LMFDB - Embedded newform 4332.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4332,2,Mod(1,4332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4332, 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("4332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4332 = 2^{2} \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4332.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:	$34.5911941556$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 228)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	4332.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.879385 q^{5} -2.18479 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.879385 q^{5} -2.18479 q^{7} +1.00000 q^{9} -0.162504 q^{11} +0.758770 q^{13} -0.879385 q^{15} +3.10607 q^{17} -2.18479 q^{21} +1.29086 q^{23} -4.22668 q^{25} +1.00000 q^{27} -6.51754 q^{29} +0.958111 q^{31} -0.162504 q^{33} +1.92127 q^{35} -1.16250 q^{37} +0.758770 q^{39} -10.5963 q^{41} -0.177052 q^{43} -0.879385 q^{45} -3.46791 q^{47} -2.22668 q^{49} +3.10607 q^{51} +8.33275 q^{53} +0.142903 q^{55} -2.78106 q^{59} +7.22668 q^{61} -2.18479 q^{63} -0.667252 q^{65} +1.06418 q^{67} +1.29086 q^{69} +5.08378 q^{71} +2.24897 q^{73} -4.22668 q^{75} +0.355037 q^{77} -5.24123 q^{79} +1.00000 q^{81} -17.1138 q^{83} -2.73143 q^{85} -6.51754 q^{87} +15.6236 q^{89} -1.65776 q^{91} +0.958111 q^{93} -9.41921 q^{97} -0.162504 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} - 9 q^{13} + 3 q^{15} - 3 q^{17} - 3 q^{21} - 12 q^{23} - 6 q^{25} + 3 q^{27} + 3 q^{29} + 6 q^{31} - 3 q^{33} - 3 q^{35} - 6 q^{37} - 9 q^{39} - 18 q^{41} - 21 q^{43} + 3 q^{45} - 15 q^{47} - 3 q^{51} + 6 q^{53} + 9 q^{59} + 15 q^{61} - 3 q^{63} - 21 q^{65} - 6 q^{67} - 12 q^{69} + 9 q^{71} - 6 q^{73} - 6 q^{75} - 24 q^{77} - 27 q^{79} + 3 q^{81} - 15 q^{83} - 18 q^{85} + 3 q^{87} + 12 q^{89} + 9 q^{91} + 6 q^{93} + 6 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9 - 3 * q^11 - 9 * q^13 + 3 * q^15 - 3 * q^17 - 3 * q^21 - 12 * q^23 - 6 * q^25 + 3 * q^27 + 3 * q^29 + 6 * q^31 - 3 * q^33 - 3 * q^35 - 6 * q^37 - 9 * q^39 - 18 * q^41 - 21 * q^43 + 3 * q^45 - 15 * q^47 - 3 * q^51 + 6 * q^53 + 9 * q^59 + 15 * q^61 - 3 * q^63 - 21 * q^65 - 6 * q^67 - 12 * q^69 + 9 * q^71 - 6 * q^73 - 6 * q^75 - 24 * q^77 - 27 * q^79 + 3 * q^81 - 15 * q^83 - 18 * q^85 + 3 * q^87 + 12 * q^89 + 9 * q^91 + 6 * q^93 + 6 * q^97 - 3 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.879385	−0.393273	−0.196637	−	0.980476i	$-0.563002\pi$
$5$	−0.879385	−0.393273	−0.196637	+	0.980476i	$0.563002\pi$
$6$	0	0
$7$	−2.18479	−0.825774	−0.412887	−	0.910782i	$-0.635480\pi$
$7$	−2.18479	−0.825774	−0.412887	+	0.910782i	$0.635480\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.162504	−0.0489967	−0.0244984	−	0.999700i	$-0.507799\pi$
$11$	−0.162504	−0.0489967	−0.0244984	+	0.999700i	$0.507799\pi$
$12$	0	0
$13$	0.758770	0.210445	0.105223	−	0.994449i	$-0.466445\pi$
$13$	0.758770	0.210445	0.105223	+	0.994449i	$0.466445\pi$
$14$	0	0
$15$	−0.879385	−0.227056
$16$	0	0
$17$	3.10607	0.753332	0.376666	−	0.926349i	$-0.377070\pi$
$17$	3.10607	0.753332	0.376666	+	0.926349i	$0.377070\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−2.18479	−0.476761
$22$	0	0
$23$	1.29086	0.269163	0.134581	−	0.990903i	$-0.457031\pi$
$23$	1.29086	0.269163	0.134581	+	0.990903i	$0.457031\pi$
$24$	0	0
$25$	−4.22668	−0.845336
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.51754	−1.21028	−0.605138	−	0.796120i	$-0.706882\pi$
$29$	−6.51754	−1.21028	−0.605138	+	0.796120i	$0.706882\pi$
$30$	0	0
$31$	0.958111	0.172082	0.0860409	−	0.996292i	$-0.472578\pi$
$31$	0.958111	0.172082	0.0860409	+	0.996292i	$0.472578\pi$
$32$	0	0
$33$	−0.162504	−0.0282883
$34$	0	0
$35$	1.92127	0.324755
$36$	0	0
$37$	−1.16250	−0.191114	−0.0955572	−	0.995424i	$-0.530463\pi$
$37$	−1.16250	−0.191114	−0.0955572	+	0.995424i	$0.530463\pi$
$38$	0	0
$39$	0.758770	0.121501
$40$	0	0
$41$	−10.5963	−1.65486	−0.827429	−	0.561570i	$-0.810198\pi$
$41$	−10.5963	−1.65486	−0.827429	+	0.561570i	$0.810198\pi$
$42$	0	0
$43$	−0.177052	−0.0270001	−0.0135001	−	0.999909i	$-0.504297\pi$
$43$	−0.177052	−0.0270001	−0.0135001	+	0.999909i	$0.504297\pi$
$44$	0	0
$45$	−0.879385	−0.131091
$46$	0	0
$47$	−3.46791	−0.505847	−0.252923	−	0.967486i	$-0.581392\pi$
$47$	−3.46791	−0.505847	−0.252923	+	0.967486i	$0.581392\pi$
$48$	0	0
$49$	−2.22668	−0.318097
$50$	0	0
$51$	3.10607	0.434936
$52$	0	0
$53$	8.33275	1.14459	0.572296	−	0.820047i	$-0.306053\pi$
$53$	8.33275	1.14459	0.572296	+	0.820047i	$0.306053\pi$
$54$	0	0
$55$	0.142903	0.0192691
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.78106	−0.362063	−0.181032	−	0.983477i	$-0.557944\pi$
$59$	−2.78106	−0.362063	−0.181032	+	0.983477i	$0.557944\pi$
$60$	0	0
$61$	7.22668	0.925282	0.462641	−	0.886546i	$-0.346902\pi$
$61$	7.22668	0.925282	0.462641	+	0.886546i	$0.346902\pi$
$62$	0	0
$63$	−2.18479	−0.275258
$64$	0	0
$65$	−0.667252	−0.0827624
$66$	0	0
$67$	1.06418	0.130010	0.0650050	−	0.997885i	$-0.479294\pi$
$67$	1.06418	0.130010	0.0650050	+	0.997885i	$0.479294\pi$
$68$	0	0
$69$	1.29086	0.155401
$70$	0	0
$71$	5.08378	0.603333	0.301667	−	0.953413i	$-0.402457\pi$
$71$	5.08378	0.603333	0.301667	+	0.953413i	$0.402457\pi$
$72$	0	0
$73$	2.24897	0.263222	0.131611	−	0.991301i	$-0.457985\pi$
$73$	2.24897	0.263222	0.131611	+	0.991301i	$0.457985\pi$
$74$	0	0
$75$	−4.22668	−0.488055
$76$	0	0
$77$	0.355037	0.0404602
$78$	0	0
$79$	−5.24123	−0.589684	−0.294842	−	0.955546i	$-0.595267\pi$
$79$	−5.24123	−0.589684	−0.294842	+	0.955546i	$0.595267\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−17.1138	−1.87848	−0.939242	−	0.343255i	$-0.888470\pi$
$83$	−17.1138	−1.87848	−0.939242	+	0.343255i	$0.888470\pi$
$84$	0	0
$85$	−2.73143	−0.296265
$86$	0	0
$87$	−6.51754	−0.698754
$88$	0	0
$89$	15.6236	1.65610	0.828050	−	0.560655i	$-0.189451\pi$
$89$	15.6236	1.65610	0.828050	+	0.560655i	$0.189451\pi$
$90$	0	0
$91$	−1.65776	−0.173780
$92$	0	0
$93$	0.958111	0.0993515
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−9.41921	−0.956376	−0.478188	−	0.878257i	$-0.658706\pi$
$97$	−9.41921	−0.956376	−0.478188	+	0.878257i	$0.658706\pi$
$98$	0	0
$99$	−0.162504	−0.0163322
$100$	0	0
$101$	−6.30541	−0.627411	−0.313706	−	0.949520i	$-0.601571\pi$
$101$	−6.30541	−0.627411	−0.313706	+	0.949520i	$0.601571\pi$
$102$	0	0
$103$	−8.61587	−0.848947	−0.424473	−	0.905440i	$-0.639541\pi$
$103$	−8.61587	−0.848947	−0.424473	+	0.905440i	$0.639541\pi$
$104$	0	0
$105$	1.92127	0.187497
$106$	0	0
$107$	−13.1480	−1.27106	−0.635530	−	0.772076i	$-0.719219\pi$
$107$	−13.1480	−1.27106	−0.635530	+	0.772076i	$0.719219\pi$
$108$	0	0
$109$	−10.0496	−0.962580	−0.481290	−	0.876561i	$-0.659832\pi$
$109$	−10.0496	−0.962580	−0.481290	+	0.876561i	$0.659832\pi$
$110$	0	0
$111$	−1.16250	−0.110340
$112$	0	0
$113$	8.04458	0.756770	0.378385	−	0.925648i	$-0.376480\pi$
$113$	8.04458	0.756770	0.378385	+	0.925648i	$0.376480\pi$
$114$	0	0
$115$	−1.13516	−0.105854
$116$	0	0
$117$	0.758770	0.0701484
$118$	0	0
$119$	−6.78611	−0.622082
$120$	0	0
$121$	−10.9736	−0.997599
$122$	0	0
$123$	−10.5963	−0.955433
$124$	0	0
$125$	8.11381	0.725721
$126$	0	0
$127$	20.0496	1.77912	0.889558	−	0.456821i	$-0.151012\pi$
$127$	20.0496	1.77912	0.889558	+	0.456821i	$0.151012\pi$
$128$	0	0
$129$	−0.177052	−0.0155885
$130$	0	0
$131$	−15.9094	−1.39001	−0.695006	−	0.719004i	$-0.744598\pi$
$131$	−15.9094	−1.39001	−0.695006	+	0.719004i	$0.744598\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.879385	−0.0756854
$136$	0	0
$137$	−5.04458	−0.430987	−0.215494	−	0.976505i	$-0.569136\pi$
$137$	−5.04458	−0.430987	−0.215494	+	0.976505i	$0.569136\pi$
$138$	0	0
$139$	−10.5544	−0.895211	−0.447605	−	0.894231i	$-0.647723\pi$
$139$	−10.5544	−0.895211	−0.447605	+	0.894231i	$0.647723\pi$
$140$	0	0
$141$	−3.46791	−0.292051
$142$	0	0
$143$	−0.123303	−0.0103111
$144$	0	0
$145$	5.73143	0.475969
$146$	0	0
$147$	−2.22668	−0.183654
$148$	0	0
$149$	13.7956	1.13018	0.565090	−	0.825029i	$-0.308841\pi$
$149$	13.7956	1.13018	0.565090	+	0.825029i	$0.308841\pi$
$150$	0	0
$151$	−20.2841	−1.65069	−0.825346	−	0.564627i	$-0.809020\pi$
$151$	−20.2841	−1.65069	−0.825346	+	0.564627i	$0.809020\pi$
$152$	0	0
$153$	3.10607	0.251111
$154$	0	0
$155$	−0.842549	−0.0676751
$156$	0	0
$157$	−13.1753	−1.05150	−0.525752	−	0.850638i	$-0.676216\pi$
$157$	−13.1753	−1.05150	−0.525752	+	0.850638i	$0.676216\pi$
$158$	0	0
$159$	8.33275	0.660830
$160$	0	0
$161$	−2.82026	−0.222268
$162$	0	0
$163$	−17.4902	−1.36994	−0.684969	−	0.728572i	$-0.740184\pi$
$163$	−17.4902	−1.36994	−0.684969	+	0.728572i	$0.740184\pi$
$164$	0	0
$165$	0.142903	0.0111250
$166$	0	0
$167$	14.9067	1.15352	0.576759	−	0.816915i	$-0.304317\pi$
$167$	14.9067	1.15352	0.576759	+	0.816915i	$0.304317\pi$
$168$	0	0
$169$	−12.4243	−0.955713
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.79561	0.136517	0.0682587	−	0.997668i	$-0.478256\pi$
$173$	1.79561	0.136517	0.0682587	+	0.997668i	$0.478256\pi$
$174$	0	0
$175$	9.23442	0.698057
$176$	0	0
$177$	−2.78106	−0.209037
$178$	0	0
$179$	7.83481	0.585601	0.292801	−	0.956174i	$-0.405413\pi$
$179$	7.83481	0.585601	0.292801	+	0.956174i	$0.405413\pi$
$180$	0	0
$181$	−0.419215	−0.0311600	−0.0155800	−	0.999879i	$-0.504959\pi$
$181$	−0.419215	−0.0311600	−0.0155800	+	0.999879i	$0.504959\pi$
$182$	0	0
$183$	7.22668	0.534212
$184$	0	0
$185$	1.02229	0.0751602
$186$	0	0
$187$	−0.504748	−0.0369108
$188$	0	0
$189$	−2.18479	−0.158920
$190$	0	0
$191$	−19.5398	−1.41385	−0.706926	−	0.707287i	$-0.749919\pi$
$191$	−19.5398	−1.41385	−0.706926	+	0.707287i	$0.749919\pi$
$192$	0	0
$193$	−16.6408	−1.19783	−0.598917	−	0.800811i	$-0.704402\pi$
$193$	−16.6408	−1.19783	−0.598917	+	0.800811i	$0.704402\pi$
$194$	0	0
$195$	−0.667252	−0.0477829
$196$	0	0
$197$	−21.9959	−1.56714	−0.783571	−	0.621302i	$-0.786604\pi$
$197$	−21.9959	−1.56714	−0.783571	+	0.621302i	$0.786604\pi$
$198$	0	0
$199$	−21.7442	−1.54141	−0.770704	−	0.637194i	$-0.780095\pi$
$199$	−21.7442	−1.54141	−0.770704	+	0.637194i	$0.780095\pi$
$200$	0	0
$201$	1.06418	0.0750613
$202$	0	0
$203$	14.2395	0.999415
$204$	0	0
$205$	9.31820	0.650811
$206$	0	0
$207$	1.29086	0.0897209
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−8.62630	−0.593859	−0.296929	−	0.954899i	$-0.595963\pi$
$211$	−8.62630	−0.593859	−0.296929	+	0.954899i	$0.595963\pi$
$212$	0	0
$213$	5.08378	0.348335
$214$	0	0
$215$	0.155697	0.0106184
$216$	0	0
$217$	−2.09327	−0.142101
$218$	0	0
$219$	2.24897	0.151971
$220$	0	0
$221$	2.35679	0.158535
$222$	0	0
$223$	15.1584	1.01508	0.507540	−	0.861628i	$-0.330555\pi$
$223$	15.1584	1.01508	0.507540	+	0.861628i	$0.330555\pi$
$224$	0	0
$225$	−4.22668	−0.281779
$226$	0	0
$227$	−5.71419	−0.379264	−0.189632	−	0.981855i	$-0.560730\pi$
$227$	−5.71419	−0.379264	−0.189632	+	0.981855i	$0.560730\pi$
$228$	0	0
$229$	22.1634	1.46460	0.732301	−	0.680982i	$-0.238447\pi$
$229$	22.1634	1.46460	0.732301	+	0.680982i	$0.238447\pi$
$230$	0	0
$231$	0.355037	0.0233597
$232$	0	0
$233$	−10.8280	−0.709366	−0.354683	−	0.934987i	$-0.615411\pi$
$233$	−10.8280	−0.709366	−0.354683	+	0.934987i	$0.615411\pi$
$234$	0	0
$235$	3.04963	0.198936
$236$	0	0
$237$	−5.24123	−0.340454
$238$	0	0
$239$	−9.86215	−0.637929	−0.318965	−	0.947767i	$-0.603335\pi$
$239$	−9.86215	−0.637929	−0.318965	+	0.947767i	$0.603335\pi$
$240$	0	0
$241$	−6.90673	−0.444901	−0.222451	−	0.974944i	$-0.571406\pi$
$241$	−6.90673	−0.444901	−0.222451	+	0.974944i	$0.571406\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.95811	0.125099
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−17.1138	−1.08454
$250$	0	0
$251$	24.8922	1.57118	0.785590	−	0.618747i	$-0.212359\pi$
$251$	24.8922	1.57118	0.785590	+	0.618747i	$0.212359\pi$
$252$	0	0
$253$	−0.209770	−0.0131881
$254$	0	0
$255$	−2.73143	−0.171049
$256$	0	0
$257$	−14.2567	−0.889309	−0.444655	−	0.895702i	$-0.646674\pi$
$257$	−14.2567	−0.889309	−0.444655	+	0.895702i	$0.646674\pi$
$258$	0	0
$259$	2.53983	0.157817
$260$	0	0
$261$	−6.51754	−0.403426
$262$	0	0
$263$	21.6168	1.33295	0.666475	−	0.745528i	$-0.267803\pi$
$263$	21.6168	1.33295	0.666475	+	0.745528i	$0.267803\pi$
$264$	0	0
$265$	−7.32770	−0.450137
$266$	0	0
$267$	15.6236	0.956149
$268$	0	0
$269$	−10.6263	−0.647897	−0.323948	−	0.946075i	$-0.605010\pi$
$269$	−10.6263	−0.647897	−0.323948	+	0.946075i	$0.605010\pi$
$270$	0	0
$271$	−12.1506	−0.738099	−0.369050	−	0.929410i	$-0.620317\pi$
$271$	−12.1506	−0.738099	−0.369050	+	0.929410i	$0.620317\pi$
$272$	0	0
$273$	−1.65776	−0.100332
$274$	0	0
$275$	0.686852	0.0414187
$276$	0	0
$277$	9.81521	0.589739	0.294869	−	0.955538i	$-0.404724\pi$
$277$	9.81521	0.589739	0.294869	+	0.955538i	$0.404724\pi$
$278$	0	0
$279$	0.958111	0.0573606
$280$	0	0
$281$	28.2148	1.68316	0.841578	−	0.540136i	$-0.181627\pi$
$281$	28.2148	1.68316	0.841578	+	0.540136i	$0.181627\pi$
$282$	0	0
$283$	−15.4192	−0.916577	−0.458289	−	0.888803i	$-0.651537\pi$
$283$	−15.4192	−0.916577	−0.458289	+	0.888803i	$0.651537\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	23.1506	1.36654
$288$	0	0
$289$	−7.35235	−0.432491
$290$	0	0
$291$	−9.41921	−0.552164
$292$	0	0
$293$	27.7769	1.62275	0.811373	−	0.584529i	$-0.198721\pi$
$293$	27.7769	1.62275	0.811373	+	0.584529i	$0.198721\pi$
$294$	0	0
$295$	2.44562	0.142390
$296$	0	0
$297$	−0.162504	−0.00942943
$298$	0	0
$299$	0.979466	0.0566440
$300$	0	0
$301$	0.386821	0.0222960
$302$	0	0
$303$	−6.30541	−0.362236
$304$	0	0
$305$	−6.35504	−0.363888
$306$	0	0
$307$	27.1908	1.55186	0.775930	−	0.630819i	$-0.217281\pi$
$307$	27.1908	1.55186	0.775930	+	0.630819i	$0.217281\pi$
$308$	0	0
$309$	−8.61587	−0.490140
$310$	0	0
$311$	−5.94356	−0.337029	−0.168514	−	0.985699i	$-0.553897\pi$
$311$	−5.94356	−0.337029	−0.168514	+	0.985699i	$0.553897\pi$
$312$	0	0
$313$	7.15064	0.404178	0.202089	−	0.979367i	$-0.435227\pi$
$313$	7.15064	0.404178	0.202089	+	0.979367i	$0.435227\pi$
$314$	0	0
$315$	1.92127	0.108252
$316$	0	0
$317$	22.2909	1.25198	0.625990	−	0.779831i	$-0.284695\pi$
$317$	22.2909	1.25198	0.625990	+	0.779831i	$0.284695\pi$
$318$	0	0
$319$	1.05913	0.0592996
$320$	0	0
$321$	−13.1480	−0.733847
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−3.20708	−0.177897
$326$	0	0
$327$	−10.0496	−0.555746
$328$	0	0
$329$	7.57667	0.417715
$330$	0	0
$331$	6.30365	0.346480	0.173240	−	0.984880i	$-0.444576\pi$
$331$	6.30365	0.346480	0.173240	+	0.984880i	$0.444576\pi$
$332$	0	0
$333$	−1.16250	−0.0637048
$334$	0	0
$335$	−0.935822	−0.0511294
$336$	0	0
$337$	−25.8726	−1.40937	−0.704685	−	0.709521i	$-0.748911\pi$
$337$	−25.8726	−1.40937	−0.704685	+	0.709521i	$0.748911\pi$
$338$	0	0
$339$	8.04458	0.436921
$340$	0	0
$341$	−0.155697	−0.00843145
$342$	0	0
$343$	20.1584	1.08845
$344$	0	0
$345$	−1.13516	−0.0611151
$346$	0	0
$347$	−18.6040	−0.998715	−0.499358	−	0.866396i	$-0.666431\pi$
$347$	−18.6040	−0.998715	−0.499358	+	0.866396i	$0.666431\pi$
$348$	0	0
$349$	−1.74329	−0.0933161	−0.0466581	−	0.998911i	$-0.514857\pi$
$349$	−1.74329	−0.0933161	−0.0466581	+	0.998911i	$0.514857\pi$
$350$	0	0
$351$	0.758770	0.0405002
$352$	0	0
$353$	−10.6723	−0.568029	−0.284015	−	0.958820i	$-0.591666\pi$
$353$	−10.6723	−0.568029	−0.284015	+	0.958820i	$0.591666\pi$
$354$	0	0
$355$	−4.47060	−0.237275
$356$	0	0
$357$	−6.78611	−0.359159
$358$	0	0
$359$	−6.81252	−0.359551	−0.179776	−	0.983708i	$-0.557537\pi$
$359$	−6.81252	−0.359551	−0.179776	+	0.983708i	$0.557537\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−10.9736	−0.575964
$364$	0	0
$365$	−1.97771	−0.103518
$366$	0	0
$367$	−17.9905	−0.939097	−0.469548	−	0.882907i	$-0.655583\pi$
$367$	−17.9905	−0.939097	−0.469548	+	0.882907i	$0.655583\pi$
$368$	0	0
$369$	−10.5963	−0.551620
$370$	0	0
$371$	−18.2053	−0.945173
$372$	0	0
$373$	3.36009	0.173979	0.0869894	−	0.996209i	$-0.472275\pi$
$373$	3.36009	0.173979	0.0869894	+	0.996209i	$0.472275\pi$
$374$	0	0
$375$	8.11381	0.418995
$376$	0	0
$377$	−4.94532	−0.254697
$378$	0	0
$379$	15.1584	0.778634	0.389317	−	0.921104i	$-0.372711\pi$
$379$	15.1584	0.778634	0.389317	+	0.921104i	$0.372711\pi$
$380$	0	0
$381$	20.0496	1.02717
$382$	0	0
$383$	7.13516	0.364590	0.182295	−	0.983244i	$-0.441647\pi$
$383$	7.13516	0.364590	0.182295	+	0.983244i	$0.441647\pi$
$384$	0	0
$385$	−0.312214	−0.0159119
$386$	0	0
$387$	−0.177052	−0.00900005
$388$	0	0
$389$	24.9358	1.26430	0.632148	−	0.774848i	$-0.282173\pi$
$389$	24.9358	1.26430	0.632148	+	0.774848i	$0.282173\pi$
$390$	0	0
$391$	4.00950	0.202769
$392$	0	0
$393$	−15.9094	−0.802524
$394$	0	0
$395$	4.60906	0.231907
$396$	0	0
$397$	20.6536	1.03658	0.518288	−	0.855206i	$-0.326569\pi$
$397$	20.6536	1.03658	0.518288	+	0.855206i	$0.326569\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	36.3218	1.81383	0.906913	−	0.421319i	$-0.138433\pi$
$401$	36.3218	1.81383	0.906913	+	0.421319i	$0.138433\pi$
$402$	0	0
$403$	0.726986	0.0362138
$404$	0	0
$405$	−0.879385	−0.0436970
$406$	0	0
$407$	0.188911	0.00936399
$408$	0	0
$409$	4.95636	0.245076	0.122538	−	0.992464i	$-0.460897\pi$
$409$	4.95636	0.245076	0.122538	+	0.992464i	$0.460897\pi$
$410$	0	0
$411$	−5.04458	−0.248831
$412$	0	0
$413$	6.07604	0.298982
$414$	0	0
$415$	15.0496	0.738757
$416$	0	0
$417$	−10.5544	−0.516850
$418$	0	0
$419$	−35.0966	−1.71458	−0.857290	−	0.514834i	$-0.827854\pi$
$419$	−35.0966	−1.71458	−0.857290	+	0.514834i	$0.827854\pi$
$420$	0	0
$421$	−18.4020	−0.896858	−0.448429	−	0.893819i	$-0.648016\pi$
$421$	−18.4020	−0.896858	−0.448429	+	0.893819i	$0.648016\pi$
$422$	0	0
$423$	−3.46791	−0.168616
$424$	0	0
$425$	−13.1284	−0.636819
$426$	0	0
$427$	−15.7888	−0.764074
$428$	0	0
$429$	−0.123303	−0.00595313
$430$	0	0
$431$	8.79385	0.423585	0.211792	−	0.977315i	$-0.432070\pi$
$431$	8.79385	0.423585	0.211792	+	0.977315i	$0.432070\pi$
$432$	0	0
$433$	17.4115	0.836742	0.418371	−	0.908276i	$-0.362601\pi$
$433$	17.4115	0.836742	0.418371	+	0.908276i	$0.362601\pi$
$434$	0	0
$435$	5.73143	0.274801
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−28.4911	−1.35981	−0.679904	−	0.733301i	$-0.737978\pi$
$439$	−28.4911	−1.35981	−0.679904	+	0.733301i	$0.737978\pi$
$440$	0	0
$441$	−2.22668	−0.106032
$442$	0	0
$443$	18.0428	0.857240	0.428620	−	0.903485i	$-0.359000\pi$
$443$	18.0428	0.857240	0.428620	+	0.903485i	$0.359000\pi$
$444$	0	0
$445$	−13.7392	−0.651299
$446$	0	0
$447$	13.7956	0.652510
$448$	0	0
$449$	9.53478	0.449974	0.224987	−	0.974362i	$-0.427766\pi$
$449$	9.53478	0.449974	0.224987	+	0.974362i	$0.427766\pi$
$450$	0	0
$451$	1.72193	0.0810827
$452$	0	0
$453$	−20.2841	−0.953028
$454$	0	0
$455$	1.45781	0.0683430
$456$	0	0
$457$	10.4584	0.489224	0.244612	−	0.969621i	$-0.421339\pi$
$457$	10.4584	0.489224	0.244612	+	0.969621i	$0.421339\pi$
$458$	0	0
$459$	3.10607	0.144979
$460$	0	0
$461$	41.6614	1.94036	0.970182	−	0.242378i	$-0.0779274\pi$
$461$	41.6614	1.94036	0.970182	+	0.242378i	$0.0779274\pi$
$462$	0	0
$463$	13.8179	0.642172	0.321086	−	0.947050i	$-0.395952\pi$
$463$	13.8179	0.642172	0.321086	+	0.947050i	$0.395952\pi$
$464$	0	0
$465$	−0.842549	−0.0390723
$466$	0	0
$467$	−7.45336	−0.344901	−0.172450	−	0.985018i	$-0.555168\pi$
$467$	−7.45336	−0.344901	−0.172450	+	0.985018i	$0.555168\pi$
$468$	0	0
$469$	−2.32501	−0.107359
$470$	0	0
$471$	−13.1753	−0.607086
$472$	0	0
$473$	0.0287716	0.00132292
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.33275	0.381530
$478$	0	0
$479$	23.7870	1.08686	0.543429	−	0.839455i	$-0.317126\pi$
$479$	23.7870	1.08686	0.543429	+	0.839455i	$0.317126\pi$
$480$	0	0
$481$	−0.882074	−0.0402191
$482$	0	0
$483$	−2.82026	−0.128326
$484$	0	0
$485$	8.28312	0.376117
$486$	0	0
$487$	4.85710	0.220096	0.110048	−	0.993926i	$-0.464900\pi$
$487$	4.85710	0.220096	0.110048	+	0.993926i	$0.464900\pi$
$488$	0	0
$489$	−17.4902	−0.790934
$490$	0	0
$491$	−2.99764	−0.135281	−0.0676407	−	0.997710i	$-0.521547\pi$
$491$	−2.99764	−0.135281	−0.0676407	+	0.997710i	$0.521547\pi$
$492$	0	0
$493$	−20.2439	−0.911740
$494$	0	0
$495$	0.142903	0.00642303
$496$	0	0
$497$	−11.1070	−0.498217
$498$	0	0
$499$	2.52435	0.113005	0.0565027	−	0.998402i	$-0.482005\pi$
$499$	2.52435	0.113005	0.0565027	+	0.998402i	$0.482005\pi$
$500$	0	0
$501$	14.9067	0.665983
$502$	0	0
$503$	18.3482	0.818107	0.409054	−	0.912510i	$-0.365859\pi$
$503$	18.3482	0.818107	0.409054	+	0.912510i	$0.365859\pi$
$504$	0	0
$505$	5.54488	0.246744
$506$	0	0
$507$	−12.4243	−0.551781
$508$	0	0
$509$	16.2736	0.721316	0.360658	−	0.932698i	$-0.382552\pi$
$509$	16.2736	0.721316	0.360658	+	0.932698i	$0.382552\pi$
$510$	0	0
$511$	−4.91353	−0.217362
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7.57667	0.333868
$516$	0	0
$517$	0.563549	0.0247848
$518$	0	0
$519$	1.79561	0.0788184
$520$	0	0
$521$	28.2909	1.23945	0.619723	−	0.784821i	$-0.287245\pi$
$521$	28.2909	1.23945	0.619723	+	0.784821i	$0.287245\pi$
$522$	0	0
$523$	4.84793	0.211985	0.105992	−	0.994367i	$-0.466198\pi$
$523$	4.84793	0.211985	0.105992	+	0.994367i	$0.466198\pi$
$524$	0	0
$525$	9.23442	0.403023
$526$	0	0
$527$	2.97596	0.129635
$528$	0	0
$529$	−21.3337	−0.927551
$530$	0	0
$531$	−2.78106	−0.120688
$532$	0	0
$533$	−8.04013	−0.348257
$534$	0	0
$535$	11.5621	0.499874
$536$	0	0
$537$	7.83481	0.338097
$538$	0	0
$539$	0.361844	0.0155857
$540$	0	0
$541$	−24.3131	−1.04530	−0.522652	−	0.852546i	$-0.675057\pi$
$541$	−24.3131	−1.04530	−0.522652	+	0.852546i	$0.675057\pi$
$542$	0	0
$543$	−0.419215	−0.0179902
$544$	0	0
$545$	8.83750	0.378557
$546$	0	0
$547$	−22.7733	−0.973717	−0.486858	−	0.873481i	$-0.661857\pi$
$547$	−22.7733	−0.973717	−0.486858	+	0.873481i	$0.661857\pi$
$548$	0	0
$549$	7.22668	0.308427
$550$	0	0
$551$	0	0
$552$	0	0
$553$	11.4510	0.486946
$554$	0	0
$555$	1.02229	0.0433937
$556$	0	0
$557$	19.1343	0.810748	0.405374	−	0.914151i	$-0.367141\pi$
$557$	19.1343	0.810748	0.405374	+	0.914151i	$0.367141\pi$
$558$	0	0
$559$	−0.134342	−0.00568205
$560$	0	0
$561$	−0.504748	−0.0213105
$562$	0	0
$563$	−21.3482	−0.899721	−0.449860	−	0.893099i	$-0.648526\pi$
$563$	−21.3482	−0.899721	−0.449860	+	0.893099i	$0.648526\pi$
$564$	0	0
$565$	−7.07428	−0.297617
$566$	0	0
$567$	−2.18479	−0.0917527
$568$	0	0
$569$	39.2121	1.64386	0.821929	−	0.569590i	$-0.192898\pi$
$569$	39.2121	1.64386	0.821929	+	0.569590i	$0.192898\pi$
$570$	0	0
$571$	−7.27900	−0.304617	−0.152308	−	0.988333i	$-0.548671\pi$
$571$	−7.27900	−0.304617	−0.152308	+	0.988333i	$0.548671\pi$
$572$	0	0
$573$	−19.5398	−0.816288
$574$	0	0
$575$	−5.45605	−0.227533
$576$	0	0
$577$	14.9923	0.624136	0.312068	−	0.950060i	$-0.398978\pi$
$577$	14.9923	0.624136	0.312068	+	0.950060i	$0.398978\pi$
$578$	0	0
$579$	−16.6408	−0.691570
$580$	0	0
$581$	37.3901	1.55120
$582$	0	0
$583$	−1.35410	−0.0560812
$584$	0	0
$585$	−0.667252	−0.0275875
$586$	0	0
$587$	−12.0104	−0.495723	−0.247862	−	0.968795i	$-0.579728\pi$
$587$	−12.0104	−0.495723	−0.247862	+	0.968795i	$0.579728\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−21.9959	−0.904790
$592$	0	0
$593$	−36.3806	−1.49397	−0.746987	−	0.664839i	$-0.768500\pi$
$593$	−36.3806	−1.49397	−0.746987	+	0.664839i	$0.768500\pi$
$594$	0	0
$595$	5.96761	0.244648
$596$	0	0
$597$	−21.7442	−0.889932
$598$	0	0
$599$	13.8060	0.564099	0.282050	−	0.959400i	$-0.408986\pi$
$599$	13.8060	0.564099	0.282050	+	0.959400i	$0.408986\pi$
$600$	0	0
$601$	−0.967606	−0.0394695	−0.0197347	−	0.999805i	$-0.506282\pi$
$601$	−0.967606	−0.0394695	−0.0197347	+	0.999805i	$0.506282\pi$
$602$	0	0
$603$	1.06418	0.0433367
$604$	0	0
$605$	9.65002	0.392329
$606$	0	0
$607$	1.53890	0.0624619	0.0312309	−	0.999512i	$-0.490057\pi$
$607$	1.53890	0.0624619	0.0312309	+	0.999512i	$0.490057\pi$
$608$	0	0
$609$	14.2395	0.577013
$610$	0	0
$611$	−2.63135	−0.106453
$612$	0	0
$613$	41.1753	1.66305	0.831527	−	0.555484i	$-0.187467\pi$
$613$	41.1753	1.66305	0.831527	+	0.555484i	$0.187467\pi$
$614$	0	0
$615$	9.31820	0.375746
$616$	0	0
$617$	30.6860	1.23537	0.617687	−	0.786424i	$-0.288070\pi$
$617$	30.6860	1.23537	0.617687	+	0.786424i	$0.288070\pi$
$618$	0	0
$619$	−11.9581	−0.480637	−0.240319	−	0.970694i	$-0.577252\pi$
$619$	−11.9581	−0.480637	−0.240319	+	0.970694i	$0.577252\pi$
$620$	0	0
$621$	1.29086	0.0518004
$622$	0	0
$623$	−34.1343	−1.36756
$624$	0	0
$625$	13.9982	0.559930
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.61081	−0.143973
$630$	0	0
$631$	−48.1762	−1.91787	−0.958933	−	0.283634i	$-0.908460\pi$
$631$	−48.1762	−1.91787	−0.958933	+	0.283634i	$0.908460\pi$
$632$	0	0
$633$	−8.62630	−0.342864
$634$	0	0
$635$	−17.6313	−0.699679
$636$	0	0
$637$	−1.68954	−0.0669420
$638$	0	0
$639$	5.08378	0.201111
$640$	0	0
$641$	6.72967	0.265806	0.132903	−	0.991129i	$-0.457570\pi$
$641$	6.72967	0.265806	0.132903	+	0.991129i	$0.457570\pi$
$642$	0	0
$643$	−25.6928	−1.01323	−0.506613	−	0.862173i	$-0.669103\pi$
$643$	−25.6928	−1.01323	−0.506613	+	0.862173i	$0.669103\pi$
$644$	0	0
$645$	0.155697	0.00613055
$646$	0	0
$647$	−50.4380	−1.98292	−0.991461	−	0.130403i	$-0.958373\pi$
$647$	−50.4380	−1.98292	−0.991461	+	0.130403i	$0.958373\pi$
$648$	0	0
$649$	0.451933	0.0177399
$650$	0	0
$651$	−2.09327	−0.0820419
$652$	0	0
$653$	−35.8607	−1.40334	−0.701669	−	0.712503i	$-0.747562\pi$
$653$	−35.8607	−1.40334	−0.701669	+	0.712503i	$0.747562\pi$
$654$	0	0
$655$	13.9905	0.546654
$656$	0	0
$657$	2.24897	0.0877407
$658$	0	0
$659$	−12.6108	−0.491248	−0.245624	−	0.969365i	$-0.578993\pi$
$659$	−12.6108	−0.491248	−0.245624	+	0.969365i	$0.578993\pi$
$660$	0	0
$661$	−44.5699	−1.73357	−0.866783	−	0.498685i	$-0.833816\pi$
$661$	−44.5699	−1.73357	−0.866783	+	0.498685i	$0.833816\pi$
$662$	0	0
$663$	2.35679	0.0915302
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.41323	−0.325762
$668$	0	0
$669$	15.1584	0.586057
$670$	0	0
$671$	−1.17436	−0.0453358
$672$	0	0
$673$	−12.8203	−0.494185	−0.247092	−	0.968992i	$-0.579475\pi$
$673$	−12.8203	−0.494185	−0.247092	+	0.968992i	$0.579475\pi$
$674$	0	0
$675$	−4.22668	−0.162685
$676$	0	0
$677$	−29.3688	−1.12873	−0.564367	−	0.825524i	$-0.690880\pi$
$677$	−29.3688	−1.12873	−0.564367	+	0.825524i	$0.690880\pi$
$678$	0	0
$679$	20.5790	0.789751
$680$	0	0
$681$	−5.71419	−0.218968
$682$	0	0
$683$	38.0033	1.45416	0.727078	−	0.686555i	$-0.240878\pi$
$683$	38.0033	1.45416	0.727078	+	0.686555i	$0.240878\pi$
$684$	0	0
$685$	4.43613	0.169496
$686$	0	0
$687$	22.1634	0.845588
$688$	0	0
$689$	6.32264	0.240874
$690$	0	0
$691$	32.2918	1.22844	0.614219	−	0.789136i	$-0.289471\pi$
$691$	32.2918	1.22844	0.614219	+	0.789136i	$0.289471\pi$
$692$	0	0
$693$	0.355037	0.0134867
$694$	0	0
$695$	9.28136	0.352062
$696$	0	0
$697$	−32.9127	−1.24666
$698$	0	0
$699$	−10.8280	−0.409553
$700$	0	0
$701$	23.1138	0.872996	0.436498	−	0.899705i	$-0.356219\pi$
$701$	23.1138	0.872996	0.436498	+	0.899705i	$0.356219\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	3.04963	0.114856
$706$	0	0
$707$	13.7760	0.518100
$708$	0	0
$709$	36.7588	1.38050	0.690252	−	0.723569i	$-0.257500\pi$
$709$	36.7588	1.38050	0.690252	+	0.723569i	$0.257500\pi$
$710$	0	0
$711$	−5.24123	−0.196561
$712$	0	0
$713$	1.23679	0.0463180
$714$	0	0
$715$	0.108431	0.00405509
$716$	0	0
$717$	−9.86215	−0.368309
$718$	0	0
$719$	32.9504	1.22884	0.614421	−	0.788979i	$-0.289390\pi$
$719$	32.9504	1.22884	0.614421	+	0.788979i	$0.289390\pi$
$720$	0	0
$721$	18.8239	0.701038
$722$	0	0
$723$	−6.90673	−0.256864
$724$	0	0
$725$	27.5476	1.02309
$726$	0	0
$727$	−7.92303	−0.293849	−0.146924	−	0.989148i	$-0.546937\pi$
$727$	−7.92303	−0.293849	−0.146924	+	0.989148i	$0.546937\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.549935	−0.0203401
$732$	0	0
$733$	−7.43470	−0.274607	−0.137303	−	0.990529i	$-0.543844\pi$
$733$	−7.43470	−0.274607	−0.137303	+	0.990529i	$0.543844\pi$
$734$	0	0
$735$	1.95811	0.0722260
$736$	0	0
$737$	−0.172933	−0.00637007
$738$	0	0
$739$	19.3500	0.711801	0.355900	−	0.934524i	$-0.384174\pi$
$739$	19.3500	0.711801	0.355900	+	0.934524i	$0.384174\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.3833	1.15134	0.575671	−	0.817681i	$-0.304741\pi$
$743$	31.3833	1.15134	0.575671	+	0.817681i	$0.304741\pi$
$744$	0	0
$745$	−12.1317	−0.444469
$746$	0	0
$747$	−17.1138	−0.626161
$748$	0	0
$749$	28.7256	1.04961
$750$	0	0
$751$	5.22256	0.190574	0.0952870	−	0.995450i	$-0.469623\pi$
$751$	5.22256	0.190574	0.0952870	+	0.995450i	$0.469623\pi$
$752$	0	0
$753$	24.8922	0.907121
$754$	0	0
$755$	17.8375	0.649173
$756$	0	0
$757$	−29.7716	−1.08207	−0.541033	−	0.841001i	$-0.681967\pi$
$757$	−29.7716	−1.08207	−0.541033	+	0.841001i	$0.681967\pi$
$758$	0	0
$759$	−0.209770	−0.00761415
$760$	0	0
$761$	−9.35267	−0.339034	−0.169517	−	0.985527i	$-0.554221\pi$
$761$	−9.35267	−0.339034	−0.169517	+	0.985527i	$0.554221\pi$
$762$	0	0
$763$	21.9564	0.794873
$764$	0	0
$765$	−2.73143	−0.0987550
$766$	0	0
$767$	−2.11019	−0.0761944
$768$	0	0
$769$	38.8334	1.40037	0.700184	−	0.713963i	$-0.253101\pi$
$769$	38.8334	1.40037	0.700184	+	0.713963i	$0.253101\pi$
$770$	0	0
$771$	−14.2567	−0.513443
$772$	0	0
$773$	−23.7674	−0.854856	−0.427428	−	0.904049i	$-0.640580\pi$
$773$	−23.7674	−0.854856	−0.427428	+	0.904049i	$0.640580\pi$
$774$	0	0
$775$	−4.04963	−0.145467
$776$	0	0
$777$	2.53983	0.0911159
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−0.826133	−0.0295614
$782$	0	0
$783$	−6.51754	−0.232918
$784$	0	0
$785$	11.5862	0.413528
$786$	0	0
$787$	−13.7561	−0.490351	−0.245176	−	0.969479i	$-0.578846\pi$
$787$	−13.7561	−0.490351	−0.245176	+	0.969479i	$0.578846\pi$
$788$	0	0
$789$	21.6168	0.769578
$790$	0	0
$791$	−17.5757	−0.624921
$792$	0	0
$793$	5.48339	0.194721
$794$	0	0
$795$	−7.32770	−0.259887
$796$	0	0
$797$	−7.77238	−0.275312	−0.137656	−	0.990480i	$-0.543957\pi$
$797$	−7.77238	−0.275312	−0.137656	+	0.990480i	$0.543957\pi$
$798$	0	0
$799$	−10.7716	−0.381071
$800$	0	0
$801$	15.6236	0.552033
$802$	0	0
$803$	−0.365466	−0.0128970
$804$	0	0
$805$	2.48009	0.0874119
$806$	0	0
$807$	−10.6263	−0.374063
$808$	0	0
$809$	13.6919	0.481382	0.240691	−	0.970602i	$-0.422626\pi$
$809$	13.6919	0.481382	0.240691	+	0.970602i	$0.422626\pi$
$810$	0	0
$811$	−22.1084	−0.776332	−0.388166	−	0.921589i	$-0.626891\pi$
$811$	−22.1084	−0.776332	−0.388166	+	0.921589i	$0.626891\pi$
$812$	0	0
$813$	−12.1506	−0.426142
$814$	0	0
$815$	15.3806	0.538760
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−1.65776	−0.0579267
$820$	0	0
$821$	−21.5776	−0.753063	−0.376532	−	0.926404i	$-0.622883\pi$
$821$	−21.5776	−0.753063	−0.376532	+	0.926404i	$0.622883\pi$
$822$	0	0
$823$	−9.35235	−0.326002	−0.163001	−	0.986626i	$-0.552117\pi$
$823$	−9.35235	−0.326002	−0.163001	+	0.986626i	$0.552117\pi$
$824$	0	0
$825$	0.686852	0.0239131
$826$	0	0
$827$	−33.5800	−1.16769	−0.583845	−	0.811865i	$-0.698452\pi$
$827$	−33.5800	−1.16769	−0.583845	+	0.811865i	$0.698452\pi$
$828$	0	0
$829$	0.650340	0.0225872	0.0112936	−	0.999936i	$-0.496405\pi$
$829$	0.650340	0.0225872	0.0112936	+	0.999936i	$0.496405\pi$
$830$	0	0
$831$	9.81521	0.340486
$832$	0	0
$833$	−6.91622	−0.239633
$834$	0	0
$835$	−13.1088	−0.453647
$836$	0	0
$837$	0.958111	0.0331172
$838$	0	0
$839$	−22.9077	−0.790860	−0.395430	−	0.918496i	$-0.629404\pi$
$839$	−22.9077	−0.790860	−0.395430	+	0.918496i	$0.629404\pi$
$840$	0	0
$841$	13.4783	0.464770
$842$	0	0
$843$	28.2148	0.971770
$844$	0	0
$845$	10.9257	0.375856
$846$	0	0
$847$	23.9750	0.823792
$848$	0	0
$849$	−15.4192	−0.529186
$850$	0	0
$851$	−1.50063	−0.0514409
$852$	0	0
$853$	30.5158	1.04484	0.522420	−	0.852688i	$-0.325029\pi$
$853$	30.5158	1.04484	0.522420	+	0.852688i	$0.325029\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.0473	0.616483	0.308241	−	0.951308i	$-0.400260\pi$
$857$	18.0473	0.616483	0.308241	+	0.951308i	$0.400260\pi$
$858$	0	0
$859$	36.8857	1.25852	0.629262	−	0.777193i	$-0.283357\pi$
$859$	36.8857	1.25852	0.629262	+	0.777193i	$0.283357\pi$
$860$	0	0
$861$	23.1506	0.788972
$862$	0	0
$863$	−28.1712	−0.958958	−0.479479	−	0.877553i	$-0.659174\pi$
$863$	−28.1712	−0.958958	−0.479479	+	0.877553i	$0.659174\pi$
$864$	0	0
$865$	−1.57903	−0.0536886
$866$	0	0
$867$	−7.35235	−0.249699
$868$	0	0
$869$	0.851720	0.0288926
$870$	0	0
$871$	0.807467	0.0273600
$872$	0	0
$873$	−9.41921	−0.318792
$874$	0	0
$875$	−17.7270	−0.599282
$876$	0	0
$877$	12.7115	0.429237	0.214619	−	0.976698i	$-0.431149\pi$
$877$	12.7115	0.429237	0.214619	+	0.976698i	$0.431149\pi$
$878$	0	0
$879$	27.7769	0.936893
$880$	0	0
$881$	36.8530	1.24161	0.620804	−	0.783966i	$-0.286806\pi$
$881$	36.8530	1.24161	0.620804	+	0.783966i	$0.286806\pi$
$882$	0	0
$883$	−39.6245	−1.33347	−0.666736	−	0.745294i	$-0.732309\pi$
$883$	−39.6245	−1.33347	−0.666736	+	0.745294i	$0.732309\pi$
$884$	0	0
$885$	2.44562	0.0822087
$886$	0	0
$887$	−47.5057	−1.59508	−0.797542	−	0.603263i	$-0.793867\pi$
$887$	−47.5057	−1.59508	−0.797542	+	0.603263i	$0.793867\pi$
$888$	0	0
$889$	−43.8043	−1.46915
$890$	0	0
$891$	−0.162504	−0.00544408
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−6.88981	−0.230301
$896$	0	0
$897$	0.979466	0.0327034
$898$	0	0
$899$	−6.24453	−0.208267
$900$	0	0
$901$	25.8821	0.862257
$902$	0	0
$903$	0.386821	0.0128726
$904$	0	0
$905$	0.368651	0.0122544
$906$	0	0
$907$	23.0060	0.763901	0.381951	−	0.924183i	$-0.375252\pi$
$907$	23.0060	0.763901	0.381951	+	0.924183i	$0.375252\pi$
$908$	0	0
$909$	−6.30541	−0.209137
$910$	0	0
$911$	13.7888	0.456843	0.228422	−	0.973562i	$-0.426644\pi$
$911$	13.7888	0.456843	0.228422	+	0.973562i	$0.426644\pi$
$912$	0	0
$913$	2.78106	0.0920396
$914$	0	0
$915$	−6.35504	−0.210091
$916$	0	0
$917$	34.7588	1.14784
$918$	0	0
$919$	54.6623	1.80314	0.901572	−	0.432630i	$-0.142414\pi$
$919$	54.6623	1.80314	0.901572	+	0.432630i	$0.142414\pi$
$920$	0	0
$921$	27.1908	0.895967
$922$	0	0
$923$	3.85742	0.126969
$924$	0	0
$925$	4.91353	0.161556
$926$	0	0
$927$	−8.61587	−0.282982
$928$	0	0
$929$	46.8084	1.53573	0.767867	−	0.640609i	$-0.221318\pi$
$929$	46.8084	1.53573	0.767867	+	0.640609i	$0.221318\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−5.94356	−0.194584
$934$	0	0
$935$	0.443868	0.0145160
$936$	0	0
$937$	−38.3209	−1.25189	−0.625944	−	0.779868i	$-0.715286\pi$
$937$	−38.3209	−1.25189	−0.625944	+	0.779868i	$0.715286\pi$
$938$	0	0
$939$	7.15064	0.233352
$940$	0	0
$941$	−29.4584	−0.960317	−0.480158	−	0.877182i	$-0.659421\pi$
$941$	−29.4584	−0.960317	−0.480158	+	0.877182i	$0.659421\pi$
$942$	0	0
$943$	−13.6783	−0.445426
$944$	0	0
$945$	1.92127	0.0624991
$946$	0	0
$947$	−37.9273	−1.23247	−0.616235	−	0.787562i	$-0.711343\pi$
$947$	−37.9273	−1.23247	−0.616235	+	0.787562i	$0.711343\pi$
$948$	0	0
$949$	1.70645	0.0553938
$950$	0	0
$951$	22.2909	0.722831
$952$	0	0
$953$	−8.23267	−0.266682	−0.133341	−	0.991070i	$-0.542571\pi$
$953$	−8.23267	−0.266682	−0.133341	+	0.991070i	$0.542571\pi$
$954$	0	0
$955$	17.1830	0.556030
$956$	0	0
$957$	1.05913	0.0342367
$958$	0	0
$959$	11.0214	0.355898
$960$	0	0
$961$	−30.0820	−0.970388
$962$	0	0
$963$	−13.1480	−0.423687
$964$	0	0
$965$	14.6337	0.471076
$966$	0	0
$967$	−16.2294	−0.521901	−0.260951	−	0.965352i	$-0.584036\pi$
$967$	−16.2294	−0.521901	−0.260951	+	0.965352i	$0.584036\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	33.1962	1.06532	0.532658	−	0.846331i	$-0.321193\pi$
$971$	33.1962	1.06532	0.532658	+	0.846331i	$0.321193\pi$
$972$	0	0
$973$	23.0591	0.739242
$974$	0	0
$975$	−3.20708	−0.102709
$976$	0	0
$977$	23.9932	0.767610	0.383805	−	0.923414i	$-0.374613\pi$
$977$	23.9932	0.767610	0.383805	+	0.923414i	$0.374613\pi$
$978$	0	0
$979$	−2.53890	−0.0811435
$980$	0	0
$981$	−10.0496	−0.320860
$982$	0	0
$983$	46.4962	1.48300	0.741499	−	0.670954i	$-0.234115\pi$
$983$	46.4962	1.48300	0.741499	+	0.670954i	$0.234115\pi$
$984$	0	0
$985$	19.3429	0.616315
$986$	0	0
$987$	7.57667	0.241168
$988$	0	0
$989$	−0.228549	−0.00726743
$990$	0	0
$991$	35.2513	1.11980	0.559898	−	0.828562i	$-0.310840\pi$
$991$	35.2513	1.11980	0.559898	+	0.828562i	$0.310840\pi$
$992$	0	0
$993$	6.30365	0.200040
$994$	0	0
$995$	19.1215	0.606194
$996$	0	0
$997$	55.5877	1.76048	0.880240	−	0.474529i	$-0.157381\pi$
$997$	55.5877	1.76048	0.880240	+	0.474529i	$0.157381\pi$
$998$	0	0
$999$	−1.16250	−0.0367800

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4332.2.a.o.1.1		3
19.6	even	9		228.2.q.a.169.1	yes	6
19.16	even	9		228.2.q.a.85.1	✓	6
19.18	odd	2		4332.2.a.n.1.1		3
57.35	odd	18		684.2.bo.a.541.1		6
57.44	odd	18		684.2.bo.a.397.1		6
76.35	odd	18		912.2.bo.e.769.1		6
76.63	odd	18		912.2.bo.e.625.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.2.q.a.85.1	✓	6	19.16	even	9
228.2.q.a.169.1	yes	6	19.6	even	9
684.2.bo.a.397.1		6	57.44	odd	18
684.2.bo.a.541.1		6	57.35	odd	18
912.2.bo.e.625.1		6	76.63	odd	18
912.2.bo.e.769.1		6	76.35	odd	18
4332.2.a.n.1.1		3	19.18	odd	2
4332.2.a.o.1.1		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4332 = 2^{2} \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4332.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.879385 q^{5} -2.18479 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.879385 q^{5} -2.18479 q^{7} +1.00000 q^{9} -0.162504 q^{11} +0.758770 q^{13} -0.879385 q^{15} +3.10607 q^{17} -2.18479 q^{21} +1.29086 q^{23} -4.22668 q^{25} +1.00000 q^{27} -6.51754 q^{29} +0.958111 q^{31} -0.162504 q^{33} +1.92127 q^{35} -1.16250 q^{37} +0.758770 q^{39} -10.5963 q^{41} -0.177052 q^{43} -0.879385 q^{45} -3.46791 q^{47} -2.22668 q^{49} +3.10607 q^{51} +8.33275 q^{53} +0.142903 q^{55} -2.78106 q^{59} +7.22668 q^{61} -2.18479 q^{63} -0.667252 q^{65} +1.06418 q^{67} +1.29086 q^{69} +5.08378 q^{71} +2.24897 q^{73} -4.22668 q^{75} +0.355037 q^{77} -5.24123 q^{79} +1.00000 q^{81} -17.1138 q^{83} -2.73143 q^{85} -6.51754 q^{87} +15.6236 q^{89} -1.65776 q^{91} +0.958111 q^{93} -9.41921 q^{97} -0.162504 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} - 9 q^{13} + 3 q^{15} - 3 q^{17} - 3 q^{21} - 12 q^{23} - 6 q^{25} + 3 q^{27} + 3 q^{29} + 6 q^{31} - 3 q^{33} - 3 q^{35} - 6 q^{37} - 9 q^{39} - 18 q^{41} - 21 q^{43} + 3 q^{45} - 15 q^{47} - 3 q^{51} + 6 q^{53} + 9 q^{59} + 15 q^{61} - 3 q^{63} - 21 q^{65} - 6 q^{67} - 12 q^{69} + 9 q^{71} - 6 q^{73} - 6 q^{75} - 24 q^{77} - 27 q^{79} + 3 q^{81} - 15 q^{83} - 18 q^{85} + 3 q^{87} + 12 q^{89} + 9 q^{91} + 6 q^{93} + 6 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9 - 3 * q^11 - 9 * q^13 + 3 * q^15 - 3 * q^17 - 3 * q^21 - 12 * q^23 - 6 * q^25 + 3 * q^27 + 3 * q^29 + 6 * q^31 - 3 * q^33 - 3 * q^35 - 6 * q^37 - 9 * q^39 - 18 * q^41 - 21 * q^43 + 3 * q^45 - 15 * q^47 - 3 * q^51 + 6 * q^53 + 9 * q^59 + 15 * q^61 - 3 * q^63 - 21 * q^65 - 6 * q^67 - 12 * q^69 + 9 * q^71 - 6 * q^73 - 6 * q^75 - 24 * q^77 - 27 * q^79 + 3 * q^81 - 15 * q^83 - 18 * q^85 + 3 * q^87 + 12 * q^89 + 9 * q^91 + 6 * q^93 + 6 * q^97 - 3 * q^99

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