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

• Label: 392.2.i
• Level: $392$
• Weight: $2$
• Character orbit: 392.i
• Rep. character: $\chi_{392}(177,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $20$
• 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.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(112\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

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

	Total	New	Old
Modular forms	144	20	124
Cusp forms	80	20	60
Eisenstein series	64	0	64

Trace form

20 * q + 2 * q^3 - 2 * q^5 - 8 * q^9 + 2 * q^11 + 8 * q^13 + 12 * q^15 - 2 * q^17 + 6 * q^19 - 14 * q^23 - 24 * q^25 - 28 * q^27 + 8 * q^29 - 6 * q^31 - 6 * q^33 - 6 * q^37 + 20 * q^39 + 24 * q^41 + 24 * q^43 - 4 * q^45 + 6 * q^47 + 22 * q^51 + 10 * q^53 + 4 * q^55 - 44 * q^57 + 18 * q^59 - 2 * q^61 + 36 * q^65 - 2 * q^67 + 4 * q^69 - 48 * q^71 + 14 * q^73 - 8 * q^75 - 14 * q^79 + 2 * q^81 - 16 * q^83 - 60 * q^85 - 20 * q^87 - 2 * q^89 + 6 * q^93 - 2 * q^95 - 8 * q^97 - 80 * 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.i.a	$392$	$2$	392.i	7.c	$3$	$2$	$1$	$3.130$	\(\Q(\sqrt{-3}) \)	None					56.2.a.b	$2$	$0$	\(0\)	\(-2\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{3}+4\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots\)
392.2.i.b	$392$	$2$	392.i	7.c	$3$	$2$	$1$	$3.130$	\(\Q(\sqrt{-3}) \)	None					56.2.i.b	$2$	$0$	\(0\)	\(-1\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+2\zeta_{6}q^{9}+\cdots\)
392.2.i.c	$392$	$2$	392.i	7.c	$3$	$2$	$1$	$3.130$	\(\Q(\sqrt{-3}) \)	None					56.2.a.a	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+\cdots\)
392.2.i.d	$392$	$2$	392.i	7.c	$3$	$2$	$1$	$3.130$	\(\Q(\sqrt{-3}) \)	None					56.2.a.a	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+\cdots\)
392.2.i.e	$392$	$2$	392.i	7.c	$3$	$2$	$1$	$3.130$	\(\Q(\sqrt{-3}) \)	None					56.2.a.b	$2$	$0$	\(0\)	\(2\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{3}-4\zeta_{6}q^{5}-\zeta_{6}q^{9}-8q^{15}+\cdots\)
392.2.i.f	$392$	$2$	392.i	7.c	$3$	$2$	$1$	$3.130$	\(\Q(\sqrt{-3}) \)	None					56.2.i.a	$2$	$0$	\(0\)	\(3\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(3-3\zeta_{6})q^{3}-\zeta_{6}q^{5}-6\zeta_{6}q^{9}+(1+\cdots)q^{11}+\cdots\)
392.2.i.g	$392$	$2$	392.i	7.c	$3$	$4$	$2$	$3.130$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			392.2.a.h	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{5}+5\beta _{2}q^{9}+\cdots\)
392.2.i.h	$392$	$2$	392.i	7.c	$3$	$4$	$2$	$3.130$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			392.2.a.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{3}+(-2\beta _{1}-2\beta _{3})q^{5}-\beta _{2}q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(392, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 2}\)