Space of modular forms of level 1568, weight 4, and trivial character

• Label: 1568.4.a
• Level: $1568$
• Weight: $4$
• Character orbit: 1568.a
• Rep. character: $\chi_{1568}(1,\cdot)$
• Character field: $\Q$
• Dimension: $123$
• Newform subspaces: $39$
• Sturm bound: $896$
• Trace bound: $25$

Defining parameters

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1568.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 39 \)
Sturm bound:			\(896\)
Trace bound:			\(25\)
Distinguishing \(T_p\):			\(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1568))\).

	Total	New	Old
Modular forms	704	123	581
Cusp forms	640	123	517
Eisenstein series	64	0	64

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

\(2\)	\(7\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(180\)	\(32\)	\(148\)		\(164\)	\(32\)	\(132\)		\(16\)	\(0\)	\(16\)
\(+\)	\(-\)	\(-\)		\(172\)	\(30\)	\(142\)		\(156\)	\(30\)	\(126\)		\(16\)	\(0\)	\(16\)
\(-\)	\(+\)	\(-\)		\(172\)	\(28\)	\(144\)		\(156\)	\(28\)	\(128\)		\(16\)	\(0\)	\(16\)
\(-\)	\(-\)	\(+\)		\(180\)	\(33\)	\(147\)		\(164\)	\(33\)	\(131\)		\(16\)	\(0\)	\(16\)
Plus space		\(+\)		\(360\)	\(65\)	\(295\)		\(328\)	\(65\)	\(263\)		\(32\)	\(0\)	\(32\)
Minus space		\(-\)		\(344\)	\(58\)	\(286\)		\(312\)	\(58\)	\(254\)		\(32\)	\(0\)	\(32\)

Trace form

123 * q + 2 * q^5 + 1127 * q^9 + 26 * q^13 - 154 * q^17 + 2973 * q^25 - 86 * q^29 - 592 * q^33 - 174 * q^37 - 2 * q^41 - 454 * q^45 - 1086 * q^53 - 720 * q^57 - 1670 * q^61 + 236 * q^65 + 720 * q^69 + 1374 * q^73 + 14035 * q^81 - 1708 * q^85 + 1934 * q^89 - 176 * q^93 + 3254 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1568))\) 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	7
1568.4.a.a	$1568$	$4$	1568.a	1.a	$1$	$1$	$1$	$92.515$	\(\Q\)	None	✓	✓			1568.4.a.a	$1$	$1$	\(0\)	\(-8\)	\(-4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}-4q^{5}+37q^{9}-40q^{11}+6^{2}q^{13}+\cdots\)
1568.4.a.b	$1568$	$4$	1568.a	1.a	$1$	$1$	$1$	$92.515$	\(\Q\)	None	✓	✓			1568.4.a.a	$1$	$0$	\(0\)	\(-8\)	\(4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}+4q^{5}+37q^{9}+40q^{11}-6^{2}q^{13}+\cdots\)
1568.4.a.c	$1568$	$4$	1568.a	1.a	$1$	$1$	$1$	$92.515$	\(\Q\)	None	✓				32.4.a.a	$1$	$1$	\(0\)	\(-8\)	\(10\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}+10q^{5}+37q^{9}-40q^{11}+\cdots\)
1568.4.a.d	$1568$	$4$	1568.a	1.a	$1$	$1$	$1$	$92.515$	\(\Q\)	None	✓	✓			1568.4.a.d	$1$	$0$	\(0\)	\(-2\)	\(-14\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-14q^{5}-23q^{9}-20q^{11}+\cdots\)
1568.4.a.e	$1568$	$4$	1568.a	1.a	$1$	$1$	$1$	$92.515$	\(\Q\)	None	✓				224.4.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-23q^{9}-20q^{11}+20q^{13}+\cdots\)
1568.4.a.f	$1568$	$4$	1568.a	1.a	$1$	$1$	$1$	$92.515$	\(\Q\)	None	✓	✓			1568.4.a.d	$1$	$1$	\(0\)	\(-2\)	\(14\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+14q^{5}-23q^{9}+20q^{11}+\cdots\)
1568.4.a.g	$1568$	$4$	1568.a	1.a	$1$	$1$	$1$	$92.515$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.4.a.b	$2$	$1$	\(0\)	\(0\)	\(-22\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-22q^{5}-3^{3}q^{9}+18q^{13}+94q^{17}+\cdots\)
1568.4.a.h	$1568$	$4$	1568.a	1.a	$1$	$1$	$1$	$92.515$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	✓			1568.4.a.h	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-4q^{5}-3^{3}q^{9}-92q^{13}-104q^{17}+\cdots\)
1568.4.a.i	$1568$	$4$	1568.a	1.a	$1$	$1$	$1$	$92.515$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	✓			1568.4.a.h	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+4q^{5}-3^{3}q^{9}+92q^{13}+104q^{17}+\cdots\)
1568.4.a.j	$1568$	$4$	1568.a	1.a	$1$	$1$	$1$	$92.515$	\(\Q\)	None	✓	✓			1568.4.a.d	$1$	$1$	\(0\)	\(2\)	\(-14\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-14q^{5}-23q^{9}+20q^{11}+\cdots\)
1568.4.a.k	$1568$	$4$	1568.a	1.a	$1$	$1$	$1$	$92.515$	\(\Q\)	None	✓				224.4.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-23q^{9}+20q^{11}+20q^{13}+\cdots\)
1568.4.a.l	$1568$	$4$	1568.a	1.a	$1$	$1$	$1$	$92.515$	\(\Q\)	None	✓	✓			1568.4.a.d	$1$	$0$	\(0\)	\(2\)	\(14\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+14q^{5}-23q^{9}-20q^{11}+\cdots\)
1568.4.a.m	$1568$	$4$	1568.a	1.a	$1$	$1$	$1$	$92.515$	\(\Q\)	None	✓	✓			1568.4.a.a	$1$	$0$	\(0\)	\(8\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}-4q^{5}+37q^{9}+40q^{11}+6^{2}q^{13}+\cdots\)
1568.4.a.n	$1568$	$4$	1568.a	1.a	$1$	$1$	$1$	$92.515$	\(\Q\)	None	✓	✓			1568.4.a.a	$1$	$1$	\(0\)	\(8\)	\(4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}+4q^{5}+37q^{9}-40q^{11}-6^{2}q^{13}+\cdots\)
1568.4.a.o	$1568$	$4$	1568.a	1.a	$1$	$1$	$1$	$92.515$	\(\Q\)	None	✓				32.4.a.a	$1$	$0$	\(0\)	\(8\)	\(10\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}+10q^{5}+37q^{9}+40q^{11}+\cdots\)
1568.4.a.p	$1568$	$4$	1568.a	1.a	$1$	$2$	$2$	$92.515$	\(\Q(\sqrt{37}) \)	None	✓				224.4.a.c	$1$	$1$	\(0\)	\(-6\)	\(6\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{3}+(3-\beta )q^{5}+(19+6\beta )q^{9}+\cdots\)
1568.4.a.q	$1568$	$4$	1568.a	1.a	$1$	$2$	$2$	$92.515$	\(\Q(\sqrt{7}) \)	None	✓				224.4.i.a	$2$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}-20q^{9}+\beta q^{11}+60q^{13}+\cdots\)
1568.4.a.r	$1568$	$4$	1568.a	1.a	$1$	$2$	$2$	$92.515$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-1}) \)	✓	✓			1568.4.a.r	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+9\beta q^{5}-3^{3}q^{9}+55\beta q^{13}+99\beta q^{17}+\cdots\)
1568.4.a.s	$1568$	$4$	1568.a	1.a	$1$	$2$	$2$	$92.515$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-1}) \)	✓	✓			1568.4.a.s	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-13\beta q^{5}-3^{3}q^{9}+37\beta q^{13}+5\beta q^{17}+\cdots\)
1568.4.a.t	$1568$	$4$	1568.a	1.a	$1$	$2$	$2$	$92.515$	\(\Q(\sqrt{7}) \)	None	✓				224.4.i.a	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}-20q^{9}-\beta q^{11}-60q^{13}+\cdots\)
1568.4.a.u	$1568$	$4$	1568.a	1.a	$1$	$2$	$2$	$92.515$	\(\Q(\sqrt{37}) \)	None	✓				224.4.a.c	$1$	$1$	\(0\)	\(6\)	\(6\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{3}+(3-\beta )q^{5}+(19+6\beta )q^{9}+\cdots\)
1568.4.a.v	$1568$	$4$	1568.a	1.a	$1$	$3$	$3$	$92.515$	3.3.2981.1	None	✓				224.4.a.e	$1$	$0$	\(0\)	\(-8\)	\(-10\)	\(0\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{3}+(-3-\beta _{2})q^{5}+(11+\cdots)q^{9}+\cdots\)
1568.4.a.w	$1568$	$4$	1568.a	1.a	$1$	$3$	$3$	$92.515$	3.3.621.1	None	✓				224.4.a.f	$1$	$0$	\(0\)	\(0\)	\(6\)	\(0\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(2+2\beta _{1}+\beta _{2})q^{5}+(5-\beta _{1}+\cdots)q^{9}+\cdots\)
1568.4.a.x	$1568$	$4$	1568.a	1.a	$1$	$3$	$3$	$92.515$	3.3.621.1	None	✓				224.4.a.f	$1$	$0$	\(0\)	\(0\)	\(6\)	\(0\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(2+2\beta _{1}+\beta _{2})q^{5}+(5-\beta _{1}+\cdots)q^{9}+\cdots\)
1568.4.a.y	$1568$	$4$	1568.a	1.a	$1$	$3$	$3$	$92.515$	3.3.2981.1	None	✓				224.4.a.e	$1$	$1$	\(0\)	\(8\)	\(-10\)	\(0\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{3}+(-3-\beta _{2})q^{5}+(11+3\beta _{1}+\cdots)q^{9}+\cdots\)
1568.4.a.z	$1568$	$4$	1568.a	1.a	$1$	$4$	$4$	$92.515$	\(\Q(\sqrt{32 +2 \sqrt{67}})\)	None	✓				224.4.i.b	$2$	$1$	\(0\)	\(0\)	\(-12\)	\(0\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}+(-3-\beta _{1})q^{5}+(26+\beta _{1}+\cdots)q^{9}+\cdots\)
1568.4.a.ba	$1568$	$4$	1568.a	1.a	$1$	$4$	$4$	$92.515$	\(\Q(\sqrt{7}, \sqrt{23})\)	None	✓	✓			1568.4.a.ba	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{2}q^{5}+q^{9}+\beta _{3}q^{11}-3\beta _{2}q^{13}+\cdots\)
1568.4.a.bb	$1568$	$4$	1568.a	1.a	$1$	$4$	$4$	$92.515$	\(\Q(\sqrt{2}, \sqrt{17})\)	None	✓	✓			1568.4.a.bb	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-4\beta _{1}q^{5}+7q^{9}+\beta _{3}q^{11}+\cdots\)
1568.4.a.bc	$1568$	$4$	1568.a	1.a	$1$	$4$	$4$	$92.515$	\(\Q(\sqrt{11}, \sqrt{19})\)	None	✓	✓			1568.4.a.bc	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+3\beta _{1}q^{5}+7^{2}q^{9}+\beta _{3}q^{11}+\cdots\)
1568.4.a.bd	$1568$	$4$	1568.a	1.a	$1$	$4$	$4$	$92.515$	\(\Q(\sqrt{32 +2 \sqrt{67}})\)	None	✓				224.4.i.b	$2$	$0$	\(0\)	\(0\)	\(12\)	\(0\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}+(3+\beta _{1})q^{5}+(26+\beta _{1})q^{9}+\cdots\)
1568.4.a.be	$1568$	$4$	1568.a	1.a	$1$	$6$	$6$	$92.515$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				224.4.i.c	$1$	$1$	\(0\)	\(-6\)	\(-10\)	\(0\)		$-$	$+$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-2-\beta _{2})q^{5}+(6+\cdots)q^{9}+\cdots\)
1568.4.a.bf	$1568$	$4$	1568.a	1.a	$1$	$6$	$6$	$92.515$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		224.4.i.c	$1$	$1$	\(0\)	\(-6\)	\(10\)	\(0\)		$+$	$-$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(2+\beta _{2})q^{5}+(6-2\beta _{1}+\cdots)q^{9}+\cdots\)
1568.4.a.bg	$1568$	$4$	1568.a	1.a	$1$	$6$	$6$	$92.515$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				224.4.i.d	$2$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)		$-$	$-$	$+$	$2^{7}\cdot 7$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{4})q^{5}+(12+\beta _{3}+\cdots)q^{9}+\cdots\)
1568.4.a.bh	$1568$	$4$	1568.a	1.a	$1$	$6$	$6$	$92.515$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				224.4.i.d	$2$	$1$	\(0\)	\(0\)	\(6\)	\(0\)		$-$	$+$	$-$	$2^{7}\cdot 7$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{4})q^{5}+(12+\beta _{3})q^{9}+\cdots\)
1568.4.a.bi	$1568$	$4$	1568.a	1.a	$1$	$6$	$6$	$92.515$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				224.4.i.c	$1$	$0$	\(0\)	\(6\)	\(-10\)	\(0\)		$+$	$+$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-2-\beta _{2})q^{5}+(6-2\beta _{1}+\cdots)q^{9}+\cdots\)
1568.4.a.bj	$1568$	$4$	1568.a	1.a	$1$	$6$	$6$	$92.515$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				224.4.i.c	$1$	$0$	\(0\)	\(6\)	\(10\)	\(0\)		$-$	$-$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(2+\beta _{2})q^{5}+(6-2\beta _{1}+\cdots)q^{9}+\cdots\)
1568.4.a.bk	$1568$	$4$	1568.a	1.a	$1$	$8$	$8$	$92.515$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			1568.4.a.bk	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{12}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{4}-\beta _{6})q^{5}+(19+2\beta _{2}+\cdots)q^{9}+\cdots\)
1568.4.a.bl	$1568$	$4$	1568.a	1.a	$1$	$10$	$10$	$92.515$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		1568.4.a.bl	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{14}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(\beta _{1}-\beta _{2})q^{5}+(13+\beta _{4})q^{9}+\cdots\)
1568.4.a.bm	$1568$	$4$	1568.a	1.a	$1$	$10$	$10$	$92.515$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		1568.4.a.bl	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{14}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-\beta _{1}+\beta _{2})q^{5}+(13+\beta _{4}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1568))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1568)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(784))\)\(^{\oplus 2}\)