# Properties

 Label 48.2 Level 48 Weight 2 Dimension 23 Nonzero newspaces 4 Newform subspaces 4 Sturm bound 256 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newform subspaces: $$4$$ Sturm bound: $$256$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(48))$$.

Total New Old
Modular forms 92 31 61
Cusp forms 37 23 14
Eisenstein series 55 8 47

## Trace form

 $$23q - q^{3} - 8q^{4} - 2q^{5} - 8q^{6} - 8q^{7} - 12q^{8} - 5q^{9} + O(q^{10})$$ $$23q - q^{3} - 8q^{4} - 2q^{5} - 8q^{6} - 8q^{7} - 12q^{8} - 5q^{9} - 8q^{10} - 12q^{11} - 10q^{13} + 12q^{14} - 10q^{15} + 16q^{16} + 2q^{17} + 8q^{18} - 16q^{19} + 16q^{20} + 4q^{21} + 16q^{22} + 8q^{23} + 28q^{24} + 9q^{25} + 20q^{26} + 11q^{27} - 10q^{29} + 20q^{30} + 16q^{31} - 8q^{33} - 8q^{34} + 24q^{35} + 16q^{36} - 34q^{37} - 8q^{38} + 18q^{39} - 24q^{40} - 6q^{41} - 44q^{42} - 40q^{44} - 14q^{45} - 48q^{46} - 64q^{48} - 45q^{49} - 36q^{50} + 34q^{51} - 32q^{52} + 14q^{53} - 48q^{54} + 32q^{55} - 8q^{57} + 16q^{58} + 28q^{59} + 8q^{60} + 54q^{61} - 12q^{62} + 8q^{63} + 64q^{64} - 12q^{65} + 52q^{66} + 16q^{67} + 32q^{68} + 28q^{69} + 72q^{70} - 8q^{71} + 36q^{72} + 30q^{73} + 52q^{74} - 19q^{75} + 64q^{76} + 16q^{77} + 48q^{78} - 16q^{79} + 8q^{80} + 7q^{81} + 24q^{82} - 36q^{83} - 8q^{84} + 12q^{85} - 16q^{86} - 54q^{87} - 32q^{88} - 6q^{89} - 24q^{90} - 64q^{91} - 16q^{92} - 16q^{93} - 40q^{94} - 56q^{95} - 56q^{96} - 34q^{97} - 40q^{98} - 64q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(48))$$

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
48.2.a $$\chi_{48}(1, \cdot)$$ 48.2.a.a 1 1
48.2.c $$\chi_{48}(47, \cdot)$$ 48.2.c.a 2 1
48.2.d $$\chi_{48}(25, \cdot)$$ None 0 1
48.2.f $$\chi_{48}(23, \cdot)$$ None 0 1
48.2.j $$\chi_{48}(13, \cdot)$$ 48.2.j.a 8 2
48.2.k $$\chi_{48}(11, \cdot)$$ 48.2.k.a 12 2

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(48))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(48)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ()()($$1 + 2 T^{2} + 4 T^{3} + 2 T^{4} + 8 T^{5} + 8 T^{6} + 16 T^{8}$$)($$1 + 2 T^{2} - 2 T^{4} - 16 T^{6} - 8 T^{8} + 32 T^{10} + 64 T^{12}$$)
$3$ ($$1 - T$$)($$1 + 3 T^{2}$$)($$( 1 + T^{4} )^{2}$$)($$1 + 2 T + 2 T^{2} - 2 T^{3} - 5 T^{4} - 20 T^{5} - 28 T^{6} - 60 T^{7} - 45 T^{8} - 54 T^{9} + 162 T^{10} + 486 T^{11} + 729 T^{12}$$)
$5$ ($$1 + 2 T + 5 T^{2}$$)($$( 1 - 5 T^{2} )^{2}$$)($$1 - 16 T^{3} - 12 T^{4} + 48 T^{5} + 128 T^{6} - 32 T^{7} - 506 T^{8} - 160 T^{9} + 3200 T^{10} + 6000 T^{11} - 7500 T^{12} - 50000 T^{13} + 390625 T^{16}$$)($$1 - 30 T^{4} - 49 T^{8} + 12796 T^{12} - 30625 T^{16} - 11718750 T^{20} + 244140625 T^{24}$$)
$7$ ($$1 + 7 T^{2}$$)($$( 1 - 4 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$)($$1 - 24 T^{2} + 292 T^{4} - 2440 T^{6} + 17222 T^{8} - 119560 T^{10} + 701092 T^{12} - 2823576 T^{14} + 5764801 T^{16}$$)($$( 1 + 2 T + 15 T^{2} + 20 T^{3} + 105 T^{4} + 98 T^{5} + 343 T^{6} )^{4}$$)
$11$ ($$1 + 4 T + 11 T^{2}$$)($$( 1 + 11 T^{2} )^{2}$$)($$1 + 8 T + 32 T^{2} + 88 T^{3} + 132 T^{4} + 344 T^{5} + 2400 T^{6} + 13000 T^{7} + 54374 T^{8} + 143000 T^{9} + 290400 T^{10} + 457864 T^{11} + 1932612 T^{12} + 14172488 T^{13} + 56689952 T^{14} + 155897368 T^{15} + 214358881 T^{16}$$)($$1 - 62 T^{4} + 11023 T^{8} - 2631620 T^{12} + 161387743 T^{16} - 13290250622 T^{20} + 3138428376721 T^{24}$$)
$13$ ($$1 + 2 T + 13 T^{2}$$)($$( 1 + 2 T + 13 T^{2} )^{2}$$)($$1 - 64 T^{3} - 4 T^{4} + 704 T^{5} + 2048 T^{6} - 1408 T^{7} - 53466 T^{8} - 18304 T^{9} + 346112 T^{10} + 1546688 T^{11} - 114244 T^{12} - 23762752 T^{13} + 815730721 T^{16}$$)($$( 1 + 2 T + 2 T^{2} - 6 T^{3} - 25 T^{4} + 412 T^{5} + 892 T^{6} + 5356 T^{7} - 4225 T^{8} - 13182 T^{9} + 57122 T^{10} + 742586 T^{11} + 4826809 T^{12} )^{2}$$)
$17$ ($$1 - 2 T + 17 T^{2}$$)($$( 1 - 17 T^{2} )^{2}$$)($$( 1 + 36 T^{2} + 64 T^{3} + 662 T^{4} + 1088 T^{5} + 10404 T^{6} + 83521 T^{8} )^{2}$$)($$( 1 - 62 T^{2} + 1903 T^{4} - 38180 T^{6} + 549967 T^{8} - 5178302 T^{10} + 24137569 T^{12} )^{2}$$)
$19$ ($$1 - 4 T + 19 T^{2}$$)($$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$1 + 8 T + 32 T^{2} + 120 T^{3} + 452 T^{4} + 2168 T^{5} + 10080 T^{6} + 37832 T^{7} + 138918 T^{8} + 718808 T^{9} + 3638880 T^{10} + 14870312 T^{11} + 58905092 T^{12} + 297131880 T^{13} + 1505468192 T^{14} + 7150973912 T^{15} + 16983563041 T^{16}$$)($$( 1 + 6 T + 18 T^{2} + 82 T^{3} + 539 T^{4} + 2636 T^{5} + 9476 T^{6} + 50084 T^{7} + 194579 T^{8} + 562438 T^{9} + 2345778 T^{10} + 14856594 T^{11} + 47045881 T^{12} )^{2}$$)
$23$ ($$1 - 8 T + 23 T^{2}$$)($$( 1 + 23 T^{2} )^{2}$$)($$( 1 - 38 T^{2} + 529 T^{4} )^{4}$$)($$( 1 - 86 T^{2} + 3791 T^{4} - 105684 T^{6} + 2005439 T^{8} - 24066326 T^{10} + 148035889 T^{12} )^{2}$$)
$29$ ($$1 - 6 T + 29 T^{2}$$)($$( 1 - 29 T^{2} )^{2}$$)($$1 + 16 T + 128 T^{2} + 928 T^{3} + 6580 T^{4} + 38208 T^{5} + 199680 T^{6} + 1073680 T^{7} + 5802054 T^{8} + 31136720 T^{9} + 167930880 T^{10} + 931854912 T^{11} + 4653908980 T^{12} + 19034346272 T^{13} + 76137385088 T^{14} + 275998020944 T^{15} + 500246412961 T^{16}$$)($$1 - 830 T^{4} + 2253679 T^{8} - 1165110596 T^{12} + 1593984336799 T^{16} - 415204522757630 T^{20} + 353814783205469041 T^{24}$$)
$31$ ($$1 + 8 T + 31 T^{2}$$)($$( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$)($$( 1 - 12 T + 164 T^{2} - 1140 T^{3} + 8218 T^{4} - 35340 T^{5} + 157604 T^{6} - 357492 T^{7} + 923521 T^{8} )^{2}$$)($$( 1 - 150 T^{2} + 10019 T^{4} - 392444 T^{6} + 9628259 T^{8} - 138528150 T^{10} + 887503681 T^{12} )^{2}$$)
$37$ ($$1 - 6 T + 37 T^{2}$$)($$( 1 + 10 T + 37 T^{2} )^{2}$$)($$1 + 16 T + 128 T^{2} + 1008 T^{3} + 5948 T^{4} + 15248 T^{5} - 9344 T^{6} - 717840 T^{7} - 7530650 T^{8} - 26560080 T^{9} - 12791936 T^{10} + 772356944 T^{11} + 11147509628 T^{12} + 69898708656 T^{13} + 328412980352 T^{14} + 1518910034128 T^{15} + 3512479453921 T^{16}$$)($$( 1 + 2 T + 2 T^{2} - 54 T^{3} + 567 T^{4} + 8764 T^{5} + 17852 T^{6} + 324268 T^{7} + 776223 T^{8} - 2735262 T^{9} + 3748322 T^{10} + 138687914 T^{11} + 2565726409 T^{12} )^{2}$$)
$41$ ($$1 + 6 T + 41 T^{2}$$)($$( 1 - 41 T^{2} )^{2}$$)($$1 - 200 T^{2} + 19452 T^{4} - 1244536 T^{6} + 58583750 T^{8} - 2092065016 T^{10} + 54966702972 T^{12} - 950020848200 T^{14} + 7984925229121 T^{16}$$)($$( 1 + 138 T^{2} + 7887 T^{4} + 320492 T^{6} + 13258047 T^{8} + 389955018 T^{10} + 4750104241 T^{12} )^{2}$$)
$43$ ($$1 + 4 T + 43 T^{2}$$)($$( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} )$$)($$1 + 8 T + 32 T^{2} + 56 T^{3} + 260 T^{4} + 504 T^{5} - 2720 T^{6} - 625528 T^{7} - 7635866 T^{8} - 26897704 T^{9} - 5029280 T^{10} + 40071528 T^{11} + 888888260 T^{12} + 8232472808 T^{13} + 202283617568 T^{14} + 2174548888856 T^{15} + 11688200277601 T^{16}$$)($$( 1 - 6 T + 18 T^{2} - 226 T^{3} + 4235 T^{4} - 18188 T^{5} + 58436 T^{6} - 782084 T^{7} + 7830515 T^{8} - 17968582 T^{9} + 61538418 T^{10} - 882050658 T^{11} + 6321363049 T^{12} )^{2}$$)
$47$ ($$1 + 47 T^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 86 T^{2} + 2209 T^{4} )^{4}$$)($$( 1 + 170 T^{2} + 15791 T^{4} + 908172 T^{6} + 34882319 T^{8} + 829545770 T^{10} + 10779215329 T^{12} )^{2}$$)
$53$ ($$1 + 2 T + 53 T^{2}$$)($$( 1 - 53 T^{2} )^{2}$$)($$1 - 16 T + 128 T^{2} - 928 T^{3} + 8564 T^{4} - 82496 T^{5} + 654336 T^{6} - 5021328 T^{7} + 38116486 T^{8} - 266130384 T^{9} + 1838029824 T^{10} - 12281756992 T^{11} + 67574079284 T^{12} - 388085417504 T^{13} + 2837038224512 T^{14} - 18795378237392 T^{15} + 62259690411361 T^{16}$$)($$1 + 7714 T^{4} + 19237903 T^{8} + 30633057916 T^{12} + 151796308101343 T^{16} + 480271251833238754 T^{20} +$$$$49\!\cdots\!41$$$$T^{24}$$)
$59$ ($$1 + 4 T + 59 T^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 - 8 T + 32 T^{2} - 472 T^{3} + 3481 T^{4} )^{4}$$)($$( 1 - 30 T + 450 T^{2} - 3458 T^{3} + 3915 T^{4} + 231740 T^{5} - 2735068 T^{6} + 13672660 T^{7} + 13628115 T^{8} - 710200582 T^{9} + 5452812450 T^{10} - 21447728970 T^{11} + 42180533641 T^{12} )( 1 + 30 T + 450 T^{2} + 3458 T^{3} + 3915 T^{4} - 231740 T^{5} - 2735068 T^{6} - 13672660 T^{7} + 13628115 T^{8} + 710200582 T^{9} + 5452812450 T^{10} + 21447728970 T^{11} + 42180533641 T^{12} )$$)
$61$ ($$1 + 2 T + 61 T^{2}$$)($$( 1 - 14 T + 61 T^{2} )^{2}$$)($$1 - 16 T + 128 T^{2} - 1392 T^{3} + 14204 T^{4} - 79760 T^{5} + 426880 T^{6} - 2945904 T^{7} + 19569574 T^{8} - 179700144 T^{9} + 1588420480 T^{10} - 18104004560 T^{11} + 196666325564 T^{12} - 1175678050992 T^{13} + 6594607918208 T^{14} - 50283885376336 T^{15} + 191707312997281 T^{16}$$)($$( 1 - 6 T + 18 T^{2} - 430 T^{3} - 121 T^{4} + 30796 T^{5} - 90148 T^{6} + 1878556 T^{7} - 450241 T^{8} - 97601830 T^{9} + 249225138 T^{10} - 5067577806 T^{11} + 51520374361 T^{12} )^{2}$$)
$67$ ($$1 - 4 T + 67 T^{2}$$)($$( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)($$1 + 16 T + 128 T^{2} + 304 T^{3} + 4388 T^{4} + 107696 T^{5} + 1207680 T^{6} + 4800272 T^{7} + 13154790 T^{8} + 321618224 T^{9} + 5421275520 T^{10} + 32390972048 T^{11} + 88423118948 T^{12} + 410438032528 T^{13} + 11578672917632 T^{14} + 96971385685168 T^{15} + 406067677556641 T^{16}$$)($$( 1 - 14 T + 98 T^{2} - 706 T^{3} + 9435 T^{4} - 112164 T^{5} + 894884 T^{6} - 7514988 T^{7} + 42353715 T^{8} - 212338678 T^{9} + 1974809858 T^{10} - 18901751498 T^{11} + 90458382169 T^{12} )^{2}$$)
$71$ ($$1 + 8 T + 71 T^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$1 - 440 T^{2} + 90844 T^{4} - 11522952 T^{6} + 984512390 T^{8} - 58087201032 T^{10} + 2308498748764 T^{12} - 56364124925240 T^{14} + 645753531245761 T^{16}$$)($$( 1 - 230 T^{2} + 30127 T^{4} - 2527028 T^{6} + 151870207 T^{8} - 5844686630 T^{10} + 128100283921 T^{12} )^{2}$$)
$73$ ($$1 - 10 T + 73 T^{2}$$)($$( 1 - 10 T + 73 T^{2} )^{2}$$)($$1 - 328 T^{2} + 45404 T^{4} - 3734648 T^{6} + 259745542 T^{8} - 19901939192 T^{10} + 1289393734364 T^{12} - 49637626222792 T^{14} + 806460091894081 T^{16}$$)($$( 1 - 166 T^{2} + 19007 T^{4} - 1414164 T^{6} + 101288303 T^{8} - 4714108006 T^{10} + 151334226289 T^{12} )^{2}$$)
$79$ ($$1 - 8 T + 79 T^{2}$$)($$( 1 - 4 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} )$$)($$( 1 + 12 T + 148 T^{2} - 44 T^{3} + 794 T^{4} - 3476 T^{5} + 923668 T^{6} + 5916468 T^{7} + 38950081 T^{8} )^{2}$$)($$( 1 - 358 T^{2} + 58915 T^{4} - 5817628 T^{6} + 367688515 T^{8} - 13944128998 T^{10} + 243087455521 T^{12} )^{2}$$)
$83$ ($$1 - 4 T + 83 T^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$1 + 40 T + 800 T^{2} + 11000 T^{3} + 122436 T^{4} + 1297720 T^{5} + 14460000 T^{6} + 161033000 T^{7} + 1597489574 T^{8} + 13365739000 T^{9} + 99614940000 T^{10} + 742019425640 T^{11} + 5810606989956 T^{12} + 43329447073000 T^{13} + 261552298695200 T^{14} + 1085442039585080 T^{15} + 2252292232139041 T^{16}$$)($$1 - 1374 T^{4} + 18563631 T^{8} - 336062521604 T^{12} + 880998758923551 T^{16} - 3094649526959042334 T^{20} +$$$$10\!\cdots\!61$$$$T^{24}$$)
$89$ ($$1 + 6 T + 89 T^{2}$$)($$( 1 - 89 T^{2} )^{2}$$)($$1 - 248 T^{2} + 36316 T^{4} - 4626504 T^{6} + 476004998 T^{8} - 36646538184 T^{10} + 2278547224156 T^{12} - 123251360158328 T^{14} + 3936588805702081 T^{16}$$)($$( 1 + 322 T^{2} + 51919 T^{4} + 5548348 T^{6} + 411250399 T^{8} + 20203001602 T^{10} + 496981290961 T^{12} )^{2}$$)
$97$ ($$1 - 2 T + 97 T^{2}$$)($$( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 + 164 T^{2} + 768 T^{3} + 13510 T^{4} + 74496 T^{5} + 1543076 T^{6} + 88529281 T^{8} )^{2}$$)($$( 1 + 2 T + 163 T^{2} - 220 T^{3} + 15811 T^{4} + 18818 T^{5} + 912673 T^{6} )^{4}$$)