# Properties

 Label 53.2.a Level 53 Weight 2 Character orbit a Rep. character $$\chi_{53}(1,\cdot)$$ Character field $$\Q$$ Dimension 4 Newform subspaces 2 Sturm bound 9 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$53$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 53.a (trivial) Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$9$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(53))$$.

Total New Old
Modular forms 5 5 0
Cusp forms 4 4 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 and the Fricke involution.

$$53$$Dim.
$$+$$$$1$$
$$-$$$$3$$

## Trace form

 $$4q - 2q^{2} - 2q^{5} + 8q^{9} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{5} + 8q^{9} - 4q^{10} - 4q^{11} - 2q^{12} - 2q^{14} - 8q^{15} - 4q^{16} - 8q^{17} - 6q^{18} + 6q^{19} + 10q^{20} + 10q^{21} + 10q^{22} + 10q^{23} - 4q^{24} - 8q^{25} + 2q^{26} + 6q^{27} + 16q^{28} - 12q^{29} + 2q^{30} + 2q^{31} - 2q^{32} - 14q^{33} + 18q^{34} + 8q^{35} - 28q^{36} + 8q^{38} + 12q^{39} - 4q^{40} - 4q^{41} - 16q^{42} + 16q^{43} - 24q^{45} - 24q^{46} - 12q^{47} + 16q^{48} + 4q^{49} + 2q^{50} + 4q^{52} + 2q^{53} + 16q^{54} + 4q^{55} - 22q^{56} + 24q^{57} - 4q^{58} + 20q^{60} - 18q^{61} + 28q^{62} - 44q^{63} - 4q^{64} - 2q^{65} + 4q^{66} - 6q^{67} - 8q^{68} - 28q^{69} - 16q^{70} - 4q^{71} + 38q^{72} + 2q^{73} + 32q^{74} + 28q^{75} + 4q^{76} - 4q^{77} - 12q^{78} - 8q^{79} - 10q^{80} + 44q^{81} - 16q^{82} + 26q^{83} - 2q^{84} - 6q^{85} - 30q^{86} + 42q^{87} - 6q^{88} - 16q^{89} + 20q^{90} + 16q^{91} + 20q^{92} - 16q^{93} - 12q^{94} - 12q^{95} + 8q^{96} - 30q^{98} - 24q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(53))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 53
53.2.a.a $$1$$ $$0.423$$ $$\Q$$ None $$-1$$ $$-3$$ $$0$$ $$-4$$ $$+$$ $$q-q^{2}-3q^{3}-q^{4}+3q^{6}-4q^{7}+3q^{8}+\cdots$$
53.2.a.b $$3$$ $$0.423$$ 3.3.148.1 None $$-1$$ $$3$$ $$-2$$ $$4$$ $$-$$ $$q-\beta _{1}q^{2}+(1-\beta _{2})q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + T + 2 T^{2}$$)($$1 + T + 3 T^{2} + 3 T^{3} + 6 T^{4} + 4 T^{5} + 8 T^{6}$$)
$3$ ($$1 + 3 T + 3 T^{2}$$)($$1 - 3 T + 8 T^{2} - 17 T^{3} + 24 T^{4} - 27 T^{5} + 27 T^{6}$$)
$5$ ($$1 + 5 T^{2}$$)($$1 + 2 T + 11 T^{2} + 16 T^{3} + 55 T^{4} + 50 T^{5} + 125 T^{6}$$)
$7$ ($$1 + 4 T + 7 T^{2}$$)($$1 - 4 T + 21 T^{2} - 52 T^{3} + 147 T^{4} - 196 T^{5} + 343 T^{6}$$)
$11$ ($$1 + 11 T^{2}$$)($$1 + 4 T + 29 T^{2} + 68 T^{3} + 319 T^{4} + 484 T^{5} + 1331 T^{6}$$)
$13$ ($$1 + 3 T + 13 T^{2}$$)($$( 1 - T + 13 T^{2} )^{3}$$)
$17$ ($$1 + 3 T + 17 T^{2}$$)($$1 + 5 T + 46 T^{2} + 153 T^{3} + 782 T^{4} + 1445 T^{5} + 4913 T^{6}$$)
$19$ ($$1 + 5 T + 19 T^{2}$$)($$1 - 11 T + 94 T^{2} - 455 T^{3} + 1786 T^{4} - 3971 T^{5} + 6859 T^{6}$$)
$23$ ($$1 - 7 T + 23 T^{2}$$)($$1 - 3 T + 38 T^{2} - 167 T^{3} + 874 T^{4} - 1587 T^{5} + 12167 T^{6}$$)
$29$ ($$1 + 7 T + 29 T^{2}$$)($$1 + 5 T + 50 T^{2} + 229 T^{3} + 1450 T^{4} + 4205 T^{5} + 24389 T^{6}$$)
$31$ ($$1 - 4 T + 31 T^{2}$$)($$1 + 2 T + 17 T^{2} + 240 T^{3} + 527 T^{4} + 1922 T^{5} + 29791 T^{6}$$)
$37$ ($$1 - 5 T + 37 T^{2}$$)($$1 + 5 T + 22 T^{2} + 17 T^{3} + 814 T^{4} + 6845 T^{5} + 50653 T^{6}$$)
$41$ ($$1 - 6 T + 41 T^{2}$$)($$1 + 10 T + 143 T^{2} + 812 T^{3} + 5863 T^{4} + 16810 T^{5} + 68921 T^{6}$$)
$43$ ($$1 + 2 T + 43 T^{2}$$)($$1 - 18 T + 153 T^{2} - 992 T^{3} + 6579 T^{4} - 33282 T^{5} + 79507 T^{6}$$)
$47$ ($$1 + 2 T + 47 T^{2}$$)($$1 + 10 T + 137 T^{2} + 932 T^{3} + 6439 T^{4} + 22090 T^{5} + 103823 T^{6}$$)
$53$ ($$1 + T$$)($$( 1 - T )^{3}$$)
$59$ ($$1 + 2 T + 59 T^{2}$$)($$1 - 2 T + 117 T^{2} - 36 T^{3} + 6903 T^{4} - 6962 T^{5} + 205379 T^{6}$$)
$61$ ($$1 + 8 T + 61 T^{2}$$)($$1 + 10 T + 127 T^{2} + 664 T^{3} + 7747 T^{4} + 37210 T^{5} + 226981 T^{6}$$)
$67$ ($$1 + 12 T + 67 T^{2}$$)($$1 - 6 T + 129 T^{2} - 912 T^{3} + 8643 T^{4} - 26934 T^{5} + 300763 T^{6}$$)
$71$ ($$1 - T + 71 T^{2}$$)($$1 + 5 T + 108 T^{2} + 987 T^{3} + 7668 T^{4} + 25205 T^{5} + 357911 T^{6}$$)
$73$ ($$1 + 4 T + 73 T^{2}$$)($$1 - 6 T + 191 T^{2} - 880 T^{3} + 13943 T^{4} - 31974 T^{5} + 389017 T^{6}$$)
$79$ ($$1 + T + 79 T^{2}$$)($$1 + 7 T + 160 T^{2} + 1237 T^{3} + 12640 T^{4} + 43687 T^{5} + 493039 T^{6}$$)
$83$ ($$1 + T + 83 T^{2}$$)($$1 - 27 T + 462 T^{2} - 4939 T^{3} + 38346 T^{4} - 186003 T^{5} + 571787 T^{6}$$)
$89$ ($$1 + 14 T + 89 T^{2}$$)($$1 + 2 T + 55 T^{2} + 1404 T^{3} + 4895 T^{4} + 15842 T^{5} + 704969 T^{6}$$)
$97$ ($$1 - T + 97 T^{2}$$)($$1 + T + 158 T^{2} + 57 T^{3} + 15326 T^{4} + 9409 T^{5} + 912673 T^{6}$$)