Space of modular forms of level 78, weight 4, and Character 78.b

• Label: 78.4.b
• Level: $78$
• Weight: $4$
• Character orbit: 78.b
• Rep. character: $\chi_{78}(25,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $2$
• Sturm bound: $56$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	78.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(56\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(78, [\chi])\).

	Total	New	Old
Modular forms	46	6	40
Cusp forms	38	6	32
Eisenstein series	8	0	8

Trace form

6 * q + 6 * q^3 - 24 * q^4 + 54 * q^9 + 104 * q^10 - 24 * q^12 + 110 * q^13 + 56 * q^14 + 96 * q^16 - 188 * q^17 - 144 * q^22 + 184 * q^23 - 314 * q^25 + 8 * q^26 + 54 * q^27 + 164 * q^29 + 120 * q^30 - 1448 * q^35 - 216 * q^36 + 456 * q^38 - 138 * q^39 - 416 * q^40 - 168 * q^42 + 760 * q^43 + 96 * q^48 - 782 * q^49 - 12 * q^51 - 440 * q^52 + 876 * q^53 + 96 * q^55 - 224 * q^56 + 2156 * q^61 + 1928 * q^62 - 384 * q^64 + 1240 * q^65 - 1152 * q^66 + 752 * q^68 + 1896 * q^69 - 704 * q^74 - 1674 * q^75 - 2064 * q^77 - 600 * q^78 - 2096 * q^79 + 486 * q^81 + 40 * q^82 + 2532 * q^87 + 576 * q^88 + 936 * q^90 + 1720 * q^91 - 736 * q^92 - 288 * q^94 - 696 * q^95

Decomposition of \(S_{4}^{\mathrm{new}}(78, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
78.4.b.a	$78$	$4$	78.b	13.b	$2$	$2$	$2$	$4.602$	\(\Q(\sqrt{-1}) \)	None		✓			78.4.b.a	$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}-3 q^{3}-4 q^{4}-4\beta q^{5}-3\beta q^{6}+\cdots\)
78.4.b.b	$78$	$4$	78.b	13.b	$2$	$4$	$4$	$4.602$	\(\Q(i, \sqrt{17})\)	None		✓	✓		78.4.b.b	$2$	$0$	\(0\)	\(12\)	\(0\)	\(0\)	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+3q^{3}-4q^{4}+(4\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(78, [\chi])\) into lower level spaces

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