Space of modular forms of level 410, weight 2, and Character 410.g

• Label: 410.2.g
• Level: $410$
• Weight: $2$
• Character orbit: 410.g
• Rep. character: $\chi_{410}(9,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $44$
• Newform subspaces: $6$
• Sturm bound: $126$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 410 = 2 \cdot 5 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	410.g (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 205 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(126\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(410, [\chi])\).

	Total	New	Old
Modular forms	132	44	88
Cusp forms	116	44	72
Eisenstein series	16	0	16

Trace form

44 * q + 44 * q^4 - 4 * q^6 - 4 * q^11 + 20 * q^15 + 44 * q^16 - 20 * q^19 - 4 * q^24 - 12 * q^25 - 8 * q^26 + 24 * q^29 + 8 * q^30 - 16 * q^31 - 20 * q^34 - 12 * q^35 + 8 * q^41 - 4 * q^44 + 4 * q^45 - 16 * q^51 - 16 * q^54 - 20 * q^55 - 8 * q^59 + 20 * q^60 + 44 * q^64 - 28 * q^65 + 48 * q^66 - 32 * q^69 - 36 * q^70 + 16 * q^71 + 16 * q^75 - 20 * q^76 - 64 * q^79 - 100 * q^81 + 28 * q^85 - 88 * q^86 - 44 * q^89 + 8 * q^94 - 4 * q^95 - 4 * q^96 - 100 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(410, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
410.2.g.a	$410$	$2$	410.g	205.j	$4$	$2$	$1$	$3.274$	\(\Q(\sqrt{-1}) \)	None		✓			410.2.g.a	$2$	$1$	\(-2\)	\(-2\)	\(-2\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-q^{2}+(-i-1)q^{3}+q^{4}+(2 i-1)q^{5}+\cdots\)
410.2.g.b	$410$	$2$	410.g	205.j	$4$	$2$	$1$	$3.274$	\(\Q(\sqrt{-1}) \)	None		✓			410.2.g.b	$2$	$0$	\(-2\)	\(4\)	\(4\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-q^{2}+(2 i+2)q^{3}+q^{4}+(i+2)q^{5}+\cdots\)
410.2.g.c	$410$	$2$	410.g	205.j	$4$	$2$	$1$	$3.274$	\(\Q(\sqrt{-1}) \)	None		✓			410.2.g.b	$2$	$1$	\(2\)	\(-4\)	\(-4\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+q^{2}+(-2 i-2)q^{3}+q^{4}+(i-2)q^{5}+\cdots\)
410.2.g.d	$410$	$2$	410.g	205.j	$4$	$2$	$1$	$3.274$	\(\Q(\sqrt{-1}) \)	None		✓			410.2.g.a	$2$	$0$	\(2\)	\(2\)	\(2\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+q^{2}+(i+1)q^{3}+q^{4}+(2 i+1)q^{5}+\cdots\)
410.2.g.e	$410$	$2$	410.g	205.j	$4$	$18$	$9$	$3.274$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		✓			410.2.g.e	$2$	$0$	\(-18\)	\(0\)	\(-2\)	\(-6\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q-q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{14}q^{5}-\beta _{1}q^{6}+\cdots\)
410.2.g.f	$410$	$2$	410.g	205.j	$4$	$18$	$9$	$3.274$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		✓			410.2.g.e	$2$	$0$	\(18\)	\(0\)	\(2\)	\(6\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q+q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{9}q^{5}-\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(410, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(410, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(205, [\chi])\)\(^{\oplus 2}\)