Properties

Label 512.2.e
Level $512$
Weight $2$
Character orbit 512.e
Rep. character $\chi_{512}(129,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $32$
Newform subspaces $10$
Sturm bound $128$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.e (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 16 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 10 \)
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 160 32 128
Cusp forms 96 32 64
Eisenstein series 64 0 64

Trace form

\( 32q + O(q^{10}) \) \( 32q - 32q^{49} + 64q^{65} - 32q^{81} + O(q^{100}) \)

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.e.a \(2\) \(4.088\) \(\Q(\sqrt{-1}) \) None \(0\) \(-4\) \(-2\) \(0\) \(q+(-2+2i)q^{3}+(-1-i)q^{5}+4iq^{7}+\cdots\)
512.2.e.b \(2\) \(4.088\) \(\Q(\sqrt{-1}) \) None \(0\) \(-4\) \(2\) \(0\) \(q+(-2+2i)q^{3}+(1+i)q^{5}-4iq^{7}+\cdots\)
512.2.e.c \(2\) \(4.088\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-6\) \(0\) \(q+(-3-3i)q^{5}+3iq^{9}+(-1+i)q^{13}+\cdots\)
512.2.e.d \(2\) \(4.088\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-2\) \(0\) \(q+(-1-i)q^{5}+3iq^{9}+(5-5i)q^{13}+\cdots\)
512.2.e.e \(2\) \(4.088\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(2\) \(0\) \(q+(1+i)q^{5}+3iq^{9}+(-5+5i)q^{13}+\cdots\)
512.2.e.f \(2\) \(4.088\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(6\) \(0\) \(q+(3+3i)q^{5}+3iq^{9}+(1-i)q^{13}+\cdots\)
512.2.e.g \(2\) \(4.088\) \(\Q(\sqrt{-1}) \) None \(0\) \(4\) \(-2\) \(0\) \(q+(2-2i)q^{3}+(-1-i)q^{5}-4iq^{7}+\cdots\)
512.2.e.h \(2\) \(4.088\) \(\Q(\sqrt{-1}) \) None \(0\) \(4\) \(2\) \(0\) \(q+(2-2i)q^{3}+(1+i)q^{5}+4iq^{7}-5iq^{9}+\cdots\)
512.2.e.i \(8\) \(4.088\) \(\Q(\zeta_{16})\) None \(0\) \(0\) \(-8\) \(0\) \(q-\zeta_{16}^{6}q^{3}+(-1+\zeta_{16}^{2}-\zeta_{16}^{4}+\cdots)q^{5}+\cdots\)
512.2.e.j \(8\) \(4.088\) \(\Q(\zeta_{16})\) None \(0\) \(0\) \(8\) \(0\) \(q-\zeta_{16}^{6}q^{3}+(1-\zeta_{16}^{2}+\zeta_{16}^{4})q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(512, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 2}\)