Space of modular forms of level 1600, weight 2, and Character 1600.d

• Label: 1600.2.d
• Level: $1600$
• Weight: $2$
• Character orbit: 1600.d
• Rep. character: $\chi_{1600}(801,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $9$
• Sturm bound: $480$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(480\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(3\), \(7\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1600, [\chi])\).

	Total	New	Old
Modular forms	276	38	238
Cusp forms	204	38	166
Eisenstein series	72	0	72

Trace form

\( 38 q - 38 q^{9} + 12 q^{17} + 24 q^{33} - 36 q^{41} + 22 q^{49} - 8 q^{57} + 28 q^{73} + 110 q^{81} - 36 q^{89} - 52 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1600, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1600.2.d.a	$1600$	$2$	1600.d	8.b	$2$	$2$	$2$	$12.776$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-2}) \)					64.2.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{3}-q^{9}+3\beta q^{11}+6 q^{17}-\beta q^{19}+\cdots\)
1600.2.d.b	$1600$	$2$	1600.d	8.b	$2$	$4$	$4$	$12.776$	\(\Q(\zeta_{12})\)	None					320.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{3} q^{3}+(\beta_1-3)q^{7}+(2\beta_1-1)q^{9}+\cdots\)
1600.2.d.c	$1600$	$2$	1600.d	8.b	$2$	$4$	$4$	$12.776$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-2}) \)		✓			1600.2.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{2}q^{3}+(-4-\beta _{3})q^{9}+(\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)
1600.2.d.d	$1600$	$2$	1600.d	8.b	$2$	$4$	$4$	$12.776$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-2}) \)		✓			1600.2.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{2}q^{3}+(-4-\beta _{3})q^{9}+(-\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
1600.2.d.e	$1600$	$2$	1600.d	8.b	$2$	$4$	$4$	$12.776$	\(\Q(\zeta_{12})\)	None		✓			1600.2.d.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{3}-\beta_{3} q^{7}+2 q^{9}+3\beta_1 q^{11}+\cdots\)
1600.2.d.f	$1600$	$2$	1600.d	8.b	$2$	$4$	$4$	$12.776$	\(\Q(\zeta_{12})\)	None		✓			1600.2.d.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{3}+\beta_{3} q^{7}+2 q^{9}-3\beta_1 q^{11}+\cdots\)
1600.2.d.g	$1600$	$2$	1600.d	8.b	$2$	$4$	$4$	$12.776$	\(\Q(i, \sqrt{5})\)	\(\Q(\sqrt{-10}) \)					320.2.f.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{3}q^{7}+3q^{9}-\beta _{1}q^{11}+\beta _{2}q^{13}+\cdots\)
1600.2.d.h	$1600$	$2$	1600.d	8.b	$2$	$4$	$4$	$12.776$	\(\Q(\zeta_{12})\)	None					320.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{3} q^{3}+(-\beta_1+3)q^{7}+(2\beta_1-1)q^{9}+\cdots\)
1600.2.d.i	$1600$	$2$	1600.d	8.b	$2$	$8$	$8$	$12.776$	\(\Q(\zeta_{24})\)	None			✓		320.2.f.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{5} q^{3}-\beta_{4} q^{7}-3 q^{9}-\beta_1 q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1600, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1600, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 2}\)