# Properties

 Label 368.2.a Level $368$ Weight $2$ Character orbit 368.a Rep. character $\chi_{368}(1,\cdot)$ Character field $\Q$ Dimension $11$ Newform subspaces $9$ Sturm bound $96$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$368 = 2^{4} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 368.a (trivial) Character field: $$\Q$$ Newform subspaces: $$9$$ Sturm bound: $$96$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(368))$$.

Total New Old
Modular forms 54 11 43
Cusp forms 43 11 32
Eisenstein series 11 0 11

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$2$$$$23$$FrickeDim.
$$+$$$$+$$$$+$$$$3$$
$$+$$$$-$$$$-$$$$3$$
$$-$$$$+$$$$-$$$$4$$
$$-$$$$-$$$$+$$$$1$$
Plus space$$+$$$$4$$
Minus space$$-$$$$7$$

## Trace form

 $$11 q + 2 q^{3} - 2 q^{5} + 4 q^{7} + 7 q^{9} + O(q^{10})$$ $$11 q + 2 q^{3} - 2 q^{5} + 4 q^{7} + 7 q^{9} - 2 q^{13} - 2 q^{17} + 4 q^{19} - 3 q^{23} + 5 q^{25} + 2 q^{27} - 2 q^{29} + 10 q^{31} - 8 q^{33} - 10 q^{37} + 14 q^{39} + 6 q^{41} - 10 q^{45} - 22 q^{47} + 11 q^{49} + 4 q^{51} - 10 q^{53} - 24 q^{55} + 8 q^{57} - 4 q^{59} - 18 q^{61} + 32 q^{63} + 12 q^{65} - 8 q^{67} - 6 q^{71} - 2 q^{73} + 6 q^{75} - 8 q^{77} + 28 q^{79} + 3 q^{81} - 20 q^{83} - 12 q^{85} - 46 q^{87} + 6 q^{89} + 4 q^{91} + 12 q^{93} + 8 q^{95} + 6 q^{97} - 8 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(368))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 23
368.2.a.a $1$ $2.938$ $$\Q$$ None $$0$$ $$-3$$ $$0$$ $$2$$ $+$ $+$ $$q-3q^{3}+2q^{7}+6q^{9}-5q^{13}-6q^{17}+\cdots$$
368.2.a.b $1$ $2.938$ $$\Q$$ None $$0$$ $$-1$$ $$0$$ $$-2$$ $-$ $-$ $$q-q^{3}-2q^{7}-2q^{9}-q^{13}-6q^{17}+\cdots$$
368.2.a.c $1$ $2.938$ $$\Q$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $+$ $+$ $$q-4q^{7}-3q^{9}-6q^{11}-2q^{13}+6q^{17}+\cdots$$
368.2.a.d $1$ $2.938$ $$\Q$$ None $$0$$ $$0$$ $$4$$ $$4$$ $-$ $+$ $$q+4q^{5}+4q^{7}-3q^{9}-2q^{11}-2q^{13}+\cdots$$
368.2.a.e $1$ $2.938$ $$\Q$$ None $$0$$ $$1$$ $$-4$$ $$-2$$ $+$ $+$ $$q+q^{3}-4q^{5}-2q^{7}-2q^{9}+4q^{11}+\cdots$$
368.2.a.f $1$ $2.938$ $$\Q$$ None $$0$$ $$1$$ $$-2$$ $$4$$ $+$ $-$ $$q+q^{3}-2q^{5}+4q^{7}-2q^{9}+2q^{11}+\cdots$$
368.2.a.g $1$ $2.938$ $$\Q$$ None $$0$$ $$3$$ $$-2$$ $$4$$ $-$ $+$ $$q+3q^{3}-2q^{5}+4q^{7}+6q^{9}-2q^{11}+\cdots$$
368.2.a.h $2$ $2.938$ $$\Q(\sqrt{5})$$ None $$0$$ $$0$$ $$-2$$ $$-2$$ $-$ $+$ $$q-\beta q^{3}+(-1-\beta )q^{5}+(-1+\beta )q^{7}+\cdots$$
368.2.a.i $2$ $2.938$ $$\Q(\sqrt{17})$$ None $$0$$ $$1$$ $$4$$ $$0$$ $+$ $-$ $$q+\beta q^{3}+2q^{5}+(1+\beta )q^{9}-2\beta q^{11}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(368))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(368)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(23))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(46))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(92))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(184))$$$$^{\oplus 2}$$