Properties

Label 800.3.b
Level $800$
Weight $3$
Character orbit 800.b
Rep. character $\chi_{800}(351,\cdot)$
Character field $\Q$
Dimension $38$
Newform subspaces $9$
Sturm bound $360$
Trace bound $17$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 800.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(360\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(3\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(800, [\chi])\).

Total New Old
Modular forms 264 38 226
Cusp forms 216 38 178
Eisenstein series 48 0 48

Trace form

\( 38 q - 122 q^{9} + O(q^{10}) \) \( 38 q - 122 q^{9} - 4 q^{13} + 12 q^{17} + 32 q^{21} + 28 q^{29} - 64 q^{33} - 100 q^{37} - 116 q^{41} - 218 q^{49} - 100 q^{53} + 160 q^{57} - 36 q^{61} + 384 q^{69} - 20 q^{73} + 32 q^{77} + 358 q^{81} - 116 q^{89} - 224 q^{93} - 340 q^{97} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(800, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
800.3.b.a 800.b 4.b $2$ $21.798$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}-2iq^{7}-7q^{9}+iq^{11}+14q^{13}+\cdots\)
800.3.b.b 800.b 4.b $4$ $21.798$ \(\Q(i, \sqrt{11})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}+2\beta _{2}q^{7}+(-6-\beta _{1})q^{9}+\cdots\)
800.3.b.c 800.b 4.b $4$ $21.798$ \(\Q(i, \sqrt{11})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}+2\beta _{2}q^{7}+(-6-\beta _{1})q^{9}+\cdots\)
800.3.b.d 800.b 4.b $4$ $21.798$ \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+(-\beta _{1}+4\beta _{2})q^{7}+(-5+3\beta _{3})q^{9}+\cdots\)
800.3.b.e 800.b 4.b $4$ $21.798$ \(\Q(i, \sqrt{11})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}+(-\beta _{1}-2\beta _{3})q^{7}-2q^{9}+\cdots\)
800.3.b.f 800.b 4.b $4$ $21.798$ \(\Q(i, \sqrt{11})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{3}+(\beta _{1}+2\beta _{3})q^{7}-2q^{9}+(2\beta _{1}+\cdots)q^{11}+\cdots\)
800.3.b.g 800.b 4.b $4$ $21.798$ \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+\beta _{1}q^{7}+(3+\beta _{3})q^{9}+(-5\beta _{1}+\cdots)q^{11}+\cdots\)
800.3.b.h 800.b 4.b $6$ $21.798$ 6.0.1827904.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}-\beta _{5}q^{7}+(-3-\beta _{2}+\beta _{4}+\cdots)q^{9}+\cdots\)
800.3.b.i 800.b 4.b $6$ $21.798$ 6.0.1827904.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}-\beta _{5}q^{7}+(-3-\beta _{2}+\beta _{4}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(800, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(800, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 2}\)