LMFDB - Embedded newform 6160.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.a (trivial)

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:	yes
Analytic conductor:	$49.1878476451$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1540)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{3} -1.00000 q^{5} -1.00000 q^{7} +4.46410 q^{9} +O(q^{10})$	$q-2.73205 q^{3} -1.00000 q^{5} -1.00000 q^{7} +4.46410 q^{9} -1.00000 q^{11} +0.732051 q^{13} +2.73205 q^{15} +4.73205 q^{17} +1.46410 q^{19} +2.73205 q^{21} +1.00000 q^{25} -4.00000 q^{27} +3.46410 q^{29} -4.19615 q^{31} +2.73205 q^{33} +1.00000 q^{35} -1.46410 q^{37} -2.00000 q^{39} -2.19615 q^{41} -2.00000 q^{43} -4.46410 q^{45} +5.66025 q^{47} +1.00000 q^{49} -12.9282 q^{51} +2.53590 q^{53} +1.00000 q^{55} -4.00000 q^{57} -10.7321 q^{59} -8.73205 q^{61} -4.46410 q^{63} -0.732051 q^{65} +4.00000 q^{67} +2.53590 q^{71} +10.1962 q^{73} -2.73205 q^{75} +1.00000 q^{77} -2.00000 q^{79} -2.46410 q^{81} -16.3923 q^{83} -4.73205 q^{85} -9.46410 q^{87} -0.928203 q^{89} -0.732051 q^{91} +11.4641 q^{93} -1.46410 q^{95} -2.39230 q^{97} -4.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{15} + 6 q^{17} - 4 q^{19} + 2 q^{21} + 2 q^{25} - 8 q^{27} + 2 q^{31} + 2 q^{33} + 2 q^{35} + 4 q^{37} - 4 q^{39} + 6 q^{41} - 4 q^{43} - 2 q^{45} - 6 q^{47} + 2 q^{49} - 12 q^{51} + 12 q^{53} + 2 q^{55} - 8 q^{57} - 18 q^{59} - 14 q^{61} - 2 q^{63} + 2 q^{65} + 8 q^{67} + 12 q^{71} + 10 q^{73} - 2 q^{75} + 2 q^{77} - 4 q^{79} + 2 q^{81} - 12 q^{83} - 6 q^{85} - 12 q^{87} + 12 q^{89} + 2 q^{91} + 16 q^{93} + 4 q^{95} + 16 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^13 + 2 * q^15 + 6 * q^17 - 4 * q^19 + 2 * q^21 + 2 * q^25 - 8 * q^27 + 2 * q^31 + 2 * q^33 + 2 * q^35 + 4 * q^37 - 4 * q^39 + 6 * q^41 - 4 * q^43 - 2 * q^45 - 6 * q^47 + 2 * q^49 - 12 * q^51 + 12 * q^53 + 2 * q^55 - 8 * q^57 - 18 * q^59 - 14 * q^61 - 2 * q^63 + 2 * q^65 + 8 * q^67 + 12 * q^71 + 10 * q^73 - 2 * q^75 + 2 * q^77 - 4 * q^79 + 2 * q^81 - 12 * q^83 - 6 * q^85 - 12 * q^87 + 12 * q^89 + 2 * q^91 + 16 * q^93 + 4 * q^95 + 16 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−2.73205	−1.57735	−0.788675	−	0.614810i	$-0.789233\pi$
$3$	−2.73205	−1.57735	−0.788675	+	0.614810i	$0.789233\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.732051	0.203034	0.101517	−	0.994834i	$-0.467630\pi$
$13$	0.732051	0.203034	0.101517	+	0.994834i	$0.467630\pi$
$14$	0	0
$15$	2.73205	0.705412
$16$	0	0
$17$	4.73205	1.14769	0.573845	−	0.818964i	$-0.305451\pi$
$17$	4.73205	1.14769	0.573845	+	0.818964i	$0.305451\pi$
$18$	0	0
$19$	1.46410	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$19$	1.46410	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$20$	0	0
$21$	2.73205	0.596182
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	−4.19615	−0.753651	−0.376826	−	0.926284i	$-0.622984\pi$
$31$	−4.19615	−0.753651	−0.376826	+	0.926284i	$0.622984\pi$
$32$	0	0
$33$	2.73205	0.475589
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−1.46410	−0.240697	−0.120348	−	0.992732i	$-0.538401\pi$
$37$	−1.46410	−0.240697	−0.120348	+	0.992732i	$0.538401\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−2.19615	−0.342981	−0.171491	−	0.985186i	$-0.554858\pi$
$41$	−2.19615	−0.342981	−0.171491	+	0.985186i	$0.554858\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	−4.46410	−0.665469
$46$	0	0
$47$	5.66025	0.825633	0.412816	−	0.910814i	$-0.364545\pi$
$47$	5.66025	0.825633	0.412816	+	0.910814i	$0.364545\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−12.9282	−1.81031
$52$	0	0
$53$	2.53590	0.348332	0.174166	−	0.984716i	$-0.444277\pi$
$53$	2.53590	0.348332	0.174166	+	0.984716i	$0.444277\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	−10.7321	−1.39719	−0.698597	−	0.715515i	$-0.746192\pi$
$59$	−10.7321	−1.39719	−0.698597	+	0.715515i	$0.746192\pi$
$60$	0	0
$61$	−8.73205	−1.11802	−0.559012	−	0.829159i	$-0.688820\pi$
$61$	−8.73205	−1.11802	−0.559012	+	0.829159i	$0.688820\pi$
$62$	0	0
$63$	−4.46410	−0.562424
$64$	0	0
$65$	−0.732051	−0.0907997
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.53590	0.300956	0.150478	−	0.988613i	$-0.451919\pi$
$71$	2.53590	0.300956	0.150478	+	0.988613i	$0.451919\pi$
$72$	0	0
$73$	10.1962	1.19337	0.596685	−	0.802476i	$-0.296484\pi$
$73$	10.1962	1.19337	0.596685	+	0.802476i	$0.296484\pi$
$74$	0	0
$75$	−2.73205	−0.315470
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	−16.3923	−1.79929	−0.899645	−	0.436623i	$-0.856174\pi$
$83$	−16.3923	−1.79929	−0.899645	+	0.436623i	$0.856174\pi$
$84$	0	0
$85$	−4.73205	−0.513263
$86$	0	0
$87$	−9.46410	−1.01466
$88$	0	0
$89$	−0.928203	−0.0983893	−0.0491947	−	0.998789i	$-0.515665\pi$
$89$	−0.928203	−0.0983893	−0.0491947	+	0.998789i	$0.515665\pi$
$90$	0	0
$91$	−0.732051	−0.0767398
$92$	0	0
$93$	11.4641	1.18877
$94$	0	0
$95$	−1.46410	−0.150214
$96$	0	0
$97$	−2.39230	−0.242902	−0.121451	−	0.992597i	$-0.538755\pi$
$97$	−2.39230	−0.242902	−0.121451	+	0.992597i	$0.538755\pi$
$98$	0	0
$99$	−4.46410	−0.448659
$100$	0	0
$101$	−11.6603	−1.16024	−0.580119	−	0.814532i	$-0.696994\pi$
$101$	−11.6603	−1.16024	−0.580119	+	0.814532i	$0.696994\pi$
$102$	0	0
$103$	12.1962	1.20172	0.600861	−	0.799353i	$-0.294824\pi$
$103$	12.1962	1.20172	0.600861	+	0.799353i	$0.294824\pi$
$104$	0	0
$105$	−2.73205	−0.266621
$106$	0	0
$107$	6.92820	0.669775	0.334887	−	0.942258i	$-0.391302\pi$
$107$	6.92820	0.669775	0.334887	+	0.942258i	$0.391302\pi$
$108$	0	0
$109$	4.53590	0.434460	0.217230	−	0.976120i	$-0.430298\pi$
$109$	4.53590	0.434460	0.217230	+	0.976120i	$0.430298\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	−12.9282	−1.21618	−0.608092	−	0.793867i	$-0.708065\pi$
$113$	−12.9282	−1.21618	−0.608092	+	0.793867i	$0.708065\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.26795	0.302122
$118$	0	0
$119$	−4.73205	−0.433786
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	10.9282	0.969721	0.484861	−	0.874591i	$-0.338870\pi$
$127$	10.9282	0.969721	0.484861	+	0.874591i	$0.338870\pi$
$128$	0	0
$129$	5.46410	0.481087
$130$	0	0
$131$	−5.07180	−0.443125	−0.221562	−	0.975146i	$-0.571116\pi$
$131$	−5.07180	−0.443125	−0.221562	+	0.975146i	$0.571116\pi$
$132$	0	0
$133$	−1.46410	−0.126954
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	−2.53590	−0.216656	−0.108328	−	0.994115i	$-0.534550\pi$
$137$	−2.53590	−0.216656	−0.108328	+	0.994115i	$0.534550\pi$
$138$	0	0
$139$	−5.46410	−0.463459	−0.231730	−	0.972780i	$-0.574438\pi$
$139$	−5.46410	−0.463459	−0.231730	+	0.972780i	$0.574438\pi$
$140$	0	0
$141$	−15.4641	−1.30231
$142$	0	0
$143$	−0.732051	−0.0612172
$144$	0	0
$145$	−3.46410	−0.287678
$146$	0	0
$147$	−2.73205	−0.225336
$148$	0	0
$149$	12.9282	1.05912	0.529560	−	0.848273i	$-0.322357\pi$
$149$	12.9282	1.05912	0.529560	+	0.848273i	$0.322357\pi$
$150$	0	0
$151$	3.07180	0.249979	0.124990	−	0.992158i	$-0.460110\pi$
$151$	3.07180	0.249979	0.124990	+	0.992158i	$0.460110\pi$
$152$	0	0
$153$	21.1244	1.70780
$154$	0	0
$155$	4.19615	0.337043
$156$	0	0
$157$	−23.8564	−1.90395	−0.951974	−	0.306178i	$-0.900950\pi$
$157$	−23.8564	−1.90395	−0.951974	+	0.306178i	$0.900950\pi$
$158$	0	0
$159$	−6.92820	−0.549442
$160$	0	0
$161$	0	0
$162$	0	0
$163$	24.7846	1.94128	0.970640	−	0.240536i	$-0.0773232\pi$
$163$	24.7846	1.94128	0.970640	+	0.240536i	$0.0773232\pi$
$164$	0	0
$165$	−2.73205	−0.212690
$166$	0	0
$167$	18.9282	1.46471	0.732354	−	0.680924i	$-0.238422\pi$
$167$	18.9282	1.46471	0.732354	+	0.680924i	$0.238422\pi$
$168$	0	0
$169$	−12.4641	−0.958777
$170$	0	0
$171$	6.53590	0.499813
$172$	0	0
$173$	0.339746	0.0258304	0.0129152	−	0.999917i	$-0.495889\pi$
$173$	0.339746	0.0258304	0.0129152	+	0.999917i	$0.495889\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	29.3205	2.20386
$178$	0	0
$179$	−20.7846	−1.55351	−0.776757	−	0.629800i	$-0.783137\pi$
$179$	−20.7846	−1.55351	−0.776757	+	0.629800i	$0.783137\pi$
$180$	0	0
$181$	−14.3923	−1.06977	−0.534886	−	0.844924i	$-0.679645\pi$
$181$	−14.3923	−1.06977	−0.534886	+	0.844924i	$0.679645\pi$
$182$	0	0
$183$	23.8564	1.76352
$184$	0	0
$185$	1.46410	0.107643
$186$	0	0
$187$	−4.73205	−0.346042
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	6.39230	0.460128	0.230064	−	0.973175i	$-0.426106\pi$
$193$	6.39230	0.460128	0.230064	+	0.973175i	$0.426106\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	8.53590	0.608158	0.304079	−	0.952647i	$-0.401651\pi$
$197$	8.53590	0.608158	0.304079	+	0.952647i	$0.401651\pi$
$198$	0	0
$199$	21.6603	1.53545	0.767727	−	0.640777i	$-0.221388\pi$
$199$	21.6603	1.53545	0.767727	+	0.640777i	$0.221388\pi$
$200$	0	0
$201$	−10.9282	−0.770816
$202$	0	0
$203$	−3.46410	−0.243132
$204$	0	0
$205$	2.19615	0.153386
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.46410	−0.101274
$210$	0	0
$211$	9.07180	0.624528	0.312264	−	0.949995i	$-0.398913\pi$
$211$	9.07180	0.624528	0.312264	+	0.949995i	$0.398913\pi$
$212$	0	0
$213$	−6.92820	−0.474713
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	4.19615	0.284853
$218$	0	0
$219$	−27.8564	−1.88236
$220$	0	0
$221$	3.46410	0.233021
$222$	0	0
$223$	−6.73205	−0.450811	−0.225406	−	0.974265i	$-0.572371\pi$
$223$	−6.73205	−0.450811	−0.225406	+	0.974265i	$0.572371\pi$
$224$	0	0
$225$	4.46410	0.297607
$226$	0	0
$227$	−18.9282	−1.25631	−0.628154	−	0.778089i	$-0.716189\pi$
$227$	−18.9282	−1.25631	−0.628154	+	0.778089i	$0.716189\pi$
$228$	0	0
$229$	6.39230	0.422415	0.211208	−	0.977441i	$-0.432260\pi$
$229$	6.39230	0.422415	0.211208	+	0.977441i	$0.432260\pi$
$230$	0	0
$231$	−2.73205	−0.179756
$232$	0	0
$233$	24.9282	1.63310	0.816550	−	0.577274i	$-0.195884\pi$
$233$	24.9282	1.63310	0.816550	+	0.577274i	$0.195884\pi$
$234$	0	0
$235$	−5.66025	−0.369234
$236$	0	0
$237$	5.46410	0.354932
$238$	0	0
$239$	18.9282	1.22436	0.612182	−	0.790717i	$-0.290292\pi$
$239$	18.9282	1.22436	0.612182	+	0.790717i	$0.290292\pi$
$240$	0	0
$241$	−4.33975	−0.279548	−0.139774	−	0.990183i	$-0.544638\pi$
$241$	−4.33975	−0.279548	−0.139774	+	0.990183i	$0.544638\pi$
$242$	0	0
$243$	18.7321	1.20166
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	1.07180	0.0681968
$248$	0	0
$249$	44.7846	2.83811
$250$	0	0
$251$	24.5885	1.55201	0.776005	−	0.630727i	$-0.217243\pi$
$251$	24.5885	1.55201	0.776005	+	0.630727i	$0.217243\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	12.9282	0.809595
$256$	0	0
$257$	4.14359	0.258470	0.129235	−	0.991614i	$-0.458748\pi$
$257$	4.14359	0.258470	0.129235	+	0.991614i	$0.458748\pi$
$258$	0	0
$259$	1.46410	0.0909748
$260$	0	0
$261$	15.4641	0.957204
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−2.53590	−0.155779
$266$	0	0
$267$	2.53590	0.155194
$268$	0	0
$269$	10.3923	0.633630	0.316815	−	0.948487i	$-0.397387\pi$
$269$	10.3923	0.633630	0.316815	+	0.948487i	$0.397387\pi$
$270$	0	0
$271$	−0.392305	−0.0238308	−0.0119154	−	0.999929i	$-0.503793\pi$
$271$	−0.392305	−0.0238308	−0.0119154	+	0.999929i	$0.503793\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−9.32051	−0.560015	−0.280008	−	0.959998i	$-0.590337\pi$
$277$	−9.32051	−0.560015	−0.280008	+	0.959998i	$0.590337\pi$
$278$	0	0
$279$	−18.7321	−1.12146
$280$	0	0
$281$	−1.60770	−0.0959071	−0.0479535	−	0.998850i	$-0.515270\pi$
$281$	−1.60770	−0.0959071	−0.0479535	+	0.998850i	$0.515270\pi$
$282$	0	0
$283$	−21.8564	−1.29923	−0.649614	−	0.760264i	$-0.725070\pi$
$283$	−21.8564	−1.29923	−0.649614	+	0.760264i	$0.725070\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	0	0
$287$	2.19615	0.129635
$288$	0	0
$289$	5.39230	0.317194
$290$	0	0
$291$	6.53590	0.383141
$292$	0	0
$293$	11.6603	0.681199	0.340600	−	0.940208i	$-0.389370\pi$
$293$	11.6603	0.681199	0.340600	+	0.940208i	$0.389370\pi$
$294$	0	0
$295$	10.7321	0.624844
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	0	0
$300$	0	0
$301$	2.00000	0.115278
$302$	0	0
$303$	31.8564	1.83010
$304$	0	0
$305$	8.73205	0.499996
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	−33.3205	−1.89554
$310$	0	0
$311$	−3.12436	−0.177166	−0.0885830	−	0.996069i	$-0.528234\pi$
$311$	−3.12436	−0.177166	−0.0885830	+	0.996069i	$0.528234\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	0	0
$315$	4.46410	0.251524
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	−3.46410	−0.193952
$320$	0	0
$321$	−18.9282	−1.05647
$322$	0	0
$323$	6.92820	0.385496
$324$	0	0
$325$	0.732051	0.0406069
$326$	0	0
$327$	−12.3923	−0.685296
$328$	0	0
$329$	−5.66025	−0.312060
$330$	0	0
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	0	0
$333$	−6.53590	−0.358165
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	10.7846	0.587475	0.293738	−	0.955886i	$-0.405101\pi$
$337$	10.7846	0.587475	0.293738	+	0.955886i	$0.405101\pi$
$338$	0	0
$339$	35.3205	1.91835
$340$	0	0
$341$	4.19615	0.227234
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	31.8564	1.71014	0.855071	−	0.518511i	$-0.173514\pi$
$347$	31.8564	1.71014	0.855071	+	0.518511i	$0.173514\pi$
$348$	0	0
$349$	−22.5885	−1.20913	−0.604566	−	0.796555i	$-0.706654\pi$
$349$	−22.5885	−1.20913	−0.604566	+	0.796555i	$0.706654\pi$
$350$	0	0
$351$	−2.92820	−0.156296
$352$	0	0
$353$	14.7846	0.786905	0.393453	−	0.919345i	$-0.371281\pi$
$353$	14.7846	0.786905	0.393453	+	0.919345i	$0.371281\pi$
$354$	0	0
$355$	−2.53590	−0.134592
$356$	0	0
$357$	12.9282	0.684233
$358$	0	0
$359$	7.85641	0.414645	0.207323	−	0.978273i	$-0.433525\pi$
$359$	7.85641	0.414645	0.207323	+	0.978273i	$0.433525\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	0	0
$363$	−2.73205	−0.143395
$364$	0	0
$365$	−10.1962	−0.533691
$366$	0	0
$367$	7.80385	0.407358	0.203679	−	0.979038i	$-0.434710\pi$
$367$	7.80385	0.407358	0.203679	+	0.979038i	$0.434710\pi$
$368$	0	0
$369$	−9.80385	−0.510368
$370$	0	0
$371$	−2.53590	−0.131657
$372$	0	0
$373$	18.3923	0.952317	0.476159	−	0.879359i	$-0.342029\pi$
$373$	18.3923	0.952317	0.476159	+	0.879359i	$0.342029\pi$
$374$	0	0
$375$	2.73205	0.141082
$376$	0	0
$377$	2.53590	0.130605
$378$	0	0
$379$	13.4641	0.691604	0.345802	−	0.938307i	$-0.387607\pi$
$379$	13.4641	0.691604	0.345802	+	0.938307i	$0.387607\pi$
$380$	0	0
$381$	−29.8564	−1.52959
$382$	0	0
$383$	17.6603	0.902397	0.451198	−	0.892424i	$-0.350997\pi$
$383$	17.6603	0.902397	0.451198	+	0.892424i	$0.350997\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	−8.92820	−0.453846
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	13.8564	0.698963
$394$	0	0
$395$	2.00000	0.100631
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	0.928203	0.0463523	0.0231761	−	0.999731i	$-0.492622\pi$
$401$	0.928203	0.0463523	0.0231761	+	0.999731i	$0.492622\pi$
$402$	0	0
$403$	−3.07180	−0.153017
$404$	0	0
$405$	2.46410	0.122442
$406$	0	0
$407$	1.46410	0.0725728
$408$	0	0
$409$	−13.1244	−0.648958	−0.324479	−	0.945893i	$-0.605189\pi$
$409$	−13.1244	−0.648958	−0.324479	+	0.945893i	$0.605189\pi$
$410$	0	0
$411$	6.92820	0.341743
$412$	0	0
$413$	10.7321	0.528090
$414$	0	0
$415$	16.3923	0.804667
$416$	0	0
$417$	14.9282	0.731037
$418$	0	0
$419$	26.4449	1.29192	0.645958	−	0.763373i	$-0.276458\pi$
$419$	26.4449	1.29192	0.645958	+	0.763373i	$0.276458\pi$
$420$	0	0
$421$	−13.4641	−0.656200	−0.328100	−	0.944643i	$-0.606408\pi$
$421$	−13.4641	−0.656200	−0.328100	+	0.944643i	$0.606408\pi$
$422$	0	0
$423$	25.2679	1.22857
$424$	0	0
$425$	4.73205	0.229538
$426$	0	0
$427$	8.73205	0.422574
$428$	0	0
$429$	2.00000	0.0965609
$430$	0	0
$431$	11.0718	0.533310	0.266655	−	0.963792i	$-0.414082\pi$
$431$	11.0718	0.533310	0.266655	+	0.963792i	$0.414082\pi$
$432$	0	0
$433$	9.60770	0.461716	0.230858	−	0.972987i	$-0.425847\pi$
$433$	9.60770	0.461716	0.230858	+	0.972987i	$0.425847\pi$
$434$	0	0
$435$	9.46410	0.453769
$436$	0	0
$437$	0	0
$438$	0	0
$439$	15.3205	0.731208	0.365604	−	0.930771i	$-0.380862\pi$
$439$	15.3205	0.731208	0.365604	+	0.930771i	$0.380862\pi$
$440$	0	0
$441$	4.46410	0.212576
$442$	0	0
$443$	28.3923	1.34896	0.674480	−	0.738294i	$-0.264368\pi$
$443$	28.3923	1.34896	0.674480	+	0.738294i	$0.264368\pi$
$444$	0	0
$445$	0.928203	0.0440011
$446$	0	0
$447$	−35.3205	−1.67060
$448$	0	0
$449$	−4.39230	−0.207286	−0.103643	−	0.994615i	$-0.533050\pi$
$449$	−4.39230	−0.207286	−0.103643	+	0.994615i	$0.533050\pi$
$450$	0	0
$451$	2.19615	0.103413
$452$	0	0
$453$	−8.39230	−0.394305
$454$	0	0
$455$	0.732051	0.0343191
$456$	0	0
$457$	9.60770	0.449429	0.224715	−	0.974425i	$-0.427855\pi$
$457$	9.60770	0.449429	0.224715	+	0.974425i	$0.427855\pi$
$458$	0	0
$459$	−18.9282	−0.883493
$460$	0	0
$461$	18.5885	0.865751	0.432875	−	0.901454i	$-0.357499\pi$
$461$	18.5885	0.865751	0.432875	+	0.901454i	$0.357499\pi$
$462$	0	0
$463$	21.0718	0.979289	0.489645	−	0.871922i	$-0.337126\pi$
$463$	21.0718	0.979289	0.489645	+	0.871922i	$0.337126\pi$
$464$	0	0
$465$	−11.4641	−0.531635
$466$	0	0
$467$	−27.1244	−1.25517	−0.627583	−	0.778550i	$-0.715956\pi$
$467$	−27.1244	−1.25517	−0.627583	+	0.778550i	$0.715956\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	65.1769	3.00319
$472$	0	0
$473$	2.00000	0.0919601
$474$	0	0
$475$	1.46410	0.0671776
$476$	0	0
$477$	11.3205	0.518330
$478$	0	0
$479$	25.1769	1.15036	0.575181	−	0.818026i	$-0.304932\pi$
$479$	25.1769	1.15036	0.575181	+	0.818026i	$0.304932\pi$
$480$	0	0
$481$	−1.07180	−0.0488697
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.39230	0.108629
$486$	0	0
$487$	−4.78461	−0.216811	−0.108406	−	0.994107i	$-0.534575\pi$
$487$	−4.78461	−0.216811	−0.108406	+	0.994107i	$0.534575\pi$
$488$	0	0
$489$	−67.7128	−3.06208
$490$	0	0
$491$	24.9282	1.12499	0.562497	−	0.826799i	$-0.309841\pi$
$491$	24.9282	1.12499	0.562497	+	0.826799i	$0.309841\pi$
$492$	0	0
$493$	16.3923	0.738272
$494$	0	0
$495$	4.46410	0.200646
$496$	0	0
$497$	−2.53590	−0.113751
$498$	0	0
$499$	−31.3205	−1.40210	−0.701049	−	0.713113i	$-0.747285\pi$
$499$	−31.3205	−1.40210	−0.701049	+	0.713113i	$0.747285\pi$
$500$	0	0
$501$	−51.7128	−2.31036
$502$	0	0
$503$	−37.1769	−1.65764	−0.828818	−	0.559518i	$-0.810986\pi$
$503$	−37.1769	−1.65764	−0.828818	+	0.559518i	$0.810986\pi$
$504$	0	0
$505$	11.6603	0.518874
$506$	0	0
$507$	34.0526	1.51233
$508$	0	0
$509$	36.2487	1.60670	0.803348	−	0.595510i	$-0.203050\pi$
$509$	36.2487	1.60670	0.803348	+	0.595510i	$0.203050\pi$
$510$	0	0
$511$	−10.1962	−0.451051
$512$	0	0
$513$	−5.85641	−0.258567
$514$	0	0
$515$	−12.1962	−0.537427
$516$	0	0
$517$	−5.66025	−0.248938
$518$	0	0
$519$	−0.928203	−0.0407436
$520$	0	0
$521$	13.6077	0.596164	0.298082	−	0.954540i	$-0.403653\pi$
$521$	13.6077	0.596164	0.298082	+	0.954540i	$0.403653\pi$
$522$	0	0
$523$	24.7846	1.08376	0.541878	−	0.840457i	$-0.317714\pi$
$523$	24.7846	1.08376	0.541878	+	0.840457i	$0.317714\pi$
$524$	0	0
$525$	2.73205	0.119236
$526$	0	0
$527$	−19.8564	−0.864959
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−47.9090	−2.07907
$532$	0	0
$533$	−1.60770	−0.0696370
$534$	0	0
$535$	−6.92820	−0.299532
$536$	0	0
$537$	56.7846	2.45044
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	15.8564	0.681720	0.340860	−	0.940114i	$-0.389282\pi$
$541$	15.8564	0.681720	0.340860	+	0.940114i	$0.389282\pi$
$542$	0	0
$543$	39.3205	1.68740
$544$	0	0
$545$	−4.53590	−0.194297
$546$	0	0
$547$	12.7846	0.546630	0.273315	−	0.961925i	$-0.411880\pi$
$547$	12.7846	0.546630	0.273315	+	0.961925i	$0.411880\pi$
$548$	0	0
$549$	−38.9808	−1.66366
$550$	0	0
$551$	5.07180	0.216066
$552$	0	0
$553$	2.00000	0.0850487
$554$	0	0
$555$	−4.00000	−0.169791
$556$	0	0
$557$	1.60770	0.0681202	0.0340601	−	0.999420i	$-0.489156\pi$
$557$	1.60770	0.0681202	0.0340601	+	0.999420i	$0.489156\pi$
$558$	0	0
$559$	−1.46410	−0.0619249
$560$	0	0
$561$	12.9282	0.545829
$562$	0	0
$563$	2.53590	0.106875	0.0534377	−	0.998571i	$-0.482982\pi$
$563$	2.53590	0.106875	0.0534377	+	0.998571i	$0.482982\pi$
$564$	0	0
$565$	12.9282	0.543894
$566$	0	0
$567$	2.46410	0.103483
$568$	0	0
$569$	−19.1769	−0.803938	−0.401969	−	0.915653i	$-0.631674\pi$
$569$	−19.1769	−0.803938	−0.401969	+	0.915653i	$0.631674\pi$
$570$	0	0
$571$	45.5692	1.90701	0.953506	−	0.301373i	$-0.0974450\pi$
$571$	45.5692	1.90701	0.953506	+	0.301373i	$0.0974450\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15.1769	0.631823	0.315912	−	0.948789i	$-0.397690\pi$
$577$	15.1769	0.631823	0.315912	+	0.948789i	$0.397690\pi$
$578$	0	0
$579$	−17.4641	−0.725783
$580$	0	0
$581$	16.3923	0.680067
$582$	0	0
$583$	−2.53590	−0.105026
$584$	0	0
$585$	−3.26795	−0.135113
$586$	0	0
$587$	−32.1962	−1.32888	−0.664439	−	0.747343i	$-0.731329\pi$
$587$	−32.1962	−1.32888	−0.664439	+	0.747343i	$0.731329\pi$
$588$	0	0
$589$	−6.14359	−0.253142
$590$	0	0
$591$	−23.3205	−0.959278
$592$	0	0
$593$	14.8756	0.610869	0.305435	−	0.952213i	$-0.401198\pi$
$593$	14.8756	0.610869	0.305435	+	0.952213i	$0.401198\pi$
$594$	0	0
$595$	4.73205	0.193995
$596$	0	0
$597$	−59.1769	−2.42195
$598$	0	0
$599$	4.39230	0.179465	0.0897324	−	0.995966i	$-0.471399\pi$
$599$	4.39230	0.179465	0.0897324	+	0.995966i	$0.471399\pi$
$600$	0	0
$601$	19.6603	0.801958	0.400979	−	0.916087i	$-0.368670\pi$
$601$	19.6603	0.801958	0.400979	+	0.916087i	$0.368670\pi$
$602$	0	0
$603$	17.8564	0.727169
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	5.85641	0.237704	0.118852	−	0.992912i	$-0.462079\pi$
$607$	5.85641	0.237704	0.118852	+	0.992912i	$0.462079\pi$
$608$	0	0
$609$	9.46410	0.383505
$610$	0	0
$611$	4.14359	0.167632
$612$	0	0
$613$	−6.78461	−0.274028	−0.137014	−	0.990569i	$-0.543751\pi$
$613$	−6.78461	−0.274028	−0.137014	+	0.990569i	$0.543751\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	35.3205	1.42195	0.710975	−	0.703217i	$-0.248254\pi$
$617$	35.3205	1.42195	0.710975	+	0.703217i	$0.248254\pi$
$618$	0	0
$619$	−20.5885	−0.827520	−0.413760	−	0.910386i	$-0.635785\pi$
$619$	−20.5885	−0.827520	−0.413760	+	0.910386i	$0.635785\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.928203	0.0371877
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.00000	0.159745
$628$	0	0
$629$	−6.92820	−0.276246
$630$	0	0
$631$	−25.0718	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$631$	−25.0718	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$632$	0	0
$633$	−24.7846	−0.985100
$634$	0	0
$635$	−10.9282	−0.433673
$636$	0	0
$637$	0.732051	0.0290049
$638$	0	0
$639$	11.3205	0.447832
$640$	0	0
$641$	−25.1769	−0.994428	−0.497214	−	0.867628i	$-0.665644\pi$
$641$	−25.1769	−0.994428	−0.497214	+	0.867628i	$0.665644\pi$
$642$	0	0
$643$	7.12436	0.280957	0.140479	−	0.990084i	$-0.455136\pi$
$643$	7.12436	0.280957	0.140479	+	0.990084i	$0.455136\pi$
$644$	0	0
$645$	−5.46410	−0.215149
$646$	0	0
$647$	−38.4449	−1.51142	−0.755712	−	0.654904i	$-0.772709\pi$
$647$	−38.4449	−1.51142	−0.755712	+	0.654904i	$0.772709\pi$
$648$	0	0
$649$	10.7321	0.421270
$650$	0	0
$651$	−11.4641	−0.449314
$652$	0	0
$653$	31.8564	1.24664	0.623319	−	0.781968i	$-0.285784\pi$
$653$	31.8564	1.24664	0.623319	+	0.781968i	$0.285784\pi$
$654$	0	0
$655$	5.07180	0.198171
$656$	0	0
$657$	45.5167	1.77577
$658$	0	0
$659$	−7.85641	−0.306042	−0.153021	−	0.988223i	$-0.548900\pi$
$659$	−7.85641	−0.306042	−0.153021	+	0.988223i	$0.548900\pi$
$660$	0	0
$661$	−30.7846	−1.19738	−0.598691	−	0.800980i	$-0.704312\pi$
$661$	−30.7846	−1.19738	−0.598691	+	0.800980i	$0.704312\pi$
$662$	0	0
$663$	−9.46410	−0.367555
$664$	0	0
$665$	1.46410	0.0567754
$666$	0	0
$667$	0	0
$668$	0	0
$669$	18.3923	0.711088
$670$	0	0
$671$	8.73205	0.337097
$672$	0	0
$673$	43.5692	1.67947	0.839735	−	0.542996i	$-0.182710\pi$
$673$	43.5692	1.67947	0.839735	+	0.542996i	$0.182710\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	35.6603	1.37053	0.685267	−	0.728292i	$-0.259685\pi$
$677$	35.6603	1.37053	0.685267	+	0.728292i	$0.259685\pi$
$678$	0	0
$679$	2.39230	0.0918082
$680$	0	0
$681$	51.7128	1.98164
$682$	0	0
$683$	44.1051	1.68764	0.843818	−	0.536630i	$-0.180303\pi$
$683$	44.1051	1.68764	0.843818	+	0.536630i	$0.180303\pi$
$684$	0	0
$685$	2.53590	0.0968917
$686$	0	0
$687$	−17.4641	−0.666297
$688$	0	0
$689$	1.85641	0.0707235
$690$	0	0
$691$	−23.1244	−0.879692	−0.439846	−	0.898073i	$-0.644967\pi$
$691$	−23.1244	−0.879692	−0.439846	+	0.898073i	$0.644967\pi$
$692$	0	0
$693$	4.46410	0.169577
$694$	0	0
$695$	5.46410	0.207265
$696$	0	0
$697$	−10.3923	−0.393637
$698$	0	0
$699$	−68.1051	−2.57597
$700$	0	0
$701$	−12.2487	−0.462627	−0.231314	−	0.972879i	$-0.574302\pi$
$701$	−12.2487	−0.462627	−0.231314	+	0.972879i	$0.574302\pi$
$702$	0	0
$703$	−2.14359	−0.0808472
$704$	0	0
$705$	15.4641	0.582412
$706$	0	0
$707$	11.6603	0.438529
$708$	0	0
$709$	−3.07180	−0.115364	−0.0576819	−	0.998335i	$-0.518371\pi$
$709$	−3.07180	−0.115364	−0.0576819	+	0.998335i	$0.518371\pi$
$710$	0	0
$711$	−8.92820	−0.334834
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0.732051	0.0273771
$716$	0	0
$717$	−51.7128	−1.93125
$718$	0	0
$719$	−38.4449	−1.43375	−0.716876	−	0.697201i	$-0.754429\pi$
$719$	−38.4449	−1.43375	−0.716876	+	0.697201i	$0.754429\pi$
$720$	0	0
$721$	−12.1962	−0.454208
$722$	0	0
$723$	11.8564	0.440945
$724$	0	0
$725$	3.46410	0.128654
$726$	0	0
$727$	−41.3731	−1.53444	−0.767221	−	0.641383i	$-0.778361\pi$
$727$	−41.3731	−1.53444	−0.767221	+	0.641383i	$0.778361\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	−9.46410	−0.350042
$732$	0	0
$733$	14.5885	0.538837	0.269418	−	0.963023i	$-0.413169\pi$
$733$	14.5885	0.538837	0.269418	+	0.963023i	$0.413169\pi$
$734$	0	0
$735$	2.73205	0.100773
$736$	0	0
$737$	−4.00000	−0.147342
$738$	0	0
$739$	−38.9282	−1.43200	−0.715999	−	0.698102i	$-0.754028\pi$
$739$	−38.9282	−1.43200	−0.715999	+	0.698102i	$0.754028\pi$
$740$	0	0
$741$	−2.92820	−0.107570
$742$	0	0
$743$	−8.78461	−0.322276	−0.161138	−	0.986932i	$-0.551516\pi$
$743$	−8.78461	−0.322276	−0.161138	+	0.986932i	$0.551516\pi$
$744$	0	0
$745$	−12.9282	−0.473653
$746$	0	0
$747$	−73.1769	−2.67740
$748$	0	0
$749$	−6.92820	−0.253151
$750$	0	0
$751$	17.1769	0.626795	0.313397	−	0.949622i	$-0.398533\pi$
$751$	17.1769	0.626795	0.313397	+	0.949622i	$0.398533\pi$
$752$	0	0
$753$	−67.1769	−2.44806
$754$	0	0
$755$	−3.07180	−0.111794
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	35.6603	1.29268	0.646342	−	0.763048i	$-0.276298\pi$
$761$	35.6603	1.29268	0.646342	+	0.763048i	$0.276298\pi$
$762$	0	0
$763$	−4.53590	−0.164211
$764$	0	0
$765$	−21.1244	−0.763753
$766$	0	0
$767$	−7.85641	−0.283678
$768$	0	0
$769$	−10.5885	−0.381830	−0.190915	−	0.981607i	$-0.561145\pi$
$769$	−10.5885	−0.381830	−0.190915	+	0.981607i	$0.561145\pi$
$770$	0	0
$771$	−11.3205	−0.407698
$772$	0	0
$773$	36.2487	1.30378	0.651888	−	0.758315i	$-0.273977\pi$
$773$	36.2487	1.30378	0.651888	+	0.758315i	$0.273977\pi$
$774$	0	0
$775$	−4.19615	−0.150730
$776$	0	0
$777$	−4.00000	−0.143499
$778$	0	0
$779$	−3.21539	−0.115203
$780$	0	0
$781$	−2.53590	−0.0907416
$782$	0	0
$783$	−13.8564	−0.495188
$784$	0	0
$785$	23.8564	0.851472
$786$	0	0
$787$	−31.3205	−1.11646	−0.558228	−	0.829688i	$-0.688518\pi$
$787$	−31.3205	−1.11646	−0.558228	+	0.829688i	$0.688518\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.9282	0.459674
$792$	0	0
$793$	−6.39230	−0.226997
$794$	0	0
$795$	6.92820	0.245718
$796$	0	0
$797$	−50.7846	−1.79888	−0.899442	−	0.437041i	$-0.856026\pi$
$797$	−50.7846	−1.79888	−0.899442	+	0.437041i	$0.856026\pi$
$798$	0	0
$799$	26.7846	0.947571
$800$	0	0
$801$	−4.14359	−0.146407
$802$	0	0
$803$	−10.1962	−0.359814
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−28.3923	−0.999456
$808$	0	0
$809$	50.7846	1.78549	0.892746	−	0.450561i	$-0.148776\pi$
$809$	50.7846	1.78549	0.892746	+	0.450561i	$0.148776\pi$
$810$	0	0
$811$	−45.8564	−1.61024	−0.805118	−	0.593115i	$-0.797898\pi$
$811$	−45.8564	−1.61024	−0.805118	+	0.593115i	$0.797898\pi$
$812$	0	0
$813$	1.07180	0.0375896
$814$	0	0
$815$	−24.7846	−0.868167
$816$	0	0
$817$	−2.92820	−0.102445
$818$	0	0
$819$	−3.26795	−0.114191
$820$	0	0
$821$	4.14359	0.144612	0.0723062	−	0.997382i	$-0.476964\pi$
$821$	4.14359	0.144612	0.0723062	+	0.997382i	$0.476964\pi$
$822$	0	0
$823$	4.67949	0.163117	0.0815584	−	0.996669i	$-0.474010\pi$
$823$	4.67949	0.163117	0.0815584	+	0.996669i	$0.474010\pi$
$824$	0	0
$825$	2.73205	0.0951178
$826$	0	0
$827$	−4.14359	−0.144087	−0.0720434	−	0.997401i	$-0.522952\pi$
$827$	−4.14359	−0.144087	−0.0720434	+	0.997401i	$0.522952\pi$
$828$	0	0
$829$	15.8564	0.550716	0.275358	−	0.961342i	$-0.411204\pi$
$829$	15.8564	0.550716	0.275358	+	0.961342i	$0.411204\pi$
$830$	0	0
$831$	25.4641	0.883340
$832$	0	0
$833$	4.73205	0.163956
$834$	0	0
$835$	−18.9282	−0.655037
$836$	0	0
$837$	16.7846	0.580161
$838$	0	0
$839$	−19.5167	−0.673790	−0.336895	−	0.941542i	$-0.609377\pi$
$839$	−19.5167	−0.673790	−0.336895	+	0.941542i	$0.609377\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	4.39230	0.151279
$844$	0	0
$845$	12.4641	0.428778
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	59.7128	2.04934
$850$	0	0
$851$	0	0
$852$	0	0
$853$	47.3731	1.62202	0.811011	−	0.585030i	$-0.198917\pi$
$853$	47.3731	1.62202	0.811011	+	0.585030i	$0.198917\pi$
$854$	0	0
$855$	−6.53590	−0.223523
$856$	0	0
$857$	−11.6603	−0.398307	−0.199153	−	0.979968i	$-0.563819\pi$
$857$	−11.6603	−0.398307	−0.199153	+	0.979968i	$0.563819\pi$
$858$	0	0
$859$	31.8038	1.08513	0.542567	−	0.840013i	$-0.317453\pi$
$859$	31.8038	1.08513	0.542567	+	0.840013i	$0.317453\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	14.5359	0.494808	0.247404	−	0.968912i	$-0.420423\pi$
$863$	14.5359	0.494808	0.247404	+	0.968912i	$0.420423\pi$
$864$	0	0
$865$	−0.339746	−0.0115517
$866$	0	0
$867$	−14.7321	−0.500327
$868$	0	0
$869$	2.00000	0.0678454
$870$	0	0
$871$	2.92820	0.0992184
$872$	0	0
$873$	−10.6795	−0.361446
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−49.7128	−1.67868	−0.839341	−	0.543605i	$-0.817059\pi$
$877$	−49.7128	−1.67868	−0.839341	+	0.543605i	$0.817059\pi$
$878$	0	0
$879$	−31.8564	−1.07449
$880$	0	0
$881$	27.4641	0.925289	0.462645	−	0.886544i	$-0.346901\pi$
$881$	27.4641	0.925289	0.462645	+	0.886544i	$0.346901\pi$
$882$	0	0
$883$	−0.392305	−0.0132021	−0.00660105	−	0.999978i	$-0.502101\pi$
$883$	−0.392305	−0.0132021	−0.00660105	+	0.999978i	$0.502101\pi$
$884$	0	0
$885$	−29.3205	−0.985598
$886$	0	0
$887$	−0.679492	−0.0228151	−0.0114076	−	0.999935i	$-0.503631\pi$
$887$	−0.679492	−0.0228151	−0.0114076	+	0.999935i	$0.503631\pi$
$888$	0	0
$889$	−10.9282	−0.366520
$890$	0	0
$891$	2.46410	0.0825505
$892$	0	0
$893$	8.28719	0.277320
$894$	0	0
$895$	20.7846	0.694753
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−14.5359	−0.484799
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	−5.46410	−0.181834
$904$	0	0
$905$	14.3923	0.478416
$906$	0	0
$907$	41.8564	1.38982	0.694910	−	0.719097i	$-0.255444\pi$
$907$	41.8564	1.38982	0.694910	+	0.719097i	$0.255444\pi$
$908$	0	0
$909$	−52.0526	−1.72647
$910$	0	0
$911$	5.75129	0.190549	0.0952743	−	0.995451i	$-0.469627\pi$
$911$	5.75129	0.190549	0.0952743	+	0.995451i	$0.469627\pi$
$912$	0	0
$913$	16.3923	0.542506
$914$	0	0
$915$	−23.8564	−0.788668
$916$	0	0
$917$	5.07180	0.167485
$918$	0	0
$919$	−16.7846	−0.553673	−0.276837	−	0.960917i	$-0.589286\pi$
$919$	−16.7846	−0.553673	−0.276837	+	0.960917i	$0.589286\pi$
$920$	0	0
$921$	54.6410	1.80048
$922$	0	0
$923$	1.85641	0.0611044
$924$	0	0
$925$	−1.46410	−0.0481394
$926$	0	0
$927$	54.4449	1.78820
$928$	0	0
$929$	−13.6077	−0.446454	−0.223227	−	0.974766i	$-0.571659\pi$
$929$	−13.6077	−0.446454	−0.223227	+	0.974766i	$0.571659\pi$
$930$	0	0
$931$	1.46410	0.0479840
$932$	0	0
$933$	8.53590	0.279453
$934$	0	0
$935$	4.73205	0.154755
$936$	0	0
$937$	−38.9808	−1.27345	−0.636723	−	0.771093i	$-0.719711\pi$
$937$	−38.9808	−1.27345	−0.636723	+	0.771093i	$0.719711\pi$
$938$	0	0
$939$	−5.46410	−0.178314
$940$	0	0
$941$	34.9808	1.14034	0.570170	−	0.821527i	$-0.306877\pi$
$941$	34.9808	1.14034	0.570170	+	0.821527i	$0.306877\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	−20.1051	−0.653329	−0.326664	−	0.945140i	$-0.605925\pi$
$947$	−20.1051	−0.653329	−0.326664	+	0.945140i	$0.605925\pi$
$948$	0	0
$949$	7.46410	0.242295
$950$	0	0
$951$	−16.3923	−0.531557
$952$	0	0
$953$	−3.46410	−0.112213	−0.0561066	−	0.998425i	$-0.517869\pi$
$953$	−3.46410	−0.112213	−0.0561066	+	0.998425i	$0.517869\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.46410	0.305931
$958$	0	0
$959$	2.53590	0.0818884
$960$	0	0
$961$	−13.3923	−0.432010
$962$	0	0
$963$	30.9282	0.996647
$964$	0	0
$965$	−6.39230	−0.205776
$966$	0	0
$967$	−20.9282	−0.673006	−0.336503	−	0.941682i	$-0.609244\pi$
$967$	−20.9282	−0.673006	−0.336503	+	0.941682i	$0.609244\pi$
$968$	0	0
$969$	−18.9282	−0.608061
$970$	0	0
$971$	−13.9474	−0.447595	−0.223797	−	0.974636i	$-0.571845\pi$
$971$	−13.9474	−0.447595	−0.223797	+	0.974636i	$0.571845\pi$
$972$	0	0
$973$	5.46410	0.175171
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	43.8564	1.40309	0.701545	−	0.712625i	$-0.252494\pi$
$977$	43.8564	1.40309	0.701545	+	0.712625i	$0.252494\pi$
$978$	0	0
$979$	0.928203	0.0296655
$980$	0	0
$981$	20.2487	0.646492
$982$	0	0
$983$	−23.4115	−0.746712	−0.373356	−	0.927688i	$-0.621793\pi$
$983$	−23.4115	−0.746712	−0.373356	+	0.927688i	$0.621793\pi$
$984$	0	0
$985$	−8.53590	−0.271976
$986$	0	0
$987$	15.4641	0.492228
$988$	0	0
$989$	0	0
$990$	0	0
$991$	23.6077	0.749923	0.374962	−	0.927040i	$-0.377656\pi$
$991$	23.6077	0.749923	0.374962	+	0.927040i	$0.377656\pi$
$992$	0	0
$993$	−43.7128	−1.38718
$994$	0	0
$995$	−21.6603	−0.686676
$996$	0	0
$997$	−15.6603	−0.495965	−0.247983	−	0.968764i	$-0.579768\pi$
$997$	−15.6603	−0.495965	−0.247983	+	0.968764i	$0.579768\pi$
$998$	0	0
$999$	5.85641	0.185289

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.s.1.1		2
4.3	odd	2		1540.2.a.e.1.2	✓	2
20.3	even	4		7700.2.e.l.1849.4		4
20.7	even	4		7700.2.e.l.1849.1		4
20.19	odd	2		7700.2.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1540.2.a.e.1.2	✓	2	4.3	odd	2
6160.2.a.s.1.1		2	1.1	even	1	trivial
7700.2.a.n.1.1		2	20.19	odd	2
7700.2.e.l.1849.1		4	20.7	even	4
7700.2.e.l.1849.4		4	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73205 q^{3} -1.00000 q^{5} -1.00000 q^{7} +4.46410 q^{9} +O(q^{10})\)	\(q-2.73205 q^{3} -1.00000 q^{5} -1.00000 q^{7} +4.46410 q^{9} -1.00000 q^{11} +0.732051 q^{13} +2.73205 q^{15} +4.73205 q^{17} +1.46410 q^{19} +2.73205 q^{21} +1.00000 q^{25} -4.00000 q^{27} +3.46410 q^{29} -4.19615 q^{31} +2.73205 q^{33} +1.00000 q^{35} -1.46410 q^{37} -2.00000 q^{39} -2.19615 q^{41} -2.00000 q^{43} -4.46410 q^{45} +5.66025 q^{47} +1.00000 q^{49} -12.9282 q^{51} +2.53590 q^{53} +1.00000 q^{55} -4.00000 q^{57} -10.7321 q^{59} -8.73205 q^{61} -4.46410 q^{63} -0.732051 q^{65} +4.00000 q^{67} +2.53590 q^{71} +10.1962 q^{73} -2.73205 q^{75} +1.00000 q^{77} -2.00000 q^{79} -2.46410 q^{81} -16.3923 q^{83} -4.73205 q^{85} -9.46410 q^{87} -0.928203 q^{89} -0.732051 q^{91} +11.4641 q^{93} -1.46410 q^{95} -2.39230 q^{97} -4.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{15} + 6 q^{17} - 4 q^{19} + 2 q^{21} + 2 q^{25} - 8 q^{27} + 2 q^{31} + 2 q^{33} + 2 q^{35} + 4 q^{37} - 4 q^{39} + 6 q^{41} - 4 q^{43} - 2 q^{45} - 6 q^{47} + 2 q^{49} - 12 q^{51} + 12 q^{53} + 2 q^{55} - 8 q^{57} - 18 q^{59} - 14 q^{61} - 2 q^{63} + 2 q^{65} + 8 q^{67} + 12 q^{71} + 10 q^{73} - 2 q^{75} + 2 q^{77} - 4 q^{79} + 2 q^{81} - 12 q^{83} - 6 q^{85} - 12 q^{87} + 12 q^{89} + 2 q^{91} + 16 q^{93} + 4 q^{95} + 16 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^13 + 2 * q^15 + 6 * q^17 - 4 * q^19 + 2 * q^21 + 2 * q^25 - 8 * q^27 + 2 * q^31 + 2 * q^33 + 2 * q^35 + 4 * q^37 - 4 * q^39 + 6 * q^41 - 4 * q^43 - 2 * q^45 - 6 * q^47 + 2 * q^49 - 12 * q^51 + 12 * q^53 + 2 * q^55 - 8 * q^57 - 18 * q^59 - 14 * q^61 - 2 * q^63 + 2 * q^65 + 8 * q^67 + 12 * q^71 + 10 * q^73 - 2 * q^75 + 2 * q^77 - 4 * q^79 + 2 * q^81 - 12 * q^83 - 6 * q^85 - 12 * q^87 + 12 * q^89 + 2 * q^91 + 16 * q^93 + 4 * q^95 + 16 * q^97 - 2 * q^99

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