Space of modular forms of level 49, weight 4, and Character 49.c

• Label: 49.4.c
• Level: $49$
• Weight: $4$
• Character orbit: 49.c
• Rep. character: $\chi_{49}(18,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $16$
• Newform subspaces: $5$
• Sturm bound: $18$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.c (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(18\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(49, [\chi])\).

	Total	New	Old
Modular forms	36	24	12
Cusp forms	20	16	4
Eisenstein series	16	8	8

Trace form

16 * q + 3 * q^2 + 7 * q^3 - 33 * q^4 + 7 * q^5 - 28 * q^6 + 54 * q^8 + 11 * q^9 + 14 * q^10 - 11 * q^11 - 28 * q^12 + 28 * q^13 - 158 * q^15 + 19 * q^16 - 21 * q^17 + 145 * q^18 + 49 * q^19 + 56 * q^20 + 12 * q^22 - 249 * q^23 - 168 * q^24 + 67 * q^25 - 28 * q^26 + 70 * q^27 - 136 * q^29 + 390 * q^30 + 147 * q^31 + 277 * q^32 - 35 * q^33 + 84 * q^34 - 550 * q^36 - 389 * q^37 + 98 * q^38 - 742 * q^39 - 168 * q^40 - 700 * q^41 + 984 * q^43 + 564 * q^44 + 154 * q^45 - 482 * q^46 + 525 * q^47 + 224 * q^48 + 3154 * q^50 - 65 * q^51 + 56 * q^52 - 585 * q^53 - 70 * q^54 + 70 * q^55 - 2202 * q^57 - 386 * q^58 - 105 * q^59 + 812 * q^60 - 413 * q^61 - 588 * q^62 - 3702 * q^64 - 42 * q^65 - 70 * q^66 + 1781 * q^67 + 84 * q^68 + 2226 * q^69 + 1936 * q^71 - 267 * q^72 - 1113 * q^73 + 1604 * q^74 - 532 * q^75 + 392 * q^76 - 2856 * q^78 + 1655 * q^79 - 112 * q^80 + 524 * q^81 + 700 * q^82 - 2184 * q^83 - 3406 * q^85 - 5216 * q^86 + 406 * q^87 - 4200 * q^88 - 329 * q^89 - 616 * q^90 + 5592 * q^92 + 4387 * q^93 + 1050 * q^94 + 985 * q^95 + 1120 * q^96 + 1764 * q^97 + 10868 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(49, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
49.4.c.a	$49$	$4$	49.c	7.c	$3$	$2$	$1$	$2.891$	\(\Q(\sqrt{-3}) \)	None					7.4.c.a	$2$	$0$	\(-2\)	\(7\)	\(7\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{2}+(7-7\zeta_{6})q^{3}+(4-4\zeta_{6})q^{4}+\cdots\)
49.4.c.b	$49$	$4$	49.c	7.c	$3$	$2$	$1$	$2.891$	\(\Q(\sqrt{-3}) \)	None					7.4.a.a	$2$	$0$	\(1\)	\(-2\)	\(16\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(7-7\zeta_{6})q^{4}+\cdots\)
49.4.c.c	$49$	$4$	49.c	7.c	$3$	$2$	$1$	$2.891$	\(\Q(\sqrt{-3}) \)	None					7.4.a.a	$2$	$0$	\(1\)	\(2\)	\(-16\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{2}+(2-2\zeta_{6})q^{3}+(7-7\zeta_{6})q^{4}+\cdots\)
49.4.c.d	$49$	$4$	49.c	7.c	$3$	$2$	$1$	$2.891$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-7}) \)		✓			49.4.a.a	$4$	$0$	\(5\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+5\zeta_{6}q^{2}+(-17+17\zeta_{6})q^{4}-45q^{8}+\cdots\)
49.4.c.e	$49$	$4$	49.c	7.c	$3$	$8$	$4$	$2.891$	\(\Q(\sqrt{2}, \sqrt{-3}, \sqrt{65})\)	None		✓	✓		49.4.a.e	$4$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{2}-\beta _{6})q^{2}+\beta _{3}q^{3}+(-8+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(49, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(49, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 2}\)