Space of modular forms of level 900, weight 4, and trivial character

• Label: 900.4.a
• Level: $900$
• Weight: $4$
• Character orbit: 900.a
• Rep. character: $\chi_{900}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $20$
• Sturm bound: $720$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	900.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(720\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(900))\).

	Total	New	Old
Modular forms	576	24	552
Cusp forms	504	24	480
Eisenstein series	72	0	72

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

\(2\)	\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(75\)	\(0\)	\(75\)		\(63\)	\(0\)	\(63\)		\(12\)	\(0\)	\(12\)
\(+\)	\(+\)	\(-\)	\(-\)		\(73\)	\(0\)	\(73\)		\(61\)	\(0\)	\(61\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(+\)	\(-\)		\(72\)	\(0\)	\(72\)		\(60\)	\(0\)	\(60\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)	\(+\)		\(74\)	\(0\)	\(74\)		\(62\)	\(0\)	\(62\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(+\)	\(-\)		\(69\)	\(4\)	\(65\)		\(63\)	\(4\)	\(59\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(71\)	\(6\)	\(65\)		\(65\)	\(6\)	\(59\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(72\)	\(7\)	\(65\)		\(66\)	\(7\)	\(59\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(70\)	\(7\)	\(63\)		\(64\)	\(7\)	\(57\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(292\)	\(13\)	\(279\)		\(256\)	\(13\)	\(243\)		\(36\)	\(0\)	\(36\)
Minus space			\(-\)		\(284\)	\(11\)	\(273\)		\(248\)	\(11\)	\(237\)		\(36\)	\(0\)	\(36\)

Trace form

24 * q + 6 * q^11 + 12 * q^13 + 60 * q^17 - 190 * q^19 - 144 * q^23 - 384 * q^29 - 44 * q^31 - 180 * q^37 + 390 * q^41 + 264 * q^43 + 936 * q^47 + 2304 * q^49 - 612 * q^53 - 1104 * q^59 + 1848 * q^61 - 408 * q^67 - 360 * q^71 + 180 * q^73 + 3072 * q^77 + 1628 * q^79 - 1584 * q^83 - 2070 * q^89 + 160 * q^91 - 1164 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(900))\) 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
900.4.a.a	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			900.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-37\)		$-$	$+$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-37q^{7}+89q^{13}-163q^{19}-17^{2}q^{31}+\cdots\)
900.4.a.b	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	None	✓				60.4.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-32\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{5}q^{7}-6^{2}q^{11}+10q^{13}-78q^{17}+\cdots\)
900.4.a.c	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	None	✓				100.4.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-26\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-26q^{7}-45q^{11}-44q^{13}+117q^{17}+\cdots\)
900.4.a.d	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	None	✓				60.4.d.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-22\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-22q^{7}+14q^{11}+30q^{13}+62q^{17}+\cdots\)
900.4.a.e	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			900.4.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-17\)		$-$	$+$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-17q^{7}+19q^{13}+107q^{19}-19q^{31}+\cdots\)
900.4.a.f	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	None	✓				300.4.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-13\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-13q^{7}-6q^{11}+5q^{13}+78q^{17}+\cdots\)
900.4.a.g	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	None	✓				12.4.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{7}-6^{2}q^{11}+10q^{13}+18q^{17}+\cdots\)
900.4.a.h	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	None	✓				300.4.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{7}+54q^{11}-55q^{13}-18q^{17}+\cdots\)
900.4.a.i	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	None	✓				180.4.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-30q^{11}+4q^{13}+90q^{17}+\cdots\)
900.4.a.j	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	None	✓				180.4.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+30q^{11}+4q^{13}-90q^{17}+\cdots\)
900.4.a.k	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	None	✓				300.4.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7q^{7}+54q^{11}+55q^{13}+18q^{17}+\cdots\)
900.4.a.l	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	None	✓				300.4.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(13\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+13q^{7}-6q^{11}-5q^{13}-78q^{17}+\cdots\)
900.4.a.m	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	None	✓				20.4.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(16\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{7}+60q^{11}-86q^{13}+18q^{17}+\cdots\)
900.4.a.n	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			900.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(17\)		$-$	$+$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+17q^{7}-19q^{13}+107q^{19}-19q^{31}+\cdots\)
900.4.a.o	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	None	✓				60.4.d.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(22\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+22q^{7}+14q^{11}-30q^{13}-62q^{17}+\cdots\)
900.4.a.p	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	None	✓				100.4.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(26\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+26q^{7}-45q^{11}+44q^{13}-117q^{17}+\cdots\)
900.4.a.q	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	None	✓				60.4.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(28\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+28q^{7}+24q^{11}+70q^{13}+102q^{17}+\cdots\)
900.4.a.r	$900$	$4$	900.a	1.a	$1$	$1$	$1$	$53.102$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			900.4.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(37\)		$-$	$+$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+37q^{7}-89q^{13}-163q^{19}-17^{2}q^{31}+\cdots\)
900.4.a.s	$900$	$4$	900.a	1.a	$1$	$2$	$2$	$53.102$	\(\Q(\sqrt{19}) \)	None	✓		✓		20.4.c.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{7}-20q^{11}-6\beta q^{13}+8\beta q^{17}+\cdots\)
900.4.a.t	$900$	$4$	900.a	1.a	$1$	$4$	$4$	$53.102$	\(\Q(\sqrt{10}, \sqrt{34})\)	None	✓		✓		180.4.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2^{7}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{7}+\beta _{2}q^{11}+3\beta _{1}q^{13}-\beta _{3}q^{17}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(900))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(900)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 2}\)