# Properties

 Label 75.6.a Level $75$ Weight $6$ Character orbit 75.a Rep. character $\chi_{75}(1,\cdot)$ Character field $\Q$ Dimension $15$ Newform subspaces $10$ Sturm bound $60$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 75.a (trivial) Character field: $$\Q$$ Newform subspaces: $$10$$ Sturm bound: $$60$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_0(75))$$.

Total New Old
Modular forms 56 15 41
Cusp forms 44 15 29
Eisenstein series 12 0 12

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

$$3$$$$5$$FrickeDim.
$$+$$$$+$$$$+$$$$3$$
$$+$$$$-$$$$-$$$$4$$
$$-$$$$+$$$$-$$$$5$$
$$-$$$$-$$$$+$$$$3$$
Plus space$$+$$$$6$$
Minus space$$-$$$$9$$

## Trace form

 $$15q + 2q^{2} + 9q^{3} + 176q^{4} + 54q^{6} + 272q^{7} + 60q^{8} + 1215q^{9} + O(q^{10})$$ $$15q + 2q^{2} + 9q^{3} + 176q^{4} + 54q^{6} + 272q^{7} + 60q^{8} + 1215q^{9} - 288q^{11} + 828q^{12} + 146q^{13} + 2268q^{14} + 3124q^{16} - 3538q^{17} + 162q^{18} + 4442q^{19} - 2214q^{21} + 3444q^{22} + 576q^{23} - 1404q^{24} + 20880q^{26} + 729q^{27} + 884q^{28} - 11526q^{29} - 15486q^{31} + 25612q^{32} + 10548q^{33} - 23908q^{34} + 14256q^{36} - 12598q^{37} - 59440q^{38} + 1944q^{39} - 23286q^{41} - 29484q^{42} + 38876q^{43} + 48720q^{44} + 3904q^{46} + 19832q^{47} + 59904q^{48} + 71693q^{49} - 33750q^{51} - 63688q^{52} - 53494q^{53} + 4374q^{54} + 16920q^{56} - 14220q^{57} + 71280q^{58} - 33432q^{59} - 87072q^{61} + 46104q^{62} + 22032q^{63} - 172340q^{64} - 124200q^{66} + 54932q^{67} - 244256q^{68} - 15012q^{69} + 210120q^{71} + 4860q^{72} - 20674q^{73} - 282372q^{74} + 23728q^{76} - 52416q^{77} + 133668q^{78} - 32320q^{79} + 98415q^{81} - 3996q^{82} - 103404q^{83} + 55512q^{84} - 203148q^{86} + 14670q^{87} + 335220q^{88} + 234762q^{89} + 29834q^{91} + 172992q^{92} + 101088q^{93} + 640304q^{94} + 107676q^{96} - 134218q^{97} - 327326q^{98} - 23328q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_0(75))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 3 5
75.6.a.a $$1$$ $$12.029$$ $$\Q$$ None $$-7$$ $$-9$$ $$0$$ $$-12$$ $$+$$ $$+$$ $$q-7q^{2}-9q^{3}+17q^{4}+63q^{6}-12q^{7}+\cdots$$
75.6.a.b $$1$$ $$12.029$$ $$\Q$$ None $$-4$$ $$-9$$ $$0$$ $$225$$ $$+$$ $$+$$ $$q-4q^{2}-9q^{3}-2^{4}q^{4}+6^{2}q^{6}+15^{2}q^{7}+\cdots$$
75.6.a.c $$1$$ $$12.029$$ $$\Q$$ None $$2$$ $$9$$ $$0$$ $$132$$ $$-$$ $$+$$ $$q+2q^{2}+9q^{3}-28q^{4}+18q^{6}+132q^{7}+\cdots$$
75.6.a.d $$1$$ $$12.029$$ $$\Q$$ None $$4$$ $$9$$ $$0$$ $$-225$$ $$-$$ $$-$$ $$q+4q^{2}+9q^{3}-2^{4}q^{4}+6^{2}q^{6}-15^{2}q^{7}+\cdots$$
75.6.a.e $$1$$ $$12.029$$ $$\Q$$ None $$6$$ $$-9$$ $$0$$ $$40$$ $$+$$ $$+$$ $$q+6q^{2}-9q^{3}+4q^{4}-54q^{6}+40q^{7}+\cdots$$
75.6.a.f $$2$$ $$12.029$$ $$\Q(\sqrt{89})$$ None $$-9$$ $$18$$ $$0$$ $$-108$$ $$-$$ $$-$$ $$q+(-4-\beta )q^{2}+9q^{3}+(6+9\beta )q^{4}+\cdots$$
75.6.a.g $$2$$ $$12.029$$ $$\Q(\sqrt{31})$$ None $$-6$$ $$-18$$ $$0$$ $$-102$$ $$+$$ $$-$$ $$q+(-3+\beta )q^{2}-9q^{3}+(8-6\beta )q^{4}+\cdots$$
75.6.a.h $$2$$ $$12.029$$ $$\Q(\sqrt{409})$$ None $$1$$ $$18$$ $$0$$ $$112$$ $$-$$ $$+$$ $$q+\beta q^{2}+9q^{3}+(70+\beta )q^{4}+9\beta q^{6}+\cdots$$
75.6.a.i $$2$$ $$12.029$$ $$\Q(\sqrt{31})$$ None $$6$$ $$18$$ $$0$$ $$102$$ $$-$$ $$+$$ $$q+(3+\beta )q^{2}+9q^{3}+(8+6\beta )q^{4}+(3^{3}+\cdots)q^{6}+\cdots$$
75.6.a.j $$2$$ $$12.029$$ $$\Q(\sqrt{89})$$ None $$9$$ $$-18$$ $$0$$ $$108$$ $$+$$ $$-$$ $$q+(5-\beta )q^{2}-9q^{3}+(15-9\beta )q^{4}+(-45+\cdots)q^{6}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_0(75))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_0(75)) \cong$$ $$S_{6}^{\mathrm{new}}(\Gamma_0(3))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(5))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(15))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(25))$$$$^{\oplus 2}$$