# Properties

 Label 75.4.b Level $75$ Weight $4$ Character orbit 75.b Rep. character $\chi_{75}(49,\cdot)$ Character field $\Q$ Dimension $8$ Newform subspaces $3$ Sturm bound $40$ Trace bound $4$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 75.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$40$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(75, [\chi])$$.

Total New Old
Modular forms 36 8 28
Cusp forms 24 8 16
Eisenstein series 12 0 12

## Trace form

 $$8 q - 36 q^{4} - 24 q^{6} - 72 q^{9} + O(q^{10})$$ $$8 q - 36 q^{4} - 24 q^{6} - 72 q^{9} + 112 q^{11} - 324 q^{14} + 180 q^{16} + 276 q^{19} - 108 q^{21} + 468 q^{24} - 764 q^{26} - 488 q^{29} + 68 q^{31} + 1088 q^{34} + 324 q^{36} + 204 q^{39} + 1688 q^{41} + 712 q^{44} - 2232 q^{46} - 1100 q^{49} - 816 q^{51} + 216 q^{54} - 480 q^{56} - 496 q^{59} + 2340 q^{61} - 3284 q^{64} + 1488 q^{66} - 504 q^{69} + 416 q^{71} - 1904 q^{74} + 784 q^{76} + 3280 q^{79} + 648 q^{81} - 936 q^{84} - 1508 q^{86} - 984 q^{89} - 2492 q^{91} + 824 q^{94} - 2748 q^{96} - 1008 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(75, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
75.4.b.a $2$ $4.425$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{2}+3iq^{3}-q^{4}-9q^{6}+20iq^{7}+\cdots$$
75.4.b.b $2$ $4.425$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-3iq^{3}+7q^{4}+3q^{6}-24iq^{7}+\cdots$$
75.4.b.c $4$ $4.425$ $$\Q(i, \sqrt{19})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{1}+\beta _{3})q^{2}-3\beta _{1}q^{3}+(-12+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(75, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(75, [\chi]) \simeq$$ $$S_{4}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 2}$$