# Properties

 Label 6900.2.f Level $6900$ Weight $2$ Character orbit 6900.f Rep. character $\chi_{6900}(6349,\cdot)$ Character field $\Q$ Dimension $68$ Newform subspaces $19$ Sturm bound $2880$ Trace bound $21$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6900.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$19$$ Sturm bound: $$2880$$ Trace bound: $$21$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(6900, [\chi])$$.

Total New Old
Modular forms 1476 68 1408
Cusp forms 1404 68 1336
Eisenstein series 72 0 72

## Trace form

 $$68 q - 68 q^{9} + O(q^{10})$$ $$68 q - 68 q^{9} + 4 q^{19} - 4 q^{21} - 32 q^{29} + 12 q^{31} + 4 q^{39} + 8 q^{41} - 96 q^{49} + 16 q^{59} + 4 q^{61} - 8 q^{69} - 40 q^{71} + 16 q^{79} + 68 q^{81} + 32 q^{89} - 20 q^{91} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(6900, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
6900.2.f.a $2$ $55.097$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{3}+3iq^{7}-q^{9}-5q^{11}-iq^{13}+\cdots$$
6900.2.f.b $2$ $55.097$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+3iq^{7}-q^{9}-2q^{11}-2iq^{13}+\cdots$$
6900.2.f.c $2$ $55.097$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-q^{9}+6iq^{13}+2iq^{17}-6q^{19}+\cdots$$
6900.2.f.d $2$ $55.097$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+4iq^{7}-q^{9}+2iq^{13}-6iq^{17}+\cdots$$
6900.2.f.e $2$ $55.097$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+iq^{7}-q^{9}-4iq^{13}+3iq^{17}+\cdots$$
6900.2.f.f $2$ $55.097$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{3}+5iq^{7}-q^{9}+4iq^{13}+3iq^{17}+\cdots$$
6900.2.f.g $2$ $55.097$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+3iq^{7}-q^{9}+q^{11}+iq^{13}+\cdots$$
6900.2.f.h $4$ $55.097$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{3}-\beta _{1}q^{7}-q^{9}+(-2+\beta _{3})q^{11}+\cdots$$
6900.2.f.i $4$ $55.097$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{3}+(\zeta_{12}-2\zeta_{12}^{2})q^{7}-q^{9}+\cdots$$
6900.2.f.j $4$ $55.097$ $$\Q(i, \sqrt{73})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}-\beta _{2}q^{7}-q^{9}+(-1+\beta _{3})q^{11}+\cdots$$
6900.2.f.k $4$ $55.097$ $$\Q(i, \sqrt{10})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(2\beta _{1}-\beta _{2})q^{7}-q^{9}-4\beta _{1}q^{13}+\cdots$$
6900.2.f.l $4$ $55.097$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}q^{3}-\zeta_{8}^{2}q^{7}-q^{9}-4\zeta_{8}^{3}q^{11}+\cdots$$
6900.2.f.m $4$ $55.097$ $$\Q(i, \sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}+(\beta _{1}+3\beta _{2})q^{7}-q^{9}+\beta _{3}q^{11}+\cdots$$
6900.2.f.n $4$ $55.097$ $$\Q(i, \sqrt{21})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}+(\beta _{1}-\beta _{2})q^{7}-q^{9}+\beta _{3}q^{11}+\cdots$$
6900.2.f.o $4$ $55.097$ $$\Q(i, \sqrt{15})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{3}+3\beta _{1}q^{7}-q^{9}+(1+\beta _{2})q^{11}+\cdots$$
6900.2.f.p $4$ $55.097$ $$\Q(i, \sqrt{17})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}+\beta _{1}q^{7}-q^{9}+(2-2\beta _{3})q^{11}+\cdots$$
6900.2.f.q $4$ $55.097$ $$\Q(i, \sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}+(\beta _{1}+3\beta _{2})q^{7}-q^{9}+(2+\beta _{3})q^{11}+\cdots$$
6900.2.f.r $6$ $55.097$ 6.0.158155776.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+(-\beta _{2}-\beta _{5})q^{7}-q^{9}+(1+\cdots)q^{11}+\cdots$$
6900.2.f.s $8$ $55.097$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{4}q^{3}+(-\beta _{4}+\beta _{5})q^{7}-q^{9}+\beta _{2}q^{11}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(6900, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(6900, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(115, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(230, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(345, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(460, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(575, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(690, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1150, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1380, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1725, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(2300, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(3450, [\chi])$$$$^{\oplus 2}$$