# Properties

 Label 900.4.d Level $900$ Weight $4$ Character orbit 900.d Rep. character $\chi_{900}(649,\cdot)$ Character field $\Q$ Dimension $22$ Newform subspaces $11$ Sturm bound $720$ Trace bound $19$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 900.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$11$$ Sturm bound: $$720$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(900, [\chi])$$.

Total New Old
Modular forms 576 22 554
Cusp forms 504 22 482
Eisenstein series 72 0 72

## Trace form

 $$22 q + O(q^{10})$$ $$22 q + 30 q^{11} + 118 q^{19} + 96 q^{29} - 508 q^{31} + 1446 q^{41} - 1830 q^{49} - 648 q^{59} - 460 q^{61} + 2328 q^{71} + 580 q^{79} - 2946 q^{89} - 3968 q^{91} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(900, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
900.4.d.a $2$ $53.102$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+26iq^{7}-45q^{11}-44iq^{13}-117iq^{17}+\cdots$$
900.4.d.b $2$ $53.102$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2^{4}iq^{7}-6^{2}q^{11}+5iq^{13}+39iq^{17}+\cdots$$
900.4.d.c $2$ $53.102$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4iq^{7}-6^{2}q^{11}+5iq^{13}-9iq^{17}+\cdots$$
900.4.d.d $2$ $53.102$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{7}-30q^{11}+2iq^{13}-45iq^{17}+\cdots$$
900.4.d.e $2$ $53.102$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+13iq^{7}-6q^{11}+5iq^{13}-78iq^{17}+\cdots$$
900.4.d.f $2$ $53.102$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+17iq^{7}+19iq^{13}-107q^{19}-19q^{31}+\cdots$$
900.4.d.g $2$ $53.102$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+37iq^{7}+89iq^{13}+163q^{19}-17^{2}q^{31}+\cdots$$
900.4.d.h $2$ $53.102$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+14iq^{7}+24q^{11}-35iq^{13}+51iq^{17}+\cdots$$
900.4.d.i $2$ $53.102$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{7}+30q^{11}+2iq^{13}+45iq^{17}+\cdots$$
900.4.d.j $2$ $53.102$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+7iq^{7}+54q^{11}-55iq^{13}+18iq^{17}+\cdots$$
900.4.d.k $2$ $53.102$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+8iq^{7}+60q^{11}+43iq^{13}+9iq^{17}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(900, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(900, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(180, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(450, [\chi])$$$$^{\oplus 2}$$