Properties

Label 25.4
Level 25
Weight 4
Dimension 59
Nonzero newspaces 4
Newform subspaces 7
Sturm bound 200
Trace bound 1

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Defining parameters

Level: \( N \) = \( 25 = 5^{2} \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 7 \)
Sturm bound: \(200\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(25))\).

Total New Old
Modular forms 89 80 9
Cusp forms 61 59 2
Eisenstein series 28 21 7

Trace form

\( 59q - 2q^{2} - 14q^{3} - 26q^{4} - 5q^{5} - 2q^{6} - 22q^{7} - 10q^{8} + 36q^{9} + O(q^{10}) \) \( 59q - 2q^{2} - 14q^{3} - 26q^{4} - 5q^{5} - 2q^{6} - 22q^{7} - 10q^{8} + 36q^{9} - 30q^{10} - 82q^{11} - 42q^{12} + 66q^{13} + 38q^{14} - 146q^{16} - 382q^{17} - 644q^{18} - 330q^{19} + 170q^{20} + 198q^{21} + 966q^{22} + 586q^{23} + 1420q^{24} + 685q^{25} + 108q^{26} + 730q^{27} + 614q^{28} + 170q^{29} - 290q^{30} - 162q^{31} - 1642q^{32} - 1098q^{33} - 1452q^{34} - 840q^{35} - 1026q^{36} - 1077q^{37} - 1700q^{38} - 2298q^{39} - 2060q^{40} - 522q^{41} - 614q^{42} - 614q^{43} + 808q^{44} + 2195q^{45} + 1578q^{46} + 2738q^{47} + 6276q^{48} + 2799q^{49} + 4500q^{50} + 2688q^{51} + 4388q^{52} + 1651q^{53} + 2960q^{54} + 910q^{55} - 110q^{56} - 970q^{57} - 2020q^{58} - 4310q^{59} - 11470q^{60} - 2802q^{61} - 11144q^{62} - 7954q^{63} - 6696q^{64} - 1955q^{65} - 1114q^{66} + 458q^{67} + 3414q^{68} + 3382q^{69} + 4190q^{70} + 1638q^{71} + 9240q^{72} + 4626q^{73} + 9508q^{74} + 5860q^{75} + 3620q^{76} + 5046q^{77} + 5312q^{78} + 950q^{79} + 5830q^{80} - 2496q^{81} - 6674q^{82} - 7354q^{83} - 13402q^{84} - 9645q^{85} - 4942q^{86} - 8470q^{87} - 12610q^{88} - 7335q^{89} - 8160q^{90} - 1362q^{91} - 3742q^{92} + 4582q^{93} + 6918q^{94} + 6810q^{95} + 4778q^{96} + 12238q^{97} + 11034q^{98} + 13332q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(25))\)

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
25.4.a \(\chi_{25}(1, \cdot)\) 25.4.a.a 1 1
25.4.a.b 1
25.4.a.c 1
25.4.b \(\chi_{25}(24, \cdot)\) 25.4.b.a 2 1
25.4.b.b 2
25.4.d \(\chi_{25}(6, \cdot)\) 25.4.d.a 28 4
25.4.e \(\chi_{25}(4, \cdot)\) 25.4.e.a 24 4

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(25))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(25)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 8 T^{2} \))(\( 1 - T + 8 T^{2} \))(\( 1 - 4 T + 8 T^{2} \))(\( ( 1 - 4 T + 8 T^{2} )( 1 + 4 T + 8 T^{2} ) \))(\( 1 - 15 T^{2} + 64 T^{4} \))
$3$ (\( 1 + 7 T + 27 T^{2} \))(\( 1 - 7 T + 27 T^{2} \))(\( 1 + 2 T + 27 T^{2} \))(\( 1 - 50 T^{2} + 729 T^{4} \))(\( 1 - 5 T^{2} + 729 T^{4} \))
$5$ 1
$7$ (\( 1 + 6 T + 343 T^{2} \))(\( 1 - 6 T + 343 T^{2} \))(\( 1 + 6 T + 343 T^{2} \))(\( 1 - 650 T^{2} + 117649 T^{4} \))(\( 1 - 650 T^{2} + 117649 T^{4} \))
$11$ (\( 1 + 43 T + 1331 T^{2} \))(\( 1 + 43 T + 1331 T^{2} \))(\( 1 - 32 T + 1331 T^{2} \))(\( ( 1 - 32 T + 1331 T^{2} )^{2} \))(\( ( 1 + 43 T + 1331 T^{2} )^{2} \))
$13$ (\( 1 - 28 T + 2197 T^{2} \))(\( 1 + 28 T + 2197 T^{2} \))(\( 1 - 38 T + 2197 T^{2} \))(\( 1 - 2950 T^{2} + 4826809 T^{4} \))(\( 1 - 3610 T^{2} + 4826809 T^{4} \))
$17$ (\( 1 + 91 T + 4913 T^{2} \))(\( 1 - 91 T + 4913 T^{2} \))(\( 1 + 26 T + 4913 T^{2} \))(\( 1 - 9150 T^{2} + 24137569 T^{4} \))(\( 1 - 1545 T^{2} + 24137569 T^{4} \))
$19$ (\( 1 + 35 T + 6859 T^{2} \))(\( 1 + 35 T + 6859 T^{2} \))(\( 1 - 100 T + 6859 T^{2} \))(\( ( 1 + 100 T + 6859 T^{2} )^{2} \))(\( ( 1 - 35 T + 6859 T^{2} )^{2} \))
$23$ (\( 1 + 162 T + 12167 T^{2} \))(\( 1 - 162 T + 12167 T^{2} \))(\( 1 - 78 T + 12167 T^{2} \))(\( 1 - 18250 T^{2} + 148035889 T^{4} \))(\( 1 + 1910 T^{2} + 148035889 T^{4} \))
$29$ (\( 1 - 160 T + 24389 T^{2} \))(\( 1 - 160 T + 24389 T^{2} \))(\( 1 + 50 T + 24389 T^{2} \))(\( ( 1 - 50 T + 24389 T^{2} )^{2} \))(\( ( 1 + 160 T + 24389 T^{2} )^{2} \))
$31$ (\( 1 - 42 T + 29791 T^{2} \))(\( 1 - 42 T + 29791 T^{2} \))(\( 1 + 108 T + 29791 T^{2} \))(\( ( 1 + 108 T + 29791 T^{2} )^{2} \))(\( ( 1 - 42 T + 29791 T^{2} )^{2} \))
$37$ (\( 1 - 314 T + 50653 T^{2} \))(\( 1 + 314 T + 50653 T^{2} \))(\( 1 + 266 T + 50653 T^{2} \))(\( 1 - 30550 T^{2} + 2565726409 T^{4} \))(\( 1 - 2710 T^{2} + 2565726409 T^{4} \))
$41$ (\( 1 + 203 T + 68921 T^{2} \))(\( 1 + 203 T + 68921 T^{2} \))(\( 1 - 22 T + 68921 T^{2} \))(\( ( 1 - 22 T + 68921 T^{2} )^{2} \))(\( ( 1 + 203 T + 68921 T^{2} )^{2} \))
$43$ (\( 1 + 92 T + 79507 T^{2} \))(\( 1 - 92 T + 79507 T^{2} \))(\( 1 + 442 T + 79507 T^{2} \))(\( 1 + 36350 T^{2} + 6321363049 T^{4} \))(\( 1 - 150550 T^{2} + 6321363049 T^{4} \))
$47$ (\( 1 + 196 T + 103823 T^{2} \))(\( 1 - 196 T + 103823 T^{2} \))(\( 1 - 514 T + 103823 T^{2} \))(\( 1 + 56550 T^{2} + 10779215329 T^{4} \))(\( 1 - 169230 T^{2} + 10779215329 T^{4} \))
$53$ (\( 1 + 82 T + 148877 T^{2} \))(\( 1 - 82 T + 148877 T^{2} \))(\( 1 + 2 T + 148877 T^{2} \))(\( 1 - 297750 T^{2} + 22164361129 T^{4} \))(\( 1 - 291030 T^{2} + 22164361129 T^{4} \))
$59$ (\( 1 + 280 T + 205379 T^{2} \))(\( 1 + 280 T + 205379 T^{2} \))(\( 1 - 500 T + 205379 T^{2} \))(\( ( 1 + 500 T + 205379 T^{2} )^{2} \))(\( ( 1 - 280 T + 205379 T^{2} )^{2} \))
$61$ (\( 1 + 518 T + 226981 T^{2} \))(\( 1 + 518 T + 226981 T^{2} \))(\( 1 + 518 T + 226981 T^{2} \))(\( ( 1 + 518 T + 226981 T^{2} )^{2} \))(\( ( 1 + 518 T + 226981 T^{2} )^{2} \))
$67$ (\( 1 + 141 T + 300763 T^{2} \))(\( 1 - 141 T + 300763 T^{2} \))(\( 1 + 126 T + 300763 T^{2} \))(\( 1 - 585650 T^{2} + 90458382169 T^{4} \))(\( 1 - 581645 T^{2} + 90458382169 T^{4} \))
$71$ (\( 1 - 412 T + 357911 T^{2} \))(\( 1 - 412 T + 357911 T^{2} \))(\( 1 - 412 T + 357911 T^{2} \))(\( ( 1 - 412 T + 357911 T^{2} )^{2} \))(\( ( 1 - 412 T + 357911 T^{2} )^{2} \))
$73$ (\( 1 - 763 T + 389017 T^{2} \))(\( 1 + 763 T + 389017 T^{2} \))(\( 1 - 878 T + 389017 T^{2} \))(\( 1 - 7150 T^{2} + 151334226289 T^{4} \))(\( 1 - 195865 T^{2} + 151334226289 T^{4} \))
$79$ (\( 1 - 510 T + 493039 T^{2} \))(\( 1 - 510 T + 493039 T^{2} \))(\( 1 - 600 T + 493039 T^{2} \))(\( ( 1 + 600 T + 493039 T^{2} )^{2} \))(\( ( 1 + 510 T + 493039 T^{2} )^{2} \))
$83$ (\( 1 + 777 T + 571787 T^{2} \))(\( 1 - 777 T + 571787 T^{2} \))(\( 1 + 282 T + 571787 T^{2} \))(\( 1 - 1064050 T^{2} + 326940373369 T^{4} \))(\( 1 - 539845 T^{2} + 326940373369 T^{4} \))
$89$ (\( 1 + 945 T + 704969 T^{2} \))(\( 1 + 945 T + 704969 T^{2} \))(\( 1 + 150 T + 704969 T^{2} \))(\( ( 1 - 150 T + 704969 T^{2} )^{2} \))(\( ( 1 - 945 T + 704969 T^{2} )^{2} \))
$97$ (\( 1 + 1246 T + 912673 T^{2} \))(\( 1 - 1246 T + 912673 T^{2} \))(\( 1 + 386 T + 912673 T^{2} \))(\( 1 - 1676350 T^{2} + 832972004929 T^{4} \))(\( 1 - 272830 T^{2} + 832972004929 T^{4} \))
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