Space of modular forms of level 338, weight 3, and Character 338.f

• Label: 338.3.f
• Level: $338$
• Weight: $3$
• Character orbit: 338.f
• Rep. character: $\chi_{338}(19,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $100$
• Newform subspaces: $12$
• Sturm bound: $136$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	338.f (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(136\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(338, [\chi])\).

	Total	New	Old
Modular forms	420	100	320
Cusp forms	308	100	208
Eisenstein series	112	0	112

Trace form

100 * q + 2 * q^2 - 6 * q^5 + 24 * q^7 + 8 * q^8 - 138 * q^9 + 30 * q^10 + 24 * q^11 - 32 * q^14 - 72 * q^15 + 200 * q^16 - 36 * q^17 - 108 * q^18 + 12 * q^19 - 12 * q^20 + 60 * q^21 + 48 * q^22 + 24 * q^23 - 120 * q^27 + 48 * q^28 + 68 * q^29 + 192 * q^30 + 96 * q^31 - 8 * q^32 - 12 * q^33 + 84 * q^34 + 52 * q^35 + 48 * q^36 + 66 * q^37 + 24 * q^40 - 132 * q^41 - 240 * q^42 - 12 * q^43 - 120 * q^44 - 276 * q^45 - 144 * q^46 - 192 * q^47 - 300 * q^49 - 232 * q^50 + 460 * q^53 + 360 * q^54 + 236 * q^55 + 96 * q^56 + 396 * q^57 + 66 * q^58 - 24 * q^59 + 96 * q^60 - 202 * q^61 + 72 * q^62 + 192 * q^63 - 160 * q^66 - 12 * q^67 - 68 * q^68 - 132 * q^69 - 240 * q^70 + 132 * q^71 - 156 * q^72 - 30 * q^73 + 166 * q^74 - 120 * q^75 - 24 * q^76 - 16 * q^79 - 24 * q^80 - 442 * q^81 + 330 * q^82 + 48 * q^83 + 168 * q^84 + 66 * q^85 + 24 * q^86 - 440 * q^87 - 234 * q^89 + 16 * q^92 - 492 * q^93 - 64 * q^94 - 672 * q^95 - 222 * q^97 + 158 * q^98 - 144 * q^99

Decomposition of \(S_{3}^{\mathrm{new}}(338, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
338.3.f.a	$338$	$3$	338.f	13.f	$12$	$4$	$1$	$9.210$	\(\Q(\zeta_{12})\)	None		✓			338.3.d.b	$4$	$0$	\(-2\)	\(-8\)	\(16\)	\(-10\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}-4\zeta_{12}^{2}q^{3}+\cdots\)
338.3.f.b	$338$	$3$	338.f	13.f	$12$	$4$	$1$	$9.210$	\(\Q(\zeta_{12})\)	None					26.3.d.a	$4$	$0$	\(-2\)	\(0\)	\(-12\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots\)
338.3.f.c	$338$	$3$	338.f	13.f	$12$	$4$	$1$	$9.210$	\(\Q(\zeta_{12})\)	None					26.3.f.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(-20\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(\zeta_{12}+\cdots)q^{3}+\cdots\)
338.3.f.d	$338$	$3$	338.f	13.f	$12$	$4$	$1$	$9.210$	\(\Q(\zeta_{12})\)	None					26.3.f.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(22\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(-\zeta_{12}+\cdots)q^{3}+\cdots\)
338.3.f.e	$338$	$3$	338.f	13.f	$12$	$4$	$1$	$9.210$	\(\Q(\zeta_{12})\)	None		✓			338.3.d.b	$4$	$0$	\(2\)	\(-8\)	\(-16\)	\(10\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}-4\zeta_{12}^{2}q^{3}+\cdots\)
338.3.f.f	$338$	$3$	338.f	13.f	$12$	$4$	$1$	$9.210$	\(\Q(\zeta_{12})\)	None					26.3.f.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(20\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(\zeta_{12}+\zeta_{12}^{3})q^{3}+\cdots\)
338.3.f.g	$338$	$3$	338.f	13.f	$12$	$4$	$1$	$9.210$	\(\Q(\zeta_{12})\)	None					26.3.d.a	$4$	$0$	\(2\)	\(0\)	\(12\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots\)
338.3.f.h	$338$	$3$	338.f	13.f	$12$	$8$	$2$	$9.210$	8.0.\(\cdots\).1	None					26.3.f.b	$2$	$0$	\(-4\)	\(0\)	\(6\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-\beta _{3}+\beta _{4})q^{2}+(\beta _{2}-\beta _{5})q^{3}-2\beta _{5}q^{4}+\cdots\)
338.3.f.i	$338$	$3$	338.f	13.f	$12$	$8$	$2$	$9.210$	8.0.\(\cdots\).1	None					26.3.f.b	$2$	$0$	\(4\)	\(0\)	\(-6\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\beta _{3}+\beta _{4})q^{2}+(\beta _{2}-\beta _{5})q^{3}+2\beta _{5}q^{4}+\cdots\)
338.3.f.j	$338$	$3$	338.f	13.f	$12$	$8$	$2$	$9.210$	8.0.\(\cdots\).1	None					26.3.f.b	$2$	$0$	\(4\)	\(0\)	\(-6\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\beta _{3}-\beta _{4})q^{2}+(\beta _{2}-\beta _{5})q^{3}-2\beta _{5}q^{4}+\cdots\)
338.3.f.k	$338$	$3$	338.f	13.f	$12$	$24$	$6$	$9.210$		None		✓			338.3.d.h	$4$	$0$	\(-12\)	\(8\)	\(-16\)	\(24\)		$\mathrm{SU}(2)[C_{12}]$
338.3.f.l	$338$	$3$	338.f	13.f	$12$	$24$	$6$	$9.210$		None		✓			338.3.d.h	$4$	$0$	\(12\)	\(8\)	\(16\)	\(-24\)		$\mathrm{SU}(2)[C_{12}]$

Decomposition of \(S_{3}^{\mathrm{old}}(338, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(338, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 2}\)