Space of modular forms of level 1024, weight 2, and Character 1024.e

• Label: 1024.2.e
• Level: $1024$
• Weight: $2$
• Character orbit: 1024.e
• Rep. character: $\chi_{1024}(257,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $56$
• Newform subspaces: $16$
• Sturm bound: $256$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1024.e (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 16 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(256\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(3\), \(5\), \(47\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1024, [\chi])\).

	Total	New	Old
Modular forms	304	72	232
Cusp forms	208	56	152
Eisenstein series	96	16	80

Trace form

\( 56 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1024, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1024.2.e.a	$1024$	$2$	1024.e	16.e	$4$	$2$	$1$	$8.177$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-2}) \)		$4$	$1$	\(0\)	\(-4\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-2+2i)q^{3}-5iq^{9}+(2+2i)q^{11}+\cdots\)
1024.2.e.b	$1024$	$2$	1024.e	16.e	$4$	$2$	$1$	$8.177$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(-2\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+i)q^{3}+(-2-2i)q^{5}-4iq^{7}+\cdots\)
1024.2.e.c	$1024$	$2$	1024.e	16.e	$4$	$2$	$1$	$8.177$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+i)q^{3}+(2+2i)q^{5}+4iq^{7}+\cdots\)
1024.2.e.d	$1024$	$2$	1024.e	16.e	$4$	$2$	$1$	$8.177$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-i)q^{3}+(-2-2i)q^{5}+4iq^{7}+\cdots\)
1024.2.e.e	$1024$	$2$	1024.e	16.e	$4$	$2$	$1$	$8.177$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-i)q^{3}+(2+2i)q^{5}-4iq^{7}+iq^{9}+\cdots\)
1024.2.e.f	$1024$	$2$	1024.e	16.e	$4$	$2$	$1$	$8.177$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(2-2i)q^{3}-5iq^{9}+(-2-2i)q^{11}+\cdots\)
1024.2.e.g	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{4}]$	\(q+(-1+\zeta_{8}-\zeta_{8}^{2})q^{3}+(-2\zeta_{8}+3\zeta_{8}^{2}+\cdots)q^{9}+\cdots\)
1024.2.e.h	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\zeta_{8}^{2})q^{3}-\zeta_{8}q^{5}+(-\zeta_{8}-\zeta_{8}^{3})q^{7}+\cdots\)
1024.2.e.i	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}q^{3}+\zeta_{8}^{3}q^{5}+4\zeta_{8}^{2}q^{7}+\zeta_{8}^{2}q^{9}+\cdots\)
1024.2.e.j	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-1}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{4}]$	\(q-\zeta_{8}q^{5}+3\zeta_{8}^{2}q^{9}-3\zeta_{8}^{3}q^{13}-2q^{17}+\cdots\)
1024.2.e.k	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-1}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{4}]$	\(q+\zeta_{8}q^{5}+3\zeta_{8}^{2}q^{9}-\zeta_{8}^{3}q^{13}+2q^{17}+\cdots\)
1024.2.e.l	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{4}]$	\(q+\zeta_{8}q^{3}+\zeta_{8}^{2}q^{9}-3\zeta_{8}^{3}q^{11}+6q^{17}+\cdots\)
1024.2.e.m	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}q^{3}-\zeta_{8}^{3}q^{5}-4\zeta_{8}^{2}q^{7}+\zeta_{8}^{2}q^{9}+\cdots\)
1024.2.e.n	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\zeta_{8}^{2})q^{3}-\zeta_{8}q^{5}+(\zeta_{8}+\zeta_{8}^{3})q^{7}+\cdots\)
1024.2.e.o	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{4}]$	\(q+(1+\zeta_{8}+\zeta_{8}^{2})q^{3}+(2\zeta_{8}+3\zeta_{8}^{2}+\cdots)q^{9}+\cdots\)
1024.2.e.p	$1024$	$2$	1024.e	16.e	$4$	$8$	$4$	$8.177$	\(\Q(\zeta_{24})\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{24}^{7}q^{3}+\zeta_{24}q^{5}-\zeta_{24}^{6}q^{7}-3\zeta_{24}^{2}q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1024, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1024, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(512, [\chi])\)\(^{\oplus 2}\)