Properties

Label 25.34.b.b
Level $25$
Weight $34$
Character orbit 25.b
Analytic conductor $172.457$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(172.457072203\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} + 2 x^{8} + 325405868686 x^{7} + 327807761582101561 x^{6} + 230788916575056333044 x^{5} + 52482256238971860814088 x^{4} - 28868477926481009935781561728 x^{3} + 5099072964771111558047480848223296 x^{2} - 596428985476749456366946929757065984 x + 34881588984360865981321269538813997568\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{10}\cdot 5^{24}\cdot 11^{2} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{2} ) q^{2} + ( 296 \beta_{1} + 472 \beta_{2} - \beta_{6} ) q^{3} + ( -228262272 + \beta_{3} + \beta_{4} ) q^{4} + ( 2585012623152 - 204 \beta_{3} + 2313 \beta_{4} + 112 \beta_{5} + 24 \beta_{7} ) q^{6} + ( -275205168 \beta_{1} - 2087730834 \beta_{2} - 372337 \beta_{6} + 147 \beta_{8} + 49 \beta_{9} ) q^{7} + ( -1828921160 \beta_{1} - 4933673680 \beta_{2} + 1240152 \beta_{6} - 1904 \beta_{8} - 140 \beta_{9} ) q^{8} + ( -2908250198081793 - 641112 \beta_{3} + 1952160 \beta_{4} - 22602 \beta_{5} - 26778 \beta_{7} ) q^{9} +O(q^{10})\) \( q +(-\beta_{1} + \beta_{2}) q^{2} +(296 \beta_{1} + 472 \beta_{2} - \beta_{6}) q^{3} +(-228262272 + \beta_{3} + \beta_{4}) q^{4} +(2585012623152 - 204 \beta_{3} + 2313 \beta_{4} + 112 \beta_{5} + 24 \beta_{7}) q^{6} +(-275205168 \beta_{1} - 2087730834 \beta_{2} - 372337 \beta_{6} + 147 \beta_{8} + 49 \beta_{9}) q^{7} +(-1828921160 \beta_{1} - 4933673680 \beta_{2} + 1240152 \beta_{6} - 1904 \beta_{8} - 140 \beta_{9}) q^{8} +(-2908250198081793 - 641112 \beta_{3} + 1952160 \beta_{4} - 22602 \beta_{5} - 26778 \beta_{7}) q^{9} +(-57560360375595328 + 7637996 \beta_{3} + 40807352 \beta_{4} - 3653091 \beta_{5} - 336003 \beta_{7}) q^{11} +(-948464380378 \beta_{1} - 12949861739948 \beta_{2} - 4880272954 \beta_{6} - 853524 \beta_{8} + 212325 \beta_{9}) q^{12} +(-10002600394128 \beta_{1} - 76456864282961 \beta_{2} + 16868648856 \beta_{6} - 1499798 \beta_{8} + 733006 \beta_{9}) q^{13} +(-2335289983254496464 + 841732668 \beta_{3} - 7746075071 \beta_{4} + 163094736 \beta_{5} - 6035960 \beta_{7}) q^{14} +(-17836017908649191424 + 6547621920 \beta_{3} - 6834298720 \beta_{4} - 1189601280 \beta_{5} - 35471616 \beta_{7}) q^{16} +(1520616954030768 \beta_{1} - 9916893565342971 \beta_{2} + 682028768568 \beta_{6} + 447660418 \beta_{8} + 1199446 \beta_{9}) q^{17} +(-1339839823931919 \beta_{1} - 17178312781175649 \beta_{2} - 4033188898512 \beta_{6} + 1588384512 \beta_{8} - 14326296 \beta_{9}) q^{18} +(-\)\(18\!\cdots\!00\)\( + 51677080452 \beta_{3} - 1240875807125 \beta_{4} - 35228687358 \beta_{5} - 318526969 \beta_{7}) q^{19} +(-\)\(21\!\cdots\!28\)\( - 683931417792 \beta_{3} + 9748997660220 \beta_{4} - 32520248628 \beta_{5} - 20758921680 \beta_{7}) q^{21} +(96643717534241032 \beta_{1} - 442673266738187920 \beta_{2} - 54913184443224 \beta_{6} + 5736943232 \beta_{8} - 1237530004 \beta_{9}) q^{22} +(-73151359548044240 \beta_{1} - 1772691797703336386 \beta_{2} + 42920072839651 \beta_{6} + 159355160675 \beta_{8} + 2258202145 \beta_{9}) q^{23} +(\)\(14\!\cdots\!60\)\( - 294628701840 \beta_{3} - 38033504642928 \beta_{4} + 1327729261056 \beta_{5} + 209711069376 \beta_{7}) q^{24} +(-\)\(84\!\cdots\!68\)\( + 10839062774288 \beta_{3} - 268044264816493 \beta_{4} - 1827696967488 \beta_{5} - 749666738208 \beta_{7}) q^{26} +(-1364712796880073360 \beta_{1} + 21773931807766844088 \beta_{2} + 3738010392824238 \beta_{6} - 854243246196 \beta_{8} - 290365734780 \beta_{9}) q^{27} +(5772692398205055974 \beta_{1} + 43551348924948968404 \beta_{2} - 2477001321794426 \beta_{6} - 124927711636 \beta_{8} - 5712459515 \beta_{9}) q^{28} +(\)\(18\!\cdots\!10\)\( + 45010439372352 \beta_{3} + 773285103602268 \beta_{4} - 9499249501908 \beta_{5} + 1337266709872 \beta_{7}) q^{29} +(-\)\(26\!\cdots\!88\)\( + 433249088292808 \beta_{3} - 2156008927368259 \beta_{4} - 113508163550925 \beta_{5} - 1903411871394 \beta_{7}) q^{31} +(41010175897357266944 \beta_{1} - 27037069370540907520 \beta_{2} + 4857573366895616 \beta_{6} - 21239811566592 \beta_{8} - 2291391179264 \beta_{9}) q^{32} +(-71147209492360387984 \beta_{1} + \)\(10\!\cdots\!80\)\( \beta_{2} + 66699708936493832 \beta_{6} - 45331662836934 \beta_{8} + 861783308094 \beta_{9}) q^{33} +(\)\(13\!\cdots\!56\)\( - 901140812434480 \beta_{3} - 3448198169639975 \beta_{4} + 137204715828672 \beta_{5} - 2226787292064 \beta_{7}) q^{34} +(-\)\(36\!\cdots\!44\)\( - 1493403977156865 \beta_{3} + 37817977320954879 \beta_{4} + 995837560737792 \beta_{5} - 75547401540096 \beta_{7}) q^{36} +(63192466522757491488 \beta_{1} - \)\(45\!\cdots\!59\)\( \beta_{2} + 311049504638197888 \beta_{6} - 26888876097092 \beta_{8} - 10073970220140 \beta_{9}) q^{37} +(\)\(38\!\cdots\!92\)\( \beta_{1} + \)\(10\!\cdots\!44\)\( \beta_{2} - 334137022995902944 \beta_{6} + 201513271412480 \beta_{8} - 29916234761232 \beta_{9}) q^{38} +(\)\(15\!\cdots\!24\)\( - 274687131302256 \beta_{3} + 61158028164159987 \beta_{4} + 2274966667053479 \beta_{5} + 481953507778812 \beta_{7}) q^{39} +(-\)\(26\!\cdots\!58\)\( - 2315959677073112 \beta_{3} - 125921818372203496 \beta_{4} - 6116592205133682 \beta_{5} + 892735108284166 \beta_{7}) q^{41} +(-\)\(22\!\cdots\!20\)\( \beta_{1} - \)\(84\!\cdots\!56\)\( \beta_{2} - 2747234421496061280 \beta_{6} + 815017790909952 \beta_{8} + 230622333284784 \beta_{9}) q^{42} +(\)\(18\!\cdots\!60\)\( \beta_{1} + \)\(74\!\cdots\!72\)\( \beta_{2} + 4681959313565541995 \beta_{6} - 1047042153810838 \beta_{8} + 533601857683662 \beta_{9}) q^{43} +(\)\(37\!\cdots\!16\)\( - 25184464128968000 \beta_{3} + 336847931500254272 \beta_{4} - 23813451407370240 \beta_{5} - 881462448148224 \beta_{7}) q^{44} +(-\)\(57\!\cdots\!68\)\( + 323669043109800108 \beta_{3} + 563377032131941853 \beta_{4} + 73157007099848208 \beta_{5} + 3265031938168616 \beta_{7}) q^{46} +(\)\(72\!\cdots\!28\)\( \beta_{1} - \)\(24\!\cdots\!10\)\( \beta_{2} + 2182816640506653395 \beta_{6} - 6224958331986491 \beta_{8} + 1338590365431959 \beta_{9}) q^{47} +(-\)\(20\!\cdots\!00\)\( \beta_{1} + \)\(23\!\cdots\!52\)\( \beta_{2} - 10280798860322047808 \beta_{6} - 13279027668066432 \beta_{8} + 1861286374163616 \beta_{9}) q^{48} +(-\)\(82\!\cdots\!37\)\( + 287587336313087608 \beta_{3} - 7473718588673269072 \beta_{4} - 87975239542139502 \beta_{5} - 19428477831767214 \beta_{7}) q^{49} +(\)\(14\!\cdots\!52\)\( + 1329935351003519568 \beta_{3} + 3179061303460659033 \beta_{4} - 756831063763060787 \beta_{5} - 48752164266519156 \beta_{7}) q^{51} +(\)\(57\!\cdots\!94\)\( \beta_{1} + \)\(15\!\cdots\!96\)\( \beta_{2} + 29317667051785437978 \beta_{6} + 17499789353194324 \beta_{8} - 6524009430505301 \beta_{9}) q^{52} +(\)\(37\!\cdots\!52\)\( \beta_{1} - \)\(35\!\cdots\!69\)\( \beta_{2} + \)\(29\!\cdots\!24\)\( \beta_{6} - 4198948412841354 \beta_{8} + 8808894819001170 \beta_{9}) q^{53} +(-\)\(12\!\cdots\!00\)\( - 2107942576365640248 \beta_{3} + 52783704804866857650 \beta_{4} - 1130017658502633120 \beta_{5} - 439765913802384 \beta_{7}) q^{54} +(\)\(28\!\cdots\!60\)\( + 1461129436904112752 \beta_{3} - 19599873059767544432 \beta_{4} + 1613119783531952640 \beta_{5} + 5823257790116032 \beta_{7}) q^{56} +(-\)\(40\!\cdots\!92\)\( \beta_{1} + \)\(66\!\cdots\!32\)\( \beta_{2} - 98940298156134052712 \beta_{6} - 166115566651863798 \beta_{8} + 42479107518861102 \beta_{9}) q^{57} +(\)\(97\!\cdots\!62\)\( \beta_{1} - \)\(67\!\cdots\!22\)\( \beta_{2} + \)\(21\!\cdots\!04\)\( \beta_{6} - 140229452754510336 \beta_{8} + 23879905529100592 \beta_{9}) q^{58} +(-\)\(51\!\cdots\!60\)\( + 4120978175839918492 \beta_{3} + 20814545047020237471 \beta_{4} - 3062949321648469872 \beta_{5} + 547033541408337905 \beta_{7}) q^{59} +(\)\(14\!\cdots\!82\)\( - 1716415096131067200 \beta_{3} + \)\(19\!\cdots\!72\)\( \beta_{4} - 5708205314743964304 \beta_{5} + 1217578806810299600 \beta_{7}) q^{61} +(\)\(27\!\cdots\!48\)\( \beta_{1} + \)\(16\!\cdots\!16\)\( \beta_{2} - \)\(71\!\cdots\!16\)\( \beta_{6} + 946494519587712 \beta_{8} - 100129685110676836 \beta_{9}) q^{62} +(-\)\(28\!\cdots\!16\)\( \beta_{1} + \)\(32\!\cdots\!78\)\( \beta_{2} + \)\(69\!\cdots\!97\)\( \beta_{6} - 507458377815291063 \beta_{8} - 121035445173046845 \beta_{9}) q^{63} +(\)\(20\!\cdots\!68\)\( - 31752890677639058432 \beta_{3} - 52938654476587062272 \beta_{4} - 23253035821101809664 \beta_{5} - 190495590188515328 \beta_{7}) q^{64} +(-\)\(66\!\cdots\!56\)\( + 2162057533246071504 \beta_{3} - 20004590752383147216 \beta_{4} - 28062609359844562496 \beta_{5} - 3429657268765039008 \beta_{7}) q^{66} +(-\)\(58\!\cdots\!76\)\( \beta_{1} - \)\(12\!\cdots\!08\)\( \beta_{2} + \)\(22\!\cdots\!75\)\( \beta_{6} + 896124675862549770 \beta_{8} - 46440558273674322 \beta_{9}) q^{67} +(-\)\(61\!\cdots\!98\)\( \beta_{1} - \)\(37\!\cdots\!80\)\( \beta_{2} + \)\(52\!\cdots\!98\)\( \beta_{6} + 5218748848455176444 \beta_{8} + 12424970658968385 \beta_{9}) q^{68} +(\)\(55\!\cdots\!04\)\( + \)\(18\!\cdots\!16\)\( \beta_{3} + \)\(29\!\cdots\!56\)\( \beta_{4} - \)\(16\!\cdots\!00\)\( \beta_{5} - 6977086754652503016 \beta_{7}) q^{69} +(-\)\(11\!\cdots\!88\)\( - \)\(40\!\cdots\!00\)\( \beta_{3} - \)\(37\!\cdots\!37\)\( \beta_{4} - 32122236274030979097 \beta_{5} - 5349512571233196200 \beta_{7}) q^{71} +(\)\(18\!\cdots\!08\)\( \beta_{1} - \)\(51\!\cdots\!04\)\( \beta_{2} - \)\(35\!\cdots\!88\)\( \beta_{6} + 9379213566592232304 \beta_{8} + 239403407195746188 \beta_{9}) q^{72} +(\)\(89\!\cdots\!36\)\( \beta_{1} + \)\(43\!\cdots\!31\)\( \beta_{2} + \)\(43\!\cdots\!64\)\( \beta_{6} + 5961738642705263070 \beta_{8} - 2938521848138564662 \beta_{9}) q^{73} +(\)\(72\!\cdots\!16\)\( - \)\(19\!\cdots\!48\)\( \beta_{3} - \)\(37\!\cdots\!47\)\( \beta_{4} - 58214007309090036864 \beta_{5} - 4278169457359479616 \beta_{7}) q^{74} +(\)\(14\!\cdots\!20\)\( + \)\(19\!\cdots\!44\)\( \beta_{3} + \)\(95\!\cdots\!80\)\( \beta_{4} - \)\(20\!\cdots\!60\)\( \beta_{5} + 23940239356113081344 \beta_{7}) q^{76} +(\)\(22\!\cdots\!24\)\( \beta_{1} + \)\(15\!\cdots\!92\)\( \beta_{2} + \)\(56\!\cdots\!44\)\( \beta_{6} - 30857876302129389422 \beta_{8} - 2573913234233619194 \beta_{9}) q^{77} +(-\)\(15\!\cdots\!20\)\( \beta_{1} - \)\(38\!\cdots\!76\)\( \beta_{2} + \)\(80\!\cdots\!32\)\( \beta_{6} - 25626338150852511360 \beta_{8} + 3992301843394655676 \beta_{9}) q^{78} +(\)\(20\!\cdots\!40\)\( - \)\(30\!\cdots\!36\)\( \beta_{3} + \)\(45\!\cdots\!46\)\( \beta_{4} - \)\(31\!\cdots\!50\)\( \beta_{5} + \)\(11\!\cdots\!36\)\( \beta_{7}) q^{79} +(\)\(15\!\cdots\!01\)\( + \)\(97\!\cdots\!44\)\( \beta_{3} - \)\(46\!\cdots\!96\)\( \beta_{4} + \)\(56\!\cdots\!18\)\( \beta_{5} + 75223021076941246470 \beta_{7}) q^{81} +(\)\(23\!\cdots\!86\)\( \beta_{1} + \)\(86\!\cdots\!74\)\( \beta_{2} + \)\(53\!\cdots\!60\)\( \beta_{6} + 27935516539281854208 \beta_{8} + 5153144720205861960 \beta_{9}) q^{82} +(\)\(30\!\cdots\!52\)\( \beta_{1} - \)\(29\!\cdots\!52\)\( \beta_{2} - \)\(52\!\cdots\!33\)\( \beta_{6} - 43033318367416525320 \beta_{8} + 4613264806019047080 \beta_{9}) q^{83} +(-\)\(34\!\cdots\!44\)\( - \)\(65\!\cdots\!48\)\( \beta_{3} - \)\(23\!\cdots\!00\)\( \beta_{4} + \)\(65\!\cdots\!56\)\( \beta_{5} - \)\(17\!\cdots\!08\)\( \beta_{7}) q^{84} +(\)\(13\!\cdots\!12\)\( - \)\(44\!\cdots\!20\)\( \beta_{3} - \)\(49\!\cdots\!15\)\( \beta_{4} - \)\(45\!\cdots\!40\)\( \beta_{5} - \)\(36\!\cdots\!32\)\( \beta_{7}) q^{86} +(\)\(19\!\cdots\!84\)\( \beta_{1} + \)\(43\!\cdots\!68\)\( \beta_{2} - \)\(44\!\cdots\!30\)\( \beta_{6} - \)\(10\!\cdots\!16\)\( \beta_{8} + 7359902008807383216 \beta_{9}) q^{87} +(\)\(22\!\cdots\!68\)\( \beta_{1} - \)\(62\!\cdots\!56\)\( \beta_{2} - \)\(74\!\cdots\!60\)\( \beta_{6} + \)\(18\!\cdots\!88\)\( \beta_{8} + 4068256326594220416 \beta_{9}) q^{88} +(\)\(12\!\cdots\!30\)\( + \)\(42\!\cdots\!76\)\( \beta_{3} - \)\(71\!\cdots\!44\)\( \beta_{4} + \)\(59\!\cdots\!32\)\( \beta_{5} - 10026730679801932284 \beta_{7}) q^{89} +(-\)\(53\!\cdots\!68\)\( + \)\(19\!\cdots\!12\)\( \beta_{3} - \)\(37\!\cdots\!09\)\( \beta_{4} + \)\(16\!\cdots\!05\)\( \beta_{5} + 84176942221259978622 \beta_{7}) q^{91} +(\)\(22\!\cdots\!06\)\( \beta_{1} - \)\(22\!\cdots\!64\)\( \beta_{2} + \)\(17\!\cdots\!22\)\( \beta_{6} + \)\(32\!\cdots\!08\)\( \beta_{8} - 22771145512410814095 \beta_{9}) q^{92} +(-\)\(13\!\cdots\!20\)\( \beta_{1} + \)\(22\!\cdots\!48\)\( \beta_{2} - \)\(17\!\cdots\!48\)\( \beta_{6} - \)\(93\!\cdots\!80\)\( \beta_{8} + \)\(16\!\cdots\!24\)\( \beta_{9}) q^{93} +(\)\(72\!\cdots\!56\)\( - \)\(82\!\cdots\!08\)\( \beta_{3} - \)\(60\!\cdots\!99\)\( \beta_{4} - \)\(17\!\cdots\!40\)\( \beta_{5} - \)\(79\!\cdots\!44\)\( \beta_{7}) q^{94} +(-\)\(68\!\cdots\!68\)\( + \)\(11\!\cdots\!60\)\( \beta_{3} - \)\(64\!\cdots\!32\)\( \beta_{4} + \)\(82\!\cdots\!56\)\( \beta_{5} + \)\(86\!\cdots\!92\)\( \beta_{7}) q^{96} +(\)\(39\!\cdots\!60\)\( \beta_{1} - \)\(27\!\cdots\!07\)\( \beta_{2} - \)\(31\!\cdots\!24\)\( \beta_{6} - \)\(67\!\cdots\!06\)\( \beta_{8} + \)\(15\!\cdots\!70\)\( \beta_{9}) q^{97} +(\)\(22\!\cdots\!45\)\( \beta_{1} + \)\(63\!\cdots\!99\)\( \beta_{2} - \)\(33\!\cdots\!80\)\( \beta_{6} + \)\(10\!\cdots\!68\)\( \beta_{8} - \)\(33\!\cdots\!00\)\( \beta_{9}) q^{98} +(\)\(35\!\cdots\!04\)\( + \)\(89\!\cdots\!08\)\( \beta_{3} - \)\(61\!\cdots\!24\)\( \beta_{4} + \)\(54\!\cdots\!55\)\( \beta_{5} + \)\(35\!\cdots\!23\)\( \beta_{7}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 2282622720q^{4} + 25850126231520q^{6} - 29082501980817930q^{9} + O(q^{10}) \) \( 10q - 2282622720q^{4} + 25850126231520q^{6} - 29082501980817930q^{9} - 575603603755953280q^{11} - 23352899832544964640q^{14} - \)\(17\!\cdots\!40\)\(q^{16} - \)\(18\!\cdots\!00\)\(q^{19} - \)\(21\!\cdots\!80\)\(q^{21} + \)\(14\!\cdots\!00\)\(q^{24} - \)\(84\!\cdots\!80\)\(q^{26} + \)\(18\!\cdots\!00\)\(q^{29} - \)\(26\!\cdots\!80\)\(q^{31} + \)\(13\!\cdots\!60\)\(q^{34} - \)\(36\!\cdots\!40\)\(q^{36} + \)\(15\!\cdots\!40\)\(q^{39} - \)\(26\!\cdots\!80\)\(q^{41} + \)\(37\!\cdots\!60\)\(q^{44} - \)\(57\!\cdots\!80\)\(q^{46} - \)\(82\!\cdots\!70\)\(q^{49} + \)\(14\!\cdots\!20\)\(q^{51} - \)\(12\!\cdots\!00\)\(q^{54} + \)\(28\!\cdots\!00\)\(q^{56} - \)\(51\!\cdots\!00\)\(q^{59} + \)\(14\!\cdots\!20\)\(q^{61} + \)\(20\!\cdots\!80\)\(q^{64} - \)\(66\!\cdots\!60\)\(q^{66} + \)\(55\!\cdots\!40\)\(q^{69} - \)\(11\!\cdots\!80\)\(q^{71} + \)\(72\!\cdots\!60\)\(q^{74} + \)\(14\!\cdots\!00\)\(q^{76} + \)\(20\!\cdots\!00\)\(q^{79} + \)\(15\!\cdots\!10\)\(q^{81} - \)\(34\!\cdots\!40\)\(q^{84} + \)\(13\!\cdots\!20\)\(q^{86} + \)\(12\!\cdots\!00\)\(q^{89} - \)\(53\!\cdots\!80\)\(q^{91} + \)\(72\!\cdots\!60\)\(q^{94} - \)\(68\!\cdots\!80\)\(q^{96} + \)\(35\!\cdots\!40\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} + 2 x^{8} + 325405868686 x^{7} + 327807761582101561 x^{6} + 230788916575056333044 x^{5} + 52482256238971860814088 x^{4} - 28868477926481009935781561728 x^{3} + 5099072964771111558047480848223296 x^{2} - 596428985476749456366946929757065984 x + 34881588984360865981321269538813997568\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(25\!\cdots\!39\)\( \nu^{9} - \)\(51\!\cdots\!52\)\( \nu^{8} - \)\(15\!\cdots\!94\)\( \nu^{7} - \)\(80\!\cdots\!30\)\( \nu^{6} - \)\(83\!\cdots\!63\)\( \nu^{5} - \)\(81\!\cdots\!98\)\( \nu^{4} - \)\(63\!\cdots\!00\)\( \nu^{3} + \)\(46\!\cdots\!36\)\( \nu^{2} - \)\(20\!\cdots\!92\)\( \nu + \)\(12\!\cdots\!56\)\(\)\()/ \)\(19\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(16\!\cdots\!35\)\( \nu^{9} - \)\(92\!\cdots\!20\)\( \nu^{8} - \)\(28\!\cdots\!50\)\( \nu^{7} - \)\(54\!\cdots\!10\)\( \nu^{6} - \)\(53\!\cdots\!75\)\( \nu^{5} - \)\(40\!\cdots\!90\)\( \nu^{4} - \)\(12\!\cdots\!40\)\( \nu^{3} + \)\(43\!\cdots\!20\)\( \nu^{2} - \)\(83\!\cdots\!80\)\( \nu + \)\(48\!\cdots\!20\)\(\)\()/ \)\(78\!\cdots\!76\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(18\!\cdots\!05\)\( \nu^{9} + \)\(42\!\cdots\!14\)\( \nu^{8} - \)\(39\!\cdots\!46\)\( \nu^{7} - \)\(50\!\cdots\!14\)\( \nu^{6} - \)\(45\!\cdots\!97\)\( \nu^{5} + \)\(16\!\cdots\!72\)\( \nu^{4} - \)\(60\!\cdots\!76\)\( \nu^{3} + \)\(18\!\cdots\!92\)\( \nu^{2} - \)\(17\!\cdots\!20\)\( \nu + \)\(57\!\cdots\!92\)\(\)\()/ \)\(65\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(12\!\cdots\!55\)\( \nu^{9} - \)\(38\!\cdots\!10\)\( \nu^{8} - \)\(13\!\cdots\!50\)\( \nu^{7} - \)\(39\!\cdots\!30\)\( \nu^{6} - \)\(42\!\cdots\!75\)\( \nu^{5} - \)\(46\!\cdots\!20\)\( \nu^{4} - \)\(53\!\cdots\!20\)\( \nu^{3} + \)\(16\!\cdots\!60\)\( \nu^{2} - \)\(15\!\cdots\!40\)\( \nu + \)\(76\!\cdots\!60\)\(\)\()/ \)\(26\!\cdots\!88\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(34\!\cdots\!51\)\( \nu^{9} - \)\(30\!\cdots\!93\)\( \nu^{8} + \)\(25\!\cdots\!40\)\( \nu^{7} + \)\(54\!\cdots\!96\)\( \nu^{6} - \)\(34\!\cdots\!75\)\( \nu^{5} - \)\(97\!\cdots\!31\)\( \nu^{4} - \)\(10\!\cdots\!16\)\( \nu^{3} + \)\(16\!\cdots\!28\)\( \nu^{2} - \)\(12\!\cdots\!52\)\( \nu - \)\(83\!\cdots\!52\)\(\)\()/ \)\(27\!\cdots\!24\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(69\!\cdots\!07\)\( \nu^{9} + \)\(39\!\cdots\!92\)\( \nu^{8} + \)\(61\!\cdots\!58\)\( \nu^{7} + \)\(22\!\cdots\!34\)\( \nu^{6} + \)\(22\!\cdots\!71\)\( \nu^{5} + \)\(17\!\cdots\!02\)\( \nu^{4} + \)\(61\!\cdots\!96\)\( \nu^{3} - \)\(21\!\cdots\!80\)\( \nu^{2} + \)\(35\!\cdots\!76\)\( \nu - \)\(20\!\cdots\!40\)\(\)\()/ \)\(17\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(86\!\cdots\!51\)\( \nu^{9} + \)\(60\!\cdots\!10\)\( \nu^{8} + \)\(11\!\cdots\!78\)\( \nu^{7} + \)\(26\!\cdots\!98\)\( \nu^{6} + \)\(27\!\cdots\!91\)\( \nu^{5} + \)\(21\!\cdots\!68\)\( \nu^{4} + \)\(34\!\cdots\!52\)\( \nu^{3} - \)\(10\!\cdots\!68\)\( \nu^{2} + \)\(99\!\cdots\!88\)\( \nu + \)\(18\!\cdots\!52\)\(\)\()/ \)\(22\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(15\!\cdots\!17\)\( \nu^{9} + \)\(83\!\cdots\!60\)\( \nu^{8} - \)\(79\!\cdots\!34\)\( \nu^{7} + \)\(49\!\cdots\!26\)\( \nu^{6} + \)\(49\!\cdots\!77\)\( \nu^{5} + \)\(37\!\cdots\!26\)\( \nu^{4} - \)\(13\!\cdots\!76\)\( \nu^{3} - \)\(43\!\cdots\!36\)\( \nu^{2} + \)\(76\!\cdots\!96\)\( \nu - \)\(44\!\cdots\!96\)\(\)\()/ \)\(17\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(89\!\cdots\!81\)\( \nu^{9} - \)\(51\!\cdots\!88\)\( \nu^{8} - \)\(37\!\cdots\!06\)\( \nu^{7} - \)\(29\!\cdots\!90\)\( \nu^{6} - \)\(29\!\cdots\!37\)\( \nu^{5} - \)\(22\!\cdots\!82\)\( \nu^{4} - \)\(17\!\cdots\!80\)\( \nu^{3} + \)\(24\!\cdots\!04\)\( \nu^{2} - \)\(45\!\cdots\!68\)\( \nu + \)\(26\!\cdots\!84\)\(\)\()/ \)\(17\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + 154 \beta_{2} - 6250 \beta_{1} + 10000\)\()/50000\)
\(\nu^{2}\)\(=\)\((\)\(5 \beta_{9} - 20 \beta_{8} + 17030 \beta_{6} - 1756118124 \beta_{2} - 3547610 \beta_{1}\)\()/40000\)
\(\nu^{3}\)\(=\)\((\)\(166525 \beta_{9} + 1258900 \beta_{8} - 1925000 \beta_{7} - 774215850 \beta_{6} - 3825600 \beta_{5} - 1914434794 \beta_{4} + 6981250 \beta_{3} - 2826687701636 \beta_{2} - 11946380726050 \beta_{1} - 19524352121240000\)\()/200000\)
\(\nu^{4}\)\(=\)\((\)\(-480675000 \beta_{7} + 20902314240 \beta_{5} - 528932034229 \beta_{4} + 375334179375 \beta_{3} - 2622467299150711480000\)\()/20000\)
\(\nu^{5}\)\(=\)\((\)\(-59152625828125 \beta_{9} - 423686085575000 \beta_{8} - 660296588887500 \beta_{7} + 258614078274706250 \beta_{6} - 1312114416271200 \beta_{5} - 514338624578034728 \beta_{4} + 2902817857056250 \beta_{3} + 1771794935374910263008 \beta_{2} + 3206577683020171168750 \beta_{1} - 11572226637451613676980000\)\()/100000\)
\(\nu^{6}\)\(=\)\((\)\(-1761178509147482735 \beta_{9} + 6259927512633570940 \beta_{8} - 12949075190939983315410 \beta_{6} + 450408130755598542286560388 \beta_{2} + 3450987268211616280582670 \beta_{1}\)\()/40000\)
\(\nu^{7}\)\(=\)\((\)\(-71870280557364120667275 \beta_{9} - 481794817461164087505900 \beta_{8} + 769275939690620570175000 \beta_{7} + 282337062242260096645861350 \beta_{6} + 1493088199530072710414400 \beta_{5} + 569873167739347938496778006 \beta_{4} - 4120603224909859903668750 \beta_{3} + 2947772588393460661452829793404 \beta_{2} + 3549986725369255533612843217550 \beta_{1} + 18975481517718094164712433194760000\)\()/200000\)
\(\nu^{8}\)\(=\)\((\)\(279419043560573776468605000 \beta_{7} - 8170600444362900368230187520 \beta_{5} + 262085122381808450821234461007 \beta_{4} - 119573024416219057963737498125 \beta_{3} + 779287612790339035414136441640768520000\)\()/20000\)
\(\nu^{9}\)\(=\)\((\)\(21176180070516978960678126885625 \beta_{9} + 132343577428546563129466829320000 \beta_{8} + 217048297710614478972179336862500 \beta_{7} - 72897674488338678543450247539461250 \beta_{6} + 406456337065586452356336321357600 \beta_{5} + 159069444366038618615378228546438692 \beta_{4} - 1387175658838184118670303657881250 \beta_{3} - 1076413462057806295700388866824933410392 \beta_{2} - 990154165526731056633145893233925808750 \beta_{1} + 6881529469577814990786420999384362469020000\)\()/100000\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
−16501.2 16501.2i
16897.1 16897.1i
−8230.22 8230.22i
7776.84 7776.84i
58.5034 + 58.5034i
58.5034 58.5034i
7776.84 + 7776.84i
−8230.22 + 8230.22i
16897.1 + 16897.1i
−16501.2 + 16501.2i
138106.i 1.45282e7i −1.04832e10 0 −2.00642e12 3.75584e13i 2.61472e14i 5.34799e15 0
24.2 129081.i 6.82881e7i −8.07187e9 0 8.81467e12 1.39338e14i 6.68716e13i 8.95800e14 0
24.3 71937.8i 1.37573e8i 3.41489e9 0 9.89671e12 1.07684e14i 8.63600e14i −1.33673e16 0
24.4 56118.7i 5.48591e7i 5.44063e9 0 −3.07862e12 7.38557e13i 7.87377e14i 2.54954e15 0
24.5 5627.97i 1.24605e8i 8.55826e9 0 −7.01272e11 7.01560e13i 9.65096e13i −9.96727e15 0
24.6 5627.97i 1.24605e8i 8.55826e9 0 −7.01272e11 7.01560e13i 9.65096e13i −9.96727e15 0
24.7 56118.7i 5.48591e7i 5.44063e9 0 −3.07862e12 7.38557e13i 7.87377e14i 2.54954e15 0
24.8 71937.8i 1.37573e8i 3.41489e9 0 9.89671e12 1.07684e14i 8.63600e14i −1.33673e16 0
24.9 129081.i 6.82881e7i −8.07187e9 0 8.81467e12 1.39338e14i 6.68716e13i 8.95800e14 0
24.10 138106.i 1.45282e7i −1.04832e10 0 −2.00642e12 3.75584e13i 2.61472e14i 5.34799e15 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.10
Significant digits:
Format:

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.b.b 10
5.b even 2 1 inner 25.34.b.b 10
5.c odd 4 1 5.34.a.a 5
5.c odd 4 1 25.34.a.b 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.34.a.a 5 5.c odd 4 1
25.34.a.b 5 5.c odd 4 1
25.34.b.b 10 1.a even 1 1 trivial
25.34.b.b 10 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 44090984320 T_{2}^{8} + \)632956927553415029760

'>\(63\!\cdots\!60\)\( T_{2}^{6} + \)3247828667956387957668094935040
'>\(32\!\cdots\!40\)\( T_{2}^{4} + \)5281572959719960176307049186292379156480'>\(52\!\cdots\!80\)\( T_{2}^{2} + \)164050629092480295701242997169331061238502785024'>\(16\!\cdots\!24\)\( \) acting on \(S_{34}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 41808361600 T^{2} + \)\(92\!\cdots\!20\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{6} + \)\(17\!\cdots\!60\)\( T^{8} - \)\(16\!\cdots\!00\)\( T^{10} + \)\(12\!\cdots\!40\)\( T^{12} - \)\(78\!\cdots\!00\)\( T^{14} + \)\(37\!\cdots\!80\)\( T^{16} - \)\(12\!\cdots\!00\)\( T^{18} + \)\(21\!\cdots\!24\)\( T^{20} \)
$3$ \( 1 - 13254051842368650 T^{2} + \)\(88\!\cdots\!45\)\( T^{4} - \)\(50\!\cdots\!00\)\( T^{6} + \)\(37\!\cdots\!10\)\( T^{8} - \)\(25\!\cdots\!00\)\( T^{10} + \)\(11\!\cdots\!90\)\( T^{12} - \)\(48\!\cdots\!00\)\( T^{14} + \)\(26\!\cdots\!05\)\( T^{16} - \)\(12\!\cdots\!50\)\( T^{18} + \)\(28\!\cdots\!49\)\( T^{20} \)
$5$ 1
$7$ \( 1 - \)\(34\!\cdots\!50\)\( T^{2} + \)\(67\!\cdots\!45\)\( T^{4} - \)\(91\!\cdots\!00\)\( T^{6} + \)\(96\!\cdots\!10\)\( T^{8} - \)\(82\!\cdots\!00\)\( T^{10} + \)\(57\!\cdots\!90\)\( T^{12} - \)\(32\!\cdots\!00\)\( T^{14} + \)\(14\!\cdots\!05\)\( T^{16} - \)\(44\!\cdots\!50\)\( T^{18} + \)\(76\!\cdots\!49\)\( T^{20} \)
$11$ \( ( 1 + 287801801877976640 T + \)\(11\!\cdots\!95\)\( T^{2} + \)\(23\!\cdots\!80\)\( T^{3} + \)\(53\!\cdots\!10\)\( T^{4} + \)\(78\!\cdots\!48\)\( T^{5} + \)\(12\!\cdots\!10\)\( T^{6} + \)\(12\!\cdots\!80\)\( T^{7} + \)\(14\!\cdots\!45\)\( T^{8} + \)\(83\!\cdots\!40\)\( T^{9} + \)\(67\!\cdots\!51\)\( T^{10} )^{2} \)
$13$ \( 1 - \)\(33\!\cdots\!50\)\( T^{2} + \)\(55\!\cdots\!45\)\( T^{4} - \)\(59\!\cdots\!00\)\( T^{6} + \)\(46\!\cdots\!10\)\( T^{8} - \)\(29\!\cdots\!00\)\( T^{10} + \)\(15\!\cdots\!90\)\( T^{12} - \)\(65\!\cdots\!00\)\( T^{14} + \)\(20\!\cdots\!05\)\( T^{16} - \)\(40\!\cdots\!50\)\( T^{18} + \)\(39\!\cdots\!49\)\( T^{20} \)
$17$ \( 1 - \)\(20\!\cdots\!50\)\( T^{2} + \)\(24\!\cdots\!45\)\( T^{4} - \)\(19\!\cdots\!00\)\( T^{6} + \)\(11\!\cdots\!10\)\( T^{8} - \)\(52\!\cdots\!00\)\( T^{10} + \)\(18\!\cdots\!90\)\( T^{12} - \)\(50\!\cdots\!00\)\( T^{14} + \)\(10\!\cdots\!05\)\( T^{16} - \)\(14\!\cdots\!50\)\( T^{18} + \)\(11\!\cdots\!49\)\( T^{20} \)
$19$ \( ( 1 + \)\(91\!\cdots\!00\)\( T + \)\(53\!\cdots\!95\)\( T^{2} + \)\(37\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!10\)\( T^{4} + \)\(82\!\cdots\!00\)\( T^{5} + \)\(22\!\cdots\!90\)\( T^{6} + \)\(93\!\cdots\!00\)\( T^{7} + \)\(21\!\cdots\!05\)\( T^{8} + \)\(57\!\cdots\!00\)\( T^{9} + \)\(98\!\cdots\!99\)\( T^{10} )^{2} \)
$23$ \( 1 - \)\(31\!\cdots\!50\)\( T^{2} + \)\(17\!\cdots\!45\)\( T^{4} - \)\(36\!\cdots\!00\)\( T^{6} + \)\(18\!\cdots\!10\)\( T^{8} - \)\(31\!\cdots\!00\)\( T^{10} + \)\(13\!\cdots\!90\)\( T^{12} - \)\(20\!\cdots\!00\)\( T^{14} + \)\(73\!\cdots\!05\)\( T^{16} - \)\(98\!\cdots\!50\)\( T^{18} + \)\(23\!\cdots\!49\)\( T^{20} \)
$29$ \( ( 1 - \)\(92\!\cdots\!50\)\( T + \)\(83\!\cdots\!45\)\( T^{2} - \)\(59\!\cdots\!00\)\( T^{3} + \)\(28\!\cdots\!10\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{5} + \)\(52\!\cdots\!90\)\( T^{6} - \)\(19\!\cdots\!00\)\( T^{7} + \)\(50\!\cdots\!05\)\( T^{8} - \)\(10\!\cdots\!50\)\( T^{9} + \)\(19\!\cdots\!49\)\( T^{10} )^{2} \)
$31$ \( ( 1 + \)\(13\!\cdots\!40\)\( T + \)\(43\!\cdots\!95\)\( T^{2} - \)\(16\!\cdots\!20\)\( T^{3} + \)\(70\!\cdots\!10\)\( T^{4} - \)\(12\!\cdots\!52\)\( T^{5} + \)\(11\!\cdots\!10\)\( T^{6} - \)\(44\!\cdots\!20\)\( T^{7} + \)\(19\!\cdots\!45\)\( T^{8} + \)\(96\!\cdots\!40\)\( T^{9} + \)\(11\!\cdots\!51\)\( T^{10} )^{2} \)
$37$ \( 1 - \)\(46\!\cdots\!50\)\( T^{2} + \)\(10\!\cdots\!45\)\( T^{4} - \)\(13\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!10\)\( T^{8} - \)\(85\!\cdots\!00\)\( T^{10} + \)\(40\!\cdots\!90\)\( T^{12} - \)\(14\!\cdots\!00\)\( T^{14} + \)\(32\!\cdots\!05\)\( T^{16} - \)\(47\!\cdots\!50\)\( T^{18} + \)\(32\!\cdots\!49\)\( T^{20} \)
$41$ \( ( 1 + \)\(13\!\cdots\!90\)\( T + \)\(13\!\cdots\!45\)\( T^{2} + \)\(98\!\cdots\!80\)\( T^{3} + \)\(56\!\cdots\!10\)\( T^{4} + \)\(25\!\cdots\!48\)\( T^{5} + \)\(94\!\cdots\!10\)\( T^{6} + \)\(27\!\cdots\!80\)\( T^{7} + \)\(64\!\cdots\!45\)\( T^{8} + \)\(10\!\cdots\!90\)\( T^{9} + \)\(12\!\cdots\!01\)\( T^{10} )^{2} \)
$43$ \( 1 - \)\(21\!\cdots\!50\)\( T^{2} + \)\(27\!\cdots\!45\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{6} + \)\(76\!\cdots\!10\)\( T^{8} + \)\(59\!\cdots\!00\)\( T^{10} + \)\(49\!\cdots\!90\)\( T^{12} - \)\(61\!\cdots\!00\)\( T^{14} + \)\(72\!\cdots\!05\)\( T^{16} - \)\(36\!\cdots\!50\)\( T^{18} + \)\(11\!\cdots\!49\)\( T^{20} \)
$47$ \( 1 - \)\(10\!\cdots\!50\)\( T^{2} + \)\(49\!\cdots\!45\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{6} + \)\(34\!\cdots\!10\)\( T^{8} - \)\(59\!\cdots\!00\)\( T^{10} + \)\(78\!\cdots\!90\)\( T^{12} - \)\(80\!\cdots\!00\)\( T^{14} + \)\(59\!\cdots\!05\)\( T^{16} - \)\(27\!\cdots\!50\)\( T^{18} + \)\(61\!\cdots\!49\)\( T^{20} \)
$53$ \( 1 - \)\(11\!\cdots\!50\)\( T^{2} + \)\(12\!\cdots\!45\)\( T^{4} - \)\(16\!\cdots\!00\)\( T^{6} + \)\(14\!\cdots\!10\)\( T^{8} - \)\(99\!\cdots\!00\)\( T^{10} + \)\(90\!\cdots\!90\)\( T^{12} - \)\(64\!\cdots\!00\)\( T^{14} + \)\(33\!\cdots\!05\)\( T^{16} - \)\(18\!\cdots\!50\)\( T^{18} + \)\(10\!\cdots\!49\)\( T^{20} \)
$59$ \( ( 1 + \)\(25\!\cdots\!00\)\( T + \)\(12\!\cdots\!95\)\( T^{2} + \)\(24\!\cdots\!00\)\( T^{3} + \)\(62\!\cdots\!10\)\( T^{4} + \)\(93\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!90\)\( T^{6} + \)\(18\!\cdots\!00\)\( T^{7} + \)\(24\!\cdots\!05\)\( T^{8} + \)\(14\!\cdots\!00\)\( T^{9} + \)\(15\!\cdots\!99\)\( T^{10} )^{2} \)
$61$ \( ( 1 - \)\(74\!\cdots\!10\)\( T + \)\(40\!\cdots\!45\)\( T^{2} - \)\(17\!\cdots\!20\)\( T^{3} + \)\(62\!\cdots\!10\)\( T^{4} - \)\(18\!\cdots\!52\)\( T^{5} + \)\(51\!\cdots\!10\)\( T^{6} - \)\(11\!\cdots\!20\)\( T^{7} + \)\(22\!\cdots\!45\)\( T^{8} - \)\(34\!\cdots\!10\)\( T^{9} + \)\(37\!\cdots\!01\)\( T^{10} )^{2} \)
$67$ \( 1 - \)\(13\!\cdots\!50\)\( T^{2} + \)\(86\!\cdots\!45\)\( T^{4} - \)\(35\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!10\)\( T^{8} - \)\(21\!\cdots\!00\)\( T^{10} + \)\(34\!\cdots\!90\)\( T^{12} - \)\(39\!\cdots\!00\)\( T^{14} + \)\(31\!\cdots\!05\)\( T^{16} - \)\(16\!\cdots\!50\)\( T^{18} + \)\(40\!\cdots\!49\)\( T^{20} \)
$71$ \( ( 1 + \)\(55\!\cdots\!40\)\( T + \)\(31\!\cdots\!95\)\( T^{2} - \)\(69\!\cdots\!20\)\( T^{3} - \)\(24\!\cdots\!90\)\( T^{4} - \)\(22\!\cdots\!52\)\( T^{5} - \)\(29\!\cdots\!90\)\( T^{6} - \)\(10\!\cdots\!20\)\( T^{7} + \)\(59\!\cdots\!45\)\( T^{8} + \)\(12\!\cdots\!40\)\( T^{9} + \)\(28\!\cdots\!51\)\( T^{10} )^{2} \)
$73$ \( 1 - \)\(80\!\cdots\!50\)\( T^{2} + \)\(48\!\cdots\!45\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{6} + \)\(94\!\cdots\!10\)\( T^{8} - \)\(31\!\cdots\!00\)\( T^{10} + \)\(90\!\cdots\!90\)\( T^{12} - \)\(21\!\cdots\!00\)\( T^{14} + \)\(42\!\cdots\!05\)\( T^{16} - \)\(66\!\cdots\!50\)\( T^{18} + \)\(78\!\cdots\!49\)\( T^{20} \)
$79$ \( ( 1 - \)\(10\!\cdots\!00\)\( T + \)\(74\!\cdots\!95\)\( T^{2} + \)\(86\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!10\)\( T^{4} + \)\(82\!\cdots\!00\)\( T^{5} + \)\(51\!\cdots\!90\)\( T^{6} + \)\(15\!\cdots\!00\)\( T^{7} + \)\(54\!\cdots\!05\)\( T^{8} - \)\(31\!\cdots\!00\)\( T^{9} + \)\(12\!\cdots\!99\)\( T^{10} )^{2} \)
$83$ \( 1 - \)\(14\!\cdots\!50\)\( T^{2} + \)\(99\!\cdots\!45\)\( T^{4} - \)\(41\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!10\)\( T^{8} - \)\(29\!\cdots\!00\)\( T^{10} + \)\(57\!\cdots\!90\)\( T^{12} - \)\(86\!\cdots\!00\)\( T^{14} + \)\(94\!\cdots\!05\)\( T^{16} - \)\(63\!\cdots\!50\)\( T^{18} + \)\(19\!\cdots\!49\)\( T^{20} \)
$89$ \( ( 1 - \)\(60\!\cdots\!50\)\( T + \)\(56\!\cdots\!45\)\( T^{2} - \)\(56\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!10\)\( T^{4} - \)\(16\!\cdots\!00\)\( T^{5} + \)\(39\!\cdots\!90\)\( T^{6} - \)\(25\!\cdots\!00\)\( T^{7} + \)\(54\!\cdots\!05\)\( T^{8} - \)\(12\!\cdots\!50\)\( T^{9} + \)\(44\!\cdots\!49\)\( T^{10} )^{2} \)
$97$ \( 1 - \)\(23\!\cdots\!50\)\( T^{2} + \)\(25\!\cdots\!45\)\( T^{4} - \)\(16\!\cdots\!00\)\( T^{6} + \)\(75\!\cdots\!10\)\( T^{8} - \)\(28\!\cdots\!00\)\( T^{10} + \)\(10\!\cdots\!90\)\( T^{12} - \)\(29\!\cdots\!00\)\( T^{14} + \)\(60\!\cdots\!05\)\( T^{16} - \)\(76\!\cdots\!50\)\( T^{18} + \)\(43\!\cdots\!49\)\( T^{20} \)
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