Space of modular forms of level 392, weight 2, and trivial character

• Label: 392.2.a
• Level: $392$
• Weight: $2$
• Character orbit: 392.a
• Rep. character: $\chi_{392}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $8$
• Sturm bound: $112$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(112\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(392))\).

	Total	New	Old
Modular forms	72	10	62
Cusp forms	41	10	31
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.

\(2\)	\(7\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(16\)	\(1\)	\(15\)		\(9\)	\(1\)	\(8\)		\(7\)	\(0\)	\(7\)
\(+\)	\(-\)	\(-\)		\(20\)	\(4\)	\(16\)		\(12\)	\(4\)	\(8\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(-\)		\(20\)	\(3\)	\(17\)		\(12\)	\(3\)	\(9\)		\(8\)	\(0\)	\(8\)
\(-\)	\(-\)	\(+\)		\(16\)	\(2\)	\(14\)		\(8\)	\(2\)	\(6\)		\(8\)	\(0\)	\(8\)
Plus space		\(+\)		\(32\)	\(3\)	\(29\)		\(17\)	\(3\)	\(14\)		\(15\)	\(0\)	\(15\)
Minus space		\(-\)		\(40\)	\(7\)	\(33\)		\(24\)	\(7\)	\(17\)		\(16\)	\(0\)	\(16\)

Trace form

10 * q - 2 * q^3 + 2 * q^5 + 14 * q^9 + 4 * q^11 - 2 * q^13 + 12 * q^15 + 8 * q^17 - 6 * q^19 - 4 * q^23 + 6 * q^25 + 4 * q^27 - 8 * q^29 - 12 * q^31 + 12 * q^37 - 8 * q^39 + 10 * q^45 + 12 * q^47 - 4 * q^51 - 16 * q^53 + 8 * q^55 - 4 * q^57 - 6 * q^59 + 2 * q^61 - 36 * q^65 + 20 * q^67 - 16 * q^69 + 4 * q^73 - 22 * q^75 - 4 * q^79 + 10 * q^81 - 14 * q^83 - 24 * q^85 - 4 * q^87 - 4 * q^89 - 12 * q^93 - 28 * q^95 + 8 * q^97 - 64 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(392))\) 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	7
392.2.a.a	$392$	$2$	392.a	1.a	$1$	$1$	$1$	$3.130$	\(\Q\)	None	✓				56.2.i.a	$1$	$1$	\(0\)	\(-3\)	\(1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+q^{5}+6q^{9}-q^{11}-2q^{13}+\cdots\)
392.2.a.b	$392$	$2$	392.a	1.a	$1$	$1$	$1$	$3.130$	\(\Q\)	None	✓				56.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+4q^{5}+q^{9}-8q^{15}+2q^{17}+\cdots\)
392.2.a.c	$392$	$2$	392.a	1.a	$1$	$1$	$1$	$3.130$	\(\Q\)	None	✓		✓	✓	56.2.i.b	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-2q^{9}+3q^{11}-6q^{13}+\cdots\)
392.2.a.d	$392$	$2$	392.a	1.a	$1$	$1$	$1$	$3.130$	\(\Q\)	None	✓				56.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-3q^{9}-4q^{11}-2q^{13}+6q^{17}+\cdots\)
392.2.a.e	$392$	$2$	392.a	1.a	$1$	$1$	$1$	$3.130$	\(\Q\)	None	✓				56.2.i.b	$1$	$0$	\(0\)	\(1\)	\(1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{9}+3q^{11}+6q^{13}+\cdots\)
392.2.a.f	$392$	$2$	392.a	1.a	$1$	$1$	$1$	$3.130$	\(\Q\)	None	✓				56.2.i.a	$1$	$0$	\(0\)	\(3\)	\(-1\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-q^{5}+6q^{9}-q^{11}+2q^{13}+\cdots\)
392.2.a.g	$392$	$2$	392.a	1.a	$1$	$2$	$2$	$3.130$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓		392.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2\beta q^{5}-q^{9}+6q^{11}-4\beta q^{13}+\cdots\)
392.2.a.h	$392$	$2$	392.a	1.a	$1$	$2$	$2$	$3.130$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓		392.2.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+\beta q^{5}+5q^{9}-4q^{11}-\beta q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(392))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(392)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)