Space of modular forms of level 2394, weight 2, and trivial character

• Label: 2394.2.a
• Level: $2394$
• Weight: $2$
• Character orbit: 2394.a
• Rep. character: $\chi_{2394}(1,\cdot)$
• Character field: $\Q$
• Dimension: $46$
• Newform subspaces: $29$
• Sturm bound: $960$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 29 \)
Sturm bound:			\(960\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2394))\).

	Total	New	Old
Modular forms	496	46	450
Cusp forms	465	46	419
Eisenstein series	31	0	31

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
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	\(2\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)	\(3\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)	\(4\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(2\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(2\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(5\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(5\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(2\)
Plus space				\(+\)	\(18\)
Minus space				\(-\)	\(28\)

Trace form

46 * q + 2 * q^2 + 46 * q^4 - 4 * q^5 + 2 * q^8 + 4 * q^10 - 4 * q^11 + 12 * q^13 + 46 * q^16 + 12 * q^17 - 4 * q^20 + 78 * q^25 + 12 * q^26 + 4 * q^29 + 2 * q^32 - 4 * q^34 - 4 * q^35 + 12 * q^37 + 4 * q^40 + 28 * q^41 - 12 * q^43 - 4 * q^44 - 24 * q^46 + 8 * q^47 + 46 * q^49 + 14 * q^50 + 12 * q^52 + 12 * q^53 + 8 * q^55 - 8 * q^58 + 24 * q^59 + 12 * q^61 + 8 * q^62 + 46 * q^64 + 16 * q^65 + 16 * q^67 + 12 * q^68 - 24 * q^71 - 68 * q^73 - 8 * q^77 - 16 * q^79 - 4 * q^80 + 20 * q^82 - 32 * q^83 + 32 * q^85 - 36 * q^89 + 16 * q^91 - 8 * q^94 + 36 * q^97 + 2 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2394))\) 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	19
2394.2.a.a	$2394$	$2$	2394.a	1.a	$1$	$1$	$1$	$19.116$	\(\Q\)	None	✓	✓			2394.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(-4\)	\(1\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-4q^{5}+q^{7}-q^{8}+4q^{10}+\cdots\)
2394.2.a.b	$2394$	$2$	2394.a	1.a	$1$	$1$	$1$	$19.116$	\(\Q\)	None	✓				798.2.a.i	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(-1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}-q^{7}-q^{8}+2q^{10}+\cdots\)
2394.2.a.c	$2394$	$2$	2394.a	1.a	$1$	$1$	$1$	$19.116$	\(\Q\)	None	✓	✓			2394.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}-6q^{11}+2q^{13}+\cdots\)
2394.2.a.d	$2394$	$2$	2394.a	1.a	$1$	$1$	$1$	$19.116$	\(\Q\)	None	✓	✓			2394.2.a.d	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}+2q^{13}-q^{14}+\cdots\)
2394.2.a.e	$2394$	$2$	2394.a	1.a	$1$	$1$	$1$	$19.116$	\(\Q\)	None	✓				798.2.a.g	$1$	$1$	\(-1\)	\(0\)	\(2\)	\(-1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{5}-q^{7}-q^{8}-2q^{10}+\cdots\)
2394.2.a.f	$2394$	$2$	2394.a	1.a	$1$	$1$	$1$	$19.116$	\(\Q\)	None	✓				798.2.a.h	$1$	$0$	\(-1\)	\(0\)	\(4\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+4q^{5}-q^{7}-q^{8}-4q^{10}+\cdots\)
2394.2.a.g	$2394$	$2$	2394.a	1.a	$1$	$1$	$1$	$19.116$	\(\Q\)	None	✓				798.2.a.f	$1$	$1$	\(1\)	\(0\)	\(-4\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-4q^{5}+q^{7}+q^{8}-4q^{10}+\cdots\)
2394.2.a.h	$2394$	$2$	2394.a	1.a	$1$	$1$	$1$	$19.116$	\(\Q\)	None	✓				798.2.a.e	$1$	$1$	\(1\)	\(0\)	\(-2\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{5}-q^{7}+q^{8}-2q^{10}+\cdots\)
2394.2.a.i	$2394$	$2$	2394.a	1.a	$1$	$1$	$1$	$19.116$	\(\Q\)	None	✓				798.2.a.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{7}+q^{8}-2q^{11}-4q^{13}+\cdots\)
2394.2.a.j	$2394$	$2$	2394.a	1.a	$1$	$1$	$1$	$19.116$	\(\Q\)	None	✓				798.2.a.d	$1$	$1$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}-6q^{11}-4q^{13}+\cdots\)
2394.2.a.k	$2394$	$2$	2394.a	1.a	$1$	$1$	$1$	$19.116$	\(\Q\)	None	✓	✓			2394.2.a.d	$1$	$0$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}+2q^{13}+q^{14}+\cdots\)
2394.2.a.l	$2394$	$2$	2394.a	1.a	$1$	$1$	$1$	$19.116$	\(\Q\)	None	✓	✓			2394.2.a.c	$1$	$0$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}+6q^{11}+2q^{13}+\cdots\)
2394.2.a.m	$2394$	$2$	2394.a	1.a	$1$	$1$	$1$	$19.116$	\(\Q\)	None	✓				798.2.a.b	$1$	$0$	\(1\)	\(0\)	\(2\)	\(-1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{5}-q^{7}+q^{8}+2q^{10}+\cdots\)
2394.2.a.n	$2394$	$2$	2394.a	1.a	$1$	$1$	$1$	$19.116$	\(\Q\)	None	✓				798.2.a.c	$1$	$0$	\(1\)	\(0\)	\(2\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{5}+q^{7}+q^{8}+2q^{10}+\cdots\)
2394.2.a.o	$2394$	$2$	2394.a	1.a	$1$	$1$	$1$	$19.116$	\(\Q\)	None	✓	✓			2394.2.a.a	$1$	$0$	\(1\)	\(0\)	\(4\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{5}+q^{7}+q^{8}+4q^{10}+\cdots\)
2394.2.a.p	$2394$	$2$	2394.a	1.a	$1$	$2$	$2$	$19.116$	\(\Q(\sqrt{5}) \)	None	✓		✓	✓	798.2.a.m	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(2\)		$+$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1-\beta )q^{5}+q^{7}-q^{8}+\cdots\)
2394.2.a.q	$2394$	$2$	2394.a	1.a	$1$	$2$	$2$	$19.116$	\(\Q(\sqrt{13}) \)	None	✓				266.2.a.c	$1$	$0$	\(-2\)	\(0\)	\(-1\)	\(2\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-\beta q^{5}+q^{7}-q^{8}+\beta q^{10}+\cdots\)
2394.2.a.r	$2394$	$2$	2394.a	1.a	$1$	$2$	$2$	$19.116$	\(\Q(\sqrt{2}) \)	None	✓				798.2.a.l	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta q^{5}+q^{7}-q^{8}-\beta q^{10}+\cdots\)
2394.2.a.s	$2394$	$2$	2394.a	1.a	$1$	$2$	$2$	$19.116$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	2394.2.a.s	$1$	$0$	\(-2\)	\(0\)	\(2\)	\(-2\)		$+$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1+\beta )q^{5}-q^{7}-q^{8}+\cdots\)
2394.2.a.t	$2394$	$2$	2394.a	1.a	$1$	$2$	$2$	$19.116$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	2394.2.a.t	$1$	$0$	\(-2\)	\(0\)	\(4\)	\(2\)		$+$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{5}+q^{7}-q^{8}-2q^{10}+\cdots\)
2394.2.a.u	$2394$	$2$	2394.a	1.a	$1$	$2$	$2$	$19.116$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	2394.2.a.t	$1$	$1$	\(2\)	\(0\)	\(-4\)	\(2\)		$-$	$+$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{5}+q^{7}+q^{8}-2q^{10}+\cdots\)
2394.2.a.v	$2394$	$2$	2394.a	1.a	$1$	$2$	$2$	$19.116$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	2394.2.a.s	$1$	$1$	\(2\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1-\beta )q^{5}-q^{7}+q^{8}+\cdots\)
2394.2.a.w	$2394$	$2$	2394.a	1.a	$1$	$2$	$2$	$19.116$	\(\Q(\sqrt{5}) \)	None	✓				266.2.a.b	$1$	$0$	\(2\)	\(0\)	\(-1\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1-3\beta )q^{5}+q^{7}+q^{8}+\cdots\)
2394.2.a.x	$2394$	$2$	2394.a	1.a	$1$	$2$	$2$	$19.116$	\(\Q(\sqrt{3}) \)	None	✓				798.2.a.k	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta q^{5}-q^{7}+q^{8}+\beta q^{10}+\cdots\)
2394.2.a.y	$2394$	$2$	2394.a	1.a	$1$	$2$	$2$	$19.116$	\(\Q(\sqrt{29}) \)	None	✓				266.2.a.a	$1$	$0$	\(2\)	\(0\)	\(1\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta q^{5}-q^{7}+q^{8}+\beta q^{10}+\cdots\)
2394.2.a.z	$2394$	$2$	2394.a	1.a	$1$	$2$	$2$	$19.116$	\(\Q(\sqrt{5}) \)	None	✓				798.2.a.j	$1$	$0$	\(2\)	\(0\)	\(2\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1+\beta )q^{5}+q^{7}+q^{8}+\cdots\)
2394.2.a.ba	$2394$	$2$	2394.a	1.a	$1$	$3$	$3$	$19.116$	3.3.469.1	None	✓		✓		266.2.a.d	$1$	$0$	\(-3\)	\(0\)	\(-5\)	\(-3\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-2+\beta _{1})q^{5}-q^{7}-q^{8}+\cdots\)
2394.2.a.bb	$2394$	$2$	2394.a	1.a	$1$	$3$	$3$	$19.116$	3.3.148.1	None	✓	✓	✓	✓	2394.2.a.bb	$1$	$1$	\(-3\)	\(0\)	\(-2\)	\(-3\)		$+$	$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1-\beta _{1})q^{5}-q^{7}-q^{8}+\cdots\)
2394.2.a.bc	$2394$	$2$	2394.a	1.a	$1$	$3$	$3$	$19.116$	3.3.148.1	None	✓	✓	✓	✓	2394.2.a.bb	$1$	$0$	\(3\)	\(0\)	\(2\)	\(-3\)		$-$	$+$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1+\beta _{1})q^{5}-q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2394))\) into lower level spaces

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