Properties

Label 99.2.j
Level 99
Weight 2
Character orbit j
Rep. character \(\chi_{99}(8,\cdot)\)
Character field \(\Q(\zeta_{10})\)
Dimension 16
Newform subspaces 1
Sturm bound 24
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.j (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 33 \)
Character field: \(\Q(\zeta_{10})\)
Newform subspaces: \( 1 \)
Sturm bound: \(24\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(99, [\chi])\).

Total New Old
Modular forms 64 16 48
Cusp forms 32 16 16
Eisenstein series 32 0 32

Trace form

\( 16q - 4q^{4} + O(q^{10}) \) \( 16q - 4q^{4} - 20q^{16} - 48q^{22} - 32q^{25} + 40q^{28} + 16q^{31} + 40q^{34} - 12q^{37} + 60q^{40} - 40q^{46} - 24q^{49} - 40q^{52} + 16q^{55} + 12q^{58} + 36q^{64} + 96q^{67} + 76q^{70} - 20q^{73} - 12q^{82} - 100q^{85} - 12q^{88} - 72q^{91} - 80q^{94} + 60q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(99, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
99.2.j.a \(16\) \(0.791\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{8}-\beta _{10}-\beta _{11}+\beta _{12}-\beta _{14}+2\beta _{15})q^{2}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(99, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(99, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(33, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 5 T^{4} - 20 T^{6} + 30 T^{8} - 106 T^{10} + 212 T^{12} - 350 T^{14} + 1105 T^{16} - 1400 T^{18} + 3392 T^{20} - 6784 T^{22} + 7680 T^{24} - 20480 T^{26} + 20480 T^{28} - 32768 T^{30} + 65536 T^{32} \)
$3$ 1
$5$ \( 1 + 26 T^{2} + 267 T^{4} + 1158 T^{6} - 345 T^{8} - 23178 T^{10} - 84123 T^{12} - 137046 T^{14} - 351264 T^{16} - 3426150 T^{18} - 52576875 T^{20} - 362156250 T^{22} - 134765625 T^{24} + 11308593750 T^{26} + 65185546875 T^{28} + 158691406250 T^{30} + 152587890625 T^{32} \)
$7$ \( ( 1 + 13 T^{2} + 20 T^{3} + 80 T^{4} + 260 T^{5} + 783 T^{6} + 2170 T^{7} + 6819 T^{8} + 15190 T^{9} + 38367 T^{10} + 89180 T^{11} + 192080 T^{12} + 336140 T^{13} + 1529437 T^{14} + 5764801 T^{16} )^{2} \)
$11$ \( 1 - 60 T^{2} + 1779 T^{4} - 33610 T^{6} + 439341 T^{8} - 4066810 T^{10} + 26046339 T^{12} - 106293660 T^{14} + 214358881 T^{16} \)
$13$ \( ( 1 + 7 T^{2} + 80 T^{3} + 170 T^{4} + 560 T^{5} + 4587 T^{6} + 13570 T^{7} + 40179 T^{8} + 176410 T^{9} + 775203 T^{10} + 1230320 T^{11} + 4855370 T^{12} + 29703440 T^{13} + 33787663 T^{14} + 815730721 T^{16} )^{2} \)
$17$ \( 1 - 58 T^{2} + 1230 T^{4} + 3270 T^{6} - 669405 T^{8} + 13275246 T^{10} - 49036968 T^{12} - 2836388280 T^{14} + 74928517905 T^{16} - 819716212920 T^{18} - 4095616604328 T^{20} + 320432166316974 T^{22} - 4669606909792605 T^{24} + 6592300054468230 T^{26} + 716625351792606030 T^{28} - 9765913940445253882 T^{30} + 48661191875666868481 T^{32} \)
$19$ \( ( 1 + 34 T^{2} + 20 T^{3} + 395 T^{4} + 680 T^{5} + 1356 T^{6} + 48280 T^{7} - 47451 T^{8} + 917320 T^{9} + 489516 T^{10} + 4664120 T^{11} + 51476795 T^{12} + 49521980 T^{13} + 1599559954 T^{14} + 16983563041 T^{16} )^{2} \)
$23$ \( ( 1 - 102 T^{2} + 5430 T^{4} - 195472 T^{6} + 5177079 T^{8} - 103404688 T^{10} + 1519536630 T^{12} - 15099660678 T^{14} + 78310985281 T^{16} )^{2} \)
$29$ \( 1 + 20 T^{2} + 338 T^{4} + 28080 T^{6} - 44437 T^{8} - 6939980 T^{10} + 33962216 T^{12} - 13624904800 T^{14} - 492015847995 T^{16} - 11458544936800 T^{18} + 24020830094696 T^{20} - 4128061951273580 T^{22} - 22229449852747957 T^{24} + 11813459111069644080 T^{26} + \)\(11\!\cdots\!58\)\( T^{28} + \)\(59\!\cdots\!20\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$ \( ( 1 - 8 T - 24 T^{2} + 88 T^{3} + 1681 T^{4} - 7064 T^{5} + 54924 T^{6} - 168376 T^{7} - 1019363 T^{8} - 5219656 T^{9} + 52781964 T^{10} - 210443624 T^{11} + 1552438801 T^{12} + 2519365288 T^{13} - 21300088344 T^{14} - 220100912888 T^{15} + 852891037441 T^{16} )^{2} \)
$37$ \( ( 1 + 6 T - 7 T^{2} - 118 T^{3} - 254 T^{4} + 3396 T^{5} + 36665 T^{6} - 93294 T^{7} - 1522501 T^{8} - 3451878 T^{9} + 50194385 T^{10} + 172017588 T^{11} - 476036894 T^{12} - 8182586926 T^{13} - 17960084863 T^{14} + 569591262798 T^{15} + 3512479453921 T^{16} )^{2} \)
$41$ \( 1 - 138 T^{2} + 4581 T^{4} + 242158 T^{6} - 19370244 T^{8} + 47020276 T^{10} + 27362707279 T^{12} - 302286948126 T^{14} - 26002484153453 T^{16} - 508144359799806 T^{18} + 77320471083414319 T^{20} + 223351212440590516 T^{22} - \)\(15\!\cdots\!24\)\( T^{24} + \)\(32\!\cdots\!58\)\( T^{26} + \)\(10\!\cdots\!61\)\( T^{28} - \)\(52\!\cdots\!18\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( ( 1 - 180 T^{2} + 12786 T^{4} - 463360 T^{6} + 14361051 T^{8} - 856752640 T^{10} + 43712789586 T^{12} - 1137845348820 T^{14} + 11688200277601 T^{16} )^{2} \)
$47$ \( 1 + 192 T^{2} + 14115 T^{4} + 332470 T^{6} - 12112005 T^{8} - 594955574 T^{10} + 21788151037 T^{12} + 2123156175840 T^{14} + 93108257741080 T^{16} + 4690051992430560 T^{18} + 106319226640379197 T^{20} - 6413154243334793846 T^{22} - \)\(28\!\cdots\!05\)\( T^{24} + \)\(17\!\cdots\!30\)\( T^{26} + \)\(16\!\cdots\!15\)\( T^{28} + \)\(49\!\cdots\!48\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 + 166 T^{2} + 9174 T^{4} - 111674 T^{6} - 30841109 T^{8} - 525703922 T^{10} + 67446933816 T^{12} + 3591171944168 T^{14} + 110175435537457 T^{16} + 10087601991167912 T^{18} + 532188749783405496 T^{20} - 11651891574139647938 T^{22} - \)\(19\!\cdots\!49\)\( T^{24} - \)\(19\!\cdots\!26\)\( T^{26} + \)\(45\!\cdots\!34\)\( T^{28} + \)\(22\!\cdots\!54\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( 1 + 170 T^{2} + 22518 T^{4} + 2498730 T^{6} + 233218683 T^{8} + 18919071570 T^{10} + 1419551700816 T^{12} + 95367637994400 T^{14} + 5833449282172425 T^{16} + 331974747858506400 T^{18} + 17201220416951466576 T^{20} + \)\(79\!\cdots\!70\)\( T^{22} + \)\(34\!\cdots\!43\)\( T^{24} + \)\(12\!\cdots\!30\)\( T^{26} + \)\(40\!\cdots\!58\)\( T^{28} + \)\(10\!\cdots\!70\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( ( 1 + 83 T^{2} + 420 T^{3} + 3413 T^{4} + 34860 T^{5} + 227941 T^{6} + 4235840 T^{7} + 8009300 T^{8} + 258386240 T^{9} + 848168461 T^{10} + 7912557660 T^{11} + 47255855333 T^{12} + 354730446420 T^{13} + 4276191071963 T^{14} + 191707312997281 T^{16} )^{2} \)
$67$ \( ( 1 - 24 T + 429 T^{2} - 5008 T^{3} + 47859 T^{4} - 335536 T^{5} + 1925781 T^{6} - 7218312 T^{7} + 20151121 T^{8} )^{4} \)
$71$ \( 1 + 338 T^{2} + 50091 T^{4} + 3709542 T^{6} + 63034791 T^{8} - 13931687706 T^{10} - 1397230758891 T^{12} - 60140798780574 T^{14} - 2187888491323968 T^{16} - 303169766652873534 T^{18} - 35505982328326005771 T^{20} - \)\(17\!\cdots\!26\)\( T^{22} + \)\(40\!\cdots\!51\)\( T^{24} + \)\(12\!\cdots\!42\)\( T^{26} + \)\(82\!\cdots\!31\)\( T^{28} + \)\(27\!\cdots\!78\)\( T^{30} + \)\(41\!\cdots\!21\)\( T^{32} \)
$73$ \( ( 1 + 10 T + 276 T^{2} + 2170 T^{3} + 24127 T^{4} + 117430 T^{5} - 172242 T^{6} - 4912540 T^{7} - 124201175 T^{8} - 358615420 T^{9} - 917877618 T^{10} + 45682266310 T^{11} + 685164360607 T^{12} + 4498565356810 T^{13} + 41768246455764 T^{14} + 110473985190970 T^{15} + 806460091894081 T^{16} )^{2} \)
$79$ \( ( 1 + 203 T^{2} - 180 T^{3} + 12338 T^{4} - 36540 T^{5} - 807689 T^{6} - 4199410 T^{7} - 150806725 T^{8} - 331753390 T^{9} - 5040787049 T^{10} - 18015645060 T^{11} + 480566099378 T^{12} - 553870151820 T^{13} + 49346753470763 T^{14} + 1517108809906561 T^{16} )^{2} \)
$83$ \( 1 - 208 T^{2} + 15045 T^{4} + 774660 T^{6} - 236047260 T^{8} + 17546406456 T^{10} - 74692835973 T^{12} - 112052982977490 T^{14} + 12843285776364315 T^{16} - 771932999731928610 T^{18} - 3544796586006981333 T^{20} + \)\(57\!\cdots\!64\)\( T^{22} - \)\(53\!\cdots\!60\)\( T^{24} + \)\(12\!\cdots\!40\)\( T^{26} + \)\(16\!\cdots\!45\)\( T^{28} - \)\(15\!\cdots\!32\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( ( 1 - 582 T^{2} + 157158 T^{4} - 25689904 T^{6} + 2778331335 T^{8} - 203489729584 T^{10} + 9860445111078 T^{12} - 289243111339302 T^{14} + 3936588805702081 T^{16} )^{2} \)
$97$ \( ( 1 - 30 T + 411 T^{2} - 3150 T^{3} + 11217 T^{4} + 58230 T^{5} - 1794727 T^{6} + 27808950 T^{7} - 324085500 T^{8} + 2697468150 T^{9} - 16886586343 T^{10} + 53144948790 T^{11} + 993032944977 T^{12} - 27050121809550 T^{13} + 342351494025819 T^{14} - 2423948534343390 T^{15} + 7837433594376961 T^{16} )^{2} \)
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