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

• Label: 1600.2.n
• Level: $1600$
• Weight: $2$
• Character orbit: 1600.n
• Rep. character: $\chi_{1600}(1343,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $68$
• Newform subspaces: $22$
• Sturm bound: $480$
• Trace bound: $21$

Defining parameters

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.n (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(480\)
Trace bound:			\(21\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(13\)

Dimensions

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

	Total	New	Old
Modular forms	552	76	476
Cusp forms	408	68	340
Eisenstein series	144	8	136

Trace form

\( 68 q - 4 q^{13} + 12 q^{17} + 8 q^{21} - 8 q^{33} - 4 q^{37} - 8 q^{41} - 52 q^{53} + 16 q^{57} + 72 q^{61} + 12 q^{73} + 72 q^{77} - 108 q^{81} + 56 q^{93} + 28 q^{97}+O(q^{100}) \)

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.n.a	$1600$	$2$	1600.n	20.e	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	None					160.2.n.a	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2 i-2)q^{3}+(-2 i+2)q^{7}+\cdots\)
1600.2.n.b	$1600$	$2$	1600.n	20.e	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-5}) \)					100.2.e.a	$4$	$0$	\(0\)	\(-2\)	\(0\)	\(-6\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-i-1)q^{3}+(3 i-3)q^{7}-i q^{9}+\cdots\)
1600.2.n.c	$1600$	$2$	1600.n	20.e	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	None					160.2.n.c	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i-1)q^{3}+(3 i-3)q^{7}-i q^{9}+\cdots\)
1600.2.n.d	$1600$	$2$	1600.n	20.e	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	None					800.2.n.b	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i-1)q^{3}+(-i+1)q^{7}-i q^{9}+\cdots\)
1600.2.n.e	$1600$	$2$	1600.n	20.e	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	None					160.2.n.b	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i-1)q^{3}+(-i+1)q^{7}-i q^{9}+\cdots\)
1600.2.n.f	$1600$	$2$	1600.n	20.e	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	None					800.2.n.b	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i-1)q^{3}+(-i+1)q^{7}-i q^{9}+\cdots\)
1600.2.n.g	$1600$	$2$	1600.n	20.e	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					80.2.n.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q-3 i q^{9}+(5 i-5)q^{13}+(5 i+5)q^{17}+\cdots\)
1600.2.n.h	$1600$	$2$	1600.n	20.e	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					20.2.e.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q-3 i q^{9}+(i-1)q^{13}+(-3 i-3)q^{17}+\cdots\)
1600.2.n.i	$1600$	$2$	1600.n	20.e	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	None					800.2.n.b	$2$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i+1)q^{3}+(i-1)q^{7}-i q^{9}-4 i q^{11}+\cdots\)
1600.2.n.j	$1600$	$2$	1600.n	20.e	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	None					160.2.n.b	$2$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i+1)q^{3}+(i-1)q^{7}-i q^{9}-6 i q^{11}+\cdots\)
1600.2.n.k	$1600$	$2$	1600.n	20.e	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	None					800.2.n.b	$2$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i+1)q^{3}+(i-1)q^{7}-i q^{9}+4 i q^{11}+\cdots\)
1600.2.n.l	$1600$	$2$	1600.n	20.e	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-5}) \)					100.2.e.a	$4$	$0$	\(0\)	\(2\)	\(0\)	\(6\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(i+1)q^{3}+(-3 i+3)q^{7}-i q^{9}+\cdots\)
1600.2.n.m	$1600$	$2$	1600.n	20.e	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	None					160.2.n.c	$2$	$0$	\(0\)	\(2\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i+1)q^{3}+(-3 i+3)q^{7}-i q^{9}+\cdots\)
1600.2.n.n	$1600$	$2$	1600.n	20.e	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	None					160.2.n.a	$2$	$0$	\(0\)	\(4\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2 i+2)q^{3}+(2 i-2)q^{7}+5 i q^{9}+\cdots\)
1600.2.n.o	$1600$	$2$	1600.n	20.e	$4$	$4$	$2$	$12.776$	\(\Q(i, \sqrt{6})\)	None					800.2.n.k	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{1}-\beta _{2})q^{3}+(-2+2\beta _{2}+\cdots)q^{7}+\cdots\)
1600.2.n.p	$1600$	$2$	1600.n	20.e	$4$	$4$	$2$	$12.776$	\(\Q(i, \sqrt{6})\)	None					800.2.n.k	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{1}-\beta _{2})q^{3}+(-2+2\beta _{2}+\cdots)q^{7}+\cdots\)
1600.2.n.q	$1600$	$2$	1600.n	20.e	$4$	$4$	$2$	$12.776$	\(\Q(i, \sqrt{5})\)	\(\Q(\sqrt{-5}) \)					400.2.n.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{4}]$	\(q-\beta _{3}q^{3}-\beta _{2}q^{7}-7\beta _{1}q^{9}-10q^{21}+\cdots\)
1600.2.n.r	$1600$	$2$	1600.n	20.e	$4$	$4$	$2$	$12.776$	\(\Q(\zeta_{12})\)	None					80.2.n.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta_{2} q^{3}-\beta_{3} q^{7}+3\beta_1 q^{9}+(\beta_{3}-\beta_{2})q^{11}+\cdots\)
1600.2.n.s	$1600$	$2$	1600.n	20.e	$4$	$4$	$2$	$12.776$	\(\Q(i, \sqrt{6})\)	None					800.2.n.k	$2$	$0$	\(0\)	\(4\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{1}+\beta _{2})q^{3}+(2-2\beta _{2})q^{7}+(2\beta _{1}+\cdots)q^{9}+\cdots\)
1600.2.n.t	$1600$	$2$	1600.n	20.e	$4$	$4$	$2$	$12.776$	\(\Q(i, \sqrt{6})\)	None					800.2.n.k	$2$	$0$	\(0\)	\(4\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{1}+\beta _{2})q^{3}+(2-2\beta _{2})q^{7}+(2\beta _{1}+\cdots)q^{9}+\cdots\)
1600.2.n.u	$1600$	$2$	1600.n	20.e	$4$	$8$	$4$	$12.776$	\(\Q(\zeta_{24})\)	None					400.2.n.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta_{5} q^{3}-4\beta_1 q^{7}+2\beta_{3} q^{9}+3\beta_{4} q^{11}+\cdots\)
1600.2.n.v	$1600$	$2$	1600.n	20.e	$4$	$8$	$4$	$12.776$	8.0.3317760000.5	None					100.2.e.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{3}+2\beta _{3}q^{9}+\beta _{7}q^{11}+2\beta _{4}q^{13}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1600, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 2}\)