Space of modular forms of level 4608, weight 2, and trivial character

• Label: 4608.2.a
• Level: $4608$
• Weight: $2$
• Character orbit: 4608.a
• Rep. character: $\chi_{4608}(1,\cdot)$
• Character field: $\Q$
• Dimension: $80$
• Newform subspaces: $29$
• Sturm bound: $1536$
• Trace bound: $25$

Defining parameters

Level:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 29 \)
Sturm bound:			\(1536\)
Trace bound:			\(25\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\), \(17\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4608))\).

	Total	New	Old
Modular forms	832	80	752
Cusp forms	705	80	625
Eisenstein series	127	0	127

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
\(+\)	\(+\)	\(+\)		\(200\)	\(16\)	\(184\)		\(169\)	\(16\)	\(153\)		\(31\)	\(0\)	\(31\)
\(+\)	\(-\)	\(-\)		\(208\)	\(26\)	\(182\)		\(176\)	\(26\)	\(150\)		\(32\)	\(0\)	\(32\)
\(-\)	\(+\)	\(-\)		\(216\)	\(16\)	\(200\)		\(184\)	\(16\)	\(168\)		\(32\)	\(0\)	\(32\)
\(-\)	\(-\)	\(+\)		\(208\)	\(22\)	\(186\)		\(176\)	\(22\)	\(154\)		\(32\)	\(0\)	\(32\)
Plus space		\(+\)		\(408\)	\(38\)	\(370\)		\(345\)	\(38\)	\(307\)		\(63\)	\(0\)	\(63\)
Minus space		\(-\)		\(424\)	\(42\)	\(382\)		\(360\)	\(42\)	\(318\)		\(64\)	\(0\)	\(64\)

Trace form

\( 80 q + 80 q^{25} + 80 q^{49} - 32 q^{65} - 32 q^{73} - 32 q^{89}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4608))\) 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
4608.2.a.a	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	None	✓				1536.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{5}+(-2-\beta )q^{7}+2q^{11}+\cdots\)
4608.2.a.b	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	None	✓				1536.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{5}-\beta q^{7}+(-2+2\beta )q^{11}+\cdots\)
4608.2.a.c	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	None	✓				512.2.a.b	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+2\beta q^{7}+3\beta q^{11}+6q^{13}+\cdots\)
4608.2.a.d	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	None	✓				1536.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{5}+\beta q^{7}+(2-2\beta )q^{11}+\cdots\)
4608.2.a.e	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	None	✓				1536.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{5}+(2+\beta )q^{7}-2q^{11}+\cdots\)
4608.2.a.f	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	None	✓				512.2.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-8\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{5}-4q^{7}-\beta q^{11}+2\beta q^{13}+\cdots\)
4608.2.a.g	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	None	✓				1536.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-3\beta q^{7}-6q^{11}-4\beta q^{13}+\cdots\)
4608.2.a.h	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	None	✓				1536.2.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-\beta q^{7}-2q^{11}-2q^{17}+4q^{19}+\cdots\)
4608.2.a.i	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-2}) \)	✓				512.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+(-2+3\beta )q^{11}+4\beta q^{17}+(6-\beta )q^{19}+\cdots\)
4608.2.a.j	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	None	✓				1536.2.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+\beta q^{7}+2q^{11}-2q^{17}-4q^{19}+\cdots\)
4608.2.a.k	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-2}) \)	✓				512.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+(2-3\beta )q^{11}+4\beta q^{17}+(-6+\beta )q^{19}+\cdots\)
4608.2.a.l	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	None	✓				1536.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+3\beta q^{7}+6q^{11}-4\beta q^{13}+\cdots\)
4608.2.a.m	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	None	✓				512.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{5}+4q^{7}+\beta q^{11}+2\beta q^{13}+\cdots\)
4608.2.a.n	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	None	✓				1536.2.a.b	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(-2+\beta )q^{7}-2q^{11}+\cdots\)
4608.2.a.o	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	None	✓				1536.2.a.a	$1$	$1$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}-\beta q^{7}+(-2-2\beta )q^{11}+\cdots\)
4608.2.a.p	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	None	✓				512.2.a.b	$2$	$1$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+2\beta q^{7}-3\beta q^{11}-6q^{13}+\cdots\)
4608.2.a.q	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	None	✓				1536.2.a.a	$1$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+\beta q^{7}+(2+2\beta )q^{11}+(-2+\cdots)q^{13}+\cdots\)
4608.2.a.r	$4608$	$2$	4608.a	1.a	$1$	$2$	$2$	$36.795$	\(\Q(\sqrt{2}) \)	None	✓				1536.2.a.b	$1$	$0$	\(0\)	\(0\)	\(4\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(2-\beta )q^{7}+2q^{11}+2\beta q^{13}+\cdots\)
4608.2.a.s	$4608$	$2$	4608.a	1.a	$1$	$4$	$4$	$36.795$	\(\Q(\zeta_{24})^+\)	\(\Q(\sqrt{-6}) \)	✓	✓			4608.2.a.s	$4$	$1$	\(0\)	\(0\)	\(-8\)	\(0\)		$+$	$+$	$+$	$2^{2}$	$N(\mathrm{U}(1))$	\(q+(-2-\beta _{2})q^{5}+(\beta _{1}+\beta _{3})q^{7}-\beta _{3}q^{11}+\cdots\)
4608.2.a.t	$4608$	$2$	4608.a	1.a	$1$	$4$	$4$	$36.795$	\(\Q(\sqrt{10 +4 \sqrt{2}})\)	None	✓		✓		1536.2.a.m	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}-\beta _{3}q^{7}+(-2-\beta _{2})q^{11}+\cdots\)
4608.2.a.u	$4608$	$2$	4608.a	1.a	$1$	$4$	$4$	$36.795$	\(\Q(\zeta_{24})^+\)	None	✓	✓			4608.2.a.u	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}-\beta _{1}q^{7}-2q^{11}-\beta _{2}q^{13}+\cdots\)
4608.2.a.v	$4608$	$2$	4608.a	1.a	$1$	$4$	$4$	$36.795$	\(\Q(\zeta_{24})^+\)	None	✓	✓			4608.2.a.v	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{5}+\beta _{1}q^{7}-\beta _{3}q^{11}-2\beta _{2}q^{13}+\cdots\)
4608.2.a.w	$4608$	$2$	4608.a	1.a	$1$	$4$	$4$	$36.795$	\(\Q(\zeta_{24})^+\)	None	✓				512.2.a.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{5}+\beta _{3}q^{7}+\beta _{1}q^{11}+\beta _{2}q^{13}+\cdots\)
4608.2.a.x	$4608$	$2$	4608.a	1.a	$1$	$4$	$4$	$36.795$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	✓			4608.2.a.x	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}-\beta _{2}q^{7}+\beta _{3}q^{11}-\beta _{3}q^{13}+\cdots\)
4608.2.a.y	$4608$	$2$	4608.a	1.a	$1$	$4$	$4$	$36.795$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	✓			4608.2.a.x	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+\beta _{2}q^{7}-\beta _{3}q^{11}-\beta _{3}q^{13}+\cdots\)
4608.2.a.z	$4608$	$2$	4608.a	1.a	$1$	$4$	$4$	$36.795$	\(\Q(\zeta_{24})^+\)	None	✓	✓			4608.2.a.v	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{5}+\beta _{1}q^{7}-\beta _{3}q^{11}+2\beta _{2}q^{13}+\cdots\)
4608.2.a.ba	$4608$	$2$	4608.a	1.a	$1$	$4$	$4$	$36.795$	\(\Q(\sqrt{10 +4 \sqrt{2}})\)	None	✓				1536.2.a.m	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+\beta _{3}q^{7}+(2+\beta _{2})q^{11}+(-\beta _{1}+\cdots)q^{13}+\cdots\)
4608.2.a.bb	$4608$	$2$	4608.a	1.a	$1$	$4$	$4$	$36.795$	\(\Q(\zeta_{24})^+\)	None	✓	✓			4608.2.a.u	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+\beta _{1}q^{7}+2q^{11}-\beta _{2}q^{13}+\cdots\)
4608.2.a.bc	$4608$	$2$	4608.a	1.a	$1$	$4$	$4$	$36.795$	\(\Q(\zeta_{24})^+\)	\(\Q(\sqrt{-6}) \)	✓	✓			4608.2.a.s	$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)		$-$	$+$	$-$	$2^{2}$	$N(\mathrm{U}(1))$	\(q+(2+\beta _{2})q^{5}+(\beta _{1}+\beta _{3})q^{7}+\beta _{3}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4608))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4608)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 14}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(384))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(512))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(576))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(768))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1152))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1536))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2304))\)\(^{\oplus 2}\)