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 $2$

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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:

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

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