Properties

Label 25.2
Level 25
Weight 2
Dimension 12
Nonzero newspaces 2
Newform subspaces 2
Sturm bound 100
Trace bound 1

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Defining parameters

Level: \( N \) = \( 25\( 25 = 5^{2} \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 2 \)
Sturm bound: \(100\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(25))\).

Total New Old
Modular forms 39 33 6
Cusp forms 12 12 0
Eisenstein series 27 21 6

Trace form

\( 12q - 7q^{2} - 6q^{3} - 3q^{4} - 5q^{5} - 6q^{6} - 2q^{7} + 5q^{8} + 3q^{9} + O(q^{10}) \) \( 12q - 7q^{2} - 6q^{3} - 3q^{4} - 5q^{5} - 6q^{6} - 2q^{7} + 5q^{8} + 3q^{9} + 5q^{10} - 6q^{11} + 18q^{12} + 4q^{13} + 14q^{14} + 10q^{15} - 3q^{16} - 2q^{17} + 4q^{18} - 10q^{19} - 10q^{20} - 6q^{21} - 14q^{22} - 6q^{23} - 20q^{24} - 15q^{25} - 6q^{26} - 14q^{28} + 10q^{30} - 6q^{31} + 18q^{32} + 18q^{33} + 19q^{34} + 20q^{35} + 17q^{36} + 23q^{37} + 20q^{38} + 6q^{39} + 15q^{40} + 4q^{41} - 14q^{42} - 6q^{43} + 4q^{44} - 25q^{45} - 6q^{46} - 2q^{47} - 36q^{48} - 8q^{49} - 25q^{50} - 16q^{51} - 22q^{52} - q^{53} - 20q^{54} + 10q^{55} + 10q^{56} - 10q^{57} + 10q^{58} - 10q^{60} + 4q^{61} - 24q^{62} + 4q^{63} + 7q^{64} + 5q^{65} + 18q^{66} + 18q^{67} - 4q^{68} + 16q^{69} + 30q^{70} + 14q^{71} + 15q^{72} + 24q^{73} - 6q^{74} + 10q^{75} + 6q^{77} + 28q^{78} + 30q^{79} + 5q^{80} + 27q^{81} - 4q^{82} - 36q^{83} + 14q^{84} - 35q^{85} - 6q^{86} - 20q^{87} - 30q^{88} - 45q^{89} - 25q^{90} - 6q^{91} + 18q^{92} - 12q^{93} - 26q^{94} + 10q^{95} + 14q^{96} - 52q^{97} + q^{98} + 16q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(25))\)

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
25.2.a \(\chi_{25}(1, \cdot)\) None 0 1
25.2.b \(\chi_{25}(24, \cdot)\) None 0 1
25.2.d \(\chi_{25}(6, \cdot)\) 25.2.d.a 4 4
25.2.e \(\chi_{25}(4, \cdot)\) 25.2.e.a 8 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} + 5 T^{3} + 11 T^{4} + 10 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 + 5 T + 15 T^{2} + 30 T^{3} + 41 T^{4} + 30 T^{5} - 20 T^{6} - 110 T^{7} - 199 T^{8} - 220 T^{9} - 80 T^{10} + 240 T^{11} + 656 T^{12} + 960 T^{13} + 960 T^{14} + 640 T^{15} + 256 T^{16} \))
$3$ (\( 1 + T - 2 T^{2} - 5 T^{3} + T^{4} - 15 T^{5} - 18 T^{6} + 27 T^{7} + 81 T^{8} \))(\( 1 + 5 T + 15 T^{2} + 30 T^{3} + 36 T^{4} + 5 T^{5} - 120 T^{6} - 400 T^{7} - 809 T^{8} - 1200 T^{9} - 1080 T^{10} + 135 T^{11} + 2916 T^{12} + 7290 T^{13} + 10935 T^{14} + 10935 T^{15} + 6561 T^{16} \))
$5$ (\( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} \))(\( 1 + 5 T^{2} - 20 T^{3} + 5 T^{4} - 100 T^{5} + 125 T^{6} + 625 T^{8} \))
$7$ (\( ( 1 + T + 13 T^{2} + 7 T^{3} + 49 T^{4} )^{2} \))(\( 1 - 35 T^{2} + 611 T^{4} - 7045 T^{6} + 57976 T^{8} - 345205 T^{10} + 1467011 T^{12} - 4117715 T^{14} + 5764801 T^{16} \))
$11$ (\( 1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 374 T^{5} + 1573 T^{6} + 2662 T^{7} + 14641 T^{8} \))(\( ( 1 + 2 T - 7 T^{2} - 36 T^{3} + 5 T^{4} - 396 T^{5} - 847 T^{6} + 2662 T^{7} + 14641 T^{8} )^{2} \))
$13$ (\( 1 - 9 T + 23 T^{2} - 15 T^{3} + 16 T^{4} - 195 T^{5} + 3887 T^{6} - 19773 T^{7} + 28561 T^{8} \))(\( 1 + 5 T + 30 T^{2} + 60 T^{3} + 346 T^{4} + 655 T^{5} + 6335 T^{6} + 13320 T^{7} + 88856 T^{8} + 173160 T^{9} + 1070615 T^{10} + 1439035 T^{11} + 9882106 T^{12} + 22277580 T^{13} + 144804270 T^{14} + 313742585 T^{15} + 815730721 T^{16} \))
$17$ (\( 1 - 8 T + 7 T^{2} + 110 T^{3} - 579 T^{4} + 1870 T^{5} + 2023 T^{6} - 39304 T^{7} + 83521 T^{8} \))(\( 1 + 10 T + 90 T^{2} + 720 T^{3} + 4451 T^{4} + 26310 T^{5} + 136780 T^{6} + 638720 T^{7} + 2802941 T^{8} + 10858240 T^{9} + 39529420 T^{10} + 129261030 T^{11} + 371751971 T^{12} + 1022297040 T^{13} + 2172381210 T^{14} + 4103386730 T^{15} + 6975757441 T^{16} \))
$19$ (\( 1 + 5 T + 21 T^{2} + 145 T^{3} + 956 T^{4} + 2755 T^{5} + 7581 T^{6} + 34295 T^{7} + 130321 T^{8} \))(\( 1 + 5 T - 8 T^{2} + 40 T^{3} + 878 T^{4} + 1705 T^{5} - 1861 T^{6} + 31550 T^{7} + 293380 T^{8} + 599450 T^{9} - 671821 T^{10} + 11694595 T^{11} + 114421838 T^{12} + 99043960 T^{13} - 376367048 T^{14} + 4469358695 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 + 11 T + 28 T^{2} - 245 T^{3} - 2259 T^{4} - 5635 T^{5} + 14812 T^{6} + 133837 T^{7} + 279841 T^{8} \))(\( 1 - 5 T + 45 T^{2} + 100 T^{3} - 104 T^{4} + 7645 T^{5} + 18890 T^{6} + 8420 T^{7} + 1238691 T^{8} + 193660 T^{9} + 9992810 T^{10} + 93016715 T^{11} - 29103464 T^{12} + 643634300 T^{13} + 6661615005 T^{14} - 17024127235 T^{15} + 78310985281 T^{16} \))
$29$ (\( 1 - 5 T - 19 T^{2} + 145 T^{3} - 4 T^{4} + 4205 T^{5} - 15979 T^{6} - 121945 T^{7} + 707281 T^{8} \))(\( 1 + 5 T - 28 T^{2} + 140 T^{3} + 1268 T^{4} - 5045 T^{5} + 48219 T^{6} + 207000 T^{7} - 1738520 T^{8} + 6003000 T^{9} + 40552179 T^{10} - 123042505 T^{11} + 896832308 T^{12} + 2871560860 T^{13} - 16655052988 T^{14} + 86249381545 T^{15} + 500246412961 T^{16} \))
$31$ (\( 1 - 3 T - 22 T^{2} + 159 T^{3} + 205 T^{4} + 4929 T^{5} - 21142 T^{6} - 89373 T^{7} + 923521 T^{8} \))(\( 1 + 9 T + 55 T^{2} + 390 T^{3} + 2980 T^{4} + 20297 T^{5} + 114748 T^{6} + 589990 T^{7} + 3582095 T^{8} + 18289690 T^{9} + 110272828 T^{10} + 604667927 T^{11} + 2752092580 T^{12} + 11165368890 T^{13} + 48812702455 T^{14} + 247613526999 T^{15} + 852891037441 T^{16} \))
$37$ (\( 1 + 7 T - 18 T^{2} - 145 T^{3} + 371 T^{4} - 5365 T^{5} - 24642 T^{6} + 354571 T^{7} + 1874161 T^{8} \))(\( 1 - 30 T + 480 T^{2} - 5675 T^{3} + 56171 T^{4} - 489630 T^{5} + 3826535 T^{6} - 26904445 T^{7} + 171416106 T^{8} - 995464465 T^{9} + 5238526415 T^{10} - 24801228390 T^{11} + 105273497531 T^{12} - 393526955975 T^{13} + 1231548676320 T^{14} - 2847956313990 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 - 8 T - 17 T^{2} + 254 T^{3} - 435 T^{4} + 10414 T^{5} - 28577 T^{6} - 551368 T^{7} + 2825761 T^{8} \))(\( 1 + 4 T - 30 T^{2} - 240 T^{3} + 195 T^{4} + 18372 T^{5} + 77388 T^{6} - 230240 T^{7} - 1438395 T^{8} - 9439840 T^{9} + 130089228 T^{10} + 1266216612 T^{11} + 551023395 T^{12} - 27805488240 T^{13} - 142503127230 T^{14} + 779017095524 T^{15} + 7984925229121 T^{16} \))
$43$ (\( ( 1 + 3 T + 77 T^{2} + 129 T^{3} + 1849 T^{4} )^{2} \))(\( 1 - 215 T^{2} + 22911 T^{4} - 1578205 T^{6} + 78597176 T^{8} - 2918101045 T^{10} + 78328149711 T^{12} - 1359093055535 T^{14} + 11688200277601 T^{16} \))
$47$ (\( 1 + 2 T - 43 T^{2} + 50 T^{3} + 2351 T^{4} + 2350 T^{5} - 94987 T^{6} + 207646 T^{7} + 4879681 T^{8} \))(\( 1 + 110 T^{2} - 90 T^{3} + 4101 T^{4} - 9900 T^{5} - 3830 T^{6} - 910260 T^{7} - 5610889 T^{8} - 42782220 T^{9} - 8460470 T^{10} - 1027847700 T^{11} + 20011571781 T^{12} - 20641050630 T^{13} + 1185713686190 T^{14} + 23811286661761 T^{16} \))
$53$ (\( 1 - 9 T + 8 T^{2} - 315 T^{3} + 5131 T^{4} - 16695 T^{5} + 22472 T^{6} - 1339893 T^{7} + 7890481 T^{8} \))(\( 1 + 10 T + 100 T^{2} + 1625 T^{3} + 11531 T^{4} + 95310 T^{5} + 857875 T^{6} + 5635035 T^{7} + 42472426 T^{8} + 298656855 T^{9} + 2409770875 T^{10} + 14189466870 T^{11} + 90985136411 T^{12} + 679567676125 T^{13} + 2216436112900 T^{14} + 11747111398370 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 + 31 T^{2} + 210 T^{3} + 2851 T^{4} + 12390 T^{5} + 107911 T^{6} + 12117361 T^{8} \))(\( 1 - 118 T^{2} + 900 T^{3} + 3393 T^{4} - 96300 T^{5} + 615034 T^{6} + 2943900 T^{7} - 65890945 T^{8} + 173690100 T^{9} + 2140933354 T^{10} - 19777997700 T^{11} + 41114205873 T^{12} + 643431869100 T^{13} - 4977302969638 T^{14} + 146830437604321 T^{16} \))
$61$ (\( 1 - 13 T + 78 T^{2} - 941 T^{3} + 11075 T^{4} - 57401 T^{5} + 290238 T^{6} - 2950753 T^{7} + 13845841 T^{8} \))(\( 1 + 9 T - 165 T^{2} - 1800 T^{3} + 7560 T^{4} + 154437 T^{5} + 526138 T^{6} - 4559670 T^{7} - 69838275 T^{8} - 278139870 T^{9} + 1957759498 T^{10} + 35054264697 T^{11} + 104674557960 T^{12} - 1520273341800 T^{13} - 8500861769565 T^{14} + 28284685524189 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 + 2 T - 3 T^{2} - 410 T^{3} + 1601 T^{4} - 27470 T^{5} - 13467 T^{6} + 601526 T^{7} + 20151121 T^{8} \))(\( 1 - 20 T + 250 T^{2} - 2600 T^{3} + 26091 T^{4} - 241820 T^{5} + 2034700 T^{6} - 15361680 T^{7} + 117317461 T^{8} - 1029232560 T^{9} + 9133768300 T^{10} - 72730508660 T^{11} + 525762898011 T^{12} - 3510325278200 T^{13} + 22614595542250 T^{14} - 121214232106460 T^{15} + 406067677556641 T^{16} \))
$71$ (\( 1 - 8 T - 37 T^{2} + 694 T^{3} - 2425 T^{4} + 49274 T^{5} - 186517 T^{6} - 2863288 T^{7} + 25411681 T^{8} \))(\( 1 - 6 T - 910 T^{3} + 4875 T^{4} + 34402 T^{5} + 474398 T^{6} - 3835500 T^{7} - 25760305 T^{8} - 272320500 T^{9} + 2391440318 T^{10} + 12312854222 T^{11} + 123881944875 T^{12} - 1641848709410 T^{13} - 54570720950346 T^{15} + 645753531245761 T^{16} \))
$73$ (\( 1 - 9 T + 8 T^{2} + 585 T^{3} - 5849 T^{4} + 42705 T^{5} + 42632 T^{6} - 3501153 T^{7} + 28398241 T^{8} \))(\( 1 - 15 T + 195 T^{2} - 1340 T^{3} + 15786 T^{4} - 132465 T^{5} + 1900340 T^{6} - 14658400 T^{7} + 154250461 T^{8} - 1070063200 T^{9} + 10126911860 T^{10} - 51531136905 T^{11} + 448294632426 T^{12} - 2777915934620 T^{13} + 29510174126355 T^{14} - 165710977786455 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 - 15 T + 21 T^{2} + 145 T^{3} + 2916 T^{4} + 11455 T^{5} + 131061 T^{6} - 7395585 T^{7} + 38950081 T^{8} \))(\( 1 - 15 T - 58 T^{2} + 2180 T^{3} - 7802 T^{4} - 148865 T^{5} + 1559169 T^{6} + 5584500 T^{7} - 169067020 T^{8} + 441175500 T^{9} + 9730773729 T^{10} - 73396250735 T^{11} - 303888531962 T^{12} + 6707982949820 T^{13} - 14099072420218 T^{14} - 288058634792385 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( 1 - 9 T - 52 T^{2} + 675 T^{3} + 121 T^{4} + 56025 T^{5} - 358228 T^{6} - 5146083 T^{7} + 47458321 T^{8} \))(\( 1 + 45 T + 1115 T^{2} + 19890 T^{3} + 284376 T^{4} + 3450645 T^{5} + 37197280 T^{6} + 367888080 T^{7} + 3431240591 T^{8} + 30534710640 T^{9} + 256252061920 T^{10} + 1973033952615 T^{11} + 13496007492696 T^{12} + 78347518389270 T^{13} + 364538516306435 T^{14} + 1221122294533215 T^{15} + 2252292232139041 T^{16} \))
$89$ (\( 1 + 20 T + 151 T^{2} + 1600 T^{3} + 21441 T^{4} + 142400 T^{5} + 1196071 T^{6} + 14099380 T^{7} + 62742241 T^{8} \))(\( 1 + 25 T + 342 T^{2} + 5000 T^{3} + 78868 T^{4} + 928525 T^{5} + 9098049 T^{6} + 101226750 T^{7} + 1076434080 T^{8} + 9009180750 T^{9} + 72065646129 T^{10} + 654581340725 T^{11} + 4948355063188 T^{12} + 27920297245000 T^{13} + 169967601508662 T^{14} + 1105783372388225 T^{15} + 3936588805702081 T^{16} \))
$97$ (\( 1 - 8 T - 63 T^{2} + 20 T^{3} + 9821 T^{4} + 1940 T^{5} - 592767 T^{6} - 7301384 T^{7} + 88529281 T^{8} \))(\( 1 + 60 T + 1830 T^{2} + 35660 T^{3} + 467711 T^{4} + 3770460 T^{5} + 6583460 T^{6} - 292271720 T^{7} - 4451764059 T^{8} - 28350356840 T^{9} + 61943775140 T^{10} + 3441197039580 T^{11} + 41406118545791 T^{12} + 306224553564620 T^{13} + 1524338769020070 T^{14} + 4847897068686780 T^{15} + 7837433594376961 T^{16} \))
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