Space of modular forms of level 49, weight 4, and trivial character

• Label: 49.4.a
• Level: $49$
• Weight: $4$
• Character orbit: 49.a
• Rep. character: $\chi_{49}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $5$
• Sturm bound: $18$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.a (trivial)
Character field:			\(\Q\)
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 \(M_{4}(\Gamma_0(49))\).

	Total	New	Old
Modular forms	18	13	5
Cusp forms	10	8	2
Eisenstein series	8	5	3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(7\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(10\)	\(7\)	\(3\)		\(6\)	\(5\)	\(1\)		\(4\)	\(2\)	\(2\)
\(-\)		\(8\)	\(6\)	\(2\)		\(4\)	\(3\)	\(1\)		\(4\)	\(3\)	\(1\)

Trace form

8 * q + 2 * q^3 + 36 * q^4 - 16 * q^5 - 2 * q^6 - 12 * q^8 + 34 * q^9 + 16 * q^10 + 14 * q^11 - 14 * q^12 - 28 * q^13 + 2 * q^15 - 76 * q^16 - 54 * q^17 - 124 * q^18 + 110 * q^19 + 112 * q^20 - 12 * q^22 + 42 * q^23 + 30 * q^24 - 274 * q^25 + 28 * q^26 - 100 * q^27 + 100 * q^29 - 324 * q^30 - 12 * q^31 + 44 * q^32 - 16 * q^33 + 54 * q^34 - 524 * q^36 + 854 * q^37 - 110 * q^38 + 700 * q^39 - 240 * q^40 - 182 * q^41 + 240 * q^43 - 600 * q^44 + 368 * q^45 + 212 * q^46 - 324 * q^47 + 82 * q^48 + 1556 * q^50 + 26 * q^51 + 196 * q^52 + 1050 * q^53 + 100 * q^54 + 128 * q^55 - 978 * q^57 + 392 * q^58 - 810 * q^59 - 1232 * q^60 + 488 * q^61 + 12 * q^62 - 1236 * q^64 - 504 * q^65 + 16 * q^66 - 1610 * q^67 + 378 * q^68 + 96 * q^69 + 1304 * q^71 + 84 * q^72 + 702 * q^73 - 1412 * q^74 + 262 * q^75 - 770 * q^76 - 1680 * q^78 - 2198 * q^79 - 656 * q^80 - 1784 * q^81 + 182 * q^82 + 1302 * q^83 - 2714 * q^85 + 5096 * q^86 - 220 * q^87 + 4440 * q^88 - 730 * q^89 - 368 * q^90 + 3768 * q^92 - 3334 * q^93 + 324 * q^94 + 1118 * q^95 - 322 * q^96 - 294 * q^97 + 5140 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(49))\) 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					A-L signs	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		7
49.4.a.a	$49$	$4$	49.a	1.a	$1$	$1$	$1$	$2.891$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓	✓			49.4.a.a	$2$	$1$	\(-5\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-5q^{2}+17q^{4}-45q^{8}-3^{3}q^{9}-68q^{11}+\cdots\)
49.4.a.b	$49$	$4$	49.a	1.a	$1$	$1$	$1$	$2.891$	\(\Q\)	None	✓				7.4.a.a	$1$	$1$	\(-1\)	\(2\)	\(-16\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2q^{3}-7q^{4}-2^{4}q^{5}-2q^{6}+\cdots\)
49.4.a.c	$49$	$4$	49.a	1.a	$1$	$1$	$1$	$2.891$	\(\Q\)	None	✓				7.4.c.a	$1$	$1$	\(2\)	\(-7\)	\(-7\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-7q^{3}-4q^{4}-7q^{5}-14q^{6}+\cdots\)
49.4.a.d	$49$	$4$	49.a	1.a	$1$	$1$	$1$	$2.891$	\(\Q\)	None	✓				7.4.c.a	$1$	$0$	\(2\)	\(7\)	\(7\)	\(0\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+7q^{3}-4q^{4}+7q^{5}+14q^{6}+\cdots\)
49.4.a.e	$49$	$4$	49.a	1.a	$1$	$4$	$4$	$2.891$	\(\Q(\sqrt{2}, \sqrt{65})\)	None	✓	✓	✓		49.4.a.e	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$7$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+\beta _{2}q^{3}+(9+\beta _{1})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(49))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(49)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)