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Properties

Label 800.2.n
Level $800$
Weight $2$
Character orbit 800.n
Rep. character $\chi_{800}(543,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $36$
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.n (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 20 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 14 \)
Sturm bound: \(240\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(800, [\chi])\).

Total New Old
Modular forms 288 36 252
Cusp forms 192 36 156
Eisenstein series 96 0 96

Trace form

\( 36 q + O(q^{10}) \) \( 36 q - 4 q^{13} + 12 q^{17} - 32 q^{21} + 16 q^{33} + 20 q^{37} + 32 q^{41} + 52 q^{53} - 32 q^{57} - 28 q^{73} - 48 q^{77} - 132 q^{81} - 80 q^{93} + 36 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(800, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
800.2.n.a 800.n 20.e $2$ $6.388$ \(\Q(\sqrt{-1}) \) None \(0\) \(-4\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2-2i)q^{3}+(-2+2i)q^{7}+5iq^{9}+\cdots\)
800.2.n.b 800.n 20.e $2$ $6.388$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{3}+(-1+i)q^{7}-iq^{9}+\cdots\)
800.2.n.c 800.n 20.e $2$ $6.388$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{3}+(-1+i)q^{7}-iq^{9}+\cdots\)
800.2.n.d 800.n 20.e $2$ $6.388$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{3}+(-1+i)q^{7}-iq^{9}+\cdots\)
800.2.n.e 800.n 20.e $2$ $6.388$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{3}+(3-3i)q^{7}-iq^{9}+\cdots\)
800.2.n.f 800.n 20.e $2$ $6.388$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{3}+(-3+3i)q^{7}-iq^{9}+\cdots\)
800.2.n.g 800.n 20.e $2$ $6.388$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{3}+(1-i)q^{7}-iq^{9}+4iq^{11}+\cdots\)
800.2.n.h 800.n 20.e $2$ $6.388$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{3}+(1-i)q^{7}-iq^{9}-6iq^{11}+\cdots\)
800.2.n.i 800.n 20.e $2$ $6.388$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{3}+(1-i)q^{7}-iq^{9}-4iq^{11}+\cdots\)
800.2.n.j 800.n 20.e $2$ $6.388$ \(\Q(\sqrt{-1}) \) None \(0\) \(4\) \(0\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2+2i)q^{3}+(2-2i)q^{7}+5iq^{9}+\cdots\)
800.2.n.k 800.n 20.e $4$ $6.388$ \(\Q(i, \sqrt{6})\) None \(0\) \(-4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{3}+(2-2\beta _{2})q^{7}+\cdots\)
800.2.n.l 800.n 20.e $4$ $6.388$ \(\Q(i, \sqrt{6})\) None \(0\) \(-4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{3}+(2-2\beta _{2})q^{7}+\cdots\)
800.2.n.m 800.n 20.e $4$ $6.388$ \(\Q(i, \sqrt{6})\) None \(0\) \(4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\beta _{1}+\beta _{2})q^{3}+(-2+2\beta _{2})q^{7}+\cdots\)
800.2.n.n 800.n 20.e $4$ $6.388$ \(\Q(i, \sqrt{6})\) None \(0\) \(4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\beta _{1}+\beta _{2})q^{3}+(-2+2\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(800, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(800, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 2}\)