Space of modular forms of level 30, weight 4, and trivial character

• Label: 30.4.a
• Level: $30$
• Weight: $4$
• Character orbit: 30.a
• Rep. character: $\chi_{30}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $24$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 30 = 2 \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	30.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(24\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(30))\).

	Total	New	Old
Modular forms	22	2	20
Cusp forms	14	2	12
Eisenstein series	8	0	8

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

\(2\)	\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(4\)	\(0\)	\(4\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(12\)	\(2\)	\(10\)		\(8\)	\(2\)	\(6\)		\(4\)	\(0\)	\(4\)
Minus space			\(-\)		\(10\)	\(0\)	\(10\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)

Trace form

2 * q + 6 * q^3 + 8 * q^4 + 28 * q^7 + 18 * q^9 - 20 * q^10 - 108 * q^11 + 24 * q^12 - 32 * q^13 - 72 * q^14 + 32 * q^16 - 72 * q^17 + 64 * q^19 + 84 * q^21 + 24 * q^22 + 72 * q^23 + 50 * q^25 + 72 * q^26 + 54 * q^27 + 112 * q^28 + 216 * q^29 - 60 * q^30 + 40 * q^31 - 324 * q^33 - 312 * q^34 + 180 * q^35 + 72 * q^36 - 200 * q^37 + 432 * q^38 - 96 * q^39 - 80 * q^40 + 36 * q^41 - 216 * q^42 - 680 * q^43 - 432 * q^44 + 144 * q^46 - 144 * q^47 + 96 * q^48 + 354 * q^49 - 216 * q^51 - 128 * q^52 + 144 * q^53 - 60 * q^55 - 288 * q^56 + 192 * q^57 + 408 * q^58 + 900 * q^59 - 188 * q^61 + 1008 * q^62 + 252 * q^63 + 128 * q^64 - 180 * q^65 + 72 * q^66 + 1408 * q^67 - 288 * q^68 + 216 * q^69 - 280 * q^70 - 648 * q^71 + 268 * q^73 - 936 * q^74 + 150 * q^75 + 256 * q^76 - 1728 * q^77 + 216 * q^78 - 488 * q^79 + 162 * q^81 - 864 * q^82 - 1152 * q^83 + 336 * q^84 + 780 * q^85 + 288 * q^86 + 648 * q^87 + 96 * q^88 - 468 * q^89 - 180 * q^90 - 1096 * q^91 + 288 * q^92 + 120 * q^93 + 1152 * q^94 - 1080 * q^95 - 1340 * q^97 - 2016 * q^98 - 972 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(30))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	5
30.4.a.a	$30$	$4$	30.a	1.a	$1$	$1$	$1$	$1.770$	\(\Q\)	None	✓	✓	✓	✓	30.4.a.a	$1$	$0$	\(-2\)	\(3\)	\(5\)	\(32\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
30.4.a.b	$30$	$4$	30.a	1.a	$1$	$1$	$1$	$1.770$	\(\Q\)	None	✓	✓	✓	✓	30.4.a.b	$1$	$0$	\(2\)	\(3\)	\(-5\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(30))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(30)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)