# Properties

 Label 320.3.p Level $320$ Weight $3$ Character orbit 320.p Rep. character $\chi_{320}(193,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $44$ Newform subspaces $14$ Sturm bound $144$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 320.p (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q(i)$$ Newform subspaces: $$14$$ Sturm bound: $$144$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$3$$, $$7$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(320, [\chi])$$.

Total New Old
Modular forms 216 52 164
Cusp forms 168 44 124
Eisenstein series 48 8 40

## Trace form

 $$44 q + 4 q^{5} + O(q^{10})$$ $$44 q + 4 q^{5} + 4 q^{13} - 20 q^{17} + 8 q^{21} + 44 q^{25} - 40 q^{33} + 4 q^{37} - 8 q^{41} - 96 q^{45} + 292 q^{53} + 32 q^{57} + 72 q^{61} - 116 q^{65} + 44 q^{73} + 296 q^{77} - 52 q^{81} - 284 q^{85} - 632 q^{93} + 76 q^{97} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(320, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
320.3.p.a $2$ $8.719$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-4$$ $$0$$ $$-4$$ $$q+(-2+2i)q^{3}+5iq^{5}+(-2-2i)q^{7}+\cdots$$
320.3.p.b $2$ $8.719$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$-10$$ $$-6$$ $$q+(-1+i)q^{3}-5q^{5}+(-3-3i)q^{7}+\cdots$$
320.3.p.c $2$ $8.719$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$6$$ $$-14$$ $$q+(-1+i)q^{3}+(3-4i)q^{5}+(-7-7i)q^{7}+\cdots$$
320.3.p.d $2$ $8.719$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$-8$$ $$0$$ $$q+(-4-3i)q^{5}+9iq^{9}+(-7+7i)q^{13}+\cdots$$
320.3.p.e $2$ $8.719$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$8$$ $$0$$ $$q+(4-3i)q^{5}+9iq^{9}+(17-17i)q^{13}+\cdots$$
320.3.p.f $2$ $8.719$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$-10$$ $$6$$ $$q+(1-i)q^{3}-5q^{5}+(3+3i)q^{7}+7iq^{9}+\cdots$$
320.3.p.g $2$ $8.719$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$6$$ $$14$$ $$q+(1-i)q^{3}+(3-4i)q^{5}+(7+7i)q^{7}+\cdots$$
320.3.p.h $2$ $8.719$ $$\Q(\sqrt{-1})$$ None $$0$$ $$4$$ $$0$$ $$4$$ $$q+(2-2i)q^{3}+5iq^{5}+(2+2i)q^{7}+\cdots$$
320.3.p.i $4$ $8.719$ $$\Q(i, \sqrt{41})$$ None $$0$$ $$-2$$ $$6$$ $$-14$$ $$q+(-1+\beta _{2})q^{3}+(1+2\beta _{1}-\beta _{3})q^{5}+\cdots$$
320.3.p.j $4$ $8.719$ $$\Q(i, \sqrt{7})$$ None $$0$$ $$0$$ $$-12$$ $$0$$ $$q-\beta _{2}q^{3}+(-3-4\beta _{1})q^{5}+3\beta _{3}q^{7}+\cdots$$
320.3.p.k $4$ $8.719$ $$\Q(i, \sqrt{15})$$ None $$0$$ $$0$$ $$20$$ $$0$$ $$q-\beta _{2}q^{3}+5q^{5}-\beta _{3}q^{7}+21\beta _{1}q^{9}+\cdots$$
320.3.p.l $4$ $8.719$ $$\Q(i, \sqrt{41})$$ None $$0$$ $$2$$ $$6$$ $$14$$ $$q+(1-\beta _{1}+\beta _{3})q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+\cdots$$
320.3.p.m $6$ $8.719$ 6.0.3534400.1 None $$0$$ $$0$$ $$-4$$ $$-8$$ $$q-\beta _{2}q^{3}+(-1-\beta _{2}-\beta _{4})q^{5}+(-2+\cdots)q^{7}+\cdots$$
320.3.p.n $6$ $8.719$ 6.0.3534400.1 None $$0$$ $$0$$ $$-4$$ $$8$$ $$q+\beta _{2}q^{3}+(-1-\beta _{2}-\beta _{4})q^{5}+(2+2\beta _{3}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(320, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(320, [\chi]) \simeq$$ $$S_{3}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(160, [\chi])$$$$^{\oplus 2}$$