# Properties

 Label 48.7.e Level 48 Weight 7 Character orbit e Rep. character $$\chi_{48}(17,\cdot)$$ Character field $$\Q$$ Dimension 11 Newform subspaces 4 Sturm bound 56 Trace bound 3

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 48.e (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$3$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$56$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{7}(48, [\chi])$$.

Total New Old
Modular forms 54 13 41
Cusp forms 42 11 31
Eisenstein series 12 2 10

## Trace form

 $$11q + q^{3} - 358q^{7} - 461q^{9} + O(q^{10})$$ $$11q + q^{3} - 358q^{7} - 461q^{9} - 2q^{13} - 32q^{15} - 7006q^{19} - 8754q^{21} - 9949q^{25} - 13031q^{27} + 34058q^{31} + 13088q^{33} + 56494q^{37} - 47174q^{39} + 55250q^{43} - 61376q^{45} + 128433q^{49} + 80000q^{51} + 189504q^{55} - 178586q^{57} + 95710q^{61} - 457782q^{63} - 310654q^{67} - 132160q^{69} + 69094q^{73} + 392009q^{75} + 896618q^{79} - 221477q^{81} - 562944q^{85} - 1841760q^{87} - 1950236q^{91} + 486430q^{93} + 381910q^{97} + 3371456q^{99} + O(q^{100})$$

## Decomposition of $$S_{7}^{\mathrm{new}}(48, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
48.7.e.a $$1$$ $$11.043$$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$27$$ $$0$$ $$286$$ $$q+3^{3}q^{3}+286q^{7}+3^{6}q^{9}+506q^{13}+\cdots$$
48.7.e.b $$2$$ $$11.043$$ $$\Q(\sqrt{-2})$$ None $$0$$ $$-42$$ $$0$$ $$-4$$ $$q+(-21+\beta )q^{3}+10\beta q^{5}-2q^{7}+(153+\cdots)q^{9}+\cdots$$
48.7.e.c $$2$$ $$11.043$$ $$\Q(\sqrt{-5})$$ None $$0$$ $$6$$ $$0$$ $$-484$$ $$q+(3+\beta )q^{3}-6\beta q^{5}-242q^{7}+(-711+\cdots)q^{9}+\cdots$$
48.7.e.d $$6$$ $$11.043$$ 6.0.1173604352.2 None $$0$$ $$10$$ $$0$$ $$-156$$ $$q+(2-\beta _{1})q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(-3^{3}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{7}^{\mathrm{old}}(48, [\chi])$$ into lower level spaces

$$S_{7}^{\mathrm{old}}(48, [\chi]) \cong$$ $$S_{7}^{\mathrm{new}}(3, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(6, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(12, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(24, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 - 27 T$$)($$1 + 42 T + 729 T^{2}$$)($$1 - 6 T + 729 T^{2}$$)($$1 - 10 T + 87 T^{2} + 11988 T^{3} + 63423 T^{4} - 5314410 T^{5} + 387420489 T^{6}$$)
$5$ ($$( 1 - 125 T )( 1 + 125 T )$$)($$1 - 2450 T^{2} + 244140625 T^{4}$$)($$1 - 5330 T^{2} + 244140625 T^{4}$$)($$1 - 57558 T^{2} + 1665299775 T^{4} - 31537483192500 T^{6} + 406567327880859375 T^{8} -$$$$34\!\cdots\!50$$$$T^{10} +$$$$14\!\cdots\!25$$$$T^{12}$$)
$7$ ($$1 - 286 T + 117649 T^{2}$$)($$( 1 + 2 T + 117649 T^{2} )^{2}$$)($$( 1 + 242 T + 117649 T^{2} )^{2}$$)($$( 1 + 78 T + 50079 T^{2} + 1090308 T^{3} + 5891744271 T^{4} + 1079620401678 T^{5} + 1628413597910449 T^{6} )^{2}$$)
$11$ ($$( 1 - 1331 T )( 1 + 1331 T )$$)($$1 - 3541970 T^{2} + 3138428376721 T^{4}$$)($$1 - 406802 T^{2} + 3138428376721 T^{4}$$)($$1 - 3337110 T^{2} + 13030923525375 T^{4} - 22216923537947614260 T^{6} +$$$$40\!\cdots\!75$$$$T^{8} -$$$$32\!\cdots\!10$$$$T^{10} +$$$$30\!\cdots\!61$$$$T^{12}$$)
$13$ ($$1 - 506 T + 4826809 T^{2}$$)($$( 1 + 2950 T + 4826809 T^{2} )^{2}$$)($$( 1 - 2618 T + 4826809 T^{2} )^{2}$$)($$( 1 - 78 T + 8645655 T^{2} - 5741493604 T^{3} + 41730925364895 T^{4} - 1817250639553518 T^{5} +$$$$11\!\cdots\!29$$$$T^{6} )^{2}$$)
$17$ ($$( 1 - 4913 T )( 1 + 4913 T )$$)($$1 - 28202690 T^{2} + 582622237229761 T^{4}$$)($$1 + 1905982 T^{2} + 582622237229761 T^{4}$$)($$1 - 63219270 T^{2} + 2233832285753679 T^{4} -$$$$57\!\cdots\!32$$$$T^{6} +$$$$13\!\cdots\!19$$$$T^{8} -$$$$21\!\cdots\!70$$$$T^{10} +$$$$19\!\cdots\!81$$$$T^{12}$$)
$19$ ($$1 - 10582 T + 47045881 T^{2}$$)($$( 1 + 5258 T + 47045881 T^{2} )^{2}$$)($$( 1 + 5786 T + 47045881 T^{2} )^{2}$$)($$( 1 - 2250 T + 64193463 T^{2} - 323410900012 T^{3} + 3020038021275903 T^{4} - 4979958567898862250 T^{5} +$$$$10\!\cdots\!41$$$$T^{6} )^{2}$$)
$23$ ($$( 1 - 12167 T )( 1 + 12167 T )$$)($$1 - 191004770 T^{2} + 21914624432020321 T^{4}$$)($$1 - 208876898 T^{2} + 21914624432020321 T^{4}$$)($$1 + 37480026 T^{2} + 40240367212774575 T^{4} +$$$$51\!\cdots\!12$$$$T^{6} +$$$$88\!\cdots\!75$$$$T^{8} +$$$$17\!\cdots\!66$$$$T^{10} +$$$$10\!\cdots\!61$$$$T^{12}$$)
$29$ ($$( 1 - 24389 T )( 1 + 24389 T )$$)($$1 - 1184779442 T^{2} + 353814783205469041 T^{4}$$)($$1 - 1035966962 T^{2} + 353814783205469041 T^{4}$$)($$1 - 1660966710 T^{2} + 1523344801209102879 T^{4} -$$$$10\!\cdots\!08$$$$T^{6} +$$$$53\!\cdots\!39$$$$T^{8} -$$$$20\!\cdots\!10$$$$T^{10} +$$$$44\!\cdots\!21$$$$T^{12}$$)
$31$ ($$1 + 35282 T + 887503681 T^{2}$$)($$( 1 + 22898 T + 887503681 T^{2} )^{2}$$)($$( 1 - 20446 T + 887503681 T^{2} )^{2}$$)($$( 1 - 37122 T + 2847433071 T^{2} - 66134143254940 T^{3} + 2527107331913634351 T^{4} -$$$$29\!\cdots\!42$$$$T^{5} +$$$$69\!\cdots\!41$$$$T^{6} )^{2}$$)
$37$ ($$1 + 89206 T + 2565726409 T^{2}$$)($$( 1 - 34058 T + 2565726409 T^{2} )^{2}$$)($$( 1 + 46774 T + 2565726409 T^{2} )^{2}$$)($$( 1 - 85566 T + 8033556999 T^{2} - 380834469871044 T^{3} + 20611909350541086591 T^{4} -$$$$56\!\cdots\!46$$$$T^{5} +$$$$16\!\cdots\!29$$$$T^{6} )^{2}$$)
$41$ ($$( 1 - 68921 T )( 1 + 68921 T )$$)($$1 - 9219079010 T^{2} + 22563490300366186081 T^{4}$$)($$1 - 9487663202 T^{2} + 22563490300366186081 T^{4}$$)($$1 - 7569104550 T^{2} + 1363608137405888943 T^{4} +$$$$15\!\cdots\!00$$$$T^{6} +$$$$30\!\cdots\!83$$$$T^{8} -$$$$38\!\cdots\!50$$$$T^{10} +$$$$11\!\cdots\!41$$$$T^{12}$$)
$43$ ($$1 + 111386 T + 6321363049 T^{2}$$)($$( 1 - 6406 T + 6321363049 T^{2} )^{2}$$)($$( 1 + 68618 T + 6321363049 T^{2} )^{2}$$)($$( 1 - 145530 T + 21837177927 T^{2} - 1677260136965132 T^{3} +$$$$13\!\cdots\!23$$$$T^{4} -$$$$58\!\cdots\!30$$$$T^{5} +$$$$25\!\cdots\!49$$$$T^{6} )^{2}$$)
$47$ ($$( 1 - 103823 T )( 1 + 103823 T )$$)($$1 + 10801249342 T^{2} +$$$$11\!\cdots\!41$$$$T^{4}$$)($$1 - 21106800578 T^{2} +$$$$11\!\cdots\!41$$$$T^{4}$$)($$1 - 52339293510 T^{2} +$$$$12\!\cdots\!23$$$$T^{4} -$$$$17\!\cdots\!20$$$$T^{6} +$$$$14\!\cdots\!43$$$$T^{8} -$$$$70\!\cdots\!10$$$$T^{10} +$$$$15\!\cdots\!21$$$$T^{12}$$)
$53$ ($$( 1 - 148877 T )( 1 + 148877 T )$$)($$1 - 7253988050 T^{2} +$$$$49\!\cdots\!41$$$$T^{4}$$)($$1 - 14819087378 T^{2} +$$$$49\!\cdots\!41$$$$T^{4}$$)($$1 - 75107476374 T^{2} +$$$$29\!\cdots\!67$$$$T^{4} -$$$$77\!\cdots\!24$$$$T^{6} +$$$$14\!\cdots\!47$$$$T^{8} -$$$$18\!\cdots\!94$$$$T^{10} +$$$$11\!\cdots\!21$$$$T^{12}$$)
$59$ ($$( 1 - 205379 T )( 1 + 205379 T )$$)($$1 + 22449655150 T^{2} +$$$$17\!\cdots\!81$$$$T^{4}$$)($$1 - 61991044562 T^{2} +$$$$17\!\cdots\!81$$$$T^{4}$$)($$1 - 180693503190 T^{2} +$$$$15\!\cdots\!31$$$$T^{4} -$$$$77\!\cdots\!36$$$$T^{6} +$$$$26\!\cdots\!11$$$$T^{8} -$$$$57\!\cdots\!90$$$$T^{10} +$$$$56\!\cdots\!41$$$$T^{12}$$)
$61$ ($$1 + 420838 T + 51520374361 T^{2}$$)($$( 1 + 62566 T + 51520374361 T^{2} )^{2}$$)($$( 1 - 24794 T + 51520374361 T^{2} )^{2}$$)($$( 1 - 296046 T + 42371190135 T^{2} - 4032498794872228 T^{3} +$$$$21\!\cdots\!35$$$$T^{4} -$$$$78\!\cdots\!66$$$$T^{5} +$$$$13\!\cdots\!81$$$$T^{6} )^{2}$$)
$67$ ($$1 + 172874 T + 90458382169 T^{2}$$)($$( 1 + 438698 T + 90458382169 T^{2} )^{2}$$)($$( 1 - 84358 T + 90458382169 T^{2} )^{2}$$)($$( 1 - 285450 T + 220735162839 T^{2} - 36790378201873708 T^{3} +$$$$19\!\cdots\!91$$$$T^{4} -$$$$23\!\cdots\!50$$$$T^{5} +$$$$74\!\cdots\!09$$$$T^{6} )^{2}$$)
$71$ ($$( 1 - 357911 T )( 1 + 357911 T )$$)($$1 - 251546372642 T^{2} +$$$$16\!\cdots\!41$$$$T^{4}$$)($$1 - 151063967522 T^{2} +$$$$16\!\cdots\!41$$$$T^{4}$$)($$1 - 402983850726 T^{2} +$$$$77\!\cdots\!75$$$$T^{4} -$$$$10\!\cdots\!48$$$$T^{6} +$$$$12\!\cdots\!75$$$$T^{8} -$$$$10\!\cdots\!06$$$$T^{10} +$$$$44\!\cdots\!21$$$$T^{12}$$)
$73$ ($$1 - 638066 T + 151334226289 T^{2}$$)($$( 1 + 730510 T + 151334226289 T^{2} )^{2}$$)($$( 1 + 113806 T + 151334226289 T^{2} )^{2}$$)($$( 1 - 559830 T + 328149232287 T^{2} - 146928200928984628 T^{3} +$$$$49\!\cdots\!43$$$$T^{4} -$$$$12\!\cdots\!30$$$$T^{5} +$$$$34\!\cdots\!69$$$$T^{6} )^{2}$$)
$79$ ($$1 - 204622 T + 243087455521 T^{2}$$)($$( 1 + 340562 T + 243087455521 T^{2} )^{2}$$)($$( 1 - 159742 T + 243087455521 T^{2} )^{2}$$)($$( 1 - 526818 T + 705484689807 T^{2} - 230000056692908636 T^{3} +$$$$17\!\cdots\!47$$$$T^{4} -$$$$31\!\cdots\!38$$$$T^{5} +$$$$14\!\cdots\!61$$$$T^{6} )^{2}$$)
$83$ ($$( 1 - 571787 T )( 1 + 571787 T )$$)($$1 - 407613512306 T^{2} +$$$$10\!\cdots\!61$$$$T^{4}$$)($$1 - 388294032818 T^{2} +$$$$10\!\cdots\!61$$$$T^{4}$$)($$1 - 1779363154038 T^{2} +$$$$13\!\cdots\!67$$$$T^{4} -$$$$58\!\cdots\!36$$$$T^{6} +$$$$14\!\cdots\!87$$$$T^{8} -$$$$20\!\cdots\!98$$$$T^{10} +$$$$12\!\cdots\!81$$$$T^{12}$$)
$89$ ($$( 1 - 704969 T )( 1 + 704969 T )$$)($$1 - 844406214050 T^{2} +$$$$24\!\cdots\!21$$$$T^{4}$$)($$1 + 583819025758 T^{2} +$$$$24\!\cdots\!21$$$$T^{4}$$)($$1 - 1464277161318 T^{2} +$$$$11\!\cdots\!39$$$$T^{4} -$$$$68\!\cdots\!36$$$$T^{6} +$$$$28\!\cdots\!19$$$$T^{8} -$$$$89\!\cdots\!38$$$$T^{10} +$$$$15\!\cdots\!61$$$$T^{12}$$)
$97$ ($$1 + 56446 T + 832972004929 T^{2}$$)($$( 1 + 281086 T + 832972004929 T^{2} )^{2}$$)($$( 1 - 899522 T + 832972004929 T^{2} )^{2}$$)($$( 1 + 399258 T + 1108436400207 T^{2} + 1034642865778948780 T^{3} +$$$$92\!\cdots\!03$$$$T^{4} +$$$$27\!\cdots\!78$$$$T^{5} +$$$$57\!\cdots\!89$$$$T^{6} )^{2}$$)