Space of modular forms of level 225, weight 4, and trivial character

• Label: 225.4.a
• Level: $225$
• Weight: $4$
• Character orbit: 225.a
• Rep. character: $\chi_{225}(1,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $15$
• Sturm bound: $120$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(120\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(225))\).

	Total	New	Old
Modular forms	102	25	77
Cusp forms	78	22	56
Eisenstein series	24	3	21

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

\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(27\)	\(5\)	\(22\)		\(21\)	\(5\)	\(16\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(25\)	\(4\)	\(21\)		\(19\)	\(4\)	\(15\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(24\)	\(7\)	\(17\)		\(18\)	\(6\)	\(12\)		\(6\)	\(1\)	\(5\)
\(-\)	\(-\)	\(+\)		\(26\)	\(9\)	\(17\)		\(20\)	\(7\)	\(13\)		\(6\)	\(2\)	\(4\)
Plus space		\(+\)		\(53\)	\(14\)	\(39\)		\(41\)	\(12\)	\(29\)		\(12\)	\(2\)	\(10\)
Minus space		\(-\)		\(49\)	\(11\)	\(38\)		\(37\)	\(10\)	\(27\)		\(12\)	\(1\)	\(11\)

Trace form

22 * q + 82 * q^4 + 38 * q^7 - 36 * q^8 + 54 * q^11 + 52 * q^13 - 48 * q^14 + 154 * q^16 + 66 * q^17 + 14 * q^19 - 352 * q^22 - 366 * q^23 + 924 * q^26 + 944 * q^28 + 30 * q^29 - 292 * q^31 + 372 * q^32 - 42 * q^34 - 192 * q^37 - 792 * q^38 - 36 * q^41 - 386 * q^43 + 582 * q^44 - 684 * q^46 - 282 * q^47 + 42 * q^49 + 504 * q^52 + 114 * q^53 + 540 * q^56 + 304 * q^58 + 60 * q^59 + 1184 * q^61 + 744 * q^62 + 2506 * q^64 + 1394 * q^67 + 360 * q^68 - 2196 * q^71 + 976 * q^73 - 3588 * q^74 - 622 * q^76 - 1536 * q^77 - 3184 * q^79 - 5024 * q^82 + 1650 * q^83 - 1116 * q^86 - 4224 * q^88 + 180 * q^89 - 164 * q^91 + 432 * q^92 - 4392 * q^94 + 3624 * q^97 + 1632 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(225))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5
225.4.a.a	$225$	$4$	225.a	1.a	$1$	$1$	$1$	$13.275$	\(\Q\)	None	✓				45.4.a.a	$1$	$0$	\(-5\)	\(0\)	\(0\)	\(30\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}+17q^{4}+30q^{7}-45q^{8}+50q^{11}+\cdots\)
225.4.a.b	$225$	$4$	225.a	1.a	$1$	$1$	$1$	$13.275$	\(\Q\)	None	✓				5.4.a.a	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+8q^{4}-6q^{7}-2^{5}q^{11}+38q^{13}+\cdots\)
225.4.a.c	$225$	$4$	225.a	1.a	$1$	$1$	$1$	$13.275$	\(\Q\)	None	✓				25.4.a.a	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-7q^{4}+6q^{7}+15q^{8}+43q^{11}+\cdots\)
225.4.a.d	$225$	$4$	225.a	1.a	$1$	$1$	$1$	$13.275$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				9.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-20\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-8q^{4}-20q^{7}+70q^{13}+2^{6}q^{16}+\cdots\)
225.4.a.e	$225$	$4$	225.a	1.a	$1$	$1$	$1$	$13.275$	\(\Q\)	None	✓				25.4.a.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-7q^{4}-6q^{7}-15q^{8}+43q^{11}+\cdots\)
225.4.a.f	$225$	$4$	225.a	1.a	$1$	$1$	$1$	$13.275$	\(\Q\)	None	✓				15.4.a.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(24\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-7q^{4}+24q^{7}-15q^{8}-52q^{11}+\cdots\)
225.4.a.g	$225$	$4$	225.a	1.a	$1$	$1$	$1$	$13.275$	\(\Q\)	None	✓				15.4.a.b	$1$	$1$	\(3\)	\(0\)	\(0\)	\(-20\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}-20q^{7}-21q^{8}+24q^{11}+\cdots\)
225.4.a.h	$225$	$4$	225.a	1.a	$1$	$1$	$1$	$13.275$	\(\Q\)	None	✓				45.4.a.a	$1$	$0$	\(5\)	\(0\)	\(0\)	\(30\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}+17q^{4}+30q^{7}+45q^{8}-50q^{11}+\cdots\)
225.4.a.i	$225$	$4$	225.a	1.a	$1$	$2$	$2$	$13.275$	\(\Q(\sqrt{41}) \)	None	✓				15.4.b.a	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(3+3\beta )q^{4}+(6-6\beta )q^{7}+\cdots\)
225.4.a.j	$225$	$4$	225.a	1.a	$1$	$2$	$2$	$13.275$	\(\Q(\sqrt{19}) \)	None	✓		✓		75.4.a.d	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-26\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(12-2\beta )q^{4}+(-13+\cdots)q^{7}+\cdots\)
225.4.a.k	$225$	$4$	225.a	1.a	$1$	$2$	$2$	$13.275$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-15}) \)	✓				45.4.b.a	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$N(\mathrm{U}(1))$	\(q-\beta q^{2}-3q^{4}+11\beta q^{8}-31q^{16}+\cdots\)
225.4.a.l	$225$	$4$	225.a	1.a	$1$	$2$	$2$	$13.275$	\(\Q(\sqrt{10}) \)	None	✓	✓			225.4.a.l	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-30\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+2q^{4}-15q^{7}-6\beta q^{8}-20\beta q^{11}+\cdots\)
225.4.a.m	$225$	$4$	225.a	1.a	$1$	$2$	$2$	$13.275$	\(\Q(\sqrt{10}) \)	None	✓	✓	✓		225.4.a.l	$2$	$0$	\(0\)	\(0\)	\(0\)	\(30\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+2q^{4}+15q^{7}-6\beta q^{8}+20\beta q^{11}+\cdots\)
225.4.a.n	$225$	$4$	225.a	1.a	$1$	$2$	$2$	$13.275$	\(\Q(\sqrt{19}) \)	None	✓				75.4.a.d	$1$	$0$	\(2\)	\(0\)	\(0\)	\(26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(12+2\beta )q^{4}+(13-4\beta )q^{7}+\cdots\)
225.4.a.o	$225$	$4$	225.a	1.a	$1$	$2$	$2$	$13.275$	\(\Q(\sqrt{41}) \)	None	✓				15.4.b.a	$1$	$0$	\(3\)	\(0\)	\(0\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(3+3\beta )q^{4}+(-6+6\beta )q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(225))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(225)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)