LMFDB - Embedded newform 4000.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(1,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4000.a (trivial)

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:	yes
Analytic conductor:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.578340050000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 21x^{6} + 129x^{4} - 220x^{2} + 80 $ x^8 - 21x^6 + 129x^4 - 220*x^2 + 80
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.33481$ of defining polynomial
Character	$\chi$	$=$	4000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.33481 q^{3} -1.78602 q^{7} +8.12097 q^{9} +O(q^{10})$	$q-3.33481 q^{3} -1.78602 q^{7} +8.12097 q^{9} -4.05479 q^{11} -4.78297 q^{13} -4.40100 q^{17} -6.94463 q^{19} +5.95604 q^{21} +2.78101 q^{23} -17.0775 q^{27} +5.16493 q^{29} -2.06103 q^{31} +13.5220 q^{33} -5.33800 q^{37} +15.9503 q^{39} -2.50293 q^{41} +3.73821 q^{43} -10.0717 q^{47} -3.81014 q^{49} +14.6765 q^{51} -12.4590 q^{53} +23.1590 q^{57} -0.658848 q^{59} +5.75217 q^{61} -14.5042 q^{63} -8.73065 q^{67} -9.27414 q^{69} -12.2000 q^{71} +5.41276 q^{73} +7.24193 q^{77} +2.04146 q^{79} +32.5872 q^{81} -1.77989 q^{83} -17.2241 q^{87} -16.7771 q^{89} +8.54247 q^{91} +6.87314 q^{93} -6.68690 q^{97} -32.9288 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 18 q^{9}+O(q^{10}) $ 8 * q + 18 * q^9	$ 8 q + 18 q^{9} - 12 q^{13} + 18 q^{21} + 24 q^{29} + 26 q^{33} - 22 q^{37} + 18 q^{41} + 26 q^{49} - 32 q^{53} + 50 q^{57} + 22 q^{61} + 32 q^{69} + 50 q^{73} - 36 q^{77} + 60 q^{81} + 10 q^{89} - 16 q^{93} + 46 q^{97}+O(q^{100}) $ 8 * q + 18 * q^9 - 12 * q^13 + 18 * q^21 + 24 * q^29 + 26 * q^33 - 22 * q^37 + 18 * q^41 + 26 * q^49 - 32 * q^53 + 50 * q^57 + 22 * q^61 + 32 * q^69 + 50 * q^73 - 36 * q^77 + 60 * q^81 + 10 * q^89 - 16 * q^93 + 46 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−3.33481	−1.92535	−0.962677	−	0.270652i	$-0.912761\pi$
$3$	−3.33481	−1.92535	−0.962677	+	0.270652i	$0.912761\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.78602	−0.675052	−0.337526	−	0.941316i	$-0.609590\pi$
$7$	−1.78602	−0.675052	−0.337526	+	0.941316i	$0.609590\pi$
$8$	0	0
$9$	8.12097	2.70699
$10$	0	0
$11$	−4.05479	−1.22257	−0.611283	−	0.791412i	$-0.709346\pi$
$11$	−4.05479	−1.22257	−0.611283	+	0.791412i	$0.709346\pi$
$12$	0	0
$13$	−4.78297	−1.32656	−0.663278	−	0.748373i	$-0.730835\pi$
$13$	−4.78297	−1.32656	−0.663278	+	0.748373i	$0.730835\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.40100	−1.06740	−0.533700	−	0.845674i	$-0.679199\pi$
$17$	−4.40100	−1.06740	−0.533700	+	0.845674i	$0.679199\pi$
$18$	0	0
$19$	−6.94463	−1.59321	−0.796604	−	0.604502i	$-0.793372\pi$
$19$	−6.94463	−1.59321	−0.796604	+	0.604502i	$0.793372\pi$
$20$	0	0
$21$	5.95604	1.29971
$22$	0	0
$23$	2.78101	0.579880	0.289940	−	0.957045i	$-0.406365\pi$
$23$	2.78101	0.579880	0.289940	+	0.957045i	$0.406365\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−17.0775	−3.28656
$28$	0	0
$29$	5.16493	0.959104	0.479552	−	0.877514i	$-0.340799\pi$
$29$	5.16493	0.959104	0.479552	+	0.877514i	$0.340799\pi$
$30$	0	0
$31$	−2.06103	−0.370171	−0.185086	−	0.982722i	$-0.559256\pi$
$31$	−2.06103	−0.370171	−0.185086	+	0.982722i	$0.559256\pi$
$32$	0	0
$33$	13.5220	2.35387
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.33800	−0.877562	−0.438781	−	0.898594i	$-0.644590\pi$
$37$	−5.33800	−0.877562	−0.438781	+	0.898594i	$0.644590\pi$
$38$	0	0
$39$	15.9503	2.55409
$40$	0	0
$41$	−2.50293	−0.390892	−0.195446	−	0.980714i	$-0.562616\pi$
$41$	−2.50293	−0.390892	−0.195446	+	0.980714i	$0.562616\pi$
$42$	0	0
$43$	3.73821	0.570072	0.285036	−	0.958517i	$-0.407994\pi$
$43$	3.73821	0.570072	0.285036	+	0.958517i	$0.407994\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.0717	−1.46911	−0.734554	−	0.678550i	$-0.762609\pi$
$47$	−10.0717	−1.46911	−0.734554	+	0.678550i	$0.762609\pi$
$48$	0	0
$49$	−3.81014	−0.544305
$50$	0	0
$51$	14.6765	2.05512
$52$	0	0
$53$	−12.4590	−1.71137	−0.855685	−	0.517496i	$-0.826864\pi$
$53$	−12.4590	−1.71137	−0.855685	+	0.517496i	$0.826864\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	23.1590	3.06749
$58$	0	0
$59$	−0.658848	−0.0857747	−0.0428873	−	0.999080i	$-0.513656\pi$
$59$	−0.658848	−0.0857747	−0.0428873	+	0.999080i	$0.513656\pi$
$60$	0	0
$61$	5.75217	0.736490	0.368245	−	0.929729i	$-0.379959\pi$
$61$	5.75217	0.736490	0.368245	+	0.929729i	$0.379959\pi$
$62$	0	0
$63$	−14.5042	−1.82736
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.73065	−1.06662	−0.533309	−	0.845920i	$-0.679052\pi$
$67$	−8.73065	−1.06662	−0.533309	+	0.845920i	$0.679052\pi$
$68$	0	0
$69$	−9.27414	−1.11647
$70$	0	0
$71$	−12.2000	−1.44787	−0.723936	−	0.689867i	$-0.757669\pi$
$71$	−12.2000	−1.44787	−0.723936	+	0.689867i	$0.757669\pi$
$72$	0	0
$73$	5.41276	0.633516	0.316758	−	0.948506i	$-0.397406\pi$
$73$	5.41276	0.633516	0.316758	+	0.948506i	$0.397406\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.24193	0.825295
$78$	0	0
$79$	2.04146	0.229683	0.114841	−	0.993384i	$-0.463364\pi$
$79$	2.04146	0.229683	0.114841	+	0.993384i	$0.463364\pi$
$80$	0	0
$81$	32.5872	3.62080
$82$	0	0
$83$	−1.77989	−0.195368	−0.0976842	−	0.995217i	$-0.531143\pi$
$83$	−1.77989	−0.195368	−0.0976842	+	0.995217i	$0.531143\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−17.2241	−1.84661
$88$	0	0
$89$	−16.7771	−1.77837	−0.889183	−	0.457552i	$-0.848726\pi$
$89$	−16.7771	−1.77837	−0.889183	+	0.457552i	$0.848726\pi$
$90$	0	0
$91$	8.54247	0.895494
$92$	0	0
$93$	6.87314	0.712711
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.68690	−0.678952	−0.339476	−	0.940615i	$-0.610250\pi$
$97$	−6.68690	−0.678952	−0.339476	+	0.940615i	$0.610250\pi$
$98$	0	0
$99$	−32.9288	−3.30947
$100$	0	0
$101$	2.63120	0.261814	0.130907	−	0.991395i	$-0.458211\pi$
$101$	2.63120	0.261814	0.130907	+	0.991395i	$0.458211\pi$
$102$	0	0
$103$	11.7489	1.15765	0.578826	−	0.815451i	$-0.303511\pi$
$103$	11.7489	1.15765	0.578826	+	0.815451i	$0.303511\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.55492	−0.150320	−0.0751600	−	0.997171i	$-0.523947\pi$
$107$	−1.55492	−0.150320	−0.0751600	+	0.997171i	$0.523947\pi$
$108$	0	0
$109$	−16.4662	−1.57718	−0.788590	−	0.614920i	$-0.789188\pi$
$109$	−16.4662	−1.57718	−0.788590	+	0.614920i	$0.789188\pi$
$110$	0	0
$111$	17.8012	1.68962
$112$	0	0
$113$	−13.4069	−1.26121	−0.630606	−	0.776103i	$-0.717194\pi$
$113$	−13.4069	−1.26121	−0.630606	+	0.776103i	$0.717194\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−38.8423	−3.59097
$118$	0	0
$119$	7.86027	0.720550
$120$	0	0
$121$	5.44134	0.494667
$122$	0	0
$123$	8.34681	0.752607
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.14407	0.633934	0.316967	−	0.948437i	$-0.397336\pi$
$127$	7.14407	0.633934	0.316967	+	0.948437i	$0.397336\pi$
$128$	0	0
$129$	−12.4662	−1.09759
$130$	0	0
$131$	−6.45575	−0.564041	−0.282021	−	0.959408i	$-0.591005\pi$
$131$	−6.45575	−0.564041	−0.282021	+	0.959408i	$0.591005\pi$
$132$	0	0
$133$	12.4032	1.07550
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.1364	1.20775	0.603876	−	0.797078i	$-0.293622\pi$
$137$	14.1364	1.20775	0.603876	+	0.797078i	$0.293622\pi$
$138$	0	0
$139$	−5.83468	−0.494891	−0.247446	−	0.968902i	$-0.579591\pi$
$139$	−5.83468	−0.494891	−0.247446	+	0.968902i	$0.579591\pi$
$140$	0	0
$141$	33.5872	2.82855
$142$	0	0
$143$	19.3939	1.62180
$144$	0	0
$145$	0	0
$146$	0	0
$147$	12.7061	1.04798
$148$	0	0
$149$	9.16493	0.750820	0.375410	−	0.926859i	$-0.377502\pi$
$149$	9.16493	0.750820	0.375410	+	0.926859i	$0.377502\pi$
$150$	0	0
$151$	−0.743331	−0.0604914	−0.0302457	−	0.999542i	$-0.509629\pi$
$151$	−0.743331	−0.0604914	−0.0302457	+	0.999542i	$0.509629\pi$
$152$	0	0
$153$	−35.7404	−2.88944
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.04983	0.562638	0.281319	−	0.959614i	$-0.409228\pi$
$157$	7.04983	0.562638	0.281319	+	0.959614i	$0.409228\pi$
$158$	0	0
$159$	41.5483	3.29500
$160$	0	0
$161$	−4.96693	−0.391449
$162$	0	0
$163$	14.8503	1.16316	0.581581	−	0.813489i	$-0.302434\pi$
$163$	14.8503	1.16316	0.581581	+	0.813489i	$0.302434\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.7439	−0.986150	−0.493075	−	0.869987i	$-0.664127\pi$
$167$	−12.7439	−0.986150	−0.493075	+	0.869987i	$0.664127\pi$
$168$	0	0
$169$	9.87676	0.759751
$170$	0	0
$171$	−56.3971	−4.31280
$172$	0	0
$173$	5.94287	0.451828	0.225914	−	0.974147i	$-0.427463\pi$
$173$	5.94287	0.451828	0.225914	+	0.974147i	$0.427463\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.19713	0.165147
$178$	0	0
$179$	4.74908	0.354963	0.177481	−	0.984124i	$-0.443205\pi$
$179$	4.74908	0.354963	0.177481	+	0.984124i	$0.443205\pi$
$180$	0	0
$181$	−17.8079	−1.32365	−0.661824	−	0.749659i	$-0.730217\pi$
$181$	−17.8079	−1.32365	−0.661824	+	0.749659i	$0.730217\pi$
$182$	0	0
$183$	−19.1824	−1.41800
$184$	0	0
$185$	0	0
$186$	0	0
$187$	17.8451	1.30497
$188$	0	0
$189$	30.5007	2.21860
$190$	0	0
$191$	17.0458	1.23339	0.616696	−	0.787202i	$-0.288471\pi$
$191$	17.0458	1.23339	0.616696	+	0.787202i	$0.288471\pi$
$192$	0	0
$193$	14.9171	1.07376	0.536878	−	0.843660i	$-0.319603\pi$
$193$	14.9171	1.07376	0.536878	+	0.843660i	$0.319603\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.87314	−0.347197	−0.173598	−	0.984817i	$-0.555539\pi$
$197$	−4.87314	−0.347197	−0.173598	+	0.984817i	$0.555539\pi$
$198$	0	0
$199$	−22.2165	−1.57489	−0.787443	−	0.616387i	$-0.788596\pi$
$199$	−22.2165	−1.57489	−0.787443	+	0.616387i	$0.788596\pi$
$200$	0	0
$201$	29.1151	2.05362
$202$	0	0
$203$	−9.22466	−0.647445
$204$	0	0
$205$	0	0
$206$	0	0
$207$	22.5845	1.56973
$208$	0	0
$209$	28.1590	1.94780
$210$	0	0
$211$	20.5211	1.41273	0.706365	−	0.707847i	$-0.250334\pi$
$211$	20.5211	1.41273	0.706365	+	0.707847i	$0.250334\pi$
$212$	0	0
$213$	40.6846	2.78767
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.68103	0.249885
$218$	0	0
$219$	−18.0505	−1.21974
$220$	0	0
$221$	21.0498	1.41596
$222$	0	0
$223$	−14.2927	−0.957108	−0.478554	−	0.878058i	$-0.658839\pi$
$223$	−14.2927	−0.957108	−0.478554	+	0.878058i	$0.658839\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.14614	0.607051	0.303525	−	0.952823i	$-0.401836\pi$
$227$	9.14614	0.607051	0.303525	+	0.952823i	$0.401836\pi$
$228$	0	0
$229$	16.1268	1.06569	0.532846	−	0.846212i	$-0.321123\pi$
$229$	16.1268	1.06569	0.532846	+	0.846212i	$0.321123\pi$
$230$	0	0
$231$	−24.1505	−1.58899
$232$	0	0
$233$	−2.54327	−0.166615	−0.0833077	−	0.996524i	$-0.526548\pi$
$233$	−2.54327	−0.166615	−0.0833077	+	0.996524i	$0.526548\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−6.80790	−0.442220
$238$	0	0
$239$	25.3020	1.63665	0.818325	−	0.574755i	$-0.194903\pi$
$239$	25.3020	1.63665	0.818325	+	0.574755i	$0.194903\pi$
$240$	0	0
$241$	−2.66786	−0.171852	−0.0859261	−	0.996302i	$-0.527385\pi$
$241$	−2.66786	−0.171852	−0.0859261	+	0.996302i	$0.527385\pi$
$242$	0	0
$243$	−57.4398	−3.68477
$244$	0	0
$245$	0	0
$246$	0	0
$247$	33.2159	2.11348
$248$	0	0
$249$	5.93560	0.376153
$250$	0	0
$251$	24.7005	1.55908	0.779541	−	0.626351i	$-0.215452\pi$
$251$	24.7005	1.55908	0.779541	+	0.626351i	$0.215452\pi$
$252$	0	0
$253$	−11.2764	−0.708942
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.87900	−0.241965	−0.120983	−	0.992655i	$-0.538605\pi$
$257$	−3.87900	−0.241965	−0.120983	+	0.992655i	$0.538605\pi$
$258$	0	0
$259$	9.53377	0.592400
$260$	0	0
$261$	41.9442	2.59628
$262$	0	0
$263$	24.9876	1.54080	0.770401	−	0.637560i	$-0.220056\pi$
$263$	24.9876	1.54080	0.770401	+	0.637560i	$0.220056\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	55.9484	3.42398
$268$	0	0
$269$	12.5161	0.763120	0.381560	−	0.924344i	$-0.375387\pi$
$269$	12.5161	0.763120	0.381560	+	0.924344i	$0.375387\pi$
$270$	0	0
$271$	−10.6890	−0.649309	−0.324654	−	0.945833i	$-0.605248\pi$
$271$	−10.6890	−0.649309	−0.324654	+	0.945833i	$0.605248\pi$
$272$	0	0
$273$	−28.4875	−1.72414
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−25.8423	−1.55272	−0.776358	−	0.630292i	$-0.782935\pi$
$277$	−25.8423	−1.55272	−0.776358	+	0.630292i	$0.782935\pi$
$278$	0	0
$279$	−16.7375	−1.00205
$280$	0	0
$281$	−2.30493	−0.137501	−0.0687504	−	0.997634i	$-0.521901\pi$
$281$	−2.30493	−0.137501	−0.0687504	+	0.997634i	$0.521901\pi$
$282$	0	0
$283$	23.3632	1.38880	0.694400	−	0.719589i	$-0.255670\pi$
$283$	23.3632	1.38880	0.694400	+	0.719589i	$0.255670\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.47029	0.263873
$288$	0	0
$289$	2.36880	0.139341
$290$	0	0
$291$	22.2995	1.30722
$292$	0	0
$293$	7.51247	0.438883	0.219442	−	0.975626i	$-0.429576\pi$
$293$	7.51247	0.438883	0.219442	+	0.975626i	$0.429576\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	69.2455	4.01803
$298$	0	0
$299$	−13.3015	−0.769243
$300$	0	0
$301$	−6.67652	−0.384828
$302$	0	0
$303$	−8.77456	−0.504085
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−19.4708	−1.11126	−0.555630	−	0.831430i	$-0.687523\pi$
$307$	−19.4708	−1.11126	−0.555630	+	0.831430i	$0.687523\pi$
$308$	0	0
$309$	−39.1803	−2.22889
$310$	0	0
$311$	−19.5223	−1.10701	−0.553505	−	0.832846i	$-0.686710\pi$
$311$	−19.5223	−1.10701	−0.553505	+	0.832846i	$0.686710\pi$
$312$	0	0
$313$	12.9443	0.731654	0.365827	−	0.930683i	$-0.380786\pi$
$313$	12.9443	0.731654	0.365827	+	0.930683i	$0.380786\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.69052	0.0949493	0.0474746	−	0.998872i	$-0.484883\pi$
$317$	1.69052	0.0949493	0.0474746	+	0.998872i	$0.484883\pi$
$318$	0	0
$319$	−20.9427	−1.17257
$320$	0	0
$321$	5.18537	0.289419
$322$	0	0
$323$	30.5633	1.70059
$324$	0	0
$325$	0	0
$326$	0	0
$327$	54.9118	3.03663
$328$	0	0
$329$	17.9882	0.991724
$330$	0	0
$331$	−26.6197	−1.46315	−0.731575	−	0.681760i	$-0.761215\pi$
$331$	−26.6197	−1.46315	−0.731575	+	0.681760i	$0.761215\pi$
$332$	0	0
$333$	−43.3497	−2.37555
$334$	0	0
$335$	0	0
$336$	0	0
$337$	23.3453	1.27170	0.635849	−	0.771813i	$-0.280650\pi$
$337$	23.3453	1.27170	0.635849	+	0.771813i	$0.280650\pi$
$338$	0	0
$339$	44.7094	2.42828
$340$	0	0
$341$	8.35704	0.452559
$342$	0	0
$343$	19.3071	1.04249
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.8473	−1.11914	−0.559572	−	0.828782i	$-0.689034\pi$
$347$	−20.8473	−1.11914	−0.559572	+	0.828782i	$0.689034\pi$
$348$	0	0
$349$	−34.9728	−1.87205	−0.936025	−	0.351932i	$-0.885525\pi$
$349$	−34.9728	−1.87205	−0.936025	+	0.351932i	$0.885525\pi$
$350$	0	0
$351$	81.6809	4.35980
$352$	0	0
$353$	31.8695	1.69624	0.848120	−	0.529804i	$-0.177735\pi$
$353$	31.8695	1.69624	0.848120	+	0.529804i	$0.177735\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−26.2125	−1.38731
$358$	0	0
$359$	−14.2488	−0.752020	−0.376010	−	0.926616i	$-0.622704\pi$
$359$	−14.2488	−0.752020	−0.376010	+	0.926616i	$0.622704\pi$
$360$	0	0
$361$	29.2279	1.53831
$362$	0	0
$363$	−18.1458	−0.952410
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.45931	−0.284974	−0.142487	−	0.989797i	$-0.545510\pi$
$367$	−5.45931	−0.284974	−0.142487	+	0.989797i	$0.545510\pi$
$368$	0	0
$369$	−20.3262	−1.05814
$370$	0	0
$371$	22.2520	1.15526
$372$	0	0
$373$	−21.6071	−1.11877	−0.559387	−	0.828907i	$-0.688963\pi$
$373$	−21.6071	−1.11877	−0.559387	+	0.828907i	$0.688963\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−24.7037	−1.27230
$378$	0	0
$379$	0.937644	0.0481635	0.0240818	−	0.999710i	$-0.492334\pi$
$379$	0.937644	0.0481635	0.0240818	+	0.999710i	$0.492334\pi$
$380$	0	0
$381$	−23.8241	−1.22055
$382$	0	0
$383$	−1.03060	−0.0526610	−0.0263305	−	0.999653i	$-0.508382\pi$
$383$	−1.03060	−0.0526610	−0.0263305	+	0.999653i	$0.508382\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	30.3579	1.54318
$388$	0	0
$389$	6.15907	0.312277	0.156139	−	0.987735i	$-0.450095\pi$
$389$	6.15907	0.312277	0.156139	+	0.987735i	$0.450095\pi$
$390$	0	0
$391$	−12.2392	−0.618964
$392$	0	0
$393$	21.5287	1.08598
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10.4794	0.525946	0.262973	−	0.964803i	$-0.415297\pi$
$397$	10.4794	0.525946	0.262973	+	0.964803i	$0.415297\pi$
$398$	0	0
$399$	−41.3625	−2.07071
$400$	0	0
$401$	−8.42455	−0.420702	−0.210351	−	0.977626i	$-0.567461\pi$
$401$	−8.42455	−0.420702	−0.210351	+	0.977626i	$0.567461\pi$
$402$	0	0
$403$	9.85782	0.491053
$404$	0	0
$405$	0	0
$406$	0	0
$407$	21.6445	1.07288
$408$	0	0
$409$	−23.0199	−1.13826	−0.569130	−	0.822248i	$-0.692720\pi$
$409$	−23.0199	−1.13826	−0.569130	+	0.822248i	$0.692720\pi$
$410$	0	0
$411$	−47.1421	−2.32535
$412$	0	0
$413$	1.17671	0.0579023
$414$	0	0
$415$	0	0
$416$	0	0
$417$	19.4576	0.952841
$418$	0	0
$419$	−21.1995	−1.03566	−0.517832	−	0.855482i	$-0.673261\pi$
$419$	−21.1995	−1.03566	−0.517832	+	0.855482i	$0.673261\pi$
$420$	0	0
$421$	−24.7412	−1.20581	−0.602907	−	0.797811i	$-0.705991\pi$
$421$	−24.7412	−1.20581	−0.602907	+	0.797811i	$0.705991\pi$
$422$	0	0
$423$	−81.7919	−3.97686
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−10.2735	−0.497169
$428$	0	0
$429$	−64.6751	−3.12254
$430$	0	0
$431$	8.12915	0.391567	0.195784	−	0.980647i	$-0.437275\pi$
$431$	8.12915	0.391567	0.195784	+	0.980647i	$0.437275\pi$
$432$	0	0
$433$	−19.6656	−0.945068	−0.472534	−	0.881312i	$-0.656661\pi$
$433$	−19.6656	−0.945068	−0.472534	+	0.881312i	$0.656661\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−19.3131	−0.923869
$438$	0	0
$439$	24.3362	1.16150	0.580752	−	0.814080i	$-0.302758\pi$
$439$	24.3362	1.16150	0.580752	+	0.814080i	$0.302758\pi$
$440$	0	0
$441$	−30.9420	−1.47343
$442$	0	0
$443$	−2.57922	−0.122543	−0.0612713	−	0.998121i	$-0.519515\pi$
$443$	−2.57922	−0.122543	−0.0612713	+	0.998121i	$0.519515\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−30.5633	−1.44560
$448$	0	0
$449$	6.48530	0.306060	0.153030	−	0.988222i	$-0.451097\pi$
$449$	6.48530	0.306060	0.153030	+	0.988222i	$0.451097\pi$
$450$	0	0
$451$	10.1489	0.477892
$452$	0	0
$453$	2.47887	0.116467
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11.0820	−0.518394	−0.259197	−	0.965824i	$-0.583458\pi$
$457$	−11.0820	−0.518394	−0.259197	+	0.965824i	$0.583458\pi$
$458$	0	0
$459$	75.1579	3.50807
$460$	0	0
$461$	−37.0380	−1.72503	−0.862517	−	0.506029i	$-0.831113\pi$
$461$	−37.0380	−1.72503	−0.862517	+	0.506029i	$0.831113\pi$
$462$	0	0
$463$	−15.0285	−0.698435	−0.349217	−	0.937042i	$-0.613553\pi$
$463$	−15.0285	−0.698435	−0.349217	+	0.937042i	$0.613553\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−0.633156	−0.0292990	−0.0146495	−	0.999893i	$-0.504663\pi$
$467$	−0.633156	−0.0292990	−0.0146495	+	0.999893i	$0.504663\pi$
$468$	0	0
$469$	15.5931	0.720023
$470$	0	0
$471$	−23.5099	−1.08328
$472$	0	0
$473$	−15.1577	−0.696951
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−101.179	−4.63266
$478$	0	0
$479$	−1.21499	−0.0555144	−0.0277572	−	0.999615i	$-0.508837\pi$
$479$	−1.21499	−0.0555144	−0.0277572	+	0.999615i	$0.508837\pi$
$480$	0	0
$481$	25.5315	1.16414
$482$	0	0
$483$	16.5638	0.753678
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−9.87838	−0.447632	−0.223816	−	0.974631i	$-0.571851\pi$
$487$	−9.87838	−0.447632	−0.223816	+	0.974631i	$0.571851\pi$
$488$	0	0
$489$	−49.5228	−2.23950
$490$	0	0
$491$	1.26035	0.0568788	0.0284394	−	0.999596i	$-0.490946\pi$
$491$	1.26035	0.0568788	0.0284394	+	0.999596i	$0.490946\pi$
$492$	0	0
$493$	−22.7309	−1.02375
$494$	0	0
$495$	0	0
$496$	0	0
$497$	21.7894	0.977388
$498$	0	0
$499$	32.0660	1.43547	0.717736	−	0.696315i	$-0.245178\pi$
$499$	32.0660	1.43547	0.717736	+	0.696315i	$0.245178\pi$
$500$	0	0
$501$	42.4984	1.89869
$502$	0	0
$503$	33.6134	1.49875	0.749373	−	0.662148i	$-0.230355\pi$
$503$	33.6134	1.49875	0.749373	+	0.662148i	$0.230355\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−32.9371	−1.46279
$508$	0	0
$509$	33.3021	1.47609	0.738046	−	0.674751i	$-0.235749\pi$
$509$	33.3021	1.47609	0.738046	+	0.674751i	$0.235749\pi$
$510$	0	0
$511$	−9.66730	−0.427656
$512$	0	0
$513$	118.597	5.23617
$514$	0	0
$515$	0	0
$516$	0	0
$517$	40.8386	1.79608
$518$	0	0
$519$	−19.8183	−0.869929
$520$	0	0
$521$	4.09690	0.179489	0.0897443	−	0.995965i	$-0.471395\pi$
$521$	4.09690	0.179489	0.0897443	+	0.995965i	$0.471395\pi$
$522$	0	0
$523$	14.2414	0.622735	0.311368	−	0.950290i	$-0.399213\pi$
$523$	14.2414	0.622735	0.311368	+	0.950290i	$0.399213\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.07058	0.395121
$528$	0	0
$529$	−15.2660	−0.663739
$530$	0	0
$531$	−5.35048	−0.232191
$532$	0	0
$533$	11.9714	0.518541
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−15.8373	−0.683430
$538$	0	0
$539$	15.4493	0.665449
$540$	0	0
$541$	2.80200	0.120467	0.0602337	−	0.998184i	$-0.480815\pi$
$541$	2.80200	0.120467	0.0602337	+	0.998184i	$0.480815\pi$
$542$	0	0
$543$	59.3859	2.54849
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.85292	0.378523	0.189262	−	0.981927i	$-0.439391\pi$
$547$	8.85292	0.378523	0.189262	+	0.981927i	$0.439391\pi$
$548$	0	0
$549$	46.7132	1.99367
$550$	0	0
$551$	−35.8685	−1.52805
$552$	0	0
$553$	−3.64609	−0.155048
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.44721	−0.188434	−0.0942171	−	0.995552i	$-0.530035\pi$
$557$	−4.44721	−0.188434	−0.0942171	+	0.995552i	$0.530035\pi$
$558$	0	0
$559$	−17.8797	−0.756233
$560$	0	0
$561$	−59.5102	−2.51252
$562$	0	0
$563$	39.3846	1.65986	0.829931	−	0.557866i	$-0.188380\pi$
$563$	39.3846	1.65986	0.829931	+	0.557866i	$0.188380\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−58.2014	−2.44423
$568$	0	0
$569$	−2.48079	−0.104000	−0.0520001	−	0.998647i	$-0.516560\pi$
$569$	−2.48079	−0.104000	−0.0520001	+	0.998647i	$0.516560\pi$
$570$	0	0
$571$	−4.32601	−0.181038	−0.0905190	−	0.995895i	$-0.528853\pi$
$571$	−4.32601	−0.181038	−0.0905190	+	0.995895i	$0.528853\pi$
$572$	0	0
$573$	−56.8445	−2.37472
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.38643	0.224240	0.112120	−	0.993695i	$-0.464236\pi$
$577$	5.38643	0.224240	0.112120	+	0.993695i	$0.464236\pi$
$578$	0	0
$579$	−49.7457	−2.06736
$580$	0	0
$581$	3.17892	0.131884
$582$	0	0
$583$	50.5185	2.09226
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.63539	−0.108774	−0.0543871	−	0.998520i	$-0.517321\pi$
$587$	−2.63539	−0.108774	−0.0543871	+	0.998520i	$0.517321\pi$
$588$	0	0
$589$	14.3131	0.589760
$590$	0	0
$591$	16.2510	0.668476
$592$	0	0
$593$	−10.8849	−0.446988	−0.223494	−	0.974705i	$-0.571746\pi$
$593$	−10.8849	−0.446988	−0.223494	+	0.974705i	$0.571746\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	74.0879	3.03221
$598$	0	0
$599$	−35.9152	−1.46746	−0.733729	−	0.679443i	$-0.762222\pi$
$599$	−35.9152	−1.46746	−0.733729	+	0.679443i	$0.762222\pi$
$600$	0	0
$601$	−25.5505	−1.04223	−0.521114	−	0.853487i	$-0.674483\pi$
$601$	−25.5505	−1.04223	−0.521114	+	0.853487i	$0.674483\pi$
$602$	0	0
$603$	−70.9013	−2.88732
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−12.3993	−0.503270	−0.251635	−	0.967822i	$-0.580968\pi$
$607$	−12.3993	−0.503270	−0.251635	+	0.967822i	$0.580968\pi$
$608$	0	0
$609$	30.7625	1.24656
$610$	0	0
$611$	48.1726	1.94885
$612$	0	0
$613$	1.45088	0.0586005	0.0293003	−	0.999571i	$-0.490672\pi$
$613$	1.45088	0.0586005	0.0293003	+	0.999571i	$0.490672\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.74714	−0.352147	−0.176073	−	0.984377i	$-0.556340\pi$
$617$	−8.74714	−0.352147	−0.176073	+	0.984377i	$0.556340\pi$
$618$	0	0
$619$	48.2337	1.93868	0.969338	−	0.245730i	$-0.0790278\pi$
$619$	48.2337	1.93868	0.969338	+	0.245730i	$0.0790278\pi$
$620$	0	0
$621$	−47.4925	−1.90581
$622$	0	0
$623$	29.9642	1.20049
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−93.9051	−3.75021
$628$	0	0
$629$	23.4925	0.936709
$630$	0	0
$631$	−45.4575	−1.80963	−0.904817	−	0.425801i	$-0.859992\pi$
$631$	−45.4575	−1.80963	−0.904817	+	0.425801i	$0.859992\pi$
$632$	0	0
$633$	−68.4340	−2.72001
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18.2238	0.722052
$638$	0	0
$639$	−99.0757	−3.91937
$640$	0	0
$641$	37.9442	1.49871	0.749353	−	0.662170i	$-0.230364\pi$
$641$	37.9442	1.49871	0.749353	+	0.662170i	$0.230364\pi$
$642$	0	0
$643$	−20.4638	−0.807012	−0.403506	−	0.914977i	$-0.632209\pi$
$643$	−20.4638	−0.807012	−0.403506	+	0.914977i	$0.632209\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−33.7257	−1.32589	−0.662947	−	0.748666i	$-0.730695\pi$
$647$	−33.7257	−1.32589	−0.662947	+	0.748666i	$0.730695\pi$
$648$	0	0
$649$	2.67149	0.104865
$650$	0	0
$651$	−12.2755	−0.481117
$652$	0	0
$653$	10.2243	0.400106	0.200053	−	0.979785i	$-0.435889\pi$
$653$	10.2243	0.400106	0.200053	+	0.979785i	$0.435889\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	43.9569	1.71492
$658$	0	0
$659$	20.2742	0.789772	0.394886	−	0.918730i	$-0.370784\pi$
$659$	20.2742	0.789772	0.394886	+	0.918730i	$0.370784\pi$
$660$	0	0
$661$	0.329864	0.0128302	0.00641510	−	0.999979i	$-0.497958\pi$
$661$	0.329864	0.0128302	0.00641510	+	0.999979i	$0.497958\pi$
$662$	0	0
$663$	−70.1972	−2.72623
$664$	0	0
$665$	0	0
$666$	0	0
$667$	14.3637	0.556165
$668$	0	0
$669$	47.6633	1.84277
$670$	0	0
$671$	−23.3238	−0.900407
$672$	0	0
$673$	29.3960	1.13313	0.566566	−	0.824017i	$-0.308272\pi$
$673$	29.3960	1.13313	0.566566	+	0.824017i	$0.308272\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.05346	0.0404876	0.0202438	−	0.999795i	$-0.493556\pi$
$677$	1.05346	0.0404876	0.0202438	+	0.999795i	$0.493556\pi$
$678$	0	0
$679$	11.9429	0.458327
$680$	0	0
$681$	−30.5007	−1.16879
$682$	0	0
$683$	−19.5661	−0.748675	−0.374337	−	0.927293i	$-0.622130\pi$
$683$	−19.5661	−0.748675	−0.374337	+	0.927293i	$0.622130\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−53.7799	−2.05183
$688$	0	0
$689$	59.5908	2.27023
$690$	0	0
$691$	46.2129	1.75802	0.879011	−	0.476802i	$-0.158204\pi$
$691$	46.2129	1.75802	0.879011	+	0.476802i	$0.158204\pi$
$692$	0	0
$693$	58.8115	2.23406
$694$	0	0
$695$	0	0
$696$	0	0
$697$	11.0154	0.417238
$698$	0	0
$699$	8.48133	0.320793
$700$	0	0
$701$	−19.2134	−0.725679	−0.362840	−	0.931852i	$-0.618193\pi$
$701$	−19.2134	−0.725679	−0.362840	+	0.931852i	$0.618193\pi$
$702$	0	0
$703$	37.0704	1.39814
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.69937	−0.176738
$708$	0	0
$709$	−4.96644	−0.186519	−0.0932594	−	0.995642i	$-0.529729\pi$
$709$	−4.96644	−0.186519	−0.0932594	+	0.995642i	$0.529729\pi$
$710$	0	0
$711$	16.5787	0.621748
$712$	0	0
$713$	−5.73173	−0.214655
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−84.3774	−3.15113
$718$	0	0
$719$	2.76975	0.103294	0.0516471	−	0.998665i	$-0.483553\pi$
$719$	2.76975	0.103294	0.0516471	+	0.998665i	$0.483553\pi$
$720$	0	0
$721$	−20.9837	−0.781475
$722$	0	0
$723$	8.89683	0.330877
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−33.9826	−1.26035	−0.630173	−	0.776455i	$-0.717016\pi$
$727$	−33.9826	−1.26035	−0.630173	+	0.776455i	$0.717016\pi$
$728$	0	0
$729$	93.7893	3.47368
$730$	0	0
$731$	−16.4519	−0.608495
$732$	0	0
$733$	9.75576	0.360337	0.180169	−	0.983636i	$-0.442336\pi$
$733$	9.75576	0.360337	0.180169	+	0.983636i	$0.442336\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	35.4010	1.30401
$738$	0	0
$739$	3.98140	0.146458	0.0732291	−	0.997315i	$-0.476670\pi$
$739$	3.98140	0.146458	0.0732291	+	0.997315i	$0.476670\pi$
$740$	0	0
$741$	−110.769	−4.06920
$742$	0	0
$743$	−39.5319	−1.45029	−0.725143	−	0.688599i	$-0.758226\pi$
$743$	−39.5319	−1.45029	−0.725143	+	0.688599i	$0.758226\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−14.4544	−0.528860
$748$	0	0
$749$	2.77712	0.101474
$750$	0	0
$751$	−2.99037	−0.109120	−0.0545600	−	0.998510i	$-0.517376\pi$
$751$	−2.99037	−0.109120	−0.0545600	+	0.998510i	$0.517376\pi$
$752$	0	0
$753$	−82.3715	−3.00178
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−14.4685	−0.525866	−0.262933	−	0.964814i	$-0.584690\pi$
$757$	−14.4685	−0.525866	−0.262933	+	0.964814i	$0.584690\pi$
$758$	0	0
$759$	37.6047	1.36496
$760$	0	0
$761$	−7.76482	−0.281474	−0.140737	−	0.990047i	$-0.544947\pi$
$761$	−7.76482	−0.281474	−0.140737	+	0.990047i	$0.544947\pi$
$762$	0	0
$763$	29.4090	1.06468
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.15125	0.113785
$768$	0	0
$769$	−41.2823	−1.48868	−0.744338	−	0.667803i	$-0.767235\pi$
$769$	−41.2823	−1.48868	−0.744338	+	0.667803i	$0.767235\pi$
$770$	0	0
$771$	12.9357	0.465869
$772$	0	0
$773$	15.3856	0.553381	0.276691	−	0.960959i	$-0.410762\pi$
$773$	15.3856	0.553381	0.276691	+	0.960959i	$0.410762\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−31.7933	−1.14058
$778$	0	0
$779$	17.3819	0.622773
$780$	0	0
$781$	49.4684	1.77012
$782$	0	0
$783$	−88.2039	−3.15215
$784$	0	0
$785$	0	0
$786$	0	0
$787$	30.3700	1.08257	0.541287	−	0.840838i	$-0.317937\pi$
$787$	30.3700	1.08257	0.541287	+	0.840838i	$0.317937\pi$
$788$	0	0
$789$	−83.3289	−2.96659
$790$	0	0
$791$	23.9449	0.851383
$792$	0	0
$793$	−27.5124	−0.976995
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11.9375	−0.422848	−0.211424	−	0.977394i	$-0.567810\pi$
$797$	−11.9375	−0.422848	−0.211424	+	0.977394i	$0.567810\pi$
$798$	0	0
$799$	44.3255	1.56812
$800$	0	0
$801$	−136.246	−4.81402
$802$	0	0
$803$	−21.9476	−0.774515
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−41.7388	−1.46928
$808$	0	0
$809$	9.92975	0.349111	0.174556	−	0.984647i	$-0.444151\pi$
$809$	9.92975	0.349111	0.174556	+	0.984647i	$0.444151\pi$
$810$	0	0
$811$	35.0971	1.23243	0.616213	−	0.787580i	$-0.288666\pi$
$811$	35.0971	1.23243	0.616213	+	0.787580i	$0.288666\pi$
$812$	0	0
$813$	35.6457	1.25015
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−25.9605	−0.908244
$818$	0	0
$819$	69.3731	2.42409
$820$	0	0
$821$	13.2442	0.462224	0.231112	−	0.972927i	$-0.425764\pi$
$821$	13.2442	0.462224	0.231112	+	0.972927i	$0.425764\pi$
$822$	0	0
$823$	10.3502	0.360786	0.180393	−	0.983595i	$-0.442263\pi$
$823$	10.3502	0.360786	0.180393	+	0.983595i	$0.442263\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−26.7372	−0.929743	−0.464871	−	0.885378i	$-0.653899\pi$
$827$	−26.7372	−0.929743	−0.464871	+	0.885378i	$0.653899\pi$
$828$	0	0
$829$	−31.2632	−1.08582	−0.542909	−	0.839792i	$-0.682677\pi$
$829$	−31.2632	−1.08582	−0.542909	+	0.839792i	$0.682677\pi$
$830$	0	0
$831$	86.1793	2.98953
$832$	0	0
$833$	16.7684	0.580991
$834$	0	0
$835$	0	0
$836$	0	0
$837$	35.1971	1.21659
$838$	0	0
$839$	−12.7498	−0.440173	−0.220087	−	0.975480i	$-0.570634\pi$
$839$	−12.7498	−0.440173	−0.220087	+	0.975480i	$0.570634\pi$
$840$	0	0
$841$	−2.32348	−0.0801200
$842$	0	0
$843$	7.68652	0.264738
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−9.71833	−0.333926
$848$	0	0
$849$	−77.9120	−2.67393
$850$	0	0
$851$	−14.8450	−0.508881
$852$	0	0
$853$	−33.5497	−1.14872	−0.574359	−	0.818603i	$-0.694749\pi$
$853$	−33.5497	−1.14872	−0.574359	+	0.818603i	$0.694749\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−8.07787	−0.275935	−0.137967	−	0.990437i	$-0.544057\pi$
$857$	−8.07787	−0.275935	−0.137967	+	0.990437i	$0.544057\pi$
$858$	0	0
$859$	−49.1730	−1.67776	−0.838880	−	0.544316i	$-0.816789\pi$
$859$	−49.1730	−1.67776	−0.838880	+	0.544316i	$0.816789\pi$
$860$	0	0
$861$	−14.9076	−0.508048
$862$	0	0
$863$	13.7683	0.468680	0.234340	−	0.972155i	$-0.424707\pi$
$863$	13.7683	0.468680	0.234340	+	0.972155i	$0.424707\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−7.89950	−0.268281
$868$	0	0
$869$	−8.27771	−0.280802
$870$	0	0
$871$	41.7584	1.41493
$872$	0	0
$873$	−54.3041	−1.83791
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−19.4377	−0.656364	−0.328182	−	0.944614i	$-0.606436\pi$
$877$	−19.4377	−0.656364	−0.328182	+	0.944614i	$0.606436\pi$
$878$	0	0
$879$	−25.0527	−0.845006
$880$	0	0
$881$	6.89720	0.232373	0.116186	−	0.993227i	$-0.462933\pi$
$881$	6.89720	0.232373	0.116186	+	0.993227i	$0.462933\pi$
$882$	0	0
$883$	18.1775	0.611721	0.305861	−	0.952076i	$-0.401056\pi$
$883$	18.1775	0.611721	0.305861	+	0.952076i	$0.401056\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−16.0469	−0.538801	−0.269400	−	0.963028i	$-0.586826\pi$
$887$	−16.0469	−0.538801	−0.269400	+	0.963028i	$0.586826\pi$
$888$	0	0
$889$	−12.7595	−0.427938
$890$	0	0
$891$	−132.134	−4.42667
$892$	0	0
$893$	69.9442	2.34059
$894$	0	0
$895$	0	0
$896$	0	0
$897$	44.3579	1.48107
$898$	0	0
$899$	−10.6451	−0.355033
$900$	0	0
$901$	54.8319	1.82672
$902$	0	0
$903$	22.2649	0.740931
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.16580	0.138323	0.0691615	−	0.997605i	$-0.477968\pi$
$907$	4.16580	0.138323	0.0691615	+	0.997605i	$0.477968\pi$
$908$	0	0
$909$	21.3679	0.708729
$910$	0	0
$911$	−39.3846	−1.30487	−0.652435	−	0.757845i	$-0.726252\pi$
$911$	−39.3846	−1.30487	−0.652435	+	0.757845i	$0.726252\pi$
$912$	0	0
$913$	7.21708	0.238851
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.5301	0.380757
$918$	0	0
$919$	−50.6597	−1.67111	−0.835555	−	0.549406i	$-0.814854\pi$
$919$	−50.6597	−1.67111	−0.835555	+	0.549406i	$0.814854\pi$
$920$	0	0
$921$	64.9316	2.13957
$922$	0	0
$923$	58.3521	1.92068
$924$	0	0
$925$	0	0
$926$	0	0
$927$	95.4123	3.13375
$928$	0	0
$929$	11.1283	0.365107	0.182553	−	0.983196i	$-0.441564\pi$
$929$	11.1283	0.365107	0.182553	+	0.983196i	$0.441564\pi$
$930$	0	0
$931$	26.4600	0.867191
$932$	0	0
$933$	65.1033	2.13139
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−32.4472	−1.06000	−0.530001	−	0.847997i	$-0.677809\pi$
$937$	−32.4472	−1.06000	−0.530001	+	0.847997i	$0.677809\pi$
$938$	0	0
$939$	−43.1667	−1.40869
$940$	0	0
$941$	−34.7793	−1.13377	−0.566887	−	0.823796i	$-0.691852\pi$
$941$	−34.7793	−1.13377	−0.566887	+	0.823796i	$0.691852\pi$
$942$	0	0
$943$	−6.96068	−0.226671
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.9403	0.745459	0.372729	−	0.927940i	$-0.378422\pi$
$947$	22.9403	0.745459	0.372729	+	0.927940i	$0.378422\pi$
$948$	0	0
$949$	−25.8891	−0.840394
$950$	0	0
$951$	−5.63758	−0.182811
$952$	0	0
$953$	−41.1854	−1.33412	−0.667062	−	0.745002i	$-0.732449\pi$
$953$	−41.1854	−1.33412	−0.667062	+	0.745002i	$0.732449\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	69.8400	2.25761
$958$	0	0
$959$	−25.2478	−0.815295
$960$	0	0
$961$	−26.7522	−0.862973
$962$	0	0
$963$	−12.6275	−0.406914
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−28.4819	−0.915916	−0.457958	−	0.888974i	$-0.651419\pi$
$967$	−28.4819	−0.915916	−0.457958	+	0.888974i	$0.651419\pi$
$968$	0	0
$969$	−101.923	−3.27424
$970$	0	0
$971$	50.1014	1.60783	0.803916	−	0.594743i	$-0.202746\pi$
$971$	50.1014	1.60783	0.803916	+	0.594743i	$0.202746\pi$
$972$	0	0
$973$	10.4209	0.334077
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14.6370	0.468280	0.234140	−	0.972203i	$-0.424773\pi$
$977$	14.6370	0.468280	0.234140	+	0.972203i	$0.424773\pi$
$978$	0	0
$979$	68.0275	2.17417
$980$	0	0
$981$	−133.722	−4.26941
$982$	0	0
$983$	19.1817	0.611800	0.305900	−	0.952064i	$-0.401043\pi$
$983$	19.1817	0.611800	0.305900	+	0.952064i	$0.401043\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−59.9874	−1.90942
$988$	0	0
$989$	10.3960	0.330574
$990$	0	0
$991$	−7.30268	−0.231977	−0.115989	−	0.993251i	$-0.537004\pi$
$991$	−7.30268	−0.231977	−0.115989	+	0.993251i	$0.537004\pi$
$992$	0	0
$993$	88.7717	2.81708
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−57.4816	−1.82046	−0.910230	−	0.414104i	$-0.864095\pi$
$997$	−57.4816	−1.82046	−0.910230	+	0.414104i	$0.864095\pi$
$998$	0	0
$999$	91.1595	2.88416

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.a.q.1.1	✓	8
4.3	odd	2	inner	4000.2.a.q.1.8	yes	8
5.2	odd	4		4000.2.c.h.1249.15		16
5.3	odd	4		4000.2.c.h.1249.1		16
5.4	even	2		4000.2.a.r.1.8	yes	8
8.3	odd	2		8000.2.a.cb.1.1		8
8.5	even	2		8000.2.a.cb.1.8		8
20.3	even	4		4000.2.c.h.1249.16		16
20.7	even	4		4000.2.c.h.1249.2		16
20.19	odd	2		4000.2.a.r.1.1	yes	8
40.19	odd	2		8000.2.a.ca.1.8		8
40.29	even	2		8000.2.a.ca.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4000.2.a.q.1.1	✓	8	1.1	even	1	trivial
4000.2.a.q.1.8	yes	8	4.3	odd	2	inner
4000.2.a.r.1.1	yes	8	20.19	odd	2
4000.2.a.r.1.8	yes	8	5.4	even	2
4000.2.c.h.1249.1		16	5.3	odd	4
4000.2.c.h.1249.2		16	20.7	even	4
4000.2.c.h.1249.15		16	5.2	odd	4
4000.2.c.h.1249.16		16	20.3	even	4
8000.2.a.ca.1.1		8	40.29	even	2
8000.2.a.ca.1.8		8	40.19	odd	2
8000.2.a.cb.1.1		8	8.3	odd	2
8000.2.a.cb.1.8		8	8.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.a (trivial)

\(f(q)\)	\(=\)	\(q-3.33481 q^{3} -1.78602 q^{7} +8.12097 q^{9} +O(q^{10})\)	\(q-3.33481 q^{3} -1.78602 q^{7} +8.12097 q^{9} -4.05479 q^{11} -4.78297 q^{13} -4.40100 q^{17} -6.94463 q^{19} +5.95604 q^{21} +2.78101 q^{23} -17.0775 q^{27} +5.16493 q^{29} -2.06103 q^{31} +13.5220 q^{33} -5.33800 q^{37} +15.9503 q^{39} -2.50293 q^{41} +3.73821 q^{43} -10.0717 q^{47} -3.81014 q^{49} +14.6765 q^{51} -12.4590 q^{53} +23.1590 q^{57} -0.658848 q^{59} +5.75217 q^{61} -14.5042 q^{63} -8.73065 q^{67} -9.27414 q^{69} -12.2000 q^{71} +5.41276 q^{73} +7.24193 q^{77} +2.04146 q^{79} +32.5872 q^{81} -1.77989 q^{83} -17.2241 q^{87} -16.7771 q^{89} +8.54247 q^{91} +6.87314 q^{93} -6.68690 q^{97} -32.9288 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 18 q^{9}+O(q^{10}) \) 8 * q + 18 * q^9	\( 8 q + 18 q^{9} - 12 q^{13} + 18 q^{21} + 24 q^{29} + 26 q^{33} - 22 q^{37} + 18 q^{41} + 26 q^{49} - 32 q^{53} + 50 q^{57} + 22 q^{61} + 32 q^{69} + 50 q^{73} - 36 q^{77} + 60 q^{81} + 10 q^{89} - 16 q^{93} + 46 q^{97}+O(q^{100}) \) 8 * q + 18 * q^9 - 12 * q^13 + 18 * q^21 + 24 * q^29 + 26 * q^33 - 22 * q^37 + 18 * q^41 + 26 * q^49 - 32 * q^53 + 50 * q^57 + 22 * q^61 + 32 * q^69 + 50 * q^73 - 36 * q^77 + 60 * q^81 + 10 * q^89 - 16 * q^93 + 46 * q^97

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