Properties

Label 6048.2.j
Level 6048
Weight 2
Character orbit j
Rep. character \(\chi_{6048}(5615,\cdot)\)
Character field \(\Q\)
Dimension 96
Newform subspaces 4
Sturm bound 2304
Trace bound 5

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Defining parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.j (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 24 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(2304\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

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

\( 96q + O(q^{10}) \) \( 96q + 16q^{19} + 96q^{25} - 32q^{43} - 96q^{49} - 64q^{67} + O(q^{100}) \)

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.j.a \(8\) \(48.294\) 8.0.56070144.2 None \(0\) \(0\) \(-8\) \(0\) \(q+(-1-\beta _{4})q^{5}-\beta _{2}q^{7}+(-\beta _{1}-\beta _{6}+\cdots)q^{11}+\cdots\)
6048.2.j.b \(8\) \(48.294\) 8.0.56070144.2 None \(0\) \(0\) \(8\) \(0\) \(q+(1+\beta _{4})q^{5}-\beta _{2}q^{7}+(\beta _{1}+\beta _{6})q^{11}+\cdots\)
6048.2.j.c \(32\) \(48.294\) None \(0\) \(0\) \(0\) \(0\)
6048.2.j.d \(48\) \(48.294\) None \(0\) \(0\) \(0\) \(0\)

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( ( 1 + 4 T + 10 T^{2} + 32 T^{3} + 87 T^{4} + 160 T^{5} + 250 T^{6} + 500 T^{7} + 625 T^{8} )^{2} \))(\( ( 1 - 4 T + 10 T^{2} - 32 T^{3} + 87 T^{4} - 160 T^{5} + 250 T^{6} - 500 T^{7} + 625 T^{8} )^{2} \))
$7$ (\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))
$11$ (\( 1 - 44 T^{2} + 994 T^{4} - 15968 T^{6} + 198763 T^{8} - 1932128 T^{10} + 14553154 T^{12} - 77948684 T^{14} + 214358881 T^{16} \))(\( 1 - 44 T^{2} + 994 T^{4} - 15968 T^{6} + 198763 T^{8} - 1932128 T^{10} + 14553154 T^{12} - 77948684 T^{14} + 214358881 T^{16} \))
$13$ (\( 1 - 8 T^{2} + 228 T^{4} - 4504 T^{6} + 20006 T^{8} - 761176 T^{10} + 6511908 T^{12} - 38614472 T^{14} + 815730721 T^{16} \))(\( 1 - 8 T^{2} + 228 T^{4} - 4504 T^{6} + 20006 T^{8} - 761176 T^{10} + 6511908 T^{12} - 38614472 T^{14} + 815730721 T^{16} \))
$17$ (\( ( 1 - 8 T + 17 T^{2} )^{4}( 1 + 8 T + 17 T^{2} )^{4} \))(\( ( 1 - 8 T + 17 T^{2} )^{4}( 1 + 8 T + 17 T^{2} )^{4} \))
$19$ (\( ( 1 + 8 T + 30 T^{2} + 112 T^{3} + 611 T^{4} + 2128 T^{5} + 10830 T^{6} + 54872 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 + 8 T + 30 T^{2} + 112 T^{3} + 611 T^{4} + 2128 T^{5} + 10830 T^{6} + 54872 T^{7} + 130321 T^{8} )^{2} \))
$23$ (\( ( 1 + 8 T + 46 T^{2} + 88 T^{3} + 231 T^{4} + 2024 T^{5} + 24334 T^{6} + 97336 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 - 8 T + 46 T^{2} - 88 T^{3} + 231 T^{4} - 2024 T^{5} + 24334 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 + 6 T + 64 T^{2} + 174 T^{3} + 841 T^{4} )^{4} \))(\( ( 1 - 6 T + 64 T^{2} - 174 T^{3} + 841 T^{4} )^{4} \))
$31$ (\( 1 - 156 T^{2} + 12106 T^{4} - 610416 T^{6} + 22035219 T^{8} - 586609776 T^{10} + 11180145226 T^{12} - 138450574236 T^{14} + 852891037441 T^{16} \))(\( 1 - 156 T^{2} + 12106 T^{4} - 610416 T^{6} + 22035219 T^{8} - 586609776 T^{10} + 11180145226 T^{12} - 138450574236 T^{14} + 852891037441 T^{16} \))
$37$ (\( 1 - 164 T^{2} + 14154 T^{4} - 829168 T^{6} + 35617331 T^{8} - 1135130992 T^{10} + 26526874794 T^{12} - 420779131076 T^{14} + 3512479453921 T^{16} \))(\( 1 - 164 T^{2} + 14154 T^{4} - 829168 T^{6} + 35617331 T^{8} - 1135130992 T^{10} + 26526874794 T^{12} - 420779131076 T^{14} + 3512479453921 T^{16} \))
$41$ (\( 1 - 116 T^{2} + 11362 T^{4} - 660608 T^{6} + 32960203 T^{8} - 1110482048 T^{10} + 32106296482 T^{12} - 551012091956 T^{14} + 7984925229121 T^{16} \))(\( 1 - 116 T^{2} + 11362 T^{4} - 660608 T^{6} + 32960203 T^{8} - 1110482048 T^{10} + 32106296482 T^{12} - 551012091956 T^{14} + 7984925229121 T^{16} \))
$43$ (\( ( 1 + 4 T + 84 T^{2} + 44 T^{3} + 3002 T^{4} + 1892 T^{5} + 155316 T^{6} + 318028 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 + 4 T + 84 T^{2} + 44 T^{3} + 3002 T^{4} + 1892 T^{5} + 155316 T^{6} + 318028 T^{7} + 3418801 T^{8} )^{2} \))
$47$ (\( ( 1 - 4 T + 76 T^{2} - 44 T^{3} + 2154 T^{4} - 2068 T^{5} + 167884 T^{6} - 415292 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 + 4 T + 76 T^{2} + 44 T^{3} + 2154 T^{4} + 2068 T^{5} + 167884 T^{6} + 415292 T^{7} + 4879681 T^{8} )^{2} \))
$53$ (\( ( 1 + 124 T^{2} - 192 T^{3} + 7734 T^{4} - 10176 T^{5} + 348316 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 + 124 T^{2} + 192 T^{3} + 7734 T^{4} + 10176 T^{5} + 348316 T^{6} + 7890481 T^{8} )^{2} \))
$59$ (\( 1 - 232 T^{2} + 28836 T^{4} - 2582456 T^{6} + 175948646 T^{8} - 8989529336 T^{10} + 349416221796 T^{12} - 9785883804712 T^{14} + 146830437604321 T^{16} \))(\( 1 - 232 T^{2} + 28836 T^{4} - 2582456 T^{6} + 175948646 T^{8} - 8989529336 T^{10} + 349416221796 T^{12} - 9785883804712 T^{14} + 146830437604321 T^{16} \))
$61$ (\( ( 1 - 92 T^{2} + 6486 T^{4} - 342332 T^{6} + 13845841 T^{8} )^{2} \))(\( ( 1 - 92 T^{2} + 6486 T^{4} - 342332 T^{6} + 13845841 T^{8} )^{2} \))
$67$ (\( ( 1 + 4 T + 228 T^{2} + 620 T^{3} + 21530 T^{4} + 41540 T^{5} + 1023492 T^{6} + 1203052 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + 4 T + 228 T^{2} + 620 T^{3} + 21530 T^{4} + 41540 T^{5} + 1023492 T^{6} + 1203052 T^{7} + 20151121 T^{8} )^{2} \))
$71$ (\( ( 1 + 16 T + 310 T^{2} + 3320 T^{3} + 33807 T^{4} + 235720 T^{5} + 1562710 T^{6} + 5726576 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 - 16 T + 310 T^{2} - 3320 T^{3} + 33807 T^{4} - 235720 T^{5} + 1562710 T^{6} - 5726576 T^{7} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 - 4 T + 156 T^{2} - 1460 T^{3} + 11402 T^{4} - 106580 T^{5} + 831324 T^{6} - 1556068 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 - 4 T + 156 T^{2} - 1460 T^{3} + 11402 T^{4} - 106580 T^{5} + 831324 T^{6} - 1556068 T^{7} + 28398241 T^{8} )^{2} \))
$79$ (\( 1 - 248 T^{2} + 37188 T^{4} - 3949096 T^{6} + 344639558 T^{8} - 24646308136 T^{10} + 1448475612228 T^{12} - 60285688969208 T^{14} + 1517108809906561 T^{16} \))(\( 1 - 248 T^{2} + 37188 T^{4} - 3949096 T^{6} + 344639558 T^{8} - 24646308136 T^{10} + 1448475612228 T^{12} - 60285688969208 T^{14} + 1517108809906561 T^{16} \))
$83$ (\( 1 - 232 T^{2} + 34596 T^{4} - 3949112 T^{6} + 355979174 T^{8} - 27205432568 T^{10} + 1641868073316 T^{12} - 75850166621608 T^{14} + 2252292232139041 T^{16} \))(\( 1 - 232 T^{2} + 34596 T^{4} - 3949112 T^{6} + 355979174 T^{8} - 27205432568 T^{10} + 1641868073316 T^{12} - 75850166621608 T^{14} + 2252292232139041 T^{16} \))
$89$ (\( 1 - 292 T^{2} + 47890 T^{4} - 6052768 T^{6} + 614802619 T^{8} - 47943975328 T^{10} + 3004725921490 T^{12} - 145118536960612 T^{14} + 3936588805702081 T^{16} \))(\( 1 - 292 T^{2} + 47890 T^{4} - 6052768 T^{6} + 614802619 T^{8} - 47943975328 T^{10} + 3004725921490 T^{12} - 145118536960612 T^{14} + 3936588805702081 T^{16} \))
$97$ (\( ( 1 + 16 T + 300 T^{2} + 4208 T^{3} + 41318 T^{4} + 408176 T^{5} + 2822700 T^{6} + 14602768 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 + 16 T + 300 T^{2} + 4208 T^{3} + 41318 T^{4} + 408176 T^{5} + 2822700 T^{6} + 14602768 T^{7} + 88529281 T^{8} )^{2} \))
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