Properties

Label 3.37.b.b
Level 3
Weight 37
Character orbit 3.b
Analytic conductor 24.627
Analytic rank 0
Dimension 10
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 37 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(24.6273775978\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{44}\cdot 3^{71}\cdot 5^{4}\cdot 7^{2}\cdot 13 \)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -55215675 - 69 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( -43520029006 + 5 \beta_{1} + 26 \beta_{2} + \beta_{3} ) q^{4} \) \( + ( -869875 \beta_{1} + 51 \beta_{2} - \beta_{5} ) q^{5} \) \( + ( 7723890958958 - 21413767 \beta_{1} - 1816 \beta_{2} - 312 \beta_{3} + 6 \beta_{5} - \beta_{6} + \beta_{7} ) q^{6} \) \( + ( -122739291389701 + 421960 \beta_{1} + 2308411 \beta_{2} - 11053 \beta_{3} + 7 \beta_{4} - 42 \beta_{6} ) q^{7} \) \( + ( 2456 - 33944682441 \beta_{1} - 10822935 \beta_{2} + 6231 \beta_{3} + 2 \beta_{4} + 318 \beta_{5} - 186 \beta_{6} + 13 \beta_{7} - 18 \beta_{8} + \beta_{9} ) q^{8} \) \( + ( 6932804017276075 + 103697837058 \beta_{1} + 48436671 \beta_{2} - 474942 \beta_{3} - 1189 \beta_{4} - 1203 \beta_{5} - \beta_{6} - 28 \beta_{7} + 143 \beta_{8} - 3 \beta_{9} ) q^{9} \) \(+O(q^{10})\) \( q\) \(+\beta_{1} q^{2}\) \(+(-55215675 - 69 \beta_{1} - \beta_{2}) q^{3}\) \(+(-43520029006 + 5 \beta_{1} + 26 \beta_{2} + \beta_{3}) q^{4}\) \(+(-869875 \beta_{1} + 51 \beta_{2} - \beta_{5}) q^{5}\) \(+(7723890958958 - 21413767 \beta_{1} - 1816 \beta_{2} - 312 \beta_{3} + 6 \beta_{5} - \beta_{6} + \beta_{7}) q^{6}\) \(+(-122739291389701 + 421960 \beta_{1} + 2308411 \beta_{2} - 11053 \beta_{3} + 7 \beta_{4} - 42 \beta_{6}) q^{7}\) \(+(2456 - 33944682441 \beta_{1} - 10822935 \beta_{2} + 6231 \beta_{3} + 2 \beta_{4} + 318 \beta_{5} - 186 \beta_{6} + 13 \beta_{7} - 18 \beta_{8} + \beta_{9}) q^{8}\) \(+(6932804017276075 + 103697837058 \beta_{1} + 48436671 \beta_{2} - 474942 \beta_{3} - 1189 \beta_{4} - 1203 \beta_{5} - \beta_{6} - 28 \beta_{7} + 143 \beta_{8} - 3 \beta_{9}) q^{9}\) \(+(97635396840185538 - 111123011 \beta_{1} - 604684808 \beta_{2} + 205893 \beta_{3} - 9474 \beta_{4} + 9232 \beta_{6} + 1361 \beta_{7} + 1361 \beta_{8}) q^{10}\) \(+(216331 - 5680628563869 \beta_{1} - 951642612 \beta_{2} + 550923 \beta_{3} - 1328 \beta_{4} - 174424 \beta_{5} - 19083 \beta_{6} - 8992 \beta_{7} + 11610 \beta_{8} + 182 \beta_{9}) q^{11}\) \(+(-1390966929653297286 + 21201604321728 \beta_{1} + 43974096911 \beta_{2} - 154665900 \beta_{3} + 153642 \beta_{4} - 53658 \beta_{5} - 398202 \beta_{6} + 5997 \beta_{7} + 137658 \beta_{8} - 531 \beta_{9}) q^{12}\) \(+(14644202138387952222 - 8159026538 \beta_{1} - 44499638042 \beta_{2} + 102935664 \beta_{3} + 142318 \beta_{4} + 1265050 \beta_{6} + 774200 \beta_{7} + 774200 \beta_{8}) q^{13}\) \(+(-79685438 + 442596300120037 \beta_{1} + 352736634096 \beta_{2} - 201679149 \beta_{3} - 417158 \beta_{4} - 81117148 \beta_{5} + 5402418 \beta_{6} - 2795947 \beta_{7} + 3674223 \beta_{8} - 10192 \beta_{9}) q^{14}\) \(+(734630691528154021 - 741333710663282 \beta_{1} + 38815958539 \beta_{2} + 272226291 \beta_{3} - 4834593 \beta_{4} + 157595520 \beta_{5} + 1597054 \beta_{6} + 1832432 \beta_{7} + 10436742 \beta_{8} + 33210 \beta_{9}) q^{15}\) \(+(\)\(81\!\cdots\!92\)\( + 707050780848 \beta_{1} + 3901587768480 \beta_{2} - 46491436864 \beta_{3} - 11013792 \beta_{4} - 74112304 \beta_{6} + 22443736 \beta_{7} + 22443736 \beta_{8}) q^{16}\) \(+(276924288 - 9840345986980836 \beta_{1} - 1084327120794 \beta_{2} + 703181268 \beta_{3} - 255408 \beta_{4} - 2684142172 \beta_{5} - 21888180 \beta_{6} - 1813152 \beta_{7} + 2153322 \beta_{8} + 231846 \beta_{9}) q^{17}\) \(+(-\)\(11\!\cdots\!34\)\( + 32988403964679174 \beta_{1} - 5229744350616 \beta_{2} + 351814850685 \beta_{3} + 28726254 \beta_{4} + 7393240872 \beta_{5} + 438424740 \beta_{6} - 30162123 \beta_{7} - 127216647 \beta_{8} - 865296 \beta_{9}) q^{18}\) \(+(\)\(25\!\cdots\!17\)\( - 19469756537341 \beta_{1} - 105591771138406 \beta_{2} - 261250049873 \beta_{3} + 380189794 \beta_{4} + 1349359503 \beta_{6} - 629163552 \beta_{7} - 629163552 \beta_{8}) q^{19}\) \(+(-2276016240 + 46548911500256810 \beta_{1} + 10833398074902 \beta_{2} - 5999134230 \beta_{3} + 157136460 \beta_{4} + 1973254708 \beta_{5} + 453017700 \beta_{6} + 1056062190 \beta_{7} - 1381286700 \beta_{8} - 2918490 \beta_{9}) q^{20}\) \(+(-\)\(33\!\cdots\!34\)\( - 490865515874336757 \beta_{1} + 277133129591731 \beta_{2} + 5156816847588 \beta_{3} + 657387948 \beta_{4} - 12789739683 \beta_{5} - 4598059200 \beta_{6} + 17058720 \beta_{7} - 1790083050 \beta_{8} + 13316058 \beta_{9}) q^{21}\) \(+(\)\(63\!\cdots\!26\)\( - 275357156455711 \beta_{1} - 1476752543985136 \beta_{2} - 17550808544511 \beta_{3} - 6968255778 \beta_{4} + 17221289624 \beta_{6} + 2537679925 \beta_{7} + 2537679925 \beta_{8}) q^{22}\) \(+(-806065926622 + 791084858107571922 \beta_{1} + 3524993633765148 \beta_{2} - 2043732935382 \beta_{3} - 1721410816 \beta_{4} + 310487937456 \beta_{5} + 58967281686 \beta_{6} - 11564002304 \beta_{7} + 15136606944 \beta_{8} + 20149792 \beta_{9}) q^{23}\) \(+(-\)\(18\!\cdots\!24\)\( + 4650804351930500299 \beta_{1} + 456606386917861 \beta_{2} + 49244785613115 \beta_{3} - 12046308390 \beta_{4} - 1144533906618 \beta_{5} + 242782909630 \beta_{6} + 2475874001 \beta_{7} + 40135743918 \beta_{8} - 133178499 \beta_{9}) q^{24}\) \(+(-\)\(72\!\cdots\!15\)\( - 1946841150162370 \beta_{1} - 10478214706677610 \beta_{2} - 92779676990440 \beta_{3} + 86070116670 \beta_{4} + 177244036690 \beta_{6} + 46192862120 \beta_{7} + 46192862120 \beta_{8}) q^{25}\) \(+(-10218213794088 + 10139533344557950262 \beta_{1} + 45026305351826016 \beta_{2} - 25940594422044 \beta_{3} + 1456584120 \beta_{4} - 115149106448 \beta_{5} + 788594731992 \beta_{6} + 9790593660 \beta_{7} - 12802600044 \beta_{8} - 30280608 \beta_{9}) q^{26}\) \(+(\)\(37\!\cdots\!66\)\( - 31532839618930209486 \beta_{1} - 6881018097337875 \beta_{2} - 406909586771163 \beta_{3} + 72158478210 \beta_{4} + 6221507481864 \beta_{5} + 1050271750953 \beta_{6} - 13305132576 \beta_{7} - 235463305914 \beta_{8} + 852027498 \beta_{9}) q^{27}\) \(+(-\)\(58\!\cdots\!96\)\( - 57425159483748070 \beta_{1} - 313820135349060844 \beta_{2} + 1222643370593442 \beta_{3} - 792087288608 \beta_{4} + 5288856363728 \beta_{6} - 562592647400 \beta_{7} - 562592647400 \beta_{8}) q^{28}\) \(+(-76282898416288 + \)\(10\!\cdots\!87\)\( \beta_{1} + 337431837699321141 \beta_{2} - 193767264580344 \beta_{3} + 89064563360 \beta_{4} - 17346994577323 \beta_{5} + 6024986880312 \beta_{6} + 598240342720 \beta_{7} - 783226423404 \beta_{8} - 871318388 \beta_{9}) q^{29}\) \(+(\)\(83\!\cdots\!10\)\( - 16921594724664076785 \beta_{1} - 130787216599961088 \beta_{2} - 2810619578981745 \beta_{3} + 25654286850 \beta_{4} + 29955005612568 \beta_{5} + 12708175159860 \beta_{6} - 74684285265 \beta_{7} - 248983451445 \beta_{8} - 2604332880 \beta_{9}) q^{30}\) \(+(-\)\(28\!\cdots\!17\)\( - 69156053470862078 \beta_{1} - 377035342545910211 \beta_{2} + 740138620776683 \beta_{3} + 5577399627217 \beta_{4} + 8687069682804 \beta_{6} + 1810945964064 \beta_{7} + 1810945964064 \beta_{8}) q^{31}\) \(+(-360605498470080 + \)\(10\!\cdots\!48\)\( \beta_{1} + 1583642526169795896 \beta_{2} - 914517464095800 \beta_{3} - 613334521104 \beta_{4} + 25635641096464 \beta_{5} + 26654944832208 \beta_{6} - 4121362771176 \beta_{7} + 5392056725136 \beta_{8} + 9850441848 \beta_{9}) q^{32}\) \(+(-\)\(18\!\cdots\!54\)\( - \)\(16\!\cdots\!98\)\( \beta_{1} + 154548941733240479 \beta_{2} + 4367219878117002 \beta_{3} - 2929139565147 \beta_{4} - 293523355885269 \beta_{5} + 38552991715877 \beta_{6} + 983583519196 \beta_{7} + 9754841383119 \beta_{8} - 10393631883 \beta_{9}) q^{33}\) \(+(\)\(11\!\cdots\!44\)\( - 368129766134545908 \beta_{1} - 1973527505096624640 \beta_{2} - 24111234380011316 \beta_{3} - 29741911095768 \beta_{4} + 21192553660384 \beta_{6} + 7163058200444 \beta_{7} + 7163058200444 \beta_{8}) q^{34}\) \(+(509314764201670 + \)\(10\!\cdots\!70\)\( \beta_{1} - 2268151451525600130 \beta_{2} + 1292097931431090 \beta_{3} + 569683658320 \beta_{4} + 546063727841800 \beta_{5} - 38146265216850 \beta_{6} + 3846000512480 \beta_{7} - 4991249028150 \beta_{8} - 51341494330 \beta_{9}) q^{35}\) \(+(-\)\(32\!\cdots\!18\)\( - \)\(24\!\cdots\!17\)\( \beta_{1} + 3956129565742613820 \beta_{2} + 56841183150239247 \beta_{3} + 17312557851044 \beta_{4} + 109653839974812 \beta_{5} - 310917424546612 \beta_{6} - 2109566041606 \beta_{7} - 44804957603284 \beta_{8} + 187635831666 \beta_{9}) q^{36}\) \(+(\)\(46\!\cdots\!26\)\( + 2668812036383775698 \beta_{1} + 14526670508767220114 \beta_{2} - 8602610988620960 \beta_{3} + 116484679428586 \beta_{4} - 266472205587714 \beta_{6} - 71636140212600 \beta_{7} - 71636140212600 \beta_{8}) q^{37}\) \(+(10126635961180046 + \)\(44\!\cdots\!59\)\( \beta_{1} - 44459336323807408032 \beta_{2} + 25689556609827861 \beta_{3} + 11584682251190 \beta_{4} - 1610988154491940 \beta_{5} - 758658883774722 \beta_{6} + 77729247303235 \beta_{7} - 101954768223735 \beta_{8} + 84631349920 \beta_{9}) q^{38}\) \(+(\)\(57\!\cdots\!66\)\( - \)\(59\!\cdots\!32\)\( \beta_{1} - 14003284466092953674 \beta_{2} + 159477692300382348 \beta_{3} - 28387939281486 \beta_{4} + 4338966634258032 \beta_{5} - 262352831951742 \beta_{6} - 19212803276496 \beta_{7} + 23449393388406 \beta_{8} - 1274124214134 \beta_{9}) q^{39}\) \(+(\)\(14\!\cdots\!64\)\( + 20987822536465570592 \beta_{1} + \)\(11\!\cdots\!76\)\( \beta_{2} + 79293646840453504 \beta_{3} - 315144100596672 \beta_{4} - 1858472033903904 \beta_{6} + 114268487355408 \beta_{7} + 114268487355408 \beta_{8}) q^{40}\) \(+(16749703295819024 + \)\(26\!\cdots\!74\)\( \beta_{1} - 73673133937752858198 \beta_{2} + 42600103646007792 \beta_{3} - 57772073648512 \beta_{4} - 8717437818999646 \beta_{5} - 1390461585749232 \beta_{6} - 388149855488768 \beta_{7} + 507948854901840 \beta_{8} + 797698593328 \beta_{9}) q^{41}\) \(+(\)\(55\!\cdots\!18\)\( - \)\(64\!\cdots\!07\)\( \beta_{1} + 12884759019081132232 \beta_{2} - 1571926589688549297 \beta_{3} - 160075393399782 \beta_{4} - 4627375786457088 \beta_{5} - 3758140262847152 \beta_{6} + 142720229798867 \beta_{7} + 409365068231763 \beta_{8} + 5388829435584 \beta_{9}) q^{42}\) \(+(-\)\(54\!\cdots\!71\)\( + 23047621140152850175 \beta_{1} + \)\(12\!\cdots\!06\)\( \beta_{2} + 1106163568848867247 \beta_{3} + 487967323780462 \beta_{4} - 976618067679037 \beta_{6} + 894661045477600 \beta_{7} + 894661045477600 \beta_{8}) q^{43}\) \(+(-41991857702292464 + \)\(13\!\cdots\!86\)\( \beta_{1} + \)\(18\!\cdots\!18\)\( \beta_{2} - 106668502961414502 \beta_{3} + 55146429232108 \beta_{4} + 41694326337137556 \beta_{5} + 3331483253454660 \beta_{6} + 373349162230142 \beta_{7} - 482165359478604 \beta_{8} - 7435610104330 \beta_{9}) q^{44}\) \(+(-\)\(75\!\cdots\!60\)\( - \)\(10\!\cdots\!65\)\( \beta_{1} + 56027988545827098699 \beta_{2} - 218851357138821180 \beta_{3} + 931397424419250 \beta_{4} - 61687663199350929 \beta_{5} + 11248048535959890 \beta_{6} - 295390238443560 \beta_{7} - 387601871568930 \beta_{8} - 14045179631670 \beta_{9}) q^{45}\) \(+(-\)\(88\!\cdots\!64\)\( - \)\(32\!\cdots\!02\)\( \beta_{1} - \)\(17\!\cdots\!80\)\( \beta_{2} + 3089888867184682726 \beta_{3} + 142591749241908 \beta_{4} + 27891978048556496 \beta_{6} - 5072630353252514 \beta_{7} - 5072630353252514 \beta_{8}) q^{46}\) \(+(-234633704798253276 + \)\(15\!\cdots\!20\)\( \beta_{1} + \)\(10\!\cdots\!76\)\( \beta_{2} - 596061635254407396 \beta_{3} + 304748139079584 \beta_{4} + 51141271355292656 \beta_{5} + 18565445086305444 \beta_{6} + 2032071007951296 \beta_{7} - 2694085552978956 \beta_{8} + 32035025294892 \beta_{9}) q^{47}\) \(+(-\)\(61\!\cdots\!84\)\( - \)\(32\!\cdots\!76\)\( \beta_{1} - \)\(43\!\cdots\!96\)\( \beta_{2} + 11503147130265519720 \beta_{3} - 385602089825520 \beta_{4} + 172063310619604176 \beta_{5} + 34220890151216928 \beta_{6} - 740474292536784 \beta_{7} - 8168513162168376 \beta_{8} + 12478947950232 \beta_{9}) q^{48}\) \(+(\)\(41\!\cdots\!67\)\( - 82057863703161940958 \beta_{1} - \)\(40\!\cdots\!70\)\( \beta_{2} - 37573283569741731896 \beta_{3} - 3896836044171518 \beta_{4} - 5196325818223666 \beta_{6} + 8752185291368344 \beta_{7} + 8752185291368344 \beta_{8}) q^{49}\) \(+(-801110649213591800 + \)\(50\!\cdots\!25\)\( \beta_{1} + \)\(35\!\cdots\!20\)\( \beta_{2} - 2033354987362406100 \beta_{3} - 132788825379800 \beta_{4} - 490478261017593520 \beta_{5} + 61433418000453000 \beta_{6} - 860543250354700 \beta_{7} + 1197554337301500 \beta_{8} - 72467082032800 \beta_{9}) q^{50}\) \(+(-\)\(23\!\cdots\!68\)\( - \)\(20\!\cdots\!56\)\( \beta_{1} + \)\(19\!\cdots\!38\)\( \beta_{2} + 10604577704060050200 \beta_{3} - 14911643855548404 \beta_{4} + 118615873230235368 \beta_{5} + 58132050995613900 \beta_{6} + 5707598741691072 \beta_{7} + 37973086923969846 \beta_{8} + 65967083929818 \beta_{9}) q^{51}\) \(+(-\)\(13\!\cdots\!00\)\( - \)\(91\!\cdots\!86\)\( \beta_{1} - \)\(50\!\cdots\!00\)\( \beta_{2} + 42337216080896604946 \beta_{3} + 17653542699670144 \beta_{4} + 105756696986261952 \beta_{6} + 1001907657511200 \beta_{7} + 1001907657511200 \beta_{8}) q^{52}\) \(+(326207998899263392 - \)\(70\!\cdots\!19\)\( \beta_{1} - \)\(14\!\cdots\!57\)\( \beta_{2} + 837809652182651232 \beta_{3} - 6854466576743168 \beta_{4} - 85955782246386437 \beta_{5} - 37141547514185568 \beta_{6} - 46012444562478592 \beta_{7} + 60304707967722912 \beta_{8} + 34473701216 \beta_{9}) q^{53}\) \(+(\)\(35\!\cdots\!64\)\( + \)\(27\!\cdots\!04\)\( \beta_{1} + \)\(91\!\cdots\!52\)\( \beta_{2} - 96758502745061644677 \beta_{3} + 77281675694568378 \beta_{4} - 182080820125545354 \beta_{5} - 369339412697916993 \beta_{6} - 13214552079185064 \beta_{7} - 60131438258457681 \beta_{8} - 323386476261984 \beta_{9}) q^{54}\) \(+(-\)\(30\!\cdots\!80\)\( + \)\(59\!\cdots\!10\)\( \beta_{1} + \)\(32\!\cdots\!30\)\( \beta_{2} + 76043704589366141320 \beta_{3} - 51278944368770910 \beta_{4} - 500211582579084970 \beta_{6} + 28202846226401440 \beta_{7} + 28202846226401440 \beta_{8}) q^{55}\) \(+(12624500981757810544 - \)\(88\!\cdots\!06\)\( \beta_{1} - \)\(55\!\cdots\!98\)\( \beta_{2} + 32007091904354066262 \beta_{3} + 27679891705228084 \beta_{4} + 4049446823049749324 \beta_{5} - 922817642882985444 \beta_{6} + 185552096187831506 \beta_{7} - 243767865248396244 \beta_{8} + 602865608568986 \beta_{9}) q^{56}\) \(+(\)\(14\!\cdots\!44\)\( + \)\(55\!\cdots\!04\)\( \beta_{1} - \)\(34\!\cdots\!69\)\( \beta_{2} - 35372422821039877890 \beta_{3} - 186014499714684567 \beta_{4} - 4854259422358010739 \beta_{5} - 424200359240236179 \beta_{6} + 12661728689883276 \beta_{7} + 6984403967071089 \beta_{8} + 615271072138227 \beta_{9}) q^{57}\) \(+(-\)\(11\!\cdots\!02\)\( + \)\(98\!\cdots\!67\)\( \beta_{1} + \)\(53\!\cdots\!72\)\( \beta_{2} + \)\(25\!\cdots\!27\)\( \beta_{3} + 46834279690224426 \beta_{4} - 966340105191260688 \beta_{6} - 342161565114401925 \beta_{7} - 342161565114401925 \beta_{8}) q^{58}\) \(+(-248473290617989641 - \)\(77\!\cdots\!41\)\( \beta_{1} + \)\(91\!\cdots\!62\)\( \beta_{2} - 574990557679976733 \beta_{3} - 38339939095917120 \beta_{4} - 883160087007581536 \beta_{5} - 47081191979851011 \beta_{6} - 256506286086645120 \beta_{7} + 338127603998229432 \beta_{8} - 2022766934476536 \beta_{9}) q^{59}\) \(+(\)\(19\!\cdots\!72\)\( + \)\(20\!\cdots\!46\)\( \beta_{1} - \)\(11\!\cdots\!22\)\( \beta_{2} + \)\(34\!\cdots\!02\)\( \beta_{3} + 105539064945390684 \beta_{4} + 14721041325292970340 \beta_{5} - 136238154466992332 \beta_{6} - 19197418429533946 \beta_{7} - 161143394070961836 \beta_{8} - 62042482517490 \beta_{9}) q^{60}\) \(+(\)\(25\!\cdots\!06\)\( - \)\(82\!\cdots\!22\)\( \beta_{1} - \)\(43\!\cdots\!54\)\( \beta_{2} - \)\(91\!\cdots\!48\)\( \beta_{3} + 335633930737767858 \beta_{4} + 956787738789134646 \beta_{6} + 960107303686220136 \beta_{7} + 960107303686220136 \beta_{8}) q^{61}\) \(+(-20479459198310457590 - \)\(31\!\cdots\!23\)\( \beta_{1} + \)\(91\!\cdots\!04\)\( \beta_{2} - 51979334642157285345 \beta_{3} - 5915420189469086 \beta_{4} - 31505072926712278924 \beta_{5} + 1563581743740720762 \beta_{6} - 40687996208409559 \beta_{7} + 51112904977204299 \beta_{8} + 2301037600887632 \beta_{9}) q^{62}\) \(+(-\)\(43\!\cdots\!95\)\( + \)\(31\!\cdots\!96\)\( \beta_{1} + \)\(33\!\cdots\!05\)\( \beta_{2} + \)\(29\!\cdots\!93\)\( \beta_{3} + 860959790160448993 \beta_{4} - 11906952381681356496 \beta_{5} + 867485274225166114 \beta_{6} + 111919274676924064 \beta_{7} + 1381021825360508296 \beta_{8} - 2217723355500072 \beta_{9}) q^{63}\) \(+(-\)\(62\!\cdots\!40\)\( - \)\(76\!\cdots\!24\)\( \beta_{1} - \)\(41\!\cdots\!96\)\( \beta_{2} - \)\(12\!\cdots\!64\)\( \beta_{3} - 1511037473622522624 \beta_{4} + 5845125335525859712 \beta_{6} - 502451063766217408 \beta_{7} - 502451063766217408 \beta_{8}) q^{64}\) \(+(-94347335797117323280 + \)\(14\!\cdots\!70\)\( \beta_{1} + \)\(41\!\cdots\!90\)\( \beta_{2} - \)\(23\!\cdots\!60\)\( \beta_{3} - 16688823013518880 \beta_{4} + 47164444060726925630 \beta_{5} + 7225498796153669400 \beta_{6} - 113933913668688320 \beta_{7} + 145015671244895100 \beta_{8} + 4478514083157220 \beta_{9}) q^{65}\) \(+(\)\(18\!\cdots\!10\)\( - \)\(44\!\cdots\!45\)\( \beta_{1} - \)\(69\!\cdots\!00\)\( \beta_{2} + \)\(67\!\cdots\!75\)\( \beta_{3} - 3289240448201562558 \beta_{4} + 31692004977865656600 \beta_{5} + 16237724479069233756 \beta_{6} - 212874758146175037 \beta_{7} - 2509885437102964833 \beta_{8} + 3551567942385936 \beta_{9}) q^{66}\) \(+(-\)\(32\!\cdots\!47\)\( - \)\(11\!\cdots\!39\)\( \beta_{1} - \)\(59\!\cdots\!08\)\( \beta_{2} + \)\(12\!\cdots\!83\)\( \beta_{3} + 1593455209932930840 \beta_{4} + 9599032125061869889 \beta_{6} - 2150383898154462400 \beta_{7} - 2150383898154462400 \beta_{8}) q^{67}\) \(+(-\)\(10\!\cdots\!76\)\( + \)\(19\!\cdots\!36\)\( \beta_{1} + \)\(44\!\cdots\!40\)\( \beta_{2} - \)\(25\!\cdots\!76\)\( \beta_{3} + 340309654995302288 \beta_{4} + 23649540937341319792 \beta_{5} + 8422378979406622896 \beta_{6} + 2292533181661976872 \beta_{7} - 2986292491974834192 \beta_{8} - 19068169354752056 \beta_{9}) q^{68}\) \(+(\)\(52\!\cdots\!72\)\( - \)\(23\!\cdots\!20\)\( \beta_{1} - \)\(73\!\cdots\!06\)\( \beta_{2} - \)\(26\!\cdots\!68\)\( \beta_{3} + 4953198772547973198 \beta_{4} - \)\(19\!\cdots\!34\)\( \beta_{5} - 1939838752027198874 \beta_{6} - 220182246785219416 \beta_{7} - 2737861444948587930 \beta_{8} + 1986433573538898 \beta_{9}) q^{69}\) \(+(-\)\(11\!\cdots\!00\)\( + \)\(24\!\cdots\!50\)\( \beta_{1} + \)\(13\!\cdots\!00\)\( \beta_{2} + \)\(11\!\cdots\!50\)\( \beta_{3} + 6673015362153081900 \beta_{4} - 14903453187079388400 \beta_{6} + 1624563177942454050 \beta_{7} + 1624563177942454050 \beta_{8}) q^{70}\) \(+(\)\(31\!\cdots\!82\)\( + \)\(34\!\cdots\!82\)\( \beta_{1} - \)\(13\!\cdots\!64\)\( \beta_{2} + \)\(80\!\cdots\!06\)\( \beta_{3} - 130575005621839456 \beta_{4} + \)\(16\!\cdots\!72\)\( \beta_{5} - 24494919855838244646 \beta_{6} - 882784966300856384 \beta_{7} + 1142829992325600180 \beta_{8} + 14723957122495084 \beta_{9}) q^{71}\) \(+(\)\(19\!\cdots\!84\)\( - \)\(44\!\cdots\!37\)\( \beta_{1} + \)\(13\!\cdots\!25\)\( \beta_{2} - \)\(21\!\cdots\!89\)\( \beta_{3} + 2447278398791851842 \beta_{4} + \)\(29\!\cdots\!06\)\( \beta_{5} - 47890138109465028762 \beta_{6} + 1245156774584375421 \beta_{7} + 13975762714303726494 \beta_{8} - 2514954992238783 \beta_{9}) q^{72}\) \(+(-\)\(43\!\cdots\!94\)\( + \)\(26\!\cdots\!88\)\( \beta_{1} + \)\(14\!\cdots\!84\)\( \beta_{2} - \)\(54\!\cdots\!60\)\( \beta_{3} - 26050794596446578384 \beta_{4} - 28554032984427941184 \beta_{6} + 4804695488927023200 \beta_{7} + 4804695488927023200 \beta_{8}) q^{73}\) \(+(\)\(67\!\cdots\!32\)\( + \)\(46\!\cdots\!82\)\( \beta_{1} - \)\(29\!\cdots\!64\)\( \beta_{2} + \)\(17\!\cdots\!56\)\( \beta_{3} - 2623152754933544056 \beta_{4} - \)\(97\!\cdots\!28\)\( \beta_{5} - 56701170640126323096 \beta_{6} - 17625806977600344284 \beta_{7} + 23061826427396203980 \beta_{8} + 40411688284151584 \beta_{9}) q^{74}\) \(+(\)\(15\!\cdots\!25\)\( - \)\(49\!\cdots\!75\)\( \beta_{1} + \)\(41\!\cdots\!15\)\( \beta_{2} + \)\(44\!\cdots\!00\)\( \beta_{3} - 27735258005943627750 \beta_{4} + 74577636280558573560 \beta_{5} - 83487234057184310550 \beta_{6} - 1641496063841836800 \beta_{7} - 7827901199768857650 \beta_{8} - 51385986055756350 \beta_{9}) q^{75}\) \(+(-\)\(32\!\cdots\!48\)\( + \)\(14\!\cdots\!26\)\( \beta_{1} + \)\(78\!\cdots\!96\)\( \beta_{2} - \)\(17\!\cdots\!42\)\( \beta_{3} + 23419660912434627616 \beta_{4} - \)\(12\!\cdots\!08\)\( \beta_{6} + 16554653824576466472 \beta_{7} + 16554653824576466472 \beta_{8}) q^{76}\) \(+(\)\(18\!\cdots\!48\)\( + \)\(15\!\cdots\!78\)\( \beta_{1} - \)\(80\!\cdots\!98\)\( \beta_{2} + \)\(46\!\cdots\!08\)\( \beta_{3} + 5530940093148899968 \beta_{4} + \)\(89\!\cdots\!62\)\( \beta_{5} - \)\(13\!\cdots\!12\)\( \beta_{6} + 37143051378582017792 \beta_{7} - 48646117103881640112 \beta_{8} - 35590770243844816 \beta_{9}) q^{77}\) \(+(\)\(66\!\cdots\!04\)\( - \)\(31\!\cdots\!78\)\( \beta_{1} - \)\(23\!\cdots\!04\)\( \beta_{2} + \)\(23\!\cdots\!92\)\( \beta_{3} + 55409171445094764600 \beta_{4} + \)\(23\!\cdots\!12\)\( \beta_{5} - 5103481485766946618 \beta_{6} + 4689979237207109342 \beta_{7} - 6860240918104138812 \beta_{8} + 143085750097793184 \beta_{9}) q^{78}\) \(+(-\)\(62\!\cdots\!53\)\( - \)\(25\!\cdots\!46\)\( \beta_{1} - \)\(12\!\cdots\!11\)\( \beta_{2} - \)\(94\!\cdots\!13\)\( \beta_{3} + 65032006173267409489 \beta_{4} - 62290711269195518732 \beta_{6} - 92244373399009114912 \beta_{7} - 92244373399009114912 \beta_{8}) q^{79}\) \(+(-\)\(35\!\cdots\!60\)\( - \)\(38\!\cdots\!60\)\( \beta_{1} + \)\(15\!\cdots\!20\)\( \beta_{2} - \)\(89\!\cdots\!20\)\( \beta_{3} - 1084354754393880160 \beta_{4} + \)\(94\!\cdots\!20\)\( \beta_{5} + \)\(26\!\cdots\!00\)\( \beta_{6} - 7144415080611920240 \beta_{7} + 9667889071345807200 \beta_{8} - 316320811125446960 \beta_{9}) q^{80}\) \(+(\)\(10\!\cdots\!05\)\( + \)\(24\!\cdots\!54\)\( \beta_{1} - \)\(89\!\cdots\!28\)\( \beta_{2} + \)\(14\!\cdots\!12\)\( \beta_{3} - 21119819647296601332 \beta_{4} - \)\(18\!\cdots\!66\)\( \beta_{5} + \)\(32\!\cdots\!88\)\( \beta_{6} - 18018164040363866448 \beta_{7} - 78993814893447473190 \beta_{8} + 46463009632401798 \beta_{9}) q^{81}\) \(+(-\)\(29\!\cdots\!96\)\( - \)\(89\!\cdots\!94\)\( \beta_{1} - \)\(48\!\cdots\!44\)\( \beta_{2} + \)\(24\!\cdots\!06\)\( \beta_{3} - \)\(20\!\cdots\!12\)\( \beta_{4} + \)\(93\!\cdots\!96\)\( \beta_{6} + 82653876803223586350 \beta_{7} + 82653876803223586350 \beta_{8}) q^{82}\) \(+(-\)\(48\!\cdots\!39\)\( - \)\(76\!\cdots\!15\)\( \beta_{1} + \)\(21\!\cdots\!84\)\( \beta_{2} - \)\(12\!\cdots\!19\)\( \beta_{3} - 1211766694465432304 \beta_{4} + \)\(96\!\cdots\!64\)\( \beta_{5} + \)\(36\!\cdots\!51\)\( \beta_{6} - 8428526881273026976 \beta_{7} + 10381459051303559586 \beta_{8} + 691418565799418798 \beta_{9}) q^{83}\) \(+(\)\(49\!\cdots\!16\)\( + \)\(11\!\cdots\!68\)\( \beta_{1} + \)\(34\!\cdots\!06\)\( \beta_{2} - \)\(72\!\cdots\!12\)\( \beta_{3} - \)\(14\!\cdots\!92\)\( \beta_{4} - \)\(20\!\cdots\!08\)\( \beta_{5} + \)\(53\!\cdots\!80\)\( \beta_{6} + 10521076055138136810 \beta_{7} + \)\(21\!\cdots\!60\)\( \beta_{8} - 899863480309265262 \beta_{9}) q^{84}\) \(+(-\)\(41\!\cdots\!32\)\( + \)\(59\!\cdots\!04\)\( \beta_{1} + \)\(34\!\cdots\!12\)\( \beta_{2} - \)\(20\!\cdots\!52\)\( \beta_{3} + \)\(16\!\cdots\!36\)\( \beta_{4} - 32137504152165464248 \beta_{6} + \)\(14\!\cdots\!96\)\( \beta_{7} + \)\(14\!\cdots\!96\)\( \beta_{8}) q^{85}\) \(+(-\)\(12\!\cdots\!78\)\( - \)\(12\!\cdots\!53\)\( \beta_{1} + \)\(54\!\cdots\!76\)\( \beta_{2} - \)\(31\!\cdots\!19\)\( \beta_{3} - 18420278467810462138 \beta_{4} - \)\(37\!\cdots\!88\)\( \beta_{5} + \)\(91\!\cdots\!38\)\( \beta_{6} - \)\(12\!\cdots\!17\)\( \beta_{7} + \)\(16\!\cdots\!73\)\( \beta_{8} + 975594285587871328 \beta_{9}) q^{86}\) \(+(\)\(50\!\cdots\!59\)\( + \)\(63\!\cdots\!42\)\( \beta_{1} - \)\(22\!\cdots\!59\)\( \beta_{2} + \)\(37\!\cdots\!37\)\( \beta_{3} + \)\(37\!\cdots\!85\)\( \beta_{4} + \)\(11\!\cdots\!12\)\( \beta_{5} + 69331013660469454802 \beta_{6} + \)\(10\!\cdots\!92\)\( \beta_{7} + 85605505119929872638 \beta_{8} + 1532946450364443234 \beta_{9}) q^{87}\) \(+(-\)\(10\!\cdots\!52\)\( - \)\(49\!\cdots\!08\)\( \beta_{1} - \)\(34\!\cdots\!28\)\( \beta_{2} + \)\(64\!\cdots\!52\)\( \beta_{3} + \)\(15\!\cdots\!76\)\( \beta_{4} + \)\(27\!\cdots\!12\)\( \beta_{6} - \)\(14\!\cdots\!00\)\( \beta_{7} - \)\(14\!\cdots\!00\)\( \beta_{8}) q^{88}\) \(+(\)\(61\!\cdots\!16\)\( + \)\(96\!\cdots\!66\)\( \beta_{1} - \)\(26\!\cdots\!12\)\( \beta_{2} + \)\(15\!\cdots\!08\)\( \beta_{3} - 2200771656255429040 \beta_{4} - \)\(16\!\cdots\!14\)\( \beta_{5} - \)\(46\!\cdots\!44\)\( \beta_{6} - 12223330615791861920 \beta_{7} + 21784545798673866858 \beta_{8} - 5992345930318517594 \beta_{9}) q^{89}\) \(+(\)\(11\!\cdots\!02\)\( + \)\(34\!\cdots\!81\)\( \beta_{1} - \)\(12\!\cdots\!12\)\( \beta_{2} + \)\(27\!\cdots\!97\)\( \beta_{3} - \)\(27\!\cdots\!46\)\( \beta_{4} - \)\(26\!\cdots\!20\)\( \beta_{5} + \)\(48\!\cdots\!28\)\( \beta_{6} - \)\(24\!\cdots\!31\)\( \beta_{7} - \)\(74\!\cdots\!31\)\( \beta_{8} + 546018460205796000 \beta_{9}) q^{90}\) \(+(-\)\(19\!\cdots\!42\)\( + \)\(17\!\cdots\!52\)\( \beta_{1} + \)\(98\!\cdots\!30\)\( \beta_{2} - \)\(10\!\cdots\!26\)\( \beta_{3} - \)\(41\!\cdots\!58\)\( \beta_{4} - \)\(23\!\cdots\!96\)\( \beta_{6} - \)\(22\!\cdots\!36\)\( \beta_{7} - \)\(22\!\cdots\!36\)\( \beta_{8}) q^{91}\) \(+(\)\(60\!\cdots\!52\)\( - \)\(15\!\cdots\!80\)\( \beta_{1} - \)\(26\!\cdots\!16\)\( \beta_{2} + \)\(15\!\cdots\!72\)\( \beta_{3} + \)\(14\!\cdots\!28\)\( \beta_{4} + \)\(42\!\cdots\!52\)\( \beta_{5} - \)\(43\!\cdots\!40\)\( \beta_{6} + \)\(97\!\cdots\!32\)\( \beta_{7} - \)\(12\!\cdots\!52\)\( \beta_{8} + 6886478907549904964 \beta_{9}) q^{92}\) \(+(\)\(70\!\cdots\!50\)\( - \)\(98\!\cdots\!83\)\( \beta_{1} + \)\(26\!\cdots\!91\)\( \beta_{2} + \)\(16\!\cdots\!80\)\( \beta_{3} - \)\(64\!\cdots\!82\)\( \beta_{4} - \)\(26\!\cdots\!03\)\( \beta_{5} - \)\(80\!\cdots\!66\)\( \beta_{6} - 33640379710835572248 \beta_{7} + \)\(40\!\cdots\!28\)\( \beta_{8} - 4906897535566497096 \beta_{9}) q^{93}\) \(+(-\)\(17\!\cdots\!72\)\( + \)\(52\!\cdots\!00\)\( \beta_{1} + \)\(28\!\cdots\!56\)\( \beta_{2} - \)\(61\!\cdots\!04\)\( \beta_{3} + \)\(63\!\cdots\!20\)\( \beta_{4} - \)\(54\!\cdots\!60\)\( \beta_{6} - \)\(85\!\cdots\!60\)\( \beta_{7} - \)\(85\!\cdots\!60\)\( \beta_{8}) q^{94}\) \(+(\)\(10\!\cdots\!10\)\( + \)\(18\!\cdots\!10\)\( \beta_{1} - \)\(47\!\cdots\!72\)\( \beta_{2} + \)\(27\!\cdots\!70\)\( \beta_{3} - \)\(17\!\cdots\!40\)\( \beta_{4} - \)\(13\!\cdots\!68\)\( \beta_{5} - \)\(85\!\cdots\!50\)\( \beta_{6} - \)\(11\!\cdots\!60\)\( \beta_{7} + \)\(15\!\cdots\!00\)\( \beta_{8} + 11434182215627035760 \beta_{9}) q^{95}\) \(+(\)\(24\!\cdots\!72\)\( - \)\(95\!\cdots\!80\)\( \beta_{1} - \)\(40\!\cdots\!76\)\( \beta_{2} - \)\(90\!\cdots\!12\)\( \beta_{3} + \)\(17\!\cdots\!76\)\( \beta_{4} - \)\(53\!\cdots\!12\)\( \beta_{5} - \)\(81\!\cdots\!56\)\( \beta_{6} + \)\(98\!\cdots\!00\)\( \beta_{7} - 23784578999839890288 \beta_{8} + 4301877076275225240 \beta_{9}) q^{96}\) \(+(-\)\(50\!\cdots\!10\)\( + \)\(35\!\cdots\!74\)\( \beta_{1} + \)\(19\!\cdots\!10\)\( \beta_{2} - \)\(68\!\cdots\!44\)\( \beta_{3} - \)\(17\!\cdots\!26\)\( \beta_{4} - \)\(19\!\cdots\!38\)\( \beta_{6} + \)\(33\!\cdots\!00\)\( \beta_{7} + \)\(33\!\cdots\!00\)\( \beta_{8}) q^{97}\) \(+(-\)\(22\!\cdots\!24\)\( + \)\(26\!\cdots\!35\)\( \beta_{1} + \)\(10\!\cdots\!24\)\( \beta_{2} - \)\(57\!\cdots\!04\)\( \beta_{3} - \)\(15\!\cdots\!84\)\( \beta_{4} + \)\(96\!\cdots\!44\)\( \beta_{5} + \)\(17\!\cdots\!56\)\( \beta_{6} - \)\(10\!\cdots\!96\)\( \beta_{7} + \)\(13\!\cdots\!56\)\( \beta_{8} - 41490019499962148192 \beta_{9}) q^{98}\) \(+(\)\(29\!\cdots\!51\)\( - \)\(11\!\cdots\!99\)\( \beta_{1} + \)\(13\!\cdots\!08\)\( \beta_{2} - \)\(20\!\cdots\!37\)\( \beta_{3} - \)\(49\!\cdots\!10\)\( \beta_{4} + \)\(40\!\cdots\!16\)\( \beta_{5} - \)\(10\!\cdots\!29\)\( \beta_{6} - \)\(15\!\cdots\!60\)\( \beta_{7} + \)\(33\!\cdots\!08\)\( \beta_{8} + 2733205171963422636 \beta_{9}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut -\mathstrut 552156750q^{3} \) \(\mathstrut -\mathstrut 435200290064q^{4} \) \(\mathstrut +\mathstrut 77238909590832q^{6} \) \(\mathstrut -\mathstrut 1227392913852700q^{7} \) \(\mathstrut +\mathstrut 69328040174658378q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut -\mathstrut 552156750q^{3} \) \(\mathstrut -\mathstrut 435200290064q^{4} \) \(\mathstrut +\mathstrut 77238909590832q^{6} \) \(\mathstrut -\mathstrut 1227392913852700q^{7} \) \(\mathstrut +\mathstrut 69328040174658378q^{9} \) \(\mathstrut +\mathstrut 976353968400999840q^{10} \) \(\mathstrut -\mathstrut 13909669295912917200q^{12} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!00\)\(q^{13} \) \(\mathstrut +\mathstrut 7346306914204243680q^{15} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!44\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!36\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!72\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!00\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!04\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!50\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!50\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!00\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(83\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!20\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!76\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!68\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!00\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!68\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!20\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!00\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!00\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(75\!\cdots\!00\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(88\!\cdots\!76\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!00\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!26\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!88\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!68\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!00\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!60\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!20\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!00\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!48\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!60\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!00\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!48\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!00\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!50\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!76\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(66\!\cdots\!00\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!84\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!90\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!00\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!28\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!60\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!00\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!60\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!84\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!00\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!84\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!76\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!00\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!20\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut +\mathstrut \) \(1732091138\) \(x^{8}\mathstrut +\mathstrut \) \(1041970113139590528\) \(x^{6}\mathstrut +\mathstrut \) \(259633704442571812012752896\) \(x^{4}\mathstrut +\mathstrut \) \(25957407434307800681948964695572480\) \(x^{2}\mathstrut +\mathstrut \) \(871827272021197959834823668999764154777600\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 18 \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(117545333724888589553925\) \(\nu^{9}\mathstrut +\mathstrut \) \(1104697757607640351180878336\) \(\nu^{8}\mathstrut -\mathstrut \) \(189508897968531290018365415863050\) \(\nu^{7}\mathstrut +\mathstrut \) \(1721340955175379871981262163066676224\) \(\nu^{6}\mathstrut -\mathstrut \) \(99702511198202418898547938279566293654400\) \(\nu^{5}\mathstrut +\mathstrut \) \(892863217557603983286848119360548259325214720\) \(\nu^{4}\mathstrut -\mathstrut \) \(18729386111550257326236335471415748973419487232000\) \(\nu^{3}\mathstrut +\mathstrut \) \(174777932919856881932667950321525593574512936051802112\) \(\nu^{2}\mathstrut -\mathstrut \) \(943772191629184136622175364640195252196373604939595776000\) \(\nu\mathstrut +\mathstrut \) \(9884794768673760636726955462874354987044134247302245835079680\)\()/\)\(35\!\cdots\!00\)
\(\beta_{3}\)\(=\)\((\)\(1528089338423551664201025\) \(\nu^{9}\mathstrut -\mathstrut \) \(14361070848899324565351418368\) \(\nu^{8}\mathstrut +\mathstrut \) \(2463615673590906770238750406219650\) \(\nu^{7}\mathstrut -\mathstrut \) \(22377432417279938335756408119866790912\) \(\nu^{6}\mathstrut +\mathstrut \) \(1296132645576631445681123197634361817507200\) \(\nu^{5}\mathstrut -\mathstrut \) \(11607221828248851782729025551687127371227791360\) \(\nu^{4}\mathstrut +\mathstrut \) \(243482019450153345241072361128404736654453334016000\) \(\nu^{3}\mathstrut -\mathstrut \) \(1693930877687874456920180684020564607658610395873017856\) \(\nu^{2}\mathstrut +\mathstrut \) \(12268877884998763146919334045136382926300409625944522752000\) \(\nu\mathstrut +\mathstrut \) \(71790538374932131599805611567804696752697736814648905129000960\)\()/\)\(17\!\cdots\!00\)
\(\beta_{4}\)\(=\)\((\)\(18\!\cdots\!25\) \(\nu^{9}\mathstrut -\mathstrut \) \(46\!\cdots\!44\) \(\nu^{8}\mathstrut +\mathstrut \) \(29\!\cdots\!50\) \(\nu^{7}\mathstrut -\mathstrut \) \(81\!\cdots\!60\) \(\nu^{6}\mathstrut +\mathstrut \) \(15\!\cdots\!00\) \(\nu^{5}\mathstrut -\mathstrut \) \(48\!\cdots\!68\) \(\nu^{4}\mathstrut +\mathstrut \) \(29\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(10\!\cdots\!32\) \(\nu^{2}\mathstrut +\mathstrut \) \(14\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(62\!\cdots\!80\)\()/\)\(50\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(1512842034757447949955748299\) \(\nu^{9}\mathstrut +\mathstrut \) \(56339585637989657910224795136\) \(\nu^{8}\mathstrut +\mathstrut \) \(2568575388174211652943164314694066838\) \(\nu^{7}\mathstrut +\mathstrut \) \(87788388713944373471044370316400487424\) \(\nu^{6}\mathstrut +\mathstrut \) \(1461866596273207114148976022897359966646475904\) \(\nu^{5}\mathstrut +\mathstrut \) \(45536024095437803147629254087387961225585950720\) \(\nu^{4}\mathstrut +\mathstrut \) \(311504670163830124432339864484387368681113574844989440\) \(\nu^{3}\mathstrut +\mathstrut \) \(8913674578912700978566065466397805272300159738641907712\) \(\nu^{2}\mathstrut +\mathstrut \) \(18689566731073686124484754616475044856570067243161781220147200\) \(\nu\mathstrut +\mathstrut \) \(504124533202361792473074728606592104339250846612414537589063680\)\()/\)\(35\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(35\!\cdots\!75\) \(\nu^{9}\mathstrut -\mathstrut \) \(32\!\cdots\!32\) \(\nu^{8}\mathstrut -\mathstrut \) \(56\!\cdots\!50\) \(\nu^{7}\mathstrut -\mathstrut \) \(50\!\cdots\!88\) \(\nu^{6}\mathstrut -\mathstrut \) \(29\!\cdots\!00\) \(\nu^{5}\mathstrut -\mathstrut \) \(25\!\cdots\!40\) \(\nu^{4}\mathstrut -\mathstrut \) \(56\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(50\!\cdots\!44\) \(\nu^{2}\mathstrut -\mathstrut \) \(28\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(28\!\cdots\!60\)\()/\)\(17\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(44\!\cdots\!49\) \(\nu^{9}\mathstrut -\mathstrut \) \(40\!\cdots\!92\) \(\nu^{8}\mathstrut -\mathstrut \) \(70\!\cdots\!70\) \(\nu^{7}\mathstrut -\mathstrut \) \(65\!\cdots\!28\) \(\nu^{6}\mathstrut -\mathstrut \) \(37\!\cdots\!48\) \(\nu^{5}\mathstrut -\mathstrut \) \(33\!\cdots\!40\) \(\nu^{4}\mathstrut -\mathstrut \) \(75\!\cdots\!32\) \(\nu^{3}\mathstrut -\mathstrut \) \(61\!\cdots\!64\) \(\nu^{2}\mathstrut -\mathstrut \) \(42\!\cdots\!80\) \(\nu\mathstrut -\mathstrut \) \(29\!\cdots\!60\)\()/\)\(44\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(98\!\cdots\!23\) \(\nu^{9}\mathstrut -\mathstrut \) \(74\!\cdots\!20\) \(\nu^{8}\mathstrut +\mathstrut \) \(15\!\cdots\!90\) \(\nu^{7}\mathstrut -\mathstrut \) \(12\!\cdots\!44\) \(\nu^{6}\mathstrut +\mathstrut \) \(84\!\cdots\!96\) \(\nu^{5}\mathstrut -\mathstrut \) \(63\!\cdots\!88\) \(\nu^{4}\mathstrut +\mathstrut \) \(16\!\cdots\!64\) \(\nu^{3}\mathstrut -\mathstrut \) \(11\!\cdots\!24\) \(\nu^{2}\mathstrut +\mathstrut \) \(93\!\cdots\!60\) \(\nu\mathstrut -\mathstrut \) \(60\!\cdots\!60\)\()/\)\(89\!\cdots\!00\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(19\!\cdots\!05\) \(\nu^{9}\mathstrut -\mathstrut \) \(29\!\cdots\!32\) \(\nu^{8}\mathstrut -\mathstrut \) \(31\!\cdots\!74\) \(\nu^{7}\mathstrut -\mathstrut \) \(47\!\cdots\!00\) \(\nu^{6}\mathstrut -\mathstrut \) \(16\!\cdots\!68\) \(\nu^{5}\mathstrut -\mathstrut \) \(24\!\cdots\!44\) \(\nu^{4}\mathstrut -\mathstrut \) \(29\!\cdots\!84\) \(\nu^{3}\mathstrut -\mathstrut \) \(46\!\cdots\!16\) \(\nu^{2}\mathstrut -\mathstrut \) \(46\!\cdots\!60\) \(\nu\mathstrut -\mathstrut \) \(23\!\cdots\!40\)\()/\)\(35\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/18\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(26\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut -\mathstrut \) \(112239505742\)\()/324\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{9}\mathstrut -\mathstrut \) \(18\) \(\beta_{8}\mathstrut +\mathstrut \) \(13\) \(\beta_{7}\mathstrut -\mathstrut \) \(186\) \(\beta_{6}\mathstrut +\mathstrut \) \(318\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(6231\) \(\beta_{3}\mathstrut -\mathstrut \) \(10822935\) \(\beta_{2}\mathstrut -\mathstrut \) \(171383635913\) \(\beta_{1}\mathstrut +\mathstrut \) \(2456\)\()/5832\)
\(\nu^{4}\)\(=\)\((\)\(2805467\) \(\beta_{8}\mathstrut +\mathstrut \) \(2805467\) \(\beta_{7}\mathstrut -\mathstrut \) \(9264038\) \(\beta_{6}\mathstrut -\mathstrut \) \(1376724\) \(\beta_{4}\mathstrut -\mathstrut \) \(31581233384\) \(\beta_{3}\mathstrut -\mathstrut \) \(182316427116\) \(\beta_{2}\mathstrut -\mathstrut \) \(40467671274\) \(\beta_{1}\mathstrut +\mathstrut \) \(2404473925951683653104\)\()/13122\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(33128433137\) \(\beta_{9}\mathstrut +\mathstrut \) \(1292482381266\) \(\beta_{8}\mathstrut -\mathstrut \) \(961846945181\) \(\beta_{7}\mathstrut +\mathstrut \) \(9722779440474\) \(\beta_{6}\mathstrut -\mathstrut \) \(7721941663966\) \(\beta_{5}\mathstrut -\mathstrut \) \(145386291874\) \(\beta_{4}\mathstrut -\mathstrut \) \(328410212782983\) \(\beta_{3}\mathstrut +\mathstrut \) \(569828530745094567\) \(\beta_{2}\mathstrut +\mathstrut \) \(4250552322821837906329\) \(\beta_{1}\mathstrut -\mathstrut \) \(129463204740568\)\()/236196\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(42781562673806489\) \(\beta_{8}\mathstrut -\mathstrut \) \(42781562673806489\) \(\beta_{7}\mathstrut +\mathstrut \) \(163072495252201346\) \(\beta_{6}\mathstrut +\mathstrut \) \(11839961676943708\) \(\beta_{4}\mathstrut +\mathstrut \) \(298106395447501993560\) \(\beta_{3}\mathstrut -\mathstrut \) \(3398542331863473547996\) \(\beta_{2}\mathstrut -\mathstrut \) \(559199299532174128882\) \(\beta_{1}\mathstrut -\mathstrut \) \(19877839227752914839508526094992\)\()/177147\)
\(\nu^{7}\)\(=\)\((\)\(35497906761239824179\) \(\beta_{9}\mathstrut -\mathstrut \) \(2079222999188009312822\) \(\beta_{8}\mathstrut +\mathstrut \) \(1560389943269118867127\) \(\beta_{7}\mathstrut -\mathstrut \) \(12901509473599665366990\) \(\beta_{6}\mathstrut +\mathstrut \) \(15230129069742632898522\) \(\beta_{5}\mathstrut +\mathstrut \) \(234701380249994875558\) \(\beta_{4}\mathstrut +\mathstrut \) \(438314414866346531966517\) \(\beta_{3}\mathstrut -\mathstrut \) \(760636313505866479059812501\) \(\beta_{2}\mathstrut -\mathstrut \) \(4125458854998582067663499984171\) \(\beta_{1}\mathstrut +\mathstrut \) \(172809389968304706792072\)\()/354294\)
\(\nu^{8}\)\(=\)\((\)\(36051063501798798897391986\) \(\beta_{8}\mathstrut +\mathstrut \) \(36051063501798798897391986\) \(\beta_{7}\mathstrut -\mathstrut \) \(157863451260028380024624004\) \(\beta_{6}\mathstrut -\mathstrut \) \(3427233323973509161111992\) \(\beta_{4}\mathstrut -\mathstrut \) \(206261551283998606844706511536\) \(\beta_{3}\mathstrut +\mathstrut \) \(5330634458244814343078142465208\) \(\beta_{2}\mathstrut +\mathstrut \) \(934616024170565231428685772388\) \(\beta_{1}\mathstrut +\mathstrut \) \(12861771605874362821401930268586013141280\)\()/177147\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(12380318898365983195110709395\) \(\beta_{9}\mathstrut +\mathstrut \) \(940982628881701748718789349110\) \(\beta_{8}\mathstrut -\mathstrut \) \(708881723401506229313329962135\) \(\beta_{7}\mathstrut +\mathstrut \) \(5069525668531828756541719207822\) \(\beta_{6}\mathstrut -\mathstrut \) \(8903897297979417121727995082010\) \(\beta_{5}\mathstrut -\mathstrut \) \(106386837964621592524972178790\) \(\beta_{4}\mathstrut -\mathstrut \) \(173064543939236735630795852091861\) \(\beta_{3}\mathstrut +\mathstrut \) \(300431494910216651121418673622626357\) \(\beta_{2}\mathstrut +\mathstrut \) \(1372027889422157333050540825484576130443\) \(\beta_{1}\mathstrut -\mathstrut \) \(68239531319271116483025116578696\)\()/177147\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
26735.5i
23576.4i
16809.5i
10236.4i
8608.85i
8608.85i
10236.4i
16809.5i
23576.4i
26735.5i
481239.i −3.00483e8 + 2.44550e8i −1.62872e11 1.10872e12i 1.17687e14 + 1.44604e14i 2.90621e15 4.53097e16i 3.04857e16 1.46966e17i −5.33560e17
2.2 424376.i 3.83451e8 5.53148e7i −1.11375e11 3.68903e12i −2.34743e13 1.62727e14i −2.28183e15 1.81020e16i 1.43975e17 4.24211e16i 1.56554e18
2.3 302571.i −5.37826e7 3.83669e8i −2.28297e10 4.03107e12i −1.16087e14 + 1.62730e13i 4.01364e14 1.38849e16i −1.44310e17 + 4.12694e16i −1.21969e18
2.4 184256.i 7.73223e7 + 3.79626e8i 3.47692e10 1.78906e12i 6.99484e13 1.42471e13i −1.12077e15 1.90684e16i −1.38137e17 + 5.87071e16i −3.29645e17
2.5 154959.i −3.82586e8 6.10115e7i 4.47071e10 6.48901e12i −9.45430e12 + 5.92853e13i −5.18677e14 1.75765e16i 1.42650e17 + 4.66843e16i 1.00553e18
2.6 154959.i −3.82586e8 + 6.10115e7i 4.47071e10 6.48901e12i −9.45430e12 5.92853e13i −5.18677e14 1.75765e16i 1.42650e17 4.66843e16i 1.00553e18
2.7 184256.i 7.73223e7 3.79626e8i 3.47692e10 1.78906e12i 6.99484e13 + 1.42471e13i −1.12077e15 1.90684e16i −1.38137e17 5.87071e16i −3.29645e17
2.8 302571.i −5.37826e7 + 3.83669e8i −2.28297e10 4.03107e12i −1.16087e14 1.62730e13i 4.01364e14 1.38849e16i −1.44310e17 4.12694e16i −1.21969e18
2.9 424376.i 3.83451e8 + 5.53148e7i −1.11375e11 3.68903e12i −2.34743e13 + 1.62727e14i −2.28183e15 1.81020e16i 1.43975e17 + 4.24211e16i 1.56554e18
2.10 481239.i −3.00483e8 2.44550e8i −1.62872e11 1.10872e12i 1.17687e14 1.44604e14i 2.90621e15 4.53097e16i 3.04857e16 + 1.46966e17i −5.33560e17
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.10
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{10} \) \(\mathstrut +\mathstrut 561197528712 T_{2}^{8} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!28\)\( T_{2}^{6} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!04\)\( T_{2}^{4} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!80\)\( T_{2}^{2} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\( \) acting on \(S_{37}^{\mathrm{new}}(3, [\chi])\).