Space of modular forms of level 220, weight 4, and trivial character

• Label: 220.4.a
• Level: $220$
• Weight: $4$
• Character orbit: 220.a
• Rep. character: $\chi_{220}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $6$
• Sturm bound: $144$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	220.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(144\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(220))\).

	Total	New	Old
Modular forms	114	10	104
Cusp forms	102	10	92
Eisenstein series	12	0	12

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

\(2\)	\(5\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(18\)	\(0\)	\(18\)		\(16\)	\(0\)	\(16\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(11\)	\(0\)	\(11\)		\(9\)	\(0\)	\(9\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(14\)	\(0\)	\(14\)		\(12\)	\(0\)	\(12\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(15\)	\(0\)	\(15\)		\(13\)	\(0\)	\(13\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(15\)	\(3\)	\(12\)		\(14\)	\(3\)	\(11\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(13\)	\(3\)	\(10\)		\(12\)	\(3\)	\(9\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(13\)	\(2\)	\(11\)		\(12\)	\(2\)	\(10\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(15\)	\(2\)	\(13\)		\(14\)	\(2\)	\(12\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(59\)	\(5\)	\(54\)		\(53\)	\(5\)	\(48\)		\(6\)	\(0\)	\(6\)
Minus space			\(-\)		\(55\)	\(5\)	\(50\)		\(49\)	\(5\)	\(44\)		\(6\)	\(0\)	\(6\)

Trace form

10 * q - 12 * q^3 - 10 * q^5 + 32 * q^7 + 118 * q^9 - 136 * q^13 + 64 * q^17 + 24 * q^19 + 80 * q^21 - 84 * q^23 + 250 * q^25 - 432 * q^27 + 60 * q^29 - 196 * q^31 - 132 * q^33 + 40 * q^35 - 436 * q^37 - 32 * q^39 + 596 * q^41 - 264 * q^43 + 110 * q^45 - 76 * q^47 - 610 * q^49 + 576 * q^51 + 284 * q^53 - 976 * q^57 - 1032 * q^59 + 1420 * q^61 + 2048 * q^63 + 160 * q^65 + 1212 * q^67 + 1800 * q^69 - 1140 * q^71 + 544 * q^73 - 300 * q^75 - 792 * q^77 - 3440 * q^79 + 154 * q^81 - 440 * q^83 - 560 * q^85 + 2840 * q^87 + 224 * q^89 + 2120 * q^91 - 2224 * q^93 - 440 * q^95 - 2796 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(220))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5	11
220.4.a.a	$220$	$4$	220.a	1.a	$1$	$1$	$1$	$12.980$	\(\Q\)	None	✓	✓			220.4.a.a	$1$	$0$	\(0\)	\(-5\)	\(5\)	\(11\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}+5q^{5}+11q^{7}-2q^{9}-11q^{11}+\cdots\)
220.4.a.b	$220$	$4$	220.a	1.a	$1$	$1$	$1$	$12.980$	\(\Q\)	None	✓	✓			220.4.a.b	$1$	$1$	\(0\)	\(5\)	\(-5\)	\(-19\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-5q^{5}-19q^{7}-2q^{9}-11q^{11}+\cdots\)
220.4.a.c	$220$	$4$	220.a	1.a	$1$	$1$	$1$	$12.980$	\(\Q\)	None	✓	✓			220.4.a.c	$1$	$0$	\(0\)	\(8\)	\(5\)	\(24\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}+5q^{5}+24q^{7}+37q^{9}-11q^{11}+\cdots\)
220.4.a.d	$220$	$4$	220.a	1.a	$1$	$2$	$2$	$12.980$	\(\Q(\sqrt{97}) \)	None	✓	✓	✓	✓	220.4.a.d	$1$	$1$	\(0\)	\(-9\)	\(10\)	\(-15\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{3}+5q^{5}+(-8+\beta )q^{7}+\cdots\)
220.4.a.e	$220$	$4$	220.a	1.a	$1$	$2$	$2$	$12.980$	\(\Q(\sqrt{6}) \)	None	✓	✓	✓		220.4.a.e	$1$	$1$	\(0\)	\(-8\)	\(-10\)	\(36\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-4+\beta )q^{3}-5q^{5}+(18-2\beta )q^{7}+\cdots\)
220.4.a.f	$220$	$4$	220.a	1.a	$1$	$3$	$3$	$12.980$	3.3.9192.1	None	✓	✓	✓	✓	220.4.a.f	$1$	$0$	\(0\)	\(-3\)	\(-15\)	\(-5\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{3}-5q^{5}+(-2-\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(220))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(220)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 2}\)