# Properties

 Label 3330.2.d Level $3330$ Weight $2$ Character orbit 3330.d Rep. character $\chi_{3330}(1999,\cdot)$ Character field $\Q$ Dimension $90$ Newform subspaces $18$ Sturm bound $1368$ Trace bound $14$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3330.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$18$$ Sturm bound: $$1368$$ Trace bound: $$14$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$17$$, $$29$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(3330, [\chi])$$.

Total New Old
Modular forms 700 90 610
Cusp forms 668 90 578
Eisenstein series 32 0 32

## Trace form

 $$90 q - 90 q^{4} - 4 q^{5} + O(q^{10})$$ $$90 q - 90 q^{4} - 4 q^{5} + 4 q^{10} + 16 q^{11} + 4 q^{14} + 90 q^{16} + 12 q^{19} + 4 q^{20} - 8 q^{25} - 16 q^{29} - 8 q^{31} - 4 q^{40} + 4 q^{41} - 16 q^{44} - 12 q^{46} - 78 q^{49} - 8 q^{50} + 20 q^{55} - 4 q^{56} - 4 q^{59} - 90 q^{64} - 24 q^{65} - 8 q^{71} + 10 q^{74} - 12 q^{76} + 32 q^{79} - 4 q^{80} - 12 q^{85} + 52 q^{86} + 60 q^{89} - 112 q^{91} + 28 q^{94} - 4 q^{95} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(3330, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
3330.2.d.a $2$ $26.590$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+iq^{2}-q^{4}+(-2+i)q^{5}+2iq^{7}+\cdots$$
3330.2.d.b $2$ $26.590$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q-iq^{2}-q^{4}+(-2+i)q^{5}+2iq^{7}+\cdots$$
3330.2.d.c $2$ $26.590$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q-iq^{2}-q^{4}+(-1-2i)q^{5}+iq^{7}+\cdots$$
3330.2.d.d $2$ $26.590$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q-iq^{2}-q^{4}+(-1+2i)q^{5}+iq^{7}+\cdots$$
3330.2.d.e $2$ $26.590$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+iq^{2}-q^{4}+(1-2i)q^{5}+5iq^{7}+\cdots$$
3330.2.d.f $2$ $26.590$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+iq^{2}-q^{4}+(2-i)q^{5}+2iq^{7}-iq^{8}+\cdots$$
3330.2.d.g $2$ $26.590$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q-iq^{2}-q^{4}+(2+i)q^{5}+2iq^{7}+iq^{8}+\cdots$$
3330.2.d.h $4$ $26.590$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q-\beta _{2}q^{2}-q^{4}+(-1-\beta _{2}-\beta _{3})q^{5}+\cdots$$
3330.2.d.i $4$ $26.590$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+\zeta_{12}q^{2}-q^{4}+(-1-2\zeta_{12})q^{5}+\cdots$$
3330.2.d.j $4$ $26.590$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-q^{4}-\beta _{2}q^{5}-2\beta _{1}q^{7}-\beta _{1}q^{8}+\cdots$$
3330.2.d.k $4$ $26.590$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{8}^{2}q^{2}-q^{4}+(\zeta_{8}+2\zeta_{8}^{3})q^{5}+(\zeta_{8}+\cdots)q^{7}+\cdots$$
3330.2.d.l $4$ $26.590$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q-\zeta_{12}q^{2}-q^{4}+(1+2\zeta_{12})q^{5}+(\zeta_{12}+\cdots)q^{7}+\cdots$$
3330.2.d.m $4$ $26.590$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$8$$ $$0$$ $$q-\beta _{1}q^{2}-q^{4}+(2-\beta _{1})q^{5}+(2\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots$$
3330.2.d.n $6$ $26.590$ 6.0.5161984.1 None $$0$$ $$0$$ $$-2$$ $$0$$ $$q+\beta _{4}q^{2}-q^{4}+(\beta _{1}+\beta _{3})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots$$
3330.2.d.o $8$ $26.590$ 8.0.$$\cdots$$.2 None $$0$$ $$0$$ $$2$$ $$0$$ $$q+\beta _{3}q^{2}-q^{4}+(-\beta _{1}-\beta _{5})q^{5}+(-\beta _{2}+\cdots)q^{7}+\cdots$$
3330.2.d.p $10$ $26.590$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$0$$ $$0$$ $$-6$$ $$0$$ $$q+\beta _{2}q^{2}-q^{4}+(-1+\beta _{1}+\beta _{3}-\beta _{4}+\cdots)q^{5}+\cdots$$
3330.2.d.q $14$ $26.590$ $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+\beta _{6}q^{2}-q^{4}+\beta _{9}q^{5}+(\beta _{6}-\beta _{7})q^{7}+\cdots$$
3330.2.d.r $14$ $26.590$ $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q-\beta _{6}q^{2}-q^{4}-\beta _{9}q^{5}+(\beta _{6}-\beta _{7})q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(3330, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(3330, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(185, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(370, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(555, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1110, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1665, [\chi])$$$$^{\oplus 2}$$