Properties

Label 28.2
Level 28
Weight 2
Dimension 8
Nonzero newspaces 3
Newform subspaces 3
Sturm bound 96
Trace bound 2

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Defining parameters

Level: \( N \) = \( 28\( 28 = 2^{2} \cdot 7 \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 3 \)
Sturm bound: \(96\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(28))\).

Total New Old
Modular forms 39 20 19
Cusp forms 10 8 2
Eisenstein series 29 12 17

Trace form

\( 8q - 3q^{2} - q^{3} - 3q^{4} - 9q^{5} - 4q^{7} - 3q^{8} - 4q^{9} + O(q^{10}) \) \( 8q - 3q^{2} - q^{3} - 3q^{4} - 9q^{5} - 4q^{7} - 3q^{8} - 4q^{9} + 6q^{10} + 3q^{11} + 12q^{12} + 4q^{13} + 9q^{14} + 6q^{15} + 9q^{16} - 9q^{17} + 3q^{18} + q^{19} + 5q^{21} - 18q^{22} - 3q^{23} - 12q^{24} + 2q^{25} - 12q^{26} - 10q^{27} - 27q^{28} - 6q^{30} + 7q^{31} - 3q^{32} + 9q^{33} + 15q^{35} + 9q^{36} + 7q^{37} + 18q^{38} - 2q^{39} + 12q^{40} + 12q^{41} + 12q^{42} - 8q^{43} + 18q^{44} + 6q^{45} + 12q^{46} + 9q^{47} - 16q^{49} + 3q^{50} - 3q^{51} - 21q^{53} - 18q^{54} - 18q^{55} + 9q^{56} - 38q^{57} - 6q^{58} - 9q^{59} - 12q^{60} - 17q^{61} - 10q^{63} + 9q^{64} + 6q^{65} - 6q^{66} + 7q^{67} + 6q^{69} + 6q^{70} - 15q^{72} + 31q^{73} - 12q^{74} - 4q^{75} + 21q^{77} + 24q^{78} + 13q^{79} - 24q^{80} + 35q^{81} + 12q^{82} + 24q^{83} + 30q^{85} - 18q^{86} + 6q^{87} + 18q^{88} + 39q^{89} - 8q^{91} - 6q^{92} + 13q^{93} - 30q^{94} + 3q^{95} - 24q^{96} - 20q^{97} - 15q^{98} + 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(28))\)

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
28.2.a \(\chi_{28}(1, \cdot)\) None 0 1
28.2.d \(\chi_{28}(27, \cdot)\) 28.2.d.a 2 1
28.2.e \(\chi_{28}(9, \cdot)\) 28.2.e.a 2 2
28.2.f \(\chi_{28}(3, \cdot)\) 28.2.f.a 4 2

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(28))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(28)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))
$3$ (\( ( 1 + 3 T^{2} )^{2} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( ( 1 - 3 T^{2} )^{2}( 1 + 3 T^{2} + 9 T^{4} ) \))
$5$ (\( ( 1 - 5 T^{2} )^{2} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( ( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} )^{2} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 2 T^{2} + 49 T^{4} \))
$11$ (\( ( 1 - 4 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} ) \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + 21 T^{2} + 320 T^{4} + 2541 T^{6} + 14641 T^{8} \))
$13$ (\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))
$17$ (\( ( 1 - 17 T^{2} )^{2} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( ( 1 + 3 T + 20 T^{2} + 51 T^{3} + 289 T^{4} )^{2} \))
$19$ (\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - 37 T^{2} + 361 T^{4} )( 1 + 26 T^{2} + 361 T^{4} ) \))
$23$ (\( ( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} ) \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 + 45 T^{2} + 1496 T^{4} + 23805 T^{6} + 279841 T^{8} \))
$29$ (\( ( 1 + 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 4 T + 29 T^{2} )^{4} \))
$31$ (\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( ( 1 - 46 T^{2} + 961 T^{4} )( 1 - 13 T^{2} + 961 T^{4} ) \))
$37$ (\( ( 1 - 6 T + 37 T^{2} )^{2} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( ( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( ( 1 - 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 70 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 - 12 T + 43 T^{2} )( 1 + 12 T + 43 T^{2} ) \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 82 T^{2} + 1849 T^{4} )^{2} \))
$47$ (\( ( 1 + 47 T^{2} )^{2} \))(\( 1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4} \))(\( 1 - 19 T^{2} - 1848 T^{4} - 41971 T^{6} + 4879681 T^{8} \))
$53$ (\( ( 1 + 10 T + 53 T^{2} )^{2} \))(\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( ( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( ( 1 + 59 T^{2} )^{2} \))(\( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 - 91 T^{2} + 4800 T^{4} - 316771 T^{6} + 12117361 T^{8} \))
$61$ (\( ( 1 - 61 T^{2} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( ( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( ( 1 - 4 T + 67 T^{2} )( 1 + 4 T + 67 T^{2} ) \))(\( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \))(\( 1 + 125 T^{2} + 11136 T^{4} + 561125 T^{6} + 20151121 T^{8} \))
$71$ (\( ( 1 - 16 T + 71 T^{2} )( 1 + 16 T + 71 T^{2} ) \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 54 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 - 73 T^{2} )^{2} \))(\( 1 - T - 72 T^{2} - 73 T^{3} + 5329 T^{4} \))(\( ( 1 - 15 T + 148 T^{2} - 1095 T^{3} + 5329 T^{4} )^{2} \))
$79$ (\( ( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( 1 + 77 T^{2} - 312 T^{4} + 480557 T^{6} + 38950081 T^{8} \))
$83$ (\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 26 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 - 89 T^{2} )^{2} \))(\( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \))(\( ( 1 - 27 T + 332 T^{2} - 2403 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 - 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 106 T^{2} + 9409 T^{4} )^{2} \))
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