Space of modular forms of level 49, weight 10, and Character 49.c

• Label: 49.10.c
• Level: $49$
• Weight: $10$
• Character orbit: 49.c
• Rep. character: $\chi_{49}(18,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $56$
• Newform subspaces: $8$
• Sturm bound: $46$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
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:			\(46\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(49, [\chi])\).

	Total	New	Old
Modular forms	92	64	28
Cusp forms	76	56	20
Eisenstein series	16	8	8

Trace form

56 * q + 35 * q^2 - 161 * q^3 - 6825 * q^4 - 1533 * q^5 + 8708 * q^6 - 68250 * q^8 - 142261 * q^9 - 4298 * q^10 - 59731 * q^11 - 135604 * q^12 + 319676 * q^13 - 181118 * q^15 - 2079973 * q^16 - 324681 * q^17 + 1711801 * q^18 + 16121 * q^19 + 350616 * q^20 + 1217244 * q^22 - 3510185 * q^23 - 8449728 * q^24 - 4701949 * q^25 - 4179252 * q^26 + 18331558 * q^27 - 21539672 * q^29 - 30056082 * q^30 - 19179237 * q^31 + 19278805 * q^32 - 1689359 * q^33 + 62909700 * q^34 - 31321990 * q^36 - 29326969 * q^37 - 67365270 * q^38 + 61073642 * q^39 - 5721744 * q^40 + 53436852 * q^41 - 3069192 * q^43 - 137957372 * q^44 - 85098230 * q^45 + 58191182 * q^46 - 32509659 * q^47 + 185141600 * q^48 - 97915678 * q^50 - 62181665 * q^51 - 103893272 * q^52 - 137763941 * q^53 - 51200926 * q^54 + 144695222 * q^55 - 245916930 * q^57 - 255256834 * q^58 - 46776513 * q^59 - 239485708 * q^60 + 113075039 * q^61 - 467465628 * q^62 + 552046362 * q^64 + 538560918 * q^65 + 836682602 * q^66 + 522906853 * q^67 - 32262636 * q^68 - 1323616182 * q^69 + 1645887824 * q^71 + 1290363501 * q^72 + 859257651 * q^73 - 426747972 * q^74 + 169061732 * q^75 - 1101475592 * q^76 - 1277033016 * q^78 + 1199023735 * q^79 + 1257352656 * q^80 - 1384909288 * q^81 + 1341703076 * q^82 + 144863208 * q^83 - 2245136390 * q^85 + 3411709392 * q^86 + 340781350 * q^87 - 1006995528 * q^88 - 1661554797 * q^89 - 1967758744 * q^90 + 7904319752 * q^92 + 640526215 * q^93 + 272580882 * q^94 + 1644723129 * q^95 + 1441922272 * q^96 - 869770188 * q^97 - 3174578428 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(49, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
49.10.c.a	$49$	$10$	49.c	7.c	$3$	$2$	$1$	$25.237$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-7}) \)		✓			49.10.a.a	$4$	$0$	\(5\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+5\zeta_{6}q^{2}+(487-487\zeta_{6})q^{4}+4995q^{8}+\cdots\)
49.10.c.b	$49$	$10$	49.c	7.c	$3$	$4$	$2$	$25.237$	\(\Q(\sqrt{-3}, \sqrt{193})\)	None					7.10.a.a	$2$	$0$	\(6\)	\(-86\)	\(-2238\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(3\beta _{1}+\beta _{2})q^{2}+(-43+43\beta _{1}-11\beta _{2}+\cdots)q^{3}+\cdots\)
49.10.c.c	$49$	$10$	49.c	7.c	$3$	$4$	$2$	$25.237$	\(\Q(\sqrt{-3}, \sqrt{193})\)	None					7.10.a.a	$2$	$0$	\(6\)	\(86\)	\(2238\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(3\beta _{1}+\beta _{2})q^{2}+(43-43\beta _{1}+11\beta _{2}+\cdots)q^{3}+\cdots\)
49.10.c.d	$49$	$10$	49.c	7.c	$3$	$6$	$3$	$25.237$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None					7.10.a.b	$2$	$0$	\(-21\)	\(-84\)	\(-1554\)	\(0\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(7\beta _{3}-\beta _{5})q^{2}+(-28+\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
49.10.c.e	$49$	$10$	49.c	7.c	$3$	$6$	$3$	$25.237$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None					7.10.a.b	$2$	$0$	\(-21\)	\(84\)	\(1554\)	\(0\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(7\beta _{3}-\beta _{5})q^{2}+(28-\beta _{1}-\beta _{2}+28\beta _{3}+\cdots)q^{3}+\cdots\)
49.10.c.f	$49$	$10$	49.c	7.c	$3$	$8$	$4$	$25.237$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			49.10.a.d	$4$	$0$	\(12\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{4}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(3-3\beta _{1}-\beta _{3}+\beta _{5})q^{2}+\beta _{2}q^{3}+\cdots\)
49.10.c.g	$49$	$10$	49.c	7.c	$3$	$10$	$5$	$25.237$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None					7.10.c.a	$2$	$0$	\(-18\)	\(-161\)	\(-1533\)	\(0\)	$2^{12}\cdot 3^{3}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}+\beta _{2}-4\beta _{3})q^{2}+(-33+\beta _{1}+\cdots)q^{3}+\cdots\)
49.10.c.h	$49$	$10$	49.c	7.c	$3$	$16$	$8$	$25.237$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓	✓		49.10.a.g	$4$	$0$	\(66\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 7^{12}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-8\beta _{1}+\beta _{2}-\beta _{6})q^{2}+(-\beta _{4}-\beta _{8}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(49, [\chi])\) into lower level spaces

\( S_{10}^{\mathrm{old}}(49, [\chi]) \simeq \) \(S_{10}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 2}\)