Space of modular forms of level 55, weight 4, and trivial character

• Label: 55.4.a
• Level: $55$
• Weight: $4$
• Character orbit: 55.a
• Rep. character: $\chi_{55}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $4$
• Sturm bound: $24$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 55 = 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	55.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(24\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(55))\).

	Total	New	Old
Modular forms	20	10	10
Cusp forms	16	10	6
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
\(+\)	\(+\)	\(+\)		\(7\)	\(3\)	\(4\)		\(6\)	\(3\)	\(3\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(5\)	\(2\)	\(3\)		\(4\)	\(2\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(5\)	\(4\)	\(1\)		\(4\)	\(4\)	\(0\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(12\)	\(7\)	\(5\)		\(10\)	\(7\)	\(3\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(8\)	\(3\)	\(5\)		\(6\)	\(3\)	\(3\)		\(2\)	\(0\)	\(2\)

Trace form

10 * q + 52 * q^4 + 10 * q^5 + 56 * q^6 - 40 * q^7 - 12 * q^8 + 62 * q^9 - 60 * q^10 + 16 * q^12 + 8 * q^13 + 64 * q^14 + 60 * q^15 + 152 * q^16 + 64 * q^17 - 512 * q^18 - 312 * q^19 + 100 * q^20 - 232 * q^21 + 44 * q^22 + 344 * q^23 - 248 * q^24 + 250 * q^25 - 212 * q^26 + 264 * q^27 + 28 * q^28 - 52 * q^29 - 280 * q^30 - 20 * q^31 - 936 * q^32 + 132 * q^33 + 232 * q^34 + 40 * q^35 + 136 * q^36 - 812 * q^37 + 984 * q^38 + 208 * q^39 - 240 * q^40 + 172 * q^41 - 232 * q^42 + 312 * q^43 - 308 * q^44 - 230 * q^45 - 176 * q^46 + 968 * q^47 - 1536 * q^48 + 1382 * q^49 + 1712 * q^51 + 1100 * q^52 + 1380 * q^53 - 504 * q^54 + 220 * q^55 - 552 * q^56 + 1664 * q^57 + 2016 * q^58 - 1848 * q^59 - 180 * q^60 + 100 * q^61 + 1232 * q^62 - 2360 * q^63 + 1524 * q^64 + 160 * q^65 - 1100 * q^66 - 56 * q^67 - 2572 * q^68 - 2160 * q^69 - 700 * q^70 + 308 * q^71 - 444 * q^72 - 1456 * q^73 + 4272 * q^74 - 5152 * q^76 + 440 * q^77 - 1160 * q^78 - 1264 * q^79 + 1840 * q^80 + 3186 * q^81 - 3440 * q^82 - 168 * q^83 - 2096 * q^84 - 800 * q^85 + 1156 * q^86 - 4816 * q^87 + 528 * q^88 + 640 * q^89 + 180 * q^90 - 4968 * q^91 + 3528 * q^92 + 768 * q^93 + 168 * q^94 - 520 * q^95 + 3464 * q^96 - 3388 * q^97 + 4920 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(55))\) 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
55.4.a.a	$55$	$4$	55.a	1.a	$1$	$1$	$1$	$3.245$	\(\Q\)	None	✓	✓	✓	✓	55.4.a.a	$1$	$1$	\(1\)	\(-3\)	\(-5\)	\(-9\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}-7q^{4}-5q^{5}-3q^{6}+\cdots\)
55.4.a.b	$55$	$4$	55.a	1.a	$1$	$2$	$2$	$3.245$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	55.4.a.b	$1$	$1$	\(-7\)	\(-3\)	\(10\)	\(-25\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{2}+(-1-\beta )q^{3}+(5+7\beta )q^{4}+\cdots\)
55.4.a.c	$55$	$4$	55.a	1.a	$1$	$3$	$3$	$3.245$	3.3.568.1	None	✓	✓	✓	✓	55.4.a.c	$1$	$0$	\(5\)	\(-3\)	\(-15\)	\(-15\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1}-\beta _{2})q^{2}+(-2-3\beta _{2})q^{3}+\cdots\)
55.4.a.d	$55$	$4$	55.a	1.a	$1$	$4$	$4$	$3.245$	4.4.1539480.1	None	✓	✓	✓	✓	55.4.a.d	$1$	$0$	\(1\)	\(9\)	\(20\)	\(9\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2-\beta _{2})q^{3}+(5+\beta _{1}+\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(55))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(55)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)