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

\( 56 q + 16 q^{17} - 16 q^{33} + 8 q^{49} - 16 q^{65} + 24 q^{81} - 16 q^{97}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	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	$4$	$2$	$1$	$8.177$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-2}) \)					128.2.b.a	$4$	$1$	\(0\)	\(-4\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(2 i-2)q^{3}-5 i q^{9}+(2 i+2)q^{11}+\cdots\)
1024.2.e.b	$1024$	$2$	1024.e	16.e	$4$	$2$	$1$	$8.177$	\(\Q(\sqrt{-1}) \)	None					512.2.a.c	$2$	$1$	\(0\)	\(-2\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i-1)q^{3}+(-2 i-2)q^{5}-4 i q^{7}+\cdots\)
1024.2.e.c	$1024$	$2$	1024.e	16.e	$4$	$2$	$1$	$8.177$	\(\Q(\sqrt{-1}) \)	None					512.2.a.c	$2$	$0$	\(0\)	\(-2\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i-1)q^{3}+(2 i+2)q^{5}+4 i q^{7}+\cdots\)
1024.2.e.d	$1024$	$2$	1024.e	16.e	$4$	$2$	$1$	$8.177$	\(\Q(\sqrt{-1}) \)	None					512.2.a.c	$2$	$0$	\(0\)	\(2\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i+1)q^{3}+(-2 i-2)q^{5}+4 i q^{7}+\cdots\)
1024.2.e.e	$1024$	$2$	1024.e	16.e	$4$	$2$	$1$	$8.177$	\(\Q(\sqrt{-1}) \)	None					512.2.a.c	$2$	$0$	\(0\)	\(2\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i+1)q^{3}+(2 i+2)q^{5}-4 i q^{7}+\cdots\)
1024.2.e.f	$1024$	$2$	1024.e	16.e	$4$	$2$	$1$	$8.177$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-2}) \)					128.2.b.a	$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-2 i+2)q^{3}-5 i q^{9}+(-2 i-2)q^{11}+\cdots\)
1024.2.e.g	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)					512.2.a.a	$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{4}]$	\(q+(-\beta_{2}+\beta_1-1)q^{3}+(-2\beta_{3}+3\beta_{2}-2\beta_1)q^{9}+\cdots\)
1024.2.e.h	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	None					512.2.a.b	$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta_{2}-1)q^{3}-\beta_1 q^{5}+(-\beta_{3}-\beta_1)q^{7}+\cdots\)
1024.2.e.i	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	None					128.2.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta_1 q^{3}+\beta_{3} q^{5}+4\beta_{2} q^{7}+\beta_{2} q^{9}+\cdots\)
1024.2.e.j	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-1}) \)					32.2.a.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{4}]$	\(q-\beta_1 q^{5}+3\beta_{2} q^{9}-3\beta_{3} q^{13}-2 q^{17}+\cdots\)
1024.2.e.k	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-1}) \)					128.2.b.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta_1 q^{5}+3\beta_{2} q^{9}-\beta_{3} q^{13}+2 q^{17}+\cdots\)
1024.2.e.l	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)					64.2.b.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta_1 q^{3}+\beta_{2} q^{9}-3\beta_{3} q^{11}+6 q^{17}+\cdots\)
1024.2.e.m	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	None					128.2.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta_1 q^{3}-\beta_{3} q^{5}-4\beta_{2} q^{7}+\beta_{2} q^{9}+\cdots\)
1024.2.e.n	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	None					512.2.a.b	$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta_{2}+1)q^{3}-\beta_1 q^{5}+(\beta_{3}+\beta_1)q^{7}+\cdots\)
1024.2.e.o	$1024$	$2$	1024.e	16.e	$4$	$4$	$2$	$8.177$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)					512.2.a.a	$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{4}]$	\(q+(\beta_{2}+\beta_1+1)q^{3}+(2\beta_{3}+3\beta_{2}+2\beta_1)q^{9}+\cdots\)
1024.2.e.p	$1024$	$2$	1024.e	16.e	$4$	$8$	$4$	$8.177$	\(\Q(\zeta_{24})\)	None			✓		512.2.a.g	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta_{7} q^{3}+\beta_1 q^{5}-\beta_{6} q^{7}-3\beta_{2} q^{9}+\cdots\)

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

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