Space of modular forms of level 322, weight 4, and trivial character

• Label: 322.4.a
• Level: $322$
• Weight: $4$
• Character orbit: 322.a
• Rep. character: $\chi_{322}(1,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $9$
• Sturm bound: $192$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 322 = 2 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	322.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(192\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(322))\).

	Total	New	Old
Modular forms	148	32	116
Cusp forms	140	32	108
Eisenstein series	8	0	8

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	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	\(5\)
\(+\)	\(-\)	\(+\)	\(-\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(5\)
\(-\)	\(-\)	\(+\)	\(+\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)	\(2\)
Plus space			\(+\)	\(18\)
Minus space			\(-\)	\(14\)

Trace form

32 * q + 4 * q^3 + 128 * q^4 + 20 * q^5 + 40 * q^6 + 168 * q^9 - 8 * q^10 + 84 * q^11 + 16 * q^12 + 76 * q^13 + 80 * q^15 + 512 * q^16 + 144 * q^17 + 256 * q^18 - 180 * q^19 + 80 * q^20 + 224 * q^21 - 72 * q^22 + 160 * q^24 + 1112 * q^25 - 72 * q^26 + 400 * q^27 + 80 * q^29 - 624 * q^30 - 480 * q^31 - 640 * q^33 + 1040 * q^34 - 140 * q^35 + 672 * q^36 - 308 * q^37 + 376 * q^38 + 472 * q^39 - 32 * q^40 + 8 * q^41 + 508 * q^43 + 336 * q^44 - 404 * q^45 - 184 * q^46 - 2040 * q^47 + 64 * q^48 + 1568 * q^49 - 272 * q^50 - 248 * q^51 + 304 * q^52 - 1068 * q^53 + 1120 * q^54 - 2984 * q^55 - 1128 * q^57 + 1104 * q^58 - 1900 * q^59 + 320 * q^60 + 1188 * q^61 - 352 * q^62 + 2048 * q^64 + 2928 * q^65 + 704 * q^66 - 204 * q^67 + 576 * q^68 - 280 * q^70 - 56 * q^71 + 1024 * q^72 + 2656 * q^73 + 728 * q^74 + 2732 * q^75 - 720 * q^76 - 1064 * q^77 + 1024 * q^78 - 3416 * q^79 + 320 * q^80 - 2448 * q^81 - 640 * q^82 - 1636 * q^83 + 896 * q^84 - 3496 * q^85 - 664 * q^86 - 752 * q^87 - 288 * q^88 - 3272 * q^89 + 504 * q^90 - 1008 * q^91 - 4368 * q^93 - 2448 * q^94 - 816 * q^95 + 640 * q^96 + 1400 * q^97 + 3588 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(322))\) 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
322.4.a.a	$322$	$4$	322.a	1.a	$1$	$2$	$2$	$18.999$	\(\Q(\sqrt{2}) \)	None	✓	✓			322.4.a.a	$1$	$0$	\(-4\)	\(-4\)	\(16\)	\(14\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-2+3\beta )q^{3}+4q^{4}+(8-3\beta )q^{5}+\cdots\)
322.4.a.b	$322$	$4$	322.a	1.a	$1$	$2$	$2$	$18.999$	\(\Q(\sqrt{73}) \)	None	✓	✓			322.4.a.b	$1$	$0$	\(-4\)	\(9\)	\(-8\)	\(14\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(5-\beta )q^{3}+4q^{4}+(-2-4\beta )q^{5}+\cdots\)
322.4.a.c	$322$	$4$	322.a	1.a	$1$	$2$	$2$	$18.999$	\(\Q(\sqrt{57}) \)	None	✓	✓	✓	✓	322.4.a.c	$1$	$1$	\(4\)	\(1\)	\(-18\)	\(14\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+\beta q^{3}+4q^{4}+(-8-2\beta )q^{5}+\cdots\)
322.4.a.d	$322$	$4$	322.a	1.a	$1$	$3$	$3$	$18.999$	3.3.15384.1	None	✓	✓	✓	✓	322.4.a.d	$1$	$0$	\(-6\)	\(-3\)	\(8\)	\(-21\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-1+\beta _{1})q^{3}+4q^{4}+(3+2\beta _{1}+\cdots)q^{5}+\cdots\)
322.4.a.e	$322$	$4$	322.a	1.a	$1$	$3$	$3$	$18.999$	3.3.3368.1	None	✓	✓	✓	✓	322.4.a.e	$1$	$1$	\(6\)	\(-7\)	\(-8\)	\(-21\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+(-2-\beta _{2})q^{3}+4q^{4}+(-3+\cdots)q^{5}+\cdots\)
322.4.a.f	$322$	$4$	322.a	1.a	$1$	$4$	$4$	$18.999$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	322.4.a.f	$1$	$1$	\(-8\)	\(-1\)	\(-2\)	\(28\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-\beta _{1}q^{3}+4q^{4}-\beta _{2}q^{5}+2\beta _{1}q^{6}+\cdots\)
322.4.a.g	$322$	$4$	322.a	1.a	$1$	$5$	$5$	$18.999$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	322.4.a.g	$1$	$1$	\(-10\)	\(-9\)	\(-2\)	\(-35\)		$+$	$+$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-2+\beta _{4})q^{3}+4q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
322.4.a.h	$322$	$4$	322.a	1.a	$1$	$5$	$5$	$18.999$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	322.4.a.h	$1$	$0$	\(10\)	\(5\)	\(22\)	\(-35\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+(4-\beta _{1}+\cdots)q^{5}+\cdots\)
322.4.a.i	$322$	$4$	322.a	1.a	$1$	$6$	$6$	$18.999$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	322.4.a.i	$1$	$0$	\(12\)	\(13\)	\(12\)	\(42\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+(2+\beta _{1})q^{3}+4q^{4}+(2-\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(322))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(322)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(161))\)\(^{\oplus 2}\)