Properties

Label 42.2
Level 42
Weight 2
Dimension 13
Nonzero newspaces 4
Newform subspaces 5
Sturm bound 192
Trace bound 4

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Defining parameters

Level: \( N \) = \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 5 \)
Sturm bound: \(192\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(42))\).

Total New Old
Modular forms 72 13 59
Cusp forms 25 13 12
Eisenstein series 47 0 47

Trace form

\( 13q + q^{2} - q^{3} - 3q^{4} - 6q^{5} - 5q^{6} - 9q^{7} + q^{8} - 7q^{9} + O(q^{10}) \) \( 13q + q^{2} - q^{3} - 3q^{4} - 6q^{5} - 5q^{6} - 9q^{7} + q^{8} - 7q^{9} - 6q^{10} - 12q^{11} - q^{12} - 2q^{13} + 7q^{14} + 6q^{15} + q^{16} + 6q^{17} + 13q^{18} + 4q^{19} + 6q^{20} + 17q^{21} + 12q^{22} + 12q^{23} + 7q^{24} + 7q^{25} + 2q^{26} - q^{27} - 5q^{28} + 6q^{29} - 6q^{30} - 8q^{31} + q^{32} - 12q^{33} - 6q^{34} - 6q^{35} - 7q^{36} - 18q^{37} - 16q^{38} - 14q^{39} - 6q^{40} - 6q^{41} - 23q^{42} - 36q^{43} - 12q^{44} - 6q^{45} - 24q^{46} + 12q^{47} - q^{48} + q^{49} - 17q^{50} + 30q^{51} + 22q^{52} + 18q^{53} + 19q^{54} + 36q^{55} - 5q^{56} - 8q^{57} + 42q^{58} + 12q^{59} + 18q^{60} + 22q^{61} + 8q^{62} + 23q^{63} - 3q^{64} + 12q^{66} + 44q^{67} + 6q^{68} + 42q^{70} - 11q^{72} + 22q^{73} + 2q^{74} - 7q^{75} + 4q^{76} - 24q^{77} - 10q^{78} - 40q^{79} - 6q^{80} - 55q^{81} - 30q^{82} - 36q^{83} - 19q^{84} - 84q^{85} - 16q^{86} - 30q^{87} - 6q^{89} - 6q^{90} - 14q^{91} + 4q^{93} - 24q^{94} + 12q^{95} + 7q^{96} - 2q^{97} + 25q^{98} + 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(42))\)

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
42.2.a \(\chi_{42}(1, \cdot)\) 42.2.a.a 1 1
42.2.d \(\chi_{42}(41, \cdot)\) 42.2.d.a 4 1
42.2.e \(\chi_{42}(25, \cdot)\) 42.2.e.a 2 2
42.2.e.b 2
42.2.f \(\chi_{42}(5, \cdot)\) 42.2.f.a 4 2

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(42))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(42)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T^{2} + T^{4} \))
$3$ (\( 1 + T \))(\( 1 + 9 T^{4} \))(\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + 3 T^{2} + 9 T^{4} \))
$5$ (\( 1 + 2 T + 5 T^{2} \))(\( ( 1 + 4 T^{2} + 25 T^{4} )^{2} \))(\( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 7 T^{2} + 24 T^{4} - 175 T^{6} + 625 T^{8} \))
$7$ (\( 1 + T \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( ( 1 + 5 T + 7 T^{2} )^{2} \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( ( 1 - 11 T^{2} )^{4} \))(\( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} \))
$13$ (\( 1 - 6 T + 13 T^{2} \))(\( ( 1 - 20 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))
$17$ (\( 1 - 2 T + 17 T^{2} \))(\( ( 1 + 10 T^{2} + 289 T^{4} )^{2} \))(\( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 22 T^{2} + 195 T^{4} - 6358 T^{6} + 83521 T^{8} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( ( 1 - 32 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2} \))
$23$ (\( 1 - 8 T + 23 T^{2} \))(\( ( 1 - 10 T^{2} + 529 T^{4} )^{2} \))(\( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} \))
$29$ (\( 1 + 2 T + 29 T^{2} \))(\( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 5 T + 29 T^{2} )^{2} \))(\( ( 1 - 9 T + 29 T^{2} )^{2} \))(\( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \))
$31$ (\( 1 + 31 T^{2} \))(\( ( 1 - 31 T^{2} )^{4} \))(\( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} \))(\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))(\( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))
$37$ (\( 1 + 10 T + 37 T^{2} \))(\( ( 1 + 2 T + 37 T^{2} )^{4} \))(\( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( ( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( ( 1 + 58 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 34 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{4} \))(\( ( 1 - 2 T + 43 T^{2} )^{2} \))(\( ( 1 + 10 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{4} \))
$47$ (\( 1 + 47 T^{2} \))(\( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 - 46 T^{2} - 93 T^{4} - 101614 T^{6} + 4879681 T^{8} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( ( 1 - 70 T^{2} + 2809 T^{4} )^{2} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \))(\( 1 + 25 T^{2} - 2184 T^{4} + 70225 T^{6} + 7890481 T^{8} \))
$59$ (\( 1 - 4 T + 59 T^{2} \))(\( ( 1 - 32 T^{2} + 3481 T^{4} )^{2} \))(\( 1 - 11 T + 62 T^{2} - 649 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 - 115 T^{2} + 9744 T^{4} - 400315 T^{6} + 12117361 T^{8} \))
$61$ (\( 1 - 6 T + 61 T^{2} \))(\( ( 1 + 28 T^{2} + 3721 T^{4} )^{2} \))(\( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( ( 1 + 61 T^{2} + 3721 T^{4} )^{2} \))
$67$ (\( 1 - 4 T + 67 T^{2} \))(\( ( 1 - 8 T + 67 T^{2} )^{4} \))(\( 1 - 2 T - 63 T^{2} - 134 T^{3} + 4489 T^{4} \))(\( 1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4} \))(\( ( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} )^{2} \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 2 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( 1 - 10 T + 73 T^{2} \))(\( ( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2} \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( ( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} )^{2} \))
$79$ (\( 1 + 79 T^{2} \))(\( ( 1 + 10 T + 79 T^{2} )^{4} \))(\( 1 + 3 T - 70 T^{2} + 237 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( ( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} )^{2} \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( ( 1 + 160 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 7 T + 83 T^{2} )^{2} \))(\( ( 1 + 9 T + 83 T^{2} )^{2} \))(\( ( 1 + 91 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} \))
$97$ (\( 1 + 14 T + 97 T^{2} \))(\( ( 1 - 170 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 7 T + 97 T^{2} )^{2} \))(\( ( 1 + T + 97 T^{2} )^{2} \))(\( ( 1 - 19 T + 97 T^{2} )^{2}( 1 + 19 T + 97 T^{2} )^{2} \))
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