Properties

Label 48.7
Level 48
Weight 7
Dimension 157
Nonzero newspaces 4
Newform subspaces 9
Sturm bound 896
Trace bound 1

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Defining parameters

Level: \( N \) = \( 48\( 48 = 2^{4} \cdot 3 \) \)
Weight: \( k \) = \( 7 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 9 \)
Sturm bound: \(896\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(\Gamma_1(48))\).

Total New Old
Modular forms 412 167 245
Cusp forms 356 157 199
Eisenstein series 56 10 46

Trace form

\( 157q - q^{3} - 184q^{4} + 132q^{5} + 508q^{6} - 358q^{7} - 1932q^{8} - 1919q^{9} + O(q^{10}) \) \( 157q - q^{3} - 184q^{4} + 132q^{5} + 508q^{6} - 358q^{7} - 1932q^{8} - 1919q^{9} + 2240q^{10} - 2720q^{11} + 2344q^{12} + 4998q^{13} - 15388q^{14} - 36q^{15} + 14096q^{16} - 7332q^{17} + 14880q^{18} - 14882q^{19} - 14000q^{20} + 2422q^{21} - 83272q^{22} + 26240q^{23} + 84844q^{24} - 42427q^{25} - 5300q^{26} - 47353q^{27} - 69120q^{28} + 121780q^{29} - 160884q^{30} + 34050q^{31} - 52960q^{32} - 524q^{33} + 250632q^{34} + 162336q^{35} + 204496q^{36} + 63942q^{37} + 16968q^{38} - 47174q^{39} - 473464q^{40} - 129924q^{41} - 10100q^{42} + 200686q^{43} - 193192q^{44} - 124704q^{45} - 299552q^{46} + 111328q^{48} + 135395q^{49} + 537764q^{50} + 545408q^{51} + 1003856q^{52} + 363796q^{53} + 230644q^{54} + 422208q^{55} - 420448q^{56} - 110546q^{57} - 1477624q^{58} - 886144q^{59} - 385608q^{60} - 1378986q^{61} - 719652q^{62} + 12810q^{63} - 443776q^{64} + 296360q^{65} + 1264828q^{66} - 65090q^{67} + 3197632q^{68} + 408108q^{69} + 989208q^{70} + 534016q^{71} - 820860q^{72} - 33902q^{73} - 4894836q^{74} + 732699q^{75} - 3522544q^{76} - 860672q^{77} + 1140968q^{78} + 1757538q^{79} + 4668072q^{80} - 2701539q^{81} + 2664760q^{82} - 2497760q^{83} + 581560q^{84} + 1261320q^{85} + 4142928q^{86} - 1841760q^{87} - 1534912q^{88} - 3515412q^{89} - 799176q^{90} + 1145764q^{91} - 9470992q^{92} - 1935934q^{93} - 6463224q^{94} + 3835400q^{96} + 645530q^{97} + 12708584q^{98} - 1337732q^{99} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(\Gamma_1(48))\)

We only show spaces with odd 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
48.7.b \(\chi_{48}(7, \cdot)\) None 0 1
48.7.e \(\chi_{48}(17, \cdot)\) 48.7.e.a 1 1
48.7.e.b 2
48.7.e.c 2
48.7.e.d 6
48.7.g \(\chi_{48}(31, \cdot)\) 48.7.g.a 2 1
48.7.g.b 2
48.7.g.c 2
48.7.h \(\chi_{48}(41, \cdot)\) None 0 1
48.7.i \(\chi_{48}(5, \cdot)\) 48.7.i.a 92 2
48.7.l \(\chi_{48}(19, \cdot)\) 48.7.l.a 48 2

Decomposition of \(S_{7}^{\mathrm{old}}(\Gamma_1(48))\) into lower level spaces

\( S_{7}^{\mathrm{old}}(\Gamma_1(48)) \cong \) \(S_{7}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 5}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 - 27 T \))(\( 1 + 42 T + 729 T^{2} \))(\( 1 - 6 T + 729 T^{2} \))(\( 1 - 10 T + 87 T^{2} + 11988 T^{3} + 63423 T^{4} - 5314410 T^{5} + 387420489 T^{6} \))(\( 1 + 243 T^{2} \))(\( 1 + 243 T^{2} \))(\( 1 + 243 T^{2} \))
$5$ (\( ( 1 - 125 T )( 1 + 125 T ) \))(\( 1 - 2450 T^{2} + 244140625 T^{4} \))(\( 1 - 5330 T^{2} + 244140625 T^{4} \))(\( 1 - 57558 T^{2} + 1665299775 T^{4} - 31537483192500 T^{6} + 406567327880859375 T^{8} - \)\(34\!\cdots\!50\)\( T^{10} + \)\(14\!\cdots\!25\)\( T^{12} \))(\( ( 1 + 90 T + 15625 T^{2} )^{2} \))(\( ( 1 - 6 T + 15625 T^{2} )^{2} \))(\( ( 1 - 150 T + 15625 T^{2} )^{2} \))
$7$ (\( 1 - 286 T + 117649 T^{2} \))(\( ( 1 + 2 T + 117649 T^{2} )^{2} \))(\( ( 1 + 242 T + 117649 T^{2} )^{2} \))(\( ( 1 + 78 T + 50079 T^{2} + 1090308 T^{3} + 5891744271 T^{4} + 1079620401678 T^{5} + 1628413597910449 T^{6} )^{2} \))(\( 1 - 200306 T^{2} + 13841287201 T^{4} \))(\( 1 - 194930 T^{2} + 13841287201 T^{4} \))(\( 1 - 129266 T^{2} + 13841287201 T^{4} \))
$11$ (\( ( 1 - 1331 T )( 1 + 1331 T ) \))(\( 1 - 3541970 T^{2} + 3138428376721 T^{4} \))(\( 1 - 406802 T^{2} + 3138428376721 T^{4} \))(\( 1 - 3337110 T^{2} + 13030923525375 T^{4} - 22216923537947614260 T^{6} + \)\(40\!\cdots\!75\)\( T^{8} - \)\(32\!\cdots\!10\)\( T^{10} + \)\(30\!\cdots\!61\)\( T^{12} \))(\( 1 - 708770 T^{2} + 3138428376721 T^{4} \))(\( 1 - 3127970 T^{2} + 3138428376721 T^{4} \))(\( 1 - 1365410 T^{2} + 3138428376721 T^{4} \))
$13$ (\( 1 - 506 T + 4826809 T^{2} \))(\( ( 1 + 2950 T + 4826809 T^{2} )^{2} \))(\( ( 1 - 2618 T + 4826809 T^{2} )^{2} \))(\( ( 1 - 78 T + 8645655 T^{2} - 5741493604 T^{3} + 41730925364895 T^{4} - 1817250639553518 T^{5} + \)\(11\!\cdots\!29\)\( T^{6} )^{2} \))(\( ( 1 - 1762 T + 4826809 T^{2} )^{2} \))(\( ( 1 + 2654 T + 4826809 T^{2} )^{2} \))(\( ( 1 - 3394 T + 4826809 T^{2} )^{2} \))
$17$ (\( ( 1 - 4913 T )( 1 + 4913 T ) \))(\( 1 - 28202690 T^{2} + 582622237229761 T^{4} \))(\( 1 + 1905982 T^{2} + 582622237229761 T^{4} \))(\( 1 - 63219270 T^{2} + 2233832285753679 T^{4} - \)\(57\!\cdots\!32\)\( T^{6} + \)\(13\!\cdots\!19\)\( T^{8} - \)\(21\!\cdots\!70\)\( T^{10} + \)\(19\!\cdots\!81\)\( T^{12} \))(\( ( 1 + 1638 T + 24137569 T^{2} )^{2} \))(\( ( 1 + 7206 T + 24137569 T^{2} )^{2} \))(\( ( 1 - 5178 T + 24137569 T^{2} )^{2} \))
$19$ (\( 1 - 10582 T + 47045881 T^{2} \))(\( ( 1 + 5258 T + 47045881 T^{2} )^{2} \))(\( ( 1 + 5786 T + 47045881 T^{2} )^{2} \))(\( ( 1 - 2250 T + 64193463 T^{2} - 323410900012 T^{3} + 3020038021275903 T^{4} - 4979958567898862250 T^{5} + \)\(10\!\cdots\!41\)\( T^{6} )^{2} \))(\( 1 + 62987326 T^{2} + 2213314919066161 T^{4} \))(\( 1 - 81557954 T^{2} + 2213314919066161 T^{4} \))(\( 1 - 47709890 T^{2} + 2213314919066161 T^{4} \))
$23$ (\( ( 1 - 12167 T )( 1 + 12167 T ) \))(\( 1 - 191004770 T^{2} + 21914624432020321 T^{4} \))(\( 1 - 208876898 T^{2} + 21914624432020321 T^{4} \))(\( 1 + 37480026 T^{2} + 40240367212774575 T^{4} + \)\(51\!\cdots\!12\)\( T^{6} + \)\(88\!\cdots\!75\)\( T^{8} + \)\(17\!\cdots\!66\)\( T^{10} + \)\(10\!\cdots\!61\)\( T^{12} \))(\( 1 - 114673250 T^{2} + 21914624432020321 T^{4} \))(\( 1 + 8747422 T^{2} + 21914624432020321 T^{4} \))(\( 1 - 280146530 T^{2} + 21914624432020321 T^{4} \))
$29$ (\( ( 1 - 24389 T )( 1 + 24389 T ) \))(\( 1 - 1184779442 T^{2} + 353814783205469041 T^{4} \))(\( 1 - 1035966962 T^{2} + 353814783205469041 T^{4} \))(\( 1 - 1660966710 T^{2} + 1523344801209102879 T^{4} - \)\(10\!\cdots\!08\)\( T^{6} + \)\(53\!\cdots\!39\)\( T^{8} - \)\(20\!\cdots\!10\)\( T^{10} + \)\(44\!\cdots\!21\)\( T^{12} \))(\( ( 1 + 16002 T + 594823321 T^{2} )^{2} \))(\( ( 1 - 11550 T + 594823321 T^{2} )^{2} \))(\( ( 1 - 32142 T + 594823321 T^{2} )^{2} \))
$31$ (\( 1 + 35282 T + 887503681 T^{2} \))(\( ( 1 + 22898 T + 887503681 T^{2} )^{2} \))(\( ( 1 - 20446 T + 887503681 T^{2} )^{2} \))(\( ( 1 - 37122 T + 2847433071 T^{2} - 66134143254940 T^{3} + 2527107331913634351 T^{4} - \)\(29\!\cdots\!42\)\( T^{5} + \)\(69\!\cdots\!41\)\( T^{6} )^{2} \))(\( 1 - 1522190162 T^{2} + 787662783788549761 T^{4} \))(\( 1 - 1702033490 T^{2} + 787662783788549761 T^{4} \))(\( 1 - 711526610 T^{2} + 787662783788549761 T^{4} \))
$37$ (\( 1 + 89206 T + 2565726409 T^{2} \))(\( ( 1 - 34058 T + 2565726409 T^{2} )^{2} \))(\( ( 1 + 46774 T + 2565726409 T^{2} )^{2} \))(\( ( 1 - 85566 T + 8033556999 T^{2} - 380834469871044 T^{3} + 20611909350541086591 T^{4} - \)\(56\!\cdots\!46\)\( T^{5} + \)\(16\!\cdots\!29\)\( T^{6} )^{2} \))(\( ( 1 - 61130 T + 2565726409 T^{2} )^{2} \))(\( ( 1 - 22346 T + 2565726409 T^{2} )^{2} \))(\( ( 1 + 76150 T + 2565726409 T^{2} )^{2} \))
$41$ (\( ( 1 - 68921 T )( 1 + 68921 T ) \))(\( 1 - 9219079010 T^{2} + 22563490300366186081 T^{4} \))(\( 1 - 9487663202 T^{2} + 22563490300366186081 T^{4} \))(\( 1 - 7569104550 T^{2} + 1363608137405888943 T^{4} + \)\(15\!\cdots\!00\)\( T^{6} + \)\(30\!\cdots\!83\)\( T^{8} - \)\(38\!\cdots\!50\)\( T^{10} + \)\(11\!\cdots\!41\)\( T^{12} \))(\( ( 1 + 98550 T + 4750104241 T^{2} )^{2} \))(\( ( 1 - 103626 T + 4750104241 T^{2} )^{2} \))(\( ( 1 + 70038 T + 4750104241 T^{2} )^{2} \))
$43$ (\( 1 + 111386 T + 6321363049 T^{2} \))(\( ( 1 - 6406 T + 6321363049 T^{2} )^{2} \))(\( ( 1 + 68618 T + 6321363049 T^{2} )^{2} \))(\( ( 1 - 145530 T + 21837177927 T^{2} - 1677260136965132 T^{3} + \)\(13\!\cdots\!23\)\( T^{4} - \)\(58\!\cdots\!30\)\( T^{5} + \)\(25\!\cdots\!49\)\( T^{6} )^{2} \))(\( 1 - 10222364450 T^{2} + 39959630797262576401 T^{4} \))(\( 1 + 3585198814 T^{2} + 39959630797262576401 T^{4} \))(\( 1 - 2451652130 T^{2} + 39959630797262576401 T^{4} \))
$47$ (\( ( 1 - 103823 T )( 1 + 103823 T ) \))(\( 1 + 10801249342 T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \))(\( 1 - 21106800578 T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \))(\( 1 - 52339293510 T^{2} + \)\(12\!\cdots\!23\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(14\!\cdots\!43\)\( T^{8} - \)\(70\!\cdots\!10\)\( T^{10} + \)\(15\!\cdots\!21\)\( T^{12} \))(\( 1 + 10288348414 T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \))(\( 1 + 4308279550 T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \))(\( 1 + 1425021694 T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \))
$53$ (\( ( 1 - 148877 T )( 1 + 148877 T ) \))(\( 1 - 7253988050 T^{2} + \)\(49\!\cdots\!41\)\( T^{4} \))(\( 1 - 14819087378 T^{2} + \)\(49\!\cdots\!41\)\( T^{4} \))(\( 1 - 75107476374 T^{2} + \)\(29\!\cdots\!67\)\( T^{4} - \)\(77\!\cdots\!24\)\( T^{6} + \)\(14\!\cdots\!47\)\( T^{8} - \)\(18\!\cdots\!94\)\( T^{10} + \)\(11\!\cdots\!21\)\( T^{12} \))(\( ( 1 + 275346 T + 22164361129 T^{2} )^{2} \))(\( ( 1 - 168462 T + 22164361129 T^{2} )^{2} \))(\( ( 1 - 66942 T + 22164361129 T^{2} )^{2} \))
$59$ (\( ( 1 - 205379 T )( 1 + 205379 T ) \))(\( 1 + 22449655150 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \))(\( 1 - 61991044562 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \))(\( 1 - 180693503190 T^{2} + \)\(15\!\cdots\!31\)\( T^{4} - \)\(77\!\cdots\!36\)\( T^{6} + \)\(26\!\cdots\!11\)\( T^{8} - \)\(57\!\cdots\!90\)\( T^{10} + \)\(56\!\cdots\!41\)\( T^{12} \))(\( 1 - 19735007330 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \))(\( 1 - 71908165730 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \))(\( 1 + 68178858910 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \))
$61$ (\( 1 + 420838 T + 51520374361 T^{2} \))(\( ( 1 + 62566 T + 51520374361 T^{2} )^{2} \))(\( ( 1 - 24794 T + 51520374361 T^{2} )^{2} \))(\( ( 1 - 296046 T + 42371190135 T^{2} - 4032498794872228 T^{3} + \)\(21\!\cdots\!35\)\( T^{4} - \)\(78\!\cdots\!66\)\( T^{5} + \)\(13\!\cdots\!81\)\( T^{6} )^{2} \))(\( ( 1 - 106634 T + 51520374361 T^{2} )^{2} \))(\( ( 1 + 260470 T + 51520374361 T^{2} )^{2} \))(\( ( 1 + 257014 T + 51520374361 T^{2} )^{2} \))
$67$ (\( 1 + 172874 T + 90458382169 T^{2} \))(\( ( 1 + 438698 T + 90458382169 T^{2} )^{2} \))(\( ( 1 - 84358 T + 90458382169 T^{2} )^{2} \))(\( ( 1 - 285450 T + 220735162839 T^{2} - 36790378201873708 T^{3} + \)\(19\!\cdots\!91\)\( T^{4} - \)\(23\!\cdots\!50\)\( T^{5} + \)\(74\!\cdots\!09\)\( T^{6} )^{2} \))(\( 1 + 123394697854 T^{2} + \)\(81\!\cdots\!61\)\( T^{4} \))(\( 1 - 80375086466 T^{2} + \)\(81\!\cdots\!61\)\( T^{4} \))(\( 1 - 77748332930 T^{2} + \)\(81\!\cdots\!61\)\( T^{4} \))
$71$ (\( ( 1 - 357911 T )( 1 + 357911 T ) \))(\( 1 - 251546372642 T^{2} + \)\(16\!\cdots\!41\)\( T^{4} \))(\( 1 - 151063967522 T^{2} + \)\(16\!\cdots\!41\)\( T^{4} \))(\( 1 - 402983850726 T^{2} + \)\(77\!\cdots\!75\)\( T^{4} - \)\(10\!\cdots\!48\)\( T^{6} + \)\(12\!\cdots\!75\)\( T^{8} - \)\(10\!\cdots\!06\)\( T^{10} + \)\(44\!\cdots\!21\)\( T^{12} \))(\( 1 - 250713262370 T^{2} + \)\(16\!\cdots\!41\)\( T^{4} \))(\( 1 + 245662064350 T^{2} + \)\(16\!\cdots\!41\)\( T^{4} \))(\( 1 - 138474492194 T^{2} + \)\(16\!\cdots\!41\)\( T^{4} \))
$73$ (\( 1 - 638066 T + 151334226289 T^{2} \))(\( ( 1 + 730510 T + 151334226289 T^{2} )^{2} \))(\( ( 1 + 113806 T + 151334226289 T^{2} )^{2} \))(\( ( 1 - 559830 T + 328149232287 T^{2} - 146928200928984628 T^{3} + \)\(49\!\cdots\!43\)\( T^{4} - \)\(12\!\cdots\!30\)\( T^{5} + \)\(34\!\cdots\!69\)\( T^{6} )^{2} \))(\( ( 1 - 100978 T + 151334226289 T^{2} )^{2} \))(\( ( 1 + 395918 T + 151334226289 T^{2} )^{2} \))(\( ( 1 - 243442 T + 151334226289 T^{2} )^{2} \))
$79$ (\( 1 - 204622 T + 243087455521 T^{2} \))(\( ( 1 + 340562 T + 243087455521 T^{2} )^{2} \))(\( ( 1 - 159742 T + 243087455521 T^{2} )^{2} \))(\( ( 1 - 526818 T + 705484689807 T^{2} - 230000056692908636 T^{3} + \)\(17\!\cdots\!47\)\( T^{4} - \)\(31\!\cdots\!38\)\( T^{5} + \)\(14\!\cdots\!61\)\( T^{6} )^{2} \))(\( 1 - 480206360594 T^{2} + \)\(59\!\cdots\!41\)\( T^{4} \))(\( 1 - 176211604754 T^{2} + \)\(59\!\cdots\!41\)\( T^{4} \))(\( 1 - 260585085842 T^{2} + \)\(59\!\cdots\!41\)\( T^{4} \))
$83$ (\( ( 1 - 571787 T )( 1 + 571787 T ) \))(\( 1 - 407613512306 T^{2} + \)\(10\!\cdots\!61\)\( T^{4} \))(\( 1 - 388294032818 T^{2} + \)\(10\!\cdots\!61\)\( T^{4} \))(\( 1 - 1779363154038 T^{2} + \)\(13\!\cdots\!67\)\( T^{4} - \)\(58\!\cdots\!36\)\( T^{6} + \)\(14\!\cdots\!87\)\( T^{8} - \)\(20\!\cdots\!98\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} \))(\( 1 - 238112486210 T^{2} + \)\(10\!\cdots\!61\)\( T^{4} \))(\( 1 - 645933881666 T^{2} + \)\(10\!\cdots\!61\)\( T^{4} \))(\( 1 + 415389237694 T^{2} + \)\(10\!\cdots\!61\)\( T^{4} \))
$89$ (\( ( 1 - 704969 T )( 1 + 704969 T ) \))(\( 1 - 844406214050 T^{2} + \)\(24\!\cdots\!21\)\( T^{4} \))(\( 1 + 583819025758 T^{2} + \)\(24\!\cdots\!21\)\( T^{4} \))(\( 1 - 1464277161318 T^{2} + \)\(11\!\cdots\!39\)\( T^{4} - \)\(68\!\cdots\!36\)\( T^{6} + \)\(28\!\cdots\!19\)\( T^{8} - \)\(89\!\cdots\!38\)\( T^{10} + \)\(15\!\cdots\!61\)\( T^{12} \))(\( ( 1 + 819054 T + 496981290961 T^{2} )^{2} \))(\( ( 1 + 251886 T + 496981290961 T^{2} )^{2} \))(\( ( 1 + 686766 T + 496981290961 T^{2} )^{2} \))
$97$ (\( 1 + 56446 T + 832972004929 T^{2} \))(\( ( 1 + 281086 T + 832972004929 T^{2} )^{2} \))(\( ( 1 - 899522 T + 832972004929 T^{2} )^{2} \))(\( ( 1 + 399258 T + 1108436400207 T^{2} + 1034642865778948780 T^{3} + \)\(92\!\cdots\!03\)\( T^{4} + \)\(27\!\cdots\!78\)\( T^{5} + \)\(57\!\cdots\!89\)\( T^{6} )^{2} \))(\( ( 1 - 557026 T + 832972004929 T^{2} )^{2} \))(\( ( 1 - 517474 T + 832972004929 T^{2} )^{2} \))(\( ( 1 + 942686 T + 832972004929 T^{2} )^{2} \))
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