Space of modular forms of level 880, weight 4, and trivial character

• Label: 880.4.a
• Level: $880$
• Weight: $4$
• Character orbit: 880.a
• Rep. character: $\chi_{880}(1,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $30$
• Sturm bound: $576$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	880.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 30 \)
Sturm bound:			\(576\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(880))\).

	Total	New	Old
Modular forms	444	60	384
Cusp forms	420	60	360
Eisenstein series	24	0	24

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	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)	\(7\)
\(+\)	\(-\)	\(+\)	\(-\)	\(8\)
\(+\)	\(-\)	\(-\)	\(+\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)	\(7\)
\(-\)	\(+\)	\(-\)	\(+\)	\(9\)
\(-\)	\(-\)	\(+\)	\(+\)	\(8\)
\(-\)	\(-\)	\(-\)	\(-\)	\(6\)
Plus space			\(+\)	\(32\)
Minus space			\(-\)	\(28\)

Trace form

\( 60 q + 12 q^{3} - 72 q^{7} + 540 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(880))\) 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
880.4.a.a	$880$	$4$	880.a	1.a	$1$	$1$	$1$	$51.922$	\(\Q\)	None	✓				220.4.a.c	$1$	$1$	\(0\)	\(-8\)	\(5\)	\(-24\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}+5q^{5}-24q^{7}+37q^{9}+11q^{11}+\cdots\)
880.4.a.b	$880$	$4$	880.a	1.a	$1$	$1$	$1$	$51.922$	\(\Q\)	None	✓				110.4.a.h	$1$	$1$	\(0\)	\(-8\)	\(5\)	\(12\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}+5q^{5}+12q^{7}+37q^{9}+11q^{11}+\cdots\)
880.4.a.c	$880$	$4$	880.a	1.a	$1$	$1$	$1$	$51.922$	\(\Q\)	None	✓				110.4.a.g	$1$	$1$	\(0\)	\(-7\)	\(-5\)	\(-11\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}-5q^{5}-11q^{7}+22q^{9}-11q^{11}+\cdots\)
880.4.a.d	$880$	$4$	880.a	1.a	$1$	$1$	$1$	$51.922$	\(\Q\)	None	✓				440.4.a.d	$1$	$1$	\(0\)	\(-6\)	\(5\)	\(32\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{3}+5q^{5}+2^{5}q^{7}+9q^{9}-11q^{11}+\cdots\)
880.4.a.e	$880$	$4$	880.a	1.a	$1$	$1$	$1$	$51.922$	\(\Q\)	None	✓				220.4.a.b	$1$	$0$	\(0\)	\(-5\)	\(-5\)	\(19\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}-5q^{5}+19q^{7}-2q^{9}+11q^{11}+\cdots\)
880.4.a.f	$880$	$4$	880.a	1.a	$1$	$1$	$1$	$51.922$	\(\Q\)	None	✓				110.4.a.b	$1$	$1$	\(0\)	\(-4\)	\(-5\)	\(30\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-5q^{5}+30q^{7}-11q^{9}-11q^{11}+\cdots\)
880.4.a.g	$880$	$4$	880.a	1.a	$1$	$1$	$1$	$51.922$	\(\Q\)	None	✓				110.4.a.c	$1$	$0$	\(0\)	\(-4\)	\(5\)	\(-20\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}+5q^{5}-20q^{7}-11q^{9}-11q^{11}+\cdots\)
880.4.a.h	$880$	$4$	880.a	1.a	$1$	$1$	$1$	$51.922$	\(\Q\)	None	✓				110.4.a.f	$1$	$1$	\(0\)	\(-1\)	\(5\)	\(-23\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+5q^{5}-23q^{7}-26q^{9}+11q^{11}+\cdots\)
880.4.a.i	$880$	$4$	880.a	1.a	$1$	$1$	$1$	$51.922$	\(\Q\)	None	✓				440.4.a.c	$1$	$0$	\(0\)	\(0\)	\(5\)	\(-16\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-2^{4}q^{7}-3^{3}q^{9}+11q^{11}+\cdots\)
880.4.a.j	$880$	$4$	880.a	1.a	$1$	$1$	$1$	$51.922$	\(\Q\)	None	✓				55.4.a.a	$1$	$1$	\(0\)	\(3\)	\(-5\)	\(9\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+9q^{7}-18q^{9}-11q^{11}+\cdots\)
880.4.a.k	$880$	$4$	880.a	1.a	$1$	$1$	$1$	$51.922$	\(\Q\)	None	✓				110.4.a.e	$1$	$0$	\(0\)	\(4\)	\(-5\)	\(22\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}-5q^{5}+22q^{7}-11q^{9}+11q^{11}+\cdots\)
880.4.a.l	$880$	$4$	880.a	1.a	$1$	$1$	$1$	$51.922$	\(\Q\)	None	✓				440.4.a.b	$1$	$1$	\(0\)	\(4\)	\(5\)	\(-8\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}+5q^{5}-8q^{7}-11q^{9}-11q^{11}+\cdots\)
880.4.a.m	$880$	$4$	880.a	1.a	$1$	$1$	$1$	$51.922$	\(\Q\)	None	✓				220.4.a.a	$1$	$1$	\(0\)	\(5\)	\(5\)	\(-11\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}+5q^{5}-11q^{7}-2q^{9}+11q^{11}+\cdots\)
880.4.a.n	$880$	$4$	880.a	1.a	$1$	$1$	$1$	$51.922$	\(\Q\)	None	✓				440.4.a.a	$1$	$1$	\(0\)	\(5\)	\(5\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}+5q^{5}-q^{7}-2q^{9}-11q^{11}+\cdots\)
880.4.a.o	$880$	$4$	880.a	1.a	$1$	$1$	$1$	$51.922$	\(\Q\)	None	✓				110.4.a.a	$1$	$0$	\(0\)	\(7\)	\(5\)	\(35\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}+5q^{5}+35q^{7}+22q^{9}-11q^{11}+\cdots\)
880.4.a.p	$880$	$4$	880.a	1.a	$1$	$1$	$1$	$51.922$	\(\Q\)	None	✓				110.4.a.d	$1$	$1$	\(0\)	\(8\)	\(-5\)	\(-26\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}-5q^{5}-26q^{7}+37q^{9}-11q^{11}+\cdots\)
880.4.a.q	$880$	$4$	880.a	1.a	$1$	$2$	$2$	$51.922$	\(\Q(\sqrt{177}) \)	None	✓				110.4.a.i	$1$	$0$	\(0\)	\(-7\)	\(-10\)	\(-27\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{3}-5q^{5}+(-13-\beta )q^{7}+\cdots\)
880.4.a.r	$880$	$4$	880.a	1.a	$1$	$2$	$2$	$51.922$	\(\Q(\sqrt{17}) \)	None	✓		✓		55.4.a.b	$1$	$1$	\(0\)	\(3\)	\(10\)	\(25\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+5q^{5}+(17-9\beta )q^{7}+(-22+\cdots)q^{9}+\cdots\)
880.4.a.s	$880$	$4$	880.a	1.a	$1$	$2$	$2$	$51.922$	\(\Q(\sqrt{145}) \)	None	✓				440.4.a.e	$1$	$0$	\(0\)	\(3\)	\(10\)	\(33\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+5q^{5}+(17-\beta )q^{7}+(10+\cdots)q^{9}+\cdots\)
880.4.a.t	$880$	$4$	880.a	1.a	$1$	$2$	$2$	$51.922$	\(\Q(\sqrt{6}) \)	None	✓				220.4.a.e	$1$	$0$	\(0\)	\(8\)	\(-10\)	\(-36\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{3}-5q^{5}+(-18-2\beta )q^{7}+\cdots\)
880.4.a.u	$880$	$4$	880.a	1.a	$1$	$2$	$2$	$51.922$	\(\Q(\sqrt{97}) \)	None	✓				220.4.a.d	$1$	$0$	\(0\)	\(9\)	\(10\)	\(15\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(5-\beta )q^{3}+5q^{5}+(7+\beta )q^{7}+(22+\cdots)q^{9}+\cdots\)
880.4.a.v	$880$	$4$	880.a	1.a	$1$	$3$	$3$	$51.922$	3.3.12188.1	None	✓				440.4.a.g	$1$	$1$	\(0\)	\(-1\)	\(-15\)	\(-9\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-5q^{5}+(-4+2\beta _{1}+\beta _{2})q^{7}+\cdots\)
880.4.a.w	$880$	$4$	880.a	1.a	$1$	$3$	$3$	$51.922$	3.3.9192.1	None	✓		✓		220.4.a.f	$1$	$1$	\(0\)	\(3\)	\(-15\)	\(5\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2})q^{3}-5q^{5}+(2+\beta _{1}-2\beta _{2})q^{7}+\cdots\)
880.4.a.x	$880$	$4$	880.a	1.a	$1$	$3$	$3$	$51.922$	3.3.568.1	None	✓		✓		55.4.a.c	$1$	$0$	\(0\)	\(3\)	\(-15\)	\(15\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2+3\beta _{2})q^{3}-5q^{5}+(8-7\beta _{1}+2\beta _{2})q^{7}+\cdots\)
880.4.a.y	$880$	$4$	880.a	1.a	$1$	$3$	$3$	$51.922$	3.3.9676.1	None	✓				440.4.a.f	$1$	$0$	\(0\)	\(7\)	\(-15\)	\(-11\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta _{2})q^{3}-5q^{5}+(-4-\beta _{1}+2\beta _{2})q^{7}+\cdots\)
880.4.a.z	$880$	$4$	880.a	1.a	$1$	$4$	$4$	$51.922$	4.4.1539480.1	None	✓		✓		55.4.a.d	$1$	$0$	\(0\)	\(-9\)	\(20\)	\(-9\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{2})q^{3}+5q^{5}+(-3+\beta _{1}+\cdots)q^{7}+\cdots\)
880.4.a.ba	$880$	$4$	880.a	1.a	$1$	$4$	$4$	$51.922$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		440.4.a.i	$1$	$0$	\(0\)	\(-4\)	\(-20\)	\(-14\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}-5q^{5}+(-3+\beta _{1}+\cdots)q^{7}+\cdots\)
880.4.a.bb	$880$	$4$	880.a	1.a	$1$	$4$	$4$	$51.922$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		440.4.a.h	$1$	$1$	\(0\)	\(4\)	\(-20\)	\(-2\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}-5q^{5}+(-1-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
880.4.a.bc	$880$	$4$	880.a	1.a	$1$	$5$	$5$	$51.922$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		440.4.a.k	$1$	$1$	\(0\)	\(-6\)	\(25\)	\(-62\)		$+$	$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+5q^{5}+(-13+\beta _{1}+\cdots)q^{7}+\cdots\)
880.4.a.bd	$880$	$4$	880.a	1.a	$1$	$5$	$5$	$51.922$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		440.4.a.j	$1$	$0$	\(0\)	\(6\)	\(25\)	\(-14\)		$+$	$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+5q^{5}+(-3+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(880))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(880)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(440))\)\(^{\oplus 2}\)