# Properties

 Label 490.4.e Level $490$ Weight $4$ Character orbit 490.e Rep. character $\chi_{490}(361,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $80$ Newform subspaces $27$ Sturm bound $336$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$490 = 2 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 490.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$27$$ Sturm bound: $$336$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$3$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(490, [\chi])$$.

Total New Old
Modular forms 536 80 456
Cusp forms 472 80 392
Eisenstein series 64 0 64

## Trace form

 $$80 q + 12 q^{3} - 160 q^{4} + 10 q^{5} + 8 q^{6} - 282 q^{9} + O(q^{10})$$ $$80 q + 12 q^{3} - 160 q^{4} + 10 q^{5} + 8 q^{6} - 282 q^{9} + 20 q^{10} + 2 q^{11} + 48 q^{12} - 16 q^{13} + 160 q^{15} - 640 q^{16} + 132 q^{17} - 112 q^{18} - 250 q^{19} - 80 q^{20} - 112 q^{22} - 168 q^{23} - 16 q^{24} - 1000 q^{25} + 156 q^{26} - 816 q^{27} + 748 q^{29} + 140 q^{30} - 96 q^{31} + 128 q^{33} + 480 q^{34} + 2256 q^{36} - 980 q^{37} + 704 q^{38} - 1816 q^{39} + 80 q^{40} - 1096 q^{41} - 464 q^{43} + 8 q^{44} + 380 q^{45} + 424 q^{46} - 500 q^{47} - 384 q^{48} + 1296 q^{51} + 32 q^{52} + 1716 q^{53} - 556 q^{54} + 480 q^{55} - 1992 q^{57} + 240 q^{58} - 1140 q^{59} - 320 q^{60} - 1146 q^{61} + 160 q^{62} + 5120 q^{64} - 1210 q^{65} - 2880 q^{66} + 1124 q^{67} + 528 q^{68} - 1684 q^{69} + 2752 q^{71} - 448 q^{72} + 972 q^{73} + 1108 q^{74} + 300 q^{75} + 2000 q^{76} - 2176 q^{78} + 6108 q^{79} + 160 q^{80} - 752 q^{81} - 1536 q^{82} - 968 q^{83} - 4000 q^{85} + 1692 q^{86} - 2580 q^{87} + 224 q^{88} + 162 q^{89} - 1800 q^{90} + 1344 q^{92} + 4684 q^{93} - 1084 q^{94} - 600 q^{95} - 64 q^{96} - 3504 q^{97} + 17396 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(490, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
490.4.e.a $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$-8$$ $$5$$ $$0$$ $$q-2\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.b $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$-7$$ $$5$$ $$0$$ $$q-2\zeta_{6}q^{2}+(-7+7\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.c $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$-5$$ $$-5$$ $$0$$ $$q-2\zeta_{6}q^{2}+(-5+5\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.d $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$-1$$ $$5$$ $$0$$ $$q-2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.e $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$1$$ $$-5$$ $$0$$ $$q-2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.f $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$1$$ $$-5$$ $$0$$ $$q-2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.g $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$5$$ $$5$$ $$0$$ $$q-2\zeta_{6}q^{2}+(5-5\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.h $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$7$$ $$-5$$ $$0$$ $$q-2\zeta_{6}q^{2}+(7-7\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.i $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$8$$ $$-5$$ $$0$$ $$q-2\zeta_{6}q^{2}+(8-8\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.j $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$2$$ $$-8$$ $$-5$$ $$0$$ $$q+2\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.k $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$2$$ $$-4$$ $$-5$$ $$0$$ $$q+2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.l $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$2$$ $$-3$$ $$5$$ $$0$$ $$q+2\zeta_{6}q^{2}+(-3+3\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.m $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$2$$ $$-1$$ $$-5$$ $$0$$ $$q+2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.n $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$2$$ $$-1$$ $$-5$$ $$0$$ $$q+2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.o $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$2$$ $$1$$ $$5$$ $$0$$ $$q+2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.p $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$2$$ $$3$$ $$-5$$ $$0$$ $$q+2\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.q $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$2$$ $$4$$ $$5$$ $$0$$ $$q+2\zeta_{6}q^{2}+(4-4\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.r $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$2$$ $$8$$ $$5$$ $$0$$ $$q+2\zeta_{6}q^{2}+(8-8\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.s $2$ $28.911$ $$\Q(\sqrt{-3})$$ None $$2$$ $$10$$ $$-5$$ $$0$$ $$q+2\zeta_{6}q^{2}+(10-10\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
490.4.e.t $4$ $28.911$ $$\Q(\sqrt{-3}, \sqrt{-59})$$ None $$4$$ $$-5$$ $$-10$$ $$0$$ $$q-2\beta _{1}q^{2}+(-3-3\beta _{1}-\beta _{3})q^{3}+(-4+\cdots)q^{4}+\cdots$$
490.4.e.u $4$ $28.911$ $$\Q(\sqrt{-3}, \sqrt{46})$$ None $$4$$ $$-2$$ $$10$$ $$0$$ $$q-2\beta _{2}q^{2}+(-1+\beta _{1}-\beta _{2})q^{3}+(-4+\cdots)q^{4}+\cdots$$
490.4.e.v $4$ $28.911$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$4$$ $$-2$$ $$10$$ $$0$$ $$q+(2+2\beta _{2})q^{2}+(3\beta _{1}+\beta _{2}+3\beta _{3})q^{3}+\cdots$$
490.4.e.w $4$ $28.911$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$4$$ $$2$$ $$-10$$ $$0$$ $$q+(2+2\beta _{2})q^{2}+(3\beta _{1}-\beta _{2}+3\beta _{3})q^{3}+\cdots$$
490.4.e.x $4$ $28.911$ $$\Q(\sqrt{-3}, \sqrt{-59})$$ None $$4$$ $$5$$ $$10$$ $$0$$ $$q-2\beta _{1}q^{2}+(3+3\beta _{1}+\beta _{3})q^{3}+(-4+\cdots)q^{4}+\cdots$$
490.4.e.y $6$ $28.911$ 6.0.$$\cdots$$.2 None $$-6$$ $$4$$ $$15$$ $$0$$ $$q+(-2-2\beta _{3})q^{2}+(\beta _{1}-\beta _{3})q^{3}+4\beta _{3}q^{4}+\cdots$$
490.4.e.z $8$ $28.911$ 8.0.$$\cdots$$.19 None $$-8$$ $$-10$$ $$-20$$ $$0$$ $$q+2\beta _{1}q^{2}+(-3-3\beta _{1}+\beta _{5})q^{3}+(-4+\cdots)q^{4}+\cdots$$
490.4.e.ba $8$ $28.911$ 8.0.$$\cdots$$.19 None $$-8$$ $$10$$ $$20$$ $$0$$ $$q+2\beta _{1}q^{2}+(3+3\beta _{1}-\beta _{5})q^{3}+(-4+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(490, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(490, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(7, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(14, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(49, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(98, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(245, [\chi])$$$$^{\oplus 2}$$