Properties

Label 276.2.i.b
Level $276$
Weight $2$
Character orbit 276.i
Analytic conductor $2.204$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 276.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.20387109579\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} + 18 x^{18} - 58 x^{17} + 169 x^{16} - 363 x^{15} + 800 x^{14} - 1682 x^{13} + 2817 x^{12} - 3731 x^{11} + 5829 x^{10} - 8312 x^{9} + 7737 x^{8} + 4348 x^{7} - 12995 x^{6} - 2574 x^{5} + 24401 x^{4} + 22582 x^{3} + 9558 x^{2} + 2829 x + 529\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{5} q^{3} + ( 1 - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{5} + ( -2 + \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{16} - \beta_{18} - \beta_{19} ) q^{7} -\beta_{13} q^{9} +O(q^{10})\) \( q -\beta_{5} q^{3} + ( 1 - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{5} + ( -2 + \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{16} - \beta_{18} - \beta_{19} ) q^{7} -\beta_{13} q^{9} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} - 5 \beta_{7} - \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + 3 \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} - 2 \beta_{18} ) q^{11} + ( \beta_{4} + \beta_{6} - \beta_{9} + \beta_{13} - \beta_{16} + \beta_{19} ) q^{13} + ( -1 - \beta_{2} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} ) q^{15} + ( 1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{14} - 2 \beta_{15} - 2 \beta_{16} - 2 \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{17} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} - 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} - \beta_{16} - \beta_{18} - 2 \beta_{19} ) q^{19} + ( 1 - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{16} - \beta_{17} + \beta_{19} ) q^{21} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + 5 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - 2 \beta_{13} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{23} + ( 2 \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{17} - \beta_{19} ) q^{25} -\beta_{10} q^{27} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} - 3 \beta_{12} - 4 \beta_{13} - \beta_{14} + 2 \beta_{15} - \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{29} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 4 \beta_{7} - 3 \beta_{10} + 3 \beta_{11} + 2 \beta_{13} - \beta_{14} + \beta_{15} + \beta_{18} - \beta_{19} ) q^{31} + ( -\beta_{1} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} - 2 \beta_{13} - 2 \beta_{17} ) q^{33} + ( -6 + \beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{6} - 4 \beta_{7} + \beta_{8} - 2 \beta_{9} - 5 \beta_{10} + 4 \beta_{11} + 8 \beta_{12} + 6 \beta_{13} - 3 \beta_{14} - \beta_{15} + 8 \beta_{16} + 3 \beta_{17} - 2 \beta_{18} ) q^{35} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} + 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} + 2 \beta_{15} + \beta_{18} + \beta_{19} ) q^{37} + ( \beta_{2} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{12} - \beta_{15} ) q^{39} + ( 3 - \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 5 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 5 \beta_{10} - 6 \beta_{11} - 4 \beta_{12} - 6 \beta_{13} + 3 \beta_{14} + 2 \beta_{15} + 3 \beta_{17} + 3 \beta_{18} + 2 \beta_{19} ) q^{41} + ( -3 - 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 3 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} + 2 \beta_{14} + 2 \beta_{16} + 4 \beta_{17} - \beta_{18} - \beta_{19} ) q^{43} + ( 1 - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{12} - \beta_{13} - \beta_{16} + \beta_{18} ) q^{45} + ( 1 + \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} + 3 \beta_{10} - \beta_{11} - 3 \beta_{12} - 3 \beta_{13} - \beta_{16} - \beta_{17} + 2 \beta_{18} ) q^{47} + ( 4 + 4 \beta_{2} + 4 \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{9} - \beta_{10} - 2 \beta_{12} - 4 \beta_{13} - \beta_{14} + \beta_{16} - \beta_{17} + 2 \beta_{19} ) q^{49} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{9} - 2 \beta_{11} + \beta_{12} - \beta_{13} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{51} + ( -2 \beta_{1} - \beta_{3} + \beta_{8} + \beta_{9} + \beta_{11} - 2 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{53} + ( 2 + \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + 6 \beta_{10} - \beta_{11} + \beta_{13} - 4 \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{55} + ( -1 - \beta_{1} + \beta_{6} + \beta_{8} + \beta_{10} - 2 \beta_{11} + \beta_{14} - \beta_{15} + 2 \beta_{17} ) q^{57} + ( -4 \beta_{1} - \beta_{2} + \beta_{4} - 4 \beta_{7} - 4 \beta_{8} - \beta_{9} - 4 \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} - 2 \beta_{15} - \beta_{16} - \beta_{18} ) q^{59} + ( \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + 3 \beta_{9} - 3 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} + 4 \beta_{16} - \beta_{18} + \beta_{19} ) q^{61} + ( \beta_{1} - \beta_{3} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} + 2 \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{63} + ( -1 + \beta_{3} + 2 \beta_{5} + 2 \beta_{7} + 3 \beta_{10} - 2 \beta_{11} - 2 \beta_{16} + \beta_{19} ) q^{65} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{11} + 4 \beta_{12} + \beta_{14} + \beta_{15} + \beta_{16} + 3 \beta_{17} - 2 \beta_{19} ) q^{67} + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} + \beta_{14} + 2 \beta_{15} + \beta_{16} + 4 \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{69} + ( -4 + \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + \beta_{4} - 6 \beta_{5} - 3 \beta_{7} - 7 \beta_{10} + 6 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} + \beta_{15} + 4 \beta_{16} + 4 \beta_{17} ) q^{71} + ( 3 - 2 \beta_{1} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{10} - 3 \beta_{11} - 2 \beta_{12} + 2 \beta_{14} - 2 \beta_{16} - 2 \beta_{17} ) q^{73} + ( \beta_{2} + 3 \beta_{3} - 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} - \beta_{12} - \beta_{13} - 3 \beta_{14} + 3 \beta_{16} - \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{75} + ( 12 - \beta_{2} - 4 \beta_{3} - 5 \beta_{4} + 12 \beta_{5} + 7 \beta_{7} + 4 \beta_{9} + 9 \beta_{10} - 6 \beta_{11} - 12 \beta_{12} - 6 \beta_{13} + 5 \beta_{14} - \beta_{15} - 6 \beta_{16} - 7 \beta_{17} + 2 \beta_{18} ) q^{77} + ( -5 + 3 \beta_{1} + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{13} - 2 \beta_{14} + 4 \beta_{15} + 4 \beta_{16} + 2 \beta_{17} + 2 \beta_{18} + 3 \beta_{19} ) q^{79} + ( -1 + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{16} + \beta_{17} ) q^{81} + ( -1 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{8} - 3 \beta_{10} + 2 \beta_{11} + \beta_{12} - 3 \beta_{13} + \beta_{16} - \beta_{17} - \beta_{18} - 3 \beta_{19} ) q^{83} + ( 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + 6 \beta_{7} + \beta_{8} + 4 \beta_{10} - 6 \beta_{11} - 3 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{18} - \beta_{19} ) q^{85} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} + \beta_{9} - 2 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{87} + ( -4 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + 4 \beta_{11} - 3 \beta_{12} + 4 \beta_{13} - 3 \beta_{14} + 3 \beta_{16} + 3 \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{89} + ( -3 - \beta_{2} + 6 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} + 6 \beta_{11} - \beta_{15} + 2 \beta_{16} - \beta_{19} ) q^{91} + ( -3 - \beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + 4 \beta_{12} + 4 \beta_{13} - 2 \beta_{15} + 3 \beta_{16} - \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{93} + ( -2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} - 6 \beta_{11} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 5 \beta_{17} - \beta_{18} ) q^{95} + ( 1 + \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} - 4 \beta_{11} - 4 \beta_{12} - 4 \beta_{13} + \beta_{14} + \beta_{15} - 4 \beta_{16} - 3 \beta_{17} + \beta_{18} + \beta_{19} ) q^{97} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} - 4 \beta_{10} + \beta_{11} + 4 \beta_{12} + 2 \beta_{13} - \beta_{15} + \beta_{16} - \beta_{18} - \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 2q^{3} - 4q^{7} - 2q^{9} + O(q^{10}) \) \( 20q + 2q^{3} - 4q^{7} - 2q^{9} + 14q^{13} - 11q^{15} + 3q^{17} + 7q^{19} + 4q^{21} + 24q^{23} + 12q^{25} + 2q^{27} - 26q^{29} + 33q^{31} - 11q^{33} - 2q^{35} - 18q^{37} - 14q^{39} + 4q^{41} - 40q^{43} - 54q^{47} + 30q^{49} - 14q^{51} - 14q^{53} + 11q^{55} - 29q^{57} + 4q^{59} + 12q^{61} - 4q^{63} - 33q^{65} + 15q^{67} - 2q^{69} - 33q^{71} + 15q^{73} + 10q^{75} + 66q^{77} - 42q^{79} - 2q^{81} - 14q^{83} - 13q^{85} + 4q^{87} - 66q^{89} - 16q^{91} - 22q^{93} - 31q^{95} - 24q^{97} + 11q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 4 x^{19} + 18 x^{18} - 58 x^{17} + 169 x^{16} - 363 x^{15} + 800 x^{14} - 1682 x^{13} + 2817 x^{12} - 3731 x^{11} + 5829 x^{10} - 8312 x^{9} + 7737 x^{8} + 4348 x^{7} - 12995 x^{6} - 2574 x^{5} + 24401 x^{4} + 22582 x^{3} + 9558 x^{2} + 2829 x + 529\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(14\!\cdots\!02\)\( \nu^{19} + \)\(24\!\cdots\!73\)\( \nu^{18} - \)\(11\!\cdots\!73\)\( \nu^{17} + \)\(50\!\cdots\!45\)\( \nu^{16} - \)\(16\!\cdots\!94\)\( \nu^{15} + \)\(49\!\cdots\!07\)\( \nu^{14} - \)\(11\!\cdots\!55\)\( \nu^{13} + \)\(26\!\cdots\!19\)\( \nu^{12} - \)\(54\!\cdots\!40\)\( \nu^{11} + \)\(99\!\cdots\!83\)\( \nu^{10} - \)\(15\!\cdots\!05\)\( \nu^{9} + \)\(23\!\cdots\!56\)\( \nu^{8} - \)\(33\!\cdots\!55\)\( \nu^{7} + \)\(37\!\cdots\!71\)\( \nu^{6} - \)\(17\!\cdots\!83\)\( \nu^{5} - \)\(17\!\cdots\!03\)\( \nu^{4} + \)\(16\!\cdots\!78\)\( \nu^{3} + \)\(29\!\cdots\!38\)\( \nu^{2} + \)\(30\!\cdots\!86\)\( \nu + \)\(72\!\cdots\!66\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(13\!\cdots\!54\)\( \nu^{19} - \)\(53\!\cdots\!14\)\( \nu^{18} + \)\(22\!\cdots\!99\)\( \nu^{17} - \)\(68\!\cdots\!59\)\( \nu^{16} + \)\(18\!\cdots\!81\)\( \nu^{15} - \)\(33\!\cdots\!08\)\( \nu^{14} + \)\(60\!\cdots\!93\)\( \nu^{13} - \)\(11\!\cdots\!73\)\( \nu^{12} + \)\(12\!\cdots\!99\)\( \nu^{11} + \)\(35\!\cdots\!66\)\( \nu^{10} - \)\(19\!\cdots\!17\)\( \nu^{9} + \)\(38\!\cdots\!57\)\( \nu^{8} - \)\(12\!\cdots\!58\)\( \nu^{7} + \)\(39\!\cdots\!47\)\( \nu^{6} - \)\(55\!\cdots\!01\)\( \nu^{5} + \)\(14\!\cdots\!87\)\( \nu^{4} + \)\(51\!\cdots\!57\)\( \nu^{3} + \)\(14\!\cdots\!50\)\( \nu^{2} - \)\(15\!\cdots\!06\)\( \nu + \)\(87\!\cdots\!80\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(14\!\cdots\!52\)\( \nu^{19} - \)\(73\!\cdots\!27\)\( \nu^{18} + \)\(32\!\cdots\!30\)\( \nu^{17} - \)\(11\!\cdots\!67\)\( \nu^{16} + \)\(34\!\cdots\!75\)\( \nu^{15} - \)\(83\!\cdots\!63\)\( \nu^{14} + \)\(18\!\cdots\!84\)\( \nu^{13} - \)\(39\!\cdots\!25\)\( \nu^{12} + \)\(74\!\cdots\!69\)\( \nu^{11} - \)\(11\!\cdots\!39\)\( \nu^{10} + \)\(17\!\cdots\!81\)\( \nu^{9} - \)\(25\!\cdots\!34\)\( \nu^{8} + \)\(30\!\cdots\!42\)\( \nu^{7} - \)\(15\!\cdots\!07\)\( \nu^{6} - \)\(13\!\cdots\!61\)\( \nu^{5} + \)\(12\!\cdots\!00\)\( \nu^{4} + \)\(37\!\cdots\!57\)\( \nu^{3} - \)\(78\!\cdots\!64\)\( \nu^{2} - \)\(42\!\cdots\!54\)\( \nu - \)\(80\!\cdots\!73\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(15\!\cdots\!37\)\( \nu^{19} - \)\(46\!\cdots\!96\)\( \nu^{18} + \)\(20\!\cdots\!39\)\( \nu^{17} - \)\(55\!\cdots\!16\)\( \nu^{16} + \)\(14\!\cdots\!86\)\( \nu^{15} - \)\(20\!\cdots\!56\)\( \nu^{14} + \)\(38\!\cdots\!37\)\( \nu^{13} - \)\(69\!\cdots\!50\)\( \nu^{12} + \)\(29\!\cdots\!04\)\( \nu^{11} + \)\(17\!\cdots\!22\)\( \nu^{10} - \)\(25\!\cdots\!66\)\( \nu^{9} + \)\(44\!\cdots\!37\)\( \nu^{8} - \)\(13\!\cdots\!65\)\( \nu^{7} + \)\(37\!\cdots\!18\)\( \nu^{6} - \)\(35\!\cdots\!22\)\( \nu^{5} - \)\(17\!\cdots\!99\)\( \nu^{4} + \)\(49\!\cdots\!37\)\( \nu^{3} + \)\(71\!\cdots\!91\)\( \nu^{2} + \)\(66\!\cdots\!82\)\( \nu + \)\(75\!\cdots\!19\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(21\!\cdots\!09\)\( \nu^{19} + \)\(94\!\cdots\!92\)\( \nu^{18} - \)\(41\!\cdots\!96\)\( \nu^{17} + \)\(13\!\cdots\!67\)\( \nu^{16} - \)\(39\!\cdots\!89\)\( \nu^{15} + \)\(87\!\cdots\!00\)\( \nu^{14} - \)\(18\!\cdots\!16\)\( \nu^{13} + \)\(39\!\cdots\!97\)\( \nu^{12} - \)\(66\!\cdots\!68\)\( \nu^{11} + \)\(86\!\cdots\!39\)\( \nu^{10} - \)\(12\!\cdots\!44\)\( \nu^{9} + \)\(17\!\cdots\!39\)\( \nu^{8} - \)\(15\!\cdots\!07\)\( \nu^{7} - \)\(14\!\cdots\!68\)\( \nu^{6} + \)\(44\!\cdots\!26\)\( \nu^{5} - \)\(14\!\cdots\!95\)\( \nu^{4} - \)\(51\!\cdots\!27\)\( \nu^{3} - \)\(29\!\cdots\!00\)\( \nu^{2} - \)\(86\!\cdots\!94\)\( \nu - \)\(53\!\cdots\!68\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(56\!\cdots\!04\)\( \nu^{19} - \)\(29\!\cdots\!37\)\( \nu^{18} + \)\(13\!\cdots\!11\)\( \nu^{17} - \)\(46\!\cdots\!18\)\( \nu^{16} + \)\(14\!\cdots\!86\)\( \nu^{15} - \)\(34\!\cdots\!85\)\( \nu^{14} + \)\(77\!\cdots\!08\)\( \nu^{13} - \)\(16\!\cdots\!70\)\( \nu^{12} + \)\(31\!\cdots\!26\)\( \nu^{11} - \)\(48\!\cdots\!53\)\( \nu^{10} + \)\(72\!\cdots\!72\)\( \nu^{9} - \)\(10\!\cdots\!99\)\( \nu^{8} + \)\(13\!\cdots\!86\)\( \nu^{7} - \)\(73\!\cdots\!51\)\( \nu^{6} - \)\(53\!\cdots\!07\)\( \nu^{5} + \)\(72\!\cdots\!15\)\( \nu^{4} + \)\(10\!\cdots\!31\)\( \nu^{3} - \)\(22\!\cdots\!89\)\( \nu^{2} - \)\(13\!\cdots\!23\)\( \nu - \)\(27\!\cdots\!81\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(61\!\cdots\!85\)\( \nu^{19} + \)\(26\!\cdots\!19\)\( \nu^{18} - \)\(11\!\cdots\!51\)\( \nu^{17} + \)\(39\!\cdots\!16\)\( \nu^{16} - \)\(11\!\cdots\!17\)\( \nu^{15} + \)\(26\!\cdots\!73\)\( \nu^{14} - \)\(57\!\cdots\!73\)\( \nu^{13} + \)\(12\!\cdots\!22\)\( \nu^{12} - \)\(21\!\cdots\!13\)\( \nu^{11} + \)\(29\!\cdots\!91\)\( \nu^{10} - \)\(44\!\cdots\!93\)\( \nu^{9} + \)\(63\!\cdots\!14\)\( \nu^{8} - \)\(65\!\cdots\!70\)\( \nu^{7} - \)\(98\!\cdots\!03\)\( \nu^{6} + \)\(89\!\cdots\!83\)\( \nu^{5} - \)\(15\!\cdots\!53\)\( \nu^{4} - \)\(15\!\cdots\!75\)\( \nu^{3} - \)\(79\!\cdots\!84\)\( \nu^{2} - \)\(18\!\cdots\!58\)\( \nu - \)\(56\!\cdots\!99\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(78\!\cdots\!10\)\( \nu^{19} - \)\(34\!\cdots\!73\)\( \nu^{18} + \)\(15\!\cdots\!11\)\( \nu^{17} - \)\(50\!\cdots\!50\)\( \nu^{16} + \)\(15\!\cdots\!23\)\( \nu^{15} - \)\(33\!\cdots\!10\)\( \nu^{14} + \)\(74\!\cdots\!87\)\( \nu^{13} - \)\(15\!\cdots\!36\)\( \nu^{12} + \)\(27\!\cdots\!71\)\( \nu^{11} - \)\(38\!\cdots\!62\)\( \nu^{10} + \)\(59\!\cdots\!89\)\( \nu^{9} - \)\(85\!\cdots\!15\)\( \nu^{8} + \)\(89\!\cdots\!92\)\( \nu^{7} + \)\(38\!\cdots\!95\)\( \nu^{6} - \)\(10\!\cdots\!12\)\( \nu^{5} + \)\(23\!\cdots\!13\)\( \nu^{4} + \)\(17\!\cdots\!28\)\( \nu^{3} + \)\(11\!\cdots\!49\)\( \nu^{2} + \)\(33\!\cdots\!11\)\( \nu + \)\(81\!\cdots\!22\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(10\!\cdots\!41\)\( \nu^{19} + \)\(45\!\cdots\!57\)\( \nu^{18} - \)\(20\!\cdots\!93\)\( \nu^{17} + \)\(69\!\cdots\!22\)\( \nu^{16} - \)\(20\!\cdots\!81\)\( \nu^{15} + \)\(47\!\cdots\!68\)\( \nu^{14} - \)\(10\!\cdots\!98\)\( \nu^{13} + \)\(22\!\cdots\!26\)\( \nu^{12} - \)\(40\!\cdots\!07\)\( \nu^{11} + \)\(60\!\cdots\!97\)\( \nu^{10} - \)\(92\!\cdots\!35\)\( \nu^{9} + \)\(13\!\cdots\!75\)\( \nu^{8} - \)\(15\!\cdots\!89\)\( \nu^{7} + \)\(43\!\cdots\!25\)\( \nu^{6} + \)\(10\!\cdots\!54\)\( \nu^{5} - \)\(32\!\cdots\!85\)\( \nu^{4} - \)\(21\!\cdots\!48\)\( \nu^{3} - \)\(11\!\cdots\!65\)\( \nu^{2} - \)\(52\!\cdots\!07\)\( \nu - \)\(16\!\cdots\!06\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(10\!\cdots\!92\)\( \nu^{19} - \)\(42\!\cdots\!77\)\( \nu^{18} + \)\(18\!\cdots\!48\)\( \nu^{17} - \)\(62\!\cdots\!32\)\( \nu^{16} + \)\(18\!\cdots\!15\)\( \nu^{15} - \)\(40\!\cdots\!85\)\( \nu^{14} + \)\(88\!\cdots\!00\)\( \nu^{13} - \)\(18\!\cdots\!60\)\( \nu^{12} + \)\(32\!\cdots\!61\)\( \nu^{11} - \)\(44\!\cdots\!20\)\( \nu^{10} + \)\(67\!\cdots\!07\)\( \nu^{9} - \)\(95\!\cdots\!48\)\( \nu^{8} + \)\(95\!\cdots\!43\)\( \nu^{7} + \)\(28\!\cdots\!09\)\( \nu^{6} - \)\(14\!\cdots\!08\)\( \nu^{5} + \)\(18\!\cdots\!18\)\( \nu^{4} + \)\(22\!\cdots\!97\)\( \nu^{3} + \)\(17\!\cdots\!17\)\( \nu^{2} + \)\(66\!\cdots\!36\)\( \nu + \)\(19\!\cdots\!74\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(10\!\cdots\!31\)\( \nu^{19} + \)\(48\!\cdots\!09\)\( \nu^{18} - \)\(21\!\cdots\!77\)\( \nu^{17} + \)\(73\!\cdots\!49\)\( \nu^{16} - \)\(22\!\cdots\!55\)\( \nu^{15} + \)\(50\!\cdots\!70\)\( \nu^{14} - \)\(11\!\cdots\!73\)\( \nu^{13} + \)\(23\!\cdots\!15\)\( \nu^{12} - \)\(42\!\cdots\!49\)\( \nu^{11} + \)\(60\!\cdots\!74\)\( \nu^{10} - \)\(91\!\cdots\!90\)\( \nu^{9} + \)\(13\!\cdots\!65\)\( \nu^{8} - \)\(14\!\cdots\!61\)\( \nu^{7} + \)\(19\!\cdots\!82\)\( \nu^{6} + \)\(14\!\cdots\!48\)\( \nu^{5} - \)\(62\!\cdots\!89\)\( \nu^{4} - \)\(24\!\cdots\!78\)\( \nu^{3} - \)\(85\!\cdots\!67\)\( \nu^{2} - \)\(22\!\cdots\!14\)\( \nu - \)\(12\!\cdots\!41\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(11\!\cdots\!59\)\( \nu^{19} - \)\(49\!\cdots\!29\)\( \nu^{18} + \)\(22\!\cdots\!18\)\( \nu^{17} - \)\(74\!\cdots\!77\)\( \nu^{16} + \)\(22\!\cdots\!00\)\( \nu^{15} - \)\(50\!\cdots\!62\)\( \nu^{14} + \)\(11\!\cdots\!12\)\( \nu^{13} - \)\(23\!\cdots\!72\)\( \nu^{12} + \)\(42\!\cdots\!56\)\( \nu^{11} - \)\(61\!\cdots\!42\)\( \nu^{10} + \)\(93\!\cdots\!76\)\( \nu^{9} - \)\(13\!\cdots\!83\)\( \nu^{8} + \)\(15\!\cdots\!79\)\( \nu^{7} - \)\(24\!\cdots\!04\)\( \nu^{6} - \)\(13\!\cdots\!63\)\( \nu^{5} + \)\(27\!\cdots\!13\)\( \nu^{4} + \)\(26\!\cdots\!86\)\( \nu^{3} + \)\(12\!\cdots\!54\)\( \nu^{2} + \)\(29\!\cdots\!24\)\( \nu + \)\(34\!\cdots\!32\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(11\!\cdots\!85\)\( \nu^{19} - \)\(56\!\cdots\!90\)\( \nu^{18} + \)\(25\!\cdots\!40\)\( \nu^{17} - \)\(88\!\cdots\!93\)\( \nu^{16} + \)\(26\!\cdots\!51\)\( \nu^{15} - \)\(64\!\cdots\!29\)\( \nu^{14} + \)\(14\!\cdots\!33\)\( \nu^{13} - \)\(30\!\cdots\!43\)\( \nu^{12} + \)\(56\!\cdots\!23\)\( \nu^{11} - \)\(87\!\cdots\!35\)\( \nu^{10} + \)\(13\!\cdots\!33\)\( \nu^{9} - \)\(19\!\cdots\!96\)\( \nu^{8} + \)\(23\!\cdots\!88\)\( \nu^{7} - \)\(11\!\cdots\!67\)\( \nu^{6} - \)\(10\!\cdots\!61\)\( \nu^{5} + \)\(10\!\cdots\!41\)\( \nu^{4} + \)\(19\!\cdots\!62\)\( \nu^{3} + \)\(46\!\cdots\!05\)\( \nu^{2} + \)\(61\!\cdots\!90\)\( \nu + \)\(22\!\cdots\!45\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(11\!\cdots\!36\)\( \nu^{19} - \)\(59\!\cdots\!25\)\( \nu^{18} + \)\(26\!\cdots\!95\)\( \nu^{17} - \)\(93\!\cdots\!64\)\( \nu^{16} + \)\(28\!\cdots\!71\)\( \nu^{15} - \)\(68\!\cdots\!82\)\( \nu^{14} + \)\(15\!\cdots\!15\)\( \nu^{13} - \)\(33\!\cdots\!78\)\( \nu^{12} + \)\(61\!\cdots\!27\)\( \nu^{11} - \)\(94\!\cdots\!72\)\( \nu^{10} + \)\(14\!\cdots\!30\)\( \nu^{9} - \)\(21\!\cdots\!12\)\( \nu^{8} + \)\(25\!\cdots\!40\)\( \nu^{7} - \)\(13\!\cdots\!84\)\( \nu^{6} - \)\(11\!\cdots\!26\)\( \nu^{5} + \)\(11\!\cdots\!59\)\( \nu^{4} + \)\(24\!\cdots\!96\)\( \nu^{3} - \)\(35\!\cdots\!41\)\( \nu^{2} - \)\(29\!\cdots\!66\)\( \nu + \)\(35\!\cdots\!90\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(13\!\cdots\!32\)\( \nu^{19} + \)\(57\!\cdots\!87\)\( \nu^{18} - \)\(25\!\cdots\!48\)\( \nu^{17} + \)\(84\!\cdots\!55\)\( \nu^{16} - \)\(24\!\cdots\!92\)\( \nu^{15} + \)\(54\!\cdots\!76\)\( \nu^{14} - \)\(11\!\cdots\!70\)\( \nu^{13} + \)\(25\!\cdots\!51\)\( \nu^{12} - \)\(43\!\cdots\!70\)\( \nu^{11} + \)\(58\!\cdots\!60\)\( \nu^{10} - \)\(90\!\cdots\!26\)\( \nu^{9} + \)\(13\!\cdots\!02\)\( \nu^{8} - \)\(12\!\cdots\!52\)\( \nu^{7} - \)\(39\!\cdots\!60\)\( \nu^{6} + \)\(19\!\cdots\!90\)\( \nu^{5} + \)\(17\!\cdots\!66\)\( \nu^{4} - \)\(34\!\cdots\!63\)\( \nu^{3} - \)\(23\!\cdots\!41\)\( \nu^{2} - \)\(70\!\cdots\!68\)\( \nu - \)\(16\!\cdots\!32\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(16\!\cdots\!60\)\( \nu^{19} + \)\(71\!\cdots\!30\)\( \nu^{18} - \)\(32\!\cdots\!80\)\( \nu^{17} + \)\(10\!\cdots\!07\)\( \nu^{16} - \)\(31\!\cdots\!50\)\( \nu^{15} + \)\(71\!\cdots\!93\)\( \nu^{14} - \)\(15\!\cdots\!25\)\( \nu^{13} + \)\(33\!\cdots\!08\)\( \nu^{12} - \)\(59\!\cdots\!16\)\( \nu^{11} + \)\(84\!\cdots\!57\)\( \nu^{10} - \)\(12\!\cdots\!63\)\( \nu^{9} + \)\(18\!\cdots\!25\)\( \nu^{8} - \)\(19\!\cdots\!00\)\( \nu^{7} + \)\(75\!\cdots\!46\)\( \nu^{6} + \)\(20\!\cdots\!64\)\( \nu^{5} - \)\(44\!\cdots\!44\)\( \nu^{4} - \)\(37\!\cdots\!84\)\( \nu^{3} - \)\(22\!\cdots\!24\)\( \nu^{2} - \)\(81\!\cdots\!15\)\( \nu - \)\(16\!\cdots\!99\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(\beta_{18}\)\(=\)\((\)\(\)\(28\!\cdots\!86\)\( \nu^{19} - \)\(11\!\cdots\!31\)\( \nu^{18} + \)\(51\!\cdots\!81\)\( \nu^{17} - \)\(16\!\cdots\!64\)\( \nu^{16} + \)\(48\!\cdots\!03\)\( \nu^{15} - \)\(10\!\cdots\!85\)\( \nu^{14} + \)\(23\!\cdots\!85\)\( \nu^{13} - \)\(48\!\cdots\!70\)\( \nu^{12} + \)\(81\!\cdots\!05\)\( \nu^{11} - \)\(10\!\cdots\!78\)\( \nu^{10} + \)\(16\!\cdots\!94\)\( \nu^{9} - \)\(23\!\cdots\!16\)\( \nu^{8} + \)\(22\!\cdots\!18\)\( \nu^{7} + \)\(12\!\cdots\!54\)\( \nu^{6} - \)\(39\!\cdots\!79\)\( \nu^{5} - \)\(61\!\cdots\!97\)\( \nu^{4} + \)\(73\!\cdots\!25\)\( \nu^{3} + \)\(59\!\cdots\!84\)\( \nu^{2} + \)\(17\!\cdots\!24\)\( \nu + \)\(26\!\cdots\!97\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(34\!\cdots\!21\)\( \nu^{19} + \)\(15\!\cdots\!25\)\( \nu^{18} - \)\(68\!\cdots\!41\)\( \nu^{17} + \)\(22\!\cdots\!01\)\( \nu^{16} - \)\(67\!\cdots\!86\)\( \nu^{15} + \)\(15\!\cdots\!52\)\( \nu^{14} - \)\(33\!\cdots\!03\)\( \nu^{13} + \)\(71\!\cdots\!04\)\( \nu^{12} - \)\(12\!\cdots\!94\)\( \nu^{11} + \)\(17\!\cdots\!28\)\( \nu^{10} - \)\(27\!\cdots\!48\)\( \nu^{9} + \)\(39\!\cdots\!11\)\( \nu^{8} - \)\(42\!\cdots\!28\)\( \nu^{7} + \)\(11\!\cdots\!21\)\( \nu^{6} + \)\(47\!\cdots\!53\)\( \nu^{5} - \)\(13\!\cdots\!56\)\( \nu^{4} - \)\(81\!\cdots\!44\)\( \nu^{3} - \)\(38\!\cdots\!17\)\( \nu^{2} - \)\(77\!\cdots\!54\)\( \nu - \)\(15\!\cdots\!70\)\(\)\()/ \)\(79\!\cdots\!93\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{18} + \beta_{16} + 2 \beta_{13} + \beta_{12} + \beta_{11} - 2 \beta_{10} - \beta_{8} - 2 \beta_{7} + 2 \beta_{5} + \beta_{4} + \beta_{3} - 2\)
\(\nu^{3}\)\(=\)\(-\beta_{17} - \beta_{16} - 2 \beta_{15} + \beta_{14} - \beta_{13} - 2 \beta_{12} - 3 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{5} + 4 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(7 \beta_{18} + 2 \beta_{17} - 3 \beta_{16} + 6 \beta_{15} - \beta_{14} - 20 \beta_{13} - 7 \beta_{12} - 4 \beta_{11} + 5 \beta_{10} + 9 \beta_{8} + 6 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} - 6 \beta_{3} + 7 \beta_{1} + 1\)
\(\nu^{5}\)\(=\)\(2 \beta_{19} - 8 \beta_{18} - 2 \beta_{17} - 3 \beta_{16} + 32 \beta_{15} - 10 \beta_{14} + 28 \beta_{12} + 38 \beta_{11} + 13 \beta_{9} + 12 \beta_{8} - 28 \beta_{7} - 8 \beta_{6} - 3 \beta_{3} + 12 \beta_{2} + 13 \beta_{1} - 38\)
\(\nu^{6}\)\(=\)\(-9 \beta_{19} - 36 \beta_{18} - 15 \beta_{17} - 64 \beta_{16} - 36 \beta_{15} + 41 \beta_{14} + 36 \beta_{13} + 21 \beta_{12} - 27 \beta_{11} + 53 \beta_{10} - 23 \beta_{9} - 77 \beta_{8} - 13 \beta_{7} + 38 \beta_{6} + 27 \beta_{5} + 13 \beta_{4} - 23 \beta_{3} - 74 \beta_{1} + 23\)
\(\nu^{7}\)\(=\)\(-85 \beta_{19} + 117 \beta_{18} + 179 \beta_{17} + 62 \beta_{16} - 174 \beta_{15} + 117 \beta_{14} - 56 \beta_{13} - 89 \beta_{12} - 145 \beta_{11} + 32 \beta_{10} - 174 \beta_{9} - 85 \beta_{8} + 174 \beta_{7} + 100 \beta_{6} - 116 \beta_{5} - 100 \beta_{4} + 31 \beta_{3} - 115 \beta_{2} - 115 \beta_{1} + 259\)
\(\nu^{8}\)\(=\)\(311 \beta_{18} + 387 \beta_{17} + 767 \beta_{16} + 441 \beta_{15} - 333 \beta_{14} + \beta_{13} + 253 \beta_{12} + 786 \beta_{11} - 345 \beta_{10} + 311 \beta_{9} + 589 \beta_{8} - 39 \beta_{7} - 130 \beta_{6} - 553 \beta_{5} - 263 \beta_{4} + 312 \beta_{3} + 48 \beta_{2} + 644 \beta_{1} - 206\)
\(\nu^{9}\)\(=\)\(691 \beta_{19} - 763 \beta_{18} - 1198 \beta_{17} - 179 \beta_{16} + 896 \beta_{15} - 436 \beta_{14} + 903 \beta_{13} + 756 \beta_{12} + 641 \beta_{11} - 202 \beta_{10} + 1471 \beta_{9} - 1108 \beta_{7} + 762 \beta_{5} + 436 \beta_{4} + 457 \beta_{3} + 896 \beta_{2} + 691 \beta_{1} - 1447\)
\(\nu^{10}\)\(=\)\(256 \beta_{19} - 1399 \beta_{18} - 2084 \beta_{17} - 3849 \beta_{16} - 4539 \beta_{15} + 3140 \beta_{14} + 970 \beta_{13} - 2659 \beta_{12} - 7280 \beta_{11} + 1112 \beta_{10} - 2601 \beta_{9} - 5083 \beta_{8} + 1112 \beta_{7} + 1202 \beta_{6} + 4058 \beta_{5} + 1655 \beta_{4} - 1399 \beta_{2} - 5083 \beta_{1} + 2194\)
\(\nu^{11}\)\(=\)\(-3164 \beta_{19} + 7685 \beta_{18} + 7909 \beta_{17} + 5601 \beta_{16} - 3164 \beta_{15} - 8847 \beta_{13} - 8847 \beta_{12} - 3388 \beta_{11} - 224 \beta_{10} - 6450 \beta_{9} + 5548 \beta_{8} + 7685 \beta_{7} - 3567 \beta_{6} - 5601 \beta_{5} - 2137 \beta_{4} - 3388 \beta_{3} - 6450 \beta_{2} + 8194\)
\(\nu^{12}\)\(=\)\(5377 \beta_{19} + 5377 \beta_{18} + 15222 \beta_{16} + 44148 \beta_{15} - 25611 \beta_{14} - 15766 \beta_{13} + 14140 \beta_{12} + 46555 \beta_{11} - 980 \beta_{10} + 30988 \beta_{9} + 44148 \beta_{8} - 15567 \beta_{7} - 14864 \beta_{6} - 15594 \beta_{5} - 5377 \beta_{4} - 8686 \beta_{3} + 20241 \beta_{2} + 43495 \beta_{1} - 28754\)
\(\nu^{13}\)\(=\)\(20599 \beta_{19} - 73401 \beta_{18} - 68326 \beta_{17} - 99139 \beta_{16} + 25334 \beta_{14} + 57891 \beta_{13} + 52802 \beta_{12} - 20259 \beta_{11} + 25738 \beta_{10} + 20599 \beta_{9} - 73401 \beta_{8} - 53409 \beta_{7} + 25334 \beta_{6} + 74533 \beta_{5} + 28881 \beta_{4} + 607 \beta_{3} + 44149 \beta_{2} - 44520 \beta_{1} - 57891\)
\(\nu^{14}\)\(=\)\(-104214 \beta_{19} + 105031 \beta_{17} - 94267 \beta_{16} - 365033 \beta_{15} + 198481 \beta_{14} + 64523 \beta_{13} - 118531 \beta_{12} - 317012 \beta_{11} + 23371 \beta_{10} - 315424 \beta_{9} - 315424 \beta_{8} + 165035 \beta_{7} + 104214 \beta_{6} + 29744 \beta_{5} + 4275 \beta_{4} + 817 \beta_{3} - 198481 \beta_{2} - 365033 \beta_{1} + 294137\)
\(\nu^{15}\)\(=\)\(-199298 \beta_{19} + 613836 \beta_{18} + 642307 \beta_{17} + 963973 \beta_{16} + 255176 \beta_{15} - 352874 \beta_{14} - 455113 \beta_{13} - 143743 \beta_{12} + 605313 \beta_{11} - 242420 \beta_{10} + 803263 \beta_{8} + 330189 \beta_{7} - 189427 \beta_{6} - 856256 \beta_{5} - 358660 \beta_{4} + 15619 \beta_{3} - 199298 \beta_{2} + 613836 \beta_{1} + 358660\)
\(\nu^{16}\)\(=\)\(935502 \beta_{19} - 440370 \beta_{18} - 1118995 \beta_{17} + 501913 \beta_{16} + 2676823 \beta_{15} - 1375872 \beta_{14} + 324362 \beta_{13} + 1547661 \beta_{12} + 2534956 \beta_{11} - 324362 \beta_{10} + 2580370 \beta_{9} + 1751617 \beta_{8} - 1547661 \beta_{7} - 440370 \beta_{6} + 501913 \beta_{3} + 1751617 \beta_{2} + 2580370 \beta_{1} - 2534956\)
\(\nu^{17}\)\(=\)\(1180538 \beta_{19} - 4594806 \beta_{18} - 4583484 \beta_{17} - 6986005 \beta_{16} - 4594806 \beta_{15} + 3803669 \beta_{14} + 4594806 \beta_{13} + 11322 \beta_{12} - 6914828 \beta_{11} + 1285496 \beta_{10} - 1690074 \beta_{9} - 8398475 \beta_{8} - 1412470 \beta_{7} + 2210594 \beta_{6} + 6914828 \beta_{5} + 2904732 \beta_{4} + 774424 \beta_{3} - 6805400 \beta_{1} - 774424\)
\(\nu^{18}\)\(=\)\(-6997327 \beta_{19} + 7813292 \beta_{18} + 10594230 \beta_{17} + 2780938 \beta_{16} - 18386211 \beta_{15} + 7813292 \beta_{14} - 6438939 \beta_{13} - 15451990 \beta_{12} - 17827823 \beta_{11} + 815965 \beta_{10} - 18386211 \beta_{9} - 6997327 \beta_{8} + 14217969 \beta_{7} + 1176068 \beta_{6} - 3529576 \beta_{5} - 1176068 \beta_{4} - 3467751 \beta_{3} - 14565829 \beta_{2} - 14565829 \beta_{1} + 21215296\)
\(\nu^{19}\)\(=\)\(31797283 \beta_{18} + 23043869 \beta_{17} + 50800630 \beta_{16} + 55446365 \beta_{15} - 38425439 \beta_{14} - 38654423 \beta_{13} + 4226990 \beta_{12} + 64275440 \beta_{11} - 8829075 \beta_{10} + 31797283 \beta_{9} + 79025245 \beta_{8} + 198285 \beta_{7} - 23649082 \beta_{6} - 46375355 \beta_{5} - 18911370 \beta_{4} - 6857140 \beta_{3} + 12885913 \beta_{2} + 70222722 \beta_{1} - 14895668\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/276\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(139\) \(185\)
\(\chi(n)\) \(\beta_{7}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
1.79494 + 0.527041i
−0.453683 0.133213i
1.61721 1.86636i
−1.25953 + 1.45357i
−0.634872 + 0.408008i
1.55029 0.996309i
−0.0489501 0.340455i
−0.410543 2.85539i
1.79494 0.527041i
−0.453683 + 0.133213i
−0.0489501 + 0.340455i
−0.410543 + 2.85539i
0.753595 + 1.65014i
−0.908456 1.98924i
−0.634872 0.408008i
1.55029 + 0.996309i
0.753595 1.65014i
−0.908456 + 1.98924i
1.61721 + 1.86636i
−1.25953 1.45357i
0 −0.841254 0.540641i 0 −0.143141 + 0.313435i 0 1.55877 + 0.457697i 0 0.415415 + 0.909632i 0
13.2 0 −0.841254 0.540641i 0 1.63926 3.58947i 0 −4.27493 1.25523i 0 0.415415 + 0.909632i 0
25.1 0 0.142315 + 0.989821i 0 −2.42853 0.713080i 0 −2.20327 + 2.54270i 0 −0.959493 + 0.281733i 0
25.2 0 0.142315 + 0.989821i 0 1.44496 + 0.424279i 0 2.15021 2.48148i 0 −0.959493 + 0.281733i 0
49.1 0 −0.415415 + 0.909632i 0 0.0154463 0.0178260i 0 −3.20986 + 2.06285i 0 −0.654861 0.755750i 0
49.2 0 −0.415415 + 0.909632i 0 0.542284 0.625829i 0 3.79055 2.43604i 0 −0.654861 0.755750i 0
73.1 0 0.959493 0.281733i 0 −3.55432 2.28422i 0 0.00534053 + 0.0371442i 0 0.841254 0.540641i 0
73.2 0 0.959493 0.281733i 0 2.17941 + 1.40062i 0 −0.529416 3.68217i 0 0.841254 0.540641i 0
85.1 0 −0.841254 + 0.540641i 0 −0.143141 0.313435i 0 1.55877 0.457697i 0 0.415415 0.909632i 0
85.2 0 −0.841254 + 0.540641i 0 1.63926 + 3.58947i 0 −4.27493 + 1.25523i 0 0.415415 0.909632i 0
121.1 0 0.959493 + 0.281733i 0 −3.55432 + 2.28422i 0 0.00534053 0.0371442i 0 0.841254 + 0.540641i 0
121.2 0 0.959493 + 0.281733i 0 2.17941 1.40062i 0 −0.529416 + 3.68217i 0 0.841254 + 0.540641i 0
133.1 0 0.654861 0.755750i 0 −0.0513473 + 0.357128i 0 1.38384 + 3.03019i 0 −0.142315 0.989821i 0
133.2 0 0.654861 0.755750i 0 0.355980 2.47589i 0 −0.671250 1.46983i 0 −0.142315 0.989821i 0
169.1 0 −0.415415 0.909632i 0 0.0154463 + 0.0178260i 0 −3.20986 2.06285i 0 −0.654861 + 0.755750i 0
169.2 0 −0.415415 0.909632i 0 0.542284 + 0.625829i 0 3.79055 + 2.43604i 0 −0.654861 + 0.755750i 0
193.1 0 0.654861 + 0.755750i 0 −0.0513473 0.357128i 0 1.38384 3.03019i 0 −0.142315 + 0.989821i 0
193.2 0 0.654861 + 0.755750i 0 0.355980 + 2.47589i 0 −0.671250 + 1.46983i 0 −0.142315 + 0.989821i 0
265.1 0 0.142315 0.989821i 0 −2.42853 + 0.713080i 0 −2.20327 2.54270i 0 −0.959493 0.281733i 0
265.2 0 0.142315 0.989821i 0 1.44496 0.424279i 0 2.15021 + 2.48148i 0 −0.959493 0.281733i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 265.2
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 276.2.i.b 20
3.b odd 2 1 828.2.q.b 20
23.c even 11 1 inner 276.2.i.b 20
23.c even 11 1 6348.2.a.r 10
23.d odd 22 1 6348.2.a.q 10
69.h odd 22 1 828.2.q.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.i.b 20 1.a even 1 1 trivial
276.2.i.b 20 23.c even 11 1 inner
828.2.q.b 20 3.b odd 2 1
828.2.q.b 20 69.h odd 22 1
6348.2.a.q 10 23.d odd 22 1
6348.2.a.r 10 23.c even 11 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{20} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(276, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$5$ \( 1 - 55 T + 1781 T^{2} + 220 T^{3} + 23904 T^{4} - 28028 T^{5} + 101650 T^{6} - 224180 T^{7} + 321751 T^{8} - 196064 T^{9} + 10438 T^{10} + 34562 T^{11} - 13639 T^{12} + 2024 T^{13} + 1781 T^{14} - 858 T^{15} + 111 T^{16} + 44 T^{17} - T^{18} + T^{20} \)
$7$ \( 1067089 - 8468534 T + 760663233 T^{2} - 266902696 T^{3} + 44851385 T^{4} - 183588130 T^{5} + 54409853 T^{6} + 11694087 T^{7} + 4854883 T^{8} + 1108873 T^{9} + 1189011 T^{10} + 292069 T^{11} + 77462 T^{12} + 21235 T^{13} + 6186 T^{14} + 1265 T^{15} + 327 T^{16} + 24 T^{17} + 4 T^{19} + T^{20} \)
$11$ \( 4097152081 - 17457430606 T + 27285948547 T^{2} - 3400767557 T^{3} + 5033473313 T^{4} + 4005076163 T^{5} + 1260045721 T^{6} + 505106514 T^{7} + 133396813 T^{8} + 23394987 T^{9} + 12644621 T^{10} + 2602105 T^{11} + 293304 T^{12} - 39567 T^{13} - 2420 T^{14} - 3278 T^{15} + 253 T^{16} - 44 T^{17} + 22 T^{18} + T^{20} \)
$13$ \( 279841 - 3808271 T + 30843345 T^{2} + 32652387 T^{3} + 12668324 T^{4} + 19938111 T^{5} + 9907083 T^{6} - 554834 T^{7} + 4551152 T^{8} - 966101 T^{9} + 731698 T^{10} - 214888 T^{11} + 93141 T^{12} - 18704 T^{13} + 6298 T^{14} - 2975 T^{15} + 961 T^{16} - 283 T^{17} + 81 T^{18} - 14 T^{19} + T^{20} \)
$17$ \( 132871729 + 428769819 T + 850446719 T^{2} + 1095457127 T^{3} + 1253048088 T^{4} + 829129627 T^{5} + 663339040 T^{6} + 3565157 T^{7} + 128111057 T^{8} + 25709126 T^{9} + 17955695 T^{10} + 1262894 T^{11} - 133673 T^{12} - 151465 T^{13} - 24195 T^{14} + 3887 T^{15} + 1561 T^{16} + 31 T^{17} - 41 T^{18} - 3 T^{19} + T^{20} \)
$19$ \( 542843401 + 987807703 T + 2284350827 T^{2} + 4387413036 T^{3} + 9399270899 T^{4} + 8467929030 T^{5} + 2639270857 T^{6} + 993260167 T^{7} + 643710760 T^{8} - 244014995 T^{9} + 44350558 T^{10} - 8416565 T^{11} + 2320835 T^{12} - 600386 T^{13} + 130343 T^{14} - 14410 T^{15} + 613 T^{16} + 2 T^{17} + 22 T^{18} - 7 T^{19} + T^{20} \)
$23$ \( 41426511213649 - 43227663875112 T + 22631874746209 T^{2} - 8328203043362 T^{3} + 2381453346343 T^{4} - 510363381842 T^{5} + 68819338243 T^{6} + 1126274856 T^{7} - 4176826887 T^{8} + 1552669372 T^{9} - 378760767 T^{10} + 67507364 T^{11} - 7895703 T^{12} + 92568 T^{13} + 245923 T^{14} - 79294 T^{15} + 16087 T^{16} - 2446 T^{17} + 289 T^{18} - 24 T^{19} + T^{20} \)
$29$ \( 1195983889 + 4794517954 T + 23710213666 T^{2} + 30765626540 T^{3} + 67266539621 T^{4} + 73321479476 T^{5} + 41829259173 T^{6} + 14929821027 T^{7} + 4489312523 T^{8} + 1237742315 T^{9} + 238941955 T^{10} + 42157505 T^{11} + 12941988 T^{12} + 2447896 T^{13} + 313604 T^{14} + 16662 T^{15} + 3797 T^{16} + 1419 T^{17} + 272 T^{18} + 26 T^{19} + T^{20} \)
$31$ \( 5844755401 + 29161163636 T + 339199389042 T^{2} + 569025657937 T^{3} + 3888434988018 T^{4} - 5041051007949 T^{5} + 2281753824158 T^{6} - 359052187989 T^{7} - 5736276707 T^{8} - 4560930506 T^{9} + 3820678157 T^{10} - 321203058 T^{11} - 43247652 T^{12} + 16887530 T^{13} - 1297800 T^{14} - 131230 T^{15} + 53146 T^{16} - 7293 T^{17} + 634 T^{18} - 33 T^{19} + T^{20} \)
$37$ \( 2582449 + 528532658 T + 38269485866 T^{2} + 14243396695 T^{3} - 14262238985 T^{4} + 3292512330 T^{5} + 212137583 T^{6} - 4156863447 T^{7} + 2580043634 T^{8} - 476443799 T^{9} + 192971756 T^{10} - 19447246 T^{11} + 13872170 T^{12} + 1412455 T^{13} + 480716 T^{14} + 84835 T^{15} + 15259 T^{16} + 1705 T^{17} + 228 T^{18} + 18 T^{19} + T^{20} \)
$41$ \( 1230449878017361 + 1184243616646573 T + 658692849593007 T^{2} + 260449991355801 T^{3} + 68109030467215 T^{4} + 11579563168520 T^{5} + 1413548728484 T^{6} + 166440802640 T^{7} + 56685713934 T^{8} + 15606526850 T^{9} + 2558384081 T^{10} + 501487058 T^{11} + 55856421 T^{12} - 1471449 T^{13} + 364130 T^{14} + 26351 T^{15} + 17878 T^{16} - 109 T^{17} + 78 T^{18} - 4 T^{19} + T^{20} \)
$43$ \( 197657083729249 + 71849401278894 T + 361915257629948 T^{2} + 857437007366461 T^{3} + 792478228391032 T^{4} + 193754478178085 T^{5} + 15185653926875 T^{6} - 2247497711633 T^{7} - 637101415731 T^{8} - 101716101582 T^{9} - 4732270346 T^{10} + 1653843183 T^{11} + 611494719 T^{12} + 127849497 T^{13} + 19855565 T^{14} + 2368619 T^{15} + 219788 T^{16} + 16212 T^{17} + 957 T^{18} + 40 T^{19} + T^{20} \)
$47$ \( ( -12485353 - 11705327 T - 493227 T^{2} + 2072556 T^{3} + 534739 T^{4} - 31807 T^{5} - 24980 T^{6} - 2215 T^{7} + 133 T^{8} + 27 T^{9} + T^{10} )^{2} \)
$53$ \( 1080239689 - 1811694774 T + 6472795557 T^{2} - 7601350467 T^{3} + 13136811451 T^{4} - 6898111429 T^{5} + 7149472112 T^{6} + 3974957769 T^{7} - 2118684013 T^{8} - 482456318 T^{9} + 253613073 T^{10} + 13212737 T^{11} - 7157543 T^{12} - 1055608 T^{13} + 171298 T^{14} + 23496 T^{15} + 6356 T^{16} + 801 T^{17} + 171 T^{18} + 14 T^{19} + T^{20} \)
$59$ \( 10883401 - 1006709644 T + 54040089660 T^{2} - 72792553867 T^{3} - 24335647605 T^{4} + 106773965339 T^{5} + 5079874490 T^{6} - 29382313411 T^{7} + 16072982632 T^{8} - 4093853254 T^{9} + 889370866 T^{10} - 237610125 T^{11} + 94677268 T^{12} - 18321347 T^{13} + 2184121 T^{14} - 334452 T^{15} + 38073 T^{16} - 2431 T^{17} + 338 T^{18} - 4 T^{19} + T^{20} \)
$61$ \( 4209025129 - 6847637596 T + 4260830003 T^{2} + 263801649 T^{3} + 23252211879 T^{4} + 43221576518 T^{5} + 38203092800 T^{6} + 18128259235 T^{7} + 7257318294 T^{8} + 2714508974 T^{9} + 1189486154 T^{10} + 451634686 T^{11} + 123059276 T^{12} + 22541463 T^{13} + 2384684 T^{14} + 68894 T^{15} - 8859 T^{16} - 988 T^{17} + 49 T^{18} - 12 T^{19} + T^{20} \)
$67$ \( 3426093642841 - 14433468346322 T + 26846740133092 T^{2} - 35108570034951 T^{3} + 31533102503679 T^{4} - 6459017454527 T^{5} + 1804936322280 T^{6} - 470441398502 T^{7} + 202325172236 T^{8} - 41082601556 T^{9} + 6711510299 T^{10} - 45250994 T^{11} + 53498005 T^{12} + 9131161 T^{13} - 279682 T^{14} + 112739 T^{15} + 14044 T^{16} - 13 T^{17} + 88 T^{18} - 15 T^{19} + T^{20} \)
$71$ \( 694529152074241 + 1028035545060667 T + 493865162465510 T^{2} + 431382187987080 T^{3} + 452412920920917 T^{4} + 111045847361031 T^{5} + 34649671213399 T^{6} + 5384922945941 T^{7} + 1913795912545 T^{8} + 454508294779 T^{9} + 78573583418 T^{10} + 8498999223 T^{11} + 709907121 T^{12} + 50990610 T^{13} + 5131368 T^{14} + 739673 T^{15} + 95117 T^{16} + 8382 T^{17} + 583 T^{18} + 33 T^{19} + T^{20} \)
$73$ \( 41412657001 - 560007889868 T + 2541930416552 T^{2} - 4145692945813 T^{3} + 4271173782494 T^{4} - 3110955775059 T^{5} + 1726542082118 T^{6} - 744372695426 T^{7} + 252650459307 T^{8} - 67936204035 T^{9} + 14676278319 T^{10} - 2566234068 T^{11} + 373045388 T^{12} - 46157154 T^{13} + 4992369 T^{14} - 492140 T^{15} + 42420 T^{16} - 2731 T^{17} + 161 T^{18} - 15 T^{19} + T^{20} \)
$79$ \( 30821664721 + 446583118189 T + 3691210404030 T^{2} + 6418739916974 T^{3} + 10088545959027 T^{4} + 11907198976489 T^{5} + 10402838016080 T^{6} + 6436534657311 T^{7} + 2849474518735 T^{8} + 845971679621 T^{9} + 171967496634 T^{10} + 22328641919 T^{11} + 1564168193 T^{12} + 55614999 T^{13} + 7400528 T^{14} + 1396942 T^{15} + 152489 T^{16} + 13420 T^{17} + 937 T^{18} + 42 T^{19} + T^{20} \)
$83$ \( 26683984696542889 + 18843914957171837 T + 7041715852998194 T^{2} + 1416026109798258 T^{3} + 302670028724479 T^{4} + 82947190592293 T^{5} + 14407088203672 T^{6} + 1865040648465 T^{7} + 427688055367 T^{8} + 111833178219 T^{9} + 22187680592 T^{10} + 3256233467 T^{11} + 367900493 T^{12} + 33341147 T^{13} + 2377326 T^{14} + 164334 T^{15} + 17747 T^{16} + 2120 T^{17} + 257 T^{18} + 14 T^{19} + T^{20} \)
$89$ \( 67730052449929 + 192609389739140 T + 3686569334166131 T^{2} + 2908928520739604 T^{3} + 2979428180517726 T^{4} + 218997539163810 T^{5} + 58277367114842 T^{6} - 5584724773661 T^{7} + 1173751802122 T^{8} + 1202709586122 T^{9} + 314038996787 T^{10} + 38486119661 T^{11} + 1213482704 T^{12} - 313790917 T^{13} - 46641873 T^{14} - 1840311 T^{15} + 189545 T^{16} + 30591 T^{17} + 1960 T^{18} + 66 T^{19} + T^{20} \)
$97$ \( 48644670794726161 - 34524493530978840 T + 6218152800818328 T^{2} + 446655095024388 T^{3} - 127100329558484 T^{4} + 3302433315764 T^{5} + 6549123177693 T^{6} - 187082098708 T^{7} + 97759352912 T^{8} + 70676023834 T^{9} + 16095634763 T^{10} + 2984231254 T^{11} + 638286213 T^{12} + 102573672 T^{13} + 13141597 T^{14} + 1548833 T^{15} + 167181 T^{16} + 11065 T^{17} + 648 T^{18} + 24 T^{19} + T^{20} \)
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