# Properties

 Label 3380.2.f Level $3380$ Weight $2$ Character orbit 3380.f Rep. character $\chi_{3380}(3041,\cdot)$ Character field $\Q$ Dimension $50$ Newform subspaces $10$ Sturm bound $1092$ Trace bound $23$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$3380 = 2^{2} \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3380.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q$$ Newform subspaces: $$10$$ Sturm bound: $$1092$$ Trace bound: $$23$$ Distinguishing $$T_p$$: $$3$$, $$19$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(3380, [\chi])$$.

Total New Old
Modular forms 588 50 538
Cusp forms 504 50 454
Eisenstein series 84 0 84

## Trace form

 $$50 q + 50 q^{9} + O(q^{10})$$ $$50 q + 50 q^{9} - 4 q^{17} + 12 q^{23} - 50 q^{25} - 12 q^{27} + 16 q^{29} - 12 q^{43} - 50 q^{49} + 4 q^{51} + 16 q^{53} - 16 q^{55} + 12 q^{61} - 4 q^{69} + 28 q^{77} + 28 q^{79} + 82 q^{81} + 44 q^{87} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(3380, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
3380.2.f.a $2$ $26.989$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-6$$ $$0$$ $$0$$ $$q-3q^{3}-iq^{5}+3iq^{7}+6q^{9}+3iq^{11}+\cdots$$
3380.2.f.b $2$ $26.989$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-4$$ $$0$$ $$0$$ $$q-2q^{3}+iq^{5}+2iq^{7}+q^{9}-2iq^{15}+\cdots$$
3380.2.f.c $2$ $26.989$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q-q^{3}+iq^{5}+5iq^{7}-2q^{9}-5iq^{11}+\cdots$$
3380.2.f.d $2$ $26.989$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+q^{3}-iq^{5}+iq^{7}-2q^{9}-3iq^{11}+\cdots$$
3380.2.f.e $2$ $26.989$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+q^{3}-iq^{5}-iq^{7}-2q^{9}+3iq^{11}+\cdots$$
3380.2.f.f $2$ $26.989$ $$\Q(\sqrt{-1})$$ None $$0$$ $$4$$ $$0$$ $$0$$ $$q+2q^{3}+iq^{5}+2iq^{7}+q^{9}+4iq^{11}+\cdots$$
3380.2.f.g $6$ $26.989$ 6.0.153664.1 None $$0$$ $$-2$$ $$0$$ $$0$$ $$q-\beta _{4}q^{3}+\beta _{5}q^{5}-\beta _{1}q^{7}+(-1-\beta _{2}+\cdots)q^{9}+\cdots$$
3380.2.f.h $6$ $26.989$ 6.0.5089536.1 None $$0$$ $$4$$ $$0$$ $$0$$ $$q+(1+\beta _{3})q^{3}+\beta _{4}q^{5}+(\beta _{4}+\beta _{5})q^{7}+\cdots$$
3380.2.f.i $8$ $26.989$ 8.0.22581504.2 None $$0$$ $$4$$ $$0$$ $$0$$ $$q+(\beta _{2}-\beta _{4})q^{3}-\beta _{5}q^{5}+(-\beta _{3}+\beta _{5}+\cdots)q^{7}+\cdots$$
3380.2.f.j $18$ $26.989$ $$\mathbb{Q}[x]/(x^{18} + \cdots)$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+\beta _{14}q^{5}+(-\beta _{7}+\beta _{10}-\beta _{16}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(3380, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(3380, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(26, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(65, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(130, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(169, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(260, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(338, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(676, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(845, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1690, [\chi])$$$$^{\oplus 2}$$