Space of modular forms of level 238, weight 2, and Character 238.e

• Label: 238.2.e
• Level: $238$
• Weight: $2$
• Character orbit: 238.e
• Rep. character: $\chi_{238}(137,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $24$
• Newform subspaces: $6$
• Sturm bound: $72$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 238 = 2 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	238.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(72\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(238, [\chi])\).

	Total	New	Old
Modular forms	80	24	56
Cusp forms	64	24	40
Eisenstein series	16	0	16

Trace form

24 * q - 12 * q^4 + 4 * q^5 - 4 * q^7 - 20 * q^9 - 4 * q^11 + 8 * q^13 - 4 * q^14 - 8 * q^15 - 12 * q^16 + 4 * q^17 + 4 * q^18 - 4 * q^19 - 8 * q^20 + 8 * q^21 + 8 * q^22 + 16 * q^23 - 16 * q^25 - 8 * q^26 + 8 * q^28 - 8 * q^29 + 4 * q^30 + 8 * q^31 - 28 * q^33 - 8 * q^34 + 4 * q^35 + 40 * q^36 - 4 * q^38 - 4 * q^39 + 8 * q^41 + 24 * q^42 - 32 * q^43 - 4 * q^44 + 32 * q^45 - 16 * q^46 + 20 * q^47 + 12 * q^49 + 24 * q^50 - 4 * q^52 - 20 * q^53 + 12 * q^54 + 40 * q^55 - 4 * q^56 + 24 * q^57 - 16 * q^58 - 8 * q^59 + 4 * q^60 - 24 * q^61 - 8 * q^62 + 36 * q^63 + 24 * q^64 - 12 * q^65 - 24 * q^66 - 4 * q^67 + 4 * q^68 - 80 * q^69 - 36 * q^70 + 56 * q^71 + 4 * q^72 + 36 * q^73 - 12 * q^74 - 52 * q^75 + 8 * q^76 + 36 * q^77 + 16 * q^78 - 4 * q^79 + 4 * q^80 - 72 * q^81 - 48 * q^83 - 16 * q^84 - 24 * q^86 + 24 * q^87 - 4 * q^88 - 28 * q^89 - 72 * q^90 - 24 * q^91 - 32 * q^92 + 28 * q^93 + 4 * q^94 + 24 * q^95 + 56 * q^97 + 24 * q^98 + 16 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(238, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
238.2.e.a	$238$	$2$	238.e	7.c	$3$	$2$	$1$	$1.900$	\(\Q(\sqrt{-3}) \)	None		✓			238.2.e.a	$2$	$0$	\(-1\)	\(0\)	\(1\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots\)
238.2.e.b	$238$	$2$	238.e	7.c	$3$	$2$	$1$	$1.900$	\(\Q(\sqrt{-3}) \)	None		✓			238.2.e.b	$2$	$0$	\(1\)	\(-3\)	\(2\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{2}+(-3+3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
238.2.e.c	$238$	$2$	238.e	7.c	$3$	$2$	$1$	$1.900$	\(\Q(\sqrt{-3}) \)	None		✓			238.2.e.c	$2$	$0$	\(1\)	\(1\)	\(2\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
238.2.e.d	$238$	$2$	238.e	7.c	$3$	$2$	$1$	$1.900$	\(\Q(\sqrt{-3}) \)	None		✓			238.2.e.d	$2$	$0$	\(1\)	\(2\)	\(-3\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{2}+(2-2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
238.2.e.e	$238$	$2$	238.e	7.c	$3$	$6$	$3$	$1.900$	6.0.4406832.1	None		✓			238.2.e.e	$2$	$0$	\(3\)	\(0\)	\(1\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{4}q^{2}+(-\beta _{1}-\beta _{3}+\beta _{5})q^{3}+(-1+\cdots)q^{4}+\cdots\)
238.2.e.f	$238$	$2$	238.e	7.c	$3$	$10$	$5$	$1.900$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		✓	✓		238.2.e.f	$2$	$0$	\(-5\)	\(0\)	\(1\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{4})q^{2}-\beta _{1}q^{3}+\beta _{4}q^{4}+\beta _{9}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(238, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(238, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(119, [\chi])\)\(^{\oplus 2}\)