Properties

Label 2736.3.o.q
Level $2736$
Weight $3$
Character orbit 2736.o
Analytic conductor $74.551$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 264 x^{18} + 28274 x^{16} - 1545308 x^{14} + 45358441 x^{12} - 637328868 x^{10} + 1825819356 x^{8} + 32794262368 x^{6} + 135580415344 x^{4} + 245217530816 x^{2} + 194396337216\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{32} \)
Twist minimal: no (minimal twist has level 1368)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{5} + ( 1 - \beta_{3} ) q^{7} +O(q^{10})\) \( q + \beta_{10} q^{5} + ( 1 - \beta_{3} ) q^{7} + \beta_{14} q^{11} + \beta_{2} q^{13} -\beta_{17} q^{17} + \beta_{6} q^{19} -\beta_{12} q^{23} + ( 3 + \beta_{1} ) q^{25} + ( -\beta_{11} + \beta_{13} ) q^{29} + ( \beta_{2} - \beta_{5} ) q^{31} + ( -\beta_{14} + \beta_{15} ) q^{35} + ( -\beta_{4} - \beta_{6} - \beta_{9} ) q^{37} -\beta_{19} q^{41} + ( 6 - \beta_{4} - \beta_{7} ) q^{43} + ( -\beta_{12} - \beta_{14} + \beta_{15} ) q^{47} + ( 6 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{7} ) q^{49} + ( \beta_{11} - \beta_{13} + \beta_{16} ) q^{53} + ( -8 + 2 \beta_{3} + \beta_{6} + \beta_{7} ) q^{55} + ( -\beta_{16} - \beta_{18} ) q^{59} + ( -6 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{61} + ( \beta_{13} - \beta_{18} - \beta_{19} ) q^{65} + ( 2 \beta_{2} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{8} - 2 \beta_{9} ) q^{67} + ( -\beta_{11} - \beta_{16} ) q^{71} + ( -5 + \beta_{1} - \beta_{3} + \beta_{6} + \beta_{7} ) q^{73} + ( -2 \beta_{10} + 2 \beta_{12} + 4 \beta_{14} - 2 \beta_{15} - 3 \beta_{17} ) q^{77} + ( -5 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{8} ) q^{79} + ( -2 \beta_{10} + 2 \beta_{12} + 3 \beta_{15} + 2 \beta_{17} ) q^{83} + ( -15 - \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{85} + ( \beta_{13} + \beta_{16} - \beta_{18} ) q^{89} + ( 4 \beta_{2} - \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{8} ) q^{91} + ( 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} - 2 \beta_{17} + \beta_{18} + \beta_{19} ) q^{95} + ( \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} + 3 \beta_{9} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 16q^{7} + O(q^{10}) \) \( 20q + 16q^{7} + 8q^{19} + 68q^{25} + 128q^{43} + 116q^{49} - 144q^{55} - 104q^{61} - 88q^{73} - 280q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 264 x^{18} + 28274 x^{16} - 1545308 x^{14} + 45358441 x^{12} - 637328868 x^{10} + 1825819356 x^{8} + 32794262368 x^{6} + 135580415344 x^{4} + 245217530816 x^{2} + 194396337216\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-129489917422785473 \nu^{18} + 33432506842079348758 \nu^{16} - 3476259767914987127990 \nu^{14} + 182074458436738505382096 \nu^{12} - 5011698785842294974419113 \nu^{10} + 61763237176902078811489826 \nu^{8} - 45391504120928525817384200 \nu^{6} - 3702829789190585699235987824 \nu^{4} - 12190837648852937460231075024 \nu^{2} - 638369885816155526430987655776\)\()/ \)\(22\!\cdots\!64\)\( \)
\(\beta_{2}\)\(=\)\((\)\(35116765958608993803719 \nu^{18} - 8076572300383004970532870 \nu^{16} + 681598682797592393217008234 \nu^{14} - 21490080044822837838555072216 \nu^{12} - 155190093936485078566858669361 \nu^{10} + 27071560611517174645767270133222 \nu^{8} - 579879831522048779568875509589512 \nu^{6} + 1990567124114139649842880700973488 \nu^{4} + 51619731317782974450435317852972208 \nu^{2} + 91625860166110757849665875519474912\)\()/ \)\(46\!\cdots\!12\)\( \)
\(\beta_{3}\)\(=\)\((\)\(101894909798153827801412 \nu^{18} - 26825934497932153760690115 \nu^{16} + 2860511881284992952029139300 \nu^{14} - 155136681280442697634744196354 \nu^{12} + 4483419030720163435170710143336 \nu^{10} - 60282145337731960211207694458523 \nu^{8} + 99992436742872930021312844686280 \nu^{6} + 4072616522793763459691495200367064 \nu^{4} + 13097705066903038980064666484716960 \nu^{2} + 17539334343458177259252063049044048\)\()/ \)\(29\!\cdots\!32\)\( \)
\(\beta_{4}\)\(=\)\((\)\(7272872906326920207294719 \nu^{18} - 1963099110138045459692325718 \nu^{16} + 217127942361528288383912198906 \nu^{14} - 12490175691325292914538412133848 \nu^{12} + 400114690951371661917986638759495 \nu^{10} - 6795226019110020291617337076289546 \nu^{8} + 47066733315704109220312484337659000 \nu^{6} + 62317443726245485271428138156671152 \nu^{4} + 10829489628938772020875887116084016 \nu^{2} - 291036387920393992772141303342024736\)\()/ \)\(18\!\cdots\!48\)\( \)
\(\beta_{5}\)\(=\)\((\)\(9884868144864782575579057 \nu^{18} - 2632940204455372338428310186 \nu^{16} + 285667130094003805329931408806 \nu^{14} - 15940868744172208182805233466216 \nu^{12} + 485092343719037908756724241483113 \nu^{10} - 7398845960161632994455974684562486 \nu^{8} + 34311420948424157864538861491569928 \nu^{6} + 257102273627856930373330265126114640 \nu^{4} + 702769014298878592625153136817937360 \nu^{2} + 605028068744822134117210981141427232\)\()/ \)\(93\!\cdots\!24\)\( \)
\(\beta_{6}\)\(=\)\((\)\(25136308373843910022990879 \nu^{18} - 6667000406846352383580434134 \nu^{16} + 718867335956038161207653261498 \nu^{14} - 39714046781846289140814496757336 \nu^{12} + 1187444513936588306257076643167591 \nu^{10} - 17395105231098351068708401417516298 \nu^{8} + 64796537549445862499099338477145976 \nu^{6} + 779658751828937211420750310034347184 \nu^{4} + 2317102809929481872238601580579988784 \nu^{2} + 2322081982538100973184965326728886240\)\()/ \)\(18\!\cdots\!48\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-42862540979321512695660255 \nu^{18} + 11332984589246257323051120854 \nu^{16} - 1216370191890664507779169566010 \nu^{14} + 66698453003183097130194587461976 \nu^{12} - 1967695859884145136853320495097383 \nu^{10} + 27899296305381582447285568918427146 \nu^{8} - 82383691335415860037626421320276344 \nu^{6} - 1490690422049141773364430371214918832 \nu^{4} - 4603012852711303811129581061787857200 \nu^{2} - 5161245572753592595080304126773217248\)\()/ \)\(18\!\cdots\!48\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-11955986190457431405787387 \nu^{18} + 3186996396873088217521099374 \nu^{16} - 346174848379449378817770584034 \nu^{14} + 19354326508754479459668786715576 \nu^{12} - 591020153809573342545678097955939 \nu^{10} + 9085323744381465326433923415246834 \nu^{8} - 43560501683008698832835915414006360 \nu^{6} - 305974170314605442363739187833205104 \nu^{4} - 686666200249957531735914581881786352 \nu^{2} - 438086896902954681044459261073095520\)\()/ \)\(46\!\cdots\!12\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-11313752589445717219287155 \nu^{18} + 3015399905540883795036341502 \nu^{16} - 327472033386409469133856991922 \nu^{14} + 18302786625553866374750467845560 \nu^{12} - 558583581394064015353896546421915 \nu^{10} + 8575339754979736734728775175416354 \nu^{8} - 40889142613257589317722997233756824 \nu^{6} - 290402509216169452951220401303620336 \nu^{4} - 676757600200028681186057414044616560 \nu^{2} - 462962330490265639477636022806911072\)\()/ \)\(31\!\cdots\!08\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-6598669986324889439703 \nu^{19} + 1737291157843822828232626 \nu^{17} - 185342418026958244586096186 \nu^{15} + 10069252782835009793231954944 \nu^{13} - 292615603501305658317542611791 \nu^{11} + 4021383035900599664880960196358 \nu^{9} - 9778674524534502263877973104376 \nu^{7} - 218066289455796552269795028273104 \nu^{5} - 1030699789578355810121140848632240 \nu^{3} - 1241433863593862580659247863739168 \nu\)\()/ \)\(16\!\cdots\!76\)\( \)
\(\beta_{11}\)\(=\)\((\)\(6598669986324889439703 \nu^{19} - 1737291157843822828232626 \nu^{17} + 185342418026958244586096186 \nu^{15} - 10069252782835009793231954944 \nu^{13} + 292615603501305658317542611791 \nu^{11} - 4021383035900599664880960196358 \nu^{9} + 9778674524534502263877973104376 \nu^{7} + 218066289455796552269795028273104 \nu^{5} + 1030699789578355810121140848632240 \nu^{3} + 2890616771626922425204699206099744 \nu\)\()/ \)\(20\!\cdots\!72\)\( \)
\(\beta_{12}\)\(=\)\((\)\(7994392052742425622925458561 \nu^{19} - 2219917099195036008455667319338 \nu^{17} + 255153667513412135058394362288646 \nu^{15} - 15509370322005736031071432075353896 \nu^{13} + 538266118149472080535465193680854841 \nu^{11} - 10411822145537500486195371522592746870 \nu^{9} + 94514506068836826621688618515083525000 \nu^{7} - 75470561301534886015671994158219321520 \nu^{5} - 2231147410865090036189350795378778114864 \nu^{3} - 3881659026972935631223065878840289132512 \nu\)\()/ \)\(11\!\cdots\!36\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-17463398905880338020608653595 \nu^{19} + 4620121452408262693918508331474 \nu^{17} - 496303124570360817478991147017554 \nu^{15} + 27251864238320358150859447062953072 \nu^{13} - 805950336915358552491712326861521683 \nu^{11} + 11495255410893977713020263994070538838 \nu^{9} - 35408250282729773565277655130971870776 \nu^{7} - 604033895521589425436724623429491430544 \nu^{5} - 1715372268232870696891863149192427740272 \nu^{3} - 1657566739816469908397528695725080853792 \nu\)\()/ \)\(42\!\cdots\!76\)\( \)
\(\beta_{14}\)\(=\)\((\)\(159456042579942782405061755609 \nu^{19} - 42624683559291875759630395700154 \nu^{17} + 4648958290601089467604753486092438 \nu^{15} - 261620416891495281296293284803055272 \nu^{13} + 8078881173990356722839670504682926417 \nu^{11} - 127246958773102932553237157142496526118 \nu^{9} + 677581301203920631413166695134395351624 \nu^{7} + 3539744027836530123420712449681586263504 \nu^{5} + 6806788029844574880516834926021293460560 \nu^{3} + 1885549379502286869696597347867836404000 \nu\)\()/ \)\(34\!\cdots\!08\)\( \)
\(\beta_{15}\)\(=\)\((\)\(214877985712369574881118269017 \nu^{19} - 57435080811762059335233345976826 \nu^{17} + 6263604187449793226898427862029462 \nu^{15} - 352424880287836846797403947902072360 \nu^{13} + 10879619770170210091138378470271956945 \nu^{11} - 171230651130510265693718747588522084710 \nu^{9} + 908540793279279260191782222332913433160 \nu^{7} + 4801336566166684115584148216924822004688 \nu^{5} + 9436231071005053435071768159113569301584 \nu^{3} + 2933958220258317637146230188637201513760 \nu\)\()/ \)\(34\!\cdots\!08\)\( \)
\(\beta_{16}\)\(=\)\((\)\(920834220946425393396379621 \nu^{19} - 243867976238205763771562466812 \nu^{17} + 26234715744277297266469461368098 \nu^{15} - 1443688843497474971779644536088144 \nu^{13} + 42843479010705155721958208912078921 \nu^{11} - 615306123233851646943088333051483516 \nu^{9} + 1966708083023646815258637980508940936 \nu^{7} + 32159101796547073840626144934005513952 \nu^{5} + 80267846897819046071829624936735903312 \nu^{3} + 26651611014769083838178510215186248000 \nu\)\()/ \)\(13\!\cdots\!68\)\( \)
\(\beta_{17}\)\(=\)\((\)\(77988139085550298007634852523 \nu^{19} - 20821300903414028396667935320942 \nu^{17} + 2266873085835542907955975972473986 \nu^{15} - 127212826489929361523815689873106424 \nu^{13} + 3910057199087814243531769997297145875 \nu^{11} - 60986765737624686357235684355214899762 \nu^{9} + 312553931720215401567510315651215525592 \nu^{7} + 1811792444391479803105891551212139187568 \nu^{5} + 4140220265790999814365233653455812354800 \nu^{3} + 2173830137189723923017470172710612954976 \nu\)\()/ \)\(85\!\cdots\!52\)\( \)
\(\beta_{18}\)\(=\)\((\)\(22073946146084221785202730395 \nu^{19} - 5838851176130178575925526851930 \nu^{17} + 627035001244097059578552318948114 \nu^{15} - 34411489415295007662276418802136352 \nu^{13} + 1016590489375649968633590095232432755 \nu^{11} - 14461665722684568802801052994979271646 \nu^{9} + 43818902401506345519835950439044402008 \nu^{7} + 761891507376417663664709098540252796304 \nu^{5} + 2280320053667534645979458227182840305648 \nu^{3} + 2740926373384809876070616486206784475552 \nu\)\()/ \)\(10\!\cdots\!44\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-101428080313796856325774464511 \nu^{19} + 26827816503635179523110956709258 \nu^{17} - 2880901980178419342990189704474090 \nu^{15} + 158096172502471542232313637244568048 \nu^{13} - 4670484110453671769322461370772709015 \nu^{11} + 66448614474167732558265872635121025854 \nu^{9} - 201597521509687400869998625157708164056 \nu^{7} - 3500044905907723231918620694605744212048 \nu^{5} - 10433826683250431800020369657949177451440 \nu^{3} - 12358928169572335902270219942006619108512 \nu\)\()/ \)\(42\!\cdots\!76\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} + 8 \beta_{10}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{9} + \beta_{8} - \beta_{5} + \beta_{2} + 4 \beta_{1} + 104\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(12 \beta_{19} + 6 \beta_{18} + 24 \beta_{17} + 12 \beta_{16} - 32 \beta_{15} - 12 \beta_{13} + 85 \beta_{11} + 400 \beta_{10}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-45 \beta_{9} + 57 \beta_{8} - 6 \beta_{7} - 12 \beta_{6} - 69 \beta_{5} + 70 \beta_{4} + 90 \beta_{3} - 19 \beta_{2} + 126 \beta_{1} + 2594\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(1800 \beta_{19} + 1190 \beta_{18} + 2168 \beta_{17} + 1040 \beta_{16} - 2800 \beta_{15} - 352 \beta_{14} - 2040 \beta_{13} + 400 \beta_{12} + 8239 \beta_{11} + 20784 \beta_{10}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-6983 \beta_{9} + 10787 \beta_{8} - 1340 \beta_{7} + 3416 \beta_{6} - 15647 \beta_{5} + 17004 \beta_{4} + 13364 \beta_{3} - 9977 \beta_{2} + 14236 \beta_{1} + 254132\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(193816 \beta_{19} + 135058 \beta_{18} + 131352 \beta_{17} + 86800 \beta_{16} - 158928 \beta_{15} - 39104 \beta_{14} - 238056 \beta_{13} + 30416 \beta_{12} + 744617 \beta_{11} + 963664 \beta_{10}\)\()/8\)
\(\nu^{8}\)\(=\)\(-139277 \beta_{9} + 237009 \beta_{8} - 19579 \beta_{7} + 200770 \beta_{6} - 383701 \beta_{5} + 366211 \beta_{4} + 149021 \beta_{3} - 280819 \beta_{2} + 146187 \beta_{1} + 2422221\)
\(\nu^{9}\)\(=\)\((\)\(17687544 \beta_{19} + 12474966 \beta_{18} + 4163672 \beta_{17} + 7095456 \beta_{16} - 4647216 \beta_{15} - 1854112 \beta_{14} - 22694952 \beta_{13} + 1127568 \beta_{12} + 63706435 \beta_{11} + 23106448 \beta_{10}\)\()/8\)
\(\nu^{10}\)\(=\)\((\)\(-44083593 \beta_{9} + 77897725 \beta_{8} - 1068972 \beta_{7} + 101882264 \beta_{6} - 133205905 \beta_{5} + 105647164 \beta_{4} + 770404 \beta_{3} - 99861687 \beta_{2} - 1853508 \beta_{1} - 73038332\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(1438379976 \beta_{19} + 1016671194 \beta_{18} - 270794024 \beta_{17} + 552352064 \beta_{16} + 348633392 \beta_{15} + 45780096 \beta_{14} - 1885121304 \beta_{13} - 50709488 \beta_{12} + 5066398789 \beta_{11} - 2545983280 \beta_{10}\)\()/8\)
\(\nu^{12}\)\(=\)\((\)\(-1650743743 \beta_{9} + 2953625547 \beta_{8} + 203506234 \beta_{7} + 5149511828 \beta_{6} - 5174256087 \beta_{5} + 3201959142 \beta_{4} - 1868925286 \beta_{3} - 3882392001 \beta_{2} - 2048193738 \beta_{1} - 36917990150\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(105024498904 \beta_{19} + 74232781038 \beta_{18} - 69629386600 \beta_{17} + 39694140512 \beta_{16} + 84003448336 \beta_{15} + 20680898464 \beta_{14} - 138926537672 \beta_{13} - 15461943536 \beta_{12} + 367288482911 \beta_{11} - 544008572912 \beta_{10}\)\()/8\)
\(\nu^{14}\)\(=\)\((\)\(-224805993819 \beta_{9} + 403674825655 \beta_{8} + 69837133684 \beta_{7} + 908375012408 \beta_{6} - 713986498739 \beta_{5} + 286772038460 \beta_{4} - 576055591836 \beta_{3} - 534498221797 \beta_{2} - 619544515844 \beta_{1} - 10956535197500\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(6723822781032 \beta_{19} + 4751208947538 \beta_{18} - 8911789447912 \beta_{17} + 2527999345856 \beta_{16} + 10619167606384 \beta_{15} + 2851654407104 \beta_{14} - 8926967630968 \beta_{13} - 2028234451568 \beta_{12} + 23466274098817 \beta_{11} - 67405260572720 \beta_{10}\)\()/8\)
\(\nu^{16}\)\(=\)\(-3277009021882 \beta_{9} + 5889328683906 \beta_{8} + 1999352894565 \beta_{7} + 17962554953202 \beta_{6} - 10453558652106 \beta_{5} + 511116759795 \beta_{4} - 16011195129939 \beta_{3} - 7812458737222 \beta_{2} - 17162170587493 \beta_{1} - 302143772553619\)
\(\nu^{17}\)\(=\)\((\)\(345929797303224 \beta_{19} + 244394114338790 \beta_{18} - 912359080292776 \beta_{17} + 129815448836512 \beta_{16} + 1082752000400464 \beta_{15} + 298669679910880 \beta_{14} - 459995509972584 \beta_{13} - 209135379254128 \beta_{12} + 1206543161462587 \beta_{11} - 6834111834877936 \beta_{10}\)\()/8\)
\(\nu^{18}\)\(=\)\((\)\(-545831763358445 \beta_{9} + 981473797990129 \beta_{8} + 767114926063572 \beta_{7} + 5085384527568728 \beta_{6} - 1746275442537493 \beta_{5} - 1566461556577028 \beta_{4} - 6081893447599196 \beta_{3} - 1303526795533651 \beta_{2} - 6514484704526020 \beta_{1} - 114539285934097340\)\()/4\)
\(\nu^{19}\)\(=\)\((\)\(9126876597089096 \beta_{19} + 6446481860506570 \beta_{18} - 81867456611436456 \beta_{17} + 3408387189172736 \beta_{16} + 97022445323936944 \beta_{15} + 27004221346790144 \beta_{14} - 12176830049213720 \beta_{13} - 18808831210708976 \beta_{12} + 31772331833389533 \beta_{11} - 611339052329188144 \beta_{10}\)\()/8\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
−8.89263 1.41421i
−8.89263 + 1.41421i
−6.04300 + 1.41421i
−6.04300 1.41421i
−5.08927 + 1.41421i
−5.08927 1.41421i
−0.585690 + 1.41421i
−0.585690 1.41421i
−0.399553 + 1.41421i
−0.399553 1.41421i
0.399553 1.41421i
0.399553 + 1.41421i
0.585690 1.41421i
0.585690 + 1.41421i
5.08927 1.41421i
5.08927 + 1.41421i
6.04300 1.41421i
6.04300 + 1.41421i
8.89263 + 1.41421i
8.89263 1.41421i
0 0 0 −8.89263 0 −4.08883 0 0 0
721.2 0 0 0 −8.89263 0 −4.08883 0 0 0
721.3 0 0 0 −6.04300 0 0.880524 0 0 0
721.4 0 0 0 −6.04300 0 0.880524 0 0 0
721.5 0 0 0 −5.08927 0 11.6968 0 0 0
721.6 0 0 0 −5.08927 0 11.6968 0 0 0
721.7 0 0 0 −0.585690 0 5.15901 0 0 0
721.8 0 0 0 −0.585690 0 5.15901 0 0 0
721.9 0 0 0 −0.399553 0 −9.64754 0 0 0
721.10 0 0 0 −0.399553 0 −9.64754 0 0 0
721.11 0 0 0 0.399553 0 −9.64754 0 0 0
721.12 0 0 0 0.399553 0 −9.64754 0 0 0
721.13 0 0 0 0.585690 0 5.15901 0 0 0
721.14 0 0 0 0.585690 0 5.15901 0 0 0
721.15 0 0 0 5.08927 0 11.6968 0 0 0
721.16 0 0 0 5.08927 0 11.6968 0 0 0
721.17 0 0 0 6.04300 0 0.880524 0 0 0
721.18 0 0 0 6.04300 0 0.880524 0 0 0
721.19 0 0 0 8.89263 0 −4.08883 0 0 0
721.20 0 0 0 8.89263 0 −4.08883 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 721.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.o.q 20
3.b odd 2 1 inner 2736.3.o.q 20
4.b odd 2 1 1368.3.o.d 20
12.b even 2 1 1368.3.o.d 20
19.b odd 2 1 inner 2736.3.o.q 20
57.d even 2 1 inner 2736.3.o.q 20
76.d even 2 1 1368.3.o.d 20
228.b odd 2 1 1368.3.o.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.3.o.d 20 4.b odd 2 1
1368.3.o.d 20 12.b even 2 1
1368.3.o.d 20 76.d even 2 1
1368.3.o.d 20 228.b odd 2 1
2736.3.o.q 20 1.a even 1 1 trivial
2736.3.o.q 20 3.b odd 2 1 inner
2736.3.o.q 20 19.b odd 2 1 inner
2736.3.o.q 20 57.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{10} - 142 T_{5}^{8} + 5953 T_{5}^{6} - 77760 T_{5}^{4} + 37920 T_{5}^{2} - 4096 \)
\( T_{7}^{5} - 4 T_{7}^{4} - 129 T_{7}^{3} + 280 T_{7}^{2} + 2236 T_{7} - 2096 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( T^{20} \)
$5$ \( ( -4096 + 37920 T^{2} - 77760 T^{4} + 5953 T^{6} - 142 T^{8} + T^{10} )^{2} \)
$7$ \( ( -2096 + 2236 T + 280 T^{2} - 129 T^{3} - 4 T^{4} + T^{5} )^{4} \)
$11$ \( ( -7953785856 + 648389632 T^{2} - 14664562 T^{4} + 141205 T^{6} - 616 T^{8} + T^{10} )^{2} \)
$13$ \( ( 55207657472 + 2429980672 T^{2} + 37826816 T^{4} + 263328 T^{6} + 840 T^{8} + T^{10} )^{2} \)
$17$ \( ( -194784526336 + 7035551744 T^{2} - 91110176 T^{4} + 513265 T^{6} - 1246 T^{8} + T^{10} )^{2} \)
$19$ \( ( 6131066257801 - 67934252164 T - 4657542219 T^{2} + 271067680 T^{3} - 4293734 T^{4} - 274360 T^{5} - 11894 T^{6} + 2080 T^{7} - 99 T^{8} - 4 T^{9} + T^{10} )^{2} \)
$23$ \( ( -218994049024 + 10978095104 T^{2} - 176305728 T^{4} + 979744 T^{6} - 1938 T^{8} + T^{10} )^{2} \)
$29$ \( ( 34662336954368 + 1035423825920 T^{2} + 7518424576 T^{4} + 11459024 T^{6} + 5962 T^{8} + T^{10} )^{2} \)
$31$ \( ( 7860327022592 + 176677388288 T^{2} + 1382245504 T^{4} + 4270624 T^{6} + 4004 T^{8} + T^{10} )^{2} \)
$37$ \( ( 549755813888 + 75040292864 T^{2} + 2312344064 T^{4} + 5932064 T^{6} + 4560 T^{8} + T^{10} )^{2} \)
$41$ \( ( 31832109857374208 + 138628300603392 T^{2} + 166449264640 T^{4} + 75505088 T^{6} + 14514 T^{8} + T^{10} )^{2} \)
$43$ \( ( -103358208 + 5031884 T + 99356 T^{2} - 5201 T^{3} - 32 T^{4} + T^{5} )^{4} \)
$47$ \( ( -1095929372070976 + 7163199332128 T^{2} - 16126470562 T^{4} + 15679989 T^{6} - 6648 T^{8} + T^{10} )^{2} \)
$53$ \( ( 72266262876520448 + 505791627919360 T^{2} + 535886170112 T^{4} + 177851520 T^{6} + 22818 T^{8} + T^{10} )^{2} \)
$59$ \( ( 5227372559204352 + 66749672194048 T^{2} + 231156334592 T^{4} + 118170624 T^{6} + 20104 T^{8} + T^{10} )^{2} \)
$61$ \( ( 2358752 + 1702852 T - 21980 T^{2} - 3747 T^{3} + 26 T^{4} + T^{5} )^{4} \)
$67$ \( ( 1587867153996972032 + 5406396715892736 T^{2} + 2672865813504 T^{4} + 494912736 T^{6} + 37788 T^{8} + T^{10} )^{2} \)
$71$ \( ( 283486331907080192 + 539131235008512 T^{2} + 375806722048 T^{4} + 120376576 T^{6} + 17896 T^{8} + T^{10} )^{2} \)
$73$ \( ( -134016 - 1159868 T - 248380 T^{2} - 6123 T^{3} + 22 T^{4} + T^{5} )^{4} \)
$79$ \( ( 1519607415111680000 + 5123632517734400 T^{2} + 3196811396864 T^{4} + 607639328 T^{6} + 43704 T^{8} + T^{10} )^{2} \)
$83$ \( ( -13704300769547649024 + 22484264745435136 T^{2} - 8243208929280 T^{4} + 1078004096 T^{6} - 56226 T^{8} + T^{10} )^{2} \)
$89$ \( ( 441645282027896832 + 2189338363297792 T^{2} + 1706488160256 T^{4} + 404656960 T^{6} + 35186 T^{8} + T^{10} )^{2} \)
$97$ \( ( 318407822234615808 + 6815419128217600 T^{2} + 6050670473216 T^{4} + 938310400 T^{6} + 52664 T^{8} + T^{10} )^{2} \)
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