# Properties

 Label 464.4.a Level $464$ Weight $4$ Character orbit 464.a Rep. character $\chi_{464}(1,\cdot)$ Character field $\Q$ Dimension $42$ Newform subspaces $14$ Sturm bound $240$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$464 = 2^{4} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 464.a (trivial) Character field: $$\Q$$ Newform subspaces: $$14$$ Sturm bound: $$240$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(464))$$.

Total New Old
Modular forms 186 42 144
Cusp forms 174 42 132
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.

$$2$$$$29$$FrickeDim.
$$+$$$$+$$$$+$$$$12$$
$$+$$$$-$$$$-$$$$9$$
$$-$$$$+$$$$-$$$$9$$
$$-$$$$-$$$$+$$$$12$$
Plus space$$+$$$$24$$
Minus space$$-$$$$18$$

## Trace form

 $$42 q - 6 q^{3} + 36 q^{7} + 398 q^{9} + O(q^{10})$$ $$42 q - 6 q^{3} + 36 q^{7} + 398 q^{9} - 66 q^{11} + 72 q^{15} - 76 q^{17} + 90 q^{19} - 84 q^{23} + 1050 q^{25} + 228 q^{27} - 318 q^{31} - 384 q^{33} + 576 q^{35} + 732 q^{39} + 276 q^{41} + 510 q^{43} + 1058 q^{47} + 1802 q^{49} - 44 q^{51} + 1364 q^{55} + 296 q^{57} + 748 q^{59} - 912 q^{61} + 1276 q^{63} + 184 q^{65} - 168 q^{67} + 512 q^{69} + 112 q^{71} + 1044 q^{73} + 18 q^{75} + 656 q^{77} - 1610 q^{79} + 4098 q^{81} + 2196 q^{83} + 1536 q^{85} + 522 q^{87} - 1796 q^{89} + 1936 q^{91} + 800 q^{93} + 208 q^{95} + 756 q^{97} + 874 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(464))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 29
464.4.a.a $$1$$ $$27.377$$ $$\Q$$ None $$0$$ $$-7$$ $$5$$ $$2$$ $$-$$ $$+$$ $$q-7q^{3}+5q^{5}+2q^{7}+22q^{9}-37q^{11}+\cdots$$
464.4.a.b $$1$$ $$27.377$$ $$\Q$$ None $$0$$ $$7$$ $$-15$$ $$18$$ $$-$$ $$+$$ $$q+7q^{3}-15q^{5}+18q^{7}+22q^{9}-3^{3}q^{11}+\cdots$$
464.4.a.c $$2$$ $$27.377$$ $$\Q(\sqrt{22})$$ None $$0$$ $$-10$$ $$30$$ $$0$$ $$-$$ $$-$$ $$q+(-5+\beta )q^{3}+15q^{5}-2\beta q^{7}+(20+\cdots)q^{9}+\cdots$$
464.4.a.d $$2$$ $$27.377$$ $$\Q(\sqrt{13})$$ None $$0$$ $$0$$ $$-10$$ $$20$$ $$-$$ $$+$$ $$q-\beta q^{3}+(-5-2\beta )q^{5}+(10+4\beta )q^{7}+\cdots$$
464.4.a.e $$2$$ $$27.377$$ $$\Q(\sqrt{6})$$ None $$0$$ $$2$$ $$-10$$ $$16$$ $$-$$ $$-$$ $$q+(1+\beta )q^{3}+(-5+6\beta )q^{5}+(8+8\beta )q^{7}+\cdots$$
464.4.a.f $$2$$ $$27.377$$ $$\Q(\sqrt{2})$$ None $$0$$ $$10$$ $$-10$$ $$16$$ $$-$$ $$-$$ $$q+(5+3\beta )q^{3}+(-5+4\beta )q^{5}+(8-10\beta )q^{7}+\cdots$$
464.4.a.g $$3$$ $$27.377$$ 3.3.229.1 None $$0$$ $$-6$$ $$4$$ $$-16$$ $$+$$ $$+$$ $$q+(-1+3\beta _{2})q^{3}+(1+3\beta _{1}-4\beta _{2})q^{5}+\cdots$$
464.4.a.h $$3$$ $$27.377$$ 3.3.4481.1 None $$0$$ $$-3$$ $$11$$ $$38$$ $$+$$ $$+$$ $$q+(-1+\beta _{1})q^{3}+(4+\beta _{1}+\beta _{2})q^{5}+\cdots$$
464.4.a.i $$3$$ $$27.377$$ 3.3.19816.1 None $$0$$ $$-2$$ $$20$$ $$-24$$ $$-$$ $$-$$ $$q+(-1+\beta _{1})q^{3}+(6+2\beta _{1}+\beta _{2})q^{5}+\cdots$$
464.4.a.j $$3$$ $$27.377$$ 3.3.148344.1 None $$0$$ $$10$$ $$-20$$ $$-8$$ $$-$$ $$-$$ $$q+(3+\beta _{1})q^{3}+(-7+\beta _{2})q^{5}+(-4+\cdots)q^{7}+\cdots$$
464.4.a.k $$4$$ $$27.377$$ 4.4.225792.1 None $$0$$ $$0$$ $$-20$$ $$8$$ $$+$$ $$-$$ $$q+\beta _{1}q^{3}+(-5-\beta _{2})q^{5}+(2-2\beta _{1}+\cdots)q^{7}+\cdots$$
464.4.a.l $$5$$ $$27.377$$ 5.5.13458092.1 None $$0$$ $$-8$$ $$10$$ $$-40$$ $$-$$ $$+$$ $$q+(-2+\beta _{3})q^{3}+(2-2\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots$$
464.4.a.m $$5$$ $$27.377$$ $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ None $$0$$ $$-4$$ $$10$$ $$-32$$ $$+$$ $$-$$ $$q+(-1+\beta _{1})q^{3}+(2+\beta _{3})q^{5}+(-6+\cdots)q^{7}+\cdots$$
464.4.a.n $$6$$ $$27.377$$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$0$$ $$5$$ $$-5$$ $$38$$ $$+$$ $$+$$ $$q+(1-\beta _{3})q^{3}+(-1-\beta _{3}-\beta _{4})q^{5}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(464))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(464)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(29))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(58))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(116))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(232))$$$$^{\oplus 2}$$