Properties

Label 1.38
Level 1
Weight 38
Dimension 2
Nonzero newspaces 1
Newform subspaces 1
Sturm bound 3
Trace bound 0

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Defining parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 38 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 1 \)
Sturm bound: \(3\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{38}(\Gamma_1(1))\).

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

Trace form

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

Decomposition of \(S_{38}^{\mathrm{new}}(\Gamma_1(1))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
1.38.a \(\chi_{1}(1, \cdot)\) 1.38.a.a 2 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 194400 T + 137474498560 T^{2} + 26718132554956800 T^{3} + \)\(18\!\cdots\!84\)\( T^{4} \)
$3$ \( 1 - 13991400 T + 899855474728862070 T^{2} - \)\(63\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!69\)\( T^{4} \)
$5$ \( 1 - 5529584385900 T + \)\(30\!\cdots\!50\)\( T^{2} - \)\(40\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 + 3448443953486000 T + \)\(26\!\cdots\!50\)\( T^{2} + \)\(64\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 + 26734036354848538056 T + \)\(70\!\cdots\!26\)\( T^{2} + \)\(90\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} \)
$13$ \( 1 - \)\(53\!\cdots\!00\)\( T + \)\(32\!\cdots\!90\)\( T^{2} - \)\(87\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!89\)\( T^{4} \)
$17$ \( 1 + \)\(89\!\cdots\!00\)\( T + \)\(86\!\cdots\!30\)\( T^{2} + \)\(30\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \)
$19$ \( 1 - \)\(37\!\cdots\!20\)\( T + \)\(20\!\cdots\!78\)\( T^{2} - \)\(76\!\cdots\!80\)\( T^{3} + \)\(42\!\cdots\!21\)\( T^{4} \)
$23$ \( 1 + \)\(26\!\cdots\!00\)\( T + \)\(48\!\cdots\!10\)\( T^{2} + \)\(63\!\cdots\!00\)\( T^{3} + \)\(58\!\cdots\!09\)\( T^{4} \)
$29$ \( 1 + \)\(12\!\cdots\!20\)\( T + \)\(21\!\cdots\!18\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!81\)\( T^{4} \)
$31$ \( 1 - \)\(26\!\cdots\!24\)\( T + \)\(24\!\cdots\!66\)\( T^{2} - \)\(39\!\cdots\!64\)\( T^{3} + \)\(22\!\cdots\!21\)\( T^{4} \)
$37$ \( 1 + \)\(68\!\cdots\!00\)\( T + \)\(13\!\cdots\!90\)\( T^{2} + \)\(71\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \)
$41$ \( 1 + \)\(12\!\cdots\!36\)\( T + \)\(12\!\cdots\!86\)\( T^{2} + \)\(59\!\cdots\!16\)\( T^{3} + \)\(22\!\cdots\!61\)\( T^{4} \)
$43$ \( 1 + \)\(25\!\cdots\!00\)\( T + \)\(44\!\cdots\!50\)\( T^{2} + \)\(70\!\cdots\!00\)\( T^{3} + \)\(75\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - \)\(42\!\cdots\!00\)\( T + \)\(14\!\cdots\!70\)\( T^{2} - \)\(31\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!69\)\( T^{4} \)
$53$ \( 1 - \)\(15\!\cdots\!00\)\( T + \)\(16\!\cdots\!70\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(39\!\cdots\!69\)\( T^{4} \)
$59$ \( 1 + \)\(23\!\cdots\!40\)\( T + \)\(64\!\cdots\!38\)\( T^{2} + \)\(79\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!61\)\( T^{4} \)
$61$ \( 1 - \)\(10\!\cdots\!44\)\( T + \)\(10\!\cdots\!26\)\( T^{2} - \)\(11\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!41\)\( T^{4} \)
$67$ \( 1 + \)\(10\!\cdots\!00\)\( T + \)\(94\!\cdots\!30\)\( T^{2} + \)\(39\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!29\)\( T^{4} \)
$71$ \( 1 + \)\(74\!\cdots\!16\)\( T + \)\(58\!\cdots\!46\)\( T^{2} + \)\(23\!\cdots\!56\)\( T^{3} + \)\(98\!\cdots\!81\)\( T^{4} \)
$73$ \( 1 - \)\(19\!\cdots\!00\)\( T + \)\(18\!\cdots\!10\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(76\!\cdots\!09\)\( T^{4} \)
$79$ \( 1 - \)\(27\!\cdots\!80\)\( T + \)\(50\!\cdots\!18\)\( T^{2} - \)\(44\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{4} \)
$83$ \( 1 + \)\(47\!\cdots\!00\)\( T + \)\(24\!\cdots\!30\)\( T^{2} + \)\(47\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!29\)\( T^{4} \)
$89$ \( 1 + \)\(13\!\cdots\!60\)\( T + \)\(23\!\cdots\!58\)\( T^{2} + \)\(17\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!41\)\( T^{4} \)
$97$ \( 1 - \)\(60\!\cdots\!00\)\( T + \)\(57\!\cdots\!70\)\( T^{2} - \)\(19\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!69\)\( T^{4} \)
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