Space of modular forms of level 980, weight 4, and trivial character

• Label: 980.4.a
• Level: $980$
• Weight: $4$
• Character orbit: 980.a
• Rep. character: $\chi_{980}(1,\cdot)$
• Character field: $\Q$
• Dimension: $41$
• Newform subspaces: $23$
• Sturm bound: $672$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	980.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(672\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(980))\).

	Total	New	Old
Modular forms	528	41	487
Cusp forms	480	41	439
Eisenstein series	48	0	48

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

\(2\)	\(5\)	\(7\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	$-$	\(9\)
\(-\)	\(+\)	\(-\)	$+$	\(12\)
\(-\)	\(-\)	\(+\)	$+$	\(11\)
\(-\)	\(-\)	\(-\)	$-$	\(9\)
Plus space			\(+\)	\(23\)
Minus space			\(-\)	\(18\)

Trace form

\( 41 q - 8 q^{3} - 5 q^{5} + 409 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(980))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	7
980.4.a.a	$980$	$4$	980.a	1.a	$1$	$1$	$1$	$57.822$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-9\)	\(-5\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}-5q^{5}+54q^{9}+55q^{11}+69q^{13}+\cdots\)
980.4.a.b	$980$	$4$	980.a	1.a	$1$	$1$	$1$	$57.822$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-8\)	\(5\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}+5q^{5}+37q^{9}+28q^{11}-82q^{13}+\cdots\)
980.4.a.c	$980$	$4$	980.a	1.a	$1$	$1$	$1$	$57.822$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-4\)	\(-5\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-5q^{5}-11q^{9}-60q^{11}-86q^{13}+\cdots\)
980.4.a.d	$980$	$4$	980.a	1.a	$1$	$1$	$1$	$57.822$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-5\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-5q^{5}-23q^{9}+3^{3}q^{11}+41q^{13}+\cdots\)
980.4.a.e	$980$	$4$	980.a	1.a	$1$	$1$	$1$	$57.822$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(5\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+5q^{5}-23q^{9}-69q^{11}+11q^{13}+\cdots\)
980.4.a.f	$980$	$4$	980.a	1.a	$1$	$1$	$1$	$57.822$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(5\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+5q^{5}-26q^{9}-21q^{11}+9q^{13}+\cdots\)
980.4.a.g	$980$	$4$	980.a	1.a	$1$	$1$	$1$	$57.822$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(5\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+5q^{5}-26q^{9}-7q^{11}+23q^{13}+\cdots\)
980.4.a.h	$980$	$4$	980.a	1.a	$1$	$1$	$1$	$57.822$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(-5\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-5q^{5}-26q^{9}-21q^{11}-9q^{13}+\cdots\)
980.4.a.i	$980$	$4$	980.a	1.a	$1$	$1$	$1$	$57.822$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-5\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-5q^{5}-23q^{9}-69q^{11}-11q^{13}+\cdots\)
980.4.a.j	$980$	$4$	980.a	1.a	$1$	$1$	$1$	$57.822$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(5\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+5q^{5}-23q^{9}+3^{3}q^{11}-41q^{13}+\cdots\)
980.4.a.k	$980$	$4$	980.a	1.a	$1$	$1$	$1$	$57.822$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(4\)	\(-5\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}-5q^{5}-11q^{9}+68q^{11}-22q^{13}+\cdots\)
980.4.a.l	$980$	$4$	980.a	1.a	$1$	$1$	$1$	$57.822$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(5\)	\(-5\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-5q^{5}-2q^{9}-15q^{11}+13q^{13}+\cdots\)
980.4.a.m	$980$	$4$	980.a	1.a	$1$	$1$	$1$	$57.822$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(5\)	\(5\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}+5q^{5}-2q^{9}+15q^{11}-17q^{13}+\cdots\)
980.4.a.n	$980$	$4$	980.a	1.a	$1$	$2$	$2$	$57.822$	\(\Q(\sqrt{22}) \)	None	✓	$1$	$0$	\(0\)	\(-10\)	\(-10\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-5+\beta )q^{3}-5q^{5}+(20-10\beta )q^{9}+\cdots\)
980.4.a.o	$980$	$4$	980.a	1.a	$1$	$2$	$2$	$57.822$	\(\Q(\sqrt{46}) \)	None	✓	$1$	$0$	\(0\)	\(-6\)	\(10\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-3+\beta )q^{3}+5q^{5}+(28-6\beta )q^{9}+\cdots\)
980.4.a.p	$980$	$4$	980.a	1.a	$1$	$2$	$2$	$57.822$	\(\Q(\sqrt{249}) \)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(10\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+5q^{5}+(35+\beta )q^{9}+(-26+\cdots)q^{11}+\cdots\)
980.4.a.q	$980$	$4$	980.a	1.a	$1$	$2$	$2$	$57.822$	\(\Q(\sqrt{37}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(0\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-5q^{5}+10q^{9}+(-24-2\beta )q^{11}+\cdots\)
980.4.a.r	$980$	$4$	980.a	1.a	$1$	$2$	$2$	$57.822$	\(\Q(\sqrt{37}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(10\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+5q^{5}+10q^{9}+(-24+2\beta )q^{11}+\cdots\)
980.4.a.s	$980$	$4$	980.a	1.a	$1$	$2$	$2$	$57.822$	\(\Q(\sqrt{249}) \)	None	✓	$1$	$0$	\(0\)	\(1\)	\(-10\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-5q^{5}+(35+\beta )q^{9}+(-26+\cdots)q^{11}+\cdots\)
980.4.a.t	$980$	$4$	980.a	1.a	$1$	$2$	$2$	$57.822$	\(\Q(\sqrt{46}) \)	None	✓	$1$	$0$	\(0\)	\(6\)	\(-10\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{3}-5q^{5}+(28+6\beta )q^{9}+(10+\cdots)q^{11}+\cdots\)
980.4.a.u	$980$	$4$	980.a	1.a	$1$	$2$	$2$	$57.822$	\(\Q(\sqrt{22}) \)	None	✓	$1$	$0$	\(0\)	\(10\)	\(10\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(5+\beta )q^{3}+5q^{5}+(20+10\beta )q^{9}+\cdots\)
980.4.a.v	$980$	$4$	980.a	1.a	$1$	$6$	$6$	$57.822$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-4\)	\(30\)	\(0\)		$-$	$-$	$+$	$+$	$2^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+5q^{5}+(12-\beta _{2}+\beta _{4}+\cdots)q^{9}+\cdots\)
980.4.a.w	$980$	$4$	980.a	1.a	$1$	$6$	$6$	$57.822$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(4\)	\(-30\)	\(0\)		$-$	$+$	$+$	$-$	$2^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}-5q^{5}+(12-\beta _{2}+\beta _{4}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(980))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(980)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(490))\)\(^{\oplus 2}\)