# Properties

 Label 42.3.b Level $42$ Weight $3$ Character orbit 42.b Rep. character $\chi_{42}(29,\cdot)$ Character field $\Q$ Dimension $4$ Newform subspaces $1$ Sturm bound $24$ Trace bound $0$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$42 = 2 \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 42.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$3$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$24$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(42, [\chi])$$.

Total New Old
Modular forms 20 4 16
Cusp forms 12 4 8
Eisenstein series 8 0 8

## Trace form

 $$4q - 8q^{4} - 8q^{6} + 20q^{9} + O(q^{10})$$ $$4q - 8q^{4} - 8q^{6} + 20q^{9} - 16q^{10} + 40q^{13} - 16q^{15} + 16q^{16} - 64q^{19} - 28q^{21} + 40q^{22} + 16q^{24} + 12q^{25} + 56q^{30} - 128q^{31} + 40q^{33} + 16q^{34} - 40q^{36} + 80q^{37} - 112q^{39} + 32q^{40} + 80q^{43} + 112q^{45} - 8q^{46} + 28q^{49} + 16q^{51} - 80q^{52} - 152q^{54} - 32q^{55} - 160q^{58} + 32q^{60} - 56q^{61} - 32q^{64} + 112q^{66} + 240q^{67} - 8q^{69} - 56q^{70} + 120q^{73} + 224q^{75} + 128q^{76} - 80q^{78} - 128q^{79} - 124q^{81} + 240q^{82} + 56q^{84} - 248q^{85} - 160q^{87} - 80q^{88} - 80q^{90} + 112q^{91} - 280q^{93} + 48q^{94} - 32q^{96} - 360q^{97} + 224q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(42, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
42.3.b.a $$4$$ $$1.144$$ $$\Q(\sqrt{-2}, \sqrt{7})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+(-\beta _{1}+\beta _{3})q^{3}-2q^{4}+(-2\beta _{1}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(42, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(42, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} )^{2}$$
$3$ $$1 - 10 T^{2} + 81 T^{4}$$
$5$ $$1 - 56 T^{2} + 1586 T^{4} - 35000 T^{6} + 390625 T^{8}$$
$7$ $$( 1 - 7 T^{2} )^{2}$$
$11$ $$1 - 272 T^{2} + 36578 T^{4} - 3982352 T^{6} + 214358881 T^{8}$$
$13$ $$( 1 - 20 T + 326 T^{2} - 3380 T^{3} + 28561 T^{4} )^{2}$$
$17$ $$1 - 440 T^{2} + 204242 T^{4} - 36749240 T^{6} + 6975757441 T^{8}$$
$19$ $$( 1 + 16 T + 361 T^{2} )^{4}$$
$23$ $$1 + 688 T^{2} + 666818 T^{4} + 192530608 T^{6} + 78310985281 T^{8}$$
$29$ $$1 - 1652 T^{2} + 1917638 T^{4} - 1168428212 T^{6} + 500246412961 T^{8}$$
$31$ $$( 1 + 64 T + 2246 T^{2} + 61504 T^{3} + 923521 T^{4} )^{2}$$
$37$ $$( 1 - 20 T + 1369 T^{2} )^{4}$$
$41$ $$1 - 856 T^{2} - 2330094 T^{4} - 2418851416 T^{6} + 7984925229121 T^{8}$$
$43$ $$( 1 - 40 T + 3090 T^{2} - 73960 T^{3} + 3418801 T^{4} )^{2}$$
$47$ $$( 1 - 4346 T^{2} + 4879681 T^{4} )^{2}$$
$53$ $$( 1 - 3026 T^{2} + 7890481 T^{4} )^{2}$$
$59$ $$1 - 10532 T^{2} + 49098278 T^{4} - 127620046052 T^{6} + 146830437604321 T^{8}$$
$61$ $$( 1 + 28 T + 4838 T^{2} + 104188 T^{3} + 13845841 T^{4} )^{2}$$
$67$ $$( 1 - 120 T + 12466 T^{2} - 538680 T^{3} + 20151121 T^{4} )^{2}$$
$71$ $$1 - 12176 T^{2} + 74167106 T^{4} - 309412627856 T^{6} + 645753531245761 T^{8}$$
$73$ $$( 1 - 60 T + 9766 T^{2} - 319740 T^{3} + 28398241 T^{4} )^{2}$$
$79$ $$( 1 + 64 T + 10706 T^{2} + 399424 T^{3} + 38950081 T^{4} )^{2}$$
$83$ $$1 - 6356 T^{2} - 6983674 T^{4} - 301645088276 T^{6} + 2252292232139041 T^{8}$$
$89$ $$1 - 27832 T^{2} + 318232338 T^{4} - 1746242051512 T^{6} + 3936588805702081 T^{8}$$
$97$ $$( 1 + 180 T + 26470 T^{2} + 1693620 T^{3} + 88529281 T^{4} )^{2}$$