Properties

Label 21.2
Level 21
Weight 2
Dimension 5
Nonzero newspaces 3
Newform subspaces 3
Sturm bound 64
Trace bound 2

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Defining parameters

Level: \( N \) = \( 21\( 21 = 3 \cdot 7 \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 3 \)
Sturm bound: \(64\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(21))\).

Total New Old
Modular forms 28 17 11
Cusp forms 5 5 0
Eisenstein series 23 12 11

Trace form

\( 5q - 3q^{2} - 3q^{3} - 5q^{4} + 3q^{6} - 5q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 5q - 3q^{2} - 3q^{3} - 5q^{4} + 3q^{6} - 5q^{7} + 3q^{8} + 3q^{9} + 6q^{10} + 6q^{11} + 3q^{12} + 3q^{14} - 6q^{15} - q^{16} - 6q^{17} - 3q^{18} - 6q^{19} - 6q^{20} - 3q^{21} - 12q^{22} + 3q^{24} + 5q^{25} + 3q^{27} + 17q^{28} + 6q^{29} + 6q^{30} + 6q^{31} + 3q^{32} + 6q^{33} + 6q^{34} - 9q^{36} + 2q^{37} - 6q^{38} - 6q^{40} - 18q^{41} - 9q^{42} - 4q^{43} + 6q^{47} - 9q^{48} - q^{49} - 3q^{50} - 6q^{51} - 6q^{52} - 6q^{53} - 3q^{54} - 3q^{56} + 24q^{57} - 6q^{58} + 24q^{59} + 6q^{60} + 36q^{62} + 15q^{63} + 7q^{64} + 6q^{65} - 2q^{67} + 6q^{68} - 18q^{70} - 12q^{71} + 3q^{72} - 30q^{73} - 12q^{74} - 15q^{75} - 12q^{77} + 6q^{78} - 2q^{79} - 6q^{80} - 9q^{81} + 18q^{82} - 3q^{84} + 12q^{85} - 6q^{86} - 6q^{87} + 12q^{88} - 30q^{89} - 6q^{90} + 6q^{91} - 24q^{93} + 12q^{94} - 6q^{95} + 3q^{96} + 6q^{97} + 3q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)

We only show spaces with even 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
21.2.a \(\chi_{21}(1, \cdot)\) 21.2.a.a 1 1
21.2.c \(\chi_{21}(20, \cdot)\) None 0 1
21.2.e \(\chi_{21}(4, \cdot)\) 21.2.e.a 2 2
21.2.g \(\chi_{21}(5, \cdot)\) 21.2.g.a 2 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 + 2 T^{2} + 4 T^{4} \))
$3$ (\( 1 - T \))(\( 1 + T + T^{2} \))(\( 1 + 3 T + 3 T^{2} \))
$5$ (\( 1 + 2 T + 5 T^{2} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))
$7$ (\( 1 + T \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( ( 1 - T + 13 T^{2} )^{2} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 + T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))
$29$ (\( 1 + 2 T + 29 T^{2} \))(\( ( 1 - 4 T + 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 + 9 T + 50 T^{2} + 279 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 - 4 T + 31 T^{2} ) \))
$37$ (\( 1 - 6 T + 37 T^{2} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))
$41$ (\( 1 - 2 T + 41 T^{2} \))(\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{2} \))(\( ( 1 + 5 T + 43 T^{2} )^{2} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))
$59$ (\( 1 - 12 T + 59 T^{2} \))(\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))
$61$ (\( 1 + 2 T + 61 T^{2} \))(\( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))
$67$ (\( 1 - 4 T + 67 T^{2} \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))
$71$ (\( 1 + 71 T^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( 1 - 3 T - 64 T^{2} - 219 T^{3} + 5329 T^{4} \))(\( ( 1 + 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))
$79$ (\( 1 + 16 T + 79 T^{2} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))
$83$ (\( 1 + 12 T + 83 T^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))
$89$ (\( 1 + 14 T + 89 T^{2} \))(\( 1 + 16 T + 167 T^{2} + 1424 T^{3} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))
$97$ (\( 1 - 18 T + 97 T^{2} \))(\( ( 1 + 6 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))
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