# Properties

 Label 92.2.e.a Level $92$ Weight $2$ Character orbit 92.e Analytic conductor $0.735$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$92 = 2^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 92.e (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.734623698596$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$2$$ over $$\Q(\zeta_{11})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 9 x^{19} + 51 x^{18} - 200 x^{17} + 633 x^{16} - 1688 x^{15} + 3957 x^{14} - 8161 x^{13} + 14788 x^{12} - 23925 x^{11} + 35080 x^{10} - 43945 x^{9} + 57269 x^{8} - 57348 x^{7} + 52821 x^{6} - 34986 x^{5} + 26231 x^{4} + 96928 x^{3} + 49370 x^{2} + 8165 x + 529$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} - \beta_{18} ) q^{3} + ( -\beta_{3} + \beta_{4} + \beta_{6} + \beta_{10} - \beta_{13} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{5} + ( 1 - \beta_{1} + \beta_{3} + \beta_{7} - \beta_{8} + \beta_{11} + \beta_{13} + \beta_{16} ) q^{7} + ( \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} - \beta_{14} - \beta_{17} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} - \beta_{18} ) q^{3} + ( -\beta_{3} + \beta_{4} + \beta_{6} + \beta_{10} - \beta_{13} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{5} + ( 1 - \beta_{1} + \beta_{3} + \beta_{7} - \beta_{8} + \beta_{11} + \beta_{13} + \beta_{16} ) q^{7} + ( \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} - \beta_{14} - \beta_{17} ) q^{9} + ( -1 + \beta_{1} + \beta_{2} - \beta_{5} - \beta_{7} - \beta_{9} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} + \beta_{18} - \beta_{19} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} - \beta_{7} + \beta_{9} + \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{16} + 2 \beta_{19} ) q^{13} + ( -2 - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} - \beta_{19} ) q^{15} + ( -\beta_{1} - \beta_{4} - \beta_{5} + 2 \beta_{8} ) q^{17} + ( -1 - \beta_{2} + \beta_{5} - \beta_{7} - \beta_{10} + \beta_{13} + 2 \beta_{14} + \beta_{16} - \beta_{17} + \beta_{18} ) q^{19} + ( -2 - \beta_{1} - \beta_{5} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{17} ) q^{21} + ( -2 + \beta_{1} - \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{23} + ( -2 + 2 \beta_{1} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{25} + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - 4 \beta_{13} - 2 \beta_{14} + \beta_{15} - 2 \beta_{16} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{27} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{6} - 2 \beta_{8} + \beta_{9} + 3 \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} + 3 \beta_{15} - \beta_{16} + 2 \beta_{17} - \beta_{19} ) q^{29} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - 3 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} + \beta_{19} ) q^{31} + ( 2 + \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 4 \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} - 4 \beta_{12} - \beta_{13} - 2 \beta_{14} - 4 \beta_{15} - \beta_{18} ) q^{33} + ( 4 - \beta_{1} - \beta_{2} + \beta_{5} - 3 \beta_{6} + \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - \beta_{12} - 4 \beta_{14} - 4 \beta_{15} + \beta_{16} + \beta_{17} - 2 \beta_{18} ) q^{35} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} + 4 \beta_{6} - \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + \beta_{10} + \beta_{11} + 3 \beta_{12} + 2 \beta_{13} + 4 \beta_{14} ) q^{37} + ( 1 + \beta_{1} + 2 \beta_{4} + 3 \beta_{6} - \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{39} + ( 6 - 2 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 4 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} ) q^{41} + ( 5 - \beta_{1} - \beta_{2} - \beta_{5} - 4 \beta_{6} + \beta_{7} + 4 \beta_{8} - \beta_{9} - 4 \beta_{10} + \beta_{11} - 3 \beta_{12} - 2 \beta_{14} + \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{43} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{7} + 4 \beta_{10} + \beta_{11} + 4 \beta_{12} + \beta_{13} + 4 \beta_{14} + 4 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} - 2 \beta_{19} ) q^{45} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - \beta_{13} - 3 \beta_{14} - \beta_{15} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{47} + ( 6 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} - 4 \beta_{9} - 5 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} + \beta_{13} - 4 \beta_{14} - 3 \beta_{15} + \beta_{16} ) q^{49} + ( 2 + \beta_{1} + 3 \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - 4 \beta_{13} + \beta_{14} - 3 \beta_{15} - \beta_{16} + \beta_{17} + \beta_{18} ) q^{51} + ( -5 + 3 \beta_{1} + 2 \beta_{4} + 3 \beta_{6} - 3 \beta_{7} - 5 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + 5 \beta_{11} + 2 \beta_{12} + \beta_{13} + 4 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} - \beta_{17} ) q^{53} + ( 3 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - 3 \beta_{6} - \beta_{8} - 3 \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} - 3 \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{55} + ( 5 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - 4 \beta_{9} - 3 \beta_{11} - 5 \beta_{12} - 3 \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{17} - \beta_{18} ) q^{57} + ( -1 - \beta_{1} + \beta_{3} + \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 5 \beta_{9} + 2 \beta_{10} + 2 \beta_{12} + 4 \beta_{13} + 2 \beta_{14} + 4 \beta_{15} - \beta_{16} + \beta_{18} - 2 \beta_{19} ) q^{59} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + 3 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} + \beta_{16} ) q^{61} + ( -3 + \beta_{1} - \beta_{2} + \beta_{4} + 3 \beta_{6} + 4 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{15} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{63} + ( -1 + \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + 6 \beta_{8} - 8 \beta_{9} - 5 \beta_{10} - 6 \beta_{11} - 4 \beta_{12} - 4 \beta_{13} - 3 \beta_{14} - 6 \beta_{15} - 3 \beta_{16} + \beta_{17} + \beta_{18} ) q^{65} + ( -4 + 2 \beta_{1} + \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - \beta_{8} + 4 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + \beta_{13} + 2 \beta_{14} - 2 \beta_{15} - \beta_{17} + \beta_{18} ) q^{67} + ( -5 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} + 4 \beta_{6} - 3 \beta_{8} + \beta_{9} + 6 \beta_{10} + 4 \beta_{11} + 5 \beta_{12} + 6 \beta_{13} + 5 \beta_{14} + 3 \beta_{15} - \beta_{16} + 3 \beta_{17} - 2 \beta_{19} ) q^{69} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} - 3 \beta_{10} - 3 \beta_{12} - 3 \beta_{13} - \beta_{15} + 2 \beta_{17} - 2 \beta_{18} ) q^{71} + ( 2 + \beta_{2} + \beta_{3} - 4 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 7 \beta_{9} - \beta_{10} - \beta_{11} + 2 \beta_{12} - 3 \beta_{14} - \beta_{15} + 4 \beta_{16} - 2 \beta_{18} + \beta_{19} ) q^{73} + ( -5 - \beta_{1} - 4 \beta_{2} - \beta_{4} + \beta_{5} + 5 \beta_{6} - 7 \beta_{8} + 8 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + 8 \beta_{13} + 3 \beta_{15} + 5 \beta_{16} - 4 \beta_{17} - \beta_{18} + 4 \beta_{19} ) q^{75} + ( -6 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - 3 \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} + 4 \beta_{12} - \beta_{13} + 6 \beta_{14} + 5 \beta_{15} - 2 \beta_{16} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{77} + ( -3 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} - 4 \beta_{16} + 4 \beta_{18} - 2 \beta_{19} ) q^{79} + ( -3 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{11} - \beta_{13} - \beta_{16} - 2 \beta_{17} + 3 \beta_{18} ) q^{81} + ( -\beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + 5 \beta_{8} + \beta_{9} - 3 \beta_{10} + 2 \beta_{12} - 5 \beta_{13} + \beta_{17} ) q^{83} + ( -3 - \beta_{1} - 2 \beta_{4} - \beta_{5} + 5 \beta_{6} - \beta_{7} - \beta_{8} + 5 \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} + 7 \beta_{13} + 5 \beta_{14} + 6 \beta_{15} + 2 \beta_{16} - 3 \beta_{17} - \beta_{19} ) q^{85} + ( 3 + 2 \beta_{1} + \beta_{2} + 2 \beta_{4} - 3 \beta_{6} - 2 \beta_{7} + 7 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - 7 \beta_{13} - 7 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} + 2 \beta_{18} ) q^{87} + ( \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} + 2 \beta_{11} - 6 \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} + \beta_{16} - 2 \beta_{17} + 3 \beta_{18} ) q^{89} + ( -4 + \beta_{1} - \beta_{7} - 3 \beta_{8} + 3 \beta_{9} + 6 \beta_{10} + 5 \beta_{11} + 7 \beta_{12} + 5 \beta_{13} + 7 \beta_{14} + 6 \beta_{15} + \beta_{16} - 2 \beta_{17} + 2 \beta_{19} ) q^{91} + ( 1 - \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + \beta_{7} - 3 \beta_{8} + 3 \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{13} + \beta_{15} + 3 \beta_{16} - \beta_{17} + \beta_{19} ) q^{93} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{9} - 2 \beta_{10} - 5 \beta_{11} + 2 \beta_{12} + 3 \beta_{14} - 3 \beta_{15} - \beta_{17} + 3 \beta_{18} - 2 \beta_{19} ) q^{95} + ( 1 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} - 4 \beta_{8} + 4 \beta_{9} + \beta_{10} + 4 \beta_{11} - 2 \beta_{12} + 5 \beta_{13} + 4 \beta_{14} - \beta_{15} + 4 \beta_{16} - 4 \beta_{17} - 2 \beta_{18} ) q^{97} + ( -3 \beta_{1} - \beta_{3} - 3 \beta_{4} + \beta_{5} + 4 \beta_{6} + 2 \beta_{7} - 5 \beta_{8} + \beta_{10} + 5 \beta_{11} + 2 \beta_{12} + 6 \beta_{13} - \beta_{15} + 2 \beta_{16} + \beta_{19} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 2q^{3} + 2q^{5} + 2q^{7} - 4q^{9} + O(q^{10})$$ $$20q + 2q^{3} + 2q^{5} + 2q^{7} - 4q^{9} - 2q^{11} + 6q^{13} - 17q^{15} - 9q^{17} - 11q^{19} - 47q^{21} - 22q^{23} - 16q^{25} - 19q^{27} - q^{29} - 13q^{31} - 5q^{33} + 14q^{35} + 34q^{37} + 30q^{39} + 28q^{41} + 44q^{43} + 78q^{45} + 26q^{47} + 60q^{49} + 62q^{51} + 14q^{53} + 26q^{55} + 3q^{57} - 10q^{59} - 56q^{61} - 27q^{63} - 87q^{65} - 44q^{67} - 51q^{69} - 37q^{71} - 12q^{73} - 53q^{75} - 47q^{77} - 6q^{79} - 10q^{81} - 25q^{83} + 8q^{85} + 48q^{87} + 10q^{89} + 26q^{91} - 14q^{93} + 29q^{95} - q^{97} - q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 9 x^{19} + 51 x^{18} - 200 x^{17} + 633 x^{16} - 1688 x^{15} + 3957 x^{14} - 8161 x^{13} + 14788 x^{12} - 23925 x^{11} + 35080 x^{10} - 43945 x^{9} + 57269 x^{8} - 57348 x^{7} + 52821 x^{6} - 34986 x^{5} + 26231 x^{4} + 96928 x^{3} + 49370 x^{2} + 8165 x + 529$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$26\!\cdots\!24$$$$\nu^{19} +$$$$31\!\cdots\!85$$$$\nu^{18} -$$$$20\!\cdots\!18$$$$\nu^{17} +$$$$93\!\cdots\!10$$$$\nu^{16} -$$$$33\!\cdots\!45$$$$\nu^{15} +$$$$10\!\cdots\!07$$$$\nu^{14} -$$$$25\!\cdots\!57$$$$\nu^{13} +$$$$59\!\cdots\!55$$$$\nu^{12} -$$$$12\!\cdots\!31$$$$\nu^{11} +$$$$21\!\cdots\!26$$$$\nu^{10} -$$$$35\!\cdots\!64$$$$\nu^{9} +$$$$52\!\cdots\!81$$$$\nu^{8} -$$$$70\!\cdots\!20$$$$\nu^{7} +$$$$87\!\cdots\!38$$$$\nu^{6} -$$$$95\!\cdots\!70$$$$\nu^{5} +$$$$90\!\cdots\!49$$$$\nu^{4} -$$$$73\!\cdots\!75$$$$\nu^{3} +$$$$28\!\cdots\!88$$$$\nu^{2} +$$$$38\!\cdots\!77$$$$\nu +$$$$24\!\cdots\!76$$$$)/$$$$20\!\cdots\!31$$ $$\beta_{3}$$ $$=$$ $$($$$$58\!\cdots\!64$$$$\nu^{19} -$$$$48\!\cdots\!94$$$$\nu^{18} +$$$$25\!\cdots\!61$$$$\nu^{17} -$$$$91\!\cdots\!80$$$$\nu^{16} +$$$$26\!\cdots\!08$$$$\nu^{15} -$$$$62\!\cdots\!97$$$$\nu^{14} +$$$$12\!\cdots\!74$$$$\nu^{13} -$$$$22\!\cdots\!54$$$$\nu^{12} +$$$$32\!\cdots\!10$$$$\nu^{11} -$$$$36\!\cdots\!60$$$$\nu^{10} +$$$$26\!\cdots\!61$$$$\nu^{9} +$$$$19\!\cdots\!08$$$$\nu^{8} -$$$$44\!\cdots\!86$$$$\nu^{7} +$$$$17\!\cdots\!70$$$$\nu^{6} -$$$$27\!\cdots\!54$$$$\nu^{5} +$$$$43\!\cdots\!53$$$$\nu^{4} -$$$$38\!\cdots\!58$$$$\nu^{3} +$$$$10\!\cdots\!15$$$$\nu^{2} +$$$$53\!\cdots\!52$$$$\nu +$$$$73\!\cdots\!27$$$$)/$$$$20\!\cdots\!31$$ $$\beta_{4}$$ $$=$$ $$($$$$95\!\cdots\!62$$$$\nu^{19} -$$$$95\!\cdots\!32$$$$\nu^{18} +$$$$57\!\cdots\!81$$$$\nu^{17} -$$$$24\!\cdots\!16$$$$\nu^{16} +$$$$82\!\cdots\!21$$$$\nu^{15} -$$$$23\!\cdots\!22$$$$\nu^{14} +$$$$57\!\cdots\!68$$$$\nu^{13} -$$$$12\!\cdots\!24$$$$\nu^{12} +$$$$24\!\cdots\!29$$$$\nu^{11} -$$$$41\!\cdots\!21$$$$\nu^{10} +$$$$64\!\cdots\!91$$$$\nu^{9} -$$$$88\!\cdots\!82$$$$\nu^{8} +$$$$11\!\cdots\!92$$$$\nu^{7} -$$$$13\!\cdots\!65$$$$\nu^{6} +$$$$13\!\cdots\!34$$$$\nu^{5} -$$$$11\!\cdots\!65$$$$\nu^{4} +$$$$86\!\cdots\!05$$$$\nu^{3} +$$$$44\!\cdots\!65$$$$\nu^{2} -$$$$28\!\cdots\!48$$$$\nu -$$$$48\!\cdots\!81$$$$)/$$$$20\!\cdots\!31$$ $$\beta_{5}$$ $$=$$ $$($$$$10\!\cdots\!69$$$$\nu^{19} -$$$$99\!\cdots\!67$$$$\nu^{18} +$$$$58\!\cdots\!26$$$$\nu^{17} -$$$$24\!\cdots\!22$$$$\nu^{16} +$$$$79\!\cdots\!33$$$$\nu^{15} -$$$$22\!\cdots\!28$$$$\nu^{14} +$$$$53\!\cdots\!79$$$$\nu^{13} -$$$$11\!\cdots\!64$$$$\nu^{12} +$$$$21\!\cdots\!44$$$$\nu^{11} -$$$$37\!\cdots\!89$$$$\nu^{10} +$$$$57\!\cdots\!66$$$$\nu^{9} -$$$$78\!\cdots\!11$$$$\nu^{8} +$$$$10\!\cdots\!09$$$$\nu^{7} -$$$$11\!\cdots\!84$$$$\nu^{6} +$$$$12\!\cdots\!28$$$$\nu^{5} -$$$$10\!\cdots\!90$$$$\nu^{4} +$$$$86\!\cdots\!44$$$$\nu^{3} +$$$$55\!\cdots\!12$$$$\nu^{2} +$$$$20\!\cdots\!18$$$$\nu +$$$$45\!\cdots\!85$$$$)/$$$$20\!\cdots\!31$$ $$\beta_{6}$$ $$=$$ $$($$$$16\!\cdots\!65$$$$\nu^{19} -$$$$17\!\cdots\!00$$$$\nu^{18} +$$$$10\!\cdots\!56$$$$\nu^{17} -$$$$46\!\cdots\!28$$$$\nu^{16} +$$$$16\!\cdots\!58$$$$\nu^{15} -$$$$45\!\cdots\!22$$$$\nu^{14} +$$$$11\!\cdots\!13$$$$\nu^{13} -$$$$25\!\cdots\!41$$$$\nu^{12} +$$$$49\!\cdots\!00$$$$\nu^{11} -$$$$86\!\cdots\!41$$$$\nu^{10} +$$$$13\!\cdots\!49$$$$\nu^{9} -$$$$18\!\cdots\!31$$$$\nu^{8} +$$$$24\!\cdots\!37$$$$\nu^{7} -$$$$29\!\cdots\!34$$$$\nu^{6} +$$$$29\!\cdots\!53$$$$\nu^{5} -$$$$25\!\cdots\!66$$$$\nu^{4} +$$$$19\!\cdots\!87$$$$\nu^{3} +$$$$43\!\cdots\!84$$$$\nu^{2} -$$$$14\!\cdots\!26$$$$\nu -$$$$35\!\cdots\!17$$$$)/$$$$20\!\cdots\!31$$ $$\beta_{7}$$ $$=$$ $$($$$$28\!\cdots\!41$$$$\nu^{19} -$$$$26\!\cdots\!48$$$$\nu^{18} +$$$$15\!\cdots\!15$$$$\nu^{17} -$$$$61\!\cdots\!25$$$$\nu^{16} +$$$$20\!\cdots\!00$$$$\nu^{15} -$$$$54\!\cdots\!58$$$$\nu^{14} +$$$$12\!\cdots\!22$$$$\nu^{13} -$$$$27\!\cdots\!41$$$$\nu^{12} +$$$$50\!\cdots\!91$$$$\nu^{11} -$$$$83\!\cdots\!75$$$$\nu^{10} +$$$$12\!\cdots\!20$$$$\nu^{9} -$$$$16\!\cdots\!90$$$$\nu^{8} +$$$$21\!\cdots\!33$$$$\nu^{7} -$$$$22\!\cdots\!91$$$$\nu^{6} +$$$$21\!\cdots\!02$$$$\nu^{5} -$$$$15\!\cdots\!91$$$$\nu^{4} +$$$$11\!\cdots\!87$$$$\nu^{3} +$$$$24\!\cdots\!69$$$$\nu^{2} +$$$$71\!\cdots\!45$$$$\nu +$$$$87\!\cdots\!95$$$$)/$$$$20\!\cdots\!31$$ $$\beta_{8}$$ $$=$$ $$($$$$30\!\cdots\!51$$$$\nu^{19} -$$$$29\!\cdots\!11$$$$\nu^{18} +$$$$17\!\cdots\!14$$$$\nu^{17} -$$$$75\!\cdots\!84$$$$\nu^{16} +$$$$25\!\cdots\!67$$$$\nu^{15} -$$$$71\!\cdots\!62$$$$\nu^{14} +$$$$17\!\cdots\!61$$$$\nu^{13} -$$$$38\!\cdots\!96$$$$\nu^{12} +$$$$73\!\cdots\!10$$$$\nu^{11} -$$$$12\!\cdots\!46$$$$\nu^{10} +$$$$19\!\cdots\!38$$$$\nu^{9} -$$$$26\!\cdots\!85$$$$\nu^{8} +$$$$35\!\cdots\!68$$$$\nu^{7} -$$$$41\!\cdots\!00$$$$\nu^{6} +$$$$42\!\cdots\!12$$$$\nu^{5} -$$$$36\!\cdots\!32$$$$\nu^{4} +$$$$28\!\cdots\!53$$$$\nu^{3} +$$$$13\!\cdots\!60$$$$\nu^{2} -$$$$59\!\cdots\!10$$$$\nu -$$$$28\!\cdots\!69$$$$)/$$$$20\!\cdots\!31$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$45\!\cdots\!44$$$$\nu^{19} +$$$$40\!\cdots\!72$$$$\nu^{18} -$$$$23\!\cdots\!59$$$$\nu^{17} +$$$$89\!\cdots\!82$$$$\nu^{16} -$$$$28\!\cdots\!42$$$$\nu^{15} +$$$$73\!\cdots\!27$$$$\nu^{14} -$$$$17\!\cdots\!01$$$$\nu^{13} +$$$$34\!\cdots\!27$$$$\nu^{12} -$$$$61\!\cdots\!17$$$$\nu^{11} +$$$$97\!\cdots\!69$$$$\nu^{10} -$$$$13\!\cdots\!94$$$$\nu^{9} +$$$$16\!\cdots\!16$$$$\nu^{8} -$$$$20\!\cdots\!55$$$$\nu^{7} +$$$$19\!\cdots\!92$$$$\nu^{6} -$$$$15\!\cdots\!86$$$$\nu^{5} +$$$$64\!\cdots\!14$$$$\nu^{4} -$$$$29\!\cdots\!15$$$$\nu^{3} -$$$$51\!\cdots\!07$$$$\nu^{2} -$$$$19\!\cdots\!92$$$$\nu +$$$$12\!\cdots\!17$$$$)/$$$$20\!\cdots\!31$$ $$\beta_{10}$$ $$=$$ $$($$$$86\!\cdots\!65$$$$\nu^{19} -$$$$79\!\cdots\!54$$$$\nu^{18} +$$$$45\!\cdots\!82$$$$\nu^{17} -$$$$17\!\cdots\!26$$$$\nu^{16} +$$$$57\!\cdots\!67$$$$\nu^{15} -$$$$15\!\cdots\!53$$$$\nu^{14} +$$$$36\!\cdots\!33$$$$\nu^{13} -$$$$76\!\cdots\!44$$$$\nu^{12} +$$$$13\!\cdots\!84$$$$\nu^{11} -$$$$22\!\cdots\!69$$$$\nu^{10} +$$$$34\!\cdots\!89$$$$\nu^{9} -$$$$43\!\cdots\!91$$$$\nu^{8} +$$$$57\!\cdots\!96$$$$\nu^{7} -$$$$60\!\cdots\!29$$$$\nu^{6} +$$$$57\!\cdots\!49$$$$\nu^{5} -$$$$42\!\cdots\!18$$$$\nu^{4} +$$$$33\!\cdots\!05$$$$\nu^{3} +$$$$75\!\cdots\!76$$$$\nu^{2} +$$$$37\!\cdots\!38$$$$\nu +$$$$50\!\cdots\!07$$$$)/$$$$20\!\cdots\!31$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$91\!\cdots\!89$$$$\nu^{19} +$$$$81\!\cdots\!39$$$$\nu^{18} -$$$$45\!\cdots\!07$$$$\nu^{17} +$$$$17\!\cdots\!19$$$$\nu^{16} -$$$$55\!\cdots\!21$$$$\nu^{15} +$$$$14\!\cdots\!11$$$$\nu^{14} -$$$$34\!\cdots\!51$$$$\nu^{13} +$$$$69\!\cdots\!61$$$$\nu^{12} -$$$$12\!\cdots\!08$$$$\nu^{11} +$$$$19\!\cdots\!96$$$$\nu^{10} -$$$$28\!\cdots\!99$$$$\nu^{9} +$$$$33\!\cdots\!14$$$$\nu^{8} -$$$$43\!\cdots\!59$$$$\nu^{7} +$$$$41\!\cdots\!80$$$$\nu^{6} -$$$$35\!\cdots\!04$$$$\nu^{5} +$$$$18\!\cdots\!20$$$$\nu^{4} -$$$$12\!\cdots\!94$$$$\nu^{3} -$$$$97\!\cdots\!97$$$$\nu^{2} -$$$$49\!\cdots\!95$$$$\nu -$$$$46\!\cdots\!37$$$$)/$$$$20\!\cdots\!31$$ $$\beta_{12}$$ $$=$$ $$($$$$12\!\cdots\!80$$$$\nu^{19} -$$$$11\!\cdots\!25$$$$\nu^{18} +$$$$66\!\cdots\!20$$$$\nu^{17} -$$$$26\!\cdots\!21$$$$\nu^{16} +$$$$83\!\cdots\!81$$$$\nu^{15} -$$$$22\!\cdots\!94$$$$\nu^{14} +$$$$53\!\cdots\!93$$$$\nu^{13} -$$$$11\!\cdots\!78$$$$\nu^{12} +$$$$20\!\cdots\!95$$$$\nu^{11} -$$$$33\!\cdots\!66$$$$\nu^{10} +$$$$49\!\cdots\!08$$$$\nu^{9} -$$$$62\!\cdots\!88$$$$\nu^{8} +$$$$81\!\cdots\!18$$$$\nu^{7} -$$$$84\!\cdots\!13$$$$\nu^{6} +$$$$79\!\cdots\!15$$$$\nu^{5} -$$$$56\!\cdots\!93$$$$\nu^{4} +$$$$42\!\cdots\!94$$$$\nu^{3} +$$$$11\!\cdots\!80$$$$\nu^{2} +$$$$50\!\cdots\!83$$$$\nu +$$$$40\!\cdots\!07$$$$)/$$$$20\!\cdots\!31$$ $$\beta_{13}$$ $$=$$ $$($$$$13\!\cdots\!63$$$$\nu^{19} -$$$$12\!\cdots\!31$$$$\nu^{18} +$$$$71\!\cdots\!07$$$$\nu^{17} -$$$$27\!\cdots\!61$$$$\nu^{16} +$$$$88\!\cdots\!59$$$$\nu^{15} -$$$$23\!\cdots\!52$$$$\nu^{14} +$$$$55\!\cdots\!88$$$$\nu^{13} -$$$$11\!\cdots\!17$$$$\nu^{12} +$$$$20\!\cdots\!98$$$$\nu^{11} -$$$$33\!\cdots\!85$$$$\nu^{10} +$$$$48\!\cdots\!00$$$$\nu^{9} -$$$$61\!\cdots\!96$$$$\nu^{8} +$$$$79\!\cdots\!39$$$$\nu^{7} -$$$$79\!\cdots\!38$$$$\nu^{6} +$$$$71\!\cdots\!53$$$$\nu^{5} -$$$$45\!\cdots\!64$$$$\nu^{4} +$$$$32\!\cdots\!00$$$$\nu^{3} +$$$$13\!\cdots\!22$$$$\nu^{2} +$$$$58\!\cdots\!95$$$$\nu +$$$$59\!\cdots\!43$$$$)/$$$$20\!\cdots\!31$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$16\!\cdots\!55$$$$\nu^{19} +$$$$15\!\cdots\!36$$$$\nu^{18} -$$$$86\!\cdots\!53$$$$\nu^{17} +$$$$34\!\cdots\!15$$$$\nu^{16} -$$$$11\!\cdots\!40$$$$\nu^{15} +$$$$29\!\cdots\!40$$$$\nu^{14} -$$$$70\!\cdots\!93$$$$\nu^{13} +$$$$14\!\cdots\!77$$$$\nu^{12} -$$$$27\!\cdots\!81$$$$\nu^{11} +$$$$44\!\cdots\!66$$$$\nu^{10} -$$$$66\!\cdots\!75$$$$\nu^{9} +$$$$85\!\cdots\!95$$$$\nu^{8} -$$$$11\!\cdots\!85$$$$\nu^{7} +$$$$11\!\cdots\!73$$$$\nu^{6} -$$$$10\!\cdots\!46$$$$\nu^{5} +$$$$79\!\cdots\!32$$$$\nu^{4} -$$$$59\!\cdots\!96$$$$\nu^{3} -$$$$14\!\cdots\!53$$$$\nu^{2} -$$$$57\!\cdots\!81$$$$\nu -$$$$62\!\cdots\!30$$$$)/$$$$20\!\cdots\!31$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$18\!\cdots\!81$$$$\nu^{19} +$$$$17\!\cdots\!42$$$$\nu^{18} -$$$$98\!\cdots\!50$$$$\nu^{17} +$$$$39\!\cdots\!77$$$$\nu^{16} -$$$$12\!\cdots\!01$$$$\nu^{15} +$$$$34\!\cdots\!85$$$$\nu^{14} -$$$$81\!\cdots\!97$$$$\nu^{13} +$$$$17\!\cdots\!74$$$$\nu^{12} -$$$$31\!\cdots\!27$$$$\nu^{11} +$$$$52\!\cdots\!25$$$$\nu^{10} -$$$$78\!\cdots\!48$$$$\nu^{9} +$$$$10\!\cdots\!61$$$$\nu^{8} -$$$$13\!\cdots\!26$$$$\nu^{7} +$$$$13\!\cdots\!95$$$$\nu^{6} -$$$$13\!\cdots\!79$$$$\nu^{5} +$$$$99\!\cdots\!27$$$$\nu^{4} -$$$$74\!\cdots\!23$$$$\nu^{3} -$$$$16\!\cdots\!81$$$$\nu^{2} -$$$$55\!\cdots\!92$$$$\nu -$$$$37\!\cdots\!88$$$$)/$$$$20\!\cdots\!31$$ $$\beta_{16}$$ $$=$$ $$($$$$20\!\cdots\!15$$$$\nu^{19} -$$$$19\!\cdots\!36$$$$\nu^{18} +$$$$11\!\cdots\!09$$$$\nu^{17} -$$$$44\!\cdots\!28$$$$\nu^{16} +$$$$14\!\cdots\!55$$$$\nu^{15} -$$$$39\!\cdots\!95$$$$\nu^{14} +$$$$93\!\cdots\!99$$$$\nu^{13} -$$$$19\!\cdots\!89$$$$\nu^{12} +$$$$36\!\cdots\!86$$$$\nu^{11} -$$$$60\!\cdots\!25$$$$\nu^{10} +$$$$90\!\cdots\!84$$$$\nu^{9} -$$$$11\!\cdots\!00$$$$\nu^{8} +$$$$15\!\cdots\!83$$$$\nu^{7} -$$$$16\!\cdots\!92$$$$\nu^{6} +$$$$15\!\cdots\!60$$$$\nu^{5} -$$$$11\!\cdots\!34$$$$\nu^{4} +$$$$85\!\cdots\!16$$$$\nu^{3} +$$$$17\!\cdots\!22$$$$\nu^{2} +$$$$37\!\cdots\!09$$$$\nu -$$$$37\!\cdots\!16$$$$)/$$$$20\!\cdots\!31$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$22\!\cdots\!50$$$$\nu^{19} +$$$$20\!\cdots\!19$$$$\nu^{18} -$$$$11\!\cdots\!23$$$$\nu^{17} +$$$$47\!\cdots\!88$$$$\nu^{16} -$$$$15\!\cdots\!02$$$$\nu^{15} +$$$$40\!\cdots\!80$$$$\nu^{14} -$$$$96\!\cdots\!24$$$$\nu^{13} +$$$$20\!\cdots\!77$$$$\nu^{12} -$$$$36\!\cdots\!45$$$$\nu^{11} +$$$$60\!\cdots\!43$$$$\nu^{10} -$$$$89\!\cdots\!85$$$$\nu^{9} +$$$$11\!\cdots\!77$$$$\nu^{8} -$$$$15\!\cdots\!87$$$$\nu^{7} +$$$$15\!\cdots\!77$$$$\nu^{6} -$$$$14\!\cdots\!77$$$$\nu^{5} +$$$$10\!\cdots\!25$$$$\nu^{4} -$$$$84\!\cdots\!59$$$$\nu^{3} -$$$$20\!\cdots\!73$$$$\nu^{2} -$$$$93\!\cdots\!14$$$$\nu -$$$$97\!\cdots\!34$$$$)/$$$$20\!\cdots\!31$$ $$\beta_{18}$$ $$=$$ $$($$$$31\!\cdots\!77$$$$\nu^{19} -$$$$29\!\cdots\!42$$$$\nu^{18} +$$$$16\!\cdots\!66$$$$\nu^{17} -$$$$65\!\cdots\!57$$$$\nu^{16} +$$$$20\!\cdots\!33$$$$\nu^{15} -$$$$56\!\cdots\!53$$$$\nu^{14} +$$$$13\!\cdots\!52$$$$\nu^{13} -$$$$27\!\cdots\!02$$$$\nu^{12} +$$$$50\!\cdots\!30$$$$\nu^{11} -$$$$81\!\cdots\!74$$$$\nu^{10} +$$$$12\!\cdots\!58$$$$\nu^{9} -$$$$15\!\cdots\!21$$$$\nu^{8} +$$$$19\!\cdots\!12$$$$\nu^{7} -$$$$20\!\cdots\!34$$$$\nu^{6} +$$$$18\!\cdots\!52$$$$\nu^{5} -$$$$12\!\cdots\!86$$$$\nu^{4} +$$$$90\!\cdots\!75$$$$\nu^{3} +$$$$30\!\cdots\!88$$$$\nu^{2} +$$$$11\!\cdots\!50$$$$\nu +$$$$90\!\cdots\!66$$$$)/$$$$20\!\cdots\!31$$ $$\beta_{19}$$ $$=$$ $$($$$$-$$$$39\!\cdots\!49$$$$\nu^{19} +$$$$36\!\cdots\!97$$$$\nu^{18} -$$$$20\!\cdots\!80$$$$\nu^{17} +$$$$83\!\cdots\!20$$$$\nu^{16} -$$$$26\!\cdots\!98$$$$\nu^{15} +$$$$72\!\cdots\!05$$$$\nu^{14} -$$$$17\!\cdots\!23$$$$\nu^{13} +$$$$36\!\cdots\!50$$$$\nu^{12} -$$$$66\!\cdots\!08$$$$\nu^{11} +$$$$10\!\cdots\!59$$$$\nu^{10} -$$$$16\!\cdots\!39$$$$\nu^{9} +$$$$21\!\cdots\!93$$$$\nu^{8} -$$$$27\!\cdots\!18$$$$\nu^{7} +$$$$29\!\cdots\!46$$$$\nu^{6} -$$$$27\!\cdots\!86$$$$\nu^{5} +$$$$20\!\cdots\!71$$$$\nu^{4} -$$$$15\!\cdots\!06$$$$\nu^{3} -$$$$34\!\cdots\!12$$$$\nu^{2} -$$$$12\!\cdots\!50$$$$\nu -$$$$10\!\cdots\!67$$$$)/$$$$20\!\cdots\!31$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{17} - \beta_{15} - 4 \beta_{12} - \beta_{11} + \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{19} - 5 \beta_{17} - 2 \beta_{15} - 2 \beta_{14} - 7 \beta_{12} - \beta_{11} - 7 \beta_{10} + \beta_{9} - \beta_{6} + \beta_{4} - 1$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{19} - 2 \beta_{18} - 7 \beta_{17} + 2 \beta_{16} - 2 \beta_{15} - 11 \beta_{14} - 3 \beta_{12} - 25 \beta_{10} + 4 \beta_{9} + 2 \beta_{8} - 11 \beta_{6} + 7 \beta_{5} + \beta_{2} - 3 \beta_{1} + 2$$ $$\nu^{5}$$ $$=$$ $$-11 \beta_{18} + 11 \beta_{16} + 12 \beta_{15} + 23 \beta_{13} + 24 \beta_{12} + 24 \beta_{11} - 8 \beta_{10} - 8 \beta_{9} - 7 \beta_{8} - 12 \beta_{6} + 30 \beta_{5} - 12 \beta_{4} - 12 \beta_{1} + 12$$ $$\nu^{6}$$ $$=$$ $$-5 \beta_{19} + 18 \beta_{17} + 18 \beta_{16} + 77 \beta_{15} + 98 \beta_{14} + 93 \beta_{13} + 114 \beta_{12} + 112 \beta_{11} + 130 \beta_{10} - 30 \beta_{9} - 72 \beta_{8} + 78 \beta_{6} + 45 \beta_{5} - 36 \beta_{4} - 5 \beta_{3} - 45 \beta_{2}$$ $$\nu^{7}$$ $$=$$ $$114 \beta_{18} + 19 \beta_{17} - 19 \beta_{16} + 148 \beta_{15} + 227 \beta_{14} - \beta_{13} + 114 \beta_{12} + 50 \beta_{11} + 353 \beta_{10} + 50 \beta_{9} - 227 \beta_{8} - 16 \beta_{7} + 353 \beta_{6} - 8 \beta_{5} + 11 \beta_{4} - 3 \beta_{3} - 197 \beta_{2} + 114 \beta_{1} - 132$$ $$\nu^{8}$$ $$=$$ $$82 \beta_{19} + 342 \beta_{18} - 211 \beta_{17} - 131 \beta_{16} - 102 \beta_{15} - 102 \beta_{14} - 787 \beta_{13} - 656 \beta_{12} - 738 \beta_{11} + 283 \beta_{9} - 174 \beta_{8} - 131 \beta_{7} + 414 \beta_{6} - 131 \beta_{5} + 500 \beta_{4} + 209 \beta_{3} - 369 \beta_{2} + 340 \beta_{1} - 305$$ $$\nu^{9}$$ $$=$$ $$67 \beta_{19} - 86 \beta_{18} - 830 \beta_{17} - 80 \beta_{16} - 1124 \beta_{15} - 2005 \beta_{14} - 2190 \beta_{13} - 2885 \beta_{12} - 2559 \beta_{11} - 3106 \beta_{10} - 166 \beta_{9} + 2055 \beta_{8} - 147 \beta_{7} - 2479 \beta_{6} - 166 \beta_{5} + 1521 \beta_{4} + 830 \beta_{3} + 80 \beta_{2} + 80 \beta_{1} + 977$$ $$\nu^{10}$$ $$=$$ $$-1761 \beta_{19} - 4000 \beta_{18} + 1035 \beta_{16} - 1908 \beta_{15} - 3683 \beta_{14} + 615 \beta_{13} - 3669 \beta_{12} - 2965 \beta_{11} - 8810 \beta_{10} - 3385 \beta_{9} + 8810 \beta_{8} + 1848 \beta_{7} - 11824 \beta_{6} + 55 \beta_{5} + 55 \beta_{3} + 2965 \beta_{2} - 2796 \beta_{1} + 6479$$ $$\nu^{11}$$ $$=$$ $$-6957 \beta_{19} - 11824 \beta_{18} + 6957 \beta_{17} + 3994 \beta_{16} + 3237 \beta_{15} + 11652 \beta_{14} + 23795 \beta_{13} + 11652 \beta_{12} + 11971 \beta_{11} + 3237 \beta_{10} - 6092 \beta_{9} + 6092 \beta_{8} + 9989 \beta_{7} - 11824 \beta_{6} - 11824 \beta_{4} - 8390 \beta_{3} + 8390 \beta_{2} - 9989 \beta_{1} + 7927$$ $$\nu^{12}$$ $$=$$ $$-939 \beta_{19} + 15349 \beta_{17} + 6292 \beta_{16} + 21527 \beta_{15} + 64536 \beta_{14} + 68375 \beta_{13} + 62706 \beta_{12} + 65099 \beta_{11} + 78946 \beta_{10} + 20588 \beta_{9} - 65099 \beta_{8} + 21641 \beta_{7} + 62083 \beta_{6} - 939 \beta_{5} - 32185 \beta_{4} - 25893 \beta_{3} - 12051 \beta_{1} - 41065$$ $$\nu^{13}$$ $$=$$ $$69465 \beta_{19} + 94268 \beta_{18} - 22527 \beta_{17} + 3016 \beta_{16} + 37992 \beta_{15} + 61882 \beta_{14} + 3016 \beta_{13} + 98874 \beta_{12} + 107457 \beta_{11} + 178476 \beta_{10} + 131347 \beta_{9} - 260035 \beta_{8} - 8091 \beta_{7} + 257019 \beta_{6} + 11107 \beta_{5} - 3016 \beta_{4} + 3016 \beta_{3} - 69465 \beta_{2} + 19511 \beta_{1} - 197987$$ $$\nu^{14}$$ $$=$$ $$219027 \beta_{19} + 260035 \beta_{18} - 180672 \beta_{17} + 10817 \beta_{15} - 353220 \beta_{14} - 501879 \beta_{13} - 101178 \beta_{12} - 134193 \beta_{11} - 101178 \beta_{10} + 227454 \beta_{9} - 260035 \beta_{8} - 206244 \beta_{7} + 229844 \beta_{6} + 89650 \beta_{5} + 219027 \beta_{4} + 260035 \beta_{3} - 206244 \beta_{2} + 89650 \beta_{1} - 227454$$ $$\nu^{15}$$ $$=$$ $$-10817 \beta_{19} + 10817 \beta_{18} - 190828 \beta_{17} - 10817 \beta_{16} + 10817 \beta_{15} - 1270139 \beta_{14} - 1345570 \beta_{13} - 668518 \beta_{12} - 1049152 \beta_{11} - 1381010 \beta_{10} - 477690 \beta_{9} + 1334753 \beta_{8} - 570281 \beta_{7} - 1270139 \beta_{6} + 190828 \beta_{5} + 570281 \beta_{4} + 761914 \beta_{3} - 68440 \beta_{2} + 79257 \beta_{1} + 1038335$$ $$\nu^{16}$$ $$=$$ $$-2107484 \beta_{19} - 1840420 \beta_{18} + 1359709 \beta_{17} - 267064 \beta_{16} + 292387 \beta_{15} + 25323 \beta_{13} - 672330 \beta_{12} - 2032039 \beta_{11} - 2010725 \beta_{10} - 3370434 \beta_{9} + 5913241 \beta_{8} - 4815018 \beta_{6} - 524829 \beta_{5} - 152547 \beta_{4} + 1359709 \beta_{2} - 152547 \beta_{1} + 4815018$$ $$\nu^{17}$$ $$=$$ $$-6205628 \beta_{19} - 4662471 \beta_{18} + 5660522 \beta_{17} - 1517834 \beta_{16} - 194412 \beta_{15} + 10748978 \beta_{14} + 9205821 \beta_{13} + 1673060 \beta_{12} + 682551 \beta_{11} + 6343073 \beta_{10} - 5590034 \beta_{9} + 6400040 \beta_{8} + 4662471 \beta_{7} - 2469628 \beta_{6} - 4427345 \beta_{5} - 4142688 \beta_{4} - 6205628 \beta_{3} + 4427345 \beta_{2} + 4662471$$ $$\nu^{18}$$ $$=$$ $$1673060 \beta_{18} + 3385306 \beta_{17} - 3385306 \beta_{16} - 8227854 \beta_{15} + 25425140 \beta_{14} + 20925100 \beta_{13} + 1673060 \beta_{12} + 12040041 \beta_{11} + 27435577 \beta_{10} + 12040041 \beta_{9} - 25425140 \beta_{8} + 13738389 \beta_{7} + 27435577 \beta_{6} - 10964830 \beta_{5} - 7579524 \beta_{4} - 17123695 \beta_{3} + 3459068 \beta_{2} + 1673060 \beta_{1} - 25983466$$ $$\nu^{19}$$ $$=$$ $$50776689 \beta_{19} + 35015101 \beta_{18} - 39163529 \beta_{17} + 4148428 \beta_{16} - 32082975 \beta_{15} - 32082975 \beta_{14} - 16208439 \beta_{13} - 20356867 \beta_{12} + 22591893 \beta_{11} + 77278974 \beta_{9} - 108467221 \beta_{8} + 4148428 \beta_{7} + 73130546 \beta_{6} + 4148428 \beta_{5} + 11936468 \beta_{4} + 1114734 \beta_{3} - 16084896 \beta_{2} - 3033694 \beta_{1} - 104318793$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/92\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$47$$ $$\chi(n)$$ $$-\beta_{13}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 −0.858865 + 1.88065i 1.20400 − 2.63640i −0.420431 − 0.123450i 2.26168 + 0.664090i 1.82483 − 2.10597i −0.967148 + 1.11615i 0.291382 − 2.02660i −0.250875 + 1.74487i −0.858865 − 1.88065i 1.20400 + 2.63640i 1.53106 − 0.983952i −0.115644 + 0.0743196i 0.291382 + 2.02660i −0.250875 − 1.74487i 1.53106 + 0.983952i −0.115644 − 0.0743196i 1.82483 + 2.10597i −0.967148 − 1.11615i −0.420431 + 0.123450i 2.26168 − 0.664090i
0 −1.35392 1.56250i 0 −0.187926 1.30705i 0 1.18148 2.58708i 0 −0.181380 + 1.26153i 0
9.2 0 1.89799 + 2.19040i 0 −0.556197 3.86843i 0 −1.06324 + 2.32817i 0 −0.768534 + 5.34527i 0
13.1 0 −0.368621 0.236898i 0 0.781296 1.71080i 0 4.72847 + 1.38840i 0 −1.16648 2.55425i 0
13.2 0 1.98297 + 1.27438i 0 −0.105327 + 0.230633i 0 −3.93129 1.15433i 0 1.06190 + 2.32523i 0
25.1 0 −0.396574 2.75823i 0 −1.50454 0.441774i 0 −0.107929 + 0.124557i 0 −4.57211 + 1.34249i 0
25.2 0 0.210181 + 1.46184i 0 0.926812 + 0.272136i 0 −1.14874 + 1.32572i 0 0.785668 0.230693i 0
29.1 0 −1.96451 0.576832i 0 3.22799 2.07450i 0 0.267813 1.86268i 0 1.00280 + 0.644459i 0
29.2 0 1.69141 + 0.496642i 0 −0.630070 + 0.404921i 0 −0.0283674 + 0.197300i 0 0.0904475 + 0.0581271i 0
41.1 0 −1.35392 + 1.56250i 0 −0.187926 + 1.30705i 0 1.18148 + 2.58708i 0 −0.181380 1.26153i 0
41.2 0 1.89799 2.19040i 0 −0.556197 + 3.86843i 0 −1.06324 2.32817i 0 −0.768534 5.34527i 0
49.1 0 −0.756044 + 1.65551i 0 −2.72780 + 3.14805i 0 1.70185 1.09371i 0 −0.204514 0.236022i 0
49.2 0 0.0571054 0.125043i 0 1.77577 2.04934i 0 −0.600040 + 0.385622i 0 1.95221 + 2.25297i 0
73.1 0 −1.96451 + 0.576832i 0 3.22799 + 2.07450i 0 0.267813 + 1.86268i 0 1.00280 0.644459i 0
73.2 0 1.69141 0.496642i 0 −0.630070 0.404921i 0 −0.0283674 0.197300i 0 0.0904475 0.0581271i 0
77.1 0 −0.756044 1.65551i 0 −2.72780 3.14805i 0 1.70185 + 1.09371i 0 −0.204514 + 0.236022i 0
77.2 0 0.0571054 + 0.125043i 0 1.77577 + 2.04934i 0 −0.600040 0.385622i 0 1.95221 2.25297i 0
81.1 0 −0.396574 + 2.75823i 0 −1.50454 + 0.441774i 0 −0.107929 0.124557i 0 −4.57211 1.34249i 0
81.2 0 0.210181 1.46184i 0 0.926812 0.272136i 0 −1.14874 1.32572i 0 0.785668 + 0.230693i 0
85.1 0 −0.368621 + 0.236898i 0 0.781296 + 1.71080i 0 4.72847 1.38840i 0 −1.16648 + 2.55425i 0
85.2 0 1.98297 1.27438i 0 −0.105327 0.230633i 0 −3.93129 + 1.15433i 0 1.06190 2.32523i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 85.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 92.2.e.a 20
3.b odd 2 1 828.2.q.a 20
4.b odd 2 1 368.2.m.d 20
23.c even 11 1 inner 92.2.e.a 20
23.c even 11 1 2116.2.a.j 10
23.d odd 22 1 2116.2.a.i 10
69.h odd 22 1 828.2.q.a 20
92.g odd 22 1 368.2.m.d 20
92.g odd 22 1 8464.2.a.ce 10
92.h even 22 1 8464.2.a.cd 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.2.e.a 20 1.a even 1 1 trivial
92.2.e.a 20 23.c even 11 1 inner
368.2.m.d 20 4.b odd 2 1
368.2.m.d 20 92.g odd 22 1
828.2.q.a 20 3.b odd 2 1
828.2.q.a 20 69.h odd 22 1
2116.2.a.i 10 23.d odd 22 1
2116.2.a.j 10 23.c even 11 1
8464.2.a.cd 10 92.h even 22 1
8464.2.a.ce 10 92.g odd 22 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(92, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$529 - 1334 T + 19010 T^{2} + 84506 T^{3} + 122998 T^{4} - 21059 T^{5} + 29325 T^{6} - 27539 T^{7} - 2978 T^{8} - 3696 T^{9} + 3554 T^{10} + 2233 T^{11} + 70 T^{12} - 254 T^{13} + 19 T^{14} + 49 T^{15} + 39 T^{16} - 9 T^{17} + 7 T^{18} - 2 T^{19} + T^{20}$$
$5$ $$14641 + 58564 T + 248897 T^{2} + 99825 T^{3} - 268862 T^{4} - 291731 T^{5} + 141328 T^{6} - 1331 T^{7} + 174185 T^{8} + 94149 T^{9} - 7325 T^{10} + 42971 T^{11} + 13159 T^{12} - 6893 T^{13} + 4552 T^{14} - 637 T^{15} + 324 T^{16} - 63 T^{17} + 15 T^{18} - 2 T^{19} + T^{20}$$
$7$ $$529 + 6141 T + 50924 T^{2} + 230365 T^{3} + 868175 T^{4} + 1379388 T^{5} + 1049881 T^{6} + 267933 T^{7} + 246848 T^{8} + 15433 T^{9} + 61865 T^{10} - 253 T^{11} + 25516 T^{12} + 2199 T^{13} + 2312 T^{14} + 109 T^{15} + 88 T^{16} + 26 T^{17} - 21 T^{18} - 2 T^{19} + T^{20}$$
$11$ $$25715041 - 61191757 T + 50107068 T^{2} + 15969822 T^{3} + 45717672 T^{4} + 46352460 T^{5} + 18462851 T^{6} + 16648423 T^{7} + 11912450 T^{8} + 4698276 T^{9} + 2351339 T^{10} + 1176676 T^{11} + 424768 T^{12} + 128256 T^{13} + 30622 T^{14} + 5180 T^{15} + 610 T^{16} - 25 T^{17} + 4 T^{18} + 2 T^{19} + T^{20}$$
$13$ $$14988860041 + 5771915205 T + 23411469106 T^{2} - 1360958011 T^{3} - 1940989369 T^{4} - 2526000632 T^{5} + 33864247 T^{6} + 278665879 T^{7} + 107902438 T^{8} + 15338807 T^{9} + 5736765 T^{10} + 2036881 T^{11} + 249658 T^{12} + 32903 T^{13} + 28408 T^{14} + 1495 T^{15} - 418 T^{16} + 130 T^{17} + 31 T^{18} - 6 T^{19} + T^{20}$$
$17$ $$139129 + 3532310 T + 34447783 T^{2} + 81693119 T^{3} + 157991042 T^{4} + 137894186 T^{5} + 97349451 T^{6} + 44716701 T^{7} + 6786441 T^{8} + 3704360 T^{9} + 3869625 T^{10} + 569932 T^{11} - 23490 T^{12} + 99556 T^{13} + 19908 T^{14} - 450 T^{15} + 1507 T^{16} + 248 T^{17} + 83 T^{18} + 9 T^{19} + T^{20}$$
$19$ $$1024 + 1588224 T + 666147584 T^{2} - 2562398080 T^{3} + 4259652736 T^{4} - 3527991456 T^{5} + 1906364960 T^{6} - 1000697720 T^{7} + 455573496 T^{8} + 153079278 T^{9} + 79261271 T^{10} + 16453679 T^{11} + 5818497 T^{12} + 1270632 T^{13} + 225784 T^{14} + 38874 T^{15} + 5692 T^{16} + 869 T^{17} + 125 T^{18} + 11 T^{19} + T^{20}$$
$23$ $$41426511213649 + 39625358552186 T + 19029569423283 T^{2} + 5730321227301 T^{3} + 1063489826576 T^{4} + 81844537588 T^{5} - 17290256026 T^{6} - 7858640966 T^{7} - 1745455073 T^{8} - 316825575 T^{9} - 61105010 T^{10} - 13775025 T^{11} - 3299537 T^{12} - 645898 T^{13} - 61786 T^{14} + 12716 T^{15} + 7184 T^{16} + 1683 T^{17} + 243 T^{18} + 22 T^{19} + T^{20}$$
$29$ $$10805664116809 - 15225543735887 T + 6567711178536 T^{2} - 1946236282114 T^{3} + 1313945215379 T^{4} - 231686326132 T^{5} + 255024990189 T^{6} - 8258941525 T^{7} + 18006959169 T^{8} - 1271364710 T^{9} + 411294412 T^{10} - 87387960 T^{11} + 7606358 T^{12} - 1580712 T^{13} + 269073 T^{14} - 4524 T^{15} + 3492 T^{16} - 481 T^{17} - 31 T^{18} + T^{19} + T^{20}$$
$31$ $$2079845655889 + 480887701816 T + 580280033262 T^{2} - 62474051023 T^{3} - 39317033084 T^{4} - 22936155795 T^{5} + 4770214646 T^{6} + 1323765781 T^{7} + 407438277 T^{8} - 139150440 T^{9} + 37334881 T^{10} + 20568636 T^{11} + 7695086 T^{12} + 435472 T^{13} - 140518 T^{14} - 44624 T^{15} - 2153 T^{16} + 431 T^{17} + 129 T^{18} + 13 T^{19} + T^{20}$$
$37$ $$1866439595329 - 1299484337391 T + 1108031337474 T^{2} - 346968234256 T^{3} + 250911844678 T^{4} - 81363417752 T^{5} + 48213433299 T^{6} - 9346729981 T^{7} + 3974715960 T^{8} - 1140149956 T^{9} + 176131649 T^{10} - 5255998 T^{11} - 8741566 T^{12} + 2617608 T^{13} - 205996 T^{14} - 56002 T^{15} + 23584 T^{16} - 4545 T^{17} + 524 T^{18} - 34 T^{19} + T^{20}$$
$41$ $$65806215082609 - 5219862468998 T + 16652988278937 T^{2} - 1949012180244 T^{3} + 1238438322846 T^{4} - 591217533560 T^{5} + 81912387487 T^{6} - 48334151016 T^{7} + 11047666910 T^{8} - 622932431 T^{9} + 955534141 T^{10} - 88254925 T^{11} + 8799360 T^{12} - 4303803 T^{13} + 802144 T^{14} + 2205 T^{15} + 2191 T^{16} - 1761 T^{17} + 316 T^{18} - 28 T^{19} + T^{20}$$
$43$ $$150230641573321 - 287752340085518 T + 368154189431033 T^{2} - 318437232647369 T^{3} + 213233893567841 T^{4} - 113648585605049 T^{5} + 46548300718132 T^{6} - 13427619003887 T^{7} + 2584333087587 T^{8} - 303949250649 T^{9} + 13571949810 T^{10} + 2314108962 T^{11} - 578104115 T^{12} + 58873243 T^{13} - 651967 T^{14} - 821480 T^{15} + 152596 T^{16} - 15675 T^{17} + 1054 T^{18} - 44 T^{19} + T^{20}$$
$47$ $$( -3761152 - 546816 T + 2479040 T^{2} + 713904 T^{3} - 220332 T^{4} - 58453 T^{5} + 8002 T^{6} + 1565 T^{7} - 139 T^{8} - 13 T^{9} + T^{10} )^{2}$$
$53$ $$130889227506489 - 317876883086106 T + 522932778990531 T^{2} - 490308462951273 T^{3} + 313068944951829 T^{4} - 138246585734511 T^{5} + 46071967321944 T^{6} - 11634613369695 T^{7} + 2191724060481 T^{8} - 291696063615 T^{9} + 23039463250 T^{10} + 118744252 T^{11} - 257470115 T^{12} + 20490385 T^{13} + 896371 T^{14} - 203930 T^{15} + 5962 T^{16} + 331 T^{17} + 52 T^{18} - 14 T^{19} + T^{20}$$
$59$ $$6274789532209 + 10487677101528 T + 16427933157746 T^{2} + 8808684531636 T^{3} + 18702985958327 T^{4} + 7966733490386 T^{5} + 5327969333500 T^{6} - 48577411564 T^{7} - 116112034049 T^{8} - 22673012219 T^{9} + 351589272 T^{10} + 158158252 T^{11} + 74316463 T^{12} + 14870371 T^{13} + 1069003 T^{14} - 38578 T^{15} + 3312 T^{16} + 1000 T^{17} + 45 T^{18} + 10 T^{19} + T^{20}$$
$61$ $$791079807551881 + 1716857785687196 T + 2105956655831243 T^{2} + 1835141836807109 T^{3} + 1161533709296499 T^{4} + 542538224484015 T^{5} + 193962481087824 T^{6} + 54307074509111 T^{7} + 12078729029219 T^{8} + 2135785820531 T^{9} + 297791958234 T^{10} + 32131444202 T^{11} + 2663486729 T^{12} + 187398207 T^{13} + 17216035 T^{14} + 2325324 T^{15} + 286132 T^{16} + 25357 T^{17} + 1514 T^{18} + 56 T^{19} + T^{20}$$
$67$ $$14858951082361 + 14796319413073 T + 5835361655568 T^{2} - 1233407508938 T^{3} - 2800332705156 T^{4} - 1340117095119 T^{5} - 76842130977 T^{6} + 286623802619 T^{7} + 219050321807 T^{8} + 91801891632 T^{9} + 27107184677 T^{10} + 6055211140 T^{11} + 1107560217 T^{12} + 168271994 T^{13} + 21163332 T^{14} + 2212782 T^{15} + 200875 T^{16} + 15829 T^{17} + 1010 T^{18} + 44 T^{19} + T^{20}$$
$71$ $$59472089161 + 72288624456 T + 80496754562 T^{2} + 147371389353 T^{3} + 269675062008 T^{4} + 340342245275 T^{5} + 323338838896 T^{6} + 247640197229 T^{7} + 150349325741 T^{8} + 68812284860 T^{9} + 23045270073 T^{10} + 5669919904 T^{11} + 1062258438 T^{12} + 159210718 T^{13} + 20106368 T^{14} + 2200098 T^{15} + 207735 T^{16} + 15819 T^{17} + 895 T^{18} + 37 T^{19} + T^{20}$$
$73$ $$12375058668184081 - 89040444149129452 T + 371077878123752868 T^{2} - 76343125077186190 T^{3} + 22741849037843611 T^{4} - 3232677823081814 T^{5} + 217298932418856 T^{6} - 28229616434018 T^{7} + 1103752859199 T^{8} + 149584260507 T^{9} + 63560087768 T^{10} - 5270465464 T^{11} + 622323351 T^{12} + 36397327 T^{13} - 399595 T^{14} + 55140 T^{15} + 38538 T^{16} + 2192 T^{17} + 35 T^{18} + 12 T^{19} + T^{20}$$
$79$ $$1214604572281 + 238567434588 T + 8533877644537 T^{2} + 7737742530942 T^{3} + 6159050960117 T^{4} + 4149524158858 T^{5} - 2329855884214 T^{6} - 1110301124138 T^{7} + 1206237027650 T^{8} - 497574261690 T^{9} + 120744249264 T^{10} - 17724099162 T^{11} + 1648922244 T^{12} - 72679708 T^{13} + 1323603 T^{14} - 209648 T^{15} + 53691 T^{16} - 2066 T^{17} - 79 T^{18} + 6 T^{19} + T^{20}$$
$83$ $$19165912096321 - 71795921762963 T + 143354036735999 T^{2} - 202834880691703 T^{3} + 188488201124729 T^{4} - 97596830120219 T^{5} + 27405873419663 T^{6} - 4167044556027 T^{7} + 533029132687 T^{8} - 126986101594 T^{9} + 18890970473 T^{10} - 440257686 T^{11} + 166630780 T^{12} - 38698610 T^{13} + 348466 T^{14} + 202784 T^{15} + 94930 T^{16} + 8330 T^{17} + 687 T^{18} + 25 T^{19} + T^{20}$$
$89$ $$153528281766409 - 1148843639916863 T + 5047735006716204 T^{2} + 121830861058962 T^{3} + 855566577597402 T^{4} - 169557864565210 T^{5} + 108590037299755 T^{6} + 26300135001657 T^{7} - 1277565906914 T^{8} + 504835025530 T^{9} + 61170201115 T^{10} - 19599755966 T^{11} + 2833634794 T^{12} - 184496916 T^{13} + 8920088 T^{14} + 243596 T^{15} - 46034 T^{16} + 4071 T^{17} - 32 T^{18} - 10 T^{19} + T^{20}$$
$97$ $$1487827650272161 + 6812702882596556 T + 13140466860655277 T^{2} + 13531510406557847 T^{3} + 7583879461936228 T^{4} + 1877714313737108 T^{5} + 388957002729405 T^{6} + 58251280814133 T^{7} + 7072134575605 T^{8} + 850536497514 T^{9} + 96738561591 T^{10} + 9495784860 T^{11} + 834214140 T^{12} + 57043064 T^{13} + 6014944 T^{14} + 444764 T^{15} + 16809 T^{16} + 188 T^{17} + 221 T^{18} + T^{19} + T^{20}$$