Properties

Label 12.3
Level 12
Weight 3
Dimension 3
Nonzero newspaces 2
Newform subspaces 2
Sturm bound 24
Trace bound 1

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Defining parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 2 \)
Sturm bound: \(24\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(12))\).

Total New Old
Modular forms 13 3 10
Cusp forms 3 3 0
Eisenstein series 10 0 10

Trace form

\( 3q - 2q^{2} - 3q^{3} - 4q^{4} - 4q^{5} + 6q^{6} + 2q^{7} + 16q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 2q^{2} - 3q^{3} - 4q^{4} - 4q^{5} + 6q^{6} + 2q^{7} + 16q^{8} + 3q^{9} + 4q^{10} - 12q^{12} - 18q^{13} - 24q^{14} - 16q^{16} + 20q^{17} + 6q^{18} + 26q^{19} + 8q^{20} + 18q^{21} + 24q^{22} - 17q^{25} - 4q^{26} - 27q^{27} + 48q^{28} - 52q^{29} - 12q^{30} - 46q^{31} - 32q^{32} - 24q^{33} - 20q^{34} + 12q^{36} + 78q^{37} + 72q^{38} + 66q^{39} - 32q^{40} + 116q^{41} - 24q^{42} - 22q^{43} - 48q^{44} + 12q^{45} - 96q^{46} + 48q^{48} - 43q^{49} + 42q^{50} - 8q^{52} - 148q^{53} - 18q^{54} - 150q^{57} + 52q^{58} + 24q^{60} + 126q^{61} + 24q^{62} + 18q^{63} + 128q^{64} - 8q^{65} + 24q^{66} + 122q^{67} - 40q^{68} + 96q^{69} + 48q^{70} - 48q^{72} - 138q^{73} - 52q^{74} - 75q^{75} - 144q^{76} + 96q^{77} + 12q^{78} - 142q^{79} + 32q^{80} + 99q^{81} - 116q^{82} - 48q^{84} - 40q^{85} - 168q^{86} + 164q^{89} - 12q^{90} - 44q^{91} + 192q^{92} + 114q^{93} + 240q^{94} - 96q^{96} + 6q^{97} - 2q^{98} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(12))\)

We only show spaces with odd 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
12.3.c \(\chi_{12}(5, \cdot)\) 12.3.c.a 1 1
12.3.d \(\chi_{12}(7, \cdot)\) 12.3.d.a 2 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 4 T^{2} \))
$3$ (\( 1 + 3 T \))(\( 1 + 3 T^{2} \))
$5$ (\( ( 1 - 5 T )( 1 + 5 T ) \))(\( ( 1 + 2 T + 25 T^{2} )^{2} \))
$7$ (\( 1 - 2 T + 49 T^{2} \))(\( 1 - 50 T^{2} + 2401 T^{4} \))
$11$ (\( ( 1 - 11 T )( 1 + 11 T ) \))(\( 1 - 194 T^{2} + 14641 T^{4} \))
$13$ (\( 1 + 22 T + 169 T^{2} \))(\( ( 1 - 2 T + 169 T^{2} )^{2} \))
$17$ (\( ( 1 - 17 T )( 1 + 17 T ) \))(\( ( 1 - 10 T + 289 T^{2} )^{2} \))
$19$ (\( 1 - 26 T + 361 T^{2} \))(\( 1 - 290 T^{2} + 130321 T^{4} \))
$23$ (\( ( 1 - 23 T )( 1 + 23 T ) \))(\( 1 - 290 T^{2} + 279841 T^{4} \))
$29$ (\( ( 1 - 29 T )( 1 + 29 T ) \))(\( ( 1 + 26 T + 841 T^{2} )^{2} \))
$31$ (\( 1 + 46 T + 961 T^{2} \))(\( 1 - 1874 T^{2} + 923521 T^{4} \))
$37$ (\( 1 - 26 T + 1369 T^{2} \))(\( ( 1 - 26 T + 1369 T^{2} )^{2} \))
$41$ (\( ( 1 - 41 T )( 1 + 41 T ) \))(\( ( 1 - 58 T + 1681 T^{2} )^{2} \))
$43$ (\( 1 + 22 T + 1849 T^{2} \))(\( 1 - 1346 T^{2} + 3418801 T^{4} \))
$47$ (\( ( 1 - 47 T )( 1 + 47 T ) \))(\( 1 + 382 T^{2} + 4879681 T^{4} \))
$53$ (\( ( 1 - 53 T )( 1 + 53 T ) \))(\( ( 1 + 74 T + 2809 T^{2} )^{2} \))
$59$ (\( ( 1 - 59 T )( 1 + 59 T ) \))(\( 1 + 1150 T^{2} + 12117361 T^{4} \))
$61$ (\( 1 - 74 T + 3721 T^{2} \))(\( ( 1 - 26 T + 3721 T^{2} )^{2} \))
$67$ (\( 1 - 122 T + 4489 T^{2} \))(\( 1 - 8930 T^{2} + 20151121 T^{4} \))
$71$ (\( ( 1 - 71 T )( 1 + 71 T ) \))(\( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \))
$73$ (\( 1 + 46 T + 5329 T^{2} \))(\( ( 1 + 46 T + 5329 T^{2} )^{2} \))
$79$ (\( 1 + 142 T + 6241 T^{2} \))(\( 1 + 1390 T^{2} + 38950081 T^{4} \))
$83$ (\( ( 1 - 83 T )( 1 + 83 T ) \))(\( 1 - 11426 T^{2} + 47458321 T^{4} \))
$89$ (\( ( 1 - 89 T )( 1 + 89 T ) \))(\( ( 1 - 82 T + 7921 T^{2} )^{2} \))
$97$ (\( 1 - 2 T + 9409 T^{2} \))(\( ( 1 - 2 T + 9409 T^{2} )^{2} \))
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