# Properties

 Label 825.4.c Level $825$ Weight $4$ Character orbit 825.c Rep. character $\chi_{825}(199,\cdot)$ Character field $\Q$ Dimension $92$ Newform subspaces $19$ Sturm bound $480$ Trace bound $14$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$19$$ Sturm bound: $$480$$ Trace bound: $$14$$ Distinguishing $$T_p$$: $$2$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(825, [\chi])$$.

Total New Old
Modular forms 372 92 280
Cusp forms 348 92 256
Eisenstein series 24 0 24

## Trace form

 $$92 q - 360 q^{4} + 24 q^{6} - 828 q^{9} + O(q^{10})$$ $$92 q - 360 q^{4} + 24 q^{6} - 828 q^{9} + 472 q^{14} + 1624 q^{16} - 24 q^{19} + 48 q^{21} - 648 q^{24} + 512 q^{26} + 976 q^{29} + 848 q^{31} - 1712 q^{34} + 3240 q^{36} + 168 q^{39} - 1984 q^{41} - 1232 q^{44} + 1848 q^{46} - 5516 q^{49} + 1488 q^{51} - 216 q^{54} + 264 q^{56} + 2624 q^{59} - 5216 q^{61} - 4888 q^{64} + 528 q^{66} - 48 q^{69} - 1152 q^{71} + 9904 q^{74} + 152 q^{76} - 1024 q^{79} + 7452 q^{81} + 3744 q^{84} + 2616 q^{86} + 4768 q^{89} + 3064 q^{91} - 4088 q^{94} + 2808 q^{96} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(825, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
825.4.c.a $2$ $48.677$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+5iq^{2}+3iq^{3}-17q^{4}-15q^{6}+\cdots$$
825.4.c.b $2$ $48.677$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+5iq^{2}-3iq^{3}-17q^{4}+15q^{6}+\cdots$$
825.4.c.c $2$ $48.677$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4iq^{2}+3iq^{3}-8q^{4}-12q^{6}-21iq^{7}+\cdots$$
825.4.c.d $2$ $48.677$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{2}-3iq^{3}-q^{4}+9q^{6}+7iq^{7}+\cdots$$
825.4.c.e $2$ $48.677$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-3iq^{3}+7q^{4}+3q^{6}+6^{2}iq^{7}+\cdots$$
825.4.c.f $2$ $48.677$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-3iq^{3}+7q^{4}+3q^{6}+26iq^{7}+\cdots$$
825.4.c.g $2$ $48.677$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{3}+8q^{4}-2iq^{7}-9q^{9}-11q^{11}+\cdots$$
825.4.c.h $4$ $48.677$ $$\Q(i, \sqrt{97})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-3\beta _{2}q^{3}+(-17+\beta _{3})q^{4}+\cdots$$
825.4.c.i $4$ $48.677$ $$\Q(i, \sqrt{33})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+3\beta _{2}q^{3}+(-1+\beta _{3})q^{4}+\cdots$$
825.4.c.j $4$ $48.677$ $$\Q(i, \sqrt{17})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-3\beta _{2}q^{3}+(3+\beta _{3})q^{4}+(-3+\cdots)q^{6}+\cdots$$
825.4.c.k $6$ $48.677$ 6.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{3})q^{2}+3\beta _{3}q^{3}+(-10+\beta _{4}+\cdots)q^{4}+\cdots$$
825.4.c.l $6$ $48.677$ 6.0.2230106176.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{2})q^{2}+3\beta _{2}q^{3}+(-7+\beta _{3}+\cdots)q^{4}+\cdots$$
825.4.c.m $6$ $48.677$ 6.0.245110336.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}+3\beta _{1}q^{3}+(-6-2\beta _{3}+\beta _{5})q^{4}+\cdots$$
825.4.c.n $6$ $48.677$ 6.0.181494784.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{2}-3\beta _{4}q^{3}+(-3+2\beta _{2})q^{4}+\cdots$$
825.4.c.o $6$ $48.677$ 6.0.9935104.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}+3\beta _{3}q^{3}+(2-\beta _{4})q^{4}+3\beta _{1}q^{6}+\cdots$$
825.4.c.p $8$ $48.677$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{2})q^{2}+3\beta _{2}q^{3}+(-7+\beta _{3}+\cdots)q^{4}+\cdots$$
825.4.c.q $8$ $48.677$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{2}+\beta _{3})q^{2}+3\beta _{3}q^{3}+(3+\beta _{1})q^{4}+\cdots$$
825.4.c.r $10$ $48.677$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{2})q^{2}+3\beta _{2}q^{3}+(-9-\beta _{3}+\cdots)q^{4}+\cdots$$
825.4.c.s $10$ $48.677$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+3\beta _{2}q^{3}+(-1+\beta _{3})q^{4}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(825, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(825, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(55, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(165, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(275, [\chi])$$$$^{\oplus 2}$$