LMFDB - Embedded newform 3200.2.f.j.449.1

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} -4.24264i q^{7} -1.00000 q^{9} +5.65685i q^{11} +2.00000 q^{13} +6.00000i q^{17} -2.82843i q^{19} +6.00000i q^{21} -7.07107i q^{23} +5.65685 q^{27} -4.00000i q^{29} -2.82843 q^{31} -8.00000i q^{33} -2.00000 q^{37} -2.82843 q^{39} -8.00000 q^{41} -1.41421 q^{43} -1.41421i q^{47} -11.0000 q^{49} -8.48528i q^{51} -2.00000 q^{53} +4.00000i q^{57} -2.82843i q^{59} +14.0000i q^{61} +4.24264i q^{63} -4.24264 q^{67} +10.0000i q^{69} +2.82843 q^{71} +6.00000i q^{73} +24.0000 q^{77} -16.9706 q^{79} -5.00000 q^{81} +12.7279 q^{83} +5.65685i q^{87} +6.00000 q^{89} -8.48528i q^{91} +4.00000 q^{93} +10.0000i q^{97} -5.65685i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9} + 8 q^{13} - 8 q^{37} - 32 q^{41} - 44 q^{49} - 8 q^{53} + 96 q^{77} - 20 q^{81} + 24 q^{89} + 16 q^{93}+O(q^{100}) $ 4 * q - 4 * q^9 + 8 * q^13 - 8 * q^37 - 32 * q^41 - 44 * q^49 - 8 * q^53 + 96 * q^77 - 20 * q^81 + 24 * q^89 + 16 * q^93

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	$-0.703885\pi$
$7$	− 4.24264i	− 1.60357i	0.597614	−	0.801784i	$-0.296115\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.65685i	1.70561i	0.522233	+	0.852803i	$0.325099\pi$
$11$	5.65685i	1.70561i	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	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
$19$	− 2.82843i	− 0.648886i	−0.945905	−	0.324443i	$-0.894823\pi$
$19$	− 2.82843i	− 0.648886i	0.945905	−	0.324443i	$-0.105177\pi$
$20$	0	0
$21$	6.00000i	1.30931i
$22$	0	0
$23$	− 7.07107i	− 1.47442i	−0.675664	−	0.737210i	$-0.736143\pi$
$23$	− 7.07107i	− 1.47442i	0.675664	−	0.737210i	$-0.263857\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	− 4.00000i	− 0.742781i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	− 4.00000i	− 0.742781i	0.928477	−	0.371391i	$-0.121119\pi$
$30$	0	0
$31$	−2.82843	−0.508001	−0.254000	−	0.967204i	$-0.581746\pi$
$31$	−2.82843	−0.508001	−0.254000	+	0.967204i	$0.581746\pi$
$32$	0	0
$33$	− 8.00000i	− 1.39262i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−2.82843	−0.452911
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	−1.41421	−0.215666	−0.107833	−	0.994169i	$-0.534391\pi$
$43$	−1.41421	−0.215666	−0.107833	+	0.994169i	$0.534391\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1.41421i	− 0.206284i	−0.994667	−	0.103142i	$-0.967110\pi$
$47$	− 1.41421i	− 0.206284i	0.994667	−	0.103142i	$-0.0328896\pi$
$48$	0	0
$49$	−11.0000	−1.57143
$50$	0	0
$51$	− 8.48528i	− 1.18818i
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.00000i	0.529813i
$58$	0	0
$59$	− 2.82843i	− 0.368230i	−0.982905	−	0.184115i	$-0.941058\pi$
$59$	− 2.82843i	− 0.368230i	0.982905	−	0.184115i	$-0.0589419\pi$
$60$	0	0
$61$	14.0000i	1.79252i	0.443533	+	0.896258i	$0.353725\pi$
$61$	14.0000i	1.79252i	−0.443533	+	0.896258i	$0.646275\pi$
$62$	0	0
$63$	4.24264i	0.534522i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.24264	−0.518321	−0.259161	−	0.965834i	$-0.583446\pi$
$67$	−4.24264	−0.518321	−0.259161	+	0.965834i	$0.583446\pi$
$68$	0	0
$69$	10.0000i	1.20386i
$70$	0	0
$71$	2.82843	0.335673	0.167836	−	0.985815i	$-0.446322\pi$
$71$	2.82843	0.335673	0.167836	+	0.985815i	$0.446322\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	24.0000	2.73505
$78$	0	0
$79$	−16.9706	−1.90934	−0.954669	−	0.297670i	$-0.903790\pi$
$79$	−16.9706	−1.90934	−0.954669	+	0.297670i	$0.903790\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	12.7279	1.39707	0.698535	−	0.715575i	$-0.253835\pi$
$83$	12.7279	1.39707	0.698535	+	0.715575i	$0.253835\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.65685i	0.606478i
$88$	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
$91$	− 8.48528i	− 0.889499i
$92$	0	0
$93$	4.00000	0.414781
$94$	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
$99$	− 5.65685i	− 0.568535i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	9.89949i	0.975426i	0.873004	+	0.487713i	$0.162169\pi$
$103$	9.89949i	0.975426i	−0.873004	+	0.487713i	$0.837831\pi$
$104$	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
$109$	6.00000i	0.574696i	0.957826	+	0.287348i	$0.0927736\pi$
$109$	6.00000i	0.574696i	−0.957826	+	0.287348i	$0.907226\pi$
$110$	0	0
$111$	2.82843	0.268462
$112$	0	0
$113$	10.0000i	0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$	10.0000i	0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	25.4558	2.33353
$120$	0	0
$121$	−21.0000	−1.90909
$122$	0	0
$123$	11.3137	1.02012
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 9.89949i	− 0.878438i	−0.898380	−	0.439219i	$-0.855255\pi$
$127$	− 9.89949i	− 0.878438i	0.898380	−	0.439219i	$-0.144745\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	0	0
$131$	11.3137i	0.988483i	0.869325	+	0.494242i	$0.164554\pi$
$131$	11.3137i	0.988483i	−0.869325	+	0.494242i	$0.835446\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.00000i	0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$	2.00000i	0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$	0	0
$139$	8.48528i	0.719712i	0.933008	+	0.359856i	$0.117174\pi$
$139$	8.48528i	0.719712i	−0.933008	+	0.359856i	$0.882826\pi$
$140$	0	0
$141$	2.00000i	0.168430i
$142$	0	0
$143$	11.3137i	0.946100i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	15.5563	1.28307
$148$	0	0
$149$	22.0000i	1.80231i	0.433497	+	0.901155i	$0.357280\pi$
$149$	22.0000i	1.80231i	−0.433497	+	0.901155i	$0.642720\pi$
$150$	0	0
$151$	14.1421	1.15087	0.575435	−	0.817847i	$-0.304833\pi$
$151$	14.1421	1.15087	0.575435	+	0.817847i	$0.304833\pi$
$152$	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	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	0	0
$159$	2.82843	0.224309
$160$	0	0
$161$	−30.0000	−2.36433
$162$	0	0
$163$	−18.3848	−1.44001	−0.720003	−	0.693971i	$-0.755860\pi$
$163$	−18.3848	−1.44001	−0.720003	+	0.693971i	$0.755860\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.89949i	0.766046i	0.923739	+	0.383023i	$0.125117\pi$
$167$	9.89949i	0.766046i	−0.923739	+	0.383023i	$0.874883\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	2.82843i	0.216295i
$172$	0	0
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.00000i	0.300658i
$178$	0	0
$179$	19.7990i	1.47985i	0.672692	+	0.739923i	$0.265138\pi$
$179$	19.7990i	1.47985i	−0.672692	+	0.739923i	$0.734862\pi$
$180$	0	0
$181$	16.0000i	1.18927i	0.803996	+	0.594635i	$0.202704\pi$
$181$	16.0000i	1.18927i	−0.803996	+	0.594635i	$0.797296\pi$
$182$	0	0
$183$	− 19.7990i	− 1.46358i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−33.9411	−2.48202
$188$	0	0
$189$	− 24.0000i	− 1.74574i
$190$	0	0
$191$	2.82843	0.204658	0.102329	−	0.994751i	$-0.467371\pi$
$191$	2.82843	0.204658	0.102329	+	0.994751i	$0.467371\pi$
$192$	0	0
$193$	− 6.00000i	− 0.431889i	−0.976406	−	0.215945i	$-0.930717\pi$
$193$	− 6.00000i	− 0.431889i	0.976406	−	0.215945i	$-0.0692831\pi$
$194$	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
$199$	−11.3137	−0.802008	−0.401004	−	0.916076i	$-0.631339\pi$
$199$	−11.3137	−0.802008	−0.401004	+	0.916076i	$0.631339\pi$
$200$	0	0
$201$	6.00000	0.423207
$202$	0	0
$203$	−16.9706	−1.19110
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.07107i	0.491473i
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	16.9706i	1.16830i	0.811645	+	0.584151i	$0.198572\pi$
$211$	16.9706i	1.16830i	−0.811645	+	0.584151i	$0.801428\pi$
$212$	0	0
$213$	−4.00000	−0.274075
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.0000i	0.814613i
$218$	0	0
$219$	− 8.48528i	− 0.573382i
$220$	0	0
$221$	12.0000i	0.807207i
$222$	0	0
$223$	− 15.5563i	− 1.04173i	−0.853639	−	0.520865i	$-0.825609\pi$
$223$	− 15.5563i	− 1.04173i	0.853639	−	0.520865i	$-0.174391\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.07107	−0.469323	−0.234662	−	0.972077i	$-0.575398\pi$
$227$	−7.07107	−0.469323	−0.234662	+	0.972077i	$0.575398\pi$
$228$	0	0
$229$	20.0000i	1.32164i	0.750546	+	0.660819i	$0.229791\pi$
$229$	20.0000i	1.32164i	−0.750546	+	0.660819i	$0.770209\pi$
$230$	0	0
$231$	−33.9411	−2.23316
$232$	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
$237$	24.0000	1.55897
$238$	0	0
$239$	11.3137	0.731823	0.365911	−	0.930650i	$-0.380757\pi$
$239$	11.3137	0.731823	0.365911	+	0.930650i	$0.380757\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 5.65685i	− 0.359937i
$248$	0	0
$249$	−18.0000	−1.14070
$250$	0	0
$251$	− 22.6274i	− 1.42823i	−0.700028	−	0.714115i	$-0.746829\pi$
$251$	− 22.6274i	− 1.42823i	0.700028	−	0.714115i	$-0.253171\pi$
$252$	0	0
$253$	40.0000	2.51478
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	0	0
$259$	8.48528i	0.527250i
$260$	0	0
$261$	4.00000i	0.247594i
$262$	0	0
$263$	7.07107i	0.436021i	0.975946	+	0.218010i	$0.0699567\pi$
$263$	7.07107i	0.436021i	−0.975946	+	0.218010i	$0.930043\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−8.48528	−0.519291
$268$	0	0
$269$	− 6.00000i	− 0.365826i	−0.983129	−	0.182913i	$-0.941447\pi$
$269$	− 6.00000i	− 0.365826i	0.983129	−	0.182913i	$-0.0585527\pi$
$270$	0	0
$271$	−25.4558	−1.54633	−0.773166	−	0.634203i	$-0.781328\pi$
$271$	−25.4558	−1.54633	−0.773166	+	0.634203i	$0.781328\pi$
$272$	0	0
$273$	12.0000i	0.726273i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−30.0000	−1.80253	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−30.0000	−1.80253	−0.901263	+	0.433273i	$0.857359\pi$
$278$	0	0
$279$	2.82843	0.169334
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	0	0
$283$	−9.89949	−0.588464	−0.294232	−	0.955734i	$-0.595064\pi$
$283$	−9.89949	−0.588464	−0.294232	+	0.955734i	$0.595064\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	33.9411i	2.00348i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	− 14.1421i	− 0.829027i
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	32.0000i	1.85683i
$298$	0	0
$299$	− 14.1421i	− 0.817861i
$300$	0	0
$301$	6.00000i	0.345834i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−15.5563	−0.887848	−0.443924	−	0.896065i	$-0.646414\pi$
$307$	−15.5563	−0.887848	−0.443924	+	0.896065i	$0.646414\pi$
$308$	0	0
$309$	− 14.0000i	− 0.796432i
$310$	0	0
$311$	2.82843	0.160385	0.0801927	−	0.996779i	$-0.474446\pi$
$311$	2.82843	0.160385	0.0801927	+	0.996779i	$0.474446\pi$
$312$	0	0
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	22.6274	1.26689
$320$	0	0
$321$	−10.0000	−0.558146
$322$	0	0
$323$	16.9706	0.944267
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 8.48528i	− 0.469237i
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	22.6274i	1.24372i	0.783130	+	0.621858i	$0.213622\pi$
$331$	22.6274i	1.24372i	−0.783130	+	0.621858i	$0.786378\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.0000i	0.980522i	0.871576	+	0.490261i	$0.163099\pi$
$337$	18.0000i	0.980522i	−0.871576	+	0.490261i	$0.836901\pi$
$338$	0	0
$339$	− 14.1421i	− 0.768095i
$340$	0	0
$341$	− 16.0000i	− 0.866449i
$342$	0	0
$343$	16.9706i	0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.24264	0.227757	0.113878	−	0.993495i	$-0.463673\pi$
$347$	4.24264	0.227757	0.113878	+	0.993495i	$0.463673\pi$
$348$	0	0
$349$	− 12.0000i	− 0.642345i	−0.947021	−	0.321173i	$-0.895923\pi$
$349$	− 12.0000i	− 0.642345i	0.947021	−	0.321173i	$-0.104077\pi$
$350$	0	0
$351$	11.3137	0.603881
$352$	0	0
$353$	− 10.0000i	− 0.532246i	−0.963939	−	0.266123i	$-0.914257\pi$
$353$	− 10.0000i	− 0.532246i	0.963939	−	0.266123i	$-0.0857428\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−36.0000	−1.90532
$358$	0	0
$359$	−28.2843	−1.49279	−0.746393	−	0.665505i	$-0.768216\pi$
$359$	−28.2843	−1.49279	−0.746393	+	0.665505i	$0.768216\pi$
$360$	0	0
$361$	11.0000	0.578947
$362$	0	0
$363$	29.6985	1.55877
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 4.24264i	− 0.221464i	−0.993850	−	0.110732i	$-0.964680\pi$
$367$	− 4.24264i	− 0.221464i	0.993850	−	0.110732i	$-0.0353195\pi$
$368$	0	0
$369$	8.00000	0.416463
$370$	0	0
$371$	8.48528i	0.440534i
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.00000i	− 0.412021i
$378$	0	0
$379$	31.1127i	1.59815i	0.601230	+	0.799076i	$0.294678\pi$
$379$	31.1127i	1.59815i	−0.601230	+	0.799076i	$0.705322\pi$
$380$	0	0
$381$	14.0000i	0.717242i
$382$	0	0
$383$	7.07107i	0.361315i	0.983546	+	0.180657i	$0.0578225\pi$
$383$	7.07107i	0.361315i	−0.983546	+	0.180657i	$0.942177\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.41421	0.0718885
$388$	0	0
$389$	14.0000i	0.709828i	0.934899	+	0.354914i	$0.115490\pi$
$389$	14.0000i	0.709828i	−0.934899	+	0.354914i	$0.884510\pi$
$390$	0	0
$391$	42.4264	2.14560
$392$	0	0
$393$	− 16.0000i	− 0.807093i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	0	0
$399$	16.9706	0.849591
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	−5.65685	−0.281788
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 11.3137i	− 0.560800i
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	0	0
$411$	− 2.82843i	− 0.139516i
$412$	0	0
$413$	−12.0000	−0.590481
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 12.0000i	− 0.587643i
$418$	0	0
$419$	− 8.48528i	− 0.414533i	−0.978285	−	0.207267i	$-0.933543\pi$
$419$	− 8.48528i	− 0.414533i	0.978285	−	0.207267i	$-0.0664567\pi$
$420$	0	0
$421$	10.0000i	0.487370i	0.969854	+	0.243685i	$0.0783563\pi$
$421$	10.0000i	0.487370i	−0.969854	+	0.243685i	$0.921644\pi$
$422$	0	0
$423$	1.41421i	0.0687614i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	59.3970	2.87442
$428$	0	0
$429$	− 16.0000i	− 0.772487i
$430$	0	0
$431$	31.1127	1.49865	0.749323	−	0.662205i	$-0.230379\pi$
$431$	31.1127	1.49865	0.749323	+	0.662205i	$0.230379\pi$
$432$	0	0
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−20.0000	−0.956730
$438$	0	0
$439$	−16.9706	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$439$	−16.9706	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$440$	0	0
$441$	11.0000	0.523810
$442$	0	0
$443$	−15.5563	−0.739104	−0.369552	−	0.929210i	$-0.620489\pi$
$443$	−15.5563	−0.739104	−0.369552	+	0.929210i	$0.620489\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 31.1127i	− 1.47158i
$448$	0	0
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	− 45.2548i	− 2.13097i
$452$	0	0
$453$	−20.0000	−0.939682
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 34.0000i	− 1.59045i	−0.606313	−	0.795226i	$-0.707352\pi$
$457$	− 34.0000i	− 1.59045i	0.606313	−	0.795226i	$-0.292648\pi$
$458$	0	0
$459$	33.9411i	1.58424i
$460$	0	0
$461$	− 24.0000i	− 1.11779i	−0.829238	−	0.558896i	$-0.811225\pi$
$461$	− 24.0000i	− 1.11779i	0.829238	−	0.558896i	$-0.188775\pi$
$462$	0	0
$463$	− 7.07107i	− 0.328620i	−0.986409	−	0.164310i	$-0.947460\pi$
$463$	− 7.07107i	− 0.328620i	0.986409	−	0.164310i	$-0.0525398\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9.89949	0.458094	0.229047	−	0.973415i	$-0.426439\pi$
$467$	9.89949	0.458094	0.229047	+	0.973415i	$0.426439\pi$
$468$	0	0
$469$	18.0000i	0.831163i
$470$	0	0
$471$	−25.4558	−1.17294
$472$	0	0
$473$	− 8.00000i	− 0.367840i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	−33.9411	−1.55081	−0.775405	−	0.631464i	$-0.782454\pi$
$479$	−33.9411	−1.55081	−0.775405	+	0.631464i	$0.782454\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	42.4264	1.93047
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.7279i	0.576757i	0.957516	+	0.288379i	$0.0931162\pi$
$487$	12.7279i	0.576757i	−0.957516	+	0.288379i	$0.906884\pi$
$488$	0	0
$489$	26.0000	1.17576
$490$	0	0
$491$	16.9706i	0.765871i	0.923775	+	0.382935i	$0.125087\pi$
$491$	16.9706i	0.765871i	−0.923775	+	0.382935i	$0.874913\pi$
$492$	0	0
$493$	24.0000	1.08091
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 12.0000i	− 0.538274i
$498$	0	0
$499$	− 25.4558i	− 1.13956i	−0.821797	−	0.569780i	$-0.807028\pi$
$499$	− 25.4558i	− 1.13956i	0.821797	−	0.569780i	$-0.192972\pi$
$500$	0	0
$501$	− 14.0000i	− 0.625474i
$502$	0	0
$503$	32.5269i	1.45030i	0.688589	+	0.725152i	$0.258230\pi$
$503$	32.5269i	1.45030i	−0.688589	+	0.725152i	$0.741770\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.7279	0.565267
$508$	0	0
$509$	− 20.0000i	− 0.886484i	−0.896402	−	0.443242i	$-0.853828\pi$
$509$	− 20.0000i	− 0.886484i	0.896402	−	0.443242i	$-0.146172\pi$
$510$	0	0
$511$	25.4558	1.12610
$512$	0	0
$513$	− 16.0000i	− 0.706417i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	25.4558	1.11739
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	12.7279	0.556553	0.278277	−	0.960501i	$-0.410237\pi$
$523$	12.7279	0.556553	0.278277	+	0.960501i	$0.410237\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 16.9706i	− 0.739249i
$528$	0	0
$529$	−27.0000	−1.17391
$530$	0	0
$531$	2.82843i	0.122743i
$532$	0	0
$533$	−16.0000	−0.693037
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 28.0000i	− 1.20829i
$538$	0	0
$539$	− 62.2254i	− 2.68024i
$540$	0	0
$541$	− 40.0000i	− 1.71973i	−0.510518	−	0.859867i	$-0.670546\pi$
$541$	− 40.0000i	− 1.71973i	0.510518	−	0.859867i	$-0.329454\pi$
$542$	0	0
$543$	− 22.6274i	− 0.971035i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.24264	0.181402	0.0907011	−	0.995878i	$-0.471089\pi$
$547$	4.24264	0.181402	0.0907011	+	0.995878i	$0.471089\pi$
$548$	0	0
$549$	− 14.0000i	− 0.597505i
$550$	0	0
$551$	−11.3137	−0.481980
$552$	0	0
$553$	72.0000i	3.06175i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	−2.82843	−0.119630
$560$	0	0
$561$	48.0000	2.02656
$562$	0	0
$563$	−35.3553	−1.49005	−0.745025	−	0.667037i	$-0.767562\pi$
$563$	−35.3553	−1.49005	−0.745025	+	0.667037i	$0.767562\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	21.2132i	0.890871i
$568$	0	0
$569$	28.0000	1.17382	0.586911	−	0.809652i	$-0.300344\pi$
$569$	28.0000	1.17382	0.586911	+	0.809652i	$0.300344\pi$
$570$	0	0
$571$	22.6274i	0.946928i	0.880813	+	0.473464i	$0.156997\pi$
$571$	22.6274i	0.946928i	−0.880813	+	0.473464i	$0.843003\pi$
$572$	0	0
$573$	−4.00000	−0.167102
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 10.0000i	− 0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$	− 10.0000i	− 0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$	0	0
$579$	8.48528i	0.352636i
$580$	0	0
$581$	− 54.0000i	− 2.24030i
$582$	0	0
$583$	− 11.3137i	− 0.468566i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.7279	−0.525338	−0.262669	−	0.964886i	$-0.584603\pi$
$587$	−12.7279	−0.525338	−0.262669	+	0.964886i	$0.584603\pi$
$588$	0	0
$589$	8.00000i	0.329634i
$590$	0	0
$591$	−8.48528	−0.349038
$592$	0	0
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.0000	0.654836
$598$	0	0
$599$	−39.5980	−1.61793	−0.808965	−	0.587857i	$-0.799972\pi$
$599$	−39.5980	−1.61793	−0.808965	+	0.587857i	$0.799972\pi$
$600$	0	0
$601$	−32.0000	−1.30531	−0.652654	−	0.757656i	$-0.726344\pi$
$601$	−32.0000	−1.30531	−0.652654	+	0.757656i	$0.726344\pi$
$602$	0	0
$603$	4.24264	0.172774
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9.89949i	0.401808i	0.979611	+	0.200904i	$0.0643879\pi$
$607$	9.89949i	0.401808i	−0.979611	+	0.200904i	$0.935612\pi$
$608$	0	0
$609$	24.0000	0.972529
$610$	0	0
$611$	− 2.82843i	− 0.114426i
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 2.00000i	− 0.0805170i	−0.999189	−	0.0402585i	$-0.987182\pi$
$617$	− 2.00000i	− 0.0805170i	0.999189	−	0.0402585i	$-0.0128181\pi$
$618$	0	0
$619$	− 19.7990i	− 0.795789i	−0.917431	−	0.397894i	$-0.869741\pi$
$619$	− 19.7990i	− 0.795789i	0.917431	−	0.397894i	$-0.130259\pi$
$620$	0	0
$621$	− 40.0000i	− 1.60514i
$622$	0	0
$623$	− 25.4558i	− 1.01987i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−22.6274	−0.903652
$628$	0	0
$629$	− 12.0000i	− 0.478471i
$630$	0	0
$631$	−36.7696	−1.46377	−0.731886	−	0.681427i	$-0.761360\pi$
$631$	−36.7696	−1.46377	−0.731886	+	0.681427i	$0.761360\pi$
$632$	0	0
$633$	− 24.0000i	− 0.953914i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−22.0000	−0.871672
$638$	0	0
$639$	−2.82843	−0.111891
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	7.07107	0.278856	0.139428	−	0.990232i	$-0.455474\pi$
$643$	7.07107	0.278856	0.139428	+	0.990232i	$0.455474\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 32.5269i	− 1.27876i	−0.768889	−	0.639382i	$-0.779190\pi$
$647$	− 32.5269i	− 1.27876i	0.768889	−	0.639382i	$-0.220810\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	− 16.9706i	− 0.665129i
$652$	0	0
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 6.00000i	− 0.234082i
$658$	0	0
$659$	25.4558i	0.991619i	0.868431	+	0.495809i	$0.165129\pi$
$659$	25.4558i	0.991619i	−0.868431	+	0.495809i	$0.834871\pi$
$660$	0	0
$661$	− 10.0000i	− 0.388955i	−0.980907	−	0.194477i	$-0.937699\pi$
$661$	− 10.0000i	− 0.388955i	0.980907	−	0.194477i	$-0.0623011\pi$
$662$	0	0
$663$	− 16.9706i	− 0.659082i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−28.2843	−1.09517
$668$	0	0
$669$	22.0000i	0.850569i
$670$	0	0
$671$	−79.1960	−3.05733
$672$	0	0
$673$	− 50.0000i	− 1.92736i	−0.267063	−	0.963679i	$-0.586053\pi$
$673$	− 50.0000i	− 1.92736i	0.267063	−	0.963679i	$-0.413947\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.0000	−0.384331	−0.192166	−	0.981363i	$-0.561551\pi$
$677$	−10.0000	−0.384331	−0.192166	+	0.981363i	$0.561551\pi$
$678$	0	0
$679$	42.4264	1.62818
$680$	0	0
$681$	10.0000	0.383201
$682$	0	0
$683$	−35.3553	−1.35283	−0.676417	−	0.736519i	$-0.736468\pi$
$683$	−35.3553	−1.35283	−0.676417	+	0.736519i	$0.736468\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 28.2843i	− 1.07911i
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	22.6274i	0.860788i	0.902641	+	0.430394i	$0.141625\pi$
$691$	22.6274i	0.860788i	−0.902641	+	0.430394i	$0.858375\pi$
$692$	0	0
$693$	−24.0000	−0.911685
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 48.0000i	− 1.81813i
$698$	0	0
$699$	8.48528i	0.320943i
$700$	0	0
$701$	18.0000i	0.679851i	0.940452	+	0.339925i	$0.110402\pi$
$701$	18.0000i	0.679851i	−0.940452	+	0.339925i	$0.889598\pi$
$702$	0	0
$703$	5.65685i	0.213352i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	28.0000i	1.05156i	0.850620	+	0.525781i	$0.176227\pi$
$709$	28.0000i	1.05156i	−0.850620	+	0.525781i	$0.823773\pi$
$710$	0	0
$711$	16.9706	0.636446
$712$	0	0
$713$	20.0000i	0.749006i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−16.0000	−0.597531
$718$	0	0
$719$	11.3137	0.421930	0.210965	−	0.977494i	$-0.432339\pi$
$719$	11.3137	0.421930	0.210965	+	0.977494i	$0.432339\pi$
$720$	0	0
$721$	42.0000	1.56416
$722$	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	$-0.974931\pi$
$727$	− 4.24264i	− 0.157351i	0.996900	−	0.0786754i	$-0.0250691\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	− 8.48528i	− 0.313839i
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 24.0000i	− 0.884051i
$738$	0	0
$739$	− 25.4558i	− 0.936408i	−0.883620	−	0.468204i	$-0.844901\pi$
$739$	− 25.4558i	− 0.936408i	0.883620	−	0.468204i	$-0.155099\pi$
$740$	0	0
$741$	8.00000i	0.293887i
$742$	0	0
$743$	− 32.5269i	− 1.19330i	−0.802503	−	0.596648i	$-0.796499\pi$
$743$	− 32.5269i	− 1.19330i	0.802503	−	0.596648i	$-0.203501\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−12.7279	−0.465690
$748$	0	0
$749$	− 30.0000i	− 1.09618i
$750$	0	0
$751$	8.48528	0.309632	0.154816	−	0.987943i	$-0.450521\pi$
$751$	8.48528	0.309632	0.154816	+	0.987943i	$0.450521\pi$
$752$	0	0
$753$	32.0000i	1.16614i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−46.0000	−1.67190	−0.835949	−	0.548807i	$-0.815082\pi$
$757$	−46.0000	−1.67190	−0.835949	+	0.548807i	$0.815082\pi$
$758$	0	0
$759$	−56.5685	−2.05331
$760$	0	0
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	0	0
$763$	25.4558	0.921563
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 5.65685i	− 0.204257i
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	25.4558i	0.916770i
$772$	0	0
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 12.0000i	− 0.430498i
$778$	0	0
$779$	22.6274i	0.810711i
$780$	0	0
$781$	16.0000i	0.572525i
$782$	0	0
$783$	− 22.6274i	− 0.808638i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	46.6690	1.66357	0.831786	−	0.555097i	$-0.187319\pi$
$787$	46.6690	1.66357	0.831786	+	0.555097i	$0.187319\pi$
$788$	0	0
$789$	− 10.0000i	− 0.356009i
$790$	0	0
$791$	42.4264	1.50851
$792$	0	0
$793$	28.0000i	0.994309i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10.0000	0.354218	0.177109	−	0.984191i	$-0.443325\pi$
$797$	10.0000	0.354218	0.177109	+	0.984191i	$0.443325\pi$
$798$	0	0
$799$	8.48528	0.300188
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	−33.9411	−1.19776
$804$	0	0
$805$	0	0
$806$	0	0
$807$	8.48528i	0.298696i
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	36.0000	1.26258
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.00000i	0.139942i
$818$	0	0
$819$	8.48528i	0.296500i
$820$	0	0
$821$	− 26.0000i	− 0.907406i	−0.891153	−	0.453703i	$-0.850103\pi$
$821$	− 26.0000i	− 0.907406i	0.891153	−	0.453703i	$-0.149897\pi$
$822$	0	0
$823$	26.8701i	0.936631i	0.883561	+	0.468316i	$0.155139\pi$
$823$	26.8701i	0.936631i	−0.883561	+	0.468316i	$0.844861\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7.07107	−0.245885	−0.122943	−	0.992414i	$-0.539233\pi$
$827$	−7.07107	−0.245885	−0.122943	+	0.992414i	$0.539233\pi$
$828$	0	0
$829$	26.0000i	0.903017i	0.892267	+	0.451509i	$0.149114\pi$
$829$	26.0000i	0.903017i	−0.892267	+	0.451509i	$0.850886\pi$
$830$	0	0
$831$	42.4264	1.47176
$832$	0	0
$833$	− 66.0000i	− 2.28676i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−16.0000	−0.553041
$838$	0	0
$839$	28.2843	0.976481	0.488241	−	0.872709i	$-0.337639\pi$
$839$	28.2843	0.976481	0.488241	+	0.872709i	$0.337639\pi$
$840$	0	0
$841$	13.0000	0.448276
$842$	0	0
$843$	−11.3137	−0.389665
$844$	0	0
$845$	0	0
$846$	0	0
$847$	89.0955i	3.06136i
$848$	0	0
$849$	14.0000	0.480479
$850$	0	0
$851$	14.1421i	0.484786i
$852$	0	0
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	0	0
$859$	− 2.82843i	− 0.0965047i	−0.998835	−	0.0482523i	$-0.984635\pi$
$859$	− 2.82843i	− 0.0965047i	0.998835	−	0.0482523i	$-0.0153652\pi$
$860$	0	0
$861$	− 48.0000i	− 1.63584i
$862$	0	0
$863$	55.1543i	1.87748i	0.344633	+	0.938738i	$0.388003\pi$
$863$	55.1543i	1.87748i	−0.344633	+	0.938738i	$0.611997\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	26.8701	0.912555
$868$	0	0
$869$	− 96.0000i	− 3.25658i
$870$	0	0
$871$	−8.48528	−0.287513
$872$	0	0
$873$	− 10.0000i	− 0.338449i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	0	0
$879$	8.48528	0.286201
$880$	0	0
$881$	48.0000	1.61716	0.808581	−	0.588386i	$-0.200236\pi$
$881$	48.0000	1.61716	0.808581	+	0.588386i	$0.200236\pi$
$882$	0	0
$883$	21.2132	0.713881	0.356941	−	0.934127i	$-0.383820\pi$
$883$	21.2132	0.713881	0.356941	+	0.934127i	$0.383820\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 26.8701i	− 0.902208i	−0.892471	−	0.451104i	$-0.851030\pi$
$887$	− 26.8701i	− 0.902208i	0.892471	−	0.451104i	$-0.148970\pi$
$888$	0	0
$889$	−42.0000	−1.40863
$890$	0	0
$891$	− 28.2843i	− 0.947559i
$892$	0	0
$893$	−4.00000	−0.133855
$894$	0	0
$895$	0	0
$896$	0	0
$897$	20.0000i	0.667781i
$898$	0	0
$899$	11.3137i	0.377333i
$900$	0	0
$901$	− 12.0000i	− 0.399778i
$902$	0	0
$903$	− 8.48528i	− 0.282372i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.41421	0.0469582	0.0234791	−	0.999724i	$-0.492526\pi$
$907$	1.41421	0.0469582	0.0234791	+	0.999724i	$0.492526\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	31.1127	1.03081	0.515405	−	0.856947i	$-0.327642\pi$
$911$	31.1127	1.03081	0.515405	+	0.856947i	$0.327642\pi$
$912$	0	0
$913$	72.0000i	2.38285i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	48.0000	1.58510
$918$	0	0
$919$	39.5980	1.30622	0.653108	−	0.757264i	$-0.273465\pi$
$919$	39.5980	1.30622	0.653108	+	0.757264i	$0.273465\pi$
$920$	0	0
$921$	22.0000	0.724925
$922$	0	0
$923$	5.65685	0.186198
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 9.89949i	− 0.325142i
$928$	0	0
$929$	28.0000	0.918650	0.459325	−	0.888268i	$-0.348091\pi$
$929$	28.0000	0.918650	0.459325	+	0.888268i	$0.348091\pi$
$930$	0	0
$931$	31.1127i	1.01968i
$932$	0	0
$933$	−4.00000	−0.130954
$934$	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
$939$	8.48528i	0.276907i
$940$	0	0
$941$	48.0000i	1.56476i	0.622804	+	0.782378i	$0.285993\pi$
$941$	48.0000i	1.56476i	−0.622804	+	0.782378i	$0.714007\pi$
$942$	0	0
$943$	56.5685i	1.84213i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−15.5563	−0.505513	−0.252757	−	0.967530i	$-0.581337\pi$
$947$	−15.5563	−0.505513	−0.252757	+	0.967530i	$0.581337\pi$
$948$	0	0
$949$	12.0000i	0.389536i
$950$	0	0
$951$	−2.82843	−0.0917180
$952$	0	0
$953$	− 54.0000i	− 1.74923i	−0.484817	−	0.874616i	$-0.661114\pi$
$953$	− 54.0000i	− 1.74923i	0.484817	−	0.874616i	$-0.338886\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−32.0000	−1.03441
$958$	0	0
$959$	8.48528	0.274004
$960$	0	0
$961$	−23.0000	−0.741935
$962$	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	$0.0217310\pi$
$967$	4.24264i	0.136434i	−0.997671	+	0.0682171i	$0.978269\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	− 28.2843i	− 0.907685i	−0.891082	−	0.453843i	$-0.850053\pi$
$971$	− 28.2843i	− 0.907685i	0.891082	−	0.453843i	$-0.149947\pi$
$972$	0	0
$973$	36.0000	1.15411
$974$	0	0
$975$	0	0
$976$	0	0
$977$	42.0000i	1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$	42.0000i	1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$	0	0
$979$	33.9411i	1.08476i
$980$	0	0
$981$	− 6.00000i	− 0.191565i
$982$	0	0
$983$	18.3848i	0.586383i	0.956054	+	0.293192i	$0.0947174\pi$
$983$	18.3848i	0.586383i	−0.956054	+	0.293192i	$0.905283\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	8.48528	0.270089
$988$	0	0
$989$	10.0000i	0.317982i
$990$	0	0
$991$	−8.48528	−0.269544	−0.134772	−	0.990877i	$-0.543030\pi$
$991$	−8.48528	−0.269544	−0.134772	+	0.990877i	$0.543030\pi$
$992$	0	0
$993$	− 32.0000i	− 1.01549i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	0	0
$999$	−11.3137	−0.357950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3200.2.f.j.449.1		4
4.3	odd	2	inner	3200.2.f.j.449.4		4
5.2	odd	4		3200.2.d.s.1601.4		4
5.3	odd	4		640.2.d.d.321.2	yes	4
5.4	even	2		3200.2.f.i.449.4		4
8.3	odd	2		3200.2.f.i.449.2		4
8.5	even	2		3200.2.f.i.449.3		4
15.8	even	4		5760.2.k.o.2881.1		4
20.3	even	4		640.2.d.d.321.4	yes	4
20.7	even	4		3200.2.d.s.1601.1		4
20.19	odd	2		3200.2.f.i.449.1		4
40.3	even	4		640.2.d.d.321.1	✓	4
40.13	odd	4		640.2.d.d.321.3	yes	4
40.19	odd	2	inner	3200.2.f.j.449.3		4
40.27	even	4		3200.2.d.s.1601.3		4
40.29	even	2	inner	3200.2.f.j.449.2		4
40.37	odd	4		3200.2.d.s.1601.2		4
60.23	odd	4		5760.2.k.o.2881.2		4
80.3	even	4		1280.2.a.j.1.1		2
80.13	odd	4		1280.2.a.j.1.2		2
80.27	even	4		6400.2.a.bl.1.1		2
80.37	odd	4		6400.2.a.bl.1.2		2
80.43	even	4		1280.2.a.f.1.2		2
80.53	odd	4		1280.2.a.f.1.1		2
80.67	even	4		6400.2.a.bn.1.2		2
80.77	odd	4		6400.2.a.bn.1.1		2
120.53	even	4		5760.2.k.o.2881.3		4
120.83	odd	4		5760.2.k.o.2881.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.d.d.321.1	✓	4	40.3	even	4
640.2.d.d.321.2	yes	4	5.3	odd	4
640.2.d.d.321.3	yes	4	40.13	odd	4
640.2.d.d.321.4	yes	4	20.3	even	4
1280.2.a.f.1.1		2	80.53	odd	4
1280.2.a.f.1.2		2	80.43	even	4
1280.2.a.j.1.1		2	80.3	even	4
1280.2.a.j.1.2		2	80.13	odd	4
3200.2.d.s.1601.1		4	20.7	even	4
3200.2.d.s.1601.2		4	40.37	odd	4
3200.2.d.s.1601.3		4	40.27	even	4
3200.2.d.s.1601.4		4	5.2	odd	4
3200.2.f.i.449.1		4	20.19	odd	2
3200.2.f.i.449.2		4	8.3	odd	2
3200.2.f.i.449.3		4	8.5	even	2
3200.2.f.i.449.4		4	5.4	even	2
3200.2.f.j.449.1		4	1.1	even	1	trivial
3200.2.f.j.449.2		4	40.29	even	2	inner
3200.2.f.j.449.3		4	40.19	odd	2	inner
3200.2.f.j.449.4		4	4.3	odd	2	inner
5760.2.k.o.2881.1		4	15.8	even	4
5760.2.k.o.2881.2		4	60.23	odd	4
5760.2.k.o.2881.3		4	120.53	even	4
5760.2.k.o.2881.4		4	120.83	odd	4
6400.2.a.bl.1.1		2	80.27	even	4
6400.2.a.bl.1.2		2	80.37	odd	4
6400.2.a.bn.1.1		2	80.77	odd	4
6400.2.a.bn.1.2		2	80.67	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} -4.24264i q^{7} -1.00000 q^{9} +5.65685i q^{11} +2.00000 q^{13} +6.00000i q^{17} -2.82843i q^{19} +6.00000i q^{21} -7.07107i q^{23} +5.65685 q^{27} -4.00000i q^{29} -2.82843 q^{31} -8.00000i q^{33} -2.00000 q^{37} -2.82843 q^{39} -8.00000 q^{41} -1.41421 q^{43} -1.41421i q^{47} -11.0000 q^{49} -8.48528i q^{51} -2.00000 q^{53} +4.00000i q^{57} -2.82843i q^{59} +14.0000i q^{61} +4.24264i q^{63} -4.24264 q^{67} +10.0000i q^{69} +2.82843 q^{71} +6.00000i q^{73} +24.0000 q^{77} -16.9706 q^{79} -5.00000 q^{81} +12.7279 q^{83} +5.65685i q^{87} +6.00000 q^{89} -8.48528i q^{91} +4.00000 q^{93} +10.0000i q^{97} -5.65685i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9} + 8 q^{13} - 8 q^{37} - 32 q^{41} - 44 q^{49} - 8 q^{53} + 96 q^{77} - 20 q^{81} + 24 q^{89} + 16 q^{93}+O(q^{100}) \) 4 * q - 4 * q^9 + 8 * q^13 - 8 * q^37 - 32 * q^41 - 44 * q^49 - 8 * q^53 + 96 * q^77 - 20 * q^81 + 24 * q^89 + 16 * q^93

Introduction

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