Properties

Label 8.8.a
Level 8
Weight 8
Character orbit a
Rep. character \(\chi_{8}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newforms 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: \(\Q\)
Newforms: \( 2 \)
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

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

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(8))\) into irreducible Hecke orbits

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

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