Properties

Label 256.6.b
Level $256$
Weight $6$
Character orbit 256.b
Rep. character $\chi_{256}(129,\cdot)$
Character field $\Q$
Dimension $38$
Newform subspaces $14$
Sturm bound $192$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 256.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(192\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(256, [\chi])\).

Total New Old
Modular forms 172 42 130
Cusp forms 148 38 110
Eisenstein series 24 4 20

Trace form

\( 38 q - 2750 q^{9} + O(q^{10}) \) \( 38 q - 2750 q^{9} - 4 q^{17} - 18746 q^{25} - 976 q^{33} + 4 q^{41} + 120950 q^{49} + 52528 q^{57} + 42872 q^{65} + 230420 q^{73} + 143366 q^{81} + 140468 q^{89} - 294756 q^{97} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(256, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
256.6.b.a 256.b 8.b $2$ $41.058$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-488\) $\mathrm{SU}(2)[C_{2}]$ \(q+3iq^{3}+47iq^{5}-244q^{7}+207q^{9}+\cdots\)
256.6.b.b 256.b 8.b $2$ $41.058$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-416\) $\mathrm{SU}(2)[C_{2}]$ \(q+4iq^{3}-7iq^{5}-208q^{7}+179q^{9}+\cdots\)
256.6.b.c 256.b 8.b $2$ $41.058$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-176\) $\mathrm{SU}(2)[C_{2}]$ \(q+6iq^{3}-3^{3}iq^{5}-88q^{7}+99q^{9}+\cdots\)
256.6.b.d 256.b 8.b $2$ $41.058$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-48\) $\mathrm{SU}(2)[C_{2}]$ \(q+10iq^{3}-37iq^{5}-24q^{7}-157q^{9}+\cdots\)
256.6.b.e 256.b 8.b $2$ $41.058$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-41iq^{5}+3^{5}q^{9}+597iq^{13}+2242q^{17}+\cdots\)
256.6.b.f 256.b 8.b $2$ $41.058$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(48\) $\mathrm{SU}(2)[C_{2}]$ \(q+10iq^{3}+37iq^{5}+24q^{7}-157q^{9}+\cdots\)
256.6.b.g 256.b 8.b $2$ $41.058$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(176\) $\mathrm{SU}(2)[C_{2}]$ \(q+6iq^{3}+3^{3}iq^{5}+88q^{7}+99q^{9}+\cdots\)
256.6.b.h 256.b 8.b $2$ $41.058$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(416\) $\mathrm{SU}(2)[C_{2}]$ \(q+4iq^{3}+7iq^{5}+208q^{7}+179q^{9}+\cdots\)
256.6.b.i 256.b 8.b $2$ $41.058$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(488\) $\mathrm{SU}(2)[C_{2}]$ \(q+3iq^{3}-47iq^{5}+244q^{7}+207q^{9}+\cdots\)
256.6.b.j 256.b 8.b $4$ $41.058$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(-368\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}+\beta _{2})q^{3}+(-5^{2}\beta _{1}+2\beta _{2})q^{5}+\cdots\)
256.6.b.k 256.b 8.b $4$ $41.058$ \(\Q(i, \sqrt{19})\) None \(0\) \(0\) \(0\) \(-272\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-5\beta _{1}+\beta _{3})q^{3}+(11\beta _{1}-2\beta _{3})q^{5}+\cdots\)
256.6.b.l 256.b 8.b $4$ $41.058$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{12}^{2}q^{3}-23\zeta_{12}q^{5}+3\zeta_{12}^{3}q^{7}+\cdots\)
256.6.b.m 256.b 8.b $4$ $41.058$ \(\Q(i, \sqrt{19})\) None \(0\) \(0\) \(0\) \(272\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-5\beta _{1}+\beta _{3})q^{3}+(-11\beta _{1}+2\beta _{3})q^{5}+\cdots\)
256.6.b.n 256.b 8.b $4$ $41.058$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(368\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}+\beta _{2})q^{3}+(5^{2}\beta _{1}-2\beta _{2})q^{5}+(92+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(256, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(256, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 2}\)