Properties

Label 252.9.z.b
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: \(x^{10} - 38255 x^{8} + 1483053595 x^{6} - 139470625170 x^{5} + 5194605060018 x^{4} - 71600654137860 x^{3} + 119846615988780 x^{2} + 4263507454127400 x + 15758720495290800\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{8}\cdot 7^{3} \)
Twist minimal: no (minimal twist has level 84)
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.

\(f(q)\) \(=\) \( q + ( -92 - 93 \beta_{1} - \beta_{5} ) q^{5} + ( 139 - 35 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{9} ) q^{7} +O(q^{10})\) \( q + ( -92 - 93 \beta_{1} - \beta_{5} ) q^{5} + ( 139 - 35 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{9} ) q^{7} + ( 6 + 170 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} - 10 \beta_{5} - 4 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{11} + ( -3044 + 6087 \beta_{1} - 9 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 9 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} - \beta_{8} + 14 \beta_{9} ) q^{13} + ( 1823 - 894 \beta_{1} - 41 \beta_{2} + 11 \beta_{3} - 6 \beta_{4} + 5 \beta_{6} + \beta_{7} - \beta_{8} - 18 \beta_{9} ) q^{17} + ( -1913 - 1984 \beta_{1} - 7 \beta_{3} - 22 \beta_{4} - 72 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + 12 \beta_{9} ) q^{19} + ( -62565 + 62712 \beta_{1} - 298 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 149 \beta_{5} + 24 \beta_{6} - 16 \beta_{7} + 27 \beta_{9} ) q^{23} + ( -228 - 4121 \beta_{1} - 207 \beta_{2} - 78 \beta_{3} + 113 \beta_{4} + 414 \beta_{5} + 99 \beta_{6} + 6 \beta_{7} + 6 \beta_{8} - 71 \beta_{9} ) q^{25} + ( 29031 + 46 \beta_{1} - 168 \beta_{2} - 208 \beta_{3} - 161 \beta_{4} - 168 \beta_{5} + 46 \beta_{6} + 22 \beta_{8} + 92 \beta_{9} ) q^{29} + ( 32395 - 16058 \beta_{1} - 396 \beta_{2} + 273 \beta_{3} + 18 \beta_{4} + 156 \beta_{6} - 39 \beta_{7} + 39 \beta_{8} - 216 \beta_{9} ) q^{31} + ( -143800 + 44997 \beta_{1} - 266 \beta_{2} - 604 \beta_{3} - 433 \beta_{4} + 1043 \beta_{5} + 148 \beta_{6} + 21 \beta_{7} - 35 \beta_{8} + 852 \beta_{9} ) q^{35} + ( 382026 - 381634 \beta_{1} - 1170 \beta_{2} + 193 \beta_{3} + 386 \beta_{4} + 585 \beta_{5} + 754 \beta_{6} + 18 \beta_{7} - 1393 \beta_{9} ) q^{37} + ( 443984 - 887650 \beta_{1} + 2338 \beta_{2} + 305 \beta_{3} - 1433 \beta_{4} - 2338 \beta_{5} - 928 \beta_{6} - 26 \beta_{7} - 13 \beta_{8} + 2230 \beta_{9} ) q^{41} + ( -86376 - 399 \beta_{1} + 828 \beta_{2} + 1962 \beta_{3} + 779 \beta_{4} + 828 \beta_{5} - 399 \beta_{6} - 33 \beta_{8} - 798 \beta_{9} ) q^{43} + ( 18739 + 20622 \beta_{1} - 1590 \beta_{3} - 2472 \beta_{4} + 1343 \beta_{5} + 1065 \beta_{6} + 15 \beta_{7} + 30 \beta_{8} + 1776 \beta_{9} ) q^{47} + ( 1161929 - 363626 \beta_{1} + 8253 \beta_{2} + 264 \beta_{3} + 1168 \beta_{4} - 2709 \beta_{5} + 290 \beta_{6} - 119 \beta_{7} - 259 \beta_{8} - 1241 \beta_{9} ) q^{49} + ( -2333 + 2133843 \beta_{1} - 2507 \beta_{2} + 801 \beta_{3} - 4617 \beta_{4} + 5014 \beta_{5} - 975 \beta_{6} + 105 \beta_{7} + 105 \beta_{8} + 4269 \beta_{9} ) q^{53} + ( -1539683 + 3079054 \beta_{1} - 6480 \beta_{2} - 238 \beta_{3} + 2645 \beta_{4} + 6480 \beta_{5} + 788 \beta_{6} + 148 \beta_{7} + 74 \beta_{8} - 4666 \beta_{9} ) q^{55} + ( -2455609 + 1223884 \beta_{1} + 10160 \beta_{2} - 4518 \beta_{3} - 232 \beta_{4} - 2199 \beta_{6} - 120 \beta_{7} + 120 \beta_{8} + 4406 \beta_{9} ) q^{59} + ( -921868 - 931566 \beta_{1} + 712 \beta_{3} + 7502 \beta_{4} - 9756 \beta_{5} - 327 \beta_{6} + 443 \beta_{7} + 886 \beta_{8} - 3693 \beta_{9} ) q^{61} + ( 2994001 - 2994468 \beta_{1} + 4254 \beta_{2} - 1660 \beta_{3} - 3320 \beta_{4} - 2127 \beta_{5} - 7530 \beta_{6} + 890 \beta_{7} + 1365 \beta_{9} ) q^{65} + ( 5459 - 4147853 \beta_{1} + 4410 \beta_{2} + 3848 \beta_{3} + 4484 \beta_{4} - 8820 \beta_{5} - 4897 \beta_{6} - 348 \beta_{7} - 348 \beta_{8} - 6582 \beta_{9} ) q^{67} + ( -11281684 + 1052 \beta_{1} - 27373 \beta_{2} - 5165 \beta_{3} + 242 \beta_{4} - 27373 \beta_{5} + 1052 \beta_{6} + 95 \beta_{8} + 2104 \beta_{9} ) q^{71} + ( 5795718 - 2890169 \beta_{1} - 18513 \beta_{2} + 5395 \beta_{3} + 2982 \beta_{4} + 2262 \beta_{6} + 871 \beta_{7} - 871 \beta_{8} - 3284 \beta_{9} ) q^{73} + ( 6548339 + 6675651 \beta_{1} - 21854 \beta_{2} - 1221 \beta_{3} - 19271 \beta_{4} + 5194 \beta_{5} - 6458 \beta_{6} - 2156 \beta_{7} - 2009 \beta_{8} + 11421 \beta_{9} ) q^{77} + ( -8441730 + 8441359 \beta_{1} + 4932 \beta_{2} - 2095 \beta_{3} - 4190 \beta_{4} - 2466 \beta_{5} - 8083 \beta_{6} - 297 \beta_{7} - 9692 \beta_{9} ) q^{79} + ( 4115919 - 8232315 \beta_{1} + 32788 \beta_{2} - 1302 \beta_{3} - 13383 \beta_{4} - 32788 \beta_{5} + 3081 \beta_{6} - 1650 \beta_{7} - 825 \beta_{8} + 27720 \beta_{9} ) q^{83} + ( 14245912 + 1332 \beta_{1} + 16596 \beta_{2} - 6093 \beta_{3} - 1223 \beta_{4} + 16596 \beta_{5} + 1332 \beta_{6} + 567 \beta_{8} + 2664 \beta_{9} ) q^{85} + ( -4526518 - 4447018 \beta_{1} + 1492 \beta_{3} - 3452 \beta_{4} + 78888 \beta_{5} - 440 \beta_{6} + 1664 \beta_{7} + 3328 \beta_{8} + 2338 \beta_{9} ) q^{89} + ( -19117109 - 10971302 \beta_{1} + 65772 \beta_{2} + 3531 \beta_{3} - 24042 \beta_{4} - 81774 \beta_{5} - 18619 \beta_{6} - 3297 \beta_{7} - 2394 \beta_{8} + 15039 \beta_{9} ) q^{91} + ( -13501 + 28151474 \beta_{1} - 14531 \beta_{2} + 6195 \beta_{3} + 9630 \beta_{4} + 29062 \beta_{5} - 7225 \beta_{6} + 2075 \beta_{7} + 2075 \beta_{8} - 11690 \beta_{9} ) q^{95} + ( -22923541 + 45846697 \beta_{1} - 114975 \beta_{2} - 1246 \beta_{3} - 39390 \beta_{4} + 114975 \beta_{5} + 2877 \beta_{6} - 1722 \beta_{7} - 861 \beta_{8} + 79550 \beta_{9} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 1389 q^{5} + 1217 q^{7} + O(q^{10}) \) \( 10 q - 1389 q^{5} + 1217 q^{7} + 879 q^{11} + 13674 q^{17} - 29268 q^{19} - 312732 q^{23} - 22052 q^{25} + 289794 q^{29} + 242787 q^{31} - 1209372 q^{35} + 1913308 q^{37} - 861848 q^{43} + 305448 q^{47} + 9821659 q^{49} + 10663233 q^{53} - 18410871 q^{59} - 13937808 q^{61} + 14966808 q^{65} - 20722822 q^{67} - 113032584 q^{71} + 43436322 q^{73} + 98823405 q^{77} - 42189637 q^{79} + 142602108 q^{85} - 67171914 q^{89} - 246091266 q^{91} + 140649894 q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 38255 x^{8} + 1483053595 x^{6} - 139470625170 x^{5} + 5194605060018 x^{4} - 71600654137860 x^{3} + 119846615988780 x^{2} + 4263507454127400 x + 15758720495290800\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(272153962454078370610521305639 \nu^{9} - 7804201098746205229207193944284 \nu^{8} - 10508963126474563853994149197651605 \nu^{7} + 295816416513335686392206334903581040 \nu^{6} + 407247873947801469805329113847737444505 \nu^{5} - 49432323370232420671816106511466809823710 \nu^{4} + 2361167606711182835213093021122334838884122 \nu^{3} - 50311808783007248050547388838766467752232152 \nu^{2} + 310007330258636047056371023027532001368945580 \nu + 2560106014199641516715702430770984676831048480\)\()/ \)\(19\!\cdots\!80\)\( \)
\(\beta_{2}\)\(=\)\((\)\(532630612913952831343276059927313052 \nu^{9} + 7870412045695935977777751606167139423 \nu^{8} - 19994495655697126596428117946280492544005 \nu^{7} - 294398589639692400218010034596205124844645 \nu^{6} + 775352444641497378675785892510643463540083910 \nu^{5} - 62876999260667221030581873809544404857848115080 \nu^{4} + 2233381193858871090009298488823372903064410792956 \nu^{3} - 40242428444367127486561034996929984578050403227196 \nu^{2} + 586448131599276055459085751241829208346571679040140 \nu - 1078735705906769675415332633077119427873704893649810\)\()/ \)\(57\!\cdots\!30\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-204614186603676854088703130386730570 \nu^{9} - 3246451082198705599763328915997444869 \nu^{8} + 7627898151160140166317081586356955394110 \nu^{7} + 118019207271863297887038018968875916038890 \nu^{6} - 296034674386757575598878888639092410056688255 \nu^{5} + 23946551205117981449732536802869245497095310730 \nu^{4} - 898604579864163378810330486014199328406070050970 \nu^{3} + 16944013800571923749735267866200265950623106507168 \nu^{2} - 263577652563944625609641101301157681568421512160460 \nu - 2598814389843358830802168363680544682375898814149870\)\()/ \)\(19\!\cdots\!10\)\( \)
\(\beta_{4}\)\(=\)\((\)\(4204770312641237494384848933 \nu^{9} + 48826521187590904857733922530 \nu^{8} - 159590479431792254941050535412205 \nu^{7} - 1820481341930148198652470129183830 \nu^{6} + 6188375721497063895916391112390776055 \nu^{5} - 515871636771755277492400915734308895140 \nu^{4} + 16872517881168933617029012438624753922319 \nu^{3} - 152333793468636799121418589411954938505920 \nu^{2} - 1819449872910862947016101531720000456316920 \nu + 33464348625032329313820696140810334992737920\)\()/ \)\(24\!\cdots\!65\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-1618439339072742924425118221464244117 \nu^{9} - 2272436844388429587685896996872381793 \nu^{8} + 62081366160749117215994564164255013429395 \nu^{7} + 95583868167182138326520184262962142427190 \nu^{6} - 2406385278346685811283500413143991438396776550 \nu^{5} + 222027676673540012619104989929351376380509513790 \nu^{4} - 7851473574291654631049349409823274193323312329526 \nu^{3} + 93300647143174493333824794251437218022775609173896 \nu^{2} + 34832055152195878846278109115358683323428396428460 \nu - 4294548229985885807395075384212369224834365652847450\)\()/ \)\(57\!\cdots\!30\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-2179483260990883745836929408241933939 \nu^{9} - 75268334323461281196264785746980750240 \nu^{8} + 81995458216232489810648481079270894506125 \nu^{7} + 2848568278746643391370152984756331778875660 \nu^{6} - 3180336698686086901112384376876310259729577085 \nu^{5} + 193500313102657780166256211917881434787130996510 \nu^{4} - 2838752209987202470948260202434074813749717371762 \nu^{3} - 86572196292660480900155655360631781758804887647640 \nu^{2} + 1020101344318079450977636648620781996729425256488420 \nu - 1485761827155484536329503772546514717139114517722680\)\()/ \)\(76\!\cdots\!40\)\( \)
\(\beta_{7}\)\(=\)\((\)\(318443371724602025613880767677021597 \nu^{9} + 36578504403459235578355632876886077432 \nu^{8} - 11582160003070321206854790597848640629635 \nu^{7} - 1399932730182171774073007736150029718527940 \nu^{6} + 448478185031561734110814343907563332462607675 \nu^{5} + 9805364357520911139214322426785574771683111590 \nu^{4} - 2525897975933825053216766986534381152420318526794 \nu^{3} + 84864340784719869974155988031904158946959730446696 \nu^{2} - 464422025971117234463332559981837535006056401378860 \nu - 1631428644599401802435045072699269149239259600518040\)\()/ \)\(10\!\cdots\!20\)\( \)
\(\beta_{8}\)\(=\)\((\)\(684070894048704959508734332969630298 \nu^{9} - 27004682297995660733339267495566312551 \nu^{8} - 27543685197067700292026033244281398806220 \nu^{7} + 991271384753550460027738691536799868051660 \nu^{6} + 1066738181227501224504906927562310518687879685 \nu^{5} - 133799452544367282982944886967264428178391771690 \nu^{4} + 5303505008538672248969271906736760067256100165424 \nu^{3} - 61307193571096435941381695732933183527777990379088 \nu^{2} - 241821570560664249202218062952866912341391278432260 \nu - 4966690042660575942850729595967244584029629695676950\)\()/ \)\(19\!\cdots\!10\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-21285468882417707370951983948134087 \nu^{9} + 166802970724034577546937084981332038 \nu^{8} + 814169950798001130867605097419409420375 \nu^{7} - 6432943521603493526685349644701775026650 \nu^{6} - 31568804114576470524035623949368973401800165 \nu^{5} + 3217746710447923589952203054212526480124886640 \nu^{4} - 133802328271531062019350797042258591132398654316 \nu^{3} + 2339759522782711753070849660808964247862078994824 \nu^{2} - 15346834890558830124294893024572179403018021941240 \nu - 30733528526416724760261931925663268401776539663600\)\()/ \)\(58\!\cdots\!70\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-7 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - 7 \beta_{4} - \beta_{3} - 18 \beta_{2} + 10 \beta_{1} - 2\)\()/252\)
\(\nu^{2}\)\(=\)\((\)\(-133 \beta_{9} - 25 \beta_{7} + 641 \beta_{6} - 612 \beta_{5} + 308 \beta_{4} + 154 \beta_{3} + 1224 \beta_{2} - 1286234 \beta_{1} + 1285468\)\()/84\)
\(\nu^{3}\)\(=\)\((\)\(-98350 \beta_{9} + 9962 \beta_{8} + 19924 \beta_{7} - 44872 \beta_{6} + 204264 \beta_{5} + 40859 \beta_{4} + 18278 \beta_{3} - 204264 \beta_{2} + 92222164 \beta_{1} - 46106924\)\()/63\)
\(\nu^{4}\)\(=\)\((\)\(7241731 \beta_{9} - 1087531 \beta_{8} - 1087531 \beta_{7} + 15842271 \beta_{6} - 31426884 \beta_{5} - 1339835 \beta_{4} - 12891323 \beta_{3} + 15713442 \beta_{2} - 25264706626 \beta_{1} + 12762494\)\()/42\)
\(\nu^{5}\)\(=\)\((\)\(-2674551586 \beta_{9} + 817640630 \beta_{8} + 408820315 \beta_{7} - 1888573513 \beta_{6} + 9578395467 \beta_{5} + 3869349974 \beta_{4} + 3037270427 \beta_{3} + 2933192406112 \beta_{1} + 2924353887244\)\()/63\)
\(\nu^{6}\)\(=\)\((\)\(17306620562 \beta_{9} - 4718798746 \beta_{8} + 8653310281 \beta_{6} - 55189236609 \beta_{5} - 28925053171 \beta_{4} - 47985350151 \beta_{3} - 55189236609 \beta_{2} + 8653310281 \beta_{1} - 77852302903406\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(38202992333786 \beta_{9} + 17818961417759 \beta_{8} - 17818961417759 \beta_{7} + 62423963401070 \beta_{6} + 127412996300408 \beta_{4} + 107028965384381 \beta_{3} + 456393445840359 \beta_{2} - 159209696525729543 \beta_{1} + 317918394603635416\)\()/63\)
\(\nu^{8}\)\(=\)\((\)\(-5175651354674584 \beta_{9} + 1771166158318955 \beta_{7} - 12355733903245207 \beta_{6} + 18814139744811489 \beta_{5} - 5292283872463126 \beta_{4} - 2646141936231563 \beta_{3} - 37628279489622978 \beta_{2} + 24692408294605392058 \beta_{1} - 24670948012924349006\)\()/21\)
\(\nu^{9}\)\(=\)\((\)\(7446890317225525670 \beta_{9} - 810177308705867941 \beta_{8} - 1620354617411735882 \beta_{7} + 8770729461200014061 \beta_{6} - 21874016157841757997 \beta_{5} - 1339986877350003442 \beta_{4} - 3193635589968627334 \beta_{3} + 21874016157841757997 \beta_{2} - 16274286664182963265697 \beta_{1} + 8135951602950850253152\)\()/63\)

Character values

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

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

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
16.3207 + 9.42278i
−190.225 109.826i
139.521 + 80.5522i
38.0902 + 21.9914i
−3.70642 2.13991i
16.3207 9.42278i
−190.225 + 109.826i
139.521 80.5522i
38.0902 21.9914i
−3.70642 + 2.13991i
0 0 0 −790.151 + 456.194i 0 −2318.52 623.902i 0 0 0
73.2 0 0 0 −656.878 + 379.249i 0 1906.96 + 1458.87i 0 0 0
73.3 0 0 0 −111.592 + 64.4274i 0 2392.99 195.963i 0 0 0
73.4 0 0 0 374.307 216.106i 0 83.7630 2399.54i 0 0 0
73.5 0 0 0 489.814 282.795i 0 −1456.69 + 1908.63i 0 0 0
145.1 0 0 0 −790.151 456.194i 0 −2318.52 + 623.902i 0 0 0
145.2 0 0 0 −656.878 379.249i 0 1906.96 1458.87i 0 0 0
145.3 0 0 0 −111.592 64.4274i 0 2392.99 + 195.963i 0 0 0
145.4 0 0 0 374.307 + 216.106i 0 83.7630 + 2399.54i 0 0 0
145.5 0 0 0 489.814 + 282.795i 0 −1456.69 1908.63i 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.b 10
3.b odd 2 1 84.9.m.a 10
7.d odd 6 1 inner 252.9.z.b 10
21.g even 6 1 84.9.m.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.9.m.a 10 3.b odd 2 1
84.9.m.a 10 21.g even 6 1
252.9.z.b 10 1.a even 1 1 trivial
252.9.z.b 10 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(38\!\cdots\!25\)\( T_{5}^{5} + \)\(38\!\cdots\!75\)\( T_{5}^{4} - \)\(86\!\cdots\!00\)\( T_{5}^{3} + \)\(94\!\cdots\!00\)\( T_{5}^{2} + \)\(50\!\cdots\!00\)\( T_{5} + \)\(47\!\cdots\!00\)\( \)">\(T_{5}^{10} + \cdots\) acting on \(S_{9}^{\mathrm{new}}(252, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \)
$3$ \( T^{10} \)
$5$ \( \)\(47\!\cdots\!00\)\( + \)\(50\!\cdots\!00\)\( T + \)\(94\!\cdots\!00\)\( T^{2} - 86555484350433360000 T^{3} + 38523776306784975 T^{4} + 381886886416725 T^{5} - 19821737736 T^{6} - 894492387 T^{7} - 876 T^{8} + 1389 T^{9} + T^{10} \)
$7$ \( \)\(63\!\cdots\!01\)\( - \)\(13\!\cdots\!17\)\( T - \)\(79\!\cdots\!85\)\( T^{2} - \)\(28\!\cdots\!22\)\( T^{3} + 26985197271477204165 T^{4} + 29595577293673965 T^{5} + 4681028412165 T^{6} - 865819122 T^{7} - 4170285 T^{8} - 1217 T^{9} + T^{10} \)
$11$ \( \)\(15\!\cdots\!00\)\( + \)\(15\!\cdots\!00\)\( T + \)\(38\!\cdots\!80\)\( T^{2} + \)\(29\!\cdots\!80\)\( T^{3} + \)\(60\!\cdots\!01\)\( T^{4} + \)\(43\!\cdots\!53\)\( T^{5} + 415614765251269170 T^{6} + 11944495101069 T^{7} + 680030694 T^{8} - 879 T^{9} + T^{10} \)
$13$ \( \)\(67\!\cdots\!12\)\( + \)\(69\!\cdots\!76\)\( T^{2} + \)\(48\!\cdots\!04\)\( T^{4} + 7777752229808322057 T^{6} + 4745027382 T^{8} + T^{10} \)
$17$ \( \)\(77\!\cdots\!00\)\( + \)\(17\!\cdots\!00\)\( T + \)\(10\!\cdots\!84\)\( T^{2} - \)\(41\!\cdots\!12\)\( T^{3} - \)\(14\!\cdots\!56\)\( T^{4} + \)\(58\!\cdots\!04\)\( T^{5} + 26673226645011929280 T^{6} + 74542852124424 T^{7} - 5389103784 T^{8} - 13674 T^{9} + T^{10} \)
$19$ \( \)\(22\!\cdots\!28\)\( - \)\(70\!\cdots\!32\)\( T - \)\(75\!\cdots\!16\)\( T^{2} + \)\(23\!\cdots\!88\)\( T^{3} + \)\(25\!\cdots\!32\)\( T^{4} - \)\(11\!\cdots\!54\)\( T^{5} + \)\(17\!\cdots\!49\)\( T^{6} - 379925119959588 T^{7} - 12695366133 T^{8} + 29268 T^{9} + T^{10} \)
$23$ \( \)\(38\!\cdots\!56\)\( - \)\(56\!\cdots\!16\)\( T + \)\(71\!\cdots\!48\)\( T^{2} - \)\(38\!\cdots\!48\)\( T^{3} + \)\(20\!\cdots\!72\)\( T^{4} - \)\(31\!\cdots\!60\)\( T^{5} + \)\(21\!\cdots\!20\)\( T^{6} - 7998210132804576 T^{7} + 254433143304 T^{8} + 312732 T^{9} + T^{10} \)
$29$ \( ( -\)\(37\!\cdots\!32\)\( + \)\(39\!\cdots\!64\)\( T + 14883790016495205 T^{2} - 753140988189 T^{3} - 144897 T^{4} + T^{5} )^{2} \)
$31$ \( \)\(46\!\cdots\!43\)\( + \)\(27\!\cdots\!17\)\( T + \)\(55\!\cdots\!11\)\( T^{2} + \)\(43\!\cdots\!30\)\( T^{3} - \)\(85\!\cdots\!25\)\( T^{4} - \)\(18\!\cdots\!17\)\( T^{5} + \)\(11\!\cdots\!13\)\( T^{6} + 899220119867033526 T^{7} - 3684092300175 T^{8} - 242787 T^{9} + T^{10} \)
$37$ \( \)\(82\!\cdots\!24\)\( - \)\(14\!\cdots\!92\)\( T + \)\(36\!\cdots\!00\)\( T^{2} + \)\(20\!\cdots\!16\)\( T^{3} + \)\(24\!\cdots\!32\)\( T^{4} - \)\(43\!\cdots\!14\)\( T^{5} + \)\(81\!\cdots\!81\)\( T^{6} - 5363809215598887792 T^{7} + 12415347113703 T^{8} - 1913308 T^{9} + T^{10} \)
$41$ \( \)\(31\!\cdots\!92\)\( + \)\(21\!\cdots\!28\)\( T^{2} + \)\(56\!\cdots\!20\)\( T^{4} + \)\(72\!\cdots\!04\)\( T^{6} + 44132018588988 T^{8} + T^{10} \)
$43$ \( ( \)\(41\!\cdots\!00\)\( + \)\(28\!\cdots\!40\)\( T - 32516554487062370230 T^{2} - 33859396405223 T^{3} + 430924 T^{4} + T^{5} )^{2} \)
$47$ \( \)\(51\!\cdots\!00\)\( + \)\(50\!\cdots\!00\)\( T + \)\(14\!\cdots\!20\)\( T^{2} - \)\(24\!\cdots\!40\)\( T^{3} - \)\(14\!\cdots\!12\)\( T^{4} + \)\(10\!\cdots\!32\)\( T^{5} + \)\(11\!\cdots\!88\)\( T^{6} + 11980067355204350688 T^{7} - 39190199566188 T^{8} - 305448 T^{9} + T^{10} \)
$53$ \( \)\(16\!\cdots\!96\)\( - \)\(13\!\cdots\!00\)\( T + \)\(76\!\cdots\!84\)\( T^{2} - \)\(24\!\cdots\!60\)\( T^{3} + \)\(62\!\cdots\!49\)\( T^{4} - \)\(10\!\cdots\!09\)\( T^{5} + \)\(13\!\cdots\!98\)\( T^{6} - \)\(12\!\cdots\!77\)\( T^{7} + 154157011850934 T^{8} - 10663233 T^{9} + T^{10} \)
$59$ \( \)\(20\!\cdots\!92\)\( + \)\(17\!\cdots\!00\)\( T + \)\(54\!\cdots\!48\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!55\)\( T^{4} + \)\(20\!\cdots\!27\)\( T^{5} + \)\(96\!\cdots\!52\)\( T^{6} - \)\(65\!\cdots\!73\)\( T^{7} - 240910458862116 T^{8} + 18410871 T^{9} + T^{10} \)
$61$ \( \)\(15\!\cdots\!00\)\( - \)\(13\!\cdots\!80\)\( T + \)\(42\!\cdots\!84\)\( T^{2} - \)\(15\!\cdots\!60\)\( T^{3} - \)\(27\!\cdots\!00\)\( T^{4} + \)\(14\!\cdots\!60\)\( T^{5} + \)\(15\!\cdots\!52\)\( T^{6} - \)\(67\!\cdots\!60\)\( T^{7} - 420490666322832 T^{8} + 13937808 T^{9} + T^{10} \)
$67$ \( \)\(62\!\cdots\!04\)\( + \)\(23\!\cdots\!60\)\( T + \)\(92\!\cdots\!00\)\( T^{2} - \)\(10\!\cdots\!52\)\( T^{3} + \)\(20\!\cdots\!16\)\( T^{4} - \)\(33\!\cdots\!68\)\( T^{5} + \)\(10\!\cdots\!89\)\( T^{6} + \)\(23\!\cdots\!86\)\( T^{7} + 673223450645571 T^{8} + 20722822 T^{9} + T^{10} \)
$71$ \( ( \)\(22\!\cdots\!80\)\( + \)\(23\!\cdots\!24\)\( T - \)\(74\!\cdots\!84\)\( T^{2} - 1035634560514308 T^{3} + 56516292 T^{4} + T^{5} )^{2} \)
$73$ \( \)\(77\!\cdots\!00\)\( + \)\(17\!\cdots\!00\)\( T + \)\(13\!\cdots\!20\)\( T^{2} - \)\(11\!\cdots\!40\)\( T^{3} - \)\(49\!\cdots\!32\)\( T^{4} - \)\(15\!\cdots\!52\)\( T^{5} + \)\(20\!\cdots\!93\)\( T^{6} + \)\(66\!\cdots\!22\)\( T^{7} - 909005573558673 T^{8} - 43436322 T^{9} + T^{10} \)
$79$ \( \)\(95\!\cdots\!25\)\( + \)\(10\!\cdots\!45\)\( T + \)\(41\!\cdots\!59\)\( T^{2} + \)\(10\!\cdots\!18\)\( T^{3} + \)\(13\!\cdots\!45\)\( T^{4} + \)\(32\!\cdots\!35\)\( T^{5} + \)\(22\!\cdots\!49\)\( T^{6} + \)\(50\!\cdots\!02\)\( T^{7} + 2553591424668207 T^{8} + 42189637 T^{9} + T^{10} \)
$83$ \( \)\(34\!\cdots\!72\)\( + \)\(44\!\cdots\!28\)\( T^{2} + \)\(16\!\cdots\!23\)\( T^{4} + \)\(20\!\cdots\!19\)\( T^{6} + 8964682477181961 T^{8} + T^{10} \)
$89$ \( \)\(26\!\cdots\!92\)\( + \)\(30\!\cdots\!28\)\( T + \)\(56\!\cdots\!04\)\( T^{2} - \)\(70\!\cdots\!20\)\( T^{3} + \)\(32\!\cdots\!32\)\( T^{4} + \)\(57\!\cdots\!56\)\( T^{5} + \)\(60\!\cdots\!52\)\( T^{6} - \)\(62\!\cdots\!72\)\( T^{7} - 7781951756367216 T^{8} + 67171914 T^{9} + T^{10} \)
$97$ \( \)\(12\!\cdots\!00\)\( + \)\(34\!\cdots\!04\)\( T^{2} + \)\(11\!\cdots\!43\)\( T^{4} + \)\(13\!\cdots\!19\)\( T^{6} + 61712953046491425 T^{8} + T^{10} \)
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