# Properties

 Label 1296.3.e Level $1296$ Weight $3$ Character orbit 1296.e Rep. character $\chi_{1296}(161,\cdot)$ Character field $\Q$ Dimension $46$ Newform subspaces $10$ Sturm bound $648$ Trace bound $13$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1296.e (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$3$$ Character field: $$\Q$$ Newform subspaces: $$10$$ Sturm bound: $$648$$ Trace bound: $$13$$ Distinguishing $$T_p$$: $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(1296, [\chi])$$.

Total New Old
Modular forms 468 50 418
Cusp forms 396 46 350
Eisenstein series 72 4 68

## Trace form

 $$46 q - 2 q^{7} + O(q^{10})$$ $$46 q - 2 q^{7} + 2 q^{13} + 4 q^{19} - 188 q^{25} - 50 q^{31} + 20 q^{37} + 94 q^{43} + 240 q^{49} + 54 q^{55} - 118 q^{61} - 98 q^{67} - 28 q^{73} + 190 q^{79} + 120 q^{85} - 286 q^{91} + 122 q^{97} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(1296, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1296.3.e.a $2$ $35.313$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-2\zeta_{6}q^{5}-2q^{7}+\zeta_{6}q^{11}-4q^{13}+\cdots$$
1296.3.e.b $4$ $35.313$ $$\Q(\sqrt{-2}, \sqrt{3})$$ None $$0$$ $$0$$ $$0$$ $$-12$$ $$q+\beta _{1}q^{5}+(-3+\beta _{2})q^{7}+(-5\beta _{1}-\beta _{3})q^{11}+\cdots$$
1296.3.e.c $4$ $35.313$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-12$$ $$q+(\beta _{1}-\beta _{2})q^{5}+(-3-\beta _{3})q^{7}+(2\beta _{1}+\cdots)q^{11}+\cdots$$
1296.3.e.d $4$ $35.313$ $$\Q(\sqrt{-2}, \sqrt{3})$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+\beta _{1}q^{5}+(-2-2\beta _{3})q^{7}+(-2\beta _{1}+\cdots)q^{11}+\cdots$$
1296.3.e.e $4$ $35.313$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$0$$ $$0$$ $$0$$ $$-2$$ $$q+\beta _{1}q^{5}+(-1+\beta _{2})q^{7}+(\beta _{1}-\beta _{3})q^{11}+\cdots$$
1296.3.e.f $4$ $35.313$ $$\Q(\sqrt{-2}, \sqrt{3})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q+\beta _{1}q^{5}+(1-\beta _{2})q^{7}+\beta _{3}q^{11}+(-1+\cdots)q^{13}+\cdots$$
1296.3.e.g $4$ $35.313$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q+\beta _{3}q^{5}+(1+\beta _{2})q^{7}+(\beta _{1}-\beta _{3})q^{11}+\cdots$$
1296.3.e.h $4$ $35.313$ $$\Q(\sqrt{-2}, \sqrt{3})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q-\beta _{1}q^{5}+(1+\beta _{3})q^{7}+(-\beta _{1}-3\beta _{2}+\cdots)q^{11}+\cdots$$
1296.3.e.i $8$ $35.313$ 8.0.$$\cdots$$.9 None $$0$$ $$0$$ $$0$$ $$12$$ $$q+(\beta _{1}+\beta _{3})q^{5}+(2-\beta _{5})q^{7}+(-2\beta _{1}+\cdots)q^{11}+\cdots$$
1296.3.e.j $8$ $35.313$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$12$$ $$q+(-\beta _{1}+\beta _{3})q^{5}+(2+\beta _{5})q^{7}+(\beta _{1}+\cdots)q^{11}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(1296, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(1296, [\chi]) \simeq$$ $$S_{3}^{\mathrm{new}}(12, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(24, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(48, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(54, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(81, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(108, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(144, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(162, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(216, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(324, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(432, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(648, [\chi])$$$$^{\oplus 2}$$