Space of modular forms of level 322, weight 2, and Character 322.e

• Label: 322.2.e
• Level: $322$
• Weight: $2$
• Character orbit: 322.e
• Rep. character: $\chi_{322}(93,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $32$
• Newform subspaces: $4$
• Sturm bound: $96$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 322 = 2 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	322.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(96\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(322, [\chi])\).

	Total	New	Old
Modular forms	104	32	72
Cusp forms	88	32	56
Eisenstein series	16	0	16

Trace form

32 * q - 16 * q^4 - 8 * q^6 - 4 * q^7 - 12 * q^9 - 4 * q^10 + 8 * q^11 + 4 * q^14 + 16 * q^15 - 16 * q^16 - 16 * q^17 + 20 * q^21 + 16 * q^22 + 4 * q^24 - 28 * q^25 + 12 * q^26 - 24 * q^27 + 8 * q^28 - 12 * q^30 + 4 * q^31 - 24 * q^33 - 12 * q^35 + 24 * q^36 - 4 * q^38 + 4 * q^39 - 4 * q^40 - 24 * q^41 - 4 * q^42 - 24 * q^43 + 8 * q^44 + 12 * q^45 + 16 * q^47 + 20 * q^49 + 4 * q^51 - 16 * q^53 + 16 * q^54 + 40 * q^55 + 4 * q^56 + 24 * q^57 + 24 * q^59 - 8 * q^60 - 28 * q^61 - 24 * q^62 + 28 * q^63 + 32 * q^64 + 28 * q^65 - 32 * q^66 + 8 * q^67 - 16 * q^68 - 16 * q^69 - 20 * q^70 + 40 * q^73 + 16 * q^74 - 24 * q^75 + 8 * q^77 + 16 * q^78 + 8 * q^79 + 40 * q^83 - 4 * q^84 - 88 * q^85 + 16 * q^86 - 8 * q^88 - 12 * q^89 - 8 * q^90 - 60 * q^91 + 48 * q^93 + 4 * q^94 + 12 * q^95 + 4 * q^96 + 72 * q^97 + 24 * q^98 - 144 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(322, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
322.2.e.a	$322$	$2$	322.e	7.c	$3$	$8$	$4$	$2.571$	8.0.310217769.2	None		✓			322.2.e.a	$2$	$0$	\(-4\)	\(-3\)	\(-7\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{5})q^{2}+(\beta _{1}+\beta _{2}-\beta _{5})q^{3}+\cdots\)
322.2.e.b	$322$	$2$	322.e	7.c	$3$	$8$	$4$	$2.571$	8.0.1767277521.3	None		✓			322.2.e.b	$2$	$0$	\(-4\)	\(5\)	\(5\)	\(1\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{5}q^{2}+(1+\beta _{2}-\beta _{5}+\beta _{6})q^{3}+(-1+\cdots)q^{4}+\cdots\)
322.2.e.c	$322$	$2$	322.e	7.c	$3$	$8$	$4$	$2.571$	8.0.1767277521.3	None		✓			322.2.e.c	$2$	$0$	\(4\)	\(-1\)	\(-3\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{6})q^{2}+(\beta _{4}+\beta _{5}-\beta _{6}-\beta _{7})q^{3}+\cdots\)
322.2.e.d	$322$	$2$	322.e	7.c	$3$	$8$	$4$	$2.571$	8.0.6498455769.2	None		✓			322.2.e.d	$2$	$0$	\(4\)	\(-1\)	\(5\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{5})q^{2}-\beta _{1}q^{3}-\beta _{5}q^{4}+(1+\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(322, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(322, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(161, [\chi])\)\(^{\oplus 2}\)