Properties

Label 3.33.b.a
Level 3
Weight 33
Character orbit 3.b
Analytic conductor 19.460
Analytic rank 0
Dimension 10
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 33 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(19.4599965427\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{61}\cdot 5^{5} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -2138715 + 35 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( -2579203486 + 5 \beta_{1} + 18 \beta_{2} + \beta_{3} ) q^{4} \) \( + ( 101591 \beta_{1} - 1092 \beta_{2} + \beta_{3} - \beta_{4} ) q^{5} \) \( + ( -242453078830 + 2722701 \beta_{1} + 619 \beta_{2} + 141 \beta_{3} - 4 \beta_{4} + \beta_{6} ) q^{6} \) \( + ( -556806241656 + 3208 \beta_{1} + 11574 \beta_{2} + 594 \beta_{3} + \beta_{5} - \beta_{6} ) q^{7} \) \( + ( -688 - 2970459932 \beta_{1} + 1835342 \beta_{2} - 1896 \beta_{3} + 172 \beta_{4} - \beta_{5} - 40 \beta_{6} - 4 \beta_{7} + \beta_{8} ) q^{8} \) \( + ( -79012360393819 + 7951416345 \beta_{1} + 3284457 \beta_{2} + 53126 \beta_{3} + 1182 \beta_{4} + 43 \beta_{5} + 38 \beta_{6} + 10 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{9} \) \(+O(q^{10})\) \( q\) \(+\beta_{1} q^{2}\) \(+(-2138715 + 35 \beta_{1} - \beta_{2}) q^{3}\) \(+(-2579203486 + 5 \beta_{1} + 18 \beta_{2} + \beta_{3}) q^{4}\) \(+(101591 \beta_{1} - 1092 \beta_{2} + \beta_{3} - \beta_{4}) q^{5}\) \(+(-242453078830 + 2722701 \beta_{1} + 619 \beta_{2} + 141 \beta_{3} - 4 \beta_{4} + \beta_{6}) q^{6}\) \(+(-556806241656 + 3208 \beta_{1} + 11574 \beta_{2} + 594 \beta_{3} + \beta_{5} - \beta_{6}) q^{7}\) \(+(-688 - 2970459932 \beta_{1} + 1835342 \beta_{2} - 1896 \beta_{3} + 172 \beta_{4} - \beta_{5} - 40 \beta_{6} - 4 \beta_{7} + \beta_{8}) q^{8}\) \(+(-79012360393819 + 7951416345 \beta_{1} + 3284457 \beta_{2} + 53126 \beta_{3} + 1182 \beta_{4} + 43 \beta_{5} + 38 \beta_{6} + 10 \beta_{7} - 2 \beta_{8} - \beta_{9}) q^{9}\) \(+(-700381278541630 + 16508370 \beta_{1} + 60956008 \beta_{2} + 300012 \beta_{3} - 4605 \beta_{4} - 341 \beta_{5} + 2824 \beta_{6} - 19 \beta_{7} + 9 \beta_{8} - 9 \beta_{9}) q^{10}\) \(+(-85268 + 112444454054 \beta_{1} + 151954249 \beta_{2} - 163991 \beta_{3} - 48806 \beta_{4} - 212 \beta_{5} - 8771 \beta_{6} + 103 \beta_{7} - 4 \beta_{8} - 54 \beta_{9}) q^{11}\) \(+(-27900742433599278 - 746753218411 \beta_{1} + 2529577112 \beta_{2} + 10311813 \beta_{3} + 815832 \beta_{4} - 1737 \beta_{5} + 2376 \beta_{6} - 1368 \beta_{7} + 369 \beta_{8} - 180 \beta_{9}) q^{12}\) \(+(56769755762652894 + 445277448 \beta_{1} + 1654993846 \beta_{2} - 13172582 \beta_{3} - 158928 \beta_{4} + 9434 \beta_{5} + 87780 \beta_{6} - 2866 \beta_{7} + 360 \beta_{8} - 360 \beta_{9}) q^{13}\) \(+(-1431350 - 3345897351216 \beta_{1} - 8121325454 \beta_{2} + 6955858 \beta_{3} - 10555739 \beta_{4} - 83 \beta_{5} - 16178 \beta_{6} - 20021 \beta_{7} + 1271 \beta_{8} + 297 \beta_{9}) q^{14}\) \(+(-2034220559489773910 + 16111160759850 \beta_{1} + 4362656816 \beta_{2} + 718118994 \beta_{3} + 23215090 \beta_{4} - 1107 \beta_{5} - 37477 \beta_{6} - 48438 \beta_{7} - 9612 \beta_{8} + 3942 \beta_{9}) q^{15}\) \(+(9345260973586445728 - 61784913560 \beta_{1} - 226620009856 \beta_{2} - 4105657336 \beta_{3} + 11941944 \beta_{4} - 120760 \beta_{5} - 4283584 \beta_{6} - 341096 \beta_{7} - 14616 \beta_{8} + 14616 \beta_{9}) q^{16}\) \(+(-39382332 + 88193183778324 \beta_{1} + 118080486780 \beta_{2} - 123656192 \beta_{3} + 23729426 \beta_{4} + 137964 \beta_{5} + 4123248 \beta_{6} - 1358886 \beta_{7} - 48324 \beta_{8} + 22410 \beta_{9}) q^{17}\) \(+(-54653243923841949750 - 338506303752045 \beta_{1} + 259951349916 \beta_{2} + 19980538008 \beta_{3} - 274809717 \beta_{4} + 929379 \beta_{5} - 9535548 \beta_{6} - 5847531 \beta_{7} + 113169 \beta_{8} - 5745 \beta_{9}) q^{18}\) \(+(57100768841239485830 - 616153984660 \beta_{1} - 2266961778411 \beta_{2} - 27124446349 \beta_{3} + 117061452 \beta_{4} + 890590 \beta_{5} + 30264473 \beta_{6} - 17603913 \beta_{7} + 175392 \beta_{8} - 175392 \beta_{9}) q^{19}\) \(+(-4792468800 - 1943920873494088 \beta_{1} + 14953878781716 \beta_{2} - 15257077648 \beta_{3} + 3226796728 \beta_{4} - 2565030 \beta_{5} - 154211280 \beta_{6} - 51387720 \beta_{7} + 741990 \beta_{8} - 455760 \beta_{9}) q^{20}\) \(+(-20280929681717183754 + 2056695405311749 \beta_{1} + 861133921564 \beta_{2} - 36041126229 \beta_{3} + 296796879 \beta_{4} - 20224008 \beta_{5} - 39188016 \beta_{6} - 122369958 \beta_{7} - 795996 \beta_{8} - 640026 \beta_{9}) q^{21}\) \(+(-\)\(77\!\cdots\!70\)\( - 13537600980510 \beta_{1} - 50184477413536 \beta_{2} + 144426788988 \beta_{3} + 3846476319 \beta_{4} - 3028369 \beta_{5} - 465120088 \beta_{6} - 292220327 \beta_{7} - 632835 \beta_{8} + 632835 \beta_{9}) q^{22}\) \(+(-33904266448 + 15746726856083428 \beta_{1} + 61682914026878 \beta_{2} - 68202285330 \beta_{3} - 17169921944 \beta_{4} + 20633408 \beta_{5} + 234924254 \beta_{6} - 593029954 \beta_{7} - 5824448 \beta_{8} + 3702240 \beta_{9}) q^{23}\) \(+(\)\(40\!\cdots\!20\)\( - 75362041757123364 \beta_{1} - 14385978051082 \beta_{2} - 1993476194016 \beta_{3} + 50984306068 \beta_{4} + 231083739 \beta_{5} - 2119193416 \beta_{6} - 1160163324 \beta_{7} + 5167125 \beta_{8} + 9145656 \beta_{9}) q^{24}\) \(+(-\)\(68\!\cdots\!75\)\( - 81220813333400 \beta_{1} - 301510497473810 \beta_{2} + 1695454041410 \beta_{3} + 24298881600 \beta_{4} - 19136630 \beta_{5} - 3310076180 \beta_{6} - 1774670170 \beta_{7} - 5591880 \beta_{8} + 5591880 \beta_{9}) q^{25}\) \(+(-194754015144 + 109134759772749770 \beta_{1} + 516942284796984 \beta_{2} - 537691903432 \beta_{3} + 48495380524 \beta_{4} - 58367412 \beta_{5} - 4688292024 \beta_{6} - 2360695788 \beta_{7} + 18998820 \beta_{8} - 9842148 \beta_{9}) q^{26}\) \(+(-\)\(72\!\cdots\!31\)\( - 125227308918747915 \beta_{1} + 71237801185272 \beta_{2} + 2238275781045 \beta_{3} - 136617890274 \beta_{4} - 1680998526 \beta_{5} + 5356446183 \beta_{6} - 1549131957 \beta_{7} - 48891132 \beta_{8} - 60426090 \beta_{9}) q^{27}\) \(+(\)\(20\!\cdots\!24\)\( + 148457176496658 \beta_{1} + 552587530503524 \beta_{2} - 5995245206486 \beta_{3} - 52451462280 \beta_{4} + 390032136 \beta_{5} + 21614035264 \beta_{6} + 987967960 \beta_{7} + 75597480 \beta_{8} - 75597480 \beta_{9}) q^{28}\) \(+(502051343816 + 1842880061075988045 \beta_{1} - 458263744227316 \beta_{2} + 594184952323 \beta_{3} + 670440040929 \beta_{4} - 362866792 \beta_{5} - 5589629824 \beta_{6} + 9135373172 \beta_{7} + 73858360 \beta_{8} - 72252108 \beta_{9}) q^{29}\) \(+(-\)\(11\!\cdots\!50\)\( - 5341730622289040178 \beta_{1} + 121747168438956 \beta_{2} + 53598049267032 \beta_{3} - 750697266447 \beta_{4} + 8167238865 \beta_{5} - 9421984260 \beta_{6} + 23541048135 \beta_{7} + 454215555 \beta_{8} + 190255005 \beta_{9}) q^{30}\) \(+(\)\(23\!\cdots\!72\)\( + 1463522705803840 \beta_{1} + 5454155030357688 \beta_{2} - 72426301019760 \beta_{3} - 466267159224 \beta_{4} - 3003341225 \beta_{5} - 35239830661 \beta_{6} + 53647321746 \beta_{7} - 330541344 \beta_{8} + 330541344 \beta_{9}) q^{31}\) \(+(8576579586432 + 16579533026720980896 \beta_{1} - 26224980711306576 \beta_{2} + 26815305603776 \beta_{3} - 5285038808480 \beta_{4} + 4428365592 \beta_{5} + 270946085568 \beta_{6} + 92872667616 \beta_{7} - 1184494872 \beta_{8} + 810967680 \beta_{9}) q^{32}\) \(+(\)\(25\!\cdots\!00\)\( - 13666410288853821537 \beta_{1} - 2085025797138005 \beta_{2} - 63260221203282 \beta_{3} + 3344329471556 \beta_{4} - 25936027335 \beta_{5} + 243518851642 \beta_{6} + 138336105636 \beta_{7} - 2893387338 \beta_{8} + 8474895 \beta_{9}) q^{33}\) \(+(-\)\(60\!\cdots\!12\)\( + 7701515070457880 \beta_{1} + 28356364329188416 \beta_{2} + 298569637608016 \beta_{3} - 1652819096364 \beta_{4} + 10731769300 \beta_{5} + 113721712864 \beta_{6} + 140283025676 \beta_{7} - 56940804 \beta_{8} + 56940804 \beta_{9}) q^{34}\) \(+(-3714994129100 + 44870492512169080820 \beta_{1} + 11516220234590230 \beta_{2} - 11447198440090 \beta_{3} + 2245454652750 \beta_{4} - 20390483060 \beta_{5} - 744168346310 \beta_{6} + 68625019060 \beta_{7} + 6697269980 \beta_{8} - 3423303270 \beta_{9}) q^{35}\) \(+(\)\(19\!\cdots\!54\)\( - \)\(11\!\cdots\!79\)\( \beta_{1} + 10907297341232622 \beta_{2} - 740714191524895 \beta_{3} - 2399238551400 \beta_{4} + 48699183214 \beta_{5} - 359063425648 \beta_{6} - 130407455912 \beta_{7} + 11025157042 \beta_{8} - 1844513536 \beta_{9}) q^{36}\) \(+(-\)\(35\!\cdots\!86\)\( - 1557927785048872 \beta_{1} - 6383616996964398 \beta_{2} + 1215832356274910 \beta_{3} + 1543075774128 \beta_{4} + 16431214318 \beta_{5} + 1659448002428 \beta_{6} - 474784639014 \beta_{7} + 7736200920 \beta_{8} - 7736200920 \beta_{9}) q^{37}\) \(+(-52044718847962 + \)\(19\!\cdots\!12\)\( \beta_{1} + 170492669711015294 \beta_{2} - 174340713222818 \beta_{3} + 43207108478547 \beta_{4} + 46325199275 \beta_{5} + 811650164354 \beta_{6} - 992209300291 \beta_{7} - 22646540015 \beta_{8} + 5919664815 \beta_{9}) q^{38}\) \(+(-\)\(31\!\cdots\!62\)\( - \)\(13\!\cdots\!94\)\( \beta_{1} - 71354259637410140 \beta_{2} + 935406980587386 \beta_{3} - 54125774431830 \beta_{4} - 57090291246 \beta_{5} - 1358337734028 \beta_{6} - 1825561307592 \beta_{7} - 15140707308 \beta_{8} - 605770506 \beta_{9}) q^{39}\) \(+(\)\(10\!\cdots\!60\)\( - 225882763188973840 \beta_{1} - 832699894638867456 \beta_{2} - 6764623867663184 \beta_{3} + 53809305921360 \beta_{4} - 410490692688 \beta_{5} - 13426256419968 \beta_{6} - 2689765575792 \beta_{7} - 40097055888 \beta_{8} + 40097055888 \beta_{9}) q^{40}\) \(+(-267805319535712 - 98073712365116399774 \beta_{1} + 672700381693062296 \beta_{2} - 705256494007394 \beta_{3} + 32209003935906 \beta_{4} - 37432320928 \beta_{5} - 4791591600784 \beta_{6} - 3518090663488 \beta_{7} + 50312775904 \beta_{8} + 3220113744 \beta_{9}) q^{41}\) \(+(-\)\(14\!\cdots\!50\)\( + \)\(14\!\cdots\!76\)\( \beta_{1} + 34017996217666904 \beta_{2} + 5777235292091844 \beta_{3} + 302463332817001 \beta_{4} + 308724464961 \beta_{5} - 4480249938664 \beta_{6} - 2778616543545 \beta_{7} - 80284639941 \beta_{8} + 58137275205 \beta_{9}) q^{42}\) \(+(\)\(20\!\cdots\!94\)\( + 112954635657412988 \beta_{1} + 419112883017953009 \beta_{2} - 1936597164703081 \beta_{3} - 38672174658180 \beta_{4} + 2267464309746 \beta_{5} + 23068466852929 \beta_{6} - 1019839399565 \beta_{7} + 94804930080 \beta_{8} - 94804930080 \beta_{9}) q^{43}\) \(+(333745590846784 - \)\(70\!\cdots\!84\)\( \beta_{1} - 1579635691732694060 \beta_{2} + 1556783806243824 \beta_{3} - 718404353025160 \beta_{4} + 103861318330 \beta_{5} + 7925293754032 \beta_{6} + 4045977921784 \beta_{7} - 79459862266 \beta_{8} + 6100364016 \beta_{9}) q^{44}\) \(+(\)\(32\!\cdots\!00\)\( + \)\(11\!\cdots\!99\)\( \beta_{1} + 2157217167500700642 \beta_{2} + 2246731841083899 \beta_{3} - 138194772582159 \beta_{4} - 2331255881190 \beta_{5} + 16505351943060 \beta_{6} + 10180958929440 \beta_{7} + 547456293420 \beta_{8} - 290497120530 \beta_{9}) q^{45}\) \(+(-\)\(10\!\cdots\!28\)\( + 1042583752210477100 \beta_{1} + 3842837382649750784 \beta_{2} + 32265141981562024 \beta_{3} - 231020250930966 \beta_{4} - 5967006387190 \beta_{5} + 8586554297456 \beta_{6} + 22442633987494 \beta_{7} - 71306065026 \beta_{8} + 71306065026 \beta_{9}) q^{46}\) \(+(1959314732793000 - \)\(43\!\cdots\!16\)\( \beta_{1} - 2362337090488904268 \beta_{2} + 2844332124888052 \beta_{3} + 2091146499928988 \beta_{4} - 1383037898472 \beta_{5} - 21743455647492 \beta_{6} + 35558599238640 \beta_{7} + 76483021752 \beta_{8} - 326638719180 \beta_{9}) q^{47}\) \(+(\)\(39\!\cdots\!92\)\( + \)\(10\!\cdots\!52\)\( \beta_{1} - 8520145578090357136 \beta_{2} - 91672099525566936 \beta_{3} - 1469772608486472 \beta_{4} + 8676218438496 \beta_{5} + 36764340963840 \beta_{6} + 49275850250520 \beta_{7} - 1493355756528 \beta_{8} + 623136344760 \beta_{9}) q^{48}\) \(+(-\)\(27\!\cdots\!29\)\( + 2413547237452651160 \beta_{1} + 8906283671909091986 \beta_{2} + 54596646647036606 \beta_{3} - 568893344202624 \beta_{4} + 538611142230 \beta_{5} + 38108021106484 \beta_{6} + 49088641718746 \beta_{7} - 37561185144 \beta_{8} + 37561185144 \beta_{9}) q^{49}\) \(+(4653138114325000 - \)\(13\!\cdots\!95\)\( \beta_{1} - 15164981517251414360 \beta_{2} + 15344982523271080 \beta_{3} - 3690174757757180 \beta_{4} + 5760715056100 \beta_{5} + 271821540880600 \beta_{6} + 29547000979900 \beta_{7} + 306901966700 \beta_{8} + 1516904255700 \beta_{9}) q^{50}\) \(+(\)\(19\!\cdots\!40\)\( + \)\(74\!\cdots\!60\)\( \beta_{1} + 731821283637244116 \beta_{2} + 116675153330778684 \beta_{3} - 4130169791772246 \beta_{4} - 12342259095048 \beta_{5} - 939779752404 \beta_{6} - 2402041579434 \beta_{7} + 1420786601028 \beta_{8} - 167957101530 \beta_{9}) q^{51}\) \(+(-\)\(50\!\cdots\!36\)\( - 5082437750495790742 \beta_{1} - 18748087001800662492 \beta_{2} - 128308425071189278 \beta_{3} + 1238352756547104 \beta_{4} + 54969282668512 \beta_{5} - 338262603261184 \beta_{6} - 68643641715552 \beta_{7} - 772121147040 \beta_{8} + 772121147040 \beta_{9}) q^{52}\) \(+(-13504913532198080 + \)\(60\!\cdots\!03\)\( \beta_{1} + 37595322118397905996 \beta_{2} - 38812799015930299 \beta_{3} + 4924018521397179 \beta_{4} - 7762745089856 \beta_{5} - 477452639831456 \beta_{6} - 139264012709120 \beta_{7} - 2451133847104 \beta_{8} - 2553469734240 \beta_{9}) q^{53}\) \(+(\)\(86\!\cdots\!12\)\( - \)\(18\!\cdots\!45\)\( \beta_{1} + 51616781087339590803 \beta_{2} + 226568271085180857 \beta_{3} + 30146868840093561 \beta_{4} - 30808586642931 \beta_{5} - 266034059093871 \beta_{6} - 305508975026805 \beta_{7} + 3750211568439 \beta_{8} - 1888166259063 \beta_{9}) q^{54}\) \(+(-\)\(11\!\cdots\!00\)\( - 14598642562523857000 \beta_{1} - 53652524066739479350 \beta_{2} - 759270352084368650 \beta_{3} + 2621614943985000 \beta_{4} - 195501972092050 \beta_{5} + 882949125886700 \beta_{6} - 392271552702950 \beta_{7} + 3883880239200 \beta_{8} - 3883880239200 \beta_{9}) q^{55}\) \(+(-45628821969275360 + \)\(31\!\cdots\!28\)\( \beta_{1} + 90004095643512613132 \beta_{2} - 97454581684150544 \beta_{3} - 17147040919623368 \beta_{4} - 15685174276106 \beta_{5} - 1162069170309776 \beta_{6} - 541681111975592 \beta_{7} + 5994429164042 \beta_{8} - 2422686278016 \beta_{9}) q^{56}\) \(+(\)\(40\!\cdots\!74\)\( - \)\(38\!\cdots\!43\)\( \beta_{1} - 67421892868571843885 \beta_{2} + 583247456438938524 \beta_{3} - 36621720037589508 \beta_{4} + 197298062874945 \beta_{5} - 443999078816526 \beta_{6} - 386204900095062 \beta_{7} - 15843432774798 \beta_{8} + 1973413733985 \beta_{9}) q^{57}\) \(+(-\)\(12\!\cdots\!10\)\( - 15460795324464357850 \beta_{1} - 58038429730853508168 \beta_{2} + 1591047815347217764 \beta_{3} + 6439491943037937 \beta_{4} + 236788497863913 \beta_{5} - 1407688621147944 \beta_{6} - 414935832068961 \beta_{7} - 2719296011565 \beta_{8} + 2719296011565 \beta_{9}) q^{58}\) \(+(33163535655336312 + \)\(11\!\cdots\!70\)\( \beta_{1} + 524264923424710113 \beta_{2} + 9453257740539121 \beta_{3} + 73420023417003068 \beta_{4} + 66401596064016 \beta_{5} + 2861946931502817 \beta_{6} + 54986207415129 \beta_{7} + 6033754506960 \beta_{8} + 18108837642744 \beta_{9}) q^{59}\) \(+(\)\(27\!\cdots\!80\)\( - \)\(28\!\cdots\!60\)\( \beta_{1} - 69878315950177390588 \beta_{2} - 4591555903966041312 \beta_{3} - 11474670019882520 \beta_{4} - 410831567134974 \beta_{5} + 509058767713136 \beta_{6} + 321703299906984 \beta_{7} + 25026416961246 \beta_{8} + 4381089557184 \beta_{9}) q^{60}\) \(+(-\)\(19\!\cdots\!38\)\( + 32311458042023255480 \beta_{1} + \)\(11\!\cdots\!42\)\( \beta_{2} - 522595291309267850 \beta_{3} - 7747211559235536 \beta_{4} + 449635992383910 \beta_{5} - 5423755183907124 \beta_{6} + 1723768066922274 \beta_{7} - 24093119634696 \beta_{8} + 24093119634696 \beta_{9}) q^{61}\) \(+(153108086405589298 + \)\(54\!\cdots\!64\)\( \beta_{1} - \)\(43\!\cdots\!74\)\( \beta_{2} + 453179182637398234 \beta_{3} - 67901401820093735 \beta_{4} - 35102907697727 \beta_{5} + 502611790353862 \beta_{6} + 2375199029070679 \beta_{7} - 70789711710733 \beta_{8} - 26473154852115 \beta_{9}) q^{62}\) \(+(\)\(34\!\cdots\!04\)\( - \)\(18\!\cdots\!20\)\( \beta_{1} - 44594297604065760018 \beta_{2} + 3863485341899555062 \beta_{3} - 56661367291427928 \beta_{4} + 63429537312503 \beta_{5} + 2694916483196401 \beta_{6} + 3962098650460640 \beta_{7} - 14318757649072 \beta_{8} + 7606512373000 \beta_{9}) q^{63}\) \(+(-\)\(73\!\cdots\!08\)\( + \)\(31\!\cdots\!20\)\( \beta_{1} + \)\(11\!\cdots\!00\)\( \beta_{2} + 9052401485771459392 \beta_{3} - 76665315742999872 \beta_{4} - 2262049199072960 \beta_{5} + 25607555710300672 \beta_{6} + 3135555834235328 \beta_{7} + 72697664305728 \beta_{8} - 72697664305728 \beta_{9}) q^{64}\) \(+(288723098498533400 - \)\(33\!\cdots\!30\)\( \beta_{1} - \)\(90\!\cdots\!20\)\( \beta_{2} + 930310994348437210 \beta_{3} - 203773676246997150 \beta_{4} - 130269934627960 \beta_{5} + 56560069957040 \beta_{6} + 4732555044652460 \beta_{7} + 132656486264680 \beta_{8} + 596637909180 \beta_{9}) q^{65}\) \(+(\)\(93\!\cdots\!54\)\( + \)\(55\!\cdots\!54\)\( \beta_{1} + \)\(74\!\cdots\!52\)\( \beta_{2} + 2423742665980781160 \beta_{3} + 300584590107793677 \beta_{4} + 1794210420936789 \beta_{5} + 4262920048348956 \beta_{6} + 1843549180864947 \beta_{7} - 26543754077769 \beta_{8} - 90975585370455 \beta_{9}) q^{66}\) \(+(-\)\(44\!\cdots\!66\)\( - 78045061144700267852 \beta_{1} - \)\(27\!\cdots\!85\)\( \beta_{2} - 29032048241091406219 \beta_{3} - 14829439395991164 \beta_{4} + 3375926100218920 \beta_{5} - 12572581775183363 \beta_{6} + 3237294837278117 \beta_{7} - 44726440691520 \beta_{8} + 44726440691520 \beta_{9}) q^{67}\) \(+(-568815496529980672 - \)\(17\!\cdots\!28\)\( \beta_{1} + \)\(15\!\cdots\!48\)\( \beta_{2} - 1585027778749299904 \beta_{3} + 156043153314827168 \beta_{4} - 171149108169544 \beta_{5} - 13121207589684160 \beta_{6} - 6948376178460256 \beta_{7} + 169155453018184 \beta_{8} - 498413787840 \beta_{9}) q^{68}\) \(+(\)\(10\!\cdots\!80\)\( + \)\(99\!\cdots\!98\)\( \beta_{1} - 1238672781137815174 \beta_{2} + 31430267792787036252 \beta_{3} - 86921178775932692 \beta_{4} - 4310122815231138 \beta_{5} + 1330822476000428 \beta_{6} - 4787812979550108 \beta_{7} + 125586557787564 \beta_{8} + 157378307076054 \beta_{9}) q^{69}\) \(+(-\)\(30\!\cdots\!00\)\( - \)\(11\!\cdots\!00\)\( \beta_{1} - \)\(40\!\cdots\!00\)\( \beta_{2} + 11022212572900207000 \beta_{3} + 317188825299879750 \beta_{4} + 421902278799750 \beta_{5} - 55586082413526000 \beta_{6} - 20920088853525750 \beta_{7} - 123180184464750 \beta_{8} + 123180184464750 \beta_{9}) q^{70}\) \(+(31703200374498584 + \)\(12\!\cdots\!88\)\( \beta_{1} + 34951997184128081618 \beta_{2} - 46840687929833662 \beta_{3} + 116915726223794468 \beta_{4} + 1593456571903256 \beta_{5} + 51252281637965498 \beta_{6} - 8634604125904594 \beta_{7} - 1154769758798408 \beta_{8} + 109671703276212 \beta_{9}) q^{71}\) \(+(\)\(54\!\cdots\!08\)\( + \)\(38\!\cdots\!44\)\( \beta_{1} - \)\(42\!\cdots\!98\)\( \beta_{2} - 82293478113169258584 \beta_{3} - 1358361664736560164 \beta_{4} + 2202577265503167 \beta_{5} - 69792357031961640 \beta_{6} - 37638630587309844 \beta_{7} - 278784464858463 \beta_{8} + 133338382924080 \beta_{9}) q^{72}\) \(+(-\)\(45\!\cdots\!06\)\( - \)\(36\!\cdots\!92\)\( \beta_{1} - \)\(13\!\cdots\!48\)\( \beta_{2} - 1732484891661846720 \beta_{3} + 85021856071306848 \beta_{4} - 10332540064161552 \beta_{5} + 37039029143001168 \beta_{6} - 13568224588481664 \beta_{7} + 144873070745760 \beta_{8} - 144873070745760 \beta_{9}) q^{73}\) \(+(-4393537516916584216 - \)\(90\!\cdots\!02\)\( \beta_{1} + \)\(13\!\cdots\!28\)\( \beta_{2} - 13905829793226035192 \beta_{3} + 2868623178780470388 \beta_{4} - 1422034491309484 \beta_{5} - 103593247059277192 \beta_{6} - 53190890585895604 \beta_{7} + 1559579745876412 \beta_{8} + 34386313641732 \beta_{9}) q^{74}\) \(+(\)\(56\!\cdots\!25\)\( + \)\(35\!\cdots\!35\)\( \beta_{1} + \)\(67\!\cdots\!05\)\( \beta_{2} + 16212657483913544010 \beta_{3} + 2700444081656365290 \beta_{4} + 9099535041747450 \beta_{5} + 32549119044257700 \beta_{6} + 13572681944455800 \beta_{7} + 140996047986900 \beta_{8} - 712323386042850 \beta_{9}) q^{75}\) \(+(-\)\(10\!\cdots\!76\)\( - \)\(51\!\cdots\!30\)\( \beta_{1} - \)\(19\!\cdots\!44\)\( \beta_{2} + \)\(15\!\cdots\!54\)\( \beta_{3} + 341232575526827208 \beta_{4} + 20728198992623800 \beta_{5} - 21839742487014208 \beta_{6} - 33678369824926872 \beta_{7} + 117146857304088 \beta_{8} - 117146857304088 \beta_{9}) q^{76}\) \(+(2950476981396551200 - \)\(31\!\cdots\!34\)\( \beta_{1} - \)\(13\!\cdots\!56\)\( \beta_{2} + 13269721557974175434 \beta_{3} - 5813203495156342554 \beta_{4} - 5807645719953824 \beta_{5} - 156965922875572064 \beta_{6} + 75286986775796080 \beta_{7} + 1776747679426784 \beta_{8} - 1007724510131760 \beta_{9}) q^{77}\) \(+(\)\(92\!\cdots\!00\)\( - \)\(71\!\cdots\!34\)\( \beta_{1} - \)\(20\!\cdots\!62\)\( \beta_{2} - 66283299513187537038 \beta_{3} - 193243893763294220 \beta_{4} - 17663231348916228 \beta_{5} + 166635284896436138 \beta_{6} - 471175642706652 \beta_{7} + 643325533690644 \beta_{8} + 485472531696300 \beta_{9}) q^{78}\) \(+(-\)\(21\!\cdots\!20\)\( + \)\(88\!\cdots\!60\)\( \beta_{1} + \)\(32\!\cdots\!84\)\( \beta_{2} - \)\(12\!\cdots\!04\)\( \beta_{3} - 2573531036674935288 \beta_{4} - 24335980027913545 \beta_{5} + 383652074275005723 \beta_{6} + 186669841223631122 \beta_{7} + 621029687134752 \beta_{8} - 621029687134752 \beta_{9}) q^{79}\) \(+(10721856803562348800 + \)\(37\!\cdots\!20\)\( \beta_{1} - \)\(28\!\cdots\!00\)\( \beta_{2} + 29680952218215632000 \beta_{3} - 2818263206097234880 \beta_{4} + 14430268033560080 \beta_{5} + 620165565557886080 \beta_{6} + 65659309129881920 \beta_{7} - 8340423299674640 \beta_{8} + 1522461183471360 \beta_{9}) q^{80}\) \(+(-\)\(45\!\cdots\!51\)\( - \)\(21\!\cdots\!34\)\( \beta_{1} + \)\(45\!\cdots\!96\)\( \beta_{2} + \)\(33\!\cdots\!66\)\( \beta_{3} + 1183354652644017336 \beta_{4} + 1000004431257792 \beta_{5} - 117031832338908744 \beta_{6} + 303901782272212662 \beta_{7} - 546824681374356 \beta_{8} + 347411890455354 \beta_{9}) q^{81}\) \(+(\)\(67\!\cdots\!20\)\( - \)\(73\!\cdots\!40\)\( \beta_{1} - \)\(27\!\cdots\!84\)\( \beta_{2} - \)\(47\!\cdots\!48\)\( \beta_{3} + 1522807167062055546 \beta_{4} - 292813975191126 \beta_{5} - 811629546859321872 \beta_{6} + 4523728233255462 \beta_{7} - 2936856435495570 \beta_{8} + 2936856435495570 \beta_{9}) q^{82}\) \(+(21935446298568291340 + \)\(13\!\cdots\!06\)\( \beta_{1} - \)\(45\!\cdots\!17\)\( \beta_{2} + 48645808663117011563 \beta_{3} + 6516093145101821562 \beta_{4} - 4497949098140708 \beta_{5} + 186705399549839407 \beta_{6} + 327812044884676225 \beta_{7} + 8118735567397868 \beta_{8} + 905196617314290 \beta_{9}) q^{83}\) \(+(-\)\(10\!\cdots\!04\)\( - \)\(31\!\cdots\!06\)\( \beta_{1} - \)\(23\!\cdots\!96\)\( \beta_{2} + \)\(21\!\cdots\!06\)\( \beta_{3} - 14284834048725828696 \beta_{4} + 12578909189673582 \beta_{5} - 239749429712221296 \beta_{6} - 187931389431226968 \beta_{7} - 2305743603177006 \beta_{8} + 1639083138929424 \beta_{9}) q^{84}\) \(+(\)\(87\!\cdots\!20\)\( + \)\(17\!\cdots\!20\)\( \beta_{1} + \)\(65\!\cdots\!88\)\( \beta_{2} + \)\(27\!\cdots\!32\)\( \beta_{3} - 4483056536886809280 \beta_{4} + 108546635387222024 \beta_{5} + 621220105600872464 \beta_{6} + 305253653927805016 \beta_{7} + 1527025493022624 \beta_{8} - 1527025493022624 \beta_{9}) q^{85}\) \(+(-51714606558990301930 + \)\(88\!\cdots\!04\)\( \beta_{1} + \)\(13\!\cdots\!06\)\( \beta_{2} - \)\(14\!\cdots\!22\)\( \beta_{3} + 15079468534506226891 \beta_{4} - 16099030188235933 \beta_{5} - 1260040857199030318 \beta_{6} - 623824775260474171 \beta_{7} + 6246683203529401 \beta_{8} - 2463086746176633 \beta_{9}) q^{86}\) \(+(-\)\(12\!\cdots\!50\)\( - \)\(19\!\cdots\!34\)\( \beta_{1} - \)\(97\!\cdots\!32\)\( \beta_{2} - \)\(48\!\cdots\!58\)\( \beta_{3} - 16481361234631524790 \beta_{4} + 55475667163178487 \beta_{5} + 2148426247452215113 \beta_{6} + 150089224511551338 \beta_{7} + 1253257742824164 \beta_{8} - 4201446955640130 \beta_{9}) q^{87}\) \(+(\)\(15\!\cdots\!40\)\( - \)\(38\!\cdots\!00\)\( \beta_{1} - \)\(14\!\cdots\!28\)\( \beta_{2} + \)\(59\!\cdots\!64\)\( \beta_{3} + 10666049875889485392 \beta_{4} - 243512657166476112 \beta_{5} + 1042078246435195776 \beta_{6} - 1212572914231718256 \beta_{7} + 7234311163670640 \beta_{8} - 7234311163670640 \beta_{9}) q^{88}\) \(+(8085314880239436548 + \)\(58\!\cdots\!30\)\( \beta_{1} - \)\(35\!\cdots\!48\)\( \beta_{2} + 35227316529869499414 \beta_{3} - 14903761348363180148 \beta_{4} + 4675359031933484 \beta_{5} + 183174870442846688 \beta_{6} + 93231188833712426 \beta_{7} - 18304792691145860 \beta_{8} - 3407358414803094 \beta_{9}) q^{89}\) \(+(-\)\(76\!\cdots\!70\)\( + \)\(11\!\cdots\!30\)\( \beta_{1} + \)\(12\!\cdots\!72\)\( \beta_{2} - \)\(11\!\cdots\!92\)\( \beta_{3} + 99046120957073588655 \beta_{4} - 90488259331788969 \beta_{5} - 5402692282354647984 \beta_{6} - 1831544764559306271 \beta_{7} + 11580573406859181 \beta_{8} - 6262360747379181 \beta_{9}) q^{90}\) \(+(\)\(59\!\cdots\!52\)\( + \)\(39\!\cdots\!80\)\( \beta_{1} + \)\(13\!\cdots\!56\)\( \beta_{2} + \)\(19\!\cdots\!76\)\( \beta_{3} + 2034089096874962736 \beta_{4} - 572618610039730 \beta_{5} - 2147560623112387866 \beta_{6} + 210100315067418356 \beta_{7} - 8482854520826784 \beta_{8} + 8482854520826784 \beta_{9}) q^{91}\) \(+(-\)\(15\!\cdots\!48\)\( - \)\(20\!\cdots\!80\)\( \beta_{1} + \)\(35\!\cdots\!00\)\( \beta_{2} - \)\(38\!\cdots\!48\)\( \beta_{3} - 12497804874156521136 \beta_{4} - 7081865555316164 \beta_{5} - 2378026798551397088 \beta_{6} - 2138866623862342064 \beta_{7} + 9827428216590404 \beta_{8} + 686390665318560 \beta_{9}) q^{92}\) \(+(-\)\(10\!\cdots\!54\)\( + \)\(13\!\cdots\!29\)\( \beta_{1} - \)\(20\!\cdots\!70\)\( \beta_{2} - \)\(25\!\cdots\!67\)\( \beta_{3} - 39035796717493142235 \beta_{4} - 280872572606144154 \beta_{5} + 449813158533771084 \beta_{6} + 1027043386332581178 \beta_{7} - 8589666743671896 \beta_{8} + 23562895271374800 \beta_{9}) q^{93}\) \(+(\)\(30\!\cdots\!48\)\( - \)\(10\!\cdots\!00\)\( \beta_{1} - \)\(40\!\cdots\!72\)\( \beta_{2} - \)\(41\!\cdots\!84\)\( \beta_{3} + 24078788203481254020 \beta_{4} + 793166375503729540 \beta_{5} - 3890370527238208160 \beta_{6} - 1797322076129465380 \beta_{7} - 4675834804334580 \beta_{8} + 4675834804334580 \beta_{9}) q^{94}\) \(+(\)\(25\!\cdots\!00\)\( - \)\(20\!\cdots\!92\)\( \beta_{1} - \)\(65\!\cdots\!86\)\( \beta_{2} + \)\(68\!\cdots\!58\)\( \beta_{3} - 32815092314298413528 \beta_{4} + 137024185419639520 \beta_{5} + 8402222663834078770 \beta_{6} + 2766229015986321730 \beta_{7} - 14741929290816160 \beta_{8} + 30570564032205840 \beta_{9}) q^{95}\) \(+(-\)\(52\!\cdots\!00\)\( + \)\(53\!\cdots\!08\)\( \beta_{1} + \)\(12\!\cdots\!56\)\( \beta_{2} + \)\(94\!\cdots\!48\)\( \beta_{3} - 37037966071099584864 \beta_{4} + 777971959883404920 \beta_{5} + 12382605861033362880 \beta_{6} - 704320499112260832 \beta_{7} - 40137609104817912 \beta_{8} - 8705706758670528 \beta_{9}) q^{96}\) \(+(\)\(39\!\cdots\!54\)\( + \)\(21\!\cdots\!08\)\( \beta_{1} + \)\(78\!\cdots\!18\)\( \beta_{2} + 24002497929615160862 \beta_{3} - 56055268203893900256 \beta_{4} - 713298456993362298 \beta_{5} + 2594067100004106916 \beta_{6} + 5206642300602263098 \beta_{7} - 11963574307883160 \beta_{8} + 11963574307883160 \beta_{9}) q^{97}\) \(+(-92520170200338013000 - \)\(57\!\cdots\!69\)\( \beta_{1} + \)\(26\!\cdots\!88\)\( \beta_{2} - \)\(26\!\cdots\!92\)\( \beta_{3} + 37089933854360944732 \beta_{4} - 214775935411891588 \beta_{5} - 8538646384949858968 \beta_{6} - 22678975589797660 \beta_{7} + 78422242449810868 \beta_{8} - 34088423240520180 \beta_{9}) q^{98}\) \(+(-\)\(33\!\cdots\!52\)\( + \)\(75\!\cdots\!70\)\( \beta_{1} - \)\(20\!\cdots\!53\)\( \beta_{2} - \)\(13\!\cdots\!01\)\( \beta_{3} - \)\(23\!\cdots\!88\)\( \beta_{4} + 13461479270827914 \beta_{5} - 14882264021146662327 \beta_{6} + 7383926773146751311 \beta_{7} + 30899637423348600 \beta_{8} - 16464564953297004 \beta_{9}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut -\mathstrut 21387150q^{3} \) \(\mathstrut -\mathstrut 25792034864q^{4} \) \(\mathstrut -\mathstrut 2424530788848q^{6} \) \(\mathstrut -\mathstrut 5568062418940q^{7} \) \(\mathstrut -\mathstrut 790123604155542q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut -\mathstrut 21387150q^{3} \) \(\mathstrut -\mathstrut 25792034864q^{4} \) \(\mathstrut -\mathstrut 2424530788848q^{6} \) \(\mathstrut -\mathstrut 5568062418940q^{7} \) \(\mathstrut -\mathstrut 790123604155542q^{9} \) \(\mathstrut -\mathstrut 7003812786596640q^{10} \) \(\mathstrut -\mathstrut 279007424380502640q^{12} \) \(\mathstrut +\mathstrut 567697557679805780q^{13} \) \(\mathstrut -\mathstrut 20342205597863242080q^{15} \) \(\mathstrut +\mathstrut 93452609752238437504q^{16} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!00\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!76\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!52\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!60\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!16\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!50\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!30\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!00\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!40\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!60\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!24\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!92\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!40\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!72\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!80\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!60\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!40\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!00\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!76\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!20\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!54\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!12\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!20\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(86\!\cdots\!48\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!00\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!40\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!80\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!40\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(73\!\cdots\!08\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(93\!\cdots\!20\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!40\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!08\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!60\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!20\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!50\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!96\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(92\!\cdots\!60\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!64\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!30\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!60\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!52\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!60\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!40\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(76\!\cdots\!60\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!16\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!20\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!56\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!84\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!00\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!60\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut +\mathstrut \) \(954745942\) \(x^{8}\mathstrut +\mathstrut \) \(302468338607088448\) \(x^{6}\mathstrut +\mathstrut \) \(37939920124077893929140224\) \(x^{4}\mathstrut +\mathstrut \) \(1938513915962148831841211918581760\) \(x^{2}\mathstrut +\mathstrut \) \(31225030372218346257929044634944667648000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(59633227320101615\) \(\nu^{9}\mathstrut +\mathstrut \) \(1090600286927067120512\) \(\nu^{8}\mathstrut -\mathstrut \) \(52709649968373927593486410\) \(\nu^{7}\mathstrut +\mathstrut \) \(948801410985586336615193128192\) \(\nu^{6}\mathstrut -\mathstrut \) \(14435713677373805300696362864355520\) \(\nu^{5}\mathstrut +\mathstrut \) \(248579350147503316782603801211147345920\) \(\nu^{4}\mathstrut -\mathstrut \) \(1323557405808714060187144410359102911283200\) \(\nu^{3}\mathstrut +\mathstrut \) \(19904574529903728581896468193614499104744800256\) \(\nu^{2}\mathstrut -\mathstrut \) \(35407556469442706594842899659709085678354269470720\) \(\nu\mathstrut +\mathstrut \) \(398640447768924348641639738540538024670178168243486720\)\()/\)\(10\!\cdots\!20\)
\(\beta_{3}\)\(=\)\((\)\(59633227320101615\) \(\nu^{9}\mathstrut -\mathstrut \) \(1090600286927067120512\) \(\nu^{8}\mathstrut +\mathstrut \) \(52709649968373927593486410\) \(\nu^{7}\mathstrut -\mathstrut \) \(948801410985586336615193128192\) \(\nu^{6}\mathstrut +\mathstrut \) \(14435713677373805300696362864355520\) \(\nu^{5}\mathstrut -\mathstrut \) \(248579350147503316782603801211147345920\) \(\nu^{4}\mathstrut +\mathstrut \) \(1323557405808714060187144410359102911283200\) \(\nu^{3}\mathstrut -\mathstrut \) \(17903598787082067170171536201916752232622063616\) \(\nu^{2}\mathstrut +\mathstrut \) \(35405888989657021876999795549716004222627500523520\) \(\nu\mathstrut -\mathstrut \) \(16555753691274039942295301922045854570789098005463040\)\()/\)\(55\!\cdots\!40\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(6748425254370034631\) \(\nu^{9}\mathstrut -\mathstrut \) \(50440263270376854323680\) \(\nu^{8}\mathstrut -\mathstrut \) \(5463625870662307811665879258\) \(\nu^{7}\mathstrut -\mathstrut \) \(43882065258083368068452682178880\) \(\nu^{6}\mathstrut -\mathstrut \) \(1232724199829376532900019904271148736\) \(\nu^{5}\mathstrut -\mathstrut \) \(11496794944322028401195425806015564748800\) \(\nu^{4}\mathstrut -\mathstrut \) \(60697171359334841226155598040137246756044800\) \(\nu^{3}\mathstrut -\mathstrut \) \(919085840200931200853917954960897273440354959360\) \(\nu^{2}\mathstrut +\mathstrut \) \(441589701534271412296948664504046080788377402081280\) \(\nu\mathstrut -\mathstrut \) \(18150557188754513393151329580036014513421198478582743040\)\()/\)\(41\!\cdots\!80\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(7253290202712871422403\) \(\nu^{9}\mathstrut +\mathstrut \) \(165168677402717555021954176\) \(\nu^{8}\mathstrut -\mathstrut \) \(6271976456514303332525528585090\) \(\nu^{7}\mathstrut +\mathstrut \) \(127832648772253674808774383620281088\) \(\nu^{6}\mathstrut -\mathstrut \) \(1624787459152076683950846846367251028416\) \(\nu^{5}\mathstrut +\mathstrut \) \(25811375565705079312323057384815178271727616\) \(\nu^{4}\mathstrut -\mathstrut \) \(126626904659742152729503815327082560036049780736\) \(\nu^{3}\mathstrut +\mathstrut \) \(646916249362852014258641421974558136561932529827840\) \(\nu^{2}\mathstrut -\mathstrut \) \(2415347634433347759661172135718291558994732589944668160\) \(\nu\mathstrut -\mathstrut \) \(39544588103070603764203024367873599544718686204926959288320\)\()/\)\(10\!\cdots\!20\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(7305886709209201046833\) \(\nu^{9}\mathstrut -\mathstrut \) \(28098994266482882343270272\) \(\nu^{8}\mathstrut -\mathstrut \) \(6318466367786409136662983598710\) \(\nu^{7}\mathstrut -\mathstrut \) \(24000625452591668769374076605900032\) \(\nu^{6}\mathstrut -\mathstrut \) \(1637519758615520380226061038413612597056\) \(\nu^{5}\mathstrut -\mathstrut \) \(6260203394463381014766128471534660574535680\) \(\nu^{4}\mathstrut -\mathstrut \) \(127794282291665438530588876697019288803801563136\) \(\nu^{3}\mathstrut -\mathstrut \) \(535006931119558709916410202926366526438892682346496\) \(\nu^{2}\mathstrut -\mathstrut \) \(2446575670542715852211575601616712900371774359983882240\) \(\nu\mathstrut -\mathstrut \) \(12876918065246757622942862380738644135772214585920480870400\)\()/\)\(10\!\cdots\!20\)
\(\beta_{7}\)\(=\)\((\)\(3821640389947315485755\) \(\nu^{9}\mathstrut +\mathstrut \) \(54616636537615807357500352\) \(\nu^{8}\mathstrut +\mathstrut \) \(3379233693605605472734703052754\) \(\nu^{7}\mathstrut +\mathstrut \) \(47404189536089680146141470677799552\) \(\nu^{6}\mathstrut +\mathstrut \) \(921225273741721497674511177558107938752\) \(\nu^{5}\mathstrut +\mathstrut \) \(12412618675190131867279992446086015811727360\) \(\nu^{4}\mathstrut +\mathstrut \) \(82645310993067251716965206094458006834180521984\) \(\nu^{3}\mathstrut +\mathstrut \) \(1000722737516932818048196146774351732340008981364736\) \(\nu^{2}\mathstrut +\mathstrut \) \(2116511143545111533986804449402519925577837908255047680\) \(\nu\mathstrut +\mathstrut \) \(20303993052819697404725739919220008228535223977273479659520\)\()/\)\(25\!\cdots\!80\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(99002483403773996372825\) \(\nu^{9}\mathstrut -\mathstrut \) \(1915552020276173336124268672\) \(\nu^{8}\mathstrut -\mathstrut \) \(83849935149810968170537835209702\) \(\nu^{7}\mathstrut -\mathstrut \) \(1666335945163834138953357575097608960\) \(\nu^{6}\mathstrut -\mathstrut \) \(20310154278569673806896997893124011934016\) \(\nu^{5}\mathstrut -\mathstrut \) \(443247729599879743787281812862166369216077824\) \(\nu^{4}\mathstrut -\mathstrut \) \(975059053403993335335691761049450666486411558912\) \(\nu^{3}\mathstrut -\mathstrut \) \(38090526493067233107840296943515728194887302768492544\) \(\nu^{2}\mathstrut +\mathstrut \) \(67352436297742361833574316646036448743954629330636636160\) \(\nu\mathstrut -\mathstrut \) \(887038493243047735583641355934340909001502983758838581166080\)\()/\)\(10\!\cdots\!20\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(405686191520795612219921\) \(\nu^{9}\mathstrut +\mathstrut \) \(1449470074400225843888100224\) \(\nu^{8}\mathstrut -\mathstrut \) \(352603011978332535838318832543158\) \(\nu^{7}\mathstrut +\mathstrut \) \(1259200438080272932053214825907850496\) \(\nu^{6}\mathstrut -\mathstrut \) \(92386196517360385958043848500554700166976\) \(\nu^{5}\mathstrut +\mathstrut \) \(334248081429324830335352309135036993546182656\) \(\nu^{4}\mathstrut -\mathstrut \) \(7399831155135833313723157917017558912255257477120\) \(\nu^{3}\mathstrut +\mathstrut \) \(28623636740156794758048238299017404042772398697611264\) \(\nu^{2}\mathstrut -\mathstrut \) \(141777561351222851802990567506599967599120298496512491520\) \(\nu\mathstrut +\mathstrut \) \(663965512849069476645870899193805129827594764178669594214400\)\()/\)\(16\!\cdots\!20\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(18\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut -\mathstrut \) \(6874170782\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{8}\mathstrut -\mathstrut \) \(4\) \(\beta_{7}\mathstrut -\mathstrut \) \(40\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(172\) \(\beta_{4}\mathstrut -\mathstrut \) \(1896\) \(\beta_{3}\mathstrut +\mathstrut \) \(1835342\) \(\beta_{2}\mathstrut -\mathstrut \) \(11560394524\) \(\beta_{1}\mathstrut -\mathstrut \) \(688\)\()/216\)
\(\nu^{4}\)\(=\)\((\)\(1827\) \(\beta_{9}\mathstrut -\mathstrut \) \(1827\) \(\beta_{8}\mathstrut -\mathstrut \) \(42637\) \(\beta_{7}\mathstrut -\mathstrut \) \(535448\) \(\beta_{6}\mathstrut -\mathstrut \) \(15095\) \(\beta_{5}\mathstrut +\mathstrut \) \(1492743\) \(\beta_{4}\mathstrut -\mathstrut \) \(2123819903\) \(\beta_{3}\mathstrut -\mathstrut \) \(57318530480\) \(\beta_{2}\mathstrut -\mathstrut \) \(15776177875\) \(\beta_{1}\mathstrut +\mathstrut \) \(9933941623412891316\)\()/162\)
\(\nu^{5}\)\(=\)\((\)\(101370960\) \(\beta_{9}\mathstrut -\mathstrut \) \(2295545507\) \(\beta_{8}\mathstrut +\mathstrut \) \(20199018044\) \(\beta_{7}\mathstrut +\mathstrut \) \(119767606616\) \(\beta_{6}\mathstrut +\mathstrut \) \(2701029347\) \(\beta_{5}\mathstrut -\mathstrut \) \(1029997038516\) \(\beta_{4}\mathstrut +\mathstrut \) \(7423542197080\) \(\beta_{3}\mathstrut -\mathstrut \) \(7219489522400938\) \(\beta_{2}\mathstrut +\mathstrut \) \(19980670805417784308\) \(\beta_{1}\mathstrut +\mathstrut \) \(2549541198128\)\()/972\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(2013405595299\) \(\beta_{9}\mathstrut +\mathstrut \) \(2013405595299\) \(\beta_{8}\mathstrut +\mathstrut \) \(54481961761549\) \(\beta_{7}\mathstrut +\mathstrut \) \(612483446138776\) \(\beta_{6}\mathstrut +\mathstrut \) \(1725271115895\) \(\beta_{5}\mathstrut -\mathstrut \) \(1734982345814151\) \(\beta_{4}\mathstrut +\mathstrut \) \(1371049052513905423\) \(\beta_{3}\mathstrut +\mathstrut \) \(46950489952321105616\) \(\beta_{2}\mathstrut +\mathstrut \) \(12871900689522271395\) \(\beta_{1}\mathstrut -\mathstrut \) \(5723497677272189911205324436\)\()/243\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(40535193998651760\) \(\beta_{9}\mathstrut +\mathstrut \) \(518072918865464337\) \(\beta_{8}\mathstrut -\mathstrut \) \(6041150546090155732\) \(\beta_{7}\mathstrut -\mathstrut \) \(31461153023021715784\) \(\beta_{6}\mathstrut -\mathstrut \) \(680213694860071377\) \(\beta_{5}\mathstrut +\mathstrut \) \(378269432631689762172\) \(\beta_{4}\mathstrut -\mathstrut \) \(2166524046173593960264\) \(\beta_{3}\mathstrut +\mathstrut \) \(2116655491056528477241390\) \(\beta_{2}\mathstrut -\mathstrut \) \(4086068679929605699602302268\) \(\beta_{1}\mathstrut -\mathstrut \) \(712966455586322920464\)\()/486\)
\(\nu^{8}\)\(=\)\((\)\(1126985179959820651554\) \(\beta_{9}\mathstrut -\mathstrut \) \(1126985179959820651554\) \(\beta_{8}\mathstrut -\mathstrut \) \(32191757431188314196734\) \(\beta_{7}\mathstrut -\mathstrut \) \(349153565221332092200976\) \(\beta_{6}\mathstrut +\mathstrut \) \(3659927761313636872230\) \(\beta_{5}\mathstrut +\mathstrut \) \(991492130812683460245306\) \(\beta_{4}\mathstrut -\mathstrut \) \(589739491791903028909307882\) \(\beta_{3}\mathstrut -\mathstrut \) \(23359804434277145551162490848\) \(\beta_{2}\mathstrut -\mathstrut \) \(6391613531488719357403506210\) \(\beta_{1}\mathstrut +\mathstrut \) \(2341021908304834252239025205583566904\)\()/243\)
\(\nu^{9}\)\(=\)\((\)\(11779630864580341864328400\) \(\beta_{9}\mathstrut -\mathstrut \) \(115007422495490230370557035\) \(\beta_{8}\mathstrut +\mathstrut \) \(1558843504876584277170250972\) \(\beta_{7}\mathstrut +\mathstrut \) \(7666309069099149661853147032\) \(\beta_{6}\mathstrut +\mathstrut \) \(162125945953811597827870635\) \(\beta_{5}\mathstrut -\mathstrut \) \(109274346344839729935902863764\) \(\beta_{4}\mathstrut +\mathstrut \) \(557791633792562005002747148696\) \(\beta_{3}\mathstrut -\mathstrut \) \(546487743005801598892421325621658\) \(\beta_{2}\mathstrut +\mathstrut \) \(861236151716353221628601874624905940\) \(\beta_{1}\mathstrut +\mathstrut \) \(178806936838867436999598270384\)\()/243\)

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
21147.4i
17303.1i
9879.94i
9002.40i
5429.47i
5429.47i
9002.40i
9879.94i
17303.1i
21147.4i
126884.i 2.43525e7 3.54961e7i −1.18046e10 1.55775e11i −4.50390e12 3.08995e12i 3.32502e12 9.52859e14i −6.66929e14 1.72884e15i −1.97653e16
2.2 103819.i −2.32547e7 + 3.62248e7i −6.48340e9 2.49832e11i 3.76082e12 + 2.41428e12i −3.04380e13 2.27200e14i −7.71458e14 1.68480e15i 2.59372e16
2.3 59279.7i −4.29008e7 3.54120e6i 7.80890e8 2.23415e11i −2.09921e11 + 2.54315e12i 4.13333e13 3.00895e14i 1.82794e15 + 3.03841e14i −1.32440e16
2.4 54014.4i 3.76547e7 + 2.08602e7i 1.37741e9 5.87306e9i 1.12675e12 2.03389e12i 1.74600e13 3.06390e14i 9.82725e14 + 1.57097e15i −3.17230e14
2.5 32576.8i −6.54523e6 4.25462e7i 3.23372e9 1.19330e11i −1.38602e12 + 2.13223e11i −3.44644e13 2.45260e14i −1.76734e15 + 5.56949e14i 3.88740e15
2.6 32576.8i −6.54523e6 + 4.25462e7i 3.23372e9 1.19330e11i −1.38602e12 2.13223e11i −3.44644e13 2.45260e14i −1.76734e15 5.56949e14i 3.88740e15
2.7 54014.4i 3.76547e7 2.08602e7i 1.37741e9 5.87306e9i 1.12675e12 + 2.03389e12i 1.74600e13 3.06390e14i 9.82725e14 1.57097e15i −3.17230e14
2.8 59279.7i −4.29008e7 + 3.54120e6i 7.80890e8 2.23415e11i −2.09921e11 2.54315e12i 4.13333e13 3.00895e14i 1.82794e15 3.03841e14i −1.32440e16
2.9 103819.i −2.32547e7 3.62248e7i −6.48340e9 2.49832e11i 3.76082e12 2.41428e12i −3.04380e13 2.27200e14i −7.71458e14 + 1.68480e15i 2.59372e16
2.10 126884.i 2.43525e7 + 3.54961e7i −1.18046e10 1.55775e11i −4.50390e12 + 3.08995e12i 3.32502e12 9.52859e14i −6.66929e14 + 1.72884e15i −1.97653e16
\(n\): 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 \(S_{33}^{\mathrm{new}}(3, [\chi])\).