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Properties

Label 42.6.e
Level $42$
Weight $6$
Character orbit 42.e
Rep. character $\chi_{42}(25,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $12$
Newform subspaces $4$
Sturm bound $48$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 42.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(48\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(42, [\chi])\).

Total New Old
Modular forms 88 12 76
Cusp forms 72 12 60
Eisenstein series 16 0 16

Trace form

\( 12 q - 96 q^{4} - 44 q^{5} - 144 q^{6} - 234 q^{7} - 486 q^{9} + O(q^{10}) \) \( 12 q - 96 q^{4} - 44 q^{5} - 144 q^{6} - 234 q^{7} - 486 q^{9} + 648 q^{10} + 296 q^{11} - 3312 q^{13} - 704 q^{14} - 396 q^{15} - 1536 q^{16} + 2612 q^{17} + 1020 q^{19} + 1408 q^{20} + 1764 q^{21} - 5232 q^{22} + 4964 q^{23} + 1152 q^{24} - 7404 q^{25} + 880 q^{26} + 9504 q^{28} + 18376 q^{29} - 4176 q^{30} - 18834 q^{31} + 6714 q^{33} - 12384 q^{34} + 200 q^{35} + 15552 q^{36} - 1512 q^{37} + 816 q^{38} + 8028 q^{39} + 10368 q^{40} + 53424 q^{41} + 20520 q^{42} + 34032 q^{43} + 4736 q^{44} - 3564 q^{45} - 18000 q^{46} - 99492 q^{47} - 24054 q^{49} - 109376 q^{50} - 16884 q^{51} + 26496 q^{52} - 10404 q^{53} + 5832 q^{54} - 119028 q^{55} + 30976 q^{56} + 98928 q^{57} - 25896 q^{58} + 73096 q^{59} + 3168 q^{60} - 47592 q^{61} - 18080 q^{62} - 29160 q^{63} + 49152 q^{64} + 58116 q^{65} + 34848 q^{66} - 26916 q^{67} + 41792 q^{68} - 197784 q^{69} + 94008 q^{70} - 616 q^{71} - 2544 q^{73} - 183696 q^{74} + 9432 q^{75} - 32640 q^{76} + 188308 q^{77} + 38304 q^{78} - 28434 q^{79} - 11264 q^{80} - 39366 q^{81} + 75264 q^{82} + 446624 q^{83} + 21312 q^{84} + 421656 q^{85} + 32496 q^{86} - 119574 q^{87} + 41856 q^{88} - 127008 q^{89} - 104976 q^{90} + 292812 q^{91} - 158848 q^{92} - 102456 q^{93} + 108192 q^{94} - 442348 q^{95} + 18432 q^{96} - 342372 q^{97} + 152544 q^{98} - 47952 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(42, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
42.6.e.a 42.e 7.c $2$ $6.736$ \(\Q(\sqrt{-3}) \) None 42.6.e.a \(-4\) \(-9\) \(6\) \(119\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-4+4\zeta_{6})q^{2}-9\zeta_{6}q^{3}-2^{4}\zeta_{6}q^{4}+\cdots\)
42.6.e.b 42.e 7.c $2$ $6.736$ \(\Q(\sqrt{-3}) \) None 42.6.e.b \(4\) \(9\) \(-86\) \(49\) $\mathrm{SU}(2)[C_{3}]$ \(q+(4-4\zeta_{6})q^{2}+9\zeta_{6}q^{3}-2^{4}\zeta_{6}q^{4}+\cdots\)
42.6.e.c 42.e 7.c $4$ $6.736$ \(\Q(\sqrt{-3}, \sqrt{9601})\) None 42.6.e.c \(-8\) \(18\) \(53\) \(6\) $\mathrm{SU}(2)[C_{3}]$ \(q-4\beta _{2}q^{2}+(9-9\beta _{2})q^{3}+(-2^{4}+2^{4}\beta _{2}+\cdots)q^{4}+\cdots\)
42.6.e.d 42.e 7.c $4$ $6.736$ \(\Q(\sqrt{-3}, \sqrt{505})\) None 42.6.e.d \(8\) \(-18\) \(-17\) \(-408\) $\mathrm{SU}(2)[C_{3}]$ \(q+(4-4\beta _{1})q^{2}-9\beta _{1}q^{3}-2^{4}\beta _{1}q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(42, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(42, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)