# Properties

 Label 273.2.t.d Level $273$ Weight $2$ Character orbit 273.t Analytic conductor $2.180$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.t (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 33 x^{18} + 455 x^{16} + 3403 x^{14} + 15006 x^{12} + 39799 x^{10} + 62505 x^{8} + 55993 x^{6} + 27166 x^{4} + 6435 x^{2} + 576$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} + \beta_{3} ) q^{2} + \beta_{11} q^{3} + ( -1 + \beta_{1} ) q^{4} -\beta_{8} q^{5} + \beta_{2} q^{6} + ( \beta_{12} + \beta_{14} ) q^{7} + ( -\beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{8} + ( -1 + \beta_{11} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} + \beta_{3} ) q^{2} + \beta_{11} q^{3} + ( -1 + \beta_{1} ) q^{4} -\beta_{8} q^{5} + \beta_{2} q^{6} + ( \beta_{12} + \beta_{14} ) q^{7} + ( -\beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{8} + ( -1 + \beta_{11} ) q^{9} + ( -1 + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} - \beta_{13} + \beta_{14} - \beta_{16} + \beta_{18} ) q^{10} + ( -1 - \beta_{3} - \beta_{4} - \beta_{7} - \beta_{10} + \beta_{16} - \beta_{18} + \beta_{19} ) q^{11} + ( -\beta_{11} - \beta_{17} ) q^{12} + ( 1 + \beta_{3} + \beta_{6} + \beta_{7} ) q^{13} + ( \beta_{2} + \beta_{4} - \beta_{6} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{18} - \beta_{19} ) q^{14} -\beta_{7} q^{15} + ( 1 - \beta_{1} - \beta_{4} + \beta_{5} - \beta_{14} - \beta_{15} - \beta_{18} + \beta_{19} ) q^{16} + ( 2 - \beta_{2} - \beta_{5} + \beta_{8} - \beta_{10} + \beta_{14} + \beta_{15} ) q^{17} -\beta_{3} q^{18} + ( \beta_{5} + \beta_{7} + \beta_{18} ) q^{19} + ( \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} - \beta_{16} ) q^{20} + \beta_{14} q^{21} + ( -1 - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{22} + ( -2 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{8} + \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{23} + ( -\beta_{2} + \beta_{5} - \beta_{18} + \beta_{19} ) q^{24} + ( 1 - \beta_{1} - \beta_{4} - \beta_{5} - \beta_{11} + \beta_{12} + \beta_{15} - \beta_{17} - \beta_{19} ) q^{25} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{26} - q^{27} + ( 1 - \beta_{1} - \beta_{4} + \beta_{8} - 2 \beta_{11} - \beta_{12} - \beta_{14} - 2 \beta_{17} - 2 \beta_{18} ) q^{28} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{16} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{29} + ( -1 + \beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} - \beta_{15} ) q^{30} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} - 2 \beta_{14} - \beta_{15} + \beta_{16} - \beta_{18} + \beta_{19} ) q^{31} + ( \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + 2 \beta_{17} - \beta_{19} ) q^{32} + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{18} ) q^{33} + ( \beta_{2} + \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} + \beta_{19} ) q^{34} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{9} - \beta_{11} - \beta_{17} - 2 \beta_{19} ) q^{35} + ( 1 - \beta_{1} - \beta_{11} - \beta_{17} ) q^{36} + ( \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{7} + \beta_{10} - 2 \beta_{11} + 2 \beta_{17} + \beta_{18} + \beta_{19} ) q^{37} + ( 1 - \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{38} + ( \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{39} + ( 2 - 2 \beta_{2} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} - 2 \beta_{18} - \beta_{19} ) q^{40} + ( 1 - \beta_{1} + \beta_{4} + \beta_{11} - \beta_{12} + \beta_{15} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{41} + ( 1 - \beta_{3} + \beta_{5} - \beta_{11} - \beta_{15} + \beta_{16} ) q^{42} + ( -\beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{13} - \beta_{14} - \beta_{16} + \beta_{17} ) q^{43} + ( 1 - \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{16} + 2 \beta_{18} - \beta_{19} ) q^{44} + ( -\beta_{7} + \beta_{8} ) q^{45} + ( -2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{8} + \beta_{14} - \beta_{15} + \beta_{18} + \beta_{19} ) q^{46} + ( 1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} - \beta_{17} - \beta_{18} ) q^{47} + ( -\beta_{4} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} + \beta_{17} + \beta_{19} ) q^{48} + ( -3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} + 2 \beta_{16} - \beta_{18} ) q^{49} + ( -2 + \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{17} ) q^{50} + ( \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} + 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{15} - \beta_{16} - \beta_{19} ) q^{51} + ( -4 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{18} - \beta_{19} ) q^{52} + ( -3 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} + 3 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{16} + \beta_{17} + \beta_{19} ) q^{53} + ( -\beta_{2} - \beta_{3} ) q^{54} + ( 1 - \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{55} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{11} + 2 \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} + 2 \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{56} + ( \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} + \beta_{18} + \beta_{19} ) q^{57} + ( 3 - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{18} - \beta_{19} ) q^{58} + ( 3 \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{8} + \beta_{9} - \beta_{11} + \beta_{14} - \beta_{15} + \beta_{16} + \beta_{19} ) q^{59} + ( -1 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{60} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{61} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{62} -\beta_{12} q^{63} + ( -3 + \beta_{1} + 3 \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} + 2 \beta_{15} - \beta_{16} + 3 \beta_{18} - 2 \beta_{19} ) q^{64} + ( -3 - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} + 2 \beta_{12} + \beta_{13} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{65} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{18} ) q^{66} + ( -2 \beta_{2} - \beta_{8} + \beta_{12} + \beta_{14} + \beta_{15} ) q^{67} + ( 1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{16} - \beta_{18} - \beta_{19} ) q^{68} + ( -\beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{9} - 2 \beta_{11} + \beta_{13} - \beta_{14} + \beta_{16} - \beta_{17} - \beta_{18} ) q^{69} + ( -3 + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - 3 \beta_{6} - 3 \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} - \beta_{18} + \beta_{19} ) q^{70} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{71} + ( \beta_{3} - \beta_{4} - \beta_{18} + \beta_{19} ) q^{72} + ( -\beta_{1} + \beta_{3} + \beta_{6} - \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} - \beta_{19} ) q^{73} + ( 1 + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} + 3 \beta_{18} - 2 \beta_{19} ) q^{74} + ( 1 - \beta_{1} + \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} + \beta_{18} - \beta_{19} ) q^{75} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - 3 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{15} + \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{76} + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{16} - \beta_{19} ) q^{77} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} - \beta_{13} + \beta_{14} ) q^{78} + ( \beta_{4} + 2 \beta_{7} - \beta_{8} + 2 \beta_{11} + \beta_{13} - \beta_{14} - 2 \beta_{17} - \beta_{19} ) q^{79} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} + \beta_{16} + \beta_{17} - \beta_{19} ) q^{80} -\beta_{11} q^{81} + ( -2 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} - \beta_{15} + \beta_{16} - 2 \beta_{17} - \beta_{18} + \beta_{19} ) q^{82} + ( -1 - \beta_{2} - \beta_{3} + 4 \beta_{4} - 4 \beta_{6} - 2 \beta_{9} + 2 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} - 2 \beta_{16} + \beta_{18} - \beta_{19} ) q^{83} + ( 2 - 2 \beta_{1} - 2 \beta_{4} + \beta_{7} - \beta_{11} - \beta_{14} - \beta_{17} - \beta_{18} ) q^{84} + ( -5 - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} + 4 \beta_{11} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{85} + ( 1 + 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{10} + \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} - \beta_{16} + 3 \beta_{18} - 2 \beta_{19} ) q^{86} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{7} + \beta_{10} - \beta_{14} - \beta_{15} + \beta_{18} - \beta_{19} ) q^{87} + ( 1 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 6 \beta_{7} + 4 \beta_{8} - \beta_{9} - \beta_{10} - 4 \beta_{11} - 2 \beta_{17} - \beta_{18} - \beta_{19} ) q^{88} + ( -\beta_{4} - \beta_{5} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{18} - \beta_{19} ) q^{89} + ( -\beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} - \beta_{18} ) q^{90} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} + \beta_{9} + \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} - \beta_{16} - \beta_{17} + 2 \beta_{18} ) q^{91} + ( 4 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} - 2 \beta_{15} - \beta_{16} + \beta_{18} ) q^{92} + ( 1 - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{16} - \beta_{18} + \beta_{19} ) q^{93} + ( 1 + 2 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{13} - \beta_{14} + \beta_{17} - 3 \beta_{18} ) q^{94} + ( -5 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} - 2 \beta_{18} + 2 \beta_{19} ) q^{95} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{16} + \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{96} + ( -2 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - \beta_{8} + 2 \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} - 2 \beta_{16} ) q^{97} + ( -1 - \beta_{1} - 3 \beta_{3} + \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - \beta_{8} + 4 \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{98} + ( 1 - \beta_{2} - \beta_{5} + \beta_{8} - \beta_{9} - \beta_{11} - \beta_{16} - \beta_{19} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 10q^{3} - 26q^{4} + 6q^{5} + 3q^{6} + 2q^{7} - 10q^{9} + O(q^{10})$$ $$20q + 10q^{3} - 26q^{4} + 6q^{5} + 3q^{6} + 2q^{7} - 10q^{9} + 2q^{10} - 12q^{11} - 13q^{12} + 8q^{13} + 2q^{14} + 6q^{15} + 42q^{16} + 16q^{17} + 3q^{18} - 9q^{19} - 5q^{21} - 9q^{22} - 36q^{23} + 3q^{24} + 12q^{25} - 16q^{26} - 20q^{27} - 2q^{28} - 3q^{29} - 2q^{30} - 18q^{31} - 12q^{33} + 18q^{35} + 13q^{36} + 9q^{38} + 7q^{39} + 5q^{40} + 21q^{41} + 16q^{42} + 16q^{43} - 6q^{44} + 21q^{47} + 21q^{48} - 24q^{49} - 54q^{50} + 8q^{51} - 41q^{52} - 26q^{53} + 17q^{55} - 6q^{56} + 42q^{58} + 4q^{62} - 7q^{63} - 46q^{64} - 50q^{65} + 9q^{66} - 3q^{67} + 6q^{68} - 18q^{69} + 15q^{71} + 3q^{72} - 9q^{73} + 12q^{74} + 24q^{75} + 75q^{76} + 20q^{77} - 32q^{78} + 3q^{79} - 24q^{80} - 10q^{81} + 15q^{82} + 41q^{84} - 78q^{85} + 3q^{86} - 6q^{87} - 22q^{88} - 4q^{90} + 4q^{91} + 142q^{92} + 36q^{94} - 84q^{95} - 24q^{96} - 15q^{97} + 81q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 33 x^{18} + 455 x^{16} + 3403 x^{14} + 15006 x^{12} + 39799 x^{10} + 62505 x^{8} + 55993 x^{6} + 27166 x^{4} + 6435 x^{2} + 576$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + 3$$ $$\beta_{2}$$ $$=$$ $$($$$$-2116 \nu^{18} - 67386 \nu^{16} - 887545 \nu^{14} - 6247121 \nu^{12} - 25326750 \nu^{10} - 59379234 \nu^{8} - 76822211 \nu^{6} - 49872829 \nu^{4} - 14373941 \nu^{2} + 39407 \nu - 1430232$$$$)/78814$$ $$\beta_{3}$$ $$=$$ $$($$$$2116 \nu^{18} + 67386 \nu^{16} + 887545 \nu^{14} + 6247121 \nu^{12} + 25326750 \nu^{10} + 59379234 \nu^{8} + 76822211 \nu^{6} + 49872829 \nu^{4} + 14373941 \nu^{2} + 39407 \nu + 1430232$$$$)/78814$$ $$\beta_{4}$$ $$=$$ $$($$$$-43788 \nu^{18} - 1443263 \nu^{16} - 19879084 \nu^{14} - 148563367 \nu^{12} - 654599666 \nu^{10} - 1731744079 \nu^{8} - 2688768170 \nu^{6} - 2301207896 \nu^{4} + 669919 \nu^{3} - 954730321 \nu^{2} + 3349595 \nu - 138397894$$$$)/1339838$$ $$\beta_{5}$$ $$=$$ $$($$$$43788 \nu^{18} + 1443263 \nu^{16} + 19879084 \nu^{14} + 148563367 \nu^{12} + 654599666 \nu^{10} + 1731744079 \nu^{8} + 2688768170 \nu^{6} + 2301207896 \nu^{4} + 669919 \nu^{3} + 954730321 \nu^{2} + 3349595 \nu + 138397894$$$$)/1339838$$ $$\beta_{6}$$ $$=$$ $$($$$$-39625 \nu^{19} - 283077 \nu^{18} - 1345812 \nu^{17} - 9180339 \nu^{16} - 19234190 \nu^{15} - 123682944 \nu^{14} - 150601135 \nu^{13} - 895927566 \nu^{12} - 704477745 \nu^{11} - 3771499899 \nu^{10} - 2011864318 \nu^{9} - 9313020981 \nu^{8} - 3429634107 \nu^{7} - 13009680222 \nu^{6} - 3240922843 \nu^{5} - 9527147499 \nu^{4} - 1443336925 \nu^{3} - 3259390377 \nu^{2} - 215895534 \nu - 394130712$$$$)/4019514$$ $$\beta_{7}$$ $$=$$ $$($$$$43633 \nu^{19} - 161830 \nu^{18} + 1428531 \nu^{17} - 5224322 \nu^{16} + 19439089 \nu^{15} - 69879294 \nu^{14} + 142303411 \nu^{13} - 500562492 \nu^{12} + 605956008 \nu^{11} - 2070866896 \nu^{10} + 1517064371 \nu^{9} - 4973995106 \nu^{8} + 2163227429 \nu^{7} - 6638969622 \nu^{6} + 1650229279 \nu^{5} - 4517188760 \nu^{4} + 618127568 \nu^{3} - 1415365554 \nu^{2} + 86613417 \nu - 163312344$$$$)/2679676$$ $$\beta_{8}$$ $$=$$ $$($$$$-43633 \nu^{19} - 161830 \nu^{18} - 1428531 \nu^{17} - 5224322 \nu^{16} - 19439089 \nu^{15} - 69879294 \nu^{14} - 142303411 \nu^{13} - 500562492 \nu^{12} - 605956008 \nu^{11} - 2070866896 \nu^{10} - 1517064371 \nu^{9} - 4973995106 \nu^{8} - 2163227429 \nu^{7} - 6638969622 \nu^{6} - 1650229279 \nu^{5} - 4517188760 \nu^{4} - 618127568 \nu^{3} - 1415365554 \nu^{2} - 86613417 \nu - 163312344$$$$)/2679676$$ $$\beta_{9}$$ $$=$$ $$($$$$-50615 \nu^{19} + 177050 \nu^{18} - 1547557 \nu^{17} + 5676836 \nu^{16} - 19031503 \nu^{15} + 75395618 \nu^{14} - 118377513 \nu^{13} + 535964342 \nu^{12} - 372728110 \nu^{11} + 2197560554 \nu^{10} - 426614751 \nu^{9} + 5215268002 \nu^{8} + 539531249 \nu^{7} + 6829125376 \nu^{6} + 1745722461 \nu^{5} + 4482267922 \nu^{4} + 1220725582 \nu^{3} + 1305556474 \nu^{2} + 215671875 \nu + 132712408$$$$)/2679676$$ $$\beta_{10}$$ $$=$$ $$($$$$-43633 \nu^{19} - 379746 \nu^{18} - 1428531 \nu^{17} - 12287124 \nu^{16} - 19439089 \nu^{15} - 164855188 \nu^{14} - 142303411 \nu^{13} - 1185603732 \nu^{12} - 605956008 \nu^{11} - 4929224596 \nu^{10} - 1517064371 \nu^{9} - 11908974300 \nu^{8} - 2163227429 \nu^{7} - 15995718404 \nu^{6} - 1650229279 \nu^{5} - 10921960114 \nu^{4} - 619467406 \nu^{3} - 3363044754 \nu^{2} - 94652445 \nu - 360716608$$$$)/2679676$$ $$\beta_{11}$$ $$=$$ $$($$$$59593 \nu^{19} + 1915785 \nu^{17} + 25497551 \nu^{15} + 181493899 \nu^{13} + 744321654 \nu^{11} + 1763899807 \nu^{9} + 2299758849 \nu^{7} + 1493057785 \nu^{5} + 421955542 \nu^{3} + 38506371 \nu + 945768$$$$)/1891536$$ $$\beta_{12}$$ $$=$$ $$($$$$347073 \nu^{19} - 375400 \nu^{18} + 11381641 \nu^{17} - 12340840 \nu^{16} + 155349919 \nu^{15} - 168898896 \nu^{14} + 1142984483 \nu^{13} - 1246257184 \nu^{12} + 4905144382 \nu^{11} - 5364298064 \nu^{10} + 12418566743 \nu^{9} - 13625130800 \nu^{8} + 17948969089 \nu^{7} - 19789044016 \nu^{6} + 13778780745 \nu^{5} - 15372041520 \nu^{4} + 4984192438 \nu^{3} - 5721815032 \nu^{2} + 642445051 \nu - 749416136$$$$)/10718704$$ $$\beta_{13}$$ $$=$$ $$($$$$347073 \nu^{19} + 375400 \nu^{18} + 11381641 \nu^{17} + 12340840 \nu^{16} + 155349919 \nu^{15} + 168898896 \nu^{14} + 1142984483 \nu^{13} + 1246257184 \nu^{12} + 4905144382 \nu^{11} + 5364298064 \nu^{10} + 12418566743 \nu^{9} + 13625130800 \nu^{8} + 17948969089 \nu^{7} + 19789044016 \nu^{6} + 13778780745 \nu^{5} + 15372041520 \nu^{4} + 4984192438 \nu^{3} + 5721815032 \nu^{2} + 642445051 \nu + 749416136$$$$)/10718704$$ $$\beta_{14}$$ $$=$$ $$($$$$1741799 \nu^{19} - 1134696 \nu^{18} + 56733111 \nu^{17} - 36066624 \nu^{16} + 768476209 \nu^{15} - 472206984 \nu^{14} + 5605555037 \nu^{13} - 3278410032 \nu^{12} + 23821454658 \nu^{11} - 12911657208 \nu^{10} + 59634435401 \nu^{9} - 28476462456 \nu^{8} + 85114772007 \nu^{7} - 32097331944 \nu^{6} + 64636511399 \nu^{5} - 14477568624 \nu^{4} + 23492280146 \nu^{3} - 831211848 \nu^{2} + 3020817237 \nu + 354108216$$$$)/32156112$$ $$\beta_{15}$$ $$=$$ $$($$$$-1741799 \nu^{19} - 1134696 \nu^{18} - 56733111 \nu^{17} - 36066624 \nu^{16} - 768476209 \nu^{15} - 472206984 \nu^{14} - 5605555037 \nu^{13} - 3278410032 \nu^{12} - 23821454658 \nu^{11} - 12911657208 \nu^{10} - 59634435401 \nu^{9} - 28476462456 \nu^{8} - 85114772007 \nu^{7} - 32097331944 \nu^{6} - 64636511399 \nu^{5} - 14477568624 \nu^{4} - 23492280146 \nu^{3} - 831211848 \nu^{2} - 3020817237 \nu + 354108216$$$$)/32156112$$ $$\beta_{16}$$ $$=$$ $$($$$$-310569 \nu^{19} - 344756 \nu^{18} - 10038565 \nu^{17} - 11208964 \nu^{16} - 134537471 \nu^{15} - 151495600 \nu^{14} - 966428131 \nu^{13} - 1101847768 \nu^{12} - 4013273958 \nu^{11} - 4662693032 \nu^{10} - 9687433523 \nu^{9} - 11594661228 \nu^{8} - 13019367253 \nu^{7} - 16362724324 \nu^{6} - 8955349613 \nu^{5} - 12197312564 \nu^{4} - 2861054306 \nu^{3} - 4344638492 \nu^{2} - 346068823 \nu - 567210144$$$$)/5359352$$ $$\beta_{17}$$ $$=$$ $$($$$$-42665 \nu^{19} - 1376697 \nu^{17} - 18397191 \nu^{15} - 131516931 \nu^{13} - 541707654 \nu^{11} - 1288865935 \nu^{9} - 1685181161 \nu^{7} - 1094075153 \nu^{5} - 306964014 \nu^{3} - 315256 \nu^{2} - 27064515 \nu - 945768$$$$)/630512$$ $$\beta_{18}$$ $$=$$ $$($$$$-2792711 \nu^{19} + 2185608 \nu^{18} - 91371423 \nu^{17} + 70704936 \nu^{16} - 1245574225 \nu^{15} + 949305000 \nu^{14} - 9171075845 \nu^{13} + 6843930840 \nu^{12} - 39531846642 \nu^{11} + 28622049192 \nu^{10} - 101196293297 \nu^{9} + 70038320352 \nu^{8} - 149645208087 \nu^{7} + 96627768024 \nu^{6} - 119865500903 \nu^{5} + 69690480072 \nu^{4} - 46421885906 \nu^{3} + 23632193160 \nu^{2} - 6454913085 \nu + 2870972904$$$$)/32156112$$ $$\beta_{19}$$ $$=$$ $$($$$$-2792711 \nu^{19} - 2185608 \nu^{18} - 91371423 \nu^{17} - 70704936 \nu^{16} - 1245574225 \nu^{15} - 949305000 \nu^{14} - 9171075845 \nu^{13} - 6843930840 \nu^{12} - 39531846642 \nu^{11} - 28622049192 \nu^{10} - 101196293297 \nu^{9} - 70038320352 \nu^{8} - 149645208087 \nu^{7} - 96627768024 \nu^{6} - 119865500903 \nu^{5} - 69690480072 \nu^{4} - 46421885906 \nu^{3} - 23632193160 \nu^{2} - 6454913085 \nu - 2870972904$$$$)/32156112$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{3} + \beta_{2}$$ $$\nu^{2}$$ $$=$$ $$\beta_{1} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} - 5 \beta_{3} - 5 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$\beta_{19} - \beta_{18} - \beta_{15} - \beta_{14} + \beta_{5} - \beta_{4} - 7 \beta_{1} + 15$$ $$\nu^{5}$$ $$=$$ $$-\beta_{19} + 2 \beta_{17} - \beta_{16} + \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} - \beta_{9} + \beta_{8} - 10 \beta_{5} - 9 \beta_{4} + 29 \beta_{3} + 28 \beta_{2} + \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-12 \beta_{19} + 13 \beta_{18} - \beta_{16} + 12 \beta_{15} + 12 \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - 11 \beta_{5} + 13 \beta_{4} - \beta_{3} + 3 \beta_{2} + 47 \beta_{1} - 89$$ $$\nu^{7}$$ $$=$$ $$10 \beta_{19} - 28 \beta_{17} + 10 \beta_{16} - 12 \beta_{15} + 12 \beta_{14} - 15 \beta_{13} - 15 \beta_{12} - 24 \beta_{11} + 2 \beta_{10} + 10 \beta_{9} - 11 \beta_{8} - \beta_{7} - 4 \beta_{6} + 81 \beta_{5} + 73 \beta_{4} - 180 \beta_{3} - 168 \beta_{2} - 14 \beta_{1} + 5$$ $$\nu^{8}$$ $$=$$ $$114 \beta_{19} - 128 \beta_{18} + 14 \beta_{16} - 111 \beta_{15} - 111 \beta_{14} + 18 \beta_{13} - 18 \beta_{12} - 14 \beta_{11} - 13 \beta_{10} - 14 \beta_{9} + 27 \beta_{8} + 97 \beta_{5} - 124 \beta_{4} + 19 \beta_{3} - 46 \beta_{2} - 322 \beta_{1} + 570$$ $$\nu^{9}$$ $$=$$ $$-75 \beta_{19} + 4 \beta_{18} + 294 \beta_{17} - 79 \beta_{16} + 103 \beta_{15} - 103 \beta_{14} + 159 \beta_{13} + 159 \beta_{12} + 311 \beta_{11} - 31 \beta_{10} - 79 \beta_{9} + 91 \beta_{8} + 19 \beta_{7} + 62 \beta_{6} - 618 \beta_{5} - 570 \beta_{4} + 1173 \beta_{3} + 1063 \beta_{2} + 147 \beta_{1} - 85$$ $$\nu^{10}$$ $$=$$ $$-1003 \beta_{19} + 1143 \beta_{18} - 140 \beta_{16} + 934 \beta_{15} + 934 \beta_{14} - 213 \beta_{13} + 213 \beta_{12} + 140 \beta_{11} + 128 \beta_{10} + 140 \beta_{9} - 262 \beta_{8} + 6 \beta_{7} - 796 \beta_{5} + 1064 \beta_{4} - 235 \beta_{3} + 503 \beta_{2} + 2255 \beta_{1} - 3823$$ $$\nu^{11}$$ $$=$$ $$502 \beta_{19} - 85 \beta_{18} - 2756 \beta_{17} + 587 \beta_{16} - 776 \beta_{15} + 776 \beta_{14} - 1478 \beta_{13} - 1478 \beta_{12} - 3239 \beta_{11} + 329 \beta_{10} + 587 \beta_{9} - 685 \beta_{8} - 231 \beta_{7} - 658 \beta_{6} + 4621 \beta_{5} + 4363 \beta_{4} - 7935 \beta_{3} - 7019 \beta_{2} - 1378 \beta_{1} + 997$$ $$\nu^{12}$$ $$=$$ $$8493 \beta_{19} - 9733 \beta_{18} + 1240 \beta_{16} - 7520 \beta_{15} - 7520 \beta_{14} + 2123 \beta_{13} - 2123 \beta_{12} - 1240 \beta_{11} - 1149 \beta_{10} - 1240 \beta_{9} + 2254 \beta_{8} - 135 \beta_{7} + 6328 \beta_{5} - 8717 \beta_{4} + 2428 \beta_{3} - 4817 \beta_{2} - 16088 \beta_{1} + 26481$$ $$\nu^{13}$$ $$=$$ $$-3141 \beta_{19} + 1151 \beta_{18} + 24322 \beta_{17} - 4292 \beta_{16} + 5486 \beta_{15} - 5486 \beta_{14} + 12870 \beta_{13} + 12870 \beta_{12} + 30364 \beta_{11} - 3002 \beta_{10} - 4292 \beta_{9} + 4945 \beta_{8} + 2349 \beta_{7} + 6004 \beta_{6} - 34359 \beta_{5} - 33069 \beta_{4} + 55248 \beta_{3} + 47954 \beta_{2} + 12161 \beta_{1} - 10034$$ $$\nu^{14}$$ $$=$$ $$-70218 \beta_{19} + 80627 \beta_{18} - 10409 \beta_{16} + 59217 \beta_{15} + 59217 \beta_{14} - 19372 \beta_{13} + 19372 \beta_{12} + 10409 \beta_{11} + 9868 \beta_{10} + 10409 \beta_{9} - 18356 \beta_{8} + 1921 \beta_{7} - 49522 \beta_{5} + 69799 \beta_{4} - 22796 \beta_{3} + 43073 \beta_{2} + 116533 \beta_{1} - 187868$$ $$\nu^{15}$$ $$=$$ $$18608 \beta_{19} - 12749 \beta_{18} - 206846 \beta_{17} + 31357 \beta_{16} - 37505 \beta_{15} + 37505 \beta_{14} - 107946 \beta_{13} - 107946 \beta_{12} - 268249 \beta_{11} + 25398 \beta_{10} + 31357 \beta_{9} - 34921 \beta_{8} - 21834 \beta_{7} - 50796 \beta_{6} + 255613 \beta_{5} + 249654 \beta_{4} - 393320 \beta_{3} - 336565 \beta_{2} - 103423 \beta_{1} + 93048$$ $$\nu^{16}$$ $$=$$ $$571047 \beta_{19} - 656179 \beta_{18} + 85132 \beta_{16} - 461045 \beta_{15} - 461045 \beta_{14} + 167927 \beta_{13} - 167927 \beta_{12} - 85132 \beta_{11} - 82548 \beta_{10} - 85132 \beta_{9} + 145451 \beta_{8} - 22229 \beta_{7} + 384434 \beta_{5} - 552114 \beta_{4} + 202013 \beta_{3} - 369693 \beta_{2} - 854581 \beta_{1} + 1357352$$ $$\nu^{17}$$ $$=$$ $$-103921 \beta_{19} + 126294 \beta_{18} + 1716384 \beta_{17} - 230215 \beta_{16} + 251956 \beta_{15} - 251956 \beta_{14} + 884425 \beta_{13} + 884425 \beta_{12} + 2283771 \beta_{11} - 206017 \beta_{10} - 230215 \beta_{9} + 243492 \beta_{8} + 192740 \beta_{7} + 412034 \beta_{6} - 1907799 \beta_{5} - 1883601 \beta_{4} + 2848680 \beta_{3} + 2412448 \beta_{2} + 858192 \beta_{1} - 820761$$ $$\nu^{18}$$ $$=$$ $$-4588963 \beta_{19} + 5275835 \beta_{18} - 686872 \beta_{16} + 3569231 \beta_{15} + 3569231 \beta_{14} - 1409476 \beta_{13} + 1409476 \beta_{12} + 686872 \beta_{11} + 678408 \beta_{10} + 686872 \beta_{9} - 1136381 \beta_{8} + 228899 \beta_{7} - 2972008 \beta_{5} + 4337288 \beta_{4} - 1723979 \beta_{3} + 3089259 \beta_{2} + 6330142 \beta_{1} - 9946338$$ $$\nu^{19}$$ $$=$$ $$534336 \beta_{19} - 1167446 \beta_{18} - 13999628 \beta_{17} + 1701782 \beta_{16} - 1679181 \beta_{15} + 1679181 \beta_{14} - 7134820 \beta_{13} - 7134820 \beta_{12} - 18968912 \beta_{11} + 1630086 \beta_{10} + 1701782 \beta_{9} - 1685029 \beta_{8} - 1646839 \beta_{7} - 3260172 \beta_{6} + 14300068 \beta_{5} + 14228372 \beta_{4} - 20909663 \beta_{3} - 17577795 \beta_{2} - 6999814 \beta_{1} + 7003479$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$\beta_{11}$$ $$-\beta_{11}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 − 2.50900i − 2.44406i − 2.04830i − 0.915396i − 0.560998i − 0.493650i 0.871638i 1.46676i 2.11978i 2.78118i − 2.78118i − 2.11978i − 1.46676i − 0.871638i 0.493650i 0.560998i 0.915396i 2.04830i 2.44406i 2.50900i
2.50900i 0.500000 + 0.866025i −4.29509 −2.97008 + 1.71478i 2.17286 1.25450i −0.0473512 + 2.64533i 5.75840i −0.500000 + 0.866025i 4.30238 + 7.45195i
4.2 2.44406i 0.500000 + 0.866025i −3.97343 3.36767 1.94432i 2.11662 1.22203i 1.82552 + 1.91507i 4.82318i −0.500000 + 0.866025i −4.75204 8.23078i
4.3 2.04830i 0.500000 + 0.866025i −2.19554 0.341406 0.197111i 1.77388 1.02415i −0.908042 2.48505i 0.400534i −0.500000 + 0.866025i −0.403742 0.699302i
4.4 0.915396i 0.500000 + 0.866025i 1.16205 −2.26729 + 1.30902i 0.792756 0.457698i 2.57477 0.608732i 2.89453i −0.500000 + 0.866025i 1.19827 + 2.07547i
4.5 0.560998i 0.500000 + 0.866025i 1.68528 0.449963 0.259786i 0.485838 0.280499i 0.680125 + 2.55684i 2.06743i −0.500000 + 0.866025i −0.145740 0.252428i
4.6 0.493650i 0.500000 + 0.866025i 1.75631 2.40375 1.38781i 0.427513 0.246825i −0.134370 2.64234i 1.85430i −0.500000 + 0.866025i −0.685090 1.18661i
4.7 0.871638i 0.500000 + 0.866025i 1.24025 1.34003 0.773665i −0.754861 + 0.435819i −2.02693 + 1.70046i 2.82432i −0.500000 + 0.866025i 0.674356 + 1.16802i
4.8 1.46676i 0.500000 + 0.866025i −0.151375 −1.88109 + 1.08605i −1.27025 + 0.733378i −2.04728 1.67590i 2.71148i −0.500000 + 0.866025i −1.59296 2.75910i
4.9 2.11978i 0.500000 + 0.866025i −2.49347 3.01977 1.74346i −1.83578 + 1.05989i 2.64461 0.0777267i 1.04605i −0.500000 + 0.866025i 3.69576 + 6.40125i
4.10 2.78118i 0.500000 + 0.866025i −5.73498 −0.804122 + 0.464260i −2.40857 + 1.39059i −1.56105 + 2.13615i 10.3876i −0.500000 + 0.866025i −1.29119 2.23641i
205.1 2.78118i 0.500000 0.866025i −5.73498 −0.804122 0.464260i −2.40857 1.39059i −1.56105 2.13615i 10.3876i −0.500000 0.866025i −1.29119 + 2.23641i
205.2 2.11978i 0.500000 0.866025i −2.49347 3.01977 + 1.74346i −1.83578 1.05989i 2.64461 + 0.0777267i 1.04605i −0.500000 0.866025i 3.69576 6.40125i
205.3 1.46676i 0.500000 0.866025i −0.151375 −1.88109 1.08605i −1.27025 0.733378i −2.04728 + 1.67590i 2.71148i −0.500000 0.866025i −1.59296 + 2.75910i
205.4 0.871638i 0.500000 0.866025i 1.24025 1.34003 + 0.773665i −0.754861 0.435819i −2.02693 1.70046i 2.82432i −0.500000 0.866025i 0.674356 1.16802i
205.5 0.493650i 0.500000 0.866025i 1.75631 2.40375 + 1.38781i 0.427513 + 0.246825i −0.134370 + 2.64234i 1.85430i −0.500000 0.866025i −0.685090 + 1.18661i
205.6 0.560998i 0.500000 0.866025i 1.68528 0.449963 + 0.259786i 0.485838 + 0.280499i 0.680125 2.55684i 2.06743i −0.500000 0.866025i −0.145740 + 0.252428i
205.7 0.915396i 0.500000 0.866025i 1.16205 −2.26729 1.30902i 0.792756 + 0.457698i 2.57477 + 0.608732i 2.89453i −0.500000 0.866025i 1.19827 2.07547i
205.8 2.04830i 0.500000 0.866025i −2.19554 0.341406 + 0.197111i 1.77388 + 1.02415i −0.908042 + 2.48505i 0.400534i −0.500000 0.866025i −0.403742 + 0.699302i
205.9 2.44406i 0.500000 0.866025i −3.97343 3.36767 + 1.94432i 2.11662 + 1.22203i 1.82552 1.91507i 4.82318i −0.500000 0.866025i −4.75204 + 8.23078i
205.10 2.50900i 0.500000 0.866025i −4.29509 −2.97008 1.71478i 2.17286 + 1.25450i −0.0473512 2.64533i 5.75840i −0.500000 0.866025i 4.30238 7.45195i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 205.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.k even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.t.d 20
3.b odd 2 1 819.2.bm.g 20
7.c even 3 1 273.2.bl.d yes 20
13.e even 6 1 273.2.bl.d yes 20
21.h odd 6 1 819.2.do.g 20
39.h odd 6 1 819.2.do.g 20
91.k even 6 1 inner 273.2.t.d 20
273.bp odd 6 1 819.2.bm.g 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.t.d 20 1.a even 1 1 trivial
273.2.t.d 20 91.k even 6 1 inner
273.2.bl.d yes 20 7.c even 3 1
273.2.bl.d yes 20 13.e even 6 1
819.2.bm.g 20 3.b odd 2 1
819.2.bm.g 20 273.bp odd 6 1
819.2.do.g 20 21.h odd 6 1
819.2.do.g 20 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{20} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$576 + 6435 T^{2} + 27166 T^{4} + 55993 T^{6} + 62505 T^{8} + 39799 T^{10} + 15006 T^{12} + 3403 T^{14} + 455 T^{16} + 33 T^{18} + T^{20}$$
$3$ $$( 1 - T + T^{2} )^{10}$$
$5$ $$46656 - 307152 T + 720900 T^{2} - 308574 T^{3} - 1071791 T^{4} + 795132 T^{5} + 1593187 T^{6} - 1283004 T^{7} - 472547 T^{8} + 544920 T^{9} + 126185 T^{10} - 145362 T^{11} - 11691 T^{12} + 22614 T^{13} + 619 T^{14} - 2484 T^{15} + 139 T^{16} + 150 T^{17} - 13 T^{18} - 6 T^{19} + T^{20}$$
$7$ $$282475249 - 80707214 T + 80707214 T^{2} - 54353838 T^{3} + 19294436 T^{4} - 13815354 T^{5} + 5735989 T^{6} - 2499441 T^{7} + 1164632 T^{8} - 435876 T^{9} + 177844 T^{10} - 62268 T^{11} + 23768 T^{12} - 7287 T^{13} + 2389 T^{14} - 822 T^{15} + 164 T^{16} - 66 T^{17} + 14 T^{18} - 2 T^{19} + T^{20}$$
$11$ $$18257414400 - 22966076160 T + 2126763648 T^{2} + 9437983104 T^{3} - 2215223312 T^{4} - 2644707408 T^{5} + 913283240 T^{6} + 414315384 T^{7} - 155427239 T^{8} - 49317555 T^{9} + 17252497 T^{10} + 4682406 T^{11} - 1164858 T^{12} - 328626 T^{13} + 52955 T^{14} + 17574 T^{15} - 1013 T^{16} - 564 T^{17} + T^{18} + 12 T^{19} + T^{20}$$
$13$ $$137858491849 - 84835994984 T + 31813498119 T^{2} - 2196198095 T^{3} - 2022432971 T^{4} + 1207444836 T^{5} - 237941691 T^{6} + 18711849 T^{7} + 10648352 T^{8} - 2908438 T^{9} + 998748 T^{10} - 223726 T^{11} + 63008 T^{12} + 8517 T^{13} - 8331 T^{14} + 3252 T^{15} - 419 T^{16} - 35 T^{17} + 39 T^{18} - 8 T^{19} + T^{20}$$
$17$ $$( 49716 - 215340 T - 207563 T^{2} + 183268 T^{3} + 2379 T^{4} - 19170 T^{5} + 1509 T^{6} + 688 T^{7} - 76 T^{8} - 8 T^{9} + T^{10} )^{2}$$
$19$ $$137768653584 + 496298535264 T + 646006855188 T^{2} + 180302867640 T^{3} - 125068246331 T^{4} - 59677016370 T^{5} + 20708676285 T^{6} + 11830141530 T^{7} - 1153160983 T^{8} - 1102874637 T^{9} + 50390960 T^{10} + 70056777 T^{11} + 809232 T^{12} - 2703951 T^{13} - 96259 T^{14} + 73419 T^{15} + 5376 T^{16} - 981 T^{17} - 82 T^{18} + 9 T^{19} + T^{20}$$
$23$ $$( -651492 - 807384 T - 75917 T^{2} + 266659 T^{3} + 117826 T^{4} + 4329 T^{5} - 6302 T^{6} - 1042 T^{7} + 36 T^{8} + 18 T^{9} + T^{10} )^{2}$$
$29$ $$1191651690384 + 963235081152 T + 1000388675844 T^{2} + 478998676896 T^{3} + 336541295245 T^{4} + 137296637114 T^{5} + 72769125955 T^{6} + 22479904394 T^{7} + 8848527113 T^{8} + 2203174061 T^{9} + 713366920 T^{10} + 136707703 T^{11} + 32614726 T^{12} + 4015173 T^{13} + 804769 T^{14} + 69255 T^{15} + 14464 T^{16} + 683 T^{17} + 142 T^{18} + 3 T^{19} + T^{20}$$
$31$ $$293380784803044 - 19074811929318 T - 83199969327681 T^{2} + 5436311187786 T^{3} + 17208797821753 T^{4} - 3611518907358 T^{5} - 1079931525099 T^{6} + 333603179175 T^{7} + 48788189633 T^{8} - 21540672588 T^{9} - 452894144 T^{10} + 688968732 T^{11} - 4521534 T^{12} - 15687621 T^{13} + 251470 T^{14} + 248784 T^{15} - 714 T^{16} - 2736 T^{17} - 44 T^{18} + 18 T^{19} + T^{20}$$
$37$ $$111696214861449 + 118999034572620 T^{2} + 32786591831686 T^{4} + 4204101433433 T^{6} + 302320470589 T^{8} + 13246119946 T^{10} + 366000213 T^{12} + 6382361 T^{14} + 67846 T^{16} + 400 T^{18} + T^{20}$$
$41$ $$28328929344 + 492798348432 T + 2998106756964 T^{2} + 2445841776846 T^{3} + 88013338093 T^{4} - 554005779522 T^{5} - 47112086920 T^{6} + 76348115376 T^{7} + 3187227676 T^{8} - 8027860269 T^{9} + 955022836 T^{10} + 252480663 T^{11} - 53225133 T^{12} - 5905374 T^{13} + 2611061 T^{14} - 222444 T^{15} - 9425 T^{16} + 2058 T^{17} + 49 T^{18} - 21 T^{19} + T^{20}$$
$43$ $$13841465449744 - 9677398039156 T + 8740240935425 T^{2} - 2787195655574 T^{3} + 1654248338850 T^{4} - 341957428624 T^{5} + 211992820266 T^{6} - 28493766286 T^{7} + 17930870094 T^{8} - 1475629485 T^{9} + 1133437538 T^{10} - 72374136 T^{11} + 43982442 T^{12} - 3577324 T^{13} + 1258563 T^{14} - 103591 T^{15} + 23991 T^{16} - 2078 T^{17} + 299 T^{18} - 16 T^{19} + T^{20}$$
$47$ $$14078614613904 + 7928003560752 T - 6695064134676 T^{2} - 4608162581484 T^{3} + 2684057695369 T^{4} + 1923181758072 T^{5} - 320211453560 T^{6} - 343910006892 T^{7} + 36968899999 T^{8} + 40997064114 T^{9} + 2251287353 T^{10} - 1579372476 T^{11} - 76798722 T^{12} + 39423126 T^{13} + 1737376 T^{14} - 622689 T^{15} + 3058 T^{16} + 4305 T^{17} - 58 T^{18} - 21 T^{19} + T^{20}$$
$53$ $$38226506256 + 222977395296 T + 923328614268 T^{2} + 1927002143712 T^{3} + 2977704582409 T^{4} + 1921365413467 T^{5} + 1222079618165 T^{6} + 550241745786 T^{7} + 290327548049 T^{8} + 108933530408 T^{9} + 32668691701 T^{10} + 7209456624 T^{11} + 1270756681 T^{12} + 172863813 T^{13} + 19372066 T^{14} + 1727263 T^{15} + 136532 T^{16} + 8988 T^{17} + 578 T^{18} + 26 T^{19} + T^{20}$$
$59$ $$3923769600 + 13366141452 T^{2} + 16993272541 T^{4} + 10584211457 T^{6} + 3469766929 T^{8} + 594997639 T^{10} + 51394587 T^{12} + 2124587 T^{14} + 41248 T^{16} + 343 T^{18} + T^{20}$$
$61$ $$23460552631824400 - 2396811894610080 T + 5006162888088396 T^{2} + 47199164258128 T^{3} + 690989280874533 T^{4} + 14250196918273 T^{5} + 49444049038818 T^{6} + 2628258475395 T^{7} + 2440705129704 T^{8} + 88188877231 T^{9} + 59840598924 T^{10} + 920082930 T^{11} + 1040032012 T^{12} + 6705342 T^{13} + 10859215 T^{14} + 22672 T^{15} + 82011 T^{16} + 54 T^{17} + 352 T^{18} + T^{20}$$
$67$ $$46656 + 101088 T - 917568 T^{2} - 2146248 T^{3} + 18604384 T^{4} + 17895822 T^{5} - 32413233 T^{6} - 36971712 T^{7} + 49753823 T^{8} + 109788216 T^{9} + 79818259 T^{10} + 25285410 T^{11} + 838377 T^{12} - 1438041 T^{13} - 109292 T^{14} + 106617 T^{15} + 21426 T^{16} - 465 T^{17} - 152 T^{18} + 3 T^{19} + T^{20}$$
$71$ $$131239686215393856 - 135526569138956880 T + 46679600383043220 T^{2} - 29369000762850 T^{3} - 2622803529343823 T^{4} + 190768061207196 T^{5} + 105526230475523 T^{6} - 15154861711716 T^{7} - 1865071327499 T^{8} + 419001390927 T^{9} + 22187063464 T^{10} - 8126436621 T^{11} - 27763860 T^{12} + 97082661 T^{13} - 1764709 T^{14} - 852483 T^{15} + 33340 T^{16} + 4395 T^{17} - 218 T^{18} - 15 T^{19} + T^{20}$$
$73$ $$29690641983744 - 22713544470960 T - 18358732906581 T^{2} + 18475480440915 T^{3} + 16181858294449 T^{4} - 160816500888 T^{5} - 2469640610259 T^{6} - 127576457475 T^{7} + 283072804385 T^{8} + 28666899630 T^{9} - 14817228986 T^{10} - 1065690768 T^{11} + 603883593 T^{12} - 19952505 T^{13} - 6728219 T^{14} + 310224 T^{15} + 58587 T^{16} - 2691 T^{17} - 272 T^{18} + 9 T^{19} + T^{20}$$
$79$ $$10220388275716096 - 12672461306953728 T + 12503991854419968 T^{2} - 5157735831754752 T^{3} + 1969048647914496 T^{4} - 395178835081216 T^{5} + 106144558703616 T^{6} - 14231434273536 T^{7} + 3955602528768 T^{8} - 318184159200 T^{9} + 90410027712 T^{10} - 2770894176 T^{11} + 1573300896 T^{12} - 14325900 T^{13} + 15314061 T^{14} - 82231 T^{15} + 106986 T^{16} - 339 T^{17} + 402 T^{18} - 3 T^{19} + T^{20}$$
$83$ $$548870434304256 + 1162967504573952 T^{2} + 481990635340384 T^{4} + 78123720309136 T^{6} + 5961775342321 T^{8} + 228192981215 T^{10} + 4556293128 T^{12} + 49406062 T^{14} + 290248 T^{16} + 860 T^{18} + T^{20}$$
$89$ $$5010940944 + 165237359258664 T^{2} + 91389293942929 T^{4} + 19220764520625 T^{6} + 1929541856039 T^{8} + 96037015256 T^{10} + 2315223831 T^{12} + 29259557 T^{14} + 200154 T^{16} + 704 T^{18} + T^{20}$$
$97$ $$2671271842542734025 + 4005093103774099785 T + 2275985078462274843 T^{2} + 411331798976613336 T^{3} - 33243769701674348 T^{4} - 17223954244007904 T^{5} + 92347778555673 T^{6} + 450682818537645 T^{7} + 15996831887819 T^{8} - 6964185574608 T^{9} - 302748161176 T^{10} + 75825093504 T^{11} + 3923393907 T^{12} - 542280837 T^{13} - 28565479 T^{14} + 2735544 T^{15} + 157380 T^{16} - 8160 T^{17} - 469 T^{18} + 15 T^{19} + T^{20}$$