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

• Label: 1600.2.f
• Level: $1600$
• Weight: $2$
• Character orbit: 1600.f
• Rep. character: $\chi_{1600}(1249,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $10$
• Sturm bound: $480$
• Trace bound: $31$

Defining parameters

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 40 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(480\)
Trace bound:			\(31\)
Distinguishing \(T_p\):			\(3\), \(31\)

Dimensions

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

	Total	New	Old
Modular forms	276	36	240
Cusp forms	204	36	168
Eisenstein series	72	0	72

Trace form

\( 36 q + 36 q^{9} + 24 q^{41} - 36 q^{49} + 132 q^{81} + 120 q^{89}+O(q^{100}) \)

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.f.a	$1600$	$2$	1600.f	40.f	$2$	$2$	$2$	$12.776$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-2}) \)					64.2.b.a	$4$	$1$	\(0\)	\(-4\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-2 q^{3}+q^{9}+3\beta q^{11}-3\beta q^{17}+\cdots\)
1600.2.f.b	$1600$	$2$	1600.f	40.f	$2$	$2$	$2$	$12.776$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-2}) \)					64.2.b.a	$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+2 q^{3}+q^{9}-3\beta q^{11}-3\beta q^{17}+\cdots\)
1600.2.f.c	$1600$	$2$	1600.f	40.f	$2$	$4$	$4$	$12.776$	\(\Q(\zeta_{12})\)	None		✓			1600.2.d.e	$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}-\beta_{2} q^{7}-2 q^{9}+3\beta_1 q^{11}+\cdots\)
1600.2.f.d	$1600$	$2$	1600.f	40.f	$2$	$4$	$4$	$12.776$	\(\Q(\zeta_{12})\)	None					320.2.d.a	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_1-1)q^{3}+(-\beta_{3}+\beta_{2})q^{7}+\cdots\)
1600.2.f.e	$1600$	$2$	1600.f	40.f	$2$	$4$	$4$	$12.776$	\(\Q(\zeta_{12})\)	None					320.2.d.a	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_1-1)q^{3}+(-\beta_{3}+\beta_{2})q^{7}+\cdots\)
1600.2.f.f	$1600$	$2$	1600.f	40.f	$2$	$4$	$4$	$12.776$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-2}) \)		✓			1600.2.d.c	$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+(-1+\beta _{3})q^{3}+(4-2\beta _{3})q^{9}+(3\beta _{1}+\cdots)q^{11}+\cdots\)
1600.2.f.g	$1600$	$2$	1600.f	40.f	$2$	$4$	$4$	$12.776$	\(\Q(\zeta_{12})\)	None		✓			1600.2.d.e	$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}+\beta_{2} q^{7}-2 q^{9}+3\beta_1 q^{11}+\cdots\)
1600.2.f.h	$1600$	$2$	1600.f	40.f	$2$	$4$	$4$	$12.776$	\(\Q(\zeta_{12})\)	None					320.2.d.a	$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta_1+1)q^{3}+(\beta_{3}-\beta_{2})q^{7}+\cdots\)
1600.2.f.i	$1600$	$2$	1600.f	40.f	$2$	$4$	$4$	$12.776$	\(\Q(\zeta_{12})\)	None					320.2.d.a	$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta_1+1)q^{3}+(\beta_{3}-\beta_{2})q^{7}+\cdots\)
1600.2.f.j	$1600$	$2$	1600.f	40.f	$2$	$4$	$4$	$12.776$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-2}) \)		✓			1600.2.d.c	$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+(1-\beta _{3})q^{3}+(4-2\beta _{3})q^{9}+(3\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)

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

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