Space of modular forms of level 285, weight 2, and trivial character

• Label: 285.2.a
• Level: $285$
• Weight: $2$
• Character orbit: 285.a
• Rep. character: $\chi_{285}(1,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $7$
• Sturm bound: $80$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 285 = 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	285.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(80\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(285))\).

	Total	New	Old
Modular forms	44	11	33
Cusp forms	37	11	26
Eisenstein series	7	0	7

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

\(3\)	\(5\)	\(19\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(2\)	\(1\)	\(1\)		\(2\)	\(1\)	\(1\)		\(0\)	\(0\)	\(0\)
\(+\)	\(+\)	\(-\)	\(-\)		\(8\)	\(2\)	\(6\)		\(7\)	\(2\)	\(5\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(9\)	\(3\)	\(6\)		\(8\)	\(3\)	\(5\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(5\)	\(2\)	\(3\)		\(4\)	\(2\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(7\)	\(1\)	\(6\)		\(6\)	\(1\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(6\)	\(0\)	\(6\)		\(5\)	\(0\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(4\)	\(2\)	\(2\)		\(3\)	\(2\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(18\)	\(2\)	\(16\)		\(15\)	\(2\)	\(13\)		\(3\)	\(0\)	\(3\)
Minus space			\(-\)		\(26\)	\(9\)	\(17\)		\(22\)	\(9\)	\(13\)		\(4\)	\(0\)	\(4\)

Trace form

11 * q + 5 * q^2 - q^3 + 13 * q^4 - q^5 - 3 * q^6 + 9 * q^8 + 11 * q^9 - 3 * q^10 + 12 * q^11 - 7 * q^12 - 6 * q^13 - q^15 + 21 * q^16 - 10 * q^17 + 5 * q^18 - q^19 + 9 * q^20 - 8 * q^21 + 4 * q^22 + 9 * q^24 + 11 * q^25 - 26 * q^26 - q^27 - 8 * q^28 + 10 * q^29 + q^30 - 7 * q^32 - 4 * q^33 - 6 * q^34 + 13 * q^36 + 2 * q^37 - 3 * q^38 - 6 * q^39 - 15 * q^40 - 18 * q^41 + 8 * q^42 + 20 * q^43 + 12 * q^44 - q^45 - 32 * q^46 + 8 * q^47 - 31 * q^48 - 13 * q^49 + 5 * q^50 + 14 * q^51 - 42 * q^52 + 10 * q^53 - 3 * q^54 + 20 * q^55 - 48 * q^56 + 3 * q^57 - 42 * q^58 + 52 * q^59 - 7 * q^60 - 22 * q^61 - 56 * q^62 + 21 * q^64 - 6 * q^65 - 20 * q^66 + 12 * q^67 - 70 * q^68 - 8 * q^69 - 8 * q^70 + 24 * q^71 + 9 * q^72 - 10 * q^73 - 26 * q^74 - q^75 - 7 * q^76 + 8 * q^77 - 2 * q^78 + q^80 + 11 * q^81 + 2 * q^82 + 4 * q^83 - 8 * q^84 - 2 * q^85 + 4 * q^86 + 10 * q^87 + 20 * q^88 - 18 * q^89 - 3 * q^90 + 48 * q^91 + 64 * q^92 - 16 * q^93 + 56 * q^94 - q^95 - 15 * q^96 - 58 * q^97 + 29 * q^98 + 12 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(285))\) 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	19
285.2.a.a	$285$	$2$	285.a	1.a	$1$	$1$	$1$	$2.276$	\(\Q\)	None	✓	✓	✓	✓	285.2.a.a	$1$	$1$	\(-1\)	\(1\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-q^{5}-q^{6}-2q^{7}+\cdots\)
285.2.a.b	$285$	$2$	285.a	1.a	$1$	$1$	$1$	$2.276$	\(\Q\)	None	✓	✓	✓	✓	285.2.a.b	$1$	$1$	\(1\)	\(-1\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{5}-q^{6}-2q^{7}+\cdots\)
285.2.a.c	$285$	$2$	285.a	1.a	$1$	$1$	$1$	$2.276$	\(\Q\)	None	✓	✓			285.2.a.c	$1$	$0$	\(1\)	\(-1\)	\(1\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}+q^{5}-q^{6}+4q^{7}+\cdots\)
285.2.a.d	$285$	$2$	285.a	1.a	$1$	$2$	$2$	$2.276$	\(\Q(\sqrt{7}) \)	None	✓	✓	✓		285.2.a.d	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}+5q^{4}+q^{5}-\beta q^{6}+(-1+\cdots)q^{7}+\cdots\)
285.2.a.e	$285$	$2$	285.a	1.a	$1$	$2$	$2$	$2.276$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	285.2.a.e	$1$	$0$	\(0\)	\(2\)	\(2\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+q^{4}+q^{5}+\beta q^{6}+(-1+\cdots)q^{7}+\cdots\)
285.2.a.f	$285$	$2$	285.a	1.a	$1$	$2$	$2$	$2.276$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	285.2.a.f	$1$	$0$	\(2\)	\(-2\)	\(-2\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-q^{3}+(1+2\beta )q^{4}-q^{5}+\cdots\)
285.2.a.g	$285$	$2$	285.a	1.a	$1$	$2$	$2$	$2.276$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	285.2.a.g	$1$	$0$	\(2\)	\(2\)	\(-2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+q^{3}+(1+2\beta )q^{4}-q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(285))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(285)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 2}\)