Properties

Label 26.2
Level 26
Weight 2
Dimension 6
Nonzero newspaces 3
Newform subspaces 4
Sturm bound 84
Trace bound 4

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Defining parameters

Level: \( N \) = \( 26\( 26 = 2 \cdot 13 \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 4 \)
Sturm bound: \(84\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(26))\).

Total New Old
Modular forms 33 6 27
Cusp forms 10 6 4
Eisenstein series 23 0 23

Trace form

\( 6q - q^{2} - 4q^{3} - q^{4} - 6q^{5} - 4q^{6} - 4q^{7} + 2q^{8} + 3q^{9} + O(q^{10}) \) \( 6q - q^{2} - 4q^{3} - q^{4} - 6q^{5} - 4q^{6} - 4q^{7} + 2q^{8} + 3q^{9} + 9q^{10} + 11q^{13} + 4q^{14} + 3q^{16} - 3q^{17} + 2q^{18} + 8q^{19} - 3q^{20} - 4q^{21} - 12q^{22} - 12q^{23} - 4q^{24} - 16q^{25} - 13q^{26} - 4q^{27} - 4q^{28} + 9q^{29} + 8q^{31} - q^{32} + 12q^{33} + 6q^{34} + 24q^{35} + 11q^{36} - 7q^{37} + 16q^{38} - 6q^{40} + 9q^{41} + 4q^{42} + 12q^{44} - 3q^{45} - 4q^{48} - 25q^{49} - 4q^{50} - 6q^{52} - 18q^{53} - 4q^{54} - 12q^{55} + 4q^{56} - 16q^{57} - 3q^{58} - 12q^{59} - 23q^{61} + 4q^{62} + 20q^{63} + 2q^{64} + 9q^{65} + 8q^{67} - 15q^{68} + 24q^{69} - 12q^{70} + 11q^{72} + 14q^{73} + q^{74} + 24q^{75} + 8q^{76} + 24q^{77} + 8q^{78} + 16q^{79} - 3q^{80} + 3q^{81} + 9q^{82} + 12q^{83} - 4q^{84} + 15q^{85} - 20q^{86} - 12q^{88} + 6q^{89} - 18q^{90} - 28q^{91} - 16q^{93} + 12q^{94} - 48q^{95} - 4q^{96} + 2q^{97} - 9q^{98} - 48q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(26))\)

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
26.2.a \(\chi_{26}(1, \cdot)\) 26.2.a.a 1 1
26.2.a.b 1
26.2.b \(\chi_{26}(25, \cdot)\) 26.2.b.a 2 1
26.2.c \(\chi_{26}(3, \cdot)\) 26.2.c.a 2 2
26.2.e \(\chi_{26}(17, \cdot)\) None 0 2

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(26))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(26)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T^{2} \))(\( 1 + T + T^{2} \))
$3$ (\( 1 - T + 3 T^{2} \))(\( 1 + 3 T + 3 T^{2} \))(\( ( 1 + T + 3 T^{2} )^{2} \))(\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 - T^{2} + 25 T^{4} \))(\( ( 1 + T + 5 T^{2} )^{2} \))
$7$ (\( 1 + T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 5 T^{2} + 49 T^{4} \))(\( ( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))
$11$ (\( 1 - 6 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} \))
$13$ (\( 1 - T \))(\( 1 + T \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 7 T + 13 T^{2} \))
$17$ (\( 1 + 3 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( ( 1 - 3 T + 17 T^{2} )^{2} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))
$19$ (\( 1 - 2 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 - 2 T^{2} + 361 T^{4} \))(\( 1 - 19 T^{2} + 361 T^{4} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( ( 1 + 6 T + 23 T^{2} )^{2} \))(\( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( ( 1 - 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))
$37$ (\( 1 + 7 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( 1 - 65 T^{2} + 1369 T^{4} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} \))
$43$ (\( 1 + T + 43 T^{2} \))(\( 1 + 5 T + 43 T^{2} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 - 13 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} ) \))
$47$ (\( 1 - 3 T + 47 T^{2} \))(\( 1 - 13 T + 47 T^{2} \))(\( 1 - 85 T^{2} + 2209 T^{4} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))
$53$ (\( 1 + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 9 T + 53 T^{2} )^{2} \))
$59$ (\( 1 + 6 T + 59 T^{2} \))(\( 1 + 10 T + 59 T^{2} \))(\( 1 - 82 T^{2} + 3481 T^{4} \))(\( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} \))
$61$ (\( 1 - 8 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( ( 1 + 8 T + 61 T^{2} )^{2} \))(\( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \))
$67$ (\( 1 - 14 T + 67 T^{2} \))(\( 1 + 2 T + 67 T^{2} \))(\( 1 + 10 T^{2} + 4489 T^{4} \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))
$71$ (\( 1 + 3 T + 71 T^{2} \))(\( 1 + 5 T + 71 T^{2} \))(\( 1 + 83 T^{2} + 5041 T^{4} \))(\( 1 - 8 T - 7 T^{2} - 568 T^{3} + 5041 T^{4} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \))(\( ( 1 - 11 T + 73 T^{2} )^{2} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{2} \))(\( ( 1 + 4 T + 79 T^{2} )^{2} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( ( 1 + 83 T^{2} )^{2} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 142 T^{2} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + 10 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 - 50 T^{2} + 9409 T^{4} \))(\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))
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