# Properties

 Label 252.9.z.c Level $252$ Weight $9$ Character orbit 252.z Analytic conductor $102.659$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 252.z (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$102.659409735$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - x^{9} + 1442 x^{8} + 59551 x^{7} + 2229058 x^{6} + 41253567 x^{5} + 582209889 x^{4} + 4552713792 x^{3} + 25685059104 x^{2} + 45666643968 x + 63214027776$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{12}\cdot 3^{9}\cdot 7^{6}$$ Twist minimal: no (minimal twist has level 28) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 112 + 56 \beta_{1} - \beta_{3} ) q^{5} + ( 178 + 50 \beta_{1} + \beta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 112 + 56 \beta_{1} - \beta_{3} ) q^{5} + ( 178 + 50 \beta_{1} + \beta_{6} ) q^{7} + ( -746 - 747 \beta_{1} + 4 \beta_{3} - 3 \beta_{4} - \beta_{5} + 4 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{11} + ( -405 - 808 \beta_{1} + 5 \beta_{2} + 9 \beta_{4} - 10 \beta_{5} - 9 \beta_{6} - 9 \beta_{7} + \beta_{8} - \beta_{9} ) q^{13} + ( -5208 + 5207 \beta_{1} + 14 \beta_{2} + 25 \beta_{3} - 26 \beta_{4} - 7 \beta_{5} - 25 \beta_{6} - 25 \beta_{7} - 26 \beta_{8} - \beta_{9} ) q^{17} + ( -12927 - 6457 \beta_{1} + 13 \beta_{2} + 168 \beta_{3} + 26 \beta_{4} + 13 \beta_{5} - 13 \beta_{6} - 39 \beta_{8} - 13 \beta_{9} ) q^{19} + ( 12 + 41748 \beta_{1} + 268 \beta_{2} + 244 \beta_{3} - 37 \beta_{4} - 268 \beta_{5} - 12 \beta_{6} - 488 \beta_{7} - 49 \beta_{8} + 12 \beta_{9} ) q^{23} + ( 73910 + 73908 \beta_{1} - 276 \beta_{3} + 44 \beta_{4} + 258 \beta_{5} - 42 \beta_{6} - 276 \beta_{7} - 44 \beta_{8} + 2 \beta_{9} ) q^{25} + ( -76081 - 1301 \beta_{2} + 342 \beta_{3} + 23 \beta_{4} + 23 \beta_{6} - 171 \beta_{7} - 37 \beta_{8} - 37 \beta_{9} ) q^{29} + ( -70434 + 70380 \beta_{1} - 748 \beta_{2} - 132 \beta_{3} - 207 \beta_{4} + 374 \beta_{5} - 153 \beta_{6} + 132 \beta_{7} - 207 \beta_{8} - 54 \beta_{9} ) q^{31} + ( 151713 + 39469 \beta_{1} + 1911 \beta_{2} - 588 \beta_{3} + 441 \beta_{4} + 1127 \beta_{5} - 50 \beta_{6} + 392 \beta_{7} + 15 \beta_{8} + 49 \beta_{9} ) q^{35} + ( 128 + 201010 \beta_{1} - 3579 \beta_{2} - 132 \beta_{3} - 299 \beta_{4} + 3579 \beta_{5} - 128 \beta_{6} + 264 \beta_{7} - 427 \beta_{8} + 128 \beta_{9} ) q^{37} + ( 313473 + 627364 \beta_{1} - 3863 \beta_{2} + 173 \beta_{4} + 7726 \beta_{5} - 173 \beta_{6} + 65 \beta_{7} + 209 \beta_{8} - 209 \beta_{9} ) q^{41} + ( 76310 + 7728 \beta_{2} + 504 \beta_{3} - 574 \beta_{4} - 574 \beta_{6} - 252 \beta_{7} - 294 \beta_{8} - 294 \beta_{9} ) q^{43} + ( -97499 - 49132 \beta_{1} - 1194 \beta_{2} + 5672 \beta_{3} + 297 \beta_{4} - 1194 \beta_{5} + 765 \beta_{6} + 468 \beta_{8} + 765 \beta_{9} ) q^{47} + ( 1042538 - 571956 \beta_{1} - 5145 \beta_{2} - 6174 \beta_{3} + 1323 \beta_{4} - 4116 \beta_{5} - 49 \beta_{6} + 5145 \beta_{7} + 91 \beta_{8} + 539 \beta_{9} ) q^{49} + ( -2102619 - 2102484 \beta_{1} - 3748 \beta_{3} - 2244 \beta_{4} - 5559 \beta_{5} + 2109 \beta_{6} - 3748 \beta_{7} + 2244 \beta_{8} - 135 \beta_{9} ) q^{53} + ( -2180559 - 4358604 \beta_{1} + 5706 \beta_{2} + 435 \beta_{4} - 11412 \beta_{5} - 435 \beta_{6} + 6612 \beta_{7} + 1257 \beta_{8} - 1257 \beta_{9} ) q^{55} + ( 2474955 - 2472419 \beta_{1} + 4582 \beta_{2} - 1636 \beta_{3} + 458 \beta_{4} - 2291 \beta_{5} - 2078 \beta_{6} + 1636 \beta_{7} + 458 \beta_{8} + 2536 \beta_{9} ) q^{59} + ( -1084054 - 542932 \beta_{1} - 61 \beta_{2} - 4740 \beta_{3} + 3723 \beta_{4} - 61 \beta_{5} + 1810 \beta_{6} - 1913 \beta_{8} + 1810 \beta_{9} ) q^{61} + ( -1596 + 4294308 \beta_{1} - 4193 \beta_{2} + 5061 \beta_{3} - 4753 \beta_{4} + 4193 \beta_{5} + 1596 \beta_{6} - 10122 \beta_{7} - 3157 \beta_{8} - 1596 \beta_{9} ) q^{65} + ( 9597593 + 9599547 \beta_{1} - 4176 \beta_{3} - 2454 \beta_{4} - 12333 \beta_{5} + 500 \beta_{6} - 4176 \beta_{7} + 2454 \beta_{8} - 1954 \beta_{9} ) q^{67} + ( 3198238 + 22652 \beta_{2} + 25984 \beta_{3} + 6436 \beta_{4} + 6436 \beta_{6} - 12992 \beta_{7} + 6778 \beta_{8} + 6778 \beta_{9} ) q^{71} + ( -8880921 + 8886477 \beta_{1} + 25520 \beta_{2} - 4518 \beta_{3} + 2776 \beta_{4} - 12760 \beta_{5} - 2780 \beta_{6} + 4518 \beta_{7} + 2776 \beta_{8} + 5556 \beta_{9} ) q^{73} + ( -19118405 - 509604 \beta_{1} - 35329 \beta_{2} - 4459 \beta_{3} - 245 \beta_{4} - 6174 \beta_{5} - 287 \beta_{6} + 23324 \beta_{7} - 281 \beta_{8} - 3479 \beta_{9} ) q^{77} + ( -4112 - 10753594 \beta_{1} + 56370 \beta_{2} - 11364 \beta_{3} - 14509 \beta_{4} - 56370 \beta_{5} + 4112 \beta_{6} + 22728 \beta_{7} - 10397 \beta_{8} - 4112 \beta_{9} ) q^{79} + ( 16104980 + 32194468 \beta_{1} + 44634 \beta_{2} + 15672 \beta_{4} - 89268 \beta_{5} - 15672 \beta_{6} - 28612 \beta_{7} - 7746 \beta_{8} + 7746 \beta_{9} ) q^{83} + ( -15766093 - 108363 \beta_{2} + 20100 \beta_{3} + 1513 \beta_{4} + 1513 \beta_{6} - 10050 \beta_{7} + 12417 \beta_{8} + 12417 \beta_{9} ) q^{85} + ( 32213550 + 16105365 \beta_{1} + 16254 \beta_{2} - 43272 \beta_{3} + 198 \beta_{4} + 16254 \beta_{5} + 2820 \beta_{6} + 2622 \beta_{8} + 2820 \beta_{9} ) q^{89} + ( 41474168 - 1210680 \beta_{1} + 39886 \beta_{2} + 59388 \beta_{3} - 9163 \beta_{4} + 68992 \beta_{5} + 1105 \beta_{6} - 38220 \beta_{7} + 236 \beta_{8} ) q^{91} + ( -69385973 - 69402498 \beta_{1} + 24476 \beta_{3} + 1578 \beta_{4} + 98174 \beta_{5} + 14947 \beta_{6} + 24476 \beta_{7} - 1578 \beta_{8} + 16525 \beta_{9} ) q^{95} + ( -29360261 - 58735684 \beta_{1} - 55897 \beta_{2} + 13475 \beta_{4} + 111794 \beta_{5} - 13475 \beta_{6} - 65157 \beta_{7} - 7581 \beta_{8} + 7581 \beta_{9} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 837 q^{5} + 1526 q^{7} + O(q^{10})$$ $$10 q + 837 q^{5} + 1526 q^{7} - 3705 q^{11} - 78003 q^{17} - 96741 q^{19} - 208533 q^{23} + 367978 q^{25} - 754764 q^{29} - 1053717 q^{31} + 1306389 q^{35} - 998075 q^{37} + 738292 q^{43} - 710883 q^{47} + 13288114 q^{49} - 10501461 q^{53} + 37089081 q^{59} - 8180481 q^{61} - 21459108 q^{65} + 48020189 q^{67} + 31918236 q^{71} - 133345593 q^{73} - 188477625 q^{77} + 53590181 q^{79} - 157179282 q^{85} + 241368273 q^{89} + 420709128 q^{91} - 347126775 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} + 1442 x^{8} + 59551 x^{7} + 2229058 x^{6} + 41253567 x^{5} + 582209889 x^{4} + 4552713792 x^{3} + 25685059104 x^{2} + 45666643968 x + 63214027776$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$1030697870430285810692 \nu^{9} - 3186494936398341761183 \nu^{8} + 1502269822242301134142915 \nu^{7} + 58163732835193296674394374 \nu^{6} + 2190542520486224422709006723 \nu^{5} + 38394539797114026420659485446 \nu^{4} + 538652733031031925437154677919 \nu^{3} + 3846903580412093833776248092389 \nu^{2} + 23670412897584825298516273226208 \nu + 10385400836607513563710951726752$$$$)/$$$$31\!\cdots\!28$$ $$\beta_{2}$$ $$=$$ $$($$$$13658650058195493821 \nu^{9} - 155548400327514803569 \nu^{8} + 31252145203919660470242 \nu^{7} + 331005582793425382879183 \nu^{6} + 40097688574391982601702242 \nu^{5} + 597075951213580638619254511 \nu^{4} + 12199176230677951701975622725 \nu^{3} + 27245381291205410009631337440 \nu^{2} + 48537063291938511490321388544 \nu - 4253349958884443336566984156480$$$$)/$$$$94\!\cdots\!76$$ $$\beta_{3}$$ $$=$$ $$($$$$1173843358015606911369110 \nu^{9} - 23475690752821862937502133 \nu^{8} + 1690424355519846781368664171 \nu^{7} + 42006693284378482289536056164 \nu^{6} + 1195100063483944827431566405163 \nu^{5} + 3221542485240245840317377075780 \nu^{4} - 100053414941989038438158131945803 \nu^{3} - 2920605148698112326427173578879271 \nu^{2} - 19292725569823446200452216032635136 \nu - 62444966247717974086077822758225184$$$$)/$$$$73\!\cdots\!32$$ $$\beta_{4}$$ $$=$$ $$($$$$-1830142209287638580557876 \nu^{9} - 26088611497512455797014791 \nu^{8} - 2066282252378723783549897483 \nu^{7} - 157010671979983386678298497592 \nu^{6} - 4885989219050623018455018165307 \nu^{5} - 115987959542693125014001148225544 \nu^{4} - 1348208193559326436953952515930183 \nu^{3} - 14362592456948612372123256018717777 \nu^{2} - 43776218156775758226152576611256160 \nu - 340863474377470395736765558005135120$$$$)/$$$$11\!\cdots\!48$$ $$\beta_{5}$$ $$=$$ $$($$$$-3772487294881347041506277 \nu^{9} + 20824593332643910062471563 \nu^{8} - 5540452570548694309716977452 \nu^{7} - 204458142028463369079938501303 \nu^{6} - 7400040628859705160083449969100 \nu^{5} - 129714720424206619533352790235735 \nu^{4} - 1810508567327659325393100691421511 \nu^{3} - 14397668204015891377599526948705506 \nu^{2} - 79288751965868852163563289526972128 \nu - 140908902774939618167886298547079168$$$$)/$$$$22\!\cdots\!96$$ $$\beta_{6}$$ $$=$$ $$($$$$347703494121554214607156 \nu^{9} + 2710449097434077532956837 \nu^{8} + 499413256765552272419110037 \nu^{7} + 24522650841230322026389978876 \nu^{6} + 960033401974483160694647504869 \nu^{5} + 20471548434386228928560743567836 \nu^{4} + 296844298798834664417552981841825 \nu^{3} + 2229845143119553881974610957440871 \nu^{2} + 6570930980161293327173120498161056 \nu - 31725482081885333342542871143531920$$$$)/$$$$15\!\cdots\!64$$ $$\beta_{7}$$ $$=$$ $$($$$$-1860167609407756545850427 \nu^{9} - 18183571878092701737843211 \nu^{8} - 2399061347732601526190140648 \nu^{7} - 139343908369701796711251290297 \nu^{6} - 5025643873897035254477627698664 \nu^{5} - 103982784157939497551773315244985 \nu^{4} - 1378319461047584518173107736327213 \nu^{3} - 10880091703859610281905001309535342 \nu^{2} - 47562110454193961375283932037053952 \nu - 56593216173906372856013102596533984$$$$)/$$$$73\!\cdots\!32$$ $$\beta_{8}$$ $$=$$ $$($$$$14157220102452586339672 \nu^{9} - 73181436628106258815207 \nu^{8} + 21450699265353932332431611 \nu^{7} + 761938413766396462137421954 \nu^{6} + 29317438782871425297298702939 \nu^{5} + 516790032530003206067288876514 \nu^{4} + 8119052430248244827601444385635 \nu^{3} + 72511604077245336800673448999401 \nu^{2} + 542989543533569507090576104983648 \nu + 1086087319368116240165041940033088$$$$)/$$$$32\!\cdots\!24$$ $$\beta_{9}$$ $$=$$ $$($$$$-271971886371848257241800 \nu^{9} + 1785009164308942036996837 \nu^{8} - 405868404758120882305760261 \nu^{7} - 13940242051771287986602803802 \nu^{6} - 533312211128327909483108123557 \nu^{5} - 8502967248620214158215036601370 \nu^{4} - 118346364453615080871322916541213 \nu^{3} - 730510311747512760541251656190687 \nu^{2} - 4198070192290174425851016223614624 \nu - 1192476841474308302702017122042816$$$$)/$$$$22\!\cdots\!68$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{9} - 2 \beta_{7} - \beta_{6} - 5 \beta_{5} + \beta_{4} + \beta_{3} + 5 \beta_{2} - 34 \beta_{1} + 1$$$$)/168$$ $$\nu^{2}$$ $$=$$ $$($$$$-16 \beta_{9} - 23 \beta_{8} - 47 \beta_{7} - 39 \beta_{6} - 179 \beta_{5} + 23 \beta_{4} - 47 \beta_{3} - 96886 \beta_{1} - 96902$$$$)/168$$ $$\nu^{3}$$ $$=$$ $$($$$$-1857 \beta_{9} - 1857 \beta_{8} + 2185 \beta_{7} - 376 \beta_{6} - 376 \beta_{4} - 4370 \beta_{3} - 9637 \beta_{2} - 3076875$$$$)/168$$ $$\nu^{4}$$ $$=$$ $$($$$$-62551 \beta_{9} - 25040 \beta_{8} + 206046 \beta_{7} + 62551 \beta_{6} + 423667 \beta_{5} - 87591 \beta_{4} - 103023 \beta_{3} - 423667 \beta_{2} + 168089654 \beta_{1} - 62551$$$$)/168$$ $$\nu^{5}$$ $$=$$ $$($$$$1031736 \beta_{9} + 3046569 \beta_{8} + 4798825 \beta_{7} + 4078305 \beta_{6} + 20419589 \beta_{5} - 3046569 \beta_{4} + 4798825 \beta_{3} + 7462628042 \beta_{1} + 7463659778$$$$)/168$$ $$\nu^{6}$$ $$=$$ $$($$$$191587143 \beta_{9} + 191587143 \beta_{8} - 225528271 \beta_{7} + 50945168 \beta_{6} + 50945168 \beta_{4} + 451056542 \beta_{3} + 944696019 \beta_{2} + 356822200957$$$$)/168$$ $$\nu^{7}$$ $$=$$ $$($$$$6622946953 \beta_{9} + 2337530360 \beta_{8} - 21085302418 \beta_{7} - 6622946953 \beta_{6} - 44474360933 \beta_{5} + 8960477313 \beta_{4} + 10542651209 \beta_{3} + 44474360933 \beta_{2} - 16577003654346 \beta_{1} + 6622946953$$$$)/168$$ $$\nu^{8}$$ $$=$$ $$($$$$-110369207632 \beta_{9} - 309539717399 \beta_{8} - 494220385071 \beta_{7} - 419908925031 \beta_{6} - 2078446940339 \beta_{5} + 309539717399 \beta_{4} - 494220385071 \beta_{3} - 778611969281590 \beta_{1} - 778722338489222$$$$)/168$$ $$\nu^{9}$$ $$=$$ $$($$$$-19661294346273 \beta_{9} - 19661294346273 \beta_{8} + 23136427328297 \beta_{7} - 5152654242744 \beta_{6} - 5152654242744 \beta_{4} - 46272854656594 \beta_{3} - 97429514717893 \beta_{2} - 36445351843598059$$$$)/168$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$1$$ $$-\beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1
 −4.78762 − 8.29240i −9.33129 − 16.1623i 23.4172 + 40.5598i −0.957903 − 1.65914i −7.84041 − 13.5800i −4.78762 + 8.29240i −9.33129 + 16.1623i 23.4172 − 40.5598i −0.957903 + 1.65914i −7.84041 + 13.5800i
0 0 0 −485.304 + 280.190i 0 −622.564 + 2318.88i 0 0 0
73.2 0 0 0 −336.492 + 194.274i 0 2329.92 579.874i 0 0 0
73.3 0 0 0 −327.308 + 188.971i 0 −894.589 2228.12i 0 0 0
73.4 0 0 0 616.084 355.696i 0 −2382.47 297.754i 0 0 0
73.5 0 0 0 951.520 549.361i 0 2332.69 + 568.626i 0 0 0
145.1 0 0 0 −485.304 280.190i 0 −622.564 2318.88i 0 0 0
145.2 0 0 0 −336.492 194.274i 0 2329.92 + 579.874i 0 0 0
145.3 0 0 0 −327.308 188.971i 0 −894.589 + 2228.12i 0 0 0
145.4 0 0 0 616.084 + 355.696i 0 −2382.47 + 297.754i 0 0 0
145.5 0 0 0 951.520 + 549.361i 0 2332.69 568.626i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 145.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.9.z.c 10
3.b odd 2 1 28.9.h.a 10
7.d odd 6 1 inner 252.9.z.c 10
12.b even 2 1 112.9.s.b 10
21.c even 2 1 196.9.h.a 10
21.g even 6 1 28.9.h.a 10
21.g even 6 1 196.9.b.a 10
21.h odd 6 1 196.9.b.a 10
21.h odd 6 1 196.9.h.a 10
84.j odd 6 1 112.9.s.b 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.9.h.a 10 3.b odd 2 1
28.9.h.a 10 21.g even 6 1
112.9.s.b 10 12.b even 2 1
112.9.s.b 10 84.j odd 6 1
196.9.b.a 10 21.g even 6 1
196.9.b.a 10 21.h odd 6 1
196.9.h.a 10 21.c even 2 1
196.9.h.a 10 21.h odd 6 1
252.9.z.c 10 1.a even 1 1 trivial
252.9.z.c 10 7.d odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$24\!\cdots\!95$$$$T_{5}^{5} -$$$$27\!\cdots\!37$$$$T_{5}^{4} +$$$$39\!\cdots\!70$$$$T_{5}^{3} +$$$$95\!\cdots\!75$$$$T_{5}^{2} +$$$$33\!\cdots\!75$$$$T_{5} +$$$$41\!\cdots\!75$$">$$T_{5}^{10} - \cdots$$ acting on $$S_{9}^{\mathrm{new}}(252, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$T^{10}$$
$5$ $$41\!\cdots\!75$$$$+$$$$33\!\cdots\!75$$$$T +$$$$95\!\cdots\!75$$$$T^{2} + 39746058814580193270 T^{3} - 277882314887427237 T^{4} - 248983749198495 T^{5} + 751173109353 T^{6} + 873652230 T^{7} - 810267 T^{8} - 837 T^{9} + T^{10}$$
$7$ $$63\!\cdots\!01$$$$-$$$$16\!\cdots\!26$$$$T -$$$$10\!\cdots\!19$$$$T^{2} -$$$$35\!\cdots\!72$$$$T^{3} + 16329836006199201318 T^{4} + 209768437334342796 T^{5} + 2832679914918 T^{6} - 10618835472 T^{7} - 5479719 T^{8} - 1526 T^{9} + T^{10}$$
$11$ $$43\!\cdots\!25$$$$-$$$$31\!\cdots\!75$$$$T +$$$$19\!\cdots\!25$$$$T^{2} -$$$$44\!\cdots\!80$$$$T^{3} +$$$$94\!\cdots\!61$$$$T^{4} +$$$$58\!\cdots\!93$$$$T^{5} + 142073667324328869 T^{6} + 2265611872032 T^{7} + 401686713 T^{8} + 3705 T^{9} + T^{10}$$
$13$ $$73\!\cdots\!92$$$$+$$$$19\!\cdots\!44$$$$T^{2} +$$$$16\!\cdots\!24$$$$T^{4} + 5182077234059079696 T^{6} + 4263799464 T^{8} + T^{10}$$
$17$ $$25\!\cdots\!43$$$$+$$$$79\!\cdots\!75$$$$T +$$$$78\!\cdots\!15$$$$T^{2} -$$$$86\!\cdots\!50$$$$T^{3} -$$$$14\!\cdots\!73$$$$T^{4} +$$$$17\!\cdots\!57$$$$T^{5} +$$$$25\!\cdots\!09$$$$T^{6} - 1502065287283914 T^{7} - 17228350635 T^{8} + 78003 T^{9} + T^{10}$$
$19$ $$38\!\cdots\!47$$$$+$$$$19\!\cdots\!61$$$$T +$$$$30\!\cdots\!77$$$$T^{2} -$$$$84\!\cdots\!52$$$$T^{3} -$$$$34\!\cdots\!85$$$$T^{4} +$$$$56\!\cdots\!21$$$$T^{5} +$$$$30\!\cdots\!21$$$$T^{6} - 5855375420599176 T^{7} - 57406699509 T^{8} + 96741 T^{9} + T^{10}$$
$23$ $$30\!\cdots\!41$$$$+$$$$15\!\cdots\!69$$$$T +$$$$17\!\cdots\!57$$$$T^{2} +$$$$37\!\cdots\!96$$$$T^{3} +$$$$49\!\cdots\!53$$$$T^{4} +$$$$11\!\cdots\!57$$$$T^{5} +$$$$66\!\cdots\!89$$$$T^{6} + 57654704482745652 T^{7} + 295796002149 T^{8} + 208533 T^{9} + T^{10}$$
$29$ $$( -$$$$35\!\cdots\!08$$$$+$$$$82\!\cdots\!68$$$$T - 365110245396917640 T^{2} - 921101790060 T^{3} + 377382 T^{4} + T^{5} )^{2}$$
$31$ $$18\!\cdots\!07$$$$+$$$$50\!\cdots\!49$$$$T -$$$$41\!\cdots\!19$$$$T^{2} -$$$$23\!\cdots\!00$$$$T^{3} +$$$$11\!\cdots\!35$$$$T^{4} +$$$$89\!\cdots\!61$$$$T^{5} +$$$$43\!\cdots\!65$$$$T^{6} - 1072697413581363528 T^{7} - 647906313621 T^{8} + 1053717 T^{9} + T^{10}$$
$37$ $$81\!\cdots\!41$$$$-$$$$50\!\cdots\!09$$$$T +$$$$28\!\cdots\!75$$$$T^{2} -$$$$20\!\cdots\!78$$$$T^{3} +$$$$24\!\cdots\!69$$$$T^{4} +$$$$44\!\cdots\!93$$$$T^{5} +$$$$91\!\cdots\!53$$$$T^{6} + 6408411825362856258 T^{7} + 11057152256943 T^{8} + 998075 T^{9} + T^{10}$$
$41$ $$31\!\cdots\!28$$$$+$$$$23\!\cdots\!68$$$$T^{2} +$$$$62\!\cdots\!24$$$$T^{4} +$$$$78\!\cdots\!36$$$$T^{6} + 45924266952552 T^{8} + T^{10}$$
$43$ $$( -$$$$17\!\cdots\!00$$$$+$$$$80\!\cdots\!00$$$$T + 64988524084922662976 T^{2} - 31935618972272 T^{3} - 369146 T^{4} + T^{5} )^{2}$$
$47$ $$30\!\cdots\!75$$$$-$$$$47\!\cdots\!25$$$$T +$$$$24\!\cdots\!25$$$$T^{2} +$$$$29\!\cdots\!20$$$$T^{3} -$$$$24\!\cdots\!93$$$$T^{4} -$$$$19\!\cdots\!21$$$$T^{5} +$$$$19\!\cdots\!17$$$$T^{6} - 35379983228249243448 T^{7} - 49600614851493 T^{8} + 710883 T^{9} + T^{10}$$
$53$ $$18\!\cdots\!41$$$$-$$$$34\!\cdots\!47$$$$T +$$$$18\!\cdots\!99$$$$T^{2} +$$$$37\!\cdots\!14$$$$T^{3} +$$$$17\!\cdots\!37$$$$T^{4} +$$$$20\!\cdots\!75$$$$T^{5} +$$$$33\!\cdots\!57$$$$T^{6} +$$$$12\!\cdots\!62$$$$T^{7} + 248554071653679 T^{8} + 10501461 T^{9} + T^{10}$$
$59$ $$16\!\cdots\!83$$$$+$$$$60\!\cdots\!91$$$$T +$$$$89\!\cdots\!73$$$$T^{2} +$$$$54\!\cdots\!08$$$$T^{3} +$$$$13\!\cdots\!59$$$$T^{4} +$$$$73\!\cdots\!79$$$$T^{5} -$$$$27\!\cdots\!91$$$$T^{6} -$$$$17\!\cdots\!48$$$$T^{7} + 463262471283495 T^{8} - 37089081 T^{9} + T^{10}$$
$61$ $$62\!\cdots\!03$$$$+$$$$14\!\cdots\!13$$$$T +$$$$42\!\cdots\!87$$$$T^{2} -$$$$15\!\cdots\!34$$$$T^{3} -$$$$72\!\cdots\!25$$$$T^{4} +$$$$14\!\cdots\!75$$$$T^{5} +$$$$11\!\cdots\!53$$$$T^{6} -$$$$32\!\cdots\!42$$$$T^{7} - 378853385942295 T^{8} + 8180481 T^{9} + T^{10}$$
$67$ $$18\!\cdots\!81$$$$-$$$$29\!\cdots\!57$$$$T +$$$$33\!\cdots\!37$$$$T^{2} -$$$$17\!\cdots\!32$$$$T^{3} +$$$$61\!\cdots\!81$$$$T^{4} -$$$$13\!\cdots\!37$$$$T^{5} +$$$$22\!\cdots\!81$$$$T^{6} -$$$$23\!\cdots\!40$$$$T^{7} + 1680785033422677 T^{8} - 48020189 T^{9} + T^{10}$$
$71$ $$($$$$10\!\cdots\!12$$$$+$$$$40\!\cdots\!28$$$$T +$$$$11\!\cdots\!32$$$$T^{2} - 1655416268261280 T^{3} - 15959118 T^{4} + T^{5} )^{2}$$
$73$ $$56\!\cdots\!75$$$$+$$$$17\!\cdots\!25$$$$T +$$$$19\!\cdots\!75$$$$T^{2} +$$$$57\!\cdots\!90$$$$T^{3} -$$$$39\!\cdots\!13$$$$T^{4} -$$$$16\!\cdots\!29$$$$T^{5} +$$$$21\!\cdots\!65$$$$T^{6} +$$$$20\!\cdots\!82$$$$T^{7} + 7454537132180457 T^{8} + 133345593 T^{9} + T^{10}$$
$79$ $$11\!\cdots\!21$$$$-$$$$13\!\cdots\!37$$$$T +$$$$22\!\cdots\!61$$$$T^{2} +$$$$43\!\cdots\!56$$$$T^{3} +$$$$50\!\cdots\!21$$$$T^{4} -$$$$43\!\cdots\!05$$$$T^{5} +$$$$26\!\cdots\!21$$$$T^{6} -$$$$13\!\cdots\!60$$$$T^{7} + 7347968697126885 T^{8} - 53590181 T^{9} + T^{10}$$
$83$ $$56\!\cdots\!68$$$$+$$$$92\!\cdots\!16$$$$T^{2} +$$$$56\!\cdots\!84$$$$T^{4} +$$$$15\!\cdots\!84$$$$T^{6} + 20533343760435360 T^{8} + T^{10}$$
$89$ $$32\!\cdots\!23$$$$+$$$$12\!\cdots\!31$$$$T +$$$$16\!\cdots\!59$$$$T^{2} +$$$$26\!\cdots\!30$$$$T^{3} -$$$$71\!\cdots\!97$$$$T^{4} -$$$$35\!\cdots\!63$$$$T^{5} +$$$$25\!\cdots\!73$$$$T^{6} -$$$$11\!\cdots\!30$$$$T^{7} + 24350366415430953 T^{8} - 241368273 T^{9} + T^{10}$$
$97$ $$24\!\cdots\!88$$$$+$$$$20\!\cdots\!68$$$$T^{2} +$$$$57\!\cdots\!24$$$$T^{4} +$$$$73\!\cdots\!76$$$$T^{6} + 43865514686507112 T^{8} + T^{10}$$