Space of modular forms of level 49, weight 10, and trivial character

• Label: 49.10.a
• Level: $49$
• Weight: $10$
• Character orbit: 49.a
• Rep. character: $\chi_{49}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $7$
• Sturm bound: $46$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	49.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(46\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(49))\).

	Total	New	Old
Modular forms	46	33	13
Cusp forms	38	28	10
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
\(+\)		\(22\)	\(15\)	\(7\)		\(18\)	\(13\)	\(5\)		\(4\)	\(2\)	\(2\)
\(-\)		\(24\)	\(18\)	\(6\)		\(20\)	\(15\)	\(5\)		\(4\)	\(3\)	\(1\)

Trace form

28 * q - 32 * q^2 + 2 * q^3 + 6828 * q^4 + 684 * q^5 - 926 * q^6 - 33660 * q^8 + 192958 * q^9 + 54476 * q^10 - 14354 * q^11 + 54250 * q^12 + 46312 * q^13 + 91970 * q^15 + 1275700 * q^16 - 552774 * q^17 - 757708 * q^18 + 702574 * q^19 + 1448328 * q^20 + 2246436 * q^22 - 1792438 * q^23 - 1851666 * q^24 + 851098 * q^25 + 12912480 * q^26 + 2246732 * q^27 + 2798180 * q^29 - 2650932 * q^30 - 5336700 * q^31 - 39521812 * q^32 + 35900576 * q^33 + 6228318 * q^34 + 61086364 * q^36 - 42371186 * q^37 - 46984530 * q^38 - 55201412 * q^39 + 115269264 * q^40 + 33524106 * q^41 + 106949472 * q^43 - 40108552 * q^44 - 52112980 * q^45 - 145817228 * q^46 + 92563980 * q^47 + 12007714 * q^48 - 32306036 * q^50 - 15811366 * q^51 + 47441660 * q^52 + 151165850 * q^53 + 65044828 * q^54 + 23377216 * q^55 - 224065866 * q^57 + 394508176 * q^58 + 49659318 * q^59 + 210479584 * q^60 - 236063228 * q^61 - 7926732 * q^62 + 24749388 * q^64 - 194751312 * q^65 + 91316416 * q^66 + 100898150 * q^67 - 909131622 * q^68 - 62722704 * q^69 - 274432424 * q^71 + 548656212 * q^72 + 154939878 * q^73 + 522115380 * q^74 - 1092259130 * q^75 - 1011351754 * q^76 - 753904032 * q^78 - 180080758 * q^79 + 1083429216 * q^80 + 1454819884 * q^81 - 952261394 * q^82 + 656493222 * q^83 - 2332902778 * q^85 - 1435248312 * q^86 + 593687252 * q^87 - 116819064 * q^88 - 1143083874 * q^89 - 2030209892 * q^90 + 2939547400 * q^92 - 1119203278 * q^93 - 1235781636 * q^94 + 650723454 * q^95 + 1381595390 * q^96 + 2280214314 * q^97 - 2046116492 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(49))\) 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
49.10.a.a	$49$	$10$	49.a	1.a	$1$	$1$	$1$	$25.237$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓	✓			49.10.a.a	$2$	$0$	\(-5\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-5q^{2}-487q^{4}+4995q^{8}-3^{9}q^{9}+\cdots\)
49.10.a.b	$49$	$10$	49.a	1.a	$1$	$2$	$2$	$25.237$	\(\Q(\sqrt{193}) \)	None	✓				7.10.a.a	$1$	$0$	\(-6\)	\(86\)	\(2238\)	\(0\)		$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{2}+(43-11\beta )q^{3}+(-310+\cdots)q^{4}+\cdots\)
49.10.a.c	$49$	$10$	49.a	1.a	$1$	$3$	$3$	$25.237$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				7.10.a.b	$1$	$0$	\(21\)	\(-84\)	\(-1554\)	\(0\)		$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(7-\beta _{2})q^{2}+(-28+\beta _{1}+\beta _{2})q^{3}+\cdots\)
49.10.a.d	$49$	$10$	49.a	1.a	$1$	$4$	$4$	$25.237$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			49.10.a.d	$2$	$0$	\(-12\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2^{5}\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{3})q^{2}-\beta _{1}q^{3}+(698-6\beta _{3})q^{4}+\cdots\)
49.10.a.e	$49$	$10$	49.a	1.a	$1$	$5$	$5$	$25.237$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				7.10.c.a	$1$	$1$	\(18\)	\(-161\)	\(-1533\)	\(0\)		$+$	$+$	$2^{6}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(4-\beta _{1})q^{2}+(-33+\beta _{1}+\beta _{2})q^{3}+\cdots\)
49.10.a.f	$49$	$10$	49.a	1.a	$1$	$5$	$5$	$25.237$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		7.10.c.a	$1$	$0$	\(18\)	\(161\)	\(1533\)	\(0\)		$-$	$-$	$2^{6}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(4-\beta _{1})q^{2}+(33-\beta _{1}-\beta _{2})q^{3}+(191+\cdots)q^{4}+\cdots\)
49.10.a.g	$49$	$10$	49.a	1.a	$1$	$8$	$8$	$25.237$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		49.10.a.g	$2$	$1$	\(-66\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$2^{3}\cdot 7^{6}$	$\mathrm{SU}(2)$	\(q+(-8+\beta _{1})q^{2}+(-\beta _{3}-\beta _{5})q^{3}+(209+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(49))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(49)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)