# Properties

 Label 289.4.a Level $289$ Weight $4$ Character orbit 289.a Rep. character $\chi_{289}(1,\cdot)$ Character field $\Q$ Dimension $60$ Newform subspaces $9$ Sturm bound $102$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$289 = 17^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 289.a (trivial) Character field: $$\Q$$ Newform subspaces: $$9$$ Sturm bound: $$102$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$, $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(289))$$.

Total New Old
Modular forms 86 75 11
Cusp forms 68 60 8
Eisenstein series 18 15 3

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

$$17$$Dim.
$$+$$$$32$$
$$-$$$$28$$

## Trace form

 $$60 q + 2 q^{2} + 4 q^{3} + 198 q^{4} + 2 q^{5} + 50 q^{6} + 6 q^{7} + 18 q^{8} + 336 q^{9} + O(q^{10})$$ $$60 q + 2 q^{2} + 4 q^{3} + 198 q^{4} + 2 q^{5} + 50 q^{6} + 6 q^{7} + 18 q^{8} + 336 q^{9} + 74 q^{10} + 52 q^{11} - 14 q^{12} + 4 q^{13} - 176 q^{14} - 72 q^{15} + 574 q^{16} + 142 q^{18} - 184 q^{19} + 162 q^{20} - 52 q^{21} - 358 q^{22} - 82 q^{23} + 834 q^{24} + 1160 q^{25} - 160 q^{26} + 100 q^{27} - 448 q^{28} + 426 q^{29} - 834 q^{30} - 58 q^{31} - 204 q^{32} + 360 q^{33} + 360 q^{35} - 596 q^{36} - 298 q^{37} - 592 q^{38} - 732 q^{39} + 298 q^{40} + 636 q^{41} + 2102 q^{42} - 280 q^{43} + 1146 q^{44} + 162 q^{45} + 524 q^{46} - 1020 q^{47} - 1342 q^{48} + 532 q^{49} - 448 q^{50} + 432 q^{52} - 312 q^{53} + 860 q^{54} - 16 q^{55} - 96 q^{56} + 1648 q^{57} + 1394 q^{58} - 744 q^{59} - 1294 q^{60} - 26 q^{61} - 1024 q^{62} - 86 q^{63} + 242 q^{64} - 60 q^{65} - 2056 q^{66} - 520 q^{67} - 1540 q^{69} + 756 q^{70} + 1110 q^{71} + 320 q^{72} - 1200 q^{73} - 1010 q^{74} + 836 q^{75} + 660 q^{76} - 572 q^{77} + 84 q^{78} + 1078 q^{79} + 466 q^{80} + 1244 q^{81} - 1384 q^{82} + 1796 q^{83} + 516 q^{84} + 1866 q^{86} - 980 q^{87} - 1334 q^{88} + 812 q^{89} + 2674 q^{90} - 608 q^{91} - 4836 q^{92} - 2100 q^{93} + 2972 q^{94} - 224 q^{95} - 318 q^{96} + 652 q^{97} - 1356 q^{98} + 3808 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(289))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 17
289.4.a.a $1$ $17.052$ $$\Q$$ None $$-3$$ $$8$$ $$-6$$ $$28$$ $+$ $$q-3q^{2}+8q^{3}+q^{4}-6q^{5}-24q^{6}+\cdots$$
289.4.a.b $3$ $17.052$ 3.3.2636.1 None $$1$$ $$-4$$ $$8$$ $$-22$$ $+$ $$q+(\beta _{1}-\beta _{2})q^{2}+(-2+\beta _{1}-2\beta _{2})q^{3}+\cdots$$
289.4.a.c $4$ $17.052$ 4.4.2555057.1 None $$1$$ $$-2$$ $$14$$ $$36$$ $+$ $$q+\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+(3+\beta _{1}+\cdots)q^{4}+\cdots$$
289.4.a.d $4$ $17.052$ 4.4.2555057.1 None $$1$$ $$2$$ $$-14$$ $$-36$$ $-$ $$q+\beta _{1}q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+(3+\beta _{1}+\cdots)q^{4}+\cdots$$
289.4.a.e $4$ $17.052$ 4.4.4669632.2 None $$2$$ $$0$$ $$0$$ $$0$$ $+$ $$q+(1+\beta _{2})q^{2}-\beta _{1}q^{3}+(1+\beta _{2})q^{4}+\cdots$$
289.4.a.f $8$ $17.052$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$8$$ $$0$$ $$0$$ $$0$$ $+$ $$q+(1-\beta _{1})q^{2}-\beta _{3}q^{3}+(5-\beta _{1}+\beta _{6}+\cdots)q^{4}+\cdots$$
289.4.a.g $12$ $17.052$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$-8$$ $$0$$ $$0$$ $$0$$ $-$ $$q+(-1+\beta _{5})q^{2}+\beta _{7}q^{3}+(2+2\beta _{1}+\cdots)q^{4}+\cdots$$
289.4.a.h $12$ $17.052$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$-18$$ $$-30$$ $$-24$$ $-$ $$q-\beta _{1}q^{2}+(-2+\beta _{5})q^{3}+(4+\beta _{2})q^{4}+\cdots$$
289.4.a.i $12$ $17.052$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$18$$ $$30$$ $$24$$ $+$ $$q-\beta _{1}q^{2}+(2-\beta _{5})q^{3}+(4+\beta _{2})q^{4}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(289))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(289)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(17))$$$$^{\oplus 2}$$