Properties

Label 504.2.t
Level 504
Weight 2
Character orbit t
Rep. character \(\chi_{504}(193,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 48
Newform subspaces 4
Sturm bound 192
Trace bound 3

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Defining parameters

Level: \( N \) = \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 504.t (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 63 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(192\)
Trace bound: \(3\)
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 48 160
Cusp forms 176 48 128
Eisenstein series 32 0 32

Trace form

\( 48q - 8q^{5} - 2q^{9} + O(q^{10}) \) \( 48q - 8q^{5} - 2q^{9} - 8q^{15} + 8q^{17} + 18q^{21} + 8q^{23} + 48q^{25} - 6q^{27} - 6q^{29} + 6q^{31} - 4q^{33} + 12q^{35} + 4q^{39} + 18q^{41} + 6q^{43} + 22q^{45} + 6q^{47} + 12q^{49} - 18q^{51} + 4q^{53} + 12q^{55} - 20q^{57} - 36q^{59} + 6q^{61} - 4q^{63} + 12q^{65} - 32q^{69} - 40q^{71} - 24q^{75} + 4q^{77} - 6q^{79} - 26q^{81} - 36q^{83} + 46q^{87} + 18q^{89} - 6q^{91} - 52q^{93} - 10q^{95} - 32q^{99} + O(q^{100}) \)

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.t.a \(2\) \(4.024\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(2\) \(-5\) \(q+(-1-\zeta_{6})q^{3}+q^{5}+(-2-\zeta_{6})q^{7}+\cdots\)
504.2.t.b \(2\) \(4.024\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-2\) \(-1\) \(q+(1+\zeta_{6})q^{3}-q^{5}+(-2+3\zeta_{6})q^{7}+\cdots\)
504.2.t.c \(22\) \(4.024\) None \(0\) \(-2\) \(-2\) \(-1\)
504.2.t.d \(22\) \(4.024\) None \(0\) \(2\) \(-6\) \(7\)

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

\( S_{2}^{\mathrm{old}}(504, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))
$5$ (\( ( 1 - T + 5 T^{2} )^{2} \))(\( ( 1 + T + 5 T^{2} )^{2} \))
$7$ (\( 1 + 5 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))
$11$ (\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( ( 1 - 3 T + 11 T^{2} )^{2} \))
$13$ (\( 1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4} \))(\( 1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4} \))
$17$ (\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - 5 T + 8 T^{2} - 85 T^{3} + 289 T^{4} \))
$19$ (\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))
$23$ (\( ( 1 - T + 23 T^{2} )^{2} \))(\( ( 1 - 5 T + 23 T^{2} )^{2} \))
$29$ (\( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} \))(\( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \))
$31$ (\( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} \))
$37$ (\( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))
$41$ (\( 1 + 7 T + 8 T^{2} + 287 T^{3} + 1681 T^{4} \))(\( 1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4} \))
$43$ (\( 1 + 3 T - 34 T^{2} + 129 T^{3} + 1849 T^{4} \))(\( 1 - 7 T + 6 T^{2} - 301 T^{3} + 1849 T^{4} \))
$47$ (\( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \))
$53$ (\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} \))
$59$ (\( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))
$61$ (\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} \))(\( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} \))
$71$ (\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))
$73$ (\( 1 - 13 T + 96 T^{2} - 949 T^{3} + 5329 T^{4} \))(\( 1 - 5 T - 48 T^{2} - 365 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 13 T + 86 T^{2} - 1079 T^{3} + 6889 T^{4} \))(\( 1 - 15 T + 142 T^{2} - 1245 T^{3} + 6889 T^{4} \))
$89$ (\( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} \))(\( 1 - 5 T - 64 T^{2} - 445 T^{3} + 7921 T^{4} \))
$97$ (\( 1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4} \))(\( 1 + 7 T - 48 T^{2} + 679 T^{3} + 9409 T^{4} \))
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