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

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

Defining parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	13.e (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(4\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

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

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

Trace form

4 * q - 3 * q^2 + 5 * q^3 - q^4 - 48 * q^6 + 15 * q^7 + q^9 + 51 * q^10 - 15 * q^11 + 76 * q^12 + 65 * q^13 - 204 * q^14 + 174 * q^15 + 79 * q^16 - 144 * q^17 - 351 * q^19 - 111 * q^20 + 102 * q^22 + 21 * q^23 + 246 * q^24 + 110 * q^25 + 351 * q^26 - 130 * q^27 + 276 * q^28 + 210 * q^29 - 330 * q^30 - 603 * q^32 - 321 * q^33 + 336 * q^35 + 203 * q^36 - 318 * q^37 + 216 * q^38 + 325 * q^39 - 462 * q^40 - 210 * q^41 - 498 * q^42 - 19 * q^43 + 597 * q^45 + 576 * q^46 - 562 * q^48 + 13 * q^49 + 768 * q^50 + 90 * q^51 - 494 * q^52 + 1038 * q^53 - 510 * q^54 + 336 * q^55 + 624 * q^56 + 9 * q^58 - 525 * q^59 - 128 * q^61 - 522 * q^62 - 1410 * q^63 - 742 * q^64 - 1209 * q^65 + 996 * q^66 - 1077 * q^67 - 477 * q^68 + 555 * q^69 + 2841 * q^71 + 1425 * q^72 - 111 * q^74 - 713 * q^75 + 378 * q^76 - 1398 * q^77 - 858 * q^78 + 64 * q^79 + 1923 * q^80 + 418 * q^81 + 1833 * q^82 + 852 * q^84 + 297 * q^85 - 201 * q^87 + 624 * q^88 - 1161 * q^89 - 1974 * q^90 - 741 * q^91 - 1236 * q^92 - 1422 * q^93 - 1710 * q^94 - 1026 * q^95 + 2487 * q^97 - 1437 * q^98

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.e.a	$13$	$4$	13.e	13.e	$6$	$2$	$1$	$0.767$	\(\Q(\sqrt{-3}) \)	None		✓			13.4.e.a	$2$	$0$	\(-6\)	\(7\)	\(0\)	\(39\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2-2\zeta_{6})q^{2}+(7-7\zeta_{6})q^{3}+4\zeta_{6}q^{4}+\cdots\)
13.4.e.b	$13$	$4$	13.e	13.e	$6$	$2$	$1$	$0.767$	\(\Q(\sqrt{-3}) \)	None		✓			13.4.e.b	$2$	$0$	\(3\)	\(-2\)	\(0\)	\(-24\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}-5\zeta_{6}q^{4}+\cdots\)