Properties

Label 512.4.a
Level $512$
Weight $4$
Character orbit 512.a
Rep. character $\chi_{512}(1,\cdot)$
Character field $\Q$
Dimension $48$
Newform subspaces $15$
Sturm bound $256$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 512.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(256\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(512))\).

Total New Old
Modular forms 208 48 160
Cusp forms 176 48 128
Eisenstein series 32 0 32

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
\(+\)\(26\)
\(-\)\(22\)

Trace form

\( 48 q + 432 q^{9} + O(q^{10}) \) \( 48 q + 432 q^{9} + 1200 q^{25} + 2352 q^{49} + 1952 q^{65} + 1184 q^{73} + 3888 q^{81} - 352 q^{89} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(512))\) 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
512.4.a.a 512.a 1.a $2$ $30.209$ \(\Q(\sqrt{2}) \) None \(0\) \(-8\) \(-16\) \(-24\) $-$ $\mathrm{SU}(2)$ \(q+(-4+3\beta )q^{3}+(-8-2\beta )q^{5}+(-12+\cdots)q^{7}+\cdots\)
512.4.a.b 512.a 1.a $2$ $30.209$ \(\Q(\sqrt{2}) \) None \(0\) \(-8\) \(16\) \(24\) $+$ $\mathrm{SU}(2)$ \(q+(-4+3\beta )q^{3}+(8+2\beta )q^{5}+(12-2^{4}\beta )q^{7}+\cdots\)
512.4.a.c 512.a 1.a $2$ $30.209$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) \(0\) \(-4\) \(0\) \(0\) $-$ $N(\mathrm{U}(1))$ \(q+(-2+5\beta )q^{3}+(3^{3}-20\beta )q^{9}+(-50+\cdots)q^{11}+\cdots\)
512.4.a.d 512.a 1.a $2$ $30.209$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-8\) $-$ $\mathrm{SU}(2)$ \(q+5\beta q^{3}-10\beta q^{5}-4q^{7}+23q^{9}+\cdots\)
512.4.a.e 512.a 1.a $2$ $30.209$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(8\) $+$ $\mathrm{SU}(2)$ \(q+5\beta q^{3}+10\beta q^{5}+4q^{7}+23q^{9}+\cdots\)
512.4.a.f 512.a 1.a $2$ $30.209$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) \(0\) \(4\) \(0\) \(0\) $+$ $N(\mathrm{U}(1))$ \(q+(2+5\beta )q^{3}+(3^{3}+20\beta )q^{9}+(50+9\beta )q^{11}+\cdots\)
512.4.a.g 512.a 1.a $2$ $30.209$ \(\Q(\sqrt{2}) \) None \(0\) \(8\) \(-16\) \(24\) $+$ $\mathrm{SU}(2)$ \(q+(4+3\beta )q^{3}+(-8+2\beta )q^{5}+(12+2^{4}\beta )q^{7}+\cdots\)
512.4.a.h 512.a 1.a $2$ $30.209$ \(\Q(\sqrt{2}) \) None \(0\) \(8\) \(16\) \(-24\) $-$ $\mathrm{SU}(2)$ \(q+(4+3\beta )q^{3}+(8-2\beta )q^{5}+(-12-2^{4}\beta )q^{7}+\cdots\)
512.4.a.i 512.a 1.a $4$ $30.209$ \(\Q(\sqrt{2}, \sqrt{11})\) None \(0\) \(-16\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+(-4-\beta _{1})q^{3}+(4\beta _{2}+\beta _{3})q^{5}+(-\beta _{2}+\cdots)q^{7}+\cdots\)
512.4.a.j 512.a 1.a $4$ $30.209$ \(\Q(\zeta_{16})^+\) None \(0\) \(-8\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+(-2-3\beta _{2})q^{3}-\beta _{1}q^{5}+(\beta _{1}-\beta _{3})q^{7}+\cdots\)
512.4.a.k 512.a 1.a $4$ $30.209$ \(\Q(\sqrt{2}, \sqrt{11})\) None \(0\) \(0\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-\beta _{3}q^{5}-\beta _{2}q^{7}-5q^{9}-7\beta _{1}q^{11}+\cdots\)
512.4.a.l 512.a 1.a $4$ $30.209$ \(\Q(\zeta_{16})^+\) None \(0\) \(8\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+(2+3\beta _{2})q^{3}+\beta _{1}q^{5}+(\beta _{1}-\beta _{3})q^{7}+\cdots\)
512.4.a.m 512.a 1.a $4$ $30.209$ \(\Q(\sqrt{2}, \sqrt{11})\) None \(0\) \(16\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+(4-\beta _{1})q^{3}+(4\beta _{2}-\beta _{3})q^{5}+(\beta _{2}+4\beta _{3})q^{7}+\cdots\)
512.4.a.n 512.a 1.a $6$ $30.209$ 6.6.33032192.1 None \(0\) \(0\) \(-20\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(-3+\beta _{2})q^{5}+(\beta _{1}-\beta _{4}+\cdots)q^{7}+\cdots\)
512.4.a.o 512.a 1.a $6$ $30.209$ 6.6.33032192.1 None \(0\) \(0\) \(20\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(3-\beta _{2})q^{5}+(-\beta _{1}+\beta _{4}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(512))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(512)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 2}\)