Properties

Label 21.2.e
Level $21$
Weight $2$
Character orbit 21.e
Rep. character $\chi_{21}(4,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $2$
Newform subspaces $1$
Sturm bound $5$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 21.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 1 \)
Sturm bound: \(5\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(21, [\chi])\).

Total New Old
Modular forms 10 2 8
Cusp forms 2 2 0
Eisenstein series 8 0 8

Trace form

\( 2q - 2q^{2} - q^{3} - 2q^{4} + 2q^{5} + 4q^{6} - 5q^{7} - q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - q^{3} - 2q^{4} + 2q^{5} + 4q^{6} - 5q^{7} - q^{9} + 4q^{10} + 2q^{11} - 2q^{12} + 2q^{13} + 2q^{14} - 4q^{15} + 4q^{16} - 2q^{18} - q^{19} - 8q^{20} + 4q^{21} - 8q^{22} + q^{25} - 2q^{26} + 2q^{27} + 8q^{28} + 8q^{29} + 4q^{30} - 9q^{31} + 8q^{32} + 2q^{33} - 2q^{35} + 4q^{36} - 3q^{37} - 2q^{38} - q^{39} - 20q^{41} - 10q^{42} + 10q^{43} + 4q^{44} + 2q^{45} + 6q^{47} - 8q^{48} + 11q^{49} - 4q^{50} - 2q^{52} - 12q^{53} - 2q^{54} + 8q^{55} + 2q^{57} - 8q^{58} + 12q^{59} + 4q^{60} - 10q^{61} + 36q^{62} + q^{63} - 16q^{64} + 2q^{65} + 4q^{66} + 5q^{67} - 16q^{70} - 12q^{71} + 3q^{73} - 6q^{74} + q^{75} + 4q^{76} - 8q^{77} + 4q^{78} + q^{79} - 8q^{80} - q^{81} + 20q^{82} + 12q^{83} + 2q^{84} - 10q^{86} - 4q^{87} - 16q^{89} - 8q^{90} - 5q^{91} - 9q^{93} + 12q^{94} + 2q^{95} + 8q^{96} - 12q^{97} + 4q^{98} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(21, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
21.2.e.a \(2\) \(0.168\) \(\Q(\sqrt{-3}) \) None \(-2\) \(-1\) \(2\) \(-5\) \(q+(-2+2\zeta_{6})q^{2}-\zeta_{6}q^{3}-2\zeta_{6}q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \)
$7$ \( 1 + 5 T + 7 T^{2} \)
$11$ \( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - T + 13 T^{2} )^{2} \)
$17$ \( 1 - 17 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 - 23 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 4 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 9 T + 50 T^{2} + 279 T^{3} + 961 T^{4} \)
$37$ \( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 10 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 5 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 3 T - 64 T^{2} - 219 T^{3} + 5329 T^{4} \)
$79$ \( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 16 T + 167 T^{2} + 1424 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 6 T + 97 T^{2} )^{2} \)
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