Properties

Label 504.2.bf
Level 504
Weight 2
Character orbit bf
Rep. character \(\chi_{504}(115,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 184
Newform subspaces 2
Sturm bound 192
Trace bound 1

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Defining parameters

Level: \( N \) = \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 504.bf (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 504 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(192\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 200 200 0
Cusp forms 184 184 0
Eisenstein series 16 16 0

Trace form

\( 184q - 2q^{2} - 6q^{3} - 2q^{4} - 6q^{6} - 8q^{8} - 2q^{9} + O(q^{10}) \) \( 184q - 2q^{2} - 6q^{3} - 2q^{4} - 6q^{6} - 8q^{8} - 2q^{9} - 6q^{10} + 2q^{11} + 12q^{12} + 2q^{14} - 2q^{16} - 12q^{17} - 4q^{18} - 12q^{19} + 24q^{20} - 6q^{22} - 12q^{24} - 74q^{25} - 5q^{30} - 42q^{32} - 6q^{33} - 6q^{34} - 18q^{35} + 2q^{36} - 33q^{38} + 12q^{40} + 10q^{42} - 4q^{43} - 21q^{44} + 2q^{46} - 9q^{48} - 2q^{49} + 19q^{50} + 6q^{51} + 21q^{52} - 51q^{54} - 2q^{56} - 20q^{57} + 5q^{58} + 25q^{60} - 8q^{64} + 36q^{65} + 36q^{66} - 4q^{67} + 12q^{68} + 21q^{70} - 4q^{72} - 12q^{73} + 47q^{74} - 6q^{75} - 12q^{76} - 39q^{78} - 63q^{80} + 14q^{81} - 12q^{82} + 60q^{83} + 7q^{84} - 31q^{86} + 9q^{88} - 36q^{89} - 33q^{90} + 20q^{91} - 32q^{92} - 45q^{96} + 27q^{98} - 26q^{99} + O(q^{100}) \)

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.bf.a \(4\) \(4.024\) \(\Q(\zeta_{12})\) None \(4\) \(0\) \(0\) \(0\) \(q+(1+\zeta_{12}^{3})q^{2}+(1-2\zeta_{12}^{2})q^{3}+2\zeta_{12}^{3}q^{4}+\cdots\)
504.2.bf.b \(180\) \(4.024\) None \(-6\) \(-6\) \(0\) \(0\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T + 2 T^{2} )^{2} \))
$3$ (\( ( 1 + 3 T^{2} )^{2} \))
$5$ (\( 1 - 7 T^{2} + 24 T^{4} - 175 T^{6} + 625 T^{8} \))
$7$ (\( 1 + 2 T^{2} + 49 T^{4} \))
$11$ (\( ( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} )^{2} \))
$13$ (\( ( 1 - 22 T^{2} + 169 T^{4} )( 1 - T^{2} + 169 T^{4} ) \))
$17$ (\( ( 1 - 3 T + 20 T^{2} - 51 T^{3} + 289 T^{4} )^{2} \))
$19$ (\( ( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 - 3 T^{2} - 520 T^{4} - 1587 T^{6} + 279841 T^{8} \))
$29$ (\( 1 + 9 T^{2} - 760 T^{4} + 7569 T^{6} + 707281 T^{8} \))
$31$ (\( ( 1 + 14 T^{2} + 961 T^{4} )^{2} \))
$37$ (\( ( 1 + 26 T^{2} + 1369 T^{4} )( 1 + 47 T^{2} + 1369 T^{4} ) \))
$41$ (\( ( 1 - 21 T + 188 T^{2} - 861 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 + 9 T + 38 T^{2} + 387 T^{3} + 1849 T^{4} )^{2} \))
$47$ (\( ( 1 + 82 T^{2} + 2209 T^{4} )^{2} \))
$53$ (\( 1 - 15 T^{2} - 2584 T^{4} - 42135 T^{6} + 7890481 T^{8} \))
$59$ (\( ( 1 - 70 T^{2} + 3481 T^{4} )^{2} \))
$61$ (\( ( 1 + 110 T^{2} + 3721 T^{4} )^{2} \))
$67$ (\( ( 1 + 8 T + 67 T^{2} )^{4} \))
$71$ (\( ( 1 - 42 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 + 15 T + 148 T^{2} + 1095 T^{3} + 5329 T^{4} )^{2} \))
$79$ (\( ( 1 - 122 T^{2} + 6241 T^{4} )^{2} \))
$83$ (\( ( 1 - 27 T + 326 T^{2} - 2241 T^{3} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 - 15 T + 164 T^{2} - 1335 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 - 3 T + 100 T^{2} - 291 T^{3} + 9409 T^{4} )^{2} \))
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