Space of modular forms of level 300, weight 4, and trivial character

• Label: 300.4.a
• Level: $300$
• Weight: $4$
• Character orbit: 300.a
• Rep. character: $\chi_{300}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $9$
• Sturm bound: $240$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	300.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(240\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(300))\).

	Total	New	Old
Modular forms	198	9	189
Cusp forms	162	9	153
Eisenstein series	36	0	36

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
\(+\)	\(+\)	\(+\)	\(+\)		\(27\)	\(0\)	\(27\)		\(21\)	\(0\)	\(21\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)		\(24\)	\(0\)	\(24\)		\(18\)	\(0\)	\(18\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(24\)	\(0\)	\(24\)		\(18\)	\(0\)	\(18\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(27\)	\(0\)	\(27\)		\(21\)	\(0\)	\(21\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(24\)	\(2\)	\(22\)		\(21\)	\(2\)	\(19\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(24\)	\(2\)	\(22\)		\(21\)	\(2\)	\(19\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(24\)	\(3\)	\(21\)		\(21\)	\(3\)	\(18\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(24\)	\(2\)	\(22\)		\(21\)	\(2\)	\(19\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(102\)	\(5\)	\(97\)		\(84\)	\(5\)	\(79\)		\(18\)	\(0\)	\(18\)
Minus space			\(-\)		\(96\)	\(4\)	\(92\)		\(78\)	\(4\)	\(74\)		\(18\)	\(0\)	\(18\)

Trace form

9 * q + 3 * q^3 - 12 * q^7 + 81 * q^9 - 76 * q^11 + 90 * q^13 - 42 * q^17 - 100 * q^19 - 84 * q^21 + 192 * q^23 + 27 * q^27 + 294 * q^29 + 256 * q^31 - 72 * q^33 + 474 * q^37 + 30 * q^39 - 190 * q^41 - 108 * q^43 - 768 * q^47 + 189 * q^49 + 186 * q^51 + 114 * q^53 + 780 * q^57 + 1444 * q^59 - 702 * q^61 - 108 * q^63 + 372 * q^67 - 1056 * q^69 - 840 * q^71 + 186 * q^73 - 2112 * q^77 + 8 * q^79 + 729 * q^81 + 2532 * q^83 + 1638 * q^87 - 670 * q^89 + 880 * q^91 - 408 * q^93 + 1146 * q^97 - 684 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(300))\) 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
300.4.a.a	$300$	$4$	300.a	1.a	$1$	$1$	$1$	$17.701$	\(\Q\)	None	✓	✓			300.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-13\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-13q^{7}+9q^{9}+6q^{11}+5q^{13}+\cdots\)
300.4.a.b	$300$	$4$	300.a	1.a	$1$	$1$	$1$	$17.701$	\(\Q\)	None	✓				12.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-8\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-8q^{7}+9q^{9}+6^{2}q^{11}+10q^{13}+\cdots\)
300.4.a.c	$300$	$4$	300.a	1.a	$1$	$1$	$1$	$17.701$	\(\Q\)	None	✓	✓			300.4.a.c	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+7q^{7}+9q^{9}-54q^{11}+55q^{13}+\cdots\)
300.4.a.d	$300$	$4$	300.a	1.a	$1$	$1$	$1$	$17.701$	\(\Q\)	None	✓				60.4.d.a	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(22\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+22q^{7}+9q^{9}-14q^{11}-30q^{13}+\cdots\)
300.4.a.e	$300$	$4$	300.a	1.a	$1$	$1$	$1$	$17.701$	\(\Q\)	None	✓				60.4.a.b	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-32\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-2^{5}q^{7}+9q^{9}+6^{2}q^{11}+10q^{13}+\cdots\)
300.4.a.f	$300$	$4$	300.a	1.a	$1$	$1$	$1$	$17.701$	\(\Q\)	None	✓				60.4.d.a	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-22\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-22q^{7}+9q^{9}-14q^{11}+30q^{13}+\cdots\)
300.4.a.g	$300$	$4$	300.a	1.a	$1$	$1$	$1$	$17.701$	\(\Q\)	None	✓	✓			300.4.a.c	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-7q^{7}+9q^{9}-54q^{11}-55q^{13}+\cdots\)
300.4.a.h	$300$	$4$	300.a	1.a	$1$	$1$	$1$	$17.701$	\(\Q\)	None	✓	✓			300.4.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(13\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+13q^{7}+9q^{9}+6q^{11}-5q^{13}+\cdots\)
300.4.a.i	$300$	$4$	300.a	1.a	$1$	$1$	$1$	$17.701$	\(\Q\)	None	✓				60.4.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(28\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+28q^{7}+9q^{9}-24q^{11}+70q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(300))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(300)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)