Space of modular forms of level 5312, weight 2, and trivial character

• Label: 5312.2.a
• Level: $5312$
• Weight: $2$
• Character orbit: 5312.a
• Rep. character: $\chi_{5312}(1,\cdot)$
• Character field: $\Q$
• Dimension: $164$
• Newform subspaces: $48$
• Sturm bound: $1344$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 5312 = 2^{6} \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5312.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 48 \)
Sturm bound:			\(1344\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5312))\).

	Total	New	Old
Modular forms	684	164	520
Cusp forms	661	164	497
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.

\(2\)	\(83\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(37\)
\(+\)	\(-\)	$-$	\(46\)
\(-\)	\(+\)	$-$	\(45\)
\(-\)	\(-\)	$+$	\(36\)
Plus space		\(+\)	\(73\)
Minus space		\(-\)	\(91\)

Trace form

\( 164 q + 164 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5312))\) 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	83
5312.2.a.a	$5312$	$2$	5312.a	1.a	$1$	$1$	$1$	$42.417$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(2\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+2q^{5}+3q^{7}+6q^{9}-3q^{11}+\cdots\)
5312.2.a.b	$5312$	$2$	5312.a	1.a	$1$	$1$	$1$	$42.417$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(4\)	\(5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+4q^{5}+5q^{7}+6q^{9}-3q^{11}+\cdots\)
5312.2.a.c	$5312$	$2$	5312.a	1.a	$1$	$1$	$1$	$42.417$	\(\Q\)	None	✓	$1$	$2$	\(0\)	\(-1\)	\(-2\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}-3q^{7}-2q^{9}-q^{11}+\cdots\)
5312.2.a.d	$5312$	$2$	5312.a	1.a	$1$	$1$	$1$	$42.417$	\(\Q\)	None	✓	$1$	$2$	\(0\)	\(-1\)	\(0\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{7}-2q^{9}-q^{11}-6q^{13}+\cdots\)
5312.2.a.e	$5312$	$2$	5312.a	1.a	$1$	$1$	$1$	$42.417$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}-2q^{9}-q^{11}+4q^{13}+\cdots\)
5312.2.a.f	$5312$	$2$	5312.a	1.a	$1$	$1$	$1$	$42.417$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-q^{7}-2q^{9}-5q^{11}+\cdots\)
5312.2.a.g	$5312$	$2$	5312.a	1.a	$1$	$1$	$1$	$42.417$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-q^{7}-2q^{9}+3q^{11}+\cdots\)
5312.2.a.h	$5312$	$2$	5312.a	1.a	$1$	$1$	$1$	$42.417$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+3q^{7}-2q^{9}+3q^{11}+\cdots\)
5312.2.a.i	$5312$	$2$	5312.a	1.a	$1$	$1$	$1$	$42.417$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+3q^{7}-2q^{9}+q^{11}+\cdots\)
5312.2.a.j	$5312$	$2$	5312.a	1.a	$1$	$1$	$1$	$42.417$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}-2q^{9}+q^{11}+4q^{13}+\cdots\)
5312.2.a.k	$5312$	$2$	5312.a	1.a	$1$	$1$	$1$	$42.417$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{7}-2q^{9}+q^{11}-6q^{13}+\cdots\)
5312.2.a.l	$5312$	$2$	5312.a	1.a	$1$	$1$	$1$	$42.417$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(2\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}-3q^{7}-2q^{9}-3q^{11}+\cdots\)
5312.2.a.m	$5312$	$2$	5312.a	1.a	$1$	$1$	$1$	$42.417$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(2\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{7}-2q^{9}-3q^{11}+\cdots\)
5312.2.a.n	$5312$	$2$	5312.a	1.a	$1$	$1$	$1$	$42.417$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(2\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{7}-2q^{9}+5q^{11}+\cdots\)
5312.2.a.o	$5312$	$2$	5312.a	1.a	$1$	$1$	$1$	$42.417$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(2\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{5}-3q^{7}+6q^{9}+3q^{11}+\cdots\)
5312.2.a.p	$5312$	$2$	5312.a	1.a	$1$	$1$	$1$	$42.417$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(4\)	\(-5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+4q^{5}-5q^{7}+6q^{9}+3q^{11}+\cdots\)
5312.2.a.q	$5312$	$2$	5312.a	1.a	$1$	$2$	$2$	$42.417$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-3\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2\beta q^{3}+(-2+\beta )q^{5}+(1+\beta )q^{7}+\cdots\)
5312.2.a.r	$5312$	$2$	5312.a	1.a	$1$	$2$	$2$	$42.417$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2\beta q^{3}-\beta q^{5}+(1-\beta )q^{7}+(1+4\beta )q^{9}+\cdots\)
5312.2.a.s	$5312$	$2$	5312.a	1.a	$1$	$2$	$2$	$42.417$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+\beta q^{5}+(3+\beta )q^{7}-2\beta q^{9}+\cdots\)
5312.2.a.t	$5312$	$2$	5312.a	1.a	$1$	$2$	$2$	$42.417$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1+\beta )q^{5}+\beta q^{7}+4q^{9}+(-2+\cdots)q^{11}+\cdots\)
5312.2.a.u	$5312$	$2$	5312.a	1.a	$1$	$2$	$2$	$42.417$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1-\beta )q^{5}+\beta q^{7}+4q^{9}+(2+\cdots)q^{11}+\cdots\)
5312.2.a.v	$5312$	$2$	5312.a	1.a	$1$	$2$	$2$	$42.417$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(-3\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}+(-2+\beta )q^{5}+(-1-\beta )q^{7}+\cdots\)
5312.2.a.w	$5312$	$2$	5312.a	1.a	$1$	$2$	$2$	$42.417$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}-\beta q^{5}+(-1+\beta )q^{7}+(1+4\beta )q^{9}+\cdots\)
5312.2.a.x	$5312$	$2$	5312.a	1.a	$1$	$2$	$2$	$42.417$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-\beta q^{5}+(-3+\beta )q^{7}+2\beta q^{9}+\cdots\)
5312.2.a.y	$5312$	$2$	5312.a	1.a	$1$	$3$	$3$	$42.417$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$0$	\(0\)	\(-4\)	\(-2\)	\(8\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+2\beta _{2}q^{5}+(2+\beta _{1}+\cdots)q^{7}+\cdots\)
5312.2.a.z	$5312$	$2$	5312.a	1.a	$1$	$3$	$3$	$42.417$	3.3.229.1	None	✓	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}+\beta _{2})q^{3}-\beta _{2}q^{5}+(1-2\beta _{1}+\cdots)q^{7}+\cdots\)
5312.2.a.ba	$5312$	$2$	5312.a	1.a	$1$	$3$	$3$	$42.417$	3.3.148.1	None	✓	$1$	$1$	\(0\)	\(-1\)	\(4\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{1}+2\beta _{2})q^{5}+(-2+\cdots)q^{7}+\cdots\)
5312.2.a.bb	$5312$	$2$	5312.a	1.a	$1$	$3$	$3$	$42.417$	3.3.229.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(1\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}-\beta _{2})q^{3}-\beta _{2}q^{5}+(-1+2\beta _{1}+\cdots)q^{7}+\cdots\)
5312.2.a.bc	$5312$	$2$	5312.a	1.a	$1$	$3$	$3$	$42.417$	3.3.148.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(4\)	\(5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{1}+2\beta _{2})q^{5}+(2-\beta _{1}+\cdots)q^{7}+\cdots\)
5312.2.a.bd	$5312$	$2$	5312.a	1.a	$1$	$3$	$3$	$42.417$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$0$	\(0\)	\(4\)	\(-2\)	\(-8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+2\beta _{2}q^{5}+(-2-\beta _{1}+\cdots)q^{7}+\cdots\)
5312.2.a.be	$5312$	$2$	5312.a	1.a	$1$	$4$	$4$	$42.417$	4.4.15529.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1+\beta _{1}+\beta _{3})q^{7}+\cdots\)
5312.2.a.bf	$5312$	$2$	5312.a	1.a	$1$	$4$	$4$	$42.417$	4.4.2225.1	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(5\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}-\beta _{1}q^{5}+(1+\beta _{1})q^{7}+\cdots\)
5312.2.a.bg	$5312$	$2$	5312.a	1.a	$1$	$4$	$4$	$42.417$	4.4.2225.1	None	✓	$1$	$1$	\(0\)	\(2\)	\(-1\)	\(-5\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}-\beta _{1}q^{5}+(-1-\beta _{1})q^{7}+\cdots\)
5312.2.a.bh	$5312$	$2$	5312.a	1.a	$1$	$4$	$4$	$42.417$	4.4.15529.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1-\beta _{1}-\beta _{3})q^{7}+(1+\cdots)q^{9}+\cdots\)
5312.2.a.bi	$5312$	$2$	5312.a	1.a	$1$	$5$	$5$	$42.417$	5.5.265504.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(-2\)	\(-3\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{3})q^{3}-\beta _{3}q^{5}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
5312.2.a.bj	$5312$	$2$	5312.a	1.a	$1$	$5$	$5$	$42.417$	5.5.356173.1	None	✓	$1$	$0$	\(0\)	\(-2\)	\(1\)	\(7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{2}+\beta _{4})q^{5}+(1-\beta _{4})q^{7}+\cdots\)
5312.2.a.bk	$5312$	$2$	5312.a	1.a	$1$	$5$	$5$	$42.417$	5.5.356173.1	None	✓	$1$	$1$	\(0\)	\(2\)	\(1\)	\(-7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{2}+\beta _{4})q^{5}+(-1+\beta _{4})q^{7}+\cdots\)
5312.2.a.bl	$5312$	$2$	5312.a	1.a	$1$	$5$	$5$	$42.417$	5.5.265504.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(-2\)	\(3\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{3})q^{3}-\beta _{3}q^{5}+(1+\beta _{1})q^{7}+\cdots\)
5312.2.a.bm	$5312$	$2$	5312.a	1.a	$1$	$6$	$6$	$42.417$	6.6.905177.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(4\)	\(5\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+(1-\beta _{2}-\beta _{3})q^{5}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
5312.2.a.bn	$5312$	$2$	5312.a	1.a	$1$	$6$	$6$	$42.417$	6.6.9059636.1	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(3\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}+(-1-\beta _{2}-\beta _{3}-\beta _{5})q^{5}+\cdots\)
5312.2.a.bo	$5312$	$2$	5312.a	1.a	$1$	$6$	$6$	$42.417$	6.6.9059636.1	None	✓	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(-3\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+(-1-\beta _{2}-\beta _{3}-\beta _{5})q^{5}+\cdots\)
5312.2.a.bp	$5312$	$2$	5312.a	1.a	$1$	$6$	$6$	$42.417$	6.6.905177.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(4\)	\(-5\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}+(1-\beta _{2}-\beta _{3})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
5312.2.a.bq	$5312$	$2$	5312.a	1.a	$1$	$8$	$8$	$42.417$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(1\)	\(-10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{3}q^{5}+(-1-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
5312.2.a.br	$5312$	$2$	5312.a	1.a	$1$	$8$	$8$	$42.417$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(-7\)	\(1\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{7})q^{5}-\beta _{2}q^{7}+(2+\cdots)q^{9}+\cdots\)
5312.2.a.bs	$5312$	$2$	5312.a	1.a	$1$	$8$	$8$	$42.417$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-7\)	\(-1\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{7})q^{5}+\beta _{2}q^{7}+(2+\cdots)q^{9}+\cdots\)
5312.2.a.bt	$5312$	$2$	5312.a	1.a	$1$	$8$	$8$	$42.417$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(1\)	\(10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{3}q^{5}+(1+\beta _{1}-\beta _{3}+\beta _{6}+\cdots)q^{7}+\cdots\)
5312.2.a.bu	$5312$	$2$	5312.a	1.a	$1$	$11$	$11$	$42.417$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-5\)	\(-10\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{3}q^{5}+(-1-\beta _{10})q^{7}+\cdots\)
5312.2.a.bv	$5312$	$2$	5312.a	1.a	$1$	$11$	$11$	$42.417$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(-5\)	\(10\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{3}q^{5}+(1+\beta _{10})q^{7}+(2+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5312))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5312)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(83))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(166))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(332))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(664))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1328))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2656))\)\(^{\oplus 2}\)