Properties

Label 5.8
Level 5
Weight 8
Dimension 5
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 16
Trace bound 1

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Defining parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(16\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(5))\).

Total New Old
Modular forms 9 7 2
Cusp forms 5 5 0
Eisenstein series 4 2 2

Trace form

\( 5q + 6q^{2} - 28q^{3} + 188q^{4} + 25q^{5} - 1040q^{6} - 1744q^{7} + 2280q^{8} + 7957q^{9} + O(q^{10}) \) \( 5q + 6q^{2} - 28q^{3} + 188q^{4} + 25q^{5} - 1040q^{6} - 1744q^{7} + 2280q^{8} + 7957q^{9} + 1550q^{10} - 8940q^{11} - 26624q^{12} + 7402q^{13} + 39576q^{14} + 8900q^{15} - 29520q^{16} - 39594q^{17} + 29582q^{18} + 26540q^{19} - 1700q^{20} + 36960q^{21} + 103832q^{22} - 111168q^{23} - 269280q^{24} - 86875q^{25} + 107460q^{26} + 305720q^{27} + 53648q^{28} - 102690q^{29} + 158800q^{30} + 314160q^{31} + 100896q^{32} - 449216q^{33} - 619604q^{34} - 419200q^{35} - 184420q^{36} - 12894q^{37} + 702120q^{38} + 1244984q^{39} + 737000q^{40} - 333990q^{41} - 1649808q^{42} - 75908q^{43} + 1281936q^{44} - 508175q^{45} - 2114440q^{46} - 502344q^{47} + 1964032q^{48} + 2007993q^{49} + 963750q^{50} + 741160q^{51} - 1415784q^{52} - 3672078q^{53} - 2605760q^{54} - 1570700q^{55} + 835200q^{56} + 2021840q^{57} + 3386980q^{58} + 420420q^{59} + 2720800q^{60} + 4095110q^{61} + 4839072q^{62} - 1831968q^{63} - 9543232q^{64} - 5423350q^{65} - 1696480q^{66} + 3719156q^{67} - 5744952q^{68} - 3256176q^{69} + 2616600q^{70} + 4761960q^{71} + 1178760q^{72} + 301582q^{73} + 5924436q^{74} + 2172500q^{75} - 711600q^{76} + 2894832q^{77} + 7266064q^{78} + 2212560q^{79} - 7307600q^{80} - 9376595q^{81} - 12540308q^{82} - 3652188q^{83} + 5293536q^{84} + 10330550q^{85} + 2516160q^{86} - 20541640q^{87} + 10712160q^{88} + 19324530q^{89} + 2522150q^{90} - 20224640q^{91} - 6316944q^{92} + 14003664q^{93} + 11291656q^{94} - 7058500q^{95} + 2476160q^{96} - 272394q^{97} - 38814042q^{98} - 12697996q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(5))\)

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
5.8.a \(\chi_{5}(1, \cdot)\) 5.8.a.a 1 1
5.8.a.b 2
5.8.b \(\chi_{5}(4, \cdot)\) 5.8.b.a 2 1

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 14 T + 128 T^{2} \))(\( 1 - 20 T + 280 T^{2} - 2560 T^{3} + 16384 T^{4} \))(\( 1 - 140 T^{2} + 16384 T^{4} \))
$3$ (\( 1 + 48 T + 2187 T^{2} \))(\( 1 - 20 T - 390 T^{2} - 43740 T^{3} + 4782969 T^{4} \))(\( 1 - 3330 T^{2} + 4782969 T^{4} \))
$5$ (\( 1 - 125 T \))(\( ( 1 + 125 T )^{2} \))(\( 1 - 150 T + 78125 T^{2} \))
$7$ (\( 1 + 1644 T + 823543 T^{2} \))(\( 1 + 100 T + 1411250 T^{2} + 82354300 T^{3} + 678223072849 T^{4} \))(\( 1 - 1470650 T^{2} + 678223072849 T^{4} \))
$11$ (\( 1 - 172 T + 19487171 T^{2} \))(\( 1 - 4544 T + 31976326 T^{2} - 88549705024 T^{3} + 379749833583241 T^{4} \))(\( ( 1 + 6828 T + 19487171 T^{2} )^{2} \))
$13$ (\( 1 - 3862 T + 62748517 T^{2} \))(\( 1 - 3540 T + 100535470 T^{2} - 222129750180 T^{3} + 3937376385699289 T^{4} \))(\( 1 - 22562810 T^{2} + 3937376385699289 T^{4} \))
$17$ (\( 1 + 12254 T + 410338673 T^{2} \))(\( 1 + 27340 T + 901005190 T^{2} + 11218659319820 T^{3} + 168377826559400929 T^{4} \))(\( 1 - 574764770 T^{2} + 168377826559400929 T^{4} \))
$19$ (\( 1 + 25940 T + 893871739 T^{2} \))(\( 1 - 38760 T + 2155545478 T^{2} - 34646468603640 T^{3} + 799006685782884121 T^{4} \))(\( ( 1 - 6860 T + 893871739 T^{2} )^{2} \))
$23$ (\( 1 - 12972 T + 3404825447 T^{2} \))(\( 1 + 124140 T + 10649684530 T^{2} + 422675030990580 T^{3} + 11592836324538749809 T^{4} \))(\( 1 - 5955848090 T^{2} + 11592836324538749809 T^{4} \))
$29$ (\( 1 + 81610 T + 17249876309 T^{2} \))(\( 1 + 72260 T + 6846819118 T^{2} + 1246476062088340 T^{3} + \)\(29\!\cdots\!81\)\( T^{4} \))(\( ( 1 - 25590 T + 17249876309 T^{2} )^{2} \))
$31$ (\( 1 + 156888 T + 27512614111 T^{2} \))(\( 1 - 306824 T + 77964629966 T^{2} - 8441530311993464 T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \))(\( ( 1 - 82112 T + 27512614111 T^{2} )^{2} \))
$37$ (\( 1 - 110126 T + 94931877133 T^{2} \))(\( 1 + 123020 T + 144088599870 T^{2} + 11678519524901660 T^{3} + \)\(90\!\cdots\!89\)\( T^{4} \))(\( 1 - 139899246410 T^{2} + \)\(90\!\cdots\!89\)\( T^{4} \))
$41$ (\( 1 - 467882 T + 194754273881 T^{2} \))(\( 1 - 264364 T + 161786388886 T^{2} - 51486018860276684 T^{3} + \)\(37\!\cdots\!61\)\( T^{4} \))(\( ( 1 + 533118 T + 194754273881 T^{2} )^{2} \))
$43$ (\( 1 + 499208 T + 271818611107 T^{2} \))(\( 1 - 423300 T + 446651231050 T^{2} - 115060818081593100 T^{3} + \)\(73\!\cdots\!49\)\( T^{4} \))(\( 1 - 41047812050 T^{2} + \)\(73\!\cdots\!49\)\( T^{4} \))
$47$ (\( 1 + 396884 T + 506623120463 T^{2} \))(\( 1 + 105460 T + 858715356610 T^{2} + 53428474284027980 T^{3} + \)\(25\!\cdots\!69\)\( T^{4} \))(\( 1 - 1013212289930 T^{2} + \)\(25\!\cdots\!69\)\( T^{4} \))
$53$ (\( 1 + 1280498 T + 1174711139837 T^{2} \))(\( 1 + 2391580 T + 3562552504510 T^{2} + 2809415667811372460 T^{3} + \)\(13\!\cdots\!69\)\( T^{4} \))(\( 1 - 2002060594730 T^{2} + \)\(13\!\cdots\!69\)\( T^{4} \))
$59$ (\( 1 + 1337420 T + 2488651484819 T^{2} \))(\( 1 + 1120120 T + 1362334883638 T^{2} + 2787588301175458280 T^{3} + \)\(61\!\cdots\!61\)\( T^{4} \))(\( ( 1 - 1438980 T + 2488651484819 T^{2} )^{2} \))
$61$ (\( 1 + 923978 T + 3142742836021 T^{2} \))(\( 1 - 2257044 T + 5613447576526 T^{2} - 7093308861584181924 T^{3} + \)\(98\!\cdots\!41\)\( T^{4} \))(\( ( 1 - 1381022 T + 3142742836021 T^{2} )^{2} \))
$67$ (\( 1 + 797304 T + 6060711605323 T^{2} \))(\( 1 - 4516460 T + 16742087664890 T^{2} - 27372961536977116580 T^{3} + \)\(36\!\cdots\!29\)\( T^{4} \))(\( 1 - 4750924642370 T^{2} + \)\(36\!\cdots\!29\)\( T^{4} \))
$71$ (\( 1 - 5103392 T + 9095120158391 T^{2} \))(\( 1 - 621784 T + 17914494152446 T^{2} - 5655200192564989544 T^{3} + \)\(82\!\cdots\!81\)\( T^{4} \))(\( ( 1 + 481608 T + 9095120158391 T^{2} )^{2} \))
$73$ (\( 1 + 4267478 T + 11047398519097 T^{2} \))(\( 1 - 4569060 T + 23424949855030 T^{2} - 50476226677665338820 T^{3} + \)\(12\!\cdots\!09\)\( T^{4} \))(\( 1 - 19886077213490 T^{2} + \)\(12\!\cdots\!09\)\( T^{4} \))
$79$ (\( 1 + 960 T + 19203908986159 T^{2} \))(\( 1 - 4333040 T + 26135588252318 T^{2} - 83211305793386393360 T^{3} + \)\(36\!\cdots\!81\)\( T^{4} \))(\( ( 1 + 1059760 T + 19203908986159 T^{2} )^{2} \))
$83$ (\( 1 - 6140832 T + 27136050989627 T^{2} \))(\( 1 + 9793020 T + 59971104320890 T^{2} + \)\(26\!\cdots\!40\)\( T^{3} + \)\(73\!\cdots\!29\)\( T^{4} \))(\( 1 - 47492314121570 T^{2} + \)\(73\!\cdots\!29\)\( T^{4} \))
$89$ (\( 1 - 2010570 T + 44231334895529 T^{2} \))(\( 1 - 6025620 T + 89865866149558 T^{2} - \)\(26\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \))(\( ( 1 - 5644170 T + 44231334895529 T^{2} )^{2} \))
$97$ (\( 1 + 4881934 T + 80798284478113 T^{2} \))(\( 1 - 4609540 T + 142930351581510 T^{2} - \)\(37\!\cdots\!20\)\( T^{3} + \)\(65\!\cdots\!69\)\( T^{4} \))(\( 1 - 17378330046530 T^{2} + \)\(65\!\cdots\!69\)\( T^{4} \))
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