# Properties

 Label 63.2.g.b Level $63$ Weight $2$ Character orbit 63.g Analytic conductor $0.503$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 63.g (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.503057532734$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: 10.0.991381711347.1 Defining polynomial: $$x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{9} + 9 \nu^{8} - 3 \nu^{7} + 95 \nu^{6} + 18 \nu^{5} + 402 \nu^{4} - 87 \nu^{3} + 936 \nu^{2} + 342 \nu + 72$$$$)/189$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{9} + \nu^{8} - 12 \nu^{7} - 8 \nu^{6} - 68 \nu^{5} - 30 \nu^{4} - 123 \nu^{3} - 204 \nu^{2} - 270 \nu - 63$$$$)/63$$ $$\beta_{4}$$ $$=$$ $$($$$$17 \nu^{9} - 24 \nu^{8} + 159 \nu^{7} - 106 \nu^{6} + 786 \nu^{5} - 417 \nu^{4} + 1893 \nu^{3} - 27 \nu^{2} + 1395 \nu + 639$$$$)/567$$ $$\beta_{5}$$ $$=$$ $$($$$$16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} - 351 \nu^{2} + 684 \nu - 180$$$$)/567$$ $$\beta_{6}$$ $$=$$ $$($$$$20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + 1620 \nu^{2} + 369 \nu + 504$$$$)/567$$ $$\beta_{7}$$ $$=$$ $$($$$$8 \nu^{9} - 12 \nu^{8} + 69 \nu^{7} - 43 \nu^{6} + 330 \nu^{5} - 219 \nu^{4} + 732 \nu^{3} - 45 \nu^{2} + 477 \nu - 306$$$$)/189$$ $$\beta_{8}$$ $$=$$ $$($$$$-71 \nu^{9} + 123 \nu^{8} - 591 \nu^{7} + 403 \nu^{6} - 2604 \nu^{5} + 1794 \nu^{4} - 5214 \nu^{3} - 1458 \nu^{2} - 1476 \nu - 234$$$$)/567$$ $$\beta_{9}$$ $$=$$ $$($$$$-82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} - 432 \nu^{2} - 2898 \nu + 720$$$$)/567$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} + 3 \beta_{6} + \beta_{4} - \beta_{2} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{9} + 4 \beta_{5} - \beta_{3} - 4 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{8} - 5 \beta_{7} - 13 \beta_{6} - \beta_{4} - \beta_{3} + 4 \beta_{2} - \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-7 \beta_{9} + \beta_{8} - 2 \beta_{7} - 7 \beta_{6} - 19 \beta_{5} - \beta_{4} + \beta_{2} + 7$$ $$\nu^{6}$$ $$=$$ $$-10 \beta_{9} + 9 \beta_{8} + 15 \beta_{7} - 10 \beta_{5} - 15 \beta_{4} + 10 \beta_{3} + 9 \beta_{2} + 10 \beta_{1} + 61$$ $$\nu^{7}$$ $$=$$ $$11 \beta_{8} + 11 \beta_{7} + 46 \beta_{6} + 19 \beta_{4} + 43 \beta_{3} + 8 \beta_{2} + 94 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$73 \beta_{9} + 56 \beta_{8} + 62 \beta_{7} + 298 \beta_{6} + 76 \beta_{5} + 118 \beta_{4} - 118 \beta_{2} - 298$$ $$\nu^{9}$$ $$=$$ $$253 \beta_{9} - 135 \beta_{8} + 48 \beta_{7} + 478 \beta_{5} - 48 \beta_{4} - 253 \beta_{3} - 135 \beta_{2} - 478 \beta_{1} - 295$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$$.

 $$n$$ $$10$$ $$29$$ $$\chi(n)$$ $$-1 + \beta_{6}$$ $$-\beta_{6}$$

## 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
 −1.02682 + 1.77851i −0.335166 + 0.580525i 0.247934 − 0.429435i 0.920620 − 1.59456i 1.19343 − 2.06709i −1.02682 − 1.77851i −0.335166 − 0.580525i 0.247934 + 0.429435i 0.920620 + 1.59456i 1.19343 + 2.06709i
−1.02682 + 1.77851i 1.09995 + 1.33795i −1.10873 1.92038i −0.146246 −3.50901 + 0.582422i 0.0802402 2.64453i 0.446582 −0.580240 + 2.94335i 0.150168 0.260099i
4.2 −0.335166 + 0.580525i −0.377302 1.69046i 0.775327 + 1.34291i 1.42494 1.10781 + 0.347551i 2.21529 1.44655i −2.38012 −2.71529 + 1.27563i −0.477591 + 0.827212i
4.3 0.247934 0.429435i 1.59836 0.667278i 0.877057 + 1.51911i −3.69258 0.109735 0.851830i −2.60948 0.436591i 1.86155 2.10948 2.13309i −0.915516 + 1.58572i
4.4 0.920620 1.59456i −1.58800 0.691567i −0.695084 1.20392i 1.33475 −2.56469 + 1.89549i −2.54347 + 0.728536i 1.12285 2.04347 + 2.19641i 1.22880 2.12835i
4.5 1.19343 2.06709i 0.266999 + 1.71135i −1.84857 3.20182i −2.92087 3.85615 + 1.49047i 2.35742 + 1.20106i −4.05086 −2.85742 + 0.913855i −3.48586 + 6.03769i
16.1 −1.02682 1.77851i 1.09995 1.33795i −1.10873 + 1.92038i −0.146246 −3.50901 0.582422i 0.0802402 + 2.64453i 0.446582 −0.580240 2.94335i 0.150168 + 0.260099i
16.2 −0.335166 0.580525i −0.377302 + 1.69046i 0.775327 1.34291i 1.42494 1.10781 0.347551i 2.21529 + 1.44655i −2.38012 −2.71529 1.27563i −0.477591 0.827212i
16.3 0.247934 + 0.429435i 1.59836 + 0.667278i 0.877057 1.51911i −3.69258 0.109735 + 0.851830i −2.60948 + 0.436591i 1.86155 2.10948 + 2.13309i −0.915516 1.58572i
16.4 0.920620 + 1.59456i −1.58800 + 0.691567i −0.695084 + 1.20392i 1.33475 −2.56469 1.89549i −2.54347 0.728536i 1.12285 2.04347 2.19641i 1.22880 + 2.12835i
16.5 1.19343 + 2.06709i 0.266999 1.71135i −1.84857 + 3.20182i −2.92087 3.85615 1.49047i 2.35742 1.20106i −4.05086 −2.85742 0.913855i −3.48586 6.03769i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 16.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.2.g.b 10
3.b odd 2 1 189.2.g.b 10
4.b odd 2 1 1008.2.t.i 10
7.b odd 2 1 441.2.g.f 10
7.c even 3 1 63.2.h.b yes 10
7.c even 3 1 441.2.f.e 10
7.d odd 6 1 441.2.f.f 10
7.d odd 6 1 441.2.h.f 10
9.c even 3 1 63.2.h.b yes 10
9.c even 3 1 567.2.e.f 10
9.d odd 6 1 189.2.h.b 10
9.d odd 6 1 567.2.e.e 10
12.b even 2 1 3024.2.t.i 10
21.c even 2 1 1323.2.g.f 10
21.g even 6 1 1323.2.f.f 10
21.g even 6 1 1323.2.h.f 10
21.h odd 6 1 189.2.h.b 10
21.h odd 6 1 1323.2.f.e 10
28.g odd 6 1 1008.2.q.i 10
36.f odd 6 1 1008.2.q.i 10
36.h even 6 1 3024.2.q.i 10
63.g even 3 1 inner 63.2.g.b 10
63.g even 3 1 3969.2.a.z 5
63.h even 3 1 441.2.f.e 10
63.h even 3 1 567.2.e.f 10
63.i even 6 1 1323.2.f.f 10
63.j odd 6 1 567.2.e.e 10
63.j odd 6 1 1323.2.f.e 10
63.k odd 6 1 441.2.g.f 10
63.k odd 6 1 3969.2.a.ba 5
63.l odd 6 1 441.2.h.f 10
63.n odd 6 1 189.2.g.b 10
63.n odd 6 1 3969.2.a.bc 5
63.o even 6 1 1323.2.h.f 10
63.s even 6 1 1323.2.g.f 10
63.s even 6 1 3969.2.a.bb 5
63.t odd 6 1 441.2.f.f 10
84.n even 6 1 3024.2.q.i 10
252.o even 6 1 3024.2.t.i 10
252.bl odd 6 1 1008.2.t.i 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b 10 1.a even 1 1 trivial
63.2.g.b 10 63.g even 3 1 inner
63.2.h.b yes 10 7.c even 3 1
63.2.h.b yes 10 9.c even 3 1
189.2.g.b 10 3.b odd 2 1
189.2.g.b 10 63.n odd 6 1
189.2.h.b 10 9.d odd 6 1
189.2.h.b 10 21.h odd 6 1
441.2.f.e 10 7.c even 3 1
441.2.f.e 10 63.h even 3 1
441.2.f.f 10 7.d odd 6 1
441.2.f.f 10 63.t odd 6 1
441.2.g.f 10 7.b odd 2 1
441.2.g.f 10 63.k odd 6 1
441.2.h.f 10 7.d odd 6 1
441.2.h.f 10 63.l odd 6 1
567.2.e.e 10 9.d odd 6 1
567.2.e.e 10 63.j odd 6 1
567.2.e.f 10 9.c even 3 1
567.2.e.f 10 63.h even 3 1
1008.2.q.i 10 28.g odd 6 1
1008.2.q.i 10 36.f odd 6 1
1008.2.t.i 10 4.b odd 2 1
1008.2.t.i 10 252.bl odd 6 1
1323.2.f.e 10 21.h odd 6 1
1323.2.f.e 10 63.j odd 6 1
1323.2.f.f 10 21.g even 6 1
1323.2.f.f 10 63.i even 6 1
1323.2.g.f 10 21.c even 2 1
1323.2.g.f 10 63.s even 6 1
1323.2.h.f 10 21.g even 6 1
1323.2.h.f 10 63.o even 6 1
3024.2.q.i 10 36.h even 6 1
3024.2.q.i 10 84.n even 6 1
3024.2.t.i 10 12.b even 2 1
3024.2.t.i 10 252.o even 6 1
3969.2.a.z 5 63.g even 3 1
3969.2.a.ba 5 63.k odd 6 1
3969.2.a.bb 5 63.s even 6 1
3969.2.a.bc 5 63.n odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T - T^{2} + 4 T^{3} - 2 T^{4} + 2 T^{5} + 6 T^{6} - 21 T^{7} + 6 T^{8} + 13 T^{9} - 5 T^{10} + 26 T^{11} + 24 T^{12} - 168 T^{13} + 96 T^{14} + 64 T^{15} - 128 T^{16} + 512 T^{17} - 256 T^{18} - 1024 T^{19} + 1024 T^{20}$$
$3$ $$1 - 2 T + 4 T^{2} - 3 T^{3} + 9 T^{5} - 27 T^{7} + 108 T^{8} - 162 T^{9} + 243 T^{10}$$
$5$ $$( 1 + 4 T + 20 T^{2} + 62 T^{3} + 193 T^{4} + 423 T^{5} + 965 T^{6} + 1550 T^{7} + 2500 T^{8} + 2500 T^{9} + 3125 T^{10} )^{2}$$
$7$ $$1 + T - 12 T^{2} + 8 T^{3} + 62 T^{4} - 75 T^{5} + 434 T^{6} + 392 T^{7} - 4116 T^{8} + 2401 T^{9} + 16807 T^{10}$$
$11$ $$( 1 + 4 T + 47 T^{2} + 161 T^{3} + 958 T^{4} + 2589 T^{5} + 10538 T^{6} + 19481 T^{7} + 62557 T^{8} + 58564 T^{9} + 161051 T^{10} )^{2}$$
$13$ $$1 + 8 T - 14 T^{2} - 182 T^{3} + 686 T^{4} + 4429 T^{5} - 12871 T^{6} - 43199 T^{7} + 305249 T^{8} + 358672 T^{9} - 3841969 T^{10} + 4662736 T^{11} + 51587081 T^{12} - 94908203 T^{13} - 367608631 T^{14} + 1644456697 T^{15} + 3311190974 T^{16} - 11420230094 T^{17} - 11420230094 T^{18} + 84835994984 T^{19} + 137858491849 T^{20}$$
$17$ $$1 - 12 T + 14 T^{2} + 192 T^{3} + 1185 T^{4} - 11847 T^{5} - 6180 T^{6} + 65736 T^{7} + 1002861 T^{8} - 2436261 T^{9} - 7749777 T^{10} - 41416437 T^{11} + 289826829 T^{12} + 322960968 T^{13} - 516159780 T^{14} - 16821045879 T^{15} + 28603019265 T^{16} + 78785025216 T^{17} + 97660604174 T^{18} - 1423054517964 T^{19} + 2015993900449 T^{20}$$
$19$ $$1 - T - 53 T^{2} + 190 T^{3} + 1262 T^{4} - 7007 T^{5} - 13111 T^{6} + 116110 T^{7} + 67964 T^{8} - 721616 T^{9} - 440023 T^{10} - 13710704 T^{11} + 24535004 T^{12} + 796398490 T^{13} - 1708638631 T^{14} - 17350025693 T^{15} + 59371901822 T^{16} + 169835630410 T^{17} - 900128841173 T^{18} - 322687697779 T^{19} + 6131066257801 T^{20}$$
$23$ $$( 1 + 3 T + 52 T^{2} + 225 T^{3} + 2023 T^{4} + 5565 T^{5} + 46529 T^{6} + 119025 T^{7} + 632684 T^{8} + 839523 T^{9} + 6436343 T^{10} )^{2}$$
$29$ $$1 - 7 T - 76 T^{2} + 419 T^{3} + 4561 T^{4} - 15146 T^{5} - 199563 T^{6} + 341373 T^{7} + 6918636 T^{8} - 2570041 T^{9} - 219913241 T^{10} - 74531189 T^{11} + 5818572876 T^{12} + 8325746097 T^{13} - 141147118203 T^{14} - 310661862754 T^{15} + 2712989167081 T^{16} + 7227698173471 T^{17} - 38018727385036 T^{18} - 101550021831083 T^{19} + 420707233300201 T^{20}$$
$31$ $$1 + 3 T - 125 T^{2} - 214 T^{3} + 9282 T^{4} + 8387 T^{5} - 503981 T^{6} - 245082 T^{7} + 21459514 T^{8} + 3619498 T^{9} - 734820027 T^{10} + 112204438 T^{11} + 20622592954 T^{12} - 7301237862 T^{13} - 465437037101 T^{14} + 240112689437 T^{15} + 8237809167042 T^{16} - 5887699419754 T^{17} - 106611379680125 T^{18} + 79318866482013 T^{19} + 819628286980801 T^{20}$$
$37$ $$1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 1225850552 T^{11} - 13418701163 T^{12} + 116326438008 T^{13} + 50845987930 T^{14} - 3144887137864 T^{15} + 11553466019727 T^{16} + 53161851194480 T^{17} - 312610671398969 T^{18} + 4808584372417849 T^{20}$$
$41$ $$1 - 5 T - 136 T^{2} + 733 T^{3} + 10507 T^{4} - 54412 T^{5} - 554055 T^{6} + 2345451 T^{7} + 23706084 T^{8} - 41392439 T^{9} - 952045937 T^{10} - 1697089999 T^{11} + 39849927204 T^{12} + 161650828371 T^{13} - 1565627010855 T^{14} - 6303967608812 T^{15} + 49909345260187 T^{16} + 142754882754773 T^{17} - 1085949831160456 T^{18} - 1636909671969805 T^{19} + 13422659310152401 T^{20}$$
$43$ $$1 + 7 T - 77 T^{2} - 66 T^{3} + 7014 T^{4} - 3843 T^{5} - 95427 T^{6} + 1632678 T^{7} - 3708600 T^{8} - 15416324 T^{9} + 670279801 T^{10} - 662901932 T^{11} - 6857201400 T^{12} + 129809329746 T^{13} - 326245923027 T^{14} - 564953446449 T^{15} + 44338040425686 T^{16} - 17940028333062 T^{17} - 899991421375277 T^{18} + 3518148283557901 T^{19} + 21611482313284249 T^{20}$$
$47$ $$1 - 27 T + 281 T^{2} - 1758 T^{3} + 13050 T^{4} - 78783 T^{5} - 25248 T^{6} + 1518381 T^{7} + 9454350 T^{8} - 53043051 T^{9} - 242331903 T^{10} - 2493023397 T^{11} + 20884659150 T^{12} + 157642870563 T^{13} - 123202185888 T^{14} - 18068487686481 T^{15} + 140668760043450 T^{16} - 890643445773954 T^{17} + 6690971551954841 T^{18} - 30216522773774709 T^{19} + 52599132235830049 T^{20}$$
$53$ $$1 + 21 T + 41 T^{2} - 924 T^{3} + 12966 T^{4} + 177027 T^{5} - 601755 T^{6} - 3783942 T^{7} + 110973258 T^{8} + 340111866 T^{9} - 4044436041 T^{10} + 18025928898 T^{11} + 311723881722 T^{12} - 563341933134 T^{13} - 4748136394155 T^{14} + 74031893539311 T^{15} + 287383106398614 T^{16} - 1085433093209388 T^{17} + 2552647306865801 T^{18} + 69295035427844793 T^{19} + 174887470365513049 T^{20}$$
$59$ $$1 - 30 T + 299 T^{2} - 1644 T^{3} + 26547 T^{4} - 344442 T^{5} + 1635267 T^{6} - 9620487 T^{7} + 170035344 T^{8} - 1056366303 T^{9} + 3109579647 T^{10} - 62325611877 T^{11} + 591893032464 T^{12} - 1975845999573 T^{13} + 19815120570387 T^{14} - 246249955396158 T^{15} + 1119766626567627 T^{16} - 4091343041042436 T^{17} + 43902300843691979 T^{18} - 259889874559648170 T^{19} + 511116753300641401 T^{20}$$
$61$ $$1 + 14 T - 143 T^{2} - 2072 T^{3} + 23777 T^{4} + 251656 T^{5} - 2164351 T^{6} - 13562879 T^{7} + 202896254 T^{8} + 466067647 T^{9} - 12461386219 T^{10} + 28430126467 T^{11} + 754976961134 T^{12} - 3078515838299 T^{13} - 29967259814191 T^{14} + 212547726724456 T^{15} + 1224999941181497 T^{16} - 6511763156235512 T^{17} - 27414145758611183 T^{18} + 163718045299677974 T^{19} + 713342911662882601 T^{20}$$
$67$ $$1 + 2 T - 128 T^{2} - 128 T^{3} + 6161 T^{4} - 2183 T^{5} + 29300 T^{6} + 394018 T^{7} - 17169907 T^{8} - 2850929 T^{9} + 1197895103 T^{10} - 191012243 T^{11} - 77075712523 T^{12} + 118506035734 T^{13} + 590427845300 T^{14} - 2947323108581 T^{15} + 557314092543209 T^{16} - 775771085481344 T^{17} - 51976662727250048 T^{18} + 54413068792589894 T^{19} + 1822837804551761449 T^{20}$$
$71$ $$( 1 + 3 T + 187 T^{2} + 285 T^{3} + 15679 T^{4} + 10143 T^{5} + 1113209 T^{6} + 1436685 T^{7} + 66929357 T^{8} + 76235043 T^{9} + 1804229351 T^{10} )^{2}$$
$73$ $$1 - 15 T - 134 T^{2} + 2501 T^{3} + 16563 T^{4} - 235276 T^{5} - 2002535 T^{6} + 9021201 T^{7} + 288508378 T^{8} - 238799411 T^{9} - 25271949561 T^{10} - 17432357003 T^{11} + 1537461146362 T^{12} + 3509400549417 T^{13} - 56868471540935 T^{14} - 487743992114668 T^{15} + 2506548790024707 T^{16} + 27629543696261597 T^{17} - 108065652313806854 T^{18} - 883073800624018695 T^{19} + 4297625829703557649 T^{20}$$
$79$ $$1 + 4 T - 284 T^{2} - 1776 T^{3} + 44175 T^{4} + 312399 T^{5} - 4187754 T^{6} - 29772300 T^{7} + 295992489 T^{8} + 1067553919 T^{9} - 20151634301 T^{10} + 84336759601 T^{11} + 1847289123849 T^{12} - 14678905019700 T^{13} - 163113357508074 T^{14} + 961269341991201 T^{15} + 10738388347640175 T^{16} - 34106142359418384 T^{17} - 430858902013463324 T^{18} + 479406383930473276 T^{19} + 9468276082626847201 T^{20}$$
$83$ $$1 - 9 T - 148 T^{2} - 297 T^{3} + 24654 T^{4} + 118125 T^{5} - 807174 T^{6} - 21382137 T^{7} - 37648479 T^{8} + 452536146 T^{9} + 15509586612 T^{10} + 37560500118 T^{11} - 259360371831 T^{12} - 12226027968819 T^{13} - 38307122794854 T^{14} + 465299175954375 T^{15} + 8060387965039326 T^{16} - 8059407143919219 T^{17} - 333339250356578068 T^{18} - 1682462297407863627 T^{19} + 15516041187205853449 T^{20}$$
$89$ $$1 - 28 T + 104 T^{2} + 1736 T^{3} + 31273 T^{4} - 611939 T^{5} - 1780638 T^{6} + 18973932 T^{7} + 740914101 T^{8} - 3271180573 T^{9} - 40614588329 T^{10} - 291135070997 T^{11} + 5868780594021 T^{12} + 13376033868108 T^{13} - 111721218529758 T^{14} - 3417103755161611 T^{15} + 15542095912223353 T^{16} + 76785597378638344 T^{17} + 409405235793016424 T^{18} - 9809979303809585852 T^{19} + 31181719929966183601 T^{20}$$
$97$ $$1 + 12 T - 197 T^{2} - 1534 T^{3} + 27813 T^{4} + 14090 T^{5} - 4545035 T^{6} - 6881349 T^{7} + 472663750 T^{8} + 908843245 T^{9} - 38512186359 T^{10} + 88157794765 T^{11} + 4447293223750 T^{12} - 6280421435877 T^{13} - 402368680669835 T^{14} + 120995624221130 T^{15} + 23167450373090277 T^{16} - 123944568389425342 T^{17} - 1543974418092261317 T^{18} + 9122772703854782604 T^{19} + 73742412689492826049 T^{20}$$