Properties

Label 8.8.a
Level $8$
Weight $8$
Character orbit 8.a
Rep. character $\chi_{8}(1,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $2$
Sturm bound $8$
Trace bound $3$

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Defining parameters

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

Dimensions

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

Total New Old
Modular forms 9 2 7
Cusp forms 5 2 3
Eisenstein series 4 0 4

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.
\(+\)\(1\)
\(-\)\(1\)

Trace form

\( 2q - 40q^{3} + 348q^{5} - 1680q^{7} + 4618q^{9} + O(q^{10}) \) \( 2q - 40q^{3} + 348q^{5} - 1680q^{7} + 4618q^{9} - 5688q^{11} - 4660q^{13} + 25808q^{15} - 27420q^{17} - 1352q^{19} - 15552q^{21} + 83280q^{23} + 35374q^{25} - 332560q^{27} + 222636q^{29} + 210880q^{31} + 72800q^{33} - 488928q^{35} - 471780q^{37} + 1174544q^{39} - 177996q^{41} + 20360q^{43} - 507188q^{45} - 397920q^{47} + 59026q^{49} + 220720q^{51} + 2109180q^{53} - 1153552q^{55} - 587360q^{57} - 1863576q^{59} + 1002860q^{61} - 1913040q^{63} + 3514536q^{65} + 7625560q^{67} - 6793792q^{69} - 5259024q^{71} - 7351180q^{73} + 10695784q^{75} + 5023680q^{77} - 1086112q^{79} + 4104658q^{81} - 6949320q^{83} - 6081800q^{85} - 2978160q^{87} + 5132628q^{89} - 2573664q^{91} + 14460160q^{93} - 2692848q^{95} + 4865860q^{97} - 11495192q^{99} + O(q^{100}) \)

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

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
8.8.a.a \(1\) \(2.499\) \(\Q\) None \(0\) \(-84\) \(-82\) \(-456\) \(-\) \(q-84q^{3}-82q^{5}-456q^{7}+4869q^{9}+\cdots\)
8.8.a.b \(1\) \(2.499\) \(\Q\) None \(0\) \(44\) \(430\) \(-1224\) \(+\) \(q+44q^{3}+430q^{5}-1224q^{7}-251q^{9}+\cdots\)

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

\( S_{8}^{\mathrm{old}}(\Gamma_0(8)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)