# Properties

 Label 503.2.a.e Level 503 Weight 2 Character orbit 503.a Self dual yes Analytic conductor 4.016 Analytic rank 1 Dimension 10 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$503$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 503.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.01647522167$$ Analytic rank: $$1$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 2 x^{9} - 9 x^{8} + 14 x^{7} + 27 x^{6} - 27 x^{5} - 34 x^{4} + 14 x^{3} + 17 x^{2} + x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} - 9 x^{8} + 14 x^{7} + 27 x^{6} - 27 x^{5} - 34 x^{4} + 14 x^{3} + 17 x^{2} + x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$3 \nu^{9} - 8 \nu^{8} - 22 \nu^{7} + 57 \nu^{6} + 46 \nu^{5} - 113 \nu^{4} - 34 \nu^{3} + 65 \nu^{2} + 12 \nu - 5$$ $$\beta_{3}$$ $$=$$ $$5 \nu^{9} - 13 \nu^{8} - 37 \nu^{7} + 92 \nu^{6} + 78 \nu^{5} - 181 \nu^{4} - 57 \nu^{3} + 104 \nu^{2} + 20 \nu - 8$$ $$\beta_{4}$$ $$=$$ $$-6 \nu^{9} + 15 \nu^{8} + 47 \nu^{7} - 108 \nu^{6} - 113 \nu^{5} + 221 \nu^{4} + 108 \nu^{3} - 139 \nu^{2} - 44 \nu + 13$$ $$\beta_{5}$$ $$=$$ $$10 \nu^{9} - 25 \nu^{8} - 78 \nu^{7} + 180 \nu^{6} + 184 \nu^{5} - 368 \nu^{4} - 165 \nu^{3} + 230 \nu^{2} + 60 \nu - 21$$ $$\beta_{6}$$ $$=$$ $$-11 \nu^{9} + 28 \nu^{8} + 84 \nu^{7} - 200 \nu^{6} - 191 \nu^{5} + 402 \nu^{4} + 165 \nu^{3} - 244 \nu^{2} - 63 \nu + 22$$ $$\beta_{7}$$ $$=$$ $$-18 \nu^{9} + 45 \nu^{8} + 139 \nu^{7} - 321 \nu^{6} - 321 \nu^{5} + 645 \nu^{4} + 278 \nu^{3} - 394 \nu^{2} - 102 \nu + 38$$ $$\beta_{8}$$ $$=$$ $$-21 \nu^{9} + 53 \nu^{8} + 162 \nu^{7} - 380 \nu^{6} - 375 \nu^{5} + 770 \nu^{4} + 331 \nu^{3} - 475 \nu^{2} - 127 \nu + 45$$ $$\beta_{9}$$ $$=$$ $$-22 \nu^{9} + 55 \nu^{8} + 170 \nu^{7} - 392 \nu^{6} - 394 \nu^{5} + 785 \nu^{4} + 346 \nu^{3} - 473 \nu^{2} - 130 \nu + 42$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{6} + \beta_{4} - \beta_{3} + \beta_{1} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{8} - 2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 5 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$-\beta_{9} + \beta_{8} + 2 \beta_{7} - 9 \beta_{6} + 2 \beta_{5} + 8 \beta_{4} - 6 \beta_{3} - 2 \beta_{2} + 10 \beta_{1} - 1$$ $$\nu^{5}$$ $$=$$ $$-\beta_{9} + 8 \beta_{8} + 3 \beta_{7} - 22 \beta_{6} + 9 \beta_{5} + 12 \beta_{4} - 9 \beta_{3} - 5 \beta_{2} + 34 \beta_{1} - 12$$ $$\nu^{6}$$ $$=$$ $$-8 \beta_{9} + 12 \beta_{8} + 19 \beta_{7} - 73 \beta_{6} + 22 \beta_{5} + 58 \beta_{4} - 37 \beta_{3} - 24 \beta_{2} + 83 \beta_{1} - 28$$ $$\nu^{7}$$ $$=$$ $$-12 \beta_{9} + 58 \beta_{8} + 37 \beta_{7} - 192 \beta_{6} + 73 \beta_{5} + 113 \beta_{4} - 71 \beta_{3} - 63 \beta_{2} + 252 \beta_{1} - 107$$ $$\nu^{8}$$ $$=$$ $$-58 \beta_{9} + 113 \beta_{8} + 152 \beta_{7} - 578 \beta_{6} + 192 \beta_{5} + 425 \beta_{4} - 247 \beta_{3} - 220 \beta_{2} + 661 \beta_{1} - 278$$ $$\nu^{9}$$ $$=$$ $$-113 \beta_{9} + 425 \beta_{8} + 345 \beta_{7} - 1565 \beta_{6} + 578 \beta_{5} + 967 \beta_{4} - 554 \beta_{3} - 591 \beta_{2} + 1920 \beta_{1} - 879$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.858231 2.78533 −0.489003 1.07636 −2.07227 1.95007 1.31567 0.208270 −1.40552 −0.510671
−2.58686 −1.85823 4.69185 1.44291 4.80698 −1.96509 −6.96343 0.453023 −3.73261
1.2 −2.03947 1.78533 2.15945 −0.701114 −3.64112 −2.02991 −0.325186 0.187388 1.42990
1.3 −1.62786 −1.48900 0.649933 −1.79865 2.42389 0.552233 2.19772 −0.782869 2.92795
1.4 −1.37178 0.0763625 −0.118218 1.17276 −0.104753 0.469303 2.90573 −2.99417 −1.60876
1.5 −0.756417 −3.07227 −1.42783 0.386144 2.32392 0.194914 2.59287 6.43884 −0.292086
1.6 −0.392284 0.950069 −1.84611 −2.28693 −0.372697 2.71022 1.50877 −2.09737 0.897127
1.7 0.0830530 0.315672 −1.99310 2.25024 0.0262175 −3.20647 −0.331639 −2.90035 0.186890
1.8 1.17266 −0.791730 −0.624870 0.178789 −0.928430 −0.0809018 −3.07808 −2.37316 0.209658
1.9 1.36113 −2.40552 −0.147314 0.590303 −3.27424 1.95900 −2.92278 2.78655 0.803481
1.10 2.15783 −1.51067 2.65622 −2.23445 −3.25977 −3.60329 1.41602 −0.717874 −4.82156
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$503$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 503.2.a.e 10
3.b odd 2 1 4527.2.a.k 10
4.b odd 2 1 8048.2.a.p 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
503.2.a.e 10 1.a even 1 1 trivial
4527.2.a.k 10 3.b odd 2 1
8048.2.a.p 10 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(503))$$:

 $$T_{2}^{10} + \cdots$$ $$T_{3}^{10} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 16 T^{2} + 41 T^{3} + 103 T^{4} + 208 T^{5} + 412 T^{6} + 707 T^{7} + 1190 T^{8} + 1794 T^{9} + 2665 T^{10} + 3588 T^{11} + 4760 T^{12} + 5656 T^{13} + 6592 T^{14} + 6656 T^{15} + 6592 T^{16} + 5248 T^{17} + 4096 T^{18} + 2048 T^{19} + 1024 T^{20}$$
$3$ $$1 + 8 T + 48 T^{2} + 206 T^{3} + 752 T^{4} + 2307 T^{5} + 6300 T^{6} + 15215 T^{7} + 33404 T^{8} + 66208 T^{9} + 120343 T^{10} + 198624 T^{11} + 300636 T^{12} + 410805 T^{13} + 510300 T^{14} + 560601 T^{15} + 548208 T^{16} + 450522 T^{17} + 314928 T^{18} + 157464 T^{19} + 59049 T^{20}$$
$5$ $$1 + T + 39 T^{2} + 38 T^{3} + 726 T^{4} + 662 T^{5} + 8471 T^{6} + 7016 T^{7} + 68463 T^{8} + 50106 T^{9} + 400081 T^{10} + 250530 T^{11} + 1711575 T^{12} + 877000 T^{13} + 5294375 T^{14} + 2068750 T^{15} + 11343750 T^{16} + 2968750 T^{17} + 15234375 T^{18} + 1953125 T^{19} + 9765625 T^{20}$$
$7$ $$1 + 5 T + 61 T^{2} + 251 T^{3} + 1709 T^{4} + 5912 T^{5} + 29171 T^{6} + 85970 T^{7} + 337948 T^{8} + 851532 T^{9} + 2785579 T^{10} + 5960724 T^{11} + 16559452 T^{12} + 29487710 T^{13} + 70039571 T^{14} + 99362984 T^{15} + 201062141 T^{16} + 206709293 T^{17} + 351652861 T^{18} + 201768035 T^{19} + 282475249 T^{20}$$
$11$ $$1 + 3 T + 69 T^{2} + 217 T^{3} + 2352 T^{4} + 7445 T^{5} + 52841 T^{6} + 159886 T^{7} + 869048 T^{8} + 2400415 T^{9} + 10894513 T^{10} + 26404565 T^{11} + 105154808 T^{12} + 212808266 T^{13} + 773645081 T^{14} + 1199024695 T^{15} + 4166711472 T^{16} + 4228716107 T^{17} + 14790762789 T^{18} + 7073843073 T^{19} + 25937424601 T^{20}$$
$13$ $$1 + 18 T + 250 T^{2} + 2413 T^{3} + 19873 T^{4} + 134672 T^{5} + 810062 T^{6} + 4230421 T^{7} + 19976351 T^{8} + 83675331 T^{9} + 319368199 T^{10} + 1087779303 T^{11} + 3376003319 T^{12} + 9294234937 T^{13} + 23136180782 T^{14} + 50002770896 T^{15} + 95923175257 T^{16} + 151412171521 T^{17} + 203932680250 T^{18} + 190880988714 T^{19} + 137858491849 T^{20}$$
$17$ $$1 + 11 T + 172 T^{2} + 1316 T^{3} + 12110 T^{4} + 72481 T^{5} + 497736 T^{6} + 2460829 T^{7} + 13750697 T^{8} + 57744215 T^{9} + 272963791 T^{10} + 981651655 T^{11} + 3973951433 T^{12} + 12090052877 T^{13} + 41571408456 T^{14} + 102912655217 T^{15} + 292305960590 T^{16} + 540005693668 T^{17} + 1199830279852 T^{18} + 1304466641467 T^{19} + 2015993900449 T^{20}$$
$19$ $$1 + 117 T^{2} + 144 T^{3} + 6545 T^{4} + 14584 T^{5} + 240551 T^{6} + 681477 T^{7} + 6591561 T^{8} + 19407794 T^{9} + 141051753 T^{10} + 368748086 T^{11} + 2379553521 T^{12} + 4674250743 T^{13} + 31348846871 T^{14} + 36111427816 T^{15} + 307915291145 T^{16} + 128717530416 T^{17} + 1987076875797 T^{18} + 6131066257801 T^{20}$$
$23$ $$1 + 2 T + 128 T^{2} + 139 T^{3} + 7117 T^{4} - 755 T^{5} + 224173 T^{6} - 407616 T^{7} + 4690668 T^{8} - 19318174 T^{9} + 93011881 T^{10} - 444318002 T^{11} + 2481363372 T^{12} - 4959463872 T^{13} + 62732796493 T^{14} - 4859438965 T^{15} + 1053571422013 T^{16} + 473270737133 T^{17} + 10023806115968 T^{18} + 3602305322926 T^{19} + 41426511213649 T^{20}$$
$29$ $$1 + 9 T + 225 T^{2} + 1647 T^{3} + 23252 T^{4} + 143611 T^{5} + 1501329 T^{6} + 8016660 T^{7} + 68262730 T^{8} + 317447419 T^{9} + 2290933469 T^{10} + 9205975151 T^{11} + 57408955930 T^{12} + 195518320740 T^{13} + 1061861476449 T^{14} + 2945626619039 T^{15} + 13830831859892 T^{16} + 28410546280923 T^{17} + 112555442916225 T^{18} + 130564313782821 T^{19} + 420707233300201 T^{20}$$
$31$ $$1 + 22 T + 437 T^{2} + 5905 T^{3} + 71196 T^{4} + 707620 T^{5} + 6360319 T^{6} + 49929045 T^{7} + 358058017 T^{8} + 2291835411 T^{9} + 13467173163 T^{10} + 71046897741 T^{11} + 344093754337 T^{12} + 1487436179595 T^{13} + 5873888163199 T^{14} + 20258559830620 T^{15} + 63186712072476 T^{16} + 162461986325455 T^{17} + 372713383361717 T^{18} + 581671687534762 T^{19} + 819628286980801 T^{20}$$
$37$ $$1 + 35 T + 807 T^{2} + 13514 T^{3} + 185245 T^{4} + 2135413 T^{5} + 21450226 T^{6} + 189788329 T^{7} + 1500972503 T^{8} + 10649984634 T^{9} + 68217693835 T^{10} + 394049431458 T^{11} + 2054831356607 T^{12} + 9613348228837 T^{13} + 40201177010386 T^{14} + 148077987249241 T^{15} + 475287988635205 T^{16} + 1282909387575362 T^{17} + 2834570919314247 T^{18} + 4548660892827695 T^{19} + 4808584372417849 T^{20}$$
$41$ $$1 + 4 T + 260 T^{2} + 1417 T^{3} + 32755 T^{4} + 218127 T^{5} + 2678340 T^{6} + 19788650 T^{7} + 159907986 T^{8} + 1182434807 T^{9} + 7385189807 T^{10} + 48479827087 T^{11} + 268805324466 T^{12} + 1363853546650 T^{13} + 7568348716740 T^{14} + 25271365555527 T^{15} + 155589664413955 T^{16} + 275966806089377 T^{17} + 2076080559571460 T^{18} + 1309527737575844 T^{19} + 13422659310152401 T^{20}$$
$43$ $$1 + 20 T + 435 T^{2} + 5094 T^{3} + 60693 T^{4} + 456684 T^{5} + 3529418 T^{6} + 14396035 T^{7} + 65477767 T^{8} - 145261781 T^{9} - 450150713 T^{10} - 6246256583 T^{11} + 121068391183 T^{12} + 1144585554745 T^{13} + 12066377787818 T^{14} + 67136403783012 T^{15} + 383662487532957 T^{16} + 1384644004979058 T^{17} + 5084367120756435 T^{18} + 10051852238736860 T^{19} + 21611482313284249 T^{20}$$
$47$ $$1 - 7 T + 322 T^{2} - 2204 T^{3} + 50553 T^{4} - 325252 T^{5} + 5105346 T^{6} - 30078097 T^{7} + 367899299 T^{8} - 1942791244 T^{9} + 19840453349 T^{10} - 91311188468 T^{11} + 812689551491 T^{12} - 3122798264831 T^{13} + 24912459874626 T^{14} - 74594922216764 T^{15} + 544921672526937 T^{16} - 1116597357500452 T^{17} + 7667234305087042 T^{18} - 7833913311719369 T^{19} + 52599132235830049 T^{20}$$
$53$ $$1 + 24 T + 570 T^{2} + 8461 T^{3} + 123312 T^{4} + 1397686 T^{5} + 15606645 T^{6} + 145668337 T^{7} + 1341977182 T^{8} + 10633420505 T^{9} + 83159919093 T^{10} + 563571286765 T^{11} + 3769613904238 T^{12} + 21686665007549 T^{13} + 123143935846245 T^{14} + 584505985829198 T^{15} + 2733131699539248 T^{16} + 9939230954160857 T^{17} + 35488023534475770 T^{18} + 79194326203251192 T^{19} + 174887470365513049 T^{20}$$
$59$ $$1 - 17 T + 467 T^{2} - 5784 T^{3} + 84135 T^{4} - 790560 T^{5} + 7647144 T^{6} - 56238779 T^{7} + 400513752 T^{8} - 2655972904 T^{9} + 18692560589 T^{10} - 156702401336 T^{11} + 1394188370712 T^{12} - 11550264192241 T^{13} + 92663204466984 T^{14} - 565190553817440 T^{15} + 3548859197885535 T^{16} - 14394360188193096 T^{17} + 68569814361217907 T^{18} - 147270928917133963 T^{19} + 511116753300641401 T^{20}$$
$61$ $$1 + 4 T + 252 T^{2} + 859 T^{3} + 35750 T^{4} + 141968 T^{5} + 3623379 T^{6} + 15927450 T^{7} + 286566685 T^{8} + 1331963236 T^{9} + 19096375159 T^{10} + 81249757396 T^{11} + 1066314634885 T^{12} + 3615228528450 T^{13} + 50168729516739 T^{14} + 119905647660368 T^{15} + 1841853383405750 T^{16} + 2699616096142039 T^{17} + 48310242875314812 T^{18} + 46776584371336564 T^{19} + 713342911662882601 T^{20}$$
$67$ $$1 + 6 T + 318 T^{2} + 2402 T^{3} + 61276 T^{4} + 443521 T^{5} + 8166786 T^{6} + 55393313 T^{7} + 807832572 T^{8} + 4959723492 T^{9} + 61619242847 T^{10} + 332301473964 T^{11} + 3626360415708 T^{12} + 16660258997819 T^{13} + 164569892867106 T^{14} + 598808837581747 T^{15} + 5542927825787644 T^{16} + 14557829275985846 T^{17} + 129129521463011838 T^{18} + 163239206377769682 T^{19} + 1822837804551761449 T^{20}$$
$71$ $$1 + T + 262 T^{2} + 1233 T^{3} + 39245 T^{4} + 293739 T^{5} + 4405465 T^{6} + 40998321 T^{7} + 397241676 T^{8} + 3994326482 T^{9} + 30391775917 T^{10} + 283597180222 T^{11} + 2002495288716 T^{12} + 14673750067431 T^{13} + 111950271236665 T^{14} + 529972525333389 T^{15} + 5027295642479645 T^{16} + 11214283155296103 T^{17} + 169187425186389382 T^{18} + 45848500718449031 T^{19} + 3255243551009881201 T^{20}$$
$73$ $$1 + 31 T + 857 T^{2} + 16625 T^{3} + 280173 T^{4} + 4020157 T^{5} + 51396432 T^{6} + 589991993 T^{7} + 6192642640 T^{8} + 59433187850 T^{9} + 529794375619 T^{10} + 4338622713050 T^{11} + 33000592628560 T^{12} + 229516915140881 T^{13} + 1459568262476112 T^{14} + 8334073276100101 T^{15} + 42399764182067997 T^{16} + 183663000379987625 T^{17} + 691136298753227417 T^{18} + 1825019187956305303 T^{19} + 4297625829703557649 T^{20}$$
$79$ $$1 + 10 T + 248 T^{2} + 2937 T^{3} + 29662 T^{4} + 273591 T^{5} + 1835094 T^{6} + 1901304 T^{7} - 54353511 T^{8} - 1412662644 T^{9} - 15400667207 T^{10} - 111600348876 T^{11} - 339220262151 T^{12} + 937417022856 T^{13} + 71477059942614 T^{14} + 841854937258809 T^{15} + 7210460105663902 T^{16} + 56401880692348983 T^{17} + 376242984856827128 T^{18} + 1198515959826183190 T^{19} + 9468276082626847201 T^{20}$$
$83$ $$1 - 22 T + 669 T^{2} - 9824 T^{3} + 169480 T^{4} - 1798297 T^{5} + 22307432 T^{6} - 175385507 T^{7} + 1816492439 T^{8} - 11655093474 T^{9} + 132417751709 T^{10} - 967372758342 T^{11} + 12513816412271 T^{12} - 100283152891009 T^{13} + 1058673268541672 T^{14} - 7083564971184971 T^{15} + 55409854478578120 T^{16} - 266584564922095648 T^{17} + 1506783503301018429 T^{18} - 4112685615885888866 T^{19} + 15516041187205853449 T^{20}$$
$89$ $$1 - T + 510 T^{2} - 62 T^{3} + 113775 T^{4} + 104240 T^{5} + 14626103 T^{6} + 33572815 T^{7} + 1283471757 T^{8} + 5086267491 T^{9} + 104337188501 T^{10} + 452677806699 T^{11} + 10166379787197 T^{12} + 23667793817735 T^{13} + 917674479316823 T^{14} + 582082356963760 T^{15} + 56544046379087775 T^{16} - 2742342763522798 T^{17} + 2007660290908061310 T^{18} - 350356403707485209 T^{19} + 31181719929966183601 T^{20}$$
$97$ $$1 + 57 T + 1999 T^{2} + 49485 T^{3} + 981253 T^{4} + 16180839 T^{5} + 233385980 T^{6} + 3001658219 T^{7} + 35423192688 T^{8} + 386010871872 T^{9} + 3937832201969 T^{10} + 37443054571584 T^{11} + 333296820001392 T^{12} + 2739532411709387 T^{13} + 20661493004880380 T^{14} + 138950370136735623 T^{15} + 817356278752596037 T^{16} + 3998303107399421805 T^{17} + 15667029755159545039 T^{18} + 43333170343310217369 T^{19} + 73742412689492826049 T^{20}$$
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