Properties

Label 189.2.h
Level 189
Weight 2
Character orbit h
Rep. character \(\chi_{189}(37,\cdot)\)
Character field \(\Q(\zeta_{3})\)
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.h (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 63 \)
Character field: \(\Q(\zeta_{3})\)
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

\( 12q + 2q^{2} + 6q^{4} - 5q^{5} + 12q^{8} + O(q^{10}) \) \( 12q + 2q^{2} + 6q^{4} - 5q^{5} + 12q^{8} - 6q^{10} + q^{11} - 3q^{13} + 16q^{14} - 6q^{16} - 9q^{17} - 4q^{20} - 6q^{22} + 3q^{25} - 16q^{26} - 6q^{28} - 8q^{29} + 6q^{31} - 14q^{32} - 4q^{35} - 3q^{37} - 19q^{38} - 6q^{40} - 10q^{41} - 6q^{43} + 5q^{44} + 54q^{47} - 6q^{49} - 23q^{50} - 15q^{52} + 12q^{53} - 6q^{55} - 6q^{56} - 9q^{58} + 60q^{59} + 12q^{62} - 36q^{64} - 32q^{65} + 12q^{67} - 30q^{68} + 39q^{70} + 30q^{71} + 12q^{73} + 39q^{74} + 6q^{76} + 14q^{77} + 24q^{79} - 19q^{80} - 18q^{83} - 3q^{85} + 7q^{86} - 3q^{88} - 41q^{89} + 21q^{91} - 30q^{92} + 6q^{94} - 26q^{95} - 3q^{97} - 61q^{98} + O(q^{100}) \)

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.h.a \(2\) \(1.509\) \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-1\) \(4\) \(q-q^{2}-q^{4}+(-1+\zeta_{6})q^{5}+(3-2\zeta_{6})q^{7}+\cdots\)
189.2.h.b \(10\) \(1.509\) 10.0.\(\cdots\).1 None \(4\) \(0\) \(-4\) \(-4\) \(q+(\beta _{1}-\beta _{5})q^{2}+(1+\beta _{3})q^{4}+(\beta _{6}-\beta _{9})q^{5}+\cdots\)

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

$p$ $F_p(T)$
$2$ (\( ( 1 + T + 2 T^{2} )^{2} \))(\( ( 1 - 2 T + 5 T^{2} - 7 T^{3} + 13 T^{4} - 15 T^{5} + 26 T^{6} - 28 T^{7} + 40 T^{8} - 32 T^{9} + 32 T^{10} )^{2} \))
$3$ 1
$5$ (\( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} \))(\( 1 + 4 T - 4 T^{2} - 44 T^{3} - 41 T^{4} + 119 T^{5} + 222 T^{6} + 456 T^{7} + 1623 T^{8} - 2021 T^{9} - 16541 T^{10} - 10105 T^{11} + 40575 T^{12} + 57000 T^{13} + 138750 T^{14} + 371875 T^{15} - 640625 T^{16} - 3437500 T^{17} - 1562500 T^{18} + 7812500 T^{19} + 9765625 T^{20} \))
$7$ (\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 12 T^{2} + 47 T^{3} + 146 T^{4} + 309 T^{5} + 1022 T^{6} + 2303 T^{7} + 4116 T^{8} + 9604 T^{9} + 16807 T^{10} \))
$11$ (\( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} \))(\( 1 + 4 T - 31 T^{2} - 134 T^{3} + 607 T^{4} + 2492 T^{5} - 8385 T^{6} - 27495 T^{7} + 98940 T^{8} + 135733 T^{9} - 1043873 T^{10} + 1493063 T^{11} + 11971740 T^{12} - 36595845 T^{13} - 122764785 T^{14} + 401339092 T^{15} + 1075337527 T^{16} - 2611280914 T^{17} - 6645125311 T^{18} + 9431790764 T^{19} + 25937424601 T^{20} \))
$13$ (\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 + 8 T - 14 T^{2} - 182 T^{3} + 686 T^{4} + 4429 T^{5} - 12871 T^{6} - 43199 T^{7} + 305249 T^{8} + 358672 T^{9} - 3841969 T^{10} + 4662736 T^{11} + 51587081 T^{12} - 94908203 T^{13} - 367608631 T^{14} + 1644456697 T^{15} + 3311190974 T^{16} - 11420230094 T^{17} - 11420230094 T^{18} + 84835994984 T^{19} + 137858491849 T^{20} \))
$17$ (\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 + 12 T + 14 T^{2} - 192 T^{3} + 1185 T^{4} + 11847 T^{5} - 6180 T^{6} - 65736 T^{7} + 1002861 T^{8} + 2436261 T^{9} - 7749777 T^{10} + 41416437 T^{11} + 289826829 T^{12} - 322960968 T^{13} - 516159780 T^{14} + 16821045879 T^{15} + 28603019265 T^{16} - 78785025216 T^{17} + 97660604174 T^{18} + 1423054517964 T^{19} + 2015993900449 T^{20} \))
$19$ (\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - T - 53 T^{2} + 190 T^{3} + 1262 T^{4} - 7007 T^{5} - 13111 T^{6} + 116110 T^{7} + 67964 T^{8} - 721616 T^{9} - 440023 T^{10} - 13710704 T^{11} + 24535004 T^{12} + 796398490 T^{13} - 1708638631 T^{14} - 17350025693 T^{15} + 59371901822 T^{16} + 169835630410 T^{17} - 900128841173 T^{18} - 322687697779 T^{19} + 6131066257801 T^{20} \))
$23$ (\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 43 T^{2} - 294 T^{3} + 6 T^{4} + 5127 T^{5} + 21792 T^{6} + 135027 T^{7} + 502362 T^{8} - 3271749 T^{9} - 33095343 T^{10} - 75250227 T^{11} + 265749498 T^{12} + 1642873509 T^{13} + 6098295072 T^{14} + 32999130561 T^{15} + 888215334 T^{16} - 1001018681418 T^{17} - 3367372367083 T^{18} + 5403457984389 T^{19} + 41426511213649 T^{20} \))
$29$ (\( 1 + T - 28 T^{2} + 29 T^{3} + 841 T^{4} \))(\( 1 + 7 T - 76 T^{2} - 419 T^{3} + 4561 T^{4} + 15146 T^{5} - 199563 T^{6} - 341373 T^{7} + 6918636 T^{8} + 2570041 T^{9} - 219913241 T^{10} + 74531189 T^{11} + 5818572876 T^{12} - 8325746097 T^{13} - 141147118203 T^{14} + 310661862754 T^{15} + 2712989167081 T^{16} - 7227698173471 T^{17} - 38018727385036 T^{18} + 101550021831083 T^{19} + 420707233300201 T^{20} \))
$31$ (\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 - 3 T + 134 T^{2} - 308 T^{3} + 7750 T^{4} - 13615 T^{5} + 240250 T^{6} - 295988 T^{7} + 3991994 T^{8} - 2770563 T^{9} + 28629151 T^{10} )^{2} \))
$37$ (\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 1225850552 T^{11} - 13418701163 T^{12} + 116326438008 T^{13} + 50845987930 T^{14} - 3144887137864 T^{15} + 11553466019727 T^{16} + 53161851194480 T^{17} - 312610671398969 T^{18} + 4808584372417849 T^{20} \))
$41$ (\( 1 + 5 T - 16 T^{2} + 205 T^{3} + 1681 T^{4} \))(\( 1 + 5 T - 136 T^{2} - 733 T^{3} + 10507 T^{4} + 54412 T^{5} - 554055 T^{6} - 2345451 T^{7} + 23706084 T^{8} + 41392439 T^{9} - 952045937 T^{10} + 1697089999 T^{11} + 39849927204 T^{12} - 161650828371 T^{13} - 1565627010855 T^{14} + 6303967608812 T^{15} + 49909345260187 T^{16} - 142754882754773 T^{17} - 1085949831160456 T^{18} + 1636909671969805 T^{19} + 13422659310152401 T^{20} \))
$43$ (\( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 + 7 T - 77 T^{2} - 66 T^{3} + 7014 T^{4} - 3843 T^{5} - 95427 T^{6} + 1632678 T^{7} - 3708600 T^{8} - 15416324 T^{9} + 670279801 T^{10} - 662901932 T^{11} - 6857201400 T^{12} + 129809329746 T^{13} - 326245923027 T^{14} - 564953446449 T^{15} + 44338040425686 T^{16} - 17940028333062 T^{17} - 899991421375277 T^{18} + 3518148283557901 T^{19} + 21611482313284249 T^{20} \))
$47$ (\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 27 T + 448 T^{2} - 5169 T^{3} + 48091 T^{4} - 359985 T^{5} + 2260277 T^{6} - 11418321 T^{7} + 46512704 T^{8} - 131751387 T^{9} + 229345007 T^{10} )^{2} \))
$53$ (\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 - 21 T + 41 T^{2} + 924 T^{3} + 12966 T^{4} - 177027 T^{5} - 601755 T^{6} + 3783942 T^{7} + 110973258 T^{8} - 340111866 T^{9} - 4044436041 T^{10} - 18025928898 T^{11} + 311723881722 T^{12} + 563341933134 T^{13} - 4748136394155 T^{14} - 74031893539311 T^{15} + 287383106398614 T^{16} + 1085433093209388 T^{17} + 2552647306865801 T^{18} - 69295035427844793 T^{19} + 174887470365513049 T^{20} \))
$59$ (\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - 30 T + 601 T^{2} - 8193 T^{3} + 88864 T^{4} - 752289 T^{5} + 5242976 T^{6} - 28519833 T^{7} + 123432779 T^{8} - 363520830 T^{9} + 714924299 T^{10} )^{2} \))
$61$ (\( ( 1 + 14 T + 61 T^{2} )^{2} \))(\( ( 1 - 14 T + 339 T^{2} - 3409 T^{3} + 43418 T^{4} - 311709 T^{5} + 2648498 T^{6} - 12684889 T^{7} + 76946559 T^{8} - 193841774 T^{9} + 844596301 T^{10} )^{2} \))
$67$ (\( ( 1 - 4 T + 67 T^{2} )^{2} \))(\( ( 1 - 2 T + 132 T^{2} - 196 T^{3} + 10871 T^{4} - 15429 T^{5} + 728357 T^{6} - 879844 T^{7} + 39700716 T^{8} - 40302242 T^{9} + 1350125107 T^{10} )^{2} \))
$71$ (\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 3 T + 187 T^{2} - 285 T^{3} + 15679 T^{4} - 10143 T^{5} + 1113209 T^{6} - 1436685 T^{7} + 66929357 T^{8} - 76235043 T^{9} + 1804229351 T^{10} )^{2} \))
$73$ (\( 1 + 3 T - 64 T^{2} + 219 T^{3} + 5329 T^{4} \))(\( 1 - 15 T - 134 T^{2} + 2501 T^{3} + 16563 T^{4} - 235276 T^{5} - 2002535 T^{6} + 9021201 T^{7} + 288508378 T^{8} - 238799411 T^{9} - 25271949561 T^{10} - 17432357003 T^{11} + 1537461146362 T^{12} + 3509400549417 T^{13} - 56868471540935 T^{14} - 487743992114668 T^{15} + 2506548790024707 T^{16} + 27629543696261597 T^{17} - 108065652313806854 T^{18} - 883073800624018695 T^{19} + 4297625829703557649 T^{20} \))
$79$ (\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 4 T + 300 T^{2} - 1488 T^{3} + 39873 T^{4} - 184983 T^{5} + 3149967 T^{6} - 9286608 T^{7} + 147911700 T^{8} - 155800324 T^{9} + 3077056399 T^{10} )^{2} \))
$83$ (\( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \))(\( 1 + 9 T - 148 T^{2} + 297 T^{3} + 24654 T^{4} - 118125 T^{5} - 807174 T^{6} + 21382137 T^{7} - 37648479 T^{8} - 452536146 T^{9} + 15509586612 T^{10} - 37560500118 T^{11} - 259360371831 T^{12} + 12226027968819 T^{13} - 38307122794854 T^{14} - 465299175954375 T^{15} + 8060387965039326 T^{16} + 8059407143919219 T^{17} - 333339250356578068 T^{18} + 1682462297407863627 T^{19} + 15516041187205853449 T^{20} \))
$89$ (\( 1 + 13 T + 80 T^{2} + 1157 T^{3} + 7921 T^{4} \))(\( 1 + 28 T + 104 T^{2} - 1736 T^{3} + 31273 T^{4} + 611939 T^{5} - 1780638 T^{6} - 18973932 T^{7} + 740914101 T^{8} + 3271180573 T^{9} - 40614588329 T^{10} + 291135070997 T^{11} + 5868780594021 T^{12} - 13376033868108 T^{13} - 111721218529758 T^{14} + 3417103755161611 T^{15} + 15542095912223353 T^{16} - 76785597378638344 T^{17} + 409405235793016424 T^{18} + 9809979303809585852 T^{19} + 31181719929966183601 T^{20} \))
$97$ (\( 1 - 9 T - 16 T^{2} - 873 T^{3} + 9409 T^{4} \))(\( 1 + 12 T - 197 T^{2} - 1534 T^{3} + 27813 T^{4} + 14090 T^{5} - 4545035 T^{6} - 6881349 T^{7} + 472663750 T^{8} + 908843245 T^{9} - 38512186359 T^{10} + 88157794765 T^{11} + 4447293223750 T^{12} - 6280421435877 T^{13} - 402368680669835 T^{14} + 120995624221130 T^{15} + 23167450373090277 T^{16} - 123944568389425342 T^{17} - 1543974418092261317 T^{18} + 9122772703854782604 T^{19} + 73742412689492826049 T^{20} \))
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