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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(19.4599965427\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 954745942 x^{8} + 302468338607088448 x^{6} + 37939920124077893929140224 x^{4} + 1938513915962148831841211918581760 x^{2} + 31225030372218346257929044634944667648000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{61}\cdot 5^{5} \)
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.

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

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

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

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 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 3.33.b.a 10
3.b odd 2 1 inner 3.33.b.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.33.b.a 10 1.a even 1 1 trivial
3.33.b.a 10 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{33}^{\mathrm{new}}(3, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 8578819048 T^{2} + 41128902250647155712 T^{4} - \)\(86\!\cdots\!08\)\( T^{6} + \)\(27\!\cdots\!52\)\( T^{8} - \)\(90\!\cdots\!68\)\( T^{10} + \)\(50\!\cdots\!32\)\( T^{12} - \)\(29\!\cdots\!48\)\( T^{14} + \)\(25\!\cdots\!52\)\( T^{16} - \)\(99\!\cdots\!28\)\( T^{18} + \)\(21\!\cdots\!76\)\( T^{20} \)
$3$ \( 1 + 21387150 T + 623766894639021 T^{2} + \)\(34\!\cdots\!60\)\( T^{3} + \)\(81\!\cdots\!78\)\( T^{4} + \)\(17\!\cdots\!20\)\( T^{5} + \)\(15\!\cdots\!98\)\( T^{6} + \)\(11\!\cdots\!60\)\( T^{7} + \)\(39\!\cdots\!41\)\( T^{8} + \)\(25\!\cdots\!50\)\( T^{9} + \)\(21\!\cdots\!01\)\( T^{10} \)
$5$ \( 1 - \)\(81\!\cdots\!50\)\( T^{2} + \)\(40\!\cdots\!25\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{6} + \)\(47\!\cdots\!50\)\( T^{8} - \)\(12\!\cdots\!00\)\( T^{10} + \)\(25\!\cdots\!50\)\( T^{12} - \)\(45\!\cdots\!00\)\( T^{14} + \)\(65\!\cdots\!25\)\( T^{16} - \)\(70\!\cdots\!50\)\( T^{18} + \)\(46\!\cdots\!25\)\( T^{20} \)
$7$ \( ( 1 + 2784031209470 T + \)\(34\!\cdots\!41\)\( T^{2} + \)\(42\!\cdots\!80\)\( T^{3} + \)\(61\!\cdots\!18\)\( T^{4} + \)\(10\!\cdots\!20\)\( T^{5} + \)\(68\!\cdots\!18\)\( T^{6} + \)\(51\!\cdots\!80\)\( T^{7} + \)\(46\!\cdots\!41\)\( T^{8} + \)\(41\!\cdots\!70\)\( T^{9} + \)\(16\!\cdots\!01\)\( T^{10} )^{2} \)
$11$ \( 1 - \)\(76\!\cdots\!70\)\( T^{2} + \)\(32\!\cdots\!25\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{6} + \)\(30\!\cdots\!70\)\( T^{8} - \)\(70\!\cdots\!52\)\( T^{10} + \)\(13\!\cdots\!70\)\( T^{12} - \)\(21\!\cdots\!00\)\( T^{14} + \)\(29\!\cdots\!25\)\( T^{16} - \)\(30\!\cdots\!70\)\( T^{18} + \)\(17\!\cdots\!01\)\( T^{20} \)
$13$ \( ( 1 - 283848778839902890 T + \)\(16\!\cdots\!61\)\( T^{2} - \)\(26\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!18\)\( T^{4} - \)\(13\!\cdots\!00\)\( T^{5} + \)\(53\!\cdots\!58\)\( T^{6} - \)\(52\!\cdots\!40\)\( T^{7} + \)\(14\!\cdots\!01\)\( T^{8} - \)\(10\!\cdots\!90\)\( T^{9} + \)\(17\!\cdots\!01\)\( T^{10} )^{2} \)
$17$ \( 1 - \)\(18\!\cdots\!38\)\( T^{2} + \)\(17\!\cdots\!77\)\( T^{4} - \)\(95\!\cdots\!28\)\( T^{6} + \)\(36\!\cdots\!42\)\( T^{8} - \)\(10\!\cdots\!08\)\( T^{10} + \)\(20\!\cdots\!82\)\( T^{12} - \)\(30\!\cdots\!48\)\( T^{14} + \)\(30\!\cdots\!97\)\( T^{16} - \)\(18\!\cdots\!78\)\( T^{18} + \)\(55\!\cdots\!01\)\( T^{20} \)
$19$ \( ( 1 - \)\(28\!\cdots\!38\)\( T + \)\(24\!\cdots\!97\)\( T^{2} - \)\(48\!\cdots\!68\)\( T^{3} + \)\(33\!\cdots\!62\)\( T^{4} - \)\(57\!\cdots\!08\)\( T^{5} + \)\(27\!\cdots\!82\)\( T^{6} - \)\(33\!\cdots\!28\)\( T^{7} + \)\(14\!\cdots\!57\)\( T^{8} - \)\(13\!\cdots\!58\)\( T^{9} + \)\(39\!\cdots\!01\)\( T^{10} )^{2} \)
$23$ \( 1 - \)\(14\!\cdots\!38\)\( T^{2} + \)\(82\!\cdots\!37\)\( T^{4} - \)\(18\!\cdots\!48\)\( T^{6} - \)\(25\!\cdots\!98\)\( T^{8} + \)\(26\!\cdots\!92\)\( T^{10} - \)\(35\!\cdots\!18\)\( T^{12} - \)\(36\!\cdots\!88\)\( T^{14} + \)\(23\!\cdots\!77\)\( T^{16} - \)\(58\!\cdots\!18\)\( T^{18} + \)\(56\!\cdots\!01\)\( T^{20} \)
$29$ \( 1 - \)\(40\!\cdots\!50\)\( T^{2} + \)\(80\!\cdots\!65\)\( T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(98\!\cdots\!50\)\( T^{8} - \)\(70\!\cdots\!52\)\( T^{10} + \)\(38\!\cdots\!50\)\( T^{12} - \)\(16\!\cdots\!40\)\( T^{14} + \)\(48\!\cdots\!65\)\( T^{16} - \)\(96\!\cdots\!50\)\( T^{18} + \)\(92\!\cdots\!01\)\( T^{20} \)
$31$ \( ( 1 - \)\(11\!\cdots\!70\)\( T + \)\(20\!\cdots\!25\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!70\)\( T^{4} - \)\(56\!\cdots\!52\)\( T^{5} + \)\(68\!\cdots\!70\)\( T^{6} - \)\(34\!\cdots\!00\)\( T^{7} + \)\(30\!\cdots\!25\)\( T^{8} - \)\(91\!\cdots\!70\)\( T^{9} + \)\(41\!\cdots\!01\)\( T^{10} )^{2} \)
$37$ \( ( 1 + \)\(17\!\cdots\!70\)\( T + \)\(55\!\cdots\!61\)\( T^{2} + \)\(85\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!18\)\( T^{4} + \)\(18\!\cdots\!80\)\( T^{5} + \)\(21\!\cdots\!58\)\( T^{6} + \)\(19\!\cdots\!60\)\( T^{7} + \)\(19\!\cdots\!01\)\( T^{8} + \)\(94\!\cdots\!70\)\( T^{9} + \)\(81\!\cdots\!01\)\( T^{10} )^{2} \)
$41$ \( 1 - \)\(30\!\cdots\!70\)\( T^{2} + \)\(43\!\cdots\!25\)\( T^{4} - \)\(40\!\cdots\!00\)\( T^{6} + \)\(25\!\cdots\!70\)\( T^{8} - \)\(12\!\cdots\!52\)\( T^{10} + \)\(42\!\cdots\!70\)\( T^{12} - \)\(11\!\cdots\!00\)\( T^{14} + \)\(19\!\cdots\!25\)\( T^{16} - \)\(22\!\cdots\!70\)\( T^{18} + \)\(12\!\cdots\!01\)\( T^{20} \)
$43$ \( ( 1 - \)\(10\!\cdots\!70\)\( T + \)\(53\!\cdots\!61\)\( T^{2} + \)\(13\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!78\)\( T^{4} + \)\(39\!\cdots\!80\)\( T^{5} + \)\(28\!\cdots\!78\)\( T^{6} + \)\(45\!\cdots\!20\)\( T^{7} + \)\(34\!\cdots\!61\)\( T^{8} - \)\(12\!\cdots\!70\)\( T^{9} + \)\(22\!\cdots\!01\)\( T^{10} )^{2} \)
$47$ \( 1 - \)\(12\!\cdots\!38\)\( T^{2} + \)\(90\!\cdots\!97\)\( T^{4} - \)\(47\!\cdots\!08\)\( T^{6} + \)\(20\!\cdots\!82\)\( T^{8} - \)\(70\!\cdots\!68\)\( T^{10} + \)\(20\!\cdots\!42\)\( T^{12} - \)\(51\!\cdots\!88\)\( T^{14} + \)\(99\!\cdots\!77\)\( T^{16} - \)\(14\!\cdots\!98\)\( T^{18} + \)\(11\!\cdots\!01\)\( T^{20} \)
$53$ \( 1 - \)\(91\!\cdots\!38\)\( T^{2} + \)\(41\!\cdots\!97\)\( T^{4} - \)\(12\!\cdots\!08\)\( T^{6} + \)\(26\!\cdots\!82\)\( T^{8} - \)\(44\!\cdots\!68\)\( T^{10} + \)\(59\!\cdots\!42\)\( T^{12} - \)\(62\!\cdots\!88\)\( T^{14} + \)\(47\!\cdots\!77\)\( T^{16} - \)\(23\!\cdots\!98\)\( T^{18} + \)\(58\!\cdots\!01\)\( T^{20} \)
$59$ \( 1 - \)\(20\!\cdots\!70\)\( T^{2} + \)\(24\!\cdots\!25\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!70\)\( T^{8} - \)\(67\!\cdots\!52\)\( T^{10} + \)\(28\!\cdots\!70\)\( T^{12} - \)\(93\!\cdots\!00\)\( T^{14} + \)\(24\!\cdots\!25\)\( T^{16} - \)\(44\!\cdots\!70\)\( T^{18} + \)\(47\!\cdots\!01\)\( T^{20} \)
$61$ \( ( 1 + \)\(98\!\cdots\!90\)\( T + \)\(77\!\cdots\!45\)\( T^{2} + \)\(45\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!10\)\( T^{4} + \)\(87\!\cdots\!48\)\( T^{5} + \)\(29\!\cdots\!10\)\( T^{6} + \)\(82\!\cdots\!80\)\( T^{7} + \)\(19\!\cdots\!45\)\( T^{8} + \)\(32\!\cdots\!90\)\( T^{9} + \)\(44\!\cdots\!01\)\( T^{10} )^{2} \)
$67$ \( ( 1 + \)\(22\!\cdots\!70\)\( T + \)\(54\!\cdots\!81\)\( T^{2} + \)\(38\!\cdots\!40\)\( T^{3} + \)\(24\!\cdots\!18\)\( T^{4} - \)\(69\!\cdots\!60\)\( T^{5} + \)\(67\!\cdots\!98\)\( T^{6} + \)\(28\!\cdots\!40\)\( T^{7} + \)\(11\!\cdots\!61\)\( T^{8} + \)\(12\!\cdots\!70\)\( T^{9} + \)\(14\!\cdots\!01\)\( T^{10} )^{2} \)
$71$ \( 1 - \)\(55\!\cdots\!50\)\( T^{2} + \)\(19\!\cdots\!65\)\( T^{4} - \)\(53\!\cdots\!40\)\( T^{6} + \)\(11\!\cdots\!50\)\( T^{8} - \)\(21\!\cdots\!52\)\( T^{10} + \)\(35\!\cdots\!50\)\( T^{12} - \)\(48\!\cdots\!40\)\( T^{14} + \)\(53\!\cdots\!65\)\( T^{16} - \)\(46\!\cdots\!50\)\( T^{18} + \)\(25\!\cdots\!01\)\( T^{20} \)
$73$ \( ( 1 + \)\(22\!\cdots\!10\)\( T + \)\(38\!\cdots\!81\)\( T^{2} + \)\(43\!\cdots\!20\)\( T^{3} + \)\(40\!\cdots\!98\)\( T^{4} + \)\(28\!\cdots\!20\)\( T^{5} + \)\(16\!\cdots\!58\)\( T^{6} + \)\(77\!\cdots\!20\)\( T^{7} + \)\(28\!\cdots\!41\)\( T^{8} + \)\(72\!\cdots\!10\)\( T^{9} + \)\(13\!\cdots\!01\)\( T^{10} )^{2} \)
$79$ \( ( 1 + \)\(10\!\cdots\!82\)\( T + \)\(11\!\cdots\!17\)\( T^{2} + \)\(49\!\cdots\!32\)\( T^{3} + \)\(61\!\cdots\!22\)\( T^{4} + \)\(63\!\cdots\!92\)\( T^{5} + \)\(32\!\cdots\!02\)\( T^{6} + \)\(13\!\cdots\!92\)\( T^{7} + \)\(17\!\cdots\!57\)\( T^{8} + \)\(83\!\cdots\!02\)\( T^{9} + \)\(41\!\cdots\!01\)\( T^{10} )^{2} \)
$83$ \( 1 - \)\(14\!\cdots\!98\)\( T^{2} + \)\(78\!\cdots\!77\)\( T^{4} - \)\(15\!\cdots\!88\)\( T^{6} - \)\(27\!\cdots\!58\)\( T^{8} + \)\(18\!\cdots\!32\)\( T^{10} - \)\(18\!\cdots\!18\)\( T^{12} - \)\(67\!\cdots\!08\)\( T^{14} + \)\(22\!\cdots\!97\)\( T^{16} - \)\(27\!\cdots\!38\)\( T^{18} + \)\(12\!\cdots\!01\)\( T^{20} \)
$89$ \( 1 - \)\(19\!\cdots\!70\)\( T^{2} + \)\(17\!\cdots\!25\)\( T^{4} - \)\(96\!\cdots\!00\)\( T^{6} + \)\(37\!\cdots\!70\)\( T^{8} - \)\(10\!\cdots\!52\)\( T^{10} + \)\(21\!\cdots\!70\)\( T^{12} - \)\(32\!\cdots\!00\)\( T^{14} + \)\(33\!\cdots\!25\)\( T^{16} - \)\(21\!\cdots\!70\)\( T^{18} + \)\(63\!\cdots\!01\)\( T^{20} \)
$97$ \( ( 1 - \)\(19\!\cdots\!50\)\( T + \)\(26\!\cdots\!61\)\( T^{2} - \)\(24\!\cdots\!60\)\( T^{3} + \)\(19\!\cdots\!98\)\( T^{4} - \)\(12\!\cdots\!20\)\( T^{5} + \)\(74\!\cdots\!18\)\( T^{6} - \)\(34\!\cdots\!60\)\( T^{7} + \)\(14\!\cdots\!81\)\( T^{8} - \)\(39\!\cdots\!50\)\( T^{9} + \)\(76\!\cdots\!01\)\( T^{10} )^{2} \)
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