# Properties

 Label 6032.2.a.z Level 6032 Weight 2 Character orbit 6032.a Self dual yes Analytic conductor 48.166 Analytic rank 1 Dimension 10 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6032 = 2^{4} \cdot 13 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6032.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.1657624992$$ Analytic rank: $$1$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 3 x^{9} - 17 x^{8} + 47 x^{7} + 104 x^{6} - 235 x^{5} - 283 x^{4} + 364 x^{3} + 330 x^{2} + 12 x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3016) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} -\beta_{2} q^{5} -\beta_{6} q^{7} + ( 1 + \beta_{1} + \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} -\beta_{2} q^{5} -\beta_{6} q^{7} + ( 1 + \beta_{1} + \beta_{5} + \beta_{6} ) q^{9} + ( -1 - \beta_{5} + \beta_{7} ) q^{11} - q^{13} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{15} + ( 1 + \beta_{2} - \beta_{5} - \beta_{8} - \beta_{9} ) q^{17} + ( -1 - \beta_{4} + \beta_{6} + \beta_{9} ) q^{19} + ( -1 + 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} ) q^{21} + ( \beta_{2} + \beta_{3} ) q^{23} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{25} + ( -2 - \beta_{1} - \beta_{4} - \beta_{6} ) q^{27} - q^{29} + ( -1 + \beta_{1} + \beta_{3} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{31} + ( 2 \beta_{1} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{33} + ( -2 + \beta_{1} + \beta_{4} - \beta_{8} ) q^{35} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{37} + \beta_{1} q^{39} + ( \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{41} + ( -3 + \beta_{2} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{43} + ( 1 - \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{45} + ( -2 + \beta_{1} - \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{7} + \beta_{9} ) q^{47} + ( -2 + \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{49} + ( -2 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} ) q^{51} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{9} ) q^{53} + ( 1 + 2 \beta_{2} + \beta_{3} + \beta_{7} - 2 \beta_{9} ) q^{55} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{57} + ( -2 + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} ) q^{59} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{7} + \beta_{9} ) q^{61} + ( -2 - 3 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{63} + \beta_{2} q^{65} + ( -3 + \beta_{1} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{8} + \beta_{9} ) q^{67} + ( 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{69} + ( 1 + \beta_{2} - 2 \beta_{4} + 3 \beta_{6} - \beta_{8} - \beta_{9} ) q^{71} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{73} + ( -2 + \beta_{2} + 3 \beta_{4} + \beta_{5} + \beta_{8} ) q^{75} + ( 2 \beta_{1} - 2 \beta_{3} + 3 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{77} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{79} + ( -2 + 3 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{9} ) q^{81} + ( -5 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{9} ) q^{83} + ( -2 + \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{85} + \beta_{1} q^{87} + ( 4 - \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{89} + \beta_{6} q^{91} + ( 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{93} + ( -2 + 2 \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{95} + ( -\beta_{1} - 2 \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{97} + ( -5 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{8} - 3 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 3q^{3} + 4q^{5} + 3q^{7} + 13q^{9} + O(q^{10})$$ $$10q - 3q^{3} + 4q^{5} + 3q^{7} + 13q^{9} - 14q^{11} - 10q^{13} - 7q^{15} + 5q^{17} - 11q^{19} - 7q^{23} + 10q^{25} - 21q^{27} - 10q^{29} - 5q^{31} + 5q^{33} - 11q^{35} + 8q^{37} + 3q^{39} + 14q^{41} - 35q^{43} + 7q^{45} - 7q^{49} - 20q^{51} - 11q^{53} - 8q^{55} + 4q^{57} - 23q^{59} - 8q^{61} - 43q^{63} - 4q^{65} - 27q^{67} + 10q^{69} - 3q^{71} + 7q^{73} - 23q^{75} + 2q^{77} - 9q^{79} - 6q^{81} - 48q^{83} - 6q^{85} + 3q^{87} + 20q^{89} - 3q^{91} - 11q^{93} - 11q^{95} + q^{97} - 54q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 3 x^{9} - 17 x^{8} + 47 x^{7} + 104 x^{6} - 235 x^{5} - 283 x^{4} + 364 x^{3} + 330 x^{2} + 12 x - 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$13 \nu^{9} - 71 \nu^{8} - 185 \nu^{7} + 1243 \nu^{6} + 828 \nu^{5} - 6935 \nu^{4} - 1999 \nu^{3} + 12100 \nu^{2} + 5022 \nu - 524$$$$)/492$$ $$\beta_{3}$$ $$=$$ $$($$$$-27 \nu^{9} + 97 \nu^{8} + 359 \nu^{7} - 1257 \nu^{6} - 1808 \nu^{5} + 5005 \nu^{4} + 4997 \nu^{3} - 5804 \nu^{2} - 6898 \nu - 1132$$$$)/492$$ $$\beta_{4}$$ $$=$$ $$($$$$-33 \nu^{9} + 73 \nu^{8} + 539 \nu^{7} - 1017 \nu^{6} - 2966 \nu^{5} + 4459 \nu^{4} + 6317 \nu^{3} - 6374 \nu^{2} - 5242 \nu + 1040$$$$)/492$$ $$\beta_{5}$$ $$=$$ $$($$$$-33 \nu^{9} + 73 \nu^{8} + 539 \nu^{7} - 1017 \nu^{6} - 2966 \nu^{5} + 4459 \nu^{4} + 5825 \nu^{3} - 5882 \nu^{2} - 2290 \nu + 56$$$$)/492$$ $$\beta_{6}$$ $$=$$ $$($$$$33 \nu^{9} - 73 \nu^{8} - 539 \nu^{7} + 1017 \nu^{6} + 2966 \nu^{5} - 4459 \nu^{4} - 5825 \nu^{3} + 6374 \nu^{2} + 1798 \nu - 2024$$$$)/492$$ $$\beta_{7}$$ $$=$$ $$($$$$-4 \nu^{9} - 57 \nu^{8} + 79 \nu^{7} + 1103 \nu^{6} - 239 \nu^{5} - 6678 \nu^{4} - 2031 \nu^{3} + 12453 \nu^{2} + 8402 \nu + 136$$$$)/246$$ $$\beta_{8}$$ $$=$$ $$($$$$45 \nu^{9} - 107 \nu^{8} - 817 \nu^{7} + 1767 \nu^{6} + 5200 \nu^{5} - 9599 \nu^{4} - 13795 \nu^{3} + 17272 \nu^{2} + 14066 \nu - 1612$$$$)/492$$ $$\beta_{9}$$ $$=$$ $$($$$$59 \nu^{9} - 51 \nu^{8} - 1073 \nu^{7} + 551 \nu^{6} + 6262 \nu^{5} - 945 \nu^{4} - 11955 \nu^{3} - 3210 \nu^{2} + 2330 \nu + 1684$$$$)/492$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{4} + 7 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{9} + 9 \beta_{6} + 9 \beta_{5} + 3 \beta_{4} - \beta_{3} + \beta_{2} + 12 \beta_{1} + 25$$ $$\nu^{5}$$ $$=$$ $$\beta_{9} + 2 \beta_{8} - \beta_{7} + 14 \beta_{6} + 6 \beta_{5} + 16 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 57 \beta_{1} + 22$$ $$\nu^{6}$$ $$=$$ $$12 \beta_{9} + 4 \beta_{8} - 3 \beta_{7} + 80 \beta_{6} + 79 \beta_{5} + 48 \beta_{4} - 15 \beta_{3} + 18 \beta_{2} + 124 \beta_{1} + 177$$ $$\nu^{7}$$ $$=$$ $$16 \beta_{9} + 33 \beta_{8} - 22 \beta_{7} + 158 \beta_{6} + 104 \beta_{5} + 194 \beta_{4} - 52 \beta_{3} + 47 \beta_{2} + 497 \beta_{1} + 224$$ $$\nu^{8}$$ $$=$$ $$119 \beta_{9} + 85 \beta_{8} - 66 \beta_{7} + 720 \beta_{6} + 718 \beta_{5} + 593 \beta_{4} - 186 \beta_{3} + 239 \beta_{2} + 1242 \beta_{1} + 1361$$ $$\nu^{9}$$ $$=$$ $$200 \beta_{9} + 424 \beta_{8} - 323 \beta_{7} + 1664 \beta_{6} + 1336 \beta_{5} + 2145 \beta_{4} - 664 \beta_{3} + 697 \beta_{2} + 4530 \beta_{1} + 2257$$

## 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
 3.16847 2.63743 2.52246 2.02548 0.131336 −0.206934 −0.696475 −1.58994 −2.43593 −2.55589
0 −3.16847 0 −1.06896 0 −3.52691 0 7.03919 0
1.2 0 −2.63743 0 3.03277 0 0.746946 0 3.95604 0
1.3 0 −2.52246 0 3.65209 0 −0.974037 0 3.36279 0
1.4 0 −2.02548 0 −3.01540 0 3.00262 0 1.10258 0
1.5 0 −0.131336 0 −0.686438 0 3.43967 0 −2.98275 0
1.6 0 0.206934 0 2.11442 0 4.22910 0 −2.95718 0
1.7 0 0.696475 0 −1.84500 0 −0.816882 0 −2.51492 0
1.8 0 1.58994 0 3.10645 0 −2.36044 0 −0.472088 0
1.9 0 2.43593 0 1.40458 0 −1.75008 0 2.93376 0
1.10 0 2.55589 0 −2.69451 0 1.01002 0 3.53259 0
 $$n$$: 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

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$13$$ $$1$$
$$29$$ $$1$$

## 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 6032.2.a.z 10
4.b odd 2 1 3016.2.a.h 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3016.2.a.h 10 4.b odd 2 1
6032.2.a.z 10 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6032))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 3 T + 13 T^{2} + 34 T^{3} + 101 T^{4} + 220 T^{5} + 545 T^{6} + 1082 T^{7} + 2280 T^{8} + 4065 T^{9} + 7696 T^{10} + 12195 T^{11} + 20520 T^{12} + 29214 T^{13} + 44145 T^{14} + 53460 T^{15} + 73629 T^{16} + 74358 T^{17} + 85293 T^{18} + 59049 T^{19} + 59049 T^{20}$$
$5$ $$1 - 4 T + 28 T^{2} - 92 T^{3} + 419 T^{4} - 1177 T^{5} + 4181 T^{6} - 10331 T^{7} + 30971 T^{8} - 67556 T^{9} + 175536 T^{10} - 337780 T^{11} + 774275 T^{12} - 1291375 T^{13} + 2613125 T^{14} - 3678125 T^{15} + 6546875 T^{16} - 7187500 T^{17} + 10937500 T^{18} - 7812500 T^{19} + 9765625 T^{20}$$
$7$ $$1 - 3 T + 43 T^{2} - 126 T^{3} + 947 T^{4} - 2559 T^{5} + 13784 T^{6} - 33570 T^{7} + 145439 T^{8} - 315844 T^{9} + 1161702 T^{10} - 2210908 T^{11} + 7126511 T^{12} - 11514510 T^{13} + 33095384 T^{14} - 43009113 T^{15} + 111413603 T^{16} - 103766418 T^{17} + 247886443 T^{18} - 121060821 T^{19} + 282475249 T^{20}$$
$11$ $$1 + 14 T + 134 T^{2} + 904 T^{3} + 5137 T^{4} + 24445 T^{5} + 105024 T^{6} + 403390 T^{7} + 1470524 T^{8} + 5041255 T^{9} + 17059432 T^{10} + 55453805 T^{11} + 177933404 T^{12} + 536912090 T^{13} + 1537656384 T^{14} + 3936891695 T^{15} + 9100508857 T^{16} + 17616402584 T^{17} + 28724090054 T^{18} + 33011267674 T^{19} + 25937424601 T^{20}$$
$13$ $$( 1 + T )^{10}$$
$17$ $$1 - 5 T + 112 T^{2} - 498 T^{3} + 6300 T^{4} - 25338 T^{5} + 231843 T^{6} - 834518 T^{7} + 6105295 T^{8} - 19432937 T^{9} + 119690082 T^{10} - 330359929 T^{11} + 1764430255 T^{12} - 4099986934 T^{13} + 19363759203 T^{14} - 35976336666 T^{15} + 152066684700 T^{16} - 204348659154 T^{17} + 781284833392 T^{18} - 592939382485 T^{19} + 2015993900449 T^{20}$$
$19$ $$1 + 11 T + 170 T^{2} + 1241 T^{3} + 10798 T^{4} + 56565 T^{5} + 353190 T^{6} + 1371510 T^{7} + 7152375 T^{8} + 22753959 T^{9} + 125558756 T^{10} + 432325221 T^{11} + 2582007375 T^{12} + 9407187090 T^{13} + 46028073990 T^{14} + 140060539935 T^{15} + 508001423038 T^{16} + 1109294828099 T^{17} + 2887205716970 T^{18} + 3549564675569 T^{19} + 6131066257801 T^{20}$$
$23$ $$1 + 7 T + 157 T^{2} + 928 T^{3} + 11663 T^{4} + 60280 T^{5} + 551638 T^{6} + 2539770 T^{7} + 18788248 T^{8} + 77346219 T^{9} + 489411626 T^{10} + 1778963037 T^{11} + 9938983192 T^{12} + 30901381590 T^{13} + 154370929558 T^{14} + 387982756040 T^{15} + 1726542573407 T^{16} + 3159678014816 T^{17} + 12294824689117 T^{18} + 12608068630241 T^{19} + 41426511213649 T^{20}$$
$29$ $$( 1 + T )^{10}$$
$31$ $$1 + 5 T + 181 T^{2} + 898 T^{3} + 17999 T^{4} + 82396 T^{5} + 1184032 T^{6} + 4929330 T^{7} + 56450384 T^{8} + 208915939 T^{9} + 2011211414 T^{10} + 6476394109 T^{11} + 54248819024 T^{12} + 146849670030 T^{13} + 1093478416672 T^{14} + 2358927525796 T^{15} + 15974178754319 T^{16} + 24706327471678 T^{17} + 154373277776821 T^{18} + 132198110803355 T^{19} + 819628286980801 T^{20}$$
$37$ $$1 - 8 T + 167 T^{2} - 1017 T^{3} + 11943 T^{4} - 44941 T^{5} + 357677 T^{6} + 319009 T^{7} - 2718752 T^{8} + 110489181 T^{9} - 470218808 T^{10} + 4088099697 T^{11} - 3721971488 T^{12} + 16158762877 T^{13} + 670344283997 T^{14} - 3116386771537 T^{15} + 30642470502687 T^{16} - 96545719044261 T^{17} + 586584068804807 T^{18} - 1039693918360616 T^{19} + 4808584372417849 T^{20}$$
$41$ $$1 - 14 T + 255 T^{2} - 2020 T^{3} + 23887 T^{4} - 157284 T^{5} + 1740012 T^{6} - 10727179 T^{7} + 100025972 T^{8} - 520487267 T^{9} + 4347974930 T^{10} - 21339977947 T^{11} + 168143658932 T^{12} - 739327903859 T^{13} + 4916858049132 T^{14} - 18222326718084 T^{15} + 113465740004767 T^{16} - 393403633239620 T^{17} + 2036155933425855 T^{18} - 4583347081515454 T^{19} + 13422659310152401 T^{20}$$
$43$ $$1 + 35 T + 891 T^{2} + 16210 T^{3} + 246169 T^{4} + 3113896 T^{5} + 34475007 T^{6} + 333387848 T^{7} + 2881378156 T^{8} + 22163752175 T^{9} + 153655853008 T^{10} + 953041343525 T^{11} + 5327668210444 T^{12} + 26506667630936 T^{13} + 117863188406607 T^{14} + 457769002623928 T^{15} + 1556123620409281 T^{16} + 4406179686044470 T^{17} + 10414186447342491 T^{18} + 17590741417789505 T^{19} + 21611482313284249 T^{20}$$
$47$ $$1 + 198 T^{2} + 306 T^{3} + 20920 T^{4} + 65701 T^{5} + 1551573 T^{6} + 7040897 T^{7} + 91003583 T^{8} + 482886134 T^{9} + 4555687258 T^{10} + 22695648298 T^{11} + 201026914847 T^{12} + 731007049231 T^{13} + 7571181288213 T^{14} + 15068196304907 T^{15} + 225501184682680 T^{16} + 155026674861678 T^{17} + 4714634759028678 T^{18} + 52599132235830049 T^{20}$$
$53$ $$1 + 11 T + 412 T^{2} + 3498 T^{3} + 75168 T^{4} + 522646 T^{5} + 8493208 T^{6} + 50592888 T^{7} + 684662055 T^{8} + 3567810117 T^{9} + 41575412808 T^{10} + 189093936201 T^{11} + 1923215712495 T^{12} + 7532117386776 T^{13} + 67015496353048 T^{14} + 218568201634478 T^{15} + 1666050697344672 T^{16} + 4109139567149826 T^{17} + 25650992449480732 T^{18} + 36297399509823463 T^{19} + 174887470365513049 T^{20}$$
$59$ $$1 + 23 T + 568 T^{2} + 8622 T^{3} + 128106 T^{4} + 1484434 T^{5} + 16728074 T^{6} + 159548494 T^{7} + 1494192265 T^{8} + 12288692963 T^{9} + 100141178180 T^{10} + 725032884817 T^{11} + 5201283274465 T^{12} + 32767910149226 T^{13} + 202700111492714 T^{14} + 1061257936861766 T^{15} + 5403579442613946 T^{16} + 21457153102109418 T^{17} + 83399688559254328 T^{18} + 199248903829063597 T^{19} + 511116753300641401 T^{20}$$
$61$ $$1 + 8 T + 417 T^{2} + 2698 T^{3} + 84049 T^{4} + 458650 T^{5} + 10933960 T^{6} + 51438837 T^{7} + 1020347998 T^{8} + 4180156915 T^{9} + 71406285838 T^{10} + 254989571815 T^{11} + 3796714900558 T^{12} + 11675638661097 T^{13} + 151389871660360 T^{14} + 387374093453650 T^{15} + 4330235944667689 T^{16} + 8479120171584658 T^{17} + 79941949519866177 T^{18} + 93553168742673128 T^{19} + 713342911662882601 T^{20}$$
$67$ $$1 + 27 T + 427 T^{2} + 4542 T^{3} + 43209 T^{4} + 408414 T^{5} + 4210168 T^{6} + 40846430 T^{7} + 376162806 T^{8} + 3172940923 T^{9} + 26530748026 T^{10} + 212587041841 T^{11} + 1688594836134 T^{12} + 12285094826090 T^{13} + 84839604798328 T^{14} + 551409995450298 T^{15} + 3908616235140321 T^{16} + 27527752111377066 T^{17} + 173390898316685707 T^{18} + 734576428699963569 T^{19} + 1822837804551761449 T^{20}$$
$71$ $$1 + 3 T + 339 T^{2} + 2438 T^{3} + 55875 T^{4} + 591768 T^{5} + 7071361 T^{6} + 74858184 T^{7} + 773733582 T^{8} + 6441590187 T^{9} + 64967003220 T^{10} + 457352903277 T^{11} + 3900390986862 T^{12} + 26792567493624 T^{13} + 179695169967841 T^{14} + 1067685194582568 T^{15} + 7157603364085875 T^{16} + 22173902946157258 T^{17} + 218910447092312979 T^{18} + 137545502155347093 T^{19} + 3255243551009881201 T^{20}$$
$73$ $$1 - 7 T + 149 T^{2} + 86 T^{3} + 19256 T^{4} - 62019 T^{5} + 2510809 T^{6} + 314926 T^{7} + 178368761 T^{8} - 208183770 T^{9} + 19040957664 T^{10} - 15197415210 T^{11} + 950527127369 T^{12} + 122511567742 T^{13} + 71302559086969 T^{14} - 128569827126267 T^{15} + 2914091861420984 T^{16} + 950076272642342 T^{17} + 120162553692218069 T^{18} - 412101106957875391 T^{19} + 4297625829703557649 T^{20}$$
$79$ $$1 + 9 T + 430 T^{2} + 3943 T^{3} + 101090 T^{4} + 890355 T^{5} + 16018218 T^{6} + 132232226 T^{7} + 1875969073 T^{8} + 14092748579 T^{9} + 168293794632 T^{10} + 1113327137741 T^{11} + 11707922984593 T^{12} + 65195644474814 T^{13} + 623910888575658 T^{14} + 2739672550131645 T^{15} + 24573710878617890 T^{16} + 75721013132424937 T^{17} + 652356788259821230 T^{18} + 1078664363843564871 T^{19} + 9468276082626847201 T^{20}$$
$83$ $$1 + 48 T + 1607 T^{2} + 39067 T^{3} + 791164 T^{4} + 13466480 T^{5} + 201308134 T^{6} + 2648331548 T^{7} + 31293429719 T^{8} + 331438073283 T^{9} + 3181189934558 T^{10} + 27509360082489 T^{11} + 215580437334191 T^{12} + 1514281550836276 T^{13} + 9553746043283014 T^{14} + 53045012038146640 T^{15} + 258663453556111516 T^{16} + 1060124104011758009 T^{17} + 3619433617047438887 T^{18} + 8973132252841939344 T^{19} + 15516041187205853449 T^{20}$$
$89$ $$1 - 20 T + 826 T^{2} - 13265 T^{3} + 309864 T^{4} - 4125997 T^{5} + 69930056 T^{6} - 784800296 T^{7} + 10554625333 T^{8} - 100327277122 T^{9} + 1115257123896 T^{10} - 8929127663858 T^{11} + 83603187262693 T^{12} - 553259879870824 T^{13} + 4387568426695496 T^{14} - 23039812534395653 T^{15} + 153996610742339304 T^{16} - 586728657389192185 T^{17} + 3251622353509918906 T^{18} - 7007128074149704180 T^{19} + 31181719929966183601 T^{20}$$
$97$ $$1 - T + 474 T^{2} - 1011 T^{3} + 124564 T^{4} - 289437 T^{5} + 22765700 T^{6} - 50389262 T^{7} + 3148321249 T^{8} - 6418058069 T^{9} + 341520385872 T^{10} - 622551632693 T^{11} + 29622554631841 T^{12} - 45988918917326 T^{13} + 2015431052461700 T^{14} - 2485494001965309 T^{15} + 103758324821975956 T^{16} - 81687065607372243 T^{17} + 3714943523734679514 T^{18} - 760231058654565217 T^{19} + 73742412689492826049 T^{20}$$