# Properties

 Label 225.2.p Level 225 Weight 2 Character orbit p Rep. character $$\chi_{225}(32,\cdot)$$ Character field $$\Q(\zeta_{12})$$ Dimension 64 Newform subspaces 3 Sturm bound 60 Trace bound 2

# Related objects

## Defining parameters

 Level: $$N$$ = $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 225.p (of order $$12$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$45$$ Character field: $$\Q(\zeta_{12})$$ Newform subspaces: $$3$$ Sturm bound: $$60$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(225, [\chi])$$.

Total New Old
Modular forms 144 80 64
Cusp forms 96 64 32
Eisenstein series 48 16 32

## Trace form

 $$64q + 6q^{2} + 6q^{3} - 24q^{6} + 2q^{7} + O(q^{10})$$ $$64q + 6q^{2} + 6q^{3} - 24q^{6} + 2q^{7} - 36q^{11} + 6q^{12} + 2q^{13} + 16q^{16} - 36q^{18} + 10q^{22} - 18q^{23} - 18q^{27} + 16q^{28} - 4q^{31} - 30q^{32} + 12q^{33} + 24q^{36} - 4q^{37} + 30q^{38} + 12q^{41} - 6q^{42} + 2q^{43} - 112q^{46} + 12q^{47} + 30q^{48} + 24q^{51} + 14q^{52} - 108q^{56} + 6q^{57} + 6q^{58} - 28q^{61} - 36q^{63} - 108q^{66} - 4q^{67} - 42q^{68} - 18q^{72} + 8q^{73} - 12q^{76} + 6q^{77} + 42q^{78} + 132q^{81} - 32q^{82} + 66q^{83} + 240q^{86} + 18q^{87} - 18q^{88} + 32q^{91} + 60q^{92} + 18q^{93} + 48q^{96} - 28q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(225, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
225.2.p.a $$16$$ $$1.797$$ 16.0.$$\cdots$$.9 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{6}-\beta _{13})q^{2}+\beta _{1}q^{3}+(-\beta _{9}+\cdots)q^{4}+\cdots$$
225.2.p.b $$16$$ $$1.797$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$6$$ $$6$$ $$0$$ $$2$$ $$q+\beta _{5}q^{2}+(1-\beta _{9}+\beta _{11}+\beta _{12}+\beta _{13}+\cdots)q^{3}+\cdots$$
225.2.p.c $$32$$ $$1.797$$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(225, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(225, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + 2 T^{4} - 5 T^{8} - 46 T^{12} - 239 T^{16} - 736 T^{20} - 1280 T^{24} + 8192 T^{28} + 65536 T^{32}$$)($$1 - 6 T + 18 T^{2} - 36 T^{3} + 58 T^{4} - 78 T^{5} + 72 T^{6} - 149 T^{8} + 354 T^{9} - 576 T^{10} + 732 T^{11} - 530 T^{12} - 480 T^{13} + 2592 T^{14} - 5496 T^{15} + 8641 T^{16} - 10992 T^{17} + 10368 T^{18} - 3840 T^{19} - 8480 T^{20} + 23424 T^{21} - 36864 T^{22} + 45312 T^{23} - 38144 T^{24} + 73728 T^{26} - 159744 T^{27} + 237568 T^{28} - 294912 T^{29} + 294912 T^{30} - 196608 T^{31} + 65536 T^{32}$$)
$3$ ($$1 - 6 T^{4} - 45 T^{8} - 486 T^{12} + 6561 T^{16}$$)($$1 - 6 T + 18 T^{2} - 30 T^{3} + 30 T^{4} - 36 T^{5} + 126 T^{6} - 396 T^{7} + 819 T^{8} - 1188 T^{9} + 1134 T^{10} - 972 T^{11} + 2430 T^{12} - 7290 T^{13} + 13122 T^{14} - 13122 T^{15} + 6561 T^{16}$$)
$5$ ()()
$7$ ($$1 + 38 T^{4} + 1681 T^{8} - 191482 T^{12} - 9175676 T^{16} - 459748282 T^{20} + 9690630481 T^{24} + 525968913638 T^{28} + 33232930569601 T^{32}$$)($$1 - 2 T + 2 T^{2} + 16 T^{3} - 66 T^{4} + 104 T^{5} + 52 T^{6} - 1072 T^{7} - 235 T^{8} - 2614 T^{9} - 2614 T^{10} - 626 T^{11} + 8430 T^{12} + 20146 T^{13} + 75264 T^{14} + 1168356 T^{15} + 4373152 T^{16} + 8178492 T^{17} + 3687936 T^{18} + 6910078 T^{19} + 20240430 T^{20} - 10521182 T^{21} - 307534486 T^{22} - 2152741402 T^{23} - 1354728235 T^{24} - 43259066704 T^{25} + 14688712948 T^{26} + 205641981272 T^{27} - 913524955266 T^{28} + 1550224166512 T^{29} + 1356446145698 T^{30} - 9495123019886 T^{31} + 33232930569601 T^{32}$$)
$11$ ($$( 1 + 4 T^{2} - 105 T^{4} + 484 T^{6} + 14641 T^{8} )^{4}$$)($$( 1 + 34 T^{2} + 652 T^{4} - 414 T^{5} + 9016 T^{6} - 9072 T^{7} + 100687 T^{8} - 99792 T^{9} + 1090936 T^{10} - 551034 T^{11} + 9545932 T^{12} + 60233074 T^{14} + 214358881 T^{16} )^{2}$$)
$13$ ($$1 + 116 T^{4} + 8266 T^{8} - 6024112 T^{12} - 1173111437 T^{16} - 172054662832 T^{20} + 6742830139786 T^{24} + 2702577874207796 T^{28} + 665416609183179841 T^{32}$$)($$1 - 2 T + 2 T^{2} - 56 T^{3} + 144 T^{4} - 658 T^{5} + 2596 T^{6} - 4150 T^{7} + 35702 T^{8} + 23108 T^{9} - 42562 T^{10} - 1025978 T^{11} - 7929096 T^{12} + 10433170 T^{13} - 40700658 T^{14} + 417980868 T^{15} - 885935405 T^{16} + 5433751284 T^{17} - 6878411202 T^{18} + 22921674490 T^{19} - 226462910856 T^{20} - 380938449554 T^{21} - 205438644658 T^{22} + 1449992730836 T^{23} + 29123218201142 T^{24} - 44008672397950 T^{25} + 357880644840004 T^{26} - 1179241539276346 T^{27} + 3354924257637264 T^{28} - 16961005969166168 T^{29} + 7874752771398578 T^{30} - 102371786028181514 T^{31} + 665416609183179841 T^{32}$$)
$17$ ($$( 1 + 460 T^{4} + 144678 T^{8} + 38419660 T^{12} + 6975757441 T^{16} )^{2}$$)($$1 + 964 T^{4} + 237772 T^{8} - 70157828 T^{12} - 47638889354 T^{16} - 5859651952388 T^{20} + 1658639798261452 T^{24} + 561647836689489604 T^{28} + 48661191875666868481 T^{32}$$)
$19$ ($$( 1 - 56 T^{2} + 1410 T^{4} - 20216 T^{6} + 130321 T^{8} )^{4}$$)($$( 1 - 92 T^{2} + 4132 T^{4} - 123032 T^{6} + 2693650 T^{8} - 44414552 T^{10} + 538486372 T^{12} - 4328221052 T^{14} + 16983563041 T^{16} )^{2}$$)
$23$ ($$1 - 58 T^{4} + 491185 T^{8} + 60755174 T^{12} + 158716957636 T^{16} + 17001788647334 T^{20} + 38465181305247985 T^{24} - 1271048217057178618 T^{28} +$$$$61\!\cdots\!61$$$$T^{32}$$)($$1 + 18 T + 162 T^{2} + 972 T^{3} + 4138 T^{4} + 11184 T^{5} + 3348 T^{6} - 119100 T^{7} - 338219 T^{8} + 2163498 T^{9} + 26656146 T^{10} + 182453766 T^{11} + 892508410 T^{12} + 1883089950 T^{13} - 11227491072 T^{14} - 141260851668 T^{15} - 837597200144 T^{16} - 3248999588364 T^{17} - 5939342777088 T^{18} + 22911555421650 T^{19} + 249760445962810 T^{20} + 1174335019617738 T^{21} + 3946066270423794 T^{22} + 7366333044933606 T^{23} - 26486263130754539 T^{24} - 214517281980243300 T^{25} + 138695959543296852 T^{26} + 10656224332509359568 T^{27} + 90682715899700088298 T^{28} +$$$$48\!\cdots\!76$$$$T^{29} +$$$$18\!\cdots\!58$$$$T^{30} +$$$$47\!\cdots\!26$$$$T^{31} +$$$$61\!\cdots\!61$$$$T^{32}$$)
$29$ ($$( 1 - 31 T^{2} + 120 T^{4} - 26071 T^{6} + 707281 T^{8} )^{4}$$)($$1 - 148 T^{2} + 10806 T^{4} - 562376 T^{6} + 24743345 T^{8} - 973959144 T^{10} + 34914297094 T^{12} - 1155748123996 T^{14} + 35134519733316 T^{16} - 971984172280636 T^{18} + 24694218962941414 T^{20} - 579333612552397224 T^{22} + 12377769580906494545 T^{24} -$$$$23\!\cdots\!76$$$$T^{26} +$$$$38\!\cdots\!46$$$$T^{28} -$$$$44\!\cdots\!88$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$)
$31$ ($$( 1 - 4 T - 44 T^{2} + 8 T^{3} + 2143 T^{4} + 248 T^{5} - 42284 T^{6} - 119164 T^{7} + 923521 T^{8} )^{4}$$)($$( 1 + 2 T - 78 T^{2} + 76 T^{3} + 3500 T^{4} - 7572 T^{5} - 100340 T^{6} + 131474 T^{7} + 2627523 T^{8} + 4075694 T^{9} - 96426740 T^{10} - 225577452 T^{11} + 3232323500 T^{12} + 2175815476 T^{13} - 69225287118 T^{14} + 55025228222 T^{15} + 852891037441 T^{16} )^{2}$$)
$37$ ($$( 1 - 4244 T^{4} + 7905606 T^{8} - 7953939284 T^{12} + 3512479453921 T^{16} )^{2}$$)($$( 1 + 2 T + 2 T^{2} - 50 T^{3} + 680 T^{4} + 6430 T^{5} + 12750 T^{6} + 67506 T^{7} - 1916078 T^{8} + 2497722 T^{9} + 17454750 T^{10} + 325698790 T^{11} + 1274429480 T^{12} - 3467197850 T^{13} + 5131452818 T^{14} + 189863754266 T^{15} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$( 1 + 18 T + 199 T^{2} + 1638 T^{3} + 11028 T^{4} + 67158 T^{5} + 334519 T^{6} + 1240578 T^{7} + 2825761 T^{8} )^{4}$$)($$( 1 + 12 T + 190 T^{2} + 1704 T^{3} + 16861 T^{4} + 130290 T^{5} + 1067482 T^{6} + 7225794 T^{7} + 51027208 T^{8} + 296257554 T^{9} + 1794437242 T^{10} + 8979717090 T^{11} + 47645156221 T^{12} + 197418966504 T^{13} + 902519805790 T^{14} + 2337051286572 T^{15} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$1 - 2284 T^{4} + 1070026 T^{8} + 6146180048 T^{12} - 12328368670157 T^{16} + 21012566494282448 T^{20} + 12506678190240287626 T^{24} -$$$$91\!\cdots\!84$$$$T^{28} +$$$$13\!\cdots\!01$$$$T^{32}$$)($$1 - 2 T + 2 T^{2} + 112 T^{3} - 3840 T^{4} + 7982 T^{5} - 2012 T^{6} - 464698 T^{7} + 6098798 T^{8} - 19506388 T^{9} - 1100914 T^{10} + 890327938 T^{11} - 7456003800 T^{12} + 38444523010 T^{13} - 1944910962 T^{14} - 1611174599916 T^{15} + 14665036034491 T^{16} - 69280507796388 T^{17} - 3596140368738 T^{18} + 3056608690956070 T^{19} - 25490593247443800 T^{20} + 130885723924780534 T^{21} - 6959277079726786 T^{22} - 5302199293874251516 T^{23} + 71283972476632423598 T^{24} -$$$$23\!\cdots\!14$$$$T^{25} - 43482302414327908988 T^{26} +$$$$74\!\cdots\!74$$$$T^{27} -$$$$15\!\cdots\!40$$$$T^{28} +$$$$19\!\cdots\!16$$$$T^{29} +$$$$14\!\cdots\!98$$$$T^{30} -$$$$63\!\cdots\!14$$$$T^{31} +$$$$13\!\cdots\!01$$$$T^{32}$$)
$47$ ($$1 - 5722 T^{4} + 16357201 T^{8} - 37906653562 T^{12} + 84859867638724 T^{16} - 184972377160073722 T^{20} +$$$$38\!\cdots\!61$$$$T^{24} -$$$$66\!\cdots\!02$$$$T^{28} +$$$$56\!\cdots\!21$$$$T^{32}$$)($$1 - 12 T + 72 T^{2} - 288 T^{3} - 134 T^{4} + 24762 T^{5} - 246024 T^{6} + 2286006 T^{7} - 18983651 T^{8} + 98916168 T^{9} - 204929766 T^{10} - 1977304884 T^{11} + 21327903874 T^{12} - 124668663210 T^{13} + 894258211950 T^{14} - 7685415723390 T^{15} + 46309407650224 T^{16} - 361214538999330 T^{17} + 1975416390197550 T^{18} - 12943474620451830 T^{19} + 104073367303784194 T^{20} - 453485002462114188 T^{21} - 2208982075035583014 T^{22} + 50113217696402345784 T^{23} -$$$$45\!\cdots\!11$$$$T^{24} +$$$$25\!\cdots\!02$$$$T^{25} -$$$$12\!\cdots\!76$$$$T^{26} +$$$$61\!\cdots\!86$$$$T^{27} -$$$$15\!\cdots\!94$$$$T^{28} -$$$$15\!\cdots\!76$$$$T^{29} +$$$$18\!\cdots\!68$$$$T^{30} -$$$$14\!\cdots\!16$$$$T^{31} +$$$$56\!\cdots\!21$$$$T^{32}$$)
$53$ ($$( 1 - 1244 T^{4} + 14286246 T^{8} - 9815758364 T^{12} + 62259690411361 T^{16} )^{2}$$)($$1 - 6680 T^{4} + 12266524 T^{8} + 25145244376 T^{12} - 161780081477882 T^{16} + 198408072989184856 T^{20} +$$$$76\!\cdots\!64$$$$T^{24} -$$$$32\!\cdots\!80$$$$T^{28} +$$$$38\!\cdots\!21$$$$T^{32}$$)
$59$ ($$( 1 - 46 T^{2} - 1365 T^{4} - 160126 T^{6} + 12117361 T^{8} )^{4}$$)($$1 - 328 T^{2} + 54732 T^{4} - 6528848 T^{6} + 640897706 T^{8} - 55004293416 T^{10} + 4215408852016 T^{12} - 289798546717912 T^{14} + 17966936709291123 T^{16} - 1008788741125051672 T^{18} + 51079630822473449776 T^{20} -$$$$23\!\cdots\!56$$$$T^{22} +$$$$94\!\cdots\!26$$$$T^{24} -$$$$33\!\cdots\!48$$$$T^{26} +$$$$97\!\cdots\!92$$$$T^{28} -$$$$20\!\cdots\!08$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$)
$61$ ($$( 1 + 10 T - 41 T^{2} + 190 T^{3} + 10060 T^{4} + 11590 T^{5} - 152561 T^{6} + 2269810 T^{7} + 13845841 T^{8} )^{4}$$)($$( 1 - 4 T - 126 T^{2} + 1444 T^{3} + 6365 T^{4} - 126018 T^{5} + 435286 T^{6} + 4572842 T^{7} - 47796264 T^{8} + 278943362 T^{9} + 1619699206 T^{10} - 28603691658 T^{11} + 88128777965 T^{12} + 1219597058644 T^{13} - 6491567169486 T^{14} - 12570971344084 T^{15} + 191707312997281 T^{16} )^{2}$$)
$67$ ($$1 - 5914 T^{4} + 8186161 T^{8} + 79915923398 T^{12} - 393198002490332 T^{16} + 1610395442219829158 T^{20} +$$$$33\!\cdots\!01$$$$T^{24} -$$$$48\!\cdots\!54$$$$T^{28} +$$$$16\!\cdots\!81$$$$T^{32}$$)($$1 + 4 T + 8 T^{2} - 1436 T^{3} - 5070 T^{4} - 45274 T^{5} + 890512 T^{6} + 2479154 T^{7} + 72952373 T^{8} - 410747644 T^{9} - 246911326 T^{10} - 51154442144 T^{11} + 220045950810 T^{12} - 1378484916350 T^{13} + 21635958751422 T^{14} - 133914238479582 T^{15} + 1373516219103616 T^{16} - 8972253978131994 T^{17} + 97123818835133358 T^{18} - 414597258896175050 T^{19} + 4434172580332358010 T^{20} - 69064896673193309408 T^{21} - 22335199089162546094 T^{22} -$$$$24\!\cdots\!12$$$$T^{23} +$$$$29\!\cdots\!93$$$$T^{24} +$$$$67\!\cdots\!38$$$$T^{25} +$$$$16\!\cdots\!88$$$$T^{26} -$$$$55\!\cdots\!42$$$$T^{27} -$$$$41\!\cdots\!70$$$$T^{28} -$$$$78\!\cdots\!32$$$$T^{29} +$$$$29\!\cdots\!32$$$$T^{30} +$$$$98\!\cdots\!72$$$$T^{31} +$$$$16\!\cdots\!81$$$$T^{32}$$)
$71$ ($$( 1 - 32 T^{2} + 2562 T^{4} - 161312 T^{6} + 25411681 T^{8} )^{4}$$)($$( 1 - 452 T^{2} + 95704 T^{4} - 12356144 T^{6} + 1062541066 T^{8} - 62287321904 T^{10} + 2431999518424 T^{12} - 57901328332292 T^{14} + 645753531245761 T^{16} )^{2}$$)
$73$ ($$( 1 + 5329 T^{4} )^{8}$$)($$( 1 - 4 T + 8 T^{2} + 292 T^{3} - 844 T^{4} - 28196 T^{5} + 162168 T^{6} + 417348 T^{7} - 41227034 T^{8} + 30466404 T^{9} + 864193272 T^{10} - 10968723332 T^{11} - 23968115404 T^{12} + 605336905156 T^{13} + 1210673810312 T^{14} - 44189594076388 T^{15} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$( 1 + 116 T^{2} - 854 T^{4} + 212048 T^{6} + 81847123 T^{8} + 1323391568 T^{10} - 33263369174 T^{12} + 28198144840436 T^{14} + 1517108809906561 T^{16} )^{2}$$)($$1 + 260 T^{2} + 23724 T^{4} + 1706392 T^{6} + 225017642 T^{8} + 14869625148 T^{10} - 181450880816 T^{12} - 19987311435172 T^{14} + 2160188956785123 T^{16} - 124740810666908452 T^{18} - 7067526505304546096 T^{20} +$$$$36\!\cdots\!08$$$$T^{22} +$$$$34\!\cdots\!62$$$$T^{24} +$$$$16\!\cdots\!92$$$$T^{26} +$$$$14\!\cdots\!84$$$$T^{28} +$$$$95\!\cdots\!60$$$$T^{30} +$$$$23\!\cdots\!21$$$$T^{32}$$)
$83$ ($$1 - 922 T^{4} - 40638479 T^{8} + 49260688838 T^{12} - 571334931683036 T^{16} + 2337829583554920998 T^{20} -$$$$91\!\cdots\!39$$$$T^{24} -$$$$98\!\cdots\!42$$$$T^{28} +$$$$50\!\cdots\!81$$$$T^{32}$$)($$1 - 66 T + 2178 T^{2} - 47916 T^{3} + 791830 T^{4} - 10596156 T^{5} + 122712084 T^{6} - 1311937572 T^{7} + 13564902253 T^{8} - 136924099746 T^{9} + 1327585690602 T^{10} - 12258029803398 T^{11} + 109901521003222 T^{12} - 994600086849090 T^{13} + 9293464416138048 T^{14} - 88105947208965780 T^{15} + 817933709355268336 T^{16} - 7312793618344159740 T^{17} + 64022676362775012672 T^{18} -$$$$56\!\cdots\!30$$$$T^{19} +$$$$52\!\cdots\!62$$$$T^{20} -$$$$48\!\cdots\!14$$$$T^{21} +$$$$43\!\cdots\!38$$$$T^{22} -$$$$37\!\cdots\!42$$$$T^{23} +$$$$30\!\cdots\!73$$$$T^{24} -$$$$24\!\cdots\!16$$$$T^{25} +$$$$19\!\cdots\!16$$$$T^{26} -$$$$13\!\cdots\!52$$$$T^{27} +$$$$84\!\cdots\!30$$$$T^{28} -$$$$42\!\cdots\!08$$$$T^{29} +$$$$16\!\cdots\!62$$$$T^{30} -$$$$40\!\cdots\!62$$$$T^{31} +$$$$50\!\cdots\!81$$$$T^{32}$$)
$89$ ($$( 1 + 158 T^{2} + 14307 T^{4} + 1251518 T^{6} + 62742241 T^{8} )^{4}$$)($$( 1 + 412 T^{2} + 69850 T^{4} + 6725680 T^{6} + 546021283 T^{8} + 53274111280 T^{10} + 4382545533850 T^{12} + 204756291875932 T^{14} + 3936588805702081 T^{16} )^{2}$$)
$97$ ($$1 + 21116 T^{4} + 182916106 T^{8} + 1814092199408 T^{12} + 21841526617008403 T^{16} +$$$$16\!\cdots\!48$$$$T^{20} +$$$$14\!\cdots\!66$$$$T^{24} +$$$$14\!\cdots\!56$$$$T^{28} +$$$$61\!\cdots\!21$$$$T^{32}$$)($$1 + 28 T + 392 T^{2} - 872 T^{3} - 117924 T^{4} - 2250436 T^{5} - 16405808 T^{6} + 121301108 T^{7} + 5089087514 T^{8} + 63345173552 T^{9} + 270560580248 T^{10} - 4797083097932 T^{11} - 105427115995440 T^{12} - 964302703147436 T^{13} - 1291090828127400 T^{14} + 85355358908186064 T^{15} + 1288991316725561923 T^{16} + 8279469814094048208 T^{17} - 12147873601850706600 T^{18} -$$$$88\!\cdots\!28$$$$T^{19} -$$$$93\!\cdots\!40$$$$T^{20} -$$$$41\!\cdots\!24$$$$T^{21} +$$$$22\!\cdots\!92$$$$T^{22} +$$$$51\!\cdots\!76$$$$T^{23} +$$$$39\!\cdots\!54$$$$T^{24} +$$$$92\!\cdots\!36$$$$T^{25} -$$$$12\!\cdots\!92$$$$T^{26} -$$$$16\!\cdots\!08$$$$T^{27} -$$$$81\!\cdots\!84$$$$T^{28} -$$$$58\!\cdots\!44$$$$T^{29} +$$$$25\!\cdots\!48$$$$T^{30} +$$$$17\!\cdots\!04$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$)