LMFDB - Embedded newform 4020.2.q.b.841.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(841,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("4020.841");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.q (of order $3$, degree $2$, 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:	$32.0998616126$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			841.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	4020.841
Dual form			4020.2.q.b.3781.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +(-1.50000 + 2.59808i) q^{7} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{3} +1.00000 q^{5} +(-1.50000 + 2.59808i) q^{7} +1.00000 q^{9} +(2.50000 - 4.33013i) q^{11} +(-2.50000 - 4.33013i) q^{13} -1.00000 q^{15} +(-1.50000 - 2.59808i) q^{17} +(0.500000 + 0.866025i) q^{19} +(1.50000 - 2.59808i) q^{21} +(-3.50000 - 6.06218i) q^{23} +1.00000 q^{25} -1.00000 q^{27} +(-2.50000 + 4.33013i) q^{29} +(-2.50000 + 4.33013i) q^{31} +(-2.50000 + 4.33013i) q^{33} +(-1.50000 + 2.59808i) q^{35} +(5.50000 + 9.52628i) q^{37} +(2.50000 + 4.33013i) q^{39} +(5.50000 - 9.52628i) q^{41} +4.00000 q^{43} +1.00000 q^{45} +(-6.50000 + 11.2583i) q^{47} +(-1.00000 - 1.73205i) q^{49} +(1.50000 + 2.59808i) q^{51} -14.0000 q^{53} +(2.50000 - 4.33013i) q^{55} +(-0.500000 - 0.866025i) q^{57} +(4.50000 + 7.79423i) q^{61} +(-1.50000 + 2.59808i) q^{63} +(-2.50000 - 4.33013i) q^{65} +(-8.00000 + 1.73205i) q^{67} +(3.50000 + 6.06218i) q^{69} +(-7.50000 + 12.9904i) q^{71} +(1.50000 + 2.59808i) q^{73} -1.00000 q^{75} +(7.50000 + 12.9904i) q^{77} +(1.50000 - 2.59808i) q^{79} +1.00000 q^{81} +(-5.50000 - 9.52628i) q^{83} +(-1.50000 - 2.59808i) q^{85} +(2.50000 - 4.33013i) q^{87} +2.00000 q^{89} +15.0000 q^{91} +(2.50000 - 4.33013i) q^{93} +(0.500000 + 0.866025i) q^{95} +(-4.50000 - 7.79423i) q^{97} +(2.50000 - 4.33013i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 - 3 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} - 3 q^{7} + 2 q^{9} + 5 q^{11} - 5 q^{13} - 2 q^{15} - 3 q^{17} + q^{19} + 3 q^{21} - 7 q^{23} + 2 q^{25} - 2 q^{27} - 5 q^{29} - 5 q^{31} - 5 q^{33} - 3 q^{35} + 11 q^{37} + 5 q^{39} + 11 q^{41} + 8 q^{43} + 2 q^{45} - 13 q^{47} - 2 q^{49} + 3 q^{51} - 28 q^{53} + 5 q^{55} - q^{57} + 9 q^{61} - 3 q^{63} - 5 q^{65} - 16 q^{67} + 7 q^{69} - 15 q^{71} + 3 q^{73} - 2 q^{75} + 15 q^{77} + 3 q^{79} + 2 q^{81} - 11 q^{83} - 3 q^{85} + 5 q^{87} + 4 q^{89} + 30 q^{91} + 5 q^{93} + q^{95} - 9 q^{97} + 5 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 3 * q^7 + 2 * q^9 + 5 * q^11 - 5 * q^13 - 2 * q^15 - 3 * q^17 + q^19 + 3 * q^21 - 7 * q^23 + 2 * q^25 - 2 * q^27 - 5 * q^29 - 5 * q^31 - 5 * q^33 - 3 * q^35 + 11 * q^37 + 5 * q^39 + 11 * q^41 + 8 * q^43 + 2 * q^45 - 13 * q^47 - 2 * q^49 + 3 * q^51 - 28 * q^53 + 5 * q^55 - q^57 + 9 * q^61 - 3 * q^63 - 5 * q^65 - 16 * q^67 + 7 * q^69 - 15 * q^71 + 3 * q^73 - 2 * q^75 + 15 * q^77 + 3 * q^79 + 2 * q^81 - 11 * q^83 - 3 * q^85 + 5 * q^87 + 4 * q^89 + 30 * q^91 + 5 * q^93 + q^95 - 9 * q^97 + 5 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times$.

$n$	$1141$	$2011$	$2681$	$3217$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	−1.50000	+	2.59808i	−0.566947	+	0.981981i	0.429919	+	0.902867i	$0.358542\pi$
$7$	−1.50000	+	2.59808i	−0.566947	+	0.981981i	−0.996866	+	0.0791130i	$0.974791\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	2.50000	−	4.33013i	0.753778	−	1.30558i	−0.192201	−	0.981356i	$-0.561563\pi$
$11$	2.50000	−	4.33013i	0.753778	−	1.30558i	0.945979	−	0.324227i	$-0.105104\pi$
$12$		0			0
$13$	−2.50000	−	4.33013i	−0.693375	−	1.20096i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−2.50000	−	4.33013i	−0.693375	−	1.20096i	0.277350	−	0.960769i	$-0.410544\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	0.624780	−	0.780801i	$-0.285189\pi$
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	−0.988583	+	0.150675i	$0.951855\pi$
$18$		0			0
$19$	0.500000	+	0.866025i	0.114708	+	0.198680i	0.917663	−	0.397360i	$-0.130073\pi$
$19$	0.500000	+	0.866025i	0.114708	+	0.198680i	−0.802955	+	0.596040i	$0.796740\pi$
$20$		0			0
$21$	1.50000	−	2.59808i	0.327327	−	0.566947i
$22$		0			0
$23$	−3.50000	−	6.06218i	−0.729800	−	1.26405i	−0.956967	−	0.290196i	$-0.906280\pi$
$23$	−3.50000	−	6.06218i	−0.729800	−	1.26405i	0.227167	−	0.973856i	$-0.427054\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−2.50000	+	4.33013i	−0.464238	+	0.804084i	−0.999167	−	0.0408130i	$-0.987005\pi$
$29$	−2.50000	+	4.33013i	−0.464238	+	0.804084i	0.534928	+	0.844897i	$0.320339\pi$
$30$		0			0
$31$	−2.50000	+	4.33013i	−0.449013	+	0.777714i	−0.998322	−	0.0579057i	$-0.981558\pi$
$31$	−2.50000	+	4.33013i	−0.449013	+	0.777714i	0.549309	+	0.835619i	$0.314891\pi$
$32$		0			0
$33$	−2.50000	+	4.33013i	−0.435194	+	0.753778i
$34$		0			0
$35$	−1.50000	+	2.59808i	−0.253546	+	0.439155i
$36$		0			0
$37$	5.50000	+	9.52628i	0.904194	+	1.56611i	0.821995	+	0.569495i	$0.192861\pi$
$37$	5.50000	+	9.52628i	0.904194	+	1.56611i	0.0821995	+	0.996616i	$0.473806\pi$
$38$		0			0
$39$	2.50000	+	4.33013i	0.400320	+	0.693375i
$40$		0			0
$41$	5.50000	−	9.52628i	0.858956	−	1.48775i	−0.0139704	−	0.999902i	$-0.504447\pi$
$41$	5.50000	−	9.52628i	0.858956	−	1.48775i	0.872926	−	0.487852i	$-0.162220\pi$
$42$		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
$45$	1.00000			0.149071
$46$		0			0
$47$	−6.50000	+	11.2583i	−0.948122	+	1.64220i	−0.198747	+	0.980051i	$0.563687\pi$
$47$	−6.50000	+	11.2583i	−0.948122	+	1.64220i	−0.749375	+	0.662145i	$0.769646\pi$
$48$		0			0
$49$	−1.00000	−	1.73205i	−0.142857	−	0.247436i
$50$		0			0
$51$	1.50000	+	2.59808i	0.210042	+	0.363803i
$52$		0			0
$53$	−14.0000			−1.92305			−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000			−1.92305			−0.961524	+	0.274721i	$0.911414\pi$
$54$		0			0
$55$	2.50000	−	4.33013i	0.337100	−	0.583874i
$56$		0			0
$57$	−0.500000	−	0.866025i	−0.0662266	−	0.114708i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	4.50000	+	7.79423i	0.576166	+	0.997949i	0.995914	+	0.0903080i	$0.0287851\pi$
$61$	4.50000	+	7.79423i	0.576166	+	0.997949i	−0.419748	+	0.907641i	$0.637882\pi$
$62$		0			0
$63$	−1.50000	+	2.59808i	−0.188982	+	0.327327i
$64$		0			0
$65$	−2.50000	−	4.33013i	−0.310087	−	0.537086i
$66$		0			0
$67$	−8.00000	+	1.73205i	−0.977356	+	0.211604i
$67$	−8.00000	+	1.73205i	−0.977356	+	0.211604i
$68$		0			0
$69$	3.50000	+	6.06218i	0.421350	+	0.729800i
$70$		0			0
$71$	−7.50000	+	12.9904i	−0.890086	+	1.54167i	−0.0503155	+	0.998733i	$0.516023\pi$
$71$	−7.50000	+	12.9904i	−0.890086	+	1.54167i	−0.839771	+	0.542941i	$0.817311\pi$
$72$		0			0
$73$	1.50000	+	2.59808i	0.175562	+	0.304082i	0.940356	−	0.340193i	$-0.110493\pi$
$73$	1.50000	+	2.59808i	0.175562	+	0.304082i	−0.764794	+	0.644275i	$0.777159\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		0			0
$77$	7.50000	+	12.9904i	0.854704	+	1.48039i
$78$		0			0
$79$	1.50000	−	2.59808i	0.168763	−	0.292306i	−0.769222	−	0.638982i	$-0.779356\pi$
$79$	1.50000	−	2.59808i	0.168763	−	0.292306i	0.937985	+	0.346675i	$0.112689\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−5.50000	−	9.52628i	−0.603703	−	1.04565i	−0.992255	−	0.124218i	$-0.960358\pi$
$83$	−5.50000	−	9.52628i	−0.603703	−	1.04565i	0.388552	−	0.921427i	$-0.372976\pi$
$84$		0			0
$85$	−1.50000	−	2.59808i	−0.162698	−	0.281801i
$86$		0			0
$87$	2.50000	−	4.33013i	0.268028	−	0.464238i
$88$		0			0
$89$	2.00000			0.212000			0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000			0.212000			0.106000	+	0.994366i	$0.466196\pi$
$90$		0			0
$91$	15.0000			1.57243
$92$		0			0
$93$	2.50000	−	4.33013i	0.259238	−	0.449013i
$94$		0			0
$95$	0.500000	+	0.866025i	0.0512989	+	0.0888523i
$96$		0			0
$97$	−4.50000	−	7.79423i	−0.456906	−	0.791384i	0.541890	−	0.840450i	$-0.317709\pi$
$97$	−4.50000	−	7.79423i	−0.456906	−	0.791384i	−0.998796	+	0.0490655i	$0.984376\pi$
$98$		0			0
$99$	2.50000	−	4.33013i	0.251259	−	0.435194i
$100$		0			0
$101$	3.50000	−	6.06218i	0.348263	−	0.603209i	−0.637678	−	0.770303i	$-0.720105\pi$
$101$	3.50000	−	6.06218i	0.348263	−	0.603209i	0.985941	+	0.167094i	$0.0534383\pi$
$102$		0			0
$103$	−5.50000	+	9.52628i	−0.541931	+	0.938652i	0.456862	+	0.889538i	$0.348973\pi$
$103$	−5.50000	+	9.52628i	−0.541931	+	0.938652i	−0.998793	+	0.0491146i	$0.984360\pi$
$104$		0			0
$105$	1.50000	−	2.59808i	0.146385	−	0.253546i
$106$		0			0
$107$	4.00000			0.386695			0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000			0.386695			0.193347	+	0.981130i	$0.438066\pi$
$108$		0			0
$109$	−18.0000			−1.72409			−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000			−1.72409			−0.862044	+	0.506834i	$0.830816\pi$
$110$		0			0
$111$	−5.50000	−	9.52628i	−0.522037	−	0.904194i
$112$		0			0
$113$	8.50000	−	14.7224i	0.799613	−	1.38497i	−0.120256	−	0.992743i	$-0.538371\pi$
$113$	8.50000	−	14.7224i	0.799613	−	1.38497i	0.919868	−	0.392227i	$-0.128295\pi$
$114$		0			0
$115$	−3.50000	−	6.06218i	−0.326377	−	0.565301i
$116$		0			0
$117$	−2.50000	−	4.33013i	−0.231125	−	0.400320i
$118$		0			0
$119$	9.00000			0.825029
$120$		0			0
$121$	−7.00000	−	12.1244i	−0.636364	−	1.10221i
$122$		0			0
$123$	−5.50000	+	9.52628i	−0.495918	+	0.858956i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	2.50000	−	4.33013i	0.221839	−	0.384237i	−0.733527	−	0.679660i	$-0.762127\pi$
$127$	2.50000	−	4.33013i	0.221839	−	0.384237i	0.955366	+	0.295423i	$0.0954607\pi$
$128$		0			0
$129$	−4.00000			−0.352180
$130$		0			0
$131$	−20.0000			−1.74741			−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000			−1.74741			−0.873704	+	0.486458i	$0.838289\pi$
$132$		0			0
$133$	−3.00000			−0.260133
$134$		0			0
$135$	−1.00000			−0.0860663
$136$		0			0
$137$	−18.0000			−1.53784			−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000			−1.53784			−0.768922	+	0.639343i	$0.779207\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$	6.50000	−	11.2583i	0.547399	−	0.948122i
$142$		0			0
$143$	−25.0000			−2.09061
$144$		0			0
$145$	−2.50000	+	4.33013i	−0.207614	+	0.359597i
$146$		0			0
$147$	1.00000	+	1.73205i	0.0824786	+	0.142857i
$148$		0			0
$149$	14.0000			1.14692			0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000			1.14692			0.573462	+	0.819232i	$0.305600\pi$
$150$		0			0
$151$	2.50000	+	4.33013i	0.203447	+	0.352381i	0.949637	−	0.313353i	$-0.101452\pi$
$151$	2.50000	+	4.33013i	0.203447	+	0.352381i	−0.746190	+	0.665733i	$0.768119\pi$
$152$		0			0
$153$	−1.50000	−	2.59808i	−0.121268	−	0.210042i
$154$		0			0
$155$	−2.50000	+	4.33013i	−0.200805	+	0.347804i
$156$		0			0
$157$	−6.50000	−	11.2583i	−0.518756	−	0.898513i	−0.999762	−	0.0217953i	$-0.993062\pi$
$157$	−6.50000	−	11.2583i	−0.518756	−	0.898513i	0.481006	−	0.876717i	$-0.340272\pi$
$158$		0			0
$159$	14.0000			1.11027
$160$		0			0
$161$	21.0000			1.65503
$162$		0			0
$163$	−9.50000	+	16.4545i	−0.744097	+	1.28881i	0.206518	+	0.978443i	$0.433787\pi$
$163$	−9.50000	+	16.4545i	−0.744097	+	1.28881i	−0.950615	+	0.310372i	$0.899546\pi$
$164$		0			0
$165$	−2.50000	+	4.33013i	−0.194625	+	0.337100i
$166$		0			0
$167$	1.50000	−	2.59808i	0.116073	−	0.201045i	−0.802135	−	0.597143i	$-0.796303\pi$
$167$	1.50000	−	2.59808i	0.116073	−	0.201045i	0.918208	+	0.396098i	$0.129636\pi$
$168$		0			0
$169$	−6.00000	+	10.3923i	−0.461538	+	0.799408i
$170$		0			0
$171$	0.500000	+	0.866025i	0.0382360	+	0.0662266i
$172$		0			0
$173$	2.50000	+	4.33013i	0.190071	+	0.329213i	0.945274	−	0.326278i	$-0.105795\pi$
$173$	2.50000	+	4.33013i	0.190071	+	0.329213i	−0.755202	+	0.655492i	$0.772461\pi$
$174$		0			0
$175$	−1.50000	+	2.59808i	−0.113389	+	0.196396i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−20.0000			−1.49487			−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000			−1.49487			−0.747435	+	0.664335i	$0.768715\pi$
$180$		0			0
$181$	−3.50000	+	6.06218i	−0.260153	+	0.450598i	−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−3.50000	+	6.06218i	−0.260153	+	0.450598i	0.706129	+	0.708083i	$0.250440\pi$
$182$		0			0
$183$	−4.50000	−	7.79423i	−0.332650	−	0.576166i
$184$		0			0
$185$	5.50000	+	9.52628i	0.404368	+	0.700386i
$186$		0			0
$187$	−15.0000			−1.09691
$188$		0			0
$189$	1.50000	−	2.59808i	0.109109	−	0.188982i
$190$		0			0
$191$	−2.50000	−	4.33013i	−0.180894	−	0.313317i	0.761291	−	0.648410i	$-0.224566\pi$
$191$	−2.50000	−	4.33013i	−0.180894	−	0.313317i	−0.942185	+	0.335093i	$0.891232\pi$
$192$		0			0
$193$	−14.0000			−1.00774			−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000			−1.00774			−0.503871	+	0.863779i	$0.668091\pi$
$194$		0			0
$195$	2.50000	+	4.33013i	0.179029	+	0.310087i
$196$		0			0
$197$	−1.50000	+	2.59808i	−0.106871	+	0.185105i	−0.914501	−	0.404584i	$-0.867416\pi$
$197$	−1.50000	+	2.59808i	−0.106871	+	0.185105i	0.807630	+	0.589689i	$0.200750\pi$
$198$		0			0
$199$	10.5000	+	18.1865i	0.744325	+	1.28921i	0.950509	+	0.310696i	$0.100562\pi$
$199$	10.5000	+	18.1865i	0.744325	+	1.28921i	−0.206184	+	0.978513i	$0.566105\pi$
$200$		0			0
$201$	8.00000	−	1.73205i	0.564276	−	0.122169i
$202$		0			0
$203$	−7.50000	−	12.9904i	−0.526397	−	0.911746i
$204$		0			0
$205$	5.50000	−	9.52628i	0.384137	−	0.665344i
$206$		0			0
$207$	−3.50000	−	6.06218i	−0.243267	−	0.421350i
$208$		0			0
$209$	5.00000			0.345857
$210$		0			0
$211$	4.50000	+	7.79423i	0.309793	+	0.536577i	0.978317	−	0.207114i	$-0.0664070\pi$
$211$	4.50000	+	7.79423i	0.309793	+	0.536577i	−0.668524	+	0.743690i	$0.733074\pi$
$212$		0			0
$213$	7.50000	−	12.9904i	0.513892	−	0.890086i
$214$		0			0
$215$	4.00000			0.272798
$216$		0			0
$217$	−7.50000	−	12.9904i	−0.509133	−	0.881845i
$218$		0			0
$219$	−1.50000	−	2.59808i	−0.101361	−	0.175562i
$220$		0			0
$221$	−7.50000	+	12.9904i	−0.504505	+	0.873828i
$222$		0			0
$223$	8.00000			0.535720			0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000			0.535720			0.267860	+	0.963458i	$0.413684\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−2.50000	+	4.33013i	−0.165931	+	0.287401i	−0.936985	−	0.349368i	$-0.886396\pi$
$227$	−2.50000	+	4.33013i	−0.165931	+	0.287401i	0.771055	+	0.636769i	$0.219730\pi$
$228$		0			0
$229$	−3.50000	−	6.06218i	−0.231287	−	0.400600i	0.726900	−	0.686743i	$-0.240960\pi$
$229$	−3.50000	−	6.06218i	−0.231287	−	0.400600i	−0.958187	+	0.286143i	$0.907627\pi$
$230$		0			0
$231$	−7.50000	−	12.9904i	−0.493464	−	0.854704i
$232$		0			0
$233$	4.50000	−	7.79423i	0.294805	−	0.510617i	−0.680135	−	0.733087i	$-0.738079\pi$
$233$	4.50000	−	7.79423i	0.294805	−	0.510617i	0.974939	+	0.222470i	$0.0714120\pi$
$234$		0			0
$235$	−6.50000	+	11.2583i	−0.424013	+	0.734412i
$236$		0			0
$237$	−1.50000	+	2.59808i	−0.0974355	+	0.168763i
$238$		0			0
$239$	−1.50000	+	2.59808i	−0.0970269	+	0.168056i	−0.910453	−	0.413613i	$-0.864267\pi$
$239$	−1.50000	+	2.59808i	−0.0970269	+	0.168056i	0.813426	+	0.581669i	$0.197600\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	−1.00000	−	1.73205i	−0.0638877	−	0.110657i
$246$		0			0
$247$	2.50000	−	4.33013i	0.159071	−	0.275519i
$248$		0			0
$249$	5.50000	+	9.52628i	0.348548	+	0.603703i
$250$		0			0
$251$	15.5000	+	26.8468i	0.978351	+	1.69455i	0.668400	+	0.743802i	$0.266979\pi$
$251$	15.5000	+	26.8468i	0.978351	+	1.69455i	0.309951	+	0.950753i	$0.399687\pi$
$252$		0			0
$253$	−35.0000			−2.20043
$254$		0			0
$255$	1.50000	+	2.59808i	0.0939336	+	0.162698i
$256$		0			0
$257$	12.5000	−	21.6506i	0.779729	−	1.35053i	−0.152370	−	0.988324i	$-0.548690\pi$
$257$	12.5000	−	21.6506i	0.779729	−	1.35053i	0.932098	−	0.362206i	$-0.117976\pi$
$258$		0			0
$259$	−33.0000			−2.05052
$260$		0			0
$261$	−2.50000	+	4.33013i	−0.154746	+	0.268028i
$262$		0			0
$263$	8.00000			0.493301			0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000			0.493301			0.246651	+	0.969104i	$0.420670\pi$
$264$		0			0
$265$	−14.0000			−0.860013
$266$		0			0
$267$	−2.00000			−0.122398
$268$		0			0
$269$	2.00000			0.121942			0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000			0.121942			0.0609711	+	0.998140i	$0.480580\pi$
$270$		0			0
$271$	−20.0000			−1.21491			−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000			−1.21491			−0.607457	+	0.794353i	$0.707810\pi$
$272$		0			0
$273$	−15.0000			−0.907841
$274$		0			0
$275$	2.50000	−	4.33013i	0.150756	−	0.261116i
$276$		0			0
$277$	−6.00000			−0.360505			−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000			−0.360505			−0.180253	+	0.983620i	$0.557691\pi$
$278$		0			0
$279$	−2.50000	+	4.33013i	−0.149671	+	0.259238i
$280$		0			0
$281$	−6.50000	−	11.2583i	−0.387757	−	0.671616i	0.604390	−	0.796689i	$-0.293417\pi$
$281$	−6.50000	−	11.2583i	−0.387757	−	0.671616i	−0.992148	+	0.125073i	$0.960084\pi$
$282$		0			0
$283$	4.00000			0.237775			0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000			0.237775			0.118888	+	0.992908i	$0.462067\pi$
$284$		0			0
$285$	−0.500000	−	0.866025i	−0.0296174	−	0.0512989i
$286$		0			0
$287$	16.5000	+	28.5788i	0.973964	+	1.68696i
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$		0			0
$291$	4.50000	+	7.79423i	0.263795	+	0.456906i
$292$		0			0
$293$	18.0000			1.05157			0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000			1.05157			0.525786	+	0.850617i	$0.323771\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−2.50000	+	4.33013i	−0.145065	+	0.251259i
$298$		0			0
$299$	−17.5000	+	30.3109i	−1.01205	+	1.75292i
$300$		0			0
$301$	−6.00000	+	10.3923i	−0.345834	+	0.599002i
$302$		0			0
$303$	−3.50000	+	6.06218i	−0.201070	+	0.348263i
$304$		0			0
$305$	4.50000	+	7.79423i	0.257669	+	0.446296i
$306$		0			0
$307$	11.5000	+	19.9186i	0.656340	+	1.13681i	0.981556	+	0.191174i	$0.0612295\pi$
$307$	11.5000	+	19.9186i	0.656340	+	1.13681i	−0.325216	+	0.945640i	$0.605437\pi$
$308$		0			0
$309$	5.50000	−	9.52628i	0.312884	−	0.541931i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−30.0000			−1.69570			−0.847850	−	0.530236i	$-0.822103\pi$
$313$	−30.0000			−1.69570			−0.847850	+	0.530236i	$0.822103\pi$
$314$		0			0
$315$	−1.50000	+	2.59808i	−0.0845154	+	0.146385i
$316$		0			0
$317$	−11.5000	−	19.9186i	−0.645904	−	1.11874i	−0.984092	−	0.177660i	$-0.943147\pi$
$317$	−11.5000	−	19.9186i	−0.645904	−	1.11874i	0.338188	−	0.941079i	$-0.390186\pi$
$318$		0			0
$319$	12.5000	+	21.6506i	0.699866	+	1.21220i
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$	1.50000	−	2.59808i	0.0834622	−	0.144561i
$324$		0			0
$325$	−2.50000	−	4.33013i	−0.138675	−	0.240192i
$326$		0			0
$327$	18.0000			0.995402
$328$		0			0
$329$	−19.5000	−	33.7750i	−1.07507	−	1.86208i
$330$		0			0
$331$	7.50000	−	12.9904i	0.412237	−	0.714016i	−0.582897	−	0.812546i	$-0.698081\pi$
$331$	7.50000	−	12.9904i	0.412237	−	0.714016i	0.995134	+	0.0985303i	$0.0314141\pi$
$332$		0			0
$333$	5.50000	+	9.52628i	0.301398	+	0.522037i
$334$		0			0
$335$	−8.00000	+	1.73205i	−0.437087	+	0.0946320i
$336$		0			0
$337$	−8.50000	−	14.7224i	−0.463025	−	0.801982i	0.536085	−	0.844164i	$-0.319902\pi$
$337$	−8.50000	−	14.7224i	−0.463025	−	0.801982i	−0.999110	+	0.0421818i	$0.986569\pi$
$338$		0			0
$339$	−8.50000	+	14.7224i	−0.461657	+	0.799613i
$340$		0			0
$341$	12.5000	+	21.6506i	0.676913	+	1.17245i
$342$		0			0
$343$	−15.0000			−0.809924
$344$		0			0
$345$	3.50000	+	6.06218i	0.188434	+	0.326377i
$346$		0			0
$347$	−12.5000	+	21.6506i	−0.671035	+	1.16227i	0.306576	+	0.951846i	$0.400817\pi$
$347$	−12.5000	+	21.6506i	−0.671035	+	1.16227i	−0.977611	+	0.210421i	$0.932517\pi$
$348$		0			0
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$	2.50000	+	4.33013i	0.133440	+	0.231125i
$352$		0			0
$353$	10.5000	+	18.1865i	0.558859	+	0.967972i	0.997592	+	0.0693543i	$0.0220939\pi$
$353$	10.5000	+	18.1865i	0.558859	+	0.967972i	−0.438733	+	0.898617i	$0.644573\pi$
$354$		0			0
$355$	−7.50000	+	12.9904i	−0.398059	+	0.689458i
$356$		0			0
$357$	−9.00000			−0.476331
$358$		0			0
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	9.00000	−	15.5885i	0.473684	−	0.820445i
$362$		0			0
$363$	7.00000	+	12.1244i	0.367405	+	0.636364i
$364$		0			0
$365$	1.50000	+	2.59808i	0.0785136	+	0.135990i
$366$		0			0
$367$	6.50000	−	11.2583i	0.339297	−	0.587680i	−0.645003	−	0.764180i	$-0.723144\pi$
$367$	6.50000	−	11.2583i	0.339297	−	0.587680i	0.984301	+	0.176500i	$0.0564774\pi$
$368$		0			0
$369$	5.50000	−	9.52628i	0.286319	−	0.495918i
$370$		0			0
$371$	21.0000	−	36.3731i	1.09027	−	1.88840i
$372$		0			0
$373$	−4.50000	+	7.79423i	−0.233001	+	0.403570i	−0.958690	−	0.284453i	$-0.908188\pi$
$373$	−4.50000	+	7.79423i	−0.233001	+	0.403570i	0.725689	+	0.688023i	$0.241521\pi$
$374$		0			0
$375$	−1.00000			−0.0516398
$376$		0			0
$377$	25.0000			1.28757
$378$		0			0
$379$	0.500000	+	0.866025i	0.0256833	+	0.0444847i	0.878581	−	0.477593i	$-0.158491\pi$
$379$	0.500000	+	0.866025i	0.0256833	+	0.0444847i	−0.852898	+	0.522077i	$0.825157\pi$
$380$		0			0
$381$	−2.50000	+	4.33013i	−0.128079	+	0.221839i
$382$		0			0
$383$	14.5000	+	25.1147i	0.740915	+	1.28330i	0.952079	+	0.305852i	$0.0989414\pi$
$383$	14.5000	+	25.1147i	0.740915	+	1.28330i	−0.211164	+	0.977451i	$0.567725\pi$
$384$		0			0
$385$	7.50000	+	12.9904i	0.382235	+	0.662051i
$386$		0			0
$387$	4.00000			0.203331
$388$		0			0
$389$	3.50000	+	6.06218i	0.177457	+	0.307365i	0.941009	−	0.338382i	$-0.109880\pi$
$389$	3.50000	+	6.06218i	0.177457	+	0.307365i	−0.763552	+	0.645747i	$0.776546\pi$
$390$		0			0
$391$	−10.5000	+	18.1865i	−0.531008	+	0.919733i
$392$		0			0
$393$	20.0000			1.00887
$394$		0			0
$395$	1.50000	−	2.59808i	0.0754732	−	0.130723i
$396$		0			0
$397$	−34.0000			−1.70641			−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000			−1.70641			−0.853206	+	0.521575i	$0.825345\pi$
$398$		0			0
$399$	3.00000			0.150188
$400$		0			0
$401$	30.0000			1.49813			0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000			1.49813			0.749064	+	0.662497i	$0.230503\pi$
$402$		0			0
$403$	25.0000			1.24534
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$	55.0000			2.72625
$408$		0			0
$409$	10.5000	−	18.1865i	0.519192	−	0.899266i	−0.480560	−	0.876962i	$-0.659566\pi$
$409$	10.5000	−	18.1865i	0.519192	−	0.899266i	0.999751	−	0.0223042i	$-0.00710022\pi$
$410$		0			0
$411$	18.0000			0.887875
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−5.50000	−	9.52628i	−0.269984	−	0.467627i
$416$		0			0
$417$	4.00000			0.195881
$418$		0			0
$419$	−10.5000	−	18.1865i	−0.512959	−	0.888470i	−0.999887	−	0.0150285i	$-0.995216\pi$
$419$	−10.5000	−	18.1865i	−0.512959	−	0.888470i	0.486928	−	0.873442i	$-0.338117\pi$
$420$		0			0
$421$	2.50000	+	4.33013i	0.121843	+	0.211037i	0.920494	−	0.390756i	$-0.127786\pi$
$421$	2.50000	+	4.33013i	0.121843	+	0.211037i	−0.798652	+	0.601793i	$0.794453\pi$
$422$		0			0
$423$	−6.50000	+	11.2583i	−0.316041	+	0.547399i
$424$		0			0
$425$	−1.50000	−	2.59808i	−0.0727607	−	0.126025i
$426$		0			0
$427$	−27.0000			−1.30662
$428$		0			0
$429$	25.0000			1.20701
$430$		0			0
$431$	0.500000	−	0.866025i	0.0240842	−	0.0417150i	−0.853732	−	0.520712i	$-0.825666\pi$
$431$	0.500000	−	0.866025i	0.0240842	−	0.0417150i	0.877816	+	0.478997i	$0.159000\pi$
$432$		0			0
$433$	−18.5000	+	32.0429i	−0.889053	+	1.53989i	−0.0480569	+	0.998845i	$0.515303\pi$
$433$	−18.5000	+	32.0429i	−0.889053	+	1.53989i	−0.840996	+	0.541041i	$0.818030\pi$
$434$		0			0
$435$	2.50000	−	4.33013i	0.119866	−	0.207614i
$436$		0			0
$437$	3.50000	−	6.06218i	0.167428	−	0.289993i
$438$		0			0
$439$	−9.50000	−	16.4545i	−0.453410	−	0.785330i	0.545185	−	0.838316i	$-0.316459\pi$
$439$	−9.50000	−	16.4545i	−0.453410	−	0.785330i	−0.998595	+	0.0529862i	$0.983126\pi$
$440$		0			0
$441$	−1.00000	−	1.73205i	−0.0476190	−	0.0824786i
$442$		0			0
$443$	−10.5000	+	18.1865i	−0.498870	+	0.864068i	−0.999999	−	0.00130426i	$-0.999585\pi$
$443$	−10.5000	+	18.1865i	−0.498870	+	0.864068i	0.501129	+	0.865373i	$0.332918\pi$
$444$		0			0
$445$	2.00000			0.0948091
$446$		0			0
$447$	−14.0000			−0.662177
$448$		0			0
$449$	−10.5000	+	18.1865i	−0.495526	+	0.858276i	−0.999987	−	0.00515887i	$-0.998358\pi$
$449$	−10.5000	+	18.1865i	−0.495526	+	0.858276i	0.504461	+	0.863434i	$0.331691\pi$
$450$		0			0
$451$	−27.5000	−	47.6314i	−1.29492	−	2.24287i
$452$		0			0
$453$	−2.50000	−	4.33013i	−0.117460	−	0.203447i
$454$		0			0
$455$	15.0000			0.703211
$456$		0			0
$457$	−16.5000	+	28.5788i	−0.771837	+	1.33686i	0.164717	+	0.986341i	$0.447329\pi$
$457$	−16.5000	+	28.5788i	−0.771837	+	1.33686i	−0.936555	+	0.350521i	$0.886005\pi$
$458$		0			0
$459$	1.50000	+	2.59808i	0.0700140	+	0.121268i
$460$		0			0
$461$	−42.0000			−1.95614			−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000			−1.95614			−0.978068	+	0.208288i	$0.933211\pi$
$462$		0			0
$463$	−2.50000	−	4.33013i	−0.116185	−	0.201238i	0.802068	−	0.597233i	$-0.203733\pi$
$463$	−2.50000	−	4.33013i	−0.116185	−	0.201238i	−0.918253	+	0.395995i	$0.870400\pi$
$464$		0			0
$465$	2.50000	−	4.33013i	0.115935	−	0.200805i
$466$		0			0
$467$	−7.50000	−	12.9904i	−0.347059	−	0.601123i	0.638667	−	0.769483i	$-0.279486\pi$
$467$	−7.50000	−	12.9904i	−0.347059	−	0.601123i	−0.985726	+	0.168360i	$0.946153\pi$
$468$		0			0
$469$	7.50000	−	23.3827i	0.346318	−	1.07971i
$470$		0			0
$471$	6.50000	+	11.2583i	0.299504	+	0.518756i
$472$		0			0
$473$	10.0000	−	17.3205i	0.459800	−	0.796398i
$474$		0			0
$475$	0.500000	+	0.866025i	0.0229416	+	0.0397360i
$476$		0			0
$477$	−14.0000			−0.641016
$478$		0			0
$479$	−10.5000	−	18.1865i	−0.479757	−	0.830964i	0.519973	−	0.854183i	$-0.325942\pi$
$479$	−10.5000	−	18.1865i	−0.479757	−	0.830964i	−0.999730	+	0.0232187i	$0.992609\pi$
$480$		0			0
$481$	27.5000	−	47.6314i	1.25389	−	2.17180i
$482$		0			0
$483$	−21.0000			−0.955533
$484$		0			0
$485$	−4.50000	−	7.79423i	−0.204334	−	0.353918i
$486$		0			0
$487$	−16.5000	−	28.5788i	−0.747686	−	1.29503i	−0.948929	−	0.315489i	$-0.897831\pi$
$487$	−16.5000	−	28.5788i	−0.747686	−	1.29503i	0.201243	−	0.979541i	$-0.435502\pi$
$488$		0			0
$489$	9.50000	−	16.4545i	0.429605	−	0.744097i
$490$		0			0
$491$	−8.00000			−0.361035			−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000			−0.361035			−0.180517	+	0.983572i	$0.557777\pi$
$492$		0			0
$493$	15.0000			0.675566
$494$		0			0
$495$	2.50000	−	4.33013i	0.112367	−	0.194625i
$496$		0			0
$497$	−22.5000	−	38.9711i	−1.00926	−	1.74809i
$498$		0			0
$499$	−19.5000	−	33.7750i	−0.872940	−	1.51198i	−0.858941	−	0.512074i	$-0.828877\pi$
$499$	−19.5000	−	33.7750i	−0.872940	−	1.51198i	−0.0139987	−	0.999902i	$-0.504456\pi$
$500$		0			0
$501$	−1.50000	+	2.59808i	−0.0670151	+	0.116073i
$502$		0			0
$503$	11.5000	−	19.9186i	0.512760	−	0.888126i	−0.487131	−	0.873329i	$-0.661957\pi$
$503$	11.5000	−	19.9186i	0.512760	−	0.888126i	0.999891	−	0.0147968i	$-0.00471014\pi$
$504$		0			0
$505$	3.50000	−	6.06218i	0.155748	−	0.269763i
$506$		0			0
$507$	6.00000	−	10.3923i	0.266469	−	0.461538i
$508$		0			0
$509$	−18.0000			−0.797836			−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000			−0.797836			−0.398918	+	0.916987i	$0.630614\pi$
$510$		0			0
$511$	−9.00000			−0.398137
$512$		0			0
$513$	−0.500000	−	0.866025i	−0.0220755	−	0.0382360i
$514$		0			0
$515$	−5.50000	+	9.52628i	−0.242359	+	0.419778i
$516$		0			0
$517$	32.5000	+	56.2917i	1.42935	+	2.47570i
$518$		0			0
$519$	−2.50000	−	4.33013i	−0.109738	−	0.190071i
$520$		0			0
$521$	10.0000			0.438108			0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000			0.438108			0.219054	+	0.975713i	$0.429703\pi$
$522$		0			0
$523$	5.50000	+	9.52628i	0.240498	+	0.416555i	0.960856	−	0.277047i	$-0.0893559\pi$
$523$	5.50000	+	9.52628i	0.240498	+	0.416555i	−0.720358	+	0.693602i	$0.756023\pi$
$524$		0			0
$525$	1.50000	−	2.59808i	0.0654654	−	0.113389i
$526$		0			0
$527$	15.0000			0.653410
$528$		0			0
$529$	−13.0000	+	22.5167i	−0.565217	+	0.978985i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−55.0000			−2.38231
$534$		0			0
$535$	4.00000			0.172935
$536$		0			0
$537$	20.0000			0.863064
$538$		0			0
$539$	−10.0000			−0.430730
$540$		0			0
$541$	10.0000			0.429934			0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000			0.429934			0.214967	+	0.976621i	$0.431036\pi$
$542$		0			0
$543$	3.50000	−	6.06218i	0.150199	−	0.260153i
$544$		0			0
$545$	−18.0000			−0.771035
$546$		0			0
$547$	16.5000	−	28.5788i	0.705489	−	1.22194i	−0.261026	−	0.965332i	$-0.584061\pi$
$547$	16.5000	−	28.5788i	0.705489	−	1.22194i	0.966515	−	0.256611i	$-0.0826059\pi$
$548$		0			0
$549$	4.50000	+	7.79423i	0.192055	+	0.332650i
$550$		0			0
$551$	−5.00000			−0.213007
$552$		0			0
$553$	4.50000	+	7.79423i	0.191359	+	0.331444i
$554$		0			0
$555$	−5.50000	−	9.52628i	−0.233462	−	0.404368i
$556$		0			0
$557$	−3.50000	+	6.06218i	−0.148300	+	0.256863i	−0.930599	−	0.366040i	$-0.880713\pi$
$557$	−3.50000	+	6.06218i	−0.148300	+	0.256863i	0.782299	+	0.622903i	$0.214047\pi$
$558$		0			0
$559$	−10.0000	−	17.3205i	−0.422955	−	0.732579i
$560$		0			0
$561$	15.0000			0.633300
$562$		0			0
$563$	−20.0000			−0.842900			−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000			−0.842900			−0.421450	+	0.906852i	$0.638479\pi$
$564$		0			0
$565$	8.50000	−	14.7224i	0.357598	−	0.619377i
$566$		0			0
$567$	−1.50000	+	2.59808i	−0.0629941	+	0.109109i
$568$		0			0
$569$	13.5000	−	23.3827i	0.565949	−	0.980253i	−0.431011	−	0.902347i	$-0.641843\pi$
$569$	13.5000	−	23.3827i	0.565949	−	0.980253i	0.996961	−	0.0779066i	$-0.0248236\pi$
$570$		0			0
$571$	−0.500000	+	0.866025i	−0.0209243	+	0.0362420i	−0.876298	−	0.481770i	$-0.839994\pi$
$571$	−0.500000	+	0.866025i	−0.0209243	+	0.0362420i	0.855374	+	0.518012i	$0.173328\pi$
$572$		0			0
$573$	2.50000	+	4.33013i	0.104439	+	0.180894i
$574$		0			0
$575$	−3.50000	−	6.06218i	−0.145960	−	0.252810i
$576$		0			0
$577$	3.50000	−	6.06218i	0.145707	−	0.252372i	−0.783930	−	0.620850i	$-0.786788\pi$
$577$	3.50000	−	6.06218i	0.145707	−	0.252372i	0.929636	+	0.368478i	$0.120121\pi$
$578$		0			0
$579$	14.0000			0.581820
$580$		0			0
$581$	33.0000			1.36907
$582$		0			0
$583$	−35.0000	+	60.6218i	−1.44955	+	2.51070i
$584$		0			0
$585$	−2.50000	−	4.33013i	−0.103362	−	0.179029i
$586$		0			0
$587$	22.5000	+	38.9711i	0.928674	+	1.60851i	0.785543	+	0.618808i	$0.212384\pi$
$587$	22.5000	+	38.9711i	0.928674	+	1.60851i	0.143132	+	0.989704i	$0.454283\pi$
$588$		0			0
$589$	−5.00000			−0.206021
$590$		0			0
$591$	1.50000	−	2.59808i	0.0617018	−	0.106871i
$592$		0			0
$593$	10.5000	+	18.1865i	0.431183	+	0.746831i	0.996976	−	0.0777165i	$-0.0247629\pi$
$593$	10.5000	+	18.1865i	0.431183	+	0.746831i	−0.565792	+	0.824548i	$0.691430\pi$
$594$		0			0
$595$	9.00000			0.368964
$596$		0			0
$597$	−10.5000	−	18.1865i	−0.429736	−	0.744325i
$598$		0			0
$599$	0.500000	−	0.866025i	0.0204294	−	0.0353848i	−0.855630	−	0.517588i	$-0.826830\pi$
$599$	0.500000	−	0.866025i	0.0204294	−	0.0353848i	0.876059	+	0.482203i	$0.160163\pi$
$600$		0			0
$601$	18.5000	+	32.0429i	0.754631	+	1.30706i	0.945558	+	0.325455i	$0.105517\pi$
$601$	18.5000	+	32.0429i	0.754631	+	1.30706i	−0.190927	+	0.981604i	$0.561149\pi$
$602$		0			0
$603$	−8.00000	+	1.73205i	−0.325785	+	0.0705346i
$604$		0			0
$605$	−7.00000	−	12.1244i	−0.284590	−	0.492925i
$606$		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
$609$	7.50000	+	12.9904i	0.303915	+	0.526397i
$610$		0			0
$611$	65.0000			2.62962
$612$		0			0
$613$	−20.5000	−	35.5070i	−0.827987	−	1.43412i	−0.899615	−	0.436684i	$-0.856153\pi$
$613$	−20.5000	−	35.5070i	−0.827987	−	1.43412i	0.0716275	−	0.997431i	$-0.477181\pi$
$614$		0			0
$615$	−5.50000	+	9.52628i	−0.221781	+	0.384137i
$616$		0			0
$617$	−2.00000			−0.0805170			−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000			−0.0805170			−0.0402585	+	0.999189i	$0.512818\pi$
$618$		0			0
$619$	10.5000	+	18.1865i	0.422031	+	0.730978i	0.996138	−	0.0878015i	$-0.0279841\pi$
$619$	10.5000	+	18.1865i	0.422031	+	0.730978i	−0.574107	+	0.818780i	$0.694651\pi$
$620$		0			0
$621$	3.50000	+	6.06218i	0.140450	+	0.243267i
$622$		0			0
$623$	−3.00000	+	5.19615i	−0.120192	+	0.208179i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−5.00000			−0.199681
$628$		0			0
$629$	16.5000	−	28.5788i	0.657898	−	1.13951i
$630$		0			0
$631$	4.50000	+	7.79423i	0.179142	+	0.310283i	0.941587	−	0.336770i	$-0.109334\pi$
$631$	4.50000	+	7.79423i	0.179142	+	0.310283i	−0.762445	+	0.647053i	$0.776001\pi$
$632$		0			0
$633$	−4.50000	−	7.79423i	−0.178859	−	0.309793i
$634$		0			0
$635$	2.50000	−	4.33013i	0.0992095	−	0.171836i
$636$		0			0
$637$	−5.00000	+	8.66025i	−0.198107	+	0.343132i
$638$		0			0
$639$	−7.50000	+	12.9904i	−0.296695	+	0.513892i
$640$		0			0
$641$	13.5000	−	23.3827i	0.533218	−	0.923561i	−0.466029	−	0.884769i	$-0.654316\pi$
$641$	13.5000	−	23.3827i	0.533218	−	0.923561i	0.999247	−	0.0387913i	$-0.0123508\pi$
$642$		0			0
$643$	−12.0000			−0.473234			−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000			−0.473234			−0.236617	+	0.971603i	$0.576039\pi$
$644$		0			0
$645$	−4.00000			−0.157500
$646$		0			0
$647$	10.5000	+	18.1865i	0.412798	+	0.714986i	0.995194	−	0.0979182i	$-0.0312184\pi$
$647$	10.5000	+	18.1865i	0.412798	+	0.714986i	−0.582397	+	0.812905i	$0.697885\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	7.50000	+	12.9904i	0.293948	+	0.509133i
$652$		0			0
$653$	−1.50000	−	2.59808i	−0.0586995	−	0.101671i	0.835182	−	0.549973i	$-0.185362\pi$
$653$	−1.50000	−	2.59808i	−0.0586995	−	0.101671i	−0.893882	+	0.448303i	$0.852029\pi$
$654$		0			0
$655$	−20.0000			−0.781465
$656$		0			0
$657$	1.50000	+	2.59808i	0.0585206	+	0.101361i
$658$		0			0
$659$	10.5000	−	18.1865i	0.409022	−	0.708447i	−0.585758	−	0.810486i	$-0.699203\pi$
$659$	10.5000	−	18.1865i	0.409022	−	0.708447i	0.994780	+	0.102039i	$0.0325366\pi$
$660$		0			0
$661$	10.0000			0.388955			0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000			0.388955			0.194477	+	0.980907i	$0.437699\pi$
$662$		0			0
$663$	7.50000	−	12.9904i	0.291276	−	0.504505i
$664$		0			0
$665$	−3.00000			−0.116335
$666$		0			0
$667$	35.0000			1.35521
$668$		0			0
$669$	−8.00000			−0.309298
$670$		0			0
$671$	45.0000			1.73721
$672$		0			0
$673$	−2.00000			−0.0770943			−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000			−0.0770943			−0.0385472	+	0.999257i	$0.512273\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$	4.50000	−	7.79423i	0.172949	−	0.299557i	−0.766501	−	0.642244i	$-0.778004\pi$
$677$	4.50000	−	7.79423i	0.172949	−	0.299557i	0.939450	+	0.342687i	$0.111337\pi$
$678$		0			0
$679$	27.0000			1.03616
$680$		0			0
$681$	2.50000	−	4.33013i	0.0958002	−	0.165931i
$682$		0			0
$683$	−3.50000	−	6.06218i	−0.133924	−	0.231963i	0.791262	−	0.611477i	$-0.209424\pi$
$683$	−3.50000	−	6.06218i	−0.133924	−	0.231963i	−0.925186	+	0.379514i	$0.876091\pi$
$684$		0			0
$685$	−18.0000			−0.687745
$686$		0			0
$687$	3.50000	+	6.06218i	0.133533	+	0.231287i
$688$		0			0
$689$	35.0000	+	60.6218i	1.33339	+	2.30951i
$690$		0			0
$691$	−2.50000	+	4.33013i	−0.0951045	+	0.164726i	−0.909652	−	0.415371i	$-0.863652\pi$
$691$	−2.50000	+	4.33013i	−0.0951045	+	0.164726i	0.814548	+	0.580097i	$0.196985\pi$
$692$		0			0
$693$	7.50000	+	12.9904i	0.284901	+	0.493464i
$694$		0			0
$695$	−4.00000			−0.151729
$696$		0			0
$697$	−33.0000			−1.24996
$698$		0			0
$699$	−4.50000	+	7.79423i	−0.170206	+	0.294805i
$700$		0			0
$701$	7.50000	−	12.9904i	0.283271	−	0.490640i	−0.688917	−	0.724840i	$-0.741914\pi$
$701$	7.50000	−	12.9904i	0.283271	−	0.490640i	0.972188	+	0.234200i	$0.0752470\pi$
$702$		0			0
$703$	−5.50000	+	9.52628i	−0.207436	+	0.359290i
$704$		0			0
$705$	6.50000	−	11.2583i	0.244804	−	0.424013i
$706$		0			0
$707$	10.5000	+	18.1865i	0.394893	+	0.683975i
$708$		0			0
$709$	2.50000	+	4.33013i	0.0938895	+	0.162621i	0.909145	−	0.416481i	$-0.136737\pi$
$709$	2.50000	+	4.33013i	0.0938895	+	0.162621i	−0.815255	+	0.579102i	$0.803403\pi$
$710$		0			0
$711$	1.50000	−	2.59808i	0.0562544	−	0.0974355i
$712$		0			0
$713$	35.0000			1.31076
$714$		0			0
$715$	−25.0000			−0.934947
$716$		0			0
$717$	1.50000	−	2.59808i	0.0560185	−	0.0970269i
$718$		0			0
$719$	3.50000	+	6.06218i	0.130528	+	0.226081i	0.923880	−	0.382682i	$-0.124999\pi$
$719$	3.50000	+	6.06218i	0.130528	+	0.226081i	−0.793352	+	0.608763i	$0.791666\pi$
$720$		0			0
$721$	−16.5000	−	28.5788i	−0.614492	−	1.06433i
$722$		0			0
$723$	−2.00000			−0.0743808
$724$		0			0
$725$	−2.50000	+	4.33013i	−0.0928477	+	0.160817i
$726$		0			0
$727$	17.5000	+	30.3109i	0.649039	+	1.12417i	0.983353	+	0.181707i	$0.0581622\pi$
$727$	17.5000	+	30.3109i	0.649039	+	1.12417i	−0.334314	+	0.942462i	$0.608504\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−6.00000	−	10.3923i	−0.221918	−	0.384373i
$732$		0			0
$733$	−12.5000	+	21.6506i	−0.461698	+	0.799684i	−0.999046	−	0.0436764i	$-0.986093\pi$
$733$	−12.5000	+	21.6506i	−0.461698	+	0.799684i	0.537348	+	0.843361i	$0.319426\pi$
$734$		0			0
$735$	1.00000	+	1.73205i	0.0368856	+	0.0638877i
$736$		0			0
$737$	−12.5000	+	38.9711i	−0.460443	+	1.43552i
$738$		0			0
$739$	−13.5000	−	23.3827i	−0.496606	−	0.860146i	0.503387	−	0.864061i	$-0.332087\pi$
$739$	−13.5000	−	23.3827i	−0.496606	−	0.860146i	−0.999992	+	0.00391517i	$0.998754\pi$
$740$		0			0
$741$	−2.50000	+	4.33013i	−0.0918398	+	0.159071i
$742$		0			0
$743$	−19.5000	−	33.7750i	−0.715386	−	1.23908i	−0.962811	−	0.270177i	$-0.912918\pi$
$743$	−19.5000	−	33.7750i	−0.715386	−	1.23908i	0.247425	−	0.968907i	$-0.420416\pi$
$744$		0			0
$745$	14.0000			0.512920
$746$		0			0
$747$	−5.50000	−	9.52628i	−0.201234	−	0.348548i
$748$		0			0
$749$	−6.00000	+	10.3923i	−0.219235	+	0.379727i
$750$		0			0
$751$	−24.0000			−0.875772			−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000			−0.875772			−0.437886	+	0.899030i	$0.644273\pi$
$752$		0			0
$753$	−15.5000	−	26.8468i	−0.564851	−	0.978351i
$754$		0			0
$755$	2.50000	+	4.33013i	0.0909843	+	0.157589i
$756$		0			0
$757$	19.5000	−	33.7750i	0.708740	−	1.22757i	−0.256585	−	0.966522i	$-0.582597\pi$
$757$	19.5000	−	33.7750i	0.708740	−	1.22757i	0.965325	−	0.261051i	$-0.0840692\pi$
$758$		0			0
$759$	35.0000			1.27042
$760$		0			0
$761$	22.0000			0.797499			0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000			0.797499			0.398750	+	0.917060i	$0.369444\pi$
$762$		0			0
$763$	27.0000	−	46.7654i	0.977466	−	1.69302i
$764$		0			0
$765$	−1.50000	−	2.59808i	−0.0542326	−	0.0939336i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−11.5000	+	19.9186i	−0.414701	+	0.718283i	−0.995397	−	0.0958377i	$-0.969447\pi$
$769$	−11.5000	+	19.9186i	−0.414701	+	0.718283i	0.580696	+	0.814120i	$0.302780\pi$
$770$		0			0
$771$	−12.5000	+	21.6506i	−0.450177	+	0.779729i
$772$		0			0
$773$	−3.50000	+	6.06218i	−0.125886	+	0.218041i	−0.922079	−	0.387002i	$-0.873511\pi$
$773$	−3.50000	+	6.06218i	−0.125886	+	0.218041i	0.796193	+	0.605043i	$0.206844\pi$
$774$		0			0
$775$	−2.50000	+	4.33013i	−0.0898027	+	0.155543i
$776$		0			0
$777$	33.0000			1.18387
$778$		0			0
$779$	11.0000			0.394116
$780$		0			0
$781$	37.5000	+	64.9519i	1.34186	+	2.32416i
$782$		0			0
$783$	2.50000	−	4.33013i	0.0893427	−	0.154746i
$784$		0			0
$785$	−6.50000	−	11.2583i	−0.231995	−	0.401827i
$786$		0			0
$787$	21.5000	+	37.2391i	0.766392	+	1.32743i	0.939507	+	0.342529i	$0.111283\pi$
$787$	21.5000	+	37.2391i	0.766392	+	1.32743i	−0.173115	+	0.984902i	$0.555383\pi$
$788$		0			0
$789$	−8.00000			−0.284808
$790$		0			0
$791$	25.5000	+	44.1673i	0.906676	+	1.57041i
$792$		0			0
$793$	22.5000	−	38.9711i	0.798998	−	1.38391i
$794$		0			0
$795$	14.0000			0.496529
$796$		0			0
$797$	12.5000	−	21.6506i	0.442773	−	0.766905i	−0.555121	−	0.831769i	$-0.687328\pi$
$797$	12.5000	−	21.6506i	0.442773	−	0.766905i	0.997894	+	0.0648645i	$0.0206615\pi$
$798$		0			0
$799$	39.0000			1.37972
$800$		0			0
$801$	2.00000			0.0706665
$802$		0			0
$803$	15.0000			0.529339
$804$		0			0
$805$	21.0000			0.740153
$806$		0			0
$807$	−2.00000			−0.0704033
$808$		0			0
$809$	−22.0000			−0.773479			−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000			−0.773479			−0.386739	+	0.922189i	$0.626399\pi$
$810$		0			0
$811$	13.5000	−	23.3827i	0.474049	−	0.821077i	−0.525509	−	0.850788i	$-0.676125\pi$
$811$	13.5000	−	23.3827i	0.474049	−	0.821077i	0.999559	+	0.0297106i	$0.00945858\pi$
$812$		0			0
$813$	20.0000			0.701431
$814$		0			0
$815$	−9.50000	+	16.4545i	−0.332770	+	0.576375i
$816$		0			0
$817$	2.00000	+	3.46410i	0.0699711	+	0.121194i
$818$		0			0
$819$	15.0000			0.524142
$820$		0			0
$821$	−16.5000	−	28.5788i	−0.575854	−	0.997408i	−0.995948	−	0.0899279i	$-0.971336\pi$
$821$	−16.5000	−	28.5788i	−0.575854	−	0.997408i	0.420094	−	0.907480i	$-0.361997\pi$
$822$		0			0
$823$	19.5000	+	33.7750i	0.679727	+	1.17732i	0.975063	+	0.221929i	$0.0712352\pi$
$823$	19.5000	+	33.7750i	0.679727	+	1.17732i	−0.295336	+	0.955394i	$0.595431\pi$
$824$		0			0
$825$	−2.50000	+	4.33013i	−0.0870388	+	0.150756i
$826$		0			0
$827$	−19.5000	−	33.7750i	−0.678081	−	1.17447i	−0.975558	−	0.219742i	$-0.929478\pi$
$827$	−19.5000	−	33.7750i	−0.678081	−	1.17447i	0.297477	−	0.954729i	$-0.403855\pi$
$828$		0			0
$829$	−14.0000			−0.486240			−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000			−0.486240			−0.243120	+	0.969996i	$0.578171\pi$
$830$		0			0
$831$	6.00000			0.208138
$832$		0			0
$833$	−3.00000	+	5.19615i	−0.103944	+	0.180036i
$834$		0			0
$835$	1.50000	−	2.59808i	0.0519096	−	0.0899101i
$836$		0			0
$837$	2.50000	−	4.33013i	0.0864126	−	0.149671i
$838$		0			0
$839$	12.5000	−	21.6506i	0.431548	−	0.747463i	−0.565459	−	0.824776i	$-0.691301\pi$
$839$	12.5000	−	21.6506i	0.431548	−	0.747463i	0.997007	+	0.0773135i	$0.0246342\pi$
$840$		0			0
$841$	2.00000	+	3.46410i	0.0689655	+	0.119452i
$842$		0			0
$843$	6.50000	+	11.2583i	0.223872	+	0.387757i
$844$		0			0
$845$	−6.00000	+	10.3923i	−0.206406	+	0.357506i
$846$		0			0
$847$	42.0000			1.44314
$848$		0			0
$849$	−4.00000			−0.137280
$850$		0			0
$851$	38.5000	−	66.6840i	1.31976	−	2.28590i
$852$		0			0
$853$	7.50000	+	12.9904i	0.256795	+	0.444782i	0.965382	−	0.260842i	$-0.0840001\pi$
$853$	7.50000	+	12.9904i	0.256795	+	0.444782i	−0.708586	+	0.705624i	$0.750667\pi$
$854$		0			0
$855$	0.500000	+	0.866025i	0.0170996	+	0.0296174i
$856$		0			0
$857$	42.0000			1.43469			0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000			1.43469			0.717346	+	0.696717i	$0.245357\pi$
$858$		0			0
$859$	−0.500000	+	0.866025i	−0.0170598	+	0.0295484i	−0.874429	−	0.485153i	$-0.838764\pi$
$859$	−0.500000	+	0.866025i	−0.0170598	+	0.0295484i	0.857369	+	0.514701i	$0.172097\pi$
$860$		0			0
$861$	−16.5000	−	28.5788i	−0.562318	−	0.973964i
$862$		0			0
$863$	48.0000			1.63394			0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000			1.63394			0.816970	+	0.576681i	$0.195652\pi$
$864$		0			0
$865$	2.50000	+	4.33013i	0.0850026	+	0.147229i
$866$		0			0
$867$	−4.00000	+	6.92820i	−0.135847	+	0.235294i
$868$		0			0
$869$	−7.50000	−	12.9904i	−0.254420	−	0.440668i
$870$		0			0
$871$	27.5000	+	30.3109i	0.931802	+	1.02705i
$872$		0			0
$873$	−4.50000	−	7.79423i	−0.152302	−	0.263795i
$874$		0			0
$875$	−1.50000	+	2.59808i	−0.0507093	+	0.0878310i
$876$		0			0
$877$	29.5000	+	51.0955i	0.996144	+	1.72537i	0.574049	+	0.818821i	$0.305372\pi$
$877$	29.5000	+	51.0955i	0.996144	+	1.72537i	0.422095	+	0.906552i	$0.361295\pi$
$878$		0			0
$879$	−18.0000			−0.607125
$880$		0			0
$881$	25.5000	+	44.1673i	0.859117	+	1.48803i	0.872772	+	0.488127i	$0.162320\pi$
$881$	25.5000	+	44.1673i	0.859117	+	1.48803i	−0.0136556	+	0.999907i	$0.504347\pi$
$882$		0			0
$883$	−9.50000	+	16.4545i	−0.319700	+	0.553737i	−0.980425	−	0.196891i	$-0.936916\pi$
$883$	−9.50000	+	16.4545i	−0.319700	+	0.553737i	0.660725	+	0.750628i	$0.270249\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	10.5000	+	18.1865i	0.352555	+	0.610644i	0.986696	−	0.162573i	$-0.0519794\pi$
$887$	10.5000	+	18.1865i	0.352555	+	0.610644i	−0.634141	+	0.773217i	$0.718646\pi$
$888$		0			0
$889$	7.50000	+	12.9904i	0.251542	+	0.435683i
$890$		0			0
$891$	2.50000	−	4.33013i	0.0837532	−	0.145065i
$892$		0			0
$893$	−13.0000			−0.435028
$894$		0			0
$895$	−20.0000			−0.668526
$896$		0			0
$897$	17.5000	−	30.3109i	0.584308	−	1.01205i
$898$		0			0
$899$	−12.5000	−	21.6506i	−0.416898	−	0.722089i
$900$		0			0
$901$	21.0000	+	36.3731i	0.699611	+	1.21176i
$902$		0			0
$903$	6.00000	−	10.3923i	0.199667	−	0.345834i
$904$		0			0
$905$	−3.50000	+	6.06218i	−0.116344	+	0.201514i
$906$		0			0
$907$	24.5000	−	42.4352i	0.813509	−	1.40904i	−0.0968843	−	0.995296i	$-0.530888\pi$
$907$	24.5000	−	42.4352i	0.813509	−	1.40904i	0.910393	−	0.413744i	$-0.135779\pi$
$908$		0			0
$909$	3.50000	−	6.06218i	0.116088	−	0.201070i
$910$		0			0
$911$	−44.0000			−1.45779			−0.728893	−	0.684628i	$-0.759965\pi$
$911$	−44.0000			−1.45779			−0.728893	+	0.684628i	$0.759965\pi$
$912$		0			0
$913$	−55.0000			−1.82023
$914$		0			0
$915$	−4.50000	−	7.79423i	−0.148765	−	0.257669i
$916$		0			0
$917$	30.0000	−	51.9615i	0.990687	−	1.71592i
$918$		0			0
$919$	14.5000	+	25.1147i	0.478311	+	0.828459i	0.999691	−	0.0248659i	$-0.00791589\pi$
$919$	14.5000	+	25.1147i	0.478311	+	0.828459i	−0.521380	+	0.853325i	$0.674583\pi$
$920$		0			0
$921$	−11.5000	−	19.9186i	−0.378938	−	0.656340i
$922$		0			0
$923$	75.0000			2.46866
$924$		0			0
$925$	5.50000	+	9.52628i	0.180839	+	0.313222i
$926$		0			0
$927$	−5.50000	+	9.52628i	−0.180644	+	0.312884i
$928$		0			0
$929$	18.0000			0.590561			0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000			0.590561			0.295280	+	0.955411i	$0.404587\pi$
$930$		0			0
$931$	1.00000	−	1.73205i	0.0327737	−	0.0567657i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−15.0000			−0.490552
$936$		0			0
$937$	−58.0000			−1.89478			−0.947389	−	0.320085i	$-0.896288\pi$
$937$	−58.0000			−1.89478			−0.947389	+	0.320085i	$0.896288\pi$
$938$		0			0
$939$	30.0000			0.979013
$940$		0			0
$941$	−14.0000			−0.456387			−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000			−0.456387			−0.228193	+	0.973616i	$0.573282\pi$
$942$		0			0
$943$	−77.0000			−2.50746
$944$		0			0
$945$	1.50000	−	2.59808i	0.0487950	−	0.0845154i
$946$		0			0
$947$	4.00000			0.129983			0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000			0.129983			0.0649913	+	0.997886i	$0.479298\pi$
$948$		0			0
$949$	7.50000	−	12.9904i	0.243460	−	0.421686i
$950$		0			0
$951$	11.5000	+	19.9186i	0.372913	+	0.645904i
$952$		0			0
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$	−2.50000	−	4.33013i	−0.0808981	−	0.140120i
$956$		0			0
$957$	−12.5000	−	21.6506i	−0.404068	−	0.699866i
$958$		0			0
$959$	27.0000	−	46.7654i	0.871875	−	1.51013i
$960$		0			0
$961$	3.00000	+	5.19615i	0.0967742	+	0.167618i
$962$		0			0
$963$	4.00000			0.128898
$964$		0			0
$965$	−14.0000			−0.450676
$966$		0			0
$967$	−9.50000	+	16.4545i	−0.305499	+	0.529140i	−0.977372	−	0.211526i	$-0.932157\pi$
$967$	−9.50000	+	16.4545i	−0.305499	+	0.529140i	0.671873	+	0.740666i	$0.265490\pi$
$968$		0			0
$969$	−1.50000	+	2.59808i	−0.0481869	+	0.0834622i
$970$		0			0
$971$	−7.50000	+	12.9904i	−0.240686	+	0.416881i	−0.960910	−	0.276861i	$-0.910706\pi$
$971$	−7.50000	+	12.9904i	−0.240686	+	0.416881i	0.720224	+	0.693742i	$0.244039\pi$
$972$		0			0
$973$	6.00000	−	10.3923i	0.192351	−	0.333162i
$974$		0			0
$975$	2.50000	+	4.33013i	0.0800641	+	0.138675i
$976$		0			0
$977$	10.5000	+	18.1865i	0.335925	+	0.581839i	0.983662	−	0.180025i	$-0.0576179\pi$
$977$	10.5000	+	18.1865i	0.335925	+	0.581839i	−0.647737	+	0.761864i	$0.724285\pi$
$978$		0			0
$979$	5.00000	−	8.66025i	0.159801	−	0.276783i
$980$		0			0
$981$	−18.0000			−0.574696
$982$		0			0
$983$	−20.0000			−0.637901			−0.318950	−	0.947771i	$-0.603330\pi$
$983$	−20.0000			−0.637901			−0.318950	+	0.947771i	$0.603330\pi$
$984$		0			0
$985$	−1.50000	+	2.59808i	−0.0477940	+	0.0827816i
$986$		0			0
$987$	19.5000	+	33.7750i	0.620692	+	1.07507i
$988$		0			0
$989$	−14.0000	−	24.2487i	−0.445174	−	0.771064i
$990$		0			0
$991$	56.0000			1.77890			0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000			1.77890			0.889449	+	0.457034i	$0.151088\pi$
$992$		0			0
$993$	−7.50000	+	12.9904i	−0.238005	+	0.412237i
$994$		0			0
$995$	10.5000	+	18.1865i	0.332872	+	0.576552i
$996$		0			0
$997$	−18.0000			−0.570066			−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000			−0.570066			−0.285033	+	0.958518i	$0.592005\pi$
$998$		0			0
$999$	−5.50000	−	9.52628i	−0.174012	−	0.301398i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.q.b.841.1	✓	2
67.29	even	3	inner	4020.2.q.b.3781.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.q.b.841.1	✓	2	1.1	even	1	trivial
4020.2.q.b.3781.1	yes	2	67.29	even	3	inner

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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +(-1.50000 + 2.59808i) q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +(-1.50000 + 2.59808i) q^{7} +1.00000 q^{9} +(2.50000 - 4.33013i) q^{11} +(-2.50000 - 4.33013i) q^{13} -1.00000 q^{15} +(-1.50000 - 2.59808i) q^{17} +(0.500000 + 0.866025i) q^{19} +(1.50000 - 2.59808i) q^{21} +(-3.50000 - 6.06218i) q^{23} +1.00000 q^{25} -1.00000 q^{27} +(-2.50000 + 4.33013i) q^{29} +(-2.50000 + 4.33013i) q^{31} +(-2.50000 + 4.33013i) q^{33} +(-1.50000 + 2.59808i) q^{35} +(5.50000 + 9.52628i) q^{37} +(2.50000 + 4.33013i) q^{39} +(5.50000 - 9.52628i) q^{41} +4.00000 q^{43} +1.00000 q^{45} +(-6.50000 + 11.2583i) q^{47} +(-1.00000 - 1.73205i) q^{49} +(1.50000 + 2.59808i) q^{51} -14.0000 q^{53} +(2.50000 - 4.33013i) q^{55} +(-0.500000 - 0.866025i) q^{57} +(4.50000 + 7.79423i) q^{61} +(-1.50000 + 2.59808i) q^{63} +(-2.50000 - 4.33013i) q^{65} +(-8.00000 + 1.73205i) q^{67} +(3.50000 + 6.06218i) q^{69} +(-7.50000 + 12.9904i) q^{71} +(1.50000 + 2.59808i) q^{73} -1.00000 q^{75} +(7.50000 + 12.9904i) q^{77} +(1.50000 - 2.59808i) q^{79} +1.00000 q^{81} +(-5.50000 - 9.52628i) q^{83} +(-1.50000 - 2.59808i) q^{85} +(2.50000 - 4.33013i) q^{87} +2.00000 q^{89} +15.0000 q^{91} +(2.50000 - 4.33013i) q^{93} +(0.500000 + 0.866025i) q^{95} +(-4.50000 - 7.79423i) q^{97} +(2.50000 - 4.33013i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 - 3 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} - 3 q^{7} + 2 q^{9} + 5 q^{11} - 5 q^{13} - 2 q^{15} - 3 q^{17} + q^{19} + 3 q^{21} - 7 q^{23} + 2 q^{25} - 2 q^{27} - 5 q^{29} - 5 q^{31} - 5 q^{33} - 3 q^{35} + 11 q^{37} + 5 q^{39} + 11 q^{41} + 8 q^{43} + 2 q^{45} - 13 q^{47} - 2 q^{49} + 3 q^{51} - 28 q^{53} + 5 q^{55} - q^{57} + 9 q^{61} - 3 q^{63} - 5 q^{65} - 16 q^{67} + 7 q^{69} - 15 q^{71} + 3 q^{73} - 2 q^{75} + 15 q^{77} + 3 q^{79} + 2 q^{81} - 11 q^{83} - 3 q^{85} + 5 q^{87} + 4 q^{89} + 30 q^{91} + 5 q^{93} + q^{95} - 9 q^{97} + 5 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 3 * q^7 + 2 * q^9 + 5 * q^11 - 5 * q^13 - 2 * q^15 - 3 * q^17 + q^19 + 3 * q^21 - 7 * q^23 + 2 * q^25 - 2 * q^27 - 5 * q^29 - 5 * q^31 - 5 * q^33 - 3 * q^35 + 11 * q^37 + 5 * q^39 + 11 * q^41 + 8 * q^43 + 2 * q^45 - 13 * q^47 - 2 * q^49 + 3 * q^51 - 28 * q^53 + 5 * q^55 - q^57 + 9 * q^61 - 3 * q^63 - 5 * q^65 - 16 * q^67 + 7 * q^69 - 15 * q^71 + 3 * q^73 - 2 * q^75 + 15 * q^77 + 3 * q^79 + 2 * q^81 - 11 * q^83 - 3 * q^85 + 5 * q^87 + 4 * q^89 + 30 * q^91 + 5 * q^93 + q^95 - 9 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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