# Properties

 Label 56.6.a Level $56$ Weight $6$ Character orbit 56.a Rep. character $\chi_{56}(1,\cdot)$ Character field $\Q$ Dimension $8$ Newform subspaces $5$ Sturm bound $48$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$56 = 2^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 56.a (trivial) Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$48$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_0(56))$$.

Total New Old
Modular forms 44 8 36
Cusp forms 36 8 28
Eisenstein series 8 0 8

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

$$2$$$$7$$FrickeDim
$$+$$$$+$$$$+$$$$2$$
$$+$$$$-$$$$-$$$$3$$
$$-$$$$+$$$$-$$$$2$$
$$-$$$$-$$$$+$$$$1$$
Plus space$$+$$$$3$$
Minus space$$-$$$$5$$

## Trace form

 $$8 q - 22 q^{3} + 98 q^{5} + 876 q^{9} + O(q^{10})$$ $$8 q - 22 q^{3} + 98 q^{5} + 876 q^{9} - 268 q^{11} - 1178 q^{13} + 3112 q^{15} + 2324 q^{17} + 894 q^{19} + 882 q^{21} - 1560 q^{23} + 6916 q^{25} + 4220 q^{27} - 2852 q^{29} + 476 q^{31} - 14256 q^{33} - 7350 q^{35} - 11812 q^{37} - 10736 q^{39} - 36 q^{41} - 12004 q^{43} - 8550 q^{45} - 28476 q^{47} + 19208 q^{49} + 49612 q^{51} - 6464 q^{53} - 56136 q^{55} - 37092 q^{57} + 33838 q^{59} + 67602 q^{61} + 21364 q^{63} + 6396 q^{65} + 68568 q^{67} + 45856 q^{69} + 9656 q^{71} - 61272 q^{73} - 164162 q^{75} + 23716 q^{77} + 109944 q^{79} + 174900 q^{81} + 74166 q^{83} + 9572 q^{85} + 349268 q^{87} - 275152 q^{89} - 76930 q^{91} + 33128 q^{93} - 74040 q^{95} + 64436 q^{97} - 377436 q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_0(56))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 7
56.6.a.a $1$ $8.981$ $$\Q$$ None $$0$$ $$-6$$ $$4$$ $$49$$ $-$ $-$ $$q-6q^{3}+4q^{5}+7^{2}q^{7}-207q^{9}-240q^{11}+\cdots$$
56.6.a.b $1$ $8.981$ $$\Q$$ None $$0$$ $$30$$ $$32$$ $$49$$ $+$ $-$ $$q+30q^{3}+2^{5}q^{5}+7^{2}q^{7}+657q^{9}+\cdots$$
56.6.a.c $2$ $8.981$ $$\Q(\sqrt{177})$$ None $$0$$ $$-26$$ $$-62$$ $$98$$ $+$ $-$ $$q+(-13-\beta )q^{3}+(-31-5\beta )q^{5}+7^{2}q^{7}+\cdots$$
56.6.a.d $2$ $8.981$ $$\Q(\sqrt{193})$$ None $$0$$ $$-14$$ $$42$$ $$-98$$ $+$ $+$ $$q+(-7-\beta )q^{3}+(21+5\beta )q^{5}-7^{2}q^{7}+\cdots$$
56.6.a.e $2$ $8.981$ $$\Q(\sqrt{345})$$ None $$0$$ $$-6$$ $$82$$ $$-98$$ $-$ $+$ $$q+(-3-\beta )q^{3}+(41-3\beta )q^{5}-7^{2}q^{7}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_0(56))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_0(56)) \simeq$$ $$S_{6}^{\mathrm{new}}(\Gamma_0(4))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(7))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(14))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(28))$$$$^{\oplus 2}$$