Space of modular forms of level 2352, weight 2, and trivial character

• Label: 2352.2.a
• Level: $2352$
• Weight: $2$
• Character orbit: 2352.a
• Rep. character: $\chi_{2352}(1,\cdot)$
• Character field: $\Q$
• Dimension: $41$
• Newform subspaces: $33$
• Sturm bound: $896$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 33 \)
Sturm bound:			\(896\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2352))\).

	Total	New	Old
Modular forms	496	41	455
Cusp forms	401	41	360
Eisenstein series	95	0	95

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

\(2\)	\(3\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(60\)	\(5\)	\(55\)		\(49\)	\(5\)	\(44\)		\(11\)	\(0\)	\(11\)
\(+\)	\(+\)	\(-\)	\(-\)		\(63\)	\(6\)	\(57\)		\(51\)	\(6\)	\(45\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(+\)	\(-\)		\(64\)	\(7\)	\(57\)		\(52\)	\(7\)	\(45\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)	\(+\)		\(61\)	\(3\)	\(58\)		\(49\)	\(3\)	\(46\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(+\)	\(-\)		\(64\)	\(4\)	\(60\)		\(52\)	\(4\)	\(48\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)	\(+\)		\(61\)	\(6\)	\(55\)		\(49\)	\(6\)	\(43\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)	\(+\)		\(60\)	\(4\)	\(56\)		\(48\)	\(4\)	\(44\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(-\)	\(-\)		\(63\)	\(6\)	\(57\)		\(51\)	\(6\)	\(45\)		\(12\)	\(0\)	\(12\)
Plus space			\(+\)		\(242\)	\(18\)	\(224\)		\(195\)	\(18\)	\(177\)		\(47\)	\(0\)	\(47\)
Minus space			\(-\)		\(254\)	\(23\)	\(231\)		\(206\)	\(23\)	\(183\)		\(48\)	\(0\)	\(48\)

Trace form

41 * q - q^3 - 2 * q^5 + 41 * q^9 + 4 * q^11 - 2 * q^13 - 2 * q^15 + 2 * q^17 - 12 * q^19 - 8 * q^23 + 47 * q^25 - q^27 - 10 * q^29 + 4 * q^33 - 10 * q^37 - 6 * q^39 + 10 * q^41 - 2 * q^45 + 24 * q^47 + 2 * q^51 - 2 * q^53 - 4 * q^57 + 28 * q^59 + 14 * q^61 + 20 * q^65 + 32 * q^67 + 8 * q^69 + 32 * q^71 + 10 * q^73 - 15 * q^75 - 12 * q^79 + 41 * q^81 - 20 * q^83 + 12 * q^85 + 18 * q^87 + 10 * q^89 + 8 * q^93 - 32 * q^95 - 14 * q^97 + 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2352))\) 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	7
2352.2.a.a	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				84.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-4\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{5}+q^{9}-2q^{11}+6q^{13}+\cdots\)
2352.2.a.b	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				294.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-4\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{5}+q^{9}+4q^{11}-4q^{13}+\cdots\)
2352.2.a.c	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				168.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+q^{9}-6q^{13}+2q^{15}+\cdots\)
2352.2.a.d	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				21.2.e.a	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+q^{9}+2q^{11}+q^{13}+\cdots\)
2352.2.a.e	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				168.2.q.b	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+q^{9}+6q^{11}+3q^{13}+\cdots\)
2352.2.a.f	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				42.2.e.a	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{9}-5q^{11}+q^{15}+4q^{17}+\cdots\)
2352.2.a.g	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				168.2.q.a	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{9}-3q^{11}+4q^{13}+\cdots\)
2352.2.a.h	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				1176.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}+4q^{13}+4q^{17}+4q^{19}+\cdots\)
2352.2.a.i	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				24.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+q^{9}-4q^{11}+2q^{13}+\cdots\)
2352.2.a.j	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				588.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+q^{9}-2q^{11}-4q^{13}+\cdots\)
2352.2.a.k	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				84.2.i.a	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+q^{9}-2q^{11}+3q^{13}+\cdots\)
2352.2.a.l	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				42.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+q^{9}+4q^{11}-6q^{13}+\cdots\)
2352.2.a.m	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				42.2.e.b	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}+q^{9}-3q^{11}-4q^{13}+\cdots\)
2352.2.a.n	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				42.2.e.b	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}+q^{9}-3q^{11}+4q^{13}+\cdots\)
2352.2.a.o	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				84.2.i.a	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+q^{9}-2q^{11}-3q^{13}+\cdots\)
2352.2.a.p	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				588.2.a.b	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+q^{9}-2q^{11}+4q^{13}+\cdots\)
2352.2.a.q	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				168.2.a.b	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+q^{9}+2q^{13}-2q^{15}+\cdots\)
2352.2.a.r	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				1176.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}-4q^{13}-4q^{17}-4q^{19}+\cdots\)
2352.2.a.s	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				84.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}+6q^{11}-2q^{13}-4q^{19}+\cdots\)
2352.2.a.t	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				42.2.e.a	$1$	$1$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+q^{9}-5q^{11}+q^{15}-4q^{17}+\cdots\)
2352.2.a.u	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				168.2.q.a	$1$	$1$	\(0\)	\(1\)	\(1\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+q^{9}-3q^{11}-4q^{13}+\cdots\)
2352.2.a.v	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				21.2.a.a	$1$	$0$	\(0\)	\(1\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{9}-4q^{11}+2q^{13}+\cdots\)
2352.2.a.w	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				21.2.e.a	$1$	$0$	\(0\)	\(1\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{9}+2q^{11}-q^{13}+\cdots\)
2352.2.a.x	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				168.2.q.b	$1$	$0$	\(0\)	\(1\)	\(2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{9}+6q^{11}-3q^{13}+\cdots\)
2352.2.a.y	$2352$	$2$	2352.a	1.a	$1$	$1$	$1$	$18.781$	\(\Q\)	None	✓				294.2.a.b	$1$	$0$	\(0\)	\(1\)	\(4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}+q^{9}+4q^{11}+4q^{13}+\cdots\)
2352.2.a.z	$2352$	$2$	2352.a	1.a	$1$	$2$	$2$	$18.781$	\(\Q(\sqrt{2}) \)	None	✓				1176.2.a.l	$1$	$1$	\(0\)	\(-2\)	\(-4\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-2+\beta )q^{5}+q^{9}+(2+2\beta )q^{11}+\cdots\)
2352.2.a.ba	$2352$	$2$	2352.a	1.a	$1$	$2$	$2$	$18.781$	\(\Q(\sqrt{57}) \)	None	✓		✓		168.2.q.c	$1$	$0$	\(0\)	\(-2\)	\(1\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{5}+q^{9}+\beta q^{11}+(-3+\beta )q^{13}+\cdots\)
2352.2.a.bb	$2352$	$2$	2352.a	1.a	$1$	$2$	$2$	$18.781$	\(\Q(\sqrt{2}) \)	None	✓				1176.2.a.j	$1$	$1$	\(0\)	\(-2\)	\(4\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(2+\beta )q^{5}+q^{9}+(-2-2\beta )q^{11}+\cdots\)
2352.2.a.bc	$2352$	$2$	2352.a	1.a	$1$	$2$	$2$	$18.781$	\(\Q(\sqrt{2}) \)	None	✓		✓		147.2.a.d	$1$	$0$	\(0\)	\(-2\)	\(4\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(2+\beta )q^{5}+q^{9}+2q^{11}+(4+\cdots)q^{13}+\cdots\)
2352.2.a.bd	$2352$	$2$	2352.a	1.a	$1$	$2$	$2$	$18.781$	\(\Q(\sqrt{2}) \)	None	✓				1176.2.a.j	$1$	$0$	\(0\)	\(2\)	\(-4\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-2+\beta )q^{5}+q^{9}+(-2+2\beta )q^{11}+\cdots\)
2352.2.a.be	$2352$	$2$	2352.a	1.a	$1$	$2$	$2$	$18.781$	\(\Q(\sqrt{2}) \)	None	✓		✓		147.2.a.d	$1$	$1$	\(0\)	\(2\)	\(-4\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-2+\beta )q^{5}+q^{9}+2q^{11}+\cdots\)
2352.2.a.bf	$2352$	$2$	2352.a	1.a	$1$	$2$	$2$	$18.781$	\(\Q(\sqrt{57}) \)	None	✓				168.2.q.c	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta q^{5}+q^{9}+\beta q^{11}+(3-\beta )q^{13}+\cdots\)
2352.2.a.bg	$2352$	$2$	2352.a	1.a	$1$	$2$	$2$	$18.781$	\(\Q(\sqrt{2}) \)	None	✓				1176.2.a.l	$1$	$0$	\(0\)	\(2\)	\(4\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(2+\beta )q^{5}+q^{9}+(2-2\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2352))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2352)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(588))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(784))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1176))\)\(^{\oplus 2}\)