# Properties

 Label 58.2.a Level $58$ Weight $2$ Character orbit 58.a Rep. character $\chi_{58}(1,\cdot)$ Character field $\Q$ Dimension $2$ Newform subspaces $2$ Sturm bound $15$ Trace bound $2$

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## Defining parameters

 Level: $$N$$ $$=$$ $$58 = 2 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 58.a (trivial) Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$15$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(58))$$.

Total New Old
Modular forms 9 2 7
Cusp forms 6 2 4
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.

$$2$$$$29$$FrickeDim
$$+$$$$+$$$+$$$1$$
$$-$$$$+$$$-$$$1$$
Plus space$$+$$$$1$$
Minus space$$-$$$$1$$

## Trace form

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

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(58))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 29
58.2.a.a $1$ $0.463$ $$\Q$$ None $$-1$$ $$-3$$ $$-3$$ $$-2$$ $+$ $+$ $$q-q^{2}-3q^{3}+q^{4}-3q^{5}+3q^{6}-2q^{7}+\cdots$$
58.2.a.b $1$ $0.463$ $$\Q$$ None $$1$$ $$-1$$ $$1$$ $$-2$$ $-$ $+$ $$q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-2q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(58))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(58)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(29))$$$$^{\oplus 2}$$