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Properties

Label 600.2.a
Level $600$
Weight $2$
Character orbit 600.a
Rep. character $\chi_{600}(1,\cdot)$
Character field $\Q$
Dimension $9$
Newform subspaces $9$
Sturm bound $240$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(240\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(600))\).

Total New Old
Modular forms 144 9 135
Cusp forms 97 9 88
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
\(+\)\(+\)\(+\)$+$\(2\)
\(+\)\(+\)\(-\)$-$\(1\)
\(+\)\(-\)\(+\)$-$\(1\)
\(+\)\(-\)\(-\)$+$\(1\)
\(-\)\(+\)\(+\)$-$\(2\)
\(-\)\(-\)\(-\)$-$\(2\)
Plus space\(+\)\(3\)
Minus space\(-\)\(6\)

Trace form

\( 9 q - q^{3} - 4 q^{7} + 9 q^{9} + O(q^{10}) \) \( 9 q - q^{3} - 4 q^{7} + 9 q^{9} - 4 q^{11} + 2 q^{13} + 6 q^{17} - 4 q^{21} + 16 q^{23} - q^{27} + 22 q^{29} - 4 q^{31} + 8 q^{33} + 2 q^{37} - 6 q^{39} + 34 q^{41} - 12 q^{43} - 16 q^{47} + 29 q^{49} + 10 q^{51} - 14 q^{53} - 4 q^{57} - 20 q^{59} + 2 q^{61} - 4 q^{63} + 4 q^{67} - 24 q^{71} + 10 q^{73} - 48 q^{79} + 9 q^{81} + 4 q^{83} + 14 q^{87} + 18 q^{89} - 20 q^{91} + 16 q^{93} - 6 q^{97} - 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(600))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5
600.2.a.a 600.a 1.a $1$ $4.791$ \(\Q\) None \(0\) \(-1\) \(0\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{7}+q^{9}+6q^{13}+2q^{17}+\cdots\)
600.2.a.b 600.a 1.a $1$ $4.791$ \(\Q\) None \(0\) \(-1\) \(0\) \(-3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{7}+q^{9}+2q^{11}+3q^{13}+\cdots\)
600.2.a.c 600.a 1.a $1$ $4.791$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{9}-4q^{11}-6q^{13}+6q^{17}+\cdots\)
600.2.a.d 600.a 1.a $1$ $4.791$ \(\Q\) None \(0\) \(-1\) \(0\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
600.2.a.e 600.a 1.a $1$ $4.791$ \(\Q\) None \(0\) \(-1\) \(0\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+5q^{7}+q^{9}-6q^{11}+3q^{13}+\cdots\)
600.2.a.f 600.a 1.a $1$ $4.791$ \(\Q\) None \(0\) \(1\) \(0\) \(-5\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-5q^{7}+q^{9}-6q^{11}-3q^{13}+\cdots\)
600.2.a.g 600.a 1.a $1$ $4.791$ \(\Q\) None \(0\) \(1\) \(0\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
600.2.a.h 600.a 1.a $1$ $4.791$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{9}+4q^{11}+2q^{13}-2q^{17}+\cdots\)
600.2.a.i 600.a 1.a $1$ $4.791$ \(\Q\) None \(0\) \(1\) \(0\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+3q^{7}+q^{9}+2q^{11}-3q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(600))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(600)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)