# Properties

 Label 156.2.a Level $156$ Weight $2$ Character orbit 156.a Rep. character $\chi_{156}(1,\cdot)$ Character field $\Q$ Dimension $2$ Newform subspaces $2$ Sturm bound $56$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$156 = 2^{2} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 156.a (trivial) Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$56$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(156))$$.

Total New Old
Modular forms 34 2 32
Cusp forms 23 2 21
Eisenstein series 11 0 11

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

$$2$$$$3$$$$13$$FrickeDim
$$-$$$$+$$$$-$$$+$$$1$$
$$-$$$$-$$$$-$$$-$$$1$$
Plus space$$+$$$$1$$
Minus space$$-$$$$1$$

## Trace form

 $$2 q - 4 q^{5} + 2 q^{9} + O(q^{10})$$ $$2 q - 4 q^{5} + 2 q^{9} - 4 q^{11} + 2 q^{13} + 4 q^{15} - 4 q^{17} + 4 q^{21} + 6 q^{25} - 12 q^{29} - 8 q^{31} + 4 q^{33} + 8 q^{35} + 12 q^{37} - 4 q^{41} - 4 q^{45} - 4 q^{47} - 6 q^{49} - 8 q^{51} - 4 q^{53} + 16 q^{55} + 4 q^{57} + 4 q^{59} - 12 q^{61} - 4 q^{65} - 8 q^{67} + 28 q^{71} + 4 q^{73} - 16 q^{75} + 8 q^{77} - 8 q^{79} + 2 q^{81} + 12 q^{83} - 8 q^{85} - 4 q^{89} + 12 q^{93} + 8 q^{95} - 12 q^{97} - 4 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(156))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 13
156.2.a.a $1$ $1.246$ $$\Q$$ None $$0$$ $$-1$$ $$-4$$ $$-2$$ $-$ $+$ $-$ $$q-q^{3}-4q^{5}-2q^{7}+q^{9}-4q^{11}+\cdots$$
156.2.a.b $1$ $1.246$ $$\Q$$ None $$0$$ $$1$$ $$0$$ $$2$$ $-$ $-$ $-$ $$q+q^{3}+2q^{7}+q^{9}+q^{13}-6q^{17}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(156))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(156)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(26))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(39))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(52))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(78))$$$$^{\oplus 2}$$