# Properties

 Label 1520.2.d Level $1520$ Weight $2$ Character orbit 1520.d Rep. character $\chi_{1520}(609,\cdot)$ Character field $\Q$ Dimension $54$ Newform subspaces $11$ Sturm bound $480$ Trace bound $15$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$11$$ Sturm bound: $$480$$ Trace bound: $$15$$ Distinguishing $$T_p$$: $$3$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1520, [\chi])$$.

Total New Old
Modular forms 252 54 198
Cusp forms 228 54 174
Eisenstein series 24 0 24

## Trace form

 $$54 q + 2 q^{5} - 54 q^{9} + O(q^{10})$$ $$54 q + 2 q^{5} - 54 q^{9} - 12 q^{11} + 12 q^{15} + 6 q^{19} - 8 q^{21} + 10 q^{25} + 4 q^{29} - 6 q^{35} - 40 q^{39} - 4 q^{41} - 10 q^{45} - 70 q^{49} + 24 q^{51} + 10 q^{55} + 16 q^{59} + 20 q^{61} - 8 q^{65} + 8 q^{69} + 8 q^{71} + 28 q^{75} + 62 q^{81} - 8 q^{85} + 12 q^{89} - 16 q^{91} + 28 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1520, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1520.2.d.a $2$ $12.137$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q+iq^{3}+(-1-i)q^{5}+iq^{7}-q^{9}+\cdots$$
1520.2.d.b $2$ $12.137$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q+(-1-i)q^{5}-iq^{7}+3q^{9}+4q^{11}+\cdots$$
1520.2.d.c $4$ $12.137$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+\beta _{1}q^{3}+(-1-\beta _{2})q^{5}+(2\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$
1520.2.d.d $4$ $12.137$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\zeta_{8}+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+(2\zeta_{8}-\zeta_{8}^{3})q^{5}+\cdots$$
1520.2.d.e $4$ $12.137$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\zeta_{8}+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+(-2\zeta_{8}-\zeta_{8}^{3})q^{5}+\cdots$$
1520.2.d.f $4$ $12.137$ $$\Q(\sqrt{-2}, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+\beta _{2}q^{5}+(\beta _{1}-\beta _{3})q^{7}+\beta _{2}q^{9}+\cdots$$
1520.2.d.g $4$ $12.137$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+\zeta_{8}^{2}q^{3}+(1-\zeta_{8})q^{5}+(-\zeta_{8}-2\zeta_{8}^{2}+\cdots)q^{7}+\cdots$$
1520.2.d.h $6$ $12.137$ 6.0.16516096.1 None $$0$$ $$0$$ $$-1$$ $$0$$ $$q-\beta _{4}q^{3}+\beta _{3}q^{5}+(\beta _{1}+\beta _{4})q^{7}+(-3+\cdots)q^{9}+\cdots$$
1520.2.d.i $6$ $12.137$ 6.0.14077504.2 None $$0$$ $$0$$ $$-1$$ $$0$$ $$q+\beta _{2}q^{3}+\beta _{5}q^{5}+(-\beta _{3}-\beta _{4})q^{7}+\cdots$$
1520.2.d.j $6$ $12.137$ 6.0.5161984.1 None $$0$$ $$0$$ $$2$$ $$0$$ $$q+\beta _{5}q^{3}+(\beta _{2}+\beta _{3})q^{5}+(-\beta _{4}+\beta _{5})q^{7}+\cdots$$
1520.2.d.k $12$ $12.137$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$6$$ $$0$$ $$q-\beta _{11}q^{3}-\beta _{6}q^{5}+(\beta _{5}+\beta _{11})q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1520, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1520, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(95, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(190, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(380, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(760, [\chi])$$$$^{\oplus 2}$$