LMFDB - Embedded newform 3200.2.f.g.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(449,3200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3200.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.f (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:	$25.5521286468$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 640)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3200.449
Dual form			3200.2.f.g.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} +4.24264i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.41421 q^{3} +4.24264i q^{7} -1.00000 q^{9} -2.82843i q^{11} -6.00000 q^{13} +6.00000i q^{17} -6.00000i q^{21} -4.24264i q^{23} +5.65685 q^{27} -8.48528 q^{31} +4.00000i q^{33} -6.00000 q^{37} +8.48528 q^{39} +4.24264 q^{43} -4.24264i q^{47} -11.0000 q^{49} -8.48528i q^{51} +6.00000 q^{53} -11.3137i q^{59} +6.00000i q^{61} -4.24264i q^{63} +12.7279 q^{67} +6.00000i q^{69} +8.48528 q^{71} -2.00000i q^{73} +12.0000 q^{77} -5.00000 q^{81} -9.89949 q^{83} +6.00000 q^{89} -25.4558i q^{91} +12.0000 q^{93} +10.0000i q^{97} +2.82843i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 24 q^{13} - 24 q^{37} - 44 q^{49} + 24 q^{53} + 48 q^{77} - 20 q^{81} + 24 q^{89} + 48 q^{93}+O(q^{100}) $ 4 * q - 4 * q^9 - 24 * q^13 - 24 * q^37 - 44 * q^49 + 24 * q^53 + 48 * q^77 - 20 * q^81 + 24 * q^89 + 48 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$.

$n$	$901$	$1151$	$2177$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.24264i	1.60357i	0.597614	+	0.801784i	$0.296115\pi$
$7$	4.24264i	1.60357i	−0.597614	+	0.801784i	$0.703885\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 2.82843i	− 0.852803i	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	− 2.82843i	− 0.852803i	0.904534	−	0.426401i	$-0.140219\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	− 6.00000i	− 1.30931i
$22$	0	0
$23$	− 4.24264i	− 0.884652i	−0.896854	−	0.442326i	$-0.854153\pi$
$23$	− 4.24264i	− 0.884652i	0.896854	−	0.442326i	$-0.145847\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−8.48528	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	−8.48528	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$32$	0	0
$33$	4.00000i	0.696311i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	8.48528	1.35873
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	4.24264	0.646997	0.323498	−	0.946229i	$-0.395141\pi$
$43$	4.24264	0.646997	0.323498	+	0.946229i	$0.395141\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 4.24264i	− 0.618853i	−0.950923	−	0.309426i	$-0.899863\pi$
$47$	− 4.24264i	− 0.618853i	0.950923	−	0.309426i	$-0.100137\pi$
$48$	0	0
$49$	−11.0000	−1.57143
$50$	0	0
$51$	− 8.48528i	− 1.18818i
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 11.3137i	− 1.47292i	−0.676481	−	0.736460i	$-0.736496\pi$
$59$	− 11.3137i	− 1.47292i	0.676481	−	0.736460i	$-0.263504\pi$
$60$	0	0
$61$	6.00000i	0.768221i	0.923287	+	0.384111i	$0.125492\pi$
$61$	6.00000i	0.768221i	−0.923287	+	0.384111i	$0.874508\pi$
$62$	0	0
$63$	− 4.24264i	− 0.534522i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.7279	1.55496	0.777482	−	0.628906i	$-0.216497\pi$
$67$	12.7279	1.55496	0.777482	+	0.628906i	$0.216497\pi$
$68$	0	0
$69$	6.00000i	0.722315i
$70$	0	0
$71$	8.48528	1.00702	0.503509	−	0.863990i	$-0.332042\pi$
$71$	8.48528	1.00702	0.503509	+	0.863990i	$0.332042\pi$
$72$	0	0
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.0000	1.36753
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−9.89949	−1.08661	−0.543305	−	0.839535i	$-0.682827\pi$
$83$	−9.89949	−1.08661	−0.543305	+	0.839535i	$0.682827\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	− 25.4558i	− 2.66850i
$92$	0	0
$93$	12.0000	1.24434
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	0	0
$99$	2.82843i	0.284268i
$100$	0	0
$101$	12.0000i	1.19404i	0.802225	+	0.597022i	$0.203650\pi$
$101$	12.0000i	1.19404i	−0.802225	+	0.597022i	$0.796350\pi$
$102$	0	0
$103$	12.7279i	1.25412i	0.778971	+	0.627060i	$0.215742\pi$
$103$	12.7279i	1.25412i	−0.778971	+	0.627060i	$0.784258\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.07107	0.683586	0.341793	−	0.939775i	$-0.388966\pi$
$107$	7.07107	0.683586	0.341793	+	0.939775i	$0.388966\pi$
$108$	0	0
$109$	− 18.0000i	− 1.72409i	−0.506834	−	0.862044i	$-0.669184\pi$
$109$	− 18.0000i	− 1.72409i	0.506834	−	0.862044i	$-0.330816\pi$
$110$	0	0
$111$	8.48528	0.805387
$112$	0	0
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.00000	0.554700
$118$	0	0
$119$	−25.4558	−2.33353
$120$	0	0
$121$	3.00000	0.272727
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.24264i	0.376473i	0.982124	+	0.188237i	$0.0602772\pi$
$127$	4.24264i	0.376473i	−0.982124	+	0.188237i	$0.939723\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	− 14.1421i	− 1.23560i	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	− 14.1421i	− 1.23560i	0.786334	−	0.617802i	$-0.211977\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	− 16.9706i	− 1.43942i	−0.694273	−	0.719712i	$-0.744274\pi$
$139$	− 16.9706i	− 1.43942i	0.694273	−	0.719712i	$-0.255726\pi$
$140$	0	0
$141$	6.00000i	0.505291i
$142$	0	0
$143$	16.9706i	1.41915i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	15.5563	1.28307
$148$	0	0
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	0	0
$151$	−8.48528	−0.690522	−0.345261	−	0.938507i	$-0.612210\pi$
$151$	−8.48528	−0.690522	−0.345261	+	0.938507i	$0.612210\pi$
$152$	0	0
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	−8.48528	−0.672927
$160$	0	0
$161$	18.0000	1.41860
$162$	0	0
$163$	4.24264	0.332309	0.166155	−	0.986100i	$-0.446865\pi$
$163$	4.24264	0.332309	0.166155	+	0.986100i	$0.446865\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 21.2132i	− 1.64153i	−0.571268	−	0.820763i	$-0.693548\pi$
$167$	− 21.2132i	− 1.64153i	0.571268	−	0.820763i	$-0.306452\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	16.0000i	1.20263i
$178$	0	0
$179$	5.65685i	0.422813i	0.977398	+	0.211407i	$0.0678044\pi$
$179$	5.65685i	0.422813i	−0.977398	+	0.211407i	$0.932196\pi$
$180$	0	0
$181$	− 12.0000i	− 0.891953i	−0.895045	−	0.445976i	$-0.852856\pi$
$181$	− 12.0000i	− 0.891953i	0.895045	−	0.445976i	$-0.147144\pi$
$182$	0	0
$183$	− 8.48528i	− 0.627250i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	16.9706	1.24101
$188$	0	0
$189$	24.0000i	1.74574i
$190$	0	0
$191$	25.4558	1.84192	0.920960	−	0.389657i	$-0.127406\pi$
$191$	25.4558	1.84192	0.920960	+	0.389657i	$0.127406\pi$
$192$	0	0
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	16.9706	1.20301	0.601506	−	0.798869i	$-0.294568\pi$
$199$	16.9706	1.20301	0.601506	+	0.798869i	$0.294568\pi$
$200$	0	0
$201$	−18.0000	−1.26962
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.24264i	0.294884i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 8.48528i	− 0.584151i	−0.956395	−	0.292075i	$-0.905654\pi$
$211$	− 8.48528i	− 0.584151i	0.956395	−	0.292075i	$-0.0943458\pi$
$212$	0	0
$213$	−12.0000	−0.822226
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 36.0000i	− 2.44384i
$218$	0	0
$219$	2.82843i	0.191127i
$220$	0	0
$221$	− 36.0000i	− 2.42162i
$222$	0	0
$223$	− 12.7279i	− 0.852325i	−0.904647	−	0.426162i	$-0.859865\pi$
$223$	− 12.7279i	− 0.852325i	0.904647	−	0.426162i	$-0.140135\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.89949	0.657053	0.328526	−	0.944495i	$-0.393448\pi$
$227$	9.89949	0.657053	0.328526	+	0.944495i	$0.393448\pi$
$228$	0	0
$229$	− 24.0000i	− 1.58596i	−0.609245	−	0.792982i	$-0.708527\pi$
$229$	− 24.0000i	− 1.58596i	0.609245	−	0.792982i	$-0.291473\pi$
$230$	0	0
$231$	−16.9706	−1.11658
$232$	0	0
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.9706	1.09773	0.548867	−	0.835910i	$-0.315059\pi$
$239$	16.9706	1.09773	0.548867	+	0.835910i	$0.315059\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	14.0000	0.887214
$250$	0	0
$251$	− 19.7990i	− 1.24970i	−0.780744	−	0.624851i	$-0.785160\pi$
$251$	− 19.7990i	− 1.24970i	0.780744	−	0.624851i	$-0.214840\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	0	0
$259$	− 25.4558i	− 1.58175i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.24264i	0.261612i	0.991408	+	0.130806i	$0.0417566\pi$
$263$	4.24264i	0.261612i	−0.991408	+	0.130806i	$0.958243\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−8.48528	−0.519291
$268$	0	0
$269$	18.0000i	1.09748i	0.835993	+	0.548740i	$0.184892\pi$
$269$	18.0000i	1.09748i	−0.835993	+	0.548740i	$0.815108\pi$
$270$	0	0
$271$	−8.48528	−0.515444	−0.257722	−	0.966219i	$-0.582972\pi$
$271$	−8.48528	−0.515444	−0.257722	+	0.966219i	$0.582972\pi$
$272$	0	0
$273$	36.0000i	2.17882i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	0	0
$279$	8.48528	0.508001
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−4.24264	−0.252199	−0.126099	−	0.992018i	$-0.540246\pi$
$283$	−4.24264	−0.252199	−0.126099	+	0.992018i	$0.540246\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	− 14.1421i	− 0.829027i
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 16.0000i	− 0.928414i
$298$	0	0
$299$	25.4558i	1.47215i
$300$	0	0
$301$	18.0000i	1.03750i
$302$	0	0
$303$	− 16.9706i	− 0.974933i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−21.2132	−1.21070	−0.605351	−	0.795959i	$-0.706967\pi$
$307$	−21.2132	−1.21070	−0.605351	+	0.795959i	$0.706967\pi$
$308$	0	0
$309$	− 18.0000i	− 1.02398i
$310$	0	0
$311$	−8.48528	−0.481156	−0.240578	−	0.970630i	$-0.577337\pi$
$311$	−8.48528	−0.481156	−0.240578	+	0.970630i	$0.577337\pi$
$312$	0	0
$313$	− 26.0000i	− 1.46961i	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	− 26.0000i	− 1.46961i	0.678280	−	0.734803i	$-0.262726\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−10.0000	−0.558146
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	25.4558i	1.40771i
$328$	0	0
$329$	18.0000	0.992372
$330$	0	0
$331$	8.48528i	0.466393i	0.972430	+	0.233197i	$0.0749186\pi$
$331$	8.48528i	0.466393i	−0.972430	+	0.233197i	$0.925081\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.0000i	0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$	14.0000i	0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	0	0
$339$	− 8.48528i	− 0.460857i
$340$	0	0
$341$	24.0000i	1.29967i
$342$	0	0
$343$	− 16.9706i	− 0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.8701	1.44246	0.721230	−	0.692696i	$-0.243577\pi$
$347$	26.8701	1.44246	0.721230	+	0.692696i	$0.243577\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	−33.9411	−1.81164
$352$	0	0
$353$	− 30.0000i	− 1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$	− 30.0000i	− 1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	36.0000	1.90532
$358$	0	0
$359$	−16.9706	−0.895672	−0.447836	−	0.894116i	$-0.647805\pi$
$359$	−16.9706	−0.895672	−0.447836	+	0.894116i	$0.647805\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	−4.24264	−0.222681
$364$	0	0
$365$	0	0
$366$	0	0
$367$	21.2132i	1.10732i	0.832743	+	0.553660i	$0.186769\pi$
$367$	21.2132i	1.10732i	−0.832743	+	0.553660i	$0.813231\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	25.4558i	1.32160i
$372$	0	0
$373$	18.0000	0.932005	0.466002	−	0.884783i	$-0.345694\pi$
$373$	18.0000	0.932005	0.466002	+	0.884783i	$0.345694\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 16.9706i	− 0.871719i	−0.900015	−	0.435860i	$-0.856444\pi$
$379$	− 16.9706i	− 0.871719i	0.900015	−	0.435860i	$-0.143556\pi$
$380$	0	0
$381$	− 6.00000i	− 0.307389i
$382$	0	0
$383$	21.2132i	1.08394i	0.840397	+	0.541972i	$0.182322\pi$
$383$	21.2132i	1.08394i	−0.840397	+	0.541972i	$0.817678\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.24264	−0.215666
$388$	0	0
$389$	6.00000i	0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$	6.00000i	0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$	0	0
$391$	25.4558	1.28736
$392$	0	0
$393$	20.0000i	1.00887i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	50.9117	2.53609
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16.9706i	0.841200i
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	25.4558i	1.25564i
$412$	0	0
$413$	48.0000	2.36193
$414$	0	0
$415$	0	0
$416$	0	0
$417$	24.0000i	1.17529i
$418$	0	0
$419$	11.3137i	0.552711i	0.961056	+	0.276355i	$0.0891267\pi$
$419$	11.3137i	0.552711i	−0.961056	+	0.276355i	$0.910873\pi$
$420$	0	0
$421$	− 6.00000i	− 0.292422i	−0.989253	−	0.146211i	$-0.953292\pi$
$421$	− 6.00000i	− 0.292422i	0.989253	−	0.146211i	$-0.0467079\pi$
$422$	0	0
$423$	4.24264i	0.206284i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−25.4558	−1.23189
$428$	0	0
$429$	− 24.0000i	− 1.15873i
$430$	0	0
$431$	−8.48528	−0.408722	−0.204361	−	0.978896i	$-0.565512\pi$
$431$	−8.48528	−0.408722	−0.204361	+	0.978896i	$0.565512\pi$
$432$	0	0
$433$	38.0000i	1.82616i	0.407777	+	0.913082i	$0.366304\pi$
$433$	38.0000i	1.82616i	−0.407777	+	0.913082i	$0.633696\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	0	0
$441$	11.0000	0.523810
$442$	0	0
$443$	−9.89949	−0.470339	−0.235170	−	0.971954i	$-0.575565\pi$
$443$	−9.89949	−0.470339	−0.235170	+	0.971954i	$0.575565\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 8.48528i	− 0.401340i
$448$	0	0
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	12.0000	0.563809
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	0	0
$459$	33.9411i	1.58424i
$460$	0	0
$461$	12.0000i	0.558896i	0.960161	+	0.279448i	$0.0901514\pi$
$461$	12.0000i	0.558896i	−0.960161	+	0.279448i	$0.909849\pi$
$462$	0	0
$463$	− 21.2132i	− 0.985861i	−0.870069	−	0.492931i	$-0.835926\pi$
$463$	− 21.2132i	− 0.985861i	0.870069	−	0.492931i	$-0.164074\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.07107	−0.327210	−0.163605	−	0.986526i	$-0.552312\pi$
$467$	−7.07107	−0.327210	−0.163605	+	0.986526i	$0.552312\pi$
$468$	0	0
$469$	54.0000i	2.49349i
$470$	0	0
$471$	25.4558	1.17294
$472$	0	0
$473$	− 12.0000i	− 0.551761i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	36.0000	1.64146
$482$	0	0
$483$	−25.4558	−1.15828
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 12.7279i	− 0.576757i	−0.957516	−	0.288379i	$-0.906884\pi$
$487$	− 12.7279i	− 0.576757i	0.957516	−	0.288379i	$-0.0931162\pi$
$488$	0	0
$489$	−6.00000	−0.271329
$490$	0	0
$491$	36.7696i	1.65939i	0.558219	+	0.829693i	$0.311485\pi$
$491$	36.7696i	1.65939i	−0.558219	+	0.829693i	$0.688515\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	36.0000i	1.61482i
$498$	0	0
$499$	− 16.9706i	− 0.759707i	−0.925047	−	0.379853i	$-0.875974\pi$
$499$	− 16.9706i	− 0.759707i	0.925047	−	0.379853i	$-0.124026\pi$
$500$	0	0
$501$	30.0000i	1.34030i
$502$	0	0
$503$	12.7279i	0.567510i	0.958897	+	0.283755i	$0.0915802\pi$
$503$	12.7279i	0.567510i	−0.958897	+	0.283755i	$0.908420\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−32.5269	−1.44457
$508$	0	0
$509$	− 24.0000i	− 1.06378i	−0.846813	−	0.531891i	$-0.821482\pi$
$509$	− 24.0000i	− 1.06378i	0.846813	−	0.531891i	$-0.178518\pi$
$510$	0	0
$511$	8.48528	0.375367
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−12.0000	−0.527759
$518$	0	0
$519$	8.48528	0.372463
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	−4.24264	−0.185518	−0.0927589	−	0.995689i	$-0.529569\pi$
$523$	−4.24264	−0.185518	−0.0927589	+	0.995689i	$0.529569\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 50.9117i	− 2.21775i
$528$	0	0
$529$	5.00000	0.217391
$530$	0	0
$531$	11.3137i	0.490973i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 8.00000i	− 0.345225i
$538$	0	0
$539$	31.1127i	1.34012i
$540$	0	0
$541$	12.0000i	0.515920i	0.966156	+	0.257960i	$0.0830503\pi$
$541$	12.0000i	0.515920i	−0.966156	+	0.257960i	$0.916950\pi$
$542$	0	0
$543$	16.9706i	0.728277i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−12.7279	−0.544207	−0.272103	−	0.962268i	$-0.587719\pi$
$547$	−12.7279	−0.544207	−0.272103	+	0.962268i	$0.587719\pi$
$548$	0	0
$549$	− 6.00000i	− 0.256074i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	0	0
$559$	−25.4558	−1.07667
$560$	0	0
$561$	−24.0000	−1.01328
$562$	0	0
$563$	−1.41421	−0.0596020	−0.0298010	−	0.999556i	$-0.509487\pi$
$563$	−1.41421	−0.0596020	−0.0298010	+	0.999556i	$0.509487\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 21.2132i	− 0.890871i
$568$	0	0
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	0	0
$571$	− 8.48528i	− 0.355098i	−0.984112	−	0.177549i	$-0.943183\pi$
$571$	− 8.48528i	− 0.355098i	0.984112	−	0.177549i	$-0.0568168\pi$
$572$	0	0
$573$	−36.0000	−1.50392
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	0	0
$579$	19.7990i	0.822818i
$580$	0	0
$581$	− 42.0000i	− 1.74245i
$582$	0	0
$583$	− 16.9706i	− 0.702849i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−35.3553	−1.45927	−0.729636	−	0.683836i	$-0.760310\pi$
$587$	−35.3553	−1.45927	−0.729636	+	0.683836i	$0.760310\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−8.48528	−0.349038
$592$	0	0
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−24.0000	−0.982255
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	0	0
$603$	−12.7279	−0.518321
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 38.1838i	− 1.54983i	−0.632065	−	0.774916i	$-0.717792\pi$
$607$	− 38.1838i	− 1.54983i	0.632065	−	0.774916i	$-0.282208\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	25.4558i	1.02983i
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	0	0
$619$	− 16.9706i	− 0.682105i	−0.940044	−	0.341052i	$-0.889217\pi$
$619$	− 16.9706i	− 0.682105i	0.940044	−	0.341052i	$-0.110783\pi$
$620$	0	0
$621$	− 24.0000i	− 0.963087i
$622$	0	0
$623$	25.4558i	1.01987i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 36.0000i	− 1.43541i
$630$	0	0
$631$	8.48528	0.337794	0.168897	−	0.985634i	$-0.445980\pi$
$631$	8.48528	0.337794	0.168897	+	0.985634i	$0.445980\pi$
$632$	0	0
$633$	12.0000i	0.476957i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	66.0000	2.61502
$638$	0	0
$639$	−8.48528	−0.335673
$640$	0	0
$641$	48.0000	1.89589	0.947943	−	0.318440i	$-0.103159\pi$
$641$	48.0000	1.89589	0.947943	+	0.318440i	$0.103159\pi$
$642$	0	0
$643$	29.6985	1.17119	0.585597	−	0.810602i	$-0.300860\pi$
$643$	29.6985	1.17119	0.585597	+	0.810602i	$0.300860\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.2132i	0.833977i	0.908912	+	0.416989i	$0.136914\pi$
$647$	21.2132i	0.833977i	−0.908912	+	0.416989i	$0.863086\pi$
$648$	0	0
$649$	−32.0000	−1.25611
$650$	0	0
$651$	50.9117i	1.99539i
$652$	0	0
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.00000i	0.0780274i
$658$	0	0
$659$	22.6274i	0.881439i	0.897645	+	0.440720i	$0.145277\pi$
$659$	22.6274i	0.881439i	−0.897645	+	0.440720i	$0.854723\pi$
$660$	0	0
$661$	− 42.0000i	− 1.63361i	−0.576913	−	0.816805i	$-0.695743\pi$
$661$	− 42.0000i	− 1.63361i	0.576913	−	0.816805i	$-0.304257\pi$
$662$	0	0
$663$	50.9117i	1.97725i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	18.0000i	0.695920i
$670$	0	0
$671$	16.9706	0.655141
$672$	0	0
$673$	− 10.0000i	− 0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$	− 10.0000i	− 0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−42.4264	−1.62818
$680$	0	0
$681$	−14.0000	−0.536481
$682$	0	0
$683$	−18.3848	−0.703474	−0.351737	−	0.936099i	$-0.614409\pi$
$683$	−18.3848	−0.703474	−0.351737	+	0.936099i	$0.614409\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	33.9411i	1.29493i
$688$	0	0
$689$	−36.0000	−1.37149
$690$	0	0
$691$	42.4264i	1.61398i	0.590567	+	0.806988i	$0.298904\pi$
$691$	42.4264i	1.61398i	−0.590567	+	0.806988i	$0.701096\pi$
$692$	0	0
$693$	−12.0000	−0.455842
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	8.48528i	0.320943i
$700$	0	0
$701$	42.0000i	1.58632i	0.609015	+	0.793159i	$0.291565\pi$
$701$	42.0000i	1.58632i	−0.609015	+	0.793159i	$0.708435\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−50.9117	−1.91473
$708$	0	0
$709$	24.0000i	0.901339i	0.892691	+	0.450669i	$0.148815\pi$
$709$	24.0000i	0.901339i	−0.892691	+	0.450669i	$0.851185\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	36.0000i	1.34821i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	−16.9706	−0.632895	−0.316448	−	0.948610i	$-0.602490\pi$
$719$	−16.9706	−0.632895	−0.316448	+	0.948610i	$0.602490\pi$
$720$	0	0
$721$	−54.0000	−2.01107
$722$	0	0
$723$	11.3137	0.420761
$724$	0	0
$725$	0	0
$726$	0	0
$727$	4.24264i	0.157351i	0.996900	+	0.0786754i	$0.0250691\pi$
$727$	4.24264i	0.157351i	−0.996900	+	0.0786754i	$0.974931\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	25.4558i	0.941518i
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 36.0000i	− 1.32608i
$738$	0	0
$739$	− 16.9706i	− 0.624272i	−0.950037	−	0.312136i	$-0.898955\pi$
$739$	− 16.9706i	− 0.624272i	0.950037	−	0.312136i	$-0.101045\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 46.6690i	− 1.71212i	−0.516875	−	0.856061i	$-0.672905\pi$
$743$	− 46.6690i	− 1.71212i	0.516875	−	0.856061i	$-0.327095\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.89949	0.362204
$748$	0	0
$749$	30.0000i	1.09618i
$750$	0	0
$751$	42.4264	1.54816	0.774081	−	0.633087i	$-0.218212\pi$
$751$	42.4264	1.54816	0.774081	+	0.633087i	$0.218212\pi$
$752$	0	0
$753$	28.0000i	1.02038i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	0	0
$759$	16.9706	0.615992
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	76.3675	2.76469
$764$	0	0
$765$	0	0
$766$	0	0
$767$	67.8823i	2.45109i
$768$	0	0
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	8.48528i	0.305590i
$772$	0	0
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	36.0000i	1.29149i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	− 24.0000i	− 0.858788i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−38.1838	−1.36110	−0.680552	−	0.732700i	$-0.738260\pi$
$787$	−38.1838	−1.36110	−0.680552	+	0.732700i	$0.738260\pi$
$788$	0	0
$789$	− 6.00000i	− 0.213606i
$790$	0	0
$791$	−25.4558	−0.905106
$792$	0	0
$793$	− 36.0000i	− 1.27840i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	25.4558	0.900563
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	−5.65685	−0.199626
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 25.4558i	− 0.896088i
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	− 8.48528i	− 0.297959i	−0.988840	−	0.148979i	$-0.952401\pi$
$811$	− 8.48528i	− 0.297959i	0.988840	−	0.148979i	$-0.0475988\pi$
$812$	0	0
$813$	12.0000	0.420858
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	25.4558i	0.889499i
$820$	0	0
$821$	− 18.0000i	− 0.628204i	−0.949389	−	0.314102i	$-0.898297\pi$
$821$	− 18.0000i	− 0.628204i	0.949389	−	0.314102i	$-0.101703\pi$
$822$	0	0
$823$	29.6985i	1.03522i	0.855615	+	0.517612i	$0.173179\pi$
$823$	29.6985i	1.03522i	−0.855615	+	0.517612i	$0.826821\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18.3848	−0.639301	−0.319651	−	0.947535i	$-0.603566\pi$
$827$	−18.3848	−0.639301	−0.319651	+	0.947535i	$0.603566\pi$
$828$	0	0
$829$	− 30.0000i	− 1.04194i	−0.853574	−	0.520972i	$-0.825570\pi$
$829$	− 30.0000i	− 1.04194i	0.853574	−	0.520972i	$-0.174430\pi$
$830$	0	0
$831$	25.4558	0.883053
$832$	0	0
$833$	− 66.0000i	− 2.28676i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−48.0000	−1.65912
$838$	0	0
$839$	−16.9706	−0.585889	−0.292944	−	0.956129i	$-0.594635\pi$
$839$	−16.9706	−0.585889	−0.292944	+	0.956129i	$0.594635\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	12.7279i	0.437337i
$848$	0	0
$849$	6.00000	0.205919
$850$	0	0
$851$	25.4558i	0.872615i
$852$	0	0
$853$	42.0000	1.43805	0.719026	−	0.694983i	$-0.244588\pi$
$853$	42.0000	1.43805	0.719026	+	0.694983i	$0.244588\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.00000i	0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$	6.00000i	0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 38.1838i	− 1.29979i	−0.760024	−	0.649895i	$-0.774813\pi$
$863$	− 38.1838i	− 1.29979i	0.760024	−	0.649895i	$-0.225187\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	26.8701	0.912555
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−76.3675	−2.58762
$872$	0	0
$873$	− 10.0000i	− 0.338449i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.0000	1.01303	0.506514	−	0.862232i	$-0.330934\pi$
$877$	30.0000	1.01303	0.506514	+	0.862232i	$0.330934\pi$
$878$	0	0
$879$	25.4558	0.858604
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−46.6690	−1.57054	−0.785269	−	0.619154i	$-0.787475\pi$
$883$	−46.6690	−1.57054	−0.785269	+	0.619154i	$0.787475\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 29.6985i	− 0.997178i	−0.866839	−	0.498589i	$-0.833852\pi$
$887$	− 29.6985i	− 0.997178i	0.866839	−	0.498589i	$-0.166148\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	14.1421i	0.473779i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 36.0000i	− 1.20201i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	36.0000i	1.19933i
$902$	0	0
$903$	− 25.4558i	− 0.847117i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−21.2132	−0.704373	−0.352186	−	0.935930i	$-0.614562\pi$
$907$	−21.2132	−0.704373	−0.352186	+	0.935930i	$0.614562\pi$
$908$	0	0
$909$	− 12.0000i	− 0.398015i
$910$	0	0
$911$	−42.4264	−1.40565	−0.702825	−	0.711363i	$-0.748078\pi$
$911$	−42.4264	−1.40565	−0.702825	+	0.711363i	$0.748078\pi$
$912$	0	0
$913$	28.0000i	0.926665i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	60.0000	1.98137
$918$	0	0
$919$	33.9411	1.11961	0.559807	−	0.828623i	$-0.310875\pi$
$919$	33.9411	1.11961	0.559807	+	0.828623i	$0.310875\pi$
$920$	0	0
$921$	30.0000	0.988534
$922$	0	0
$923$	−50.9117	−1.67578
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 12.7279i	− 0.418040i
$928$	0	0
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	12.0000	0.392862
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 2.00000i	− 0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$	− 2.00000i	− 0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$	0	0
$939$	36.7696i	1.19993i
$940$	0	0
$941$	12.0000i	0.391189i	0.980685	+	0.195594i	$0.0626636\pi$
$941$	12.0000i	0.391189i	−0.980685	+	0.195594i	$0.937336\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9.89949	−0.321690	−0.160845	−	0.986980i	$-0.551422\pi$
$947$	−9.89949	−0.321690	−0.160845	+	0.986980i	$0.551422\pi$
$948$	0	0
$949$	12.0000i	0.389536i
$950$	0	0
$951$	8.48528	0.275154
$952$	0	0
$953$	30.0000i	0.971795i	0.874016	+	0.485898i	$0.161507\pi$
$953$	30.0000i	0.971795i	−0.874016	+	0.485898i	$0.838493\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	76.3675	2.46604
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	−7.07107	−0.227862
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 4.24264i	− 0.136434i	−0.997671	−	0.0682171i	$-0.978269\pi$
$967$	− 4.24264i	− 0.136434i	0.997671	−	0.0682171i	$-0.0217310\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 19.7990i	− 0.635380i	−0.948195	−	0.317690i	$-0.897093\pi$
$971$	− 19.7990i	− 0.635380i	0.948195	−	0.317690i	$-0.102907\pi$
$972$	0	0
$973$	72.0000	2.30821
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 54.0000i	− 1.72761i	−0.503824	−	0.863807i	$-0.668074\pi$
$977$	− 54.0000i	− 1.72761i	0.503824	−	0.863807i	$-0.331926\pi$
$978$	0	0
$979$	− 16.9706i	− 0.542382i
$980$	0	0
$981$	18.0000i	0.574696i
$982$	0	0
$983$	− 29.6985i	− 0.947235i	−0.880731	−	0.473617i	$-0.842948\pi$
$983$	− 29.6985i	− 0.947235i	0.880731	−	0.473617i	$-0.157052\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−25.4558	−0.810268
$988$	0	0
$989$	− 18.0000i	− 0.572367i
$990$	0	0
$991$	−42.4264	−1.34772	−0.673860	−	0.738859i	$-0.735365\pi$
$991$	−42.4264	−1.34772	−0.673860	+	0.738859i	$0.735365\pi$
$992$	0	0
$993$	− 12.0000i	− 0.380808i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	0	0
$999$	−33.9411	−1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3200.2.f.g.449.2		4
4.3	odd	2	inner	3200.2.f.g.449.3		4
5.2	odd	4		640.2.d.b.321.4	yes	4
5.3	odd	4		3200.2.d.v.1601.2		4
5.4	even	2		3200.2.f.l.449.3		4
8.3	odd	2		3200.2.f.l.449.1		4
8.5	even	2		3200.2.f.l.449.4		4
15.2	even	4		5760.2.k.v.2881.1		4
20.3	even	4		3200.2.d.v.1601.3		4
20.7	even	4		640.2.d.b.321.2	yes	4
20.19	odd	2		3200.2.f.l.449.2		4
40.3	even	4		3200.2.d.v.1601.1		4
40.13	odd	4		3200.2.d.v.1601.4		4
40.19	odd	2	inner	3200.2.f.g.449.4		4
40.27	even	4		640.2.d.b.321.3	yes	4
40.29	even	2	inner	3200.2.f.g.449.1		4
40.37	odd	4		640.2.d.b.321.1	✓	4
60.47	odd	4		5760.2.k.v.2881.2		4
80.3	even	4		6400.2.a.bp.1.1		2
80.13	odd	4		6400.2.a.bp.1.2		2
80.27	even	4		1280.2.a.g.1.1		2
80.37	odd	4		1280.2.a.g.1.2		2
80.43	even	4		6400.2.a.bk.1.2		2
80.53	odd	4		6400.2.a.bk.1.1		2
80.67	even	4		1280.2.a.i.1.2		2
80.77	odd	4		1280.2.a.i.1.1		2
120.77	even	4		5760.2.k.v.2881.3		4
120.107	odd	4		5760.2.k.v.2881.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.d.b.321.1	✓	4	40.37	odd	4
640.2.d.b.321.2	yes	4	20.7	even	4
640.2.d.b.321.3	yes	4	40.27	even	4
640.2.d.b.321.4	yes	4	5.2	odd	4
1280.2.a.g.1.1		2	80.27	even	4
1280.2.a.g.1.2		2	80.37	odd	4
1280.2.a.i.1.1		2	80.77	odd	4
1280.2.a.i.1.2		2	80.67	even	4
3200.2.d.v.1601.1		4	40.3	even	4
3200.2.d.v.1601.2		4	5.3	odd	4
3200.2.d.v.1601.3		4	20.3	even	4
3200.2.d.v.1601.4		4	40.13	odd	4
3200.2.f.g.449.1		4	40.29	even	2	inner
3200.2.f.g.449.2		4	1.1	even	1	trivial
3200.2.f.g.449.3		4	4.3	odd	2	inner
3200.2.f.g.449.4		4	40.19	odd	2	inner
3200.2.f.l.449.1		4	8.3	odd	2
3200.2.f.l.449.2		4	20.19	odd	2
3200.2.f.l.449.3		4	5.4	even	2
3200.2.f.l.449.4		4	8.5	even	2
5760.2.k.v.2881.1		4	15.2	even	4
5760.2.k.v.2881.2		4	60.47	odd	4
5760.2.k.v.2881.3		4	120.77	even	4
5760.2.k.v.2881.4		4	120.107	odd	4
6400.2.a.bk.1.1		2	80.53	odd	4
6400.2.a.bk.1.2		2	80.43	even	4
6400.2.a.bp.1.1		2	80.3	even	4
6400.2.a.bp.1.2		2	80.13	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:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} +4.24264i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{3} +4.24264i q^{7} -1.00000 q^{9} -2.82843i q^{11} -6.00000 q^{13} +6.00000i q^{17} -6.00000i q^{21} -4.24264i q^{23} +5.65685 q^{27} -8.48528 q^{31} +4.00000i q^{33} -6.00000 q^{37} +8.48528 q^{39} +4.24264 q^{43} -4.24264i q^{47} -11.0000 q^{49} -8.48528i q^{51} +6.00000 q^{53} -11.3137i q^{59} +6.00000i q^{61} -4.24264i q^{63} +12.7279 q^{67} +6.00000i q^{69} +8.48528 q^{71} -2.00000i q^{73} +12.0000 q^{77} -5.00000 q^{81} -9.89949 q^{83} +6.00000 q^{89} -25.4558i q^{91} +12.0000 q^{93} +10.0000i q^{97} +2.82843i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 24 q^{13} - 24 q^{37} - 44 q^{49} + 24 q^{53} + 48 q^{77} - 20 q^{81} + 24 q^{89} + 48 q^{93}+O(q^{100}) \) 4 * q - 4 * q^9 - 24 * q^13 - 24 * q^37 - 44 * q^49 + 24 * q^53 + 48 * q^77 - 20 * q^81 + 24 * q^89 + 48 * q^93

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