Properties

Label 200.2.a
Level $200$
Weight $2$
Character orbit 200.a
Rep. character $\chi_{200}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $5$
Sturm bound $60$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 200.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(60\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(200))\).

Total New Old
Modular forms 42 5 37
Cusp forms 19 5 14
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.

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

Trace form

\( 5 q + 4 q^{7} + 11 q^{9} + O(q^{10}) \) \( 5 q + 4 q^{7} + 11 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{17} - 2 q^{19} - 4 q^{21} - 4 q^{23} - 14 q^{29} + 12 q^{31} - 6 q^{37} - 8 q^{39} - 8 q^{41} + 8 q^{43} - 4 q^{47} - 3 q^{49} - 30 q^{51} - 6 q^{53} - 12 q^{59} - 2 q^{61} - 12 q^{63} - 8 q^{67} + 4 q^{69} - 8 q^{71} + 6 q^{73} + 16 q^{77} + 44 q^{79} + 5 q^{81} + 16 q^{83} - 12 q^{89} + 40 q^{91} + 14 q^{97} - 8 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(200))\) 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 5
200.2.a.a 200.a 1.a $1$ $1.597$ \(\Q\) None \(0\) \(-3\) \(0\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+2q^{7}+6q^{9}+q^{11}+4q^{13}+\cdots\)
200.2.a.b 200.a 1.a $1$ $1.597$ \(\Q\) None \(0\) \(-2\) \(0\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-2q^{7}+q^{9}-4q^{11}-4q^{13}+\cdots\)
200.2.a.c 200.a 1.a $1$ $1.597$ \(\Q\) None \(0\) \(0\) \(0\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{7}-3q^{9}+4q^{11}+2q^{13}-2q^{17}+\cdots\)
200.2.a.d 200.a 1.a $1$ $1.597$ \(\Q\) None \(0\) \(2\) \(0\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+2q^{7}+q^{9}-4q^{11}+4q^{13}+\cdots\)
200.2.a.e 200.a 1.a $1$ $1.597$ \(\Q\) None \(0\) \(3\) \(0\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-2q^{7}+6q^{9}+q^{11}-4q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(200))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(200)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)