LMFDB - Embedded newform 288.4.d.a.145.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,4,Mod(145,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("288.145");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	288.d (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:	$16.9925500817$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 2 $ x^2 - x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			145.1
Root			$0.500000 + 1.32288i$ of defining polynomial
Character	$\chi$	$=$	288.145
Dual form			288.4.d.a.145.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.5830i q^{5} +8.00000 q^{7} +O(q^{10})$	$q-10.5830i q^{5} +8.00000 q^{7} +15.8745i q^{11} -52.9150i q^{13} +14.0000 q^{17} -37.0405i q^{19} -152.000 q^{23} +13.0000 q^{25} -158.745i q^{29} -224.000 q^{31} -84.6640i q^{35} -243.409i q^{37} +70.0000 q^{41} -439.195i q^{43} +336.000 q^{47} -279.000 q^{49} +31.7490i q^{53} +168.000 q^{55} -534.442i q^{59} -95.2470i q^{61} -560.000 q^{65} +174.620i q^{67} -72.0000 q^{71} -294.000 q^{73} +126.996i q^{77} +464.000 q^{79} +545.025i q^{83} -148.162i q^{85} -266.000 q^{89} -423.320i q^{91} -392.000 q^{95} +994.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{7}+O(q^{10}) $ 2 * q + 16 * q^7	$ 2 q + 16 q^{7} + 28 q^{17} - 304 q^{23} + 26 q^{25} - 448 q^{31} + 140 q^{41} + 672 q^{47} - 558 q^{49} + 336 q^{55} - 1120 q^{65} - 144 q^{71} - 588 q^{73} + 928 q^{79} - 532 q^{89} - 784 q^{95} + 1988 q^{97}+O(q^{100}) $ 2 * q + 16 * q^7 + 28 * q^17 - 304 * q^23 + 26 * q^25 - 448 * q^31 + 140 * q^41 + 672 * q^47 - 558 * q^49 + 336 * q^55 - 1120 * q^65 - 144 * q^71 - 588 * q^73 + 928 * q^79 - 532 * q^89 - 784 * q^95 + 1988 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 10.5830i	− 0.946573i	−0.880909	−	0.473286i	$-0.843068\pi$
$5$	− 10.5830i	− 0.946573i	0.880909	−	0.473286i	$-0.156932\pi$
$6$	0	0
$7$	8.00000	0.431959	0.215980	−	0.976398i	$-0.430705\pi$
$7$	8.00000	0.431959	0.215980	+	0.976398i	$0.430705\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	15.8745i	0.435122i	0.976047	+	0.217561i	$0.0698101\pi$
$11$	15.8745i	0.435122i	−0.976047	+	0.217561i	$0.930190\pi$
$12$	0	0
$13$	− 52.9150i	− 1.12892i	−0.825460	−	0.564461i	$-0.809084\pi$
$13$	− 52.9150i	− 1.12892i	0.825460	−	0.564461i	$-0.190916\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	14.0000	0.199735	0.0998676	−	0.995001i	$-0.468158\pi$
$17$	14.0000	0.199735	0.0998676	+	0.995001i	$0.468158\pi$
$18$	0	0
$19$	− 37.0405i	− 0.447246i	−0.974676	−	0.223623i	$-0.928212\pi$
$19$	− 37.0405i	− 0.447246i	0.974676	−	0.223623i	$-0.0717885\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−152.000	−1.37801	−0.689004	−	0.724757i	$-0.741952\pi$
$23$	−152.000	−1.37801	−0.689004	+	0.724757i	$0.741952\pi$
$24$	0	0
$25$	13.0000	0.104000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 158.745i	− 1.01649i	−0.861212	−	0.508245i	$-0.830294\pi$
$29$	− 158.745i	− 1.01649i	0.861212	−	0.508245i	$-0.169706\pi$
$30$	0	0
$31$	−224.000	−1.29779	−0.648897	−	0.760877i	$-0.724769\pi$
$31$	−224.000	−1.29779	−0.648897	+	0.760877i	$0.724769\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 84.6640i	− 0.408881i
$36$	0	0
$37$	− 243.409i	− 1.08152i	−0.841177	−	0.540760i	$-0.818137\pi$
$37$	− 243.409i	− 1.08152i	0.841177	−	0.540760i	$-0.181863\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	70.0000	0.266638	0.133319	−	0.991073i	$-0.457436\pi$
$41$	70.0000	0.266638	0.133319	+	0.991073i	$0.457436\pi$
$42$	0	0
$43$	− 439.195i	− 1.55759i	−0.627276	−	0.778797i	$-0.715830\pi$
$43$	− 439.195i	− 1.55759i	0.627276	−	0.778797i	$-0.284170\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	336.000	1.04278	0.521390	−	0.853319i	$-0.325414\pi$
$47$	336.000	1.04278	0.521390	+	0.853319i	$0.325414\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	0	0
$51$	0	0
$52$	0	0
$53$	31.7490i	0.0822842i	0.999153	+	0.0411421i	$0.0130996\pi$
$53$	31.7490i	0.0822842i	−0.999153	+	0.0411421i	$0.986900\pi$
$54$	0	0
$55$	168.000	0.411875
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 534.442i	− 1.17929i	−0.807661	−	0.589647i	$-0.799267\pi$
$59$	− 534.442i	− 1.17929i	0.807661	−	0.589647i	$-0.200733\pi$
$60$	0	0
$61$	− 95.2470i	− 0.199920i	−0.994991	−	0.0999601i	$-0.968128\pi$
$61$	− 95.2470i	− 0.199920i	0.994991	−	0.0999601i	$-0.0318715\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−560.000	−1.06861
$66$	0	0
$67$	174.620i	0.318406i	0.987246	+	0.159203i	$0.0508924\pi$
$67$	174.620i	0.318406i	−0.987246	+	0.159203i	$0.949108\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−72.0000	−0.120350	−0.0601748	−	0.998188i	$-0.519166\pi$
$71$	−72.0000	−0.120350	−0.0601748	+	0.998188i	$0.519166\pi$
$72$	0	0
$73$	−294.000	−0.471371	−0.235686	−	0.971829i	$-0.575734\pi$
$73$	−294.000	−0.471371	−0.235686	+	0.971829i	$0.575734\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	126.996i	0.187955i
$78$	0	0
$79$	464.000	0.660811	0.330406	−	0.943839i	$-0.392814\pi$
$79$	464.000	0.660811	0.330406	+	0.943839i	$0.392814\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	545.025i	0.720774i	0.932803	+	0.360387i	$0.117355\pi$
$83$	545.025i	0.720774i	−0.932803	+	0.360387i	$0.882645\pi$
$84$	0	0
$85$	− 148.162i	− 0.189064i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−266.000	−0.316808	−0.158404	−	0.987374i	$-0.550635\pi$
$89$	−266.000	−0.316808	−0.158404	+	0.987374i	$0.550635\pi$
$90$	0	0
$91$	− 423.320i	− 0.487649i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−392.000	−0.423351
$96$	0	0
$97$	994.000	1.04047	0.520234	−	0.854024i	$-0.325845\pi$
$97$	994.000	1.04047	0.520234	+	0.854024i	$0.325845\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	751.393i	0.740262i	0.928980	+	0.370131i	$0.120687\pi$
$101$	751.393i	0.740262i	−0.928980	+	0.370131i	$0.879313\pi$
$102$	0	0
$103$	−1176.00	−1.12500	−0.562499	−	0.826798i	$-0.690160\pi$
$103$	−1176.00	−1.12500	−0.562499	+	0.826798i	$0.690160\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	269.867i	0.243822i	0.992541	+	0.121911i	$0.0389023\pi$
$107$	269.867i	0.243822i	−0.992541	+	0.121911i	$0.961098\pi$
$108$	0	0
$109$	1894.36i	1.66465i	0.554290	+	0.832324i	$0.312990\pi$
$109$	1894.36i	1.66465i	−0.554290	+	0.832324i	$0.687010\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1710.00	1.42357	0.711784	−	0.702398i	$-0.247887\pi$
$113$	1710.00	1.42357	0.711784	+	0.702398i	$0.247887\pi$
$114$	0	0
$115$	1608.62i	1.30439i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	112.000	0.0862775
$120$	0	0
$121$	1079.00	0.810669
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1460.45i	− 1.04502i
$126$	0	0
$127$	1664.00	1.16265	0.581323	−	0.813673i	$-0.302535\pi$
$127$	1664.00	1.16265	0.581323	+	0.813673i	$0.302535\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	672.021i	0.448204i	0.974566	+	0.224102i	$0.0719449\pi$
$131$	672.021i	0.448204i	−0.974566	+	0.224102i	$0.928055\pi$
$132$	0	0
$133$	− 296.324i	− 0.193192i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1062.00	0.662283	0.331142	−	0.943581i	$-0.392566\pi$
$137$	1062.00	0.662283	0.331142	+	0.943581i	$0.392566\pi$
$138$	0	0
$139$	2693.37i	1.64352i	0.569835	+	0.821759i	$0.307007\pi$
$139$	2693.37i	1.64352i	−0.569835	+	0.821759i	$0.692993\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	840.000	0.491219
$144$	0	0
$145$	−1680.00	−0.962182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	793.725i	0.436406i	0.975903	+	0.218203i	$0.0700195\pi$
$149$	793.725i	0.436406i	−0.975903	+	0.218203i	$0.929980\pi$
$150$	0	0
$151$	−744.000	−0.400966	−0.200483	−	0.979697i	$-0.564251\pi$
$151$	−744.000	−0.400966	−0.200483	+	0.979697i	$0.564251\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2370.59i	1.22846i
$156$	0	0
$157$	− 179.911i	− 0.0914552i	−0.998954	−	0.0457276i	$-0.985439\pi$
$157$	− 179.911i	− 0.0914552i	0.998954	−	0.0457276i	$-0.0145606\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1216.00	−0.595244
$162$	0	0
$163$	− 1772.65i	− 0.851809i	−0.904768	−	0.425905i	$-0.859956\pi$
$163$	− 1772.65i	− 0.851809i	0.904768	−	0.425905i	$-0.140044\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1960.00	−0.908200	−0.454100	−	0.890951i	$-0.650039\pi$
$167$	−1960.00	−0.908200	−0.454100	+	0.890951i	$0.650039\pi$
$168$	0	0
$169$	−603.000	−0.274465
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2000.19i	0.879026i	0.898236	+	0.439513i	$0.144849\pi$
$173$	2000.19i	0.879026i	−0.898236	+	0.439513i	$0.855151\pi$
$174$	0	0
$175$	104.000	0.0449238
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 3264.86i	− 1.36328i	−0.731688	−	0.681639i	$-0.761267\pi$
$179$	− 3264.86i	− 1.36328i	0.731688	−	0.681639i	$-0.238733\pi$
$180$	0	0
$181$	2338.84i	0.960469i	0.877140	+	0.480235i	$0.159448\pi$
$181$	2338.84i	0.960469i	−0.877140	+	0.480235i	$0.840552\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2576.00	−1.02374
$186$	0	0
$187$	222.243i	0.0869092i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3904.00	1.47897	0.739486	−	0.673172i	$-0.235069\pi$
$191$	3904.00	1.47897	0.739486	+	0.673172i	$0.235069\pi$
$192$	0	0
$193$	3330.00	1.24196	0.620981	−	0.783826i	$-0.286734\pi$
$193$	3330.00	1.24196	0.620981	+	0.783826i	$0.286734\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1195.88i	− 0.432502i	−0.976338	−	0.216251i	$-0.930617\pi$
$197$	− 1195.88i	− 0.432502i	0.976338	−	0.216251i	$-0.0693829\pi$
$198$	0	0
$199$	1736.00	0.618401	0.309200	−	0.950997i	$-0.399939\pi$
$199$	1736.00	0.618401	0.309200	+	0.950997i	$0.399939\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1269.96i	− 0.439083i
$204$	0	0
$205$	− 740.810i	− 0.252392i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	588.000	0.194607
$210$	0	0
$211$	− 2915.62i	− 0.951277i	−0.879641	−	0.475638i	$-0.842217\pi$
$211$	− 2915.62i	− 0.951277i	0.879641	−	0.475638i	$-0.157783\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4648.00	−1.47438
$216$	0	0
$217$	−1792.00	−0.560594
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 740.810i	− 0.225486i
$222$	0	0
$223$	−1568.00	−0.470857	−0.235428	−	0.971892i	$-0.575649\pi$
$223$	−1568.00	−0.470857	−0.235428	+	0.971892i	$0.575649\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1264.67i	0.369775i	0.982760	+	0.184888i	$0.0591922\pi$
$227$	1264.67i	0.369775i	−0.982760	+	0.184888i	$0.940808\pi$
$228$	0	0
$229$	− 5153.92i	− 1.48725i	−0.668595	−	0.743626i	$-0.733104\pi$
$229$	− 5153.92i	− 1.48725i	0.668595	−	0.743626i	$-0.266896\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	838.000	0.235619	0.117809	−	0.993036i	$-0.462413\pi$
$233$	838.000	0.235619	0.117809	+	0.993036i	$0.462413\pi$
$234$	0	0
$235$	− 3555.89i	− 0.987067i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6288.00	1.70183	0.850914	−	0.525305i	$-0.176049\pi$
$239$	6288.00	1.70183	0.850914	+	0.525305i	$0.176049\pi$
$240$	0	0
$241$	−2926.00	−0.782076	−0.391038	−	0.920375i	$-0.627884\pi$
$241$	−2926.00	−0.782076	−0.391038	+	0.920375i	$0.627884\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2952.66i	0.769953i
$246$	0	0
$247$	−1960.00	−0.504906
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 5444.96i	− 1.36925i	−0.728894	−	0.684627i	$-0.759965\pi$
$251$	− 5444.96i	− 1.36925i	0.728894	−	0.684627i	$-0.240035\pi$
$252$	0	0
$253$	− 2412.93i	− 0.599602i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2562.00	−0.621841	−0.310921	−	0.950436i	$-0.600637\pi$
$257$	−2562.00	−0.621841	−0.310921	+	0.950436i	$0.600637\pi$
$258$	0	0
$259$	− 1947.27i	− 0.467172i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5896.00	−1.38237	−0.691184	−	0.722679i	$-0.742911\pi$
$263$	−5896.00	−1.38237	−0.691184	+	0.722679i	$0.742911\pi$
$264$	0	0
$265$	336.000	0.0778880
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 5365.58i	− 1.21615i	−0.793878	−	0.608077i	$-0.791941\pi$
$269$	− 5365.58i	− 1.21615i	0.793878	−	0.608077i	$-0.208059\pi$
$270$	0	0
$271$	1680.00	0.376578	0.188289	−	0.982114i	$-0.439706\pi$
$271$	1680.00	0.376578	0.188289	+	0.982114i	$0.439706\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	206.369i	0.0452527i
$276$	0	0
$277$	1576.87i	0.342039i	0.985268	+	0.171019i	$0.0547061\pi$
$277$	1576.87i	0.342039i	−0.985268	+	0.171019i	$0.945294\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2742.00	0.582114	0.291057	−	0.956706i	$-0.405993\pi$
$281$	2742.00	0.582114	0.291057	+	0.956706i	$0.405993\pi$
$282$	0	0
$283$	2989.70i	0.627983i	0.949426	+	0.313991i	$0.101666\pi$
$283$	2989.70i	0.627983i	−0.949426	+	0.313991i	$0.898334\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	560.000	0.115177
$288$	0	0
$289$	−4717.00	−0.960106
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 9238.96i	− 1.84214i	−0.389401	−	0.921068i	$-0.627318\pi$
$293$	− 9238.96i	− 1.84214i	0.389401	−	0.921068i	$-0.372682\pi$
$294$	0	0
$295$	−5656.00	−1.11629
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8043.08i	1.55566i
$300$	0	0
$301$	− 3513.56i	− 0.672818i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1008.00	−0.189239
$306$	0	0
$307$	2587.54i	0.481039i	0.970644	+	0.240520i	$0.0773178\pi$
$307$	2587.54i	0.481039i	−0.970644	+	0.240520i	$0.922682\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2744.00	−0.500315	−0.250157	−	0.968205i	$-0.580482\pi$
$311$	−2744.00	−0.500315	−0.250157	+	0.968205i	$0.580482\pi$
$312$	0	0
$313$	2282.00	0.412097	0.206048	−	0.978542i	$-0.433940\pi$
$313$	2282.00	0.412097	0.206048	+	0.978542i	$0.433940\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9577.62i	1.69695i	0.529237	+	0.848474i	$0.322478\pi$
$317$	9577.62i	1.69695i	−0.529237	+	0.848474i	$0.677522\pi$
$318$	0	0
$319$	2520.00	0.442298
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 518.567i	− 0.0893308i
$324$	0	0
$325$	− 687.895i	− 0.117408i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2688.00	0.450438
$330$	0	0
$331$	− 4249.08i	− 0.705590i	−0.935701	−	0.352795i	$-0.885231\pi$
$331$	− 4249.08i	− 0.705590i	0.935701	−	0.352795i	$-0.114769\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1848.00	0.301394
$336$	0	0
$337$	6130.00	0.990868	0.495434	−	0.868646i	$-0.335009\pi$
$337$	6130.00	0.990868	0.495434	+	0.868646i	$0.335009\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 3555.89i	− 0.564699i
$342$	0	0
$343$	−4976.00	−0.783320
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2481.71i	− 0.383935i	−0.981401	−	0.191967i	$-0.938513\pi$
$347$	− 2481.71i	− 0.383935i	0.981401	−	0.191967i	$-0.0614868\pi$
$348$	0	0
$349$	328.073i	0.0503191i	0.999683	+	0.0251595i	$0.00800937\pi$
$349$	328.073i	0.0503191i	−0.999683	+	0.0251595i	$0.991991\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10206.0	1.53884	0.769420	−	0.638743i	$-0.220545\pi$
$353$	10206.0	1.53884	0.769420	+	0.638743i	$0.220545\pi$
$354$	0	0
$355$	761.976i	0.113920i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3176.00	−0.466916	−0.233458	−	0.972367i	$-0.575004\pi$
$359$	−3176.00	−0.466916	−0.233458	+	0.972367i	$0.575004\pi$
$360$	0	0
$361$	5487.00	0.799971
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3111.40i	0.446187i
$366$	0	0
$367$	11760.0	1.67266	0.836331	−	0.548225i	$-0.184696\pi$
$367$	11760.0	1.67266	0.836331	+	0.548225i	$0.184696\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	253.992i	0.0355434i
$372$	0	0
$373$	10974.6i	1.52344i	0.647908	+	0.761719i	$0.275644\pi$
$373$	10974.6i	1.52344i	−0.647908	+	0.761719i	$0.724356\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8400.00	−1.14754
$378$	0	0
$379$	3074.36i	0.416674i	0.978057	+	0.208337i	$0.0668051\pi$
$379$	3074.36i	0.416674i	−0.978057	+	0.208337i	$0.933195\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2688.00	0.358617	0.179309	−	0.983793i	$-0.442614\pi$
$383$	2688.00	0.358617	0.179309	+	0.983793i	$0.442614\pi$
$384$	0	0
$385$	1344.00	0.177913
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10487.8i	1.36697i	0.729966	+	0.683484i	$0.239536\pi$
$389$	10487.8i	1.36697i	−0.729966	+	0.683484i	$0.760464\pi$
$390$	0	0
$391$	−2128.00	−0.275237
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 4910.51i	− 0.625506i
$396$	0	0
$397$	5704.24i	0.721127i	0.932735	+	0.360564i	$0.117416\pi$
$397$	5704.24i	0.721127i	−0.932735	+	0.360564i	$0.882584\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12402.0	−1.54445	−0.772227	−	0.635346i	$-0.780857\pi$
$401$	−12402.0	−1.54445	−0.772227	+	0.635346i	$0.780857\pi$
$402$	0	0
$403$	11853.0i	1.46511i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3864.00	0.470593
$408$	0	0
$409$	−12278.0	−1.48437	−0.742186	−	0.670194i	$-0.766211\pi$
$409$	−12278.0	−1.48437	−0.742186	+	0.670194i	$0.766211\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 4275.53i	− 0.509407i
$414$	0	0
$415$	5768.00	0.682265
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8207.12i	0.956907i	0.878113	+	0.478454i	$0.158802\pi$
$419$	8207.12i	0.956907i	−0.878113	+	0.478454i	$0.841198\pi$
$420$	0	0
$421$	1449.87i	0.167844i	0.996472	+	0.0839221i	$0.0267447\pi$
$421$	1449.87i	0.167844i	−0.996472	+	0.0839221i	$0.973255\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	182.000	0.0207725
$426$	0	0
$427$	− 761.976i	− 0.0863574i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7632.00	0.852948	0.426474	−	0.904500i	$-0.359756\pi$
$431$	7632.00	0.852948	0.426474	+	0.904500i	$0.359756\pi$
$432$	0	0
$433$	3794.00	0.421081	0.210540	−	0.977585i	$-0.432478\pi$
$433$	3794.00	0.421081	0.210540	+	0.977585i	$0.432478\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5630.16i	0.616309i
$438$	0	0
$439$	1848.00	0.200912	0.100456	−	0.994942i	$-0.467970\pi$
$439$	1848.00	0.200912	0.100456	+	0.994942i	$0.467970\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12334.5i	1.32287i	0.750004	+	0.661433i	$0.230051\pi$
$443$	12334.5i	1.32287i	−0.750004	+	0.661433i	$0.769949\pi$
$444$	0	0
$445$	2815.08i	0.299882i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3582.00	0.376492	0.188246	−	0.982122i	$-0.439720\pi$
$449$	3582.00	0.376492	0.188246	+	0.982122i	$0.439720\pi$
$450$	0	0
$451$	1111.22i	0.116020i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4480.00	−0.461595
$456$	0	0
$457$	2714.00	0.277802	0.138901	−	0.990306i	$-0.455643\pi$
$457$	2714.00	0.277802	0.138901	+	0.990306i	$0.455643\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8349.99i	0.843596i	0.906690	+	0.421798i	$0.138601\pi$
$461$	8349.99i	0.843596i	−0.906690	+	0.421798i	$0.861399\pi$
$462$	0	0
$463$	−2224.00	−0.223236	−0.111618	−	0.993751i	$-0.535603\pi$
$463$	−2224.00	−0.223236	−0.111618	+	0.993751i	$0.535603\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 10292.0i	− 1.01982i	−0.860228	−	0.509910i	$-0.829679\pi$
$467$	− 10292.0i	− 1.01982i	0.860228	−	0.509910i	$-0.170321\pi$
$468$	0	0
$469$	1396.96i	0.137538i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6972.00	0.677744
$474$	0	0
$475$	− 481.527i	− 0.0465136i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	17696.0	1.68800	0.843999	−	0.536345i	$-0.180195\pi$
$479$	17696.0	1.68800	0.843999	+	0.536345i	$0.180195\pi$
$480$	0	0
$481$	−12880.0	−1.22095
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 10519.5i	− 0.984879i
$486$	0	0
$487$	−1304.00	−0.121334	−0.0606672	−	0.998158i	$-0.519323\pi$
$487$	−1304.00	−0.121334	−0.0606672	+	0.998158i	$0.519323\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 16662.9i	− 1.53154i	−0.643112	−	0.765772i	$-0.722357\pi$
$491$	− 16662.9i	− 1.53154i	0.643112	−	0.765772i	$-0.277643\pi$
$492$	0	0
$493$	− 2222.43i	− 0.203029i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−576.000	−0.0519862
$498$	0	0
$499$	3095.53i	0.277705i	0.990313	+	0.138853i	$0.0443414\pi$
$499$	3095.53i	0.277705i	−0.990313	+	0.138853i	$0.955659\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−19320.0	−1.71260	−0.856298	−	0.516481i	$-0.827242\pi$
$503$	−19320.0	−1.71260	−0.856298	+	0.516481i	$0.827242\pi$
$504$	0	0
$505$	7952.00	0.700712
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 4476.61i	− 0.389828i	−0.980820	−	0.194914i	$-0.937557\pi$
$509$	− 4476.61i	− 0.389828i	0.980820	−	0.194914i	$-0.0624427\pi$
$510$	0	0
$511$	−2352.00	−0.203613
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12445.6i	1.06489i
$516$	0	0
$517$	5333.83i	0.453737i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2982.00	0.250756	0.125378	−	0.992109i	$-0.459986\pi$
$521$	2982.00	0.250756	0.125378	+	0.992109i	$0.459986\pi$
$522$	0	0
$523$	2016.06i	0.168559i	0.996442	+	0.0842794i	$0.0268588\pi$
$523$	2016.06i	0.168559i	−0.996442	+	0.0842794i	$0.973141\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3136.00	−0.259215
$528$	0	0
$529$	10937.0	0.898907
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 3704.05i	− 0.301014i
$534$	0	0
$535$	2856.00	0.230796
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 4428.99i	− 0.353933i
$540$	0	0
$541$	− 15419.4i	− 1.22539i	−0.790321	−	0.612693i	$-0.790086\pi$
$541$	− 15419.4i	− 1.22539i	0.790321	−	0.612693i	$-0.209914\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	20048.0	1.57571
$546$	0	0
$547$	− 12609.7i	− 0.985649i	−0.870129	−	0.492824i	$-0.835965\pi$
$547$	− 12609.7i	− 0.985649i	0.870129	−	0.492824i	$-0.164035\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5880.00	−0.454621
$552$	0	0
$553$	3712.00	0.285444
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7143.53i	− 0.543413i	−0.962380	−	0.271706i	$-0.912412\pi$
$557$	− 7143.53i	− 0.543413i	0.962380	−	0.271706i	$-0.0875880\pi$
$558$	0	0
$559$	−23240.0	−1.75840
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7572.14i	0.566834i	0.958997	+	0.283417i	$0.0914681\pi$
$563$	7572.14i	0.566834i	−0.958997	+	0.283417i	$0.908532\pi$
$564$	0	0
$565$	− 18096.9i	− 1.34751i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−15594.0	−1.14892	−0.574459	−	0.818533i	$-0.694788\pi$
$569$	−15594.0	−1.14892	−0.574459	+	0.818533i	$0.694788\pi$
$570$	0	0
$571$	− 16737.0i	− 1.22666i	−0.789827	−	0.613330i	$-0.789830\pi$
$571$	− 16737.0i	− 1.22666i	0.789827	−	0.613330i	$-0.210170\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1976.00	−0.143313
$576$	0	0
$577$	6594.00	0.475757	0.237879	−	0.971295i	$-0.423548\pi$
$577$	6594.00	0.475757	0.237879	+	0.971295i	$0.423548\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4360.20i	0.311345i
$582$	0	0
$583$	−504.000	−0.0358037
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23213.8i	1.63226i	0.577868	+	0.816130i	$0.303885\pi$
$587$	23213.8i	1.63226i	−0.577868	+	0.816130i	$0.696115\pi$
$588$	0	0
$589$	8297.08i	0.580433i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14322.0	−0.991794	−0.495897	−	0.868381i	$-0.665161\pi$
$593$	−14322.0	−0.991794	−0.495897	+	0.868381i	$0.665161\pi$
$594$	0	0
$595$	− 1185.30i	− 0.0816679i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−16088.0	−1.09739	−0.548696	−	0.836022i	$-0.684876\pi$
$599$	−16088.0	−1.09739	−0.548696	+	0.836022i	$0.684876\pi$
$600$	0	0
$601$	−21238.0	−1.44146	−0.720729	−	0.693217i	$-0.756193\pi$
$601$	−21238.0	−1.44146	−0.720729	+	0.693217i	$0.756193\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 11419.1i	− 0.767357i
$606$	0	0
$607$	13664.0	0.913681	0.456841	−	0.889549i	$-0.348981\pi$
$607$	13664.0	0.913681	0.456841	+	0.889549i	$0.348981\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 17779.4i	− 1.17722i
$612$	0	0
$613$	− 20393.5i	− 1.34369i	−0.740690	−	0.671846i	$-0.765501\pi$
$613$	− 20393.5i	− 1.34369i	0.740690	−	0.671846i	$-0.234499\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3782.00	0.246771	0.123385	−	0.992359i	$-0.460625\pi$
$617$	3782.00	0.246771	0.123385	+	0.992359i	$0.460625\pi$
$618$	0	0
$619$	5825.94i	0.378295i	0.981949	+	0.189147i	$0.0605724\pi$
$619$	5825.94i	0.378295i	−0.981949	+	0.189147i	$0.939428\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2128.00	−0.136848
$624$	0	0
$625$	−13831.0	−0.885184
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 3407.73i	− 0.216017i
$630$	0	0
$631$	−2056.00	−0.129712	−0.0648558	−	0.997895i	$-0.520659\pi$
$631$	−2056.00	−0.129712	−0.0648558	+	0.997895i	$0.520659\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 17610.1i	− 1.10053i
$636$	0	0
$637$	14763.3i	0.918278i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−11842.0	−0.729689	−0.364845	−	0.931068i	$-0.618878\pi$
$641$	−11842.0	−0.729689	−0.364845	+	0.931068i	$0.618878\pi$
$642$	0	0
$643$	− 16250.2i	− 0.996649i	−0.866991	−	0.498325i	$-0.833949\pi$
$643$	− 16250.2i	− 0.996649i	0.866991	−	0.498325i	$-0.166051\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19320.0	1.17395	0.586976	−	0.809604i	$-0.300318\pi$
$647$	19320.0	1.17395	0.586976	+	0.809604i	$0.300318\pi$
$648$	0	0
$649$	8484.00	0.513137
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2317.68i	− 0.138894i	−0.997586	−	0.0694470i	$-0.977877\pi$
$653$	− 2317.68i	− 0.138894i	0.997586	−	0.0694470i	$-0.0221235\pi$
$654$	0	0
$655$	7112.00	0.424258
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 27732.8i	− 1.63932i	−0.572847	−	0.819662i	$-0.694161\pi$
$659$	− 27732.8i	− 1.63932i	0.572847	−	0.819662i	$-0.305839\pi$
$660$	0	0
$661$	− 22467.7i	− 1.32208i	−0.750352	−	0.661039i	$-0.770116\pi$
$661$	− 22467.7i	− 1.32208i	0.750352	−	0.661039i	$-0.229884\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3136.00	−0.182870
$666$	0	0
$667$	24129.3i	1.40073i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1512.00	0.0869897
$672$	0	0
$673$	−10078.0	−0.577234	−0.288617	−	0.957445i	$-0.593195\pi$
$673$	−10078.0	−0.577234	−0.288617	+	0.957445i	$0.593195\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16160.2i	0.917413i	0.888588	+	0.458707i	$0.151687\pi$
$677$	16160.2i	0.917413i	−0.888588	+	0.458707i	$0.848313\pi$
$678$	0	0
$679$	7952.00	0.449440
$680$	0	0
$681$	0	0
$682$	0	0
$683$	16356.0i	0.916320i	0.888870	+	0.458160i	$0.151491\pi$
$683$	16356.0i	0.916320i	−0.888870	+	0.458160i	$0.848509\pi$
$684$	0	0
$685$	− 11239.2i	− 0.626899i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1680.00	0.0928925
$690$	0	0
$691$	− 29246.1i	− 1.61009i	−0.593211	−	0.805047i	$-0.702140\pi$
$691$	− 29246.1i	− 1.61009i	0.593211	−	0.805047i	$-0.297860\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	28504.0	1.55571
$696$	0	0
$697$	980.000	0.0532570
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2465.84i	0.132858i	0.997791	+	0.0664290i	$0.0211606\pi$
$701$	2465.84i	0.132858i	−0.997791	+	0.0664290i	$0.978839\pi$
$702$	0	0
$703$	−9016.00	−0.483705
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6011.15i	0.319763i
$708$	0	0
$709$	31674.9i	1.67782i	0.544267	+	0.838912i	$0.316808\pi$
$709$	31674.9i	1.67782i	−0.544267	+	0.838912i	$0.683192\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	34048.0	1.78837
$714$	0	0
$715$	− 8889.72i	− 0.464975i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9296.00	−0.482173	−0.241086	−	0.970504i	$-0.577504\pi$
$719$	−9296.00	−0.482173	−0.241086	+	0.970504i	$0.577504\pi$
$720$	0	0
$721$	−9408.00	−0.485953
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2063.69i	− 0.105715i
$726$	0	0
$727$	−21672.0	−1.10560	−0.552799	−	0.833315i	$-0.686440\pi$
$727$	−21672.0	−1.10560	−0.552799	+	0.833315i	$0.686440\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 6148.73i	− 0.311106i
$732$	0	0
$733$	9471.79i	0.477283i	0.971108	+	0.238642i	$0.0767021\pi$
$733$	9471.79i	0.477283i	−0.971108	+	0.238642i	$0.923298\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2772.00	−0.138545
$738$	0	0
$739$	6863.08i	0.341627i	0.985303	+	0.170814i	$0.0546396\pi$
$739$	6863.08i	0.341627i	−0.985303	+	0.170814i	$0.945360\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17432.0	0.860724	0.430362	−	0.902656i	$-0.358386\pi$
$743$	17432.0	0.860724	0.430362	+	0.902656i	$0.358386\pi$
$744$	0	0
$745$	8400.00	0.413090
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2158.93i	0.105321i
$750$	0	0
$751$	11632.0	0.565190	0.282595	−	0.959239i	$-0.408805\pi$
$751$	11632.0	0.565190	0.282595	+	0.959239i	$0.408805\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7873.76i	0.379543i
$756$	0	0
$757$	16731.7i	0.803336i	0.915785	+	0.401668i	$0.131569\pi$
$757$	16731.7i	0.803336i	−0.915785	+	0.401668i	$0.868431\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−39466.0	−1.87995	−0.939975	−	0.341244i	$-0.889152\pi$
$761$	−39466.0	−1.87995	−0.939975	+	0.341244i	$0.889152\pi$
$762$	0	0
$763$	15154.9i	0.719060i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−28280.0	−1.33133
$768$	0	0
$769$	35266.0	1.65374	0.826869	−	0.562395i	$-0.190120\pi$
$769$	35266.0	1.65374	0.826869	+	0.562395i	$0.190120\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	16244.9i	0.755872i	0.925832	+	0.377936i	$0.123366\pi$
$773$	16244.9i	0.755872i	−0.925832	+	0.377936i	$0.876634\pi$
$774$	0	0
$775$	−2912.00	−0.134970
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2592.84i	− 0.119253i
$780$	0	0
$781$	− 1142.96i	− 0.0523668i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1904.00	−0.0865690
$786$	0	0
$787$	34844.5i	1.57824i	0.614240	+	0.789119i	$0.289463\pi$
$787$	34844.5i	1.57824i	−0.614240	+	0.789119i	$0.710537\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	13680.0	0.614924
$792$	0	0
$793$	−5040.00	−0.225694
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2550.50i	0.113354i	0.998393	+	0.0566772i	$0.0180506\pi$
$797$	2550.50i	0.113354i	−0.998393	+	0.0566772i	$0.981949\pi$
$798$	0	0
$799$	4704.00	0.208280
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 4667.11i	− 0.205104i
$804$	0	0
$805$	12868.9i	0.563441i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24390.0	1.05996	0.529979	−	0.848010i	$-0.322200\pi$
$809$	24390.0	1.05996	0.529979	+	0.848010i	$0.322200\pi$
$810$	0	0
$811$	− 9582.91i	− 0.414922i	−0.978243	−	0.207461i	$-0.933480\pi$
$811$	− 9582.91i	− 0.414922i	0.978243	−	0.207461i	$-0.0665200\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−18760.0	−0.806300
$816$	0	0
$817$	−16268.0	−0.696628
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 8773.31i	− 0.372948i	−0.982460	−	0.186474i	$-0.940294\pi$
$821$	− 8773.31i	− 0.372948i	0.982460	−	0.186474i	$-0.0597061\pi$
$822$	0	0
$823$	21688.0	0.918586	0.459293	−	0.888285i	$-0.348103\pi$
$823$	21688.0	0.918586	0.459293	+	0.888285i	$0.348103\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19446.3i	0.817670i	0.912608	+	0.408835i	$0.134065\pi$
$827$	19446.3i	0.817670i	−0.912608	+	0.408835i	$0.865935\pi$
$828$	0	0
$829$	19546.8i	0.818925i	0.912327	+	0.409462i	$0.134284\pi$
$829$	19546.8i	0.818925i	−0.912327	+	0.409462i	$0.865716\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3906.00	−0.162467
$834$	0	0
$835$	20742.7i	0.859677i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−18760.0	−0.771951	−0.385976	−	0.922509i	$-0.626135\pi$
$839$	−18760.0	−0.771951	−0.385976	+	0.922509i	$0.626135\pi$
$840$	0	0
$841$	−811.000	−0.0332527
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6381.55i	0.259801i
$846$	0	0
$847$	8632.00	0.350176
$848$	0	0
$849$	0	0
$850$	0	0
$851$	36998.2i	1.49034i
$852$	0	0
$853$	− 28732.9i	− 1.15333i	−0.816979	−	0.576667i	$-0.804353\pi$
$853$	− 28732.9i	− 1.15333i	0.816979	−	0.576667i	$-0.195647\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−8778.00	−0.349884	−0.174942	−	0.984579i	$-0.555974\pi$
$857$	−8778.00	−0.349884	−0.174942	+	0.984579i	$0.555974\pi$
$858$	0	0
$859$	− 5646.03i	− 0.224261i	−0.993693	−	0.112130i	$-0.964233\pi$
$859$	− 5646.03i	− 0.224261i	0.993693	−	0.112130i	$-0.0357675\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9312.00	−0.367305	−0.183652	−	0.982991i	$-0.558792\pi$
$863$	−9312.00	−0.367305	−0.183652	+	0.982991i	$0.558792\pi$
$864$	0	0
$865$	21168.0	0.832062
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7365.77i	0.287534i
$870$	0	0
$871$	9240.00	0.359455
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 11683.6i	− 0.451405i
$876$	0	0
$877$	− 137.579i	− 0.00529728i	−0.999996	−	0.00264864i	$-0.999157\pi$
$877$	− 137.579i	− 0.00529728i	0.999996	−	0.00264864i	$-0.000843089\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	31150.0	1.19123	0.595613	−	0.803272i	$-0.296909\pi$
$881$	31150.0	1.19123	0.595613	+	0.803272i	$0.296909\pi$
$882$	0	0
$883$	12577.9i	0.479366i	0.970851	+	0.239683i	$0.0770435\pi$
$883$	12577.9i	0.479366i	−0.970851	+	0.239683i	$0.922957\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37128.0	1.40545	0.702726	−	0.711460i	$-0.251966\pi$
$887$	37128.0	1.40545	0.702726	+	0.711460i	$0.251966\pi$
$888$	0	0
$889$	13312.0	0.502216
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 12445.6i	− 0.466379i
$894$	0	0
$895$	−34552.0	−1.29044
$896$	0	0
$897$	0	0
$898$	0	0
$899$	35558.9i	1.31919i
$900$	0	0
$901$	444.486i	0.0164351i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24752.0	0.909154
$906$	0	0
$907$	35204.4i	1.28880i	0.764688	+	0.644400i	$0.222893\pi$
$907$	35204.4i	1.28880i	−0.764688	+	0.644400i	$0.777107\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−10512.0	−0.382303	−0.191152	−	0.981561i	$-0.561222\pi$
$911$	−10512.0	−0.382303	−0.191152	+	0.981561i	$0.561222\pi$
$912$	0	0
$913$	−8652.00	−0.313625
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5376.17i	0.193606i
$918$	0	0
$919$	46104.0	1.65488	0.827438	−	0.561557i	$-0.189798\pi$
$919$	46104.0	1.65488	0.827438	+	0.561557i	$0.189798\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3809.88i	0.135865i
$924$	0	0
$925$	− 3164.32i	− 0.112478i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	5726.00	0.202222	0.101111	−	0.994875i	$-0.467760\pi$
$929$	5726.00	0.202222	0.101111	+	0.994875i	$0.467760\pi$
$930$	0	0
$931$	10334.3i	0.363795i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2352.00	0.0822659
$936$	0	0
$937$	1274.00	0.0444181	0.0222091	−	0.999753i	$-0.492930\pi$
$937$	1274.00	0.0444181	0.0222091	+	0.999753i	$0.492930\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 26446.9i	− 0.916201i	−0.888900	−	0.458101i	$-0.848530\pi$
$941$	− 26446.9i	− 0.916201i	0.888900	−	0.458101i	$-0.151470\pi$
$942$	0	0
$943$	−10640.0	−0.367430
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 23922.9i	− 0.820897i	−0.911884	−	0.410448i	$-0.865372\pi$
$947$	− 23922.9i	− 0.820897i	0.911884	−	0.410448i	$-0.134628\pi$
$948$	0	0
$949$	15557.0i	0.532141i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−38250.0	−1.30015	−0.650073	−	0.759872i	$-0.725262\pi$
$953$	−38250.0	−1.30015	−0.650073	+	0.759872i	$0.725262\pi$
$954$	0	0
$955$	− 41316.1i	− 1.39995i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8496.00	0.286079
$960$	0	0
$961$	20385.0	0.684267
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 35241.4i	− 1.17561i
$966$	0	0
$967$	−4664.00	−0.155103	−0.0775513	−	0.996988i	$-0.524710\pi$
$967$	−4664.00	−0.155103	−0.0775513	+	0.996988i	$0.524710\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 30971.2i	− 1.02360i	−0.859106	−	0.511798i	$-0.828980\pi$
$971$	− 30971.2i	− 1.02360i	0.859106	−	0.511798i	$-0.171020\pi$
$972$	0	0
$973$	21547.0i	0.709933i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4814.00	0.157639	0.0788196	−	0.996889i	$-0.474885\pi$
$977$	4814.00	0.157639	0.0788196	+	0.996889i	$0.474885\pi$
$978$	0	0
$979$	− 4222.62i	− 0.137850i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−12376.0	−0.401560	−0.200780	−	0.979636i	$-0.564348\pi$
$983$	−12376.0	−0.401560	−0.200780	+	0.979636i	$0.564348\pi$
$984$	0	0
$985$	−12656.0	−0.409395
$986$	0	0
$987$	0	0
$988$	0	0
$989$	66757.6i	2.14638i
$990$	0	0
$991$	−45344.0	−1.45348	−0.726740	−	0.686912i	$-0.758966\pi$
$991$	−45344.0	−1.45348	−0.726740	+	0.686912i	$0.758966\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 18372.1i	− 0.585361i
$996$	0	0
$997$	26002.4i	0.825984i	0.910735	+	0.412992i	$0.135516\pi$
$997$	26002.4i	0.825984i	−0.910735	+	0.412992i	$0.864484\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.4.d.a.145.1		2
3.2	odd	2		32.4.b.a.17.2		2
4.3	odd	2		72.4.d.b.37.1		2
8.3	odd	2		72.4.d.b.37.2		2
8.5	even	2	inner	288.4.d.a.145.2		2
12.11	even	2		8.4.b.a.5.2	yes	2
15.2	even	4		800.4.f.a.49.4		4
15.8	even	4		800.4.f.a.49.1		4
15.14	odd	2		800.4.d.a.401.1		2
16.3	odd	4		2304.4.a.bn.1.1		2
16.5	even	4		2304.4.a.v.1.2		2
16.11	odd	4		2304.4.a.bn.1.2		2
16.13	even	4		2304.4.a.v.1.1		2
24.5	odd	2		32.4.b.a.17.1		2
24.11	even	2		8.4.b.a.5.1	✓	2
48.5	odd	4		256.4.a.j.1.2		2
48.11	even	4		256.4.a.l.1.1		2
48.29	odd	4		256.4.a.j.1.1		2
48.35	even	4		256.4.a.l.1.2		2
60.23	odd	4		200.4.f.a.149.4		4
60.47	odd	4		200.4.f.a.149.1		4
60.59	even	2		200.4.d.a.101.1		2
120.29	odd	2		800.4.d.a.401.2		2
120.53	even	4		800.4.f.a.49.3		4
120.59	even	2		200.4.d.a.101.2		2
120.77	even	4		800.4.f.a.49.2		4
120.83	odd	4		200.4.f.a.149.2		4
120.107	odd	4		200.4.f.a.149.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.4.b.a.5.1	✓	2	24.11	even	2
8.4.b.a.5.2	yes	2	12.11	even	2
32.4.b.a.17.1		2	24.5	odd	2
32.4.b.a.17.2		2	3.2	odd	2
72.4.d.b.37.1		2	4.3	odd	2
72.4.d.b.37.2		2	8.3	odd	2
200.4.d.a.101.1		2	60.59	even	2
200.4.d.a.101.2		2	120.59	even	2
200.4.f.a.149.1		4	60.47	odd	4
200.4.f.a.149.2		4	120.83	odd	4
200.4.f.a.149.3		4	120.107	odd	4
200.4.f.a.149.4		4	60.23	odd	4
256.4.a.j.1.1		2	48.29	odd	4
256.4.a.j.1.2		2	48.5	odd	4
256.4.a.l.1.1		2	48.11	even	4
256.4.a.l.1.2		2	48.35	even	4
288.4.d.a.145.1		2	1.1	even	1	trivial
288.4.d.a.145.2		2	8.5	even	2	inner
800.4.d.a.401.1		2	15.14	odd	2
800.4.d.a.401.2		2	120.29	odd	2
800.4.f.a.49.1		4	15.8	even	4
800.4.f.a.49.2		4	120.77	even	4
800.4.f.a.49.3		4	120.53	even	4
800.4.f.a.49.4		4	15.2	even	4
2304.4.a.v.1.1		2	16.13	even	4
2304.4.a.v.1.2		2	16.5	even	4
2304.4.a.bn.1.1		2	16.3	odd	4
2304.4.a.bn.1.2		2	16.11	odd	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:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	288.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-10.5830i q^{5} +8.00000 q^{7} +O(q^{10})\)	\(q-10.5830i q^{5} +8.00000 q^{7} +15.8745i q^{11} -52.9150i q^{13} +14.0000 q^{17} -37.0405i q^{19} -152.000 q^{23} +13.0000 q^{25} -158.745i q^{29} -224.000 q^{31} -84.6640i q^{35} -243.409i q^{37} +70.0000 q^{41} -439.195i q^{43} +336.000 q^{47} -279.000 q^{49} +31.7490i q^{53} +168.000 q^{55} -534.442i q^{59} -95.2470i q^{61} -560.000 q^{65} +174.620i q^{67} -72.0000 q^{71} -294.000 q^{73} +126.996i q^{77} +464.000 q^{79} +545.025i q^{83} -148.162i q^{85} -266.000 q^{89} -423.320i q^{91} -392.000 q^{95} +994.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{7}+O(q^{10}) \) 2 * q + 16 * q^7	\( 2 q + 16 q^{7} + 28 q^{17} - 304 q^{23} + 26 q^{25} - 448 q^{31} + 140 q^{41} + 672 q^{47} - 558 q^{49} + 336 q^{55} - 1120 q^{65} - 144 q^{71} - 588 q^{73} + 928 q^{79} - 532 q^{89} - 784 q^{95} + 1988 q^{97}+O(q^{100}) \) 2 * q + 16 * q^7 + 28 * q^17 - 304 * q^23 + 26 * q^25 - 448 * q^31 + 140 * q^41 + 672 * q^47 - 558 * q^49 + 336 * q^55 - 1120 * q^65 - 144 * q^71 - 588 * q^73 + 928 * q^79 - 532 * q^89 - 784 * q^95 + 1988 * q^97

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