Space of modular forms of level 78, weight 3, and Character 78.f

• Label: 78.3.f
• Level: $78$
• Weight: $3$
• Character orbit: 78.f
• Rep. character: $\chi_{78}(31,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $12$
• Newform subspaces: $2$
• Sturm bound: $42$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	78.f (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(42\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(78, [\chi])\).

	Total	New	Old
Modular forms	64	12	52
Cusp forms	48	12	36
Eisenstein series	16	0	16

Trace form

12 * q - 4 * q^2 + 12 * q^5 + 8 * q^8 + 36 * q^9 + 24 * q^11 - 24 * q^13 - 16 * q^14 - 24 * q^15 - 48 * q^16 - 12 * q^18 - 24 * q^20 + 24 * q^21 - 48 * q^22 + 52 * q^26 - 96 * q^29 - 48 * q^31 + 16 * q^32 + 72 * q^33 + 72 * q^34 - 48 * q^35 - 84 * q^37 - 96 * q^39 - 48 * q^40 + 132 * q^41 + 48 * q^44 + 36 * q^45 - 216 * q^47 + 164 * q^50 + 24 * q^52 + 264 * q^53 + 288 * q^55 - 72 * q^57 + 240 * q^58 - 72 * q^59 + 48 * q^60 + 120 * q^61 + 84 * q^65 - 192 * q^66 - 144 * q^67 + 48 * q^68 - 48 * q^70 - 408 * q^71 + 24 * q^72 - 36 * q^73 - 104 * q^74 + 96 * q^78 - 48 * q^79 - 48 * q^80 + 108 * q^81 + 72 * q^83 + 48 * q^84 + 216 * q^85 - 48 * q^86 + 96 * q^87 - 156 * q^89 - 144 * q^91 + 192 * q^92 - 168 * q^93 - 144 * q^94 + 60 * q^97 - 508 * q^98 + 72 * q^99

Decomposition of \(S_{3}^{\mathrm{new}}(78, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
78.3.f.a	$78$	$3$	78.f	13.d	$4$	$4$	$2$	$2.125$	\(\Q(\zeta_{12})\)	None		✓			78.3.f.a	$2$	$0$	\(4\)	\(0\)	\(12\)	\(-4\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta_1+1)q^{2}+\beta_{3} q^{3}+2\beta_1 q^{4}+\cdots\)
78.3.f.b	$78$	$3$	78.f	13.d	$4$	$8$	$4$	$2.125$	8.0.\(\cdots\).1	None		✓	✓		78.3.f.b	$2$	$0$	\(-8\)	\(0\)	\(0\)	\(4\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\beta _{6})q^{2}+\beta _{1}q^{3}+2\beta _{6}q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(78, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(78, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)