Space of modular forms of level 150, weight 4, and trivial character

• Label: 150.4.a
• Level: $150$
• Weight: $4$
• Character orbit: 150.a
• Rep. character: $\chi_{150}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $9$
• Sturm bound: $120$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(120\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(150))\).

	Total	New	Old
Modular forms	102	9	93
Cusp forms	78	9	69
Eisenstein series	24	0	24

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
\(+\)	\(+\)	\(+\)	\(+\)		\(15\)	\(1\)	\(14\)		\(12\)	\(1\)	\(11\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(11\)	\(1\)	\(10\)		\(8\)	\(1\)	\(7\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(12\)	\(1\)	\(11\)		\(9\)	\(1\)	\(8\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(13\)	\(1\)	\(12\)		\(10\)	\(1\)	\(9\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(12\)	\(1\)	\(11\)		\(9\)	\(1\)	\(8\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(13\)	\(2\)	\(11\)		\(10\)	\(2\)	\(8\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(12\)	\(2\)	\(10\)		\(9\)	\(2\)	\(7\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(14\)	\(0\)	\(14\)		\(11\)	\(0\)	\(11\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(53\)	\(6\)	\(47\)		\(41\)	\(6\)	\(35\)		\(12\)	\(0\)	\(12\)
Minus space			\(-\)		\(49\)	\(3\)	\(46\)		\(37\)	\(3\)	\(34\)		\(12\)	\(0\)	\(12\)

Trace form

9 * q + 2 * q^2 - 3 * q^3 + 36 * q^4 - 6 * q^6 - 12 * q^7 + 8 * q^8 + 81 * q^9 + 68 * q^11 - 12 * q^12 - 6 * q^13 + 48 * q^14 + 144 * q^16 + 198 * q^17 + 18 * q^18 + 200 * q^19 + 12 * q^21 - 240 * q^23 - 24 * q^24 + 596 * q^26 - 27 * q^27 - 48 * q^28 - 210 * q^29 - 452 * q^31 + 32 * q^32 + 360 * q^33 - 52 * q^34 + 324 * q^36 - 54 * q^37 - 392 * q^38 - 306 * q^39 - 622 * q^41 + 312 * q^42 + 732 * q^43 + 272 * q^44 + 656 * q^46 + 240 * q^47 - 48 * q^48 - 723 * q^49 - 498 * q^51 - 24 * q^52 - 342 * q^53 - 54 * q^54 + 192 * q^56 - 132 * q^57 - 348 * q^58 + 340 * q^59 - 2742 * q^61 - 1184 * q^62 - 108 * q^63 + 576 * q^64 + 168 * q^66 - 2292 * q^67 + 792 * q^68 + 384 * q^69 - 72 * q^71 + 72 * q^72 - 486 * q^73 - 732 * q^74 + 800 * q^76 + 1920 * q^77 - 444 * q^78 + 2240 * q^79 + 729 * q^81 + 948 * q^82 + 1644 * q^83 + 48 * q^84 - 1224 * q^86 - 558 * q^87 - 4990 * q^89 - 452 * q^91 - 960 * q^92 - 384 * q^93 - 1472 * q^94 - 96 * q^96 + 186 * q^97 + 1842 * q^98 + 612 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(150))\) 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
150.4.a.a	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓	✓	✓	✓	150.4.a.a	$1$	$1$	\(-2\)	\(-3\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}-q^{7}+\cdots\)
150.4.a.b	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓		✓	✓	30.4.a.b	$1$	$0$	\(-2\)	\(-3\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}+4q^{7}+\cdots\)
150.4.a.c	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓	✓	✓	✓	150.4.a.c	$1$	$1$	\(-2\)	\(3\)	\(0\)	\(-23\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}-23q^{7}+\cdots\)
150.4.a.d	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓		✓	✓	30.4.c.a	$1$	$0$	\(-2\)	\(3\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}+2q^{7}+\cdots\)
150.4.a.e	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓		✓	✓	30.4.a.a	$1$	$1$	\(2\)	\(-3\)	\(0\)	\(-32\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}-2^{5}q^{7}+\cdots\)
150.4.a.f	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓				30.4.c.a	$1$	$0$	\(2\)	\(-3\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}-2q^{7}+\cdots\)
150.4.a.g	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓	✓			150.4.a.c	$1$	$0$	\(2\)	\(-3\)	\(0\)	\(23\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+23q^{7}+\cdots\)
150.4.a.h	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓	✓			150.4.a.a	$1$	$0$	\(2\)	\(3\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}+q^{7}+\cdots\)
150.4.a.i	$150$	$4$	150.a	1.a	$1$	$1$	$1$	$8.850$	\(\Q\)	None	✓				6.4.a.a	$1$	$0$	\(2\)	\(3\)	\(0\)	\(16\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}+2^{4}q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(150))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(150)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)