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

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 $$T_{3}^{20} + 1095980 T_{3}^{16} + 297452922160 T_{3}^{12} +$$12246534533136960

'>$$12\!\cdots\!60$$$$T_{3}^{8} +$$108964362492176302080
'>$$10\!\cdots\!80$$$$T_{3}^{4} +$$216763542247808434176'>$$21\!\cdots\!76$$ acting on $$S_{6}^{\mathrm{new}}(80, [\chi])$$.

Hecke characteristic polynomials

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