Space of modular forms of level 15, weight 6, and trivial character

• Label: 15.6.a
• Level: $15$
• Weight: $6$
• Character orbit: 15.a
• Rep. character: $\chi_{15}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $3$
• Sturm bound: $12$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	15.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(12\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(15))\).

	Total	New	Old
Modular forms	12	4	8
Cusp forms	8	4	4
Eisenstein series	4	0	4

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

\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(2\)	\(1\)	\(1\)		\(1\)	\(1\)	\(0\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(4\)	\(2\)	\(2\)		\(3\)	\(2\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(5\)	\(1\)	\(4\)		\(3\)	\(1\)	\(2\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(7\)	\(3\)	\(4\)		\(5\)	\(3\)	\(2\)		\(2\)	\(0\)	\(2\)

Trace form

4 * q + 4 * q^2 - 18 * q^3 + 130 * q^4 + 90 * q^6 - 232 * q^7 - 228 * q^8 + 324 * q^9 - 150 * q^10 + 832 * q^11 - 864 * q^12 - 784 * q^13 - 2868 * q^14 - 450 * q^15 + 2962 * q^16 + 2656 * q^17 + 324 * q^18 + 704 * q^19 + 3800 * q^20 + 2304 * q^21 - 6828 * q^22 + 264 * q^23 + 162 * q^24 + 2500 * q^25 - 2612 * q^26 - 1458 * q^27 - 724 * q^28 - 2032 * q^29 - 1800 * q^30 + 4256 * q^31 - 27052 * q^32 - 5472 * q^33 + 1020 * q^34 + 200 * q^35 + 10530 * q^36 + 15008 * q^37 + 56104 * q^38 - 10476 * q^39 - 6450 * q^40 - 18424 * q^41 + 27324 * q^42 - 48520 * q^43 + 12716 * q^44 + 13776 * q^46 - 1160 * q^47 - 49680 * q^48 + 8964 * q^49 + 2500 * q^50 + 15372 * q^51 + 61136 * q^52 + 19744 * q^53 + 7290 * q^54 - 8400 * q^55 - 131100 * q^56 + 19224 * q^57 - 43452 * q^58 + 8176 * q^59 - 41850 * q^60 - 3736 * q^61 - 19704 * q^62 - 18792 * q^63 + 7018 * q^64 + 63400 * q^65 + 75564 * q^66 - 31864 * q^67 + 240728 * q^68 + 55512 * q^69 - 89100 * q^70 - 146032 * q^71 - 18468 * q^72 + 61784 * q^73 - 125548 * q^74 - 11250 * q^75 + 145856 * q^76 + 29856 * q^77 - 99216 * q^78 + 102800 * q^79 + 105200 * q^80 + 26244 * q^81 - 37224 * q^82 + 20952 * q^83 + 10188 * q^84 + 35400 * q^85 - 110408 * q^86 - 56412 * q^87 - 240468 * q^88 - 20136 * q^89 - 12150 * q^90 - 22544 * q^91 - 169632 * q^92 - 140688 * q^93 - 281976 * q^94 + 55600 * q^95 + 142794 * q^96 + 84776 * q^97 + 236084 * q^98 + 67392 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(15))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5
15.6.a.a	$15$	$6$	15.a	1.a	$1$	$1$	$1$	$2.406$	\(\Q\)	None	✓	✓	✓	✓	15.6.a.a	$1$	$1$	\(-2\)	\(-9\)	\(-25\)	\(-132\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-9q^{3}-28q^{4}-5^{2}q^{5}+18q^{6}+\cdots\)
15.6.a.b	$15$	$6$	15.a	1.a	$1$	$1$	$1$	$2.406$	\(\Q\)	None	✓	✓	✓	✓	15.6.a.b	$1$	$0$	\(7\)	\(9\)	\(-25\)	\(12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7q^{2}+9q^{3}+17q^{4}-5^{2}q^{5}+63q^{6}+\cdots\)
15.6.a.c	$15$	$6$	15.a	1.a	$1$	$2$	$2$	$2.406$	\(\Q(\sqrt{409}) \)	None	✓	✓	✓	✓	15.6.a.c	$1$	$0$	\(-1\)	\(-18\)	\(50\)	\(-112\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-9q^{3}+(70+\beta )q^{4}+5^{2}q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(15))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(15)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)