Properties

Label 3.35.b.a
Level 3
Weight 35
Character orbit 3.b
Analytic conductor 21.968
Analytic rank 0
Dimension 10
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 35 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(21.9676962128\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{54}\cdot 3^{65}\cdot 5^{4}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 11936911 + 41 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( -5558253328 + 5 \beta_{1} - 15 \beta_{2} + \beta_{3} ) q^{4} \) \( + ( -179 + 11341 \beta_{1} + 448 \beta_{2} + \beta_{4} ) q^{5} \) \( + ( -925022485773 + 931029 \beta_{1} - 634 \beta_{2} - 98 \beta_{3} + \beta_{5} - \beta_{7} ) q^{6} \) \( + ( -12356977038690 + 93855 \beta_{1} - 294796 \beta_{2} - 75 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} \) \( + ( 1041214 - 2180472708 \beta_{1} - 2605108 \beta_{2} - 2028 \beta_{3} - 178 \beta_{4} - 37 \beta_{5} - 14 \beta_{7} + \beta_{8} ) q^{8} \) \( + ( 478750119881211 - 19601898386 \beta_{1} - 12820206 \beta_{2} - 2481 \beta_{3} - 1033 \beta_{4} - 504 \beta_{5} - 27 \beta_{6} - 93 \beta_{7} + 9 \beta_{8} + \beta_{9} ) q^{9} \) \(+O(q^{10})\) \( q\) \(+\beta_{1} q^{2}\) \(+(11936911 + 41 \beta_{1} - \beta_{2}) q^{3}\) \(+(-5558253328 + 5 \beta_{1} - 15 \beta_{2} + \beta_{3}) q^{4}\) \(+(-179 + 11341 \beta_{1} + 448 \beta_{2} + \beta_{4}) q^{5}\) \(+(-925022485773 + 931029 \beta_{1} - 634 \beta_{2} - 98 \beta_{3} + \beta_{5} - \beta_{7}) q^{6}\) \(+(-12356977038690 + 93855 \beta_{1} - 294796 \beta_{2} - 75 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + \beta_{6} + \beta_{7}) q^{7}\) \(+(1041214 - 2180472708 \beta_{1} - 2605108 \beta_{2} - 2028 \beta_{3} - 178 \beta_{4} - 37 \beta_{5} - 14 \beta_{7} + \beta_{8}) q^{8}\) \(+(478750119881211 - 19601898386 \beta_{1} - 12820206 \beta_{2} - 2481 \beta_{3} - 1033 \beta_{4} - 504 \beta_{5} - 27 \beta_{6} - 93 \beta_{7} + 9 \beta_{8} + \beta_{9}) q^{9}\) \(+(-261120544837442 + 8882677 \beta_{1} - 27617834 \beta_{2} + 395040 \beta_{3} - 825 \beta_{4} - 678 \beta_{5} + 256 \beta_{6} + 1894 \beta_{7} + 21 \beta_{9}) q^{10}\) \(+(-313697505 + 364529872817 \beta_{1} + 784846370 \beta_{2} + 666006 \beta_{3} - 98065 \beta_{4} + 21703 \beta_{5} - 14476 \beta_{7} - 154 \beta_{8} + 198 \beta_{9}) q^{11}\) \(+(183908803496146678 + 1100365474673 \beta_{1} + 5814671753 \beta_{2} + 6182361 \beta_{3} + 804714 \beta_{4} - 2457 \beta_{5} - 5184 \beta_{6} - 4086 \beta_{7} - 459 \beta_{8} + 1164 \beta_{9}) q^{12}\) \(+(363987432663962900 - 2915528772 \beta_{1} + 9248554560 \beta_{2} + 131859478 \beta_{3} - 400778 \beta_{4} - 368186 \beta_{5} + 3722 \beta_{6} + 366890 \beta_{7} + 4656 \beta_{9}) q^{13}\) \(+(2988787188 - 14612583471891 \beta_{1} - 7481926306 \beta_{2} + 4011300 \beta_{3} - 27775543 \beta_{4} + 643444 \beta_{5} - 1111180 \beta_{7} + 7496 \beta_{8} + 11979 \beta_{9}) q^{14}\) \(+(7384035128905791360 - 19950874291149 \beta_{1} + 3927410218 \beta_{2} - 1616615035 \beta_{3} + 93051738 \beta_{4} + 142058 \beta_{5} + 224289 \beta_{6} + 70621 \beta_{7} + 10530 \beta_{8} + 12834 \beta_{9}) q^{15}\) \(+(-45891356811047886016 - 4425926352 \beta_{1} + 18475685264 \beta_{2} + 6518390864 \beta_{3} + 666336 \beta_{4} + 324288 \beta_{5} - 437632 \beta_{6} - 4249024 \beta_{7} - 48864 \beta_{9}) q^{16}\) \(+(364204533760 + 422431919267738 \beta_{1} - 911303847620 \beta_{2} - 548591448 \beta_{3} - 511433078 \beta_{4} - 25822674 \beta_{5} + 27758724 \beta_{7} - 200226 \beta_{8} - 297090 \beta_{9}) q^{17}\) \(+(\)\(44\!\cdots\!34\)\( + 377916126705654 \beta_{1} + 1262297450430 \beta_{2} - 97049515752 \beta_{3} + 604712991 \beta_{4} - 48999114 \beta_{5} - 3836160 \beta_{6} + 201618 \beta_{7} - 138456 \beta_{8} - 822387 \beta_{9}) q^{18}\) \(+(-\)\(18\!\cdots\!69\)\( + 65559742801 \beta_{1} - 57730533052 \beta_{2} + 210993236792 \beta_{3} - 14802653 \beta_{4} - 22152989 \beta_{5} + 10351366 \beta_{6} - 71552378 \beta_{7} - 1050048 \beta_{9}) q^{19}\) \(+(13042205891684 - 5742885592694736 \beta_{1} - 32630175593608 \beta_{2} - 28575794760 \beta_{3} + 6587001884 \beta_{4} - 552316950 \beta_{5} - 137179140 \beta_{7} + 3523710 \beta_{8} + 1045800 \beta_{9}) q^{20}\) \(+(\)\(47\!\cdots\!89\)\( + 4833706875798193 \beta_{1} + 16016083363252 \beta_{2} - 974852681292 \beta_{3} - 24603257055 \beta_{4} - 1474204482 \beta_{5} + 32458644 \beta_{6} - 8001216 \beta_{7} + 1047114 \beta_{8} + 9466986 \beta_{9}) q^{21}\) \(+(-\)\(82\!\cdots\!50\)\( - 106933058580271 \beta_{1} + 336880895600942 \beta_{2} + 1518552843984 \beta_{3} - 6960834837 \beta_{4} - 6796177806 \beta_{5} - 126539776 \beta_{6} + 1708209998 \beta_{7} + 23522433 \beta_{9}) q^{22}\) \(+(180604432160266 + 121513681280307690 \beta_{1} - 451861802916652 \beta_{2} - 373289881308 \beta_{3} + 28867438382 \beta_{4} - 7189823266 \beta_{5} - 1858098368 \beta_{7} - 44562848 \beta_{8} + 29547360 \beta_{9}) q^{23}\) \(+(-\)\(40\!\cdots\!78\)\( + 175653020879206428 \beta_{1} - 19361431907444 \beta_{2} + 2490405490868 \beta_{3} + 28884462822 \beta_{4} - 26528602873 \beta_{5} - 80652672 \beta_{6} + 17233402 \beta_{7} - 2686203 \beta_{8} - 22370400 \beta_{9}) q^{24}\) \(+(-\)\(30\!\cdots\!53\)\( - 954373812050092 \beta_{1} + 2982478651466464 \beta_{2} - 20879018865250 \beta_{3} - 38027531370 \beta_{4} - 39202119162 \beta_{5} + 882686194 \beta_{6} - 12205577774 \beta_{7} - 167798256 \beta_{9}) q^{25}\) \(+(3460833054615344 - 1327327227886759002 \beta_{1} - 8659017884548648 \beta_{2} - 6638157242352 \beta_{3} - 882315658444 \beta_{4} - 158421445344 \beta_{5} + 28301141232 \beta_{7} + 424251216 \beta_{8} - 407626884 \beta_{9}) q^{26}\) \(+(-\)\(51\!\cdots\!94\)\( + 2759811678927875052 \beta_{1} - 695740467256731 \beta_{2} + 53598728810484 \beta_{3} + 955599732387 \beta_{4} - 195751898181 \beta_{5} - 1177744698 \beta_{6} + 421442514 \beta_{7} - 34696998 \beta_{8} - 397830150 \beta_{9}) q^{27}\) \(+(\)\(12\!\cdots\!08\)\( - 6517567715668838 \beta_{1} + 20399994716977042 \beta_{2} - 96729518497550 \beta_{3} - 322651877984 \beta_{4} - 321447357632 \beta_{5} - 2647105920 \beta_{6} + 10774692288 \beta_{7} + 172074336 \beta_{9}) q^{28}\) \(+(3431331374080751 - 4006854062731420973 \beta_{1} - 8584849308767592 \beta_{2} - 7429134708528 \beta_{3} + 1490870330799 \beta_{4} - 95167702516 \beta_{5} - 133211852312 \beta_{7} - 3113052596 \beta_{8} + 2104697484 \beta_{9}) q^{29}\) \(+(\)\(45\!\cdots\!34\)\( + 29013691278964334991 \beta_{1} + 2206628787593178 \beta_{2} - 84213420189480 \beta_{3} - 7292081300955 \beta_{4} - 294060977694 \beta_{5} + 14615728128 \beta_{6} - 2550488058 \beta_{7} + 482059080 \beta_{8} + 3872545503 \beta_{9}) q^{30}\) \(+(-\)\(16\!\cdots\!84\)\( - 1292788608630117 \beta_{1} + 4460185966823444 \beta_{2} + 570163781804727 \beta_{3} - 436835090306 \beta_{4} - 400880266562 \beta_{5} - 12407207345 \beta_{6} + 388232257231 \beta_{7} + 5136403392 \beta_{9}) q^{31}\) \(+(-10088497325217568 - \)\(17\!\cdots\!00\)\( \beta_{1} + 25243444020629696 \beta_{2} + 14529202836288 \beta_{3} + 15979707255008 \beta_{4} + 360882972720 \beta_{5} - 86982940128 \beta_{7} + 17701571472 \beta_{8} - 1914750720 \beta_{9}) q^{32}\) \(+(\)\(12\!\cdots\!70\)\( + \)\(24\!\cdots\!84\)\( \beta_{1} + 22761755033701970 \beta_{2} - 1765382201634623 \beta_{3} - 4804524529101 \beta_{4} + 3282608910658 \beta_{5} - 73416609981 \beta_{6} - 7327264351 \beta_{7} - 2915969463 \beta_{8} - 11127578319 \beta_{9}) q^{33}\) \(+(-\)\(95\!\cdots\!00\)\( + 197813754023941596 \beta_{1} - 619879478122915832 \beta_{2} + 1906883083862464 \beta_{3} + 11346635479476 \beta_{4} + 11093327394936 \beta_{5} + 192726940672 \beta_{6} - 2629848858488 \beta_{7} - 36186869220 \beta_{9}) q^{34}\) \(+(-339823896113075726 - \)\(76\!\cdots\!96\)\( \beta_{1} + 850219302527371512 \beta_{2} + 703248369493860 \beta_{3} - 56778994410336 \beta_{4} + 13460678834200 \beta_{5} + 3641325647540 \beta_{7} - 76663306810 \beta_{8} - 30571897050 \beta_{9}) q^{35}\) \(+(-\)\(37\!\cdots\!80\)\( + \)\(14\!\cdots\!57\)\( \beta_{1} - 104901808359841311 \beta_{2} + 2324810768114865 \beta_{3} + 189146285083076 \beta_{4} + 19864781029638 \beta_{5} + 84073400064 \beta_{6} + 119094450468 \beta_{7} + 7340806098 \beta_{8} - 36739164584 \beta_{9}) q^{36}\) \(+(\)\(11\!\cdots\!36\)\( + 270533020239228772 \beta_{1} - 853486701369210880 \beta_{2} - 5550984917527342 \beta_{3} + 12800147149090 \beta_{4} + 13421426655826 \beta_{5} - 1141494108434 \beta_{6} + 5781334680910 \beta_{7} + 88754215248 \beta_{9}) q^{37}\) \(+(-392333075067413092 - \)\(46\!\cdots\!29\)\( \beta_{1} + 981582645407270274 \beta_{2} + 841444117024188 \beta_{3} - 146463738951897 \beta_{4} + 24869704462388 \beta_{5} - 13113721319300 \beta_{7} + 236674475440 \beta_{8} + 116669984085 \beta_{9}) q^{38}\) \(+(-\)\(14\!\cdots\!32\)\( + \)\(62\!\cdots\!12\)\( \beta_{1} - 120039024127554662 \beta_{2} + 33283389752061786 \beta_{3} - 278892372904356 \beta_{4} + 13060120920156 \beta_{5} + 1317868813998 \beta_{6} - 275451628950 \beta_{7} + 29689459362 \beta_{8} + 403372205346 \beta_{9}) q^{39}\) \(+(\)\(12\!\cdots\!84\)\( - 464231640737822944 \beta_{1} + 1433035100084305248 \beta_{2} - 35395636466508320 \beta_{3} - 22512214860480 \beta_{4} - 21952761945984 \beta_{5} + 3977789998848 \beta_{6} + 10211693903232 \beta_{7} + 79921844928 \beta_{9}) q^{40}\) \(+(1538011003332489230 - \)\(11\!\cdots\!22\)\( \beta_{1} - 3847829950368702720 \beta_{2} - 3626222755320096 \beta_{3} + 1489541240094390 \beta_{4} - 69458206351648 \beta_{5} - 18814055487584 \beta_{7} - 394503267536 \beta_{8} + 289727395632 \beta_{9}) q^{41}\) \(+(-\)\(11\!\cdots\!06\)\( + \)\(18\!\cdots\!57\)\( \beta_{1} + 941768432275634050 \beta_{2} - 34542990631804960 \beta_{3} - 1557532111930155 \beta_{4} - 251534646110386 \beta_{5} - 10090628234496 \beta_{6} - 1720053863918 \beta_{7} - 394321548384 \beta_{8} - 1287836685441 \beta_{9}) q^{42}\) \(+(-\)\(96\!\cdots\!29\)\( - 7147740588393122991 \beta_{1} + 22492703870639864548 \beta_{2} + 65213735589962900 \beta_{3} - 357276023336969 \beta_{4} - 361724888576585 \beta_{5} - 7717150593894 \beta_{6} - 57290220406758 \beta_{7} - 635552177088 \beta_{9}) q^{43}\) \(+(15533881521734926196 - \)\(18\!\cdots\!16\)\( \beta_{1} - 38865548870251551272 \beta_{2} - 30500546554286952 \beta_{3} - 2011671705614516 \beta_{4} - 817036035592238 \beta_{5} + 307865691659372 \beta_{7} - 550043697610 \beta_{8} - 3573393808248 \beta_{9}) q^{44}\) \(+(\)\(67\!\cdots\!91\)\( + \)\(16\!\cdots\!67\)\( \beta_{1} - 6111023212303107084 \beta_{2} - 354140863363212030 \beta_{3} + 2291156087314389 \beta_{4} - 590510283358932 \beta_{5} + 36739352785734 \beta_{6} + 11852354792466 \beta_{7} + 1992795701130 \beta_{8} + 1171319545434 \beta_{9}) q^{45}\) \(+(-\)\(27\!\cdots\!68\)\( - 7766664404189521346 \beta_{1} + 24599481923445521252 \beta_{2} + 297481243704266464 \beta_{3} - 375725173776054 \beta_{4} - 393582902814756 \beta_{5} + 3588474392576 \beta_{6} - 195397649181532 \beta_{7} - 2551104148386 \beta_{9}) q^{46}\) \(+(3763789081773117196 + \)\(84\!\cdots\!88\)\( \beta_{1} - 9418386530132042912 \beta_{2} - 3787120755147240 \beta_{3} - 10428723750360920 \beta_{4} + 455114206592568 \beta_{5} - 1073799764550072 \beta_{7} + 6244918438428 \beta_{8} + 11742510790620 \beta_{9}) q^{47}\) \(+(-\)\(83\!\cdots\!68\)\( - \)\(56\!\cdots\!20\)\( \beta_{1} + 42644477975297580176 \beta_{2} + 88751140492977360 \beta_{3} + 19258727425397568 \beta_{4} + 488846804236272 \beta_{5} - 64940297692032 \beta_{6} - 15768441247392 \beta_{7} - 5417328344688 \beta_{8} + 1146109984416 \beta_{9}) q^{48}\) \(+(\)\(12\!\cdots\!85\)\( + 17270618437546619628 \beta_{1} - 54220721721128844896 \beta_{2} + 23359976606091314 \beta_{3} - 239226403761254 \beta_{4} - 63017684839382 \beta_{5} + 11530381145598 \beta_{6} + 1974998963417886 \beta_{7} + 25172674131696 \beta_{9}) q^{49}\) \(+(-99487172539493805360 + \)\(28\!\cdots\!65\)\( \beta_{1} + \)\(24\!\cdots\!20\)\( \beta_{2} + 196451045380791600 \beta_{3} + 9571963687676540 \beta_{4} + 3518504567032000 \beta_{5} + 1500668070112400 \beta_{7} - 21117421183600 \beta_{8} - 14345525875500 \beta_{9}) q^{50}\) \(+(-\)\(15\!\cdots\!36\)\( - \)\(44\!\cdots\!66\)\( \beta_{1} - 35626580158813542996 \beta_{2} + 2748006102416857116 \beta_{3} - 73378571999699934 \beta_{4} + 6131590279959066 \beta_{5} - 32613113274156 \beta_{6} - 91080383870688 \beta_{7} + 2785582728954 \beta_{8} + 17569753473114 \beta_{9}) q^{51}\) \(+(\)\(36\!\cdots\!24\)\( + \)\(20\!\cdots\!90\)\( \beta_{1} - \)\(64\!\cdots\!66\)\( \beta_{2} - 4657454974314861526 \beta_{3} + 15338713926116992 \beta_{4} + 14822372626357504 \beta_{5} + 32967166970368 \beta_{6} - 5720550173206784 \beta_{7} - 73763042822784 \beta_{9}) q^{52}\) \(+(-\)\(23\!\cdots\!63\)\( + \)\(75\!\cdots\!65\)\( \beta_{1} + \)\(57\!\cdots\!36\)\( \beta_{2} + 409943550258594240 \beta_{3} + 154277185547323605 \beta_{4} + 9065511052411456 \beta_{5} - 301822922666944 \beta_{7} + 32007732748256 \beta_{8} - 1741492092960 \beta_{9}) q^{53}\) \(+(-\)\(62\!\cdots\!51\)\( - \)\(12\!\cdots\!84\)\( \beta_{1} - \)\(56\!\cdots\!36\)\( \beta_{2} + 1903977800661357486 \beta_{3} + 72281654791380441 \beta_{4} + 9643534158442887 \beta_{5} + 466371723257856 \beta_{6} + 389040768969129 \beta_{7} + 42514131364272 \beta_{8} - 128480264346645 \beta_{9}) q^{54}\) \(+(\)\(61\!\cdots\!94\)\( + 53402364838544631316 \beta_{1} - \)\(16\!\cdots\!72\)\( \beta_{2} + 5161360192450635250 \beta_{3} - 1384052908194990 \beta_{4} - 906621201950574 \beta_{5} - 237703410713362 \beta_{6} + 5082249887438702 \beta_{7} + 68204529463488 \beta_{9}) q^{55}\) \(+(\)\(26\!\cdots\!84\)\( + \)\(14\!\cdots\!72\)\( \beta_{1} - \)\(67\!\cdots\!08\)\( \beta_{2} - 397848831288592920 \beta_{3} - 397738113121379524 \beta_{4} - 10872492975667898 \beta_{5} + 4453371104689220 \beta_{7} + 35422453930418 \beta_{8} - 58920064996608 \beta_{9}) q^{56}\) \(+(-\)\(20\!\cdots\!34\)\( - \)\(90\!\cdots\!48\)\( \beta_{1} + \)\(17\!\cdots\!94\)\( \beta_{2} - 16857530192793023451 \beta_{3} - 3339661646124693 \beta_{4} - 15590706019095924 \beta_{5} - 982963164641625 \beta_{6} - 219191419223703 \beta_{7} - 188714928833577 \beta_{8} + 295334356452543 \beta_{9}) q^{57}\) \(+(\)\(91\!\cdots\!46\)\( - \)\(84\!\cdots\!53\)\( \beta_{1} + \)\(26\!\cdots\!86\)\( \beta_{2} + 15058460302942410208 \beta_{3} - 52782787419960459 \beta_{4} - 52079524443213618 \beta_{5} - 20455728979200 \beta_{6} + 7815903154771314 \beta_{7} + 100466139535263 \beta_{9}) q^{58}\) \(+(\)\(82\!\cdots\!27\)\( + \)\(10\!\cdots\!19\)\( \beta_{1} - \)\(20\!\cdots\!34\)\( \beta_{2} - 1711903909972411386 \beta_{3} + 129591226200490573 \beta_{4} - 21958255653023187 \beta_{5} - 30569587147515024 \beta_{7} - 319088614875192 \beta_{8} + 417105092330568 \beta_{9}) q^{59}\) \(+(-\)\(53\!\cdots\!80\)\( + \)\(13\!\cdots\!44\)\( \beta_{1} - \)\(31\!\cdots\!08\)\( \beta_{2} - 12076862862416058760 \beta_{3} + 849098736512603532 \beta_{4} - 112350455564996398 \beta_{5} + 335256579507456 \beta_{6} - 1404236597811476 \beta_{7} + 330387364550070 \beta_{8} - 18296755980984 \beta_{9}) q^{60}\) \(+(\)\(37\!\cdots\!16\)\( - \)\(16\!\cdots\!76\)\( \beta_{1} + \)\(50\!\cdots\!92\)\( \beta_{2} + 21290804310289689418 \beta_{3} - 89247111346908294 \beta_{4} - 89872975335258006 \beta_{5} + 3091069551798966 \beta_{6} - 3882843461240682 \beta_{7} - 89409141192816 \beta_{9}) q^{61}\) \(+(\)\(41\!\cdots\!28\)\( - \)\(93\!\cdots\!55\)\( \beta_{1} - \)\(10\!\cdots\!66\)\( \beta_{2} - 8147883472193222028 \beta_{3} - 594051794043218323 \beta_{4} - 196720859317383500 \beta_{5} + 39318858205070948 \beta_{7} + 764420577553208 \beta_{8} - 595485074890665 \beta_{9}) q^{62}\) \(+(-\)\(17\!\cdots\!58\)\( + \)\(12\!\cdots\!41\)\( \beta_{1} - \)\(53\!\cdots\!24\)\( \beta_{2} + 616604195413586715 \beta_{3} - 3628406040163105124 \beta_{4} - 73304131630926132 \beta_{5} + 1147183255751391 \beta_{6} - 667934540810385 \beta_{7} + 253061300733048 \beta_{8} - 1151574299312968 \beta_{9}) q^{63}\) \(+(\)\(31\!\cdots\!72\)\( - \)\(16\!\cdots\!08\)\( \beta_{1} + \)\(49\!\cdots\!76\)\( \beta_{2} - \)\(18\!\cdots\!84\)\( \beta_{3} + 739364034682368 \beta_{4} - 3839276339524608 \beta_{5} - 9102676499535872 \beta_{6} - 60121812097842176 \beta_{7} - 654091482029568 \beta_{9}) q^{64}\) \(+(-\)\(35\!\cdots\!34\)\( - \)\(25\!\cdots\!14\)\( \beta_{1} + \)\(87\!\cdots\!08\)\( \beta_{2} + 5110984293032028240 \beta_{3} + 5338063023755647326 \beta_{4} + 43824894522515300 \beta_{5} + 134483398514734360 \beta_{7} - 480244789358540 \beta_{8} - 1520952041234700 \beta_{9}) q^{65}\) \(+(-\)\(56\!\cdots\!34\)\( + \)\(36\!\cdots\!75\)\( \beta_{1} + \)\(64\!\cdots\!70\)\( \beta_{2} + \)\(31\!\cdots\!52\)\( \beta_{3} + 3270923594035655217 \beta_{4} + 407286043543280682 \beta_{5} + 3482392728013056 \beta_{6} + 19428605311505742 \beta_{7} - 2794654113063912 \beta_{8} + 1617419031330531 \beta_{9}) q^{66}\) \(+(\)\(64\!\cdots\!17\)\( + \)\(11\!\cdots\!69\)\( \beta_{1} - \)\(37\!\cdots\!24\)\( \beta_{2} + 65234014652262481526 \beta_{3} + 624308190638358321 \beta_{4} + 622153995474783153 \beta_{5} + 3337531205567176 \beta_{6} - 20666357759984696 \beta_{7} - 307742166225024 \beta_{9}) q^{67}\) \(+(-\)\(21\!\cdots\!24\)\( - \)\(35\!\cdots\!52\)\( \beta_{1} + \)\(54\!\cdots\!28\)\( \beta_{2} + 47394268113855638688 \beta_{3} - 9020807265978798512 \beta_{4} + 1267945327080724472 \beta_{5} - 475699031473161776 \beta_{7} - 2465198006258456 \beta_{8} + 6073950042390240 \beta_{9}) q^{68}\) \(+(-\)\(75\!\cdots\!92\)\( + \)\(19\!\cdots\!44\)\( \beta_{1} - \)\(44\!\cdots\!52\)\( \beta_{2} - \)\(39\!\cdots\!14\)\( \beta_{3} + 2709965171166417630 \beta_{4} + 700827823484211536 \beta_{5} - 15701419633805910 \beta_{6} - 24860142030974714 \beta_{7} + 6125186604962466 \beta_{8} + 138262022979858 \beta_{9}) q^{69}\) \(+(\)\(17\!\cdots\!72\)\( + \)\(13\!\cdots\!58\)\( \beta_{1} - \)\(41\!\cdots\!36\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} + 253625936443023630 \beta_{4} + 314258260432919988 \beta_{5} + 37153582376454144 \beta_{6} + 712770906835299276 \beta_{7} + 8661760569985194 \beta_{9}) q^{70}\) \(+(-\)\(12\!\cdots\!70\)\( - \)\(28\!\cdots\!26\)\( \beta_{1} + \)\(31\!\cdots\!60\)\( \beta_{2} + 26514392714724107772 \beta_{3} - 3981815906048220170 \beta_{4} + 424923808845230966 \beta_{5} + 304431298092204808 \beta_{7} + 8505587069306812 \beta_{8} - 5041779965029764 \beta_{9}) q^{71}\) \(+(-\)\(25\!\cdots\!62\)\( - \)\(27\!\cdots\!84\)\( \beta_{1} + \)\(21\!\cdots\!12\)\( \beta_{2} - 917962054046293932 \beta_{3} + 9993892592743989054 \beta_{4} + 295418999149839891 \beta_{5} - 10781108761353984 \beta_{6} - 119274153237371070 \beta_{7} - 1312936581803607 \beta_{8} + 4520572081856832 \beta_{9}) q^{72}\) \(+(\)\(21\!\cdots\!78\)\( + \)\(19\!\cdots\!24\)\( \beta_{1} - \)\(60\!\cdots\!52\)\( \beta_{2} + \)\(15\!\cdots\!72\)\( \beta_{3} + 1440613470265303200 \beta_{4} + 1322736940051594080 \beta_{5} - 68501902518515568 \beta_{6} - 1381983239185560048 \beta_{7} - 16839504316244160 \beta_{9}) q^{73}\) \(+(\)\(36\!\cdots\!20\)\( + \)\(17\!\cdots\!46\)\( \beta_{1} - \)\(91\!\cdots\!60\)\( \beta_{2} - 79341959514231207312 \beta_{3} + 16895557758588184620 \beta_{4} - 1959999215698412896 \beta_{5} + 468553225281271312 \beta_{7} - 10320056606636432 \beta_{8} - 3858005152242396 \beta_{9}) q^{74}\) \(+(-\)\(49\!\cdots\!59\)\( - \)\(16\!\cdots\!81\)\( \beta_{1} - \)\(14\!\cdots\!73\)\( \beta_{2} - \)\(22\!\cdots\!50\)\( \beta_{3} - 22340730413445728700 \beta_{4} - 254009758316930676 \beta_{5} + 145374717611234862 \beta_{6} + 367231954690685898 \beta_{7} - 23878645489407150 \beta_{8} - 23905137795644238 \beta_{9}) q^{75}\) \(+(\)\(75\!\cdots\!24\)\( - \)\(12\!\cdots\!38\)\( \beta_{1} + \)\(39\!\cdots\!26\)\( \beta_{2} - \)\(27\!\cdots\!98\)\( \beta_{3} - 5643111612512309408 \beta_{4} - 5668913266239480128 \beta_{5} - 53992345731629696 \beta_{6} - 341496487262960576 \beta_{7} - 3685950532452960 \beta_{9}) q^{76}\) \(+(\)\(92\!\cdots\!62\)\( + \)\(45\!\cdots\!74\)\( \beta_{1} - \)\(23\!\cdots\!64\)\( \beta_{2} - \)\(19\!\cdots\!80\)\( \beta_{3} + 20374072856006132010 \beta_{4} - 4608208871556457904 \beta_{5} + 813454555868832416 \beta_{7} - 9815655767343184 \beta_{8} - 8048040180071760 \beta_{9}) q^{77}\) \(+(-\)\(14\!\cdots\!98\)\( - \)\(60\!\cdots\!02\)\( \beta_{1} - \)\(13\!\cdots\!40\)\( \beta_{2} + \)\(55\!\cdots\!36\)\( \beta_{3} - 82604829266812733148 \beta_{4} - 13866413481887971414 \beta_{5} - 246791751923736576 \beta_{6} + 208313185513869958 \beta_{7} + 53993953836482544 \beta_{8} + 14731440363587340 \beta_{9}) q^{78}\) \(+(\)\(12\!\cdots\!48\)\( - \)\(88\!\cdots\!61\)\( \beta_{1} + \)\(27\!\cdots\!72\)\( \beta_{2} + \)\(52\!\cdots\!27\)\( \beta_{3} - 6682164741069185378 \beta_{4} - 6419490038895083426 \beta_{5} + 227989329610609055 \beta_{6} + 3154936010979173663 \beta_{7} + 37524957453443136 \beta_{9}) q^{79}\) \(+(\)\(27\!\cdots\!56\)\( + \)\(52\!\cdots\!76\)\( \beta_{1} - \)\(69\!\cdots\!72\)\( \beta_{2} - \)\(55\!\cdots\!60\)\( \beta_{3} - 2292402270677115584 \beta_{4} - 10210335684570815200 \beta_{5} - 3801029616562015040 \beta_{7} + 61231649617502560 \beta_{8} + 35045077641868800 \beta_{9}) q^{80}\) \(+(-\)\(17\!\cdots\!65\)\( - \)\(49\!\cdots\!96\)\( \beta_{1} + \)\(36\!\cdots\!08\)\( \beta_{2} - \)\(55\!\cdots\!76\)\( \beta_{3} + \)\(18\!\cdots\!00\)\( \beta_{4} + 4887048661025575338 \beta_{5} - 129459017712914172 \beta_{6} - 2421985494731523192 \beta_{7} - 19818774520661826 \beta_{8} + 80848127326685310 \beta_{9}) q^{81}\) \(+(\)\(26\!\cdots\!00\)\( - \)\(13\!\cdots\!14\)\( \beta_{1} + \)\(41\!\cdots\!28\)\( \beta_{2} - \)\(96\!\cdots\!44\)\( \beta_{3} - 4822735546057982358 \beta_{4} - 4820950398647708004 \beta_{5} + 326730503697225216 \beta_{6} + 346622146268853732 \beta_{7} + 255021058610622 \beta_{9}) q^{82}\) \(+(-\)\(54\!\cdots\!19\)\( + \)\(26\!\cdots\!43\)\( \beta_{1} + \)\(13\!\cdots\!18\)\( \beta_{2} + \)\(12\!\cdots\!50\)\( \beta_{3} - \)\(27\!\cdots\!55\)\( \beta_{4} + 28304149492823644753 \beta_{5} - 4363823536997273692 \beta_{7} - 96818473274930482 \beta_{8} + 68086692414836910 \beta_{9}) q^{83}\) \(+(-\)\(33\!\cdots\!80\)\( + \)\(46\!\cdots\!98\)\( \beta_{1} - \)\(33\!\cdots\!74\)\( \beta_{2} + \)\(34\!\cdots\!02\)\( \beta_{3} + 5356412002353063060 \beta_{4} + 13634490246254729118 \beta_{5} + 888395375200425984 \beta_{6} + 2001688897545940788 \beta_{7} - 119030972427964134 \beta_{8} - 131596718379489288 \beta_{9}) q^{84}\) \(+(\)\(30\!\cdots\!24\)\( + \)\(81\!\cdots\!76\)\( \beta_{1} - \)\(25\!\cdots\!92\)\( \beta_{2} + \)\(20\!\cdots\!40\)\( \beta_{3} + 39566478262110832440 \beta_{4} + 39511068869173103736 \beta_{5} - 1757621152270206712 \beta_{6} - 2375040102147755128 \beta_{7} - 7915627562532672 \beta_{9}) q^{85}\) \(+(-\)\(14\!\cdots\!88\)\( - \)\(16\!\cdots\!89\)\( \beta_{1} + \)\(36\!\cdots\!06\)\( \beta_{2} + \)\(17\!\cdots\!60\)\( \beta_{3} + \)\(32\!\cdots\!43\)\( \beta_{4} - 9377981377587558124 \beta_{5} + 26205649223630554300 \beta_{7} - 11653536794096816 \beta_{8} - 310029758434680939 \beta_{9}) q^{86}\) \(+(-\)\(13\!\cdots\!92\)\( + \)\(18\!\cdots\!37\)\( \beta_{1} + \)\(11\!\cdots\!90\)\( \beta_{2} - \)\(25\!\cdots\!21\)\( \beta_{3} + \)\(12\!\cdots\!78\)\( \beta_{4} + 50139338092880212594 \beta_{5} - 673367925639132909 \beta_{6} + 10959751192482092447 \beta_{7} + 203086996795375866 \beta_{8} - 28773401724012870 \beta_{9}) q^{87}\) \(+(\)\(28\!\cdots\!96\)\( + \)\(53\!\cdots\!92\)\( \beta_{1} - \)\(16\!\cdots\!04\)\( \beta_{2} + \)\(33\!\cdots\!28\)\( \beta_{3} + 42125369190811976256 \beta_{4} + 39759763151626945152 \beta_{5} + 587464213982257920 \beta_{6} - 25772145936936660096 \beta_{7} - 337943719883575872 \beta_{9}) q^{88}\) \(+(-\)\(19\!\cdots\!22\)\( - \)\(57\!\cdots\!24\)\( \beta_{1} + \)\(48\!\cdots\!24\)\( \beta_{2} + \)\(38\!\cdots\!76\)\( \beta_{3} + \)\(18\!\cdots\!72\)\( \beta_{4} + 92832053486490562862 \beta_{5} - 19776384314035595036 \beta_{7} + 336853783286399662 \beta_{8} + 179290849381261902 \beta_{9}) q^{89}\) \(+(-\)\(37\!\cdots\!18\)\( + \)\(54\!\cdots\!93\)\( \beta_{1} - \)\(42\!\cdots\!06\)\( \beta_{2} - \)\(57\!\cdots\!60\)\( \beta_{3} - \)\(10\!\cdots\!25\)\( \beta_{4} - 5906614852093968342 \beta_{5} - 495766276381714176 \beta_{6} - 29195567527375357434 \beta_{7} - 7209128868699600 \beta_{8} - 77771566050405111 \beta_{9}) q^{90}\) \(+(\)\(33\!\cdots\!84\)\( + \)\(18\!\cdots\!54\)\( \beta_{1} - \)\(56\!\cdots\!68\)\( \beta_{2} + \)\(17\!\cdots\!46\)\( \beta_{3} + 56555966544706058044 \beta_{4} + 60679871686376202364 \beta_{5} + 7496669524660362758 \beta_{6} + 53448755388984828038 \beta_{7} + 589129305952877760 \beta_{9}) q^{91}\) \(+(\)\(20\!\cdots\!60\)\( - \)\(44\!\cdots\!44\)\( \beta_{1} - \)\(50\!\cdots\!20\)\( \beta_{2} - \)\(43\!\cdots\!12\)\( \beta_{3} + \)\(79\!\cdots\!68\)\( \beta_{4} - 84372859889472438116 \beta_{5} - 21428682913590894424 \beta_{7} - 557280800865123244 \beta_{8} + 347983501496459760 \beta_{9}) q^{92}\) \(+(-\)\(91\!\cdots\!13\)\( + \)\(71\!\cdots\!35\)\( \beta_{1} + \)\(19\!\cdots\!52\)\( \beta_{2} + \)\(55\!\cdots\!26\)\( \beta_{3} + \)\(43\!\cdots\!41\)\( \beta_{4} + 35595808240573813554 \beta_{5} - 1374109258633384698 \beta_{6} - 2280407560943239818 \beta_{7} - 139483224382955976 \beta_{8} + 781962232612512072 \beta_{9}) q^{93}\) \(+(-\)\(19\!\cdots\!00\)\( - \)\(75\!\cdots\!32\)\( \beta_{1} + \)\(23\!\cdots\!84\)\( \beta_{2} - \)\(58\!\cdots\!84\)\( \beta_{3} - \)\(38\!\cdots\!20\)\( \beta_{4} - \)\(38\!\cdots\!12\)\( \beta_{5} - 14647248602276335616 \beta_{6} + 17955421798261411816 \beta_{7} + 417982953853048044 \beta_{9}) q^{94}\) \(+(\)\(36\!\cdots\!78\)\( - \)\(39\!\cdots\!62\)\( \beta_{1} - \)\(91\!\cdots\!36\)\( \beta_{2} - 66572689583907349860 \beta_{3} - \)\(18\!\cdots\!02\)\( \beta_{4} + 4449868723824776050 \beta_{5} - 11804734440036963040 \beta_{7} - 60560310250775440 \beta_{8} + 150625937899378800 \beta_{9}) q^{95}\) \(+(\)\(57\!\cdots\!32\)\( + \)\(13\!\cdots\!92\)\( \beta_{1} + \)\(49\!\cdots\!68\)\( \beta_{2} + \)\(43\!\cdots\!72\)\( \beta_{3} + \)\(21\!\cdots\!12\)\( \beta_{4} - \)\(55\!\cdots\!32\)\( \beta_{5} + 5024813519174658048 \beta_{6} + \)\(13\!\cdots\!92\)\( \beta_{7} - 461123414828109360 \beta_{8} - 109265850453033216 \beta_{9}) q^{96}\) \(+(-\)\(11\!\cdots\!00\)\( + \)\(51\!\cdots\!44\)\( \beta_{1} - \)\(17\!\cdots\!64\)\( \beta_{2} - \)\(15\!\cdots\!66\)\( \beta_{3} + \)\(12\!\cdots\!02\)\( \beta_{4} + \)\(11\!\cdots\!18\)\( \beta_{5} - 115304699948229330 \beta_{6} - \)\(10\!\cdots\!66\)\( \beta_{7} - 1343820413585016912 \beta_{9}) q^{97}\) \(+(\)\(16\!\cdots\!56\)\( + \)\(11\!\cdots\!63\)\( \beta_{1} - \)\(41\!\cdots\!32\)\( \beta_{2} - \)\(31\!\cdots\!40\)\( \beta_{3} - \)\(45\!\cdots\!20\)\( \beta_{4} - \)\(71\!\cdots\!12\)\( \beta_{5} + 53342144017012306288 \beta_{7} + 1448872063255698928 \beta_{8} - 876504201221334420 \beta_{9}) q^{98}\) \(+(\)\(25\!\cdots\!17\)\( - \)\(17\!\cdots\!13\)\( \beta_{1} - \)\(11\!\cdots\!58\)\( \beta_{2} + \)\(42\!\cdots\!16\)\( \beta_{3} - \)\(62\!\cdots\!93\)\( \beta_{4} + \)\(43\!\cdots\!63\)\( \beta_{5} + 5167720315144821258 \beta_{6} - \)\(24\!\cdots\!54\)\( \beta_{7} + 668664769215000492 \beta_{8} - 2884502179525317780 \beta_{9}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut +\mathstrut 119369106q^{3} \) \(\mathstrut -\mathstrut 55582533344q^{4} \) \(\mathstrut -\mathstrut 9250224859872q^{6} \) \(\mathstrut -\mathstrut 123569771565772q^{7} \) \(\mathstrut +\mathstrut 4787501147541018q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut +\mathstrut 119369106q^{3} \) \(\mathstrut -\mathstrut 55582533344q^{4} \) \(\mathstrut -\mathstrut 9250224859872q^{6} \) \(\mathstrut -\mathstrut 123569771565772q^{7} \) \(\mathstrut +\mathstrut 4787501147541018q^{9} \) \(\mathstrut -\mathstrut 2611205560422720q^{10} \) \(\mathstrut +\mathstrut 1839088058193784224q^{12} \) \(\mathstrut +\mathstrut 3639874363106470052q^{13} \) \(\mathstrut +\mathstrut 73840351311049043520q^{15} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!64\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!20\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!44\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!32\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(82\!\cdots\!40\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!04\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!30\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!66\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!52\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!40\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!20\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!80\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(95\!\cdots\!84\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!72\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!88\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!72\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!40\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!28\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!20\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!84\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(83\!\cdots\!44\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!66\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!92\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!48\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!72\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!40\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!12\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(91\!\cdots\!20\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!20\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!20\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!68\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!32\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!40\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!48\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(75\!\cdots\!52\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!20\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!60\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!92\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!90\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(75\!\cdots\!16\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!60\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!76\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!10\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!40\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!32\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!40\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!40\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!20\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!80\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!24\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!68\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!24\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!64\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!92\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!80\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut +\mathstrut \) \(789518143\) \(x^{8}\mathstrut +\mathstrut \) \(211496483076151936\) \(x^{6}\mathstrut +\mathstrut \) \(21382790524640936160081920\) \(x^{4}\mathstrut +\mathstrut \) \(613809329098098496707904510361600\) \(x^{2}\mathstrut +\mathstrut \) \(5042460246515433703013776627104481280000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 12 \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(813769774277032030625\) \(\nu^{9}\mathstrut -\mathstrut \) \(9780550086447533942804992\) \(\nu^{8}\mathstrut -\mathstrut \) \(673192836087077207641203625375\) \(\nu^{7}\mathstrut -\mathstrut \) \(6254358931783500043872998745959936\) \(\nu^{6}\mathstrut -\mathstrut \) \(180540950325777997810512352021141392000\) \(\nu^{5}\mathstrut -\mathstrut \) \(1303964481766876975585250068903589849137152\) \(\nu^{4}\mathstrut -\mathstrut \) \(16429212943982368694771692507902970357584640000\) \(\nu^{3}\mathstrut -\mathstrut \) \(96589184880043036232672717705317890797525789573120\) \(\nu^{2}\mathstrut -\mathstrut \) \(248516745081595419090079697918785349647543467769856000\) \(\nu\mathstrut -\mathstrut \) \(1423250952868181532781553879867871070838188895673253888000\)\()/\)\(26\!\cdots\!00\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(813769774277032030625\) \(\nu^{9}\mathstrut -\mathstrut \) \(9780550086447533942804992\) \(\nu^{8}\mathstrut -\mathstrut \) \(673192836087077207641203625375\) \(\nu^{7}\mathstrut -\mathstrut \) \(6254358931783500043872998745959936\) \(\nu^{6}\mathstrut -\mathstrut \) \(180540950325777997810512352021141392000\) \(\nu^{5}\mathstrut -\mathstrut \) \(1303964481766876975585250068903589849137152\) \(\nu^{4}\mathstrut -\mathstrut \) \(16429212943982368694771692507902970357584640000\) \(\nu^{3}\mathstrut -\mathstrut \) \(71219808228439283328465415712003895254592672235520\) \(\nu^{2}\mathstrut -\mathstrut \) \(248527315655200253987123117627949230479019689902080000\) \(\nu\mathstrut +\mathstrut \) \(2582665674612641208988178293527652787074555394912996556800\)\()/\)\(17\!\cdots\!00\)
\(\beta_{4}\)\(=\)\((\)\(4368554465180769376546483\) \(\nu^{9}\mathstrut +\mathstrut \) \(2190843219364247603188318208\) \(\nu^{8}\mathstrut +\mathstrut \) \(3421766662062795928682451024376589\) \(\nu^{7}\mathstrut +\mathstrut \) \(1400976400719504009827551719095025664\) \(\nu^{6}\mathstrut +\mathstrut \) \(897092484532198968240742535017828383205248\) \(\nu^{5}\mathstrut +\mathstrut \) \(292088043915780442531096015434404126206722048\) \(\nu^{4}\mathstrut +\mathstrut \) \(84192161445796041505267707534414684604591691540480\) \(\nu^{3}\mathstrut +\mathstrut \) \(21635977413129640116118688765991207538645776864378880\) \(\nu^{2}\mathstrut +\mathstrut \) \(1539616518526272562891381651676708386847307504258383872000\) \(\nu\mathstrut +\mathstrut \) \(318808213678989247751248890436919112409587916911279079424000\)\()/\)\(13\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(397993241367760976813189\) \(\nu^{9}\mathstrut +\mathstrut \) \(3936671409795132411979009280\) \(\nu^{8}\mathstrut -\mathstrut \) \(326572317431689099871307961440187\) \(\nu^{7}\mathstrut +\mathstrut \) \(2517379470042858767658881995248874240\) \(\nu^{6}\mathstrut -\mathstrut \) \(87296349263958934901326162237344350381184\) \(\nu^{5}\mathstrut +\mathstrut \) \(524845703911167982673063152733694914277703680\) \(\nu^{4}\mathstrut -\mathstrut \) \(7979472524429858822232345422677483818426462115840\) \(\nu^{3}\mathstrut +\mathstrut \) \(38755585317761720767651442221005821484027575782604800\) \(\nu^{2}\mathstrut -\mathstrut \) \(124401390942017340152318981301124638493389515707645952000\) \(\nu\mathstrut +\mathstrut \) \(553663491776167648624065597281029038153678423301378015232000\)\()/\)\(18\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(42149854040599894893256165\) \(\nu^{9}\mathstrut -\mathstrut \) \(1230064829635123441164610249216\) \(\nu^{8}\mathstrut +\mathstrut \) \(27393982486954866732362799513685595\) \(\nu^{7}\mathstrut -\mathstrut \) \(1642087562995834802903866852939494153728\) \(\nu^{6}\mathstrut +\mathstrut \) \(5839496218051792430857984548829240015885440\) \(\nu^{5}\mathstrut -\mathstrut \) \(592487084059182078335499676852034290645222096896\) \(\nu^{4}\mathstrut +\mathstrut \) \(444812318984433674832628665616167607060488776652800\) \(\nu^{3}\mathstrut -\mathstrut \) \(67097906120459715146011584958479185672946550166399221760\) \(\nu^{2}\mathstrut +\mathstrut \) \(6870963178207946886759492783868162181535464179816726528000\) \(\nu\mathstrut -\mathstrut \) \(1155994652306469325727391992423865913636377425311097864323072000\)\()/\)\(88\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(21620682023656139072790515\) \(\nu^{9}\mathstrut +\mathstrut \) \(27372117443529517936628135936\) \(\nu^{8}\mathstrut -\mathstrut \) \(15082790395554752557883404871389645\) \(\nu^{7}\mathstrut +\mathstrut \) \(34310469311062950501152118564612671488\) \(\nu^{6}\mathstrut -\mathstrut \) \(3479798739465896678177581275046022894471040\) \(\nu^{5}\mathstrut +\mathstrut \) \(11247348841107330184949388010445543596867977216\) \(\nu^{4}\mathstrut -\mathstrut \) \(284193399755534650182466339036593063723920242124800\) \(\nu^{3}\mathstrut +\mathstrut \) \(1092865698714889059139659795444103738841259102919720960\) \(\nu^{2}\mathstrut -\mathstrut \) \(4352366552121897753159309617622347349061651910799392768000\) \(\nu\mathstrut +\mathstrut \) \(15453710206187692216158040725309439042469777100990108925952000\)\()/\)\(33\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(1282883003178340459042926431\) \(\nu^{9}\mathstrut -\mathstrut \) \(239177749269363147637850251264\) \(\nu^{8}\mathstrut -\mathstrut \) \(996329867197679719721630926962053473\) \(\nu^{7}\mathstrut +\mathstrut \) \(317644972779510273521413080744538021888\) \(\nu^{6}\mathstrut -\mathstrut \) \(252827053233821958857498637709238272291403136\) \(\nu^{5}\mathstrut +\mathstrut \) \(180857355393662022056809793208083726783417942016\) \(\nu^{4}\mathstrut -\mathstrut \) \(20776260960654516632012949338761102416319246262978560\) \(\nu^{3}\mathstrut +\mathstrut \) \(20700339994346974430537389555531766786868079036947496960\) \(\nu^{2}\mathstrut -\mathstrut \) \(24598093431109979691355109838257224789050792450619080704000\) \(\nu\mathstrut +\mathstrut \) \(291268148772949878751989541204000538972662083598489021841408000\)\()/\)\(66\!\cdots\!00\)
\(\beta_{9}\)\(=\)\((\)\(2927669529521947263121010843\) \(\nu^{9}\mathstrut -\mathstrut \) \(1359604661424192937691389591808\) \(\nu^{8}\mathstrut +\mathstrut \) \(1983985851105389202691257723814232869\) \(\nu^{7}\mathstrut +\mathstrut \) \(1954125343015237235716980606291682017536\) \(\nu^{6}\mathstrut +\mathstrut \) \(443834836809686034352400084015315759027956608\) \(\nu^{5}\mathstrut +\mathstrut \) \(1095204870321044471645986756155128008181525348352\) \(\nu^{4}\mathstrut +\mathstrut \) \(35332873873566059710041814197103564600804266479380480\) \(\nu^{3}\mathstrut +\mathstrut \) \(124944990578125174291516145029466582648057920470576005120\) \(\nu^{2}\mathstrut +\mathstrut \) \(543736505914215842681043947098183728832446980684282593280000\) \(\nu\mathstrut +\mathstrut \) \(1758229452876134569507387939654538800586866852062262359752704000\)\()/\)\(66\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/12\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(15\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut -\mathstrut \) \(22738122512\)\()/144\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{8}\mathstrut -\mathstrut \) \(14\) \(\beta_{7}\mathstrut -\mathstrut \) \(37\) \(\beta_{5}\mathstrut -\mathstrut \) \(178\) \(\beta_{4}\mathstrut -\mathstrut \) \(2028\) \(\beta_{3}\mathstrut -\mathstrut \) \(2605108\) \(\beta_{2}\mathstrut -\mathstrut \) \(36540211076\) \(\beta_{1}\mathstrut +\mathstrut \) \(1041214\)\()/1728\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(3054\) \(\beta_{9}\mathstrut -\mathstrut \) \(265564\) \(\beta_{7}\mathstrut -\mathstrut \) \(27352\) \(\beta_{6}\mathstrut +\mathstrut \) \(20268\) \(\beta_{5}\mathstrut +\mathstrut \) \(41646\) \(\beta_{4}\mathstrut -\mathstrut \) \(2813826043\) \(\beta_{3}\mathstrut +\mathstrut \) \(49473112409\) \(\beta_{2}\mathstrut -\mathstrut \) \(16382747757\) \(\beta_{1}\mathstrut +\mathstrut \) \(51929665546710981172\)\()/1296\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(119671920\) \(\beta_{9}\mathstrut -\mathstrut \) \(3188619079\) \(\beta_{8}\mathstrut +\mathstrut \) \(54693108386\) \(\beta_{7}\mathstrut +\mathstrut \) \(181468975747\) \(\beta_{5}\mathstrut +\mathstrut \) \(1763235882126\) \(\beta_{4}\mathstrut +\mathstrut \) \(9618268853556\) \(\beta_{3}\mathstrut +\mathstrut \) \(12766568913837324\) \(\beta_{2}\mathstrut +\mathstrut \) \(90844859202895131948\) \(\beta_{1}\mathstrut -\mathstrut \) \(5102511160963442\)\()/15552\)
\(\nu^{6}\)\(=\)\((\)\(4613664266934\) \(\beta_{9}\mathstrut +\mathstrut \) \(396961680078828\) \(\beta_{7}\mathstrut +\mathstrut \) \(37095867257976\) \(\beta_{6}\mathstrut -\mathstrut \) \(41270056548476\) \(\beta_{5}\mathstrut -\mathstrut \) \(73565706417014\) \(\beta_{4}\mathstrut +\mathstrut \) \(2493262785889896655\) \(\beta_{3}\mathstrut -\mathstrut \) \(47497973816040933173\) \(\beta_{2}\mathstrut +\mathstrut \) \(15682139254663210489\) \(\beta_{1}\mathstrut -\mathstrut \) \(43036123593875731562871796580\)\()/3888\)
\(\nu^{7}\)\(=\)\((\)\(44911600291889040\) \(\beta_{9}\mathstrut +\mathstrut \) \(1009062848660696313\) \(\beta_{8}\mathstrut -\mathstrut \) \(17899454305768427742\) \(\beta_{7}\mathstrut -\mathstrut \) \(70480278887364115133\) \(\beta_{5}\mathstrut -\mathstrut \) \(744604967380393692146\) \(\beta_{4}\mathstrut -\mathstrut \) \(3657816529246065919436\) \(\beta_{3}\mathstrut -\mathstrut \) \(4888181049637394552195060\) \(\beta_{2}\mathstrut -\mathstrut \) \(25852409317510429716922691540\) \(\beta_{1}\mathstrut +\mathstrut \) \(1953692153234519473481230\)\()/15552\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(1728797417808879535050\) \(\beta_{9}\mathstrut -\mathstrut \) \(147627970380695507690772\) \(\beta_{7}\mathstrut -\mathstrut \) \(12781771791602903956872\) \(\beta_{6}\mathstrut +\mathstrut \) \(27179667185517467902084\) \(\beta_{5}\mathstrut +\mathstrut \) \(39281249110179624647434\) \(\beta_{4}\mathstrut -\mathstrut \) \(735160969549507359049890161\) \(\beta_{3}\mathstrut +\mathstrut \) \(14048428228502531576080612747\) \(\beta_{2}\mathstrut -\mathstrut \) \(4637783463541261541652243911\) \(\beta_{1}\mathstrut +\mathstrut \) \(12247304739615256814581122918054493084\)\()/3888\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(10603103829042081750514800\) \(\beta_{9}\mathstrut -\mathstrut \) \(309031481108638520231415495\) \(\beta_{8}\mathstrut +\mathstrut \) \(5217101457160474150283060130\) \(\beta_{7}\mathstrut +\mathstrut \) \(24340008205788417128491792387\) \(\beta_{5}\mathstrut +\mathstrut \) \(256704108783639068645051984782\) \(\beta_{4}\mathstrut +\mathstrut \) \(1240869836905392647448058494772\) \(\beta_{3}\mathstrut +\mathstrut \) \(1660067608204160261720377485132556\) \(\beta_{2}\mathstrut +\mathstrut \) \(7475452530494627887154033131787510572\) \(\beta_{1}\mathstrut -\mathstrut \) \(663490136069339733085540563690098\)\()/15552\)

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
17330.4i
16987.2i
12705.0i
4945.08i
3839.21i
3839.21i
4945.08i
12705.0i
16987.2i
17330.4i
207965.i 7.87239e7 + 1.02371e8i −2.60695e10 2.95819e11i 2.12895e13 1.63718e13i 4.71207e13 1.84873e15i −4.28228e15 + 1.61180e16i −6.15199e16
2.2 203846.i −1.15096e8 5.85668e7i −2.43733e10 1.79877e11i −1.19386e13 + 2.34619e13i −2.97937e14 1.46635e15i 9.81704e15 + 1.34816e16i −3.66671e16
2.3 152461.i 6.59892e7 1.11007e8i −6.06435e9 1.02364e12i −1.69242e13 1.00608e13i 3.85840e14 1.69468e15i −7.96802e15 1.46506e16i 1.56064e17
2.4 59341.0i −8.15029e7 + 1.00172e8i 1.36585e10 7.02280e10i 5.94431e12 + 4.83646e12i 9.28487e13 1.82998e15i −3.39173e15 1.63286e16i 4.16740e15
2.5 46070.5i 1.11570e8 6.50325e7i 1.50574e10 1.37507e12i −2.99608e12 5.14011e12i −2.89657e14 1.48519e15i 8.21874e15 1.45114e16i −6.33504e16
2.6 46070.5i 1.11570e8 + 6.50325e7i 1.50574e10 1.37507e12i −2.99608e12 + 5.14011e12i −2.89657e14 1.48519e15i 8.21874e15 + 1.45114e16i −6.33504e16
2.7 59341.0i −8.15029e7 1.00172e8i 1.36585e10 7.02280e10i 5.94431e12 4.83646e12i 9.28487e13 1.82998e15i −3.39173e15 + 1.63286e16i 4.16740e15
2.8 152461.i 6.59892e7 + 1.11007e8i −6.06435e9 1.02364e12i −1.69242e13 + 1.00608e13i 3.85840e14 1.69468e15i −7.96802e15 + 1.46506e16i 1.56064e17
2.9 203846.i −1.15096e8 + 5.85668e7i −2.43733e10 1.79877e11i −1.19386e13 2.34619e13i −2.97937e14 1.46635e15i 9.81704e15 1.34816e16i −3.66671e16
2.10 207965.i 7.87239e7 1.02371e8i −2.60695e10 2.95819e11i 2.12895e13 + 1.63718e13i 4.71207e13 1.84873e15i −4.28228e15 1.61180e16i −6.15199e16
\(n\): 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

There are no other newforms in \(S_{35}^{\mathrm{new}}(3, [\chi])\).