Space of modular forms of level 13, weight 6, and trivial character

• Label: 13.6.a
• Level: $13$
• Weight: $6$
• Character orbit: 13.a
• Rep. character: $\chi_{13}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $2$
• Sturm bound: $7$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	13.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(7\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(13))\).

	Total	New	Old
Modular forms	7	5	2
Cusp forms	5	5	0
Eisenstein series	2	0	2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(13\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(3\)	\(2\)	\(1\)		\(2\)	\(2\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)		\(4\)	\(3\)	\(1\)		\(3\)	\(3\)	\(0\)		\(1\)	\(0\)	\(1\)

Trace form

5 * q + 2 * q^2 - 20 * q^3 + 78 * q^4 + 14 * q^5 - 180 * q^6 - 96 * q^7 + 552 * q^8 + 21 * q^9 - 846 * q^10 + 180 * q^11 - 234 * q^12 + 169 * q^13 - 1298 * q^14 + 520 * q^15 + 3250 * q^16 - 1722 * q^17 + 1874 * q^18 - 164 * q^19 - 1564 * q^20 + 5464 * q^21 - 3184 * q^22 + 1000 * q^23 - 14436 * q^24 + 10255 * q^25 + 2028 * q^26 - 9008 * q^27 - 2912 * q^28 - 9570 * q^29 + 1658 * q^30 + 5112 * q^31 + 29696 * q^32 + 7720 * q^33 + 21476 * q^34 - 18696 * q^35 - 16376 * q^36 - 37490 * q^37 + 34804 * q^38 + 6084 * q^39 - 62174 * q^40 - 3158 * q^41 + 12666 * q^42 + 4452 * q^43 + 61444 * q^44 + 38750 * q^45 + 15188 * q^46 + 44232 * q^47 - 101858 * q^48 - 35887 * q^49 - 99898 * q^50 + 56816 * q^51 + 27716 * q^52 - 56290 * q^53 + 54420 * q^54 + 14648 * q^55 - 28942 * q^56 + 4656 * q^57 - 16688 * q^58 + 25012 * q^59 + 160128 * q^60 - 10770 * q^61 - 63448 * q^62 - 105424 * q^63 + 120610 * q^64 + 16562 * q^65 - 116640 * q^66 - 13796 * q^67 + 12810 * q^68 - 55592 * q^69 + 105284 * q^70 + 144976 * q^71 - 5904 * q^72 + 31386 * q^73 - 105014 * q^74 - 6020 * q^75 - 94120 * q^76 + 35696 * q^77 - 36842 * q^78 - 117168 * q^79 - 146948 * q^80 - 15459 * q^81 + 106068 * q^82 + 22380 * q^83 - 29108 * q^84 - 43516 * q^85 - 35844 * q^86 + 147992 * q^87 + 144984 * q^88 + 150202 * q^89 - 24360 * q^90 - 4056 * q^91 + 297100 * q^92 + 20160 * q^93 - 34282 * q^94 - 180488 * q^95 - 202148 * q^96 + 41578 * q^97 - 54926 * q^98 - 220140 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(13))\) 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					A-L signs	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		13
13.6.a.a	$13$	$6$	13.a	1.a	$1$	$2$	$2$	$2.085$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	13.6.a.a	$1$	$1$	\(-5\)	\(-28\)	\(-42\)	\(-36\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{2}+(-17+6\beta )q^{3}+(-24+\cdots)q^{4}+\cdots\)
13.6.a.b	$13$	$6$	13.a	1.a	$1$	$3$	$3$	$2.085$	3.3.168897.1	None	✓	✓	✓	✓	13.6.a.b	$1$	$0$	\(7\)	\(8\)	\(56\)	\(-60\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{2}+(3-\beta _{1}-\beta _{2})q^{3}+(40+\cdots)q^{4}+\cdots\)