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

• Label: 78.4.e
• Level: $78$
• Weight: $4$
• Character orbit: 78.e
• Rep. character: $\chi_{78}(55,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $16$
• Newform subspaces: $4$
• Sturm bound: $56$
• Trace bound: $2$

Defining parameters

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

Dimensions

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

	Total	New	Old
Modular forms	92	16	76
Cusp forms	76	16	60
Eisenstein series	16	0	16

Trace form

16 * q + 4 * q^2 - 6 * q^3 - 32 * q^4 + 20 * q^5 - 26 * q^7 - 32 * q^8 - 72 * q^9 - 60 * q^10 + 68 * q^11 + 48 * q^12 + 108 * q^13 + 96 * q^14 + 24 * q^15 - 128 * q^16 - 70 * q^17 - 72 * q^18 + 104 * q^19 - 40 * q^20 - 252 * q^21 - 32 * q^22 - 204 * q^23 + 1148 * q^25 - 380 * q^26 + 108 * q^27 - 104 * q^28 + 102 * q^29 - 120 * q^30 - 132 * q^31 + 64 * q^32 - 192 * q^33 + 440 * q^34 + 260 * q^35 - 288 * q^36 - 550 * q^37 - 288 * q^38 - 96 * q^39 + 480 * q^40 + 362 * q^41 - 168 * q^42 + 214 * q^43 - 544 * q^44 - 90 * q^45 + 832 * q^46 + 968 * q^47 - 96 * q^48 - 882 * q^49 + 768 * q^50 + 960 * q^51 - 648 * q^52 - 4180 * q^53 + 888 * q^55 - 192 * q^56 + 504 * q^57 - 236 * q^58 + 556 * q^59 - 192 * q^60 - 8 * q^61 + 1232 * q^62 - 234 * q^63 + 1024 * q^64 + 2606 * q^65 + 1680 * q^66 + 1470 * q^67 - 280 * q^68 - 1080 * q^69 - 288 * q^70 - 956 * q^71 + 144 * q^72 - 376 * q^73 - 1572 * q^74 - 738 * q^75 + 416 * q^76 - 8488 * q^77 - 864 * q^78 - 2908 * q^79 - 160 * q^80 - 648 * q^81 - 940 * q^82 + 2928 * q^83 + 504 * q^84 - 1486 * q^85 + 3936 * q^86 + 504 * q^87 - 128 * q^88 + 2756 * q^89 + 1080 * q^90 + 1386 * q^91 + 1632 * q^92 + 1806 * q^93 - 1088 * q^94 + 4856 * q^95 - 462 * q^97 + 3348 * q^98 - 1224 * q^99

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.e.a	$78$	$4$	78.e	13.c	$3$	$2$	$1$	$4.602$	\(\Q(\sqrt{-3}) \)	None		✓			78.4.e.a	$2$	$0$	\(-2\)	\(3\)	\(14\)	\(-16\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots\)
78.4.e.b	$78$	$4$	78.e	13.c	$3$	$4$	$2$	$4.602$	\(\Q(\sqrt{-3}, \sqrt{673})\)	None		✓			78.4.e.b	$2$	$0$	\(-4\)	\(-6\)	\(26\)	\(-9\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\beta _{2})q^{2}+(-3+3\beta _{2})q^{3}+\cdots\)
78.4.e.c	$78$	$4$	78.e	13.c	$3$	$4$	$2$	$4.602$	\(\Q(\sqrt{-3}, \sqrt{61})\)	None		✓			78.4.e.c	$2$	$0$	\(4\)	\(6\)	\(4\)	\(-18\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\beta _{1})q^{2}+(3-3\beta _{1})q^{3}-4\beta _{1}q^{4}+\cdots\)
78.4.e.d	$78$	$4$	78.e	13.c	$3$	$6$	$3$	$4.602$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		✓	✓		78.4.e.d	$2$	$0$	\(6\)	\(-9\)	\(-24\)	\(17\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2+2\beta _{2})q^{2}+(-3-3\beta _{2})q^{3}+4\beta _{2}q^{4}+\cdots\)

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

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