Space of modular forms of level 980, weight 2, and trivial character

• Label: 980.2.a
• Level: $980$
• Weight: $2$
• Character orbit: 980.a
• Rep. character: $\chi_{980}(1,\cdot)$
• Character field: $\Q$
• Dimension: $13$
• Newform subspaces: $11$
• Sturm bound: $336$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(336\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(980))\).

	Total	New	Old
Modular forms	192	13	179
Cusp forms	145	13	132
Eisenstein series	47	0	47

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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(22\)	\(0\)	\(22\)		\(15\)	\(0\)	\(15\)		\(7\)	\(0\)	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)		\(27\)	\(0\)	\(27\)		\(19\)	\(0\)	\(19\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)		\(26\)	\(0\)	\(26\)		\(18\)	\(0\)	\(18\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)	\(+\)		\(25\)	\(0\)	\(25\)		\(17\)	\(0\)	\(17\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)		\(26\)	\(4\)	\(22\)		\(22\)	\(4\)	\(18\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(21\)	\(2\)	\(19\)		\(17\)	\(2\)	\(15\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(22\)	\(2\)	\(20\)		\(18\)	\(2\)	\(16\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(23\)	\(5\)	\(18\)		\(19\)	\(5\)	\(14\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(90\)	\(4\)	\(86\)		\(67\)	\(4\)	\(63\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(102\)	\(9\)	\(93\)		\(78\)	\(9\)	\(69\)		\(24\)	\(0\)	\(24\)

Trace form

13 * q - 2 * q^3 + q^5 + 9 * q^9 + 2 * q^13 - 2 * q^15 + 10 * q^17 - 4 * q^19 - 6 * q^23 + 13 * q^25 - 8 * q^27 + 10 * q^29 + 12 * q^33 + 34 * q^37 + 44 * q^39 - 2 * q^41 + 34 * q^43 + 9 * q^45 + 10 * q^47 - 4 * q^51 - 6 * q^53 - 8 * q^55 - 8 * q^57 - 16 * q^59 - 2 * q^61 - 2 * q^65 + 14 * q^67 + 4 * q^71 - 18 * q^73 - 2 * q^75 + 17 * q^81 + 10 * q^83 - 10 * q^85 + 48 * q^87 - 10 * q^89 + 32 * q^93 + 4 * q^95 - 6 * q^97 - 48 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(980))\) 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	7
980.2.a.a	$980$	$2$	980.a	1.a	$1$	$1$	$1$	$7.825$	\(\Q\)	None	✓				140.2.i.b	$1$	$0$	\(0\)	\(-3\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{5}+6q^{9}-2q^{11}-6q^{13}+\cdots\)
980.2.a.b	$980$	$2$	980.a	1.a	$1$	$1$	$1$	$7.825$	\(\Q\)	None	✓				140.2.a.b	$1$	$0$	\(0\)	\(-3\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+q^{5}+6q^{9}-5q^{11}+3q^{13}+\cdots\)
980.2.a.c	$980$	$2$	980.a	1.a	$1$	$1$	$1$	$7.825$	\(\Q\)	None	✓				140.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-2q^{9}+3q^{11}+q^{13}+\cdots\)
980.2.a.d	$980$	$2$	980.a	1.a	$1$	$1$	$1$	$7.825$	\(\Q\)	None	✓	✓			980.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-2q^{9}-q^{11}+5q^{13}+\cdots\)
980.2.a.e	$980$	$2$	980.a	1.a	$1$	$1$	$1$	$7.825$	\(\Q\)	None	✓				140.2.i.a	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-2q^{9}+6q^{11}-2q^{13}+\cdots\)
980.2.a.f	$980$	$2$	980.a	1.a	$1$	$1$	$1$	$7.825$	\(\Q\)	None	✓	✓			980.2.a.d	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{9}-q^{11}-5q^{13}+\cdots\)
980.2.a.g	$980$	$2$	980.a	1.a	$1$	$1$	$1$	$7.825$	\(\Q\)	None	✓				140.2.i.a	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{9}+6q^{11}+2q^{13}+\cdots\)
980.2.a.h	$980$	$2$	980.a	1.a	$1$	$1$	$1$	$7.825$	\(\Q\)	None	✓				20.2.a.a	$1$	$0$	\(0\)	\(2\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+q^{9}-2q^{13}+2q^{15}+\cdots\)
980.2.a.i	$980$	$2$	980.a	1.a	$1$	$1$	$1$	$7.825$	\(\Q\)	None	✓				140.2.i.b	$1$	$0$	\(0\)	\(3\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{5}+6q^{9}-2q^{11}+6q^{13}+\cdots\)
980.2.a.j	$980$	$2$	980.a	1.a	$1$	$2$	$2$	$7.825$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	980.2.a.j	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+q^{5}-2\beta q^{9}+(-1+\cdots)q^{11}+\cdots\)
980.2.a.k	$980$	$2$	980.a	1.a	$1$	$2$	$2$	$7.825$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓		980.2.a.j	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-q^{5}+2\beta q^{9}+(-1+2\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(980))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(980)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(490))\)\(^{\oplus 2}\)