Properties

Label 432.2.a
Level 432
Weight 2
Character orbit a
Rep. character \(\chi_{432}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 8
Sturm bound 144
Trace bound 7

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Defining parameters

Level: \( N \) = \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 432.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(144\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(432))\).

Total New Old
Modular forms 90 8 82
Cusp forms 55 8 47
Eisenstein series 35 0 35

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

Trace form

\( 8q - 2q^{7} + O(q^{10}) \) \( 8q - 2q^{7} + 2q^{19} + 12q^{25} + 20q^{31} - 16q^{37} + 24q^{43} + 8q^{49} - 4q^{55} - 24q^{61} - 46q^{67} - 42q^{79} - 16q^{85} + 14q^{91} - 8q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(432))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3
432.2.a.a \(1\) \(3.450\) \(\Q\) None \(0\) \(0\) \(-4\) \(3\) \(+\) \(-\) \(q-4q^{5}+3q^{7}+4q^{11}+q^{13}+4q^{17}+\cdots\)
432.2.a.b \(1\) \(3.450\) \(\Q\) None \(0\) \(0\) \(-3\) \(1\) \(-\) \(-\) \(q-3q^{5}+q^{7}-3q^{11}-4q^{13}-2q^{19}+\cdots\)
432.2.a.c \(1\) \(3.450\) \(\Q\) None \(0\) \(0\) \(-1\) \(-3\) \(+\) \(+\) \(q-q^{5}-3q^{7}-5q^{11}+4q^{13}-8q^{17}+\cdots\)
432.2.a.d \(1\) \(3.450\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-5\) \(-\) \(-\) \(q-5q^{7}-7q^{13}+q^{19}-5q^{25}+4q^{31}+\cdots\)
432.2.a.e \(1\) \(3.450\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(1\) \(-\) \(+\) \(q+q^{7}+5q^{13}+7q^{19}-5q^{25}+4q^{31}+\cdots\)
432.2.a.f \(1\) \(3.450\) \(\Q\) None \(0\) \(0\) \(1\) \(-3\) \(+\) \(-\) \(q+q^{5}-3q^{7}+5q^{11}+4q^{13}+8q^{17}+\cdots\)
432.2.a.g \(1\) \(3.450\) \(\Q\) None \(0\) \(0\) \(3\) \(1\) \(-\) \(+\) \(q+3q^{5}+q^{7}+3q^{11}-4q^{13}-2q^{19}+\cdots\)
432.2.a.h \(1\) \(3.450\) \(\Q\) None \(0\) \(0\) \(4\) \(3\) \(+\) \(-\) \(q+4q^{5}+3q^{7}-4q^{11}+q^{13}-4q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(432))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(432)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$3$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))
$7$ (\( 1 - 3 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + 3 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + 3 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 3 T + 7 T^{2} \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 5 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))
$13$ (\( 1 - T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 7 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))
$17$ (\( 1 - 4 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 8 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 8 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))
$19$ (\( 1 - T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( 1 - 7 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))
$23$ (\( 1 - 4 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 2 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))
$29$ (\( 1 + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))
$31$ (\( 1 - 4 T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 - 7 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 7 T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))
$37$ (\( 1 + 9 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( 1 - 11 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 9 T + 37 T^{2} \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))
$43$ (\( 1 - 8 T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 - 2 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 2 T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))
$47$ (\( 1 + 12 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))
$53$ (\( 1 - 8 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 - 5 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 5 T + 53 T^{2} \))(\( 1 - 9 T + 53 T^{2} \))(\( 1 + 8 T + 53 T^{2} \))
$59$ (\( 1 - 4 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))
$61$ (\( 1 + 5 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 + 13 T + 61 T^{2} \))(\( 1 + T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 5 T + 61 T^{2} \))
$67$ (\( 1 + 11 T + 67 T^{2} \))(\( 1 + 14 T + 67 T^{2} \))(\( 1 - 10 T + 67 T^{2} \))(\( 1 + 11 T + 67 T^{2} \))(\( 1 + 5 T + 67 T^{2} \))(\( 1 - 10 T + 67 T^{2} \))(\( 1 + 14 T + 67 T^{2} \))(\( 1 + 11 T + 67 T^{2} \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))
$73$ (\( 1 - T + 73 T^{2} \))(\( 1 + 7 T + 73 T^{2} \))(\( 1 - T + 73 T^{2} \))(\( 1 - 17 T + 73 T^{2} \))(\( 1 + 7 T + 73 T^{2} \))(\( 1 - T + 73 T^{2} \))(\( 1 + 7 T + 73 T^{2} \))(\( 1 - T + 73 T^{2} \))
$79$ (\( 1 - 5 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 - 13 T + 79 T^{2} \))(\( 1 + 17 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 5 T + 79 T^{2} \))
$83$ (\( 1 - 8 T + 83 T^{2} \))(\( 1 + 3 T + 83 T^{2} \))(\( 1 - 11 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 11 T + 83 T^{2} \))(\( 1 - 3 T + 83 T^{2} \))(\( 1 + 8 T + 83 T^{2} \))
$89$ (\( 1 + 12 T + 89 T^{2} \))(\( 1 - 18 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 18 T + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))
$97$ (\( 1 - 5 T + 97 T^{2} \))(\( 1 + T + 97 T^{2} \))(\( 1 + T + 97 T^{2} \))(\( 1 - 5 T + 97 T^{2} \))(\( 1 + 19 T + 97 T^{2} \))(\( 1 + T + 97 T^{2} \))(\( 1 + T + 97 T^{2} \))(\( 1 - 5 T + 97 T^{2} \))
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