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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(21\)	\(0\)	\(21\)		\(15\)	\(0\)	\(15\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(22\)	\(0\)	\(22\)		\(16\)	\(0\)	\(16\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(21\)	\(4\)	\(17\)		\(18\)	\(4\)	\(14\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(20\)	\(4\)	\(16\)		\(17\)	\(4\)	\(13\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(41\)	\(4\)	\(37\)		\(32\)	\(4\)	\(28\)		\(9\)	\(0\)	\(9\)
Minus space		\(-\)		\(43\)	\(4\)	\(39\)		\(34\)	\(4\)	\(30\)		\(9\)	\(0\)	\(9\)

Trace form

8 * q - 10 * q^3 - 130 * q^7 + 968 * q^9 - 260 * q^11 + 1040 * q^13 - 780 * q^17 - 672 * q^19 - 7492 * q^21 + 7290 * q^23 - 4060 * q^27 - 3672 * q^29 - 8452 * q^31 + 17040 * q^33 - 5320 * q^37 + 32628 * q^39 - 36872 * q^41 + 12470 * q^43 - 14850 * q^47 + 11316 * q^49 + 57356 * q^51 - 15120 * q^53 + 36520 * q^57 - 40864 * q^59 + 10756 * q^61 - 61250 * q^63 - 10270 * q^67 - 168076 * q^69 + 4284 * q^71 - 101260 * q^73 + 152160 * q^77 + 81384 * q^79 + 326012 * q^81 - 29250 * q^83 - 128700 * q^87 + 52972 * q^89 - 191148 * q^91 - 154760 * q^93 + 27740 * q^97 + 313540 * q^99

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	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	5
100.6.a.a	$100$	$6$	100.a	1.a	$1$	$1$	$1$	$16.038$	\(\Q\)	None	✓				20.6.a.a	$1$	$0$	\(0\)	\(-22\)	\(0\)	\(-218\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-22q^{3}-218q^{7}+241q^{9}-480q^{11}+\cdots\)
100.6.a.b	$100$	$6$	100.a	1.a	$1$	$1$	$1$	$16.038$	\(\Q\)	None	✓				4.6.a.a	$1$	$0$	\(0\)	\(12\)	\(0\)	\(88\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+12q^{3}+88q^{7}-99q^{9}+540q^{11}+\cdots\)
100.6.a.c	$100$	$6$	100.a	1.a	$1$	$2$	$2$	$16.038$	\(\Q(\sqrt{409}) \)	None	✓	✓			100.6.a.c	$1$	$1$	\(0\)	\(-20\)	\(0\)	\(40\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-10-\beta )q^{3}+(20+6\beta )q^{7}+(266+\cdots)q^{9}+\cdots\)
100.6.a.d	$100$	$6$	100.a	1.a	$1$	$2$	$2$	$16.038$	\(\Q(\sqrt{31}) \)	None	✓				20.6.c.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-11\beta q^{7}-119q^{9}-10^{2}q^{11}+\cdots\)
100.6.a.e	$100$	$6$	100.a	1.a	$1$	$2$	$2$	$16.038$	\(\Q(\sqrt{409}) \)	None	✓	✓	✓		100.6.a.c	$1$	$0$	\(0\)	\(20\)	\(0\)	\(-40\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(10-\beta )q^{3}+(-20+6\beta )q^{7}+(266+\cdots)q^{9}+\cdots\)

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

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