# Properties

 Label 28.3.h Level 28 Weight 3 Character orbit h Rep. character $$\chi_{28}(5,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 2 Newform subspaces 1 Sturm bound 12 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$28 = 2^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 28.h (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$1$$ Sturm bound: $$12$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(28, [\chi])$$.

Total New Old
Modular forms 22 2 20
Cusp forms 10 2 8
Eisenstein series 12 0 12

## Trace form

 $$2q + 3q^{3} + 3q^{5} - 14q^{7} - 6q^{9} + O(q^{10})$$ $$2q + 3q^{3} + 3q^{5} - 14q^{7} - 6q^{9} - 15q^{11} + 6q^{15} + 51q^{17} + 27q^{19} - 21q^{21} + 9q^{23} - 22q^{25} - 12q^{29} - 21q^{31} - 45q^{33} - 21q^{35} - 31q^{37} + 24q^{39} + 20q^{43} - 18q^{45} + 75q^{47} + 98q^{49} + 51q^{51} + 57q^{53} + 54q^{57} - 141q^{59} - 141q^{61} + 42q^{63} - 24q^{65} + 49q^{67} - 252q^{71} - 45q^{73} - 66q^{75} + 105q^{77} + 73q^{79} - 9q^{81} + 102q^{85} - 18q^{87} + 99q^{89} - 21q^{93} + 27q^{95} + 180q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(28, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
28.3.h.a $$2$$ $$0.763$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$3$$ $$-14$$ $$q+(1+\zeta_{6})q^{3}+(2-\zeta_{6})q^{5}-7q^{7}-6\zeta_{6}q^{9}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(28, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(28, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(14, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 3 T + 12 T^{2} - 27 T^{3} + 81 T^{4}$$
$5$ $$1 - 3 T + 28 T^{2} - 75 T^{3} + 625 T^{4}$$
$7$ $$( 1 + 7 T )^{2}$$
$11$ $$1 + 15 T + 104 T^{2} + 1815 T^{3} + 14641 T^{4}$$
$13$ $$( 1 - 22 T + 169 T^{2} )( 1 + 22 T + 169 T^{2} )$$
$17$ $$( 1 - 17 T )^{2}( 1 - 17 T + 289 T^{2} )$$
$19$ $$1 - 27 T + 604 T^{2} - 9747 T^{3} + 130321 T^{4}$$
$23$ $$1 - 9 T - 448 T^{2} - 4761 T^{3} + 279841 T^{4}$$
$29$ $$( 1 + 6 T + 841 T^{2} )^{2}$$
$31$ $$1 + 21 T + 1108 T^{2} + 20181 T^{3} + 923521 T^{4}$$
$37$ $$1 + 31 T - 408 T^{2} + 42439 T^{3} + 1874161 T^{4}$$
$41$ $$1 - 290 T^{2} + 2825761 T^{4}$$
$43$ $$( 1 - 10 T + 1849 T^{2} )^{2}$$
$47$ $$1 - 75 T + 4084 T^{2} - 165675 T^{3} + 4879681 T^{4}$$
$53$ $$1 - 57 T + 440 T^{2} - 160113 T^{3} + 7890481 T^{4}$$
$59$ $$1 + 141 T + 10108 T^{2} + 490821 T^{3} + 12117361 T^{4}$$
$61$ $$1 + 141 T + 10348 T^{2} + 524661 T^{3} + 13845841 T^{4}$$
$67$ $$1 - 49 T - 2088 T^{2} - 219961 T^{3} + 20151121 T^{4}$$
$71$ $$( 1 + 126 T + 5041 T^{2} )^{2}$$
$73$ $$1 + 45 T + 6004 T^{2} + 239805 T^{3} + 28398241 T^{4}$$
$79$ $$1 - 73 T - 912 T^{2} - 455593 T^{3} + 38950081 T^{4}$$
$83$ $$1 - 13586 T^{2} + 47458321 T^{4}$$
$89$ $$1 - 99 T + 11188 T^{2} - 784179 T^{3} + 62742241 T^{4}$$
$97$ $$1 - 18050 T^{2} + 88529281 T^{4}$$