# Properties

 Label 460.2.e.a Level $460$ Weight $2$ Character orbit 460.e Analytic conductor $3.673$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$460 = 2^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 460.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.67311849298$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: 16.0.7465802011608416256.3 Defining polynomial: $$x^{16} - x^{14} + x^{12} + 8 x^{10} - 20 x^{8} + 32 x^{6} + 16 x^{4} - 64 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{8}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{2} + \beta_{8} q^{3} + ( -\beta_{2} + \beta_{8} ) q^{4} + \beta_{3} q^{5} + ( 1 + \beta_{2} + \beta_{6} - \beta_{8} + \beta_{10} ) q^{6} + ( 2 \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{7} + ( \beta_{2} - \beta_{8} + 2 \beta_{10} ) q^{8} + ( -\beta_{2} - \beta_{4} - \beta_{6} + \beta_{8} - \beta_{10} + \beta_{14} ) q^{9} +O(q^{10})$$ $$q -\beta_{4} q^{2} + \beta_{8} q^{3} + ( -\beta_{2} + \beta_{8} ) q^{4} + \beta_{3} q^{5} + ( 1 + \beta_{2} + \beta_{6} - \beta_{8} + \beta_{10} ) q^{6} + ( 2 \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{7} + ( \beta_{2} - \beta_{8} + 2 \beta_{10} ) q^{8} + ( -\beta_{2} - \beta_{4} - \beta_{6} + \beta_{8} - \beta_{10} + \beta_{14} ) q^{9} + \beta_{12} q^{10} + ( \beta_{12} + \beta_{13} - \beta_{15} ) q^{11} + ( -2 - \beta_{2} - \beta_{4} - \beta_{9} - \beta_{10} ) q^{12} + ( -\beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{13} + ( -\beta_{1} - 3 \beta_{3} - \beta_{5} - \beta_{7} + 2 \beta_{11} + \beta_{12} + \beta_{15} ) q^{14} + ( -\beta_{5} + \beta_{11} + \beta_{15} ) q^{15} + ( -\beta_{2} - \beta_{8} - 2 \beta_{9} ) q^{16} + ( 4 \beta_{1} + \beta_{5} + \beta_{7} + \beta_{11} - \beta_{12} + 2 \beta_{15} ) q^{17} + ( 2 + \beta_{2} - \beta_{4} + 2 \beta_{8} + \beta_{9} + \beta_{10} ) q^{18} + ( -2 \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{19} + ( -\beta_{1} - \beta_{5} + \beta_{11} + \beta_{12} ) q^{20} + ( \beta_{1} + 3 \beta_{3} + \beta_{5} + \beta_{7} + \beta_{11} - \beta_{12} - \beta_{15} ) q^{21} + ( -2 \beta_{1} - 3 \beta_{3} - 2 \beta_{5} + \beta_{7} + \beta_{11} + 2 \beta_{12} - \beta_{15} ) q^{22} + ( -\beta_{2} - \beta_{4} + \beta_{6} - \beta_{10} + \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{23} + ( \beta_{2} + \beta_{4} + 2 \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{14} ) q^{24} - q^{25} + ( -3 + 2 \beta_{2} - \beta_{6} - 2 \beta_{8} + \beta_{10} ) q^{26} + ( -\beta_{2} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{9} - \beta_{10} - \beta_{14} ) q^{27} + ( 3 \beta_{5} + 3 \beta_{7} - \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{28} + ( -3 - \beta_{4} - \beta_{6} ) q^{29} + ( \beta_{1} + \beta_{3} + \beta_{5} + \beta_{7} - \beta_{15} ) q^{30} + ( -3 \beta_{2} + 4 \beta_{8} + \beta_{9} - 3 \beta_{10} - 3 \beta_{14} ) q^{31} + ( 3 \beta_{2} - 2 \beta_{4} + \beta_{8} + 2 \beta_{10} + 4 \beta_{14} ) q^{32} + ( 5 \beta_{1} + 3 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{11} - 2 \beta_{12} + \beta_{15} ) q^{33} + ( -3 \beta_{1} - 3 \beta_{7} - 3 \beta_{11} - 2 \beta_{12} + \beta_{13} - 4 \beta_{15} ) q^{34} + ( \beta_{4} - \beta_{6} - \beta_{9} ) q^{35} + ( 2 - \beta_{2} - \beta_{4} + 2 \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{14} ) q^{36} + ( \beta_{1} + 3 \beta_{3} + \beta_{5} + \beta_{7} + \beta_{11} + 2 \beta_{12} + 2 \beta_{15} ) q^{37} + ( \beta_{1} + 3 \beta_{3} + \beta_{5} + \beta_{7} - 2 \beta_{11} - \beta_{12} - \beta_{15} ) q^{38} + ( -2 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{14} ) q^{39} + ( \beta_{1} + \beta_{5} + 2 \beta_{7} + \beta_{11} + \beta_{12} ) q^{40} + ( -3 - 3 \beta_{2} + 2 \beta_{4} + \beta_{8} - \beta_{10} + \beta_{14} ) q^{41} + ( -3 \beta_{3} + 3 \beta_{5} - 3 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} ) q^{42} + ( 3 \beta_{5} - 3 \beta_{7} - 3 \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{15} ) q^{43} + ( 3 \beta_{1} + 3 \beta_{7} + 3 \beta_{11} - 2 \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{44} + ( -2 \beta_{1} - \beta_{5} - \beta_{7} - \beta_{11} + \beta_{12} ) q^{45} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} + 2 \beta_{15} ) q^{46} + ( -3 \beta_{2} + 5 \beta_{4} - 5 \beta_{6} - 6 \beta_{8} - 3 \beta_{10} - 3 \beta_{14} ) q^{47} + ( 2 + 3 \beta_{2} - \beta_{4} + 2 \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{14} ) q^{48} + ( 5 + 3 \beta_{2} - 3 \beta_{4} + 3 \beta_{6} + 3 \beta_{8} - 3 \beta_{10} + 3 \beta_{14} ) q^{49} + \beta_{4} q^{50} + ( -3 \beta_{5} + 3 \beta_{7} + 3 \beta_{11} + 2 \beta_{12} - 4 \beta_{13} + \beta_{15} ) q^{51} + ( 2 - 2 \beta_{2} + 3 \beta_{4} + \beta_{8} - \beta_{9} - 3 \beta_{10} ) q^{52} + ( -4 \beta_{1} - 4 \beta_{5} - 4 \beta_{7} - 4 \beta_{11} - 2 \beta_{12} - 2 \beta_{15} ) q^{53} + ( -3 + \beta_{2} - \beta_{4} + \beta_{6} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + 4 \beta_{14} ) q^{54} + ( 2 \beta_{4} - 2 \beta_{6} + \beta_{9} ) q^{55} + ( 2 \beta_{1} + 5 \beta_{5} - \beta_{7} - 4 \beta_{11} - 5 \beta_{12} + 3 \beta_{13} + 4 \beta_{15} ) q^{56} + ( -\beta_{1} - 3 \beta_{3} - \beta_{5} - \beta_{7} - \beta_{11} + \beta_{12} + \beta_{15} ) q^{57} + ( 2 - \beta_{2} + 3 \beta_{4} + \beta_{8} ) q^{58} + ( -4 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} - 4 \beta_{10} - 4 \beta_{14} ) q^{59} + ( -\beta_{1} - 2 \beta_{3} - \beta_{7} - \beta_{11} + \beta_{13} ) q^{60} + ( \beta_{1} - 3 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{11} - \beta_{12} + 2 \beta_{15} ) q^{61} + ( -\beta_{2} + 4 \beta_{4} - 4 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - 2 \beta_{14} ) q^{62} + ( -6 \beta_{5} + 6 \beta_{11} + 5 \beta_{12} - 4 \beta_{13} + \beta_{15} ) q^{63} + ( 4 + 3 \beta_{2} - 4 \beta_{4} + 4 \beta_{6} + 3 \beta_{8} - 2 \beta_{9} ) q^{64} + ( -\beta_{1} - \beta_{12} - 2 \beta_{15} ) q^{65} + ( -3 \beta_{1} - 3 \beta_{3} + 3 \beta_{5} - 3 \beta_{7} - 6 \beta_{11} - \beta_{12} + 2 \beta_{13} - 5 \beta_{15} ) q^{66} + ( 3 \beta_{5} + 3 \beta_{7} - 3 \beta_{11} + 3 \beta_{13} - 3 \beta_{15} ) q^{67} + ( \beta_{1} - 6 \beta_{3} - 2 \beta_{5} + \beta_{7} + \beta_{11} + 2 \beta_{12} - 3 \beta_{13} + 2 \beta_{15} ) q^{68} + ( 3 - 4 \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{14} - 2 \beta_{15} ) q^{69} + ( 3 + 3 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{8} + \beta_{10} + 2 \beta_{14} ) q^{70} + ( 3 \beta_{2} + \beta_{4} - \beta_{6} + 3 \beta_{9} + 3 \beta_{10} + 3 \beta_{14} ) q^{71} + ( -6 - 3 \beta_{2} - \beta_{4} - 2 \beta_{6} - \beta_{9} + \beta_{10} + 2 \beta_{14} ) q^{72} + ( -2 \beta_{2} + 4 \beta_{4} - \beta_{6} - 3 \beta_{8} + 3 \beta_{10} - 3 \beta_{14} ) q^{73} + ( -3 \beta_{1} + 3 \beta_{3} + 4 \beta_{12} + \beta_{13} - \beta_{15} ) q^{74} -\beta_{8} q^{75} + ( -3 \beta_{5} - 3 \beta_{7} + \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{76} + ( 6 + 6 \beta_{2} - 3 \beta_{4} + 3 \beta_{6} ) q^{77} + ( 2 + 2 \beta_{2} + \beta_{4} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{14} ) q^{78} + ( -3 \beta_{5} - 3 \beta_{7} + 3 \beta_{11} + 3 \beta_{13} + 3 \beta_{15} ) q^{79} + ( -\beta_{1} + \beta_{5} - \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{80} + ( 3 + 2 \beta_{2} - 3 \beta_{4} - \beta_{6} ) q^{81} + ( -2 + 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} - \beta_{8} + \beta_{9} + 3 \beta_{10} ) q^{82} + ( -6 \beta_{7} - 6 \beta_{12} + 6 \beta_{15} ) q^{83} + ( -5 \beta_{1} - 5 \beta_{5} - 2 \beta_{7} + \beta_{11} - \beta_{12} + 2 \beta_{15} ) q^{84} + ( -3 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{8} - \beta_{10} + \beta_{14} ) q^{85} + ( -5 \beta_{1} - 3 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{11} + 2 \beta_{12} - 3 \beta_{13} - \beta_{15} ) q^{86} + ( \beta_{2} - \beta_{4} + \beta_{6} - 4 \beta_{8} + \beta_{10} + \beta_{14} ) q^{87} + ( \beta_{1} + 4 \beta_{5} + \beta_{7} - 5 \beta_{11} - 4 \beta_{12} + 3 \beta_{13} - 4 \beta_{15} ) q^{88} + ( -\beta_{1} + 9 \beta_{3} - \beta_{5} - \beta_{7} - \beta_{11} - 2 \beta_{12} - 2 \beta_{15} ) q^{89} + ( \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + \beta_{7} + 3 \beta_{11} + 2 \beta_{12} - \beta_{13} + 2 \beta_{15} ) q^{90} + ( 6 \beta_{5} + 6 \beta_{7} - 6 \beta_{11} - \beta_{12} - \beta_{13} - 5 \beta_{15} ) q^{91} + ( -3 \beta_{1} + 4 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{10} - 3 \beta_{11} + \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - 4 \beta_{15} ) q^{92} + ( -8 - 2 \beta_{2} - 3 \beta_{4} - 4 \beta_{6} + \beta_{8} - \beta_{10} + \beta_{14} ) q^{93} + ( 1 - 4 \beta_{2} + 3 \beta_{4} - 9 \beta_{6} + \beta_{8} + 3 \beta_{9} - 6 \beta_{10} ) q^{94} + ( -\beta_{4} + \beta_{6} + \beta_{9} ) q^{95} + ( -2 - 3 \beta_{2} - \beta_{4} + 2 \beta_{6} - \beta_{9} - 3 \beta_{10} + 2 \beta_{14} ) q^{96} + ( \beta_{1} - 3 \beta_{3} + 4 \beta_{5} + 4 \beta_{7} + 4 \beta_{11} + 5 \beta_{12} + 2 \beta_{15} ) q^{97} + ( 3 \beta_{2} - 8 \beta_{4} + 6 \beta_{6} + 6 \beta_{8} + 3 \beta_{9} - 3 \beta_{10} ) q^{98} + ( -9 \beta_{5} + 3 \beta_{7} + 9 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} + 5 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{4} + 14q^{6} + 4q^{9} + O(q^{10})$$ $$16q + 4q^{4} + 14q^{6} + 4q^{9} - 30q^{12} + 4q^{13} + 4q^{16} + 30q^{18} + 2q^{24} - 16q^{25} - 54q^{26} - 48q^{29} + 34q^{36} - 36q^{41} - 40q^{46} + 18q^{48} + 68q^{49} + 34q^{52} - 40q^{54} + 36q^{58} + 6q^{62} + 52q^{64} + 40q^{69} + 42q^{70} - 78q^{72} + 8q^{73} + 72q^{77} + 32q^{78} + 40q^{81} - 42q^{82} + 12q^{85} - 120q^{93} + 20q^{94} - 22q^{96} - 18q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{14} + x^{12} + 8 x^{10} - 20 x^{8} + 32 x^{6} + 16 x^{4} - 64 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{15} + 5 \nu^{13} + 3 \nu^{11} + 4 \nu^{9} + 172 \nu^{7} - 96 \nu^{5} - 112 \nu^{3} - 256 \nu$$$$)/1152$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{14} - \nu^{12} - 15 \nu^{10} - 2 \nu^{8} + 28 \nu^{6} - 96 \nu^{4} + 80 \nu^{2} - 160$$$$)/288$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{15} + 7 \nu^{13} + 9 \nu^{11} - 16 \nu^{9} + 44 \nu^{7} + 144 \nu^{5} - 80 \nu^{3} + 640 \nu$$$$)/1152$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{14} - 3 \nu^{13} + 4 \nu^{12} + 15 \nu^{11} - 12 \nu^{10} - 27 \nu^{9} - 22 \nu^{8} + 24 \nu^{7} + 32 \nu^{6} + 24 \nu^{5} - 120 \nu^{4} - 216 \nu^{3} - 128 \nu^{2} + 576 \nu + 64$$$$)/576$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{15} + 6 \nu^{14} + 11 \nu^{13} - 6 \nu^{12} - 15 \nu^{11} + 30 \nu^{10} + 22 \nu^{9} + 24 \nu^{8} - 32 \nu^{7} + 96 \nu^{6} - 168 \nu^{5} + 192 \nu^{4} + 176 \nu^{3} - 576 \nu^{2} + 128 \nu + 768$$$$)/1152$$ $$\beta_{6}$$ $$=$$ $$($$$$2 \nu^{14} - 3 \nu^{13} - 4 \nu^{12} + 15 \nu^{11} + 12 \nu^{10} - 27 \nu^{9} + 22 \nu^{8} + 24 \nu^{7} - 32 \nu^{6} + 24 \nu^{5} + 120 \nu^{4} - 216 \nu^{3} + 128 \nu^{2} + 576 \nu - 64$$$$)/576$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{15} - 9 \nu^{13} + 9 \nu^{11} - 16 \nu^{9} - 36 \nu^{7} + 144 \nu^{5} - 80 \nu^{3}$$$$)/384$$ $$\beta_{8}$$ $$=$$ $$($$$$-3 \nu^{15} - 4 \nu^{14} - 3 \nu^{13} + 8 \nu^{12} + 3 \nu^{11} - 24 \nu^{10} - 54 \nu^{9} - 44 \nu^{8} - 12 \nu^{7} + 64 \nu^{6} + 48 \nu^{5} - 240 \nu^{4} - 672 \nu^{3} - 256 \nu^{2} + 128$$$$)/1152$$ $$\beta_{9}$$ $$=$$ $$($$$$2 \nu^{14} + 9 \nu^{13} - 4 \nu^{12} - 9 \nu^{11} + 12 \nu^{10} - 27 \nu^{9} + 22 \nu^{8} + 36 \nu^{7} - 32 \nu^{6} - 144 \nu^{5} + 120 \nu^{4} + 216 \nu^{3} + 128 \nu^{2} - 64$$$$)/576$$ $$\beta_{10}$$ $$=$$ $$($$$$-3 \nu^{15} + 8 \nu^{14} + 3 \nu^{13} - 4 \nu^{12} - 39 \nu^{11} + 36 \nu^{10} - 12 \nu^{9} - 44 \nu^{8} + 48 \nu^{7} - 80 \nu^{6} - 120 \nu^{5} + 288 \nu^{4} - 448 \nu^{2} + 896$$$$)/1152$$ $$\beta_{11}$$ $$=$$ $$($$$$-\nu^{15} - 6 \nu^{14} + 11 \nu^{13} + 30 \nu^{12} - 15 \nu^{11} - 6 \nu^{10} + 22 \nu^{9} - 48 \nu^{8} - 32 \nu^{7} + 336 \nu^{6} - 168 \nu^{5} - 96 \nu^{4} + 176 \nu^{3} + 192 \nu^{2} + 128 \nu + 1152$$$$)/1152$$ $$\beta_{12}$$ $$=$$ $$($$$$\nu^{15} - 4 \nu^{12} + 8 \nu^{10} + 5 \nu^{9} - 8 \nu^{8} - 12 \nu^{6} + 32 \nu^{4} - 8 \nu^{3} - 128 \nu^{2} + 64$$$$)/192$$ $$\beta_{13}$$ $$=$$ $$($$$$\nu^{15} + 4 \nu^{12} + 5 \nu^{9} + 20 \nu^{6} - 96 \nu^{4} - 8 \nu^{3} + 96 \nu^{2} + 64$$$$)/192$$ $$\beta_{14}$$ $$=$$ $$($$$$3 \nu^{15} + 8 \nu^{14} - 3 \nu^{13} - 4 \nu^{12} + 39 \nu^{11} + 36 \nu^{10} + 12 \nu^{9} - 44 \nu^{8} - 48 \nu^{7} - 80 \nu^{6} + 120 \nu^{5} + 288 \nu^{4} - 448 \nu^{2} + 896$$$$)/1152$$ $$\beta_{15}$$ $$=$$ $$($$$$-\nu^{15} - 4 \nu^{12} + 8 \nu^{10} - 5 \nu^{9} - 8 \nu^{8} - 12 \nu^{6} + 32 \nu^{4} + 8 \nu^{3} - 128 \nu^{2} + 64$$$$)/192$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{14} + \beta_{10} - \beta_{9} - 2 \beta_{8} - \beta_{7} + \beta_{6} + 2 \beta_{4} + \beta_{3} - 2 \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{11} + \beta_{10} - \beta_{7} + 4 \beta_{6} - 4 \beta_{5} - 4 \beta_{4} + 3 \beta_{2} - \beta_{1} + 1$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{15} - \beta_{12} + \beta_{9} - 3 \beta_{8} + \beta_{7} - 2 \beta_{6} + 2 \beta_{4}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-5 \beta_{15} + \beta_{14} - 7 \beta_{13} + 2 \beta_{12} + 4 \beta_{11} + \beta_{10} + \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{2} + \beta_{1} - 1$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{15} - \beta_{14} - 2 \beta_{12} + \beta_{10} - 3 \beta_{9} + 5 \beta_{7} - 3 \beta_{6} - 6 \beta_{4} + 17 \beta_{3} + 2 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$\beta_{15} + \beta_{12} + 5 \beta_{11} + 5 \beta_{7} + 3 \beta_{6} + 5 \beta_{5} - 3 \beta_{4} + 6 \beta_{2} + 5 \beta_{1} - 5$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-5 \beta_{14} + 5 \beta_{10} - \beta_{9} - 10 \beta_{8} + 7 \beta_{7} - 3 \beta_{6} + 6 \beta_{4} + 5 \beta_{3} + 26 \beta_{1}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-9 \beta_{15} - 25 \beta_{14} - \beta_{13} - 8 \beta_{12} - 18 \beta_{11} - 25 \beta_{10} + 9 \beta_{7} - 4 \beta_{6} + 36 \beta_{5} + 4 \beta_{4} - 27 \beta_{2} + 9 \beta_{1} + 23$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$-\beta_{15} + \beta_{12} - 21 \beta_{9} - 9 \beta_{8} - 13 \beta_{7} + 6 \beta_{6} - 6 \beta_{4}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$49 \beta_{15} - 13 \beta_{14} + 43 \beta_{13} + 6 \beta_{12} + 12 \beta_{11} - 13 \beta_{10} + 3 \beta_{7} + 22 \beta_{6} - 6 \beta_{5} - 22 \beta_{4} - 35 \beta_{2} + 3 \beta_{1} - 35$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$22 \beta_{15} + 61 \beta_{14} - 22 \beta_{12} - 61 \beta_{10} + 7 \beta_{9} - 17 \beta_{7} + 7 \beta_{6} + 14 \beta_{4} - 29 \beta_{3} + 22 \beta_{1}$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$-29 \beta_{15} - 29 \beta_{12} - \beta_{11} - \beta_{7} - 39 \beta_{6} - \beta_{5} + 39 \beta_{4} - 78 \beta_{2} - \beta_{1} - 31$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$65 \beta_{14} - 65 \beta_{10} + 45 \beta_{9} + 130 \beta_{8} - 91 \beta_{7} - 25 \beta_{6} - 110 \beta_{4} + 223 \beta_{3} - 50 \beta_{1}$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$-43 \beta_{15} + 229 \beta_{14} + 109 \beta_{13} - 152 \beta_{12} - 86 \beta_{11} + 229 \beta_{10} + 43 \beta_{7} + 52 \beta_{6} + 172 \beta_{5} - 52 \beta_{4} + 255 \beta_{2} + 43 \beta_{1} - 203$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$-179 \beta_{15} + 179 \beta_{12} + 113 \beta_{9} + 21 \beta_{8} + 73 \beta_{7} - 46 \beta_{6} + 46 \beta_{4}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/460\mathbb{Z}\right)^\times$$.

 $$n$$ $$231$$ $$277$$ $$281$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1
 1.18353 − 0.774115i −0.0786378 + 1.41203i −0.0786378 − 1.41203i 1.18353 + 0.774115i 1.37379 − 0.335728i 0.977642 − 1.02187i 0.977642 + 1.02187i 1.37379 + 0.335728i −0.977642 + 1.02187i −1.37379 + 0.335728i −1.37379 − 0.335728i −0.977642 − 1.02187i 0.0786378 − 1.41203i −1.18353 + 0.774115i −1.18353 − 0.774115i 0.0786378 + 1.41203i
−1.35760 0.396143i 1.47175i 1.68614 + 1.07561i 1.00000i 0.583024 1.99804i −3.53986 −1.86301 2.12819i 0.833952 −0.396143 + 1.35760i
91.2 −1.35760 0.396143i 1.47175i 1.68614 + 1.07561i 1.00000i 0.583024 1.99804i 3.53986 −1.86301 2.12819i 0.833952 0.396143 1.35760i
91.3 −1.35760 + 0.396143i 1.47175i 1.68614 1.07561i 1.00000i 0.583024 + 1.99804i 3.53986 −1.86301 + 2.12819i 0.833952 0.396143 + 1.35760i
91.4 −1.35760 + 0.396143i 1.47175i 1.68614 1.07561i 1.00000i 0.583024 + 1.99804i −3.53986 −1.86301 + 2.12819i 0.833952 −0.396143 1.35760i
91.5 −0.637910 1.26217i 2.87247i −1.18614 + 1.61030i 1.00000i 3.62554 1.83238i −2.68161 2.78912 + 0.469882i −5.25109 −1.26217 + 0.637910i
91.6 −0.637910 1.26217i 2.87247i −1.18614 + 1.61030i 1.00000i 3.62554 1.83238i 2.68161 2.78912 + 0.469882i −5.25109 1.26217 0.637910i
91.7 −0.637910 + 1.26217i 2.87247i −1.18614 1.61030i 1.00000i 3.62554 + 1.83238i 2.68161 2.78912 0.469882i −5.25109 1.26217 + 0.637910i
91.8 −0.637910 + 1.26217i 2.87247i −1.18614 1.61030i 1.00000i 3.62554 + 1.83238i −2.68161 2.78912 0.469882i −5.25109 −1.26217 0.637910i
91.9 0.637910 1.26217i 0.348132i −1.18614 1.61030i 1.00000i −0.439402 0.222077i −4.89140 −2.78912 + 0.469882i 2.87880 −1.26217 0.637910i
91.10 0.637910 1.26217i 0.348132i −1.18614 1.61030i 1.00000i −0.439402 0.222077i 4.89140 −2.78912 + 0.469882i 2.87880 1.26217 + 0.637910i
91.11 0.637910 + 1.26217i 0.348132i −1.18614 + 1.61030i 1.00000i −0.439402 + 0.222077i 4.89140 −2.78912 0.469882i 2.87880 1.26217 0.637910i
91.12 0.637910 + 1.26217i 0.348132i −1.18614 + 1.61030i 1.00000i −0.439402 + 0.222077i −4.89140 −2.78912 0.469882i 2.87880 −1.26217 + 0.637910i
91.13 1.35760 0.396143i 0.679463i 1.68614 1.07561i 1.00000i −0.269165 0.922437i 1.16300 1.86301 2.12819i 2.53833 −0.396143 1.35760i
91.14 1.35760 0.396143i 0.679463i 1.68614 1.07561i 1.00000i −0.269165 0.922437i −1.16300 1.86301 2.12819i 2.53833 0.396143 + 1.35760i
91.15 1.35760 + 0.396143i 0.679463i 1.68614 + 1.07561i 1.00000i −0.269165 + 0.922437i −1.16300 1.86301 + 2.12819i 2.53833 0.396143 1.35760i
91.16 1.35760 + 0.396143i 0.679463i 1.68614 + 1.07561i 1.00000i −0.269165 + 0.922437i 1.16300 1.86301 + 2.12819i 2.53833 −0.396143 + 1.35760i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.e.a 16
4.b odd 2 1 inner 460.2.e.a 16
23.b odd 2 1 inner 460.2.e.a 16
92.b even 2 1 inner 460.2.e.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.e.a 16 1.a even 1 1 trivial
460.2.e.a 16 4.b odd 2 1 inner
460.2.e.a 16 23.b odd 2 1 inner
460.2.e.a 16 92.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 11 T_{3}^{6} + 24 T_{3}^{4} + 11 T_{3}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(460, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16 - 4 T^{2} - T^{6} + T^{8} )^{2}$$
$3$ $$( 1 + 11 T^{2} + 24 T^{4} + 11 T^{6} + T^{8} )^{2}$$
$5$ $$( 1 + T^{2} )^{8}$$
$7$ $$( 2916 - 2916 T^{2} + 621 T^{4} - 45 T^{6} + T^{8} )^{2}$$
$11$ $$( 2916 - 2916 T^{2} + 621 T^{4} - 45 T^{6} + T^{8} )^{2}$$
$13$ $$( 31 - 25 T - 24 T^{2} - T^{3} + T^{4} )^{4}$$
$17$ $$( 11664 + 21384 T^{2} + 2889 T^{4} + 99 T^{6} + T^{8} )^{2}$$
$19$ $$( 2916 - 2916 T^{2} + 621 T^{4} - 45 T^{6} + T^{8} )^{2}$$
$23$ $$( 279841 - 8464 T^{2} - 66 T^{4} - 16 T^{6} + T^{8} )^{2}$$
$29$ $$( 12 + 54 T + 45 T^{2} + 12 T^{3} + T^{4} )^{4}$$
$31$ $$( 4124961 + 430191 T^{2} + 14892 T^{4} + 207 T^{6} + T^{8} )^{2}$$
$37$ $$( 46656 + 19440 T^{2} + 2484 T^{4} + 108 T^{6} + T^{8} )^{2}$$
$41$ $$( -681 - 387 T - 30 T^{2} + 9 T^{3} + T^{4} )^{4}$$
$43$ $$( 46656 - 89424 T^{2} + 7668 T^{4} - 180 T^{6} + T^{8} )^{2}$$
$47$ $$( 163216 + 1530020 T^{2} + 41457 T^{4} + 362 T^{6} + T^{8} )^{2}$$
$53$ $$( 6912 + 180 T^{2} + T^{4} )^{4}$$
$59$ $$( 256 + 76 T^{2} + T^{4} )^{4}$$
$61$ $$( 13089924 + 879660 T^{2} + 22005 T^{4} + 243 T^{6} + T^{8} )^{2}$$
$67$ $$( 6718464 - 793152 T^{2} + 26244 T^{4} - 288 T^{6} + T^{8} )^{2}$$
$71$ $$( 6889 + 21359 T^{2} + 8940 T^{4} + 215 T^{6} + T^{8} )^{2}$$
$73$ $$( -908 + 718 T - 123 T^{2} - 2 T^{3} + T^{4} )^{4}$$
$79$ $$( 3779136 - 524880 T^{2} + 22356 T^{4} - 324 T^{6} + T^{8} )^{2}$$
$83$ $$( 5184 - 252 T^{2} + T^{4} )^{4}$$
$89$ $$( 13483584 + 1839024 T^{2} + 46548 T^{4} + 396 T^{6} + T^{8} )^{2}$$
$97$ $$( 29746116 + 4740444 T^{2} + 88749 T^{4} + 531 T^{6} + T^{8} )^{2}$$