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

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