Space of modular forms of level 855, weight 2, and trivial character

• Label: 855.2.a
• Level: $855$
• Weight: $2$
• Character orbit: 855.a
• Rep. character: $\chi_{855}(1,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $13$
• Sturm bound: $240$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(240\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(855))\).

	Total	New	Old
Modular forms	128	30	98
Cusp forms	113	30	83
Eisenstein series	15	0	15

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	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(7\)
Plus space			\(+\)	\(11\)
Minus space			\(-\)	\(19\)

Trace form

30 * q - 4 * q^2 + 26 * q^4 + 2 * q^5 - 4 * q^7 - 8 * q^11 - 4 * q^13 + 12 * q^14 + 10 * q^16 + 4 * q^17 - 2 * q^20 + 16 * q^22 + 12 * q^23 + 30 * q^25 + 28 * q^26 + 12 * q^28 - 4 * q^29 + 8 * q^31 + 16 * q^32 + 24 * q^34 + 4 * q^35 - 8 * q^37 + 6 * q^38 + 4 * q^41 + 4 * q^43 - 24 * q^44 + 4 * q^46 + 4 * q^47 + 14 * q^49 - 4 * q^50 - 28 * q^52 - 16 * q^53 + 8 * q^55 + 44 * q^56 - 24 * q^58 - 32 * q^59 + 4 * q^61 + 8 * q^62 - 38 * q^64 - 4 * q^67 + 68 * q^68 - 20 * q^70 + 4 * q^73 - 4 * q^74 + 24 * q^77 - 16 * q^79 + 6 * q^80 - 16 * q^82 + 28 * q^83 - 20 * q^85 + 20 * q^86 + 44 * q^88 + 12 * q^89 + 16 * q^91 - 44 * q^92 - 4 * q^94 + 8 * q^95 + 16 * q^97 - 52 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(855))\) 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
855.2.a.a	$855$	$2$	855.a	1.a	$1$	$1$	$1$	$6.827$	\(\Q\)	None	✓				285.2.a.c	$1$	$0$	\(-1\)	\(0\)	\(-1\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-q^{5}+4q^{7}+3q^{8}+q^{10}+\cdots\)
855.2.a.b	$855$	$2$	855.a	1.a	$1$	$1$	$1$	$6.827$	\(\Q\)	None	✓				285.2.a.b	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+q^{5}-2q^{7}+3q^{8}-q^{10}+\cdots\)
855.2.a.c	$855$	$2$	855.a	1.a	$1$	$1$	$1$	$6.827$	\(\Q\)	None	✓				285.2.a.a	$1$	$0$	\(1\)	\(0\)	\(1\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+q^{5}-2q^{7}-3q^{8}+q^{10}+\cdots\)
855.2.a.d	$855$	$2$	855.a	1.a	$1$	$2$	$2$	$6.827$	\(\Q(\sqrt{2}) \)	None	✓		✓		285.2.a.g	$1$	$1$	\(-2\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+q^{5}+\beta q^{7}+\cdots\)
855.2.a.e	$855$	$2$	855.a	1.a	$1$	$2$	$2$	$6.827$	\(\Q(\sqrt{2}) \)	None	✓				285.2.a.f	$1$	$0$	\(-2\)	\(0\)	\(2\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+q^{5}+(2+\cdots)q^{7}+\cdots\)
855.2.a.f	$855$	$2$	855.a	1.a	$1$	$2$	$2$	$6.827$	\(\Q(\sqrt{3}) \)	None	✓		✓	✓	285.2.a.e	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{4}-q^{5}+(-1-\beta )q^{7}-\beta q^{8}+\cdots\)
855.2.a.g	$855$	$2$	855.a	1.a	$1$	$2$	$2$	$6.827$	\(\Q(\sqrt{7}) \)	None	✓				285.2.a.d	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+5q^{4}-q^{5}+(-1+\beta )q^{7}+\cdots\)
855.2.a.h	$855$	$2$	855.a	1.a	$1$	$3$	$3$	$6.827$	3.3.148.1	None	✓	✓	✓	✓	855.2.a.h	$1$	$1$	\(-1\)	\(0\)	\(-3\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}-q^{5}-\beta _{2}q^{7}+\cdots\)
855.2.a.i	$855$	$2$	855.a	1.a	$1$	$3$	$3$	$6.827$	3.3.148.1	None	✓		✓		95.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(-3\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}-q^{5}+2\beta _{2}q^{7}+\cdots\)
855.2.a.j	$855$	$2$	855.a	1.a	$1$	$3$	$3$	$6.827$	3.3.148.1	None	✓	✓	✓	✓	855.2.a.j	$1$	$1$	\(-1\)	\(0\)	\(3\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+q^{5}+(-2+\cdots)q^{7}+\cdots\)
855.2.a.k	$855$	$2$	855.a	1.a	$1$	$3$	$3$	$6.827$	3.3.148.1	None	✓	✓	✓	✓	855.2.a.j	$1$	$0$	\(1\)	\(0\)	\(-3\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}-q^{5}+(-2+\cdots)q^{7}+\cdots\)
855.2.a.l	$855$	$2$	855.a	1.a	$1$	$3$	$3$	$6.827$	3.3.148.1	None	✓	✓	✓	✓	855.2.a.h	$1$	$0$	\(1\)	\(0\)	\(3\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+q^{5}-\beta _{2}q^{7}+\cdots\)
855.2.a.m	$855$	$2$	855.a	1.a	$1$	$4$	$4$	$6.827$	4.4.11344.1	None	✓		✓		95.2.a.b	$1$	$0$	\(2\)	\(0\)	\(4\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+q^{5}+(2+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(855))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(855)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(285))\)\(^{\oplus 2}\)