Space of modular forms of level 392, weight 2, and Character 392.q

• Label: 392.2.q
• Level: $392$
• Weight: $2$
• Character orbit: 392.q
• Rep. character: $\chi_{392}(57,\cdot)$
• Character field: $\Q(\zeta_{7})$
• Dimension: $84$
• Newform subspaces: $2$
• Sturm bound: $112$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.q (of order \(7\) and degree \(6\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 49 \)
Character field:			\(\Q(\zeta_{7})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(112\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(392, [\chi])\).

	Total	New	Old
Modular forms	360	84	276
Cusp forms	312	84	228
Eisenstein series	48	0	48

Trace form

84 * q - 2 * q^3 + 2 * q^5 - 24 * q^9 + 4 * q^11 - 2 * q^13 - 10 * q^15 + 8 * q^17 + 36 * q^19 - 2 * q^21 - 12 * q^23 - 26 * q^25 + 4 * q^27 + 34 * q^29 - 12 * q^31 - 8 * q^35 + 18 * q^37 + 22 * q^39 - 8 * q^43 - 46 * q^45 - 30 * q^47 - 2 * q^49 + 12 * q^51 - 4 * q^53 - 48 * q^55 + 12 * q^57 - 20 * q^59 + 16 * q^61 + 52 * q^63 + 12 * q^67 + 68 * q^69 + 16 * q^71 + 4 * q^73 + 62 * q^75 - 4 * q^77 - 12 * q^79 + 8 * q^81 - 42 * q^83 - 52 * q^85 - 74 * q^87 - 32 * q^89 - 96 * q^91 - 88 * q^93 - 76 * q^95 - 104 * q^97 - 16 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(392, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
392.2.q.a	$392$	$2$	392.q	49.e	$7$	$42$	$7$	$3.130$		None		✓			392.2.q.a	$2$	$0$	\(0\)	\(-7\)	\(-2\)	\(1\)		$\mathrm{SU}(2)[C_{7}]$
392.2.q.b	$392$	$2$	392.q	49.e	$7$	$42$	$7$	$3.130$		None		✓			392.2.q.b	$2$	$0$	\(0\)	\(5\)	\(4\)	\(-1\)		$\mathrm{SU}(2)[C_{7}]$

Decomposition of \(S_{2}^{\mathrm{old}}(392, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(392, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 2}\)