# Properties

 Label 6048.2.c Level 6048 Weight 2 Character orbit c Rep. character $$\chi_{6048}(3025,\cdot)$$ Character field $$\Q$$ Dimension 96 Newform subspaces 7 Sturm bound 2304 Trace bound 31

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$6048 = 2^{5} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6048.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$7$$ Sturm bound: $$2304$$ Trace bound: $$31$$ Distinguishing $$T_p$$: $$5$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(6048, [\chi])$$.

Total New Old
Modular forms 1200 96 1104
Cusp forms 1104 96 1008
Eisenstein series 96 0 96

## Trace form

 $$96q + O(q^{10})$$ $$96q - 96q^{25} + 16q^{31} + 96q^{49} - 32q^{55} - 80q^{79} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(6048, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
6048.2.c.a $$2$$ $$48.294$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+2iq^{5}+q^{7}-5iq^{13}-q^{17}+4iq^{19}+\cdots$$
6048.2.c.b $$2$$ $$48.294$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+2iq^{5}+q^{7}+5iq^{13}+q^{17}-4iq^{19}+\cdots$$
6048.2.c.c $$8$$ $$48.294$$ 8.0.3317760000.5 None $$0$$ $$0$$ $$0$$ $$8$$ $$q+\beta _{6}q^{5}+q^{7}+(2\beta _{2}+\beta _{6})q^{11}+(-\beta _{1}+\cdots)q^{13}+\cdots$$
6048.2.c.d $$16$$ $$48.294$$ $$\Q(\zeta_{40})$$ None $$0$$ $$0$$ $$0$$ $$16$$ $$q+(\zeta_{40}^{11}+\zeta_{40}^{12})q^{5}+q^{7}-\zeta_{40}^{12}q^{11}+\cdots$$
6048.2.c.e $$20$$ $$48.294$$ $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$20$$ $$q+\beta _{12}q^{5}+q^{7}+(-\beta _{4}-\beta _{10}+\beta _{12}+\cdots)q^{11}+\cdots$$
6048.2.c.f $$24$$ $$48.294$$ None $$0$$ $$0$$ $$0$$ $$-24$$
6048.2.c.g $$24$$ $$48.294$$ None $$0$$ $$0$$ $$0$$ $$-24$$

## Decomposition of $$S_{2}^{\mathrm{old}}(6048, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(6048, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(24, [\chi])$$$$^{\oplus 18}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(96, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(216, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(224, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(288, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(504, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(672, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(864, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1512, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(2016, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ ($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 12 T^{2} + 71 T^{4} - 300 T^{6} + 625 T^{8} )^{2}$$)($$( 1 - 12 T^{2} + 54 T^{4} - 48 T^{6} - 469 T^{8} - 1200 T^{10} + 33750 T^{12} - 187500 T^{14} + 390625 T^{16} )^{2}$$)($$( 1 - 18 T^{2} + 187 T^{4} - 1608 T^{6} + 10781 T^{8} - 57906 T^{10} + 269525 T^{12} - 1005000 T^{14} + 2921875 T^{16} - 7031250 T^{18} + 9765625 T^{20} )^{2}$$)
$7$ ($$( 1 - T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{8}$$)($$( 1 - T )^{16}$$)($$( 1 - T )^{20}$$)
$11$ ($$( 1 - 11 T^{2} )^{2}$$)($$( 1 - 11 T^{2} )^{2}$$)($$( 1 + 4 T^{2} + 111 T^{4} + 484 T^{6} + 14641 T^{8} )^{2}$$)($$( 1 - 60 T^{2} + 1734 T^{4} - 31920 T^{6} + 413051 T^{8} - 3862320 T^{10} + 25387494 T^{12} - 106293660 T^{14} + 214358881 T^{16} )^{2}$$)($$( 1 - 34 T^{2} + 589 T^{4} - 8472 T^{6} + 122234 T^{8} - 1528172 T^{10} + 14790314 T^{12} - 124038552 T^{14} + 1043449429 T^{16} - 7288201954 T^{18} + 25937424601 T^{20} )^{2}$$)
$13$ ($$1 - T^{2} + 169 T^{4}$$)($$1 - T^{2} + 169 T^{4}$$)($$( 1 - 20 T^{2} + 378 T^{4} - 3380 T^{6} + 28561 T^{8} )^{2}$$)($$( 1 - 40 T^{2} + 936 T^{4} - 16040 T^{6} + 219326 T^{8} - 2710760 T^{10} + 26733096 T^{12} - 193072360 T^{14} + 815730721 T^{16} )^{2}$$)($$( 1 - 83 T^{2} + 3533 T^{4} - 98556 T^{6} + 1975738 T^{8} - 29558418 T^{10} + 333899722 T^{12} - 2814857916 T^{14} + 17053116197 T^{16} - 67705649843 T^{18} + 137858491849 T^{20} )^{2}$$)
$17$ ($$( 1 + T + 17 T^{2} )^{2}$$)($$( 1 - T + 17 T^{2} )^{2}$$)($$( 1 + 24 T^{2} + 289 T^{4} )^{4}$$)($$( 1 + 24 T^{2} + 289 T^{4} )^{8}$$)($$( 1 + 39 T^{2} + 455 T^{4} - 2382 T^{6} - 47987 T^{8} + 145305 T^{10} - 13868243 T^{12} - 198947022 T^{14} + 10982593895 T^{16} + 272054540199 T^{18} + 2015993900449 T^{20} )^{2}$$)
$19$ ($$1 - 22 T^{2} + 361 T^{4}$$)($$1 - 22 T^{2} + 361 T^{4}$$)($$( 1 - 14 T^{2} - 189 T^{4} - 5054 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 - 66 T^{2} + 1791 T^{4} - 23826 T^{6} + 130321 T^{8} )^{4}$$)($$( 1 - 70 T^{2} + 3325 T^{4} - 107448 T^{6} + 2823098 T^{8} - 58351460 T^{10} + 1019138378 T^{12} - 14002730808 T^{14} + 156427554325 T^{16} - 1188849412870 T^{18} + 6131066257801 T^{20} )^{2}$$)
$23$ ($$( 1 + 5 T + 23 T^{2} )^{2}$$)($$( 1 - 5 T + 23 T^{2} )^{2}$$)($$( 1 + 84 T^{2} + 2807 T^{4} + 44436 T^{6} + 279841 T^{8} )^{2}$$)($$( 1 + 52 T^{2} + 2430 T^{4} + 71312 T^{6} + 1906499 T^{8} + 37724048 T^{10} + 680013630 T^{12} + 7697866228 T^{14} + 78310985281 T^{16} )^{2}$$)($$( 1 + 139 T^{2} + 10081 T^{4} + 485484 T^{6} + 16965806 T^{8} + 447075730 T^{10} + 8974911374 T^{12} + 135858328044 T^{14} + 1492349797009 T^{16} + 10885226954059 T^{18} + 41426511213649 T^{20} )^{2}$$)
$29$ ($$1 + 23 T^{2} + 841 T^{4}$$)($$1 + 23 T^{2} + 841 T^{4}$$)($$( 1 - 52 T^{2} + 841 T^{4} )^{4}$$)($$( 1 - 44 T^{2} + 960 T^{4} - 18676 T^{6} + 833774 T^{8} - 15706516 T^{10} + 678989760 T^{12} - 26172226124 T^{14} + 500246412961 T^{16} )^{2}$$)($$( 1 - 189 T^{2} + 17145 T^{4} - 999612 T^{6} + 42327774 T^{8} - 1383568894 T^{10} + 35597657934 T^{12} - 707006574972 T^{14} + 10198245838545 T^{16} - 94546572049629 T^{18} + 420707233300201 T^{20} )^{2}$$)
$31$ ($$( 1 - 7 T + 31 T^{2} )^{2}$$)($$( 1 - 7 T + 31 T^{2} )^{2}$$)($$( 1 - 7 T + 31 T^{2} )^{8}$$)($$( 1 + 10 T + 114 T^{2} + 800 T^{3} + 5351 T^{4} + 24800 T^{5} + 109554 T^{6} + 297910 T^{7} + 923521 T^{8} )^{4}$$)($$( 1 + 9 T + 171 T^{2} + 1092 T^{3} + 11058 T^{4} + 50402 T^{5} + 342798 T^{6} + 1049412 T^{7} + 5094261 T^{8} + 8311689 T^{9} + 28629151 T^{10} )^{4}$$)
$37$ ($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 110 T^{2} + 5523 T^{4} - 150590 T^{6} + 1874161 T^{8} )^{2}$$)($$( 1 - 192 T^{2} + 17010 T^{4} - 960672 T^{6} + 40219139 T^{8} - 1315159968 T^{10} + 31879478610 T^{12} - 492619470528 T^{14} + 3512479453921 T^{16} )^{2}$$)($$( 1 - 22 T^{2} + 1771 T^{4} + 13920 T^{6} + 526445 T^{8} + 112600450 T^{10} + 720703205 T^{12} + 26088321120 T^{14} + 4543901470339 T^{16} - 77274547986262 T^{18} + 4808584372417849 T^{20} )^{2}$$)
$41$ ($$( 1 + 2 T + 41 T^{2} )^{2}$$)($$( 1 - 2 T + 41 T^{2} )^{2}$$)($$( 1 + 12 T^{2} + 2663 T^{4} + 20172 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 + 220 T^{2} + 24294 T^{4} + 1713680 T^{6} + 83598011 T^{8} + 2880696080 T^{10} + 68649037734 T^{12} + 1045022933020 T^{14} + 7984925229121 T^{16} )^{2}$$)($$( 1 + 190 T^{2} + 16651 T^{4} + 899952 T^{6} + 36306749 T^{8} + 1400716342 T^{10} + 61031645069 T^{12} + 2543049263472 T^{14} + 79093985716891 T^{16} + 1517135793532990 T^{18} + 13422659310152401 T^{20} )^{2}$$)
$43$ ($$1 - 61 T^{2} + 1849 T^{4}$$)($$1 - 61 T^{2} + 1849 T^{4}$$)($$( 1 - 92 T^{2} + 4314 T^{4} - 170108 T^{6} + 3418801 T^{8} )^{2}$$)($$( 1 + 32 T^{2} + 4920 T^{4} + 158992 T^{6} + 11619614 T^{8} + 293976208 T^{10} + 16820500920 T^{12} + 202283617568 T^{14} + 11688200277601 T^{16} )^{2}$$)($$( 1 - 131 T^{2} + 5615 T^{4} - 93162 T^{6} + 5954173 T^{8} - 455929005 T^{10} + 11009265877 T^{12} - 318502338762 T^{14} + 35494453520135 T^{16} - 1531154236365731 T^{18} + 21611482313284249 T^{20} )^{2}$$)
$47$ ($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 40 T^{2} + 2209 T^{4} )^{4}$$)($$( 1 + 124 T^{2} + 7312 T^{4} + 372292 T^{6} + 18784750 T^{8} + 822393028 T^{10} + 35680227472 T^{12} + 1336622700796 T^{14} + 23811286661761 T^{16} )^{2}$$)($$( 1 + 110 T^{2} + 11187 T^{4} + 742128 T^{6} + 47768349 T^{8} + 2326511142 T^{10} + 105520282941 T^{12} + 3621347901168 T^{14} + 120587081885523 T^{16} + 2619241532793710 T^{18} + 52599132235830049 T^{20} )^{2}$$)
$53$ ($$1 - 25 T^{2} + 2809 T^{4}$$)($$1 - 25 T^{2} + 2809 T^{4}$$)($$( 1 - 53 T^{2} )^{8}$$)($$( 1 - 216 T^{2} + 24972 T^{4} - 2015208 T^{6} + 122805830 T^{8} - 5660719272 T^{10} + 197041091532 T^{12} - 4787502003864 T^{14} + 62259690411361 T^{16} )^{2}$$)($$( 1 - 193 T^{2} + 20557 T^{4} - 1818204 T^{6} + 128583722 T^{8} - 7326778022 T^{10} + 361191675098 T^{12} - 14346504116124 T^{14} + 455632771728853 T^{16} - 12016120249392673 T^{18} + 174887470365513049 T^{20} )^{2}$$)
$59$ ($$1 - 117 T^{2} + 3481 T^{4}$$)($$1 - 117 T^{2} + 3481 T^{4}$$)($$( 1 - 24 T^{2} + 3266 T^{4} - 83544 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 - 380 T^{2} + 67344 T^{4} - 7257700 T^{6} + 519439886 T^{8} - 25264053700 T^{10} + 816031559184 T^{12} - 16028602783580 T^{14} + 146830437604321 T^{16} )^{2}$$)($$( 1 - 449 T^{2} + 96439 T^{4} - 13053078 T^{6} + 1232082461 T^{8} - 84671750375 T^{10} + 4288879046741 T^{12} - 158168858287158 T^{14} + 4067848483804399 T^{16} - 65926866484340129 T^{18} + 511116753300641401 T^{20} )^{2}$$)
$61$ ($$1 - 86 T^{2} + 3721 T^{4}$$)($$1 - 86 T^{2} + 3721 T^{4}$$)($$( 1 - 52 T^{2} - 522 T^{4} - 193492 T^{6} + 13845841 T^{8} )^{2}$$)($$( 1 - 304 T^{2} + 46620 T^{4} - 4694096 T^{6} + 336476774 T^{8} - 17466731216 T^{10} + 645493107420 T^{12} - 15662193805744 T^{14} + 191707312997281 T^{16} )^{2}$$)($$( 1 - 186 T^{2} + 23517 T^{4} - 2152152 T^{6} + 175128378 T^{8} - 11500636220 T^{10} + 651652694538 T^{12} - 29798354399832 T^{14} + 1211604643847637 T^{16} - 35657560217494266 T^{18} + 713342911662882601 T^{20} )^{2}$$)
$67$ ($$1 - 53 T^{2} + 4489 T^{4}$$)($$1 - 53 T^{2} + 4489 T^{4}$$)($$( 1 + 20 T^{2} + 4218 T^{4} + 89780 T^{6} + 20151121 T^{8} )^{2}$$)($$( 1 - 232 T^{2} + 32520 T^{4} - 3258632 T^{6} + 244468094 T^{8} - 14627999048 T^{10} + 655314454920 T^{12} - 20986344663208 T^{14} + 406067677556641 T^{16} )^{2}$$)($$( 1 - 495 T^{2} + 120093 T^{4} - 18487716 T^{6} + 1986970986 T^{8} - 155229602234 T^{10} + 8919512756154 T^{12} - 372548202129636 T^{14} + 10863418489821717 T^{16} - 201003500390537295 T^{18} + 1822837804551761449 T^{20} )^{2}$$)
$71$ ($$( 1 + 15 T + 71 T^{2} )^{2}$$)($$( 1 - 15 T + 71 T^{2} )^{2}$$)($$( 1 + 236 T^{2} + 23871 T^{4} + 1189676 T^{6} + 25411681 T^{8} )^{2}$$)($$( 1 + 396 T^{2} + 77350 T^{4} + 9537264 T^{6} + 808347579 T^{8} + 48077347824 T^{10} + 1965593525350 T^{12} + 50727712432716 T^{14} + 645753531245761 T^{16} )^{2}$$)($$( 1 + 343 T^{2} + 52613 T^{4} + 4802116 T^{6} + 309151594 T^{8} + 19321038474 T^{10} + 1558433185354 T^{12} + 122029839916996 T^{14} + 6739740237935573 T^{16} + 221493461217296023 T^{18} + 3255243551009881201 T^{20} )^{2}$$)
$73$ ($$( 1 + 73 T^{2} )^{2}$$)($$( 1 + 73 T^{2} )^{2}$$)($$( 1 + 6 T + 140 T^{2} + 438 T^{3} + 5329 T^{4} )^{4}$$)($$( 1 - 6 T + 238 T^{2} - 990 T^{3} + 23766 T^{4} - 72270 T^{5} + 1268302 T^{6} - 2334102 T^{7} + 28398241 T^{8} )^{4}$$)($$( 1 + 149 T^{2} + 432 T^{3} + 14410 T^{4} + 51408 T^{5} + 1051930 T^{6} + 2302128 T^{7} + 57963533 T^{8} + 2073071593 T^{10} )^{4}$$)
$79$ ($$( 1 - 14 T + 79 T^{2} )^{2}$$)($$( 1 - 14 T + 79 T^{2} )^{2}$$)($$( 1 + 2 T + 24 T^{2} + 158 T^{3} + 6241 T^{4} )^{4}$$)($$( 1 + 6 T + 82 T^{2} + 18 T^{3} + 6630 T^{4} + 1422 T^{5} + 511762 T^{6} + 2958234 T^{7} + 38950081 T^{8} )^{4}$$)($$( 1 + 16 T + 335 T^{2} + 3202 T^{3} + 40117 T^{4} + 296322 T^{5} + 3169243 T^{6} + 19983682 T^{7} + 165168065 T^{8} + 623201296 T^{9} + 3077056399 T^{10} )^{4}$$)
$83$ ($$1 - 150 T^{2} + 6889 T^{4}$$)($$1 - 150 T^{2} + 6889 T^{4}$$)($$( 1 - 120 T^{2} + 13538 T^{4} - 826680 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 - 252 T^{2} + 47280 T^{4} - 5629572 T^{6} + 552317774 T^{8} - 38782121508 T^{10} + 2243829416880 T^{12} - 82388974088988 T^{14} + 2252292232139041 T^{16} )^{2}$$)($$( 1 - 138 T^{2} + 13627 T^{4} - 1173792 T^{6} + 82503245 T^{8} - 3517947618 T^{10} + 568364854805 T^{12} - 55706197523232 T^{14} + 4455216467899363 T^{16} - 310816328035187658 T^{18} + 15516041187205853449 T^{20} )^{2}$$)
$89$ ($$( 1 - 13 T + 89 T^{2} )^{2}$$)($$( 1 + 13 T + 89 T^{2} )^{2}$$)($$( 1 + 204 T^{2} + 25511 T^{4} + 1615884 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 + 364 T^{2} + 76230 T^{4} + 10653296 T^{6} + 1102200539 T^{8} + 84384757616 T^{10} + 4782841031430 T^{12} + 180901189909804 T^{14} + 3936588805702081 T^{16} )^{2}$$)($$( 1 + 547 T^{2} + 153205 T^{4} + 28413396 T^{6} + 3828024122 T^{8} + 389779473874 T^{10} + 30321779070362 T^{12} + 1782720139460436 T^{14} + 76140018681680005 T^{16} + 2153314076719038307 T^{18} + 31181719929966183601 T^{20} )^{2}$$)
$97$ ($$( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 + 4 T + 138 T^{2} + 388 T^{3} + 9409 T^{4} )^{4}$$)($$( 1 - 2 T + 70 T^{2} - 194 T^{3} + 9409 T^{4} )^{8}$$)($$( 1 - 14 T + 341 T^{2} - 3480 T^{3} + 53038 T^{4} - 434964 T^{5} + 5144686 T^{6} - 32743320 T^{7} + 311221493 T^{8} - 1239409934 T^{9} + 8587340257 T^{10} )^{4}$$)