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

• Label: 338.3.d
• Level: $338$
• Weight: $3$
• Character orbit: 338.d
• Rep. character: $\chi_{338}(99,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $54$
• Newform subspaces: $9$
• Sturm bound: $136$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	338.d (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(136\)
Trace bound:			\(3\)
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	210	54	156
Cusp forms	154	54	100
Eisenstein series	56	0	56

Trace form

54 * q - 2 * q^2 + 6 * q^5 - 4 * q^7 + 4 * q^8 + 186 * q^9 - 12 * q^11 - 16 * q^14 - 216 * q^16 + 18 * q^18 - 52 * q^19 - 12 * q^20 - 48 * q^22 - 24 * q^27 - 8 * q^28 + 88 * q^29 + 28 * q^31 + 8 * q^32 + 12 * q^34 + 8 * q^35 - 74 * q^37 - 24 * q^40 + 18 * q^41 + 96 * q^42 - 24 * q^44 - 54 * q^45 - 48 * q^46 - 84 * q^47 - 14 * q^50 + 20 * q^53 - 8 * q^55 + 96 * q^58 + 108 * q^59 + 124 * q^61 + 36 * q^63 - 32 * q^66 + 44 * q^67 + 56 * q^68 + 24 * q^70 - 12 * q^71 - 36 * q^72 - 34 * q^73 - 244 * q^74 + 104 * q^76 + 256 * q^79 - 24 * q^80 + 310 * q^81 - 156 * q^83 - 36 * q^85 + 72 * q^86 + 176 * q^87 + 18 * q^89 - 112 * q^92 - 176 * q^94 + 94 * q^97 + 82 * q^98 + 108 * 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.d.a	$338$	$3$	338.d	13.d	$4$	$2$	$1$	$9.210$	\(\Q(\sqrt{-1}) \)	None					26.3.d.a	$2$	$0$	\(-2\)	\(0\)	\(6\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i-1)q^{2}+2 i q^{4}+(3 i+3)q^{5}+\cdots\)
338.3.d.b	$338$	$3$	338.d	13.d	$4$	$2$	$1$	$9.210$	\(\Q(\sqrt{-1}) \)	None		✓			338.3.d.b	$2$	$0$	\(-2\)	\(8\)	\(-8\)	\(-10\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i-1)q^{2}+4 q^{3}+2 i q^{4}+(-4 i-4)q^{5}+\cdots\)
338.3.d.c	$338$	$3$	338.d	13.d	$4$	$2$	$1$	$9.210$	\(\Q(\sqrt{-1}) \)	None		✓			338.3.d.b	$2$	$0$	\(2\)	\(8\)	\(8\)	\(10\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i+1)q^{2}+4 q^{3}+2 i q^{4}+(4 i+4)q^{5}+\cdots\)
338.3.d.d	$338$	$3$	338.d	13.d	$4$	$4$	$2$	$9.210$	\(\Q(\zeta_{12})\)	None					26.3.f.a	$2$	$0$	\(-4\)	\(0\)	\(0\)	\(2\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta_{2}-1)q^{2}+(\beta_{3}-\beta_{2}+\beta_1)q^{3}+\cdots\)
338.3.d.e	$338$	$3$	338.d	13.d	$4$	$4$	$2$	$9.210$	\(\Q(\zeta_{12})\)	None					26.3.f.a	$2$	$0$	\(4\)	\(0\)	\(0\)	\(-2\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta_{2}+1)q^{2}+(\beta_{3}-\beta_{2}+\beta_1)q^{3}+\cdots\)
338.3.d.f	$338$	$3$	338.d	13.d	$4$	$8$	$4$	$9.210$	8.0.\(\cdots\).1	None					26.3.f.b	$2$	$0$	\(-8\)	\(0\)	\(-6\)	\(-10\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{5})q^{2}+(-1+\beta _{1}+\beta _{3}+\beta _{5}+\cdots)q^{3}+\cdots\)
338.3.d.g	$338$	$3$	338.d	13.d	$4$	$8$	$4$	$9.210$	8.0.\(\cdots\).1	None					26.3.f.b	$2$	$0$	\(8\)	\(0\)	\(6\)	\(10\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\beta _{5})q^{2}+(-1+\beta _{1}+\beta _{3}+\beta _{5}+\cdots)q^{3}+\cdots\)
338.3.d.h	$338$	$3$	338.d	13.d	$4$	$12$	$6$	$9.210$	12.0.\(\cdots\).2	None		✓			338.3.d.h	$2$	$0$	\(-12\)	\(-8\)	\(8\)	\(24\)	$13^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{1})q^{2}+(-1+\beta _{4}+\beta _{7}-\beta _{9}+\cdots)q^{3}+\cdots\)
338.3.d.i	$338$	$3$	338.d	13.d	$4$	$12$	$6$	$9.210$	12.0.\(\cdots\).2	None		✓			338.3.d.h	$2$	$0$	\(12\)	\(-8\)	\(-8\)	\(-24\)	$13^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\beta _{1})q^{2}+(-1+\beta _{4}+\beta _{7}-\beta _{9}+\cdots)q^{3}+\cdots\)

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

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