# Properties

 Label 48.2.k.a Level $48$ Weight $2$ Character orbit 48.k Analytic conductor $0.383$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 48.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.383281929702$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: 12.0.163368480538624.2 Defining polynomial: $$x^{12} - 2 x^{10} - 2 x^{8} + 16 x^{6} - 8 x^{4} - 32 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{8} - 2 \nu^{4} + 4 \nu^{2} + 8$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{11} + 2 \nu^{9} + 2 \nu^{7} + 8 \nu^{3}$$$$)/32$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{11} - 2 \nu^{9} - 2 \nu^{7} + 16 \nu^{5} - 8 \nu^{3} - 32 \nu$$$$)/32$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{11} - \nu^{10} - 6 \nu^{7} + 10 \nu^{6} + 4 \nu^{5} - 12 \nu^{4} + 8 \nu^{3} - 32 \nu + 64$$$$)/32$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{11} + \nu^{10} + 8 \nu^{7} - 2 \nu^{6} - 16 \nu^{5} - 4 \nu^{4} - 24 \nu^{3} + 16 \nu^{2} + 32 \nu$$$$)/32$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{11} + 2 \nu^{9} + 10 \nu^{7} - 16 \nu^{5} - 8 \nu^{3} + 64 \nu$$$$)/32$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{11} + \nu^{10} + 6 \nu^{7} - 10 \nu^{6} - 4 \nu^{5} + 12 \nu^{4} - 8 \nu^{3} + 32 \nu^{2} + 32 \nu - 64$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{9} + 2 \nu^{5} - 4 \nu^{3} - 16 \nu$$$$)/8$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{11} - \nu^{10} + 6 \nu^{7} + 10 \nu^{6} - 4 \nu^{5} - 12 \nu^{4} - 8 \nu^{3} - 32 \nu^{2} + 32 \nu + 64$$$$)/32$$ $$\beta_{10}$$ $$=$$ $$($$$$-2 \nu^{11} - \nu^{10} + 8 \nu^{7} + 2 \nu^{6} - 16 \nu^{5} + 4 \nu^{4} - 24 \nu^{3} - 16 \nu^{2} + 32 \nu$$$$)/32$$ $$\beta_{11}$$ $$=$$ $$($$$$2 \nu^{11} + 3 \nu^{10} - 8 \nu^{7} - 6 \nu^{6} + 16 \nu^{5} + 20 \nu^{4} + 24 \nu^{3} + 16 \nu^{2} - 32 \nu - 32$$$$)/32$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{4}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - 2 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$\beta_{11} + 2 \beta_{10} + \beta_{7} - \beta_{5} + \beta_{4} + 1$$ $$\nu^{5}$$ $$=$$ $$\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{3} + \beta_{2}$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{7} + 4 \beta_{4} - 6$$ $$\nu^{7}$$ $$=$$ $$-2 \beta_{10} + 4 \beta_{8} + 6 \beta_{6} - 2 \beta_{5} + 2 \beta_{2}$$ $$\nu^{8}$$ $$=$$ $$2 \beta_{11} + 4 \beta_{10} - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + 8 \beta_{1} - 6$$ $$\nu^{9}$$ $$=$$ $$4 \beta_{10} - 10 \beta_{9} - 6 \beta_{8} - 10 \beta_{7} + 6 \beta_{6} + 4 \beta_{5} + 12 \beta_{3} + 10 \beta_{2}$$ $$\nu^{10}$$ $$=$$ $$8 \beta_{11} - 4 \beta_{10} + 4 \beta_{9} - 8 \beta_{7} + 12 \beta_{5} - 4 \beta_{4} - 8$$ $$\nu^{11}$$ $$=$$ $$-4 \beta_{10} - 12 \beta_{9} + 4 \beta_{8} - 12 \beta_{7} + 24 \beta_{6} - 4 \beta_{5} + 8 \beta_{3} - 8 \beta_{2}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$31$$ $$37$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 0.204810 − 1.39930i 1.27715 − 0.607364i −1.35164 − 0.416001i 1.35164 + 0.416001i −1.27715 + 0.607364i −0.204810 + 1.39930i 0.204810 + 1.39930i 1.27715 + 0.607364i −1.35164 + 0.416001i 1.35164 − 0.416001i −1.27715 − 0.607364i −0.204810 − 1.39930i
−1.39930 + 0.204810i −0.814141 1.52878i 1.91611 0.573183i 2.08397 2.08397i 1.45234 + 1.97249i −1.14637 −2.56382 + 1.19449i −1.67435 + 2.48929i −2.48929 + 3.34292i
11.2 −0.607364 + 1.27715i 1.73003 + 0.0835731i −1.26222 1.55139i −0.431733 + 0.431733i −1.15749 + 2.15875i −3.10278 2.74798 0.669785i 2.98603 + 0.289169i −0.289169 0.813607i
11.3 −0.416001 1.35164i 0.966579 1.43726i −1.65389 + 1.12457i −1.57184 + 1.57184i −2.34477 0.708570i 2.24914 2.20804 + 1.76765i −1.13145 2.77846i 2.77846 + 1.47068i
11.4 0.416001 + 1.35164i −1.43726 + 0.966579i −1.65389 + 1.12457i 1.57184 1.57184i −1.90437 1.54057i 2.24914 −2.20804 1.76765i 1.13145 2.77846i 2.77846 + 1.47068i
11.5 0.607364 1.27715i 0.0835731 + 1.73003i −1.26222 1.55139i 0.431733 0.431733i 2.26027 + 0.944024i −3.10278 −2.74798 + 0.669785i −2.98603 + 0.289169i −0.289169 0.813607i
11.6 1.39930 0.204810i −1.52878 0.814141i 1.91611 0.573183i −2.08397 + 2.08397i −2.30598 0.826122i −1.14637 2.56382 1.19449i 1.67435 + 2.48929i −2.48929 + 3.34292i
35.1 −1.39930 0.204810i −0.814141 + 1.52878i 1.91611 + 0.573183i 2.08397 + 2.08397i 1.45234 1.97249i −1.14637 −2.56382 1.19449i −1.67435 2.48929i −2.48929 3.34292i
35.2 −0.607364 1.27715i 1.73003 0.0835731i −1.26222 + 1.55139i −0.431733 0.431733i −1.15749 2.15875i −3.10278 2.74798 + 0.669785i 2.98603 0.289169i −0.289169 + 0.813607i
35.3 −0.416001 + 1.35164i 0.966579 + 1.43726i −1.65389 1.12457i −1.57184 1.57184i −2.34477 + 0.708570i 2.24914 2.20804 1.76765i −1.13145 + 2.77846i 2.77846 1.47068i
35.4 0.416001 1.35164i −1.43726 0.966579i −1.65389 1.12457i 1.57184 + 1.57184i −1.90437 + 1.54057i 2.24914 −2.20804 + 1.76765i 1.13145 + 2.77846i 2.77846 1.47068i
35.5 0.607364 + 1.27715i 0.0835731 1.73003i −1.26222 + 1.55139i 0.431733 + 0.431733i 2.26027 0.944024i −3.10278 −2.74798 0.669785i −2.98603 0.289169i −0.289169 + 0.813607i
35.6 1.39930 + 0.204810i −1.52878 + 0.814141i 1.91611 + 0.573183i −2.08397 2.08397i −2.30598 + 0.826122i −1.14637 2.56382 + 1.19449i 1.67435 2.48929i −2.48929 3.34292i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 35.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.2.k.a 12
3.b odd 2 1 inner 48.2.k.a 12
4.b odd 2 1 192.2.k.a 12
8.b even 2 1 384.2.k.b 12
8.d odd 2 1 384.2.k.a 12
12.b even 2 1 192.2.k.a 12
16.e even 4 1 192.2.k.a 12
16.e even 4 1 384.2.k.a 12
16.f odd 4 1 inner 48.2.k.a 12
16.f odd 4 1 384.2.k.b 12
24.f even 2 1 384.2.k.a 12
24.h odd 2 1 384.2.k.b 12
48.i odd 4 1 192.2.k.a 12
48.i odd 4 1 384.2.k.a 12
48.k even 4 1 inner 48.2.k.a 12
48.k even 4 1 384.2.k.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.k.a 12 1.a even 1 1 trivial
48.2.k.a 12 3.b odd 2 1 inner
48.2.k.a 12 16.f odd 4 1 inner
48.2.k.a 12 48.k even 4 1 inner
192.2.k.a 12 4.b odd 2 1
192.2.k.a 12 12.b even 2 1
192.2.k.a 12 16.e even 4 1
192.2.k.a 12 48.i odd 4 1
384.2.k.a 12 8.d odd 2 1
384.2.k.a 12 16.e even 4 1
384.2.k.a 12 24.f even 2 1
384.2.k.a 12 48.i odd 4 1
384.2.k.b 12 8.b even 2 1
384.2.k.b 12 16.f odd 4 1
384.2.k.b 12 24.h odd 2 1
384.2.k.b 12 48.k even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 + 32 T^{2} - 8 T^{4} - 16 T^{6} - 2 T^{8} + 2 T^{10} + T^{12}$$
$3$ $$729 + 486 T + 162 T^{2} - 54 T^{3} - 45 T^{4} - 60 T^{5} - 28 T^{6} - 20 T^{7} - 5 T^{8} - 2 T^{9} + 2 T^{10} + 2 T^{11} + T^{12}$$
$5$ $$256 + 1856 T^{4} + 100 T^{8} + T^{12}$$
$7$ $$( -8 - 6 T + 2 T^{2} + T^{3} )^{4}$$
$11$ $$65536 + 12288 T^{4} + 356 T^{8} + T^{12}$$
$13$ $$( 8 - 56 T + 196 T^{2} - 32 T^{3} + 2 T^{4} + 2 T^{5} + T^{6} )^{2}$$
$17$ $$( 512 + 288 T^{2} + 40 T^{4} + T^{6} )^{2}$$
$19$ $$( 128 + 128 T + 64 T^{2} - 32 T^{3} + 18 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$23$ $$( 2048 + 640 T^{2} + 52 T^{4} + T^{6} )^{2}$$
$29$ $$71639296 + 548160 T^{4} + 1316 T^{8} + T^{12}$$
$31$ $$( 16 + 68 T^{2} + 36 T^{4} + T^{6} )^{2}$$
$37$ $$( 2312 - 2040 T + 900 T^{2} - 128 T^{3} + 2 T^{4} + 2 T^{5} + T^{6} )^{2}$$
$41$ $$( -128 + 384 T^{2} - 108 T^{4} + T^{6} )^{2}$$
$43$ $$( 128 - 128 T + 64 T^{2} + 32 T^{3} + 18 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$47$ $$( -32768 + 3712 T^{2} - 112 T^{4} + T^{6} )^{2}$$
$53$ $$256 + 87360 T^{4} + 24356 T^{8} + T^{12}$$
$59$ $$4096 + 24418304 T^{4} + 27748 T^{8} + T^{12}$$
$61$ $$( 95048 - 27032 T + 3844 T^{2} - 64 T^{3} + 18 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$67$ $$( 32 + 128 T + 256 T^{2} + 232 T^{3} + 98 T^{4} - 14 T^{5} + T^{6} )^{2}$$
$71$ $$( 147968 + 10176 T^{2} + 196 T^{4} + T^{6} )^{2}$$
$73$ $$( 369664 + 18496 T^{2} + 272 T^{4} + T^{6} )^{2}$$
$79$ $$( 8464 + 1956 T^{2} + 116 T^{4} + T^{6} )^{2}$$
$83$ $$5473632256 + 32944128 T^{4} + 19044 T^{8} + T^{12}$$
$89$ $$( -2048 + 8576 T^{2} - 212 T^{4} + T^{6} )^{2}$$
$97$ $$( -608 - 128 T + 2 T^{2} + T^{3} )^{4}$$