Properties

 Label 576.2.k Level 576 Weight 2 Character orbit k Rep. character $$\chi_{576}(145,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 18 Newform subspaces 3 Sturm bound 192 Trace bound 11

Related objects

Defining parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 576.k (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$16$$ Character field: $$\Q(i)$$ Newform subspaces: $$3$$ Sturm bound: $$192$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$5$$

Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(576, [\chi])$$.

Total New Old
Modular forms 224 22 202
Cusp forms 160 18 142
Eisenstein series 64 4 60

Trace form

 $$18q + 2q^{5} + O(q^{10})$$ $$18q + 2q^{5} - 6q^{11} - 2q^{13} + 4q^{17} + 10q^{19} + 10q^{29} + 16q^{31} + 20q^{35} + 6q^{37} + 22q^{43} + 16q^{47} - 10q^{49} - 6q^{53} + 26q^{59} + 14q^{61} + 12q^{65} - 6q^{67} - 20q^{77} - 32q^{79} - 42q^{83} - 28q^{85} - 52q^{91} - 60q^{95} - 4q^{97} + O(q^{100})$$

Decomposition of $$S_{2}^{\mathrm{new}}(576, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
576.2.k.a $$2$$ $$4.599$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+(1+i)q^{5}+2iq^{7}+(1+i)q^{11}+(-1+\cdots)q^{13}+\cdots$$
576.2.k.b $$8$$ $$4.599$$ 8.0.18939904.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{4}+\beta _{6})q^{5}+(\beta _{1}+\beta _{3}-\beta _{4}-\beta _{5}+\cdots)q^{7}+\cdots$$
576.2.k.c $$8$$ $$4.599$$ 8.0.629407744.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{5}+(\beta _{3}-\beta _{7})q^{7}+(-\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots$$

Decomposition of $$S_{2}^{\mathrm{old}}(576, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(576, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(16, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(48, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(64, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(144, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(192, [\chi])$$$$^{\oplus 2}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ ($$( 1 - 4 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )$$)($$1 + 16 T^{3} - 12 T^{4} - 48 T^{5} + 128 T^{6} + 32 T^{7} - 506 T^{8} + 160 T^{9} + 3200 T^{10} - 6000 T^{11} - 7500 T^{12} + 50000 T^{13} + 390625 T^{16}$$)($$1 - 12 T^{4} - 506 T^{8} - 7500 T^{12} + 390625 T^{16}$$)
$7$ ($$1 - 10 T^{2} + 49 T^{4}$$)($$1 - 24 T^{2} + 292 T^{4} - 2440 T^{6} + 17222 T^{8} - 119560 T^{10} + 701092 T^{12} - 2823576 T^{14} + 5764801 T^{16}$$)($$( 1 - 12 T^{2} + 106 T^{4} - 588 T^{6} + 2401 T^{8} )^{2}$$)
$11$ ($$1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4}$$)($$1 + 8 T + 32 T^{2} + 88 T^{3} + 132 T^{4} + 344 T^{5} + 2400 T^{6} + 13000 T^{7} + 54374 T^{8} + 143000 T^{9} + 290400 T^{10} + 457864 T^{11} + 1932612 T^{12} + 14172488 T^{13} + 56689952 T^{14} + 155897368 T^{15} + 214358881 T^{16}$$)($$1 - 60 T^{4} - 14618 T^{8} - 878460 T^{12} + 214358881 T^{16}$$)
$13$ ($$( 1 - 4 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$)($$1 - 64 T^{3} - 4 T^{4} + 704 T^{5} + 2048 T^{6} - 1408 T^{7} - 53466 T^{8} - 18304 T^{9} + 346112 T^{10} + 1546688 T^{11} - 114244 T^{12} - 23762752 T^{13} + 815730721 T^{16}$$)($$( 1 - 194 T^{4} + 28561 T^{8} )^{2}$$)
$17$ ($$( 1 - 2 T + 17 T^{2} )^{2}$$)($$( 1 + 36 T^{2} - 64 T^{3} + 662 T^{4} - 1088 T^{5} + 10404 T^{6} + 83521 T^{8} )^{2}$$)($$( 1 + 28 T^{2} + 662 T^{4} + 8092 T^{6} + 83521 T^{8} )^{2}$$)
$19$ ($$1 + 6 T + 18 T^{2} + 114 T^{3} + 361 T^{4}$$)($$1 - 8 T + 32 T^{2} - 120 T^{3} + 452 T^{4} - 2168 T^{5} + 10080 T^{6} - 37832 T^{7} + 138918 T^{8} - 718808 T^{9} + 3638880 T^{10} - 14870312 T^{11} + 58905092 T^{12} - 297131880 T^{13} + 1505468192 T^{14} - 7150973912 T^{15} + 16983563041 T^{16}$$)($$( 1 - 4 T + 8 T^{2} - 28 T^{3} - 46 T^{4} - 532 T^{5} + 2888 T^{6} - 27436 T^{7} + 130321 T^{8} )^{2}$$)
$23$ ($$1 - 10 T^{2} + 529 T^{4}$$)($$( 1 - 38 T^{2} + 529 T^{4} )^{4}$$)($$( 1 - 12 T^{2} + 646 T^{4} - 6348 T^{6} + 279841 T^{8} )^{2}$$)
$29$ ($$( 1 - 4 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )$$)($$1 - 16 T + 128 T^{2} - 928 T^{3} + 6580 T^{4} - 38208 T^{5} + 199680 T^{6} - 1073680 T^{7} + 5802054 T^{8} - 31136720 T^{9} + 167930880 T^{10} - 931854912 T^{11} + 4653908980 T^{12} - 19034346272 T^{13} + 76137385088 T^{14} - 275998020944 T^{15} + 500246412961 T^{16}$$)($$1 + 180 T^{4} - 772538 T^{8} + 127310580 T^{12} + 500246412961 T^{16}$$)
$31$ ($$( 1 - 8 T + 31 T^{2} )^{2}$$)($$( 1 + 12 T + 164 T^{2} + 1140 T^{3} + 8218 T^{4} + 35340 T^{5} + 157604 T^{6} + 357492 T^{7} + 923521 T^{8} )^{2}$$)($$( 1 - 6 T + 64 T^{2} - 186 T^{3} + 961 T^{4} )^{4}$$)
$37$ ($$1 - 6 T + 18 T^{2} - 222 T^{3} + 1369 T^{4}$$)($$1 + 16 T + 128 T^{2} + 1008 T^{3} + 5948 T^{4} + 15248 T^{5} - 9344 T^{6} - 717840 T^{7} - 7530650 T^{8} - 26560080 T^{9} - 12791936 T^{10} + 772356944 T^{11} + 11147509628 T^{12} + 69898708656 T^{13} + 328412980352 T^{14} + 1518910034128 T^{15} + 3512479453921 T^{16}$$)($$( 1 - 8 T + 32 T^{2} - 248 T^{3} + 1886 T^{4} - 9176 T^{5} + 43808 T^{6} - 405224 T^{7} + 1874161 T^{8} )^{2}$$)
$41$ ($$( 1 - 41 T^{2} )^{2}$$)($$1 - 200 T^{2} + 19452 T^{4} - 1244536 T^{6} + 58583750 T^{8} - 2092065016 T^{10} + 54966702972 T^{12} - 950020848200 T^{14} + 7984925229121 T^{16}$$)($$( 1 - 60 T^{2} + 4150 T^{4} - 100860 T^{6} + 2825761 T^{8} )^{2}$$)
$43$ ($$1 + 10 T + 50 T^{2} + 430 T^{3} + 1849 T^{4}$$)($$1 - 8 T + 32 T^{2} - 56 T^{3} + 260 T^{4} - 504 T^{5} - 2720 T^{6} + 625528 T^{7} - 7635866 T^{8} + 26897704 T^{9} - 5029280 T^{10} - 40071528 T^{11} + 888888260 T^{12} - 8232472808 T^{13} + 202283617568 T^{14} - 2174548888856 T^{15} + 11688200277601 T^{16}$$)($$( 1 - 12 T + 72 T^{2} - 564 T^{3} + 4402 T^{4} - 24252 T^{5} + 133128 T^{6} - 954084 T^{7} + 3418801 T^{8} )^{2}$$)
$47$ ($$( 1 - 8 T + 47 T^{2} )^{2}$$)($$( 1 + 86 T^{2} + 2209 T^{4} )^{4}$$)($$( 1 - 20 T^{2} + 4070 T^{4} - 44180 T^{6} + 4879681 T^{8} )^{2}$$)
$53$ ($$( 1 - 14 T + 53 T^{2} )( 1 + 4 T + 53 T^{2} )$$)($$1 + 16 T + 128 T^{2} + 928 T^{3} + 8564 T^{4} + 82496 T^{5} + 654336 T^{6} + 5021328 T^{7} + 38116486 T^{8} + 266130384 T^{9} + 1838029824 T^{10} + 12281756992 T^{11} + 67574079284 T^{12} + 388085417504 T^{13} + 2837038224512 T^{14} + 18795378237392 T^{15} + 62259690411361 T^{16}$$)($$1 - 5772 T^{4} + 23807110 T^{8} - 45543856332 T^{12} + 62259690411361 T^{16}$$)
$59$ ($$1 + 6 T + 18 T^{2} + 354 T^{3} + 3481 T^{4}$$)($$( 1 - 8 T + 32 T^{2} - 472 T^{3} + 3481 T^{4} )^{4}$$)($$1 - 7452 T^{4} + 27767206 T^{8} - 90298574172 T^{12} + 146830437604321 T^{16}$$)
$61$ ($$1 + 18 T + 162 T^{2} + 1098 T^{3} + 3721 T^{4}$$)($$1 - 16 T + 128 T^{2} - 1392 T^{3} + 14204 T^{4} - 79760 T^{5} + 426880 T^{6} - 2945904 T^{7} + 19569574 T^{8} - 179700144 T^{9} + 1588420480 T^{10} - 18104004560 T^{11} + 196666325564 T^{12} - 1175678050992 T^{13} + 6594607918208 T^{14} - 50283885376336 T^{15} + 191707312997281 T^{16}$$)($$( 1 - 8 T + 32 T^{2} - 440 T^{3} + 6014 T^{4} - 26840 T^{5} + 119072 T^{6} - 1815848 T^{7} + 13845841 T^{8} )^{2}$$)
$67$ ($$1 - 10 T + 50 T^{2} - 670 T^{3} + 4489 T^{4}$$)($$1 - 16 T + 128 T^{2} - 304 T^{3} + 4388 T^{4} - 107696 T^{5} + 1207680 T^{6} - 4800272 T^{7} + 13154790 T^{8} - 321618224 T^{9} + 5421275520 T^{10} - 32390972048 T^{11} + 88423118948 T^{12} - 410438032528 T^{13} + 11578672917632 T^{14} - 96971385685168 T^{15} + 406067677556641 T^{16}$$)($$( 1 + 8 T + 32 T^{2} + 536 T^{3} + 4489 T^{4} )^{4}$$)
$71$ ($$1 - 42 T^{2} + 5041 T^{4}$$)($$1 - 440 T^{2} + 90844 T^{4} - 11522952 T^{6} + 984512390 T^{8} - 58087201032 T^{10} + 2308498748764 T^{12} - 56364124925240 T^{14} + 645753531245761 T^{16}$$)($$( 1 - 92 T^{2} + 5030 T^{4} - 463772 T^{6} + 25411681 T^{8} )^{2}$$)
$73$ ($$1 - 130 T^{2} + 5329 T^{4}$$)($$1 - 328 T^{2} + 45404 T^{4} - 3734648 T^{6} + 259745542 T^{8} - 19901939192 T^{10} + 1289393734364 T^{12} - 49637626222792 T^{14} + 806460091894081 T^{16}$$)($$( 1 - 228 T^{2} + 23206 T^{4} - 1215012 T^{6} + 28398241 T^{8} )^{2}$$)
$79$ ($$( 1 + 79 T^{2} )^{2}$$)($$( 1 - 12 T + 148 T^{2} + 44 T^{3} + 794 T^{4} + 3476 T^{5} + 923668 T^{6} - 5916468 T^{7} + 38950081 T^{8} )^{2}$$)($$( 1 + 14 T + 200 T^{2} + 1106 T^{3} + 6241 T^{4} )^{4}$$)
$83$ ($$1 + 2 T + 2 T^{2} + 166 T^{3} + 6889 T^{4}$$)($$1 + 40 T + 800 T^{2} + 11000 T^{3} + 122436 T^{4} + 1297720 T^{5} + 14460000 T^{6} + 161033000 T^{7} + 1597489574 T^{8} + 13365739000 T^{9} + 99614940000 T^{10} + 742019425640 T^{11} + 5810606989956 T^{12} + 43329447073000 T^{13} + 261552298695200 T^{14} + 1085442039585080 T^{15} + 2252292232139041 T^{16}$$)($$1 + 20100 T^{4} + 185294374 T^{8} + 953912252100 T^{12} + 2252292232139041 T^{16}$$)
$89$ ($$1 - 162 T^{2} + 7921 T^{4}$$)($$1 - 248 T^{2} + 36316 T^{4} - 4626504 T^{6} + 476004998 T^{8} - 36646538184 T^{10} + 2278547224156 T^{12} - 123251360158328 T^{14} + 3936588805702081 T^{16}$$)($$( 1 - 260 T^{2} + 30950 T^{4} - 2059460 T^{6} + 62742241 T^{8} )^{2}$$)
$97$ ($$( 1 + 2 T + 97 T^{2} )^{2}$$)($$( 1 + 164 T^{2} + 768 T^{3} + 13510 T^{4} + 74496 T^{5} + 1543076 T^{6} + 88529281 T^{8} )^{2}$$)($$( 1 + 82 T^{2} + 9409 T^{4} )^{4}$$)