Properties

Label 216.2.a
Level $216$
Weight $2$
Character orbit 216.a
Rep. character $\chi_{216}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $4$
Sturm bound $72$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 216.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(72\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(216))\).

Total New Old
Modular forms 48 4 44
Cusp forms 25 4 21
Eisenstein series 23 0 23

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

Trace form

\( 4q + O(q^{10}) \) \( 4q + 10q^{13} + 2q^{19} + 14q^{25} - 22q^{31} - 30q^{37} - 20q^{43} + 8q^{49} + 22q^{55} - 26q^{61} + 2q^{67} + 4q^{73} + 22q^{79} - 16q^{85} + 18q^{91} + 8q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(216))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3
216.2.a.a \(1\) \(1.725\) \(\Q\) None \(0\) \(0\) \(-4\) \(-3\) \(+\) \(+\) \(q-4q^{5}-3q^{7}-4q^{11}+q^{13}+4q^{17}+\cdots\)
216.2.a.b \(1\) \(1.725\) \(\Q\) None \(0\) \(0\) \(-1\) \(3\) \(+\) \(-\) \(q-q^{5}+3q^{7}+5q^{11}+4q^{13}-8q^{17}+\cdots\)
216.2.a.c \(1\) \(1.725\) \(\Q\) None \(0\) \(0\) \(1\) \(3\) \(-\) \(+\) \(q+q^{5}+3q^{7}-5q^{11}+4q^{13}+8q^{17}+\cdots\)
216.2.a.d \(1\) \(1.725\) \(\Q\) None \(0\) \(0\) \(4\) \(-3\) \(-\) \(+\) \(q+4q^{5}-3q^{7}+4q^{11}+q^{13}-4q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(216))\) into lower level spaces

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

Hecke characteristic polynomials

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