# Properties

 Label 13.4.b Level 13 Weight 4 Character orbit b Rep. character $$\chi_{13}(12,\cdot)$$ Character field $$\Q$$ Dimension 2 Newform subspaces 1 Sturm bound 4 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 13.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$4$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(13, [\chi])$$.

Total New Old
Modular forms 4 4 0
Cusp forms 2 2 0
Eisenstein series 2 2 0

## Trace form

 $$2q - 2q^{3} - 2q^{4} - 52q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{4} - 52q^{9} + 54q^{10} + 2q^{12} + 52q^{13} + 90q^{14} - 142q^{16} - 90q^{17} - 288q^{22} + 324q^{23} + 88q^{25} + 234q^{26} + 106q^{27} - 288q^{29} - 54q^{30} - 270q^{35} + 52q^{36} - 36q^{38} - 52q^{39} + 378q^{40} - 90q^{42} - 194q^{43} + 142q^{48} + 236q^{49} + 90q^{51} - 52q^{52} - 828q^{53} + 864q^{55} + 630q^{56} + 752q^{61} - 1584q^{62} - 866q^{64} - 702q^{65} + 288q^{66} + 90q^{68} - 324q^{69} + 1818q^{74} - 88q^{75} + 1440q^{77} - 234q^{78} - 1660q^{79} + 1298q^{81} + 1152q^{82} + 288q^{87} - 2016q^{88} - 1404q^{90} - 1170q^{91} - 324q^{92} + 666q^{94} + 108q^{95} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(13, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
13.4.b.a $$2$$ $$0.767$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q+iq^{2}-q^{3}-q^{4}-3iq^{5}-iq^{6}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 7 T^{2} + 64 T^{4}$$
$3$ $$( 1 + T + 27 T^{2} )^{2}$$
$5$ $$1 - 169 T^{2} + 15625 T^{4}$$
$7$ $$1 - 461 T^{2} + 117649 T^{4}$$
$11$ $$1 - 358 T^{2} + 1771561 T^{4}$$
$13$ $$1 - 52 T + 2197 T^{2}$$
$17$ $$( 1 + 45 T + 4913 T^{2} )^{2}$$
$19$ $$1 - 13682 T^{2} + 47045881 T^{4}$$
$23$ $$( 1 - 162 T + 12167 T^{2} )^{2}$$
$29$ $$( 1 + 144 T + 24389 T^{2} )^{2}$$
$31$ $$1 + 10114 T^{2} + 887503681 T^{4}$$
$37$ $$1 - 9497 T^{2} + 2565726409 T^{4}$$
$41$ $$1 - 100978 T^{2} + 4750104241 T^{4}$$
$43$ $$( 1 + 97 T + 79507 T^{2} )^{2}$$
$47$ $$1 - 195325 T^{2} + 10779215329 T^{4}$$
$53$ $$( 1 + 414 T + 148877 T^{2} )^{2}$$
$59$ $$1 - 138274 T^{2} + 42180533641 T^{4}$$
$61$ $$( 1 - 376 T + 226981 T^{2} )^{2}$$
$67$ $$1 - 600230 T^{2} + 90458382169 T^{4}$$
$71$ $$1 - 588373 T^{2} + 128100283921 T^{4}$$
$73$ $$( 1 - 592 T + 389017 T^{2} )( 1 + 592 T + 389017 T^{2} )$$
$79$ $$( 1 + 830 T + 493039 T^{2} )^{2}$$
$83$ $$1 - 951730 T^{2} + 326940373369 T^{4}$$
$89$ $$1 - 1218094 T^{2} + 496981290961 T^{4}$$
$97$ $$1 - 1099442 T^{2} + 832972004929 T^{4}$$