# Properties

 Label 353.2.a Level $353$ Weight $2$ Character orbit 353.a Rep. character $\chi_{353}(1,\cdot)$ Character field $\Q$ Dimension $29$ Newform subspaces $4$ Sturm bound $59$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$353$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 353.a (trivial) Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$59$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(353))$$.

Total New Old
Modular forms 30 30 0
Cusp forms 29 29 0
Eisenstein series 1 1 0

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

$$353$$Dim.
$$+$$$$11$$
$$-$$$$18$$

## Trace form

 $$29q - q^{2} - 2q^{3} + 27q^{4} - 4q^{5} - 6q^{6} + 4q^{7} - 3q^{8} + 23q^{9} + O(q^{10})$$ $$29q - q^{2} - 2q^{3} + 27q^{4} - 4q^{5} - 6q^{6} + 4q^{7} - 3q^{8} + 23q^{9} - 6q^{10} - 4q^{11} - 22q^{12} + 2q^{13} + 8q^{14} - 2q^{15} + 27q^{16} - 6q^{17} - 7q^{18} + 2q^{20} + 2q^{21} + 6q^{22} - 10q^{23} + 19q^{25} + 12q^{26} - 8q^{27} + 18q^{28} + 6q^{29} + 2q^{30} + 2q^{31} + 3q^{32} - 18q^{33} - 26q^{34} - 4q^{35} + 27q^{36} + 6q^{37} + 12q^{38} - 30q^{39} - 36q^{40} - 10q^{41} - 20q^{42} - 14q^{43} - 10q^{44} - 44q^{45} - 2q^{46} - 10q^{47} - 60q^{48} + 59q^{49} - 3q^{50} - 22q^{51} - 2q^{52} - 8q^{53} - 40q^{54} + 4q^{55} + 58q^{56} + 28q^{57} - 16q^{58} - 2q^{59} + 16q^{61} + 14q^{62} + 24q^{63} + 5q^{64} - 8q^{65} + 24q^{66} + 24q^{67} - 20q^{68} - 10q^{69} - 24q^{70} + 10q^{71} + 21q^{72} + 10q^{73} - 2q^{74} + 46q^{75} - 12q^{76} - 32q^{77} + 6q^{78} + 12q^{79} - 12q^{80} + 21q^{81} - 18q^{82} - 8q^{83} - 44q^{84} - 52q^{85} - 16q^{86} + 24q^{87} - 22q^{88} - 2q^{89} - 40q^{90} + 46q^{91} + 34q^{92} - 20q^{93} - 16q^{94} - 12q^{95} - 18q^{96} - 8q^{97} - 81q^{98} - 38q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(353))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 353
353.2.a.a $$1$$ $$2.819$$ $$\Q$$ None $$-1$$ $$2$$ $$2$$ $$-2$$ $$-$$ $$q-q^{2}+2q^{3}-q^{4}+2q^{5}-2q^{6}-2q^{7}+\cdots$$
353.2.a.b $$3$$ $$2.819$$ 3.3.229.1 None $$1$$ $$3$$ $$2$$ $$2$$ $$-$$ $$q+(\beta _{1}-\beta _{2})q^{2}+(1+\beta _{1})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots$$
353.2.a.c $$11$$ $$2.819$$ $$\mathbb{Q}[x]/(x^{11} - \cdots)$$ None $$-5$$ $$-5$$ $$-4$$ $$-25$$ $$+$$ $$q-\beta _{1}q^{2}+(-1+\beta _{1}-\beta _{2}-\beta _{5}-\beta _{8}+\cdots)q^{3}+\cdots$$
353.2.a.d $$14$$ $$2.819$$ $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ None $$4$$ $$-2$$ $$-4$$ $$29$$ $$-$$ $$q+\beta _{1}q^{2}+\beta _{5}q^{3}+(1+\beta _{2})q^{4}+\beta _{12}q^{5}+\cdots$$