# Properties

 Label 675.4.b Level $675$ Weight $4$ Character orbit 675.b Rep. character $\chi_{675}(649,\cdot)$ Character field $\Q$ Dimension $72$ Newform subspaces $17$ Sturm bound $360$ Trace bound $19$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 675.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$17$$ Sturm bound: $$360$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$2$$, $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(675, [\chi])$$.

Total New Old
Modular forms 288 72 216
Cusp forms 252 72 180
Eisenstein series 36 0 36

## Trace form

 $$72 q - 276 q^{4} + O(q^{10})$$ $$72 q - 276 q^{4} + 1116 q^{16} + 12 q^{19} - 264 q^{31} - 2460 q^{34} + 528 q^{46} - 1620 q^{49} + 936 q^{61} - 1920 q^{64} - 7668 q^{76} - 4296 q^{79} + 6660 q^{91} - 4980 q^{94} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(675, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
675.4.b.a $2$ $39.826$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{2}-q^{4}-5^{2}iq^{7}+21iq^{8}+\cdots$$
675.4.b.b $2$ $39.826$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{2}-q^{4}+5^{2}iq^{7}+21iq^{8}+\cdots$$
675.4.b.c $2$ $39.826$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{2}+4q^{4}+24iq^{8}-10q^{11}+\cdots$$
675.4.b.d $2$ $39.826$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{2}+4q^{4}+24iq^{8}+10q^{11}+\cdots$$
675.4.b.e $2$ $39.826$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+7q^{4}-6iq^{7}+15iq^{8}-47q^{11}+\cdots$$
675.4.b.f $2$ $39.826$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+7q^{4}+6iq^{7}+15iq^{8}+47q^{11}+\cdots$$
675.4.b.g $2$ $39.826$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+8q^{4}-17iq^{7}-70iq^{13}+2^{6}q^{16}+\cdots$$
675.4.b.h $2$ $39.826$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+8q^{4}-37iq^{7}+70iq^{13}+2^{6}q^{16}+\cdots$$
675.4.b.i $4$ $39.826$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{2}q^{2}-10q^{4}-11\zeta_{8}q^{7}-2\zeta_{8}^{2}q^{8}+\cdots$$
675.4.b.j $4$ $39.826$ $$\Q(i, \sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}-5q^{4}+9\beta _{1}q^{7}+3\beta _{2}q^{8}+\cdots$$
675.4.b.k $6$ $39.826$ 6.0.2033649216.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-7-\beta _{2}+\beta _{4})q^{4}+(2\beta _{1}+\cdots)q^{7}+\cdots$$
675.4.b.l $6$ $39.826$ 6.0.2033649216.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-7-\beta _{2}+\beta _{4})q^{4}+(-2\beta _{1}+\cdots)q^{7}+\cdots$$
675.4.b.m $6$ $39.826$ 6.0.12559936.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(2\beta _{2}-\beta _{4})q^{2}+(-7+3\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots$$
675.4.b.n $6$ $39.826$ 6.0.12559936.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(2\beta _{2}-\beta _{4})q^{2}+(-7+3\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots$$
675.4.b.o $8$ $39.826$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}+(-7+\beta _{6})q^{4}+(-7\beta _{1}-\beta _{4}+\cdots)q^{7}+\cdots$$
675.4.b.p $8$ $39.826$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}+(-5+\beta _{5})q^{4}+(\beta _{1}-3\beta _{2}+\cdots)q^{7}+\cdots$$
675.4.b.q $8$ $39.826$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}+(-5+\beta _{5})q^{4}+(-\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(675, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(675, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(135, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 2}$$