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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				Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
409.2.a.a	$409$	$2$	409.a	1.a	$1$	$13$	$13$	$3.266$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	✓	✓	✓	409.2.a.a	$1$	$1$	\(-6\)	\(-3\)	\(-10\)	\(-9\)	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
409.2.a.b	$409$	$2$	409.a	1.a	$1$	$20$	$20$	$3.266$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	✓	✓	✓	409.2.a.b	$1$	$0$	\(5\)	\(1\)	\(8\)	\(5\)	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{10}q^{3}+(1+\beta _{2})q^{4}+\beta _{4}q^{5}+\cdots\)
409.2.b.a	$409$	$2$	409.b	409.b	$2$	$32$	$32$	$3.266$		None		✓	✓	✓	409.2.b.a	$2$	$0$	\(-2\)	\(-2\)	\(0\)	\(0\)			$\mathrm{SU}(2)[C_{2}]$
409.2.c.a	$409$	$2$	409.c	409.c	$3$	$66$	$33$	$3.266$		None		✓	✓	✓	409.2.c.a	$2$	$0$	\(1\)	\(-1\)	\(-4\)	\(-2\)			$\mathrm{SU}(2)[C_{3}]$
409.2.d.a	$409$	$2$	409.d	409.d	$4$	$66$	$33$	$3.266$		None		✓	✓	✓	409.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)			$\mathrm{SU}(2)[C_{4}]$
409.2.e.a	$409$	$2$	409.e	409.e	$6$	$66$	$33$	$3.266$		None		✓	✓	✓	409.2.e.a	$2$	$0$	\(-1\)	\(-3\)	\(-12\)	\(6\)			$\mathrm{SU}(2)[C_{6}]$
409.2.g.a	$409$	$2$	409.g	409.g	$12$	$4$	$1$	$3.266$	\(\Q(\zeta_{12})\)	None		✓			409.2.g.a	$2$	$1$	\(-6\)	\(-6\)	\(0\)	\(-4\)		$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-2+\zeta_{12}^{2})q^{2}+(-1+\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
409.2.g.b	$409$	$2$	409.g	409.g	$12$	$132$	$33$	$3.266$		None		✓	✓		409.2.g.b	$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)			$\mathrm{SU}(2)[C_{12}]$
409.2.h.a	$409$	$2$	409.h	409.h	$17$	$512$	$32$	$3.266$		None		✓	✓	✓	409.2.h.a	$2$	$0$	\(-13\)	\(-11\)	\(-9\)	\(-22\)			$\mathrm{SU}(2)[C_{17}]$
409.2.j.a	$409$	$2$	409.j	409.j	$34$	$512$	$32$	$3.266$		None		✓	✓	✓	409.2.j.a	$2$	$0$	\(-15\)	\(-15\)	\(-17\)	\(0\)			$\mathrm{SU}(2)[C_{34}]$
409.2.k.a	$409$	$2$	409.k	409.k	$51$	$1056$	$33$	$3.266$		None		✓	✓	✓	409.2.k.a	$2$	$0$	\(-35\)	\(-33\)	\(-30\)	\(-15\)			$\mathrm{SU}(2)[C_{51}]$
409.2.l.a	$409$	$2$	409.l	409.l	$68$	$1056$	$33$	$3.266$		None		✓	✓	✓	409.2.l.a	$2$	$0$	\(-34\)	\(-34\)	\(-34\)	\(-34\)			$\mathrm{SU}(2)[C_{68}]$
409.2.m.a	$409$	$2$	409.m	409.m	$102$	$1056$	$33$	$3.266$		None		✓	✓	✓	409.2.m.a	$2$	$0$	\(-33\)	\(-31\)	\(-22\)	\(-57\)			$\mathrm{SU}(2)[C_{102}]$
409.2.o.a	$409$	$2$	409.o	409.o	$204$	$2176$	$34$	$3.266$		None		✓	✓	✓	409.2.o.a	$2$	$0$	\(-62\)	\(-56\)	\(-68\)	\(-64\)			$\mathrm{SU}(2)[C_{204}]$
409.3.f.a	$409$	$3$	409.f	409.f	$8$	$272$	$68$	$11.144$		None		✓	✓	✓	409.3.f.a	$2$	$0$	\(0\)	\(-12\)	\(-8\)	\(-20\)			$\mathrm{SU}(2)[C_{8}]$
409.3.i.a	$409$	$3$	409.i	409.i	$24$	$536$	$67$	$11.144$		None		✓	✓	✓	409.3.i.a	$2$	$0$	\(-12\)	\(-24\)	\(-16\)	\(8\)			$\mathrm{SU}(2)[C_{24}]$
409.3.n.a	$409$	$3$	409.n	409.n	$136$	$4352$	$68$	$11.144$		None		✓	✓	✓	409.3.n.a	$2$	$0$	\(-68\)	\(-56\)	\(-60\)	\(-48\)			$\mathrm{SU}(2)[C_{136}]$
409.3.p.a	$409$	$3$	409.p	409.p	$408$	$8576$	$67$	$11.144$		None		✓	✓	✓	409.3.p.a	$2$	$0$	\(-124\)	\(-112\)	\(-120\)	\(-144\)			$\mathrm{SU}(2)[C_{408}]$
409.4.a.a	$409$	$4$	409.a	1.a	$1$	$47$	$47$	$24.132$		None	✓	✓	✓	✓	409.4.a.a	$1$	$1$	\(-15\)	\(-14\)	\(-60\)	\(-82\)	$-$		$\mathrm{SU}(2)$
409.4.a.b	$409$	$4$	409.a	1.a	$1$	$55$	$55$	$24.132$		None	✓	✓	✓	✓	409.4.a.b	$1$	$0$	\(13\)	\(10\)	\(60\)	\(72\)	$+$		$\mathrm{SU}(2)$
409.6.a.a	$409$	$6$	409.a	1.a	$1$	$81$	$81$	$65.597$		None	✓	✓	✓	✓	409.6.a.a	$1$	$1$	\(-29\)	\(-35\)	\(-318\)	\(-549\)	$+$		$\mathrm{SU}(2)$
409.6.a.b	$409$	$6$	409.a	1.a	$1$	$89$	$89$	$65.597$		None	✓	✓	✓	✓	409.6.a.b	$1$	$0$	\(27\)	\(37\)	\(282\)	\(529\)	$-$		$\mathrm{SU}(2)$
409.8.a.a	$409$	$8$	409.a	1.a	$1$	$115$	$115$	$127.765$		None	✓	✓	✓	✓	409.8.a.a	$1$	$1$	\(-49\)	\(-95\)	\(-1570\)	\(-3919\)	$-$		$\mathrm{SU}(2)$
409.8.a.b	$409$	$8$	409.a	1.a	$1$	$123$	$123$	$127.765$		None	✓	✓	✓	✓	409.8.a.b	$1$	$0$	\(63\)	\(121\)	\(1430\)	\(3627\)	$+$		$\mathrm{SU}(2)$

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