Properties

Label 512.2.g
Level $512$
Weight $2$
Character orbit 512.g
Rep. character $\chi_{512}(65,\cdot)$
Character field $\Q(\zeta_{8})$
Dimension $48$
Newform subspaces $8$
Sturm bound $128$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.g (of order \(8\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 32 \)
Character field: \(\Q(\zeta_{8})\)
Newform subspaces: \( 8 \)
Sturm bound: \(128\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(512, [\chi])\).

Total New Old
Modular forms 320 80 240
Cusp forms 192 48 144
Eisenstein series 128 32 96

Trace form

\( 48q + 16q^{9} + O(q^{10}) \) \( 48q + 16q^{9} + 16q^{25} - 32q^{33} + 16q^{41} + 16q^{57} - 32q^{65} + 16q^{73} + 16q^{89} - 32q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(512, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
512.2.g.a \(4\) \(4.088\) \(\Q(\zeta_{8})\) None \(0\) \(-4\) \(-8\) \(-4\) \(q+(-1-\zeta_{8}^{3})q^{3}+(-2-2\zeta_{8}+\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
512.2.g.b \(4\) \(4.088\) \(\Q(\zeta_{8})\) None \(0\) \(-4\) \(8\) \(4\) \(q+(-1-\zeta_{8}^{3})q^{3}+(2+2\zeta_{8}-\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
512.2.g.c \(4\) \(4.088\) \(\Q(\zeta_{8})\) None \(0\) \(4\) \(-8\) \(4\) \(q+(1+\zeta_{8}^{3})q^{3}+(-2-2\zeta_{8}+\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
512.2.g.d \(4\) \(4.088\) \(\Q(\zeta_{8})\) None \(0\) \(4\) \(8\) \(-4\) \(q+(1+\zeta_{8}^{3})q^{3}+(2+2\zeta_{8}-\zeta_{8}^{2}+\zeta_{8}^{3})q^{5}+\cdots\)
512.2.g.e \(8\) \(4.088\) 8.0.18939904.2 None \(0\) \(-4\) \(-8\) \(8\) \(q+(-1+\beta _{4}-\beta _{5})q^{3}+(-1-\beta _{6})q^{5}+\cdots\)
512.2.g.f \(8\) \(4.088\) 8.0.18939904.2 None \(0\) \(-4\) \(8\) \(-8\) \(q+(-1+\beta _{4}-\beta _{5})q^{3}+(1+\beta _{6})q^{5}+\cdots\)
512.2.g.g \(8\) \(4.088\) 8.0.18939904.2 None \(0\) \(4\) \(-8\) \(-8\) \(q+(1+\beta _{3}-\beta _{4}+\beta _{7})q^{3}+(-1-\beta _{4}+\cdots)q^{5}+\cdots\)
512.2.g.h \(8\) \(4.088\) 8.0.18939904.2 None \(0\) \(4\) \(8\) \(8\) \(q+(1+\beta _{3}-\beta _{4}+\beta _{7})q^{3}+(1+\beta _{4})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(512, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(512, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 2}\)