Space of modular forms of level 55, weight 6, and trivial character

• Label: 55.6.a
• Level: $55$
• Weight: $6$
• Character orbit: 55.a
• Rep. character: $\chi_{55}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $4$
• Sturm bound: $36$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 55 = 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	55.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(36\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(55))\).

	Total	New	Old
Modular forms	32	18	14
Cusp forms	28	18	10
Eisenstein series	4	0	4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(5\)	\(11\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(6\)	\(4\)	\(2\)		\(5\)	\(4\)	\(1\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(10\)	\(6\)	\(4\)		\(9\)	\(6\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(8\)	\(5\)	\(3\)		\(7\)	\(5\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(8\)	\(3\)	\(5\)		\(7\)	\(3\)	\(4\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(14\)	\(7\)	\(7\)		\(12\)	\(7\)	\(5\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(18\)	\(11\)	\(7\)		\(16\)	\(11\)	\(5\)		\(2\)	\(0\)	\(2\)

Trace form

18 * q - 36 * q^3 + 332 * q^4 - 50 * q^5 + 32 * q^6 - 188 * q^7 + 852 * q^8 + 2162 * q^9 + 100 * q^10 - 2336 * q^12 - 300 * q^13 + 1480 * q^14 - 900 * q^15 + 6200 * q^16 - 956 * q^17 + 7072 * q^18 - 1912 * q^19 - 500 * q^20 + 128 * q^21 - 968 * q^22 - 5140 * q^23 + 8104 * q^24 + 11250 * q^25 - 18764 * q^26 - 4200 * q^27 - 18180 * q^28 + 13004 * q^29 - 7600 * q^30 + 12280 * q^31 + 46104 * q^32 - 4356 * q^33 - 34632 * q^34 + 3100 * q^35 + 2800 * q^36 - 29116 * q^37 + 6504 * q^38 - 24104 * q^39 + 15600 * q^40 - 22268 * q^41 - 10360 * q^42 + 20172 * q^43 - 2420 * q^44 + 11350 * q^45 + 47384 * q^46 - 3580 * q^47 - 115296 * q^48 + 3906 * q^49 - 20248 * q^51 - 99828 * q^52 - 18396 * q^53 - 72216 * q^54 - 12100 * q^55 + 81528 * q^56 + 52400 * q^57 + 41856 * q^58 + 37920 * q^59 - 83700 * q^60 - 138532 * q^61 - 167776 * q^62 + 274324 * q^63 + 258996 * q^64 + 100 * q^65 + 99220 * q^66 - 44 * q^67 - 142732 * q^68 + 83040 * q^69 + 20500 * q^70 + 101600 * q^71 + 155076 * q^72 + 223908 * q^73 - 53568 * q^74 - 22500 * q^75 - 10048 * q^76 - 17908 * q^77 - 59528 * q^78 - 195328 * q^79 - 45200 * q^80 + 189282 * q^81 + 343952 * q^82 + 328860 * q^83 - 535856 * q^84 + 130500 * q^85 - 59708 * q^86 + 11048 * q^87 - 46464 * q^88 + 118804 * q^89 - 38700 * q^90 + 185440 * q^91 + 314328 * q^92 - 434112 * q^93 + 218032 * q^94 + 91400 * q^95 - 691192 * q^96 + 30852 * q^97 - 476136 * q^98

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(55))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	11
55.6.a.a	$55$	$6$	55.a	1.a	$1$	$3$	$3$	$8.821$	3.3.21865.1	None	✓	✓	✓	✓	55.6.a.a	$1$	$1$	\(-7\)	\(-36\)	\(75\)	\(-102\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{2})q^{2}+(-11-3\beta _{1})q^{3}+\cdots\)
55.6.a.b	$55$	$6$	55.a	1.a	$1$	$4$	$4$	$8.821$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	55.6.a.b	$1$	$1$	\(-5\)	\(0\)	\(-100\)	\(-90\)		$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(\beta _{1}-\beta _{3})q^{3}+(15+\cdots)q^{4}+\cdots\)
55.6.a.c	$55$	$6$	55.a	1.a	$1$	$5$	$5$	$8.821$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	55.6.a.c	$1$	$0$	\(9\)	\(0\)	\(125\)	\(70\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+(\beta _{1}-\beta _{3})q^{3}+(24-3\beta _{1}+\cdots)q^{4}+\cdots\)
55.6.a.d	$55$	$6$	55.a	1.a	$1$	$6$	$6$	$8.821$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	55.6.a.d	$1$	$0$	\(3\)	\(0\)	\(-150\)	\(-66\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(-\beta _{1}-\beta _{3})q^{3}+(20+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(55))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(55)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)