# Properties

 Label 42.4.f Level $42$ Weight $4$ Character orbit 42.f Rep. character $\chi_{42}(5,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $16$ Newform subspaces $1$ Sturm bound $32$ Trace bound $0$

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## Defining parameters

 Level: $$N$$ $$=$$ $$42 = 2 \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 42.f (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$21$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$1$$ Sturm bound: $$32$$ Trace bound: $$0$$

## Dimensions

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

Total New Old
Modular forms 56 16 40
Cusp forms 40 16 24
Eisenstein series 16 0 16

## Trace form

 $$16q + 32q^{4} + 80q^{7} + 18q^{9} + O(q^{10})$$ $$16q + 32q^{4} + 80q^{7} + 18q^{9} - 36q^{10} - 128q^{16} - 48q^{18} - 342q^{19} - 450q^{21} + 24q^{22} - 48q^{24} - 194q^{25} + 88q^{28} + 360q^{30} + 804q^{31} + 1332q^{33} + 144q^{36} - 962q^{37} + 594q^{39} - 144q^{40} - 180q^{42} + 1732q^{43} - 2394q^{45} + 168q^{46} + 820q^{49} + 1638q^{51} + 744q^{52} + 180q^{54} - 2664q^{57} - 780q^{58} - 4620q^{61} - 2016q^{63} - 1024q^{64} - 2016q^{66} - 706q^{67} - 60q^{70} + 192q^{72} + 3294q^{73} + 6174q^{75} + 2832q^{78} - 2656q^{79} + 126q^{81} + 432q^{82} - 432q^{84} + 5232q^{85} + 1026q^{87} + 48q^{88} + 4098q^{91} + 2016q^{93} + 3888q^{94} - 192q^{96} - 4284q^{99} + O(q^{100})$$

## 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.f.a $$16$$ $$2.478$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$80$$ $$q-\beta _{2}q^{2}+\beta _{1}q^{3}+(4+4\beta _{5})q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots$$

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

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