Space of modular forms of level 196, weight 4, and Character 196.e

• Label: 196.4.e
• Level: $196$
• Weight: $4$
• Character orbit: 196.e
• Rep. character: $\chi_{196}(165,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $20$
• Newform subspaces: $8$
• Sturm bound: $112$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	196.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(112\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(196, [\chi])\).

	Total	New	Old
Modular forms	192	20	172
Cusp forms	144	20	124
Eisenstein series	48	0	48

Trace form

20 * q - 14 * q^5 - 72 * q^9 + 36 * q^11 + 56 * q^13 + 424 * q^15 - 154 * q^17 - 224 * q^19 + 164 * q^23 - 160 * q^25 + 664 * q^29 - 196 * q^31 - 518 * q^33 - 50 * q^37 - 136 * q^39 + 840 * q^41 + 1656 * q^43 - 140 * q^45 + 84 * q^47 - 340 * q^51 + 342 * q^53 - 1624 * q^55 - 1644 * q^57 - 56 * q^59 + 98 * q^61 - 764 * q^65 + 1456 * q^67 + 1036 * q^69 - 2128 * q^71 + 966 * q^73 + 2072 * q^75 + 1212 * q^79 - 3358 * q^81 - 784 * q^83 - 2452 * q^85 + 2072 * q^87 + 294 * q^89 + 2338 * q^93 + 1940 * q^95 + 840 * q^97 - 9848 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(196, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
196.4.e.a	$196$	$4$	196.e	7.c	$3$	$2$	$1$	$11.564$	\(\Q(\sqrt{-3}) \)	None					28.4.a.a	$2$	$0$	\(0\)	\(-10\)	\(-8\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-10+10\zeta_{6})q^{3}-8\zeta_{6}q^{5}-73\zeta_{6}q^{9}+\cdots\)
196.4.e.b	$196$	$4$	196.e	7.c	$3$	$2$	$1$	$11.564$	\(\Q(\sqrt{-3}) \)	None		✓			196.4.a.a	$2$	$0$	\(0\)	\(-4\)	\(-20\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-4+4\zeta_{6})q^{3}-20\zeta_{6}q^{5}+11\zeta_{6}q^{9}+\cdots\)
196.4.e.c	$196$	$4$	196.e	7.c	$3$	$2$	$1$	$11.564$	\(\Q(\sqrt{-3}) \)	None					28.4.a.b	$2$	$0$	\(0\)	\(-4\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-4+4\zeta_{6})q^{3}-6\zeta_{6}q^{5}+11\zeta_{6}q^{9}+\cdots\)
196.4.e.d	$196$	$4$	196.e	7.c	$3$	$2$	$1$	$11.564$	\(\Q(\sqrt{-3}) \)	None					28.4.a.b	$2$	$0$	\(0\)	\(4\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4-4\zeta_{6})q^{3}+6\zeta_{6}q^{5}+11\zeta_{6}q^{9}+\cdots\)
196.4.e.e	$196$	$4$	196.e	7.c	$3$	$2$	$1$	$11.564$	\(\Q(\sqrt{-3}) \)	None		✓			196.4.a.a	$2$	$0$	\(0\)	\(4\)	\(20\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4-4\zeta_{6})q^{3}+20\zeta_{6}q^{5}+11\zeta_{6}q^{9}+\cdots\)
196.4.e.f	$196$	$4$	196.e	7.c	$3$	$2$	$1$	$11.564$	\(\Q(\sqrt{-3}) \)	None					28.4.a.a	$2$	$0$	\(0\)	\(10\)	\(8\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(10-10\zeta_{6})q^{3}+8\zeta_{6}q^{5}-73\zeta_{6}q^{9}+\cdots\)
196.4.e.g	$196$	$4$	196.e	7.c	$3$	$4$	$2$	$11.564$	\(\Q(\sqrt{-3}, \sqrt{37})\)	None					28.4.e.a	$2$	$0$	\(0\)	\(0\)	\(-14\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{3}+(-7+7\beta _{1}-2\beta _{2}-2\beta _{3})q^{5}+\cdots\)
196.4.e.h	$196$	$4$	196.e	7.c	$3$	$4$	$2$	$11.564$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			196.4.a.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{3}+(2\beta _{1}+2\beta _{3})q^{5}-5^{2}\beta _{2}q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(196, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(196, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 2}\)