Space of modular forms of level 10, weight 9, and Character 10.c

• Label: 10.9.c
• Level: $10$
• Weight: $9$
• Character orbit: 10.c
• Rep. character: $\chi_{10}(3,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $8$
• Newform subspaces: $2$
• Sturm bound: $13$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
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:			\(13\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(10, [\chi])\).

	Total	New	Old
Modular forms	28	8	20
Cusp forms	20	8	12
Eisenstein series	8	0	8

Trace form

8 * q + 140 * q^3 - 780 * q^5 + 512 * q^6 + 4540 * q^7 - 6400 * q^10 - 34584 * q^11 + 17920 * q^12 + 119280 * q^13 - 252580 * q^15 - 131072 * q^16 + 202560 * q^17 + 158720 * q^18 + 53760 * q^20 + 110696 * q^21 + 40960 * q^22 + 174540 * q^23 - 368000 * q^25 - 450048 * q^26 - 704560 * q^27 - 581120 * q^28 + 1697280 * q^30 + 2342456 * q^31 - 3190840 * q^33 + 4359900 * q^35 - 618496 * q^36 - 6531240 * q^37 - 5076480 * q^38 + 1146880 * q^40 + 6454056 * q^41 + 13150720 * q^42 + 12059820 * q^43 - 14797820 * q^45 - 17410048 * q^46 - 8748660 * q^47 - 2293760 * q^48 + 5414400 * q^50 + 4713976 * q^51 + 15267840 * q^52 + 30279480 * q^53 - 31155960 * q^55 - 14155776 * q^56 - 21729760 * q^57 - 24194560 * q^58 + 18675200 * q^60 + 41805096 * q^61 + 56017920 * q^62 + 21701900 * q^63 - 18428280 * q^65 - 94667776 * q^66 - 47676980 * q^67 - 25927680 * q^68 + 105582080 * q^70 + 133410936 * q^71 + 20316160 * q^72 - 72484120 * q^73 + 1104700 * q^75 - 44615680 * q^76 - 200406840 * q^77 - 100792320 * q^78 + 12779520 * q^80 + 131272288 * q^81 + 120186880 * q^82 + 135315900 * q^83 - 3822800 * q^85 - 35731968 * q^86 + 66654080 * q^87 - 5242880 * q^88 - 21256960 * q^90 - 291657384 * q^91 + 22341120 * q^92 + 115691240 * q^93 - 242319600 * q^95 - 8388608 * q^96 + 173745000 * q^97 + 115169280 * q^98

Decomposition of \(S_{9}^{\mathrm{new}}(10, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
10.9.c.a	$10$	$9$	10.c	5.c	$4$	$4$	$2$	$4.074$	\(\Q(i, \sqrt{249})\)	None		✓			10.9.c.a	$2$	$0$	\(-32\)	\(54\)	\(90\)	\(-1186\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-8+8\beta _{1})q^{2}+(13+13\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
10.9.c.b	$10$	$9$	10.c	5.c	$4$	$4$	$2$	$4.074$	\(\Q(i, \sqrt{601})\)	None		✓			10.9.c.b	$2$	$0$	\(32\)	\(86\)	\(-870\)	\(5726\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(8+8\beta _{1})q^{2}+(22-21\beta _{1}+\beta _{3})q^{3}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(10, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(10, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 2}\)