Space of modular forms of level 512, weight 4, and Character 512.e

• Label: 512.4.e
• Level: $512$
• Weight: $4$
• Character orbit: 512.e
• Rep. character: $\chi_{512}(129,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $96$
• Newform subspaces: $18$
• Sturm bound: $256$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 512 = 2^{9} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	512.e (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 16 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(256\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(512, [\chi])\).

	Total	New	Old
Modular forms	416	96	320
Cusp forms	352	96	256
Eisenstein series	64	0	64

Trace form

\( 96 q + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(512, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
512.4.e.a	$512$	$4$	512.e	16.e	$4$	$2$	$1$	$30.209$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(-8\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-4+4i)q^{3}+(-1-i)q^{5}+8iq^{7}+\cdots\)
512.4.e.b	$512$	$4$	512.e	16.e	$4$	$2$	$1$	$30.209$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-8\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-4+4i)q^{3}+(1+i)q^{5}-8iq^{7}+\cdots\)
512.4.e.c	$512$	$4$	512.e	16.e	$4$	$2$	$1$	$30.209$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(-26\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-13-13i)q^{5}+3^{3}iq^{9}+(-55+\cdots)q^{13}+\cdots\)
512.4.e.d	$512$	$4$	512.e	16.e	$4$	$2$	$1$	$30.209$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(-18\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-9-9i)q^{5}+3^{3}iq^{9}+(37-37i)q^{13}+\cdots\)
512.4.e.e	$512$	$4$	512.e	16.e	$4$	$2$	$1$	$30.209$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(18\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(9+9i)q^{5}+3^{3}iq^{9}+(-37+37i)q^{13}+\cdots\)
512.4.e.f	$512$	$4$	512.e	16.e	$4$	$2$	$1$	$30.209$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(26\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(13+13i)q^{5}+3^{3}iq^{9}+(55-55i)q^{13}+\cdots\)
512.4.e.g	$512$	$4$	512.e	16.e	$4$	$2$	$1$	$30.209$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(8\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(4-4i)q^{3}+(-1-i)q^{5}-8iq^{7}+\cdots\)
512.4.e.h	$512$	$4$	512.e	16.e	$4$	$2$	$1$	$30.209$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(8\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(4-4i)q^{3}+(1+i)q^{5}+8iq^{7}-5iq^{9}+\cdots\)
512.4.e.i	$512$	$4$	512.e	16.e	$4$	$4$	$2$	$30.209$	\(\Q(i, \sqrt{17})\)	None		$2$	$1$	\(0\)	\(-12\)	\(-12\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-3+3\beta _{1}+\beta _{3})q^{3}+(-3-3\beta _{1}+\cdots)q^{5}+\cdots\)
512.4.e.j	$512$	$4$	512.e	16.e	$4$	$4$	$2$	$30.209$	\(\Q(i, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(-12\)	\(12\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-3+3\beta _{1}+\beta _{3})q^{3}+(3+3\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots\)
512.4.e.k	$512$	$4$	512.e	16.e	$4$	$4$	$2$	$30.209$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(-44\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{3}+(-11-11\beta _{1})q^{5}+(-3\beta _{2}+\cdots)q^{7}+\cdots\)
512.4.e.l	$512$	$4$	512.e	16.e	$4$	$4$	$2$	$30.209$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(-28\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{3}+(-7-7\beta _{1})q^{5}+(\beta _{2}+\beta _{3})q^{7}+\cdots\)
512.4.e.m	$512$	$4$	512.e	16.e	$4$	$4$	$2$	$30.209$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(28\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{3}+(7+7\beta _{1})q^{5}+(-\beta _{2}-\beta _{3})q^{7}+\cdots\)
512.4.e.n	$512$	$4$	512.e	16.e	$4$	$4$	$2$	$30.209$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(44\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{3}+(11+11\beta _{1})q^{5}+(3\beta _{2}+3\beta _{3})q^{7}+\cdots\)
512.4.e.o	$512$	$4$	512.e	16.e	$4$	$4$	$2$	$30.209$	\(\Q(i, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(12\)	\(-12\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(3-3\beta _{1}+\beta _{3})q^{3}+(-3-3\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots\)
512.4.e.p	$512$	$4$	512.e	16.e	$4$	$4$	$2$	$30.209$	\(\Q(i, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(12\)	\(12\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(3-3\beta _{1}+\beta _{3})q^{3}+(3+3\beta _{1}-2\beta _{2}+\cdots)q^{5}+\cdots\)
512.4.e.q	$512$	$4$	512.e	16.e	$4$	$24$	$12$	$30.209$		None		$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$
512.4.e.r	$512$	$4$	512.e	16.e	$4$	$24$	$12$	$30.209$		None		$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{4}^{\mathrm{old}}(512, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(512, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 2}\)