Properties

Label 200.6.a
Level $200$
Weight $6$
Character orbit 200.a
Rep. character $\chi_{200}(1,\cdot)$
Character field $\Q$
Dimension $24$
Newform subspaces $11$
Sturm bound $180$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 200.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(180\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(200))\).

Total New Old
Modular forms 162 24 138
Cusp forms 138 24 114
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.
\(+\)\(+\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(7\)
\(-\)\(+\)\(-\)\(6\)
\(-\)\(-\)\(+\)\(6\)
Plus space\(+\)\(11\)
Minus space\(-\)\(13\)

Trace form

\( 24 q + 20 q^{3} - 100 q^{7} + 1802 q^{9} + O(q^{10}) \) \( 24 q + 20 q^{3} - 100 q^{7} + 1802 q^{9} - 582 q^{11} - 700 q^{13} - 380 q^{17} - 702 q^{19} + 844 q^{21} - 1260 q^{23} + 7640 q^{27} + 8756 q^{29} + 5756 q^{31} + 13760 q^{33} - 10780 q^{37} - 752 q^{39} + 21254 q^{41} - 21180 q^{43} + 5260 q^{47} + 98984 q^{49} - 25310 q^{51} + 5620 q^{53} - 63760 q^{57} - 41144 q^{59} + 35264 q^{61} - 28900 q^{63} + 146380 q^{67} + 178276 q^{69} + 69088 q^{71} + 96340 q^{73} - 225920 q^{77} - 64644 q^{79} + 78656 q^{81} - 80380 q^{83} + 399800 q^{87} + 275778 q^{89} - 72968 q^{91} + 118960 q^{93} - 338700 q^{97} - 270180 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(200))\) 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
200.6.a.a 200.a 1.a $1$ $32.077$ \(\Q\) None \(0\) \(-20\) \(0\) \(24\) $+$ $+$ $\mathrm{SU}(2)$ \(q-20q^{3}+24q^{7}+157q^{9}+124q^{11}+\cdots\)
200.6.a.b 200.a 1.a $1$ $32.077$ \(\Q\) None \(0\) \(2\) \(0\) \(62\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+62q^{7}-239q^{9}-12^{2}q^{11}+\cdots\)
200.6.a.c 200.a 1.a $1$ $32.077$ \(\Q\) None \(0\) \(8\) \(0\) \(108\) $+$ $+$ $\mathrm{SU}(2)$ \(q+8q^{3}+108q^{7}-179q^{9}-604q^{11}+\cdots\)
200.6.a.d 200.a 1.a $1$ $32.077$ \(\Q\) None \(0\) \(18\) \(0\) \(-242\) $+$ $+$ $\mathrm{SU}(2)$ \(q+18q^{3}-242q^{7}+3^{4}q^{9}+656q^{11}+\cdots\)
200.6.a.e 200.a 1.a $2$ $32.077$ \(\Q(\sqrt{241}) \) None \(0\) \(-8\) \(0\) \(8\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-4-\beta )q^{3}+(4+2\beta )q^{7}+(14+8\beta )q^{9}+\cdots\)
200.6.a.f 200.a 1.a $2$ $32.077$ \(\Q(\sqrt{241}) \) None \(0\) \(8\) \(0\) \(-8\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(4+\beta )q^{3}+(-4-2\beta )q^{7}+(14+8\beta )q^{9}+\cdots\)
200.6.a.g 200.a 1.a $2$ $32.077$ \(\Q(\sqrt{129}) \) None \(0\) \(12\) \(0\) \(-52\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(6+\beta )q^{3}+(-26+3\beta )q^{7}+(309+\cdots)q^{9}+\cdots\)
200.6.a.h 200.a 1.a $3$ $32.077$ 3.3.47217.1 None \(0\) \(-1\) \(0\) \(70\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(24-\beta _{1}-\beta _{2})q^{7}+(50+5\beta _{1}+\cdots)q^{9}+\cdots\)
200.6.a.i 200.a 1.a $3$ $32.077$ 3.3.47217.1 None \(0\) \(1\) \(0\) \(-70\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-24+\beta _{1}+\beta _{2})q^{7}+(50+\cdots)q^{9}+\cdots\)
200.6.a.j 200.a 1.a $4$ $32.077$ 4.4.1595208.1 None \(0\) \(-4\) \(0\) \(-148\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{3}+(-37-\beta _{1}+\beta _{3})q^{7}+\cdots\)
200.6.a.k 200.a 1.a $4$ $32.077$ 4.4.1595208.1 None \(0\) \(4\) \(0\) \(148\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{3}+(37+\beta _{1}-\beta _{3})q^{7}+(5^{3}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(200))\) into lower level spaces

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