Properties

Label 4.21
Level 4
Weight 21
Dimension 9
Nonzero newspaces 1
Newform subspaces 2
Sturm bound 21
Trace bound 0

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Defining parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 21 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 2 \)
Sturm bound: \(21\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{21}(\Gamma_1(4))\).

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

Trace form

\( 9q - 628q^{2} - 267504q^{4} - 738494q^{5} + 99460608q^{6} + 1103532992q^{8} - 6924577767q^{9} + O(q^{10}) \) \( 9q - 628q^{2} - 267504q^{4} - 738494q^{5} + 99460608q^{6} + 1103532992q^{8} - 6924577767q^{9} + 1611884376q^{10} - 41779799040q^{12} + 147660923874q^{13} - 200556776448q^{14} + 70138216704q^{16} - 1684556709806q^{17} + 1635822350412q^{18} + 241483768096q^{20} + 33281721747456q^{21} - 31324969489920q^{22} - 47563142934528q^{24} - 567413012949q^{25} - 63229698776360q^{26} + 385881741772800q^{28} + 303916348382242q^{29} - 1063857826698240q^{30} + 454535173225472q^{32} + 1131041167426560q^{33} - 4199565255965160q^{34} + 16320708718416912q^{36} - 7415248775263806q^{37} - 19214136907706880q^{38} + 42169382228654976q^{40} - 5755597456531022q^{41} - 92697489416232960q^{42} + 157933848933319680q^{44} + 22280783580364386q^{45} - 271527229329687552q^{46} + 485296685862666240q^{48} - 37006326729654807q^{49} - 623084159258016924q^{50} + 813052968459434784q^{52} + 132739492344115714q^{53} - 1607547577815069696q^{54} + 1749313578676543488q^{56} + 440441203792112640q^{57} - 1972695284657517096q^{58} + 2279505537583872000q^{60} - 1172450092804974942q^{61} - 1243411198213386240q^{62} + 1333510659266973696q^{64} - 828610552028041948q^{65} - 21993146403409920q^{66} - 1641594099654542816q^{68} + 3450619355851659264q^{69} + 8506721176491632640q^{70} - 19477455141526114368q^{72} - 4955973119248806606q^{73} + 23508341140743382360q^{74} - 26576303558589158400q^{76} - 5280312525070141440q^{77} + 39300720744848010240q^{78} - 53478714049403538944q^{80} + 27869130702616624425q^{81} + 59306974273770046104q^{82} - 85636060864270565376q^{84} + 5624508021574984452q^{85} + 60721004056878217728q^{86} - 36088003765030440960q^{88} - 30717207683107545998q^{89} + 14470051757664399576q^{90} + 64932970344704317440q^{92} - 53255995615294218240q^{93} - 49076400009934399488q^{94} + 148896006651003076608q^{96} + 21227709255069480594q^{97} - 274458557005261496308q^{98} + O(q^{100}) \)

Decomposition of \(S_{21}^{\mathrm{new}}(\Gamma_1(4))\)

We only show spaces with odd 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
4.21.b \(\chi_{4}(3, \cdot)\) 4.21.b.a 1 1
4.21.b.b 8

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 1024 T \))(\( 1 - 396 T + 736448 T^{2} - 996691968 T^{3} + 813872185344 T^{4} - 1045107277037568 T^{5} + 809733139252379648 T^{6} - \)\(45\!\cdots\!96\)\( T^{7} + \)\(12\!\cdots\!76\)\( T^{8} \))
$3$ (\( ( 1 - 59049 T )( 1 + 59049 T ) \))(\( 1 - 8741456520 T^{2} + 43354208534771316636 T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(30\!\cdots\!66\)\( T^{8} - \)\(13\!\cdots\!60\)\( T^{10} + \)\(64\!\cdots\!36\)\( T^{12} - \)\(15\!\cdots\!20\)\( T^{14} + \)\(21\!\cdots\!01\)\( T^{16} \))
$5$ (\( 1 + 19306574 T + 95367431640625 T^{2} \))(\( ( 1 - 9284040 T + 303317507889500 T^{2} - \)\(26\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!50\)\( T^{4} - \)\(24\!\cdots\!00\)\( T^{5} + \)\(27\!\cdots\!00\)\( T^{6} - \)\(80\!\cdots\!00\)\( T^{7} + \)\(82\!\cdots\!25\)\( T^{8} )^{2} \))
$7$ (\( ( 1 - 282475249 T )( 1 + 282475249 T ) \))(\( 1 - 260769768676814600 T^{2} + \)\(41\!\cdots\!96\)\( T^{4} - \)\(46\!\cdots\!00\)\( T^{6} + \)\(41\!\cdots\!06\)\( T^{8} - \)\(29\!\cdots\!00\)\( T^{10} + \)\(16\!\cdots\!96\)\( T^{12} - \)\(67\!\cdots\!00\)\( T^{14} + \)\(16\!\cdots\!01\)\( T^{16} \))
$11$ (\( ( 1 - 25937424601 T )( 1 + 25937424601 T ) \))(\( 1 - \)\(28\!\cdots\!08\)\( T^{2} + \)\(43\!\cdots\!08\)\( T^{4} - \)\(45\!\cdots\!76\)\( T^{6} + \)\(35\!\cdots\!50\)\( T^{8} - \)\(20\!\cdots\!76\)\( T^{10} + \)\(90\!\cdots\!08\)\( T^{12} - \)\(26\!\cdots\!08\)\( T^{14} + \)\(41\!\cdots\!01\)\( T^{16} \))
$13$ (\( 1 - 190840318802 T + \)\(19\!\cdots\!01\)\( T^{2} \))(\( ( 1 + 21589697464 T + \)\(27\!\cdots\!08\)\( T^{2} + \)\(34\!\cdots\!12\)\( T^{3} + \)\(31\!\cdots\!14\)\( T^{4} + \)\(65\!\cdots\!12\)\( T^{5} + \)\(98\!\cdots\!08\)\( T^{6} + \)\(14\!\cdots\!64\)\( T^{7} + \)\(13\!\cdots\!01\)\( T^{8} )^{2} \))
$17$ (\( 1 - 750325121602 T + \)\(40\!\cdots\!01\)\( T^{2} \))(\( ( 1 + 1217440915704 T + \)\(11\!\cdots\!08\)\( T^{2} + \)\(11\!\cdots\!72\)\( T^{3} + \)\(69\!\cdots\!14\)\( T^{4} + \)\(48\!\cdots\!72\)\( T^{5} + \)\(19\!\cdots\!08\)\( T^{6} + \)\(81\!\cdots\!04\)\( T^{7} + \)\(27\!\cdots\!01\)\( T^{8} )^{2} \))
$19$ (\( ( 1 - 6131066257801 T )( 1 + 6131066257801 T ) \))(\( 1 - \)\(14\!\cdots\!88\)\( T^{2} + \)\(10\!\cdots\!88\)\( T^{4} - \)\(53\!\cdots\!76\)\( T^{6} + \)\(21\!\cdots\!50\)\( T^{8} - \)\(75\!\cdots\!76\)\( T^{10} + \)\(21\!\cdots\!88\)\( T^{12} - \)\(42\!\cdots\!88\)\( T^{14} + \)\(39\!\cdots\!01\)\( T^{16} \))
$23$ (\( ( 1 - 41426511213649 T )( 1 + 41426511213649 T ) \))(\( 1 - \)\(39\!\cdots\!60\)\( T^{2} + \)\(14\!\cdots\!56\)\( T^{4} - \)\(30\!\cdots\!20\)\( T^{6} + \)\(64\!\cdots\!26\)\( T^{8} - \)\(90\!\cdots\!20\)\( T^{10} + \)\(12\!\cdots\!56\)\( T^{12} - \)\(10\!\cdots\!60\)\( T^{14} + \)\(75\!\cdots\!01\)\( T^{16} \))
$29$ (\( 1 - 203154876160402 T + \)\(17\!\cdots\!01\)\( T^{2} \))(\( ( 1 - 50380736110920 T + \)\(46\!\cdots\!76\)\( T^{2} - \)\(48\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!06\)\( T^{4} - \)\(85\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!76\)\( T^{6} - \)\(27\!\cdots\!20\)\( T^{7} + \)\(98\!\cdots\!01\)\( T^{8} )^{2} \))
$31$ (\( ( 1 - 819628286980801 T )( 1 + 819628286980801 T ) \))(\( 1 - \)\(31\!\cdots\!88\)\( T^{2} + \)\(46\!\cdots\!88\)\( T^{4} - \)\(46\!\cdots\!76\)\( T^{6} + \)\(34\!\cdots\!50\)\( T^{8} - \)\(20\!\cdots\!76\)\( T^{10} + \)\(95\!\cdots\!88\)\( T^{12} - \)\(28\!\cdots\!88\)\( T^{14} + \)\(41\!\cdots\!01\)\( T^{16} \))
$37$ (\( 1 + 9492206529013198 T + \)\(23\!\cdots\!01\)\( T^{2} \))(\( ( 1 - 1038478876874696 T + \)\(41\!\cdots\!48\)\( T^{2} + \)\(76\!\cdots\!92\)\( T^{3} + \)\(76\!\cdots\!94\)\( T^{4} + \)\(17\!\cdots\!92\)\( T^{5} + \)\(22\!\cdots\!48\)\( T^{6} - \)\(12\!\cdots\!96\)\( T^{7} + \)\(28\!\cdots\!01\)\( T^{8} )^{2} \))
$41$ (\( 1 - 16082418088944802 T + \)\(18\!\cdots\!01\)\( T^{2} \))(\( ( 1 + 10919007772737912 T + \)\(30\!\cdots\!28\)\( T^{2} + \)\(38\!\cdots\!64\)\( T^{3} + \)\(43\!\cdots\!90\)\( T^{4} + \)\(70\!\cdots\!64\)\( T^{5} + \)\(98\!\cdots\!28\)\( T^{6} + \)\(63\!\cdots\!12\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} )^{2} \))
$43$ (\( ( 1 - 21611482313284249 T )( 1 + 21611482313284249 T ) \))(\( 1 - \)\(26\!\cdots\!00\)\( T^{2} + \)\(34\!\cdots\!96\)\( T^{4} - \)\(28\!\cdots\!00\)\( T^{6} + \)\(15\!\cdots\!06\)\( T^{8} - \)\(61\!\cdots\!00\)\( T^{10} + \)\(16\!\cdots\!96\)\( T^{12} - \)\(27\!\cdots\!00\)\( T^{14} + \)\(22\!\cdots\!01\)\( T^{16} \))
$47$ (\( ( 1 - 52599132235830049 T )( 1 + 52599132235830049 T ) \))(\( 1 - \)\(13\!\cdots\!00\)\( T^{2} + \)\(88\!\cdots\!76\)\( T^{4} - \)\(38\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!86\)\( T^{8} - \)\(29\!\cdots\!80\)\( T^{10} + \)\(52\!\cdots\!76\)\( T^{12} - \)\(60\!\cdots\!00\)\( T^{14} + \)\(34\!\cdots\!01\)\( T^{16} \))
$53$ (\( 1 - 263609364120076402 T + \)\(30\!\cdots\!01\)\( T^{2} \))(\( ( 1 + 65434935887980344 T + \)\(10\!\cdots\!68\)\( T^{2} + \)\(52\!\cdots\!92\)\( T^{3} + \)\(45\!\cdots\!34\)\( T^{4} + \)\(16\!\cdots\!92\)\( T^{5} + \)\(96\!\cdots\!68\)\( T^{6} + \)\(18\!\cdots\!44\)\( T^{7} + \)\(87\!\cdots\!01\)\( T^{8} )^{2} \))
$59$ (\( ( 1 - 511116753300641401 T )( 1 + 511116753300641401 T ) \))(\( 1 - \)\(96\!\cdots\!48\)\( T^{2} + \)\(43\!\cdots\!48\)\( T^{4} - \)\(13\!\cdots\!76\)\( T^{6} + \)\(37\!\cdots\!50\)\( T^{8} - \)\(93\!\cdots\!76\)\( T^{10} + \)\(20\!\cdots\!48\)\( T^{12} - \)\(30\!\cdots\!48\)\( T^{14} + \)\(21\!\cdots\!01\)\( T^{16} \))
$61$ (\( 1 - 342453856112605202 T + \)\(50\!\cdots\!01\)\( T^{2} \))(\( ( 1 + 757451974458790072 T + \)\(11\!\cdots\!68\)\( T^{2} + \)\(40\!\cdots\!64\)\( T^{3} + \)\(60\!\cdots\!90\)\( T^{4} + \)\(20\!\cdots\!64\)\( T^{5} + \)\(30\!\cdots\!68\)\( T^{6} + \)\(99\!\cdots\!72\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \))
$67$ (\( ( 1 - 1822837804551761449 T )( 1 + 1822837804551761449 T ) \))(\( 1 - \)\(13\!\cdots\!20\)\( T^{2} + \)\(10\!\cdots\!36\)\( T^{4} - \)\(54\!\cdots\!60\)\( T^{6} + \)\(21\!\cdots\!66\)\( T^{8} - \)\(60\!\cdots\!60\)\( T^{10} + \)\(12\!\cdots\!36\)\( T^{12} - \)\(18\!\cdots\!20\)\( T^{14} + \)\(14\!\cdots\!01\)\( T^{16} \))
$71$ (\( ( 1 - 3255243551009881201 T )( 1 + 3255243551009881201 T ) \))(\( 1 - \)\(49\!\cdots\!28\)\( T^{2} + \)\(12\!\cdots\!28\)\( T^{4} - \)\(21\!\cdots\!76\)\( T^{6} + \)\(26\!\cdots\!50\)\( T^{8} - \)\(24\!\cdots\!76\)\( T^{10} + \)\(15\!\cdots\!28\)\( T^{12} - \)\(70\!\cdots\!28\)\( T^{14} + \)\(15\!\cdots\!01\)\( T^{16} \))
$73$ (\( 1 - 5395059597962887202 T + \)\(18\!\cdots\!01\)\( T^{2} \))(\( ( 1 + 5175516358605846904 T + \)\(58\!\cdots\!68\)\( T^{2} + \)\(28\!\cdots\!92\)\( T^{3} + \)\(14\!\cdots\!34\)\( T^{4} + \)\(53\!\cdots\!92\)\( T^{5} + \)\(19\!\cdots\!68\)\( T^{6} + \)\(32\!\cdots\!04\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} )^{2} \))
$79$ (\( ( 1 - 9468276082626847201 T )( 1 + 9468276082626847201 T ) \))(\( 1 - \)\(43\!\cdots\!88\)\( T^{2} + \)\(90\!\cdots\!88\)\( T^{4} - \)\(12\!\cdots\!76\)\( T^{6} + \)\(12\!\cdots\!50\)\( T^{8} - \)\(97\!\cdots\!76\)\( T^{10} + \)\(58\!\cdots\!88\)\( T^{12} - \)\(22\!\cdots\!88\)\( T^{14} + \)\(41\!\cdots\!01\)\( T^{16} \))
$83$ (\( ( 1 - 15516041187205853449 T )( 1 + 15516041187205853449 T ) \))(\( 1 - \)\(10\!\cdots\!60\)\( T^{2} + \)\(59\!\cdots\!56\)\( T^{4} - \)\(22\!\cdots\!20\)\( T^{6} + \)\(63\!\cdots\!26\)\( T^{8} - \)\(13\!\cdots\!20\)\( T^{10} + \)\(19\!\cdots\!56\)\( T^{12} - \)\(20\!\cdots\!60\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} \))
$89$ (\( 1 - 10944684939688527202 T + \)\(97\!\cdots\!01\)\( T^{2} \))(\( ( 1 + 20830946311398036600 T + \)\(30\!\cdots\!96\)\( T^{2} + \)\(41\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!06\)\( T^{4} + \)\(39\!\cdots\!00\)\( T^{5} + \)\(29\!\cdots\!96\)\( T^{6} + \)\(19\!\cdots\!00\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} )^{2} \))
$97$ (\( 1 + 72063723240789129598 T + \)\(54\!\cdots\!01\)\( T^{2} \))(\( ( 1 - 46645716247929305096 T + \)\(18\!\cdots\!48\)\( T^{2} - \)\(80\!\cdots\!48\)\( T^{3} + \)\(13\!\cdots\!94\)\( T^{4} - \)\(43\!\cdots\!48\)\( T^{5} + \)\(53\!\cdots\!48\)\( T^{6} - \)\(75\!\cdots\!96\)\( T^{7} + \)\(87\!\cdots\!01\)\( T^{8} )^{2} \))
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