Properties

Label 25.34.a.f
Level $25$
Weight $34$
Character orbit 25.a
Self dual yes
Analytic conductor $172.457$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(172.457072203\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 26286285043 x^{14} + 277176279803774573548 x^{12} - 1496006322727341104924267746816 x^{10} + 4376228902975645752284567792282218577920 x^{8} - 6785210496827316795155770011788917433647451078656 x^{6} + 5093319366938751307797338793282286802764384084972313509888 x^{4} - 1608229292592013531797229664939538769264311330074914543142623510528 x^{2} + 163532475457876517394407745867705361011086521692855999713211555842344615936\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: multiple of \( 2^{85}\cdot 3^{26}\cdot 5^{66}\cdot 7^{4}\cdot 11^{8} \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 52 \beta_{1} + \beta_{9} ) q^{3} + ( 4553207930 + \beta_{2} ) q^{4} + ( 687611084267 + 175 \beta_{2} + \beta_{3} ) q^{6} + ( 49478169 \beta_{1} + 168943 \beta_{9} + \beta_{10} ) q^{7} + ( 3608565526 \beta_{1} + 1699587 \beta_{9} + \beta_{10} + \beta_{11} ) q^{8} + ( 1720947285919771 + 15746 \beta_{2} + 303 \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})\) \( q +\beta_{1} q^{2} +(52 \beta_{1} + \beta_{9}) q^{3} +(4553207930 + \beta_{2}) q^{4} +(687611084267 + 175 \beta_{2} + \beta_{3}) q^{6} +(49478169 \beta_{1} + 168943 \beta_{9} + \beta_{10}) q^{7} +(3608565526 \beta_{1} + 1699587 \beta_{9} + \beta_{10} + \beta_{11}) q^{8} +(1720947285919771 + 15746 \beta_{2} + 303 \beta_{3} - \beta_{4}) q^{9} +(13748411922429715 + 2051065 \beta_{2} - 2479 \beta_{3} + 5 \beta_{4} + \beta_{5} - \beta_{6}) q^{11} +(1573497495065 \beta_{1} + 4644056378 \beta_{9} + 1731 \beta_{10} - 86 \beta_{11} + 4 \beta_{12} + \beta_{14}) q^{12} +(106980619285 \beta_{1} + 8191769048 \beta_{9} + 2347 \beta_{10} - 489 \beta_{11} - 6 \beta_{12} - \beta_{13} + \beta_{14}) q^{13} +(651002723925203171 + 156736803 \beta_{2} + 349224 \beta_{3} - 79 \beta_{4} + 56 \beta_{5} - 16 \beta_{6} - \beta_{8}) q^{14} +(8323216025195010208 + 1641870484 \beta_{2} + 69225 \beta_{3} - 945 \beta_{4} + 171 \beta_{5} - 359 \beta_{6} + \beta_{7}) q^{16} +(404358354648 \beta_{1} + 325023482806 \beta_{9} + 461895 \beta_{10} + 10149 \beta_{11} - 1519 \beta_{12} - 37 \beta_{13} + 57 \beta_{14} + 4 \beta_{15}) q^{17} +(1839702622430586 \beta_{1} + 3948865413358 \beta_{9} + 801498 \beta_{10} - 5510 \beta_{11} + 7385 \beta_{12} + \beta_{13} + 727 \beta_{14} - 13 \beta_{15}) q^{18} +(\)\(18\!\cdots\!67\)\( - 32361519743 \beta_{2} + 7913495 \beta_{3} - 65701 \beta_{4} + 7115 \beta_{5} - 1015 \beta_{6} - 38 \beta_{7} + 28 \beta_{8}) q^{19} +(\)\(12\!\cdots\!88\)\( + 193034351198 \beta_{2} + 147712984 \beta_{3} - 575163 \beta_{4} - 22655 \beta_{5} - 22624 \beta_{6} + 180 \beta_{7} + 248 \beta_{8}) q^{21} +(29442713940904854 \beta_{1} - 28601468287552 \beta_{9} + 40578980 \beta_{10} + 14324616 \beta_{11} - 87714 \beta_{12} + 3774 \beta_{13} - 6402 \beta_{14} + 306 \beta_{15}) q^{22} +(19185971618479201 \beta_{1} - 61348390785803 \beta_{9} + 86110215 \beta_{10} + 13172918 \beta_{11} - 5594 \beta_{12} - 1798 \beta_{13} - 27006 \beta_{14} - 404 \beta_{15}) q^{23} +(\)\(14\!\cdots\!88\)\( + 243865259420 \beta_{2} + 4137828341 \beta_{3} - 8804125 \beta_{4} + 113095 \beta_{5} + 67373 \beta_{6} - 171 \beta_{7} - 2248 \beta_{8}) q^{24} +(\)\(14\!\cdots\!14\)\( - 1772014823318 \beta_{2} + 16058837532 \beta_{3} - 5589210 \beta_{4} - 629383 \beta_{5} + 989395 \beta_{6} - 3213 \beta_{7} - 5425 \beta_{8}) q^{26} +(1981249022167055106 \beta_{1} + 2222954719973490 \beta_{9} + 1680977826 \beta_{10} + 391289328 \beta_{11} + 3456312 \beta_{12} - 103008 \beta_{13} + 127656 \beta_{14} - 8088 \beta_{15}) q^{27} +(1422414133979912337 \beta_{1} + 3332984264283326 \beta_{9} + 10264636935 \beta_{10} + 157236582 \beta_{11} - 5969668 \beta_{12} + 49624 \beta_{13} + 500497 \beta_{14} + 16120 \beta_{15}) q^{28} +(-\)\(13\!\cdots\!02\)\( + 23246722207156 \beta_{2} - 4262080154 \beta_{3} + 98885628 \beta_{4} + 796334 \beta_{5} - 984904 \beta_{6} + 23436 \beta_{7} + 64776 \beta_{8}) q^{29} +(\)\(30\!\cdots\!24\)\( + 124985206414232 \beta_{2} + 398460293598 \beta_{3} + 262734858 \beta_{4} - 16820910 \beta_{5} - 15135130 \beta_{6} - 11098 \beta_{7} + 34596 \beta_{8}) q^{31} +(-10121918732004650848 \beta_{1} - 10913046728393300 \beta_{9} + 14637217820 \beta_{10} - 1672789340 \beta_{11} + 64858672 \beta_{12} + 1487696 \beta_{13} - 659544 \beta_{14} + 42640 \beta_{15}) q^{32} +(-14218623885209309426 \beta_{1} - 17714031301371282 \beta_{9} + 36584212485 \beta_{10} - 4525847741 \beta_{11} - 103844919 \beta_{12} - 422095 \beta_{13} - 6032717 \beta_{14} - 168428 \beta_{15}) q^{33} +(\)\(66\!\cdots\!36\)\( + 145889210148328 \beta_{2} + 725049199324 \beta_{3} + 1404538756 \beta_{4} + 355863862 \beta_{5} - 464878 \beta_{6} - 336942 \beta_{7} - 918502 \beta_{8}) q^{34} +(\)\(94\!\cdots\!46\)\( + 2218328984053449 \beta_{2} + 7232607399018 \beta_{3} - 3806253114 \beta_{4} - 1087612746 \beta_{5} + 109701042 \beta_{6} + 729378 \beta_{7} + 58776 \beta_{8}) q^{36} +(-\)\(20\!\cdots\!79\)\( \beta_{1} - 84177802143992094 \beta_{9} - 206968881066 \beta_{10} - 29560330584 \beta_{11} + 1035595487 \beta_{12} - 14484834 \beta_{13} - 10584262 \beta_{14} + 635128 \beta_{15}) q^{37} +(-63129783132497343910 \beta_{1} + 47510705797364744 \beta_{9} - 647262006836 \beta_{10} - 52114419264 \beta_{11} - 2286801102 \beta_{12} - 707726 \beta_{13} + 48174810 \beta_{14} + 227438 \beta_{15}) q^{38} +(\)\(59\!\cdots\!38\)\( + 6906768376500350 \beta_{2} + 5282617321798 \beta_{3} - 18518225754 \beta_{4} + 3060633814 \beta_{5} + 165248210 \beta_{6} + 1610784 \beta_{7} + 7723328 \beta_{8}) q^{39} +(\)\(60\!\cdots\!84\)\( + 5805198271458926 \beta_{2} - 607433046241 \beta_{3} - 1543016635 \beta_{4} - 7181946138 \beta_{5} - 175402392 \beta_{6} - 7767144 \beta_{7} - 363376 \beta_{8}) q^{41} +(\)\(27\!\cdots\!35\)\( \beta_{1} + 2240231871979291662 \beta_{9} - 1764833157546 \beta_{10} + 655267653970 \beta_{11} + 18500367195 \beta_{12} + 111624227 \beta_{13} + 227295289 \beta_{14} - 11177951 \beta_{15}) q^{42} +(-88995216443066151602 \beta_{1} - 1359687048945782111 \beta_{9} - 2331395578690 \beta_{10} + 58104703884 \beta_{11} - 21010931564 \beta_{12} + 38700964 \beta_{13} - 211583268 \beta_{14} + 9610112 \beta_{15}) q^{43} +(\)\(26\!\cdots\!60\)\( + 102537220631266732 \beta_{2} - 75048259103740 \beta_{3} + 126215286492 \beta_{4} + 24896716796 \beta_{5} - 2121653868 \beta_{6} + 6726132 \beta_{7} - 42635984 \beta_{8}) q^{44} +(\)\(25\!\cdots\!81\)\( + 102432024126997109 \beta_{2} - 271972884164106 \beta_{3} + 40541456865 \beta_{4} - 31018044300 \beta_{5} - 3955353532 \beta_{6} + 33356372 \beta_{7} - 30055845 \beta_{8}) q^{46} +(-\)\(51\!\cdots\!07\)\( \beta_{1} + 1245256354575354411 \beta_{9} + 8957782047311 \beta_{10} + 457353024214 \beta_{11} - 13060196722 \beta_{12} - 745333782 \beta_{13} - 2186147862 \beta_{14} + 72970404 \beta_{15}) q^{47} +(\)\(31\!\cdots\!76\)\( \beta_{1} + 14072283428329894468 \beta_{9} + 30378903060900 \beta_{10} - 760745620436 \beta_{11} + 8817466672 \beta_{12} - 270631344 \beta_{13} - 172479320 \beta_{14} - 88868976 \beta_{15}) q^{48} +(\)\(30\!\cdots\!59\)\( + 635567334399907938 \beta_{2} - 361836554951417 \beta_{3} + 274450291339 \beta_{4} - 228508364 \beta_{5} + 12702707808 \beta_{6} - 131164376 \beta_{7} + 177651824 \beta_{8}) q^{49} +(\)\(23\!\cdots\!68\)\( + 393371872574794264 \beta_{2} - 410533667230430 \beta_{3} - 507285506454 \beta_{4} + 57568776846 \beta_{5} + 40128765654 \beta_{6} + 30066822 \beta_{7} + 457057956 \beta_{8}) q^{51} +(-\)\(13\!\cdots\!18\)\( \beta_{1} + \)\(13\!\cdots\!40\)\( \beta_{9} + 34487289052866 \beta_{10} - 17095257365544 \beta_{11} - 69017843408 \beta_{12} + 4206037320 \beta_{13} + 13863534226 \beta_{14} - 189124312 \beta_{15}) q^{52} +(-\)\(27\!\cdots\!33\)\( \beta_{1} + 97164681110105680096 \beta_{9} + 77241438049533 \beta_{10} - 2032206413663 \beta_{11} + 331886069838 \beta_{12} + 171108905 \beta_{13} + 9574082319 \beta_{14} + 277935064 \beta_{15}) q^{53} +(\)\(26\!\cdots\!94\)\( + 4872857348862075402 \beta_{2} + 2850699008064180 \beta_{3} - 5218746162630 \beta_{4} - 678356089512 \beta_{5} - 24599361000 \beta_{6} + 735441336 \beta_{7} - 740215506 \beta_{8}) q^{54} +(\)\(13\!\cdots\!32\)\( + 2374257130895764204 \beta_{2} + 5402322634798393 \beta_{3} + 3143166280063 \beta_{4} + 1660342903251 \beta_{5} - 179969471823 \beta_{6} - 1064061063 \beta_{7} - 3316901960 \beta_{8}) q^{56} +(\)\(22\!\cdots\!90\)\( \beta_{1} + \)\(54\!\cdots\!86\)\( \beta_{9} - 248316633278331 \beta_{10} + 12377432841187 \beta_{11} - 1777404095495 \beta_{12} - 17780493135 \beta_{13} - 65700525941 \beta_{14} - 350672244 \beta_{15}) q^{57} +(\)\(39\!\cdots\!14\)\( \beta_{1} - 15863185088726310800 \beta_{9} - 914039664094040 \beta_{10} + 62808485732352 \beta_{11} + 1326285598684 \beta_{12} + 9939117660 \beta_{13} - 67231433780 \beta_{14} + 684045284 \beta_{15}) q^{58} +(\)\(12\!\cdots\!77\)\( + 5508867296954199587 \beta_{2} + 334561831078357 \beta_{3} + 13219872285809 \beta_{4} - 1517930384943 \beta_{5} - 204163370053 \beta_{6} - 1669078386 \beta_{7} + 3408235348 \beta_{8}) q^{59} +(-\)\(39\!\cdots\!18\)\( + 3998848735136572154 \beta_{2} + 9042104819484846 \beta_{3} - 1280560460115 \beta_{4} - 342373815093 \beta_{5} + 251774917256 \beta_{6} + 5190006752 \beta_{7} + 14235107520 \beta_{8}) q^{61} +(\)\(12\!\cdots\!36\)\( \beta_{1} + \)\(53\!\cdots\!08\)\( \beta_{9} + 380063641909016 \beta_{10} + 163700279504856 \beta_{11} - 219498928228 \beta_{12} + 43587293692 \beta_{13} + 252702390372 \beta_{14} + 3296168372 \beta_{15}) q^{62} +(\)\(11\!\cdots\!43\)\( \beta_{1} + \)\(37\!\cdots\!55\)\( \beta_{9} - 599796990356787 \beta_{10} - 36398633961714 \beta_{11} + 2198889501150 \beta_{12} - 82598482782 \beta_{13} + 217103459466 \beta_{14} - 9421433892 \beta_{15}) q^{63} +(-\)\(20\!\cdots\!00\)\( - 34943644764392534224 \beta_{2} - 7168427549047668 \beta_{3} + 17575031763732 \beta_{4} + 2367905801124 \beta_{5} + 1985391861260 \beta_{6} - 1417944340 \beta_{7} - 11361274176 \beta_{8}) q^{64} +(-\)\(18\!\cdots\!36\)\( - 41782098602496329612 \beta_{2} - 52267903011772156 \beta_{3} + 1073515173336 \beta_{4} - 10610503505092 \beta_{5} + 1806711406228 \beta_{6} - 9643139820 \beta_{7} - 34708144268 \beta_{8}) q^{66} +(-\)\(22\!\cdots\!26\)\( \beta_{1} + \)\(12\!\cdots\!07\)\( \beta_{9} + 468104253773006 \beta_{10} - 480281372164020 \beta_{11} + 14403385122340 \beta_{12} + 20332540292 \beta_{13} - 828696705940 \beta_{14} + 1061992880 \beta_{15}) q^{67} +(\)\(11\!\cdots\!48\)\( \beta_{1} + \)\(68\!\cdots\!96\)\( \beta_{9} + 13298706331541128 \beta_{10} - 505074879116456 \beta_{11} - 33433955437968 \beta_{12} + 360315898800 \beta_{13} - 6354719424 \beta_{14} + 35459603184 \beta_{15}) q^{68} +(-\)\(43\!\cdots\!04\)\( - \)\(17\!\cdots\!10\)\( \beta_{2} - 40974482086013542 \beta_{3} - 164350093092777 \beta_{4} + 38617914719049 \beta_{5} - 8589375224520 \beta_{6} + 10461146220 \beta_{7} + 1422538824 \beta_{8}) q^{69} +(-\)\(54\!\cdots\!26\)\( - \)\(10\!\cdots\!02\)\( \beta_{2} - 168554584372588000 \beta_{3} - 183196897895164 \beta_{4} - 31925871727348 \beta_{5} - 12904692101564 \beta_{6} - 4063553838 \beta_{7} + 24421204780 \beta_{8}) q^{71} +(\)\(10\!\cdots\!34\)\( \beta_{1} + \)\(63\!\cdots\!43\)\( \beta_{9} - 1436312924030535 \beta_{10} - 396368222930903 \beta_{11} + 78245302343840 \beta_{12} - 639069897632 \beta_{13} + 2250415109680 \beta_{14} - 81980263456 \beta_{15}) q^{72} +(\)\(74\!\cdots\!70\)\( \beta_{1} + \)\(11\!\cdots\!66\)\( \beta_{9} - 4575006498632919 \beta_{10} - 1450286821808217 \beta_{11} - 48027393044839 \beta_{12} - 942872336091 \beta_{13} - 3042194524177 \beta_{14} - 52472237948 \beta_{15}) q^{73} +(-\)\(26\!\cdots\!98\)\( - \)\(42\!\cdots\!02\)\( \beta_{2} - 114289019019228524 \beta_{3} + 214410438396270 \beta_{4} - 4662235552817 \beta_{5} + 21247021675365 \beta_{6} + 54273330693 \beta_{7} + 207051020837 \beta_{8}) q^{74} +(-\)\(24\!\cdots\!20\)\( - \)\(16\!\cdots\!44\)\( \beta_{2} + 221776903252425324 \beta_{3} + 768877428900660 \beta_{4} - 24198906585324 \beta_{5} + 38386731255580 \beta_{6} - 11315614916 \beta_{7} + 122600084880 \beta_{8}) q^{76} +(\)\(29\!\cdots\!54\)\( \beta_{1} + \)\(47\!\cdots\!74\)\( \beta_{9} + 7379822591419149 \beta_{10} + 7467576721069163 \beta_{11} - 186541016145895 \beta_{12} + 2458379349273 \beta_{13} - 4230676680849 \beta_{14} + 373494814440 \beta_{15}) q^{77} +(\)\(11\!\cdots\!56\)\( \beta_{1} + \)\(80\!\cdots\!16\)\( \beta_{9} - 78738040160109384 \beta_{10} + 8782230787216912 \beta_{11} + 189151621394532 \beta_{12} + 1130876742692 \beta_{13} + 12386649223876 \beta_{14} - 4287651524 \beta_{15}) q^{78} +(\)\(66\!\cdots\!52\)\( + \)\(22\!\cdots\!72\)\( \beta_{2} + 467983061623022838 \beta_{3} + 1533312300708834 \beta_{4} - 46909540066350 \beta_{5} - 25536343070082 \beta_{6} - 496735009350 \beta_{7} - 1078764074340 \beta_{8}) q^{79} +(\)\(78\!\cdots\!07\)\( + \)\(13\!\cdots\!42\)\( \beta_{2} + 2925942467647441815 \beta_{3} - 2352197864483271 \beta_{4} + 369172800525186 \beta_{5} - 42038374397688 \beta_{6} + 513571300992 \beta_{7} - 394388085888 \beta_{8}) q^{81} +(\)\(10\!\cdots\!17\)\( \beta_{1} + \)\(20\!\cdots\!70\)\( \beta_{9} - 99585095821251938 \beta_{10} - 3511767231981978 \beta_{11} - 290141727898085 \beta_{12} - 3670878906893 \beta_{13} + 1541710862541 \beta_{14} - 716758405735 \beta_{15}) q^{82} +(\)\(19\!\cdots\!82\)\( \beta_{1} - \)\(12\!\cdots\!63\)\( \beta_{9} - 80355878785996854 \beta_{10} - 7852942283535332 \beta_{11} + 660757534222868 \beta_{12} + 2130422399860 \beta_{13} - 13199784904164 \beta_{14} - 229047782704 \beta_{15}) q^{83} +(\)\(25\!\cdots\!16\)\( + \)\(53\!\cdots\!32\)\( \beta_{2} + 2000896123867937094 \beta_{3} - 5171626249754358 \beta_{4} - 1039028163546534 \beta_{5} - 7203368940546 \beta_{6} + 1469465439246 \beta_{7} + 2442808716648 \beta_{8}) q^{84} +(-\)\(11\!\cdots\!65\)\( - 71420327425648554693 \beta_{2} - 4306557265995095917 \beta_{3} + 9336288227681754 \beta_{4} + 1169979419452944 \beta_{5} - 88900074669184 \beta_{6} - 2323453736160 \beta_{7} + 331447309286 \beta_{8}) q^{86} +(-\)\(64\!\cdots\!88\)\( \beta_{1} - \)\(65\!\cdots\!86\)\( \beta_{9} + 367838532970648656 \beta_{10} - 87637737810230248 \beta_{11} - 580253204382384 \beta_{12} - 3875101732328 \beta_{13} + 21753148250432 \beta_{14} + 187581095480 \beta_{15}) q^{87} +(\)\(80\!\cdots\!96\)\( \beta_{1} - \)\(55\!\cdots\!64\)\( \beta_{9} + 664880816926509836 \beta_{10} + 29609511994095468 \beta_{11} - 980339408598464 \beta_{12} - 15719802986560 \beta_{13} - 69362591920672 \beta_{14} + 2379190045376 \beta_{15}) q^{88} +(\)\(51\!\cdots\!38\)\( + \)\(70\!\cdots\!24\)\( \beta_{2} - 2440266881275532050 \beta_{3} - 7329207711803388 \beta_{4} + 242262492222038 \beta_{5} + 72571894978872 \beta_{6} - 1371820441176 \beta_{7} - 253175009552 \beta_{8}) q^{89} +(\)\(35\!\cdots\!58\)\( + \)\(38\!\cdots\!42\)\( \beta_{2} - 8205520778241257132 \beta_{3} - 25050502278969452 \beta_{4} - 2341210126663400 \beta_{5} + 338304411751716 \beta_{6} + 3758637468946 \beta_{7} + 808138551788 \beta_{8}) q^{91} +(\)\(87\!\cdots\!83\)\( \beta_{1} - \)\(28\!\cdots\!90\)\( \beta_{9} - 456237393735875599 \beta_{10} + 119928228653257442 \beta_{11} + 1850727809041428 \beta_{12} + 23118450329208 \beta_{13} - 82719303609969 \beta_{14} + 946904760600 \beta_{15}) q^{92} +(\)\(30\!\cdots\!32\)\( \beta_{1} + \)\(28\!\cdots\!56\)\( \beta_{9} + 713545698672411000 \beta_{10} - 176805918539167768 \beta_{11} + 943955165056568 \beta_{12} + 46622564258808 \beta_{13} + 316736229322976 \beta_{14} - 6659602297608 \beta_{15}) q^{93} +(-\)\(67\!\cdots\!17\)\( - \)\(13\!\cdots\!61\)\( \beta_{2} - 15338719854445255564 \beta_{3} + 48761864582677985 \beta_{4} + 5729751786693500 \beta_{5} - 254318379174628 \beta_{6} - 2649340042964 \beta_{7} - 13544342634557 \beta_{8}) q^{94} +(-\)\(86\!\cdots\!52\)\( + \)\(56\!\cdots\!28\)\( \beta_{2} - 15640845203765396140 \beta_{3} + 38541602808816652 \beta_{4} - 6985959119001220 \beta_{5} - 103764366927788 \beta_{6} + 3402883671156 \beta_{7} - 5740873089344 \beta_{8}) q^{96} +(\)\(30\!\cdots\!92\)\( \beta_{1} - \)\(34\!\cdots\!38\)\( \beta_{9} + 1460907446626202239 \beta_{10} + 287509869803294373 \beta_{11} + 6470414349823229 \beta_{12} - 11386022658797 \beta_{13} + 160655064282889 \beta_{14} + 2804180631148 \beta_{15}) q^{97} +(\)\(79\!\cdots\!06\)\( \beta_{1} - \)\(36\!\cdots\!14\)\( \beta_{9} - 4420487159975767230 \beta_{10} + 452741154613264370 \beta_{11} - 11026163454818995 \beta_{12} - 65310941492139 \beta_{13} - 419247597304557 \beta_{14} + 1720531516431 \beta_{15}) q^{98} +(-\)\(21\!\cdots\!85\)\( - \)\(50\!\cdots\!55\)\( \beta_{2} - 9809976407090121429 \beta_{3} - 22599571996742457 \beta_{4} + 3327824842230423 \beta_{5} + 1385467676150397 \beta_{6} + 5367885323310 \beta_{7} + 31716988069524 \beta_{8}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 72851326872q^{4} + 11001777346872q^{6} + 27535156574590368q^{9} + O(q^{10}) \) \( 16q + 72851326872q^{4} + 11001777346872q^{6} + 27535156574590368q^{9} + 219974590742466912q^{11} + 10416043581549356184q^{14} + \)\(13\!\cdots\!76\)\(q^{16} + \)\(29\!\cdots\!00\)\(q^{19} + \)\(20\!\cdots\!92\)\(q^{21} + \)\(23\!\cdots\!00\)\(q^{24} + \)\(23\!\cdots\!32\)\(q^{26} - \)\(22\!\cdots\!00\)\(q^{29} + \)\(48\!\cdots\!72\)\(q^{31} + \)\(10\!\cdots\!64\)\(q^{34} + \)\(15\!\cdots\!56\)\(q^{36} + \)\(95\!\cdots\!16\)\(q^{39} + \)\(96\!\cdots\!52\)\(q^{41} + \)\(43\!\cdots\!04\)\(q^{44} + \)\(40\!\cdots\!92\)\(q^{46} + \)\(48\!\cdots\!12\)\(q^{49} + \)\(37\!\cdots\!32\)\(q^{51} + \)\(41\!\cdots\!00\)\(q^{54} + \)\(20\!\cdots\!00\)\(q^{56} + \)\(20\!\cdots\!00\)\(q^{59} - \)\(62\!\cdots\!88\)\(q^{61} - \)\(32\!\cdots\!08\)\(q^{64} - \)\(29\!\cdots\!96\)\(q^{66} - \)\(68\!\cdots\!04\)\(q^{69} - \)\(87\!\cdots\!08\)\(q^{71} - \)\(43\!\cdots\!76\)\(q^{74} - \)\(38\!\cdots\!00\)\(q^{76} + \)\(10\!\cdots\!00\)\(q^{79} + \)\(12\!\cdots\!36\)\(q^{81} + \)\(40\!\cdots\!64\)\(q^{84} - \)\(18\!\cdots\!88\)\(q^{86} + \)\(81\!\cdots\!00\)\(q^{89} + \)\(57\!\cdots\!52\)\(q^{91} - \)\(10\!\cdots\!96\)\(q^{94} - \)\(13\!\cdots\!08\)\(q^{96} - \)\(34\!\cdots\!24\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 26286285043 x^{14} + 277176279803774573548 x^{12} - 1496006322727341104924267746816 x^{10} + 4376228902975645752284567792282218577920 x^{8} - 6785210496827316795155770011788917433647451078656 x^{6} + 5093319366938751307797338793282286802764384084972313509888 x^{4} - 1608229292592013531797229664939538769264311330074914543142623510528 x^{2} + 163532475457876517394407745867705361011086521692855999713211555842344615936\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( 4 \nu^{2} - 13143142522 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(23\!\cdots\!27\)\( \nu^{14} + \)\(48\!\cdots\!85\)\( \nu^{12} - \)\(37\!\cdots\!60\)\( \nu^{10} + \)\(14\!\cdots\!00\)\( \nu^{8} - \)\(26\!\cdots\!00\)\( \nu^{6} + \)\(22\!\cdots\!32\)\( \nu^{4} - \)\(85\!\cdots\!00\)\( \nu^{2} + \)\(10\!\cdots\!20\)\(\)\()/ \)\(17\!\cdots\!40\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(39\!\cdots\!63\)\( \nu^{14} - \)\(40\!\cdots\!33\)\( \nu^{12} - \)\(15\!\cdots\!28\)\( \nu^{10} + \)\(32\!\cdots\!20\)\( \nu^{8} - \)\(13\!\cdots\!00\)\( \nu^{6} + \)\(23\!\cdots\!12\)\( \nu^{4} - \)\(14\!\cdots\!92\)\( \nu^{2} + \)\(22\!\cdots\!08\)\(\)\()/ \)\(57\!\cdots\!80\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(77\!\cdots\!41\)\( \nu^{14} + \)\(66\!\cdots\!15\)\( \nu^{12} + \)\(68\!\cdots\!84\)\( \nu^{10} - \)\(10\!\cdots\!20\)\( \nu^{8} + \)\(47\!\cdots\!80\)\( \nu^{6} - \)\(87\!\cdots\!24\)\( \nu^{4} + \)\(62\!\cdots\!40\)\( \nu^{2} - \)\(11\!\cdots\!84\)\(\)\()/ \)\(42\!\cdots\!60\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(40\!\cdots\!83\)\( \nu^{14} + \)\(77\!\cdots\!25\)\( \nu^{12} - \)\(52\!\cdots\!28\)\( \nu^{10} + \)\(14\!\cdots\!20\)\( \nu^{8} - \)\(99\!\cdots\!40\)\( \nu^{6} - \)\(32\!\cdots\!12\)\( \nu^{4} + \)\(66\!\cdots\!80\)\( \nu^{2} - \)\(25\!\cdots\!72\)\(\)\()/ \)\(57\!\cdots\!80\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(19\!\cdots\!57\)\( \nu^{14} + \)\(58\!\cdots\!35\)\( \nu^{12} - \)\(67\!\cdots\!32\)\( \nu^{10} + \)\(38\!\cdots\!20\)\( \nu^{8} - \)\(10\!\cdots\!00\)\( \nu^{6} + \)\(13\!\cdots\!92\)\( \nu^{4} - \)\(42\!\cdots\!20\)\( \nu^{2} + \)\(11\!\cdots\!92\)\(\)\()/ \)\(28\!\cdots\!40\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(13\!\cdots\!33\)\( \nu^{14} - \)\(26\!\cdots\!83\)\( \nu^{12} + \)\(20\!\cdots\!96\)\( \nu^{10} - \)\(73\!\cdots\!00\)\( \nu^{8} + \)\(13\!\cdots\!20\)\( \nu^{6} - \)\(11\!\cdots\!88\)\( \nu^{4} + \)\(45\!\cdots\!28\)\( \nu^{2} - \)\(52\!\cdots\!56\)\(\)\()/ \)\(17\!\cdots\!40\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(37\!\cdots\!75\)\( \nu^{15} + \)\(83\!\cdots\!69\)\( \nu^{13} - \)\(73\!\cdots\!20\)\( \nu^{11} + \)\(32\!\cdots\!20\)\( \nu^{9} - \)\(76\!\cdots\!00\)\( \nu^{7} + \)\(91\!\cdots\!00\)\( \nu^{5} - \)\(48\!\cdots\!04\)\( \nu^{3} + \)\(74\!\cdots\!40\)\( \nu\)\()/ \)\(22\!\cdots\!40\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(29\!\cdots\!35\)\( \nu^{15} + \)\(60\!\cdots\!73\)\( \nu^{13} - \)\(48\!\cdots\!32\)\( \nu^{11} + \)\(18\!\cdots\!40\)\( \nu^{9} - \)\(38\!\cdots\!60\)\( \nu^{7} + \)\(40\!\cdots\!00\)\( \nu^{5} - \)\(19\!\cdots\!28\)\( \nu^{3} + \)\(29\!\cdots\!72\)\( \nu\)\()/ \)\(33\!\cdots\!60\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(25\!\cdots\!45\)\( \nu^{15} - \)\(54\!\cdots\!55\)\( \nu^{13} + \)\(47\!\cdots\!84\)\( \nu^{11} - \)\(20\!\cdots\!00\)\( \nu^{9} + \)\(46\!\cdots\!20\)\( \nu^{7} - \)\(54\!\cdots\!00\)\( \nu^{5} + \)\(29\!\cdots\!60\)\( \nu^{3} - \)\(71\!\cdots\!84\)\( \nu\)\()/ \)\(66\!\cdots\!20\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(33\!\cdots\!81\)\( \nu^{15} - \)\(10\!\cdots\!71\)\( \nu^{13} + \)\(12\!\cdots\!64\)\( \nu^{11} - \)\(80\!\cdots\!80\)\( \nu^{9} + \)\(30\!\cdots\!00\)\( \nu^{7} - \)\(62\!\cdots\!36\)\( \nu^{5} + \)\(59\!\cdots\!76\)\( \nu^{3} - \)\(14\!\cdots\!24\)\( \nu\)\()/ \)\(33\!\cdots\!60\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(39\!\cdots\!53\)\( \nu^{15} + \)\(10\!\cdots\!39\)\( \nu^{13} - \)\(12\!\cdots\!24\)\( \nu^{11} + \)\(68\!\cdots\!20\)\( \nu^{9} - \)\(21\!\cdots\!80\)\( \nu^{7} + \)\(38\!\cdots\!28\)\( \nu^{5} - \)\(36\!\cdots\!84\)\( \nu^{3} + \)\(13\!\cdots\!44\)\( \nu\)\()/ \)\(22\!\cdots\!40\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(24\!\cdots\!05\)\( \nu^{15} + \)\(26\!\cdots\!47\)\( \nu^{13} + \)\(54\!\cdots\!56\)\( \nu^{11} - \)\(54\!\cdots\!40\)\( \nu^{9} + \)\(20\!\cdots\!40\)\( \nu^{7} - \)\(34\!\cdots\!60\)\( \nu^{5} + \)\(22\!\cdots\!88\)\( \nu^{3} - \)\(38\!\cdots\!76\)\( \nu\)\()/ \)\(11\!\cdots\!20\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(23\!\cdots\!37\)\( \nu^{15} - \)\(70\!\cdots\!03\)\( \nu^{13} + \)\(87\!\cdots\!48\)\( \nu^{11} - \)\(54\!\cdots\!80\)\( \nu^{9} + \)\(18\!\cdots\!60\)\( \nu^{7} - \)\(31\!\cdots\!72\)\( \nu^{5} + \)\(23\!\cdots\!28\)\( \nu^{3} - \)\(49\!\cdots\!88\)\( \nu\)\()/ \)\(66\!\cdots\!20\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 13143142522\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{11} + \beta_{10} + 1699587 \beta_{9} + 20788434710 \beta_{1}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{7} - 359 \beta_{6} + 171 \beta_{5} - 945 \beta_{4} + 69225 \beta_{3} + 27411674260 \beta_{2} + 273232443522298566816\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(10660 \beta_{15} - 164886 \beta_{14} + 371924 \beta_{13} + 16214668 \beta_{12} + 8171737257 \beta_{11} + 12249239047 \beta_{10} + 11871079481315179 \beta_{9} + 120700582524832670760 \beta_{1}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-2840318544 \beta_{8} + 10382932155 \beta_{7} - 3358385182845 \beta_{6} + 2428074969321 \beta_{5} - 5753102295867 \beta_{4} - 1048809109597917 \beta_{3} + 174914235554907059148 \beta_{2} + 1586434433353082161402564961600\)\()/16\)
\(\nu^{7}\)\(=\)\((\)\(136499115736620 \beta_{15} - 2262561078032226 \beta_{14} + 3589556647778748 \beta_{13} + 151880834833615716 \beta_{12} + 56412146521554039423 \beta_{11} + 109049727225684170937 \beta_{10} + 70927051967169655700039373 \beta_{9} + 730927623461536879752318017072 \beta_{1}\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(-42691027079367431664 \beta_{8} + 76825092896935810965 \beta_{7} - 24794474173663719316659 \beta_{6} + 23064061721645823913095 \beta_{5} - 15723412210689612002517 \beta_{4} - 19980035675571457222652307 \beta_{3} + 1105908827300269014731724189236 \beta_{2} + 9606981528873716449358067115346890688576\)\()/16\)
\(\nu^{9}\)\(=\)\((\)\(1285956383284095994631316 \beta_{15} - 21581235824005445096340414 \beta_{14} + 26844994590558708833320452 \beta_{13} + 995808552908047073449661148 \beta_{12} + 370513904015703366576893949377 \beta_{11} + 856109094579464109234386829863 \beta_{10} + 405251060054157320877267801104266995 \beta_{9} + 4515521212752738369496992352516297277104 \beta_{1}\)\()/8\)
\(\nu^{10}\)\(=\)\((\)\(-435597954259987487783700103440 \beta_{8} + 504012570392205859971069629195 \beta_{7} - 170279745747183611733858702570925 \beta_{6} + 192952894812687188095946491595865 \beta_{5} + 91732231719854797901186880862005 \beta_{4} - 214976503847936690621817920333784525 \beta_{3} + 6983035218668743874777697541147412172620 \beta_{2} + 59349827811894327294518702110279884467978794892736\)\()/16\)
\(\nu^{11}\)\(=\)\((\)\(10824024885239361938904064750136300 \beta_{15} - 179383935512029617559390314805343490 \beta_{14} + 185451869889771912903597551458338940 \beta_{13} + 5546135297841507326171932376632828580 \beta_{12} + 2387271776097740250934266927043208804895 \beta_{11} + 6322351216945087886365980174812970628985 \beta_{10} + 2271487433513329961689128191638580429756094125 \beta_{9} + 28184581556298296235184373414621834400296875560144 \beta_{1}\)\()/8\)
\(\nu^{12}\)\(=\)\((\)\(-3795371731956497608802273238972309166320 \beta_{8} + 3133050498262966763141349212257903137045 \beta_{7} - 1136977301752459513097751193999965326310195 \beta_{6} + 1520358054382431846847595666145506076071175 \beta_{5} + 2004263603050295589409807290094948847837355 \beta_{4} - 1909659920193502742384199090074158841641945875 \beta_{3} + 44111844391491637011147400663770371867494622427444 \beta_{2} + 370443439117400007061879612015861946285617800367874118910528\)\()/16\)
\(\nu^{13}\)\(=\)\((\)\(85857633759031760845010351228834443860534420 \beta_{15} - 1394745061941545882985146620951088071862378430 \beta_{14} + 1242326920214686755310007269372683110897409540 \beta_{13} + 27143798172317056165803141598174855765958320860 \beta_{12} + 15261437208015334006080882433939842203146470116481 \beta_{11} + 45175768197474572953077747819210402869860072658151 \beta_{10} + 12563871981596298436539785894926681829666141699918052147 \beta_{9} + 176925407400280176500970829161051137281968718940209951305520 \beta_{1}\)\()/8\)
\(\nu^{14}\)\(=\)\((\)\(-30471108549707188574009628893058269172133319139600 \beta_{8} + 18942248543119431499429482381852975952028382047691 \beta_{7} - 7501377810073741345301720240460856076386079102073069 \beta_{6} + 11549348949902495794281058976076650003033498826091161 \beta_{5} + 22484594157256890276437490449356601690592750663948405 \beta_{4} - 15473808450817914385027938899439943700318987579919256845 \beta_{3} + 278923885031473985484620348809447589373640948600998090991180 \beta_{2} + 2325408207216806167099603488648725719588214373058343073710618215216576\)\()/16\)
\(\nu^{15}\)\(=\)\((\)\(\)\(65\!\cdots\!60\)\( \beta_{15} - \)\(10\!\cdots\!06\)\( \beta_{14} + \)\(82\!\cdots\!84\)\( \beta_{13} + \)\(11\!\cdots\!68\)\( \beta_{12} + \)\(97\!\cdots\!47\)\( \beta_{11} + \)\(31\!\cdots\!77\)\( \beta_{10} + \)\(68\!\cdots\!49\)\( \beta_{9} + \)\(11\!\cdots\!20\)\( \beta_{1}\)\()/8\)

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
−81004.3
−78261.9
−78034.2
−57471.0
−51228.2
−31348.8
−20202.3
−13863.7
13863.7
20202.3
31348.8
51228.2
57471.0
78034.2
78261.9
81004.3
−162009. 1.50582e7 1.76568e10 0 −2.43956e12 −1.33679e14 −1.46891e15 −5.33231e15 0
1.2 −156524. −1.41317e8 1.59098e10 0 2.21195e13 −8.25005e13 −1.14573e15 1.44114e16 0
1.3 −156068. 6.96239e7 1.57674e10 0 −1.08661e13 1.55193e14 −1.12018e15 −7.11573e14 0
1.4 −114942. −6.13071e7 4.62174e9 0 7.04677e12 1.47106e14 4.56113e14 −1.80050e15 0
1.5 −102456. 8.77372e7 1.90737e9 0 −8.98924e12 −1.11815e14 6.84671e14 2.13876e15 0
1.6 −62697.7 −3.24591e7 −4.65894e9 0 2.03511e12 −2.03055e13 8.30673e14 −4.50547e15 0
1.7 −40404.6 1.28550e8 −6.95740e9 0 −5.19400e12 4.50276e13 6.28184e14 1.09659e16 0
1.8 −27727.3 −6.45007e7 −7.82113e9 0 1.78843e12 −3.09117e13 4.55035e14 −1.39871e15 0
1.9 27727.3 6.45007e7 −7.82113e9 0 1.78843e12 3.09117e13 −4.55035e14 −1.39871e15 0
1.10 40404.6 −1.28550e8 −6.95740e9 0 −5.19400e12 −4.50276e13 −6.28184e14 1.09659e16 0
1.11 62697.7 3.24591e7 −4.65894e9 0 2.03511e12 2.03055e13 −8.30673e14 −4.50547e15 0
1.12 102456. −8.77372e7 1.90737e9 0 −8.98924e12 1.11815e14 −6.84671e14 2.13876e15 0
1.13 114942. 6.13071e7 4.62174e9 0 7.04677e12 −1.47106e14 −4.56113e14 −1.80050e15 0
1.14 156068. −6.96239e7 1.57674e10 0 −1.08661e13 −1.55193e14 1.12018e15 −7.11573e14 0
1.15 156524. 1.41317e8 1.59098e10 0 2.21195e13 8.25005e13 1.14573e15 1.44114e16 0
1.16 162009. −1.50582e7 1.76568e10 0 −2.43956e12 1.33679e14 1.46891e15 −5.33231e15 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.34.a.f 16
5.b even 2 1 inner 25.34.a.f 16
5.c odd 4 2 5.34.b.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.34.b.a 16 5.c odd 4 2
25.34.a.f 16 1.a even 1 1 trivial
25.34.a.f 16 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(44\!\cdots\!68\)\( T_{2}^{12} - \)\(95\!\cdots\!24\)\( T_{2}^{10} + \)\(11\!\cdots\!20\)\( T_{2}^{8} - \)\(69\!\cdots\!44\)\( T_{2}^{6} + \)\(20\!\cdots\!48\)\( T_{2}^{4} - \)\(26\!\cdots\!52\)\( T_{2}^{2} + \)\(10\!\cdots\!96\)\( \)">\(T_{2}^{16} - \cdots\) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(25))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( \)\(10\!\cdots\!96\)\( - \)\(26\!\cdots\!52\)\( T^{2} + \)\(20\!\cdots\!48\)\( T^{4} - \)\(69\!\cdots\!44\)\( T^{6} + \)\(11\!\cdots\!20\)\( T^{8} - \)\(95\!\cdots\!24\)\( T^{10} + \)\(44\!\cdots\!68\)\( T^{12} - 105145140172 T^{14} + T^{16} \)
$3$ \( \)\(46\!\cdots\!36\)\( - \)\(29\!\cdots\!28\)\( T^{2} + \)\(44\!\cdots\!88\)\( T^{4} - \)\(27\!\cdots\!56\)\( T^{6} + \)\(85\!\cdots\!20\)\( T^{8} - \)\(14\!\cdots\!16\)\( T^{10} + \)\(13\!\cdots\!48\)\( T^{12} - 58240062819739368 T^{14} + T^{16} \)
$5$ \( T^{16} \)
$7$ \( \)\(63\!\cdots\!56\)\( - \)\(27\!\cdots\!32\)\( T^{2} + \)\(36\!\cdots\!08\)\( T^{4} - \)\(19\!\cdots\!44\)\( T^{6} + \)\(43\!\cdots\!20\)\( T^{8} - \)\(49\!\cdots\!64\)\( T^{10} + \)\(29\!\cdots\!88\)\( T^{12} - \)\(86\!\cdots\!12\)\( T^{14} + T^{16} \)
$11$ \( ( \)\(26\!\cdots\!76\)\( - \)\(14\!\cdots\!44\)\( T - \)\(15\!\cdots\!28\)\( T^{2} - \)\(48\!\cdots\!92\)\( T^{3} + \)\(24\!\cdots\!20\)\( T^{4} + \)\(51\!\cdots\!92\)\( T^{5} - \)\(94\!\cdots\!28\)\( T^{6} - 109987295371233456 T^{7} + T^{8} )^{2} \)
$13$ \( \)\(13\!\cdots\!16\)\( - \)\(26\!\cdots\!88\)\( T^{2} + \)\(36\!\cdots\!68\)\( T^{4} - \)\(20\!\cdots\!56\)\( T^{6} + \)\(61\!\cdots\!20\)\( T^{8} - \)\(10\!\cdots\!96\)\( T^{10} + \)\(10\!\cdots\!08\)\( T^{12} - \)\(51\!\cdots\!48\)\( T^{14} + T^{16} \)
$17$ \( \)\(58\!\cdots\!76\)\( - \)\(21\!\cdots\!92\)\( T^{2} + \)\(69\!\cdots\!28\)\( T^{4} - \)\(63\!\cdots\!44\)\( T^{6} + \)\(16\!\cdots\!20\)\( T^{8} - \)\(10\!\cdots\!44\)\( T^{10} + \)\(25\!\cdots\!28\)\( T^{12} - \)\(26\!\cdots\!92\)\( T^{14} + T^{16} \)
$19$ \( ( -\)\(29\!\cdots\!00\)\( - \)\(53\!\cdots\!00\)\( T - \)\(21\!\cdots\!00\)\( T^{2} - \)\(64\!\cdots\!00\)\( T^{3} + \)\(79\!\cdots\!00\)\( T^{4} + \)\(60\!\cdots\!00\)\( T^{5} - \)\(59\!\cdots\!00\)\( T^{6} - \)\(14\!\cdots\!00\)\( T^{7} + T^{8} )^{2} \)
$23$ \( \)\(77\!\cdots\!96\)\( - \)\(25\!\cdots\!48\)\( T^{2} + \)\(24\!\cdots\!48\)\( T^{4} - \)\(10\!\cdots\!56\)\( T^{6} + \)\(24\!\cdots\!20\)\( T^{8} - \)\(31\!\cdots\!76\)\( T^{10} + \)\(21\!\cdots\!68\)\( T^{12} - \)\(74\!\cdots\!28\)\( T^{14} + T^{16} \)
$29$ \( ( -\)\(78\!\cdots\!00\)\( - \)\(13\!\cdots\!00\)\( T - \)\(34\!\cdots\!00\)\( T^{2} + \)\(28\!\cdots\!00\)\( T^{3} + \)\(49\!\cdots\!00\)\( T^{4} - \)\(33\!\cdots\!00\)\( T^{5} - \)\(49\!\cdots\!00\)\( T^{6} + \)\(11\!\cdots\!00\)\( T^{7} + T^{8} )^{2} \)
$31$ \( ( \)\(28\!\cdots\!16\)\( + \)\(18\!\cdots\!16\)\( T - \)\(17\!\cdots\!68\)\( T^{2} - \)\(38\!\cdots\!92\)\( T^{3} + \)\(24\!\cdots\!20\)\( T^{4} + \)\(19\!\cdots\!72\)\( T^{5} - \)\(94\!\cdots\!08\)\( T^{6} - \)\(24\!\cdots\!36\)\( T^{7} + T^{8} )^{2} \)
$37$ \( \)\(41\!\cdots\!16\)\( - \)\(11\!\cdots\!12\)\( T^{2} + \)\(99\!\cdots\!68\)\( T^{4} - \)\(38\!\cdots\!44\)\( T^{6} + \)\(59\!\cdots\!20\)\( T^{8} - \)\(25\!\cdots\!04\)\( T^{10} + \)\(45\!\cdots\!08\)\( T^{12} - \)\(35\!\cdots\!52\)\( T^{14} + T^{16} \)
$41$ \( ( -\)\(88\!\cdots\!64\)\( + \)\(77\!\cdots\!96\)\( T + \)\(17\!\cdots\!12\)\( T^{2} - \)\(27\!\cdots\!92\)\( T^{3} + \)\(29\!\cdots\!20\)\( T^{4} + \)\(22\!\cdots\!12\)\( T^{5} - \)\(40\!\cdots\!48\)\( T^{6} - \)\(48\!\cdots\!76\)\( T^{7} + T^{8} )^{2} \)
$43$ \( \)\(12\!\cdots\!56\)\( - \)\(42\!\cdots\!68\)\( T^{2} + \)\(28\!\cdots\!08\)\( T^{4} - \)\(55\!\cdots\!56\)\( T^{6} + \)\(40\!\cdots\!20\)\( T^{8} - \)\(10\!\cdots\!36\)\( T^{10} + \)\(11\!\cdots\!88\)\( T^{12} - \)\(54\!\cdots\!88\)\( T^{14} + T^{16} \)
$47$ \( \)\(27\!\cdots\!36\)\( - \)\(16\!\cdots\!72\)\( T^{2} + \)\(27\!\cdots\!88\)\( T^{4} - \)\(94\!\cdots\!44\)\( T^{6} + \)\(14\!\cdots\!20\)\( T^{8} - \)\(11\!\cdots\!84\)\( T^{10} + \)\(50\!\cdots\!48\)\( T^{12} - \)\(11\!\cdots\!32\)\( T^{14} + T^{16} \)
$53$ \( \)\(14\!\cdots\!36\)\( - \)\(75\!\cdots\!28\)\( T^{2} + \)\(15\!\cdots\!88\)\( T^{4} - \)\(15\!\cdots\!56\)\( T^{6} + \)\(84\!\cdots\!20\)\( T^{8} - \)\(25\!\cdots\!16\)\( T^{10} + \)\(40\!\cdots\!48\)\( T^{12} - \)\(31\!\cdots\!68\)\( T^{14} + T^{16} \)
$59$ \( ( \)\(47\!\cdots\!00\)\( + \)\(29\!\cdots\!00\)\( T - \)\(41\!\cdots\!00\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(87\!\cdots\!00\)\( T^{4} + \)\(89\!\cdots\!00\)\( T^{5} - \)\(70\!\cdots\!00\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} + T^{8} )^{2} \)
$61$ \( ( \)\(85\!\cdots\!76\)\( + \)\(59\!\cdots\!56\)\( T - \)\(26\!\cdots\!28\)\( T^{2} - \)\(68\!\cdots\!92\)\( T^{3} + \)\(14\!\cdots\!20\)\( T^{4} + \)\(45\!\cdots\!92\)\( T^{5} - \)\(22\!\cdots\!28\)\( T^{6} + \)\(31\!\cdots\!44\)\( T^{7} + T^{8} )^{2} \)
$67$ \( \)\(15\!\cdots\!76\)\( - \)\(11\!\cdots\!92\)\( T^{2} + \)\(99\!\cdots\!28\)\( T^{4} - \)\(29\!\cdots\!44\)\( T^{6} + \)\(40\!\cdots\!20\)\( T^{8} - \)\(28\!\cdots\!44\)\( T^{10} + \)\(10\!\cdots\!28\)\( T^{12} - \)\(16\!\cdots\!92\)\( T^{14} + T^{16} \)
$71$ \( ( \)\(18\!\cdots\!96\)\( - \)\(12\!\cdots\!64\)\( T - \)\(22\!\cdots\!48\)\( T^{2} + \)\(27\!\cdots\!08\)\( T^{3} + \)\(49\!\cdots\!20\)\( T^{4} - \)\(16\!\cdots\!68\)\( T^{5} - \)\(39\!\cdots\!68\)\( T^{6} + \)\(43\!\cdots\!04\)\( T^{7} + T^{8} )^{2} \)
$73$ \( \)\(16\!\cdots\!96\)\( - \)\(18\!\cdots\!48\)\( T^{2} + \)\(20\!\cdots\!48\)\( T^{4} - \)\(89\!\cdots\!56\)\( T^{6} + \)\(18\!\cdots\!20\)\( T^{8} - \)\(18\!\cdots\!76\)\( T^{10} + \)\(82\!\cdots\!68\)\( T^{12} - \)\(15\!\cdots\!28\)\( T^{14} + T^{16} \)
$79$ \( ( -\)\(17\!\cdots\!00\)\( + \)\(15\!\cdots\!00\)\( T - \)\(11\!\cdots\!00\)\( T^{2} - \)\(97\!\cdots\!00\)\( T^{3} + \)\(88\!\cdots\!00\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{5} - \)\(17\!\cdots\!00\)\( T^{6} - \)\(52\!\cdots\!00\)\( T^{7} + T^{8} )^{2} \)
$83$ \( \)\(22\!\cdots\!76\)\( - \)\(14\!\cdots\!08\)\( T^{2} + \)\(44\!\cdots\!28\)\( T^{4} - \)\(45\!\cdots\!56\)\( T^{6} + \)\(20\!\cdots\!20\)\( T^{8} - \)\(41\!\cdots\!56\)\( T^{10} + \)\(36\!\cdots\!28\)\( T^{12} - \)\(11\!\cdots\!08\)\( T^{14} + T^{16} \)
$89$ \( ( \)\(11\!\cdots\!00\)\( - \)\(50\!\cdots\!00\)\( T + \)\(56\!\cdots\!00\)\( T^{2} + \)\(42\!\cdots\!00\)\( T^{3} - \)\(12\!\cdots\!00\)\( T^{4} + \)\(59\!\cdots\!00\)\( T^{5} + \)\(36\!\cdots\!00\)\( T^{6} - \)\(40\!\cdots\!00\)\( T^{7} + T^{8} )^{2} \)
$97$ \( \)\(13\!\cdots\!36\)\( - \)\(77\!\cdots\!72\)\( T^{2} + \)\(15\!\cdots\!88\)\( T^{4} - \)\(13\!\cdots\!44\)\( T^{6} + \)\(69\!\cdots\!20\)\( T^{8} - \)\(20\!\cdots\!84\)\( T^{10} + \)\(32\!\cdots\!48\)\( T^{12} - \)\(28\!\cdots\!32\)\( T^{14} + T^{16} \)
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