Properties

Label 5.10.a
Level 5
Weight 10
Character orbit a
Rep. character \(\chi_{5}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 2
Sturm bound 5
Trace bound 1

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 5.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(5\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(5))\).

Total New Old
Modular forms 5 3 2
Cusp forms 3 3 0
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(5\)Dim.
\(+\)\(1\)
\(-\)\(2\)

Trace form

\( 3q - 18q^{2} + 146q^{3} + 596q^{4} + 625q^{5} - 4424q^{6} + 5942q^{7} - 12600q^{8} - 4181q^{9} + O(q^{10}) \) \( 3q - 18q^{2} + 146q^{3} + 596q^{4} + 625q^{5} - 4424q^{6} + 5942q^{7} - 12600q^{8} - 4181q^{9} - 1250q^{10} - 22224q^{11} + 227152q^{12} - 914q^{13} - 474288q^{14} + 233750q^{15} - 144112q^{16} + 907662q^{17} - 1008394q^{18} - 1305260q^{19} + 932500q^{20} + 601116q^{21} + 4083944q^{22} - 1581774q^{23} - 3754080q^{24} + 1171875q^{25} - 2375964q^{26} + 313100q^{27} + 3305504q^{28} + 529410q^{29} - 3905000q^{30} - 1751884q^{31} - 4067808q^{32} + 717232q^{33} + 19920572q^{34} - 1588750q^{35} + 14797508q^{36} - 36053298q^{37} + 3821400q^{38} + 33625748q^{39} - 17475000q^{40} - 15901014q^{41} - 61126176q^{42} + 46492906q^{43} - 5121168q^{44} + 5745625q^{45} + 48651696q^{46} + 22853022q^{47} - 38239424q^{48} - 9204929q^{49} - 7031250q^{50} - 54632204q^{51} + 139262232q^{52} + 703446q^{53} - 72617360q^{54} + 43870000q^{55} - 10570560q^{56} + 87911000q^{57} - 231081500q^{58} - 140181180q^{59} + 78130000q^{60} - 228831074q^{61} + 331039104q^{62} + 198326886q^{63} - 213219264q^{64} + 144346250q^{65} + 430656992q^{66} - 45604738q^{67} - 265631256q^{68} - 153977772q^{69} - 254010000q^{70} - 197098404q^{71} - 139728600q^{72} + 533029126q^{73} + 639347892q^{74} + 57031250q^{75} + 354443280q^{76} - 996146736q^{77} - 794572208q^{78} + 101918360q^{79} - 299990000q^{80} - 486443657q^{81} - 268068116q^{82} + 1664055066q^{83} + 1378576512q^{84} - 51263750q^{85} - 161213784q^{86} - 523405300q^{87} - 368023200q^{88} + 810150030q^{89} - 697116250q^{90} + 189838876q^{91} - 700560288q^{92} - 31047288q^{93} - 1063675648q^{94} + 445137500q^{95} + 1404784256q^{96} + 618891222q^{97} - 621046626q^{98} - 1654739152q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(5))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5
5.10.a.a \(1\) \(2.575\) \(\Q\) None \(-8\) \(-114\) \(-625\) \(4242\) \(+\) \(q-8q^{2}-114q^{3}-448q^{4}-5^{4}q^{5}+\cdots\)
5.10.a.b \(2\) \(2.575\) \(\Q(\sqrt{1009}) \) None \(-10\) \(260\) \(1250\) \(1700\) \(-\) \(q+(-5-\beta )q^{2}+(130+2\beta )q^{3}+(522+\cdots)q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 8 T + 512 T^{2} \))(\( 1 + 10 T + 40 T^{2} + 5120 T^{3} + 262144 T^{4} \))
$3$ (\( 1 + 114 T + 19683 T^{2} \))(\( 1 - 260 T + 52230 T^{2} - 5117580 T^{3} + 387420489 T^{4} \))
$5$ (\( 1 + 625 T \))(\( ( 1 - 625 T )^{2} \))
$7$ (\( 1 - 4242 T + 40353607 T^{2} \))(\( 1 - 1700 T + 35221550 T^{2} - 68601131900 T^{3} + 1628413597910449 T^{4} \))
$11$ (\( 1 + 46208 T + 2357947691 T^{2} \))(\( 1 - 23984 T + 1217213446 T^{2} - 56553017420944 T^{3} + 5559917313492231481 T^{4} \))
$13$ (\( 1 + 115934 T + 10604499373 T^{2} \))(\( 1 - 115020 T + 22672043710 T^{2} - 1219729517882460 T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \))
$17$ (\( 1 - 494842 T + 118587876497 T^{2} \))(\( 1 - 412820 T + 113016614470 T^{2} - 48955447175491540 T^{3} + \)\(14\!\cdots\!09\)\( T^{4} \))
$19$ (\( 1 + 1008740 T + 322687697779 T^{2} \))(\( 1 + 296520 T + 659218232758 T^{2} + 95683356145429080 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} \))
$23$ (\( 1 + 532554 T + 1801152661463 T^{2} \))(\( 1 + 1049220 T + 3497852029390 T^{2} + 1889805395460208860 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \))
$29$ (\( 1 - 4196390 T + 14507145975869 T^{2} \))(\( 1 + 3666980 T + 20832571957438 T^{2} + 53197414150592105620 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))
$31$ (\( 1 + 3365028 T + 26439622160671 T^{2} \))(\( 1 - 1613144 T + 29382323902526 T^{2} - 42650917850753459624 T^{3} + \)\(69\!\cdots\!41\)\( T^{4} \))
$37$ (\( 1 + 14931358 T + 129961739795077 T^{2} \))(\( 1 + 21121940 T + 328931801286510 T^{2} + \)\(27\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!29\)\( T^{4} \))
$41$ (\( 1 - 11056262 T + 327381934393961 T^{2} \))(\( 1 + 26957276 T + 811945448362966 T^{2} + \)\(88\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} \))
$43$ (\( 1 + 6396794 T + 502592611936843 T^{2} \))(\( 1 - 52889700 T + 1703843788760950 T^{2} - \)\(26\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!49\)\( T^{4} \))
$47$ (\( 1 + 35559158 T + 1119130473102767 T^{2} \))(\( 1 - 58412180 T + 2814913257457630 T^{2} - \)\(65\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!89\)\( T^{4} \))
$53$ (\( 1 - 39738586 T + 3299763591802133 T^{2} \))(\( 1 + 39035140 T + 5675030678030830 T^{2} + \)\(12\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!89\)\( T^{4} \))
$59$ (\( 1 + 85185620 T + 8662995818654939 T^{2} \))(\( 1 + 54995560 T + 15674484224932678 T^{2} + \)\(47\!\cdots\!40\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \))
$61$ (\( 1 - 45748642 T + 11694146092834141 T^{2} \))(\( 1 + 274579716 T + 41753623519328446 T^{2} + \)\(32\!\cdots\!56\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} \))
$67$ (\( 1 + 45286158 T + 27206534396294947 T^{2} \))(\( 1 + 318580 T + 48520062064444070 T^{2} + \)\(86\!\cdots\!60\)\( T^{3} + \)\(74\!\cdots\!09\)\( T^{4} \))
$71$ (\( 1 + 189967468 T + 45848500718449031 T^{2} \))(\( 1 + 7130936 T + 51935375688707086 T^{2} + \)\(32\!\cdots\!16\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))
$73$ (\( 1 - 412170946 T + 58871586708267913 T^{2} \))(\( 1 - 120858180 T + 42707263689423190 T^{2} - \)\(71\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!69\)\( T^{4} \))
$79$ (\( 1 - 95040840 T + 119851595982618319 T^{2} \))(\( 1 - 6877520 T - 115982362290712162 T^{2} - \)\(82\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} \))
$83$ (\( 1 - 261706326 T + 186940255267540403 T^{2} \))(\( 1 - 1402348740 T + 857904310704391270 T^{2} - \)\(26\!\cdots\!20\)\( T^{3} + \)\(34\!\cdots\!09\)\( T^{4} \))
$89$ (\( 1 + 19938630 T + 350356403707485209 T^{2} \))(\( 1 - 830088660 T + 692293619421117718 T^{2} - \)\(29\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} \))
$97$ (\( 1 + 19503358 T + 760231058654565217 T^{2} \))(\( 1 - 638394580 T + 1615411126351062630 T^{2} - \)\(48\!\cdots\!60\)\( T^{3} + \)\(57\!\cdots\!89\)\( T^{4} \))
show more
show less