Space of modular forms of level 546, weight 8, and trivial character

• Label: 546.8.a
• Level: $546$
• Weight: $8$
• Character orbit: 546.a
• Rep. character: $\chi_{546}(1,\cdot)$
• Character field: $\Q$
• Dimension: $84$
• Newform subspaces: $19$
• Sturm bound: $896$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(896\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(546))\).

	Total	New	Old
Modular forms	792	84	708
Cusp forms	776	84	692
Eisenstein series	16	0	16

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

\(2\)	\(3\)	\(7\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(6\)
\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(5\)
\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(6\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(5\)
\(+\)	\(-\)	\(+\)	\(-\)	$+$	\(5\)
\(+\)	\(-\)	\(-\)	\(+\)	$+$	\(6\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(5\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(5\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(6\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(6\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(4\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(7\)
Plus space				\(+\)	\(48\)
Minus space				\(-\)	\(36\)

Trace form

\( 84 q + 5376 q^{4} + 61236 q^{9} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\) 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	7	13
546.8.a.a	$546$	$8$	546.a	1.a	$1$	$1$	$1$	$170.562$	\(\Q\)	None	✓	$1$	$1$	\(8\)	\(-27\)	\(-135\)	\(343\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}-135q^{5}+\cdots\)
546.8.a.b	$546$	$8$	546.a	1.a	$1$	$1$	$1$	$170.562$	\(\Q\)	None	✓	$1$	$0$	\(8\)	\(27\)	\(-390\)	\(-343\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}-390q^{5}+\cdots\)
546.8.a.c	$546$	$8$	546.a	1.a	$1$	$1$	$1$	$170.562$	\(\Q\)	None	✓	$1$	$1$	\(8\)	\(27\)	\(-10\)	\(-343\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}-10q^{5}+6^{3}q^{6}+\cdots\)
546.8.a.d	$546$	$8$	546.a	1.a	$1$	$3$	$3$	$170.562$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(24\)	\(-81\)	\(351\)	\(1029\)		$-$	$+$	$-$	$-$	$-$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(117+\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.e	$546$	$8$	546.a	1.a	$1$	$3$	$3$	$170.562$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(24\)	\(81\)	\(-378\)	\(-1029\)		$-$	$-$	$+$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(-126+\beta _{2})q^{5}+\cdots\)
546.8.a.f	$546$	$8$	546.a	1.a	$1$	$4$	$4$	$170.562$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(-32\)	\(-108\)	\(-134\)	\(1372\)		$+$	$+$	$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(-34-2\beta _{2}+\cdots)q^{5}+\cdots\)
546.8.a.g	$546$	$8$	546.a	1.a	$1$	$4$	$4$	$170.562$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(32\)	\(108\)	\(-328\)	\(1372\)		$-$	$-$	$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(-82+\beta _{2}+\cdots)q^{5}+\cdots\)
546.8.a.h	$546$	$8$	546.a	1.a	$1$	$5$	$5$	$170.562$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-40\)	\(-135\)	\(168\)	\(-1715\)		$+$	$+$	$+$	$-$	$-$	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(34-\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.i	$546$	$8$	546.a	1.a	$1$	$5$	$5$	$170.562$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-40\)	\(135\)	\(-340\)	\(1715\)		$+$	$-$	$-$	$-$	$-$	$2^{2}\cdot 3^{2}\cdot 5\cdot 13$	$\mathrm{SU}(2)$	\(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(-68-\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.j	$546$	$8$	546.a	1.a	$1$	$5$	$5$	$170.562$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-40\)	\(135\)	\(56\)	\(-1715\)		$+$	$-$	$+$	$+$	$-$	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(11-\beta _{2}+\cdots)q^{5}+\cdots\)
546.8.a.k	$546$	$8$	546.a	1.a	$1$	$5$	$5$	$170.562$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(-40\)	\(135\)	\(509\)	\(-1715\)		$+$	$-$	$+$	$-$	$+$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	\(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(102-\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.l	$546$	$8$	546.a	1.a	$1$	$5$	$5$	$170.562$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(40\)	\(-135\)	\(-250\)	\(-1715\)		$-$	$+$	$+$	$+$	$-$	$2^{2}\cdot 3^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(-50+\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.m	$546$	$8$	546.a	1.a	$1$	$5$	$5$	$170.562$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(40\)	\(135\)	\(299\)	\(-1715\)		$-$	$-$	$+$	$+$	$+$	$2^{2}\cdot 3^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(60-\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.n	$546$	$8$	546.a	1.a	$1$	$6$	$6$	$170.562$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(-48\)	\(-162\)	\(-181\)	\(2058\)		$+$	$+$	$-$	$-$	$+$	$2^{5}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(-30-\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.o	$546$	$8$	546.a	1.a	$1$	$6$	$6$	$170.562$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(-48\)	\(-162\)	\(-35\)	\(-2058\)		$+$	$+$	$+$	$+$	$+$	$2^{3}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(-6+\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.p	$546$	$8$	546.a	1.a	$1$	$6$	$6$	$170.562$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(-48\)	\(162\)	\(-43\)	\(2058\)		$+$	$-$	$-$	$+$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(-7-\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.q	$546$	$8$	546.a	1.a	$1$	$6$	$6$	$170.562$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(48\)	\(-162\)	\(13\)	\(2058\)		$-$	$+$	$-$	$+$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(2+\beta _{1})q^{5}+\cdots\)
546.8.a.r	$546$	$8$	546.a	1.a	$1$	$6$	$6$	$170.562$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(48\)	\(-162\)	\(203\)	\(-2058\)		$-$	$+$	$+$	$-$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(34-\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.s	$546$	$8$	546.a	1.a	$1$	$7$	$7$	$170.562$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(56\)	\(189\)	\(625\)	\(2401\)		$-$	$-$	$-$	$-$	$+$	$2^{6}\cdot 3^{6}\cdot 7$	$\mathrm{SU}(2)$	\(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(89+\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(546))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(546)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(182))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(273))\)\(^{\oplus 2}\)