Space of modular forms of level 7, weight 10, and trivial character

• Label: 7.10.a
• Level: $7$
• Weight: $10$
• Character orbit: 7.a
• Rep. character: $\chi_{7}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $2$
• Sturm bound: $6$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	7.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(6\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(7))\).

	Total	New	Old
Modular forms	7	5	2
Cusp forms	5	5	0
Eisenstein series	2	0	2

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
\(+\)		\(3\)	\(2\)	\(1\)		\(2\)	\(2\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)		\(4\)	\(3\)	\(1\)		\(3\)	\(3\)	\(0\)		\(1\)	\(0\)	\(1\)

Trace form

5 * q + 15 * q^2 - 2 * q^3 + 937 * q^4 - 684 * q^5 + 926 * q^6 + 2401 * q^7 + 16671 * q^8 - 14963 * q^9 - 54476 * q^10 + 31872 * q^11 - 54250 * q^12 - 46312 * q^13 + 64827 * q^14 - 106832 * q^15 + 482209 * q^16 + 552774 * q^17 + 58775 * q^18 - 702574 * q^19 - 1448328 * q^20 + 408170 * q^21 - 1669160 * q^22 + 2663760 * q^23 + 1851666 * q^24 + 5154943 * q^25 - 12912480 * q^26 - 2246732 * q^27 + 5226977 * q^28 - 5921766 * q^29 + 12149072 * q^30 + 5336700 * q^31 + 26506527 * q^32 - 35900576 * q^33 - 6228318 * q^34 + 9104592 * q^35 - 41009095 * q^36 + 32131170 * q^37 + 46984530 * q^38 + 38427256 * q^39 - 115269264 * q^40 - 33524106 * q^41 + 21373702 * q^42 - 57566072 * q^43 + 78361008 * q^44 + 52112980 * q^45 + 102115248 * q^46 - 92563980 * q^47 - 12007714 * q^48 + 28824005 * q^49 - 14987571 * q^50 + 33824628 * q^51 - 47441660 * q^52 + 12312798 * q^53 - 65044828 * q^54 - 23377216 * q^55 + 27465039 * q^56 + 40252228 * q^57 - 92605826 * q^58 - 49659318 * q^59 - 103385632 * q^60 + 236063228 * q^61 + 7926732 * q^62 - 88930639 * q^63 + 155269313 * q^64 - 5696040 * q^65 - 91316416 * q^66 - 497097124 * q^67 + 909131622 * q^68 + 62722704 * q^69 - 339126844 * q^70 + 503008320 * q^71 - 888461985 * q^72 - 154939878 * q^73 - 687124938 * q^74 + 1092259130 * q^75 + 1011351754 * q^76 - 93062760 * q^77 + 375103736 * q^78 - 491877560 * q^79 - 1083429216 * q^80 - 621573311 * q^81 + 952261394 * q^82 - 656493222 * q^83 - 380611322 * q^84 + 611654472 * q^85 - 1297812864 * q^86 - 593687252 * q^87 + 220113984 * q^88 + 1143083874 * q^89 + 2030209892 * q^90 + 16201948 * q^91 + 361692000 * q^92 + 901734552 * q^93 + 1235781636 * q^94 - 1098323088 * q^95 - 1381595390 * q^96 - 2280214314 * q^97 + 86472015 * q^98 - 491561312 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(7))\) 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
7.10.a.a	$7$	$10$	7.a	1.a	$1$	$2$	$2$	$3.605$	\(\Q(\sqrt{193}) \)	None	✓	✓	✓	✓	7.10.a.a	$1$	$1$	\(-6\)	\(-86\)	\(-2238\)	\(-4802\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{2}+(-43+11\beta )q^{3}+(-310+\cdots)q^{4}+\cdots\)
7.10.a.b	$7$	$10$	7.a	1.a	$1$	$3$	$3$	$3.605$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	7.10.a.b	$1$	$0$	\(21\)	\(84\)	\(1554\)	\(7203\)		$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(7-\beta _{2})q^{2}+(28-\beta _{1}-\beta _{2})q^{3}+(519+\cdots)q^{4}+\cdots\)