# Properties

 Label 55.4.a Level $55$ Weight $4$ Character orbit 55.a Rep. character $\chi_{55}(1,\cdot)$ Character field $\Q$ Dimension $10$ Newform subspaces $4$ Sturm bound $24$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 55.a (trivial) Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$24$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(55))$$.

Total New Old
Modular forms 20 10 10
Cusp forms 16 10 6
Eisenstein series 4 0 4

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

$$5$$$$11$$FrickeDim
$$+$$$$+$$$+$$$3$$
$$+$$$$-$$$-$$$1$$
$$-$$$$+$$$-$$$2$$
$$-$$$$-$$$+$$$4$$
Plus space$$+$$$$7$$
Minus space$$-$$$$3$$

## Trace form

 $$10 q + 52 q^{4} + 10 q^{5} + 56 q^{6} - 40 q^{7} - 12 q^{8} + 62 q^{9} + O(q^{10})$$ $$10 q + 52 q^{4} + 10 q^{5} + 56 q^{6} - 40 q^{7} - 12 q^{8} + 62 q^{9} - 60 q^{10} + 16 q^{12} + 8 q^{13} + 64 q^{14} + 60 q^{15} + 152 q^{16} + 64 q^{17} - 512 q^{18} - 312 q^{19} + 100 q^{20} - 232 q^{21} + 44 q^{22} + 344 q^{23} - 248 q^{24} + 250 q^{25} - 212 q^{26} + 264 q^{27} + 28 q^{28} - 52 q^{29} - 280 q^{30} - 20 q^{31} - 936 q^{32} + 132 q^{33} + 232 q^{34} + 40 q^{35} + 136 q^{36} - 812 q^{37} + 984 q^{38} + 208 q^{39} - 240 q^{40} + 172 q^{41} - 232 q^{42} + 312 q^{43} - 308 q^{44} - 230 q^{45} - 176 q^{46} + 968 q^{47} - 1536 q^{48} + 1382 q^{49} + 1712 q^{51} + 1100 q^{52} + 1380 q^{53} - 504 q^{54} + 220 q^{55} - 552 q^{56} + 1664 q^{57} + 2016 q^{58} - 1848 q^{59} - 180 q^{60} + 100 q^{61} + 1232 q^{62} - 2360 q^{63} + 1524 q^{64} + 160 q^{65} - 1100 q^{66} - 56 q^{67} - 2572 q^{68} - 2160 q^{69} - 700 q^{70} + 308 q^{71} - 444 q^{72} - 1456 q^{73} + 4272 q^{74} - 5152 q^{76} + 440 q^{77} - 1160 q^{78} - 1264 q^{79} + 1840 q^{80} + 3186 q^{81} - 3440 q^{82} - 168 q^{83} - 2096 q^{84} - 800 q^{85} + 1156 q^{86} - 4816 q^{87} + 528 q^{88} + 640 q^{89} + 180 q^{90} - 4968 q^{91} + 3528 q^{92} + 768 q^{93} + 168 q^{94} - 520 q^{95} + 3464 q^{96} - 3388 q^{97} + 4920 q^{98} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(55))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 11
55.4.a.a $1$ $3.245$ $$\Q$$ None $$1$$ $$-3$$ $$-5$$ $$-9$$ $+$ $-$ $$q+q^{2}-3q^{3}-7q^{4}-5q^{5}-3q^{6}+\cdots$$
55.4.a.b $2$ $3.245$ $$\Q(\sqrt{17})$$ None $$-7$$ $$-3$$ $$10$$ $$-25$$ $-$ $+$ $$q+(-3-\beta )q^{2}+(-1-\beta )q^{3}+(5+7\beta )q^{4}+\cdots$$
55.4.a.c $3$ $3.245$ 3.3.568.1 None $$5$$ $$-3$$ $$-15$$ $$-15$$ $+$ $+$ $$q+(1+\beta _{1}-\beta _{2})q^{2}+(-2-3\beta _{2})q^{3}+\cdots$$
55.4.a.d $4$ $3.245$ 4.4.1539480.1 None $$1$$ $$9$$ $$20$$ $$9$$ $-$ $-$ $$q+\beta _{1}q^{2}+(2-\beta _{2})q^{3}+(5+\beta _{1}+\beta _{3})q^{4}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(55))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(55)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(5))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(11))$$$$^{\oplus 2}$$