Properties

Label 20.10
Level 20
Weight 10
Dimension 57
Nonzero newspaces 3
Newform subspaces 5
Sturm bound 240
Trace bound 1

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Defining parameters

Level: \( N \) = \( 20\( 20 = 2^{2} \cdot 5 \) \)
Weight: \( k \) = \( 10 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 5 \)
Sturm bound: \(240\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_1(20))\).

Total New Old
Modular forms 118 65 53
Cusp forms 98 57 41
Eisenstein series 20 8 12

Trace form

\( 57q - 2q^{2} - 308q^{3} + 31q^{5} + 6152q^{6} - 912q^{7} - 716q^{8} + 8459q^{9} + O(q^{10}) \) \( 57q - 2q^{2} - 308q^{3} + 31q^{5} + 6152q^{6} - 912q^{7} - 716q^{8} + 8459q^{9} + 11446q^{10} + 34740q^{11} - 155360q^{12} - 6704q^{13} + 32740q^{15} + 3000q^{16} - 637428q^{17} - 679086q^{18} - 447636q^{19} - 334324q^{20} + 1944376q^{21} + 3176920q^{22} - 808416q^{23} + 5274381q^{25} - 11222732q^{26} - 8154584q^{27} + 12731840q^{28} + 899874q^{29} + 15498680q^{30} + 7989712q^{31} - 29092792q^{32} - 13035520q^{33} - 8209160q^{35} + 44773372q^{36} + 6991724q^{37} - 26115120q^{38} + 27695864q^{39} - 55995604q^{40} + 552302q^{41} + 82830200q^{42} - 32882700q^{43} - 54538499q^{45} - 92534488q^{46} + 78552936q^{47} + 46448320q^{48} + 216766143q^{49} - 54429014q^{50} - 114424456q^{51} + 88926156q^{52} - 274039236q^{53} - 282892300q^{55} - 177356448q^{56} + 270379472q^{57} + 82723048q^{58} + 466689828q^{59} + 166923520q^{60} + 366448194q^{61} - 292810200q^{62} - 723019072q^{63} - 526945428q^{65} + 614341200q^{66} - 26145732q^{67} - 289868412q^{68} + 1021502632q^{69} - 203227600q^{70} + 224888616q^{71} + 930217668q^{72} - 44807784q^{73} - 1158654100q^{75} - 1061841600q^{76} - 195453840q^{77} + 49362600q^{78} + 2245791312q^{79} + 210150736q^{80} - 552782283q^{81} - 591442904q^{82} - 1909553076q^{83} - 538201268q^{85} - 874588728q^{86} + 585010632q^{87} + 865939360q^{88} + 1748967126q^{89} + 1779829206q^{90} + 781823568q^{91} - 1106673600q^{92} - 4005699248q^{93} - 2253979460q^{95} + 1870685312q^{96} + 5101092544q^{97} - 3885590026q^{98} + 4822342340q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_1(20))\)

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
20.10.a \(\chi_{20}(1, \cdot)\) 20.10.a.a 1 1
20.10.a.b 2
20.10.c \(\chi_{20}(9, \cdot)\) 20.10.c.a 4 1
20.10.e \(\chi_{20}(3, \cdot)\) 20.10.e.a 2 2
20.10.e.b 48

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_1(20))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_1(20)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 32 T + 512 T^{2} \))
$3$ (\( 1 + 48 T + 19683 T^{2} \))(\( 1 + 260 T + 36042 T^{2} + 5117580 T^{3} + 387420489 T^{4} \))(\( 1 - 34844 T^{2} + 897243462 T^{4} - 13499279518716 T^{6} + 150094635296999121 T^{8} \))(\( 1 + 387420489 T^{4} \))
$5$ (\( 1 - 625 T \))(\( ( 1 + 625 T )^{2} \))(\( 1 - 660 T + 318750 T^{2} - 1289062500 T^{3} + 3814697265625 T^{4} \))(\( 1 - 1436 T + 1953125 T^{2} \))
$7$ (\( 1 + 532 T + 40353607 T^{2} \))(\( 1 + 380 T - 15543150 T^{2} + 15334370660 T^{3} + 1628413597910449 T^{4} \))(\( 1 - 153156620 T^{2} + 9120528185156598 T^{4} - \)\(24\!\cdots\!80\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \))(\( 1 + 1628413597910449 T^{4} \))
$11$ (\( 1 + 33180 T + 2357947691 T^{2} \))(\( 1 - 102720 T + 7335543382 T^{2} - 242208386819520 T^{3} + 5559917313492231481 T^{4} \))(\( ( 1 + 17400 T + 2292818982 T^{2} + 41028289823400 T^{3} + 5559917313492231481 T^{4} )^{2} \))(\( ( 1 - 2357947691 T^{2} )^{2} \))
$13$ (\( 1 + 99682 T + 10604499373 T^{2} \))(\( 1 - 179140 T + 22798610142 T^{2} - 1899690017679220 T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \))(\( 1 - 14000436724 T^{2} + 83025702170976147702 T^{4} - \)\(15\!\cdots\!96\)\( T^{6} + \)\(12\!\cdots\!41\)\( T^{8} \))(\( ( 1 + 112806 T + 10604499373 T^{2} )( 1 + 172316 T + 10604499373 T^{2} ) \))
$17$ (\( 1 + 443454 T + 118587876497 T^{2} \))(\( 1 - 316020 T + 259693705798 T^{2} - 37476140730581940 T^{3} + \)\(14\!\cdots\!09\)\( T^{4} \))(\( 1 - 181978394436 T^{2} + \)\(34\!\cdots\!42\)\( T^{4} - \)\(25\!\cdots\!24\)\( T^{6} + \)\(19\!\cdots\!81\)\( T^{8} \))(\( ( 1 + 407992 T + 118587876497 T^{2} )( 1 + 554882 T + 118587876497 T^{2} ) \))
$19$ (\( 1 + 357244 T + 322687697779 T^{2} \))(\( 1 - 137272 T + 111610161654 T^{2} - 44295985649518888 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} \))(\( ( 1 + 113832 T + 402567657014 T^{2} + 36732186013579128 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} )^{2} \))(\( ( 1 + 322687697779 T^{2} )^{2} \))
$23$ (\( 1 + 142956 T + 1801152661463 T^{2} \))(\( 1 + 665460 T + 2886450615250 T^{2} + 1198595050097167980 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \))(\( 1 + 820282526580 T^{2} + \)\(11\!\cdots\!38\)\( T^{4} + \)\(26\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 + \)\(32\!\cdots\!69\)\( T^{4} \))
$29$ (\( 1 - 1527966 T + 14507145975869 T^{2} \))(\( 1 + 6893748 T + 40195999658014 T^{2} + \)\(10\!\cdots\!12\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))(\( ( 1 - 3132828 T + 12854589500734 T^{2} - 45448393113289727532 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 - 7314710 T + 14507145975869 T^{2} )( 1 + 7314710 T + 14507145975869 T^{2} ) \))
$31$ (\( 1 - 7323416 T + 26439622160671 T^{2} \))(\( 1 - 291832 T + 38964935800398 T^{2} - 7715927814392939272 T^{3} + \)\(69\!\cdots\!41\)\( T^{4} \))(\( ( 1 - 187232 T + 40099942836798 T^{2} - 4950343336386752672 T^{3} + \)\(69\!\cdots\!41\)\( T^{4} )^{2} \))(\( ( 1 - 26439622160671 T^{2} )^{2} \))
$37$ (\( 1 + 2666842 T + 129961739795077 T^{2} \))(\( 1 - 11261380 T + 218879982937230 T^{2} - \)\(14\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!29\)\( T^{4} \))(\( 1 - 462603258659540 T^{2} + \)\(87\!\cdots\!58\)\( T^{4} - \)\(78\!\cdots\!60\)\( T^{6} + \)\(28\!\cdots\!41\)\( T^{8} \))(\( ( 1 + 1923372 T + 129961739795077 T^{2} )( 1 + 22718882 T + 129961739795077 T^{2} ) \))
$41$ (\( 1 + 7939014 T + 327381934393961 T^{2} \))(\( 1 - 29773452 T + 771012402449398 T^{2} - \)\(97\!\cdots\!72\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} \))(\( ( 1 + 8824068 T + 671552976228678 T^{2} + \)\(28\!\cdots\!48\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} )^{2} \))(\( ( 1 - 7561912 T + 327381934393961 T^{2} )^{2} \))
$43$ (\( 1 + 21174520 T + 502592611936843 T^{2} \))(\( 1 + 11708180 T + 838769843899386 T^{2} + \)\(58\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!49\)\( T^{4} \))(\( 1 - 358707990293372 T^{2} + \)\(21\!\cdots\!94\)\( T^{4} - \)\(90\!\cdots\!28\)\( T^{6} + \)\(63\!\cdots\!01\)\( T^{8} \))(\( 1 + \)\(25\!\cdots\!49\)\( T^{4} \))
$47$ (\( 1 - 16059636 T + 1119130473102767 T^{2} \))(\( 1 - 62493300 T + 3177958884734338 T^{2} - \)\(69\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!89\)\( T^{4} \))(\( 1 - 1037666802860076 T^{2} + \)\(16\!\cdots\!22\)\( T^{4} - \)\(12\!\cdots\!64\)\( T^{6} + \)\(15\!\cdots\!21\)\( T^{8} \))(\( 1 + \)\(12\!\cdots\!89\)\( T^{4} \))
$53$ (\( 1 + 87822234 T + 3299763591802133 T^{2} \))(\( 1 - 9417780 T + 5708185761526990 T^{2} - \)\(31\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!89\)\( T^{4} \))(\( 1 - 5310844341928340 T^{2} + \)\(28\!\cdots\!78\)\( T^{4} - \)\(57\!\cdots\!60\)\( T^{6} + \)\(11\!\cdots\!21\)\( T^{8} \))(\( ( 1 - 68323684 T + 3299763591802133 T^{2} )( 1 + 92363026 T + 3299763591802133 T^{2} ) \))
$59$ (\( 1 - 120625212 T + 8662995818654939 T^{2} \))(\( 1 + 92930856 T + 16656477955483462 T^{2} + \)\(80\!\cdots\!84\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \))(\( ( 1 - 219497736 T + 28852098295168902 T^{2} - \)\(19\!\cdots\!04\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \))(\( ( 1 + 8662995818654939 T^{2} )^{2} \))
$61$ (\( 1 - 93576542 T + 11694146092834141 T^{2} \))(\( 1 - 195673924 T + 22434263296171326 T^{2} - \)\(22\!\cdots\!84\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} \))(\( ( 1 + 51522236 T - 2665246277981394 T^{2} + \)\(60\!\cdots\!76\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} )^{2} \))(\( ( 1 - 216178092 T + 11694146092834141 T^{2} )^{2} \))
$67$ (\( 1 - 193621688 T + 27206534396294947 T^{2} \))(\( 1 + 219767420 T + 65652945987990090 T^{2} + \)\(59\!\cdots\!40\)\( T^{3} + \)\(74\!\cdots\!09\)\( T^{4} \))(\( 1 - 83417417389239260 T^{2} + \)\(31\!\cdots\!18\)\( T^{4} - \)\(61\!\cdots\!40\)\( T^{6} + \)\(54\!\cdots\!81\)\( T^{8} \))(\( 1 + \)\(74\!\cdots\!09\)\( T^{4} \))
$71$ (\( 1 - 417763488 T + 45848500718449031 T^{2} \))(\( 1 - 311207016 T + 76405636625293726 T^{2} - \)\(14\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))(\( ( 1 + 252040944 T + 101042798798695246 T^{2} + \)\(11\!\cdots\!64\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 - 45848500718449031 T^{2} )^{2} \))
$73$ (\( 1 + 450372742 T + 58871586708267913 T^{2} \))(\( 1 + 99224060 T + 35402447061205782 T^{2} + \)\(58\!\cdots\!80\)\( T^{3} + \)\(34\!\cdots\!69\)\( T^{4} \))(\( 1 - 79546297979572004 T^{2} + \)\(52\!\cdots\!42\)\( T^{4} - \)\(27\!\cdots\!76\)\( T^{6} + \)\(12\!\cdots\!61\)\( T^{8} \))(\( ( 1 - 483419504 T + 58871586708267913 T^{2} )( 1 - 42331194 T + 58871586708267913 T^{2} ) \))
$79$ (\( 1 + 91425472 T + 119851595982618319 T^{2} \))(\( 1 - 542261776 T + 313115996157615582 T^{2} - \)\(64\!\cdots\!44\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} \))(\( ( 1 - 897477504 T + 421888354983619742 T^{2} - \)\(10\!\cdots\!76\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 + 119851595982618319 T^{2} )^{2} \))
$83$ (\( 1 + 652637376 T + 186940255267540403 T^{2} \))(\( 1 + 1256915700 T + 768086791626261130 T^{2} + \)\(23\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!09\)\( T^{4} \))(\( 1 - 186261173343564060 T^{2} + \)\(18\!\cdots\!18\)\( T^{4} - \)\(65\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!81\)\( T^{8} \))(\( 1 + \)\(34\!\cdots\!09\)\( T^{4} \))
$89$ (\( 1 + 170059206 T + 350356403707485209 T^{2} \))(\( 1 + 462291852 T + 159603168035249494 T^{2} + \)\(16\!\cdots\!68\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} \))(\( ( 1 - 1190659092 T + 825013495390166934 T^{2} - \)\(41\!\cdots\!28\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} )^{2} \))(\( ( 1 - 1125568310 T + 350356403707485209 T^{2} )( 1 + 1125568310 T + 350356403707485209 T^{2} ) \))
$97$ (\( 1 + 10947022 T + 760231058654565217 T^{2} \))(\( 1 - 1671716740 T + 2048690578856969670 T^{2} - \)\(12\!\cdots\!80\)\( T^{3} + \)\(57\!\cdots\!89\)\( T^{4} \))(\( 1 - 899946211039106180 T^{2} + \)\(23\!\cdots\!78\)\( T^{4} - \)\(52\!\cdots\!20\)\( T^{6} + \)\(33\!\cdots\!21\)\( T^{8} \))(\( ( 1 + 1016663992 T + 760231058654565217 T^{2} )( 1 + 1416798702 T + 760231058654565217 T^{2} ) \))
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