Space of modular forms of level 2376, weight 2, and trivial character

• Label: 2376.2.a
• Level: $2376$
• Weight: $2$
• Character orbit: 2376.a
• Rep. character: $\chi_{2376}(1,\cdot)$
• Character field: $\Q$
• Dimension: $40$
• Newform subspaces: $18$
• Sturm bound: $864$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 2376 = 2^{3} \cdot 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2376.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(864\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\), \(17\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2376))\).

	Total	New	Old
Modular forms	456	40	416
Cusp forms	409	40	369
Eisenstein series	47	0	47

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

\(2\)	\(3\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(52\)	\(3\)	\(49\)		\(47\)	\(3\)	\(44\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(58\)	\(6\)	\(52\)		\(52\)	\(6\)	\(46\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(61\)	\(7\)	\(54\)		\(55\)	\(7\)	\(48\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(55\)	\(4\)	\(51\)		\(49\)	\(4\)	\(45\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(62\)	\(5\)	\(57\)		\(56\)	\(5\)	\(51\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(56\)	\(4\)	\(52\)		\(50\)	\(4\)	\(46\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(53\)	\(5\)	\(48\)		\(47\)	\(5\)	\(42\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(59\)	\(6\)	\(53\)		\(53\)	\(6\)	\(47\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(216\)	\(16\)	\(200\)		\(193\)	\(16\)	\(177\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(240\)	\(24\)	\(216\)		\(216\)	\(24\)	\(192\)		\(24\)	\(0\)	\(24\)

Trace form

\( 40 q - 4 q^{7} - 20 q^{13} - 4 q^{19} + 24 q^{25} + 32 q^{31} + 48 q^{37} + 40 q^{43} + 40 q^{49} + 44 q^{61} + 12 q^{67} - 4 q^{73} - 20 q^{79} + 48 q^{85} + 20 q^{91} - 8 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2376))\) 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	11
2376.2.a.a	$2376$	$2$	2376.a	1.a	$1$	$1$	$1$	$18.972$	\(\Q\)	None	✓	✓			2376.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+4q^{7}+q^{11}+3q^{13}+3q^{17}+\cdots\)
2376.2.a.b	$2376$	$2$	2376.a	1.a	$1$	$1$	$1$	$18.972$	\(\Q\)	None	✓	✓			2376.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-q^{11}-3q^{13}-5q^{19}+4q^{23}+\cdots\)
2376.2.a.c	$2376$	$2$	2376.a	1.a	$1$	$1$	$1$	$18.972$	\(\Q\)	None	✓	✓			2376.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+q^{11}-3q^{13}-5q^{19}-4q^{23}+\cdots\)
2376.2.a.d	$2376$	$2$	2376.a	1.a	$1$	$1$	$1$	$18.972$	\(\Q\)	None	✓	✓			2376.2.a.a	$1$	$0$	\(0\)	\(0\)	\(3\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+4q^{7}-q^{11}+3q^{13}-3q^{17}+\cdots\)
2376.2.a.e	$2376$	$2$	2376.a	1.a	$1$	$2$	$2$	$18.972$	\(\Q(\sqrt{2}) \)	None	✓	✓			2376.2.a.e	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{5}+(-1+\beta )q^{7}-q^{11}+(-4+\cdots)q^{13}+\cdots\)
2376.2.a.f	$2376$	$2$	2376.a	1.a	$1$	$2$	$2$	$18.972$	\(\Q(\sqrt{2}) \)	None	✓	✓			2376.2.a.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+(-1-\beta )q^{7}-q^{11}+(-1+\cdots)q^{13}+\cdots\)
2376.2.a.g	$2376$	$2$	2376.a	1.a	$1$	$2$	$2$	$18.972$	\(\Q(\sqrt{2}) \)	None	✓	✓			2376.2.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{5}+(-1-\beta )q^{7}+q^{11}+(-4+\cdots)q^{13}+\cdots\)
2376.2.a.h	$2376$	$2$	2376.a	1.a	$1$	$2$	$2$	$18.972$	\(\Q(\sqrt{2}) \)	None	✓	✓			2376.2.a.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+(-1+\beta )q^{7}+q^{11}+(-1+\cdots)q^{13}+\cdots\)
2376.2.a.i	$2376$	$2$	2376.a	1.a	$1$	$2$	$2$	$18.972$	\(\Q(\sqrt{7}) \)	None	✓	✓	✓		2376.2.a.i	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{7}-q^{11}+(-2+\beta )q^{17}+2\beta q^{19}+\cdots\)
2376.2.a.j	$2376$	$2$	2376.a	1.a	$1$	$2$	$2$	$18.972$	\(\Q(\sqrt{7}) \)	None	✓	✓			2376.2.a.i	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{7}+q^{11}+(2+\beta )q^{17}-2\beta q^{19}+\cdots\)
2376.2.a.k	$2376$	$2$	2376.a	1.a	$1$	$3$	$3$	$18.972$	3.3.564.1	None	✓	✓	✓		2376.2.a.k	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{5}+(-1+\beta _{1}+\beta _{2})q^{7}+\cdots\)
2376.2.a.l	$2376$	$2$	2376.a	1.a	$1$	$3$	$3$	$18.972$	3.3.788.1	None	✓	✓	✓		2376.2.a.l	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{5}+\beta _{1}q^{7}-q^{11}+(1+\cdots)q^{13}+\cdots\)
2376.2.a.m	$2376$	$2$	2376.a	1.a	$1$	$3$	$3$	$18.972$	3.3.1016.1	None	✓	✓	✓		2376.2.a.m	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(\beta _{1}+\beta _{2})q^{7}-q^{11}+\cdots\)
2376.2.a.n	$2376$	$2$	2376.a	1.a	$1$	$3$	$3$	$18.972$	3.3.7032.1	None	✓	✓			2376.2.a.n	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}+(-1+\beta _{1})q^{7}-q^{11}+(-1+\cdots)q^{13}+\cdots\)
2376.2.a.o	$2376$	$2$	2376.a	1.a	$1$	$3$	$3$	$18.972$	3.3.7032.1	None	✓	✓	✓		2376.2.a.n	$1$	$0$	\(0\)	\(0\)	\(1\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+(-1+\beta _{1})q^{7}+q^{11}+(-1+\cdots)q^{13}+\cdots\)
2376.2.a.p	$2376$	$2$	2376.a	1.a	$1$	$3$	$3$	$18.972$	3.3.564.1	None	✓	✓			2376.2.a.k	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{5}+(-1+\beta _{1}+\beta _{2})q^{7}+\cdots\)
2376.2.a.q	$2376$	$2$	2376.a	1.a	$1$	$3$	$3$	$18.972$	3.3.788.1	None	✓	✓			2376.2.a.l	$1$	$0$	\(0\)	\(0\)	\(2\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2})q^{5}+\beta _{1}q^{7}+q^{11}+(1-2\beta _{1}+\cdots)q^{13}+\cdots\)
2376.2.a.r	$2376$	$2$	2376.a	1.a	$1$	$3$	$3$	$18.972$	3.3.1016.1	None	✓	✓			2376.2.a.m	$1$	$0$	\(0\)	\(0\)	\(2\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{5}+(\beta _{1}+\beta _{2})q^{7}+q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2376))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2376)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(264))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(297))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(396))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(594))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(792))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1188))\)\(^{\oplus 2}\)