Space of modular forms of level 1656, weight 4, and trivial character

• Label: 1656.4.a
• Level: $1656$
• Weight: $4$
• Character orbit: 1656.a
• Rep. character: $\chi_{1656}(1,\cdot)$
• Character field: $\Q$
• Dimension: $82$
• Newform subspaces: $21$
• Sturm bound: $1152$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1656 = 2^{3} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1656.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(1152\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1656))\).

	Total	New	Old
Modular forms	880	82	798
Cusp forms	848	82	766
Eisenstein series	32	0	32

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

\(2\)	\(3\)	\(23\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(118\)	\(9\)	\(109\)		\(114\)	\(9\)	\(105\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)		\(102\)	\(7\)	\(95\)		\(98\)	\(7\)	\(91\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(108\)	\(12\)	\(96\)		\(104\)	\(12\)	\(92\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(112\)	\(13\)	\(99\)		\(108\)	\(13\)	\(95\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(114\)	\(7\)	\(107\)		\(110\)	\(7\)	\(103\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(106\)	\(9\)	\(97\)		\(102\)	\(9\)	\(93\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(112\)	\(13\)	\(99\)		\(108\)	\(13\)	\(95\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(108\)	\(12\)	\(96\)		\(104\)	\(12\)	\(92\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(448\)	\(44\)	\(404\)		\(432\)	\(44\)	\(388\)		\(16\)	\(0\)	\(16\)
Minus space			\(-\)		\(432\)	\(38\)	\(394\)		\(416\)	\(38\)	\(378\)		\(16\)	\(0\)	\(16\)

Trace form

82 * q + 14 * q^5 + 48 * q^7 - 50 * q^11 - 128 * q^13 + 196 * q^17 + 90 * q^19 + 1754 * q^25 + 36 * q^29 - 156 * q^31 - 432 * q^35 + 458 * q^37 + 236 * q^41 + 174 * q^43 - 172 * q^47 + 4634 * q^49 - 174 * q^53 - 1048 * q^55 - 960 * q^59 + 1138 * q^61 - 976 * q^65 + 1406 * q^67 + 1892 * q^71 + 3284 * q^73 - 504 * q^77 - 1340 * q^79 - 1046 * q^83 - 1588 * q^85 + 1428 * q^89 - 2516 * q^91 + 752 * q^95 - 1236 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1656))\) 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	3	23
1656.4.a.a	$1656$	$4$	1656.a	1.a	$1$	$1$	$1$	$97.707$	\(\Q\)	None	✓				184.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-22\)	\(8\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-22q^{5}+8q^{7}+20q^{11}+22q^{13}+\cdots\)
1656.4.a.b	$1656$	$4$	1656.a	1.a	$1$	$1$	$1$	$97.707$	\(\Q\)	None	✓				552.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(-22\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{5}-22q^{7}+4q^{11}-14q^{13}+\cdots\)
1656.4.a.c	$1656$	$4$	1656.a	1.a	$1$	$1$	$1$	$97.707$	\(\Q\)	None	✓				184.4.a.b	$1$	$1$	\(0\)	\(0\)	\(4\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}-4q^{7}-26q^{11}+70q^{13}+\cdots\)
1656.4.a.d	$1656$	$4$	1656.a	1.a	$1$	$1$	$1$	$97.707$	\(\Q\)	None	✓				552.4.a.a	$1$	$1$	\(0\)	\(0\)	\(14\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+14q^{5}+2q^{7}-58q^{11}-50q^{13}+\cdots\)
1656.4.a.e	$1656$	$4$	1656.a	1.a	$1$	$2$	$2$	$97.707$	\(\Q(\sqrt{93}) \)	None	✓				552.4.a.d	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(10\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{5}+(5-\beta )q^{7}+(8-4\beta )q^{11}+\cdots\)
1656.4.a.f	$1656$	$4$	1656.a	1.a	$1$	$2$	$2$	$97.707$	\(\Q(\sqrt{2}) \)	None	✓				552.4.a.f	$1$	$1$	\(0\)	\(0\)	\(4\)	\(-8\)		$+$	$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(-4-3\beta )q^{7}+(2^{5}-4\beta )q^{11}+\cdots\)
1656.4.a.g	$1656$	$4$	1656.a	1.a	$1$	$2$	$2$	$97.707$	\(\Q(\sqrt{2}) \)	None	✓				552.4.a.c	$1$	$1$	\(0\)	\(0\)	\(8\)	\(4\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(4+4\beta )q^{5}+(2-9\beta )q^{7}+(-20+7\beta )q^{11}+\cdots\)
1656.4.a.h	$1656$	$4$	1656.a	1.a	$1$	$2$	$2$	$97.707$	\(\Q(\sqrt{5}) \)	None	✓				552.4.a.e	$1$	$1$	\(0\)	\(0\)	\(14\)	\(10\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(7+\beta )q^{5}+(5+\beta )q^{7}+(10-2\beta )q^{11}+\cdots\)
1656.4.a.i	$1656$	$4$	1656.a	1.a	$1$	$3$	$3$	$97.707$	3.3.761.1	None	✓				184.4.a.d	$1$	$1$	\(0\)	\(0\)	\(-16\)	\(18\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-5-2\beta _{1}-\beta _{2})q^{5}+(5+3\beta _{1})q^{7}+\cdots\)
1656.4.a.j	$1656$	$4$	1656.a	1.a	$1$	$3$	$3$	$97.707$	3.3.761.1	None	✓				184.4.a.c	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-28\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}+\beta _{2})q^{5}+(-10+2\beta _{1}+\cdots)q^{7}+\cdots\)
1656.4.a.k	$1656$	$4$	1656.a	1.a	$1$	$4$	$4$	$97.707$	4.4.2239521.1	None	✓				552.4.a.h	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(50\)		$+$	$-$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{3})q^{5}+(12-\beta _{2})q^{7}+(-5+\cdots)q^{11}+\cdots\)
1656.4.a.l	$1656$	$4$	1656.a	1.a	$1$	$4$	$4$	$97.707$	4.4.2822449.1	None	✓		✓		184.4.a.f	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-32\)		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1}-\beta _{3})q^{5}+(-8-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1656.4.a.m	$1656$	$4$	1656.a	1.a	$1$	$4$	$4$	$97.707$	4.4.1623576.1	None	✓				552.4.a.g	$1$	$0$	\(0\)	\(0\)	\(14\)	\(14\)		$-$	$-$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{3})q^{5}+(4-\beta _{1}+\beta _{2})q^{7}+(5+\cdots)q^{11}+\cdots\)
1656.4.a.n	$1656$	$4$	1656.a	1.a	$1$	$4$	$4$	$97.707$	4.4.167313.1	None	✓				184.4.a.e	$1$	$0$	\(0\)	\(0\)	\(20\)	\(-10\)		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(5-\beta _{2}-\beta _{3})q^{5}+(-1-\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots\)
1656.4.a.o	$1656$	$4$	1656.a	1.a	$1$	$5$	$5$	$97.707$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		552.4.a.j	$1$	$1$	\(0\)	\(0\)	\(-16\)	\(0\)		$-$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{2})q^{5}-\beta _{3}q^{7}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
1656.4.a.p	$1656$	$4$	1656.a	1.a	$1$	$5$	$5$	$97.707$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		552.4.a.i	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(-14\)		$+$	$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(-3-\beta _{2})q^{7}+(4+\cdots)q^{11}+\cdots\)
1656.4.a.q	$1656$	$4$	1656.a	1.a	$1$	$6$	$6$	$97.707$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		552.4.a.k	$1$	$0$	\(0\)	\(0\)	\(6\)	\(2\)		$-$	$-$	$+$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2})q^{5}+(-\beta _{1}+\beta _{2})q^{7}+(-4+\cdots)q^{11}+\cdots\)
1656.4.a.r	$1656$	$4$	1656.a	1.a	$1$	$7$	$7$	$97.707$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	1656.4.a.r	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-2\)		$-$	$+$	$+$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{6})q^{5}-\beta _{3}q^{7}+(7+\beta _{1}+\beta _{6})q^{11}+\cdots\)
1656.4.a.s	$1656$	$4$	1656.a	1.a	$1$	$7$	$7$	$97.707$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	1656.4.a.r	$1$	$1$	\(0\)	\(0\)	\(4\)	\(-2\)		$+$	$+$	$-$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{6})q^{5}-\beta _{3}q^{7}+(-7-\beta _{1}-\beta _{6})q^{11}+\cdots\)
1656.4.a.t	$1656$	$4$	1656.a	1.a	$1$	$9$	$9$	$97.707$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓	✓	1656.4.a.t	$1$	$0$	\(0\)	\(0\)	\(-26\)	\(26\)		$+$	$+$	$+$	$+$	$2^{9}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{5}+(3+\beta _{3})q^{7}+(-3+\cdots)q^{11}+\cdots\)
1656.4.a.u	$1656$	$4$	1656.a	1.a	$1$	$9$	$9$	$97.707$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓	✓	1656.4.a.t	$1$	$0$	\(0\)	\(0\)	\(26\)	\(26\)		$-$	$+$	$-$	$+$	$2^{9}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{5}+(3+\beta _{3})q^{7}+(3-\beta _{5}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1656))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1656)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(184))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(207))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(414))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(552))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(828))\)\(^{\oplus 2}\)