Properties

Label 256.8.a
Level $256$
Weight $8$
Character orbit 256.a
Rep. character $\chi_{256}(1,\cdot)$
Character field $\Q$
Dimension $54$
Newform subspaces $18$
Sturm bound $256$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 256.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(256\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(256))\).

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

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)Dim.
\(+\)\(28\)
\(-\)\(26\)

Trace form

\( 54 q + 36454 q^{9} + O(q^{10}) \) \( 54 q + 36454 q^{9} - 4 q^{17} + 718754 q^{25} + 8744 q^{33} + 4 q^{41} + 6985446 q^{49} - 6213128 q^{57} - 2541280 q^{65} + 302356 q^{73} + 20203502 q^{81} + 9562100 q^{89} + 30397212 q^{97} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(256))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2
256.8.a.a 256.a 1.a $1$ $79.971$ \(\Q\) \(\Q(\sqrt{-2}) \) \(0\) \(-86\) \(0\) \(0\) $+$ $N(\mathrm{U}(1))$ \(q-86q^{3}+5209q^{9}+8814q^{11}-22182q^{17}+\cdots\)
256.8.a.b 256.a 1.a $1$ $79.971$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-556\) \(0\) $-$ $N(\mathrm{U}(1))$ \(q-556q^{5}-3^{7}q^{9}+13108q^{13}+40094q^{17}+\cdots\)
256.8.a.c 256.a 1.a $1$ $79.971$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(556\) \(0\) $+$ $N(\mathrm{U}(1))$ \(q+556q^{5}-3^{7}q^{9}-13108q^{13}+40094q^{17}+\cdots\)
256.8.a.d 256.a 1.a $1$ $79.971$ \(\Q\) \(\Q(\sqrt{-2}) \) \(0\) \(86\) \(0\) \(0\) $-$ $N(\mathrm{U}(1))$ \(q+86q^{3}+5209q^{9}-8814q^{11}-22182q^{17}+\cdots\)
256.8.a.e 256.a 1.a $2$ $79.971$ \(\Q(\sqrt{435}) \) None \(0\) \(-60\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-30q^{3}+\beta q^{5}-2\beta q^{7}-1287q^{9}+\cdots\)
256.8.a.f 256.a 1.a $2$ $79.971$ \(\Q(\sqrt{46}) \) None \(0\) \(0\) \(-360\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-180q^{5}+4\beta q^{7}+757q^{9}+\cdots\)
256.8.a.g 256.a 1.a $2$ $79.971$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) $-$ $N(\mathrm{U}(1))$ \(q+13\beta q^{3}-835q^{9}-181\beta q^{11}+22182q^{17}+\cdots\)
256.8.a.h 256.a 1.a $2$ $79.971$ \(\Q(\sqrt{46}) \) None \(0\) \(0\) \(360\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+180q^{5}-4\beta q^{7}+757q^{9}+\cdots\)
256.8.a.i 256.a 1.a $2$ $79.971$ \(\Q(\sqrt{435}) \) None \(0\) \(60\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+30q^{3}+\beta q^{5}+2\beta q^{7}-1287q^{9}+\cdots\)
256.8.a.j 256.a 1.a $4$ $79.971$ \(\Q(\sqrt{3}, \sqrt{601})\) None \(0\) \(-80\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+(-20-\beta _{1})q^{3}+(\beta _{2}+\beta _{3})q^{5}+(7\beta _{2}+\cdots)q^{7}+\cdots\)
256.8.a.k 256.a 1.a $4$ $79.971$ 4.4.3484860.2 None \(0\) \(0\) \(-304\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+(-76+\beta _{1})q^{5}+(-\beta _{2}+\beta _{3})q^{7}+\cdots\)
256.8.a.l 256.a 1.a $4$ $79.971$ \(\Q(\sqrt{2}, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-3\beta _{1}q^{3}-17\beta _{3}q^{5}+29\beta _{2}q^{7}-1827q^{9}+\cdots\)
256.8.a.m 256.a 1.a $4$ $79.971$ \(\Q(\sqrt{2}, \sqrt{85})\) None \(0\) \(0\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+21\beta _{1}q^{3}-\beta _{2}q^{5}+\beta _{3}q^{7}+1341q^{9}+\cdots\)
256.8.a.n 256.a 1.a $4$ $79.971$ \(\Q(\sqrt{2}, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}-\beta _{3}q^{5}-71\beta _{1}q^{7}+4573q^{9}+\cdots\)
256.8.a.o 256.a 1.a $4$ $79.971$ 4.4.3484860.2 None \(0\) \(0\) \(304\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+(76-\beta _{1})q^{5}+(\beta _{2}-\beta _{3})q^{7}+\cdots\)
256.8.a.p 256.a 1.a $4$ $79.971$ \(\Q(\sqrt{3}, \sqrt{601})\) None \(0\) \(80\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+(20+\beta _{1})q^{3}+(-\beta _{2}-\beta _{3})q^{5}+(7\beta _{2}+\cdots)q^{7}+\cdots\)
256.8.a.q 256.a 1.a $6$ $79.971$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(-688\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(\beta _{1}+\beta _{2})q^{5}+(-115+\beta _{4}+\cdots)q^{7}+\cdots\)
256.8.a.r 256.a 1.a $6$ $79.971$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(688\) $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(115-\beta _{4}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(256))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(256)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 2}\)