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

• Label: 392.2.p
• Level: $392$
• Weight: $2$
• Character orbit: 392.p
• Rep. character: $\chi_{392}(165,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $72$
• Newform subspaces: $8$
• Sturm bound: $112$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.p (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 56 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(112\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(3\), \(17\)

Dimensions

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

	Total	New	Old
Modular forms	128	88	40
Cusp forms	96	72	24
Eisenstein series	32	16	16

Trace form

72 * q + 3 * q^2 - q^4 + 8 * q^6 - 18 * q^8 + 30 * q^9 + 8 * q^10 + 2 * q^12 - 4 * q^15 - 13 * q^16 + 2 * q^17 + q^18 - 8 * q^20 - 32 * q^22 - 6 * q^23 - 18 * q^24 + 22 * q^25 + 2 * q^26 - 42 * q^30 - 10 * q^31 + 23 * q^32 + 14 * q^33 - 32 * q^34 + 30 * q^36 - 18 * q^38 - 4 * q^39 - 10 * q^40 + 8 * q^41 + 12 * q^44 + 28 * q^46 - 30 * q^47 + 56 * q^48 - 2 * q^50 + 32 * q^52 - 2 * q^54 - 4 * q^55 - 20 * q^57 + 30 * q^58 - 58 * q^60 + 28 * q^62 - 46 * q^64 - 8 * q^65 + 38 * q^66 - 4 * q^68 - 64 * q^71 - 47 * q^72 + 10 * q^73 - 50 * q^74 - 52 * q^76 + 4 * q^78 + 42 * q^79 - 36 * q^80 - 16 * q^81 + 26 * q^82 - 2 * q^86 + 20 * q^87 - 20 * q^88 + 10 * q^89 - 52 * q^90 - 100 * q^92 - 42 * q^94 - 26 * q^95 - 16 * q^96 - 40 * q^97

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.p.a	$392$	$2$	392.p	56.p	$6$	$4$	$2$	$3.130$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					56.2.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+(-\beta _{1}+\beta _{3})q^{3}+2\beta _{2}q^{4}+\cdots\)
392.2.p.b	$392$	$2$	392.p	56.p	$6$	$4$	$2$	$3.130$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					56.2.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+(\beta _{1}-\beta _{3})q^{3}+2\beta _{2}q^{4}+\beta _{1}q^{5}+\cdots\)
392.2.p.c	$392$	$2$	392.p	56.p	$6$	$4$	$2$	$3.130$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	\(\Q(\sqrt{-7}) \)		✓			392.2.b.a	$8$	$0$	\(1\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+\beta _{3}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(3-\beta _{1}+\cdots)q^{8}+\cdots\)
392.2.p.d	$392$	$2$	392.p	56.p	$6$	$8$	$4$	$3.130$	8.0.\(\cdots\).10	\(\Q(\sqrt{-14}) \)		✓			392.2.b.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{6}]$	\(q-\beta _{5}q^{2}-\beta _{6}q^{3}+(-2+2\beta _{2})q^{4}+\cdots\)
392.2.p.e	$392$	$2$	392.p	56.p	$6$	$8$	$4$	$3.130$	8.0.432972864.2	None					56.2.b.b	$4$	$0$	\(1\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{1}-\beta _{6})q^{2}+(\beta _{5}+\beta _{6})q^{3}+\beta _{5}q^{4}+\cdots\)
392.2.p.f	$392$	$2$	392.p	56.p	$6$	$8$	$4$	$3.130$	8.0.432972864.2	None					56.2.b.b	$4$	$0$	\(1\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{1}-\beta _{6})q^{2}+(-\beta _{5}-\beta _{6})q^{3}+\cdots\)
392.2.p.g	$392$	$2$	392.p	56.p	$6$	$12$	$6$	$3.130$	12.0.\(\cdots\).1	None					56.2.p.a	$4$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{1}+\beta _{5})q^{2}-\beta _{6}q^{3}-\beta _{3}q^{4}+\cdots\)
392.2.p.h	$392$	$2$	392.p	56.p	$6$	$24$	$12$	$3.130$		None		✓	✓		392.2.b.g	$8$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

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

\( S_{2}^{\mathrm{old}}(392, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 2}\)