# Properties

 Label 900.6.d Level $900$ Weight $6$ Character orbit 900.d Rep. character $\chi_{900}(649,\cdot)$ Character field $\Q$ Dimension $38$ Newform subspaces $14$ Sturm bound $1080$ Trace bound $19$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 900.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$14$$ Sturm bound: $$1080$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(900, [\chi])$$.

Total New Old
Modular forms 936 38 898
Cusp forms 864 38 826
Eisenstein series 72 0 72

## Trace form

 $$38 q + O(q^{10})$$ $$38 q + 276 q^{11} - 1156 q^{19} + 3924 q^{29} + 1060 q^{31} - 18948 q^{41} - 82746 q^{49} + 9012 q^{59} + 60928 q^{61} - 68664 q^{71} - 25312 q^{79} + 162852 q^{89} + 224924 q^{91} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(900, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
900.6.d.a $2$ $144.345$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+44iq^{7}-540q^{11}-209iq^{13}+\cdots$$
900.6.d.b $2$ $144.345$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+22iq^{7}-6^{3}q^{11}-385iq^{13}-267iq^{17}+\cdots$$
900.6.d.c $2$ $144.345$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+28iq^{7}-156q^{11}-175iq^{13}+\cdots$$
900.6.d.d $2$ $144.345$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+211iq^{7}-427iq^{13}-3143q^{19}+\cdots$$
900.6.d.e $2$ $144.345$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+118iq^{7}-601iq^{13}+1432q^{19}+\cdots$$
900.6.d.f $2$ $144.345$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+122iq^{7}+12^{2}q^{11}+5^{2}iq^{13}+\cdots$$
900.6.d.g $2$ $144.345$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+91iq^{7}+174q^{11}+785iq^{13}+\cdots$$
900.6.d.h $2$ $144.345$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+109iq^{7}+480q^{11}+311iq^{13}+\cdots$$
900.6.d.i $2$ $144.345$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+8iq^{7}+564q^{11}-185iq^{13}-543iq^{17}+\cdots$$
900.6.d.j $4$ $144.345$ $$\Q(i, \sqrt{7})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-11\beta _{1}+\beta _{3})q^{7}+(-186-3\beta _{2}+\cdots)q^{11}+\cdots$$
900.6.d.k $4$ $144.345$ $$\Q(i, \sqrt{241})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+4\beta _{2})q^{7}+(-120+\beta _{3})q^{11}+\cdots$$
900.6.d.l $4$ $144.345$ $$\Q(i, \sqrt{94})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+71\beta _{1}q^{7}+\beta _{3}q^{11}-137\beta _{1}q^{13}+\cdots$$
900.6.d.m $4$ $144.345$ $$\Q(i, \sqrt{409})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-2\beta _{1}+4\beta _{2})q^{7}+(30+\beta _{3})q^{11}+\cdots$$
900.6.d.n $4$ $144.345$ $$\Q(i, \sqrt{241})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+4\beta _{2})q^{7}+(120-\beta _{3})q^{11}+(8\beta _{1}+\cdots)q^{13}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(900, [\chi])$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(900, [\chi]) \cong$$ $$S_{6}^{\mathrm{new}}(5, [\chi])$$$$^{\oplus 18}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(180, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(450, [\chi])$$$$^{\oplus 2}$$