Properties

Label 12.4
Level 12
Weight 4
Dimension 5
Nonzero newspaces 2
Newform subspaces 2
Sturm bound 32
Trace bound 1

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Defining parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 2 \)
Sturm bound: \(32\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(12))\).

Total New Old
Modular forms 17 9 8
Cusp forms 7 5 2
Eisenstein series 10 4 6

Trace form

\( 5q + 3q^{3} - 8q^{4} - 18q^{5} - 24q^{6} + 8q^{7} - 3q^{9} + O(q^{10}) \) \( 5q + 3q^{3} - 8q^{4} - 18q^{5} - 24q^{6} + 8q^{7} - 3q^{9} + 80q^{10} + 36q^{11} + 120q^{12} - 50q^{13} - 54q^{15} - 224q^{16} + 18q^{17} - 240q^{18} - 100q^{19} + 144q^{21} + 240q^{22} + 72q^{23} + 288q^{24} + 379q^{25} + 27q^{27} - 240q^{28} - 234q^{29} - 240q^{30} - 16q^{31} - 372q^{33} + 320q^{34} - 144q^{35} + 24q^{36} - 746q^{37} - 30q^{39} + 320q^{40} + 90q^{41} + 240q^{42} + 452q^{43} + 798q^{45} - 672q^{46} + 432q^{47} - 480q^{48} + 853q^{49} + 54q^{51} + 80q^{52} + 414q^{53} + 792q^{54} - 648q^{55} - 1380q^{57} - 1360q^{58} - 684q^{59} - 960q^{60} - 1346q^{61} + 72q^{63} + 1408q^{64} + 180q^{65} + 1200q^{66} + 332q^{67} + 1560q^{69} + 480q^{70} - 360q^{71} - 960q^{72} + 1666q^{73} + 597q^{75} + 2160q^{76} + 288q^{77} + 240q^{78} + 512q^{79} - 2763q^{81} - 1120q^{82} - 1188q^{83} - 240q^{84} - 1604q^{85} - 702q^{87} - 2880q^{88} - 630q^{89} - 240q^{90} - 80q^{91} + 3432q^{93} - 1344q^{94} + 1800q^{95} + 384q^{96} + 2026q^{97} + 324q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(12))\)

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
12.4.a \(\chi_{12}(1, \cdot)\) 12.4.a.a 1 1
12.4.b \(\chi_{12}(11, \cdot)\) 12.4.b.a 4 1

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(12))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(12)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 4 T^{2} + 64 T^{4} \))
$3$ (\( 1 - 3 T \))(\( 1 + 6 T^{2} + 729 T^{4} \))
$5$ (\( 1 + 18 T + 125 T^{2} \))(\( ( 1 - 170 T^{2} + 15625 T^{4} )^{2} \))
$7$ (\( 1 - 8 T + 343 T^{2} \))(\( ( 1 - 626 T^{2} + 117649 T^{4} )^{2} \))
$11$ (\( 1 - 36 T + 1331 T^{2} \))(\( ( 1 + 1462 T^{2} + 1771561 T^{4} )^{2} \))
$13$ (\( 1 + 10 T + 2197 T^{2} \))(\( ( 1 + 10 T + 2197 T^{2} )^{4} \))
$17$ (\( 1 - 18 T + 4913 T^{2} \))(\( ( 1 - 8546 T^{2} + 24137569 T^{4} )^{2} \))
$19$ (\( 1 + 100 T + 6859 T^{2} \))(\( ( 1 - 8858 T^{2} + 47045881 T^{4} )^{2} \))
$23$ (\( 1 - 72 T + 12167 T^{2} \))(\( ( 1 + 14926 T^{2} + 148035889 T^{4} )^{2} \))
$29$ (\( 1 + 234 T + 24389 T^{2} \))(\( ( 1 - 25658 T^{2} + 594823321 T^{4} )^{2} \))
$31$ (\( 1 + 16 T + 29791 T^{2} \))(\( ( 1 - 9122 T^{2} + 887503681 T^{4} )^{2} \))
$37$ (\( 1 + 226 T + 50653 T^{2} \))(\( ( 1 + 130 T + 50653 T^{2} )^{4} \))
$41$ (\( 1 - 90 T + 68921 T^{2} \))(\( ( 1 - 122162 T^{2} + 4750104241 T^{4} )^{2} \))
$43$ (\( 1 - 452 T + 79507 T^{2} \))(\( ( 1 - 108554 T^{2} + 6321363049 T^{4} )^{2} \))
$47$ (\( 1 - 432 T + 103823 T^{2} \))(\( ( 1 + 170014 T^{2} + 10779215329 T^{4} )^{2} \))
$53$ (\( 1 - 414 T + 148877 T^{2} \))(\( ( 1 - 74 T^{2} + 22164361129 T^{4} )^{2} \))
$59$ (\( 1 + 684 T + 205379 T^{2} \))(\( ( 1 + 380758 T^{2} + 42180533641 T^{4} )^{2} \))
$61$ (\( 1 - 422 T + 226981 T^{2} \))(\( ( 1 + 442 T + 226981 T^{2} )^{4} \))
$67$ (\( 1 - 332 T + 300763 T^{2} \))(\( ( 1 - 60026 T^{2} + 90458382169 T^{4} )^{2} \))
$71$ (\( 1 + 360 T + 357911 T^{2} \))(\( ( 1 - 364178 T^{2} + 128100283921 T^{4} )^{2} \))
$73$ (\( 1 - 26 T + 389017 T^{2} \))(\( ( 1 - 410 T + 389017 T^{2} )^{4} \))
$79$ (\( 1 - 512 T + 493039 T^{2} \))(\( ( 1 - 978818 T^{2} + 243087455521 T^{4} )^{2} \))
$83$ (\( 1 + 1188 T + 571787 T^{2} \))(\( ( 1 - 428954 T^{2} + 326940373369 T^{4} )^{2} \))
$89$ (\( 1 + 630 T + 704969 T^{2} \))(\( ( 1 - 703058 T^{2} + 496981290961 T^{4} )^{2} \))
$97$ (\( 1 + 1054 T + 912673 T^{2} \))(\( ( 1 - 770 T + 912673 T^{2} )^{4} \))
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