# Properties

 Label 63.4.f Level $63$ Weight $4$ Character orbit 63.f Rep. character $\chi_{63}(22,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $36$ Newform subspaces $3$ Sturm bound $32$ Trace bound $1$

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## Defining parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 63.f (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$3$$ Sturm bound: $$32$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(63, [\chi])$$.

Total New Old
Modular forms 52 36 16
Cusp forms 44 36 8
Eisenstein series 8 0 8

## Trace form

 $$36q + 4q^{2} + 2q^{3} - 72q^{4} + 8q^{5} + 10q^{6} - 108q^{8} - 34q^{9} + O(q^{10})$$ $$36q + 4q^{2} + 2q^{3} - 72q^{4} + 8q^{5} + 10q^{6} - 108q^{8} - 34q^{9} + 134q^{11} + 268q^{12} + 56q^{14} + 20q^{15} - 288q^{16} - 228q^{17} - 470q^{18} + 180q^{19} - 26q^{20} - 56q^{21} - 36q^{22} + 96q^{23} - 366q^{24} - 594q^{25} + 1864q^{26} + 704q^{27} + 20q^{29} + 40q^{30} - 36q^{31} + 266q^{32} - 610q^{33} + 576q^{34} - 560q^{35} + 952q^{36} - 144q^{37} - 1016q^{38} - 832q^{39} + 180q^{40} - 326q^{41} + 140q^{42} + 342q^{43} - 464q^{44} + 308q^{45} + 72q^{46} - 12q^{47} - 476q^{48} - 882q^{49} - 610q^{50} - 1518q^{51} - 918q^{52} - 240q^{53} + 1918q^{54} - 792q^{55} + 672q^{56} + 158q^{57} - 594q^{58} + 678q^{59} - 68q^{60} + 1848q^{62} - 476q^{63} + 4212q^{64} + 1640q^{65} + 3166q^{66} + 1386q^{67} - 1146q^{68} + 1200q^{69} - 3744q^{71} - 1926q^{72} + 2484q^{73} + 2076q^{74} - 5570q^{75} - 2196q^{76} + 616q^{77} + 4q^{78} - 468q^{79} - 4616q^{80} - 2170q^{81} - 7236q^{82} - 1644q^{83} + 2282q^{84} + 1296q^{85} + 3782q^{86} + 1916q^{87} + 1656q^{88} + 5600q^{89} - 3416q^{90} + 504q^{91} - 3726q^{92} + 4332q^{93} + 3006q^{94} + 380q^{95} - 280q^{96} + 1998q^{97} - 392q^{98} + 4472q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(63, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
63.4.f.a $$2$$ $$3.717$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$-9$$ $$14$$ $$7$$ $$q+\zeta_{6}q^{2}+(-6+3\zeta_{6})q^{3}+(7-7\zeta_{6})q^{4}+\cdots$$
63.4.f.b $$16$$ $$3.717$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$-3$$ $$2$$ $$-30$$ $$56$$ $$q-\beta _{1}q^{2}+(-\beta _{3}-\beta _{7})q^{3}+(5\beta _{3}-\beta _{5}+\cdots)q^{4}+\cdots$$
63.4.f.c $$18$$ $$3.717$$ $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ None $$6$$ $$9$$ $$24$$ $$-63$$ $$q+(1-\beta _{2}-\beta _{3}-\beta _{5})q^{2}+(1+\beta _{8})q^{3}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(63, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(63, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 2}$$