Properties

Label 99.2.g
Level 99
Weight 2
Character orbit g
Rep. character \(\chi_{99}(32,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 20
Newform subspaces 2
Sturm bound 24
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.g (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 99 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(24\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

Total New Old
Modular forms 28 28 0
Cusp forms 20 20 0
Eisenstein series 8 8 0

Trace form

\( 20q - 7q^{3} - 10q^{4} - 9q^{5} + 11q^{9} + O(q^{10}) \) \( 20q - 7q^{3} - 10q^{4} - 9q^{5} + 11q^{9} - 12q^{11} + 14q^{12} - 6q^{14} - 20q^{15} - 10q^{16} + 18q^{20} + 6q^{22} + 12q^{23} - 3q^{25} + 2q^{27} + q^{31} - 4q^{33} + 2q^{36} - 14q^{37} + 66q^{38} - 54q^{42} - 53q^{45} + 6q^{47} + 50q^{48} - 4q^{49} - 2q^{55} - 120q^{56} - 6q^{58} + 9q^{59} + 52q^{60} + 8q^{64} + 54q^{66} - 5q^{67} + 85q^{69} + 15q^{75} + 72q^{77} - 42q^{78} + 23q^{81} + 12q^{82} - 72q^{86} - 6q^{88} - 12q^{91} + 18q^{92} - 71q^{93} + 13q^{97} - 25q^{99} + O(q^{100}) \)

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.g.a \(4\) \(0.791\) \(\Q(\sqrt{-3}, \sqrt{-11})\) \(\Q(\sqrt{-11}) \) \(0\) \(-1\) \(-9\) \(0\) \(q-\beta _{1}q^{3}+(2-2\beta _{2})q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
99.2.g.b \(16\) \(0.791\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(-6\) \(0\) \(0\) \(q+\beta _{1}q^{2}+(\beta _{6}-\beta _{7}+\beta _{9}+\beta _{12}+\beta _{13}+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \))(\( 1 - T^{2} - 6 T^{4} + T^{6} + 8 T^{8} + 30 T^{10} - 35 T^{12} - 100 T^{14} + 393 T^{16} - 400 T^{18} - 560 T^{20} + 1920 T^{22} + 2048 T^{24} + 1024 T^{26} - 24576 T^{28} - 16384 T^{30} + 65536 T^{32} \))
$3$ (\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( ( 1 + 3 T + 3 T^{2} - 3 T^{3} - 15 T^{4} - 9 T^{5} + 27 T^{6} + 81 T^{7} + 81 T^{8} )^{2} \))
$5$ (\( ( 1 + 3 T + 5 T^{2} )^{2}( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} ) \))(\( ( 1 + 13 T^{2} + 88 T^{4} + 72 T^{5} + 430 T^{6} + 729 T^{7} + 1846 T^{8} + 3645 T^{9} + 10750 T^{10} + 9000 T^{11} + 55000 T^{12} + 203125 T^{14} + 390625 T^{16} )^{2} \))
$7$ (\( ( 1 + 7 T^{2} + 49 T^{4} )^{2} \))(\( 1 + 23 T^{2} + 219 T^{4} + 922 T^{6} + 284 T^{8} - 9168 T^{10} + 53788 T^{12} + 1337633 T^{14} + 12397533 T^{16} + 65544017 T^{18} + 129144988 T^{20} - 1078606032 T^{22} + 1637203484 T^{24} + 260442179578 T^{26} + 3031241897019 T^{28} + 15599130675527 T^{30} + 33232930569601 T^{32} \))
$11$ (\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 + 12 T + 88 T^{2} + 480 T^{3} + 2146 T^{4} + 8004 T^{5} + 25888 T^{6} + 79956 T^{7} + 254659 T^{8} + 879516 T^{9} + 3132448 T^{10} + 10653324 T^{11} + 31419586 T^{12} + 77304480 T^{13} + 155897368 T^{14} + 233846052 T^{15} + 214358881 T^{16} \))
$13$ (\( ( 1 + 13 T^{2} + 169 T^{4} )^{2} \))(\( 1 + 53 T^{2} + 1500 T^{4} + 25933 T^{6} + 263753 T^{8} + 468144 T^{10} - 38632730 T^{12} - 890236186 T^{14} - 13299736152 T^{16} - 150449915434 T^{18} - 1103389401530 T^{20} + 2259641672496 T^{22} + 215151424855913 T^{24} + 3575084269120117 T^{26} + 34947127683721500 T^{28} + 208680948442062317 T^{30} + 665416609183179841 T^{32} \))
$17$ (\( ( 1 + 17 T^{2} )^{4} \))(\( ( 1 + 43 T^{2} + 1684 T^{4} + 38266 T^{6} + 797440 T^{8} + 11058874 T^{10} + 140649364 T^{12} + 1037915467 T^{14} + 6975757441 T^{16} )^{2} \))
$19$ (\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 - 89 T^{2} + 4078 T^{4} - 125462 T^{6} + 2786848 T^{8} - 45291782 T^{10} + 531449038 T^{12} - 4187083409 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( ( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} )( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} ) \))(\( ( 1 - 6 T + 67 T^{2} - 330 T^{3} + 1765 T^{4} - 8928 T^{5} + 54913 T^{6} - 259650 T^{7} + 1729831 T^{8} - 5971950 T^{9} + 29048977 T^{10} - 108626976 T^{11} + 493919365 T^{12} - 2123993190 T^{13} + 9918404563 T^{14} - 20428952682 T^{15} + 78310985281 T^{16} )^{2} \))
$29$ (\( ( 1 - 29 T^{2} + 841 T^{4} )^{2} \))(\( 1 - 202 T^{2} + 22161 T^{4} - 1720736 T^{6} + 104609393 T^{8} - 5221521276 T^{10} + 219859920283 T^{12} - 7931725055308 T^{14} + 247189558914249 T^{16} - 6670580771514028 T^{18} + 155502744277680523 T^{20} - 3105882626062477596 T^{22} + 52330473610277542673 T^{24} - \)\(72\!\cdots\!36\)\( T^{26} + \)\(78\!\cdots\!01\)\( T^{28} - \)\(60\!\cdots\!62\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( ( 1 - 5 T + 31 T^{2} )^{2}( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} ) \))(\( ( 1 + 2 T - 105 T^{2} - 140 T^{3} + 6686 T^{4} + 5712 T^{5} - 296792 T^{6} - 74779 T^{7} + 10300464 T^{8} - 2318149 T^{9} - 285217112 T^{10} + 170166192 T^{11} + 6174661406 T^{12} - 4008081140 T^{13} - 93187886505 T^{14} + 55025228222 T^{15} + 852891037441 T^{16} )^{2} \))
$37$ (\( ( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 7 T + 97 T^{2} + 829 T^{3} + 4471 T^{4} + 30673 T^{5} + 132793 T^{6} + 354571 T^{7} + 1874161 T^{8} )^{4} \))
$41$ (\( ( 1 - 41 T^{2} + 1681 T^{4} )^{2} \))(\( 1 - 196 T^{2} + 19095 T^{4} - 1280834 T^{6} + 68413043 T^{8} - 3073917978 T^{10} + 120877869925 T^{12} - 4533161979472 T^{14} + 178294470855873 T^{16} - 7620245287492432 T^{18} + 341571970597137925 T^{20} - 14601430823783944698 T^{22} + \)\(54\!\cdots\!03\)\( T^{24} - \)\(17\!\cdots\!34\)\( T^{26} + \)\(43\!\cdots\!95\)\( T^{28} - \)\(74\!\cdots\!56\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( ( 1 + 43 T^{2} + 1849 T^{4} )^{2} \))(\( 1 + 161 T^{2} + 12045 T^{4} + 451720 T^{6} + 4744646 T^{8} - 258318474 T^{10} + 8737088662 T^{12} + 2425239363077 T^{14} + 153815372427309 T^{16} + 4484267582329373 T^{18} + 29870367454734262 T^{20} - 1632924856417667226 T^{22} + 55456372694318474246 T^{24} + \)\(97\!\cdots\!80\)\( T^{26} + \)\(48\!\cdots\!45\)\( T^{28} + \)\(11\!\cdots\!89\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( ( 1 - 12 T + 47 T^{2} )^{2}( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} ) \))(\( ( 1 + 15 T + 238 T^{2} + 2445 T^{3} + 25363 T^{4} + 223068 T^{5} + 1824427 T^{6} + 13850433 T^{7} + 95884210 T^{8} + 650970351 T^{9} + 4030159243 T^{10} + 23159588964 T^{11} + 123763349203 T^{12} + 560748542115 T^{13} + 2565453248302 T^{14} + 7599346806945 T^{15} + 23811286661761 T^{16} )^{2} \))
$53$ (\( ( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} )( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} ) \))(\( ( 1 - 278 T^{2} + 37843 T^{4} - 3330905 T^{6} + 207574231 T^{8} - 9356512145 T^{10} + 298599472483 T^{12} - 6161692393862 T^{14} + 62259690411361 T^{16} )^{2} \))
$59$ (\( ( 1 - 15 T + 59 T^{2} )^{2}( 1 - 15 T + 166 T^{2} - 885 T^{3} + 3481 T^{4} ) \))(\( ( 1 + 18 T + 307 T^{2} + 3582 T^{3} + 38122 T^{4} + 338940 T^{5} + 2896270 T^{6} + 22582341 T^{7} + 177100792 T^{8} + 1332358119 T^{9} + 10081915870 T^{10} + 69611158260 T^{11} + 461938036042 T^{12} + 2560858839018 T^{13} + 12949423827787 T^{14} + 44795726726742 T^{15} + 146830437604321 T^{16} )^{2} \))
$61$ (\( ( 1 + 61 T^{2} + 3721 T^{4} )^{2} \))(\( 1 + 11 T^{2} - 10329 T^{4} - 236306 T^{6} + 57256364 T^{8} + 1388511372 T^{10} - 209614040984 T^{12} - 2716193993095 T^{14} + 695408199928701 T^{16} - 10106957848306495 T^{18} - 2902282682831947544 T^{20} + 71536625689945733292 T^{22} + \)\(10\!\cdots\!84\)\( T^{24} - \)\(16\!\cdots\!06\)\( T^{26} - \)\(27\!\cdots\!09\)\( T^{28} + \)\(10\!\cdots\!51\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( ( 1 + 13 T + 67 T^{2} )^{2}( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} ) \))(\( ( 1 - 4 T - 165 T^{2} + 1330 T^{3} + 13514 T^{4} - 133584 T^{5} - 335648 T^{6} + 4908017 T^{7} + 931644 T^{8} + 328837139 T^{9} - 1506723872 T^{10} - 40177124592 T^{11} + 272322249194 T^{12} + 1795666392310 T^{13} - 14925633057885 T^{14} - 24242846421292 T^{15} + 406067677556641 T^{16} )^{2} \))
$71$ (\( ( 1 - 3 T - 62 T^{2} - 213 T^{3} + 5041 T^{4} )( 1 + 3 T - 62 T^{2} + 213 T^{3} + 5041 T^{4} ) \))(\( ( 1 - 515 T^{2} + 119305 T^{4} - 16242779 T^{6} + 1421466937 T^{8} - 81879848939 T^{10} + 3031740601705 T^{12} - 65971646219315 T^{14} + 645753531245761 T^{16} )^{2} \))
$73$ (\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 176 T^{2} + 28675 T^{4} - 2789687 T^{6} + 246500362 T^{8} - 14866242023 T^{10} + 814319560675 T^{12} - 26634823826864 T^{14} + 806460091894081 T^{16} )^{2} \))
$79$ (\( ( 1 + 79 T^{2} + 6241 T^{4} )^{2} \))(\( 1 + 353 T^{2} + 60567 T^{4} + 6638158 T^{6} + 550474730 T^{8} + 44065155330 T^{10} + 4241918300434 T^{12} + 431883347780411 T^{14} + 37912769819643561 T^{16} + 2695383973497545051 T^{18} + \)\(16\!\cdots\!54\)\( T^{20} + \)\(10\!\cdots\!30\)\( T^{22} + \)\(83\!\cdots\!30\)\( T^{24} + \)\(62\!\cdots\!58\)\( T^{26} + \)\(35\!\cdots\!47\)\( T^{28} + \)\(13\!\cdots\!93\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( ( 1 - 83 T^{2} + 6889 T^{4} )^{2} \))(\( 1 - 370 T^{2} + 62193 T^{4} - 7697168 T^{6} + 924406361 T^{8} - 98104079436 T^{10} + 8874940013659 T^{12} - 811506409257412 T^{14} + 72427176409933185 T^{16} - 5590467653374311268 T^{18} + \)\(42\!\cdots\!39\)\( T^{20} - \)\(32\!\cdots\!84\)\( T^{22} + \)\(20\!\cdots\!01\)\( T^{24} - \)\(11\!\cdots\!32\)\( T^{26} + \)\(66\!\cdots\!73\)\( T^{28} - \)\(27\!\cdots\!30\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( ( 1 - 9 T + 89 T^{2} )^{2}( 1 + 9 T + 89 T^{2} )^{2} \))(\( ( 1 - 404 T^{2} + 80980 T^{4} - 10737836 T^{6} + 1074473494 T^{8} - 85054398956 T^{10} + 5080866676180 T^{12} - 200780441548244 T^{14} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( ( 1 - 17 T + 97 T^{2} )^{2}( 1 + 17 T + 192 T^{2} + 1649 T^{3} + 9409 T^{4} ) \))(\( ( 1 + 2 T - 264 T^{2} - 224 T^{3} + 37139 T^{4} + 3000 T^{5} - 3977996 T^{6} + 3110 T^{7} + 379878912 T^{8} + 301670 T^{9} - 37428964364 T^{10} + 2738019000 T^{11} + 3287888967059 T^{12} - 1923564217568 T^{13} - 219904609301256 T^{14} + 161596568956226 T^{15} + 7837433594376961 T^{16} )^{2} \))
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