LMFDB - Embedded newform 1600.3.h.k.1599.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,3,Mod(1599,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1600.1599");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1600.h (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:	$43.5968422976$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1599.3
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	1600.1599
Dual form			1600.3.h.k.1599.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.23607 q^{3} +3.23607 q^{7} +1.47214 q^{9} +O(q^{10})$	$q+3.23607 q^{3} +3.23607 q^{7} +1.47214 q^{9} -15.4164i q^{11} +21.4164i q^{13} +10.9443i q^{17} +5.88854i q^{19} +10.4721 q^{21} +21.1246 q^{23} -24.3607 q^{27} +38.9443 q^{29} +25.5279i q^{31} -49.8885i q^{33} +43.3050i q^{37} +69.3050i q^{39} +58.1378 q^{41} +67.2361 q^{43} -21.4853 q^{47} -38.5279 q^{49} +35.4164i q^{51} -18.1378i q^{53} +19.0557i q^{57} -108.721i q^{59} +97.1935 q^{61} +4.76393 q^{63} +71.0132 q^{67} +68.3607 q^{69} +39.6393i q^{71} +18.7214i q^{73} -49.8885i q^{77} -36.4984i q^{79} -92.0820 q^{81} +129.348 q^{83} +126.026 q^{87} -117.777 q^{89} +69.3050i q^{91} +82.6099i q^{93} +81.3313i q^{97} -22.6950i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{7} - 12 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^7 - 12 * q^9	$ 4 q + 4 q^{3} + 4 q^{7} - 12 q^{9} + 24 q^{21} + 4 q^{23} - 8 q^{27} + 120 q^{29} + 260 q^{43} + 84 q^{47} - 172 q^{49} + 192 q^{61} + 28 q^{63} + 132 q^{67} + 184 q^{69} - 100 q^{81} + 580 q^{83} + 200 q^{87} - 328 q^{89}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^7 - 12 * q^9 + 24 * q^21 + 4 * q^23 - 8 * q^27 + 120 * q^29 + 260 * q^43 + 84 * q^47 - 172 * q^49 + 192 * q^61 + 28 * q^63 + 132 * q^67 + 184 * q^69 - 100 * q^81 + 580 * q^83 + 200 * q^87 - 328 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	3.23607	1.07869	0.539345	−	0.842085i	$-0.318672\pi$
$3$	3.23607	1.07869	0.539345	+	0.842085i	$0.318672\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.23607	0.462295	0.231148	−	0.972919i	$-0.425752\pi$
$7$	3.23607	0.462295	0.231148	+	0.972919i	$0.425752\pi$
$8$	0	0
$9$	1.47214	0.163571
$10$	0	0
$11$	− 15.4164i	− 1.40149i	−0.713411	−	0.700746i	$-0.752851\pi$
$11$	− 15.4164i	− 1.40149i	0.713411	−	0.700746i	$-0.247149\pi$
$12$	0	0
$13$	21.4164i	1.64742i	0.567014	+	0.823708i	$0.308098\pi$
$13$	21.4164i	1.64742i	−0.567014	+	0.823708i	$0.691902\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	10.9443i	0.643781i	0.946777	+	0.321890i	$0.104318\pi$
$17$	10.9443i	0.643781i	−0.946777	+	0.321890i	$0.895682\pi$
$18$	0	0
$19$	5.88854i	0.309923i	0.987920	+	0.154962i	$0.0495254\pi$
$19$	5.88854i	0.309923i	−0.987920	+	0.154962i	$0.950475\pi$
$20$	0	0
$21$	10.4721	0.498673
$22$	0	0
$23$	21.1246	0.918461	0.459231	−	0.888317i	$-0.348125\pi$
$23$	21.1246	0.918461	0.459231	+	0.888317i	$0.348125\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−24.3607	−0.902247
$28$	0	0
$29$	38.9443	1.34291	0.671453	−	0.741047i	$-0.265671\pi$
$29$	38.9443	1.34291	0.671453	+	0.741047i	$0.265671\pi$
$30$	0	0
$31$	25.5279i	0.823479i	0.911301	+	0.411740i	$0.135079\pi$
$31$	25.5279i	0.823479i	−0.911301	+	0.411740i	$0.864921\pi$
$32$	0	0
$33$	− 49.8885i	− 1.51177i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	43.3050i	1.17040i	0.810888	+	0.585202i	$0.198985\pi$
$37$	43.3050i	1.17040i	−0.810888	+	0.585202i	$0.801015\pi$
$38$	0	0
$39$	69.3050i	1.77705i
$40$	0	0
$41$	58.1378	1.41799	0.708997	−	0.705211i	$-0.249148\pi$
$41$	58.1378	1.41799	0.708997	+	0.705211i	$0.249148\pi$
$42$	0	0
$43$	67.2361	1.56363	0.781815	−	0.623511i	$-0.214294\pi$
$43$	67.2361	1.56363	0.781815	+	0.623511i	$0.214294\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−21.4853	−0.457134	−0.228567	−	0.973528i	$-0.573404\pi$
$47$	−21.4853	−0.457134	−0.228567	+	0.973528i	$0.573404\pi$
$48$	0	0
$49$	−38.5279	−0.786283
$50$	0	0
$51$	35.4164i	0.694439i
$52$	0	0
$53$	− 18.1378i	− 0.342222i	−0.985252	−	0.171111i	$-0.945264\pi$
$53$	− 18.1378i	− 0.342222i	0.985252	−	0.171111i	$-0.0547357\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	19.0557i	0.334311i
$58$	0	0
$59$	− 108.721i	− 1.84273i	−0.388693	−	0.921367i	$-0.627073\pi$
$59$	− 108.721i	− 1.84273i	0.388693	−	0.921367i	$-0.372927\pi$
$60$	0	0
$61$	97.1935	1.59334	0.796668	−	0.604417i	$-0.206594\pi$
$61$	97.1935	1.59334	0.796668	+	0.604417i	$0.206594\pi$
$62$	0	0
$63$	4.76393	0.0756180
$64$	0	0
$65$	0	0
$66$	0	0
$67$	71.0132	1.05990	0.529949	−	0.848030i	$-0.322211\pi$
$67$	71.0132	1.05990	0.529949	+	0.848030i	$0.322211\pi$
$68$	0	0
$69$	68.3607	0.990734
$70$	0	0
$71$	39.6393i	0.558300i	0.960247	+	0.279150i	$0.0900527\pi$
$71$	39.6393i	0.558300i	−0.960247	+	0.279150i	$0.909947\pi$
$72$	0	0
$73$	18.7214i	0.256457i	0.991745	+	0.128228i	$0.0409291\pi$
$73$	18.7214i	0.256457i	−0.991745	+	0.128228i	$0.959071\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 49.8885i	− 0.647903i
$78$	0	0
$79$	− 36.4984i	− 0.462006i	−0.972953	−	0.231003i	$-0.925799\pi$
$79$	− 36.4984i	− 0.462006i	0.972953	−	0.231003i	$-0.0742007\pi$
$80$	0	0
$81$	−92.0820	−1.13682
$82$	0	0
$83$	129.348	1.55840	0.779202	−	0.626773i	$-0.215625\pi$
$83$	129.348	1.55840	0.779202	+	0.626773i	$0.215625\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	126.026	1.44858
$88$	0	0
$89$	−117.777	−1.32334	−0.661669	−	0.749796i	$-0.730152\pi$
$89$	−117.777	−1.32334	−0.661669	+	0.749796i	$0.730152\pi$
$90$	0	0
$91$	69.3050i	0.761593i
$92$	0	0
$93$	82.6099i	0.888279i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	81.3313i	0.838467i	0.907879	+	0.419233i	$0.137701\pi$
$97$	81.3313i	0.838467i	−0.907879	+	0.419233i	$0.862299\pi$
$98$	0	0
$99$	− 22.6950i	− 0.229243i
$100$	0	0
$101$	−21.7771	−0.215615	−0.107807	−	0.994172i	$-0.534383\pi$
$101$	−21.7771	−0.215615	−0.107807	+	0.994172i	$0.534383\pi$
$102$	0	0
$103$	−138.705	−1.34665	−0.673326	−	0.739346i	$-0.735135\pi$
$103$	−138.705	−1.34665	−0.673326	+	0.739346i	$0.735135\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	96.9017	0.905623	0.452812	−	0.891606i	$-0.350421\pi$
$107$	96.9017	0.905623	0.452812	+	0.891606i	$0.350421\pi$
$108$	0	0
$109$	−30.8591	−0.283111	−0.141556	−	0.989930i	$-0.545210\pi$
$109$	−30.8591	−0.283111	−0.141556	+	0.989930i	$0.545210\pi$
$110$	0	0
$111$	140.138i	1.26250i
$112$	0	0
$113$	− 50.9443i	− 0.450834i	−0.974262	−	0.225417i	$-0.927626\pi$
$113$	− 50.9443i	− 0.450834i	0.974262	−	0.225417i	$-0.0723745\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	31.5279i	0.269469i
$118$	0	0
$119$	35.4164i	0.297617i
$120$	0	0
$121$	−116.666	−0.964179
$122$	0	0
$123$	188.138	1.52958
$124$	0	0
$125$	0	0
$126$	0	0
$127$	71.7345	0.564839	0.282419	−	0.959291i	$-0.408863\pi$
$127$	71.7345	0.564839	0.282419	+	0.959291i	$0.408863\pi$
$128$	0	0
$129$	217.580	1.68667
$130$	0	0
$131$	179.193i	1.36789i	0.729534	+	0.683945i	$0.239737\pi$
$131$	179.193i	1.36789i	−0.729534	+	0.683945i	$0.760263\pi$
$132$	0	0
$133$	19.0557i	0.143276i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 76.1641i	− 0.555942i	−0.960589	−	0.277971i	$-0.910338\pi$
$137$	− 76.1641i	− 0.555942i	0.960589	−	0.277971i	$-0.0896619\pi$
$138$	0	0
$139$	− 118.610i	− 0.853309i	−0.904415	−	0.426654i	$-0.859692\pi$
$139$	− 118.610i	− 0.853309i	0.904415	−	0.426654i	$-0.140308\pi$
$140$	0	0
$141$	−69.5279	−0.493105
$142$	0	0
$143$	330.164	2.30884
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−124.679	−0.848155
$148$	0	0
$149$	23.8034	0.159754	0.0798772	−	0.996805i	$-0.474547\pi$
$149$	23.8034	0.159754	0.0798772	+	0.996805i	$0.474547\pi$
$150$	0	0
$151$	294.748i	1.95197i	0.217835	+	0.975986i	$0.430101\pi$
$151$	294.748i	1.95197i	−0.217835	+	0.975986i	$0.569899\pi$
$152$	0	0
$153$	16.1115i	0.105304i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 179.803i	− 1.14524i	−0.819819	−	0.572622i	$-0.805926\pi$
$157$	− 179.803i	− 1.14524i	0.819819	−	0.572622i	$-0.194074\pi$
$158$	0	0
$159$	− 58.6950i	− 0.369151i
$160$	0	0
$161$	68.3607	0.424600
$162$	0	0
$163$	−146.705	−0.900031	−0.450016	−	0.893021i	$-0.648582\pi$
$163$	−146.705	−0.900031	−0.450016	+	0.893021i	$0.648582\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−74.7051	−0.447336	−0.223668	−	0.974665i	$-0.571803\pi$
$167$	−74.7051	−0.447336	−0.223668	+	0.974665i	$0.571803\pi$
$168$	0	0
$169$	−289.663	−1.71398
$170$	0	0
$171$	8.66874i	0.0506944i
$172$	0	0
$173$	− 91.3050i	− 0.527774i	−0.964554	−	0.263887i	$-0.914995\pi$
$173$	− 91.3050i	− 0.527774i	0.964554	−	0.263887i	$-0.0850047\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 351.830i	− 1.98774i
$178$	0	0
$179$	256.498i	1.43295i	0.697612	+	0.716476i	$0.254246\pi$
$179$	256.498i	1.43295i	−0.697612	+	0.716476i	$0.745754\pi$
$180$	0	0
$181$	−180.885	−0.999367	−0.499684	−	0.866208i	$-0.666550\pi$
$181$	−180.885	−0.999367	−0.499684	+	0.866208i	$0.666550\pi$
$182$	0	0
$183$	314.525	1.71871
$184$	0	0
$185$	0	0
$186$	0	0
$187$	168.721	0.902253
$188$	0	0
$189$	−78.8328	−0.417105
$190$	0	0
$191$	− 259.967i	− 1.36109i	−0.732708	−	0.680543i	$-0.761744\pi$
$191$	− 259.967i	− 1.36109i	0.732708	−	0.680543i	$-0.238256\pi$
$192$	0	0
$193$	− 28.3870i	− 0.147083i	−0.997292	−	0.0735414i	$-0.976570\pi$
$193$	− 28.3870i	− 0.147083i	0.997292	−	0.0735414i	$-0.0234301\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	103.082i	0.523259i	0.965168	+	0.261630i	$0.0842599\pi$
$197$	103.082i	0.523259i	−0.965168	+	0.261630i	$0.915740\pi$
$198$	0	0
$199$	− 52.2229i	− 0.262427i	−0.991354	−	0.131213i	$-0.958113\pi$
$199$	− 52.2229i	− 0.262427i	0.991354	−	0.131213i	$-0.0418873\pi$
$200$	0	0
$201$	229.803	1.14330
$202$	0	0
$203$	126.026	0.620819
$204$	0	0
$205$	0	0
$206$	0	0
$207$	31.0983	0.150233
$208$	0	0
$209$	90.7802	0.434355
$210$	0	0
$211$	− 46.2492i	− 0.219191i	−0.993976	−	0.109595i	$-0.965045\pi$
$211$	− 46.2492i	− 0.219191i	0.993976	−	0.109595i	$-0.0349555\pi$
$212$	0	0
$213$	128.276i	0.602233i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	82.6099i	0.380691i
$218$	0	0
$219$	60.5836i	0.276637i
$220$	0	0
$221$	−234.387	−1.06057
$222$	0	0
$223$	279.013	1.25118	0.625590	−	0.780152i	$-0.284858\pi$
$223$	279.013	1.25118	0.625590	+	0.780152i	$0.284858\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	57.7933	0.254596	0.127298	−	0.991865i	$-0.459370\pi$
$227$	57.7933	0.254596	0.127298	+	0.991865i	$0.459370\pi$
$228$	0	0
$229$	−288.610	−1.26031	−0.630153	−	0.776471i	$-0.717008\pi$
$229$	−288.610	−1.26031	−0.630153	+	0.776471i	$0.717008\pi$
$230$	0	0
$231$	− 161.443i	− 0.698886i
$232$	0	0
$233$	239.220i	1.02669i	0.858181	+	0.513347i	$0.171595\pi$
$233$	239.220i	1.02669i	−0.858181	+	0.513347i	$0.828405\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 118.111i	− 0.498361i
$238$	0	0
$239$	− 127.279i	− 0.532547i	−0.963898	−	0.266273i	$-0.914208\pi$
$239$	− 127.279i	− 0.532547i	0.963898	−	0.266273i	$-0.0857924\pi$
$240$	0	0
$241$	−254.859	−1.05751	−0.528753	−	0.848776i	$-0.677340\pi$
$241$	−254.859	−1.05751	−0.528753	+	0.848776i	$0.677340\pi$
$242$	0	0
$243$	−78.7376	−0.324023
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−126.111	−0.510573
$248$	0	0
$249$	418.577	1.68103
$250$	0	0
$251$	− 30.5248i	− 0.121613i	−0.998150	−	0.0608063i	$-0.980633\pi$
$251$	− 30.5248i	− 0.121613i	0.998150	−	0.0608063i	$-0.0193672\pi$
$252$	0	0
$253$	− 325.666i	− 1.28722i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	139.941i	0.544518i	0.962224	+	0.272259i	$0.0877708\pi$
$257$	139.941i	0.544518i	−0.962224	+	0.272259i	$0.912229\pi$
$258$	0	0
$259$	140.138i	0.541072i
$260$	0	0
$261$	57.3313	0.219660
$262$	0	0
$263$	−69.7608	−0.265250	−0.132625	−	0.991166i	$-0.542341\pi$
$263$	−69.7608	−0.265250	−0.132625	+	0.991166i	$0.542341\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−381.135	−1.42747
$268$	0	0
$269$	−13.4164	−0.0498751	−0.0249376	−	0.999689i	$-0.507939\pi$
$269$	−13.4164	−0.0498751	−0.0249376	+	0.999689i	$0.507939\pi$
$270$	0	0
$271$	271.469i	1.00173i	0.865525	+	0.500865i	$0.166985\pi$
$271$	271.469i	1.00173i	−0.865525	+	0.500865i	$0.833015\pi$
$272$	0	0
$273$	224.276i	0.821522i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	305.141i	1.10159i	0.834640	+	0.550796i	$0.185676\pi$
$277$	305.141i	1.10159i	−0.834640	+	0.550796i	$0.814324\pi$
$278$	0	0
$279$	37.5805i	0.134697i
$280$	0	0
$281$	−217.299	−0.773305	−0.386653	−	0.922225i	$-0.626369\pi$
$281$	−217.299	−0.773305	−0.386653	+	0.922225i	$0.626369\pi$
$282$	0	0
$283$	116.233	0.410717	0.205359	−	0.978687i	$-0.434164\pi$
$283$	116.233	0.410717	0.205359	+	0.978687i	$0.434164\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	188.138	0.655532
$288$	0	0
$289$	169.223	0.585546
$290$	0	0
$291$	263.193i	0.904445i
$292$	0	0
$293$	− 199.528i	− 0.680982i	−0.940248	−	0.340491i	$-0.889407\pi$
$293$	− 199.528i	− 0.680982i	0.940248	−	0.340491i	$-0.110593\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	375.554i	1.26449i
$298$	0	0
$299$	452.413i	1.51309i
$300$	0	0
$301$	217.580	0.722859
$302$	0	0
$303$	−70.4721	−0.232581
$304$	0	0
$305$	0	0
$306$	0	0
$307$	311.564	1.01487	0.507434	−	0.861691i	$-0.330594\pi$
$307$	311.564	1.01487	0.507434	+	0.861691i	$0.330594\pi$
$308$	0	0
$309$	−448.859	−1.45262
$310$	0	0
$311$	− 92.3081i	− 0.296810i	−0.988927	−	0.148405i	$-0.952586\pi$
$311$	− 92.3081i	− 0.296810i	0.988927	−	0.148405i	$-0.0474140\pi$
$312$	0	0
$313$	− 46.5511i	− 0.148725i	−0.997231	−	0.0743627i	$-0.976308\pi$
$313$	− 46.5511i	− 0.148725i	0.997231	−	0.0743627i	$-0.0236923\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 216.918i	− 0.684284i	−0.939648	−	0.342142i	$-0.888848\pi$
$317$	− 216.918i	− 0.684284i	0.939648	−	0.342142i	$-0.111152\pi$
$318$	0	0
$319$	− 600.381i	− 1.88207i
$320$	0	0
$321$	313.580	0.976886
$322$	0	0
$323$	−64.4458	−0.199523
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−99.8622	−0.305389
$328$	0	0
$329$	−69.5279	−0.211331
$330$	0	0
$331$	218.853i	0.661187i	0.943773	+	0.330594i	$0.107249\pi$
$331$	218.853i	0.661187i	−0.943773	+	0.330594i	$0.892751\pi$
$332$	0	0
$333$	63.7508i	0.191444i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 535.220i	− 1.58819i	−0.607794	−	0.794095i	$-0.707945\pi$
$337$	− 535.220i	− 1.58819i	0.607794	−	0.794095i	$-0.292055\pi$
$338$	0	0
$339$	− 164.859i	− 0.486310i
$340$	0	0
$341$	393.548	1.15410
$342$	0	0
$343$	−283.246	−0.825790
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.09830	−0.0204562	−0.0102281	−	0.999948i	$-0.503256\pi$
$347$	−7.09830	−0.0204562	−0.0102281	+	0.999948i	$0.503256\pi$
$348$	0	0
$349$	607.050	1.73940	0.869698	−	0.493583i	$-0.164313\pi$
$349$	607.050	1.73940	0.869698	+	0.493583i	$0.164313\pi$
$350$	0	0
$351$	− 521.718i	− 1.48638i
$352$	0	0
$353$	− 203.666i	− 0.576956i	−0.957486	−	0.288478i	$-0.906851\pi$
$353$	− 203.666i	− 0.576956i	0.957486	−	0.288478i	$-0.0931493\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	114.610i	0.321036i
$358$	0	0
$359$	336.997i	0.938710i	0.883010	+	0.469355i	$0.155514\pi$
$359$	336.997i	0.938710i	−0.883010	+	0.469355i	$0.844486\pi$
$360$	0	0
$361$	326.325	0.903948
$362$	0	0
$363$	−377.538	−1.04005
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−470.652	−1.28243	−0.641216	−	0.767361i	$-0.721570\pi$
$367$	−470.652	−1.28243	−0.641216	+	0.767361i	$0.721570\pi$
$368$	0	0
$369$	85.5867	0.231942
$370$	0	0
$371$	− 58.6950i	− 0.158208i
$372$	0	0
$373$	− 234.689i	− 0.629193i	−0.949226	−	0.314596i	$-0.898131\pi$
$373$	− 234.689i	− 0.629193i	0.949226	−	0.314596i	$-0.101869\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	834.046i	2.21232i
$378$	0	0
$379$	− 698.768i	− 1.84371i	−0.387530	−	0.921857i	$-0.626672\pi$
$379$	− 698.768i	− 1.84371i	0.387530	−	0.921857i	$-0.373328\pi$
$380$	0	0
$381$	232.138	0.609285
$382$	0	0
$383$	−495.649	−1.29412	−0.647062	−	0.762438i	$-0.724002\pi$
$383$	−495.649	−1.29412	−0.647062	+	0.762438i	$0.724002\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	98.9806	0.255764
$388$	0	0
$389$	−507.633	−1.30497	−0.652485	−	0.757802i	$-0.726273\pi$
$389$	−507.633	−1.30497	−0.652485	+	0.757802i	$0.726273\pi$
$390$	0	0
$391$	231.193i	0.591288i
$392$	0	0
$393$	579.882i	1.47553i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	304.354i	0.766636i	0.923616	+	0.383318i	$0.125219\pi$
$397$	304.354i	0.766636i	−0.923616	+	0.383318i	$0.874781\pi$
$398$	0	0
$399$	61.6656i	0.154550i
$400$	0	0
$401$	692.545	1.72704	0.863522	−	0.504311i	$-0.168253\pi$
$401$	692.545	1.72704	0.863522	+	0.504311i	$0.168253\pi$
$402$	0	0
$403$	−546.715	−1.35661
$404$	0	0
$405$	0	0
$406$	0	0
$407$	667.607	1.64031
$408$	0	0
$409$	−75.6455	−0.184952	−0.0924762	−	0.995715i	$-0.529478\pi$
$409$	−75.6455	−0.184952	−0.0924762	+	0.995715i	$0.529478\pi$
$410$	0	0
$411$	− 246.472i	− 0.599689i
$412$	0	0
$413$	− 351.830i	− 0.851888i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 383.830i	− 0.920455i
$418$	0	0
$419$	314.387i	0.750327i	0.926959	+	0.375163i	$0.122413\pi$
$419$	314.387i	0.750327i	−0.926959	+	0.375163i	$0.877587\pi$
$420$	0	0
$421$	242.085	0.575024	0.287512	−	0.957777i	$-0.407172\pi$
$421$	242.085	0.575024	0.287512	+	0.957777i	$0.407172\pi$
$422$	0	0
$423$	−31.6293	−0.0747737
$424$	0	0
$425$	0	0
$426$	0	0
$427$	314.525	0.736592
$428$	0	0
$429$	1068.43	2.49052
$430$	0	0
$431$	56.4659i	0.131011i	0.997852	+	0.0655057i	$0.0208661\pi$
$431$	56.4659i	0.131011i	−0.997852	+	0.0655057i	$0.979134\pi$
$432$	0	0
$433$	− 213.449i	− 0.492954i	−0.969149	−	0.246477i	$-0.920727\pi$
$433$	− 213.449i	− 0.492954i	0.969149	−	0.246477i	$-0.0792729\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	124.393i	0.284653i
$438$	0	0
$439$	− 68.7740i	− 0.156661i	−0.996927	−	0.0783303i	$-0.975041\pi$
$439$	− 68.7740i	− 0.156661i	0.996927	−	0.0783303i	$-0.0249589\pi$
$440$	0	0
$441$	−56.7183	−0.128613
$442$	0	0
$443$	−420.594	−0.949421	−0.474711	−	0.880142i	$-0.657447\pi$
$443$	−420.594	−0.949421	−0.474711	+	0.880142i	$0.657447\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	77.0294	0.172325
$448$	0	0
$449$	−18.8591	−0.0420025	−0.0210013	−	0.999779i	$-0.506685\pi$
$449$	−18.8591	−0.0420025	−0.0210013	+	0.999779i	$0.506685\pi$
$450$	0	0
$451$	− 896.276i	− 1.98731i
$452$	0	0
$453$	953.823i	2.10557i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 249.214i	− 0.545325i	−0.962110	−	0.272663i	$-0.912096\pi$
$457$	− 249.214i	− 0.545325i	0.962110	−	0.272663i	$-0.0879043\pi$
$458$	0	0
$459$	− 266.610i	− 0.580849i
$460$	0	0
$461$	−905.214	−1.96359	−0.981793	−	0.189951i	$-0.939167\pi$
$461$	−905.214	−1.96359	−0.981793	+	0.189951i	$0.939167\pi$
$462$	0	0
$463$	−267.767	−0.578331	−0.289165	−	0.957279i	$-0.593378\pi$
$463$	−267.767	−0.578331	−0.289165	+	0.957279i	$0.593378\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−268.148	−0.574192	−0.287096	−	0.957902i	$-0.592690\pi$
$467$	−268.148	−0.574192	−0.287096	+	0.957902i	$0.592690\pi$
$468$	0	0
$469$	229.803	0.489986
$470$	0	0
$471$	− 581.856i	− 1.23536i
$472$	0	0
$473$	− 1036.54i	− 2.19141i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 26.7013i	− 0.0559775i
$478$	0	0
$479$	597.390i	1.24716i	0.781759	+	0.623580i	$0.214323\pi$
$479$	597.390i	1.24716i	−0.781759	+	0.623580i	$0.785677\pi$
$480$	0	0
$481$	−927.437	−1.92814
$482$	0	0
$483$	221.220	0.458012
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−531.597	−1.09157	−0.545787	−	0.837924i	$-0.683769\pi$
$487$	−531.597	−1.09157	−0.545787	+	0.837924i	$0.683769\pi$
$488$	0	0
$489$	−474.748	−0.970854
$490$	0	0
$491$	− 255.967i	− 0.521319i	−0.965431	−	0.260659i	$-0.916060\pi$
$491$	− 255.967i	− 0.521319i	0.965431	−	0.260659i	$-0.0839399\pi$
$492$	0	0
$493$	426.217i	0.864537i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	128.276i	0.258100i
$498$	0	0
$499$	− 376.669i	− 0.754847i	−0.926041	−	0.377424i	$-0.876810\pi$
$499$	− 376.669i	− 0.754847i	0.926041	−	0.377424i	$-0.123190\pi$
$500$	0	0
$501$	−241.751	−0.482536
$502$	0	0
$503$	−257.157	−0.511247	−0.255623	−	0.966776i	$-0.582281\pi$
$503$	−257.157	−0.511247	−0.255623	+	0.966776i	$0.582281\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−937.368	−1.84885
$508$	0	0
$509$	−325.489	−0.639468	−0.319734	−	0.947507i	$-0.603594\pi$
$509$	−325.489	−0.639468	−0.319734	+	0.947507i	$0.603594\pi$
$510$	0	0
$511$	60.5836i	0.118559i
$512$	0	0
$513$	− 143.449i	− 0.279628i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	331.226i	0.640669i
$518$	0	0
$519$	− 295.469i	− 0.569304i
$520$	0	0
$521$	−393.882	−0.756012	−0.378006	−	0.925803i	$-0.623390\pi$
$521$	−393.882	−0.756012	−0.378006	+	0.925803i	$0.623390\pi$
$522$	0	0
$523$	−416.987	−0.797298	−0.398649	−	0.917104i	$-0.630521\pi$
$523$	−416.987	−0.797298	−0.398649	+	0.917104i	$0.630521\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−279.384	−0.530140
$528$	0	0
$529$	−82.7508	−0.156429
$530$	0	0
$531$	− 160.053i	− 0.301417i
$532$	0	0
$533$	1245.10i	2.33603i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	830.046i	1.54571i
$538$	0	0
$539$	593.961i	1.10197i
$540$	0	0
$541$	513.226	0.948662	0.474331	−	0.880347i	$-0.342690\pi$
$541$	513.226	0.948662	0.474331	+	0.880347i	$0.342690\pi$
$542$	0	0
$543$	−585.358	−1.07801
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−957.420	−1.75031	−0.875156	−	0.483842i	$-0.839241\pi$
$547$	−957.420	−1.75031	−0.875156	+	0.483842i	$0.839241\pi$
$548$	0	0
$549$	143.082	0.260623
$550$	0	0
$551$	229.325i	0.416198i
$552$	0	0
$553$	− 118.111i	− 0.213583i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1025.74i	− 1.84155i	−0.390092	−	0.920776i	$-0.627557\pi$
$557$	− 1025.74i	− 1.84155i	0.390092	−	0.920776i	$-0.372443\pi$
$558$	0	0
$559$	1439.96i	2.57595i
$560$	0	0
$561$	545.994	0.973251
$562$	0	0
$563$	866.555	1.53917	0.769587	−	0.638542i	$-0.220462\pi$
$563$	866.555	1.53917	0.769587	+	0.638542i	$0.220462\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−297.984	−0.525545
$568$	0	0
$569$	−630.085	−1.10736	−0.553678	−	0.832731i	$-0.686776\pi$
$569$	−630.085	−1.10736	−0.553678	+	0.832731i	$0.686776\pi$
$570$	0	0
$571$	− 837.358i	− 1.46648i	−0.679972	−	0.733238i	$-0.738008\pi$
$571$	− 837.358i	− 1.46648i	0.679972	−	0.733238i	$-0.261992\pi$
$572$	0	0
$573$	− 841.272i	− 1.46819i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1104.83i	− 1.91479i	−0.288785	−	0.957394i	$-0.593251\pi$
$577$	− 1104.83i	− 1.91479i	0.288785	−	0.957394i	$-0.406749\pi$
$578$	0	0
$579$	− 91.8622i	− 0.158657i
$580$	0	0
$581$	418.577	0.720443
$582$	0	0
$583$	−279.619	−0.479621
$584$	0	0
$585$	0	0
$586$	0	0
$587$	254.292	0.433206	0.216603	−	0.976260i	$-0.430502\pi$
$587$	254.292	0.433206	0.216603	+	0.976260i	$0.430502\pi$
$588$	0	0
$589$	−150.322	−0.255216
$590$	0	0
$591$	333.580i	0.564434i
$592$	0	0
$593$	332.440i	0.560606i	0.959912	+	0.280303i	$0.0904350\pi$
$593$	332.440i	0.560606i	−0.959912	+	0.280303i	$0.909565\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 168.997i	− 0.283077i
$598$	0	0
$599$	− 106.715i	− 0.178156i	−0.996025	−	0.0890778i	$-0.971608\pi$
$599$	− 106.715i	− 0.178156i	0.996025	−	0.0890778i	$-0.0283920\pi$
$600$	0	0
$601$	−600.748	−0.999580	−0.499790	−	0.866147i	$-0.666590\pi$
$601$	−600.748	−0.999580	−0.499790	+	0.866147i	$0.666590\pi$
$602$	0	0
$603$	104.541	0.173368
$604$	0	0
$605$	0	0
$606$	0	0
$607$	129.728	0.213720	0.106860	−	0.994274i	$-0.465920\pi$
$607$	129.728	0.213720	0.106860	+	0.994274i	$0.465920\pi$
$608$	0	0
$609$	407.830	0.669671
$610$	0	0
$611$	− 460.138i	− 0.753090i
$612$	0	0
$613$	− 708.958i	− 1.15654i	−0.815846	−	0.578269i	$-0.803728\pi$
$613$	− 708.958i	− 1.15654i	0.815846	−	0.578269i	$-0.196272\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 375.876i	− 0.609200i	−0.952480	−	0.304600i	$-0.901477\pi$
$617$	− 375.876i	− 0.609200i	0.952480	−	0.304600i	$-0.0985227\pi$
$618$	0	0
$619$	106.322i	0.171764i	0.996305	+	0.0858820i	$0.0273708\pi$
$619$	106.322i	0.171764i	−0.996305	+	0.0858820i	$0.972629\pi$
$620$	0	0
$621$	−514.610	−0.828679
$622$	0	0
$623$	−381.135	−0.611773
$624$	0	0
$625$	0	0
$626$	0	0
$627$	293.771	0.468534
$628$	0	0
$629$	−473.941	−0.753484
$630$	0	0
$631$	− 1129.67i	− 1.79029i	−0.445775	−	0.895145i	$-0.647072\pi$
$631$	− 1129.67i	− 1.79029i	0.445775	−	0.895145i	$-0.352928\pi$
$632$	0	0
$633$	− 149.666i	− 0.236439i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 825.128i	− 1.29534i
$638$	0	0
$639$	58.3545i	0.0913215i
$640$	0	0
$641$	−817.404	−1.27520	−0.637601	−	0.770367i	$-0.720073\pi$
$641$	−817.404	−1.27520	−0.637601	+	0.770367i	$0.720073\pi$
$642$	0	0
$643$	−9.60296	−0.0149346	−0.00746731	−	0.999972i	$-0.502377\pi$
$643$	−9.60296	−0.0149346	−0.00746731	+	0.999972i	$0.502377\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	500.849	0.774110	0.387055	−	0.922057i	$-0.373492\pi$
$647$	500.849	0.774110	0.387055	+	0.922057i	$0.373492\pi$
$648$	0	0
$649$	−1676.09	−2.58258
$650$	0	0
$651$	267.331i	0.410647i
$652$	0	0
$653$	99.1471i	0.151833i	0.997114	+	0.0759166i	$0.0241883\pi$
$653$	99.1471i	0.151833i	−0.997114	+	0.0759166i	$0.975812\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	27.5604i	0.0419488i
$658$	0	0
$659$	− 953.260i	− 1.44653i	−0.690573	−	0.723263i	$-0.742642\pi$
$659$	− 953.260i	− 1.44653i	0.690573	−	0.723263i	$-0.257358\pi$
$660$	0	0
$661$	−24.3545	−0.0368449	−0.0184224	−	0.999830i	$-0.505864\pi$
$661$	−24.3545	−0.0368449	−0.0184224	+	0.999830i	$0.505864\pi$
$662$	0	0
$663$	−758.492	−1.14403
$664$	0	0
$665$	0	0
$666$	0	0
$667$	822.683	1.23341
$668$	0	0
$669$	902.906	1.34963
$670$	0	0
$671$	− 1498.37i	− 2.23305i
$672$	0	0
$673$	− 393.437i	− 0.584601i	−0.956327	−	0.292301i	$-0.905579\pi$
$673$	− 393.437i	− 0.584601i	0.956327	−	0.292301i	$-0.0944208\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	471.082i	0.695838i	0.937525	+	0.347919i	$0.113112\pi$
$677$	471.082i	0.695838i	−0.937525	+	0.347919i	$0.886888\pi$
$678$	0	0
$679$	263.193i	0.387619i
$680$	0	0
$681$	187.023	0.274630
$682$	0	0
$683$	−0.200440	−0.000293470 0	−0.000146735	−	1.00000i	$-0.500047\pi$
$683$	−0.200440	−0.000293470 0	−0.000146735		1.00000i	$0.500047\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−933.961	−1.35948
$688$	0	0
$689$	388.446	0.563782
$690$	0	0
$691$	224.689i	0.325165i	0.986695	+	0.162582i	$0.0519823\pi$
$691$	224.689i	0.325165i	−0.986695	+	0.162582i	$0.948018\pi$
$692$	0	0
$693$	− 73.4427i	− 0.105978i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	636.276i	0.912877i
$698$	0	0
$699$	774.132i	1.10748i
$700$	0	0
$701$	−237.915	−0.339394	−0.169697	−	0.985496i	$-0.554279\pi$
$701$	−237.915	−0.339394	−0.169697	+	0.985496i	$0.554279\pi$
$702$	0	0
$703$	−255.003	−0.362736
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−70.4721	−0.0996777
$708$	0	0
$709$	70.6037	0.0995821	0.0497910	−	0.998760i	$-0.484144\pi$
$709$	70.6037	0.0995821	0.0497910	+	0.998760i	$0.484144\pi$
$710$	0	0
$711$	− 53.7307i	− 0.0755706i
$712$	0	0
$713$	539.266i	0.756334i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 411.882i	− 0.574452i
$718$	0	0
$719$	49.6130i	0.0690028i	0.999405	+	0.0345014i	$0.0109843\pi$
$719$	49.6130i	0.0690028i	−0.999405	+	0.0345014i	$0.989016\pi$
$720$	0	0
$721$	−448.859	−0.622551
$722$	0	0
$723$	−824.741	−1.14072
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1181.87	−1.62568	−0.812838	−	0.582490i	$-0.802078\pi$
$727$	−1181.87	−1.62568	−0.812838	+	0.582490i	$0.802078\pi$
$728$	0	0
$729$	573.938	0.787295
$730$	0	0
$731$	735.850i	1.00663i
$732$	0	0
$733$	41.5743i	0.0567180i	0.999598	+	0.0283590i	$0.00902816\pi$
$733$	41.5743i	0.0567180i	−0.999598	+	0.0283590i	$0.990972\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1094.77i	− 1.48544i
$738$	0	0
$739$	− 1180.49i	− 1.59741i	−0.601723	−	0.798705i	$-0.705519\pi$
$739$	− 1180.49i	− 1.59741i	0.601723	−	0.798705i	$-0.294481\pi$
$740$	0	0
$741$	−408.105	−0.550749
$742$	0	0
$743$	1349.95	1.81689	0.908446	−	0.418002i	$-0.137269\pi$
$743$	1349.95	1.81689	0.908446	+	0.418002i	$0.137269\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	190.417	0.254909
$748$	0	0
$749$	313.580	0.418666
$750$	0	0
$751$	1421.52i	1.89283i	0.322953	+	0.946415i	$0.395324\pi$
$751$	1421.52i	1.89283i	−0.322953	+	0.946415i	$0.604676\pi$
$752$	0	0
$753$	− 98.7802i	− 0.131182i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1340.34i	1.77060i	0.465023	+	0.885299i	$0.346046\pi$
$757$	1340.34i	1.77060i	−0.465023	+	0.885299i	$0.653954\pi$
$758$	0	0
$759$	− 1053.88i	− 1.38851i
$760$	0	0
$761$	484.334	0.636445	0.318222	−	0.948016i	$-0.396914\pi$
$761$	484.334	0.636445	0.318222	+	0.948016i	$0.396914\pi$
$762$	0	0
$763$	−99.8622	−0.130881
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2328.42	3.03575
$768$	0	0
$769$	572.217	0.744105	0.372052	−	0.928212i	$-0.378654\pi$
$769$	572.217	0.744105	0.372052	+	0.928212i	$0.378654\pi$
$770$	0	0
$771$	452.859i	0.587366i
$772$	0	0
$773$	− 861.193i	− 1.11409i	−0.830481	−	0.557046i	$-0.811935\pi$
$773$	− 861.193i	− 1.11409i	0.830481	−	0.557046i	$-0.188065\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	453.495i	0.583649i
$778$	0	0
$779$	342.347i	0.439470i
$780$	0	0
$781$	611.096	0.782453
$782$	0	0
$783$	−948.709	−1.21163
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−597.801	−0.759595	−0.379797	−	0.925070i	$-0.624006\pi$
$787$	−597.801	−0.759595	−0.379797	+	0.925070i	$0.624006\pi$
$788$	0	0
$789$	−225.751	−0.286123
$790$	0	0
$791$	− 164.859i	− 0.208419i
$792$	0	0
$793$	2081.54i	2.62489i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 170.150i	− 0.213488i	−0.994287	−	0.106744i	$-0.965957\pi$
$797$	− 170.150i	− 0.213488i	0.994287	−	0.106744i	$-0.0340426\pi$
$798$	0	0
$799$	− 235.141i	− 0.294294i
$800$	0	0
$801$	−173.384	−0.216459
$802$	0	0
$803$	288.616	0.359422
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−43.4164	−0.0537998
$808$	0	0
$809$	662.105	0.818424	0.409212	−	0.912439i	$-0.365804\pi$
$809$	662.105	0.818424	0.409212	+	0.912439i	$0.365804\pi$
$810$	0	0
$811$	− 792.964i	− 0.977761i	−0.872351	−	0.488881i	$-0.837405\pi$
$811$	− 792.964i	− 0.977761i	0.872351	−	0.488881i	$-0.162595\pi$
$812$	0	0
$813$	878.492i	1.08056i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	395.923i	0.484605i
$818$	0	0
$819$	102.026i	0.124574i
$820$	0	0
$821$	−683.240	−0.832205	−0.416102	−	0.909318i	$-0.636604\pi$
$821$	−683.240	−0.832205	−0.416102	+	0.909318i	$0.636604\pi$
$822$	0	0
$823$	418.580	0.508602	0.254301	−	0.967125i	$-0.418155\pi$
$823$	418.580	0.508602	0.254301	+	0.967125i	$0.418155\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	381.019	0.460725	0.230362	−	0.973105i	$-0.426009\pi$
$827$	381.019	0.460725	0.230362	+	0.973105i	$0.426009\pi$
$828$	0	0
$829$	−1274.06	−1.53686	−0.768432	−	0.639931i	$-0.778963\pi$
$829$	−1274.06	−1.53686	−0.768432	+	0.639931i	$0.778963\pi$
$830$	0	0
$831$	987.457i	1.18828i
$832$	0	0
$833$	− 421.659i	− 0.506194i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 621.876i	− 0.742982i
$838$	0	0
$839$	− 409.272i	− 0.487810i	−0.969799	−	0.243905i	$-0.921572\pi$
$839$	− 409.272i	− 0.487810i	0.969799	−	0.243905i	$-0.0784285\pi$
$840$	0	0
$841$	675.656	0.803396
$842$	0	0
$843$	−703.193	−0.834156
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−377.538	−0.445735
$848$	0	0
$849$	376.138	0.443036
$850$	0	0
$851$	914.800i	1.07497i
$852$	0	0
$853$	744.577i	0.872893i	0.899730	+	0.436446i	$0.143763\pi$
$853$	744.577i	0.872893i	−0.899730	+	0.436446i	$0.856237\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 156.715i	− 0.182865i	−0.995811	−	0.0914324i	$-0.970855\pi$
$857$	− 156.715i	− 0.182865i	0.995811	−	0.0914324i	$-0.0291445\pi$
$858$	0	0
$859$	721.155i	0.839528i	0.907633	+	0.419764i	$0.137887\pi$
$859$	721.155i	0.839528i	−0.907633	+	0.419764i	$0.862113\pi$
$860$	0	0
$861$	608.827	0.707116
$862$	0	0
$863$	1011.51	1.17209	0.586044	−	0.810279i	$-0.300685\pi$
$863$	1011.51	1.17209	0.586044	+	0.810279i	$0.300685\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	547.617	0.631623
$868$	0	0
$869$	−562.675	−0.647497
$870$	0	0
$871$	1520.85i	1.74609i
$872$	0	0
$873$	119.731i	0.137149i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	896.865i	1.02265i	0.859387	+	0.511326i	$0.170845\pi$
$877$	896.865i	1.02265i	−0.859387	+	0.511326i	$0.829155\pi$
$878$	0	0
$879$	− 645.686i	− 0.734569i
$880$	0	0
$881$	754.243	0.856121	0.428061	−	0.903750i	$-0.359197\pi$
$881$	754.243	0.856121	0.428061	+	0.903750i	$0.359197\pi$
$882$	0	0
$883$	360.456	0.408217	0.204109	−	0.978948i	$-0.434570\pi$
$883$	360.456	0.408217	0.204109	+	0.978948i	$0.434570\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	910.227	1.02619	0.513093	−	0.858333i	$-0.328500\pi$
$887$	910.227	1.02619	0.513093	+	0.858333i	$0.328500\pi$
$888$	0	0
$889$	232.138	0.261122
$890$	0	0
$891$	1419.57i	1.59324i
$892$	0	0
$893$	− 126.517i	− 0.141676i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1464.04i	1.63215i
$898$	0	0
$899$	994.164i	1.10586i
$900$	0	0
$901$	198.505	0.220316
$902$	0	0
$903$	704.105	0.779740
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1217.47	−1.34231	−0.671154	−	0.741318i	$-0.734201\pi$
$907$	−1217.47	−1.34231	−0.671154	+	0.741318i	$0.734201\pi$
$908$	0	0
$909$	−32.0588	−0.0352682
$910$	0	0
$911$	234.630i	0.257552i	0.991674	+	0.128776i	$0.0411048\pi$
$911$	234.630i	0.257552i	−0.991674	+	0.128776i	$0.958895\pi$
$912$	0	0
$913$	− 1994.07i	− 2.18409i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	579.882i	0.632369i
$918$	0	0
$919$	− 748.839i	− 0.814841i	−0.913241	−	0.407421i	$-0.866428\pi$
$919$	− 748.839i	− 0.814841i	0.913241	−	0.407421i	$-0.133572\pi$
$920$	0	0
$921$	1008.24	1.09473
$922$	0	0
$923$	−848.932	−0.919753
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−204.193	−0.220273
$928$	0	0
$929$	251.698	0.270935	0.135467	−	0.990782i	$-0.456746\pi$
$929$	251.698	0.270935	0.135467	+	0.990782i	$0.456746\pi$
$930$	0	0
$931$	− 226.873i	− 0.243687i
$932$	0	0
$933$	− 298.715i	− 0.320166i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 906.210i	− 0.967140i	−0.875306	−	0.483570i	$-0.839340\pi$
$937$	− 906.210i	− 0.967140i	0.875306	−	0.483570i	$-0.160660\pi$
$938$	0	0
$939$	− 150.642i	− 0.160429i
$940$	0	0
$941$	−249.344	−0.264977	−0.132489	−	0.991185i	$-0.542297\pi$
$941$	−249.344	−0.264977	−0.132489	+	0.991185i	$0.542297\pi$
$942$	0	0
$943$	1228.14	1.30237
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1318.88	1.39270	0.696348	−	0.717704i	$-0.254807\pi$
$947$	1318.88	1.39270	0.696348	+	0.717704i	$0.254807\pi$
$948$	0	0
$949$	−400.944	−0.422491
$950$	0	0
$951$	− 701.961i	− 0.738130i
$952$	0	0
$953$	− 369.777i	− 0.388014i	−0.981000	−	0.194007i	$-0.937852\pi$
$953$	− 369.777i	− 0.388014i	0.981000	−	0.194007i	$-0.0621484\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1942.87i	− 2.03017i
$958$	0	0
$959$	− 246.472i	− 0.257010i
$960$	0	0
$961$	309.328	0.321882
$962$	0	0
$963$	142.652	0.148133
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−244.764	−0.253117	−0.126558	−	0.991959i	$-0.540393\pi$
$967$	−244.764	−0.253117	−0.126558	+	0.991959i	$0.540393\pi$
$968$	0	0
$969$	−208.551	−0.215223
$970$	0	0
$971$	− 63.2461i	− 0.0651350i	−0.999470	−	0.0325675i	$-0.989632\pi$
$971$	− 63.2461i	− 0.0651350i	0.999470	−	0.0325675i	$-0.0103684\pi$
$972$	0	0
$973$	− 383.830i	− 0.394481i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 316.885i	− 0.324345i	−0.986762	−	0.162173i	$-0.948150\pi$
$977$	− 316.885i	− 0.324345i	0.986762	−	0.162173i	$-0.0518502\pi$
$978$	0	0
$979$	1815.70i	1.85465i
$980$	0	0
$981$	−45.4288	−0.0463087
$982$	0	0
$983$	535.564	0.544826	0.272413	−	0.962180i	$-0.412178\pi$
$983$	535.564	0.544826	0.272413	+	0.962180i	$0.412178\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−224.997	−0.227960
$988$	0	0
$989$	1420.34	1.43613
$990$	0	0
$991$	− 1022.45i	− 1.03173i	−0.856669	−	0.515866i	$-0.827470\pi$
$991$	− 1022.45i	− 1.03173i	0.856669	−	0.515866i	$-0.172530\pi$
$992$	0	0
$993$	708.223i	0.713215i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1952.22i	1.95810i	0.203623	+	0.979049i	$0.434728\pi$
$997$	1952.22i	1.95810i	−0.203623	+	0.979049i	$0.565272\pi$
$998$	0	0
$999$	− 1054.94i	− 1.05599i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.3.h.k.1599.3		4
4.3	odd	2		1600.3.h.f.1599.2		4
5.2	odd	4		320.3.b.d.191.1		4
5.3	odd	4		1600.3.b.u.1151.4		4
5.4	even	2		1600.3.h.f.1599.1		4
8.3	odd	2		800.3.h.h.799.3		4
8.5	even	2		800.3.h.e.799.2		4
15.2	even	4		2880.3.e.h.2431.2		4
20.3	even	4		1600.3.b.u.1151.1		4
20.7	even	4		320.3.b.d.191.4		4
20.19	odd	2	inner	1600.3.h.k.1599.4		4
40.3	even	4		800.3.b.g.351.4		4
40.13	odd	4		800.3.b.g.351.1		4
40.19	odd	2		800.3.h.e.799.1		4
40.27	even	4		160.3.b.b.31.1	✓	4
40.29	even	2		800.3.h.h.799.4		4
40.37	odd	4		160.3.b.b.31.4	yes	4
60.47	odd	4		2880.3.e.h.2431.1		4
80.27	even	4		1280.3.g.c.1151.4		4
80.37	odd	4		1280.3.g.b.1151.2		4
80.67	even	4		1280.3.g.b.1151.1		4
80.77	odd	4		1280.3.g.c.1151.3		4
120.77	even	4		1440.3.e.a.991.4		4
120.107	odd	4		1440.3.e.a.991.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.b.b.31.1	✓	4	40.27	even	4
160.3.b.b.31.4	yes	4	40.37	odd	4
320.3.b.d.191.1		4	5.2	odd	4
320.3.b.d.191.4		4	20.7	even	4
800.3.b.g.351.1		4	40.13	odd	4
800.3.b.g.351.4		4	40.3	even	4
800.3.h.e.799.1		4	40.19	odd	2
800.3.h.e.799.2		4	8.5	even	2
800.3.h.h.799.3		4	8.3	odd	2
800.3.h.h.799.4		4	40.29	even	2
1280.3.g.b.1151.1		4	80.67	even	4
1280.3.g.b.1151.2		4	80.37	odd	4
1280.3.g.c.1151.3		4	80.77	odd	4
1280.3.g.c.1151.4		4	80.27	even	4
1440.3.e.a.991.3		4	120.107	odd	4
1440.3.e.a.991.4		4	120.77	even	4
1600.3.b.u.1151.1		4	20.3	even	4
1600.3.b.u.1151.4		4	5.3	odd	4
1600.3.h.f.1599.1		4	5.4	even	2
1600.3.h.f.1599.2		4	4.3	odd	2
1600.3.h.k.1599.3		4	1.1	even	1	trivial
1600.3.h.k.1599.4		4	20.19	odd	2	inner
2880.3.e.h.2431.1		4	60.47	odd	4
2880.3.e.h.2431.2		4	15.2	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1600.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.23607 q^{3} +3.23607 q^{7} +1.47214 q^{9} +O(q^{10})\)	\(q+3.23607 q^{3} +3.23607 q^{7} +1.47214 q^{9} -15.4164i q^{11} +21.4164i q^{13} +10.9443i q^{17} +5.88854i q^{19} +10.4721 q^{21} +21.1246 q^{23} -24.3607 q^{27} +38.9443 q^{29} +25.5279i q^{31} -49.8885i q^{33} +43.3050i q^{37} +69.3050i q^{39} +58.1378 q^{41} +67.2361 q^{43} -21.4853 q^{47} -38.5279 q^{49} +35.4164i q^{51} -18.1378i q^{53} +19.0557i q^{57} -108.721i q^{59} +97.1935 q^{61} +4.76393 q^{63} +71.0132 q^{67} +68.3607 q^{69} +39.6393i q^{71} +18.7214i q^{73} -49.8885i q^{77} -36.4984i q^{79} -92.0820 q^{81} +129.348 q^{83} +126.026 q^{87} -117.777 q^{89} +69.3050i q^{91} +82.6099i q^{93} +81.3313i q^{97} -22.6950i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{7} - 12 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^7 - 12 * q^9	\( 4 q + 4 q^{3} + 4 q^{7} - 12 q^{9} + 24 q^{21} + 4 q^{23} - 8 q^{27} + 120 q^{29} + 260 q^{43} + 84 q^{47} - 172 q^{49} + 192 q^{61} + 28 q^{63} + 132 q^{67} + 184 q^{69} - 100 q^{81} + 580 q^{83} + 200 q^{87} - 328 q^{89}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^7 - 12 * q^9 + 24 * q^21 + 4 * q^23 - 8 * q^27 + 120 * q^29 + 260 * q^43 + 84 * q^47 - 172 * q^49 + 192 * q^61 + 28 * q^63 + 132 * q^67 + 184 * q^69 - 100 * q^81 + 580 * q^83 + 200 * q^87 - 328 * q^89

Introduction

L-functions

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