LMFDB - Embedded newform 1280.2.a.f.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1280,2,Mod(1,1280)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1280.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1280.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2,0,0,0,-2,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$10.2208514587$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
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:	no (minimal twist has level 640)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1280.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{3} -1.00000 q^{5} -4.24264 q^{7} -1.00000 q^{9} +5.65685 q^{11} +2.00000 q^{13} -1.41421 q^{15} +6.00000 q^{17} -2.82843 q^{19} -6.00000 q^{21} +7.07107 q^{23} +1.00000 q^{25} -5.65685 q^{27} +4.00000 q^{29} +2.82843 q^{31} +8.00000 q^{33} +4.24264 q^{35} -2.00000 q^{37} +2.82843 q^{39} +8.00000 q^{41} -1.41421 q^{43} +1.00000 q^{45} +1.41421 q^{47} +11.0000 q^{49} +8.48528 q^{51} +2.00000 q^{53} -5.65685 q^{55} -4.00000 q^{57} +2.82843 q^{59} +14.0000 q^{61} +4.24264 q^{63} -2.00000 q^{65} -4.24264 q^{67} +10.0000 q^{69} +2.82843 q^{71} +6.00000 q^{73} +1.41421 q^{75} -24.0000 q^{77} -16.9706 q^{79} -5.00000 q^{81} -12.7279 q^{83} -6.00000 q^{85} +5.65685 q^{87} +6.00000 q^{89} -8.48528 q^{91} +4.00000 q^{93} +2.82843 q^{95} +10.0000 q^{97} -5.65685 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{9} + 4 q^{13} + 12 q^{17} - 12 q^{21} + 2 q^{25} + 8 q^{29} + 16 q^{33} - 4 q^{37} + 16 q^{41} + 2 q^{45} + 22 q^{49} + 4 q^{53} - 8 q^{57} + 28 q^{61} - 4 q^{65} + 20 q^{69} + 12 q^{73}+ \cdots + 20 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^9 + 4 * q^13 + 12 * q^17 - 12 * q^21 + 2 * q^25 + 8 * q^29 + 16 * q^33 - 4 * q^37 + 16 * q^41 + 2 * q^45 + 22 * q^49 + 4 * q^53 - 8 * q^57 + 28 * q^61 - 4 * q^65 + 20 * q^69 + 12 * q^73 - 48 * q^77 - 10 * q^81 - 12 * q^85 + 12 * q^89 + 8 * q^93 + 20 * q^97

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	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.24264	−1.60357	−0.801784	−	0.597614i	$-0.796115\pi$
$7$	−4.24264	−1.60357	−0.801784	+	0.597614i	$0.796115\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.65685	1.70561	0.852803	−	0.522233i	$-0.174901\pi$
$11$	5.65685	1.70561	0.852803	+	0.522233i	$0.174901\pi$
$12$	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
$15$	−1.41421	−0.365148
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	−6.00000	−1.30931
$22$	0	0
$23$	7.07107	1.47442	0.737210	−	0.675664i	$-0.236143\pi$
$23$	7.07107	1.47442	0.737210	+	0.675664i	$0.236143\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	2.82843	0.508001	0.254000	−	0.967204i	$-0.418254\pi$
$31$	2.82843	0.508001	0.254000	+	0.967204i	$0.418254\pi$
$32$	0	0
$33$	8.00000	1.39262
$34$	0	0
$35$	4.24264	0.717137
$36$	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	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	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
$45$	1.00000	0.149071
$46$	0	0
$47$	1.41421	0.206284	0.103142	−	0.994667i	$-0.467110\pi$
$47$	1.41421	0.206284	0.103142	+	0.994667i	$0.467110\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	0	0
$51$	8.48528	1.18818
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−5.65685	−0.762770
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	2.82843	0.368230	0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843	0.368230	0.184115	+	0.982905i	$0.441058\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	4.24264	0.534522
$64$	0	0
$65$	−2.00000	−0.248069
$66$	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
$69$	10.0000	1.20386
$70$	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
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	1.41421	0.163299
$76$	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	$-0.746165\pi$
$83$	−12.7279	−1.39707	−0.698535	+	0.715575i	$0.746165\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	0	0
$87$	5.65685	0.606478
$88$	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
$91$	−8.48528	−0.889499
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	2.82843	0.290191
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−5.65685	−0.568535
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−9.89949	−0.975426	−0.487713	−	0.873004i	$-0.662169\pi$
$103$	−9.89949	−0.975426	−0.487713	+	0.873004i	$0.662169\pi$
$104$	0	0
$105$	6.00000	0.585540
$106$	0	0
$107$	−7.07107	−0.683586	−0.341793	−	0.939775i	$-0.611034\pi$
$107$	−7.07107	−0.683586	−0.341793	+	0.939775i	$0.611034\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	−2.82843	−0.268462
$112$	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
$115$	−7.07107	−0.659380
$116$	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
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.89949	0.878438	0.439219	−	0.898380i	$-0.355255\pi$
$127$	9.89949	0.878438	0.439219	+	0.898380i	$0.355255\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−11.3137	−0.988483	−0.494242	−	0.869325i	$-0.664554\pi$
$131$	−11.3137	−0.988483	−0.494242	+	0.869325i	$0.664554\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	0	0
$135$	5.65685	0.486864
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	−8.48528	−0.719712	−0.359856	−	0.933008i	$-0.617174\pi$
$139$	−8.48528	−0.719712	−0.359856	+	0.933008i	$0.617174\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	0	0
$143$	11.3137	0.946100
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	15.5563	1.28307
$148$	0	0
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	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
$153$	−6.00000	−0.485071
$154$	0	0
$155$	−2.82843	−0.227185
$156$	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
$159$	2.82843	0.224309
$160$	0	0
$161$	−30.0000	−2.36433
$162$	0	0
$163$	18.3848	1.44001	0.720003	−	0.693971i	$-0.244140\pi$
$163$	18.3848	1.44001	0.720003	+	0.693971i	$0.244140\pi$
$164$	0	0
$165$	−8.00000	−0.622799
$166$	0	0
$167$	9.89949	0.766046	0.383023	−	0.923739i	$-0.374883\pi$
$167$	9.89949	0.766046	0.383023	+	0.923739i	$0.374883\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	2.82843	0.216295
$172$	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
$175$	−4.24264	−0.320713
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	19.7990	1.47985	0.739923	−	0.672692i	$-0.234862\pi$
$179$	19.7990	1.47985	0.739923	+	0.672692i	$0.234862\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	0	0
$183$	19.7990	1.46358
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	33.9411	2.48202
$188$	0	0
$189$	24.0000	1.74574
$190$	0	0
$191$	−2.82843	−0.204658	−0.102329	−	0.994751i	$-0.532629\pi$
$191$	−2.82843	−0.204658	−0.102329	+	0.994751i	$0.532629\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	−2.82843	−0.202548
$196$	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	$-0.368661\pi$
$199$	11.3137	0.802008	0.401004	+	0.916076i	$0.368661\pi$
$200$	0	0
$201$	−6.00000	−0.423207
$202$	0	0
$203$	−16.9706	−1.19110
$204$	0	0
$205$	−8.00000	−0.558744
$206$	0	0
$207$	−7.07107	−0.491473
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	−16.9706	−1.16830	−0.584151	−	0.811645i	$-0.698572\pi$
$211$	−16.9706	−1.16830	−0.584151	+	0.811645i	$0.698572\pi$
$212$	0	0
$213$	4.00000	0.274075
$214$	0	0
$215$	1.41421	0.0964486
$216$	0	0
$217$	−12.0000	−0.814613
$218$	0	0
$219$	8.48528	0.573382
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	−15.5563	−1.04173	−0.520865	−	0.853639i	$-0.674391\pi$
$223$	−15.5563	−1.04173	−0.520865	+	0.853639i	$0.674391\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	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
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	0	0
$231$	−33.9411	−2.23316
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−1.41421	−0.0922531
$236$	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
$245$	−11.0000	−0.702764
$246$	0	0
$247$	−5.65685	−0.359937
$248$	0	0
$249$	−18.0000	−1.14070
$250$	0	0
$251$	−22.6274	−1.42823	−0.714115	−	0.700028i	$-0.753171\pi$
$251$	−22.6274	−1.42823	−0.714115	+	0.700028i	$0.753171\pi$
$252$	0	0
$253$	40.0000	2.51478
$254$	0	0
$255$	−8.48528	−0.531369
$256$	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
$259$	8.48528	0.527250
$260$	0	0
$261$	−4.00000	−0.247594
$262$	0	0
$263$	−7.07107	−0.436021	−0.218010	−	0.975946i	$-0.569957\pi$
$263$	−7.07107	−0.436021	−0.218010	+	0.975946i	$0.569957\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	8.48528	0.519291
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	25.4558	1.54633	0.773166	−	0.634203i	$-0.218672\pi$
$271$	25.4558	1.54633	0.773166	+	0.634203i	$0.218672\pi$
$272$	0	0
$273$	−12.0000	−0.726273
$274$	0	0
$275$	5.65685	0.341121
$276$	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	$-0.576695\pi$
$281$	−8.00000	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$282$	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
$285$	4.00000	0.236940
$286$	0	0
$287$	−33.9411	−2.00348
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	14.1421	0.829027
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−2.82843	−0.164677
$296$	0	0
$297$	−32.0000	−1.85683
$298$	0	0
$299$	14.1421	0.817861
$300$	0	0
$301$	6.00000	0.345834
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−14.0000	−0.801638
$306$	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
$309$	−14.0000	−0.796432
$310$	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
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	−4.24264	−0.239046
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	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
$325$	2.00000	0.110940
$326$	0	0
$327$	−8.48528	−0.469237
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	22.6274	1.24372	0.621858	−	0.783130i	$-0.286378\pi$
$331$	22.6274	1.24372	0.621858	+	0.783130i	$0.286378\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	4.24264	0.231800
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	−14.1421	−0.768095
$340$	0	0
$341$	16.0000	0.866449
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	−10.0000	−0.538382
$346$	0	0
$347$	−4.24264	−0.227757	−0.113878	−	0.993495i	$-0.536327\pi$
$347$	−4.24264	−0.227757	−0.113878	+	0.993495i	$0.536327\pi$
$348$	0	0
$349$	12.0000	0.642345	0.321173	−	0.947021i	$-0.395923\pi$
$349$	12.0000	0.642345	0.321173	+	0.947021i	$0.395923\pi$
$350$	0	0
$351$	−11.3137	−0.603881
$352$	0	0
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	0	0
$355$	−2.82843	−0.150117
$356$	0	0
$357$	−36.0000	−1.90532
$358$	0	0
$359$	28.2843	1.49279	0.746393	−	0.665505i	$-0.231784\pi$
$359$	28.2843	1.49279	0.746393	+	0.665505i	$0.231784\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	29.6985	1.55877
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	4.24264	0.221464	0.110732	−	0.993850i	$-0.464680\pi$
$367$	4.24264	0.221464	0.110732	+	0.993850i	$0.464680\pi$
$368$	0	0
$369$	−8.00000	−0.416463
$370$	0	0
$371$	−8.48528	−0.440534
$372$	0	0
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	−1.41421	−0.0730297
$376$	0	0
$377$	8.00000	0.412021
$378$	0	0
$379$	−31.1127	−1.59815	−0.799076	−	0.601230i	$-0.794678\pi$
$379$	−31.1127	−1.59815	−0.799076	+	0.601230i	$0.794678\pi$
$380$	0	0
$381$	14.0000	0.717242
$382$	0	0
$383$	7.07107	0.361315	0.180657	−	0.983546i	$-0.442177\pi$
$383$	7.07107	0.361315	0.180657	+	0.983546i	$0.442177\pi$
$384$	0	0
$385$	24.0000	1.22315
$386$	0	0
$387$	1.41421	0.0718885
$388$	0	0
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	42.4264	2.14560
$392$	0	0
$393$	−16.0000	−0.807093
$394$	0	0
$395$	16.9706	0.853882
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	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
$405$	5.00000	0.248452
$406$	0	0
$407$	−11.3137	−0.560800
$408$	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
$411$	−2.82843	−0.139516
$412$	0	0
$413$	−12.0000	−0.590481
$414$	0	0
$415$	12.7279	0.624789
$416$	0	0
$417$	−12.0000	−0.587643
$418$	0	0
$419$	−8.48528	−0.414533	−0.207267	−	0.978285i	$-0.566457\pi$
$419$	−8.48528	−0.414533	−0.207267	+	0.978285i	$0.566457\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	−1.41421	−0.0687614
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	−59.3970	−2.87442
$428$	0	0
$429$	16.0000	0.772487
$430$	0	0
$431$	−31.1127	−1.49865	−0.749323	−	0.662205i	$-0.769621\pi$
$431$	−31.1127	−1.49865	−0.749323	+	0.662205i	$0.769621\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	−5.65685	−0.271225
$436$	0	0
$437$	−20.0000	−0.956730
$438$	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
$443$	−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
$445$	−6.00000	−0.284427
$446$	0	0
$447$	31.1127	1.47158
$448$	0	0
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	45.2548	2.13097
$452$	0	0
$453$	20.0000	0.939682
$454$	0	0
$455$	8.48528	0.397796
$456$	0	0
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	0	0
$459$	−33.9411	−1.58424
$460$	0	0
$461$	−24.0000	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$462$	0	0
$463$	−7.07107	−0.328620	−0.164310	−	0.986409i	$-0.552540\pi$
$463$	−7.07107	−0.328620	−0.164310	+	0.986409i	$0.552540\pi$
$464$	0	0
$465$	−4.00000	−0.185496
$466$	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
$469$	18.0000	0.831163
$470$	0	0
$471$	−25.4558	−1.17294
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	−2.82843	−0.129777
$476$	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
$485$	−10.0000	−0.454077
$486$	0	0
$487$	12.7279	0.576757	0.288379	−	0.957516i	$-0.406884\pi$
$487$	12.7279	0.576757	0.288379	+	0.957516i	$0.406884\pi$
$488$	0	0
$489$	26.0000	1.17576
$490$	0	0
$491$	16.9706	0.765871	0.382935	−	0.923775i	$-0.374913\pi$
$491$	16.9706	0.765871	0.382935	+	0.923775i	$0.374913\pi$
$492$	0	0
$493$	24.0000	1.08091
$494$	0	0
$495$	5.65685	0.254257
$496$	0	0
$497$	−12.0000	−0.538274
$498$	0	0
$499$	−25.4558	−1.13956	−0.569780	−	0.821797i	$-0.692972\pi$
$499$	−25.4558	−1.13956	−0.569780	+	0.821797i	$0.692972\pi$
$500$	0	0
$501$	14.0000	0.625474
$502$	0	0
$503$	−32.5269	−1.45030	−0.725152	−	0.688589i	$-0.758230\pi$
$503$	−32.5269	−1.45030	−0.725152	+	0.688589i	$0.758230\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−12.7279	−0.565267
$508$	0	0
$509$	20.0000	0.886484	0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000	0.886484	0.443242	+	0.896402i	$0.353828\pi$
$510$	0	0
$511$	−25.4558	−1.12610
$512$	0	0
$513$	16.0000	0.706417
$514$	0	0
$515$	9.89949	0.436224
$516$	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	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	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
$525$	−6.00000	−0.261861
$526$	0	0
$527$	16.9706	0.739249
$528$	0	0
$529$	27.0000	1.17391
$530$	0	0
$531$	−2.82843	−0.122743
$532$	0	0
$533$	16.0000	0.693037
$534$	0	0
$535$	7.07107	0.305709
$536$	0	0
$537$	28.0000	1.20829
$538$	0	0
$539$	62.2254	2.68024
$540$	0	0
$541$	−40.0000	−1.71973	−0.859867	−	0.510518i	$-0.829454\pi$
$541$	−40.0000	−1.71973	−0.859867	+	0.510518i	$0.829454\pi$
$542$	0	0
$543$	−22.6274	−0.971035
$544$	0	0
$545$	6.00000	0.257012
$546$	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
$549$	−14.0000	−0.597505
$550$	0	0
$551$	−11.3137	−0.481980
$552$	0	0
$553$	72.0000	3.06175
$554$	0	0
$555$	2.82843	0.120060
$556$	0	0
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	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	$-0.232438\pi$
$563$	35.3553	1.49005	0.745025	+	0.667037i	$0.232438\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	0	0
$567$	21.2132	0.890871
$568$	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
$571$	22.6274	0.946928	0.473464	−	0.880813i	$-0.343003\pi$
$571$	22.6274	0.946928	0.473464	+	0.880813i	$0.343003\pi$
$572$	0	0
$573$	−4.00000	−0.167102
$574$	0	0
$575$	7.07107	0.294884
$576$	0	0
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	0	0
$579$	8.48528	0.352636
$580$	0	0
$581$	54.0000	2.24030
$582$	0	0
$583$	11.3137	0.468566
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	12.7279	0.525338	0.262669	−	0.964886i	$-0.415397\pi$
$587$	12.7279	0.525338	0.262669	+	0.964886i	$0.415397\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	8.48528	0.349038
$592$	0	0
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	25.4558	1.04359
$596$	0	0
$597$	16.0000	0.654836
$598$	0	0
$599$	39.5980	1.61793	0.808965	−	0.587857i	$-0.200028\pi$
$599$	39.5980	1.61793	0.808965	+	0.587857i	$0.200028\pi$
$600$	0	0
$601$	32.0000	1.30531	0.652654	−	0.757656i	$-0.273656\pi$
$601$	32.0000	1.30531	0.652654	+	0.757656i	$0.273656\pi$
$602$	0	0
$603$	4.24264	0.172774
$604$	0	0
$605$	−21.0000	−0.853771
$606$	0	0
$607$	−9.89949	−0.401808	−0.200904	−	0.979611i	$-0.564388\pi$
$607$	−9.89949	−0.401808	−0.200904	+	0.979611i	$0.564388\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	2.82843	0.114426
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	0	0
$615$	−11.3137	−0.456213
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	0	0
$619$	19.7990	0.795789	0.397894	−	0.917431i	$-0.369741\pi$
$619$	19.7990	0.795789	0.397894	+	0.917431i	$0.369741\pi$
$620$	0	0
$621$	−40.0000	−1.60514
$622$	0	0
$623$	−25.4558	−1.01987
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−22.6274	−0.903652
$628$	0	0
$629$	−12.0000	−0.478471
$630$	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
$633$	−24.0000	−0.953914
$634$	0	0
$635$	−9.89949	−0.392849
$636$	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	$-0.544526\pi$
$643$	−7.07107	−0.278856	−0.139428	+	0.990232i	$0.544526\pi$
$644$	0	0
$645$	2.00000	0.0787499
$646$	0	0
$647$	−32.5269	−1.27876	−0.639382	−	0.768889i	$-0.720810\pi$
$647$	−32.5269	−1.27876	−0.639382	+	0.768889i	$0.720810\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	−16.9706	−0.665129
$652$	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
$655$	11.3137	0.442063
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	25.4558	0.991619	0.495809	−	0.868431i	$-0.334871\pi$
$659$	25.4558	0.991619	0.495809	+	0.868431i	$0.334871\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	16.9706	0.659082
$664$	0	0
$665$	−12.0000	−0.465340
$666$	0	0
$667$	28.2843	1.09517
$668$	0	0
$669$	−22.0000	−0.850569
$670$	0	0
$671$	79.1960	3.05733
$672$	0	0
$673$	50.0000	1.92736	0.963679	−	0.267063i	$-0.0860531\pi$
$673$	50.0000	1.92736	0.963679	+	0.267063i	$0.0860531\pi$
$674$	0	0
$675$	−5.65685	−0.217732
$676$	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
$685$	2.00000	0.0764161
$686$	0	0
$687$	28.2843	1.07911
$688$	0	0
$689$	4.00000	0.152388
$690$	0	0
$691$	−22.6274	−0.860788	−0.430394	−	0.902641i	$-0.641625\pi$
$691$	−22.6274	−0.860788	−0.430394	+	0.902641i	$0.641625\pi$
$692$	0	0
$693$	24.0000	0.911685
$694$	0	0
$695$	8.48528	0.321865
$696$	0	0
$697$	48.0000	1.81813
$698$	0	0
$699$	−8.48528	−0.320943
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	5.65685	0.213352
$704$	0	0
$705$	−2.00000	−0.0753244
$706$	0	0
$707$	0	0
$708$	0	0
$709$	28.0000	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$709$	28.0000	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$710$	0	0
$711$	16.9706	0.636446
$712$	0	0
$713$	20.0000	0.749006
$714$	0	0
$715$	−11.3137	−0.423109
$716$	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
$725$	4.00000	0.148556
$726$	0	0
$727$	−4.24264	−0.157351	−0.0786754	−	0.996900i	$-0.525069\pi$
$727$	−4.24264	−0.157351	−0.0786754	+	0.996900i	$0.525069\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	−8.48528	−0.313839
$732$	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
$735$	−15.5563	−0.573805
$736$	0	0
$737$	−24.0000	−0.884051
$738$	0	0
$739$	−25.4558	−0.936408	−0.468204	−	0.883620i	$-0.655099\pi$
$739$	−25.4558	−0.936408	−0.468204	+	0.883620i	$0.655099\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	32.5269	1.19330	0.596648	−	0.802503i	$-0.296499\pi$
$743$	32.5269	1.19330	0.596648	+	0.802503i	$0.296499\pi$
$744$	0	0
$745$	−22.0000	−0.806018
$746$	0	0
$747$	12.7279	0.465690
$748$	0	0
$749$	30.0000	1.09618
$750$	0	0
$751$	−8.48528	−0.309632	−0.154816	−	0.987943i	$-0.549479\pi$
$751$	−8.48528	−0.309632	−0.154816	+	0.987943i	$0.549479\pi$
$752$	0	0
$753$	−32.0000	−1.16614
$754$	0	0
$755$	−14.1421	−0.514685
$756$	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	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	0	0
$763$	25.4558	0.921563
$764$	0	0
$765$	6.00000	0.216930
$766$	0	0
$767$	5.65685	0.204257
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	−25.4558	−0.916770
$772$	0	0
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	0	0
$775$	2.82843	0.101600
$776$	0	0
$777$	12.0000	0.430498
$778$	0	0
$779$	−22.6274	−0.810711
$780$	0	0
$781$	16.0000	0.572525
$782$	0	0
$783$	−22.6274	−0.808638
$784$	0	0
$785$	18.0000	0.642448
$786$	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
$789$	−10.0000	−0.356009
$790$	0	0
$791$	42.4264	1.50851
$792$	0	0
$793$	28.0000	0.994309
$794$	0	0
$795$	−2.82843	−0.100314
$796$	0	0
$797$	−10.0000	−0.354218	−0.177109	−	0.984191i	$-0.556675\pi$
$797$	−10.0000	−0.354218	−0.177109	+	0.984191i	$0.556675\pi$
$798$	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
$805$	30.0000	1.05736
$806$	0	0
$807$	8.48528	0.298696
$808$	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		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	36.0000	1.26258
$814$	0	0
$815$	−18.3848	−0.643991
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	8.48528	0.296500
$820$	0	0
$821$	26.0000	0.907406	0.453703	−	0.891153i	$-0.350103\pi$
$821$	26.0000	0.907406	0.453703	+	0.891153i	$0.350103\pi$
$822$	0	0
$823$	−26.8701	−0.936631	−0.468316	−	0.883561i	$-0.655139\pi$
$823$	−26.8701	−0.936631	−0.468316	+	0.883561i	$0.655139\pi$
$824$	0	0
$825$	8.00000	0.278524
$826$	0	0
$827$	7.07107	0.245885	0.122943	−	0.992414i	$-0.460767\pi$
$827$	7.07107	0.245885	0.122943	+	0.992414i	$0.460767\pi$
$828$	0	0
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	0	0
$831$	−42.4264	−1.47176
$832$	0	0
$833$	66.0000	2.28676
$834$	0	0
$835$	−9.89949	−0.342586
$836$	0	0
$837$	−16.0000	−0.553041
$838$	0	0
$839$	−28.2843	−0.976481	−0.488241	−	0.872709i	$-0.662361\pi$
$839$	−28.2843	−0.976481	−0.488241	+	0.872709i	$0.662361\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	−11.3137	−0.389665
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	−89.0955	−3.06136
$848$	0	0
$849$	−14.0000	−0.480479
$850$	0	0
$851$	−14.1421	−0.484786
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	−2.82843	−0.0967302
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	2.82843	0.0965047	0.0482523	−	0.998835i	$-0.484635\pi$
$859$	2.82843	0.0965047	0.0482523	+	0.998835i	$0.484635\pi$
$860$	0	0
$861$	−48.0000	−1.63584
$862$	0	0
$863$	55.1543	1.87748	0.938738	−	0.344633i	$-0.111997\pi$
$863$	55.1543	1.87748	0.938738	+	0.344633i	$0.111997\pi$
$864$	0	0
$865$	18.0000	0.612018
$866$	0	0
$867$	26.8701	0.912555
$868$	0	0
$869$	−96.0000	−3.25658
$870$	0	0
$871$	−8.48528	−0.287513
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	4.24264	0.143427
$876$	0	0
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	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	$-0.616180\pi$
$883$	−21.2132	−0.713881	−0.356941	+	0.934127i	$0.616180\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	0	0
$887$	−26.8701	−0.902208	−0.451104	−	0.892471i	$-0.648970\pi$
$887$	−26.8701	−0.902208	−0.451104	+	0.892471i	$0.648970\pi$
$888$	0	0
$889$	−42.0000	−1.40863
$890$	0	0
$891$	−28.2843	−0.947559
$892$	0	0
$893$	−4.00000	−0.133855
$894$	0	0
$895$	−19.7990	−0.661807
$896$	0	0
$897$	20.0000	0.667781
$898$	0	0
$899$	11.3137	0.377333
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	8.48528	0.282372
$904$	0	0
$905$	16.0000	0.531858
$906$	0	0
$907$	−1.41421	−0.0469582	−0.0234791	−	0.999724i	$-0.507474\pi$
$907$	−1.41421	−0.0469582	−0.0234791	+	0.999724i	$0.507474\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−31.1127	−1.03081	−0.515405	−	0.856947i	$-0.672358\pi$
$911$	−31.1127	−1.03081	−0.515405	+	0.856947i	$0.672358\pi$
$912$	0	0
$913$	−72.0000	−2.38285
$914$	0	0
$915$	−19.7990	−0.654534
$916$	0	0
$917$	48.0000	1.58510
$918$	0	0
$919$	−39.5980	−1.30622	−0.653108	−	0.757264i	$-0.726535\pi$
$919$	−39.5980	−1.30622	−0.653108	+	0.757264i	$0.726535\pi$
$920$	0	0
$921$	−22.0000	−0.724925
$922$	0	0
$923$	5.65685	0.186198
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	9.89949	0.325142
$928$	0	0
$929$	−28.0000	−0.918650	−0.459325	−	0.888268i	$-0.651909\pi$
$929$	−28.0000	−0.918650	−0.459325	+	0.888268i	$0.651909\pi$
$930$	0	0
$931$	−31.1127	−1.01968
$932$	0	0
$933$	4.00000	0.130954
$934$	0	0
$935$	−33.9411	−1.10999
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	−8.48528	−0.276907
$940$	0	0
$941$	48.0000	1.56476	0.782378	−	0.622804i	$-0.214007\pi$
$941$	48.0000	1.56476	0.782378	+	0.622804i	$0.214007\pi$
$942$	0	0
$943$	56.5685	1.84213
$944$	0	0
$945$	−24.0000	−0.780720
$946$	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
$949$	12.0000	0.389536
$950$	0	0
$951$	−2.82843	−0.0917180
$952$	0	0
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	0	0
$955$	2.82843	0.0915258
$956$	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
$965$	−6.00000	−0.193147
$966$	0	0
$967$	4.24264	0.136434	0.0682171	−	0.997671i	$-0.478269\pi$
$967$	4.24264	0.136434	0.0682171	+	0.997671i	$0.478269\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−28.2843	−0.907685	−0.453843	−	0.891082i	$-0.649947\pi$
$971$	−28.2843	−0.907685	−0.453843	+	0.891082i	$0.649947\pi$
$972$	0	0
$973$	36.0000	1.15411
$974$	0	0
$975$	2.82843	0.0905822
$976$	0	0
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	33.9411	1.08476
$980$	0	0
$981$	6.00000	0.191565
$982$	0	0
$983$	−18.3848	−0.586383	−0.293192	−	0.956054i	$-0.594717\pi$
$983$	−18.3848	−0.586383	−0.293192	+	0.956054i	$0.594717\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	−8.48528	−0.270089
$988$	0	0
$989$	−10.0000	−0.317982
$990$	0	0
$991$	8.48528	0.269544	0.134772	−	0.990877i	$-0.456970\pi$
$991$	8.48528	0.269544	0.134772	+	0.990877i	$0.456970\pi$
$992$	0	0
$993$	32.0000	1.01549
$994$	0	0
$995$	−11.3137	−0.358669
$996$	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	1280.2.a.f.1.2		2
4.3	odd	2	inner	1280.2.a.f.1.1		2
5.4	even	2		6400.2.a.bl.1.1		2
8.3	odd	2		1280.2.a.j.1.2		2
8.5	even	2		1280.2.a.j.1.1		2
16.3	odd	4		640.2.d.d.321.2	yes	4
16.5	even	4		640.2.d.d.321.1	✓	4
16.11	odd	4		640.2.d.d.321.3	yes	4
16.13	even	4		640.2.d.d.321.4	yes	4
20.19	odd	2		6400.2.a.bl.1.2		2
40.19	odd	2		6400.2.a.bn.1.1		2
40.29	even	2		6400.2.a.bn.1.2		2
48.5	odd	4		5760.2.k.o.2881.4		4
48.11	even	4		5760.2.k.o.2881.3		4
48.29	odd	4		5760.2.k.o.2881.2		4
48.35	even	4		5760.2.k.o.2881.1		4
80.3	even	4		3200.2.f.i.449.4		4
80.13	odd	4		3200.2.f.i.449.1		4
80.19	odd	4		3200.2.d.s.1601.4		4
80.27	even	4		3200.2.f.i.449.3		4
80.29	even	4		3200.2.d.s.1601.1		4
80.37	odd	4		3200.2.f.i.449.2		4
80.43	even	4		3200.2.f.j.449.2		4
80.53	odd	4		3200.2.f.j.449.3		4
80.59	odd	4		3200.2.d.s.1601.2		4
80.67	even	4		3200.2.f.j.449.1		4
80.69	even	4		3200.2.d.s.1601.3		4
80.77	odd	4		3200.2.f.j.449.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.d.d.321.1	✓	4	16.5	even	4
640.2.d.d.321.2	yes	4	16.3	odd	4
640.2.d.d.321.3	yes	4	16.11	odd	4
640.2.d.d.321.4	yes	4	16.13	even	4
1280.2.a.f.1.1		2	4.3	odd	2	inner
1280.2.a.f.1.2		2	1.1	even	1	trivial
1280.2.a.j.1.1		2	8.5	even	2
1280.2.a.j.1.2		2	8.3	odd	2
3200.2.d.s.1601.1		4	80.29	even	4
3200.2.d.s.1601.2		4	80.59	odd	4
3200.2.d.s.1601.3		4	80.69	even	4
3200.2.d.s.1601.4		4	80.19	odd	4
3200.2.f.i.449.1		4	80.13	odd	4
3200.2.f.i.449.2		4	80.37	odd	4
3200.2.f.i.449.3		4	80.27	even	4
3200.2.f.i.449.4		4	80.3	even	4
3200.2.f.j.449.1		4	80.67	even	4
3200.2.f.j.449.2		4	80.43	even	4
3200.2.f.j.449.3		4	80.53	odd	4
3200.2.f.j.449.4		4	80.77	odd	4
5760.2.k.o.2881.1		4	48.35	even	4
5760.2.k.o.2881.2		4	48.29	odd	4
5760.2.k.o.2881.3		4	48.11	even	4
5760.2.k.o.2881.4		4	48.5	odd	4
6400.2.a.bl.1.1		2	5.4	even	2
6400.2.a.bl.1.2		2	20.19	odd	2
6400.2.a.bn.1.1		2	40.19	odd	2
6400.2.a.bn.1.2		2	40.29	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1280.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} -1.00000 q^{5} -4.24264 q^{7} -1.00000 q^{9} +5.65685 q^{11} +2.00000 q^{13} -1.41421 q^{15} +6.00000 q^{17} -2.82843 q^{19} -6.00000 q^{21} +7.07107 q^{23} +1.00000 q^{25} -5.65685 q^{27} +4.00000 q^{29} +2.82843 q^{31} +8.00000 q^{33} +4.24264 q^{35} -2.00000 q^{37} +2.82843 q^{39} +8.00000 q^{41} -1.41421 q^{43} +1.00000 q^{45} +1.41421 q^{47} +11.0000 q^{49} +8.48528 q^{51} +2.00000 q^{53} -5.65685 q^{55} -4.00000 q^{57} +2.82843 q^{59} +14.0000 q^{61} +4.24264 q^{63} -2.00000 q^{65} -4.24264 q^{67} +10.0000 q^{69} +2.82843 q^{71} +6.00000 q^{73} +1.41421 q^{75} -24.0000 q^{77} -16.9706 q^{79} -5.00000 q^{81} -12.7279 q^{83} -6.00000 q^{85} +5.65685 q^{87} +6.00000 q^{89} -8.48528 q^{91} +4.00000 q^{93} +2.82843 q^{95} +10.0000 q^{97} -5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{9} + 4 q^{13} + 12 q^{17} - 12 q^{21} + 2 q^{25} + 8 q^{29} + 16 q^{33} - 4 q^{37} + 16 q^{41} + 2 q^{45} + 22 q^{49} + 4 q^{53} - 8 q^{57} + 28 q^{61} - 4 q^{65} + 20 q^{69} + 12 q^{73}+ \cdots + 20 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^9 + 4 * q^13 + 12 * q^17 - 12 * q^21 + 2 * q^25 + 8 * q^29 + 16 * q^33 - 4 * q^37 + 16 * q^41 + 2 * q^45 + 22 * q^49 + 4 * q^53 - 8 * q^57 + 28 * q^61 - 4 * q^65 + 20 * q^69 + 12 * q^73 - 48 * q^77 - 10 * q^81 - 12 * q^85 + 12 * q^89 + 8 * q^93 + 20 * q^97

Introduction

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