# Properties

 Label 405.4.e Level $405$ Weight $4$ Character orbit 405.e Rep. character $\chi_{405}(136,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $96$ Newform subspaces $24$ Sturm bound $216$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 405.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$24$$ Sturm bound: $$216$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$2$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(405, [\chi])$$.

Total New Old
Modular forms 348 96 252
Cusp forms 300 96 204
Eisenstein series 48 0 48

## Trace form

 $$96q - 192q^{4} - 60q^{7} + O(q^{10})$$ $$96q - 192q^{4} - 60q^{7} + 120q^{13} - 768q^{16} - 960q^{19} + 72q^{22} - 1200q^{25} + 1920q^{28} - 600q^{31} - 1350q^{34} - 1680q^{37} - 450q^{40} + 156q^{43} - 1188q^{46} - 756q^{49} + 3756q^{52} + 1188q^{58} + 84q^{61} + 1644q^{64} - 1464q^{67} - 360q^{70} - 5064q^{73} + 9750q^{76} + 3036q^{79} + 6984q^{82} + 720q^{85} + 864q^{88} - 14088q^{91} - 3924q^{94} + 2532q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(405, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
405.4.e.a $$2$$ $$23.896$$ $$\Q(\sqrt{-3})$$ None $$-5$$ $$0$$ $$5$$ $$-9$$ $$q+(-5+5\zeta_{6})q^{2}-17\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots$$
405.4.e.b $$2$$ $$23.896$$ $$\Q(\sqrt{-3})$$ None $$-5$$ $$0$$ $$5$$ $$30$$ $$q+(-5+5\zeta_{6})q^{2}-17\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots$$
405.4.e.c $$2$$ $$23.896$$ $$\Q(\sqrt{-3})$$ None $$-4$$ $$0$$ $$-5$$ $$-6$$ $$q+(-4+4\zeta_{6})q^{2}-8\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
405.4.e.d $$2$$ $$23.896$$ $$\Q(\sqrt{-3})$$ None $$-3$$ $$0$$ $$5$$ $$-20$$ $$q+(-3+3\zeta_{6})q^{2}-\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots$$
405.4.e.e $$2$$ $$23.896$$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$5$$ $$0$$ $$q+(-2+2\zeta_{6})q^{2}+4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots$$
405.4.e.f $$2$$ $$23.896$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-5$$ $$6$$ $$q+(-1+\zeta_{6})q^{2}+7\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
405.4.e.g $$2$$ $$23.896$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-5$$ $$24$$ $$q+(-1+\zeta_{6})q^{2}+7\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
405.4.e.h $$2$$ $$23.896$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$5$$ $$6$$ $$q+(1-\zeta_{6})q^{2}+7\zeta_{6}q^{4}+5\zeta_{6}q^{5}+(6+\cdots)q^{7}+\cdots$$
405.4.e.i $$2$$ $$23.896$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$5$$ $$24$$ $$q+(1-\zeta_{6})q^{2}+7\zeta_{6}q^{4}+5\zeta_{6}q^{5}+(24+\cdots)q^{7}+\cdots$$
405.4.e.j $$2$$ $$23.896$$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$-5$$ $$0$$ $$q+(2-2\zeta_{6})q^{2}+4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
405.4.e.k $$2$$ $$23.896$$ $$\Q(\sqrt{-3})$$ None $$3$$ $$0$$ $$-5$$ $$-20$$ $$q+(3-3\zeta_{6})q^{2}-\zeta_{6}q^{4}-5\zeta_{6}q^{5}+(-20+\cdots)q^{7}+\cdots$$
405.4.e.l $$2$$ $$23.896$$ $$\Q(\sqrt{-3})$$ None $$4$$ $$0$$ $$5$$ $$-6$$ $$q+(4-4\zeta_{6})q^{2}-8\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots$$
405.4.e.m $$2$$ $$23.896$$ $$\Q(\sqrt{-3})$$ None $$5$$ $$0$$ $$-5$$ $$-9$$ $$q+(5-5\zeta_{6})q^{2}-17\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
405.4.e.n $$2$$ $$23.896$$ $$\Q(\sqrt{-3})$$ None $$5$$ $$0$$ $$-5$$ $$30$$ $$q+(5-5\zeta_{6})q^{2}-17\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
405.4.e.o $$4$$ $$23.896$$ $$\Q(\zeta_{12})$$ None $$-2$$ $$0$$ $$-10$$ $$0$$ $$q+(-\zeta_{12}+\zeta_{12}^{2})q^{2}+(4-4\zeta_{12}-2\zeta_{12}^{2}+\cdots)q^{4}+\cdots$$
405.4.e.p $$4$$ $$23.896$$ $$\Q(\zeta_{12})$$ None $$2$$ $$0$$ $$10$$ $$0$$ $$q+(\zeta_{12}-\zeta_{12}^{2})q^{2}+(4-4\zeta_{12}-2\zeta_{12}^{2}+\cdots)q^{4}+\cdots$$
405.4.e.q $$6$$ $$23.896$$ 6.0.84779568.3 None $$-5$$ $$0$$ $$-15$$ $$4$$ $$q+(-\beta _{1}-2\beta _{3}+\beta _{5})q^{2}+(-7+7\beta _{3}+\cdots)q^{4}+\cdots$$
405.4.e.r $$6$$ $$23.896$$ 6.0.95327307.1 None $$-1$$ $$0$$ $$15$$ $$-44$$ $$q+\beta _{4}q^{2}+(-\beta _{1}-7\beta _{3}-\beta _{4}+\beta _{5})q^{4}+\cdots$$
405.4.e.s $$6$$ $$23.896$$ 6.0.148347072.2 None $$-1$$ $$0$$ $$15$$ $$25$$ $$q+(-\beta _{1}+\beta _{3})q^{2}+(-2-\beta _{1}-2\beta _{2}+\cdots)q^{4}+\cdots$$
405.4.e.t $$6$$ $$23.896$$ 6.0.95327307.1 None $$1$$ $$0$$ $$-15$$ $$-44$$ $$q+(\beta _{1}+\beta _{4})q^{2}+(-7-\beta _{2}+7\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots$$
405.4.e.u $$6$$ $$23.896$$ 6.0.148347072.2 None $$1$$ $$0$$ $$-15$$ $$25$$ $$q-\beta _{1}q^{2}+(\beta _{1}+2\beta _{2}-\beta _{3}+\beta _{4})q^{4}+\cdots$$
405.4.e.v $$6$$ $$23.896$$ 6.0.84779568.3 None $$5$$ $$0$$ $$15$$ $$4$$ $$q+(2-2\beta _{3}+\beta _{5})q^{2}+(-3\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots$$
405.4.e.w $$12$$ $$23.896$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$-4$$ $$0$$ $$-30$$ $$-40$$ $$q+(-\beta _{2}-\beta _{3})q^{2}+(-6+6\beta _{2}+\beta _{5}+\cdots)q^{4}+\cdots$$
405.4.e.x $$12$$ $$23.896$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$4$$ $$0$$ $$30$$ $$-40$$ $$q+(1-\beta _{2}-\beta _{9})q^{2}+(-6\beta _{2}-\beta _{3}-\beta _{5}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(405, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(405, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(81, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(135, [\chi])$$$$^{\oplus 2}$$