Properties

Label 288.2.a
Level 288
Weight 2
Character orbit a
Rep. character \(\chi_{288}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newform subspaces 5
Sturm bound 96
Trace bound 7

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Defining parameters

Level: \( N \) = \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 288.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(96\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(288))\).

Total New Old
Modular forms 64 5 59
Cusp forms 33 5 28
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.

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

Trace form

\( 5q - 2q^{5} + O(q^{10}) \) \( 5q - 2q^{5} - 10q^{13} + 10q^{17} + 19q^{25} + 6q^{29} - 10q^{37} - 14q^{41} - 3q^{49} - 34q^{53} - 18q^{61} + 20q^{65} - 6q^{73} + 32q^{77} + 36q^{85} - 30q^{89} - 46q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(288))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3
288.2.a.a \(1\) \(2.300\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-4\) \(0\) \(+\) \(+\) \(q-4q^{5}-6q^{13}-8q^{17}+11q^{25}+\cdots\)
288.2.a.b \(1\) \(2.300\) \(\Q\) None \(0\) \(0\) \(-2\) \(-4\) \(-\) \(-\) \(q-2q^{5}-4q^{7}-4q^{11}-2q^{13}+6q^{17}+\cdots\)
288.2.a.c \(1\) \(2.300\) \(\Q\) None \(0\) \(0\) \(-2\) \(4\) \(+\) \(-\) \(q-2q^{5}+4q^{7}+4q^{11}-2q^{13}+6q^{17}+\cdots\)
288.2.a.d \(1\) \(2.300\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(2\) \(0\) \(+\) \(-\) \(q+2q^{5}+6q^{13}-2q^{17}-q^{25}+10q^{29}+\cdots\)
288.2.a.e \(1\) \(2.300\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(4\) \(0\) \(-\) \(+\) \(q+4q^{5}-6q^{13}+8q^{17}+11q^{25}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(288))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(288)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

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