Space of modular forms of level 49, weight 6, and trivial character

• Label: 49.6.a
• Level: $49$
• Weight: $6$
• Character orbit: 49.a
• Rep. character: $\chi_{49}(1,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $7$
• Sturm bound: $28$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	49.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(28\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(49))\).

	Total	New	Old
Modular forms	28	19	9
Cusp forms	20	14	6
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.

\(7\)	Dim
\(+\)	\(6\)
\(-\)	\(8\)

Trace form

14 * q + 20 * q^3 + 140 * q^4 + 74 * q^5 + 58 * q^6 + 420 * q^8 + 490 * q^9 - 764 * q^10 - 1120 * q^11 + 2506 * q^12 + 490 * q^13 + 560 * q^15 + 1204 * q^16 - 78 * q^17 - 5740 * q^18 + 3364 * q^19 + 1288 * q^20 + 4004 * q^22 - 5208 * q^23 - 3186 * q^24 + 5138 * q^25 - 3416 * q^26 + 2312 * q^27 - 8792 * q^29 + 14028 * q^30 + 7664 * q^31 + 16940 * q^32 - 16768 * q^33 - 23154 * q^34 - 10052 * q^36 - 25340 * q^37 + 14230 * q^38 + 13384 * q^39 - 23376 * q^40 + 24010 * q^41 + 11984 * q^43 + 7896 * q^44 - 7870 * q^45 - 6636 * q^46 - 21024 * q^47 - 21278 * q^48 - 47236 * q^50 - 43960 * q^51 - 21924 * q^52 - 32256 * q^53 + 78508 * q^54 + 51896 * q^55 + 69132 * q^57 + 79968 * q^58 + 46668 * q^59 - 108416 * q^60 + 39106 * q^61 - 91740 * q^62 + 59724 * q^64 + 56028 * q^65 - 99296 * q^66 + 100184 * q^67 + 132090 * q^68 + 128928 * q^69 - 50344 * q^71 - 22764 * q^72 - 85534 * q^73 + 23044 * q^74 - 40340 * q^75 + 19446 * q^76 + 43008 * q^78 - 18704 * q^79 + 150016 * q^80 + 24766 * q^81 - 121282 * q^82 - 97020 * q^83 + 202552 * q^85 - 112504 * q^86 + 111032 * q^87 - 126168 * q^88 - 33694 * q^89 - 67532 * q^90 - 482328 * q^92 + 17948 * q^93 - 66228 * q^94 + 67424 * q^95 - 293986 * q^96 + 37730 * q^97 - 401744 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(49))\) 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
49.6.a.a	$49$	$6$	49.a	1.a	$1$	$1$	$1$	$7.859$	\(\Q\)	None	✓				7.6.a.a	$1$	$0$	\(-10\)	\(14\)	\(56\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-10q^{2}+14q^{3}+68q^{4}+56q^{5}+\cdots\)
49.6.a.b	$49$	$6$	49.a	1.a	$1$	$1$	$1$	$7.859$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓	✓			49.6.a.b	$2$	$0$	\(11\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+11q^{2}+89q^{4}+627q^{8}-3^{5}q^{9}+\cdots\)
49.6.a.c	$49$	$6$	49.a	1.a	$1$	$2$	$2$	$7.859$	\(\Q(\sqrt{39}) \)	None	✓	✓			49.6.a.c	$2$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+\beta q^{3}-28q^{4}+3\beta q^{5}-2\beta q^{6}+\cdots\)
49.6.a.d	$49$	$6$	49.a	1.a	$1$	$2$	$2$	$7.859$	\(\Q(\sqrt{37}) \)	None	✓				7.6.c.a	$1$	$1$	\(2\)	\(-8\)	\(-38\)	\(0\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(-4-\beta )q^{3}+(6+2\beta )q^{4}+\cdots\)
49.6.a.e	$49$	$6$	49.a	1.a	$1$	$2$	$2$	$7.859$	\(\Q(\sqrt{37}) \)	None	✓				7.6.c.a	$1$	$0$	\(2\)	\(8\)	\(38\)	\(0\)		$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(4+\beta )q^{3}+(6+2\beta )q^{4}+\cdots\)
49.6.a.f	$49$	$6$	49.a	1.a	$1$	$2$	$2$	$7.859$	\(\Q(\sqrt{57}) \)	None	✓				7.6.a.b	$1$	$0$	\(9\)	\(6\)	\(18\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(5-\beta )q^{2}+(6-6\beta )q^{3}+(7-9\beta )q^{4}+\cdots\)
49.6.a.g	$49$	$6$	49.a	1.a	$1$	$4$	$4$	$7.859$	\(\Q(\sqrt{2}, \sqrt{113})\)	None	✓	✓	✓		49.6.a.g	$2$	$1$	\(-10\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$7$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(2\beta _{2}-\beta _{3})q^{3}-5\beta _{1}q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(49))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(49)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)