# Properties

 Label 525.4.d Level $525$ Weight $4$ Character orbit 525.d Rep. character $\chi_{525}(274,\cdot)$ Character field $\Q$ Dimension $56$ Newform subspaces $15$ Sturm bound $320$ Trace bound $11$

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## Defining parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$15$$ Sturm bound: $$320$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$2$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(525, [\chi])$$.

Total New Old
Modular forms 252 56 196
Cusp forms 228 56 172
Eisenstein series 24 0 24

## Trace form

 $$56 q - 216 q^{4} + 24 q^{6} - 504 q^{9} + O(q^{10})$$ $$56 q - 216 q^{4} + 24 q^{6} - 504 q^{9} + 1016 q^{16} + 96 q^{19} + 84 q^{21} - 648 q^{24} + 184 q^{26} + 560 q^{29} + 96 q^{31} + 160 q^{34} + 1944 q^{36} - 576 q^{39} - 2112 q^{41} - 1580 q^{44} + 2044 q^{46} - 2744 q^{49} + 2304 q^{51} - 216 q^{54} + 84 q^{56} + 112 q^{59} - 4208 q^{61} - 5988 q^{64} - 1296 q^{66} + 2304 q^{69} - 1616 q^{71} + 8708 q^{74} - 8952 q^{76} - 6904 q^{79} + 4536 q^{81} - 1008 q^{84} - 1460 q^{86} + 4288 q^{89} + 1456 q^{91} + 1216 q^{94} + 7968 q^{96} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(525, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
525.4.d.a $2$ $30.976$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+5iq^{2}+3iq^{3}-17q^{4}-15q^{6}+\cdots$$
525.4.d.b $2$ $30.976$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4iq^{2}+3iq^{3}-8q^{4}-12q^{6}-7iq^{7}+\cdots$$
525.4.d.c $2$ $30.976$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{2}-3iq^{3}-q^{4}+9q^{6}-7iq^{7}+\cdots$$
525.4.d.d $2$ $30.976$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{2}-3iq^{3}-q^{4}+9q^{6}-7iq^{7}+\cdots$$
525.4.d.e $2$ $30.976$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{2}+3iq^{3}+4q^{4}-6q^{6}+7iq^{7}+\cdots$$
525.4.d.f $2$ $30.976$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{3}+8q^{4}-7iq^{7}-9q^{9}+42q^{11}+\cdots$$
525.4.d.g $4$ $30.976$ $$\Q(i, \sqrt{57})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{2})q^{2}-3\beta _{2}q^{3}+(-10+3\beta _{3})q^{4}+\cdots$$
525.4.d.h $4$ $30.976$ $$\Q(i, \sqrt{65})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+3\beta _{2}q^{3}+(-9+\beta _{3})q^{4}+\cdots$$
525.4.d.i $4$ $30.976$ $$\Q(i, \sqrt{17})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+4\beta _{2})q^{2}-3\beta _{2}q^{3}+(-5-7\beta _{3})q^{4}+\cdots$$
525.4.d.j $4$ $30.976$ $$\Q(i, \sqrt{41})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{2})q^{2}+3\beta _{2}q^{3}+(-6+3\beta _{3})q^{4}+\cdots$$
525.4.d.k $4$ $30.976$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-2\beta _{1}+\beta _{2})q^{2}-3\beta _{1}q^{3}+(-1+\cdots)q^{4}+\cdots$$
525.4.d.l $4$ $30.976$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\zeta_{8}+\zeta_{8}^{2})q^{2}-3\zeta_{8}q^{3}+(-1-2\zeta_{8}^{3})q^{4}+\cdots$$
525.4.d.m $4$ $30.976$ $$\Q(i, \sqrt{17})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{2})q^{2}-3\beta _{2}q^{3}+3\beta _{3}q^{4}+\cdots$$
525.4.d.n $8$ $30.976$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-3\beta _{3}q^{3}+(-4+\beta _{2})q^{4}+\cdots$$
525.4.d.o $8$ $30.976$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{4})q^{2}-3\beta _{4}q^{3}+(-5+\beta _{2}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(525, [\chi])$$ into lower level spaces

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