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

• Label: 1600.2.s
• Level: $1600$
• Weight: $2$
• Character orbit: 1600.s
• Rep. character: $\chi_{1600}(207,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $68$
• Newform subspaces: $5$
• Sturm bound: $480$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.s (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 80 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(480\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	528	76	452
Cusp forms	432	68	364
Eisenstein series	96	8	88

Trace form

68 * q - 4 * q^3 - 4 * q^7 + 60 * q^9 + 4 * q^11 + 4 * q^17 - 16 * q^19 - 4 * q^21 - 4 * q^23 - 16 * q^27 + 4 * q^33 + 24 * q^47 + 20 * q^51 + 4 * q^53 + 12 * q^57 + 32 * q^59 - 36 * q^61 - 12 * q^63 + 20 * q^69 + 72 * q^71 + 8 * q^73 + 32 * q^77 + 28 * q^81 + 36 * q^83 + 52 * q^87 + 36 * q^91 + 4 * q^97 + 76 * q^99

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.s.a	$1600$	$2$	1600.s	80.s	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	None					80.2.j.a	$2$	$1$	\(0\)	\(-4\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-2 q^{3}+(-3 i-3)q^{7}+q^{9}+(-i+1)q^{11}+\cdots\)
1600.2.s.b	$1600$	$2$	1600.s	80.s	$4$	$8$	$4$	$12.776$	\(\Q(\zeta_{24})\)	None					400.2.j.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta_{7}+\beta_{5}+\beta_1)q^{3}+(-\beta_{7}+2\beta_{5}+\cdots-\beta_1)q^{7}+\cdots\)
1600.2.s.c	$1600$	$2$	1600.s	80.s	$4$	$16$	$8$	$12.776$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					400.2.j.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{4}q^{3}-\beta _{7}q^{7}+(1-\beta _{12})q^{9}+(-1+\cdots)q^{11}+\cdots\)
1600.2.s.d	$1600$	$2$	1600.s	80.s	$4$	$18$	$9$	$12.776$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None					80.2.j.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$2^{15}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{3}+\beta _{11}q^{7}+(1+\beta _{3})q^{9}+(-\beta _{13}+\cdots)q^{11}+\cdots\)
1600.2.s.e	$1600$	$2$	1600.s	80.s	$4$	$24$	$12$	$12.776$		None			✓		400.2.j.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

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

\( S_{2}^{\mathrm{old}}(1600, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 3}\)