Space of modular forms of level 5800, weight 2, and trivial character

• Label: 5800.2.a
• Level: $5800$
• Weight: $2$
• Character orbit: 5800.a
• Rep. character: $\chi_{5800}(1,\cdot)$
• Character field: $\Q$
• Dimension: $133$
• Newform subspaces: $34$
• Sturm bound: $1800$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 5800 = 2^{3} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5800.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 34 \)
Sturm bound:			\(1800\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5800))\).

	Total	New	Old
Modular forms	924	133	791
Cusp forms	877	133	744
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.

\(2\)	\(5\)	\(29\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(114\)	\(15\)	\(99\)		\(109\)	\(15\)	\(94\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(117\)	\(18\)	\(99\)		\(111\)	\(18\)	\(93\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(117\)	\(19\)	\(98\)		\(111\)	\(19\)	\(92\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(114\)	\(15\)	\(99\)		\(108\)	\(15\)	\(93\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(117\)	\(15\)	\(102\)		\(111\)	\(15\)	\(96\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(114\)	\(15\)	\(99\)		\(108\)	\(15\)	\(93\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(114\)	\(19\)	\(95\)		\(108\)	\(19\)	\(89\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(117\)	\(17\)	\(100\)		\(111\)	\(17\)	\(94\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(456\)	\(64\)	\(392\)		\(433\)	\(64\)	\(369\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(468\)	\(69\)	\(399\)		\(444\)	\(69\)	\(375\)		\(24\)	\(0\)	\(24\)

Trace form

133 * q - 4 * q^7 + 127 * q^9 + 4 * q^11 - 4 * q^13 - 2 * q^17 + 4 * q^19 + 4 * q^23 - 3 * q^29 - 16 * q^31 - 10 * q^33 - 10 * q^37 - 12 * q^39 + 2 * q^41 - 16 * q^43 - 8 * q^47 + 141 * q^49 + 16 * q^51 + 24 * q^53 - 16 * q^57 + 12 * q^59 - 10 * q^61 - 24 * q^63 - 12 * q^67 - 20 * q^69 - 16 * q^71 - 30 * q^73 - 24 * q^77 - 72 * q^79 + 157 * q^81 + 16 * q^83 - 6 * q^89 - 108 * q^91 - 6 * q^93 - 30 * q^97 + 48 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5800))\) 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	29
5800.2.a.a	$5800$	$2$	5800.a	1.a	$1$	$1$	$1$	$46.313$	\(\Q\)	None	✓				1160.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-4q^{7}+q^{9}-6q^{13}+8q^{21}+\cdots\)
5800.2.a.b	$5800$	$2$	5800.a	1.a	$1$	$1$	$1$	$46.313$	\(\Q\)	None	✓	✓			5800.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{7}+q^{9}-6q^{11}-6q^{13}+\cdots\)
5800.2.a.c	$5800$	$2$	5800.a	1.a	$1$	$1$	$1$	$46.313$	\(\Q\)	None	✓				1160.2.d.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
5800.2.a.d	$5800$	$2$	5800.a	1.a	$1$	$1$	$1$	$46.313$	\(\Q\)	None	✓				1160.2.a.c	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+4q^{7}+q^{9}+2q^{13}+4q^{17}+\cdots\)
5800.2.a.e	$5800$	$2$	5800.a	1.a	$1$	$1$	$1$	$46.313$	\(\Q\)	None	✓				232.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}-2q^{9}+3q^{11}+q^{13}+\cdots\)
5800.2.a.f	$5800$	$2$	5800.a	1.a	$1$	$1$	$1$	$46.313$	\(\Q\)	None	✓	✓			5800.2.a.f	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}+5q^{11}-4q^{13}-3q^{17}+\cdots\)
5800.2.a.g	$5800$	$2$	5800.a	1.a	$1$	$1$	$1$	$46.313$	\(\Q\)	None	✓				1160.2.d.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-3q^{9}+6q^{17}-4q^{19}+6q^{23}+\cdots\)
5800.2.a.h	$5800$	$2$	5800.a	1.a	$1$	$1$	$1$	$46.313$	\(\Q\)	None	✓				1160.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}+2q^{13}+6q^{17}-8q^{19}+q^{29}+\cdots\)
5800.2.a.i	$5800$	$2$	5800.a	1.a	$1$	$1$	$1$	$46.313$	\(\Q\)	None	✓				1160.2.d.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-3q^{9}-6q^{17}-4q^{19}-6q^{23}+\cdots\)
5800.2.a.j	$5800$	$2$	5800.a	1.a	$1$	$1$	$1$	$46.313$	\(\Q\)	None	✓				232.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}-2q^{9}-3q^{11}+5q^{13}+\cdots\)
5800.2.a.k	$5800$	$2$	5800.a	1.a	$1$	$1$	$1$	$46.313$	\(\Q\)	None	✓	✓			5800.2.a.f	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}+5q^{11}+4q^{13}+3q^{17}+\cdots\)
5800.2.a.l	$5800$	$2$	5800.a	1.a	$1$	$1$	$1$	$46.313$	\(\Q\)	None	✓	✓			5800.2.a.b	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{7}+q^{9}-6q^{11}+6q^{13}+\cdots\)
5800.2.a.m	$5800$	$2$	5800.a	1.a	$1$	$1$	$1$	$46.313$	\(\Q\)	None	✓				1160.2.d.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{7}+q^{9}+4q^{11}-4q^{13}+\cdots\)
5800.2.a.n	$5800$	$2$	5800.a	1.a	$1$	$1$	$1$	$46.313$	\(\Q\)	None	✓				1160.2.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}+4q^{11}+2q^{13}+4q^{19}+\cdots\)
5800.2.a.o	$5800$	$2$	5800.a	1.a	$1$	$2$	$2$	$46.313$	\(\Q(\sqrt{2}) \)	None	✓				232.2.a.c	$1$	$0$	\(0\)	\(2\)	\(0\)	\(8\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+4q^{7}+2\beta q^{9}+(-1+\beta )q^{11}+\cdots\)
5800.2.a.p	$5800$	$2$	5800.a	1.a	$1$	$3$	$3$	$46.313$	3.3.568.1	None	✓				232.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(2+\beta _{2})q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
5800.2.a.q	$5800$	$2$	5800.a	1.a	$1$	$3$	$3$	$46.313$	3.3.229.1	None	✓				1160.2.a.g	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(1-\beta _{1})q^{7}+(\beta _{1}-2\beta _{2})q^{9}+\cdots\)
5800.2.a.r	$5800$	$2$	5800.a	1.a	$1$	$3$	$3$	$46.313$	3.3.148.1	None	✓				1160.2.a.e	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1+\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
5800.2.a.s	$5800$	$2$	5800.a	1.a	$1$	$3$	$3$	$46.313$	3.3.148.1	None	✓				1160.2.a.f	$1$	$1$	\(0\)	\(2\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1+\beta _{1})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
5800.2.a.t	$5800$	$2$	5800.a	1.a	$1$	$5$	$5$	$46.313$	5.5.580484.1	None	✓				1160.2.a.j	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-7\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{3})q^{3}+(-1+\beta _{2}-\beta _{3})q^{7}+\cdots\)
5800.2.a.u	$5800$	$2$	5800.a	1.a	$1$	$5$	$5$	$46.313$	5.5.3145252.1	None	✓				1160.2.a.h	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-7\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+(-1+\beta _{2})q^{7}+(2-\beta _{2}+\beta _{4})q^{9}+\cdots\)
5800.2.a.v	$5800$	$2$	5800.a	1.a	$1$	$5$	$5$	$46.313$	5.5.6083172.1	None	✓				1160.2.a.i	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(\beta _{2}-\beta _{3})q^{7}+(2+\beta _{2})q^{9}+\cdots\)
5800.2.a.w	$5800$	$2$	5800.a	1.a	$1$	$6$	$6$	$46.313$	6.6.15523969.1	None	✓	✓	✓		5800.2.a.w	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+(-\beta _{1}+\beta _{3}+\beta _{4}+\beta _{5})q^{7}+\cdots\)
5800.2.a.x	$5800$	$2$	5800.a	1.a	$1$	$6$	$6$	$46.313$	6.6.16196689.1	None	✓	✓			5800.2.a.x	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-1+\beta _{1}-\beta _{3})q^{7}+(1+\beta _{3}+\cdots)q^{9}+\cdots\)
5800.2.a.y	$5800$	$2$	5800.a	1.a	$1$	$6$	$6$	$46.313$	6.6.16196689.1	None	✓	✓	✓		5800.2.a.x	$1$	$1$	\(0\)	\(1\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(1-\beta _{1}+\beta _{3})q^{7}+(1+\beta _{3}+\cdots)q^{9}+\cdots\)
5800.2.a.z	$5800$	$2$	5800.a	1.a	$1$	$6$	$6$	$46.313$	6.6.15523969.1	None	✓	✓			5800.2.a.w	$1$	$1$	\(0\)	\(2\)	\(0\)	\(4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}+(\beta _{1}-\beta _{3}-\beta _{4}-\beta _{5})q^{7}+\cdots\)
5800.2.a.ba	$5800$	$2$	5800.a	1.a	$1$	$7$	$7$	$46.313$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓		5800.2.a.ba	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(8\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{1}+\beta _{2})q^{7}+(1+\beta _{4}+\cdots)q^{9}+\cdots\)
5800.2.a.bb	$5800$	$2$	5800.a	1.a	$1$	$7$	$7$	$46.313$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓		5800.2.a.bb	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{6})q^{7}+(2+\beta _{2})q^{9}+\cdots\)
5800.2.a.bc	$5800$	$2$	5800.a	1.a	$1$	$7$	$7$	$46.313$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			5800.2.a.bb	$1$	$0$	\(0\)	\(0\)	\(0\)	\(6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{6})q^{7}+(2+\beta _{2})q^{9}+\cdots\)
5800.2.a.bd	$5800$	$2$	5800.a	1.a	$1$	$7$	$7$	$46.313$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			5800.2.a.ba	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-8\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{1}-\beta _{2})q^{7}+(1+\beta _{4}+\cdots)q^{9}+\cdots\)
5800.2.a.be	$5800$	$2$	5800.a	1.a	$1$	$8$	$8$	$46.313$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		1160.2.d.c	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-3\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{6}q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
5800.2.a.bf	$5800$	$2$	5800.a	1.a	$1$	$8$	$8$	$46.313$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		1160.2.d.c	$1$	$0$	\(0\)	\(3\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{6}q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
5800.2.a.bg	$5800$	$2$	5800.a	1.a	$1$	$11$	$11$	$46.313$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		1160.2.d.d	$1$	$1$	\(0\)	\(-5\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{10}q^{7}+(\beta _{1}+\beta _{3}+\beta _{4}+\cdots)q^{9}+\cdots\)
5800.2.a.bh	$5800$	$2$	5800.a	1.a	$1$	$11$	$11$	$46.313$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		1160.2.d.d	$1$	$0$	\(0\)	\(5\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{10}q^{7}+(\beta _{1}+\beta _{3}+\beta _{4}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5800))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5800)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(145))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(232))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(290))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(580))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(725))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1160))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1450))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2900))\)\(^{\oplus 2}\)