Space of modular forms of level 100, weight 4, and Character 100.e

• Label: 100.4.e
• Level: $100$
• Weight: $4$
• Character orbit: 100.e
• Rep. character: $\chi_{100}(7,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $50$
• Newform subspaces: $6$
• Sturm bound: $60$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	100.e (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(60\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(100, [\chi])\).

	Total	New	Old
Modular forms	102	58	44
Cusp forms	78	50	28
Eisenstein series	24	8	16

Trace form

50 * q + 2 * q^2 - 28 * q^6 + 44 * q^8 + 80 * q^12 - 42 * q^13 - 380 * q^16 + 134 * q^17 - 306 * q^18 + 264 * q^21 - 360 * q^22 + 908 * q^26 + 880 * q^28 + 632 * q^32 - 80 * q^33 - 1028 * q^36 - 326 * q^37 - 1600 * q^38 - 600 * q^41 - 1160 * q^42 + 632 * q^46 + 2720 * q^48 + 1524 * q^52 + 698 * q^53 - 4528 * q^56 + 960 * q^57 - 2712 * q^58 + 3640 * q^61 - 2440 * q^62 + 5940 * q^66 + 2428 * q^68 + 2172 * q^72 - 1942 * q^73 - 2380 * q^76 - 3120 * q^77 - 3720 * q^78 - 7626 * q^81 - 536 * q^82 + 112 * q^86 + 2400 * q^88 + 1840 * q^92 + 3280 * q^93 + 8252 * q^96 + 5994 * q^97 - 326 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(100, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
100.4.e.a	$100$	$4$	100.e	20.e	$4$	$2$	$1$	$5.900$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					20.4.e.a	$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-2 i-2)q^{2}+8 i q^{4}+(-16 i+16)q^{8}+\cdots\)
100.4.e.b	$100$	$4$	100.e	20.e	$4$	$2$	$1$	$5.900$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-5}) \)		✓			100.4.e.b	$4$	$0$	\(-4\)	\(14\)	\(0\)	\(-18\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(2 i-2)q^{2}+(7 i+7)q^{3}-8 i q^{4}+\cdots\)
100.4.e.c	$100$	$4$	100.e	20.e	$4$	$2$	$1$	$5.900$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-5}) \)		✓			100.4.e.b	$4$	$0$	\(4\)	\(-14\)	\(0\)	\(18\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-2 i+2)q^{2}+(-7 i-7)q^{3}+\cdots\)
100.4.e.d	$100$	$4$	100.e	20.e	$4$	$8$	$4$	$5.900$	8.0.2342560000.1	None		✓			100.4.e.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{5}q^{2}+(-\beta _{3}-\beta _{6})q^{3}+(-\beta _{1}-3\beta _{4}+\cdots)q^{4}+\cdots\)
100.4.e.e	$100$	$4$	100.e	20.e	$4$	$12$	$6$	$5.900$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					20.4.e.b	$4$	$0$	\(6\)	\(0\)	\(0\)	\(0\)	$2^{15}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta _{3}+\beta _{5})q^{2}+\beta _{9}q^{3}+(-2\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
100.4.e.f	$100$	$4$	100.e	20.e	$4$	$24$	$12$	$5.900$		None		✓	✓		100.4.e.f	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{4}^{\mathrm{old}}(100, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(100, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 2}\)