# Properties

 Label 1305.2.f Level $1305$ Weight $2$ Character orbit 1305.f Rep. character $\chi_{1305}(289,\cdot)$ Character field $\Q$ Dimension $72$ Newform subspaces $13$ Sturm bound $360$ Trace bound $10$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1305 = 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1305.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$145$$ Character field: $$\Q$$ Newform subspaces: $$13$$ Sturm bound: $$360$$ Trace bound: $$10$$ Distinguishing $$T_p$$: $$2$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1305, [\chi])$$.

Total New Old
Modular forms 188 76 112
Cusp forms 172 72 100
Eisenstein series 16 4 12

## Trace form

 $$72 q + 64 q^{4} + 6 q^{5} + O(q^{10})$$ $$72 q + 64 q^{4} + 6 q^{5} + 48 q^{16} + 38 q^{20} + 10 q^{25} - 32 q^{34} - 40 q^{49} + 16 q^{64} - 26 q^{65} - 56 q^{71} + 56 q^{74} + 74 q^{80} - 20 q^{86} + 8 q^{91} - 28 q^{94} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1305, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1305.2.f.a $2$ $10.420$ $$\Q(\sqrt{-1})$$ None $$-4$$ $$0$$ $$-4$$ $$0$$ $$q-2q^{2}+2q^{4}+(-2-i)q^{5}+4iq^{7}+\cdots$$
1305.2.f.b $2$ $10.420$ $$\Q(\sqrt{-1})$$ None $$-2$$ $$0$$ $$2$$ $$0$$ $$q-q^{2}-q^{4}+(1-i)q^{5}+iq^{7}+3q^{8}+\cdots$$
1305.2.f.c $2$ $10.420$ $$\Q(\sqrt{-1})$$ None $$2$$ $$0$$ $$2$$ $$0$$ $$q+q^{2}-q^{4}+(1-i)q^{5}+iq^{7}-3q^{8}+\cdots$$
1305.2.f.d $2$ $10.420$ $$\Q(\sqrt{-1})$$ None $$4$$ $$0$$ $$-4$$ $$0$$ $$q+2q^{2}+2q^{4}+(-2-i)q^{5}+4iq^{7}+\cdots$$
1305.2.f.e $4$ $10.420$ $$\Q(\sqrt{-3}, \sqrt{17})$$ None $$-4$$ $$0$$ $$-3$$ $$0$$ $$q-q^{2}-q^{4}+(-1+\beta _{1})q^{5}-\beta _{3}q^{7}+\cdots$$
1305.2.f.f $4$ $10.420$ $$\Q(\sqrt{5}, \sqrt{-6})$$ None $$-4$$ $$0$$ $$0$$ $$0$$ $$q-q^{2}-q^{4}+\beta _{2}q^{5}+\beta _{1}q^{7}+3q^{8}+\cdots$$
1305.2.f.g $4$ $10.420$ $$\Q(\sqrt{-5}, \sqrt{-29})$$ $$\Q(\sqrt{-435})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-2q^{4}-\beta _{1}q^{5}-\beta _{2}q^{11}+4q^{16}+\cdots$$
1305.2.f.h $4$ $10.420$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+\beta _{3}q^{2}+3q^{4}+(1+\beta _{1})q^{5}+\beta _{1}q^{7}+\cdots$$
1305.2.f.i $4$ $10.420$ $$\Q(\sqrt{-3}, \sqrt{17})$$ None $$4$$ $$0$$ $$-3$$ $$0$$ $$q+q^{2}-q^{4}+(-1+\beta _{1})q^{5}-\beta _{3}q^{7}+\cdots$$
1305.2.f.j $4$ $10.420$ $$\Q(\sqrt{5}, \sqrt{-6})$$ None $$4$$ $$0$$ $$0$$ $$0$$ $$q+q^{2}-q^{4}-\beta _{2}q^{5}-\beta _{1}q^{7}-3q^{8}+\cdots$$
1305.2.f.k $12$ $10.420$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$6$$ $$0$$ $$q-\beta _{5}q^{2}+(1-\beta _{3})q^{4}+(1-\beta _{1})q^{5}+\cdots$$
1305.2.f.l $12$ $10.420$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$6$$ $$0$$ $$q+\beta _{5}q^{2}+(1-\beta _{3})q^{4}+(1-\beta _{1})q^{5}+\cdots$$
1305.2.f.m $16$ $10.420$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(2+\beta _{3})q^{4}-\beta _{14}q^{5}+\beta _{5}q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1305, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1305, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(145, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(435, [\chi])$$$$^{\oplus 2}$$