Properties

Label 189.2.i
Level 189
Weight 2
Character orbit i
Rep. character \(\chi_{189}(143,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 12
Newform subspaces 2
Sturm bound 48
Trace bound 1

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Defining parameters

Level: \( N \) = \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 189.i (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 63 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(48\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(189, [\chi])\).

Total New Old
Modular forms 60 20 40
Cusp forms 36 12 24
Eisenstein series 24 8 16

Trace form

\( 12q - 10q^{4} + 3q^{5} - 2q^{7} + O(q^{10}) \) \( 12q - 10q^{4} + 3q^{5} - 2q^{7} - 6q^{10} + 9q^{11} - 3q^{13} - 18q^{14} + 2q^{16} - 9q^{17} - 6q^{19} - 6q^{20} + 2q^{22} + 6q^{23} + 3q^{25} + 6q^{26} - 2q^{28} + 24q^{29} + 6q^{34} - q^{37} - 27q^{38} + 24q^{40} - 6q^{41} + 2q^{43} + 27q^{44} - 4q^{46} - 30q^{47} + 6q^{49} + 9q^{50} - 15q^{52} - 24q^{53} + 24q^{56} - q^{58} + 36q^{59} + 24q^{62} + 4q^{64} + 12q^{67} + 24q^{68} + 15q^{70} - 6q^{73} + 51q^{74} - 48q^{77} - 24q^{79} - 45q^{80} - 30q^{83} + 9q^{85} - 57q^{86} - 11q^{88} + 27q^{89} - 15q^{91} - 30q^{92} - 3q^{97} + 21q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(189, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
189.2.i.a \(2\) \(1.509\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(3\) \(4\) \(q+(-1+2\zeta_{6})q^{2}-q^{4}+(3-3\zeta_{6})q^{5}+\cdots\)
189.2.i.b \(10\) \(1.509\) 10.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(-6\) \(q+(-\beta _{3}-\beta _{5})q^{2}+(-1-2\beta _{2}-\beta _{4}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(189, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(189, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T^{2} + 4 T^{4} \))(\( 1 - 6 T^{2} + 19 T^{4} - 47 T^{6} + 103 T^{8} - 209 T^{10} + 412 T^{12} - 752 T^{14} + 1216 T^{16} - 1536 T^{18} + 1024 T^{20} \))
$3$ 1
$5$ (\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - 16 T^{2} - 12 T^{3} + 129 T^{4} + 147 T^{5} - 726 T^{6} - 720 T^{7} + 3663 T^{8} + 1257 T^{9} - 18069 T^{10} + 6285 T^{11} + 91575 T^{12} - 90000 T^{13} - 453750 T^{14} + 459375 T^{15} + 2015625 T^{16} - 937500 T^{17} - 6250000 T^{18} + 9765625 T^{20} \))
$7$ (\( 1 - 4 T + 7 T^{2} \))(\( 1 + 6 T + 16 T^{2} + T^{3} - 122 T^{4} - 473 T^{5} - 854 T^{6} + 49 T^{7} + 5488 T^{8} + 14406 T^{9} + 16807 T^{10} \))
$11$ (\( 1 + 3 T + 14 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 12 T + 105 T^{2} - 684 T^{3} + 3745 T^{4} - 17382 T^{5} + 72307 T^{6} - 270561 T^{7} + 952558 T^{8} - 3194589 T^{9} + 10673917 T^{10} - 35140479 T^{11} + 115259518 T^{12} - 360116691 T^{13} + 1058646787 T^{14} - 2799388482 T^{15} + 6634495945 T^{16} - 13329224964 T^{17} + 22507682505 T^{18} - 28295372292 T^{19} + 25937424601 T^{20} \))
$13$ (\( ( 1 - 5 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 + 6 T + 68 T^{2} + 336 T^{3} + 2292 T^{4} + 9399 T^{5} + 51837 T^{6} + 187401 T^{7} + 909867 T^{8} + 3004662 T^{9} + 13054461 T^{10} + 39060606 T^{11} + 153767523 T^{12} + 411719997 T^{13} + 1480516557 T^{14} + 3489782907 T^{15} + 11063046228 T^{16} + 21083501712 T^{17} + 55469689028 T^{18} + 63626996238 T^{19} + 137858491849 T^{20} \))
$17$ (\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 + 12 T + 26 T^{2} - 36 T^{3} + 1143 T^{4} + 5247 T^{5} - 21540 T^{6} - 73476 T^{7} + 337539 T^{8} - 599625 T^{9} - 13374333 T^{10} - 10193625 T^{11} + 97548771 T^{12} - 360987588 T^{13} - 1799042340 T^{14} + 7449989679 T^{15} + 27589241367 T^{16} - 14772192228 T^{17} + 181369693466 T^{18} + 1423054517964 T^{19} + 2015993900449 T^{20} \))
$19$ (\( ( 1 + T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 3 T + 71 T^{2} - 204 T^{3} + 2646 T^{4} - 8547 T^{5} + 74607 T^{6} - 258954 T^{7} + 1743726 T^{8} - 5989488 T^{9} + 35261703 T^{10} - 113800272 T^{11} + 629485086 T^{12} - 1776165486 T^{13} + 9722858847 T^{14} - 21163218153 T^{15} + 124483401126 T^{16} - 182349834756 T^{17} + 1205832975911 T^{18} - 968063093337 T^{19} + 6131066257801 T^{20} \))
$23$ (\( 1 + 9 T + 50 T^{2} + 207 T^{3} + 529 T^{4} \))(\( 1 - 15 T + 195 T^{2} - 1800 T^{3} + 15028 T^{4} - 107121 T^{5} + 712306 T^{6} - 4264599 T^{7} + 24178468 T^{8} - 125954103 T^{9} + 628044835 T^{10} - 2896944369 T^{11} + 12790409572 T^{12} - 51887376033 T^{13} + 199332423346 T^{14} - 689467498503 T^{15} + 2224683339892 T^{16} - 6128685804600 T^{17} + 15270642129795 T^{18} - 27017289921945 T^{19} + 41426511213649 T^{20} \))
$29$ (\( 1 - 9 T + 56 T^{2} - 261 T^{3} + 841 T^{4} \))(\( 1 - 15 T + 150 T^{2} - 1125 T^{3} + 6691 T^{4} - 30108 T^{5} + 81631 T^{6} + 232971 T^{7} - 5137202 T^{8} + 44535417 T^{9} - 275752187 T^{10} + 1291527093 T^{11} - 4320386882 T^{12} + 5681929719 T^{13} + 57736055311 T^{14} - 617549674092 T^{15} + 3979962840811 T^{16} - 19406110847625 T^{17} + 75036961944150 T^{18} - 217607189638035 T^{19} + 420707233300201 T^{20} \))
$31$ (\( 1 - 50 T^{2} + 961 T^{4} \))(\( 1 - 217 T^{2} + 22896 T^{4} - 1551294 T^{6} + 74760504 T^{8} - 2674186965 T^{10} + 71844844344 T^{12} - 1432652586174 T^{14} + 20320284280176 T^{16} - 185077355124697 T^{18} + 819628286980801 T^{20} \))
$37$ (\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( 1 - 6 T - 97 T^{2} + 194 T^{3} + 7179 T^{4} + 3556 T^{5} - 323794 T^{6} - 533292 T^{7} + 10739317 T^{8} + 10946526 T^{9} - 345629139 T^{10} + 405021462 T^{11} + 14702124973 T^{12} - 27012839676 T^{13} - 606842086834 T^{14} + 246587111092 T^{15} + 18419349890211 T^{16} + 18416784163802 T^{17} - 340710507030337 T^{18} - 779770438770462 T^{19} + 4808584372417849 T^{20} \))
$41$ (\( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \))(\( 1 + 9 T - 34 T^{2} - 747 T^{3} - 2085 T^{4} + 20394 T^{5} + 110775 T^{6} - 623979 T^{7} - 5992218 T^{8} + 18494757 T^{9} + 381591615 T^{10} + 758285037 T^{11} - 10072918458 T^{12} - 43005256659 T^{13} + 313023674775 T^{14} + 2362771363194 T^{15} - 9903967342485 T^{16} - 145481442589107 T^{17} - 271487457790114 T^{18} + 2946437409545649 T^{19} + 13422659310152401 T^{20} \))
$43$ (\( 1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4} \))(\( 1 - 3 T - 79 T^{2} + 1100 T^{3} + 1674 T^{4} - 79931 T^{5} + 324899 T^{6} + 3229902 T^{7} - 28512986 T^{8} - 52724394 T^{9} + 1438527201 T^{10} - 2267148942 T^{11} - 52720511114 T^{12} + 256799818314 T^{13} + 1110765026099 T^{14} - 11750531857433 T^{15} + 10581961744026 T^{16} + 299000472217700 T^{17} - 923367821930479 T^{18} - 1507777835810529 T^{19} + 21611482313284249 T^{20} \))
$47$ (\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 15 T + 274 T^{2} + 2589 T^{3} + 26755 T^{4} + 176529 T^{5} + 1257485 T^{6} + 5719101 T^{7} + 28447502 T^{8} + 73195215 T^{9} + 229345007 T^{10} )^{2} \))
$53$ (\( 1 + 15 T + 128 T^{2} + 795 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 237 T^{2} + 1890 T^{3} + 28720 T^{4} + 208689 T^{5} + 2523571 T^{6} + 16852668 T^{7} + 178203742 T^{8} + 1088604978 T^{9} + 10361882797 T^{10} + 57696063834 T^{11} + 500574311278 T^{12} + 2508974653836 T^{13} + 19912189027651 T^{14} + 87272799238677 T^{15} + 636560451624880 T^{16} + 2220204054291930 T^{17} + 14755546627492557 T^{18} + 29697872326219197 T^{19} + 174887470365513049 T^{20} \))
$59$ (\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - 18 T + 379 T^{2} - 4179 T^{3} + 48448 T^{4} - 365781 T^{5} + 2858432 T^{6} - 14547099 T^{7} + 77838641 T^{8} - 218112498 T^{9} + 714924299 T^{10} )^{2} \))
$61$ (\( 1 + 70 T^{2} + 3721 T^{4} \))(\( 1 - 358 T^{2} + 54069 T^{4} - 4372581 T^{6} + 210774216 T^{8} - 9493721499 T^{10} + 784290857736 T^{12} - 60542061285621 T^{14} + 2785655121324909 T^{16} - 68631218053026598 T^{18} + 713342911662882601 T^{20} \))
$67$ (\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 - 10 T + 282 T^{2} - 2634 T^{3} + 34317 T^{4} - 263157 T^{5} + 2299239 T^{6} - 11824026 T^{7} + 84815166 T^{8} - 201511210 T^{9} + 1350125107 T^{10} )^{2} \))
$71$ (\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 351 T^{2} + 63037 T^{4} - 7800935 T^{6} + 747809113 T^{8} - 58386380555 T^{10} + 3769705738633 T^{12} - 198234871721735 T^{14} + 8075057597528077 T^{16} - 226659489467262111 T^{18} + 3255243551009881201 T^{20} \))
$73$ (\( 1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4} \))(\( 1 - 3 T + 260 T^{2} - 771 T^{3} + 36639 T^{4} - 155724 T^{5} + 3663555 T^{6} - 21043473 T^{7} + 293486934 T^{8} - 2074196103 T^{9} + 21799556757 T^{10} - 151416315519 T^{11} + 1563991871286 T^{12} - 8186268736041 T^{13} + 104038517806755 T^{14} - 322827000748332 T^{15} + 5544734717002671 T^{16} - 8517544258223787 T^{17} + 209679623892461060 T^{18} - 176614760124803739 T^{19} + 4297625829703557649 T^{20} \))
$79$ (\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 20 T + 354 T^{2} + 3468 T^{3} + 34803 T^{4} + 265629 T^{5} + 2749437 T^{6} + 21643788 T^{7} + 174535806 T^{8} + 779001620 T^{9} + 3077056399 T^{10} )^{2} \))
$83$ (\( 1 + 15 T + 142 T^{2} + 1245 T^{3} + 6889 T^{4} \))(\( 1 + 15 T - 136 T^{2} - 1773 T^{3} + 22674 T^{4} + 93717 T^{5} - 3687774 T^{6} - 10067337 T^{7} + 346135869 T^{8} + 496605294 T^{9} - 27460905396 T^{10} + 41218239402 T^{11} + 2384530001541 T^{12} - 5756372421219 T^{13} - 175015562267454 T^{14} + 369155071940031 T^{15} + 7413046025768706 T^{16} - 48112218404608671 T^{17} - 306311743570909576 T^{18} + 2804103829013106045 T^{19} + 15516041187205853449 T^{20} \))
$89$ (\( 1 - 3 T - 80 T^{2} - 267 T^{3} + 7921 T^{4} \))(\( 1 - 24 T + 44 T^{2} + 2592 T^{3} - 1287 T^{4} - 278721 T^{5} - 235110 T^{6} + 13705920 T^{7} + 186157425 T^{8} - 904992183 T^{9} - 14602879521 T^{10} - 80544304287 T^{11} + 1474552963425 T^{12} + 9662248716480 T^{13} - 14751328281510 T^{14} - 1556394633684729 T^{15} - 639614921466807 T^{16} + 114647620049211168 T^{17} + 173209907450891564 T^{18} - 8408553688979645016 T^{19} + 31181719929966183601 T^{20} \))
$97$ (\( 1 - 3 T + 100 T^{2} - 291 T^{3} + 9409 T^{4} \))(\( 1 + 6 T + 311 T^{2} + 1794 T^{3} + 51903 T^{4} + 385032 T^{5} + 6010353 T^{6} + 63309837 T^{7} + 574248354 T^{8} + 8264282925 T^{9} + 54719955099 T^{10} + 801635443725 T^{11} + 5403102762786 T^{12} + 57781178864301 T^{13} + 532092229646193 T^{14} + 3306400793833224 T^{15} + 43233745971829887 T^{16} + 144952122353734722 T^{17} + 2437441847851234871 T^{18} + 4561386351927391302 T^{19} + 73742412689492826049 T^{20} \))
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