Properties

Label 5.10
Level 5
Weight 10
Dimension 7
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 20
Trace bound 1

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Defining parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 10 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(20\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_1(5))\).

Total New Old
Modular forms 11 9 2
Cusp forms 7 7 0
Eisenstein series 4 2 2

Trace form

\( 7q - 18q^{2} + 146q^{3} - 772q^{4} + 1765q^{5} - 1616q^{6} + 5942q^{7} - 12600q^{8} + 7447q^{9} + O(q^{10}) \) \( 7q - 18q^{2} + 146q^{3} - 772q^{4} + 1765q^{5} - 1616q^{6} + 5942q^{7} - 12600q^{8} + 7447q^{9} - 70410q^{10} + 87744q^{11} + 227152q^{12} - 914q^{13} - 898824q^{14} - 162970q^{15} + 1487152q^{16} + 907662q^{17} - 1008394q^{18} - 1942140q^{19} - 2369780q^{20} + 4125084q^{21} + 4083944q^{22} - 1581774q^{23} - 6189120q^{24} - 166025q^{25} + 4242804q^{26} + 313100q^{27} + 3305504q^{28} - 3002310q^{29} - 192320q^{30} - 12339596q^{31} - 4067808q^{32} + 717232q^{33} + 46355196q^{34} + 12041090q^{35} - 41602468q^{36} - 36053298q^{37} + 3821400q^{38} + 35312564q^{39} + 26103400q^{40} - 32689566q^{41} - 61126176q^{42} + 46492906q^{43} + 15517776q^{44} + 61482805q^{45} - 12598376q^{46} + 22853022q^{47} - 38239424q^{48} - 56125957q^{49} - 157123650q^{50} + 29384884q^{51} + 139262232q^{52} + 703446q^{53} + 43238560q^{54} + 16962880q^{55} + 304607520q^{56} + 87911000q^{57} - 231081500q^{58} - 601010220q^{59} - 228876560q^{60} + 131659494q^{61} + 331039104q^{62} + 198326886q^{63} - 78224192q^{64} + 328241930q^{65} + 449606528q^{66} - 45604738q^{67} - 265631256q^{68} - 440502636q^{69} - 762351960q^{70} - 244710276q^{71} - 139728600q^{72} + 533029126q^{73} + 1816209636q^{74} + 716270450q^{75} - 814046160q^{76} - 996146736q^{77} - 794572208q^{78} - 626125160q^{79} + 665853040q^{80} - 829831493q^{81} - 268068116q^{82} + 1664055066q^{83} + 2497474656q^{84} + 1224156090q^{85} - 2537118336q^{86} - 523405300q^{87} - 368023200q^{88} - 772550730q^{89} - 1894205170q^{90} + 663161404q^{91} - 700560288q^{92} - 31047288q^{93} + 2263426056q^{94} + 1649929100q^{95} + 1804123904q^{96} + 618891222q^{97} - 621046626q^{98} - 2383526176q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_1(5))\)

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
5.10.a \(\chi_{5}(1, \cdot)\) 5.10.a.a 1 1
5.10.a.b 2
5.10.b \(\chi_{5}(4, \cdot)\) 5.10.b.a 4 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 8 T + 512 T^{2} \))(\( 1 + 10 T + 40 T^{2} + 5120 T^{3} + 262144 T^{4} \))(\( 1 - 340 T^{2} - 174912 T^{4} - 89128960 T^{6} + 68719476736 T^{8} \))
$3$ (\( 1 + 114 T + 19683 T^{2} \))(\( 1 - 260 T + 52230 T^{2} - 5117580 T^{3} + 387420489 T^{4} \))(\( 1 - 45180 T^{2} + 1049244678 T^{4} - 17503657693020 T^{6} + 150094635296999121 T^{8} \))
$5$ (\( 1 + 625 T \))(\( ( 1 - 625 T )^{2} \))(\( 1 - 1140 T + 1318750 T^{2} - 2226562500 T^{3} + 3814697265625 T^{4} \))
$7$ (\( 1 - 4242 T + 40353607 T^{2} \))(\( 1 - 1700 T + 35221550 T^{2} - 68601131900 T^{3} + 1628413597910449 T^{4} \))(\( 1 - 57246700 T^{2} + 3508143545353398 T^{4} - \)\(93\!\cdots\!00\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \))
$11$ (\( 1 + 46208 T + 2357947691 T^{2} \))(\( 1 - 23984 T + 1217213446 T^{2} - 56553017420944 T^{3} + 5559917313492231481 T^{4} \))(\( ( 1 - 54984 T + 5180465446 T^{2} - 129649395841944 T^{3} + 5559917313492231481 T^{4} )^{2} \))
$13$ (\( 1 + 115934 T + 10604499373 T^{2} \))(\( 1 - 115020 T + 22672043710 T^{2} - 1219729517882460 T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \))(\( 1 - 35613791860 T^{2} + \)\(53\!\cdots\!58\)\( T^{4} - \)\(40\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!41\)\( T^{8} \))
$17$ (\( 1 - 494842 T + 118587876497 T^{2} \))(\( 1 - 412820 T + 113016614470 T^{2} - 48955447175491540 T^{3} + \)\(14\!\cdots\!09\)\( T^{4} \))(\( 1 - 285780369220 T^{2} + \)\(48\!\cdots\!18\)\( T^{4} - \)\(40\!\cdots\!80\)\( T^{6} + \)\(19\!\cdots\!81\)\( T^{8} \))
$19$ (\( 1 + 1008740 T + 322687697779 T^{2} \))(\( 1 + 296520 T + 659218232758 T^{2} + 95683356145429080 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} \))(\( ( 1 + 318440 T + 505756418358 T^{2} + 102756670480744760 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} )^{2} \))
$23$ (\( 1 + 532554 T + 1801152661463 T^{2} \))(\( 1 + 1049220 T + 3497852029390 T^{2} + 1889805395460208860 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \))(\( 1 - 5779790962540 T^{2} + \)\(14\!\cdots\!38\)\( T^{4} - \)\(18\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))
$29$ (\( 1 - 4196390 T + 14507145975869 T^{2} \))(\( 1 + 3666980 T + 20832571957438 T^{2} + 53197414150592105620 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))(\( ( 1 + 1765860 T + 26950935551038 T^{2} + 25617588792948032340 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \))
$31$ (\( 1 + 3365028 T + 26439622160671 T^{2} \))(\( 1 - 1613144 T + 29382323902526 T^{2} - 42650917850753459624 T^{3} + \)\(69\!\cdots\!41\)\( T^{4} \))(\( ( 1 + 5293856 T + 59464921598526 T^{2} + \)\(13\!\cdots\!76\)\( T^{3} + \)\(69\!\cdots\!41\)\( T^{4} )^{2} \))
$37$ (\( 1 + 14931358 T + 129961739795077 T^{2} \))(\( 1 + 21121940 T + 328931801286510 T^{2} + \)\(27\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!29\)\( T^{4} \))(\( 1 - 231603274936660 T^{2} + \)\(44\!\cdots\!58\)\( T^{4} - \)\(39\!\cdots\!40\)\( T^{6} + \)\(28\!\cdots\!41\)\( T^{8} \))
$41$ (\( 1 - 11056262 T + 327381934393961 T^{2} \))(\( 1 + 26957276 T + 811945448362966 T^{2} + \)\(88\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} \))(\( ( 1 + 8394276 T + 221313076168966 T^{2} + \)\(27\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} )^{2} \))
$43$ (\( 1 + 6396794 T + 502592611936843 T^{2} \))(\( 1 - 52889700 T + 1703843788760950 T^{2} - \)\(26\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!49\)\( T^{4} \))(\( 1 - 614109141147100 T^{2} + \)\(57\!\cdots\!98\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{6} + \)\(63\!\cdots\!01\)\( T^{8} \))
$47$ (\( 1 + 35559158 T + 1119130473102767 T^{2} \))(\( 1 - 58412180 T + 2814913257457630 T^{2} - \)\(65\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!89\)\( T^{4} \))(\( 1 - 1368976020813580 T^{2} + \)\(29\!\cdots\!78\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!21\)\( T^{8} \))
$53$ (\( 1 - 39738586 T + 3299763591802133 T^{2} \))(\( 1 + 39035140 T + 5675030678030830 T^{2} + \)\(12\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!89\)\( T^{4} \))(\( 1 - 7684297973864980 T^{2} + \)\(36\!\cdots\!78\)\( T^{4} - \)\(83\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!21\)\( T^{8} \))
$59$ (\( 1 + 85185620 T + 8662995818654939 T^{2} \))(\( 1 + 54995560 T + 15674484224932678 T^{2} + \)\(47\!\cdots\!40\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \))(\( ( 1 + 230414520 T + 28555631923987078 T^{2} + \)\(19\!\cdots\!80\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \))
$61$ (\( 1 - 45748642 T + 11694146092834141 T^{2} \))(\( 1 + 274579716 T + 41753623519328446 T^{2} + \)\(32\!\cdots\!56\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} \))(\( ( 1 - 180245284 T + 30154717014478446 T^{2} - \)\(21\!\cdots\!44\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} )^{2} \))
$67$ (\( 1 + 45286158 T + 27206534396294947 T^{2} \))(\( 1 + 318580 T + 48520062064444070 T^{2} + \)\(86\!\cdots\!60\)\( T^{3} + \)\(74\!\cdots\!09\)\( T^{4} \))(\( 1 + 41160407446058180 T^{2} + \)\(18\!\cdots\!18\)\( T^{4} + \)\(30\!\cdots\!20\)\( T^{6} + \)\(54\!\cdots\!81\)\( T^{8} \))
$71$ (\( 1 + 189967468 T + 45848500718449031 T^{2} \))(\( 1 + 7130936 T + 51935375688707086 T^{2} + \)\(32\!\cdots\!16\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))(\( ( 1 + 23805936 T + 85782754020107086 T^{2} + \)\(10\!\cdots\!16\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \))
$73$ (\( 1 - 412170946 T + 58871586708267913 T^{2} \))(\( 1 - 120858180 T + 42707263689423190 T^{2} - \)\(71\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!69\)\( T^{4} \))(\( 1 - 229489314868712740 T^{2} + \)\(20\!\cdots\!38\)\( T^{4} - \)\(79\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!61\)\( T^{8} \))
$79$ (\( 1 - 95040840 T + 119851595982618319 T^{2} \))(\( 1 - 6877520 T - 115982362290712162 T^{2} - \)\(82\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} \))(\( ( 1 + 364021760 T + 220545463862625438 T^{2} + \)\(43\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} )^{2} \))
$83$ (\( 1 - 261706326 T + 186940255267540403 T^{2} \))(\( 1 - 1402348740 T + 857904310704391270 T^{2} - \)\(26\!\cdots\!20\)\( T^{3} + \)\(34\!\cdots\!09\)\( T^{4} \))(\( 1 - 434569632367965820 T^{2} + \)\(10\!\cdots\!18\)\( T^{4} - \)\(15\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!81\)\( T^{8} \))
$89$ (\( 1 + 19938630 T + 350356403707485209 T^{2} \))(\( 1 - 830088660 T + 692293619421117718 T^{2} - \)\(29\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} \))(\( ( 1 + 791350380 T + 832192702699668118 T^{2} + \)\(27\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} )^{2} \))
$97$ (\( 1 + 19503358 T + 760231058654565217 T^{2} \))(\( 1 - 638394580 T + 1615411126351062630 T^{2} - \)\(48\!\cdots\!60\)\( T^{3} + \)\(57\!\cdots\!89\)\( T^{4} \))(\( 1 - 2561123777205326980 T^{2} + \)\(27\!\cdots\!78\)\( T^{4} - \)\(14\!\cdots\!20\)\( T^{6} + \)\(33\!\cdots\!21\)\( T^{8} \))
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