# Properties

 Label 1155.2.c.e Level 1155 Weight 2 Character orbit 1155.c Analytic conductor 9.223 Analytic rank 0 Dimension 20 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1155 = 3 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1155.c (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$9.22272143346$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ \beta_{13} q^{3}$$ $$+ ( -1 + \beta_{2} ) q^{4}$$ $$+ \beta_{7} q^{5}$$ $$-\beta_{3} q^{6}$$ $$-\beta_{13} q^{7}$$ $$+ ( -\beta_{1} + \beta_{17} + \beta_{18} ) q^{8}$$ $$- q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ \beta_{13} q^{3}$$ $$+ ( -1 + \beta_{2} ) q^{4}$$ $$+ \beta_{7} q^{5}$$ $$-\beta_{3} q^{6}$$ $$-\beta_{13} q^{7}$$ $$+ ( -\beta_{1} + \beta_{17} + \beta_{18} ) q^{8}$$ $$- q^{9}$$ $$+ \beta_{5} q^{10}$$ $$- q^{11}$$ $$+ ( -\beta_{13} - \beta_{17} ) q^{12}$$ $$+ ( -\beta_{4} - \beta_{10} - \beta_{19} ) q^{13}$$ $$+ \beta_{3} q^{14}$$ $$+ \beta_{6} q^{15}$$ $$+ ( -\beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} + \beta_{14} - \beta_{15} - \beta_{16} ) q^{16}$$ $$+ ( \beta_{4} + \beta_{5} + \beta_{10} - \beta_{13} - \beta_{15} + \beta_{19} ) q^{17}$$ $$-\beta_{1} q^{18}$$ $$+ ( \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{19}$$ $$+ ( 1 + \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{16} + \beta_{18} ) q^{20}$$ $$+ q^{21}$$ $$-\beta_{1} q^{22}$$ $$+ ( 2 \beta_{7} + 2 \beta_{12} + 2 \beta_{13} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{23}$$ $$+ ( \beta_{2} + \beta_{3} - \beta_{8} ) q^{24}$$ $$+ ( -\beta_{2} + \beta_{9} - \beta_{11} + \beta_{18} ) q^{25}$$ $$+ ( 1 + \beta_{4} - \beta_{7} - 2 \beta_{8} - \beta_{10} + \beta_{14} ) q^{26}$$ $$-\beta_{13} q^{27}$$ $$+ ( \beta_{13} + \beta_{17} ) q^{28}$$ $$+ ( -3 + \beta_{2} - 2 \beta_{3} - \beta_{7} + \beta_{9} + \beta_{14} ) q^{29}$$ $$+ ( \beta_{6} - \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} ) q^{30}$$ $$+ ( -\beta_{2} - \beta_{7} + \beta_{8} + \beta_{12} - \beta_{15} - \beta_{16} ) q^{31}$$ $$+ ( \beta_{1} + \beta_{5} + 2 \beta_{7} + 2 \beta_{12} - \beta_{13} - \beta_{16} - 2 \beta_{17} - \beta_{18} + \beta_{19} ) q^{32}$$ $$-\beta_{13} q^{33}$$ $$+ ( -1 + 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{14} ) q^{34}$$ $$-\beta_{6} q^{35}$$ $$+ ( 1 - \beta_{2} ) q^{36}$$ $$+ ( 2 \beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{16} + \beta_{17} ) q^{37}$$ $$+ ( \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{10} + \beta_{11} + 2 \beta_{13} - \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{38}$$ $$+ ( -\beta_{9} - \beta_{15} - \beta_{16} ) q^{39}$$ $$+ ( -\beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{7} + \beta_{13} + 2 \beta_{16} + \beta_{17} - \beta_{19} ) q^{40}$$ $$+ ( 1 + \beta_{2} - \beta_{4} + \beta_{8} + \beta_{10} + \beta_{12} - \beta_{14} ) q^{41}$$ $$+ \beta_{1} q^{42}$$ $$+ ( -\beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{10} + \beta_{11} - \beta_{15} - \beta_{16} - \beta_{18} ) q^{43}$$ $$+ ( 1 - \beta_{2} ) q^{44}$$ $$-\beta_{7} q^{45}$$ $$+ ( 2 + \beta_{2} - 2 \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{10} + 2 \beta_{11} + \beta_{12} - 2 \beta_{14} + \beta_{15} + \beta_{16} ) q^{46}$$ $$+ ( \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} - \beta_{12} - 3 \beta_{13} - \beta_{16} - \beta_{18} - \beta_{19} ) q^{47}$$ $$+ ( -2 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{10} - \beta_{11} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{48}$$ $$- q^{49}$$ $$+ ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{15} - \beta_{16} - \beta_{17} - 2 \beta_{18} ) q^{50}$$ $$+ ( 1 + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} + \beta_{16} ) q^{51}$$ $$+ ( 2 \beta_{1} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} + \beta_{17} - \beta_{19} ) q^{52}$$ $$+ ( -2 \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} - \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{53}$$ $$+ \beta_{3} q^{54}$$ $$-\beta_{7} q^{55}$$ $$+ ( -\beta_{2} - \beta_{3} + \beta_{8} ) q^{56}$$ $$+ ( -\beta_{7} - \beta_{12} - \beta_{15} + \beta_{16} - \beta_{18} - \beta_{19} ) q^{57}$$ $$+ ( -4 \beta_{1} + \beta_{7} + \beta_{12} - 4 \beta_{13} - 2 \beta_{17} + \beta_{19} ) q^{58}$$ $$+ ( -1 + 2 \beta_{2} - 2 \beta_{3} + \beta_{7} - \beta_{8} - 2 \beta_{12} + \beta_{14} ) q^{59}$$ $$+ ( 1 + \beta_{1} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{16} + \beta_{19} ) q^{60}$$ $$+ ( 4 + \beta_{2} - \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} - 2 \beta_{14} ) q^{61}$$ $$+ ( 2 \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} + \beta_{12} + 3 \beta_{13} + \beta_{16} - 2 \beta_{17} - 3 \beta_{18} ) q^{62}$$ $$+ \beta_{13} q^{63}$$ $$+ ( 1 + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{11} + 3 \beta_{12} - 2 \beta_{14} ) q^{64}$$ $$+ ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + \beta_{13} + \beta_{14} - \beta_{15} + 2 \beta_{17} + \beta_{19} ) q^{65}$$ $$+ \beta_{3} q^{66}$$ $$+ ( -\beta_{4} - \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{16} + \beta_{17} + \beta_{19} ) q^{67}$$ $$+ ( -2 \beta_{1} - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} - \beta_{12} + 2 \beta_{13} + 2 \beta_{16} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{68}$$ $$+ ( -2 - \beta_{2} + \beta_{4} + 2 \beta_{6} - \beta_{10} - 2 \beta_{11} ) q^{69}$$ $$+ ( -\beta_{6} + \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} ) q^{70}$$ $$+ ( 3 - \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{14} ) q^{71}$$ $$+ ( \beta_{1} - \beta_{17} - \beta_{18} ) q^{72}$$ $$+ ( -2 \beta_{1} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{10} - \beta_{12} - \beta_{13} + \beta_{16} - \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{73}$$ $$+ ( -4 + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{74}$$ $$+ ( -\beta_{8} - \beta_{12} + \beta_{17} - \beta_{19} ) q^{75}$$ $$+ ( 1 - 2 \beta_{3} - \beta_{4} - 2 \beta_{7} + 2 \beta_{8} + \beta_{10} + 3 \beta_{12} - \beta_{14} ) q^{76}$$ $$+ \beta_{13} q^{77}$$ $$+ ( -\beta_{5} - \beta_{6} + \beta_{7} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{16} - \beta_{18} ) q^{78}$$ $$+ ( -2 - \beta_{2} - 2 \beta_{3} - \beta_{7} + 3 \beta_{8} + \beta_{12} - \beta_{15} - \beta_{16} ) q^{79}$$ $$+ ( -3 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} + \beta_{13} + 3 \beta_{14} + 2 \beta_{15} + \beta_{18} ) q^{80}$$ $$+ q^{81}$$ $$+ ( -2 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} + 2 \beta_{16} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{82}$$ $$+ ( -2 \beta_{1} - 2 \beta_{7} - 2 \beta_{12} + 4 \beta_{13} - \beta_{15} + \beta_{16} + \beta_{17} - 2 \beta_{18} - 2 \beta_{19} ) q^{83}$$ $$+ ( -1 + \beta_{2} ) q^{84}$$ $$+ ( -1 + 2 \beta_{1} - 2 \beta_{3} - \beta_{6} - \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} - \beta_{19} ) q^{85}$$ $$+ ( 5 + \beta_{2} + 2 \beta_{3} - \beta_{4} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} + 4 \beta_{11} + 3 \beta_{12} - \beta_{14} ) q^{86}$$ $$+ ( -2 \beta_{1} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{11} + \beta_{12} - 3 \beta_{13} + \beta_{15} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{87}$$ $$+ ( \beta_{1} - \beta_{17} - \beta_{18} ) q^{88}$$ $$+ ( \beta_{4} + 2 \beta_{6} - \beta_{7} - 3 \beta_{8} - \beta_{10} - 2 \beta_{11} - \beta_{12} + 2 \beta_{14} + \beta_{15} + \beta_{16} ) q^{89}$$ $$-\beta_{5} q^{90}$$ $$+ ( \beta_{9} + \beta_{15} + \beta_{16} ) q^{91}$$ $$+ ( -2 \beta_{1} - \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} + \beta_{12} - 3 \beta_{13} + 2 \beta_{15} + \beta_{16} + 3 \beta_{18} - 2 \beta_{19} ) q^{92}$$ $$+ ( \beta_{4} - \beta_{6} + \beta_{10} - \beta_{11} + \beta_{17} + \beta_{18} ) q^{93}$$ $$+ ( 2 + \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{15} - \beta_{16} ) q^{94}$$ $$+ ( 1 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{7} + 3 \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{95}$$ $$+ ( 1 - 2 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{6} + \beta_{9} - \beta_{10} - 3 \beta_{11} - \beta_{12} + \beta_{14} ) q^{96}$$ $$+ ( -\beta_{4} - 3 \beta_{6} + 2 \beta_{7} - \beta_{10} - 3 \beta_{11} + 2 \beta_{12} + \beta_{15} - \beta_{16} + \beta_{18} ) q^{97}$$ $$-\beta_{1} q^{98}$$ $$+ q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q$$ $$\mathstrut -\mathstrut 26q^{4}$$ $$\mathstrut -\mathstrut 2q^{5}$$ $$\mathstrut +\mathstrut 6q^{6}$$ $$\mathstrut -\mathstrut 20q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$20q$$ $$\mathstrut -\mathstrut 26q^{4}$$ $$\mathstrut -\mathstrut 2q^{5}$$ $$\mathstrut +\mathstrut 6q^{6}$$ $$\mathstrut -\mathstrut 20q^{9}$$ $$\mathstrut -\mathstrut 2q^{10}$$ $$\mathstrut -\mathstrut 20q^{11}$$ $$\mathstrut -\mathstrut 6q^{14}$$ $$\mathstrut +\mathstrut 2q^{15}$$ $$\mathstrut +\mathstrut 38q^{16}$$ $$\mathstrut +\mathstrut 2q^{19}$$ $$\mathstrut +\mathstrut 4q^{20}$$ $$\mathstrut +\mathstrut 20q^{21}$$ $$\mathstrut -\mathstrut 18q^{24}$$ $$\mathstrut +\mathstrut 12q^{25}$$ $$\mathstrut +\mathstrut 20q^{26}$$ $$\mathstrut -\mathstrut 38q^{29}$$ $$\mathstrut +\mathstrut 6q^{30}$$ $$\mathstrut +\mathstrut 20q^{31}$$ $$\mathstrut -\mathstrut 32q^{34}$$ $$\mathstrut -\mathstrut 2q^{35}$$ $$\mathstrut +\mathstrut 26q^{36}$$ $$\mathstrut +\mathstrut 2q^{40}$$ $$\mathstrut +\mathstrut 12q^{41}$$ $$\mathstrut +\mathstrut 26q^{44}$$ $$\mathstrut +\mathstrut 2q^{45}$$ $$\mathstrut +\mathstrut 8q^{46}$$ $$\mathstrut -\mathstrut 20q^{49}$$ $$\mathstrut -\mathstrut 6q^{50}$$ $$\mathstrut +\mathstrut 26q^{51}$$ $$\mathstrut -\mathstrut 6q^{54}$$ $$\mathstrut +\mathstrut 2q^{55}$$ $$\mathstrut +\mathstrut 18q^{56}$$ $$\mathstrut -\mathstrut 22q^{59}$$ $$\mathstrut +\mathstrut 16q^{60}$$ $$\mathstrut +\mathstrut 34q^{61}$$ $$\mathstrut -\mathstrut 26q^{64}$$ $$\mathstrut -\mathstrut 28q^{65}$$ $$\mathstrut -\mathstrut 6q^{66}$$ $$\mathstrut -\mathstrut 26q^{69}$$ $$\mathstrut -\mathstrut 6q^{70}$$ $$\mathstrut +\mathstrut 72q^{71}$$ $$\mathstrut -\mathstrut 72q^{74}$$ $$\mathstrut -\mathstrut 8q^{75}$$ $$\mathstrut +\mathstrut 44q^{76}$$ $$\mathstrut +\mathstrut 4q^{79}$$ $$\mathstrut -\mathstrut 8q^{80}$$ $$\mathstrut +\mathstrut 20q^{81}$$ $$\mathstrut -\mathstrut 26q^{84}$$ $$\mathstrut -\mathstrut 16q^{85}$$ $$\mathstrut +\mathstrut 52q^{86}$$ $$\mathstrut +\mathstrut 6q^{89}$$ $$\mathstrut +\mathstrut 2q^{90}$$ $$\mathstrut +\mathstrut 16q^{94}$$ $$\mathstrut +\mathstrut 14q^{95}$$ $$\mathstrut +\mathstrut 62q^{96}$$ $$\mathstrut +\mathstrut 20q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20}\mathstrut +\mathstrut$$ $$33$$ $$x^{18}\mathstrut +\mathstrut$$ $$456$$ $$x^{16}\mathstrut +\mathstrut$$ $$3426$$ $$x^{14}\mathstrut +\mathstrut$$ $$15194$$ $$x^{12}\mathstrut +\mathstrut$$ $$40320$$ $$x^{10}\mathstrut +\mathstrut$$ $$61593$$ $$x^{8}\mathstrut +\mathstrut$$ $$48545$$ $$x^{6}\mathstrut +\mathstrut$$ $$15624$$ $$x^{4}\mathstrut +\mathstrut$$ $$2116$$ $$x^{2}\mathstrut +\mathstrut$$ $$100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 3$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{18} - 65 \nu^{16} - 883 \nu^{14} - 6504 \nu^{12} - 28141 \nu^{10} - 72149 \nu^{8} - 104134 \nu^{6} - 72770 \nu^{4} - 15852 \nu^{2} - 1000$$$$)/20$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$5$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$2$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$167$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$72$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$2337$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$1090$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$17770$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$8984$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$79489$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$43540$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$210785$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$124604$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$314269$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$198808$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$225998$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$150428$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$50148$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$33220$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$3220$$ $$\nu\mathstrut -\mathstrut$$ $$2040$$$$)/20$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$10$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$337$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$23$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$4762$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$157$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$36585$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$198$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$165399$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$7591$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$443130$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$38433$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$666509$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$83507$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$481448$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$75668$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$105408$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$17612$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$6620$$ $$\nu\mathstrut +\mathstrut$$ $$1100$$$$)/40$$ $$\beta_{6}$$ $$=$$ $$($$$$6$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$5$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$194$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$167$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$2614$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$2337$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$19028$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$17770$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$81026$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$79489$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$203584$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$210785$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$286860$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$314269$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$195158$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$225998$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$41326$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$50148$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$2564$$ $$\nu\mathstrut +\mathstrut$$ $$3220$$$$)/20$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$11$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$10$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$364$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$337$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$5039$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$4762$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$37843$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$36585$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$166936$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$165399$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$435929$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$443130$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$639090$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$666509$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$450488$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$481448$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$96196$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$105408$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$5664$$ $$\nu\mathstrut -\mathstrut$$ $$6620$$$$)/40$$ $$\beta_{8}$$ $$=$$ $$($$$$-9 \nu^{18} - 296 \nu^{16} - 4067 \nu^{14} - 30273 \nu^{12} - 132214 \nu^{10} - 341693 \nu^{8} - 496350 \nu^{6} - 348454 \nu^{4} - 76008 \nu^{2} - 4740$$$$)/20$$ $$\beta_{9}$$ $$=$$ $$($$$$13 \nu^{18} + 413 \nu^{16} + 5445 \nu^{14} + 38586 \nu^{12} + 159025 \nu^{10} + 384351 \nu^{8} + 518257 \nu^{6} + 337002 \nu^{4} + 69240 \nu^{2} + 4100$$$$)/20$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$5$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$2$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$167$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$72$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$2337$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$1090$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$17770$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$8984$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$79489$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$43540$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$210785$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$124604$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$314269$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$198808$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$225998$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$150428$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$50148$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$33220$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$3220$$ $$\nu\mathstrut +\mathstrut$$ $$2040$$$$)/20$$ $$\beta_{11}$$ $$=$$ $$($$$$6$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$5$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$194$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$167$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$2614$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$2337$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$19028$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$17770$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$81026$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$79489$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$203584$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$210785$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$286860$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$314269$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$195158$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$225998$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$41326$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$50148$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$2564$$ $$\nu\mathstrut -\mathstrut$$ $$3220$$$$)/20$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$11$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$10$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$364$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$337$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$5039$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$4762$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$37843$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$36585$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$166936$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$165399$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$435929$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$443130$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$639090$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$666509$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$450488$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$481448$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$96196$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$105408$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$5664$$ $$\nu\mathstrut +\mathstrut$$ $$6620$$$$)/40$$ $$\beta_{13}$$ $$=$$ $$($$$$-10 \nu^{19} - 328 \nu^{17} - 4495 \nu^{15} - 33377 \nu^{13} - 145436 \nu^{11} - 375059 \nu^{9} - 543781 \nu^{7} - 381316 \nu^{5} - 83470 \nu^{3} - 5308 \nu$$$$)/20$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$11$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$32$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$364$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$1023$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$5039$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$13596$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$37843$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$97349$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$166936$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$406577$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$435929$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$999584$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$639090$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$1377875$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$450488$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$923432$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$96196$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$201024$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$5664$$ $$\nu\mathstrut -\mathstrut$$ $$12860$$$$)/40$$ $$\beta_{15}$$ $$=$$ $$($$$$24$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$795$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$23$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$11018$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$157$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$82855$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$198$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$366117$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$7591$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$958442$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$38433$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$1411273$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$83507$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$1004904$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$75668$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$223552$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$17612$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$14468$$ $$\nu\mathstrut +\mathstrut$$ $$1100$$$$)/40$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$24$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$795$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$23$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$11018$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$157$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$82855$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$198$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$366117$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$7591$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$958442$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$38433$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$1411273$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$83507$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$1004904$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$75668$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$223552$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$17612$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$14468$$ $$\nu\mathstrut +\mathstrut$$ $$1100$$$$)/40$$ $$\beta_{17}$$ $$=$$ $$($$$$28 \nu^{19} + 919 \nu^{17} + 12602 \nu^{15} + 93627 \nu^{13} + 408167 \nu^{11} + 1053028 \nu^{9} + 1527209 \nu^{7} + 1071178 \nu^{5} + 234558 \nu^{3} + 14924 \nu$$$$)/20$$ $$\beta_{18}$$ $$=$$ $$($$$$-28 \nu^{19} - 919 \nu^{17} - 12602 \nu^{15} - 93627 \nu^{13} - 408167 \nu^{11} - 1053028 \nu^{9} - 1527209 \nu^{7} - 1071178 \nu^{5} - 234538 \nu^{3} - 14824 \nu$$$$)/20$$ $$\beta_{19}$$ $$=$$ $$($$$$33 \nu^{19} + 1090 \nu^{17} + 15057 \nu^{15} + 112801 \nu^{13} + 496244 \nu^{11} + 1292171 \nu^{9} + 1889226 \nu^{7} + 1329130 \nu^{5} + 284588 \nu^{3} + 17340 \nu$$$$)/20$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$3$$ $$\nu^{3}$$ $$=$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-$$$$\beta_{16}\mathstrut -\mathstrut$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$14$$ $$\nu^{5}$$ $$=$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$9$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$10$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$29$$ $$\beta_{1}$$ $$\nu^{6}$$ $$=$$ $$10$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$10$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$12$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$13$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$11$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$9$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$13$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$11$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$22$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$48$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$75$$ $$\nu^{7}$$ $$=$$ $$-$$$$12$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$69$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$81$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$14$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$12$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$22$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$22$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$16$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$183$$ $$\beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-$$$$77$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$77$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$111$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$48$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$128$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$97$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$16$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$16$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$63$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$128$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$97$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$190$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$333$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$438$$ $$\nu^{9}$$ $$=$$ $$111$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$503$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$618$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$149$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$30$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$111$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$188$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$14$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$15$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$188$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$14$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$179$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$15$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$1215$$ $$\beta_{1}$$ $$\nu^{10}$$ $$=$$ $$542$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$542$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$940$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$535$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$1134$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$800$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$179$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$181$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$405$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$1134$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$800$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$1526$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$2336$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$2703$$ $$\nu^{11}$$ $$=$$ $$-$$$$940$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$3599$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$4621$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$1412$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$316$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$956$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$1482$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$142$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$153$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$1482$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$142$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$1728$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$153$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$8315$$ $$\beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-$$$$3669$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$3669$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$7647$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$5152$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$9528$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$6412$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$1728$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$1780$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$2495$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$9528$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$6412$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$11926$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$16535$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$17276$$ $$\nu^{13}$$ $$=$$ $$7647$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$25601$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$34328$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$12527$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$2902$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$8021$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$11316$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$1287$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$1336$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$11316$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$1287$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$15429$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$1336$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$57993$$ $$\beta_{1}$$ $$\nu^{14}$$ $$=$$ $$24390$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$24390$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$60876$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$45959$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$77641$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$50606$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$15429$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$16229$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$14917$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$77641$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$50606$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$92106$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$117922$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$113005$$ $$\nu^{15}$$ $$=$$ $$-$$$$60876$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$182131$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$254675$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$106552$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$24902$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$66250$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$85266$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$11070$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$10806$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$85266$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$11070$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$131454$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$10806$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$409725$$ $$\beta_{1}$$ $$\nu^{16}$$ $$=$$ $$-$$$$160845$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$160845$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$478251$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$391600$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$620511$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$395499$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$131454$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$141088$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$86651$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$620511$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$395499$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$707054$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$846531$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$751452$$ $$\nu^{17}$$ $$=$$ $$478251$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$1299675$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$1890748$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$880665$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$205648$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$539857$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$639096$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$92386$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$83924$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$639096$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$92386$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$1086313$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$83924$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$2922765$$ $$\beta_{1}$$ $$\nu^{18}$$ $$=$$ $$1058106$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$1058106$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$3724028$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$3238063$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$4894265$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$3069467$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$1086313$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$1187181$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$485965$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$4894265$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$3069467$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$5407666$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$6113188$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$5061191$$ $$\nu^{19}$$ $$=$$ $$-$$$$3724028$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$9315713$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$14057701$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$7131710$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$1658454$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$4344692$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$4782134$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$755429$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$637617$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$4782134$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$755429$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$8790164$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$637617$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$21011595$$ $$\beta_{1}$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$232$$ $$386$$ $$661$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
694.1
 − 2.74619i − 2.51841i − 2.51027i − 2.02181i − 1.76958i − 1.76011i − 1.41336i − 0.445112i − 0.437361i − 0.332438i 0.332438i 0.437361i 0.445112i 1.41336i 1.76011i 1.76958i 2.02181i 2.51027i 2.51841i 2.74619i
2.74619i 1.00000i −5.54154 2.23561 + 0.0452669i 2.74619 1.00000i 9.72574i −1.00000 0.124311 6.13940i
694.2 2.51841i 1.00000i −4.34238 −2.17741 0.508790i 2.51841 1.00000i 5.89907i −1.00000 −1.28134 + 5.48362i
694.3 2.51027i 1.00000i −4.30144 −1.80593 1.31856i −2.51027 1.00000i 5.77723i −1.00000 −3.30994 + 4.53337i
694.4 2.02181i 1.00000i −2.08773 −0.549616 + 2.16747i 2.02181 1.00000i 0.177380i −1.00000 4.38222 + 1.11122i
694.5 1.76958i 1.00000i −1.13141 −1.09533 + 1.94942i −1.76958 1.00000i 1.53703i −1.00000 3.44966 + 1.93828i
694.6 1.76011i 1.00000i −1.09799 2.22415 0.230563i −1.76011 1.00000i 1.58764i −1.00000 −0.405817 3.91475i
694.7 1.41336i 1.00000i 0.00241967 1.69889 1.45388i 1.41336 1.00000i 2.83014i −1.00000 −2.05485 2.40114i
694.8 0.445112i 1.00000i 1.80187 −2.13786 0.655413i 0.445112 1.00000i 1.69226i −1.00000 −0.291732 + 0.951587i
694.9 0.437361i 1.00000i 1.80872 0.978157 2.01077i −0.437361 1.00000i 1.66578i −1.00000 −0.879433 0.427807i
694.10 0.332438i 1.00000i 1.88948 −0.370655 2.20513i 0.332438 1.00000i 1.29301i −1.00000 −0.733071 + 0.123220i
694.11 0.332438i 1.00000i 1.88948 −0.370655 + 2.20513i 0.332438 1.00000i 1.29301i −1.00000 −0.733071 0.123220i
694.12 0.437361i 1.00000i 1.80872 0.978157 + 2.01077i −0.437361 1.00000i 1.66578i −1.00000 −0.879433 + 0.427807i
694.13 0.445112i 1.00000i 1.80187 −2.13786 + 0.655413i 0.445112 1.00000i 1.69226i −1.00000 −0.291732 0.951587i
694.14 1.41336i 1.00000i 0.00241967 1.69889 + 1.45388i 1.41336 1.00000i 2.83014i −1.00000 −2.05485 + 2.40114i
694.15 1.76011i 1.00000i −1.09799 2.22415 + 0.230563i −1.76011 1.00000i 1.58764i −1.00000 −0.405817 + 3.91475i
694.16 1.76958i 1.00000i −1.13141 −1.09533 1.94942i −1.76958 1.00000i 1.53703i −1.00000 3.44966 1.93828i
694.17 2.02181i 1.00000i −2.08773 −0.549616 2.16747i 2.02181 1.00000i 0.177380i −1.00000 4.38222 1.11122i
694.18 2.51027i 1.00000i −4.30144 −1.80593 + 1.31856i −2.51027 1.00000i 5.77723i −1.00000 −3.30994 4.53337i
694.19 2.51841i 1.00000i −4.34238 −2.17741 + 0.508790i 2.51841 1.00000i 5.89907i −1.00000 −1.28134 5.48362i
694.20 2.74619i 1.00000i −5.54154 2.23561 0.0452669i 2.74619 1.00000i 9.72574i −1.00000 0.124311 + 6.13940i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 694.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.b Even 1 yes

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1155, [\chi])$$:

 $$T_{2}^{20} + \cdots$$ $$T_{13}^{20} + \cdots$$