Space of modular forms of level 320, weight 5, and Character 320.p

• Label: 320.5.p
• Level: $320$
• Weight: $5$
• Character orbit: 320.p
• Rep. character: $\chi_{320}(193,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $92$
• Newform subspaces: $20$
• Sturm bound: $240$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	320.p (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(240\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(320, [\chi])\).

	Total	New	Old
Modular forms	408	100	308
Cusp forms	360	92	268
Eisenstein series	48	8	40

Trace form

92 * q + 4 * q^5 + 4 * q^13 + 476 * q^17 + 8 * q^21 - 676 * q^25 - 328 * q^33 + 4 * q^37 - 8 * q^41 - 2496 * q^45 - 20156 * q^53 + 320 * q^57 - 7544 * q^61 - 7396 * q^65 - 5284 * q^73 - 4792 * q^77 - 53860 * q^81 + 38596 * q^85 - 6392 * q^93 + 30876 * q^97

Decomposition of \(S_{5}^{\mathrm{new}}(320, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
320.5.p.a	$320$	$5$	320.p	5.c	$4$	$2$	$1$	$33.078$	\(\Q(\sqrt{-1}) \)	None					40.5.l.a	$2$	$0$	\(0\)	\(-20\)	\(-40\)	\(-84\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(10 i-10)q^{3}+(-15 i-20)q^{5}+\cdots\)
320.5.p.b	$320$	$5$	320.p	5.c	$4$	$2$	$1$	$33.078$	\(\Q(\sqrt{-1}) \)	None					10.5.c.a	$2$	$0$	\(0\)	\(-18\)	\(30\)	\(58\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(9 i-9)q^{3}+(-20 i+15)q^{5}+\cdots\)
320.5.p.c	$320$	$5$	320.p	5.c	$4$	$2$	$1$	$33.078$	\(\Q(\sqrt{-1}) \)	None					5.5.c.a	$2$	$0$	\(0\)	\(-12\)	\(-40\)	\(52\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(6 i-6)q^{3}+(-15 i-20)q^{5}+\cdots\)
320.5.p.d	$320$	$5$	320.p	5.c	$4$	$2$	$1$	$33.078$	\(\Q(\sqrt{-1}) \)	None					10.5.c.b	$2$	$0$	\(0\)	\(-2\)	\(30\)	\(-38\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i-1)q^{3}+(20 i+15)q^{5}+(-19 i-19)q^{7}+\cdots\)
320.5.p.e	$320$	$5$	320.p	5.c	$4$	$2$	$1$	$33.078$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					160.5.p.b	$4$	$0$	\(0\)	\(0\)	\(-48\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-7 i-24)q^{5}+81 i q^{9}+(-i+1)q^{13}+\cdots\)
320.5.p.f	$320$	$5$	320.p	5.c	$4$	$2$	$1$	$33.078$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					160.5.p.a	$4$	$0$	\(0\)	\(0\)	\(48\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-7 i+24)q^{5}+81 i q^{9}+(239 i-239)q^{13}+\cdots\)
320.5.p.g	$320$	$5$	320.p	5.c	$4$	$2$	$1$	$33.078$	\(\Q(\sqrt{-1}) \)	None					10.5.c.b	$2$	$0$	\(0\)	\(2\)	\(30\)	\(38\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i+1)q^{3}+(20 i+15)q^{5}+(19 i+19)q^{7}+\cdots\)
320.5.p.h	$320$	$5$	320.p	5.c	$4$	$2$	$1$	$33.078$	\(\Q(\sqrt{-1}) \)	None					5.5.c.a	$2$	$0$	\(0\)	\(12\)	\(-40\)	\(-52\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-6 i+6)q^{3}+(-15 i-20)q^{5}+\cdots\)
320.5.p.i	$320$	$5$	320.p	5.c	$4$	$2$	$1$	$33.078$	\(\Q(\sqrt{-1}) \)	None					10.5.c.a	$2$	$0$	\(0\)	\(18\)	\(30\)	\(-58\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-9 i+9)q^{3}+(-20 i+15)q^{5}+\cdots\)
320.5.p.j	$320$	$5$	320.p	5.c	$4$	$2$	$1$	$33.078$	\(\Q(\sqrt{-1}) \)	None					40.5.l.a	$2$	$0$	\(0\)	\(20\)	\(-40\)	\(84\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-10 i+10)q^{3}+(-15 i-20)q^{5}+\cdots\)
320.5.p.k	$320$	$5$	320.p	5.c	$4$	$4$	$2$	$33.078$	\(\Q(i, \sqrt{29})\)	None					40.5.l.b	$2$	$0$	\(0\)	\(-12\)	\(12\)	\(-44\)	$2^{4}\cdot 5$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-3+3\beta _{1})q^{3}+(3+6\beta _{1}-\beta _{3})q^{5}+\cdots\)
320.5.p.l	$320$	$5$	320.p	5.c	$4$	$4$	$2$	$33.078$	\(\Q(i, \sqrt{241})\)	None					20.5.f.a	$2$	$0$	\(0\)	\(-10\)	\(6\)	\(-110\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2+2\beta _{1}+\beta _{3})q^{3}+(6\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
320.5.p.m	$320$	$5$	320.p	5.c	$4$	$4$	$2$	$33.078$	\(\Q(i, \sqrt{241})\)	None					20.5.f.a	$2$	$0$	\(0\)	\(10\)	\(6\)	\(110\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(3-3\beta _{1}+\beta _{3})q^{3}+(3+3\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
320.5.p.n	$320$	$5$	320.p	5.c	$4$	$4$	$2$	$33.078$	\(\Q(i, \sqrt{29})\)	None					40.5.l.b	$2$	$0$	\(0\)	\(12\)	\(12\)	\(44\)	$2^{4}\cdot 5$	$\mathrm{SU}(2)[C_{4}]$	\(q+(3-3\beta _{1})q^{3}+(3+6\beta _{1}-\beta _{3})q^{5}+\cdots\)
320.5.p.o	$320$	$5$	320.p	5.c	$4$	$6$	$3$	$33.078$	6.0.313431616.3	None					40.5.l.c	$2$	$0$	\(0\)	\(-8\)	\(4\)	\(40\)	$2^{5}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\beta _{1}-\beta _{3})q^{3}+(1-7\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
320.5.p.p	$320$	$5$	320.p	5.c	$4$	$6$	$3$	$33.078$	6.0.313431616.3	None					40.5.l.c	$2$	$0$	\(0\)	\(8\)	\(4\)	\(-40\)	$2^{5}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{1}+\beta _{3})q^{3}+(1-7\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
320.5.p.q	$320$	$5$	320.p	5.c	$4$	$8$	$4$	$33.078$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					160.5.p.c	$4$	$0$	\(0\)	\(0\)	\(40\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{5}q^{3}+(5-5\beta _{4})q^{5}+(\beta _{3}+\beta _{7})q^{7}+\cdots\)
320.5.p.r	$320$	$5$	320.p	5.c	$4$	$12$	$6$	$33.078$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					160.5.p.e	$2$	$0$	\(0\)	\(0\)	\(-24\)	\(-48\)	$2^{24}\cdot 5^{5}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{8}q^{3}+(-2-\beta _{2}+\beta _{6})q^{5}+(-4+\cdots)q^{7}+\cdots\)
320.5.p.s	$320$	$5$	320.p	5.c	$4$	$12$	$6$	$33.078$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					160.5.p.e	$2$	$0$	\(0\)	\(0\)	\(-24\)	\(48\)	$2^{24}\cdot 5^{5}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{8}q^{3}+(-2-\beta _{2}+\beta _{6})q^{5}+(4+4\beta _{2}+\cdots)q^{7}+\cdots\)
320.5.p.t	$320$	$5$	320.p	5.c	$4$	$12$	$6$	$33.078$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					160.5.p.d	$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{24}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{6}q^{3}+(1-\beta _{5})q^{5}+(-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(320, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(320, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 2}\)