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Properties

Label 200.4.a
Level $200$
Weight $4$
Character orbit 200.a
Rep. character $\chi_{200}(1,\cdot)$
Character field $\Q$
Dimension $14$
Newform subspaces $12$
Sturm bound $120$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 200.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(120\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(200))\).

Total New Old
Modular forms 102 14 88
Cusp forms 78 14 64
Eisenstein series 24 0 24

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
\(+\)\(+\)$+$\(4\)
\(+\)\(-\)$-$\(3\)
\(-\)\(+\)$-$\(3\)
\(-\)\(-\)$+$\(4\)
Plus space\(+\)\(8\)
Minus space\(-\)\(6\)

Trace form

\( 14 q - 4 q^{3} + 12 q^{7} + 116 q^{9} + O(q^{10}) \) \( 14 q - 4 q^{3} + 12 q^{7} + 116 q^{9} - 6 q^{11} - 32 q^{13} + 136 q^{17} + 226 q^{19} + 196 q^{21} + 132 q^{23} - 568 q^{27} - 208 q^{29} - 100 q^{31} - 64 q^{33} + 656 q^{37} + 928 q^{39} + 38 q^{41} - 4 q^{43} - 980 q^{47} - 178 q^{49} + 970 q^{51} - 800 q^{53} + 1904 q^{57} + 1408 q^{59} - 780 q^{61} + 2060 q^{63} - 748 q^{67} - 2324 q^{69} - 1328 q^{71} - 1208 q^{73} + 736 q^{77} - 484 q^{79} + 1046 q^{81} - 1876 q^{83} - 2872 q^{87} - 2454 q^{89} + 632 q^{91} + 880 q^{93} - 40 q^{97} - 3324 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(200))\) 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
200.4.a.a 200.a 1.a $1$ $11.800$ \(\Q\) None \(0\) \(-10\) \(0\) \(18\) $-$ $+$ $\mathrm{SU}(2)$ \(q-10q^{3}+18q^{7}+73q^{9}-2^{4}q^{11}+\cdots\)
200.4.a.b 200.a 1.a $1$ $11.800$ \(\Q\) None \(0\) \(-9\) \(0\) \(-26\) $+$ $+$ $\mathrm{SU}(2)$ \(q-9q^{3}-26q^{7}+54q^{9}-59q^{11}+\cdots\)
200.4.a.c 200.a 1.a $1$ $11.800$ \(\Q\) None \(0\) \(-5\) \(0\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-5q^{3}-2q^{7}-2q^{9}+39q^{11}-84q^{13}+\cdots\)
200.4.a.d 200.a 1.a $1$ $11.800$ \(\Q\) None \(0\) \(-4\) \(0\) \(-16\) $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{3}-2^{4}q^{7}-11q^{9}+6^{2}q^{11}+\cdots\)
200.4.a.e 200.a 1.a $1$ $11.800$ \(\Q\) None \(0\) \(-1\) \(0\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+6q^{7}-26q^{9}-19q^{11}-12q^{13}+\cdots\)
200.4.a.f 200.a 1.a $1$ $11.800$ \(\Q\) None \(0\) \(1\) \(0\) \(-6\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-6q^{7}-26q^{9}-19q^{11}+12q^{13}+\cdots\)
200.4.a.g 200.a 1.a $1$ $11.800$ \(\Q\) None \(0\) \(4\) \(0\) \(-24\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{3}-24q^{7}-11q^{9}-44q^{11}+\cdots\)
200.4.a.h 200.a 1.a $1$ $11.800$ \(\Q\) None \(0\) \(5\) \(0\) \(2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+5q^{3}+2q^{7}-2q^{9}+39q^{11}+84q^{13}+\cdots\)
200.4.a.i 200.a 1.a $1$ $11.800$ \(\Q\) None \(0\) \(6\) \(0\) \(34\) $+$ $+$ $\mathrm{SU}(2)$ \(q+6q^{3}+34q^{7}+9q^{9}+2^{4}q^{11}-58q^{13}+\cdots\)
200.4.a.j 200.a 1.a $1$ $11.800$ \(\Q\) None \(0\) \(9\) \(0\) \(26\) $-$ $-$ $\mathrm{SU}(2)$ \(q+9q^{3}+26q^{7}+54q^{9}-59q^{11}+\cdots\)
200.4.a.k 200.a 1.a $2$ $11.800$ \(\Q(\sqrt{6}) \) None \(0\) \(-4\) \(0\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta )q^{3}+(-2-3\beta )q^{7}+(1-4\beta )q^{9}+\cdots\)
200.4.a.l 200.a 1.a $2$ $11.800$ \(\Q(\sqrt{6}) \) None \(0\) \(4\) \(0\) \(4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(2+\beta )q^{3}+(2-3\beta )q^{7}+(1+4\beta )q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(200))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(200)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)