# Properties

 Label 525.6.d Level $525$ Weight $6$ Character orbit 525.d Rep. character $\chi_{525}(274,\cdot)$ Character field $\Q$ Dimension $88$ Newform subspaces $16$ Sturm bound $480$ Trace bound $6$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 525.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$16$$ Sturm bound: $$480$$ Trace bound: $$6$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(525, [\chi])$$.

Total New Old
Modular forms 412 88 324
Cusp forms 388 88 300
Eisenstein series 24 0 24

## Trace form

 $$88 q - 1384 q^{4} - 360 q^{6} - 7128 q^{9} + O(q^{10})$$ $$88 q - 1384 q^{4} - 360 q^{6} - 7128 q^{9} + 28280 q^{16} + 1208 q^{19} - 1764 q^{21} + 1080 q^{24} + 6856 q^{26} + 14120 q^{29} - 4640 q^{31} - 53136 q^{34} + 112104 q^{36} + 46656 q^{39} + 59232 q^{41} - 19724 q^{44} - 32884 q^{46} - 211288 q^{49} - 20736 q^{51} + 29160 q^{54} + 32340 q^{56} - 60536 q^{59} + 227152 q^{61} - 552804 q^{64} - 221616 q^{66} + 115128 q^{69} + 214192 q^{71} - 471436 q^{74} + 266888 q^{76} + 262304 q^{79} + 577368 q^{81} + 84672 q^{84} - 92180 q^{86} - 348776 q^{89} - 132496 q^{91} + 232352 q^{94} - 249120 q^{96} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(525, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
525.6.d.a $2$ $84.202$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+10iq^{2}-9iq^{3}-68q^{4}+90q^{6}+\cdots$$
525.6.d.b $2$ $84.202$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+6iq^{2}-9iq^{3}-4q^{4}+54q^{6}-7^{2}iq^{7}+\cdots$$
525.6.d.c $2$ $84.202$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+5iq^{2}-9iq^{3}+7q^{4}+45q^{6}-7^{2}iq^{7}+\cdots$$
525.6.d.d $2$ $84.202$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+9iq^{3}+31q^{4}-9q^{6}-7^{2}iq^{7}+\cdots$$
525.6.d.e $4$ $84.202$ $$\Q(i, \sqrt{65})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+7\beta _{2})q^{2}+9\beta _{2}q^{3}+(-20-13\beta _{3})q^{4}+\cdots$$
525.6.d.f $4$ $84.202$ $$\Q(i, \sqrt{233})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-9\beta _{2}q^{3}+(-3^{3}+\beta _{3})q^{4}+\cdots$$
525.6.d.g $4$ $84.202$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-4\beta _{1}+\beta _{2})q^{2}-9\beta _{1}q^{3}+(-4+\cdots)q^{4}+\cdots$$
525.6.d.h $4$ $84.202$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(2\zeta_{8}+\zeta_{8}^{2})q^{2}-9\zeta_{8}q^{3}+(-4-4\zeta_{8}^{3})q^{4}+\cdots$$
525.6.d.i $4$ $84.202$ $$\Q(i, \sqrt{65})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{2})q^{2}-9\beta _{2}q^{3}+(12+3\beta _{3})q^{4}+\cdots$$
525.6.d.j $4$ $84.202$ $$\Q(i, \sqrt{73})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-9\beta _{2}q^{3}+(13+\beta _{3})q^{4}+(-9+\cdots)q^{6}+\cdots$$
525.6.d.k $8$ $84.202$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-2\beta _{3})q^{2}-9\beta _{3}q^{3}+(-29+\beta _{2}+\cdots)q^{4}+\cdots$$
525.6.d.l $8$ $84.202$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-2\beta _{2})q^{2}+9\beta _{2}q^{3}+(-30+3\beta _{3}+\cdots)q^{4}+\cdots$$
525.6.d.m $8$ $84.202$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-4\beta _{2})q^{2}-9\beta _{2}q^{3}+(-14+\beta _{3}+\cdots)q^{4}+\cdots$$
525.6.d.n $8$ $84.202$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{4})q^{2}-9\beta _{4}q^{3}+(-12-\beta _{3}+\cdots)q^{4}+\cdots$$
525.6.d.o $12$ $84.202$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{5})q^{2}+9\beta _{5}q^{3}+(-22+\beta _{3}+\cdots)q^{4}+\cdots$$
525.6.d.p $12$ $84.202$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{6})q^{2}-9\beta _{6}q^{3}+(-22-\beta _{2}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(525, [\chi])$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(525, [\chi]) \simeq$$ $$S_{6}^{\mathrm{new}}(5, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 2}$$