# Properties

 Label 8046.2.a.h Level 8046 Weight 2 Character orbit 8046.a Self dual yes Analytic conductor 64.248 Analytic rank 1 Dimension 9 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$8046 = 2 \cdot 3^{3} \cdot 149$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 8046.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.2476334663$$ Analytic rank: $$1$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + q^{8} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{10} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{11} + ( -1 + \beta_{5} - \beta_{6} ) q^{13} + ( -\beta_{1} + \beta_{3} ) q^{14} + q^{16} + ( -1 + 2 \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{17} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{19} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{20} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{22} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{23} + ( \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{25} + ( -1 + \beta_{5} - \beta_{6} ) q^{26} + ( -\beta_{1} + \beta_{3} ) q^{28} + ( -1 + 2 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{29} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{31} + q^{32} + ( -1 + 2 \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{34} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{8} ) q^{35} + ( -2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{37} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{38} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{40} + ( 2 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} ) q^{41} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{43} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{44} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{46} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{47} + ( -2 + 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{49} + ( \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{50} + ( -1 + \beta_{5} - \beta_{6} ) q^{52} + ( 1 - 5 \beta_{1} + 2 \beta_{4} - \beta_{8} ) q^{53} + ( -4 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{55} + ( -\beta_{1} + \beta_{3} ) q^{56} + ( -1 + 2 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{58} + ( -2 + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{59} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{61} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{62} + q^{64} + ( 1 + 4 \beta_{1} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{65} + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} - 3 \beta_{8} ) q^{67} + ( -1 + 2 \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{68} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{8} ) q^{70} + ( -3 - 3 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{71} + ( 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{73} + ( -2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{74} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{76} + ( -2 + \beta_{1} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{77} + ( -5 + \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{79} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{80} + ( 2 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} ) q^{82} + ( -3 + 5 \beta_{1} + \beta_{2} - \beta_{5} + 4 \beta_{7} + \beta_{8} ) q^{83} + ( -4 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{85} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{86} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{88} + ( -1 - 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} ) q^{89} + ( -3 + \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{8} ) q^{91} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{92} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{94} + ( -6 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} ) q^{95} + ( -3 - \beta_{2} + \beta_{4} + 2 \beta_{7} + 2 \beta_{8} ) q^{97} + ( -2 + 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9q + 9q^{2} + 9q^{4} - 4q^{5} - 4q^{7} + 9q^{8} + O(q^{10})$$ $$9q + 9q^{2} + 9q^{4} - 4q^{5} - 4q^{7} + 9q^{8} - 4q^{10} - 4q^{11} - 8q^{13} - 4q^{14} + 9q^{16} - q^{17} - 10q^{19} - 4q^{20} - 4q^{22} - 8q^{23} - 3q^{25} - 8q^{26} - 4q^{28} - 4q^{29} - 17q^{31} + 9q^{32} - q^{34} - 10q^{35} - 11q^{37} - 10q^{38} - 4q^{40} - 16q^{43} - 4q^{44} - 8q^{46} - 7q^{47} - 5q^{49} - 3q^{50} - 8q^{52} - 12q^{53} - 23q^{55} - 4q^{56} - 4q^{58} - 6q^{59} - 13q^{61} - 17q^{62} + 9q^{64} + 24q^{65} - 14q^{67} - q^{68} - 10q^{70} - 30q^{71} - 12q^{73} - 11q^{74} - 10q^{76} - 12q^{77} - 35q^{79} - 4q^{80} - 5q^{83} - 27q^{85} - 16q^{86} - 4q^{88} - 23q^{89} - 28q^{91} - 8q^{92} - 7q^{94} - 32q^{95} - 21q^{97} - 5q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 3 x^{8} - 9 x^{7} + 25 x^{6} + 29 x^{5} - 58 x^{4} - 43 x^{3} + 34 x^{2} + 25 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$15 \nu^{8} - 69 \nu^{7} - 40 \nu^{6} + 516 \nu^{5} - 329 \nu^{4} - 975 \nu^{3} + 992 \nu^{2} + 355 \nu - 347$$$$)/77$$ $$\beta_{4}$$ $$=$$ $$($$$$-19 \nu^{8} + 72 \nu^{7} + 102 \nu^{6} - 515 \nu^{5} - 35 \nu^{4} + 850 \nu^{3} - 235 \nu^{2} - 116 \nu + 34$$$$)/77$$ $$\beta_{5}$$ $$=$$ $$($$$$-24 \nu^{8} + 95 \nu^{7} + 141 \nu^{6} - 764 \nu^{5} - 105 \nu^{4} + 1714 \nu^{3} - 232 \nu^{2} - 1030 \nu - 30$$$$)/77$$ $$\beta_{6}$$ $$=$$ $$($$$$24 \nu^{8} - 95 \nu^{7} - 141 \nu^{6} + 764 \nu^{5} + 105 \nu^{4} - 1637 \nu^{3} + 155 \nu^{2} + 722 \nu + 107$$$$)/77$$ $$\beta_{7}$$ $$=$$ $$($$$$-32 \nu^{8} + 101 \nu^{7} + 265 \nu^{6} - 839 \nu^{5} - 679 \nu^{4} + 1926 \nu^{3} + 589 \nu^{2} - 1168 \nu - 194$$$$)/77$$ $$\beta_{8}$$ $$=$$ $$($$$$47 \nu^{8} - 170 \nu^{7} - 305 \nu^{6} + 1355 \nu^{5} + 427 \nu^{4} - 2978 \nu^{3} - 59 \nu^{2} + 1600 \nu + 232$$$$)/77$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{2} + 5 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + 7 \beta_{2} + 10 \beta_{1} + 15$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{8} + \beta_{7} + 8 \beta_{6} + 9 \beta_{5} + 2 \beta_{4} + 11 \beta_{2} + 33 \beta_{1} + 21$$ $$\nu^{6}$$ $$=$$ $$13 \beta_{8} + 10 \beta_{7} + 10 \beta_{6} + 16 \beta_{5} + 3 \beta_{4} - 6 \beta_{3} + 48 \beta_{2} + 82 \beta_{1} + 94$$ $$\nu^{7}$$ $$=$$ $$32 \beta_{8} + 13 \beta_{7} + 51 \beta_{6} + 80 \beta_{5} + 23 \beta_{4} + 3 \beta_{3} + 95 \beta_{2} + 240 \beta_{1} + 185$$ $$\nu^{8}$$ $$=$$ $$135 \beta_{8} + 74 \beta_{7} + 73 \beta_{6} + 188 \beta_{5} + 45 \beta_{4} - 19 \beta_{3} + 339 \beta_{2} + 642 \beta_{1} + 663$$

## 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
 1.05829 2.58022 −2.13687 −0.836154 −0.584549 1.67390 2.89499 −0.0929510 −1.55687
1.00000 0 1.00000 −3.80777 0 0.515673 1.00000 0 −3.80777
1.2 1.00000 0 1.00000 −2.72526 0 −2.79628 1.00000 0 −2.72526
1.3 1.00000 0 1.00000 −1.65274 0 0.0358535 1.00000 0 −1.65274
1.4 1.00000 0 1.00000 −1.63508 0 4.18187 1.00000 0 −1.63508
1.5 1.00000 0 1.00000 −0.683695 0 −0.639069 1.00000 0 −0.683695
1.6 1.00000 0 1.00000 0.102080 0 0.350570 1.00000 0 0.102080
1.7 1.00000 0 1.00000 1.72525 0 −2.60583 1.00000 0 1.72525
1.8 1.00000 0 1.00000 1.96687 0 −4.72097 1.00000 0 1.96687
1.9 1.00000 0 1.00000 2.71034 0 1.67819 1.00000 0 2.71034
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 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 8046.2.a.h yes 9
3.b odd 2 1 8046.2.a.g 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8046.2.a.g 9 3.b odd 2 1
8046.2.a.h yes 9 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$149$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8046))$$:

 $$T_{5}^{9} + \cdots$$ $$T_{11}^{9} + \cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{9}$$
$3$ 
$5$ $$1 + 4 T + 32 T^{2} + 104 T^{3} + 488 T^{4} + 1360 T^{5} + 4759 T^{6} + 11470 T^{7} + 32617 T^{8} + 67718 T^{9} + 163085 T^{10} + 286750 T^{11} + 594875 T^{12} + 850000 T^{13} + 1525000 T^{14} + 1625000 T^{15} + 2500000 T^{16} + 1562500 T^{17} + 1953125 T^{18}$$
$7$ $$1 + 4 T + 42 T^{2} + 141 T^{3} + 794 T^{4} + 2265 T^{5} + 9183 T^{6} + 23107 T^{7} + 77577 T^{8} + 179045 T^{9} + 543039 T^{10} + 1132243 T^{11} + 3149769 T^{12} + 5438265 T^{13} + 13344758 T^{14} + 16588509 T^{15} + 34588806 T^{16} + 23059204 T^{17} + 40353607 T^{18}$$
$11$ $$1 + 4 T + 68 T^{2} + 284 T^{3} + 2307 T^{4} + 9201 T^{5} + 50752 T^{6} + 180614 T^{7} + 781453 T^{8} + 2385860 T^{9} + 8595983 T^{10} + 21854294 T^{11} + 67550912 T^{12} + 134711841 T^{13} + 371544657 T^{14} + 503123324 T^{15} + 1325127628 T^{16} + 857435524 T^{17} + 2357947691 T^{18}$$
$13$ $$1 + 8 T + 113 T^{2} + 684 T^{3} + 5484 T^{4} + 26690 T^{5} + 155755 T^{6} + 628402 T^{7} + 2920764 T^{8} + 9865738 T^{9} + 37969932 T^{10} + 106199938 T^{11} + 342193735 T^{12} + 762293090 T^{13} + 2036170812 T^{14} + 3301537356 T^{15} + 7090582421 T^{16} + 6525845768 T^{17} + 10604499373 T^{18}$$
$17$ $$1 + T + 93 T^{2} + 19 T^{3} + 4351 T^{4} - 1013 T^{5} + 135092 T^{6} - 59947 T^{7} + 3044452 T^{8} - 1420770 T^{9} + 51755684 T^{10} - 17324683 T^{11} + 663706996 T^{12} - 84606773 T^{13} + 6177797807 T^{14} + 458613811 T^{15} + 38161496589 T^{16} + 6975757441 T^{17} + 118587876497 T^{18}$$
$19$ $$1 + 10 T + 154 T^{2} + 1087 T^{3} + 9648 T^{4} + 54358 T^{5} + 361242 T^{6} + 1729463 T^{7} + 9424332 T^{8} + 38786652 T^{9} + 179062308 T^{10} + 624336143 T^{11} + 2477758878 T^{12} + 7083988918 T^{13} + 23889403152 T^{14} + 51138872647 T^{15} + 137656247806 T^{16} + 169835630410 T^{17} + 322687697779 T^{18}$$
$23$ $$1 + 8 T + 177 T^{2} + 1015 T^{3} + 12770 T^{4} + 54339 T^{5} + 524816 T^{6} + 1728021 T^{7} + 14993524 T^{8} + 42219669 T^{9} + 344851052 T^{10} + 914123109 T^{11} + 6385436272 T^{12} + 15206280099 T^{13} + 82192100110 T^{14} + 150256427335 T^{15} + 602654104119 T^{16} + 626487882248 T^{17} + 1801152661463 T^{18}$$
$29$ $$1 + 4 T + 140 T^{2} + 228 T^{3} + 8814 T^{4} + 1232 T^{5} + 406708 T^{6} - 76890 T^{7} + 15758047 T^{8} - 990911 T^{9} + 456983363 T^{10} - 64664490 T^{11} + 9919201412 T^{12} + 871370192 T^{13} + 180785267286 T^{14} + 135619717188 T^{15} + 2414982683260 T^{16} + 2000985651844 T^{17} + 14507145975869 T^{18}$$
$31$ $$1 + 17 T + 294 T^{2} + 3288 T^{3} + 35058 T^{4} + 299254 T^{5} + 2415185 T^{6} + 16688137 T^{7} + 109028017 T^{8} + 624140757 T^{9} + 3379868527 T^{10} + 16037299657 T^{11} + 71950776335 T^{12} + 276367353334 T^{13} + 1003680775758 T^{14} + 2918112103128 T^{15} + 8088708548634 T^{16} + 14499147636497 T^{17} + 26439622160671 T^{18}$$
$37$ $$1 + 11 T + 232 T^{2} + 1750 T^{3} + 21288 T^{4} + 115727 T^{5} + 1094681 T^{6} + 4383700 T^{7} + 40610911 T^{8} + 143060408 T^{9} + 1502603707 T^{10} + 6001285300 T^{11} + 55448876693 T^{12} + 216891030047 T^{13} + 1476194156616 T^{14} + 4490021215750 T^{15} + 22024195494856 T^{16} + 38637273993131 T^{17} + 129961739795077 T^{18}$$
$41$ $$1 + 219 T^{2} - 187 T^{3} + 24091 T^{4} - 33174 T^{5} + 1783023 T^{6} - 2683062 T^{7} + 97188163 T^{8} - 134191801 T^{9} + 3984714683 T^{10} - 4510227222 T^{11} + 122887728183 T^{12} - 93741795414 T^{13} + 2791091738291 T^{14} - 888269493067 T^{15} + 42651185979939 T^{16} + 327381934393961 T^{18}$$
$43$ $$1 + 16 T + 299 T^{2} + 2907 T^{3} + 31033 T^{4} + 203017 T^{5} + 1564829 T^{6} + 6998481 T^{7} + 50274985 T^{8} + 203934498 T^{9} + 2161824355 T^{10} + 12940191369 T^{11} + 124414859303 T^{12} + 694074722617 T^{13} + 4562113011619 T^{14} + 18376202383443 T^{15} + 81273764720993 T^{16} + 187011204441616 T^{17} + 502592611936843 T^{18}$$
$47$ $$1 + 7 T + 233 T^{2} + 1150 T^{3} + 24491 T^{4} + 90119 T^{5} + 1667498 T^{6} + 4659039 T^{7} + 87538662 T^{8} + 208700788 T^{9} + 4114317114 T^{10} + 10291817151 T^{11} + 173124644854 T^{12} + 439751972039 T^{13} + 5616888566437 T^{14} + 12396097628350 T^{15} + 118043187067879 T^{16} + 166679006632327 T^{17} + 1119130473102767 T^{18}$$
$53$ $$1 + 12 T + 251 T^{2} + 2123 T^{3} + 28110 T^{4} + 185955 T^{5} + 2056930 T^{6} + 11875405 T^{7} + 123270464 T^{8} + 660472201 T^{9} + 6533334592 T^{10} + 33358012645 T^{11} + 306229567610 T^{12} + 1467274394355 T^{13} + 11755475308230 T^{14} + 47054938676867 T^{15} + 294852496099087 T^{16} + 747116284936332 T^{17} + 3299763591802133 T^{18}$$
$59$ $$1 + 6 T + 286 T^{2} + 1739 T^{3} + 43076 T^{4} + 249230 T^{5} + 4459908 T^{6} + 23858923 T^{7} + 343771964 T^{8} + 1647933976 T^{9} + 20282545876 T^{10} + 83052910963 T^{11} + 915971445132 T^{12} + 3020009882030 T^{13} + 30796079103724 T^{14} + 73351948001699 T^{15} + 711754324658234 T^{16} + 880982625625926 T^{17} + 8662995818654939 T^{18}$$
$61$ $$1 + 13 T + 484 T^{2} + 5224 T^{3} + 107984 T^{4} + 976962 T^{5} + 14476899 T^{6} + 110318977 T^{7} + 1285534845 T^{8} + 8199304700 T^{9} + 78417625545 T^{10} + 410496913417 T^{11} + 3285981011919 T^{12} + 13526860515042 T^{13} + 91202886967184 T^{14} + 269142435661864 T^{15} + 1521087532634164 T^{16} + 2492195068964653 T^{17} + 11694146092834141 T^{18}$$
$67$ $$1 + 14 T + 404 T^{2} + 4765 T^{3} + 76380 T^{4} + 793798 T^{5} + 9308190 T^{6} + 86206581 T^{7} + 824351120 T^{8} + 6730075334 T^{9} + 55231525040 T^{10} + 386981342109 T^{11} + 2799559148970 T^{12} + 15995919547558 T^{13} + 103122555672660 T^{14} + 431034191035285 T^{15} + 2448527488550492 T^{16} + 5684947485792974 T^{17} + 27206534396294947 T^{18}$$
$71$ $$1 + 30 T + 690 T^{2} + 12048 T^{3} + 178293 T^{4} + 2311374 T^{5} + 26830140 T^{6} + 282338634 T^{7} + 2711601060 T^{8} + 23874389457 T^{9} + 192523675260 T^{10} + 1423269053994 T^{11} + 9602802237540 T^{12} + 58735898759694 T^{13} + 321681463677843 T^{14} + 1543352220680208 T^{15} + 6275632909289790 T^{16} + 19372605937372830 T^{17} + 45848500718449031 T^{18}$$
$73$ $$1 + 12 T + 377 T^{2} + 4929 T^{3} + 78527 T^{4} + 927103 T^{5} + 11214517 T^{6} + 111105985 T^{7} + 1134999931 T^{8} + 9491974800 T^{9} + 82854994963 T^{10} + 592083794065 T^{11} + 4362637759789 T^{12} + 26328094425823 T^{13} + 162792092983511 T^{14} + 745926401378481 T^{15} + 4164869241699569 T^{16} + 9677521102728972 T^{17} + 58871586708267913 T^{18}$$
$79$ $$1 + 35 T + 974 T^{2} + 19318 T^{3} + 329779 T^{4} + 4739591 T^{5} + 60921526 T^{6} + 691590981 T^{7} + 7153951615 T^{8} + 66417803924 T^{9} + 565162177585 T^{10} + 4316219312421 T^{11} + 30036688257514 T^{12} + 184607453356871 T^{13} + 1014748582205821 T^{14} + 4695963465754678 T^{15} + 18704607352518866 T^{16} + 53098808346729635 T^{17} + 119851595982618319 T^{18}$$
$83$ $$1 + 5 T + 419 T^{2} + 2365 T^{3} + 86890 T^{4} + 462426 T^{5} + 11913060 T^{6} + 56786404 T^{7} + 1216257923 T^{8} + 5289477148 T^{9} + 100949407609 T^{10} + 391201537156 T^{11} + 6811732838220 T^{12} + 21945961546746 T^{13} + 342263241470270 T^{14} + 773213983017685 T^{15} + 11370005364653713 T^{16} + 11261461160695205 T^{17} + 186940255267540403 T^{18}$$
$89$ $$1 + 23 T + 858 T^{2} + 15157 T^{3} + 314746 T^{4} + 4452798 T^{5} + 66088684 T^{6} + 764102797 T^{7} + 8849517519 T^{8} + 83953359151 T^{9} + 787607059191 T^{10} + 6052458255037 T^{11} + 46590473470796 T^{12} + 279378525240318 T^{13} + 1757560375334954 T^{14} + 7532745427095877 T^{15} + 37950485340363882 T^{16} + 90541542531147863 T^{17} + 350356403707485209 T^{18}$$
$97$ $$1 + 21 T + 846 T^{2} + 14729 T^{3} + 324320 T^{4} + 4641521 T^{5} + 73149346 T^{6} + 860391843 T^{7} + 10636874310 T^{8} + 102739172088 T^{9} + 1031776808070 T^{10} + 8095426850787 T^{11} + 66761433061858 T^{12} + 410910516876401 T^{13} + 2785046192150240 T^{14} + 12268844660599241 T^{15} + 68355348668483598 T^{16} + 164586105481916181 T^{17} + 760231058654565217 T^{18}$$