# Properties

 Label 5700.2.f Level $5700$ Weight $2$ Character orbit 5700.f Rep. character $\chi_{5700}(3649,\cdot)$ Character field $\Q$ Dimension $56$ Newform subspaces $18$ Sturm bound $2400$ Trace bound $21$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5700.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$18$$ Sturm bound: $$2400$$ Trace bound: $$21$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$13$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(5700, [\chi])$$.

Total New Old
Modular forms 1236 56 1180
Cusp forms 1164 56 1108
Eisenstein series 72 0 72

## Trace form

 $$56 q - 56 q^{9} + O(q^{10})$$ $$56 q - 56 q^{9} - 4 q^{11} - 4 q^{19} - 40 q^{29} + 8 q^{31} + 16 q^{39} + 16 q^{41} - 108 q^{49} + 16 q^{51} - 32 q^{59} + 76 q^{61} - 16 q^{69} + 32 q^{71} - 8 q^{79} + 56 q^{81} - 8 q^{89} + 4 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(5700, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
5700.2.f.a $2$ $45.515$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+iq^{7}-q^{9}-5q^{11}+6iq^{13}+\cdots$$
5700.2.f.b $2$ $45.515$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+iq^{7}-q^{9}-4q^{11}-4iq^{17}+\cdots$$
5700.2.f.c $2$ $45.515$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+iq^{7}-q^{9}-4q^{11}+6iq^{17}+\cdots$$
5700.2.f.d $2$ $45.515$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+3iq^{7}-q^{9}-2q^{11}-2iq^{13}+\cdots$$
5700.2.f.e $2$ $45.515$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+iq^{7}-q^{9}-4iq^{13}-q^{19}+\cdots$$
5700.2.f.f $2$ $45.515$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{3}+2iq^{7}-q^{9}+4iq^{13}+6iq^{17}+\cdots$$
5700.2.f.g $2$ $45.515$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+2iq^{7}-q^{9}-4iq^{13}+2iq^{17}+\cdots$$
5700.2.f.h $2$ $45.515$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{3}+iq^{7}-q^{9}-2iq^{17}+q^{19}+\cdots$$
5700.2.f.i $2$ $45.515$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+4iq^{7}-q^{9}+2q^{11}+6iq^{13}+\cdots$$
5700.2.f.j $2$ $45.515$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+5iq^{7}-q^{9}+2q^{11}-2iq^{13}+\cdots$$
5700.2.f.k $2$ $45.515$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{3}-q^{9}+2q^{11}+2iq^{13}-6iq^{17}+\cdots$$
5700.2.f.l $2$ $45.515$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{3}+2iq^{7}-q^{9}+4q^{11}+2iq^{17}+\cdots$$
5700.2.f.m $4$ $45.515$ $$\Q(i, \sqrt{33})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+\beta _{1}q^{7}-q^{9}+(-1-\beta _{3})q^{11}+\cdots$$
5700.2.f.n $4$ $45.515$ $$\Q(i, \sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{3}+(-\beta _{1}+\beta _{2})q^{7}-q^{9}+(-1+\cdots)q^{11}+\cdots$$
5700.2.f.o $4$ $45.515$ $$\Q(i, \sqrt{10})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{3}+4\beta _{1}q^{7}-q^{9}+(1+\beta _{3})q^{11}+\cdots$$
5700.2.f.p $6$ $45.515$ 6.0.37161216.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+\beta _{5}q^{7}-q^{9}+(-1-\beta _{3}+\cdots)q^{11}+\cdots$$
5700.2.f.q $6$ $45.515$ 6.0.5089536.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{4}q^{3}+(\beta _{1}-\beta _{4})q^{7}-q^{9}+(2-\beta _{2}+\cdots)q^{11}+\cdots$$
5700.2.f.r $8$ $45.515$ 8.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}+(-\beta _{3}+\beta _{5})q^{7}-q^{9}+(-\beta _{2}+\cdots)q^{11}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(5700, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(5700, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(95, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(190, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(285, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(380, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(475, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(570, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(950, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1140, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1425, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1900, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(2850, [\chi])$$$$^{\oplus 2}$$