# Properties

 Label 21.3 Level 21 Weight 3 Dimension 18 Nonzero newspaces 4 Newform subspaces 7 Sturm bound 96 Trace bound 2

## Defining parameters

 Level: $$N$$ = $$21\( 21 = 3 \cdot 7$$ \) Weight: $$k$$ = $$3$$ Nonzero newspaces: $$4$$ Newform subspaces: $$7$$ Sturm bound: $$96$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(21))$$.

Total New Old
Modular forms 44 26 18
Cusp forms 20 18 2
Eisenstein series 24 8 16

## Trace form

 $$18q - 6q^{3} - 22q^{4} - 6q^{5} - 6q^{6} + 2q^{7} - 6q^{8} - 24q^{9} + O(q^{10})$$ $$18q - 6q^{3} - 22q^{4} - 6q^{5} - 6q^{6} + 2q^{7} - 6q^{8} - 24q^{9} - 12q^{10} + 6q^{11} + 18q^{12} + 2q^{13} + 24q^{14} + 48q^{15} + 90q^{16} + 48q^{17} + 96q^{18} + 2q^{19} - 24q^{21} - 108q^{22} - 96q^{23} - 150q^{24} - 132q^{25} - 126q^{26} - 132q^{27} - 102q^{28} - 12q^{29} + 24q^{30} + 86q^{31} + 78q^{32} + 186q^{33} + 240q^{34} + 210q^{35} + 198q^{36} + 160q^{37} + 174q^{38} + 66q^{39} + 96q^{40} - 72q^{42} - 188q^{43} - 168q^{44} - 198q^{45} - 204q^{46} - 222q^{47} - 210q^{48} - 120q^{49} - 114q^{50} - 168q^{51} + 152q^{52} - 12q^{53} - 84q^{54} + 156q^{55} + 18q^{56} + 108q^{57} + 84q^{59} + 96q^{60} + 38q^{61} + 78q^{63} - 34q^{64} + 18q^{65} - 36q^{66} + 102q^{67} + 36q^{68} + 288q^{69} - 60q^{70} + 120q^{71} + 174q^{72} - 100q^{73} + 90q^{74} + 180q^{75} - 292q^{76} + 72q^{77} + 24q^{78} + 162q^{79} + 48q^{80} + 108q^{82} + 102q^{84} - 168q^{85} + 210q^{86} + 12q^{87} - 24q^{88} - 60q^{89} - 132q^{90} - 148q^{91} - 84q^{92} - 264q^{93} - 324q^{94} - 438q^{95} - 402q^{96} - 376q^{97} - 270q^{98} - 432q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(21))$$

We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
21.3.b $$\chi_{21}(8, \cdot)$$ 21.3.b.a 4 1
21.3.d $$\chi_{21}(13, \cdot)$$ 21.3.d.a 2 1
21.3.f $$\chi_{21}(10, \cdot)$$ 21.3.f.a 2 2
21.3.f.b 2
21.3.f.c 2
21.3.h $$\chi_{21}(2, \cdot)$$ 21.3.h.a 2 2
21.3.h.b 4

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(21))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(21)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 - 2 T^{2} + 5 T^{4} - 32 T^{6} + 256 T^{8}$$)($$( 1 - T + 4 T^{2} )^{2}$$)($$1 + 3 T + 5 T^{2} + 12 T^{3} + 16 T^{4}$$)($$1 + T - 3 T^{2} + 4 T^{3} + 16 T^{4}$$)($$( 1 - 2 T )^{2}( 1 + 2 T + 4 T^{2} )$$)($$( 1 - 2 T + 4 T^{2} )( 1 + 2 T + 4 T^{2} )$$)($$1 + 3 T^{2} - 7 T^{4} + 48 T^{6} + 256 T^{8}$$)
$3$ ($$1 + 2 T + 12 T^{2} + 18 T^{3} + 81 T^{4}$$)($$1 + 3 T^{2}$$)($$1 + 3 T + 3 T^{2}$$)($$1 - 3 T + 3 T^{2}$$)($$1 + 3 T + 3 T^{2}$$)($$1 - 3 T + 9 T^{2}$$)($$1 + 4 T + 7 T^{2} + 36 T^{3} + 81 T^{4}$$)
$5$ ($$1 - 44 T^{2} + 1034 T^{4} - 27500 T^{6} + 390625 T^{8}$$)($$1 - 2 T^{2} + 625 T^{4}$$)($$1 - 9 T + 52 T^{2} - 225 T^{3} + 625 T^{4}$$)($$1 + 9 T + 52 T^{2} + 225 T^{3} + 625 T^{4}$$)($$1 + 6 T + 37 T^{2} + 150 T^{3} + 625 T^{4}$$)($$( 1 - 5 T + 25 T^{2} )( 1 + 5 T + 25 T^{2} )$$)($$1 + 45 T^{2} + 1400 T^{4} + 28125 T^{6} + 390625 T^{8}$$)
$7$ ($$( 1 - 7 T^{2} )^{2}$$)($$1 - 2 T + 49 T^{2}$$)($$1 - 13 T + 49 T^{2}$$)($$1 + 7 T + 49 T^{2}$$)($$1 + 7 T + 49 T^{2}$$)($$1 + 13 T + 49 T^{2}$$)($$( 1 - 7 T + 49 T^{2} )^{2}$$)
$11$ ($$1 - 428 T^{2} + 74630 T^{4} - 6266348 T^{6} + 214358881 T^{8}$$)($$( 1 - 10 T + 121 T^{2} )^{2}$$)($$1 + 15 T + 104 T^{2} + 1815 T^{3} + 14641 T^{4}$$)($$( 1 - 11 T )^{2}( 1 + 11 T + 121 T^{2} )$$)($$1 + 10 T - 21 T^{2} + 1210 T^{3} + 14641 T^{4}$$)($$( 1 - 11 T + 121 T^{2} )( 1 + 11 T + 121 T^{2} )$$)($$1 + 117 T^{2} - 952 T^{4} + 1712997 T^{6} + 214358881 T^{8}$$)
$13$ ($$( 1 + 18 T + 412 T^{2} + 3042 T^{3} + 28561 T^{4} )^{2}$$)($$1 - 290 T^{2} + 28561 T^{4}$$)($$( 1 - 22 T + 169 T^{2} )( 1 + 22 T + 169 T^{2} )$$)($$1 - 290 T^{2} + 28561 T^{4}$$)($$( 1 - 23 T + 169 T^{2} )( 1 + 23 T + 169 T^{2} )$$)($$( 1 - 23 T + 169 T^{2} )^{2}$$)($$( 1 + 2 T + 169 T^{2} )^{4}$$)
$17$ ($$1 - 988 T^{2} + 407046 T^{4} - 82518748 T^{6} + 6975757441 T^{8}$$)($$( 1 - 17 T )^{2}( 1 + 17 T )^{2}$$)($$1 - 18 T + 397 T^{2} - 5202 T^{3} + 83521 T^{4}$$)($$1 - 42 T + 877 T^{2} - 12138 T^{3} + 83521 T^{4}$$)($$1 + 12 T + 337 T^{2} + 3468 T^{3} + 83521 T^{4}$$)($$( 1 - 17 T + 289 T^{2} )( 1 + 17 T + 289 T^{2} )$$)($$1 - 142 T^{2} - 63357 T^{4} - 11859982 T^{6} + 6975757441 T^{8}$$)
$19$ ($$( 1 - 6 T + 556 T^{2} - 2166 T^{3} + 130321 T^{4} )^{2}$$)($$1 - 290 T^{2} + 130321 T^{4}$$)($$1 + 18 T + 469 T^{2} + 6498 T^{3} + 130321 T^{4}$$)($$1 + 6 T + 373 T^{2} + 2166 T^{3} + 130321 T^{4}$$)($$( 1 - 19 T )^{2}( 1 - 19 T + 361 T^{2} )$$)($$( 1 - 26 T + 361 T^{2} )( 1 + 37 T + 361 T^{2} )$$)($$( 1 + 16 T - 105 T^{2} + 5776 T^{3} + 130321 T^{4} )^{2}$$)
$23$ ($$1 - 1444 T^{2} + 980166 T^{4} - 404090404 T^{6} + 78310985281 T^{8}$$)($$( 1 + 14 T + 529 T^{2} )^{2}$$)($$1 - 529 T^{2} + 279841 T^{4}$$)($$1 + 28 T + 255 T^{2} + 14812 T^{3} + 279841 T^{4}$$)($$1 + 40 T + 1071 T^{2} + 21160 T^{3} + 279841 T^{4}$$)($$( 1 - 23 T + 529 T^{2} )( 1 + 23 T + 529 T^{2} )$$)($$( 1 - 44 T + 1407 T^{2} - 23276 T^{3} + 279841 T^{4} )( 1 + 44 T + 1407 T^{2} + 23276 T^{3} + 279841 T^{4} )$$)
$29$ ($$1 - 2972 T^{2} + 3611558 T^{4} - 2102039132 T^{6} + 500246412961 T^{8}$$)($$( 1 + 38 T + 841 T^{2} )^{2}$$)($$( 1 + 9 T + 841 T^{2} )^{2}$$)($$( 1 - 25 T + 841 T^{2} )^{2}$$)($$( 1 - 16 T + 841 T^{2} )^{2}$$)($$( 1 - 29 T )^{2}( 1 + 29 T )^{2}$$)($$( 1 - 1437 T^{2} + 707281 T^{4} )^{2}$$)
$31$ ($$( 1 - 68 T + 3050 T^{2} - 65348 T^{3} + 923521 T^{4} )^{2}$$)($$1 - 1154 T^{2} + 923521 T^{4}$$)($$1 + 21 T + 1108 T^{2} + 20181 T^{3} + 923521 T^{4}$$)($$1 + 57 T + 2044 T^{2} + 54777 T^{3} + 923521 T^{4}$$)($$1 - 9 T + 988 T^{2} - 8649 T^{3} + 923521 T^{4}$$)($$( 1 - 59 T + 961 T^{2} )( 1 + 46 T + 961 T^{2} )$$)($$( 1 - 3 T - 952 T^{2} - 2883 T^{3} + 923521 T^{4} )^{2}$$)
$37$ ($$( 1 - 8 T + 1382 T^{2} - 10952 T^{3} + 1874161 T^{4} )^{2}$$)($$( 1 - 26 T + 1369 T^{2} )^{2}$$)($$1 + 10 T - 1269 T^{2} + 13690 T^{3} + 1874161 T^{4}$$)($$1 - 58 T + 1995 T^{2} - 79402 T^{3} + 1874161 T^{4}$$)($$1 + 5 T - 1344 T^{2} + 6845 T^{3} + 1874161 T^{4}$$)($$( 1 - 47 T + 1369 T^{2} )( 1 - 26 T + 1369 T^{2} )$$)($$( 1 + 12 T - 1225 T^{2} + 16428 T^{3} + 1874161 T^{4} )^{2}$$)
$41$ ($$1 - 1292 T^{2} + 2832038 T^{4} - 3650883212 T^{6} + 7984925229121 T^{8}$$)($$1 + 1438 T^{2} + 2825761 T^{4}$$)($$1 - 3254 T^{2} + 2825761 T^{4}$$)($$1 - 3350 T^{2} + 2825761 T^{4}$$)($$1 - 2774 T^{2} + 2825761 T^{4}$$)($$( 1 - 41 T )^{2}( 1 + 41 T )^{2}$$)($$( 1 - 2382 T^{2} + 2825761 T^{4} )^{2}$$)
$43$ ($$( 1 + 80 T + 5046 T^{2} + 147920 T^{3} + 3418801 T^{4} )^{2}$$)($$( 1 - 26 T + 1849 T^{2} )^{2}$$)($$( 1 + 74 T + 1849 T^{2} )^{2}$$)($$( 1 - 26 T + 1849 T^{2} )^{2}$$)($$( 1 + 19 T + 1849 T^{2} )^{2}$$)($$( 1 + 61 T + 1849 T^{2} )^{2}$$)($$( 1 - 44 T + 1849 T^{2} )^{4}$$)
$47$ ($$1 - 6148 T^{2} + 19144326 T^{4} - 30000278788 T^{6} + 23811286661761 T^{8}$$)($$1 - 3650 T^{2} + 4879681 T^{4}$$)($$( 1 - 47 T + 2209 T^{2} )( 1 + 47 T + 2209 T^{2} )$$)($$1 + 132 T + 8017 T^{2} + 291588 T^{3} + 4879681 T^{4}$$)($$1 + 90 T + 4909 T^{2} + 198810 T^{3} + 4879681 T^{4}$$)($$( 1 - 47 T + 2209 T^{2} )( 1 + 47 T + 2209 T^{2} )$$)($$1 + 4238 T^{2} + 13080963 T^{4} + 20680088078 T^{6} + 23811286661761 T^{8}$$)
$53$ ($$1 + 20 T^{2} - 13350138 T^{4} + 157809620 T^{6} + 62259690411361 T^{8}$$)($$( 1 - 10 T + 2809 T^{2} )^{2}$$)($$1 + 33 T - 1720 T^{2} + 92697 T^{3} + 7890481 T^{4}$$)($$1 + 31 T - 1848 T^{2} + 87079 T^{3} + 7890481 T^{4}$$)($$1 - 32 T - 1785 T^{2} - 89888 T^{3} + 7890481 T^{4}$$)($$( 1 - 53 T + 2809 T^{2} )( 1 + 53 T + 2809 T^{2} )$$)($$1 + 5213 T^{2} + 19284888 T^{4} + 41133077453 T^{6} + 62259690411361 T^{8}$$)
$59$ ($$1 - 3676 T^{2} + 15964266 T^{4} - 44543419036 T^{6} + 146830437604321 T^{8}$$)($$1 - 1154 T^{2} + 12117361 T^{4}$$)($$1 - 27 T + 3724 T^{2} - 93987 T^{3} + 12117361 T^{4}$$)($$1 + 15 T + 3556 T^{2} + 52215 T^{3} + 12117361 T^{4}$$)($$1 - 72 T + 5209 T^{2} - 250632 T^{3} + 12117361 T^{4}$$)($$( 1 - 59 T + 3481 T^{2} )( 1 + 59 T + 3481 T^{2} )$$)($$1 + 6557 T^{2} + 30876888 T^{4} + 79453536077 T^{6} + 146830437604321 T^{8}$$)
$61$ ($$( 1 + 78 T + 8620 T^{2} + 290238 T^{3} + 13845841 T^{4} )^{2}$$)($$1 - 6242 T^{2} + 13845841 T^{4}$$)($$1 - 156 T + 11833 T^{2} - 580476 T^{3} + 13845841 T^{4}$$)($$1 - 24 T + 3913 T^{2} - 89304 T^{3} + 13845841 T^{4}$$)($$1 - 36 T + 4153 T^{2} - 133956 T^{3} + 13845841 T^{4}$$)($$( 1 - 47 T + 3721 T^{2} )( 1 + 121 T + 3721 T^{2} )$$)($$( 1 - 26 T - 3045 T^{2} - 96746 T^{3} + 13845841 T^{4} )^{2}$$)
$67$ ($$( 1 + 12 T + 8566 T^{2} + 53868 T^{3} + 20151121 T^{4} )^{2}$$)($$( 1 - 74 T + 4489 T^{2} )^{2}$$)($$1 - 76 T + 1287 T^{2} - 341164 T^{3} + 20151121 T^{4}$$)($$1 - 52 T - 1785 T^{2} - 233428 T^{3} + 20151121 T^{4}$$)($$1 + 59 T - 1008 T^{2} + 264851 T^{3} + 20151121 T^{4}$$)($$( 1 - 122 T + 4489 T^{2} )( 1 + 109 T + 4489 T^{2} )$$)($$( 1 + 52 T - 1785 T^{2} + 233428 T^{3} + 20151121 T^{4} )^{2}$$)
$71$ ($$1 - 10588 T^{2} + 78813510 T^{4} - 269058878428 T^{6} + 645753531245761 T^{8}$$)($$( 1 + 62 T + 5041 T^{2} )^{2}$$)($$( 1 - 84 T + 5041 T^{2} )^{2}$$)($$( 1 - 64 T + 5041 T^{2} )^{2}$$)($$( 1 + 26 T + 5041 T^{2} )^{2}$$)($$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$)($$( 1 - 1262 T^{2} + 25411681 T^{4} )^{2}$$)
$73$ ($$( 1 + 16 T + 5990 T^{2} + 85264 T^{3} + 28398241 T^{4} )^{2}$$)($$1 - 8930 T^{2} + 28398241 T^{4}$$)($$1 + 108 T + 9217 T^{2} + 575532 T^{3} + 28398241 T^{4}$$)($$1 - 12 T + 5377 T^{2} - 63948 T^{3} + 28398241 T^{4}$$)($$1 + 33 T + 5692 T^{2} + 175857 T^{3} + 28398241 T^{4}$$)($$( 1 - 143 T + 5329 T^{2} )( 1 + 46 T + 5329 T^{2} )$$)($$( 1 + 18 T - 5005 T^{2} + 95922 T^{3} + 28398241 T^{4} )^{2}$$)
$79$ ($$( 1 - 64 T + 4434 T^{2} - 399424 T^{3} + 38950081 T^{4} )^{2}$$)($$( 1 + 46 T + 6241 T^{2} )^{2}$$)($$1 - 43 T - 4392 T^{2} - 268363 T^{3} + 38950081 T^{4}$$)($$1 + 17 T - 5952 T^{2} + 106097 T^{3} + 38950081 T^{4}$$)($$1 + 47 T - 4032 T^{2} + 293327 T^{3} + 38950081 T^{4}$$)($$( 1 - 131 T + 6241 T^{2} )( 1 + 142 T + 6241 T^{2} )$$)($$( 1 - 79 T )^{4}( 1 + 79 T + 6241 T^{2} )^{2}$$)
$83$ ($$1 - 13948 T^{2} + 141899946 T^{4} - 661948661308 T^{6} + 2252292232139041 T^{8}$$)($$1 - 5666 T^{2} + 47458321 T^{4}$$)($$1 + 505 T^{2} + 47458321 T^{4}$$)($$1 - 10895 T^{2} + 47458321 T^{4}$$)($$1 - 13190 T^{2} + 47458321 T^{4}$$)($$( 1 - 83 T )^{2}( 1 + 83 T )^{2}$$)($$( 1 + 6067 T^{2} + 47458321 T^{4} )^{2}$$)
$89$ ($$1 - 11468 T^{2} + 120945830 T^{4} - 719528019788 T^{6} + 3936588805702081 T^{8}$$)($$1 - 14114 T^{2} + 62742241 T^{4}$$)($$1 + 126 T + 13213 T^{2} + 998046 T^{3} + 62742241 T^{4}$$)($$1 + 138 T + 14269 T^{2} + 1093098 T^{3} + 62742241 T^{4}$$)($$1 - 204 T + 21793 T^{2} - 1615884 T^{3} + 62742241 T^{4}$$)($$( 1 - 89 T + 7921 T^{2} )( 1 + 89 T + 7921 T^{2} )$$)($$1 + 13422 T^{2} + 117407843 T^{4} + 842126358702 T^{6} + 3936588805702081 T^{8}$$)
$97$ ($$( 1 + 4 T + 18374 T^{2} + 37636 T^{3} + 88529281 T^{4} )^{2}$$)($$1 - 15746 T^{2} + 88529281 T^{4}$$)($$1 + 15529 T^{2} + 88529281 T^{4}$$)($$1 - 10391 T^{2} + 88529281 T^{4}$$)($$1 - 16466 T^{2} + 88529281 T^{4}$$)($$( 1 - 2 T + 9409 T^{2} )^{2}$$)($$( 1 + 93 T + 9409 T^{2} )^{4}$$)