# Properties

 Label 21.3.f Level 21 Weight 3 Character orbit f Rep. character $$\chi_{21}(10,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 6 Newform subspaces 3 Sturm bound 8 Trace bound 2

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 21.f (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$3$$ Sturm bound: $$8$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(21, [\chi])$$.

Total New Old
Modular forms 14 6 8
Cusp forms 6 6 0
Eisenstein series 8 0 8

## Trace form

 $$6q - 2q^{2} - 3q^{3} - 2q^{4} - 6q^{5} - q^{7} + 8q^{8} + 9q^{9} + O(q^{10})$$ $$6q - 2q^{2} - 3q^{3} - 2q^{4} - 6q^{5} - q^{7} + 8q^{8} + 9q^{9} - 30q^{10} - 14q^{11} + 24q^{12} + 22q^{14} - 24q^{15} + 22q^{16} + 48q^{17} + 6q^{18} + 33q^{19} - 36q^{21} + 28q^{22} - 68q^{23} - 54q^{24} - 9q^{25} - 126q^{26} - 34q^{28} + 64q^{29} + 48q^{30} - 69q^{31} + 12q^{32} + 108q^{33} + 114q^{35} - 12q^{36} + 43q^{37} + 174q^{38} + 9q^{39} + 42q^{40} - 48q^{42} - 134q^{43} - 108q^{44} - 18q^{45} + 52q^{46} - 222q^{47} - 27q^{49} - 68q^{50} + 36q^{51} + 84q^{52} - 32q^{53} + 32q^{56} - 90q^{57} + 34q^{58} + 84q^{59} + 18q^{60} + 216q^{61} - 15q^{63} - 28q^{64} - 78q^{65} - 108q^{66} + 69q^{67} + 36q^{68} - 198q^{70} + 244q^{71} + 12q^{72} - 129q^{73} + 38q^{74} + 39q^{75} + 52q^{77} + 204q^{78} - 21q^{79} + 48q^{80} - 27q^{81} - 144q^{82} + 120q^{84} - 96q^{85} + 158q^{86} + 54q^{87} - 202q^{88} - 60q^{89} + 303q^{91} - 168q^{92} - 45q^{93} - 48q^{94} - 150q^{95} - 234q^{96} - 176q^{98} - 84q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(21, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
21.3.f.a $$2$$ $$0.572$$ $$\Q(\sqrt{-3})$$ None $$-3$$ $$-3$$ $$9$$ $$13$$ $$q-3\zeta_{6}q^{2}+(-1-\zeta_{6})q^{3}+(-5+5\zeta_{6})q^{4}+\cdots$$
21.3.f.b $$2$$ $$0.572$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$3$$ $$-9$$ $$-7$$ $$q-\zeta_{6}q^{2}+(1+\zeta_{6})q^{3}+(3-3\zeta_{6})q^{4}+\cdots$$
21.3.f.c $$2$$ $$0.572$$ $$\Q(\sqrt{-3})$$ None $$2$$ $$-3$$ $$-6$$ $$-7$$ $$q+2\zeta_{6}q^{2}+(-1-\zeta_{6})q^{3}+(-4+2\zeta_{6})q^{5}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 3 T + 5 T^{2} + 12 T^{3} + 16 T^{4}$$)($$1 + T - 3 T^{2} + 4 T^{3} + 16 T^{4}$$)($$( 1 - 2 T )^{2}( 1 + 2 T + 4 T^{2} )$$)
$3$ ($$1 + 3 T + 3 T^{2}$$)($$1 - 3 T + 3 T^{2}$$)($$1 + 3 T + 3 T^{2}$$)
$5$ ($$1 - 9 T + 52 T^{2} - 225 T^{3} + 625 T^{4}$$)($$1 + 9 T + 52 T^{2} + 225 T^{3} + 625 T^{4}$$)($$1 + 6 T + 37 T^{2} + 150 T^{3} + 625 T^{4}$$)
$7$ ($$1 - 13 T + 49 T^{2}$$)($$1 + 7 T + 49 T^{2}$$)($$1 + 7 T + 49 T^{2}$$)
$11$ ($$1 + 15 T + 104 T^{2} + 1815 T^{3} + 14641 T^{4}$$)($$( 1 - 11 T )^{2}( 1 + 11 T + 121 T^{2} )$$)($$1 + 10 T - 21 T^{2} + 1210 T^{3} + 14641 T^{4}$$)
$13$ ($$( 1 - 22 T + 169 T^{2} )( 1 + 22 T + 169 T^{2} )$$)($$1 - 290 T^{2} + 28561 T^{4}$$)($$( 1 - 23 T + 169 T^{2} )( 1 + 23 T + 169 T^{2} )$$)
$17$ ($$1 - 18 T + 397 T^{2} - 5202 T^{3} + 83521 T^{4}$$)($$1 - 42 T + 877 T^{2} - 12138 T^{3} + 83521 T^{4}$$)($$1 + 12 T + 337 T^{2} + 3468 T^{3} + 83521 T^{4}$$)
$19$ ($$1 + 18 T + 469 T^{2} + 6498 T^{3} + 130321 T^{4}$$)($$1 + 6 T + 373 T^{2} + 2166 T^{3} + 130321 T^{4}$$)($$( 1 - 19 T )^{2}( 1 - 19 T + 361 T^{2} )$$)
$23$ ($$1 - 529 T^{2} + 279841 T^{4}$$)($$1 + 28 T + 255 T^{2} + 14812 T^{3} + 279841 T^{4}$$)($$1 + 40 T + 1071 T^{2} + 21160 T^{3} + 279841 T^{4}$$)
$29$ ($$( 1 + 9 T + 841 T^{2} )^{2}$$)($$( 1 - 25 T + 841 T^{2} )^{2}$$)($$( 1 - 16 T + 841 T^{2} )^{2}$$)
$31$ ($$1 + 21 T + 1108 T^{2} + 20181 T^{3} + 923521 T^{4}$$)($$1 + 57 T + 2044 T^{2} + 54777 T^{3} + 923521 T^{4}$$)($$1 - 9 T + 988 T^{2} - 8649 T^{3} + 923521 T^{4}$$)
$37$ ($$1 + 10 T - 1269 T^{2} + 13690 T^{3} + 1874161 T^{4}$$)($$1 - 58 T + 1995 T^{2} - 79402 T^{3} + 1874161 T^{4}$$)($$1 + 5 T - 1344 T^{2} + 6845 T^{3} + 1874161 T^{4}$$)
$41$ ($$1 - 3254 T^{2} + 2825761 T^{4}$$)($$1 - 3350 T^{2} + 2825761 T^{4}$$)($$1 - 2774 T^{2} + 2825761 T^{4}$$)
$43$ ($$( 1 + 74 T + 1849 T^{2} )^{2}$$)($$( 1 - 26 T + 1849 T^{2} )^{2}$$)($$( 1 + 19 T + 1849 T^{2} )^{2}$$)
$47$ ($$( 1 - 47 T + 2209 T^{2} )( 1 + 47 T + 2209 T^{2} )$$)($$1 + 132 T + 8017 T^{2} + 291588 T^{3} + 4879681 T^{4}$$)($$1 + 90 T + 4909 T^{2} + 198810 T^{3} + 4879681 T^{4}$$)
$53$ ($$1 + 33 T - 1720 T^{2} + 92697 T^{3} + 7890481 T^{4}$$)($$1 + 31 T - 1848 T^{2} + 87079 T^{3} + 7890481 T^{4}$$)($$1 - 32 T - 1785 T^{2} - 89888 T^{3} + 7890481 T^{4}$$)
$59$ ($$1 - 27 T + 3724 T^{2} - 93987 T^{3} + 12117361 T^{4}$$)($$1 + 15 T + 3556 T^{2} + 52215 T^{3} + 12117361 T^{4}$$)($$1 - 72 T + 5209 T^{2} - 250632 T^{3} + 12117361 T^{4}$$)
$61$ ($$1 - 156 T + 11833 T^{2} - 580476 T^{3} + 13845841 T^{4}$$)($$1 - 24 T + 3913 T^{2} - 89304 T^{3} + 13845841 T^{4}$$)($$1 - 36 T + 4153 T^{2} - 133956 T^{3} + 13845841 T^{4}$$)
$67$ ($$1 - 76 T + 1287 T^{2} - 341164 T^{3} + 20151121 T^{4}$$)($$1 - 52 T - 1785 T^{2} - 233428 T^{3} + 20151121 T^{4}$$)($$1 + 59 T - 1008 T^{2} + 264851 T^{3} + 20151121 T^{4}$$)
$71$ ($$( 1 - 84 T + 5041 T^{2} )^{2}$$)($$( 1 - 64 T + 5041 T^{2} )^{2}$$)($$( 1 + 26 T + 5041 T^{2} )^{2}$$)
$73$ ($$1 + 108 T + 9217 T^{2} + 575532 T^{3} + 28398241 T^{4}$$)($$1 - 12 T + 5377 T^{2} - 63948 T^{3} + 28398241 T^{4}$$)($$1 + 33 T + 5692 T^{2} + 175857 T^{3} + 28398241 T^{4}$$)
$79$ ($$1 - 43 T - 4392 T^{2} - 268363 T^{3} + 38950081 T^{4}$$)($$1 + 17 T - 5952 T^{2} + 106097 T^{3} + 38950081 T^{4}$$)($$1 + 47 T - 4032 T^{2} + 293327 T^{3} + 38950081 T^{4}$$)
$83$ ($$1 + 505 T^{2} + 47458321 T^{4}$$)($$1 - 10895 T^{2} + 47458321 T^{4}$$)($$1 - 13190 T^{2} + 47458321 T^{4}$$)
$89$ ($$1 + 126 T + 13213 T^{2} + 998046 T^{3} + 62742241 T^{4}$$)($$1 + 138 T + 14269 T^{2} + 1093098 T^{3} + 62742241 T^{4}$$)($$1 - 204 T + 21793 T^{2} - 1615884 T^{3} + 62742241 T^{4}$$)
$97$ ($$1 + 15529 T^{2} + 88529281 T^{4}$$)($$1 - 10391 T^{2} + 88529281 T^{4}$$)($$1 - 16466 T^{2} + 88529281 T^{4}$$)