# Properties

 Label 605.2.a Level $605$ Weight $2$ Character orbit 605.a Rep. character $\chi_{605}(1,\cdot)$ Character field $\Q$ Dimension $37$ Newform subspaces $13$ Sturm bound $132$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.a (trivial) Character field: $$\Q$$ Newform subspaces: $$13$$ Sturm bound: $$132$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$, $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(605))$$.

Total New Old
Modular forms 78 37 41
Cusp forms 55 37 18
Eisenstein series 23 0 23

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.
$$+$$$$+$$$$+$$$$6$$
$$+$$$$-$$$$-$$$$12$$
$$-$$$$+$$$$-$$$$12$$
$$-$$$$-$$$$+$$$$7$$
Plus space$$+$$$$13$$
Minus space$$-$$$$24$$

## Trace form

 $$37q - 3q^{2} + 39q^{4} + q^{5} + 8q^{6} + 4q^{7} - 3q^{8} + 33q^{9} + O(q^{10})$$ $$37q - 3q^{2} + 39q^{4} + q^{5} + 8q^{6} + 4q^{7} - 3q^{8} + 33q^{9} + q^{10} + 16q^{12} + 6q^{13} + 31q^{16} - 14q^{17} - 7q^{18} + 4q^{19} - q^{20} - 8q^{23} + 8q^{24} + 37q^{25} - 10q^{26} + 4q^{28} - 10q^{29} - 8q^{30} + 4q^{31} + q^{32} - 34q^{34} - 4q^{35} + 7q^{36} - 2q^{37} - 20q^{38} + 16q^{39} + 9q^{40} - 14q^{41} - 44q^{42} + 8q^{43} + 13q^{45} + 4q^{46} + 4q^{47} - 20q^{48} + 41q^{49} - 3q^{50} + 16q^{51} - 6q^{52} - 14q^{53} + 16q^{54} - 20q^{56} - 6q^{58} - 8q^{59} - 16q^{60} + 6q^{61} + 8q^{62} + 20q^{63} + 39q^{64} - 10q^{65} + 4q^{67} - 18q^{68} - 24q^{69} - 8q^{70} - 20q^{71} - 39q^{72} - 6q^{73} + 22q^{74} - 4q^{76} - 52q^{78} - 16q^{79} - q^{80} + 13q^{81} - 22q^{82} + 16q^{83} - 32q^{84} + 2q^{85} - 16q^{86} - 32q^{87} - 22q^{89} + 13q^{90} + 48q^{91} - 12q^{92} + 56q^{93} + 4q^{94} + 4q^{95} + 8q^{96} + 46q^{97} + 13q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(605))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 5 11
605.2.a.a $$1$$ $$4.831$$ $$\Q$$ None $$-1$$ $$-3$$ $$1$$ $$-3$$ $$-$$ $$-$$ $$q-q^{2}-3q^{3}-q^{4}+q^{5}+3q^{6}-3q^{7}+\cdots$$
605.2.a.b $$1$$ $$4.831$$ $$\Q$$ None $$-1$$ $$0$$ $$1$$ $$0$$ $$-$$ $$-$$ $$q-q^{2}-q^{4}+q^{5}+3q^{8}-3q^{9}-q^{10}+\cdots$$
605.2.a.c $$1$$ $$4.831$$ $$\Q$$ None $$1$$ $$-3$$ $$1$$ $$3$$ $$-$$ $$-$$ $$q+q^{2}-3q^{3}-q^{4}+q^{5}-3q^{6}+3q^{7}+\cdots$$
605.2.a.d $$2$$ $$4.831$$ $$\Q(\sqrt{2})$$ None $$-2$$ $$0$$ $$-2$$ $$4$$ $$+$$ $$-$$ $$q+(-1+\beta )q^{2}+2\beta q^{3}+(1-2\beta )q^{4}+\cdots$$
605.2.a.e $$2$$ $$4.831$$ $$\Q(\sqrt{3})$$ None $$0$$ $$-2$$ $$-2$$ $$0$$ $$+$$ $$+$$ $$q+\beta q^{2}-q^{3}+q^{4}-q^{5}-\beta q^{6}-\beta q^{7}+\cdots$$
605.2.a.f $$2$$ $$4.831$$ $$\Q(\sqrt{3})$$ None $$0$$ $$4$$ $$2$$ $$0$$ $$-$$ $$+$$ $$q+\beta q^{2}+2q^{3}+q^{4}+q^{5}+2\beta q^{6}+\cdots$$
605.2.a.g $$3$$ $$4.831$$ 3.3.404.1 None $$-1$$ $$1$$ $$-3$$ $$1$$ $$+$$ $$-$$ $$q-\beta _{2}q^{2}+\beta _{1}q^{3}+(3+\beta _{1}-\beta _{2})q^{4}+\cdots$$
605.2.a.h $$3$$ $$4.831$$ 3.3.404.1 None $$1$$ $$1$$ $$-3$$ $$-1$$ $$+$$ $$-$$ $$q+\beta _{2}q^{2}+\beta _{1}q^{3}+(3+\beta _{1}-\beta _{2})q^{4}+\cdots$$
605.2.a.i $$4$$ $$4.831$$ 4.4.2525.1 None $$-3$$ $$-2$$ $$4$$ $$-11$$ $$-$$ $$-$$ $$q+(-1-\beta _{3})q^{2}-\beta _{1}q^{3}+(1+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots$$
605.2.a.j $$4$$ $$4.831$$ 4.4.725.1 None $$-1$$ $$0$$ $$-4$$ $$-3$$ $$+$$ $$+$$ $$q-\beta _{1}q^{2}+(-1+\beta _{2}+\beta _{3})q^{3}+(-1+\cdots)q^{4}+\cdots$$
605.2.a.k $$4$$ $$4.831$$ 4.4.725.1 None $$1$$ $$0$$ $$-4$$ $$3$$ $$+$$ $$-$$ $$q+\beta _{1}q^{2}+(-1+\beta _{2}+\beta _{3})q^{3}+(-1+\cdots)q^{4}+\cdots$$
605.2.a.l $$4$$ $$4.831$$ 4.4.2525.1 None $$3$$ $$-2$$ $$4$$ $$11$$ $$-$$ $$+$$ $$q+(1+\beta _{2}-\beta _{3})q^{2}+(-1+\beta _{1})q^{3}+\cdots$$
605.2.a.m $$6$$ $$4.831$$ 6.6.27433728.1 None $$0$$ $$6$$ $$6$$ $$0$$ $$-$$ $$+$$ $$q+\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(605))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(605)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(11))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(55))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(121))$$$$^{\oplus 2}$$