Properties

Label 504.2.bm
Level 504
Weight 2
Character orbit bm
Rep. character \(\chi_{504}(107,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 64
Newform subspaces 3
Sturm bound 192
Trace bound 2

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Defining parameters

Level: \( N \) = \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 504.bm (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 168 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(192\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(504, [\chi])\).

Total New Old
Modular forms 208 64 144
Cusp forms 176 64 112
Eisenstein series 32 0 32

Trace form

\( 64q + O(q^{10}) \) \( 64q + 4q^{16} - 16q^{22} - 32q^{25} + 20q^{28} + 80q^{34} - 52q^{40} + 24q^{46} + 16q^{49} + 28q^{52} + 36q^{58} - 24q^{64} + 16q^{67} - 12q^{70} - 16q^{73} - 120q^{76} - 28q^{82} - 52q^{88} - 48q^{91} + 12q^{94} + 32q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(504, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
504.2.bm.a \(8\) \(4.024\) \(\Q(\zeta_{24})\) None \(-4\) \(0\) \(4\) \(0\) \(q+(-1-\zeta_{24}+\zeta_{24}^{2})q^{2}+(2\zeta_{24}-2\zeta_{24}^{3}+\cdots)q^{4}+\cdots\)
504.2.bm.b \(8\) \(4.024\) \(\Q(\zeta_{24})\) None \(4\) \(0\) \(-4\) \(0\) \(q+(1-\zeta_{24}-\zeta_{24}^{2})q^{2}+(-2\zeta_{24}+2\zeta_{24}^{3}+\cdots)q^{4}+\cdots\)
504.2.bm.c \(48\) \(4.024\) None \(0\) \(0\) \(0\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(504, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(504, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2} \))
$3$ 1
$5$ (\( ( 1 - 2 T + T^{2} + 14 T^{3} - 36 T^{4} + 70 T^{5} + 25 T^{6} - 250 T^{7} + 625 T^{8} )^{2} \))(\( ( 1 + 2 T + T^{2} - 14 T^{3} - 36 T^{4} - 70 T^{5} + 25 T^{6} + 250 T^{7} + 625 T^{8} )^{2} \))
$7$ (\( 1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8} \))(\( 1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8} \))
$11$ (\( 1 + 22 T^{2} + 193 T^{4} + 1078 T^{6} + 11476 T^{8} + 130438 T^{10} + 2825713 T^{12} + 38974342 T^{14} + 214358881 T^{16} \))(\( 1 + 22 T^{2} + 193 T^{4} + 1078 T^{6} + 11476 T^{8} + 130438 T^{10} + 2825713 T^{12} + 38974342 T^{14} + 214358881 T^{16} \))
$13$ (\( ( 1 - 16 T^{2} + 274 T^{4} - 2704 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - 16 T^{2} + 274 T^{4} - 2704 T^{6} + 28561 T^{8} )^{2} \))
$17$ (\( 1 + 24 T^{2} + 142 T^{4} - 3456 T^{6} - 61629 T^{8} - 998784 T^{10} + 11859982 T^{12} + 579301656 T^{14} + 6975757441 T^{16} \))(\( 1 + 24 T^{2} + 142 T^{4} - 3456 T^{6} - 61629 T^{8} - 998784 T^{10} + 11859982 T^{12} + 579301656 T^{14} + 6975757441 T^{16} \))
$19$ (\( ( 1 - 8 T + 12 T^{2} - 112 T^{3} + 1127 T^{4} - 2128 T^{5} + 4332 T^{6} - 54872 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 - 8 T + 12 T^{2} - 112 T^{3} + 1127 T^{4} - 2128 T^{5} + 4332 T^{6} - 54872 T^{7} + 130321 T^{8} )^{2} \))
$23$ (\( ( 1 + 4 T - 32 T^{2} + 8 T^{3} + 1407 T^{4} + 184 T^{5} - 16928 T^{6} + 48668 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 - 4 T - 32 T^{2} - 8 T^{3} + 1407 T^{4} - 184 T^{5} - 16928 T^{6} - 48668 T^{7} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 + 3 T + 29 T^{2} )^{8} \))(\( ( 1 - 3 T + 29 T^{2} )^{8} \))
$31$ (\( 1 + 102 T^{2} + 5953 T^{4} + 257958 T^{6} + 8889636 T^{8} + 247897638 T^{10} + 5497720513 T^{12} + 90525375462 T^{14} + 852891037441 T^{16} \))(\( 1 + 102 T^{2} + 5953 T^{4} + 257958 T^{6} + 8889636 T^{8} + 247897638 T^{10} + 5497720513 T^{12} + 90525375462 T^{14} + 852891037441 T^{16} \))
$37$ (\( 1 + 40 T^{2} - 738 T^{4} - 16000 T^{6} + 1401683 T^{8} - 21904000 T^{10} - 1383130818 T^{12} + 102629056360 T^{14} + 3512479453921 T^{16} \))(\( 1 + 40 T^{2} - 738 T^{4} - 16000 T^{6} + 1401683 T^{8} - 21904000 T^{10} - 1383130818 T^{12} + 102629056360 T^{14} + 3512479453921 T^{16} \))
$41$ (\( ( 1 - 76 T^{2} + 3654 T^{4} - 127756 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 76 T^{2} + 3654 T^{4} - 127756 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 - 20 T + 184 T^{2} - 860 T^{3} + 1849 T^{4} )^{4} \))(\( ( 1 - 20 T + 184 T^{2} - 860 T^{3} + 1849 T^{4} )^{4} \))
$47$ (\( ( 1 - 8 T - 44 T^{2} - 112 T^{3} + 6447 T^{4} - 5264 T^{5} - 97196 T^{6} - 830584 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 + 8 T - 44 T^{2} + 112 T^{3} + 6447 T^{4} + 5264 T^{5} - 97196 T^{6} + 830584 T^{7} + 4879681 T^{8} )^{2} \))
$53$ (\( ( 1 - 18 T + 145 T^{2} - 1314 T^{3} + 12060 T^{4} - 69642 T^{5} + 407305 T^{6} - 2679786 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 + 18 T + 145 T^{2} + 1314 T^{3} + 12060 T^{4} + 69642 T^{5} + 407305 T^{6} + 2679786 T^{7} + 7890481 T^{8} )^{2} \))
$59$ (\( 1 + 134 T^{2} + 6705 T^{4} + 574726 T^{6} + 51962804 T^{8} + 2000621206 T^{10} + 81246905505 T^{12} + 5652191507894 T^{14} + 146830437604321 T^{16} \))(\( 1 + 134 T^{2} + 6705 T^{4} + 574726 T^{6} + 51962804 T^{8} + 2000621206 T^{10} + 81246905505 T^{12} + 5652191507894 T^{14} + 146830437604321 T^{16} \))
$61$ (\( 1 + 200 T^{2} + 22846 T^{4} + 1942400 T^{6} + 131573875 T^{8} + 7227670400 T^{10} + 316322083486 T^{12} + 10304074872200 T^{14} + 191707312997281 T^{16} \))(\( 1 + 200 T^{2} + 22846 T^{4} + 1942400 T^{6} + 131573875 T^{8} + 7227670400 T^{10} + 316322083486 T^{12} + 10304074872200 T^{14} + 191707312997281 T^{16} \))
$67$ (\( ( 1 - 4 T - 90 T^{2} + 112 T^{3} + 5675 T^{4} + 7504 T^{5} - 404010 T^{6} - 1203052 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 - 4 T - 90 T^{2} + 112 T^{3} + 5675 T^{4} + 7504 T^{5} - 404010 T^{6} - 1203052 T^{7} + 20151121 T^{8} )^{2} \))
$71$ (\( ( 1 - 8 T + 60 T^{2} - 568 T^{3} + 5041 T^{4} )^{4} \))(\( ( 1 + 8 T + 60 T^{2} + 568 T^{3} + 5041 T^{4} )^{4} \))
$73$ (\( ( 1 + 8 T + 30 T^{2} - 896 T^{3} - 8845 T^{4} - 65408 T^{5} + 159870 T^{6} + 3112136 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 + 8 T + 30 T^{2} - 896 T^{3} - 8845 T^{4} - 65408 T^{5} + 159870 T^{6} + 3112136 T^{7} + 28398241 T^{8} )^{2} \))
$79$ (\( 1 + 214 T^{2} + 22065 T^{4} + 2407286 T^{6} + 238397444 T^{8} + 15023871926 T^{10} + 859433537265 T^{12} + 52020715481494 T^{14} + 1517108809906561 T^{16} \))(\( 1 + 214 T^{2} + 22065 T^{4} + 2407286 T^{6} + 238397444 T^{8} + 15023871926 T^{10} + 859433537265 T^{12} + 52020715481494 T^{14} + 1517108809906561 T^{16} \))
$83$ (\( ( 1 - 230 T^{2} + 26803 T^{4} - 1584470 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 230 T^{2} + 26803 T^{4} - 1584470 T^{6} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 + 89 T^{2} + 7921 T^{4} )^{4} \))(\( ( 1 + 89 T^{2} + 7921 T^{4} )^{4} \))
$97$ (\( ( 1 + 26 T + 355 T^{2} + 2522 T^{3} + 9409 T^{4} )^{4} \))(\( ( 1 + 26 T + 355 T^{2} + 2522 T^{3} + 9409 T^{4} )^{4} \))
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