LMFDB - Embedded newform 1620.4.a.k.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(1,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1620.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.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:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 167x^{5} + 940x^{4} + 1153x^{3} - 13196x^{2} + 16629x + 714 $ x^7 - 2x^6 - 167x^5 + 940x^4 + 1153x^3 - 13196x^2 + 16629x + 714
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{7} $
Twist minimal:	no (minimal twist has level 180)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$3.13781$ of defining polynomial
Character	$\chi$	$=$	1620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} +27.5902 q^{7} +O(q^{10})$	$q-5.00000 q^{5} +27.5902 q^{7} -40.6355 q^{11} -67.8121 q^{13} -91.1147 q^{17} +37.9141 q^{19} -57.7036 q^{23} +25.0000 q^{25} +49.8335 q^{29} +226.034 q^{31} -137.951 q^{35} -144.062 q^{37} +348.991 q^{41} +455.982 q^{43} +179.256 q^{47} +418.217 q^{49} -300.579 q^{53} +203.177 q^{55} -439.704 q^{59} -191.659 q^{61} +339.060 q^{65} +462.629 q^{67} +671.597 q^{71} -315.669 q^{73} -1121.14 q^{77} +1009.06 q^{79} +1206.59 q^{83} +455.573 q^{85} -981.691 q^{89} -1870.95 q^{91} -189.570 q^{95} -1843.86 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 35 q^{5} + 8 q^{7}+O(q^{10}) $ 7 * q - 35 * q^5 + 8 * q^7	$ 7 q - 35 q^{5} + 8 q^{7} - 27 q^{11} + 32 q^{13} - 123 q^{17} - 67 q^{19} - 42 q^{23} + 175 q^{25} - 324 q^{29} + 98 q^{31} - 40 q^{35} + 356 q^{37} - 339 q^{41} + 119 q^{43} + 96 q^{47} + 813 q^{49} - 858 q^{53} + 135 q^{55} - 549 q^{59} + 260 q^{61} - 160 q^{65} + 881 q^{67} - 342 q^{71} + 737 q^{73} + 456 q^{77} + 1886 q^{79} + 132 q^{83} + 615 q^{85} - 396 q^{89} + 1264 q^{91} + 335 q^{95} + 1991 q^{97}+O(q^{100}) $ 7 * q - 35 * q^5 + 8 * q^7 - 27 * q^11 + 32 * q^13 - 123 * q^17 - 67 * q^19 - 42 * q^23 + 175 * q^25 - 324 * q^29 + 98 * q^31 - 40 * q^35 + 356 * q^37 - 339 * q^41 + 119 * q^43 + 96 * q^47 + 813 * q^49 - 858 * q^53 + 135 * q^55 - 549 * q^59 + 260 * q^61 - 160 * q^65 + 881 * q^67 - 342 * q^71 + 737 * q^73 + 456 * q^77 + 1886 * q^79 + 132 * q^83 + 615 * q^85 - 396 * q^89 + 1264 * q^91 + 335 * q^95 + 1991 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	27.5902	1.48973	0.744865	−	0.667216i	$-0.232514\pi$
$7$	27.5902	1.48973	0.744865	+	0.667216i	$0.232514\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−40.6355	−1.11382	−0.556912	−	0.830572i	$-0.688014\pi$
$11$	−40.6355	−1.11382	−0.556912	+	0.830572i	$0.688014\pi$
$12$	0	0
$13$	−67.8121	−1.44674	−0.723372	−	0.690458i	$-0.757409\pi$
$13$	−67.8121	−1.44674	−0.723372	+	0.690458i	$0.757409\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−91.1147	−1.29991	−0.649957	−	0.759971i	$-0.725213\pi$
$17$	−91.1147	−1.29991	−0.649957	+	0.759971i	$0.725213\pi$
$18$	0	0
$19$	37.9141	0.457794	0.228897	−	0.973451i	$-0.426488\pi$
$19$	37.9141	0.457794	0.228897	+	0.973451i	$0.426488\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−57.7036	−0.523132	−0.261566	−	0.965186i	$-0.584239\pi$
$23$	−57.7036	−0.523132	−0.261566	+	0.965186i	$0.584239\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	49.8335	0.319098	0.159549	−	0.987190i	$-0.448996\pi$
$29$	49.8335	0.319098	0.159549	+	0.987190i	$0.448996\pi$
$30$	0	0
$31$	226.034	1.30958	0.654790	−	0.755811i	$-0.272757\pi$
$31$	226.034	1.30958	0.654790	+	0.755811i	$0.272757\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−137.951	−0.666227
$36$	0	0
$37$	−144.062	−0.640097	−0.320048	−	0.947401i	$-0.603699\pi$
$37$	−144.062	−0.640097	−0.320048	+	0.947401i	$0.603699\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	348.991	1.32935	0.664674	−	0.747134i	$-0.268570\pi$
$41$	348.991	1.32935	0.664674	+	0.747134i	$0.268570\pi$
$42$	0	0
$43$	455.982	1.61713	0.808565	−	0.588406i	$-0.200244\pi$
$43$	455.982	1.61713	0.808565	+	0.588406i	$0.200244\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	179.256	0.556324	0.278162	−	0.960534i	$-0.410275\pi$
$47$	179.256	0.556324	0.278162	+	0.960534i	$0.410275\pi$
$48$	0	0
$49$	418.217	1.21929
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−300.579	−0.779012	−0.389506	−	0.921024i	$-0.627354\pi$
$53$	−300.579	−0.779012	−0.389506	+	0.921024i	$0.627354\pi$
$54$	0	0
$55$	203.177	0.498117
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−439.704	−0.970248	−0.485124	−	0.874446i	$-0.661225\pi$
$59$	−439.704	−0.970248	−0.485124	+	0.874446i	$0.661225\pi$
$60$	0	0
$61$	−191.659	−0.402285	−0.201142	−	0.979562i	$-0.564465\pi$
$61$	−191.659	−0.402285	−0.201142	+	0.979562i	$0.564465\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	339.060	0.647004
$66$	0	0
$67$	462.629	0.843570	0.421785	−	0.906696i	$-0.361404\pi$
$67$	462.629	0.843570	0.421785	+	0.906696i	$0.361404\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	671.597	1.12259	0.561295	−	0.827616i	$-0.310303\pi$
$71$	671.597	1.12259	0.561295	+	0.827616i	$0.310303\pi$
$72$	0	0
$73$	−315.669	−0.506114	−0.253057	−	0.967451i	$-0.581436\pi$
$73$	−315.669	−0.506114	−0.253057	+	0.967451i	$0.581436\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1121.14	−1.65929
$78$	0	0
$79$	1009.06	1.43706	0.718529	−	0.695497i	$-0.244816\pi$
$79$	1009.06	1.43706	0.718529	+	0.695497i	$0.244816\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1206.59	1.59566	0.797832	−	0.602879i	$-0.205980\pi$
$83$	1206.59	1.59566	0.797832	+	0.602879i	$0.205980\pi$
$84$	0	0
$85$	455.573	0.581340
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−981.691	−1.16920	−0.584602	−	0.811321i	$-0.698749\pi$
$89$	−981.691	−1.16920	−0.584602	+	0.811321i	$0.698749\pi$
$90$	0	0
$91$	−1870.95	−2.15526
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−189.570	−0.204732
$96$	0	0
$97$	−1843.86	−1.93006	−0.965029	−	0.262142i	$-0.915571\pi$
$97$	−1843.86	−1.93006	−0.965029	+	0.262142i	$0.915571\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1334.53	1.31476	0.657379	−	0.753560i	$-0.271665\pi$
$101$	1334.53	1.31476	0.657379	+	0.753560i	$0.271665\pi$
$102$	0	0
$103$	629.671	0.602363	0.301181	−	0.953567i	$-0.402619\pi$
$103$	629.671	0.602363	0.301181	+	0.953567i	$0.402619\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1576.35	−1.42422	−0.712110	−	0.702067i	$-0.752261\pi$
$107$	−1576.35	−1.42422	−0.712110	+	0.702067i	$0.752261\pi$
$108$	0	0
$109$	−970.642	−0.852941	−0.426471	−	0.904501i	$-0.640243\pi$
$109$	−970.642	−0.852941	−0.426471	+	0.904501i	$0.640243\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	888.447	0.739629	0.369814	−	0.929106i	$-0.379421\pi$
$113$	888.447	0.739629	0.369814	+	0.929106i	$0.379421\pi$
$114$	0	0
$115$	288.518	0.233952
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2513.87	−1.93652
$120$	0	0
$121$	320.241	0.240602
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	2213.09	1.54630	0.773150	−	0.634223i	$-0.218680\pi$
$127$	2213.09	1.54630	0.773150	+	0.634223i	$0.218680\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1360.28	0.907240	0.453620	−	0.891195i	$-0.350132\pi$
$131$	1360.28	0.907240	0.453620	+	0.891195i	$0.350132\pi$
$132$	0	0
$133$	1046.06	0.681989
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2560.31	1.59666	0.798330	−	0.602221i	$-0.205717\pi$
$137$	2560.31	1.59666	0.798330	+	0.602221i	$0.205717\pi$
$138$	0	0
$139$	−1084.30	−0.661648	−0.330824	−	0.943692i	$-0.607327\pi$
$139$	−1084.30	−0.661648	−0.330824	+	0.943692i	$0.607327\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2755.57	1.61142
$144$	0	0
$145$	−249.168	−0.142705
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−712.441	−0.391714	−0.195857	−	0.980632i	$-0.562749\pi$
$149$	−712.441	−0.391714	−0.195857	+	0.980632i	$0.562749\pi$
$150$	0	0
$151$	1841.99	0.992710	0.496355	−	0.868120i	$-0.334671\pi$
$151$	1841.99	0.992710	0.496355	+	0.868120i	$0.334671\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1130.17	−0.585662
$156$	0	0
$157$	−43.8854	−0.0223085	−0.0111543	−	0.999938i	$-0.503551\pi$
$157$	−43.8854	−0.0223085	−0.0111543	+	0.999938i	$0.503551\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1592.05	−0.779325
$162$	0	0
$163$	2692.78	1.29396	0.646978	−	0.762509i	$-0.276033\pi$
$163$	2692.78	1.29396	0.646978	+	0.762509i	$0.276033\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−358.991	−0.166345	−0.0831724	−	0.996535i	$-0.526505\pi$
$167$	−358.991	−0.166345	−0.0831724	+	0.996535i	$0.526505\pi$
$168$	0	0
$169$	2401.48	1.09307
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2413.47	−1.06065	−0.530325	−	0.847794i	$-0.677930\pi$
$173$	−2413.47	−1.06065	−0.530325	+	0.847794i	$0.677930\pi$
$174$	0	0
$175$	689.754	0.297946
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1877.35	0.783908	0.391954	−	0.919985i	$-0.371799\pi$
$179$	1877.35	0.783908	0.391954	+	0.919985i	$0.371799\pi$
$180$	0	0
$181$	2739.27	1.12491	0.562455	−	0.826828i	$-0.309857\pi$
$181$	2739.27	1.12491	0.562455	+	0.826828i	$0.309857\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	720.308	0.286260
$186$	0	0
$187$	3702.49	1.44788
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2091.20	−0.792221	−0.396111	−	0.918203i	$-0.629640\pi$
$191$	−2091.20	−0.792221	−0.396111	+	0.918203i	$0.629640\pi$
$192$	0	0
$193$	4664.21	1.73957	0.869786	−	0.493429i	$-0.164257\pi$
$193$	4664.21	1.73957	0.869786	+	0.493429i	$0.164257\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1904.63	0.688829	0.344414	−	0.938818i	$-0.388077\pi$
$197$	1904.63	0.688829	0.344414	+	0.938818i	$0.388077\pi$
$198$	0	0
$199$	4453.57	1.58646	0.793228	−	0.608924i	$-0.208399\pi$
$199$	4453.57	1.58646	0.793228	+	0.608924i	$0.208399\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1374.91	0.475370
$204$	0	0
$205$	−1744.96	−0.594502
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1540.66	−0.509902
$210$	0	0
$211$	1649.86	0.538297	0.269149	−	0.963099i	$-0.413258\pi$
$211$	1649.86	0.538297	0.269149	+	0.963099i	$0.413258\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2279.91	−0.723203
$216$	0	0
$217$	6236.33	1.95092
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6178.67	1.88064
$222$	0	0
$223$	651.300	0.195580	0.0977899	−	0.995207i	$-0.468823\pi$
$223$	651.300	0.195580	0.0977899	+	0.995207i	$0.468823\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5342.53	1.56210	0.781049	−	0.624469i	$-0.214685\pi$
$227$	5342.53	1.56210	0.781049	+	0.624469i	$0.214685\pi$
$228$	0	0
$229$	147.933	0.0426887	0.0213444	−	0.999772i	$-0.493205\pi$
$229$	147.933	0.0426887	0.0213444	+	0.999772i	$0.493205\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1000.28	0.281246	0.140623	−	0.990063i	$-0.455089\pi$
$233$	1000.28	0.281246	0.140623	+	0.990063i	$0.455089\pi$
$234$	0	0
$235$	−896.281	−0.248796
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5837.35	−1.57986	−0.789931	−	0.613196i	$-0.789883\pi$
$239$	−5837.35	−1.57986	−0.789931	+	0.613196i	$0.789883\pi$
$240$	0	0
$241$	3537.47	0.945512	0.472756	−	0.881193i	$-0.343259\pi$
$241$	3537.47	0.945512	0.472756	+	0.881193i	$0.343259\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2091.09	−0.545284
$246$	0	0
$247$	−2571.03	−0.662311
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4731.86	1.18993	0.594965	−	0.803751i	$-0.297166\pi$
$251$	4731.86	1.18993	0.594965	+	0.803751i	$0.297166\pi$
$252$	0	0
$253$	2344.81	0.582676
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3315.64	−0.804763	−0.402382	−	0.915472i	$-0.631817\pi$
$257$	−3315.64	−0.804763	−0.402382	+	0.915472i	$0.631817\pi$
$258$	0	0
$259$	−3974.68	−0.953571
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1729.31	−0.405453	−0.202726	−	0.979235i	$-0.564980\pi$
$263$	−1729.31	−0.405453	−0.202726	+	0.979235i	$0.564980\pi$
$264$	0	0
$265$	1502.89	0.348385
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3978.37	−0.901730	−0.450865	−	0.892592i	$-0.648884\pi$
$269$	−3978.37	−0.901730	−0.450865	+	0.892592i	$0.648884\pi$
$270$	0	0
$271$	−3766.06	−0.844176	−0.422088	−	0.906555i	$-0.638703\pi$
$271$	−3766.06	−0.844176	−0.422088	+	0.906555i	$0.638703\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1015.89	−0.222765
$276$	0	0
$277$	−5674.49	−1.23086	−0.615428	−	0.788193i	$-0.711017\pi$
$277$	−5674.49	−1.23086	−0.615428	+	0.788193i	$0.711017\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−8143.51	−1.72883	−0.864415	−	0.502779i	$-0.832311\pi$
$281$	−8143.51	−1.72883	−0.864415	+	0.502779i	$0.832311\pi$
$282$	0	0
$283$	−7837.87	−1.64634	−0.823168	−	0.567797i	$-0.807796\pi$
$283$	−7837.87	−1.64634	−0.823168	+	0.567797i	$0.807796\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9628.73	1.98037
$288$	0	0
$289$	3388.88	0.689778
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3532.41	−0.704320	−0.352160	−	0.935940i	$-0.614553\pi$
$293$	−3532.41	−0.704320	−0.352160	+	0.935940i	$0.614553\pi$
$294$	0	0
$295$	2198.52	0.433908
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3913.00	0.756838
$300$	0	0
$301$	12580.6	2.40909
$302$	0	0
$303$	0	0
$304$	0	0
$305$	958.293	0.179907
$306$	0	0
$307$	6377.34	1.18558	0.592791	−	0.805356i	$-0.298026\pi$
$307$	6377.34	1.18558	0.592791	+	0.805356i	$0.298026\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1114.21	0.203154	0.101577	−	0.994828i	$-0.467611\pi$
$311$	1114.21	0.203154	0.101577	+	0.994828i	$0.467611\pi$
$312$	0	0
$313$	5120.97	0.924774	0.462387	−	0.886678i	$-0.346993\pi$
$313$	5120.97	0.924774	0.462387	+	0.886678i	$0.346993\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8199.22	−1.45273	−0.726363	−	0.687312i	$-0.758791\pi$
$317$	−8199.22	−1.45273	−0.726363	+	0.687312i	$0.758791\pi$
$318$	0	0
$319$	−2025.01	−0.355419
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3454.53	−0.595093
$324$	0	0
$325$	−1695.30	−0.289349
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4945.71	0.828772
$330$	0	0
$331$	1277.72	0.212174	0.106087	−	0.994357i	$-0.466168\pi$
$331$	1277.72	0.212174	0.106087	+	0.994357i	$0.466168\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2313.15	−0.377256
$336$	0	0
$337$	−1000.19	−0.161674	−0.0808369	−	0.996727i	$-0.525759\pi$
$337$	−1000.19	−0.161674	−0.0808369	+	0.996727i	$0.525759\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−9185.01	−1.45864
$342$	0	0
$343$	2075.26	0.326687
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−500.737	−0.0774668	−0.0387334	−	0.999250i	$-0.512332\pi$
$347$	−500.737	−0.0774668	−0.0387334	+	0.999250i	$0.512332\pi$
$348$	0	0
$349$	−939.809	−0.144146	−0.0720728	−	0.997399i	$-0.522961\pi$
$349$	−939.809	−0.144146	−0.0720728	+	0.997399i	$0.522961\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9222.42	1.39054	0.695269	−	0.718750i	$-0.255285\pi$
$353$	9222.42	1.39054	0.695269	+	0.718750i	$0.255285\pi$
$354$	0	0
$355$	−3357.98	−0.502038
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3421.30	0.502978	0.251489	−	0.967860i	$-0.419080\pi$
$359$	3421.30	0.502978	0.251489	+	0.967860i	$0.419080\pi$
$360$	0	0
$361$	−5421.52	−0.790425
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1578.35	0.226341
$366$	0	0
$367$	7103.18	1.01031	0.505154	−	0.863029i	$-0.331436\pi$
$367$	7103.18	1.01031	0.505154	+	0.863029i	$0.331436\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8293.01	−1.16052
$372$	0	0
$373$	−5541.24	−0.769207	−0.384604	−	0.923082i	$-0.625662\pi$
$373$	−5541.24	−0.769207	−0.384604	+	0.923082i	$0.625662\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3379.31	−0.461654
$378$	0	0
$379$	6767.47	0.917208	0.458604	−	0.888641i	$-0.348350\pi$
$379$	6767.47	0.917208	0.458604	+	0.888641i	$0.348350\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2817.75	−0.375928	−0.187964	−	0.982176i	$-0.560189\pi$
$383$	−2817.75	−0.375928	−0.187964	+	0.982176i	$0.560189\pi$
$384$	0	0
$385$	5605.70	0.742059
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10613.3	1.38333	0.691667	−	0.722216i	$-0.256876\pi$
$389$	10613.3	1.38333	0.691667	+	0.722216i	$0.256876\pi$
$390$	0	0
$391$	5257.64	0.680027
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5045.28	−0.642672
$396$	0	0
$397$	−18.0434	−0.00228104	−0.00114052	−	0.999999i	$-0.500363\pi$
$397$	−18.0434	−0.00228104	−0.00114052	+	0.999999i	$0.500363\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13901.1	1.73114	0.865568	−	0.500792i	$-0.166958\pi$
$401$	13901.1	1.73114	0.865568	+	0.500792i	$0.166958\pi$
$402$	0	0
$403$	−15327.9	−1.89463
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5854.01	0.712955
$408$	0	0
$409$	10340.1	1.25008	0.625041	−	0.780592i	$-0.285082\pi$
$409$	10340.1	1.25008	0.625041	+	0.780592i	$0.285082\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−12131.5	−1.44541
$414$	0	0
$415$	−6032.94	−0.713603
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4320.01	−0.503690	−0.251845	−	0.967768i	$-0.581037\pi$
$419$	−4320.01	−0.503690	−0.251845	+	0.967768i	$0.581037\pi$
$420$	0	0
$421$	7446.00	0.861985	0.430993	−	0.902355i	$-0.358163\pi$
$421$	7446.00	0.861985	0.430993	+	0.902355i	$0.358163\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2277.87	−0.259983
$426$	0	0
$427$	−5287.89	−0.599295
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1987.46	0.222118	0.111059	−	0.993814i	$-0.464576\pi$
$431$	1987.46	0.222118	0.111059	+	0.993814i	$0.464576\pi$
$432$	0	0
$433$	−8131.93	−0.902531	−0.451265	−	0.892390i	$-0.649027\pi$
$433$	−8131.93	−0.902531	−0.451265	+	0.892390i	$0.649027\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2187.78	−0.239487
$438$	0	0
$439$	4402.69	0.478653	0.239327	−	0.970939i	$-0.423073\pi$
$439$	4402.69	0.478653	0.239327	+	0.970939i	$0.423073\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−355.972	−0.0381777	−0.0190888	−	0.999818i	$-0.506077\pi$
$443$	−355.972	−0.0381777	−0.0190888	+	0.999818i	$0.506077\pi$
$444$	0	0
$445$	4908.46	0.522883
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11811.1	1.24143	0.620715	−	0.784036i	$-0.286842\pi$
$449$	11811.1	1.24143	0.620715	+	0.784036i	$0.286842\pi$
$450$	0	0
$451$	−14181.4	−1.48066
$452$	0	0
$453$	0	0
$454$	0	0
$455$	9354.73	0.963861
$456$	0	0
$457$	14187.7	1.45224	0.726119	−	0.687569i	$-0.241322\pi$
$457$	14187.7	1.45224	0.726119	+	0.687569i	$0.241322\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15762.5	1.59248	0.796238	−	0.604983i	$-0.206820\pi$
$461$	15762.5	1.59248	0.796238	+	0.604983i	$0.206820\pi$
$462$	0	0
$463$	7776.62	0.780583	0.390292	−	0.920691i	$-0.372374\pi$
$463$	7776.62	0.780583	0.390292	+	0.920691i	$0.372374\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8631.61	0.855296	0.427648	−	0.903945i	$-0.359342\pi$
$467$	8631.61	0.855296	0.427648	+	0.903945i	$0.359342\pi$
$468$	0	0
$469$	12764.0	1.25669
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−18529.0	−1.80120
$474$	0	0
$475$	947.852	0.0915588
$476$	0	0
$477$	0	0
$478$	0	0
$479$	15912.3	1.51786	0.758928	−	0.651174i	$-0.225723\pi$
$479$	15912.3	1.51786	0.758928	+	0.651174i	$0.225723\pi$
$480$	0	0
$481$	9769.11	0.926057
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9219.30	0.863148
$486$	0	0
$487$	−3428.12	−0.318979	−0.159490	−	0.987200i	$-0.550985\pi$
$487$	−3428.12	−0.318979	−0.159490	+	0.987200i	$0.550985\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3238.02	0.297617	0.148808	−	0.988866i	$-0.452456\pi$
$491$	3238.02	0.297617	0.148808	+	0.988866i	$0.452456\pi$
$492$	0	0
$493$	−4540.56	−0.414801
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18529.5	1.67236
$498$	0	0
$499$	−7583.94	−0.680368	−0.340184	−	0.940359i	$-0.610489\pi$
$499$	−7583.94	−0.680368	−0.340184	+	0.940359i	$0.610489\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11565.0	−1.02517	−0.512584	−	0.858637i	$-0.671312\pi$
$503$	−11565.0	−1.02517	−0.512584	+	0.858637i	$0.671312\pi$
$504$	0	0
$505$	−6672.64	−0.587978
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−727.066	−0.0633136	−0.0316568	−	0.999499i	$-0.510078\pi$
$509$	−727.066	−0.0633136	−0.0316568	+	0.999499i	$0.510078\pi$
$510$	0	0
$511$	−8709.37	−0.753972
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3148.36	−0.269385
$516$	0	0
$517$	−7284.16	−0.619646
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5535.95	0.465517	0.232758	−	0.972535i	$-0.425225\pi$
$521$	5535.95	0.465517	0.232758	+	0.972535i	$0.425225\pi$
$522$	0	0
$523$	240.008	0.0200666	0.0100333	−	0.999950i	$-0.496806\pi$
$523$	240.008	0.0200666	0.0100333	+	0.999950i	$0.496806\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20595.0	−1.70234
$528$	0	0
$529$	−8837.30	−0.726333
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−23665.8	−1.92323
$534$	0	0
$535$	7881.76	0.636931
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−16994.5	−1.35808
$540$	0	0
$541$	−9081.20	−0.721685	−0.360842	−	0.932627i	$-0.617511\pi$
$541$	−9081.20	−0.721685	−0.360842	+	0.932627i	$0.617511\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4853.21	0.381447
$546$	0	0
$547$	−8343.91	−0.652212	−0.326106	−	0.945333i	$-0.605737\pi$
$547$	−8343.91	−0.652212	−0.326106	+	0.945333i	$0.605737\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1889.39	0.146081
$552$	0	0
$553$	27840.0	2.14083
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18639.2	−1.41789	−0.708947	−	0.705262i	$-0.750829\pi$
$557$	−18639.2	−1.41789	−0.708947	+	0.705262i	$0.750829\pi$
$558$	0	0
$559$	−30921.1	−2.33958
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12055.5	−0.902446	−0.451223	−	0.892411i	$-0.649012\pi$
$563$	−12055.5	−0.902446	−0.451223	+	0.892411i	$0.649012\pi$
$564$	0	0
$565$	−4442.23	−0.330772
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10144.5	0.747419	0.373709	−	0.927546i	$-0.378086\pi$
$569$	10144.5	0.747419	0.373709	+	0.927546i	$0.378086\pi$
$570$	0	0
$571$	−9602.09	−0.703739	−0.351870	−	0.936049i	$-0.614454\pi$
$571$	−9602.09	−0.703739	−0.351870	+	0.936049i	$0.614454\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1442.59	−0.104626
$576$	0	0
$577$	7252.47	0.523266	0.261633	−	0.965167i	$-0.415739\pi$
$577$	7252.47	0.523266	0.261633	+	0.965167i	$0.415739\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	33289.9	2.37711
$582$	0	0
$583$	12214.2	0.867682
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9277.45	−0.652336	−0.326168	−	0.945312i	$-0.605757\pi$
$587$	−9277.45	−0.652336	−0.326168	+	0.945312i	$0.605757\pi$
$588$	0	0
$589$	8569.89	0.599518
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−5029.91	−0.348320	−0.174160	−	0.984717i	$-0.555721\pi$
$593$	−5029.91	−0.348320	−0.174160	+	0.984717i	$0.555721\pi$
$594$	0	0
$595$	12569.3	0.866038
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4744.92	−0.323660	−0.161830	−	0.986819i	$-0.551740\pi$
$599$	−4744.92	−0.323660	−0.161830	+	0.986819i	$0.551740\pi$
$600$	0	0
$601$	−3089.13	−0.209664	−0.104832	−	0.994490i	$-0.533431\pi$
$601$	−3089.13	−0.209664	−0.104832	+	0.994490i	$0.533431\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1601.21	−0.107600
$606$	0	0
$607$	−3723.74	−0.248998	−0.124499	−	0.992220i	$-0.539732\pi$
$607$	−3723.74	−0.248998	−0.124499	+	0.992220i	$0.539732\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12155.7	−0.804858
$612$	0	0
$613$	5546.21	0.365431	0.182716	−	0.983166i	$-0.441511\pi$
$613$	5546.21	0.365431	0.182716	+	0.983166i	$0.441511\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14923.2	−0.973722	−0.486861	−	0.873480i	$-0.661858\pi$
$617$	−14923.2	−0.973722	−0.486861	+	0.873480i	$0.661858\pi$
$618$	0	0
$619$	−744.177	−0.0483215	−0.0241608	−	0.999708i	$-0.507691\pi$
$619$	−744.177	−0.0483215	−0.0241608	+	0.999708i	$0.507691\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−27085.0	−1.74180
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13126.1	0.832071
$630$	0	0
$631$	−22782.4	−1.43733	−0.718664	−	0.695358i	$-0.755246\pi$
$631$	−22782.4	−1.43733	−0.718664	+	0.695358i	$0.755246\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−11065.5	−0.691526
$636$	0	0
$637$	−28360.2	−1.76401
$638$	0	0
$639$	0	0
$640$	0	0
$641$	19382.1	1.19430	0.597151	−	0.802128i	$-0.296299\pi$
$641$	19382.1	1.19430	0.597151	+	0.802128i	$0.296299\pi$
$642$	0	0
$643$	−30460.3	−1.86817	−0.934087	−	0.357045i	$-0.883784\pi$
$643$	−30460.3	−1.86817	−0.934087	+	0.357045i	$0.883784\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	29231.7	1.77623	0.888113	−	0.459625i	$-0.152016\pi$
$647$	29231.7	1.77623	0.888113	+	0.459625i	$0.152016\pi$
$648$	0	0
$649$	17867.6	1.08068
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12970.2	−0.777278	−0.388639	−	0.921390i	$-0.627055\pi$
$653$	−12970.2	−0.777278	−0.388639	+	0.921390i	$0.627055\pi$
$654$	0	0
$655$	−6801.41	−0.405730
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−974.473	−0.0576026	−0.0288013	−	0.999585i	$-0.509169\pi$
$659$	−974.473	−0.0576026	−0.0288013	+	0.999585i	$0.509169\pi$
$660$	0	0
$661$	−28411.8	−1.67185	−0.835924	−	0.548845i	$-0.815068\pi$
$661$	−28411.8	−1.67185	−0.835924	+	0.548845i	$0.815068\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5230.28	−0.304995
$666$	0	0
$667$	−2875.57	−0.166930
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7788.13	0.448074
$672$	0	0
$673$	17557.3	1.00562	0.502812	−	0.864396i	$-0.332299\pi$
$673$	17557.3	1.00562	0.502812	+	0.864396i	$0.332299\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3641.15	0.206707	0.103354	−	0.994645i	$-0.467043\pi$
$677$	3641.15	0.206707	0.103354	+	0.994645i	$0.467043\pi$
$678$	0	0
$679$	−50872.4	−2.87526
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1265.70	−0.0709088	−0.0354544	−	0.999371i	$-0.511288\pi$
$683$	−1265.70	−0.0709088	−0.0354544	+	0.999371i	$0.511288\pi$
$684$	0	0
$685$	−12801.6	−0.714048
$686$	0	0
$687$	0	0
$688$	0	0
$689$	20382.9	1.12703
$690$	0	0
$691$	14471.4	0.796695	0.398347	−	0.917235i	$-0.369584\pi$
$691$	14471.4	0.796695	0.398347	+	0.917235i	$0.369584\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5421.50	0.295898
$696$	0	0
$697$	−31798.2	−1.72804
$698$	0	0
$699$	0	0
$700$	0	0
$701$	29212.0	1.57393	0.786964	−	0.616999i	$-0.211652\pi$
$701$	29212.0	1.57393	0.786964	+	0.616999i	$0.211652\pi$
$702$	0	0
$703$	−5461.96	−0.293032
$704$	0	0
$705$	0	0
$706$	0	0
$707$	36819.9	1.95863
$708$	0	0
$709$	−19489.9	−1.03238	−0.516191	−	0.856474i	$-0.672651\pi$
$709$	−19489.9	−1.03238	−0.516191	+	0.856474i	$0.672651\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−13043.0	−0.685083
$714$	0	0
$715$	−13777.9	−0.720648
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10719.9	0.556029	0.278015	−	0.960577i	$-0.410324\pi$
$719$	10719.9	0.556029	0.278015	+	0.960577i	$0.410324\pi$
$720$	0	0
$721$	17372.7	0.897357
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1245.84	0.0638197
$726$	0	0
$727$	−29124.6	−1.48579	−0.742897	−	0.669406i	$-0.766549\pi$
$727$	−29124.6	−1.48579	−0.742897	+	0.669406i	$0.766549\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−41546.7	−2.10213
$732$	0	0
$733$	−18383.3	−0.926334	−0.463167	−	0.886271i	$-0.653287\pi$
$733$	−18383.3	−0.926334	−0.463167	+	0.886271i	$0.653287\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−18799.2	−0.939588
$738$	0	0
$739$	−16918.9	−0.842184	−0.421092	−	0.907018i	$-0.638353\pi$
$739$	−16918.9	−0.842184	−0.421092	+	0.907018i	$0.638353\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31448.8	1.55282	0.776409	−	0.630229i	$-0.217039\pi$
$743$	31448.8	1.55282	0.776409	+	0.630229i	$0.217039\pi$
$744$	0	0
$745$	3562.20	0.175180
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−43491.8	−2.12170
$750$	0	0
$751$	20854.7	1.01331	0.506657	−	0.862148i	$-0.330881\pi$
$751$	20854.7	1.01331	0.506657	+	0.862148i	$0.330881\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−9209.97	−0.443954
$756$	0	0
$757$	9620.01	0.461883	0.230941	−	0.972968i	$-0.425819\pi$
$757$	9620.01	0.461883	0.230941	+	0.972968i	$0.425819\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−33751.6	−1.60775	−0.803873	−	0.594802i	$-0.797231\pi$
$761$	−33751.6	−1.60775	−0.803873	+	0.594802i	$0.797231\pi$
$762$	0	0
$763$	−26780.2	−1.27065
$764$	0	0
$765$	0	0
$766$	0	0
$767$	29817.2	1.40370
$768$	0	0
$769$	−7545.40	−0.353828	−0.176914	−	0.984226i	$-0.556611\pi$
$769$	−7545.40	−0.353828	−0.176914	+	0.984226i	$0.556611\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−5074.04	−0.236094	−0.118047	−	0.993008i	$-0.537663\pi$
$773$	−5074.04	−0.236094	−0.118047	+	0.993008i	$0.537663\pi$
$774$	0	0
$775$	5650.86	0.261916
$776$	0	0
$777$	0	0
$778$	0	0
$779$	13231.7	0.608568
$780$	0	0
$781$	−27290.7	−1.25037
$782$	0	0
$783$	0	0
$784$	0	0
$785$	219.427	0.00997668
$786$	0	0
$787$	20651.2	0.935370	0.467685	−	0.883895i	$-0.345088\pi$
$787$	20651.2	0.935370	0.467685	+	0.883895i	$0.345088\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	24512.4	1.10185
$792$	0	0
$793$	12996.8	0.582003
$794$	0	0
$795$	0	0
$796$	0	0
$797$	19808.3	0.880357	0.440178	−	0.897910i	$-0.354915\pi$
$797$	19808.3	0.880357	0.440178	+	0.897910i	$0.354915\pi$
$798$	0	0
$799$	−16332.9	−0.723173
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12827.4	0.563721
$804$	0	0
$805$	7960.26	0.348525
$806$	0	0
$807$	0	0
$808$	0	0
$809$	22926.4	0.996351	0.498176	−	0.867076i	$-0.334004\pi$
$809$	22926.4	0.996351	0.498176	+	0.867076i	$0.334004\pi$
$810$	0	0
$811$	−10506.7	−0.454922	−0.227461	−	0.973787i	$-0.573042\pi$
$811$	−10506.7	−0.454922	−0.227461	+	0.973787i	$0.573042\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−13463.9	−0.578674
$816$	0	0
$817$	17288.1	0.740313
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3939.95	−0.167485	−0.0837425	−	0.996487i	$-0.526687\pi$
$821$	−3939.95	−0.167485	−0.0837425	+	0.996487i	$0.526687\pi$
$822$	0	0
$823$	9372.15	0.396953	0.198477	−	0.980106i	$-0.436401\pi$
$823$	9372.15	0.396953	0.198477	+	0.980106i	$0.436401\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1550.64	0.0652006	0.0326003	−	0.999468i	$-0.489621\pi$
$827$	1550.64	0.0652006	0.0326003	+	0.999468i	$0.489621\pi$
$828$	0	0
$829$	−25282.7	−1.05923	−0.529617	−	0.848237i	$-0.677664\pi$
$829$	−25282.7	−1.05923	−0.529617	+	0.848237i	$0.677664\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−38105.7	−1.58498
$834$	0	0
$835$	1794.96	0.0743917
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−13131.9	−0.540360	−0.270180	−	0.962810i	$-0.587083\pi$
$839$	−13131.9	−0.540360	−0.270180	+	0.962810i	$0.587083\pi$
$840$	0	0
$841$	−21905.6	−0.898176
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12007.4	−0.488836
$846$	0	0
$847$	8835.51	0.358432
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8312.87	0.334855
$852$	0	0
$853$	12884.3	0.517176	0.258588	−	0.965988i	$-0.416743\pi$
$853$	12884.3	0.517176	0.258588	+	0.965988i	$0.416743\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−19594.0	−0.781000	−0.390500	−	0.920603i	$-0.627698\pi$
$857$	−19594.0	−0.781000	−0.390500	+	0.920603i	$0.627698\pi$
$858$	0	0
$859$	16755.0	0.665510	0.332755	−	0.943013i	$-0.392022\pi$
$859$	16755.0	0.665510	0.332755	+	0.943013i	$0.392022\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30825.4	−1.21588	−0.607942	−	0.793981i	$-0.708005\pi$
$863$	−30825.4	−1.21588	−0.607942	+	0.793981i	$0.708005\pi$
$864$	0	0
$865$	12067.3	0.474337
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−41003.4	−1.60063
$870$	0	0
$871$	−31371.9	−1.22043
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3448.77	−0.133245
$876$	0	0
$877$	27797.1	1.07029	0.535143	−	0.844762i	$-0.320258\pi$
$877$	27797.1	1.07029	0.535143	+	0.844762i	$0.320258\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	10623.1	0.406245	0.203123	−	0.979153i	$-0.434891\pi$
$881$	10623.1	0.406245	0.203123	+	0.979153i	$0.434891\pi$
$882$	0	0
$883$	3165.81	0.120654	0.0603272	−	0.998179i	$-0.480786\pi$
$883$	3165.81	0.120654	0.0603272	+	0.998179i	$0.480786\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7248.96	−0.274404	−0.137202	−	0.990543i	$-0.543811\pi$
$887$	−7248.96	−0.274404	−0.137202	+	0.990543i	$0.543811\pi$
$888$	0	0
$889$	61059.6	2.30357
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6796.34	0.254682
$894$	0	0
$895$	−9386.73	−0.350574
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11264.1	0.417885
$900$	0	0
$901$	27387.1	1.01265
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13696.4	−0.503075
$906$	0	0
$907$	−36264.2	−1.32760	−0.663800	−	0.747910i	$-0.731057\pi$
$907$	−36264.2	−1.32760	−0.663800	+	0.747910i	$0.731057\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	38234.8	1.39053	0.695267	−	0.718752i	$-0.255286\pi$
$911$	38234.8	1.39053	0.695267	+	0.718752i	$0.255286\pi$
$912$	0	0
$913$	−49030.2	−1.77729
$914$	0	0
$915$	0	0
$916$	0	0
$917$	37530.4	1.35154
$918$	0	0
$919$	28415.2	1.01995	0.509973	−	0.860190i	$-0.329655\pi$
$919$	28415.2	1.01995	0.509973	+	0.860190i	$0.329655\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−45542.4	−1.62410
$924$	0	0
$925$	−3601.54	−0.128019
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−32644.9	−1.15290	−0.576451	−	0.817132i	$-0.695563\pi$
$929$	−32644.9	−1.15290	−0.576451	+	0.817132i	$0.695563\pi$
$930$	0	0
$931$	15856.3	0.558185
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−18512.4	−0.647509
$936$	0	0
$937$	−23610.7	−0.823190	−0.411595	−	0.911367i	$-0.635028\pi$
$937$	−23610.7	−0.823190	−0.411595	+	0.911367i	$0.635028\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	48967.6	1.69638	0.848192	−	0.529689i	$-0.177691\pi$
$941$	48967.6	1.69638	0.848192	+	0.529689i	$0.177691\pi$
$942$	0	0
$943$	−20138.0	−0.695424
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28161.0	0.966325	0.483163	−	0.875531i	$-0.339488\pi$
$947$	28161.0	0.966325	0.483163	+	0.875531i	$0.339488\pi$
$948$	0	0
$949$	21406.2	0.732217
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22797.7	0.774910	0.387455	−	0.921889i	$-0.373354\pi$
$953$	22797.7	0.774910	0.387455	+	0.921889i	$0.373354\pi$
$954$	0	0
$955$	10456.0	0.354292
$956$	0	0
$957$	0	0
$958$	0	0
$959$	70639.5	2.37859
$960$	0	0
$961$	21300.6	0.715000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−23321.1	−0.777960
$966$	0	0
$967$	−9433.99	−0.313730	−0.156865	−	0.987620i	$-0.550139\pi$
$967$	−9433.99	−0.313730	−0.156865	+	0.987620i	$0.550139\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1008.70	0.0333374	0.0166687	−	0.999861i	$-0.494694\pi$
$971$	1008.70	0.0333374	0.0166687	+	0.999861i	$0.494694\pi$
$972$	0	0
$973$	−29916.0	−0.985677
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5323.59	−0.174326	−0.0871631	−	0.996194i	$-0.527780\pi$
$977$	−5323.59	−0.174326	−0.0871631	+	0.996194i	$0.527780\pi$
$978$	0	0
$979$	39891.5	1.30229
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−34288.7	−1.11255	−0.556276	−	0.830997i	$-0.687770\pi$
$983$	−34288.7	−1.11255	−0.556276	+	0.830997i	$0.687770\pi$
$984$	0	0
$985$	−9523.15	−0.308054
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−26311.8	−0.845972
$990$	0	0
$991$	56564.4	1.81315	0.906573	−	0.422050i	$-0.138689\pi$
$991$	56564.4	1.81315	0.906573	+	0.422050i	$0.138689\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−22267.8	−0.709485
$996$	0	0
$997$	43860.0	1.39324	0.696620	−	0.717440i	$-0.254686\pi$
$997$	43860.0	1.39324	0.696620	+	0.717440i	$0.254686\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.a.k.1.7		7
3.2	odd	2		1620.4.a.l.1.7		7
9.2	odd	6		180.4.i.c.121.7	yes	14
9.4	even	3		540.4.i.c.181.1		14
9.5	odd	6		180.4.i.c.61.7	✓	14
9.7	even	3		540.4.i.c.361.1		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.4.i.c.61.7	✓	14	9.5	odd	6
180.4.i.c.121.7	yes	14	9.2	odd	6
540.4.i.c.181.1		14	9.4	even	3
540.4.i.c.361.1		14	9.7	even	3
1620.4.a.k.1.7		7	1.1	even	1	trivial
1620.4.a.l.1.7		7	3.2	odd	2

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} +27.5902 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} +27.5902 q^{7} -40.6355 q^{11} -67.8121 q^{13} -91.1147 q^{17} +37.9141 q^{19} -57.7036 q^{23} +25.0000 q^{25} +49.8335 q^{29} +226.034 q^{31} -137.951 q^{35} -144.062 q^{37} +348.991 q^{41} +455.982 q^{43} +179.256 q^{47} +418.217 q^{49} -300.579 q^{53} +203.177 q^{55} -439.704 q^{59} -191.659 q^{61} +339.060 q^{65} +462.629 q^{67} +671.597 q^{71} -315.669 q^{73} -1121.14 q^{77} +1009.06 q^{79} +1206.59 q^{83} +455.573 q^{85} -981.691 q^{89} -1870.95 q^{91} -189.570 q^{95} -1843.86 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 35 q^{5} + 8 q^{7}+O(q^{10}) \) 7 * q - 35 * q^5 + 8 * q^7	\( 7 q - 35 q^{5} + 8 q^{7} - 27 q^{11} + 32 q^{13} - 123 q^{17} - 67 q^{19} - 42 q^{23} + 175 q^{25} - 324 q^{29} + 98 q^{31} - 40 q^{35} + 356 q^{37} - 339 q^{41} + 119 q^{43} + 96 q^{47} + 813 q^{49} - 858 q^{53} + 135 q^{55} - 549 q^{59} + 260 q^{61} - 160 q^{65} + 881 q^{67} - 342 q^{71} + 737 q^{73} + 456 q^{77} + 1886 q^{79} + 132 q^{83} + 615 q^{85} - 396 q^{89} + 1264 q^{91} + 335 q^{95} + 1991 q^{97}+O(q^{100}) \) 7 * q - 35 * q^5 + 8 * q^7 - 27 * q^11 + 32 * q^13 - 123 * q^17 - 67 * q^19 - 42 * q^23 + 175 * q^25 - 324 * q^29 + 98 * q^31 - 40 * q^35 + 356 * q^37 - 339 * q^41 + 119 * q^43 + 96 * q^47 + 813 * q^49 - 858 * q^53 + 135 * q^55 - 549 * q^59 + 260 * q^61 - 160 * q^65 + 881 * q^67 - 342 * q^71 + 737 * q^73 + 456 * q^77 + 1886 * q^79 + 132 * q^83 + 615 * q^85 - 396 * q^89 + 1264 * q^91 + 335 * q^95 + 1991 * q^97

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