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Properties

Label 16.13
Level 16
Weight 13
Dimension 52
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 208
Trace bound 1

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Defining parameters

Level: \( N \) = \( 16 = 2^{4} \)
Weight: \( k \) = \( 13 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(208\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{13}(\Gamma_1(16))\).

Total New Old
Modular forms 103 56 47
Cusp forms 89 52 37
Eisenstein series 14 4 10

Trace form

\( 52 q - 2 q^{2} - 2 q^{3} - 4232 q^{4} - 15446 q^{5} + 72928 q^{6} - 4 q^{7} - 478172 q^{8} - 154842 q^{9} + O(q^{10}) \) \( 52 q - 2 q^{2} - 2 q^{3} - 4232 q^{4} - 15446 q^{5} + 72928 q^{6} - 4 q^{7} - 478172 q^{8} - 154842 q^{9} + 2806780 q^{10} + 2668318 q^{11} + 7885108 q^{12} + 3779818 q^{13} - 26829652 q^{14} + 49704280 q^{16} - 21338104 q^{17} + 15306190 q^{18} + 51868606 q^{19} - 72195860 q^{20} + 34562332 q^{21} + 47303908 q^{22} + 298270076 q^{23} - 131729232 q^{24} + 77780658 q^{25} + 526315336 q^{26} + 970053760 q^{27} + 711713336 q^{28} + 434242186 q^{29} - 6112100060 q^{30} + 7182362248 q^{32} - 504915844 q^{33} - 6346386988 q^{34} - 3815032900 q^{35} - 11830365748 q^{36} + 3771682378 q^{37} + 14395577488 q^{38} + 15553507196 q^{39} - 19774396184 q^{40} - 7230859956 q^{41} + 1899889176 q^{42} + 363863518 q^{43} - 12173301164 q^{44} + 15010653294 q^{45} - 10837581892 q^{46} + 7988573928 q^{48} + 29967133392 q^{49} + 1530058950 q^{50} + 33806024892 q^{51} - 9789537692 q^{52} + 56218524298 q^{53} - 21820351456 q^{54} - 74491808260 q^{55} + 61228226264 q^{56} - 81539195520 q^{57} - 33805733192 q^{58} + 104334793054 q^{59} - 76265037280 q^{60} + 52062788650 q^{61} + 280885397040 q^{62} - 234402872384 q^{64} - 480916432012 q^{65} + 148565995972 q^{66} - 43778233922 q^{67} + 232865553776 q^{68} + 440218309084 q^{69} - 411016751104 q^{70} - 188251854340 q^{71} + 632336686420 q^{72} - 771384436020 q^{73} - 207892484228 q^{74} + 308961520610 q^{75} + 21519708868 q^{76} + 754686629276 q^{77} - 115706090484 q^{78} + 477844371304 q^{80} - 2530595017516 q^{81} - 114119320560 q^{82} - 1025936323202 q^{83} - 718734617432 q^{84} + 2245049106932 q^{85} + 2196044404900 q^{86} + 2368412421756 q^{87} - 2058716398472 q^{88} - 2189816386740 q^{89} + 1566205087544 q^{90} - 2028231531652 q^{91} + 151792482776 q^{92} + 4129782030400 q^{93} - 1756088243280 q^{94} + 1830456305584 q^{96} - 2349008982904 q^{97} + 707523027114 q^{98} + 4950023059646 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{13}^{\mathrm{new}}(\Gamma_1(16))\)

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
16.13.c \(\chi_{16}(15, \cdot)\) 16.13.c.a 2 1
16.13.c.b 4
16.13.d \(\chi_{16}(7, \cdot)\) None 0 1
16.13.f \(\chi_{16}(3, \cdot)\) 16.13.f.a 46 2

Decomposition of \(S_{13}^{\mathrm{old}}(\Gamma_1(16))\) into lower level spaces

\( S_{13}^{\mathrm{old}}(\Gamma_1(16)) \cong \) \(S_{13}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 2}\)