Properties

Label 10.10
Level 10
Weight 10
Dimension 7
Nonzero newspaces 2
Newform subspaces 4
Sturm bound 60
Trace bound 1

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Defining parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 10 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 4 \)
Sturm bound: \(60\)
Trace bound: \(1\)

Dimensions

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

Total New Old
Modular forms 31 7 24
Cusp forms 23 7 16
Eisenstein series 8 0 8

Trace form

\( 7q - 16q^{2} + 16q^{3} - 256q^{4} - 3205q^{5} + 448q^{6} - 228q^{7} - 4096q^{8} - 57013q^{9} + O(q^{10}) \) \( 7q - 16q^{2} + 16q^{3} - 256q^{4} - 3205q^{5} + 448q^{6} - 228q^{7} - 4096q^{8} - 57013q^{9} + 45040q^{10} - 6276q^{11} + 4096q^{12} - 232974q^{13} + 337792q^{14} + 379240q^{15} + 458752q^{16} - 934458q^{17} + 99632q^{18} + 1689660q^{19} + 500480q^{20} - 5379896q^{21} - 629952q^{22} + 2695956q^{23} + 2605056q^{24} + 770775q^{25} - 7416032q^{26} - 3754160q^{27} - 58368q^{28} + 10515330q^{29} - 4269120q^{30} + 15017464q^{31} - 1048576q^{32} - 10293648q^{33} - 7060768q^{34} + 13444420q^{35} + 22254336q^{36} + 28529802q^{37} - 31419200q^{38} - 59400976q^{39} - 16650240q^{40} + 20113374q^{41} + 38291968q^{42} - 39518424q^{43} + 51452928q^{44} - 13091985q^{45} + 6640768q^{46} - 27549708q^{47} + 1048576q^{48} - 71368737q^{49} - 28253200q^{50} - 122250496q^{51} - 59641344q^{52} + 181541706q^{53} + 311556480q^{54} + 237238940q^{55} - 8290304q^{56} - 239106400q^{57} - 180687840q^{58} + 287111460q^{59} - 232765440q^{60} - 395018086q^{61} + 142425088q^{62} + 349738556q^{63} - 16777216q^{64} + 403570090q^{65} - 103172864q^{66} - 56692488q^{67} - 239221248q^{68} - 1240048136q^{69} - 674616960q^{70} + 829596144q^{71} + 25505792q^{72} + 283392846q^{73} + 968831392q^{74} + 285810800q^{75} + 269982720q^{76} + 594093984q^{77} - 865437056q^{78} - 2219911200q^{79} - 210042880q^{80} + 721923927q^{81} + 733577568q^{82} + 560234736q^{83} + 981854208q^{84} + 731884670q^{85} + 1316223808q^{86} - 845703360q^{87} - 161267712q^{88} - 1226641290q^{89} - 2422234320q^{90} - 2017529736q^{91} + 690164736q^{92} + 2509309712q^{93} + 2064301312q^{94} + 528928300q^{95} + 29360128q^{96} - 961141338q^{97} - 1182674832q^{98} - 1384367956q^{99} + O(q^{100}) \)

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

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
10.10.a \(\chi_{10}(1, \cdot)\) 10.10.a.a 1 1
10.10.a.b 1
10.10.a.c 1
10.10.b \(\chi_{10}(9, \cdot)\) 10.10.b.a 4 1

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 16 T \))(\( 1 + 16 T \))(\( 1 - 16 T \))(\( ( 1 + 256 T^{2} )^{2} \))
$3$ (\( 1 + 204 T + 19683 T^{2} \))(\( 1 - 46 T + 19683 T^{2} \))(\( 1 - 174 T + 19683 T^{2} \))(\( 1 - 3380 T^{2} + 40679478 T^{4} - 1309481252820 T^{6} + 150094635296999121 T^{8} \))
$5$ (\( 1 - 625 T \))(\( 1 + 625 T \))(\( 1 + 625 T \))(\( 1 + 2580 T + 3528750 T^{2} + 5039062500 T^{3} + 3814697265625 T^{4} \))
$7$ (\( 1 - 5432 T + 40353607 T^{2} \))(\( 1 + 10318 T + 40353607 T^{2} \))(\( 1 - 4658 T + 40353607 T^{2} \))(\( 1 - 26720900 T^{2} + 1462069499709798 T^{4} - \)\(43\!\cdots\!00\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \))
$11$ (\( 1 - 73932 T + 2357947691 T^{2} \))(\( 1 + 5568 T + 2357947691 T^{2} \))(\( 1 - 28992 T + 2357947691 T^{2} \))(\( ( 1 + 51816 T + 3027030246 T^{2} + 122179417556856 T^{3} + 5559917313492231481 T^{4} )^{2} \))
$13$ (\( 1 + 114514 T + 10604499373 T^{2} \))(\( 1 - 45986 T + 10604499373 T^{2} \))(\( 1 + 164446 T + 10604499373 T^{2} \))(\( 1 - 22383928660 T^{2} + \)\(29\!\cdots\!58\)\( T^{4} - \)\(25\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!41\)\( T^{8} \))
$17$ (\( 1 - 41682 T + 118587876497 T^{2} \))(\( 1 + 381318 T + 118587876497 T^{2} \))(\( 1 + 594822 T + 118587876497 T^{2} \))(\( 1 - 468221105220 T^{2} + \)\(82\!\cdots\!18\)\( T^{4} - \)\(65\!\cdots\!80\)\( T^{6} + \)\(19\!\cdots\!81\)\( T^{8} \))
$19$ (\( 1 - 1057460 T + 322687697779 T^{2} \))(\( 1 - 610460 T + 322687697779 T^{2} \))(\( 1 + 295780 T + 322687697779 T^{2} \))(\( ( 1 - 158760 T + 642783370358 T^{2} - 51229898899394040 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} )^{2} \))
$23$ (\( 1 - 1599336 T + 1801152661463 T^{2} \))(\( 1 + 1447914 T + 1801152661463 T^{2} \))(\( 1 - 2544534 T + 1801152661463 T^{2} \))(\( 1 - 1626654345540 T^{2} + \)\(46\!\cdots\!38\)\( T^{4} - \)\(52\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))
$29$ (\( 1 - 2184510 T + 14507145975869 T^{2} \))(\( 1 - 5385510 T + 14507145975869 T^{2} \))(\( 1 + 3722970 T + 14507145975869 T^{2} \))(\( ( 1 - 3334140 T + 25948591194238 T^{2} - 48368855683983867660 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \))
$31$ (\( 1 + 9619648 T + 26439622160671 T^{2} \))(\( 1 - 3053852 T + 26439622160671 T^{2} \))(\( 1 - 2335772 T + 26439622160671 T^{2} \))(\( ( 1 - 9623744 T + 71729043019326 T^{2} - \)\(25\!\cdots\!24\)\( T^{3} + \)\(69\!\cdots\!41\)\( T^{4} )^{2} \))
$37$ (\( 1 - 4799942 T + 129961739795077 T^{2} \))(\( 1 - 12889442 T + 129961739795077 T^{2} \))(\( 1 - 10840418 T + 129961739795077 T^{2} \))(\( 1 + 73233430078540 T^{2} + \)\(20\!\cdots\!58\)\( T^{4} + \)\(12\!\cdots\!60\)\( T^{6} + \)\(28\!\cdots\!41\)\( T^{8} \))
$41$ (\( 1 - 9531882 T + 327381934393961 T^{2} \))(\( 1 + 33786618 T + 327381934393961 T^{2} \))(\( 1 - 21593862 T + 327381934393961 T^{2} \))(\( ( 1 - 11387124 T + 602060396517366 T^{2} - \)\(37\!\cdots\!64\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} )^{2} \))
$43$ (\( 1 + 13464484 T + 502592611936843 T^{2} \))(\( 1 + 36886234 T + 502592611936843 T^{2} \))(\( 1 - 10832294 T + 502592611936843 T^{2} \))(\( 1 - 1938214042750100 T^{2} + \)\(14\!\cdots\!98\)\( T^{4} - \)\(48\!\cdots\!00\)\( T^{6} + \)\(63\!\cdots\!01\)\( T^{8} \))
$47$ (\( 1 - 11441952 T + 1119130473102767 T^{2} \))(\( 1 + 44163798 T + 1119130473102767 T^{2} \))(\( 1 - 5172138 T + 1119130473102767 T^{2} \))(\( 1 - 3434853059674980 T^{2} + \)\(54\!\cdots\!78\)\( T^{4} - \)\(43\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!21\)\( T^{8} \))
$53$ (\( 1 - 53615766 T + 3299763591802133 T^{2} \))(\( 1 - 29746266 T + 3299763591802133 T^{2} \))(\( 1 - 98179674 T + 3299763591802133 T^{2} \))(\( 1 - 3945786642536180 T^{2} + \)\(65\!\cdots\!78\)\( T^{4} - \)\(42\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!21\)\( T^{8} \))
$59$ (\( 1 - 81862620 T + 8662995818654939 T^{2} \))(\( 1 + 65575380 T + 8662995818654939 T^{2} \))(\( 1 - 16162860 T + 8662995818654939 T^{2} \))(\( ( 1 - 127330680 T + 20590638486439878 T^{2} - \)\(11\!\cdots\!20\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \))
$61$ (\( 1 + 104691298 T + 11694146092834141 T^{2} \))(\( 1 - 40183202 T + 11694146092834141 T^{2} \))(\( 1 + 43928158 T + 11694146092834141 T^{2} \))(\( ( 1 + 143290916 T + 28088617153288446 T^{2} + \)\(16\!\cdots\!56\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} )^{2} \))
$67$ (\( 1 - 140571092 T + 27206534396294947 T^{2} \))(\( 1 + 115706158 T + 27206534396294947 T^{2} \))(\( 1 + 81557422 T + 27206534396294947 T^{2} \))(\( 1 - 105856746688500020 T^{2} + \)\(42\!\cdots\!18\)\( T^{4} - \)\(78\!\cdots\!80\)\( T^{6} + \)\(54\!\cdots\!81\)\( T^{8} \))
$71$ (\( 1 - 97098792 T + 45848500718449031 T^{2} \))(\( 1 + 231681708 T + 45848500718449031 T^{2} \))(\( 1 - 161307732 T + 45848500718449031 T^{2} \))(\( ( 1 - 401435664 T + 127868030128292686 T^{2} - \)\(18\!\cdots\!84\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \))
$73$ (\( 1 - 171848906 T + 58871586708267913 T^{2} \))(\( 1 - 358691906 T + 58871586708267913 T^{2} \))(\( 1 + 247147966 T + 58871586708267913 T^{2} \))(\( 1 - 30314907095525540 T^{2} + \)\(69\!\cdots\!38\)\( T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!61\)\( T^{8} \))
$79$ (\( 1 + 117380080 T + 119851595982618319 T^{2} \))(\( 1 + 486017080 T + 119851595982618319 T^{2} \))(\( 1 + 583345720 T + 119851595982618319 T^{2} \))(\( ( 1 + 516584160 T + 303933580876193438 T^{2} + \)\(61\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} )^{2} \))
$83$ (\( 1 - 323637636 T + 186940255267540403 T^{2} \))(\( 1 - 251168886 T + 186940255267540403 T^{2} \))(\( 1 + 14571786 T + 186940255267540403 T^{2} \))(\( 1 - 597414961848210420 T^{2} + \)\(15\!\cdots\!18\)\( T^{4} - \)\(20\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!81\)\( T^{8} \))
$89$ (\( 1 + 894379110 T + 350356403707485209 T^{2} \))(\( 1 + 526039110 T + 350356403707485209 T^{2} \))(\( 1 - 470133690 T + 350356403707485209 T^{2} \))(\( ( 1 + 138178380 T + 45471108987586518 T^{2} + \)\(48\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} )^{2} \))
$97$ (\( 1 - 232678562 T + 760231058654565217 T^{2} \))(\( 1 + 1075981438 T + 760231058654565217 T^{2} \))(\( 1 + 117838462 T + 760231058654565217 T^{2} \))(\( 1 - 228500609440802180 T^{2} - \)\(20\!\cdots\!22\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(33\!\cdots\!21\)\( T^{8} \))
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