# Properties

 Label 309.2.a Level $309$ Weight $2$ Character orbit 309.a Rep. character $\chi_{309}(1,\cdot)$ Character field $\Q$ Dimension $17$ Newform subspaces $4$ Sturm bound $69$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$309 = 3 \cdot 103$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 309.a (trivial) Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$69$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(309))$$.

Total New Old
Modular forms 36 17 19
Cusp forms 33 17 16
Eisenstein series 3 0 3

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

$$3$$$$103$$FrickeDim.
$$+$$$$+$$$$+$$$$5$$
$$+$$$$-$$$$-$$$$3$$
$$-$$$$+$$$$-$$$$8$$
$$-$$$$-$$$$+$$$$1$$
Plus space$$+$$$$6$$
Minus space$$-$$$$11$$

## Trace form

 $$17q - 3q^{2} + q^{3} + 13q^{4} - 6q^{5} - q^{6} - 3q^{8} + 17q^{9} + O(q^{10})$$ $$17q - 3q^{2} + q^{3} + 13q^{4} - 6q^{5} - q^{6} - 3q^{8} + 17q^{9} + 10q^{10} + 7q^{12} + 2q^{13} - 4q^{14} + 2q^{15} + q^{16} - 10q^{17} - 3q^{18} + 4q^{19} - 30q^{20} + 8q^{21} - 8q^{23} + 3q^{24} + 15q^{25} - 16q^{26} + q^{27} + 2q^{28} - 18q^{29} - 2q^{30} + 16q^{31} - 9q^{32} + 8q^{33} - 22q^{34} - 8q^{35} + 13q^{36} - 6q^{37} + 26q^{38} + 6q^{39} + 10q^{40} + 2q^{41} - 4q^{42} + 4q^{43} + 4q^{44} - 6q^{45} - 14q^{46} + 12q^{47} + 15q^{48} + 37q^{49} - 17q^{50} + 2q^{51} + 46q^{52} - 22q^{53} - q^{54} - 16q^{55} - 26q^{56} + 12q^{57} - 18q^{58} - 4q^{59} - 14q^{60} - 38q^{61} - 32q^{62} - 55q^{64} - 32q^{65} - 4q^{66} + 24q^{67} - 12q^{68} - 12q^{69} - 12q^{70} + 4q^{71} - 3q^{72} + 34q^{73} + 22q^{74} + 7q^{75} + 4q^{76} - 32q^{77} - 2q^{78} + 8q^{79} + 30q^{80} + 17q^{81} - 24q^{82} - 16q^{83} - 8q^{84} - 4q^{85} - 12q^{86} + 14q^{87} - 4q^{88} + 26q^{89} + 10q^{90} - 16q^{91} - 36q^{92} + 28q^{93} + 4q^{95} - 25q^{96} + 6q^{97} - 57q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(309))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 3 103
309.2.a.a $$1$$ $$2.467$$ $$\Q$$ None $$-1$$ $$1$$ $$-1$$ $$-2$$ $$-$$ $$-$$ $$q-q^{2}+q^{3}-q^{4}-q^{5}-q^{6}-2q^{7}+\cdots$$
309.2.a.b $$3$$ $$2.467$$ 3.3.148.1 None $$1$$ $$-3$$ $$1$$ $$-2$$ $$+$$ $$-$$ $$q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{5}+\cdots$$
309.2.a.c $$5$$ $$2.467$$ 5.5.81509.1 None $$-2$$ $$-5$$ $$-5$$ $$-2$$ $$+$$ $$+$$ $$q+\beta _{3}q^{2}-q^{3}+(-\beta _{3}-\beta _{4})q^{4}+(-1+\cdots)q^{5}+\cdots$$
309.2.a.d $$8$$ $$2.467$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$-1$$ $$8$$ $$-1$$ $$6$$ $$-$$ $$+$$ $$q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{7}q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(309))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(309)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(103))$$$$^{\oplus 2}$$