Space of modular forms of level 1352, weight 4, and trivial character

• Label: 1352.4.a
• Level: $1352$
• Weight: $4$
• Character orbit: 1352.a
• Rep. character: $\chi_{1352}(1,\cdot)$
• Character field: $\Q$
• Dimension: $116$
• Newform subspaces: $20$
• Sturm bound: $728$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1352.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(728\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1352))\).

	Total	New	Old
Modular forms	574	116	458
Cusp forms	518	116	402
Eisenstein series	56	0	56

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

\(2\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(31\)
\(+\)	\(-\)	$-$	\(27\)
\(-\)	\(+\)	$-$	\(28\)
\(-\)	\(-\)	$+$	\(30\)
Plus space		\(+\)	\(61\)
Minus space		\(-\)	\(55\)

Trace form

\( 116 q + 2 q^{3} + 12 q^{5} - 12 q^{7} + 1006 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1352))\) 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	13
1352.4.a.a	$1352$	$4$	1352.a	1.a	$1$	$1$	$1$	$79.771$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-4\)	\(2\)	\(-24\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}+2q^{5}-24q^{7}-11q^{9}+44q^{11}+\cdots\)
1352.4.a.b	$1352$	$4$	1352.a	1.a	$1$	$1$	$1$	$79.771$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(7\)	\(21\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+7q^{5}+21q^{7}-26q^{9}-6q^{11}+\cdots\)
1352.4.a.c	$1352$	$4$	1352.a	1.a	$1$	$1$	$1$	$79.771$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(5\)	\(-19\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-19q^{5}+3q^{7}-2q^{9}+2q^{11}+\cdots\)
1352.4.a.d	$1352$	$4$	1352.a	1.a	$1$	$1$	$1$	$79.771$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(8\)	\(-9\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}-9q^{5}-4q^{7}+37q^{9}-20q^{11}+\cdots\)
1352.4.a.e	$1352$	$4$	1352.a	1.a	$1$	$1$	$1$	$79.771$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(8\)	\(9\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}+9q^{5}+4q^{7}+37q^{9}+20q^{11}+\cdots\)
1352.4.a.f	$1352$	$4$	1352.a	1.a	$1$	$2$	$2$	$79.771$	\(\Q(\sqrt{73}) \)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(3\)	\(25\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(3-3\beta )q^{5}+(13-\beta )q^{7}+\cdots\)
1352.4.a.g	$1352$	$4$	1352.a	1.a	$1$	$2$	$2$	$79.771$	\(\Q(\sqrt{321}) \)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(11\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(6-\beta )q^{5}+(-2+3\beta )q^{7}+\cdots\)
1352.4.a.h	$1352$	$4$	1352.a	1.a	$1$	$3$	$3$	$79.771$	3.3.18257.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(8\)	\(-36\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(3+2\beta _{1}+\beta _{2})q^{5}+(-12+\cdots)q^{7}+\cdots\)
1352.4.a.i	$1352$	$4$	1352.a	1.a	$1$	$4$	$4$	$79.771$	4.4.4968857.2	None	✓	$1$	$0$	\(0\)	\(-11\)	\(-7\)	\(15\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{3}+(-2-\beta _{2})q^{5}+(4+\cdots)q^{7}+\cdots\)
1352.4.a.j	$1352$	$4$	1352.a	1.a	$1$	$4$	$4$	$79.771$	4.4.4968857.2	None	✓	$1$	$1$	\(0\)	\(-11\)	\(7\)	\(-15\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{3}+(2+\beta _{2})q^{5}+(-4+\cdots)q^{7}+\cdots\)
1352.4.a.k	$1352$	$4$	1352.a	1.a	$1$	$5$	$5$	$79.771$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(-9\)	\(17\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-2-\beta _{3})q^{5}+(4-\beta _{1}+\cdots)q^{7}+\cdots\)
1352.4.a.l	$1352$	$4$	1352.a	1.a	$1$	$5$	$5$	$79.771$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(3\)	\(9\)	\(-17\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(2+\beta _{3})q^{5}+(-4+\beta _{1}+\cdots)q^{7}+\cdots\)
1352.4.a.m	$1352$	$4$	1352.a	1.a	$1$	$6$	$6$	$79.771$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(-9\)	\(-1\)		$-$	$+$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-2+\beta _{1}+\beta _{2})q^{5}-\beta _{3}q^{7}+\cdots\)
1352.4.a.n	$1352$	$4$	1352.a	1.a	$1$	$6$	$6$	$79.771$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(9\)	\(1\)		$+$	$+$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(2-\beta _{1}-\beta _{2})q^{5}+\beta _{3}q^{7}+\cdots\)
1352.4.a.o	$1352$	$4$	1352.a	1.a	$1$	$10$	$10$	$79.771$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(6\)	\(-24\)	\(-40\)		$+$	$-$	$-$	$2^{9}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-2-\beta _{5})q^{5}+(-4+\cdots)q^{7}+\cdots\)
1352.4.a.p	$1352$	$4$	1352.a	1.a	$1$	$10$	$10$	$79.771$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(6\)	\(24\)	\(40\)		$-$	$-$	$+$	$2^{9}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(2+\beta _{5})q^{5}+(4+2\beta _{2}+\cdots)q^{7}+\cdots\)
1352.4.a.q	$1352$	$4$	1352.a	1.a	$1$	$12$	$12$	$79.771$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-13\)	\(-18\)	\(29\)		$+$	$-$	$-$	$2^{9}\cdot 13^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(-2+\beta _{8})q^{5}+(2+\cdots)q^{7}+\cdots\)
1352.4.a.r	$1352$	$4$	1352.a	1.a	$1$	$12$	$12$	$79.771$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-13\)	\(18\)	\(-29\)		$-$	$+$	$-$	$2^{9}\cdot 13^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(2-\beta _{8})q^{5}+(-2+\cdots)q^{7}+\cdots\)
1352.4.a.s	$1352$	$4$	1352.a	1.a	$1$	$15$	$15$	$79.771$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(12\)	\(-12\)	\(-23\)		$-$	$-$	$+$	$2^{12}\cdot 13^{4}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1-\beta _{3})q^{5}+(-2+\cdots)q^{7}+\cdots\)
1352.4.a.t	$1352$	$4$	1352.a	1.a	$1$	$15$	$15$	$79.771$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(12\)	\(12\)	\(23\)		$+$	$+$	$+$	$2^{12}\cdot 13^{4}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1+\beta _{3})q^{5}+(2+\beta _{4}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1352))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1352)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(676))\)\(^{\oplus 2}\)