Properties

Label 504.2.a
Level 504
Weight 2
Character orbit a
Rep. character \(\chi_{504}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 8
Sturm bound 192
Trace bound 11

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Defining parameters

Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(192\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(504))\).

Total New Old
Modular forms 112 8 104
Cusp forms 81 8 73
Eisenstein series 31 0 31

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(3\)
Minus space\(-\)\(5\)

Trace form

\( 8q - 2q^{5} + O(q^{10}) \) \( 8q - 2q^{5} + 4q^{11} - 2q^{13} + 4q^{17} + 6q^{19} + 4q^{25} - 4q^{29} - 4q^{31} + 6q^{35} - 4q^{37} + 12q^{41} - 4q^{43} + 12q^{47} + 8q^{49} + 8q^{53} + 24q^{55} - 22q^{59} - 6q^{61} - 12q^{65} + 24q^{67} + 16q^{71} - 24q^{73} - 4q^{77} - 24q^{79} - 30q^{83} + 20q^{85} - 8q^{89} - 10q^{91} - 24q^{95} - 28q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(504))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 7
504.2.a.a \(1\) \(4.024\) \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) \(+\) \(+\) \(+\) \(q-2q^{5}-q^{7}-2q^{11}+2q^{13}-6q^{17}+\cdots\)
504.2.a.b \(1\) \(4.024\) \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) \(-\) \(-\) \(+\) \(q-2q^{5}-q^{7}-2q^{13}-6q^{17}-4q^{19}+\cdots\)
504.2.a.c \(1\) \(4.024\) \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) \(+\) \(-\) \(+\) \(q-2q^{5}-q^{7}+4q^{11}+2q^{13}+6q^{17}+\cdots\)
504.2.a.d \(1\) \(4.024\) \(\Q\) None \(0\) \(0\) \(-2\) \(1\) \(-\) \(+\) \(-\) \(q-2q^{5}+q^{7}-6q^{11}-6q^{13}+2q^{17}+\cdots\)
504.2.a.e \(1\) \(4.024\) \(\Q\) None \(0\) \(0\) \(-2\) \(1\) \(-\) \(-\) \(-\) \(q-2q^{5}+q^{7}+6q^{13}+2q^{17}+4q^{19}+\cdots\)
504.2.a.f \(1\) \(4.024\) \(\Q\) None \(0\) \(0\) \(2\) \(-1\) \(-\) \(+\) \(+\) \(q+2q^{5}-q^{7}+2q^{11}+2q^{13}+6q^{17}+\cdots\)
504.2.a.g \(1\) \(4.024\) \(\Q\) None \(0\) \(0\) \(2\) \(1\) \(+\) \(+\) \(-\) \(q+2q^{5}+q^{7}+6q^{11}-6q^{13}-2q^{17}+\cdots\)
504.2.a.h \(1\) \(4.024\) \(\Q\) None \(0\) \(0\) \(4\) \(1\) \(-\) \(-\) \(-\) \(q+4q^{5}+q^{7}+2q^{17}-2q^{19}-8q^{23}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(504))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(504)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))
$7$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))
$11$ (\( 1 + 2 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 8 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))
$23$ (\( 1 + 6 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 2 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))
$29$ (\( 1 + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))
$37$ (\( 1 - 10 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))
$41$ (\( 1 + 2 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 12 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))
$47$ (\( 1 + 4 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))
$53$ (\( 1 - 12 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 4 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))
$59$ (\( 1 + 12 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))
$61$ (\( 1 - 6 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))
$71$ (\( 1 - 14 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 14 T + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))
$73$ (\( 1 + 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))
$79$ (\( 1 + 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))
$83$ (\( 1 - 16 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 8 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 16 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))
$89$ (\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 - 2 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))
$97$ (\( 1 + 18 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 18 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))
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