# Properties

 Label 288.2.p Level 288 Weight 2 Character orbit p Rep. character $$\chi_{288}(47,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 20 Newform subspaces 2 Sturm bound 96 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 288.p (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$72$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$2$$ Sturm bound: $$96$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(288, [\chi])$$.

Total New Old
Modular forms 112 28 84
Cusp forms 80 20 60
Eisenstein series 32 8 24

## Trace form

 $$20q + 4q^{3} - 4q^{9} + O(q^{10})$$ $$20q + 4q^{3} - 4q^{9} + 6q^{11} + 8q^{19} - 4q^{25} + 16q^{27} - 2q^{33} - 18q^{41} + 2q^{43} - 4q^{49} - 28q^{51} - 20q^{57} - 30q^{59} - 6q^{65} + 2q^{67} - 8q^{73} - 68q^{75} + 8q^{81} - 54q^{83} + 36q^{91} - 2q^{97} - 10q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(288, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
288.2.p.a $$4$$ $$2.300$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$-2$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{2}-\beta _{3})q^{3}+(1-2\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots$$
288.2.p.b $$16$$ $$2.300$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$6$$ $$0$$ $$0$$ $$q+(-\beta _{1}+\beta _{3}-\beta _{8})q^{3}-\beta _{9}q^{5}-\beta _{4}q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(288, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(288, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 3}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + 2 T + T^{2} + 6 T^{3} + 9 T^{4}$$)($$( 1 - 3 T + 6 T^{2} - 15 T^{3} + 30 T^{4} - 45 T^{5} + 54 T^{6} - 81 T^{7} + 81 T^{8} )^{2}$$)
$5$ ($$( 1 - 5 T^{2} + 25 T^{4} )^{2}$$)($$1 - 13 T^{2} + 48 T^{4} + 103 T^{6} - 1099 T^{8} + 3648 T^{10} - 5222 T^{12} - 177298 T^{14} + 1667616 T^{16} - 4432450 T^{18} - 3263750 T^{20} + 57000000 T^{22} - 429296875 T^{24} + 1005859375 T^{26} + 11718750000 T^{28} - 79345703125 T^{30} + 152587890625 T^{32}$$)
$7$ ($$( 1 + 7 T^{2} + 49 T^{4} )^{2}$$)($$1 + 23 T^{2} + 204 T^{4} + 1399 T^{6} + 12029 T^{8} + 45000 T^{10} - 343382 T^{12} - 4039462 T^{14} - 24407256 T^{16} - 197933638 T^{18} - 824460182 T^{20} + 5294205000 T^{22} + 69344791229 T^{24} + 395182873351 T^{26} + 2823622589004 T^{28} + 15599130675527 T^{30} + 33232930569601 T^{32}$$)
$11$ ($$( 1 - 6 T + 11 T^{2} )^{2}( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )$$)($$( 1 - 3 T + 38 T^{2} - 87 T^{3} + 606 T^{4} - 957 T^{5} + 4598 T^{6} - 3993 T^{7} + 14641 T^{8} )^{2}( 1 + 9 T + 41 T^{2} + 108 T^{3} + 276 T^{4} + 1188 T^{5} + 4961 T^{6} + 11979 T^{7} + 14641 T^{8} )^{2}$$)
$13$ ($$( 1 + 13 T^{2} + 169 T^{4} )^{2}$$)($$1 + 47 T^{2} + 1116 T^{4} + 17239 T^{6} + 168353 T^{8} + 463056 T^{10} - 26916578 T^{12} - 784598686 T^{14} - 12569281176 T^{16} - 132597177934 T^{18} - 768764384258 T^{20} + 2235082868304 T^{22} + 137330714072513 T^{24} + 2376542540984911 T^{26} + 26000662996688796 T^{28} + 185056690127866583 T^{30} + 665416609183179841 T^{32}$$)
$17$ ($$( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} )( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} )$$)($$( 1 - 101 T^{2} + 4882 T^{4} - 146891 T^{6} + 2998666 T^{8} - 42451499 T^{10} + 407749522 T^{12} - 2437894469 T^{14} + 6975757441 T^{16} )^{2}$$)
$19$ ($$( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} )^{2}$$)($$( 1 - T + 64 T^{2} - 49 T^{3} + 1726 T^{4} - 931 T^{5} + 23104 T^{6} - 6859 T^{7} + 130321 T^{8} )^{4}$$)
$23$ ($$( 1 - 23 T^{2} + 529 T^{4} )^{2}$$)($$1 - 85 T^{2} + 3432 T^{4} - 80441 T^{6} + 1017209 T^{8} + 1764696 T^{10} - 386280290 T^{12} + 9323832998 T^{14} - 183562178736 T^{16} + 4932307655942 T^{18} - 108097062633890 T^{20} + 261238341174744 T^{22} + 79658639026700729 T^{24} - 3332389988537139209 T^{26} + 75210991050693741672 T^{28} -$$$$98\!\cdots\!65$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$)
$29$ ($$( 1 - 29 T^{2} + 841 T^{4} )^{2}$$)($$1 - 97 T^{2} + 3780 T^{4} - 80561 T^{6} + 1632089 T^{8} - 54786000 T^{10} + 1038848902 T^{12} + 18974156858 T^{14} - 1329857348520 T^{16} + 15957265917578 T^{18} + 734758090255462 T^{20} - 32587990464306000 T^{22} + 816446667883105529 T^{24} - 33892595421897492761 T^{26} +$$$$13\!\cdots\!80$$$$T^{28} -$$$$28\!\cdots\!57$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$)
$31$ ($$( 1 + 31 T^{2} + 961 T^{4} )^{2}$$)($$1 + 131 T^{2} + 7152 T^{4} + 312151 T^{6} + 16524593 T^{8} + 706250232 T^{10} + 22761450214 T^{12} + 818587141454 T^{14} + 29243993208000 T^{16} + 786662242937294 T^{18} + 21020677263083494 T^{20} + 626799680607103992 T^{22} + 14093677267060286513 T^{24} +$$$$25\!\cdots\!51$$$$T^{26} +$$$$56\!\cdots\!72$$$$T^{28} +$$$$99\!\cdots\!51$$$$T^{30} +$$$$72\!\cdots\!81$$$$T^{32}$$)
$37$ ($$( 1 - 37 T^{2} )^{4}$$)($$( 1 - 140 T^{2} + 8596 T^{4} - 317540 T^{6} + 10470934 T^{8} - 434712260 T^{10} + 16110287956 T^{12} - 359201697260 T^{14} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$( 1 - 6 T + 41 T^{2} )^{2}( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} )$$)($$( 1 + 18 T + 292 T^{2} + 3312 T^{3} + 34963 T^{4} + 303732 T^{5} + 2503888 T^{6} + 17908938 T^{7} + 122450608 T^{8} + 734266458 T^{9} + 4209035728 T^{10} + 20933513172 T^{11} + 98797081843 T^{12} + 383715737712 T^{13} + 1387030438372 T^{14} + 3505576929858 T^{15} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$( 1 - 10 T + 43 T^{2} )^{2}( 1 + 10 T + 57 T^{2} + 430 T^{3} + 1849 T^{4} )$$)($$( 1 + 4 T - 96 T^{2} - 868 T^{3} + 4061 T^{4} + 55182 T^{5} + 78652 T^{6} - 1341518 T^{7} - 8451684 T^{8} - 57685274 T^{9} + 145427548 T^{10} + 4387355274 T^{11} + 13883750861 T^{12} - 127603328524 T^{13} - 606850852704 T^{14} + 1087274444428 T^{15} + 11688200277601 T^{16} )^{2}$$)
$47$ ($$( 1 - 47 T^{2} + 2209 T^{4} )^{2}$$)($$1 - 265 T^{2} + 38172 T^{4} - 3663305 T^{6} + 256619789 T^{8} - 13483362840 T^{10} + 543517455226 T^{12} - 17957421673750 T^{14} + 672442507889160 T^{16} - 39667944477313750 T^{18} + 2652191799434662906 T^{20} -$$$$14\!\cdots\!60$$$$T^{22} +$$$$61\!\cdots\!29$$$$T^{24} -$$$$19\!\cdots\!45$$$$T^{26} +$$$$44\!\cdots\!52$$$$T^{28} -$$$$68\!\cdots\!85$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$)
$53$ ($$( 1 + 53 T^{2} )^{4}$$)($$( 1 + 196 T^{2} + 21556 T^{4} + 1665484 T^{6} + 98315734 T^{8} + 4678344556 T^{10} + 170087208436 T^{12} + 4344214781284 T^{14} + 62259690411361 T^{16} )^{2}$$)
$59$ ($$( 1 + 6 T + 59 T^{2} )^{2}( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} )$$)($$( 1 + 6 T + 178 T^{2} + 996 T^{3} + 15871 T^{4} + 89238 T^{5} + 1194346 T^{6} + 6437052 T^{7} + 79012984 T^{8} + 379786068 T^{9} + 4157518426 T^{10} + 18327611202 T^{11} + 192314636431 T^{12} + 712064601804 T^{13} + 7508134988098 T^{14} + 14931908908914 T^{15} + 146830437604321 T^{16} )^{2}$$)
$61$ ($$( 1 + 61 T^{2} + 3721 T^{4} )^{2}$$)($$1 + 299 T^{2} + 43416 T^{4} + 4465735 T^{6} + 393470093 T^{8} + 31461221184 T^{10} + 2332722797890 T^{12} + 164022774523334 T^{14} + 10603540284680400 T^{16} + 610328744001325814 T^{18} + 32298508956660075490 T^{20} +$$$$16\!\cdots\!24$$$$T^{22} +$$$$75\!\cdots\!33$$$$T^{24} +$$$$31\!\cdots\!35$$$$T^{26} +$$$$11\!\cdots\!36$$$$T^{28} +$$$$29\!\cdots\!59$$$$T^{30} +$$$$36\!\cdots\!61$$$$T^{32}$$)
$67$ ($$( 1 + 14 T + 67 T^{2} )^{2}( 1 - 14 T + 129 T^{2} - 938 T^{3} + 4489 T^{4} )$$)($$( 1 - 8 T - 138 T^{2} + 1052 T^{3} + 11279 T^{4} - 57198 T^{5} - 930218 T^{6} + 1298482 T^{7} + 71382744 T^{8} + 86998294 T^{9} - 4175748602 T^{10} - 17203042074 T^{11} + 227284493759 T^{12} + 1420331612564 T^{13} - 12483256739322 T^{14} - 48485692842584 T^{15} + 406067677556641 T^{16} )^{2}$$)
$71$ ($$( 1 + 71 T^{2} )^{4}$$)($$( 1 + 400 T^{2} + 73324 T^{4} + 8374048 T^{6} + 685441990 T^{8} + 42213575968 T^{10} + 1863286097644 T^{12} + 51240113568400 T^{14} + 645753531245761 T^{16} )^{2}$$)
$73$ ($$( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 + T + 214 T^{2} - 5 T^{3} + 20758 T^{4} - 365 T^{5} + 1140406 T^{6} + 389017 T^{7} + 28398241 T^{8} )^{4}$$)
$79$ ($$( 1 + 79 T^{2} + 6241 T^{4} )^{2}$$)($$1 + 383 T^{2} + 74724 T^{4} + 9593503 T^{6} + 916548293 T^{8} + 73505234952 T^{10} + 5745220732498 T^{12} + 475438917658202 T^{14} + 38877973133064792 T^{16} + 2967214285104838682 T^{18} +$$$$22\!\cdots\!38$$$$T^{20} +$$$$17\!\cdots\!92$$$$T^{22} +$$$$13\!\cdots\!73$$$$T^{24} +$$$$90\!\cdots\!03$$$$T^{26} +$$$$44\!\cdots\!84$$$$T^{28} +$$$$14\!\cdots\!23$$$$T^{30} +$$$$23\!\cdots\!21$$$$T^{32}$$)
$83$ ($$( 1 - 18 T + 241 T^{2} - 1494 T^{3} + 6889 T^{4} )( 1 + 18 T + 241 T^{2} + 1494 T^{3} + 6889 T^{4} )$$)($$( 1 + 27 T + 544 T^{2} + 8127 T^{3} + 107593 T^{4} + 1304076 T^{5} + 14403022 T^{6} + 148987620 T^{7} + 1398163588 T^{8} + 12365972460 T^{9} + 99222418558 T^{10} + 745653703812 T^{11} + 5106183131353 T^{12} + 32012583305661 T^{13} + 177855563112736 T^{14} + 732673376719929 T^{15} + 2252292232139041 T^{16} )^{2}$$)
$89$ ($$( 1 - 18 T + 89 T^{2} )^{2}( 1 + 18 T + 89 T^{2} )^{2}$$)($$( 1 - 440 T^{2} + 100348 T^{4} - 14946776 T^{6} + 1566592486 T^{8} - 118393412696 T^{10} + 6296058399868 T^{12} - 218671768022840 T^{14} + 3936588805702081 T^{16} )^{2}$$)
$97$ ($$( 1 + 10 T + 97 T^{2} )^{2}( 1 - 10 T + 3 T^{2} - 970 T^{3} + 9409 T^{4} )$$)($$( 1 - 4 T - 198 T^{2} - 320 T^{3} + 21017 T^{4} + 106308 T^{5} - 1078406 T^{6} - 6931984 T^{7} + 64836612 T^{8} - 672402448 T^{9} - 10146722054 T^{10} + 97024441284 T^{11} + 1860619898777 T^{12} - 2747948882240 T^{13} - 164928456975942 T^{14} - 323193137912452 T^{15} + 7837433594376961 T^{16} )^{2}$$)