Properties

Label 42.2
Level 42
Weight 2
Dimension 13
Nonzero newspaces 4
Newforms 5
Sturm bound 192
Trace bound 4

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Defining parameters

Level: \( N \) = \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newforms: \( 5 \)
Sturm bound: \(192\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(42))\).

Total New Old
Modular forms 72 13 59
Cusp forms 25 13 12
Eisenstein series 47 0 47

Trace form

\(13q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 5q^{6} \) \(\mathstrut -\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut q^{8} \) \(\mathstrut -\mathstrut 7q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(13q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 5q^{6} \) \(\mathstrut -\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut q^{8} \) \(\mathstrut -\mathstrut 7q^{9} \) \(\mathstrut -\mathstrut 6q^{10} \) \(\mathstrut -\mathstrut 12q^{11} \) \(\mathstrut -\mathstrut q^{12} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 7q^{14} \) \(\mathstrut +\mathstrut 6q^{15} \) \(\mathstrut +\mathstrut q^{16} \) \(\mathstrut +\mathstrut 6q^{17} \) \(\mathstrut +\mathstrut 13q^{18} \) \(\mathstrut +\mathstrut 4q^{19} \) \(\mathstrut +\mathstrut 6q^{20} \) \(\mathstrut +\mathstrut 17q^{21} \) \(\mathstrut +\mathstrut 12q^{22} \) \(\mathstrut +\mathstrut 12q^{23} \) \(\mathstrut +\mathstrut 7q^{24} \) \(\mathstrut +\mathstrut 7q^{25} \) \(\mathstrut +\mathstrut 2q^{26} \) \(\mathstrut -\mathstrut q^{27} \) \(\mathstrut -\mathstrut 5q^{28} \) \(\mathstrut +\mathstrut 6q^{29} \) \(\mathstrut -\mathstrut 6q^{30} \) \(\mathstrut -\mathstrut 8q^{31} \) \(\mathstrut +\mathstrut q^{32} \) \(\mathstrut -\mathstrut 12q^{33} \) \(\mathstrut -\mathstrut 6q^{34} \) \(\mathstrut -\mathstrut 6q^{35} \) \(\mathstrut -\mathstrut 7q^{36} \) \(\mathstrut -\mathstrut 18q^{37} \) \(\mathstrut -\mathstrut 16q^{38} \) \(\mathstrut -\mathstrut 14q^{39} \) \(\mathstrut -\mathstrut 6q^{40} \) \(\mathstrut -\mathstrut 6q^{41} \) \(\mathstrut -\mathstrut 23q^{42} \) \(\mathstrut -\mathstrut 36q^{43} \) \(\mathstrut -\mathstrut 12q^{44} \) \(\mathstrut -\mathstrut 6q^{45} \) \(\mathstrut -\mathstrut 24q^{46} \) \(\mathstrut +\mathstrut 12q^{47} \) \(\mathstrut -\mathstrut q^{48} \) \(\mathstrut +\mathstrut q^{49} \) \(\mathstrut -\mathstrut 17q^{50} \) \(\mathstrut +\mathstrut 30q^{51} \) \(\mathstrut +\mathstrut 22q^{52} \) \(\mathstrut +\mathstrut 18q^{53} \) \(\mathstrut +\mathstrut 19q^{54} \) \(\mathstrut +\mathstrut 36q^{55} \) \(\mathstrut -\mathstrut 5q^{56} \) \(\mathstrut -\mathstrut 8q^{57} \) \(\mathstrut +\mathstrut 42q^{58} \) \(\mathstrut +\mathstrut 12q^{59} \) \(\mathstrut +\mathstrut 18q^{60} \) \(\mathstrut +\mathstrut 22q^{61} \) \(\mathstrut +\mathstrut 8q^{62} \) \(\mathstrut +\mathstrut 23q^{63} \) \(\mathstrut -\mathstrut 3q^{64} \) \(\mathstrut +\mathstrut 12q^{66} \) \(\mathstrut +\mathstrut 44q^{67} \) \(\mathstrut +\mathstrut 6q^{68} \) \(\mathstrut +\mathstrut 42q^{70} \) \(\mathstrut -\mathstrut 11q^{72} \) \(\mathstrut +\mathstrut 22q^{73} \) \(\mathstrut +\mathstrut 2q^{74} \) \(\mathstrut -\mathstrut 7q^{75} \) \(\mathstrut +\mathstrut 4q^{76} \) \(\mathstrut -\mathstrut 24q^{77} \) \(\mathstrut -\mathstrut 10q^{78} \) \(\mathstrut -\mathstrut 40q^{79} \) \(\mathstrut -\mathstrut 6q^{80} \) \(\mathstrut -\mathstrut 55q^{81} \) \(\mathstrut -\mathstrut 30q^{82} \) \(\mathstrut -\mathstrut 36q^{83} \) \(\mathstrut -\mathstrut 19q^{84} \) \(\mathstrut -\mathstrut 84q^{85} \) \(\mathstrut -\mathstrut 16q^{86} \) \(\mathstrut -\mathstrut 30q^{87} \) \(\mathstrut -\mathstrut 6q^{89} \) \(\mathstrut -\mathstrut 6q^{90} \) \(\mathstrut -\mathstrut 14q^{91} \) \(\mathstrut +\mathstrut 4q^{93} \) \(\mathstrut -\mathstrut 24q^{94} \) \(\mathstrut +\mathstrut 12q^{95} \) \(\mathstrut +\mathstrut 7q^{96} \) \(\mathstrut -\mathstrut 2q^{97} \) \(\mathstrut +\mathstrut 25q^{98} \) \(\mathstrut +\mathstrut 12q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(42))\)

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
42.2.a \(\chi_{42}(1, \cdot)\) 42.2.a.a 1 1
42.2.d \(\chi_{42}(41, \cdot)\) 42.2.d.a 4 1
42.2.e \(\chi_{42}(25, \cdot)\) 42.2.e.a 2 2
42.2.e.b 2
42.2.f \(\chi_{42}(5, \cdot)\) 42.2.f.a 4 2

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(42))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(42)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 2}\)