Space of modular forms of level 450, weight 4, and trivial character

• Label: 450.4.a
• Level: $450$
• Weight: $4$
• Character orbit: 450.a
• Rep. character: $\chi_{450}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $22$
• Sturm bound: $360$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(360\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(450))\).

	Total	New	Old
Modular forms	294	24	270
Cusp forms	246	24	222
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.

\(2\)	\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(39\)	\(2\)	\(37\)		\(33\)	\(2\)	\(31\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)		\(35\)	\(3\)	\(32\)		\(29\)	\(3\)	\(26\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(36\)	\(3\)	\(33\)		\(30\)	\(3\)	\(27\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(37\)	\(4\)	\(33\)		\(31\)	\(4\)	\(27\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(36\)	\(2\)	\(34\)		\(30\)	\(2\)	\(28\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(38\)	\(3\)	\(35\)		\(32\)	\(3\)	\(29\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(36\)	\(4\)	\(32\)		\(30\)	\(4\)	\(26\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(37\)	\(3\)	\(34\)		\(31\)	\(3\)	\(28\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(150\)	\(13\)	\(137\)		\(126\)	\(13\)	\(113\)		\(24\)	\(0\)	\(24\)
Minus space			\(-\)		\(144\)	\(11\)	\(133\)		\(120\)	\(11\)	\(109\)		\(24\)	\(0\)	\(24\)

Trace form

24 * q + 96 * q^4 - 36 * q^7 - 78 * q^11 - 84 * q^13 - 8 * q^14 + 384 * q^16 - 132 * q^17 + 110 * q^19 + 372 * q^23 - 416 * q^26 - 144 * q^28 + 360 * q^29 + 148 * q^31 + 188 * q^34 + 588 * q^37 + 192 * q^38 - 138 * q^41 + 492 * q^43 - 312 * q^44 + 536 * q^46 - 444 * q^47 + 672 * q^49 - 336 * q^52 + 564 * q^53 - 32 * q^56 + 336 * q^58 + 960 * q^59 - 1272 * q^61 + 1488 * q^62 + 1536 * q^64 - 732 * q^67 - 528 * q^68 + 3312 * q^71 - 1788 * q^73 - 728 * q^74 + 440 * q^76 - 1968 * q^77 + 1340 * q^79 - 96 * q^82 - 1572 * q^83 + 304 * q^86 + 3690 * q^89 - 992 * q^91 + 1488 * q^92 + 1888 * q^94 - 828 * q^97 - 2496 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(450))\) 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
450.4.a.a	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓				50.4.a.a	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-34\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-34q^{7}-8q^{8}-3^{3}q^{11}+\cdots\)
450.4.a.b	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓				30.4.a.a	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-32\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-2^{5}q^{7}-8q^{8}+60q^{11}+\cdots\)
450.4.a.c	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓				90.4.a.b	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-14\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-14q^{7}-8q^{8}-6q^{11}+\cdots\)
450.4.a.d	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓	✓			450.4.a.d	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-11\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-11q^{7}-8q^{8}+6^{2}q^{11}+\cdots\)
450.4.a.e	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓				30.4.c.a	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-2q^{7}-8q^{8}-70q^{11}+\cdots\)
450.4.a.f	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓				150.4.a.a	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+q^{7}-8q^{8}-42q^{11}+\cdots\)
450.4.a.g	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓	✓			450.4.a.d	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(11\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+11q^{7}-8q^{8}-6^{2}q^{11}+\cdots\)
450.4.a.h	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓				6.4.a.a	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(16\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+2^{4}q^{7}-8q^{8}-12q^{11}+\cdots\)
450.4.a.i	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓				150.4.a.c	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(23\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+23q^{7}-8q^{8}+30q^{11}+\cdots\)
450.4.a.j	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓				10.4.b.a	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(26\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+26q^{7}-8q^{8}+28q^{11}+\cdots\)
450.4.a.k	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓				10.4.b.a	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-26\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-26q^{7}+8q^{8}+28q^{11}+\cdots\)
450.4.a.l	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓				150.4.a.c	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-23\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-23q^{7}+8q^{8}+30q^{11}+\cdots\)
450.4.a.m	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓				90.4.a.b	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-14\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-14q^{7}+8q^{8}+6q^{11}+\cdots\)
450.4.a.n	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓	✓			450.4.a.d	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-11\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-11q^{7}+8q^{8}-6^{2}q^{11}+\cdots\)
450.4.a.o	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓				150.4.a.a	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-q^{7}+8q^{8}-42q^{11}+\cdots\)
450.4.a.p	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓				30.4.c.a	$1$	$1$	\(2\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+2q^{7}+8q^{8}-70q^{11}+\cdots\)
450.4.a.q	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓				10.4.a.a	$1$	$0$	\(2\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+4q^{7}+8q^{8}-12q^{11}+\cdots\)
450.4.a.r	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓				30.4.a.b	$1$	$0$	\(2\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+4q^{7}+8q^{8}+48q^{11}+\cdots\)
450.4.a.s	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓	✓			450.4.a.d	$1$	$0$	\(2\)	\(0\)	\(0\)	\(11\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+11q^{7}+8q^{8}+6^{2}q^{11}+\cdots\)
450.4.a.t	$450$	$4$	450.a	1.a	$1$	$1$	$1$	$26.551$	\(\Q\)	None	✓				50.4.a.a	$1$	$0$	\(2\)	\(0\)	\(0\)	\(34\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+34q^{7}+8q^{8}-3^{3}q^{11}+\cdots\)
450.4.a.u	$450$	$4$	450.a	1.a	$1$	$2$	$2$	$26.551$	\(\Q(\sqrt{31}) \)	None	✓		✓		90.4.c.c	$2$	$1$	\(-4\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+\beta q^{7}-8q^{8}-\beta q^{11}+\cdots\)
450.4.a.v	$450$	$4$	450.a	1.a	$1$	$2$	$2$	$26.551$	\(\Q(\sqrt{31}) \)	None	✓		✓		90.4.c.c	$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+\beta q^{7}+8q^{8}+\beta q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(450))\) into lower level spaces

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