Space of modular forms of level 13, weight 4, and Character 13.c

• Label: 13.4.c
• Level: $13$
• Weight: $4$
• Character orbit: 13.c
• Rep. character: $\chi_{13}(3,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $6$
• Newform subspaces: $2$
• Sturm bound: $4$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	13.c (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(4\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(13, [\chi])\).

	Total	New	Old
Modular forms	10	10	0
Cusp forms	6	6	0
Eisenstein series	4	4	0

Trace form

6 * q + q^2 - 7 * q^3 - 13 * q^4 + 4 * q^5 + 30 * q^6 - 35 * q^7 - 30 * q^8 - 12 * q^9 - 63 * q^10 + 15 * q^11 + 312 * q^12 + 34 * q^13 + 68 * q^14 - 124 * q^15 + 7 * q^16 - 57 * q^17 - 614 * q^18 + 111 * q^19 - 311 * q^20 + 206 * q^21 + 298 * q^22 - 223 * q^23 + 216 * q^24 + 478 * q^25 + 237 * q^26 + 470 * q^27 - 240 * q^28 - 231 * q^29 + 4 * q^30 - 428 * q^31 + 151 * q^32 - 361 * q^33 - 182 * q^34 - 270 * q^35 - 541 * q^36 + 417 * q^37 + 860 * q^38 - 51 * q^39 - 370 * q^40 - 373 * q^41 + 210 * q^42 - 299 * q^43 + 848 * q^44 + 16 * q^45 - 230 * q^46 - 204 * q^47 + 368 * q^48 + 508 * q^49 - 1106 * q^50 - 518 * q^51 - 102 * q^52 + 1276 * q^53 + 1314 * q^54 + 34 * q^55 + 172 * q^56 - 330 * q^57 - 193 * q^58 + 1673 * q^59 + 144 * q^60 - 647 * q^61 + 796 * q^62 + 70 * q^63 - 3566 * q^64 - 2102 * q^65 - 3708 * q^66 - 387 * q^67 + 609 * q^68 + 323 * q^69 + 2560 * q^70 - 781 * q^71 + 1155 * q^72 + 1600 * q^73 + 59 * q^74 + 1397 * q^75 - 100 * q^76 - 770 * q^77 + 3278 * q^78 + 328 * q^79 + 2153 * q^80 - 543 * q^81 + 2175 * q^82 + 1776 * q^83 - 1540 * q^84 + 1426 * q^85 - 5172 * q^86 - 2009 * q^87 + 1020 * q^88 - 655 * q^89 - 4578 * q^90 - 767 * q^91 - 832 * q^92 - 2052 * q^93 - 392 * q^94 - 1780 * q^95 + 2816 * q^96 + 177 * q^97 - 1513 * q^98 + 5892 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(13, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
13.4.c.a	$13$	$4$	13.c	13.c	$3$	$2$	$1$	$0.767$	\(\Q(\sqrt{-3}) \)	None		✓			13.4.c.a	$2$	$0$	\(-4\)	\(-2\)	\(34\)	\(-20\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-4+4\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}+\cdots\)
13.4.c.b	$13$	$4$	13.c	13.c	$3$	$4$	$2$	$0.767$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None		✓	✓		13.4.c.b	$2$	$0$	\(5\)	\(-5\)	\(-30\)	\(-15\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-\beta _{1}-2\beta _{2}-\beta _{3})q^{2}+(-1+3\beta _{1}+\cdots)q^{3}+\cdots\)