# Properties

 Label 73.2.a Level 73 Weight 2 Character orbit a Rep. character $$\chi_{73}(1,\cdot)$$ Character field $$\Q$$ Dimension 5 Newform subspaces 3 Sturm bound 12 Trace bound 2

# Related objects

## Defining parameters

 Level: $$N$$ = $$73$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 73.a (trivial) Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$12$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(73))$$.

Total New Old
Modular forms 6 6 0
Cusp forms 5 5 0
Eisenstein series 1 1 0

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

$$73$$Dim.
$$+$$$$2$$
$$-$$$$3$$

## Trace form

 $$5q - q^{2} - 2q^{3} + 5q^{4} - 2q^{5} - 4q^{6} - 6q^{7} - 3q^{8} - q^{9} + O(q^{10})$$ $$5q - q^{2} - 2q^{3} + 5q^{4} - 2q^{5} - 4q^{6} - 6q^{7} - 3q^{8} - q^{9} + 2q^{10} + 2q^{11} - 2q^{12} - 6q^{13} + 10q^{14} + 8q^{15} + 9q^{16} - 2q^{17} - 3q^{18} - 4q^{19} - 22q^{20} + 8q^{21} + 10q^{22} + 2q^{23} + 2q^{24} - 7q^{25} + 6q^{26} + 4q^{27} - 14q^{28} + 10q^{29} + 6q^{31} - 17q^{32} + 4q^{33} - 2q^{34} + 14q^{35} - 23q^{36} - 2q^{37} - 2q^{38} - 16q^{39} + 10q^{40} - 6q^{41} + 2q^{43} + 22q^{44} + 6q^{45} + 42q^{46} + 18q^{47} - 20q^{48} - 11q^{49} + q^{50} - 10q^{52} + 14q^{53} + 10q^{54} - 12q^{55} + 6q^{56} - 10q^{57} - 42q^{58} - 18q^{59} + 12q^{60} - 16q^{61} + 26q^{62} - 10q^{63} + 7q^{64} - 12q^{65} - 24q^{66} + 28q^{67} + 50q^{68} + 20q^{69} - 10q^{70} - 18q^{71} + 39q^{72} + q^{73} - 24q^{74} + 4q^{75} - 26q^{76} - 2q^{77} + 4q^{78} - 22q^{79} - 34q^{80} - 3q^{81} + 10q^{82} - 10q^{83} - 4q^{84} - 22q^{85} - 22q^{86} + 28q^{87} + 26q^{88} + 18q^{89} - 22q^{90} - 14q^{91} - 8q^{92} - 28q^{93} + 14q^{94} + 20q^{95} + 18q^{96} - 24q^{97} - 15q^{98} - 6q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(73))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 73
73.2.a.a $$1$$ $$0.583$$ $$\Q$$ None $$1$$ $$0$$ $$2$$ $$2$$ $$-$$ $$q+q^{2}-q^{4}+2q^{5}+2q^{7}-3q^{8}-3q^{9}+\cdots$$
73.2.a.b $$2$$ $$0.583$$ $$\Q(\sqrt{5})$$ None $$-3$$ $$-3$$ $$-3$$ $$-6$$ $$+$$ $$q+(-1-\beta )q^{2}+(-2+\beta )q^{3}+3\beta q^{4}+\cdots$$
73.2.a.c $$2$$ $$0.583$$ $$\Q(\sqrt{13})$$ None $$1$$ $$1$$ $$-1$$ $$-2$$ $$-$$ $$q+\beta q^{2}+(1-\beta )q^{3}+(1+\beta )q^{4}-\beta q^{5}+\cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - T + 2 T^{2}$$)($$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$)($$1 - T + T^{2} - 2 T^{3} + 4 T^{4}$$)
$3$ ($$1 + 3 T^{2}$$)($$1 + 3 T + 7 T^{2} + 9 T^{3} + 9 T^{4}$$)($$1 - T + 3 T^{2} - 3 T^{3} + 9 T^{4}$$)
$5$ ($$1 - 2 T + 5 T^{2}$$)($$1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4}$$)($$1 + T + 7 T^{2} + 5 T^{3} + 25 T^{4}$$)
$7$ ($$1 - 2 T + 7 T^{2}$$)($$( 1 + 3 T + 7 T^{2} )^{2}$$)($$( 1 + T + 7 T^{2} )^{2}$$)
$11$ ($$1 + 2 T + 11 T^{2}$$)($$1 + 3 T + 23 T^{2} + 33 T^{3} + 121 T^{4}$$)($$1 - 7 T + 31 T^{2} - 77 T^{3} + 121 T^{4}$$)
$13$ ($$1 + 6 T + 13 T^{2}$$)($$1 - T + 15 T^{2} - 13 T^{3} + 169 T^{4}$$)($$1 + T + 23 T^{2} + 13 T^{3} + 169 T^{4}$$)
$17$ ($$1 - 2 T + 17 T^{2}$$)($$1 - 11 T^{2} + 289 T^{4}$$)($$1 + 4 T + 25 T^{2} + 68 T^{3} + 289 T^{4}$$)
$19$ ($$1 - 8 T + 19 T^{2}$$)($$( 1 - T + 19 T^{2} )^{2}$$)($$( 1 + 7 T + 19 T^{2} )^{2}$$)
$23$ ($$1 - 4 T + 23 T^{2}$$)($$1 + 15 T + 101 T^{2} + 345 T^{3} + 529 T^{4}$$)($$1 - 13 T + 85 T^{2} - 299 T^{3} + 529 T^{4}$$)
$29$ ($$1 - 2 T + 29 T^{2}$$)($$1 - 6 T + 47 T^{2} - 174 T^{3} + 841 T^{4}$$)($$1 - 2 T + 7 T^{2} - 58 T^{3} + 841 T^{4}$$)
$31$ ($$1 + 2 T + 31 T^{2}$$)($$1 - 2 T + 18 T^{2} - 62 T^{3} + 961 T^{4}$$)($$1 - 6 T + 58 T^{2} - 186 T^{3} + 961 T^{4}$$)
$37$ ($$1 + 6 T + 37 T^{2}$$)($$1 + 4 T + 33 T^{2} + 148 T^{3} + 1369 T^{4}$$)($$1 - 8 T + 77 T^{2} - 296 T^{3} + 1369 T^{4}$$)
$41$ ($$1 - 6 T + 41 T^{2}$$)($$1 + 62 T^{2} + 1681 T^{4}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)
$43$ ($$1 + 2 T + 43 T^{2}$$)($$( 1 + T + 43 T^{2} )^{2}$$)($$1 - 6 T + 43 T^{2} - 258 T^{3} + 1849 T^{4}$$)
$47$ ($$1 - 6 T + 47 T^{2}$$)($$1 + 6 T + 83 T^{2} + 282 T^{3} + 2209 T^{4}$$)($$( 1 - 9 T + 47 T^{2} )^{2}$$)
$53$ ($$1 - 10 T + 53 T^{2}$$)($$1 - 6 T + 35 T^{2} - 318 T^{3} + 2809 T^{4}$$)($$1 + 2 T + 55 T^{2} + 106 T^{3} + 2809 T^{4}$$)
$59$ ($$1 + 6 T + 59 T^{2}$$)($$1 + 12 T + 134 T^{2} + 708 T^{3} + 3481 T^{4}$$)($$( 1 + 59 T^{2} )^{2}$$)
$61$ ($$1 + 14 T + 61 T^{2}$$)($$1 - 7 T + 123 T^{2} - 427 T^{3} + 3721 T^{4}$$)($$1 + 9 T + 139 T^{2} + 549 T^{3} + 3721 T^{4}$$)
$67$ ($$1 - 8 T + 67 T^{2}$$)($$1 - 16 T + 153 T^{2} - 1072 T^{3} + 4489 T^{4}$$)($$1 - 4 T + 21 T^{2} - 268 T^{3} + 4489 T^{4}$$)
$71$ ($$1 + 71 T^{2}$$)($$1 + 21 T + 251 T^{2} + 1491 T^{3} + 5041 T^{4}$$)($$1 - 3 T + 115 T^{2} - 213 T^{3} + 5041 T^{4}$$)
$73$ ($$1 - T$$)($$( 1 + T )^{2}$$)($$( 1 - T )^{2}$$)
$79$ ($$1 + 4 T + 79 T^{2}$$)($$1 + 19 T + 237 T^{2} + 1501 T^{3} + 6241 T^{4}$$)($$1 - T + 129 T^{2} - 79 T^{3} + 6241 T^{4}$$)
$83$ ($$1 + 14 T + 83 T^{2}$$)($$1 + 3 T + 157 T^{2} + 249 T^{3} + 6889 T^{4}$$)($$1 - 7 T + 97 T^{2} - 581 T^{3} + 6889 T^{4}$$)
$89$ ($$1 + 6 T + 89 T^{2}$$)($$1 - 12 T + 209 T^{2} - 1068 T^{3} + 7921 T^{4}$$)($$1 - 12 T + 97 T^{2} - 1068 T^{3} + 7921 T^{4}$$)
$97$ ($$1 + 10 T + 97 T^{2}$$)($$1 + 9 T + 203 T^{2} + 873 T^{3} + 9409 T^{4}$$)($$1 + 5 T + 171 T^{2} + 485 T^{3} + 9409 T^{4}$$)