Properties

Label 256.8.b
Level $256$
Weight $8$
Character orbit 256.b
Rep. character $\chi_{256}(129,\cdot)$
Character field $\Q$
Dimension $54$
Newform subspaces $14$
Sturm bound $256$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(256\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(256, [\chi])\).

Total New Old
Modular forms 236 58 178
Cusp forms 212 54 158
Eisenstein series 24 4 20

Trace form

\( 54 q - 36446 q^{9} + O(q^{10}) \) \( 54 q - 36446 q^{9} - 4 q^{17} - 718746 q^{25} - 8752 q^{33} + 4 q^{41} + 2897062 q^{49} - 6195632 q^{57} - 3166280 q^{65} - 10438860 q^{73} + 20186006 q^{81} - 9562092 q^{89} - 30397220 q^{97} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(256, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
256.8.b.a 256.b 8.b $2$ $79.971$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-2448\) $\mathrm{SU}(2)[C_{2}]$ \(q+22iq^{3}+215iq^{5}-1224q^{7}+251q^{9}+\cdots\)
256.8.b.b 256.b 8.b $2$ $79.971$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-2032\) $\mathrm{SU}(2)[C_{2}]$ \(q+6iq^{3}+105iq^{5}-1016q^{7}+2043q^{9}+\cdots\)
256.8.b.c 256.b 8.b $2$ $79.971$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-912\) $\mathrm{SU}(2)[C_{2}]$ \(q+42iq^{3}+41iq^{5}-456q^{7}-4869q^{9}+\cdots\)
256.8.b.d 256.b 8.b $2$ $79.971$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+29iq^{5}+3^{7}q^{9}-4449iq^{13}-40094q^{17}+\cdots\)
256.8.b.e 256.b 8.b $2$ $79.971$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(912\) $\mathrm{SU}(2)[C_{2}]$ \(q+42iq^{3}-41iq^{5}+456q^{7}-4869q^{9}+\cdots\)
256.8.b.f 256.b 8.b $2$ $79.971$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(2032\) $\mathrm{SU}(2)[C_{2}]$ \(q+6iq^{3}-105iq^{5}+1016q^{7}+2043q^{9}+\cdots\)
256.8.b.g 256.b 8.b $2$ $79.971$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(2448\) $\mathrm{SU}(2)[C_{2}]$ \(q+22iq^{3}-215iq^{5}+1224q^{7}+251q^{9}+\cdots\)
256.8.b.h 256.b 8.b $4$ $79.971$ \(\Q(i, \sqrt{10})\) None \(0\) \(0\) \(0\) \(-2496\) $\mathrm{SU}(2)[C_{2}]$ \(q+(4\beta _{1}+\beta _{2})q^{3}+(-45\beta _{1}-8\beta _{2})q^{5}+\cdots\)
256.8.b.i 256.b 8.b $4$ $79.971$ \(\Q(i, \sqrt{15})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{3}+35\beta _{1}q^{5}+9\beta _{2}q^{7}-1653q^{9}+\cdots\)
256.8.b.j 256.b 8.b $4$ $79.971$ \(\Q(i, \sqrt{10})\) None \(0\) \(0\) \(0\) \(2496\) $\mathrm{SU}(2)[C_{2}]$ \(q+(4\beta _{1}+\beta _{2})q^{3}+(45\beta _{1}+8\beta _{2})q^{5}+\cdots\)
256.8.b.k 256.b 8.b $6$ $79.971$ 6.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(-24\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{1}+\beta _{2})q^{3}+(\beta _{1}-\beta _{4})q^{5}+(-3+\cdots)q^{7}+\cdots\)
256.8.b.l 256.b 8.b $6$ $79.971$ 6.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(24\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{1}+\beta _{2})q^{3}+(-\beta _{1}+\beta _{4})q^{5}+\cdots\)
256.8.b.m 256.b 8.b $8$ $79.971$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(-2720\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{1}-\beta _{2})q^{3}+(-8\beta _{1}-2\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
256.8.b.n 256.b 8.b $8$ $79.971$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(2720\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{1}-\beta _{2})q^{3}+(8\beta _{1}+2\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(256, [\chi])\) into lower level spaces

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