Properties

Label 1.38.a
Level 1
Weight 38
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newforms 1
Sturm bound 3
Trace bound 0

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Defining parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 38 \)
Character orbit: \([\chi]\) = 1.a (trivial)
Character field: \(\Q\)
Newforms: \( 1 \)
Sturm bound: \(3\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{38}(\Gamma_0(1))\).

Total New Old
Modular forms 3 3 0
Cusp forms 2 2 0
Eisenstein series 1 1 0

Trace form

\(2q \) \(\mathstrut -\mathstrut 194400q^{2} \) \(\mathstrut +\mathstrut 13991400q^{3} \) \(\mathstrut +\mathstrut 37720269824q^{4} \) \(\mathstrut +\mathstrut 5529584385900q^{5} \) \(\mathstrut -\mathstrut 22506543847296q^{6} \) \(\mathstrut -\mathstrut 3448443953486000q^{7} \) \(\mathstrut -\mathstrut 34044043043635200q^{8} \) \(\mathstrut -\mathstrut 898947378401769414q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 194400q^{2} \) \(\mathstrut +\mathstrut 13991400q^{3} \) \(\mathstrut +\mathstrut 37720269824q^{4} \) \(\mathstrut +\mathstrut 5529584385900q^{5} \) \(\mathstrut -\mathstrut 22506543847296q^{6} \) \(\mathstrut -\mathstrut 3448443953486000q^{7} \) \(\mathstrut -\mathstrut 34044043043635200q^{8} \) \(\mathstrut -\mathstrut 898947378401769414q^{9} \) \(\mathstrut -\mathstrut 9015609355013275200q^{10} \) \(\mathstrut -\mathstrut 26734036354848538056q^{11} \) \(\mathstrut +\mathstrut 4374774798370099200q^{12} \) \(\mathstrut +\mathstrut 530581741653933178300q^{13} \) \(\mathstrut +\mathstrut 3169419119584065186048q^{14} \) \(\mathstrut +\mathstrut 649108943683113884400q^{15} \) \(\mathstrut -\mathstrut 31152340106193176363008q^{16} \) \(\mathstrut -\mathstrut 89439073881931767332700q^{17} \) \(\mathstrut +\mathstrut 87081814924495841786400q^{18} \) \(\mathstrut +\mathstrut 373276466572513713706120q^{19} \) \(\mathstrut +\mathstrut 1752437909050980457420800q^{20} \) \(\mathstrut -\mathstrut 228188865811378281195456q^{21} \) \(\mathstrut -\mathstrut 6854650238581457086684800q^{22} \) \(\mathstrut -\mathstrut 26241180149933881945462800q^{23} \) \(\mathstrut +\mathstrut 1869795529093100955893760q^{24} \) \(\mathstrut +\mathstrut 114502199615774855754278750q^{25} \) \(\mathstrut +\mathstrut 161198290271390767044721344q^{26} \) \(\mathstrut -\mathstrut 12567565757525316335881200q^{27} \) \(\mathstrut -\mathstrut 616012501603352544390963200q^{28} \) \(\mathstrut -\mathstrut 1270673827125882417951629220q^{29} \) \(\mathstrut -\mathstrut 180869768268003018176083200q^{30} \) \(\mathstrut +\mathstrut 261420819791418895481545024q^{31} \) \(\mathstrut +\mathstrut 13400499670935825918040473600q^{32} \) \(\mathstrut +\mathstrut 493606999076023001294056800q^{33} \) \(\mathstrut +\mathstrut 15928395480391611359115494208q^{34} \) \(\mathstrut -\mathstrut 91348258395203301140779687200q^{35} \) \(\mathstrut -\mathstrut 16896751657649331450099821568q^{36} \) \(\mathstrut -\mathstrut 68025363793820786588733238100q^{37} \) \(\mathstrut +\mathstrut 343467860448446492149172419200q^{38} \) \(\mathstrut -\mathstrut 11607709470242600846801346768q^{39} \) \(\mathstrut +\mathstrut 751002277177488834411651072000q^{40} \) \(\mathstrut -\mathstrut 1260483295466373133974684841836q^{41} \) \(\mathstrut +\mathstrut 78468780705143776426791244800q^{42} \) \(\mathstrut -\mathstrut 2570241023157918831169581605000q^{43} \) \(\mathstrut +\mathstrut 1333494270055063548163396988928q^{44} \) \(\mathstrut -\mathstrut 2476861984515775772921684271300q^{45} \) \(\mathstrut +\mathstrut 12543759397110220887548257622784q^{46} \) \(\mathstrut +\mathstrut 4252206875568934025158407583200q^{47} \) \(\mathstrut -\mathstrut 627865474342240811187752140800q^{48} \) \(\mathstrut -\mathstrut 3828013658149768338307153838286q^{49} \) \(\mathstrut -\mathstrut 58010169823175993824907262180000q^{50} \) \(\mathstrut -\mathstrut 1146602989088714802084517292976q^{51} \) \(\mathstrut -\mathstrut 31355807203456647433862622464000q^{52} \) \(\mathstrut +\mathstrut 159799736258590810071505678893900q^{53} \) \(\mathstrut +\mathstrut 20246293327921369338050721342720q^{54} \) \(\mathstrut +\mathstrut 198965756266726963295068969294800q^{55} \) \(\mathstrut -\mathstrut 223825491222026073030759257210880q^{56} \) \(\mathstrut -\mathstrut 24730693798747893221767420706400q^{57} \) \(\mathstrut -\mathstrut 554449751247961885353825667003200q^{58} \) \(\mathstrut -\mathstrut 237962459606090128758899369700840q^{59} \) \(\mathstrut +\mathstrut 35138010316774983406295554252800q^{60} \) \(\mathstrut +\mathstrut 101798841373038700200106255199644q^{61} \) \(\mathstrut +\mathstrut 1873125324903360954810571368883200q^{62} \) \(\mathstrut +\mathstrut 1547129676521036250657950915523600q^{63} \) \(\mathstrut +\mathstrut 1874672702019432167416499183550464q^{64} \) \(\mathstrut -\mathstrut 4674979790566994364848214315456600q^{65} \) \(\mathstrut +\mathstrut 168556374577560093117347449548288q^{66} \) \(\mathstrut -\mathstrut 10891923280981358108643809352546200q^{67} \) \(\mathstrut -\mathstrut 3093300961636357481932215727257600q^{68} \) \(\mathstrut -\mathstrut 903079825405531082422221799280448q^{69} \) \(\mathstrut +\mathstrut 31333293246739547087546996653401600q^{70} \) \(\mathstrut -\mathstrut 746133089793832492689962158868016q^{71} \) \(\mathstrut +\mathstrut 15331394897876624646558687016550400q^{72} \) \(\mathstrut +\mathstrut 19639851676496426023909382507487700q^{73} \) \(\mathstrut -\mathstrut 67111659095025202293405215279979072q^{74} \) \(\mathstrut +\mathstrut 4176423071329708963282786324335000q^{75} \) \(\mathstrut -\mathstrut 66783420216030647132134009134018560q^{76} \) \(\mathstrut -\mathstrut 45127994201424729507574115685000000q^{77} \) \(\mathstrut -\mathstrut 2993244266642124681941208094368000q^{78} \) \(\mathstrut +\mathstrut 272401638034095839035647945144674080q^{79} \) \(\mathstrut -\mathstrut 250481015549362947010655598516633600q^{80} \) \(\mathstrut +\mathstrut 403323837550038341003235827350619442q^{81} \) \(\mathstrut +\mathstrut 293555739501144005521394805048475200q^{82} \) \(\mathstrut -\mathstrut 470460275390909929308746217601073400q^{83} \) \(\mathstrut -\mathstrut 15246219975764064504961836760399872q^{84} \) \(\mathstrut -\mathstrut 456126475708535200141442602834276200q^{85} \) \(\mathstrut -\mathstrut 1006428830786982641255996134716471936q^{86} \) \(\mathstrut +\mathstrut 39923812909186495628003917423124400q^{87} \) \(\mathstrut +\mathstrut 1397391845312343171300827514362265600q^{88} \) \(\mathstrut -\mathstrut 1332868921711238117914343978794942860q^{89} \) \(\mathstrut +\mathstrut 4050631020486902745715661556073886400q^{90} \) \(\mathstrut +\mathstrut 1138398786311531726477856383010207584q^{91} \) \(\mathstrut -\mathstrut 2437574081740573687511448195442483200q^{92} \) \(\mathstrut -\mathstrut 134865729229255425577378938731155200q^{93} \) \(\mathstrut +\mathstrut 703460887213042587108786164575999488q^{94} \) \(\mathstrut -\mathstrut 9929993151899196579173099213128626000q^{95} \) \(\mathstrut +\mathstrut 173258625718217609590386311471038464q^{96} \) \(\mathstrut +\mathstrut 6002061888473229973039130090661237700q^{97} \) \(\mathstrut -\mathstrut 9401601645347953932912572443881928800q^{98} \) \(\mathstrut +\mathstrut 12025768918384639461946415841980309592q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{38}^{\mathrm{new}}(\Gamma_0(1))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.38.a.a \(2\) \(8.671\) \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(-194400\) \(13991400\) \(55\!\cdots\!00\) \(-3\!\cdots\!00\) \(+\) \(q+(-97200-\beta )q^{2}+(6995700+72\beta )q^{3}+\cdots\)