# Properties

 Label 432.2.u.d Level $432$ Weight $2$ Character orbit 432.u Analytic conductor $3.450$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 432.u (of order $$9$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.44953736732$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$3$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ Defining polynomial: $$x^{18} - 6 x^{17} + 24 x^{16} - 66 x^{15} + 153 x^{14} - 315 x^{13} + 651 x^{12} - 1350 x^{11} + 2700 x^{10} - 4941 x^{9} + 8100 x^{8} - 12150 x^{7} + 17577 x^{6} - 25515 x^{5} + 37179 x^{4} - 48114 x^{3} + 52488 x^{2} - 39366 x + 19683$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: no (minimal twist has level 108) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{12} q^{3} + ( -1 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{16} ) q^{5} + ( -\beta_{1} - \beta_{2} - \beta_{7} - \beta_{9} - \beta_{13} - \beta_{14} + \beta_{16} ) q^{7} + ( \beta_{3} - \beta_{4} - \beta_{8} - \beta_{9} - \beta_{12} + \beta_{13} + \beta_{15} + \beta_{17} ) q^{9} +O(q^{10})$$ $$q -\beta_{12} q^{3} + ( -1 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{16} ) q^{5} + ( -\beta_{1} - \beta_{2} - \beta_{7} - \beta_{9} - \beta_{13} - \beta_{14} + \beta_{16} ) q^{7} + ( \beta_{3} - \beta_{4} - \beta_{8} - \beta_{9} - \beta_{12} + \beta_{13} + \beta_{15} + \beta_{17} ) q^{9} + ( \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{9} - \beta_{10} + \beta_{13} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{11} + ( -\beta_{1} + \beta_{4} + \beta_{5} - \beta_{10} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{13} + ( -1 + \beta_{1} + 2 \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{16} ) q^{15} + ( -1 - 2 \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{17} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - 2 \beta_{16} + 2 \beta_{17} ) q^{19} + ( \beta_{4} + 2 \beta_{5} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} ) q^{21} + ( 2 + \beta_{1} + 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{17} ) q^{23} + ( 1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{5} - 2 \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{16} + \beta_{17} ) q^{25} + ( 2 + \beta_{1} - \beta_{3} + 2 \beta_{5} + \beta_{6} + 4 \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{27} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{15} - \beta_{16} ) q^{29} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{16} - \beta_{17} ) q^{31} + ( -1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + 5 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} + 2 \beta_{12} - \beta_{13} - 2 \beta_{14} - 2 \beta_{15} + \beta_{16} - \beta_{17} ) q^{33} + ( -3 \beta_{1} - \beta_{2} - 4 \beta_{3} + 4 \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{15} + \beta_{16} ) q^{35} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - 3 \beta_{6} + \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{14} + \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{37} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + \beta_{13} - \beta_{14} + \beta_{16} - \beta_{17} ) q^{39} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + \beta_{6} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - \beta_{14} + \beta_{17} ) q^{41} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} - \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{43} + ( \beta_{1} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + 4 \beta_{9} + \beta_{10} + 4 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} - 2 \beta_{16} ) q^{45} + ( -5 + 3 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + 4 \beta_{11} - 2 \beta_{12} - \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{47} + ( -2 - \beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} - 2 \beta_{14} - \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{49} + ( -2 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} - 3 \beta_{9} + 3 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} + \beta_{16} ) q^{51} + ( 5 + \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} - 4 \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + 3 \beta_{15} - \beta_{17} ) q^{53} + ( 3 \beta_{2} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - 4 \beta_{8} + \beta_{9} + 2 \beta_{10} - 4 \beta_{12} + 2 \beta_{13} + \beta_{14} - 3 \beta_{16} + 3 \beta_{17} ) q^{55} + ( 4 + 2 \beta_{1} + \beta_{2} + 4 \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} + 3 \beta_{14} + \beta_{15} - \beta_{17} ) q^{57} + ( -3 + \beta_{1} - \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + 5 \beta_{9} - 2 \beta_{11} - \beta_{12} + \beta_{14} + \beta_{16} - 2 \beta_{17} ) q^{59} + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - \beta_{13} - 2 \beta_{14} - \beta_{15} - \beta_{16} ) q^{61} + ( -4 - \beta_{1} - 3 \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{6} + 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{63} + ( \beta_{2} + 5 \beta_{3} + \beta_{4} + \beta_{5} + 4 \beta_{6} + 3 \beta_{7} - 5 \beta_{8} - \beta_{9} + 3 \beta_{11} + 2 \beta_{13} - \beta_{15} - 2 \beta_{16} + 2 \beta_{17} ) q^{65} + ( 4 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - 3 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{67} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - 4 \beta_{5} + \beta_{6} - \beta_{9} - \beta_{10} - 4 \beta_{11} - \beta_{13} + \beta_{14} - 2 \beta_{15} - \beta_{16} - 2 \beta_{17} ) q^{69} + ( 2 - 2 \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{8} - 5 \beta_{9} - \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} - 3 \beta_{17} ) q^{71} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{73} + ( 3 + 6 \beta_{3} - 6 \beta_{5} - 6 \beta_{8} - 6 \beta_{9} + \beta_{10} - 2 \beta_{12} + 3 \beta_{15} + 2 \beta_{16} ) q^{75} + ( -5 + \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} - 5 \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} + 3 \beta_{13} + 2 \beta_{14} + \beta_{15} - 2 \beta_{16} ) q^{77} + ( 3 - 3 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} + \beta_{16} + \beta_{17} ) q^{79} + ( -5 + 2 \beta_{1} + \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{9} + 2 \beta_{11} - 3 \beta_{12} + \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{81} + ( -3 - 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - \beta_{8} + 5 \beta_{9} - \beta_{10} + 6 \beta_{11} + 2 \beta_{13} - \beta_{15} - 2 \beta_{16} + 2 \beta_{17} ) q^{83} + ( -1 - \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} - 2 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} - 3 \beta_{13} - \beta_{15} + 2 \beta_{16} ) q^{85} + ( 5 - 3 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 6 \beta_{11} + 2 \beta_{12} - 2 \beta_{15} + \beta_{16} ) q^{87} + ( 2 - 4 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} - 4 \beta_{10} - 6 \beta_{11} + 3 \beta_{12} - \beta_{13} - 2 \beta_{14} - \beta_{15} + 3 \beta_{16} - 2 \beta_{17} ) q^{89} + ( -2 + \beta_{3} - \beta_{5} - 3 \beta_{6} - 2 \beta_{8} + \beta_{9} + 2 \beta_{11} + 3 \beta_{13} + 3 \beta_{14} ) q^{91} + ( -3 - \beta_{1} - 3 \beta_{2} + 3 \beta_{4} + \beta_{7} - 3 \beta_{8} - \beta_{10} + 6 \beta_{11} + 2 \beta_{12} + 2 \beta_{16} - 3 \beta_{17} ) q^{93} + ( -2 + \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{5} + \beta_{6} + 3 \beta_{7} - \beta_{8} - 3 \beta_{9} + 2 \beta_{10} + 5 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{16} - \beta_{17} ) q^{95} + ( 1 - 3 \beta_{1} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{17} ) q^{97} + ( 1 + 3 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{7} - 3 \beta_{8} + \beta_{9} + \beta_{10} - 4 \beta_{11} + \beta_{12} + \beta_{15} - 3 \beta_{16} - \beta_{17} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18q + 3q^{5} + 6q^{9} + O(q^{10})$$ $$18q + 3q^{5} + 6q^{9} - 3q^{11} + 9q^{15} - 12q^{17} - 30q^{21} + 30q^{23} + 9q^{25} + 27q^{27} - 24q^{29} - 9q^{31} - 18q^{33} + 21q^{35} - 3q^{39} + 21q^{41} + 9q^{43} + 45q^{45} - 45q^{47} - 18q^{49} - 63q^{51} + 66q^{53} + 54q^{57} - 60q^{59} - 18q^{61} - 57q^{63} + 33q^{65} + 27q^{67} - 9q^{69} + 12q^{71} + 9q^{73} + 33q^{75} - 75q^{77} + 36q^{79} - 54q^{81} + 45q^{83} - 36q^{85} + 63q^{87} - 48q^{89} - 9q^{91} - 33q^{93} - 6q^{95} - 27q^{97} - 27q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 6 x^{17} + 24 x^{16} - 66 x^{15} + 153 x^{14} - 315 x^{13} + 651 x^{12} - 1350 x^{11} + 2700 x^{10} - 4941 x^{9} + 8100 x^{8} - 12150 x^{7} + 17577 x^{6} - 25515 x^{5} + 37179 x^{4} - 48114 x^{3} + 52488 x^{2} - 39366 x + 19683$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-157 \nu^{17} - 5979 \nu^{16} + 22788 \nu^{15} - 73797 \nu^{14} + 149931 \nu^{13} - 307746 \nu^{12} + 623499 \nu^{11} - 1350459 \nu^{10} + 2829996 \nu^{9} - 5104836 \nu^{8} + 8231571 \nu^{7} - 11833533 \nu^{6} + 16667937 \nu^{5} - 26003430 \nu^{4} + 37329174 \nu^{3} - 52566732 \nu^{2} + 44096481 \nu - 32102973$$$$)/1174419$$ $$\beta_{3}$$ $$=$$ $$($$$$256 \nu^{17} - 252 \nu^{16} + 819 \nu^{15} - 2946 \nu^{14} + 8433 \nu^{13} - 18684 \nu^{12} + 33978 \nu^{11} - 68400 \nu^{10} + 137493 \nu^{9} - 269973 \nu^{8} + 501201 \nu^{7} - 754920 \nu^{6} + 1160973 \nu^{5} - 1495179 \nu^{4} + 2248965 \nu^{3} - 3514509 \nu^{2} + 4599261 \nu - 3726648$$$$)/1174419$$ $$\beta_{4}$$ $$=$$ $$($$$$491 \nu^{17} - 3462 \nu^{16} + 16353 \nu^{15} - 43899 \nu^{14} + 97020 \nu^{13} - 191322 \nu^{12} + 396267 \nu^{11} - 838116 \nu^{10} + 1674207 \nu^{9} - 3008475 \nu^{8} + 4739580 \nu^{7} - 6817527 \nu^{6} + 9704286 \nu^{5} - 14081607 \nu^{4} + 21038211 \nu^{3} - 25679754 \nu^{2} + 24065748 \nu - 11881971$$$$)/1174419$$ $$\beta_{5}$$ $$=$$ $$($$$$-463 \nu^{17} - 1365 \nu^{16} + 5913 \nu^{15} - 26250 \nu^{14} + 52794 \nu^{13} - 112482 \nu^{12} + 214746 \nu^{11} - 471456 \nu^{10} + 1038627 \nu^{9} - 1951695 \nu^{8} + 3283119 \nu^{7} - 4760937 \nu^{6} + 6449301 \nu^{5} - 10175868 \nu^{4} + 14715594 \nu^{3} - 23180013 \nu^{2} + 20312856 \nu - 17445699$$$$)/1174419$$ $$\beta_{6}$$ $$=$$ $$($$$$160 \nu^{17} + 1006 \nu^{16} - 2643 \nu^{15} + 13329 \nu^{14} - 26211 \nu^{13} + 61623 \nu^{12} - 113685 \nu^{11} + 238638 \nu^{10} - 513963 \nu^{9} + 949905 \nu^{8} - 1676133 \nu^{7} + 2437641 \nu^{6} - 3339549 \nu^{5} + 5149656 \nu^{4} - 6733773 \nu^{3} + 11553921 \nu^{2} - 10093005 \nu + 11763873$$$$)/391473$$ $$\beta_{7}$$ $$=$$ $$($$$$428 \nu^{17} - 1775 \nu^{16} + 4650 \nu^{15} - 10245 \nu^{14} + 20652 \nu^{13} - 44226 \nu^{12} + 92400 \nu^{11} - 184569 \nu^{10} + 331641 \nu^{9} - 524133 \nu^{8} + 785160 \nu^{7} - 1112913 \nu^{6} + 1678887 \nu^{5} - 2422953 \nu^{4} + 2934225 \nu^{3} - 2945889 \nu^{2} + 2117016 \nu - 1679616$$$$)/391473$$ $$\beta_{8}$$ $$=$$ $$($$$$-1330 \nu^{17} + 5085 \nu^{16} - 18540 \nu^{15} + 38388 \nu^{14} - 80928 \nu^{13} + 154008 \nu^{12} - 336846 \nu^{11} + 701523 \nu^{10} - 1287747 \nu^{9} + 2121012 \nu^{8} - 2980530 \nu^{7} + 4118688 \nu^{6} - 6206625 \nu^{5} + 8889669 \nu^{4} - 13340700 \nu^{3} + 10952496 \nu^{2} - 8122518 \nu - 4494285$$$$)/1174419$$ $$\beta_{9}$$ $$=$$ $$($$$$-1537 \nu^{17} + 10986 \nu^{16} - 39195 \nu^{15} + 107463 \nu^{14} - 225909 \nu^{13} + 467478 \nu^{12} - 948396 \nu^{11} + 2011095 \nu^{10} - 3987693 \nu^{9} + 7047351 \nu^{8} - 11172384 \nu^{7} + 16001631 \nu^{6} - 23041665 \nu^{5} + 34234569 \nu^{4} - 48964014 \nu^{3} + 62552574 \nu^{2} - 54849960 \nu + 36577575$$$$)/1174419$$ $$\beta_{10}$$ $$=$$ $$($$$$-965 \nu^{17} + 4460 \nu^{16} - 16464 \nu^{15} + 40854 \nu^{14} - 88314 \nu^{13} + 176328 \nu^{12} - 364659 \nu^{11} + 767751 \nu^{10} - 1483506 \nu^{9} + 2597490 \nu^{8} - 4013604 \nu^{7} + 5723595 \nu^{6} - 8348427 \nu^{5} + 12035790 \nu^{4} - 17679708 \nu^{3} + 20562174 \nu^{2} - 18950355 \nu + 8726130$$$$)/391473$$ $$\beta_{11}$$ $$=$$ $$($$$$1334 \nu^{17} - 5391 \nu^{16} + 17655 \nu^{15} - 38283 \nu^{14} + 79776 \nu^{13} - 161046 \nu^{12} + 342663 \nu^{11} - 713466 \nu^{10} + 1311669 \nu^{9} - 2164725 \nu^{8} + 3182814 \nu^{7} - 4494771 \nu^{6} + 6815826 \nu^{5} - 9813555 \nu^{4} + 13725855 \nu^{3} - 13178862 \nu^{2} + 10267965 \nu - 2267919$$$$)/391473$$ $$\beta_{12}$$ $$=$$ $$($$$$1381 \nu^{17} - 5675 \nu^{16} + 18936 \nu^{15} - 41211 \nu^{14} + 86109 \nu^{13} - 172053 \nu^{12} + 365502 \nu^{11} - 762909 \nu^{10} + 1413126 \nu^{9} - 2344473 \nu^{8} + 3462129 \nu^{7} - 4862484 \nu^{6} + 7329771 \nu^{5} - 10643157 \nu^{4} + 15152265 \nu^{3} - 14871600 \nu^{2} + 11890719 \nu - 3037743$$$$)/391473$$ $$\beta_{13}$$ $$=$$ $$($$$$1631 \nu^{17} - 6363 \nu^{16} + 21888 \nu^{15} - 45120 \nu^{14} + 94122 \nu^{13} - 183402 \nu^{12} + 393171 \nu^{11} - 818496 \nu^{10} + 1493793 \nu^{9} - 2433690 \nu^{8} + 3466584 \nu^{7} - 4789881 \nu^{6} + 7308711 \nu^{5} - 10419030 \nu^{4} + 15105366 \nu^{3} - 12828213 \nu^{2} + 9552816 \nu + 1421550$$$$)/391473$$ $$\beta_{14}$$ $$=$$ $$($$$$5449 \nu^{17} - 26718 \nu^{16} + 90324 \nu^{15} - 219036 \nu^{14} + 462159 \nu^{13} - 942084 \nu^{12} + 1959078 \nu^{11} - 4112343 \nu^{10} + 7848423 \nu^{9} - 13469004 \nu^{8} + 20649627 \nu^{7} - 29469744 \nu^{6} + 43331436 \nu^{5} - 63418383 \nu^{4} + 90022752 \nu^{3} - 101712996 \nu^{2} + 87294105 \nu - 43906212$$$$)/1174419$$ $$\beta_{15}$$ $$=$$ $$($$$$-6344 \nu^{17} + 41754 \nu^{16} - 141876 \nu^{15} + 376377 \nu^{14} - 781137 \nu^{13} + 1613871 \nu^{12} - 3305337 \nu^{11} + 6968646 \nu^{10} - 13646412 \nu^{9} + 23676300 \nu^{8} - 37052829 \nu^{7} + 52885629 \nu^{6} - 76490649 \nu^{5} + 114092388 \nu^{4} - 160096419 \nu^{3} + 198015354 \nu^{2} - 165980178 \nu + 104976000$$$$)/1174419$$ $$\beta_{16}$$ $$=$$ $$($$$$871 \nu^{17} - 4787 \nu^{16} + 16587 \nu^{15} - 41442 \nu^{14} + 86388 \nu^{13} - 175257 \nu^{12} + 362478 \nu^{11} - 763377 \nu^{10} + 1475523 \nu^{9} - 2540862 \nu^{8} + 3904443 \nu^{7} - 5543964 \nu^{6} + 8074485 \nu^{5} - 11956977 \nu^{4} + 17001738 \nu^{3} - 19917009 \nu^{2} + 16748775 \nu - 8752374$$$$)/130491$$ $$\beta_{17}$$ $$=$$ $$($$$$8860 \nu^{17} - 46278 \nu^{16} + 162486 \nu^{15} - 411825 \nu^{14} + 874917 \nu^{13} - 1784511 \nu^{12} + 3663291 \nu^{11} - 7696872 \nu^{10} + 14887080 \nu^{9} - 25895187 \nu^{8} + 40269852 \nu^{7} - 57501333 \nu^{6} + 83545020 \nu^{5} - 122274198 \nu^{4} + 174478131 \nu^{3} - 208278945 \nu^{2} + 183675195 \nu - 103873752$$$$)/1174419$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{17} + \beta_{16} + \beta_{14} - \beta_{11} + \beta_{8} + \beta_{4} + \beta_{3} - \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{16} + 2 \beta_{14} + \beta_{13} - \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - 1$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} + 2 \beta_{11} - 3 \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + 4 \beta_{5} - \beta_{4} - 3 \beta_{3} - 3$$ $$\nu^{5}$$ $$=$$ $$\beta_{17} - 2 \beta_{16} + \beta_{15} + 2 \beta_{14} + 4 \beta_{13} - 6 \beta_{12} + 3 \beta_{11} + \beta_{9} - \beta_{8} - 3 \beta_{7} - 4 \beta_{6} - 6 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} + \beta_{1} + 2$$ $$\nu^{6}$$ $$=$$ $$3 \beta_{17} - \beta_{16} - 3 \beta_{14} - \beta_{12} - 3 \beta_{11} + 2 \beta_{10} - 6 \beta_{9} - 12 \beta_{8} - 7 \beta_{7} - 5 \beta_{6} - 24 \beta_{5} - 9 \beta_{4} + 6 \beta_{2} + 8 \beta_{1} - 6$$ $$\nu^{7}$$ $$=$$ $$-12 \beta_{17} + 9 \beta_{16} - 6 \beta_{15} - 9 \beta_{14} - 12 \beta_{13} + 27 \beta_{12} - 15 \beta_{11} - 18 \beta_{10} - 9 \beta_{9} + 15 \beta_{8} - 15 \beta_{5} + 6 \beta_{4} - 18 \beta_{3} - 15 \beta_{2} - 9 \beta_{1} + 18$$ $$\nu^{8}$$ $$=$$ $$-3 \beta_{17} - 18 \beta_{16} - 18 \beta_{15} + 6 \beta_{13} + 33 \beta_{12} - 9 \beta_{11} - 12 \beta_{10} + 36 \beta_{9} + 27 \beta_{8} - 9 \beta_{7} + 39 \beta_{5} + 15 \beta_{4} + 3 \beta_{3} - 9 \beta_{2} + 3 \beta_{1} + 24$$ $$\nu^{9}$$ $$=$$ $$-18 \beta_{17} - 18 \beta_{16} - 36 \beta_{15} - 24 \beta_{14} + 15 \beta_{13} + 6 \beta_{12} - 6 \beta_{11} - 15 \beta_{10} - 30 \beta_{9} + 42 \beta_{8} + 54 \beta_{7} + 60 \beta_{6} + 66 \beta_{5} + 27 \beta_{4} + 39 \beta_{3} - 27 \beta_{2} + 15 \beta_{1} - 24$$ $$\nu^{10}$$ $$=$$ $$-27 \beta_{17} + 36 \beta_{16} + 39 \beta_{15} + 66 \beta_{14} + 51 \beta_{13} - 90 \beta_{12} - 51 \beta_{11} + 90 \beta_{10} - 96 \beta_{9} - 60 \beta_{8} + 87 \beta_{7} - 39 \beta_{5} + 66 \beta_{4} + 180 \beta_{3} + 18 \beta_{2} - 36 \beta_{1} + 108$$ $$\nu^{11}$$ $$=$$ $$69 \beta_{17} - 66 \beta_{16} + 69 \beta_{15} + 48 \beta_{14} + 6 \beta_{13} - 45 \beta_{12} + 18 \beta_{11} + 117 \beta_{10} - 21 \beta_{9} - 366 \beta_{8} - 141 \beta_{6} - 144 \beta_{5} - 12 \beta_{4} + 66 \beta_{3} + 147 \beta_{2} + 87 \beta_{1} - 114$$ $$\nu^{12}$$ $$=$$ $$-81 \beta_{17} + 39 \beta_{16} - 36 \beta_{15} - 90 \beta_{14} - 144 \beta_{13} + 210 \beta_{12} + 27 \beta_{11} - 321 \beta_{10} + 189 \beta_{9} + 162 \beta_{8} + 12 \beta_{7} - 57 \beta_{6} - 117 \beta_{5} + 45 \beta_{4} - 324 \beta_{3} + 27 \beta_{2} - 78 \beta_{1} - 306$$ $$\nu^{13}$$ $$=$$ $$243 \beta_{17} - 207 \beta_{16} - 117 \beta_{15} + 72 \beta_{14} + 180 \beta_{13} - 9 \beta_{12} - 243 \beta_{11} - 36 \beta_{10} + 702 \beta_{9} + 243 \beta_{8} - 153 \beta_{7} + 18 \beta_{6} - 261 \beta_{5} - 207 \beta_{4} - 72 \beta_{3} + 324 \beta_{2} - 180 \beta_{1} - 45$$ $$\nu^{14}$$ $$=$$ $$513 \beta_{17} - 621 \beta_{16} - 306 \beta_{15} - 720 \beta_{14} - 27 \beta_{13} + 234 \beta_{12} - 162 \beta_{11} + 99 \beta_{10} - 873 \beta_{9} - 45 \beta_{8} + 684 \beta_{7} + 918 \beta_{6} - 333 \beta_{5} - 459 \beta_{4} - 810 \beta_{3} + 252 \beta_{2} + 99 \beta_{1} + 18$$ $$\nu^{15}$$ $$=$$ $$-54 \beta_{17} + 441 \beta_{16} + 513 \beta_{15} - 495 \beta_{14} - 99 \beta_{13} + 315 \beta_{12} + 234 \beta_{11} + 405 \beta_{10} - 1476 \beta_{9} + 1062 \beta_{8} + 819 \beta_{7} + 351 \beta_{6} - 63 \beta_{5} + 837 \beta_{4} - 549 \beta_{3} - 81 \beta_{2} - 387 \beta_{1} + 2151$$ $$\nu^{16}$$ $$=$$ $$270 \beta_{17} + 675 \beta_{16} + 261 \beta_{15} - 603 \beta_{14} - 423 \beta_{13} - 54 \beta_{12} + 1152 \beta_{11} + 1566 \beta_{10} + 504 \beta_{9} - 819 \beta_{8} - 1341 \beta_{7} - 486 \beta_{6} + 225 \beta_{5} + 1152 \beta_{4} + 2322 \beta_{3} - 378 \beta_{2} + 594 \beta_{1} - 1782$$ $$\nu^{17}$$ $$=$$ $$-3366 \beta_{17} + 4329 \beta_{16} - 693 \beta_{15} - 2061 \beta_{14} - 3339 \beta_{13} + 2160 \beta_{12} - 135 \beta_{11} - 2106 \beta_{10} - 2178 \beta_{9} + 2799 \beta_{8} + 2187 \beta_{7} + 2880 \beta_{6} + 2619 \beta_{5} + 2493 \beta_{4} + 4392 \beta_{3} - 4950 \beta_{2} - 4149 \beta_{1} - 3276$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$$.

 $$n$$ $$271$$ $$325$$ $$353$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{5} - \beta_{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.219955 + 1.71803i 0.472963 − 1.66622i 1.68668 − 0.393823i −0.219955 − 1.71803i 0.472963 + 1.66622i 1.68668 + 0.393823i 1.16555 + 1.28120i 0.960398 − 1.44140i −1.29960 + 1.14501i −1.34999 + 1.08514i 1.20201 + 1.24706i 0.381933 − 1.68942i −1.34999 − 1.08514i 1.20201 − 1.24706i 0.381933 + 1.68942i 1.16555 − 1.28120i 0.960398 + 1.44140i −1.29960 − 1.14501i
0 −1.27282 + 1.17470i 0 0.0952805 0.0346793i 0 −0.165033 0.935950i 0 0.240153 2.99037i 0
49.2 0 1.43334 0.972387i 0 −3.94709 + 1.43662i 0 −0.610312 3.46125i 0 1.10893 2.78752i 0
49.3 0 1.54522 + 0.782494i 0 2.29878 0.836687i 0 0.775345 + 4.39720i 0 1.77541 + 2.41825i 0
97.1 0 −1.27282 1.17470i 0 0.0952805 + 0.0346793i 0 −0.165033 + 0.935950i 0 0.240153 + 2.99037i 0
97.2 0 1.43334 + 0.972387i 0 −3.94709 1.43662i 0 −0.610312 + 3.46125i 0 1.10893 + 2.78752i 0
97.3 0 1.54522 0.782494i 0 2.29878 + 0.836687i 0 0.775345 4.39720i 0 1.77541 2.41825i 0
193.1 0 −1.53346 0.805294i 0 0.583982 3.31193i 0 1.47262 1.23568i 0 1.70300 + 2.46977i 0
193.2 0 −0.409491 + 1.68295i 0 −0.103132 + 0.584890i 0 −2.18780 + 1.83578i 0 −2.66463 1.37830i 0
193.3 0 0.829611 1.52044i 0 −0.399598 + 2.26623i 0 0.715176 0.600104i 0 −1.62349 2.52275i 0
241.1 0 −1.30308 1.14105i 0 −0.761786 + 0.639214i 0 −1.35240 + 0.492232i 0 0.396022 + 2.97375i 0
241.2 0 −1.01939 + 1.40030i 0 1.46957 1.23312i 0 3.86125 1.40538i 0 −0.921685 2.85491i 0
241.3 0 1.73007 + 0.0827666i 0 2.26400 1.89972i 0 −2.50885 + 0.913148i 0 2.98630 + 0.286384i 0
337.1 0 −1.30308 + 1.14105i 0 −0.761786 0.639214i 0 −1.35240 0.492232i 0 0.396022 2.97375i 0
337.2 0 −1.01939 1.40030i 0 1.46957 + 1.23312i 0 3.86125 + 1.40538i 0 −0.921685 + 2.85491i 0
337.3 0 1.73007 0.0827666i 0 2.26400 + 1.89972i 0 −2.50885 0.913148i 0 2.98630 0.286384i 0
385.1 0 −1.53346 + 0.805294i 0 0.583982 + 3.31193i 0 1.47262 + 1.23568i 0 1.70300 2.46977i 0
385.2 0 −0.409491 1.68295i 0 −0.103132 0.584890i 0 −2.18780 1.83578i 0 −2.66463 + 1.37830i 0
385.3 0 0.829611 + 1.52044i 0 −0.399598 2.26623i 0 0.715176 + 0.600104i 0 −1.62349 + 2.52275i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 385.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.u.d 18
4.b odd 2 1 108.2.i.a 18
12.b even 2 1 324.2.i.a 18
27.e even 9 1 inner 432.2.u.d 18
36.f odd 6 1 972.2.i.a 18
36.f odd 6 1 972.2.i.c 18
36.h even 6 1 972.2.i.b 18
36.h even 6 1 972.2.i.d 18
108.j odd 18 1 108.2.i.a 18
108.j odd 18 1 972.2.i.a 18
108.j odd 18 1 972.2.i.c 18
108.j odd 18 1 2916.2.a.d 9
108.j odd 18 2 2916.2.e.c 18
108.l even 18 1 324.2.i.a 18
108.l even 18 1 972.2.i.b 18
108.l even 18 1 972.2.i.d 18
108.l even 18 1 2916.2.a.c 9
108.l even 18 2 2916.2.e.d 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.i.a 18 4.b odd 2 1
108.2.i.a 18 108.j odd 18 1
324.2.i.a 18 12.b even 2 1
324.2.i.a 18 108.l even 18 1
432.2.u.d 18 1.a even 1 1 trivial
432.2.u.d 18 27.e even 9 1 inner
972.2.i.a 18 36.f odd 6 1
972.2.i.a 18 108.j odd 18 1
972.2.i.b 18 36.h even 6 1
972.2.i.b 18 108.l even 18 1
972.2.i.c 18 36.f odd 6 1
972.2.i.c 18 108.j odd 18 1
972.2.i.d 18 36.h even 6 1
972.2.i.d 18 108.l even 18 1
2916.2.a.c 9 108.l even 18 1
2916.2.a.d 9 108.j odd 18 1
2916.2.e.c 18 108.j odd 18 2
2916.2.e.d 18 108.l even 18 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{18} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(432, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18}$$
$3$ $$19683 - 6561 T^{2} - 6561 T^{3} + 4374 T^{4} + 1458 T^{5} + 81 T^{6} - 1215 T^{7} + 243 T^{8} + 108 T^{9} + 81 T^{10} - 135 T^{11} + 3 T^{12} + 18 T^{13} + 18 T^{14} - 9 T^{15} - 3 T^{16} + T^{18}$$
$5$ $$729 - 13122 T + 65610 T^{2} + 1701 T^{3} + 195372 T^{4} - 37665 T^{5} + 7452 T^{6} - 84807 T^{7} + 116964 T^{8} - 72711 T^{9} + 34182 T^{10} - 12636 T^{11} + 2772 T^{12} + 27 T^{13} - 171 T^{14} + 48 T^{15} - 3 T^{17} + T^{18}$$
$7$ $$1456849 + 54315 T + 544968 T^{2} + 274575 T^{3} + 351396 T^{4} + 1457190 T^{5} + 908187 T^{6} - 105147 T^{7} - 11907 T^{8} + 98359 T^{9} + 49284 T^{10} + 9558 T^{11} + 915 T^{12} - 423 T^{13} - 162 T^{14} - 6 T^{15} + 9 T^{16} + T^{18}$$
$11$ $$23357889 + 36798462 T + 157549293 T^{2} + 104237280 T^{3} + 17891118 T^{4} + 494019 T^{5} - 928584 T^{6} - 138267 T^{7} + 1480194 T^{8} + 155088 T^{9} - 149850 T^{10} + 2187 T^{11} + 10872 T^{12} - 81 T^{13} + 45 T^{14} - 102 T^{15} + 3 T^{17} + T^{18}$$
$13$ $$4068289 + 44456697 T + 163132866 T^{2} + 197048472 T^{3} + 198727038 T^{4} + 128316231 T^{5} + 65384862 T^{6} + 23024529 T^{7} + 6132105 T^{8} + 825770 T^{9} + 109116 T^{10} + 405 T^{11} + 1725 T^{12} - 468 T^{13} + 1053 T^{14} + 33 T^{15} - 45 T^{16} + T^{18}$$
$17$ $$5247698481 - 58677210 T + 2875839390 T^{2} - 345993120 T^{3} + 1104080706 T^{4} - 111260952 T^{5} + 200310894 T^{6} - 3235059 T^{7} + 24516594 T^{8} + 920700 T^{9} + 2095146 T^{10} + 223398 T^{11} + 129753 T^{12} + 16659 T^{13} + 5769 T^{14} + 798 T^{15} + 153 T^{16} + 12 T^{17} + T^{18}$$
$19$ $$49014001 - 326575647 T + 2074057056 T^{2} - 1056361713 T^{3} + 1474984368 T^{4} + 300138966 T^{5} + 616337016 T^{6} + 55225710 T^{7} + 62754345 T^{8} + 3074488 T^{9} + 4390776 T^{10} + 112293 T^{11} + 181824 T^{12} + 1332 T^{13} + 5481 T^{14} + 12 T^{15} + 90 T^{16} + T^{18}$$
$23$ $$5597583489 + 3375518589 T - 2765907648 T^{2} + 1025621595 T^{3} + 1693288395 T^{4} - 2078424684 T^{5} + 1458449226 T^{6} - 574251039 T^{7} + 173163096 T^{8} - 47773125 T^{9} + 11619855 T^{10} - 2502738 T^{11} + 635382 T^{12} - 172287 T^{13} + 35118 T^{14} - 4785 T^{15} + 459 T^{16} - 30 T^{17} + T^{18}$$
$29$ $$451152679041 - 834533618661 T + 650948532435 T^{2} - 175653471942 T^{3} + 5064830289 T^{4} - 5658914745 T^{5} + 2614462434 T^{6} + 362237670 T^{7} + 477233856 T^{8} + 84579741 T^{9} + 20666583 T^{10} + 3980178 T^{11} + 476136 T^{12} + 69606 T^{13} + 9792 T^{14} + 1209 T^{15} + 216 T^{16} + 24 T^{17} + T^{18}$$
$31$ $$7983601201 - 5914589445 T + 498465891 T^{2} - 2367008934 T^{3} + 2471476689 T^{4} - 861159366 T^{5} + 1536450135 T^{6} - 414363879 T^{7} + 287013852 T^{8} - 59591918 T^{9} + 8642277 T^{10} - 161586 T^{11} - 244893 T^{12} + 34920 T^{13} + 4788 T^{14} - 816 T^{15} - 45 T^{16} + 9 T^{17} + T^{18}$$
$37$ $$9420061249 + 46633365018 T + 290089928232 T^{2} - 320069115984 T^{3} + 303489272520 T^{4} - 110713915692 T^{5} + 39640117686 T^{6} - 7645807071 T^{7} + 1983326472 T^{8} - 278564290 T^{9} + 67713390 T^{10} - 6165504 T^{11} + 1355325 T^{12} - 69075 T^{13} + 20061 T^{14} - 498 T^{15} + 171 T^{16} + T^{18}$$
$41$ $$29274867801 - 48894619032 T + 366142339314 T^{2} - 993983544180 T^{3} + 1271305510803 T^{4} - 943409691144 T^{5} + 453232129104 T^{6} - 147541511259 T^{7} + 32979939912 T^{8} - 4943026836 T^{9} + 455442912 T^{10} - 16220898 T^{11} - 1681785 T^{12} + 300537 T^{13} - 13617 T^{14} - 1491 T^{15} + 270 T^{16} - 21 T^{17} + T^{18}$$
$43$ $$5055621837841 - 8978252629608 T + 6998595237927 T^{2} - 2542300090917 T^{3} + 346038412563 T^{4} + 25728733683 T^{5} - 13077347277 T^{6} + 2212516440 T^{7} + 194828580 T^{8} - 140815721 T^{9} + 32885667 T^{10} - 4975812 T^{11} + 644568 T^{12} - 63378 T^{13} + 6606 T^{14} - 816 T^{15} + 99 T^{16} - 9 T^{17} + T^{18}$$
$47$ $$941480149401 + 15320449703889 T + 107289832893309 T^{2} + 50280258634155 T^{3} + 11531522434905 T^{4} + 1857114492453 T^{5} + 192676368681 T^{6} - 3120621552 T^{7} - 4087640781 T^{8} - 424718316 T^{9} + 70205616 T^{10} + 38210535 T^{11} + 8783154 T^{12} + 1369467 T^{13} + 163377 T^{14} + 15084 T^{15} + 1026 T^{16} + 45 T^{17} + T^{18}$$
$53$ $$( 9249336 - 5813532 T - 905094 T^{2} + 861291 T^{3} - 6210 T^{4} - 31653 T^{5} + 2463 T^{6} + 234 T^{7} - 33 T^{8} + T^{9} )^{2}$$
$59$ $$14164767759321 - 14872228773435 T + 3613868674650 T^{2} - 775181890233 T^{3} + 893265966576 T^{4} + 425340219582 T^{5} + 231921757251 T^{6} + 58557015153 T^{7} + 13084921683 T^{8} + 1067931243 T^{9} - 39207726 T^{10} - 38822166 T^{11} - 5076657 T^{12} + 303669 T^{13} + 171198 T^{14} + 22890 T^{15} + 1593 T^{16} + 60 T^{17} + T^{18}$$
$61$ $$173474749009 + 348268146516 T + 100468039086 T^{2} - 110032310145 T^{3} + 47340702441 T^{4} - 26571475053 T^{5} + 9885571782 T^{6} - 1788939288 T^{7} + 354085362 T^{8} - 43847170 T^{9} + 2540979 T^{10} + 1547226 T^{11} + 106521 T^{12} + 7740 T^{13} + 1791 T^{14} + 321 T^{15} + 144 T^{16} + 18 T^{17} + T^{18}$$
$67$ $$3568321 - 46820754 T + 7189055910 T^{2} + 19059869622 T^{3} + 16026308433 T^{4} + 3392981136 T^{5} + 729941982 T^{6} + 174501 T^{7} + 52876908 T^{8} + 39440014 T^{9} + 14087844 T^{10} - 931878 T^{11} - 478569 T^{12} + 90909 T^{13} + 13365 T^{14} - 3507 T^{15} + 378 T^{16} - 27 T^{17} + T^{18}$$
$71$ $$135419769 + 104813016012 T + 81076683544182 T^{2} + 36503093872872 T^{3} + 25611294223818 T^{4} - 896949481590 T^{5} + 1520508888594 T^{6} - 19636008903 T^{7} + 45283321026 T^{8} - 843673860 T^{9} + 878149674 T^{10} - 9177948 T^{11} + 10748367 T^{12} - 249723 T^{13} + 81747 T^{14} - 1986 T^{15} + 423 T^{16} - 12 T^{17} + T^{18}$$
$73$ $$13254226129 - 38888519076 T + 95286448350 T^{2} - 65131003524 T^{3} + 45095175747 T^{4} - 15219004698 T^{5} + 8088201096 T^{6} - 2387321109 T^{7} + 916431966 T^{8} - 190233394 T^{9} + 52530543 T^{10} - 8391960 T^{11} + 1967802 T^{12} - 221958 T^{13} + 34596 T^{14} - 1425 T^{15} + 234 T^{16} - 9 T^{17} + T^{18}$$
$79$ $$1826490081529 - 1137973366494 T - 29869592958 T^{2} + 408685469178 T^{3} + 188530809822 T^{4} - 61626679218 T^{5} - 3587682063 T^{6} + 4410148464 T^{7} - 913219236 T^{8} - 86052512 T^{9} + 107594838 T^{10} - 32790996 T^{11} + 6588915 T^{12} - 977832 T^{13} + 110664 T^{14} - 9861 T^{15} + 702 T^{16} - 36 T^{17} + T^{18}$$
$83$ $$289679140521369 - 361047385532673 T + 300312480958515 T^{2} - 144637176949308 T^{3} + 53704781887653 T^{4} - 5966326194642 T^{5} + 1408627076049 T^{6} - 363467911689 T^{7} + 43995638430 T^{8} - 9754014168 T^{9} + 2338056819 T^{10} - 250252578 T^{11} + 11609487 T^{12} - 704538 T^{13} + 122148 T^{14} - 13104 T^{15} + 945 T^{16} - 45 T^{17} + T^{18}$$
$89$ $$76986883963089 + 11969093668257 T + 30929640313518 T^{2} - 8815415247 T^{3} + 9346422037590 T^{4} + 478535879808 T^{5} + 717107163174 T^{6} + 113884567596 T^{7} + 45231614757 T^{8} + 7033398534 T^{9} + 1642261716 T^{10} + 272219859 T^{11} + 44001864 T^{12} + 5025402 T^{13} + 465741 T^{14} + 30444 T^{15} + 1530 T^{16} + 48 T^{17} + T^{18}$$
$97$ $$736742449 - 98913760596 T + 5642550696429 T^{2} + 5834447083563 T^{3} + 3440730089133 T^{4} + 1532081912013 T^{5} + 554459359965 T^{6} + 152587943784 T^{7} + 30685482990 T^{8} + 4591887221 T^{9} + 523915029 T^{10} + 38962782 T^{11} + 1121874 T^{12} - 170298 T^{13} - 25002 T^{14} - 1506 T^{15} + 171 T^{16} + 27 T^{17} + T^{18}$$