Properties

Label 99.2.f
Level 99
Weight 2
Character orbit f
Rep. character \(\chi_{99}(37,\cdot)\)
Character field \(\Q(\zeta_{5})\)
Dimension 16
Newform subspaces 3
Sturm bound 24
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.f (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 3 \)
Sturm bound: \(24\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

Total New Old
Modular forms 64 24 40
Cusp forms 32 16 16
Eisenstein series 32 8 24

Trace form

\( 16q + 4q^{2} - 4q^{4} + 4q^{5} - 8q^{7} - 8q^{8} + O(q^{10}) \) \( 16q + 4q^{2} - 4q^{4} + 4q^{5} - 8q^{7} - 8q^{8} - 24q^{10} + 2q^{11} - 8q^{13} - 2q^{14} + 12q^{16} - 14q^{17} - 8q^{19} - 6q^{20} + 24q^{22} + 12q^{23} + 8q^{25} - 10q^{26} + 24q^{28} + 4q^{29} + 8q^{31} + 48q^{32} - 8q^{34} - 12q^{37} + 10q^{38} - 12q^{40} - 20q^{41} - 16q^{43} - 34q^{44} - 32q^{46} - 14q^{47} - 40q^{49} - 18q^{50} + 16q^{52} - 10q^{53} + 8q^{55} - 12q^{56} + 4q^{58} + 26q^{59} + 24q^{61} + 28q^{62} + 52q^{64} + 28q^{65} + 28q^{68} + 20q^{70} + 12q^{71} + 44q^{73} + 30q^{74} + 8q^{76} + 2q^{77} + 24q^{79} - 22q^{80} - 4q^{82} - 34q^{83} + 44q^{85} - 8q^{86} - 76q^{88} + 4q^{89} + 8q^{91} - 10q^{92} - 8q^{94} - 30q^{95} - 84q^{97} - 40q^{98} + O(q^{100}) \)

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.f.a \(4\) \(0.791\) \(\Q(\zeta_{10})\) None \(1\) \(0\) \(3\) \(1\) \(q+(-1+2\zeta_{10}-2\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
99.2.f.b \(4\) \(0.791\) \(\Q(\zeta_{10})\) None \(3\) \(0\) \(1\) \(-3\) \(q+(1-\zeta_{10}^{3})q^{2}+(1-\zeta_{10})q^{4}+(1-2\zeta_{10}+\cdots)q^{5}+\cdots\)
99.2.f.c \(8\) \(0.791\) 8.0.484000000.9 None \(0\) \(0\) \(0\) \(-6\) \(q+\beta _{1}q^{2}+(1+2\beta _{2}+\beta _{3})q^{4}+(\beta _{4}+\beta _{7})q^{5}+\cdots\)

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

$p$ $F_p(T)$
$2$ (\( 1 - T + 4 T^{2} - 2 T^{3} + 9 T^{4} - 4 T^{5} + 16 T^{6} - 8 T^{7} + 16 T^{8} \))(\( 1 - 3 T + 2 T^{2} + T^{4} + 8 T^{6} - 24 T^{7} + 16 T^{8} \))(\( 1 - 3 T^{2} + 2 T^{6} + 9 T^{8} + 8 T^{10} - 192 T^{14} + 256 T^{16} \))
$3$ 1
$5$ (\( 1 - 3 T - T^{2} + 3 T^{3} + 16 T^{4} + 15 T^{5} - 25 T^{6} - 375 T^{7} + 625 T^{8} \))(\( 1 - T + T^{2} - 11 T^{3} + 36 T^{4} - 55 T^{5} + 25 T^{6} - 125 T^{7} + 625 T^{8} \))(\( 1 - 9 T^{2} + 21 T^{4} + 161 T^{6} - 1644 T^{8} + 4025 T^{10} + 13125 T^{12} - 140625 T^{14} + 390625 T^{16} \))
$7$ (\( 1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} + 91 T^{5} - 294 T^{6} - 343 T^{7} + 2401 T^{8} \))(\( 1 + 3 T + 2 T^{2} - 15 T^{3} - 59 T^{4} - 105 T^{5} + 98 T^{6} + 1029 T^{7} + 2401 T^{8} \))(\( ( 1 + 3 T + 12 T^{2} + 35 T^{3} + 141 T^{4} + 245 T^{5} + 588 T^{6} + 1029 T^{7} + 2401 T^{8} )^{2} \))
$11$ (\( 1 - 11 T + 51 T^{2} - 121 T^{3} + 121 T^{4} \))(\( 1 + 9 T + 41 T^{2} + 99 T^{3} + 121 T^{4} \))(\( 1 - 19 T^{2} + 231 T^{4} - 2299 T^{6} + 14641 T^{8} \))
$13$ (\( 1 - 7 T + 6 T^{2} + 49 T^{3} - 181 T^{4} + 637 T^{5} + 1014 T^{6} - 15379 T^{7} + 28561 T^{8} \))(\( 1 + 9 T + 18 T^{2} - 115 T^{3} - 789 T^{4} - 1495 T^{5} + 3042 T^{6} + 19773 T^{7} + 28561 T^{8} \))(\( ( 1 + 3 T + 6 T^{2} + 59 T^{3} + 339 T^{4} + 767 T^{5} + 1014 T^{6} + 6591 T^{7} + 28561 T^{8} )^{2} \))
$17$ (\( 1 + 12 T + 37 T^{2} - 180 T^{3} - 1619 T^{4} - 3060 T^{5} + 10693 T^{6} + 58956 T^{7} + 83521 T^{8} \))(\( 1 + 2 T - 13 T^{2} + 20 T^{3} + 341 T^{4} + 340 T^{5} - 3757 T^{6} + 9826 T^{7} + 83521 T^{8} \))(\( 1 - 279 T^{4} - 1420 T^{6} + 82101 T^{8} - 410380 T^{10} - 23302359 T^{12} + 6975757441 T^{16} \))
$19$ (\( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 1330 T^{5} + 7581 T^{6} + 68590 T^{7} + 130321 T^{8} \))(\( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 1330 T^{5} + 7581 T^{6} + 68590 T^{7} + 130321 T^{8} \))(\( ( 1 - 6 T - 3 T^{2} + 122 T^{3} - 525 T^{4} + 2318 T^{5} - 1083 T^{6} - 41154 T^{7} + 130321 T^{8} )^{2} \))
$23$ (\( ( 1 - 4 T + 45 T^{2} - 92 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 2 T + 27 T^{2} - 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 20 T^{2} + 753 T^{4} + 10580 T^{6} + 279841 T^{8} )^{2} \))
$29$ (\( 1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 3828 T^{5} + 5887 T^{6} + 146334 T^{7} + 707281 T^{8} \))(\( 1 - 10 T + 31 T^{2} - 200 T^{3} + 1821 T^{4} - 5800 T^{5} + 26071 T^{6} - 243890 T^{7} + 707281 T^{8} \))(\( 1 + 18 T^{2} + 1683 T^{4} + 10856 T^{6} + 1510005 T^{8} + 9129896 T^{10} + 1190353923 T^{12} + 10706819778 T^{14} + 500246412961 T^{16} \))
$31$ (\( 1 + 12 T + 63 T^{2} + 434 T^{3} + 3255 T^{4} + 13454 T^{5} + 60543 T^{6} + 357492 T^{7} + 923521 T^{8} \))(\( 1 - 8 T + 3 T^{2} - 46 T^{3} + 1175 T^{4} - 1426 T^{5} + 2883 T^{6} - 238328 T^{7} + 923521 T^{8} \))(\( ( 1 - 6 T - 15 T^{2} + 206 T^{3} - 561 T^{4} + 6386 T^{5} - 14415 T^{6} - 178746 T^{7} + 923521 T^{8} )^{2} \))
$37$ (\( 1 - 9 T - 6 T^{2} + 307 T^{3} - 1581 T^{4} + 11359 T^{5} - 8214 T^{6} - 455877 T^{7} + 1874161 T^{8} \))(\( 1 + 3 T - 18 T^{2} + 155 T^{3} + 1851 T^{4} + 5735 T^{5} - 24642 T^{6} + 151959 T^{7} + 1874161 T^{8} \))(\( ( 1 + 9 T - 6 T^{2} - 307 T^{3} - 1581 T^{4} - 11359 T^{5} - 8214 T^{6} + 455877 T^{7} + 1874161 T^{8} )^{2} \))
$41$ (\( 1 - 3 T - 22 T^{2} - 171 T^{3} + 2215 T^{4} - 7011 T^{5} - 36982 T^{6} - 206763 T^{7} + 2825761 T^{8} \))(\( 1 + 23 T + 208 T^{2} + 961 T^{3} + 3975 T^{4} + 39401 T^{5} + 349648 T^{6} + 1585183 T^{7} + 2825761 T^{8} \))(\( 1 - 87 T^{2} + 2538 T^{4} + 70841 T^{6} - 7729545 T^{8} + 119083721 T^{10} + 7171781418 T^{12} - 413259068967 T^{14} + 7984925229121 T^{16} \))
$43$ (\( ( 1 + 41 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 8 T + 97 T^{2} - 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 8 T + 57 T^{2} + 344 T^{3} + 1849 T^{4} )^{4} \))
$47$ (\( 1 + 17 T + 67 T^{2} - 335 T^{3} - 4344 T^{4} - 15745 T^{5} + 148003 T^{6} + 1764991 T^{7} + 4879681 T^{8} \))(\( 1 - 3 T - 43 T^{2} + 45 T^{3} + 2116 T^{4} + 2115 T^{5} - 94987 T^{6} - 311469 T^{7} + 4879681 T^{8} \))(\( 1 - 75 T^{2} + 5541 T^{4} - 375925 T^{6} + 17183556 T^{8} - 830418325 T^{10} + 27038312421 T^{12} - 808441149675 T^{14} + 23811286661761 T^{16} \))
$53$ (\( 1 + 4 T - 47 T^{2} - 160 T^{3} + 2121 T^{4} - 8480 T^{5} - 132023 T^{6} + 595508 T^{7} + 7890481 T^{8} \))(\( 1 + 6 T + 23 T^{2} - 120 T^{3} - 1319 T^{4} - 6360 T^{5} + 64607 T^{6} + 893262 T^{7} + 7890481 T^{8} \))(\( 1 + 24 T^{2} + 5817 T^{4} - 20908 T^{6} + 15682005 T^{8} - 58730572 T^{10} + 45898927977 T^{12} + 531944667096 T^{14} + 62259690411361 T^{16} \))
$59$ (\( 1 - 6 T + 17 T^{2} + 222 T^{3} - 1685 T^{4} + 13098 T^{5} + 59177 T^{6} - 1232274 T^{7} + 12117361 T^{8} \))(\( 1 - 20 T + 131 T^{2} - 530 T^{3} + 3851 T^{4} - 31270 T^{5} + 456011 T^{6} - 4107580 T^{7} + 12117361 T^{8} \))(\( 1 - 72 T^{2} - 327 T^{4} + 271796 T^{6} - 16865475 T^{8} + 946121876 T^{10} - 3962377047 T^{12} - 3036998422152 T^{14} + 146830437604321 T^{16} \))
$61$ (\( 1 + 21 T + 245 T^{2} + 2559 T^{3} + 23404 T^{4} + 156099 T^{5} + 911645 T^{6} + 4766601 T^{7} + 13845841 T^{8} \))(\( 1 - 3 T - 7 T^{2} - 441 T^{3} + 4900 T^{4} - 26901 T^{5} - 26047 T^{6} - 680943 T^{7} + 13845841 T^{8} \))(\( ( 1 - 21 T + 135 T^{2} - 259 T^{3} + 144 T^{4} - 15799 T^{5} + 502335 T^{6} - 4766601 T^{7} + 13845841 T^{8} )^{2} \))
$67$ (\( ( 1 + 3 T + 125 T^{2} + 201 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - T + 33 T^{2} - 67 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - T + 123 T^{2} - 67 T^{3} + 4489 T^{4} )^{4} \))
$71$ (\( 1 + 15 T + 119 T^{2} + 1455 T^{3} + 17296 T^{4} + 103305 T^{5} + 599879 T^{6} + 5368665 T^{7} + 25411681 T^{8} \))(\( 1 - 27 T + 253 T^{2} - 819 T^{3} + 100 T^{4} - 58149 T^{5} + 1275373 T^{6} - 9663597 T^{7} + 25411681 T^{8} \))(\( 1 - 117 T^{2} + 273 T^{4} + 566981 T^{6} - 45119220 T^{8} + 2858151221 T^{10} + 6937388913 T^{12} - 14987733218757 T^{14} + 645753531245761 T^{16} \))
$73$ (\( 1 - 14 T + 63 T^{2} - 850 T^{3} + 12521 T^{4} - 62050 T^{5} + 335727 T^{6} - 5446238 T^{7} + 28398241 T^{8} \))(\( 1 - 6 T - 57 T^{2} + 130 T^{3} + 4761 T^{4} + 9490 T^{5} - 303753 T^{6} - 2334102 T^{7} + 28398241 T^{8} \))(\( ( 1 - 12 T - 9 T^{2} + 164 T^{3} + 3249 T^{4} + 11972 T^{5} - 47961 T^{6} - 4668204 T^{7} + 28398241 T^{8} )^{2} \))
$79$ (\( 1 + 11 T + 42 T^{2} - 407 T^{3} - 7795 T^{4} - 32153 T^{5} + 262122 T^{6} + 5423429 T^{7} + 38950081 T^{8} \))(\( 1 - 5 T + 6 T^{2} - 715 T^{3} + 9821 T^{4} - 56485 T^{5} + 37446 T^{6} - 2465195 T^{7} + 38950081 T^{8} \))(\( ( 1 - 15 T + 6 T^{2} + 815 T^{3} - 5979 T^{4} + 64385 T^{5} + 37446 T^{6} - 7395585 T^{7} + 38950081 T^{8} )^{2} \))
$83$ (\( 1 + 13 T - 14 T^{2} - 421 T^{3} + 1449 T^{4} - 34943 T^{5} - 96446 T^{6} + 7433231 T^{7} + 47458321 T^{8} \))(\( 1 + 21 T + 88 T^{2} - 915 T^{3} - 13199 T^{4} - 75945 T^{5} + 606232 T^{6} + 12007527 T^{7} + 47458321 T^{8} \))(\( 1 - 177 T^{2} + 9810 T^{4} + 724223 T^{6} - 144542001 T^{8} + 4989172247 T^{10} + 465566129010 T^{12} - 57868446086313 T^{14} + 2252292232139041 T^{16} \))
$89$ (\( ( 1 - 12 T + 209 T^{2} - 1068 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 10 T + 183 T^{2} + 890 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 4 T^{2} + 5721 T^{4} - 31684 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 - 3 T - 43 T^{2} - 765 T^{3} + 11236 T^{4} - 74205 T^{5} - 404587 T^{6} - 2738019 T^{7} + 88529281 T^{8} \))(\( 1 + 33 T + 537 T^{2} + 6655 T^{3} + 71196 T^{4} + 645535 T^{5} + 5052633 T^{6} + 30118209 T^{7} + 88529281 T^{8} \))(\( ( 1 + 27 T + 357 T^{2} + 4325 T^{3} + 49476 T^{4} + 419525 T^{5} + 3359013 T^{6} + 24642171 T^{7} + 88529281 T^{8} )^{2} \))
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