# Properties

 Label 8752.2.a.s Level $8752$ Weight $2$ Character orbit 8752.a Self dual yes Analytic conductor $69.885$ Analytic rank $1$ Dimension $18$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8752 = 2^{4} \cdot 547$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8752.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$69.8850718490$$ Analytic rank: $$1$$ Dimension: $$18$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ Defining polynomial: $$x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} - 137 x^{10} - 7703 x^{9} + 2068 x^{8} + 11068 x^{7} - 4274 x^{6} - 9021 x^{5} + 4048 x^{4} + 3834 x^{3} - 1851 x^{2} - 654 x + 328$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 547) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{9} ) q^{3} + ( -2 - \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{7} + ( 1 - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{9} - 3 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{17} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{9} ) q^{3} + ( -2 - \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{7} + ( 1 - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{9} - 3 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{17} ) q^{9} + ( -\beta_{3} + \beta_{4} + \beta_{6} - 2 \beta_{7} + 3 \beta_{9} - \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{11} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} - \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{14} - \beta_{16} ) q^{13} + ( -2 - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{12} + \beta_{17} ) q^{15} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} - \beta_{17} ) q^{17} + ( -1 - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} + 2 \beta_{12} - \beta_{14} - \beta_{16} - 2 \beta_{17} ) q^{19} + ( -\beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{21} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} + 2 \beta_{11} - 3 \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} + \beta_{16} + 2 \beta_{17} ) q^{23} + ( 3 - \beta_{1} + 5 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} - \beta_{16} ) q^{25} + ( 3 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{9} + \beta_{10} - 3 \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} ) q^{27} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{9} + \beta_{11} + \beta_{16} + \beta_{17} ) q^{29} + ( -\beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{15} ) q^{31} + ( -1 + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{33} + ( 1 + 2 \beta_{1} - \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + 5 \beta_{9} - 2 \beta_{10} + 4 \beta_{11} + 5 \beta_{12} - 2 \beta_{13} + \beta_{14} + 3 \beta_{15} + 3 \beta_{16} - 2 \beta_{17} ) q^{35} + ( -\beta_{1} + \beta_{2} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + 4 \beta_{9} - \beta_{10} + \beta_{11} + 5 \beta_{12} - 2 \beta_{14} + \beta_{15} + \beta_{16} - 3 \beta_{17} ) q^{37} + ( -1 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} + 4 \beta_{9} - \beta_{11} + 6 \beta_{12} - 2 \beta_{14} + \beta_{15} - 3 \beta_{17} ) q^{39} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{9} - 2 \beta_{12} + 3 \beta_{14} - \beta_{15} + \beta_{17} ) q^{41} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{8} + 2 \beta_{9} + \beta_{11} + 4 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{16} - \beta_{17} ) q^{43} + ( -3 - \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - \beta_{7} + 3 \beta_{8} + 6 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} + 6 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} - \beta_{16} - 2 \beta_{17} ) q^{45} + ( 3 + \beta_{1} + \beta_{3} + 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + \beta_{8} + 4 \beta_{9} + \beta_{10} + \beta_{11} + 5 \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} - 3 \beta_{17} ) q^{47} + ( 2 - 2 \beta_{1} + 5 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{10} - 2 \beta_{11} + 6 \beta_{12} - \beta_{14} - 5 \beta_{16} - 3 \beta_{17} ) q^{49} + ( -1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} + 4 \beta_{9} - \beta_{10} + 2 \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{17} ) q^{51} + ( -2 - \beta_{2} - \beta_{4} + 3 \beta_{6} - \beta_{7} + 3 \beta_{9} - \beta_{10} + 3 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} + \beta_{16} ) q^{53} + ( \beta_{2} + 4 \beta_{3} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - 4 \beta_{11} + 4 \beta_{12} + \beta_{13} - 3 \beta_{14} - \beta_{15} - 3 \beta_{16} - 2 \beta_{17} ) q^{55} + ( -1 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} + 4 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} - 4 \beta_{17} ) q^{57} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} + 3 \beta_{8} + 4 \beta_{9} - \beta_{10} + \beta_{11} + 4 \beta_{12} - \beta_{13} + \beta_{14} + 3 \beta_{15} + 2 \beta_{16} - 2 \beta_{17} ) q^{59} + ( -4 - \beta_{1} - \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{16} - \beta_{17} ) q^{61} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - 5 \beta_{9} + \beta_{10} - 3 \beta_{11} - 7 \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} + 5 \beta_{17} ) q^{63} + ( 2 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + 7 \beta_{9} - 5 \beta_{10} + 7 \beta_{11} + 3 \beta_{12} - \beta_{13} + 2 \beta_{14} + 3 \beta_{16} - \beta_{17} ) q^{65} + ( -4 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} - 4 \beta_{12} - \beta_{15} - 2 \beta_{16} ) q^{67} + ( 1 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + 6 \beta_{9} - 4 \beta_{10} + 7 \beta_{11} + 5 \beta_{12} - 3 \beta_{13} + 2 \beta_{14} + \beta_{15} + \beta_{16} ) q^{69} + ( 4 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} + 4 \beta_{9} - \beta_{10} + 4 \beta_{11} + \beta_{12} + \beta_{14} + 3 \beta_{16} + 2 \beta_{17} ) q^{71} + ( 4 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - 3 \beta_{10} + \beta_{11} - 6 \beta_{12} + \beta_{13} + 2 \beta_{14} - 2 \beta_{15} + 3 \beta_{16} + 5 \beta_{17} ) q^{73} + ( 3 + 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - \beta_{8} + 5 \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{17} ) q^{75} + ( -6 + \beta_{1} - \beta_{2} - 2 \beta_{3} - 5 \beta_{6} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 4 \beta_{12} - \beta_{14} + \beta_{15} + 3 \beta_{16} + \beta_{17} ) q^{77} + ( -2 - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} - \beta_{11} - 3 \beta_{12} + \beta_{13} + 2 \beta_{17} ) q^{79} + ( 4 - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + 4 \beta_{5} - \beta_{7} + \beta_{8} - 3 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{16} - \beta_{17} ) q^{81} + ( 3 + 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + \beta_{4} - 2 \beta_{5} - 6 \beta_{6} - \beta_{7} - 2 \beta_{8} - 5 \beta_{9} - 3 \beta_{10} - 8 \beta_{12} - 2 \beta_{13} - \beta_{14} - 3 \beta_{15} + \beta_{16} + 4 \beta_{17} ) q^{83} + ( 4 - \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - 7 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} - 10 \beta_{12} + 5 \beta_{13} - 3 \beta_{15} + 2 \beta_{16} + 5 \beta_{17} ) q^{85} + ( -1 - \beta_{1} + \beta_{2} - 4 \beta_{3} - 4 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} + 4 \beta_{11} - 5 \beta_{12} + 3 \beta_{14} - 2 \beta_{15} + 4 \beta_{17} ) q^{87} + ( -2 - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{8} + 3 \beta_{9} + \beta_{11} + 3 \beta_{12} - \beta_{13} + 3 \beta_{15} + \beta_{16} - \beta_{17} ) q^{89} + ( 1 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} ) q^{91} + ( -4 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} + 4 \beta_{17} ) q^{93} + ( -1 + 3 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} - 5 \beta_{5} - 5 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} - 9 \beta_{12} - \beta_{13} + 4 \beta_{14} + 2 \beta_{16} + 3 \beta_{17} ) q^{95} + ( -3 - 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - 5 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 6 \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{15} - \beta_{16} + 3 \beta_{17} ) q^{97} + ( 2 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + 7 \beta_{9} - \beta_{10} + 6 \beta_{11} + 5 \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} - 3 \beta_{17} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18q + 10q^{3} - 27q^{5} + 11q^{7} + 14q^{9} + O(q^{10})$$ $$18q + 10q^{3} - 27q^{5} + 11q^{7} + 14q^{9} - 2q^{11} - 25q^{13} - 9q^{15} - 30q^{17} - 4q^{19} - 16q^{21} + 26q^{23} + 31q^{25} + 37q^{27} - 18q^{29} + 5q^{31} - 10q^{33} + 9q^{35} - 18q^{37} - 7q^{39} - 17q^{41} - 8q^{43} - 44q^{45} + 52q^{47} + 29q^{49} - 19q^{51} - 60q^{53} - 11q^{55} + 4q^{57} + 8q^{59} - 26q^{61} + q^{63} - 6q^{65} - 12q^{67} - 38q^{69} + q^{71} - 2q^{73} + 17q^{75} - 73q^{77} - 18q^{79} + 18q^{81} + 43q^{83} + 51q^{85} - 3q^{87} - 28q^{89} + q^{91} - 60q^{93} + 18q^{95} - 34q^{97} - 15q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} - 137 x^{10} - 7703 x^{9} + 2068 x^{8} + 11068 x^{7} - 4274 x^{6} - 9021 x^{5} + 4048 x^{4} + 3834 x^{3} - 1851 x^{2} - 654 x + 328$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$73030826 \nu^{17} - 256120405 \nu^{16} - 1356797574 \nu^{15} + 5113465236 \nu^{14} + 9822519058 \nu^{13} - 40388724797 \nu^{12} - 37177126598 \nu^{11} + 163882987462 \nu^{10} + 92262525211 \nu^{9} - 377723445077 \nu^{8} - 191420819452 \nu^{7} + 529318619097 \nu^{6} + 308337281661 \nu^{5} - 473115271532 \nu^{4} - 270072920673 \nu^{3} + 252013415040 \nu^{2} + 84141702217 \nu - 59655036146$$$$)/ 2338032863$$ $$\beta_{4}$$ $$=$$ $$($$$$-197952229 \nu^{17} - 3918595304 \nu^{16} + 22600044356 \nu^{15} + 63480744292 \nu^{14} - 406133708186 \nu^{13} - 324605514650 \nu^{12} + 3091850363118 \nu^{11} + 212070080346 \nu^{10} - 12110742676833 \nu^{9} + 3292313272555 \nu^{8} + 25761837332094 \nu^{7} - 11293715285784 \nu^{6} - 29356638606926 \nu^{5} + 15052674500601 \nu^{4} + 16464410672688 \nu^{3} - 8838109885626 \nu^{2} - 3494627834537 \nu + 1867879847532$$$$)/ 4676065726$$ $$\beta_{5}$$ $$=$$ $$($$$$377442591 \nu^{17} - 3691417402 \nu^{16} + 2195758068 \nu^{15} + 67864507870 \nu^{14} - 136147624382 \nu^{13} - 463002782106 \nu^{12} + 1307571395468 \nu^{11} + 1393851573734 \nu^{10} - 5634532502087 \nu^{9} - 1407900557235 \nu^{8} + 12578032685850 \nu^{7} - 1636390467968 \nu^{6} - 14752318588402 \nu^{5} + 4796920139089 \nu^{4} + 8479774846946 \nu^{3} - 3614898714428 \nu^{2} - 1849586438863 \nu + 875081238676$$$$)/ 4676065726$$ $$\beta_{6}$$ $$=$$ $$($$$$280914143 \nu^{17} - 701368576 \nu^{16} - 6767277373 \nu^{15} + 16268509010 \nu^{14} + 67383774797 \nu^{13} - 153950006775 \nu^{12} - 359494769204 \nu^{11} + 767990167544 \nu^{10} + 1112789084625 \nu^{9} - 2178751678709 \nu^{8} - 2022050999427 \nu^{7} + 3538625257235 \nu^{6} + 2083530847860 \nu^{5} - 3158025449217 \nu^{4} - 1112397543996 \nu^{3} + 1407786874034 \nu^{2} + 239108215524 \nu - 245184253071$$$$)/ 2338032863$$ $$\beta_{7}$$ $$=$$ $$($$$$918896525 \nu^{17} - 5543735306 \nu^{16} - 8092976468 \nu^{15} + 105501268198 \nu^{14} - 60933186040 \nu^{13} - 765818676414 \nu^{12} + 1039858492108 \nu^{11} + 2637470974952 \nu^{10} - 5149492335763 \nu^{9} - 4229517789413 \nu^{8} + 12152589498648 \nu^{7} + 1974660952940 \nu^{6} - 14594279986318 \nu^{5} + 2221552590097 \nu^{4} + 8435391103120 \nu^{3} - 2660699161670 \nu^{2} - 1826393345453 \nu + 742560783748$$$$)/ 4676065726$$ $$\beta_{8}$$ $$=$$ $$($$$$1002969289 \nu^{17} - 5357577374 \nu^{16} - 11353710084 \nu^{15} + 102147035436 \nu^{14} - 15077828450 \nu^{13} - 742460530448 \nu^{12} + 719668444006 \nu^{11} + 2554781465752 \nu^{10} - 3915968147965 \nu^{9} - 4059181425139 \nu^{8} + 9456059751276 \nu^{7} + 1761924457730 \nu^{6} - 11330964484020 \nu^{5} + 2327689707151 \nu^{4} + 6443984989758 \nu^{3} - 2595584128412 \nu^{2} - 1361950041733 \nu + 677849390104$$$$)/ 4676065726$$ $$\beta_{9}$$ $$=$$ $$($$$$-685736770 \nu^{17} + 2668474422 \nu^{16} + 11910611780 \nu^{15} - 53542188699 \nu^{14} - 72646310628 \nu^{13} + 421926099202 \nu^{12} + 159397792583 \nu^{11} - 1669988944760 \nu^{10} + 105700448546 \nu^{9} + 3530406219995 \nu^{8} - 1020369463139 \nu^{7} - 3904262274773 \nu^{6} + 1630638051902 \nu^{5} + 2002598616766 \nu^{4} - 1055239106380 \nu^{3} - 288673423392 \nu^{2} + 241733576401 \nu - 42018281445$$$$)/ 2338032863$$ $$\beta_{10}$$ $$=$$ $$($$$$-1386154783 \nu^{17} + 6430527688 \nu^{16} + 19874972614 \nu^{15} - 125741499426 \nu^{14} - 62775758566 \nu^{13} + 953649814168 \nu^{12} - 338251157070 \nu^{11} - 3555019516198 \nu^{10} + 2820111137503 \nu^{9} + 6789169061169 \nu^{8} - 7571956262232 \nu^{7} - 6130162137078 \nu^{6} + 9466541681870 \nu^{5} + 1619918836209 \nu^{4} - 5505720453480 \nu^{3} + 800314810142 \nu^{2} + 1176511343261 \nu - 374413808574$$$$)/ 4676065726$$ $$\beta_{11}$$ $$=$$ $$($$$$1665088329 \nu^{17} - 6609375872 \nu^{16} - 28840683926 \nu^{15} + 133345072732 \nu^{14} + 175522841140 \nu^{13} - 1060816401736 \nu^{12} - 388928496010 \nu^{11} + 4271913813578 \nu^{10} - 195622849629 \nu^{9} - 9337524855313 \nu^{8} + 2171494179692 \nu^{7} + 11072418009430 \nu^{6} - 3313170350702 \nu^{5} - 6711664631477 \nu^{4} + 1978357909406 \nu^{3} + 1751178628584 \nu^{2} - 412599628507 \nu - 107954465010$$$$)/ 4676065726$$ $$\beta_{12}$$ $$=$$ $$($$$$1832524509 \nu^{17} - 6639698786 \nu^{16} - 33972458906 \nu^{15} + 135523924258 \nu^{14} + 236411293988 \nu^{13} - 1095061846114 \nu^{12} - 751627852732 \nu^{11} + 4503367193936 \nu^{10} + 969054521373 \nu^{9} - 10127247887873 \nu^{8} + 172575097992 \nu^{7} + 12485993487236 \nu^{6} - 1620756524402 \nu^{5} - 8006299422797 \nu^{4} + 1401444986280 \nu^{3} + 2295956452378 \nu^{2} - 373320812211 \nu - 180269480248$$$$)/ 4676065726$$ $$\beta_{13}$$ $$=$$ $$($$$$1216895826 \nu^{17} - 5844979198 \nu^{16} - 17002403395 \nu^{15} + 114758539026 \nu^{14} + 46183633539 \nu^{13} - 876187343173 \nu^{12} + 367014413536 \nu^{11} + 3304628929445 \nu^{10} - 2751485250448 \nu^{9} - 6454959292553 \nu^{8} + 7227541456661 \nu^{7} + 6146488078100 \nu^{6} - 8956043701859 \nu^{5} - 2059176049323 \nu^{4} + 5183979508206 \nu^{3} - 465598650622 \nu^{2} - 1108167717886 \nu + 306757426577$$$$)/ 2338032863$$ $$\beta_{14}$$ $$=$$ $$($$$$-2694174873 \nu^{17} + 8898905406 \nu^{16} + 54157234818 \nu^{15} - 186122321980 \nu^{14} - 433999335340 \nu^{13} + 1556547553040 \nu^{12} + 1809525369456 \nu^{11} - 6725442689912 \nu^{10} - 4359336517881 \nu^{9} + 16254679484487 \nu^{8} + 6453241055578 \nu^{7} - 22305169708038 \nu^{6} - 5981629649466 \nu^{5} + 16869186956217 \nu^{4} + 3171431762500 \nu^{3} - 6422460662454 \nu^{2} - 696794541829 \nu + 952441749148$$$$)/ 4676065726$$ $$\beta_{15}$$ $$=$$ $$($$$$1410676829 \nu^{17} - 5054336188 \nu^{16} - 26608639913 \nu^{15} + 103627914575 \nu^{14} + 192515702928 \nu^{13} - 843825093768 \nu^{12} - 677013696028 \nu^{11} + 3518625852588 \nu^{10} + 1225025755998 \nu^{9} - 8120029442208 \nu^{8} - 1160492675804 \nu^{7} + 10522113418083 \nu^{6} + 652018872805 \nu^{5} - 7446708263691 \nu^{4} - 293835473034 \nu^{3} + 2636488005199 \nu^{2} + 81529793478 \nu - 361693721127$$$$)/ 2338032863$$ $$\beta_{16}$$ $$=$$ $$($$$$1422023073 \nu^{17} - 4606595223 \nu^{16} - 28914727965 \nu^{15} + 96834645667 \nu^{14} + 234949310539 \nu^{13} - 814866392181 \nu^{12} - 993274871485 \nu^{11} + 3546564720314 \nu^{10} + 2406800470543 \nu^{9} - 8638012215790 \nu^{8} - 3497637260481 \nu^{7} + 11927671515175 \nu^{6} + 3064559023173 \nu^{5} - 9027042005382 \nu^{4} - 1501425031058 \nu^{3} + 3394560268895 \nu^{2} + 308210347484 \nu - 485286282032$$$$)/ 2338032863$$ $$\beta_{17}$$ $$=$$ $$($$$$3038182645 \nu^{17} - 11663900722 \nu^{16} - 54084465652 \nu^{15} + 236680213784 \nu^{14} + 347894361946 \nu^{13} - 1897444758324 \nu^{12} - 908861840806 \nu^{11} + 7720593308970 \nu^{10} + 328593632691 \nu^{9} - 17114585269829 \nu^{8} + 2814844117860 \nu^{7} + 20702464661638 \nu^{6} - 5221618643208 \nu^{5} - 12970037049775 \nu^{4} + 3467300081708 \nu^{3} + 3665906661752 \nu^{2} - 785496755045 \nu - 317577900352$$$$)/ 4676065726$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{14} - \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} + \beta_{2} + 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{15} + 2 \beta_{11} - \beta_{10} + \beta_{9} - \beta_{5} - \beta_{3} + 8 \beta_{2} + 15$$ $$\nu^{5}$$ $$=$$ $$-\beta_{17} - \beta_{16} - \beta_{15} - 9 \beta_{14} - \beta_{13} + \beta_{12} - 5 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} + 7 \beta_{7} - 6 \beta_{6} - 2 \beta_{5} + \beta_{4} - 3 \beta_{3} + 10 \beta_{2} + 21 \beta_{1} + 1$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{17} + \beta_{16} - 11 \beta_{15} - 2 \beta_{13} - 3 \beta_{12} + 24 \beta_{11} - 13 \beta_{10} + 12 \beta_{9} - \beta_{8} - 2 \beta_{6} - 12 \beta_{5} + \beta_{4} - 15 \beta_{3} + 57 \beta_{2} + 2 \beta_{1} + 87$$ $$\nu^{7}$$ $$=$$ $$-13 \beta_{17} - 14 \beta_{16} - 11 \beta_{15} - 69 \beta_{14} - 16 \beta_{13} + 18 \beta_{12} - 15 \beta_{11} + 6 \beta_{10} + 31 \beta_{9} + 2 \beta_{8} + 41 \beta_{7} - 31 \beta_{6} - 21 \beta_{5} + 13 \beta_{4} - 38 \beta_{3} + 84 \beta_{2} + 122 \beta_{1} + 16$$ $$\nu^{8}$$ $$=$$ $$19 \beta_{17} + 10 \beta_{16} - 88 \beta_{15} - 5 \beta_{14} - 35 \beta_{13} - 27 \beta_{12} + 218 \beta_{11} - 125 \beta_{10} + 117 \beta_{9} - 10 \beta_{8} - 5 \beta_{7} - 22 \beta_{6} - 104 \beta_{5} + 19 \beta_{4} - 154 \beta_{3} + 401 \beta_{2} + 31 \beta_{1} + 532$$ $$\nu^{9}$$ $$=$$ $$-129 \beta_{17} - 142 \beta_{16} - 90 \beta_{15} - 504 \beta_{14} - 177 \beta_{13} + 215 \beta_{12} + 34 \beta_{11} - 82 \beta_{10} + 337 \beta_{9} + 38 \beta_{8} + 222 \beta_{7} - 142 \beta_{6} - 170 \beta_{5} + 128 \beta_{4} - 358 \beta_{3} + 665 \beta_{2} + 746 \beta_{1} + 177$$ $$\nu^{10}$$ $$=$$ $$110 \beta_{17} + 52 \beta_{16} - 631 \beta_{15} - 100 \beta_{14} - 405 \beta_{13} - 123 \beta_{12} + 1793 \beta_{11} - 1069 \beta_{10} + 1050 \beta_{9} - 61 \beta_{8} - 96 \beta_{7} - 157 \beta_{6} - 807 \beta_{5} + 233 \beta_{4} - 1363 \beta_{3} + 2824 \beta_{2} + 329 \beta_{1} + 3349$$ $$\nu^{11}$$ $$=$$ $$-1143 \beta_{17} - 1271 \beta_{16} - 668 \beta_{15} - 3612 \beta_{14} - 1672 \beta_{13} + 2131 \beta_{12} + 1154 \beta_{11} - 1273 \beta_{10} + 3144 \beta_{9} + 452 \beta_{8} + 1111 \beta_{7} - 513 \beta_{6} - 1286 \beta_{5} + 1138 \beta_{4} - 3044 \beta_{3} + 5110 \beta_{2} + 4710 \beta_{1} + 1668$$ $$\nu^{12}$$ $$=$$ $$363 \beta_{17} + 9 \beta_{16} - 4324 \beta_{15} - 1303 \beta_{14} - 3942 \beta_{13} + 112 \beta_{12} + 14074 \beta_{11} - 8613 \beta_{10} + 8939 \beta_{9} - 226 \beta_{8} - 1201 \beta_{7} - 839 \beta_{6} - 5980 \beta_{5} + 2372 \beta_{4} - 11202 \beta_{3} + 19938 \beta_{2} + 2999 \beta_{1} + 21527$$ $$\nu^{13}$$ $$=$$ $$-9516 \beta_{17} - 10684 \beta_{16} - 4773 \beta_{15} - 25720 \beta_{14} - 14522 \beta_{13} + 19102 \beta_{12} + 14051 \beta_{11} - 12989 \beta_{10} + 27005 \beta_{9} + 4393 \beta_{8} + 4867 \beta_{7} - 555 \beta_{6} - 9572 \beta_{5} + 9620 \beta_{4} - 24686 \beta_{3} + 38571 \beta_{2} + 30427 \beta_{1} + 14428$$ $$\nu^{14}$$ $$=$$ $$-1271 \beta_{17} - 3636 \beta_{16} - 29054 \beta_{15} - 14050 \beta_{14} - 35039 \beta_{13} + 9835 \beta_{12} + 107749 \beta_{11} - 67069 \beta_{10} + 73347 \beta_{9} + 373 \beta_{8} - 12477 \beta_{7} - 2680 \beta_{6} - 43413 \beta_{5} + 21875 \beta_{4} - 88303 \beta_{3} + 141110 \beta_{2} + 25288 \beta_{1} + 140724$$ $$\nu^{15}$$ $$=$$ $$-76316 \beta_{17} - 86672 \beta_{16} - 33603 \beta_{15} - 183102 \beta_{14} - 120050 \beta_{13} + 161457 \beta_{12} + 137444 \beta_{11} - 115631 \beta_{10} + 220799 \beta_{9} + 38344 \beta_{8} + 14983 \beta_{7} + 15344 \beta_{6} - 71221 \beta_{5} + 78982 \beta_{4} - 195039 \beta_{3} + 287784 \beta_{2} + 200115 \beta_{1} + 118635$$ $$\nu^{16}$$ $$=$$ $$-39322 \beta_{17} - 57635 \beta_{16} - 193666 \beta_{15} - 136266 \beta_{14} - 295219 \beta_{13} + 142040 \beta_{12} + 813230 \beta_{11} - 511554 \beta_{10} + 586247 \beta_{9} + 18257 \beta_{8} - 117170 \beta_{7} + 10258 \beta_{6} - 312410 \beta_{5} + 190191 \beta_{4} - 678773 \beta_{3} + 1001004 \beta_{2} + 203810 \beta_{1} + 933162$$ $$\nu^{17}$$ $$=$$ $$-597910 \beta_{17} - 688126 \beta_{16} - 235431 \beta_{15} - 1307213 \beta_{14} - 962160 \beta_{13} + 1316142 \beta_{12} + 1220338 \beta_{11} - 964358 \beta_{10} + 1749909 \beta_{9} + 314188 \beta_{8} - 21463 \beta_{7} + 230986 \beta_{6} - 531815 \beta_{5} + 636273 \beta_{4} - 1515957 \beta_{3} + 2130566 \beta_{2} + 1335686 \beta_{1} + 944762$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0.826129 −0.924759 2.59964 0.763493 −0.957552 −0.735255 1.35726 1.98431 −1.15793 −2.24960 −1.87675 1.74487 2.72204 −1.52216 1.04467 −2.50138 0.523506 2.35947
0 −2.26041 0 −0.786316 0 5.06179 0 2.10946 0
1.2 0 −2.22306 0 −3.95421 0 3.08028 0 1.94201 0
1.3 0 −2.09484 0 −3.02323 0 0.561390 0 1.38836 0
1.4 0 −1.83524 0 −1.51218 0 −1.20167 0 0.368111 0
1.5 0 −0.564538 0 −4.10274 0 −4.97202 0 −2.68130 0
1.6 0 −0.544167 0 0.962787 0 3.25298 0 −2.70388 0
1.7 0 −0.387927 0 −1.46929 0 −0.194696 0 −2.84951 0
1.8 0 0.150114 0 2.87852 0 2.68467 0 −2.97747 0
1.9 0 0.220625 0 −0.421419 0 0.645304 0 −2.95132 0
1.10 0 0.790850 0 −3.96974 0 4.97706 0 −2.37456 0
1.11 0 1.13919 0 −1.30620 0 1.71403 0 −1.70224 0
1.12 0 1.27304 0 −3.61409 0 −4.28084 0 −1.37936 0
1.13 0 1.76734 0 0.469688 0 −1.03831 0 0.123492 0
1.14 0 2.58636 0 1.24712 0 0.899316 0 3.68924 0
1.15 0 2.71791 0 0.714085 0 2.03236 0 4.38705 0
1.16 0 3.08733 0 −3.57921 0 −1.44216 0 6.53160 0
1.17 0 3.08736 0 −1.35183 0 −3.60927 0 6.53179 0
1.18 0 3.09007 0 −4.18174 0 2.82979 0 6.54852 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$547$$ $$-1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8752.2.a.s 18
4.b odd 2 1 547.2.a.b 18
12.b even 2 1 4923.2.a.l 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.2.a.b 18 4.b odd 2 1
4923.2.a.l 18 12.b even 2 1
8752.2.a.s 18 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8752))$$:

 $$T_{3}^{18} - \cdots$$ $$T_{5}^{18} + \cdots$$ $$T_{7}^{18} - \cdots$$ $$T_{11}^{18} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 10 T + 70 T^{2} - 369 T^{3} + 1646 T^{4} - 6380 T^{5} + 22262 T^{6} - 70831 T^{7} + 208802 T^{8} - 574237 T^{9} + 1485851 T^{10} - 3631572 T^{11} + 8425641 T^{12} - 18599141 T^{13} + 39182807 T^{14} - 78875280 T^{15} + 151998095 T^{16} - 280521922 T^{17} + 496314970 T^{18} - 841565766 T^{19} + 1367982855 T^{20} - 2129632560 T^{21} + 3173807367 T^{22} - 4519591263 T^{23} + 6142292289 T^{24} - 7942247964 T^{25} + 9748668411 T^{26} - 11302706871 T^{27} + 12329549298 T^{28} - 12547499157 T^{29} + 11830939542 T^{30} - 10171780740 T^{31} + 7872766974 T^{32} - 5294746683 T^{33} + 3013270470 T^{34} - 1291401630 T^{35} + 387420489 T^{36}$$
$5$ $$1 + 27 T + 394 T^{2} + 4066 T^{3} + 33017 T^{4} + 223247 T^{5} + 1301478 T^{6} + 6696595 T^{7} + 30923803 T^{8} + 129765410 T^{9} + 499584606 T^{10} + 1777920166 T^{11} + 5884067150 T^{12} + 18197020803 T^{13} + 52790609192 T^{14} + 144100415815 T^{15} + 370957947126 T^{16} + 902062261929 T^{17} + 2074042607038 T^{18} + 4510311309645 T^{19} + 9273948678150 T^{20} + 18012551976875 T^{21} + 32994130745000 T^{22} + 56865690009375 T^{23} + 91938549218750 T^{24} + 138900012968750 T^{25} + 195150236718750 T^{26} + 253448066406250 T^{27} + 301990263671875 T^{28} + 326982177734375 T^{29} + 317743652343750 T^{30} + 272518310546875 T^{31} + 201519775390625 T^{32} + 124084472656250 T^{33} + 60119628906250 T^{34} + 20599365234375 T^{35} + 3814697265625 T^{36}$$
$7$ $$1 - 11 T + 109 T^{2} - 718 T^{3} + 4269 T^{4} - 20685 T^{5} + 92163 T^{6} - 360502 T^{7} + 1322737 T^{8} - 4452817 T^{9} + 14363197 T^{10} - 43877097 T^{11} + 130743026 T^{12} - 375593700 T^{13} + 1062872977 T^{14} - 2919179505 T^{15} + 7963303304 T^{16} - 21214026883 T^{17} + 56674382450 T^{18} - 148498188181 T^{19} + 390201861896 T^{20} - 1001278570215 T^{21} + 2551958017777 T^{22} - 6312603315900 T^{23} + 15381786265874 T^{24} - 36134676094671 T^{25} + 82800972428797 T^{26} - 179687227260919 T^{27} + 373640463436513 T^{28} - 712830245504986 T^{29} + 1275654552305763 T^{30} - 2004149180268795 T^{31} + 2895334297992381 T^{32} - 3408749164139074 T^{33} + 3622389432086509 T^{34} - 2558935653859277 T^{35} + 1628413597910449 T^{36}$$
$11$ $$1 + 2 T + 85 T^{2} + 150 T^{3} + 3546 T^{4} + 5526 T^{5} + 98861 T^{6} + 141478 T^{7} + 2126155 T^{8} + 2969051 T^{9} + 38313335 T^{10} + 54369337 T^{11} + 603595390 T^{12} + 870619543 T^{13} + 8472071478 T^{14} + 12151342611 T^{15} + 107165383120 T^{16} + 149528311671 T^{17} + 1233316997354 T^{18} + 1644811428381 T^{19} + 12967011357520 T^{20} + 16173437015241 T^{21} + 124039598509398 T^{22} + 140214148019693 T^{23} + 1069306052703790 T^{24} + 1059504567275627 T^{25} + 8212803617978135 T^{26} + 7000866949911241 T^{27} + 55146985002539155 T^{28} + 40365324534703058 T^{29} + 310268167751014781 T^{30} + 190772507307362706 T^{31} + 1346592909886172586 T^{32} + 626587225412347650 T^{33} + 3905727038403633685 T^{34} + 1010894056998587542 T^{35} + 5559917313492231481 T^{36}$$
$13$ $$1 + 25 T + 419 T^{2} + 5077 T^{3} + 50838 T^{4} + 430037 T^{5} + 3215885 T^{6} + 21488134 T^{7} + 131328937 T^{8} + 738626108 T^{9} + 3880155769 T^{10} + 19109203885 T^{11} + 89183382015 T^{12} + 395302024323 T^{13} + 1678118625851 T^{14} + 6828960297885 T^{15} + 26814195655203 T^{16} + 101525943915242 T^{17} + 372390103391781 T^{18} + 1319837270898146 T^{19} + 4531599065729307 T^{20} + 15003225774453345 T^{21} + 47928746072930411 T^{22} + 146772874516959639 T^{23} + 430471150960440135 T^{24} + 1199074204834388545 T^{25} + 3165162263038679449 T^{26} + 7832760099167430284 T^{27} + 18104809190952334513 T^{28} + 38510182696559856958 T^{29} + 74923962474109810685 T^{30} +$$$$13\!\cdots\!61$$$$T^{31} +$$$$20\!\cdots\!82$$$$T^{32} +$$$$25\!\cdots\!89$$$$T^{33} +$$$$27\!\cdots\!79$$$$T^{34} +$$$$21\!\cdots\!25$$$$T^{35} +$$$$11\!\cdots\!29$$$$T^{36}$$
$17$ $$1 + 30 T + 584 T^{2} + 8365 T^{3} + 98434 T^{4} + 985682 T^{5} + 8696277 T^{6} + 68768626 T^{7} + 495900647 T^{8} + 3296047078 T^{9} + 20410580082 T^{10} + 118632779583 T^{11} + 651877922855 T^{12} + 3403006356186 T^{13} + 16950930016609 T^{14} + 80772988719250 T^{15} + 368969665979574 T^{16} + 1617067599179214 T^{17} + 6804533522448758 T^{18} + 27490149186046638 T^{19} + 106632233468096886 T^{20} + 396837693577675250 T^{21} + 1415758625917200289 T^{22} + 4831782395875185402 T^{23} + 15734748342489239495 T^{24} + 48679617348389713359 T^{25} +$$$$14\!\cdots\!62$$$$T^{26} +$$$$39\!\cdots\!66$$$$T^{27} +$$$$99\!\cdots\!03$$$$T^{28} +$$$$23\!\cdots\!58$$$$T^{29} +$$$$50\!\cdots\!97$$$$T^{30} +$$$$97\!\cdots\!34$$$$T^{31} +$$$$16\!\cdots\!86$$$$T^{32} +$$$$23\!\cdots\!45$$$$T^{33} +$$$$28\!\cdots\!04$$$$T^{34} +$$$$24\!\cdots\!10$$$$T^{35} +$$$$14\!\cdots\!09$$$$T^{36}$$
$19$ $$1 + 4 T + 130 T^{2} + 516 T^{3} + 8958 T^{4} + 36501 T^{5} + 441125 T^{6} + 1838457 T^{7} + 17317003 T^{8} + 72845222 T^{9} + 571934352 T^{10} + 2393239283 T^{11} + 16359891658 T^{12} + 67164262668 T^{13} + 412288983795 T^{14} + 1640265737352 T^{15} + 9248689375227 T^{16} + 35240518894600 T^{17} + 185627099738938 T^{18} + 669569858997400 T^{19} + 3338776864456947 T^{20} + 11250582692497368 T^{21} + 53729912657148195 T^{22} + 166305363627972132 T^{23} + 769665516115160698 T^{24} + 2139248959738323137 T^{25} + 9713483122505484432 T^{26} + 23506256981380161938 T^{27} +$$$$10\!\cdots\!03$$$$T^{28} +$$$$21\!\cdots\!83$$$$T^{29} +$$$$97\!\cdots\!25$$$$T^{30} +$$$$15\!\cdots\!59$$$$T^{31} +$$$$71\!\cdots\!18$$$$T^{32} +$$$$78\!\cdots\!84$$$$T^{33} +$$$$37\!\cdots\!30$$$$T^{34} +$$$$21\!\cdots\!56$$$$T^{35} +$$$$10\!\cdots\!41$$$$T^{36}$$
$23$ $$1 - 26 T + 516 T^{2} - 7395 T^{3} + 90928 T^{4} - 951516 T^{5} + 8960211 T^{6} - 75889920 T^{7} + 593605359 T^{8} - 4293455106 T^{9} + 29167458828 T^{10} - 186453853585 T^{11} + 1133632710313 T^{12} - 6568187578396 T^{13} + 36553007673681 T^{14} - 195688621929534 T^{15} + 1013582264018988 T^{16} - 5081312191544094 T^{17} + 24743429994041086 T^{18} - 116870180405514162 T^{19} + 536185017666044652 T^{20} - 2380943463016640178 T^{21} + 10229030220410564721 T^{22} - 42275108142896045828 T^{23} +$$$$16\!\cdots\!57$$$$T^{24} -$$$$63\!\cdots\!95$$$$T^{25} +$$$$22\!\cdots\!68$$$$T^{26} -$$$$77\!\cdots\!78$$$$T^{27} +$$$$24\!\cdots\!91$$$$T^{28} -$$$$72\!\cdots\!40$$$$T^{29} +$$$$19\!\cdots\!31$$$$T^{30} -$$$$47\!\cdots\!28$$$$T^{31} +$$$$10\!\cdots\!52$$$$T^{32} -$$$$19\!\cdots\!65$$$$T^{33} +$$$$31\!\cdots\!76$$$$T^{34} -$$$$36\!\cdots\!78$$$$T^{35} +$$$$32\!\cdots\!69$$$$T^{36}$$
$29$ $$1 + 18 T + 531 T^{2} + 7242 T^{3} + 124429 T^{4} + 1388519 T^{5} + 17800604 T^{6} + 169607253 T^{7} + 1778477154 T^{8} + 14850454505 T^{9} + 133353152337 T^{10} + 992168503525 T^{11} + 7832929693192 T^{12} + 52481764441320 T^{13} + 370156555204319 T^{14} + 2247849800307197 T^{15} + 14304279024344391 T^{16} + 78983107272313558 T^{17} + 456054513586518578 T^{18} + 2290510110897093182 T^{19} + 12029898659473632831 T^{20} + 54822808779692227633 T^{21} +$$$$26\!\cdots\!39$$$$T^{22} +$$$$10\!\cdots\!80$$$$T^{23} +$$$$46\!\cdots\!32$$$$T^{24} +$$$$17\!\cdots\!25$$$$T^{25} +$$$$66\!\cdots\!57$$$$T^{26} +$$$$21\!\cdots\!45$$$$T^{27} +$$$$74\!\cdots\!54$$$$T^{28} +$$$$20\!\cdots\!37$$$$T^{29} +$$$$62\!\cdots\!64$$$$T^{30} +$$$$14\!\cdots\!91$$$$T^{31} +$$$$37\!\cdots\!49$$$$T^{32} +$$$$62\!\cdots\!58$$$$T^{33} +$$$$13\!\cdots\!51$$$$T^{34} +$$$$13\!\cdots\!62$$$$T^{35} +$$$$21\!\cdots\!61$$$$T^{36}$$
$31$ $$1 - 5 T + 226 T^{2} - 736 T^{3} + 25466 T^{4} - 48427 T^{5} + 1947515 T^{6} - 1181005 T^{7} + 114803460 T^{8} + 68748799 T^{9} + 5711179330 T^{10} + 8784337491 T^{11} + 254611322915 T^{12} + 519109476169 T^{13} + 10413960551250 T^{14} + 22094105896997 T^{15} + 386758640569635 T^{16} + 776800932680999 T^{17} + 12771866096424756 T^{18} + 24080828913110969 T^{19} + 371675053587419235 T^{20} + 658205508777437627 T^{21} + 9617511262250951250 T^{22} + 14861663578773202519 T^{23} +$$$$22\!\cdots\!15$$$$T^{24} +$$$$24\!\cdots\!01$$$$T^{25} +$$$$48\!\cdots\!30$$$$T^{26} +$$$$18\!\cdots\!29$$$$T^{27} +$$$$94\!\cdots\!60$$$$T^{28} -$$$$30\!\cdots\!55$$$$T^{29} +$$$$15\!\cdots\!15$$$$T^{30} -$$$$11\!\cdots\!57$$$$T^{31} +$$$$19\!\cdots\!86$$$$T^{32} -$$$$17\!\cdots\!36$$$$T^{33} +$$$$16\!\cdots\!06$$$$T^{34} -$$$$11\!\cdots\!55$$$$T^{35} +$$$$69\!\cdots\!41$$$$T^{36}$$
$37$ $$1 + 18 T + 477 T^{2} + 6182 T^{3} + 98542 T^{4} + 1025448 T^{5} + 12472914 T^{6} + 109789868 T^{7} + 1115943998 T^{8} + 8578978218 T^{9} + 76645978734 T^{10} + 526915335411 T^{11} + 4285047049335 T^{12} + 26892539266031 T^{13} + 204473204412967 T^{14} + 1194720790323650 T^{15} + 8656318678483359 T^{16} + 47928410626044380 T^{17} + 333914579074632830 T^{18} + 1773351193163642060 T^{19} + 11850500270843718471 T^{20} + 60516192192263843450 T^{21} +$$$$38\!\cdots\!87$$$$T^{22} +$$$$18\!\cdots\!67$$$$T^{23} +$$$$10\!\cdots\!15$$$$T^{24} +$$$$50\!\cdots\!63$$$$T^{25} +$$$$26\!\cdots\!14$$$$T^{26} +$$$$11\!\cdots\!86$$$$T^{27} +$$$$53\!\cdots\!02$$$$T^{28} +$$$$19\!\cdots\!84$$$$T^{29} +$$$$82\!\cdots\!34$$$$T^{30} +$$$$24\!\cdots\!56$$$$T^{31} +$$$$88\!\cdots\!38$$$$T^{32} +$$$$20\!\cdots\!26$$$$T^{33} +$$$$58\!\cdots\!57$$$$T^{34} +$$$$82\!\cdots\!06$$$$T^{35} +$$$$16\!\cdots\!29$$$$T^{36}$$
$41$ $$1 + 17 T + 500 T^{2} + 6935 T^{3} + 118746 T^{4} + 1411049 T^{5} + 18245075 T^{6} + 191387044 T^{7} + 2060476099 T^{8} + 19451816951 T^{9} + 182957350222 T^{10} + 1574307116444 T^{11} + 13284361461403 T^{12} + 105065275805631 T^{13} + 807813171251088 T^{14} + 5903001862494918 T^{15} + 41738761877644030 T^{16} + 282544368422135749 T^{17} + 1846894667477774252 T^{18} + 11584319105307565709 T^{19} + 70162858716319614430 T^{20} +$$$$40\!\cdots\!78$$$$T^{21} +$$$$22\!\cdots\!68$$$$T^{22} +$$$$12\!\cdots\!31$$$$T^{23} +$$$$63\!\cdots\!23$$$$T^{24} +$$$$30\!\cdots\!64$$$$T^{25} +$$$$14\!\cdots\!62$$$$T^{26} +$$$$63\!\cdots\!11$$$$T^{27} +$$$$27\!\cdots\!99$$$$T^{28} +$$$$10\!\cdots\!04$$$$T^{29} +$$$$41\!\cdots\!75$$$$T^{30} +$$$$13\!\cdots\!29$$$$T^{31} +$$$$45\!\cdots\!06$$$$T^{32} +$$$$10\!\cdots\!35$$$$T^{33} +$$$$31\!\cdots\!00$$$$T^{34} +$$$$44\!\cdots\!77$$$$T^{35} +$$$$10\!\cdots\!21$$$$T^{36}$$
$43$ $$1 + 8 T + 492 T^{2} + 3407 T^{3} + 117668 T^{4} + 715275 T^{5} + 18347951 T^{6} + 99049565 T^{7} + 2107507666 T^{8} + 10209475962 T^{9} + 190684760633 T^{10} + 837060417712 T^{11} + 14161353767533 T^{12} + 56841478497834 T^{13} + 886000874851046 T^{14} + 3276977381712634 T^{15} + 47444588926667291 T^{16} + 162652673998347287 T^{17} + 2192977866661315750 T^{18} + 6994064981928933341 T^{19} + 87725044925407821059 T^{20} +$$$$26\!\cdots\!38$$$$T^{21} +$$$$30\!\cdots\!46$$$$T^{22} +$$$$83\!\cdots\!62$$$$T^{23} +$$$$89\!\cdots\!17$$$$T^{24} +$$$$22\!\cdots\!84$$$$T^{25} +$$$$22\!\cdots\!33$$$$T^{26} +$$$$51\!\cdots\!66$$$$T^{27} +$$$$45\!\cdots\!34$$$$T^{28} +$$$$92\!\cdots\!55$$$$T^{29} +$$$$73\!\cdots\!51$$$$T^{30} +$$$$12\!\cdots\!25$$$$T^{31} +$$$$86\!\cdots\!32$$$$T^{32} +$$$$10\!\cdots\!49$$$$T^{33} +$$$$67\!\cdots\!92$$$$T^{34} +$$$$46\!\cdots\!44$$$$T^{35} +$$$$25\!\cdots\!49$$$$T^{36}$$
$47$ $$1 - 52 T + 1865 T^{2} - 49131 T^{3} + 1077176 T^{4} - 20125023 T^{5} + 332676442 T^{6} - 4927129277 T^{7} + 66491991801 T^{8} - 823722681768 T^{9} + 9453432637832 T^{10} - 100973005595782 T^{11} + 1009215441829873 T^{12} - 9466179137600663 T^{13} + 83603559634490998 T^{14} - 696394998474434729 T^{15} + 5481817752006743685 T^{16} - 40807789859381398323 T^{17} +$$$$28\!\cdots\!97$$$$T^{18} -$$$$19\!\cdots\!81$$$$T^{19} +$$$$12\!\cdots\!65$$$$T^{20} -$$$$72\!\cdots\!67$$$$T^{21} +$$$$40\!\cdots\!38$$$$T^{22} -$$$$21\!\cdots\!41$$$$T^{23} +$$$$10\!\cdots\!17$$$$T^{24} -$$$$51\!\cdots\!66$$$$T^{25} +$$$$22\!\cdots\!52$$$$T^{26} -$$$$92\!\cdots\!56$$$$T^{27} +$$$$34\!\cdots\!49$$$$T^{28} -$$$$12\!\cdots\!31$$$$T^{29} +$$$$38\!\cdots\!22$$$$T^{30} -$$$$10\!\cdots\!21$$$$T^{31} +$$$$27\!\cdots\!44$$$$T^{32} -$$$$59\!\cdots\!33$$$$T^{33} +$$$$10\!\cdots\!65$$$$T^{34} -$$$$13\!\cdots\!24$$$$T^{35} +$$$$12\!\cdots\!89$$$$T^{36}$$
$53$ $$1 + 60 T + 2292 T^{2} + 64439 T^{3} + 1482802 T^{4} + 29045053 T^{5} + 500611675 T^{6} + 7729740473 T^{7} + 108589762152 T^{8} + 1401743191155 T^{9} + 16770166121672 T^{10} + 187053090421054 T^{11} + 1955182548973064 T^{12} + 19221087500660183 T^{13} + 178277342948315803 T^{14} + 1563393906027613236 T^{15} + 12985863111074957420 T^{16} +$$$$10\!\cdots\!72$$$$T^{17} +$$$$76\!\cdots\!76$$$$T^{18} +$$$$54\!\cdots\!16$$$$T^{19} +$$$$36\!\cdots\!80$$$$T^{20} +$$$$23\!\cdots\!72$$$$T^{21} +$$$$14\!\cdots\!43$$$$T^{22} +$$$$80\!\cdots\!19$$$$T^{23} +$$$$43\!\cdots\!56$$$$T^{24} +$$$$21\!\cdots\!98$$$$T^{25} +$$$$10\!\cdots\!92$$$$T^{26} +$$$$46\!\cdots\!15$$$$T^{27} +$$$$18\!\cdots\!48$$$$T^{28} +$$$$71\!\cdots\!81$$$$T^{29} +$$$$24\!\cdots\!75$$$$T^{30} +$$$$75\!\cdots\!69$$$$T^{31} +$$$$20\!\cdots\!38$$$$T^{32} +$$$$47\!\cdots\!23$$$$T^{33} +$$$$88\!\cdots\!32$$$$T^{34} +$$$$12\!\cdots\!80$$$$T^{35} +$$$$10\!\cdots\!89$$$$T^{36}$$
$59$ $$1 - 8 T + 547 T^{2} - 4886 T^{3} + 151709 T^{4} - 1426288 T^{5} + 28528730 T^{6} - 268965857 T^{7} + 4083691371 T^{8} - 37250203024 T^{9} + 471936791901 T^{10} - 4078707707888 T^{11} + 45548114306989 T^{12} - 370272133366719 T^{13} + 3754286226421140 T^{14} - 28722234855750938 T^{15} + 268673863454487823 T^{16} - 1936169927933708248 T^{17} + 16882310523288642930 T^{18} -$$$$11\!\cdots\!32$$$$T^{19} +$$$$93\!\cdots\!63$$$$T^{20} -$$$$58\!\cdots\!02$$$$T^{21} +$$$$45\!\cdots\!40$$$$T^{22} -$$$$26\!\cdots\!81$$$$T^{23} +$$$$19\!\cdots\!49$$$$T^{24} -$$$$10\!\cdots\!72$$$$T^{25} +$$$$69\!\cdots\!21$$$$T^{26} -$$$$32\!\cdots\!36$$$$T^{27} +$$$$20\!\cdots\!71$$$$T^{28} -$$$$81\!\cdots\!63$$$$T^{29} +$$$$50\!\cdots\!30$$$$T^{30} -$$$$14\!\cdots\!52$$$$T^{31} +$$$$93\!\cdots\!49$$$$T^{32} -$$$$17\!\cdots\!14$$$$T^{33} +$$$$11\!\cdots\!27$$$$T^{34} -$$$$10\!\cdots\!52$$$$T^{35} +$$$$75\!\cdots\!21$$$$T^{36}$$
$61$ $$1 + 26 T + 1092 T^{2} + 21198 T^{3} + 517593 T^{4} + 8124611 T^{5} + 147632317 T^{6} + 1959197923 T^{7} + 29069634275 T^{8} + 335550327593 T^{9} + 4266607342662 T^{10} + 43706374725751 T^{11} + 490749289239130 T^{12} + 4530429185717696 T^{13} + 45852509636769219 T^{14} + 386091138768948475 T^{15} + 3572270707620359676 T^{16} + 27670900627019165169 T^{17} +$$$$23\!\cdots\!26$$$$T^{18} +$$$$16\!\cdots\!09$$$$T^{19} +$$$$13\!\cdots\!96$$$$T^{20} +$$$$87\!\cdots\!75$$$$T^{21} +$$$$63\!\cdots\!79$$$$T^{22} +$$$$38\!\cdots\!96$$$$T^{23} +$$$$25\!\cdots\!30$$$$T^{24} +$$$$13\!\cdots\!71$$$$T^{25} +$$$$81\!\cdots\!22$$$$T^{26} +$$$$39\!\cdots\!13$$$$T^{27} +$$$$20\!\cdots\!75$$$$T^{28} +$$$$85\!\cdots\!03$$$$T^{29} +$$$$39\!\cdots\!57$$$$T^{30} +$$$$13\!\cdots\!91$$$$T^{31} +$$$$51\!\cdots\!13$$$$T^{32} +$$$$12\!\cdots\!98$$$$T^{33} +$$$$40\!\cdots\!12$$$$T^{34} +$$$$58\!\cdots\!46$$$$T^{35} +$$$$13\!\cdots\!81$$$$T^{36}$$
$67$ $$1 + 12 T + 743 T^{2} + 8061 T^{3} + 273924 T^{4} + 2736258 T^{5} + 67007649 T^{6} + 622982958 T^{7} + 12221255375 T^{8} + 106355514112 T^{9} + 1765865096326 T^{10} + 14415408887134 T^{11} + 209411947253484 T^{12} + 1603069573249209 T^{13} + 20828295096643948 T^{14} + 149173768364533151 T^{15} + 1760455572360289685 T^{16} + 11748171421911845627 T^{17} +$$$$12\!\cdots\!21$$$$T^{18} +$$$$78\!\cdots\!09$$$$T^{19} +$$$$79\!\cdots\!65$$$$T^{20} +$$$$44\!\cdots\!13$$$$T^{21} +$$$$41\!\cdots\!08$$$$T^{22} +$$$$21\!\cdots\!63$$$$T^{23} +$$$$18\!\cdots\!96$$$$T^{24} +$$$$87\!\cdots\!82$$$$T^{25} +$$$$71\!\cdots\!66$$$$T^{26} +$$$$28\!\cdots\!64$$$$T^{27} +$$$$22\!\cdots\!75$$$$T^{28} +$$$$76\!\cdots\!14$$$$T^{29} +$$$$54\!\cdots\!89$$$$T^{30} +$$$$15\!\cdots\!46$$$$T^{31} +$$$$10\!\cdots\!96$$$$T^{32} +$$$$19\!\cdots\!23$$$$T^{33} +$$$$12\!\cdots\!83$$$$T^{34} +$$$$13\!\cdots\!24$$$$T^{35} +$$$$74\!\cdots\!09$$$$T^{36}$$
$71$ $$1 - T + 570 T^{2} + 395 T^{3} + 165367 T^{4} + 402481 T^{5} + 32659200 T^{6} + 135249850 T^{7} + 4976102531 T^{8} + 28212173634 T^{9} + 627081345260 T^{10} + 4301680526387 T^{11} + 68002974751364 T^{12} + 517699284019560 T^{13} + 6477452704212298 T^{14} + 51294377851106202 T^{15} + 547109894053610156 T^{16} + 4280850522490499670 T^{17} + 41143559904963586258 T^{18} +$$$$30\!\cdots\!70$$$$T^{19} +$$$$27\!\cdots\!96$$$$T^{20} +$$$$18\!\cdots\!22$$$$T^{21} +$$$$16\!\cdots\!38$$$$T^{22} +$$$$93\!\cdots\!60$$$$T^{23} +$$$$87\!\cdots\!44$$$$T^{24} +$$$$39\!\cdots\!17$$$$T^{25} +$$$$40\!\cdots\!60$$$$T^{26} +$$$$12\!\cdots\!54$$$$T^{27} +$$$$16\!\cdots\!31$$$$T^{28} +$$$$31\!\cdots\!50$$$$T^{29} +$$$$53\!\cdots\!00$$$$T^{30} +$$$$46\!\cdots\!91$$$$T^{31} +$$$$13\!\cdots\!27$$$$T^{32} +$$$$23\!\cdots\!45$$$$T^{33} +$$$$23\!\cdots\!70$$$$T^{34} -$$$$29\!\cdots\!91$$$$T^{35} +$$$$21\!\cdots\!61$$$$T^{36}$$
$73$ $$1 + 2 T + 402 T^{2} + 2100 T^{3} + 91054 T^{4} + 664143 T^{5} + 16192289 T^{6} + 128526512 T^{7} + 2389630448 T^{8} + 19163681612 T^{9} + 298745214234 T^{10} + 2361654820993 T^{11} + 32513178733431 T^{12} + 249000864582652 T^{13} + 3122389883437745 T^{14} + 23022354077637360 T^{15} + 267483657127421832 T^{16} + 1885230323459498199 T^{17} + 20590875702168070970 T^{18} +$$$$13\!\cdots\!27$$$$T^{19} +$$$$14\!\cdots\!28$$$$T^{20} +$$$$89\!\cdots\!20$$$$T^{21} +$$$$88\!\cdots\!45$$$$T^{22} +$$$$51\!\cdots\!36$$$$T^{23} +$$$$49\!\cdots\!59$$$$T^{24} +$$$$26\!\cdots\!21$$$$T^{25} +$$$$24\!\cdots\!54$$$$T^{26} +$$$$11\!\cdots\!56$$$$T^{27} +$$$$10\!\cdots\!52$$$$T^{28} +$$$$40\!\cdots\!24$$$$T^{29} +$$$$37\!\cdots\!69$$$$T^{30} +$$$$11\!\cdots\!19$$$$T^{31} +$$$$11\!\cdots\!86$$$$T^{32} +$$$$18\!\cdots\!00$$$$T^{33} +$$$$26\!\cdots\!22$$$$T^{34} +$$$$94\!\cdots\!06$$$$T^{35} +$$$$34\!\cdots\!69$$$$T^{36}$$
$79$ $$1 + 18 T + 1297 T^{2} + 20031 T^{3} + 786188 T^{4} + 10631617 T^{5} + 298506186 T^{6} + 3585561802 T^{7} + 80086048911 T^{8} + 863338284200 T^{9} + 16209659796490 T^{10} + 157996153097097 T^{11} + 2576626887472372 T^{12} + 22823417890305456 T^{13} + 330077405985329133 T^{14} + 2664890406520132266 T^{15} + 34639279636509263037 T^{16} +$$$$25\!\cdots\!59$$$$T^{17} +$$$$30\!\cdots\!66$$$$T^{18} +$$$$20\!\cdots\!61$$$$T^{19} +$$$$21\!\cdots\!17$$$$T^{20} +$$$$13\!\cdots\!74$$$$T^{21} +$$$$12\!\cdots\!73$$$$T^{22} +$$$$70\!\cdots\!44$$$$T^{23} +$$$$62\!\cdots\!12$$$$T^{24} +$$$$30\!\cdots\!23$$$$T^{25} +$$$$24\!\cdots\!90$$$$T^{26} +$$$$10\!\cdots\!00$$$$T^{27} +$$$$75\!\cdots\!11$$$$T^{28} +$$$$26\!\cdots\!58$$$$T^{29} +$$$$17\!\cdots\!26$$$$T^{30} +$$$$49\!\cdots\!63$$$$T^{31} +$$$$28\!\cdots\!28$$$$T^{32} +$$$$58\!\cdots\!69$$$$T^{33} +$$$$29\!\cdots\!37$$$$T^{34} +$$$$32\!\cdots\!62$$$$T^{35} +$$$$14\!\cdots\!61$$$$T^{36}$$
$83$ $$1 - 43 T + 1214 T^{2} - 27029 T^{3} + 517233 T^{4} - 8734852 T^{5} + 134157792 T^{6} - 1904363732 T^{7} + 25324885272 T^{8} - 318326987644 T^{9} + 3816043239532 T^{10} - 43881446080752 T^{11} + 486363584266611 T^{12} - 5212562566603327 T^{13} + 54157832160852686 T^{14} - 546073362822746053 T^{15} + 5347456077270647030 T^{16} - 50872297755235833738 T^{17} +$$$$47\!\cdots\!82$$$$T^{18} -$$$$42\!\cdots\!54$$$$T^{19} +$$$$36\!\cdots\!70$$$$T^{20} -$$$$31\!\cdots\!11$$$$T^{21} +$$$$25\!\cdots\!06$$$$T^{22} -$$$$20\!\cdots\!61$$$$T^{23} +$$$$15\!\cdots\!59$$$$T^{24} -$$$$11\!\cdots\!04$$$$T^{25} +$$$$85\!\cdots\!12$$$$T^{26} -$$$$59\!\cdots\!32$$$$T^{27} +$$$$39\!\cdots\!28$$$$T^{28} -$$$$24\!\cdots\!44$$$$T^{29} +$$$$14\!\cdots\!12$$$$T^{30} -$$$$77\!\cdots\!76$$$$T^{31} +$$$$38\!\cdots\!57$$$$T^{32} -$$$$16\!\cdots\!03$$$$T^{33} +$$$$61\!\cdots\!34$$$$T^{34} -$$$$18\!\cdots\!89$$$$T^{35} +$$$$34\!\cdots\!09$$$$T^{36}$$
$89$ $$1 + 28 T + 1278 T^{2} + 27910 T^{3} + 750883 T^{4} + 13674851 T^{5} + 278327874 T^{6} + 4395094780 T^{7} + 74207320481 T^{8} + 1041079823079 T^{9} + 15262444839605 T^{10} + 193236416956553 T^{11} + 2524149327752443 T^{12} + 29133832029172985 T^{13} + 344418211589959806 T^{14} + 3646576704043225682 T^{15} + 39387762497694988269 T^{16} +$$$$38\!\cdots\!80$$$$T^{17} +$$$$38\!\cdots\!72$$$$T^{18} +$$$$34\!\cdots\!20$$$$T^{19} +$$$$31\!\cdots\!49$$$$T^{20} +$$$$25\!\cdots\!58$$$$T^{21} +$$$$21\!\cdots\!46$$$$T^{22} +$$$$16\!\cdots\!65$$$$T^{23} +$$$$12\!\cdots\!23$$$$T^{24} +$$$$85\!\cdots\!37$$$$T^{25} +$$$$60\!\cdots\!05$$$$T^{26} +$$$$36\!\cdots\!11$$$$T^{27} +$$$$23\!\cdots\!81$$$$T^{28} +$$$$12\!\cdots\!20$$$$T^{29} +$$$$68\!\cdots\!54$$$$T^{30} +$$$$30\!\cdots\!19$$$$T^{31} +$$$$14\!\cdots\!03$$$$T^{32} +$$$$48\!\cdots\!90$$$$T^{33} +$$$$19\!\cdots\!58$$$$T^{34} +$$$$38\!\cdots\!12$$$$T^{35} +$$$$12\!\cdots\!81$$$$T^{36}$$
$97$ $$1 + 34 T + 1734 T^{2} + 44982 T^{3} + 1350738 T^{4} + 28687000 T^{5} + 647807944 T^{6} + 11728222048 T^{7} + 217762436551 T^{8} + 3446769515305 T^{9} + 54984423618660 T^{10} + 773527117230350 T^{11} + 10868994800485399 T^{12} + 137384482180690541 T^{13} + 1726110187080262341 T^{14} + 19737225141617642237 T^{15} +$$$$22\!\cdots\!37$$$$T^{16} +$$$$23\!\cdots\!18$$$$T^{17} +$$$$23\!\cdots\!52$$$$T^{18} +$$$$22\!\cdots\!46$$$$T^{19} +$$$$21\!\cdots\!33$$$$T^{20} +$$$$18\!\cdots\!01$$$$T^{21} +$$$$15\!\cdots\!21$$$$T^{22} +$$$$11\!\cdots\!37$$$$T^{23} +$$$$90\!\cdots\!71$$$$T^{24} +$$$$62\!\cdots\!50$$$$T^{25} +$$$$43\!\cdots\!60$$$$T^{26} +$$$$26\!\cdots\!85$$$$T^{27} +$$$$16\!\cdots\!99$$$$T^{28} +$$$$83\!\cdots\!44$$$$T^{29} +$$$$44\!\cdots\!04$$$$T^{30} +$$$$19\!\cdots\!00$$$$T^{31} +$$$$88\!\cdots\!22$$$$T^{32} +$$$$28\!\cdots\!26$$$$T^{33} +$$$$10\!\cdots\!14$$$$T^{34} +$$$$20\!\cdots\!58$$$$T^{35} +$$$$57\!\cdots\!89$$$$T^{36}$$