# Properties

 Label 1339.2.a Level $1339$ Weight $2$ Character orbit 1339.a Rep. character $\chi_{1339}(1,\cdot)$ Character field $\Q$ Dimension $103$ Newform subspaces $7$ Sturm bound $242$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1339 = 13 \cdot 103$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1339.a (trivial) Character field: $$\Q$$ Newform subspaces: $$7$$ Sturm bound: $$242$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$, $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(1339))$$.

Total New Old
Modular forms 122 103 19
Cusp forms 119 103 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.

$$13$$$$103$$FrickeDim.
$$+$$$$+$$$$+$$$$22$$
$$+$$$$-$$$$-$$$$30$$
$$-$$$$+$$$$-$$$$29$$
$$-$$$$-$$$$+$$$$22$$
Plus space$$+$$$$44$$
Minus space$$-$$$$59$$

## Trace form

 $$103q - 3q^{2} + 97q^{4} + 2q^{5} - 3q^{8} + 103q^{9} + O(q^{10})$$ $$103q - 3q^{2} + 97q^{4} + 2q^{5} - 3q^{8} + 103q^{9} + 10q^{10} - 4q^{11} + 16q^{12} - q^{13} + 16q^{14} + 4q^{15} + 81q^{16} - 14q^{17} + 9q^{18} - 16q^{19} + 26q^{20} + 8q^{21} - 4q^{22} - 16q^{23} + 24q^{24} + 125q^{25} - 3q^{26} - 12q^{27} - 20q^{28} + 6q^{29} - 12q^{30} + 16q^{31} - 19q^{32} - 12q^{33} - 54q^{34} - 16q^{35} + 73q^{36} + 6q^{37} + 36q^{38} + 4q^{39} + 10q^{40} - 18q^{41} - 48q^{42} - 20q^{43} - 36q^{44} + 14q^{45} - 24q^{46} + 24q^{47} + 20q^{48} + 111q^{49} - 53q^{50} - 24q^{51} + q^{52} + 2q^{53} - 52q^{54} - 4q^{55} - 12q^{56} - 36q^{57} - 14q^{58} - 16q^{59} - 60q^{60} - 14q^{61} - 88q^{62} - 12q^{63} + 5q^{64} + 2q^{65} - 84q^{66} - 16q^{67} - 18q^{68} + 32q^{69} + 48q^{70} + 16q^{71} - 39q^{72} + 10q^{73} + 6q^{74} + 4q^{75} - 64q^{76} - 40q^{77} + 12q^{78} + 4q^{79} + 86q^{80} + 111q^{81} + 22q^{82} - 4q^{83} - 100q^{84} - 24q^{86} + 20q^{87} - 16q^{88} + 78q^{89} + 22q^{90} - 8q^{91} - 76q^{92} + 76q^{93} - 36q^{94} + 16q^{95} - 4q^{96} - 6q^{97} + 17q^{98} - 28q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(1339))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 13 103
1339.2.a.a $$1$$ $$10.692$$ $$\Q$$ None $$1$$ $$-1$$ $$1$$ $$4$$ $$+$$ $$+$$ $$q+q^{2}-q^{3}-q^{4}+q^{5}-q^{6}+4q^{7}+\cdots$$
1339.2.a.b $$1$$ $$10.692$$ $$\Q$$ None $$1$$ $$0$$ $$0$$ $$-4$$ $$-$$ $$+$$ $$q+q^{2}-q^{4}-4q^{7}-3q^{8}-3q^{9}+6q^{11}+\cdots$$
1339.2.a.c $$3$$ $$10.692$$ 3.3.148.1 None $$-1$$ $$-1$$ $$-7$$ $$2$$ $$-$$ $$-$$ $$q-\beta _{1}q^{2}+(-\beta _{1}-\beta _{2})q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots$$
1339.2.a.d $$19$$ $$10.692$$ $$\mathbb{Q}[x]/(x^{19} - \cdots)$$ None $$-9$$ $$-2$$ $$-18$$ $$-8$$ $$-$$ $$-$$ $$q-\beta _{1}q^{2}-\beta _{14}q^{3}+(\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots$$
1339.2.a.e $$21$$ $$10.692$$ None $$-1$$ $$-6$$ $$-18$$ $$-2$$ $$+$$ $$+$$
1339.2.a.f $$28$$ $$10.692$$ None $$6$$ $$5$$ $$27$$ $$6$$ $$-$$ $$+$$
1339.2.a.g $$30$$ $$10.692$$ None $$0$$ $$5$$ $$17$$ $$2$$ $$+$$ $$-$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(1339))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(1339)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(103))$$$$^{\oplus 2}$$