Properties

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

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$2.20387109579$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$2$$ over $$\Q(\zeta_{11})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 8 x^{19} + 43 x^{18} - 165 x^{17} + 538 x^{16} - 1433 x^{15} + 3444 x^{14} - 7370 x^{13} + 15500 x^{12} - 28190 x^{11} + 41920 x^{10} - 33520 x^{9} - 13837 x^{8} + 78980 x^{7} - 92652 x^{6} - 52852 x^{5} + 177374 x^{4} + 151360 x^{3} + 115323 x^{2} + 12834 x + 529$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{17} q^{3} + ( \beta_{3} + \beta_{4} - \beta_{9} - \beta_{10} + \beta_{15} - \beta_{16} - \beta_{18} ) q^{5} + ( \beta_{3} - \beta_{5} - \beta_{9} - \beta_{10} - \beta_{12} + \beta_{15} - \beta_{18} ) q^{7} -\beta_{14} q^{9} +O(q^{10})$$ $$q + \beta_{17} q^{3} + ( \beta_{3} + \beta_{4} - \beta_{9} - \beta_{10} + \beta_{15} - \beta_{16} - \beta_{18} ) q^{5} + ( \beta_{3} - \beta_{5} - \beta_{9} - \beta_{10} - \beta_{12} + \beta_{15} - \beta_{18} ) q^{7} -\beta_{14} q^{9} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} + 2 \beta_{15} - 2 \beta_{16} ) q^{11} + ( -3 + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{11} - \beta_{12} + \beta_{14} + 3 \beta_{16} + \beta_{18} ) q^{13} + ( 1 - \beta_{2} + \beta_{4} + \beta_{11} + \beta_{15} - \beta_{16} - \beta_{18} ) q^{15} + ( -\beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{9} - \beta_{11} - \beta_{12} + 2 \beta_{16} + 2 \beta_{17} - \beta_{18} + \beta_{19} ) q^{17} + ( 2 + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{19} ) q^{19} + ( 1 + \beta_{4} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} + 2 \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{21} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{23} + ( -1 + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{8} + \beta_{10} - 3 \beta_{11} + 2 \beta_{14} - 4 \beta_{15} + 2 \beta_{16} - 2 \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{25} -\beta_{10} q^{27} + ( 3 - \beta_{3} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + 3 \beta_{11} + \beta_{12} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{29} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{10} + 2 \beta_{11} + \beta_{13} - \beta_{14} - 2 \beta_{16} - \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{31} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{9} - \beta_{11} + \beta_{14} - \beta_{16} ) q^{33} + ( -3 + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 3 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{14} - 4 \beta_{15} + 2 \beta_{16} + 3 \beta_{18} - 2 \beta_{19} ) q^{35} + ( -2 - 3 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{5} + \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} + 3 \beta_{14} - 3 \beta_{15} + 2 \beta_{16} + 3 \beta_{18} + \beta_{19} ) q^{37} + ( -2 + \beta_{1} + \beta_{4} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{17} + \beta_{19} ) q^{39} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} + 4 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{41} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{8} - 3 \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} - 2 \beta_{16} - 5 \beta_{17} - \beta_{18} ) q^{43} + ( 1 - \beta_{1} - \beta_{2} + \beta_{11} + \beta_{12} - \beta_{16} + \beta_{17} ) q^{45} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} - 4 \beta_{14} + 3 \beta_{15} - \beta_{16} - \beta_{18} - \beta_{19} ) q^{47} + ( -3 + \beta_{2} - \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - \beta_{6} + 3 \beta_{10} + \beta_{13} + 2 \beta_{14} - 5 \beta_{15} + 3 \beta_{16} - \beta_{17} + 4 \beta_{18} - \beta_{19} ) q^{49} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} + 3 \beta_{11} + \beta_{12} - 3 \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{51} + ( -3 - 2 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{7} + 2 \beta_{9} + 4 \beta_{10} - \beta_{11} - 2 \beta_{12} + 2 \beta_{14} - 4 \beta_{15} + 3 \beta_{16} - 3 \beta_{17} + 3 \beta_{18} + \beta_{19} ) q^{53} + ( -5 - \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 5 \beta_{10} - 6 \beta_{11} - 3 \beta_{12} + 5 \beta_{14} - 3 \beta_{15} + 6 \beta_{16} - 5 \beta_{17} + 5 \beta_{18} ) q^{55} + ( -2 - \beta_{1} + 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} + \beta_{16} + 3 \beta_{18} ) q^{57} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + 3 \beta_{11} - 4 \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{17} + 4 \beta_{18} ) q^{59} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 3 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + 5 \beta_{16} - 3 \beta_{17} - \beta_{18} - \beta_{19} ) q^{61} + ( 1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{16} - \beta_{18} - \beta_{19} ) q^{63} + ( 2 + \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + 6 \beta_{9} + 3 \beta_{10} + 2 \beta_{12} + 2 \beta_{13} - \beta_{14} - 8 \beta_{15} - \beta_{16} - \beta_{17} - \beta_{19} ) q^{65} + ( 5 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{9} - 4 \beta_{10} + 3 \beta_{11} + 4 \beta_{12} - \beta_{13} - \beta_{14} + 3 \beta_{15} - 3 \beta_{16} + 2 \beta_{17} - 3 \beta_{18} ) q^{67} + ( 3 + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} + \beta_{13} - 3 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + 3 \beta_{17} - 2 \beta_{18} ) q^{69} + ( 5 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} - \beta_{11} + 5 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} - 4 \beta_{16} + 2 \beta_{17} - 4 \beta_{18} + 3 \beta_{19} ) q^{71} + ( 1 - 3 \beta_{3} - \beta_{4} - \beta_{7} + \beta_{8} - \beta_{9} + 5 \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{14} + \beta_{15} + \beta_{16} + 2 \beta_{17} - \beta_{18} + \beta_{19} ) q^{73} + ( -3 + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + 4 \beta_{10} - 3 \beta_{11} - 4 \beta_{12} - 2 \beta_{13} + 4 \beta_{14} + 4 \beta_{16} - \beta_{17} + \beta_{18} ) q^{75} + ( -4 + \beta_{6} + 2 \beta_{9} - 3 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - 4 \beta_{15} - \beta_{16} - \beta_{17} + 3 \beta_{18} - 2 \beta_{19} ) q^{77} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{7} - 2 \beta_{8} + 3 \beta_{9} - \beta_{10} - 5 \beta_{11} - 4 \beta_{12} + \beta_{13} + 3 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} - 2 \beta_{17} + 4 \beta_{18} ) q^{79} -\beta_{18} q^{81} + ( -2 + \beta_{1} + \beta_{5} - \beta_{7} + 3 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + 4 \beta_{14} - 2 \beta_{15} - 2 \beta_{16} - 3 \beta_{17} + 2 \beta_{18} ) q^{83} + ( 1 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} + 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{12} - 3 \beta_{13} - \beta_{14} + \beta_{15} - 4 \beta_{16} + \beta_{17} - \beta_{18} ) q^{85} + ( -\beta_{3} - \beta_{4} + \beta_{11} + 2 \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{17} - \beta_{19} ) q^{87} + ( 3 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - 4 \beta_{9} - 5 \beta_{10} + 2 \beta_{11} - 3 \beta_{14} + 4 \beta_{15} - 3 \beta_{16} + 3 \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{89} + ( \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - \beta_{6} - 4 \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} + 7 \beta_{11} + 6 \beta_{12} + 3 \beta_{13} - \beta_{14} - 2 \beta_{15} - 8 \beta_{16} + 7 \beta_{17} - \beta_{18} - \beta_{19} ) q^{91} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{15} + \beta_{16} + \beta_{18} + \beta_{19} ) q^{93} + ( -4 + 3 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - 3 \beta_{8} - 2 \beta_{10} - \beta_{11} + \beta_{13} + 3 \beta_{14} + 4 \beta_{15} - \beta_{16} - 7 \beta_{17} - 4 \beta_{18} ) q^{95} + ( 2 \beta_{1} + \beta_{2} + \beta_{5} - 2 \beta_{6} - \beta_{7} - 4 \beta_{10} + 4 \beta_{11} - \beta_{12} + \beta_{13} - 4 \beta_{14} - 3 \beta_{15} - \beta_{17} - 4 \beta_{18} - \beta_{19} ) q^{97} + ( 1 - \beta_{1} - \beta_{2} + \beta_{8} + \beta_{10} - \beta_{12} - \beta_{15} + \beta_{17} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 2q^{3} - 4q^{5} - 2q^{9} + O(q^{10})$$ $$20q - 2q^{3} - 4q^{5} - 2q^{9} - 22q^{13} + 7q^{15} + 7q^{17} + 19q^{19} + 20q^{23} + 20q^{25} - 2q^{27} + 32q^{29} - 3q^{31} + 11q^{33} - 26q^{35} - 10q^{37} - 22q^{39} - 40q^{41} + 8q^{43} - 4q^{45} - 18q^{47} - 34q^{49} - 26q^{51} - 34q^{53} - 17q^{55} - 3q^{57} - 32q^{59} + 32q^{61} + 49q^{65} + 35q^{67} - 2q^{69} + 33q^{71} - q^{73} - 2q^{75} - 50q^{77} + 22q^{79} - 2q^{81} - 14q^{83} - 9q^{85} - 12q^{87} + 10q^{89} - 72q^{91} + 30q^{93} - 51q^{95} - 4q^{97} + 11q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 8 x^{19} + 43 x^{18} - 165 x^{17} + 538 x^{16} - 1433 x^{15} + 3444 x^{14} - 7370 x^{13} + 15500 x^{12} - 28190 x^{11} + 41920 x^{10} - 33520 x^{9} - 13837 x^{8} + 78980 x^{7} - 92652 x^{6} - 52852 x^{5} + 177374 x^{4} + 151360 x^{3} + 115323 x^{2} + 12834 x + 529$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$17\!\cdots\!28$$$$\nu^{19} +$$$$15\!\cdots\!67$$$$\nu^{18} -$$$$86\!\cdots\!95$$$$\nu^{17} +$$$$35\!\cdots\!61$$$$\nu^{16} -$$$$11\!\cdots\!03$$$$\nu^{15} +$$$$33\!\cdots\!61$$$$\nu^{14} -$$$$85\!\cdots\!69$$$$\nu^{13} +$$$$19\!\cdots\!30$$$$\nu^{12} -$$$$40\!\cdots\!61$$$$\nu^{11} +$$$$78\!\cdots\!34$$$$\nu^{10} -$$$$13\!\cdots\!04$$$$\nu^{9} +$$$$15\!\cdots\!49$$$$\nu^{8} -$$$$82\!\cdots\!06$$$$\nu^{7} -$$$$93\!\cdots\!48$$$$\nu^{6} +$$$$24\!\cdots\!11$$$$\nu^{5} -$$$$97\!\cdots\!60$$$$\nu^{4} -$$$$23\!\cdots\!36$$$$\nu^{3} -$$$$42\!\cdots\!06$$$$\nu^{2} -$$$$19\!\cdots\!98$$$$\nu -$$$$23\!\cdots\!38$$$$)/$$$$18\!\cdots\!47$$ $$\beta_{3}$$ $$=$$ $$($$$$28\!\cdots\!78$$$$\nu^{19} -$$$$23\!\cdots\!79$$$$\nu^{18} +$$$$13\!\cdots\!20$$$$\nu^{17} -$$$$51\!\cdots\!26$$$$\nu^{16} +$$$$17\!\cdots\!57$$$$\nu^{15} -$$$$46\!\cdots\!20$$$$\nu^{14} +$$$$11\!\cdots\!04$$$$\nu^{13} -$$$$24\!\cdots\!58$$$$\nu^{12} +$$$$51\!\cdots\!87$$$$\nu^{11} -$$$$96\!\cdots\!60$$$$\nu^{10} +$$$$14\!\cdots\!28$$$$\nu^{9} -$$$$14\!\cdots\!09$$$$\nu^{8} +$$$$26\!\cdots\!43$$$$\nu^{7} +$$$$22\!\cdots\!11$$$$\nu^{6} -$$$$30\!\cdots\!58$$$$\nu^{5} -$$$$12\!\cdots\!44$$$$\nu^{4} +$$$$64\!\cdots\!97$$$$\nu^{3} +$$$$24\!\cdots\!48$$$$\nu^{2} +$$$$12\!\cdots\!27$$$$\nu -$$$$62\!\cdots\!66$$$$)/$$$$18\!\cdots\!47$$ $$\beta_{4}$$ $$=$$ $$($$$$32\!\cdots\!78$$$$\nu^{19} -$$$$25\!\cdots\!20$$$$\nu^{18} +$$$$13\!\cdots\!66$$$$\nu^{17} -$$$$51\!\cdots\!92$$$$\nu^{16} +$$$$16\!\cdots\!38$$$$\nu^{15} -$$$$44\!\cdots\!86$$$$\nu^{14} +$$$$10\!\cdots\!88$$$$\nu^{13} -$$$$22\!\cdots\!04$$$$\nu^{12} +$$$$48\!\cdots\!38$$$$\nu^{11} -$$$$88\!\cdots\!42$$$$\nu^{10} +$$$$13\!\cdots\!14$$$$\nu^{9} -$$$$11\!\cdots\!75$$$$\nu^{8} -$$$$14\!\cdots\!94$$$$\nu^{7} +$$$$17\!\cdots\!78$$$$\nu^{6} -$$$$17\!\cdots\!39$$$$\nu^{5} -$$$$24\!\cdots\!28$$$$\nu^{4} +$$$$48\!\cdots\!43$$$$\nu^{3} +$$$$67\!\cdots\!90$$$$\nu^{2} +$$$$42\!\cdots\!45$$$$\nu +$$$$22\!\cdots\!33$$$$)/$$$$18\!\cdots\!47$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$39\!\cdots\!69$$$$\nu^{19} +$$$$30\!\cdots\!46$$$$\nu^{18} -$$$$16\!\cdots\!59$$$$\nu^{17} +$$$$61\!\cdots\!59$$$$\nu^{16} -$$$$19\!\cdots\!92$$$$\nu^{15} +$$$$51\!\cdots\!83$$$$\nu^{14} -$$$$12\!\cdots\!75$$$$\nu^{13} +$$$$25\!\cdots\!26$$$$\nu^{12} -$$$$52\!\cdots\!23$$$$\nu^{11} +$$$$92\!\cdots\!49$$$$\nu^{10} -$$$$12\!\cdots\!95$$$$\nu^{9} +$$$$68\!\cdots\!61$$$$\nu^{8} +$$$$14\!\cdots\!67$$$$\nu^{7} -$$$$37\!\cdots\!12$$$$\nu^{6} +$$$$36\!\cdots\!58$$$$\nu^{5} +$$$$30\!\cdots\!36$$$$\nu^{4} -$$$$75\!\cdots\!88$$$$\nu^{3} -$$$$69\!\cdots\!01$$$$\nu^{2} -$$$$33\!\cdots\!76$$$$\nu -$$$$10\!\cdots\!84$$$$)/$$$$18\!\cdots\!47$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$39\!\cdots\!27$$$$\nu^{19} +$$$$33\!\cdots\!00$$$$\nu^{18} -$$$$18\!\cdots\!73$$$$\nu^{17} +$$$$72\!\cdots\!73$$$$\nu^{16} -$$$$24\!\cdots\!57$$$$\nu^{15} +$$$$66\!\cdots\!76$$$$\nu^{14} -$$$$16\!\cdots\!69$$$$\nu^{13} +$$$$35\!\cdots\!21$$$$\nu^{12} -$$$$75\!\cdots\!05$$$$\nu^{11} +$$$$14\!\cdots\!71$$$$\nu^{10} -$$$$22\!\cdots\!60$$$$\nu^{9} +$$$$21\!\cdots\!33$$$$\nu^{8} -$$$$28\!\cdots\!24$$$$\nu^{7} -$$$$31\!\cdots\!92$$$$\nu^{6} +$$$$52\!\cdots\!07$$$$\nu^{5} -$$$$35\!\cdots\!45$$$$\nu^{4} -$$$$64\!\cdots\!38$$$$\nu^{3} -$$$$27\!\cdots\!02$$$$\nu^{2} -$$$$34\!\cdots\!72$$$$\nu -$$$$14\!\cdots\!85$$$$)/$$$$18\!\cdots\!47$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$44\!\cdots\!45$$$$\nu^{19} +$$$$35\!\cdots\!52$$$$\nu^{18} -$$$$18\!\cdots\!95$$$$\nu^{17} +$$$$72\!\cdots\!61$$$$\nu^{16} -$$$$23\!\cdots\!41$$$$\nu^{15} +$$$$61\!\cdots\!97$$$$\nu^{14} -$$$$14\!\cdots\!04$$$$\nu^{13} +$$$$31\!\cdots\!12$$$$\nu^{12} -$$$$65\!\cdots\!64$$$$\nu^{11} +$$$$11\!\cdots\!52$$$$\nu^{10} -$$$$17\!\cdots\!95$$$$\nu^{9} +$$$$12\!\cdots\!38$$$$\nu^{8} +$$$$10\!\cdots\!13$$$$\nu^{7} -$$$$39\!\cdots\!56$$$$\nu^{6} +$$$$43\!\cdots\!91$$$$\nu^{5} +$$$$26\!\cdots\!06$$$$\nu^{4} -$$$$81\!\cdots\!56$$$$\nu^{3} -$$$$71\!\cdots\!18$$$$\nu^{2} -$$$$34\!\cdots\!58$$$$\nu -$$$$28\!\cdots\!56$$$$)/$$$$18\!\cdots\!47$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$48\!\cdots\!89$$$$\nu^{19} +$$$$39\!\cdots\!13$$$$\nu^{18} -$$$$21\!\cdots\!50$$$$\nu^{17} +$$$$82\!\cdots\!99$$$$\nu^{16} -$$$$27\!\cdots\!01$$$$\nu^{15} +$$$$72\!\cdots\!06$$$$\nu^{14} -$$$$17\!\cdots\!74$$$$\nu^{13} +$$$$37\!\cdots\!61$$$$\nu^{12} -$$$$79\!\cdots\!48$$$$\nu^{11} +$$$$14\!\cdots\!05$$$$\nu^{10} -$$$$21\!\cdots\!17$$$$\nu^{9} +$$$$18\!\cdots\!34$$$$\nu^{8} +$$$$59\!\cdots\!66$$$$\nu^{7} -$$$$41\!\cdots\!73$$$$\nu^{6} +$$$$50\!\cdots\!13$$$$\nu^{5} +$$$$23\!\cdots\!87$$$$\nu^{4} -$$$$10\!\cdots\!40$$$$\nu^{3} -$$$$52\!\cdots\!60$$$$\nu^{2} -$$$$39\!\cdots\!32$$$$\nu -$$$$25\!\cdots\!91$$$$)/$$$$18\!\cdots\!47$$ $$\beta_{9}$$ $$=$$ $$($$$$11\!\cdots\!54$$$$\nu^{19} -$$$$91\!\cdots\!54$$$$\nu^{18} +$$$$48\!\cdots\!43$$$$\nu^{17} -$$$$18\!\cdots\!90$$$$\nu^{16} +$$$$58\!\cdots\!26$$$$\nu^{15} -$$$$15\!\cdots\!25$$$$\nu^{14} +$$$$36\!\cdots\!56$$$$\nu^{13} -$$$$75\!\cdots\!76$$$$\nu^{12} +$$$$15\!\cdots\!42$$$$\nu^{11} -$$$$28\!\cdots\!73$$$$\nu^{10} +$$$$39\!\cdots\!20$$$$\nu^{9} -$$$$24\!\cdots\!52$$$$\nu^{8} -$$$$30\!\cdots\!07$$$$\nu^{7} +$$$$93\!\cdots\!63$$$$\nu^{6} -$$$$87\!\cdots\!97$$$$\nu^{5} -$$$$92\!\cdots\!66$$$$\nu^{4} +$$$$19\!\cdots\!52$$$$\nu^{3} +$$$$24\!\cdots\!37$$$$\nu^{2} +$$$$16\!\cdots\!90$$$$\nu +$$$$27\!\cdots\!63$$$$)/$$$$18\!\cdots\!47$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$19\!\cdots\!96$$$$\nu^{19} +$$$$16\!\cdots\!37$$$$\nu^{18} -$$$$87\!\cdots\!74$$$$\nu^{17} +$$$$33\!\cdots\!99$$$$\nu^{16} -$$$$11\!\cdots\!07$$$$\nu^{15} +$$$$30\!\cdots\!60$$$$\nu^{14} -$$$$72\!\cdots\!07$$$$\nu^{13} +$$$$15\!\cdots\!95$$$$\nu^{12} -$$$$32\!\cdots\!26$$$$\nu^{11} +$$$$60\!\cdots\!63$$$$\nu^{10} -$$$$91\!\cdots\!69$$$$\nu^{9} +$$$$78\!\cdots\!15$$$$\nu^{8} +$$$$20\!\cdots\!91$$$$\nu^{7} -$$$$16\!\cdots\!47$$$$\nu^{6} +$$$$21\!\cdots\!04$$$$\nu^{5} +$$$$67\!\cdots\!34$$$$\nu^{4} -$$$$37\!\cdots\!40$$$$\nu^{3} -$$$$22\!\cdots\!72$$$$\nu^{2} -$$$$15\!\cdots\!07$$$$\nu +$$$$88\!\cdots\!12$$$$)/$$$$18\!\cdots\!47$$ $$\beta_{11}$$ $$=$$ $$($$$$24\!\cdots\!93$$$$\nu^{19} -$$$$19\!\cdots\!65$$$$\nu^{18} +$$$$10\!\cdots\!97$$$$\nu^{17} -$$$$40\!\cdots\!91$$$$\nu^{16} +$$$$13\!\cdots\!17$$$$\nu^{15} -$$$$35\!\cdots\!08$$$$\nu^{14} +$$$$85\!\cdots\!30$$$$\nu^{13} -$$$$18\!\cdots\!44$$$$\nu^{12} +$$$$38\!\cdots\!11$$$$\nu^{11} -$$$$70\!\cdots\!17$$$$\nu^{10} +$$$$10\!\cdots\!09$$$$\nu^{9} -$$$$84\!\cdots\!68$$$$\nu^{8} -$$$$33\!\cdots\!86$$$$\nu^{7} +$$$$19\!\cdots\!44$$$$\nu^{6} -$$$$23\!\cdots\!26$$$$\nu^{5} -$$$$11\!\cdots\!44$$$$\nu^{4} +$$$$43\!\cdots\!82$$$$\nu^{3} +$$$$34\!\cdots\!24$$$$\nu^{2} +$$$$29\!\cdots\!43$$$$\nu +$$$$33\!\cdots\!91$$$$)/$$$$18\!\cdots\!47$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$27\!\cdots\!65$$$$\nu^{19} +$$$$22\!\cdots\!47$$$$\nu^{18} -$$$$12\!\cdots\!95$$$$\nu^{17} +$$$$47\!\cdots\!98$$$$\nu^{16} -$$$$15\!\cdots\!43$$$$\nu^{15} +$$$$42\!\cdots\!02$$$$\nu^{14} -$$$$10\!\cdots\!36$$$$\nu^{13} +$$$$22\!\cdots\!19$$$$\nu^{12} -$$$$46\!\cdots\!21$$$$\nu^{11} +$$$$86\!\cdots\!55$$$$\nu^{10} -$$$$13\!\cdots\!71$$$$\nu^{9} +$$$$11\!\cdots\!60$$$$\nu^{8} +$$$$16\!\cdots\!72$$$$\nu^{7} -$$$$21\!\cdots\!76$$$$\nu^{6} +$$$$29\!\cdots\!72$$$$\nu^{5} +$$$$95\!\cdots\!73$$$$\nu^{4} -$$$$49\!\cdots\!65$$$$\nu^{3} -$$$$35\!\cdots\!62$$$$\nu^{2} -$$$$29\!\cdots\!93$$$$\nu -$$$$15\!\cdots\!38$$$$)/$$$$18\!\cdots\!47$$ $$\beta_{13}$$ $$=$$ $$($$$$32\!\cdots\!22$$$$\nu^{19} -$$$$24\!\cdots\!27$$$$\nu^{18} +$$$$13\!\cdots\!07$$$$\nu^{17} -$$$$49\!\cdots\!24$$$$\nu^{16} +$$$$15\!\cdots\!92$$$$\nu^{15} -$$$$41\!\cdots\!17$$$$\nu^{14} +$$$$98\!\cdots\!78$$$$\nu^{13} -$$$$20\!\cdots\!68$$$$\nu^{12} +$$$$43\!\cdots\!49$$$$\nu^{11} -$$$$78\!\cdots\!11$$$$\nu^{10} +$$$$11\!\cdots\!50$$$$\nu^{9} -$$$$73\!\cdots\!27$$$$\nu^{8} -$$$$70\!\cdots\!23$$$$\nu^{7} +$$$$23\!\cdots\!01$$$$\nu^{6} -$$$$21\!\cdots\!32$$$$\nu^{5} -$$$$26\!\cdots\!49$$$$\nu^{4} +$$$$54\!\cdots\!24$$$$\nu^{3} +$$$$64\!\cdots\!16$$$$\nu^{2} +$$$$44\!\cdots\!56$$$$\nu +$$$$75\!\cdots\!61$$$$)/$$$$18\!\cdots\!47$$ $$\beta_{14}$$ $$=$$ $$($$$$39\!\cdots\!53$$$$\nu^{19} -$$$$31\!\cdots\!35$$$$\nu^{18} +$$$$17\!\cdots\!54$$$$\nu^{17} -$$$$65\!\cdots\!29$$$$\nu^{16} +$$$$21\!\cdots\!18$$$$\nu^{15} -$$$$56\!\cdots\!04$$$$\nu^{14} +$$$$13\!\cdots\!80$$$$\nu^{13} -$$$$29\!\cdots\!59$$$$\nu^{12} +$$$$61\!\cdots\!28$$$$\nu^{11} -$$$$11\!\cdots\!75$$$$\nu^{10} +$$$$16\!\cdots\!86$$$$\nu^{9} -$$$$13\!\cdots\!73$$$$\nu^{8} -$$$$57\!\cdots\!44$$$$\nu^{7} +$$$$31\!\cdots\!03$$$$\nu^{6} -$$$$36\!\cdots\!38$$$$\nu^{5} -$$$$22\!\cdots\!55$$$$\nu^{4} +$$$$72\!\cdots\!84$$$$\nu^{3} +$$$$59\!\cdots\!52$$$$\nu^{2} +$$$$42\!\cdots\!20$$$$\nu +$$$$36\!\cdots\!49$$$$)/$$$$18\!\cdots\!47$$ $$\beta_{15}$$ $$=$$ $$($$$$42\!\cdots\!77$$$$\nu^{19} -$$$$34\!\cdots\!94$$$$\nu^{18} +$$$$18\!\cdots\!31$$$$\nu^{17} -$$$$71\!\cdots\!71$$$$\nu^{16} +$$$$23\!\cdots\!18$$$$\nu^{15} -$$$$62\!\cdots\!79$$$$\nu^{14} +$$$$15\!\cdots\!74$$$$\nu^{13} -$$$$32\!\cdots\!78$$$$\nu^{12} +$$$$68\!\cdots\!04$$$$\nu^{11} -$$$$12\!\cdots\!68$$$$\nu^{10} +$$$$18\!\cdots\!82$$$$\nu^{9} -$$$$15\!\cdots\!54$$$$\nu^{8} -$$$$47\!\cdots\!74$$$$\nu^{7} +$$$$33\!\cdots\!54$$$$\nu^{6} -$$$$41\!\cdots\!82$$$$\nu^{5} -$$$$20\!\cdots\!65$$$$\nu^{4} +$$$$77\!\cdots\!26$$$$\nu^{3} +$$$$59\!\cdots\!77$$$$\nu^{2} +$$$$42\!\cdots\!81$$$$\nu +$$$$12\!\cdots\!73$$$$)/$$$$18\!\cdots\!47$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$43\!\cdots\!22$$$$\nu^{19} +$$$$35\!\cdots\!04$$$$\nu^{18} -$$$$18\!\cdots\!13$$$$\nu^{17} +$$$$73\!\cdots\!25$$$$\nu^{16} -$$$$23\!\cdots\!97$$$$\nu^{15} +$$$$63\!\cdots\!29$$$$\nu^{14} -$$$$15\!\cdots\!29$$$$\nu^{13} +$$$$33\!\cdots\!09$$$$\nu^{12} -$$$$69\!\cdots\!30$$$$\nu^{11} +$$$$12\!\cdots\!41$$$$\nu^{10} -$$$$19\!\cdots\!74$$$$\nu^{9} +$$$$15\!\cdots\!44$$$$\nu^{8} +$$$$45\!\cdots\!65$$$$\nu^{7} -$$$$33\!\cdots\!54$$$$\nu^{6} +$$$$41\!\cdots\!92$$$$\nu^{5} +$$$$20\!\cdots\!33$$$$\nu^{4} -$$$$76\!\cdots\!68$$$$\nu^{3} -$$$$63\!\cdots\!84$$$$\nu^{2} -$$$$50\!\cdots\!00$$$$\nu -$$$$37\!\cdots\!50$$$$)/$$$$18\!\cdots\!47$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$47\!\cdots\!79$$$$\nu^{19} +$$$$38\!\cdots\!21$$$$\nu^{18} -$$$$20\!\cdots\!10$$$$\nu^{17} +$$$$80\!\cdots\!85$$$$\nu^{16} -$$$$26\!\cdots\!01$$$$\nu^{15} +$$$$70\!\cdots\!08$$$$\nu^{14} -$$$$17\!\cdots\!82$$$$\nu^{13} +$$$$36\!\cdots\!04$$$$\nu^{12} -$$$$77\!\cdots\!61$$$$\nu^{11} +$$$$14\!\cdots\!58$$$$\nu^{10} -$$$$21\!\cdots\!85$$$$\nu^{9} +$$$$18\!\cdots\!97$$$$\nu^{8} +$$$$47\!\cdots\!89$$$$\nu^{7} -$$$$38\!\cdots\!86$$$$\nu^{6} +$$$$48\!\cdots\!81$$$$\nu^{5} +$$$$20\!\cdots\!95$$$$\nu^{4} -$$$$86\!\cdots\!33$$$$\nu^{3} -$$$$61\!\cdots\!00$$$$\nu^{2} -$$$$49\!\cdots\!57$$$$\nu -$$$$21\!\cdots\!54$$$$)/$$$$18\!\cdots\!47$$ $$\beta_{18}$$ $$=$$ $$($$$$-$$$$55\!\cdots\!03$$$$\nu^{19} +$$$$44\!\cdots\!46$$$$\nu^{18} -$$$$24\!\cdots\!99$$$$\nu^{17} +$$$$92\!\cdots\!19$$$$\nu^{16} -$$$$30\!\cdots\!38$$$$\nu^{15} +$$$$80\!\cdots\!24$$$$\nu^{14} -$$$$19\!\cdots\!03$$$$\nu^{13} +$$$$41\!\cdots\!24$$$$\nu^{12} -$$$$87\!\cdots\!77$$$$\nu^{11} +$$$$15\!\cdots\!44$$$$\nu^{10} -$$$$23\!\cdots\!40$$$$\nu^{9} +$$$$19\!\cdots\!64$$$$\nu^{8} +$$$$75\!\cdots\!14$$$$\nu^{7} -$$$$44\!\cdots\!18$$$$\nu^{6} +$$$$52\!\cdots\!52$$$$\nu^{5} +$$$$28\!\cdots\!50$$$$\nu^{4} -$$$$99\!\cdots\!40$$$$\nu^{3} -$$$$82\!\cdots\!42$$$$\nu^{2} -$$$$61\!\cdots\!87$$$$\nu -$$$$32\!\cdots\!97$$$$)/$$$$20\!\cdots\!83$$ $$\beta_{19}$$ $$=$$ $$($$$$39\!\cdots\!80$$$$\nu^{19} -$$$$32\!\cdots\!71$$$$\nu^{18} +$$$$17\!\cdots\!39$$$$\nu^{17} -$$$$66\!\cdots\!01$$$$\nu^{16} +$$$$21\!\cdots\!27$$$$\nu^{15} -$$$$57\!\cdots\!64$$$$\nu^{14} +$$$$13\!\cdots\!04$$$$\nu^{13} -$$$$29\!\cdots\!54$$$$\nu^{12} +$$$$62\!\cdots\!00$$$$\nu^{11} -$$$$11\!\cdots\!58$$$$\nu^{10} +$$$$17\!\cdots\!93$$$$\nu^{9} -$$$$13\!\cdots\!87$$$$\nu^{8} -$$$$50\!\cdots\!93$$$$\nu^{7} +$$$$31\!\cdots\!08$$$$\nu^{6} -$$$$38\!\cdots\!80$$$$\nu^{5} -$$$$19\!\cdots\!63$$$$\nu^{4} +$$$$70\!\cdots\!84$$$$\nu^{3} +$$$$58\!\cdots\!08$$$$\nu^{2} +$$$$45\!\cdots\!57$$$$\nu +$$$$34\!\cdots\!99$$$$)/$$$$62\!\cdots\!49$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{18} + 4 \beta_{17} - \beta_{16} + \beta_{15} + 2 \beta_{11} - \beta_{10} - 2 \beta_{9} - \beta_{8} - \beta_{6} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{19} + 2 \beta_{18} + 2 \beta_{16} + \beta_{14} - \beta_{13} - 2 \beta_{12} - \beta_{11} + 2 \beta_{10} + 3 \beta_{9} - 5 \beta_{8} - \beta_{5} + \beta_{4} - \beta_{2} - \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{19} - 3 \beta_{18} - 2 \beta_{17} - 3 \beta_{16} + 13 \beta_{15} - 29 \beta_{14} + 6 \beta_{13} + 13 \beta_{12} + 3 \beta_{11} - 4 \beta_{10} - 6 \beta_{9} - 6 \beta_{8} - 7 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - 2 \beta_{2} - \beta_{1} + 5$$ $$\nu^{5}$$ $$=$$ $$-\beta_{19} - 11 \beta_{18} + 10 \beta_{17} - 24 \beta_{16} - 58 \beta_{14} + 41 \beta_{13} + 23 \beta_{12} + 7 \beta_{11} - 58 \beta_{10} - 61 \beta_{9} - \beta_{8} - 11 \beta_{7} - 10 \beta_{6} + \beta_{5} - 10 \beta_{4} - 11 \beta_{3} + \beta_{1} + 6$$ $$\nu^{6}$$ $$=$$ $$-15 \beta_{19} + 92 \beta_{18} - 54 \beta_{17} - 15 \beta_{16} - 86 \beta_{15} + 44 \beta_{14} + 63 \beta_{13} - 59 \beta_{12} - 39 \beta_{11} - 124 \beta_{10} + 23 \beta_{9} - 15 \beta_{8} - 15 \beta_{7} - 27 \beta_{6} + 63 \beta_{5} - 15 \beta_{4} - 76 \beta_{3} + 27 \beta_{2} - 59$$ $$\nu^{7}$$ $$=$$ $$2 \beta_{19} + 81 \beta_{18} - 180 \beta_{17} + 216 \beta_{16} - 39 \beta_{15} + 127 \beta_{14} - 2 \beta_{13} - 180 \beta_{12} - 127 \beta_{11} + 81 \beta_{10} + 216 \beta_{9} - 69 \beta_{7} + 20 \beta_{6} + 275 \beta_{5} - 80 \beta_{4} - 80 \beta_{3} + 69 \beta_{2} - 20 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-332 \beta_{19} - 1181 \beta_{18} - 190 \beta_{16} - 142 \beta_{15} - 95 \beta_{14} + 95 \beta_{12} + 4 \beta_{11} + 328 \beta_{10} - 446 \beta_{9} + 162 \beta_{8} - 107 \beta_{7} + 162 \beta_{6} + 332 \beta_{5} - 454 \beta_{4} + 9 \beta_{3} - 9 \beta_{2} + 107 \beta_{1} + 446$$ $$\nu^{9}$$ $$=$$ $$-1972 \beta_{19} - 2836 \beta_{18} + 1640 \beta_{17} - 3235 \beta_{16} - 203 \beta_{15} - 45 \beta_{14} + 45 \beta_{13} + 571 \beta_{12} + 2836 \beta_{11} - 616 \beta_{10} - 1685 \beta_{9} + 212 \beta_{8} + 212 \beta_{7} - 45 \beta_{6} - 640 \beta_{4} - 549 \beta_{3} + 640 \beta_{2} + 549 \beta_{1} - 203$$ $$\nu^{10}$$ $$=$$ $$3473 \beta_{18} + 2239 \beta_{17} + 1891 \beta_{16} + 2239 \beta_{15} + 5064 \beta_{14} - 3855 \beta_{13} - 3473 \beta_{12} + 5039 \beta_{11} + 3855 \beta_{10} + 5064 \beta_{9} + 1454 \beta_{8} + 3855 \beta_{7} + 2311 \beta_{6} - 2311 \beta_{5} + 2490 \beta_{4} + 1454 \beta_{3} + 1039 \beta_{2} - 2490 \beta_{1} - 1891$$ $$\nu^{11}$$ $$=$$ $$14352 \beta_{19} + 14352 \beta_{18} - 6649 \beta_{17} + 21001 \beta_{16} + 8645 \beta_{15} + 7806 \beta_{14} - 16451 \beta_{13} + 1768 \beta_{12} - 12584 \beta_{11} + 21793 \beta_{10} + 21793 \beta_{9} + 10094 \beta_{8} + 13698 \beta_{7} + 14352 \beta_{6} - 13698 \beta_{5} + 10094 \beta_{4} + 16451 \beta_{3} - 9420 \beta_{2} - 9420 \beta_{1} + 5425$$ $$\nu^{12}$$ $$=$$ $$16140 \beta_{19} - 12711 \beta_{18} + 13271 \beta_{17} - 26763 \beta_{16} + 15857 \beta_{15} - 33309 \beta_{14} - 9594 \beta_{13} + 63071 \beta_{12} - 16140 \beta_{11} - 6263 \beta_{10} - 19257 \beta_{9} + 18511 \beta_{8} + 16140 \beta_{7} - 29938 \beta_{5} + 9594 \beta_{4} + 29938 \beta_{3} - 18511 \beta_{2} + 8589 \beta_{1} + 13271$$ $$\nu^{13}$$ $$=$$ $$-7830 \beta_{19} + 49326 \beta_{18} + 32552 \beta_{17} - 66024 \beta_{16} - 57156 \beta_{15} + 32974 \beta_{14} + 33309 \beta_{13} - 1853 \beta_{11} - 91503 \beta_{10} - 23626 \beta_{9} - 38000 \beta_{8} - 106311 \beta_{6} - 18962 \beta_{5} + 18962 \beta_{4} - 7830 \beta_{3} + 33309 \beta_{2} + 38000 \beta_{1} - 66283$$ $$\nu^{14}$$ $$=$$ $$118706 \beta_{19} + 481841 \beta_{18} - 365341 \beta_{17} + 481841 \beta_{16} - 235616 \beta_{15} + 391771 \beta_{14} - 26430 \beta_{13} - 475963 \beta_{12} - 354322 \beta_{11} + 175573 \beta_{10} + 502393 \beta_{9} - 183407 \beta_{8} - 112855 \beta_{7} - 112855 \beta_{6} + 48915 \beta_{5} + 118706 \beta_{4} + 48915 \beta_{2} - 26430 \beta_{1} - 149143$$ $$\nu^{15}$$ $$=$$ $$246225 \beta_{19} - 234914 \beta_{18} - 521536 \beta_{17} + 699115 \beta_{16} - 43151 \beta_{15} - 496051 \beta_{14} + 209898 \beta_{13} - 43151 \beta_{12} - 699115 \beta_{11} + 311638 \beta_{10} - 209898 \beta_{9} - 209898 \beta_{8} - 714524 \beta_{7} + 84131 \beta_{6} + 178749 \beta_{5} + 84131 \beta_{3} - 246225 \beta_{2} + 178749 \beta_{1} + 481139$$ $$\nu^{16}$$ $$=$$ $$-1359003 \beta_{19} - 3875898 \beta_{18} + 2516895 \beta_{17} - 3358624 \beta_{16} - 3284227 \beta_{14} + 2365813 \beta_{13} + 1999621 \beta_{12} + 2174313 \beta_{11} - 3284227 \beta_{10} - 6007240 \beta_{9} - 728320 \beta_{8} - 1785631 \beta_{7} - 996807 \beta_{6} + 728320 \beta_{5} - 996807 \beta_{4} - 1785631 \beta_{3} + 1359003 \beta_{1} + 815310$$ $$\nu^{17}$$ $$=$$ $$-3426712 \beta_{19} + 593231 \beta_{18} + 4013236 \beta_{17} - 3426712 \beta_{16} + 84365 \beta_{15} + 396078 \beta_{14} + 3623865 \beta_{13} - 3472966 \beta_{12} + 7439948 \beta_{11} - 6203736 \beta_{10} - 3708230 \beta_{9} - 3426712 \beta_{8} - 1336445 \beta_{7} - 2181118 \beta_{6} + 3623865 \beta_{5} - 1336445 \beta_{4} - 7404189 \beta_{3} + 2181118 \beta_{2} - 3472966$$ $$\nu^{18}$$ $$=$$ $$5253987 \beta_{19} + 14241220 \beta_{18} - 10610971 \beta_{17} + 21841851 \beta_{16} + 11560191 \beta_{15} + 2296270 \beta_{14} - 5253987 \beta_{13} - 10610971 \beta_{12} - 2296270 \beta_{11} + 14241220 \beta_{10} + 21841851 \beta_{9} + 2676791 \beta_{7} + 10912914 \beta_{6} + 7610724 \beta_{5} - 2827798 \beta_{4} - 2827798 \beta_{3} - 2676791 \beta_{2} - 10912914 \beta_{1}$$ $$\nu^{19}$$ $$=$$ $$-1170363 \beta_{19} - 50581146 \beta_{18} - 28971841 \beta_{16} + 27801478 \beta_{15} - 59664219 \beta_{14} + 59664219 \beta_{12} - 3358630 \beta_{11} + 4528993 \beta_{10} - 29086355 \beta_{9} + 27463738 \beta_{8} + 8616644 \beta_{7} + 27463738 \beta_{6} + 1170363 \beta_{5} - 29799572 \beta_{4} + 16717783 \beta_{3} - 16717783 \beta_{2} - 8616644 \beta_{1} + 29086355$$

Character values

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

 $$n$$ $$97$$ $$139$$ $$185$$ $$\chi(n)$$ $$-1 + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} + \beta_{18}$$ $$1$$ $$1$$

Embeddings

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

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

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 1.84381 + 0.541390i −0.962045 − 0.282482i −1.54238 + 1.78001i 1.74521 − 2.01408i −0.0616736 + 0.0396352i 2.31834 − 1.48991i 0.302381 + 2.10310i −0.404188 − 2.81119i 1.84381 − 0.541390i −0.962045 + 0.282482i 0.302381 − 2.10310i −0.404188 + 2.81119i −0.262998 − 0.575885i 1.02355 + 2.24127i −0.0616736 − 0.0396352i 2.31834 + 1.48991i −0.262998 + 0.575885i 1.02355 − 2.24127i −1.54238 − 1.78001i 1.74521 + 2.01408i
0 0.841254 + 0.540641i 0 −1.21471 + 2.65985i 0 0.960219 + 0.281946i 0 0.415415 + 0.909632i 0
13.2 0 0.841254 + 0.540641i 0 1.11647 2.44474i 0 0.161591 + 0.0474474i 0 0.415415 + 0.909632i 0
25.1 0 −0.142315 0.989821i 0 −2.38954 0.701632i 0 2.64891 3.05701i 0 −0.959493 + 0.281733i 0
25.2 0 −0.142315 0.989821i 0 3.91931 + 1.15081i 0 −0.0825209 + 0.0952342i 0 −0.959493 + 0.281733i 0
49.1 0 0.415415 0.909632i 0 −2.38152 + 2.74842i 0 −3.67577 + 2.36227i 0 −0.654861 0.755750i 0
49.2 0 0.415415 0.909632i 0 0.735636 0.848969i 0 0.891451 0.572901i 0 −0.654861 0.755750i 0
73.1 0 −0.959493 + 0.281733i 0 −1.52130 0.977682i 0 0.485296 + 3.37531i 0 0.841254 0.540641i 0
73.2 0 −0.959493 + 0.281733i 0 −0.332496 0.213682i 0 −0.440112 3.06105i 0 0.841254 0.540641i 0
85.1 0 0.841254 0.540641i 0 −1.21471 2.65985i 0 0.960219 0.281946i 0 0.415415 0.909632i 0
85.2 0 0.841254 0.540641i 0 1.11647 + 2.44474i 0 0.161591 0.0474474i 0 0.415415 0.909632i 0
121.1 0 −0.959493 0.281733i 0 −1.52130 + 0.977682i 0 0.485296 3.37531i 0 0.841254 + 0.540641i 0
121.2 0 −0.959493 0.281733i 0 −0.332496 + 0.213682i 0 −0.440112 + 3.06105i 0 0.841254 + 0.540641i 0
133.1 0 −0.654861 + 0.755750i 0 −0.149019 + 1.03645i 0 0.607780 + 1.33085i 0 −0.142315 0.989821i 0
133.2 0 −0.654861 + 0.755750i 0 0.217172 1.51046i 0 −1.55685 3.40903i 0 −0.142315 0.989821i 0
169.1 0 0.415415 + 0.909632i 0 −2.38152 2.74842i 0 −3.67577 2.36227i 0 −0.654861 + 0.755750i 0
169.2 0 0.415415 + 0.909632i 0 0.735636 + 0.848969i 0 0.891451 + 0.572901i 0 −0.654861 + 0.755750i 0
193.1 0 −0.654861 0.755750i 0 −0.149019 1.03645i 0 0.607780 1.33085i 0 −0.142315 + 0.989821i 0
193.2 0 −0.654861 0.755750i 0 0.217172 + 1.51046i 0 −1.55685 + 3.40903i 0 −0.142315 + 0.989821i 0
265.1 0 −0.142315 + 0.989821i 0 −2.38954 + 0.701632i 0 2.64891 + 3.05701i 0 −0.959493 0.281733i 0
265.2 0 −0.142315 + 0.989821i 0 3.91931 1.15081i 0 −0.0825209 0.0952342i 0 −0.959493 0.281733i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 265.2 Significant digits: Format: 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 276.2.i.a 20
3.b odd 2 1 828.2.q.c 20
23.c even 11 1 inner 276.2.i.a 20
23.c even 11 1 6348.2.a.s 10
23.d odd 22 1 6348.2.a.t 10
69.h odd 22 1 828.2.q.c 20

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}$$
$5$ $$139129 + 659837 T + 1370675 T^{2} + 1372212 T^{3} + 1879946 T^{4} + 1476498 T^{5} + 1518254 T^{6} + 1324154 T^{7} + 944325 T^{8} + 576264 T^{9} + 312388 T^{10} + 117918 T^{11} + 32177 T^{12} + 7254 T^{13} + 1043 T^{14} - 188 T^{15} - 199 T^{16} - 62 T^{17} + 3 T^{18} + 4 T^{19} + T^{20}$$
$7$ $$529 - 2530 T - 5263 T^{2} - 152240 T^{3} + 1813793 T^{4} - 5575526 T^{5} + 8765093 T^{6} - 8532689 T^{7} + 5696271 T^{8} - 2768359 T^{9} + 1036815 T^{10} - 337579 T^{11} + 105066 T^{12} - 15873 T^{13} + 7686 T^{14} - 231 T^{15} + 455 T^{16} + 24 T^{18} + T^{20}$$
$11$ $$36590401 - 22357104 T - 15648931 T^{2} - 6714697 T^{3} + 13901977 T^{4} - 905597 T^{5} + 6461861 T^{6} - 2908906 T^{7} + 1511093 T^{8} - 1024793 T^{9} + 285229 T^{10} - 140525 T^{11} + 59566 T^{12} - 8591 T^{13} + 8678 T^{14} + 330 T^{15} + 595 T^{16} + 66 T^{17} + 10 T^{18} + T^{20}$$
$13$ $$746546329 - 3899675175 T + 7491372877 T^{2} - 6423196593 T^{3} + 2968964780 T^{4} - 595120405 T^{5} + 50964803 T^{6} + 119149558 T^{7} + 22526164 T^{8} + 16471851 T^{9} + 13758942 T^{10} + 6486040 T^{11} + 2508837 T^{12} + 893288 T^{13} + 290806 T^{14} + 75185 T^{15} + 15621 T^{16} + 2453 T^{17} + 285 T^{18} + 22 T^{19} + T^{20}$$
$17$ $$181252369 - 425848153 T + 296568609 T^{2} + 439386189 T^{3} - 867895682 T^{4} + 360956995 T^{5} + 424708084 T^{6} - 767459429 T^{7} + 637891631 T^{8} - 349625082 T^{9} + 141436283 T^{10} - 43334088 T^{11} + 10151453 T^{12} - 1754049 T^{13} + 217607 T^{14} - 16943 T^{15} + 1289 T^{16} - 187 T^{17} + 63 T^{18} - 7 T^{19} + T^{20}$$
$19$ $$512524321 - 3123978249 T + 26405217311 T^{2} - 21295938332 T^{3} + 80259141223 T^{4} - 75137752954 T^{5} + 19751255693 T^{6} - 688924957 T^{7} + 1218325768 T^{8} - 451626915 T^{9} + 58792986 T^{10} - 13803937 T^{11} + 5269935 T^{12} - 1151262 T^{13} + 195803 T^{14} - 36190 T^{15} + 7465 T^{16} - 1358 T^{17} + 198 T^{18} - 19 T^{19} + T^{20}$$
$23$ $$41426511213649 - 36023053229260 T + 20909033070027 T^{2} - 8900213718458 T^{3} + 3127258155125 T^{4} - 934659985088 T^{5} + 247062384147 T^{6} - 59304415734 T^{7} + 13279102417 T^{8} - 2853515424 T^{9} + 598445177 T^{10} - 124065888 T^{11} + 25102273 T^{12} - 4874202 T^{13} + 882867 T^{14} - 145216 T^{15} + 21125 T^{16} - 2614 T^{17} + 267 T^{18} - 20 T^{19} + T^{20}$$
$29$ $$94249 + 519444 T + 7821200 T^{2} + 23219658 T^{3} + 55182717 T^{4} + 89529410 T^{5} + 94828913 T^{6} + 86003885 T^{7} - 24528037 T^{8} - 118089797 T^{9} + 114501529 T^{10} - 48304863 T^{11} + 12578894 T^{12} - 2739088 T^{13} + 688692 T^{14} - 176564 T^{15} + 35791 T^{16} - 5181 T^{17} + 512 T^{18} - 32 T^{19} + T^{20}$$
$31$ $$1366115521 - 1978004876 T + 14271050678 T^{2} + 25924290097 T^{3} + 28775697562 T^{4} + 6940917303 T^{5} + 1899672010 T^{6} + 940463831 T^{7} - 359440379 T^{8} - 120133918 T^{9} + 29345997 T^{10} - 6790926 T^{11} + 1962384 T^{12} + 250094 T^{13} + 132664 T^{14} + 17126 T^{15} + 2558 T^{16} - 197 T^{17} - 10 T^{18} + 3 T^{19} + T^{20}$$
$37$ $$41662309043667841 + 41731143250060526 T + 22356141446478578 T^{2} + 8075906329777803 T^{3} + 2177449951797787 T^{4} + 444412175882650 T^{5} + 70750852182783 T^{6} + 9064644220885 T^{7} + 1012395922334 T^{8} + 103024379921 T^{9} + 9793034624 T^{10} + 743266086 T^{11} + 32345102 T^{12} - 1764653 T^{13} - 220832 T^{14} - 2681 T^{15} - 1125 T^{16} + 77 T^{17} + 72 T^{18} + 10 T^{19} + T^{20}$$
$41$ $$389752131263881 + 669630738999991 T + 707953600583513 T^{2} + 461805388747059 T^{3} + 205218730340075 T^{4} + 66531457171570 T^{5} + 17142140167902 T^{6} + 3598689054226 T^{7} + 625389775992 T^{8} + 84863330246 T^{9} + 7549111603 T^{10} - 70954108 T^{11} - 163923425 T^{12} - 28078887 T^{13} - 1519224 T^{14} + 275017 T^{15} + 76244 T^{16} + 9659 T^{17} + 782 T^{18} + 40 T^{19} + T^{20}$$
$43$ $$577670697717961 - 463909200184418 T + 171338146440788 T^{2} - 54905861248707 T^{3} + 21299481357280 T^{4} - 6810443215059 T^{5} + 1474974796583 T^{6} - 296529410945 T^{7} + 82779882905 T^{8} - 18687977058 T^{9} + 2443057990 T^{10} - 268560425 T^{11} + 44026667 T^{12} - 6573659 T^{13} + 1135437 T^{14} - 172293 T^{15} + 19380 T^{16} - 1740 T^{17} + 237 T^{18} - 8 T^{19} + T^{20}$$
$47$ $$( -337853 - 112793 T + 691965 T^{2} + 815546 T^{3} + 361609 T^{4} + 63239 T^{5} - 1114 T^{6} - 1665 T^{7} - 125 T^{8} + 9 T^{9} + T^{10} )^{2}$$
$53$ $$11896440939631249 + 11128281313853122 T + 5587027563331885 T^{2} + 1697746128917211 T^{3} + 315796719707061 T^{4} + 46237293611795 T^{5} + 13028620028722 T^{6} + 5127709123725 T^{7} + 1514610526385 T^{8} + 329258348550 T^{9} + 57187682551 T^{10} + 8609367447 T^{11} + 1201045131 T^{12} + 158550866 T^{13} + 19247658 T^{14} + 2040236 T^{15} + 181290 T^{16} + 13133 T^{17} + 759 T^{18} + 34 T^{19} + T^{20}$$
$59$ $$1812553308721 - 8949419274204 T - 5693007031864 T^{2} + 41298924756925 T^{3} + 93785300758035 T^{4} + 99417124595049 T^{5} + 65713808784726 T^{6} + 29908664797847 T^{7} + 9846622188804 T^{8} + 2420978984634 T^{9} + 458543484068 T^{10} + 69203826067 T^{11} + 8577394846 T^{12} + 882744313 T^{13} + 77386621 T^{14} + 5717924 T^{15} + 375515 T^{16} + 19965 T^{17} + 996 T^{18} + 32 T^{19} + T^{20}$$
$61$ $$16910706981169 + 13521351030728 T + 21181259906611 T^{2} + 9841256141673 T^{3} + 6007237816983 T^{4} + 848503312286 T^{5} + 222488054748 T^{6} - 266115937989 T^{7} + 94408392138 T^{8} - 33711904722 T^{9} + 16246854502 T^{10} - 5657273338 T^{11} + 1230757372 T^{12} - 168619605 T^{13} + 13914476 T^{14} - 517190 T^{15} - 7791 T^{16} - 44 T^{17} + 325 T^{18} - 32 T^{19} + T^{20}$$
$67$ $$24475292457001 - 60492442340986 T + 86621718869308 T^{2} - 80656859182347 T^{3} + 54227478592951 T^{4} - 26887554432959 T^{5} + 10283893312668 T^{6} - 3112931992534 T^{7} + 770962969712 T^{8} - 158363476676 T^{9} + 27292918179 T^{10} - 4043670382 T^{11} + 512642541 T^{12} - 54006967 T^{13} + 4731822 T^{14} - 282469 T^{15} + 13060 T^{16} - 2749 T^{17} + 472 T^{18} - 35 T^{19} + T^{20}$$
$71$ $$14637426640321 - 136783569942789 T + 501278829617156 T^{2} - 1715569725290344 T^{3} + 5756998707742049 T^{4} - 3342206430928013 T^{5} + 974602274541561 T^{6} - 177912397479575 T^{7} + 29255301652229 T^{8} - 4256202216101 T^{9} + 542325677266 T^{10} - 57807076855 T^{11} + 5637296701 T^{12} - 498105432 T^{13} + 42031926 T^{14} - 3243427 T^{15} + 227001 T^{16} - 12496 T^{17} + 623 T^{18} - 33 T^{19} + T^{20}$$
$73$ $$41788181960853241 - 15955582587444860 T + 2794954307710516 T^{2} - 247841669832781 T^{3} + 53783136580094 T^{4} + 43932000343253 T^{5} - 5330147265514 T^{6} - 611684060090 T^{7} + 230744116727 T^{8} - 9377005991 T^{9} + 1316336031 T^{10} - 714559816 T^{11} + 71257576 T^{12} + 1289334 T^{13} + 749861 T^{14} - 142120 T^{15} + 23864 T^{16} - 315 T^{17} + 261 T^{18} + T^{19} + T^{20}$$
$79$ $$684944324016481 + 1007307698589533 T + 725468196081510 T^{2} + 342451363704934 T^{3} + 120765990289091 T^{4} + 35536820378609 T^{5} + 9892003594800 T^{6} + 2685392761623 T^{7} + 663475474095 T^{8} + 136476335369 T^{9} + 22517102486 T^{10} + 2784334663 T^{11} + 263299925 T^{12} + 14781987 T^{13} + 649788 T^{14} + 17842 T^{15} + 9705 T^{16} - 308 T^{17} + 105 T^{18} - 22 T^{19} + T^{20}$$
$83$ $$1251350934564721 + 3235983602956341 T + 3122798983535958 T^{2} + 503910130702060 T^{3} + 120454263688919 T^{4} + 23737763861453 T^{5} + 4049123011748 T^{6} + 1012207001133 T^{7} + 141016540959 T^{8} + 9760198929 T^{9} + 1124967876 T^{10} - 68876847 T^{11} + 25858491 T^{12} - 523297 T^{13} + 92020 T^{14} - 49328 T^{15} + 1143 T^{16} + 618 T^{17} + 179 T^{18} + 14 T^{19} + T^{20}$$
$89$ $$6872782557649009 - 37488648447839868 T + 115032853652783955 T^{2} - 70824988505802990 T^{3} + 21897041167232970 T^{4} - 4270498507709518 T^{5} + 621137417849182 T^{6} - 72213650609129 T^{7} + 6413842447184 T^{8} - 437657925376 T^{9} + 24791006899 T^{10} + 760176289 T^{11} - 278929550 T^{12} + 6284913 T^{13} + 337071 T^{14} + 14593 T^{15} + 41819 T^{16} - 1709 T^{17} + 246 T^{18} - 10 T^{19} + T^{20}$$
$97$ $$58917139761417961 + 23869085839650988 T + 9607634304263708 T^{2} + 1190266860993040 T^{3} + 871800404775500 T^{4} + 104153120045272 T^{5} + 25398571393977 T^{6} + 8302568888676 T^{7} + 1021825258220 T^{8} - 17799470814 T^{9} + 7724302783 T^{10} + 1242332430 T^{11} - 9391303 T^{12} + 38333828 T^{13} + 5241069 T^{14} - 113103 T^{15} - 10979 T^{16} - 4143 T^{17} + 124 T^{18} + 4 T^{19} + T^{20}$$