Properties

Label 25.2.e
Level 25
Weight 2
Character orbit e
Rep. character \(\chi_{25}(4,\cdot)\)
Character field \(\Q(\zeta_{10})\)
Dimension 8
Newform subspaces 1
Sturm bound 5
Trace bound 0

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Defining parameters

Level: \( N \) = \( 25 = 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 25.e (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 25 \)
Character field: \(\Q(\zeta_{10})\)
Newform subspaces: \( 1 \)
Sturm bound: \(5\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(25, [\chi])\).

Total New Old
Modular forms 16 16 0
Cusp forms 8 8 0
Eisenstein series 8 8 0

Trace form

\( 8q - 5q^{2} - 5q^{3} - q^{4} - 9q^{6} + 10q^{8} + q^{9} + O(q^{10}) \) \( 8q - 5q^{2} - 5q^{3} - q^{4} - 9q^{6} + 10q^{8} + q^{9} - 5q^{10} - 4q^{11} + 15q^{12} - 5q^{13} + 13q^{14} + 15q^{15} + 3q^{16} - 10q^{17} - 5q^{19} - 15q^{20} - 4q^{21} + 5q^{23} - 20q^{24} - 10q^{25} + 6q^{26} - 5q^{27} - 15q^{28} - 5q^{29} + 15q^{30} - 9q^{31} + 10q^{33} + 13q^{34} + 15q^{35} + 23q^{36} + 30q^{37} + 15q^{38} - 3q^{39} + 10q^{40} - 4q^{41} - 15q^{42} - 2q^{44} - 15q^{45} - 19q^{46} - 30q^{48} + 14q^{49} - 15q^{50} - 4q^{51} - 10q^{52} - 10q^{53} - 5q^{54} - 10q^{55} + 10q^{56} + 20q^{58} - 10q^{60} - 9q^{61} - 30q^{62} + 10q^{63} + 4q^{64} + 5q^{65} + 12q^{66} + 20q^{67} + 17q^{69} + 30q^{70} + 6q^{71} + 5q^{72} + 15q^{73} - 12q^{74} - 10q^{75} - 20q^{76} + 10q^{77} + 25q^{78} + 15q^{79} + 20q^{80} + 28q^{81} - 45q^{83} + 18q^{84} - 15q^{85} - 9q^{86} - 20q^{87} - 20q^{88} - 25q^{89} - 25q^{90} + 6q^{91} + 30q^{92} - 27q^{94} + 15q^{95} + 16q^{96} - 60q^{97} - 10q^{98} - 8q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(25, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
25.2.e.a \(8\) \(0.200\) 8.0.58140625.2 None \(-5\) \(-5\) \(0\) \(0\) \(q+(-1+\beta _{1})q^{2}+(-1+\beta _{2}-\beta _{3}+\beta _{7})q^{3}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 5 T + 15 T^{2} + 30 T^{3} + 41 T^{4} + 30 T^{5} - 20 T^{6} - 110 T^{7} - 199 T^{8} - 220 T^{9} - 80 T^{10} + 240 T^{11} + 656 T^{12} + 960 T^{13} + 960 T^{14} + 640 T^{15} + 256 T^{16} \)
$3$ \( 1 + 5 T + 15 T^{2} + 30 T^{3} + 36 T^{4} + 5 T^{5} - 120 T^{6} - 400 T^{7} - 809 T^{8} - 1200 T^{9} - 1080 T^{10} + 135 T^{11} + 2916 T^{12} + 7290 T^{13} + 10935 T^{14} + 10935 T^{15} + 6561 T^{16} \)
$5$ \( 1 + 5 T^{2} - 20 T^{3} + 5 T^{4} - 100 T^{5} + 125 T^{6} + 625 T^{8} \)
$7$ \( 1 - 35 T^{2} + 611 T^{4} - 7045 T^{6} + 57976 T^{8} - 345205 T^{10} + 1467011 T^{12} - 4117715 T^{14} + 5764801 T^{16} \)
$11$ \( ( 1 + 2 T - 7 T^{2} - 36 T^{3} + 5 T^{4} - 396 T^{5} - 847 T^{6} + 2662 T^{7} + 14641 T^{8} )^{2} \)
$13$ \( 1 + 5 T + 30 T^{2} + 60 T^{3} + 346 T^{4} + 655 T^{5} + 6335 T^{6} + 13320 T^{7} + 88856 T^{8} + 173160 T^{9} + 1070615 T^{10} + 1439035 T^{11} + 9882106 T^{12} + 22277580 T^{13} + 144804270 T^{14} + 313742585 T^{15} + 815730721 T^{16} \)
$17$ \( 1 + 10 T + 90 T^{2} + 720 T^{3} + 4451 T^{4} + 26310 T^{5} + 136780 T^{6} + 638720 T^{7} + 2802941 T^{8} + 10858240 T^{9} + 39529420 T^{10} + 129261030 T^{11} + 371751971 T^{12} + 1022297040 T^{13} + 2172381210 T^{14} + 4103386730 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 + 5 T - 8 T^{2} + 40 T^{3} + 878 T^{4} + 1705 T^{5} - 1861 T^{6} + 31550 T^{7} + 293380 T^{8} + 599450 T^{9} - 671821 T^{10} + 11694595 T^{11} + 114421838 T^{12} + 99043960 T^{13} - 376367048 T^{14} + 4469358695 T^{15} + 16983563041 T^{16} \)
$23$ \( 1 - 5 T + 45 T^{2} + 100 T^{3} - 104 T^{4} + 7645 T^{5} + 18890 T^{6} + 8420 T^{7} + 1238691 T^{8} + 193660 T^{9} + 9992810 T^{10} + 93016715 T^{11} - 29103464 T^{12} + 643634300 T^{13} + 6661615005 T^{14} - 17024127235 T^{15} + 78310985281 T^{16} \)
$29$ \( 1 + 5 T - 28 T^{2} + 140 T^{3} + 1268 T^{4} - 5045 T^{5} + 48219 T^{6} + 207000 T^{7} - 1738520 T^{8} + 6003000 T^{9} + 40552179 T^{10} - 123042505 T^{11} + 896832308 T^{12} + 2871560860 T^{13} - 16655052988 T^{14} + 86249381545 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 + 9 T + 55 T^{2} + 390 T^{3} + 2980 T^{4} + 20297 T^{5} + 114748 T^{6} + 589990 T^{7} + 3582095 T^{8} + 18289690 T^{9} + 110272828 T^{10} + 604667927 T^{11} + 2752092580 T^{12} + 11165368890 T^{13} + 48812702455 T^{14} + 247613526999 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 - 30 T + 480 T^{2} - 5675 T^{3} + 56171 T^{4} - 489630 T^{5} + 3826535 T^{6} - 26904445 T^{7} + 171416106 T^{8} - 995464465 T^{9} + 5238526415 T^{10} - 24801228390 T^{11} + 105273497531 T^{12} - 393526955975 T^{13} + 1231548676320 T^{14} - 2847956313990 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 + 4 T - 30 T^{2} - 240 T^{3} + 195 T^{4} + 18372 T^{5} + 77388 T^{6} - 230240 T^{7} - 1438395 T^{8} - 9439840 T^{9} + 130089228 T^{10} + 1266216612 T^{11} + 551023395 T^{12} - 27805488240 T^{13} - 142503127230 T^{14} + 779017095524 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 - 215 T^{2} + 22911 T^{4} - 1578205 T^{6} + 78597176 T^{8} - 2918101045 T^{10} + 78328149711 T^{12} - 1359093055535 T^{14} + 11688200277601 T^{16} \)
$47$ \( 1 + 110 T^{2} - 90 T^{3} + 4101 T^{4} - 9900 T^{5} - 3830 T^{6} - 910260 T^{7} - 5610889 T^{8} - 42782220 T^{9} - 8460470 T^{10} - 1027847700 T^{11} + 20011571781 T^{12} - 20641050630 T^{13} + 1185713686190 T^{14} + 23811286661761 T^{16} \)
$53$ \( 1 + 10 T + 100 T^{2} + 1625 T^{3} + 11531 T^{4} + 95310 T^{5} + 857875 T^{6} + 5635035 T^{7} + 42472426 T^{8} + 298656855 T^{9} + 2409770875 T^{10} + 14189466870 T^{11} + 90985136411 T^{12} + 679567676125 T^{13} + 2216436112900 T^{14} + 11747111398370 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 - 118 T^{2} + 900 T^{3} + 3393 T^{4} - 96300 T^{5} + 615034 T^{6} + 2943900 T^{7} - 65890945 T^{8} + 173690100 T^{9} + 2140933354 T^{10} - 19777997700 T^{11} + 41114205873 T^{12} + 643431869100 T^{13} - 4977302969638 T^{14} + 146830437604321 T^{16} \)
$61$ \( 1 + 9 T - 165 T^{2} - 1800 T^{3} + 7560 T^{4} + 154437 T^{5} + 526138 T^{6} - 4559670 T^{7} - 69838275 T^{8} - 278139870 T^{9} + 1957759498 T^{10} + 35054264697 T^{11} + 104674557960 T^{12} - 1520273341800 T^{13} - 8500861769565 T^{14} + 28284685524189 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 - 20 T + 250 T^{2} - 2600 T^{3} + 26091 T^{4} - 241820 T^{5} + 2034700 T^{6} - 15361680 T^{7} + 117317461 T^{8} - 1029232560 T^{9} + 9133768300 T^{10} - 72730508660 T^{11} + 525762898011 T^{12} - 3510325278200 T^{13} + 22614595542250 T^{14} - 121214232106460 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 - 6 T - 910 T^{3} + 4875 T^{4} + 34402 T^{5} + 474398 T^{6} - 3835500 T^{7} - 25760305 T^{8} - 272320500 T^{9} + 2391440318 T^{10} + 12312854222 T^{11} + 123881944875 T^{12} - 1641848709410 T^{13} - 54570720950346 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 - 15 T + 195 T^{2} - 1340 T^{3} + 15786 T^{4} - 132465 T^{5} + 1900340 T^{6} - 14658400 T^{7} + 154250461 T^{8} - 1070063200 T^{9} + 10126911860 T^{10} - 51531136905 T^{11} + 448294632426 T^{12} - 2777915934620 T^{13} + 29510174126355 T^{14} - 165710977786455 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 - 15 T - 58 T^{2} + 2180 T^{3} - 7802 T^{4} - 148865 T^{5} + 1559169 T^{6} + 5584500 T^{7} - 169067020 T^{8} + 441175500 T^{9} + 9730773729 T^{10} - 73396250735 T^{11} - 303888531962 T^{12} + 6707982949820 T^{13} - 14099072420218 T^{14} - 288058634792385 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 + 45 T + 1115 T^{2} + 19890 T^{3} + 284376 T^{4} + 3450645 T^{5} + 37197280 T^{6} + 367888080 T^{7} + 3431240591 T^{8} + 30534710640 T^{9} + 256252061920 T^{10} + 1973033952615 T^{11} + 13496007492696 T^{12} + 78347518389270 T^{13} + 364538516306435 T^{14} + 1221122294533215 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 + 25 T + 342 T^{2} + 5000 T^{3} + 78868 T^{4} + 928525 T^{5} + 9098049 T^{6} + 101226750 T^{7} + 1076434080 T^{8} + 9009180750 T^{9} + 72065646129 T^{10} + 654581340725 T^{11} + 4948355063188 T^{12} + 27920297245000 T^{13} + 169967601508662 T^{14} + 1105783372388225 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 + 60 T + 1830 T^{2} + 35660 T^{3} + 467711 T^{4} + 3770460 T^{5} + 6583460 T^{6} - 292271720 T^{7} - 4451764059 T^{8} - 28350356840 T^{9} + 61943775140 T^{10} + 3441197039580 T^{11} + 41406118545791 T^{12} + 306224553564620 T^{13} + 1524338769020070 T^{14} + 4847897068686780 T^{15} + 7837433594376961 T^{16} \)
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