# Properties

 Label 50.4.b Level $50$ Weight $4$ Character orbit 50.b Rep. character $\chi_{50}(49,\cdot)$ Character field $\Q$ Dimension $4$ Newform subspaces $2$ Sturm bound $30$ Trace bound $6$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 50.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$30$$ Trace bound: $$6$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(50, [\chi])$$.

Total New Old
Modular forms 28 4 24
Cusp forms 16 4 12
Eisenstein series 12 0 12

## Trace form

 $$4 q - 16 q^{4} - 4 q^{6} - 118 q^{9} + O(q^{10})$$ $$4 q - 16 q^{4} - 4 q^{6} - 118 q^{9} + 78 q^{11} + 152 q^{14} + 64 q^{16} + 130 q^{19} - 412 q^{21} + 16 q^{24} - 344 q^{26} + 420 q^{29} + 668 q^{31} - 348 q^{34} + 472 q^{36} - 536 q^{39} - 162 q^{41} - 312 q^{44} + 216 q^{46} - 972 q^{49} - 762 q^{51} + 460 q^{54} - 608 q^{56} + 840 q^{59} + 1328 q^{61} - 256 q^{64} + 372 q^{66} + 3204 q^{69} - 552 q^{71} - 448 q^{74} - 520 q^{76} - 980 q^{79} - 2396 q^{81} + 1648 q^{84} - 464 q^{86} - 3090 q^{89} + 2368 q^{91} + 1152 q^{94} - 64 q^{96} - 2076 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(50, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
50.4.b.a $2$ $2.950$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+4iq^{3}-4q^{4}-2^{4}q^{6}-2iq^{7}+\cdots$$
50.4.b.b $2$ $2.950$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{2}-7iq^{3}-4q^{4}+14q^{6}-34iq^{7}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(50, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(50, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 2}$$