LMFDB - Embedded newform 4600.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(1,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.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:	$36.7311849298$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.15529.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} - x + 2 $ x^4 - x^3 - 6*x^2 - x + 2
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			1.2
Root			$0.491317$ of defining polynomial
Character	$\chi$	$=$	4600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.329452 q^{3} +4.07069 q^{7} -2.89146 q^{9} -3.08806 q^{11} +6.32062 q^{13} -0.982634 q^{17} -7.08806 q^{19} +1.34110 q^{21} -1.00000 q^{23} -1.94095 q^{27} -7.63270 q^{29} -5.92047 q^{31} -1.01737 q^{33} -11.4825 q^{37} +2.08234 q^{39} -5.74124 q^{41} -1.74696 q^{43} -2.24992 q^{47} +9.57054 q^{49} -0.323730 q^{51} -1.31781 q^{53} -2.33517 q^{57} +7.92901 q^{59} +9.51722 q^{61} -11.7703 q^{63} +3.34110 q^{67} -0.329452 q^{69} -3.92619 q^{71} +1.66744 q^{73} -12.5705 q^{77} -15.9231 q^{79} +8.03493 q^{81} +11.9116 q^{83} -2.51461 q^{87} -10.4998 q^{89} +25.7293 q^{91} -1.95051 q^{93} -2.84125 q^{97} +8.92901 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{7} + 6 q^{9} + q^{11} - 3 q^{13} - 2 q^{17} - 15 q^{19} + 8 q^{21} - 4 q^{23} - 27 q^{27} + q^{29} - 12 q^{31} - 6 q^{33} - 18 q^{37} - 3 q^{39} - 9 q^{41} + 9 q^{43} + 4 q^{47} - 3 q^{49} - 2 q^{51}+ \cdots - q^{99}+O(q^{100}) $ 4 * q + q^7 + 6 * q^9 + q^11 - 3 * q^13 - 2 * q^17 - 15 * q^19 + 8 * q^21 - 4 * q^23 - 27 * q^27 + q^29 - 12 * q^31 - 6 * q^33 - 18 * q^37 - 3 * q^39 - 9 * q^41 + 9 * q^43 + 4 * q^47 - 3 * q^49 - 2 * q^51 - 6 * q^57 - 5 * q^59 + 14 * q^61 - 39 * q^63 + 16 * q^67 - 2 * q^71 - 21 * q^73 - 9 * q^77 - 21 * q^79 + 44 * q^81 + 9 * q^83 - 17 * q^87 - 16 * q^89 + 33 * q^91 + 29 * q^93 - 40 * q^97 - q^99

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
$3$	0.329452	0.190209	0.0951045	−	0.995467i	$-0.469681\pi$
$3$	0.329452	0.190209	0.0951045	+	0.995467i	$0.469681\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.07069	1.53858	0.769289	−	0.638901i	$-0.220611\pi$
$7$	4.07069	1.53858	0.769289	+	0.638901i	$0.220611\pi$
$8$	0	0
$9$	−2.89146	−0.963821
$10$	0	0
$11$	−3.08806	−0.931085	−0.465542	−	0.885026i	$-0.654141\pi$
$11$	−3.08806	−0.931085	−0.465542	+	0.885026i	$0.654141\pi$
$12$	0	0
$13$	6.32062	1.75302	0.876512	−	0.481380i	$-0.159864\pi$
$13$	6.32062	1.75302	0.876512	+	0.481380i	$0.159864\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.982634	−0.238324	−0.119162	−	0.992875i	$-0.538021\pi$
$17$	−0.982634	−0.238324	−0.119162	+	0.992875i	$0.538021\pi$
$18$	0	0
$19$	−7.08806	−1.62611	−0.813056	−	0.582185i	$-0.802198\pi$
$19$	−7.08806	−1.62611	−0.813056	+	0.582185i	$0.802198\pi$
$20$	0	0
$21$	1.34110	0.292651
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.94095	−0.373536
$28$	0	0
$29$	−7.63270	−1.41736	−0.708679	−	0.705531i	$-0.750708\pi$
$29$	−7.63270	−1.41736	−0.708679	+	0.705531i	$0.750708\pi$
$30$	0	0
$31$	−5.92047	−1.06335	−0.531674	−	0.846949i	$-0.678437\pi$
$31$	−5.92047	−1.06335	−0.531674	+	0.846949i	$0.678437\pi$
$32$	0	0
$33$	−1.01737	−0.177101
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11.4825	−1.88771	−0.943854	−	0.330362i	$-0.892829\pi$
$37$	−11.4825	−1.88771	−0.943854	+	0.330362i	$0.892829\pi$
$38$	0	0
$39$	2.08234	0.333441
$40$	0	0
$41$	−5.74124	−0.896631	−0.448316	−	0.893875i	$-0.647976\pi$
$41$	−5.74124	−0.896631	−0.448316	+	0.893875i	$0.647976\pi$
$42$	0	0
$43$	−1.74696	−0.266409	−0.133205	−	0.991089i	$-0.542527\pi$
$43$	−1.74696	−0.266409	−0.133205	+	0.991089i	$0.542527\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.24992	−0.328185	−0.164093	−	0.986445i	$-0.552470\pi$
$47$	−2.24992	−0.328185	−0.164093	+	0.986445i	$0.552470\pi$
$48$	0	0
$49$	9.57054	1.36722
$50$	0	0
$51$	−0.323730	−0.0453313
$52$	0	0
$53$	−1.31781	−0.181015	−0.0905073	−	0.995896i	$-0.528849\pi$
$53$	−1.31781	−0.181015	−0.0905073	+	0.995896i	$0.528849\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.33517	−0.309301
$58$	0	0
$59$	7.92901	1.03227	0.516134	−	0.856508i	$-0.327371\pi$
$59$	7.92901	1.03227	0.516134	+	0.856508i	$0.327371\pi$
$60$	0	0
$61$	9.51722	1.21855	0.609277	−	0.792957i	$-0.291460\pi$
$61$	9.51722	1.21855	0.609277	+	0.792957i	$0.291460\pi$
$62$	0	0
$63$	−11.7703	−1.48291
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.34110	0.408180	0.204090	−	0.978952i	$-0.434576\pi$
$67$	3.34110	0.408180	0.204090	+	0.978952i	$0.434576\pi$
$68$	0	0
$69$	−0.329452	−0.0396613
$70$	0	0
$71$	−3.92619	−0.465954	−0.232977	−	0.972482i	$-0.574847\pi$
$71$	−3.92619	−0.465954	−0.232977	+	0.972482i	$0.574847\pi$
$72$	0	0
$73$	1.66744	0.195159	0.0975793	−	0.995228i	$-0.468890\pi$
$73$	1.66744	0.195159	0.0975793	+	0.995228i	$0.468890\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−12.5705	−1.43255
$78$	0	0
$79$	−15.9231	−1.79149	−0.895743	−	0.444572i	$-0.853356\pi$
$79$	−15.9231	−1.79149	−0.895743	+	0.444572i	$0.853356\pi$
$80$	0	0
$81$	8.03493	0.892771
$82$	0	0
$83$	11.9116	1.30747	0.653736	−	0.756723i	$-0.273201\pi$
$83$	11.9116	1.30747	0.653736	+	0.756723i	$0.273201\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.51461	−0.269594
$88$	0	0
$89$	−10.4998	−1.11298	−0.556491	−	0.830854i	$-0.687853\pi$
$89$	−10.4998	−1.11298	−0.556491	+	0.830854i	$0.687853\pi$
$90$	0	0
$91$	25.7293	2.69716
$92$	0	0
$93$	−1.95051	−0.202258
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.84125	−0.288485	−0.144242	−	0.989542i	$-0.546075\pi$
$97$	−2.84125	−0.288485	−0.144242	+	0.989542i	$0.546075\pi$
$98$	0	0
$99$	8.92901	0.897399
$100$	0	0
$101$	4.74696	0.472340	0.236170	−	0.971712i	$-0.424108\pi$
$101$	4.74696	0.472340	0.236170	+	0.971712i	$0.424108\pi$
$102$	0	0
$103$	−5.05333	−0.497919	−0.248960	−	0.968514i	$-0.580089\pi$
$103$	−5.05333	−0.497919	−0.248960	+	0.968514i	$0.580089\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	10.1067	0.968042	0.484021	−	0.875056i	$-0.339176\pi$
$109$	10.1067	0.968042	0.484021	+	0.875056i	$0.339176\pi$
$110$	0	0
$111$	−3.78292	−0.359059
$112$	0	0
$113$	−0.335173	−0.0315304	−0.0157652	−	0.999876i	$-0.505018\pi$
$113$	−0.335173	−0.0315304	−0.0157652	+	0.999876i	$0.505018\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−18.2758	−1.68960
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−1.46389	−0.133081
$122$	0	0
$123$	−1.89146	−0.170547
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−5.92047	−0.525357	−0.262679	−	0.964883i	$-0.584606\pi$
$127$	−5.92047	−0.525357	−0.262679	+	0.964883i	$0.584606\pi$
$128$	0	0
$129$	−0.575540	−0.0506735
$130$	0	0
$131$	7.02309	0.613610	0.306805	−	0.951772i	$-0.400740\pi$
$131$	7.02309	0.613610	0.306805	+	0.951772i	$0.400740\pi$
$132$	0	0
$133$	−28.8533	−2.50190
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.1414	−1.03731	−0.518654	−	0.854984i	$-0.673567\pi$
$137$	−12.1414	−1.03731	−0.518654	+	0.854984i	$0.673567\pi$
$138$	0	0
$139$	−14.2499	−1.20866	−0.604331	−	0.796733i	$-0.706560\pi$
$139$	−14.2499	−1.20866	−0.604331	+	0.796733i	$0.706560\pi$
$140$	0	0
$141$	−0.741241	−0.0624238
$142$	0	0
$143$	−19.5184	−1.63221
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.15303	0.260058
$148$	0	0
$149$	−18.7826	−1.53873	−0.769366	−	0.638808i	$-0.779428\pi$
$149$	−18.7826	−1.53873	−0.769366	+	0.638808i	$0.779428\pi$
$150$	0	0
$151$	12.3566	1.00556	0.502782	−	0.864413i	$-0.332310\pi$
$151$	12.3566	1.00556	0.502782	+	0.864413i	$0.332310\pi$
$152$	0	0
$153$	2.84125	0.229701
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.9997	1.03749	0.518744	−	0.854929i	$-0.326400\pi$
$157$	12.9997	1.03749	0.518744	+	0.854929i	$0.326400\pi$
$158$	0	0
$159$	−0.434154	−0.0344306
$160$	0	0
$161$	−4.07069	−0.320816
$162$	0	0
$163$	12.5384	0.982085	0.491042	−	0.871136i	$-0.336616\pi$
$163$	12.5384	0.982085	0.491042	+	0.871136i	$0.336616\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.2121	0.867617	0.433808	−	0.901005i	$-0.357169\pi$
$167$	11.2121	0.867617	0.433808	+	0.901005i	$0.357169\pi$
$168$	0	0
$169$	26.9502	2.07309
$170$	0	0
$171$	20.4949	1.56728
$172$	0	0
$173$	−17.2133	−1.30870	−0.654352	−	0.756190i	$-0.727058\pi$
$173$	−17.2133	−1.30870	−0.654352	+	0.756190i	$0.727058\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.61222	0.196347
$178$	0	0
$179$	−25.1859	−1.88248	−0.941240	−	0.337737i	$-0.890339\pi$
$179$	−25.1859	−1.88248	−0.941240	+	0.337737i	$0.890339\pi$
$180$	0	0
$181$	15.3411	1.14029	0.570147	−	0.821542i	$-0.306886\pi$
$181$	15.3411	1.14029	0.570147	+	0.821542i	$0.306886\pi$
$182$	0	0
$183$	3.13546	0.231780
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.03443	0.221900
$188$	0	0
$189$	−7.90102	−0.574715
$190$	0	0
$191$	−17.2232	−1.24623	−0.623114	−	0.782131i	$-0.714133\pi$
$191$	−17.2232	−1.24623	−0.623114	+	0.782131i	$0.714133\pi$
$192$	0	0
$193$	9.30064	0.669475	0.334737	−	0.942311i	$-0.391352\pi$
$193$	9.30064	0.669475	0.334737	+	0.942311i	$0.391352\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.8938	1.63111	0.815557	−	0.578677i	$-0.196431\pi$
$197$	22.8938	1.63111	0.815557	+	0.578677i	$0.196431\pi$
$198$	0	0
$199$	−6.35969	−0.450827	−0.225413	−	0.974263i	$-0.572373\pi$
$199$	−6.35969	−0.450827	−0.225413	+	0.974263i	$0.572373\pi$
$200$	0	0
$201$	1.10073	0.0776395
$202$	0	0
$203$	−31.0704	−2.18071
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.89146	0.200970
$208$	0	0
$209$	21.8883	1.51405
$210$	0	0
$211$	−10.3656	−0.713598	−0.356799	−	0.934181i	$-0.616132\pi$
$211$	−10.3656	−0.713598	−0.356799	+	0.934181i	$0.616132\pi$
$212$	0	0
$213$	−1.29349	−0.0886286
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−24.1004	−1.63604
$218$	0	0
$219$	0.549339	0.0371209
$220$	0	0
$221$	−6.21085	−0.417787
$222$	0	0
$223$	5.75289	0.385242	0.192621	−	0.981273i	$-0.438301\pi$
$223$	5.75289	0.385242	0.192621	+	0.981273i	$0.438301\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.49393	−0.630134	−0.315067	−	0.949069i	$-0.602027\pi$
$227$	−9.49393	−0.630134	−0.315067	+	0.949069i	$0.602027\pi$
$228$	0	0
$229$	1.83532	0.121282	0.0606408	−	0.998160i	$-0.480686\pi$
$229$	1.83532	0.121282	0.0606408	+	0.998160i	$0.480686\pi$
$230$	0	0
$231$	−4.14139	−0.272483
$232$	0	0
$233$	20.5687	1.34750	0.673749	−	0.738960i	$-0.264683\pi$
$233$	20.5687	1.34750	0.673749	+	0.738960i	$0.264683\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−5.24589	−0.340757
$238$	0	0
$239$	−12.4014	−0.802178	−0.401089	−	0.916039i	$-0.631368\pi$
$239$	−12.4014	−0.802178	−0.401089	+	0.916039i	$0.631368\pi$
$240$	0	0
$241$	15.7829	1.01667	0.508334	−	0.861160i	$-0.330262\pi$
$241$	15.7829	1.01667	0.508334	+	0.861160i	$0.330262\pi$
$242$	0	0
$243$	8.46998	0.543349
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−44.8009	−2.85061
$248$	0	0
$249$	3.92431	0.248693
$250$	0	0
$251$	−9.71662	−0.613308	−0.306654	−	0.951821i	$-0.599209\pi$
$251$	−9.71662	−0.613308	−0.306654	+	0.951821i	$0.599209\pi$
$252$	0	0
$253$	3.08806	0.194145
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.3115	1.20462	0.602309	−	0.798263i	$-0.294248\pi$
$257$	19.3115	1.20462	0.602309	+	0.798263i	$0.294248\pi$
$258$	0	0
$259$	−46.7417	−2.90439
$260$	0	0
$261$	22.0697	1.36608
$262$	0	0
$263$	1.74819	0.107798	0.0538990	−	0.998546i	$-0.482835\pi$
$263$	1.74819	0.107798	0.0538990	+	0.998546i	$0.482835\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3.45919	−0.211699
$268$	0	0
$269$	11.7079	0.713843	0.356921	−	0.934134i	$-0.383826\pi$
$269$	11.7079	0.713843	0.356921	+	0.934134i	$0.383826\pi$
$270$	0	0
$271$	10.1051	0.613843	0.306921	−	0.951735i	$-0.400701\pi$
$271$	10.1051	0.613843	0.306921	+	0.951735i	$0.400701\pi$
$272$	0	0
$273$	8.47656	0.513025
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.29319	0.318037	0.159018	−	0.987276i	$-0.449167\pi$
$277$	5.29319	0.318037	0.159018	+	0.987276i	$0.449167\pi$
$278$	0	0
$279$	17.1188	1.02488
$280$	0	0
$281$	−13.8062	−0.823610	−0.411805	−	0.911272i	$-0.635101\pi$
$281$	−13.8062	−0.823610	−0.411805	+	0.911272i	$0.635101\pi$
$282$	0	0
$283$	7.79437	0.463327	0.231663	−	0.972796i	$-0.425583\pi$
$283$	7.79437	0.463327	0.231663	+	0.972796i	$0.425583\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−23.3708	−1.37954
$288$	0	0
$289$	−16.0344	−0.943202
$290$	0	0
$291$	−0.936054	−0.0548724
$292$	0	0
$293$	−24.0171	−1.40309	−0.701546	−	0.712624i	$-0.747506\pi$
$293$	−24.0171	−1.40309	−0.701546	+	0.712624i	$0.747506\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.99377	0.347794
$298$	0	0
$299$	−6.32062	−0.365531
$300$	0	0
$301$	−7.11135	−0.409891
$302$	0	0
$303$	1.56389	0.0898434
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	−1.66483	−0.0947087
$310$	0	0
$311$	−26.8507	−1.52256	−0.761282	−	0.648421i	$-0.775430\pi$
$311$	−26.8507	−1.52256	−0.761282	+	0.648421i	$0.775430\pi$
$312$	0	0
$313$	31.5110	1.78111	0.890553	−	0.454879i	$-0.150318\pi$
$313$	31.5110	1.78111	0.890553	+	0.454879i	$0.150318\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−22.1110	−1.24188	−0.620940	−	0.783858i	$-0.713249\pi$
$317$	−22.1110	−1.24188	−0.620940	+	0.783858i	$0.713249\pi$
$318$	0	0
$319$	23.5702	1.31968
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.96497	0.387541
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.32965	0.184130
$328$	0	0
$329$	−9.15875	−0.504938
$330$	0	0
$331$	23.8953	1.31340	0.656702	−	0.754150i	$-0.271951\pi$
$331$	23.8953	1.31340	0.656702	+	0.754150i	$0.271951\pi$
$332$	0	0
$333$	33.2012	1.81941
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−20.1647	−1.09844	−0.549220	−	0.835678i	$-0.685075\pi$
$337$	−20.1647	−1.09844	−0.549220	+	0.835678i	$0.685075\pi$
$338$	0	0
$339$	−0.110423	−0.00599737
$340$	0	0
$341$	18.2828	0.990068
$342$	0	0
$343$	10.4639	0.564997
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−32.2465	−1.73108	−0.865542	−	0.500837i	$-0.833026\pi$
$347$	−32.2465	−1.73108	−0.865542	+	0.500837i	$0.833026\pi$
$348$	0	0
$349$	35.3203	1.89065	0.945327	−	0.326125i	$-0.105743\pi$
$349$	35.3203	1.89065	0.945327	+	0.326125i	$0.105743\pi$
$350$	0	0
$351$	−12.2680	−0.654818
$352$	0	0
$353$	−30.7409	−1.63618	−0.818088	−	0.575094i	$-0.804966\pi$
$353$	−30.7409	−1.63618	−0.818088	+	0.575094i	$0.804966\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.31781	−0.0697457
$358$	0	0
$359$	−34.4053	−1.81584	−0.907920	−	0.419143i	$-0.862331\pi$
$359$	−34.4053	−1.81584	−0.907920	+	0.419143i	$0.862331\pi$
$360$	0	0
$361$	31.2406	1.64424
$362$	0	0
$363$	−0.482281	−0.0253132
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−10.2524	−0.535173	−0.267586	−	0.963534i	$-0.586226\pi$
$367$	−10.2524	−0.535173	−0.267586	+	0.963534i	$0.586226\pi$
$368$	0	0
$369$	16.6006	0.864192
$370$	0	0
$371$	−5.36439	−0.278505
$372$	0	0
$373$	22.6993	1.17532	0.587662	−	0.809107i	$-0.300049\pi$
$373$	22.6993	1.17532	0.587662	+	0.809107i	$0.300049\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−48.2434	−2.48466
$378$	0	0
$379$	−1.03503	−0.0531661	−0.0265831	−	0.999647i	$-0.508463\pi$
$379$	−1.03503	−0.0531661	−0.0265831	+	0.999647i	$0.508463\pi$
$380$	0	0
$381$	−1.95051	−0.0999276
$382$	0	0
$383$	−25.8707	−1.32193	−0.660965	−	0.750417i	$-0.729853\pi$
$383$	−25.8707	−1.32193	−0.660965	+	0.750417i	$0.729853\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.05128	0.256771
$388$	0	0
$389$	−9.21678	−0.467309	−0.233655	−	0.972320i	$-0.575068\pi$
$389$	−9.21678	−0.467309	−0.233655	+	0.972320i	$0.575068\pi$
$390$	0	0
$391$	0.982634	0.0496939
$392$	0	0
$393$	2.31377	0.116714
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−8.84825	−0.444081	−0.222040	−	0.975037i	$-0.571272\pi$
$397$	−8.84825	−0.444081	−0.222040	+	0.975037i	$0.571272\pi$
$398$	0	0
$399$	−9.50577	−0.475884
$400$	0	0
$401$	11.7281	0.585672	0.292836	−	0.956163i	$-0.405401\pi$
$401$	11.7281	0.585672	0.292836	+	0.956163i	$0.405401\pi$
$402$	0	0
$403$	−37.4210	−1.86408
$404$	0	0
$405$	0	0
$406$	0	0
$407$	35.4586	1.75762
$408$	0	0
$409$	−22.8709	−1.13089	−0.565446	−	0.824785i	$-0.691296\pi$
$409$	−22.8709	−1.13089	−0.565446	+	0.824785i	$0.691296\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	0	0
$413$	32.2765	1.58823
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.69466	−0.229899
$418$	0	0
$419$	2.41771	0.118113	0.0590565	−	0.998255i	$-0.481191\pi$
$419$	2.41771	0.118113	0.0590565	+	0.998255i	$0.481191\pi$
$420$	0	0
$421$	−16.6583	−0.811876	−0.405938	−	0.913901i	$-0.633055\pi$
$421$	−16.6583	−0.811876	−0.405938	+	0.913901i	$0.633055\pi$
$422$	0	0
$423$	6.50557	0.316312
$424$	0	0
$425$	0	0
$426$	0	0
$427$	38.7417	1.87484
$428$	0	0
$429$	−6.43038	−0.310462
$430$	0	0
$431$	7.51722	0.362092	0.181046	−	0.983475i	$-0.442052\pi$
$431$	7.51722	0.362092	0.181046	+	0.983475i	$0.442052\pi$
$432$	0	0
$433$	28.6003	1.37444	0.687221	−	0.726449i	$-0.258830\pi$
$433$	28.6003	1.37444	0.687221	+	0.726449i	$0.258830\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.08806	0.339068
$438$	0	0
$439$	−8.52886	−0.407060	−0.203530	−	0.979069i	$-0.565241\pi$
$439$	−8.52886	−0.407060	−0.203530	+	0.979069i	$0.565241\pi$
$440$	0	0
$441$	−27.6729	−1.31776
$442$	0	0
$443$	−35.9224	−1.70673	−0.853363	−	0.521317i	$-0.825441\pi$
$443$	−35.9224	−1.70673	−0.853363	+	0.521317i	$0.825441\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−6.18797	−0.292681
$448$	0	0
$449$	5.11165	0.241234	0.120617	−	0.992699i	$-0.461513\pi$
$449$	5.11165	0.241234	0.120617	+	0.992699i	$0.461513\pi$
$450$	0	0
$451$	17.7293	0.834840
$452$	0	0
$453$	4.07090	0.191267
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11.7944	−0.551717	−0.275859	−	0.961198i	$-0.588962\pi$
$457$	−11.7944	−0.551717	−0.275859	+	0.961198i	$0.588962\pi$
$458$	0	0
$459$	1.90724	0.0890226
$460$	0	0
$461$	15.7079	0.731589	0.365795	−	0.930696i	$-0.380797\pi$
$461$	15.7079	0.731589	0.365795	+	0.930696i	$0.380797\pi$
$462$	0	0
$463$	−35.7752	−1.66261	−0.831307	−	0.555814i	$-0.812407\pi$
$463$	−35.7752	−1.66261	−0.831307	+	0.555814i	$0.812407\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.2118	0.981564	0.490782	−	0.871282i	$-0.336711\pi$
$467$	21.2118	0.981564	0.490782	+	0.871282i	$0.336711\pi$
$468$	0	0
$469$	13.6006	0.628016
$470$	0	0
$471$	4.28277	0.197340
$472$	0	0
$473$	5.39472	0.248050
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.81039	0.174466
$478$	0	0
$479$	24.6077	1.12436	0.562178	−	0.827016i	$-0.309964\pi$
$479$	24.6077	1.12436	0.562178	+	0.827016i	$0.309964\pi$
$480$	0	0
$481$	−72.5764	−3.30920
$482$	0	0
$483$	−1.34110	−0.0610220
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−15.3507	−0.695605	−0.347802	−	0.937568i	$-0.613072\pi$
$487$	−15.3507	−0.695605	−0.347802	+	0.937568i	$0.613072\pi$
$488$	0	0
$489$	4.13080	0.186801
$490$	0	0
$491$	−13.1430	−0.593134	−0.296567	−	0.955012i	$-0.595842\pi$
$491$	−13.1430	−0.593134	−0.296567	+	0.955012i	$0.595842\pi$
$492$	0	0
$493$	7.50015	0.337790
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−15.9823	−0.716906
$498$	0	0
$499$	−20.6841	−0.925949	−0.462975	−	0.886371i	$-0.653218\pi$
$499$	−20.6841	−0.925949	−0.462975	+	0.886371i	$0.653218\pi$
$500$	0	0
$501$	3.69384	0.165029
$502$	0	0
$503$	−35.7956	−1.59605	−0.798023	−	0.602627i	$-0.794121\pi$
$503$	−35.7956	−1.59605	−0.798023	+	0.602627i	$0.794121\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.87879	0.394321
$508$	0	0
$509$	33.9242	1.50366	0.751832	−	0.659355i	$-0.229170\pi$
$509$	33.9242	1.50366	0.751832	+	0.659355i	$0.229170\pi$
$510$	0	0
$511$	6.78762	0.300267
$512$	0	0
$513$	13.7576	0.607412
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.94790	0.305568
$518$	0	0
$519$	−5.67095	−0.248927
$520$	0	0
$521$	12.3346	0.540387	0.270194	−	0.962806i	$-0.412912\pi$
$521$	12.3346	0.540387	0.270194	+	0.962806i	$0.412912\pi$
$522$	0	0
$523$	5.25274	0.229686	0.114843	−	0.993384i	$-0.463363\pi$
$523$	5.25274	0.229686	0.114843	+	0.993384i	$0.463363\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.81766	0.253421
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−22.9264	−0.994922
$532$	0	0
$533$	−36.2882	−1.57182
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−8.29753	−0.358065
$538$	0	0
$539$	−29.5544	−1.27300
$540$	0	0
$541$	−28.4715	−1.22408	−0.612042	−	0.790825i	$-0.709652\pi$
$541$	−28.4715	−1.22408	−0.612042	+	0.790825i	$0.709652\pi$
$542$	0	0
$543$	5.05415	0.216894
$544$	0	0
$545$	0	0
$546$	0	0
$547$	25.6025	1.09468	0.547341	−	0.836910i	$-0.315640\pi$
$547$	25.6025	1.09468	0.547341	+	0.836910i	$0.315640\pi$
$548$	0	0
$549$	−27.5187	−1.17447
$550$	0	0
$551$	54.1011	2.30478
$552$	0	0
$553$	−64.8180	−2.75634
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15.7938	0.669203	0.334602	−	0.942360i	$-0.391398\pi$
$557$	15.7938	0.669203	0.334602	+	0.942360i	$0.391398\pi$
$558$	0	0
$559$	−11.0419	−0.467022
$560$	0	0
$561$	0.999698	0.0422073
$562$	0	0
$563$	41.8996	1.76586	0.882929	−	0.469507i	$-0.155568\pi$
$563$	41.8996	1.76586	0.882929	+	0.469507i	$0.155568\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	32.7078	1.37360
$568$	0	0
$569$	24.5485	1.02913	0.514563	−	0.857453i	$-0.327954\pi$
$569$	24.5485	1.02913	0.514563	+	0.857453i	$0.327954\pi$
$570$	0	0
$571$	−38.3231	−1.60377	−0.801886	−	0.597476i	$-0.796170\pi$
$571$	−38.3231	−1.60377	−0.801886	+	0.597476i	$0.796170\pi$
$572$	0	0
$573$	−5.67422	−0.237044
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.9288	0.996169	0.498085	−	0.867128i	$-0.334037\pi$
$577$	23.9288	0.996169	0.498085	+	0.867128i	$0.334037\pi$
$578$	0	0
$579$	3.06411	0.127340
$580$	0	0
$581$	48.4886	2.01165
$582$	0	0
$583$	4.06947	0.168540
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−34.8130	−1.43689	−0.718443	−	0.695585i	$-0.755145\pi$
$587$	−34.8130	−1.43689	−0.718443	+	0.695585i	$0.755145\pi$
$588$	0	0
$589$	41.9647	1.72912
$590$	0	0
$591$	7.54239	0.310252
$592$	0	0
$593$	−15.5702	−0.639393	−0.319697	−	0.947520i	$-0.603581\pi$
$593$	−15.5702	−0.639393	−0.319697	+	0.947520i	$0.603581\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.09521	−0.0857513
$598$	0	0
$599$	31.2121	1.27529	0.637645	−	0.770330i	$-0.279908\pi$
$599$	31.2121	1.27529	0.637645	+	0.770330i	$0.279908\pi$
$600$	0	0
$601$	31.2553	1.27493	0.637466	−	0.770478i	$-0.279983\pi$
$601$	31.2553	1.27493	0.637466	+	0.770478i	$0.279983\pi$
$602$	0	0
$603$	−9.66065	−0.393412
$604$	0	0
$605$	0	0
$606$	0	0
$607$	30.6469	1.24392	0.621959	−	0.783050i	$-0.286337\pi$
$607$	30.6469	1.24392	0.621959	+	0.783050i	$0.286337\pi$
$608$	0	0
$609$	−10.2362	−0.414791
$610$	0	0
$611$	−14.2209	−0.575317
$612$	0	0
$613$	0.459494	0.0185588	0.00927940	−	0.999957i	$-0.497046\pi$
$613$	0.459494	0.0185588	0.00927940	+	0.999957i	$0.497046\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−23.6010	−0.950140	−0.475070	−	0.879948i	$-0.657577\pi$
$617$	−23.6010	−0.950140	−0.475070	+	0.879948i	$0.657577\pi$
$618$	0	0
$619$	24.8893	1.00038	0.500192	−	0.865914i	$-0.333263\pi$
$619$	24.8893	1.00038	0.500192	+	0.865914i	$0.333263\pi$
$620$	0	0
$621$	1.94095	0.0778877
$622$	0	0
$623$	−42.7417	−1.71241
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.21115	0.287986
$628$	0	0
$629$	11.2831	0.449886
$630$	0	0
$631$	−20.8918	−0.831690	−0.415845	−	0.909435i	$-0.636514\pi$
$631$	−20.8918	−0.831690	−0.415845	+	0.909435i	$0.636514\pi$
$632$	0	0
$633$	−3.41497	−0.135733
$634$	0	0
$635$	0	0
$636$	0	0
$637$	60.4917	2.39677
$638$	0	0
$639$	11.3524	0.449096
$640$	0	0
$641$	−16.0056	−0.632184	−0.316092	−	0.948729i	$-0.602371\pi$
$641$	−16.0056	−0.632184	−0.316092	+	0.948729i	$0.602371\pi$
$642$	0	0
$643$	15.0242	0.592497	0.296249	−	0.955111i	$-0.404264\pi$
$643$	15.0242	0.592497	0.296249	+	0.955111i	$0.404264\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	45.3791	1.78404	0.892018	−	0.452001i	$-0.149290\pi$
$647$	45.3791	1.78404	0.892018	+	0.452001i	$0.149290\pi$
$648$	0	0
$649$	−24.4852	−0.961130
$650$	0	0
$651$	−7.93993	−0.311190
$652$	0	0
$653$	14.2506	0.557671	0.278835	−	0.960339i	$-0.410052\pi$
$653$	14.2506	0.557671	0.278835	+	0.960339i	$0.410052\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4.82133	−0.188098
$658$	0	0
$659$	−10.3535	−0.403314	−0.201657	−	0.979456i	$-0.564633\pi$
$659$	−10.3535	−0.403314	−0.201657	+	0.979456i	$0.564633\pi$
$660$	0	0
$661$	26.4651	1.02937	0.514687	−	0.857378i	$-0.327908\pi$
$661$	26.4651	1.02937	0.514687	+	0.857378i	$0.327908\pi$
$662$	0	0
$663$	−2.04618	−0.0794669
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.63270	0.295539
$668$	0	0
$669$	1.89530	0.0732765
$670$	0	0
$671$	−29.3897	−1.13458
$672$	0	0
$673$	12.0587	0.464831	0.232415	−	0.972617i	$-0.425337\pi$
$673$	12.0587	0.464831	0.232415	+	0.972617i	$0.425337\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.54878	−0.213257	−0.106629	−	0.994299i	$-0.534006\pi$
$677$	−5.54878	−0.213257	−0.106629	+	0.994299i	$0.534006\pi$
$678$	0	0
$679$	−11.5658	−0.443856
$680$	0	0
$681$	−3.12779	−0.119857
$682$	0	0
$683$	0.397032	0.0151920	0.00759601	−	0.999971i	$-0.497582\pi$
$683$	0.397032	0.0151920	0.00759601	+	0.999971i	$0.497582\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0.604651	0.0230689
$688$	0	0
$689$	−8.32935	−0.317323
$690$	0	0
$691$	21.4949	0.817703	0.408851	−	0.912601i	$-0.365929\pi$
$691$	21.4949	0.817703	0.408851	+	0.912601i	$0.365929\pi$
$692$	0	0
$693$	36.3472	1.38072
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.64154	0.213688
$698$	0	0
$699$	6.77638	0.256306
$700$	0	0
$701$	44.6027	1.68462	0.842311	−	0.538992i	$-0.181195\pi$
$701$	44.6027	1.68462	0.842311	+	0.538992i	$0.181195\pi$
$702$	0	0
$703$	81.3885	3.06963
$704$	0	0
$705$	0	0
$706$	0	0
$707$	19.3234	0.726732
$708$	0	0
$709$	29.7476	1.11719	0.558597	−	0.829439i	$-0.311340\pi$
$709$	29.7476	1.11719	0.558597	+	0.829439i	$0.311340\pi$
$710$	0	0
$711$	46.0410	1.72667
$712$	0	0
$713$	5.92047	0.221723
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−4.08565	−0.152582
$718$	0	0
$719$	35.1758	1.31184	0.655918	−	0.754832i	$-0.272282\pi$
$719$	35.1758	1.31184	0.655918	+	0.754832i	$0.272282\pi$
$720$	0	0
$721$	−20.5705	−0.766087
$722$	0	0
$723$	5.19971	0.193379
$724$	0	0
$725$	0	0
$726$	0	0
$727$	27.6030	1.02374	0.511870	−	0.859063i	$-0.328953\pi$
$727$	27.6030	1.02374	0.511870	+	0.859063i	$0.328953\pi$
$728$	0	0
$729$	−21.3144	−0.789421
$730$	0	0
$731$	1.71662	0.0634916
$732$	0	0
$733$	−51.0939	−1.88720	−0.943598	−	0.331093i	$-0.892583\pi$
$733$	−51.0939	−1.88720	−0.943598	+	0.331093i	$0.892583\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.3175	−0.380050
$738$	0	0
$739$	13.5558	0.498658	0.249329	−	0.968419i	$-0.419790\pi$
$739$	13.5558	0.498658	0.249329	+	0.968419i	$0.419790\pi$
$740$	0	0
$741$	−14.7597	−0.542212
$742$	0	0
$743$	53.8006	1.97375	0.986877	−	0.161477i	$-0.0516257\pi$
$743$	53.8006	1.97375	0.986877	+	0.161477i	$0.0516257\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−34.4420	−1.26017
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−44.6539	−1.62944	−0.814722	−	0.579852i	$-0.803111\pi$
$751$	−44.6539	−1.62944	−0.814722	+	0.579852i	$0.803111\pi$
$752$	0	0
$753$	−3.20116	−0.116657
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−25.1581	−0.914388	−0.457194	−	0.889367i	$-0.651146\pi$
$757$	−25.1581	−0.914388	−0.457194	+	0.889367i	$0.651146\pi$
$758$	0	0
$759$	1.01737	0.0369281
$760$	0	0
$761$	−22.9387	−0.831526	−0.415763	−	0.909473i	$-0.636485\pi$
$761$	−22.9387	−0.831526	−0.415763	+	0.909473i	$0.636485\pi$
$762$	0	0
$763$	41.1411	1.48941
$764$	0	0
$765$	0	0
$766$	0	0
$767$	50.1162	1.80959
$768$	0	0
$769$	−35.7337	−1.28859	−0.644295	−	0.764777i	$-0.722849\pi$
$769$	−35.7337	−1.28859	−0.644295	+	0.764777i	$0.722849\pi$
$770$	0	0
$771$	6.36220	0.229129
$772$	0	0
$773$	10.9587	0.394158	0.197079	−	0.980388i	$-0.436854\pi$
$773$	10.9587	0.394158	0.197079	+	0.980388i	$0.436854\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−15.3991	−0.552440
$778$	0	0
$779$	40.6943	1.45802
$780$	0	0
$781$	12.1243	0.433842
$782$	0	0
$783$	14.8147	0.529435
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−3.56216	−0.126977	−0.0634887	−	0.997983i	$-0.520223\pi$
$787$	−3.56216	−0.126977	−0.0634887	+	0.997983i	$0.520223\pi$
$788$	0	0
$789$	0.575944	0.0205042
$790$	0	0
$791$	−1.36439	−0.0485120
$792$	0	0
$793$	60.1547	2.13616
$794$	0	0
$795$	0	0
$796$	0	0
$797$	44.7128	1.58381	0.791903	−	0.610646i	$-0.209090\pi$
$797$	44.7128	1.58381	0.791903	+	0.610646i	$0.209090\pi$
$798$	0	0
$799$	2.21085	0.0782143
$800$	0	0
$801$	30.3599	1.07271
$802$	0	0
$803$	−5.14914	−0.181709
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.85718	0.135779
$808$	0	0
$809$	9.68159	0.340387	0.170193	−	0.985411i	$-0.445561\pi$
$809$	9.68159	0.340387	0.170193	+	0.985411i	$0.445561\pi$
$810$	0	0
$811$	−35.5981	−1.25002	−0.625009	−	0.780618i	$-0.714905\pi$
$811$	−35.5981	−1.25002	−0.625009	+	0.780618i	$0.714905\pi$
$812$	0	0
$813$	3.32915	0.116758
$814$	0	0
$815$	0	0
$816$	0	0
$817$	12.3826	0.433212
$818$	0	0
$819$	−74.3953	−2.59958
$820$	0	0
$821$	8.17888	0.285445	0.142722	−	0.989763i	$-0.454414\pi$
$821$	8.17888	0.285445	0.142722	+	0.989763i	$0.454414\pi$
$822$	0	0
$823$	1.50587	0.0524914	0.0262457	−	0.999656i	$-0.491645\pi$
$823$	1.50587	0.0524914	0.0262457	+	0.999656i	$0.491645\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.3652	0.882035	0.441017	−	0.897499i	$-0.354618\pi$
$827$	25.3652	0.882035	0.441017	+	0.897499i	$0.354618\pi$
$828$	0	0
$829$	38.2130	1.32719	0.663596	−	0.748091i	$-0.269029\pi$
$829$	38.2130	1.32719	0.663596	+	0.748091i	$0.269029\pi$
$830$	0	0
$831$	1.74385	0.0604935
$832$	0	0
$833$	−9.40434	−0.325841
$834$	0	0
$835$	0	0
$836$	0	0
$837$	11.4914	0.397199
$838$	0	0
$839$	−13.1575	−0.454248	−0.227124	−	0.973866i	$-0.572932\pi$
$839$	−13.1575	−0.454248	−0.227124	+	0.973866i	$0.572932\pi$
$840$	0	0
$841$	29.2582	1.00890
$842$	0	0
$843$	−4.54848	−0.156658
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5.95904	−0.204755
$848$	0	0
$849$	2.56787	0.0881290
$850$	0	0
$851$	11.4825	0.393614
$852$	0	0
$853$	−13.2756	−0.454549	−0.227274	−	0.973831i	$-0.572981\pi$
$853$	−13.2756	−0.454549	−0.227274	+	0.973831i	$0.572981\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.18894	−0.0747729	−0.0373864	−	0.999301i	$-0.511903\pi$
$857$	−2.18894	−0.0747729	−0.0373864	+	0.999301i	$0.511903\pi$
$858$	0	0
$859$	21.4092	0.730472	0.365236	−	0.930915i	$-0.380988\pi$
$859$	21.4092	0.730472	0.365236	+	0.930915i	$0.380988\pi$
$860$	0	0
$861$	−7.69956	−0.262400
$862$	0	0
$863$	−14.2923	−0.486514	−0.243257	−	0.969962i	$-0.578216\pi$
$863$	−14.2923	−0.486514	−0.243257	+	0.969962i	$0.578216\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−5.28257	−0.179405
$868$	0	0
$869$	49.1714	1.66803
$870$	0	0
$871$	21.1178	0.715549
$872$	0	0
$873$	8.21536	0.278048
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−22.7835	−0.769346	−0.384673	−	0.923053i	$-0.625686\pi$
$877$	−22.7835	−0.769346	−0.384673	+	0.923053i	$0.625686\pi$
$878$	0	0
$879$	−7.91246	−0.266881
$880$	0	0
$881$	−43.1007	−1.45210	−0.726050	−	0.687642i	$-0.758646\pi$
$881$	−43.1007	−1.45210	−0.726050	+	0.687642i	$0.758646\pi$
$882$	0	0
$883$	−9.35101	−0.314687	−0.157343	−	0.987544i	$-0.550293\pi$
$883$	−9.35101	−0.314687	−0.157343	+	0.987544i	$0.550293\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22.9902	−0.771936	−0.385968	−	0.922512i	$-0.626132\pi$
$887$	−22.9902	−0.771936	−0.385968	+	0.922512i	$0.626132\pi$
$888$	0	0
$889$	−24.1004	−0.808302
$890$	0	0
$891$	−24.8124	−0.831245
$892$	0	0
$893$	15.9476	0.533666
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.08234	−0.0695272
$898$	0	0
$899$	45.1892	1.50714
$900$	0	0
$901$	1.29492	0.0431401
$902$	0	0
$903$	−2.34285	−0.0779650
$904$	0	0
$905$	0	0
$906$	0	0
$907$	29.3070	0.973123	0.486561	−	0.873646i	$-0.338251\pi$
$907$	29.3070	0.973123	0.486561	+	0.873646i	$0.338251\pi$
$908$	0	0
$909$	−13.7257	−0.455251
$910$	0	0
$911$	−52.8533	−1.75111	−0.875554	−	0.483120i	$-0.839504\pi$
$911$	−52.8533	−1.75111	−0.875554	+	0.483120i	$0.839504\pi$
$912$	0	0
$913$	−36.7838	−1.21737
$914$	0	0
$915$	0	0
$916$	0	0
$917$	28.5888	0.944086
$918$	0	0
$919$	−27.3696	−0.902840	−0.451420	−	0.892312i	$-0.649082\pi$
$919$	−27.3696	−0.902840	−0.451420	+	0.892312i	$0.649082\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−24.8160	−0.816828
$924$	0	0
$925$	0	0
$926$	0	0
$927$	14.6115	0.479905
$928$	0	0
$929$	4.64485	0.152393	0.0761963	−	0.997093i	$-0.475722\pi$
$929$	4.64485	0.152393	0.0761963	+	0.997093i	$0.475722\pi$
$930$	0	0
$931$	−67.8366	−2.22325
$932$	0	0
$933$	−8.84601	−0.289605
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−19.6704	−0.642606	−0.321303	−	0.946977i	$-0.604121\pi$
$937$	−19.6704	−0.642606	−0.321303	+	0.946977i	$0.604121\pi$
$938$	0	0
$939$	10.3813	0.338783
$940$	0	0
$941$	−7.31259	−0.238384	−0.119192	−	0.992871i	$-0.538030\pi$
$941$	−7.31259	−0.238384	−0.119192	+	0.992871i	$0.538030\pi$
$942$	0	0
$943$	5.74124	0.186961
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−36.3846	−1.18234	−0.591170	−	0.806547i	$-0.701334\pi$
$947$	−36.3846	−1.18234	−0.591170	+	0.806547i	$0.701334\pi$
$948$	0	0
$949$	10.5392	0.342118
$950$	0	0
$951$	−7.28452	−0.236217
$952$	0	0
$953$	22.2456	0.720605	0.360303	−	0.932835i	$-0.382673\pi$
$953$	22.2456	0.720605	0.360303	+	0.932835i	$0.382673\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7.76526	0.251015
$958$	0	0
$959$	−49.4239	−1.59598
$960$	0	0
$961$	4.05200	0.130710
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−49.4661	−1.59072	−0.795361	−	0.606136i	$-0.792719\pi$
$967$	−49.4661	−1.59072	−0.795361	+	0.606136i	$0.792719\pi$
$968$	0	0
$969$	2.29462	0.0737138
$970$	0	0
$971$	28.9113	0.927809	0.463904	−	0.885885i	$-0.346448\pi$
$971$	28.9113	0.927809	0.463904	+	0.885885i	$0.346448\pi$
$972$	0	0
$973$	−58.0071	−1.85962
$974$	0	0
$975$	0	0
$976$	0	0
$977$	55.1228	1.76354	0.881768	−	0.471684i	$-0.156354\pi$
$977$	55.1228	1.76354	0.881768	+	0.471684i	$0.156354\pi$
$978$	0	0
$979$	32.4242	1.03628
$980$	0	0
$981$	−29.2230	−0.933019
$982$	0	0
$983$	−26.6824	−0.851037	−0.425518	−	0.904950i	$-0.639908\pi$
$983$	−26.6824	−0.851037	−0.425518	+	0.904950i	$0.639908\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3.01737	−0.0960438
$988$	0	0
$989$	1.74696	0.0555502
$990$	0	0
$991$	−25.7063	−0.816588	−0.408294	−	0.912851i	$-0.633876\pi$
$991$	−25.7063	−0.816588	−0.408294	+	0.912851i	$0.633876\pi$
$992$	0	0
$993$	7.87235	0.249821
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.95372	0.0618750	0.0309375	−	0.999521i	$-0.490151\pi$
$997$	1.95372	0.0618750	0.0309375	+	0.999521i	$0.490151\pi$
$998$	0	0
$999$	22.2869	0.705128

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.a.bb.1.2	yes	4
4.3	odd	2		9200.2.a.cn.1.3		4
5.2	odd	4		4600.2.e.t.4049.4		8
5.3	odd	4		4600.2.e.t.4049.5		8
5.4	even	2		4600.2.a.ba.1.3	✓	4
20.19	odd	2		9200.2.a.cp.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.ba.1.3	✓	4	5.4	even	2
4600.2.a.bb.1.2	yes	4	1.1	even	1	trivial
4600.2.e.t.4049.4		8	5.2	odd	4
4600.2.e.t.4049.5		8	5.3	odd	4
9200.2.a.cn.1.3		4	4.3	odd	2
9200.2.a.cp.1.2		4	20.19	odd	2

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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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.a (trivial)

\(f(q)\)	\(=\)	\(q+0.329452 q^{3} +4.07069 q^{7} -2.89146 q^{9} -3.08806 q^{11} +6.32062 q^{13} -0.982634 q^{17} -7.08806 q^{19} +1.34110 q^{21} -1.00000 q^{23} -1.94095 q^{27} -7.63270 q^{29} -5.92047 q^{31} -1.01737 q^{33} -11.4825 q^{37} +2.08234 q^{39} -5.74124 q^{41} -1.74696 q^{43} -2.24992 q^{47} +9.57054 q^{49} -0.323730 q^{51} -1.31781 q^{53} -2.33517 q^{57} +7.92901 q^{59} +9.51722 q^{61} -11.7703 q^{63} +3.34110 q^{67} -0.329452 q^{69} -3.92619 q^{71} +1.66744 q^{73} -12.5705 q^{77} -15.9231 q^{79} +8.03493 q^{81} +11.9116 q^{83} -2.51461 q^{87} -10.4998 q^{89} +25.7293 q^{91} -1.95051 q^{93} -2.84125 q^{97} +8.92901 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{7} + 6 q^{9} + q^{11} - 3 q^{13} - 2 q^{17} - 15 q^{19} + 8 q^{21} - 4 q^{23} - 27 q^{27} + q^{29} - 12 q^{31} - 6 q^{33} - 18 q^{37} - 3 q^{39} - 9 q^{41} + 9 q^{43} + 4 q^{47} - 3 q^{49} - 2 q^{51}+ \cdots - q^{99}+O(q^{100}) \) 4 * q + q^7 + 6 * q^9 + q^11 - 3 * q^13 - 2 * q^17 - 15 * q^19 + 8 * q^21 - 4 * q^23 - 27 * q^27 + q^29 - 12 * q^31 - 6 * q^33 - 18 * q^37 - 3 * q^39 - 9 * q^41 + 9 * q^43 + 4 * q^47 - 3 * q^49 - 2 * q^51 - 6 * q^57 - 5 * q^59 + 14 * q^61 - 39 * q^63 + 16 * q^67 - 2 * q^71 - 21 * q^73 - 9 * q^77 - 21 * q^79 + 44 * q^81 + 9 * q^83 - 17 * q^87 - 16 * q^89 + 33 * q^91 + 29 * q^93 - 40 * q^97 - q^99

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