Properties

 Label 45.2.l Level 45 Weight 2 Character orbit l Rep. character $$\chi_{45}(2,\cdot)$$ Character field $$\Q(\zeta_{12})$$ Dimension 16 Newform subspaces 1 Sturm bound 12 Trace bound 0

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Defining parameters

 Level: $$N$$ = $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 45.l (of order $$12$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$45$$ Character field: $$\Q(\zeta_{12})$$ Newform subspaces: $$1$$ Sturm bound: $$12$$ Trace bound: $$0$$

Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(45, [\chi])$$.

Total New Old
Modular forms 32 32 0
Cusp forms 16 16 0
Eisenstein series 16 16 0

Trace form

 $$16q - 6q^{2} - 6q^{3} - 6q^{5} - 2q^{7} + O(q^{10})$$ $$16q - 6q^{2} - 6q^{3} - 6q^{5} - 2q^{7} - 8q^{10} - 6q^{12} - 2q^{13} - 6q^{15} - 8q^{16} + 36q^{18} + 18q^{20} - 12q^{21} - 10q^{22} + 18q^{23} + 4q^{25} + 18q^{27} - 16q^{28} + 30q^{30} - 4q^{31} + 30q^{32} - 12q^{33} - 48q^{36} + 4q^{37} - 30q^{38} + 6q^{40} - 24q^{41} + 6q^{42} - 2q^{43} - 36q^{45} + 32q^{46} - 12q^{47} - 30q^{48} - 54q^{50} + 36q^{51} - 14q^{52} - 16q^{55} + 36q^{56} - 6q^{57} - 6q^{58} + 18q^{60} + 8q^{61} + 36q^{63} + 66q^{65} + 36q^{66} + 4q^{67} + 42q^{68} + 18q^{70} + 18q^{72} - 8q^{73} + 42q^{75} + 24q^{76} - 6q^{77} - 42q^{78} - 48q^{81} + 32q^{82} - 66q^{83} + 22q^{85} - 48q^{86} - 18q^{87} + 18q^{88} - 66q^{90} - 40q^{91} - 60q^{92} - 18q^{93} - 36q^{95} - 24q^{96} + 28q^{97} + O(q^{100})$$

Decomposition of $$S_{2}^{\mathrm{new}}(45, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
45.2.l.a $$16$$ $$0.359$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$-6$$ $$-6$$ $$-6$$ $$-2$$ $$q-\beta _{5}q^{2}+(-1+\beta _{9}-\beta _{11}-\beta _{12}+\cdots)q^{3}+\cdots$$

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 6 T + 18 T^{2} + 36 T^{3} + 58 T^{4} + 78 T^{5} + 72 T^{6} - 149 T^{8} - 354 T^{9} - 576 T^{10} - 732 T^{11} - 530 T^{12} + 480 T^{13} + 2592 T^{14} + 5496 T^{15} + 8641 T^{16} + 10992 T^{17} + 10368 T^{18} + 3840 T^{19} - 8480 T^{20} - 23424 T^{21} - 36864 T^{22} - 45312 T^{23} - 38144 T^{24} + 73728 T^{26} + 159744 T^{27} + 237568 T^{28} + 294912 T^{29} + 294912 T^{30} + 196608 T^{31} + 65536 T^{32}$$
$3$ $$1 + 6 T + 18 T^{2} + 30 T^{3} + 30 T^{4} + 36 T^{5} + 126 T^{6} + 396 T^{7} + 819 T^{8} + 1188 T^{9} + 1134 T^{10} + 972 T^{11} + 2430 T^{12} + 7290 T^{13} + 13122 T^{14} + 13122 T^{15} + 6561 T^{16}$$
$5$ $$1 + 6 T + 16 T^{2} + 24 T^{3} + 22 T^{4} + 30 T^{5} + 136 T^{6} + 630 T^{7} + 1831 T^{8} + 3150 T^{9} + 3400 T^{10} + 3750 T^{11} + 13750 T^{12} + 75000 T^{13} + 250000 T^{14} + 468750 T^{15} + 390625 T^{16}$$
$7$ $$1 + 2 T + 2 T^{2} - 16 T^{3} - 66 T^{4} - 104 T^{5} + 52 T^{6} + 1072 T^{7} - 235 T^{8} + 2614 T^{9} - 2614 T^{10} + 626 T^{11} + 8430 T^{12} - 20146 T^{13} + 75264 T^{14} - 1168356 T^{15} + 4373152 T^{16} - 8178492 T^{17} + 3687936 T^{18} - 6910078 T^{19} + 20240430 T^{20} + 10521182 T^{21} - 307534486 T^{22} + 2152741402 T^{23} - 1354728235 T^{24} + 43259066704 T^{25} + 14688712948 T^{26} - 205641981272 T^{27} - 913524955266 T^{28} - 1550224166512 T^{29} + 1356446145698 T^{30} + 9495123019886 T^{31} + 33232930569601 T^{32}$$
$11$ $$( 1 + 34 T^{2} + 652 T^{4} - 414 T^{5} + 9016 T^{6} - 9072 T^{7} + 100687 T^{8} - 99792 T^{9} + 1090936 T^{10} - 551034 T^{11} + 9545932 T^{12} + 60233074 T^{14} + 214358881 T^{16} )^{2}$$
$13$ $$1 + 2 T + 2 T^{2} + 56 T^{3} + 144 T^{4} + 658 T^{5} + 2596 T^{6} + 4150 T^{7} + 35702 T^{8} - 23108 T^{9} - 42562 T^{10} + 1025978 T^{11} - 7929096 T^{12} - 10433170 T^{13} - 40700658 T^{14} - 417980868 T^{15} - 885935405 T^{16} - 5433751284 T^{17} - 6878411202 T^{18} - 22921674490 T^{19} - 226462910856 T^{20} + 380938449554 T^{21} - 205438644658 T^{22} - 1449992730836 T^{23} + 29123218201142 T^{24} + 44008672397950 T^{25} + 357880644840004 T^{26} + 1179241539276346 T^{27} + 3354924257637264 T^{28} + 16961005969166168 T^{29} + 7874752771398578 T^{30} + 102371786028181514 T^{31} + 665416609183179841 T^{32}$$
$17$ $$1 + 964 T^{4} + 237772 T^{8} - 70157828 T^{12} - 47638889354 T^{16} - 5859651952388 T^{20} + 1658639798261452 T^{24} + 561647836689489604 T^{28} + 48661191875666868481 T^{32}$$
$19$ $$( 1 - 92 T^{2} + 4132 T^{4} - 123032 T^{6} + 2693650 T^{8} - 44414552 T^{10} + 538486372 T^{12} - 4328221052 T^{14} + 16983563041 T^{16} )^{2}$$
$23$ $$1 - 18 T + 162 T^{2} - 972 T^{3} + 4138 T^{4} - 11184 T^{5} + 3348 T^{6} + 119100 T^{7} - 338219 T^{8} - 2163498 T^{9} + 26656146 T^{10} - 182453766 T^{11} + 892508410 T^{12} - 1883089950 T^{13} - 11227491072 T^{14} + 141260851668 T^{15} - 837597200144 T^{16} + 3248999588364 T^{17} - 5939342777088 T^{18} - 22911555421650 T^{19} + 249760445962810 T^{20} - 1174335019617738 T^{21} + 3946066270423794 T^{22} - 7366333044933606 T^{23} - 26486263130754539 T^{24} + 214517281980243300 T^{25} + 138695959543296852 T^{26} - 10656224332509359568 T^{27} + 90682715899700088298 T^{28} -$$$$48\!\cdots\!76$$$$T^{29} +$$$$18\!\cdots\!58$$$$T^{30} -$$$$47\!\cdots\!26$$$$T^{31} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$1 - 148 T^{2} + 10806 T^{4} - 562376 T^{6} + 24743345 T^{8} - 973959144 T^{10} + 34914297094 T^{12} - 1155748123996 T^{14} + 35134519733316 T^{16} - 971984172280636 T^{18} + 24694218962941414 T^{20} - 579333612552397224 T^{22} + 12377769580906494545 T^{24} -$$$$23\!\cdots\!76$$$$T^{26} +$$$$38\!\cdots\!46$$$$T^{28} -$$$$44\!\cdots\!88$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$
$31$ $$( 1 + 2 T - 78 T^{2} + 76 T^{3} + 3500 T^{4} - 7572 T^{5} - 100340 T^{6} + 131474 T^{7} + 2627523 T^{8} + 4075694 T^{9} - 96426740 T^{10} - 225577452 T^{11} + 3232323500 T^{12} + 2175815476 T^{13} - 69225287118 T^{14} + 55025228222 T^{15} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 - 2 T + 2 T^{2} + 50 T^{3} + 680 T^{4} - 6430 T^{5} + 12750 T^{6} - 67506 T^{7} - 1916078 T^{8} - 2497722 T^{9} + 17454750 T^{10} - 325698790 T^{11} + 1274429480 T^{12} + 3467197850 T^{13} + 5131452818 T^{14} - 189863754266 T^{15} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 + 12 T + 190 T^{2} + 1704 T^{3} + 16861 T^{4} + 130290 T^{5} + 1067482 T^{6} + 7225794 T^{7} + 51027208 T^{8} + 296257554 T^{9} + 1794437242 T^{10} + 8979717090 T^{11} + 47645156221 T^{12} + 197418966504 T^{13} + 902519805790 T^{14} + 2337051286572 T^{15} + 7984925229121 T^{16} )^{2}$$
$43$ $$1 + 2 T + 2 T^{2} - 112 T^{3} - 3840 T^{4} - 7982 T^{5} - 2012 T^{6} + 464698 T^{7} + 6098798 T^{8} + 19506388 T^{9} - 1100914 T^{10} - 890327938 T^{11} - 7456003800 T^{12} - 38444523010 T^{13} - 1944910962 T^{14} + 1611174599916 T^{15} + 14665036034491 T^{16} + 69280507796388 T^{17} - 3596140368738 T^{18} - 3056608690956070 T^{19} - 25490593247443800 T^{20} - 130885723924780534 T^{21} - 6959277079726786 T^{22} + 5302199293874251516 T^{23} + 71283972476632423598 T^{24} +$$$$23\!\cdots\!14$$$$T^{25} - 43482302414327908988 T^{26} -$$$$74\!\cdots\!74$$$$T^{27} -$$$$15\!\cdots\!40$$$$T^{28} -$$$$19\!\cdots\!16$$$$T^{29} +$$$$14\!\cdots\!98$$$$T^{30} +$$$$63\!\cdots\!14$$$$T^{31} +$$$$13\!\cdots\!01$$$$T^{32}$$
$47$ $$1 + 12 T + 72 T^{2} + 288 T^{3} - 134 T^{4} - 24762 T^{5} - 246024 T^{6} - 2286006 T^{7} - 18983651 T^{8} - 98916168 T^{9} - 204929766 T^{10} + 1977304884 T^{11} + 21327903874 T^{12} + 124668663210 T^{13} + 894258211950 T^{14} + 7685415723390 T^{15} + 46309407650224 T^{16} + 361214538999330 T^{17} + 1975416390197550 T^{18} + 12943474620451830 T^{19} + 104073367303784194 T^{20} + 453485002462114188 T^{21} - 2208982075035583014 T^{22} - 50113217696402345784 T^{23} -$$$$45\!\cdots\!11$$$$T^{24} -$$$$25\!\cdots\!02$$$$T^{25} -$$$$12\!\cdots\!76$$$$T^{26} -$$$$61\!\cdots\!86$$$$T^{27} -$$$$15\!\cdots\!94$$$$T^{28} +$$$$15\!\cdots\!76$$$$T^{29} +$$$$18\!\cdots\!68$$$$T^{30} +$$$$14\!\cdots\!16$$$$T^{31} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 - 6680 T^{4} + 12266524 T^{8} + 25145244376 T^{12} - 161780081477882 T^{16} + 198408072989184856 T^{20} +$$$$76\!\cdots\!64$$$$T^{24} -$$$$32\!\cdots\!80$$$$T^{28} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$1 - 328 T^{2} + 54732 T^{4} - 6528848 T^{6} + 640897706 T^{8} - 55004293416 T^{10} + 4215408852016 T^{12} - 289798546717912 T^{14} + 17966936709291123 T^{16} - 1008788741125051672 T^{18} + 51079630822473449776 T^{20} -$$$$23\!\cdots\!56$$$$T^{22} +$$$$94\!\cdots\!26$$$$T^{24} -$$$$33\!\cdots\!48$$$$T^{26} +$$$$97\!\cdots\!92$$$$T^{28} -$$$$20\!\cdots\!08$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$( 1 - 4 T - 126 T^{2} + 1444 T^{3} + 6365 T^{4} - 126018 T^{5} + 435286 T^{6} + 4572842 T^{7} - 47796264 T^{8} + 278943362 T^{9} + 1619699206 T^{10} - 28603691658 T^{11} + 88128777965 T^{12} + 1219597058644 T^{13} - 6491567169486 T^{14} - 12570971344084 T^{15} + 191707312997281 T^{16} )^{2}$$
$67$ $$1 - 4 T + 8 T^{2} + 1436 T^{3} - 5070 T^{4} + 45274 T^{5} + 890512 T^{6} - 2479154 T^{7} + 72952373 T^{8} + 410747644 T^{9} - 246911326 T^{10} + 51154442144 T^{11} + 220045950810 T^{12} + 1378484916350 T^{13} + 21635958751422 T^{14} + 133914238479582 T^{15} + 1373516219103616 T^{16} + 8972253978131994 T^{17} + 97123818835133358 T^{18} + 414597258896175050 T^{19} + 4434172580332358010 T^{20} + 69064896673193309408 T^{21} - 22335199089162546094 T^{22} +$$$$24\!\cdots\!12$$$$T^{23} +$$$$29\!\cdots\!93$$$$T^{24} -$$$$67\!\cdots\!38$$$$T^{25} +$$$$16\!\cdots\!88$$$$T^{26} +$$$$55\!\cdots\!42$$$$T^{27} -$$$$41\!\cdots\!70$$$$T^{28} +$$$$78\!\cdots\!32$$$$T^{29} +$$$$29\!\cdots\!32$$$$T^{30} -$$$$98\!\cdots\!72$$$$T^{31} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$( 1 - 452 T^{2} + 95704 T^{4} - 12356144 T^{6} + 1062541066 T^{8} - 62287321904 T^{10} + 2431999518424 T^{12} - 57901328332292 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 + 4 T + 8 T^{2} - 292 T^{3} - 844 T^{4} + 28196 T^{5} + 162168 T^{6} - 417348 T^{7} - 41227034 T^{8} - 30466404 T^{9} + 864193272 T^{10} + 10968723332 T^{11} - 23968115404 T^{12} - 605336905156 T^{13} + 1210673810312 T^{14} + 44189594076388 T^{15} + 806460091894081 T^{16} )^{2}$$
$79$ $$1 + 260 T^{2} + 23724 T^{4} + 1706392 T^{6} + 225017642 T^{8} + 14869625148 T^{10} - 181450880816 T^{12} - 19987311435172 T^{14} + 2160188956785123 T^{16} - 124740810666908452 T^{18} - 7067526505304546096 T^{20} +$$$$36\!\cdots\!08$$$$T^{22} +$$$$34\!\cdots\!62$$$$T^{24} +$$$$16\!\cdots\!92$$$$T^{26} +$$$$14\!\cdots\!84$$$$T^{28} +$$$$95\!\cdots\!60$$$$T^{30} +$$$$23\!\cdots\!21$$$$T^{32}$$
$83$ $$1 + 66 T + 2178 T^{2} + 47916 T^{3} + 791830 T^{4} + 10596156 T^{5} + 122712084 T^{6} + 1311937572 T^{7} + 13564902253 T^{8} + 136924099746 T^{9} + 1327585690602 T^{10} + 12258029803398 T^{11} + 109901521003222 T^{12} + 994600086849090 T^{13} + 9293464416138048 T^{14} + 88105947208965780 T^{15} + 817933709355268336 T^{16} + 7312793618344159740 T^{17} + 64022676362775012672 T^{18} +$$$$56\!\cdots\!30$$$$T^{19} +$$$$52\!\cdots\!62$$$$T^{20} +$$$$48\!\cdots\!14$$$$T^{21} +$$$$43\!\cdots\!38$$$$T^{22} +$$$$37\!\cdots\!42$$$$T^{23} +$$$$30\!\cdots\!73$$$$T^{24} +$$$$24\!\cdots\!16$$$$T^{25} +$$$$19\!\cdots\!16$$$$T^{26} +$$$$13\!\cdots\!52$$$$T^{27} +$$$$84\!\cdots\!30$$$$T^{28} +$$$$42\!\cdots\!08$$$$T^{29} +$$$$16\!\cdots\!62$$$$T^{30} +$$$$40\!\cdots\!62$$$$T^{31} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$( 1 + 412 T^{2} + 69850 T^{4} + 6725680 T^{6} + 546021283 T^{8} + 53274111280 T^{10} + 4382545533850 T^{12} + 204756291875932 T^{14} + 3936588805702081 T^{16} )^{2}$$
$97$ $$1 - 28 T + 392 T^{2} + 872 T^{3} - 117924 T^{4} + 2250436 T^{5} - 16405808 T^{6} - 121301108 T^{7} + 5089087514 T^{8} - 63345173552 T^{9} + 270560580248 T^{10} + 4797083097932 T^{11} - 105427115995440 T^{12} + 964302703147436 T^{13} - 1291090828127400 T^{14} - 85355358908186064 T^{15} + 1288991316725561923 T^{16} - 8279469814094048208 T^{17} - 12147873601850706600 T^{18} +$$$$88\!\cdots\!28$$$$T^{19} -$$$$93\!\cdots\!40$$$$T^{20} +$$$$41\!\cdots\!24$$$$T^{21} +$$$$22\!\cdots\!92$$$$T^{22} -$$$$51\!\cdots\!76$$$$T^{23} +$$$$39\!\cdots\!54$$$$T^{24} -$$$$92\!\cdots\!36$$$$T^{25} -$$$$12\!\cdots\!92$$$$T^{26} +$$$$16\!\cdots\!08$$$$T^{27} -$$$$81\!\cdots\!84$$$$T^{28} +$$$$58\!\cdots\!44$$$$T^{29} +$$$$25\!\cdots\!48$$$$T^{30} -$$$$17\!\cdots\!04$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$
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