Properties

Label 80.6.n.d
Level 80
Weight 6
Character orbit 80.n
Analytic conductor 12.831
Analytic rank 0
Dimension 20
CM no
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 80.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.8307055850\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \(x^{20} + 271 x^{18} + 109637 x^{16} + 25993614 x^{14} + 5522961902 x^{12} + 881545050522 x^{10} + 133816049059481 x^{8} + 14779507781220031 x^{6} + 824105698447750789 x^{4} + 12044868290803250652 x^{2} + 579398322543528055824\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{65}\cdot 3^{4}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + ( -2 + 10 \beta_{2} - \beta_{5} ) q^{5} + ( \beta_{8} + \beta_{16} ) q^{7} + ( 1 + 103 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{9} + \beta_{18} ) q^{9} +O(q^{10})\) \( q + \beta_{7} q^{3} + ( -2 + 10 \beta_{2} - \beta_{5} ) q^{5} + ( \beta_{8} + \beta_{16} ) q^{7} + ( 1 + 103 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{9} + \beta_{18} ) q^{9} + ( \beta_{1} + \beta_{7} + \beta_{8} - \beta_{15} - \beta_{16} ) q^{11} + ( 39 - 39 \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{9} - \beta_{12} - \beta_{18} ) q^{13} + ( \beta_{1} - 7 \beta_{7} + 20 \beta_{8} + \beta_{13} + 5 \beta_{15} + 2 \beta_{16} + \beta_{17} + \beta_{19} ) q^{15} + ( -111 - 114 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} - \beta_{6} + 2 \beta_{9} + \beta_{10} - 2 \beta_{18} ) q^{17} + ( \beta_{1} - 4 \beta_{7} + 4 \beta_{8} + \beta_{11} + 3 \beta_{13} - 2 \beta_{14} - 3 \beta_{15} + 5 \beta_{16} + 2 \beta_{17} + 3 \beta_{19} ) q^{19} + ( -235 - 10 \beta_{2} + 7 \beta_{3} + 2 \beta_{4} + 28 \beta_{5} - 10 \beta_{6} + 11 \beta_{9} - 5 \beta_{10} + 5 \beta_{12} ) q^{21} + ( 3 \beta_{1} + 61 \beta_{7} - 2 \beta_{11} + 2 \beta_{13} + \beta_{14} + 3 \beta_{15} + 2 \beta_{16} - \beta_{17} + 4 \beta_{19} ) q^{23} + ( 40 + 3 \beta_{2} + 13 \beta_{3} - 5 \beta_{4} + \beta_{5} - 11 \beta_{6} + 3 \beta_{9} + 10 \beta_{10} + 5 \beta_{12} ) q^{25} + ( 9 \beta_{1} + 140 \beta_{8} + 4 \beta_{11} + \beta_{13} - 7 \beta_{14} + 2 \beta_{15} + 2 \beta_{16} + 7 \beta_{17} + 2 \beta_{19} ) q^{27} + ( 17 + 963 \beta_{2} - 23 \beta_{3} - 33 \beta_{5} - 3 \beta_{6} + 17 \beta_{9} - 10 \beta_{10} - 10 \beta_{12} - 2 \beta_{18} ) q^{29} + ( 7 \beta_{1} - 95 \beta_{7} - 95 \beta_{8} - 3 \beta_{11} + 4 \beta_{13} + 5 \beta_{14} + 21 \beta_{15} + 23 \beta_{16} + \beta_{17} + \beta_{19} ) q^{31} + ( -531 + 578 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} - 58 \beta_{5} + 28 \beta_{6} + 19 \beta_{9} - 21 \beta_{12} + 8 \beta_{18} ) q^{33} + ( 6 \beta_{1} - 80 \beta_{7} + 253 \beta_{8} + \beta_{11} + 3 \beta_{13} + 10 \beta_{14} - 31 \beta_{15} + 13 \beta_{16} - 3 \beta_{17} + \beta_{19} ) q^{35} + ( 2227 + 2220 \beta_{2} - 11 \beta_{3} - 14 \beta_{4} - 34 \beta_{5} - 18 \beta_{6} - 11 \beta_{9} + 20 \beta_{10} + 14 \beta_{18} ) q^{37} + ( 225 \beta_{7} - 225 \beta_{8} + 11 \beta_{13} - 11 \beta_{14} + 5 \beta_{15} - 5 \beta_{16} ) q^{39} + ( -361 - 20 \beta_{2} + 19 \beta_{3} - 11 \beta_{4} + 31 \beta_{5} - 20 \beta_{6} + 37 \beta_{9} - 35 \beta_{10} + 35 \beta_{12} ) q^{41} + ( -8 \beta_{1} + 321 \beta_{7} + 2 \beta_{11} - 37 \beta_{13} - \beta_{14} + 12 \beta_{15} - 2 \beta_{16} + 6 \beta_{17} - 4 \beta_{19} ) q^{43} + ( -4672 - 2682 \beta_{2} + 83 \beta_{3} + 35 \beta_{4} - 15 \beta_{5} - 26 \beta_{6} - 82 \beta_{9} + 35 \beta_{10} + 30 \beta_{12} - 5 \beta_{18} ) q^{45} + ( -23 \beta_{1} + 101 \beta_{8} - 8 \beta_{11} - 2 \beta_{13} - 21 \beta_{14} - 4 \beta_{15} + 7 \beta_{16} - 19 \beta_{17} - 4 \beta_{19} ) q^{47} + ( 12 - 174 \beta_{2} - 107 \beta_{3} + 133 \beta_{5} + 85 \beta_{6} + 12 \beta_{9} - 25 \beta_{10} - 25 \beta_{12} - 23 \beta_{18} ) q^{49} + ( -27 \beta_{1} - 571 \beta_{7} - 571 \beta_{8} + 9 \beta_{11} + 16 \beta_{13} + 13 \beta_{14} - 62 \beta_{15} - 68 \beta_{16} - 3 \beta_{17} - 3 \beta_{19} ) q^{51} + ( 9106 - 9206 \beta_{2} - 58 \beta_{3} - 25 \beta_{4} + 167 \beta_{5} - 8 \beta_{6} - 92 \beta_{9} - 43 \beta_{12} - 25 \beta_{18} ) q^{53} + ( -30 \beta_{1} - 483 \beta_{7} - 23 \beta_{8} - 4 \beta_{11} + 9 \beta_{13} + 79 \beta_{14} + 67 \beta_{15} - 107 \beta_{16} + 8 \beta_{17} - 11 \beta_{19} ) q^{55} + ( -1829 - 1646 \beta_{2} + 266 \beta_{3} + 27 \beta_{4} + 14 \beta_{5} + 110 \beta_{6} - 73 \beta_{9} + 23 \beta_{10} - 27 \beta_{18} ) q^{57} + ( -11 \beta_{1} + 48 \beta_{7} - 48 \beta_{8} - 11 \beta_{11} + 31 \beta_{13} - 42 \beta_{14} + 59 \beta_{15} - 81 \beta_{16} - 26 \beta_{17} - 33 \beta_{19} ) q^{59} + ( -1964 - 47 \beta_{2} - 232 \beta_{3} + 8 \beta_{4} - 3 \beta_{5} - 47 \beta_{6} - 80 \beta_{9} - 25 \beta_{10} + 25 \beta_{12} ) q^{61} + ( -5 \beta_{1} - 183 \beta_{7} + 20 \beta_{11} - 195 \beta_{13} - 10 \beta_{14} - 159 \beta_{15} - 20 \beta_{16} - 15 \beta_{17} - 40 \beta_{19} ) q^{63} + ( -7760 - 4737 \beta_{2} - 51 \beta_{3} - 65 \beta_{4} - 21 \beta_{5} + 106 \beta_{6} + 27 \beta_{9} + 15 \beta_{10} - 5 \beta_{12} + 40 \beta_{18} ) q^{65} + ( -42 \beta_{1} - 1083 \beta_{8} - 32 \beta_{11} - 8 \beta_{13} - 84 \beta_{14} - 16 \beta_{15} - 30 \beta_{16} - 26 \beta_{17} - 16 \beta_{19} ) q^{67} + ( -119 + 20174 \beta_{2} + 430 \beta_{3} - 95 \beta_{5} - 112 \beta_{6} - 119 \beta_{9} + 10 \beta_{10} + 10 \beta_{12} + 90 \beta_{18} ) q^{69} + ( -35 \beta_{1} + 525 \beta_{7} + 525 \beta_{8} + 15 \beta_{11} + 67 \beta_{13} + 62 \beta_{14} + 5 \beta_{15} - 5 \beta_{16} - 5 \beta_{17} - 5 \beta_{19} ) q^{71} + ( 13451 - 13144 \beta_{2} - 374 \beta_{3} + 39 \beta_{4} - 354 \beta_{5} + 221 \beta_{6} + 86 \beta_{9} + 57 \beta_{12} + 39 \beta_{18} ) q^{73} + ( -22 \beta_{1} + 1097 \beta_{7} - 156 \beta_{8} - \beta_{11} + 42 \beta_{13} + 237 \beta_{14} - 20 \beta_{15} + 196 \beta_{16} - 32 \beta_{17} - 15 \beta_{19} ) q^{75} + ( 20178 + 20041 \beta_{2} - 353 \beta_{3} + 19 \beta_{4} - 377 \beta_{5} - 197 \beta_{6} - 60 \beta_{9} - 79 \beta_{10} - 19 \beta_{18} ) q^{77} + ( -3 \beta_{1} + 160 \beta_{7} - 160 \beta_{8} - 3 \beta_{11} + 163 \beta_{13} - 166 \beta_{14} - 224 \beta_{15} + 218 \beta_{16} + 21 \beta_{17} - 9 \beta_{19} ) q^{79} + ( -26098 + 292 \beta_{2} + 493 \beta_{3} + 33 \beta_{4} - 383 \beta_{5} + 292 \beta_{6} + 33 \beta_{9} + 135 \beta_{10} - 135 \beta_{12} ) q^{81} + ( 9 \beta_{1} - 147 \beta_{7} - 16 \beta_{11} - 364 \beta_{13} + 8 \beta_{14} + 400 \beta_{15} + 16 \beta_{16} + 7 \beta_{17} + 32 \beta_{19} ) q^{83} + ( -17197 - 23562 \beta_{2} - 194 \beta_{3} - 65 \beta_{4} + 44 \beta_{5} - 150 \beta_{6} + 55 \beta_{9} - 150 \beta_{10} - 125 \beta_{12} - 95 \beta_{18} ) q^{85} + ( 108 \beta_{1} + 1716 \beta_{8} + 68 \beta_{11} + 17 \beta_{13} - 359 \beta_{14} + 34 \beta_{15} - 320 \beta_{16} + 74 \beta_{17} + 34 \beta_{19} ) q^{87} + ( 82 + 28686 \beta_{2} - 394 \beta_{3} + 156 \beta_{5} + 54 \beta_{6} + 82 \beta_{9} + 130 \beta_{10} + 130 \beta_{12} - 46 \beta_{18} ) q^{89} + ( 89 \beta_{1} + 593 \beta_{7} + 593 \beta_{8} - 69 \beta_{11} + 198 \beta_{13} + 221 \beta_{14} + 46 \beta_{16} + 23 \beta_{17} + 23 \beta_{19} ) q^{91} + ( 34995 - 35757 \beta_{2} + 1171 \beta_{3} - 20 \beta_{4} + 781 \beta_{5} - 723 \beta_{6} - 39 \beta_{9} + 216 \beta_{12} - 20 \beta_{18} ) q^{93} + ( 143 \beta_{1} - 256 \beta_{7} - 2324 \beta_{8} + 25 \beta_{11} + 141 \beta_{13} + 456 \beta_{14} - 158 \beta_{15} - 46 \beta_{16} - 9 \beta_{17} + 44 \beta_{19} ) q^{95} + ( 18544 + 18029 \beta_{2} - 198 \beta_{3} - 99 \beta_{4} + 1468 \beta_{5} + 218 \beta_{6} + 733 \beta_{9} - 117 \beta_{10} + 99 \beta_{18} ) q^{97} + ( 47 \beta_{1} - 3515 \beta_{7} + 3515 \beta_{8} + 47 \beta_{11} + 461 \beta_{13} - 414 \beta_{14} + 171 \beta_{15} - 77 \beta_{16} + 46 \beta_{17} + 141 \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 44q^{5} + O(q^{10}) \) \( 20q - 44q^{5} + 804q^{13} - 2236q^{17} - 4520q^{21} + 948q^{25} - 11096q^{33} + 44260q^{37} - 6760q^{41} - 92816q^{45} + 182452q^{53} - 34288q^{57} - 41080q^{61} - 155772q^{65} + 264372q^{73} + 399304q^{77} - 520220q^{81} - 344796q^{85} + 713496q^{93} + 374772q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 271 x^{18} + 109637 x^{16} + 25993614 x^{14} + 5522961902 x^{12} + 881545050522 x^{10} + 133816049059481 x^{8} + 14779507781220031 x^{6} + 824105698447750789 x^{4} + 12044868290803250652 x^{2} + 579398322543528055824\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(\)\(12\!\cdots\!45\)\( \nu^{18} - \)\(82\!\cdots\!16\)\( \nu^{16} + \)\(12\!\cdots\!04\)\( \nu^{14} - \)\(77\!\cdots\!26\)\( \nu^{12} - \)\(92\!\cdots\!08\)\( \nu^{10} - \)\(32\!\cdots\!56\)\( \nu^{8} - \)\(33\!\cdots\!39\)\( \nu^{6} - \)\(73\!\cdots\!00\)\( \nu^{4} - \)\(46\!\cdots\!32\)\( \nu^{2} - \)\(59\!\cdots\!24\)\(\)\()/ \)\(11\!\cdots\!60\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(10\!\cdots\!57\)\( \nu^{19} + \)\(48\!\cdots\!81\)\( \nu^{17} + \)\(74\!\cdots\!38\)\( \nu^{15} + \)\(49\!\cdots\!50\)\( \nu^{13} + \)\(11\!\cdots\!18\)\( \nu^{11} + \)\(26\!\cdots\!44\)\( \nu^{9} + \)\(39\!\cdots\!11\)\( \nu^{7} + \)\(65\!\cdots\!81\)\( \nu^{5} + \)\(77\!\cdots\!98\)\( \nu^{3} + \)\(27\!\cdots\!16\)\( \nu\)\()/ \)\(16\!\cdots\!20\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(31\!\cdots\!07\)\( \nu^{19} - \)\(28\!\cdots\!25\)\( \nu^{18} + \)\(13\!\cdots\!87\)\( \nu^{17} - \)\(32\!\cdots\!90\)\( \nu^{16} + \)\(50\!\cdots\!48\)\( \nu^{15} - \)\(25\!\cdots\!40\)\( \nu^{14} + \)\(14\!\cdots\!30\)\( \nu^{13} - \)\(40\!\cdots\!50\)\( \nu^{12} + \)\(32\!\cdots\!30\)\( \nu^{11} - \)\(91\!\cdots\!00\)\( \nu^{10} + \)\(59\!\cdots\!60\)\( \nu^{9} - \)\(13\!\cdots\!00\)\( \nu^{8} + \)\(92\!\cdots\!87\)\( \nu^{7} - \)\(18\!\cdots\!85\)\( \nu^{6} + \)\(12\!\cdots\!59\)\( \nu^{5} - \)\(17\!\cdots\!10\)\( \nu^{4} + \)\(10\!\cdots\!48\)\( \nu^{3} - \)\(16\!\cdots\!60\)\( \nu^{2} + \)\(38\!\cdots\!12\)\( \nu - \)\(10\!\cdots\!40\)\(\)\()/ \)\(43\!\cdots\!80\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(43\!\cdots\!47\)\( \nu^{19} + \)\(11\!\cdots\!81\)\( \nu^{18} - \)\(11\!\cdots\!51\)\( \nu^{17} + \)\(23\!\cdots\!34\)\( \nu^{16} - \)\(25\!\cdots\!76\)\( \nu^{15} + \)\(10\!\cdots\!04\)\( \nu^{14} - \)\(12\!\cdots\!10\)\( \nu^{13} + \)\(22\!\cdots\!26\)\( \nu^{12} - \)\(30\!\cdots\!34\)\( \nu^{11} + \)\(45\!\cdots\!00\)\( \nu^{10} + \)\(43\!\cdots\!88\)\( \nu^{9} + \)\(65\!\cdots\!52\)\( \nu^{8} + \)\(71\!\cdots\!69\)\( \nu^{7} + \)\(97\!\cdots\!17\)\( \nu^{6} + \)\(28\!\cdots\!61\)\( \nu^{5} + \)\(90\!\cdots\!62\)\( \nu^{4} + \)\(56\!\cdots\!44\)\( \nu^{3} + \)\(83\!\cdots\!80\)\( \nu^{2} + \)\(18\!\cdots\!04\)\( \nu - \)\(13\!\cdots\!08\)\(\)\()/ \)\(57\!\cdots\!40\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(43\!\cdots\!47\)\( \nu^{19} + \)\(68\!\cdots\!95\)\( \nu^{18} + \)\(11\!\cdots\!51\)\( \nu^{17} + \)\(15\!\cdots\!30\)\( \nu^{16} + \)\(25\!\cdots\!76\)\( \nu^{15} + \)\(67\!\cdots\!60\)\( \nu^{14} + \)\(12\!\cdots\!10\)\( \nu^{13} + \)\(14\!\cdots\!50\)\( \nu^{12} + \)\(30\!\cdots\!34\)\( \nu^{11} + \)\(29\!\cdots\!80\)\( \nu^{10} - \)\(43\!\cdots\!88\)\( \nu^{9} + \)\(42\!\cdots\!40\)\( \nu^{8} - \)\(71\!\cdots\!69\)\( \nu^{7} + \)\(58\!\cdots\!15\)\( \nu^{6} - \)\(28\!\cdots\!61\)\( \nu^{5} + \)\(51\!\cdots\!50\)\( \nu^{4} - \)\(56\!\cdots\!44\)\( \nu^{3} + \)\(50\!\cdots\!60\)\( \nu^{2} - \)\(18\!\cdots\!04\)\( \nu - \)\(23\!\cdots\!80\)\(\)\()/ \)\(57\!\cdots\!40\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(46\!\cdots\!17\)\( \nu^{19} - \)\(17\!\cdots\!15\)\( \nu^{18} - \)\(16\!\cdots\!91\)\( \nu^{17} - \)\(35\!\cdots\!30\)\( \nu^{16} - \)\(14\!\cdots\!48\)\( \nu^{15} - \)\(16\!\cdots\!40\)\( \nu^{14} - \)\(16\!\cdots\!70\)\( \nu^{13} - \)\(33\!\cdots\!50\)\( \nu^{12} - \)\(36\!\cdots\!58\)\( \nu^{11} - \)\(70\!\cdots\!60\)\( \nu^{10} - \)\(87\!\cdots\!44\)\( \nu^{9} - \)\(10\!\cdots\!80\)\( \nu^{8} - \)\(13\!\cdots\!91\)\( \nu^{7} - \)\(14\!\cdots\!35\)\( \nu^{6} - \)\(21\!\cdots\!51\)\( \nu^{5} - \)\(12\!\cdots\!30\)\( \nu^{4} - \)\(25\!\cdots\!08\)\( \nu^{3} - \)\(12\!\cdots\!00\)\( \nu^{2} - \)\(88\!\cdots\!96\)\( \nu + \)\(33\!\cdots\!40\)\(\)\()/ \)\(57\!\cdots\!40\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(42\!\cdots\!31\)\( \nu^{19} - \)\(18\!\cdots\!83\)\( \nu^{18} + \)\(11\!\cdots\!49\)\( \nu^{17} - \)\(53\!\cdots\!64\)\( \nu^{16} + \)\(45\!\cdots\!44\)\( \nu^{15} - \)\(21\!\cdots\!68\)\( \nu^{14} + \)\(10\!\cdots\!86\)\( \nu^{13} - \)\(52\!\cdots\!46\)\( \nu^{12} + \)\(22\!\cdots\!30\)\( \nu^{11} - \)\(11\!\cdots\!08\)\( \nu^{10} + \)\(35\!\cdots\!92\)\( \nu^{9} - \)\(18\!\cdots\!56\)\( \nu^{8} + \)\(52\!\cdots\!07\)\( \nu^{7} - \)\(28\!\cdots\!07\)\( \nu^{6} + \)\(55\!\cdots\!57\)\( \nu^{5} - \)\(31\!\cdots\!36\)\( \nu^{4} + \)\(26\!\cdots\!00\)\( \nu^{3} - \)\(18\!\cdots\!44\)\( \nu^{2} - \)\(17\!\cdots\!08\)\( \nu - \)\(25\!\cdots\!12\)\(\)\()/ \)\(26\!\cdots\!80\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(42\!\cdots\!31\)\( \nu^{19} - \)\(18\!\cdots\!83\)\( \nu^{18} - \)\(11\!\cdots\!49\)\( \nu^{17} - \)\(53\!\cdots\!64\)\( \nu^{16} - \)\(45\!\cdots\!44\)\( \nu^{15} - \)\(21\!\cdots\!68\)\( \nu^{14} - \)\(10\!\cdots\!86\)\( \nu^{13} - \)\(52\!\cdots\!46\)\( \nu^{12} - \)\(22\!\cdots\!30\)\( \nu^{11} - \)\(11\!\cdots\!08\)\( \nu^{10} - \)\(35\!\cdots\!92\)\( \nu^{9} - \)\(18\!\cdots\!56\)\( \nu^{8} - \)\(52\!\cdots\!07\)\( \nu^{7} - \)\(28\!\cdots\!07\)\( \nu^{6} - \)\(55\!\cdots\!57\)\( \nu^{5} - \)\(31\!\cdots\!36\)\( \nu^{4} - \)\(26\!\cdots\!00\)\( \nu^{3} - \)\(18\!\cdots\!44\)\( \nu^{2} + \)\(17\!\cdots\!08\)\( \nu - \)\(25\!\cdots\!12\)\(\)\()/ \)\(26\!\cdots\!80\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(12\!\cdots\!29\)\( \nu^{19} - \)\(51\!\cdots\!55\)\( \nu^{18} - \)\(37\!\cdots\!13\)\( \nu^{17} + \)\(68\!\cdots\!90\)\( \nu^{16} - \)\(15\!\cdots\!24\)\( \nu^{15} + \)\(22\!\cdots\!20\)\( \nu^{14} - \)\(38\!\cdots\!90\)\( \nu^{13} + \)\(36\!\cdots\!50\)\( \nu^{12} - \)\(87\!\cdots\!14\)\( \nu^{11} + \)\(55\!\cdots\!80\)\( \nu^{10} - \)\(15\!\cdots\!72\)\( \nu^{9} + \)\(76\!\cdots\!40\)\( \nu^{8} - \)\(23\!\cdots\!93\)\( \nu^{7} + \)\(10\!\cdots\!05\)\( \nu^{6} - \)\(28\!\cdots\!73\)\( \nu^{5} + \)\(56\!\cdots\!90\)\( \nu^{4} - \)\(21\!\cdots\!04\)\( \nu^{3} + \)\(81\!\cdots\!00\)\( \nu^{2} - \)\(80\!\cdots\!08\)\( \nu - \)\(52\!\cdots\!60\)\(\)\()/ \)\(57\!\cdots\!40\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(15\!\cdots\!75\)\( \nu^{19} - \)\(26\!\cdots\!07\)\( \nu^{18} + \)\(55\!\cdots\!51\)\( \nu^{17} - \)\(62\!\cdots\!78\)\( \nu^{16} + \)\(19\!\cdots\!44\)\( \nu^{15} - \)\(27\!\cdots\!28\)\( \nu^{14} + \)\(54\!\cdots\!78\)\( \nu^{13} - \)\(60\!\cdots\!42\)\( \nu^{12} + \)\(11\!\cdots\!02\)\( \nu^{11} - \)\(12\!\cdots\!20\)\( \nu^{10} + \)\(20\!\cdots\!52\)\( \nu^{9} - \)\(17\!\cdots\!04\)\( \nu^{8} + \)\(31\!\cdots\!75\)\( \nu^{7} - \)\(26\!\cdots\!99\)\( \nu^{6} + \)\(38\!\cdots\!95\)\( \nu^{5} - \)\(23\!\cdots\!94\)\( \nu^{4} + \)\(28\!\cdots\!32\)\( \nu^{3} - \)\(22\!\cdots\!40\)\( \nu^{2} + \)\(10\!\cdots\!60\)\( \nu + \)\(53\!\cdots\!76\)\(\)\()/ \)\(57\!\cdots\!40\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(23\!\cdots\!39\)\( \nu^{19} + \)\(75\!\cdots\!22\)\( \nu^{18} - \)\(86\!\cdots\!91\)\( \nu^{17} + \)\(16\!\cdots\!72\)\( \nu^{16} - \)\(29\!\cdots\!96\)\( \nu^{15} + \)\(30\!\cdots\!20\)\( \nu^{14} - \)\(79\!\cdots\!74\)\( \nu^{13} + \)\(14\!\cdots\!68\)\( \nu^{12} - \)\(16\!\cdots\!10\)\( \nu^{11} + \)\(26\!\cdots\!36\)\( \nu^{10} - \)\(27\!\cdots\!28\)\( \nu^{9} + \)\(55\!\cdots\!84\)\( \nu^{8} - \)\(39\!\cdots\!83\)\( \nu^{7} + \)\(70\!\cdots\!66\)\( \nu^{6} - \)\(48\!\cdots\!83\)\( \nu^{5} + \)\(12\!\cdots\!44\)\( \nu^{4} - \)\(22\!\cdots\!20\)\( \nu^{3} + \)\(79\!\cdots\!48\)\( \nu^{2} + \)\(16\!\cdots\!72\)\( \nu + \)\(98\!\cdots\!56\)\(\)\()/ \)\(67\!\cdots\!20\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(24\!\cdots\!13\)\( \nu^{19} + \)\(16\!\cdots\!87\)\( \nu^{18} + \)\(75\!\cdots\!33\)\( \nu^{17} + \)\(43\!\cdots\!78\)\( \nu^{16} + \)\(28\!\cdots\!44\)\( \nu^{15} + \)\(17\!\cdots\!48\)\( \nu^{14} + \)\(74\!\cdots\!78\)\( \nu^{13} + \)\(40\!\cdots\!42\)\( \nu^{12} + \)\(16\!\cdots\!26\)\( \nu^{11} + \)\(81\!\cdots\!40\)\( \nu^{10} + \)\(27\!\cdots\!44\)\( \nu^{9} + \)\(11\!\cdots\!64\)\( \nu^{8} + \)\(42\!\cdots\!37\)\( \nu^{7} + \)\(17\!\cdots\!79\)\( \nu^{6} + \)\(51\!\cdots\!01\)\( \nu^{5} + \)\(16\!\cdots\!14\)\( \nu^{4} + \)\(36\!\cdots\!12\)\( \nu^{3} + \)\(15\!\cdots\!00\)\( \nu^{2} + \)\(13\!\cdots\!12\)\( \nu - \)\(43\!\cdots\!16\)\(\)\()/ \)\(57\!\cdots\!40\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(22\!\cdots\!60\)\( \nu^{19} - \)\(72\!\cdots\!68\)\( \nu^{18} + \)\(54\!\cdots\!60\)\( \nu^{17} - \)\(27\!\cdots\!16\)\( \nu^{16} + \)\(23\!\cdots\!40\)\( \nu^{15} - \)\(89\!\cdots\!92\)\( \nu^{14} + \)\(51\!\cdots\!20\)\( \nu^{13} - \)\(25\!\cdots\!36\)\( \nu^{12} + \)\(11\!\cdots\!80\)\( \nu^{11} - \)\(48\!\cdots\!40\)\( \nu^{10} + \)\(16\!\cdots\!20\)\( \nu^{9} - \)\(86\!\cdots\!92\)\( \nu^{8} + \)\(26\!\cdots\!20\)\( \nu^{7} - \)\(12\!\cdots\!44\)\( \nu^{6} + \)\(26\!\cdots\!00\)\( \nu^{5} - \)\(15\!\cdots\!16\)\( \nu^{4} + \)\(12\!\cdots\!40\)\( \nu^{3} - \)\(90\!\cdots\!28\)\( \nu^{2} - \)\(88\!\cdots\!40\)\( \nu - \)\(12\!\cdots\!64\)\(\)\()/ \)\(20\!\cdots\!05\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(22\!\cdots\!60\)\( \nu^{19} - \)\(72\!\cdots\!68\)\( \nu^{18} - \)\(54\!\cdots\!60\)\( \nu^{17} - \)\(27\!\cdots\!16\)\( \nu^{16} - \)\(23\!\cdots\!40\)\( \nu^{15} - \)\(89\!\cdots\!92\)\( \nu^{14} - \)\(51\!\cdots\!20\)\( \nu^{13} - \)\(25\!\cdots\!36\)\( \nu^{12} - \)\(11\!\cdots\!80\)\( \nu^{11} - \)\(48\!\cdots\!40\)\( \nu^{10} - \)\(16\!\cdots\!20\)\( \nu^{9} - \)\(86\!\cdots\!92\)\( \nu^{8} - \)\(26\!\cdots\!20\)\( \nu^{7} - \)\(12\!\cdots\!44\)\( \nu^{6} - \)\(26\!\cdots\!00\)\( \nu^{5} - \)\(15\!\cdots\!16\)\( \nu^{4} - \)\(12\!\cdots\!40\)\( \nu^{3} - \)\(90\!\cdots\!28\)\( \nu^{2} + \)\(88\!\cdots\!40\)\( \nu - \)\(12\!\cdots\!64\)\(\)\()/ \)\(20\!\cdots\!05\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(25\!\cdots\!11\)\( \nu^{19} - \)\(16\!\cdots\!12\)\( \nu^{18} + \)\(71\!\cdots\!04\)\( \nu^{17} - \)\(41\!\cdots\!63\)\( \nu^{16} + \)\(27\!\cdots\!84\)\( \nu^{15} - \)\(17\!\cdots\!68\)\( \nu^{14} + \)\(67\!\cdots\!46\)\( \nu^{13} - \)\(39\!\cdots\!72\)\( \nu^{12} + \)\(13\!\cdots\!00\)\( \nu^{11} - \)\(86\!\cdots\!50\)\( \nu^{10} + \)\(22\!\cdots\!12\)\( \nu^{9} - \)\(13\!\cdots\!84\)\( \nu^{8} + \)\(32\!\cdots\!27\)\( \nu^{7} - \)\(20\!\cdots\!24\)\( \nu^{6} + \)\(36\!\cdots\!52\)\( \nu^{5} - \)\(21\!\cdots\!39\)\( \nu^{4} + \)\(17\!\cdots\!00\)\( \nu^{3} - \)\(12\!\cdots\!80\)\( \nu^{2} - \)\(11\!\cdots\!68\)\( \nu - \)\(17\!\cdots\!64\)\(\)\()/ \)\(22\!\cdots\!40\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(25\!\cdots\!11\)\( \nu^{19} - \)\(16\!\cdots\!12\)\( \nu^{18} - \)\(71\!\cdots\!04\)\( \nu^{17} - \)\(41\!\cdots\!63\)\( \nu^{16} - \)\(27\!\cdots\!84\)\( \nu^{15} - \)\(17\!\cdots\!68\)\( \nu^{14} - \)\(67\!\cdots\!46\)\( \nu^{13} - \)\(39\!\cdots\!72\)\( \nu^{12} - \)\(13\!\cdots\!00\)\( \nu^{11} - \)\(86\!\cdots\!50\)\( \nu^{10} - \)\(22\!\cdots\!12\)\( \nu^{9} - \)\(13\!\cdots\!84\)\( \nu^{8} - \)\(32\!\cdots\!27\)\( \nu^{7} - \)\(20\!\cdots\!24\)\( \nu^{6} - \)\(36\!\cdots\!52\)\( \nu^{5} - \)\(21\!\cdots\!39\)\( \nu^{4} - \)\(17\!\cdots\!00\)\( \nu^{3} - \)\(12\!\cdots\!80\)\( \nu^{2} + \)\(11\!\cdots\!68\)\( \nu - \)\(17\!\cdots\!64\)\(\)\()/ \)\(22\!\cdots\!40\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(39\!\cdots\!47\)\( \nu^{19} - \)\(80\!\cdots\!13\)\( \nu^{17} - \)\(40\!\cdots\!28\)\( \nu^{15} - \)\(80\!\cdots\!22\)\( \nu^{13} - \)\(19\!\cdots\!10\)\( \nu^{11} - \)\(26\!\cdots\!24\)\( \nu^{9} - \)\(43\!\cdots\!59\)\( \nu^{7} - \)\(39\!\cdots\!69\)\( \nu^{5} - \)\(19\!\cdots\!00\)\( \nu^{3} + \)\(14\!\cdots\!56\)\( \nu\)\()/ \)\(30\!\cdots\!40\)\( \)
\(\beta_{18}\)\(=\)\((\)\(\)\(10\!\cdots\!21\)\( \nu^{19} + \)\(36\!\cdots\!25\)\( \nu^{18} + \)\(28\!\cdots\!21\)\( \nu^{17} + \)\(42\!\cdots\!70\)\( \nu^{16} + \)\(11\!\cdots\!00\)\( \nu^{15} + \)\(32\!\cdots\!20\)\( \nu^{14} + \)\(27\!\cdots\!74\)\( \nu^{13} + \)\(52\!\cdots\!50\)\( \nu^{12} + \)\(59\!\cdots\!98\)\( \nu^{11} + \)\(11\!\cdots\!00\)\( \nu^{10} + \)\(96\!\cdots\!52\)\( \nu^{9} + \)\(17\!\cdots\!00\)\( \nu^{8} + \)\(14\!\cdots\!73\)\( \nu^{7} + \)\(24\!\cdots\!05\)\( \nu^{6} + \)\(16\!\cdots\!77\)\( \nu^{5} + \)\(22\!\cdots\!30\)\( \nu^{4} + \)\(10\!\cdots\!64\)\( \nu^{3} + \)\(20\!\cdots\!80\)\( \nu^{2} + \)\(40\!\cdots\!28\)\( \nu + \)\(14\!\cdots\!20\)\(\)\()/ \)\(57\!\cdots\!40\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(24\!\cdots\!10\)\( \nu^{19} + \)\(36\!\cdots\!89\)\( \nu^{18} + \)\(52\!\cdots\!50\)\( \nu^{17} + \)\(74\!\cdots\!54\)\( \nu^{16} + \)\(24\!\cdots\!80\)\( \nu^{15} + \)\(34\!\cdots\!76\)\( \nu^{14} + \)\(51\!\cdots\!80\)\( \nu^{13} + \)\(75\!\cdots\!90\)\( \nu^{12} + \)\(11\!\cdots\!20\)\( \nu^{11} + \)\(15\!\cdots\!20\)\( \nu^{10} + \)\(16\!\cdots\!00\)\( \nu^{9} + \)\(24\!\cdots\!60\)\( \nu^{8} + \)\(26\!\cdots\!30\)\( \nu^{7} + \)\(37\!\cdots\!49\)\( \nu^{6} + \)\(24\!\cdots\!30\)\( \nu^{5} + \)\(34\!\cdots\!58\)\( \nu^{4} + \)\(12\!\cdots\!00\)\( \nu^{3} + \)\(17\!\cdots\!16\)\( \nu^{2} - \)\(80\!\cdots\!60\)\( \nu + \)\(26\!\cdots\!24\)\(\)\()/ \)\(67\!\cdots\!20\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-36 \beta_{19} + 80 \beta_{18} + 4 \beta_{17} + 72 \beta_{16} - 96 \beta_{15} - 19 \beta_{14} + 7 \beta_{13} + 60 \beta_{12} - 12 \beta_{11} + 60 \beta_{10} + 132 \beta_{9} - 480 \beta_{8} + 480 \beta_{7} + 36 \beta_{6} - 280 \beta_{3} - 444 \beta_{2} - 12 \beta_{1} + 132\)\()/3840\)
\(\nu^{2}\)\(=\)\((\)\(24 \beta_{19} + 24 \beta_{17} - 1968 \beta_{16} - 2016 \beta_{15} + 101 \beta_{14} + 77 \beta_{13} - 240 \beta_{12} - 72 \beta_{11} + 240 \beta_{10} + 240 \beta_{9} + 13440 \beta_{8} + 13440 \beta_{7} - 1200 \beta_{6} + 2160 \beta_{5} - 720 \beta_{4} - 480 \beta_{3} - 1200 \beta_{2} - 632 \beta_{1} - 104400\)\()/3840\)
\(\nu^{3}\)\(=\)\((\)\(1017 \beta_{19} + 120 \beta_{18} - 133 \beta_{17} - 354 \beta_{16} + 1032 \beta_{15} + 1588 \beta_{14} - 1249 \beta_{13} - 90 \beta_{12} + 339 \beta_{11} - 90 \beta_{10} + 2382 \beta_{9} + 3840 \beta_{8} - 3840 \beta_{7} + 1566 \beta_{6} + 660 \beta_{5} - 6120 \beta_{3} + 660486 \beta_{2} + 339 \beta_{1} + 2382\)\()/480\)
\(\nu^{4}\)\(=\)\((\)\(-38844 \beta_{19} - 38844 \beta_{17} - 54912 \beta_{16} + 22776 \beta_{15} - 2211 \beta_{14} + 36633 \beta_{13} + 79020 \beta_{12} + 116532 \beta_{11} - 79020 \beta_{10} - 327252 \beta_{9} + 706080 \beta_{8} + 706080 \beta_{7} + 240564 \beta_{6} - 809560 \beta_{5} + 77840 \beta_{4} - 492960 \beta_{3} + 240564 \beta_{2} + 91212 \beta_{1} - 55641012\)\()/3840\)
\(\nu^{5}\)\(=\)\((\)\(-151344 \beta_{19} + 648560 \beta_{18} - 1533904 \beta_{17} - 4981392 \beta_{16} + 4880496 \beta_{15} + 362479 \beta_{14} - 412927 \beta_{13} - 1941240 \beta_{12} - 50448 \beta_{11} - 1941240 \beta_{10} - 1520808 \beta_{9} + 39860160 \beta_{8} - 39860160 \beta_{7} + 111576 \beta_{6} - 197280 \beta_{5} + 5519840 \beta_{3} - 179262504 \beta_{2} - 50448 \beta_{1} - 1520808\)\()/3840\)
\(\nu^{6}\)\(=\)\((\)\(726150 \beta_{19} + 726150 \beta_{17} + 6352440 \beta_{16} + 4900140 \beta_{15} - 4553905 \beta_{14} - 5280055 \beta_{13} + 139395 \beta_{12} - 2178450 \beta_{11} - 139395 \beta_{10} + 678411 \beta_{9} - 40194480 \beta_{8} - 40194480 \beta_{7} + 9725613 \beta_{6} - 16697610 \beta_{5} + 2214600 \beta_{4} + 10943040 \beta_{3} + 9725613 \beta_{2} + 2853970 \beta_{1} - 420549309\)\()/480\)
\(\nu^{7}\)\(=\)\((\)\(50802300 \beta_{19} - 97402960 \beta_{18} + 374650340 \beta_{17} + 356054760 \beta_{16} - 322186560 \beta_{15} - 532797245 \beta_{14} + 549731345 \beta_{13} - 34112340 \beta_{12} + 16934100 \beta_{11} - 34112340 \beta_{10} - 406138668 \beta_{9} + 17241120 \beta_{8} - 17241120 \beta_{7} - 1113411084 \beta_{6} - 1854795840 \beta_{5} + 1862397800 \beta_{3} - 220843698924 \beta_{2} + 16934100 \beta_{1} - 406138668\)\()/3840\)
\(\nu^{8}\)\(=\)\((\)\(728816424 \beta_{19} + 728816424 \beta_{17} + 7550452272 \beta_{16} + 6092819424 \beta_{15} + 153357531 \beta_{14} - 575458893 \beta_{13} - 8852128560 \beta_{12} - 2186449272 \beta_{11} + 8852128560 \beta_{10} + 4782094896 \beta_{9} - 65084244480 \beta_{8} - 65084244480 \beta_{7} - 9965868912 \beta_{6} + 34223347280 \beta_{5} + 657386000 \beta_{4} + 8450449440 \beta_{3} - 9965868912 \beta_{2} - 1839046152 \beta_{1} + 1263465976176\)\()/3840\)
\(\nu^{9}\)\(=\)\((\)\(-1277012079 \beta_{19} - 7813909080 \beta_{18} + 12278449931 \beta_{17} + 20408007798 \beta_{16} - 21259349184 \beta_{15} - 7781297711 \beta_{14} + 7355627018 \beta_{13} + 8239378860 \beta_{12} - 425670693 \beta_{11} + 8239378860 \beta_{10} - 16910733408 \beta_{9} - 166583140080 \beta_{8} + 166583140080 \beta_{7} + 10622622816 \beta_{6} + 46395357900 \beta_{5} + 7145556060 \beta_{3} + 1277433099816 \beta_{2} - 425670693 \beta_{1} - 16910733408\)\()/480\)
\(\nu^{10}\)\(=\)\((\)\(-172680569364 \beta_{19} - 172680569364 \beta_{17} - 2132499788352 \beta_{16} - 1787138649624 \beta_{15} + 2844126258059 \beta_{14} + 3016806827423 \beta_{13} + 305046798660 \beta_{12} + 518041708092 \beta_{11} - 305046798660 \beta_{10} + 1459808245380 \beta_{9} + 5514275174880 \beta_{8} + 5514275174880 \beta_{7} - 1672259724900 \beta_{6} + 3988404759480 \beta_{5} - 510876137040 \beta_{4} + 942309967200 \beta_{3} - 1672259724900 \beta_{2} - 1377072172028 \beta_{1} + 360137469933540\)\()/3840\)
\(\nu^{11}\)\(=\)\((\)\(-6921312769440 \beta_{19} - 1703697317680 \beta_{18} - 23877482320160 \beta_{17} - 9711105054480 \beta_{16} + 5096896541520 \beta_{15} + 15593726881505 \beta_{14} - 17900831137985 \beta_{13} + 18496362809160 \beta_{12} - 2307104256480 \beta_{11} + 18496362809160 \beta_{10} + 18855140337432 \beta_{9} - 30066165355200 \beta_{8} + 30066165355200 \beta_{7} + 31248309629976 \beta_{6} + 62137841731680 \beta_{5} - 89158650431680 \beta_{3} + 6369320153114136 \beta_{2} - 2307104256480 \beta_{1} + 18855140337432\)\()/3840\)
\(\nu^{12}\)\(=\)\((\)\(-13367968300 \beta_{19} - 13367968300 \beta_{17} - 12410747120880 \beta_{16} - 12384011184280 \beta_{15} - 1641814684690 \beta_{14} - 1628446716390 \beta_{13} + 18156210673345 \beta_{12} + 40103904900 \beta_{11} - 18156210673345 \beta_{10} - 12606411569367 \beta_{9} + 124107916747360 \beta_{8} + 124107916747360 \beta_{7} + 1106879061279 \beta_{6} - 39718126907310 \beta_{5} - 6741746542040 \beta_{4} - 42262154750800 \beta_{3} + 1106879061279 \beta_{2} - 4721153675140 \beta_{1} - 2370195249863567\)\()/160\)
\(\nu^{13}\)\(=\)\((\)\(1321368664628628 \beta_{19} + 2622145591363600 \beta_{18} - 4613824394860852 \beta_{17} - 6282143870337096 \beta_{16} + 7163056313422848 \beta_{15} + 7963850873682997 \beta_{14} - 7523394652140121 \beta_{13} - 3119829047575740 \beta_{12} + 440456221542876 \beta_{11} - 3119829047575740 \beta_{10} + 5675537698909692 \beta_{9} + 29532414422819040 \beta_{8} - 29532414422819040 \beta_{7} - 2944618866357924 \beta_{6} - 14684604479201280 \beta_{5} - 2664481892522120 \beta_{3} - 259425800917526724 \beta_{2} + 440456221542876 \beta_{1} + 5675537698909692\)\()/3840\)
\(\nu^{14}\)\(=\)\((\)\(133874082937224 \beta_{19} + 133874082937224 \beta_{17} + 84205040731630992 \beta_{16} + 83937292565756544 \beta_{15} - 120413619463719899 \beta_{14} - 120547493546657123 \beta_{13} + 7030156399557600 \beta_{12} - 401622248811672 \beta_{11} - 7030156399557600 \beta_{10} - 76418546964960576 \beta_{9} - 209834313845064960 \beta_{8} - 209834313845064960 \beta_{7} + 21702646541893632 \beta_{6} - 123810325233776880 \beta_{5} + 3043671214528560 \beta_{4} - 138164603787585120 \beta_{3} + 21702646541893632 \beta_{2} + 97322502335015768 \beta_{1} - 14946807762370958976\)\()/3840\)
\(\nu^{15}\)\(=\)\((\)\(10462981965544053 \beta_{19} + 47398577306303160 \beta_{18} + 89620689409742623 \beta_{17} + 41952685567644054 \beta_{16} - 34977364257281352 \beta_{15} - 110700788835411238 \beta_{14} + 114188449490592589 \beta_{13} - 173080483479383190 \beta_{12} + 3487660655181351 \beta_{11} - 173080483479383190 \beta_{10} - 69639904534014198 \beta_{9} + 215673673563408480 \beta_{8} - 215673673563408480 \beta_{7} - 122558699455385094 \beta_{6} - 348557977856139180 \beta_{5} + 482317569309099840 \beta_{3} - 25103735483032342974 \beta_{2} + 3487660655181351 \beta_{1} - 69639904534014198\)\()/480\)
\(\nu^{16}\)\(=\)\((\)\(113158507977357156 \beta_{19} + 113158507977357156 \beta_{17} + 12251622931276270848 \beta_{16} + 12025305915321556536 \beta_{15} - 8005543059103831251 \beta_{14} - 8118701567081188407 \beta_{13} - 18517924754091579060 \beta_{12} - 339475523932071468 \beta_{11} + 18517924754091579060 \beta_{10} + 20989144858731674316 \beta_{9} - 74965120409693960160 \beta_{8} - 74965120409693960160 \beta_{7} + 9153526402022194068 \beta_{6} + 30580534129237748200 \beta_{5} + 9380517320458882960 \beta_{4} + 69649740873577121760 \beta_{3} + 9153526402022194068 \beta_{2} + 8613945296668073772 \beta_{1} + 3374034790992514545516\)\()/3840\)
\(\nu^{17}\)\(=\)\((\)\(-45708321341830198608 \beta_{19} - 63969575176698141200 \beta_{18} + 278156120416789822672 \beta_{17} + 286748512401635098416 \beta_{16} - 317220726629521897488 \beta_{15} - 441689693809038574417 \beta_{14} + 426453586695095174881 \beta_{13} + 137634910590601483560 \beta_{12} - 15236107113943399536 \beta_{11} + 137634910590601483560 \beta_{10} - 45137160680300718408 \beta_{9} - 887619907041211659840 \beta_{8} + 887619907041211659840 \beta_{7} + 40377204871899328056 \beta_{6} + 263526481014700858080 \beta_{5} - 151707369278597516000 \beta_{3} + 5000154642073208158776 \beta_{2} - 15236107113943399536 \beta_{1} - 45137160680300718408\)\()/3840\)
\(\nu^{18}\)\(=\)\((\)\(6152838546166078170 \beta_{19} + 6152838546166078170 \beta_{17} - 526163551596594027960 \beta_{16} - 538469228688926184300 \beta_{15} + 665908285687355650645 \beta_{14} + 659755447141189572475 \beta_{13} - 259147863864720893985 \beta_{12} - 18458515638498234510 \beta_{11} + 259147863864720893985 \beta_{10} + 270876389699314025703 \beta_{9} + 1714657170296179454640 \beta_{8} + 1714657170296179454640 \beta_{7} - 60828125318245908111 \beta_{6} + 735212982640930576110 \beta_{5} + 83532478440403840200 \beta_{4} + 740072517945103037280 \beta_{3} - 60828125318245908111 \beta_{2} - 591902906034684341170 \beta_{1} + 52145209660918683104703\)\()/480\)
\(\nu^{19}\)\(=\)\((\)\(5412331396873201445532 \beta_{19} - 25647352402046987464400 \beta_{18} - 15455864286546470837948 \beta_{17} - 14598060915750246006744 \beta_{16} + 18206281846999046970432 \beta_{15} + 34101527580668764092083 \beta_{14} - 32297417115044363610239 \beta_{13} + 68841569159995305234060 \beta_{12} + 1804110465624400481844 \beta_{11} + 68841569159995305234060 \beta_{10} + 648529338334104111732 \beta_{9} + 12299443413769771883040 \beta_{8} - 12299443413769771883040 \beta_{7} + 62034323203756087405716 \beta_{6} + 192261686229173375933760 \beta_{5} - 157820303442466588327640 \beta_{3} + 11098874450310481979049396 \beta_{2} + 1804110465624400481844 \beta_{1} + 648529338334104111732\)\()/3840\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−11.4741 + 7.80740i
3.75557 + 3.81117i
−10.8505 + 10.2794i
5.50401 11.9953i
−1.99079 10.4027i
1.99079 10.4027i
−5.50401 11.9953i
10.8505 + 10.2794i
−3.75557 + 3.81117i
11.4741 + 7.80740i
−11.4741 7.80740i
3.75557 3.81117i
−10.8505 10.2794i
5.50401 + 11.9953i
−1.99079 + 10.4027i
1.99079 + 10.4027i
−5.50401 + 11.9953i
10.8505 10.2794i
−3.75557 3.81117i
11.4741 7.80740i
0 −20.3843 20.3843i 0 −46.4503 + 31.1026i 0 −76.9082 + 76.9082i 0 588.037i 0
47.2 0 −17.2921 17.2921i 0 46.1930 + 31.4834i 0 154.079 154.079i 0 355.037i 0
47.3 0 −9.68301 9.68301i 0 −49.1893 26.5597i 0 −48.6629 + 48.6629i 0 55.4787i 0
47.4 0 −7.48311 7.48311i 0 34.7301 43.8043i 0 −19.2260 + 19.2260i 0 131.006i 0
47.5 0 −0.839817 0.839817i 0 3.71634 + 55.7780i 0 99.3589 99.3589i 0 241.589i 0
47.6 0 0.839817 + 0.839817i 0 3.71634 + 55.7780i 0 −99.3589 + 99.3589i 0 241.589i 0
47.7 0 7.48311 + 7.48311i 0 34.7301 43.8043i 0 19.2260 19.2260i 0 131.006i 0
47.8 0 9.68301 + 9.68301i 0 −49.1893 26.5597i 0 48.6629 48.6629i 0 55.4787i 0
47.9 0 17.2921 + 17.2921i 0 46.1930 + 31.4834i 0 −154.079 + 154.079i 0 355.037i 0
47.10 0 20.3843 + 20.3843i 0 −46.4503 + 31.1026i 0 76.9082 76.9082i 0 588.037i 0
63.1 0 −20.3843 + 20.3843i 0 −46.4503 31.1026i 0 −76.9082 76.9082i 0 588.037i 0
63.2 0 −17.2921 + 17.2921i 0 46.1930 31.4834i 0 154.079 + 154.079i 0 355.037i 0
63.3 0 −9.68301 + 9.68301i 0 −49.1893 + 26.5597i 0 −48.6629 48.6629i 0 55.4787i 0
63.4 0 −7.48311 + 7.48311i 0 34.7301 + 43.8043i 0 −19.2260 19.2260i 0 131.006i 0
63.5 0 −0.839817 + 0.839817i 0 3.71634 55.7780i 0 99.3589 + 99.3589i 0 241.589i 0
63.6 0 0.839817 0.839817i 0 3.71634 55.7780i 0 −99.3589 99.3589i 0 241.589i 0
63.7 0 7.48311 7.48311i 0 34.7301 + 43.8043i 0 19.2260 + 19.2260i 0 131.006i 0
63.8 0 9.68301 9.68301i 0 −49.1893 + 26.5597i 0 48.6629 + 48.6629i 0 55.4787i 0
63.9 0 17.2921 17.2921i 0 46.1930 31.4834i 0 −154.079 154.079i 0 355.037i 0
63.10 0 20.3843 20.3843i 0 −46.4503 31.1026i 0 76.9082 + 76.9082i 0 588.037i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.6.n.d 20
4.b odd 2 1 inner 80.6.n.d 20
5.b even 2 1 400.6.n.g 20
5.c odd 4 1 inner 80.6.n.d 20
5.c odd 4 1 400.6.n.g 20
20.d odd 2 1 400.6.n.g 20
20.e even 4 1 inner 80.6.n.d 20
20.e even 4 1 400.6.n.g 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.6.n.d 20 1.a even 1 1 trivial
80.6.n.d 20 4.b odd 2 1 inner
80.6.n.d 20 5.c odd 4 1 inner
80.6.n.d 20 20.e even 4 1 inner
400.6.n.g 20 5.b even 2 1
400.6.n.g 20 5.c odd 4 1
400.6.n.g 20 20.d odd 2 1
400.6.n.g 20 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 1095980 T_{3}^{16} + 297452922160 T_{3}^{12} + \)\(12\!\cdots\!60\)\( T_{3}^{8} + \)\(10\!\cdots\!80\)\( T_{3}^{4} + \)\(21\!\cdots\!76\)\( \) acting on \(S_{6}^{\mathrm{new}}(80, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 4910 T^{4} + 4404762445 T^{8} + 138021311000520 T^{12} - 6849972091873491390 T^{16} + \)\(85\!\cdots\!52\)\( T^{20} - \)\(23\!\cdots\!90\)\( T^{24} + \)\(16\!\cdots\!20\)\( T^{28} + \)\(18\!\cdots\!45\)\( T^{32} + \)\(72\!\cdots\!10\)\( T^{36} + \)\(51\!\cdots\!01\)\( T^{40} \)
$5$ \( ( 1 + 22 T + 5 T^{2} + 137760 T^{3} + 11738750 T^{4} - 4607500 T^{5} + 36683593750 T^{6} + 1345312500000 T^{7} + 152587890625 T^{8} + 2098083496093750 T^{9} + 298023223876953125 T^{10} )^{2} \)
$7$ \( 1 - 50878370 T^{4} + 88119764714863965 T^{8} + \)\(17\!\cdots\!80\)\( T^{12} + \)\(80\!\cdots\!70\)\( T^{16} + \)\(16\!\cdots\!76\)\( T^{20} + \)\(63\!\cdots\!70\)\( T^{24} + \)\(10\!\cdots\!80\)\( T^{28} + \)\(44\!\cdots\!65\)\( T^{32} - \)\(20\!\cdots\!70\)\( T^{36} + \)\(32\!\cdots\!01\)\( T^{40} \)
$11$ \( ( 1 - 630250 T^{2} + 230184799605 T^{4} - 63113737617167000 T^{6} + \)\(13\!\cdots\!10\)\( T^{8} - \)\(24\!\cdots\!00\)\( T^{10} + \)\(35\!\cdots\!10\)\( T^{12} - \)\(42\!\cdots\!00\)\( T^{14} + \)\(40\!\cdots\!05\)\( T^{16} - \)\(28\!\cdots\!50\)\( T^{18} + \)\(11\!\cdots\!01\)\( T^{20} )^{2} \)
$13$ \( ( 1 - 402 T + 80802 T^{2} - 412195466 T^{3} + 274076716325 T^{4} + 98802411059208 T^{5} + 23088035017184312 T^{6} - 33522742681396790296 T^{7} - \)\(55\!\cdots\!50\)\( T^{8} + \)\(34\!\cdots\!68\)\( T^{9} + \)\(30\!\cdots\!32\)\( T^{10} + \)\(12\!\cdots\!24\)\( T^{11} - \)\(75\!\cdots\!50\)\( T^{12} - \)\(17\!\cdots\!72\)\( T^{13} + \)\(43\!\cdots\!12\)\( T^{14} + \)\(69\!\cdots\!44\)\( T^{15} + \)\(71\!\cdots\!25\)\( T^{16} - \)\(40\!\cdots\!62\)\( T^{17} + \)\(29\!\cdots\!02\)\( T^{18} - \)\(53\!\cdots\!86\)\( T^{19} + \)\(49\!\cdots\!49\)\( T^{20} )^{2} \)
$17$ \( ( 1 + 1118 T + 624962 T^{2} + 1188702526 T^{3} - 1739300659939 T^{4} - 3459207301051384 T^{5} - 2073890095879259656 T^{6} - \)\(50\!\cdots\!88\)\( T^{7} + \)\(54\!\cdots\!26\)\( T^{8} + \)\(82\!\cdots\!84\)\( T^{9} + \)\(52\!\cdots\!56\)\( T^{10} + \)\(11\!\cdots\!88\)\( T^{11} + \)\(10\!\cdots\!74\)\( T^{12} - \)\(14\!\cdots\!84\)\( T^{13} - \)\(84\!\cdots\!56\)\( T^{14} - \)\(19\!\cdots\!88\)\( T^{15} - \)\(14\!\cdots\!11\)\( T^{16} + \)\(13\!\cdots\!18\)\( T^{17} + \)\(10\!\cdots\!62\)\( T^{18} + \)\(26\!\cdots\!26\)\( T^{19} + \)\(33\!\cdots\!49\)\( T^{20} )^{2} \)
$19$ \( ( 1 + 8249550 T^{2} + 36376777568405 T^{4} + \)\(13\!\cdots\!00\)\( T^{6} + \)\(42\!\cdots\!10\)\( T^{8} + \)\(11\!\cdots\!00\)\( T^{10} + \)\(26\!\cdots\!10\)\( T^{12} + \)\(49\!\cdots\!00\)\( T^{14} + \)\(83\!\cdots\!05\)\( T^{16} + \)\(11\!\cdots\!50\)\( T^{18} + \)\(86\!\cdots\!01\)\( T^{20} )^{2} \)
$23$ \( 1 - 207859623483490 T^{4} + \)\(17\!\cdots\!45\)\( T^{8} - \)\(53\!\cdots\!80\)\( T^{12} - \)\(11\!\cdots\!90\)\( T^{16} + \)\(13\!\cdots\!52\)\( T^{20} - \)\(19\!\cdots\!90\)\( T^{24} - \)\(15\!\cdots\!80\)\( T^{28} + \)\(86\!\cdots\!45\)\( T^{32} - \)\(18\!\cdots\!90\)\( T^{36} + \)\(14\!\cdots\!01\)\( T^{40} \)
$29$ \( ( 1 - 99799410 T^{2} + 5177263888867445 T^{4} - \)\(18\!\cdots\!20\)\( T^{6} + \)\(49\!\cdots\!10\)\( T^{8} - \)\(10\!\cdots\!52\)\( T^{10} + \)\(20\!\cdots\!10\)\( T^{12} - \)\(32\!\cdots\!20\)\( T^{14} + \)\(38\!\cdots\!45\)\( T^{16} - \)\(31\!\cdots\!10\)\( T^{18} + \)\(13\!\cdots\!01\)\( T^{20} )^{2} \)
$31$ \( ( 1 - 101180050 T^{2} + 5842226806223805 T^{4} - \)\(25\!\cdots\!00\)\( T^{6} + \)\(95\!\cdots\!10\)\( T^{8} - \)\(29\!\cdots\!00\)\( T^{10} + \)\(77\!\cdots\!10\)\( T^{12} - \)\(17\!\cdots\!00\)\( T^{14} + \)\(32\!\cdots\!05\)\( T^{16} - \)\(45\!\cdots\!50\)\( T^{18} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \)
$37$ \( ( 1 - 22130 T + 244868450 T^{2} - 1308308732410 T^{3} + 13809729158765845 T^{4} - \)\(24\!\cdots\!80\)\( T^{5} + \)\(28\!\cdots\!00\)\( T^{6} - \)\(15\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!10\)\( T^{8} - \)\(15\!\cdots\!80\)\( T^{9} + \)\(18\!\cdots\!00\)\( T^{10} - \)\(10\!\cdots\!60\)\( T^{11} + \)\(50\!\cdots\!90\)\( T^{12} - \)\(51\!\cdots\!80\)\( T^{13} + \)\(64\!\cdots\!00\)\( T^{14} - \)\(38\!\cdots\!60\)\( T^{15} + \)\(15\!\cdots\!05\)\( T^{16} - \)\(10\!\cdots\!30\)\( T^{17} + \)\(13\!\cdots\!50\)\( T^{18} - \)\(82\!\cdots\!10\)\( T^{19} + \)\(25\!\cdots\!49\)\( T^{20} )^{2} \)
$41$ \( ( 1 + 1690 T + 164119545 T^{2} - 1028151329120 T^{3} + 25291870422416110 T^{4} - 93379183076463398452 T^{5} + \)\(29\!\cdots\!10\)\( T^{6} - \)\(13\!\cdots\!20\)\( T^{7} + \)\(25\!\cdots\!45\)\( T^{8} + \)\(30\!\cdots\!90\)\( T^{9} + \)\(20\!\cdots\!01\)\( T^{10} )^{4} \)
$43$ \( 1 + 22292551011039310 T^{4} + \)\(40\!\cdots\!45\)\( T^{8} + \)\(23\!\cdots\!20\)\( T^{12} + \)\(67\!\cdots\!10\)\( T^{16} + \)\(21\!\cdots\!52\)\( T^{20} + \)\(31\!\cdots\!10\)\( T^{24} + \)\(51\!\cdots\!20\)\( T^{28} + \)\(41\!\cdots\!45\)\( T^{32} + \)\(10\!\cdots\!10\)\( T^{36} + \)\(22\!\cdots\!01\)\( T^{40} \)
$47$ \( 1 + 74279974827999230 T^{4} + \)\(48\!\cdots\!65\)\( T^{8} + \)\(19\!\cdots\!80\)\( T^{12} + \)\(14\!\cdots\!70\)\( T^{16} - \)\(25\!\cdots\!24\)\( T^{20} + \)\(39\!\cdots\!70\)\( T^{24} + \)\(15\!\cdots\!80\)\( T^{28} + \)\(10\!\cdots\!65\)\( T^{32} + \)\(43\!\cdots\!30\)\( T^{36} + \)\(16\!\cdots\!01\)\( T^{40} \)
$53$ \( ( 1 - 91226 T + 4161091538 T^{2} - 141622957033378 T^{3} + 3893612180777232821 T^{4} - \)\(81\!\cdots\!12\)\( T^{5} + \)\(12\!\cdots\!56\)\( T^{6} - \)\(10\!\cdots\!76\)\( T^{7} - \)\(19\!\cdots\!54\)\( T^{8} + \)\(10\!\cdots\!72\)\( T^{9} - \)\(26\!\cdots\!96\)\( T^{10} + \)\(45\!\cdots\!96\)\( T^{11} - \)\(34\!\cdots\!46\)\( T^{12} - \)\(80\!\cdots\!32\)\( T^{13} + \)\(39\!\cdots\!56\)\( T^{14} - \)\(10\!\cdots\!16\)\( T^{15} + \)\(20\!\cdots\!29\)\( T^{16} - \)\(31\!\cdots\!46\)\( T^{17} + \)\(38\!\cdots\!38\)\( T^{18} - \)\(35\!\cdots\!18\)\( T^{19} + \)\(16\!\cdots\!49\)\( T^{20} )^{2} \)
$59$ \( ( 1 + 4265422750 T^{2} + 8278744468566020805 T^{4} + \)\(98\!\cdots\!00\)\( T^{6} + \)\(85\!\cdots\!10\)\( T^{8} + \)\(63\!\cdots\!00\)\( T^{10} + \)\(43\!\cdots\!10\)\( T^{12} + \)\(25\!\cdots\!00\)\( T^{14} + \)\(11\!\cdots\!05\)\( T^{16} + \)\(29\!\cdots\!50\)\( T^{18} + \)\(34\!\cdots\!01\)\( T^{20} )^{2} \)
$61$ \( ( 1 + 10270 T + 3306972765 T^{2} + 29698468018720 T^{3} + 4981733552475787870 T^{4} + \)\(35\!\cdots\!24\)\( T^{5} + \)\(42\!\cdots\!70\)\( T^{6} + \)\(21\!\cdots\!20\)\( T^{7} + \)\(19\!\cdots\!65\)\( T^{8} + \)\(52\!\cdots\!70\)\( T^{9} + \)\(42\!\cdots\!01\)\( T^{10} )^{4} \)
$67$ \( 1 - 2038648142805486290 T^{4} - \)\(25\!\cdots\!55\)\( T^{8} + \)\(14\!\cdots\!20\)\( T^{12} + \)\(73\!\cdots\!10\)\( T^{16} - \)\(57\!\cdots\!48\)\( T^{20} + \)\(24\!\cdots\!10\)\( T^{24} + \)\(15\!\cdots\!20\)\( T^{28} - \)\(93\!\cdots\!55\)\( T^{32} - \)\(24\!\cdots\!90\)\( T^{36} + \)\(40\!\cdots\!01\)\( T^{40} \)
$71$ \( ( 1 - 14421599170 T^{2} + 96751044929093627565 T^{4} - \)\(40\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!70\)\( T^{8} - \)\(24\!\cdots\!24\)\( T^{10} + \)\(37\!\cdots\!70\)\( T^{12} - \)\(42\!\cdots\!20\)\( T^{14} + \)\(33\!\cdots\!65\)\( T^{16} - \)\(16\!\cdots\!70\)\( T^{18} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \)
$73$ \( ( 1 - 132186 T + 8736569298 T^{2} - 431249337673738 T^{3} + 10750357541220477581 T^{4} + \)\(12\!\cdots\!88\)\( T^{5} - \)\(17\!\cdots\!84\)\( T^{6} + \)\(58\!\cdots\!44\)\( T^{7} + \)\(47\!\cdots\!06\)\( T^{8} - \)\(60\!\cdots\!08\)\( T^{9} + \)\(32\!\cdots\!04\)\( T^{10} - \)\(12\!\cdots\!44\)\( T^{11} + \)\(20\!\cdots\!94\)\( T^{12} + \)\(52\!\cdots\!08\)\( T^{13} - \)\(32\!\cdots\!84\)\( T^{14} + \)\(47\!\cdots\!84\)\( T^{15} + \)\(85\!\cdots\!69\)\( T^{16} - \)\(70\!\cdots\!66\)\( T^{17} + \)\(29\!\cdots\!98\)\( T^{18} - \)\(93\!\cdots\!98\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} )^{2} \)
$79$ \( ( 1 + 18614864790 T^{2} + \)\(16\!\cdots\!45\)\( T^{4} + \)\(10\!\cdots\!80\)\( T^{6} + \)\(44\!\cdots\!10\)\( T^{8} + \)\(15\!\cdots\!48\)\( T^{10} + \)\(42\!\cdots\!10\)\( T^{12} + \)\(90\!\cdots\!80\)\( T^{14} + \)\(14\!\cdots\!45\)\( T^{16} + \)\(14\!\cdots\!90\)\( T^{18} + \)\(76\!\cdots\!01\)\( T^{20} )^{2} \)
$83$ \( 1 - 18609684090616711570 T^{4} - \)\(46\!\cdots\!35\)\( T^{8} + \)\(34\!\cdots\!80\)\( T^{12} + \)\(25\!\cdots\!70\)\( T^{16} - \)\(17\!\cdots\!24\)\( T^{20} + \)\(60\!\cdots\!70\)\( T^{24} + \)\(20\!\cdots\!80\)\( T^{28} - \)\(64\!\cdots\!35\)\( T^{32} - \)\(62\!\cdots\!70\)\( T^{36} + \)\(80\!\cdots\!01\)\( T^{40} \)
$89$ \( ( 1 - 33120686970 T^{2} + \)\(41\!\cdots\!65\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{6} - \)\(32\!\cdots\!30\)\( T^{8} + \)\(73\!\cdots\!76\)\( T^{10} - \)\(10\!\cdots\!30\)\( T^{12} - \)\(20\!\cdots\!20\)\( T^{14} + \)\(12\!\cdots\!65\)\( T^{16} - \)\(31\!\cdots\!70\)\( T^{18} + \)\(29\!\cdots\!01\)\( T^{20} )^{2} \)
$97$ \( ( 1 - 187386 T + 17556756498 T^{2} - 497237634071002 T^{3} - \)\(23\!\cdots\!35\)\( T^{4} + \)\(42\!\cdots\!24\)\( T^{5} - \)\(37\!\cdots\!32\)\( T^{6} + \)\(18\!\cdots\!68\)\( T^{7} + \)\(11\!\cdots\!50\)\( T^{8} - \)\(33\!\cdots\!76\)\( T^{9} + \)\(36\!\cdots\!68\)\( T^{10} - \)\(29\!\cdots\!32\)\( T^{11} + \)\(87\!\cdots\!50\)\( T^{12} + \)\(11\!\cdots\!24\)\( T^{13} - \)\(20\!\cdots\!32\)\( T^{14} + \)\(19\!\cdots\!68\)\( T^{15} - \)\(93\!\cdots\!15\)\( T^{16} - \)\(17\!\cdots\!86\)\( T^{17} + \)\(51\!\cdots\!98\)\( T^{18} - \)\(47\!\cdots\!02\)\( T^{19} + \)\(21\!\cdots\!49\)\( T^{20} )^{2} \)
show more
show less