Space of modular forms of level 49, weight 7, and Character 49.d

• Label: 49.7.d
• Level: $49$
• Weight: $7$
• Character orbit: 49.d
• Rep. character: $\chi_{49}(19,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $36$
• Newform subspaces: $5$
• Sturm bound: $32$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	49.d (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(32\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(49, [\chi])\).

	Total	New	Old
Modular forms	64	44	20
Cusp forms	48	36	12
Eisenstein series	16	8	8

Trace form

36 * q - 9 * q^2 + 3 * q^3 - 601 * q^4 - 165 * q^5 + 1770 * q^8 + 3187 * q^9 - 2580 * q^10 + 565 * q^11 - 5964 * q^12 + 32114 * q^15 - 21757 * q^16 - 8229 * q^17 - 1429 * q^18 - 29985 * q^19 + 103508 * q^22 - 18391 * q^23 - 28872 * q^24 + 6015 * q^25 - 33768 * q^26 + 43584 * q^29 + 888 * q^30 + 39399 * q^31 + 59971 * q^32 + 36711 * q^33 - 362966 * q^36 + 80141 * q^37 - 60960 * q^38 - 275660 * q^39 + 137280 * q^40 - 83704 * q^43 + 248306 * q^44 + 370530 * q^45 + 214694 * q^46 + 407823 * q^47 - 1665718 * q^50 + 65673 * q^51 - 128856 * q^52 + 283365 * q^53 + 44496 * q^54 + 1149314 * q^57 - 613666 * q^58 + 436299 * q^59 + 222012 * q^60 - 141201 * q^61 + 579226 * q^64 + 78484 * q^65 - 1025532 * q^66 - 1743639 * q^67 - 799596 * q^68 + 2536984 * q^71 - 758461 * q^72 + 421599 * q^73 + 2324410 * q^74 - 826050 * q^75 + 1794800 * q^78 + 482025 * q^79 - 2654040 * q^80 - 650262 * q^81 - 2585016 * q^82 - 1246254 * q^85 + 2255382 * q^86 + 1909746 * q^87 - 2434418 * q^88 + 2990823 * q^89 + 7711196 * q^92 - 2093687 * q^93 - 223584 * q^94 - 1628635 * q^95 - 972720 * q^96 - 8793216 * q^99

Decomposition of \(S_{7}^{\mathrm{new}}(49, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
49.7.d.a	$49$	$7$	49.d	7.d	$6$	$2$	$1$	$11.273$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-7}) \)					7.7.b.a	$4$	$0$	\(-9\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q-9\zeta_{6}q^{2}+(-17+17\zeta_{6})q^{4}-423q^{8}+\cdots\)
49.7.d.b	$49$	$7$	49.d	7.d	$6$	$2$	$1$	$11.273$	\(\Q(\sqrt{-3}) \)	None					7.7.d.a	$2$	$0$	\(12\)	\(21\)	\(-315\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+12\zeta_{6}q^{2}+(7+7\zeta_{6})q^{3}+(-80+80\zeta_{6})q^{4}+\cdots\)
49.7.d.c	$49$	$7$	49.d	7.d	$6$	$4$	$2$	$11.273$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					7.7.d.b	$2$	$0$	\(-8\)	\(-18\)	\(150\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-4-4\beta _{1}+\beta _{2}+\beta _{3})q^{2}+(-6+\cdots)q^{3}+\cdots\)
49.7.d.d	$49$	$7$	49.d	7.d	$6$	$4$	$2$	$11.273$	\(\Q(\sqrt{-3}, \sqrt{170})\)	None					7.7.b.b	$4$	$0$	\(16\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{6}]$	\(q+(8+8\beta _{1})q^{2}+\beta _{3}q^{3}+(-\beta _{2}+\beta _{3})q^{5}+\cdots\)
49.7.d.e	$49$	$7$	49.d	7.d	$6$	$24$	$12$	$11.273$		None		✓	✓		49.7.b.c	$4$	$0$	\(-20\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{7}^{\mathrm{old}}(49, [\chi])\) into lower level spaces

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