LMFDB - Embedded newform 2016.4.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,4,Mod(1,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2016.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2016.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:	$118.947850572$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.777569.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 41x^{2} + 58x + 244 $ x^4 - x^3 - 41x^2 + 58x + 244
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.93229$ of defining polynomial
Character	$\chi$	$=$	2016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.25902 q^{5} +7.00000 q^{7} +O(q^{10})$	$q-3.25902 q^{5} +7.00000 q^{7} +57.6310 q^{11} -6.25703 q^{13} +75.3700 q^{17} -108.487 q^{19} -113.888 q^{23} -114.379 q^{25} -191.048 q^{29} -156.623 q^{31} -22.8131 q^{35} +215.145 q^{37} +133.740 q^{41} -122.158 q^{43} -194.748 q^{47} +49.0000 q^{49} -444.655 q^{53} -187.821 q^{55} -321.225 q^{59} +424.034 q^{61} +20.3918 q^{65} +1028.02 q^{67} +127.851 q^{71} +676.311 q^{73} +403.417 q^{77} +935.511 q^{79} +849.516 q^{83} -245.632 q^{85} -21.8424 q^{89} -43.7992 q^{91} +353.561 q^{95} -1844.18 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{5} + 28 q^{7}+O(q^{10}) $ 4 * q - 10 * q^5 + 28 * q^7	$ 4 q - 10 q^{5} + 28 q^{7} - 22 q^{11} + 20 q^{13} + 10 q^{17} + 20 q^{19} - 158 q^{23} - 40 q^{25} - 124 q^{29} + 44 q^{31} - 70 q^{35} + 268 q^{37} - 298 q^{41} - 104 q^{43} - 364 q^{47} + 196 q^{49} - 176 q^{53} - 268 q^{55} - 300 q^{59} + 556 q^{61} - 628 q^{65} + 512 q^{67} - 266 q^{71} - 320 q^{73} - 154 q^{77} - 184 q^{79} + 728 q^{83} + 1100 q^{85} - 1714 q^{89} + 140 q^{91} + 304 q^{95} - 1216 q^{97}+O(q^{100}) $ 4 * q - 10 * q^5 + 28 * q^7 - 22 * q^11 + 20 * q^13 + 10 * q^17 + 20 * q^19 - 158 * q^23 - 40 * q^25 - 124 * q^29 + 44 * q^31 - 70 * q^35 + 268 * q^37 - 298 * q^41 - 104 * q^43 - 364 * q^47 + 196 * q^49 - 176 * q^53 - 268 * q^55 - 300 * q^59 + 556 * q^61 - 628 * q^65 + 512 * q^67 - 266 * q^71 - 320 * q^73 - 154 * q^77 - 184 * q^79 + 728 * q^83 + 1100 * q^85 - 1714 * q^89 + 140 * q^91 + 304 * q^95 - 1216 * 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
$5$	−3.25902	−0.291496	−0.145748	−	0.989322i	$-0.546559\pi$
$5$	−3.25902	−0.291496	−0.145748	+	0.989322i	$0.546559\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	57.6310	1.57967	0.789837	−	0.613317i	$-0.210165\pi$
$11$	57.6310	1.57967	0.789837	+	0.613317i	$0.210165\pi$
$12$	0	0
$13$	−6.25703	−0.133491	−0.0667457	−	0.997770i	$-0.521262\pi$
$13$	−6.25703	−0.133491	−0.0667457	+	0.997770i	$0.521262\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	75.3700	1.07529	0.537645	−	0.843171i	$-0.319314\pi$
$17$	75.3700	1.07529	0.537645	+	0.843171i	$0.319314\pi$
$18$	0	0
$19$	−108.487	−1.30993	−0.654964	−	0.755660i	$-0.727316\pi$
$19$	−108.487	−1.30993	−0.654964	+	0.755660i	$0.727316\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−113.888	−1.03249	−0.516246	−	0.856440i	$-0.672671\pi$
$23$	−113.888	−1.03249	−0.516246	+	0.856440i	$0.672671\pi$
$24$	0	0
$25$	−114.379	−0.915030
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−191.048	−1.22333	−0.611667	−	0.791115i	$-0.709501\pi$
$29$	−191.048	−1.22333	−0.611667	+	0.791115i	$0.709501\pi$
$30$	0	0
$31$	−156.623	−0.907432	−0.453716	−	0.891146i	$-0.649902\pi$
$31$	−156.623	−0.907432	−0.453716	+	0.891146i	$0.649902\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−22.8131	−0.110175
$36$	0	0
$37$	215.145	0.955937	0.477969	−	0.878377i	$-0.341373\pi$
$37$	215.145	0.955937	0.477969	+	0.878377i	$0.341373\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	133.740	0.509433	0.254716	−	0.967016i	$-0.418018\pi$
$41$	133.740	0.509433	0.254716	+	0.967016i	$0.418018\pi$
$42$	0	0
$43$	−122.158	−0.433230	−0.216615	−	0.976257i	$-0.569502\pi$
$43$	−122.158	−0.433230	−0.216615	+	0.976257i	$0.569502\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−194.748	−0.604403	−0.302201	−	0.953244i	$-0.597721\pi$
$47$	−194.748	−0.604403	−0.302201	+	0.953244i	$0.597721\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−444.655	−1.15242	−0.576208	−	0.817303i	$-0.695468\pi$
$53$	−444.655	−1.15242	−0.576208	+	0.817303i	$0.695468\pi$
$54$	0	0
$55$	−187.821	−0.460468
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−321.225	−0.708812	−0.354406	−	0.935092i	$-0.615317\pi$
$59$	−321.225	−0.708812	−0.354406	+	0.935092i	$0.615317\pi$
$60$	0	0
$61$	424.034	0.890032	0.445016	−	0.895523i	$-0.353198\pi$
$61$	424.034	0.890032	0.445016	+	0.895523i	$0.353198\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	20.3918	0.0389122
$66$	0	0
$67$	1028.02	1.87452	0.937259	−	0.348635i	$-0.113355\pi$
$67$	1028.02	1.87452	0.937259	+	0.348635i	$0.113355\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	127.851	0.213706	0.106853	−	0.994275i	$-0.465923\pi$
$71$	127.851	0.213706	0.106853	+	0.994275i	$0.465923\pi$
$72$	0	0
$73$	676.311	1.08433	0.542166	−	0.840271i	$-0.317604\pi$
$73$	676.311	1.08433	0.542166	+	0.840271i	$0.317604\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	403.417	0.597061
$78$	0	0
$79$	935.511	1.33232	0.666159	−	0.745810i	$-0.267937\pi$
$79$	935.511	1.33232	0.666159	+	0.745810i	$0.267937\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	849.516	1.12345	0.561726	−	0.827323i	$-0.310137\pi$
$83$	849.516	1.12345	0.561726	+	0.827323i	$0.310137\pi$
$84$	0	0
$85$	−245.632	−0.313442
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−21.8424	−0.0260145	−0.0130072	−	0.999915i	$-0.504140\pi$
$89$	−21.8424	−0.0260145	−0.0130072	+	0.999915i	$0.504140\pi$
$90$	0	0
$91$	−43.7992	−0.0504550
$92$	0	0
$93$	0	0
$94$	0	0
$95$	353.561	0.381838
$96$	0	0
$97$	−1844.18	−1.93040	−0.965198	−	0.261522i	$-0.915776\pi$
$97$	−1844.18	−1.93040	−0.965198	+	0.261522i	$0.915776\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1709.18	−1.68386	−0.841929	−	0.539589i	$-0.818580\pi$
$101$	−1709.18	−1.68386	−0.841929	+	0.539589i	$0.818580\pi$
$102$	0	0
$103$	−12.3192	−0.0117850	−0.00589248	−	0.999983i	$-0.501876\pi$
$103$	−12.3192	−0.0117850	−0.00589248	+	0.999983i	$0.501876\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−762.119	−0.688569	−0.344284	−	0.938865i	$-0.611878\pi$
$107$	−762.119	−0.688569	−0.344284	+	0.938865i	$0.611878\pi$
$108$	0	0
$109$	−1203.23	−1.05732	−0.528662	−	0.848832i	$-0.677306\pi$
$109$	−1203.23	−1.05732	−0.528662	+	0.848832i	$0.677306\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1791.80	−1.49166	−0.745832	−	0.666134i	$-0.767948\pi$
$113$	−1791.80	−1.49166	−0.745832	+	0.666134i	$0.767948\pi$
$114$	0	0
$115$	371.163	0.300967
$116$	0	0
$117$	0	0
$118$	0	0
$119$	527.590	0.406421
$120$	0	0
$121$	1990.34	1.49537
$122$	0	0
$123$	0	0
$124$	0	0
$125$	780.140	0.558223
$126$	0	0
$127$	−1796.25	−1.25505	−0.627524	−	0.778597i	$-0.715932\pi$
$127$	−1796.25	−1.25505	−0.627524	+	0.778597i	$0.715932\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	893.702	0.596054	0.298027	−	0.954557i	$-0.403671\pi$
$131$	893.702	0.596054	0.298027	+	0.954557i	$0.403671\pi$
$132$	0	0
$133$	−759.409	−0.495106
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1512.43	−0.943183	−0.471591	−	0.881817i	$-0.656320\pi$
$137$	−1512.43	−0.943183	−0.471591	+	0.881817i	$0.656320\pi$
$138$	0	0
$139$	−1650.32	−1.00704	−0.503518	−	0.863985i	$-0.667961\pi$
$139$	−1650.32	−1.00704	−0.503518	+	0.863985i	$0.667961\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−360.599	−0.210873
$144$	0	0
$145$	622.629	0.356597
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−741.262	−0.407561	−0.203780	−	0.979017i	$-0.565323\pi$
$149$	−741.262	−0.407561	−0.203780	+	0.979017i	$0.565323\pi$
$150$	0	0
$151$	235.154	0.126732	0.0633661	−	0.997990i	$-0.479816\pi$
$151$	235.154	0.126732	0.0633661	+	0.997990i	$0.479816\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	510.439	0.264512
$156$	0	0
$157$	3732.95	1.89759	0.948796	−	0.315888i	$-0.102302\pi$
$157$	3732.95	1.89759	0.948796	+	0.315888i	$0.102302\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−797.217	−0.390245
$162$	0	0
$163$	−1677.90	−0.806278	−0.403139	−	0.915139i	$-0.632081\pi$
$163$	−1677.90	−0.806278	−0.403139	+	0.915139i	$0.632081\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3423.75	−1.58645	−0.793227	−	0.608927i	$-0.791600\pi$
$167$	−3423.75	−1.58645	−0.793227	+	0.608927i	$0.791600\pi$
$168$	0	0
$169$	−2157.85	−0.982180
$170$	0	0
$171$	0	0
$172$	0	0
$173$	572.695	0.251683	0.125842	−	0.992050i	$-0.459837\pi$
$173$	572.695	0.251683	0.125842	+	0.992050i	$0.459837\pi$
$174$	0	0
$175$	−800.652	−0.345849
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−377.517	−0.157637	−0.0788184	−	0.996889i	$-0.525115\pi$
$179$	−377.517	−0.157637	−0.0788184	+	0.996889i	$0.525115\pi$
$180$	0	0
$181$	−247.484	−0.101632	−0.0508158	−	0.998708i	$-0.516182\pi$
$181$	−247.484	−0.101632	−0.0508158	+	0.998708i	$0.516182\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−701.163	−0.278652
$186$	0	0
$187$	4343.65	1.69861
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2860.82	1.08378	0.541889	−	0.840450i	$-0.317709\pi$
$191$	2860.82	1.08378	0.541889	+	0.840450i	$0.317709\pi$
$192$	0	0
$193$	−3081.82	−1.14940	−0.574700	−	0.818364i	$-0.694881\pi$
$193$	−3081.82	−1.14940	−0.574700	+	0.818364i	$0.694881\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5013.63	−1.81323	−0.906616	−	0.421957i	$-0.861343\pi$
$197$	−5013.63	−1.81323	−0.906616	+	0.421957i	$0.861343\pi$
$198$	0	0
$199$	−4724.50	−1.68297	−0.841485	−	0.540281i	$-0.818318\pi$
$199$	−4724.50	−1.68297	−0.841485	+	0.540281i	$0.818318\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1337.34	−0.462377
$204$	0	0
$205$	−435.862	−0.148497
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6252.22	−2.06926
$210$	0	0
$211$	1815.60	0.592375	0.296187	−	0.955130i	$-0.404285\pi$
$211$	1815.60	0.592375	0.296187	+	0.955130i	$0.404285\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	398.115	0.126285
$216$	0	0
$217$	−1096.36	−0.342977
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−471.593	−0.143542
$222$	0	0
$223$	2655.11	0.797307	0.398653	−	0.917102i	$-0.369478\pi$
$223$	2655.11	0.797307	0.398653	+	0.917102i	$0.369478\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2550.52	0.745745	0.372873	−	0.927883i	$-0.378373\pi$
$227$	2550.52	0.745745	0.372873	+	0.927883i	$0.378373\pi$
$228$	0	0
$229$	−176.872	−0.0510395	−0.0255198	−	0.999674i	$-0.508124\pi$
$229$	−176.872	−0.0510395	−0.0255198	+	0.999674i	$0.508124\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2936.85	−0.825748	−0.412874	−	0.910788i	$-0.635475\pi$
$233$	−2936.85	−0.825748	−0.412874	+	0.910788i	$0.635475\pi$
$234$	0	0
$235$	634.688	0.176181
$236$	0	0
$237$	0	0
$238$	0	0
$239$	467.238	0.126457	0.0632283	−	0.997999i	$-0.479860\pi$
$239$	467.238	0.126457	0.0632283	+	0.997999i	$0.479860\pi$
$240$	0	0
$241$	5354.07	1.43106	0.715532	−	0.698580i	$-0.246185\pi$
$241$	5354.07	1.43106	0.715532	+	0.698580i	$0.246185\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−159.692	−0.0416422
$246$	0	0
$247$	678.807	0.174864
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3846.72	0.967341	0.483671	−	0.875250i	$-0.339303\pi$
$251$	3846.72	0.967341	0.483671	+	0.875250i	$0.339303\pi$
$252$	0	0
$253$	−6563.49	−1.63100
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7024.70	−1.70501	−0.852507	−	0.522716i	$-0.824919\pi$
$257$	−7024.70	−1.70501	−0.852507	+	0.522716i	$0.824919\pi$
$258$	0	0
$259$	1506.02	0.361310
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4632.00	−1.08601	−0.543007	−	0.839728i	$-0.682714\pi$
$263$	−4632.00	−1.08601	−0.543007	+	0.839728i	$0.682714\pi$
$264$	0	0
$265$	1449.14	0.335924
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5340.99	−1.21058	−0.605290	−	0.796005i	$-0.706943\pi$
$269$	−5340.99	−1.21058	−0.605290	+	0.796005i	$0.706943\pi$
$270$	0	0
$271$	−1854.55	−0.415705	−0.207853	−	0.978160i	$-0.566647\pi$
$271$	−1854.55	−0.415705	−0.207853	+	0.978160i	$0.566647\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6591.77	−1.44545
$276$	0	0
$277$	544.592	0.118128	0.0590638	−	0.998254i	$-0.481188\pi$
$277$	544.592	0.118128	0.0590638	+	0.998254i	$0.481188\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6538.19	1.38803	0.694014	−	0.719962i	$-0.255841\pi$
$281$	6538.19	1.38803	0.694014	+	0.719962i	$0.255841\pi$
$282$	0	0
$283$	4819.82	1.01240	0.506199	−	0.862417i	$-0.331050\pi$
$283$	4819.82	1.01240	0.506199	+	0.862417i	$0.331050\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	936.183	0.192547
$288$	0	0
$289$	767.644	0.156247
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5020.79	−1.00108	−0.500542	−	0.865712i	$-0.666866\pi$
$293$	−5020.79	−1.00108	−0.500542	+	0.865712i	$0.666866\pi$
$294$	0	0
$295$	1046.88	0.206616
$296$	0	0
$297$	0	0
$298$	0	0
$299$	712.602	0.137829
$300$	0	0
$301$	−855.105	−0.163746
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1381.93	−0.259440
$306$	0	0
$307$	6993.55	1.30014	0.650070	−	0.759874i	$-0.274739\pi$
$307$	6993.55	1.30014	0.650070	+	0.759874i	$0.274739\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1134.17	0.206794	0.103397	−	0.994640i	$-0.467029\pi$
$311$	1134.17	0.206794	0.103397	+	0.994640i	$0.467029\pi$
$312$	0	0
$313$	2509.13	0.453113	0.226556	−	0.973998i	$-0.427253\pi$
$313$	2509.13	0.453113	0.226556	+	0.973998i	$0.427253\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7187.64	−1.27350	−0.636748	−	0.771072i	$-0.719721\pi$
$317$	−7187.64	−1.27350	−0.636748	+	0.771072i	$0.719721\pi$
$318$	0	0
$319$	−11010.3	−1.93247
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8176.67	−1.40855
$324$	0	0
$325$	715.672	0.122149
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1363.24	−0.228443
$330$	0	0
$331$	2352.10	0.390583	0.195291	−	0.980745i	$-0.437435\pi$
$331$	2352.10	0.390583	0.195291	+	0.980745i	$0.437435\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3350.34	−0.546413
$336$	0	0
$337$	272.321	0.0440186	0.0220093	−	0.999758i	$-0.492994\pi$
$337$	272.321	0.0440186	0.0220093	+	0.999758i	$0.492994\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−9026.37	−1.43345
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4160.64	0.643674	0.321837	−	0.946795i	$-0.395700\pi$
$347$	4160.64	0.643674	0.321837	+	0.946795i	$0.395700\pi$
$348$	0	0
$349$	3814.79	0.585104	0.292552	−	0.956250i	$-0.405496\pi$
$349$	3814.79	0.585104	0.292552	+	0.956250i	$0.405496\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−773.859	−0.116681	−0.0583405	−	0.998297i	$-0.518581\pi$
$353$	−773.859	−0.116681	−0.0583405	+	0.998297i	$0.518581\pi$
$354$	0	0
$355$	−416.669	−0.0622943
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5559.10	0.817265	0.408632	−	0.912699i	$-0.366006\pi$
$359$	5559.10	0.817265	0.408632	+	0.912699i	$0.366006\pi$
$360$	0	0
$361$	4910.43	0.715911
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2204.11	−0.316078
$366$	0	0
$367$	−850.861	−0.121021	−0.0605103	−	0.998168i	$-0.519273\pi$
$367$	−850.861	−0.121021	−0.0605103	+	0.998168i	$0.519273\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3112.58	−0.435572
$372$	0	0
$373$	2757.78	0.382822	0.191411	−	0.981510i	$-0.438694\pi$
$373$	2757.78	0.382822	0.191411	+	0.981510i	$0.438694\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1195.39	0.163305
$378$	0	0
$379$	−8485.68	−1.15008	−0.575040	−	0.818126i	$-0.695013\pi$
$379$	−8485.68	−1.15008	−0.575040	+	0.818126i	$0.695013\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7544.05	1.00648	0.503241	−	0.864146i	$-0.332141\pi$
$383$	7544.05	1.00648	0.503241	+	0.864146i	$0.332141\pi$
$384$	0	0
$385$	−1314.74	−0.174041
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9772.25	−1.27371	−0.636854	−	0.770984i	$-0.719765\pi$
$389$	−9772.25	−1.27371	−0.636854	+	0.770984i	$0.719765\pi$
$390$	0	0
$391$	−8583.75	−1.11023
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3048.85	−0.388365
$396$	0	0
$397$	−4913.99	−0.621224	−0.310612	−	0.950537i	$-0.600534\pi$
$397$	−4913.99	−0.621224	−0.310612	+	0.950537i	$0.600534\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6840.39	−0.851852	−0.425926	−	0.904758i	$-0.640052\pi$
$401$	−6840.39	−0.851852	−0.425926	+	0.904758i	$0.640052\pi$
$402$	0	0
$403$	979.998	0.121134
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12399.1	1.51007
$408$	0	0
$409$	−7226.59	−0.873672	−0.436836	−	0.899541i	$-0.643901\pi$
$409$	−7226.59	−0.873672	−0.436836	+	0.899541i	$0.643901\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2248.57	−0.267906
$414$	0	0
$415$	−2768.59	−0.327481
$416$	0	0
$417$	0	0
$418$	0	0
$419$	15977.9	1.86294	0.931471	−	0.363816i	$-0.118526\pi$
$419$	15977.9	1.86294	0.931471	+	0.363816i	$0.118526\pi$
$420$	0	0
$421$	12919.5	1.49563	0.747815	−	0.663908i	$-0.231103\pi$
$421$	12919.5	1.49563	0.747815	+	0.663908i	$0.231103\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−8620.73	−0.983922
$426$	0	0
$427$	2968.24	0.336401
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1790.50	−0.200105	−0.100053	−	0.994982i	$-0.531901\pi$
$431$	−1790.50	−0.200105	−0.100053	+	0.994982i	$0.531901\pi$
$432$	0	0
$433$	−1572.99	−0.174579	−0.0872897	−	0.996183i	$-0.527821\pi$
$433$	−1572.99	−0.174579	−0.0872897	+	0.996183i	$0.527821\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12355.4	1.35249
$438$	0	0
$439$	−15439.4	−1.67855	−0.839276	−	0.543706i	$-0.817021\pi$
$439$	−15439.4	−1.67855	−0.839276	+	0.543706i	$0.817021\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3779.22	−0.405319	−0.202660	−	0.979249i	$-0.564958\pi$
$443$	−3779.22	−0.405319	−0.202660	+	0.979249i	$0.564958\pi$
$444$	0	0
$445$	71.1848	0.00758311
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2905.25	−0.305361	−0.152680	−	0.988276i	$-0.548791\pi$
$449$	−2905.25	−0.305361	−0.152680	+	0.988276i	$0.548791\pi$
$450$	0	0
$451$	7707.60	0.804738
$452$	0	0
$453$	0	0
$454$	0	0
$455$	142.743	0.0147074
$456$	0	0
$457$	−12218.8	−1.25071	−0.625354	−	0.780341i	$-0.715045\pi$
$457$	−12218.8	−1.25071	−0.625354	+	0.780341i	$0.715045\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−9695.95	−0.979578	−0.489789	−	0.871841i	$-0.662926\pi$
$461$	−9695.95	−0.979578	−0.489789	+	0.871841i	$0.662926\pi$
$462$	0	0
$463$	−8738.42	−0.877125	−0.438563	−	0.898701i	$-0.644512\pi$
$463$	−8738.42	−0.877125	−0.438563	+	0.898701i	$0.644512\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8166.27	−0.809186	−0.404593	−	0.914497i	$-0.632587\pi$
$467$	−8166.27	−0.809186	−0.404593	+	0.914497i	$0.632587\pi$
$468$	0	0
$469$	7196.14	0.708501
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−7040.08	−0.684362
$474$	0	0
$475$	12408.6	1.19862
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15291.2	−1.45861	−0.729305	−	0.684188i	$-0.760157\pi$
$479$	−15291.2	−1.45861	−0.729305	+	0.684188i	$0.760157\pi$
$480$	0	0
$481$	−1346.17	−0.127609
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6010.23	0.562702
$486$	0	0
$487$	1709.77	0.159091	0.0795454	−	0.996831i	$-0.474653\pi$
$487$	1709.77	0.159091	0.0795454	+	0.996831i	$0.474653\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6896.90	0.633916	0.316958	−	0.948440i	$-0.397339\pi$
$491$	6896.90	0.633916	0.316958	+	0.948440i	$0.397339\pi$
$492$	0	0
$493$	−14399.3	−1.31544
$494$	0	0
$495$	0	0
$496$	0	0
$497$	894.956	0.0807732
$498$	0	0
$499$	13884.4	1.24560	0.622798	−	0.782382i	$-0.285996\pi$
$499$	13884.4	1.24560	0.622798	+	0.782382i	$0.285996\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16445.5	−1.45779	−0.728896	−	0.684625i	$-0.759966\pi$
$503$	−16445.5	−1.45779	−0.728896	+	0.684625i	$0.759966\pi$
$504$	0	0
$505$	5570.25	0.490837
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16417.7	1.42967	0.714833	−	0.699295i	$-0.246503\pi$
$509$	16417.7	1.42967	0.714833	+	0.699295i	$0.246503\pi$
$510$	0	0
$511$	4734.18	0.409839
$512$	0	0
$513$	0	0
$514$	0	0
$515$	40.1487	0.00343526
$516$	0	0
$517$	−11223.5	−0.954759
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−18778.2	−1.57905	−0.789526	−	0.613717i	$-0.789674\pi$
$521$	−18778.2	−1.57905	−0.789526	+	0.613717i	$0.789674\pi$
$522$	0	0
$523$	13705.5	1.14589	0.572945	−	0.819594i	$-0.305801\pi$
$523$	13705.5	1.14589	0.572945	+	0.819594i	$0.305801\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11804.7	−0.975752
$528$	0	0
$529$	803.496	0.0660389
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−836.818	−0.0680049
$534$	0	0
$535$	2483.76	0.200715
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2823.92	0.225668
$540$	0	0
$541$	−1331.61	−0.105823	−0.0529115	−	0.998599i	$-0.516850\pi$
$541$	−1331.61	−0.105823	−0.0529115	+	0.998599i	$0.516850\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3921.34	0.308205
$546$	0	0
$547$	21740.6	1.69938	0.849690	−	0.527282i	$-0.176789\pi$
$547$	21740.6	1.69938	0.849690	+	0.527282i	$0.176789\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	20726.2	1.60248
$552$	0	0
$553$	6548.57	0.503569
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3777.90	0.287387	0.143694	−	0.989622i	$-0.454102\pi$
$557$	3777.90	0.287387	0.143694	+	0.989622i	$0.454102\pi$
$558$	0	0
$559$	764.346	0.0578325
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1306.12	0.0977736	0.0488868	−	0.998804i	$-0.484433\pi$
$563$	1306.12	0.0977736	0.0488868	+	0.998804i	$0.484433\pi$
$564$	0	0
$565$	5839.50	0.434813
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1220.99	0.0899587	0.0449794	−	0.998988i	$-0.485678\pi$
$569$	1220.99	0.0899587	0.0449794	+	0.998988i	$0.485678\pi$
$570$	0	0
$571$	7741.89	0.567405	0.283702	−	0.958912i	$-0.408437\pi$
$571$	7741.89	0.567405	0.283702	+	0.958912i	$0.408437\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	13026.4	0.944761
$576$	0	0
$577$	−14054.9	−1.01406	−0.507031	−	0.861928i	$-0.669257\pi$
$577$	−14054.9	−1.01406	−0.507031	+	0.861928i	$0.669257\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5946.61	0.424625
$582$	0	0
$583$	−25625.9	−1.82044
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2584.06	−0.181696	−0.0908481	−	0.995865i	$-0.528958\pi$
$587$	−2584.06	−0.181696	−0.0908481	+	0.995865i	$0.528958\pi$
$588$	0	0
$589$	16991.6	1.18867
$590$	0	0
$591$	0	0
$592$	0	0
$593$	10755.5	0.744815	0.372407	−	0.928069i	$-0.378532\pi$
$593$	10755.5	0.744815	0.372407	+	0.928069i	$0.378532\pi$
$594$	0	0
$595$	−1719.43	−0.118470
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−23282.4	−1.58814	−0.794068	−	0.607829i	$-0.792040\pi$
$599$	−23282.4	−1.58814	−0.794068	+	0.607829i	$0.792040\pi$
$600$	0	0
$601$	15779.7	1.07100	0.535498	−	0.844537i	$-0.320124\pi$
$601$	15779.7	1.07100	0.535498	+	0.844537i	$0.320124\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6486.55	−0.435894
$606$	0	0
$607$	14052.7	0.939675	0.469837	−	0.882753i	$-0.344313\pi$
$607$	14052.7	0.939675	0.469837	+	0.882753i	$0.344313\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1218.55	0.0806826
$612$	0	0
$613$	8466.67	0.557856	0.278928	−	0.960312i	$-0.410021\pi$
$613$	8466.67	0.557856	0.278928	+	0.960312i	$0.410021\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−15413.9	−1.00574	−0.502870	−	0.864362i	$-0.667723\pi$
$617$	−15413.9	−1.00574	−0.502870	+	0.864362i	$0.667723\pi$
$618$	0	0
$619$	−26874.3	−1.74502	−0.872511	−	0.488594i	$-0.837510\pi$
$619$	−26874.3	−1.74502	−0.872511	+	0.488594i	$0.837510\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−152.897	−0.00983255
$624$	0	0
$625$	11754.9	0.752311
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16215.5	1.02791
$630$	0	0
$631$	−21492.7	−1.35596	−0.677979	−	0.735081i	$-0.737144\pi$
$631$	−21492.7	−1.35596	−0.677979	+	0.735081i	$0.737144\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5854.01	0.365841
$636$	0	0
$637$	−306.595	−0.0190702
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24078.9	1.48371	0.741856	−	0.670559i	$-0.233946\pi$
$641$	24078.9	1.48371	0.741856	+	0.670559i	$0.233946\pi$
$642$	0	0
$643$	−1936.98	−0.118798	−0.0593989	−	0.998234i	$-0.518918\pi$
$643$	−1936.98	−0.118798	−0.0593989	+	0.998234i	$0.518918\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2109.98	−0.128210	−0.0641050	−	0.997943i	$-0.520419\pi$
$647$	−2109.98	−0.128210	−0.0641050	+	0.997943i	$0.520419\pi$
$648$	0	0
$649$	−18512.5	−1.11969
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5446.09	−0.326374	−0.163187	−	0.986595i	$-0.552177\pi$
$653$	−5446.09	−0.326374	−0.163187	+	0.986595i	$0.552177\pi$
$654$	0	0
$655$	−2912.59	−0.173747
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−8978.96	−0.530760	−0.265380	−	0.964144i	$-0.585497\pi$
$659$	−8978.96	−0.530760	−0.265380	+	0.964144i	$0.585497\pi$
$660$	0	0
$661$	−32568.0	−1.91641	−0.958206	−	0.286080i	$-0.907648\pi$
$661$	−32568.0	−1.91641	−0.958206	+	0.286080i	$0.907648\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2474.93	0.144321
$666$	0	0
$667$	21758.1	1.26308
$668$	0	0
$669$	0	0
$670$	0	0
$671$	24437.5	1.40596
$672$	0	0
$673$	−4724.96	−0.270630	−0.135315	−	0.990803i	$-0.543205\pi$
$673$	−4724.96	−0.270630	−0.135315	+	0.990803i	$0.543205\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9223.30	−0.523604	−0.261802	−	0.965122i	$-0.584317\pi$
$677$	−9223.30	−0.523604	−0.261802	+	0.965122i	$0.584317\pi$
$678$	0	0
$679$	−12909.3	−0.729621
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−251.013	−0.0140626	−0.00703129	−	0.999975i	$-0.502238\pi$
$683$	−251.013	−0.0140626	−0.00703129	+	0.999975i	$0.502238\pi$
$684$	0	0
$685$	4929.05	0.274934
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2782.22	0.153838
$690$	0	0
$691$	3374.37	0.185770	0.0928850	−	0.995677i	$-0.470391\pi$
$691$	3374.37	0.185770	0.0928850	+	0.995677i	$0.470391\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5378.41	0.293546
$696$	0	0
$697$	10080.0	0.547788
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2256.91	−0.121601	−0.0608004	−	0.998150i	$-0.519365\pi$
$701$	−2256.91	−0.121601	−0.0608004	+	0.998150i	$0.519365\pi$
$702$	0	0
$703$	−23340.5	−1.25221
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−11964.2	−0.636438
$708$	0	0
$709$	−25769.6	−1.36502	−0.682509	−	0.730877i	$-0.739111\pi$
$709$	−25769.6	−1.36502	−0.682509	+	0.730877i	$0.739111\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	17837.5	0.936916
$714$	0	0
$715$	1175.20	0.0614685
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9753.61	0.505909	0.252954	−	0.967478i	$-0.418598\pi$
$719$	9753.61	0.505909	0.252954	+	0.967478i	$0.418598\pi$
$720$	0	0
$721$	−86.2347	−0.00445430
$722$	0	0
$723$	0	0
$724$	0	0
$725$	21851.8	1.11939
$726$	0	0
$727$	−29515.8	−1.50575	−0.752874	−	0.658164i	$-0.771333\pi$
$727$	−29515.8	−1.50575	−0.752874	+	0.658164i	$0.771333\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9207.04	−0.465848
$732$	0	0
$733$	8628.26	0.434778	0.217389	−	0.976085i	$-0.430246\pi$
$733$	8628.26	0.434778	0.217389	+	0.976085i	$0.430246\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	59245.9	2.96113
$738$	0	0
$739$	6756.54	0.336324	0.168162	−	0.985759i	$-0.446217\pi$
$739$	6756.54	0.336324	0.168162	+	0.985759i	$0.446217\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17676.1	−0.872776	−0.436388	−	0.899758i	$-0.643743\pi$
$743$	−17676.1	−0.872776	−0.436388	+	0.899758i	$0.643743\pi$
$744$	0	0
$745$	2415.79	0.118802
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5334.84	−0.260255
$750$	0	0
$751$	−6735.88	−0.327292	−0.163646	−	0.986519i	$-0.552325\pi$
$751$	−6735.88	−0.327292	−0.163646	+	0.986519i	$0.552325\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−766.372	−0.0369419
$756$	0	0
$757$	−3666.75	−0.176051	−0.0880254	−	0.996118i	$-0.528056\pi$
$757$	−3666.75	−0.176051	−0.0880254	+	0.996118i	$0.528056\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	41149.1	1.96012	0.980062	−	0.198690i	$-0.0636688\pi$
$761$	41149.1	1.96012	0.980062	+	0.198690i	$0.0636688\pi$
$762$	0	0
$763$	−8422.60	−0.399631
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2009.92	0.0946204
$768$	0	0
$769$	−15503.9	−0.727029	−0.363515	−	0.931589i	$-0.618423\pi$
$769$	−15503.9	−0.727029	−0.363515	+	0.931589i	$0.618423\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10699.7	0.497857	0.248928	−	0.968522i	$-0.419922\pi$
$773$	10699.7	0.497857	0.248928	+	0.968522i	$0.419922\pi$
$774$	0	0
$775$	17914.4	0.830328
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−14509.1	−0.667320
$780$	0	0
$781$	7368.18	0.337586
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−12165.8	−0.553140
$786$	0	0
$787$	39278.6	1.77908	0.889538	−	0.456861i	$-0.151026\pi$
$787$	39278.6	1.77908	0.889538	+	0.456861i	$0.151026\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12542.6	−0.563796
$792$	0	0
$793$	−2653.19	−0.118812
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1606.83	−0.0714139	−0.0357069	−	0.999362i	$-0.511368\pi$
$797$	−1606.83	−0.0714139	−0.0357069	+	0.999362i	$0.511368\pi$
$798$	0	0
$799$	−14678.2	−0.649908
$800$	0	0
$801$	0	0
$802$	0	0
$803$	38976.5	1.71289
$804$	0	0
$805$	2598.14	0.113755
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−40506.4	−1.76036	−0.880178	−	0.474644i	$-0.842577\pi$
$809$	−40506.4	−1.76036	−0.880178	+	0.474644i	$0.842577\pi$
$810$	0	0
$811$	16847.2	0.729452	0.364726	−	0.931115i	$-0.381163\pi$
$811$	16847.2	0.729452	0.364726	+	0.931115i	$0.381163\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5468.31	0.235026
$816$	0	0
$817$	13252.5	0.567500
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−13054.0	−0.554918	−0.277459	−	0.960738i	$-0.589492\pi$
$821$	−13054.0	−0.554918	−0.277459	+	0.960738i	$0.589492\pi$
$822$	0	0
$823$	−3575.58	−0.151442	−0.0757210	−	0.997129i	$-0.524126\pi$
$823$	−3575.58	−0.151442	−0.0757210	+	0.997129i	$0.524126\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30011.7	1.26192	0.630961	−	0.775814i	$-0.282661\pi$
$827$	30011.7	1.26192	0.630961	+	0.775814i	$0.282661\pi$
$828$	0	0
$829$	26174.0	1.09657	0.548287	−	0.836290i	$-0.315280\pi$
$829$	26174.0	1.09657	0.548287	+	0.836290i	$0.315280\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3693.13	0.153613
$834$	0	0
$835$	11158.1	0.462444
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2643.15	−0.108762	−0.0543812	−	0.998520i	$-0.517319\pi$
$839$	−2643.15	−0.108762	−0.0543812	+	0.998520i	$0.517319\pi$
$840$	0	0
$841$	12110.3	0.496548
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7032.47	0.286301
$846$	0	0
$847$	13932.4	0.565197
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−24502.5	−0.986997
$852$	0	0
$853$	46934.1	1.88393	0.941965	−	0.335710i	$-0.108976\pi$
$853$	46934.1	1.88393	0.941965	+	0.335710i	$0.108976\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	42059.1	1.67644	0.838221	−	0.545330i	$-0.183596\pi$
$857$	42059.1	1.67644	0.838221	+	0.545330i	$0.183596\pi$
$858$	0	0
$859$	−6551.58	−0.260229	−0.130115	−	0.991499i	$-0.541535\pi$
$859$	−6551.58	−0.260229	−0.130115	+	0.991499i	$0.541535\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−37772.6	−1.48991	−0.744956	−	0.667114i	$-0.767530\pi$
$863$	−37772.6	−1.48991	−0.744956	+	0.667114i	$0.767530\pi$
$864$	0	0
$865$	−1866.42	−0.0733645
$866$	0	0
$867$	0	0
$868$	0	0
$869$	53914.5	2.10463
$870$	0	0
$871$	−6432.36	−0.250232
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5460.98	0.210988
$876$	0	0
$877$	6794.58	0.261615	0.130808	−	0.991408i	$-0.458243\pi$
$877$	6794.58	0.261615	0.130808	+	0.991408i	$0.458243\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	8218.27	0.314280	0.157140	−	0.987576i	$-0.449773\pi$
$881$	8218.27	0.314280	0.157140	+	0.987576i	$0.449773\pi$
$882$	0	0
$883$	21929.6	0.835774	0.417887	−	0.908499i	$-0.362771\pi$
$883$	21929.6	0.835774	0.417887	+	0.908499i	$0.362771\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13387.9	0.506791	0.253395	−	0.967363i	$-0.418453\pi$
$887$	13387.9	0.506791	0.253395	+	0.967363i	$0.418453\pi$
$888$	0	0
$889$	−12573.7	−0.474364
$890$	0	0
$891$	0	0
$892$	0	0
$893$	21127.6	0.791724
$894$	0	0
$895$	1230.34	0.0459504
$896$	0	0
$897$	0	0
$898$	0	0
$899$	29922.6	1.11009
$900$	0	0
$901$	−33513.7	−1.23918
$902$	0	0
$903$	0	0
$904$	0	0
$905$	806.554	0.0296252
$906$	0	0
$907$	19642.6	0.719099	0.359549	−	0.933126i	$-0.382930\pi$
$907$	19642.6	0.719099	0.359549	+	0.933126i	$0.382930\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	14315.7	0.520637	0.260318	−	0.965523i	$-0.416172\pi$
$911$	14315.7	0.520637	0.260318	+	0.965523i	$0.416172\pi$
$912$	0	0
$913$	48958.5	1.77469
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6255.91	0.225287
$918$	0	0
$919$	2695.06	0.0967376	0.0483688	−	0.998830i	$-0.484598\pi$
$919$	2695.06	0.0967376	0.0483688	+	0.998830i	$0.484598\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−799.968	−0.0285279
$924$	0	0
$925$	−24608.1	−0.874712
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−11047.4	−0.390156	−0.195078	−	0.980788i	$-0.562496\pi$
$929$	−11047.4	−0.390156	−0.195078	+	0.980788i	$0.562496\pi$
$930$	0	0
$931$	−5315.86	−0.187133
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−14156.1	−0.495136
$936$	0	0
$937$	−5317.02	−0.185378	−0.0926892	−	0.995695i	$-0.529546\pi$
$937$	−5317.02	−0.185378	−0.0926892	+	0.995695i	$0.529546\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	26867.9	0.930783	0.465392	−	0.885105i	$-0.345913\pi$
$941$	26867.9	0.930783	0.465392	+	0.885105i	$0.345913\pi$
$942$	0	0
$943$	−15231.4	−0.525985
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6878.65	−0.236036	−0.118018	−	0.993011i	$-0.537654\pi$
$947$	−6878.65	−0.236036	−0.118018	+	0.993011i	$0.537654\pi$
$948$	0	0
$949$	−4231.70	−0.144749
$950$	0	0
$951$	0	0
$952$	0	0
$953$	44653.1	1.51779	0.758896	−	0.651211i	$-0.225739\pi$
$953$	44653.1	1.51779	0.758896	+	0.651211i	$0.225739\pi$
$954$	0	0
$955$	−9323.47	−0.315917
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10587.0	−0.356490
$960$	0	0
$961$	−5260.11	−0.176567
$962$	0	0
$963$	0	0
$964$	0	0
$965$	10043.7	0.335045
$966$	0	0
$967$	21798.6	0.724919	0.362460	−	0.932000i	$-0.381937\pi$
$967$	21798.6	0.724919	0.362460	+	0.932000i	$0.381937\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	46011.4	1.52068	0.760338	−	0.649528i	$-0.225033\pi$
$971$	46011.4	1.52068	0.760338	+	0.649528i	$0.225033\pi$
$972$	0	0
$973$	−11552.2	−0.380624
$974$	0	0
$975$	0	0
$976$	0	0
$977$	37478.5	1.22727	0.613635	−	0.789590i	$-0.289707\pi$
$977$	37478.5	1.22727	0.613635	+	0.789590i	$0.289707\pi$
$978$	0	0
$979$	−1258.80	−0.0410944
$980$	0	0
$981$	0	0
$982$	0	0
$983$	13597.1	0.441180	0.220590	−	0.975367i	$-0.429202\pi$
$983$	13597.1	0.441180	0.220590	+	0.975367i	$0.429202\pi$
$984$	0	0
$985$	16339.5	0.528549
$986$	0	0
$987$	0	0
$988$	0	0
$989$	13912.3	0.447306
$990$	0	0
$991$	13562.8	0.434751	0.217375	−	0.976088i	$-0.430250\pi$
$991$	13562.8	0.434751	0.217375	+	0.976088i	$0.430250\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	15397.2	0.490578
$996$	0	0
$997$	7641.94	0.242751	0.121375	−	0.992607i	$-0.461270\pi$
$997$	7641.94	0.242751	0.121375	+	0.992607i	$0.461270\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.4.a.bb.1.2	yes	4
3.2	odd	2		2016.4.a.bd.1.3	yes	4
4.3	odd	2		2016.4.a.ba.1.2	✓	4
12.11	even	2		2016.4.a.bc.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.4.a.ba.1.2	✓	4	4.3	odd	2
2016.4.a.bb.1.2	yes	4	1.1	even	1	trivial
2016.4.a.bc.1.3	yes	4	12.11	even	2
2016.4.a.bd.1.3	yes	4	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:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2016.a (trivial)

\(f(q)\)	\(=\)	\(q-3.25902 q^{5} +7.00000 q^{7} +O(q^{10})\)	\(q-3.25902 q^{5} +7.00000 q^{7} +57.6310 q^{11} -6.25703 q^{13} +75.3700 q^{17} -108.487 q^{19} -113.888 q^{23} -114.379 q^{25} -191.048 q^{29} -156.623 q^{31} -22.8131 q^{35} +215.145 q^{37} +133.740 q^{41} -122.158 q^{43} -194.748 q^{47} +49.0000 q^{49} -444.655 q^{53} -187.821 q^{55} -321.225 q^{59} +424.034 q^{61} +20.3918 q^{65} +1028.02 q^{67} +127.851 q^{71} +676.311 q^{73} +403.417 q^{77} +935.511 q^{79} +849.516 q^{83} -245.632 q^{85} -21.8424 q^{89} -43.7992 q^{91} +353.561 q^{95} -1844.18 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{5} + 28 q^{7}+O(q^{10}) \) 4 * q - 10 * q^5 + 28 * q^7	\( 4 q - 10 q^{5} + 28 q^{7} - 22 q^{11} + 20 q^{13} + 10 q^{17} + 20 q^{19} - 158 q^{23} - 40 q^{25} - 124 q^{29} + 44 q^{31} - 70 q^{35} + 268 q^{37} - 298 q^{41} - 104 q^{43} - 364 q^{47} + 196 q^{49} - 176 q^{53} - 268 q^{55} - 300 q^{59} + 556 q^{61} - 628 q^{65} + 512 q^{67} - 266 q^{71} - 320 q^{73} - 154 q^{77} - 184 q^{79} + 728 q^{83} + 1100 q^{85} - 1714 q^{89} + 140 q^{91} + 304 q^{95} - 1216 q^{97}+O(q^{100}) \) 4 * q - 10 * q^5 + 28 * q^7 - 22 * q^11 + 20 * q^13 + 10 * q^17 + 20 * q^19 - 158 * q^23 - 40 * q^25 - 124 * q^29 + 44 * q^31 - 70 * q^35 + 268 * q^37 - 298 * q^41 - 104 * q^43 - 364 * q^47 + 196 * q^49 - 176 * q^53 - 268 * q^55 - 300 * q^59 + 556 * q^61 - 628 * q^65 + 512 * q^67 - 266 * q^71 - 320 * q^73 - 154 * q^77 - 184 * q^79 + 728 * q^83 + 1100 * q^85 - 1714 * q^89 + 140 * q^91 + 304 * q^95 - 1216 * q^97

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