# Properties

 Label 84.2.e.a Level $84$ Weight $2$ Character orbit 84.e Analytic conductor $0.671$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 84.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.670743376979$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.2593100598870016.2 Defining polynomial: $$x^{12} - 2 x^{10} + x^{8} + 4 x^{6} + 4 x^{4} - 32 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{10} + x^{8} + 4 x^{6} + 4 x^{4} - 32 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{8} + \nu^{4} + 14 \nu^{2} + 8$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{8} - \nu^{4} + 2 \nu^{2} - 8$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{9} - \nu^{5} + 2 \nu^{3} - 8 \nu$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{11} + 2 \nu^{10} + 2 \nu^{9} + 4 \nu^{8} + 9 \nu^{7} - 14 \nu^{6} + 8 \nu^{5} - 16 \nu^{4} + 4 \nu^{3} + 72 \nu^{2} + 48 \nu - 32$$$$)/128$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{11} - 2 \nu^{10} + 2 \nu^{9} - 4 \nu^{8} + 9 \nu^{7} + 14 \nu^{6} + 8 \nu^{5} + 16 \nu^{4} + 4 \nu^{3} - 72 \nu^{2} + 48 \nu + 32$$$$)/128$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{11} + 6 \nu^{10} - 2 \nu^{9} - 12 \nu^{8} - 9 \nu^{7} - 10 \nu^{6} + 24 \nu^{5} + 24 \nu^{4} + 28 \nu^{3} + 8 \nu^{2} + 16 \nu - 160$$$$)/128$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{10} - \nu^{6} + 2 \nu^{4} - 8 \nu^{2} + 16$$$$)/16$$ $$\beta_{9}$$ $$=$$ $$($$$$-3 \nu^{11} + 2 \nu^{10} + 2 \nu^{9} - 4 \nu^{8} + 5 \nu^{7} - 14 \nu^{6} - 16 \nu^{5} + 40 \nu^{4} + 4 \nu^{3} + 24 \nu^{2} + 112 \nu - 96$$$$)/128$$ $$\beta_{10}$$ $$=$$ $$($$$$3 \nu^{11} - 2 \nu^{10} + 6 \nu^{9} + 4 \nu^{8} - 5 \nu^{7} - 18 \nu^{6} - 8 \nu^{5} - 8 \nu^{4} + 76 \nu^{3} - 24 \nu^{2} + 16 \nu - 32$$$$)/128$$ $$\beta_{11}$$ $$=$$ $$($$$$-3 \nu^{11} - 2 \nu^{10} + 2 \nu^{9} + 4 \nu^{8} + 5 \nu^{7} + 14 \nu^{6} - 16 \nu^{5} - 40 \nu^{4} + 4 \nu^{3} - 24 \nu^{2} + 112 \nu + 96$$$$)/128$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{10} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{11} + \beta_{9} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$\beta_{11} - \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_{1} - 2$$ $$\nu^{6}$$ $$=$$ $$\beta_{11} - \beta_{9} - 2 \beta_{8} + 3 \beta_{6} - 3 \beta_{5} + \beta_{3} + 3 \beta_{2} - 2$$ $$\nu^{7}$$ $$=$$ $$\beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{7} + 5 \beta_{6} + 6 \beta_{5} + 3 \beta_{4} - \beta_{2} - 5 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$\beta_{11} - \beta_{9} - \beta_{6} + \beta_{5} - 13 \beta_{3} + \beta_{2} - 8$$ $$\nu^{9}$$ $$=$$ $$\beta_{11} + 3 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{5} - 13 \beta_{4} + 2 \beta_{3} + \beta_{2} - 9 \beta_{1} + 2$$ $$\nu^{10}$$ $$=$$ $$-3 \beta_{11} + 3 \beta_{9} - 14 \beta_{8} - \beta_{6} + \beta_{5} - 11 \beta_{3} - 9 \beta_{2} + 18$$ $$\nu^{11}$$ $$=$$ $$-23 \beta_{11} + 7 \beta_{10} - 6 \beta_{9} - 12 \beta_{8} - 17 \beta_{7} + 5 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + 12 \beta_{3} - 5 \beta_{2} + 27 \beta_{1} + 12$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$43$$ $$73$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
71.1
 0.430469 − 1.34711i 0.430469 + 1.34711i 1.19877 − 0.750295i 1.19877 + 0.750295i −1.37027 + 0.349801i −1.37027 − 0.349801i 1.37027 + 0.349801i 1.37027 − 0.349801i −1.19877 − 0.750295i −1.19877 + 0.750295i −0.430469 − 1.34711i −0.430469 + 1.34711i
−1.34711 0.430469i 0.916638 1.46962i 1.62939 + 1.15978i 0.348612i −1.86743 + 1.58515i 1.00000i −1.69572 2.26374i −1.31955 2.69421i −0.150067 + 0.469617i
71.2 −1.34711 + 0.430469i 0.916638 + 1.46962i 1.62939 1.15978i 0.348612i −1.86743 1.58515i 1.00000i −1.69572 + 2.26374i −1.31955 + 2.69421i −0.150067 0.469617i
71.3 −0.750295 1.19877i −0.448478 + 1.67298i −0.874114 + 1.79887i 3.56257i 2.34202 0.717607i 1.00000i 2.81228 0.301817i −2.59774 1.50059i 4.27072 2.67298i
71.4 −0.750295 + 1.19877i −0.448478 1.67298i −0.874114 1.79887i 3.56257i 2.34202 + 0.717607i 1.00000i 2.81228 + 0.301817i −2.59774 + 1.50059i 4.27072 + 2.67298i
71.5 −0.349801 1.37027i 1.72007 + 0.203364i −1.75528 + 0.958643i 2.27740i −0.323018 2.42810i 1.00000i 1.92760 + 2.06987i 2.91729 + 0.699602i −3.12065 + 0.796636i
71.6 −0.349801 + 1.37027i 1.72007 0.203364i −1.75528 0.958643i 2.27740i −0.323018 + 2.42810i 1.00000i 1.92760 2.06987i 2.91729 0.699602i −3.12065 0.796636i
71.7 0.349801 1.37027i −1.72007 0.203364i −1.75528 0.958643i 2.27740i −0.880346 + 2.28582i 1.00000i −1.92760 + 2.06987i 2.91729 + 0.699602i −3.12065 0.796636i
71.8 0.349801 + 1.37027i −1.72007 + 0.203364i −1.75528 + 0.958643i 2.27740i −0.880346 2.28582i 1.00000i −1.92760 2.06987i 2.91729 0.699602i −3.12065 + 0.796636i
71.9 0.750295 1.19877i 0.448478 1.67298i −0.874114 1.79887i 3.56257i −1.66903 1.79285i 1.00000i −2.81228 0.301817i −2.59774 1.50059i 4.27072 + 2.67298i
71.10 0.750295 + 1.19877i 0.448478 + 1.67298i −0.874114 + 1.79887i 3.56257i −1.66903 + 1.79285i 1.00000i −2.81228 + 0.301817i −2.59774 + 1.50059i 4.27072 2.67298i
71.11 1.34711 0.430469i −0.916638 + 1.46962i 1.62939 1.15978i 0.348612i −0.602184 + 2.37432i 1.00000i 1.69572 2.26374i −1.31955 2.69421i −0.150067 0.469617i
71.12 1.34711 + 0.430469i −0.916638 1.46962i 1.62939 + 1.15978i 0.348612i −0.602184 2.37432i 1.00000i 1.69572 + 2.26374i −1.31955 + 2.69421i −0.150067 + 0.469617i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 71.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.2.e.a 12
3.b odd 2 1 inner 84.2.e.a 12
4.b odd 2 1 inner 84.2.e.a 12
7.b odd 2 1 588.2.e.c 12
7.c even 3 2 588.2.n.f 24
7.d odd 6 2 588.2.n.g 24
8.b even 2 1 1344.2.h.h 12
8.d odd 2 1 1344.2.h.h 12
12.b even 2 1 inner 84.2.e.a 12
21.c even 2 1 588.2.e.c 12
21.g even 6 2 588.2.n.g 24
21.h odd 6 2 588.2.n.f 24
24.f even 2 1 1344.2.h.h 12
24.h odd 2 1 1344.2.h.h 12
28.d even 2 1 588.2.e.c 12
28.f even 6 2 588.2.n.g 24
28.g odd 6 2 588.2.n.f 24
84.h odd 2 1 588.2.e.c 12
84.j odd 6 2 588.2.n.g 24
84.n even 6 2 588.2.n.f 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.e.a 12 1.a even 1 1 trivial
84.2.e.a 12 3.b odd 2 1 inner
84.2.e.a 12 4.b odd 2 1 inner
84.2.e.a 12 12.b even 2 1 inner
588.2.e.c 12 7.b odd 2 1
588.2.e.c 12 21.c even 2 1
588.2.e.c 12 28.d even 2 1
588.2.e.c 12 84.h odd 2 1
588.2.n.f 24 7.c even 3 2
588.2.n.f 24 21.h odd 6 2
588.2.n.f 24 28.g odd 6 2
588.2.n.f 24 84.n even 6 2
588.2.n.g 24 7.d odd 6 2
588.2.n.g 24 21.g even 6 2
588.2.n.g 24 28.f even 6 2
588.2.n.g 24 84.j odd 6 2
1344.2.h.h 12 8.b even 2 1
1344.2.h.h 12 8.d odd 2 1
1344.2.h.h 12 24.f even 2 1
1344.2.h.h 12 24.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(84, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 + 32 T^{2} + 4 T^{4} - 4 T^{6} + T^{8} + 2 T^{10} + T^{12}$$
$3$ $$729 + 162 T^{2} - 45 T^{4} - 44 T^{6} - 5 T^{8} + 2 T^{10} + T^{12}$$
$5$ $$( 8 + 68 T^{2} + 18 T^{4} + T^{6} )^{2}$$
$7$ $$( 1 + T^{2} )^{6}$$
$11$ $$( -32 + 288 T^{2} - 34 T^{4} + T^{6} )^{2}$$
$13$ $$( -4 - 10 T + T^{3} )^{4}$$
$17$ $$( 32 + 288 T^{2} + 34 T^{4} + T^{6} )^{2}$$
$19$ $$( 3136 + 1124 T^{2} + 64 T^{4} + T^{6} )^{2}$$
$23$ $$( -128 + 128 T^{2} - 26 T^{4} + T^{6} )^{2}$$
$29$ $$( 46208 + 5136 T^{2} + 144 T^{4} + T^{6} )^{2}$$
$31$ $$( 6400 + 1344 T^{2} + 68 T^{4} + T^{6} )^{2}$$
$37$ $$( 16 - 28 T + 4 T^{2} + T^{3} )^{4}$$
$41$ $$( 800 + 512 T^{2} + 66 T^{4} + T^{6} )^{2}$$
$43$ $$( 43264 + 7952 T^{2} + 184 T^{4} + T^{6} )^{2}$$
$47$ $$( -8192 + 2048 T^{2} - 104 T^{4} + T^{6} )^{2}$$
$53$ $$( 2048 + 848 T^{2} + 56 T^{4} + T^{6} )^{2}$$
$59$ $$( -204800 + 17428 T^{2} - 280 T^{4} + T^{6} )^{2}$$
$61$ $$( -52 - 146 T + 4 T^{2} + T^{3} )^{4}$$
$67$ $$( 256 + 656 T^{2} + 72 T^{4} + T^{6} )^{2}$$
$71$ $$( -366368 + 21312 T^{2} - 274 T^{4} + T^{6} )^{2}$$
$73$ $$( 104 - 28 T - 6 T^{2} + T^{3} )^{4}$$
$79$ $$( 16 + T^{2} )^{6}$$
$83$ $$( -1568 + 3860 T^{2} - 128 T^{4} + T^{6} )^{2}$$
$89$ $$( 326432 + 26368 T^{2} + 322 T^{4} + T^{6} )^{2}$$
$97$ $$( 2 + T )^{12}$$