# 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: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ 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

$p$ $F_p(T)$
$2$ 
$3$ $$1 - 3 T^{2} - 9 T^{3} + 18 T^{4} + 18 T^{5} + 3 T^{6} - 135 T^{7} + 81 T^{8} + 108 T^{9} + 243 T^{10} - 1215 T^{11} + 81 T^{12} + 1458 T^{13} + 4374 T^{14} - 6561 T^{15} - 6561 T^{16} + 19683 T^{18}$$
$5$ $$1 - 3 T + 18 T^{3} - 36 T^{4} + 42 T^{5} + 12 T^{6} - 66 T^{7} + 207 T^{8} - 306 T^{9} - 1881 T^{10} + 8988 T^{11} + 3777 T^{12} - 51780 T^{13} + 130257 T^{14} - 141489 T^{15} - 166455 T^{16} + 864168 T^{17} - 1249796 T^{18} + 4320840 T^{19} - 4161375 T^{20} - 17686125 T^{21} + 81410625 T^{22} - 161812500 T^{23} + 59015625 T^{24} + 702187500 T^{25} - 734765625 T^{26} - 597656250 T^{27} + 2021484375 T^{28} - 3222656250 T^{29} + 2929687500 T^{30} + 51269531250 T^{31} - 219726562500 T^{32} + 549316406250 T^{33} - 2288818359375 T^{35} + 3814697265625 T^{36}$$
$7$ $$1 + 9 T^{2} - 6 T^{3} + 90 T^{4} + 144 T^{5} + 642 T^{6} + 1305 T^{7} + 3294 T^{8} + 16123 T^{9} + 41769 T^{10} + 164808 T^{11} + 268632 T^{12} + 1051596 T^{13} + 2182869 T^{14} + 10020276 T^{15} + 22667607 T^{16} + 57261276 T^{17} + 121068222 T^{18} + 400828932 T^{19} + 1110712743 T^{20} + 3436954668 T^{21} + 5241068469 T^{22} + 17674173972 T^{23} + 31604286168 T^{24} + 135726474744 T^{25} + 240789972969 T^{26} + 650621205661 T^{27} + 930473470206 T^{28} + 2580411399615 T^{29} + 8886106383042 T^{30} + 13952017498608 T^{31} + 61040076556410 T^{32} - 28485369059658 T^{33} + 299096375126409 T^{34} + 1628413597910449 T^{36}$$
$11$ $$1 + 3 T - 36 T^{3} - 252 T^{4} + 183 T^{5} + 3414 T^{6} + 14331 T^{7} + 19242 T^{8} - 82413 T^{9} - 362790 T^{10} - 885057 T^{11} + 3067980 T^{12} + 12803547 T^{13} + 53216793 T^{14} + 119847402 T^{15} - 27497151 T^{16} - 185418060 T^{17} - 3750422330 T^{18} - 2039598660 T^{19} - 3327155271 T^{20} + 159516892062 T^{21} + 779147066313 T^{22} + 2062024047897 T^{23} + 5435113716780 T^{24} - 17247257103747 T^{25} - 77767258437990 T^{26} - 194325543058383 T^{27} + 499087924172442 T^{28} + 4088801551526241 T^{29} + 10714594478125494 T^{30} + 6317656322339373 T^{31} - 95696958062976732 T^{32} - 150380934098963436 T^{33} + 1516341085497881313 T^{35} + 5559917313492231481 T^{36}$$
$13$ $$1 - 45 T^{2} + 33 T^{3} + 819 T^{4} - 2223 T^{5} - 6621 T^{6} + 67680 T^{7} + 2295 T^{8} - 1105627 T^{9} + 769527 T^{10} + 10632708 T^{11} - 24128055 T^{12} - 87782535 T^{13} + 521321076 T^{14} + 1050449085 T^{15} - 7341713262 T^{16} - 6958867761 T^{17} + 89217898098 T^{18} - 90465280893 T^{19} - 1240749541278 T^{20} + 2307836639745 T^{21} + 14889451251636 T^{22} - 32593040767755 T^{23} - 116461513026495 T^{24} + 667186658694036 T^{25} + 627726814538967 T^{26} - 11724620828271871 T^{27} + 316385238793455 T^{28} + 121293415468424160 T^{29} - 154256621595946701 T^{30} - 673291361954578419 T^{31} + 3224711259887717691 T^{32} + 1689134469464994981 T^{33} - 29943747413243092845 T^{34} +$$$$11\!\cdots\!29$$$$T^{36}$$
$17$ $$1 + 12 T - 630 T^{3} - 2493 T^{4} + 9264 T^{5} + 71715 T^{6} - 20382 T^{7} - 663750 T^{8} + 1631844 T^{9} + 5926023 T^{10} - 55869558 T^{11} - 116813226 T^{12} + 478010541 T^{13} - 1415343366 T^{14} - 8092973394 T^{15} + 53937481662 T^{16} + 118392015135 T^{17} - 620668817804 T^{18} + 2012664257295 T^{19} + 15587932200318 T^{20} - 39760778284722 T^{21} - 118210893271686 T^{22} + 678706612712637 T^{23} - 2819587302687594 T^{24} - 22925440290816534 T^{25} + 41338499037787143 T^{26} + 193516914734370468 T^{27} - 1338115951423023750 T^{28} - 698529790542175806 T^{29} + 41782753742932310115 T^{30} + 91756010896840600368 T^{31} -$$$$41\!\cdots\!97$$$$T^{32} -$$$$18\!\cdots\!90$$$$T^{33} +$$$$99\!\cdots\!24$$$$T^{35} +$$$$14\!\cdots\!09$$$$T^{36}$$
$19$ $$1 - 81 T^{2} + 12 T^{3} + 2916 T^{4} - 378 T^{5} - 63789 T^{6} - 26217 T^{7} + 1135107 T^{8} + 2182495 T^{9} - 24267384 T^{10} - 68364540 T^{11} + 564078960 T^{12} + 1060991325 T^{13} - 9783469926 T^{14} - 6143751819 T^{15} + 107000768556 T^{16} - 9549612666 T^{17} - 1139182247424 T^{18} - 181442640654 T^{19} + 38627277448716 T^{20} - 42139993726521 T^{21} - 1274991584226246 T^{22} + 2627119558841175 T^{23} + 26537591626763760 T^{24} - 61109130255735060 T^{25} - 412146646004154744 T^{26} + 704264286964178605 T^{27} + 6959416226693719707 T^{28} - 3054025117534607523 T^{29} -$$$$14\!\cdots\!29$$$$T^{30} - 15896027748733168302 T^{31} +$$$$23\!\cdots\!36$$$$T^{32} +$$$$18\!\cdots\!88$$$$T^{33} -$$$$23\!\cdots\!61$$$$T^{34} +$$$$10\!\cdots\!41$$$$T^{36}$$
$23$ $$1 - 30 T + 459 T^{2} - 4716 T^{3} + 35739 T^{4} - 203682 T^{5} + 823269 T^{6} - 1638444 T^{7} - 6429924 T^{8} + 76742136 T^{9} - 307889172 T^{10} - 367007538 T^{11} + 14732471886 T^{12} - 121578320496 T^{13} + 628848268830 T^{14} - 2040092984925 T^{15} + 1341049467978 T^{16} + 31755560848485 T^{17} - 226915587058772 T^{18} + 730377899515155 T^{19} + 709415168560362 T^{20} - 24821811347582475 T^{21} + 175977528397656030 T^{22} - 782519772076186128 T^{23} + 2180934572811516654 T^{24} - 1249596604623219486 T^{25} - 24111104416671277332 T^{26} +$$$$13\!\cdots\!68$$$$T^{27} -$$$$26\!\cdots\!76$$$$T^{28} -$$$$15\!\cdots\!88$$$$T^{29} +$$$$18\!\cdots\!49$$$$T^{30} -$$$$10\!\cdots\!06$$$$T^{31} +$$$$41\!\cdots\!51$$$$T^{32} -$$$$12\!\cdots\!12$$$$T^{33} +$$$$28\!\cdots\!99$$$$T^{34} -$$$$42\!\cdots\!90$$$$T^{35} +$$$$32\!\cdots\!69$$$$T^{36}$$
$29$ $$1 + 24 T + 216 T^{2} + 513 T^{3} - 5085 T^{4} - 38622 T^{5} - 39165 T^{6} + 335661 T^{7} - 972927 T^{8} - 15344370 T^{9} + 17037873 T^{10} + 849699627 T^{11} + 5794706544 T^{12} + 21856511310 T^{13} + 65301140610 T^{14} + 312703277274 T^{15} + 367683018633 T^{16} - 19987658397597 T^{17} - 182924253089192 T^{18} - 579642093530313 T^{19} + 309221418670353 T^{20} + 7626520229435586 T^{21} + 46186256031781410 T^{22} + 448302160099595190 T^{23} + 3446826590722512624 T^{24} + 14657213465553436743 T^{25} + 8523134852735071953 T^{26} -$$$$22\!\cdots\!30$$$$T^{27} -$$$$40\!\cdots\!27$$$$T^{28} +$$$$40\!\cdots\!69$$$$T^{29} -$$$$13\!\cdots\!65$$$$T^{30} -$$$$39\!\cdots\!58$$$$T^{31} -$$$$15\!\cdots\!85$$$$T^{32} +$$$$44\!\cdots\!37$$$$T^{33} +$$$$54\!\cdots\!36$$$$T^{34} +$$$$17\!\cdots\!16$$$$T^{35} +$$$$21\!\cdots\!61$$$$T^{36}$$
$31$ $$1 + 9 T - 45 T^{2} - 816 T^{3} - 1629 T^{4} + 21807 T^{5} + 131943 T^{6} - 1719 T^{7} - 1559916 T^{8} + 2236900 T^{9} - 3018924 T^{10} - 598265892 T^{11} - 3251038833 T^{12} + 8698989627 T^{13} + 152835071832 T^{14} + 400071404862 T^{15} - 1915318217007 T^{16} - 9512206426008 T^{17} - 6801570693948 T^{18} - 294878399206248 T^{19} - 1840620806543727 T^{20} + 11918527222243842 T^{21} + 141146398373360472 T^{22} + 249044687578816677 T^{23} - 2885308931361444273 T^{24} - 16459858622369202012 T^{25} - 2574813222315533484 T^{26} + 59142790811204959900 T^{27} -$$$$12\!\cdots\!16$$$$T^{28} - 43677171784919904489 T^{29} +$$$$10\!\cdots\!23$$$$T^{30} +$$$$53\!\cdots\!37$$$$T^{31} -$$$$12\!\cdots\!09$$$$T^{32} -$$$$19\!\cdots\!16$$$$T^{33} -$$$$32\!\cdots\!45$$$$T^{34} +$$$$20\!\cdots\!99$$$$T^{35} +$$$$69\!\cdots\!41$$$$T^{36}$$
$37$ $$1 - 162 T^{2} - 498 T^{3} + 12069 T^{4} + 69120 T^{5} - 436659 T^{6} - 3993678 T^{7} + 5314518 T^{8} + 92943608 T^{9} - 67375773 T^{10} + 822064248 T^{11} + 22823309772 T^{12} - 67304973087 T^{13} - 1588732176582 T^{14} - 666327415302 T^{15} + 53054144866422 T^{16} + 47616232092741 T^{17} - 1542710676723420 T^{18} + 1761800587431417 T^{19} + 72631124322131718 T^{20} - 33751482567292206 T^{21} - 2977539884795097702 T^{22} - 4667193159631085259 T^{23} + 58558368622808168748 T^{24} + 78040102186568040984 T^{25} -$$$$23\!\cdots\!33$$$$T^{26} +$$$$12\!\cdots\!16$$$$T^{27} +$$$$25\!\cdots\!82$$$$T^{28} -$$$$71\!\cdots\!14$$$$T^{29} -$$$$28\!\cdots\!79$$$$T^{30} +$$$$16\!\cdots\!40$$$$T^{31} +$$$$10\!\cdots\!41$$$$T^{32} -$$$$16\!\cdots\!14$$$$T^{33} -$$$$19\!\cdots\!42$$$$T^{34} +$$$$16\!\cdots\!29$$$$T^{36}$$
$41$ $$1 - 21 T + 270 T^{2} - 2844 T^{3} + 28449 T^{4} - 225534 T^{5} + 1364187 T^{6} - 5783241 T^{7} + 7120458 T^{8} + 266586381 T^{9} - 3961333476 T^{10} + 35030856111 T^{11} - 226766681787 T^{12} + 1184373682962 T^{13} - 3751533361044 T^{14} - 14056953699150 T^{15} + 350489698231473 T^{16} - 3267580417540392 T^{17} + 22906077409966138 T^{18} - 133970797119156072 T^{19} + 589173182727106113 T^{20} - 968819305899117150 T^{21} - 10600936661837054484 T^{22} +$$$$13\!\cdots\!62$$$$T^{23} -$$$$10\!\cdots\!67$$$$T^{24} +$$$$68\!\cdots\!91$$$$T^{25} -$$$$31\!\cdots\!96$$$$T^{26} +$$$$87\!\cdots\!41$$$$T^{27} +$$$$95\!\cdots\!58$$$$T^{28} -$$$$31\!\cdots\!81$$$$T^{29} +$$$$30\!\cdots\!47$$$$T^{30} -$$$$20\!\cdots\!14$$$$T^{31} +$$$$10\!\cdots\!89$$$$T^{32} -$$$$44\!\cdots\!44$$$$T^{33} +$$$$17\!\cdots\!70$$$$T^{34} -$$$$54\!\cdots\!01$$$$T^{35} +$$$$10\!\cdots\!21$$$$T^{36}$$
$43$ $$1 - 9 T + 99 T^{2} - 816 T^{3} + 7380 T^{4} - 44028 T^{5} + 425784 T^{6} - 2654973 T^{7} + 17329428 T^{8} - 77745686 T^{9} + 612406224 T^{10} - 1498691268 T^{11} + 14593371894 T^{12} - 17177157162 T^{13} + 121626443133 T^{14} + 4355914982778 T^{15} - 8046633177891 T^{16} + 188327320360398 T^{17} - 573554182033272 T^{18} + 8098074775497114 T^{19} - 14878224745920459 T^{20} + 346325732535730446 T^{21} + 415816605409543533 T^{22} - 2525187129551918766 T^{23} + 92250001851046744806 T^{24} -$$$$40\!\cdots\!76$$$$T^{25} +$$$$71\!\cdots\!24$$$$T^{26} -$$$$39\!\cdots\!98$$$$T^{27} +$$$$37\!\cdots\!72$$$$T^{28} -$$$$24\!\cdots\!11$$$$T^{29} +$$$$17\!\cdots\!84$$$$T^{30} -$$$$75\!\cdots\!04$$$$T^{31} +$$$$54\!\cdots\!20$$$$T^{32} -$$$$25\!\cdots\!12$$$$T^{33} +$$$$13\!\cdots\!99$$$$T^{34} -$$$$52\!\cdots\!87$$$$T^{35} +$$$$25\!\cdots\!49$$$$T^{36}$$
$47$ $$1 + 45 T + 1026 T^{2} + 15507 T^{3} + 173529 T^{4} + 1550934 T^{5} + 12137826 T^{6} + 94180626 T^{7} + 777562713 T^{8} + 6408653373 T^{9} + 47766713094 T^{10} + 314746073955 T^{11} + 2010959274003 T^{12} + 14589090065835 T^{13} + 120231976929849 T^{14} + 958518905650704 T^{15} + 6721325237600946 T^{16} + 42688545402868929 T^{17} + 277542048358340170 T^{18} + 2006361633934839663 T^{19} + 14847407449860489714 T^{20} + 99516308341373041392 T^{21} +$$$$58\!\cdots\!69$$$$T^{22} +$$$$33\!\cdots\!45$$$$T^{23} +$$$$21\!\cdots\!87$$$$T^{24} +$$$$15\!\cdots\!65$$$$T^{25} +$$$$11\!\cdots\!34$$$$T^{26} +$$$$71\!\cdots\!91$$$$T^{27} +$$$$40\!\cdots\!37$$$$T^{28} +$$$$23\!\cdots\!78$$$$T^{29} +$$$$14\!\cdots\!66$$$$T^{30} +$$$$84\!\cdots\!18$$$$T^{31} +$$$$44\!\cdots\!01$$$$T^{32} +$$$$18\!\cdots\!01$$$$T^{33} +$$$$58\!\cdots\!46$$$$T^{34} +$$$$11\!\cdots\!15$$$$T^{35} +$$$$12\!\cdots\!89$$$$T^{36}$$
$53$ $$( 1 - 33 T + 711 T^{2} - 11529 T^{3} + 156285 T^{4} - 1818492 T^{5} + 18782340 T^{6} - 173567805 T^{7} + 1455502203 T^{8} - 11084684058 T^{9} + 77141616759 T^{10} - 487551964245 T^{11} + 2796258432180 T^{12} - 14348776574652 T^{13} + 65357682623505 T^{14} - 255532919456241 T^{15} + 835219620424107 T^{16} - 2054569783574913 T^{17} + 3299763591802133 T^{18} )^{2}$$
$59$ $$1 + 60 T + 1593 T^{2} + 24660 T^{3} + 247662 T^{4} + 1708872 T^{5} + 8490216 T^{6} + 33550125 T^{7} + 148813002 T^{8} + 1136539629 T^{9} + 11730436263 T^{10} + 113179328610 T^{11} + 651096766104 T^{12} - 3048081644244 T^{13} - 120313325069001 T^{14} - 1538872617278898 T^{15} - 13453209049931613 T^{16} - 98005926759519540 T^{17} - 714266619705703766 T^{18} - 5782349678811652860 T^{19} - 46830620702811944853 T^{20} -$$$$31\!\cdots\!42$$$$T^{21} -$$$$14\!\cdots\!61$$$$T^{22} -$$$$21\!\cdots\!56$$$$T^{23} +$$$$27\!\cdots\!64$$$$T^{24} +$$$$28\!\cdots\!90$$$$T^{25} +$$$$17\!\cdots\!23$$$$T^{26} +$$$$98\!\cdots\!31$$$$T^{27} +$$$$76\!\cdots\!02$$$$T^{28} +$$$$10\!\cdots\!75$$$$T^{29} +$$$$15\!\cdots\!96$$$$T^{30} +$$$$17\!\cdots\!88$$$$T^{31} +$$$$15\!\cdots\!82$$$$T^{32} +$$$$90\!\cdots\!40$$$$T^{33} +$$$$34\!\cdots\!13$$$$T^{34} +$$$$76\!\cdots\!40$$$$T^{35} +$$$$75\!\cdots\!21$$$$T^{36}$$
$61$ $$1 + 18 T + 144 T^{2} + 870 T^{3} + 693 T^{4} + 46719 T^{5} + 1363548 T^{6} + 13321629 T^{7} + 112641282 T^{8} + 631876487 T^{9} + 4072259232 T^{10} + 57229816275 T^{11} + 499398580341 T^{12} + 4512706586946 T^{13} + 40501364633055 T^{14} + 302618732115951 T^{15} + 2515574747971161 T^{16} + 17759529627029685 T^{17} + 116077839022099230 T^{18} + 1083331307248810785 T^{19} + 9360453637200690081 T^{20} + 68688702434410673931 T^{21} +$$$$56\!\cdots\!55$$$$T^{22} +$$$$38\!\cdots\!46$$$$T^{23} +$$$$25\!\cdots\!01$$$$T^{24} +$$$$17\!\cdots\!75$$$$T^{25} +$$$$78\!\cdots\!92$$$$T^{26} +$$$$73\!\cdots\!67$$$$T^{27} +$$$$80\!\cdots\!82$$$$T^{28} +$$$$57\!\cdots\!69$$$$T^{29} +$$$$36\!\cdots\!08$$$$T^{30} +$$$$75\!\cdots\!39$$$$T^{31} +$$$$68\!\cdots\!13$$$$T^{32} +$$$$52\!\cdots\!70$$$$T^{33} +$$$$52\!\cdots\!84$$$$T^{34} +$$$$40\!\cdots\!78$$$$T^{35} +$$$$13\!\cdots\!81$$$$T^{36}$$
$67$ $$1 - 27 T + 378 T^{2} - 4110 T^{3} + 42309 T^{4} - 359532 T^{5} + 1903683 T^{6} + 2087343 T^{7} - 177145164 T^{8} + 2600725729 T^{9} - 29272869126 T^{10} + 274304636919 T^{11} - 1972009698093 T^{12} + 9826793958816 T^{13} - 14621223469968 T^{14} - 409049170442514 T^{15} + 7291627922092167 T^{16} - 85616450880794064 T^{17} + 791533857384890682 T^{18} - 5736302209013202288 T^{19} + 32732117742271737663 T^{20} -$$$$12\!\cdots\!82$$$$T^{21} -$$$$29\!\cdots\!28$$$$T^{22} +$$$$13\!\cdots\!12$$$$T^{23} -$$$$17\!\cdots\!17$$$$T^{24} +$$$$16\!\cdots\!37$$$$T^{25} -$$$$11\!\cdots\!66$$$$T^{26} +$$$$70\!\cdots\!63$$$$T^{27} -$$$$32\!\cdots\!36$$$$T^{28} +$$$$25\!\cdots\!69$$$$T^{29} +$$$$15\!\cdots\!63$$$$T^{30} -$$$$19\!\cdots\!84$$$$T^{31} +$$$$15\!\cdots\!61$$$$T^{32} -$$$$10\!\cdots\!30$$$$T^{33} +$$$$62\!\cdots\!18$$$$T^{34} -$$$$29\!\cdots\!29$$$$T^{35} +$$$$74\!\cdots\!09$$$$T^{36}$$
$71$ $$1 - 12 T - 216 T^{2} + 3978 T^{3} + 8901 T^{4} - 529392 T^{5} + 1798959 T^{6} + 37173408 T^{7} - 298164978 T^{8} - 1281013128 T^{9} + 23327409465 T^{10} - 43122203202 T^{11} - 995587443048 T^{12} + 12405448357179 T^{13} - 32462907484212 T^{14} - 1158165262802520 T^{15} + 11716337115395100 T^{16} + 40103807685557901 T^{17} - 1167130398411512264 T^{18} + 2847370345674610971 T^{19} + 59062055398706699100 T^{20} -$$$$41\!\cdots\!20$$$$T^{21} -$$$$82\!\cdots\!72$$$$T^{22} +$$$$22\!\cdots\!29$$$$T^{23} -$$$$12\!\cdots\!08$$$$T^{24} -$$$$39\!\cdots\!82$$$$T^{25} +$$$$15\!\cdots\!65$$$$T^{26} -$$$$58\!\cdots\!68$$$$T^{27} -$$$$97\!\cdots\!78$$$$T^{28} +$$$$85\!\cdots\!68$$$$T^{29} +$$$$29\!\cdots\!19$$$$T^{30} -$$$$61\!\cdots\!12$$$$T^{31} +$$$$73\!\cdots\!81$$$$T^{32} +$$$$23\!\cdots\!78$$$$T^{33} -$$$$90\!\cdots\!36$$$$T^{34} -$$$$35\!\cdots\!92$$$$T^{35} +$$$$21\!\cdots\!61$$$$T^{36}$$
$73$ $$1 - 9 T - 423 T^{2} + 3174 T^{3} + 104238 T^{4} - 633897 T^{5} - 17768040 T^{6} + 86261373 T^{7} + 2286597285 T^{8} - 8821811794 T^{9} - 229412589363 T^{10} + 696472394517 T^{11} + 18405723457890 T^{12} - 43263862214751 T^{13} - 1209322407919761 T^{14} + 2007285699967488 T^{15} + 70978846328739252 T^{16} - 49678641613188693 T^{17} - 4558841577782266638 T^{18} - 3626540837762774589 T^{19} +$$$$37\!\cdots\!08$$$$T^{20} +$$$$78\!\cdots\!96$$$$T^{21} -$$$$34\!\cdots\!01$$$$T^{22} -$$$$89\!\cdots\!43$$$$T^{23} +$$$$27\!\cdots\!10$$$$T^{24} +$$$$76\!\cdots\!49$$$$T^{25} -$$$$18\!\cdots\!03$$$$T^{26} -$$$$51\!\cdots\!22$$$$T^{27} +$$$$98\!\cdots\!65$$$$T^{28} +$$$$27\!\cdots\!21$$$$T^{29} -$$$$40\!\cdots\!40$$$$T^{30} -$$$$10\!\cdots\!01$$$$T^{31} +$$$$12\!\cdots\!42$$$$T^{32} +$$$$28\!\cdots\!18$$$$T^{33} -$$$$27\!\cdots\!03$$$$T^{34} -$$$$42\!\cdots\!77$$$$T^{35} +$$$$34\!\cdots\!69$$$$T^{36}$$
$79$ $$1 - 36 T + 702 T^{2} - 9861 T^{3} + 119196 T^{4} - 1409409 T^{5} + 16913820 T^{6} - 201585240 T^{7} + 2283030162 T^{8} - 24444793301 T^{9} + 255412728198 T^{10} - 2659995762120 T^{11} + 27368820023094 T^{12} - 271885146597855 T^{13} + 2605507077760911 T^{14} - 24385423616176365 T^{15} + 226920837770375571 T^{16} - 2103229216817817381 T^{17} + 19027927100722312626 T^{18} -$$$$16\!\cdots\!99$$$$T^{19} +$$$$14\!\cdots\!11$$$$T^{20} -$$$$12\!\cdots\!35$$$$T^{21} +$$$$10\!\cdots\!91$$$$T^{22} -$$$$83\!\cdots\!45$$$$T^{23} +$$$$66\!\cdots\!74$$$$T^{24} -$$$$51\!\cdots\!80$$$$T^{25} +$$$$38\!\cdots\!78$$$$T^{26} -$$$$29\!\cdots\!19$$$$T^{27} +$$$$21\!\cdots\!62$$$$T^{28} -$$$$15\!\cdots\!60$$$$T^{29} +$$$$99\!\cdots\!20$$$$T^{30} -$$$$65\!\cdots\!51$$$$T^{31} +$$$$43\!\cdots\!76$$$$T^{32} -$$$$28\!\cdots\!39$$$$T^{33} +$$$$16\!\cdots\!42$$$$T^{34} -$$$$65\!\cdots\!24$$$$T^{35} +$$$$14\!\cdots\!61$$$$T^{36}$$
$83$ $$1 - 45 T + 945 T^{2} - 10116 T^{3} + 12339 T^{4} + 1380339 T^{5} - 22826715 T^{6} + 161300349 T^{7} + 246197610 T^{8} - 18368963478 T^{9} + 201770388396 T^{10} - 948024221238 T^{11} - 3270167174595 T^{12} + 95828229532791 T^{13} - 847092215711700 T^{14} + 3777334618062672 T^{15} + 7299207689386761 T^{16} - 335349191198370492 T^{17} + 3919398090577201708 T^{18} - 27833982869464750836 T^{19} + 50284241772185396529 T^{20} +$$$$21\!\cdots\!64$$$$T^{21} -$$$$40\!\cdots\!00$$$$T^{22} +$$$$37\!\cdots\!13$$$$T^{23} -$$$$10\!\cdots\!55$$$$T^{24} -$$$$25\!\cdots\!26$$$$T^{25} +$$$$45\!\cdots\!36$$$$T^{26} -$$$$34\!\cdots\!34$$$$T^{27} +$$$$38\!\cdots\!90$$$$T^{28} +$$$$20\!\cdots\!83$$$$T^{29} -$$$$24\!\cdots\!15$$$$T^{30} +$$$$12\!\cdots\!57$$$$T^{31} +$$$$90\!\cdots\!31$$$$T^{32} -$$$$61\!\cdots\!12$$$$T^{33} +$$$$47\!\cdots\!45$$$$T^{34} -$$$$18\!\cdots\!35$$$$T^{35} +$$$$34\!\cdots\!09$$$$T^{36}$$
$89$ $$1 + 48 T + 729 T^{2} + 540 T^{3} - 60516 T^{4} + 309648 T^{5} + 9164505 T^{6} - 43924695 T^{7} - 296290449 T^{8} + 17649268605 T^{9} + 60514001892 T^{10} - 1616101715796 T^{11} + 6831214397718 T^{12} + 217593105310137 T^{13} - 956051155825812 T^{14} - 10434643262054325 T^{15} + 213339340235303568 T^{16} + 707450617777750446 T^{17} - 16906270535571748808 T^{18} + 62963104982219789694 T^{19} +$$$$16\!\cdots\!28$$$$T^{20} -$$$$73\!\cdots\!25$$$$T^{21} -$$$$59\!\cdots\!92$$$$T^{22} +$$$$12\!\cdots\!13$$$$T^{23} +$$$$33\!\cdots\!98$$$$T^{24} -$$$$71\!\cdots\!84$$$$T^{25} +$$$$23\!\cdots\!52$$$$T^{26} +$$$$61\!\cdots\!45$$$$T^{27} -$$$$92\!\cdots\!49$$$$T^{28} -$$$$12\!\cdots\!55$$$$T^{29} +$$$$22\!\cdots\!05$$$$T^{30} +$$$$68\!\cdots\!12$$$$T^{31} -$$$$11\!\cdots\!56$$$$T^{32} +$$$$94\!\cdots\!60$$$$T^{33} +$$$$11\!\cdots\!69$$$$T^{34} +$$$$66\!\cdots\!92$$$$T^{35} +$$$$12\!\cdots\!81$$$$T^{36}$$
$97$ $$1 + 27 T + 171 T^{2} - 1506 T^{3} - 5796 T^{4} + 238266 T^{5} - 1047822 T^{6} - 47407473 T^{7} + 141058260 T^{8} + 5136199520 T^{9} - 34758298530 T^{10} - 613734906312 T^{11} + 4119485195664 T^{12} + 37279586300094 T^{13} - 763210465522713 T^{14} - 3540249166293810 T^{15} + 84441145547637873 T^{16} + 225577194346293930 T^{17} - 6812284288441068552 T^{18} + 21880987851590511210 T^{19} +$$$$79\!\cdots\!57$$$$T^{20} -$$$$32\!\cdots\!30$$$$T^{21} -$$$$67\!\cdots\!53$$$$T^{22} +$$$$32\!\cdots\!58$$$$T^{23} +$$$$34\!\cdots\!56$$$$T^{24} -$$$$49\!\cdots\!56$$$$T^{25} -$$$$27\!\cdots\!30$$$$T^{26} +$$$$39\!\cdots\!40$$$$T^{27} +$$$$10\!\cdots\!40$$$$T^{28} -$$$$33\!\cdots\!69$$$$T^{29} -$$$$72\!\cdots\!02$$$$T^{30} +$$$$16\!\cdots\!82$$$$T^{31} -$$$$37\!\cdots\!24$$$$T^{32} -$$$$95\!\cdots\!58$$$$T^{33} +$$$$10\!\cdots\!91$$$$T^{34} +$$$$16\!\cdots\!99$$$$T^{35} +$$$$57\!\cdots\!89$$$$T^{36}$$