Space of modular forms of level 49, weight 8, and Character 49.c

• Label: 49.8.c
• Level: $49$
• Weight: $8$
• Character orbit: 49.c
• Rep. character: $\chi_{49}(18,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $42$
• Newform subspaces: $8$
• Sturm bound: $37$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
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:			\(37\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(49, [\chi])\).

	Total	New	Old
Modular forms	74	50	24
Cusp forms	58	42	16
Eisenstein series	16	8	8

Trace form

42 * q - 13 * q^2 + 28 * q^3 - 969 * q^4 + 252 * q^5 - 2044 * q^6 + 3174 * q^8 - 11269 * q^9 + 4774 * q^10 - 7528 * q^11 - 5404 * q^12 + 2352 * q^13 + 36712 * q^15 + 16571 * q^16 + 56364 * q^17 - 85967 * q^18 + 41748 * q^19 - 324744 * q^20 + 456956 * q^22 + 186964 * q^23 + 389592 * q^24 - 354483 * q^25 + 113652 * q^26 - 378056 * q^27 - 155036 * q^29 + 891990 * q^30 + 401212 * q^31 - 773219 * q^32 + 24052 * q^33 - 164220 * q^34 + 1713386 * q^36 - 727614 * q^37 + 31794 * q^38 + 610232 * q^39 - 443688 * q^40 - 820848 * q^41 - 2672936 * q^43 - 2105612 * q^44 - 1209656 * q^45 + 2478606 * q^46 + 1470084 * q^47 + 4037600 * q^48 - 182686 * q^50 - 2591252 * q^51 - 5269768 * q^52 + 1459798 * q^53 - 4567318 * q^54 + 6126680 * q^55 - 12247224 * q^57 + 3636894 * q^58 + 752220 * q^59 + 5131532 * q^60 + 1325772 * q^61 + 6032628 * q^62 + 3744250 * q^64 - 1298472 * q^65 - 6013294 * q^66 - 3056504 * q^67 + 2453556 * q^68 - 9830520 * q^69 - 7262080 * q^71 + 8102733 * q^72 + 6706588 * q^73 + 7250268 * q^74 + 7223888 * q^75 - 5846008 * q^76 + 38814552 * q^78 - 7678572 * q^79 + 10695888 * q^80 - 17868949 * q^81 - 17563252 * q^82 - 19084128 * q^83 - 13789016 * q^85 + 6967368 * q^86 + 17836840 * q^87 - 33667608 * q^88 + 16165212 * q^89 - 6831496 * q^90 + 35033816 * q^92 - 10441220 * q^93 + 5720442 * q^94 - 5805180 * q^95 - 8448608 * q^96 - 3066224 * q^97 + 35531048 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(49, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
49.8.c.a	$49$	$8$	49.c	7.c	$3$	$2$	$1$	$15.307$	\(\Q(\sqrt{-3}) \)	None					7.8.a.a	$2$	$0$	\(6\)	\(-42\)	\(-84\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+6\zeta_{6}q^{2}+(-42+42\zeta_{6})q^{3}+(92+\cdots)q^{4}+\cdots\)
49.8.c.b	$49$	$8$	49.c	7.c	$3$	$2$	$1$	$15.307$	\(\Q(\sqrt{-3}) \)	None					7.8.a.a	$2$	$0$	\(6\)	\(42\)	\(84\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+6\zeta_{6}q^{2}+(42-42\zeta_{6})q^{3}+(92-92\zeta_{6})q^{4}+\cdots\)
49.8.c.c	$49$	$8$	49.c	7.c	$3$	$2$	$1$	$15.307$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-7}) \)		✓			49.8.a.a	$4$	$0$	\(13\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+13\zeta_{6}q^{2}+(-41+41\zeta_{6})q^{4}+1131q^{8}+\cdots\)
49.8.c.d	$49$	$8$	49.c	7.c	$3$	$4$	$2$	$15.307$	\(\Q(\sqrt{-3}, \sqrt{-230})\)	None		✓			49.8.a.d	$4$	$0$	\(-20\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-10+10\beta _{1})q^{2}-\beta _{2}q^{3}+28\beta _{1}q^{4}+\cdots\)
49.8.c.e	$49$	$8$	49.c	7.c	$3$	$4$	$2$	$15.307$	\(\Q(\sqrt{-3}, \sqrt{865})\)	None					7.8.a.b	$2$	$0$	\(3\)	\(-94\)	\(-330\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+\beta _{2})q^{2}+(-48+2\beta _{1}+46\beta _{2}+\cdots)q^{3}+\cdots\)
49.8.c.f	$49$	$8$	49.c	7.c	$3$	$4$	$2$	$15.307$	\(\Q(\sqrt{-3}, \sqrt{865})\)	None					7.8.a.b	$2$	$0$	\(3\)	\(94\)	\(330\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+\beta _{2})q^{2}+(48-2\beta _{1}-46\beta _{2}+\cdots)q^{3}+\cdots\)
49.8.c.g	$49$	$8$	49.c	7.c	$3$	$8$	$4$	$15.307$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					7.8.c.a	$2$	$0$	\(6\)	\(28\)	\(252\)	\(0\)	$2^{8}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+2\beta _{2}+\beta _{3}-\beta _{4})q^{2}+(6-6\beta _{2}+\cdots)q^{3}+\cdots\)
49.8.c.h	$49$	$8$	49.c	7.c	$3$	$16$	$8$	$15.307$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓	✓		49.8.a.g	$4$	$0$	\(-30\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 7^{12}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-4-4\beta _{1}-\beta _{2})q^{2}+(-\beta _{7}-\beta _{10}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(49, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(49, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 2}\)