Properties

Label 42.4.e
Level $42$
Weight $4$
Character orbit 42.e
Rep. character $\chi_{42}(25,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $8$
Newform subspaces $3$
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.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 3 \)
Sturm bound: \(32\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(42, [\chi])\).

Total New Old
Modular forms 56 8 48
Cusp forms 40 8 32
Eisenstein series 16 0 16

Trace form

\( 8q - 6q^{3} - 16q^{4} + 16q^{5} + 24q^{6} + 28q^{7} - 36q^{9} + O(q^{10}) \) \( 8q - 6q^{3} - 16q^{4} + 16q^{5} + 24q^{6} + 28q^{7} - 36q^{9} - 52q^{10} - 28q^{11} - 24q^{12} + 12q^{13} - 32q^{14} + 84q^{15} - 64q^{16} + 260q^{17} - 50q^{19} - 128q^{20} - 36q^{21} + 424q^{22} - 148q^{23} - 48q^{24} - 446q^{25} - 152q^{26} + 108q^{27} - 104q^{28} - 104q^{29} + 24q^{30} + 156q^{31} - 138q^{33} + 304q^{34} + 860q^{35} + 288q^{36} + 94q^{37} - 72q^{38} - 546q^{39} - 208q^{40} - 936q^{41} + 252q^{42} - 212q^{43} - 112q^{44} + 144q^{45} - 296q^{46} - 72q^{47} + 192q^{48} + 1220q^{49} + 1696q^{50} + 276q^{51} - 24q^{52} + 372q^{53} - 108q^{54} - 380q^{55} - 224q^{56} - 948q^{57} - 412q^{58} + 112q^{59} - 168q^{60} - 1764q^{61} - 3536q^{62} - 18q^{63} + 512q^{64} - 432q^{65} - 528q^{66} + 2q^{67} + 1040q^{68} + 1896q^{69} + 844q^{70} + 2432q^{71} + 746q^{73} - 216q^{74} - 738q^{75} + 400q^{76} + 532q^{77} - 1200q^{78} + 2280q^{79} + 256q^{80} - 324q^{81} + 768q^{82} - 3256q^{83} + 648q^{84} - 2312q^{85} + 216q^{86} + 462q^{87} - 848q^{88} + 864q^{89} + 936q^{90} - 2106q^{91} + 1184q^{92} + 1578q^{93} - 2160q^{94} + 2888q^{95} - 192q^{96} + 4340q^{97} - 240q^{98} + 504q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(42, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
42.4.e.a \(2\) \(2.478\) \(\Q(\sqrt{-3}) \) None \(2\) \(-3\) \(6\) \(-7\) \(q+(2-2\zeta_{6})q^{2}-3\zeta_{6}q^{3}-4\zeta_{6}q^{4}+\cdots\)
42.4.e.b \(2\) \(2.478\) \(\Q(\sqrt{-3}) \) None \(2\) \(3\) \(15\) \(35\) \(q+(2-2\zeta_{6})q^{2}+3\zeta_{6}q^{3}-4\zeta_{6}q^{4}+\cdots\)
42.4.e.c \(4\) \(2.478\) \(\Q(\sqrt{-3}, \sqrt{1345})\) None \(-4\) \(-6\) \(-5\) \(0\) \(q-2\beta _{2}q^{2}+(-3+3\beta _{2})q^{3}+(-4+4\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(42, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(42, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)