Space of modular forms of level 414, weight 2, and Character 414.e

• Label: 414.2.e
• Level: $414$
• Weight: $2$
• Character orbit: 414.e
• Rep. character: $\chi_{414}(139,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $44$
• Newform subspaces: $5$
• Sturm bound: $144$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	414.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(144\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(414, [\chi])\).

	Total	New	Old
Modular forms	152	44	108
Cusp forms	136	44	92
Eisenstein series	16	0	16

Trace form

44 * q + 2 * q^2 + 2 * q^3 - 22 * q^4 + 2 * q^6 + 4 * q^7 - 4 * q^8 - 14 * q^9 - 10 * q^11 - 4 * q^12 + 4 * q^13 + 20 * q^15 - 22 * q^16 + 28 * q^17 - 4 * q^18 + 4 * q^19 - 16 * q^21 - 6 * q^22 + 2 * q^24 - 22 * q^25 - 24 * q^26 + 20 * q^27 - 8 * q^28 + 28 * q^29 - 12 * q^30 + 4 * q^31 + 2 * q^32 + 30 * q^33 - 6 * q^34 + 10 * q^36 - 8 * q^37 - 14 * q^38 - 48 * q^39 + 14 * q^41 - 20 * q^42 - 2 * q^43 + 20 * q^44 - 16 * q^45 - 20 * q^47 + 2 * q^48 - 18 * q^49 - 2 * q^50 + 22 * q^51 + 4 * q^52 - 80 * q^53 - 10 * q^54 - 24 * q^55 + 10 * q^57 + 12 * q^58 - 2 * q^59 - 16 * q^60 + 28 * q^61 - 8 * q^62 + 48 * q^63 + 44 * q^64 + 20 * q^65 + 22 * q^67 - 14 * q^68 + 12 * q^70 + 72 * q^71 + 2 * q^72 - 44 * q^73 - 2 * q^75 - 2 * q^76 - 16 * q^77 - 4 * q^78 - 8 * q^79 - 14 * q^81 - 36 * q^82 - 12 * q^83 - 4 * q^84 - 30 * q^86 + 52 * q^87 - 6 * q^88 + 72 * q^89 + 28 * q^90 - 16 * q^91 + 36 * q^93 - 12 * q^95 - 4 * q^96 + 22 * q^97 - 4 * q^98 + 12 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(414, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
414.2.e.a	$414$	$2$	414.e	9.c	$3$	$2$	$1$	$3.306$	\(\Q(\sqrt{-3}) \)	None		✓			414.2.e.a	$2$	$0$	\(1\)	\(3\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
414.2.e.b	$414$	$2$	414.e	9.c	$3$	$10$	$5$	$3.306$	10.0.\(\cdots\).1	None		✓			414.2.e.b	$2$	$0$	\(-5\)	\(1\)	\(-5\)	\(-3\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{3}q^{2}-\beta _{7}q^{3}+(-1+\beta _{3})q^{4}+(-1+\cdots)q^{5}+\cdots\)
414.2.e.c	$414$	$2$	414.e	9.c	$3$	$10$	$5$	$3.306$	10.0.\(\cdots\).1	None		✓			414.2.e.c	$2$	$0$	\(-5\)	\(1\)	\(5\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{4})q^{2}+(1+\beta _{5}-\beta _{6}-\beta _{9})q^{3}+\cdots\)
414.2.e.d	$414$	$2$	414.e	9.c	$3$	$10$	$5$	$3.306$	10.0.\(\cdots\).1	None		✓			414.2.e.d	$2$	$0$	\(5\)	\(-3\)	\(-1\)	\(5\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{7})q^{2}+\beta _{1}q^{3}+\beta _{7}q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
414.2.e.e	$414$	$2$	414.e	9.c	$3$	$12$	$6$	$3.306$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓	✓		414.2.e.e	$2$	$0$	\(6\)	\(0\)	\(5\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(-1+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(414, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(414, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(207, [\chi])\)\(^{\oplus 2}\)