Properties

Label 21.4.e
Level 21
Weight 4
Character orbit e
Rep. character \(\chi_{21}(4,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 8
Newform subspaces 2
Sturm bound 10
Trace bound 1

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Defining parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 21.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(10\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(21, [\chi])\).

Total New Old
Modular forms 20 8 12
Cusp forms 12 8 4
Eisenstein series 8 0 8

Trace form

\( 8q + 2q^{2} + 6q^{3} - 26q^{4} - 8q^{5} - 24q^{6} - 20q^{7} + 120q^{8} - 36q^{9} + O(q^{10}) \) \( 8q + 2q^{2} + 6q^{3} - 26q^{4} - 8q^{5} - 24q^{6} - 20q^{7} + 120q^{8} - 36q^{9} + 46q^{10} - 20q^{11} + 72q^{12} - 4q^{13} - 242q^{14} - 84q^{15} - 170q^{16} - 132q^{17} + 18q^{18} + 218q^{19} + 872q^{20} + 108q^{21} + 76q^{22} - 132q^{23} + 54q^{24} - 14q^{25} - 466q^{26} - 108q^{27} - 166q^{28} - 488q^{29} - 192q^{30} + 348q^{31} - 728q^{32} + 150q^{33} - 552q^{34} + 140q^{35} + 468q^{36} + 54q^{37} + 350q^{38} + 378q^{39} + 42q^{40} + 1208q^{41} - 156q^{42} + 772q^{43} + 920q^{44} - 72q^{45} + 804q^{46} + 240q^{47} - 1872q^{48} - 940q^{49} - 2060q^{50} - 108q^{51} - 260q^{52} - 756q^{53} + 108q^{54} - 1972q^{55} + 1152q^{56} + 1116q^{57} + 358q^{58} - 1128q^{59} + 1326q^{60} + 188q^{61} + 3636q^{62} - 126q^{63} + 68q^{64} + 280q^{65} - 156q^{66} + 998q^{67} - 2028q^{68} - 1800q^{69} + 3566q^{70} - 48q^{71} - 540q^{72} - 1350q^{73} - 1950q^{74} + 738q^{75} - 4712q^{76} + 548q^{77} - 492q^{78} - 1328q^{79} - 388q^{80} - 324q^{81} + 956q^{82} + 1992q^{83} + 1536q^{84} + 3096q^{85} + 3286q^{86} + 1050q^{87} + 1206q^{88} - 2672q^{89} - 828q^{90} - 206q^{91} - 1512q^{92} + 474q^{93} + 3204q^{94} + 688q^{95} + 1914q^{96} + 1044q^{97} - 5884q^{98} + 360q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(21, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
21.4.e.a \(2\) \(1.239\) \(\Q(\sqrt{-3}) \) None \(3\) \(-3\) \(3\) \(-7\) \(q+(3-3\zeta_{6})q^{2}-3\zeta_{6}q^{3}-\zeta_{6}q^{4}+(3+\cdots)q^{5}+\cdots\)
21.4.e.b \(6\) \(1.239\) 6.0.9924270768.1 None \(-1\) \(9\) \(-11\) \(-13\) \(q-\beta _{1}q^{2}+(3-3\beta _{4})q^{3}+(-8+\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(21, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(21, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 3 T + T^{2} - 24 T^{3} + 64 T^{4} \))(\( 1 + T + T^{2} - 20 T^{3} - 10 T^{4} + 64 T^{5} + 1060 T^{6} + 512 T^{7} - 640 T^{8} - 10240 T^{9} + 4096 T^{10} + 32768 T^{11} + 262144 T^{12} \))
$3$ (\( 1 + 3 T + 9 T^{2} \))(\( ( 1 - 3 T + 9 T^{2} )^{3} \))
$5$ (\( 1 - 3 T - 116 T^{2} - 375 T^{3} + 15625 T^{4} \))(\( 1 + 11 T - 62 T^{2} - 1015 T^{3} - 6040 T^{4} - 54313 T^{5} + 121696 T^{6} - 6789125 T^{7} - 94375000 T^{8} - 1982421875 T^{9} - 15136718750 T^{10} + 335693359375 T^{11} + 3814697265625 T^{12} \))
$7$ (\( 1 + 7 T + 343 T^{2} \))(\( 1 + 13 T + 236 T^{2} + 12145 T^{3} + 80948 T^{4} + 1529437 T^{5} + 40353607 T^{6} \))
$11$ (\( 1 - 15 T - 1106 T^{2} - 19965 T^{3} + 1771561 T^{4} \))(\( 1 + 35 T - 1400 T^{2} - 113593 T^{3} - 198940 T^{4} + 87110135 T^{5} + 3928586038 T^{6} + 115943589685 T^{7} - 352434345340 T^{8} - 267846352063763 T^{9} - 4393799727409400 T^{10} + 146203685929547785 T^{11} + 5559917313492231481 T^{12} \))
$13$ (\( ( 1 + 64 T + 2197 T^{2} )^{2} \))(\( ( 1 - 62 T + 7016 T^{2} - 253976 T^{3} + 15414152 T^{4} - 299262158 T^{5} + 10604499373 T^{6} )^{2} \))
$17$ (\( 1 + 84 T + 2143 T^{2} + 412692 T^{3} + 24137569 T^{4} \))(\( 1 + 48 T - 10035 T^{2} - 125232 T^{3} + 74409318 T^{4} - 234420432 T^{5} - 437742983351 T^{6} - 1151707582416 T^{7} + 1796060047467942 T^{8} - 14850996949472304 T^{9} - 5846614150600651635 T^{10} + \)\(13\!\cdots\!64\)\( T^{11} + \)\(14\!\cdots\!09\)\( T^{12} \))
$19$ (\( 1 - 16 T - 6603 T^{2} - 109744 T^{3} + 47045881 T^{4} \))(\( 1 - 202 T + 7946 T^{2} - 627636 T^{3} + 247297462 T^{4} - 17185599794 T^{5} + 349471935958 T^{6} - 117876028987046 T^{7} + 11634326968854022 T^{8} - 202530415883220444 T^{9} + 17587000346899715306 T^{10} - \)\(30\!\cdots\!98\)\( T^{11} + \)\(10\!\cdots\!41\)\( T^{12} \))
$23$ (\( 1 - 84 T - 5111 T^{2} - 1022028 T^{3} + 148035889 T^{4} \))(\( 1 + 216 T + 10827 T^{2} + 387864 T^{3} + 53856198 T^{4} - 24653558952 T^{5} - 5413409425505 T^{6} - 299959851768984 T^{7} + 7972650149090022 T^{8} + 698602275885685032 T^{9} + \)\(23\!\cdots\!67\)\( T^{10} + \)\(57\!\cdots\!12\)\( T^{11} + \)\(32\!\cdots\!69\)\( T^{12} \))
$29$ (\( ( 1 + 297 T + 24389 T^{2} )^{2} \))(\( ( 1 - 53 T + 52695 T^{2} - 3410210 T^{3} + 1285178355 T^{4} - 31525636013 T^{5} + 14507145975869 T^{6} )^{2} \))
$31$ (\( 1 - 253 T + 34218 T^{2} - 7537123 T^{3} + 887503681 T^{4} \))(\( 1 - 95 T - 70347 T^{2} + 3756594 T^{3} + 3398738767 T^{4} - 83374434539 T^{5} - 110906046363338 T^{6} - 2483807779351349 T^{7} + 3016393166469901327 T^{8} + 99322925971043714574 T^{9} - \)\(55\!\cdots\!67\)\( T^{10} - \)\(22\!\cdots\!45\)\( T^{11} + \)\(69\!\cdots\!41\)\( T^{12} \))
$37$ (\( 1 - 316 T + 49203 T^{2} - 16006348 T^{3} + 2565726409 T^{4} \))(\( 1 + 262 T - 97404 T^{2} - 9678072 T^{3} + 12194182072 T^{4} + 680381910454 T^{5} - 605701122868778 T^{6} + 34463384910226462 T^{7} + 31286934978284739448 T^{8} - \)\(12\!\cdots\!44\)\( T^{9} - \)\(64\!\cdots\!24\)\( T^{10} + \)\(87\!\cdots\!66\)\( T^{11} + \)\(16\!\cdots\!29\)\( T^{12} \))
$41$ (\( ( 1 - 360 T + 68921 T^{2} )^{2} \))(\( ( 1 - 244 T + 187983 T^{2} - 33933832 T^{3} + 12955976343 T^{4} - 1159025434804 T^{5} + 327381934393961 T^{6} )^{2} \))
$43$ (\( ( 1 - 26 T + 79507 T^{2} )^{2} \))(\( ( 1 - 360 T + 166158 T^{2} - 38975294 T^{3} + 13210724106 T^{4} - 2275690697640 T^{5} + 502592611936843 T^{6} )^{2} \))
$47$ (\( 1 - 30 T - 102923 T^{2} - 3114690 T^{3} + 10779215329 T^{4} \))(\( 1 - 210 T - 20853 T^{2} + 83809446 T^{3} - 12756928590 T^{4} - 2596137940074 T^{5} + 3698984470026571 T^{6} - 269538829352302902 T^{7} - \)\(13\!\cdots\!10\)\( T^{8} + \)\(93\!\cdots\!82\)\( T^{9} - \)\(24\!\cdots\!73\)\( T^{10} - \)\(25\!\cdots\!30\)\( T^{11} + \)\(12\!\cdots\!89\)\( T^{12} \))
$53$ (\( 1 + 363 T - 17108 T^{2} + 54042351 T^{3} + 22164361129 T^{4} \))(\( 1 + 393 T - 211446 T^{2} - 23899125 T^{3} + 46453564620 T^{4} - 3425920762143 T^{5} - 9724787230272680 T^{6} - 510040805305563411 T^{7} + \)\(10\!\cdots\!80\)\( T^{8} - \)\(78\!\cdots\!25\)\( T^{9} - \)\(10\!\cdots\!86\)\( T^{10} + \)\(28\!\cdots\!01\)\( T^{11} + \)\(10\!\cdots\!89\)\( T^{12} \))
$59$ (\( 1 - 15 T - 205154 T^{2} - 3080685 T^{3} + 42180533641 T^{4} \))(\( 1 + 1143 T + 557208 T^{2} + 118327563 T^{3} - 14314666608 T^{4} - 27063102119841 T^{5} - 16891447327378130 T^{6} - 5558192850270824739 T^{7} - \)\(60\!\cdots\!28\)\( T^{8} + \)\(10\!\cdots\!57\)\( T^{9} + \)\(99\!\cdots\!48\)\( T^{10} + \)\(41\!\cdots\!57\)\( T^{11} + \)\(75\!\cdots\!21\)\( T^{12} \))
$61$ (\( 1 - 118 T - 213057 T^{2} - 26783758 T^{3} + 51520374361 T^{4} \))(\( 1 - 70 T - 335143 T^{2} - 129510330 T^{3} + 42145697866 T^{4} + 25171752927730 T^{5} - 316289217432887 T^{6} + 5713509651289083130 T^{7} + \)\(21\!\cdots\!26\)\( T^{8} - \)\(15\!\cdots\!30\)\( T^{9} - \)\(88\!\cdots\!03\)\( T^{10} - \)\(42\!\cdots\!70\)\( T^{11} + \)\(13\!\cdots\!81\)\( T^{12} \))
$67$ (\( 1 - 370 T - 163863 T^{2} - 111282310 T^{3} + 90458382169 T^{4} \))(\( 1 - 628 T - 202942 T^{2} + 436381932 T^{3} - 77667044702 T^{4} - 73528811914784 T^{5} + 76060129771959310 T^{6} - 22114746057926180192 T^{7} - \)\(70\!\cdots\!38\)\( T^{8} + \)\(11\!\cdots\!04\)\( T^{9} - \)\(16\!\cdots\!62\)\( T^{10} - \)\(15\!\cdots\!04\)\( T^{11} + \)\(74\!\cdots\!09\)\( T^{12} \))
$71$ (\( ( 1 + 342 T + 357911 T^{2} )^{2} \))(\( ( 1 - 318 T + 742929 T^{2} - 256167372 T^{3} + 265902461319 T^{4} - 40735890286878 T^{5} + 45848500718449031 T^{6} )^{2} \))
$73$ (\( 1 + 362 T - 257973 T^{2} + 140824154 T^{3} + 151334226289 T^{4} \))(\( 1 + 988 T - 186552 T^{2} - 102237300 T^{3} + 281568890272 T^{4} - 16988127696596 T^{5} - 164639785652996186 T^{6} - 6608670472146686132 T^{7} + \)\(42\!\cdots\!08\)\( T^{8} - \)\(60\!\cdots\!00\)\( T^{9} - \)\(42\!\cdots\!92\)\( T^{10} + \)\(88\!\cdots\!16\)\( T^{11} + \)\(34\!\cdots\!69\)\( T^{12} \))
$79$ (\( 1 + 467 T - 274950 T^{2} + 230249213 T^{3} + 243087455521 T^{4} \))(\( 1 + 861 T - 479895 T^{2} - 258646666 T^{3} + 325257480351 T^{4} - 27564282842211 T^{5} - 246706047980056146 T^{6} - 13590266448240869229 T^{7} + \)\(79\!\cdots\!71\)\( T^{8} - \)\(30\!\cdots\!54\)\( T^{9} - \)\(28\!\cdots\!95\)\( T^{10} + \)\(25\!\cdots\!39\)\( T^{11} + \)\(14\!\cdots\!61\)\( T^{12} \))
$83$ (\( ( 1 - 477 T + 571787 T^{2} )^{2} \))(\( ( 1 - 519 T + 1583745 T^{2} - 545598870 T^{3} + 905564802315 T^{4} - 169682053778511 T^{5} + 186940255267540403 T^{6} )^{2} \))
$89$ (\( 1 + 906 T + 115867 T^{2} + 638701914 T^{3} + 496981290961 T^{4} \))(\( 1 + 1766 T + 725929 T^{2} - 728159446 T^{3} - 335534377858 T^{4} + 846551335831238 T^{5} + 1249625385561159997 T^{6} + \)\(59\!\cdots\!22\)\( T^{7} - \)\(16\!\cdots\!38\)\( T^{8} - \)\(25\!\cdots\!14\)\( T^{9} + \)\(17\!\cdots\!09\)\( T^{10} + \)\(30\!\cdots\!34\)\( T^{11} + \)\(12\!\cdots\!81\)\( T^{12} \))
$97$ (\( ( 1 - 503 T + 912673 T^{2} )^{2} \))(\( ( 1 - 19 T + 2168419 T^{2} + 10094878 T^{3} + 1979057473987 T^{4} - 15826468093651 T^{5} + 760231058654565217 T^{6} )^{2} \))
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