Properties

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

Related objects

Newspace parameters

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

Newform invariants

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

$q$-expansion

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

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

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

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

Character values

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

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

Embeddings

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

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

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

Inner twists

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

Twists

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

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

Hecke kernels

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

Hecke characteristic polynomials

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