LMFDB - Embedded newform 4600.2.e.t.4049.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.61734359296.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 13x^{6} + 38x^{4} + 25x^{2} + 4 $ x^8 + 13x^6 + 38x^4 + 25*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.5
Root			$0.491317i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.t.4049.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.329452i q^{3} -4.07069i q^{7} +2.89146 q^{9} -3.08806 q^{11} +6.32062i q^{13} +0.982634i q^{17} +7.08806 q^{19} +1.34110 q^{21} -1.00000i q^{23} +1.94095i q^{27} +7.63270 q^{29} -5.92047 q^{31} -1.01737i q^{33} +11.4825i q^{37} -2.08234 q^{39} -5.74124 q^{41} -1.74696i q^{43} +2.24992i q^{47} -9.57054 q^{49} -0.323730 q^{51} -1.31781i q^{53} +2.33517i q^{57} -7.92901 q^{59} +9.51722 q^{61} -11.7703i q^{63} -3.34110i q^{67} +0.329452 q^{69} -3.92619 q^{71} +1.66744i q^{73} +12.5705i q^{77} +15.9231 q^{79} +8.03493 q^{81} +11.9116i q^{83} +2.51461i q^{87} +10.4998 q^{89} +25.7293 q^{91} -1.95051i q^{93} +2.84125i q^{97} -8.92901 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{9} + 2 q^{11} + 30 q^{19} + 16 q^{21} - 2 q^{29} - 24 q^{31} + 6 q^{39} - 18 q^{41} + 6 q^{49} - 4 q^{51} + 10 q^{59} + 28 q^{61} - 4 q^{71} + 42 q^{79} + 88 q^{81} + 32 q^{89} + 66 q^{91} + 2 q^{99}+O(q^{100}) $ 8 * q - 12 * q^9 + 2 * q^11 + 30 * q^19 + 16 * q^21 - 2 * q^29 - 24 * q^31 + 6 * q^39 - 18 * q^41 + 6 * q^49 - 4 * q^51 + 10 * q^59 + 28 * q^61 - 4 * q^71 + 42 * q^79 + 88 * q^81 + 32 * q^89 + 66 * q^91 + 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	0.329452i	0.190209i	0.995467	+	0.0951045i	$0.0303185\pi$
$3$	0.329452i	0.190209i	−0.995467	+	0.0951045i	$0.969681\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.07069i	− 1.53858i	−0.638901	−	0.769289i	$-0.720611\pi$
$7$	− 4.07069i	− 1.53858i	0.638901	−	0.769289i	$-0.279389\pi$
$8$	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
$13$	6.32062i	1.75302i	0.481380	+	0.876512i	$0.340136\pi$
$13$	6.32062i	1.75302i	−0.481380	+	0.876512i	$0.659864\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.982634i	0.238324i	0.992875	+	0.119162i	$0.0380207\pi$
$17$	0.982634i	0.238324i	−0.992875	+	0.119162i	$0.961979\pi$
$18$	0	0
$19$	7.08806	1.62611	0.813056	−	0.582185i	$-0.197802\pi$
$19$	7.08806	1.62611	0.813056	+	0.582185i	$0.197802\pi$
$20$	0	0
$21$	1.34110	0.292651
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.94095i	0.373536i
$28$	0	0
$29$	7.63270	1.41736	0.708679	−	0.705531i	$-0.249292\pi$
$29$	7.63270	1.41736	0.708679	+	0.705531i	$0.249292\pi$
$30$	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
$33$	− 1.01737i	− 0.177101i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.4825i	1.88771i	0.330362	+	0.943854i	$0.392829\pi$
$37$	11.4825i	1.88771i	−0.330362	+	0.943854i	$0.607171\pi$
$38$	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
$43$	− 1.74696i	− 0.266409i	−0.991089	−	0.133205i	$-0.957473\pi$
$43$	− 1.74696i	− 0.266409i	0.991089	−	0.133205i	$-0.0425268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.24992i	0.328185i	0.986445	+	0.164093i	$0.0524696\pi$
$47$	2.24992i	0.328185i	−0.986445	+	0.164093i	$0.947530\pi$
$48$	0	0
$49$	−9.57054	−1.36722
$50$	0	0
$51$	−0.323730	−0.0453313
$52$	0	0
$53$	− 1.31781i	− 0.181015i	−0.995896	−	0.0905073i	$-0.971151\pi$
$53$	− 1.31781i	− 0.181015i	0.995896	−	0.0905073i	$-0.0288489\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.33517i	0.309301i
$58$	0	0
$59$	−7.92901	−1.03227	−0.516134	−	0.856508i	$-0.672629\pi$
$59$	−7.92901	−1.03227	−0.516134	+	0.856508i	$0.672629\pi$
$60$	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
$63$	− 11.7703i	− 1.48291i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 3.34110i	− 0.408180i	−0.978952	−	0.204090i	$-0.934576\pi$
$67$	− 3.34110i	− 0.408180i	0.978952	−	0.204090i	$-0.0654235\pi$
$68$	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
$73$	1.66744i	0.195159i	0.995228	+	0.0975793i	$0.0311100\pi$
$73$	1.66744i	0.195159i	−0.995228	+	0.0975793i	$0.968890\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.5705i	1.43255i
$78$	0	0
$79$	15.9231	1.79149	0.895743	−	0.444572i	$-0.146644\pi$
$79$	15.9231	1.79149	0.895743	+	0.444572i	$0.146644\pi$
$80$	0	0
$81$	8.03493	0.892771
$82$	0	0
$83$	11.9116i	1.30747i	0.756723	+	0.653736i	$0.226799\pi$
$83$	11.9116i	1.30747i	−0.756723	+	0.653736i	$0.773201\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.51461i	0.269594i
$88$	0	0
$89$	10.4998	1.11298	0.556491	−	0.830854i	$-0.312147\pi$
$89$	10.4998	1.11298	0.556491	+	0.830854i	$0.312147\pi$
$90$	0	0
$91$	25.7293	2.69716
$92$	0	0
$93$	− 1.95051i	− 0.202258i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.84125i	0.288485i	0.989542	+	0.144242i	$0.0460745\pi$
$97$	2.84125i	0.288485i	−0.989542	+	0.144242i	$0.953925\pi$
$98$	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
$103$	− 5.05333i	− 0.497919i	−0.968514	−	0.248960i	$-0.919911\pi$
$103$	− 5.05333i	− 0.497919i	0.968514	−	0.248960i	$-0.0800886\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	−10.1067	−0.968042	−0.484021	−	0.875056i	$-0.660824\pi$
$109$	−10.1067	−0.968042	−0.484021	+	0.875056i	$0.660824\pi$
$110$	0	0
$111$	−3.78292	−0.359059
$112$	0	0
$113$	− 0.335173i	− 0.0315304i	−0.999876	−	0.0157652i	$-0.994982\pi$
$113$	− 0.335173i	− 0.0315304i	0.999876	−	0.0157652i	$-0.00501843\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	18.2758i	1.68960i
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−1.46389	−0.133081
$122$	0	0
$123$	− 1.89146i	− 0.170547i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	5.92047i	0.525357i	0.964883	+	0.262679i	$0.0846059\pi$
$127$	5.92047i	0.525357i	−0.964883	+	0.262679i	$0.915394\pi$
$128$	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
$133$	− 28.8533i	− 2.50190i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.1414i	1.03731i	0.854984	+	0.518654i	$0.173567\pi$
$137$	12.1414i	1.03731i	−0.854984	+	0.518654i	$0.826433\pi$
$138$	0	0
$139$	14.2499	1.20866	0.604331	−	0.796733i	$-0.293440\pi$
$139$	14.2499	1.20866	0.604331	+	0.796733i	$0.293440\pi$
$140$	0	0
$141$	−0.741241	−0.0624238
$142$	0	0
$143$	− 19.5184i	− 1.63221i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 3.15303i	− 0.260058i
$148$	0	0
$149$	18.7826	1.53873	0.769366	−	0.638808i	$-0.220572\pi$
$149$	18.7826	1.53873	0.769366	+	0.638808i	$0.220572\pi$
$150$	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
$153$	2.84125i	0.229701i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 12.9997i	− 1.03749i	−0.854929	−	0.518744i	$-0.826400\pi$
$157$	− 12.9997i	− 1.03749i	0.854929	−	0.518744i	$-0.173600\pi$
$158$	0	0
$159$	0.434154	0.0344306
$160$	0	0
$161$	−4.07069	−0.320816
$162$	0	0
$163$	12.5384i	0.982085i	0.871136	+	0.491042i	$0.163384\pi$
$163$	12.5384i	0.982085i	−0.871136	+	0.491042i	$0.836616\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 11.2121i	− 0.867617i	−0.901005	−	0.433808i	$-0.857169\pi$
$167$	− 11.2121i	− 0.867617i	0.901005	−	0.433808i	$-0.142831\pi$
$168$	0	0
$169$	−26.9502	−2.07309
$170$	0	0
$171$	20.4949	1.56728
$172$	0	0
$173$	− 17.2133i	− 1.30870i	−0.756190	−	0.654352i	$-0.772942\pi$
$173$	− 17.2133i	− 1.30870i	0.756190	−	0.654352i	$-0.227058\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 2.61222i	− 0.196347i
$178$	0	0
$179$	25.1859	1.88248	0.941240	−	0.337737i	$-0.109661\pi$
$179$	25.1859	1.88248	0.941240	+	0.337737i	$0.109661\pi$
$180$	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
$183$	3.13546i	0.231780i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 3.03443i	− 0.221900i
$188$	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
$193$	9.30064i	0.669475i	0.942311	+	0.334737i	$0.108648\pi$
$193$	9.30064i	0.669475i	−0.942311	+	0.334737i	$0.891352\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 22.8938i	− 1.63111i	−0.578677	−	0.815557i	$-0.696431\pi$
$197$	− 22.8938i	− 1.63111i	0.578677	−	0.815557i	$-0.303569\pi$
$198$	0	0
$199$	6.35969	0.450827	0.225413	−	0.974263i	$-0.427627\pi$
$199$	6.35969	0.450827	0.225413	+	0.974263i	$0.427627\pi$
$200$	0	0
$201$	1.10073	0.0776395
$202$	0	0
$203$	− 31.0704i	− 2.18071i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.89146i	− 0.200970i
$208$	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
$213$	− 1.29349i	− 0.0886286i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	24.1004i	1.63604i
$218$	0	0
$219$	−0.549339	−0.0371209
$220$	0	0
$221$	−6.21085	−0.417787
$222$	0	0
$223$	5.75289i	0.385242i	0.981273	+	0.192621i	$0.0616988\pi$
$223$	5.75289i	0.385242i	−0.981273	+	0.192621i	$0.938301\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.49393i	0.630134i	0.949069	+	0.315067i	$0.102027\pi$
$227$	9.49393i	0.630134i	−0.949069	+	0.315067i	$0.897973\pi$
$228$	0	0
$229$	−1.83532	−0.121282	−0.0606408	−	0.998160i	$-0.519314\pi$
$229$	−1.83532	−0.121282	−0.0606408	+	0.998160i	$0.519314\pi$
$230$	0	0
$231$	−4.14139	−0.272483
$232$	0	0
$233$	20.5687i	1.34750i	0.738960	+	0.673749i	$0.235317\pi$
$233$	20.5687i	1.34750i	−0.738960	+	0.673749i	$0.764683\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.24589i	0.340757i
$238$	0	0
$239$	12.4014	0.802178	0.401089	−	0.916039i	$-0.368632\pi$
$239$	12.4014	0.802178	0.401089	+	0.916039i	$0.368632\pi$
$240$	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
$243$	8.46998i	0.543349i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	44.8009i	2.85061i
$248$	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
$253$	3.08806i	0.194145i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 19.3115i	− 1.20462i	−0.798263	−	0.602309i	$-0.794248\pi$
$257$	− 19.3115i	− 1.20462i	0.798263	−	0.602309i	$-0.205752\pi$
$258$	0	0
$259$	46.7417	2.90439
$260$	0	0
$261$	22.0697	1.36608
$262$	0	0
$263$	1.74819i	0.107798i	0.998546	+	0.0538990i	$0.0171649\pi$
$263$	1.74819i	0.107798i	−0.998546	+	0.0538990i	$0.982835\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.45919i	0.211699i
$268$	0	0
$269$	−11.7079	−0.713843	−0.356921	−	0.934134i	$-0.616174\pi$
$269$	−11.7079	−0.713843	−0.356921	+	0.934134i	$0.616174\pi$
$270$	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
$273$	8.47656i	0.513025i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 5.29319i	− 0.318037i	−0.987276	−	0.159018i	$-0.949167\pi$
$277$	− 5.29319i	− 0.318037i	0.987276	−	0.159018i	$-0.0508329\pi$
$278$	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
$283$	7.79437i	0.463327i	0.972796	+	0.231663i	$0.0744168\pi$
$283$	7.79437i	0.463327i	−0.972796	+	0.231663i	$0.925583\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	23.3708i	1.37954i
$288$	0	0
$289$	16.0344	0.943202
$290$	0	0
$291$	−0.936054	−0.0548724
$292$	0	0
$293$	− 24.0171i	− 1.40309i	−0.712624	−	0.701546i	$-0.752494\pi$
$293$	− 24.0171i	− 1.40309i	0.712624	−	0.701546i	$-0.247506\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.99377i	− 0.347794i
$298$	0	0
$299$	6.32062	0.365531
$300$	0	0
$301$	−7.11135	−0.409891
$302$	0	0
$303$	1.56389i	0.0898434i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	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
$313$	31.5110i	1.78111i	0.454879	+	0.890553i	$0.349682\pi$
$313$	31.5110i	1.78111i	−0.454879	+	0.890553i	$0.650318\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.1110i	1.24188i	0.783858	+	0.620940i	$0.213249\pi$
$317$	22.1110i	1.24188i	−0.783858	+	0.620940i	$0.786751\pi$
$318$	0	0
$319$	−23.5702	−1.31968
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.96497i	0.387541i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 3.32965i	− 0.184130i
$328$	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
$333$	33.2012i	1.81941i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	20.1647i	1.09844i	0.835678	+	0.549220i	$0.185075\pi$
$337$	20.1647i	1.09844i	−0.835678	+	0.549220i	$0.814925\pi$
$338$	0	0
$339$	0.110423	0.00599737
$340$	0	0
$341$	18.2828	0.990068
$342$	0	0
$343$	10.4639i	0.564997i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	32.2465i	1.73108i	0.500837	+	0.865542i	$0.333026\pi$
$347$	32.2465i	1.73108i	−0.500837	+	0.865542i	$0.666974\pi$
$348$	0	0
$349$	−35.3203	−1.89065	−0.945327	−	0.326125i	$-0.894257\pi$
$349$	−35.3203	−1.89065	−0.945327	+	0.326125i	$0.894257\pi$
$350$	0	0
$351$	−12.2680	−0.654818
$352$	0	0
$353$	− 30.7409i	− 1.63618i	−0.575094	−	0.818088i	$-0.695034\pi$
$353$	− 30.7409i	− 1.63618i	0.575094	−	0.818088i	$-0.304966\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.31781i	0.0697457i
$358$	0	0
$359$	34.4053	1.81584	0.907920	−	0.419143i	$-0.137669\pi$
$359$	34.4053	1.81584	0.907920	+	0.419143i	$0.137669\pi$
$360$	0	0
$361$	31.2406	1.64424
$362$	0	0
$363$	− 0.482281i	− 0.0253132i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.2524i	0.535173i	0.963534	+	0.267586i	$0.0862261\pi$
$367$	10.2524i	0.535173i	−0.963534	+	0.267586i	$0.913774\pi$
$368$	0	0
$369$	−16.6006	−0.864192
$370$	0	0
$371$	−5.36439	−0.278505
$372$	0	0
$373$	22.6993i	1.17532i	0.809107	+	0.587662i	$0.199951\pi$
$373$	22.6993i	1.17532i	−0.809107	+	0.587662i	$0.800049\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	48.2434i	2.48466i
$378$	0	0
$379$	1.03503	0.0531661	0.0265831	−	0.999647i	$-0.491537\pi$
$379$	1.03503	0.0531661	0.0265831	+	0.999647i	$0.491537\pi$
$380$	0	0
$381$	−1.95051	−0.0999276
$382$	0	0
$383$	− 25.8707i	− 1.32193i	−0.750417	−	0.660965i	$-0.770147\pi$
$383$	− 25.8707i	− 1.32193i	0.750417	−	0.660965i	$-0.229853\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 5.05128i	− 0.256771i
$388$	0	0
$389$	9.21678	0.467309	0.233655	−	0.972320i	$-0.424932\pi$
$389$	9.21678	0.467309	0.233655	+	0.972320i	$0.424932\pi$
$390$	0	0
$391$	0.982634	0.0496939
$392$	0	0
$393$	2.31377i	0.116714i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.84825i	0.444081i	0.975037	+	0.222040i	$0.0712717\pi$
$397$	8.84825i	0.444081i	−0.975037	+	0.222040i	$0.928728\pi$
$398$	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
$403$	− 37.4210i	− 1.86408i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 35.4586i	− 1.75762i
$408$	0	0
$409$	22.8709	1.13089	0.565446	−	0.824785i	$-0.308704\pi$
$409$	22.8709	1.13089	0.565446	+	0.824785i	$0.308704\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	0	0
$413$	32.2765i	1.58823i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.69466i	0.229899i
$418$	0	0
$419$	−2.41771	−0.118113	−0.0590565	−	0.998255i	$-0.518809\pi$
$419$	−2.41771	−0.118113	−0.0590565	+	0.998255i	$0.518809\pi$
$420$	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
$423$	6.50557i	0.316312i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 38.7417i	− 1.87484i
$428$	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
$433$	28.6003i	1.37444i	0.726449	+	0.687221i	$0.241170\pi$
$433$	28.6003i	1.37444i	−0.726449	+	0.687221i	$0.758830\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7.08806i	− 0.339068i
$438$	0	0
$439$	8.52886	0.407060	0.203530	−	0.979069i	$-0.434759\pi$
$439$	8.52886	0.407060	0.203530	+	0.979069i	$0.434759\pi$
$440$	0	0
$441$	−27.6729	−1.31776
$442$	0	0
$443$	− 35.9224i	− 1.70673i	−0.521317	−	0.853363i	$-0.674559\pi$
$443$	− 35.9224i	− 1.70673i	0.521317	−	0.853363i	$-0.325441\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.18797i	0.292681i
$448$	0	0
$449$	−5.11165	−0.241234	−0.120617	−	0.992699i	$-0.538487\pi$
$449$	−5.11165	−0.241234	−0.120617	+	0.992699i	$0.538487\pi$
$450$	0	0
$451$	17.7293	0.834840
$452$	0	0
$453$	4.07090i	0.191267i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11.7944i	0.551717i	0.961198	+	0.275859i	$0.0889621\pi$
$457$	11.7944i	0.551717i	−0.961198	+	0.275859i	$0.911038\pi$
$458$	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
$463$	− 35.7752i	− 1.66261i	−0.555814	−	0.831307i	$-0.687593\pi$
$463$	− 35.7752i	− 1.66261i	0.555814	−	0.831307i	$-0.312407\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 21.2118i	− 0.981564i	−0.871282	−	0.490782i	$-0.836711\pi$
$467$	− 21.2118i	− 0.981564i	0.871282	−	0.490782i	$-0.163289\pi$
$468$	0	0
$469$	−13.6006	−0.628016
$470$	0	0
$471$	4.28277	0.197340
$472$	0	0
$473$	5.39472i	0.248050i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 3.81039i	− 0.174466i
$478$	0	0
$479$	−24.6077	−1.12436	−0.562178	−	0.827016i	$-0.690036\pi$
$479$	−24.6077	−1.12436	−0.562178	+	0.827016i	$0.690036\pi$
$480$	0	0
$481$	−72.5764	−3.30920
$482$	0	0
$483$	− 1.34110i	− 0.0610220i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	15.3507i	0.695605i	0.937568	+	0.347802i	$0.113072\pi$
$487$	15.3507i	0.695605i	−0.937568	+	0.347802i	$0.886928\pi$
$488$	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
$493$	7.50015i	0.337790i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.9823i	0.716906i
$498$	0	0
$499$	20.6841	0.925949	0.462975	−	0.886371i	$-0.346782\pi$
$499$	20.6841	0.925949	0.462975	+	0.886371i	$0.346782\pi$
$500$	0	0
$501$	3.69384	0.165029
$502$	0	0
$503$	− 35.7956i	− 1.59605i	−0.602627	−	0.798023i	$-0.705879\pi$
$503$	− 35.7956i	− 1.59605i	0.602627	−	0.798023i	$-0.294121\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 8.87879i	− 0.394321i
$508$	0	0
$509$	−33.9242	−1.50366	−0.751832	−	0.659355i	$-0.770830\pi$
$509$	−33.9242	−1.50366	−0.751832	+	0.659355i	$0.770830\pi$
$510$	0	0
$511$	6.78762	0.300267
$512$	0	0
$513$	13.7576i	0.607412i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 6.94790i	− 0.305568i
$518$	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
$523$	5.25274i	0.229686i	0.993384	+	0.114843i	$0.0366365\pi$
$523$	5.25274i	0.229686i	−0.993384	+	0.114843i	$0.963363\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 5.81766i	− 0.253421i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−22.9264	−0.994922
$532$	0	0
$533$	− 36.2882i	− 1.57182i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.29753i	0.358065i
$538$	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
$543$	5.05415i	0.216894i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 25.6025i	− 1.09468i	−0.836910	−	0.547341i	$-0.815640\pi$
$547$	− 25.6025i	− 1.09468i	0.836910	−	0.547341i	$-0.184360\pi$
$548$	0	0
$549$	27.5187	1.17447
$550$	0	0
$551$	54.1011	2.30478
$552$	0	0
$553$	− 64.8180i	− 2.75634i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 15.7938i	− 0.669203i	−0.942360	−	0.334602i	$-0.891398\pi$
$557$	− 15.7938i	− 0.669203i	0.942360	−	0.334602i	$-0.108602\pi$
$558$	0	0
$559$	11.0419	0.467022
$560$	0	0
$561$	0.999698	0.0422073
$562$	0	0
$563$	41.8996i	1.76586i	0.469507	+	0.882929i	$0.344432\pi$
$563$	41.8996i	1.76586i	−0.469507	+	0.882929i	$0.655568\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 32.7078i	− 1.37360i
$568$	0	0
$569$	−24.5485	−1.02913	−0.514563	−	0.857453i	$-0.672046\pi$
$569$	−24.5485	−1.02913	−0.514563	+	0.857453i	$0.672046\pi$
$570$	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
$573$	− 5.67422i	− 0.237044i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 23.9288i	− 0.996169i	−0.867128	−	0.498085i	$-0.834037\pi$
$577$	− 23.9288i	− 0.996169i	0.867128	−	0.498085i	$-0.165963\pi$
$578$	0	0
$579$	−3.06411	−0.127340
$580$	0	0
$581$	48.4886	2.01165
$582$	0	0
$583$	4.06947i	0.168540i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.8130i	1.43689i	0.695585	+	0.718443i	$0.255145\pi$
$587$	34.8130i	1.43689i	−0.695585	+	0.718443i	$0.744855\pi$
$588$	0	0
$589$	−41.9647	−1.72912
$590$	0	0
$591$	7.54239	0.310252
$592$	0	0
$593$	− 15.5702i	− 0.639393i	−0.947520	−	0.319697i	$-0.896419\pi$
$593$	− 15.5702i	− 0.639393i	0.947520	−	0.319697i	$-0.103581\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.09521i	0.0857513i
$598$	0	0
$599$	−31.2121	−1.27529	−0.637645	−	0.770330i	$-0.720092\pi$
$599$	−31.2121	−1.27529	−0.637645	+	0.770330i	$0.720092\pi$
$600$	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
$603$	− 9.66065i	− 0.393412i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 30.6469i	− 1.24392i	−0.783050	−	0.621959i	$-0.786337\pi$
$607$	− 30.6469i	− 1.24392i	0.783050	−	0.621959i	$-0.213663\pi$
$608$	0	0
$609$	10.2362	0.414791
$610$	0	0
$611$	−14.2209	−0.575317
$612$	0	0
$613$	0.459494i	0.0185588i	0.999957	+	0.00927940i	$0.00295377\pi$
$613$	0.459494i	0.0185588i	−0.999957	+	0.00927940i	$0.997046\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	23.6010i	0.950140i	0.879948	+	0.475070i	$0.157577\pi$
$617$	23.6010i	0.950140i	−0.879948	+	0.475070i	$0.842423\pi$
$618$	0	0
$619$	−24.8893	−1.00038	−0.500192	−	0.865914i	$-0.666737\pi$
$619$	−24.8893	−1.00038	−0.500192	+	0.865914i	$0.666737\pi$
$620$	0	0
$621$	1.94095	0.0778877
$622$	0	0
$623$	− 42.7417i	− 1.71241i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 7.21115i	− 0.287986i
$628$	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
$633$	− 3.41497i	− 0.135733i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 60.4917i	− 2.39677i
$638$	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
$643$	15.0242i	0.592497i	0.955111	+	0.296249i	$0.0957357\pi$
$643$	15.0242i	0.592497i	−0.955111	+	0.296249i	$0.904264\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 45.3791i	− 1.78404i	−0.452001	−	0.892018i	$-0.649290\pi$
$647$	− 45.3791i	− 1.78404i	0.452001	−	0.892018i	$-0.350710\pi$
$648$	0	0
$649$	24.4852	0.961130
$650$	0	0
$651$	−7.93993	−0.311190
$652$	0	0
$653$	14.2506i	0.557671i	0.960339	+	0.278835i	$0.0899484\pi$
$653$	14.2506i	0.557671i	−0.960339	+	0.278835i	$0.910052\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.82133i	0.188098i
$658$	0	0
$659$	10.3535	0.403314	0.201657	−	0.979456i	$-0.435367\pi$
$659$	10.3535	0.403314	0.201657	+	0.979456i	$0.435367\pi$
$660$	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
$663$	− 2.04618i	− 0.0794669i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 7.63270i	− 0.295539i
$668$	0	0
$669$	−1.89530	−0.0732765
$670$	0	0
$671$	−29.3897	−1.13458
$672$	0	0
$673$	12.0587i	0.464831i	0.972617	+	0.232415i	$0.0746628\pi$
$673$	12.0587i	0.464831i	−0.972617	+	0.232415i	$0.925337\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.54878i	0.213257i	0.994299	+	0.106629i	$0.0340056\pi$
$677$	5.54878i	0.213257i	−0.994299	+	0.106629i	$0.965994\pi$
$678$	0	0
$679$	11.5658	0.443856
$680$	0	0
$681$	−3.12779	−0.119857
$682$	0	0
$683$	0.397032i	0.0151920i	0.999971	+	0.00759601i	$0.00241791\pi$
$683$	0.397032i	0.0151920i	−0.999971	+	0.00759601i	$0.997582\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 0.604651i	− 0.0230689i
$688$	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
$693$	36.3472i	1.38072i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 5.64154i	− 0.213688i
$698$	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
$703$	81.3885i	3.06963i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 19.3234i	− 0.726732i
$708$	0	0
$709$	−29.7476	−1.11719	−0.558597	−	0.829439i	$-0.688660\pi$
$709$	−29.7476	−1.11719	−0.558597	+	0.829439i	$0.688660\pi$
$710$	0	0
$711$	46.0410	1.72667
$712$	0	0
$713$	5.92047i	0.221723i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.08565i	0.152582i
$718$	0	0
$719$	−35.1758	−1.31184	−0.655918	−	0.754832i	$-0.727718\pi$
$719$	−35.1758	−1.31184	−0.655918	+	0.754832i	$0.727718\pi$
$720$	0	0
$721$	−20.5705	−0.766087
$722$	0	0
$723$	5.19971i	0.193379i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 27.6030i	− 1.02374i	−0.859063	−	0.511870i	$-0.828953\pi$
$727$	− 27.6030i	− 1.02374i	0.859063	−	0.511870i	$-0.171047\pi$
$728$	0	0
$729$	21.3144	0.789421
$730$	0	0
$731$	1.71662	0.0634916
$732$	0	0
$733$	− 51.0939i	− 1.88720i	−0.331093	−	0.943598i	$-0.607417\pi$
$733$	− 51.0939i	− 1.88720i	0.331093	−	0.943598i	$-0.392583\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.3175i	0.380050i
$738$	0	0
$739$	−13.5558	−0.498658	−0.249329	−	0.968419i	$-0.580210\pi$
$739$	−13.5558	−0.498658	−0.249329	+	0.968419i	$0.580210\pi$
$740$	0	0
$741$	−14.7597	−0.542212
$742$	0	0
$743$	53.8006i	1.97375i	0.161477	+	0.986877i	$0.448374\pi$
$743$	53.8006i	1.97375i	−0.161477	+	0.986877i	$0.551626\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	34.4420i	1.26017i
$748$	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
$753$	− 3.20116i	− 0.116657i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.1581i	0.914388i	0.889367	+	0.457194i	$0.151146\pi$
$757$	25.1581i	0.914388i	−0.889367	+	0.457194i	$0.848854\pi$
$758$	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
$763$	41.1411i	1.48941i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 50.1162i	− 1.80959i
$768$	0	0
$769$	35.7337	1.28859	0.644295	−	0.764777i	$-0.277151\pi$
$769$	35.7337	1.28859	0.644295	+	0.764777i	$0.277151\pi$
$770$	0	0
$771$	6.36220	0.229129
$772$	0	0
$773$	10.9587i	0.394158i	0.980388	+	0.197079i	$0.0631456\pi$
$773$	10.9587i	0.394158i	−0.980388	+	0.197079i	$0.936854\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	15.3991i	0.552440i
$778$	0	0
$779$	−40.6943	−1.45802
$780$	0	0
$781$	12.1243	0.433842
$782$	0	0
$783$	14.8147i	0.529435i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3.56216i	0.126977i	0.997983	+	0.0634887i	$0.0202227\pi$
$787$	3.56216i	0.126977i	−0.997983	+	0.0634887i	$0.979777\pi$
$788$	0	0
$789$	−0.575944	−0.0205042
$790$	0	0
$791$	−1.36439	−0.0485120
$792$	0	0
$793$	60.1547i	2.13616i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 44.7128i	− 1.58381i	−0.610646	−	0.791903i	$-0.709090\pi$
$797$	− 44.7128i	− 1.58381i	0.610646	−	0.791903i	$-0.290910\pi$
$798$	0	0
$799$	−2.21085	−0.0782143
$800$	0	0
$801$	30.3599	1.07271
$802$	0	0
$803$	− 5.14914i	− 0.181709i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 3.85718i	− 0.135779i
$808$	0	0
$809$	−9.68159	−0.340387	−0.170193	−	0.985411i	$-0.554439\pi$
$809$	−9.68159	−0.340387	−0.170193	+	0.985411i	$0.554439\pi$
$810$	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
$813$	3.32915i	0.116758i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 12.3826i	− 0.433212i
$818$	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
$823$	1.50587i	0.0524914i	0.999656	+	0.0262457i	$0.00835523\pi$
$823$	1.50587i	0.0524914i	−0.999656	+	0.0262457i	$0.991645\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 25.3652i	− 0.882035i	−0.897499	−	0.441017i	$-0.854618\pi$
$827$	− 25.3652i	− 0.882035i	0.897499	−	0.441017i	$-0.145382\pi$
$828$	0	0
$829$	−38.2130	−1.32719	−0.663596	−	0.748091i	$-0.730971\pi$
$829$	−38.2130	−1.32719	−0.663596	+	0.748091i	$0.730971\pi$
$830$	0	0
$831$	1.74385	0.0604935
$832$	0	0
$833$	− 9.40434i	− 0.325841i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 11.4914i	− 0.397199i
$838$	0	0
$839$	13.1575	0.454248	0.227124	−	0.973866i	$-0.427068\pi$
$839$	13.1575	0.454248	0.227124	+	0.973866i	$0.427068\pi$
$840$	0	0
$841$	29.2582	1.00890
$842$	0	0
$843$	− 4.54848i	− 0.156658i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.95904i	0.204755i
$848$	0	0
$849$	−2.56787	−0.0881290
$850$	0	0
$851$	11.4825	0.393614
$852$	0	0
$853$	− 13.2756i	− 0.454549i	−0.973831	−	0.227274i	$-0.927019\pi$
$853$	− 13.2756i	− 0.454549i	0.973831	−	0.227274i	$-0.0729814\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.18894i	0.0747729i	0.999301	+	0.0373864i	$0.0119032\pi$
$857$	2.18894i	0.0747729i	−0.999301	+	0.0373864i	$0.988097\pi$
$858$	0	0
$859$	−21.4092	−0.730472	−0.365236	−	0.930915i	$-0.619012\pi$
$859$	−21.4092	−0.730472	−0.365236	+	0.930915i	$0.619012\pi$
$860$	0	0
$861$	−7.69956	−0.262400
$862$	0	0
$863$	− 14.2923i	− 0.486514i	−0.969962	−	0.243257i	$-0.921784\pi$
$863$	− 14.2923i	− 0.486514i	0.969962	−	0.243257i	$-0.0782159\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5.28257i	0.179405i
$868$	0	0
$869$	−49.1714	−1.66803
$870$	0	0
$871$	21.1178	0.715549
$872$	0	0
$873$	8.21536i	0.278048i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.7835i	0.769346i	0.923053	+	0.384673i	$0.125686\pi$
$877$	22.7835i	0.769346i	−0.923053	+	0.384673i	$0.874314\pi$
$878$	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
$883$	− 9.35101i	− 0.314687i	−0.987544	−	0.157343i	$-0.949707\pi$
$883$	− 9.35101i	− 0.314687i	0.987544	−	0.157343i	$-0.0502929\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22.9902i	0.771936i	0.922512	+	0.385968i	$0.126132\pi$
$887$	22.9902i	0.771936i	−0.922512	+	0.385968i	$0.873868\pi$
$888$	0	0
$889$	24.1004	0.808302
$890$	0	0
$891$	−24.8124	−0.831245
$892$	0	0
$893$	15.9476i	0.533666i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.08234i	0.0695272i
$898$	0	0
$899$	−45.1892	−1.50714
$900$	0	0
$901$	1.29492	0.0431401
$902$	0	0
$903$	− 2.34285i	− 0.0779650i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 29.3070i	− 0.973123i	−0.873646	−	0.486561i	$-0.838251\pi$
$907$	− 29.3070i	− 0.973123i	0.873646	−	0.486561i	$-0.161749\pi$
$908$	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
$913$	− 36.7838i	− 1.21737i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 28.5888i	− 0.944086i
$918$	0	0
$919$	27.3696	0.902840	0.451420	−	0.892312i	$-0.350918\pi$
$919$	27.3696	0.902840	0.451420	+	0.892312i	$0.350918\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 24.8160i	− 0.816828i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 14.6115i	− 0.479905i
$928$	0	0
$929$	−4.64485	−0.152393	−0.0761963	−	0.997093i	$-0.524278\pi$
$929$	−4.64485	−0.152393	−0.0761963	+	0.997093i	$0.524278\pi$
$930$	0	0
$931$	−67.8366	−2.22325
$932$	0	0
$933$	− 8.84601i	− 0.289605i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19.6704i	0.642606i	0.946977	+	0.321303i	$0.104121\pi$
$937$	19.6704i	0.642606i	−0.946977	+	0.321303i	$0.895879\pi$
$938$	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
$943$	5.74124i	0.186961i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.3846i	1.18234i	0.806547	+	0.591170i	$0.201334\pi$
$947$	36.3846i	1.18234i	−0.806547	+	0.591170i	$0.798666\pi$
$948$	0	0
$949$	−10.5392	−0.342118
$950$	0	0
$951$	−7.28452	−0.236217
$952$	0	0
$953$	22.2456i	0.720605i	0.932835	+	0.360303i	$0.117327\pi$
$953$	22.2456i	0.720605i	−0.932835	+	0.360303i	$0.882673\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 7.76526i	− 0.251015i
$958$	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
$967$	49.4661i	1.59072i	0.606136	+	0.795361i	$0.292719\pi$
$967$	49.4661i	1.59072i	−0.606136	+	0.795361i	$0.707281\pi$
$968$	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
$973$	− 58.0071i	− 1.85962i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 55.1228i	− 1.76354i	−0.471684	−	0.881768i	$-0.656354\pi$
$977$	− 55.1228i	− 1.76354i	0.471684	−	0.881768i	$-0.343646\pi$
$978$	0	0
$979$	−32.4242	−1.03628
$980$	0	0
$981$	−29.2230	−0.933019
$982$	0	0
$983$	− 26.6824i	− 0.851037i	−0.904950	−	0.425518i	$-0.860092\pi$
$983$	− 26.6824i	− 0.851037i	0.904950	−	0.425518i	$-0.139908\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.01737i	0.0960438i
$988$	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
$993$	7.87235i	0.249821i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1.95372i	− 0.0618750i	−0.999521	−	0.0309375i	$-0.990151\pi$
$997$	− 1.95372i	− 0.0618750i	0.999521	−	0.0309375i	$-0.00984928\pi$
$998$	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.e.t.4049.5		8
5.2	odd	4		4600.2.a.bb.1.2	yes	4
5.3	odd	4		4600.2.a.ba.1.3	✓	4
5.4	even	2	inner	4600.2.e.t.4049.4		8
20.3	even	4		9200.2.a.cp.1.2		4
20.7	even	4		9200.2.a.cn.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.ba.1.3	✓	4	5.3	odd	4
4600.2.a.bb.1.2	yes	4	5.2	odd	4
4600.2.e.t.4049.4		8	5.4	even	2	inner
4600.2.e.t.4049.5		8	1.1	even	1	trivial
9200.2.a.cn.1.3		4	20.7	even	4
9200.2.a.cp.1.2		4	20.3	even	4

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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.329452i q^{3} -4.07069i q^{7} +2.89146 q^{9} -3.08806 q^{11} +6.32062i q^{13} +0.982634i q^{17} +7.08806 q^{19} +1.34110 q^{21} -1.00000i q^{23} +1.94095i q^{27} +7.63270 q^{29} -5.92047 q^{31} -1.01737i q^{33} +11.4825i q^{37} -2.08234 q^{39} -5.74124 q^{41} -1.74696i q^{43} +2.24992i q^{47} -9.57054 q^{49} -0.323730 q^{51} -1.31781i q^{53} +2.33517i q^{57} -7.92901 q^{59} +9.51722 q^{61} -11.7703i q^{63} -3.34110i q^{67} +0.329452 q^{69} -3.92619 q^{71} +1.66744i q^{73} +12.5705i q^{77} +15.9231 q^{79} +8.03493 q^{81} +11.9116i q^{83} +2.51461i q^{87} +10.4998 q^{89} +25.7293 q^{91} -1.95051i q^{93} +2.84125i q^{97} -8.92901 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{9} + 2 q^{11} + 30 q^{19} + 16 q^{21} - 2 q^{29} - 24 q^{31} + 6 q^{39} - 18 q^{41} + 6 q^{49} - 4 q^{51} + 10 q^{59} + 28 q^{61} - 4 q^{71} + 42 q^{79} + 88 q^{81} + 32 q^{89} + 66 q^{91} + 2 q^{99}+O(q^{100}) \) 8 * q - 12 * q^9 + 2 * q^11 + 30 * q^19 + 16 * q^21 - 2 * q^29 - 24 * q^31 + 6 * q^39 - 18 * q^41 + 6 * q^49 - 4 * q^51 + 10 * q^59 + 28 * q^61 - 4 * q^71 + 42 * q^79 + 88 * q^81 + 32 * q^89 + 66 * q^91 + 2 * q^99

Introduction

L-functions

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