# Properties

 Label 1849.2.a.q Level $1849$ Weight $2$ Character orbit 1849.a Self dual yes Analytic conductor $14.764$ Analytic rank $1$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1849 = 43^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1849.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.7643393337$$ Analytic rank: $$1$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 27 x^{18} + 300 x^{16} - 1775 x^{14} + 6037 x^{12} - 11859 x^{10} + 12809 x^{8} - 6823 x^{6} + 1500 x^{4} - 75 x^{2} + 1$$ 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 $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 27 x^{18} + 300 x^{16} - 1775 x^{14} + 6037 x^{12} - 11859 x^{10} + 12809 x^{8} - 6823 x^{6} + 1500 x^{4} - 75 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-27659 \nu^{19} + 533429 \nu^{17} - 3048022 \nu^{15} - 2821968 \nu^{13} + 98515291 \nu^{11} - 424283337 \nu^{9} + 818927614 \nu^{7} - 747265236 \nu^{5} + 275318380 \nu^{3} - 26526277 \nu$$$$)/1963853$$ $$\beta_{3}$$ $$=$$ $$($$$$-32741 \nu^{18} + 969056 \nu^{16} - 11941032 \nu^{14} + 79319430 \nu^{12} - 307062428 \nu^{10} + 698608976 \nu^{8} - 895324181 \nu^{6} + 580699660 \nu^{4} - 145773292 \nu^{2} + 2418746$$$$)/1963853$$ $$\beta_{4}$$ $$=$$ $$($$$$-104687 \nu^{18} + 3273453 \nu^{16} - 42032783 \nu^{14} + 285881978 \nu^{12} - 1107993096 \nu^{10} + 2450520491 \nu^{8} - 2934073585 \nu^{6} + 1697907062 \nu^{4} - 387773159 \nu^{2} + 14278584$$$$)/1963853$$ $$\beta_{5}$$ $$=$$ $$($$$$-131444 \nu^{19} + 3602040 \nu^{17} - 40943432 \nu^{15} + 250857133 \nu^{13} - 900255456 \nu^{11} + 1920911813 \nu^{9} - 2355356698 \nu^{7} + 1510642955 \nu^{5} - 411757298 \nu^{3} + 27579461 \nu$$$$)/1963853$$ $$\beta_{6}$$ $$=$$ $$($$$$-152197 \nu^{18} + 3956200 \nu^{16} - 42164100 \nu^{14} + 239188604 \nu^{12} - 785122399 \nu^{10} + 1517964126 \nu^{8} - 1681080551 \nu^{6} + 976052115 \nu^{4} - 232881548 \nu^{2} + 2656569$$$$)/1963853$$ $$\beta_{7}$$ $$=$$ $$($$$$-152197 \nu^{18} + 3956200 \nu^{16} - 42164100 \nu^{14} + 239188604 \nu^{12} - 785122399 \nu^{10} + 1517964126 \nu^{8} - 1681080551 \nu^{6} + 976052115 \nu^{4} - 234845401 \nu^{2} + 8548128$$$$)/1963853$$ $$\beta_{8}$$ $$=$$ $$($$$$-159103 \nu^{18} + 4135469 \nu^{16} - 43991454 \nu^{14} + 248035165 \nu^{12} - 801740165 \nu^{10} + 1496628476 \nu^{8} - 1536429084 \nu^{6} + 763377719 \nu^{4} - 136438918 \nu^{2} + 1053184$$$$)/1963853$$ $$\beta_{9}$$ $$=$$ $$($$$$212899 \nu^{18} - 6006257 \nu^{16} + 69816397 \nu^{14} - 431405640 \nu^{12} + 1522703970 \nu^{10} - 3062673744 \nu^{8} + 3302046534 \nu^{6} - 1678562036 \nu^{4} + 329474479 \nu^{2} - 9133342$$$$)/1963853$$ $$\beta_{10}$$ $$=$$ $$($$$$-285767 \nu^{18} + 7849454 \nu^{16} - 88705119 \nu^{14} + 532470474 \nu^{12} - 1825452336 \nu^{10} + 3565031955 \nu^{8} - 3722103403 \nu^{6} + 1813167422 \nu^{4} - 337531739 \nu^{2} + 8669543$$$$)/1963853$$ $$\beta_{11}$$ $$=$$ $$($$$$286689 \nu^{19} - 7388254 \nu^{17} + 77502090 \nu^{15} - 426972760 \nu^{13} + 1327270034 \nu^{11} - 2315478651 \nu^{9} + 2103410868 \nu^{7} - 832529259 \nu^{5} + 117336103 \nu^{3} - 10092877 \nu$$$$)/1963853$$ $$\beta_{12}$$ $$=$$ $$($$$$-319430 \nu^{18} + 8357310 \nu^{16} - 89443122 \nu^{14} + 506292190 \nu^{12} - 1634332462 \nu^{10} + 3014087627 \nu^{8} - 2998735049 \nu^{6} + 1413228919 \nu^{4} - 263109395 \nu^{2} + 12511623$$$$)/1963853$$ $$\beta_{13}$$ $$=$$ $$($$$$463799 \nu^{19} - 12595441 \nu^{17} + 140982897 \nu^{15} - 842131947 \nu^{13} + 2901019397 \nu^{11} - 5802940707 \nu^{9} + 6443159602 \nu^{7} - 3584557446 \nu^{5} + 830303886 \nu^{3} - 42842185 \nu$$$$)/1963853$$ $$\beta_{14}$$ $$=$$ $$($$$$-596780 \nu^{18} + 15875946 \nu^{16} - 173276438 \nu^{14} + 1003467048 \nu^{12} - 3325698762 \nu^{10} + 6325178486 \nu^{8} - 6535167537 \nu^{6} + 3233121230 \nu^{4} - 612215946 \nu^{2} + 14696711$$$$)/1963853$$ $$\beta_{15}$$ $$=$$ $$($$$$-651785 \nu^{19} + 17350711 \nu^{17} - 189482587 \nu^{15} + 1097567004 \nu^{13} - 3635096403 \nu^{11} + 6896116521 \nu^{9} - 7086537953 \nu^{7} + 3487143410 \nu^{5} - 681655983 \nu^{3} + 27774347 \nu$$$$)/1963853$$ $$\beta_{16}$$ $$=$$ $$($$$$808455 \nu^{19} - 21795831 \nu^{17} + 241632621 \nu^{15} - 1424650828 \nu^{13} + 4817550600 \nu^{11} - 9367021555 \nu^{9} + 9911337190 \nu^{7} - 5025209785 \nu^{5} + 951458866 \nu^{3} - 11460945 \nu$$$$)/1963853$$ $$\beta_{17}$$ $$=$$ $$($$$$1897282 \nu^{19} - 51413396 \nu^{17} + 573187456 \nu^{15} - 3399374624 \nu^{13} + 11561404040 \nu^{11} - 22595687461 \nu^{9} + 24019751632 \nu^{7} - 12284174748 \nu^{5} + 2439434596 \nu^{3} - 68776600 \nu$$$$)/1963853$$ $$\beta_{18}$$ $$=$$ $$($$$$-2180923 \nu^{19} + 58971636 \nu^{17} - 656294988 \nu^{15} + 3889420361 \nu^{13} - 13246781895 \nu^{11} + 26034563400 \nu^{9} - 28056188881 \nu^{7} + 14770869818 \nu^{5} - 3086037295 \nu^{3} + 104904189 \nu$$$$)/1963853$$ $$\beta_{19}$$ $$=$$ $$($$$$-2201676 \nu^{19} + 59325796 \nu^{17} - 657515656 \nu^{15} + 3877751832 \nu^{13} - 13131648838 \nu^{11} + 25631615713 \nu^{9} - 27381912734 \nu^{7} + 14236278978 \nu^{5} - 2907161545 \nu^{3} + 78017444 \nu$$$$)/1963853$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} + \beta_{6} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{19} - 2 \beta_{18} - \beta_{17} + 2 \beta_{5} + 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{12} - 2 \beta_{10} - \beta_{9} - 7 \beta_{7} + 7 \beta_{6} + \beta_{4} - 2 \beta_{3} + 14$$ $$\nu^{5}$$ $$=$$ $$7 \beta_{19} - 17 \beta_{18} - 9 \beta_{17} - \beta_{16} - \beta_{15} - 4 \beta_{13} - \beta_{11} + 17 \beta_{5} + 2 \beta_{2} + 19 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-\beta_{14} + 8 \beta_{12} - 20 \beta_{10} - 12 \beta_{9} + 4 \beta_{8} - 44 \beta_{7} + 46 \beta_{6} + 10 \beta_{4} - 24 \beta_{3} + 73$$ $$\nu^{7}$$ $$=$$ $$46 \beta_{19} - 123 \beta_{18} - 67 \beta_{17} - 9 \beta_{16} - 14 \beta_{15} - 47 \beta_{13} - 11 \beta_{11} + 122 \beta_{5} + 26 \beta_{2} + 98 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-17 \beta_{14} + 52 \beta_{12} - 152 \beta_{10} - 103 \beta_{9} + 54 \beta_{8} - 270 \beta_{7} + 304 \beta_{6} + 81 \beta_{4} - 220 \beta_{3} + 405$$ $$\nu^{9}$$ $$=$$ $$304 \beta_{19} - 846 \beta_{18} - 471 \beta_{17} - 54 \beta_{16} - 146 \beta_{15} - 408 \beta_{13} - 100 \beta_{11} + 830 \beta_{5} + 240 \beta_{2} + 540 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-186 \beta_{14} + 313 \beta_{12} - 1052 \beta_{10} - 779 \beta_{9} + 526 \beta_{8} - 1655 \beta_{7} + 2030 \beta_{6} + 617 \beta_{4} - 1819 \beta_{3} + 2349$$ $$\nu^{11}$$ $$=$$ $$2030 \beta_{19} - 5702 \beta_{18} - 3237 \beta_{17} - 249 \beta_{16} - 1344 \beta_{15} - 3179 \beta_{13} - 854 \beta_{11} + 5525 \beta_{5} + 1950 \beta_{2} + 3141 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-1701 \beta_{14} + 1798 \beta_{12} - 7003 \beta_{10} - 5562 \beta_{9} + 4505 \beta_{8} - 10202 \beta_{7} + 13677 \beta_{6} + 4581 \beta_{4} - 14274 \beta_{3} + 14085$$ $$\nu^{13}$$ $$=$$ $$13677 \beta_{19} - 38145 \beta_{18} - 22046 \beta_{17} - 721 \beta_{16} - 11495 \beta_{15} - 23602 \beta_{13} - 7011 \beta_{11} + 36456 \beta_{5} + 14910 \beta_{2} + 19058 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-14189 \beta_{14} + 9926 \beta_{12} - 45869 \beta_{10} - 38637 \beta_{9} + 36110 \beta_{8} - 63371 \beta_{7} + 92801 \beta_{6} + 33541 \beta_{4} - 108627 \beta_{3} + 86607$$ $$\nu^{15}$$ $$=$$ $$92801 \beta_{19} - 254867 \beta_{18} - 149738 \beta_{17} + 1861 \beta_{16} - 93605 \beta_{15} - 171087 \beta_{13} - 55801 \beta_{11} + 240006 \beta_{5} + 110357 \beta_{2} + 119350 \beta_{1}$$ $$\nu^{16}$$ $$=$$ $$-112218 \beta_{14} + 52464 \beta_{12} - 298875 \beta_{10} - 265024 \beta_{9} + 278376 \beta_{8} - 396876 \beta_{7} + 633176 \beta_{6} + 243343 \beta_{4} - 810400 \beta_{3} + 542927$$ $$\nu^{17}$$ $$=$$ $$633176 \beta_{19} - 1706169 \beta_{18} - 1017461 \beta_{17} + 56686 \beta_{16} - 736255 \beta_{15} - 1224561 \beta_{13} - 433158 \beta_{11} + 1582170 \beta_{5} + 801687 \beta_{2} + 765010 \beta_{1}$$ $$\nu^{18}$$ $$=$$ $$-858373 \beta_{14} + 261612 \beta_{12} - 1948358 \beta_{10} - 1808864 \beta_{9} + 2093218 \beta_{8} - 2506268 \beta_{7} + 4339200 \beta_{6} + 1753716 \beta_{4} - 5962953 \beta_{3} + 3455823$$ $$\nu^{19}$$ $$=$$ $$4339200 \beta_{19} - 11461063 \beta_{18} - 6927221 \beta_{17} + 663731 \beta_{16} - 5646193 \beta_{15} - 8702714 \beta_{13} - 3295716 \beta_{11} + 10465029 \beta_{5} + 5758322 \beta_{2} + 4988292 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.64610 −2.42479 −2.02007 −1.88727 −1.86133 −1.36200 −0.805985 −0.666460 −0.200809 −0.149497 0.149497 0.200809 0.666460 0.805985 1.36200 1.86133 1.88727 2.02007 2.42479 2.64610
−2.64610 0.666460 5.00186 2.47875 −1.76352 1.01239 −7.94323 −2.55583 −6.55903
1.2 −2.42479 1.86133 3.87962 −2.06206 −4.51334 4.60517 −4.55770 0.464543 5.00007
1.3 −2.02007 0.200809 2.08069 1.28791 −0.405649 −0.549923 −0.163010 −2.95968 −2.60166
1.4 −1.88727 0.805985 1.56178 0.126968 −1.52111 1.51778 0.827047 −2.35039 −0.239623
1.5 −1.86133 2.42479 1.46454 −1.27440 −4.51334 −1.26870 0.996660 2.87962 2.37209
1.6 −1.36200 −0.149497 −0.144951 −3.99648 0.203616 3.68871 2.92143 −2.97765 5.44321
1.7 −0.805985 1.88727 −1.35039 −1.04963 −1.52111 −0.595112 2.70036 0.561775 0.845990
1.8 −0.666460 2.64610 −1.55583 −3.91294 −1.76352 0.421793 2.36982 4.00186 2.60781
1.9 −0.200809 2.02007 −1.95968 1.96610 −0.405649 −2.70408 0.795139 1.08069 −0.394811
1.10 −0.149497 −1.36200 −1.97765 3.17090 0.203616 −3.47866 0.594648 −1.14495 −0.474041
1.11 0.149497 1.36200 −1.97765 −3.17090 0.203616 3.47866 −0.594648 −1.14495 −0.474041
1.12 0.200809 −2.02007 −1.95968 −1.96610 −0.405649 2.70408 −0.795139 1.08069 −0.394811
1.13 0.666460 −2.64610 −1.55583 3.91294 −1.76352 −0.421793 −2.36982 4.00186 2.60781
1.14 0.805985 −1.88727 −1.35039 1.04963 −1.52111 0.595112 −2.70036 0.561775 0.845990
1.15 1.36200 0.149497 −0.144951 3.99648 0.203616 −3.68871 −2.92143 −2.97765 5.44321
1.16 1.86133 −2.42479 1.46454 1.27440 −4.51334 1.26870 −0.996660 2.87962 2.37209
1.17 1.88727 −0.805985 1.56178 −0.126968 −1.52111 −1.51778 −0.827047 −2.35039 −0.239623
1.18 2.02007 −0.200809 2.08069 −1.28791 −0.405649 0.549923 0.163010 −2.95968 −2.60166
1.19 2.42479 −1.86133 3.87962 2.06206 −4.51334 −4.60517 4.55770 0.464543 5.00007
1.20 2.64610 −0.666460 5.00186 −2.47875 −1.76352 −1.01239 7.94323 −2.55583 −6.55903
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$43$$ $$1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1849.2.a.q 20
43.b odd 2 1 inner 1849.2.a.q 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1849.2.a.q 20 1.a even 1 1 trivial
1849.2.a.q 20 43.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{20} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1849))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 13 T^{2} + 88 T^{4} + 421 T^{6} + 1601 T^{8} + 5137 T^{10} + 14485 T^{12} + 36989 T^{14} + 87208 T^{16} + 191889 T^{18} + 396021 T^{20} + 767556 T^{22} + 1395328 T^{24} + 2367296 T^{26} + 3708160 T^{28} + 5260288 T^{30} + 6557696 T^{32} + 6897664 T^{34} + 5767168 T^{36} + 3407872 T^{38} + 1048576 T^{40}$$
$3$ $$1 + 33 T^{2} + 552 T^{4} + 6226 T^{6} + 53068 T^{8} + 363000 T^{10} + 2065709 T^{12} + 10006931 T^{14} + 41887653 T^{16} + 152934639 T^{18} + 489574513 T^{20} + 1376411751 T^{22} + 3392899893 T^{24} + 7295052699 T^{26} + 13553116749 T^{28} + 21434787000 T^{30} + 28202510988 T^{32} + 29778764994 T^{34} + 23761789992 T^{36} + 12784876137 T^{38} + 3486784401 T^{40}$$
$5$ $$1 + 40 T^{2} + 816 T^{4} + 11301 T^{6} + 120252 T^{8} + 1058397 T^{10} + 8091780 T^{12} + 55414621 T^{14} + 345351343 T^{16} + 1969780626 T^{18} + 10291287491 T^{20} + 49244515650 T^{22} + 215844589375 T^{24} + 865853453125 T^{26} + 3160851562500 T^{28} + 10335908203125 T^{30} + 29358398437500 T^{32} + 68975830078125 T^{34} + 124511718750000 T^{36} + 152587890625000 T^{38} + 95367431640625 T^{40}$$
$7$ $$1 + 80 T^{2} + 3128 T^{4} + 80110 T^{6} + 1519434 T^{8} + 22863527 T^{10} + 285235077 T^{12} + 3041104389 T^{14} + 28315900111 T^{16} + 233769578782 T^{18} + 1727104571291 T^{20} + 11454709360318 T^{22} + 67986476166511 T^{24} + 357782890261461 T^{26} + 1644323457124677 T^{28} + 6458380482343223 T^{30} + 21030922376964234 T^{32} + 54332450365933390 T^{34} + 103952606821711928 T^{36} + 130273087832835920 T^{38} + 79792266297612001 T^{40}$$
$11$ $$( 1 + 10 T + 89 T^{2} + 560 T^{3} + 3312 T^{4} + 16609 T^{5} + 78717 T^{6} + 333233 T^{7} + 1335269 T^{8} + 4868709 T^{9} + 16857633 T^{10} + 53555799 T^{11} + 161567549 T^{12} + 443533123 T^{13} + 1152495597 T^{14} + 2674896059 T^{15} + 5867410032 T^{16} + 10912815760 T^{17} + 19077940409 T^{18} + 23579476910 T^{19} + 25937424601 T^{20} )^{2}$$
$13$ $$( 1 - 3 T + 62 T^{2} - 191 T^{3} + 1868 T^{4} - 5937 T^{5} + 40013 T^{6} - 126293 T^{7} + 713982 T^{8} - 2075216 T^{9} + 10448043 T^{10} - 26977808 T^{11} + 120662958 T^{12} - 277465721 T^{13} + 1142811293 T^{14} - 2204366541 T^{15} + 9016479212 T^{16} - 11984966747 T^{17} + 50575304702 T^{18} - 31813498119 T^{19} + 137858491849 T^{20} )^{2}$$
$17$ $$( 1 + 4 T + 120 T^{2} + 423 T^{3} + 6791 T^{4} + 21150 T^{5} + 244131 T^{6} + 671314 T^{7} + 6280349 T^{8} + 15217026 T^{9} + 121947849 T^{10} + 258689442 T^{11} + 1815020861 T^{12} + 3298165682 T^{13} + 20390065251 T^{14} + 30029975550 T^{15} + 163918231079 T^{16} + 173573258679 T^{17} + 837090892920 T^{18} + 474351505988 T^{19} + 2015993900449 T^{20} )^{2}$$
$19$ $$1 + 107 T^{2} + 7227 T^{4} + 360211 T^{6} + 14713794 T^{8} + 510883835 T^{10} + 15543473190 T^{12} + 420092793969 T^{14} + 10206899333785 T^{16} + 224136189553522 T^{18} + 4468613587347447 T^{20} + 80913164428821442 T^{22} + 1330173328078194985 T^{24} + 19763635594023091689 T^{26} +$$$$26\!\cdots\!90$$$$T^{28} +$$$$31\!\cdots\!35$$$$T^{30} +$$$$32\!\cdots\!34$$$$T^{32} +$$$$28\!\cdots\!31$$$$T^{34} +$$$$20\!\cdots\!87$$$$T^{36} +$$$$11\!\cdots\!87$$$$T^{38} +$$$$37\!\cdots\!01$$$$T^{40}$$
$23$ $$( 1 + 23 T + 396 T^{2} + 4883 T^{3} + 50830 T^{4} + 443047 T^{5} + 3415281 T^{6} + 23180741 T^{7} + 142307418 T^{8} + 785568948 T^{9} + 3960909963 T^{10} + 18068085804 T^{11} + 75280624122 T^{12} + 282040075747 T^{13} + 955735650321 T^{14} + 2851602457121 T^{15} + 7524664237870 T^{16} + 16625762657701 T^{17} + 31011150171276 T^{18} + 41426511213649 T^{19} + 41426511213649 T^{20} )^{2}$$
$29$ $$1 + 355 T^{2} + 61637 T^{4} + 7010484 T^{6} + 590034042 T^{8} + 39313078799 T^{10} + 2162612821331 T^{12} + 100939658183434 T^{14} + 4068834289792210 T^{16} + 143205948796309798 T^{18} + 4427082547596807219 T^{20} +$$$$12\!\cdots\!18$$$$T^{22} +$$$$28\!\cdots\!10$$$$T^{24} +$$$$60\!\cdots\!14$$$$T^{26} +$$$$10\!\cdots\!91$$$$T^{28} +$$$$16\!\cdots\!99$$$$T^{30} +$$$$20\!\cdots\!22$$$$T^{32} +$$$$20\!\cdots\!04$$$$T^{34} +$$$$15\!\cdots\!77$$$$T^{36} +$$$$74\!\cdots\!55$$$$T^{38} +$$$$17\!\cdots\!01$$$$T^{40}$$
$31$ $$( 1 + 19 T + 341 T^{2} + 3954 T^{3} + 42555 T^{4} + 364364 T^{5} + 2954570 T^{6} + 20513682 T^{7} + 137929044 T^{8} + 823585701 T^{9} + 4835787010 T^{10} + 25531156731 T^{11} + 132549811284 T^{12} + 611123100462 T^{13} + 2728607440970 T^{14} + 10431431974964 T^{15} + 37767719144955 T^{16} + 108784876194894 T^{17} + 290835843767381 T^{18} + 502352821052749 T^{19} + 819628286980801 T^{20} )^{2}$$
$37$ $$1 + 544 T^{2} + 145924 T^{4} + 25610022 T^{6} + 3292942466 T^{8} + 329356598647 T^{10} + 26564739745913 T^{12} + 1768153360037473 T^{14} + 98591681997173059 T^{16} + 4649349777875033374 T^{18} +$$$$18\!\cdots\!27$$$$T^{20} +$$$$63\!\cdots\!06$$$$T^{22} +$$$$18\!\cdots\!99$$$$T^{24} +$$$$45\!\cdots\!57$$$$T^{26} +$$$$93\!\cdots\!73$$$$T^{28} +$$$$15\!\cdots\!03$$$$T^{30} +$$$$21\!\cdots\!46$$$$T^{32} +$$$$23\!\cdots\!58$$$$T^{34} +$$$$18\!\cdots\!84$$$$T^{36} +$$$$91\!\cdots\!76$$$$T^{38} +$$$$23\!\cdots\!01$$$$T^{40}$$
$41$ $$( 1 + 28 T + 644 T^{2} + 10141 T^{3} + 139888 T^{4} + 1586005 T^{5} + 16258560 T^{6} + 145249239 T^{7} + 1191564631 T^{8} + 8716554418 T^{9} + 58920438831 T^{10} + 357378731138 T^{11} + 2003020144711 T^{12} + 10010722801119 T^{13} + 45942804764160 T^{14} + 183748514067005 T^{15} + 664482582065008 T^{16} + 1975003091427221 T^{17} + 5142291847553924 T^{18} + 9166694163030908 T^{19} + 13422659310152401 T^{20} )^{2}$$
$43$ 1
$47$ $$( 1 + 12 T + 339 T^{2} + 3526 T^{3} + 56816 T^{4} + 511010 T^{5} + 6080868 T^{6} + 47412308 T^{7} + 456152128 T^{8} + 3078600974 T^{9} + 24961460139 T^{10} + 144694245778 T^{11} + 1007640050752 T^{12} + 4922488053484 T^{13} + 29672696043108 T^{14} + 117197592027070 T^{15} + 612431898132464 T^{16} + 1786353122752538 T^{17} + 8072026178336979 T^{18} + 13429565677233204 T^{19} + 52599132235830049 T^{20} )^{2}$$
$53$ $$( 1 + 21 T + 498 T^{2} + 6309 T^{3} + 87582 T^{4} + 825969 T^{5} + 8991377 T^{6} + 72779787 T^{7} + 699192874 T^{8} + 5071930248 T^{9} + 42659895983 T^{10} + 268812303144 T^{11} + 1964032783066 T^{12} + 10835236349199 T^{13} + 70946289382337 T^{14} + 345416513157717 T^{15} + 1941199076400078 T^{16} + 7411252581231633 T^{17} + 31005325824857778 T^{18} + 69295035427844793 T^{19} + 174887470365513049 T^{20} )^{2}$$
$59$ $$( 1 + 45 T + 1251 T^{2} + 25722 T^{3} + 430204 T^{4} + 6101381 T^{5} + 75494762 T^{6} + 828309220 T^{7} + 8153878001 T^{8} + 72482217211 T^{9} + 584374584815 T^{10} + 4276450815449 T^{11} + 28383649321481 T^{12} + 170117319294380 T^{13} + 914797284763082 T^{14} + 4362025534356919 T^{15} + 18146234294492764 T^{16} + 64013093492514318 T^{17} + 183684877443005571 T^{18} + 389834811839472255 T^{19} + 511116753300641401 T^{20} )^{2}$$
$61$ $$1 + 790 T^{2} + 309299 T^{4} + 80006711 T^{6} + 15356484539 T^{8} + 2325961684078 T^{10} + 288434648480662 T^{12} + 29976458677407681 T^{14} + 2650762140208233663 T^{16} +$$$$20\!\cdots\!90$$$$T^{18} +$$$$13\!\cdots\!81$$$$T^{20} +$$$$74\!\cdots\!90$$$$T^{22} +$$$$36\!\cdots\!83$$$$T^{24} +$$$$15\!\cdots\!41$$$$T^{26} +$$$$55\!\cdots\!22$$$$T^{28} +$$$$16\!\cdots\!78$$$$T^{30} +$$$$40\!\cdots\!19$$$$T^{32} +$$$$79\!\cdots\!51$$$$T^{34} +$$$$11\!\cdots\!39$$$$T^{36} +$$$$10\!\cdots\!90$$$$T^{38} +$$$$50\!\cdots\!01$$$$T^{40}$$
$67$ $$( 1 + 31 T + 839 T^{2} + 14659 T^{3} + 233414 T^{4} + 2954891 T^{5} + 35492860 T^{6} + 366766920 T^{7} + 3681988473 T^{8} + 32750116631 T^{9} + 284096631401 T^{10} + 2194257814277 T^{11} + 16528446255297 T^{12} + 110309919159960 T^{13} + 715220916496060 T^{14} + 3989472527548337 T^{15} + 21114252815594966 T^{16} + 88843971422429857 T^{17} + 340690781470021799 T^{18} + 843402566285143357 T^{19} + 1822837804551761449 T^{20} )^{2}$$
$71$ $$1 + 883 T^{2} + 375953 T^{4} + 102911488 T^{6} + 20417028278 T^{8} + 3145772941707 T^{10} + 394935886833835 T^{12} + 41937626625065310 T^{14} + 3880901102881147810 T^{16} +$$$$32\!\cdots\!10$$$$T^{18} +$$$$23\!\cdots\!51$$$$T^{20} +$$$$16\!\cdots\!10$$$$T^{22} +$$$$98\!\cdots\!10$$$$T^{24} +$$$$53\!\cdots\!10$$$$T^{26} +$$$$25\!\cdots\!35$$$$T^{28} +$$$$10\!\cdots\!07$$$$T^{30} +$$$$33\!\cdots\!98$$$$T^{32} +$$$$85\!\cdots\!28$$$$T^{34} +$$$$15\!\cdots\!13$$$$T^{36} +$$$$18\!\cdots\!63$$$$T^{38} +$$$$10\!\cdots\!01$$$$T^{40}$$
$73$ $$1 + 686 T^{2} + 236929 T^{4} + 55292228 T^{6} + 9832089880 T^{8} + 1421510767666 T^{10} + 173988005462946 T^{12} + 18523591874249588 T^{14} + 1747963825372995868 T^{16} +$$$$14\!\cdots\!98$$$$T^{18} +$$$$11\!\cdots\!53$$$$T^{20} +$$$$78\!\cdots\!42$$$$T^{22} +$$$$49\!\cdots\!88$$$$T^{24} +$$$$28\!\cdots\!32$$$$T^{26} +$$$$14\!\cdots\!26$$$$T^{28} +$$$$61\!\cdots\!34$$$$T^{30} +$$$$22\!\cdots\!80$$$$T^{32} +$$$$67\!\cdots\!52$$$$T^{34} +$$$$15\!\cdots\!69$$$$T^{36} +$$$$23\!\cdots\!34$$$$T^{38} +$$$$18\!\cdots\!01$$$$T^{40}$$
$79$ $$( 1 + 5 T + 507 T^{2} + 1337 T^{3} + 119796 T^{4} + 92825 T^{5} + 18138748 T^{6} - 10048960 T^{7} + 2022931171 T^{8} - 2401078215 T^{9} + 178030821825 T^{10} - 189685178985 T^{11} + 12625113438211 T^{12} - 4954529189440 T^{13} + 706505703838588 T^{14} + 285627760237175 T^{15} + 29120904821593716 T^{16} + 25675626314494583 T^{17} + 769174166622626427 T^{18} + 599257979913091595 T^{19} + 9468276082626847201 T^{20} )^{2}$$
$83$ $$( 1 + 34 T + 1205 T^{2} + 26070 T^{3} + 543584 T^{4} + 8719411 T^{5} + 133183791 T^{6} + 1688637709 T^{7} + 20356364785 T^{8} + 209959755927 T^{9} + 2059284856595 T^{10} + 17426659741941 T^{11} + 140234997003865 T^{12} + 965541089715983 T^{13} + 6320679105274911 T^{14} + 34346114312021273 T^{15} + 177719555917414496 T^{16} + 707436849299575890 T^{17} + 2714012139727544405 T^{18} + 6355968679096373702 T^{19} + 15516041187205853449 T^{20} )^{2}$$
$89$ $$1 + 732 T^{2} + 290380 T^{4} + 81076078 T^{6} + 17694037714 T^{8} + 3187529936855 T^{10} + 489734789644473 T^{12} + 65519895614350941 T^{14} + 7737018844136455179 T^{16} +$$$$81\!\cdots\!50$$$$T^{18} +$$$$76\!\cdots\!55$$$$T^{20} +$$$$64\!\cdots\!50$$$$T^{22} +$$$$48\!\cdots\!39$$$$T^{24} +$$$$32\!\cdots\!01$$$$T^{26} +$$$$19\!\cdots\!13$$$$T^{28} +$$$$99\!\cdots\!55$$$$T^{30} +$$$$43\!\cdots\!94$$$$T^{32} +$$$$15\!\cdots\!98$$$$T^{34} +$$$$44\!\cdots\!80$$$$T^{36} +$$$$89\!\cdots\!92$$$$T^{38} +$$$$97\!\cdots\!01$$$$T^{40}$$
$97$ $$( 1 + 22 T + 937 T^{2} + 16324 T^{3} + 392836 T^{4} + 5615225 T^{5} + 98111241 T^{6} + 1173560157 T^{7} + 16271043365 T^{8} + 164104753461 T^{9} + 1878652272815 T^{10} + 15918161085717 T^{11} + 153094247021285 T^{12} + 1071076669169661 T^{13} + 8685717623747721 T^{14} + 48219847694612825 T^{15} + 327221390528288644 T^{16} + 1318951195820716612 T^{17} + 7343675277931212457 T^{18} + 16725083290400434774 T^{19} + 73742412689492826049 T^{20} )^{2}$$