cdots\!29\)\( \nu^{3} - \)\(75\!\cdots\!76\)\( \nu^{2} - \)\(11\!\cdots\!80\)\( \nu + \)\(60\!\cdots\!60\)\(\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(24\!\cdots\!01\)\( \nu^{19} - \)\(89\!\cdots\!98\)\( \nu^{18} + \)\(50\!\cdots\!13\)\( \nu^{17} - \)\(16\!\cdots\!70\)\( \nu^{16} + \)\(52\!\cdots\!18\)\( \nu^{15} - \)\(12\!\cdots\!98\)\( \nu^{14} + \)\(26\!\cdots\!91\)\( \nu^{13} - \)\(41\!\cdots\!67\)\( \nu^{12} + \)\(72\!\cdots\!92\)\( \nu^{11} - \)\(82\!\cdots\!46\)\( \nu^{10} + \)\(11\!\cdots\!95\)\( \nu^{9} - \)\(12\!\cdots\!47\)\( \nu^{8} + \)\(21\!\cdots\!85\)\( \nu^{7} - \)\(29\!\cdots\!43\)\( \nu^{6} + \)\(35\!\cdots\!26\)\( \nu^{5} - \)\(17\!\cdots\!01\)\( \nu^{4} + \)\(51\!\cdots\!39\)\( \nu^{3} + \)\(14\!\cdots\!25\)\( \nu^{2} + \)\(14\!\cdots\!15\)\( \nu - \)\(51\!\cdots\!50\)\(\)\()/ \)\(27\!\cdots\!05\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(10\!\cdots\!02\)\( \nu^{19} + \)\(61\!\cdots\!07\)\( \nu^{18} - \)\(95\!\cdots\!30\)\( \nu^{17} + \)\(18\!\cdots\!02\)\( \nu^{16} - \)\(15\!\cdots\!84\)\( \nu^{15} - \)\(16\!\cdots\!76\)\( \nu^{14} + \)\(39\!\cdots\!92\)\( \nu^{13} - \)\(15\!\cdots\!02\)\( \nu^{12} + \)\(19\!\cdots\!91\)\( \nu^{11} - \)\(50\!\cdots\!89\)\( \nu^{10} + \)\(47\!\cdots\!72\)\( \nu^{9} - \)\(69\!\cdots\!80\)\( \nu^{8} + \)\(45\!\cdots\!13\)\( \nu^{7} - \)\(12\!\cdots\!66\)\( \nu^{6} + \)\(19\!\cdots\!30\)\( \nu^{5} - \)\(30\!\cdots\!89\)\( \nu^{4} + \)\(10\!\cdots\!95\)\( \nu^{3} + \)\(37\!\cdots\!85\)\( \nu^{2} - \)\(38\!\cdots\!50\)\( \nu - \)\(18\!\cdots\!80\)\(\)\()/ \)\(10\!\cdots\!20\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(61\!\cdots\!76\)\( \nu^{19} - \)\(14\!\cdots\!43\)\( \nu^{18} + \)\(97\!\cdots\!68\)\( \nu^{17} - \)\(22\!\cdots\!18\)\( \nu^{16} + \)\(77\!\cdots\!64\)\( \nu^{15} - \)\(12\!\cdots\!38\)\( \nu^{14} + \)\(26\!\cdots\!32\)\( \nu^{13} - \)\(13\!\cdots\!84\)\( \nu^{12} + \)\(47\!\cdots\!23\)\( \nu^{11} + \)\(27\!\cdots\!95\)\( \nu^{10} + \)\(20\!\cdots\!44\)\( \nu^{9} + \)\(23\!\cdots\!96\)\( \nu^{8} + \)\(17\!\cdots\!13\)\( \nu^{7} - \)\(52\!\cdots\!84\)\( \nu^{6} - \)\(50\!\cdots\!94\)\( \nu^{5} + \)\(62\!\cdots\!83\)\( \nu^{4} - \)\(22\!\cdots\!89\)\( \nu^{3} - \)\(84\!\cdots\!55\)\( \nu^{2} + \)\(90\!\cdots\!90\)\( \nu + \)\(14\!\cdots\!50\)\(\)\()/ \)\(54\!\cdots\!10\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(28\!\cdots\!51\)\( \nu^{19} - \)\(13\!\cdots\!98\)\( \nu^{18} + \)\(67\!\cdots\!34\)\( \nu^{17} - \)\(23\!\cdots\!40\)\( \nu^{16} + \)\(75\!\cdots\!04\)\( \nu^{15} - \)\(18\!\cdots\!40\)\( \nu^{14} + \)\(40\!\cdots\!36\)\( \nu^{13} - \)\(69\!\cdots\!31\)\( \nu^{12} + \)\(11\!\cdots\!91\)\( \nu^{11} - \)\(15\!\cdots\!28\)\( \nu^{10} + \)\(18\!\cdots\!58\)\( \nu^{9} - \)\(22\!\cdots\!35\)\( \nu^{8} + \)\(33\!\cdots\!30\)\( \nu^{7} - \)\(51\!\cdots\!64\)\( \nu^{6} + \)\(60\!\cdots\!73\)\( \nu^{5} - \)\(42\!\cdots\!71\)\( \nu^{4} + \)\(70\!\cdots\!41\)\( \nu^{3} - \)\(25\!\cdots\!30\)\( \nu^{2} + \)\(12\!\cdots\!20\)\( \nu - \)\(40\!\cdots\!00\)\(\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(23\!\cdots\!45\)\( \nu^{19} - \)\(11\!\cdots\!77\)\( \nu^{18} + \)\(58\!\cdots\!42\)\( \nu^{17} - \)\(20\!\cdots\!74\)\( \nu^{16} + \)\(66\!\cdots\!64\)\( \nu^{15} - \)\(16\!\cdots\!72\)\( \nu^{14} + \)\(36\!\cdots\!92\)\( \nu^{13} - \)\(63\!\cdots\!57\)\( \nu^{12} + \)\(10\!\cdots\!52\)\( \nu^{11} - \)\(14\!\cdots\!29\)\( \nu^{10} + \)\(16\!\cdots\!60\)\( \nu^{9} - \)\(20\!\cdots\!71\)\( \nu^{8} + \)\(29\!\cdots\!45\)\( \nu^{7} - \)\(47\!\cdots\!24\)\( \nu^{6} + \)\(56\!\cdots\!07\)\( \nu^{5} - \)\(42\!\cdots\!98\)\( \nu^{4} + \)\(91\!\cdots\!60\)\( \nu^{3} + \)\(70\!\cdots\!59\)\( \nu^{2} + \)\(17\!\cdots\!00\)\( \nu - \)\(38\!\cdots\!00\)\(\)\()/ \)\(10\!\cdots\!20\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(49\!\cdots\!95\)\( \nu^{19} + \)\(19\!\cdots\!62\)\( \nu^{18} - \)\(10\!\cdots\!34\)\( \nu^{17} + \)\(33\!\cdots\!96\)\( \nu^{16} - \)\(10\!\cdots\!68\)\( \nu^{15} + \)\(24\!\cdots\!72\)\( \nu^{14} - \)\(50\!\cdots\!76\)\( \nu^{13} + \)\(76\!\cdots\!79\)\( \nu^{12} - \)\(12\!\cdots\!51\)\( \nu^{11} + \)\(14\!\cdots\!00\)\( \nu^{10} - \)\(14\!\cdots\!46\)\( \nu^{9} + \)\(20\!\cdots\!19\)\( \nu^{8} - \)\(33\!\cdots\!54\)\( \nu^{7} + \)\(54\!\cdots\!88\)\( \nu^{6} - \)\(48\!\cdots\!37\)\( \nu^{5} + \)\(86\!\cdots\!47\)\( \nu^{4} + \)\(27\!\cdots\!47\)\( \nu^{3} + \)\(10\!\cdots\!38\)\( \nu^{2} - \)\(52\!\cdots\!40\)\( \nu - \)\(48\!\cdots\!20\)\(\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(58\!\cdots\!99\)\( \nu^{19} + \)\(19\!\cdots\!24\)\( \nu^{18} - \)\(11\!\cdots\!66\)\( \nu^{17} + \)\(34\!\cdots\!52\)\( \nu^{16} - \)\(11\!\cdots\!28\)\( \nu^{15} + \)\(25\!\cdots\!36\)\( \nu^{14} - \)\(56\!\cdots\!72\)\( \nu^{13} + \)\(81\!\cdots\!43\)\( \nu^{12} - \)\(15\!\cdots\!65\)\( \nu^{11} + \)\(15\!\cdots\!94\)\( \nu^{10} - \)\(23\!\cdots\!62\)\( \nu^{9} + \)\(23\!\cdots\!79\)\( \nu^{8} - \)\(48\!\cdots\!84\)\( \nu^{7} + \)\(56\!\cdots\!28\)\( \nu^{6} - \)\(69\!\cdots\!53\)\( \nu^{5} + \)\(22\!\cdots\!13\)\( \nu^{4} - \)\(15\!\cdots\!35\)\( \nu^{3} + \)\(51\!\cdots\!72\)\( \nu^{2} - \)\(11\!\cdots\!60\)\( \nu - \)\(27\!\cdots\!80\)\(\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{18}\)\(=\)\((\)\(\)\(71\!\cdots\!25\)\( \nu^{19} - \)\(27\!\cdots\!52\)\( \nu^{18} + \)\(15\!\cdots\!86\)\( \nu^{17} - \)\(48\!\cdots\!76\)\( \nu^{16} + \)\(15\!\cdots\!32\)\( \nu^{15} - \)\(35\!\cdots\!60\)\( \nu^{14} + \)\(77\!\cdots\!96\)\( \nu^{13} - \)\(11\!\cdots\!41\)\( \nu^{12} + \)\(20\!\cdots\!07\)\( \nu^{11} - \)\(22\!\cdots\!38\)\( \nu^{10} + \)\(28\!\cdots\!18\)\( \nu^{9} - \)\(31\!\cdots\!61\)\( \nu^{8} + \)\(58\!\cdots\!72\)\( \nu^{7} - \)\(81\!\cdots\!72\)\( \nu^{6} + \)\(92\!\cdots\!95\)\( \nu^{5} - \)\(30\!\cdots\!27\)\( \nu^{4} - \)\(14\!\cdots\!99\)\( \nu^{3} + \)\(14\!\cdots\!64\)\( \nu^{2} + \)\(79\!\cdots\!20\)\( \nu - \)\(52\!\cdots\!40\)\(\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(10\!\cdots\!79\)\( \nu^{19} + \)\(40\!\cdots\!02\)\( \nu^{18} - \)\(22\!\cdots\!58\)\( \nu^{17} + \)\(71\!\cdots\!76\)\( \nu^{16} - \)\(23\!\cdots\!64\)\( \nu^{15} + \)\(52\!\cdots\!32\)\( \nu^{14} - \)\(11\!\cdots\!16\)\( \nu^{13} + \)\(17\!\cdots\!39\)\( \nu^{12} - \)\(29\!\cdots\!55\)\( \nu^{11} + \)\(32\!\cdots\!32\)\( \nu^{10} - \)\(42\!\cdots\!98\)\( \nu^{9} + \)\(47\!\cdots\!87\)\( \nu^{8} - \)\(86\!\cdots\!42\)\( \nu^{7} + \)\(11\!\cdots\!24\)\( \nu^{6} - \)\(13\!\cdots\!09\)\( \nu^{5} + \)\(44\!\cdots\!11\)\( \nu^{4} + \)\(14\!\cdots\!95\)\( \nu^{3} - \)\(13\!\cdots\!94\)\( \nu^{2} - \)\(29\!\cdots\!60\)\( \nu + \)\(67\!\cdots\!20\)\(\)\()/ \)\(21\!\cdots\!40\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} + 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{18} + \beta_{17} - \beta_{16} - \beta_{15} - \beta_{10} - \beta_{9} - \beta_{6} - 5 \beta_{5} - \beta_{4} - \beta_{2} - \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{19} - 2 \beta_{18} + 2 \beta_{17} + 2 \beta_{16} + 9 \beta_{14} + \beta_{13} - 2 \beta_{12} + 9 \beta_{11} + 16 \beta_{10} + \beta_{9} + 8 \beta_{8} - 8 \beta_{7} - \beta_{6} + \beta_{3} - 2 \beta_{2} + 10\)
\(\nu^{5}\)\(=\)\(9 \beta_{19} - 11 \beta_{17} + 10 \beta_{16} + 10 \beta_{15} + 10 \beta_{14} - 6 \beta_{13} + 2 \beta_{12} - 11 \beta_{11} + 9 \beta_{9} + 32 \beta_{7} + 21 \beta_{6} + 33 \beta_{5} + 11 \beta_{4} + 11 \beta_{3} + 33 \beta_{2} + 11 \beta_{1} - 42\)
\(\nu^{6}\)\(=\)\(13 \beta_{19} + 28 \beta_{18} - 11 \beta_{17} - 28 \beta_{16} - 5 \beta_{15} - 75 \beta_{14} - 47 \beta_{13} + 5 \beta_{12} - 27 \beta_{11} - 108 \beta_{10} - 17 \beta_{9} - 59 \beta_{8} + 80 \beta_{7} - 27 \beta_{5} - 47 \beta_{4} - 28 \beta_{3} - 48 \beta_{1} - 47\)
\(\nu^{7}\)\(=\)\(-110 \beta_{19} - 87 \beta_{18} + 110 \beta_{17} - 30 \beta_{15} - 26 \beta_{14} - 87 \beta_{12} + 249 \beta_{11} + 106 \beta_{10} + 18 \beta_{8} - 249 \beta_{7} - 185 \beta_{6} - 107 \beta_{5} - 78 \beta_{4} - 18 \beta_{3} - 249 \beta_{2} - 142 \beta_{1} + 355\)
\(\nu^{8}\)\(=\)\(-215 \beta_{18} - 37 \beta_{17} + 294 \beta_{16} + 215 \beta_{15} + 406 \beta_{14} + 325 \beta_{13} + 482 \beta_{10} + 196 \beta_{9} + 325 \beta_{8} - 238 \beta_{7} + 196 \beta_{6} + 629 \beta_{5} + 482 \beta_{4} + 406 \beta_{3} + 285 \beta_{2} + 285 \beta_{1} - 285\)
\(\nu^{9}\)\(=\)\(946 \beta_{19} + 1062 \beta_{18} - 736 \beta_{17} - 736 \beta_{16} - 965 \beta_{14} + 59 \beta_{13} + 736 \beta_{12} - 2004 \beta_{11} - 1669 \beta_{10} - 619 \beta_{9} - 170 \beta_{8} + 1248 \beta_{7} + 946 \beta_{6} - 795 \beta_{3} + 995 \beta_{2} - 1984\)
\(\nu^{10}\)\(=\)\(-1291 \beta_{19} + 1348 \beta_{17} - 1911 \beta_{16} - 1911 \beta_{15} - 1911 \beta_{14} - 976 \beta_{13} - 886 \beta_{12} + 2754 \beta_{11} - 1291 \beta_{9} - 3325 \beta_{7} - 3259 \beta_{6} - 5351 \beta_{5} - 3937 \beta_{4} - 4248 \beta_{3} - 5351 \beta_{2} - 2754 \beta_{1} + 7959\)
\(\nu^{11}\)\(=\)\(-5134 \beta_{19} - 9369 \beta_{18} + 1348 \beta_{17} + 9369 \beta_{16} + 3153 \beta_{15} + 11869 \beta_{14} + 2500 \beta_{13} - 3153 \beta_{12} + 9015 \beta_{11} + 15021 \beta_{10} + 8021 \beta_{9} + 2974 \beta_{8} - 5652 \beta_{7} + 9015 \beta_{5} + 5967 \beta_{4} + 9369 \beta_{3} + 7648 \beta_{1} + 5967\)
\(\nu^{12}\)\(=\)\(18559 \beta_{19} + 16737 \beta_{18} - 18559 \beta_{17} + 8761 \beta_{15} - 8320 \beta_{14} + 16737 \beta_{12} - 45991 \beta_{11} - 32871 \beta_{10} - 17475 \beta_{8} + 45991 \beta_{7} + 30428 \beta_{6} + 25482 \beta_{5} + 21203 \beta_{4} + 17475 \beta_{3} + 45991 \beta_{2} + 20509 \beta_{1} - 78862\)
\(\nu^{13}\)\(=\)\(52771 \beta_{18} + 29282 \beta_{17} - 81721 \beta_{16} - 52771 \beta_{15} - 91550 \beta_{14} - 25126 \beta_{13} - 81696 \beta_{10} - 68030 \beta_{9} - 25126 \beta_{8} - 27912 \beta_{7} - 68030 \beta_{6} - 141029 \beta_{5} - 81696 \beta_{4} - 91550 \beta_{3} - 80364 \beta_{2} - 80364 \beta_{1} + 80364\)
\(\nu^{14}\)\(=\)\(-169447 \beta_{19} - 227679 \beta_{18} + 145927 \beta_{17} + 145927 \beta_{16} + 281283 \beta_{14} + 70497 \beta_{13} - 145927 \beta_{12} + 397640 \beta_{11} + 454836 \beta_{10} + 106847 \beta_{9} + 205853 \beta_{8} - 362431 \beta_{7} - 169447 \beta_{6} + 75430 \beta_{3} - 230121 \beta_{2} + 508358\)
\(\nu^{15}\)\(=\)\(363035 \beta_{19} - 491764 \beta_{17} + 450730 \beta_{16} + 450730 \beta_{15} + 450730 \beta_{14} + 118285 \beta_{13} + 259096 \beta_{12} - 709222 \beta_{11} + 363035 \beta_{9} + 927471 \beta_{7} + 942494 \beta_{6} + 1205894 \beta_{5} + 729153 \beta_{4} + 691132 \beta_{3} + 1205894 \beta_{2} + 709222 \beta_{1} - 1667114\)
\(\nu^{16}\)\(=\)\(950228 \beta_{19} + 2011007 \beta_{18} - 491764 \beta_{17} - 2011007 \beta_{16} - 740416 \beta_{15} - 3090882 \beta_{14} - 1079875 \beta_{13} + 740416 \beta_{12} - 2048424 \beta_{11} - 3868440 \beta_{10} - 1519243 \beta_{9} - 1677424 \beta_{8} + 1857433 \beta_{7} - 2048424 \beta_{5} - 1492754 \beta_{4} - 2011007 \beta_{3} - 1400372 \beta_{1} - 1492754\)
\(\nu^{17}\)\(=\)\(-4959022 \beta_{19} - 3869709 \beta_{18} + 4959022 \beta_{17} - 2288193 \beta_{15} + 1217201 \beta_{14} - 3869709 \beta_{12} + 10374818 \beta_{11} + 6456331 \beta_{10} + 2229836 \beta_{8} - 10374818 \beta_{7} - 8049904 \beta_{6} - 6220539 \beta_{5} - 4046479 \beta_{4} - 2229836 \beta_{3} - 10374818 \beta_{2} - 4154279 \beta_{1} + 16831149\)
\(\nu^{18}\)\(=\)\(-11058567 \beta_{18} - 4207543 \beta_{17} + 17654843 \beta_{16} + 11058567 \beta_{15} + 22734859 \beta_{14} + 8819783 \beta_{13} + 20399815 \beta_{10} + 13474648 \beta_{9} + 8819783 \beta_{8} + 2591659 \beta_{7} + 13474648 \beta_{6} + 29958488 \beta_{5} + 20399815 \beta_{4} + 22734859 \beta_{3} + 18074266 \beta_{2} + 18074266 \beta_{1} - 18074266\)
\(\nu^{19}\)\(=\)\(42613209 \beta_{19} + 53417028 \beta_{18} - 33352998 \beta_{17} - 33352998 \beta_{16} - 53619230 \beta_{14} - 11670564 \beta_{13} + 33352998 \beta_{12} - 89597912 \beta_{11} - 92278923 \beta_{10} - 26474626 \beta_{9} - 31936796 \beta_{8} + 77851727 \beta_{7} + 42613209 \beta_{6} - 21682434 \beta_{3} + 54357916 \beta_{2} - 111204725\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/572\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(353\) \(365\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−0.911154 + 2.80424i
−0.601689 + 1.85181i
0.170304 0.524141i
0.465638 1.43309i
0.758867 2.33555i
−1.11876 + 0.812829i
−0.384533 + 0.279379i
0.695167 0.505068i
1.06177 0.771421i
1.86439 1.35456i
−0.911154 2.80424i
−0.601689 1.85181i
0.170304 + 0.524141i
0.465638 + 1.43309i
0.758867 + 2.33555i
−1.11876 0.812829i
−0.384533 0.279379i
0.695167 + 0.505068i
1.06177 + 0.771421i
1.86439 + 1.35456i
0 −2.38543 + 1.73312i 0 −1.16829 3.59562i 0 2.62868 + 1.90985i 0 1.75954 5.41530i 0
53.2 0 −1.57524 + 1.14448i 0 −0.125488 0.386211i 0 −1.28300 0.932153i 0 0.244502 0.752499i 0
53.3 0 0.445861 0.323937i 0 0.710005 + 2.18517i 0 3.17229 + 2.30481i 0 −0.833194 + 2.56431i 0
53.4 0 1.21906 0.885697i 0 −1.09133 3.35878i 0 −3.27072 2.37632i 0 −0.225410 + 0.693741i 0
53.5 0 1.98674 1.44345i 0 0.248052 + 0.763424i 0 0.0617615 + 0.0448724i 0 0.936532 2.88235i 0
157.1 0 −0.427329 1.31518i 0 −1.65997 1.20604i 0 0.287529 0.884923i 0 0.879951 0.639322i 0
157.2 0 −0.146878 0.452045i 0 3.01285 + 2.18896i 0 0.736420 2.26647i 0 2.24428 1.63056i 0
157.3 0 0.265530 + 0.817217i 0 −1.45473 1.05692i 0 0.157953 0.486129i 0 1.82971 1.32936i 0
157.4 0 0.405560 + 1.24819i 0 1.75793 + 1.27721i 0 −1.12323 + 3.45694i 0 1.03356 0.750928i 0
157.5 0 0.712135 + 2.19173i 0 0.270974 + 0.196874i 0 0.132310 0.407207i 0 −1.86947 + 1.35825i 0
313.1 0 −2.38543 1.73312i 0 −1.16829 + 3.59562i 0 2.62868 1.90985i 0 1.75954 + 5.41530i 0
313.2 0 −1.57524 1.14448i 0 −0.125488 + 0.386211i 0 −1.28300 + 0.932153i 0 0.244502 + 0.752499i 0
313.3 0 0.445861 + 0.323937i 0 0.710005 2.18517i 0 3.17229 2.30481i 0 −0.833194 2.56431i 0
313.4 0 1.21906 + 0.885697i 0 −1.09133 + 3.35878i 0 −3.27072 + 2.37632i 0 −0.225410 0.693741i 0
313.5 0 1.98674 + 1.44345i 0 0.248052 0.763424i 0 0.0617615 0.0448724i 0 0.936532 + 2.88235i 0
521.1 0 −0.427329 + 1.31518i 0 −1.65997 + 1.20604i 0 0.287529 + 0.884923i 0 0.879951 + 0.639322i 0
521.2 0 −0.146878 + 0.452045i 0 3.01285 2.18896i 0 0.736420 + 2.26647i 0 2.24428 + 1.63056i 0
521.3 0 0.265530 0.817217i 0 −1.45473 + 1.05692i 0 0.157953 + 0.486129i 0 1.82971 + 1.32936i 0
521.4 0 0.405560 1.24819i 0 1.75793 1.27721i 0 −1.12323 3.45694i 0 1.03356 + 0.750928i 0
521.5 0 0.712135 2.19173i 0 0.270974 0.196874i 0 0.132310 + 0.407207i 0 −1.86947 1.35825i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 521.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.n.a 20
11.c even 5 1 inner 572.2.n.a 20
11.c even 5 1 6292.2.a.w 10
11.d odd 10 1 6292.2.a.x 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.n.a 20 1.a even 1 1 trivial
572.2.n.a 20 11.c even 5 1 inner
6292.2.a.w 10 11.c even 5 1
6292.2.a.x 10 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{20} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(572, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( 400 - 1200 T + 3560 T^{2} - 8210 T^{3} + 16971 T^{4} - 22315 T^{5} + 26909 T^{6} - 20244 T^{7} + 15917 T^{8} - 8202 T^{9} + 4908 T^{10} - 1715 T^{11} + 1022 T^{12} - 369 T^{13} + 422 T^{14} - 129 T^{15} + 46 T^{16} + 2 T^{17} + 2 T^{18} - T^{19} + T^{20} \)
$5$ \( 10000 - 35000 T + 95500 T^{2} - 206750 T^{3} + 646525 T^{4} - 257850 T^{5} + 601795 T^{6} + 367450 T^{7} + 191836 T^{8} - 16434 T^{9} + 61439 T^{10} + 6420 T^{11} + 8740 T^{12} + 1106 T^{13} + 2681 T^{14} - 116 T^{15} + 255 T^{16} - 55 T^{17} + 19 T^{18} - T^{19} + T^{20} \)
$7$ \( 121 - 2915 T + 29720 T^{2} - 94628 T^{3} + 310263 T^{4} - 440384 T^{5} + 733944 T^{6} - 310353 T^{7} + 349008 T^{8} + 105169 T^{9} + 118642 T^{10} - 57564 T^{11} + 46374 T^{12} - 12346 T^{13} + 3117 T^{14} - 469 T^{15} + 276 T^{16} - 25 T^{17} + 7 T^{18} - 3 T^{19} + T^{20} \)
$11$ \( 25937424601 + 2357947691 T + 6002048668 T^{2} - 233846052 T^{3} + 666106936 T^{4} - 112735700 T^{5} + 44127974 T^{6} - 20561288 T^{7} + 1228997 T^{8} - 2224926 T^{9} + 38073 T^{10} - 202266 T^{11} + 10157 T^{12} - 15448 T^{13} + 3014 T^{14} - 700 T^{15} + 376 T^{16} - 12 T^{17} + 28 T^{18} + T^{19} + T^{20} \)
$13$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
$17$ \( 5606265625 - 11493312500 T + 15883256250 T^{2} - 14208162500 T^{3} + 9716790000 T^{4} - 5031351250 T^{5} + 2329603875 T^{6} - 859436250 T^{7} + 297985475 T^{8} - 75876900 T^{9} + 20197585 T^{10} - 2812510 T^{11} + 1193156 T^{12} + 27601 T^{13} + 55599 T^{14} - 5228 T^{15} + 2099 T^{16} - 6 T^{17} + 6 T^{18} + 3 T^{19} + T^{20} \)
$19$ \( 87025 - 954325 T + 3788435 T^{2} + 7448745 T^{3} + 24360196 T^{4} + 79932303 T^{5} + 176467264 T^{6} + 218893919 T^{7} + 188238197 T^{8} + 108499703 T^{9} + 43396413 T^{10} + 10779149 T^{11} + 2611706 T^{12} + 331736 T^{13} + 80368 T^{14} + 8787 T^{15} + 2432 T^{16} + 291 T^{17} + 59 T^{18} + 7 T^{19} + T^{20} \)
$23$ \( ( 2384644 + 41990 T - 1057917 T^{2} - 111931 T^{3} + 142459 T^{4} + 27928 T^{5} - 4834 T^{6} - 1516 T^{7} - 38 T^{8} + 15 T^{9} + T^{10} )^{2} \)
$29$ \( 7055832001 - 18740428897 T + 25210953869 T^{2} - 20893301592 T^{3} + 22308461031 T^{4} - 16631064740 T^{5} + 8638316476 T^{6} - 1946439738 T^{7} + 4328945032 T^{8} + 1349434206 T^{9} + 518458024 T^{10} + 92593621 T^{11} + 16758850 T^{12} + 2147051 T^{13} + 365129 T^{14} + 40376 T^{15} + 6913 T^{16} + 892 T^{17} + 142 T^{18} + 14 T^{19} + T^{20} \)
$31$ \( 87658037041 - 567742492819 T + 6067890611983 T^{2} - 9304683372215 T^{3} + 5454396818649 T^{4} + 850591229644 T^{5} + 574431881554 T^{6} + 46017010031 T^{7} + 22254875546 T^{8} + 199987032 T^{9} + 819331296 T^{10} - 43787165 T^{11} + 23299793 T^{12} - 1196268 T^{13} + 496897 T^{14} - 22206 T^{15} + 6487 T^{16} - 530 T^{17} + 121 T^{18} - 11 T^{19} + T^{20} \)
$37$ \( 1362906814096 - 2883632296416 T + 4657254155408 T^{2} - 6945662062498 T^{3} + 8791364808687 T^{4} - 6926871444177 T^{5} + 4047549432461 T^{6} - 1959516049689 T^{7} + 818617725285 T^{8} - 275247157708 T^{9} + 73503052030 T^{10} - 15595798602 T^{11} + 2657103185 T^{12} - 367460424 T^{13} + 41973828 T^{14} - 4019158 T^{15} + 324963 T^{16} - 21803 T^{17} + 1167 T^{18} - 45 T^{19} + T^{20} \)
$41$ \( 453666314709136 - 10579391235912 T + 103475595408788 T^{2} + 9951647964710 T^{3} + 11989625686597 T^{4} - 1726198957214 T^{5} + 1072877126621 T^{6} - 270434787154 T^{7} + 148967321212 T^{8} - 21351784704 T^{9} + 5214927656 T^{10} - 590258440 T^{11} + 98633146 T^{12} - 9387321 T^{13} + 1160816 T^{14} - 94554 T^{15} + 20531 T^{16} - 1780 T^{17} + 149 T^{18} - 9 T^{19} + T^{20} \)
$43$ \( ( 28117204 + 16866350 T - 765589 T^{2} - 1975416 T^{3} - 169132 T^{4} + 79584 T^{5} + 10640 T^{6} - 1202 T^{7} - 203 T^{8} + 4 T^{9} + T^{10} )^{2} \)
$47$ \( 63646550123417041 - 7446265305833277 T + 17028798453873360 T^{2} + 3546816172101378 T^{3} + 847813067229418 T^{4} - 39707836756253 T^{5} + 37681069054942 T^{6} - 4583646447165 T^{7} + 1228532322377 T^{8} - 151570492283 T^{9} + 28553368782 T^{10} - 3511478350 T^{11} + 604610060 T^{12} - 81990680 T^{13} + 11278160 T^{14} - 1305198 T^{15} + 139956 T^{16} - 12065 T^{17} + 846 T^{18} - 39 T^{19} + T^{20} \)
$53$ \( 28251040697650681 - 26958112189356260 T + 13399506094386532 T^{2} - 4061221387559994 T^{3} + 1070487172454910 T^{4} - 217388022204220 T^{5} + 46221048312673 T^{6} - 8585570451192 T^{7} + 1716238172137 T^{8} - 294856095106 T^{9} + 56051363658 T^{10} - 8506710089 T^{11} + 1337999010 T^{12} - 166549634 T^{13} + 19904098 T^{14} - 2013840 T^{15} + 173378 T^{16} - 11916 T^{17} + 760 T^{18} - 36 T^{19} + T^{20} \)
$59$ \( 844561 - 10681537 T + 782570792 T^{2} + 6225338580 T^{3} + 19450952234 T^{4} + 11248922815 T^{5} + 5896899719 T^{6} + 2000715291 T^{7} + 826618192 T^{8} + 382214282 T^{9} + 185763106 T^{10} + 77443840 T^{11} + 27789636 T^{12} + 8379783 T^{13} + 2143134 T^{14} + 444098 T^{15} + 70764 T^{16} + 8057 T^{17} + 626 T^{18} + 31 T^{19} + T^{20} \)
$61$ \( 287193860772025 - 809846999677900 T + 3031838292623665 T^{2} - 2822823809719520 T^{3} + 1483299756347071 T^{4} - 502347642823492 T^{5} + 131718387100934 T^{6} - 25549100336581 T^{7} + 4137136426547 T^{8} - 516954143422 T^{9} + 56680427808 T^{10} - 4580230376 T^{11} + 347701066 T^{12} - 11480639 T^{13} + 1634443 T^{14} - 110293 T^{15} + 29302 T^{16} - 874 T^{17} - 11 T^{18} + 7 T^{19} + T^{20} \)
$67$ \( ( 1577216 + 3665808 T + 2389881 T^{2} + 33793 T^{3} - 316891 T^{4} - 38270 T^{5} + 12874 T^{6} + 1593 T^{7} - 171 T^{8} - 13 T^{9} + T^{10} )^{2} \)
$71$ \( 156383344908062281 + 21017245713862020 T + 29655664070211378 T^{2} - 3594295584880707 T^{3} + 1031283398019995 T^{4} - 47429472179530 T^{5} + 43443859799212 T^{6} - 17895479813116 T^{7} + 4709959090032 T^{8} - 638040654572 T^{9} + 70634981427 T^{10} - 7573070677 T^{11} + 1082576435 T^{12} - 141066278 T^{13} + 18083997 T^{14} - 1905590 T^{15} + 181383 T^{16} - 13542 T^{17} + 835 T^{18} - 33 T^{19} + T^{20} \)
$73$ \( 5706011979515232256 + 5846285653317619712 T + 4150003269650567680 T^{2} + 1788200153650831232 T^{3} + 504125909795737593 T^{4} + 98538831859403958 T^{5} + 14784463235741942 T^{6} + 1856673133993350 T^{7} + 209002898147527 T^{8} + 20869633252498 T^{9} + 1946248745537 T^{10} + 171078671280 T^{11} + 14987356540 T^{12} + 1254363060 T^{13} + 102171765 T^{14} + 7375148 T^{15} + 497091 T^{16} + 28080 T^{17} + 1331 T^{18} + 44 T^{19} + T^{20} \)
$79$ \( 5891433798216976 + 28658799125488736 T + 54773204229140960 T^{2} + 5921276703209398 T^{3} + 26903719349858003 T^{4} + 8150429434901401 T^{5} + 1470530125236021 T^{6} + 76379391678655 T^{7} + 31101914309499 T^{8} - 923383367248 T^{9} + 402502539046 T^{10} - 24375346934 T^{11} + 3169738662 T^{12} - 132941509 T^{13} + 19483896 T^{14} - 938544 T^{15} + 101682 T^{16} - 5647 T^{17} + 492 T^{18} - 21 T^{19} + T^{20} \)
$83$ \( 9965038995025 + 29717123918250 T + 56413555499165 T^{2} + 65772439486465 T^{3} + 51736797088576 T^{4} + 25443372448495 T^{5} + 12008882644803 T^{6} + 4497078195805 T^{7} + 1336877662028 T^{8} + 302343556500 T^{9} + 53246204971 T^{10} + 7203509075 T^{11} + 814140345 T^{12} + 78667165 T^{13} + 6827646 T^{14} + 450570 T^{15} + 37483 T^{16} + 3030 T^{17} + 348 T^{18} + 25 T^{19} + T^{20} \)
$89$ \( ( 723464500 - 639462500 T + 113576075 T^{2} + 19821525 T^{3} - 5445245 T^{4} - 169705 T^{5} + 81255 T^{6} + 290 T^{7} - 486 T^{8} + T^{9} + T^{10} )^{2} \)
$97$ \( 10588408982718405136 - 3580474924676460400 T + 1887227571889918920 T^{2} - 790582760773661654 T^{3} + 231328865620804239 T^{4} - 45934150656749043 T^{5} + 8069494280099301 T^{6} - 1208667816679122 T^{7} + 153885460992305 T^{8} - 16212263041738 T^{9} + 1575997479386 T^{10} - 142904259101 T^{11} + 13038387483 T^{12} - 1147391126 T^{13} + 99225297 T^{14} - 7909995 T^{15} + 574734 T^{16} - 34211 T^{17} + 1611 T^{18} - 52 T^{19} + T^{20} \)
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