# Properties

 Label 57.2.i.b Level $57$ Weight $2$ Character orbit 57.i Analytic conductor $0.455$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$57 = 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 57.i (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.455147291521$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$6 \nu^{11} - 33 \nu^{10} + 127 \nu^{9} - 324 \nu^{8} + 438 \nu^{7} - 252 \nu^{6} - 1278 \nu^{5} + 3234 \nu^{4} - 5701 \nu^{3} + 5358 \nu^{2} - 3477 \nu + 1060$$$$)/218$$ $$\beta_{2}$$ $$=$$ $$($$$$36 \nu^{11} - 89 \nu^{10} + 544 \nu^{9} - 745 \nu^{8} + 2301 \nu^{7} - 1512 \nu^{6} + 3777 \nu^{5} - 2069 \nu^{4} + 5579 \nu^{3} - 6002 \nu^{2} + 5080 \nu - 1706$$$$)/218$$ $$\beta_{3}$$ $$=$$ $$($$$$-27 \nu^{11} + 94 \nu^{10} - 408 \nu^{9} + 586 \nu^{8} - 445 \nu^{7} - 2572 \nu^{6} + 9021 \nu^{5} - 18150 \nu^{4} + 24401 \nu^{3} - 20623 \nu^{2} + 10469 \nu - 2263$$$$)/218$$ $$\beta_{4}$$ $$=$$ $$($$$$26 \nu^{11} - 34 \nu^{10} + 187 \nu^{9} + 449 \nu^{8} - 1590 \nu^{7} + 6865 \nu^{6} - 12623 \nu^{5} + 20118 \nu^{4} - 19981 \nu^{3} + 12972 \nu^{2} - 4712 \nu + 524$$$$)/218$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{11} + 120 \nu^{10} - 551 \nu^{9} + 2615 \nu^{8} - 6686 \nu^{7} + 15780 \nu^{6} - 23990 \nu^{5} + 31404 \nu^{4} - 25822 \nu^{3} + 15545 \nu^{2} - 4618 \nu + 555$$$$)/218$$ $$\beta_{6}$$ $$=$$ $$($$$$-27 \nu^{11} + 203 \nu^{10} - 953 \nu^{9} + 3311 \nu^{8} - 8075 \nu^{7} + 16285 \nu^{6} - 23134 \nu^{5} + 26758 \nu^{4} - 19635 \nu^{3} + 9570 \nu^{2} - 1957 \nu - 83$$$$)/218$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{10} - 5 \nu^{9} + 25 \nu^{8} - 70 \nu^{7} + 173 \nu^{6} - 295 \nu^{5} + 412 \nu^{4} - 404 \nu^{3} + 279 \nu^{2} - 116 \nu + 26$$$$)/2$$ $$\beta_{8}$$ $$=$$ $$($$$$2 \nu^{11} + 98 \nu^{10} - 539 \nu^{9} + 2726 \nu^{8} - 8138 \nu^{7} + 20190 \nu^{6} - 36614 \nu^{5} + 52308 \nu^{4} - 55710 \nu^{3} + 41571 \nu^{2} - 19689 \nu + 4350$$$$)/218$$ $$\beta_{9}$$ $$=$$ $$($$$$26 \nu^{11} - 252 \nu^{10} + 1277 \nu^{9} - 4892 \nu^{8} + 13234 \nu^{7} - 28887 \nu^{6} + 47327 \nu^{5} - 60760 \nu^{4} + 56973 \nu^{3} - 36296 \nu^{2} + 13927 \nu - 2201$$$$)/218$$ $$\beta_{10}$$ $$=$$ $$($$$$36 \nu^{11} - 307 \nu^{10} + 1634 \nu^{9} - 6086 \nu^{8} + 17125 \nu^{7} - 37373 \nu^{6} + 64054 \nu^{5} - 84255 \nu^{4} + 84604 \nu^{3} - 58431 \nu^{2} + 25899 \nu - 5194$$$$)/218$$ $$\beta_{11}$$ $$=$$ $$($$$$91 \nu^{11} - 664 \nu^{10} + 3325 \nu^{9} - 11563 \nu^{8} + 30405 \nu^{7} - 62791 \nu^{6} + 99754 \nu^{5} - 123498 \nu^{4} + 112914 \nu^{3} - 71337 \nu^{2} + 28743 \nu - 5469$$$$)/218$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + 4 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 5 \beta_{3} - \beta_{2} - \beta_{1} - 7$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-7 \beta_{11} + 2 \beta_{10} + 5 \beta_{9} - 5 \beta_{8} + \beta_{7} - 13 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_{1} - 10$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-16 \beta_{11} + 11 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 20 \beta_{7} - 7 \beta_{6} + 14 \beta_{5} - 2 \beta_{4} - 31 \beta_{3} - \beta_{2} - 7 \beta_{1} + 32$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$23 \beta_{11} + 14 \beta_{10} - 34 \beta_{9} + 25 \beta_{8} - 26 \beta_{7} + 65 \beta_{6} - 10 \beta_{5} + 4 \beta_{4} - 10 \beta_{3} - 16 \beta_{2} + 29 \beta_{1} + 77$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$110 \beta_{11} - 49 \beta_{10} - 88 \beta_{9} + 67 \beta_{8} + 85 \beta_{7} + 101 \beta_{6} - 94 \beta_{5} - 17 \beta_{4} + 161 \beta_{3} + 17 \beta_{2} + 104 \beta_{1} - 121$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-25 \beta_{11} - 187 \beta_{10} + 158 \beta_{9} - 101 \beta_{8} + 229 \beta_{7} - 271 \beta_{6} - 46 \beta_{5} - 56 \beta_{4} + 212 \beta_{3} + 149 \beta_{2} - 82 \beta_{1} - 535$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$-652 \beta_{11} + 41 \beta_{10} + 761 \beta_{9} - 509 \beta_{8} - 263 \beta_{7} - 826 \beta_{6} + 515 \beta_{5} + 151 \beta_{4} - 724 \beta_{3} - 19 \beta_{2} - 829 \beta_{1} + 257$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$-478 \beta_{11} + 1304 \beta_{10} - 268 \beta_{9} + 160 \beta_{8} - 1571 \beta_{7} + 821 \beta_{6} + 812 \beta_{5} + 640 \beta_{4} - 1951 \beta_{3} - 1108 \beta_{2} - 406 \beta_{1} + 3359$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$3353 \beta_{11} + 1451 \beta_{10} - 5224 \beta_{9} + 3352 \beta_{8} + 25 \beta_{7} + 5630 \beta_{6} - 2308 \beta_{5} - 521 \beta_{4} + 2375 \beta_{3} - 913 \beta_{2} + 4949 \beta_{1} + 1451$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$6041 \beta_{11} - 6547 \beta_{10} - 3685 \beta_{9} + 2323 \beta_{8} + 9241 \beta_{7} + 326 \beta_{6} - 7273 \beta_{5} - 5168 \beta_{4} + 14021 \beta_{3} + 6587 \beta_{2} + 7973 \beta_{1} - 18736$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/57\mathbb{Z}\right)^\times$$.

 $$n$$ $$20$$ $$40$$ $$\chi(n)$$ $$1$$ $$-\beta_{3} + \beta_{9}$$

## 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
 0.5 − 1.80139i 0.5 − 0.168222i 0.5 + 1.74095i 0.5 − 2.42499i 0.5 − 1.74095i 0.5 + 2.42499i 0.5 − 0.677980i 0.5 + 1.96356i 0.5 + 1.80139i 0.5 + 0.168222i 0.5 + 0.677980i 0.5 − 1.96356i
−2.23121 0.812094i 0.173648 + 0.984808i 2.78672 + 2.33833i 2.16379 1.81563i 0.412311 2.33833i 1.05756 1.83175i −1.94440 3.36780i −0.939693 + 0.342020i −6.30233 + 2.29386i
4.2 0.791518 + 0.288089i 0.173648 + 0.984808i −0.988583 0.829520i 1.30800 1.09754i −0.146267 + 0.829520i −1.96517 + 3.40377i −1.38582 2.40031i −0.939693 + 0.342020i 1.35149 0.491903i
16.1 −0.958464 0.804247i −0.939693 + 0.342020i −0.0754558 0.427931i 0.755623 4.28535i 1.17573 + 0.427931i 0.898157 + 1.55565i −1.52303 + 2.63796i 0.766044 0.642788i −4.17072 + 3.49965i
16.2 1.22451 + 1.02748i −0.939693 + 0.342020i 0.0964003 + 0.546713i −0.174371 + 0.988909i −1.50208 0.546713i −1.28482 2.22537i 1.15479 2.00015i 0.766044 0.642788i −1.22961 + 1.03176i
25.1 −0.958464 + 0.804247i −0.939693 0.342020i −0.0754558 + 0.427931i 0.755623 + 4.28535i 1.17573 0.427931i 0.898157 1.55565i −1.52303 2.63796i 0.766044 + 0.642788i −4.17072 3.49965i
25.2 1.22451 1.02748i −0.939693 0.342020i 0.0964003 0.546713i −0.174371 0.988909i −1.50208 + 0.546713i −1.28482 + 2.22537i 1.15479 + 2.00015i 0.766044 + 0.642788i −1.22961 1.03176i
28.1 −0.458021 2.59757i 0.766044 0.642788i −4.65819 + 1.69544i 1.91800 + 0.698096i −2.02055 1.69544i −1.30802 + 2.26556i 3.89993 + 6.75488i 0.173648 0.984808i 0.934866 5.30189i
28.2 0.131669 + 0.746734i 0.766044 0.642788i 1.33911 0.487396i −2.97104 1.08137i 0.580856 + 0.487396i −1.89771 + 3.28694i 1.29853 + 2.24912i 0.173648 0.984808i 0.416301 2.36096i
43.1 −2.23121 + 0.812094i 0.173648 0.984808i 2.78672 2.33833i 2.16379 + 1.81563i 0.412311 + 2.33833i 1.05756 + 1.83175i −1.94440 + 3.36780i −0.939693 0.342020i −6.30233 2.29386i
43.2 0.791518 0.288089i 0.173648 0.984808i −0.988583 + 0.829520i 1.30800 + 1.09754i −0.146267 0.829520i −1.96517 3.40377i −1.38582 + 2.40031i −0.939693 0.342020i 1.35149 + 0.491903i
55.1 −0.458021 + 2.59757i 0.766044 + 0.642788i −4.65819 1.69544i 1.91800 0.698096i −2.02055 + 1.69544i −1.30802 2.26556i 3.89993 6.75488i 0.173648 + 0.984808i 0.934866 + 5.30189i
55.2 0.131669 0.746734i 0.766044 + 0.642788i 1.33911 + 0.487396i −2.97104 + 1.08137i 0.580856 0.487396i −1.89771 3.28694i 1.29853 2.24912i 0.173648 + 0.984808i 0.416301 + 2.36096i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 55.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.2.i.b 12
3.b odd 2 1 171.2.u.e 12
4.b odd 2 1 912.2.bo.j 12
19.e even 9 1 inner 57.2.i.b 12
19.e even 9 1 1083.2.a.q 6
19.f odd 18 1 1083.2.a.p 6
57.j even 18 1 3249.2.a.bg 6
57.l odd 18 1 171.2.u.e 12
57.l odd 18 1 3249.2.a.bh 6
76.l odd 18 1 912.2.bo.j 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.i.b 12 1.a even 1 1 trivial
57.2.i.b 12 19.e even 9 1 inner
171.2.u.e 12 3.b odd 2 1
171.2.u.e 12 57.l odd 18 1
912.2.bo.j 12 4.b odd 2 1
912.2.bo.j 12 76.l odd 18 1
1083.2.a.p 6 19.f odd 18 1
1083.2.a.q 6 19.e even 9 1
3249.2.a.bg 6 57.j even 18 1
3249.2.a.bh 6 57.l odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(57, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 - 96 T + 96 T^{2} - 64 T^{3} - 36 T^{4} + 18 T^{5} + 99 T^{6} - 9 T^{7} - 9 T^{8} + 8 T^{9} + 6 T^{10} + 3 T^{11} + T^{12}$$
$3$ $$( 1 + T^{3} + T^{6} )^{2}$$
$5$ $$18496 - 27744 T + 32640 T^{2} - 33184 T^{3} + 22608 T^{4} - 7380 T^{5} - 243 T^{6} + 927 T^{7} - 198 T^{8} - 28 T^{9} + 24 T^{10} - 6 T^{11} + T^{12}$$
$7$ $$145161 + 37719 T + 74952 T^{2} + 31077 T^{3} + 29763 T^{4} + 11232 T^{5} + 6405 T^{6} + 2232 T^{7} + 963 T^{8} + 261 T^{9} + 66 T^{10} + 9 T^{11} + T^{12}$$
$11$ $$207936 + 426816 T + 572400 T^{2} + 467424 T^{3} + 286236 T^{4} + 121014 T^{5} + 40749 T^{6} + 10026 T^{7} + 2241 T^{8} + 396 T^{9} + 75 T^{10} + 9 T^{11} + T^{12}$$
$13$ $$3249 - 4104 T + 8586 T^{2} - 5463 T^{3} + 1602 T^{4} + 63 T^{5} - 264 T^{6} + 72 T^{7} + 63 T^{8} - 21 T^{9} - 6 T^{10} + 3 T^{11} + T^{12}$$
$17$ $$41062464 - 24914304 T + 21580992 T^{2} - 12634272 T^{3} + 4728456 T^{4} - 1138374 T^{5} + 160947 T^{6} - 2403 T^{7} - 3780 T^{8} + 570 T^{9} + 18 T^{10} - 12 T^{11} + T^{12}$$
$19$ $$47045881 + 22284891 T + 9383112 T^{2} + 3690142 T^{3} + 1153395 T^{4} + 308655 T^{5} + 78033 T^{6} + 16245 T^{7} + 3195 T^{8} + 538 T^{9} + 72 T^{10} + 9 T^{11} + T^{12}$$
$23$ $$2483776 - 2893536 T + 2369952 T^{2} - 1393360 T^{3} + 686808 T^{4} - 228132 T^{5} + 55245 T^{6} - 6147 T^{7} - 765 T^{8} + 416 T^{9} - 18 T^{10} - 9 T^{11} + T^{12}$$
$29$ $$87616 - 195360 T + 213696 T^{2} - 163880 T^{3} + 104076 T^{4} - 51300 T^{5} + 17217 T^{6} - 3780 T^{7} + 846 T^{8} - 125 T^{9} - 6 T^{10} + T^{12}$$
$31$ $$185761 + 77580 T + 168165 T^{2} - 48080 T^{3} + 85509 T^{4} - 9810 T^{5} + 10578 T^{6} - 540 T^{7} + 981 T^{8} - 20 T^{9} + 36 T^{10} + T^{12}$$
$37$ $$( 2467 - 1146 T - 2415 T^{2} - 862 T^{3} - 84 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$41$ $$87616 + 3786432 T + 69992640 T^{2} + 34553384 T^{3} + 10604664 T^{4} + 2290680 T^{5} + 280881 T^{6} + 24885 T^{7} + 6084 T^{8} + 1349 T^{9} + 183 T^{10} + 18 T^{11} + T^{12}$$
$43$ $$587917009 - 142281396 T + 857427150 T^{2} + 171228403 T^{3} - 4778694 T^{4} - 5593545 T^{5} + 402930 T^{6} + 57456 T^{7} - 8451 T^{8} + 343 T^{9} + 90 T^{10} - 15 T^{11} + T^{12}$$
$47$ $$46656 + 139968 T + 128304 T^{2} + 40824 T^{3} + 56376 T^{4} + 11178 T^{5} + 459 T^{6} + 2997 T^{7} + 1674 T^{8} + 378 T^{9} + 63 T^{10} + 9 T^{11} + T^{12}$$
$53$ $$1871424 + 4875552 T + 6262272 T^{2} + 5620320 T^{3} + 4036392 T^{4} + 2274480 T^{5} + 947457 T^{6} + 274239 T^{7} + 53028 T^{8} + 6531 T^{9} + 531 T^{10} + 30 T^{11} + T^{12}$$
$59$ $$1617984 + 10761120 T + 26204256 T^{2} + 27039384 T^{3} + 16121844 T^{4} + 6036804 T^{5} + 1604913 T^{6} + 320535 T^{7} + 49104 T^{8} + 5730 T^{9} + 498 T^{10} + 30 T^{11} + T^{12}$$
$61$ $$5329 - 75993 T + 348843 T^{2} + 4160081 T^{3} + 8575632 T^{4} + 1329138 T^{5} + 229773 T^{6} - 2970 T^{7} + 12024 T^{8} - 2761 T^{9} + 285 T^{10} - 21 T^{11} + T^{12}$$
$67$ $$1254293056 - 2060786208 T + 857960448 T^{2} + 80549632 T^{3} + 39637476 T^{4} + 3938382 T^{5} + 1276461 T^{6} - 40950 T^{7} + 12474 T^{8} - 890 T^{9} - 66 T^{10} - 3 T^{11} + T^{12}$$
$71$ $$241118784 + 122422752 T + 148443840 T^{2} + 33201216 T^{3} + 1499076 T^{4} - 346050 T^{5} - 108615 T^{6} - 33228 T^{7} - 396 T^{8} + 1869 T^{9} + 390 T^{10} + 30 T^{11} + T^{12}$$
$73$ $$1056705049 + 401981562 T + 88609584 T^{2} + 10395746 T^{3} + 2441322 T^{4} - 662904 T^{5} - 162576 T^{6} + 12366 T^{7} + 4338 T^{8} + 908 T^{9} + 261 T^{10} + 24 T^{11} + T^{12}$$
$79$ $$4606201161 + 3190318083 T + 828240876 T^{2} - 36650988 T^{3} - 27313038 T^{4} - 2208708 T^{5} - 72495 T^{6} + 144558 T^{7} + 30888 T^{8} - 5376 T^{9} + 513 T^{10} - 24 T^{11} + T^{12}$$
$83$ $$75809912896 - 13632436032 T + 8648150160 T^{2} + 1113766400 T^{3} + 430576812 T^{4} + 28134126 T^{5} + 5809521 T^{6} + 184824 T^{7} + 53667 T^{8} + 830 T^{9} + 285 T^{10} - 3 T^{11} + T^{12}$$
$89$ $$21486869056 - 9197852832 T + 6328237152 T^{2} - 1314646264 T^{3} - 14903244 T^{4} + 9430164 T^{5} + 3951117 T^{6} - 341937 T^{7} + 15147 T^{8} - 1036 T^{9} - 18 T^{10} - 3 T^{11} + T^{12}$$
$97$ $$128881 + 818520 T - 1154538 T^{2} - 10936207 T^{3} + 22476942 T^{4} + 604305 T^{5} + 3824802 T^{6} + 1031418 T^{7} + 141561 T^{8} + 10529 T^{9} + 606 T^{10} + 27 T^{11} + T^{12}$$
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