LMFDB - Embedded newform 3640.2.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, 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("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.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:	$29.0655463357$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.93543 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.745898 q^{9} +O(q^{10})$	$q+1.93543 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.745898 q^{9} -3.18953 q^{11} +1.00000 q^{13} -1.93543 q^{15} +1.44364 q^{17} +2.93543 q^{19} -1.93543 q^{21} -0.173127 q^{23} +1.00000 q^{25} -4.36266 q^{27} -7.63317 q^{29} +2.06457 q^{31} -6.17313 q^{33} +1.00000 q^{35} -7.42723 q^{37} +1.93543 q^{39} +2.76231 q^{41} -7.87086 q^{43} -0.745898 q^{45} +9.53579 q^{47} +1.00000 q^{49} +2.79406 q^{51} -8.37907 q^{53} +3.18953 q^{55} +5.68133 q^{57} -11.9794 q^{59} -2.81047 q^{61} -0.745898 q^{63} -1.00000 q^{65} +3.74590 q^{67} -0.335076 q^{69} -10.2499 q^{71} +1.15672 q^{73} +1.93543 q^{75} +3.18953 q^{77} -2.12497 q^{79} -10.6813 q^{81} -4.98359 q^{83} -1.44364 q^{85} -14.7735 q^{87} +0.108560 q^{89} -1.00000 q^{91} +3.99583 q^{93} -2.93543 q^{95} +10.4231 q^{97} -2.37907 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q - 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} - q^{11} + 3 q^{13} + 2 q^{15} - 5 q^{17} + q^{19} + 2 q^{21} + 5 q^{23} + 3 q^{25} + q^{27} - 5 q^{29} + 14 q^{31} - 13 q^{33} + 3 q^{35} - 16 q^{37} - 2 q^{39} + 6 q^{41} - 8 q^{43} - 3 q^{45} + 9 q^{47} + 3 q^{49} + 20 q^{51} - 8 q^{53} + q^{55} + 10 q^{57} - 7 q^{59} - 17 q^{61} - 3 q^{63} - 3 q^{65} + 12 q^{67} - 5 q^{69} + 2 q^{71} + q^{73} - 2 q^{75} + q^{77} + 10 q^{79} - 25 q^{81} - 18 q^{83} + 5 q^{85} - 27 q^{87} - 13 q^{89} - 3 q^{91} - 20 q^{93} - q^{95} - 7 q^{97} + 10 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 - q^11 + 3 * q^13 + 2 * q^15 - 5 * q^17 + q^19 + 2 * q^21 + 5 * q^23 + 3 * q^25 + q^27 - 5 * q^29 + 14 * q^31 - 13 * q^33 + 3 * q^35 - 16 * q^37 - 2 * q^39 + 6 * q^41 - 8 * q^43 - 3 * q^45 + 9 * q^47 + 3 * q^49 + 20 * q^51 - 8 * q^53 + q^55 + 10 * q^57 - 7 * q^59 - 17 * q^61 - 3 * q^63 - 3 * q^65 + 12 * q^67 - 5 * q^69 + 2 * q^71 + q^73 - 2 * q^75 + q^77 + 10 * q^79 - 25 * q^81 - 18 * q^83 + 5 * q^85 - 27 * q^87 - 13 * q^89 - 3 * q^91 - 20 * q^93 - q^95 - 7 * q^97 + 10 * 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
$3$	1.93543	1.11742	0.558711	−	0.829362i	$-0.311296\pi$
$3$	1.93543	1.11742	0.558711	+	0.829362i	$0.311296\pi$
$4$	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
$9$	0.745898	0.248633
$10$	0	0
$11$	−3.18953	−0.961681	−0.480840	−	0.876808i	$-0.659668\pi$
$11$	−3.18953	−0.961681	−0.480840	+	0.876808i	$0.659668\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.93543	−0.499726
$16$	0	0
$17$	1.44364	0.350133	0.175067	−	0.984557i	$-0.443986\pi$
$17$	1.44364	0.350133	0.175067	+	0.984557i	$0.443986\pi$
$18$	0	0
$19$	2.93543	0.673434	0.336717	−	0.941606i	$-0.390683\pi$
$19$	2.93543	0.673434	0.336717	+	0.941606i	$0.390683\pi$
$20$	0	0
$21$	−1.93543	−0.422346
$22$	0	0
$23$	−0.173127	−0.0360995	−0.0180498	−	0.999837i	$-0.505746\pi$
$23$	−0.173127	−0.0360995	−0.0180498	+	0.999837i	$0.505746\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.36266	−0.839595
$28$	0	0
$29$	−7.63317	−1.41744	−0.708722	−	0.705488i	$-0.750728\pi$
$29$	−7.63317	−1.41744	−0.708722	+	0.705488i	$0.750728\pi$
$30$	0	0
$31$	2.06457	0.370807	0.185404	−	0.982662i	$-0.440641\pi$
$31$	2.06457	0.370807	0.185404	+	0.982662i	$0.440641\pi$
$32$	0	0
$33$	−6.17313	−1.07460
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−7.42723	−1.22103	−0.610514	−	0.792005i	$-0.709037\pi$
$37$	−7.42723	−1.22103	−0.610514	+	0.792005i	$0.709037\pi$
$38$	0	0
$39$	1.93543	0.309917
$40$	0	0
$41$	2.76231	0.431400	0.215700	−	0.976460i	$-0.430797\pi$
$41$	2.76231	0.431400	0.215700	+	0.976460i	$0.430797\pi$
$42$	0	0
$43$	−7.87086	−1.20030	−0.600148	−	0.799889i	$-0.704892\pi$
$43$	−7.87086	−1.20030	−0.600148	+	0.799889i	$0.704892\pi$
$44$	0	0
$45$	−0.745898	−0.111192
$46$	0	0
$47$	9.53579	1.39094	0.695469	−	0.718556i	$-0.255197\pi$
$47$	9.53579	1.39094	0.695469	+	0.718556i	$0.255197\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.79406	0.391247
$52$	0	0
$53$	−8.37907	−1.15095	−0.575477	−	0.817818i	$-0.695183\pi$
$53$	−8.37907	−1.15095	−0.575477	+	0.817818i	$0.695183\pi$
$54$	0	0
$55$	3.18953	0.430077
$56$	0	0
$57$	5.68133	0.752511
$58$	0	0
$59$	−11.9794	−1.55959	−0.779794	−	0.626036i	$-0.784676\pi$
$59$	−11.9794	−1.55959	−0.779794	+	0.626036i	$0.784676\pi$
$60$	0	0
$61$	−2.81047	−0.359843	−0.179922	−	0.983681i	$-0.557584\pi$
$61$	−2.81047	−0.359843	−0.179922	+	0.983681i	$0.557584\pi$
$62$	0	0
$63$	−0.745898	−0.0939744
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	3.74590	0.457634	0.228817	−	0.973469i	$-0.426514\pi$
$67$	3.74590	0.457634	0.228817	+	0.973469i	$0.426514\pi$
$68$	0	0
$69$	−0.335076	−0.0403384
$70$	0	0
$71$	−10.2499	−1.21644	−0.608222	−	0.793767i	$-0.708117\pi$
$71$	−10.2499	−1.21644	−0.608222	+	0.793767i	$0.708117\pi$
$72$	0	0
$73$	1.15672	0.135384	0.0676919	−	0.997706i	$-0.478436\pi$
$73$	1.15672	0.135384	0.0676919	+	0.997706i	$0.478436\pi$
$74$	0	0
$75$	1.93543	0.223484
$76$	0	0
$77$	3.18953	0.363481
$78$	0	0
$79$	−2.12497	−0.239077	−0.119539	−	0.992830i	$-0.538142\pi$
$79$	−2.12497	−0.239077	−0.119539	+	0.992830i	$0.538142\pi$
$80$	0	0
$81$	−10.6813	−1.18681
$82$	0	0
$83$	−4.98359	−0.547020	−0.273510	−	0.961869i	$-0.588185\pi$
$83$	−4.98359	−0.547020	−0.273510	+	0.961869i	$0.588185\pi$
$84$	0	0
$85$	−1.44364	−0.156584
$86$	0	0
$87$	−14.7735	−1.58388
$88$	0	0
$89$	0.108560	0.0115073	0.00575365	−	0.999983i	$-0.498169\pi$
$89$	0.108560	0.0115073	0.00575365	+	0.999983i	$0.498169\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	3.99583	0.414348
$94$	0	0
$95$	−2.93543	−0.301169
$96$	0	0
$97$	10.4231	1.05830	0.529151	−	0.848528i	$-0.322511\pi$
$97$	10.4231	1.05830	0.529151	+	0.848528i	$0.322511\pi$
$98$	0	0
$99$	−2.37907	−0.239105
$100$	0	0
$101$	1.15672	0.115098	0.0575490	−	0.998343i	$-0.481671\pi$
$101$	1.15672	0.115098	0.0575490	+	0.998343i	$0.481671\pi$
$102$	0	0
$103$	−13.6608	−1.34603	−0.673017	−	0.739627i	$-0.735002\pi$
$103$	−13.6608	−1.34603	−0.673017	+	0.739627i	$0.735002\pi$
$104$	0	0
$105$	1.93543	0.188879
$106$	0	0
$107$	3.23353	0.312597	0.156298	−	0.987710i	$-0.450044\pi$
$107$	3.23353	0.312597	0.156298	+	0.987710i	$0.450044\pi$
$108$	0	0
$109$	−11.7089	−1.12151	−0.560755	−	0.827982i	$-0.689489\pi$
$109$	−11.7089	−1.12151	−0.560755	+	0.827982i	$0.689489\pi$
$110$	0	0
$111$	−14.3749	−1.36441
$112$	0	0
$113$	−18.7693	−1.76567	−0.882834	−	0.469685i	$-0.844368\pi$
$113$	−18.7693	−1.76567	−0.882834	+	0.469685i	$0.844368\pi$
$114$	0	0
$115$	0.173127	0.0161442
$116$	0	0
$117$	0.745898	0.0689583
$118$	0	0
$119$	−1.44364	−0.132338
$120$	0	0
$121$	−0.826873	−0.0751702
$122$	0	0
$123$	5.34625	0.482056
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−13.9477	−1.23766	−0.618828	−	0.785527i	$-0.712392\pi$
$127$	−13.9477	−1.23766	−0.618828	+	0.785527i	$0.712392\pi$
$128$	0	0
$129$	−15.2335	−1.34124
$130$	0	0
$131$	−19.3627	−1.69172	−0.845862	−	0.533402i	$-0.820913\pi$
$131$	−19.3627	−1.69172	−0.845862	+	0.533402i	$0.820913\pi$
$132$	0	0
$133$	−2.93543	−0.254534
$134$	0	0
$135$	4.36266	0.375478
$136$	0	0
$137$	18.3585	1.56847	0.784236	−	0.620463i	$-0.213055\pi$
$137$	18.3585	1.56847	0.784236	+	0.620463i	$0.213055\pi$
$138$	0	0
$139$	20.5962	1.74695	0.873473	−	0.486873i	$-0.161862\pi$
$139$	20.5962	1.74695	0.873473	+	0.486873i	$0.161862\pi$
$140$	0	0
$141$	18.4559	1.55426
$142$	0	0
$143$	−3.18953	−0.266722
$144$	0	0
$145$	7.63317	0.633900
$146$	0	0
$147$	1.93543	0.159632
$148$	0	0
$149$	−17.8820	−1.46495	−0.732477	−	0.680792i	$-0.761636\pi$
$149$	−17.8820	−1.46495	−0.732477	+	0.680792i	$0.761636\pi$
$150$	0	0
$151$	2.16195	0.175937	0.0879685	−	0.996123i	$-0.471963\pi$
$151$	2.16195	0.175937	0.0879685	+	0.996123i	$0.471963\pi$
$152$	0	0
$153$	1.07681	0.0870546
$154$	0	0
$155$	−2.06457	−0.165830
$156$	0	0
$157$	0.660755	0.0527340	0.0263670	−	0.999652i	$-0.491606\pi$
$157$	0.660755	0.0527340	0.0263670	+	0.999652i	$0.491606\pi$
$158$	0	0
$159$	−16.2171	−1.28610
$160$	0	0
$161$	0.173127	0.0136443
$162$	0	0
$163$	14.9630	1.17199	0.585997	−	0.810313i	$-0.300703\pi$
$163$	14.9630	1.17199	0.585997	+	0.810313i	$0.300703\pi$
$164$	0	0
$165$	6.17313	0.480577
$166$	0	0
$167$	12.9753	1.00406	0.502028	−	0.864852i	$-0.332588\pi$
$167$	12.9753	1.00406	0.502028	+	0.864852i	$0.332588\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.18953	0.167438
$172$	0	0
$173$	5.34209	0.406151	0.203076	−	0.979163i	$-0.434906\pi$
$173$	5.34209	0.406151	0.203076	+	0.979163i	$0.434906\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−23.1854	−1.74272
$178$	0	0
$179$	5.57277	0.416528	0.208264	−	0.978073i	$-0.433219\pi$
$179$	5.57277	0.416528	0.208264	+	0.978073i	$0.433219\pi$
$180$	0	0
$181$	11.0932	0.824552	0.412276	−	0.911059i	$-0.364734\pi$
$181$	11.0932	0.824552	0.412276	+	0.911059i	$0.364734\pi$
$182$	0	0
$183$	−5.43947	−0.402097
$184$	0	0
$185$	7.42723	0.546061
$186$	0	0
$187$	−4.60453	−0.336716
$188$	0	0
$189$	4.36266	0.317337
$190$	0	0
$191$	−8.23769	−0.596059	−0.298029	−	0.954557i	$-0.596329\pi$
$191$	−8.23769	−0.596059	−0.298029	+	0.954557i	$0.596329\pi$
$192$	0	0
$193$	5.28169	0.380184	0.190092	−	0.981766i	$-0.439121\pi$
$193$	5.28169	0.380184	0.190092	+	0.981766i	$0.439121\pi$
$194$	0	0
$195$	−1.93543	−0.138599
$196$	0	0
$197$	−12.9190	−0.920442	−0.460221	−	0.887804i	$-0.652230\pi$
$197$	−12.9190	−0.920442	−0.460221	+	0.887804i	$0.652230\pi$
$198$	0	0
$199$	−5.14554	−0.364758	−0.182379	−	0.983228i	$-0.558380\pi$
$199$	−5.14554	−0.364758	−0.182379	+	0.983228i	$0.558380\pi$
$200$	0	0
$201$	7.24993	0.511371
$202$	0	0
$203$	7.63317	0.535743
$204$	0	0
$205$	−2.76231	−0.192928
$206$	0	0
$207$	−0.129135	−0.00897553
$208$	0	0
$209$	−9.36266	−0.647629
$210$	0	0
$211$	25.8227	1.77771	0.888854	−	0.458190i	$-0.151502\pi$
$211$	25.8227	1.77771	0.888854	+	0.458190i	$0.151502\pi$
$212$	0	0
$213$	−19.8381	−1.35928
$214$	0	0
$215$	7.87086	0.536789
$216$	0	0
$217$	−2.06457	−0.140152
$218$	0	0
$219$	2.23875	0.151281
$220$	0	0
$221$	1.44364	0.0971094
$222$	0	0
$223$	7.69774	0.515479	0.257739	−	0.966214i	$-0.417022\pi$
$223$	7.69774	0.515479	0.257739	+	0.966214i	$0.417022\pi$
$224$	0	0
$225$	0.745898	0.0497266
$226$	0	0
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	0	0
$229$	−4.68550	−0.309627	−0.154813	−	0.987944i	$-0.549478\pi$
$229$	−4.68550	−0.309627	−0.154813	+	0.987944i	$0.549478\pi$
$230$	0	0
$231$	6.17313	0.406162
$232$	0	0
$233$	15.8820	1.04047	0.520234	−	0.854024i	$-0.325845\pi$
$233$	15.8820	1.04047	0.520234	+	0.854024i	$0.325845\pi$
$234$	0	0
$235$	−9.53579	−0.622046
$236$	0	0
$237$	−4.11273	−0.267150
$238$	0	0
$239$	6.75814	0.437147	0.218574	−	0.975820i	$-0.429860\pi$
$239$	6.75814	0.437147	0.218574	+	0.975820i	$0.429860\pi$
$240$	0	0
$241$	−8.33091	−0.536641	−0.268320	−	0.963330i	$-0.586469\pi$
$241$	−8.33091	−0.536641	−0.268320	+	0.963330i	$0.586469\pi$
$242$	0	0
$243$	−7.58501	−0.486579
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	2.93543	0.186777
$248$	0	0
$249$	−9.64541	−0.611253
$250$	0	0
$251$	13.7857	0.870147	0.435074	−	0.900395i	$-0.356722\pi$
$251$	13.7857	0.870147	0.435074	+	0.900395i	$0.356722\pi$
$252$	0	0
$253$	0.552195	0.0347162
$254$	0	0
$255$	−2.79406	−0.174971
$256$	0	0
$257$	−14.3463	−0.894895	−0.447447	−	0.894310i	$-0.647667\pi$
$257$	−14.3463	−0.894895	−0.447447	+	0.894310i	$0.647667\pi$
$258$	0	0
$259$	7.42723	0.461506
$260$	0	0
$261$	−5.69357	−0.352423
$262$	0	0
$263$	−23.1044	−1.42468	−0.712339	−	0.701836i	$-0.752364\pi$
$263$	−23.1044	−1.42468	−0.712339	+	0.701836i	$0.752364\pi$
$264$	0	0
$265$	8.37907	0.514722
$266$	0	0
$267$	0.210110	0.0128585
$268$	0	0
$269$	−6.23875	−0.380384	−0.190192	−	0.981747i	$-0.560911\pi$
$269$	−6.23875	−0.380384	−0.190192	+	0.981747i	$0.560911\pi$
$270$	0	0
$271$	26.9313	1.63596	0.817979	−	0.575248i	$-0.195095\pi$
$271$	26.9313	1.63596	0.817979	+	0.575248i	$0.195095\pi$
$272$	0	0
$273$	−1.93543	−0.117138
$274$	0	0
$275$	−3.18953	−0.192336
$276$	0	0
$277$	−12.2583	−0.736528	−0.368264	−	0.929721i	$-0.620048\pi$
$277$	−12.2583	−0.736528	−0.368264	+	0.929721i	$0.620048\pi$
$278$	0	0
$279$	1.53996	0.0921948
$280$	0	0
$281$	1.14554	0.0683373	0.0341687	−	0.999416i	$-0.489122\pi$
$281$	1.14554	0.0683373	0.0341687	+	0.999416i	$0.489122\pi$
$282$	0	0
$283$	−30.4629	−1.81083	−0.905415	−	0.424527i	$-0.860440\pi$
$283$	−30.4629	−1.81083	−0.905415	+	0.424527i	$0.860440\pi$
$284$	0	0
$285$	−5.68133	−0.336533
$286$	0	0
$287$	−2.76231	−0.163054
$288$	0	0
$289$	−14.9159	−0.877407
$290$	0	0
$291$	20.1731	1.18257
$292$	0	0
$293$	−7.26634	−0.424504	−0.212252	−	0.977215i	$-0.568080\pi$
$293$	−7.26634	−0.424504	−0.212252	+	0.977215i	$0.568080\pi$
$294$	0	0
$295$	11.9794	0.697469
$296$	0	0
$297$	13.9149	0.807422
$298$	0	0
$299$	−0.173127	−0.0100122
$300$	0	0
$301$	7.87086	0.453669
$302$	0	0
$303$	2.23875	0.128613
$304$	0	0
$305$	2.81047	0.160927
$306$	0	0
$307$	15.1044	0.862053	0.431027	−	0.902339i	$-0.358152\pi$
$307$	15.1044	0.862053	0.431027	+	0.902339i	$0.358152\pi$
$308$	0	0
$309$	−26.4395	−1.50409
$310$	0	0
$311$	21.8709	1.24018	0.620091	−	0.784530i	$-0.287095\pi$
$311$	21.8709	1.24018	0.620091	+	0.784530i	$0.287095\pi$
$312$	0	0
$313$	16.8667	0.953362	0.476681	−	0.879076i	$-0.341840\pi$
$313$	16.8667	0.953362	0.476681	+	0.879076i	$0.341840\pi$
$314$	0	0
$315$	0.745898	0.0420266
$316$	0	0
$317$	14.6402	0.822274	0.411137	−	0.911573i	$-0.365132\pi$
$317$	14.6402	0.822274	0.411137	+	0.911573i	$0.365132\pi$
$318$	0	0
$319$	24.3463	1.36313
$320$	0	0
$321$	6.25827	0.349303
$322$	0	0
$323$	4.23769	0.235792
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−22.6618	−1.25320
$328$	0	0
$329$	−9.53579	−0.525725
$330$	0	0
$331$	20.0357	1.10126	0.550630	−	0.834750i	$-0.314388\pi$
$331$	20.0357	1.10126	0.550630	+	0.834750i	$0.314388\pi$
$332$	0	0
$333$	−5.53996	−0.303588
$334$	0	0
$335$	−3.74590	−0.204660
$336$	0	0
$337$	13.4478	0.732549	0.366274	−	0.930507i	$-0.380633\pi$
$337$	13.4478	0.732549	0.366274	+	0.930507i	$0.380633\pi$
$338$	0	0
$339$	−36.3267	−1.97300
$340$	0	0
$341$	−6.58501	−0.356598
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0.335076	0.0180399
$346$	0	0
$347$	18.8461	1.01171	0.505856	−	0.862618i	$-0.331177\pi$
$347$	18.8461	1.01171	0.505856	+	0.862618i	$0.331177\pi$
$348$	0	0
$349$	2.39964	0.128450	0.0642250	−	0.997935i	$-0.479542\pi$
$349$	2.39964	0.128450	0.0642250	+	0.997935i	$0.479542\pi$
$350$	0	0
$351$	−4.36266	−0.232862
$352$	0	0
$353$	7.56860	0.402836	0.201418	−	0.979505i	$-0.435445\pi$
$353$	7.56860	0.402836	0.201418	+	0.979505i	$0.435445\pi$
$354$	0	0
$355$	10.2499	0.544010
$356$	0	0
$357$	−2.79406	−0.147877
$358$	0	0
$359$	33.4423	1.76502	0.882509	−	0.470296i	$-0.155853\pi$
$359$	33.4423	1.76502	0.882509	+	0.470296i	$0.155853\pi$
$360$	0	0
$361$	−10.3832	−0.546486
$362$	0	0
$363$	−1.60036	−0.0839969
$364$	0	0
$365$	−1.15672	−0.0605455
$366$	0	0
$367$	−3.24186	−0.169224	−0.0846120	−	0.996414i	$-0.526965\pi$
$367$	−3.24186	−0.169224	−0.0846120	+	0.996414i	$0.526965\pi$
$368$	0	0
$369$	2.06040	0.107260
$370$	0	0
$371$	8.37907	0.435020
$372$	0	0
$373$	−13.2007	−0.683507	−0.341753	−	0.939790i	$-0.611021\pi$
$373$	−13.2007	−0.683507	−0.341753	+	0.939790i	$0.611021\pi$
$374$	0	0
$375$	−1.93543	−0.0999453
$376$	0	0
$377$	−7.63317	−0.393128
$378$	0	0
$379$	−5.07158	−0.260509	−0.130255	−	0.991481i	$-0.541580\pi$
$379$	−5.07158	−0.260509	−0.130255	+	0.991481i	$0.541580\pi$
$380$	0	0
$381$	−26.9948	−1.38298
$382$	0	0
$383$	18.9396	0.967768	0.483884	−	0.875132i	$-0.339226\pi$
$383$	18.9396	0.967768	0.483884	+	0.875132i	$0.339226\pi$
$384$	0	0
$385$	−3.18953	−0.162554
$386$	0	0
$387$	−5.87086	−0.298433
$388$	0	0
$389$	−35.8144	−1.81586	−0.907930	−	0.419121i	$-0.862338\pi$
$389$	−35.8144	−1.81586	−0.907930	+	0.419121i	$0.862338\pi$
$390$	0	0
$391$	−0.249933	−0.0126396
$392$	0	0
$393$	−37.4751	−1.89037
$394$	0	0
$395$	2.12497	0.106919
$396$	0	0
$397$	4.62900	0.232323	0.116161	−	0.993230i	$-0.462941\pi$
$397$	4.62900	0.232323	0.116161	+	0.993230i	$0.462941\pi$
$398$	0	0
$399$	−5.68133	−0.284422
$400$	0	0
$401$	32.0245	1.59923	0.799613	−	0.600516i	$-0.205038\pi$
$401$	32.0245	1.59923	0.799613	+	0.600516i	$0.205038\pi$
$402$	0	0
$403$	2.06457	0.102843
$404$	0	0
$405$	10.6813	0.530760
$406$	0	0
$407$	23.6894	1.17424
$408$	0	0
$409$	−24.4014	−1.20657	−0.603286	−	0.797525i	$-0.706142\pi$
$409$	−24.4014	−1.20657	−0.603286	+	0.797525i	$0.706142\pi$
$410$	0	0
$411$	35.5316	1.75265
$412$	0	0
$413$	11.9794	0.589469
$414$	0	0
$415$	4.98359	0.244635
$416$	0	0
$417$	39.8625	1.95208
$418$	0	0
$419$	9.14838	0.446928	0.223464	−	0.974712i	$-0.428264\pi$
$419$	9.14838	0.446928	0.223464	+	0.974712i	$0.428264\pi$
$420$	0	0
$421$	−4.84328	−0.236047	−0.118023	−	0.993011i	$-0.537656\pi$
$421$	−4.84328	−0.236047	−0.118023	+	0.993011i	$0.537656\pi$
$422$	0	0
$423$	7.11273	0.345833
$424$	0	0
$425$	1.44364	0.0700266
$426$	0	0
$427$	2.81047	0.136008
$428$	0	0
$429$	−6.17313	−0.298041
$430$	0	0
$431$	−31.3871	−1.51187	−0.755933	−	0.654649i	$-0.772816\pi$
$431$	−31.3871	−1.51187	−0.755933	+	0.654649i	$0.772816\pi$
$432$	0	0
$433$	22.1361	1.06380	0.531898	−	0.846809i	$-0.321479\pi$
$433$	22.1361	1.06380	0.531898	+	0.846809i	$0.321479\pi$
$434$	0	0
$435$	14.7735	0.708334
$436$	0	0
$437$	−0.508203	−0.0243107
$438$	0	0
$439$	27.0716	1.29206	0.646028	−	0.763314i	$-0.276429\pi$
$439$	27.0716	1.29206	0.646028	+	0.763314i	$0.276429\pi$
$440$	0	0
$441$	0.745898	0.0355190
$442$	0	0
$443$	19.4506	0.924128	0.462064	−	0.886847i	$-0.347109\pi$
$443$	19.4506	0.924128	0.462064	+	0.886847i	$0.347109\pi$
$444$	0	0
$445$	−0.108560	−0.00514622
$446$	0	0
$447$	−34.6095	−1.63697
$448$	0	0
$449$	−19.9037	−0.939313	−0.469656	−	0.882849i	$-0.655622\pi$
$449$	−19.9037	−0.939313	−0.469656	+	0.882849i	$0.655622\pi$
$450$	0	0
$451$	−8.81047	−0.414869
$452$	0	0
$453$	4.18431	0.196596
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	−25.8831	−1.21076	−0.605380	−	0.795936i	$-0.706979\pi$
$457$	−25.8831	−1.21076	−0.605380	+	0.795936i	$0.706979\pi$
$458$	0	0
$459$	−6.29809	−0.293970
$460$	0	0
$461$	16.6719	0.776489	0.388245	−	0.921556i	$-0.373082\pi$
$461$	16.6719	0.776489	0.388245	+	0.921556i	$0.373082\pi$
$462$	0	0
$463$	11.2817	0.524304	0.262152	−	0.965027i	$-0.415568\pi$
$463$	11.2817	0.524304	0.262152	+	0.965027i	$0.415568\pi$
$464$	0	0
$465$	−3.99583	−0.185302
$466$	0	0
$467$	2.58918	0.119813	0.0599064	−	0.998204i	$-0.480920\pi$
$467$	2.58918	0.119813	0.0599064	+	0.998204i	$0.480920\pi$
$468$	0	0
$469$	−3.74590	−0.172970
$470$	0	0
$471$	1.27885	0.0589261
$472$	0	0
$473$	25.1044	1.15430
$474$	0	0
$475$	2.93543	0.134687
$476$	0	0
$477$	−6.24993	−0.286165
$478$	0	0
$479$	−7.76231	−0.354669	−0.177334	−	0.984151i	$-0.556747\pi$
$479$	−7.76231	−0.354669	−0.177334	+	0.984151i	$0.556747\pi$
$480$	0	0
$481$	−7.42723	−0.338652
$482$	0	0
$483$	0.335076	0.0152465
$484$	0	0
$485$	−10.4231	−0.473287
$486$	0	0
$487$	−33.9055	−1.53640	−0.768202	−	0.640208i	$-0.778848\pi$
$487$	−33.9055	−1.53640	−0.768202	+	0.640208i	$0.778848\pi$
$488$	0	0
$489$	28.9599	1.30961
$490$	0	0
$491$	23.7417	1.07145	0.535725	−	0.844393i	$-0.320039\pi$
$491$	23.7417	1.07145	0.535725	+	0.844393i	$0.320039\pi$
$492$	0	0
$493$	−11.0195	−0.496294
$494$	0	0
$495$	2.37907	0.106931
$496$	0	0
$497$	10.2499	0.459772
$498$	0	0
$499$	27.9477	1.25111	0.625555	−	0.780180i	$-0.284873\pi$
$499$	27.9477	1.25111	0.625555	+	0.780180i	$0.284873\pi$
$500$	0	0
$501$	25.1127	1.12195
$502$	0	0
$503$	−3.93437	−0.175425	−0.0877125	−	0.996146i	$-0.527956\pi$
$503$	−3.93437	−0.175425	−0.0877125	+	0.996146i	$0.527956\pi$
$504$	0	0
$505$	−1.15672	−0.0514734
$506$	0	0
$507$	1.93543	0.0859556
$508$	0	0
$509$	−19.8227	−0.878626	−0.439313	−	0.898334i	$-0.644778\pi$
$509$	−19.8227	−0.878626	−0.439313	+	0.898334i	$0.644778\pi$
$510$	0	0
$511$	−1.15672	−0.0511703
$512$	0	0
$513$	−12.8063	−0.565412
$514$	0	0
$515$	13.6608	0.601965
$516$	0	0
$517$	−30.4147	−1.33764
$518$	0	0
$519$	10.3392	0.453842
$520$	0	0
$521$	−43.3132	−1.89758	−0.948792	−	0.315901i	$-0.897693\pi$
$521$	−43.3132	−1.89758	−0.948792	+	0.315901i	$0.897693\pi$
$522$	0	0
$523$	−21.3103	−0.931836	−0.465918	−	0.884828i	$-0.654276\pi$
$523$	−21.3103	−0.931836	−0.465918	+	0.884828i	$0.654276\pi$
$524$	0	0
$525$	−1.93543	−0.0844692
$526$	0	0
$527$	2.98048	0.129832
$528$	0	0
$529$	−22.9700	−0.998697
$530$	0	0
$531$	−8.93543	−0.387765
$532$	0	0
$533$	2.76231	0.119649
$534$	0	0
$535$	−3.23353	−0.139798
$536$	0	0
$537$	10.7857	0.465438
$538$	0	0
$539$	−3.18953	−0.137383
$540$	0	0
$541$	−2.16195	−0.0929494	−0.0464747	−	0.998919i	$-0.514799\pi$
$541$	−2.16195	−0.0929494	−0.0464747	+	0.998919i	$0.514799\pi$
$542$	0	0
$543$	21.4702	0.921373
$544$	0	0
$545$	11.7089	0.501555
$546$	0	0
$547$	4.37907	0.187235	0.0936177	−	0.995608i	$-0.470157\pi$
$547$	4.37907	0.187235	0.0936177	+	0.995608i	$0.470157\pi$
$548$	0	0
$549$	−2.09632	−0.0894688
$550$	0	0
$551$	−22.4067	−0.954556
$552$	0	0
$553$	2.12497	0.0903628
$554$	0	0
$555$	14.3749	0.610180
$556$	0	0
$557$	−21.9302	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$557$	−21.9302	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$558$	0	0
$559$	−7.87086	−0.332902
$560$	0	0
$561$	−8.91175	−0.376254
$562$	0	0
$563$	−21.1086	−0.889620	−0.444810	−	0.895625i	$-0.646729\pi$
$563$	−21.1086	−0.889620	−0.444810	+	0.895625i	$0.646729\pi$
$564$	0	0
$565$	18.7693	0.789631
$566$	0	0
$567$	10.6813	0.448574
$568$	0	0
$569$	−22.9742	−0.963128	−0.481564	−	0.876411i	$-0.659931\pi$
$569$	−22.9742	−0.963128	−0.481564	+	0.876411i	$0.659931\pi$
$570$	0	0
$571$	45.7540	1.91474	0.957372	−	0.288858i	$-0.0932755\pi$
$571$	45.7540	1.91474	0.957372	+	0.288858i	$0.0932755\pi$
$572$	0	0
$573$	−15.9435	−0.666049
$574$	0	0
$575$	−0.173127	−0.00721991
$576$	0	0
$577$	−14.1208	−0.587856	−0.293928	−	0.955827i	$-0.594963\pi$
$577$	−14.1208	−0.587856	−0.293928	+	0.955827i	$0.594963\pi$
$578$	0	0
$579$	10.2223	0.424826
$580$	0	0
$581$	4.98359	0.206754
$582$	0	0
$583$	26.7253	1.10685
$584$	0	0
$585$	−0.745898	−0.0308391
$586$	0	0
$587$	−25.2663	−1.04285	−0.521427	−	0.853296i	$-0.674600\pi$
$587$	−25.2663	−1.04285	−0.521427	+	0.853296i	$0.674600\pi$
$588$	0	0
$589$	6.06040	0.249714
$590$	0	0
$591$	−25.0039	−1.02852
$592$	0	0
$593$	−5.36266	−0.220218	−0.110109	−	0.993920i	$-0.535120\pi$
$593$	−5.36266	−0.220218	−0.110109	+	0.993920i	$0.535120\pi$
$594$	0	0
$595$	1.44364	0.0591833
$596$	0	0
$597$	−9.95885	−0.407589
$598$	0	0
$599$	−16.4866	−0.673623	−0.336811	−	0.941572i	$-0.609348\pi$
$599$	−16.4866	−0.673623	−0.336811	+	0.941572i	$0.609348\pi$
$600$	0	0
$601$	7.40381	0.302008	0.151004	−	0.988533i	$-0.451749\pi$
$601$	7.40381	0.302008	0.151004	+	0.988533i	$0.451749\pi$
$602$	0	0
$603$	2.79406	0.113783
$604$	0	0
$605$	0.826873	0.0336172
$606$	0	0
$607$	−1.57800	−0.0640490	−0.0320245	−	0.999487i	$-0.510195\pi$
$607$	−1.57800	−0.0640490	−0.0320245	+	0.999487i	$0.510195\pi$
$608$	0	0
$609$	14.7735	0.598652
$610$	0	0
$611$	9.53579	0.385777
$612$	0	0
$613$	10.8105	0.436631	0.218315	−	0.975878i	$-0.429944\pi$
$613$	10.8105	0.436631	0.218315	+	0.975878i	$0.429944\pi$
$614$	0	0
$615$	−5.34625	−0.215582
$616$	0	0
$617$	19.7571	0.795390	0.397695	−	0.917518i	$-0.369810\pi$
$617$	19.7571	0.795390	0.397695	+	0.917518i	$0.369810\pi$
$618$	0	0
$619$	0.713085	0.0286613	0.0143306	−	0.999897i	$-0.495438\pi$
$619$	0.713085	0.0286613	0.0143306	+	0.999897i	$0.495438\pi$
$620$	0	0
$621$	0.755296	0.0303090
$622$	0	0
$623$	−0.108560	−0.00434935
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−18.1208	−0.723675
$628$	0	0
$629$	−10.7222	−0.427523
$630$	0	0
$631$	−21.4423	−0.853605	−0.426802	−	0.904345i	$-0.640360\pi$
$631$	−21.4423	−0.853605	−0.426802	+	0.904345i	$0.640360\pi$
$632$	0	0
$633$	49.9781	1.98645
$634$	0	0
$635$	13.9477	0.553496
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−7.64541	−0.302448
$640$	0	0
$641$	42.0479	1.66079	0.830396	−	0.557174i	$-0.188114\pi$
$641$	42.0479	1.66079	0.830396	+	0.557174i	$0.188114\pi$
$642$	0	0
$643$	−0.225457	−0.00889116	−0.00444558	−	0.999990i	$-0.501415\pi$
$643$	−0.225457	−0.00889116	−0.00444558	+	0.999990i	$0.501415\pi$
$644$	0	0
$645$	15.2335	0.599819
$646$	0	0
$647$	−28.5839	−1.12375	−0.561876	−	0.827222i	$-0.689920\pi$
$647$	−28.5839	−1.12375	−0.561876	+	0.827222i	$0.689920\pi$
$648$	0	0
$649$	38.2088	1.49983
$650$	0	0
$651$	−3.99583	−0.156609
$652$	0	0
$653$	−38.5962	−1.51039	−0.755193	−	0.655503i	$-0.772457\pi$
$653$	−38.5962	−1.51039	−0.755193	+	0.655503i	$0.772457\pi$
$654$	0	0
$655$	19.3627	0.756562
$656$	0	0
$657$	0.862796	0.0336609
$658$	0	0
$659$	14.8615	0.578921	0.289460	−	0.957190i	$-0.406524\pi$
$659$	14.8615	0.578921	0.289460	+	0.957190i	$0.406524\pi$
$660$	0	0
$661$	19.4988	0.758416	0.379208	−	0.925312i	$-0.376197\pi$
$661$	19.4988	0.758416	0.379208	+	0.925312i	$0.376197\pi$
$662$	0	0
$663$	2.79406	0.108512
$664$	0	0
$665$	2.93543	0.113831
$666$	0	0
$667$	1.32151	0.0511691
$668$	0	0
$669$	14.8984	0.576007
$670$	0	0
$671$	8.96408	0.346054
$672$	0	0
$673$	−49.2280	−1.89760	−0.948801	−	0.315876i	$-0.897702\pi$
$673$	−49.2280	−1.89760	−0.948801	+	0.315876i	$0.897702\pi$
$674$	0	0
$675$	−4.36266	−0.167919
$676$	0	0
$677$	−17.5358	−0.673955	−0.336978	−	0.941513i	$-0.609405\pi$
$677$	−17.5358	−0.673955	−0.336978	+	0.941513i	$0.609405\pi$
$678$	0	0
$679$	−10.4231	−0.400000
$680$	0	0
$681$	−11.6126	−0.444996
$682$	0	0
$683$	−36.1718	−1.38408	−0.692038	−	0.721861i	$-0.743287\pi$
$683$	−36.1718	−1.38408	−0.692038	+	0.721861i	$0.743287\pi$
$684$	0	0
$685$	−18.3585	−0.701442
$686$	0	0
$687$	−9.06847	−0.345984
$688$	0	0
$689$	−8.37907	−0.319217
$690$	0	0
$691$	5.31973	0.202372	0.101186	−	0.994868i	$-0.467736\pi$
$691$	5.31973	0.202372	0.101186	+	0.994868i	$0.467736\pi$
$692$	0	0
$693$	2.37907	0.0903733
$694$	0	0
$695$	−20.5962	−0.781258
$696$	0	0
$697$	3.98776	0.151047
$698$	0	0
$699$	30.7386	1.16264
$700$	0	0
$701$	−9.78989	−0.369759	−0.184880	−	0.982761i	$-0.559189\pi$
$701$	−9.78989	−0.369759	−0.184880	+	0.982761i	$0.559189\pi$
$702$	0	0
$703$	−21.8021	−0.822283
$704$	0	0
$705$	−18.4559	−0.695088
$706$	0	0
$707$	−1.15672	−0.0435030
$708$	0	0
$709$	2.51938	0.0946174	0.0473087	−	0.998880i	$-0.484936\pi$
$709$	2.51938	0.0946174	0.0473087	+	0.998880i	$0.484936\pi$
$710$	0	0
$711$	−1.58501	−0.0594425
$712$	0	0
$713$	−0.357433	−0.0133860
$714$	0	0
$715$	3.18953	0.119282
$716$	0	0
$717$	13.0799	0.488478
$718$	0	0
$719$	−17.1784	−0.640645	−0.320322	−	0.947309i	$-0.603791\pi$
$719$	−17.1784	−0.640645	−0.320322	+	0.947309i	$0.603791\pi$
$720$	0	0
$721$	13.6608	0.508753
$722$	0	0
$723$	−16.1239	−0.599655
$724$	0	0
$725$	−7.63317	−0.283489
$726$	0	0
$727$	53.2598	1.97530	0.987648	−	0.156689i	$-0.0500820\pi$
$727$	53.2598	1.97530	0.987648	+	0.156689i	$0.0500820\pi$
$728$	0	0
$729$	17.3637	0.643101
$730$	0	0
$731$	−11.3627	−0.420263
$732$	0	0
$733$	−0.379068	−0.0140012	−0.00700060	−	0.999975i	$-0.502228\pi$
$733$	−0.379068	−0.0140012	−0.00700060	+	0.999975i	$0.502228\pi$
$734$	0	0
$735$	−1.93543	−0.0713895
$736$	0	0
$737$	−11.9477	−0.440098
$738$	0	0
$739$	27.8381	1.02404	0.512020	−	0.858974i	$-0.328897\pi$
$739$	27.8381	1.02404	0.512020	+	0.858974i	$0.328897\pi$
$740$	0	0
$741$	5.68133	0.208709
$742$	0	0
$743$	6.24292	0.229031	0.114515	−	0.993421i	$-0.463468\pi$
$743$	6.24292	0.229031	0.114515	+	0.993421i	$0.463468\pi$
$744$	0	0
$745$	17.8820	0.655147
$746$	0	0
$747$	−3.71725	−0.136007
$748$	0	0
$749$	−3.23353	−0.118150
$750$	0	0
$751$	−17.9354	−0.654473	−0.327237	−	0.944942i	$-0.606117\pi$
$751$	−17.9354	−0.654473	−0.327237	+	0.944942i	$0.606117\pi$
$752$	0	0
$753$	26.6813	0.972322
$754$	0	0
$755$	−2.16195	−0.0786814
$756$	0	0
$757$	15.6209	0.567752	0.283876	−	0.958861i	$-0.408380\pi$
$757$	15.6209	0.567752	0.283876	+	0.958861i	$0.408380\pi$
$758$	0	0
$759$	1.06874	0.0387927
$760$	0	0
$761$	27.0398	0.980193	0.490096	−	0.871668i	$-0.336962\pi$
$761$	27.0398	0.980193	0.490096	+	0.871668i	$0.336962\pi$
$762$	0	0
$763$	11.7089	0.423891
$764$	0	0
$765$	−1.07681	−0.0389320
$766$	0	0
$767$	−11.9794	−0.432552
$768$	0	0
$769$	8.49702	0.306411	0.153205	−	0.988194i	$-0.451040\pi$
$769$	8.49702	0.306411	0.153205	+	0.988194i	$0.451040\pi$
$770$	0	0
$771$	−27.7662	−0.999975
$772$	0	0
$773$	−26.2583	−0.944444	−0.472222	−	0.881480i	$-0.656548\pi$
$773$	−26.2583	−0.944444	−0.472222	+	0.881480i	$0.656548\pi$
$774$	0	0
$775$	2.06457	0.0741615
$776$	0	0
$777$	14.3749	0.515697
$778$	0	0
$779$	8.10856	0.290519
$780$	0	0
$781$	32.6925	1.16983
$782$	0	0
$783$	33.3009	1.19008
$784$	0	0
$785$	−0.660755	−0.0235834
$786$	0	0
$787$	43.5058	1.55081	0.775407	−	0.631461i	$-0.217545\pi$
$787$	43.5058	1.55081	0.775407	+	0.631461i	$0.217545\pi$
$788$	0	0
$789$	−44.7170	−1.59197
$790$	0	0
$791$	18.7693	0.667360
$792$	0	0
$793$	−2.81047	−0.0998026
$794$	0	0
$795$	16.2171	0.575162
$796$	0	0
$797$	−8.70191	−0.308237	−0.154119	−	0.988052i	$-0.549254\pi$
$797$	−8.70191	−0.308237	−0.154119	+	0.988052i	$0.549254\pi$
$798$	0	0
$799$	13.7662	0.487013
$800$	0	0
$801$	0.0809744	0.00286109
$802$	0	0
$803$	−3.68940	−0.130196
$804$	0	0
$805$	−0.173127	−0.00610193
$806$	0	0
$807$	−12.0747	−0.425049
$808$	0	0
$809$	−27.2182	−0.956940	−0.478470	−	0.878104i	$-0.658808\pi$
$809$	−27.2182	−0.956940	−0.478470	+	0.878104i	$0.658808\pi$
$810$	0	0
$811$	22.7253	0.797994	0.398997	−	0.916952i	$-0.369358\pi$
$811$	22.7253	0.797994	0.398997	+	0.916952i	$0.369358\pi$
$812$	0	0
$813$	52.1236	1.82806
$814$	0	0
$815$	−14.9630	−0.524132
$816$	0	0
$817$	−23.1044	−0.808320
$818$	0	0
$819$	−0.745898	−0.0260638
$820$	0	0
$821$	2.22758	0.0777429	0.0388715	−	0.999244i	$-0.487624\pi$
$821$	2.22758	0.0777429	0.0388715	+	0.999244i	$0.487624\pi$
$822$	0	0
$823$	23.6238	0.823473	0.411736	−	0.911303i	$-0.364922\pi$
$823$	23.6238	0.823473	0.411736	+	0.911303i	$0.364922\pi$
$824$	0	0
$825$	−6.17313	−0.214921
$826$	0	0
$827$	−38.2744	−1.33093	−0.665466	−	0.746428i	$-0.731767\pi$
$827$	−38.2744	−1.33093	−0.665466	+	0.746428i	$0.731767\pi$
$828$	0	0
$829$	−39.8953	−1.38562	−0.692811	−	0.721119i	$-0.743628\pi$
$829$	−39.8953	−1.38562	−0.692811	+	0.721119i	$0.743628\pi$
$830$	0	0
$831$	−23.7251	−0.823013
$832$	0	0
$833$	1.44364	0.0500190
$834$	0	0
$835$	−12.9753	−0.449027
$836$	0	0
$837$	−9.00701	−0.311328
$838$	0	0
$839$	20.2716	0.699852	0.349926	−	0.936777i	$-0.386207\pi$
$839$	20.2716	0.699852	0.349926	+	0.936777i	$0.386207\pi$
$840$	0	0
$841$	29.2653	1.00915
$842$	0	0
$843$	2.21712	0.0763616
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	0.826873	0.0284117
$848$	0	0
$849$	−58.9588	−2.02346
$850$	0	0
$851$	1.28586	0.0440786
$852$	0	0
$853$	−31.2252	−1.06913	−0.534565	−	0.845127i	$-0.679525\pi$
$853$	−31.2252	−1.06913	−0.534565	+	0.845127i	$0.679525\pi$
$854$	0	0
$855$	−2.18953	−0.0748805
$856$	0	0
$857$	10.0534	0.343417	0.171709	−	0.985148i	$-0.445071\pi$
$857$	10.0534	0.343417	0.171709	+	0.985148i	$0.445071\pi$
$858$	0	0
$859$	47.9617	1.63643	0.818216	−	0.574911i	$-0.194963\pi$
$859$	47.9617	1.63643	0.818216	+	0.574911i	$0.194963\pi$
$860$	0	0
$861$	−5.34625	−0.182200
$862$	0	0
$863$	3.72426	0.126775	0.0633877	−	0.997989i	$-0.479810\pi$
$863$	3.72426	0.126775	0.0633877	+	0.997989i	$0.479810\pi$
$864$	0	0
$865$	−5.34209	−0.181636
$866$	0	0
$867$	−28.8687	−0.980434
$868$	0	0
$869$	6.77765	0.229916
$870$	0	0
$871$	3.74590	0.126925
$872$	0	0
$873$	7.77454	0.263128
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−23.7787	−0.802950	−0.401475	−	0.915870i	$-0.631502\pi$
$877$	−23.7787	−0.802950	−0.401475	+	0.915870i	$0.631502\pi$
$878$	0	0
$879$	−14.0635	−0.474350
$880$	0	0
$881$	55.6678	1.87549	0.937747	−	0.347318i	$-0.112908\pi$
$881$	55.6678	1.87549	0.937747	+	0.347318i	$0.112908\pi$
$882$	0	0
$883$	35.3955	1.19115	0.595576	−	0.803299i	$-0.296924\pi$
$883$	35.3955	1.19115	0.595576	+	0.803299i	$0.296924\pi$
$884$	0	0
$885$	23.1854	0.779368
$886$	0	0
$887$	23.5920	0.792142	0.396071	−	0.918220i	$-0.370373\pi$
$887$	23.5920	0.792142	0.396071	+	0.918220i	$0.370373\pi$
$888$	0	0
$889$	13.9477	0.467790
$890$	0	0
$891$	34.0685	1.14134
$892$	0	0
$893$	27.9917	0.936705
$894$	0	0
$895$	−5.57277	−0.186277
$896$	0	0
$897$	−0.335076	−0.0111879
$898$	0	0
$899$	−15.7592	−0.525599
$900$	0	0
$901$	−12.0963	−0.402987
$902$	0	0
$903$	15.2335	0.506940
$904$	0	0
$905$	−11.0932	−0.368751
$906$	0	0
$907$	9.53295	0.316536	0.158268	−	0.987396i	$-0.449409\pi$
$907$	9.53295	0.316536	0.158268	+	0.987396i	$0.449409\pi$
$908$	0	0
$909$	0.862796	0.0286171
$910$	0	0
$911$	35.0234	1.16038	0.580189	−	0.814482i	$-0.302979\pi$
$911$	35.0234	1.16038	0.580189	+	0.814482i	$0.302979\pi$
$912$	0	0
$913$	15.8953	0.526059
$914$	0	0
$915$	5.43947	0.179823
$916$	0	0
$917$	19.3627	0.639411
$918$	0	0
$919$	3.21011	0.105892	0.0529459	−	0.998597i	$-0.483139\pi$
$919$	3.21011	0.105892	0.0529459	+	0.998597i	$0.483139\pi$
$920$	0	0
$921$	29.2335	0.963277
$922$	0	0
$923$	−10.2499	−0.337381
$924$	0	0
$925$	−7.42723	−0.244206
$926$	0	0
$927$	−10.1895	−0.334668
$928$	0	0
$929$	38.4577	1.26175	0.630877	−	0.775883i	$-0.282695\pi$
$929$	38.4577	1.26175	0.630877	+	0.775883i	$0.282695\pi$
$930$	0	0
$931$	2.93543	0.0962049
$932$	0	0
$933$	42.3296	1.38581
$934$	0	0
$935$	4.60453	0.150584
$936$	0	0
$937$	14.1138	0.461077	0.230539	−	0.973063i	$-0.425951\pi$
$937$	14.1138	0.461077	0.230539	+	0.973063i	$0.425951\pi$
$938$	0	0
$939$	32.6443	1.06531
$940$	0	0
$941$	34.5397	1.12596	0.562981	−	0.826470i	$-0.309654\pi$
$941$	34.5397	1.12596	0.562981	+	0.826470i	$0.309654\pi$
$942$	0	0
$943$	−0.478230	−0.0155733
$944$	0	0
$945$	−4.36266	−0.141917
$946$	0	0
$947$	−53.0562	−1.72410	−0.862048	−	0.506827i	$-0.830818\pi$
$947$	−53.0562	−1.72410	−0.862048	+	0.506827i	$0.830818\pi$
$948$	0	0
$949$	1.15672	0.0375487
$950$	0	0
$951$	28.3351	0.918828
$952$	0	0
$953$	28.1536	0.911985	0.455992	−	0.889984i	$-0.349284\pi$
$953$	28.1536	0.911985	0.455992	+	0.889984i	$0.349284\pi$
$954$	0	0
$955$	8.23769	0.266566
$956$	0	0
$957$	47.1205	1.52319
$958$	0	0
$959$	−18.3585	−0.592827
$960$	0	0
$961$	−26.7376	−0.862502
$962$	0	0
$963$	2.41188	0.0777218
$964$	0	0
$965$	−5.28169	−0.170024
$966$	0	0
$967$	0.778712	0.0250417	0.0125208	−	0.999922i	$-0.496014\pi$
$967$	0.778712	0.0250417	0.0125208	+	0.999922i	$0.496014\pi$
$968$	0	0
$969$	8.20177	0.263479
$970$	0	0
$971$	−16.5522	−0.531185	−0.265593	−	0.964085i	$-0.585568\pi$
$971$	−16.5522	−0.531185	−0.265593	+	0.964085i	$0.585568\pi$
$972$	0	0
$973$	−20.5962	−0.660283
$974$	0	0
$975$	1.93543	0.0619834
$976$	0	0
$977$	−5.37801	−0.172058	−0.0860289	−	0.996293i	$-0.527418\pi$
$977$	−5.37801	−0.172058	−0.0860289	+	0.996293i	$0.527418\pi$
$978$	0	0
$979$	−0.346255	−0.0110663
$980$	0	0
$981$	−8.73366	−0.278844
$982$	0	0
$983$	−15.3627	−0.489993	−0.244996	−	0.969524i	$-0.578787\pi$
$983$	−15.3627	−0.489993	−0.244996	+	0.969524i	$0.578787\pi$
$984$	0	0
$985$	12.9190	0.411634
$986$	0	0
$987$	−18.4559	−0.587457
$988$	0	0
$989$	1.36266	0.0433301
$990$	0	0
$991$	−4.09215	−0.129992	−0.0649958	−	0.997886i	$-0.520703\pi$
$991$	−4.09215	−0.129992	−0.0649958	+	0.997886i	$0.520703\pi$
$992$	0	0
$993$	38.7777	1.23057
$994$	0	0
$995$	5.14554	0.163125
$996$	0	0
$997$	−12.9711	−0.410798	−0.205399	−	0.978678i	$-0.565849\pi$
$997$	−12.9711	−0.410798	−0.205399	+	0.978678i	$0.565849\pi$
$998$	0	0
$999$	32.4025	1.02517

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.o.1.3	✓	3
4.3	odd	2		7280.2.a.bp.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.o.1.3	✓	3	1.1	even	1	trivial
7280.2.a.bp.1.1		3	4.3	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:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q+1.93543 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.745898 q^{9} +O(q^{10})\)	\(q+1.93543 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.745898 q^{9} -3.18953 q^{11} +1.00000 q^{13} -1.93543 q^{15} +1.44364 q^{17} +2.93543 q^{19} -1.93543 q^{21} -0.173127 q^{23} +1.00000 q^{25} -4.36266 q^{27} -7.63317 q^{29} +2.06457 q^{31} -6.17313 q^{33} +1.00000 q^{35} -7.42723 q^{37} +1.93543 q^{39} +2.76231 q^{41} -7.87086 q^{43} -0.745898 q^{45} +9.53579 q^{47} +1.00000 q^{49} +2.79406 q^{51} -8.37907 q^{53} +3.18953 q^{55} +5.68133 q^{57} -11.9794 q^{59} -2.81047 q^{61} -0.745898 q^{63} -1.00000 q^{65} +3.74590 q^{67} -0.335076 q^{69} -10.2499 q^{71} +1.15672 q^{73} +1.93543 q^{75} +3.18953 q^{77} -2.12497 q^{79} -10.6813 q^{81} -4.98359 q^{83} -1.44364 q^{85} -14.7735 q^{87} +0.108560 q^{89} -1.00000 q^{91} +3.99583 q^{93} -2.93543 q^{95} +10.4231 q^{97} -2.37907 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q - 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} - q^{11} + 3 q^{13} + 2 q^{15} - 5 q^{17} + q^{19} + 2 q^{21} + 5 q^{23} + 3 q^{25} + q^{27} - 5 q^{29} + 14 q^{31} - 13 q^{33} + 3 q^{35} - 16 q^{37} - 2 q^{39} + 6 q^{41} - 8 q^{43} - 3 q^{45} + 9 q^{47} + 3 q^{49} + 20 q^{51} - 8 q^{53} + q^{55} + 10 q^{57} - 7 q^{59} - 17 q^{61} - 3 q^{63} - 3 q^{65} + 12 q^{67} - 5 q^{69} + 2 q^{71} + q^{73} - 2 q^{75} + q^{77} + 10 q^{79} - 25 q^{81} - 18 q^{83} + 5 q^{85} - 27 q^{87} - 13 q^{89} - 3 q^{91} - 20 q^{93} - q^{95} - 7 q^{97} + 10 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 - q^11 + 3 * q^13 + 2 * q^15 - 5 * q^17 + q^19 + 2 * q^21 + 5 * q^23 + 3 * q^25 + q^27 - 5 * q^29 + 14 * q^31 - 13 * q^33 + 3 * q^35 - 16 * q^37 - 2 * q^39 + 6 * q^41 - 8 * q^43 - 3 * q^45 + 9 * q^47 + 3 * q^49 + 20 * q^51 - 8 * q^53 + q^55 + 10 * q^57 - 7 * q^59 - 17 * q^61 - 3 * q^63 - 3 * q^65 + 12 * q^67 - 5 * q^69 + 2 * q^71 + q^73 - 2 * q^75 + q^77 + 10 * q^79 - 25 * q^81 - 18 * q^83 + 5 * q^85 - 27 * q^87 - 13 * q^89 - 3 * q^91 - 20 * q^93 - q^95 - 7 * q^97 + 10 * q^99

Introduction

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