Properties

Label 20.4.e
Level 20
Weight 4
Character orbit e
Rep. character \(\chi_{20}(3,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 14
Newform subspaces 2
Sturm bound 12
Trace bound 1

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Defining parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 20.e (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 20 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(12\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(20, [\chi])\).

Total New Old
Modular forms 22 22 0
Cusp forms 14 14 0
Eisenstein series 8 8 0

Trace form

\( 14q - 2q^{2} - 4q^{5} + 8q^{6} - 44q^{8} + O(q^{10}) \) \( 14q - 2q^{2} - 4q^{5} + 8q^{6} - 44q^{8} - 74q^{10} - 80q^{12} + 42q^{13} + 184q^{16} - 134q^{17} + 306q^{18} + 316q^{20} - 144q^{21} + 360q^{22} + 106q^{25} - 460q^{26} - 880q^{28} - 1240q^{30} - 632q^{32} + 80q^{33} + 892q^{36} + 326q^{37} + 1600q^{38} + 1836q^{40} + 288q^{41} + 1160q^{42} + 586q^{45} - 1432q^{46} - 2720q^{48} - 2214q^{50} - 1524q^{52} - 698q^{53} + 2048q^{56} - 960q^{57} + 2712q^{58} + 3280q^{60} - 1832q^{61} + 2440q^{62} - 1778q^{65} - 1680q^{66} - 2428q^{68} - 3040q^{70} - 2172q^{72} + 1942q^{73} + 800q^{76} + 3120q^{77} + 3720q^{78} + 2096q^{80} + 4530q^{81} + 536q^{82} + 2282q^{85} - 2552q^{86} - 2400q^{88} - 2154q^{90} - 1840q^{92} - 3280q^{93} + 1088q^{96} - 5994q^{97} + 326q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(20, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
20.4.e.a \(2\) \(1.180\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(4\) \(0\) \(-4\) \(0\) \(q+(2+2i)q^{2}+8iq^{4}+(-2-11i)q^{5}+\cdots\)
20.4.e.b \(12\) \(1.180\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-6\) \(0\) \(0\) \(0\) \(q+(\beta _{1}-\beta _{5})q^{2}-\beta _{9}q^{3}+(2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 4 T + 8 T^{2} \))(\( 1 + 6 T + 18 T^{2} + 40 T^{3} - 256 T^{5} - 736 T^{6} - 2048 T^{7} + 20480 T^{9} + 73728 T^{10} + 196608 T^{11} + 262144 T^{12} \))
$3$ (\( 1 + 729 T^{4} \))(\( 1 - 2226 T^{4} + 2588175 T^{8} - 2136402140 T^{12} + 1375462310175 T^{16} - 628688148206706 T^{20} + 150094635296999121 T^{24} \))
$5$ (\( 1 + 4 T + 125 T^{2} \))(\( ( 1 - 85 T^{2} - 400 T^{3} - 10625 T^{4} + 1953125 T^{6} )^{2} \))
$7$ (\( 1 + 117649 T^{4} \))(\( 1 - 163746 T^{4} - 12219131425 T^{8} + 4249089576371140 T^{12} - \)\(16\!\cdots\!25\)\( T^{16} - \)\(31\!\cdots\!46\)\( T^{20} + \)\(26\!\cdots\!01\)\( T^{24} \))
$11$ (\( ( 1 - 1331 T^{2} )^{2} \))(\( ( 1 - 4986 T^{2} + 12380375 T^{4} - 19918383340 T^{6} + 21932589515375 T^{8} - 15648203886330906 T^{10} + 5559917313492231481 T^{12} )^{2} \))
$13$ (\( ( 1 - 18 T + 2197 T^{2} )( 1 + 92 T + 2197 T^{2} ) \))(\( ( 1 - 58 T + 1682 T^{2} - 72450 T^{3} + 7507335 T^{4} - 469328492 T^{5} + 17218216316 T^{6} - 1031114696924 T^{7} + 36236472144015 T^{8} - 768295979573850 T^{9} + 39187379176013042 T^{10} - 2968781794817263906 T^{11} + \)\(11\!\cdots\!29\)\( T^{12} )^{2} \))
$17$ (\( ( 1 - 104 T + 4913 T^{2} )( 1 - 94 T + 4913 T^{2} ) \))(\( ( 1 + 166 T + 13778 T^{2} + 1478470 T^{3} + 180009375 T^{4} + 13146146484 T^{5} + 795027918044 T^{6} + 64587017675892 T^{7} + 4344988709709375 T^{8} + 175328617764519590 T^{9} + 8027369184551647058 T^{10} + \)\(47\!\cdots\!38\)\( T^{11} + \)\(14\!\cdots\!09\)\( T^{12} )^{2} \))
$19$ (\( ( 1 + 6859 T^{2} )^{2} \))(\( ( 1 + 16994 T^{2} + 156289815 T^{4} + 1042328090300 T^{6} + 7352792038002015 T^{8} + 37613073734610340034 T^{10} + \)\(10\!\cdots\!41\)\( T^{12} )^{2} \))
$23$ (\( 1 + 148035889 T^{4} \))(\( 1 + 86914334 T^{4} + 54372416869500575 T^{8} + \)\(33\!\cdots\!60\)\( T^{12} + \)\(11\!\cdots\!75\)\( T^{16} + \)\(41\!\cdots\!94\)\( T^{20} + \)\(10\!\cdots\!61\)\( T^{24} \))
$29$ (\( ( 1 - 130 T + 24389 T^{2} )( 1 + 130 T + 24389 T^{2} ) \))(\( ( 1 - 80686 T^{2} + 2793486135 T^{4} - 69052346800420 T^{6} + 1661630699988154335 T^{8} - \)\(28\!\cdots\!26\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \))
$31$ (\( ( 1 - 29791 T^{2} )^{2} \))(\( ( 1 - 103106 T^{2} + 5864634815 T^{4} - 214293224466300 T^{6} + 5204884986033254015 T^{8} - \)\(81\!\cdots\!66\)\( T^{10} + \)\(69\!\cdots\!41\)\( T^{12} )^{2} \))
$37$ (\( ( 1 - 214 T + 50653 T^{2} )( 1 + 396 T + 50653 T^{2} ) \))(\( ( 1 - 254 T + 32258 T^{2} + 4711770 T^{3} - 385957065 T^{4} - 961296674116 T^{5} + 267719946494684 T^{6} - 48692560433997748 T^{7} - 990260234410629585 T^{8} + \)\(61\!\cdots\!90\)\( T^{9} + \)\(21\!\cdots\!98\)\( T^{10} - \)\(84\!\cdots\!22\)\( T^{11} + \)\(16\!\cdots\!29\)\( T^{12} )^{2} \))
$41$ (\( ( 1 - 472 T + 68921 T^{2} )^{2} \))(\( ( 1 + 164 T + 188335 T^{2} + 20815080 T^{3} + 12980236535 T^{4} + 779017095524 T^{5} + 327381934393961 T^{6} )^{4} \))
$43$ (\( 1 + 6321363049 T^{4} \))(\( 1 + 9995711534 T^{4} - 4601157531678236945 T^{8} - \)\(44\!\cdots\!80\)\( T^{12} - \)\(18\!\cdots\!45\)\( T^{16} + \)\(15\!\cdots\!34\)\( T^{20} + \)\(63\!\cdots\!01\)\( T^{24} \))
$47$ (\( 1 + 10779215329 T^{4} \))(\( 1 - 14480422466 T^{4} + \)\(18\!\cdots\!35\)\( T^{8} - \)\(27\!\cdots\!80\)\( T^{12} + \)\(21\!\cdots\!35\)\( T^{16} - \)\(19\!\cdots\!46\)\( T^{20} + \)\(15\!\cdots\!21\)\( T^{24} \))
$53$ (\( ( 1 - 518 T + 148877 T^{2} )( 1 + 572 T + 148877 T^{2} ) \))(\( ( 1 + 322 T + 51842 T^{2} + 24976890 T^{3} + 1487618135 T^{4} + 291824210108 T^{5} + 328768813336156 T^{6} + 43445912928248716 T^{7} + 32972105566189474415 T^{8} + \)\(82\!\cdots\!70\)\( T^{9} + \)\(25\!\cdots\!22\)\( T^{10} + \)\(23\!\cdots\!54\)\( T^{11} + \)\(10\!\cdots\!89\)\( T^{12} )^{2} \))
$59$ (\( ( 1 + 205379 T^{2} )^{2} \))(\( ( 1 + 543794 T^{2} + 146689289095 T^{4} + 30962597079669980 T^{6} + \)\(61\!\cdots\!95\)\( T^{8} + \)\(96\!\cdots\!14\)\( T^{10} + \)\(75\!\cdots\!21\)\( T^{12} )^{2} \))
$61$ (\( ( 1 + 468 T + 226981 T^{2} )^{2} \))(\( ( 1 + 224 T + 625475 T^{2} + 89988720 T^{3} + 141970940975 T^{4} + 11540563856864 T^{5} + 11694146092834141 T^{6} )^{4} \))
$67$ (\( 1 + 90458382169 T^{4} \))(\( 1 + 152118288974 T^{4} - \)\(67\!\cdots\!65\)\( T^{8} - \)\(22\!\cdots\!80\)\( T^{12} - \)\(55\!\cdots\!65\)\( T^{16} + \)\(10\!\cdots\!54\)\( T^{20} + \)\(54\!\cdots\!81\)\( T^{24} \))
$71$ (\( ( 1 - 357911 T^{2} )^{2} \))(\( ( 1 - 1431506 T^{2} + 1032027741935 T^{4} - 460759845466007580 T^{6} + \)\(13\!\cdots\!35\)\( T^{8} - \)\(23\!\cdots\!46\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \))
$73$ (\( ( 1 - 1098 T + 389017 T^{2} )( 1 + 592 T + 389017 T^{2} ) \))(\( ( 1 - 718 T + 257762 T^{2} - 319815230 T^{3} + 531457295055 T^{4} - 223706294377092 T^{5} + 74772514744761596 T^{6} - 87025551519693198564 T^{7} + \)\(80\!\cdots\!95\)\( T^{8} - \)\(18\!\cdots\!90\)\( T^{9} + \)\(59\!\cdots\!02\)\( T^{10} - \)\(63\!\cdots\!26\)\( T^{11} + \)\(34\!\cdots\!69\)\( T^{12} )^{2} \))
$79$ (\( ( 1 + 493039 T^{2} )^{2} \))(\( ( 1 + 574874 T^{2} + 335775077615 T^{4} + 287028695918457900 T^{6} + \)\(81\!\cdots\!15\)\( T^{8} + \)\(33\!\cdots\!34\)\( T^{10} + \)\(14\!\cdots\!61\)\( T^{12} )^{2} \))
$83$ (\( 1 + 326940373369 T^{4} \))(\( 1 - 1040598079986 T^{4} + \)\(60\!\cdots\!75\)\( T^{8} - \)\(23\!\cdots\!40\)\( T^{12} + \)\(64\!\cdots\!75\)\( T^{16} - \)\(11\!\cdots\!06\)\( T^{20} + \)\(12\!\cdots\!81\)\( T^{24} \))
$89$ (\( ( 1 - 1670 T + 704969 T^{2} )( 1 + 1670 T + 704969 T^{2} ) \))(\( ( 1 - 2798646 T^{2} + 3790036251455 T^{4} - 3254227731213032180 T^{6} + \)\(18\!\cdots\!55\)\( T^{8} - \)\(69\!\cdots\!66\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} )^{2} \))
$97$ (\( ( 1 - 594 T + 912673 T^{2} )( 1 + 1816 T + 912673 T^{2} ) \))(\( ( 1 + 2386 T + 2846498 T^{2} + 2507112050 T^{3} + 1418732317695 T^{4} + 602523284624124 T^{5} + 542007267886579004 T^{6} + \)\(54\!\cdots\!52\)\( T^{7} + \)\(11\!\cdots\!55\)\( T^{8} + \)\(19\!\cdots\!50\)\( T^{9} + \)\(19\!\cdots\!18\)\( T^{10} + \)\(15\!\cdots\!98\)\( T^{11} + \)\(57\!\cdots\!89\)\( T^{12} )^{2} \))
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