# Properties

 Label 338.2.a Level $338$ Weight $2$ Character orbit 338.a Rep. character $\chi_{338}(1,\cdot)$ Character field $\Q$ Dimension $12$ Newform subspaces $8$ Sturm bound $91$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 338.a (trivial) Character field: $$\Q$$ Newform subspaces: $$8$$ Sturm bound: $$91$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(338))$$.

Total New Old
Modular forms 59 12 47
Cusp forms 32 12 20
Eisenstein series 27 0 27

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

$$2$$$$13$$FrickeDim.
$$+$$$$+$$$$+$$$$2$$
$$+$$$$-$$$$-$$$$4$$
$$-$$$$+$$$$-$$$$5$$
$$-$$$$-$$$$+$$$$1$$
Plus space$$+$$$$3$$
Minus space$$-$$$$9$$

## Trace form

 $$12q + 2q^{3} + 12q^{4} + 4q^{5} + 4q^{6} + 10q^{9} + O(q^{10})$$ $$12q + 2q^{3} + 12q^{4} + 4q^{5} + 4q^{6} + 10q^{9} - 2q^{10} - 4q^{11} + 2q^{12} - 4q^{14} + 12q^{16} + 4q^{17} - 8q^{18} - 8q^{19} + 4q^{20} + 4q^{21} + 6q^{22} + 4q^{24} + 10q^{25} + 8q^{27} - 14q^{29} - 12q^{30} - 12q^{33} - 12q^{35} + 10q^{36} + 4q^{37} - 6q^{38} - 2q^{40} - 4q^{42} + 2q^{43} - 4q^{44} + 4q^{46} - 16q^{47} + 2q^{48} + 16q^{49} + 8q^{50} - 20q^{51} - 18q^{53} + 4q^{54} + 8q^{55} - 4q^{56} + 16q^{57} + 4q^{58} + 16q^{59} - 10q^{61} - 16q^{62} - 8q^{63} + 12q^{64} - 8q^{66} - 12q^{67} + 4q^{68} - 28q^{69} + 4q^{70} + 8q^{71} - 8q^{72} + 8q^{73} - 18q^{74} - 38q^{75} - 8q^{76} - 12q^{77} - 20q^{79} + 4q^{80} - 20q^{81} - 4q^{82} - 12q^{83} + 4q^{84} - 12q^{85} + 4q^{86} - 20q^{87} + 6q^{88} - 2q^{90} + 16q^{93} - 16q^{94} - 12q^{95} + 4q^{96} - 4q^{97} + 24q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(338))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 13
338.2.a.a $$1$$ $$2.699$$ $$\Q$$ None $$-1$$ $$-3$$ $$1$$ $$-1$$ $$+$$ $$+$$ $$q-q^{2}-3q^{3}+q^{4}+q^{5}+3q^{6}-q^{7}+\cdots$$
338.2.a.b $$1$$ $$2.699$$ $$\Q$$ None $$-1$$ $$-1$$ $$3$$ $$3$$ $$+$$ $$-$$ $$q-q^{2}-q^{3}+q^{4}+3q^{5}+q^{6}+3q^{7}+\cdots$$
338.2.a.c $$1$$ $$2.699$$ $$\Q$$ None $$-1$$ $$0$$ $$1$$ $$-4$$ $$+$$ $$+$$ $$q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-3q^{9}+\cdots$$
338.2.a.d $$1$$ $$2.699$$ $$\Q$$ None $$1$$ $$-1$$ $$-3$$ $$-3$$ $$-$$ $$-$$ $$q+q^{2}-q^{3}+q^{4}-3q^{5}-q^{6}-3q^{7}+\cdots$$
338.2.a.e $$1$$ $$2.699$$ $$\Q$$ None $$1$$ $$0$$ $$-1$$ $$4$$ $$-$$ $$+$$ $$q+q^{2}+q^{4}-q^{5}+4q^{7}+q^{8}-3q^{9}+\cdots$$
338.2.a.f $$1$$ $$2.699$$ $$\Q$$ None $$1$$ $$1$$ $$3$$ $$1$$ $$-$$ $$+$$ $$q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}+q^{7}+\cdots$$
338.2.a.g $$3$$ $$2.699$$ $$\Q(\zeta_{14})^+$$ None $$-3$$ $$3$$ $$-2$$ $$4$$ $$+$$ $$-$$ $$q-q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}-2\beta _{1}q^{5}+\cdots$$
338.2.a.h $$3$$ $$2.699$$ $$\Q(\zeta_{14})^+$$ None $$3$$ $$3$$ $$2$$ $$-4$$ $$-$$ $$+$$ $$q+q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}+2\beta _{1}q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(338))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(338)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(26))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(169))$$$$^{\oplus 2}$$