Properties

Label 200.8.a
Level $200$
Weight $8$
Character orbit 200.a
Rep. character $\chi_{200}(1,\cdot)$
Character field $\Q$
Dimension $33$
Newform subspaces $18$
Sturm bound $240$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 200.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(240\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(200))\).

Total New Old
Modular forms 222 33 189
Cusp forms 198 33 165
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
\(+\)\(+\)$+$\(9\)
\(+\)\(-\)$-$\(8\)
\(-\)\(+\)$-$\(7\)
\(-\)\(-\)$+$\(9\)
Plus space\(+\)\(18\)
Minus space\(-\)\(15\)

Trace form

\( 33 q - 40 q^{3} - 332 q^{7} + 24251 q^{9} + O(q^{10}) \) \( 33 q - 40 q^{3} - 332 q^{7} + 24251 q^{9} + 5326 q^{11} + 12586 q^{13} - 9554 q^{17} + 18414 q^{19} - 80492 q^{21} - 78292 q^{23} - 17200 q^{27} - 322982 q^{29} + 124972 q^{31} + 53120 q^{33} - 43982 q^{37} - 74936 q^{39} + 462148 q^{41} + 723504 q^{43} + 152732 q^{47} + 3849577 q^{49} - 1217750 q^{51} - 821838 q^{53} + 2194400 q^{57} + 374268 q^{59} + 4636318 q^{61} - 1173916 q^{63} - 3059536 q^{67} + 399628 q^{69} + 2800936 q^{71} + 2810918 q^{73} + 9632272 q^{77} + 578508 q^{79} + 18530657 q^{81} - 589592 q^{83} + 633680 q^{87} + 4822824 q^{89} + 22137704 q^{91} - 2857760 q^{93} + 30108798 q^{97} + 79695120 q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(200))\) 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
200.8.a.a 200.a 1.a $1$ $62.477$ \(\Q\) None \(0\) \(-69\) \(0\) \(174\) $-$ $-$ $\mathrm{SU}(2)$ \(q-69q^{3}+174q^{7}+2574q^{9}+7111q^{11}+\cdots\)
200.8.a.b 200.a 1.a $1$ $62.477$ \(\Q\) None \(0\) \(-44\) \(0\) \(1224\) $-$ $+$ $\mathrm{SU}(2)$ \(q-44q^{3}+1224q^{7}-251q^{9}-3164q^{11}+\cdots\)
200.8.a.c 200.a 1.a $1$ $62.477$ \(\Q\) None \(0\) \(-34\) \(0\) \(-106\) $-$ $-$ $\mathrm{SU}(2)$ \(q-34q^{3}-106q^{7}-1031q^{9}-1324q^{11}+\cdots\)
200.8.a.d 200.a 1.a $1$ $62.477$ \(\Q\) None \(0\) \(-9\) \(0\) \(694\) $-$ $+$ $\mathrm{SU}(2)$ \(q-9q^{3}+694q^{7}-2106q^{9}+4901q^{11}+\cdots\)
200.8.a.e 200.a 1.a $1$ $62.477$ \(\Q\) None \(0\) \(9\) \(0\) \(-694\) $+$ $-$ $\mathrm{SU}(2)$ \(q+9q^{3}-694q^{7}-2106q^{9}+4901q^{11}+\cdots\)
200.8.a.f 200.a 1.a $1$ $62.477$ \(\Q\) None \(0\) \(34\) \(0\) \(106\) $+$ $-$ $\mathrm{SU}(2)$ \(q+34q^{3}+106q^{7}-1031q^{9}-1324q^{11}+\cdots\)
200.8.a.g 200.a 1.a $1$ $62.477$ \(\Q\) None \(0\) \(36\) \(0\) \(-776\) $-$ $+$ $\mathrm{SU}(2)$ \(q+6^{2}q^{3}-776q^{7}-891q^{9}-124q^{11}+\cdots\)
200.8.a.h 200.a 1.a $1$ $62.477$ \(\Q\) None \(0\) \(69\) \(0\) \(-174\) $+$ $+$ $\mathrm{SU}(2)$ \(q+69q^{3}-174q^{7}+2574q^{9}+7111q^{11}+\cdots\)
200.8.a.i 200.a 1.a $1$ $62.477$ \(\Q\) None \(0\) \(84\) \(0\) \(456\) $+$ $+$ $\mathrm{SU}(2)$ \(q+84q^{3}+456q^{7}+4869q^{9}-2524q^{11}+\cdots\)
200.8.a.j 200.a 1.a $2$ $62.477$ \(\Q(\sqrt{601}) \) None \(0\) \(-76\) \(0\) \(-796\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-38-\beta )q^{3}+(-398-7\beta )q^{7}+\cdots\)
200.8.a.k 200.a 1.a $2$ $62.477$ \(\Q(\sqrt{6}) \) None \(0\) \(-36\) \(0\) \(404\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-18+\beta )q^{3}+(202-63\beta )q^{7}+(-1479+\cdots)q^{9}+\cdots\)
200.8.a.l 200.a 1.a $2$ $62.477$ \(\Q(\sqrt{3889}) \) None \(0\) \(-4\) \(0\) \(-1144\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta )q^{3}+(-572-2\beta )q^{7}+(1706+\cdots)q^{9}+\cdots\)
200.8.a.m 200.a 1.a $2$ $62.477$ \(\Q(\sqrt{46}) \) None \(0\) \(-4\) \(0\) \(-844\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta )q^{3}+(-422+17\beta )q^{7}+\cdots\)
200.8.a.n 200.a 1.a $2$ $62.477$ \(\Q(\sqrt{3889}) \) None \(0\) \(4\) \(0\) \(1144\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(2+\beta )q^{3}+(572+2\beta )q^{7}+(1706+\cdots)q^{9}+\cdots\)
200.8.a.o 200.a 1.a $3$ $62.477$ 3.3.40101.1 None \(0\) \(-97\) \(0\) \(1630\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-2^{5}-\beta _{1})q^{3}+(543+2\beta _{1}+\beta _{2})q^{7}+\cdots\)
200.8.a.p 200.a 1.a $3$ $62.477$ 3.3.40101.1 None \(0\) \(97\) \(0\) \(-1630\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(2^{5}+\beta _{1})q^{3}+(-543-2\beta _{1}-\beta _{2})q^{7}+\cdots\)
200.8.a.q 200.a 1.a $4$ $62.477$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-32\) \(0\) \(-1632\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-8-\beta _{1})q^{3}+(-408+3\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
200.8.a.r 200.a 1.a $4$ $62.477$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(32\) \(0\) \(1632\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(8+\beta _{1})q^{3}+(408-3\beta _{1}+2\beta _{2}-\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(200))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(200)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)