Space of modular forms of level 13, weight 6, and Character 13.e

• Label: 13.6.e
• Level: $13$
• Weight: $6$
• Character orbit: 13.e
• Rep. character: $\chi_{13}(4,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $10$
• Newform subspaces: $1$
• Sturm bound: $7$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	13.e (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(7\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(13, [\chi])\).

	Total	New	Old
Modular forms	14	14	0
Cusp forms	10	10	0
Eisenstein series	4	4	0

Trace form

10 * q - 3 * q^2 - 10 * q^3 + 63 * q^4 + 168 * q^6 - 276 * q^7 - 215 * q^9 + 115 * q^10 - 240 * q^11 - 164 * q^12 - 2015 * q^13 + 3444 * q^14 - 330 * q^15 - 2137 * q^16 + 1851 * q^17 + 4626 * q^19 + 2625 * q^20 - 962 * q^22 - 1374 * q^23 - 6018 * q^24 - 8784 * q^25 - 4017 * q^26 + 8888 * q^27 - 28668 * q^28 - 4071 * q^29 + 20358 * q^30 + 56637 * q^32 + 14142 * q^33 - 17160 * q^35 - 35869 * q^36 + 13491 * q^37 - 26184 * q^38 - 58526 * q^39 + 63074 * q^40 - 23997 * q^41 - 28626 * q^42 + 38120 * q^43 + 130107 * q^45 + 17160 * q^46 - 45442 * q^48 - 15485 * q^49 - 80376 * q^50 - 82860 * q^51 - 128934 * q^52 + 114270 * q^53 - 62070 * q^54 - 68120 * q^55 + 146832 * q^56 + 254625 * q^58 + 18060 * q^59 + 12071 * q^61 - 51450 * q^62 - 40704 * q^63 - 113750 * q^64 - 84747 * q^65 + 166932 * q^66 - 141618 * q^67 + 32763 * q^68 + 104046 * q^69 - 13854 * q^71 + 141825 * q^72 - 87255 * q^74 - 33716 * q^75 - 3822 * q^76 - 45396 * q^77 - 71370 * q^78 + 392 * q^79 - 151365 * q^80 - 79241 * q^81 - 153167 * q^82 + 157380 * q^84 + 76287 * q^85 - 105234 * q^87 - 62608 * q^88 + 234522 * q^89 + 309066 * q^90 + 175968 * q^91 - 207444 * q^92 + 347328 * q^93 + 116866 * q^94 + 42390 * q^95 - 376278 * q^97 - 488517 * q^98

Decomposition of \(S_{6}^{\mathrm{new}}(13, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
13.6.e.a	$13$	$6$	13.e	13.e	$6$	$10$	$5$	$2.085$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		✓	✓	✓	13.6.e.a	$2$	$0$	\(-3\)	\(-10\)	\(0\)	\(-276\)	$2^{6}\cdot 3\cdot 13^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{3}q^{2}+(-2-2\beta _{2}-\beta _{5}-\beta _{6})q^{3}+\cdots\)