# Properties

 Label 676.2.a Level $676$ Weight $2$ Character orbit 676.a Rep. character $\chi_{676}(1,\cdot)$ Character field $\Q$ Dimension $13$ Newform subspaces $8$ Sturm bound $182$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$676 = 2^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 676.a (trivial) Character field: $$\Q$$ Newform subspaces: $$8$$ Sturm bound: $$182$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(676))$$.

Total New Old
Modular forms 112 13 99
Cusp forms 71 13 58
Eisenstein series 41 0 41

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

$$2$$$$13$$FrickeDim
$$-$$$$+$$$-$$$8$$
$$-$$$$-$$$+$$$5$$
Plus space$$+$$$$5$$
Minus space$$-$$$$8$$

## Trace form

 $$13 q - 2 q^{5} + 2 q^{7} + 17 q^{9} + O(q^{10})$$ $$13 q - 2 q^{5} + 2 q^{7} + 17 q^{9} + 2 q^{11} - 8 q^{17} + 6 q^{19} - 6 q^{23} + 17 q^{25} - 6 q^{27} - 2 q^{29} - 10 q^{31} + 6 q^{37} + 6 q^{41} - 2 q^{43} + 6 q^{45} + 2 q^{47} + 19 q^{49} - 2 q^{51} - 6 q^{53} + 8 q^{55} + 10 q^{59} + 4 q^{61} - 6 q^{63} - 10 q^{67} - 4 q^{69} - 10 q^{71} - 2 q^{73} + 2 q^{75} + 6 q^{79} + 5 q^{81} + 6 q^{83} - 12 q^{85} + 4 q^{87} + 6 q^{89} + 18 q^{95} - 2 q^{97} - 6 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(676))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 13
676.2.a.a $1$ $5.398$ $$\Q$$ None $$0$$ $$-2$$ $$-3$$ $$-4$$ $-$ $+$ $$q-2q^{3}-3q^{5}-4q^{7}+q^{9}+6q^{15}+\cdots$$
676.2.a.b $1$ $5.398$ $$\Q$$ None $$0$$ $$-2$$ $$3$$ $$4$$ $-$ $+$ $$q-2q^{3}+3q^{5}+4q^{7}+q^{9}-6q^{15}+\cdots$$
676.2.a.c $1$ $5.398$ $$\Q$$ None $$0$$ $$0$$ $$-2$$ $$2$$ $-$ $+$ $$q-2q^{5}+2q^{7}-3q^{9}+2q^{11}+6q^{17}+\cdots$$
676.2.a.d $1$ $5.398$ $$\Q$$ None $$0$$ $$3$$ $$-2$$ $$-1$$ $-$ $+$ $$q+3q^{3}-2q^{5}-q^{7}+6q^{9}+5q^{11}+\cdots$$
676.2.a.e $1$ $5.398$ $$\Q$$ None $$0$$ $$3$$ $$2$$ $$1$$ $-$ $+$ $$q+3q^{3}+2q^{5}+q^{7}+6q^{9}-5q^{11}+\cdots$$
676.2.a.f $2$ $5.398$ $$\Q(\sqrt{3})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $-$ $-$ $$q-q^{3}+\beta q^{7}-2q^{9}-3\beta q^{11}-3q^{17}+\cdots$$
676.2.a.g $3$ $5.398$ $$\Q(\zeta_{14})^+$$ None $$0$$ $$0$$ $$-8$$ $$1$$ $-$ $-$ $$q+(-\beta _{1}-\beta _{2})q^{3}+(-2-\beta _{1}+\beta _{2})q^{5}+\cdots$$
676.2.a.h $3$ $5.398$ $$\Q(\zeta_{14})^+$$ None $$0$$ $$0$$ $$8$$ $$-1$$ $-$ $+$ $$q+(-\beta _{1}-\beta _{2})q^{3}+(2+\beta _{1}-\beta _{2})q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(676))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(676)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(26))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(52))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(169))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(338))$$$$^{\oplus 2}$$