Space of modular forms of level 10, weight 5, and Character 10.c

• Label: 10.5.c
• Level: $10$
• Weight: $5$
• Character orbit: 10.c
• Rep. character: $\chi_{10}(3,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $4$
• Newform subspaces: $2$
• Sturm bound: $7$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	10.c (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(7\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(10, [\chi])\).

	Total	New	Old
Modular forms	16	4	12
Cusp forms	8	4	4
Eisenstein series	8	0	8

Trace form

4 * q + 20 * q^3 - 60 * q^5 - 64 * q^6 + 20 * q^7 + 160 * q^10 + 168 * q^11 + 160 * q^12 - 60 * q^13 + 20 * q^15 - 256 * q^16 - 1020 * q^17 - 640 * q^18 + 968 * q^21 + 1280 * q^22 + 1620 * q^23 - 700 * q^25 - 1344 * q^26 + 320 * q^27 - 160 * q^28 + 960 * q^30 - 1112 * q^31 - 1720 * q^33 - 2220 * q^35 + 32 * q^36 - 1020 * q^37 + 960 * q^38 + 1280 * q^40 + 5448 * q^41 - 2240 * q^42 + 660 * q^43 + 6400 * q^45 + 2176 * q^46 + 1620 * q^47 - 1280 * q^48 - 4800 * q^50 - 10712 * q^51 - 480 * q^52 - 4860 * q^53 - 2520 * q^55 + 3072 * q^56 + 5120 * q^57 + 3520 * q^58 - 4960 * q^60 - 7032 * q^61 - 3840 * q^62 + 7700 * q^63 + 7620 * q^65 + 10112 * q^66 + 8660 * q^67 + 8160 * q^68 + 1600 * q^70 + 5928 * q^71 - 5120 * q^72 - 12860 * q^73 - 13100 * q^75 - 5120 * q^76 - 14520 * q^77 - 5760 * q^78 + 3840 * q^80 + 964 * q^81 - 2560 * q^82 - 300 * q^83 + 16580 * q^85 - 14784 * q^86 + 11840 * q^87 - 10240 * q^88 + 9440 * q^90 + 15528 * q^91 + 12960 * q^92 + 2120 * q^93 - 9600 * q^95 + 4096 * q^96 - 13500 * q^97 + 3840 * q^98

Decomposition of \(S_{5}^{\mathrm{new}}(10, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
10.5.c.a	$10$	$5$	10.c	5.c	$4$	$2$	$1$	$1.034$	\(\Q(\sqrt{-1}) \)	None		✓			10.5.c.a	$2$	$0$	\(-4\)	\(18\)	\(-30\)	\(58\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2 i-2)q^{2}+(-9 i+9)q^{3}+\cdots\)
10.5.c.b	$10$	$5$	10.c	5.c	$4$	$2$	$1$	$1.034$	\(\Q(\sqrt{-1}) \)	None		✓			10.5.c.b	$2$	$0$	\(4\)	\(2\)	\(-30\)	\(-38\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2 i+2)q^{2}+(-i+1)q^{3}+8 i q^{4}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(10, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(10, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 2}\)