Space of modular forms of level 31, weight 6, and trivial character

• Label: 31.6.a
• Level: $31$
• Weight: $6$
• Character orbit: 31.a
• Rep. character: $\chi_{31}(1,\cdot)$
• Character field: $\Q$
• Dimension: $13$
• Newform subspaces: $2$
• Sturm bound: $16$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 31 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	31.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(16\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(31))\).

	Total	New	Old
Modular forms	15	13	2
Cusp forms	13	13	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.

\(31\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(6\)	\(5\)	\(1\)		\(5\)	\(5\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)		\(9\)	\(8\)	\(1\)		\(8\)	\(8\)	\(0\)		\(1\)	\(0\)	\(1\)

Trace form

13 * q - 2 * q^2 - 22 * q^3 + 202 * q^4 + 56 * q^5 - 72 * q^6 - 20 * q^7 + 237 * q^8 + 989 * q^9 + 797 * q^10 - 664 * q^11 - 758 * q^12 - 288 * q^13 - 795 * q^14 + 78 * q^15 + 2930 * q^16 + 648 * q^17 - 2018 * q^18 - 1140 * q^19 + 2405 * q^20 + 446 * q^21 - 1672 * q^22 - 8130 * q^23 - 9164 * q^24 + 3153 * q^25 + 10766 * q^26 - 1228 * q^27 - 8089 * q^28 - 2708 * q^29 - 21534 * q^30 + 2883 * q^31 + 6262 * q^32 + 25284 * q^33 - 32460 * q^34 + 5932 * q^35 + 8646 * q^36 + 6258 * q^37 + 2429 * q^38 + 17028 * q^39 + 46764 * q^40 + 17344 * q^41 - 3326 * q^42 + 32658 * q^43 - 24288 * q^44 - 23988 * q^45 + 37314 * q^46 - 8628 * q^47 - 54788 * q^48 + 18235 * q^49 + 61573 * q^50 + 51276 * q^51 - 46472 * q^52 - 16366 * q^53 - 16292 * q^54 - 27732 * q^55 - 169716 * q^56 + 45118 * q^57 + 30148 * q^58 + 55172 * q^59 - 90652 * q^60 - 123612 * q^61 + 15376 * q^62 - 94408 * q^63 + 104973 * q^64 - 7778 * q^65 - 104084 * q^66 + 10588 * q^67 - 15546 * q^68 + 45324 * q^69 - 104263 * q^70 + 145812 * q^71 + 180589 * q^72 + 104382 * q^73 + 171566 * q^74 - 260452 * q^75 - 211861 * q^76 + 42800 * q^77 - 88660 * q^78 - 64738 * q^79 + 520557 * q^80 + 90369 * q^81 + 228711 * q^82 - 90784 * q^83 + 177240 * q^84 - 149618 * q^85 + 155546 * q^86 - 406168 * q^87 + 30034 * q^88 - 46428 * q^89 + 398961 * q^90 + 93978 * q^91 - 376500 * q^92 + 17298 * q^93 - 260948 * q^94 + 51424 * q^95 - 248242 * q^96 - 209068 * q^97 + 127745 * q^98 - 49280 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(31))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		31
31.6.a.a	$31$	$6$	31.a	1.a	$1$	$5$	$5$	$4.972$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	31.6.a.a	$1$	$1$	\(-9\)	\(-20\)	\(-72\)	\(-108\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(-4-\beta _{1}-\beta _{3})q^{3}+\cdots\)
31.6.a.b	$31$	$6$	31.a	1.a	$1$	$8$	$8$	$4.972$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	31.6.a.b	$1$	$0$	\(7\)	\(-2\)	\(128\)	\(88\)		$-$	$-$	$2^{2}\cdot 5\cdot 13$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+\beta _{6}q^{3}+(19-3\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)