Space of modular forms of level 15, weight 4, and trivial character

• Label: 15.4.a
• Level: $15$
• Weight: $4$
• Character orbit: 15.a
• Rep. character: $\chi_{15}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $8$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	15.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(8\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(15))\).

	Total	New	Old
Modular forms	8	2	6
Cusp forms	4	2	2
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
\(+\)	\(+\)	\(+\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(1\)	\(0\)	\(1\)		\(0\)	\(0\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(2\)	\(1\)	\(1\)		\(1\)	\(1\)	\(0\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(5\)	\(2\)	\(3\)		\(3\)	\(2\)	\(1\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)		\(2\)	\(0\)	\(2\)

Trace form

2 * q + 4 * q^2 - 6 * q^4 - 6 * q^6 - 4 * q^7 - 36 * q^8 + 18 * q^9 - 10 * q^10 + 28 * q^11 - 24 * q^12 + 96 * q^13 + 36 * q^14 + 30 * q^15 - 30 * q^16 + 40 * q^17 + 36 * q^18 - 144 * q^19 - 40 * q^20 - 132 * q^21 - 20 * q^22 - 288 * q^23 + 18 * q^24 + 50 * q^25 + 244 * q^26 + 188 * q^28 + 152 * q^29 + 60 * q^30 - 88 * q^31 + 116 * q^32 + 228 * q^33 + 148 * q^34 - 220 * q^35 - 54 * q^36 - 104 * q^37 - 392 * q^38 - 156 * q^39 + 30 * q^40 + 452 * q^41 - 252 * q^42 - 96 * q^43 - 388 * q^44 - 528 * q^46 + 232 * q^47 + 336 * q^48 + 290 * q^49 + 100 * q^50 - 204 * q^51 - 80 * q^52 + 112 * q^53 - 54 * q^54 + 380 * q^55 - 60 * q^56 + 312 * q^57 - 4 * q^58 + 124 * q^59 - 90 * q^60 + 420 * q^61 + 312 * q^62 - 36 * q^63 + 266 * q^64 - 260 * q^65 + 372 * q^66 - 280 * q^67 + 152 * q^68 - 144 * q^69 - 420 * q^70 - 616 * q^71 - 324 * q^72 - 468 * q^73 - 244 * q^74 + 16 * q^76 - 1728 * q^77 - 600 * q^78 - 760 * q^79 + 560 * q^80 + 162 * q^81 + 1112 * q^82 + 1368 * q^83 + 444 * q^84 - 340 * q^85 + 88 * q^86 + 924 * q^87 - 276 * q^88 + 1356 * q^89 - 90 * q^90 + 952 * q^91 + 1056 * q^92 - 1464 * q^93 + 184 * q^94 + 520 * q^95 + 618 * q^96 + 580 * q^97 + 404 * q^98 + 252 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(15))\) 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
15.4.a.a	$15$	$4$	15.a	1.a	$1$	$1$	$1$	$0.885$	\(\Q\)	None	✓	✓	✓	✓	15.4.a.a	$1$	$0$	\(1\)	\(3\)	\(5\)	\(-24\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+3q^{3}-7q^{4}+5q^{5}+3q^{6}+\cdots\)
15.4.a.b	$15$	$4$	15.a	1.a	$1$	$1$	$1$	$0.885$	\(\Q\)	None	✓	✓	✓	✓	15.4.a.b	$1$	$0$	\(3\)	\(-3\)	\(-5\)	\(20\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}-3q^{3}+q^{4}-5q^{5}-9q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(15))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(15)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)