Properties

Label 3.28.a
Level $3$
Weight $28$
Character orbit 3.a
Rep. character $\chi_{3}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $2$
Sturm bound $9$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(9\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{28}(\Gamma_0(3))\).

Total New Old
Modular forms 10 4 6
Cusp forms 8 4 4
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)Dim.
\(+\)\(2\)
\(-\)\(2\)

Trace form

\( 4 q + 24750 q^{2} + 207468868 q^{4} - 6678011160 q^{5} - 29357863722 q^{6} + 218008110320 q^{7} + 4861742850360 q^{8} + 10167463313316 q^{9} + O(q^{10}) \) \( 4 q + 24750 q^{2} + 207468868 q^{4} - 6678011160 q^{5} - 29357863722 q^{6} + 218008110320 q^{7} + 4861742850360 q^{8} + 10167463313316 q^{9} - 75348605607660 q^{10} + 116617327795296 q^{11} - 315259548718860 q^{12} - 520418977815880 q^{13} + 1893324310934112 q^{14} - 4996797942664080 q^{15} + 34250136116550160 q^{16} - 39104040524016600 q^{17} + 62911179251142750 q^{18} - 235283005797086368 q^{19} - 267764747545855080 q^{20} - 831156116543376624 q^{21} + 5632683675098639400 q^{22} - 395117797888703520 q^{23} - 6372344648786250024 q^{24} - 5792329097957588900 q^{25} - 14055144185643612012 q^{26} + 110056945325754109760 q^{28} - 36907265348702078616 q^{29} - 72524593073276944380 q^{30} - 121003948830502004176 q^{31} + 271436460104968915680 q^{32} - 55653580676636607600 q^{33} - 880410388185699637284 q^{34} + 1324381929988226487840 q^{35} + 527358026011299961572 q^{36} - 3951239795883744917320 q^{37} + 4686476030227649721240 q^{38} - 501218193933047983392 q^{39} + 2527173123985898110800 q^{40} + 6557177394087821445384 q^{41} - 23356211857571164027680 q^{42} - 26083886844235085382880 q^{43} + 58371506068574610606384 q^{44} - 16974608368803706151640 q^{45} - 17144833914696545480016 q^{46} + 138013143478894681840800 q^{47} - 162947659054905975466800 q^{48} + 34410425434541912676900 q^{49} + 119629441545433561382850 q^{50} - 172963739695775378014368 q^{51} - 558547472740744707374440 q^{52} + 776057204698796247562440 q^{53} - 74623750587691428980538 q^{54} - 686749354564631862510720 q^{55} + 2420356285265504680160640 q^{56} - 731035293748236396968400 q^{57} - 1365834674776412884337820 q^{58} - 1205282318137912810484352 q^{59} - 142977939630775444551240 q^{60} + 626550204745612142535320 q^{61} - 199451003062435512466320 q^{62} + 554147365920986813255280 q^{63} + 5731982819128482563324992 q^{64} - 2376070365553934871341520 q^{65} + 217805339507733417051144 q^{66} + 2699918437921534069354880 q^{67} - 21448315035862189477394040 q^{68} + 8593988244855876653476032 q^{69} + 15805663059712570285631040 q^{70} - 9167162511517313641144992 q^{71} + 12357898017452914895848440 q^{72} + 9974357517783138862164200 q^{73} - 75233409689197428466094748 q^{74} + 33038618079341125367812800 q^{75} + 35575699369940789237736272 q^{76} - 18896460515387801393060160 q^{77} + 69135949936584802084977540 q^{78} - 79504862784473960913241360 q^{79} + 50660510542950588003462240 q^{80} + 25844327556906693195728964 q^{81} - 32292302343488885307107700 q^{82} - 51495943289489356792384800 q^{83} - 238645620633735758451298752 q^{84} + 127641879082698664682647440 q^{85} + 97050661022955846773837736 q^{86} - 154372509168389619245336880 q^{87} + 205195854251612782034025120 q^{88} + 730895907196143325561381992 q^{89} - 191526045806349820287400140 q^{90} - 834055189060345452612435296 q^{91} + 531670308004769001850807200 q^{92} - 225977548956049419627053520 q^{93} - 365856759063230970127868352 q^{94} + 104244857600296119696633600 q^{95} - 1139993596596830643986323104 q^{96} - 685457851768216560321461560 q^{97} + 2696477550115894198217563230 q^{98} + 296425600513904582389740384 q^{99} + O(q^{100}) \)

Decomposition of \(S_{28}^{\mathrm{new}}(\Gamma_0(3))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3
3.28.a.a 3.a 1.a $2$ $13.856$ \(\Q(\sqrt{6469}) \) None \(3168\) \(3188646\) \(-4906065060\) \(-151657089584\) $-$ $\mathrm{SU}(2)$ \(q+(1584-\beta )q^{2}+3^{13}q^{3}+(2432512+\cdots)q^{4}+\cdots\)
3.28.a.b 3.a 1.a $2$ $13.856$ \(\Q(\sqrt{30001}) \) None \(21582\) \(-3188646\) \(-1771946100\) \(369665199904\) $+$ $\mathrm{SU}(2)$ \(q+(10791-\beta )q^{2}-3^{13}q^{3}+(101301922+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{28}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{28}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{28}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)