Space of modular forms of level 49, weight 6, and Character 49.c

• Label: 49.6.c
• Level: $49$
• Weight: $6$
• Character orbit: 49.c
• Rep. character: $\chi_{49}(18,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $30$
• Newform subspaces: $8$
• Sturm bound: $28$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	49.c (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(28\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(49, [\chi])\).

	Total	New	Old
Modular forms	54	38	16
Cusp forms	38	30	8
Eisenstein series	16	8	8

Trace form

30 * q + 3 * q^2 - 8 * q^3 - 201 * q^4 - 38 * q^5 + 164 * q^6 - 90 * q^8 - 1381 * q^9 - 778 * q^10 + 916 * q^11 - 196 * q^12 + 1848 * q^13 - 2408 * q^15 - 4805 * q^16 - 2346 * q^17 + 1945 * q^18 - 360 * q^19 + 3416 * q^20 - 4388 * q^22 + 1284 * q^23 + 1392 * q^24 - 39 * q^25 + 1148 * q^26 - 5744 * q^27 + 16748 * q^29 + 5838 * q^30 + 3548 * q^31 - 16187 * q^32 - 3398 * q^33 - 14844 * q^34 - 3046 * q^36 + 40596 * q^37 + 20138 * q^38 + 6776 * q^39 + 15936 * q^40 - 7000 * q^41 + 65048 * q^43 + 13092 * q^44 + 1300 * q^45 - 5826 * q^46 - 22956 * q^47 - 22432 * q^48 - 194078 * q^50 + 11872 * q^51 - 1400 * q^52 + 23568 * q^53 + 32546 * q^54 + 50152 * q^55 + 204876 * q^57 - 141954 * q^58 - 65808 * q^59 + 67172 * q^60 - 42486 * q^61 + 98724 * q^62 - 129766 * q^64 - 106092 * q^65 - 25894 * q^66 - 34252 * q^67 + 5460 * q^68 - 10308 * q^69 - 18560 * q^71 - 35379 * q^72 - 50506 * q^73 - 217828 * q^74 - 35608 * q^75 - 77672 * q^76 - 154392 * q^78 - 55332 * q^79 + 68816 * q^80 + 132905 * q^81 + 67732 * q^82 + 208656 * q^83 + 628460 * q^85 + 458128 * q^86 + 80008 * q^87 + 229848 * q^88 - 26666 * q^89 - 261304 * q^90 - 394200 * q^92 + 6034 * q^93 + 98034 * q^94 - 295316 * q^95 + 54880 * q^96 - 418264 * q^97 - 724888 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(49, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
49.6.c.a	$49$	$6$	49.c	7.c	$3$	$2$	$1$	$7.859$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-7}) \)		✓			49.6.a.b	$4$	$0$	\(-11\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q-11\zeta_{6}q^{2}+(-89+89\zeta_{6})q^{4}+627q^{8}+\cdots\)
49.6.c.b	$49$	$6$	49.c	7.c	$3$	$2$	$1$	$7.859$	\(\Q(\sqrt{-3}) \)	None					7.6.a.a	$2$	$0$	\(10\)	\(-14\)	\(-56\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+10\zeta_{6}q^{2}+(-14+14\zeta_{6})q^{3}+(-68+\cdots)q^{4}+\cdots\)
49.6.c.c	$49$	$6$	49.c	7.c	$3$	$2$	$1$	$7.859$	\(\Q(\sqrt{-3}) \)	None					7.6.a.a	$2$	$0$	\(10\)	\(14\)	\(56\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+10\zeta_{6}q^{2}+(14-14\zeta_{6})q^{3}+(-68+\cdots)q^{4}+\cdots\)
49.6.c.d	$49$	$6$	49.c	7.c	$3$	$4$	$2$	$7.859$	\(\Q(\sqrt{-3}, \sqrt{-19})\)	None					7.6.a.b	$2$	$0$	\(-9\)	\(-6\)	\(-18\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(5\beta _{1}+\beta _{2}-\beta _{3})q^{2}+(-6-6\beta _{1}+\cdots)q^{3}+\cdots\)
49.6.c.e	$49$	$6$	49.c	7.c	$3$	$4$	$2$	$7.859$	\(\Q(\sqrt{-3}, \sqrt{-19})\)	None					7.6.a.b	$2$	$0$	\(-9\)	\(6\)	\(18\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(5\beta _{1}+\beta _{2}-\beta _{3})q^{2}+(6+6\beta _{1}-6\beta _{3})q^{3}+\cdots\)
49.6.c.f	$49$	$6$	49.c	7.c	$3$	$4$	$2$	$7.859$	\(\Q(\sqrt{-3}, \sqrt{37})\)	None					7.6.c.a	$2$	$0$	\(-2\)	\(-8\)	\(-38\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-\beta _{2})q^{2}+(-4+4\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
49.6.c.g	$49$	$6$	49.c	7.c	$3$	$4$	$2$	$7.859$	\(\Q(\sqrt{-3}, \sqrt{-13})\)	None		✓			49.6.a.c	$4$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\beta _{1})q^{2}+\beta _{2}q^{3}+28\beta _{1}q^{4}+\cdots\)
49.6.c.h	$49$	$6$	49.c	7.c	$3$	$8$	$4$	$7.859$	\(\Q(\sqrt{2}, \sqrt{-3}, \sqrt{113})\)	None		✓	✓		49.6.a.g	$4$	$0$	\(10\)	\(0\)	\(0\)	\(0\)	$7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(3-2\beta _{1}-\beta _{3}-\beta _{6})q^{2}+(-2\beta _{2}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(49, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(49, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 2}\)