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Properties

Label 16.38
Level 16
Weight 38
Dimension 164
Nonzero newspaces 2
Newform subspaces 7
Sturm bound 608
Trace bound 1

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Defining parameters

Level: \( N \) = \( 16 = 2^{4} \)
Weight: \( k \) = \( 38 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 7 \)
Sturm bound: \(608\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{38}(\Gamma_1(16))\).

Total New Old
Modular forms 303 169 134
Cusp forms 289 164 125
Eisenstein series 14 5 9

Trace form

\( 164 q - 2 q^{2} - 387420490 q^{3} + 207641071080 q^{4} - 526646293942 q^{5} + 152752427013424 q^{6} - 3130502550805008 q^{7} + 105164002093662964 q^{8} + 2869998420059843338 q^{9} + O(q^{10}) \) \( 164 q - 2 q^{2} - 387420490 q^{3} + 207641071080 q^{4} - 526646293942 q^{5} + 152752427013424 q^{6} - 3130502550805008 q^{7} + 105164002093662964 q^{8} + 2869998420059843338 q^{9} + 10202502832193478924 q^{10} - 873986500095347638 q^{11} + 150524454601337448964 q^{12} + 47214993718320365338 q^{13} + 897558897134910808988 q^{14} + 11299766864660459030484 q^{15} - 2298981488062604471144 q^{16} + 25815940198529793318592 q^{17} - 410787380368525353609858 q^{18} - 109977735913095349401762 q^{19} + 6407168914373989923144092 q^{20} - 728451343759942496092892 q^{21} - 19648486488874701166132860 q^{22} + 34930278825313644138487760 q^{23} - 95078909284035549570555728 q^{24} + 251711721158935069958237262 q^{25} - 310389170440570966700427000 q^{26} - 310513378390196143520498008 q^{27} + 1632020431978379176262592344 q^{28} + 253857262888598000160559010 q^{29} + 11307703976618381825484182676 q^{30} - 10138846695121601232070015312 q^{31} + 8821409874177676080973032008 q^{32} - 1663440470678137283571901604 q^{33} - 14907477366732834757770829772 q^{34} - 40756912917241407655097594788 q^{35} - 20952395794706147310595592468 q^{36} + 159895217098115482075546264818 q^{37} - 302663250216665296574274420032 q^{38} + 234042659597787979840359533968 q^{39} + 431052358704395959394097641992 q^{40} + 245673784184262057216601417524 q^{41} - 3582238340623701744507926211080 q^{42} + 51112940573699595040822954066 q^{43} + 589771484310208190326034452804 q^{44} + 2890443543602959646090102385438 q^{45} + 16495644000695420453479012953228 q^{46} + 29676172438637078107717812697648 q^{47} + 93701017375324813821632085233128 q^{48} - 336672839398455714852638933052936 q^{49} + 6944369788182128351300677682310 q^{50} - 62691581878223765225868437179852 q^{51} - 318942527765016445306225513685068 q^{52} + 159819578629091870938206727335106 q^{53} - 911784181670554382668083490174688 q^{54} + 373256246054007318889689135620976 q^{55} - 819271515987417310944177851067752 q^{56} + 435129045195894068058512073278368 q^{57} + 391108488502457257976134996690776 q^{58} - 560175086359230180835963122121694 q^{59} - 2143715286474332256293887628099328 q^{60} + 364696651902039723163177255428138 q^{61} - 336522427427715213022460140866512 q^{62} + 293185891975006226625490843515060 q^{63} - 6519040551852871670605770456988800 q^{64} + 1693705991144104333825665617379340 q^{65} + 1735814545461716594661571612074484 q^{66} - 1070655288685928226876009614983570 q^{67} + 14462764742810896191418668742067792 q^{68} - 9365862952685585153181271039660332 q^{69} + 3526896096017158715404607701508480 q^{70} + 50925905946749555034513126839764336 q^{71} + 22485729658775042440974973216237284 q^{72} - 28202976031724607514055123720211084 q^{73} + 127587848459030770574104864801147276 q^{74} - 280753243030771658915586052250007762 q^{75} - 186027239939928471531355518384436140 q^{76} + 199533660392683396410210537939319844 q^{77} + 62484010335248735059938393161946140 q^{78} + 868521212399764735420133716966167264 q^{79} + 311100012749756472184486661085295144 q^{80} - 2632923093103551112211756929767090180 q^{81} - 386130911448215781590824528964091936 q^{82} - 445205585100752505022663455406301162 q^{83} + 1005595381758383898162899719119268264 q^{84} - 32276536277205147880812995331772484 q^{85} - 3721207224023633774199635645333238268 q^{86} - 370688587150964555575250914786511088 q^{87} - 6246886134498857020479054806680968712 q^{88} - 740520654367189780058265496407230700 q^{89} + 8368503750910381084157592969437442072 q^{90} + 5474061822500829028812853757158820644 q^{91} - 6761510149941964077925336113945990856 q^{92} - 377021822596674451559641223894515504 q^{93} + 10107363120089728040210766993005188496 q^{94} - 32488530723385800066875066996223148028 q^{95} + 3710702930449505683434179348615029424 q^{96} + 7120702780499189030716906590027886304 q^{97} - 39133364646643876685731093181760367846 q^{98} + 32034910019485147816030154814103102150 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{38}^{\mathrm{new}}(\Gamma_1(16))\)

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 available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
16.38.a \(\chi_{16}(1, \cdot)\) 16.38.a.a 2 1
16.38.a.b 2
16.38.a.c 2
16.38.a.d 3
16.38.a.e 4
16.38.a.f 5
16.38.b \(\chi_{16}(9, \cdot)\) None 0 1
16.38.e \(\chi_{16}(5, \cdot)\) 16.38.e.a 146 2

Decomposition of \(S_{38}^{\mathrm{old}}(\Gamma_1(16))\) into lower level spaces

\( S_{38}^{\mathrm{old}}(\Gamma_1(16)) \cong \) \(S_{38}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 5}\)\(\oplus\)\(S_{38}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{38}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{38}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 2}\)