Space of modular forms of level 406, weight 2, and Character 406.e

• Label: 406.2.e
• Level: $406$
• Weight: $2$
• Character orbit: 406.e
• Rep. character: $\chi_{406}(233,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $40$
• Newform subspaces: $4$
• Sturm bound: $120$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 406 = 2 \cdot 7 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	406.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(120\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(406, [\chi])\).

	Total	New	Old
Modular forms	128	40	88
Cusp forms	112	40	72
Eisenstein series	16	0	16

Trace form

40 * q - 20 * q^4 - 4 * q^5 - 4 * q^7 - 24 * q^9 + 4 * q^10 - 8 * q^11 + 8 * q^13 + 4 * q^14 - 16 * q^15 - 20 * q^16 + 16 * q^17 + 8 * q^20 + 16 * q^21 - 16 * q^22 + 4 * q^23 - 24 * q^25 + 24 * q^27 - 4 * q^28 + 4 * q^30 + 12 * q^31 - 28 * q^33 - 4 * q^35 + 48 * q^36 + 20 * q^37 - 4 * q^39 + 4 * q^40 - 24 * q^41 + 16 * q^43 - 8 * q^44 + 4 * q^45 + 4 * q^46 - 4 * q^47 - 20 * q^49 + 16 * q^50 - 20 * q^51 - 4 * q^52 - 20 * q^53 - 40 * q^55 - 8 * q^56 - 48 * q^57 + 8 * q^59 + 8 * q^60 + 16 * q^61 + 48 * q^62 - 4 * q^63 + 40 * q^64 - 20 * q^65 + 16 * q^66 + 16 * q^68 + 72 * q^69 + 8 * q^70 - 64 * q^71 + 4 * q^73 - 4 * q^74 + 44 * q^75 - 32 * q^77 - 16 * q^78 - 16 * q^79 - 4 * q^80 - 12 * q^81 - 24 * q^82 + 88 * q^83 - 32 * q^84 + 16 * q^85 + 12 * q^86 + 8 * q^88 + 28 * q^89 - 32 * q^90 - 12 * q^91 - 8 * q^92 - 28 * q^93 - 36 * q^94 + 28 * q^95 + 72 * q^97 - 24 * q^98 + 80 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(406, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
406.2.e.a	$406$	$2$	406.e	7.c	$3$	$10$	$5$	$3.242$	10.0.\(\cdots\).1	None		✓			406.2.e.a	$2$	$0$	\(-5\)	\(-3\)	\(-7\)	\(-3\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{4})q^{2}+(-\beta _{3}-\beta _{4})q^{3}-\beta _{4}q^{4}+\cdots\)
406.2.e.b	$406$	$2$	406.e	7.c	$3$	$10$	$5$	$3.242$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		✓			406.2.e.b	$2$	$0$	\(-5\)	\(3\)	\(7\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{6})q^{2}+(-\beta _{1}-\beta _{6})q^{3}+\beta _{6}q^{4}+\cdots\)
406.2.e.c	$406$	$2$	406.e	7.c	$3$	$10$	$5$	$3.242$	10.0.\(\cdots\).1	None		✓			406.2.e.c	$2$	$0$	\(5\)	\(-3\)	\(-5\)	\(1\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{6}q^{2}+(-\beta _{1}-\beta _{7})q^{3}+(-1+\beta _{6}+\cdots)q^{4}+\cdots\)
406.2.e.d	$406$	$2$	406.e	7.c	$3$	$10$	$5$	$3.242$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		✓			406.2.e.d	$2$	$0$	\(5\)	\(3\)	\(1\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{6}q^{2}+(1-\beta _{5}+\beta _{6})q^{3}+(-1-\beta _{6}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(406, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(406, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(203, [\chi])\)\(^{\oplus 2}\)