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 $1$

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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

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