Space of modular forms of level 100, weight 6, and trivial character

• Label: 100.6.a
• Level: $100$
• Weight: $6$
• Character orbit: 100.a
• Rep. character: $\chi_{100}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $5$
• Sturm bound: $90$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	100.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(90\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(100))\).

	Total	New	Old
Modular forms	84	8	76
Cusp forms	66	8	58
Eisenstein series	18	0	18

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

\(2\)	\(5\)	Fricke	Dim
\(-\)	\(+\)	$-$	\(4\)
\(-\)	\(-\)	$+$	\(4\)
Plus space		\(+\)	\(4\)
Minus space		\(-\)	\(4\)

Trace form

\( 8 q - 10 q^{3} - 130 q^{7} + 968 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(100))\) 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	5
100.6.a.a	$100$	$6$	100.a	1.a	$1$	$1$	$1$	$16.038$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-22\)	\(0\)	\(-218\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-22q^{3}-218q^{7}+241q^{9}-480q^{11}+\cdots\)
100.6.a.b	$100$	$6$	100.a	1.a	$1$	$1$	$1$	$16.038$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(12\)	\(0\)	\(88\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+12q^{3}+88q^{7}-99q^{9}+540q^{11}+\cdots\)
100.6.a.c	$100$	$6$	100.a	1.a	$1$	$2$	$2$	$16.038$	\(\Q(\sqrt{409}) \)	None	✓	$1$	$1$	\(0\)	\(-20\)	\(0\)	\(40\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-10-\beta )q^{3}+(20+6\beta )q^{7}+(266+\cdots)q^{9}+\cdots\)
100.6.a.d	$100$	$6$	100.a	1.a	$1$	$2$	$2$	$16.038$	\(\Q(\sqrt{31}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-11\beta q^{7}-119q^{9}-10^{2}q^{11}+\cdots\)
100.6.a.e	$100$	$6$	100.a	1.a	$1$	$2$	$2$	$16.038$	\(\Q(\sqrt{409}) \)	None	✓	$1$	$0$	\(0\)	\(20\)	\(0\)	\(-40\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(10-\beta )q^{3}+(-20+6\beta )q^{7}+(266+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(100))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(100)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)