Space of modular forms of level 2304, weight 2, and trivial character

• Label: 2304.2.a
• Level: $2304$
• Weight: $2$
• Character orbit: 2304.a
• Rep. character: $\chi_{2304}(1,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $26$
• Sturm bound: $768$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2304.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 26 \)
Sturm bound:			\(768\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\), \(13\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2304))\).

	Total	New	Old
Modular forms	432	42	390
Cusp forms	337	38	299
Eisenstein series	95	4	91

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

\(2\)	\(3\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(104\)	\(6\)	\(98\)		\(81\)	\(6\)	\(75\)		\(23\)	\(0\)	\(23\)
\(+\)	\(-\)	\(-\)		\(112\)	\(14\)	\(98\)		\(88\)	\(12\)	\(76\)		\(24\)	\(2\)	\(22\)
\(-\)	\(+\)	\(-\)		\(112\)	\(10\)	\(102\)		\(88\)	\(10\)	\(78\)		\(24\)	\(0\)	\(24\)
\(-\)	\(-\)	\(+\)		\(104\)	\(12\)	\(92\)		\(80\)	\(10\)	\(70\)		\(24\)	\(2\)	\(22\)
Plus space		\(+\)		\(208\)	\(18\)	\(190\)		\(161\)	\(16\)	\(145\)		\(47\)	\(2\)	\(45\)
Minus space		\(-\)		\(224\)	\(24\)	\(200\)		\(176\)	\(22\)	\(154\)		\(48\)	\(2\)	\(46\)

Trace form

\( 38 q + 4 q^{17} + 34 q^{25} - 4 q^{41} + 54 q^{49} + 52 q^{73} + 12 q^{89} + 28 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2304))\) 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
2304.2.a.a	$2304$	$2$	2304.a	1.a	$1$	$1$	$1$	$18.398$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				128.2.b.b	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-4q^{5}+4q^{13}+2q^{17}+11q^{25}+\cdots\)
2304.2.a.b	$2304$	$2$	2304.a	1.a	$1$	$1$	$1$	$18.398$	\(\Q\)	None	✓				24.2.d.a	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-2q^{7}+4q^{13}+2q^{17}+4q^{19}+\cdots\)
2304.2.a.c	$2304$	$2$	2304.a	1.a	$1$	$1$	$1$	$18.398$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				1152.2.d.b	$2$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{5}-4q^{13}+8q^{17}-q^{25}+10q^{29}+\cdots\)
2304.2.a.d	$2304$	$2$	2304.a	1.a	$1$	$1$	$1$	$18.398$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				1152.2.d.b	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{5}+4q^{13}-8q^{17}-q^{25}+10q^{29}+\cdots\)
2304.2.a.e	$2304$	$2$	2304.a	1.a	$1$	$1$	$1$	$18.398$	\(\Q\)	None	✓				24.2.d.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+2q^{7}+4q^{13}+2q^{17}-4q^{19}+\cdots\)
2304.2.a.f	$2304$	$2$	2304.a	1.a	$1$	$1$	$1$	$18.398$	\(\Q\)	None	✓				384.2.d.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}-4q^{11}+4q^{13}+2q^{17}-4q^{19}+\cdots\)
2304.2.a.g	$2304$	$2$	2304.a	1.a	$1$	$1$	$1$	$18.398$	\(\Q\)	None	✓				384.2.d.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}+4q^{11}-4q^{13}+2q^{17}+4q^{19}+\cdots\)
2304.2.a.h	$2304$	$2$	2304.a	1.a	$1$	$1$	$1$	$18.398$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓				64.2.b.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-6q^{11}+6q^{17}+2q^{19}-5q^{25}+\cdots\)
2304.2.a.i	$2304$	$2$	2304.a	1.a	$1$	$1$	$1$	$18.398$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓				64.2.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+6q^{11}+6q^{17}-2q^{19}-5q^{25}+\cdots\)
2304.2.a.j	$2304$	$2$	2304.a	1.a	$1$	$1$	$1$	$18.398$	\(\Q\)	None	✓				384.2.d.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}-4q^{11}-4q^{13}+2q^{17}-4q^{19}+\cdots\)
2304.2.a.k	$2304$	$2$	2304.a	1.a	$1$	$1$	$1$	$18.398$	\(\Q\)	None	✓				384.2.d.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}+4q^{11}+4q^{13}+2q^{17}+4q^{19}+\cdots\)
2304.2.a.l	$2304$	$2$	2304.a	1.a	$1$	$1$	$1$	$18.398$	\(\Q\)	None	✓				24.2.d.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-2q^{7}-4q^{13}+2q^{17}-4q^{19}+\cdots\)
2304.2.a.m	$2304$	$2$	2304.a	1.a	$1$	$1$	$1$	$18.398$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				1152.2.d.b	$2$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+2q^{5}-4q^{13}-8q^{17}-q^{25}-10q^{29}+\cdots\)
2304.2.a.n	$2304$	$2$	2304.a	1.a	$1$	$1$	$1$	$18.398$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				1152.2.d.b	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+2q^{5}+4q^{13}+8q^{17}-q^{25}-10q^{29}+\cdots\)
2304.2.a.o	$2304$	$2$	2304.a	1.a	$1$	$1$	$1$	$18.398$	\(\Q\)	None	✓				24.2.d.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+2q^{7}-4q^{13}+2q^{17}+4q^{19}+\cdots\)
2304.2.a.p	$2304$	$2$	2304.a	1.a	$1$	$1$	$1$	$18.398$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				128.2.b.b	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+4q^{5}-4q^{13}+2q^{17}+11q^{25}+\cdots\)
2304.2.a.q	$2304$	$2$	2304.a	1.a	$1$	$2$	$2$	$18.398$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-6}) \)	✓				72.2.d.a	$4$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$+$	$2$	$N(\mathrm{U}(1))$	\(q+\beta q^{5}-2q^{7}-2\beta q^{11}+3q^{25}-\beta q^{29}+\cdots\)
2304.2.a.r	$2304$	$2$	2304.a	1.a	$1$	$2$	$2$	$18.398$	\(\Q(\sqrt{2}) \)	None	✓				384.2.d.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-\beta q^{7}-4q^{11}+2\beta q^{13}+2q^{17}+\cdots\)
2304.2.a.s	$2304$	$2$	2304.a	1.a	$1$	$2$	$2$	$18.398$	\(\Q(\sqrt{3}) \)	None	✓				192.2.d.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-\beta q^{7}-6q^{17}-4q^{19}-2\beta q^{23}+\cdots\)
2304.2.a.t	$2304$	$2$	2304.a	1.a	$1$	$2$	$2$	$18.398$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-2}) \)	✓				128.2.b.a	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$N(\mathrm{U}(1))$	\(q+\beta q^{11}-6q^{17}-3\beta q^{19}-5q^{25}+\cdots\)
2304.2.a.u	$2304$	$2$	2304.a	1.a	$1$	$2$	$2$	$18.398$	\(\Q(\sqrt{3}) \)	None	✓				192.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+\beta q^{7}-6q^{17}+4q^{19}+2\beta q^{23}+\cdots\)
2304.2.a.v	$2304$	$2$	2304.a	1.a	$1$	$2$	$2$	$18.398$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-3}) \)	✓				576.2.d.c	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$N(\mathrm{U}(1))$	\(q+\beta q^{7}-2\beta q^{13}-8q^{19}-5q^{25}+\cdots\)
2304.2.a.w	$2304$	$2$	2304.a	1.a	$1$	$2$	$2$	$18.398$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-3}) \)	✓				576.2.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$N(\mathrm{U}(1))$	\(q-\beta q^{7}-2\beta q^{13}+8q^{19}-5q^{25}+\cdots\)
2304.2.a.x	$2304$	$2$	2304.a	1.a	$1$	$2$	$2$	$18.398$	\(\Q(\sqrt{2}) \)	None	✓				384.2.d.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+\beta q^{7}+4q^{11}+2\beta q^{13}+2q^{17}+\cdots\)
2304.2.a.y	$2304$	$2$	2304.a	1.a	$1$	$2$	$2$	$18.398$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-6}) \)	✓				72.2.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+\beta q^{5}+2q^{7}+2\beta q^{11}+3q^{25}-\beta q^{29}+\cdots\)
2304.2.a.z	$2304$	$2$	2304.a	1.a	$1$	$4$	$4$	$18.398$	\(\Q(\zeta_{24})^+\)	\(\Q(\sqrt{-6}) \)	✓		✓		1152.2.d.g	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{5}$	$N(\mathrm{U}(1))$	\(q+\beta _{2}q^{5}+\beta _{1}q^{7}+\beta _{3}q^{11}+7q^{25}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2304))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2304)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(384))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(576))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(768))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1152))\)\(^{\oplus 2}\)