# 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

# Related objects

## 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

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