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

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$21.9676962128$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 789518143 x^{8} + 211496483076151936 x^{6} + 21382790524640936160081920 x^{4} + 613809329098098496707904510361600 x^{2} + 5042460246515433703013776627104481280000$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{54}\cdot 3^{65}\cdot 5^{4}\cdot 7^{2}$$ Twist minimal: yes 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} +($$$$44\!\cdots\!34$$$$+ 377916126705654 \beta_{1} + 1262297450430 \beta_{2} - 97049515752 \beta_{3} + 604712991 \beta_{4} - 48999114 \beta_{5} - 3836160 \beta_{6} + 201618 \beta_{7} - 138456 \beta_{8} - 822387 \beta_{9}) q^{18} +(-$$$$18\!\cdots\!69$$$$+ 65559742801 \beta_{1} - 57730533052 \beta_{2} + 210993236792 \beta_{3} - 14802653 \beta_{4} - 22152989 \beta_{5} + 10351366 \beta_{6} - 71552378 \beta_{7} - 1050048 \beta_{9}) q^{19} +(13042205891684 - 5742885592694736 \beta_{1} - 32630175593608 \beta_{2} - 28575794760 \beta_{3} + 6587001884 \beta_{4} - 552316950 \beta_{5} - 137179140 \beta_{7} + 3523710 \beta_{8} + 1045800 \beta_{9}) q^{20} +($$$$47\!\cdots\!89$$$$+ 4833706875798193 \beta_{1} + 16016083363252 \beta_{2} - 974852681292 \beta_{3} - 24603257055 \beta_{4} - 1474204482 \beta_{5} + 32458644 \beta_{6} - 8001216 \beta_{7} + 1047114 \beta_{8} + 9466986 \beta_{9}) q^{21} +(-$$$$82\!\cdots\!50$$$$- 106933058580271 \beta_{1} + 336880895600942 \beta_{2} + 1518552843984 \beta_{3} - 6960834837 \beta_{4} - 6796177806 \beta_{5} - 126539776 \beta_{6} + 1708209998 \beta_{7} + 23522433 \beta_{9}) q^{22} +(180604432160266 + 121513681280307690 \beta_{1} - 451861802916652 \beta_{2} - 373289881308 \beta_{3} + 28867438382 \beta_{4} - 7189823266 \beta_{5} - 1858098368 \beta_{7} - 44562848 \beta_{8} + 29547360 \beta_{9}) q^{23} +(-$$$$40\!\cdots\!78$$$$+ 175653020879206428 \beta_{1} - 19361431907444 \beta_{2} + 2490405490868 \beta_{3} + 28884462822 \beta_{4} - 26528602873 \beta_{5} - 80652672 \beta_{6} + 17233402 \beta_{7} - 2686203 \beta_{8} - 22370400 \beta_{9}) q^{24} +(-$$$$30\!\cdots\!53$$$$- 954373812050092 \beta_{1} + 2982478651466464 \beta_{2} - 20879018865250 \beta_{3} - 38027531370 \beta_{4} - 39202119162 \beta_{5} + 882686194 \beta_{6} - 12205577774 \beta_{7} - 167798256 \beta_{9}) q^{25} +(3460833054615344 - 1327327227886759002 \beta_{1} - 8659017884548648 \beta_{2} - 6638157242352 \beta_{3} - 882315658444 \beta_{4} - 158421445344 \beta_{5} + 28301141232 \beta_{7} + 424251216 \beta_{8} - 407626884 \beta_{9}) q^{26} +(-$$$$51\!\cdots\!94$$$$+ 2759811678927875052 \beta_{1} - 695740467256731 \beta_{2} + 53598728810484 \beta_{3} + 955599732387 \beta_{4} - 195751898181 \beta_{5} - 1177744698 \beta_{6} + 421442514 \beta_{7} - 34696998 \beta_{8} - 397830150 \beta_{9}) q^{27} +($$$$12\!\cdots\!08$$$$- 6517567715668838 \beta_{1} + 20399994716977042 \beta_{2} - 96729518497550 \beta_{3} - 322651877984 \beta_{4} - 321447357632 \beta_{5} - 2647105920 \beta_{6} + 10774692288 \beta_{7} + 172074336 \beta_{9}) q^{28} +(3431331374080751 - 4006854062731420973 \beta_{1} - 8584849308767592 \beta_{2} - 7429134708528 \beta_{3} + 1490870330799 \beta_{4} - 95167702516 \beta_{5} - 133211852312 \beta_{7} - 3113052596 \beta_{8} + 2104697484 \beta_{9}) q^{29} +($$$$45\!\cdots\!34$$$$+ 29013691278964334991 \beta_{1} + 2206628787593178 \beta_{2} - 84213420189480 \beta_{3} - 7292081300955 \beta_{4} - 294060977694 \beta_{5} + 14615728128 \beta_{6} - 2550488058 \beta_{7} + 482059080 \beta_{8} + 3872545503 \beta_{9}) q^{30} +(-$$$$16\!\cdots\!84$$$$- 1292788608630117 \beta_{1} + 4460185966823444 \beta_{2} + 570163781804727 \beta_{3} - 436835090306 \beta_{4} - 400880266562 \beta_{5} - 12407207345 \beta_{6} + 388232257231 \beta_{7} + 5136403392 \beta_{9}) q^{31} +(-10088497325217568 -$$$$17\!\cdots\!00$$$$\beta_{1} + 25243444020629696 \beta_{2} + 14529202836288 \beta_{3} + 15979707255008 \beta_{4} + 360882972720 \beta_{5} - 86982940128 \beta_{7} + 17701571472 \beta_{8} - 1914750720 \beta_{9}) q^{32} +($$$$12\!\cdots\!70$$$$+$$$$24\!\cdots\!84$$$$\beta_{1} + 22761755033701970 \beta_{2} - 1765382201634623 \beta_{3} - 4804524529101 \beta_{4} + 3282608910658 \beta_{5} - 73416609981 \beta_{6} - 7327264351 \beta_{7} - 2915969463 \beta_{8} - 11127578319 \beta_{9}) q^{33} +(-$$$$95\!\cdots\!00$$$$+ 197813754023941596 \beta_{1} - 619879478122915832 \beta_{2} + 1906883083862464 \beta_{3} + 11346635479476 \beta_{4} + 11093327394936 \beta_{5} + 192726940672 \beta_{6} - 2629848858488 \beta_{7} - 36186869220 \beta_{9}) q^{34} +(-339823896113075726 -$$$$76\!\cdots\!96$$$$\beta_{1} + 850219302527371512 \beta_{2} + 703248369493860 \beta_{3} - 56778994410336 \beta_{4} + 13460678834200 \beta_{5} + 3641325647540 \beta_{7} - 76663306810 \beta_{8} - 30571897050 \beta_{9}) q^{35} +(-$$$$37\!\cdots\!80$$$$+$$$$14\!\cdots\!57$$$$\beta_{1} - 104901808359841311 \beta_{2} + 2324810768114865 \beta_{3} + 189146285083076 \beta_{4} + 19864781029638 \beta_{5} + 84073400064 \beta_{6} + 119094450468 \beta_{7} + 7340806098 \beta_{8} - 36739164584 \beta_{9}) q^{36} +($$$$11\!\cdots\!36$$$$+ 270533020239228772 \beta_{1} - 853486701369210880 \beta_{2} - 5550984917527342 \beta_{3} + 12800147149090 \beta_{4} + 13421426655826 \beta_{5} - 1141494108434 \beta_{6} + 5781334680910 \beta_{7} + 88754215248 \beta_{9}) q^{37} +(-392333075067413092 -$$$$46\!\cdots\!29$$$$\beta_{1} + 981582645407270274 \beta_{2} + 841444117024188 \beta_{3} - 146463738951897 \beta_{4} + 24869704462388 \beta_{5} - 13113721319300 \beta_{7} + 236674475440 \beta_{8} + 116669984085 \beta_{9}) q^{38} +(-$$$$14\!\cdots\!32$$$$+$$$$62\!\cdots\!12$$$$\beta_{1} - 120039024127554662 \beta_{2} + 33283389752061786 \beta_{3} - 278892372904356 \beta_{4} + 13060120920156 \beta_{5} + 1317868813998 \beta_{6} - 275451628950 \beta_{7} + 29689459362 \beta_{8} + 403372205346 \beta_{9}) q^{39} +($$$$12\!\cdots\!84$$$$- 464231640737822944 \beta_{1} + 1433035100084305248 \beta_{2} - 35395636466508320 \beta_{3} - 22512214860480 \beta_{4} - 21952761945984 \beta_{5} + 3977789998848 \beta_{6} + 10211693903232 \beta_{7} + 79921844928 \beta_{9}) q^{40} +(1538011003332489230 -$$$$11\!\cdots\!22$$$$\beta_{1} - 3847829950368702720 \beta_{2} - 3626222755320096 \beta_{3} + 1489541240094390 \beta_{4} - 69458206351648 \beta_{5} - 18814055487584 \beta_{7} - 394503267536 \beta_{8} + 289727395632 \beta_{9}) q^{41} +(-$$$$11\!\cdots\!06$$$$+$$$$18\!\cdots\!57$$$$\beta_{1} + 941768432275634050 \beta_{2} - 34542990631804960 \beta_{3} - 1557532111930155 \beta_{4} - 251534646110386 \beta_{5} - 10090628234496 \beta_{6} - 1720053863918 \beta_{7} - 394321548384 \beta_{8} - 1287836685441 \beta_{9}) q^{42} +(-$$$$96\!\cdots\!29$$$$- 7147740588393122991 \beta_{1} + 22492703870639864548 \beta_{2} + 65213735589962900 \beta_{3} - 357276023336969 \beta_{4} - 361724888576585 \beta_{5} - 7717150593894 \beta_{6} - 57290220406758 \beta_{7} - 635552177088 \beta_{9}) q^{43} +(15533881521734926196 -$$$$18\!\cdots\!16$$$$\beta_{1} - 38865548870251551272 \beta_{2} - 30500546554286952 \beta_{3} - 2011671705614516 \beta_{4} - 817036035592238 \beta_{5} + 307865691659372 \beta_{7} - 550043697610 \beta_{8} - 3573393808248 \beta_{9}) q^{44} +($$$$67\!\cdots\!91$$$$+$$$$16\!\cdots\!67$$$$\beta_{1} - 6111023212303107084 \beta_{2} - 354140863363212030 \beta_{3} + 2291156087314389 \beta_{4} - 590510283358932 \beta_{5} + 36739352785734 \beta_{6} + 11852354792466 \beta_{7} + 1992795701130 \beta_{8} + 1171319545434 \beta_{9}) q^{45} +(-$$$$27\!\cdots\!68$$$$- 7766664404189521346 \beta_{1} + 24599481923445521252 \beta_{2} + 297481243704266464 \beta_{3} - 375725173776054 \beta_{4} - 393582902814756 \beta_{5} + 3588474392576 \beta_{6} - 195397649181532 \beta_{7} - 2551104148386 \beta_{9}) q^{46} +(3763789081773117196 +$$$$84\!\cdots\!88$$$$\beta_{1} - 9418386530132042912 \beta_{2} - 3787120755147240 \beta_{3} - 10428723750360920 \beta_{4} + 455114206592568 \beta_{5} - 1073799764550072 \beta_{7} + 6244918438428 \beta_{8} + 11742510790620 \beta_{9}) q^{47} +(-$$$$83\!\cdots\!68$$$$-$$$$56\!\cdots\!20$$$$\beta_{1} + 42644477975297580176 \beta_{2} + 88751140492977360 \beta_{3} + 19258727425397568 \beta_{4} + 488846804236272 \beta_{5} - 64940297692032 \beta_{6} - 15768441247392 \beta_{7} - 5417328344688 \beta_{8} + 1146109984416 \beta_{9}) q^{48} +($$$$12\!\cdots\!85$$$$+ 17270618437546619628 \beta_{1} - 54220721721128844896 \beta_{2} + 23359976606091314 \beta_{3} - 239226403761254 \beta_{4} - 63017684839382 \beta_{5} + 11530381145598 \beta_{6} + 1974998963417886 \beta_{7} + 25172674131696 \beta_{9}) q^{49} +(-99487172539493805360 +$$$$28\!\cdots\!65$$$$\beta_{1} +$$$$24\!\cdots\!20$$$$\beta_{2} + 196451045380791600 \beta_{3} + 9571963687676540 \beta_{4} + 3518504567032000 \beta_{5} + 1500668070112400 \beta_{7} - 21117421183600 \beta_{8} - 14345525875500 \beta_{9}) q^{50} +(-$$$$15\!\cdots\!36$$$$-$$$$44\!\cdots\!66$$$$\beta_{1} - 35626580158813542996 \beta_{2} + 2748006102416857116 \beta_{3} - 73378571999699934 \beta_{4} + 6131590279959066 \beta_{5} - 32613113274156 \beta_{6} - 91080383870688 \beta_{7} + 2785582728954 \beta_{8} + 17569753473114 \beta_{9}) q^{51} +($$$$36\!\cdots\!24$$$$+$$$$20\!\cdots\!90$$$$\beta_{1} -$$$$64\!\cdots\!66$$$$\beta_{2} - 4657454974314861526 \beta_{3} + 15338713926116992 \beta_{4} + 14822372626357504 \beta_{5} + 32967166970368 \beta_{6} - 5720550173206784 \beta_{7} - 73763042822784 \beta_{9}) q^{52} +(-$$$$23\!\cdots\!63$$$$+$$$$75\!\cdots\!65$$$$\beta_{1} +$$$$57\!\cdots\!36$$$$\beta_{2} + 409943550258594240 \beta_{3} + 154277185547323605 \beta_{4} + 9065511052411456 \beta_{5} - 301822922666944 \beta_{7} + 32007732748256 \beta_{8} - 1741492092960 \beta_{9}) q^{53} +(-$$$$62\!\cdots\!51$$$$-$$$$12\!\cdots\!84$$$$\beta_{1} -$$$$56\!\cdots\!36$$$$\beta_{2} + 1903977800661357486 \beta_{3} + 72281654791380441 \beta_{4} + 9643534158442887 \beta_{5} + 466371723257856 \beta_{6} + 389040768969129 \beta_{7} + 42514131364272 \beta_{8} - 128480264346645 \beta_{9}) q^{54} +($$$$61\!\cdots\!94$$$$+ 53402364838544631316 \beta_{1} -$$$$16\!\cdots\!72$$$$\beta_{2} + 5161360192450635250 \beta_{3} - 1384052908194990 \beta_{4} - 906621201950574 \beta_{5} - 237703410713362 \beta_{6} + 5082249887438702 \beta_{7} + 68204529463488 \beta_{9}) q^{55} +($$$$26\!\cdots\!84$$$$+$$$$14\!\cdots\!72$$$$\beta_{1} -$$$$67\!\cdots\!08$$$$\beta_{2} - 397848831288592920 \beta_{3} - 397738113121379524 \beta_{4} - 10872492975667898 \beta_{5} + 4453371104689220 \beta_{7} + 35422453930418 \beta_{8} - 58920064996608 \beta_{9}) q^{56} +(-$$$$20\!\cdots\!34$$$$-$$$$90\!\cdots\!48$$$$\beta_{1} +$$$$17\!\cdots\!94$$$$\beta_{2} - 16857530192793023451 \beta_{3} - 3339661646124693 \beta_{4} - 15590706019095924 \beta_{5} - 982963164641625 \beta_{6} - 219191419223703 \beta_{7} - 188714928833577 \beta_{8} + 295334356452543 \beta_{9}) q^{57} +($$$$91\!\cdots\!46$$$$-$$$$84\!\cdots\!53$$$$\beta_{1} +$$$$26\!\cdots\!86$$$$\beta_{2} + 15058460302942410208 \beta_{3} - 52782787419960459 \beta_{4} - 52079524443213618 \beta_{5} - 20455728979200 \beta_{6} + 7815903154771314 \beta_{7} + 100466139535263 \beta_{9}) q^{58} +($$$$82\!\cdots\!27$$$$+$$$$10\!\cdots\!19$$$$\beta_{1} -$$$$20\!\cdots\!34$$$$\beta_{2} - 1711903909972411386 \beta_{3} + 129591226200490573 \beta_{4} - 21958255653023187 \beta_{5} - 30569587147515024 \beta_{7} - 319088614875192 \beta_{8} + 417105092330568 \beta_{9}) q^{59} +(-$$$$53\!\cdots\!80$$$$+$$$$13\!\cdots\!44$$$$\beta_{1} -$$$$31\!\cdots\!08$$$$\beta_{2} - 12076862862416058760 \beta_{3} + 849098736512603532 \beta_{4} - 112350455564996398 \beta_{5} + 335256579507456 \beta_{6} - 1404236597811476 \beta_{7} + 330387364550070 \beta_{8} - 18296755980984 \beta_{9}) q^{60} +($$$$37\!\cdots\!16$$$$-$$$$16\!\cdots\!76$$$$\beta_{1} +$$$$50\!\cdots\!92$$$$\beta_{2} + 21290804310289689418 \beta_{3} - 89247111346908294 \beta_{4} - 89872975335258006 \beta_{5} + 3091069551798966 \beta_{6} - 3882843461240682 \beta_{7} - 89409141192816 \beta_{9}) q^{61} +($$$$41\!\cdots\!28$$$$-$$$$93\!\cdots\!55$$$$\beta_{1} -$$$$10\!\cdots\!66$$$$\beta_{2} - 8147883472193222028 \beta_{3} - 594051794043218323 \beta_{4} - 196720859317383500 \beta_{5} + 39318858205070948 \beta_{7} + 764420577553208 \beta_{8} - 595485074890665 \beta_{9}) q^{62} +(-$$$$17\!\cdots\!58$$$$+$$$$12\!\cdots\!41$$$$\beta_{1} -$$$$53\!\cdots\!24$$$$\beta_{2} + 616604195413586715 \beta_{3} - 3628406040163105124 \beta_{4} - 73304131630926132 \beta_{5} + 1147183255751391 \beta_{6} - 667934540810385 \beta_{7} + 253061300733048 \beta_{8} - 1151574299312968 \beta_{9}) q^{63} +($$$$31\!\cdots\!72$$$$-$$$$16\!\cdots\!08$$$$\beta_{1} +$$$$49\!\cdots\!76$$$$\beta_{2} -$$$$18\!\cdots\!84$$$$\beta_{3} + 739364034682368 \beta_{4} - 3839276339524608 \beta_{5} - 9102676499535872 \beta_{6} - 60121812097842176 \beta_{7} - 654091482029568 \beta_{9}) q^{64} +(-$$$$35\!\cdots\!34$$$$-$$$$25\!\cdots\!14$$$$\beta_{1} +$$$$87\!\cdots\!08$$$$\beta_{2} + 5110984293032028240 \beta_{3} + 5338063023755647326 \beta_{4} + 43824894522515300 \beta_{5} + 134483398514734360 \beta_{7} - 480244789358540 \beta_{8} - 1520952041234700 \beta_{9}) q^{65} +(-$$$$56\!\cdots\!34$$$$+$$$$36\!\cdots\!75$$$$\beta_{1} +$$$$64\!\cdots\!70$$$$\beta_{2} +$$$$31\!\cdots\!52$$$$\beta_{3} + 3270923594035655217 \beta_{4} + 407286043543280682 \beta_{5} + 3482392728013056 \beta_{6} + 19428605311505742 \beta_{7} - 2794654113063912 \beta_{8} + 1617419031330531 \beta_{9}) q^{66} +($$$$64\!\cdots\!17$$$$+$$$$11\!\cdots\!69$$$$\beta_{1} -$$$$37\!\cdots\!24$$$$\beta_{2} + 65234014652262481526 \beta_{3} + 624308190638358321 \beta_{4} + 622153995474783153 \beta_{5} + 3337531205567176 \beta_{6} - 20666357759984696 \beta_{7} - 307742166225024 \beta_{9}) q^{67} +(-$$$$21\!\cdots\!24$$$$-$$$$35\!\cdots\!52$$$$\beta_{1} +$$$$54\!\cdots\!28$$$$\beta_{2} + 47394268113855638688 \beta_{3} - 9020807265978798512 \beta_{4} + 1267945327080724472 \beta_{5} - 475699031473161776 \beta_{7} - 2465198006258456 \beta_{8} + 6073950042390240 \beta_{9}) q^{68} +(-$$$$75\!\cdots\!92$$$$+$$$$19\!\cdots\!44$$$$\beta_{1} -$$$$44\!\cdots\!52$$$$\beta_{2} -$$$$39\!\cdots\!14$$$$\beta_{3} + 2709965171166417630 \beta_{4} + 700827823484211536 \beta_{5} - 15701419633805910 \beta_{6} - 24860142030974714 \beta_{7} + 6125186604962466 \beta_{8} + 138262022979858 \beta_{9}) q^{69} +($$$$17\!\cdots\!72$$$$+$$$$13\!\cdots\!58$$$$\beta_{1} -$$$$41\!\cdots\!36$$$$\beta_{2} +$$$$10\!\cdots\!00$$$$\beta_{3} + 253625936443023630 \beta_{4} + 314258260432919988 \beta_{5} + 37153582376454144 \beta_{6} + 712770906835299276 \beta_{7} + 8661760569985194 \beta_{9}) q^{70} +(-$$$$12\!\cdots\!70$$$$-$$$$28\!\cdots\!26$$$$\beta_{1} +$$$$31\!\cdots\!60$$$$\beta_{2} + 26514392714724107772 \beta_{3} - 3981815906048220170 \beta_{4} + 424923808845230966 \beta_{5} + 304431298092204808 \beta_{7} + 8505587069306812 \beta_{8} - 5041779965029764 \beta_{9}) q^{71} +(-$$$$25\!\cdots\!62$$$$-$$$$27\!\cdots\!84$$$$\beta_{1} +$$$$21\!\cdots\!12$$$$\beta_{2} - 917962054046293932 \beta_{3} + 9993892592743989054 \beta_{4} + 295418999149839891 \beta_{5} - 10781108761353984 \beta_{6} - 119274153237371070 \beta_{7} - 1312936581803607 \beta_{8} + 4520572081856832 \beta_{9}) q^{72} +($$$$21\!\cdots\!78$$$$+$$$$19\!\cdots\!24$$$$\beta_{1} -$$$$60\!\cdots\!52$$$$\beta_{2} +$$$$15\!\cdots\!72$$$$\beta_{3} + 1440613470265303200 \beta_{4} + 1322736940051594080 \beta_{5} - 68501902518515568 \beta_{6} - 1381983239185560048 \beta_{7} - 16839504316244160 \beta_{9}) q^{73} +($$$$36\!\cdots\!20$$$$+$$$$17\!\cdots\!46$$$$\beta_{1} -$$$$91\!\cdots\!60$$$$\beta_{2} - 79341959514231207312 \beta_{3} + 16895557758588184620 \beta_{4} - 1959999215698412896 \beta_{5} + 468553225281271312 \beta_{7} - 10320056606636432 \beta_{8} - 3858005152242396 \beta_{9}) q^{74} +(-$$$$49\!\cdots\!59$$$$-$$$$16\!\cdots\!81$$$$\beta_{1} -$$$$14\!\cdots\!73$$$$\beta_{2} -$$$$22\!\cdots\!50$$$$\beta_{3} - 22340730413445728700 \beta_{4} - 254009758316930676 \beta_{5} + 145374717611234862 \beta_{6} + 367231954690685898 \beta_{7} - 23878645489407150 \beta_{8} - 23905137795644238 \beta_{9}) q^{75} +($$$$75\!\cdots\!24$$$$-$$$$12\!\cdots\!38$$$$\beta_{1} +$$$$39\!\cdots\!26$$$$\beta_{2} -$$$$27\!\cdots\!98$$$$\beta_{3} - 5643111612512309408 \beta_{4} - 5668913266239480128 \beta_{5} - 53992345731629696 \beta_{6} - 341496487262960576 \beta_{7} - 3685950532452960 \beta_{9}) q^{76} +($$$$92\!\cdots\!62$$$$+$$$$45\!\cdots\!74$$$$\beta_{1} -$$$$23\!\cdots\!64$$$$\beta_{2} -$$$$19\!\cdots\!80$$$$\beta_{3} + 20374072856006132010 \beta_{4} - 4608208871556457904 \beta_{5} + 813454555868832416 \beta_{7} - 9815655767343184 \beta_{8} - 8048040180071760 \beta_{9}) q^{77} +(-$$$$14\!\cdots\!98$$$$-$$$$60\!\cdots\!02$$$$\beta_{1} -$$$$13\!\cdots\!40$$$$\beta_{2} +$$$$55\!\cdots\!36$$$$\beta_{3} - 82604829266812733148 \beta_{4} - 13866413481887971414 \beta_{5} - 246791751923736576 \beta_{6} + 208313185513869958 \beta_{7} + 53993953836482544 \beta_{8} + 14731440363587340 \beta_{9}) q^{78} +($$$$12\!\cdots\!48$$$$-$$$$88\!\cdots\!61$$$$\beta_{1} +$$$$27\!\cdots\!72$$$$\beta_{2} +$$$$52\!\cdots\!27$$$$\beta_{3} - 6682164741069185378 \beta_{4} - 6419490038895083426 \beta_{5} + 227989329610609055 \beta_{6} + 3154936010979173663 \beta_{7} + 37524957453443136 \beta_{9}) q^{79} +($$$$27\!\cdots\!56$$$$+$$$$52\!\cdots\!76$$$$\beta_{1} -$$$$69\!\cdots\!72$$$$\beta_{2} -$$$$55\!\cdots\!60$$$$\beta_{3} - 2292402270677115584 \beta_{4} - 10210335684570815200 \beta_{5} - 3801029616562015040 \beta_{7} + 61231649617502560 \beta_{8} + 35045077641868800 \beta_{9}) q^{80} +(-$$$$17\!\cdots\!65$$$$-$$$$49\!\cdots\!96$$$$\beta_{1} +$$$$36\!\cdots\!08$$$$\beta_{2} -$$$$55\!\cdots\!76$$$$\beta_{3} +$$$$18\!\cdots\!00$$$$\beta_{4} + 4887048661025575338 \beta_{5} - 129459017712914172 \beta_{6} - 2421985494731523192 \beta_{7} - 19818774520661826 \beta_{8} + 80848127326685310 \beta_{9}) q^{81} +($$$$26\!\cdots\!00$$$$-$$$$13\!\cdots\!14$$$$\beta_{1} +$$$$41\!\cdots\!28$$$$\beta_{2} -$$$$96\!\cdots\!44$$$$\beta_{3} - 4822735546057982358 \beta_{4} - 4820950398647708004 \beta_{5} + 326730503697225216 \beta_{6} + 346622146268853732 \beta_{7} + 255021058610622 \beta_{9}) q^{82} +(-$$$$54\!\cdots\!19$$$$+$$$$26\!\cdots\!43$$$$\beta_{1} +$$$$13\!\cdots\!18$$$$\beta_{2} +$$$$12\!\cdots\!50$$$$\beta_{3} -$$$$27\!\cdots\!55$$$$\beta_{4} + 28304149492823644753 \beta_{5} - 4363823536997273692 \beta_{7} - 96818473274930482 \beta_{8} + 68086692414836910 \beta_{9}) q^{83} +(-$$$$33\!\cdots\!80$$$$+$$$$46\!\cdots\!98$$$$\beta_{1} -$$$$33\!\cdots\!74$$$$\beta_{2} +$$$$34\!\cdots\!02$$$$\beta_{3} + 5356412002353063060 \beta_{4} + 13634490246254729118 \beta_{5} + 888395375200425984 \beta_{6} + 2001688897545940788 \beta_{7} - 119030972427964134 \beta_{8} - 131596718379489288 \beta_{9}) q^{84} +($$$$30\!\cdots\!24$$$$+$$$$81\!\cdots\!76$$$$\beta_{1} -$$$$25\!\cdots\!92$$$$\beta_{2} +$$$$20\!\cdots\!40$$$$\beta_{3} + 39566478262110832440 \beta_{4} + 39511068869173103736 \beta_{5} - 1757621152270206712 \beta_{6} - 2375040102147755128 \beta_{7} - 7915627562532672 \beta_{9}) q^{85} +(-$$$$14\!\cdots\!88$$$$-$$$$16\!\cdots\!89$$$$\beta_{1} +$$$$36\!\cdots\!06$$$$\beta_{2} +$$$$17\!\cdots\!60$$$$\beta_{3} +$$$$32\!\cdots\!43$$$$\beta_{4} - 9377981377587558124 \beta_{5} + 26205649223630554300 \beta_{7} - 11653536794096816 \beta_{8} - 310029758434680939 \beta_{9}) q^{86} +(-$$$$13\!\cdots\!92$$$$+$$$$18\!\cdots\!37$$$$\beta_{1} +$$$$11\!\cdots\!90$$$$\beta_{2} -$$$$25\!\cdots\!21$$$$\beta_{3} +$$$$12\!\cdots\!78$$$$\beta_{4} + 50139338092880212594 \beta_{5} - 673367925639132909 \beta_{6} + 10959751192482092447 \beta_{7} + 203086996795375866 \beta_{8} - 28773401724012870 \beta_{9}) q^{87} +($$$$28\!\cdots\!96$$$$+$$$$53\!\cdots\!92$$$$\beta_{1} -$$$$16\!\cdots\!04$$$$\beta_{2} +$$$$33\!\cdots\!28$$$$\beta_{3} + 42125369190811976256 \beta_{4} + 39759763151626945152 \beta_{5} + 587464213982257920 \beta_{6} - 25772145936936660096 \beta_{7} - 337943719883575872 \beta_{9}) q^{88} +(-$$$$19\!\cdots\!22$$$$-$$$$57\!\cdots\!24$$$$\beta_{1} +$$$$48\!\cdots\!24$$$$\beta_{2} +$$$$38\!\cdots\!76$$$$\beta_{3} +$$$$18\!\cdots\!72$$$$\beta_{4} + 92832053486490562862 \beta_{5} - 19776384314035595036 \beta_{7} + 336853783286399662 \beta_{8} + 179290849381261902 \beta_{9}) q^{89} +(-$$$$37\!\cdots\!18$$$$+$$$$54\!\cdots\!93$$$$\beta_{1} -$$$$42\!\cdots\!06$$$$\beta_{2} -$$$$57\!\cdots\!60$$$$\beta_{3} -$$$$10\!\cdots\!25$$$$\beta_{4} - 5906614852093968342 \beta_{5} - 495766276381714176 \beta_{6} - 29195567527375357434 \beta_{7} - 7209128868699600 \beta_{8} - 77771566050405111 \beta_{9}) q^{90} +($$$$33\!\cdots\!84$$$$+$$$$18\!\cdots\!54$$$$\beta_{1} -$$$$56\!\cdots\!68$$$$\beta_{2} +$$$$17\!\cdots\!46$$$$\beta_{3} + 56555966544706058044 \beta_{4} + 60679871686376202364 \beta_{5} + 7496669524660362758 \beta_{6} + 53448755388984828038 \beta_{7} + 589129305952877760 \beta_{9}) q^{91} +($$$$20\!\cdots\!60$$$$-$$$$44\!\cdots\!44$$$$\beta_{1} -$$$$50\!\cdots\!20$$$$\beta_{2} -$$$$43\!\cdots\!12$$$$\beta_{3} +$$$$79\!\cdots\!68$$$$\beta_{4} - 84372859889472438116 \beta_{5} - 21428682913590894424 \beta_{7} - 557280800865123244 \beta_{8} + 347983501496459760 \beta_{9}) q^{92} +(-$$$$91\!\cdots\!13$$$$+$$$$71\!\cdots\!35$$$$\beta_{1} +$$$$19\!\cdots\!52$$$$\beta_{2} +$$$$55\!\cdots\!26$$$$\beta_{3} +$$$$43\!\cdots\!41$$$$\beta_{4} + 35595808240573813554 \beta_{5} - 1374109258633384698 \beta_{6} - 2280407560943239818 \beta_{7} - 139483224382955976 \beta_{8} + 781962232612512072 \beta_{9}) q^{93} +(-$$$$19\!\cdots\!00$$$$-$$$$75\!\cdots\!32$$$$\beta_{1} +$$$$23\!\cdots\!84$$$$\beta_{2} -$$$$58\!\cdots\!84$$$$\beta_{3} -$$$$38\!\cdots\!20$$$$\beta_{4} -$$$$38\!\cdots\!12$$$$\beta_{5} - 14647248602276335616 \beta_{6} + 17955421798261411816 \beta_{7} + 417982953853048044 \beta_{9}) q^{94} +($$$$36\!\cdots\!78$$$$-$$$$39\!\cdots\!62$$$$\beta_{1} -$$$$91\!\cdots\!36$$$$\beta_{2} - 66572689583907349860 \beta_{3} -$$$$18\!\cdots\!02$$$$\beta_{4} + 4449868723824776050 \beta_{5} - 11804734440036963040 \beta_{7} - 60560310250775440 \beta_{8} + 150625937899378800 \beta_{9}) q^{95} +($$$$57\!\cdots\!32$$$$+$$$$13\!\cdots\!92$$$$\beta_{1} +$$$$49\!\cdots\!68$$$$\beta_{2} +$$$$43\!\cdots\!72$$$$\beta_{3} +$$$$21\!\cdots\!12$$$$\beta_{4} -$$$$55\!\cdots\!32$$$$\beta_{5} + 5024813519174658048 \beta_{6} +$$$$13\!\cdots\!92$$$$\beta_{7} - 461123414828109360 \beta_{8} - 109265850453033216 \beta_{9}) q^{96} +(-$$$$11\!\cdots\!00$$$$+$$$$51\!\cdots\!44$$$$\beta_{1} -$$$$17\!\cdots\!64$$$$\beta_{2} -$$$$15\!\cdots\!66$$$$\beta_{3} +$$$$12\!\cdots\!02$$$$\beta_{4} +$$$$11\!\cdots\!18$$$$\beta_{5} - 115304699948229330 \beta_{6} -$$$$10\!\cdots\!66$$$$\beta_{7} - 1343820413585016912 \beta_{9}) q^{97} +($$$$16\!\cdots\!56$$$$+$$$$11\!\cdots\!63$$$$\beta_{1} -$$$$41\!\cdots\!32$$$$\beta_{2} -$$$$31\!\cdots\!40$$$$\beta_{3} -$$$$45\!\cdots\!20$$$$\beta_{4} -$$$$71\!\cdots\!12$$$$\beta_{5} + 53342144017012306288 \beta_{7} + 1448872063255698928 \beta_{8} - 876504201221334420 \beta_{9}) q^{98} +($$$$25\!\cdots\!17$$$$-$$$$17\!\cdots\!13$$$$\beta_{1} -$$$$11\!\cdots\!58$$$$\beta_{2} +$$$$42\!\cdots\!16$$$$\beta_{3} -$$$$62\!\cdots\!93$$$$\beta_{4} +$$$$43\!\cdots\!63$$$$\beta_{5} + 5167720315144821258 \beta_{6} -$$$$24\!\cdots\!54$$$$\beta_{7} + 668664769215000492 \beta_{8} - 2884502179525317780 \beta_{9}) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 119369106q^{3} - 55582533344q^{4} - 9250224859872q^{6} - 123569771565772q^{7} + 4787501147541018q^{9} + O(q^{10})$$ $$10q + 119369106q^{3} - 55582533344q^{4} - 9250224859872q^{6} - 123569771565772q^{7} + 4787501147541018q^{9} - 2611205560422720q^{10} + 1839088058193784224q^{12} + 3639874363106470052q^{13} + 73840351311049043520q^{15} -$$$$45\!\cdots\!64$$$$q^{16} +$$$$44\!\cdots\!20$$$$q^{18} -$$$$18\!\cdots\!44$$$$q^{19} +$$$$47\!\cdots\!32$$$$q^{21} -$$$$82\!\cdots\!40$$$$q^{22} -$$$$40\!\cdots\!04$$$$q^{24} -$$$$30\!\cdots\!30$$$$q^{25} -$$$$51\!\cdots\!66$$$$q^{27} +$$$$12\!\cdots\!52$$$$q^{28} +$$$$45\!\cdots\!40$$$$q^{30} -$$$$16\!\cdots\!20$$$$q^{31} +$$$$12\!\cdots\!80$$$$q^{33} -$$$$95\!\cdots\!84$$$$q^{34} -$$$$37\!\cdots\!72$$$$q^{36} +$$$$11\!\cdots\!88$$$$q^{37} -$$$$14\!\cdots\!72$$$$q^{39} +$$$$12\!\cdots\!40$$$$q^{40} -$$$$11\!\cdots\!00$$$$q^{42} -$$$$96\!\cdots\!28$$$$q^{43} +$$$$67\!\cdots\!20$$$$q^{45} -$$$$27\!\cdots\!84$$$$q^{46} -$$$$83\!\cdots\!44$$$$q^{48} +$$$$12\!\cdots\!66$$$$q^{49} -$$$$15\!\cdots\!92$$$$q^{51} +$$$$36\!\cdots\!48$$$$q^{52} -$$$$62\!\cdots\!72$$$$q^{54} +$$$$61\!\cdots\!40$$$$q^{55} -$$$$20\!\cdots\!12$$$$q^{57} +$$$$91\!\cdots\!20$$$$q^{58} -$$$$53\!\cdots\!20$$$$q^{60} +$$$$37\!\cdots\!20$$$$q^{61} -$$$$17\!\cdots\!68$$$$q^{63} +$$$$31\!\cdots\!32$$$$q^{64} -$$$$56\!\cdots\!40$$$$q^{66} +$$$$64\!\cdots\!48$$$$q^{67} -$$$$75\!\cdots\!52$$$$q^{69} +$$$$17\!\cdots\!20$$$$q^{70} -$$$$25\!\cdots\!60$$$$q^{72} +$$$$21\!\cdots\!92$$$$q^{73} -$$$$49\!\cdots\!90$$$$q^{75} +$$$$75\!\cdots\!16$$$$q^{76} -$$$$14\!\cdots\!60$$$$q^{78} +$$$$12\!\cdots\!76$$$$q^{79} -$$$$17\!\cdots\!10$$$$q^{81} +$$$$26\!\cdots\!40$$$$q^{82} -$$$$33\!\cdots\!32$$$$q^{84} +$$$$30\!\cdots\!40$$$$q^{85} -$$$$13\!\cdots\!40$$$$q^{87} +$$$$28\!\cdots\!20$$$$q^{88} -$$$$37\!\cdots\!80$$$$q^{90} +$$$$33\!\cdots\!24$$$$q^{91} -$$$$91\!\cdots\!68$$$$q^{93} -$$$$19\!\cdots\!24$$$$q^{94} +$$$$57\!\cdots\!64$$$$q^{96} -$$$$11\!\cdots\!92$$$$q^{97} +$$$$25\!\cdots\!80$$$$q^{99} + O(q^{100})$$

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

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

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.35.b.a 10
3.b odd 2 1 inner 3.35.b.a 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.35.b.a 10 1.a even 1 1 trivial
3.35.b.a 10 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace $$S_{35}^{\mathrm{new}}(3, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 58108079248 T^{2} +$$$$20\!\cdots\!92$$$$T^{4} -$$$$57\!\cdots\!28$$$$T^{6} +$$$$13\!\cdots\!12$$$$T^{8} -$$$$24\!\cdots\!08$$$$T^{10} +$$$$38\!\cdots\!72$$$$T^{12} -$$$$49\!\cdots\!08$$$$T^{14} +$$$$52\!\cdots\!72$$$$T^{16} -$$$$44\!\cdots\!08$$$$T^{18} +$$$$22\!\cdots\!76$$$$T^{20}$$
$3$ $$1 - 119369106 T + 4730741159849109 T^{2} +$$$$17\!\cdots\!40$$$$T^{3} +$$$$22\!\cdots\!78$$$$T^{4} -$$$$49\!\cdots\!48$$$$T^{5} +$$$$37\!\cdots\!82$$$$T^{6} +$$$$48\!\cdots\!40$$$$T^{7} +$$$$21\!\cdots\!81$$$$T^{8} -$$$$92\!\cdots\!26$$$$T^{9} +$$$$12\!\cdots\!49$$$$T^{10}$$
$5$ $$1 -$$$$27\!\cdots\!10$$$$T^{2} +$$$$33\!\cdots\!25$$$$T^{4} -$$$$25\!\cdots\!00$$$$T^{6} +$$$$16\!\cdots\!50$$$$T^{8} -$$$$98\!\cdots\!00$$$$T^{10} +$$$$55\!\cdots\!50$$$$T^{12} -$$$$29\!\cdots\!00$$$$T^{14} +$$$$12\!\cdots\!25$$$$T^{16} -$$$$36\!\cdots\!50$$$$T^{18} +$$$$44\!\cdots\!25$$$$T^{20}$$
$7$ $$( 1 + 61784885782886 T +$$$$10\!\cdots\!29$$$$T^{2} +$$$$61\!\cdots\!80$$$$T^{3} +$$$$66\!\cdots\!58$$$$T^{4} -$$$$44\!\cdots\!32$$$$T^{5} +$$$$36\!\cdots\!42$$$$T^{6} +$$$$17\!\cdots\!80$$$$T^{7} +$$$$16\!\cdots\!21$$$$T^{8} +$$$$52\!\cdots\!86$$$$T^{9} +$$$$46\!\cdots\!49$$$$T^{10} )^{2}$$
$11$ $$1 -$$$$13\!\cdots\!90$$$$T^{2} +$$$$99\!\cdots\!85$$$$T^{4} -$$$$47\!\cdots\!60$$$$T^{6} +$$$$17\!\cdots\!90$$$$T^{8} -$$$$48\!\cdots\!52$$$$T^{10} +$$$$11\!\cdots\!90$$$$T^{12} -$$$$20\!\cdots\!60$$$$T^{14} +$$$$27\!\cdots\!85$$$$T^{16} -$$$$25\!\cdots\!90$$$$T^{18} +$$$$11\!\cdots\!01$$$$T^{20}$$
$13$ $$( 1 - 1819937181553235026 T +$$$$10\!\cdots\!09$$$$T^{2} +$$$$68\!\cdots\!40$$$$T^{3} -$$$$12\!\cdots\!62$$$$T^{4} +$$$$11\!\cdots\!12$$$$T^{5} -$$$$92\!\cdots\!18$$$$T^{6} +$$$$38\!\cdots\!40$$$$T^{7} +$$$$42\!\cdots\!21$$$$T^{8} -$$$$57\!\cdots\!66$$$$T^{9} +$$$$23\!\cdots\!49$$$$T^{10} )^{2}$$
$17$ $$1 -$$$$28\!\cdots\!78$$$$T^{2} +$$$$39\!\cdots\!17$$$$T^{4} -$$$$41\!\cdots\!28$$$$T^{6} +$$$$37\!\cdots\!22$$$$T^{8} -$$$$28\!\cdots\!68$$$$T^{10} +$$$$17\!\cdots\!02$$$$T^{12} -$$$$91\!\cdots\!68$$$$T^{14} +$$$$40\!\cdots\!57$$$$T^{16} -$$$$13\!\cdots\!58$$$$T^{18} +$$$$22\!\cdots\!01$$$$T^{20}$$
$19$ $$( 1 +$$$$90\!\cdots\!22$$$$T +$$$$12\!\cdots\!77$$$$T^{2} +$$$$72\!\cdots\!32$$$$T^{3} +$$$$55\!\cdots\!42$$$$T^{4} +$$$$26\!\cdots\!52$$$$T^{5} +$$$$16\!\cdots\!82$$$$T^{6} +$$$$65\!\cdots\!12$$$$T^{7} +$$$$32\!\cdots\!97$$$$T^{8} +$$$$73\!\cdots\!82$$$$T^{9} +$$$$24\!\cdots\!01$$$$T^{10} )^{2}$$
$23$ $$1 -$$$$12\!\cdots\!78$$$$T^{2} +$$$$69\!\cdots\!97$$$$T^{4} -$$$$23\!\cdots\!48$$$$T^{6} +$$$$56\!\cdots\!82$$$$T^{8} -$$$$11\!\cdots\!08$$$$T^{10} +$$$$22\!\cdots\!42$$$$T^{12} -$$$$36\!\cdots\!28$$$$T^{14} +$$$$42\!\cdots\!77$$$$T^{16} -$$$$30\!\cdots\!38$$$$T^{18} +$$$$97\!\cdots\!01$$$$T^{20}$$
$29$ $$1 -$$$$33\!\cdots\!30$$$$T^{2} +$$$$48\!\cdots\!05$$$$T^{4} -$$$$41\!\cdots\!80$$$$T^{6} +$$$$24\!\cdots\!30$$$$T^{8} -$$$$12\!\cdots\!52$$$$T^{10} +$$$$68\!\cdots\!30$$$$T^{12} -$$$$31\!\cdots\!80$$$$T^{14} +$$$$10\!\cdots\!05$$$$T^{16} -$$$$19\!\cdots\!30$$$$T^{18} +$$$$16\!\cdots\!01$$$$T^{20}$$
$31$ $$( 1 +$$$$81\!\cdots\!10$$$$T +$$$$19\!\cdots\!85$$$$T^{2} +$$$$19\!\cdots\!40$$$$T^{3} +$$$$17\!\cdots\!90$$$$T^{4} +$$$$16\!\cdots\!48$$$$T^{5} +$$$$89\!\cdots\!90$$$$T^{6} +$$$$51\!\cdots\!40$$$$T^{7} +$$$$26\!\cdots\!85$$$$T^{8} +$$$$54\!\cdots\!10$$$$T^{9} +$$$$34\!\cdots\!01$$$$T^{10} )^{2}$$
$37$ $$( 1 -$$$$55\!\cdots\!94$$$$T +$$$$80\!\cdots\!89$$$$T^{2} -$$$$30\!\cdots\!40$$$$T^{3} +$$$$26\!\cdots\!18$$$$T^{4} -$$$$76\!\cdots\!72$$$$T^{5} +$$$$55\!\cdots\!02$$$$T^{6} -$$$$13\!\cdots\!40$$$$T^{7} +$$$$72\!\cdots\!41$$$$T^{8} -$$$$10\!\cdots\!54$$$$T^{9} +$$$$39\!\cdots\!49$$$$T^{10} )^{2}$$
$41$ $$1 -$$$$40\!\cdots\!90$$$$T^{2} +$$$$87\!\cdots\!85$$$$T^{4} -$$$$12\!\cdots\!60$$$$T^{6} +$$$$13\!\cdots\!90$$$$T^{8} -$$$$10\!\cdots\!52$$$$T^{10} +$$$$61\!\cdots\!90$$$$T^{12} -$$$$27\!\cdots\!60$$$$T^{14} +$$$$89\!\cdots\!85$$$$T^{16} -$$$$19\!\cdots\!90$$$$T^{18} +$$$$22\!\cdots\!01$$$$T^{20}$$
$43$ $$( 1 +$$$$48\!\cdots\!14$$$$T +$$$$12\!\cdots\!69$$$$T^{2} +$$$$51\!\cdots\!80$$$$T^{3} +$$$$71\!\cdots\!38$$$$T^{4} +$$$$25\!\cdots\!12$$$$T^{5} +$$$$24\!\cdots\!62$$$$T^{6} +$$$$61\!\cdots\!80$$$$T^{7} +$$$$49\!\cdots\!81$$$$T^{8} +$$$$68\!\cdots\!14$$$$T^{9} +$$$$48\!\cdots\!49$$$$T^{10} )^{2}$$
$47$ $$1 -$$$$22\!\cdots\!78$$$$T^{2} +$$$$19\!\cdots\!77$$$$T^{4} -$$$$10\!\cdots\!48$$$$T^{6} +$$$$11\!\cdots\!82$$$$T^{8} -$$$$10\!\cdots\!68$$$$T^{10} +$$$$56\!\cdots\!02$$$$T^{12} -$$$$27\!\cdots\!08$$$$T^{14} +$$$$24\!\cdots\!37$$$$T^{16} -$$$$14\!\cdots\!98$$$$T^{18} +$$$$32\!\cdots\!01$$$$T^{20}$$
$53$ $$1 -$$$$22\!\cdots\!78$$$$T^{2} +$$$$27\!\cdots\!77$$$$T^{4} -$$$$22\!\cdots\!48$$$$T^{6} +$$$$13\!\cdots\!82$$$$T^{8} -$$$$65\!\cdots\!68$$$$T^{10} +$$$$24\!\cdots\!02$$$$T^{12} -$$$$70\!\cdots\!08$$$$T^{14} +$$$$15\!\cdots\!37$$$$T^{16} -$$$$22\!\cdots\!98$$$$T^{18} +$$$$17\!\cdots\!01$$$$T^{20}$$
$59$ $$1 -$$$$10\!\cdots\!90$$$$T^{2} +$$$$57\!\cdots\!85$$$$T^{4} -$$$$19\!\cdots\!60$$$$T^{6} +$$$$48\!\cdots\!90$$$$T^{8} -$$$$90\!\cdots\!52$$$$T^{10} +$$$$12\!\cdots\!90$$$$T^{12} -$$$$13\!\cdots\!60$$$$T^{14} +$$$$10\!\cdots\!85$$$$T^{16} -$$$$51\!\cdots\!90$$$$T^{18} +$$$$12\!\cdots\!01$$$$T^{20}$$
$61$ $$( 1 -$$$$18\!\cdots\!10$$$$T +$$$$22\!\cdots\!45$$$$T^{2} -$$$$35\!\cdots\!20$$$$T^{3} +$$$$21\!\cdots\!10$$$$T^{4} -$$$$25\!\cdots\!52$$$$T^{5} +$$$$10\!\cdots\!10$$$$T^{6} -$$$$88\!\cdots\!20$$$$T^{7} +$$$$28\!\cdots\!45$$$$T^{8} -$$$$11\!\cdots\!10$$$$T^{9} +$$$$32\!\cdots\!01$$$$T^{10} )^{2}$$
$67$ $$( 1 -$$$$32\!\cdots\!74$$$$T +$$$$90\!\cdots\!49$$$$T^{2} -$$$$15\!\cdots\!60$$$$T^{3} +$$$$25\!\cdots\!78$$$$T^{4} -$$$$28\!\cdots\!12$$$$T^{5} +$$$$30\!\cdots\!62$$$$T^{6} -$$$$23\!\cdots\!60$$$$T^{7} +$$$$16\!\cdots\!61$$$$T^{8} -$$$$71\!\cdots\!94$$$$T^{9} +$$$$27\!\cdots\!49$$$$T^{10} )^{2}$$
$71$ $$1 -$$$$72\!\cdots\!30$$$$T^{2} +$$$$24\!\cdots\!05$$$$T^{4} -$$$$51\!\cdots\!80$$$$T^{6} +$$$$73\!\cdots\!30$$$$T^{8} -$$$$75\!\cdots\!52$$$$T^{10} +$$$$56\!\cdots\!30$$$$T^{12} -$$$$30\!\cdots\!80$$$$T^{14} +$$$$11\!\cdots\!05$$$$T^{16} -$$$$25\!\cdots\!30$$$$T^{18} +$$$$26\!\cdots\!01$$$$T^{20}$$
$73$ $$( 1 -$$$$10\!\cdots\!46$$$$T +$$$$97\!\cdots\!69$$$$T^{2} -$$$$60\!\cdots\!20$$$$T^{3} +$$$$38\!\cdots\!18$$$$T^{4} -$$$$19\!\cdots\!08$$$$T^{5} +$$$$87\!\cdots\!62$$$$T^{6} -$$$$30\!\cdots\!20$$$$T^{7} +$$$$11\!\cdots\!01$$$$T^{8} -$$$$28\!\cdots\!06$$$$T^{9} +$$$$58\!\cdots\!49$$$$T^{10} )^{2}$$
$79$ $$( 1 -$$$$64\!\cdots\!38$$$$T +$$$$29\!\cdots\!97$$$$T^{2} -$$$$93\!\cdots\!08$$$$T^{3} +$$$$23\!\cdots\!82$$$$T^{4} -$$$$47\!\cdots\!68$$$$T^{5} +$$$$78\!\cdots\!42$$$$T^{6} -$$$$10\!\cdots\!88$$$$T^{7} +$$$$10\!\cdots\!77$$$$T^{8} -$$$$76\!\cdots\!98$$$$T^{9} +$$$$39\!\cdots\!01$$$$T^{10} )^{2}$$
$83$ $$1 -$$$$99\!\cdots\!98$$$$T^{2} +$$$$47\!\cdots\!37$$$$T^{4} -$$$$15\!\cdots\!08$$$$T^{6} +$$$$38\!\cdots\!02$$$$T^{8} -$$$$77\!\cdots\!68$$$$T^{10} +$$$$12\!\cdots\!82$$$$T^{12} -$$$$15\!\cdots\!48$$$$T^{14} +$$$$14\!\cdots\!77$$$$T^{16} -$$$$96\!\cdots\!78$$$$T^{18} +$$$$30\!\cdots\!01$$$$T^{20}$$
$89$ $$1 -$$$$85\!\cdots\!90$$$$T^{2} +$$$$29\!\cdots\!85$$$$T^{4} -$$$$36\!\cdots\!60$$$$T^{6} -$$$$51\!\cdots\!10$$$$T^{8} +$$$$23\!\cdots\!48$$$$T^{10} -$$$$18\!\cdots\!10$$$$T^{12} -$$$$47\!\cdots\!60$$$$T^{14} +$$$$13\!\cdots\!85$$$$T^{16} -$$$$14\!\cdots\!90$$$$T^{18} +$$$$62\!\cdots\!01$$$$T^{20}$$
$97$ $$( 1 +$$$$59\!\cdots\!46$$$$T +$$$$15\!\cdots\!29$$$$T^{2} +$$$$59\!\cdots\!40$$$$T^{3} +$$$$93\!\cdots\!78$$$$T^{4} +$$$$27\!\cdots\!08$$$$T^{5} +$$$$33\!\cdots\!82$$$$T^{6} +$$$$75\!\cdots\!40$$$$T^{7} +$$$$67\!\cdots\!61$$$$T^{8} +$$$$95\!\cdots\!66$$$$T^{9} +$$$$56\!\cdots\!49$$$$T^{10} )^{2}$$