# Properties

 Label 912.2.f Level $912$ Weight $2$ Character orbit 912.f Rep. character $\chi_{912}(113,\cdot)$ Character field $\Q$ Dimension $38$ Newform subspaces $9$ Sturm bound $320$ Trace bound $7$

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## Defining parameters

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$57$$ Character field: $$\Q$$ Newform subspaces: $$9$$ Sturm bound: $$320$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$7$$, $$29$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(912, [\chi])$$.

Total New Old
Modular forms 172 42 130
Cusp forms 148 38 110
Eisenstein series 24 4 20

## Trace form

 $$38 q + 8 q^{7} + 2 q^{9} + O(q^{10})$$ $$38 q + 8 q^{7} + 2 q^{9} - 34 q^{25} - 4 q^{39} - 24 q^{43} + 8 q^{45} + 14 q^{49} + 8 q^{55} + 14 q^{57} - 20 q^{61} + 52 q^{63} - 36 q^{73} + 10 q^{81} + 16 q^{85} + 36 q^{87} + 4 q^{93} - 32 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(912, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
912.2.f.a $2$ $7.282$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$0$$ $$-2$$ $$q+(-1-\zeta_{6})q^{3}+(2-4\zeta_{6})q^{5}-q^{7}+\cdots$$
912.2.f.b $2$ $7.282$ $$\Q(\sqrt{-2})$$ None $$0$$ $$-2$$ $$0$$ $$8$$ $$q+(-1+\beta )q^{3}+\beta q^{5}+4q^{7}+(-1+\cdots)q^{9}+\cdots$$
912.2.f.c $2$ $7.282$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-8$$ $$q+\zeta_{6}q^{3}-4q^{7}-3q^{9}-4\zeta_{6}q^{13}+\cdots$$
912.2.f.d $2$ $7.282$ $$\Q(\sqrt{-2})$$ None $$0$$ $$2$$ $$0$$ $$8$$ $$q+(1-\beta )q^{3}+\beta q^{5}+4q^{7}+(-1-2\beta )q^{9}+\cdots$$
912.2.f.e $2$ $7.282$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$0$$ $$-2$$ $$q+(1+\zeta_{6})q^{3}+(2-4\zeta_{6})q^{5}-q^{7}+3\zeta_{6}q^{9}+\cdots$$
912.2.f.f $4$ $7.282$ $$\Q(\sqrt{2}, \sqrt{-5})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q+(-\beta _{1}-\beta _{2})q^{3}-\beta _{3}q^{5}-q^{7}+(-2+\cdots)q^{9}+\cdots$$
912.2.f.g $4$ $7.282$ $$\Q(\sqrt{-2}, \sqrt{-5})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q-\beta _{1}q^{3}-\beta _{2}q^{5}+q^{7}+(2-\beta _{2})q^{9}+\cdots$$
912.2.f.h $10$ $7.282$ 10.0.$$\cdots$$.1 None $$0$$ $$-1$$ $$0$$ $$2$$ $$q-\beta _{1}q^{3}-\beta _{6}q^{5}+\beta _{4}q^{7}+\beta _{2}q^{9}+\cdots$$
912.2.f.i $10$ $7.282$ 10.0.$$\cdots$$.1 None $$0$$ $$1$$ $$0$$ $$2$$ $$q+\beta _{1}q^{3}-\beta _{6}q^{5}+\beta _{4}q^{7}+\beta _{2}q^{9}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(912, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(912, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(57, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(114, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(228, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(456, [\chi])$$$$^{\oplus 2}$$