# Properties

 Label 19.3.b Level 19 Weight 3 Character orbit b Rep. character $$\chi_{19}(18,\cdot)$$ Character field $$\Q$$ Dimension 3 Newform subspaces 2 Sturm bound 5 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 19.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$19$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$5$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(19, [\chi])$$.

Total New Old
Modular forms 5 5 0
Cusp forms 3 3 0
Eisenstein series 2 2 0

## Trace form

 $$3q - 14q^{4} - q^{5} + 26q^{6} - 15q^{7} + q^{9} + O(q^{10})$$ $$3q - 14q^{4} - q^{5} + 26q^{6} - 15q^{7} + q^{9} - 17q^{11} + 74q^{16} + 45q^{17} - 31q^{19} - 108q^{20} + 40q^{23} - 130q^{24} + 38q^{25} - 26q^{26} + 70q^{28} + 104q^{30} + 5q^{35} + 108q^{36} - 130q^{38} + 26q^{39} - 130q^{42} - 125q^{43} + 192q^{44} - 113q^{45} + 95q^{47} - 72q^{49} + 130q^{54} - 107q^{55} + 130q^{57} - 130q^{58} + 23q^{61} + 260q^{62} - 5q^{63} + 62q^{64} - 260q^{66} - 210q^{68} + 185q^{73} + 156q^{74} + 32q^{76} + 85q^{77} + 88q^{80} - 121q^{81} - 260q^{82} + 10q^{83} - 15q^{85} + 130q^{87} - 750q^{92} - 260q^{93} + 123q^{95} + 234q^{96} + 107q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(19, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
19.3.b.a $$1$$ $$0.518$$ $$\Q$$ $$\Q(\sqrt{-19})$$ $$0$$ $$0$$ $$-9$$ $$-5$$ $$q+4q^{4}-9q^{5}-5q^{7}+9q^{9}+3q^{11}+\cdots$$
19.3.b.b $$2$$ $$0.518$$ $$\Q(\sqrt{-13})$$ None $$0$$ $$0$$ $$8$$ $$-10$$ $$q+\beta q^{2}-\beta q^{3}-9q^{4}+4q^{5}+13q^{6}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - 2 T )( 1 + 2 T )$$)($$1 + 5 T^{2} + 16 T^{4}$$)
$3$ ($$( 1 - 3 T )( 1 + 3 T )$$)($$1 - 5 T^{2} + 81 T^{4}$$)
$5$ ($$1 + 9 T + 25 T^{2}$$)($$( 1 - 4 T + 25 T^{2} )^{2}$$)
$7$ ($$1 + 5 T + 49 T^{2}$$)($$( 1 + 5 T + 49 T^{2} )^{2}$$)
$11$ ($$1 - 3 T + 121 T^{2}$$)($$( 1 + 10 T + 121 T^{2} )^{2}$$)
$13$ ($$( 1 - 13 T )( 1 + 13 T )$$)($$1 - 325 T^{2} + 28561 T^{4}$$)
$17$ ($$1 - 15 T + 289 T^{2}$$)($$( 1 - 15 T + 289 T^{2} )^{2}$$)
$19$ ($$1 + 19 T$$)($$1 + 12 T + 361 T^{2}$$)
$23$ ($$1 + 30 T + 529 T^{2}$$)($$( 1 - 35 T + 529 T^{2} )^{2}$$)
$29$ ($$( 1 - 29 T )( 1 + 29 T )$$)($$1 - 1357 T^{2} + 707281 T^{4}$$)
$31$ ($$( 1 - 31 T )( 1 + 31 T )$$)($$1 - 622 T^{2} + 923521 T^{4}$$)
$37$ ($$( 1 - 37 T )( 1 + 37 T )$$)($$1 - 2270 T^{2} + 1874161 T^{4}$$)
$41$ ($$( 1 - 41 T )( 1 + 41 T )$$)($$1 - 2062 T^{2} + 2825761 T^{4}$$)
$43$ ($$1 + 85 T + 1849 T^{2}$$)($$( 1 + 20 T + 1849 T^{2} )^{2}$$)
$47$ ($$1 - 75 T + 2209 T^{2}$$)($$( 1 - 10 T + 2209 T^{2} )^{2}$$)
$53$ ($$( 1 - 53 T )( 1 + 53 T )$$)($$1 + 115 T^{2} + 7890481 T^{4}$$)
$59$ ($$( 1 - 59 T )( 1 + 59 T )$$)($$1 - 6637 T^{2} + 12117361 T^{4}$$)
$61$ ($$1 - 103 T + 3721 T^{2}$$)($$( 1 + 40 T + 3721 T^{2} )^{2}$$)
$67$ ($$( 1 - 67 T )( 1 + 67 T )$$)($$1 - 7405 T^{2} + 20151121 T^{4}$$)
$71$ ($$( 1 - 71 T )( 1 + 71 T )$$)($$( 1 - 92 T + 5041 T^{2} )( 1 + 92 T + 5041 T^{2} )$$)
$73$ ($$1 + 25 T + 5329 T^{2}$$)($$( 1 - 105 T + 5329 T^{2} )^{2}$$)
$79$ ($$( 1 - 79 T )( 1 + 79 T )$$)($$1 - 11182 T^{2} + 38950081 T^{4}$$)
$83$ ($$1 - 90 T + 6889 T^{2}$$)($$( 1 + 40 T + 6889 T^{2} )^{2}$$)
$89$ ($$( 1 - 89 T )( 1 + 89 T )$$)($$( 1 - 89 T )^{2}( 1 + 89 T )^{2}$$)
$97$ ($$( 1 - 97 T )( 1 + 97 T )$$)($$1 - 3790 T^{2} + 88529281 T^{4}$$)