# Properties

 Label 289.10.a.a Level $289$ Weight $10$ Character orbit 289.a Self dual yes Analytic conductor $148.845$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$289 = 17^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 289.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$148.845356651$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ Defining polynomial: $$x^{5} - 2 x^{4} - 1596 x^{3} + 5754 x^{2} + 488987 x - 2711704$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 17) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -7 + \beta_{1} ) q^{2} + ( 48 - 2 \beta_{1} - \beta_{4} ) q^{3} + ( 177 - 17 \beta_{1} + \beta_{3} - 3 \beta_{4} ) q^{4} + ( -309 + 26 \beta_{1} - \beta_{2} + 6 \beta_{3} + 6 \beta_{4} ) q^{5} + ( -1574 + 154 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} + 12 \beta_{4} ) q^{6} + ( 2625 + 44 \beta_{1} - 13 \beta_{2} - 12 \beta_{3} - 13 \beta_{4} ) q^{7} + ( -8547 + 179 \beta_{1} + 28 \beta_{2} - 9 \beta_{3} + 67 \beta_{4} ) q^{8} + ( 2404 - 606 \beta_{1} - 73 \beta_{2} + 50 \beta_{3} - 14 \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( -7 + \beta_{1} ) q^{2} + ( 48 - 2 \beta_{1} - \beta_{4} ) q^{3} + ( 177 - 17 \beta_{1} + \beta_{3} - 3 \beta_{4} ) q^{4} + ( -309 + 26 \beta_{1} - \beta_{2} + 6 \beta_{3} + 6 \beta_{4} ) q^{5} + ( -1574 + 154 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} + 12 \beta_{4} ) q^{6} + ( 2625 + 44 \beta_{1} - 13 \beta_{2} - 12 \beta_{3} - 13 \beta_{4} ) q^{7} + ( -8547 + 179 \beta_{1} + 28 \beta_{2} - 9 \beta_{3} + 67 \beta_{4} ) q^{8} + ( 2404 - 606 \beta_{1} - 73 \beta_{2} + 50 \beta_{3} - 14 \beta_{4} ) q^{9} + ( 18048 - 560 \beta_{1} + 104 \beta_{3} - 104 \beta_{4} ) q^{10} + ( 13582 + 138 \beta_{1} + 54 \beta_{2} - 48 \beta_{3} + 139 \beta_{4} ) q^{11} + ( 82450 - 3538 \beta_{1} - 304 \beta_{2} + 266 \beta_{3} - 190 \beta_{4} ) q^{12} + ( -31799 - 150 \beta_{1} - 37 \beta_{2} + 198 \beta_{3} - 546 \beta_{4} ) q^{13} + ( 16212 + 2388 \beta_{1} + 368 \beta_{2} - 384 \beta_{3} + 256 \beta_{4} ) q^{14} + ( -139040 + 4156 \beta_{1} + 548 \beta_{2} + 56 \beta_{3} - 354 \beta_{4} ) q^{15} + ( 73093 - 8421 \beta_{1} - 1244 \beta_{2} + 179 \beta_{3} - \beta_{4} ) q^{16} + ( -387255 + 14625 \beta_{1} + 2064 \beta_{2} - 1248 \beta_{3} + 3408 \beta_{4} ) q^{18} + ( -75406 + 4968 \beta_{1} + 234 \beta_{2} - 1832 \beta_{3} + 858 \beta_{4} ) q^{19} + ( -336512 + 28224 \beta_{1} + 1760 \beta_{2} - 2384 \beta_{3} - 976 \beta_{4} ) q^{20} + ( 361375 - 11086 \beta_{1} - 685 \beta_{2} - 754 \beta_{3} - 3310 \beta_{4} ) q^{21} + ( -23518 - 4366 \beta_{1} - 2600 \beta_{2} + 716 \beta_{3} - 2340 \beta_{4} ) q^{22} + ( -329681 + 6084 \beta_{1} - 167 \beta_{2} - 4900 \beta_{3} + 2325 \beta_{4} ) q^{23} + ( -1965510 + 80934 \beta_{1} + 5784 \beta_{2} - 3618 \beta_{3} + 11766 \beta_{4} ) q^{24} + ( 642933 + 31460 \beta_{1} - 2518 \beta_{2} - 4932 \beta_{3} - 1648 \beta_{4} ) q^{25} + ( 134226 + 34018 \beta_{1} + 6048 \beta_{2} + 864 \beta_{3} + 4144 \beta_{4} ) q^{26} + ( 643184 - 117456 \beta_{1} - 6956 \beta_{2} + 5152 \beta_{3} - 472 \beta_{4} ) q^{27} + ( -6524 - 88548 \beta_{1} - 5760 \beta_{2} + 10292 \beta_{3} - 9372 \beta_{4} ) q^{28} + ( -721211 - 24298 \beta_{1} + 8977 \beta_{2} - 2494 \beta_{3} - 5182 \beta_{4} ) q^{29} + ( 3462160 - 150272 \beta_{1} - 10096 \beta_{2} + 14096 \beta_{3} - 22512 \beta_{4} ) q^{30} + ( 1493895 - 87056 \beta_{1} + 10641 \beta_{2} - 11180 \beta_{3} - 34533 \beta_{4} ) q^{31} + ( -1145387 + 92331 \beta_{1} + 16244 \beta_{2} - 23701 \beta_{3} + 17975 \beta_{4} ) q^{32} + ( -2290557 + 72690 \beta_{1} + 9963 \beta_{2} - 8766 \beta_{3} - 7002 \beta_{4} ) q^{33} + ( -5284272 - 77964 \beta_{1} + 9164 \beta_{2} + 41232 \beta_{3} + 38410 \beta_{4} ) q^{35} + ( 10198225 - 642225 \beta_{1} - 44416 \beta_{2} + 15521 \beta_{3} - 100067 \beta_{4} ) q^{36} + ( 6287685 - 22746 \beta_{1} - 15899 \beta_{2} + 5938 \beta_{3} + 12574 \beta_{4} ) q^{37} + ( 3848092 - 358532 \beta_{1} - 19808 \beta_{2} - 14752 \beta_{3} - 21536 \beta_{4} ) q^{38} + ( 8522696 - 135080 \beta_{1} - 32468 \beta_{2} + 37048 \beta_{3} + 36940 \beta_{4} ) q^{39} + ( 10976352 - 468960 \beta_{1} - 43968 \beta_{2} - 28672 \beta_{3} - 59520 \beta_{4} ) q^{40} + ( 1421988 + 387124 \beta_{1} + 17594 \beta_{2} + 65172 \beta_{3} + 41192 \beta_{4} ) q^{41} + ( -9158968 + 698216 \beta_{1} + 39904 \beta_{2} - 40592 \beta_{3} + 69696 \beta_{4} ) q^{42} + ( -11492294 + 231296 \beta_{1} - 64242 \beta_{2} + 13184 \beta_{3} + 123526 \beta_{4} ) q^{43} + ( -8739126 + 235702 \beta_{1} + 56336 \beta_{2} - 21246 \beta_{3} + 11738 \beta_{4} ) q^{44} + ( -2788251 + 697662 \beta_{1} + 43881 \beta_{2} - 104862 \beta_{3} + 146550 \beta_{4} ) q^{45} + ( 6834352 - 1010368 \beta_{1} - 34192 \beta_{2} - 60872 \beta_{3} - 18728 \beta_{4} ) q^{46} + ( -3131634 - 723308 \beta_{1} + 2726 \beta_{2} + 102088 \beta_{3} - 22048 \beta_{4} ) q^{47} + ( 21467242 - 2338474 \beta_{1} - 91768 \beta_{2} + 21734 \beta_{3} - 336130 \beta_{4} ) q^{48} + ( -2354982 + 213486 \beta_{1} + 16821 \beta_{2} - 209650 \beta_{3} - 102102 \beta_{4} ) q^{49} + ( 17202051 + 68571 \beta_{1} + 53888 \beta_{2} - 86208 \beta_{3} - 19232 \beta_{4} ) q^{50} + ( 34854146 - 465666 \beta_{1} - 155904 \beta_{2} + 61890 \beta_{3} + 17850 \beta_{4} ) q^{52} + ( -16615966 + 152616 \beta_{1} + 258816 \beta_{2} - 164784 \beta_{3} + 55220 \beta_{4} ) q^{53} + ( -78104580 + 2364012 \beta_{1} + 191328 \beta_{2} - 171480 \beta_{3} + 497928 \beta_{4} ) q^{54} + ( 1354508 - 394556 \beta_{1} - 121312 \beta_{2} + 129312 \beta_{3} + 110278 \beta_{4} ) q^{55} + ( -64080268 + 1399244 \beta_{1} + 65968 \beta_{2} + 129724 \beta_{3} + 296940 \beta_{4} ) q^{56} + ( -27808062 + 1439252 \beta_{1} + 87834 \beta_{2} - 183132 \beta_{3} + 211960 \beta_{4} ) q^{57} + ( -12857952 - 327856 \beta_{1} - 183968 \beta_{2} + 92008 \beta_{3} - 88520 \beta_{4} ) q^{58} + ( -7188478 - 1081736 \beta_{1} - 101866 \beta_{2} + 28696 \beta_{3} + 838738 \beta_{4} ) q^{59} + ( -46939264 + 6076160 \beta_{1} + 198208 \beta_{2} - 208352 \beta_{3} + 961056 \beta_{4} ) q^{60} + ( 15322337 + 601494 \beta_{1} - 252183 \beta_{2} - 190702 \beta_{3} + 679618 \beta_{4} ) q^{61} + ( -66638204 + 4277044 \beta_{1} - 23840 \beta_{2} - 121104 \beta_{3} + 256624 \beta_{4} ) q^{62} + ( 39986065 - 4362816 \beta_{1} - 37441 \beta_{2} + 351572 \beta_{3} - 85811 \beta_{4} ) q^{63} + ( 26922421 - 1487477 \beta_{1} + 8468 \beta_{2} - 2789 \beta_{3} - 694297 \beta_{4} ) q^{64} + ( 7317776 + 2449816 \beta_{1} + 380232 \beta_{2} - 393496 \beta_{3} - 683460 \beta_{4} ) q^{65} + ( 60821316 - 3258828 \beta_{1} - 218160 \beta_{2} + 115296 \beta_{3} - 377712 \beta_{4} ) q^{66} + ( -61385498 + 1506284 \beta_{1} - 104058 \beta_{2} - 346368 \beta_{3} - 351296 \beta_{4} ) q^{67} + ( -67446455 + 2471206 \beta_{1} + 163157 \beta_{2} - 478126 \beta_{3} + 523578 \beta_{4} ) q^{69} + ( -23188592 - 4344416 \beta_{1} - 362288 \beta_{2} + 741056 \beta_{3} - 280640 \beta_{4} ) q^{70} + ( 96618793 - 3613408 \beta_{1} - 572997 \beta_{2} + 81388 \beta_{3} + 671613 \beta_{4} ) q^{71} + ( -267640899 + 19472787 \beta_{1} + 871836 \beta_{2} - 785577 \beta_{3} + 1728291 \beta_{4} ) q^{72} + ( 58202062 + 413264 \beta_{1} + 551952 \beta_{2} - 652200 \beta_{3} + 1040600 \beta_{4} ) q^{73} + ( -54751612 + 6087540 \beta_{1} + 304736 \beta_{2} - 200648 \beta_{3} + 330696 \beta_{4} ) q^{74} + ( 30423556 + 1292378 \beta_{1} + 92588 \beta_{2} - 462088 \beta_{3} - 542743 \beta_{4} ) q^{75} + ( -208411364 + 5651492 \beta_{1} + 468864 \beta_{2} - 26692 \beta_{3} + 1230796 \beta_{4} ) q^{76} + ( -30391837 + 4369154 \beta_{1} - 318833 \beta_{2} - 241802 \beta_{3} + 84822 \beta_{4} ) q^{77} + ( -142222076 + 10174996 \beta_{1} + 631904 \beta_{2} - 126952 \beta_{3} + 823800 \beta_{4} ) q^{78} + ( 164909903 + 468556 \beta_{1} - 671111 \beta_{2} + 1041436 \beta_{3} - 399047 \beta_{4} ) q^{79} + ( -183928544 + 4263904 \beta_{1} + 515584 \beta_{2} - 560224 \beta_{3} + 3288352 \beta_{4} ) q^{80} + ( 183259756 - 12049086 \beta_{1} + 346643 \beta_{2} - 262678 \beta_{3} - 2418818 \beta_{4} ) q^{81} + ( 221363594 - 545318 \beta_{1} - 491104 \beta_{2} + 1698608 \beta_{3} - 1925936 \beta_{4} ) q^{82} + ( 41276982 - 6790888 \beta_{1} + 96290 \beta_{2} + 987152 \beta_{3} + 1443942 \beta_{4} ) q^{83} + ( 315652680 - 20309000 \beta_{1} - 1326912 \beta_{2} + 1373640 \beta_{3} - 1614808 \beta_{4} ) q^{84} + ( 242346664 - 22716488 \beta_{1} + 606336 \beta_{2} - 493432 \beta_{3} - 48088 \beta_{4} ) q^{86} + ( 29721792 + 4296804 \beta_{1} - 317700 \beta_{2} + 314808 \beta_{3} + 2494482 \beta_{4} ) q^{87} + ( 208194130 - 12204402 \beta_{1} - 199752 \beta_{2} + 609190 \beta_{3} - 776354 \beta_{4} ) q^{88} + ( 66779497 + 20899250 \beta_{1} + 460683 \beta_{2} + 806402 \beta_{3} + 1206630 \beta_{4} ) q^{89} + ( 460075968 - 31781232 \beta_{1} - 2644992 \beta_{2} + 312552 \beta_{3} - 3727944 \beta_{4} ) q^{90} + ( -34422318 - 12123668 \beta_{1} + 679922 \beta_{2} + 1247592 \beta_{3} - 374208 \beta_{4} ) q^{91} + ( -505380000 + 10506368 \beta_{1} + 812448 \beta_{2} - 6688 \beta_{3} + 2827040 \beta_{4} ) q^{92} + ( 703328359 - 13352182 \beta_{1} - 2485549 \beta_{2} + 1206470 \beta_{3} + 2059406 \beta_{4} ) q^{93} + ( -455043760 + 14752608 \beta_{1} + 519312 \beta_{2} + 710896 \beta_{3} + 2038064 \beta_{4} ) q^{94} + ( -290829908 - 24844352 \beta_{1} - 687740 \beta_{2} + 2235320 \beta_{3} + 886660 \beta_{4} ) q^{95} + ( -599610166 + 35015990 \beta_{1} + 2017000 \beta_{2} - 2505866 \beta_{3} + 4983438 \beta_{4} ) q^{96} + ( -125658398 - 33525784 \beta_{1} - 236504 \beta_{2} + 1535904 \beta_{3} + 5543956 \beta_{4} ) q^{97} + ( 180869073 - 13890359 \beta_{1} - 425488 \beta_{2} - 2623040 \beta_{3} + 21392 \beta_{4} ) q^{98} + ( -413521590 + 19380798 \beta_{1} - 707514 \beta_{2} + 516216 \beta_{3} + 2435865 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 33q^{2} + 236q^{3} + 853q^{4} - 1480q^{5} - 7578q^{6} + 13202q^{7} - 42423q^{8} + 10981q^{9} + O(q^{10})$$ $$5q - 33q^{2} + 236q^{3} + 853q^{4} - 1480q^{5} - 7578q^{6} + 13202q^{7} - 42423q^{8} + 10981q^{9} + 89328q^{10} + 68036q^{11} + 406010q^{12} - 158862q^{13} + 84700q^{14} - 687324q^{15} + 350225q^{16} - 1911585q^{18} - 370992q^{19} - 1632640q^{20} + 1783880q^{21} - 122290q^{22} - 1645870q^{23} - 9678702q^{24} + 3270239q^{25} + 734846q^{26} + 2998268q^{27} - 183372q^{28} - 3668616q^{29} + 17048544q^{30} + 7262362q^{31} - 5605919q^{32} - 11334900q^{33} - 26503988q^{35} + 49782133q^{36} + 31420708q^{37} + 18513700q^{38} + 42449884q^{39} + 53930464q^{40} + 7996938q^{41} - 44519496q^{42} - 56908268q^{43} - 43323054q^{44} - 12799536q^{45} + 32063472q^{46} - 16903336q^{47} + 102794498q^{48} - 11784059q^{49} + 85921093q^{50} + 173619082q^{52} - 83362982q^{53} - 386329164q^{54} + 6363364q^{55} - 317409372q^{56} - 136615904q^{57} - 64577488q^{58} - 37946604q^{59} - 223158912q^{60} + 77685452q^{61} - 324855300q^{62} + 191945278q^{63} + 131623105q^{64} + 40321288q^{65} + 298037676q^{66} - 304503600q^{67} - 333409272q^{69} - 122787392q^{70} + 476602922q^{71} - 1301701911q^{72} + 289980486q^{73} - 262289012q^{74} + 153685772q^{75} - 1031276084q^{76} - 143385648q^{77} - 691646196q^{78} + 828240610q^{79} - 912750944q^{80} + 891328609q^{81} + 1109615654q^{82} + 194681148q^{83} + 1541719592q^{84} + 1164707144q^{86} + 158149884q^{87} + 1017979978q^{88} + 376848106q^{89} + 2240087472q^{90} - 194543664q^{91} - 2506713088q^{92} + 3494835920q^{93} - 2244811104q^{94} - 1498679864q^{95} - 2935047582q^{96} - 692035246q^{97} + 871744055q^{98} - 2027106408q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 1596 x^{3} + 5754 x^{2} + 488987 x - 2711704$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{4} + 207 \nu^{3} + 6301 \nu^{2} - 209023 \nu - 509736$$$$)/8096$$ $$\beta_{3}$$ $$=$$ $$($$$$21 \nu^{4} + 345 \nu^{3} - 24845 \nu^{2} - 323145 \nu + 3450824$$$$)/16192$$ $$\beta_{4}$$ $$=$$ $$($$$$7 \nu^{4} + 115 \nu^{3} - 13679 \nu^{2} - 123907 \nu + 4604568$$$$)/16192$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-3 \beta_{4} + \beta_{3} - 3 \beta_{1} + 640$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{4} + 12 \beta_{3} + 28 \beta_{2} + 993 \beta_{1} - 1932$$ $$\nu^{4}$$ $$=$$ $$-3615 \beta_{4} + 1757 \beta_{3} - 460 \beta_{2} - 4475 \beta_{1} + 624596$$

## 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
 −35.0613 −21.1654 5.77274 18.8209 33.6330
−42.0613 256.773 1257.15 −1407.25 −10800.2 5138.64 −31342.2 46249.6 59190.7
1.2 −28.1654 3.02373 281.287 −762.851 −85.1644 −5573.11 6498.11 −19673.9 21486.0
1.3 −1.22726 −177.437 −510.494 1620.18 217.762 1834.42 1254.87 11801.0 −1988.39
1.4 11.8209 67.6654 −372.266 −2390.67 799.867 11355.8 −10452.8 −15104.4 −28259.9
1.5 26.6330 85.9747 197.318 1460.58 2289.77 446.232 −8380.94 −12291.3 38899.6
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$17$$ $$1$$

## 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 289.10.a.a 5
17.b even 2 1 17.10.a.a 5
51.c odd 2 1 153.10.a.c 5
68.d odd 2 1 272.10.a.f 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.10.a.a 5 17.b even 2 1
153.10.a.c 5 51.c odd 2 1
272.10.a.f 5 68.d odd 2 1
289.10.a.a 5 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 $$S_{10}^{\mathrm{new}}(\Gamma_0(289))$$:

 $$T_{2}^{5} + 33 T_{2}^{4} - 1162 T_{2}^{3} - 24920 T_{2}^{2} + 344192 T_{2} + 457728$$ $$T_{3}^{5} - 236 T_{3}^{4} - 26850 T_{3}^{3} + 6621804 T_{3}^{2} - 284823432 T_{3} + 801447696$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$457728 + 344192 T - 24920 T^{2} - 1162 T^{3} + 33 T^{4} + T^{5}$$
$3$ $$801447696 - 284823432 T + 6621804 T^{2} - 26850 T^{3} - 236 T^{4} + T^{5}$$
$5$ $$6073215799787520 + 6910711315456 T - 5931144800 T^{2} - 5422732 T^{3} + 1480 T^{4} + T^{5}$$
$7$ $$266210160692281344 - 769174397227488 T + 392836554656 T^{2} - 7845586 T^{3} - 13202 T^{4} + T^{5}$$
$11$ $$-$$$$18\!\cdots\!36$$$$- 583150347803390760 T + 51654458131068 T^{2} + 161240150 T^{3} - 68036 T^{4} + T^{5}$$
$13$ $$46\!\cdots\!08$$$$+ 15605144350350783296 T - 1715294166968552 T^{2} - 8325650340 T^{3} + 158862 T^{4} + T^{5}$$
$17$ $$T^{5}$$
$19$ $$-$$$$23\!\cdots\!16$$$$-$$$$17\!\cdots\!04$$$$T - 176324676443571104 T^{2} - 413150145000 T^{3} + 370992 T^{4} + T^{5}$$
$23$ $$-$$$$28\!\cdots\!00$$$$-$$$$13\!\cdots\!96$$$$T - 4397519995817681960 T^{2} - 2266461594322 T^{3} + 1645870 T^{4} + T^{5}$$
$29$ $$-$$$$14\!\cdots\!00$$$$-$$$$47\!\cdots\!52$$$$T - 43135444415107067232 T^{2} - 8671259026316 T^{3} + 3668616 T^{4} + T^{5}$$
$31$ $$63\!\cdots\!72$$$$-$$$$15\!\cdots\!16$$$$T +$$$$61\!\cdots\!12$$$$T^{2} - 55088066425306 T^{3} - 7262362 T^{4} + T^{5}$$
$37$ $$-$$$$12\!\cdots\!48$$$$+$$$$31\!\cdots\!60$$$$T -$$$$16\!\cdots\!40$$$$T^{2} + 352949672725220 T^{3} - 31420708 T^{4} + T^{5}$$
$41$ $$-$$$$84\!\cdots\!52$$$$+$$$$16\!\cdots\!32$$$$T +$$$$34\!\cdots\!32$$$$T^{2} - 944595954515528 T^{3} - 7996938 T^{4} + T^{5}$$
$43$ $$-$$$$35\!\cdots\!88$$$$-$$$$76\!\cdots\!72$$$$T -$$$$42\!\cdots\!64$$$$T^{2} + 104989881324440 T^{3} + 56908268 T^{4} + T^{5}$$
$47$ $$-$$$$53\!\cdots\!68$$$$+$$$$13\!\cdots\!68$$$$T +$$$$13\!\cdots\!84$$$$T^{2} - 2082901585389616 T^{3} + 16903336 T^{4} + T^{5}$$
$53$ $$46\!\cdots\!44$$$$-$$$$15\!\cdots\!36$$$$T -$$$$92\!\cdots\!16$$$$T^{2} - 9379424557396600 T^{3} + 83362982 T^{4} + T^{5}$$
$59$ $$43\!\cdots\!28$$$$+$$$$40\!\cdots\!72$$$$T -$$$$23\!\cdots\!92$$$$T^{2} - 31008104213596424 T^{3} + 37946604 T^{4} + T^{5}$$
$61$ $$81\!\cdots\!20$$$$+$$$$24\!\cdots\!92$$$$T +$$$$36\!\cdots\!32$$$$T^{2} - 33463527209865196 T^{3} - 77685452 T^{4} + T^{5}$$
$67$ $$-$$$$17\!\cdots\!00$$$$-$$$$58\!\cdots\!40$$$$T -$$$$55\!\cdots\!76$$$$T^{2} + 7020713481287504 T^{3} + 304503600 T^{4} + T^{5}$$
$71$ $$-$$$$27\!\cdots\!64$$$$-$$$$34\!\cdots\!28$$$$T +$$$$14\!\cdots\!84$$$$T^{2} + 9603673423875238 T^{3} - 476602922 T^{4} + T^{5}$$
$73$ $$-$$$$39\!\cdots\!92$$$$+$$$$18\!\cdots\!28$$$$T +$$$$24\!\cdots\!16$$$$T^{2} - 108975991091496536 T^{3} - 289980486 T^{4} + T^{5}$$
$79$ $$-$$$$88\!\cdots\!72$$$$-$$$$60\!\cdots\!44$$$$T +$$$$47\!\cdots\!40$$$$T^{2} + 104201235865665870 T^{3} - 828240610 T^{4} + T^{5}$$
$83$ $$-$$$$48\!\cdots\!16$$$$-$$$$32\!\cdots\!40$$$$T +$$$$93\!\cdots\!08$$$$T^{2} - 285056400431908168 T^{3} - 194681148 T^{4} + T^{5}$$
$89$ $$26\!\cdots\!96$$$$+$$$$21\!\cdots\!28$$$$T +$$$$27\!\cdots\!24$$$$T^{2} - 834395900612571220 T^{3} - 376848106 T^{4} + T^{5}$$
$97$ $$-$$$$10\!\cdots\!28$$$$+$$$$30\!\cdots\!16$$$$T -$$$$49\!\cdots\!44$$$$T^{2} - 3215802378794508088 T^{3} + 692035246 T^{4} + T^{5}$$