Properties

Label 63.2.h
Level 63
Weight 2
Character orbit h
Rep. character \(\chi_{63}(25,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 12
Newform subspaces 2
Sturm bound 16
Trace bound 1

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Defining parameters

Level: \( N \) = \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 63.h (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 63 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(16\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(63, [\chi])\).

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

Trace form

\( 12q - 2q^{2} - q^{3} + 6q^{4} + 5q^{5} - 2q^{6} - 12q^{8} + 5q^{9} + O(q^{10}) \) \( 12q - 2q^{2} - q^{3} + 6q^{4} + 5q^{5} - 2q^{6} - 12q^{8} + 5q^{9} - 6q^{10} - q^{11} - 20q^{12} - 3q^{13} - 16q^{14} - 16q^{15} - 6q^{16} + 9q^{17} - 2q^{18} + 4q^{20} + 19q^{21} - 6q^{22} + 6q^{24} + 3q^{25} + 16q^{26} - 7q^{27} - 6q^{28} + 8q^{29} + 19q^{30} + 6q^{31} + 14q^{32} + 29q^{33} + 4q^{35} + 40q^{36} - 3q^{37} + 19q^{38} - 13q^{39} - 6q^{40} + 10q^{41} + 2q^{42} - 6q^{43} - 5q^{44} - 19q^{45} - 54q^{47} - 5q^{48} - 6q^{49} + 23q^{50} - 18q^{51} - 15q^{52} - 12q^{53} + q^{54} - 6q^{55} + 6q^{56} - q^{57} - 9q^{58} - 60q^{59} + 7q^{60} - 12q^{62} - 71q^{63} - 36q^{64} + 32q^{65} + 34q^{66} + 12q^{67} + 30q^{68} + 6q^{69} + 39q^{70} - 30q^{71} - 18q^{72} + 12q^{73} - 39q^{74} - 26q^{75} + 6q^{76} - 14q^{77} - 35q^{78} + 24q^{79} + 19q^{80} + 41q^{81} + 18q^{83} + 29q^{84} - 3q^{85} - 7q^{86} + 5q^{87} - 3q^{88} + 41q^{89} + 25q^{90} + 21q^{91} + 30q^{92} - 6q^{93} + 6q^{94} + 26q^{95} + 59q^{96} - 3q^{97} + 61q^{98} + 50q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(63, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
63.2.h.a \(2\) \(0.503\) \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(1\) \(4\) \(q+q^{2}+(1-2\zeta_{6})q^{3}-q^{4}+\zeta_{6}q^{5}+\cdots\)
63.2.h.b \(10\) \(0.503\) 10.0.\(\cdots\).1 None \(-4\) \(-1\) \(4\) \(-4\) \(q+(-\beta _{1}+\beta _{5})q^{2}-\beta _{8}q^{3}+(1-\beta _{4}+\cdots)q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - T + 2 T^{2} )^{2} \))(\( ( 1 + 2 T + 5 T^{2} + 7 T^{3} + 13 T^{4} + 15 T^{5} + 26 T^{6} + 28 T^{7} + 40 T^{8} + 32 T^{9} + 32 T^{10} )^{2} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 + T - 5 T^{2} - 3 T^{3} + 9 T^{4} + 9 T^{5} + 27 T^{6} - 27 T^{7} - 135 T^{8} + 81 T^{9} + 243 T^{10} \))
$5$ (\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 - 4 T - 4 T^{2} + 44 T^{3} - 41 T^{4} - 119 T^{5} + 222 T^{6} - 456 T^{7} + 1623 T^{8} + 2021 T^{9} - 16541 T^{10} + 10105 T^{11} + 40575 T^{12} - 57000 T^{13} + 138750 T^{14} - 371875 T^{15} - 640625 T^{16} + 3437500 T^{17} - 1562500 T^{18} - 7812500 T^{19} + 9765625 T^{20} \))
$7$ (\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 12 T^{2} + 47 T^{3} + 146 T^{4} + 309 T^{5} + 1022 T^{6} + 2303 T^{7} + 4116 T^{8} + 9604 T^{9} + 16807 T^{10} \))
$11$ (\( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} \))(\( 1 - 4 T - 31 T^{2} + 134 T^{3} + 607 T^{4} - 2492 T^{5} - 8385 T^{6} + 27495 T^{7} + 98940 T^{8} - 135733 T^{9} - 1043873 T^{10} - 1493063 T^{11} + 11971740 T^{12} + 36595845 T^{13} - 122764785 T^{14} - 401339092 T^{15} + 1075337527 T^{16} + 2611280914 T^{17} - 6645125311 T^{18} - 9431790764 T^{19} + 25937424601 T^{20} \))
$13$ (\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 + 8 T - 14 T^{2} - 182 T^{3} + 686 T^{4} + 4429 T^{5} - 12871 T^{6} - 43199 T^{7} + 305249 T^{8} + 358672 T^{9} - 3841969 T^{10} + 4662736 T^{11} + 51587081 T^{12} - 94908203 T^{13} - 367608631 T^{14} + 1644456697 T^{15} + 3311190974 T^{16} - 11420230094 T^{17} - 11420230094 T^{18} + 84835994984 T^{19} + 137858491849 T^{20} \))
$17$ (\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - 12 T + 14 T^{2} + 192 T^{3} + 1185 T^{4} - 11847 T^{5} - 6180 T^{6} + 65736 T^{7} + 1002861 T^{8} - 2436261 T^{9} - 7749777 T^{10} - 41416437 T^{11} + 289826829 T^{12} + 322960968 T^{13} - 516159780 T^{14} - 16821045879 T^{15} + 28603019265 T^{16} + 78785025216 T^{17} + 97660604174 T^{18} - 1423054517964 T^{19} + 2015993900449 T^{20} \))
$19$ (\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - T - 53 T^{2} + 190 T^{3} + 1262 T^{4} - 7007 T^{5} - 13111 T^{6} + 116110 T^{7} + 67964 T^{8} - 721616 T^{9} - 440023 T^{10} - 13710704 T^{11} + 24535004 T^{12} + 796398490 T^{13} - 1708638631 T^{14} - 17350025693 T^{15} + 59371901822 T^{16} + 169835630410 T^{17} - 900128841173 T^{18} - 322687697779 T^{19} + 6131066257801 T^{20} \))
$23$ (\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 43 T^{2} + 294 T^{3} + 6 T^{4} - 5127 T^{5} + 21792 T^{6} - 135027 T^{7} + 502362 T^{8} + 3271749 T^{9} - 33095343 T^{10} + 75250227 T^{11} + 265749498 T^{12} - 1642873509 T^{13} + 6098295072 T^{14} - 32999130561 T^{15} + 888215334 T^{16} + 1001018681418 T^{17} - 3367372367083 T^{18} - 5403457984389 T^{19} + 41426511213649 T^{20} \))
$29$ (\( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \))(\( 1 - 7 T - 76 T^{2} + 419 T^{3} + 4561 T^{4} - 15146 T^{5} - 199563 T^{6} + 341373 T^{7} + 6918636 T^{8} - 2570041 T^{9} - 219913241 T^{10} - 74531189 T^{11} + 5818572876 T^{12} + 8325746097 T^{13} - 141147118203 T^{14} - 310661862754 T^{15} + 2712989167081 T^{16} + 7227698173471 T^{17} - 38018727385036 T^{18} - 101550021831083 T^{19} + 420707233300201 T^{20} \))
$31$ (\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 - 3 T + 134 T^{2} - 308 T^{3} + 7750 T^{4} - 13615 T^{5} + 240250 T^{6} - 295988 T^{7} + 3991994 T^{8} - 2770563 T^{9} + 28629151 T^{10} )^{2} \))
$37$ (\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 1225850552 T^{11} - 13418701163 T^{12} + 116326438008 T^{13} + 50845987930 T^{14} - 3144887137864 T^{15} + 11553466019727 T^{16} + 53161851194480 T^{17} - 312610671398969 T^{18} + 4808584372417849 T^{20} \))
$41$ (\( 1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4} \))(\( 1 - 5 T - 136 T^{2} + 733 T^{3} + 10507 T^{4} - 54412 T^{5} - 554055 T^{6} + 2345451 T^{7} + 23706084 T^{8} - 41392439 T^{9} - 952045937 T^{10} - 1697089999 T^{11} + 39849927204 T^{12} + 161650828371 T^{13} - 1565627010855 T^{14} - 6303967608812 T^{15} + 49909345260187 T^{16} + 142754882754773 T^{17} - 1085949831160456 T^{18} - 1636909671969805 T^{19} + 13422659310152401 T^{20} \))
$43$ (\( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 + 7 T - 77 T^{2} - 66 T^{3} + 7014 T^{4} - 3843 T^{5} - 95427 T^{6} + 1632678 T^{7} - 3708600 T^{8} - 15416324 T^{9} + 670279801 T^{10} - 662901932 T^{11} - 6857201400 T^{12} + 129809329746 T^{13} - 326245923027 T^{14} - 564953446449 T^{15} + 44338040425686 T^{16} - 17940028333062 T^{17} - 899991421375277 T^{18} + 3518148283557901 T^{19} + 21611482313284249 T^{20} \))
$47$ (\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 27 T + 448 T^{2} + 5169 T^{3} + 48091 T^{4} + 359985 T^{5} + 2260277 T^{6} + 11418321 T^{7} + 46512704 T^{8} + 131751387 T^{9} + 229345007 T^{10} )^{2} \))
$53$ (\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 + 21 T + 41 T^{2} - 924 T^{3} + 12966 T^{4} + 177027 T^{5} - 601755 T^{6} - 3783942 T^{7} + 110973258 T^{8} + 340111866 T^{9} - 4044436041 T^{10} + 18025928898 T^{11} + 311723881722 T^{12} - 563341933134 T^{13} - 4748136394155 T^{14} + 74031893539311 T^{15} + 287383106398614 T^{16} - 1085433093209388 T^{17} + 2552647306865801 T^{18} + 69295035427844793 T^{19} + 174887470365513049 T^{20} \))
$59$ (\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 30 T + 601 T^{2} + 8193 T^{3} + 88864 T^{4} + 752289 T^{5} + 5242976 T^{6} + 28519833 T^{7} + 123432779 T^{8} + 363520830 T^{9} + 714924299 T^{10} )^{2} \))
$61$ (\( ( 1 + 14 T + 61 T^{2} )^{2} \))(\( ( 1 - 14 T + 339 T^{2} - 3409 T^{3} + 43418 T^{4} - 311709 T^{5} + 2648498 T^{6} - 12684889 T^{7} + 76946559 T^{8} - 193841774 T^{9} + 844596301 T^{10} )^{2} \))
$67$ (\( ( 1 - 4 T + 67 T^{2} )^{2} \))(\( ( 1 - 2 T + 132 T^{2} - 196 T^{3} + 10871 T^{4} - 15429 T^{5} + 728357 T^{6} - 879844 T^{7} + 39700716 T^{8} - 40302242 T^{9} + 1350125107 T^{10} )^{2} \))
$71$ (\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 3 T + 187 T^{2} + 285 T^{3} + 15679 T^{4} + 10143 T^{5} + 1113209 T^{6} + 1436685 T^{7} + 66929357 T^{8} + 76235043 T^{9} + 1804229351 T^{10} )^{2} \))
$73$ (\( 1 + 3 T - 64 T^{2} + 219 T^{3} + 5329 T^{4} \))(\( 1 - 15 T - 134 T^{2} + 2501 T^{3} + 16563 T^{4} - 235276 T^{5} - 2002535 T^{6} + 9021201 T^{7} + 288508378 T^{8} - 238799411 T^{9} - 25271949561 T^{10} - 17432357003 T^{11} + 1537461146362 T^{12} + 3509400549417 T^{13} - 56868471540935 T^{14} - 487743992114668 T^{15} + 2506548790024707 T^{16} + 27629543696261597 T^{17} - 108065652313806854 T^{18} - 883073800624018695 T^{19} + 4297625829703557649 T^{20} \))
$79$ (\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 4 T + 300 T^{2} - 1488 T^{3} + 39873 T^{4} - 184983 T^{5} + 3149967 T^{6} - 9286608 T^{7} + 147911700 T^{8} - 155800324 T^{9} + 3077056399 T^{10} )^{2} \))
$83$ (\( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \))(\( 1 - 9 T - 148 T^{2} - 297 T^{3} + 24654 T^{4} + 118125 T^{5} - 807174 T^{6} - 21382137 T^{7} - 37648479 T^{8} + 452536146 T^{9} + 15509586612 T^{10} + 37560500118 T^{11} - 259360371831 T^{12} - 12226027968819 T^{13} - 38307122794854 T^{14} + 465299175954375 T^{15} + 8060387965039326 T^{16} - 8059407143919219 T^{17} - 333339250356578068 T^{18} - 1682462297407863627 T^{19} + 15516041187205853449 T^{20} \))
$89$ (\( 1 - 13 T + 80 T^{2} - 1157 T^{3} + 7921 T^{4} \))(\( 1 - 28 T + 104 T^{2} + 1736 T^{3} + 31273 T^{4} - 611939 T^{5} - 1780638 T^{6} + 18973932 T^{7} + 740914101 T^{8} - 3271180573 T^{9} - 40614588329 T^{10} - 291135070997 T^{11} + 5868780594021 T^{12} + 13376033868108 T^{13} - 111721218529758 T^{14} - 3417103755161611 T^{15} + 15542095912223353 T^{16} + 76785597378638344 T^{17} + 409405235793016424 T^{18} - 9809979303809585852 T^{19} + 31181719929966183601 T^{20} \))
$97$ (\( 1 - 9 T - 16 T^{2} - 873 T^{3} + 9409 T^{4} \))(\( 1 + 12 T - 197 T^{2} - 1534 T^{3} + 27813 T^{4} + 14090 T^{5} - 4545035 T^{6} - 6881349 T^{7} + 472663750 T^{8} + 908843245 T^{9} - 38512186359 T^{10} + 88157794765 T^{11} + 4447293223750 T^{12} - 6280421435877 T^{13} - 402368680669835 T^{14} + 120995624221130 T^{15} + 23167450373090277 T^{16} - 123944568389425342 T^{17} - 1543974418092261317 T^{18} + 9122772703854782604 T^{19} + 73742412689492826049 T^{20} \))
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