LMFDB - Embedded newform 3584.2.b.a.1793.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,2,Mod(1793,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3584.1793");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3584.b (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:	$28.6183840844$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1793.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	3584.1793
Dual form			3584.2.b.a.1793.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.41421i q^{3} -1.00000 q^{7} +1.00000 q^{9} -1.41421i q^{11} +1.41421i q^{19} +1.41421i q^{21} +4.00000 q^{23} +5.00000 q^{25} -5.65685i q^{27} +2.82843i q^{29} -4.00000 q^{31} -2.00000 q^{33} -2.82843i q^{37} +6.00000 q^{41} -4.24264i q^{43} +4.00000 q^{47} +1.00000 q^{49} +8.48528i q^{53} +2.00000 q^{57} -7.07107i q^{59} -5.65685i q^{61} -1.00000 q^{63} -1.41421i q^{67} -5.65685i q^{69} -8.00000 q^{71} +8.00000 q^{73} -7.07107i q^{75} +1.41421i q^{77} +4.00000 q^{79} -5.00000 q^{81} -7.07107i q^{83} +4.00000 q^{87} +8.00000 q^{89} +5.65685i q^{93} -8.00000 q^{97} -1.41421i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{7} + 2 q^{9} + 8 q^{23} + 10 q^{25} - 8 q^{31} - 4 q^{33} + 12 q^{41} + 8 q^{47} + 2 q^{49} + 4 q^{57} - 2 q^{63} - 16 q^{71} + 16 q^{73} + 8 q^{79} - 10 q^{81} + 8 q^{87} + 16 q^{89} - 16 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 + 2 * q^9 + 8 * q^23 + 10 * q^25 - 8 * q^31 - 4 * q^33 + 12 * q^41 + 8 * q^47 + 2 * q^49 + 4 * q^57 - 2 * q^63 - 16 * q^71 + 16 * q^73 + 8 * q^79 - 10 * q^81 + 8 * q^87 + 16 * q^89 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.41421i	− 0.816497i	−0.912871	−	0.408248i	$-0.866140\pi$
$3$	− 1.41421i	− 0.816497i	0.912871	−	0.408248i	$-0.133860\pi$
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 1.41421i	− 0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$	− 1.41421i	− 0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	1.41421i	0.324443i	0.986754	+	0.162221i	$0.0518659\pi$
$19$	1.41421i	0.324443i	−0.986754	+	0.162221i	$0.948134\pi$
$20$	0	0
$21$	1.41421i	0.308607i
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	− 5.65685i	− 1.08866i
$28$	0	0
$29$	2.82843i	0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$	2.82843i	0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.82843i	− 0.464991i	−0.972598	−	0.232495i	$-0.925311\pi$
$37$	− 2.82843i	− 0.464991i	0.972598	−	0.232495i	$-0.0746890\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	− 4.24264i	− 0.646997i	−0.946229	−	0.323498i	$-0.895141\pi$
$43$	− 4.24264i	− 0.646997i	0.946229	−	0.323498i	$-0.104859\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.48528i	1.16554i	0.812636	+	0.582772i	$0.198032\pi$
$53$	8.48528i	1.16554i	−0.812636	+	0.582772i	$0.801968\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	− 7.07107i	− 0.920575i	−0.887770	−	0.460287i	$-0.847746\pi$
$59$	− 7.07107i	− 0.920575i	0.887770	−	0.460287i	$-0.152254\pi$
$60$	0	0
$61$	− 5.65685i	− 0.724286i	−0.932123	−	0.362143i	$-0.882045\pi$
$61$	− 5.65685i	− 0.724286i	0.932123	−	0.362143i	$-0.117955\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1.41421i	− 0.172774i	−0.996262	−	0.0863868i	$-0.972468\pi$
$67$	− 1.41421i	− 0.172774i	0.996262	−	0.0863868i	$-0.0275321\pi$
$68$	0	0
$69$	− 5.65685i	− 0.681005i
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	0	0
$75$	− 7.07107i	− 0.816497i
$76$	0	0
$77$	1.41421i	0.161165i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	− 7.07107i	− 0.776151i	−0.921628	−	0.388075i	$-0.873140\pi$
$83$	− 7.07107i	− 0.776151i	0.921628	−	0.388075i	$-0.126860\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.00000	0.428845
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.65685i	0.586588i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	− 1.41421i	− 0.142134i
$100$	0	0
$101$	− 11.3137i	− 1.12576i	−0.826540	−	0.562878i	$-0.809694\pi$
$101$	− 11.3137i	− 1.12576i	0.826540	−	0.562878i	$-0.190306\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.07107i	− 0.683586i	−0.939775	−	0.341793i	$-0.888966\pi$
$107$	− 7.07107i	− 0.683586i	0.939775	−	0.341793i	$-0.111034\pi$
$108$	0	0
$109$	− 8.48528i	− 0.812743i	−0.913708	−	0.406371i	$-0.866794\pi$
$109$	− 8.48528i	− 0.812743i	0.913708	−	0.406371i	$-0.133206\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	− 8.48528i	− 0.765092i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	− 9.89949i	− 0.864923i	−0.901652	−	0.432461i	$-0.857645\pi$
$131$	− 9.89949i	− 0.864923i	0.901652	−	0.432461i	$-0.142355\pi$
$132$	0	0
$133$	− 1.41421i	− 0.122628i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	− 1.41421i	− 0.119952i	−0.998200	−	0.0599760i	$-0.980898\pi$
$139$	− 1.41421i	− 0.119952i	0.998200	−	0.0599760i	$-0.0191024\pi$
$140$	0	0
$141$	− 5.65685i	− 0.476393i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.41421i	− 0.116642i
$148$	0	0
$149$	19.7990i	1.62200i	0.585049	+	0.810998i	$0.301075\pi$
$149$	19.7990i	1.62200i	−0.585049	+	0.810998i	$0.698925\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.9706i	1.35440i	0.735800	+	0.677199i	$0.236806\pi$
$157$	16.9706i	1.35440i	−0.735800	+	0.677199i	$0.763194\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	− 9.89949i	− 0.775388i	−0.921788	−	0.387694i	$-0.873272\pi$
$163$	− 9.89949i	− 0.775388i	0.921788	−	0.387694i	$-0.126728\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	1.41421i	0.108148i
$172$	0	0
$173$	− 11.3137i	− 0.860165i	−0.902790	−	0.430083i	$-0.858484\pi$
$173$	− 11.3137i	− 0.860165i	0.902790	−	0.430083i	$-0.141516\pi$
$174$	0	0
$175$	−5.00000	−0.377964
$176$	0	0
$177$	−10.0000	−0.751646
$178$	0	0
$179$	− 15.5563i	− 1.16274i	−0.813641	−	0.581368i	$-0.802518\pi$
$179$	− 15.5563i	− 1.16274i	0.813641	−	0.581368i	$-0.197482\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	5.65685i	0.411476i
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 14.1421i	− 1.00759i	−0.863825	−	0.503793i	$-0.831938\pi$
$197$	− 14.1421i	− 1.00759i	0.863825	−	0.503793i	$-0.168062\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	0	0
$203$	− 2.82843i	− 0.198517i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	− 15.5563i	− 1.07094i	−0.844553	−	0.535472i	$-0.820134\pi$
$211$	− 15.5563i	− 1.07094i	0.844553	−	0.535472i	$-0.179866\pi$
$212$	0	0
$213$	11.3137i	0.775203i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	− 11.3137i	− 0.764510i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	5.00000	0.333333
$226$	0	0
$227$	− 18.3848i	− 1.22024i	−0.792309	−	0.610120i	$-0.791121\pi$
$227$	− 18.3848i	− 1.22024i	0.792309	−	0.610120i	$-0.208879\pi$
$228$	0	0
$229$	− 16.9706i	− 1.12145i	−0.828003	−	0.560723i	$-0.810523\pi$
$229$	− 16.9706i	− 1.12145i	0.828003	−	0.560723i	$-0.189477\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	−4.00000	−0.262049	−0.131024	−	0.991379i	$-0.541827\pi$
$233$	−4.00000	−0.262049	−0.131024	+	0.991379i	$0.541827\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 5.65685i	− 0.367452i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−16.0000	−1.03065	−0.515325	−	0.856995i	$-0.672329\pi$
$241$	−16.0000	−1.03065	−0.515325	+	0.856995i	$0.672329\pi$
$242$	0	0
$243$	− 9.89949i	− 0.635053i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−10.0000	−0.633724
$250$	0	0
$251$	9.89949i	0.624851i	0.949942	+	0.312425i	$0.101141\pi$
$251$	9.89949i	0.624851i	−0.949942	+	0.312425i	$0.898859\pi$
$252$	0	0
$253$	− 5.65685i	− 0.355643i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	2.82843i	0.175750i
$260$	0	0
$261$	2.82843i	0.175075i
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 11.3137i	− 0.692388i
$268$	0	0
$269$	11.3137i	0.689809i	0.938638	+	0.344904i	$0.112089\pi$
$269$	11.3137i	0.689809i	−0.938638	+	0.344904i	$0.887911\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 7.07107i	− 0.426401i
$276$	0	0
$277$	31.1127i	1.86938i	0.355463	+	0.934690i	$0.384323\pi$
$277$	31.1127i	1.86938i	−0.355463	+	0.934690i	$0.615677\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	0	0
$283$	1.41421i	0.0840663i	0.999116	+	0.0420331i	$0.0133835\pi$
$283$	1.41421i	0.0840663i	−0.999116	+	0.0420331i	$0.986616\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	11.3137i	0.663221i
$292$	0	0
$293$	− 28.2843i	− 1.65238i	−0.563388	−	0.826192i	$-0.690502\pi$
$293$	− 28.2843i	− 1.65238i	0.563388	−	0.826192i	$-0.309498\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−8.00000	−0.464207
$298$	0	0
$299$	0	0
$300$	0	0
$301$	4.24264i	0.244542i
$302$	0	0
$303$	−16.0000	−0.919176
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.41421i	0.0807134i	0.999185	+	0.0403567i	$0.0128494\pi$
$307$	1.41421i	0.0807134i	−0.999185	+	0.0403567i	$0.987151\pi$
$308$	0	0
$309$	5.65685i	0.321807i
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.82843i	0.158860i	0.996840	+	0.0794301i	$0.0253101\pi$
$317$	2.82843i	0.158860i	−0.996840	+	0.0794301i	$0.974690\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	−10.0000	−0.558146
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−12.0000	−0.663602
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	21.2132i	1.16598i	0.812478	+	0.582992i	$0.198118\pi$
$331$	21.2132i	1.16598i	−0.812478	+	0.582992i	$0.801882\pi$
$332$	0	0
$333$	− 2.82843i	− 0.154997i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	0	0
$339$	14.1421i	0.768095i
$340$	0	0
$341$	5.65685i	0.306336i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.07107i	0.379595i	0.981823	+	0.189797i	$0.0607831\pi$
$347$	7.07107i	0.379595i	−0.981823	+	0.189797i	$0.939217\pi$
$348$	0	0
$349$	16.9706i	0.908413i	0.890896	+	0.454207i	$0.150077\pi$
$349$	16.9706i	0.908413i	−0.890896	+	0.454207i	$0.849923\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	− 12.7279i	− 0.668043i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	− 8.48528i	− 0.440534i
$372$	0	0
$373$	8.48528i	0.439351i	0.975573	+	0.219676i	$0.0704999\pi$
$373$	8.48528i	0.439351i	−0.975573	+	0.219676i	$0.929500\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	7.07107i	0.363216i	0.983371	+	0.181608i	$0.0581303\pi$
$379$	7.07107i	0.363216i	−0.983371	+	0.181608i	$0.941870\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.24264i	− 0.215666i
$388$	0	0
$389$	19.7990i	1.00385i	0.864912	+	0.501924i	$0.167374\pi$
$389$	19.7990i	1.00385i	−0.864912	+	0.501924i	$0.832626\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−14.0000	−0.706207
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 39.5980i	− 1.98737i	−0.112225	−	0.993683i	$-0.535798\pi$
$397$	− 39.5980i	− 1.98737i	0.112225	−	0.993683i	$-0.464202\pi$
$398$	0	0
$399$	−2.00000	−0.100125
$400$	0	0
$401$	20.0000	0.998752	0.499376	−	0.866385i	$-0.333563\pi$
$401$	20.0000	0.998752	0.499376	+	0.866385i	$0.333563\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.00000	−0.198273
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	− 2.82843i	− 0.139516i
$412$	0	0
$413$	7.07107i	0.347945i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.00000	−0.0979404
$418$	0	0
$419$	− 4.24264i	− 0.207267i	−0.994616	−	0.103633i	$-0.966953\pi$
$419$	− 4.24264i	− 0.207267i	0.994616	−	0.103633i	$-0.0330468\pi$
$420$	0	0
$421$	− 36.7696i	− 1.79204i	−0.444015	−	0.896019i	$-0.646446\pi$
$421$	− 36.7696i	− 1.79204i	0.444015	−	0.896019i	$-0.353554\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.65685i	0.273754i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.65685i	0.270604i
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	− 7.07107i	− 0.335957i	−0.985791	−	0.167978i	$-0.946276\pi$
$443$	− 7.07107i	− 0.335957i	0.985791	−	0.167978i	$-0.0537239\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	28.0000	1.32435
$448$	0	0
$449$	28.0000	1.32140	0.660701	−	0.750649i	$-0.270259\pi$
$449$	28.0000	1.32140	0.660701	+	0.750649i	$0.270259\pi$
$450$	0	0
$451$	− 8.48528i	− 0.399556i
$452$	0	0
$453$	− 28.2843i	− 1.32891i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.65685i	0.263466i	0.991285	+	0.131733i	$0.0420541\pi$
$461$	5.65685i	0.263466i	−0.991285	+	0.131733i	$0.957946\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.0416i	1.11251i	0.831010	+	0.556257i	$0.187763\pi$
$467$	24.0416i	1.11251i	−0.831010	+	0.556257i	$0.812237\pi$
$468$	0	0
$469$	1.41421i	0.0653023i
$470$	0	0
$471$	24.0000	1.10586
$472$	0	0
$473$	−6.00000	−0.275880
$474$	0	0
$475$	7.07107i	0.324443i
$476$	0	0
$477$	8.48528i	0.388514i
$478$	0	0
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	5.65685i	0.257396i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	28.0000	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$487$	28.0000	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$488$	0	0
$489$	−14.0000	−0.633102
$490$	0	0
$491$	12.7279i	0.574403i	0.957870	+	0.287202i	$0.0927249\pi$
$491$	12.7279i	0.574403i	−0.957870	+	0.287202i	$0.907275\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	29.6985i	1.32949i	0.747072	+	0.664743i	$0.231459\pi$
$499$	29.6985i	1.32949i	−0.747072	+	0.664743i	$0.768541\pi$
$500$	0	0
$501$	16.9706i	0.758189i
$502$	0	0
$503$	−40.0000	−1.78351	−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000	−1.78351	−0.891756	+	0.452517i	$0.850526\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 18.3848i	− 0.816497i
$508$	0	0
$509$	− 39.5980i	− 1.75515i	−0.479440	−	0.877575i	$-0.659160\pi$
$509$	− 39.5980i	− 1.75515i	0.479440	−	0.877575i	$-0.340840\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	8.00000	0.353209
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 5.65685i	− 0.248788i
$518$	0	0
$519$	−16.0000	−0.702322
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	12.7279i	0.556553i	0.960501	+	0.278277i	$0.0897632\pi$
$523$	12.7279i	0.556553i	−0.960501	+	0.278277i	$0.910237\pi$
$524$	0	0
$525$	7.07107i	0.308607i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	− 7.07107i	− 0.306858i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−22.0000	−0.949370
$538$	0	0
$539$	− 1.41421i	− 0.0609145i
$540$	0	0
$541$	19.7990i	0.851225i	0.904906	+	0.425613i	$0.139941\pi$
$541$	19.7990i	0.851225i	−0.904906	+	0.425613i	$0.860059\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	38.1838i	1.63262i	0.577614	+	0.816310i	$0.303984\pi$
$547$	38.1838i	1.63262i	−0.577614	+	0.816310i	$0.696016\pi$
$548$	0	0
$549$	− 5.65685i	− 0.241429i
$550$	0	0
$551$	−4.00000	−0.170406
$552$	0	0
$553$	−4.00000	−0.170097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.1421i	0.599222i	0.954062	+	0.299611i	$0.0968568\pi$
$557$	14.1421i	0.599222i	−0.954062	+	0.299611i	$0.903143\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	29.6985i	1.25164i	0.779967	+	0.625821i	$0.215236\pi$
$563$	29.6985i	1.25164i	−0.779967	+	0.625821i	$0.784764\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5.00000	0.209980
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	21.2132i	0.887745i	0.896090	+	0.443872i	$0.146396\pi$
$571$	21.2132i	0.887745i	−0.896090	+	0.443872i	$0.853604\pi$
$572$	0	0
$573$	5.65685i	0.236318i
$574$	0	0
$575$	20.0000	0.834058
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	0	0
$579$	5.65685i	0.235091i
$580$	0	0
$581$	7.07107i	0.293357i
$582$	0	0
$583$	12.0000	0.496989
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 15.5563i	− 0.642079i	−0.947066	−	0.321040i	$-0.895968\pi$
$587$	− 15.5563i	− 0.642079i	0.947066	−	0.321040i	$-0.104032\pi$
$588$	0	0
$589$	− 5.65685i	− 0.233087i
$590$	0	0
$591$	−20.0000	−0.822690
$592$	0	0
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.65685i	0.231520i
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	24.0000	0.978980	0.489490	−	0.872009i	$-0.337183\pi$
$601$	24.0000	0.978980	0.489490	+	0.872009i	$0.337183\pi$
$602$	0	0
$603$	− 1.41421i	− 0.0575912i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	0	0
$609$	−4.00000	−0.162088
$610$	0	0
$611$	0	0
$612$	0	0
$613$	14.1421i	0.571195i	0.958350	+	0.285598i	$0.0921921\pi$
$613$	14.1421i	0.571195i	−0.958350	+	0.285598i	$0.907808\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	35.3553i	1.42105i	0.703671	+	0.710526i	$0.251543\pi$
$619$	35.3553i	1.42105i	−0.703671	+	0.710526i	$0.748457\pi$
$620$	0	0
$621$	− 22.6274i	− 0.908007i
$622$	0	0
$623$	−8.00000	−0.320513
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	− 2.82843i	− 0.112956i
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	0	0
$633$	−22.0000	−0.874421
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	36.0000	1.42191	0.710957	−	0.703235i	$-0.248262\pi$
$641$	36.0000	1.42191	0.710957	+	0.703235i	$0.248262\pi$
$642$	0	0
$643$	29.6985i	1.17119i	0.810602	+	0.585597i	$0.199140\pi$
$643$	29.6985i	1.17119i	−0.810602	+	0.585597i	$0.800860\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	−10.0000	−0.392534
$650$	0	0
$651$	− 5.65685i	− 0.221710i
$652$	0	0
$653$	8.48528i	0.332055i	0.986121	+	0.166027i	$0.0530940\pi$
$653$	8.48528i	0.332055i	−0.986121	+	0.166027i	$0.946906\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.00000	0.312110
$658$	0	0
$659$	− 9.89949i	− 0.385630i	−0.981235	−	0.192815i	$-0.938238\pi$
$659$	− 9.89949i	− 0.385630i	0.981235	−	0.192815i	$-0.0617617\pi$
$660$	0	0
$661$	11.3137i	0.440052i	0.975494	+	0.220026i	$0.0706143\pi$
$661$	11.3137i	0.440052i	−0.975494	+	0.220026i	$0.929386\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11.3137i	0.438069i
$668$	0	0
$669$	22.6274i	0.874826i
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	36.0000	1.38770	0.693849	−	0.720121i	$-0.255914\pi$
$673$	36.0000	1.38770	0.693849	+	0.720121i	$0.255914\pi$
$674$	0	0
$675$	− 28.2843i	− 1.08866i
$676$	0	0
$677$	− 28.2843i	− 1.08705i	−0.839392	−	0.543526i	$-0.817089\pi$
$677$	− 28.2843i	− 1.08705i	0.839392	−	0.543526i	$-0.182911\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	−26.0000	−0.996322
$682$	0	0
$683$	− 32.5269i	− 1.24461i	−0.782776	−	0.622304i	$-0.786197\pi$
$683$	− 32.5269i	− 1.24461i	0.782776	−	0.622304i	$-0.213803\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−24.0000	−0.915657
$688$	0	0
$689$	0	0
$690$	0	0
$691$	15.5563i	0.591791i	0.955220	+	0.295896i	$0.0956181\pi$
$691$	15.5563i	0.591791i	−0.955220	+	0.295896i	$0.904382\pi$
$692$	0	0
$693$	1.41421i	0.0537215i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	5.65685i	0.213962i
$700$	0	0
$701$	− 8.48528i	− 0.320485i	−0.987078	−	0.160242i	$-0.948772\pi$
$701$	− 8.48528i	− 0.320485i	0.987078	−	0.160242i	$-0.0512276\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.3137i	0.425496i
$708$	0	0
$709$	25.4558i	0.956014i	0.878356	+	0.478007i	$0.158641\pi$
$709$	25.4558i	0.956014i	−0.878356	+	0.478007i	$0.841359\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	−16.0000	−0.599205
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	22.6274i	0.841523i
$724$	0	0
$725$	14.1421i	0.525226i
$726$	0	0
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	0	0
$729$	−29.0000	−1.07407
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 16.9706i	− 0.626822i	−0.949618	−	0.313411i	$-0.898528\pi$
$733$	− 16.9706i	− 0.626822i	0.949618	−	0.313411i	$-0.101472\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.00000	−0.0736709
$738$	0	0
$739$	46.6690i	1.71675i	0.513024	+	0.858374i	$0.328525\pi$
$739$	46.6690i	1.71675i	−0.513024	+	0.858374i	$0.671475\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.00000	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 7.07107i	− 0.258717i
$748$	0	0
$749$	7.07107i	0.258371i
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	0	0
$753$	14.0000	0.510188
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 42.4264i	− 1.54201i	−0.636827	−	0.771007i	$-0.719753\pi$
$757$	− 42.4264i	− 1.54201i	0.636827	−	0.771007i	$-0.280247\pi$
$758$	0	0
$759$	−8.00000	−0.290382
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	8.48528i	0.307188i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−16.0000	−0.576975	−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000	−0.576975	−0.288487	+	0.957484i	$0.593152\pi$
$770$	0	0
$771$	25.4558i	0.916770i
$772$	0	0
$773$	5.65685i	0.203463i	0.994812	+	0.101731i	$0.0324382\pi$
$773$	5.65685i	0.203463i	−0.994812	+	0.101731i	$0.967562\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	8.48528i	0.304017i
$780$	0	0
$781$	11.3137i	0.404836i
$782$	0	0
$783$	16.0000	0.571793
$784$	0	0
$785$	0	0
$786$	0	0
$787$	49.4975i	1.76439i	0.470880	+	0.882197i	$0.343936\pi$
$787$	49.4975i	1.76439i	−0.470880	+	0.882197i	$0.656064\pi$
$788$	0	0
$789$	− 33.9411i	− 1.20834i
$790$	0	0
$791$	10.0000	0.355559
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 11.3137i	− 0.400752i	−0.979719	−	0.200376i	$-0.935784\pi$
$797$	− 11.3137i	− 0.400752i	0.979719	−	0.200376i	$-0.0642164\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	8.00000	0.282666
$802$	0	0
$803$	− 11.3137i	− 0.399252i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16.0000	0.563227
$808$	0	0
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	46.6690i	1.63877i	0.573242	+	0.819386i	$0.305685\pi$
$811$	46.6690i	1.63877i	−0.573242	+	0.819386i	$0.694315\pi$
$812$	0	0
$813$	− 33.9411i	− 1.19037i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6.00000	0.209913
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.1127i	1.08584i	0.839784	+	0.542920i	$0.182681\pi$
$821$	31.1127i	1.08584i	−0.839784	+	0.542920i	$0.817319\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	−10.0000	−0.348155
$826$	0	0
$827$	15.5563i	0.540947i	0.962727	+	0.270474i	$0.0871803\pi$
$827$	15.5563i	0.540947i	−0.962727	+	0.270474i	$0.912820\pi$
$828$	0	0
$829$	45.2548i	1.57177i	0.618376	+	0.785883i	$0.287791\pi$
$829$	45.2548i	1.57177i	−0.618376	+	0.785883i	$0.712209\pi$
$830$	0	0
$831$	44.0000	1.52634
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	22.6274i	0.782118i
$838$	0	0
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	0	0
$841$	21.0000	0.724138
$842$	0	0
$843$	5.65685i	0.194832i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−9.00000	−0.309244
$848$	0	0
$849$	2.00000	0.0686398
$850$	0	0
$851$	− 11.3137i	− 0.387829i
$852$	0	0
$853$	− 5.65685i	− 0.193687i	−0.995300	−	0.0968435i	$-0.969125\pi$
$853$	− 5.65685i	− 0.193687i	0.995300	−	0.0968435i	$-0.0308746\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	21.2132i	0.723785i	0.932220	+	0.361893i	$0.117869\pi$
$859$	21.2132i	0.723785i	−0.932220	+	0.361893i	$0.882131\pi$
$860$	0	0
$861$	8.48528i	0.289178i
$862$	0	0
$863$	−44.0000	−1.49778	−0.748889	−	0.662696i	$-0.769412\pi$
$863$	−44.0000	−1.49778	−0.748889	+	0.662696i	$0.769412\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	24.0416i	0.816497i
$868$	0	0
$869$	− 5.65685i	− 0.191896i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−8.00000	−0.270759
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.48528i	0.286528i	0.989685	+	0.143264i	$0.0457597\pi$
$877$	8.48528i	0.286528i	−0.989685	+	0.143264i	$0.954240\pi$
$878$	0	0
$879$	−40.0000	−1.34917
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	− 12.7279i	− 0.428329i	−0.976798	−	0.214164i	$-0.931297\pi$
$883$	− 12.7279i	− 0.428329i	0.976798	−	0.214164i	$-0.0687028\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−44.0000	−1.47738	−0.738688	−	0.674048i	$-0.764554\pi$
$887$	−44.0000	−1.47738	−0.738688	+	0.674048i	$0.764554\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	7.07107i	0.236890i
$892$	0	0
$893$	5.65685i	0.189299i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 11.3137i	− 0.377333i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	6.00000	0.199667
$904$	0	0
$905$	0	0
$906$	0	0
$907$	35.3553i	1.17395i	0.809603	+	0.586977i	$0.199682\pi$
$907$	35.3553i	1.17395i	−0.809603	+	0.586977i	$0.800318\pi$
$908$	0	0
$909$	− 11.3137i	− 0.375252i
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	0	0
$913$	−10.0000	−0.330952
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.89949i	0.326910i
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	2.00000	0.0659022
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 14.1421i	− 0.464991i
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	8.00000	0.262471	0.131236	−	0.991351i	$-0.458106\pi$
$929$	8.00000	0.262471	0.131236	+	0.991351i	$0.458106\pi$
$930$	0	0
$931$	1.41421i	0.0463490i
$932$	0	0
$933$	11.3137i	0.370394i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	− 8.48528i	− 0.276907i
$940$	0	0
$941$	39.5980i	1.29086i	0.763821	+	0.645429i	$0.223321\pi$
$941$	39.5980i	1.29086i	−0.763821	+	0.645429i	$0.776679\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 7.07107i	− 0.229779i	−0.993378	−	0.114889i	$-0.963349\pi$
$947$	− 7.07107i	− 0.229779i	0.993378	−	0.114889i	$-0.0366514\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	4.00000	0.129709
$952$	0	0
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 5.65685i	− 0.182860i
$958$	0	0
$959$	−2.00000	−0.0645834
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	− 7.07107i	− 0.227862i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	38.1838i	1.22538i	0.790325	+	0.612688i	$0.209912\pi$
$971$	38.1838i	1.22538i	−0.790325	+	0.612688i	$0.790088\pi$
$972$	0	0
$973$	1.41421i	0.0453376i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	0	0
$979$	− 11.3137i	− 0.361588i
$980$	0	0
$981$	− 8.48528i	− 0.270914i
$982$	0	0
$983$	44.0000	1.40338	0.701691	−	0.712481i	$-0.252429\pi$
$983$	44.0000	1.40338	0.701691	+	0.712481i	$0.252429\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.65685i	0.180060i
$988$	0	0
$989$	− 16.9706i	− 0.539633i
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	0	0
$993$	30.0000	0.952021
$994$	0	0
$995$	0	0
$996$	0	0
$997$	45.2548i	1.43323i	0.697466	+	0.716617i	$0.254311\pi$
$997$	45.2548i	1.43323i	−0.697466	+	0.716617i	$0.745689\pi$
$998$	0	0
$999$	−16.0000	−0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.b.a.1793.1		2
4.3	odd	2		3584.2.b.b.1793.2		2
8.3	odd	2		3584.2.b.b.1793.1		2
8.5	even	2	inner	3584.2.b.a.1793.2		2
16.3	odd	4		3584.2.a.a.1.1	✓	2
16.5	even	4		3584.2.a.b.1.1	yes	2
16.11	odd	4		3584.2.a.a.1.2	yes	2
16.13	even	4		3584.2.a.b.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.a.1.1	✓	2	16.3	odd	4
3584.2.a.a.1.2	yes	2	16.11	odd	4
3584.2.a.b.1.1	yes	2	16.5	even	4
3584.2.a.b.1.2	yes	2	16.13	even	4
3584.2.b.a.1793.1		2	1.1	even	1	trivial
3584.2.b.a.1793.2		2	8.5	even	2	inner
3584.2.b.b.1793.1		2	8.3	odd	2
3584.2.b.b.1793.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3584.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.41421i q^{3} -1.00000 q^{7} +1.00000 q^{9} -1.41421i q^{11} +1.41421i q^{19} +1.41421i q^{21} +4.00000 q^{23} +5.00000 q^{25} -5.65685i q^{27} +2.82843i q^{29} -4.00000 q^{31} -2.00000 q^{33} -2.82843i q^{37} +6.00000 q^{41} -4.24264i q^{43} +4.00000 q^{47} +1.00000 q^{49} +8.48528i q^{53} +2.00000 q^{57} -7.07107i q^{59} -5.65685i q^{61} -1.00000 q^{63} -1.41421i q^{67} -5.65685i q^{69} -8.00000 q^{71} +8.00000 q^{73} -7.07107i q^{75} +1.41421i q^{77} +4.00000 q^{79} -5.00000 q^{81} -7.07107i q^{83} +4.00000 q^{87} +8.00000 q^{89} +5.65685i q^{93} -8.00000 q^{97} -1.41421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{7} + 2 q^{9} + 8 q^{23} + 10 q^{25} - 8 q^{31} - 4 q^{33} + 12 q^{41} + 8 q^{47} + 2 q^{49} + 4 q^{57} - 2 q^{63} - 16 q^{71} + 16 q^{73} + 8 q^{79} - 10 q^{81} + 8 q^{87} + 16 q^{89} - 16 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 + 2 * q^9 + 8 * q^23 + 10 * q^25 - 8 * q^31 - 4 * q^33 + 12 * q^41 + 8 * q^47 + 2 * q^49 + 4 * q^57 - 2 * q^63 - 16 * q^71 + 16 * q^73 + 8 * q^79 - 10 * q^81 + 8 * q^87 + 16 * q^89 - 16 * q^97

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