Space of modular forms of level 2600, weight 2, and Character 2600.f

• Label: 2600.2.f
• Level: $2600$
• Weight: $2$
• Character orbit: 2600.f
• Rep. character: $\chi_{2600}(649,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $8$
• Sturm bound: $840$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2600.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 65 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(840\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(2600, [\chi])\).

	Total	New	Old
Modular forms	444	64	380
Cusp forms	396	64	332
Eisenstein series	48	0	48

Trace form

\( 64 q - 68 q^{9} + 4 q^{29} + 40 q^{39} + 48 q^{49} + 84 q^{51} + 52 q^{61} + 32 q^{69} + 8 q^{79} + 48 q^{81} - 52 q^{91}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(2600, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
2600.2.f.a	$2600$	$2$	2600.f	65.d	$2$	$4$	$4$	$20.761$	\(\Q(i, \sqrt{17})\)	None					104.2.f.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-10\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-3+\beta _{3})q^{7}+(-2+\beta _{3})q^{9}+\cdots\)
2600.2.f.b	$2600$	$2$	2600.f	65.d	$2$	$4$	$4$	$20.761$	\(\Q(i, \sqrt{17})\)	None					104.2.f.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(10\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(3-\beta _{3})q^{7}+(-2+\beta _{3})q^{9}+\cdots\)
2600.2.f.c	$2600$	$2$	2600.f	65.d	$2$	$6$	$6$	$20.761$	6.0.350464.1	None					520.2.k.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+(-1+\beta _{1}+\beta _{3})q^{7}+(\beta _{1}+\cdots)q^{9}+\cdots\)
2600.2.f.d	$2600$	$2$	2600.f	65.d	$2$	$6$	$6$	$20.761$	6.0.350464.1	None					520.2.k.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+(1-\beta _{1}-\beta _{3})q^{7}+(\beta _{1}+\beta _{3}+\cdots)q^{9}+\cdots\)
2600.2.f.e	$2600$	$2$	2600.f	65.d	$2$	$8$	$8$	$20.761$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					520.2.k.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{3}+(-1+\beta _{1}+\beta _{2})q^{7}+(-2+\cdots)q^{9}+\cdots\)
2600.2.f.f	$2600$	$2$	2600.f	65.d	$2$	$8$	$8$	$20.761$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					520.2.k.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{3}+(1-\beta _{1}-\beta _{2})q^{7}+(-2-\beta _{7})q^{9}+\cdots\)
2600.2.f.g	$2600$	$2$	2600.f	65.d	$2$	$14$	$14$	$20.761$	\(\mathbb{Q}[x]/(x^{14} + \cdots)\)	None		✓			2600.2.k.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+\beta _{5}q^{7}+(-1+\beta _{2})q^{9}+(-\beta _{8}+\cdots)q^{11}+\cdots\)
2600.2.f.h	$2600$	$2$	2600.f	65.d	$2$	$14$	$14$	$20.761$	\(\mathbb{Q}[x]/(x^{14} + \cdots)\)	None		✓			2600.2.k.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-\beta _{5}q^{7}+(-1+\beta _{2})q^{9}+(\beta _{8}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(2600, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(2600, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(325, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(520, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(650, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1300, [\chi])\)\(^{\oplus 2}\)