Space of modular forms of level 980, weight 4, and Character 980.i

• Label: 980.4.i
• Level: $980$
• Weight: $4$
• Character orbit: 980.i
• Rep. character: $\chi_{980}(361,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $80$
• Newform subspaces: $25$
• 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.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 25 \)
Sturm bound:			\(672\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(980, [\chi])\).

	Total	New	Old
Modular forms	1056	80	976
Cusp forms	960	80	880
Eisenstein series	96	0	96

Trace form

\( 80 q + 10 q^{5} - 378 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(980, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
980.4.i.a	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-9\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-9+9\zeta_{6})q^{3}-5\zeta_{6}q^{5}-54\zeta_{6}q^{9}+\cdots\)
980.4.i.b	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-8\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-8+8\zeta_{6})q^{3}+5\zeta_{6}q^{5}-37\zeta_{6}q^{9}+\cdots\)
980.4.i.c	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-5\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-5+5\zeta_{6})q^{3}-5\zeta_{6}q^{5}+2\zeta_{6}q^{9}+\cdots\)
980.4.i.d	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-5\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-5+5\zeta_{6})q^{3}+5\zeta_{6}q^{5}+2\zeta_{6}q^{9}+\cdots\)
980.4.i.e	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-4\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-4+4\zeta_{6})q^{3}-5\zeta_{6}q^{5}+11\zeta_{6}q^{9}+\cdots\)
980.4.i.f	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-4\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-4+4\zeta_{6})q^{3}+5\zeta_{6}q^{5}+11\zeta_{6}q^{9}+\cdots\)
980.4.i.g	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{3}-5\zeta_{6}q^{5}+23\zeta_{6}q^{9}+\cdots\)
980.4.i.h	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{3}+5\zeta_{6}q^{5}+23\zeta_{6}q^{9}+\cdots\)
980.4.i.i	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-1\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+5\zeta_{6}q^{5}+26\zeta_{6}q^{9}+\cdots\)
980.4.i.j	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-1\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+5\zeta_{6}q^{5}+26\zeta_{6}q^{9}+\cdots\)
980.4.i.k	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(1\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}-5\zeta_{6}q^{5}+26\zeta_{6}q^{9}+\cdots\)
980.4.i.l	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(1\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}-5\zeta_{6}q^{5}+26\zeta_{6}q^{9}+\cdots\)
980.4.i.m	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(4\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4-4\zeta_{6})q^{3}-5\zeta_{6}q^{5}+11\zeta_{6}q^{9}+\cdots\)
980.4.i.n	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(4\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4-4\zeta_{6})q^{3}+5\zeta_{6}q^{5}+11\zeta_{6}q^{9}+\cdots\)
980.4.i.o	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(5\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(5-5\zeta_{6})q^{3}-5\zeta_{6}q^{5}+2\zeta_{6}q^{9}+\cdots\)
980.4.i.p	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(5\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(5-5\zeta_{6})q^{3}+5\zeta_{6}q^{5}+2\zeta_{6}q^{9}+\cdots\)
980.4.i.q	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(8\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(8-8\zeta_{6})q^{3}-5\zeta_{6}q^{5}-37\zeta_{6}q^{9}+\cdots\)
980.4.i.r	$980$	$4$	980.i	7.c	$3$	$2$	$1$	$57.822$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(9\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(9-9\zeta_{6})q^{3}+5\zeta_{6}q^{5}-54\zeta_{6}q^{9}+\cdots\)
980.4.i.s	$980$	$4$	980.i	7.c	$3$	$4$	$2$	$57.822$	\(\Q(\sqrt{-3}, \sqrt{46})\)	None		$2$	$0$	\(0\)	\(-6\)	\(10\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-3+\beta _{1}-3\beta _{2})q^{3}-5\beta _{2}q^{5}+(-6\beta _{1}+\cdots)q^{9}+\cdots\)
980.4.i.t	$980$	$4$	980.i	7.c	$3$	$4$	$2$	$57.822$	\(\Q(\sqrt{-3}, \sqrt{-83})\)	None		$2$	$0$	\(0\)	\(-1\)	\(10\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{3}q^{3}-5\beta _{1}q^{5}+(35\beta _{1}+\beta _{2}-\beta _{3})q^{9}+\cdots\)
980.4.i.u	$980$	$4$	980.i	7.c	$3$	$4$	$2$	$57.822$	\(\Q(\sqrt{-3}, \sqrt{37})\)	None		$2$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{3}+(-5+5\beta _{1})q^{5}+(-10+10\beta _{1}+\cdots)q^{9}+\cdots\)
980.4.i.v	$980$	$4$	980.i	7.c	$3$	$4$	$2$	$57.822$	\(\Q(\sqrt{-3}, \sqrt{-83})\)	None		$2$	$0$	\(0\)	\(1\)	\(-10\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}+\beta _{3})q^{3}+5\beta _{1}q^{5}+(6^{2}\beta _{1}+\cdots)q^{9}+\cdots\)
980.4.i.w	$980$	$4$	980.i	7.c	$3$	$4$	$2$	$57.822$	\(\Q(\sqrt{-3}, \sqrt{22})\)	None		$2$	$0$	\(0\)	\(10\)	\(10\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(5+\beta _{1}+5\beta _{2})q^{3}-5\beta _{2}q^{5}+(10\beta _{1}+\cdots)q^{9}+\cdots\)
980.4.i.x	$980$	$4$	980.i	7.c	$3$	$12$	$6$	$57.822$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(-4\)	\(30\)	\(0\)	$2^{4}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}+\beta _{2}-\beta _{5})q^{3}+(5-5\beta _{5})q^{5}+\cdots\)
980.4.i.y	$980$	$4$	980.i	7.c	$3$	$12$	$6$	$57.822$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(4\)	\(-30\)	\(0\)	$2^{4}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}-\beta _{2}+\beta _{5})q^{3}+(-5+5\beta _{5})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(980, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(980, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 2}\)