# Properties

 Label 1728.3.e Level $1728$ Weight $3$ Character orbit 1728.e Rep. character $\chi_{1728}(1025,\cdot)$ Character field $\Q$ Dimension $64$ Newform subspaces $22$ Sturm bound $864$ Trace bound $13$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.e (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$3$$ Character field: $$\Q$$ Newform subspaces: $$22$$ Sturm bound: $$864$$ Trace bound: $$13$$ Distinguishing $$T_p$$: $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(1728, [\chi])$$.

Total New Old
Modular forms 612 64 548
Cusp forms 540 64 476
Eisenstein series 72 0 72

## Trace form

 $$64 q + O(q^{10})$$ $$64 q - 16 q^{13} - 320 q^{25} + 80 q^{37} + 448 q^{49} - 80 q^{61} - 160 q^{85} + 32 q^{97} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(1728, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1728.3.e.a $1$ $47.085$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-13$$ $$q-13q^{7}+q^{13}-11q^{19}+5^{2}q^{25}+\cdots$$
1728.3.e.b $1$ $47.085$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-11$$ $$q-11q^{7}-23q^{13}-37q^{19}+5^{2}q^{25}+\cdots$$
1728.3.e.c $1$ $47.085$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$11$$ $$q+11q^{7}-23q^{13}+37q^{19}+5^{2}q^{25}+\cdots$$
1728.3.e.d $1$ $47.085$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$13$$ $$q+13q^{7}+q^{13}+11q^{19}+5^{2}q^{25}+\cdots$$
1728.3.e.e $2$ $47.085$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-14$$ $$q+iq^{5}-7q^{7}+iq^{11}-14q^{13}-2iq^{17}+\cdots$$
1728.3.e.f $2$ $47.085$ $$\Q(\sqrt{-2})$$ None $$0$$ $$0$$ $$0$$ $$-10$$ $$q+\beta q^{5}-5q^{7}+\beta q^{11}+q^{13}+3\beta q^{17}+\cdots$$
1728.3.e.g $2$ $47.085$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-10$$ $$q+iq^{5}-5q^{7}-5iq^{11}+10q^{13}+\cdots$$
1728.3.e.h $2$ $47.085$ $$\Q(\sqrt{-2})$$ None $$0$$ $$0$$ $$0$$ $$-6$$ $$q+\beta q^{5}-3q^{7}+7\beta q^{11}-7q^{13}-5\beta q^{17}+\cdots$$
1728.3.e.i $2$ $47.085$ $$\Q(\sqrt{-2})$$ None $$0$$ $$0$$ $$0$$ $$-6$$ $$q+\beta q^{5}-3q^{7}-\beta q^{11}+17q^{13}-5\beta q^{17}+\cdots$$
1728.3.e.j $2$ $47.085$ $$\Q(\sqrt{-2})$$ None $$0$$ $$0$$ $$0$$ $$6$$ $$q+\beta q^{5}+3q^{7}-7\beta q^{11}-7q^{13}-5\beta q^{17}+\cdots$$
1728.3.e.k $2$ $47.085$ $$\Q(\sqrt{-2})$$ None $$0$$ $$0$$ $$0$$ $$6$$ $$q+\beta q^{5}+3q^{7}+\beta q^{11}+17q^{13}-5\beta q^{17}+\cdots$$
1728.3.e.l $2$ $47.085$ $$\Q(\sqrt{-2})$$ None $$0$$ $$0$$ $$0$$ $$10$$ $$q+\beta q^{5}+5q^{7}-\beta q^{11}+q^{13}+3\beta q^{17}+\cdots$$
1728.3.e.m $2$ $47.085$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$10$$ $$q+iq^{5}+5q^{7}+5iq^{11}+10q^{13}+\cdots$$
1728.3.e.n $2$ $47.085$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$14$$ $$q+iq^{5}+7q^{7}-iq^{11}-14q^{13}-2iq^{17}+\cdots$$
1728.3.e.o $4$ $47.085$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$-12$$ $$q+(\zeta_{8}+\zeta_{8}^{2})q^{5}+(-3-\zeta_{8}^{3})q^{7}+(-\zeta_{8}+\cdots)q^{11}+\cdots$$
1728.3.e.p $4$ $47.085$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$-12$$ $$q+(-\zeta_{8}+\zeta_{8}^{2})q^{5}+(-3+\zeta_{8}^{3})q^{7}+\cdots$$
1728.3.e.q $4$ $47.085$ $$\Q(\sqrt{-2}, \sqrt{-5})$$ None $$0$$ $$0$$ $$0$$ $$-12$$ $$q-\beta _{3}q^{5}+(-3+\beta _{1})q^{7}+(-\beta _{2}-3\beta _{3})q^{11}+\cdots$$
1728.3.e.r $4$ $47.085$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$12$$ $$q+(\zeta_{8}+\zeta_{8}^{2})q^{5}+(3+\zeta_{8}^{3})q^{7}+(\zeta_{8}+\cdots)q^{11}+\cdots$$
1728.3.e.s $4$ $47.085$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$12$$ $$q+(-\zeta_{8}+\zeta_{8}^{2})q^{5}+(3-\zeta_{8}^{3})q^{7}+(-3\zeta_{8}+\cdots)q^{11}+\cdots$$
1728.3.e.t $4$ $47.085$ $$\Q(\sqrt{-2}, \sqrt{-5})$$ None $$0$$ $$0$$ $$0$$ $$12$$ $$q+\beta _{3}q^{5}+(3+\beta _{1})q^{7}+(\beta _{2}-3\beta _{3})q^{11}+\cdots$$
1728.3.e.u $8$ $47.085$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{24}^{2}q^{5}+\zeta_{24}^{4}q^{7}+\zeta_{24}^{6}q^{11}+\cdots$$
1728.3.e.v $8$ $47.085$ 8.0.2441150464.4 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{6}q^{5}-\beta _{4}q^{7}+(-\beta _{1}-\beta _{2})q^{11}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(1728, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(1728, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(12, [\chi])$$$$^{\oplus 15}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(24, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 7}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(48, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(54, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(96, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(108, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(144, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(192, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(216, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(288, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(432, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(576, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(864, [\chi])$$$$^{\oplus 2}$$