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

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

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
42.4.e.a 42.e 7.c $2$ $2.478$ \(\Q(\sqrt{-3}) \) None \(2\) \(-3\) \(6\) \(-7\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-3\zeta_{6}q^{3}-4\zeta_{6}q^{4}+\cdots\)
42.4.e.b 42.e 7.c $2$ $2.478$ \(\Q(\sqrt{-3}) \) None \(2\) \(3\) \(15\) \(35\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}+3\zeta_{6}q^{3}-4\zeta_{6}q^{4}+\cdots\)
42.4.e.c 42.e 7.c $4$ $2.478$ \(\Q(\sqrt{-3}, \sqrt{1345})\) None \(-4\) \(-6\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\beta _{2}q^{2}+(-3+3\beta _{2})q^{3}+(-4+4\beta _{2}+\cdots)q^{4}+\cdots\)

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

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