Properties

Label 22.2
Level 22
Weight 2
Dimension 4
Nonzero newspaces 1
Newform subspaces 1
Sturm bound 60
Trace bound 0

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Defining parameters

Level: \( N \) = \( 22\( 22 = 2 \cdot 11 \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 1 \)
Sturm bound: \(60\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(22))\).

Total New Old
Modular forms 25 4 21
Cusp forms 6 4 2
Eisenstein series 19 0 19

Trace form

\( 4q - q^{2} - 4q^{3} - q^{4} - 6q^{5} + q^{6} + 2q^{7} - q^{8} + 7q^{9} + O(q^{10}) \) \( 4q - q^{2} - 4q^{3} - q^{4} - 6q^{5} + q^{6} + 2q^{7} - q^{8} + 7q^{9} + 4q^{10} - q^{11} + 6q^{12} - 4q^{13} + 2q^{14} + 6q^{15} - q^{16} + 2q^{17} - 8q^{18} - 5q^{19} - 6q^{20} - 12q^{21} - 11q^{22} - 4q^{23} + q^{24} + 9q^{25} + 6q^{26} + 5q^{27} + 2q^{28} + 10q^{29} + 6q^{30} - 2q^{31} + 4q^{32} + 11q^{33} + 2q^{34} + 12q^{35} - 8q^{36} - 18q^{37} + 10q^{38} - 6q^{39} + 4q^{40} - 2q^{41} - 2q^{42} + 6q^{43} + 4q^{44} - 28q^{45} - 4q^{46} - 8q^{47} - 4q^{48} + 3q^{49} - 11q^{50} - 7q^{51} + 6q^{52} - 4q^{53} + 4q^{55} - 8q^{56} + 5q^{57} + 10q^{58} + 5q^{59} - 4q^{60} + 8q^{61} - 2q^{62} + 16q^{63} - q^{64} + 16q^{65} + 16q^{66} + 22q^{67} + 2q^{68} + 14q^{69} - 8q^{70} + 8q^{71} + 7q^{72} - 14q^{73} - 18q^{74} - 9q^{75} - 10q^{76} - 8q^{77} - 16q^{78} - 30q^{79} + 4q^{80} + 14q^{81} - 7q^{82} - 19q^{83} + 8q^{84} + 2q^{85} + 21q^{86} - 20q^{87} - q^{88} - 10q^{89} + 12q^{90} - 12q^{91} + 6q^{92} + 2q^{93} + 12q^{94} - 10q^{95} - 4q^{96} - 3q^{97} - 12q^{98} - 13q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(22))\)

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
22.2.a \(\chi_{22}(1, \cdot)\) None 0 1
22.2.c \(\chi_{22}(3, \cdot)\) 22.2.c.a 4 4

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(22))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(22)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(11))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$3$ \( 1 + 4 T + 3 T^{2} - 10 T^{3} - 29 T^{4} - 30 T^{5} + 27 T^{6} + 108 T^{7} + 81 T^{8} \)
$5$ \( 1 + 6 T + 11 T^{2} + 6 T^{3} + T^{4} + 30 T^{5} + 275 T^{6} + 750 T^{7} + 625 T^{8} \)
$7$ \( 1 - 2 T - 3 T^{2} + 20 T^{3} - 19 T^{4} + 140 T^{5} - 147 T^{6} - 686 T^{7} + 2401 T^{8} \)
$11$ \( 1 + T + 21 T^{2} + 11 T^{3} + 121 T^{4} \)
$13$ \( 1 + 4 T + 3 T^{2} + 50 T^{3} + 341 T^{4} + 650 T^{5} + 507 T^{6} + 8788 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 2 T - 13 T^{2} - 20 T^{3} + 341 T^{4} - 340 T^{5} - 3757 T^{6} - 9826 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 5 T + 21 T^{2} + 145 T^{3} + 956 T^{4} + 2755 T^{5} + 7581 T^{6} + 34295 T^{7} + 130321 T^{8} \)
$23$ \( ( 1 + 2 T + 42 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \)
$29$ \( 1 - 10 T + 31 T^{2} - 200 T^{3} + 1821 T^{4} - 5800 T^{5} + 26071 T^{6} - 243890 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 2 T - 27 T^{2} - 116 T^{3} + 605 T^{4} - 3596 T^{5} - 25947 T^{6} + 59582 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 18 T + 107 T^{2} + 210 T^{3} + T^{4} + 7770 T^{5} + 146483 T^{6} + 911754 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 2 T - 17 T^{2} - 236 T^{3} + 525 T^{4} - 9676 T^{5} - 28577 T^{6} + 137842 T^{7} + 2825761 T^{8} \)
$43$ \( ( 1 - 3 T - 13 T^{2} - 129 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 8 T + 17 T^{2} + 380 T^{3} + 4721 T^{4} + 17860 T^{5} + 37553 T^{6} + 830584 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 4 T + 43 T^{2} + 380 T^{3} + 4761 T^{4} + 20140 T^{5} + 120787 T^{6} + 595508 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 5 T + T^{2} + 335 T^{3} - 1164 T^{4} + 19765 T^{5} + 3481 T^{6} - 1026895 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 8 T + 3 T^{2} - 436 T^{3} + 6905 T^{4} - 26596 T^{5} + 11163 T^{6} - 1815848 T^{7} + 13845841 T^{8} \)
$67$ \( ( 1 - 11 T + 133 T^{2} - 737 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( 1 - 8 T - 47 T^{2} + 434 T^{3} + 1365 T^{4} + 30814 T^{5} - 236927 T^{6} - 2863288 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 14 T + 63 T^{2} - 500 T^{3} - 10579 T^{4} - 36500 T^{5} + 335727 T^{6} + 5446238 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 30 T + 461 T^{2} + 5400 T^{3} + 52861 T^{4} + 426600 T^{5} + 2877101 T^{6} + 14791170 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 19 T + 103 T^{2} + 695 T^{3} + 10536 T^{4} + 57685 T^{5} + 709567 T^{6} + 10863953 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 + 5 T + 153 T^{2} + 445 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 + 3 T + 47 T^{2} - 255 T^{3} + 3496 T^{4} - 24735 T^{5} + 442223 T^{6} + 2738019 T^{7} + 88529281 T^{8} \)
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