Space of modular forms of level 900, weight 2, and trivial character

• Label: 900.2.a
• Level: $900$
• Weight: $2$
• Character orbit: 900.a
• Rep. character: $\chi_{900}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $8$
• Sturm bound: $360$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(360\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(900))\).

	Total	New	Old
Modular forms	216	8	208
Cusp forms	145	8	137
Eisenstein series	71	0	71

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
\(+\)	\(+\)	\(+\)	\(+\)		\(27\)	\(0\)	\(27\)		\(16\)	\(0\)	\(16\)		\(11\)	\(0\)	\(11\)
\(+\)	\(+\)	\(-\)	\(-\)		\(29\)	\(0\)	\(29\)		\(17\)	\(0\)	\(17\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(+\)	\(-\)		\(30\)	\(0\)	\(30\)		\(18\)	\(0\)	\(18\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)	\(+\)		\(28\)	\(0\)	\(28\)		\(16\)	\(0\)	\(16\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(+\)	\(-\)		\(27\)	\(2\)	\(25\)		\(21\)	\(2\)	\(19\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(25\)	\(1\)	\(24\)		\(19\)	\(1\)	\(18\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(24\)	\(2\)	\(22\)		\(18\)	\(2\)	\(16\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(26\)	\(3\)	\(23\)		\(20\)	\(3\)	\(17\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(104\)	\(3\)	\(101\)		\(69\)	\(3\)	\(66\)		\(35\)	\(0\)	\(35\)
Minus space			\(-\)		\(112\)	\(5\)	\(107\)		\(76\)	\(5\)	\(71\)		\(36\)	\(0\)	\(36\)

Trace form

8 * q + 2 * q^7 - 4 * q^11 - 4 * q^13 - 6 * q^17 + 6 * q^23 + 18 * q^29 + 20 * q^31 + 8 * q^37 + 14 * q^41 + 2 * q^43 - 6 * q^47 - 6 * q^53 - 8 * q^59 - 8 * q^61 + 14 * q^67 + 12 * q^71 + 8 * q^73 - 12 * q^79 + 6 * q^83 + 26 * q^89 - 16 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(900))\) 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
900.2.a.a	$900$	$2$	900.a	1.a	$1$	$1$	$1$	$7.187$	\(\Q\)	None	✓				60.2.d.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}+4q^{11}+4q^{17}+4q^{23}+6q^{29}+\cdots\)
900.2.a.b	$900$	$2$	900.a	1.a	$1$	$1$	$1$	$7.187$	\(\Q\)	None	✓				20.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-2q^{13}-6q^{17}-4q^{19}+6q^{23}+\cdots\)
900.2.a.c	$900$	$2$	900.a	1.a	$1$	$1$	$1$	$7.187$	\(\Q\)	None	✓				300.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-6q^{11}+5q^{13}+6q^{17}+5q^{19}+\cdots\)
900.2.a.d	$900$	$2$	900.a	1.a	$1$	$1$	$1$	$7.187$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓	✓	✓	900.2.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-q^{7}-7q^{13}-7q^{19}+11q^{31}-10q^{37}+\cdots\)
900.2.a.e	$900$	$2$	900.a	1.a	$1$	$1$	$1$	$7.187$	\(\Q\)	None	✓				300.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-6q^{11}-5q^{13}-6q^{17}+5q^{19}+\cdots\)
900.2.a.f	$900$	$2$	900.a	1.a	$1$	$1$	$1$	$7.187$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			900.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+q^{7}+7q^{13}-7q^{19}+11q^{31}+10q^{37}+\cdots\)
900.2.a.g	$900$	$2$	900.a	1.a	$1$	$1$	$1$	$7.187$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				36.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+4q^{7}-2q^{13}+8q^{19}-4q^{31}+10q^{37}+\cdots\)
900.2.a.h	$900$	$2$	900.a	1.a	$1$	$1$	$1$	$7.187$	\(\Q\)	None	✓				60.2.d.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}+4q^{11}-4q^{17}-4q^{23}+6q^{29}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(900))\) into lower level spaces

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