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Properties

Label 42.5.g
Level $42$
Weight $5$
Character orbit 42.g
Rep. character $\chi_{42}(19,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $12$
Newform subspaces $2$
Sturm bound $40$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 42.g (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(40\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(42, [\chi])\).

Total New Old
Modular forms 72 12 60
Cusp forms 56 12 44
Eisenstein series 16 0 16

Trace form

\( 12 q + 18 q^{3} - 48 q^{4} - 108 q^{5} + 6 q^{7} + 162 q^{9} + O(q^{10}) \) \( 12 q + 18 q^{3} - 48 q^{4} - 108 q^{5} + 6 q^{7} + 162 q^{9} + 96 q^{10} - 204 q^{11} - 144 q^{12} - 384 q^{14} + 144 q^{15} - 384 q^{16} + 864 q^{17} + 42 q^{19} + 648 q^{21} - 384 q^{22} - 840 q^{23} + 2622 q^{25} + 1152 q^{26} - 1728 q^{28} - 2112 q^{29} + 576 q^{30} - 2466 q^{31} + 1080 q^{33} + 3828 q^{35} - 2592 q^{36} + 582 q^{37} + 8640 q^{38} + 1530 q^{39} - 768 q^{40} - 3744 q^{42} - 6108 q^{43} - 1632 q^{44} - 2916 q^{45} - 4032 q^{46} - 13932 q^{47} + 3162 q^{49} + 10752 q^{50} + 3816 q^{51} + 624 q^{52} - 2016 q^{53} + 1536 q^{56} + 5580 q^{57} + 5856 q^{58} - 5112 q^{59} - 576 q^{60} + 15936 q^{61} - 5670 q^{63} + 6144 q^{64} - 20028 q^{65} + 1794 q^{67} - 6912 q^{68} - 15360 q^{70} - 5304 q^{71} + 4926 q^{73} + 8064 q^{74} + 32166 q^{75} + 24360 q^{77} - 12672 q^{78} + 11694 q^{79} + 6912 q^{80} - 4374 q^{81} - 768 q^{82} - 5328 q^{84} + 5184 q^{85} - 3264 q^{86} - 34020 q^{87} + 1536 q^{88} - 27576 q^{89} - 8490 q^{91} + 13440 q^{92} + 15318 q^{93} - 42240 q^{94} - 29724 q^{95} + 17664 q^{98} - 11016 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{5}^{\mathrm{new}}(42, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
42.5.g.a 42.g 7.d $4$ $4.342$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(-18\) \(-66\) \(-70\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+(-6-3\beta _{2})q^{3}+8\beta _{2}q^{4}+\cdots\)
42.5.g.b 42.g 7.d $8$ $4.342$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(36\) \(-42\) \(76\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}+(6+3\beta _{1})q^{3}+8\beta _{1}q^{4}+(-3+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(42, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(42, [\chi]) \cong \) \(S_{5}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)