LMFDB - Embedded newform 1680.2.bg.k.961.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(961,1680)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1680.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.bg (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1680.961
Dual form			1680.2.bg.k.1201.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-2.00000 - 1.73205i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-2.00000 - 1.73205i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(-0.500000 - 0.866025i) q^{11} +7.00000 q^{13} -1.00000 q^{15} +(2.00000 + 3.46410i) q^{17} +(0.500000 - 0.866025i) q^{19} +(0.500000 - 2.59808i) q^{21} +(0.500000 - 0.866025i) q^{23} +(-0.500000 - 0.866025i) q^{25} -1.00000 q^{27} -8.00000 q^{29} +(3.00000 + 5.19615i) q^{31} +(0.500000 - 0.866025i) q^{33} +(2.50000 - 0.866025i) q^{35} +(1.50000 - 2.59808i) q^{37} +(3.50000 + 6.06218i) q^{39} +9.00000 q^{41} +4.00000 q^{43} +(-0.500000 - 0.866025i) q^{45} +(-1.50000 + 2.59808i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-2.00000 + 3.46410i) q^{51} +(0.500000 + 0.866025i) q^{53} +1.00000 q^{55} +1.00000 q^{57} +(6.00000 + 10.3923i) q^{59} +(2.00000 - 3.46410i) q^{61} +(2.50000 - 0.866025i) q^{63} +(-3.50000 + 6.06218i) q^{65} +(6.00000 + 10.3923i) q^{67} +1.00000 q^{69} +14.0000 q^{71} +(7.00000 + 12.1244i) q^{73} +(0.500000 - 0.866025i) q^{75} +(-0.500000 + 2.59808i) q^{77} +(2.00000 - 3.46410i) q^{79} +(-0.500000 - 0.866025i) q^{81} -12.0000 q^{83} -4.00000 q^{85} +(-4.00000 - 6.92820i) q^{87} +(1.00000 - 1.73205i) q^{89} +(-14.0000 - 12.1244i) q^{91} +(-3.00000 + 5.19615i) q^{93} +(0.500000 + 0.866025i) q^{95} -16.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - q^{5} - 4 q^{7} - q^{9}+O(q^{10}) $ 2 * q + q^3 - q^5 - 4 * q^7 - q^9	$ 2 q + q^{3} - q^{5} - 4 q^{7} - q^{9} - q^{11} + 14 q^{13} - 2 q^{15} + 4 q^{17} + q^{19} + q^{21} + q^{23} - q^{25} - 2 q^{27} - 16 q^{29} + 6 q^{31} + q^{33} + 5 q^{35} + 3 q^{37} + 7 q^{39} + 18 q^{41} + 8 q^{43} - q^{45} - 3 q^{47} + 2 q^{49} - 4 q^{51} + q^{53} + 2 q^{55} + 2 q^{57} + 12 q^{59} + 4 q^{61} + 5 q^{63} - 7 q^{65} + 12 q^{67} + 2 q^{69} + 28 q^{71} + 14 q^{73} + q^{75} - q^{77} + 4 q^{79} - q^{81} - 24 q^{83} - 8 q^{85} - 8 q^{87} + 2 q^{89} - 28 q^{91} - 6 q^{93} + q^{95} - 32 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + q^3 - q^5 - 4 * q^7 - q^9 - q^11 + 14 * q^13 - 2 * q^15 + 4 * q^17 + q^19 + q^21 + q^23 - q^25 - 2 * q^27 - 16 * q^29 + 6 * q^31 + q^33 + 5 * q^35 + 3 * q^37 + 7 * q^39 + 18 * q^41 + 8 * q^43 - q^45 - 3 * q^47 + 2 * q^49 - 4 * q^51 + q^53 + 2 * q^55 + 2 * q^57 + 12 * q^59 + 4 * q^61 + 5 * q^63 - 7 * q^65 + 12 * q^67 + 2 * q^69 + 28 * q^71 + 14 * q^73 + q^75 - q^77 + 4 * q^79 - q^81 - 24 * q^83 - 8 * q^85 - 8 * q^87 + 2 * q^89 - 28 * q^91 - 6 * q^93 + q^95 - 32 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$	$1$

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			0
$2$		0			0
$3$	0.500000	+	0.866025i	0.288675	+	0.500000i
$3$	0.500000	+	0.866025i	0.288675	+	0.500000i
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$	−0.500000	−	0.866025i	−0.150756	−	0.261116i	0.780750	−	0.624844i	$-0.214837\pi$
$11$	−0.500000	−	0.866025i	−0.150756	−	0.261116i	−0.931505	+	0.363727i	$0.881504\pi$
$12$		0			0
$13$	7.00000			1.94145			0.970725	−	0.240192i	$-0.0772105\pi$
$13$	7.00000			1.94145			0.970725	+	0.240192i	$0.0772105\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$	2.00000	+	3.46410i	0.485071	+	0.840168i	0.999853	−	0.0171533i	$-0.00546033\pi$
$17$	2.00000	+	3.46410i	0.485071	+	0.840168i	−0.514782	+	0.857321i	$0.672127\pi$
$18$		0			0
$19$	0.500000	−	0.866025i	0.114708	−	0.198680i	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	0.500000	−	0.866025i	0.114708	−	0.198680i	0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$	0.500000	−	2.59808i	0.109109	−	0.566947i
$22$		0			0
$23$	0.500000	−	0.866025i	0.104257	−	0.180579i	−0.809177	−	0.587565i	$-0.800087\pi$
$23$	0.500000	−	0.866025i	0.104257	−	0.180579i	0.913434	+	0.406986i	$0.133420\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−8.00000			−1.48556			−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000			−1.48556			−0.742781	+	0.669534i	$0.766494\pi$
$30$		0			0
$31$	3.00000	+	5.19615i	0.538816	+	0.933257i	0.998968	+	0.0454165i	$0.0144615\pi$
$31$	3.00000	+	5.19615i	0.538816	+	0.933257i	−0.460152	+	0.887840i	$0.652205\pi$
$32$		0			0
$33$	0.500000	−	0.866025i	0.0870388	−	0.150756i
$34$		0			0
$35$	2.50000	−	0.866025i	0.422577	−	0.146385i
$36$		0			0
$37$	1.50000	−	2.59808i	0.246598	−	0.427121i	−0.715981	−	0.698119i	$-0.754020\pi$
$37$	1.50000	−	2.59808i	0.246598	−	0.427121i	0.962580	+	0.270998i	$0.0873538\pi$
$38$		0			0
$39$	3.50000	+	6.06218i	0.560449	+	0.970725i
$40$		0			0
$41$	9.00000			1.40556			0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000			1.40556			0.702782	+	0.711405i	$0.251941\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$	−0.500000	−	0.866025i	−0.0745356	−	0.129099i
$46$		0			0
$47$	−1.50000	+	2.59808i	−0.218797	+	0.378968i	−0.954441	−	0.298401i	$-0.903547\pi$
$47$	−1.50000	+	2.59808i	−0.218797	+	0.378968i	0.735643	+	0.677369i	$0.236880\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$	−2.00000	+	3.46410i	−0.280056	+	0.485071i
$52$		0			0
$53$	0.500000	+	0.866025i	0.0686803	+	0.118958i	0.898321	−	0.439340i	$-0.144788\pi$
$53$	0.500000	+	0.866025i	0.0686803	+	0.118958i	−0.829640	+	0.558298i	$0.811454\pi$
$54$		0			0
$55$	1.00000			0.134840
$56$		0			0
$57$	1.00000			0.132453
$58$		0			0
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	0.931282	+	0.364299i	$0.118692\pi$
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	−0.150148	+	0.988663i	$0.547975\pi$
$60$		0			0
$61$	2.00000	−	3.46410i	0.256074	−	0.443533i	−0.709113	−	0.705095i	$-0.750904\pi$
$61$	2.00000	−	3.46410i	0.256074	−	0.443533i	0.965187	+	0.261562i	$0.0842377\pi$
$62$		0			0
$63$	2.50000	−	0.866025i	0.314970	−	0.109109i
$64$		0			0
$65$	−3.50000	+	6.06218i	−0.434122	+	0.751921i
$66$		0			0
$67$	6.00000	+	10.3923i	0.733017	+	1.26962i	0.955588	+	0.294706i	$0.0952216\pi$
$67$	6.00000	+	10.3923i	0.733017	+	1.26962i	−0.222571	+	0.974916i	$0.571445\pi$
$68$		0			0
$69$	1.00000			0.120386
$70$		0			0
$71$	14.0000			1.66149			0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000			1.66149			0.830747	+	0.556650i	$0.187914\pi$
$72$		0			0
$73$	7.00000	+	12.1244i	0.819288	+	1.41905i	0.906208	+	0.422833i	$0.138964\pi$
$73$	7.00000	+	12.1244i	0.819288	+	1.41905i	−0.0869195	+	0.996215i	$0.527702\pi$
$74$		0			0
$75$	0.500000	−	0.866025i	0.0577350	−	0.100000i
$76$		0			0
$77$	−0.500000	+	2.59808i	−0.0569803	+	0.296078i
$78$		0			0
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	0.956325	+	0.292306i	$0.0944227\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$	−4.00000			−0.433861
$86$		0			0
$87$	−4.00000	−	6.92820i	−0.428845	−	0.742781i
$88$		0			0
$89$	1.00000	−	1.73205i	0.106000	−	0.183597i	−0.808146	−	0.588982i	$-0.799529\pi$
$89$	1.00000	−	1.73205i	0.106000	−	0.183597i	0.914146	+	0.405385i	$0.132862\pi$
$90$		0			0
$91$	−14.0000	−	12.1244i	−1.46760	−	1.27098i
$92$		0			0
$93$	−3.00000	+	5.19615i	−0.311086	+	0.538816i
$94$		0			0
$95$	0.500000	+	0.866025i	0.0512989	+	0.0888523i
$96$		0			0
$97$	−16.0000			−1.62455			−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000			−1.62455			−0.812277	+	0.583272i	$0.801772\pi$
$98$		0			0
$99$	1.00000			0.100504
$100$		0			0
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	8.00000	−	13.8564i	0.788263	−	1.36531i	−0.138767	−	0.990325i	$-0.544314\pi$
$103$	8.00000	−	13.8564i	0.788263	−	1.36531i	0.927030	−	0.374987i	$-0.122353\pi$
$104$		0			0
$105$	2.00000	+	1.73205i	0.195180	+	0.169031i
$106$		0			0
$107$	−9.00000	+	15.5885i	−0.870063	+	1.50699i	−0.00813215	+	0.999967i	$0.502589\pi$
$107$	−9.00000	+	15.5885i	−0.870063	+	1.50699i	−0.861931	+	0.507026i	$0.830745\pi$
$108$		0			0
$109$	5.00000	+	8.66025i	0.478913	+	0.829502i	0.999708	−	0.0241802i	$-0.00769755\pi$
$109$	5.00000	+	8.66025i	0.478913	+	0.829502i	−0.520794	+	0.853682i	$0.674364\pi$
$110$		0			0
$111$	3.00000			0.284747
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$	0.500000	+	0.866025i	0.0466252	+	0.0807573i
$116$		0			0
$117$	−3.50000	+	6.06218i	−0.323575	+	0.560449i
$118$		0			0
$119$	2.00000	−	10.3923i	0.183340	−	0.952661i
$120$		0			0
$121$	5.00000	−	8.66025i	0.454545	−	0.787296i
$122$		0			0
$123$	4.50000	+	7.79423i	0.405751	+	0.702782i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−5.00000			−0.443678			−0.221839	−	0.975083i	$-0.571206\pi$
$127$	−5.00000			−0.443678			−0.221839	+	0.975083i	$0.571206\pi$
$128$		0			0
$129$	2.00000	+	3.46410i	0.176090	+	0.304997i
$130$		0			0
$131$	−6.50000	+	11.2583i	−0.567908	+	0.983645i	0.428865	+	0.903369i	$0.358914\pi$
$131$	−6.50000	+	11.2583i	−0.567908	+	0.983645i	−0.996773	+	0.0802763i	$0.974420\pi$
$132$		0			0
$133$	−2.50000	+	0.866025i	−0.216777	+	0.0750939i
$134$		0			0
$135$	0.500000	−	0.866025i	0.0430331	−	0.0745356i
$136$		0			0
$137$	1.00000	+	1.73205i	0.0854358	+	0.147979i	0.905577	−	0.424182i	$-0.139438\pi$
$137$	1.00000	+	1.73205i	0.0854358	+	0.147979i	−0.820141	+	0.572161i	$0.806105\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$	−3.00000			−0.252646
$142$		0			0
$143$	−3.50000	−	6.06218i	−0.292685	−	0.506945i
$144$		0			0
$145$	4.00000	−	6.92820i	0.332182	−	0.575356i
$146$		0			0
$147$	−5.50000	+	4.33013i	−0.453632	+	0.357143i
$148$		0			0
$149$	2.00000	−	3.46410i	0.163846	−	0.283790i	−0.772399	−	0.635138i	$-0.780943\pi$
$149$	2.00000	−	3.46410i	0.163846	−	0.283790i	0.936245	+	0.351348i	$0.114277\pi$
$150$		0			0
$151$	−1.00000	−	1.73205i	−0.0813788	−	0.140952i	0.822464	−	0.568818i	$-0.192599\pi$
$151$	−1.00000	−	1.73205i	−0.0813788	−	0.140952i	−0.903842	+	0.427865i	$0.859266\pi$
$152$		0			0
$153$	−4.00000			−0.323381
$154$		0			0
$155$	−6.00000			−0.481932
$156$		0			0
$157$	−7.50000	−	12.9904i	−0.598565	−	1.03675i	−0.993033	−	0.117836i	$-0.962404\pi$
$157$	−7.50000	−	12.9904i	−0.598565	−	1.03675i	0.394468	−	0.918910i	$-0.370929\pi$
$158$		0			0
$159$	−0.500000	+	0.866025i	−0.0396526	+	0.0686803i
$160$		0			0
$161$	−2.50000	+	0.866025i	−0.197028	+	0.0682524i
$162$		0			0
$163$	4.00000	−	6.92820i	0.313304	−	0.542659i	−0.665771	−	0.746156i	$-0.731897\pi$
$163$	4.00000	−	6.92820i	0.313304	−	0.542659i	0.979076	+	0.203497i	$0.0652307\pi$
$164$		0			0
$165$	0.500000	+	0.866025i	0.0389249	+	0.0674200i
$166$		0			0
$167$	5.00000			0.386912			0.193456	−	0.981109i	$-0.438030\pi$
$167$	5.00000			0.386912			0.193456	+	0.981109i	$0.438030\pi$
$168$		0			0
$169$	36.0000			2.76923
$170$		0			0
$171$	0.500000	+	0.866025i	0.0382360	+	0.0662266i
$172$		0			0
$173$	10.5000	−	18.1865i	0.798300	−	1.38270i	−0.122422	−	0.992478i	$-0.539066\pi$
$173$	10.5000	−	18.1865i	0.798300	−	1.38270i	0.920722	−	0.390218i	$-0.127601\pi$
$174$		0			0
$175$	−0.500000	+	2.59808i	−0.0377964	+	0.196396i
$176$		0			0
$177$	−6.00000	+	10.3923i	−0.450988	+	0.781133i
$178$		0			0
$179$	6.50000	+	11.2583i	0.485833	+	0.841487i	0.999867	−	0.0162823i	$-0.00518305\pi$
$179$	6.50000	+	11.2583i	0.485833	+	0.841487i	−0.514035	+	0.857769i	$0.671850\pi$
$180$		0			0
$181$	−12.0000			−0.891953			−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000			−0.891953			−0.445976	+	0.895045i	$0.647144\pi$
$182$		0			0
$183$	4.00000			0.295689
$184$		0			0
$185$	1.50000	+	2.59808i	0.110282	+	0.191014i
$186$		0			0
$187$	2.00000	−	3.46410i	0.146254	−	0.253320i
$188$		0			0
$189$	2.00000	+	1.73205i	0.145479	+	0.125988i
$190$		0			0
$191$	−5.00000	+	8.66025i	−0.361787	+	0.626634i	−0.988255	−	0.152813i	$-0.951167\pi$
$191$	−5.00000	+	8.66025i	−0.361787	+	0.626634i	0.626468	+	0.779447i	$0.284500\pi$
$192$		0			0
$193$	−13.0000	−	22.5167i	−0.935760	−	1.62078i	−0.773272	−	0.634074i	$-0.781381\pi$
$193$	−13.0000	−	22.5167i	−0.935760	−	1.62078i	−0.162488	−	0.986710i	$-0.551952\pi$
$194$		0			0
$195$	−7.00000			−0.501280
$196$		0			0
$197$	−3.00000			−0.213741			−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000			−0.213741			−0.106871	+	0.994273i	$0.534083\pi$
$198$		0			0
$199$	−6.00000	−	10.3923i	−0.425329	−	0.736691i	0.571122	−	0.820865i	$-0.306508\pi$
$199$	−6.00000	−	10.3923i	−0.425329	−	0.736691i	−0.996451	+	0.0841740i	$0.973175\pi$
$200$		0			0
$201$	−6.00000	+	10.3923i	−0.423207	+	0.733017i
$202$		0			0
$203$	16.0000	+	13.8564i	1.12298	+	0.972529i
$204$		0			0
$205$	−4.50000	+	7.79423i	−0.314294	+	0.544373i
$206$		0			0
$207$	0.500000	+	0.866025i	0.0347524	+	0.0601929i
$208$		0			0
$209$	−1.00000			−0.0691714
$210$		0			0
$211$	15.0000			1.03264			0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000			1.03264			0.516321	+	0.856395i	$0.327301\pi$
$212$		0			0
$213$	7.00000	+	12.1244i	0.479632	+	0.830747i
$214$		0			0
$215$	−2.00000	+	3.46410i	−0.136399	+	0.236250i
$216$		0			0
$217$	3.00000	−	15.5885i	0.203653	−	1.05821i
$218$		0			0
$219$	−7.00000	+	12.1244i	−0.473016	+	0.819288i
$220$		0			0
$221$	14.0000	+	24.2487i	0.941742	+	1.63114i
$222$		0			0
$223$	4.00000			0.267860			0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000			0.267860			0.133930	+	0.990991i	$0.457240\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−10.0000	−	17.3205i	−0.663723	−	1.14960i	−0.979630	−	0.200812i	$-0.935642\pi$
$227$	−10.0000	−	17.3205i	−0.663723	−	1.14960i	0.315906	−	0.948790i	$-0.397691\pi$
$228$		0			0
$229$	11.0000	−	19.0526i	0.726900	−	1.25903i	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	11.0000	−	19.0526i	0.726900	−	1.25903i	0.958187	−	0.286143i	$-0.0923732\pi$
$230$		0			0
$231$	−2.50000	+	0.866025i	−0.164488	+	0.0569803i
$232$		0			0
$233$	−13.0000	+	22.5167i	−0.851658	+	1.47512i	0.0280525	+	0.999606i	$0.491069\pi$
$233$	−13.0000	+	22.5167i	−0.851658	+	1.47512i	−0.879711	+	0.475509i	$0.842264\pi$
$234$		0			0
$235$	−1.50000	−	2.59808i	−0.0978492	−	0.169480i
$236$		0			0
$237$	4.00000			0.259828
$238$		0			0
$239$	6.00000			0.388108			0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000			0.388108			0.194054	+	0.980991i	$0.437836\pi$
$240$		0			0
$241$	−3.50000	−	6.06218i	−0.225455	−	0.390499i	0.731001	−	0.682376i	$-0.239053\pi$
$241$	−3.50000	−	6.06218i	−0.225455	−	0.390499i	−0.956456	+	0.291877i	$0.905720\pi$
$242$		0			0
$243$	0.500000	−	0.866025i	0.0320750	−	0.0555556i
$244$		0			0
$245$	−6.50000	−	2.59808i	−0.415270	−	0.165985i
$246$		0			0
$247$	3.50000	−	6.06218i	0.222700	−	0.385727i
$248$		0			0
$249$	−6.00000	−	10.3923i	−0.380235	−	0.658586i
$250$		0			0
$251$	3.00000			0.189358			0.0946792	−	0.995508i	$-0.469817\pi$
$251$	3.00000			0.189358			0.0946792	+	0.995508i	$0.469817\pi$
$252$		0			0
$253$	−1.00000			−0.0628695
$254$		0			0
$255$	−2.00000	−	3.46410i	−0.125245	−	0.216930i
$256$		0			0
$257$	4.00000	−	6.92820i	0.249513	−	0.432169i	−0.713878	−	0.700270i	$-0.753063\pi$
$257$	4.00000	−	6.92820i	0.249513	−	0.432169i	0.963391	+	0.268101i	$0.0863961\pi$
$258$		0			0
$259$	−7.50000	+	2.59808i	−0.466027	+	0.161437i
$260$		0			0
$261$	4.00000	−	6.92820i	0.247594	−	0.428845i
$262$		0			0
$263$	−8.00000	−	13.8564i	−0.493301	−	0.854423i	0.506669	−	0.862141i	$-0.330877\pi$
$263$	−8.00000	−	13.8564i	−0.493301	−	0.854423i	−0.999970	+	0.00771799i	$0.997543\pi$
$264$		0			0
$265$	−1.00000			−0.0614295
$266$		0			0
$267$	2.00000			0.122398
$268$		0			0
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	−0.513905	−	0.857847i	$-0.671801\pi$
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	0.999870	+	0.0161307i	$0.00513477\pi$
$272$		0			0
$273$	3.50000	−	18.1865i	0.211830	−	1.10070i
$274$		0			0
$275$	−0.500000	+	0.866025i	−0.0301511	+	0.0522233i
$276$		0			0
$277$	1.00000	+	1.73205i	0.0600842	+	0.104069i	0.894503	−	0.447062i	$-0.147530\pi$
$277$	1.00000	+	1.73205i	0.0600842	+	0.104069i	−0.834419	+	0.551131i	$0.814196\pi$
$278$		0			0
$279$	−6.00000			−0.359211
$280$		0			0
$281$	3.00000			0.178965			0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000			0.178965			0.0894825	+	0.995988i	$0.471479\pi$
$282$		0			0
$283$	−1.00000	−	1.73205i	−0.0594438	−	0.102960i	0.834772	−	0.550596i	$-0.185599\pi$
$283$	−1.00000	−	1.73205i	−0.0594438	−	0.102960i	−0.894216	+	0.447636i	$0.852266\pi$
$284$		0			0
$285$	−0.500000	+	0.866025i	−0.0296174	+	0.0512989i
$286$		0			0
$287$	−18.0000	−	15.5885i	−1.06251	−	0.920158i
$288$		0			0
$289$	0.500000	−	0.866025i	0.0294118	−	0.0509427i
$290$		0			0
$291$	−8.00000	−	13.8564i	−0.468968	−	0.812277i
$292$		0			0
$293$	9.00000			0.525786			0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000			0.525786			0.262893	+	0.964825i	$0.415323\pi$
$294$		0			0
$295$	−12.0000			−0.698667
$296$		0			0
$297$	0.500000	+	0.866025i	0.0290129	+	0.0502519i
$298$		0			0
$299$	3.50000	−	6.06218i	0.202410	−	0.350585i
$300$		0			0
$301$	−8.00000	−	6.92820i	−0.461112	−	0.399335i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.00000	+	3.46410i	0.114520	+	0.198354i
$306$		0			0
$307$	−8.00000			−0.456584			−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000			−0.456584			−0.228292	+	0.973593i	$0.573314\pi$
$308$		0			0
$309$	16.0000			0.910208
$310$		0			0
$311$	8.00000	+	13.8564i	0.453638	+	0.785725i	0.998609	−	0.0527306i	$-0.0167924\pi$
$311$	8.00000	+	13.8564i	0.453638	+	0.785725i	−0.544970	+	0.838455i	$0.683459\pi$
$312$		0			0
$313$	12.0000	−	20.7846i	0.678280	−	1.17482i	−0.297218	−	0.954810i	$-0.596059\pi$
$313$	12.0000	−	20.7846i	0.678280	−	1.17482i	0.975499	−	0.220006i	$-0.0706077\pi$
$314$		0			0
$315$	−0.500000	+	2.59808i	−0.0281718	+	0.146385i
$316$		0			0
$317$	−5.00000	+	8.66025i	−0.280828	+	0.486408i	−0.971589	−	0.236675i	$-0.923942\pi$
$317$	−5.00000	+	8.66025i	−0.280828	+	0.486408i	0.690761	+	0.723083i	$0.257276\pi$
$318$		0			0
$319$	4.00000	+	6.92820i	0.223957	+	0.387905i
$320$		0			0
$321$	−18.0000			−1.00466
$322$		0			0
$323$	4.00000			0.222566
$324$		0			0
$325$	−3.50000	−	6.06218i	−0.194145	−	0.336269i
$326$		0			0
$327$	−5.00000	+	8.66025i	−0.276501	+	0.478913i
$328$		0			0
$329$	7.50000	−	2.59808i	0.413488	−	0.143237i
$330$		0			0
$331$	−4.50000	+	7.79423i	−0.247342	+	0.428410i	−0.962788	−	0.270259i	$-0.912891\pi$
$331$	−4.50000	+	7.79423i	−0.247342	+	0.428410i	0.715445	+	0.698669i	$0.246224\pi$
$332$		0			0
$333$	1.50000	+	2.59808i	0.0821995	+	0.142374i
$334$		0			0
$335$	−12.0000			−0.655630
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$	−3.00000	−	5.19615i	−0.162938	−	0.282216i
$340$		0			0
$341$	3.00000	−	5.19615i	0.162459	−	0.281387i
$342$		0			0
$343$	10.0000	−	15.5885i	0.539949	−	0.841698i
$344$		0			0
$345$	−0.500000	+	0.866025i	−0.0269191	+	0.0466252i
$346$		0			0
$347$	−17.0000	−	29.4449i	−0.912608	−	1.58068i	−0.810366	−	0.585923i	$-0.800732\pi$
$347$	−17.0000	−	29.4449i	−0.912608	−	1.58068i	−0.102241	−	0.994760i	$-0.532601\pi$
$348$		0			0
$349$	−28.0000			−1.49881			−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000			−1.49881			−0.749403	+	0.662114i	$0.769659\pi$
$350$		0			0
$351$	−7.00000			−0.373632
$352$		0			0
$353$	4.00000	+	6.92820i	0.212899	+	0.368751i	0.952620	−	0.304162i	$-0.0983763\pi$
$353$	4.00000	+	6.92820i	0.212899	+	0.368751i	−0.739722	+	0.672913i	$0.765043\pi$
$354$		0			0
$355$	−7.00000	+	12.1244i	−0.371521	+	0.643494i
$356$		0			0
$357$	10.0000	−	3.46410i	0.529256	−	0.183340i
$358$		0			0
$359$	18.0000	−	31.1769i	0.950004	−	1.64545i	0.204595	−	0.978847i	$-0.434412\pi$
$359$	18.0000	−	31.1769i	0.950004	−	1.64545i	0.745409	−	0.666608i	$-0.232254\pi$
$360$		0			0
$361$	9.00000	+	15.5885i	0.473684	+	0.820445i
$362$		0			0
$363$	10.0000			0.524864
$364$		0			0
$365$	−14.0000			−0.732793
$366$		0			0
$367$	9.50000	+	16.4545i	0.495896	+	0.858917i	0.999989	−	0.00473247i	$-0.00150640\pi$
$367$	9.50000	+	16.4545i	0.495896	+	0.858917i	−0.504093	+	0.863649i	$0.668173\pi$
$368$		0			0
$369$	−4.50000	+	7.79423i	−0.234261	+	0.405751i
$370$		0			0
$371$	0.500000	−	2.59808i	0.0259587	−	0.134885i
$372$		0			0
$373$	−13.0000	+	22.5167i	−0.673114	+	1.16587i	0.303902	+	0.952703i	$0.401711\pi$
$373$	−13.0000	+	22.5167i	−0.673114	+	1.16587i	−0.977016	+	0.213165i	$0.931623\pi$
$374$		0			0
$375$	0.500000	+	0.866025i	0.0258199	+	0.0447214i
$376$		0			0
$377$	−56.0000			−2.88415
$378$		0			0
$379$	−1.00000			−0.0513665			−0.0256833	−	0.999670i	$-0.508176\pi$
$379$	−1.00000			−0.0513665			−0.0256833	+	0.999670i	$0.508176\pi$
$380$		0			0
$381$	−2.50000	−	4.33013i	−0.128079	−	0.221839i
$382$		0			0
$383$	6.50000	−	11.2583i	0.332134	−	0.575274i	−0.650796	−	0.759253i	$-0.725565\pi$
$383$	6.50000	−	11.2583i	0.332134	−	0.575274i	0.982930	+	0.183979i	$0.0588979\pi$
$384$		0			0
$385$	−2.00000	−	1.73205i	−0.101929	−	0.0882735i
$386$		0			0
$387$	−2.00000	+	3.46410i	−0.101666	+	0.176090i
$388$		0			0
$389$	7.00000	+	12.1244i	0.354914	+	0.614729i	0.987103	−	0.160085i	$-0.0511768\pi$
$389$	7.00000	+	12.1244i	0.354914	+	0.614729i	−0.632189	+	0.774814i	$0.717843\pi$
$390$		0			0
$391$	4.00000			0.202289
$392$		0			0
$393$	−13.0000			−0.655763
$394$		0			0
$395$	2.00000	+	3.46410i	0.100631	+	0.174298i
$396$		0			0
$397$	−9.00000	+	15.5885i	−0.451697	+	0.782362i	−0.998492	−	0.0549046i	$-0.982515\pi$
$397$	−9.00000	+	15.5885i	−0.451697	+	0.782362i	0.546795	+	0.837267i	$0.315848\pi$
$398$		0			0
$399$	−2.00000	−	1.73205i	−0.100125	−	0.0867110i
$400$		0			0
$401$	8.50000	−	14.7224i	0.424470	−	0.735203i	−0.571901	−	0.820323i	$-0.693794\pi$
$401$	8.50000	−	14.7224i	0.424470	−	0.735203i	0.996371	+	0.0851195i	$0.0271272\pi$
$402$		0			0
$403$	21.0000	+	36.3731i	1.04608	+	1.81187i
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$	−3.00000			−0.148704
$408$		0			0
$409$	−5.00000	−	8.66025i	−0.247234	−	0.428222i	0.715523	−	0.698589i	$-0.246188\pi$
$409$	−5.00000	−	8.66025i	−0.247234	−	0.428222i	−0.962757	+	0.270367i	$0.912855\pi$
$410$		0			0
$411$	−1.00000	+	1.73205i	−0.0493264	+	0.0854358i
$412$		0			0
$413$	6.00000	−	31.1769i	0.295241	−	1.53412i
$414$		0			0
$415$	6.00000	−	10.3923i	0.294528	−	0.510138i
$416$		0			0
$417$	2.00000	+	3.46410i	0.0979404	+	0.169638i
$418$		0			0
$419$	−11.0000			−0.537385			−0.268693	−	0.963226i	$-0.586592\pi$
$419$	−11.0000			−0.537385			−0.268693	+	0.963226i	$0.586592\pi$
$420$		0			0
$421$	−14.0000			−0.682318			−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000			−0.682318			−0.341159	+	0.940006i	$0.610819\pi$
$422$		0			0
$423$	−1.50000	−	2.59808i	−0.0729325	−	0.126323i
$424$		0			0
$425$	2.00000	−	3.46410i	0.0970143	−	0.168034i
$426$		0			0
$427$	−10.0000	+	3.46410i	−0.483934	+	0.167640i
$428$		0			0
$429$	3.50000	−	6.06218i	0.168982	−	0.292685i
$430$		0			0
$431$	−6.00000	−	10.3923i	−0.289010	−	0.500580i	0.684564	−	0.728953i	$-0.259993\pi$
$431$	−6.00000	−	10.3923i	−0.289010	−	0.500580i	−0.973574	+	0.228373i	$0.926659\pi$
$432$		0			0
$433$	40.0000			1.92228			0.961139	−	0.276066i	$-0.0890309\pi$
$433$	40.0000			1.92228			0.961139	+	0.276066i	$0.0890309\pi$
$434$		0			0
$435$	8.00000			0.383571
$436$		0			0
$437$	−0.500000	−	0.866025i	−0.0239182	−	0.0414276i
$438$		0			0
$439$	8.00000	−	13.8564i	0.381819	−	0.661330i	−0.609503	−	0.792784i	$-0.708631\pi$
$439$	8.00000	−	13.8564i	0.381819	−	0.661330i	0.991322	+	0.131453i	$0.0419644\pi$
$440$		0			0
$441$	−6.50000	−	2.59808i	−0.309524	−	0.123718i
$442$		0			0
$443$	−18.0000	+	31.1769i	−0.855206	+	1.48126i	0.0212481	+	0.999774i	$0.493236\pi$
$443$	−18.0000	+	31.1769i	−0.855206	+	1.48126i	−0.876454	+	0.481486i	$0.840097\pi$
$444$		0			0
$445$	1.00000	+	1.73205i	0.0474045	+	0.0821071i
$446$		0			0
$447$	4.00000			0.189194
$448$		0			0
$449$	−25.0000			−1.17982			−0.589911	−	0.807468i	$-0.700837\pi$
$449$	−25.0000			−1.17982			−0.589911	+	0.807468i	$0.700837\pi$
$450$		0			0
$451$	−4.50000	−	7.79423i	−0.211897	−	0.367016i
$452$		0			0
$453$	1.00000	−	1.73205i	0.0469841	−	0.0813788i
$454$		0			0
$455$	17.5000	−	6.06218i	0.820413	−	0.284199i
$456$		0			0
$457$	−5.00000	+	8.66025i	−0.233890	+	0.405110i	−0.958950	−	0.283577i	$-0.908479\pi$
$457$	−5.00000	+	8.66025i	−0.233890	+	0.405110i	0.725059	+	0.688686i	$0.241812\pi$
$458$		0			0
$459$	−2.00000	−	3.46410i	−0.0933520	−	0.161690i
$460$		0			0
$461$	−28.0000			−1.30409			−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000			−1.30409			−0.652045	+	0.758180i	$0.726089\pi$
$462$		0			0
$463$	−33.0000			−1.53364			−0.766820	−	0.641862i	$-0.778162\pi$
$463$	−33.0000			−1.53364			−0.766820	+	0.641862i	$0.778162\pi$
$464$		0			0
$465$	−3.00000	−	5.19615i	−0.139122	−	0.240966i
$466$		0			0
$467$	6.00000	−	10.3923i	0.277647	−	0.480899i	−0.693153	−	0.720791i	$-0.743779\pi$
$467$	6.00000	−	10.3923i	0.277647	−	0.480899i	0.970799	+	0.239892i	$0.0771121\pi$
$468$		0			0
$469$	6.00000	−	31.1769i	0.277054	−	1.43962i
$470$		0			0
$471$	7.50000	−	12.9904i	0.345582	−	0.598565i
$472$		0			0
$473$	−2.00000	−	3.46410i	−0.0919601	−	0.159280i
$474$		0			0
$475$	−1.00000			−0.0458831
$476$		0			0
$477$	−1.00000			−0.0457869
$478$		0			0
$479$	13.0000	+	22.5167i	0.593985	+	1.02881i	0.993689	+	0.112168i	$0.0357796\pi$
$479$	13.0000	+	22.5167i	0.593985	+	1.02881i	−0.399704	+	0.916644i	$0.630887\pi$
$480$		0			0
$481$	10.5000	−	18.1865i	0.478759	−	0.829235i
$482$		0			0
$483$	−2.00000	−	1.73205i	−0.0910032	−	0.0788110i
$484$		0			0
$485$	8.00000	−	13.8564i	0.363261	−	0.629187i
$486$		0			0
$487$	−4.00000	−	6.92820i	−0.181257	−	0.313947i	0.761052	−	0.648691i	$-0.224683\pi$
$487$	−4.00000	−	6.92820i	−0.181257	−	0.313947i	−0.942309	+	0.334744i	$0.891350\pi$
$488$		0			0
$489$	8.00000			0.361773
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$	−16.0000	−	27.7128i	−0.720604	−	1.24812i
$494$		0			0
$495$	−0.500000	+	0.866025i	−0.0224733	+	0.0389249i
$496$		0			0
$497$	−28.0000	−	24.2487i	−1.25597	−	1.08770i
$498$		0			0
$499$	12.0000	−	20.7846i	0.537194	−	0.930447i	−0.461860	−	0.886953i	$-0.652818\pi$
$499$	12.0000	−	20.7846i	0.537194	−	0.930447i	0.999054	−	0.0434940i	$-0.0138489\pi$
$500$		0			0
$501$	2.50000	+	4.33013i	0.111692	+	0.193456i
$502$		0			0
$503$	−28.0000			−1.24846			−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000			−1.24846			−0.624229	+	0.781241i	$0.714587\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	18.0000	+	31.1769i	0.799408	+	1.38462i
$508$		0			0
$509$	−15.0000	+	25.9808i	−0.664863	+	1.15158i	0.314459	+	0.949271i	$0.398177\pi$
$509$	−15.0000	+	25.9808i	−0.664863	+	1.15158i	−0.979322	+	0.202306i	$0.935156\pi$
$510$		0			0
$511$	7.00000	−	36.3731i	0.309662	−	1.60905i
$512$		0			0
$513$	−0.500000	+	0.866025i	−0.0220755	+	0.0382360i
$514$		0			0
$515$	8.00000	+	13.8564i	0.352522	+	0.610586i
$516$		0			0
$517$	3.00000			0.131940
$518$		0			0
$519$	21.0000			0.921798
$520$		0			0
$521$	10.5000	+	18.1865i	0.460013	+	0.796766i	0.998961	−	0.0455727i	$-0.0145113\pi$
$521$	10.5000	+	18.1865i	0.460013	+	0.796766i	−0.538948	+	0.842339i	$0.681178\pi$
$522$		0			0
$523$	−7.00000	+	12.1244i	−0.306089	+	0.530161i	−0.977503	−	0.210921i	$-0.932354\pi$
$523$	−7.00000	+	12.1244i	−0.306089	+	0.530161i	0.671414	+	0.741082i	$0.265687\pi$
$524$		0			0
$525$	−2.50000	+	0.866025i	−0.109109	+	0.0377964i
$526$		0			0
$527$	−12.0000	+	20.7846i	−0.522728	+	0.905392i
$528$		0			0
$529$	11.0000	+	19.0526i	0.478261	+	0.828372i
$530$		0			0
$531$	−12.0000			−0.520756
$532$		0			0
$533$	63.0000			2.72883
$534$		0			0
$535$	−9.00000	−	15.5885i	−0.389104	−	0.673948i
$536$		0			0
$537$	−6.50000	+	11.2583i	−0.280496	+	0.485833i
$538$		0			0
$539$	5.50000	−	4.33013i	0.236902	−	0.186512i
$540$		0			0
$541$	−1.00000	+	1.73205i	−0.0429934	+	0.0744667i	−0.886721	−	0.462304i	$-0.847023\pi$
$541$	−1.00000	+	1.73205i	−0.0429934	+	0.0744667i	0.843728	+	0.536771i	$0.180356\pi$
$542$		0			0
$543$	−6.00000	−	10.3923i	−0.257485	−	0.445976i
$544$		0			0
$545$	−10.0000			−0.428353
$546$		0			0
$547$	−36.0000			−1.53925			−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000			−1.53925			−0.769624	+	0.638497i	$0.779557\pi$
$548$		0			0
$549$	2.00000	+	3.46410i	0.0853579	+	0.147844i
$550$		0			0
$551$	−4.00000	+	6.92820i	−0.170406	+	0.295151i
$552$		0			0
$553$	−10.0000	+	3.46410i	−0.425243	+	0.147309i
$554$		0			0
$555$	−1.50000	+	2.59808i	−0.0636715	+	0.110282i
$556$		0			0
$557$	−22.5000	−	38.9711i	−0.953356	−	1.65126i	−0.738087	−	0.674705i	$-0.764271\pi$
$557$	−22.5000	−	38.9711i	−0.953356	−	1.65126i	−0.215268	−	0.976555i	$-0.569063\pi$
$558$		0			0
$559$	28.0000			1.18427
$560$		0			0
$561$	4.00000			0.168880
$562$		0			0
$563$	7.00000	+	12.1244i	0.295015	+	0.510981i	0.974988	−	0.222256i	$-0.0713421\pi$
$563$	7.00000	+	12.1244i	0.295015	+	0.510981i	−0.679974	+	0.733237i	$0.738009\pi$
$564$		0			0
$565$	3.00000	−	5.19615i	0.126211	−	0.218604i
$566$		0			0
$567$	−0.500000	+	2.59808i	−0.0209980	+	0.109109i
$568$		0			0
$569$	18.5000	−	32.0429i	0.775560	−	1.34331i	−0.158919	−	0.987292i	$-0.550801\pi$
$569$	18.5000	−	32.0429i	0.775560	−	1.34331i	0.934479	−	0.356018i	$-0.115866\pi$
$570$		0			0
$571$	−4.00000	−	6.92820i	−0.167395	−	0.289936i	0.770108	−	0.637913i	$-0.220202\pi$
$571$	−4.00000	−	6.92820i	−0.167395	−	0.289936i	−0.937503	+	0.347977i	$0.886869\pi$
$572$		0			0
$573$	−10.0000			−0.417756
$574$		0			0
$575$	−1.00000			−0.0417029
$576$		0			0
$577$	7.00000	+	12.1244i	0.291414	+	0.504744i	0.974144	−	0.225927i	$-0.0725410\pi$
$577$	7.00000	+	12.1244i	0.291414	+	0.504744i	−0.682730	+	0.730670i	$0.739208\pi$
$578$		0			0
$579$	13.0000	−	22.5167i	0.540262	−	0.935760i
$580$		0			0
$581$	24.0000	+	20.7846i	0.995688	+	0.862291i
$582$		0			0
$583$	0.500000	−	0.866025i	0.0207079	−	0.0358671i
$584$		0			0
$585$	−3.50000	−	6.06218i	−0.144707	−	0.250640i
$586$		0			0
$587$	−42.0000			−1.73353			−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000			−1.73353			−0.866763	+	0.498721i	$0.833803\pi$
$588$		0			0
$589$	6.00000			0.247226
$590$		0			0
$591$	−1.50000	−	2.59808i	−0.0617018	−	0.106871i
$592$		0			0
$593$	−6.00000	+	10.3923i	−0.246390	+	0.426761i	−0.962522	−	0.271205i	$-0.912578\pi$
$593$	−6.00000	+	10.3923i	−0.246390	+	0.426761i	0.716131	+	0.697966i	$0.245911\pi$
$594$		0			0
$595$	8.00000	+	6.92820i	0.327968	+	0.284029i
$596$		0			0
$597$	6.00000	−	10.3923i	0.245564	−	0.425329i
$598$		0			0
$599$	3.00000	+	5.19615i	0.122577	+	0.212309i	0.920783	−	0.390075i	$-0.127551\pi$
$599$	3.00000	+	5.19615i	0.122577	+	0.212309i	−0.798206	+	0.602384i	$0.794218\pi$
$600$		0			0
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$		0			0
$603$	−12.0000			−0.488678
$604$		0			0
$605$	5.00000	+	8.66025i	0.203279	+	0.352089i
$606$		0			0
$607$	12.5000	−	21.6506i	0.507359	−	0.878772i	−0.492604	−	0.870253i	$-0.663955\pi$
$607$	12.5000	−	21.6506i	0.507359	−	0.878772i	0.999964	−	0.00851879i	$-0.00271165\pi$
$608$		0			0
$609$	−4.00000	+	20.7846i	−0.162088	+	0.842235i
$610$		0			0
$611$	−10.5000	+	18.1865i	−0.424785	+	0.735748i
$612$		0			0
$613$	7.50000	+	12.9904i	0.302922	+	0.524677i	0.976797	−	0.214169i	$-0.0687045\pi$
$613$	7.50000	+	12.9904i	0.302922	+	0.524677i	−0.673874	+	0.738846i	$0.735371\pi$
$614$		0			0
$615$	−9.00000			−0.362915
$616$		0			0
$617$	8.00000			0.322068			0.161034	−	0.986949i	$-0.448517\pi$
$617$	8.00000			0.322068			0.161034	+	0.986949i	$0.448517\pi$
$618$		0			0
$619$	−3.50000	−	6.06218i	−0.140677	−	0.243659i	0.787075	−	0.616858i	$-0.211595\pi$
$619$	−3.50000	−	6.06218i	−0.140677	−	0.243659i	−0.927752	+	0.373198i	$0.878261\pi$
$620$		0			0
$621$	−0.500000	+	0.866025i	−0.0200643	+	0.0347524i
$622$		0			0
$623$	−5.00000	+	1.73205i	−0.200321	+	0.0693932i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	−0.500000	−	0.866025i	−0.0199681	−	0.0345857i
$628$		0			0
$629$	12.0000			0.478471
$630$		0			0
$631$	−14.0000			−0.557331			−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000			−0.557331			−0.278666	+	0.960388i	$0.589892\pi$
$632$		0			0
$633$	7.50000	+	12.9904i	0.298098	+	0.516321i
$634$		0			0
$635$	2.50000	−	4.33013i	0.0992095	−	0.171836i
$636$		0			0
$637$	7.00000	+	48.4974i	0.277350	+	1.92154i
$638$		0			0
$639$	−7.00000	+	12.1244i	−0.276916	+	0.479632i
$640$		0			0
$641$	11.5000	+	19.9186i	0.454223	+	0.786737i	0.998643	−	0.0520757i	$-0.0165837\pi$
$641$	11.5000	+	19.9186i	0.454223	+	0.786737i	−0.544420	+	0.838812i	$0.683250\pi$
$642$		0			0
$643$	−26.0000			−1.02534			−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000			−1.02534			−0.512670	+	0.858586i	$0.671344\pi$
$644$		0			0
$645$	−4.00000			−0.157500
$646$		0			0
$647$	−7.50000	−	12.9904i	−0.294855	−	0.510705i	0.680096	−	0.733123i	$-0.261938\pi$
$647$	−7.50000	−	12.9904i	−0.294855	−	0.510705i	−0.974951	+	0.222419i	$0.928605\pi$
$648$		0			0
$649$	6.00000	−	10.3923i	0.235521	−	0.407934i
$650$		0			0
$651$	15.0000	−	5.19615i	0.587896	−	0.203653i
$652$		0			0
$653$	−14.5000	+	25.1147i	−0.567429	+	0.982816i	0.429390	+	0.903119i	$0.358728\pi$
$653$	−14.5000	+	25.1147i	−0.567429	+	0.982816i	−0.996819	+	0.0796966i	$0.974605\pi$
$654$		0			0
$655$	−6.50000	−	11.2583i	−0.253976	−	0.439899i
$656$		0			0
$657$	−14.0000			−0.546192
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$	−4.00000	−	6.92820i	−0.155582	−	0.269476i	0.777689	−	0.628649i	$-0.216392\pi$
$661$	−4.00000	−	6.92820i	−0.155582	−	0.269476i	−0.933271	+	0.359174i	$0.883059\pi$
$662$		0			0
$663$	−14.0000	+	24.2487i	−0.543715	+	0.941742i
$664$		0			0
$665$	0.500000	−	2.59808i	0.0193892	−	0.100749i
$666$		0			0
$667$	−4.00000	+	6.92820i	−0.154881	+	0.268261i
$668$		0			0
$669$	2.00000	+	3.46410i	0.0773245	+	0.133930i
$670$		0			0
$671$	−4.00000			−0.154418
$672$		0			0
$673$	−12.0000			−0.462566			−0.231283	−	0.972887i	$-0.574292\pi$
$673$	−12.0000			−0.462566			−0.231283	+	0.972887i	$0.574292\pi$
$674$		0			0
$675$	0.500000	+	0.866025i	0.0192450	+	0.0333333i
$676$		0			0
$677$	−0.500000	+	0.866025i	−0.0192166	+	0.0332841i	−0.875474	−	0.483266i	$-0.839451\pi$
$677$	−0.500000	+	0.866025i	−0.0192166	+	0.0332841i	0.856257	+	0.516550i	$0.172784\pi$
$678$		0			0
$679$	32.0000	+	27.7128i	1.22805	+	1.06352i
$680$		0			0
$681$	10.0000	−	17.3205i	0.383201	−	0.663723i
$682$		0			0
$683$	6.00000	+	10.3923i	0.229584	+	0.397650i	0.957685	−	0.287819i	$-0.0929302\pi$
$683$	6.00000	+	10.3923i	0.229584	+	0.397650i	−0.728101	+	0.685470i	$0.759597\pi$
$684$		0			0
$685$	−2.00000			−0.0764161
$686$		0			0
$687$	22.0000			0.839352
$688$		0			0
$689$	3.50000	+	6.06218i	0.133339	+	0.230951i
$690$		0			0
$691$	6.00000	−	10.3923i	0.228251	−	0.395342i	−0.729039	−	0.684472i	$-0.760033\pi$
$691$	6.00000	−	10.3923i	0.228251	−	0.395342i	0.957290	+	0.289130i	$0.0933661\pi$
$692$		0			0
$693$	−2.00000	−	1.73205i	−0.0759737	−	0.0657952i
$694$		0			0
$695$	−2.00000	+	3.46410i	−0.0758643	+	0.131401i
$696$		0			0
$697$	18.0000	+	31.1769i	0.681799	+	1.18091i
$698$		0			0
$699$	−26.0000			−0.983410
$700$		0			0
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$		0			0
$703$	−1.50000	−	2.59808i	−0.0565736	−	0.0979883i
$704$		0			0
$705$	1.50000	−	2.59808i	0.0564933	−	0.0978492i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−2.00000	+	3.46410i	−0.0751116	+	0.130097i	−0.901135	−	0.433539i	$-0.857265\pi$
$709$	−2.00000	+	3.46410i	−0.0751116	+	0.130097i	0.826023	+	0.563636i	$0.190598\pi$
$710$		0			0
$711$	2.00000	+	3.46410i	0.0750059	+	0.129914i
$712$		0			0
$713$	6.00000			0.224702
$714$		0			0
$715$	7.00000			0.261785
$716$		0			0
$717$	3.00000	+	5.19615i	0.112037	+	0.194054i
$718$		0			0
$719$	13.0000	−	22.5167i	0.484818	−	0.839730i	−0.515030	−	0.857172i	$-0.672219\pi$
$719$	13.0000	−	22.5167i	0.484818	−	0.839730i	0.999848	+	0.0174426i	$0.00555244\pi$
$720$		0			0
$721$	−40.0000	+	13.8564i	−1.48968	+	0.516040i
$722$		0			0
$723$	3.50000	−	6.06218i	0.130166	−	0.225455i
$724$		0			0
$725$	4.00000	+	6.92820i	0.148556	+	0.257307i
$726$		0			0
$727$	−17.0000			−0.630495			−0.315248	−	0.949009i	$-0.602088\pi$
$727$	−17.0000			−0.630495			−0.315248	+	0.949009i	$0.602088\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	8.00000	+	13.8564i	0.295891	+	0.512498i
$732$		0			0
$733$	−18.5000	+	32.0429i	−0.683313	+	1.18353i	0.290651	+	0.956829i	$0.406128\pi$
$733$	−18.5000	+	32.0429i	−0.683313	+	1.18353i	−0.973964	+	0.226704i	$0.927205\pi$
$734$		0			0
$735$	−1.00000	−	6.92820i	−0.0368856	−	0.255551i
$736$		0			0
$737$	6.00000	−	10.3923i	0.221013	−	0.382805i
$738$		0			0
$739$	20.5000	+	35.5070i	0.754105	+	1.30615i	0.945818	+	0.324697i	$0.105262\pi$
$739$	20.5000	+	35.5070i	0.754105	+	1.30615i	−0.191714	+	0.981451i	$0.561404\pi$
$740$		0			0
$741$	7.00000			0.257151
$742$		0			0
$743$	9.00000			0.330178			0.165089	−	0.986279i	$-0.447209\pi$
$743$	9.00000			0.330178			0.165089	+	0.986279i	$0.447209\pi$
$744$		0			0
$745$	2.00000	+	3.46410i	0.0732743	+	0.126915i
$746$		0			0
$747$	6.00000	−	10.3923i	0.219529	−	0.380235i
$748$		0			0
$749$	45.0000	−	15.5885i	1.64426	−	0.569590i
$750$		0			0
$751$	−13.0000	+	22.5167i	−0.474377	+	0.821645i	−0.999570	−	0.0293387i	$-0.990660\pi$
$751$	−13.0000	+	22.5167i	−0.474377	+	0.821645i	0.525193	+	0.850983i	$0.323993\pi$
$752$		0			0
$753$	1.50000	+	2.59808i	0.0546630	+	0.0946792i
$754$		0			0
$755$	2.00000			0.0727875
$756$		0			0
$757$	−2.00000			−0.0726912			−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000			−0.0726912			−0.0363456	+	0.999339i	$0.511572\pi$
$758$		0			0
$759$	−0.500000	−	0.866025i	−0.0181489	−	0.0314347i
$760$		0			0
$761$	8.50000	−	14.7224i	0.308125	−	0.533688i	−0.669827	−	0.742517i	$-0.733632\pi$
$761$	8.50000	−	14.7224i	0.308125	−	0.533688i	0.977952	+	0.208829i	$0.0669652\pi$
$762$		0			0
$763$	5.00000	−	25.9808i	0.181012	−	0.940567i
$764$		0			0
$765$	2.00000	−	3.46410i	0.0723102	−	0.125245i
$766$		0			0
$767$	42.0000	+	72.7461i	1.51653	+	2.62671i
$768$		0			0
$769$	−29.0000			−1.04577			−0.522883	−	0.852404i	$-0.675144\pi$
$769$	−29.0000			−1.04577			−0.522883	+	0.852404i	$0.675144\pi$
$770$		0			0
$771$	8.00000			0.288113
$772$		0			0
$773$	−21.5000	−	37.2391i	−0.773301	−	1.33940i	−0.935744	−	0.352679i	$-0.885271\pi$
$773$	−21.5000	−	37.2391i	−0.773301	−	1.33940i	0.162443	−	0.986718i	$-0.448063\pi$
$774$		0			0
$775$	3.00000	−	5.19615i	0.107763	−	0.186651i
$776$		0			0
$777$	−6.00000	−	5.19615i	−0.215249	−	0.186411i
$778$		0			0
$779$	4.50000	−	7.79423i	0.161229	−	0.279257i
$780$		0			0
$781$	−7.00000	−	12.1244i	−0.250480	−	0.433844i
$782$		0			0
$783$	8.00000			0.285897
$784$		0			0
$785$	15.0000			0.535373
$786$		0			0
$787$	−11.0000	−	19.0526i	−0.392108	−	0.679150i	0.600620	−	0.799535i	$-0.294921\pi$
$787$	−11.0000	−	19.0526i	−0.392108	−	0.679150i	−0.992727	+	0.120384i	$0.961587\pi$
$788$		0			0
$789$	8.00000	−	13.8564i	0.284808	−	0.493301i
$790$		0			0
$791$	12.0000	+	10.3923i	0.426671	+	0.369508i
$792$		0			0
$793$	14.0000	−	24.2487i	0.497155	−	0.861097i
$794$		0			0
$795$	−0.500000	−	0.866025i	−0.0177332	−	0.0307148i
$796$		0			0
$797$	−6.00000			−0.212531			−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000			−0.212531			−0.106265	+	0.994338i	$0.533889\pi$
$798$		0			0
$799$	−12.0000			−0.424529
$800$		0			0
$801$	1.00000	+	1.73205i	0.0353333	+	0.0611990i
$802$		0			0
$803$	7.00000	−	12.1244i	0.247025	−	0.427859i
$804$		0			0
$805$	0.500000	−	2.59808i	0.0176227	−	0.0915702i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−26.5000	−	45.8993i	−0.931690	−	1.61374i	−0.780432	−	0.625241i	$-0.785001\pi$
$809$	−26.5000	−	45.8993i	−0.931690	−	1.61374i	−0.151259	−	0.988494i	$-0.548333\pi$
$810$		0			0
$811$	−37.0000			−1.29925			−0.649623	−	0.760257i	$-0.725073\pi$
$811$	−37.0000			−1.29925			−0.649623	+	0.760257i	$0.725073\pi$
$812$		0			0
$813$	16.0000			0.561144
$814$		0			0
$815$	4.00000	+	6.92820i	0.140114	+	0.242684i
$816$		0			0
$817$	2.00000	−	3.46410i	0.0699711	−	0.121194i
$818$		0			0
$819$	17.5000	−	6.06218i	0.611499	−	0.211830i
$820$		0			0
$821$	−17.0000	+	29.4449i	−0.593304	+	1.02763i	0.400480	+	0.916306i	$0.368843\pi$
$821$	−17.0000	+	29.4449i	−0.593304	+	1.02763i	−0.993784	+	0.111327i	$0.964490\pi$
$822$		0			0
$823$	24.0000	+	41.5692i	0.836587	+	1.44901i	0.892731	+	0.450589i	$0.148786\pi$
$823$	24.0000	+	41.5692i	0.836587	+	1.44901i	−0.0561440	+	0.998423i	$0.517881\pi$
$824$		0			0
$825$	−1.00000			−0.0348155
$826$		0			0
$827$	−10.0000			−0.347734			−0.173867	−	0.984769i	$-0.555626\pi$
$827$	−10.0000			−0.347734			−0.173867	+	0.984769i	$0.555626\pi$
$828$		0			0
$829$	−6.00000	−	10.3923i	−0.208389	−	0.360940i	0.742818	−	0.669493i	$-0.233489\pi$
$829$	−6.00000	−	10.3923i	−0.208389	−	0.360940i	−0.951207	+	0.308553i	$0.900155\pi$
$830$		0			0
$831$	−1.00000	+	1.73205i	−0.0346896	+	0.0600842i
$832$		0			0
$833$	−22.0000	+	17.3205i	−0.762255	+	0.600120i
$834$		0			0
$835$	−2.50000	+	4.33013i	−0.0865161	+	0.149850i
$836$		0			0
$837$	−3.00000	−	5.19615i	−0.103695	−	0.179605i
$838$		0			0
$839$	−44.0000			−1.51905			−0.759524	−	0.650479i	$-0.774568\pi$
$839$	−44.0000			−1.51905			−0.759524	+	0.650479i	$0.774568\pi$
$840$		0			0
$841$	35.0000			1.20690
$842$		0			0
$843$	1.50000	+	2.59808i	0.0516627	+	0.0894825i
$844$		0			0
$845$	−18.0000	+	31.1769i	−0.619219	+	1.07252i
$846$		0			0
$847$	−25.0000	+	8.66025i	−0.859010	+	0.297570i
$848$		0			0
$849$	1.00000	−	1.73205i	0.0343199	−	0.0594438i
$850$		0			0
$851$	−1.50000	−	2.59808i	−0.0514193	−	0.0890609i
$852$		0			0
$853$	1.00000			0.0342393			0.0171197	−	0.999853i	$-0.494550\pi$
$853$	1.00000			0.0342393			0.0171197	+	0.999853i	$0.494550\pi$
$854$		0			0
$855$	−1.00000			−0.0341993
$856$		0			0
$857$	11.0000	+	19.0526i	0.375753	+	0.650823i	0.990439	−	0.137948i	$-0.0440508\pi$
$857$	11.0000	+	19.0526i	0.375753	+	0.650823i	−0.614687	+	0.788771i	$0.710717\pi$
$858$		0			0
$859$	−10.0000	+	17.3205i	−0.341196	+	0.590968i	−0.984655	−	0.174512i	$-0.944165\pi$
$859$	−10.0000	+	17.3205i	−0.341196	+	0.590968i	0.643459	+	0.765480i	$0.277499\pi$
$860$		0			0
$861$	4.50000	−	23.3827i	0.153360	−	0.796880i
$862$		0			0
$863$	14.5000	−	25.1147i	0.493586	−	0.854916i	−0.506387	−	0.862306i	$-0.669019\pi$
$863$	14.5000	−	25.1147i	0.493586	−	0.854916i	0.999973	+	0.00739078i	$0.00235258\pi$
$864$		0			0
$865$	10.5000	+	18.1865i	0.357011	+	0.618361i
$866$		0			0
$867$	1.00000			0.0339618
$868$		0			0
$869$	−4.00000			−0.135691
$870$		0			0
$871$	42.0000	+	72.7461i	1.42312	+	2.46491i
$872$		0			0
$873$	8.00000	−	13.8564i	0.270759	−	0.468968i
$874$		0			0
$875$	−2.00000	−	1.73205i	−0.0676123	−	0.0585540i
$876$		0			0
$877$	−11.5000	+	19.9186i	−0.388327	+	0.672603i	−0.992225	−	0.124459i	$-0.960280\pi$
$877$	−11.5000	+	19.9186i	−0.388327	+	0.672603i	0.603897	+	0.797062i	$0.293614\pi$
$878$		0			0
$879$	4.50000	+	7.79423i	0.151781	+	0.262893i
$880$		0			0
$881$	−25.0000			−0.842271			−0.421136	−	0.906998i	$-0.638368\pi$
$881$	−25.0000			−0.842271			−0.421136	+	0.906998i	$0.638368\pi$
$882$		0			0
$883$	58.0000			1.95186			0.975928	−	0.218094i	$-0.0699840\pi$
$883$	58.0000			1.95186			0.975928	+	0.218094i	$0.0699840\pi$
$884$		0			0
$885$	−6.00000	−	10.3923i	−0.201688	−	0.349334i
$886$		0			0
$887$	10.0000	−	17.3205i	0.335767	−	0.581566i	−0.647865	−	0.761755i	$-0.724338\pi$
$887$	10.0000	−	17.3205i	0.335767	−	0.581566i	0.983632	+	0.180190i	$0.0576711\pi$
$888$		0			0
$889$	10.0000	+	8.66025i	0.335389	+	0.290456i
$890$		0			0
$891$	−0.500000	+	0.866025i	−0.0167506	+	0.0290129i
$892$		0			0
$893$	1.50000	+	2.59808i	0.0501956	+	0.0869413i
$894$		0			0
$895$	−13.0000			−0.434542
$896$		0			0
$897$	7.00000			0.233723
$898$		0			0
$899$	−24.0000	−	41.5692i	−0.800445	−	1.38641i
$900$		0			0
$901$	−2.00000	+	3.46410i	−0.0666297	+	0.115406i
$902$		0			0
$903$	2.00000	−	10.3923i	0.0665558	−	0.345834i
$904$		0			0
$905$	6.00000	−	10.3923i	0.199447	−	0.345452i
$906$		0			0
$907$	23.0000	+	39.8372i	0.763702	+	1.32277i	0.940930	+	0.338602i	$0.109954\pi$
$907$	23.0000	+	39.8372i	0.763702	+	1.32277i	−0.177227	+	0.984170i	$0.556713\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	58.0000			1.92163			0.960813	−	0.277198i	$-0.0894057\pi$
$911$	58.0000			1.92163			0.960813	+	0.277198i	$0.0894057\pi$
$912$		0			0
$913$	6.00000	+	10.3923i	0.198571	+	0.343935i
$914$		0			0
$915$	−2.00000	+	3.46410i	−0.0661180	+	0.114520i
$916$		0			0
$917$	32.5000	−	11.2583i	1.07324	−	0.371783i
$918$		0			0
$919$	−22.0000	+	38.1051i	−0.725713	+	1.25697i	0.232967	+	0.972485i	$0.425157\pi$
$919$	−22.0000	+	38.1051i	−0.725713	+	1.25697i	−0.958680	+	0.284487i	$0.908177\pi$
$920$		0			0
$921$	−4.00000	−	6.92820i	−0.131804	−	0.228292i
$922$		0			0
$923$	98.0000			3.22571
$924$		0			0
$925$	−3.00000			−0.0986394
$926$		0			0
$927$	8.00000	+	13.8564i	0.262754	+	0.455104i
$928$		0			0
$929$	−15.5000	+	26.8468i	−0.508539	+	0.880815i	0.491413	+	0.870927i	$0.336481\pi$
$929$	−15.5000	+	26.8468i	−0.508539	+	0.880815i	−0.999951	+	0.00988764i	$0.996853\pi$
$930$		0			0
$931$	6.50000	+	2.59808i	0.213029	+	0.0851485i
$932$		0			0
$933$	−8.00000	+	13.8564i	−0.261908	+	0.453638i
$934$		0			0
$935$	2.00000	+	3.46410i	0.0654070	+	0.113288i
$936$		0			0
$937$	16.0000			0.522697			0.261349	−	0.965244i	$-0.415833\pi$
$937$	16.0000			0.522697			0.261349	+	0.965244i	$0.415833\pi$
$938$		0			0
$939$	24.0000			0.783210
$940$		0			0
$941$	15.0000	+	25.9808i	0.488986	+	0.846949i	0.999920	−	0.0126715i	$-0.00403357\pi$
$941$	15.0000	+	25.9808i	0.488986	+	0.846949i	−0.510934	+	0.859620i	$0.670700\pi$
$942$		0			0
$943$	4.50000	−	7.79423i	0.146540	−	0.253815i
$944$		0			0
$945$	−2.50000	+	0.866025i	−0.0813250	+	0.0281718i
$946$		0			0
$947$	23.0000	−	39.8372i	0.747400	−	1.29453i	−0.201666	−	0.979454i	$-0.564635\pi$
$947$	23.0000	−	39.8372i	0.747400	−	1.29453i	0.949065	−	0.315080i	$-0.102031\pi$
$948$		0			0
$949$	49.0000	+	84.8705i	1.59061	+	2.75501i
$950$		0			0
$951$	−10.0000			−0.324272
$952$		0			0
$953$	44.0000			1.42530			0.712650	−	0.701520i	$-0.247495\pi$
$953$	44.0000			1.42530			0.712650	+	0.701520i	$0.247495\pi$
$954$		0			0
$955$	−5.00000	−	8.66025i	−0.161796	−	0.280239i
$956$		0			0
$957$	−4.00000	+	6.92820i	−0.129302	+	0.223957i
$958$		0			0
$959$	1.00000	−	5.19615i	0.0322917	−	0.167793i
$960$		0			0
$961$	−2.50000	+	4.33013i	−0.0806452	+	0.139682i
$962$		0			0
$963$	−9.00000	−	15.5885i	−0.290021	−	0.502331i
$964$		0			0
$965$	26.0000			0.836970
$966$		0			0
$967$	20.0000			0.643157			0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000			0.643157			0.321578	+	0.946883i	$0.395787\pi$
$968$		0			0
$969$	2.00000	+	3.46410i	0.0642493	+	0.111283i
$970$		0			0
$971$	21.5000	−	37.2391i	0.689968	−	1.19506i	−0.281880	−	0.959450i	$-0.590958\pi$
$971$	21.5000	−	37.2391i	0.689968	−	1.19506i	0.971848	−	0.235610i	$-0.0757087\pi$
$972$		0			0
$973$	−8.00000	−	6.92820i	−0.256468	−	0.222108i
$974$		0			0
$975$	3.50000	−	6.06218i	0.112090	−	0.194145i
$976$		0			0
$977$	−9.00000	−	15.5885i	−0.287936	−	0.498719i	0.685381	−	0.728184i	$-0.259636\pi$
$977$	−9.00000	−	15.5885i	−0.287936	−	0.498719i	−0.973317	+	0.229465i	$0.926302\pi$
$978$		0			0
$979$	−2.00000			−0.0639203
$980$		0			0
$981$	−10.0000			−0.319275
$982$		0			0
$983$	−16.5000	−	28.5788i	−0.526268	−	0.911523i	−0.999532	−	0.0306024i	$-0.990257\pi$
$983$	−16.5000	−	28.5788i	−0.526268	−	0.911523i	0.473263	−	0.880921i	$-0.343076\pi$
$984$		0			0
$985$	1.50000	−	2.59808i	0.0477940	−	0.0827816i
$986$		0			0
$987$	6.00000	+	5.19615i	0.190982	+	0.165395i
$988$		0			0
$989$	2.00000	−	3.46410i	0.0635963	−	0.110152i
$990$		0			0
$991$	5.00000	+	8.66025i	0.158830	+	0.275102i	0.934447	−	0.356102i	$-0.115894\pi$
$991$	5.00000	+	8.66025i	0.158830	+	0.275102i	−0.775617	+	0.631204i	$0.782561\pi$
$992$		0			0
$993$	−9.00000			−0.285606
$994$		0			0
$995$	12.0000			0.380426
$996$		0			0
$997$	−5.00000	−	8.66025i	−0.158352	−	0.274273i	0.775923	−	0.630828i	$-0.217285\pi$
$997$	−5.00000	−	8.66025i	−0.158352	−	0.274273i	−0.934274	+	0.356555i	$0.883951\pi$
$998$		0			0
$999$	−1.50000	+	2.59808i	−0.0474579	+	0.0821995i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.bg.k.961.1		2
4.3	odd	2		210.2.i.a.121.1	✓	2
7.4	even	3	inner	1680.2.bg.k.1201.1		2
12.11	even	2		630.2.k.h.541.1		2
20.3	even	4		1050.2.o.j.499.2		4
20.7	even	4		1050.2.o.j.499.1		4
20.19	odd	2		1050.2.i.s.751.1		2
28.3	even	6		1470.2.i.i.361.1		2
28.11	odd	6		210.2.i.a.151.1	yes	2
28.19	even	6		1470.2.a.k.1.1		1
28.23	odd	6		1470.2.a.r.1.1		1
28.27	even	2		1470.2.i.i.961.1		2
84.11	even	6		630.2.k.h.361.1		2
84.23	even	6		4410.2.a.g.1.1		1
84.47	odd	6		4410.2.a.q.1.1		1
140.19	even	6		7350.2.a.ba.1.1		1
140.39	odd	6		1050.2.i.s.151.1		2
140.67	even	12		1050.2.o.j.949.2		4
140.79	odd	6		7350.2.a.j.1.1		1
140.123	even	12		1050.2.o.j.949.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.i.a.121.1	✓	2	4.3	odd	2
210.2.i.a.151.1	yes	2	28.11	odd	6
630.2.k.h.361.1		2	84.11	even	6
630.2.k.h.541.1		2	12.11	even	2
1050.2.i.s.151.1		2	140.39	odd	6
1050.2.i.s.751.1		2	20.19	odd	2
1050.2.o.j.499.1		4	20.7	even	4
1050.2.o.j.499.2		4	20.3	even	4
1050.2.o.j.949.1		4	140.123	even	12
1050.2.o.j.949.2		4	140.67	even	12
1470.2.a.k.1.1		1	28.19	even	6
1470.2.a.r.1.1		1	28.23	odd	6
1470.2.i.i.361.1		2	28.3	even	6
1470.2.i.i.961.1		2	28.27	even	2
1680.2.bg.k.961.1		2	1.1	even	1	trivial
1680.2.bg.k.1201.1		2	7.4	even	3	inner
4410.2.a.g.1.1		1	84.23	even	6
4410.2.a.q.1.1		1	84.47	odd	6
7350.2.a.j.1.1		1	140.79	odd	6
7350.2.a.ba.1.1		1	140.19	even	6

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.bg (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-2.00000 - 1.73205i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-2.00000 - 1.73205i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(-0.500000 - 0.866025i) q^{11} +7.00000 q^{13} -1.00000 q^{15} +(2.00000 + 3.46410i) q^{17} +(0.500000 - 0.866025i) q^{19} +(0.500000 - 2.59808i) q^{21} +(0.500000 - 0.866025i) q^{23} +(-0.500000 - 0.866025i) q^{25} -1.00000 q^{27} -8.00000 q^{29} +(3.00000 + 5.19615i) q^{31} +(0.500000 - 0.866025i) q^{33} +(2.50000 - 0.866025i) q^{35} +(1.50000 - 2.59808i) q^{37} +(3.50000 + 6.06218i) q^{39} +9.00000 q^{41} +4.00000 q^{43} +(-0.500000 - 0.866025i) q^{45} +(-1.50000 + 2.59808i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-2.00000 + 3.46410i) q^{51} +(0.500000 + 0.866025i) q^{53} +1.00000 q^{55} +1.00000 q^{57} +(6.00000 + 10.3923i) q^{59} +(2.00000 - 3.46410i) q^{61} +(2.50000 - 0.866025i) q^{63} +(-3.50000 + 6.06218i) q^{65} +(6.00000 + 10.3923i) q^{67} +1.00000 q^{69} +14.0000 q^{71} +(7.00000 + 12.1244i) q^{73} +(0.500000 - 0.866025i) q^{75} +(-0.500000 + 2.59808i) q^{77} +(2.00000 - 3.46410i) q^{79} +(-0.500000 - 0.866025i) q^{81} -12.0000 q^{83} -4.00000 q^{85} +(-4.00000 - 6.92820i) q^{87} +(1.00000 - 1.73205i) q^{89} +(-14.0000 - 12.1244i) q^{91} +(-3.00000 + 5.19615i) q^{93} +(0.500000 + 0.866025i) q^{95} -16.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) 2 * q + q^3 - q^5 - 4 * q^7 - q^9	\( 2 q + q^{3} - q^{5} - 4 q^{7} - q^{9} - q^{11} + 14 q^{13} - 2 q^{15} + 4 q^{17} + q^{19} + q^{21} + q^{23} - q^{25} - 2 q^{27} - 16 q^{29} + 6 q^{31} + q^{33} + 5 q^{35} + 3 q^{37} + 7 q^{39} + 18 q^{41} + 8 q^{43} - q^{45} - 3 q^{47} + 2 q^{49} - 4 q^{51} + q^{53} + 2 q^{55} + 2 q^{57} + 12 q^{59} + 4 q^{61} + 5 q^{63} - 7 q^{65} + 12 q^{67} + 2 q^{69} + 28 q^{71} + 14 q^{73} + q^{75} - q^{77} + 4 q^{79} - q^{81} - 24 q^{83} - 8 q^{85} - 8 q^{87} + 2 q^{89} - 28 q^{91} - 6 q^{93} + q^{95} - 32 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + q^3 - q^5 - 4 * q^7 - q^9 - q^11 + 14 * q^13 - 2 * q^15 + 4 * q^17 + q^19 + q^21 + q^23 - q^25 - 2 * q^27 - 16 * q^29 + 6 * q^31 + q^33 + 5 * q^35 + 3 * q^37 + 7 * q^39 + 18 * q^41 + 8 * q^43 - q^45 - 3 * q^47 + 2 * q^49 - 4 * q^51 + q^53 + 2 * q^55 + 2 * q^57 + 12 * q^59 + 4 * q^61 + 5 * q^63 - 7 * q^65 + 12 * q^67 + 2 * q^69 + 28 * q^71 + 14 * q^73 + q^75 - q^77 + 4 * q^79 - q^81 - 24 * q^83 - 8 * q^85 - 8 * q^87 + 2 * q^89 - 28 * q^91 - 6 * q^93 + q^95 - 32 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more