Space of modular forms of level 100, weight 8, and trivial character

• Label: 100.8.a
• Level: $100$
• Weight: $8$
• Character orbit: 100.a
• Rep. character: $\chi_{100}(1,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $5$
• Sturm bound: $120$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	100.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(120\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(100))\).

	Total	New	Old
Modular forms	114	11	103
Cusp forms	96	11	85
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
\(+\)	\(+\)	\(+\)		\(30\)	\(0\)	\(30\)		\(24\)	\(0\)	\(24\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(29\)	\(0\)	\(29\)		\(23\)	\(0\)	\(23\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(27\)	\(5\)	\(22\)		\(24\)	\(5\)	\(19\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(28\)	\(6\)	\(22\)		\(25\)	\(6\)	\(19\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(58\)	\(6\)	\(52\)		\(49\)	\(6\)	\(43\)		\(9\)	\(0\)	\(9\)
Minus space		\(-\)		\(56\)	\(5\)	\(51\)		\(47\)	\(5\)	\(42\)		\(9\)	\(0\)	\(9\)

Trace form

11 * q + 26 * q^3 - 954 * q^7 + 7679 * q^9 + 1680 * q^11 - 9126 * q^13 - 6318 * q^17 + 58844 * q^19 + 41156 * q^21 - 43518 * q^23 + 159452 * q^27 + 55554 * q^29 + 329196 * q^31 - 799920 * q^33 + 191946 * q^37 + 146364 * q^39 + 984966 * q^41 - 2083230 * q^43 + 1832862 * q^47 + 1596903 * q^49 + 3213452 * q^51 - 3235902 * q^53 + 1347304 * q^57 + 4496508 * q^59 + 2491222 * q^61 - 7176506 * q^63 - 290634 * q^67 + 5324252 * q^69 + 5222388 * q^71 - 3433686 * q^73 + 1616880 * q^77 - 5862088 * q^79 - 6922801 * q^81 + 5962722 * q^83 + 3939564 * q^87 - 16194414 * q^89 - 2075756 * q^91 + 26418136 * q^93 - 18000534 * q^97 - 25335680 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(100))\) 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
100.8.a.a	$100$	$8$	100.a	1.a	$1$	$1$	$1$	$31.239$	\(\Q\)	None	✓				20.8.a.a	$1$	$1$	\(0\)	\(6\)	\(0\)	\(706\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{3}+706q^{7}-2151q^{9}-3840q^{11}+\cdots\)
100.8.a.b	$100$	$8$	100.a	1.a	$1$	$2$	$2$	$31.239$	\(\Q(\sqrt{2521}) \)	None	✓	✓			100.8.a.b	$1$	$1$	\(0\)	\(-40\)	\(0\)	\(-280\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-20-\beta )q^{3}+(-140-6\beta )q^{7}+\cdots\)
100.8.a.c	$100$	$8$	100.a	1.a	$1$	$2$	$2$	$31.239$	\(\Q(\sqrt{1129}) \)	None	✓				20.8.a.b	$1$	$1$	\(0\)	\(20\)	\(0\)	\(-1660\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(10-\beta )q^{3}+(-830+9\beta )q^{7}+(2429+\cdots)q^{9}+\cdots\)
100.8.a.d	$100$	$8$	100.a	1.a	$1$	$2$	$2$	$31.239$	\(\Q(\sqrt{2521}) \)	None	✓	✓			100.8.a.b	$1$	$0$	\(0\)	\(40\)	\(0\)	\(280\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(20-\beta )q^{3}+(140-6\beta )q^{7}+(734+\cdots)q^{9}+\cdots\)
100.8.a.e	$100$	$8$	100.a	1.a	$1$	$4$	$4$	$31.239$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		20.8.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{8}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(6\beta _{1}+\beta _{2})q^{7}+(509+\beta _{3})q^{9}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(100))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(100)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)