Space of modular forms of level 48, weight 8, and trivial character

• Label: 48.8.a
• Level: $48$
• Weight: $8$
• Character orbit: 48.a
• Rep. character: $\chi_{48}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $7$
• Sturm bound: $64$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(64\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(48))\).

	Total	New	Old
Modular forms	62	7	55
Cusp forms	50	7	43
Eisenstein series	12	0	12

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

\(2\)	\(3\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(2\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(4\)
Minus space		\(-\)	\(3\)

Trace form

7 * q - 27 * q^3 - 278 * q^5 - 320 * q^7 + 5103 * q^9 - 1204 * q^11 + 6554 * q^13 + 6750 * q^15 + 1454 * q^17 - 61868 * q^19 + 230984 * q^23 + 127681 * q^25 - 19683 * q^27 - 112110 * q^29 + 366520 * q^31 - 128412 * q^33 - 733824 * q^35 + 456962 * q^37 + 631854 * q^39 + 33702 * q^41 - 1012532 * q^43 - 202662 * q^45 + 97920 * q^47 + 35391 * q^49 - 47574 * q^51 - 1230326 * q^53 - 2254392 * q^55 + 775548 * q^57 + 1303820 * q^59 - 2876230 * q^61 - 233280 * q^63 - 2389988 * q^65 + 9185428 * q^67 + 2395224 * q^69 - 9291464 * q^71 + 6447830 * q^73 - 137997 * q^75 - 7964160 * q^77 - 734648 * q^79 + 3720087 * q^81 + 2956372 * q^83 + 3518868 * q^85 + 2114262 * q^87 + 17795334 * q^89 + 1967744 * q^91 - 4062744 * q^93 - 4930568 * q^95 - 9925426 * q^97 - 877716 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(48))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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
48.8.a.a	$48$	$8$	48.a	1.a	$1$	$1$	$1$	$14.994$	\(\Q\)	None	✓				24.8.a.b	$1$	$0$	\(0\)	\(-27\)	\(-530\)	\(-120\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-530q^{5}-120q^{7}+3^{6}q^{9}+\cdots\)
48.8.a.b	$48$	$8$	48.a	1.a	$1$	$1$	$1$	$14.994$	\(\Q\)	None	✓				6.8.a.a	$1$	$1$	\(0\)	\(-27\)	\(-114\)	\(1576\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-114q^{5}+1576q^{7}+3^{6}q^{9}+\cdots\)
48.8.a.c	$48$	$8$	48.a	1.a	$1$	$1$	$1$	$14.994$	\(\Q\)	None	✓				24.8.a.c	$1$	$0$	\(0\)	\(-27\)	\(110\)	\(-504\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+110q^{5}-504q^{7}+3^{6}q^{9}+\cdots\)
48.8.a.d	$48$	$8$	48.a	1.a	$1$	$1$	$1$	$14.994$	\(\Q\)	None	✓				12.8.a.b	$1$	$1$	\(0\)	\(-27\)	\(270\)	\(-1112\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+270q^{5}-1112q^{7}+3^{6}q^{9}+\cdots\)
48.8.a.e	$48$	$8$	48.a	1.a	$1$	$1$	$1$	$14.994$	\(\Q\)	None	✓				12.8.a.a	$1$	$0$	\(0\)	\(27\)	\(-378\)	\(832\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-378q^{5}+832q^{7}+3^{6}q^{9}+\cdots\)
48.8.a.f	$48$	$8$	48.a	1.a	$1$	$1$	$1$	$14.994$	\(\Q\)	None	✓		✓	✓	24.8.a.a	$1$	$1$	\(0\)	\(27\)	\(-26\)	\(-1056\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-26q^{5}-1056q^{7}+3^{6}q^{9}+\cdots\)
48.8.a.g	$48$	$8$	48.a	1.a	$1$	$1$	$1$	$14.994$	\(\Q\)	None	✓				3.8.a.a	$1$	$0$	\(0\)	\(27\)	\(390\)	\(64\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+390q^{5}+2^{6}q^{7}+3^{6}q^{9}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(48))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(48)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)