# Properties

 Label 552.2.f.d Level $552$ Weight $2$ Character orbit 552.f Analytic conductor $4.408$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$552 = 2^{3} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 552.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.40774219157$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 2 x^{19} + 2 x^{18} - 2 x^{17} + x^{16} - 4 x^{15} + 16 x^{14} - 24 x^{13} + 32 x^{12} - 16 x^{11} - 16 x^{10} - 32 x^{9} + 128 x^{8} - 192 x^{7} + 256 x^{6} - 128 x^{5} + 64 x^{4} - 256 x^{3} + 512 x^{2} - 1024 x + 1024$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}$$ 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 $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 2 x^{19} + 2 x^{18} - 2 x^{17} + x^{16} - 4 x^{15} + 16 x^{14} - 24 x^{13} + 32 x^{12} - 16 x^{11} - 16 x^{10} - 32 x^{9} + 128 x^{8} - 192 x^{7} + 256 x^{6} - 128 x^{5} + 64 x^{4} - 256 x^{3} + 512 x^{2} - 1024 x + 1024$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{17} + 4 \nu^{15} - 2 \nu^{14} + 3 \nu^{13} + 6 \nu^{12} - 14 \nu^{11} - 4 \nu^{10} + 20 \nu^{9} - 24 \nu^{8} + 24 \nu^{7} + 80 \nu^{6} - 80 \nu^{5} - 32 \nu^{4} + 96 \nu^{3} - 192 \nu^{2} + 128 \nu + 256$$$$)/256$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{19} + 2 \nu^{18} - 2 \nu^{17} - 2 \nu^{16} - \nu^{15} + 20 \nu^{14} - 24 \nu^{13} + 36 \nu^{12} - 8 \nu^{11} - 40 \nu^{10} + 112 \nu^{8} - 224 \nu^{7} + 288 \nu^{6} + 64 \nu^{5} - 192 \nu^{4} - 192 \nu^{3} + 640 \nu^{2} - 1280 \nu + 1536$$$$)/1024$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{19} - 2 \nu^{16} + 13 \nu^{15} - 14 \nu^{14} + 10 \nu^{13} - 8 \nu^{12} - 4 \nu^{11} - 16 \nu^{10} + 88 \nu^{9} - 192 \nu^{8} + 144 \nu^{7} - 128 \nu^{6} - 96 \nu^{5} - 256 \nu^{4} + 512 \nu^{3} - 896 \nu^{2} + 1280 \nu - 1024$$$$)/512$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{18} + 2 \nu^{15} - 5 \nu^{14} + 6 \nu^{13} - 10 \nu^{12} + 8 \nu^{11} - 4 \nu^{10} - 8 \nu^{9} - 56 \nu^{8} + 64 \nu^{7} - 80 \nu^{6} - 96 \nu^{4} + 192 \nu^{3} - 256 \nu^{2} + 384 \nu - 768$$$$)/256$$ $$\beta_{8}$$ $$=$$ $$($$$$-3 \nu^{19} + 2 \nu^{18} - 2 \nu^{17} + 2 \nu^{16} + 5 \nu^{15} + 16 \nu^{14} - 44 \nu^{13} - 12 \nu^{12} - 32 \nu^{11} + 24 \nu^{10} + 80 \nu^{9} + 176 \nu^{8} - 128 \nu^{7} + 32 \nu^{6} - 256 \nu^{5} + 64 \nu^{4} - 192 \nu^{3} + 896 \nu^{2} + 256 \nu + 512$$$$)/1024$$ $$\beta_{9}$$ $$=$$ $$($$$$3 \nu^{19} - 2 \nu^{17} + 6 \nu^{16} - \nu^{15} + 2 \nu^{14} + 12 \nu^{13} - 16 \nu^{12} + 80 \nu^{10} - 48 \nu^{9} - 64 \nu^{6} + 128 \nu^{5} - 128 \nu^{4} - 320 \nu^{3} + 1152 \nu^{2} + 256 \nu - 512$$$$)/1024$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{18} + 4 \nu^{16} - 2 \nu^{15} + 3 \nu^{14} + 6 \nu^{13} - 14 \nu^{12} - 4 \nu^{11} + 20 \nu^{10} - 24 \nu^{9} + 24 \nu^{8} + 80 \nu^{7} - 80 \nu^{6} - 32 \nu^{5} + 96 \nu^{4} - 192 \nu^{3} + 128 \nu^{2} + 256 \nu$$$$)/256$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{19} - 8 \nu^{18} + 6 \nu^{17} + 2 \nu^{16} - 11 \nu^{15} + 6 \nu^{14} + 32 \nu^{13} - 40 \nu^{12} + 8 \nu^{11} - 32 \nu^{10} - 64 \nu^{9} + 32 \nu^{8} + 96 \nu^{7} - 256 \nu^{6} + 64 \nu^{5} - 64 \nu^{3} - 896 \nu^{2} - 768 \nu + 1536$$$$)/1024$$ $$\beta_{12}$$ $$=$$ $$($$$$-\nu^{19} - 2 \nu^{18} + 2 \nu^{17} - 6 \nu^{16} + 7 \nu^{15} + 8 \nu^{14} - 4 \nu^{13} - 16 \nu^{12} - 8 \nu^{11} - 48 \nu^{10} + 16 \nu^{9} + 32 \nu^{8} + 32 \nu^{7} + 64 \nu^{6} + 192 \nu^{5} - 384 \nu^{4} - 64 \nu^{3} - 512 \nu^{2} + 512 \nu + 512$$$$)/512$$ $$\beta_{13}$$ $$=$$ $$($$$$3 \nu^{19} - 4 \nu^{18} + 2 \nu^{17} + 6 \nu^{16} - 9 \nu^{15} - 10 \nu^{14} + 24 \nu^{13} - 80 \nu^{12} + 88 \nu^{11} + 80 \nu^{10} - 160 \nu^{9} - 128 \nu^{8} + 160 \nu^{7} - 704 \nu^{6} + 448 \nu^{5} + 640 \nu^{4} + 64 \nu^{3} + 896 \nu^{2} + 1280 \nu - 4608$$$$)/1024$$ $$\beta_{14}$$ $$=$$ $$($$$$\nu^{19} - 2 \nu^{18} - 6 \nu^{17} - 2 \nu^{16} + \nu^{15} - 4 \nu^{14} + 24 \nu^{13} - 8 \nu^{12} - 48 \nu^{11} - 32 \nu^{9} - 128 \nu^{8} + 192 \nu^{7} + 128 \nu^{6} - 320 \nu^{3} - 1024 \nu^{2} + 512 \nu$$$$)/512$$ $$\beta_{15}$$ $$=$$ $$($$$$\nu^{19} - 3 \nu^{18} + 2 \nu^{16} - 9 \nu^{15} + 11 \nu^{14} + 28 \nu^{13} - 58 \nu^{12} + 16 \nu^{11} + 44 \nu^{10} - 128 \nu^{9} + 40 \nu^{8} + 192 \nu^{7} - 304 \nu^{6} + 192 \nu^{5} + 32 \nu^{4} - 960 \nu^{3} + 1024 \nu - 1024$$$$)/512$$ $$\beta_{16}$$ $$=$$ $$($$$$\nu^{19} - 5 \nu^{18} + 6 \nu^{17} - 10 \nu^{16} + 3 \nu^{15} - 3 \nu^{14} + 14 \nu^{13} - 46 \nu^{12} + 60 \nu^{11} - 76 \nu^{10} + 24 \nu^{9} - 8 \nu^{8} + 80 \nu^{7} - 336 \nu^{6} + 480 \nu^{5} - 416 \nu^{4} + 384 \nu^{3} - 768 \nu^{2} + 768 \nu - 1024$$$$)/512$$ $$\beta_{17}$$ $$=$$ $$($$$$3 \nu^{19} - 4 \nu^{18} + 2 \nu^{17} - 2 \nu^{16} + 7 \nu^{15} - 10 \nu^{14} + 8 \nu^{13} - 24 \nu^{12} + 24 \nu^{11} - 32 \nu^{10} + 32 \nu^{9} - 96 \nu^{8} + 160 \nu^{7} - 128 \nu^{6} + 192 \nu^{5} - 512 \nu^{4} + 576 \nu^{3} - 384 \nu^{2} + 256 \nu - 512$$$$)/512$$ $$\beta_{18}$$ $$=$$ $$($$$$7 \nu^{19} - 2 \nu^{18} + 6 \nu^{17} - 10 \nu^{16} - \nu^{15} - 32 \nu^{14} + 72 \nu^{13} - 44 \nu^{12} + 104 \nu^{11} + 24 \nu^{10} + 64 \nu^{9} - 336 \nu^{8} + 352 \nu^{7} - 352 \nu^{6} + 960 \nu^{5} + 320 \nu^{4} + 832 \nu^{3} - 1664 \nu^{2} + 1280 \nu - 2560$$$$)/1024$$ $$\beta_{19}$$ $$=$$ $$($$$$-2 \nu^{19} + 4 \nu^{18} - 3 \nu^{17} + 2 \nu^{16} - 2 \nu^{15} + 10 \nu^{14} - 23 \nu^{13} + 24 \nu^{12} - 34 \nu^{11} + 8 \nu^{10} + 12 \nu^{9} + 96 \nu^{8} - 216 \nu^{7} + 224 \nu^{6} - 240 \nu^{5} + 128 \nu^{4} - 352 \nu^{3} + 640 \nu^{2} - 640 \nu + 1024$$$$)/256$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3}$$ $$\nu^{4}$$ $$=$$ $$\beta_{13} - \beta_{9} - \beta_{7} + \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$-\beta_{17} + \beta_{16} - 2 \beta_{15} + \beta_{14} + \beta_{12} - \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - \beta_{8} - 2 \beta_{4} - \beta_{2} - \beta_{1} + 1$$ $$\nu^{6}$$ $$=$$ $$\beta_{19} + 2 \beta_{18} + 2 \beta_{17} - 2 \beta_{16} + \beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} - \beta_{10} - 2 \beta_{9} + \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_{1} - 3$$ $$\nu^{7}$$ $$=$$ $$-\beta_{19} + \beta_{17} - \beta_{16} + \beta_{15} - \beta_{13} - 2 \beta_{11} + \beta_{10} - 3 \beta_{9} - \beta_{7} - \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-2 \beta_{18} + 4 \beta_{16} - \beta_{13} - 2 \beta_{12} - 4 \beta_{11} + 2 \beta_{10} + \beta_{9} - 2 \beta_{8} - 3 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 2 \beta_{1} - 2$$ $$\nu^{9}$$ $$=$$ $$4 \beta_{18} - \beta_{17} + \beta_{16} - 2 \beta_{15} + \beta_{14} - 4 \beta_{13} - 3 \beta_{12} + 3 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 7 \beta_{8} + 4 \beta_{7} + 4 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + 3 \beta_{1} - 7$$ $$\nu^{10}$$ $$=$$ $$-7 \beta_{19} + 2 \beta_{18} - 6 \beta_{17} - 6 \beta_{16} + \beta_{15} - \beta_{14} - 4 \beta_{13} + \beta_{12} + 5 \beta_{11} - 9 \beta_{10} + 10 \beta_{9} + 13 \beta_{8} + 6 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_{1} + 5$$ $$\nu^{11}$$ $$=$$ $$-\beta_{19} + 8 \beta_{18} + 5 \beta_{17} - 5 \beta_{16} + 9 \beta_{15} - 8 \beta_{14} - \beta_{13} - 8 \beta_{12} + 2 \beta_{11} - 7 \beta_{10} - 11 \beta_{9} + 12 \beta_{8} + 15 \beta_{7} - 8 \beta_{6} + 16 \beta_{5} + 15 \beta_{4} - 11 \beta_{3} - 2 \beta_{2} + 10 \beta_{1} + 16$$ $$\nu^{12}$$ $$=$$ $$-16 \beta_{19} - 10 \beta_{18} - 16 \beta_{17} + 4 \beta_{16} - 8 \beta_{15} + 4 \beta_{14} - 5 \beta_{13} - 6 \beta_{12} - 4 \beta_{11} + 2 \beta_{10} + 13 \beta_{9} - 2 \beta_{8} - 7 \beta_{7} + 6 \beta_{6} + 12 \beta_{5} - 9 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 6 \beta_{1} - 6$$ $$\nu^{13}$$ $$=$$ $$4 \beta_{19} + 4 \beta_{18} - 13 \beta_{17} + 13 \beta_{16} + 2 \beta_{15} - 3 \beta_{14} - 8 \beta_{13} - 7 \beta_{12} + 3 \beta_{11} + 6 \beta_{10} + 14 \beta_{9} - 9 \beta_{8} + 8 \beta_{7} + 4 \beta_{5} + 18 \beta_{4} + 4 \beta_{3} + 11 \beta_{2} - 13 \beta_{1} - 11$$ $$\nu^{14}$$ $$=$$ $$-11 \beta_{19} + 10 \beta_{18} - 6 \beta_{17} - 22 \beta_{16} + 21 \beta_{15} - 9 \beta_{14} + 4 \beta_{13} + \beta_{12} + 25 \beta_{11} - 13 \beta_{10} - 14 \beta_{9} + 25 \beta_{8} - 2 \beta_{7} + 22 \beta_{6} + 64 \beta_{5} + 6 \beta_{4} + 15 \beta_{3} + 9 \beta_{2} + 11 \beta_{1} - 43$$ $$\nu^{15}$$ $$=$$ $$-9 \beta_{19} - 24 \beta_{18} - 15 \beta_{17} + 15 \beta_{16} + \beta_{15} - 16 \beta_{14} + 39 \beta_{13} + 16 \beta_{12} - 26 \beta_{11} + \beta_{10} + 13 \beta_{9} - 24 \beta_{8} - 33 \beta_{7} + 8 \beta_{6} + 71 \beta_{4} - 7 \beta_{3} - 38 \beta_{2} - 82 \beta_{1} + 56$$ $$\nu^{16}$$ $$=$$ $$40 \beta_{19} + 14 \beta_{18} + 16 \beta_{17} + 20 \beta_{16} - 56 \beta_{15} + 7 \beta_{13} + 14 \beta_{12} - 12 \beta_{11} + 106 \beta_{10} + 25 \beta_{9} - 58 \beta_{8} + 37 \beta_{7} - 2 \beta_{6} + 28 \beta_{5} - 21 \beta_{4} - 30 \beta_{3} - 68 \beta_{2} + 22 \beta_{1} - 18$$ $$\nu^{17}$$ $$=$$ $$4 \beta_{18} + 15 \beta_{17} - 79 \beta_{16} + 14 \beta_{15} - 55 \beta_{14} + 12 \beta_{13} + 101 \beta_{12} + 51 \beta_{11} - 30 \beta_{10} - 30 \beta_{9} - 57 \beta_{8} - 60 \beta_{7} + 32 \beta_{6} - 92 \beta_{5} - 34 \beta_{4} + 12 \beta_{3} + 51 \beta_{2} + 19 \beta_{1} - 159$$ $$\nu^{18}$$ $$=$$ $$97 \beta_{19} + 2 \beta_{18} + 58 \beta_{17} + 26 \beta_{16} + 57 \beta_{15} - 33 \beta_{14} - 4 \beta_{13} + 33 \beta_{12} - 131 \beta_{11} + 79 \beta_{10} - 86 \beta_{9} - 187 \beta_{8} - 90 \beta_{7} + 62 \beta_{6} - 96 \beta_{5} - 34 \beta_{4} + 127 \beta_{3} - 27 \beta_{2} - 169 \beta_{1} + 69$$ $$\nu^{19}$$ $$=$$ $$31 \beta_{19} - 56 \beta_{18} + 173 \beta_{17} - 13 \beta_{16} + 105 \beta_{15} - 24 \beta_{14} + 95 \beta_{13} - 56 \beta_{12} - 22 \beta_{11} - 71 \beta_{10} - 107 \beta_{9} - 92 \beta_{8} - 289 \beta_{7} + 56 \beta_{6} - 48 \beta_{5} + 111 \beta_{4} + 77 \beta_{3} - 106 \beta_{2} + 114 \beta_{1} - 192$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/552\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$185$$ $$277$$ $$415$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
277.1
 1.38506 + 0.285679i 1.38506 − 0.285679i 1.32157 + 0.503453i 1.32157 − 0.503453i 1.19161 + 0.761625i 1.19161 − 0.761625i 0.585586 + 1.28728i 0.585586 − 1.28728i 0.459619 + 1.33744i 0.459619 − 1.33744i −0.133806 + 1.40787i −0.133806 − 1.40787i −0.238838 + 1.39390i −0.238838 − 1.39390i −1.08648 + 0.905299i −1.08648 − 0.905299i −1.14290 + 0.832928i −1.14290 − 0.832928i −1.34141 + 0.447910i −1.34141 − 0.447910i
−1.38506 0.285679i 1.00000i 1.83677 + 0.791365i 2.43947i 0.285679 1.38506i −3.50435 −2.31796 1.62082i −1.00000 −0.696905 + 3.37881i
277.2 −1.38506 + 0.285679i 1.00000i 1.83677 0.791365i 2.43947i 0.285679 + 1.38506i −3.50435 −2.31796 + 1.62082i −1.00000 −0.696905 3.37881i
277.3 −1.32157 0.503453i 1.00000i 1.49307 + 1.33069i 2.69719i −0.503453 + 1.32157i 1.19970 −1.30325 2.51029i −1.00000 −1.35791 + 3.56451i
277.4 −1.32157 + 0.503453i 1.00000i 1.49307 1.33069i 2.69719i −0.503453 1.32157i 1.19970 −1.30325 + 2.51029i −1.00000 −1.35791 3.56451i
277.5 −1.19161 0.761625i 1.00000i 0.839855 + 1.81512i 1.20415i 0.761625 1.19161i 3.87524 0.381661 2.80256i −1.00000 0.917114 1.43488i
277.6 −1.19161 + 0.761625i 1.00000i 0.839855 1.81512i 1.20415i 0.761625 + 1.19161i 3.87524 0.381661 + 2.80256i −1.00000 0.917114 + 1.43488i
277.7 −0.585586 1.28728i 1.00000i −1.31418 + 1.50763i 1.66345i 1.28728 0.585586i −0.540858 2.71030 + 0.808871i −1.00000 −2.14133 + 0.974093i
277.8 −0.585586 + 1.28728i 1.00000i −1.31418 1.50763i 1.66345i 1.28728 + 0.585586i −0.540858 2.71030 0.808871i −1.00000 −2.14133 0.974093i
277.9 −0.459619 1.33744i 1.00000i −1.57750 + 1.22943i 2.04946i −1.33744 + 0.459619i −4.47395 2.36933 + 1.54475i −1.00000 −2.74104 + 0.941972i
277.10 −0.459619 + 1.33744i 1.00000i −1.57750 1.22943i 2.04946i −1.33744 0.459619i −4.47395 2.36933 1.54475i −1.00000 −2.74104 0.941972i
277.11 0.133806 1.40787i 1.00000i −1.96419 0.376764i 2.81880i −1.40787 0.133806i 1.52994 −0.793255 + 2.71491i −1.00000 3.96850 + 0.377173i
277.12 0.133806 + 1.40787i 1.00000i −1.96419 + 0.376764i 2.81880i −1.40787 + 0.133806i 1.52994 −0.793255 2.71491i −1.00000 3.96850 0.377173i
277.13 0.238838 1.39390i 1.00000i −1.88591 0.665833i 4.08024i 1.39390 + 0.238838i −1.92513 −1.37853 + 2.46975i −1.00000 −5.68744 0.974516i
277.14 0.238838 + 1.39390i 1.00000i −1.88591 + 0.665833i 4.08024i 1.39390 0.238838i −1.92513 −1.37853 2.46975i −1.00000 −5.68744 + 0.974516i
277.15 1.08648 0.905299i 1.00000i 0.360869 1.96717i 3.71494i 0.905299 + 1.08648i 1.13753 −1.38880 2.46399i −1.00000 3.36313 + 4.03621i
277.16 1.08648 + 0.905299i 1.00000i 0.360869 + 1.96717i 3.71494i 0.905299 1.08648i 1.13753 −1.38880 + 2.46399i −1.00000 3.36313 4.03621i
277.17 1.14290 0.832928i 1.00000i 0.612462 1.90391i 1.03589i −0.832928 1.14290i −3.71260 −0.885838 2.68613i −1.00000 −0.862822 1.18392i
277.18 1.14290 + 0.832928i 1.00000i 0.612462 + 1.90391i 1.03589i −0.832928 + 1.14290i −3.71260 −0.885838 + 2.68613i −1.00000 −0.862822 + 1.18392i
277.19 1.34141 0.447910i 1.00000i 1.59875 1.20166i 1.69969i 0.447910 + 1.34141i 2.41449 1.60635 2.32801i −1.00000 −0.761306 2.27997i
277.20 1.34141 + 0.447910i 1.00000i 1.59875 + 1.20166i 1.69969i 0.447910 1.34141i 2.41449 1.60635 + 2.32801i −1.00000 −0.761306 + 2.27997i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 277.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.2.f.d 20
4.b odd 2 1 2208.2.f.d 20
8.b even 2 1 inner 552.2.f.d 20
8.d odd 2 1 2208.2.f.d 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.f.d 20 1.a even 1 1 trivial
552.2.f.d 20 8.b even 2 1 inner
2208.2.f.d 20 4.b odd 2 1
2208.2.f.d 20 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{20} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(552, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1024 + 1024 T + 512 T^{2} + 256 T^{3} + 64 T^{4} + 128 T^{5} + 256 T^{6} + 192 T^{7} + 128 T^{8} + 32 T^{9} - 16 T^{10} + 16 T^{11} + 32 T^{12} + 24 T^{13} + 16 T^{14} + 4 T^{15} + T^{16} + 2 T^{17} + 2 T^{18} + 2 T^{19} + T^{20}$$
$3$ $$( 1 + T^{2} )^{10}$$
$5$ $$4129024 + 12928256 T^{2} + 16682432 T^{4} + 11705728 T^{6} + 4965808 T^{8} + 1335504 T^{10} + 231268 T^{12} + 25520 T^{14} + 1720 T^{16} + 64 T^{18} + T^{20}$$
$7$ $$( -1184 - 128 T + 3560 T^{2} - 1544 T^{3} - 1692 T^{4} + 764 T^{5} + 322 T^{6} - 104 T^{7} - 30 T^{8} + 4 T^{9} + T^{10} )^{2}$$
$11$ $$12390400 + 716079104 T^{2} + 940416000 T^{4} + 506841088 T^{6} + 146333248 T^{8} + 24928704 T^{10} + 2600128 T^{12} + 165904 T^{14} + 6224 T^{16} + 124 T^{18} + T^{20}$$
$13$ $$262144 + 196542464 T^{2} + 878563328 T^{4} + 713488384 T^{6} + 245989120 T^{8} + 43791232 T^{10} + 4384016 T^{12} + 253920 T^{14} + 8360 T^{16} + 144 T^{18} + T^{20}$$
$17$ $$( 320 - 7744 T - 37160 T^{2} - 10408 T^{3} + 29996 T^{4} - 4604 T^{5} - 2574 T^{6} + 612 T^{7} + 30 T^{8} - 16 T^{9} + T^{10} )^{2}$$
$19$ $$45741232384 + 112906211584 T^{2} + 83663326144 T^{4} + 24636119680 T^{6} + 3800534704 T^{8} + 347260304 T^{10} + 19809028 T^{12} + 712448 T^{14} + 15672 T^{16} + 192 T^{18} + T^{20}$$
$23$ $$( -1 + T )^{20}$$
$29$ $$4194304 + 508559360 T^{2} + 917831680 T^{4} + 652378112 T^{6} + 235196416 T^{8} + 47081472 T^{10} + 5397504 T^{12} + 349312 T^{14} + 11920 T^{16} + 184 T^{18} + T^{20}$$
$31$ $$( 149504 - 251904 T + 13056 T^{2} + 105472 T^{3} - 26688 T^{4} - 9792 T^{5} + 3264 T^{6} + 208 T^{7} - 108 T^{8} + T^{10} )^{2}$$
$37$ $$1507276754944 + 2672577167360 T^{2} + 1380999842816 T^{4} + 296561499136 T^{6} + 33465336384 T^{8} + 2210044096 T^{10} + 89584576 T^{12} + 2250384 T^{14} + 34064 T^{16} + 284 T^{18} + T^{20}$$
$41$ $$( -32768 + 147456 T + 864256 T^{2} + 914432 T^{3} + 335744 T^{4} + 24832 T^{5} - 10896 T^{6} - 2032 T^{7} + 4 T^{8} + 20 T^{9} + T^{10} )^{2}$$
$43$ $$2766181017856 + 4209077871872 T^{2} + 2470130786752 T^{4} + 723790284672 T^{6} + 112782455600 T^{8} + 9138182480 T^{10} + 359416100 T^{12} + 7343728 T^{14} + 80376 T^{16} + 448 T^{18} + T^{20}$$
$47$ $$( -5888 - 125952 T - 380736 T^{2} - 159616 T^{3} + 255728 T^{4} - 39712 T^{5} - 7932 T^{6} + 1872 T^{7} + 8 T^{8} - 20 T^{9} + T^{10} )^{2}$$
$53$ $$17615950602496 + 39536098671872 T^{2} + 24239488895936 T^{4} + 5645934150784 T^{6} + 580490843056 T^{8} + 30017628496 T^{10} + 851322884 T^{12} + 13681792 T^{14} + 122456 T^{16} + 560 T^{18} + T^{20}$$
$59$ $$1504920469504 + 23487267471360 T^{2} + 70149568172032 T^{4} + 15452783310848 T^{6} + 1358023031552 T^{8} + 60012256128 T^{10} + 1456775952 T^{12} + 20073824 T^{14} + 155368 T^{16} + 624 T^{18} + T^{20}$$
$61$ $$425864306790400 + 521964678938624 T^{2} + 155561590139904 T^{4} + 18865629218816 T^{6} + 1179498428992 T^{8} + 42747974080 T^{10} + 949368128 T^{12} + 13106448 T^{14} + 109616 T^{16} + 508 T^{18} + T^{20}$$
$67$ $$259375405578496 + 1651153562672384 T^{2} + 454372717136832 T^{4} + 51891917397120 T^{6} + 3149004645936 T^{8} + 110358394320 T^{10} + 2292127364 T^{12} + 28047392 T^{14} + 195256 T^{16} + 704 T^{18} + T^{20}$$
$71$ $$( 512000 + 8601600 T - 6584320 T^{2} - 2001408 T^{3} + 460864 T^{4} + 129216 T^{5} - 6784 T^{6} - 2864 T^{7} - 48 T^{8} + 20 T^{9} + T^{10} )^{2}$$
$73$ $$( 507904 + 516096 T - 683008 T^{2} - 443392 T^{3} + 100864 T^{4} + 82304 T^{5} + 6544 T^{6} - 1840 T^{7} - 204 T^{8} + 8 T^{9} + T^{10} )^{2}$$
$79$ $$( -37745440 + 24696448 T + 11756168 T^{2} - 3633416 T^{3} - 877644 T^{4} + 132404 T^{5} + 23106 T^{6} - 1780 T^{7} - 254 T^{8} + 8 T^{9} + T^{10} )^{2}$$
$83$ $$56023360385191936 + 50758956130852864 T^{2} + 14886551957539840 T^{4} + 1628781428454400 T^{6} + 73645246452288 T^{8} + 1652714039488 T^{10} + 20309965248 T^{12} + 142418832 T^{14} + 565776 T^{16} + 1180 T^{18} + T^{20}$$
$89$ $$( -272960 + 1237504 T + 4981432 T^{2} - 2681048 T^{3} - 205588 T^{4} + 172996 T^{5} + 4850 T^{6} - 3416 T^{7} - 122 T^{8} + 20 T^{9} + T^{10} )^{2}$$
$97$ $$( -1078295552 + 1707026432 T - 606350080 T^{2} + 39846400 T^{3} + 14153856 T^{4} - 2425216 T^{5} + 48032 T^{6} + 12256 T^{7} - 588 T^{8} - 16 T^{9} + T^{10} )^{2}$$