# Properties

 Label 363.2.e Level $363$ Weight $2$ Character orbit 363.e Rep. character $\chi_{363}(124,\cdot)$ Character field $\Q(\zeta_{5})$ Dimension $72$ Newform subspaces $14$ Sturm bound $88$ Trace bound $4$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 363.e (of order $$5$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$11$$ Character field: $$\Q(\zeta_{5})$$ Newform subspaces: $$14$$ Sturm bound: $$88$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(363, [\chi])$$.

Total New Old
Modular forms 224 72 152
Cusp forms 128 72 56
Eisenstein series 96 0 96

## Trace form

 $$72q + 4q^{2} - 14q^{4} + 4q^{5} + 2q^{6} + 2q^{7} - 8q^{8} - 18q^{9} + O(q^{10})$$ $$72q + 4q^{2} - 14q^{4} + 4q^{5} + 2q^{6} + 2q^{7} - 8q^{8} - 18q^{9} - 4q^{10} + 8q^{12} + 2q^{13} + 6q^{14} - 2q^{15} - 22q^{16} - 14q^{17} - 6q^{18} + 20q^{19} + 2q^{20} - 16q^{21} - 4q^{23} - 12q^{24} - 20q^{25} + 2q^{26} + 12q^{28} + 4q^{29} + 8q^{31} + 48q^{32} - 56q^{34} - 14q^{36} - 2q^{37} + 14q^{38} + 16q^{39} - 4q^{40} - 20q^{41} + 14q^{42} - 16q^{43} + 4q^{45} - 10q^{46} - 6q^{47} - 8q^{48} - 16q^{49} - 18q^{50} - 10q^{52} - 6q^{53} - 8q^{54} - 108q^{56} + 44q^{58} + 46q^{59} + 12q^{60} + 18q^{61} + 28q^{62} + 2q^{63} + 2q^{64} + 28q^{65} - 44q^{67} + 28q^{68} + 10q^{69} + 34q^{70} + 32q^{71} + 2q^{72} - 20q^{73} + 30q^{74} - 40q^{76} - 64q^{78} + 6q^{79} + 14q^{80} - 18q^{81} + 46q^{82} - 34q^{83} + 2q^{85} + 24q^{86} - 24q^{87} - 60q^{89} - 4q^{90} - 18q^{91} + 22q^{92} - 56q^{93} - 18q^{94} - 30q^{95} + 8q^{96} + 10q^{97} - 40q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(363, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
363.2.e.a $$4$$ $$2.899$$ $$\Q(\zeta_{10})$$ None $$-5$$ $$-1$$ $$-2$$ $$-10$$ $$q+(-1-\zeta_{10}+\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots$$
363.2.e.b $$4$$ $$2.899$$ $$\Q(\zeta_{10})$$ None $$-4$$ $$1$$ $$2$$ $$-1$$ $$q+(-2+\zeta_{10}-\zeta_{10}^{2}+2\zeta_{10}^{3})q^{2}+\cdots$$
363.2.e.c $$4$$ $$2.899$$ $$\Q(\zeta_{10})$$ None $$-2$$ $$-1$$ $$4$$ $$3$$ $$q+(-\zeta_{10}+\zeta_{10}^{2})q^{2}+\zeta_{10}^{2}q^{3}+(-1+\cdots)q^{4}+\cdots$$
363.2.e.d $$4$$ $$2.899$$ $$\Q(\zeta_{10})$$ None $$-2$$ $$1$$ $$-4$$ $$1$$ $$q+(-2+2\zeta_{10}-2\zeta_{10}^{2}+2\zeta_{10}^{3})q^{2}+\cdots$$
363.2.e.e $$4$$ $$2.899$$ $$\Q(\zeta_{10})$$ None $$-1$$ $$1$$ $$2$$ $$-4$$ $$q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots$$
363.2.e.f $$4$$ $$2.899$$ $$\Q(\zeta_{10})$$ None $$1$$ $$1$$ $$-3$$ $$-1$$ $$q+(-1+2\zeta_{10}-2\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots$$
363.2.e.g $$4$$ $$2.899$$ $$\Q(\zeta_{10})$$ None $$1$$ $$1$$ $$2$$ $$4$$ $$q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}-\zeta_{10}^{2}q^{3}+\cdots$$
363.2.e.h $$4$$ $$2.899$$ $$\Q(\zeta_{10})$$ None $$2$$ $$-1$$ $$4$$ $$-3$$ $$q+(\zeta_{10}-\zeta_{10}^{2})q^{2}+\zeta_{10}^{2}q^{3}+(-1+\cdots)q^{4}+\cdots$$
363.2.e.i $$4$$ $$2.899$$ $$\Q(\zeta_{10})$$ None $$2$$ $$1$$ $$-4$$ $$-1$$ $$q+(2-2\zeta_{10}+2\zeta_{10}^{2}-2\zeta_{10}^{3})q^{2}+\cdots$$
363.2.e.j $$4$$ $$2.899$$ $$\Q(\zeta_{10})$$ None $$3$$ $$-1$$ $$-1$$ $$3$$ $$q+(1-\zeta_{10}^{3})q^{2}+\zeta_{10}^{2}q^{3}+(1-\zeta_{10}+\cdots)q^{4}+\cdots$$
363.2.e.k $$4$$ $$2.899$$ $$\Q(\zeta_{10})$$ None $$4$$ $$1$$ $$2$$ $$1$$ $$q+(2-\zeta_{10}+\zeta_{10}^{2}-2\zeta_{10}^{3})q^{2}-\zeta_{10}^{2}q^{3}+\cdots$$
363.2.e.l $$4$$ $$2.899$$ $$\Q(\zeta_{10})$$ None $$5$$ $$-1$$ $$-2$$ $$10$$ $$q+(1+\zeta_{10}-\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+\zeta_{10}^{2}q^{3}+\cdots$$
363.2.e.m $$8$$ $$2.899$$ 8.0.324000000.3 None $$0$$ $$2$$ $$6$$ $$0$$ $$q+\beta _{7}q^{2}-\beta _{6}q^{3}+\beta _{4}q^{4}+(3+3\beta _{2}+\cdots)q^{5}+\cdots$$
363.2.e.n $$16$$ $$2.899$$ 16.0.$$\cdots$$.3 None $$0$$ $$-4$$ $$-2$$ $$0$$ $$q-\beta _{12}q^{2}+\beta _{4}q^{3}+(\beta _{3}-\beta _{6}+\beta _{8}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(363, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(363, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(33, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(121, [\chi])$$$$^{\oplus 2}$$