Space of modular forms of level 1856, weight 4, and trivial character

• Label: 1856.4.a
• Level: $1856$
• Weight: $4$
• Character orbit: 1856.a
• Rep. character: $\chi_{1856}(1,\cdot)$
• Character field: $\Q$
• Dimension: $168$
• Newform subspaces: $38$
• Sturm bound: $960$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 38 \)
Sturm bound:			\(960\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1856))\).

	Total	New	Old
Modular forms	732	168	564
Cusp forms	708	168	540
Eisenstein series	24	0	24

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	Dim
\(+\)	\(+\)	$+$	\(45\)
\(+\)	\(-\)	$-$	\(39\)
\(-\)	\(+\)	$-$	\(39\)
\(-\)	\(-\)	$+$	\(45\)
Plus space		\(+\)	\(90\)
Minus space		\(-\)	\(78\)

Trace form

\( 168 q + 1512 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\) 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	29
1856.4.a.a	$1856$	$4$	1856.a	1.a	$1$	$1$	$1$	$109.508$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-7\)	\(-5\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}-5q^{5}-2q^{7}+22q^{9}-37q^{11}+\cdots\)
1856.4.a.b	$1856$	$4$	1856.a	1.a	$1$	$1$	$1$	$109.508$	\(\Q\)	None	✓	$1$	$2$	\(0\)	\(-7\)	\(13\)	\(-16\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}+13q^{5}-2^{4}q^{7}+22q^{9}-45q^{11}+\cdots\)
1856.4.a.c	$1856$	$4$	1856.a	1.a	$1$	$1$	$1$	$109.508$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-7\)	\(15\)	\(18\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}+15q^{5}+18q^{7}+22q^{9}+3^{3}q^{11}+\cdots\)
1856.4.a.d	$1856$	$4$	1856.a	1.a	$1$	$1$	$1$	$109.508$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(7\)	\(-5\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}-5q^{5}+2q^{7}+22q^{9}+37q^{11}+\cdots\)
1856.4.a.e	$1856$	$4$	1856.a	1.a	$1$	$1$	$1$	$109.508$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(7\)	\(13\)	\(16\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}+13q^{5}+2^{4}q^{7}+22q^{9}+45q^{11}+\cdots\)
1856.4.a.f	$1856$	$4$	1856.a	1.a	$1$	$1$	$1$	$109.508$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(7\)	\(15\)	\(-18\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}+15q^{5}-18q^{7}+22q^{9}-3^{3}q^{11}+\cdots\)
1856.4.a.g	$1856$	$4$	1856.a	1.a	$1$	$2$	$2$	$109.508$	\(\Q(\sqrt{22}) \)	None	✓	$1$	$0$	\(0\)	\(-10\)	\(-30\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-5+\beta )q^{3}-15q^{5}+2\beta q^{7}+(20+\cdots)q^{9}+\cdots\)
1856.4.a.h	$1856$	$4$	1856.a	1.a	$1$	$2$	$2$	$109.508$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(-10\)	\(10\)	\(16\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-5+3\beta )q^{3}+(5+4\beta )q^{5}+(8+10\beta )q^{7}+\cdots\)
1856.4.a.i	$1856$	$4$	1856.a	1.a	$1$	$2$	$2$	$109.508$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(10\)	\(16\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(5+6\beta )q^{5}+(8-8\beta )q^{7}+\cdots\)
1856.4.a.j	$1856$	$4$	1856.a	1.a	$1$	$2$	$2$	$109.508$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(10\)	\(-20\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(5+2\beta )q^{5}+(-10-4\beta )q^{7}+\cdots\)
1856.4.a.k	$1856$	$4$	1856.a	1.a	$1$	$2$	$2$	$109.508$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(10\)	\(20\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(5-2\beta )q^{5}+(10-4\beta )q^{7}+\cdots\)
1856.4.a.l	$1856$	$4$	1856.a	1.a	$1$	$2$	$2$	$109.508$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(10\)	\(-16\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(5-6\beta )q^{5}+(-8-8\beta )q^{7}+\cdots\)
1856.4.a.m	$1856$	$4$	1856.a	1.a	$1$	$2$	$2$	$109.508$	\(\Q(\sqrt{22}) \)	None	✓	$1$	$1$	\(0\)	\(10\)	\(-30\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(5+\beta )q^{3}-15q^{5}+2\beta q^{7}+(20+10\beta )q^{9}+\cdots\)
1856.4.a.n	$1856$	$4$	1856.a	1.a	$1$	$2$	$2$	$109.508$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(10\)	\(10\)	\(-16\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(5+3\beta )q^{3}+(5-4\beta )q^{5}+(-8+10\beta )q^{7}+\cdots\)
1856.4.a.o	$1856$	$4$	1856.a	1.a	$1$	$3$	$3$	$109.508$	3.3.148344.1	None	✓	$1$	$1$	\(0\)	\(-10\)	\(20\)	\(-8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{3}+(7-\beta _{2})q^{5}+(-4+\cdots)q^{7}+\cdots\)
1856.4.a.p	$1856$	$4$	1856.a	1.a	$1$	$3$	$3$	$109.508$	3.3.229.1	None	✓	$1$	$1$	\(0\)	\(-6\)	\(-4\)	\(16\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+3\beta _{2})q^{3}+(-1-3\beta _{1}+4\beta _{2})q^{5}+\cdots\)
1856.4.a.q	$1856$	$4$	1856.a	1.a	$1$	$3$	$3$	$109.508$	3.3.4481.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(-11\)	\(-38\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-4-\beta _{1}-\beta _{2})q^{5}+\cdots\)
1856.4.a.r	$1856$	$4$	1856.a	1.a	$1$	$3$	$3$	$109.508$	3.3.19816.1	None	✓	$1$	$0$	\(0\)	\(-2\)	\(-20\)	\(24\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-6-2\beta _{1}-\beta _{2})q^{5}+\cdots\)
1856.4.a.s	$1856$	$4$	1856.a	1.a	$1$	$3$	$3$	$109.508$	3.3.19816.1	None	✓	$1$	$1$	\(0\)	\(2\)	\(-20\)	\(-24\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-6-2\beta _{1}-\beta _{2})q^{5}+\cdots\)
1856.4.a.t	$1856$	$4$	1856.a	1.a	$1$	$3$	$3$	$109.508$	3.3.4481.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(-11\)	\(38\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-4-\beta _{1}-\beta _{2})q^{5}+\cdots\)
1856.4.a.u	$1856$	$4$	1856.a	1.a	$1$	$3$	$3$	$109.508$	3.3.229.1	None	✓	$1$	$0$	\(0\)	\(6\)	\(-4\)	\(-16\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-3\beta _{2})q^{3}+(-1-3\beta _{1}+4\beta _{2})q^{5}+\cdots\)
1856.4.a.v	$1856$	$4$	1856.a	1.a	$1$	$3$	$3$	$109.508$	3.3.148344.1	None	✓	$1$	$0$	\(0\)	\(10\)	\(20\)	\(8\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{3}+(7-\beta _{2})q^{5}+(4-2\beta _{1}+\cdots)q^{7}+\cdots\)
1856.4.a.w	$1856$	$4$	1856.a	1.a	$1$	$4$	$4$	$109.508$	4.4.225792.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(20\)	\(-8\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(5+\beta _{2})q^{5}+(-2+2\beta _{1}+\cdots)q^{7}+\cdots\)
1856.4.a.x	$1856$	$4$	1856.a	1.a	$1$	$4$	$4$	$109.508$	4.4.225792.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(20\)	\(8\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(\beta _{1}-\beta _{3})q^{3}+(5-\beta _{2})q^{5}+(2+2\beta _{1}+\cdots)q^{7}+\cdots\)
1856.4.a.y	$1856$	$4$	1856.a	1.a	$1$	$5$	$5$	$109.508$	5.5.13458092.1	None	✓	$1$	$1$	\(0\)	\(-8\)	\(-10\)	\(40\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{3})q^{3}+(-2+2\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
1856.4.a.z	$1856$	$4$	1856.a	1.a	$1$	$5$	$5$	$109.508$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-4\)	\(-10\)	\(32\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-2-\beta _{3})q^{5}+(6+\cdots)q^{7}+\cdots\)
1856.4.a.ba	$1856$	$4$	1856.a	1.a	$1$	$5$	$5$	$109.508$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(4\)	\(-10\)	\(-32\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-2-\beta _{3})q^{5}+(-6+\cdots)q^{7}+\cdots\)
1856.4.a.bb	$1856$	$4$	1856.a	1.a	$1$	$5$	$5$	$109.508$	5.5.13458092.1	None	✓	$1$	$0$	\(0\)	\(8\)	\(-10\)	\(-40\)		$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{3})q^{3}+(-2+2\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
1856.4.a.bc	$1856$	$4$	1856.a	1.a	$1$	$6$	$6$	$109.508$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-5\)	\(5\)	\(38\)		$-$	$-$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{3})q^{3}+(1+\beta _{3}+\beta _{4})q^{5}+\cdots\)
1856.4.a.bd	$1856$	$4$	1856.a	1.a	$1$	$6$	$6$	$109.508$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(5\)	\(5\)	\(-38\)		$+$	$-$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{3})q^{3}+(1+\beta _{3}+\beta _{4})q^{5}+(-6+\cdots)q^{7}+\cdots\)
1856.4.a.be	$1856$	$4$	1856.a	1.a	$1$	$8$	$8$	$109.508$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(20\)	\(0\)		$-$	$+$	$-$	$2^{10}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(3+\beta _{2})q^{5}-\beta _{4}q^{7}+(5+\beta _{3}+\cdots)q^{9}+\cdots\)
1856.4.a.bf	$1856$	$4$	1856.a	1.a	$1$	$9$	$9$	$109.508$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-4\)	\(-10\)	\(12\)		$+$	$-$	$-$	$2^{12}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{1}+\beta _{4})q^{5}+(1+\beta _{6}+\cdots)q^{7}+\cdots\)
1856.4.a.bg	$1856$	$4$	1856.a	1.a	$1$	$9$	$9$	$109.508$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(4\)	\(-10\)	\(-12\)		$+$	$-$	$-$	$2^{12}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{1}+\beta _{4})q^{5}+(-1+\cdots)q^{7}+\cdots\)
1856.4.a.bh	$1856$	$4$	1856.a	1.a	$1$	$10$	$10$	$109.508$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(-40\)	\(0\)		$-$	$+$	$-$	$2^{11}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-4-\beta _{3})q^{5}+\beta _{8}q^{7}+(6+\cdots)q^{9}+\cdots\)
1856.4.a.bi	$1856$	$4$	1856.a	1.a	$1$	$10$	$10$	$109.508$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$-$	$+$	$2^{14}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{2}q^{5}-\beta _{6}q^{7}+(5-\beta _{5}+\cdots)q^{9}+\cdots\)
1856.4.a.bj	$1856$	$4$	1856.a	1.a	$1$	$12$	$12$	$109.508$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-14\)	\(10\)	\(44\)		$+$	$+$	$+$	$2^{17}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(1-\beta _{4})q^{5}+(4-\beta _{6}+\cdots)q^{7}+\cdots\)
1856.4.a.bk	$1856$	$4$	1856.a	1.a	$1$	$12$	$12$	$109.508$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)		$-$	$-$	$+$	$2^{19}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{2})q^{5}-\beta _{7}q^{7}+(14+\cdots)q^{9}+\cdots\)
1856.4.a.bl	$1856$	$4$	1856.a	1.a	$1$	$12$	$12$	$109.508$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(14\)	\(10\)	\(-44\)		$+$	$+$	$+$	$2^{17}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(1-\beta _{4})q^{5}+(-4+\beta _{6}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1856))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1856)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(232))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(464))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(928))\)\(^{\oplus 2}\)