Properties

Label 21.30.a.d
Level $21$
Weight $30$
Character orbit 21.a
Self dual yes
Analytic conductor $111.884$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 30 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(111.883889004\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 3 x^{7} - 2857317313 x^{6} - 1405020216555 x^{5} + 2511494407820476041 x^{4} + 1309688035004575248975 x^{3} - 671343000872604695849378947 x^{2} - 963948141617979543405904380129 x - 192489451427205341333847862033094\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{16}\cdot 5^{2}\cdot 7^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2433 + \beta_{1} ) q^{2} + 4782969 q^{3} + ( 183377628 - 4124 \beta_{1} + \beta_{2} ) q^{4} + ( -1062304521 + 107837 \beta_{1} - 11 \beta_{2} + \beta_{4} ) q^{5} + ( -11636963577 + 4782969 \beta_{1} ) q^{6} -678223072849 q^{7} + ( -2086191207574 + 37982789 \beta_{1} - 4595 \beta_{2} + 3 \beta_{3} + 40 \beta_{4} - \beta_{6} + \beta_{7} ) q^{8} + 22876792454961 q^{9} +O(q^{10})\) \( q +(-2433 + \beta_{1}) q^{2} +4782969 q^{3} +(183377628 - 4124 \beta_{1} + \beta_{2}) q^{4} +(-1062304521 + 107837 \beta_{1} - 11 \beta_{2} + \beta_{4}) q^{5} +(-11636963577 + 4782969 \beta_{1}) q^{6} -678223072849 q^{7} +(-2086191207574 + 37982789 \beta_{1} - 4595 \beta_{2} + 3 \beta_{3} + 40 \beta_{4} - \beta_{6} + \beta_{7}) q^{8} +22876792454961 q^{9} +(79619887570591 - 5584220711 \beta_{1} + 145081 \beta_{2} - 101 \beta_{3} + 4554 \beta_{4} - 72 \beta_{5} - 3 \beta_{6} + 14 \beta_{7}) q^{10} +(49463862715753 - 738795798 \beta_{1} + 962292 \beta_{2} + 9 \beta_{3} + 10681 \beta_{4} - 57 \beta_{5} - 112 \beta_{6} + 258 \beta_{7}) q^{11} +(877089510017532 - 19724964156 \beta_{1} + 4782969 \beta_{2}) q^{12} +(2851104134997523 + 162219377684 \beta_{1} + 3357019 \beta_{2} - 1390 \beta_{3} - 211762 \beta_{4} - 2065 \beta_{5} - 1416 \beta_{6} + 1959 \beta_{7}) q^{13} +(1650116736241617 - 678223072849 \beta_{1}) q^{14} +(-5080969592502849 + 515781028053 \beta_{1} - 52612659 \beta_{2} + 4782969 \beta_{4}) q^{15} +(-66240589731806310 - 1706798620529 \beta_{1} - 174889489 \beta_{2} + 22913 \beta_{3} - 1384364 \beta_{4} - 31184 \beta_{5} + 15723 \beta_{6} - 15967 \beta_{7}) q^{16} +(-109472285177333584 + 4458263176625 \beta_{1} - 325878086 \beta_{2} + 92256 \beta_{3} - 1714023 \beta_{4} + 114219 \beta_{5} - 17432 \beta_{6} + 16449 \beta_{7}) q^{17} +(-55659236042920113 + 22876792454961 \beta_{1}) q^{18} +(178346131006237236 + 41259279276303 \beta_{1} + 1760260189 \beta_{2} + 512119 \beta_{3} - 31655180 \beta_{4} - 403559 \beta_{5} - 613872 \beta_{6} + 53502 \beta_{7}) q^{19} +(-3612414586363959288 + 86427678238816 \beta_{1} - 4088836558 \beta_{2} + 387720 \beta_{3} + 35602728 \beta_{4} + 1374240 \beta_{5} + 651460 \beta_{6} + 261820 \beta_{7}) q^{20} -3243919932521508681 q^{21} +(-648419164882820589 + 419560387952613 \beta_{1} + 20530071695 \beta_{2} + 2236673 \beta_{3} + 1614569318 \beta_{4} - 880888 \beta_{5} + 3884115 \beta_{6} - 1365522 \beta_{7}) q^{22} +(3946519701719987484 - 114067167744405 \beta_{1} - 18870608110 \beta_{2} - 18617610 \beta_{3} + 892258701 \beta_{4} - 2841093 \beta_{5} - 4167560 \beta_{6} + 789631 \beta_{7}) q^{23} +(-9978187873899007206 + 181670502320541 \beta_{1} - 21977742555 \beta_{2} + 14348907 \beta_{3} + 191318760 \beta_{4} - 4782969 \beta_{6} + 4782969 \beta_{7}) q^{24} +(60616428863091106601 + 786669841300523 \beta_{1} - 45797986959 \beta_{2} - 65484195 \beta_{3} + 1017597994 \beta_{4} - 34603865 \beta_{5} - 11481360 \beta_{6} - 4050170 \beta_{7}) q^{25} +(\)\(10\!\cdots\!78\)\( + 3890703071765906 \beta_{1} + 275575308384 \beta_{2} + 46746468 \beta_{3} + 19197778848 \beta_{4} + 53704704 \beta_{5} + 59349832 \beta_{6} - 26798644 \beta_{7}) q^{26} +\)\(10\!\cdots\!09\)\( q^{27} +(-\)\(12\!\cdots\!72\)\( + 2796991952429276 \beta_{1} - 678223072849 \beta_{2}) q^{28} +(50478298123520835140 - 2357875971198557 \beta_{1} + 620737774791 \beta_{2} + 556621875 \beta_{3} + 41751541812 \beta_{4} + 146992263 \beta_{5} - 214583296 \beta_{6} + 51117748 \beta_{7}) q^{29} +(\)\(38\!\cdots\!79\)\( - 26709154549870959 \beta_{1} + 693917925489 \beta_{2} - 483079869 \beta_{3} + 21781640826 \beta_{4} - 344373768 \beta_{5} - 14348907 \beta_{6} + 66961566 \beta_{7}) q^{30} +(\)\(75\!\cdots\!34\)\( + 35179816195732505 \beta_{1} + 4373921610421 \beta_{2} + 962712321 \beta_{3} + 202173958628 \beta_{4} + 564461933 \beta_{5} + 239892000 \beta_{6} + 34858508 \beta_{7}) q^{31} +(62023119416659231154 - 150838299043974809 \beta_{1} + 2076143646367 \beta_{2} - 853559271 \beta_{3} - 47079199060 \beta_{4} - 24524784 \beta_{5} + 36660783 \beta_{6} - 153073723 \beta_{7}) q^{32} +(\)\(23\!\cdots\!57\)\( - 3533637399164262 \beta_{1} + 4602612804948 \beta_{2} + 43046721 \beta_{3} + 51086891889 \beta_{4} - 272629233 \beta_{5} - 535692528 \beta_{6} + 1234006002 \beta_{7}) q^{33} +(\)\(34\!\cdots\!13\)\( - 242096710274072825 \beta_{1} + 9503805138515 \beta_{2} - 2472302755 \beta_{3} - 547106252594 \beta_{4} - 1762292504 \beta_{5} + 343518279 \beta_{6} - 1122975658 \beta_{7}) q^{34} +(\)\(72\!\cdots\!29\)\( - 73137541506817613 \beta_{1} + 7460453801339 \beta_{2} - 678223072849 \beta_{4}) q^{35} +(\)\(41\!\cdots\!08\)\( - 94343892084259164 \beta_{1} + 22876792454961 \beta_{2}) q^{36} +(\)\(31\!\cdots\!30\)\( + 92896293312268021 \beta_{1} + 43402569492573 \beta_{2} - 15735144197 \beta_{3} - 490124573962 \beta_{4} - 1758982357 \beta_{5} + 1363776096 \beta_{6} - 5167398968 \beta_{7}) q^{37} +(\)\(29\!\cdots\!02\)\( + 789039096273281170 \beta_{1} + 114705187660406 \beta_{2} + 26470458558 \beta_{3} + 208905000476 \beta_{4} + 8670546000 \beta_{5} + 9250447782 \beta_{6} - 6332637816 \beta_{7}) q^{38} +(\)\(13\!\cdots\!87\)\( + 775890254661863796 \beta_{1} + 16056517809411 \beta_{2} - 6648326910 \beta_{3} - 1012851081378 \beta_{4} - 9876830985 \beta_{5} - 6772684104 \beta_{6} + 9369836271 \beta_{7}) q^{39} +(\)\(27\!\cdots\!36\)\( - 2336561737113315706 \beta_{1} - 8624891411674 \beta_{2} + 6711371754 \beta_{3} - 6356590697616 \beta_{4} + 7896338688 \beta_{5} - 8675233038 \beta_{6} - 6766773906 \beta_{7}) q^{40} +(-\)\(14\!\cdots\!16\)\( + 5526008790078057121 \beta_{1} - 205327171405276 \beta_{2} + 27808173342 \beta_{3} - 4004849941309 \beta_{4} + 60761126727 \beta_{5} - 6021868104 \beta_{6} + 37294448715 \beta_{7}) q^{41} +(\)\(78\!\cdots\!73\)\( - 3243919932521508681 \beta_{1}) q^{42} +(\)\(11\!\cdots\!25\)\( + 3794924624797976685 \beta_{1} + 40789037285432 \beta_{2} - 5471209607 \beta_{3} + 5531758882076 \beta_{4} - 95621406310 \beta_{5} + 29309181432 \beta_{6} - 4716805301 \beta_{7}) q^{43} +(\)\(27\!\cdots\!20\)\( + 6754893790289469512 \beta_{1} - 65209420785728 \beta_{2} + 64944600 \beta_{3} - 3252651981352 \beta_{4} - 153774530592 \beta_{5} - 100484360548 \beta_{6} + 9230245476 \beta_{7}) q^{44} +(-\)\(24\!\cdots\!81\)\( + 2466964667965629357 \beta_{1} - 251644717004571 \beta_{2} + 22876792454961 \beta_{4}) q^{45} +(-\)\(91\!\cdots\!65\)\( - 3218054062648571343 \beta_{1} - 1221167159691777 \beta_{2} - 309366600799 \beta_{3} + 54162494400582 \beta_{4} + 236486364552 \beta_{5} + 235873286691 \beta_{6} - 115259772674 \beta_{7}) q^{46} +(\)\(21\!\cdots\!58\)\( + 4801942509444217214 \beta_{1} - 1568205700386262 \beta_{2} + 153784966656 \beta_{3} - 15666573172154 \beta_{4} + 120360660 \beta_{5} - 209876366144 \beta_{6} + 49940995740 \beta_{7}) q^{47} +(-\)\(31\!\cdots\!90\)\( - 8163564891232970601 \beta_{1} - 836491004312841 \beta_{2} + 109592168697 \beta_{3} - 6621370096716 \beta_{4} - 149152105296 \beta_{5} + 75202621587 \beta_{6} - 76369666023 \beta_{7}) q^{48} +\)\(45\!\cdots\!01\)\( q^{49} +(\)\(41\!\cdots\!49\)\( + 41550784000630192571 \beta_{1} - 2909908387306616 \beta_{2} - 221500787364 \beta_{3} + 234526065268656 \beta_{4} + 1194029184192 \beta_{5} + 682362577808 \beta_{6} - 212438544004 \beta_{7}) q^{50} +(-\)\(52\!\cdots\!96\)\( + 21323734567638899625 \beta_{1} - 1558664783117334 \beta_{2} + 441257588064 \beta_{3} - 8198118874287 \beta_{4} + 546305936211 \beta_{5} - 83376715608 \beta_{6} + 78675057081 \beta_{7}) q^{51} +(\)\(98\!\cdots\!84\)\( + \)\(11\!\cdots\!24\)\( \beta_{1} + 892759296987102 \beta_{2} - 393591532256 \beta_{3} - 178489581880640 \beta_{4} - 1958533006592 \beta_{5} - 1680109024320 \beta_{6} + 699963545280 \beta_{7}) q^{52} +(\)\(26\!\cdots\!70\)\( + 59630980208504736685 \beta_{1} - 1326632734843091 \beta_{2} + 1416825289287 \beta_{3} - 173893853388350 \beta_{4} - 2772181373793 \beta_{5} + 242153877568 \beta_{6} - 140373294208 \beta_{7}) q^{53} +(-\)\(26\!\cdots\!97\)\( + \)\(10\!\cdots\!09\)\( \beta_{1}) q^{54} +(-\)\(90\!\cdots\!24\)\( + \)\(41\!\cdots\!41\)\( \beta_{1} + 4287733839073991 \beta_{2} - 2129322374543 \beta_{3} + 61722236137044 \beta_{4} + 5983683140379 \beta_{5} + 1568404822896 \beta_{6} + 893200901702 \beta_{7}) q^{55} +(\)\(14\!\cdots\!26\)\( - 25760803870955195861 \beta_{1} + 3116435019741155 \beta_{2} - 2034669218547 \beta_{3} - 27128922913960 \beta_{4} + 678223072849 \beta_{6} - 678223072849 \beta_{7}) q^{56} +(\)\(85\!\cdots\!84\)\( + \)\(19\!\cdots\!07\)\( \beta_{1} + 8419269915921141 \beta_{2} + 2449449301311 \beta_{3} - 151405744629420 \beta_{4} - 1930210186671 \beta_{5} - 2936130745968 \beta_{6} + 255898407438 \beta_{7}) q^{57} +(-\)\(18\!\cdots\!00\)\( + \)\(28\!\cdots\!84\)\( \beta_{1} + 46543728139402618 \beta_{2} + 3655502687698 \beta_{3} - 1309738007030204 \beta_{4} - 7843650692048 \beta_{5} + 381759039114 \beta_{6} - 1519306208808 \beta_{7}) q^{58} +(\)\(52\!\cdots\!20\)\( + \)\(35\!\cdots\!80\)\( \beta_{1} + 24423288526658618 \beta_{2} - 3768910367718 \beta_{3} + 195214631884790 \beta_{4} + 7380369742512 \beta_{5} - 496651677712 \beta_{6} - 1745430348878 \beta_{7}) q^{59} +(-\)\(17\!\cdots\!72\)\( + \)\(41\!\cdots\!04\)\( \beta_{1} - 19556778502980702 \beta_{2} + 1854452740680 \beta_{3} + 170286744339432 \beta_{4} + 6572947318560 \beta_{5} + 3115912984740 \beta_{6} + 1252276943580 \beta_{7}) q^{60} +(-\)\(12\!\cdots\!27\)\( + \)\(38\!\cdots\!44\)\( \beta_{1} + 62485526571798683 \beta_{2} + 4603989145132 \beta_{3} - 1793475439382896 \beta_{4} + 4010353345061 \beta_{5} + 10631328286104 \beta_{6} - 8843539427765 \beta_{7}) q^{61} +(\)\(23\!\cdots\!82\)\( + \)\(23\!\cdots\!50\)\( \beta_{1} + 89170978617744486 \beta_{2} - 1847183233794 \beta_{3} - 2506020943320836 \beta_{4} - 34540457076528 \beta_{5} - 18586297951818 \beta_{6} + 13295812046072 \beta_{7}) q^{62} -\)\(15\!\cdots\!89\)\( q^{63} +(-\)\(72\!\cdots\!50\)\( + \)\(20\!\cdots\!95\)\( \beta_{1} - 133931162540277341 \beta_{2} - 13674914505179 \beta_{3} + 1636148034031724 \beta_{4} + 31127305916048 \beta_{5} - 6292964399397 \beta_{6} + 8198453308441 \beta_{7}) q^{64} +(-\)\(43\!\cdots\!32\)\( + \)\(43\!\cdots\!02\)\( \beta_{1} - 33237503689813112 \beta_{2} - 12662533158828 \beta_{3} + 8288718518168242 \beta_{4} + 43531841005434 \beta_{5} + 21225828440816 \beta_{6} + 419144629042 \beta_{7}) q^{65} +(-\)\(31\!\cdots\!41\)\( + \)\(20\!\cdots\!97\)\( \beta_{1} + 98194696484962455 \beta_{2} + 10697937622137 \beta_{3} + 7722434996345142 \beta_{4} - 4213259996472 \beta_{5} + 18577601637435 \beta_{6} - 6531249394818 \beta_{7}) q^{66} +(-\)\(95\!\cdots\!19\)\( + \)\(36\!\cdots\!41\)\( \beta_{1} - 272740202166422 \beta_{2} + 5648724444831 \beta_{3} - 5173918675323502 \beta_{4} - 84030003337030 \beta_{5} - 28613983298328 \beta_{6} - 16879044171799 \beta_{7}) q^{67} +(-\)\(12\!\cdots\!64\)\( + \)\(51\!\cdots\!48\)\( \beta_{1} - 299043770004953450 \beta_{2} + 31725839914872 \beta_{3} + 4155728784801400 \beta_{4} + 25833435611232 \beta_{5} + 4571308487468 \beta_{6} - 92827797548 \beta_{7}) q^{68} +(\)\(18\!\cdots\!96\)\( - \)\(54\!\cdots\!45\)\( \beta_{1} - 90257533601278590 \beta_{2} - 89047451484090 \beta_{3} + 4267645706863269 \beta_{4} - 13588859745117 \beta_{5} - 19933310285640 \beta_{6} + 3776780594439 \beta_{7}) q^{69} +(-\)\(54\!\cdots\!59\)\( + \)\(37\!\cdots\!39\)\( \beta_{1} - 98397281632005769 \beta_{2} + 68500530357749 \beta_{3} - 3088627873754346 \beta_{4} + 48832061245128 \beta_{5} + 2034669218547 \beta_{6} - 9495123019886 \beta_{7}) q^{70} +(-\)\(13\!\cdots\!54\)\( + \)\(11\!\cdots\!03\)\( \beta_{1} - 498764755508353482 \beta_{2} + 20871158590272 \beta_{3} + 13448838137928355 \beta_{4} - 132942065103663 \beta_{5} + 13581772998168 \beta_{6} - 24386248506253 \beta_{7}) q^{71} +(-\)\(47\!\cdots\!14\)\( + \)\(86\!\cdots\!29\)\( \beta_{1} - 105118861330545795 \beta_{2} + 68630377364883 \beta_{3} + 915071698198440 \beta_{4} - 22876792454961 \beta_{6} + 22876792454961 \beta_{7}) q^{72} +(-\)\(30\!\cdots\!86\)\( + \)\(27\!\cdots\!88\)\( \beta_{1} - 100557234355543646 \beta_{2} - 24533899859342 \beta_{3} - 7542533692876702 \beta_{4} + 103265899241204 \beta_{5} + 92592243982512 \beta_{6} + 10824984465046 \beta_{7}) q^{73} +(-\)\(94\!\cdots\!74\)\( + \)\(47\!\cdots\!62\)\( \beta_{1} - 1435855386402171484 \beta_{2} + 11941597292064 \beta_{3} + 10926733528720648 \beta_{4} + 258073602088416 \beta_{5} - 10579310075268 \beta_{6} + 36220564750036 \beta_{7}) q^{74} +(\)\(28\!\cdots\!69\)\( + \)\(37\!\cdots\!87\)\( \beta_{1} - 219050351887301271 \beta_{2} - 313208874674955 \beta_{3} + 4867139659764186 \beta_{4} - 165509213575185 \beta_{5} - 54914988957840 \beta_{6} - 19371837554730 \beta_{7}) q^{75} +(\)\(39\!\cdots\!12\)\( + \)\(49\!\cdots\!08\)\( \beta_{1} + 860480561747153628 \beta_{2} + 238165318913040 \beta_{3} - 92146257716235728 \beta_{4} - 270802771669568 \beta_{5} - 238166894346792 \beta_{6} + 311769090029224 \beta_{7}) q^{76} +(-\)\(33\!\cdots\!97\)\( + \)\(50\!\cdots\!02\)\( \beta_{1} - 652648637218009908 \beta_{2} - 6104007655641 \beta_{3} - 7244100641100169 \beta_{4} + 38658715152393 \beta_{5} + 75960984159088 \beta_{6} - 174981552795042 \beta_{7}) q^{77} +(\)\(52\!\cdots\!82\)\( + \)\(18\!\cdots\!14\)\( \beta_{1} + 1318068157166112096 \beta_{2} + 223586907303492 \beta_{3} + 91822381098839712 \beta_{4} + 256867934386176 \beta_{5} + 283868406611208 \beta_{6} - 128177083494036 \beta_{7}) q^{78} +(\)\(30\!\cdots\!50\)\( - \)\(41\!\cdots\!06\)\( \beta_{1} + 2539754503722582538 \beta_{2} + 115837098457052 \beta_{3} - 127251516570234546 \beta_{4} + 367939990244004 \beta_{5} - 131063898331008 \beta_{6} + 216541955303032 \beta_{7}) q^{79} +(\)\(20\!\cdots\!16\)\( - \)\(17\!\cdots\!82\)\( \beta_{1} - 191948468002993694 \beta_{2} + 169095637207230 \beta_{3} - 136125061162683496 \beta_{4} - 229596215295840 \beta_{5} - 164761559381110 \beta_{6} - 464704953955170 \beta_{7}) q^{80} +\)\(52\!\cdots\!21\)\( q^{81} +(\)\(39\!\cdots\!23\)\( - \)\(10\!\cdots\!79\)\( \beta_{1} + 8076985007155398701 \beta_{2} - 1720425007940965 \beta_{3} - 105957578681253774 \beta_{4} - 760855241902824 \beta_{5} + 616385016075369 \beta_{6} - 817940572462958 \beta_{7}) q^{82} +(-\)\(16\!\cdots\!94\)\( - \)\(63\!\cdots\!52\)\( \beta_{1} - 3213575061888880994 \beta_{2} - 872819532446376 \beta_{3} + 272766285159352728 \beta_{4} + 291456743528154 \beta_{5} + 670769330989520 \beta_{6} + 303086096509478 \beta_{7}) q^{83} +(-\)\(59\!\cdots\!68\)\( + \)\(13\!\cdots\!44\)\( \beta_{1} - 3243919932521508681 \beta_{2}) q^{84} +(\)\(11\!\cdots\!52\)\( - \)\(22\!\cdots\!99\)\( \beta_{1} + 5167118796445455757 \beta_{2} + 210337457360855 \beta_{3} - 194017327786105262 \beta_{4} - 965857684796465 \beta_{5} - 778350444363360 \beta_{6} + 622252791639680 \beta_{7}) q^{85} +(\)\(24\!\cdots\!28\)\( + \)\(12\!\cdots\!76\)\( \beta_{1} + 5327870189604857740 \beta_{2} + 2030740723040796 \beta_{3} + 379406384602801272 \beta_{4} + 27324193697184 \beta_{5} - 854115119985812 \beta_{6} + 1263475911941904 \beta_{7}) q^{86} +(\)\(24\!\cdots\!60\)\( - \)\(11\!\cdots\!33\)\( \beta_{1} + 2968969533954334479 \beta_{2} + 2662305172846875 \beta_{3} + 199696330188999828 \beta_{4} + 703059437168847 \beta_{5} - 1026345252685824 \beta_{6} + 244494604033812 \beta_{7}) q^{87} +(\)\(45\!\cdots\!52\)\( + \)\(13\!\cdots\!80\)\( \beta_{1} + 4413559770472549604 \beta_{2} + 2937155023289132 \beta_{3} - 343916675419233824 \beta_{4} + 3450006647147776 \beta_{5} + 282742010246940 \beta_{6} - 357697417407484 \beta_{7}) q^{88} +(\)\(24\!\cdots\!26\)\( - \)\(22\!\cdots\!87\)\( \beta_{1} - 11408849280348630042 \beta_{2} - 2287072096352286 \beta_{3} + 889158487893931151 \beta_{4} - 2096165779074711 \beta_{5} - 405757754868856 \beta_{6} + 288238521149957 \beta_{7}) q^{89} +(\)\(18\!\cdots\!51\)\( - \)\(12\!\cdots\!71\)\( \beta_{1} + 3318987926158196841 \beta_{2} - 2310556037951061 \beta_{3} + 104180912839892394 \beta_{4} - 1647129056757192 \beta_{5} - 68630377364883 \beta_{6} + 320275094369454 \beta_{7}) q^{90} +(-\)\(19\!\cdots\!27\)\( - \)\(11\!\cdots\!16\)\( \beta_{1} - 2276807741792477131 \beta_{2} + 942730071260110 \beta_{3} + 143621874352649938 \beta_{4} + 1400530645433185 \beta_{5} + 960363871154184 \beta_{6} - 1328638999711191 \beta_{7}) q^{91} +(-\)\(41\!\cdots\!88\)\( - \)\(49\!\cdots\!88\)\( \beta_{1} - 36445669828267027732 \beta_{2} - 7690397597107752 \beta_{3} - 949087908640194472 \beta_{4} - 3352258363133472 \beta_{5} - 1464489797838244 \beta_{6} + 2038153847510692 \beta_{7}) q^{92} +(\)\(36\!\cdots\!46\)\( + \)\(16\!\cdots\!45\)\( \beta_{1} + 20920331471073719949 \beta_{2} + 4604623187261049 \beta_{3} + 966991776725006532 \beta_{4} + 2699803927219077 \beta_{5} + 1147395999348000 \beta_{6} + 166727163150252 \beta_{7}) q^{93} +(\)\(29\!\cdots\!46\)\( - \)\(39\!\cdots\!70\)\( \beta_{1} + 26630417815011808462 \beta_{2} - 1152084474825798 \beta_{3} - 322144493441165972 \beta_{4} + 2166936548072080 \beta_{5} + 5957049856640550 \beta_{6} - 5441856503494412 \beta_{7}) q^{94} +(-\)\(88\!\cdots\!60\)\( - \)\(42\!\cdots\!08\)\( \beta_{1} - 64049147524449682160 \beta_{2} + 9217369820984808 \beta_{3} - 33720666410583840 \beta_{4} + 2569076480712576 \beta_{5} + 2233939529025024 \beta_{6} - 3494703339037112 \beta_{7}) q^{95} +(\)\(29\!\cdots\!26\)\( - \)\(72\!\cdots\!21\)\( \beta_{1} + 9930130700120323623 \beta_{2} - 4082547532855599 \beta_{3} - 225178349648809140 \beta_{4} - 117301281603696 \beta_{5} + 175347388604727 \beta_{6} - 732146871823587 \beta_{7}) q^{96} +(-\)\(67\!\cdots\!24\)\( - \)\(10\!\cdots\!34\)\( \beta_{1} + 3100578117804075658 \beta_{2} - 8565147979863572 \beta_{3} + 679142590437112694 \beta_{4} - 3768419424454708 \beta_{5} - 3162616153045632 \beta_{6} - 1036502888764544 \beta_{7}) q^{97} +(-\)\(11\!\cdots\!33\)\( + \)\(45\!\cdots\!01\)\( \beta_{1}) q^{98} +(\)\(11\!\cdots\!33\)\( - \)\(16\!\cdots\!78\)\( \beta_{1} + 22014154365069330612 \beta_{2} + 205891132094649 \beta_{3} + 244347020211438441 \beta_{4} - 1303977169932777 \beta_{5} - 2562200754955632 \beta_{6} + 5902212453379938 \beta_{7}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 19461q^{2} + 38263752q^{3} + 1467008653q^{4} - 8498112672q^{5} - 93081359709q^{6} - 5425784582792q^{7} - 16689415716987q^{8} + 183014339639688q^{9} + O(q^{10}) \) \( 8q - 19461q^{2} + 38263752q^{3} + 1467008653q^{4} - 8498112672q^{5} - 93081359709q^{6} - 5425784582792q^{7} - 16689415716987q^{8} + 183014339639688q^{9} + 636942348029770q^{10} + 395708686257744q^{11} + 7016656910030757q^{12} + 22809319742321168q^{13} + 13198899220714389q^{14} - 40646209468683168q^{15} - 529929838419631103q^{16} - 875764906948588224q^{17} - 445205257965996021q^{18} + 1426892827775457824q^{19} - 28899057412113817626q^{20} - 25951359460172069448q^{21} - 5186094623833295684q^{22} + 31571815389865907376q^{23} - 79824958002461594403q^{24} + \)\(48\!\cdots\!92\)\(q^{25} + \)\(87\!\cdots\!58\)\(q^{26} + \)\(87\!\cdots\!72\)\(q^{27} - \)\(99\!\cdots\!97\)\(q^{28} + \)\(40\!\cdots\!64\)\(q^{29} + \)\(30\!\cdots\!30\)\(q^{30} + \)\(60\!\cdots\!64\)\(q^{31} + \)\(49\!\cdots\!25\)\(q^{32} + \)\(18\!\cdots\!36\)\(q^{33} + \)\(27\!\cdots\!54\)\(q^{34} + \)\(57\!\cdots\!28\)\(q^{35} + \)\(33\!\cdots\!33\)\(q^{36} + \)\(24\!\cdots\!44\)\(q^{37} + \)\(23\!\cdots\!96\)\(q^{38} + \)\(10\!\cdots\!92\)\(q^{39} + \)\(22\!\cdots\!70\)\(q^{40} - \)\(11\!\cdots\!80\)\(q^{41} + \)\(63\!\cdots\!41\)\(q^{42} + \)\(95\!\cdots\!56\)\(q^{43} + \)\(21\!\cdots\!28\)\(q^{44} - \)\(19\!\cdots\!92\)\(q^{45} - \)\(72\!\cdots\!20\)\(q^{46} + \)\(17\!\cdots\!96\)\(q^{47} - \)\(25\!\cdots\!07\)\(q^{48} + \)\(36\!\cdots\!08\)\(q^{49} + \)\(33\!\cdots\!05\)\(q^{50} - \)\(41\!\cdots\!56\)\(q^{51} + \)\(78\!\cdots\!02\)\(q^{52} + \)\(21\!\cdots\!24\)\(q^{53} - \)\(21\!\cdots\!49\)\(q^{54} - \)\(72\!\cdots\!24\)\(q^{55} + \)\(11\!\cdots\!63\)\(q^{56} + \)\(68\!\cdots\!56\)\(q^{57} - \)\(14\!\cdots\!18\)\(q^{58} + \)\(42\!\cdots\!28\)\(q^{59} - \)\(13\!\cdots\!94\)\(q^{60} - \)\(10\!\cdots\!16\)\(q^{61} + \)\(18\!\cdots\!00\)\(q^{62} - \)\(12\!\cdots\!12\)\(q^{63} - \)\(57\!\cdots\!19\)\(q^{64} - \)\(35\!\cdots\!00\)\(q^{65} - \)\(24\!\cdots\!96\)\(q^{66} - \)\(76\!\cdots\!92\)\(q^{67} - \)\(98\!\cdots\!18\)\(q^{68} + \)\(15\!\cdots\!44\)\(q^{69} - \)\(43\!\cdots\!30\)\(q^{70} - \)\(10\!\cdots\!20\)\(q^{71} - \)\(38\!\cdots\!07\)\(q^{72} - \)\(24\!\cdots\!48\)\(q^{73} - \)\(75\!\cdots\!14\)\(q^{74} + \)\(23\!\cdots\!48\)\(q^{75} + \)\(31\!\cdots\!28\)\(q^{76} - \)\(26\!\cdots\!56\)\(q^{77} + \)\(41\!\cdots\!02\)\(q^{78} + \)\(24\!\cdots\!12\)\(q^{79} + \)\(16\!\cdots\!22\)\(q^{80} + \)\(41\!\cdots\!68\)\(q^{81} + \)\(31\!\cdots\!58\)\(q^{82} - \)\(12\!\cdots\!44\)\(q^{83} - \)\(47\!\cdots\!93\)\(q^{84} + \)\(89\!\cdots\!24\)\(q^{85} + \)\(19\!\cdots\!56\)\(q^{86} + \)\(19\!\cdots\!16\)\(q^{87} + \)\(36\!\cdots\!12\)\(q^{88} + \)\(19\!\cdots\!36\)\(q^{89} + \)\(14\!\cdots\!70\)\(q^{90} - \)\(15\!\cdots\!32\)\(q^{91} - \)\(33\!\cdots\!52\)\(q^{92} + \)\(28\!\cdots\!16\)\(q^{93} + \)\(23\!\cdots\!80\)\(q^{94} - \)\(70\!\cdots\!84\)\(q^{95} + \)\(23\!\cdots\!25\)\(q^{96} - \)\(54\!\cdots\!56\)\(q^{97} - \)\(89\!\cdots\!61\)\(q^{98} + \)\(90\!\cdots\!84\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} - 2857317313 x^{6} - 1405020216555 x^{5} + 2511494407820476041 x^{4} + 1309688035004575248975 x^{3} - 671343000872604695849378947 x^{2} - 963948141617979543405904380129 x - 192489451427205341333847862033094\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 742 \nu - 714329051 \)
\(\beta_{3}\)\(=\)\((\)\(-1028970383747405965 \nu^{7} + 1205555825822863949958234 \nu^{6} + 2344172410433501056843792167 \nu^{5} - 2145658775843115688677965573086506 \nu^{4} - 6789183257377814067106277219163696159 \nu^{3} + 663454246280814049800014016910347620427294 \nu^{2} + 3061282529563229194098878130052055735046230725 \nu + 65130583995488253399325176915107975445484082748786\)\()/ \)\(31\!\cdots\!40\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-11972612448220757317 \nu^{7} - 47267574306949444990006 \nu^{6} + 31473338439621532613335411023 \nu^{5} + 123172510697898350724894796955622 \nu^{4} - 24359557141514762499794522450846284167 \nu^{3} - 58202800821201681335303446100052311849490 \nu^{2} + 5497891368400283225396070464770455655242447037 \nu + 3716791546636622228348167606730501007942503990050\)\()/ \)\(18\!\cdots\!40\)\( \)
\(\beta_{5}\)\(=\)\((\)\(59886713579652137405 \nu^{7} + 4411686167735532915190982 \nu^{6} - 155500556813286652781454662199 \nu^{5} - 9611561255738091208366202792472438 \nu^{4} + 90055930614591652607979093080807946543 \nu^{3} + 4825272431484660900179105838859031605138242 \nu^{2} - 3627529751816647470199157265464348392848779765 \nu - 163330919573516685820881300394874133871952999499602\)\()/ \)\(47\!\cdots\!60\)\( \)
\(\beta_{6}\)\(=\)\((\)\(29046089003854318983 \nu^{7} - 80488115899203587695822 \nu^{6} - 87563925771686912023533017365 \nu^{5} + 194985777532911638948692108367166 \nu^{4} + 81713149123941404843548158313388614909 \nu^{3} - 126354794490386639358833510748878798276506 \nu^{2} - 22966676887273377364500134033066787970823895343 \nu - 12398580381970162599552020600983973539815347521814\)\()/ \)\(13\!\cdots\!60\)\( \)
\(\beta_{7}\)\(=\)\((\)\(2340031390695652766801 \nu^{7} - 15699795291009444963923170 \nu^{6} - 6955166976047789827472679600243 \nu^{5} + 30886454750999712187909626021102066 \nu^{4} + 6525916944437362806130551467190334337739 \nu^{3} - 14159500830563190817800391189129682455130358 \nu^{2} - 1894566747290794683751041688265976237616798952681 \nu - 1421046448309898672297077182519616125555193609623002\)\()/ \)\(94\!\cdots\!20\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 742 \beta_{1} + 714329051\)
\(\nu^{3}\)\(=\)\(\beta_{7} - \beta_{6} + 40 \beta_{4} + 3 \beta_{3} + 2704 \beta_{2} + 1099382004 \beta_{1} + 529684794620\)
\(\nu^{4}\)\(=\)\(-6235 \beta_{7} + 5991 \beta_{6} - 31184 \beta_{5} - 995084 \beta_{4} + 52109 \beta_{3} + 1426521641 \beta_{2} + 2381475021055 \beta_{1} + 785319580038458471\)
\(\nu^{5}\)\(=\)\(1859366260 \beta_{7} - 1978747460 \beta_{6} - 403878144 \beta_{5} + 24347154400 \beta_{4} + 6045212988 \beta_{3} + 9546050231382 \beta_{2} + 1335705508789975633 \beta_{1} + 1700660903663116576620\)
\(\nu^{6}\)\(=\)\(-6677944744134 \beta_{7} + 6207389712678 \beta_{6} - 55708519654464 \beta_{5} - 1624678496866176 \beta_{4} + 132316967946630 \beta_{3} + 2003619140600021361 \beta_{2} + 6361049714251973372748 \beta_{1} + 954129942913854147992584819\)
\(\nu^{7}\)\(=\)\(2815473751753956891 \beta_{7} - 3129952903643911587 \beta_{6} - 1162586574649583328 \beta_{5} - 36952776254523519936 \beta_{4} + 9801407807974586373 \beta_{3} + 21497151348620032348392 \beta_{2} + 1729448659115320839040451710 \beta_{1} + 4543186031837481381226206771684\)

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
−35401.8
−33437.6
−21121.7
−1200.71
−239.721
25358.2
27042.2
39004.2
−37834.8 4.78297e6 8.94601e8 −2.52890e9 −1.80963e11 −6.78223e11 −1.35346e13 2.28768e13 9.56804e13
1.2 −35870.6 4.78297e6 7.49832e8 −2.32963e10 −1.71568e11 −6.78223e11 −7.63907e12 2.28768e13 8.35653e14
1.3 −23554.7 4.78297e6 1.79529e7 8.07378e9 −1.12661e11 −6.78223e11 1.22230e13 2.28768e13 −1.90175e14
1.4 −3633.71 4.78297e6 −5.23667e8 −8.87892e9 −1.73799e10 −6.78223e11 3.85369e12 2.28768e13 3.22634e13
1.5 −2672.72 4.78297e6 −5.29727e8 1.94732e10 −1.27835e10 −6.78223e11 2.85072e12 2.28768e13 −5.20466e13
1.6 22925.2 4.78297e6 −1.13081e7 −1.77814e10 1.09650e11 −6.78223e11 −1.25671e13 2.28768e13 −4.07642e14
1.7 24609.2 4.78297e6 6.87414e7 2.32433e10 1.17705e11 −6.78223e11 −1.15203e13 2.28768e13 5.71998e14
1.8 36571.2 4.78297e6 8.00584e8 −6.80286e9 1.74919e11 −6.78223e11 9.64430e12 2.28768e13 −2.48789e14
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.30.a.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.30.a.d 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(42\!\cdots\!64\)\( T_{2}^{5} + \)\(22\!\cdots\!96\)\( T_{2}^{4} + \)\(24\!\cdots\!20\)\( T_{2}^{3} - \)\(57\!\cdots\!00\)\( T_{2}^{2} - \)\(40\!\cdots\!00\)\( T_{2} - \)\(64\!\cdots\!00\)\( \)">\(T_{2}^{8} + \cdots\) acting on \(S_{30}^{\mathrm{new}}(\Gamma_0(21))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -\)\(64\!\cdots\!00\)\( - \)\(40\!\cdots\!00\)\( T - \)\(57\!\cdots\!00\)\( T^{2} + \)\(24\!\cdots\!20\)\( T^{3} + 2243145773108109696 T^{4} - 42309992742264 T^{5} - 2691622714 T^{6} + 19461 T^{7} + T^{8} \)
$3$ \( ( -4782969 + T )^{8} \)
$5$ \( -\)\(23\!\cdots\!00\)\( - \)\(12\!\cdots\!00\)\( T - \)\(82\!\cdots\!00\)\( T^{2} + \)\(24\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!00\)\( T^{4} - \)\(86\!\cdots\!00\)\( T^{5} - \)\(95\!\cdots\!04\)\( T^{6} + 8498112672 T^{7} + T^{8} \)
$7$ \( ( 678223072849 + T )^{8} \)
$11$ \( -\)\(76\!\cdots\!44\)\( + \)\(28\!\cdots\!32\)\( T - \)\(15\!\cdots\!48\)\( T^{2} - \)\(30\!\cdots\!60\)\( T^{3} + \)\(26\!\cdots\!36\)\( T^{4} + \)\(76\!\cdots\!40\)\( T^{5} - \)\(93\!\cdots\!52\)\( T^{6} - 395708686257744 T^{7} + T^{8} \)
$13$ \( -\)\(12\!\cdots\!00\)\( + \)\(58\!\cdots\!00\)\( T + \)\(66\!\cdots\!00\)\( T^{2} - \)\(23\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!16\)\( T^{4} + \)\(18\!\cdots\!28\)\( T^{5} - \)\(67\!\cdots\!36\)\( T^{6} - 22809319742321168 T^{7} + T^{8} \)
$17$ \( \)\(87\!\cdots\!56\)\( + \)\(23\!\cdots\!28\)\( T + \)\(22\!\cdots\!12\)\( T^{2} + \)\(74\!\cdots\!40\)\( T^{3} - \)\(31\!\cdots\!44\)\( T^{4} - \)\(58\!\cdots\!80\)\( T^{5} - \)\(56\!\cdots\!12\)\( T^{6} + 875764906948588224 T^{7} + T^{8} \)
$19$ \( -\)\(41\!\cdots\!44\)\( + \)\(11\!\cdots\!32\)\( T - \)\(66\!\cdots\!08\)\( T^{2} - \)\(16\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!36\)\( T^{4} + \)\(75\!\cdots\!60\)\( T^{5} - \)\(64\!\cdots\!32\)\( T^{6} - 1426892827775457824 T^{7} + T^{8} \)
$23$ \( -\)\(18\!\cdots\!00\)\( + \)\(25\!\cdots\!00\)\( T - \)\(83\!\cdots\!00\)\( T^{2} - \)\(16\!\cdots\!60\)\( T^{3} + \)\(71\!\cdots\!16\)\( T^{4} + \)\(40\!\cdots\!84\)\( T^{5} - \)\(16\!\cdots\!44\)\( T^{6} - 31571815389865907376 T^{7} + T^{8} \)
$29$ \( -\)\(35\!\cdots\!04\)\( - \)\(56\!\cdots\!76\)\( T - \)\(73\!\cdots\!68\)\( T^{2} + \)\(33\!\cdots\!92\)\( T^{3} + \)\(64\!\cdots\!20\)\( T^{4} - \)\(19\!\cdots\!32\)\( T^{5} - \)\(14\!\cdots\!88\)\( T^{6} - \)\(40\!\cdots\!64\)\( T^{7} + T^{8} \)
$31$ \( \)\(38\!\cdots\!00\)\( + \)\(20\!\cdots\!00\)\( T - \)\(15\!\cdots\!80\)\( T^{2} - \)\(58\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!36\)\( T^{4} + \)\(41\!\cdots\!56\)\( T^{5} - \)\(84\!\cdots\!84\)\( T^{6} - \)\(60\!\cdots\!64\)\( T^{7} + T^{8} \)
$37$ \( \)\(26\!\cdots\!76\)\( - \)\(10\!\cdots\!88\)\( T + \)\(82\!\cdots\!12\)\( T^{2} + \)\(39\!\cdots\!00\)\( T^{3} - \)\(75\!\cdots\!44\)\( T^{4} + \)\(60\!\cdots\!60\)\( T^{5} + \)\(15\!\cdots\!08\)\( T^{6} - \)\(24\!\cdots\!44\)\( T^{7} + T^{8} \)
$41$ \( -\)\(60\!\cdots\!04\)\( + \)\(43\!\cdots\!80\)\( T - \)\(33\!\cdots\!36\)\( T^{2} - \)\(30\!\cdots\!60\)\( T^{3} + \)\(65\!\cdots\!16\)\( T^{4} - \)\(17\!\cdots\!60\)\( T^{5} - \)\(43\!\cdots\!76\)\( T^{6} + \)\(11\!\cdots\!80\)\( T^{7} + T^{8} \)
$43$ \( \)\(25\!\cdots\!00\)\( - \)\(55\!\cdots\!00\)\( T + \)\(40\!\cdots\!00\)\( T^{2} - \)\(93\!\cdots\!40\)\( T^{3} - \)\(10\!\cdots\!24\)\( T^{4} + \)\(64\!\cdots\!64\)\( T^{5} - \)\(39\!\cdots\!84\)\( T^{6} - \)\(95\!\cdots\!56\)\( T^{7} + T^{8} \)
$47$ \( \)\(40\!\cdots\!00\)\( + \)\(67\!\cdots\!00\)\( T + \)\(66\!\cdots\!80\)\( T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + \)\(55\!\cdots\!36\)\( T^{4} + \)\(13\!\cdots\!04\)\( T^{5} - \)\(70\!\cdots\!04\)\( T^{6} - \)\(17\!\cdots\!96\)\( T^{7} + T^{8} \)
$53$ \( \)\(17\!\cdots\!96\)\( - \)\(26\!\cdots\!68\)\( T - \)\(11\!\cdots\!68\)\( T^{2} - \)\(93\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!96\)\( T^{4} + \)\(95\!\cdots\!20\)\( T^{5} - \)\(68\!\cdots\!92\)\( T^{6} - \)\(21\!\cdots\!24\)\( T^{7} + T^{8} \)
$59$ \( \)\(68\!\cdots\!00\)\( + \)\(19\!\cdots\!00\)\( T - \)\(72\!\cdots\!00\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(38\!\cdots\!36\)\( T^{4} + \)\(19\!\cdots\!88\)\( T^{5} - \)\(57\!\cdots\!56\)\( T^{6} - \)\(42\!\cdots\!28\)\( T^{7} + T^{8} \)
$61$ \( \)\(13\!\cdots\!76\)\( - \)\(12\!\cdots\!16\)\( T - \)\(21\!\cdots\!08\)\( T^{2} + \)\(92\!\cdots\!92\)\( T^{3} + \)\(10\!\cdots\!20\)\( T^{4} - \)\(19\!\cdots\!92\)\( T^{5} - \)\(20\!\cdots\!08\)\( T^{6} + \)\(10\!\cdots\!16\)\( T^{7} + T^{8} \)
$67$ \( -\)\(31\!\cdots\!04\)\( - \)\(75\!\cdots\!28\)\( T - \)\(38\!\cdots\!72\)\( T^{2} + \)\(12\!\cdots\!44\)\( T^{3} + \)\(84\!\cdots\!20\)\( T^{4} - \)\(61\!\cdots\!76\)\( T^{5} - \)\(52\!\cdots\!52\)\( T^{6} + \)\(76\!\cdots\!92\)\( T^{7} + T^{8} \)
$71$ \( \)\(13\!\cdots\!00\)\( - \)\(76\!\cdots\!00\)\( T - \)\(89\!\cdots\!00\)\( T^{2} + \)\(63\!\cdots\!00\)\( T^{3} + \)\(44\!\cdots\!00\)\( T^{4} - \)\(17\!\cdots\!00\)\( T^{5} - \)\(14\!\cdots\!80\)\( T^{6} + \)\(10\!\cdots\!20\)\( T^{7} + T^{8} \)
$73$ \( \)\(20\!\cdots\!76\)\( + \)\(23\!\cdots\!04\)\( T - \)\(17\!\cdots\!32\)\( T^{2} - \)\(57\!\cdots\!20\)\( T^{3} + \)\(43\!\cdots\!76\)\( T^{4} + \)\(48\!\cdots\!60\)\( T^{5} - \)\(37\!\cdots\!88\)\( T^{6} + \)\(24\!\cdots\!48\)\( T^{7} + T^{8} \)
$79$ \( -\)\(77\!\cdots\!64\)\( - \)\(18\!\cdots\!52\)\( T + \)\(85\!\cdots\!08\)\( T^{2} - \)\(87\!\cdots\!44\)\( T^{3} + \)\(13\!\cdots\!20\)\( T^{4} + \)\(12\!\cdots\!16\)\( T^{5} - \)\(34\!\cdots\!32\)\( T^{6} - \)\(24\!\cdots\!12\)\( T^{7} + T^{8} \)
$83$ \( \)\(29\!\cdots\!56\)\( - \)\(70\!\cdots\!84\)\( T - \)\(13\!\cdots\!28\)\( T^{2} + \)\(68\!\cdots\!08\)\( T^{3} + \)\(25\!\cdots\!20\)\( T^{4} - \)\(20\!\cdots\!48\)\( T^{5} - \)\(13\!\cdots\!08\)\( T^{6} + \)\(12\!\cdots\!44\)\( T^{7} + T^{8} \)
$89$ \( -\)\(12\!\cdots\!04\)\( + \)\(56\!\cdots\!48\)\( T + \)\(18\!\cdots\!12\)\( T^{2} - \)\(56\!\cdots\!40\)\( T^{3} + \)\(23\!\cdots\!56\)\( T^{4} + \)\(21\!\cdots\!60\)\( T^{5} - \)\(10\!\cdots\!72\)\( T^{6} - \)\(19\!\cdots\!36\)\( T^{7} + T^{8} \)
$97$ \( -\)\(22\!\cdots\!04\)\( + \)\(49\!\cdots\!92\)\( T - \)\(20\!\cdots\!48\)\( T^{2} - \)\(19\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!16\)\( T^{4} - \)\(20\!\cdots\!60\)\( T^{5} - \)\(80\!\cdots\!12\)\( T^{6} + \)\(54\!\cdots\!56\)\( T^{7} + T^{8} \)
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