# Properties

 Label 363.2.f Level $363$ Weight $2$ Character orbit 363.f Rep. character $\chi_{363}(161,\cdot)$ Character field $\Q(\zeta_{10})$ Dimension $112$ Newform subspaces $9$ Sturm bound $88$ Trace bound $6$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 363.f (of order $$10$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$33$$ Character field: $$\Q(\zeta_{10})$$ Newform subspaces: $$9$$ Sturm bound: $$88$$ Trace bound: $$6$$ Distinguishing $$T_p$$: $$2$$, $$5$$, $$7$$

## Dimensions

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

Total New Old
Modular forms 224 176 48
Cusp forms 128 112 16
Eisenstein series 96 64 32

## Trace form

 $$112 q + 7 q^{3} - 14 q^{4} + 10 q^{7} - 11 q^{9} + O(q^{10})$$ $$112 q + 7 q^{3} - 14 q^{4} + 10 q^{7} - 11 q^{9} - 68 q^{12} + 10 q^{13} + 9 q^{15} - 2 q^{16} - 20 q^{19} + 10 q^{24} - 28 q^{25} + 16 q^{27} + 20 q^{30} + 26 q^{31} - 24 q^{34} + 16 q^{36} + 8 q^{37} - 20 q^{39} - 60 q^{42} - 100 q^{45} - 30 q^{46} - 6 q^{48} - 50 q^{49} - 30 q^{51} - 10 q^{52} + 30 q^{57} - 28 q^{58} + 44 q^{60} + 10 q^{61} + 30 q^{63} + 86 q^{64} - 68 q^{67} + 11 q^{69} + 34 q^{70} + 20 q^{72} + 24 q^{75} - 12 q^{78} - 50 q^{79} + 5 q^{81} + 6 q^{82} - 10 q^{85} - 40 q^{90} + 54 q^{91} - 13 q^{93} + 30 q^{94} - 10 q^{96} - 52 q^{97} + O(q^{100})$$

## 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.f.a $8$ $2.899$ 8.0.64000000.1 None $$-2$$ $$-2$$ $$0$$ $$0$$ $$q-\beta _{6}q^{2}+(-1+\beta _{2}-\beta _{3}-\beta _{4}+\beta _{6}+\cdots)q^{3}+\cdots$$
363.2.f.b $8$ $2.899$ $$\Q(\zeta_{20})$$ None $$0$$ $$-6$$ $$0$$ $$10$$ $$q+(-\zeta_{20}^{5}-\zeta_{20}^{7})q^{2}+(-1-\zeta_{20}+\cdots)q^{3}+\cdots$$
363.2.f.c $8$ $2.899$ 8.0.228765625.1 $$\Q(\sqrt{-11})$$ $$0$$ $$-1$$ $$0$$ $$0$$ $$q-\beta _{1}q^{3}-2\beta _{7}q^{4}+(-\beta _{4}-2\beta _{5})q^{5}+\cdots$$
363.2.f.d $8$ $2.899$ $$\Q(\zeta_{20})$$ None $$0$$ $$4$$ $$0$$ $$10$$ $$q+(-\zeta_{20}-\zeta_{20}^{7})q^{2}+(1+\zeta_{20}-\zeta_{20}^{2}+\cdots)q^{3}+\cdots$$
363.2.f.e $8$ $2.899$ $$\Q(\zeta_{20})$$ None $$0$$ $$4$$ $$0$$ $$-10$$ $$q+(-\zeta_{20}-\zeta_{20}^{7})q^{2}+(1-\zeta_{20}-\zeta_{20}^{2}+\cdots)q^{3}+\cdots$$
363.2.f.f $8$ $2.899$ 8.0.64000000.1 None $$2$$ $$-2$$ $$0$$ $$0$$ $$q+(1-\beta _{2}+\beta _{4}-\beta _{6})q^{2}+(\beta _{1}-\beta _{3}+\cdots)q^{3}+\cdots$$
363.2.f.g $16$ $2.899$ 16.0.$$\cdots$$.7 $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{5}-\beta _{10}-\beta _{13})q^{3}+2\beta _{4}q^{4}+\cdots$$
363.2.f.h $16$ $2.899$ 16.0.$$\cdots$$.1 None $$0$$ $$6$$ $$0$$ $$0$$ $$q+(\beta _{2}-2\beta _{6}-\beta _{8}+\beta _{12}+\beta _{13})q^{2}+\cdots$$
363.2.f.i $32$ $2.899$ None $$0$$ $$4$$ $$0$$ $$0$$

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

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