LMFDB - Embedded newform 812.1.l.c.307.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [812,1,Mod(307,812)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("812.307"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(812, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 812 = 2^{2} \cdot 7 \cdot 29 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	812.l (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.405240790258$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.4780244.1

Embedding invariants

Embedding label			307.2
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	812.307
Dual form			812.1.l.c.447.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.707107 + 0.707107i) q^{2} +(0.707107 - 0.707107i) q^{3} +1.00000i q^{4} -1.00000 q^{5} +1.00000 q^{6} +(0.707107 - 0.707107i) q^{7} +(-0.707107 + 0.707107i) q^{8} +(-0.707107 - 0.707107i) q^{10} +(0.707107 + 0.707107i) q^{11} +(0.707107 + 0.707107i) q^{12} +1.00000 q^{13} +1.00000 q^{14} +(-0.707107 + 0.707107i) q^{15} -1.00000 q^{16} +(-1.00000 - 1.00000i) q^{17} -1.00000i q^{20} -1.00000i q^{21} +1.00000i q^{22} +1.00000i q^{24} +(0.707107 + 0.707107i) q^{26} +(0.707107 + 0.707107i) q^{27} +(0.707107 + 0.707107i) q^{28} -1.00000i q^{29} -1.00000 q^{30} +(-0.707107 + 0.707107i) q^{31} +(-0.707107 - 0.707107i) q^{32} +1.00000 q^{33} -1.41421i q^{34} +(-0.707107 + 0.707107i) q^{35} +(-1.00000 - 1.00000i) q^{37} +(0.707107 - 0.707107i) q^{39} +(0.707107 - 0.707107i) q^{40} +(-1.00000 + 1.00000i) q^{41} +(0.707107 - 0.707107i) q^{42} +(0.707107 + 0.707107i) q^{43} +(-0.707107 + 0.707107i) q^{44} +(-0.707107 - 0.707107i) q^{47} +(-0.707107 + 0.707107i) q^{48} -1.00000i q^{49} -1.41421 q^{51} +1.00000i q^{52} -1.00000 q^{53} +1.00000i q^{54} +(-0.707107 - 0.707107i) q^{55} +1.00000i q^{56} +(0.707107 - 0.707107i) q^{58} -1.41421 q^{59} +(-0.707107 - 0.707107i) q^{60} -1.00000 q^{62} -1.00000i q^{64} -1.00000 q^{65} +(0.707107 + 0.707107i) q^{66} +1.41421 q^{67} +(1.00000 - 1.00000i) q^{68} -1.00000 q^{70} -1.41421i q^{74} +1.00000 q^{77} +1.00000 q^{78} +(-0.707107 - 0.707107i) q^{79} +1.00000 q^{80} +1.00000 q^{81} -1.41421 q^{82} +1.41421 q^{83} +1.00000 q^{84} +(1.00000 + 1.00000i) q^{85} +1.00000i q^{86} +(-0.707107 - 0.707107i) q^{87} -1.00000 q^{88} +(0.707107 - 0.707107i) q^{91} +1.00000i q^{93} -1.00000i q^{94} -1.00000 q^{96} +(-1.00000 + 1.00000i) q^{97} +(0.707107 - 0.707107i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} + 4 q^{6} + 4 q^{13} + 4 q^{14} - 4 q^{16} - 4 q^{17} - 4 q^{30} + 4 q^{33} - 4 q^{37} - 4 q^{41} - 4 q^{53} - 4 q^{62} - 4 q^{65} + 4 q^{68} - 4 q^{70} + 4 q^{77} + 4 q^{78} + 4 q^{80} + 4 q^{81}+ \cdots - 4 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 + 4 * q^6 + 4 * q^13 + 4 * q^14 - 4 * q^16 - 4 * q^17 - 4 * q^30 + 4 * q^33 - 4 * q^37 - 4 * q^41 - 4 * q^53 - 4 * q^62 - 4 * q^65 + 4 * q^68 - 4 * q^70 + 4 * q^77 + 4 * q^78 + 4 * q^80 + 4 * q^81 + 4 * q^84 + 4 * q^85 - 4 * q^88 - 4 * q^96 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/812\mathbb{Z}\right)^\times$.

$n$	$407$	$465$	$785$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.707107	+	0.707107i	0.707107	+	0.707107i
$2$	0.707107	+	0.707107i	0.707107	+	0.707107i
$3$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$3$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$4$			1.00000i			1.00000i
$5$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$5$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$6$	1.00000			1.00000
$7$	0.707107	−	0.707107i	0.707107	−	0.707107i
$7$	0.707107	−	0.707107i	0.707107	−	0.707107i
$8$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$9$		0			0
$10$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$11$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$11$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$12$	0.707107	+	0.707107i	0.707107	+	0.707107i
$13$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$13$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$14$	1.00000			1.00000
$15$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$16$	−1.00000			−1.00000
$17$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$17$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$18$		0			0
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$		−	1.00000i		−	1.00000i
$21$		−	1.00000i		−	1.00000i
$22$			1.00000i			1.00000i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$			1.00000i			1.00000i
$25$		0			0
$26$	0.707107	+	0.707107i	0.707107	+	0.707107i
$27$	0.707107	+	0.707107i	0.707107	+	0.707107i
$28$	0.707107	+	0.707107i	0.707107	+	0.707107i
$29$		−	1.00000i		−	1.00000i
$29$		−	1.00000i		−	1.00000i
$30$	−1.00000			−1.00000
$31$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$31$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$32$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$33$	1.00000			1.00000
$34$		−	1.41421i		−	1.41421i
$35$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$36$		0			0
$37$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$37$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$38$		0			0
$39$	0.707107	−	0.707107i	0.707107	−	0.707107i
$40$	0.707107	−	0.707107i	0.707107	−	0.707107i
$41$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$41$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$42$	0.707107	−	0.707107i	0.707107	−	0.707107i
$43$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$43$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$44$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$45$		0			0
$46$		0			0
$47$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$47$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$48$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$49$		−	1.00000i		−	1.00000i
$50$		0			0
$51$	−1.41421			−1.41421
$52$			1.00000i			1.00000i
$53$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$53$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$54$			1.00000i			1.00000i
$55$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$56$			1.00000i			1.00000i
$57$		0			0
$58$	0.707107	−	0.707107i	0.707107	−	0.707107i
$59$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$59$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$60$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$61$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$61$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$62$	−1.00000			−1.00000
$63$		0			0
$64$		−	1.00000i		−	1.00000i
$65$	−1.00000			−1.00000
$66$	0.707107	+	0.707107i	0.707107	+	0.707107i
$67$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$67$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$68$	1.00000	−	1.00000i	1.00000	−	1.00000i
$69$		0			0
$70$	−1.00000			−1.00000
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		−	1.41421i		−	1.41421i
$75$		0			0
$76$		0			0
$77$	1.00000			1.00000
$78$	1.00000			1.00000
$79$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$79$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$80$	1.00000			1.00000
$81$	1.00000			1.00000
$82$	−1.41421			−1.41421
$83$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$83$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$84$	1.00000			1.00000
$85$	1.00000	+	1.00000i	1.00000	+	1.00000i
$86$			1.00000i			1.00000i
$87$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$88$	−1.00000			−1.00000
$89$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$89$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$90$		0			0
$91$	0.707107	−	0.707107i	0.707107	−	0.707107i
$92$		0			0
$93$			1.00000i			1.00000i
$94$		−	1.00000i		−	1.00000i
$95$		0			0
$96$	−1.00000			−1.00000
$97$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$97$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$98$	0.707107	−	0.707107i	0.707107	−	0.707107i
$99$		0			0
$100$		0			0
$101$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$101$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$102$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$105$			1.00000i			1.00000i
$106$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$109$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$109$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$110$		−	1.00000i		−	1.00000i
$111$	−1.41421			−1.41421
$112$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$	1.00000			1.00000
$117$		0			0
$118$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$119$	−1.41421			−1.41421
$120$		−	1.00000i		−	1.00000i
$121$		0			0
$122$		0			0
$123$			1.41421i			1.41421i
$124$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$125$	1.00000			1.00000
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	0.707107	−	0.707107i	0.707107	−	0.707107i
$129$	1.00000			1.00000
$130$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$131$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$131$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$132$			1.00000i			1.00000i
$133$		0			0
$134$	1.00000	+	1.00000i	1.00000	+	1.00000i
$135$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$136$	1.41421			1.41421
$137$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$137$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$141$	−1.00000			−1.00000
$142$		0			0
$143$	0.707107	+	0.707107i	0.707107	+	0.707107i
$144$		0			0
$145$			1.00000i			1.00000i
$146$		0			0
$147$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$148$	1.00000	−	1.00000i	1.00000	−	1.00000i
$149$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$149$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		0			0
$154$	0.707107	+	0.707107i	0.707107	+	0.707107i
$155$	0.707107	−	0.707107i	0.707107	−	0.707107i
$156$	0.707107	+	0.707107i	0.707107	+	0.707107i
$157$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$157$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$158$		−	1.00000i		−	1.00000i
$159$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$160$	0.707107	+	0.707107i	0.707107	+	0.707107i
$161$		0			0
$162$	0.707107	+	0.707107i	0.707107	+	0.707107i
$163$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$163$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$164$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$165$	−1.00000			−1.00000
$166$	1.00000	+	1.00000i	1.00000	+	1.00000i
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$	0.707107	+	0.707107i	0.707107	+	0.707107i
$169$		0			0
$170$			1.41421i			1.41421i
$171$		0			0
$172$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		−	1.00000i		−	1.00000i
$175$		0			0
$176$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$177$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$181$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$182$	1.00000			1.00000
$183$		0			0
$184$		0			0
$185$	1.00000	+	1.00000i	1.00000	+	1.00000i
$186$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$187$		−	1.41421i		−	1.41421i
$188$	0.707107	−	0.707107i	0.707107	−	0.707107i
$189$	1.00000			1.00000
$190$		0			0
$191$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$191$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$192$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$193$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$193$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$194$	−1.41421			−1.41421
$195$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$196$	1.00000			1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$199$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$200$		0			0
$201$	1.00000	−	1.00000i	1.00000	−	1.00000i
$202$		0			0
$203$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$204$		−	1.41421i		−	1.41421i
$205$	1.00000	−	1.00000i	1.00000	−	1.00000i
$206$		0			0
$207$		0			0
$208$	−1.00000			−1.00000
$209$		0			0
$210$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$211$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$211$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$212$		−	1.00000i		−	1.00000i
$213$		0			0
$214$		0			0
$215$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$216$	−1.00000			−1.00000
$217$			1.00000i			1.00000i
$218$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$219$		0			0
$220$	0.707107	−	0.707107i	0.707107	−	0.707107i
$221$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$222$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−1.00000			−1.00000
$225$		0			0
$226$		0			0
$227$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$227$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$229$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$230$		0			0
$231$	0.707107	−	0.707107i	0.707107	−	0.707107i
$232$	0.707107	+	0.707107i	0.707107	+	0.707107i
$233$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$233$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$	0.707107	+	0.707107i	0.707107	+	0.707107i
$236$		−	1.41421i		−	1.41421i
$237$	−1.00000			−1.00000
$238$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$239$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$239$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$240$	0.707107	−	0.707107i	0.707107	−	0.707107i
$241$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$241$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			1.00000i			1.00000i
$246$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$247$		0			0
$248$		−	1.00000i		−	1.00000i
$249$	1.00000	−	1.00000i	1.00000	−	1.00000i
$250$	0.707107	+	0.707107i	0.707107	+	0.707107i
$251$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$251$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	1.41421			1.41421
$256$	1.00000			1.00000
$257$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$257$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$258$	0.707107	+	0.707107i	0.707107	+	0.707107i
$259$	−1.41421			−1.41421
$260$		−	1.00000i		−	1.00000i
$261$		0			0
$262$			2.00000i			2.00000i
$263$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$263$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$264$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$265$	1.00000			1.00000
$266$		0			0
$267$		0			0
$268$			1.41421i			1.41421i
$269$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$269$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$270$		−	1.00000i		−	1.00000i
$271$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$271$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$272$	1.00000	+	1.00000i	1.00000	+	1.00000i
$273$		−	1.00000i		−	1.00000i
$274$	1.41421			1.41421
$275$		0			0
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		−	1.00000i		−	1.00000i
$281$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$281$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$282$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$			1.00000i			1.00000i
$287$			1.41421i			1.41421i
$288$		0			0
$289$			1.00000i			1.00000i
$290$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$291$			1.41421i			1.41421i
$292$		0			0
$293$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$293$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$294$		−	1.00000i		−	1.00000i
$295$	1.41421			1.41421
$296$	1.41421			1.41421
$297$			1.00000i			1.00000i
$298$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$299$		0			0
$300$		0			0
$301$	1.00000			1.00000
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$307$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$308$			1.00000i			1.00000i
$309$		0			0
$310$	1.00000			1.00000
$311$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$311$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$312$			1.00000i			1.00000i
$313$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$313$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$314$	1.41421			1.41421
$315$		0			0
$316$	0.707107	−	0.707107i	0.707107	−	0.707107i
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$	−1.00000			−1.00000
$319$	0.707107	−	0.707107i	0.707107	−	0.707107i
$320$			1.00000i			1.00000i
$321$		0			0
$322$		0			0
$323$		0			0
$324$			1.00000i			1.00000i
$325$		0			0
$326$	1.00000			1.00000
$327$	0.707107	+	0.707107i	0.707107	+	0.707107i
$328$		−	1.41421i		−	1.41421i
$329$	−1.00000			−1.00000
$330$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$331$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$331$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$332$			1.41421i			1.41421i
$333$		0			0
$334$		0			0
$335$	−1.41421			−1.41421
$336$			1.00000i			1.00000i
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$		0			0
$339$		0			0
$340$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$341$	−1.00000			−1.00000
$342$		0			0
$343$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$344$	−1.00000			−1.00000
$345$		0			0
$346$		0			0
$347$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$347$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$348$	0.707107	−	0.707107i	0.707107	−	0.707107i
$349$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$349$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$350$		0			0
$351$	0.707107	+	0.707107i	0.707107	+	0.707107i
$352$		−	1.00000i		−	1.00000i
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$	−1.41421			−1.41421
$355$		0			0
$356$		0			0
$357$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$358$		0			0
$359$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$359$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$360$		0			0
$361$			1.00000i			1.00000i
$362$	0.707107	−	0.707107i	0.707107	−	0.707107i
$363$		0			0
$364$	0.707107	+	0.707107i	0.707107	+	0.707107i
$365$		0			0
$366$		0			0
$367$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$367$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$		0			0
$370$			1.41421i			1.41421i
$371$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$372$	−1.00000			−1.00000
$373$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$373$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$374$	1.00000	−	1.00000i	1.00000	−	1.00000i
$375$	0.707107	−	0.707107i	0.707107	−	0.707107i
$376$	1.00000			1.00000
$377$		−	1.00000i		−	1.00000i
$378$	0.707107	+	0.707107i	0.707107	+	0.707107i
$379$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$379$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		−	1.00000i		−	1.00000i
$385$	−1.00000			−1.00000
$386$	−1.41421			−1.41421
$387$		0			0
$388$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$389$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$389$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$390$	−1.00000			−1.00000
$391$		0			0
$392$	0.707107	+	0.707107i	0.707107	+	0.707107i
$393$	2.00000			2.00000
$394$		0			0
$395$	0.707107	+	0.707107i	0.707107	+	0.707107i
$396$		0			0
$397$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$397$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$398$	1.00000	+	1.00000i	1.00000	+	1.00000i
$399$		0			0
$400$		0			0
$401$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$401$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$402$	1.41421			1.41421
$403$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$404$		0			0
$405$	−1.00000			−1.00000
$406$		−	1.00000i		−	1.00000i
$407$		−	1.41421i		−	1.41421i
$408$	1.00000	−	1.00000i	1.00000	−	1.00000i
$409$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$409$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$410$	1.41421			1.41421
$411$		−	1.41421i		−	1.41421i
$412$		0			0
$413$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$414$		0			0
$415$	−1.41421			−1.41421
$416$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$	−1.00000			−1.00000
$421$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$421$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$422$	−1.00000			−1.00000
$423$		0			0
$424$	0.707107	−	0.707107i	0.707107	−	0.707107i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	1.00000			1.00000
$430$		−	1.00000i		−	1.00000i
$431$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$431$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$432$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$433$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$433$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$434$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$435$	0.707107	+	0.707107i	0.707107	+	0.707107i
$436$	−1.00000			−1.00000
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$	1.00000			1.00000
$441$		0			0
$442$		−	1.41421i		−	1.41421i
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		−	1.41421i		−	1.41421i
$445$		0			0
$446$		0			0
$447$	0.707107	+	0.707107i	0.707107	+	0.707107i
$448$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$449$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$449$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$450$		0			0
$451$	−1.41421			−1.41421
$452$		0			0
$453$		0			0
$454$	1.00000	+	1.00000i	1.00000	+	1.00000i
$455$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$456$		0			0
$457$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$457$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$458$		0			0
$459$		−	1.41421i		−	1.41421i
$460$		0			0
$461$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$461$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$462$	1.00000			1.00000
$463$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$463$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$464$			1.00000i			1.00000i
$465$		−	1.00000i		−	1.00000i
$466$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$467$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$467$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$468$		0			0
$469$	1.00000	−	1.00000i	1.00000	−	1.00000i
$470$			1.00000i			1.00000i
$471$		−	1.41421i		−	1.41421i
$472$	1.00000	−	1.00000i	1.00000	−	1.00000i
$473$			1.00000i			1.00000i
$474$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$475$		0			0
$476$		−	1.41421i		−	1.41421i
$477$		0			0
$478$	1.00000	−	1.00000i	1.00000	−	1.00000i
$479$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$479$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$480$	1.00000			1.00000
$481$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$482$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$483$		0			0
$484$		0			0
$485$	1.00000	−	1.00000i	1.00000	−	1.00000i
$486$		0			0
$487$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$487$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$488$		0			0
$489$		−	1.00000i		−	1.00000i
$490$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$491$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$491$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$492$	−1.41421			−1.41421
$493$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$494$		0			0
$495$		0			0
$496$	0.707107	−	0.707107i	0.707107	−	0.707107i
$497$		0			0
$498$	1.41421			1.41421
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$			1.00000i			1.00000i
$501$		0			0
$502$	1.00000			1.00000
$503$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$503$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$509$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$510$	1.00000	+	1.00000i	1.00000	+	1.00000i
$511$		0			0
$512$	0.707107	+	0.707107i	0.707107	+	0.707107i
$513$		0			0
$514$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$515$		0			0
$516$			1.00000i			1.00000i
$517$		−	1.00000i		−	1.00000i
$518$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$519$		0			0
$520$	0.707107	−	0.707107i	0.707107	−	0.707107i
$521$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$521$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$523$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$524$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$525$		0			0
$526$			1.00000i			1.00000i
$527$	1.41421			1.41421
$528$	−1.00000			−1.00000
$529$	1.00000			1.00000
$530$	0.707107	+	0.707107i	0.707107	+	0.707107i
$531$		0			0
$532$		0			0
$533$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$534$		0			0
$535$		0			0
$536$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$537$		0			0
$538$	−1.41421			−1.41421
$539$	0.707107	−	0.707107i	0.707107	−	0.707107i
$540$	0.707107	−	0.707107i	0.707107	−	0.707107i
$541$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$541$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$542$			1.00000i			1.00000i
$543$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$544$			1.41421i			1.41421i
$545$		−	1.00000i		−	1.00000i
$546$	0.707107	−	0.707107i	0.707107	−	0.707107i
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	1.00000	+	1.00000i	1.00000	+	1.00000i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.00000			−1.00000
$554$		0			0
$555$	1.41421			1.41421
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$	0.707107	+	0.707107i	0.707107	+	0.707107i
$560$	0.707107	−	0.707107i	0.707107	−	0.707107i
$561$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$562$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$563$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$563$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$564$		−	1.00000i		−	1.00000i
$565$		0			0
$566$		0			0
$567$	0.707107	−	0.707107i	0.707107	−	0.707107i
$568$		0			0
$569$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$569$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$570$		0			0
$571$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$571$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$572$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$573$		0			0
$574$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$575$		0			0
$576$		0			0
$577$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$577$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$578$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$579$			1.41421i			1.41421i
$580$	−1.00000			−1.00000
$581$	1.00000	−	1.00000i	1.00000	−	1.00000i
$582$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$583$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$584$		0			0
$585$		0			0
$586$			1.41421i			1.41421i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	0.707107	−	0.707107i	0.707107	−	0.707107i
$589$		0			0
$590$	1.00000	+	1.00000i	1.00000	+	1.00000i
$591$		0			0
$592$	1.00000	+	1.00000i	1.00000	+	1.00000i
$593$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$593$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$594$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$595$	1.41421			1.41421
$596$	−1.00000			−1.00000
$597$	1.00000	−	1.00000i	1.00000	−	1.00000i
$598$		0			0
$599$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$599$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$600$		0			0
$601$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$601$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$602$	0.707107	+	0.707107i	0.707107	+	0.707107i
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$607$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$608$		0			0
$609$	−1.00000			−1.00000
$610$		0			0
$611$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$612$		0			0
$613$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$613$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$614$	1.00000			1.00000
$615$		−	1.41421i		−	1.41421i
$616$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$619$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$620$	0.707107	+	0.707107i	0.707107	+	0.707107i
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$625$	−1.00000			−1.00000
$626$	0.707107	−	0.707107i	0.707107	−	0.707107i
$627$		0			0
$628$	1.00000	+	1.00000i	1.00000	+	1.00000i
$629$			2.00000i			2.00000i
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$	1.00000			1.00000
$633$			1.00000i			1.00000i
$634$		0			0
$635$		0			0
$636$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$637$		−	1.00000i		−	1.00000i
$638$	1.00000			1.00000
$639$		0			0
$640$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$641$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$641$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$642$		0			0
$643$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$643$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$	−1.00000			−1.00000
$646$		0			0
$647$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$647$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$648$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$649$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$650$		0			0
$651$	0.707107	+	0.707107i	0.707107	+	0.707107i
$652$	0.707107	+	0.707107i	0.707107	+	0.707107i
$653$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$653$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$654$			1.00000i			1.00000i
$655$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$656$	1.00000	−	1.00000i	1.00000	−	1.00000i
$657$		0			0
$658$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$659$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$659$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$660$		−	1.00000i		−	1.00000i
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−1.00000			−1.00000
$663$	−1.41421			−1.41421
$664$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$671$		0			0
$672$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$673$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$673$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$			1.41421i			1.41421i
$680$	−1.41421			−1.41421
$681$	1.00000	−	1.00000i	1.00000	−	1.00000i
$682$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$686$		−	1.00000i		−	1.00000i
$687$		0			0
$688$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$689$	−1.00000			−1.00000
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$695$		0			0
$696$	1.00000			1.00000
$697$	2.00000			2.00000
$698$	0.707107	−	0.707107i	0.707107	−	0.707107i
$699$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$700$		0			0
$701$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$701$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$702$			1.00000i			1.00000i
$703$		0			0
$704$	0.707107	−	0.707107i	0.707107	−	0.707107i
$705$	1.00000			1.00000
$706$		0			0
$707$		0			0
$708$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$709$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$709$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$	−1.41421			−1.41421
$715$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$716$		0			0
$717$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$718$			1.00000i			1.00000i
$719$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$719$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$720$		0			0
$721$		0			0
$722$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$723$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$724$	1.00000			1.00000
$725$		0			0
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$			1.00000i			1.00000i
$729$			1.00000i			1.00000i
$730$		0			0
$731$		−	1.41421i		−	1.41421i
$732$		0			0
$733$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$733$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$734$	−2.00000			−2.00000
$735$	0.707107	+	0.707107i	0.707107	+	0.707107i
$736$		0			0
$737$	1.00000	+	1.00000i	1.00000	+	1.00000i
$738$		0			0
$739$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$739$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$740$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$741$		0			0
$742$	−1.00000			−1.00000
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$745$		−	1.00000i		−	1.00000i
$746$	0.707107	+	0.707107i	0.707107	+	0.707107i
$747$		0			0
$748$	1.41421			1.41421
$749$		0			0
$750$	1.00000			1.00000
$751$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$751$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$752$	0.707107	+	0.707107i	0.707107	+	0.707107i
$753$		−	1.00000i		−	1.00000i
$754$	0.707107	−	0.707107i	0.707107	−	0.707107i
$755$		0			0
$756$			1.00000i			1.00000i
$757$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$757$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	0.707107	+	0.707107i	0.707107	+	0.707107i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.41421			−1.41421
$768$	0.707107	−	0.707107i	0.707107	−	0.707107i
$769$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$769$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$770$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$771$	0.707107	+	0.707107i	0.707107	+	0.707107i
$772$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$773$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$773$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		−	1.41421i		−	1.41421i
$777$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$778$		0			0
$779$		0			0
$780$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$781$		0			0
$782$		0			0
$783$	0.707107	−	0.707107i	0.707107	−	0.707107i
$784$			1.00000i			1.00000i
$785$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$786$	1.41421	+	1.41421i	1.41421	+	1.41421i
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		0			0
$789$	1.00000			1.00000
$790$			1.00000i			1.00000i
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$795$	0.707107	−	0.707107i	0.707107	−	0.707107i
$796$			1.41421i			1.41421i
$797$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$797$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$798$		0			0
$799$			1.41421i			1.41421i
$800$		0			0
$801$		0			0
$802$	0.707107	+	0.707107i	0.707107	+	0.707107i
$803$		0			0
$804$	1.00000	+	1.00000i	1.00000	+	1.00000i
$805$		0			0
$806$	−1.00000			−1.00000
$807$			1.41421i			1.41421i
$808$		0			0
$809$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$809$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$810$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	0.707107	−	0.707107i	0.707107	−	0.707107i
$813$	1.00000			1.00000
$814$	1.00000	−	1.00000i	1.00000	−	1.00000i
$815$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$816$	1.41421			1.41421
$817$		0			0
$818$		0			0
$819$		0			0
$820$	1.00000	+	1.00000i	1.00000	+	1.00000i
$821$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$821$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$822$	1.00000	−	1.00000i	1.00000	−	1.00000i
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$	−1.41421			−1.41421
$827$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$827$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$828$		0			0
$829$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$829$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$830$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$831$		0			0
$832$		−	1.00000i		−	1.00000i
$833$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.00000			−1.00000
$838$		0			0
$839$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$839$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$840$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$841$	−1.00000			−1.00000
$842$		0			0
$843$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$844$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	1.00000			1.00000
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$853$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$857$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$858$	0.707107	+	0.707107i	0.707107	+	0.707107i
$859$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$859$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$860$	0.707107	−	0.707107i	0.707107	−	0.707107i
$861$	1.00000	+	1.00000i	1.00000	+	1.00000i
$862$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$863$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$863$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$864$		−	1.00000i		−	1.00000i
$865$		0			0
$866$	1.41421			1.41421
$867$	0.707107	+	0.707107i	0.707107	+	0.707107i
$868$	−1.00000			−1.00000
$869$		−	1.00000i		−	1.00000i
$870$			1.00000i			1.00000i
$871$	1.41421			1.41421
$872$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$873$		0			0
$874$		0			0
$875$	0.707107	−	0.707107i	0.707107	−	0.707107i
$876$		0			0
$877$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$877$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$	1.41421			1.41421
$880$	0.707107	+	0.707107i	0.707107	+	0.707107i
$881$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$881$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$882$		0			0
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$	1.00000	−	1.00000i	1.00000	−	1.00000i
$885$	1.00000	−	1.00000i	1.00000	−	1.00000i
$886$		0			0
$887$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$887$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$888$	1.00000	−	1.00000i	1.00000	−	1.00000i
$889$		0			0
$890$		0			0
$891$	0.707107	+	0.707107i	0.707107	+	0.707107i
$892$		0			0
$893$		0			0
$894$			1.00000i			1.00000i
$895$		0			0
$896$		−	1.00000i		−	1.00000i
$897$		0			0
$898$		0			0
$899$	0.707107	+	0.707107i	0.707107	+	0.707107i
$900$		0			0
$901$	1.00000	+	1.00000i	1.00000	+	1.00000i
$902$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$903$	0.707107	−	0.707107i	0.707107	−	0.707107i
$904$		0			0
$905$			1.00000i			1.00000i
$906$		0			0
$907$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$907$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$908$			1.41421i			1.41421i
$909$		0			0
$910$	−1.00000			−1.00000
$911$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$911$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$912$		0			0
$913$	1.00000	+	1.00000i	1.00000	+	1.00000i
$914$	1.41421	−	1.41421i	1.41421	−	1.41421i
$915$		0			0
$916$		0			0
$917$	2.00000			2.00000
$918$	1.00000	−	1.00000i	1.00000	−	1.00000i
$919$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$919$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$920$		0			0
$921$		−	1.00000i		−	1.00000i
$922$		0			0
$923$		0			0
$924$	0.707107	+	0.707107i	0.707107	+	0.707107i
$925$		0			0
$926$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$927$		0			0
$928$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$	0.707107	−	0.707107i	0.707107	−	0.707107i
$931$		0			0
$932$		−	1.00000i		−	1.00000i
$933$		0			0
$934$	−1.00000			−1.00000
$935$			1.41421i			1.41421i
$936$		0			0
$937$	2.00000			2.00000			1.00000			$0$
$937$	2.00000			2.00000			1.00000			$0$
$938$	1.41421			1.41421
$939$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$940$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$941$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$941$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$942$	1.00000	−	1.00000i	1.00000	−	1.00000i
$943$		0			0
$944$	1.41421			1.41421
$945$	−1.00000			−1.00000
$946$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$947$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$947$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$948$		−	1.00000i		−	1.00000i
$949$		0			0
$950$		0			0
$951$		0			0
$952$	1.00000	−	1.00000i	1.00000	−	1.00000i
$953$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$953$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$954$		0			0
$955$		0			0
$956$	1.41421			1.41421
$957$		−	1.00000i		−	1.00000i
$958$		−	1.00000i		−	1.00000i
$959$		−	1.41421i		−	1.41421i
$960$	0.707107	+	0.707107i	0.707107	+	0.707107i
$961$		0			0
$962$		−	1.41421i		−	1.41421i
$963$		0			0
$964$		−	1.00000i		−	1.00000i
$965$	1.00000	−	1.00000i	1.00000	−	1.00000i
$966$		0			0
$967$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$967$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$968$		0			0
$969$		0			0
$970$	1.41421			1.41421
$971$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$971$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$972$		0			0
$973$		0			0
$974$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$975$		0			0
$976$		0			0
$977$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$977$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$978$	0.707107	−	0.707107i	0.707107	−	0.707107i
$979$		0			0
$980$	−1.00000			−1.00000
$981$		0			0
$982$	1.00000			1.00000
$983$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$983$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$984$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$985$		0			0
$986$	−1.41421			−1.41421
$987$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$991$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$992$	1.00000			1.00000
$993$			1.00000i			1.00000i
$994$		0			0
$995$	−1.41421			−1.41421
$996$	1.00000	+	1.00000i	1.00000	+	1.00000i
$997$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$997$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$998$		0			0
$999$		−	1.41421i		−	1.41421i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	812.1.l.c.307.2	yes	4
4.3	odd	2	inner	812.1.l.c.307.1	✓	4
7.6	odd	2		812.1.l.d.307.2	yes	4
28.27	even	2		812.1.l.d.307.1	yes	4
29.12	odd	4		812.1.l.d.447.1	yes	4
116.99	even	4		812.1.l.d.447.2	yes	4
203.41	even	4	inner	812.1.l.c.447.1	yes	4
812.447	odd	4	inner	812.1.l.c.447.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
812.1.l.c.307.1	✓	4	4.3	odd	2	inner
812.1.l.c.307.2	yes	4	1.1	even	1	trivial
812.1.l.c.447.1	yes	4	203.41	even	4	inner
812.1.l.c.447.2	yes	4	812.447	odd	4	inner
812.1.l.d.307.1	yes	4	28.27	even	2
812.1.l.d.307.2	yes	4	7.6	odd	2
812.1.l.d.447.1	yes	4	29.12	odd	4
812.1.l.d.447.2	yes	4	116.99	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 812 = 2^{2} \cdot 7 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	812.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{2} +(0.707107 - 0.707107i) q^{3} +1.00000i q^{4} -1.00000 q^{5} +1.00000 q^{6} +(0.707107 - 0.707107i) q^{7} +(-0.707107 + 0.707107i) q^{8} +(-0.707107 - 0.707107i) q^{10} +(0.707107 + 0.707107i) q^{11} +(0.707107 + 0.707107i) q^{12} +1.00000 q^{13} +1.00000 q^{14} +(-0.707107 + 0.707107i) q^{15} -1.00000 q^{16} +(-1.00000 - 1.00000i) q^{17} -1.00000i q^{20} -1.00000i q^{21} +1.00000i q^{22} +1.00000i q^{24} +(0.707107 + 0.707107i) q^{26} +(0.707107 + 0.707107i) q^{27} +(0.707107 + 0.707107i) q^{28} -1.00000i q^{29} -1.00000 q^{30} +(-0.707107 + 0.707107i) q^{31} +(-0.707107 - 0.707107i) q^{32} +1.00000 q^{33} -1.41421i q^{34} +(-0.707107 + 0.707107i) q^{35} +(-1.00000 - 1.00000i) q^{37} +(0.707107 - 0.707107i) q^{39} +(0.707107 - 0.707107i) q^{40} +(-1.00000 + 1.00000i) q^{41} +(0.707107 - 0.707107i) q^{42} +(0.707107 + 0.707107i) q^{43} +(-0.707107 + 0.707107i) q^{44} +(-0.707107 - 0.707107i) q^{47} +(-0.707107 + 0.707107i) q^{48} -1.00000i q^{49} -1.41421 q^{51} +1.00000i q^{52} -1.00000 q^{53} +1.00000i q^{54} +(-0.707107 - 0.707107i) q^{55} +1.00000i q^{56} +(0.707107 - 0.707107i) q^{58} -1.41421 q^{59} +(-0.707107 - 0.707107i) q^{60} -1.00000 q^{62} -1.00000i q^{64} -1.00000 q^{65} +(0.707107 + 0.707107i) q^{66} +1.41421 q^{67} +(1.00000 - 1.00000i) q^{68} -1.00000 q^{70} -1.41421i q^{74} +1.00000 q^{77} +1.00000 q^{78} +(-0.707107 - 0.707107i) q^{79} +1.00000 q^{80} +1.00000 q^{81} -1.41421 q^{82} +1.41421 q^{83} +1.00000 q^{84} +(1.00000 + 1.00000i) q^{85} +1.00000i q^{86} +(-0.707107 - 0.707107i) q^{87} -1.00000 q^{88} +(0.707107 - 0.707107i) q^{91} +1.00000i q^{93} -1.00000i q^{94} -1.00000 q^{96} +(-1.00000 + 1.00000i) q^{97} +(0.707107 - 0.707107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} + 4 q^{6} + 4 q^{13} + 4 q^{14} - 4 q^{16} - 4 q^{17} - 4 q^{30} + 4 q^{33} - 4 q^{37} - 4 q^{41} - 4 q^{53} - 4 q^{62} - 4 q^{65} + 4 q^{68} - 4 q^{70} + 4 q^{77} + 4 q^{78} + 4 q^{80} + 4 q^{81}+ \cdots - 4 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 + 4 * q^6 + 4 * q^13 + 4 * q^14 - 4 * q^16 - 4 * q^17 - 4 * q^30 + 4 * q^33 - 4 * q^37 - 4 * q^41 - 4 * q^53 - 4 * q^62 - 4 * q^65 + 4 * q^68 - 4 * q^70 + 4 * q^77 + 4 * q^78 + 4 * q^80 + 4 * q^81 + 4 * q^84 + 4 * q^85 - 4 * q^88 - 4 * q^96 - 4 * q^97

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