# Properties

 Label 4018.2.a.bt Level 4018 Weight 2 Character orbit 4018.a Self dual yes Analytic conductor 32.084 Analytic rank 0 Dimension 10 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4018 = 2 \cdot 7^{2} \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4018.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0838915322$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 574) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} -\beta_{1} q^{3} + q^{4} -\beta_{4} q^{5} -\beta_{1} q^{6} + q^{8} + ( 2 + \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})$$ $$q + q^{2} -\beta_{1} q^{3} + q^{4} -\beta_{4} q^{5} -\beta_{1} q^{6} + q^{8} + ( 2 + \beta_{3} - \beta_{4} ) q^{9} -\beta_{4} q^{10} + ( 1 + \beta_{7} ) q^{11} -\beta_{1} q^{12} + \beta_{2} q^{13} + ( 1 - \beta_{1} - \beta_{5} + \beta_{9} ) q^{15} + q^{16} + ( -1 - \beta_{3} ) q^{17} + ( 2 + \beta_{3} - \beta_{4} ) q^{18} + ( -\beta_{1} + \beta_{6} ) q^{19} -\beta_{4} q^{20} + ( 1 + \beta_{7} ) q^{22} + ( \beta_{1} - \beta_{3} + \beta_{6} + \beta_{7} ) q^{23} -\beta_{1} q^{24} + ( 3 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{25} + \beta_{2} q^{26} + ( -1 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{7} ) q^{27} + ( 2 - \beta_{7} - \beta_{9} ) q^{29} + ( 1 - \beta_{1} - \beta_{5} + \beta_{9} ) q^{30} + ( 1 + \beta_{5} - \beta_{6} + \beta_{9} ) q^{31} + q^{32} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{33} + ( -1 - \beta_{3} ) q^{34} + ( 2 + \beta_{3} - \beta_{4} ) q^{36} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} - \beta_{9} ) q^{37} + ( -\beta_{1} + \beta_{6} ) q^{38} + ( -\beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{39} -\beta_{4} q^{40} - q^{41} + ( 2 - \beta_{1} - \beta_{2} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{43} + ( 1 + \beta_{7} ) q^{44} + ( 4 + \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{45} + ( \beta_{1} - \beta_{3} + \beta_{6} + \beta_{7} ) q^{46} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{8} + \beta_{9} ) q^{47} -\beta_{1} q^{48} + ( 3 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{50} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{7} + \beta_{9} ) q^{51} + \beta_{2} q^{52} + ( 3 - \beta_{1} + 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{53} + ( -1 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{7} ) q^{54} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{9} ) q^{55} + ( 3 - \beta_{1} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{57} + ( 2 - \beta_{7} - \beta_{9} ) q^{58} + ( 2 + \beta_{3} - \beta_{8} ) q^{59} + ( 1 - \beta_{1} - \beta_{5} + \beta_{9} ) q^{60} + ( -\beta_{1} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{61} + ( 1 + \beta_{5} - \beta_{6} + \beta_{9} ) q^{62} + q^{64} + ( 2 + 2 \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{65} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{66} + ( \beta_{1} - \beta_{3} + \beta_{7} ) q^{67} + ( -1 - \beta_{3} ) q^{68} + ( -4 - \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{69} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{71} + ( 2 + \beta_{3} - \beta_{4} ) q^{72} + ( -1 + \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{73} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} - \beta_{9} ) q^{74} + ( -1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + \beta_{6} - 2 \beta_{8} + \beta_{9} ) q^{75} + ( -\beta_{1} + \beta_{6} ) q^{76} + ( -\beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{78} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{79} -\beta_{4} q^{80} + ( \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{8} - 2 \beta_{9} ) q^{81} - q^{82} + ( -2 - \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{83} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{9} ) q^{85} + ( 2 - \beta_{1} - \beta_{2} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{86} + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{9} ) q^{87} + ( 1 + \beta_{7} ) q^{88} + ( -\beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{89} + ( 4 + \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{90} + ( \beta_{1} - \beta_{3} + \beta_{6} + \beta_{7} ) q^{92} + ( 1 + 2 \beta_{2} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{93} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{8} + \beta_{9} ) q^{94} + ( 2 - \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{95} -\beta_{1} q^{96} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{7} + 2 \beta_{9} ) q^{97} + ( 5 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 10q^{2} - q^{3} + 10q^{4} + 2q^{5} - q^{6} + 10q^{8} + 17q^{9} + O(q^{10})$$ $$10q + 10q^{2} - q^{3} + 10q^{4} + 2q^{5} - q^{6} + 10q^{8} + 17q^{9} + 2q^{10} + 11q^{11} - q^{12} - 4q^{13} + 4q^{15} + 10q^{16} - 5q^{17} + 17q^{18} + q^{19} + 2q^{20} + 11q^{22} + 9q^{23} - q^{24} + 24q^{25} - 4q^{26} - 7q^{27} + 23q^{29} + 4q^{30} + 5q^{31} + 10q^{32} + 5q^{33} - 5q^{34} + 17q^{36} + 16q^{37} + q^{38} - 7q^{39} + 2q^{40} - 10q^{41} + 20q^{43} + 11q^{44} + 42q^{45} + 9q^{46} + 16q^{47} - q^{48} + 24q^{50} + 13q^{51} - 4q^{52} + 26q^{53} - 7q^{54} - 7q^{55} + 37q^{57} + 23q^{58} + 10q^{59} + 4q^{60} + 5q^{62} + 10q^{64} + 18q^{65} + 5q^{66} + 7q^{67} - 5q^{68} - 39q^{69} + 5q^{71} + 17q^{72} + 13q^{73} + 16q^{74} - 19q^{75} + q^{76} - 7q^{78} - q^{79} + 2q^{80} + 18q^{81} - 10q^{82} - 21q^{83} + 34q^{85} + 20q^{86} - 2q^{87} + 11q^{88} - 6q^{89} + 42q^{90} + 9q^{92} - 5q^{93} + 16q^{94} + 24q^{95} - q^{96} - 29q^{97} + 48q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} - 23 x^{8} + 19 x^{7} + 181 x^{6} - 109 x^{5} - 579 x^{4} + 231 x^{3} + 608 x^{2} - 204 x - 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-19 \nu^{9} + 132 \nu^{8} + 513 \nu^{7} - 2594 \nu^{6} - 4759 \nu^{5} + 15220 \nu^{4} + 17781 \nu^{3} - 26574 \nu^{2} - 20680 \nu + 4512$$$$)/1636$$ $$\beta_{3}$$ $$=$$ $$($$$$-39 \nu^{9} + 314 \nu^{8} + 235 \nu^{7} - 5712 \nu^{6} + 4525 \nu^{5} + 29562 \nu^{4} - 33721 \nu^{3} - 41760 \nu^{2} + 56056 \nu - 16484$$$$)/3272$$ $$\beta_{4}$$ $$=$$ $$($$$$-39 \nu^{9} + 314 \nu^{8} + 235 \nu^{7} - 5712 \nu^{6} + 4525 \nu^{5} + 29562 \nu^{4} - 33721 \nu^{3} - 45032 \nu^{2} + 56056 \nu - 124$$$$)/3272$$ $$\beta_{5}$$ $$=$$ $$($$$$-42 \nu^{9} + 55 \nu^{8} + 725 \nu^{7} - 740 \nu^{6} - 3244 \nu^{5} + 1161 \nu^{4} + 2603 \nu^{3} + 6310 \nu^{2} + 3108 \nu - 7936$$$$)/1636$$ $$\beta_{6}$$ $$=$$ $$($$$$-101 \nu^{9} + 142 \nu^{8} + 1909 \nu^{7} - 2208 \nu^{6} - 11521 \nu^{5} + 7846 \nu^{4} + 30673 \nu^{3} - 3192 \nu^{2} - 35880 \nu - 4516$$$$)/3272$$ $$\beta_{7}$$ $$=$$ $$($$$$-61 \nu^{9} + 187 \nu^{8} + 1238 \nu^{7} - 3334 \nu^{6} - 8003 \nu^{5} + 16381 \nu^{4} + 18748 \nu^{3} - 20264 \nu^{2} - 6120 \nu - 1788$$$$)/1636$$ $$\beta_{8}$$ $$=$$ $$($$$$91 \nu^{9} - 51 \nu^{8} - 2048 \nu^{7} + 1058 \nu^{6} + 15345 \nu^{5} - 6401 \nu^{4} - 44154 \nu^{3} + 10732 \nu^{2} + 38256 \nu - 256$$$$)/1636$$ $$\beta_{9}$$ $$=$$ $$($$$$191 \nu^{9} - 552 \nu^{8} - 3521 \nu^{7} + 10104 \nu^{6} + 18823 \nu^{5} - 53980 \nu^{4} - 30817 \nu^{3} + 92388 \nu^{2} + 1408 \nu - 19612$$$$)/3272$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{4} + \beta_{3} + 5$$ $$\nu^{3}$$ $$=$$ $$-\beta_{7} + \beta_{5} + \beta_{2} + 7 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{9} + \beta_{8} - \beta_{5} - 12 \beta_{4} + 10 \beta_{3} - \beta_{2} + \beta_{1} + 36$$ $$\nu^{5}$$ $$=$$ $$-\beta_{9} - 13 \beta_{7} - 4 \beta_{6} + 16 \beta_{5} - \beta_{3} + 13 \beta_{2} + 59 \beta_{1} + 11$$ $$\nu^{6}$$ $$=$$ $$-30 \beta_{9} + 18 \beta_{8} + 4 \beta_{7} - 12 \beta_{5} - 131 \beta_{4} + 97 \beta_{3} - 16 \beta_{2} + 10 \beta_{1} + 297$$ $$\nu^{7}$$ $$=$$ $$-18 \beta_{9} + 2 \beta_{8} - 139 \beta_{7} - 70 \beta_{6} + 195 \beta_{5} - 2 \beta_{4} - 22 \beta_{3} + 145 \beta_{2} + 547 \beta_{1} + 79$$ $$\nu^{8}$$ $$=$$ $$-358 \beta_{9} + 239 \beta_{8} + 80 \beta_{7} - 2 \beta_{6} - 123 \beta_{5} - 1388 \beta_{4} + 956 \beta_{3} - 191 \beta_{2} + 63 \beta_{1} + 2670$$ $$\nu^{9}$$ $$=$$ $$-229 \beta_{9} + 58 \beta_{8} - 1423 \beta_{7} - 902 \beta_{6} + 2176 \beta_{5} - 26 \beta_{4} - 333 \beta_{3} + 1565 \beta_{2} + 5327 \beta_{1} + 397$$

## 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
 3.20135 2.71108 2.41684 1.31428 0.386194 −0.0511860 −1.56572 −1.97715 −2.19478 −3.24091
1.00000 −3.20135 1.00000 4.32862 −3.20135 0 1.00000 7.24863 4.32862
1.2 1.00000 −2.71108 1.00000 −0.432605 −2.71108 0 1.00000 4.34997 −0.432605
1.3 1.00000 −2.41684 1.00000 −2.64605 −2.41684 0 1.00000 2.84113 −2.64605
1.4 1.00000 −1.31428 1.00000 0.106745 −1.31428 0 1.00000 −1.27267 0.106745
1.5 1.00000 −0.386194 1.00000 −4.13931 −0.386194 0 1.00000 −2.85085 −4.13931
1.6 1.00000 0.0511860 1.00000 0.949433 0.0511860 0 1.00000 −2.99738 0.949433
1.7 1.00000 1.56572 1.00000 2.99546 1.56572 0 1.00000 −0.548522 2.99546
1.8 1.00000 1.97715 1.00000 −3.36998 1.97715 0 1.00000 0.909140 −3.36998
1.9 1.00000 2.19478 1.00000 2.75416 2.19478 0 1.00000 1.81704 2.75416
1.10 1.00000 3.24091 1.00000 1.45353 3.24091 0 1.00000 7.50351 1.45353
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 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 4018.2.a.bt 10
7.b odd 2 1 4018.2.a.bu 10
7.c even 3 2 574.2.e.h 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.e.h 20 7.c even 3 2
4018.2.a.bt 10 1.a even 1 1 trivial
4018.2.a.bu 10 7.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$1$$
$$41$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4018))$$:

 $$T_{3}^{10} + \cdots$$ $$T_{5}^{10} - \cdots$$ $$T_{11}^{10} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{10}$$
$3$ $$1 + T + 7 T^{2} + 8 T^{3} + 34 T^{4} + 34 T^{5} + 123 T^{6} + 81 T^{7} + 329 T^{8} + 186 T^{9} + 936 T^{10} + 558 T^{11} + 2961 T^{12} + 2187 T^{13} + 9963 T^{14} + 8262 T^{15} + 24786 T^{16} + 17496 T^{17} + 45927 T^{18} + 19683 T^{19} + 59049 T^{20}$$
$5$ $$1 - 2 T + 15 T^{2} - 19 T^{3} + 98 T^{4} - 123 T^{5} + 553 T^{6} - 962 T^{7} + 2643 T^{8} - 5170 T^{9} + 10864 T^{10} - 25850 T^{11} + 66075 T^{12} - 120250 T^{13} + 345625 T^{14} - 384375 T^{15} + 1531250 T^{16} - 1484375 T^{17} + 5859375 T^{18} - 3906250 T^{19} + 9765625 T^{20}$$
$7$ 1
$11$ $$1 - 11 T + 104 T^{2} - 674 T^{3} + 4072 T^{4} - 20390 T^{5} + 97692 T^{6} - 410335 T^{7} + 1657881 T^{8} - 5996712 T^{9} + 20946088 T^{10} - 65963832 T^{11} + 200603601 T^{12} - 546155885 T^{13} + 1430308572 T^{14} - 3283829890 T^{15} + 7213796392 T^{16} - 13134353254 T^{17} + 22293323624 T^{18} - 25937424601 T^{19} + 25937424601 T^{20}$$
$13$ $$1 + 4 T + 31 T^{2} + 162 T^{3} + 1170 T^{4} + 4680 T^{5} + 23727 T^{6} + 95988 T^{7} + 468201 T^{8} + 1570950 T^{9} + 6225084 T^{10} + 20422350 T^{11} + 79125969 T^{12} + 210885636 T^{13} + 677666847 T^{14} + 1737651240 T^{15} + 5647366530 T^{16} + 10165259754 T^{17} + 25287652351 T^{18} + 42417997492 T^{19} + 137858491849 T^{20}$$
$17$ $$1 + 5 T + 118 T^{2} + 490 T^{3} + 6751 T^{4} + 24171 T^{5} + 248640 T^{6} + 778673 T^{7} + 6536080 T^{8} + 17936741 T^{9} + 127993012 T^{10} + 304924597 T^{11} + 1888927120 T^{12} + 3825620449 T^{13} + 20766661440 T^{14} + 34319363547 T^{15} + 162952728319 T^{16} + 201065949770 T^{17} + 823139378038 T^{18} + 592939382485 T^{19} + 2015993900449 T^{20}$$
$19$ $$1 - T + 120 T^{2} - 160 T^{3} + 7208 T^{4} - 10476 T^{5} + 284244 T^{6} - 412117 T^{7} + 8125861 T^{8} - 11020860 T^{9} + 176144404 T^{10} - 209396340 T^{11} + 2933435821 T^{12} - 2826710503 T^{13} + 37042962324 T^{14} - 25939613124 T^{15} + 339106710248 T^{16} - 143019478240 T^{17} + 2038027564920 T^{18} - 322687697779 T^{19} + 6131066257801 T^{20}$$
$23$ $$1 - 9 T + 168 T^{2} - 1090 T^{3} + 12175 T^{4} - 62579 T^{5} + 541580 T^{6} - 2324207 T^{7} + 17335456 T^{8} - 65014125 T^{9} + 439664024 T^{10} - 1495324875 T^{11} + 9170456224 T^{12} - 28278626569 T^{13} + 151556288780 T^{14} - 402779908597 T^{15} + 1802336948575 T^{16} - 3711259737230 T^{17} + 13156245527208 T^{18} - 16210373953167 T^{19} + 41426511213649 T^{20}$$
$29$ $$1 - 23 T + 408 T^{2} - 5128 T^{3} + 55655 T^{4} - 506987 T^{5} + 4175621 T^{6} - 30484005 T^{7} + 205212340 T^{8} - 1249333725 T^{9} + 7057521630 T^{10} - 36230678025 T^{11} + 172583577940 T^{12} - 743474397945 T^{13} + 2953337396501 T^{14} - 10398885898063 T^{15} + 33104891930255 T^{16} - 88457365712552 T^{17} + 204100536488088 T^{18} - 333664357444987 T^{19} + 420707233300201 T^{20}$$
$31$ $$1 - 5 T + 157 T^{2} - 710 T^{3} + 11877 T^{4} - 47207 T^{5} + 617020 T^{6} - 2225635 T^{7} + 25547802 T^{8} - 86657629 T^{9} + 872978638 T^{10} - 2686386499 T^{11} + 24551437722 T^{12} - 66303892285 T^{13} + 569830927420 T^{14} - 1351496331257 T^{15} + 10540881219237 T^{16} - 19533956018810 T^{17} + 133903892878237 T^{18} - 132198110803355 T^{19} + 819628286980801 T^{20}$$
$37$ $$1 - 16 T + 297 T^{2} - 2617 T^{3} + 27363 T^{4} - 156623 T^{5} + 1351272 T^{6} - 6377993 T^{7} + 61637924 T^{8} - 295553271 T^{9} + 2657103486 T^{10} - 10935471027 T^{11} + 84382317956 T^{12} - 323064479429 T^{13} + 2532501282792 T^{14} - 10860858577211 T^{15} + 70205971729467 T^{16} - 248436722457061 T^{17} + 1043206397814537 T^{18} - 2079387836721232 T^{19} + 4808584372417849 T^{20}$$
$41$ $$( 1 + T )^{10}$$
$43$ $$1 - 20 T + 387 T^{2} - 4356 T^{3} + 50352 T^{4} - 426716 T^{5} + 3988561 T^{6} - 29462196 T^{7} + 243795075 T^{8} - 1609225832 T^{9} + 11807019104 T^{10} - 69196710776 T^{11} + 450777093675 T^{12} - 2342450817372 T^{13} + 13636096335361 T^{14} - 62730854763188 T^{15} + 318293272243248 T^{16} - 1184041869982092 T^{17} + 4523333507431587 T^{18} - 10051852238736860 T^{19} + 21611482313284249 T^{20}$$
$47$ $$1 - 16 T + 241 T^{2} - 2509 T^{3} + 25991 T^{4} - 214893 T^{5} + 1859544 T^{6} - 13689933 T^{7} + 107568832 T^{8} - 741660077 T^{9} + 5459961710 T^{10} - 34858023619 T^{11} + 237619549888 T^{12} - 1421329913859 T^{13} + 9073981525464 T^{14} - 49284636589251 T^{15} + 280162585616039 T^{16} - 1271117409241667 T^{17} + 5738520085484401 T^{18} - 17906087569644272 T^{19} + 52599132235830049 T^{20}$$
$53$ $$1 - 26 T + 601 T^{2} - 9515 T^{3} + 137765 T^{4} - 1642903 T^{5} + 18256856 T^{6} - 177050001 T^{7} + 1613772770 T^{8} - 13131416435 T^{9} + 100857559838 T^{10} - 695965071055 T^{11} + 4533087710930 T^{12} - 26358672998877 T^{13} + 144055375387736 T^{14} - 687054630036179 T^{15} + 3053473210936685 T^{16} - 11177376495549055 T^{17} + 37418073937227961 T^{18} - 85793853386855458 T^{19} + 174887470365513049 T^{20}$$
$59$ $$1 - 10 T + 464 T^{2} - 3933 T^{3} + 100547 T^{4} - 739002 T^{5} + 13587671 T^{6} - 87375608 T^{7} + 1277936808 T^{8} - 7168651661 T^{9} + 87686002282 T^{10} - 422950447999 T^{11} + 4448498028648 T^{12} - 17945114995432 T^{13} + 164646714656231 T^{14} - 528330486809598 T^{15} + 4241126116001627 T^{16} - 9787866289793127 T^{17} + 68129323048404944 T^{18} - 86629958186549390 T^{19} + 511116753300641401 T^{20}$$
$61$ $$1 + 325 T^{2} + 341 T^{3} + 55126 T^{4} + 87657 T^{5} + 6440407 T^{6} + 11319704 T^{7} + 568806163 T^{8} + 962936366 T^{9} + 39149474996 T^{10} + 58739118326 T^{11} + 2116527732523 T^{12} + 2569357733624 T^{13} + 89172851297287 T^{14} + 74034777956757 T^{15} + 2840112157024486 T^{16} + 1071675307083161 T^{17} + 62304876724116325 T^{18} + 713342911662882601 T^{20}$$
$67$ $$1 - 7 T + 602 T^{2} - 3866 T^{3} + 167009 T^{4} - 969997 T^{5} + 28116436 T^{6} - 145123217 T^{7} + 3181082822 T^{8} - 14267922051 T^{9} + 252734277332 T^{10} - 955950777417 T^{11} + 14279880787958 T^{12} - 43647694114571 T^{13} + 566577703924756 T^{14} - 1309617303414679 T^{15} + 15107363947662521 T^{16} - 23430711066178718 T^{17} + 244452741889097882 T^{18} - 190445740774064629 T^{19} + 1822837804551761449 T^{20}$$
$71$ $$1 - 5 T + 312 T^{2} - 1090 T^{3} + 56870 T^{4} - 195450 T^{5} + 7498233 T^{6} - 22581087 T^{7} + 744547841 T^{8} - 2080097600 T^{9} + 59734234259 T^{10} - 147686929600 T^{11} + 3753265666481 T^{12} - 8082019429257 T^{13} + 190542705059673 T^{14} - 352636626652950 T^{15} + 7285063146587270 T^{16} - 9913680972646190 T^{17} + 201475101748677432 T^{18} - 229242503592245155 T^{19} + 3255243551009881201 T^{20}$$
$73$ $$1 - 13 T + 250 T^{2} - 2110 T^{3} + 24489 T^{4} - 122685 T^{5} + 1653191 T^{6} - 9421251 T^{7} + 193642338 T^{8} - 1495731037 T^{9} + 19158703894 T^{10} - 109188365701 T^{11} + 1031920019202 T^{12} - 3665026800267 T^{13} + 46947716437031 T^{14} - 254334788387205 T^{15} + 3706023867591321 T^{16} - 23310010875294670 T^{17} + 201615022973520250 T^{18} - 765330627207482869 T^{19} + 4297625829703557649 T^{20}$$
$79$ $$1 + T + 364 T^{2} - 252 T^{3} + 70672 T^{4} - 126376 T^{5} + 9900124 T^{6} - 21591815 T^{7} + 1095541909 T^{8} - 2270815516 T^{9} + 97073937704 T^{10} - 179394425764 T^{11} + 6837277054069 T^{12} - 10645606875785 T^{13} + 385610631710044 T^{14} - 388866079480024 T^{15} + 17179476656580112 T^{16} - 4839385064512068 T^{17} + 552227606805988204 T^{18} + 119851595982618319 T^{19} + 9468276082626847201 T^{20}$$
$83$ $$1 + 21 T + 584 T^{2} + 7822 T^{3} + 137615 T^{4} + 1481099 T^{5} + 21400343 T^{6} + 200504223 T^{7} + 2508803676 T^{8} + 20728996337 T^{9} + 231117196858 T^{10} + 1720506695971 T^{11} + 17283148523964 T^{12} + 114645708156501 T^{13} + 1015624347604103 T^{14} + 5834109157306657 T^{15} + 44991899481174935 T^{16} + 212258190840862394 T^{17} + 1315338663569199944 T^{18} + 3925745360618348463 T^{19} + 15516041187205853449 T^{20}$$
$89$ $$1 + 6 T + 363 T^{2} + 2853 T^{3} + 66271 T^{4} + 646201 T^{5} + 9034652 T^{6} + 90887475 T^{7} + 1070787904 T^{8} + 9430571081 T^{9} + 106341932306 T^{10} + 839320826209 T^{11} + 8481710987584 T^{12} + 64072852363275 T^{13} + 566854313135132 T^{14} + 3608424800003249 T^{15} + 32935447133276431 T^{16} + 126191998456944237 T^{17} + 1428981736469855403 T^{18} + 2102138422244911254 T^{19} + 31181719929966183601 T^{20}$$
$97$ $$1 + 29 T + 668 T^{2} + 11220 T^{3} + 178105 T^{4} + 2387957 T^{5} + 30935208 T^{6} + 361493303 T^{7} + 4156434566 T^{8} + 43540141251 T^{9} + 447164066424 T^{10} + 4223393701347 T^{11} + 39107892831494 T^{12} + 329925177328919 T^{13} + 2738671721825448 T^{14} + 20506199278084949 T^{15} + 148356478937879545 T^{16} + 906556751844427860 T^{17} + 5235405641043809948 T^{18} + 22046700700982391293 T^{19} + 73742412689492826049 T^{20}$$