# Properties

 Label 21.4.e Level $21$ Weight $4$ Character orbit 21.e Rep. character $\chi_{21}(4,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $8$ Newform subspaces $2$ Sturm bound $10$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 21.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$2$$ Sturm bound: $$10$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(21, [\chi])$$.

Total New Old
Modular forms 20 8 12
Cusp forms 12 8 4
Eisenstein series 8 0 8

## Trace form

 $$8 q + 2 q^{2} + 6 q^{3} - 26 q^{4} - 8 q^{5} - 24 q^{6} - 20 q^{7} + 120 q^{8} - 36 q^{9} + O(q^{10})$$ $$8 q + 2 q^{2} + 6 q^{3} - 26 q^{4} - 8 q^{5} - 24 q^{6} - 20 q^{7} + 120 q^{8} - 36 q^{9} + 46 q^{10} - 20 q^{11} + 72 q^{12} - 4 q^{13} - 242 q^{14} - 84 q^{15} - 170 q^{16} - 132 q^{17} + 18 q^{18} + 218 q^{19} + 872 q^{20} + 108 q^{21} + 76 q^{22} - 132 q^{23} + 54 q^{24} - 14 q^{25} - 466 q^{26} - 108 q^{27} - 166 q^{28} - 488 q^{29} - 192 q^{30} + 348 q^{31} - 728 q^{32} + 150 q^{33} - 552 q^{34} + 140 q^{35} + 468 q^{36} + 54 q^{37} + 350 q^{38} + 378 q^{39} + 42 q^{40} + 1208 q^{41} - 156 q^{42} + 772 q^{43} + 920 q^{44} - 72 q^{45} + 804 q^{46} + 240 q^{47} - 1872 q^{48} - 940 q^{49} - 2060 q^{50} - 108 q^{51} - 260 q^{52} - 756 q^{53} + 108 q^{54} - 1972 q^{55} + 1152 q^{56} + 1116 q^{57} + 358 q^{58} - 1128 q^{59} + 1326 q^{60} + 188 q^{61} + 3636 q^{62} - 126 q^{63} + 68 q^{64} + 280 q^{65} - 156 q^{66} + 998 q^{67} - 2028 q^{68} - 1800 q^{69} + 3566 q^{70} - 48 q^{71} - 540 q^{72} - 1350 q^{73} - 1950 q^{74} + 738 q^{75} - 4712 q^{76} + 548 q^{77} - 492 q^{78} - 1328 q^{79} - 388 q^{80} - 324 q^{81} + 956 q^{82} + 1992 q^{83} + 1536 q^{84} + 3096 q^{85} + 3286 q^{86} + 1050 q^{87} + 1206 q^{88} - 2672 q^{89} - 828 q^{90} - 206 q^{91} - 1512 q^{92} + 474 q^{93} + 3204 q^{94} + 688 q^{95} + 1914 q^{96} + 1044 q^{97} - 5884 q^{98} + 360 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(21, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
21.4.e.a $2$ $1.239$ $$\Q(\sqrt{-3})$$ None $$3$$ $$-3$$ $$3$$ $$-7$$ $$q+(3-3\zeta_{6})q^{2}-3\zeta_{6}q^{3}-\zeta_{6}q^{4}+(3+\cdots)q^{5}+\cdots$$
21.4.e.b $6$ $1.239$ 6.0.9924270768.1 None $$-1$$ $$9$$ $$-11$$ $$-13$$ $$q-\beta _{1}q^{2}+(3-3\beta _{4})q^{3}+(-8+\beta _{1}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(21, [\chi])$$ into lower level spaces

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