Space of modular forms of level 1856, weight 2, and trivial character

• Label: 1856.2.a
• Level: $1856$
• Weight: $2$
• Character orbit: 1856.a
• Rep. character: $\chi_{1856}(1,\cdot)$
• Character field: $\Q$
• Dimension: $56$
• Newform subspaces: $29$
• Sturm bound: $480$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 29 \)
Sturm bound:			\(480\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(3\), \(5\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1856))\).

	Total	New	Old
Modular forms	252	56	196
Cusp forms	229	56	173
Eisenstein series	23	0	23

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

\(2\)	\(29\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(56\)	\(11\)	\(45\)		\(51\)	\(11\)	\(40\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(68\)	\(17\)	\(51\)		\(62\)	\(17\)	\(45\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(70\)	\(17\)	\(53\)		\(64\)	\(17\)	\(47\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(58\)	\(11\)	\(47\)		\(52\)	\(11\)	\(41\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(114\)	\(22\)	\(92\)		\(103\)	\(22\)	\(81\)		\(11\)	\(0\)	\(11\)
Minus space		\(-\)		\(138\)	\(34\)	\(104\)		\(126\)	\(34\)	\(92\)		\(12\)	\(0\)	\(12\)

Trace form

\( 56 q + 56 q^{9} + 16 q^{13} + 16 q^{21} + 56 q^{25} - 16 q^{33} + 16 q^{37} - 16 q^{41} + 48 q^{45} + 56 q^{49} + 48 q^{53} - 16 q^{57} - 16 q^{61} + 16 q^{69} + 48 q^{77} + 40 q^{81} + 16 q^{85} + 64 q^{93}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1856))\) 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	29
1856.2.a.a	$1856$	$2$	1856.a	1.a	$1$	$1$	$1$	$14.820$	\(\Q\)	None	✓				116.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(-3\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-3q^{5}-4q^{7}+6q^{9}-q^{11}+\cdots\)
1856.2.a.b	$1856$	$2$	1856.a	1.a	$1$	$1$	$1$	$14.820$	\(\Q\)	None	✓				58.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(3\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+3q^{5}+2q^{7}+6q^{9}-q^{11}+\cdots\)
1856.2.a.c	$1856$	$2$	1856.a	1.a	$1$	$1$	$1$	$14.820$	\(\Q\)	None	✓				116.2.a.c	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{5}+4q^{7}+q^{9}+6q^{11}+\cdots\)
1856.2.a.d	$1856$	$2$	1856.a	1.a	$1$	$1$	$1$	$14.820$	\(\Q\)	None	✓				116.2.a.b	$1$	$2$	\(0\)	\(-1\)	\(-3\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}-4q^{7}-2q^{9}-3q^{11}+\cdots\)
1856.2.a.e	$1856$	$2$	1856.a	1.a	$1$	$1$	$1$	$14.820$	\(\Q\)	None	✓				232.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+2q^{7}-2q^{9}-3q^{11}+\cdots\)
1856.2.a.f	$1856$	$2$	1856.a	1.a	$1$	$1$	$1$	$14.820$	\(\Q\)	None	✓				58.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+2q^{7}-2q^{9}-3q^{11}+\cdots\)
1856.2.a.g	$1856$	$2$	1856.a	1.a	$1$	$1$	$1$	$14.820$	\(\Q\)	None	✓				928.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-2q^{9}+5q^{11}-q^{13}+\cdots\)
1856.2.a.h	$1856$	$2$	1856.a	1.a	$1$	$1$	$1$	$14.820$	\(\Q\)	None	✓				232.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(3\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}-2q^{7}-2q^{9}-3q^{11}+\cdots\)
1856.2.a.i	$1856$	$2$	1856.a	1.a	$1$	$1$	$1$	$14.820$	\(\Q\)	None	✓				116.2.a.b	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}+4q^{7}-2q^{9}+3q^{11}+\cdots\)
1856.2.a.j	$1856$	$2$	1856.a	1.a	$1$	$1$	$1$	$14.820$	\(\Q\)	None	✓				232.2.a.b	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{7}-2q^{9}+3q^{11}+\cdots\)
1856.2.a.k	$1856$	$2$	1856.a	1.a	$1$	$1$	$1$	$14.820$	\(\Q\)	None	✓				58.2.a.b	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{7}-2q^{9}+3q^{11}+\cdots\)
1856.2.a.l	$1856$	$2$	1856.a	1.a	$1$	$1$	$1$	$14.820$	\(\Q\)	None	✓				928.2.a.a	$1$	$1$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{9}-5q^{11}-q^{13}+\cdots\)
1856.2.a.m	$1856$	$2$	1856.a	1.a	$1$	$1$	$1$	$14.820$	\(\Q\)	None	✓				232.2.a.a	$1$	$0$	\(0\)	\(1\)	\(3\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{5}+2q^{7}-2q^{9}+3q^{11}+\cdots\)
1856.2.a.n	$1856$	$2$	1856.a	1.a	$1$	$1$	$1$	$14.820$	\(\Q\)	None	✓				116.2.a.c	$1$	$1$	\(0\)	\(2\)	\(2\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{5}-4q^{7}+q^{9}-6q^{11}+\cdots\)
1856.2.a.o	$1856$	$2$	1856.a	1.a	$1$	$1$	$1$	$14.820$	\(\Q\)	None	✓				116.2.a.a	$1$	$0$	\(0\)	\(3\)	\(-3\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-3q^{5}+4q^{7}+6q^{9}+q^{11}+\cdots\)
1856.2.a.p	$1856$	$2$	1856.a	1.a	$1$	$1$	$1$	$14.820$	\(\Q\)	None	✓				58.2.a.a	$1$	$0$	\(0\)	\(3\)	\(3\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+3q^{5}-2q^{7}+6q^{9}+q^{11}+\cdots\)
1856.2.a.q	$1856$	$2$	1856.a	1.a	$1$	$2$	$2$	$14.820$	\(\Q(\sqrt{2}) \)	None	✓				928.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-q^{5}+2q^{7}-2\beta q^{9}+\cdots\)
1856.2.a.r	$1856$	$2$	1856.a	1.a	$1$	$2$	$2$	$14.820$	\(\Q(\sqrt{2}) \)	None	✓				29.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+q^{5}+2\beta q^{7}-2\beta q^{9}+\cdots\)
1856.2.a.s	$1856$	$2$	1856.a	1.a	$1$	$2$	$2$	$14.820$	\(\Q(\sqrt{2}) \)	None	✓				232.2.a.c	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(1+2\beta )q^{5}+4q^{7}+\cdots\)
1856.2.a.t	$1856$	$2$	1856.a	1.a	$1$	$2$	$2$	$14.820$	\(\Q(\sqrt{5}) \)	None	✓		✓		928.2.a.d	$2$	$1$	\(0\)	\(0\)	\(-6\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-3q^{5}+2q^{9}+\beta q^{11}-q^{13}+\cdots\)
1856.2.a.u	$1856$	$2$	1856.a	1.a	$1$	$2$	$2$	$14.820$	\(\Q(\sqrt{2}) \)	None	✓				928.2.a.c	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-q^{5}-2q^{7}+2\beta q^{9}+(1+\cdots)q^{11}+\cdots\)
1856.2.a.v	$1856$	$2$	1856.a	1.a	$1$	$2$	$2$	$14.820$	\(\Q(\sqrt{2}) \)	None	✓				232.2.a.c	$1$	$1$	\(0\)	\(2\)	\(2\)	\(-8\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1-2\beta )q^{5}-4q^{7}+2\beta q^{9}+\cdots\)
1856.2.a.w	$1856$	$2$	1856.a	1.a	$1$	$2$	$2$	$14.820$	\(\Q(\sqrt{2}) \)	None	✓				29.2.a.a	$1$	$0$	\(0\)	\(2\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+q^{5}+2\beta q^{7}+2\beta q^{9}+\cdots\)
1856.2.a.x	$1856$	$2$	1856.a	1.a	$1$	$3$	$3$	$14.820$	3.3.568.1	None	✓		✓		232.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(-4\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1+\beta _{2})q^{5}+(2+\cdots)q^{9}+\cdots\)
1856.2.a.y	$1856$	$2$	1856.a	1.a	$1$	$3$	$3$	$14.820$	3.3.568.1	None	✓				232.2.a.d	$1$	$0$	\(0\)	\(2\)	\(-4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1+\beta _{2})q^{5}+(2+\beta _{2})q^{9}+\cdots\)
1856.2.a.z	$1856$	$2$	1856.a	1.a	$1$	$4$	$4$	$14.820$	\(\Q(\zeta_{20})^+\)	None	✓				928.2.a.f	$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(2-\beta _{3})q^{5}+(-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
1856.2.a.ba	$1856$	$2$	1856.a	1.a	$1$	$5$	$5$	$14.820$	5.5.230224.1	None	✓				928.2.a.g	$1$	$0$	\(0\)	\(-4\)	\(2\)	\(4\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}-\beta _{4}q^{5}+(1+\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
1856.2.a.bb	$1856$	$2$	1856.a	1.a	$1$	$5$	$5$	$14.820$	5.5.230224.1	None	✓				928.2.a.g	$1$	$0$	\(0\)	\(4\)	\(2\)	\(-4\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}-\beta _{4}q^{5}+(-1-\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
1856.2.a.bc	$1856$	$2$	1856.a	1.a	$1$	$6$	$6$	$14.820$	6.6.68772992.1	None	✓		✓		928.2.a.i	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{3}+(-1-\beta _{3})q^{5}-\beta _{1}q^{7}+(2+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1856))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1856)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(232))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(464))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(928))\)\(^{\oplus 2}\)