# Properties

 Label 6039.2.a.h Level 6039 Weight 2 Character orbit 6039.a Self dual yes Analytic conductor 48.222 Analytic rank 1 Dimension 13 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6039 = 3^{2} \cdot 11 \cdot 61$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6039.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.2216577807$$ Analytic rank: $$1$$ Dimension: $$13$$ Coefficient field: $$\mathbb{Q}[x]/(x^{13} - \cdots)$$ Defining polynomial: $$x^{13} - 4 x^{12} - 11 x^{11} + 57 x^{10} + 28 x^{9} - 290 x^{8} + 51 x^{7} + 644 x^{6} - 259 x^{5} - 640 x^{4} + 274 x^{3} + 256 x^{2} - 74 x - 35$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$5$$ Twist minimal: no (minimal twist has level 2013) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{12}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{13} - 4 x^{12} - 11 x^{11} + 57 x^{10} + 28 x^{9} - 290 x^{8} + 51 x^{7} + 644 x^{6} - 259 x^{5} - 640 x^{4} + 274 x^{3} + 256 x^{2} - 74 x - 35$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 6 \nu^{2} - \nu + 4$$ $$\beta_{5}$$ $$=$$ $$($$$$-8 \nu^{12} + 90 \nu^{11} - 26 \nu^{10} - 1165 \nu^{9} + 1132 \nu^{8} + 4883 \nu^{7} - 5564 \nu^{6} - 6816 \nu^{5} + 8767 \nu^{4} + 884 \nu^{3} - 4652 \nu^{2} + 1882 \nu + 410$$$$)/359$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{12} + 11 \nu^{11} - 67 \nu^{10} - 282 \nu^{9} - 93 \nu^{8} + 2324 \nu^{7} + 2894 \nu^{6} - 8037 \nu^{5} - 10588 \nu^{4} + 11612 \nu^{3} + 12866 \nu^{2} - 5506 \nu - 3500$$$$)/359$$ $$\beta_{7}$$ $$=$$ $$($$$$29 \nu^{12} - 57 \nu^{11} - 534 \nu^{10} + 1037 \nu^{9} + 3615 \nu^{8} - 6886 \nu^{7} - 10884 \nu^{6} + 19682 \nu^{5} + 14037 \nu^{4} - 21334 \nu^{3} - 7010 \nu^{2} + 5294 \nu + 1655$$$$)/359$$ $$\beta_{8}$$ $$=$$ $$($$$$-66 \nu^{12} + 204 \nu^{11} + 683 \nu^{10} - 2521 \nu^{9} - 1431 \nu^{8} + 10039 \nu^{7} - 3900 \nu^{6} - 13511 \nu^{5} + 13003 \nu^{4} + 3344 \nu^{3} - 8582 \nu^{2} + 2064 \nu + 690$$$$)/359$$ $$\beta_{9}$$ $$=$$ $$($$$$-102 \nu^{12} + 250 \nu^{11} + 1284 \nu^{10} - 3276 \nu^{9} - 4953 \nu^{8} + 14601 \nu^{7} + 4449 \nu^{6} - 25874 \nu^{5} + 6682 \nu^{4} + 17733 \nu^{3} - 8694 \nu^{2} - 3109 \nu + 381$$$$)/359$$ $$\beta_{10}$$ $$=$$ $$($$$$-71 \nu^{12} + 350 \nu^{11} + 577 \nu^{10} - 4730 \nu^{9} + 533 \nu^{8} + 22021 \nu^{7} - 13660 \nu^{6} - 41824 \nu^{5} + 31900 \nu^{4} + 32078 \nu^{3} - 22798 \nu^{2} - 7440 \nu + 3190$$$$)/359$$ $$\beta_{11}$$ $$=$$ $$($$$$96 \nu^{12} - 362 \nu^{11} - 1124 \nu^{10} + 5005 \nu^{9} + 4007 \nu^{8} - 24132 \nu^{7} - 4314 \nu^{6} + 48405 \nu^{5} + 2496 \nu^{4} - 39687 \nu^{3} - 4488 \nu^{2} + 10085 \nu + 1901$$$$)/359$$ $$\beta_{12}$$ $$=$$ $$($$$$-196 \nu^{12} + 410 \nu^{11} + 2953 \nu^{10} - 5746 \nu^{9} - 16423 \nu^{8} + 28627 \nu^{7} + 42464 \nu^{6} - 61446 \nu^{5} - 54279 \nu^{4} + 56481 \nu^{3} + 31780 \nu^{2} - 17434 \nu - 6469$$$$)/359$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 6 \beta_{2} + \beta_{1} + 14$$ $$\nu^{5}$$ $$=$$ $$\beta_{11} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 7 \beta_{3} + 27 \beta_{1} + 2$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 10 \beta_{4} + \beta_{3} + 32 \beta_{2} + 12 \beta_{1} + 74$$ $$\nu^{7}$$ $$=$$ $$\beta_{12} + 15 \beta_{11} + 2 \beta_{10} + 12 \beta_{8} - 11 \beta_{7} + 15 \beta_{6} + 12 \beta_{5} + 14 \beta_{4} + 42 \beta_{3} + 153 \beta_{1} + 25$$ $$\nu^{8}$$ $$=$$ $$\beta_{12} + 31 \beta_{11} + 2 \beta_{10} + 14 \beta_{9} + 13 \beta_{8} - 14 \beta_{7} + 42 \beta_{6} + 30 \beta_{5} + 80 \beta_{4} + 14 \beta_{3} + 166 \beta_{2} + 109 \beta_{1} + 414$$ $$\nu^{9}$$ $$=$$ $$15 \beta_{12} + 153 \beta_{11} + 28 \beta_{10} + 4 \beta_{9} + 104 \beta_{8} - 91 \beta_{7} + 153 \beta_{6} + 115 \beta_{5} + 137 \beta_{4} + 246 \beta_{3} - 2 \beta_{2} + 900 \beta_{1} + 233$$ $$\nu^{10}$$ $$=$$ $$19 \beta_{12} + 328 \beta_{11} + 34 \beta_{10} + 134 \beta_{9} + 123 \beta_{8} - 137 \beta_{7} + 415 \beta_{6} + 312 \beta_{5} + 599 \beta_{4} + 135 \beta_{3} + 852 \beta_{2} + 890 \beta_{1} + 2406$$ $$\nu^{11}$$ $$=$$ $$153 \beta_{12} + 1334 \beta_{11} + 273 \beta_{10} + 74 \beta_{9} + 797 \beta_{8} - 686 \beta_{7} + 1337 \beta_{6} + 1001 \beta_{5} + 1165 \beta_{4} + 1451 \beta_{3} - 36 \beta_{2} + 5466 \beta_{1} + 1943$$ $$\nu^{12}$$ $$=$$ $$227 \beta_{12} + 2960 \beta_{11} + 387 \beta_{10} + 1100 \beta_{9} + 1038 \beta_{8} - 1168 \beta_{7} + 3572 \beta_{6} + 2782 \beta_{5} + 4363 \beta_{4} + 1129 \beta_{3} + 4344 \beta_{2} + 6882 \beta_{1} + 14427$$

## Embeddings

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

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

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.65453 2.30577 2.25716 1.88247 1.31287 0.931518 0.669508 −0.312603 −0.638829 −1.17594 −1.38103 −2.17077 −2.33467
−2.65453 0 5.04652 −2.16368 0 −2.02198 −8.08709 0 5.74356
1.2 −2.30577 0 3.31659 2.09861 0 3.60849 −3.03576 0 −4.83892
1.3 −2.25716 0 3.09478 −1.69786 0 0.963803 −2.47110 0 3.83236
1.4 −1.88247 0 1.54370 −3.12160 0 2.89419 0.858967 0 5.87632
1.5 −1.31287 0 −0.276371 3.61438 0 −0.837447 2.98858 0 −4.74521
1.6 −0.931518 0 −1.13227 −1.13816 0 −2.87358 2.91777 0 1.06022
1.7 −0.669508 0 −1.55176 −2.40049 0 3.10183 2.37793 0 1.60715
1.8 0.312603 0 −1.90228 0.566394 0 3.64999 −1.21987 0 0.177056
1.9 0.638829 0 −1.59190 2.55169 0 −2.98273 −2.29461 0 1.63009
1.10 1.17594 0 −0.617176 −3.63888 0 −0.818481 −3.07763 0 −4.27908
1.11 1.38103 0 −0.0927606 0.547866 0 1.24123 −2.89016 0 0.756618
1.12 2.17077 0 2.71223 −2.18891 0 3.77754 1.54609 0 −4.75162
1.13 2.33467 0 3.45069 −0.0293558 0 −2.70286 3.38688 0 −0.0685361
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.13 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 6039.2.a.h 13
3.b odd 2 1 2013.2.a.g 13

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.g 13 3.b odd 2 1
6039.2.a.h 13 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$1$$
$$61$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{13} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6039))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 15 T^{2} + 39 T^{3} + 98 T^{4} + 206 T^{5} + 423 T^{6} + 776 T^{7} + 1407 T^{8} + 2352 T^{9} + 3888 T^{10} + 6000 T^{11} + 9138 T^{12} + 13027 T^{13} + 18276 T^{14} + 24000 T^{15} + 31104 T^{16} + 37632 T^{17} + 45024 T^{18} + 49664 T^{19} + 54144 T^{20} + 52736 T^{21} + 50176 T^{22} + 39936 T^{23} + 30720 T^{24} + 16384 T^{25} + 8192 T^{26}$$
$3$ 1
$5$ $$1 + 7 T + 56 T^{2} + 258 T^{3} + 1255 T^{4} + 4503 T^{5} + 16859 T^{6} + 50885 T^{7} + 160423 T^{8} + 425008 T^{9} + 1180717 T^{10} + 2815444 T^{11} + 7070565 T^{12} + 15390246 T^{13} + 35352825 T^{14} + 70386100 T^{15} + 147589625 T^{16} + 265630000 T^{17} + 501321875 T^{18} + 795078125 T^{19} + 1317109375 T^{20} + 1758984375 T^{21} + 2451171875 T^{22} + 2519531250 T^{23} + 2734375000 T^{24} + 1708984375 T^{25} + 1220703125 T^{26}$$
$7$ $$1 - 7 T + 70 T^{2} - 369 T^{3} + 2288 T^{4} - 9950 T^{5} + 47642 T^{6} - 178052 T^{7} + 711086 T^{8} - 2332387 T^{9} + 8053671 T^{10} - 23432188 T^{11} + 71259100 T^{12} - 184614114 T^{13} + 498813700 T^{14} - 1148177212 T^{15} + 2762409153 T^{16} - 5600061187 T^{17} + 11951222402 T^{18} - 20947639748 T^{19} + 39235235606 T^{20} - 57359769950 T^{21} + 92329052816 T^{22} - 104233366881 T^{23} + 138412872010 T^{24} - 96889010407 T^{25} + 96889010407 T^{26}$$
$11$ $$( 1 + T )^{13}$$
$13$ $$1 - 9 T + 142 T^{2} - 963 T^{3} + 8890 T^{4} - 49469 T^{5} + 343406 T^{6} - 1641094 T^{7} + 9394370 T^{8} - 39521641 T^{9} + 194950678 T^{10} - 730889714 T^{11} + 3178238861 T^{12} - 10658263028 T^{13} + 41317105193 T^{14} - 123520361666 T^{15} + 428306639566 T^{16} - 1128777588601 T^{17} + 3488063820410 T^{18} - 7921247289046 T^{19} + 21548217228902 T^{20} - 40353383037149 T^{21} + 94273999425970 T^{22} - 132757727650587 T^{23} + 254486775953254 T^{24} - 209682766102329 T^{25} + 302875106592253 T^{26}$$
$17$ $$1 + 19 T + 286 T^{2} + 3030 T^{3} + 27921 T^{4} + 216056 T^{5} + 1519141 T^{6} + 9542471 T^{7} + 55860051 T^{8} + 300909000 T^{9} + 1531481120 T^{10} + 7271209038 T^{11} + 32798268168 T^{12} + 138656460496 T^{13} + 557570558856 T^{14} + 2101379411982 T^{15} + 7524166742560 T^{16} + 25132220589000 T^{17} + 79313284432707 T^{18} + 230332052192999 T^{19} + 623362302039893 T^{20} + 1507154249672696 T^{21} + 3311092099672737 T^{22} + 6108461518360470 T^{23} + 9801762343983038 T^{24} + 11069822507365459 T^{25} + 9904578032905937 T^{26}$$
$19$ $$1 - 14 T + 248 T^{2} - 2424 T^{3} + 25701 T^{4} - 198294 T^{5} + 1592044 T^{6} - 10285272 T^{7} + 68058462 T^{8} - 379790666 T^{9} + 2155393863 T^{10} - 10559591841 T^{11} + 52449577423 T^{12} - 227152092088 T^{13} + 996541971037 T^{14} - 3812012654601 T^{15} + 14783846506317 T^{16} - 49494699383786 T^{17} + 168519489699738 T^{18} - 483879682564632 T^{19} + 1423083138844516 T^{20} - 3367738649652054 T^{21} + 8293396520618079 T^{22} - 14861704608909624 T^{23} + 28889584206758312 T^{24} - 30986408866926254 T^{25} + 42052983462257059 T^{26}$$
$23$ $$1 + 5 T + 215 T^{2} + 1100 T^{3} + 22269 T^{4} + 114329 T^{5} + 1478099 T^{6} + 7463909 T^{7} + 70504439 T^{8} + 342737121 T^{9} + 2566070728 T^{10} + 11715614720 T^{11} + 73674321281 T^{12} + 306620883056 T^{13} + 1694509389463 T^{14} + 6197560186880 T^{15} + 31221382547576 T^{16} + 95911898677761 T^{17} + 453790752426577 T^{18} + 1104926404230101 T^{19} + 5032669088385253 T^{20} + 8953216636191449 T^{21} + 40109868618119547 T^{22} + 45569162335013900 T^{23} + 204854097951494305 T^{24} + 109573122160101605 T^{25} + 504036361936467383 T^{26}$$
$29$ $$1 + 10 T + 271 T^{2} + 2320 T^{3} + 34320 T^{4} + 250863 T^{5} + 2687247 T^{6} + 16896957 T^{7} + 146990240 T^{8} + 807948642 T^{9} + 6107409596 T^{10} + 30083037390 T^{11} + 206972882469 T^{12} + 938460561856 T^{13} + 6002213591601 T^{14} + 25299834444990 T^{15} + 148953612636844 T^{16} + 571446723462402 T^{17} + 3014938714185760 T^{18} + 10050704077534197 T^{19} + 46354678361731323 T^{20} + 125493315894635343 T^{21} + 497885249891824080 T^{22} + 976040781256466320 T^{23} + 3306338146506279659 T^{24} + 3538147832054690410 T^{25} + 10260628712958602189 T^{26}$$
$31$ $$1 + T + 288 T^{2} + 128 T^{3} + 40625 T^{4} + 2854 T^{5} + 3714170 T^{6} - 635768 T^{7} + 245031489 T^{8} - 75359216 T^{9} + 12295404095 T^{10} - 4425200321 T^{11} + 482112034156 T^{12} - 167128670114 T^{13} + 14945473058836 T^{14} - 4252617508481 T^{15} + 366292383394145 T^{16} - 69595818519536 T^{17} + 7015043498335839 T^{18} - 564246440262008 T^{19} + 102186525952652870 T^{20} + 2434151020856614 T^{21} + 1074109650277259375 T^{22} + 104912420733542528 T^{23} + 7317641346164591328 T^{24} + 787662783788549761 T^{25} + 24417546297445042591 T^{26}$$
$37$ $$1 + 8 T + 275 T^{2} + 1701 T^{3} + 34615 T^{4} + 165433 T^{5} + 2730912 T^{6} + 9899925 T^{7} + 157492443 T^{8} + 426504573 T^{9} + 7393508744 T^{10} + 15442255673 T^{11} + 303399072842 T^{12} + 552624316190 T^{13} + 11225765695154 T^{14} + 21140448016337 T^{15} + 374503398409832 T^{16} + 799338237038253 T^{17} + 10921149195216951 T^{18} + 25400499019619325 T^{19} + 259250602445035296 T^{20} + 581080013500512793 T^{21} + 4498625623006590355 T^{22} + 8179402017482761149 T^{23} + 48927345989351613575 T^{24} + 52663616046720282248 T^{25} +$$$$24\!\cdots\!97$$$$T^{26}$$
$41$ $$1 + 21 T + 540 T^{2} + 7636 T^{3} + 116825 T^{4} + 1280643 T^{5} + 14810923 T^{6} + 135199551 T^{7} + 1296033361 T^{8} + 10264106178 T^{9} + 85277910821 T^{10} + 598271745082 T^{11} + 4396906526603 T^{12} + 27518809751956 T^{13} + 180273167590723 T^{14} + 1005694803482842 T^{15} + 5877438891694141 T^{16} + 29003910937651458 T^{17} + 150153501574721561 T^{18} + 642211960586395791 T^{19} + 2884490554372402163 T^{20} + 10225838600197204803 T^{21} + 38246394485574493825 T^{22} +$$$$10\!\cdots\!36$$$$T^{23} +$$$$29\!\cdots\!40$$$$T^{24} +$$$$47\!\cdots\!01$$$$T^{25} +$$$$92\!\cdots\!21$$$$T^{26}$$
$43$ $$1 - 11 T + 343 T^{2} - 2734 T^{3} + 54061 T^{4} - 355546 T^{5} + 5728635 T^{6} - 33015751 T^{7} + 460695872 T^{8} - 2369020056 T^{9} + 29408375578 T^{10} - 136593716802 T^{11} + 1532816968042 T^{12} - 6469732775516 T^{13} + 65911129625806 T^{14} - 252561782366898 T^{15} + 2338171717080046 T^{16} - 8099208136472856 T^{17} + 67726182839247296 T^{18} - 208704548406384799 T^{19} + 1557149609238948945 T^{20} - 4155692855899925146 T^{21} + 27170659193917669423 T^{22} - 59085792644519136766 T^{23} +$$$$31\!\cdots\!01$$$$T^{24} -$$$$43\!\cdots\!11$$$$T^{25} +$$$$17\!\cdots\!43$$$$T^{26}$$
$47$ $$1 + 22 T + 642 T^{2} + 10096 T^{3} + 174563 T^{4} + 2175831 T^{5} + 28224583 T^{6} + 293657148 T^{7} + 3111592527 T^{8} + 27806395138 T^{9} + 250636904182 T^{10} + 1953859409661 T^{11} + 15297994055552 T^{12} + 104712788542460 T^{13} + 719005720610944 T^{14} + 4316075435941149 T^{15} + 26021875302887786 T^{16} + 135686338033390978 T^{17} + 713628209885962689 T^{18} + 3165393631192021692 T^{19} + 14299226313226941929 T^{20} + 51809335668546098391 T^{21} +$$$$19\!\cdots\!21$$$$T^{22} +$$$$53\!\cdots\!04$$$$T^{23} +$$$$15\!\cdots\!26$$$$T^{24} +$$$$25\!\cdots\!02$$$$T^{25} +$$$$54\!\cdots\!27$$$$T^{26}$$
$53$ $$1 + 16 T + 703 T^{2} + 9307 T^{3} + 224125 T^{4} + 2520330 T^{5} + 43323009 T^{6} + 420329087 T^{7} + 5696644844 T^{8} + 48096154711 T^{9} + 539874071107 T^{10} + 3978515326043 T^{11} + 38008878473513 T^{12} + 243974894241370 T^{13} + 2014470559096189 T^{14} + 11175649550854787 T^{15} + 80374832084196839 T^{16} + 379501794920205991 T^{17} + 2382311198982488092 T^{18} + 9316325677290859223 T^{19} + 50892021283558609533 T^{20} +$$$$15\!\cdots\!30$$$$T^{21} +$$$$73\!\cdots\!25$$$$T^{22} +$$$$16\!\cdots\!43$$$$T^{23} +$$$$65\!\cdots\!91$$$$T^{24} +$$$$78\!\cdots\!56$$$$T^{25} +$$$$26\!\cdots\!73$$$$T^{26}$$
$59$ $$1 + 19 T + 588 T^{2} + 8387 T^{3} + 150323 T^{4} + 1725136 T^{5} + 23246896 T^{6} + 226730256 T^{7} + 2557136489 T^{8} + 22130373950 T^{9} + 220805866149 T^{10} + 1736747698856 T^{11} + 15678767927188 T^{12} + 112576920082032 T^{13} + 925047307704092 T^{14} + 6045618739717736 T^{15} + 45348887983815471 T^{16} + 268161730217145950 T^{17} + 1828159011845646211 T^{18} + 9563603190640542096 T^{19} + 57853422247832871824 T^{20} +$$$$25\!\cdots\!56$$$$T^{21} +$$$$13\!\cdots\!97$$$$T^{22} +$$$$42\!\cdots\!87$$$$T^{23} +$$$$17\!\cdots\!92$$$$T^{24} +$$$$33\!\cdots\!39$$$$T^{25} +$$$$10\!\cdots\!79$$$$T^{26}$$
$61$ $$( 1 - T )^{13}$$
$67$ $$1 - 12 T + 476 T^{2} - 5497 T^{3} + 116752 T^{4} - 1262381 T^{5} + 19411279 T^{6} - 192068545 T^{7} + 2419350272 T^{8} - 21779688147 T^{9} + 238485194313 T^{10} - 1959319395389 T^{11} + 19212910079827 T^{12} - 144341874973268 T^{13} + 1287264975348409 T^{14} - 8795384765901221 T^{15} + 71727522497160819 T^{16} - 438885131192462787 T^{17} + 3266425544854479104 T^{18} - 17374209846253774105 T^{19} +$$$$11\!\cdots\!17$$$$T^{20} -$$$$51\!\cdots\!21$$$$T^{21} +$$$$31\!\cdots\!44$$$$T^{22} -$$$$10\!\cdots\!53$$$$T^{23} +$$$$58\!\cdots\!08$$$$T^{24} -$$$$98\!\cdots\!32$$$$T^{25} +$$$$54\!\cdots\!87$$$$T^{26}$$
$71$ $$1 + 5 T + 667 T^{2} + 2790 T^{3} + 210426 T^{4} + 721315 T^{5} + 41963243 T^{6} + 115239987 T^{7} + 5974373412 T^{8} + 12982883096 T^{9} + 650776182024 T^{10} + 1141711263752 T^{11} + 56611739724711 T^{12} + 85847839551638 T^{13} + 4019433520454481 T^{14} + 5755366480573832 T^{15} + 232919954084391864 T^{16} + 329916883695844376 T^{17} + 10779139863764415612 T^{18} + 14762275053752349027 T^{19} +$$$$38\!\cdots\!13$$$$T^{20} +$$$$46\!\cdots\!15$$$$T^{21} +$$$$96\!\cdots\!06$$$$T^{22} +$$$$90\!\cdots\!90$$$$T^{23} +$$$$15\!\cdots\!57$$$$T^{24} +$$$$82\!\cdots\!05$$$$T^{25} +$$$$11\!\cdots\!11$$$$T^{26}$$
$73$ $$1 - 18 T + 722 T^{2} - 11164 T^{3} + 249956 T^{4} - 3378036 T^{5} + 55268985 T^{6} - 658789573 T^{7} + 8728233621 T^{8} - 92269947656 T^{9} + 1040846671509 T^{10} - 9788272584286 T^{11} + 96495170021146 T^{12} - 807214488838654 T^{13} + 7044147411543658 T^{14} - 52161704601660094 T^{15} + 404907049610416653 T^{16} - 2620304210592473096 T^{17} + 18094253176762628253 T^{18} - 99697410317215684597 T^{19} +$$$$61\!\cdots\!45$$$$T^{20} -$$$$27\!\cdots\!16$$$$T^{21} +$$$$14\!\cdots\!28$$$$T^{22} -$$$$47\!\cdots\!36$$$$T^{23} +$$$$22\!\cdots\!94$$$$T^{24} -$$$$41\!\cdots\!78$$$$T^{25} +$$$$16\!\cdots\!33$$$$T^{26}$$
$79$ $$1 + T + 592 T^{2} + 1682 T^{3} + 175369 T^{4} + 744808 T^{5} + 34871614 T^{6} + 181072780 T^{7} + 5201860224 T^{8} + 29584351201 T^{9} + 613761565080 T^{10} + 3530100791245 T^{11} + 58889113102736 T^{12} + 318873705758666 T^{13} + 4652239935116144 T^{14} + 22031359038160045 T^{15} + 302608388285478120 T^{16} + 1152312875611397281 T^{17} + 16006417288962773376 T^{18} + 44016521354313818380 T^{19} +$$$$66\!\cdots\!26$$$$T^{20} +$$$$11\!\cdots\!88$$$$T^{21} +$$$$21\!\cdots\!11$$$$T^{22} +$$$$15\!\cdots\!82$$$$T^{23} +$$$$44\!\cdots\!68$$$$T^{24} +$$$$59\!\cdots\!41$$$$T^{25} +$$$$46\!\cdots\!39$$$$T^{26}$$
$83$ $$1 + 48 T + 1946 T^{2} + 54074 T^{3} + 1322286 T^{4} + 26651308 T^{5} + 485319219 T^{6} + 7734534804 T^{7} + 113100793526 T^{8} + 1486614970798 T^{9} + 18090592348036 T^{10} + 200478795669543 T^{11} + 2066582253204772 T^{12} + 19506872219703810 T^{13} + 171526327015996076 T^{14} + 1381098423367481727 T^{15} + 10343965526906460332 T^{16} + 70552250487537110158 T^{17} +$$$$44\!\cdots\!18$$$$T^{18} +$$$$25\!\cdots\!76$$$$T^{19} +$$$$13\!\cdots\!13$$$$T^{20} +$$$$60\!\cdots\!28$$$$T^{21} +$$$$24\!\cdots\!58$$$$T^{22} +$$$$83\!\cdots\!26$$$$T^{23} +$$$$25\!\cdots\!82$$$$T^{24} +$$$$51\!\cdots\!28$$$$T^{25} +$$$$88\!\cdots\!63$$$$T^{26}$$
$89$ $$1 + 15 T + 786 T^{2} + 11565 T^{3} + 310984 T^{4} + 4296636 T^{5} + 81310600 T^{6} + 1023800184 T^{7} + 15480612414 T^{8} + 175057827139 T^{9} + 2244424047975 T^{10} + 22673108801304 T^{11} + 253614549212220 T^{12} + 2281130993837004 T^{13} + 22571694879887580 T^{14} + 179593694815128984 T^{15} + 1582249376676887775 T^{16} + 10983520379291478499 T^{17} + 86444660026703399886 T^{18} +$$$$50\!\cdots\!24$$$$T^{19} +$$$$35\!\cdots\!00$$$$T^{20} +$$$$16\!\cdots\!16$$$$T^{21} +$$$$10\!\cdots\!56$$$$T^{22} +$$$$36\!\cdots\!65$$$$T^{23} +$$$$21\!\cdots\!54$$$$T^{24} +$$$$37\!\cdots\!15$$$$T^{25} +$$$$21\!\cdots\!69$$$$T^{26}$$
$97$ $$1 + 17 T + 938 T^{2} + 14141 T^{3} + 424719 T^{4} + 5744714 T^{5} + 123323034 T^{6} + 1505382982 T^{7} + 25668744301 T^{8} + 283066250203 T^{9} + 4047350409404 T^{10} + 40194221467436 T^{11} + 497446404474603 T^{12} + 4416178399171106 T^{13} + 48252301234036491 T^{14} + 378187429787105324 T^{15} + 3693907440201976892 T^{16} + 25059651605837694043 T^{17} +$$$$22\!\cdots\!57$$$$T^{18} +$$$$12\!\cdots\!78$$$$T^{19} +$$$$99\!\cdots\!42$$$$T^{20} +$$$$45\!\cdots\!54$$$$T^{21} +$$$$32\!\cdots\!23$$$$T^{22} +$$$$10\!\cdots\!09$$$$T^{23} +$$$$67\!\cdots\!14$$$$T^{24} +$$$$11\!\cdots\!97$$$$T^{25} +$$$$67\!\cdots\!77$$$$T^{26}$$