Space of modular forms of level 49, weight 8, and trivial character

• Label: 49.8.a
• Level: $49$
• Weight: $8$
• Character orbit: 49.a
• Rep. character: $\chi_{49}(1,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $7$
• Sturm bound: $37$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	49.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(37\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(49))\).

	Total	New	Old
Modular forms	36	27	9
Cusp forms	28	22	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.

\(7\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(19\)	\(14\)	\(5\)		\(15\)	\(12\)	\(3\)		\(4\)	\(2\)	\(2\)
\(-\)		\(17\)	\(13\)	\(4\)		\(13\)	\(10\)	\(3\)		\(4\)	\(3\)	\(1\)

Trace form

22 * q + 16 * q^2 - 52 * q^3 + 1228 * q^4 - 246 * q^5 + 754 * q^6 + 468 * q^8 + 11926 * q^9 + 4316 * q^10 + 6280 * q^11 - 14966 * q^12 + 2618 * q^13 - 25888 * q^15 + 72500 * q^16 + 15474 * q^17 + 131492 * q^18 - 88180 * q^19 - 50568 * q^20 - 3788 * q^22 + 42632 * q^23 + 82590 * q^24 + 416086 * q^25 - 408576 * q^26 - 150040 * q^27 - 105712 * q^29 + 116796 * q^30 + 241504 * q^31 + 375500 * q^32 - 267184 * q^33 + 53982 * q^34 + 53212 * q^36 + 491192 * q^37 + 10350 * q^38 + 182056 * q^39 + 163680 * q^40 - 224406 * q^41 - 637376 * q^43 + 20648 * q^44 - 1141342 * q^45 - 1636140 * q^46 + 1608144 * q^47 + 1700578 * q^48 + 2217196 * q^50 + 68720 * q^51 + 441084 * q^52 - 4901812 * q^53 + 3415420 * q^54 - 435272 * q^55 - 1238904 * q^57 - 9385344 * q^58 - 2139420 * q^59 - 5908112 * q^60 + 3853394 * q^61 + 1398948 * q^62 + 1769100 * q^64 - 606564 * q^65 + 325168 * q^66 + 684832 * q^67 - 1572438 * q^68 - 3565872 * q^69 + 5742424 * q^71 - 4618764 * q^72 - 2480222 * q^73 + 7854852 * q^74 - 3087548 * q^75 + 3237990 * q^76 - 13381200 * q^78 + 19702496 * q^79 + 5520624 * q^80 + 6402454 * q^81 - 12216722 * q^82 - 12355812 * q^83 - 2219044 * q^85 + 12083208 * q^86 + 8891240 * q^87 + 36314712 * q^88 - 15330750 * q^89 + 19124332 * q^90 - 9789896 * q^92 + 45554888 * q^93 + 11211372 * q^94 + 5485008 * q^95 - 26043346 * q^96 - 20639486 * q^97 + 37533568 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(49))\) 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
49.8.a.a	$49$	$8$	49.a	1.a	$1$	$1$	$1$	$15.307$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓	✓			49.8.a.a	$2$	$1$	\(-13\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-13q^{2}+41q^{4}+1131q^{8}-3^{7}q^{9}+\cdots\)
49.8.a.b	$49$	$8$	49.a	1.a	$1$	$1$	$1$	$15.307$	\(\Q\)	None	✓				7.8.a.a	$1$	$1$	\(-6\)	\(42\)	\(84\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{2}+42q^{3}-92q^{4}+84q^{5}-252q^{6}+\cdots\)
49.8.a.c	$49$	$8$	49.a	1.a	$1$	$2$	$2$	$15.307$	\(\Q(\sqrt{865}) \)	None	✓				7.8.a.b	$1$	$1$	\(-3\)	\(-94\)	\(-330\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(-46-2\beta )q^{3}+(89+\cdots)q^{4}+\cdots\)
49.8.a.d	$49$	$8$	49.a	1.a	$1$	$2$	$2$	$15.307$	\(\Q(\sqrt{690}) \)	None	✓	✓			49.8.a.d	$2$	$1$	\(20\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+10q^{2}-\beta q^{3}-28q^{4}+9\beta q^{5}-10\beta q^{6}+\cdots\)
49.8.a.e	$49$	$8$	49.a	1.a	$1$	$4$	$4$	$15.307$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		7.8.c.a	$1$	$1$	\(-6\)	\(-28\)	\(-252\)	\(0\)		$-$	$-$	$2^{4}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-8-\beta _{1}+\beta _{2})q^{3}+\cdots\)
49.8.a.f	$49$	$8$	49.a	1.a	$1$	$4$	$4$	$15.307$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				7.8.c.a	$1$	$0$	\(-6\)	\(28\)	\(252\)	\(0\)		$+$	$+$	$2^{4}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(8+\beta _{1}-\beta _{2})q^{3}+\cdots\)
49.8.a.g	$49$	$8$	49.a	1.a	$1$	$8$	$8$	$15.307$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		49.8.a.g	$2$	$0$	\(30\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$2^{6}\cdot 7^{6}$	$\mathrm{SU}(2)$	\(q+(4+\beta _{2})q^{2}+\beta _{6}q^{3}+(59+5\beta _{2}+\beta _{5}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(49))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(49)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)