Properties

Label 48.13
Level 48
Weight 13
Dimension 319
Nonzero newspaces 4
Sturm bound 1664
Trace bound 1

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Defining parameters

Level: \( N \) = \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) = \( 13 \)
Nonzero newspaces: \( 4 \)
Sturm bound: \(1664\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{13}(\Gamma_1(48))\).

Total New Old
Modular forms 796 329 467
Cusp forms 740 319 421
Eisenstein series 56 10 46

Trace form

\( 319 q - q^{3} + 8456 q^{4} + 30888 q^{5} - 72932 q^{6} + 102962 q^{7} + 956340 q^{8} - 2428445 q^{9} + O(q^{10}) \) \( 319 q - q^{3} + 8456 q^{4} + 30888 q^{5} - 72932 q^{6} + 102962 q^{7} + 956340 q^{8} - 2428445 q^{9} - 5613568 q^{10} - 5336640 q^{11} - 7885112 q^{12} - 8834766 q^{13} + 53642916 q^{14} + 4135932 q^{15} - 84360304 q^{16} + 7258680 q^{17} + 126892416 q^{18} - 76338242 q^{19} - 204750000 q^{20} + 112483630 q^{21} + 464638136 q^{22} - 596540160 q^{23} - 1049622164 q^{24} - 440355925 q^{25} - 185886900 q^{26} + 145611599 q^{27} - 507536640 q^{28} - 3174544440 q^{29} + 1780624044 q^{30} + 2458372746 q^{31} - 8105479200 q^{32} - 1456179284 q^{33} + 7612152840 q^{34} + 6653503296 q^{35} + 1760728912 q^{36} - 3743969166 q^{37} - 21090497400 q^{38} - 1081717070 q^{39} + 22514320136 q^{40} + 10244732472 q^{41} + 34792057420 q^{42} - 5807801762 q^{43} + 4908132504 q^{44} - 1115107164 q^{45} - 13027012832 q^{46} + 62166702688 q^{48} - 71014593523 q^{49} + 61034284836 q^{50} - 27079298560 q^{51} - 64068993904 q^{52} - 38097482040 q^{53} - 79042825484 q^{54} + 61646971392 q^{55} + 27212429664 q^{56} + 88377803038 q^{57} - 80668117624 q^{58} - 3525080832 q^{59} - 120451499400 q^{60} + 151058513874 q^{61} - 79006924260 q^{62} + 86634217362 q^{63} - 47640225280 q^{64} + 482614363728 q^{65} - 517188496676 q^{66} + 153962582398 q^{67} + 48655728000 q^{68} + 268703467356 q^{69} + 1711270202904 q^{70} + 376503708672 q^{71} - 997069163964 q^{72} + 320376221158 q^{73} - 519546348468 q^{74} + 121613696883 q^{75} + 713672119376 q^{76} - 515153272320 q^{77} + 1359963835976 q^{78} - 427873100214 q^{79} - 27385362072 q^{80} - 2876972060673 q^{81} - 2462892430088 q^{82} + 686759112000 q^{83} - 1856369792648 q^{84} - 1597379085744 q^{85} - 1041179853744 q^{86} - 39159659520 q^{87} + 3003351350336 q^{88} + 2444968318104 q^{89} - 1473847220424 q^{90} + 2920297848388 q^{91} - 4930795743120 q^{92} - 1136906380726 q^{93} + 1527481373832 q^{94} + 1297466398472 q^{96} + 3749074618430 q^{97} + 4487249871720 q^{98} - 847039805828 q^{99} + O(q^{100}) \)

Decomposition of \(S_{13}^{\mathrm{new}}(\Gamma_1(48))\)

We only show spaces with odd 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
48.13.b \(\chi_{48}(7, \cdot)\) None 0 1
48.13.e \(\chi_{48}(17, \cdot)\) 48.13.e.a 1 1
48.13.e.b 2
48.13.e.c 4
48.13.e.d 4
48.13.e.e 12
48.13.g \(\chi_{48}(31, \cdot)\) 48.13.g.a 4 1
48.13.g.b 4
48.13.g.c 4
48.13.h \(\chi_{48}(41, \cdot)\) None 0 1
48.13.i \(\chi_{48}(5, \cdot)\) n/a 188 2
48.13.l \(\chi_{48}(19, \cdot)\) 48.13.l.a 96 2

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{13}^{\mathrm{old}}(\Gamma_1(48))\) into lower level spaces

\( S_{13}^{\mathrm{old}}(\Gamma_1(48)) \cong \) \(S_{13}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 10}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 5}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 1}\)