Properties

Label 256.9.d
Level $256$
Weight $9$
Character orbit 256.d
Rep. character $\chi_{256}(127,\cdot)$
Character field $\Q$
Dimension $62$
Newform subspaces $10$
Sturm bound $288$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 256.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(288\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(256, [\chi])\).

Total New Old
Modular forms 268 66 202
Cusp forms 244 62 182
Eisenstein series 24 4 20

Trace form

\( 62 q + 126850 q^{9} + O(q^{10}) \) \( 62 q + 126850 q^{9} - 4 q^{17} - 4218746 q^{25} + 26240 q^{33} + 4 q^{41} - 65445698 q^{49} + 6173824 q^{57} - 31504136 q^{65} - 140965756 q^{73} + 220042814 q^{81} - 121779836 q^{89} + 16857340 q^{97} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(256, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
256.9.d.a 256.d 8.d $2$ $104.289$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) 4.9.b.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-527iq^{5}-3^{8}q^{9}+239iq^{13}-63358q^{17}+\cdots\)
256.9.d.b 256.d 8.d $4$ $104.289$ \(\Q(i, \sqrt{39})\) None 32.9.c.a \(0\) \(-144\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-6^{2}+\beta _{2})q^{3}+(91\beta _{1}-2\beta _{3})q^{5}+\cdots\)
256.9.d.c 256.d 8.d $4$ $104.289$ \(\Q(i, \sqrt{19})\) None 32.9.c.b \(0\) \(-80\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-20-\beta _{2})q^{3}+(-133\beta _{1}-6\beta _{3})q^{5}+\cdots\)
256.9.d.d 256.d 8.d $4$ $104.289$ \(\Q(\zeta_{12})\) None 16.9.c.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{12}^{2}q^{3}-129\zeta_{12}q^{5}-119\zeta_{12}^{3}q^{7}+\cdots\)
256.9.d.e 256.d 8.d $4$ $104.289$ \(\Q(i, \sqrt{39})\) None 4.9.b.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}-305\beta _{1}q^{5}-7\beta _{3}q^{7}+3423q^{9}+\cdots\)
256.9.d.f 256.d 8.d $4$ $104.289$ \(\Q(i, \sqrt{35})\) None 16.9.c.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}-255\beta _{1}q^{5}-9\beta _{3}q^{7}+13599q^{9}+\cdots\)
256.9.d.g 256.d 8.d $4$ $104.289$ \(\Q(i, \sqrt{19})\) None 32.9.c.b \(0\) \(80\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(20+\beta _{2})q^{3}+(133\beta _{1}+6\beta _{3})q^{5}+\cdots\)
256.9.d.h 256.d 8.d $4$ $104.289$ \(\Q(i, \sqrt{39})\) None 32.9.c.a \(0\) \(144\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(6^{2}+\beta _{2})q^{3}+(91\beta _{1}+2\beta _{3})q^{5}+(236\beta _{1}+\cdots)q^{7}+\cdots\)
256.9.d.i 256.d 8.d $16$ $104.289$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 128.9.c.a \(0\) \(-224\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-14+\beta _{1})q^{3}-\beta _{5}q^{5}+(-\beta _{5}+\beta _{12}+\cdots)q^{7}+\cdots\)
256.9.d.j 256.d 8.d $16$ $104.289$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 128.9.c.a \(0\) \(224\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(14-\beta _{1})q^{3}-\beta _{5}q^{5}+(\beta _{5}-\beta _{12}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(256, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(256, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 2}\)