# Properties

 Label 21.3.h Level $21$ Weight $3$ Character orbit 21.h Rep. character $\chi_{21}(2,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $6$ Newform subspaces $2$ Sturm bound $8$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 21.h (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$21$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$2$$ Sturm bound: $$8$$ Trace bound: $$1$$ 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 14 0
Cusp forms 6 6 0
Eisenstein series 8 8 0

## Trace form

 $$6q - q^{3} - 2q^{4} - 20q^{6} + q^{7} - 7q^{9} + O(q^{10})$$ $$6q - q^{3} - 2q^{4} - 20q^{6} + q^{7} - 7q^{9} - 10q^{10} + 16q^{12} + 38q^{13} + 20q^{15} + 22q^{16} + 40q^{18} - 43q^{19} + 22q^{21} - 100q^{22} + 30q^{24} - 65q^{25} - 142q^{27} - 6q^{28} - 20q^{30} + 19q^{31} + 50q^{33} + 240q^{34} + 76q^{36} + 49q^{37} + 77q^{39} - 30q^{40} - 70q^{42} + 54q^{43} - 40q^{45} - 60q^{46} - 248q^{48} - 27q^{49} - 120q^{51} - 96q^{52} + 70q^{54} + 100q^{55} + 62q^{57} - 70q^{58} + 10q^{60} - 22q^{61} + 127q^{63} - 36q^{64} + 100q^{66} - 91q^{67} + 120q^{69} + 70q^{70} + 120q^{72} + 61q^{73} - 5q^{75} + 24q^{76} + 40q^{78} + 147q^{79} + 77q^{81} - 140q^{82} - 76q^{84} - 240q^{85} - 70q^{87} + 150q^{88} - 20q^{90} - 327q^{91} - 27q^{93} + 60q^{94} - 70q^{96} - 368q^{97} - 400q^{99} + O(q^{100})$$

## 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.h.a $$2$$ $$0.572$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$3$$ $$0$$ $$-13$$ $$q+3\zeta_{6}q^{3}-4\zeta_{6}q^{4}+(-5-3\zeta_{6})q^{7}+\cdots$$
21.3.h.b $$4$$ $$0.572$$ $$\Q(\sqrt{-3}, \sqrt{-5})$$ None $$0$$ $$-4$$ $$0$$ $$14$$ $$q+\beta _{1}q^{2}+(-\beta _{1}-2\beta _{2}+\beta _{3})q^{3}+\beta _{2}q^{4}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - 2 T + 4 T^{2} )( 1 + 2 T + 4 T^{2} )$$)($$1 + 3 T^{2} - 7 T^{4} + 48 T^{6} + 256 T^{8}$$)
$3$ ($$1 - 3 T + 9 T^{2}$$)($$1 + 4 T + 7 T^{2} + 36 T^{3} + 81 T^{4}$$)
$5$ ($$( 1 - 5 T + 25 T^{2} )( 1 + 5 T + 25 T^{2} )$$)($$1 + 45 T^{2} + 1400 T^{4} + 28125 T^{6} + 390625 T^{8}$$)
$7$ ($$1 + 13 T + 49 T^{2}$$)($$( 1 - 7 T + 49 T^{2} )^{2}$$)
$11$ ($$( 1 - 11 T + 121 T^{2} )( 1 + 11 T + 121 T^{2} )$$)($$1 + 117 T^{2} - 952 T^{4} + 1712997 T^{6} + 214358881 T^{8}$$)
$13$ ($$( 1 - 23 T + 169 T^{2} )^{2}$$)($$( 1 + 2 T + 169 T^{2} )^{4}$$)
$17$ ($$( 1 - 17 T + 289 T^{2} )( 1 + 17 T + 289 T^{2} )$$)($$1 - 142 T^{2} - 63357 T^{4} - 11859982 T^{6} + 6975757441 T^{8}$$)
$19$ ($$( 1 - 26 T + 361 T^{2} )( 1 + 37 T + 361 T^{2} )$$)($$( 1 + 16 T - 105 T^{2} + 5776 T^{3} + 130321 T^{4} )^{2}$$)
$23$ ($$( 1 - 23 T + 529 T^{2} )( 1 + 23 T + 529 T^{2} )$$)($$( 1 - 44 T + 1407 T^{2} - 23276 T^{3} + 279841 T^{4} )( 1 + 44 T + 1407 T^{2} + 23276 T^{3} + 279841 T^{4} )$$)
$29$ ($$( 1 - 29 T )^{2}( 1 + 29 T )^{2}$$)($$( 1 - 1437 T^{2} + 707281 T^{4} )^{2}$$)
$31$ ($$( 1 - 59 T + 961 T^{2} )( 1 + 46 T + 961 T^{2} )$$)($$( 1 - 3 T - 952 T^{2} - 2883 T^{3} + 923521 T^{4} )^{2}$$)
$37$ ($$( 1 - 47 T + 1369 T^{2} )( 1 - 26 T + 1369 T^{2} )$$)($$( 1 + 12 T - 1225 T^{2} + 16428 T^{3} + 1874161 T^{4} )^{2}$$)
$41$ ($$( 1 - 41 T )^{2}( 1 + 41 T )^{2}$$)($$( 1 - 2382 T^{2} + 2825761 T^{4} )^{2}$$)
$43$ ($$( 1 + 61 T + 1849 T^{2} )^{2}$$)($$( 1 - 44 T + 1849 T^{2} )^{4}$$)
$47$ ($$( 1 - 47 T + 2209 T^{2} )( 1 + 47 T + 2209 T^{2} )$$)($$1 + 4238 T^{2} + 13080963 T^{4} + 20680088078 T^{6} + 23811286661761 T^{8}$$)
$53$ ($$( 1 - 53 T + 2809 T^{2} )( 1 + 53 T + 2809 T^{2} )$$)($$1 + 5213 T^{2} + 19284888 T^{4} + 41133077453 T^{6} + 62259690411361 T^{8}$$)
$59$ ($$( 1 - 59 T + 3481 T^{2} )( 1 + 59 T + 3481 T^{2} )$$)($$1 + 6557 T^{2} + 30876888 T^{4} + 79453536077 T^{6} + 146830437604321 T^{8}$$)
$61$ ($$( 1 - 47 T + 3721 T^{2} )( 1 + 121 T + 3721 T^{2} )$$)($$( 1 - 26 T - 3045 T^{2} - 96746 T^{3} + 13845841 T^{4} )^{2}$$)
$67$ ($$( 1 - 122 T + 4489 T^{2} )( 1 + 109 T + 4489 T^{2} )$$)($$( 1 + 52 T - 1785 T^{2} + 233428 T^{3} + 20151121 T^{4} )^{2}$$)
$71$ ($$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$)($$( 1 - 1262 T^{2} + 25411681 T^{4} )^{2}$$)
$73$ ($$( 1 - 143 T + 5329 T^{2} )( 1 + 46 T + 5329 T^{2} )$$)($$( 1 + 18 T - 5005 T^{2} + 95922 T^{3} + 28398241 T^{4} )^{2}$$)
$79$ ($$( 1 - 131 T + 6241 T^{2} )( 1 + 142 T + 6241 T^{2} )$$)($$( 1 - 79 T )^{4}( 1 + 79 T + 6241 T^{2} )^{2}$$)
$83$ ($$( 1 - 83 T )^{2}( 1 + 83 T )^{2}$$)($$( 1 + 6067 T^{2} + 47458321 T^{4} )^{2}$$)
$89$ ($$( 1 - 89 T + 7921 T^{2} )( 1 + 89 T + 7921 T^{2} )$$)($$1 + 13422 T^{2} + 117407843 T^{4} + 842126358702 T^{6} + 3936588805702081 T^{8}$$)
$97$ ($$( 1 - 2 T + 9409 T^{2} )^{2}$$)($$( 1 + 93 T + 9409 T^{2} )^{4}$$)