Properties

Label 800.2.a
Level $800$
Weight $2$
Character orbit 800.a
Rep. character $\chi_{800}(1,\cdot)$
Character field $\Q$
Dimension $19$
Newform subspaces $14$
Sturm bound $240$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(240\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(800))\).

Total New Old
Modular forms 144 19 125
Cusp forms 97 19 78
Eisenstein series 47 0 47

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.
\(+\)\(+\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(6\)
\(-\)\(+\)\(-\)\(6\)
\(-\)\(-\)\(+\)\(4\)
Plus space\(+\)\(7\)
Minus space\(-\)\(12\)

Trace form

\( 19 q + 15 q^{9} + O(q^{10}) \) \( 19 q + 15 q^{9} + 10 q^{13} - 10 q^{17} + 16 q^{21} + 10 q^{29} + 16 q^{33} + 18 q^{37} + 14 q^{41} + 43 q^{49} - 30 q^{53} + 32 q^{57} + 10 q^{61} - 32 q^{69} - 2 q^{73} - 48 q^{77} + 11 q^{81} - 34 q^{89} - 16 q^{93} - 42 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(800))\) 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
800.2.a.a 800.a 1.a $1$ $6.388$ \(\Q\) None \(0\) \(-2\) \(0\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-2q^{7}+q^{9}+4q^{11}+6q^{13}+\cdots\)
800.2.a.b 800.a 1.a $1$ $6.388$ \(\Q\) None \(0\) \(-1\) \(0\) \(2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{7}-2q^{9}-5q^{11}+5q^{17}+\cdots\)
800.2.a.c 800.a 1.a $1$ $6.388$ \(\Q\) None \(0\) \(-1\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{7}-2q^{9}+5q^{11}-5q^{17}+\cdots\)
800.2.a.d 800.a 1.a $1$ $6.388$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-3q^{9}-6q^{13}-2q^{17}-10q^{29}+\cdots\)
800.2.a.e 800.a 1.a $1$ $6.388$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q-3q^{9}-4q^{13}-8q^{17}+10q^{29}+\cdots\)
800.2.a.f 800.a 1.a $1$ $6.388$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-3q^{9}+4q^{13}+8q^{17}+10q^{29}+\cdots\)
800.2.a.g 800.a 1.a $1$ $6.388$ \(\Q\) None \(0\) \(1\) \(0\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{7}-2q^{9}-5q^{11}-5q^{17}+\cdots\)
800.2.a.h 800.a 1.a $1$ $6.388$ \(\Q\) None \(0\) \(1\) \(0\) \(-2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{7}-2q^{9}+5q^{11}+5q^{17}+\cdots\)
800.2.a.i 800.a 1.a $1$ $6.388$ \(\Q\) None \(0\) \(2\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+2q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
800.2.a.j 800.a 1.a $2$ $6.388$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-5}) \) \(0\) \(-2\) \(0\) \(-6\) $-$ $-$ $N(\mathrm{U}(1))$ \(q+(-1-\beta )q^{3}+(-3+\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
800.2.a.k 800.a 1.a $2$ $6.388$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}-2\beta q^{7}+2q^{9}+\beta q^{11}-4q^{13}+\cdots\)
800.2.a.l 800.a 1.a $2$ $6.388$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}-2\beta q^{7}+2q^{9}-\beta q^{11}+4q^{13}+\cdots\)
800.2.a.m 800.a 1.a $2$ $6.388$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-\beta q^{7}+5q^{9}+2\beta q^{11}+2q^{13}+\cdots\)
800.2.a.n 800.a 1.a $2$ $6.388$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-5}) \) \(0\) \(2\) \(0\) \(6\) $+$ $-$ $N(\mathrm{U}(1))$ \(q+(1+\beta )q^{3}+(3-\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(800))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(800)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 2}\)