# Properties

 Label 209.4.a Level $209$ Weight $4$ Character orbit 209.a Rep. character $\chi_{209}(1,\cdot)$ Character field $\Q$ Dimension $46$ Newform subspaces $4$ Sturm bound $80$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$209 = 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 209.a (trivial) Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$80$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(209))$$.

Total New Old
Modular forms 62 46 16
Cusp forms 58 46 12
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.

$$11$$$$19$$FrickeDim.
$$+$$$$+$$$$+$$$$13$$
$$+$$$$-$$$$-$$$$8$$
$$-$$$$+$$$$-$$$$10$$
$$-$$$$-$$$$+$$$$15$$
Plus space$$+$$$$28$$
Minus space$$-$$$$18$$

## Trace form

 $$46 q - 4 q^{2} + 4 q^{3} + 192 q^{4} - 4 q^{5} + 76 q^{6} - 60 q^{8} + 362 q^{9} + O(q^{10})$$ $$46 q - 4 q^{2} + 4 q^{3} + 192 q^{4} - 4 q^{5} + 76 q^{6} - 60 q^{8} + 362 q^{9} + 8 q^{10} + 44 q^{11} + 128 q^{12} - 168 q^{13} + 28 q^{14} - 76 q^{15} + 888 q^{16} + 88 q^{17} - 692 q^{18} - 136 q^{20} + 64 q^{21} + 160 q^{23} + 172 q^{24} + 850 q^{25} - 460 q^{26} + 628 q^{27} + 864 q^{28} - 240 q^{29} + 1608 q^{30} + 212 q^{31} - 604 q^{32} - 176 q^{33} - 768 q^{34} + 56 q^{35} + 396 q^{36} + 704 q^{37} + 228 q^{38} - 180 q^{39} + 1368 q^{40} - 1784 q^{41} - 420 q^{42} + 760 q^{43} - 176 q^{44} - 1144 q^{45} + 2328 q^{46} + 568 q^{47} + 1088 q^{48} + 3342 q^{49} + 2692 q^{50} + 464 q^{51} + 1856 q^{52} + 276 q^{53} + 4 q^{54} + 44 q^{55} - 268 q^{56} - 1676 q^{58} - 1236 q^{59} - 3876 q^{60} - 1736 q^{61} - 2356 q^{62} + 1252 q^{63} + 5488 q^{64} + 1520 q^{65} + 528 q^{66} + 588 q^{67} - 2868 q^{68} - 2772 q^{69} - 6700 q^{70} - 76 q^{71} - 10460 q^{72} + 816 q^{73} - 2148 q^{74} - 1872 q^{75} + 440 q^{77} + 1212 q^{78} + 1088 q^{79} + 2576 q^{80} + 8958 q^{81} - 1760 q^{82} - 1008 q^{83} - 4448 q^{84} - 2008 q^{85} - 7168 q^{86} - 2948 q^{87} + 472 q^{89} - 5480 q^{90} - 208 q^{91} + 508 q^{92} + 2644 q^{93} - 1048 q^{94} - 6708 q^{96} - 1464 q^{97} + 2700 q^{98} + 968 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(209))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 11 19
209.4.a.a $8$ $12.331$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$-1$$ $$-12$$ $$-59$$ $+$ $-$ $$q-\beta _{1}q^{2}+\beta _{3}q^{3}+(3+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots$$
209.4.a.b $10$ $12.331$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$-6$$ $$-9$$ $$-10$$ $$-53$$ $-$ $+$ $$q+(-1+\beta _{1})q^{2}+(-1+\beta _{4})q^{3}+(2+\cdots)q^{4}+\cdots$$
209.4.a.c $13$ $12.331$ $$\mathbb{Q}[x]/(x^{13} - \cdots)$$ None $$-2$$ $$11$$ $$8$$ $$39$$ $+$ $+$ $$q-\beta _{1}q^{2}+(1+\beta _{5})q^{3}+(6+\beta _{2})q^{4}+\cdots$$
209.4.a.d $15$ $12.331$ $$\mathbb{Q}[x]/(x^{15} - \cdots)$$ None $$4$$ $$3$$ $$10$$ $$73$$ $-$ $-$ $$q+\beta _{1}q^{2}+\beta _{3}q^{3}+(5+\beta _{2})q^{4}+(1+\beta _{6}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(209))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(209)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(11))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(19))$$$$^{\oplus 2}$$