Space of modular forms of level 8352, weight 2, and trivial character

• Label: 8352.2.a
• Level: $8352$
• Weight: $2$
• Character orbit: 8352.a
• Rep. character: $\chi_{8352}(1,\cdot)$
• Character field: $\Q$
• Dimension: $140$
• Newform subspaces: $41$
• Sturm bound: $2880$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 8352 = 2^{5} \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8352.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 41 \)
Sturm bound:			\(2880\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8352))\).

	Total	New	Old
Modular forms	1472	140	1332
Cusp forms	1409	140	1269
Eisenstein series	63	0	63

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

\(2\)	\(3\)	\(29\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(13\)
\(+\)	\(+\)	\(-\)	$-$	\(15\)
\(+\)	\(-\)	\(+\)	$-$	\(21\)
\(+\)	\(-\)	\(-\)	$+$	\(20\)
\(-\)	\(+\)	\(+\)	$-$	\(15\)
\(-\)	\(+\)	\(-\)	$+$	\(13\)
\(-\)	\(-\)	\(+\)	$+$	\(21\)
\(-\)	\(-\)	\(-\)	$-$	\(22\)
Plus space			\(+\)	\(67\)
Minus space			\(-\)	\(73\)

Trace form

\( 140 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8352))\) 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	3	29
8352.2.a.a	$8352$	$2$	8352.a	1.a	$1$	$1$	$1$	$66.691$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-4q^{11}-2q^{13}-6q^{17}+4q^{19}+\cdots\)
8352.2.a.b	$8352$	$2$	8352.a	1.a	$1$	$1$	$1$	$66.691$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+4q^{11}-2q^{13}-6q^{17}-4q^{19}+\cdots\)
8352.2.a.c	$8352$	$2$	8352.a	1.a	$1$	$1$	$1$	$66.691$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{7}-4q^{11}-2q^{13}+3q^{17}+\cdots\)
8352.2.a.d	$8352$	$2$	8352.a	1.a	$1$	$1$	$1$	$66.691$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+3q^{7}+4q^{11}-2q^{13}+3q^{17}+\cdots\)
8352.2.a.e	$8352$	$2$	8352.a	1.a	$1$	$1$	$1$	$66.691$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{7}+4q^{11}-2q^{13}-3q^{17}+\cdots\)
8352.2.a.f	$8352$	$2$	8352.a	1.a	$1$	$1$	$1$	$66.691$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-q^{7}+6q^{11}+4q^{13}-3q^{17}+\cdots\)
8352.2.a.g	$8352$	$2$	8352.a	1.a	$1$	$1$	$1$	$66.691$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-5q^{11}+q^{13}+6q^{17}-4q^{19}+\cdots\)
8352.2.a.h	$8352$	$2$	8352.a	1.a	$1$	$1$	$1$	$66.691$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+5q^{11}+q^{13}+6q^{17}+4q^{19}+\cdots\)
8352.2.a.i	$8352$	$2$	8352.a	1.a	$1$	$1$	$1$	$66.691$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{7}-6q^{11}+4q^{13}-3q^{17}+\cdots\)
8352.2.a.j	$8352$	$2$	8352.a	1.a	$1$	$1$	$1$	$66.691$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+3q^{7}-4q^{11}-2q^{13}-3q^{17}+\cdots\)
8352.2.a.k	$8352$	$2$	8352.a	1.a	$1$	$2$	$2$	$66.691$	\(\Q(\sqrt{5}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(-6\)	\(0\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3q^{5}+\beta q^{11}+q^{13}-2q^{17}-2\beta q^{23}+\cdots\)
8352.2.a.l	$8352$	$2$	8352.a	1.a	$1$	$2$	$2$	$66.691$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}+(1-\beta )q^{11}+(1+2\beta )q^{13}+\cdots\)
8352.2.a.m	$8352$	$2$	8352.a	1.a	$1$	$2$	$2$	$66.691$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}-\beta q^{7}+(-2+\beta )q^{11}+\cdots\)
8352.2.a.n	$8352$	$2$	8352.a	1.a	$1$	$2$	$2$	$66.691$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}+\beta q^{7}+(2-\beta )q^{11}+\cdots\)
8352.2.a.o	$8352$	$2$	8352.a	1.a	$1$	$2$	$2$	$66.691$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}+(-1+\beta )q^{11}+(1+2\beta )q^{13}+\cdots\)
8352.2.a.p	$8352$	$2$	8352.a	1.a	$1$	$2$	$2$	$66.691$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+(-1+\beta )q^{7}+(1-\beta )q^{11}+\cdots\)
8352.2.a.q	$8352$	$2$	8352.a	1.a	$1$	$2$	$2$	$66.691$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+(1-\beta )q^{7}+(-1+\beta )q^{11}+\cdots\)
8352.2.a.r	$8352$	$2$	8352.a	1.a	$1$	$2$	$2$	$66.691$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(4\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(-1-\beta )q^{7}+(-1-\beta )q^{11}+\cdots\)
8352.2.a.s	$8352$	$2$	8352.a	1.a	$1$	$2$	$2$	$66.691$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(4\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(1+\beta )q^{7}+(1+\beta )q^{11}+\cdots\)
8352.2.a.t	$8352$	$2$	8352.a	1.a	$1$	$3$	$3$	$66.691$	3.3.316.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-5\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+(-2+\beta _{1}-\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{11}+\cdots\)
8352.2.a.u	$8352$	$2$	8352.a	1.a	$1$	$3$	$3$	$66.691$	3.3.316.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(5\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+(2-\beta _{1}+\beta _{2})q^{7}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
8352.2.a.v	$8352$	$2$	8352.a	1.a	$1$	$3$	$3$	$66.691$	3.3.1708.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(5\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{5}+(-\beta _{1}-\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{11}+\cdots\)
8352.2.a.w	$8352$	$2$	8352.a	1.a	$1$	$3$	$3$	$66.691$	3.3.1708.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(5\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{5}+(\beta _{1}+\beta _{2})q^{7}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
8352.2.a.x	$8352$	$2$	8352.a	1.a	$1$	$4$	$4$	$66.691$	4.4.4352.1	None	✓	$2$	$1$	\(0\)	\(0\)	\(-8\)	\(0\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{5}-\beta _{2}q^{7}-\beta _{2}q^{11}+\cdots\)
8352.2.a.y	$8352$	$2$	8352.a	1.a	$1$	$4$	$4$	$66.691$	4.4.54764.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(-5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(-1-\beta _{2})q^{7}+(2+\cdots)q^{11}+\cdots\)
8352.2.a.z	$8352$	$2$	8352.a	1.a	$1$	$4$	$4$	$66.691$	4.4.54764.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(1+\beta _{2})q^{7}+(-2+\cdots)q^{11}+\cdots\)
8352.2.a.ba	$8352$	$2$	8352.a	1.a	$1$	$4$	$4$	$66.691$	\(\Q(\zeta_{20})^+\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{3})q^{5}+(-\beta _{1}+\beta _{2})q^{7}+(\beta _{1}+\cdots)q^{11}+\cdots\)
8352.2.a.bb	$8352$	$2$	8352.a	1.a	$1$	$4$	$4$	$66.691$	4.4.4352.1	None	✓	$2$	$1$	\(0\)	\(0\)	\(8\)	\(0\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{5}+\beta _{2}q^{7}-\beta _{2}q^{11}+(-1+\cdots)q^{13}+\cdots\)
8352.2.a.bc	$8352$	$2$	8352.a	1.a	$1$	$5$	$5$	$66.691$	5.5.1544456.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-10\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(-2-\beta _{2})q^{7}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
8352.2.a.bd	$8352$	$2$	8352.a	1.a	$1$	$5$	$5$	$66.691$	5.5.1544456.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(10\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(2+\beta _{2})q^{7}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
8352.2.a.be	$8352$	$2$	8352.a	1.a	$1$	$5$	$5$	$66.691$	5.5.4591816.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}+\beta _{4}q^{7}+(-\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{11}+\cdots\)
8352.2.a.bf	$8352$	$2$	8352.a	1.a	$1$	$5$	$5$	$66.691$	5.5.4591816.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}-\beta _{4}q^{7}+(\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4})q^{11}+\cdots\)
8352.2.a.bg	$8352$	$2$	8352.a	1.a	$1$	$5$	$5$	$66.691$	5.5.230224.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-4\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{5}+(-1-\beta _{1}-\beta _{3}-\beta _{4})q^{7}+\cdots\)
8352.2.a.bh	$8352$	$2$	8352.a	1.a	$1$	$5$	$5$	$66.691$	5.5.230224.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(2\)	\(4\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{5}+(1+\beta _{1}+\beta _{3}+\beta _{4})q^{7}+(-2+\cdots)q^{11}+\cdots\)
8352.2.a.bi	$8352$	$2$	8352.a	1.a	$1$	$6$	$6$	$66.691$	6.6.68772992.1	None	✓	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{3})q^{5}-\beta _{1}q^{7}-\beta _{4}q^{11}+\cdots\)
8352.2.a.bj	$8352$	$2$	8352.a	1.a	$1$	$7$	$7$	$66.691$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-8\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{3})q^{5}+(-1-\beta _{1})q^{7}-\beta _{5}q^{11}+\cdots\)
8352.2.a.bk	$8352$	$2$	8352.a	1.a	$1$	$7$	$7$	$66.691$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(8\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{3})q^{5}+(1+\beta _{1})q^{7}+\beta _{5}q^{11}+\cdots\)
8352.2.a.bl	$8352$	$2$	8352.a	1.a	$1$	$7$	$7$	$66.691$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(4\)	\(-8\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{3})q^{5}+(-1-\beta _{1})q^{7}+\beta _{5}q^{11}+\cdots\)
8352.2.a.bm	$8352$	$2$	8352.a	1.a	$1$	$7$	$7$	$66.691$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(4\)	\(8\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{3})q^{5}+(1+\beta _{1})q^{7}-\beta _{5}q^{11}+\cdots\)
8352.2.a.bn	$8352$	$2$	8352.a	1.a	$1$	$8$	$8$	$66.691$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+\beta _{3}q^{7}-\beta _{6}q^{11}+(2-\beta _{4}+\cdots)q^{13}+\cdots\)
8352.2.a.bo	$8352$	$2$	8352.a	1.a	$1$	$8$	$8$	$66.691$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}-\beta _{3}q^{7}-\beta _{6}q^{11}+(2-\beta _{4}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8352))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8352)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(174))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(232))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(261))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(348))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(464))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(522))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(696))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(928))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1044))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1392))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2088))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2784))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4176))\)\(^{\oplus 2}\)