Space of modular forms of level 7, weight 6, and trivial character

• Label: 7.6.a
• Level: $7$
• Weight: $6$
• Character orbit: 7.a
• Rep. character: $\chi_{7}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $4$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	7.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(4\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(7))\).

	Total	New	Old
Modular forms	5	3	2
Cusp forms	3	3	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
\(+\)		\(2\)	\(1\)	\(1\)		\(1\)	\(1\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)		\(3\)	\(2\)	\(1\)		\(2\)	\(2\)	\(0\)		\(1\)	\(0\)	\(1\)

Trace form

3 * q - q^2 - 20 * q^3 + 73 * q^4 - 74 * q^5 - 58 * q^6 + 49 * q^7 - 369 * q^8 + 511 * q^9 + 764 * q^10 + 628 * q^11 - 2506 * q^12 - 490 * q^13 + 931 * q^14 - 872 * q^15 + 1537 * q^16 + 78 * q^17 + 4007 * q^18 - 3364 * q^19 - 1288 * q^20 + 392 * q^21 - 4072 * q^22 + 3912 * q^23 + 3186 * q^24 - 3227 * q^25 + 3416 * q^26 - 2312 * q^27 - 3087 * q^28 + 10114 * q^29 - 14608 * q^30 - 7664 * q^31 - 8849 * q^32 + 16768 * q^33 + 23154 * q^34 + 1862 * q^35 + 7433 * q^36 - 4166 * q^37 - 14230 * q^38 - 18536 * q^39 + 23376 * q^40 - 24010 * q^41 - 16562 * q^42 + 7860 * q^43 - 15040 * q^44 + 7870 * q^45 + 7344 * q^46 + 21024 * q^47 + 21278 * q^48 + 7203 * q^49 - 19811 * q^50 + 31704 * q^51 + 21924 * q^52 + 11730 * q^53 - 78508 * q^54 - 51896 * q^55 + 17199 * q^56 + 14248 * q^57 + 3134 * q^58 - 46668 * q^59 + 55328 * q^60 - 39106 * q^61 + 91740 * q^62 + 29645 * q^63 - 89151 * q^64 + 46900 * q^65 + 99296 * q^66 - 23620 * q^67 - 132090 * q^68 - 128928 * q^69 - 17444 * q^70 + 38856 * q^71 + 25695 * q^72 + 85534 * q^73 + 147414 * q^74 + 40340 * q^75 - 19446 * q^76 + 8036 * q^77 - 92680 * q^78 + 83040 * q^79 - 150016 * q^80 - 106493 * q^81 + 121282 * q^82 + 97020 * q^83 - 29498 * q^84 + 58572 * q^85 - 258848 * q^86 - 111032 * q^87 - 124176 * q^88 + 33694 * q^89 + 67532 * q^90 - 10290 * q^91 + 274944 * q^92 + 14736 * q^93 + 66228 * q^94 + 29752 * q^95 + 293986 * q^96 - 37730 * q^97 - 2401 * q^98 - 27644 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(7))\) 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
7.6.a.a	$7$	$6$	7.a	1.a	$1$	$1$	$1$	$1.123$	\(\Q\)	None	✓	✓	✓	✓	7.6.a.a	$1$	$1$	\(-10\)	\(-14\)	\(-56\)	\(-49\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-10q^{2}-14q^{3}+68q^{4}-56q^{5}+\cdots\)
7.6.a.b	$7$	$6$	7.a	1.a	$1$	$2$	$2$	$1.123$	\(\Q(\sqrt{57}) \)	None	✓	✓	✓	✓	7.6.a.b	$1$	$0$	\(9\)	\(-6\)	\(-18\)	\(98\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(5-\beta )q^{2}+(-6+6\beta )q^{3}+(7-9\beta )q^{4}+\cdots\)