LMFDB - Embedded newform 3640.2.a.x.1.2

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.4913.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + x + 1 $ x^4 - x^3 - 6*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.487928$ 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)

$f(q)$	$=$	$q+0.512072 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.73778 q^{9} +O(q^{10})$	$q+0.512072 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.73778 q^{9} -3.81141 q^{11} +1.00000 q^{13} -0.512072 q^{15} -0.761926 q^{17} -1.04948 q^{19} -0.512072 q^{21} +5.81141 q^{23} +1.00000 q^{25} -2.93816 q^{27} +4.32348 q^{29} +4.61103 q^{31} -1.95171 q^{33} +1.00000 q^{35} -5.92273 q^{37} +0.512072 q^{39} -7.43600 q^{41} -2.78726 q^{43} +2.73778 q^{45} -3.12311 q^{47} +1.00000 q^{49} -0.390161 q^{51} +7.12311 q^{53} +3.81141 q^{55} -0.537410 q^{57} +14.0866 q^{59} +8.24621 q^{61} +2.73778 q^{63} -1.00000 q^{65} +2.21393 q^{67} +2.97586 q^{69} +4.58205 q^{71} -3.07482 q^{73} +0.512072 q^{75} +3.81141 q^{77} +9.11439 q^{79} +6.70880 q^{81} -11.3210 q^{83} +0.761926 q^{85} +2.21393 q^{87} +17.3500 q^{89} -1.00000 q^{91} +2.36118 q^{93} +1.04948 q^{95} +11.7972 q^{97} +10.4348 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) $ 4 * q + 3 * q^3 - 4 * q^5 - 4 * q^7 + 3 * q^9	$ 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9} + 6 q^{11} + 4 q^{13} - 3 q^{15} + 9 q^{17} + 5 q^{19} - 3 q^{21} + 2 q^{23} + 4 q^{25} + 6 q^{27} - 3 q^{29} + q^{31} - 4 q^{33} + 4 q^{35} - 11 q^{37} + 3 q^{39} - 5 q^{41} + 12 q^{43} - 3 q^{45} + 4 q^{47} + 4 q^{49} - 6 q^{51} + 12 q^{53} - 6 q^{55} + 8 q^{57} + 11 q^{59} - 3 q^{63} - 4 q^{65} + 19 q^{67} + 10 q^{69} - 8 q^{71} + 8 q^{73} + 3 q^{75} - 6 q^{77} + 13 q^{79} + 4 q^{81} + 8 q^{83} - 9 q^{85} + 19 q^{87} + 25 q^{89} - 4 q^{91} + 5 q^{93} - 5 q^{95} + 18 q^{97} + 30 q^{99}+O(q^{100}) $ 4 * q + 3 * q^3 - 4 * q^5 - 4 * q^7 + 3 * q^9 + 6 * q^11 + 4 * q^13 - 3 * q^15 + 9 * q^17 + 5 * q^19 - 3 * q^21 + 2 * q^23 + 4 * q^25 + 6 * q^27 - 3 * q^29 + q^31 - 4 * q^33 + 4 * q^35 - 11 * q^37 + 3 * q^39 - 5 * q^41 + 12 * q^43 - 3 * q^45 + 4 * q^47 + 4 * q^49 - 6 * q^51 + 12 * q^53 - 6 * q^55 + 8 * q^57 + 11 * q^59 - 3 * q^63 - 4 * q^65 + 19 * q^67 + 10 * q^69 - 8 * q^71 + 8 * q^73 + 3 * q^75 - 6 * q^77 + 13 * q^79 + 4 * q^81 + 8 * q^83 - 9 * q^85 + 19 * q^87 + 25 * q^89 - 4 * q^91 + 5 * q^93 - 5 * q^95 + 18 * q^97 + 30 * 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
$3$	0.512072	0.295645	0.147822	−	0.989014i	$-0.452774\pi$
$3$	0.512072	0.295645	0.147822	+	0.989014i	$0.452774\pi$
$4$	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
$9$	−2.73778	−0.912594
$10$	0	0
$11$	−3.81141	−1.14918	−0.574591	−	0.818441i	$-0.694839\pi$
$11$	−3.81141	−1.14918	−0.574591	+	0.818441i	$0.694839\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.512072	−0.132216
$16$	0	0
$17$	−0.761926	−0.184794	−0.0923971	−	0.995722i	$-0.529453\pi$
$17$	−0.761926	−0.184794	−0.0923971	+	0.995722i	$0.529453\pi$
$18$	0	0
$19$	−1.04948	−0.240767	−0.120384	−	0.992727i	$-0.538413\pi$
$19$	−1.04948	−0.240767	−0.120384	+	0.992727i	$0.538413\pi$
$20$	0	0
$21$	−0.512072	−0.111743
$22$	0	0
$23$	5.81141	1.21176	0.605881	−	0.795555i	$-0.292821\pi$
$23$	5.81141	1.21176	0.605881	+	0.795555i	$0.292821\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.93816	−0.565448
$28$	0	0
$29$	4.32348	0.802850	0.401425	−	0.915892i	$-0.368515\pi$
$29$	4.32348	0.802850	0.401425	+	0.915892i	$0.368515\pi$
$30$	0	0
$31$	4.61103	0.828166	0.414083	−	0.910239i	$-0.364102\pi$
$31$	4.61103	0.828166	0.414083	+	0.910239i	$0.364102\pi$
$32$	0	0
$33$	−1.95171	−0.339750
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−5.92273	−0.973691	−0.486846	−	0.873488i	$-0.661853\pi$
$37$	−5.92273	−0.973691	−0.486846	+	0.873488i	$0.661853\pi$
$38$	0	0
$39$	0.512072	0.0819971
$40$	0	0
$41$	−7.43600	−1.16131	−0.580654	−	0.814150i	$-0.697203\pi$
$41$	−7.43600	−1.16131	−0.580654	+	0.814150i	$0.697203\pi$
$42$	0	0
$43$	−2.78726	−0.425054	−0.212527	−	0.977155i	$-0.568169\pi$
$43$	−2.78726	−0.425054	−0.212527	+	0.977155i	$0.568169\pi$
$44$	0	0
$45$	2.73778	0.408125
$46$	0	0
$47$	−3.12311	−0.455552	−0.227776	−	0.973714i	$-0.573145\pi$
$47$	−3.12311	−0.455552	−0.227776	+	0.973714i	$0.573145\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.390161	−0.0546334
$52$	0	0
$53$	7.12311	0.978434	0.489217	−	0.872162i	$-0.337283\pi$
$53$	7.12311	0.978434	0.489217	+	0.872162i	$0.337283\pi$
$54$	0	0
$55$	3.81141	0.513930
$56$	0	0
$57$	−0.537410	−0.0711816
$58$	0	0
$59$	14.0866	1.83392	0.916960	−	0.398980i	$-0.130636\pi$
$59$	14.0866	1.83392	0.916960	+	0.398980i	$0.130636\pi$
$60$	0	0
$61$	8.24621	1.05582	0.527910	−	0.849301i	$-0.322976\pi$
$61$	8.24621	1.05582	0.527910	+	0.849301i	$0.322976\pi$
$62$	0	0
$63$	2.73778	0.344928
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	2.21393	0.270475	0.135237	−	0.990813i	$-0.456820\pi$
$67$	2.21393	0.270475	0.135237	+	0.990813i	$0.456820\pi$
$68$	0	0
$69$	2.97586	0.358251
$70$	0	0
$71$	4.58205	0.543790	0.271895	−	0.962327i	$-0.412350\pi$
$71$	4.58205	0.543790	0.271895	+	0.962327i	$0.412350\pi$
$72$	0	0
$73$	−3.07482	−0.359880	−0.179940	−	0.983678i	$-0.557590\pi$
$73$	−3.07482	−0.359880	−0.179940	+	0.983678i	$0.557590\pi$
$74$	0	0
$75$	0.512072	0.0591289
$76$	0	0
$77$	3.81141	0.434350
$78$	0	0
$79$	9.11439	1.02545	0.512724	−	0.858553i	$-0.328636\pi$
$79$	9.11439	1.02545	0.512724	+	0.858553i	$0.328636\pi$
$80$	0	0
$81$	6.70880	0.745422
$82$	0	0
$83$	−11.3210	−1.24264	−0.621322	−	0.783555i	$-0.713404\pi$
$83$	−11.3210	−1.24264	−0.621322	+	0.783555i	$0.713404\pi$
$84$	0	0
$85$	0.761926	0.0826425
$86$	0	0
$87$	2.21393	0.237358
$88$	0	0
$89$	17.3500	1.83910	0.919549	−	0.392976i	$-0.128554\pi$
$89$	17.3500	1.83910	0.919549	+	0.392976i	$0.128554\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	2.36118	0.244843
$94$	0	0
$95$	1.04948	0.107674
$96$	0	0
$97$	11.7972	1.19782	0.598911	−	0.800816i	$-0.295600\pi$
$97$	11.7972	1.19782	0.598911	+	0.800816i	$0.295600\pi$
$98$	0	0
$99$	10.4348	1.04874
$100$	0	0
$101$	4.25350	0.423239	0.211619	−	0.977352i	$-0.432126\pi$
$101$	4.25350	0.423239	0.211619	+	0.977352i	$0.432126\pi$
$102$	0	0
$103$	12.1243	1.19464	0.597321	−	0.802002i	$-0.296232\pi$
$103$	12.1243	1.19464	0.597321	+	0.802002i	$0.296232\pi$
$104$	0	0
$105$	0.512072	0.0499731
$106$	0	0
$107$	15.2870	1.47785	0.738924	−	0.673789i	$-0.235334\pi$
$107$	15.2870	1.47785	0.738924	+	0.673789i	$0.235334\pi$
$108$	0	0
$109$	−17.2727	−1.65443	−0.827214	−	0.561886i	$-0.810076\pi$
$109$	−17.2727	−1.65443	−0.827214	+	0.561886i	$0.810076\pi$
$110$	0	0
$111$	−3.03286	−0.287867
$112$	0	0
$113$	−0.598671	−0.0563182	−0.0281591	−	0.999603i	$-0.508965\pi$
$113$	−0.598671	−0.0563182	−0.0281591	+	0.999603i	$0.508965\pi$
$114$	0	0
$115$	−5.81141	−0.541917
$116$	0	0
$117$	−2.73778	−0.253108
$118$	0	0
$119$	0.761926	0.0698456
$120$	0	0
$121$	3.52682	0.320620
$122$	0	0
$123$	−3.80776	−0.343335
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.0093	−0.888185	−0.444092	−	0.895981i	$-0.646474\pi$
$127$	−10.0093	−0.888185	−0.444092	+	0.895981i	$0.646474\pi$
$128$	0	0
$129$	−1.42728	−0.125665
$130$	0	0
$131$	0.876894	0.0766146	0.0383073	−	0.999266i	$-0.487803\pi$
$131$	0.876894	0.0766146	0.0383073	+	0.999266i	$0.487803\pi$
$132$	0	0
$133$	1.04948	0.0910016
$134$	0	0
$135$	2.93816	0.252876
$136$	0	0
$137$	7.29934	0.623624	0.311812	−	0.950144i	$-0.399064\pi$
$137$	7.29934	0.623624	0.311812	+	0.950144i	$0.399064\pi$
$138$	0	0
$139$	−15.6735	−1.32941	−0.664704	−	0.747107i	$-0.731442\pi$
$139$	−15.6735	−1.32941	−0.664704	+	0.747107i	$0.731442\pi$
$140$	0	0
$141$	−1.59925	−0.134681
$142$	0	0
$143$	−3.81141	−0.318726
$144$	0	0
$145$	−4.32348	−0.359045
$146$	0	0
$147$	0.512072	0.0422350
$148$	0	0
$149$	−7.12311	−0.583548	−0.291774	−	0.956487i	$-0.594245\pi$
$149$	−7.12311	−0.583548	−0.291774	+	0.956487i	$0.594245\pi$
$150$	0	0
$151$	3.61587	0.294255	0.147128	−	0.989118i	$-0.452997\pi$
$151$	3.61587	0.294255	0.147128	+	0.989118i	$0.452997\pi$
$152$	0	0
$153$	2.08599	0.168642
$154$	0	0
$155$	−4.61103	−0.370367
$156$	0	0
$157$	−1.82616	−0.145743	−0.0728717	−	0.997341i	$-0.523216\pi$
$157$	−1.82616	−0.145743	−0.0728717	+	0.997341i	$0.523216\pi$
$158$	0	0
$159$	3.64754	0.289269
$160$	0	0
$161$	−5.81141	−0.458003
$162$	0	0
$163$	22.5721	1.76798	0.883991	−	0.467504i	$-0.154847\pi$
$163$	22.5721	1.76798	0.883991	+	0.467504i	$0.154847\pi$
$164$	0	0
$165$	1.95171	0.151941
$166$	0	0
$167$	−1.82564	−0.141272	−0.0706360	−	0.997502i	$-0.522503\pi$
$167$	−1.82564	−0.141272	−0.0706360	+	0.997502i	$0.522503\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.87325	0.219723
$172$	0	0
$173$	−8.22691	−0.625480	−0.312740	−	0.949839i	$-0.601247\pi$
$173$	−8.22691	−0.625480	−0.312740	+	0.949839i	$0.601247\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	7.21335	0.542188
$178$	0	0
$179$	26.1312	1.95314	0.976570	−	0.215198i	$-0.0690398\pi$
$179$	26.1312	1.95314	0.976570	+	0.215198i	$0.0690398\pi$
$180$	0	0
$181$	−8.40075	−0.624423	−0.312211	−	0.950013i	$-0.601070\pi$
$181$	−8.40075	−0.624423	−0.312211	+	0.950013i	$0.601070\pi$
$182$	0	0
$183$	4.22265	0.312147
$184$	0	0
$185$	5.92273	0.435448
$186$	0	0
$187$	2.90401	0.212362
$188$	0	0
$189$	2.93816	0.213719
$190$	0	0
$191$	−6.17623	−0.446896	−0.223448	−	0.974716i	$-0.571731\pi$
$191$	−6.17623	−0.446896	−0.223448	+	0.974716i	$0.571731\pi$
$192$	0	0
$193$	3.26163	0.234778	0.117389	−	0.993086i	$-0.462548\pi$
$193$	3.26163	0.234778	0.117389	+	0.993086i	$0.462548\pi$
$194$	0	0
$195$	−0.512072	−0.0366702
$196$	0	0
$197$	21.1684	1.50818	0.754092	−	0.656769i	$-0.228077\pi$
$197$	21.1684	1.50818	0.754092	+	0.656769i	$0.228077\pi$
$198$	0	0
$199$	25.9270	1.83792	0.918958	−	0.394356i	$-0.129032\pi$
$199$	25.9270	1.83792	0.918958	+	0.394356i	$0.129032\pi$
$200$	0	0
$201$	1.13369	0.0799644
$202$	0	0
$203$	−4.32348	−0.303449
$204$	0	0
$205$	7.43600	0.519353
$206$	0	0
$207$	−15.9104	−1.10585
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	−18.6098	−1.28115	−0.640575	−	0.767896i	$-0.721304\pi$
$211$	−18.6098	−1.28115	−0.640575	+	0.767896i	$0.721304\pi$
$212$	0	0
$213$	2.34634	0.160769
$214$	0	0
$215$	2.78726	0.190090
$216$	0	0
$217$	−4.61103	−0.313017
$218$	0	0
$219$	−1.57453	−0.106397
$220$	0	0
$221$	−0.761926	−0.0512527
$222$	0	0
$223$	1.29747	0.0868850	0.0434425	−	0.999056i	$-0.486167\pi$
$223$	1.29747	0.0868850	0.0434425	+	0.999056i	$0.486167\pi$
$224$	0	0
$225$	−2.73778	−0.182519
$226$	0	0
$227$	3.63010	0.240938	0.120469	−	0.992717i	$-0.461560\pi$
$227$	3.63010	0.240938	0.120469	+	0.992717i	$0.461560\pi$
$228$	0	0
$229$	−27.4089	−1.81123	−0.905615	−	0.424101i	$-0.860590\pi$
$229$	−27.4089	−1.81123	−0.905615	+	0.424101i	$0.860590\pi$
$230$	0	0
$231$	1.95171	0.128413
$232$	0	0
$233$	−17.3210	−1.13474	−0.567369	−	0.823464i	$-0.692039\pi$
$233$	−17.3210	−1.13474	−0.567369	+	0.823464i	$0.692039\pi$
$234$	0	0
$235$	3.12311	0.203729
$236$	0	0
$237$	4.66722	0.303168
$238$	0	0
$239$	0.236879	0.0153225	0.00766123	−	0.999971i	$-0.497561\pi$
$239$	0.236879	0.0153225	0.00766123	+	0.999971i	$0.497561\pi$
$240$	0	0
$241$	−12.6654	−0.815847	−0.407924	−	0.913016i	$-0.633747\pi$
$241$	−12.6654	−0.815847	−0.407924	+	0.913016i	$0.633747\pi$
$242$	0	0
$243$	12.2499	0.785829
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−1.04948	−0.0667769
$248$	0	0
$249$	−5.79718	−0.367381
$250$	0	0
$251$	17.3234	1.09344	0.546722	−	0.837314i	$-0.315875\pi$
$251$	17.3234	1.09344	0.546722	+	0.837314i	$0.315875\pi$
$252$	0	0
$253$	−22.1496	−1.39254
$254$	0	0
$255$	0.390161	0.0244328
$256$	0	0
$257$	−11.6155	−0.724557	−0.362278	−	0.932070i	$-0.618001\pi$
$257$	−11.6155	−0.724557	−0.362278	+	0.932070i	$0.618001\pi$
$258$	0	0
$259$	5.92273	0.368021
$260$	0	0
$261$	−11.8367	−0.732676
$262$	0	0
$263$	24.1874	1.49146	0.745730	−	0.666248i	$-0.232101\pi$
$263$	24.1874	1.49146	0.745730	+	0.666248i	$0.232101\pi$
$264$	0	0
$265$	−7.12311	−0.437569
$266$	0	0
$267$	8.88445	0.543719
$268$	0	0
$269$	14.4997	0.884063	0.442031	−	0.897000i	$-0.354258\pi$
$269$	14.4997	0.884063	0.442031	+	0.897000i	$0.354258\pi$
$270$	0	0
$271$	−22.8591	−1.38859	−0.694296	−	0.719690i	$-0.744284\pi$
$271$	−22.8591	−1.38859	−0.694296	+	0.719690i	$0.744284\pi$
$272$	0	0
$273$	−0.512072	−0.0309920
$274$	0	0
$275$	−3.81141	−0.229836
$276$	0	0
$277$	8.10135	0.486763	0.243382	−	0.969931i	$-0.421743\pi$
$277$	8.10135	0.486763	0.243382	+	0.969931i	$0.421743\pi$
$278$	0	0
$279$	−12.6240	−0.755780
$280$	0	0
$281$	13.8714	0.827499	0.413750	−	0.910391i	$-0.364219\pi$
$281$	13.8714	0.827499	0.413750	+	0.910391i	$0.364219\pi$
$282$	0	0
$283$	27.5728	1.63903	0.819515	−	0.573058i	$-0.194243\pi$
$283$	27.5728	1.63903	0.819515	+	0.573058i	$0.194243\pi$
$284$	0	0
$285$	0.537410	0.0318334
$286$	0	0
$287$	7.43600	0.438933
$288$	0	0
$289$	−16.4195	−0.965851
$290$	0	0
$291$	6.04100	0.354130
$292$	0	0
$293$	−28.1461	−1.64431	−0.822156	−	0.569262i	$-0.807229\pi$
$293$	−28.1461	−1.64431	−0.822156	+	0.569262i	$0.807229\pi$
$294$	0	0
$295$	−14.0866	−0.820154
$296$	0	0
$297$	11.1985	0.649803
$298$	0	0
$299$	5.81141	0.336082
$300$	0	0
$301$	2.78726	0.160655
$302$	0	0
$303$	2.17810	0.125128
$304$	0	0
$305$	−8.24621	−0.472177
$306$	0	0
$307$	10.4514	0.596494	0.298247	−	0.954489i	$-0.403598\pi$
$307$	10.4514	0.596494	0.298247	+	0.954489i	$0.403598\pi$
$308$	0	0
$309$	6.20851	0.353190
$310$	0	0
$311$	−17.2148	−0.976161	−0.488080	−	0.872799i	$-0.662303\pi$
$311$	−17.2148	−0.976161	−0.488080	+	0.872799i	$0.662303\pi$
$312$	0	0
$313$	18.4460	1.04263	0.521315	−	0.853364i	$-0.325442\pi$
$313$	18.4460	1.04263	0.521315	+	0.853364i	$0.325442\pi$
$314$	0	0
$315$	−2.73778	−0.154257
$316$	0	0
$317$	31.5932	1.77445	0.887225	−	0.461337i	$-0.152630\pi$
$317$	31.5932	1.77445	0.887225	+	0.461337i	$0.152630\pi$
$318$	0	0
$319$	−16.4785	−0.922621
$320$	0	0
$321$	7.82802	0.436918
$322$	0	0
$323$	0.799627	0.0444924
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−8.84488	−0.489123
$328$	0	0
$329$	3.12311	0.172182
$330$	0	0
$331$	5.56520	0.305891	0.152945	−	0.988235i	$-0.451124\pi$
$331$	5.56520	0.305891	0.152945	+	0.988235i	$0.451124\pi$
$332$	0	0
$333$	16.2152	0.888585
$334$	0	0
$335$	−2.21393	−0.120960
$336$	0	0
$337$	−8.84978	−0.482078	−0.241039	−	0.970515i	$-0.577488\pi$
$337$	−8.84978	−0.482078	−0.241039	+	0.970515i	$0.577488\pi$
$338$	0	0
$339$	−0.306562	−0.0166502
$340$	0	0
$341$	−17.5745	−0.951714
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−2.97586	−0.160215
$346$	0	0
$347$	0.982799	0.0527594	0.0263797	−	0.999652i	$-0.491602\pi$
$347$	0.982799	0.0527594	0.0263797	+	0.999652i	$0.491602\pi$
$348$	0	0
$349$	−17.7248	−0.948787	−0.474394	−	0.880313i	$-0.657333\pi$
$349$	−17.7248	−0.948787	−0.474394	+	0.880313i	$0.657333\pi$
$350$	0	0
$351$	−2.93816	−0.156827
$352$	0	0
$353$	12.0483	0.641266	0.320633	−	0.947204i	$-0.396104\pi$
$353$	12.0483	0.641266	0.320633	+	0.947204i	$0.396104\pi$
$354$	0	0
$355$	−4.58205	−0.243190
$356$	0	0
$357$	0.390161	0.0206495
$358$	0	0
$359$	23.4107	1.23557	0.617784	−	0.786348i	$-0.288031\pi$
$359$	23.4107	1.23557	0.617784	+	0.786348i	$0.288031\pi$
$360$	0	0
$361$	−17.8986	−0.942031
$362$	0	0
$363$	1.80599	0.0947897
$364$	0	0
$365$	3.07482	0.160943
$366$	0	0
$367$	31.4058	1.63937	0.819684	−	0.572816i	$-0.194149\pi$
$367$	31.4058	1.63937	0.819684	+	0.572816i	$0.194149\pi$
$368$	0	0
$369$	20.3581	1.05980
$370$	0	0
$371$	−7.12311	−0.369813
$372$	0	0
$373$	−19.7701	−1.02366	−0.511828	−	0.859088i	$-0.671031\pi$
$373$	−19.7701	−1.02366	−0.511828	+	0.859088i	$0.671031\pi$
$374$	0	0
$375$	−0.512072	−0.0264433
$376$	0	0
$377$	4.32348	0.222670
$378$	0	0
$379$	16.1132	0.827679	0.413840	−	0.910350i	$-0.364187\pi$
$379$	16.1132	0.827679	0.413840	+	0.910350i	$0.364187\pi$
$380$	0	0
$381$	−5.12549	−0.262587
$382$	0	0
$383$	22.2003	1.13438	0.567192	−	0.823586i	$-0.308030\pi$
$383$	22.2003	1.13438	0.567192	+	0.823586i	$0.308030\pi$
$384$	0	0
$385$	−3.81141	−0.194247
$386$	0	0
$387$	7.63092	0.387902
$388$	0	0
$389$	−29.6377	−1.50269	−0.751344	−	0.659910i	$-0.770594\pi$
$389$	−29.6377	−1.50269	−0.751344	+	0.659910i	$0.770594\pi$
$390$	0	0
$391$	−4.42786	−0.223927
$392$	0	0
$393$	0.449033	0.0226507
$394$	0	0
$395$	−9.11439	−0.458595
$396$	0	0
$397$	1.75618	0.0881400	0.0440700	−	0.999028i	$-0.485968\pi$
$397$	1.75618	0.0881400	0.0440700	+	0.999028i	$0.485968\pi$
$398$	0	0
$399$	0.537410	0.0269041
$400$	0	0
$401$	−26.5716	−1.32692	−0.663460	−	0.748212i	$-0.730913\pi$
$401$	−26.5716	−1.32692	−0.663460	+	0.748212i	$0.730913\pi$
$402$	0	0
$403$	4.61103	0.229692
$404$	0	0
$405$	−6.70880	−0.333363
$406$	0	0
$407$	22.5739	1.11895
$408$	0	0
$409$	36.2239	1.79116	0.895578	−	0.444905i	$-0.146763\pi$
$409$	36.2239	1.79116	0.895578	+	0.444905i	$0.146763\pi$
$410$	0	0
$411$	3.73778	0.184371
$412$	0	0
$413$	−14.0866	−0.693156
$414$	0	0
$415$	11.3210	0.555728
$416$	0	0
$417$	−8.02595	−0.393032
$418$	0	0
$419$	20.9203	1.02202	0.511011	−	0.859574i	$-0.329271\pi$
$419$	20.9203	1.02202	0.511011	+	0.859574i	$0.329271\pi$
$420$	0	0
$421$	−29.6638	−1.44573	−0.722863	−	0.690991i	$-0.757174\pi$
$421$	−29.6638	−1.44573	−0.722863	+	0.690991i	$0.757174\pi$
$422$	0	0
$423$	8.55038	0.415734
$424$	0	0
$425$	−0.761926	−0.0369588
$426$	0	0
$427$	−8.24621	−0.399062
$428$	0	0
$429$	−1.95171	−0.0942296
$430$	0	0
$431$	−26.0812	−1.25629	−0.628143	−	0.778098i	$-0.716185\pi$
$431$	−26.0812	−1.25629	−0.628143	+	0.778098i	$0.716185\pi$
$432$	0	0
$433$	10.9907	0.528179	0.264090	−	0.964498i	$-0.414929\pi$
$433$	10.9907	0.528179	0.264090	+	0.964498i	$0.414929\pi$
$434$	0	0
$435$	−2.21393	−0.106150
$436$	0	0
$437$	−6.09896	−0.291753
$438$	0	0
$439$	−11.8207	−0.564173	−0.282087	−	0.959389i	$-0.591027\pi$
$439$	−11.8207	−0.564173	−0.282087	+	0.959389i	$0.591027\pi$
$440$	0	0
$441$	−2.73778	−0.130371
$442$	0	0
$443$	−19.2290	−0.913598	−0.456799	−	0.889570i	$-0.651004\pi$
$443$	−19.2290	−0.913598	−0.456799	+	0.889570i	$0.651004\pi$
$444$	0	0
$445$	−17.3500	−0.822469
$446$	0	0
$447$	−3.64754	−0.172523
$448$	0	0
$449$	−17.8455	−0.842180	−0.421090	−	0.907019i	$-0.638352\pi$
$449$	−17.8455	−0.842180	−0.421090	+	0.907019i	$0.638352\pi$
$450$	0	0
$451$	28.3416	1.33455
$452$	0	0
$453$	1.85159	0.0869951
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	16.3953	0.766941	0.383470	−	0.923553i	$-0.374729\pi$
$457$	16.3953	0.766941	0.383470	+	0.923553i	$0.374729\pi$
$458$	0	0
$459$	2.23866	0.104492
$460$	0	0
$461$	16.8739	0.785894	0.392947	−	0.919561i	$-0.371456\pi$
$461$	16.8739	0.785894	0.392947	+	0.919561i	$0.371456\pi$
$462$	0	0
$463$	−8.51082	−0.395531	−0.197766	−	0.980249i	$-0.563368\pi$
$463$	−8.51082	−0.395531	−0.197766	+	0.980249i	$0.563368\pi$
$464$	0	0
$465$	−2.36118	−0.109497
$466$	0	0
$467$	−7.09777	−0.328446	−0.164223	−	0.986423i	$-0.552512\pi$
$467$	−7.09777	−0.328446	−0.164223	+	0.986423i	$0.552512\pi$
$468$	0	0
$469$	−2.21393	−0.102230
$470$	0	0
$471$	−0.935124	−0.0430883
$472$	0	0
$473$	10.6234	0.488464
$474$	0	0
$475$	−1.04948	−0.0481535
$476$	0	0
$477$	−19.5015	−0.892913
$478$	0	0
$479$	7.46081	0.340893	0.170447	−	0.985367i	$-0.445479\pi$
$479$	7.46081	0.340893	0.170447	+	0.985367i	$0.445479\pi$
$480$	0	0
$481$	−5.92273	−0.270053
$482$	0	0
$483$	−2.97586	−0.135406
$484$	0	0
$485$	−11.7972	−0.535682
$486$	0	0
$487$	−7.20037	−0.326280	−0.163140	−	0.986603i	$-0.552162\pi$
$487$	−7.20037	−0.326280	−0.163140	+	0.986603i	$0.552162\pi$
$488$	0	0
$489$	11.5585	0.522694
$490$	0	0
$491$	12.2945	0.554843	0.277421	−	0.960748i	$-0.410520\pi$
$491$	12.2945	0.554843	0.277421	+	0.960748i	$0.410520\pi$
$492$	0	0
$493$	−3.29417	−0.148362
$494$	0	0
$495$	−10.4348	−0.469010
$496$	0	0
$497$	−4.58205	−0.205533
$498$	0	0
$499$	−5.66655	−0.253670	−0.126835	−	0.991924i	$-0.540482\pi$
$499$	−5.66655	−0.253670	−0.126835	+	0.991924i	$0.540482\pi$
$500$	0	0
$501$	−0.934856	−0.0417663
$502$	0	0
$503$	11.2363	0.501002	0.250501	−	0.968116i	$-0.419405\pi$
$503$	11.2363	0.501002	0.250501	+	0.968116i	$0.419405\pi$
$504$	0	0
$505$	−4.25350	−0.189278
$506$	0	0
$507$	0.512072	0.0227419
$508$	0	0
$509$	21.9864	0.974529	0.487265	−	0.873254i	$-0.337995\pi$
$509$	21.9864	0.974529	0.487265	+	0.873254i	$0.337995\pi$
$510$	0	0
$511$	3.07482	0.136022
$512$	0	0
$513$	3.08354	0.136142
$514$	0	0
$515$	−12.1243	−0.534261
$516$	0	0
$517$	11.9034	0.523512
$518$	0	0
$519$	−4.21277	−0.184920
$520$	0	0
$521$	39.2727	1.72057	0.860285	−	0.509813i	$-0.170286\pi$
$521$	39.2727	1.72057	0.860285	+	0.509813i	$0.170286\pi$
$522$	0	0
$523$	14.1132	0.617127	0.308563	−	0.951204i	$-0.400152\pi$
$523$	14.1132	0.617127	0.308563	+	0.951204i	$0.400152\pi$
$524$	0	0
$525$	−0.512072	−0.0223486
$526$	0	0
$527$	−3.51327	−0.153040
$528$	0	0
$529$	10.7725	0.468367
$530$	0	0
$531$	−38.5660	−1.67362
$532$	0	0
$533$	−7.43600	−0.322089
$534$	0	0
$535$	−15.2870	−0.660913
$536$	0	0
$537$	13.3811	0.577436
$538$	0	0
$539$	−3.81141	−0.164169
$540$	0	0
$541$	−31.4967	−1.35415	−0.677075	−	0.735914i	$-0.736753\pi$
$541$	−31.4967	−1.35415	−0.677075	+	0.735914i	$0.736753\pi$
$542$	0	0
$543$	−4.30178	−0.184607
$544$	0	0
$545$	17.2727	0.739883
$546$	0	0
$547$	−19.5096	−0.834171	−0.417086	−	0.908867i	$-0.636948\pi$
$547$	−19.5096	−0.834171	−0.417086	+	0.908867i	$0.636948\pi$
$548$	0	0
$549$	−22.5763	−0.963534
$550$	0	0
$551$	−4.53741	−0.193300
$552$	0	0
$553$	−9.11439	−0.387583
$554$	0	0
$555$	3.03286	0.128738
$556$	0	0
$557$	−34.7559	−1.47265	−0.736327	−	0.676626i	$-0.763441\pi$
$557$	−34.7559	−1.47265	−0.736327	+	0.676626i	$0.763441\pi$
$558$	0	0
$559$	−2.78726	−0.117889
$560$	0	0
$561$	1.48706	0.0627838
$562$	0	0
$563$	−21.7508	−0.916689	−0.458344	−	0.888775i	$-0.651557\pi$
$563$	−21.7508	−0.916689	−0.458344	+	0.888775i	$0.651557\pi$
$564$	0	0
$565$	0.598671	0.0251863
$566$	0	0
$567$	−6.70880	−0.281743
$568$	0	0
$569$	−0.898589	−0.0376708	−0.0188354	−	0.999823i	$-0.505996\pi$
$569$	−0.898589	−0.0376708	−0.0188354	+	0.999823i	$0.505996\pi$
$570$	0	0
$571$	−13.9536	−0.583939	−0.291970	−	0.956428i	$-0.594311\pi$
$571$	−13.9536	−0.583939	−0.291970	+	0.956428i	$0.594311\pi$
$572$	0	0
$573$	−3.16267	−0.132123
$574$	0	0
$575$	5.81141	0.242352
$576$	0	0
$577$	1.43218	0.0596222	0.0298111	−	0.999556i	$-0.490509\pi$
$577$	1.43218	0.0596222	0.0298111	+	0.999556i	$0.490509\pi$
$578$	0	0
$579$	1.67019	0.0694107
$580$	0	0
$581$	11.3210	0.469675
$582$	0	0
$583$	−27.1491	−1.12440
$584$	0	0
$585$	2.73778	0.113193
$586$	0	0
$587$	−7.82429	−0.322943	−0.161472	−	0.986877i	$-0.551624\pi$
$587$	−7.82429	−0.322943	−0.161472	+	0.986877i	$0.551624\pi$
$588$	0	0
$589$	−4.83919	−0.199395
$590$	0	0
$591$	10.8397	0.445886
$592$	0	0
$593$	22.0259	0.904497	0.452249	−	0.891892i	$-0.350622\pi$
$593$	22.0259	0.904497	0.452249	+	0.891892i	$0.350622\pi$
$594$	0	0
$595$	−0.761926	−0.0312359
$596$	0	0
$597$	13.2765	0.543370
$598$	0	0
$599$	28.9028	1.18094	0.590469	−	0.807060i	$-0.298943\pi$
$599$	28.9028	1.18094	0.590469	+	0.807060i	$0.298943\pi$
$600$	0	0
$601$	−13.5510	−0.552755	−0.276378	−	0.961049i	$-0.589134\pi$
$601$	−13.5510	−0.552755	−0.276378	+	0.961049i	$0.589134\pi$
$602$	0	0
$603$	−6.06126	−0.246834
$604$	0	0
$605$	−3.52682	−0.143386
$606$	0	0
$607$	33.7504	1.36989	0.684944	−	0.728596i	$-0.259827\pi$
$607$	33.7504	1.36989	0.684944	+	0.728596i	$0.259827\pi$
$608$	0	0
$609$	−2.21393	−0.0897130
$610$	0	0
$611$	−3.12311	−0.126347
$612$	0	0
$613$	−6.95230	−0.280801	−0.140400	−	0.990095i	$-0.544839\pi$
$613$	−6.95230	−0.280801	−0.140400	+	0.990095i	$0.544839\pi$
$614$	0	0
$615$	3.80776	0.153544
$616$	0	0
$617$	−22.0003	−0.885697	−0.442849	−	0.896596i	$-0.646032\pi$
$617$	−22.0003	−0.885697	−0.442849	+	0.896596i	$0.646032\pi$
$618$	0	0
$619$	−6.11610	−0.245827	−0.122913	−	0.992417i	$-0.539224\pi$
$619$	−6.11610	−0.245827	−0.122913	+	0.992417i	$0.539224\pi$
$620$	0	0
$621$	−17.0748	−0.685189
$622$	0	0
$623$	−17.3500	−0.695114
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.04829	0.0818007
$628$	0	0
$629$	4.51268	0.179932
$630$	0	0
$631$	34.6278	1.37851	0.689256	−	0.724518i	$-0.257937\pi$
$631$	34.6278	1.37851	0.689256	+	0.724518i	$0.257937\pi$
$632$	0	0
$633$	−9.52954	−0.378765
$634$	0	0
$635$	10.0093	0.397208
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−12.5447	−0.496259
$640$	0	0
$641$	−25.8153	−1.01964	−0.509822	−	0.860280i	$-0.670289\pi$
$641$	−25.8153	−1.01964	−0.509822	+	0.860280i	$0.670289\pi$
$642$	0	0
$643$	−23.0737	−0.909936	−0.454968	−	0.890508i	$-0.650349\pi$
$643$	−23.0737	−0.909936	−0.454968	+	0.890508i	$0.650349\pi$
$644$	0	0
$645$	1.42728	0.0561990
$646$	0	0
$647$	7.48554	0.294287	0.147143	−	0.989115i	$-0.452992\pi$
$647$	7.48554	0.294287	0.147143	+	0.989115i	$0.452992\pi$
$648$	0	0
$649$	−53.6898	−2.10751
$650$	0	0
$651$	−2.36118	−0.0925419
$652$	0	0
$653$	21.7972	0.852990	0.426495	−	0.904490i	$-0.359748\pi$
$653$	21.7972	0.852990	0.426495	+	0.904490i	$0.359748\pi$
$654$	0	0
$655$	−0.876894	−0.0342631
$656$	0	0
$657$	8.41819	0.328425
$658$	0	0
$659$	11.0329	0.429779	0.214890	−	0.976638i	$-0.431061\pi$
$659$	11.0329	0.429779	0.214890	+	0.976638i	$0.431061\pi$
$660$	0	0
$661$	33.5633	1.30546	0.652731	−	0.757590i	$-0.273623\pi$
$661$	33.5633	1.30546	0.652731	+	0.757590i	$0.273623\pi$
$662$	0	0
$663$	−0.390161	−0.0151526
$664$	0	0
$665$	−1.04948	−0.0406971
$666$	0	0
$667$	25.1255	0.972863
$668$	0	0
$669$	0.664398	0.0256871
$670$	0	0
$671$	−31.4297	−1.21333
$672$	0	0
$673$	38.8129	1.49613	0.748063	−	0.663628i	$-0.230984\pi$
$673$	38.8129	1.49613	0.748063	+	0.663628i	$0.230984\pi$
$674$	0	0
$675$	−2.93816	−0.113090
$676$	0	0
$677$	27.9371	1.07371	0.536856	−	0.843674i	$-0.319612\pi$
$677$	27.9371	1.07371	0.536856	+	0.843674i	$0.319612\pi$
$678$	0	0
$679$	−11.7972	−0.452734
$680$	0	0
$681$	1.85887	0.0712321
$682$	0	0
$683$	−43.7564	−1.67429	−0.837147	−	0.546978i	$-0.815778\pi$
$683$	−43.7564	−1.67429	−0.837147	+	0.546978i	$0.815778\pi$
$684$	0	0
$685$	−7.29934	−0.278893
$686$	0	0
$687$	−14.0353	−0.535481
$688$	0	0
$689$	7.12311	0.271369
$690$	0	0
$691$	19.0281	0.723861	0.361931	−	0.932205i	$-0.382118\pi$
$691$	19.0281	0.723861	0.361931	+	0.932205i	$0.382118\pi$
$692$	0	0
$693$	−10.4348	−0.396385
$694$	0	0
$695$	15.6735	0.594529
$696$	0	0
$697$	5.66568	0.214603
$698$	0	0
$699$	−8.86961	−0.335479
$700$	0	0
$701$	−42.7087	−1.61309	−0.806543	−	0.591175i	$-0.798664\pi$
$701$	−42.7087	−1.61309	−0.806543	+	0.591175i	$0.798664\pi$
$702$	0	0
$703$	6.21580	0.234433
$704$	0	0
$705$	1.59925	0.0602314
$706$	0	0
$707$	−4.25350	−0.159969
$708$	0	0
$709$	−45.9988	−1.72752	−0.863761	−	0.503901i	$-0.831898\pi$
$709$	−45.9988	−1.72752	−0.863761	+	0.503901i	$0.831898\pi$
$710$	0	0
$711$	−24.9532	−0.935818
$712$	0	0
$713$	26.7966	1.00354
$714$	0	0
$715$	3.81141	0.142539
$716$	0	0
$717$	0.121299	0.00453000
$718$	0	0
$719$	−0.524435	−0.0195581	−0.00977906	−	0.999952i	$-0.503113\pi$
$719$	−0.524435	−0.0195581	−0.00977906	+	0.999952i	$0.503113\pi$
$720$	0	0
$721$	−12.1243	−0.451533
$722$	0	0
$723$	−6.48557	−0.241201
$724$	0	0
$725$	4.32348	0.160570
$726$	0	0
$727$	2.78913	0.103443	0.0517215	−	0.998662i	$-0.483529\pi$
$727$	2.78913	0.103443	0.0517215	+	0.998662i	$0.483529\pi$
$728$	0	0
$729$	−13.8536	−0.513096
$730$	0	0
$731$	2.12369	0.0785475
$732$	0	0
$733$	−24.8546	−0.918024	−0.459012	−	0.888430i	$-0.651797\pi$
$733$	−24.8546	−0.918024	−0.459012	+	0.888430i	$0.651797\pi$
$734$	0	0
$735$	−0.512072	−0.0188880
$736$	0	0
$737$	−8.43819	−0.310825
$738$	0	0
$739$	45.6817	1.68043	0.840213	−	0.542256i	$-0.182430\pi$
$739$	45.6817	1.68043	0.840213	+	0.542256i	$0.182430\pi$
$740$	0	0
$741$	−0.537410	−0.0197422
$742$	0	0
$743$	−15.0376	−0.551678	−0.275839	−	0.961204i	$-0.588956\pi$
$743$	−15.0376	−0.551678	−0.275839	+	0.961204i	$0.588956\pi$
$744$	0	0
$745$	7.12311	0.260970
$746$	0	0
$747$	30.9945	1.13403
$748$	0	0
$749$	−15.2870	−0.558574
$750$	0	0
$751$	7.45626	0.272083	0.136041	−	0.990703i	$-0.456562\pi$
$751$	7.45626	0.272083	0.136041	+	0.990703i	$0.456562\pi$
$752$	0	0
$753$	8.87083	0.323271
$754$	0	0
$755$	−3.61587	−0.131595
$756$	0	0
$757$	−10.6778	−0.388091	−0.194046	−	0.980992i	$-0.562161\pi$
$757$	−10.6778	−0.388091	−0.194046	+	0.980992i	$0.562161\pi$
$758$	0	0
$759$	−11.3422	−0.411696
$760$	0	0
$761$	8.29636	0.300743	0.150371	−	0.988630i	$-0.451953\pi$
$761$	8.29636	0.300743	0.150371	+	0.988630i	$0.451953\pi$
$762$	0	0
$763$	17.2727	0.625315
$764$	0	0
$765$	−2.08599	−0.0754190
$766$	0	0
$767$	14.0866	0.508638
$768$	0	0
$769$	3.22446	0.116277	0.0581385	−	0.998309i	$-0.481484\pi$
$769$	3.22446	0.116277	0.0581385	+	0.998309i	$0.481484\pi$
$770$	0	0
$771$	−5.94798	−0.214211
$772$	0	0
$773$	−3.57453	−0.128567	−0.0642834	−	0.997932i	$-0.520476\pi$
$773$	−3.57453	−0.128567	−0.0642834	+	0.997932i	$0.520476\pi$
$774$	0	0
$775$	4.61103	0.165633
$776$	0	0
$777$	3.03286	0.108803
$778$	0	0
$779$	7.80394	0.279605
$780$	0	0
$781$	−17.4641	−0.624914
$782$	0	0
$783$	−12.7031	−0.453970
$784$	0	0
$785$	1.82616	0.0651784
$786$	0	0
$787$	−21.1641	−0.754419	−0.377209	−	0.926128i	$-0.623116\pi$
$787$	−21.1641	−0.754419	−0.377209	+	0.926128i	$0.623116\pi$
$788$	0	0
$789$	12.3857	0.440942
$790$	0	0
$791$	0.598671	0.0212863
$792$	0	0
$793$	8.24621	0.292832
$794$	0	0
$795$	−3.64754	−0.129365
$796$	0	0
$797$	14.6856	0.520191	0.260096	−	0.965583i	$-0.416246\pi$
$797$	14.6856	0.520191	0.260096	+	0.965583i	$0.416246\pi$
$798$	0	0
$799$	2.37958	0.0841833
$800$	0	0
$801$	−47.5006	−1.67835
$802$	0	0
$803$	11.7194	0.413568
$804$	0	0
$805$	5.81141	0.204825
$806$	0	0
$807$	7.42489	0.261368
$808$	0	0
$809$	16.0160	0.563093	0.281546	−	0.959548i	$-0.409153\pi$
$809$	16.0160	0.563093	0.281546	+	0.959548i	$0.409153\pi$
$810$	0	0
$811$	30.6243	1.07536	0.537682	−	0.843148i	$-0.319300\pi$
$811$	30.6243	1.07536	0.537682	+	0.843148i	$0.319300\pi$
$812$	0	0
$813$	−11.7055	−0.410530
$814$	0	0
$815$	−22.5721	−0.790665
$816$	0	0
$817$	2.92518	0.102339
$818$	0	0
$819$	2.73778	0.0956659
$820$	0	0
$821$	12.8449	0.448289	0.224145	−	0.974556i	$-0.428041\pi$
$821$	12.8449	0.448289	0.224145	+	0.974556i	$0.428041\pi$
$822$	0	0
$823$	−2.30634	−0.0803939	−0.0401969	−	0.999192i	$-0.512799\pi$
$823$	−2.30634	−0.0803939	−0.0401969	+	0.999192i	$0.512799\pi$
$824$	0	0
$825$	−1.95171	−0.0679499
$826$	0	0
$827$	44.6409	1.55232	0.776158	−	0.630538i	$-0.217166\pi$
$827$	44.6409	1.55232	0.776158	+	0.630538i	$0.217166\pi$
$828$	0	0
$829$	31.9583	1.10996	0.554979	−	0.831864i	$-0.312726\pi$
$829$	31.9583	1.10996	0.554979	+	0.831864i	$0.312726\pi$
$830$	0	0
$831$	4.14847	0.143909
$832$	0	0
$833$	−0.761926	−0.0263992
$834$	0	0
$835$	1.82564	0.0631787
$836$	0	0
$837$	−13.5479	−0.468285
$838$	0	0
$839$	9.15955	0.316223	0.158111	−	0.987421i	$-0.449459\pi$
$839$	9.15955	0.316223	0.158111	+	0.987421i	$0.449459\pi$
$840$	0	0
$841$	−10.3075	−0.355432
$842$	0	0
$843$	7.10316	0.244646
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−3.52682	−0.121183
$848$	0	0
$849$	14.1192	0.484570
$850$	0	0
$851$	−34.4194	−1.17988
$852$	0	0
$853$	−51.8950	−1.77685	−0.888425	−	0.459022i	$-0.848200\pi$
$853$	−51.8950	−1.77685	−0.888425	+	0.459022i	$0.848200\pi$
$854$	0	0
$855$	−2.87325	−0.0982631
$856$	0	0
$857$	25.5479	0.872701	0.436350	−	0.899777i	$-0.356271\pi$
$857$	25.5479	0.872701	0.436350	+	0.899777i	$0.356271\pi$
$858$	0	0
$859$	30.7399	1.04883	0.524415	−	0.851463i	$-0.324284\pi$
$859$	30.7399	1.04883	0.524415	+	0.851463i	$0.324284\pi$
$860$	0	0
$861$	3.80776	0.129768
$862$	0	0
$863$	33.3171	1.13413	0.567063	−	0.823674i	$-0.308079\pi$
$863$	33.3171	1.13413	0.567063	+	0.823674i	$0.308079\pi$
$864$	0	0
$865$	8.22691	0.279723
$866$	0	0
$867$	−8.40794	−0.285549
$868$	0	0
$869$	−34.7386	−1.17843
$870$	0	0
$871$	2.21393	0.0750162
$872$	0	0
$873$	−32.2981	−1.09313
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	21.1627	0.714613	0.357306	−	0.933987i	$-0.383695\pi$
$877$	21.1627	0.714613	0.357306	+	0.933987i	$0.383695\pi$
$878$	0	0
$879$	−14.4128	−0.486132
$880$	0	0
$881$	29.5364	0.995106	0.497553	−	0.867433i	$-0.334232\pi$
$881$	29.5364	0.995106	0.497553	+	0.867433i	$0.334232\pi$
$882$	0	0
$883$	−1.09144	−0.0367298	−0.0183649	−	0.999831i	$-0.505846\pi$
$883$	−1.09144	−0.0367298	−0.0183649	+	0.999831i	$0.505846\pi$
$884$	0	0
$885$	−7.21335	−0.242474
$886$	0	0
$887$	−22.7793	−0.764855	−0.382428	−	0.923985i	$-0.624912\pi$
$887$	−22.7793	−0.764855	−0.382428	+	0.923985i	$0.624912\pi$
$888$	0	0
$889$	10.0093	0.335702
$890$	0	0
$891$	−25.5700	−0.856626
$892$	0	0
$893$	3.27764	0.109682
$894$	0	0
$895$	−26.1312	−0.873471
$896$	0	0
$897$	2.97586	0.0993610
$898$	0	0
$899$	19.9357	0.664893
$900$	0	0
$901$	−5.42728	−0.180809
$902$	0	0
$903$	1.42728	0.0474969
$904$	0	0
$905$	8.40075	0.279250
$906$	0	0
$907$	−32.8156	−1.08962	−0.544812	−	0.838558i	$-0.683399\pi$
$907$	−32.8156	−1.08962	−0.544812	+	0.838558i	$0.683399\pi$
$908$	0	0
$909$	−11.6452	−0.386245
$910$	0	0
$911$	−46.6062	−1.54413	−0.772067	−	0.635542i	$-0.780777\pi$
$911$	−46.6062	−1.54413	−0.772067	+	0.635542i	$0.780777\pi$
$912$	0	0
$913$	43.1491	1.42803
$914$	0	0
$915$	−4.22265	−0.139597
$916$	0	0
$917$	−0.876894	−0.0289576
$918$	0	0
$919$	30.7266	1.01358	0.506789	−	0.862070i	$-0.330832\pi$
$919$	30.7266	1.01358	0.506789	+	0.862070i	$0.330832\pi$
$920$	0	0
$921$	5.35188	0.176350
$922$	0	0
$923$	4.58205	0.150820
$924$	0	0
$925$	−5.92273	−0.194738
$926$	0	0
$927$	−33.1937	−1.09022
$928$	0	0
$929$	−47.3451	−1.55334	−0.776671	−	0.629906i	$-0.783093\pi$
$929$	−47.3451	−1.55334	−0.776671	+	0.629906i	$0.783093\pi$
$930$	0	0
$931$	−1.04948	−0.0343954
$932$	0	0
$933$	−8.81520	−0.288597
$934$	0	0
$935$	−2.90401	−0.0949713
$936$	0	0
$937$	32.0396	1.04669	0.523344	−	0.852122i	$-0.324684\pi$
$937$	32.0396	1.04669	0.523344	+	0.852122i	$0.324684\pi$
$938$	0	0
$939$	9.44567	0.308248
$940$	0	0
$941$	−17.8676	−0.582467	−0.291233	−	0.956652i	$-0.594066\pi$
$941$	−17.8676	−0.582467	−0.291233	+	0.956652i	$0.594066\pi$
$942$	0	0
$943$	−43.2136	−1.40723
$944$	0	0
$945$	−2.93816	−0.0955782
$946$	0	0
$947$	−32.3426	−1.05099	−0.525496	−	0.850796i	$-0.676120\pi$
$947$	−32.3426	−1.05099	−0.525496	+	0.850796i	$0.676120\pi$
$948$	0	0
$949$	−3.07482	−0.0998129
$950$	0	0
$951$	16.1780	0.524607
$952$	0	0
$953$	16.5195	0.535120	0.267560	−	0.963541i	$-0.413783\pi$
$953$	16.5195	0.535120	0.267560	+	0.963541i	$0.413783\pi$
$954$	0	0
$955$	6.17623	0.199858
$956$	0	0
$957$	−8.43819	−0.272768
$958$	0	0
$959$	−7.29934	−0.235708
$960$	0	0
$961$	−9.73837	−0.314141
$962$	0	0
$963$	−41.8524	−1.34867
$964$	0	0
$965$	−3.26163	−0.104996
$966$	0	0
$967$	−37.9970	−1.22190	−0.610950	−	0.791669i	$-0.709212\pi$
$967$	−37.9970	−1.22190	−0.610950	+	0.791669i	$0.709212\pi$
$968$	0	0
$969$	0.409466	0.0131540
$970$	0	0
$971$	−16.1695	−0.518903	−0.259451	−	0.965756i	$-0.583542\pi$
$971$	−16.1695	−0.518903	−0.259451	+	0.965756i	$0.583542\pi$
$972$	0	0
$973$	15.6735	0.502469
$974$	0	0
$975$	0.512072	0.0163994
$976$	0	0
$977$	31.1318	0.995995	0.497998	−	0.867178i	$-0.334069\pi$
$977$	31.1318	0.995995	0.497998	+	0.867178i	$0.334069\pi$
$978$	0	0
$979$	−66.1280	−2.11346
$980$	0	0
$981$	47.2890	1.50982
$982$	0	0
$983$	51.5871	1.64537	0.822686	−	0.568496i	$-0.192475\pi$
$983$	51.5871	1.64537	0.822686	+	0.568496i	$0.192475\pi$
$984$	0	0
$985$	−21.1684	−0.674480
$986$	0	0
$987$	1.59925	0.0509048
$988$	0	0
$989$	−16.1979	−0.515064
$990$	0	0
$991$	−35.0341	−1.11289	−0.556447	−	0.830883i	$-0.687836\pi$
$991$	−35.0341	−1.11289	−0.556447	+	0.830883i	$0.687836\pi$
$992$	0	0
$993$	2.84978	0.0904350
$994$	0	0
$995$	−25.9270	−0.821941
$996$	0	0
$997$	25.1675	0.797061	0.398531	−	0.917155i	$-0.369520\pi$
$997$	25.1675	0.797061	0.398531	+	0.917155i	$0.369520\pi$
$998$	0	0
$999$	17.4019	0.550572

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.x.1.2	✓	4
4.3	odd	2		7280.2.a.br.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.x.1.2	✓	4	1.1	even	1	trivial
7280.2.a.br.1.3		4	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)

\(f(q)\)	\(=\)	\(q+0.512072 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.73778 q^{9} +O(q^{10})\)	\(q+0.512072 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.73778 q^{9} -3.81141 q^{11} +1.00000 q^{13} -0.512072 q^{15} -0.761926 q^{17} -1.04948 q^{19} -0.512072 q^{21} +5.81141 q^{23} +1.00000 q^{25} -2.93816 q^{27} +4.32348 q^{29} +4.61103 q^{31} -1.95171 q^{33} +1.00000 q^{35} -5.92273 q^{37} +0.512072 q^{39} -7.43600 q^{41} -2.78726 q^{43} +2.73778 q^{45} -3.12311 q^{47} +1.00000 q^{49} -0.390161 q^{51} +7.12311 q^{53} +3.81141 q^{55} -0.537410 q^{57} +14.0866 q^{59} +8.24621 q^{61} +2.73778 q^{63} -1.00000 q^{65} +2.21393 q^{67} +2.97586 q^{69} +4.58205 q^{71} -3.07482 q^{73} +0.512072 q^{75} +3.81141 q^{77} +9.11439 q^{79} +6.70880 q^{81} -11.3210 q^{83} +0.761926 q^{85} +2.21393 q^{87} +17.3500 q^{89} -1.00000 q^{91} +2.36118 q^{93} +1.04948 q^{95} +11.7972 q^{97} +10.4348 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) 4 * q + 3 * q^3 - 4 * q^5 - 4 * q^7 + 3 * q^9	\( 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9} + 6 q^{11} + 4 q^{13} - 3 q^{15} + 9 q^{17} + 5 q^{19} - 3 q^{21} + 2 q^{23} + 4 q^{25} + 6 q^{27} - 3 q^{29} + q^{31} - 4 q^{33} + 4 q^{35} - 11 q^{37} + 3 q^{39} - 5 q^{41} + 12 q^{43} - 3 q^{45} + 4 q^{47} + 4 q^{49} - 6 q^{51} + 12 q^{53} - 6 q^{55} + 8 q^{57} + 11 q^{59} - 3 q^{63} - 4 q^{65} + 19 q^{67} + 10 q^{69} - 8 q^{71} + 8 q^{73} + 3 q^{75} - 6 q^{77} + 13 q^{79} + 4 q^{81} + 8 q^{83} - 9 q^{85} + 19 q^{87} + 25 q^{89} - 4 q^{91} + 5 q^{93} - 5 q^{95} + 18 q^{97} + 30 q^{99}+O(q^{100}) \) 4 * q + 3 * q^3 - 4 * q^5 - 4 * q^7 + 3 * q^9 + 6 * q^11 + 4 * q^13 - 3 * q^15 + 9 * q^17 + 5 * q^19 - 3 * q^21 + 2 * q^23 + 4 * q^25 + 6 * q^27 - 3 * q^29 + q^31 - 4 * q^33 + 4 * q^35 - 11 * q^37 + 3 * q^39 - 5 * q^41 + 12 * q^43 - 3 * q^45 + 4 * q^47 + 4 * q^49 - 6 * q^51 + 12 * q^53 - 6 * q^55 + 8 * q^57 + 11 * q^59 - 3 * q^63 - 4 * q^65 + 19 * q^67 + 10 * q^69 - 8 * q^71 + 8 * q^73 + 3 * q^75 - 6 * q^77 + 13 * q^79 + 4 * q^81 + 8 * q^83 - 9 * q^85 + 19 * q^87 + 25 * q^89 - 4 * q^91 + 5 * q^93 - 5 * q^95 + 18 * q^97 + 30 * q^99

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