Properties

Label 42.4.a
Level $42$
Weight $4$
Character orbit 42.a
Rep. character $\chi_{42}(1,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $2$
Sturm bound $32$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 42.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(32\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(42))\).

Total New Old
Modular forms 28 2 26
Cusp forms 20 2 18
Eisenstein series 8 0 8

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

\(2\)\(3\)\(7\)FrickeDim.
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(0\)

Trace form

\( 2q + 4q^{2} + 8q^{4} + 20q^{5} + 16q^{8} + 18q^{9} + O(q^{10}) \) \( 2q + 4q^{2} + 8q^{4} + 20q^{5} + 16q^{8} + 18q^{9} + 40q^{10} - 80q^{11} - 76q^{13} - 48q^{15} + 32q^{16} + 4q^{17} + 36q^{18} - 32q^{19} + 80q^{20} - 42q^{21} - 160q^{22} - 104q^{23} + 78q^{25} - 152q^{26} + 140q^{29} - 96q^{30} - 16q^{31} + 64q^{32} + 192q^{33} + 8q^{34} + 112q^{35} + 72q^{36} + 364q^{37} - 64q^{38} - 24q^{39} + 160q^{40} + 468q^{41} - 84q^{42} + 376q^{43} - 320q^{44} + 180q^{45} - 208q^{46} - 96q^{47} + 98q^{49} + 156q^{50} - 24q^{51} - 304q^{52} + 492q^{53} - 1312q^{55} - 648q^{57} + 280q^{58} - 1264q^{59} - 192q^{60} - 220q^{61} - 32q^{62} + 128q^{64} - 696q^{65} + 384q^{66} - 888q^{67} + 16q^{68} + 768q^{69} + 224q^{70} - 200q^{71} + 144q^{72} + 1428q^{73} + 728q^{74} - 960q^{75} - 128q^{76} - 448q^{77} - 48q^{78} + 608q^{79} + 320q^{80} + 162q^{81} + 936q^{82} + 1264q^{83} - 168q^{84} + 104q^{85} + 752q^{86} + 1104q^{87} - 640q^{88} + 420q^{89} + 360q^{90} + 56q^{91} - 416q^{92} - 384q^{93} - 192q^{94} + 1408q^{95} - 1260q^{97} + 196q^{98} - 720q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(42))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 7
42.4.a.a \(1\) \(2.478\) \(\Q\) None \(2\) \(-3\) \(18\) \(7\) \(-\) \(+\) \(-\) \(q+2q^{2}-3q^{3}+4q^{4}+18q^{5}-6q^{6}+\cdots\)
42.4.a.b \(1\) \(2.478\) \(\Q\) None \(2\) \(3\) \(2\) \(-7\) \(-\) \(-\) \(+\) \(q+2q^{2}+3q^{3}+4q^{4}+2q^{5}+6q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(42))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(42)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)