Properties

Label 1200.2.a
Level $1200$
Weight $2$
Character orbit 1200.a
Rep. character $\chi_{1200}(1,\cdot)$
Character field $\Q$
Dimension $19$
Newform subspaces $19$
Sturm bound $480$
Trace bound $13$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 19 \)
Sturm bound: \(480\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1200))\).

Total New Old
Modular forms 276 19 257
Cusp forms 205 19 186
Eisenstein series 71 0 71

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
\(+\)\(+\)\(+\)$+$\(1\)
\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(-\)\(+\)$-$\(4\)
\(+\)\(-\)\(-\)$+$\(1\)
\(-\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(-\)$+$\(3\)
\(-\)\(-\)\(+\)$+$\(2\)
\(-\)\(-\)\(-\)$-$\(3\)
Plus space\(+\)\(7\)
Minus space\(-\)\(12\)

Trace form

\( 19 q + q^{3} + 19 q^{9} + O(q^{10}) \) \( 19 q + q^{3} + 19 q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{17} - 16 q^{23} + q^{27} + 10 q^{29} + 12 q^{31} + 4 q^{33} + 10 q^{37} + 6 q^{39} + 6 q^{41} + 12 q^{43} + 24 q^{47} + 35 q^{49} - 2 q^{51} + 2 q^{53} + 4 q^{57} + 28 q^{59} - 14 q^{61} + 4 q^{67} - 8 q^{69} + 24 q^{71} - 2 q^{73} + 48 q^{79} + 19 q^{81} + 20 q^{83} - 18 q^{87} - 10 q^{89} + 20 q^{91} + 8 q^{93} - 10 q^{97} - 4 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1200))\) 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
1200.2.a.a 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(-1\) \(0\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{7}+q^{9}+4q^{11}+4q^{17}+\cdots\)
1200.2.a.b 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(-1\) \(0\) \(-3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{7}+q^{9}-2q^{11}-3q^{13}+\cdots\)
1200.2.a.c 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(-1\) \(0\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{7}+q^{9}-2q^{11}-q^{13}+\cdots\)
1200.2.a.d 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{9}-4q^{11}+2q^{13}-2q^{17}+\cdots\)
1200.2.a.e 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{9}+4q^{11}+2q^{13}-2q^{17}+\cdots\)
1200.2.a.f 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}+q^{9}-6q^{11}+5q^{13}+\cdots\)
1200.2.a.g 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(-1\) \(0\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{7}+q^{9}-2q^{11}-6q^{13}+\cdots\)
1200.2.a.h 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(-1\) \(0\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
1200.2.a.i 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(-1\) \(0\) \(5\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+5q^{7}+q^{9}+6q^{11}-3q^{13}+\cdots\)
1200.2.a.j 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(1\) \(0\) \(-5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-5q^{7}+q^{9}+6q^{11}+3q^{13}+\cdots\)
1200.2.a.k 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(1\) \(0\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{7}+q^{9}-2q^{13}-6q^{17}+\cdots\)
1200.2.a.l 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(1\) \(0\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
1200.2.a.m 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(1\) \(0\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{7}+q^{9}-2q^{11}+6q^{13}+\cdots\)
1200.2.a.n 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}+q^{9}-6q^{11}-5q^{13}+\cdots\)
1200.2.a.o 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{9}+4q^{11}-6q^{13}+6q^{17}+\cdots\)
1200.2.a.p 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(1\) \(0\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+3q^{7}+q^{9}-2q^{11}+q^{13}+\cdots\)
1200.2.a.q 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(1\) \(0\) \(3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+3q^{7}+q^{9}-2q^{11}+3q^{13}+\cdots\)
1200.2.a.r 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(1\) \(0\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{7}+q^{9}+6q^{13}+2q^{17}+\cdots\)
1200.2.a.s 1200.a 1.a $1$ $9.582$ \(\Q\) None \(0\) \(1\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{7}+q^{9}+4q^{11}-4q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1200))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1200)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 2}\)