Space of modular forms of level 900, weight 6, and trivial character

• Label: 900.6.a
• Level: $900$
• Weight: $6$
• Character orbit: 900.a
• Rep. character: $\chi_{900}(1,\cdot)$
• Character field: $\Q$
• Dimension: $39$
• Newform subspaces: $24$
• Sturm bound: $1080$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	900.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(1080\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(900))\).

	Total	New	Old
Modular forms	936	39	897
Cusp forms	864	39	825
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	Dim
\(-\)	\(+\)	\(+\)	$-$	\(8\)
\(-\)	\(+\)	\(-\)	$+$	\(7\)
\(-\)	\(-\)	\(+\)	$+$	\(11\)
\(-\)	\(-\)	\(-\)	$-$	\(13\)
Plus space			\(+\)	\(18\)
Minus space			\(-\)	\(21\)

Trace form

\( 39 q - 46 q^{7} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(900))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
900.6.a.a	$900$	$6$	900.a	1.a	$1$	$1$	$1$	$144.345$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-236\)		$-$	$+$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-236q^{7}-1202q^{13}-1432q^{19}+\cdots\)
900.6.a.b	$900$	$6$	900.a	1.a	$1$	$1$	$1$	$144.345$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-218\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-218q^{7}+480q^{11}+622q^{13}+186q^{17}+\cdots\)
900.6.a.c	$900$	$6$	900.a	1.a	$1$	$1$	$1$	$144.345$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-211\)		$-$	$+$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-211q^{7}-427q^{13}+3143q^{19}+\cdots\)
900.6.a.d	$900$	$6$	900.a	1.a	$1$	$1$	$1$	$144.345$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-91\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-91q^{7}+174q^{11}+785q^{13}-1794q^{17}+\cdots\)
900.6.a.e	$900$	$6$	900.a	1.a	$1$	$1$	$1$	$144.345$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-56\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-56q^{7}-156q^{11}-350q^{13}+786q^{17}+\cdots\)
900.6.a.f	$900$	$6$	900.a	1.a	$1$	$1$	$1$	$144.345$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-44\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-44q^{7}-6^{3}q^{11}-770q^{13}+534q^{17}+\cdots\)
900.6.a.g	$900$	$6$	900.a	1.a	$1$	$1$	$1$	$144.345$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(16\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{7}+564q^{11}+370q^{13}-1086q^{17}+\cdots\)
900.6.a.h	$900$	$6$	900.a	1.a	$1$	$1$	$1$	$144.345$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(88\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+88q^{7}-540q^{11}+418q^{13}+594q^{17}+\cdots\)
900.6.a.i	$900$	$6$	900.a	1.a	$1$	$1$	$1$	$144.345$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(91\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+91q^{7}+174q^{11}-785q^{13}+1794q^{17}+\cdots\)
900.6.a.j	$900$	$6$	900.a	1.a	$1$	$1$	$1$	$144.345$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(211\)		$-$	$+$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+211q^{7}+427q^{13}+3143q^{19}+\cdots\)
900.6.a.k	$900$	$6$	900.a	1.a	$1$	$1$	$1$	$144.345$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(244\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+244q^{7}+12^{2}q^{11}-50q^{13}-1914q^{17}+\cdots\)
900.6.a.l	$900$	$6$	900.a	1.a	$1$	$2$	$2$	$144.345$	\(\Q(\sqrt{94}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-142\)		$-$	$+$	$+$	$-$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-71q^{7}+\beta q^{11}-137q^{13}+\beta q^{17}+\cdots\)
900.6.a.m	$900$	$6$	900.a	1.a	$1$	$2$	$2$	$144.345$	\(\Q(\sqrt{409}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-40\)		$-$	$-$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-20-2\beta )q^{7}+(30-5\beta )q^{11}+(460+\cdots)q^{13}+\cdots\)
900.6.a.n	$900$	$6$	900.a	1.a	$1$	$2$	$2$	$144.345$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-22\)		$-$	$-$	$+$	$+$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-11+\beta )q^{7}+(-186-3\beta )q^{11}+\cdots\)
900.6.a.o	$900$	$6$	900.a	1.a	$1$	$2$	$2$	$144.345$	\(\Q(\sqrt{61}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-8\beta q^{7}-80q^{11}+24\beta q^{13}+35\beta q^{17}+\cdots\)
900.6.a.p	$900$	$6$	900.a	1.a	$1$	$2$	$2$	$144.345$	\(\Q(\sqrt{61}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-8\beta q^{7}+80q^{11}+24\beta q^{13}-35\beta q^{17}+\cdots\)
900.6.a.q	$900$	$6$	900.a	1.a	$1$	$2$	$2$	$144.345$	\(\Q(\sqrt{31}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+11\beta q^{7}+10^{2}q^{11}+66\beta q^{13}+88\beta q^{17}+\cdots\)
900.6.a.r	$900$	$6$	900.a	1.a	$1$	$2$	$2$	$144.345$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(22\)		$-$	$-$	$-$	$-$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(11+\beta )q^{7}+(-186+3\beta )q^{11}+(5+\cdots)q^{13}+\cdots\)
900.6.a.s	$900$	$6$	900.a	1.a	$1$	$2$	$2$	$144.345$	\(\Q(\sqrt{409}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(40\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(20+2\beta )q^{7}+(30-5\beta )q^{11}+(-460+\cdots)q^{13}+\cdots\)
900.6.a.t	$900$	$6$	900.a	1.a	$1$	$2$	$2$	$144.345$	\(\Q(\sqrt{241}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(80\)		$-$	$+$	$+$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(40-\beta )q^{7}+(-120-5\beta )q^{11}+(340+\cdots)q^{13}+\cdots\)
900.6.a.u	$900$	$6$	900.a	1.a	$1$	$2$	$2$	$144.345$	\(\Q(\sqrt{241}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(80\)		$-$	$+$	$+$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(40-\beta )q^{7}+(120+5\beta )q^{11}+(340+\cdots)q^{13}+\cdots\)
900.6.a.v	$900$	$6$	900.a	1.a	$1$	$2$	$2$	$144.345$	\(\Q(\sqrt{94}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(142\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+71q^{7}-\beta q^{11}+137q^{13}+\beta q^{17}+\cdots\)
900.6.a.w	$900$	$6$	900.a	1.a	$1$	$3$	$3$	$144.345$	3.3.535753.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-88\)		$-$	$-$	$-$	$-$	$2^{3}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-29+\beta _{2})q^{7}+(-48+\beta _{1}+4\beta _{2})q^{11}+\cdots\)
900.6.a.x	$900$	$6$	900.a	1.a	$1$	$3$	$3$	$144.345$	3.3.535753.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(88\)		$-$	$-$	$-$	$-$	$2^{3}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(29-\beta _{2})q^{7}+(-48+\beta _{1}+4\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(900))\) into lower level spaces

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