# Properties

 Label 273.2.t.c Level $273$ Weight $2$ Character orbit 273.t Analytic conductor $2.180$ Analytic rank $0$ Dimension $12$ 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: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.2346760387617129.1 Defining polynomial: $$x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + 16 x^{2} - 96 x + 64$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + 16 x^{2} - 96 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 72 \nu^{8} - 91 \nu^{7} - 164 \nu^{6} + 313 \nu^{5} + 42 \nu^{4} - 620 \nu^{3} + 344 \nu^{2} + 608 \nu - 800$$$$)/224$$ $$\beta_{3}$$ $$=$$ $$($$$$-9 \nu^{11} + 5 \nu^{10} + 25 \nu^{9} - 32 \nu^{8} - 21 \nu^{7} + 132 \nu^{6} - 73 \nu^{5} - 154 \nu^{4} + 260 \nu^{3} + 40 \nu^{2} - 320 \nu + 256$$$$)/224$$ $$\beta_{4}$$ $$=$$ $$($$$$-11 \nu^{11} + 17 \nu^{10} + 29 \nu^{9} - 78 \nu^{8} + 21 \nu^{7} + 166 \nu^{6} - 167 \nu^{5} - 140 \nu^{4} + 380 \nu^{3} - 88 \nu^{2} - 304 \nu + 288$$$$)/224$$ $$\beta_{5}$$ $$=$$ $$($$$$-13 \nu^{11} + 29 \nu^{10} + 5 \nu^{9} - 96 \nu^{8} + 91 \nu^{7} + 200 \nu^{6} - 289 \nu^{5} - 126 \nu^{4} + 584 \nu^{3} - 160 \nu^{2} - 512 \nu + 544$$$$)/224$$ $$\beta_{6}$$ $$=$$ $$($$$$8 \nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 51 \nu^{8} - 42 \nu^{7} - 101 \nu^{6} + 194 \nu^{5} + 7 \nu^{4} - 340 \nu^{3} + 260 \nu^{2} + 216 \nu - 464$$$$)/112$$ $$\beta_{7}$$ $$=$$ $$($$$$13 \nu^{11} - 57 \nu^{10} - 5 \nu^{9} + 208 \nu^{8} - 231 \nu^{7} - 396 \nu^{6} + 821 \nu^{5} + 42 \nu^{4} - 1452 \nu^{3} + 720 \nu^{2} + 1184 \nu - 1664$$$$)/224$$ $$\beta_{8}$$ $$=$$ $$($$$$2 \nu^{11} - 5 \nu^{10} - 4 \nu^{9} + 18 \nu^{8} - 7 \nu^{7} - 41 \nu^{6} + 45 \nu^{5} + 35 \nu^{4} - 99 \nu^{3} + 16 \nu^{2} + 96 \nu - 88$$$$)/28$$ $$\beta_{9}$$ $$=$$ $$($$$$3 \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 20 \nu^{8} - 44 \nu^{6} + 43 \nu^{5} + 56 \nu^{4} - 82 \nu^{3} + 3 \nu^{2} + 102 \nu - 48$$$$)/28$$ $$\beta_{10}$$ $$=$$ $$($$$$-15 \nu^{11} + 20 \nu^{10} + 30 \nu^{9} - 121 \nu^{8} + 21 \nu^{7} + 269 \nu^{6} - 271 \nu^{5} - 273 \nu^{4} + 634 \nu^{3} - 64 \nu^{2} - 664 \nu + 464$$$$)/112$$ $$\beta_{11}$$ $$=$$ $$($$$$-17 \nu^{11} + 39 \nu^{10} + 13 \nu^{9} - 160 \nu^{8} + 133 \nu^{7} + 310 \nu^{6} - 547 \nu^{5} - 168 \nu^{4} + 1062 \nu^{3} - 500 \nu^{2} - 872 \nu + 1056$$$$)/112$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{9} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{2} - \beta_{1} - 1$$ $$\nu^{5}$$ $$=$$ $$\beta_{10} + 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{3} - \beta_{2} - \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-4 \beta_{11} + 2 \beta_{10} - 3 \beta_{8} + \beta_{7} - 5 \beta_{6} + 4 \beta_{5} - 7 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 3 \beta_{1} - 6$$ $$\nu^{7}$$ $$=$$ $$-\beta_{11} - \beta_{10} - \beta_{9} + 3 \beta_{8} + \beta_{7} + \beta_{6} + 6 \beta_{5} + 4 \beta_{4} - \beta_{3} - 4 \beta_{2} + \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-4 \beta_{10} - 2 \beta_{9} - \beta_{8} + 2 \beta_{5} - 4 \beta_{4} + 8 \beta_{3} - 2 \beta_{2} + 3 \beta_{1} - 6$$ $$\nu^{9}$$ $$=$$ $$2 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 6 \beta_{8} - 3 \beta_{7} + 7 \beta_{6} - 4 \beta_{5} + 21 \beta_{4} + 6 \beta_{3} - 3 \beta_{1} + 4$$ $$\nu^{10}$$ $$=$$ $$5 \beta_{11} - 9 \beta_{10} + \beta_{9} - 16 \beta_{8} + \beta_{7} + 3 \beta_{6} - 8 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 7 \beta_{2} - 6 \beta_{1} + 1$$ $$\nu^{11}$$ $$=$$ $$-2 \beta_{11} - \beta_{10} - 19 \beta_{8} + \beta_{7} + 4 \beta_{6} - 15 \beta_{5} - 5 \beta_{4} - 13 \beta_{3} - 14 \beta_{2} + 9 \beta_{1} - 5$$

## Character values

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

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$1 + \beta_{4}$$ $$-1 - \beta_{4}$$

## 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
 0.655911 − 1.25291i −1.18541 − 0.771231i 1.21245 − 0.727987i −1.38488 + 0.286553i 1.32725 + 0.488273i 0.874681 + 1.11128i 0.874681 − 1.11128i 1.32725 − 0.488273i −1.38488 − 0.286553i 1.21245 + 0.727987i −1.18541 + 0.771231i 0.655911 + 1.25291i
2.50582i −0.500000 0.866025i −4.27912 2.61265 1.50841i −2.17010 + 1.25291i −2.46263 0.967177i 5.71107i −0.500000 + 0.866025i −3.77981 6.54683i
4.2 1.54246i −0.500000 0.866025i −0.379188 1.27069 0.733632i −1.33581 + 0.771231i 2.63491 0.239300i 2.50004i −0.500000 + 0.866025i −1.13160 1.95999i
4.3 1.45597i −0.500000 0.866025i −0.119863 −3.67267 + 2.12042i −1.26091 + 0.727987i −0.357777 2.62145i 2.73743i −0.500000 + 0.866025i 3.08727 + 5.34732i
4.4 0.573107i −0.500000 0.866025i 1.67155 −2.74304 + 1.58369i 0.496325 0.286553i 0.0699870 + 2.64483i 2.10419i −0.500000 + 0.866025i −0.907626 1.57205i
4.5 0.976547i −0.500000 0.866025i 1.04636 0.233786 0.134976i 0.845714 0.488273i 1.06153 2.42346i 2.97491i −0.500000 + 0.866025i 0.131811 + 0.228303i
4.6 2.22255i −0.500000 0.866025i −2.93973 −0.701414 + 0.404962i 1.92478 1.11128i −2.44601 + 1.00849i 2.08860i −0.500000 + 0.866025i −0.900048 1.55893i
205.1 2.22255i −0.500000 + 0.866025i −2.93973 −0.701414 0.404962i 1.92478 + 1.11128i −2.44601 1.00849i 2.08860i −0.500000 0.866025i −0.900048 + 1.55893i
205.2 0.976547i −0.500000 + 0.866025i 1.04636 0.233786 + 0.134976i 0.845714 + 0.488273i 1.06153 + 2.42346i 2.97491i −0.500000 0.866025i 0.131811 0.228303i
205.3 0.573107i −0.500000 + 0.866025i 1.67155 −2.74304 1.58369i 0.496325 + 0.286553i 0.0699870 2.64483i 2.10419i −0.500000 0.866025i −0.907626 + 1.57205i
205.4 1.45597i −0.500000 + 0.866025i −0.119863 −3.67267 2.12042i −1.26091 0.727987i −0.357777 + 2.62145i 2.73743i −0.500000 0.866025i 3.08727 5.34732i
205.5 1.54246i −0.500000 + 0.866025i −0.379188 1.27069 + 0.733632i −1.33581 0.771231i 2.63491 + 0.239300i 2.50004i −0.500000 0.866025i −1.13160 + 1.95999i
205.6 2.50582i −0.500000 + 0.866025i −4.27912 2.61265 + 1.50841i −2.17010 1.25291i −2.46263 + 0.967177i 5.71107i −0.500000 0.866025i −3.77981 + 6.54683i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 205.6 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.c 12
3.b odd 2 1 819.2.bm.e 12
7.c even 3 1 273.2.bl.c yes 12
13.e even 6 1 273.2.bl.c yes 12
21.h odd 6 1 819.2.do.f 12
39.h odd 6 1 819.2.do.f 12
91.k even 6 1 inner 273.2.t.c 12
273.bp odd 6 1 819.2.bm.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.t.c 12 1.a even 1 1 trivial
273.2.t.c 12 91.k even 6 1 inner
273.2.bl.c yes 12 7.c even 3 1
273.2.bl.c yes 12 13.e even 6 1
819.2.bm.e 12 3.b odd 2 1
819.2.bm.e 12 273.bp odd 6 1
819.2.do.f 12 21.h odd 6 1
819.2.do.f 12 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 17 T_{2}^{10} + 107 T_{2}^{8} + 312 T_{2}^{6} + 435 T_{2}^{4} + 262 T_{2}^{2} + 49$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$49 + 262 T^{2} + 435 T^{4} + 312 T^{6} + 107 T^{8} + 17 T^{10} + T^{12}$$
$3$ $$( 1 + T + T^{2} )^{6}$$
$5$ $$169 - 858 T + 828 T^{2} + 3168 T^{3} + 736 T^{4} - 2946 T^{5} + 545 T^{6} + 816 T^{7} + 22 T^{8} - 84 T^{9} - 2 T^{10} + 6 T^{11} + T^{12}$$
$7$ $$117649 + 50421 T + 14406 T^{2} + 5145 T^{3} - 1911 T^{4} - 1806 T^{5} - 475 T^{6} - 258 T^{7} - 39 T^{8} + 15 T^{9} + 6 T^{10} + 3 T^{11} + T^{12}$$
$11$ $$169 + 975 T + 1953 T^{2} + 450 T^{3} - 1508 T^{4} - 252 T^{5} + 2441 T^{6} + 3072 T^{7} + 1861 T^{8} + 666 T^{9} + 145 T^{10} + 18 T^{11} + T^{12}$$
$13$ $$4826809 + 371293 T - 57122 T^{2} + 61516 T^{3} - 5746 T^{4} - 7475 T^{5} - 1057 T^{6} - 575 T^{7} - 34 T^{8} + 28 T^{9} - 2 T^{10} + T^{11} + T^{12}$$
$17$ $$( 1 + 22 T + 28 T^{2} - 4 T^{3} - 11 T^{4} + T^{6} )^{2}$$
$19$ $$17689 + 23940 T - 5426 T^{2} - 21960 T^{3} + 3882 T^{4} + 11979 T^{5} - 534 T^{6} - 3492 T^{7} + 710 T^{8} + 306 T^{9} - 7 T^{10} - 9 T^{11} + T^{12}$$
$23$ $$( -1499 + 1675 T - 445 T^{2} - 126 T^{3} + 87 T^{4} - 16 T^{5} + T^{6} )^{2}$$
$29$ $$63001 - 122990 T + 316906 T^{2} + 65604 T^{3} + 165916 T^{4} - 13463 T^{5} + 37512 T^{6} + 4150 T^{7} + 2134 T^{8} + 136 T^{9} + 65 T^{10} + 5 T^{11} + T^{12}$$
$31$ $$82369 + 1323357 T + 7168328 T^{2} + 1304913 T^{3} - 794317 T^{4} - 167313 T^{5} + 76209 T^{6} + 17277 T^{7} - 1719 T^{8} - 570 T^{9} + 37 T^{10} + 15 T^{11} + T^{12}$$
$37$ $$1104964081 + 294470944 T^{2} + 29121616 T^{4} + 1322093 T^{6} + 28216 T^{8} + 276 T^{10} + T^{12}$$
$41$ $$5098959649 - 2328153828 T - 24759491 T^{2} + 173094636 T^{3} - 1026044 T^{4} - 8070993 T^{5} + 187799 T^{6} + 220530 T^{7} + 7690 T^{8} - 2205 T^{9} - 72 T^{10} + 15 T^{11} + T^{12}$$
$43$ $$3455881 + 7095803 T + 10676743 T^{2} + 6632010 T^{3} + 3030571 T^{4} + 917819 T^{5} + 228021 T^{6} + 42209 T^{7} + 7381 T^{8} + 1031 T^{9} + 146 T^{10} + 13 T^{11} + T^{12}$$
$47$ $$14341369 + 11906328 T - 18794659 T^{2} - 18338952 T^{3} + 32583665 T^{4} + 4778142 T^{5} - 661599 T^{6} - 122256 T^{7} + 14796 T^{8} + 1368 T^{9} - 125 T^{10} - 9 T^{11} + T^{12}$$
$53$ $$823747401 - 743154993 T + 433061478 T^{2} - 161465967 T^{3} + 46103418 T^{4} - 9717246 T^{5} + 1673217 T^{6} - 225045 T^{7} + 27396 T^{8} - 2754 T^{9} + 273 T^{10} - 18 T^{11} + T^{12}$$
$59$ $$48177481 + 91350259 T^{2} + 16806018 T^{4} + 1072800 T^{6} + 28445 T^{8} + 293 T^{10} + T^{12}$$
$61$ $$59540392081 - 13112311633 T + 7572637969 T^{2} - 711937914 T^{3} + 555030094 T^{4} - 72473932 T^{5} + 12298785 T^{6} - 969958 T^{7} + 112627 T^{8} - 7744 T^{9} + 653 T^{10} - 26 T^{11} + T^{12}$$
$67$ $$31267787929 - 18544024317 T + 2544538713 T^{2} + 665091882 T^{3} - 99257564 T^{4} - 16665726 T^{5} + 2269235 T^{6} + 346818 T^{7} - 13949 T^{8} - 3288 T^{9} + 55 T^{10} + 24 T^{11} + T^{12}$$
$71$ $$1583801209 + 2939406420 T + 2116114760 T^{2} + 552472800 T^{3} - 757156 T^{4} - 20389035 T^{5} + 299118 T^{6} + 487092 T^{7} + 24234 T^{8} - 3180 T^{9} - 137 T^{10} + 15 T^{11} + T^{12}$$
$73$ $$1836722449 + 2983361484 T + 2114775183 T^{2} + 811327860 T^{3} + 168045811 T^{4} + 13288896 T^{5} - 1264900 T^{6} - 252192 T^{7} + 10801 T^{8} + 3060 T^{9} - 62 T^{10} - 18 T^{11} + T^{12}$$
$79$ $$59969536 + 40888320 T + 35560448 T^{2} + 9878528 T^{3} + 7128576 T^{4} + 2350848 T^{5} + 819968 T^{6} + 138144 T^{7} + 21280 T^{8} + 1440 T^{9} + 144 T^{10} + 4 T^{11} + T^{12}$$
$83$ $$184063489 + 115982203 T^{2} + 18539692 T^{4} + 1189694 T^{6} + 32656 T^{8} + 348 T^{10} + T^{12}$$
$89$ $$1666190761 + 485357567 T^{2} + 47450648 T^{4} + 1913253 T^{6} + 35853 T^{8} + 310 T^{10} + T^{12}$$
$97$ $$325035674161 - 122008316595 T + 5057495861 T^{2} + 3831973530 T^{3} - 168513649 T^{4} - 95693607 T^{5} + 10779870 T^{6} + 281775 T^{7} - 83649 T^{8} - 450 T^{9} + 685 T^{10} - 45 T^{11} + T^{12}$$