Space of modular forms of level 2394, weight 4, and trivial character

• Label: 2394.4.a
• Level: $2394$
• Weight: $4$
• Character orbit: 2394.a
• Rep. character: $\chi_{2394}(1,\cdot)$
• Character field: $\Q$
• Dimension: $134$
• Newform subspaces: $36$
• Sturm bound: $1920$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2394.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 36 \)
Sturm bound:			\(1920\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2394))\).

	Total	New	Old
Modular forms	1456	134	1322
Cusp forms	1424	134	1290
Eisenstein series	32	0	32

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

\(2\)	\(3\)	\(7\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(6\)
\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(7\)
\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(6\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(9\)
\(+\)	\(-\)	\(+\)	\(-\)	$+$	\(11\)
\(+\)	\(-\)	\(-\)	\(+\)	$+$	\(11\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(9\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(7\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(7\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(12\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(9\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(9\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(12\)
Plus space				\(+\)	\(72\)
Minus space				\(-\)	\(62\)

Trace form

\( 134 q + 4 q^{2} + 536 q^{4} - 12 q^{5} + 16 q^{8} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2394))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	19
2394.4.a.a	$2394$	$4$	2394.a	1.a	$1$	$1$	$1$	$141.251$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(7\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+7q^{7}-8q^{8}+42q^{11}+\cdots\)
2394.4.a.b	$2394$	$4$	2394.a	1.a	$1$	$1$	$1$	$141.251$	\(\Q\)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(10\)	\(-7\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+10q^{5}-7q^{7}-8q^{8}+\cdots\)
2394.4.a.c	$2394$	$4$	2394.a	1.a	$1$	$1$	$1$	$141.251$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(10\)	\(7\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+10q^{5}+7q^{7}-8q^{8}+\cdots\)
2394.4.a.d	$2394$	$4$	2394.a	1.a	$1$	$1$	$1$	$141.251$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(0\)	\(-10\)	\(-7\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-10q^{5}-7q^{7}+8q^{8}+\cdots\)
2394.4.a.e	$2394$	$4$	2394.a	1.a	$1$	$1$	$1$	$141.251$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(12\)	\(7\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+12q^{5}+7q^{7}+8q^{8}+\cdots\)
2394.4.a.f	$2394$	$4$	2394.a	1.a	$1$	$2$	$2$	$141.251$	\(\Q(\sqrt{37}) \)	None	✓	$1$	$1$	\(-4\)	\(0\)	\(4\)	\(-14\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(1+2\beta )q^{5}-7q^{7}+\cdots\)
2394.4.a.g	$2394$	$4$	2394.a	1.a	$1$	$2$	$2$	$141.251$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(-4\)	\(0\)	\(14\)	\(14\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(5+4\beta )q^{5}+7q^{7}+\cdots\)
2394.4.a.h	$2394$	$4$	2394.a	1.a	$1$	$2$	$2$	$141.251$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(-4\)	\(0\)	\(20\)	\(14\)		$+$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(10+3\beta )q^{5}+7q^{7}+\cdots\)
2394.4.a.i	$2394$	$4$	2394.a	1.a	$1$	$2$	$2$	$141.251$	\(\Q(\sqrt{85}) \)	None	✓	$1$	$1$	\(4\)	\(0\)	\(-12\)	\(-14\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-5-2\beta )q^{5}-7q^{7}+\cdots\)
2394.4.a.j	$2394$	$4$	2394.a	1.a	$1$	$2$	$2$	$141.251$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(4\)	\(0\)	\(-2\)	\(14\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(1-4\beta )q^{5}+7q^{7}+\cdots\)
2394.4.a.k	$2394$	$4$	2394.a	1.a	$1$	$3$	$3$	$141.251$	3.3.57553.1	None	✓	$1$	$1$	\(-6\)	\(0\)	\(-10\)	\(-21\)		$+$	$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-3+\beta _{1})q^{5}-7q^{7}+\cdots\)
2394.4.a.l	$2394$	$4$	2394.a	1.a	$1$	$3$	$3$	$141.251$	3.3.3221.1	None	✓	$1$	$0$	\(-6\)	\(0\)	\(0\)	\(-21\)		$+$	$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-\beta _{2}q^{5}-7q^{7}-8q^{8}+\cdots\)
2394.4.a.m	$2394$	$4$	2394.a	1.a	$1$	$3$	$3$	$141.251$	3.3.12092.1	None	✓	$1$	$0$	\(-6\)	\(0\)	\(20\)	\(-21\)		$+$	$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(7-\beta _{1})q^{5}-7q^{7}+\cdots\)
2394.4.a.n	$2394$	$4$	2394.a	1.a	$1$	$3$	$3$	$141.251$	3.3.93944.1	None	✓	$1$	$1$	\(6\)	\(0\)	\(-10\)	\(21\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-3+\beta _{2})q^{5}+7q^{7}+\cdots\)
2394.4.a.o	$2394$	$4$	2394.a	1.a	$1$	$3$	$3$	$141.251$	3.3.22397.1	None	✓	$1$	$1$	\(6\)	\(0\)	\(0\)	\(-21\)		$-$	$-$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-\beta _{2}q^{5}-7q^{7}+8q^{8}+\cdots\)
2394.4.a.p	$2394$	$4$	2394.a	1.a	$1$	$3$	$3$	$141.251$	3.3.42440.1	None	✓	$1$	$0$	\(6\)	\(0\)	\(20\)	\(21\)		$-$	$-$	$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(7+\beta _{1})q^{5}+7q^{7}+\cdots\)
2394.4.a.q	$2394$	$4$	2394.a	1.a	$1$	$4$	$4$	$141.251$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(-8\)	\(0\)	\(10\)	\(-28\)		$+$	$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(2-\beta _{1})q^{5}-7q^{7}+\cdots\)
2394.4.a.r	$2394$	$4$	2394.a	1.a	$1$	$4$	$4$	$141.251$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-8\)	\(0\)	\(10\)	\(28\)		$+$	$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(2+\beta _{1})q^{5}+7q^{7}+\cdots\)
2394.4.a.s	$2394$	$4$	2394.a	1.a	$1$	$4$	$4$	$141.251$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(8\)	\(0\)	\(-12\)	\(28\)		$-$	$-$	$-$	$-$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-3+\beta _{1})q^{5}+7q^{7}+\cdots\)
2394.4.a.t	$2394$	$4$	2394.a	1.a	$1$	$4$	$4$	$141.251$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(8\)	\(0\)	\(-10\)	\(-28\)		$-$	$-$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-3-\beta _{2})q^{5}-7q^{7}+\cdots\)
2394.4.a.u	$2394$	$4$	2394.a	1.a	$1$	$4$	$4$	$141.251$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(8\)	\(0\)	\(-10\)	\(28\)		$-$	$-$	$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-2-\beta _{1})q^{5}+7q^{7}+\cdots\)
2394.4.a.v	$2394$	$4$	2394.a	1.a	$1$	$4$	$4$	$141.251$	4.4.6939601.2	None	✓	$1$	$0$	\(8\)	\(0\)	\(-7\)	\(-28\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-2-2\beta _{1}+\beta _{2}-\beta _{3})q^{5}+\cdots\)
2394.4.a.w	$2394$	$4$	2394.a	1.a	$1$	$4$	$4$	$141.251$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(8\)	\(0\)	\(-7\)	\(28\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-2+\beta _{1}-\beta _{3})q^{5}+\cdots\)
2394.4.a.x	$2394$	$4$	2394.a	1.a	$1$	$4$	$4$	$141.251$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(8\)	\(0\)	\(0\)	\(-28\)		$-$	$-$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-\beta _{1}q^{5}-7q^{7}+8q^{8}+\cdots\)
2394.4.a.y	$2394$	$4$	2394.a	1.a	$1$	$4$	$4$	$141.251$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(8\)	\(0\)	\(10\)	\(-28\)		$-$	$-$	$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(2+\beta _{1})q^{5}-7q^{7}+\cdots\)
2394.4.a.z	$2394$	$4$	2394.a	1.a	$1$	$5$	$5$	$141.251$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(-10\)	\(0\)	\(-21\)	\(-35\)		$+$	$-$	$+$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-4+\beta _{1}+\beta _{4})q^{5}+\cdots\)
2394.4.a.ba	$2394$	$4$	2394.a	1.a	$1$	$5$	$5$	$141.251$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(-10\)	\(0\)	\(-21\)	\(35\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-4+\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
2394.4.a.bb	$2394$	$4$	2394.a	1.a	$1$	$5$	$5$	$141.251$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-10\)	\(0\)	\(-20\)	\(35\)		$+$	$-$	$-$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-4-\beta _{2})q^{5}+7q^{7}+\cdots\)
2394.4.a.bc	$2394$	$4$	2394.a	1.a	$1$	$5$	$5$	$141.251$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(-10\)	\(0\)	\(-8\)	\(-35\)		$+$	$+$	$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-2+\beta _{2})q^{5}-7q^{7}+\cdots\)
2394.4.a.bd	$2394$	$4$	2394.a	1.a	$1$	$5$	$5$	$141.251$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(10\)	\(0\)	\(8\)	\(-35\)		$-$	$+$	$+$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(2-\beta _{2})q^{5}-7q^{7}+\cdots\)
2394.4.a.be	$2394$	$4$	2394.a	1.a	$1$	$6$	$6$	$141.251$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(-12\)	\(0\)	\(12\)	\(42\)		$+$	$+$	$-$	$-$	$+$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(2-\beta _{1})q^{5}+7q^{7}+\cdots\)
2394.4.a.bf	$2394$	$4$	2394.a	1.a	$1$	$6$	$6$	$141.251$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(12\)	\(0\)	\(-12\)	\(42\)		$-$	$+$	$-$	$-$	$-$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-2+\beta _{1})q^{5}+7q^{7}+\cdots\)
2394.4.a.bg	$2394$	$4$	2394.a	1.a	$1$	$7$	$7$	$141.251$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-14\)	\(0\)	\(-28\)	\(49\)		$+$	$+$	$-$	$+$	$-$	$2^{9}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-4+\beta _{2})q^{5}+7q^{7}+\cdots\)
2394.4.a.bh	$2394$	$4$	2394.a	1.a	$1$	$7$	$7$	$141.251$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-14\)	\(0\)	\(-18\)	\(-49\)		$+$	$+$	$+$	$-$	$-$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-3+\beta _{1})q^{5}-7q^{7}+\cdots\)
2394.4.a.bi	$2394$	$4$	2394.a	1.a	$1$	$7$	$7$	$141.251$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(14\)	\(0\)	\(18\)	\(-49\)		$-$	$+$	$+$	$-$	$+$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(3-\beta _{1})q^{5}-7q^{7}+\cdots\)
2394.4.a.bj	$2394$	$4$	2394.a	1.a	$1$	$7$	$7$	$141.251$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(14\)	\(0\)	\(28\)	\(49\)		$-$	$+$	$-$	$+$	$+$	$2^{9}$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(4-\beta _{2})q^{5}+7q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2394))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2394)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(133))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(266))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(342))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(399))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(798))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1197))\)\(^{\oplus 2}\)