Space of modular forms of level 6348, weight 2, and trivial character

• Label: 6348.2.a
• Level: $6348$
• Weight: $2$
• Character orbit: 6348.a
• Rep. character: $\chi_{6348}(1,\cdot)$
• Character field: $\Q$
• Dimension: $84$
• Newform subspaces: $20$
• Sturm bound: $2208$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 6348 = 2^{2} \cdot 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6348.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(2208\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6348))\).

	Total	New	Old
Modular forms	1176	84	1092
Cusp forms	1033	84	949
Eisenstein series	143	0	143

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	Dim
\(-\)	\(+\)	\(+\)	$-$	\(20\)
\(-\)	\(+\)	\(-\)	$+$	\(22\)
\(-\)	\(-\)	\(+\)	$+$	\(16\)
\(-\)	\(-\)	\(-\)	$-$	\(26\)
Plus space			\(+\)	\(38\)
Minus space			\(-\)	\(46\)

Trace form

\( 84 q - 4 q^{5} - 4 q^{7} + 84 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6348))\) 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	23
6348.2.a.a	$6348$	$2$	6348.a	1.a	$1$	$1$	$1$	$50.689$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-4\)	\(3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{5}+3q^{7}+q^{9}+q^{13}+4q^{15}+\cdots\)
6348.2.a.b	$6348$	$2$	6348.a	1.a	$1$	$1$	$1$	$50.689$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+2q^{7}+q^{9}+q^{13}+q^{15}+\cdots\)
6348.2.a.c	$6348$	$2$	6348.a	1.a	$1$	$1$	$1$	$50.689$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-2q^{7}+q^{9}+q^{13}-q^{15}+\cdots\)
6348.2.a.d	$6348$	$2$	6348.a	1.a	$1$	$1$	$1$	$50.689$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(4\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}-3q^{7}+q^{9}+q^{13}-4q^{15}+\cdots\)
6348.2.a.e	$6348$	$2$	6348.a	1.a	$1$	$2$	$2$	$50.689$	\(\Q(\sqrt{10}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{5}+(-2-\beta )q^{7}+q^{9}+4q^{13}+\cdots\)
6348.2.a.f	$6348$	$2$	6348.a	1.a	$1$	$2$	$2$	$50.689$	\(\Q(\sqrt{2}) \)	None	✓	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{5}-2\beta q^{7}+q^{9}-2\beta q^{11}+\cdots\)
6348.2.a.g	$6348$	$2$	6348.a	1.a	$1$	$2$	$2$	$50.689$	\(\Q(\sqrt{2}) \)	None	✓	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{5}-\beta q^{7}+q^{9}+4\beta q^{11}+\cdots\)
6348.2.a.h	$6348$	$2$	6348.a	1.a	$1$	$2$	$2$	$50.689$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-2+\beta )q^{5}+\beta q^{7}+q^{9}-4\beta q^{11}+\cdots\)
6348.2.a.i	$6348$	$2$	6348.a	1.a	$1$	$2$	$2$	$50.689$	\(\Q(\sqrt{2}) \)	None	✓	$2$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{5}+q^{9}-2\beta q^{11}-2q^{13}+\cdots\)
6348.2.a.j	$6348$	$2$	6348.a	1.a	$1$	$2$	$2$	$50.689$	\(\Q(\sqrt{6}) \)	None	✓	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{5}+\beta q^{7}+q^{9}+4q^{13}+\cdots\)
6348.2.a.k	$6348$	$2$	6348.a	1.a	$1$	$3$	$3$	$50.689$	3.3.2024.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(-1\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta _{1}q^{5}+(1+\beta _{2})q^{7}+q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
6348.2.a.l	$6348$	$2$	6348.a	1.a	$1$	$3$	$3$	$50.689$	3.3.2024.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(1\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{1}q^{5}+(-1-\beta _{2})q^{7}+q^{9}+\cdots\)
6348.2.a.m	$6348$	$2$	6348.a	1.a	$1$	$4$	$4$	$50.689$	\(\Q(\sqrt{3}, \sqrt{11})\)	None	✓	$2$	$1$	\(0\)	\(-4\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{1}q^{5}+(-\beta _{1}+\beta _{2})q^{7}+q^{9}+\cdots\)
6348.2.a.n	$6348$	$2$	6348.a	1.a	$1$	$4$	$4$	$50.689$	\(\Q(\zeta_{24})^+\)	None	✓	$2$	$1$	\(0\)	\(4\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(\beta _{1}+\beta _{3})q^{5}+(-2\beta _{1}-\beta _{3})q^{7}+\cdots\)
6348.2.a.o	$6348$	$2$	6348.a	1.a	$1$	$6$	$6$	$50.689$	6.6.10784448.1	None	✓	$2$	$0$	\(0\)	\(6\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{2}q^{5}-\beta _{5}q^{7}+q^{9}+(\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)
6348.2.a.p	$6348$	$2$	6348.a	1.a	$1$	$8$	$8$	$50.689$	8.8.\(\cdots\).1	None	✓	$2$	$0$	\(0\)	\(-8\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}+(-\beta _{2}-\beta _{6})q^{5}-\beta _{5}q^{7}+q^{9}+\cdots\)
6348.2.a.q	$6348$	$2$	6348.a	1.a	$1$	$10$	$10$	$50.689$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-10\)	\(0\)	\(-9\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(\beta _{5}-\beta _{7})q^{5}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
6348.2.a.r	$6348$	$2$	6348.a	1.a	$1$	$10$	$10$	$50.689$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-10\)	\(0\)	\(9\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-\beta _{5}+\beta _{7})q^{5}+(1-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
6348.2.a.s	$6348$	$2$	6348.a	1.a	$1$	$10$	$10$	$50.689$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(10\)	\(-2\)	\(-11\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{4}q^{5}+(-2-\beta _{2}-\beta _{3}+\beta _{6}+\cdots)q^{7}+\cdots\)
6348.2.a.t	$6348$	$2$	6348.a	1.a	$1$	$10$	$10$	$50.689$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(10\)	\(2\)	\(11\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta _{4}q^{5}+(2+\beta _{2}+\beta _{3}-\beta _{6}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6348))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6348)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(529))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1058))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1587))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2116))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3174))\)\(^{\oplus 2}\)