Properties

Label 16.48
Level 16
Weight 48
Dimension 209
Nonzero newspaces 2
Newform subspaces 7
Sturm bound 768
Trace bound 1

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Defining parameters

Level: \( N \) = \( 16 = 2^{4} \)
Weight: \( k \) = \( 48 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 7 \)
Sturm bound: \(768\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{48}(\Gamma_1(16))\).

Total New Old
Modular forms 383 214 169
Cusp forms 369 209 160
Eisenstein series 14 5 9

Trace form

\( 209 q - 2 q^{2} + 94143178826 q^{3} + 118204588132040 q^{4} + 13063095838313536 q^{5} + 4705034946079171792 q^{6} - 98284592784472971016 q^{7} + 1720184907786008053396 q^{8} + 213132560493416036629459 q^{9} + O(q^{10}) \) \( 209 q - 2 q^{2} + 94143178826 q^{3} + 118204588132040 q^{4} + 13063095838313536 q^{5} + 4705034946079171792 q^{6} - 98284592784472971016 q^{7} + 1720184907786008053396 q^{8} + 213132560493416036629459 q^{9} + 501635537084280466441772 q^{10} - 989125861477272921181834 q^{11} - 90877298891777940118923356 q^{12} + 44844593209717838242931720 q^{13} + 610244160938325274396661372 q^{14} - 3895328481905296061699543892 q^{15} + 5931468773181476844862177816 q^{16} - 37921963394608935092945611078 q^{17} - 22103406046073875915304064546 q^{18} + 3510331056862266531019612339810 q^{19} + 4158127038493769892405965623804 q^{20} + 6442770854438716097237223525172 q^{21} - 96597715942111005079646700683068 q^{22} + 19800399242294138908048440650088 q^{23} + 537588790103566566465934424619184 q^{24} + 3633073046879264936987822819848049 q^{25} + 1398504788028015941456221590188680 q^{26} + 1488571653936950263574552934274928 q^{27} + 22847269273449057754581311407689112 q^{28} + 22683950701134981705348020802699504 q^{29} - 174988002608843240509366539731060364 q^{30} + 63375043424567177368296641632631632 q^{31} - 1247971835761580428125627240792863032 q^{32} - 642122070944104119265645688803024820 q^{33} + 2398453961749505530702163897443150580 q^{34} - 2370701287429192322490245536277938516 q^{35} - 9212543125850301283523343814127096276 q^{36} - 3926773479211312942454128595249183880 q^{37} + 29649199184874249705978302004748516128 q^{38} - 88316723824979138206915224517803853880 q^{39} + 104169296583581999145763697105736919624 q^{40} - 46076242037062523744145631439018681130 q^{41} + 113995823029382858861021973355781788600 q^{42} + 700335584463186055594254244016869317822 q^{43} - 768580350330008806255151276927701494748 q^{44} + 742053625572037961069305452357467315916 q^{45} - 3635543723122972804259650695503589693908 q^{46} - 2135354278193676479832264269113357944800 q^{47} - 19921453655045760525342632323244978002712 q^{48} - 115435062451674902830359833852070435555139 q^{49} - 14141030810789631242276076828047200934394 q^{50} - 69923890294400903276549335155606134652484 q^{51} + 8941297137812537510328926710103813786964 q^{52} - 48906240933086297939605651158976958068264 q^{53} + 284378186719038150431851633230877833654304 q^{54} + 196387249649643558529822949433395265385272 q^{55} + 260873638780029171621567558056637890399512 q^{56} - 245358434724275222404236175177680516122960 q^{57} + 1049597321335947980252338419204136026690968 q^{58} + 1651645349752315903393886090282672011575198 q^{59} - 3398342607059080016327708211561954009502272 q^{60} + 1987473694299111806844040733502065862716040 q^{61} + 2422205988618316892445196480192964148487216 q^{62} + 2212962705692880463762532292818598334252956 q^{63} + 7918943990565597518589542322248590585998080 q^{64} - 5794492198224042558189855349169829441368576 q^{65} + 33850821156089579066075382030608831039673172 q^{66} + 5013673953316634730244150844359291897594466 q^{67} - 12304511955823783528034052777361432001430000 q^{68} + 6084890100146429493686440239697599172242372 q^{69} + 107970031335373328024310805548734226890128000 q^{70} + 64033041808239277153413814508372701603421880 q^{71} + 175254354018971766623172827215133550490323972 q^{72} - 83230607270438025139551735959185457153402378 q^{73} + 359004642323437726626166746730627045354857132 q^{74} - 55291511410433977913830559103068369690717718 q^{75} - 747759244575726273775142086836875505690186892 q^{76} + 685809894894537920756430279701611342019958580 q^{77} - 3815533414940383618311501326883898858938893764 q^{78} - 2035447503861442454407895000460471632967483792 q^{79} - 6813536073982003397106367665949796110869584728 q^{80} - 10719529178895994369490104780498272261200196063 q^{81} - 507387783344449988863386100088765710102079360 q^{82} + 3258820625574791818097860854758779351327140346 q^{83} + 3967136919022670228323163870626261455003197608 q^{84} + 704543426975339819474030963923902781900160888 q^{85} + 3406531061335245613200041699368889825064913092 q^{86} - 3405549717053152967619093582511389904739733240 q^{87} + 24726921722549745740625332110175908814714287608 q^{88} + 3058139401280398148252178171787377597205071750 q^{89} + 19770896484087599998330403986782460079267259096 q^{90} - 49587470314454185679923570351959625728533174316 q^{91} - 15828531526246871859573957606939695224879945224 q^{92} - 8854741217223274294978701839545379516516535760 q^{93} + 5205382805138789552268551343485051890310821456 q^{94} + 40131696128342550630440456436345055487681108028 q^{95} - 44590066347643379513817676405429236738747031888 q^{96} - 9363777488742546430993561438680108381592885878 q^{97} + 7415521981459883382480069090572640795310570746 q^{98} + 70358352828010112912915398422221292559013237938 q^{99} + O(q^{100}) \)

Decomposition of \(S_{48}^{\mathrm{new}}(\Gamma_1(16))\)

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 the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
16.48.a \(\chi_{16}(1, \cdot)\) 16.48.a.a 1 1
16.48.a.b 2
16.48.a.c 4
16.48.a.d 4
16.48.a.e 6
16.48.a.f 6
16.48.b \(\chi_{16}(9, \cdot)\) None 0 1
16.48.e \(\chi_{16}(5, \cdot)\) 16.48.e.a 186 2

Decomposition of \(S_{48}^{\mathrm{old}}(\Gamma_1(16))\) into lower level spaces

\( S_{48}^{\mathrm{old}}(\Gamma_1(16)) \cong \) \(S_{48}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 5}\)\(\oplus\)\(S_{48}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{48}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{48}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 2}\)