Properties

Label 720.2.a
Level 720
Weight 2
Character orbit a
Rep. character \(\chi_{720}(1,\cdot)\)
Character field \(\Q\)
Dimension 10
Newform subspaces 10
Sturm bound 288
Trace bound 13

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Defining parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(288\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(720))\).

Total New Old
Modular forms 168 10 158
Cusp forms 121 10 111
Eisenstein series 47 0 47

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

\(2\)\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(4\)
Minus space\(-\)\(6\)

Trace form

\( 10q - 6q^{7} + O(q^{10}) \) \( 10q - 6q^{7} - 4q^{11} + 4q^{17} + 2q^{23} + 10q^{25} + 12q^{29} + 12q^{31} - 6q^{35} + 8q^{37} - 8q^{41} + 10q^{43} + 22q^{47} - 2q^{49} - 4q^{55} + 16q^{59} - 16q^{61} - 4q^{65} - 2q^{67} - 12q^{71} - 20q^{73} + 16q^{77} + 8q^{79} + 14q^{83} - 8q^{85} - 12q^{89} + 20q^{91} + 8q^{95} - 28q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(720))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5
720.2.a.a \(1\) \(5.749\) \(\Q\) None \(0\) \(0\) \(-1\) \(-2\) \(+\) \(+\) \(+\) \(q-q^{5}-2q^{7}-2q^{11}+4q^{13}-2q^{17}+\cdots\)
720.2.a.b \(1\) \(5.749\) \(\Q\) None \(0\) \(0\) \(-1\) \(-2\) \(-\) \(+\) \(+\) \(q-q^{5}-2q^{7}+6q^{11}-4q^{13}+6q^{17}+\cdots\)
720.2.a.c \(1\) \(5.749\) \(\Q\) None \(0\) \(0\) \(-1\) \(0\) \(-\) \(-\) \(+\) \(q-q^{5}-4q^{11}-2q^{13}-2q^{17}-4q^{19}+\cdots\)
720.2.a.d \(1\) \(5.749\) \(\Q\) None \(0\) \(0\) \(-1\) \(0\) \(+\) \(-\) \(+\) \(q-q^{5}-4q^{11}+6q^{13}+6q^{17}+4q^{19}+\cdots\)
720.2.a.e \(1\) \(5.749\) \(\Q\) None \(0\) \(0\) \(-1\) \(4\) \(+\) \(-\) \(+\) \(q-q^{5}+4q^{7}+4q^{11}-2q^{13}-2q^{17}+\cdots\)
720.2.a.f \(1\) \(5.749\) \(\Q\) None \(0\) \(0\) \(1\) \(-4\) \(+\) \(-\) \(-\) \(q+q^{5}-4q^{7}-6q^{13}+2q^{17}-4q^{19}+\cdots\)
720.2.a.g \(1\) \(5.749\) \(\Q\) None \(0\) \(0\) \(1\) \(-2\) \(-\) \(+\) \(-\) \(q+q^{5}-2q^{7}-6q^{11}-4q^{13}-6q^{17}+\cdots\)
720.2.a.h \(1\) \(5.749\) \(\Q\) None \(0\) \(0\) \(1\) \(-2\) \(-\) \(-\) \(-\) \(q+q^{5}-2q^{7}+2q^{13}+6q^{17}+4q^{19}+\cdots\)
720.2.a.i \(1\) \(5.749\) \(\Q\) None \(0\) \(0\) \(1\) \(-2\) \(+\) \(+\) \(-\) \(q+q^{5}-2q^{7}+2q^{11}+4q^{13}+2q^{17}+\cdots\)
720.2.a.j \(1\) \(5.749\) \(\Q\) None \(0\) \(0\) \(1\) \(4\) \(-\) \(-\) \(-\) \(q+q^{5}+4q^{7}+2q^{13}-6q^{17}+4q^{19}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(720))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(720)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

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