Space of modular forms of level 322, weight 2, and trivial character

• Label: 322.2.a
• Level: $322$
• Weight: $2$
• Character orbit: 322.a
• Rep. character: $\chi_{322}(1,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $7$
• Sturm bound: $96$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 322 = 2 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	322.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(96\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(322))\).

	Total	New	Old
Modular forms	52	11	41
Cusp forms	45	11	34
Eisenstein series	7	0	7

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(7\)	\(23\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(4\)	\(2\)	\(2\)		\(4\)	\(2\)	\(2\)		\(0\)	\(0\)	\(0\)
\(+\)	\(+\)	\(-\)	\(-\)		\(7\)	\(0\)	\(7\)		\(6\)	\(0\)	\(6\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(9\)	\(1\)	\(8\)		\(8\)	\(1\)	\(7\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(6\)	\(3\)	\(3\)		\(5\)	\(3\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(7\)	\(1\)	\(6\)		\(6\)	\(1\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(7\)	\(0\)	\(7\)		\(6\)	\(0\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(6\)	\(3\)	\(3\)		\(5\)	\(3\)	\(2\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(24\)	\(4\)	\(20\)		\(21\)	\(4\)	\(17\)		\(3\)	\(0\)	\(3\)
Minus space			\(-\)		\(28\)	\(7\)	\(21\)		\(24\)	\(7\)	\(17\)		\(4\)	\(0\)	\(4\)

Trace form

11 * q + 3 * q^2 + 11 * q^4 - 2 * q^5 - q^7 + 3 * q^8 + 15 * q^9 + 6 * q^10 - 2 * q^13 - q^14 - 16 * q^15 + 11 * q^16 - 10 * q^17 + 7 * q^18 - 8 * q^19 - 2 * q^20 + 4 * q^21 - q^23 - 3 * q^25 - 2 * q^26 - 24 * q^27 - q^28 + 2 * q^29 - 16 * q^31 + 3 * q^32 + 14 * q^34 - 2 * q^35 + 15 * q^36 - 10 * q^37 + 8 * q^38 - 40 * q^39 + 6 * q^40 - 18 * q^41 - 4 * q^42 - 8 * q^43 + 6 * q^45 + 3 * q^46 + 8 * q^47 + 11 * q^49 + 5 * q^50 + 32 * q^51 - 2 * q^52 - 18 * q^53 + 24 * q^54 - 8 * q^55 - q^56 - 14 * q^58 - 16 * q^60 + 14 * q^61 - 16 * q^62 - 13 * q^63 + 11 * q^64 + 36 * q^65 - 16 * q^66 - 10 * q^68 - 4 * q^69 - 2 * q^70 - 24 * q^71 + 7 * q^72 - 10 * q^73 - 10 * q^74 - 24 * q^75 - 8 * q^76 + 12 * q^77 - 8 * q^78 - 32 * q^79 - 2 * q^80 + 3 * q^81 + 30 * q^82 + 16 * q^83 + 4 * q^84 + 44 * q^85 + 24 * q^86 + 24 * q^87 + 6 * q^89 - 34 * q^90 + 10 * q^91 - q^92 + 24 * q^93 + 8 * q^94 - 8 * q^95 + 6 * q^97 + 3 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(322))\) 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	23
322.2.a.a	$322$	$2$	322.a	1.a	$1$	$1$	$1$	$2.571$	\(\Q\)	None	✓	✓	✓	✓	322.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
322.2.a.b	$322$	$2$	322.a	1.a	$1$	$1$	$1$	$2.571$	\(\Q\)	None	✓	✓	✓	✓	322.2.a.b	$1$	$0$	\(-1\)	\(2\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2q^{3}+q^{4}-2q^{6}+q^{7}-q^{8}+\cdots\)
322.2.a.c	$322$	$2$	322.a	1.a	$1$	$1$	$1$	$2.571$	\(\Q\)	None	✓	✓	✓	✓	322.2.a.c	$1$	$1$	\(1\)	\(-2\)	\(-2\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}+q^{4}-2q^{5}-2q^{6}-q^{7}+\cdots\)
322.2.a.d	$322$	$2$	322.a	1.a	$1$	$1$	$1$	$2.571$	\(\Q\)	None	✓	✓			322.2.a.d	$1$	$0$	\(1\)	\(2\)	\(-2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}-2q^{5}+2q^{6}+q^{7}+\cdots\)
322.2.a.e	$322$	$2$	322.a	1.a	$1$	$2$	$2$	$2.571$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	322.2.a.e	$1$	$1$	\(-2\)	\(-2\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1-\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots\)
322.2.a.f	$322$	$2$	322.a	1.a	$1$	$2$	$2$	$2.571$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓		322.2.a.f	$1$	$0$	\(2\)	\(-2\)	\(2\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta )q^{3}+q^{4}+(1+\beta )q^{5}+\cdots\)
322.2.a.g	$322$	$2$	322.a	1.a	$1$	$3$	$3$	$2.571$	3.3.316.1	None	✓	✓	✓	✓	322.2.a.g	$1$	$0$	\(3\)	\(2\)	\(4\)	\(-3\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta _{2})q^{3}+q^{4}+(1-\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(322))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(322)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(161))\)\(^{\oplus 2}\)