# Properties

 Label 273.2.bl.c Level $273$ Weight $2$ Character orbit 273.bl 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.bl (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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

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

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 $$\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
88.1
 0.655911 + 1.25291i −1.18541 + 0.771231i 1.21245 + 0.727987i −1.38488 − 0.286553i 1.32725 − 0.488273i 0.874681 − 1.11128i 0.655911 − 1.25291i −1.18541 − 0.771231i 1.21245 − 0.727987i −1.38488 + 0.286553i 1.32725 + 0.488273i 0.874681 + 1.11128i
−2.17010 1.25291i 1.00000 2.13956 + 3.70583i −2.61265 + 1.50841i −2.17010 1.25291i −0.393717 2.61629i 5.71107i 1.00000 7.55962
88.2 −1.33581 0.771231i 1.00000 0.189594 + 0.328387i −1.27069 + 0.733632i −1.33581 0.771231i 1.52469 + 2.16225i 2.50004i 1.00000 2.26320
88.3 −1.26091 0.727987i 1.00000 0.0599314 + 0.103804i 3.67267 2.12042i −1.26091 0.727987i 2.09135 1.62057i 2.73743i 1.00000 −6.17455
88.4 0.496325 + 0.286553i 1.00000 −0.835774 1.44760i 2.74304 1.58369i 0.496325 + 0.286553i −2.25549 + 1.38302i 2.10419i 1.00000 1.81525
88.5 0.845714 + 0.488273i 1.00000 −0.523178 0.906171i −0.233786 + 0.134976i 0.845714 + 0.488273i 2.62954 0.292422i 2.97491i 1.00000 −0.263621
88.6 1.92478 + 1.11128i 1.00000 1.46986 + 2.54588i 0.701414 0.404962i 1.92478 + 1.11128i −2.09638 1.61406i 2.08860i 1.00000 1.80010
121.1 −2.17010 + 1.25291i 1.00000 2.13956 3.70583i −2.61265 1.50841i −2.17010 + 1.25291i −0.393717 + 2.61629i 5.71107i 1.00000 7.55962
121.2 −1.33581 + 0.771231i 1.00000 0.189594 0.328387i −1.27069 0.733632i −1.33581 + 0.771231i 1.52469 2.16225i 2.50004i 1.00000 2.26320
121.3 −1.26091 + 0.727987i 1.00000 0.0599314 0.103804i 3.67267 + 2.12042i −1.26091 + 0.727987i 2.09135 + 1.62057i 2.73743i 1.00000 −6.17455
121.4 0.496325 0.286553i 1.00000 −0.835774 + 1.44760i 2.74304 + 1.58369i 0.496325 0.286553i −2.25549 1.38302i 2.10419i 1.00000 1.81525
121.5 0.845714 0.488273i 1.00000 −0.523178 + 0.906171i −0.233786 0.134976i 0.845714 0.488273i 2.62954 + 0.292422i 2.97491i 1.00000 −0.263621
121.6 1.92478 1.11128i 1.00000 1.46986 2.54588i 0.701414 + 0.404962i 1.92478 1.11128i −2.09638 + 1.61406i 2.08860i 1.00000 1.80010
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 121.6 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.u 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.bl.c yes 12
3.b odd 2 1 819.2.do.f 12
7.c even 3 1 273.2.t.c 12
13.e even 6 1 273.2.t.c 12
21.h odd 6 1 819.2.bm.e 12
39.h odd 6 1 819.2.bm.e 12
91.u even 6 1 inner 273.2.bl.c yes 12
273.x odd 6 1 819.2.do.f 12

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$49 - 126 T + 31 T^{2} + 198 T^{3} - 36 T^{4} - 177 T^{5} + 27 T^{6} + 108 T^{7} + 20 T^{8} - 21 T^{9} - 4 T^{10} + 3 T^{11} + T^{12}$$
$3$ $$( -1 + T )^{12}$$
$5$ $$169 + 858 T + 828 T^{2} - 3168 T^{3} + 736 T^{4} + 2946 T^{5} + 545 T^{6} - 816 T^{7} + 22 T^{8} + 84 T^{9} - 2 T^{10} - 6 T^{11} + T^{12}$$
$7$ $$117649 - 50421 T + 1029 T^{3} + 2205 T^{4} - 1218 T^{5} + 317 T^{6} - 174 T^{7} + 45 T^{8} + 3 T^{9} - 3 T^{11} + T^{12}$$
$11$ $$169 + 1719 T^{2} + 3124 T^{4} + 1757 T^{6} + 385 T^{8} + 34 T^{10} + T^{12}$$
$13$ $$4826809 + 371293 T - 57122 T^{2} + 61516 T^{3} - 5746 T^{4} - 7475 T^{5} - 1057 T^{6} - 575 T^{7} - 34 T^{8} + 28 T^{9} - 2 T^{10} + T^{11} + T^{12}$$
$17$ $$1 - 22 T + 456 T^{2} - 624 T^{3} + 883 T^{4} - 372 T^{5} + 326 T^{6} - 66 T^{7} + 93 T^{8} - 8 T^{9} + 11 T^{10} + T^{12}$$
$19$ $$17689 + 43252 T^{2} + 36888 T^{4} + 13530 T^{6} + 2048 T^{8} + 95 T^{10} + T^{12}$$
$23$ $$2247001 + 2510825 T + 2138570 T^{2} + 1123123 T^{3} + 539488 T^{4} + 211396 T^{5} + 78393 T^{6} + 23527 T^{7} + 5998 T^{8} + 1140 T^{9} + 169 T^{10} + 16 T^{11} + T^{12}$$
$29$ $$63001 - 122990 T + 316906 T^{2} + 65604 T^{3} + 165916 T^{4} - 13463 T^{5} + 37512 T^{6} + 4150 T^{7} + 2134 T^{8} + 136 T^{9} + 65 T^{10} + 5 T^{11} + T^{12}$$
$31$ $$82369 - 1323357 T + 7168328 T^{2} - 1304913 T^{3} - 794317 T^{4} + 167313 T^{5} + 76209 T^{6} - 17277 T^{7} - 1719 T^{8} + 570 T^{9} + 37 T^{10} - 15 T^{11} + T^{12}$$
$37$ $$1104964081 + 170127438 T - 134138510 T^{2} - 21997164 T^{3} + 13148398 T^{4} + 2160156 T^{5} - 410623 T^{6} - 67350 T^{7} + 12028 T^{8} + 792 T^{9} - 120 T^{10} - 6 T^{11} + T^{12}$$
$41$ $$5098959649 - 2328153828 T - 24759491 T^{2} + 173094636 T^{3} - 1026044 T^{4} - 8070993 T^{5} + 187799 T^{6} + 220530 T^{7} + 7690 T^{8} - 2205 T^{9} - 72 T^{10} + 15 T^{11} + T^{12}$$
$43$ $$3455881 + 7095803 T + 10676743 T^{2} + 6632010 T^{3} + 3030571 T^{4} + 917819 T^{5} + 228021 T^{6} + 42209 T^{7} + 7381 T^{8} + 1031 T^{9} + 146 T^{10} + 13 T^{11} + T^{12}$$
$47$ $$14341369 - 11906328 T - 18794659 T^{2} + 18338952 T^{3} + 32583665 T^{4} - 4778142 T^{5} - 661599 T^{6} + 122256 T^{7} + 14796 T^{8} - 1368 T^{9} - 125 T^{10} + 9 T^{11} + T^{12}$$
$53$ $$823747401 - 743154993 T + 433061478 T^{2} - 161465967 T^{3} + 46103418 T^{4} - 9717246 T^{5} + 1673217 T^{6} - 225045 T^{7} + 27396 T^{8} - 2754 T^{9} + 273 T^{10} - 18 T^{11} + T^{12}$$
$59$ $$48177481 + 98263737 T + 54535195 T^{2} - 25029576 T^{3} - 3714273 T^{4} + 2242611 T^{5} + 419979 T^{6} - 63087 T^{7} - 12385 T^{8} + 1155 T^{9} + 398 T^{10} + 33 T^{11} + T^{12}$$
$61$ $$( 244009 + 53737 T - 19200 T^{2} - 3573 T^{3} + 23 T^{4} + 26 T^{5} + T^{6} )^{2}$$
$67$ $$31267787929 + 5908849215 T^{2} + 319178020 T^{4} + 7392701 T^{6} + 84205 T^{8} + 466 T^{10} + T^{12}$$
$71$ $$1583801209 + 2939406420 T + 2116114760 T^{2} + 552472800 T^{3} - 757156 T^{4} - 20389035 T^{5} + 299118 T^{6} + 487092 T^{7} + 24234 T^{8} - 3180 T^{9} - 137 T^{10} + 15 T^{11} + T^{12}$$
$73$ $$1836722449 - 2983361484 T + 2114775183 T^{2} - 811327860 T^{3} + 168045811 T^{4} - 13288896 T^{5} - 1264900 T^{6} + 252192 T^{7} + 10801 T^{8} - 3060 T^{9} - 62 T^{10} + 18 T^{11} + T^{12}$$
$79$ $$59969536 + 40888320 T + 35560448 T^{2} + 9878528 T^{3} + 7128576 T^{4} + 2350848 T^{5} + 819968 T^{6} + 138144 T^{7} + 21280 T^{8} + 1440 T^{9} + 144 T^{10} + 4 T^{11} + T^{12}$$
$83$ $$184063489 + 115982203 T^{2} + 18539692 T^{4} + 1189694 T^{6} + 32656 T^{8} + 348 T^{10} + T^{12}$$
$89$ $$1666190761 + 472561563 T - 175665319 T^{2} - 62492646 T^{3} + 19982282 T^{4} + 4837998 T^{5} - 347085 T^{6} - 117720 T^{7} + 7815 T^{8} + 1572 T^{9} - 83 T^{10} - 12 T^{11} + T^{12}$$
$97$ $$325035674161 - 122008316595 T + 5057495861 T^{2} + 3831973530 T^{3} - 168513649 T^{4} - 95693607 T^{5} + 10779870 T^{6} + 281775 T^{7} - 83649 T^{8} - 450 T^{9} + 685 T^{10} - 45 T^{11} + T^{12}$$