Space of modular forms of level 1408, weight 2, and Character 1408.g

• Label: 1408.2.g
• Level: $1408$
• Weight: $2$
• Character orbit: 1408.g
• Rep. character: $\chi_{1408}(703,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $7$
• Sturm bound: $384$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 1408 = 2^{7} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1408.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 88 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(384\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1408, [\chi])\).

	Total	New	Old
Modular forms	208	48	160
Cusp forms	176	48	128
Eisenstein series	32	0	32

Trace form

\( 48 q + 48 q^{9} - 48 q^{25} + 16 q^{33} + 48 q^{49} + 16 q^{81} + 64 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1408, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1408.2.g.a	$1408$	$2$	1408.g	88.g	$2$	$4$	$4$	$11.243$	\(\Q(\sqrt{-2}, \sqrt{-7})\)	None		✓			1408.2.g.a	$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}+\beta _{3}q^{5}+\beta _{1}q^{7}-2q^{9}+(3-\beta _{2}+\cdots)q^{11}+\cdots\)
1408.2.g.b	$1408$	$2$	1408.g	88.g	$2$	$4$	$4$	$11.243$	\(\Q(\sqrt{2}, \sqrt{-11})\)	\(\Q(\sqrt{-22}) \)		✓			1408.2.g.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q-3q^{9}+\beta _{1}q^{11}-\beta _{2}q^{13}-2\beta _{1}q^{19}+\cdots\)
1408.2.g.c	$1408$	$2$	1408.g	88.g	$2$	$4$	$4$	$11.243$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)		✓			1408.2.g.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q+(-\beta_{2}-\beta_1)q^{3}+5 q^{9}+(-2\beta_{2}+\beta_1)q^{11}+\cdots\)
1408.2.g.d	$1408$	$2$	1408.g	88.g	$2$	$4$	$4$	$11.243$	\(\Q(\sqrt{-2}, \sqrt{-7})\)	None		✓			1408.2.g.a	$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}-\beta _{3}q^{5}+\beta _{1}q^{7}-2q^{9}+(-3+\cdots)q^{11}+\cdots\)
1408.2.g.e	$1408$	$2$	1408.g	88.g	$2$	$8$	$8$	$11.243$	\(\Q(\sqrt{2}, \sqrt{-3}, \sqrt{5})\)	None		✓			1408.2.g.e	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{5}q^{5}-\beta _{7}q^{7}+2q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots\)
1408.2.g.f	$1408$	$2$	1408.g	88.g	$2$	$12$	$12$	$11.243$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			1408.2.g.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+\beta _{10}q^{5}+(-1+\beta _{3})q^{7}+\cdots\)
1408.2.g.g	$1408$	$2$	1408.g	88.g	$2$	$12$	$12$	$11.243$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			1408.2.g.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-\beta _{10}q^{5}+(1-\beta _{3})q^{7}+(2+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1408, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1408, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(88, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(352, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(704, [\chi])\)\(^{\oplus 2}\)