# Properties

 Label 3600.2.f Level $3600$ Weight $2$ Character orbit 3600.f Rep. character $\chi_{3600}(2449,\cdot)$ Character field $\Q$ Dimension $44$ Newform subspaces $22$ Sturm bound $1440$ Trace bound $29$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3600.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$22$$ Sturm bound: $$1440$$ Trace bound: $$29$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$13$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(3600, [\chi])$$.

Total New Old
Modular forms 792 46 746
Cusp forms 648 44 604
Eisenstein series 144 2 142

## Trace form

 $$44 q + O(q^{10})$$ $$44 q + 4 q^{11} - 4 q^{19} - 12 q^{29} + 16 q^{31} - 8 q^{41} - 20 q^{49} - 16 q^{61} - 56 q^{71} - 32 q^{79} + 24 q^{89} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(3600, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
3600.2.f.a $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{7}-6q^{11}-2iq^{13}+3iq^{17}+\cdots$$
3600.2.f.b $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+5iq^{7}-6q^{11}+3iq^{13}+2iq^{17}+\cdots$$
3600.2.f.c $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4q^{11}-3iq^{13}+3iq^{17}-4q^{19}+\cdots$$
3600.2.f.d $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{7}-4q^{11}-iq^{13}-4iq^{17}+\cdots$$
3600.2.f.e $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4q^{11}+iq^{13}-iq^{17}+4q^{19}+\cdots$$
3600.2.f.f $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{7}-3q^{11}-4iq^{13}-3iq^{17}+\cdots$$
3600.2.f.g $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{7}-2q^{11}+2iq^{13}+iq^{17}+\cdots$$
3600.2.f.h $2$ $28.746$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{7}+7iq^{13}-7q^{19}-11q^{31}+\cdots$$
3600.2.f.i $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{7}-iq^{13}-3iq^{17}-4q^{19}+\cdots$$
3600.2.f.j $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{7}+iq^{13}-3iq^{17}-4q^{19}+\cdots$$
3600.2.f.k $2$ $28.746$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{7}+iq^{13}-q^{19}+7q^{31}+2iq^{37}+\cdots$$
3600.2.f.l $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{7}-3iq^{13}-iq^{17}+4q^{19}+\cdots$$
3600.2.f.m $2$ $28.746$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{7}-iq^{13}+8q^{19}+4q^{31}+\cdots$$
3600.2.f.n $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{7}+q^{11}+4iq^{13}+5iq^{17}+\cdots$$
3600.2.f.o $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{7}+2q^{11}-3iq^{13}+6iq^{17}+\cdots$$
3600.2.f.p $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{7}+2q^{11}-iq^{13}-2iq^{17}+\cdots$$
3600.2.f.q $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{7}+2q^{11}+2iq^{13}-iq^{17}+\cdots$$
3600.2.f.r $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4q^{11}-iq^{13}+iq^{17}-4q^{19}+\cdots$$
3600.2.f.s $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{7}+4q^{11}-iq^{13}+4iq^{17}+\cdots$$
3600.2.f.t $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{7}+4q^{11}+iq^{13}-iq^{17}+\cdots$$
3600.2.f.u $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{7}+6q^{11}-2iq^{13}-3iq^{17}+\cdots$$
3600.2.f.v $2$ $28.746$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{7}+6q^{11}-5iq^{13}+6iq^{17}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(3600, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(3600, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(180, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(240, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(360, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(400, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(450, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(600, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(720, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(900, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1200, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1800, [\chi])$$$$^{\oplus 2}$$