# Properties

 Label 4400.2.b Level $4400$ Weight $2$ Character orbit 4400.b Rep. character $\chi_{4400}(4049,\cdot)$ Character field $\Q$ Dimension $90$ Newform subspaces $29$ Sturm bound $1440$ Trace bound $19$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$29$$ Sturm bound: $$1440$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$3$$, $$7$$, $$13$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(4400, [\chi])$$.

Total New Old
Modular forms 756 90 666
Cusp forms 684 90 594
Eisenstein series 72 0 72

## Trace form

 $$90 q - 90 q^{9} + O(q^{10})$$ $$90 q - 90 q^{9} + 6 q^{11} - 16 q^{19} + 16 q^{21} - 12 q^{29} + 4 q^{31} + 4 q^{41} - 90 q^{49} + 16 q^{51} - 52 q^{59} - 52 q^{61} + 16 q^{69} - 12 q^{71} - 40 q^{79} + 42 q^{81} - 4 q^{89} + 64 q^{91} - 18 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(4400, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
4400.2.b.a $2$ $35.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}-iq^{7}-6q^{9}+q^{11}+6iq^{13}+\cdots$$
4400.2.b.b $2$ $35.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}-2iq^{7}-6q^{9}+q^{11}+6iq^{17}+\cdots$$
4400.2.b.c $2$ $35.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-q^{9}-q^{11}-2iq^{17}-4q^{19}+\cdots$$
4400.2.b.d $2$ $35.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{3}-q^{9}-q^{11}-3iq^{13}-4iq^{17}+\cdots$$
4400.2.b.e $2$ $35.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{3}-4iq^{7}-q^{9}+q^{11}+5iq^{13}+\cdots$$
4400.2.b.f $2$ $35.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-2iq^{7}-q^{9}+q^{11}-2iq^{13}+\cdots$$
4400.2.b.g $2$ $35.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-5iq^{7}+2q^{9}-q^{11}-2iq^{13}+\cdots$$
4400.2.b.h $2$ $35.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-2iq^{7}+2q^{9}-q^{11}+4iq^{13}+\cdots$$
4400.2.b.i $2$ $35.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+3iq^{7}+2q^{9}-q^{11}-6iq^{13}+\cdots$$
4400.2.b.j $2$ $35.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+iq^{7}+2q^{9}+q^{11}-2iq^{13}+\cdots$$
4400.2.b.k $2$ $35.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-2iq^{7}+2q^{9}+q^{11}+4iq^{13}+\cdots$$
4400.2.b.l $2$ $35.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{7}+3q^{9}-q^{11}-2iq^{13}+2iq^{17}+\cdots$$
4400.2.b.m $2$ $35.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{7}+3q^{9}+q^{11}-8q^{19}+4iq^{23}+\cdots$$
4400.2.b.n $2$ $35.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3q^{9}+q^{11}-iq^{13}+3iq^{17}-4q^{19}+\cdots$$
4400.2.b.o $2$ $35.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-2iq^{7}+3q^{9}+q^{11}-3iq^{13}-3iq^{17}+\cdots$$
4400.2.b.p $4$ $35.134$ $$\Q(i, \sqrt{33})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+\beta _{1}q^{7}+(-6+\beta _{3})q^{9}+q^{11}+\cdots$$
4400.2.b.q $4$ $35.134$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{2}q^{3}-\zeta_{8}q^{7}-5q^{9}-q^{11}+(-2\zeta_{8}+\cdots)q^{13}+\cdots$$
4400.2.b.r $4$ $35.134$ $$\Q(i, \sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{2})q^{3}+(-\beta _{1}+2\beta _{2})q^{7}+(-4+\cdots)q^{9}+\cdots$$
4400.2.b.s $4$ $35.134$ $$\Q(i, \sqrt{21})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(\beta _{1}+3\beta _{2})q^{7}+(-3+\beta _{3})q^{9}+\cdots$$
4400.2.b.t $4$ $35.134$ $$\Q(i, \sqrt{17})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}-\beta _{1}q^{7}+(-2+\beta _{3})q^{9}-q^{11}+\cdots$$
4400.2.b.u $4$ $35.134$ $$\Q(i, \sqrt{17})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(\beta _{1}+\beta _{2})q^{7}+(-2+\beta _{3})q^{9}+\cdots$$
4400.2.b.v $4$ $35.134$ $$\Q(i, \sqrt{17})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+2\beta _{1}q^{7}+(-2+\beta _{3})q^{9}+\cdots$$
4400.2.b.w $4$ $35.134$ $$\Q(i, \sqrt{17})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(-\beta _{1}+\beta _{2})q^{7}+(-2+\beta _{3})q^{9}+\cdots$$
4400.2.b.x $4$ $35.134$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{3})q^{3}+(3\beta _{1}+2\beta _{3})q^{7}+(1+\cdots)q^{9}+\cdots$$
4400.2.b.y $4$ $35.134$ $$\Q(i, \sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(\beta _{1}+3\beta _{2})q^{7}+(-1+\beta _{3})q^{9}+\cdots$$
4400.2.b.z $4$ $35.134$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(\beta _{1}-\beta _{3})q^{7}+(2+\beta _{2})q^{9}+\cdots$$
4400.2.b.ba $4$ $35.134$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots$$
4400.2.b.bb $6$ $35.134$ 6.0.96668224.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(-\beta _{2}-\beta _{4})q^{7}+(-2+\beta _{3}+\cdots)q^{9}+\cdots$$
4400.2.b.bc $6$ $35.134$ 6.0.44836416.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{3})q^{3}+(-\beta _{3}-\beta _{5})q^{7}+(-2+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(4400, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(4400, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(55, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(110, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(220, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(275, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(400, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(440, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(550, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(880, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1100, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(2200, [\chi])$$$$^{\oplus 2}$$