Properties

Label 48.2.j
Level 48
Weight 2
Character orbit j
Rep. character \(\chi_{48}(13,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 8
Newform subspaces 1
Sturm bound 16
Trace bound 0

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Defining parameters

Level: \( N \) = \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 48.j (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 16 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(16\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(48, [\chi])\).

Total New Old
Modular forms 20 8 12
Cusp forms 12 8 4
Eisenstein series 8 0 8

Trace form

\( 8q - 4q^{4} - 12q^{8} + O(q^{10}) \) \( 8q - 4q^{4} - 12q^{8} - 8q^{10} - 8q^{11} + 8q^{12} + 12q^{14} - 8q^{15} + 4q^{18} - 8q^{19} + 16q^{20} + 4q^{24} + 20q^{26} + 8q^{28} - 16q^{29} - 8q^{30} + 24q^{31} + 24q^{35} - 4q^{36} - 16q^{37} - 8q^{38} + 16q^{40} - 20q^{42} - 8q^{43} - 40q^{44} - 8q^{46} - 16q^{48} - 8q^{49} - 36q^{50} + 8q^{51} - 16q^{52} + 16q^{53} + 4q^{54} - 16q^{58} + 32q^{59} + 24q^{60} + 16q^{61} - 12q^{62} + 8q^{63} + 8q^{64} - 16q^{65} + 24q^{66} - 16q^{67} + 32q^{68} + 16q^{69} + 32q^{70} - 4q^{72} + 52q^{74} + 16q^{75} + 8q^{76} + 16q^{77} - 12q^{78} - 24q^{79} + 8q^{80} - 8q^{81} + 40q^{82} - 40q^{83} - 24q^{84} - 16q^{85} - 16q^{86} + 32q^{88} - 8q^{90} - 8q^{91} - 16q^{92} + 8q^{94} - 48q^{95} - 40q^{98} - 8q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(48, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
48.2.j.a \(8\) \(0.383\) 8.0.18939904.2 None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{4}+\beta _{5})q^{2}-\beta _{5}q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(48, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(48, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + 4 T^{3} + 2 T^{4} + 8 T^{5} + 8 T^{6} + 16 T^{8} \)
$3$ \( ( 1 + T^{4} )^{2} \)
$5$ \( 1 - 16 T^{3} - 12 T^{4} + 48 T^{5} + 128 T^{6} - 32 T^{7} - 506 T^{8} - 160 T^{9} + 3200 T^{10} + 6000 T^{11} - 7500 T^{12} - 50000 T^{13} + 390625 T^{16} \)
$7$ \( 1 - 24 T^{2} + 292 T^{4} - 2440 T^{6} + 17222 T^{8} - 119560 T^{10} + 701092 T^{12} - 2823576 T^{14} + 5764801 T^{16} \)
$11$ \( 1 + 8 T + 32 T^{2} + 88 T^{3} + 132 T^{4} + 344 T^{5} + 2400 T^{6} + 13000 T^{7} + 54374 T^{8} + 143000 T^{9} + 290400 T^{10} + 457864 T^{11} + 1932612 T^{12} + 14172488 T^{13} + 56689952 T^{14} + 155897368 T^{15} + 214358881 T^{16} \)
$13$ \( 1 - 64 T^{3} - 4 T^{4} + 704 T^{5} + 2048 T^{6} - 1408 T^{7} - 53466 T^{8} - 18304 T^{9} + 346112 T^{10} + 1546688 T^{11} - 114244 T^{12} - 23762752 T^{13} + 815730721 T^{16} \)
$17$ \( ( 1 + 36 T^{2} + 64 T^{3} + 662 T^{4} + 1088 T^{5} + 10404 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( 1 + 8 T + 32 T^{2} + 120 T^{3} + 452 T^{4} + 2168 T^{5} + 10080 T^{6} + 37832 T^{7} + 138918 T^{8} + 718808 T^{9} + 3638880 T^{10} + 14870312 T^{11} + 58905092 T^{12} + 297131880 T^{13} + 1505468192 T^{14} + 7150973912 T^{15} + 16983563041 T^{16} \)
$23$ \( ( 1 - 38 T^{2} + 529 T^{4} )^{4} \)
$29$ \( 1 + 16 T + 128 T^{2} + 928 T^{3} + 6580 T^{4} + 38208 T^{5} + 199680 T^{6} + 1073680 T^{7} + 5802054 T^{8} + 31136720 T^{9} + 167930880 T^{10} + 931854912 T^{11} + 4653908980 T^{12} + 19034346272 T^{13} + 76137385088 T^{14} + 275998020944 T^{15} + 500246412961 T^{16} \)
$31$ \( ( 1 - 12 T + 164 T^{2} - 1140 T^{3} + 8218 T^{4} - 35340 T^{5} + 157604 T^{6} - 357492 T^{7} + 923521 T^{8} )^{2} \)
$37$ \( 1 + 16 T + 128 T^{2} + 1008 T^{3} + 5948 T^{4} + 15248 T^{5} - 9344 T^{6} - 717840 T^{7} - 7530650 T^{8} - 26560080 T^{9} - 12791936 T^{10} + 772356944 T^{11} + 11147509628 T^{12} + 69898708656 T^{13} + 328412980352 T^{14} + 1518910034128 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 - 200 T^{2} + 19452 T^{4} - 1244536 T^{6} + 58583750 T^{8} - 2092065016 T^{10} + 54966702972 T^{12} - 950020848200 T^{14} + 7984925229121 T^{16} \)
$43$ \( 1 + 8 T + 32 T^{2} + 56 T^{3} + 260 T^{4} + 504 T^{5} - 2720 T^{6} - 625528 T^{7} - 7635866 T^{8} - 26897704 T^{9} - 5029280 T^{10} + 40071528 T^{11} + 888888260 T^{12} + 8232472808 T^{13} + 202283617568 T^{14} + 2174548888856 T^{15} + 11688200277601 T^{16} \)
$47$ \( ( 1 + 86 T^{2} + 2209 T^{4} )^{4} \)
$53$ \( 1 - 16 T + 128 T^{2} - 928 T^{3} + 8564 T^{4} - 82496 T^{5} + 654336 T^{6} - 5021328 T^{7} + 38116486 T^{8} - 266130384 T^{9} + 1838029824 T^{10} - 12281756992 T^{11} + 67574079284 T^{12} - 388085417504 T^{13} + 2837038224512 T^{14} - 18795378237392 T^{15} + 62259690411361 T^{16} \)
$59$ \( ( 1 - 8 T + 32 T^{2} - 472 T^{3} + 3481 T^{4} )^{4} \)
$61$ \( 1 - 16 T + 128 T^{2} - 1392 T^{3} + 14204 T^{4} - 79760 T^{5} + 426880 T^{6} - 2945904 T^{7} + 19569574 T^{8} - 179700144 T^{9} + 1588420480 T^{10} - 18104004560 T^{11} + 196666325564 T^{12} - 1175678050992 T^{13} + 6594607918208 T^{14} - 50283885376336 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 + 16 T + 128 T^{2} + 304 T^{3} + 4388 T^{4} + 107696 T^{5} + 1207680 T^{6} + 4800272 T^{7} + 13154790 T^{8} + 321618224 T^{9} + 5421275520 T^{10} + 32390972048 T^{11} + 88423118948 T^{12} + 410438032528 T^{13} + 11578672917632 T^{14} + 96971385685168 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 - 440 T^{2} + 90844 T^{4} - 11522952 T^{6} + 984512390 T^{8} - 58087201032 T^{10} + 2308498748764 T^{12} - 56364124925240 T^{14} + 645753531245761 T^{16} \)
$73$ \( 1 - 328 T^{2} + 45404 T^{4} - 3734648 T^{6} + 259745542 T^{8} - 19901939192 T^{10} + 1289393734364 T^{12} - 49637626222792 T^{14} + 806460091894081 T^{16} \)
$79$ \( ( 1 + 12 T + 148 T^{2} - 44 T^{3} + 794 T^{4} - 3476 T^{5} + 923668 T^{6} + 5916468 T^{7} + 38950081 T^{8} )^{2} \)
$83$ \( 1 + 40 T + 800 T^{2} + 11000 T^{3} + 122436 T^{4} + 1297720 T^{5} + 14460000 T^{6} + 161033000 T^{7} + 1597489574 T^{8} + 13365739000 T^{9} + 99614940000 T^{10} + 742019425640 T^{11} + 5810606989956 T^{12} + 43329447073000 T^{13} + 261552298695200 T^{14} + 1085442039585080 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 - 248 T^{2} + 36316 T^{4} - 4626504 T^{6} + 476004998 T^{8} - 36646538184 T^{10} + 2278547224156 T^{12} - 123251360158328 T^{14} + 3936588805702081 T^{16} \)
$97$ \( ( 1 + 164 T^{2} + 768 T^{3} + 13510 T^{4} + 74496 T^{5} + 1543076 T^{6} + 88529281 T^{8} )^{2} \)
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