# 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

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

## 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

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