Space of modular forms of level 2100, weight 2, and Character 2100.f

• Label: 2100.2.f
• Level: $2100$
• Weight: $2$
• Character orbit: 2100.f
• Rep. character: $\chi_{2100}(1049,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $9$
• Sturm bound: $960$
• Trace bound: $21$

Defining parameters

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 105 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(960\)
Trace bound:			\(21\)
Distinguishing \(T_p\):			\(11\), \(13\), \(23\), \(41\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(2100, [\chi])\).

	Total	New	Old
Modular forms	516	48	468
Cusp forms	444	48	396
Eisenstein series	72	0	72

Trace form

\( 48 q - 12 q^{9} + 2 q^{21} - 34 q^{49} + 28 q^{51} + 88 q^{79} + 20 q^{81} + 10 q^{91} + 60 q^{99}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(2100, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
2100.2.f.a	$2100$	$2$	2100.f	105.g	$2$	$4$	$4$	$16.769$	\(\Q(i, \sqrt{5})\)	None					420.2.d.c	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(-6\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-1+\beta _{2}-\beta _{3})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
2100.2.f.b	$2100$	$2$	2100.f	105.g	$2$	$4$	$4$	$16.769$	\(\Q(i, \sqrt{5})\)	None					420.2.d.c	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(-6\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-2-\beta _{1}-\beta _{2})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
2100.2.f.c	$2100$	$2$	2100.f	105.g	$2$	$4$	$4$	$16.769$	\(\Q(\zeta_{12})\)	None					420.2.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\zeta_{12}+2\zeta_{12}^{3})q^{3}+(-2\zeta_{12}+3\zeta_{12}^{3})q^{7}+\cdots\)
2100.2.f.d	$2100$	$2$	2100.f	105.g	$2$	$4$	$4$	$16.769$	\(\Q(\zeta_{12})\)	None					420.2.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\zeta_{12}+2\zeta_{12}^{3})q^{3}+(-2\zeta_{12}-\zeta_{12}^{3})q^{7}+\cdots\)
2100.2.f.e	$2100$	$2$	2100.f	105.g	$2$	$4$	$4$	$16.769$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)					84.2.f.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta_1 q^{3}+(-\beta_{3}-\beta_1)q^{7}+3 q^{9}+\cdots\)
2100.2.f.f	$2100$	$2$	2100.f	105.g	$2$	$4$	$4$	$16.769$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		✓			2100.2.d.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(2\zeta_{12}-\zeta_{12}^{3})q^{3}+(3\zeta_{12}-\zeta_{12}^{3})q^{7}+\cdots\)
2100.2.f.g	$2100$	$2$	2100.f	105.g	$2$	$4$	$4$	$16.769$	\(\Q(i, \sqrt{5})\)	None					420.2.d.c	$2$	$0$	\(0\)	\(2\)	\(0\)	\(6\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(1-\beta _{2}+\beta _{3})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
2100.2.f.h	$2100$	$2$	2100.f	105.g	$2$	$4$	$4$	$16.769$	\(\Q(i, \sqrt{5})\)	None					420.2.d.c	$2$	$0$	\(0\)	\(2\)	\(0\)	\(6\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(2+\beta _{1}+\beta _{2})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
2100.2.f.i	$2100$	$2$	2100.f	105.g	$2$	$16$	$16$	$16.769$	16.0.\(\cdots\).1	None		✓	✓		2100.2.d.k	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+(\beta _{1}-\beta _{8})q^{7}+(-1-\beta _{7}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(2100, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(2100, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1050, [\chi])\)\(^{\oplus 2}\)