Space of modular forms of level 10, weight 13, and Character 10.c

• Label: 10.13.c
• Level: $10$
• Weight: $13$
• Character orbit: 10.c
• Rep. character: $\chi_{10}(3,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $12$
• Newform subspaces: $2$
• Sturm bound: $19$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 13 \)
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:			\(19\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{13}(10, [\chi])\).

	Total	New	Old
Modular forms	40	12	28
Cusp forms	32	12	20
Eisenstein series	8	0	8

Trace form

12 * q - 640 * q^3 - 1800 * q^5 + 78848 * q^6 - 367440 * q^7 + 867840 * q^10 - 1765296 * q^11 - 1310720 * q^12 + 3460020 * q^13 - 15082480 * q^15 - 50331648 * q^16 + 61972740 * q^17 + 27822080 * q^18 - 54190080 * q^20 - 55913296 * q^21 + 162263040 * q^22 + 346973760 * q^23 - 682458300 * q^25 - 816233472 * q^26 + 2064641360 * q^27 + 752517120 * q^28 - 2464435200 * q^30 - 4376225376 * q^31 + 10797394640 * q^33 - 11582581680 * q^35 + 2894667776 * q^36 + 1005341340 * q^37 + 1802695680 * q^38 - 2249195520 * q^40 - 2026801296 * q^41 + 9285114880 * q^42 - 1357559520 * q^43 + 8988051460 * q^45 - 8660477952 * q^46 - 10068243840 * q^47 + 2684354560 * q^48 + 12452428800 * q^50 + 31168538464 * q^51 + 7086120960 * q^52 - 86723398980 * q^53 + 113239460400 * q^55 + 36276535296 * q^56 - 144146964640 * q^57 - 107285130240 * q^58 + 99082895360 * q^60 + 90518069424 * q^61 - 30656286720 * q^62 - 147686550400 * q^63 + 173117966220 * q^65 - 130203817984 * q^66 + 104416726080 * q^67 - 126920171520 * q^68 + 21297884160 * q^70 + 358733348064 * q^71 + 56979619840 * q^72 + 722920298220 * q^73 - 1108294984400 * q^75 - 387391979520 * q^76 - 427325430960 * q^77 + 342313912320 * q^78 + 7549747200 * q^80 + 78995180452 * q^81 + 1519071928320 * q^82 + 796544103600 * q^83 - 1316786650380 * q^85 - 1947103368192 * q^86 - 333007377280 * q^87 - 332314705920 * q^88 + 1253888330240 * q^90 + 222319671744 * q^91 + 710602260480 * q^92 + 2824159805360 * q^93 - 2482880641200 * q^95 - 330712481792 * q^96 - 1532104852500 * q^97 - 3288563527680 * q^98

Decomposition of \(S_{13}^{\mathrm{new}}(10, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
10.13.c.a	$10$	$13$	10.c	5.c	$4$	$6$	$3$	$9.140$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		✓			10.13.c.a	$2$	$0$	\(-192\)	\(-936\)	\(-16260\)	\(-45336\)	$2^{5}\cdot 3^{2}\cdot 5^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2^{5}-2^{5}\beta _{1})q^{2}+(-156+156\beta _{1}+\cdots)q^{3}+\cdots\)
10.13.c.b	$10$	$13$	10.c	5.c	$4$	$6$	$3$	$9.140$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		✓			10.13.c.b	$2$	$0$	\(192\)	\(296\)	\(14460\)	\(-322104\)	$2^{5}\cdot 5^{5}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2^{5}+2^{5}\beta _{1})q^{2}+(7^{2}-7^{2}\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{13}^{\mathrm{old}}(10, [\chi])\) into lower level spaces

\( S_{13}^{\mathrm{old}}(10, [\chi]) \simeq \) \(S_{13}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 2}\)