# Properties

 Label 8037.2.a.o Level 8037 Weight 2 Character orbit 8037.a Self dual Yes Analytic conductor 64.176 Analytic rank 0 Dimension 18 CM No Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$8037 = 3^{2} \cdot 19 \cdot 47$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 8037.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$64.1757681045$$ Analytic rank: $$0$$ Dimension: $$18$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$-\beta_{1} q^{2}$$ $$+ ( 1 + \beta_{2} ) q^{4}$$ $$-\beta_{10} q^{5}$$ $$+ ( \beta_{14} + \beta_{17} ) q^{7}$$ $$+ ( -1 - \beta_{3} - \beta_{10} - \beta_{13} + \beta_{14} + \beta_{15} + \beta_{17} ) q^{8}$$ $$+O(q^{10})$$ $$q$$ $$-\beta_{1} q^{2}$$ $$+ ( 1 + \beta_{2} ) q^{4}$$ $$-\beta_{10} q^{5}$$ $$+ ( \beta_{14} + \beta_{17} ) q^{7}$$ $$+ ( -1 - \beta_{3} - \beta_{10} - \beta_{13} + \beta_{14} + \beta_{15} + \beta_{17} ) q^{8}$$ $$+ ( -\beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{12} - \beta_{16} ) q^{10}$$ $$-\beta_{11} q^{11}$$ $$+ ( 1 - \beta_{7} ) q^{13}$$ $$+ ( 1 + \beta_{2} - \beta_{4} + \beta_{6} + \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{15} - \beta_{17} ) q^{14}$$ $$+ ( 2 + 2 \beta_{2} + \beta_{5} - \beta_{9} - \beta_{11} - 2 \beta_{12} - \beta_{17} ) q^{16}$$ $$+ ( -\beta_{1} + \beta_{2} - \beta_{9} ) q^{17}$$ $$- q^{19}$$ $$+ ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{16} ) q^{20}$$ $$+ ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{22}$$ $$+ ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{13} + \beta_{16} ) q^{23}$$ $$+ ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{12} + \beta_{15} ) q^{25}$$ $$+ ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} + \beta_{17} ) q^{26}$$ $$+ ( -1 - \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} + 2 \beta_{17} ) q^{28}$$ $$+ ( -1 - \beta_{1} - \beta_{3} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{16} - \beta_{17} ) q^{29}$$ $$+ ( 2 - \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{31}$$ $$+ ( -\beta_{2} + \beta_{9} - 2 \beta_{10} + 2 \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} + 2 \beta_{17} ) q^{32}$$ $$+ ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{16} + 2 \beta_{17} ) q^{34}$$ $$+ ( -2 - \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{11} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{35}$$ $$+ ( -\beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{12} + \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{37}$$ $$+ \beta_{1} q^{38}$$ $$+ ( 1 - \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{16} ) q^{40}$$ $$+ ( -2 - \beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{12} - \beta_{13} + 2 \beta_{15} ) q^{41}$$ $$+ ( 2 + \beta_{1} + \beta_{3} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{15} + \beta_{16} ) q^{43}$$ $$+ ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} + \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{16} + \beta_{17} ) q^{44}$$ $$+ ( -2 + \beta_{1} - 3 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{12} + 2 \beta_{17} ) q^{46}$$ $$- q^{47}$$ $$+ ( 1 + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{9} - \beta_{10} - 2 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{17} ) q^{49}$$ $$+ ( 3 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} - 3 \beta_{12} - \beta_{13} + \beta_{16} + \beta_{17} ) q^{50}$$ $$+ ( 1 + 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{12} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{52}$$ $$+ ( \beta_{1} + \beta_{3} + 2 \beta_{6} - \beta_{7} + \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} + \beta_{16} ) q^{53}$$ $$+ ( 3 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{55}$$ $$+ ( 5 - \beta_{1} + 4 \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{7} + \beta_{8} - 3 \beta_{9} + \beta_{10} - \beta_{11} - 3 \beta_{12} - 2 \beta_{14} - \beta_{15} - 3 \beta_{17} ) q^{56}$$ $$+ ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{7} + \beta_{9} + \beta_{12} + 2 \beta_{13} - 2 \beta_{14} - \beta_{15} - \beta_{16} ) q^{58}$$ $$+ ( \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{13} + \beta_{14} + \beta_{16} + 2 \beta_{17} ) q^{59}$$ $$+ ( -1 + \beta_{3} - \beta_{4} + 3 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{13} + 3 \beta_{14} - 2 \beta_{15} + \beta_{17} ) q^{61}$$ $$+ ( -4 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + 3 \beta_{12} + \beta_{14} + \beta_{15} + 2 \beta_{17} ) q^{62}$$ $$+ ( 2 - \beta_{1} + 4 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - 3 \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{15} - 2 \beta_{17} ) q^{64}$$ $$+ ( 1 - \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{10} + \beta_{11} + 3 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} - \beta_{16} ) q^{65}$$ $$+ ( 4 - \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{11} + \beta_{13} + \beta_{16} ) q^{67}$$ $$+ ( 4 - 2 \beta_{1} + 5 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} - 3 \beta_{9} + \beta_{10} - \beta_{11} - 4 \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} - 3 \beta_{17} ) q^{68}$$ $$+ ( -\beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{15} - \beta_{16} ) q^{70}$$ $$+ ( 1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{6} + \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{14} - \beta_{16} - \beta_{17} ) q^{71}$$ $$+ ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{7} + 2 \beta_{9} - \beta_{11} + 2 \beta_{14} - \beta_{15} + \beta_{16} + 2 \beta_{17} ) q^{73}$$ $$+ ( 2 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 3 \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{14} + 2 \beta_{15} + 3 \beta_{17} ) q^{74}$$ $$+ ( -1 - \beta_{2} ) q^{76}$$ $$+ ( -\beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{14} - \beta_{15} ) q^{77}$$ $$+ ( 3 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{15} + \beta_{16} + 2 \beta_{17} ) q^{79}$$ $$+ ( 3 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{10} + 2 \beta_{11} - \beta_{14} - \beta_{15} - \beta_{16} ) q^{80}$$ $$+ ( 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{11} - 3 \beta_{12} - \beta_{13} + \beta_{16} ) q^{82}$$ $$+ ( 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{8} - 3 \beta_{10} + 2 \beta_{12} - \beta_{13} + 2 \beta_{15} + 2 \beta_{17} ) q^{83}$$ $$+ ( -2 \beta_{1} - \beta_{5} - 2 \beta_{6} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{13} - 3 \beta_{14} - 2 \beta_{16} - \beta_{17} ) q^{85}$$ $$+ ( -2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} + \beta_{17} ) q^{86}$$ $$+ ( 1 - 4 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} + \beta_{16} ) q^{88}$$ $$+ ( 2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} + \beta_{16} + \beta_{17} ) q^{89}$$ $$+ ( 1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{91}$$ $$+ ( -1 + 3 \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} - 3 \beta_{9} + 3 \beta_{10} + \beta_{11} - 3 \beta_{12} + \beta_{13} - 2 \beta_{14} - 3 \beta_{15} - \beta_{16} - 5 \beta_{17} ) q^{92}$$ $$+ \beta_{1} q^{94}$$ $$+ \beta_{10} q^{95}$$ $$+ ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{11} + 4 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{17} ) q^{97}$$ $$+ ( -5 + 3 \beta_{1} - 6 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} + 4 \beta_{9} - \beta_{10} - \beta_{11} + 6 \beta_{12} - \beta_{13} + 3 \beta_{15} + 2 \beta_{17} ) q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18q$$ $$\mathstrut -\mathstrut q^{2}$$ $$\mathstrut +\mathstrut 21q^{4}$$ $$\mathstrut -\mathstrut 5q^{5}$$ $$\mathstrut +\mathstrut 3q^{7}$$ $$\mathstrut -\mathstrut 6q^{8}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$18q$$ $$\mathstrut -\mathstrut q^{2}$$ $$\mathstrut +\mathstrut 21q^{4}$$ $$\mathstrut -\mathstrut 5q^{5}$$ $$\mathstrut +\mathstrut 3q^{7}$$ $$\mathstrut -\mathstrut 6q^{8}$$ $$\mathstrut +\mathstrut 7q^{10}$$ $$\mathstrut -\mathstrut 6q^{11}$$ $$\mathstrut +\mathstrut 21q^{13}$$ $$\mathstrut +\mathstrut 9q^{14}$$ $$\mathstrut +\mathstrut 23q^{16}$$ $$\mathstrut +\mathstrut 4q^{17}$$ $$\mathstrut -\mathstrut 18q^{19}$$ $$\mathstrut -\mathstrut 9q^{20}$$ $$\mathstrut +\mathstrut 28q^{22}$$ $$\mathstrut -\mathstrut 9q^{23}$$ $$\mathstrut +\mathstrut 11q^{25}$$ $$\mathstrut +\mathstrut q^{26}$$ $$\mathstrut +\mathstrut 7q^{28}$$ $$\mathstrut -\mathstrut 12q^{29}$$ $$\mathstrut +\mathstrut 26q^{31}$$ $$\mathstrut +\mathstrut 7q^{32}$$ $$\mathstrut +\mathstrut 26q^{34}$$ $$\mathstrut -\mathstrut 9q^{35}$$ $$\mathstrut +\mathstrut 8q^{37}$$ $$\mathstrut +\mathstrut q^{38}$$ $$\mathstrut +\mathstrut 16q^{40}$$ $$\mathstrut -\mathstrut 12q^{41}$$ $$\mathstrut +\mathstrut 28q^{43}$$ $$\mathstrut -\mathstrut 2q^{44}$$ $$\mathstrut -\mathstrut 33q^{46}$$ $$\mathstrut -\mathstrut 18q^{47}$$ $$\mathstrut +\mathstrut 17q^{49}$$ $$\mathstrut +\mathstrut 29q^{50}$$ $$\mathstrut +\mathstrut 30q^{52}$$ $$\mathstrut -\mathstrut 5q^{53}$$ $$\mathstrut +\mathstrut 28q^{55}$$ $$\mathstrut +\mathstrut 77q^{56}$$ $$\mathstrut -\mathstrut 6q^{58}$$ $$\mathstrut -\mathstrut 30q^{59}$$ $$\mathstrut -\mathstrut 16q^{61}$$ $$\mathstrut -\mathstrut 16q^{62}$$ $$\mathstrut +\mathstrut 28q^{64}$$ $$\mathstrut +\mathstrut 22q^{65}$$ $$\mathstrut +\mathstrut 45q^{67}$$ $$\mathstrut +\mathstrut 96q^{68}$$ $$\mathstrut -\mathstrut 2q^{70}$$ $$\mathstrut +\mathstrut q^{71}$$ $$\mathstrut -\mathstrut 24q^{73}$$ $$\mathstrut +\mathstrut 19q^{74}$$ $$\mathstrut -\mathstrut 21q^{76}$$ $$\mathstrut -\mathstrut 2q^{77}$$ $$\mathstrut +\mathstrut 33q^{79}$$ $$\mathstrut +\mathstrut 25q^{80}$$ $$\mathstrut +\mathstrut 18q^{82}$$ $$\mathstrut +\mathstrut 13q^{83}$$ $$\mathstrut -\mathstrut 7q^{85}$$ $$\mathstrut -\mathstrut 3q^{86}$$ $$\mathstrut +\mathstrut 27q^{88}$$ $$\mathstrut +\mathstrut 6q^{89}$$ $$\mathstrut +\mathstrut 42q^{91}$$ $$\mathstrut +\mathstrut 11q^{92}$$ $$\mathstrut +\mathstrut q^{94}$$ $$\mathstrut +\mathstrut 5q^{95}$$ $$\mathstrut +\mathstrut 44q^{97}$$ $$\mathstrut -\mathstrut 58q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18}\mathstrut -\mathstrut$$ $$x^{17}\mathstrut -\mathstrut$$ $$28$$ $$x^{16}\mathstrut +\mathstrut$$ $$25$$ $$x^{15}\mathstrut +\mathstrut$$ $$322$$ $$x^{14}\mathstrut -\mathstrut$$ $$247$$ $$x^{13}\mathstrut -\mathstrut$$ $$1971$$ $$x^{12}\mathstrut +\mathstrut$$ $$1231$$ $$x^{11}\mathstrut +\mathstrut$$ $$6953$$ $$x^{10}\mathstrut -\mathstrut$$ $$3283$$ $$x^{9}\mathstrut -\mathstrut$$ $$14235$$ $$x^{8}\mathstrut +\mathstrut$$ $$4562$$ $$x^{7}\mathstrut +\mathstrut$$ $$15962$$ $$x^{6}\mathstrut -\mathstrut$$ $$2882$$ $$x^{5}\mathstrut -\mathstrut$$ $$8159$$ $$x^{4}\mathstrut +\mathstrut$$ $$606$$ $$x^{3}\mathstrut +\mathstrut$$ $$890$$ $$x^{2}\mathstrut -\mathstrut$$ $$179$$ $$x\mathstrut +\mathstrut$$ $$9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$1375$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$5084$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$36270$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$122133$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$385293$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$1142502$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$2121571$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$5271936$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$6496241$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$12574330$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$10966627$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$14789013$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$9398813$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$6930877$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$3224026$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$357518$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$175528$$ $$\nu\mathstrut +\mathstrut$$ $$15159$$$$)/25060$$ $$\beta_{4}$$ $$=$$ $$($$$$1245$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$14641$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$40655$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$363742$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$548917$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$3562623$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$3965914$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$17502894$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$16545409$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$45330325$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$39853058$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$58721567$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$51873662$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$29528923$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$29497239$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$499933$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$2194037$$ $$\nu\mathstrut -\mathstrut$$ $$125826$$$$)/25060$$ $$\beta_{5}$$ $$=$$ $$($$$$3027$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$6697$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$74329$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$168948$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$714237$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$1691399$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$3386660$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$8576416$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$8111547$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$23254771$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$8377838$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$32405463$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$180072$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$19092659$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$4021219$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$1587523$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$104647$$ $$\nu\mathstrut +\mathstrut$$ $$23292$$$$)/25060$$ $$\beta_{6}$$ $$=$$ $$($$$$2588$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$5615$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$79161$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$139029$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$1002284$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$1353473$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$6792347$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$6572488$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$26570780$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$16607951$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$60094951$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$20184294$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$73715119$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$7638688$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$40208151$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$2452891$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$3890843$$ $$\nu\mathstrut -\mathstrut$$ $$358605$$$$)/25060$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$1240$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$3931$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$38655$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$100262$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$492753$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$1019383$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$3303026$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$5294509$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$12477886$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$14969960$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$26490527$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$22628692$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$29599058$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$16352508$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$14323951$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$3935337$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$1330428$$ $$\nu\mathstrut +\mathstrut$$ $$66324$$$$)/12530$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$1023$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$7752$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$35806$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$193633$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$510113$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$1911944$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$3828075$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$9513046$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$16327263$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$25165856$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$39653957$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$33965223$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$51556133$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$19134419$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$29575684$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$1483328$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$3129092$$ $$\nu\mathstrut +\mathstrut$$ $$361577$$$$)/12530$$ $$\beta_{9}$$ $$=$$ $$($$$$1455$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$6528$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$49475$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$157256$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$679971$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$1476119$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$4872722$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$6830792$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$19578362$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$16240180$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$44099559$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$18552046$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$52477321$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$7241889$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$27430832$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$969549$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$2832801$$ $$\nu\mathstrut -\mathstrut$$ $$270523$$$$)/12530$$ $$\beta_{10}$$ $$=$$ $$($$$$2073$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$1488$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$54846$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$36877$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$589003$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$363546$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$3325055$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$1834374$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$10698493$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$5102844$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$19844447$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$7933067$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$20060543$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$6557961$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$9138204$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$2305702$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$809712$$ $$\nu\mathstrut -\mathstrut$$ $$13747$$$$)/12530$$ $$\beta_{11}$$ $$=$$ $$($$$$4595$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$9469$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$115125$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$236848$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$1140117$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$2345227$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$5679104$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$11727536$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$14932039$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$31280355$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$19577278$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$42921283$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$9713092$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$25344387$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$374051$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$2682883$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$434997$$ $$\nu\mathstrut -\mathstrut$$ $$33996$$$$)/25060$$ $$\beta_{12}$$ $$=$$ $$($$$$6239$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$4103$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$153743$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$100284$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$1492971$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$964945$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$7287684$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$4649798$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$18950143$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$11770167$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$25628064$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$14928519$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$15819256$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$7398175$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$3077931$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$255031$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$441793$$ $$\nu\mathstrut -\mathstrut$$ $$142672$$$$)/25060$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$3067$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$290$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$84064$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$8371$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$943021$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$94912$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$5611143$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$535462$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$19185335$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$1570704$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$38006829$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$2201281$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$41184681$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$1060727$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$20386644$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$7476$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$2317232$$ $$\nu\mathstrut +\mathstrut$$ $$261015$$$$)/12530$$ $$\beta_{14}$$ $$=$$ $$($$$$3252$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$2949$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$82884$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$73532$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$844628$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$726240$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$4427692$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$3626404$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$12846929$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$9675021$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$20754167$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$13366767$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$17725613$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$8147890$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$7023393$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$1092042$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$1032434$$ $$\nu\mathstrut -\mathstrut$$ $$74371$$$$)/12530$$ $$\beta_{15}$$ $$=$$ $$($$$$7089$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$664$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$186758$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$19521$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$1989179$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$238418$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$11064125$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$1547932$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$34713719$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$5656942$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$61854961$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$11592941$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$59027859$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$12461833$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$24922062$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$5524666$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$1933636$$ $$\nu\mathstrut +\mathstrut$$ $$299$$$$)/25060$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$1383$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$1899$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$41081$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$46402$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$503867$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$442805$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$3299938$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$2083314$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$12447731$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$4986871$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$27095998$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$5408617$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$31944222$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$1121065$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$16760177$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$1331003$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$1611611$$ $$\nu\mathstrut +\mathstrut$$ $$121854$$$$)/3580$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$8478$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$1039$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$223616$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$23022$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$2385882$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$184260$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$13306628$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$605434$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$41938751$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$457221$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$75327343$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$1634383$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$73063087$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$3294680$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$32332347$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$1362438$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$3544416$$ $$\nu\mathstrut +\mathstrut$$ $$341599$$$$)/12530$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$3$$ $$\nu^{3}$$ $$=$$ $$-$$$$\beta_{17}\mathstrut -\mathstrut$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$1$$ $$\nu^{4}$$ $$=$$ $$-$$$$\beta_{17}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$8$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$16$$ $$\nu^{5}$$ $$=$$ $$-$$$$10$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$10$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$9$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$9$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$10$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$8$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$20$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$8$$ $$\nu^{6}$$ $$=$$ $$-$$$$12$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$23$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$10$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$12$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$11$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$60$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$98$$ $$\nu^{7}$$ $$=$$ $$-$$$$82$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$81$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$69$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$67$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$27$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$79$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$15$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$53$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$14$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$114$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$54$$ $$\nu^{8}$$ $$=$$ $$-$$$$112$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$30$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$16$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$13$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$208$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$80$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$12$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$111$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$29$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$45$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$94$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$13$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$15$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$441$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$16$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$639$$ $$\nu^{9}$$ $$=$$ $$-$$$$637$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$620$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$507$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$476$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$272$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$20$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$587$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$158$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$18$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$16$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$36$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$338$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$146$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$699$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$358$$ $$\nu^{10}$$ $$=$$ $$-$$$$955$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$321$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$180$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$116$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$1722$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$601$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$99$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$932$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$298$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$19$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$469$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$735$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$118$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$154$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$3216$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$174$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$4302$$ $$\nu^{11}$$ $$=$$ $$-$$$$4845$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$41$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$4645$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$3680$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$3342$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$2440$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$254$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$4280$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$1444$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$212$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$175$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$431$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$84$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$38$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$2154$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$1357$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$4472$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$2427$$ $$\nu^{12}$$ $$=$$ $$-$$$$7786$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$67$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$3009$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$1748$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$880$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$13644$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$4417$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$679$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$7457$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$2677$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$237$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$4219$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$5515$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$932$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$1354$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$23384$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$1596$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$29520$$ $$\nu^{13}$$ $$=$$ $$-$$$$36459$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$538$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$34447$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$26632$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$23438$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$20592$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$2657$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$31024$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$12270$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$2088$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$1636$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$4335$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$1132$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$470$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$13866$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$11874$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$29412$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$17018$$ $$\nu^{14}$$ $$=$$ $$-$$$$61837$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$934$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$26339$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$15697$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$6050$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$105420$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$32186$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$4009$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$58034$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$22463$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$2459$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$35200$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$40495$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$6877$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$10967$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$169875$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$13303$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$205229$$ $$\nu^{15}$$ $$=$$ $$-$$$$272599$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$5784$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$254027$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$192725$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$164755$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$167527$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$25047$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$224540$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$99951$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$18714$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$14093$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$39671$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$12501$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$4830$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$90388$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$100171$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$197322$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$123317$$ $$\nu^{16}$$ $$=$$ $$-$$$$482938$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$10494$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$221286$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$134481$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$38421$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$801916$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$233707$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$19374$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$443929$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$181158$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$23040$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$281274$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$293813$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$48888$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$84514$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$1234044$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$104350$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$1440869$$ $$\nu^{17}$$ $$=$$ $$-$$$$2029601$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$55621$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$1867082$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$1395823$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$1161951$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$1330972$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$221695$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$1625134$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$793015$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$158636$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$115727$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$342661$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$123433$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$44918$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$596467$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$824853$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$1344049$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$917721$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.72555 2.37327 2.33471 2.33085 1.52472 1.43818 1.35283 0.230018 0.146947 0.0898969 −0.494084 −1.15272 −1.40846 −1.48566 −1.61470 −2.13101 −2.60234 −2.65800
−2.72555 0 5.42860 0.150400 0 −4.90968 −9.34479 0 −0.409922
1.2 −2.37327 0 3.63241 2.08614 0 4.17378 −3.87415 0 −4.95098
1.3 −2.33471 0 3.45089 −4.12904 0 −0.743390 −3.38741 0 9.64013
1.4 −2.33085 0 3.43286 −1.49796 0 0.334615 −3.33977 0 3.49151
1.5 −1.52472 0 0.324761 1.59729 0 −1.34206 2.55426 0 −2.43541
1.6 −1.43818 0 0.0683575 −1.27662 0 4.10362 2.77805 0 1.83600
1.7 −1.35283 0 −0.169850 −0.632563 0 −0.940374 2.93544 0 0.855750
1.8 −0.230018 0 −1.94709 2.49148 0 −0.810750 0.907901 0 −0.573085
1.9 −0.146947 0 −1.97841 −2.45635 0 −1.38215 0.584613 0 0.360953
1.10 −0.0898969 0 −1.99192 −2.73841 0 4.94566 0.358861 0 0.246175
1.11 0.494084 0 −1.75588 3.82270 0 −3.40967 −1.85572 0 1.88874
1.12 1.15272 0 −0.671235 1.54521 0 2.91562 −3.07919 0 1.78119
1.13 1.40846 0 −0.0162457 −0.976908 0 −4.66797 −2.83980 0 −1.37593
1.14 1.48566 0 0.207175 −2.74559 0 0.513695 −2.66352 0 −4.07900
1.15 1.61470 0 0.607250 −2.02448 0 −1.29797 −2.24887 0 −3.26892
1.16 2.13101 0 2.54118 1.76024 0 2.12155 1.15327 0 3.75109
1.17 2.60234 0 4.77220 −3.17530 0 1.94962 7.21421 0 −8.26324
1.18 2.65800 0 5.06495 3.19976 0 1.44585 8.14662 0 8.50495
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$19$$ $$1$$
$$47$$ $$1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8037))$$:

 $$T_{2}^{18} + \cdots$$ $$T_{5}^{18} + \cdots$$