Space of modular forms of level 1980, weight 4, and trivial character

• Label: 1980.4.a
• Level: $1980$
• Weight: $4$
• Character orbit: 1980.a
• Rep. character: $\chi_{1980}(1,\cdot)$
• Character field: $\Q$
• Dimension: $50$
• Newform subspaces: $18$
• Sturm bound: $1728$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1980.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(1728\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1980))\).

	Total	New	Old
Modular forms	1320	50	1270
Cusp forms	1272	50	1222
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.

\(2\)	\(3\)	\(5\)	\(11\)	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(90\)	\(0\)	\(90\)		\(86\)	\(0\)	\(86\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(76\)	\(0\)	\(76\)		\(72\)	\(0\)	\(72\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(82\)	\(0\)	\(82\)		\(78\)	\(0\)	\(78\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(84\)	\(0\)	\(84\)		\(80\)	\(0\)	\(80\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(79\)	\(0\)	\(79\)		\(75\)	\(0\)	\(75\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(87\)	\(0\)	\(87\)		\(83\)	\(0\)	\(83\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(87\)	\(0\)	\(87\)		\(83\)	\(0\)	\(83\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(79\)	\(0\)	\(79\)		\(75\)	\(0\)	\(75\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(84\)	\(5\)	\(79\)		\(82\)	\(5\)	\(77\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(80\)	\(5\)	\(75\)		\(78\)	\(5\)	\(73\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(80\)	\(5\)	\(75\)		\(78\)	\(5\)	\(73\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(84\)	\(5\)	\(79\)		\(82\)	\(5\)	\(77\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(80\)	\(7\)	\(73\)		\(78\)	\(7\)	\(71\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(84\)	\(7\)	\(77\)		\(82\)	\(7\)	\(75\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(84\)	\(8\)	\(76\)		\(82\)	\(8\)	\(74\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(80\)	\(8\)	\(72\)		\(78\)	\(8\)	\(70\)		\(2\)	\(0\)	\(2\)
Plus space				\(+\)		\(668\)	\(25\)	\(643\)		\(644\)	\(25\)	\(619\)		\(24\)	\(0\)	\(24\)
Minus space				\(-\)		\(652\)	\(25\)	\(627\)		\(628\)	\(25\)	\(603\)		\(24\)	\(0\)	\(24\)

Trace form

50 * q + 10 * q^5 + 16 * q^7 + 40 * q^13 + 64 * q^17 - 184 * q^19 - 84 * q^23 + 1250 * q^25 + 4 * q^29 + 172 * q^31 + 40 * q^35 - 884 * q^37 - 740 * q^41 + 872 * q^43 + 452 * q^47 + 1862 * q^49 + 148 * q^53 - 680 * q^59 + 828 * q^61 + 160 * q^65 - 1012 * q^67 - 820 * q^71 + 144 * q^73 + 440 * q^77 + 1056 * q^79 - 2808 * q^83 - 560 * q^85 - 1992 * q^89 - 888 * q^91 + 1640 * q^95 - 2380 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1980))\) 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	3	5	11
1980.4.a.a	$1980$	$4$	1980.a	1.a	$1$	$1$	$1$	$116.824$	\(\Q\)	None	✓				660.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}-11q^{11}-42q^{13}+14q^{17}+\cdots\)
1980.4.a.b	$1980$	$4$	1980.a	1.a	$1$	$1$	$1$	$116.824$	\(\Q\)	None	✓				220.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(11\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+11q^{7}+11q^{11}-22q^{13}+\cdots\)
1980.4.a.c	$1980$	$4$	1980.a	1.a	$1$	$1$	$1$	$116.824$	\(\Q\)	None	✓				220.4.a.c	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(24\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+24q^{7}+11q^{11}-22q^{13}+\cdots\)
1980.4.a.d	$1980$	$4$	1980.a	1.a	$1$	$1$	$1$	$116.824$	\(\Q\)	None	✓				220.4.a.b	$1$	$0$	\(0\)	\(0\)	\(5\)	\(-19\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-19q^{7}+11q^{11}-62q^{13}+\cdots\)
1980.4.a.e	$1980$	$4$	1980.a	1.a	$1$	$2$	$2$	$116.824$	\(\Q(\sqrt{21}) \)	None	✓				660.4.a.d	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(-20\)		$-$	$-$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-5q^{5}+(-10-\beta )q^{7}+11q^{11}-22q^{13}+\cdots\)
1980.4.a.f	$1980$	$4$	1980.a	1.a	$1$	$2$	$2$	$116.824$	\(\Q(\sqrt{97}) \)	None	✓				220.4.a.d	$1$	$0$	\(0\)	\(0\)	\(-10\)	\(-15\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+(-7-\beta )q^{7}-11q^{11}+(2+\cdots)q^{13}+\cdots\)
1980.4.a.g	$1980$	$4$	1980.a	1.a	$1$	$2$	$2$	$116.824$	\(\Q(\sqrt{34}) \)	None	✓				660.4.a.b	$1$	$0$	\(0\)	\(0\)	\(10\)	\(-4\)		$-$	$-$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+5q^{5}+(-2+\beta )q^{7}+11q^{11}+(40+\cdots)q^{13}+\cdots\)
1980.4.a.h	$1980$	$4$	1980.a	1.a	$1$	$2$	$2$	$116.824$	\(\Q(\sqrt{3}) \)	None	✓				660.4.a.c	$1$	$1$	\(0\)	\(0\)	\(10\)	\(4\)		$-$	$-$	$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+5q^{5}+(2+2\beta )q^{7}-11q^{11}+(2-3\beta )q^{13}+\cdots\)
1980.4.a.i	$1980$	$4$	1980.a	1.a	$1$	$2$	$2$	$116.824$	\(\Q(\sqrt{6}) \)	None	✓				220.4.a.e	$1$	$0$	\(0\)	\(0\)	\(10\)	\(36\)		$-$	$-$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+5q^{5}+(18+2\beta )q^{7}+11q^{11}+(-12+\cdots)q^{13}+\cdots\)
1980.4.a.j	$1980$	$4$	1980.a	1.a	$1$	$3$	$3$	$116.824$	3.3.953556.1	None	✓		✓		660.4.a.e	$1$	$1$	\(0\)	\(0\)	\(-15\)	\(-8\)		$-$	$-$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-5q^{5}+(-3-\beta _{1})q^{7}+11q^{11}+(28+\cdots)q^{13}+\cdots\)
1980.4.a.k	$1980$	$4$	1980.a	1.a	$1$	$3$	$3$	$116.824$	3.3.44040.1	None	✓				660.4.a.f	$1$	$1$	\(0\)	\(0\)	\(15\)	\(-8\)		$-$	$-$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+5q^{5}+(-3-\beta _{2})q^{7}-11q^{11}+(-2^{4}+\cdots)q^{13}+\cdots\)
1980.4.a.l	$1980$	$4$	1980.a	1.a	$1$	$3$	$3$	$116.824$	3.3.9192.1	None	✓				220.4.a.f	$1$	$1$	\(0\)	\(0\)	\(15\)	\(-5\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+5q^{5}+(-2-\beta _{1}+2\beta _{2})q^{7}-11q^{11}+\cdots\)
1980.4.a.m	$1980$	$4$	1980.a	1.a	$1$	$3$	$3$	$116.824$	3.3.2321692.1	None	✓		✓		660.4.a.g	$1$	$0$	\(0\)	\(0\)	\(15\)	\(12\)		$-$	$-$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+5q^{5}+(4+\beta _{1})q^{7}+11q^{11}+(15+\cdots)q^{13}+\cdots\)
1980.4.a.n	$1980$	$4$	1980.a	1.a	$1$	$4$	$4$	$116.824$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		660.4.a.h	$1$	$0$	\(0\)	\(0\)	\(-20\)	\(16\)		$-$	$-$	$+$	$+$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q-5q^{5}+(4-\beta _{1})q^{7}-11q^{11}+(3+\beta _{2}+\cdots)q^{13}+\cdots\)
1980.4.a.o	$1980$	$4$	1980.a	1.a	$1$	$5$	$5$	$116.824$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	1980.4.a.o	$1$	$1$	\(0\)	\(0\)	\(-25\)	\(-16\)		$-$	$+$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-5q^{5}+(-3+\beta _{3})q^{7}-11q^{11}+(-1+\cdots)q^{13}+\cdots\)
1980.4.a.p	$1980$	$4$	1980.a	1.a	$1$	$5$	$5$	$116.824$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	1980.4.a.p	$1$	$0$	\(0\)	\(0\)	\(-25\)	\(12\)		$-$	$+$	$+$	$-$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-5q^{5}+(2+\beta _{1})q^{7}+11q^{11}+(10+\cdots)q^{13}+\cdots\)
1980.4.a.q	$1980$	$4$	1980.a	1.a	$1$	$5$	$5$	$116.824$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	1980.4.a.o	$1$	$1$	\(0\)	\(0\)	\(25\)	\(-16\)		$-$	$+$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+5q^{5}+(-3+\beta _{3})q^{7}+11q^{11}+(-1+\cdots)q^{13}+\cdots\)
1980.4.a.r	$1980$	$4$	1980.a	1.a	$1$	$5$	$5$	$116.824$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	1980.4.a.p	$1$	$0$	\(0\)	\(0\)	\(25\)	\(12\)		$-$	$+$	$-$	$+$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+5q^{5}+(2+\beta _{1})q^{7}-11q^{11}+(10+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1980))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1980)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(330))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(396))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(495))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(660))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(990))\)\(^{\oplus 2}\)