LMFDB - Embedded newform 7803.2.a.bi.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7803,2,Mod(1,7803)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7803, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7803.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7803 = 3^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7803.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:	$62.3072686972$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.29952.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} + 13 $ x^4 - 8*x^2 + 13
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 459)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.50597$ of defining polynomial
Character	$\chi$	$=$	7803.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.50597 q^{2} +0.267949 q^{4} -1.73205 q^{5} -4.11439 q^{7} -2.60842 q^{8} +O(q^{10})$	$q+1.50597 q^{2} +0.267949 q^{4} -1.73205 q^{5} -4.11439 q^{7} -2.60842 q^{8} -2.60842 q^{10} +2.00000 q^{11} -4.46410 q^{13} -6.19615 q^{14} -4.46410 q^{16} -7.19615 q^{19} -0.464102 q^{20} +3.01194 q^{22} -8.46410 q^{23} -2.00000 q^{25} -6.72281 q^{26} -1.10245 q^{28} +1.73205 q^{29} +6.02388 q^{31} -1.50597 q^{32} +7.12633 q^{35} +9.33123 q^{37} -10.8372 q^{38} +4.51791 q^{40} -6.66025 q^{41} -0.267949 q^{43} +0.535898 q^{44} -12.7467 q^{46} -9.33123 q^{47} +9.92820 q^{49} -3.01194 q^{50} -1.19615 q^{52} -1.90949 q^{53} -3.46410 q^{55} +10.7321 q^{56} +2.60842 q^{58} -7.12633 q^{59} -1.10245 q^{61} +9.07180 q^{62} +6.66025 q^{64} +7.73205 q^{65} +4.26795 q^{67} +10.7321 q^{70} +5.53590 q^{71} +7.12633 q^{73} +14.0526 q^{74} -1.92820 q^{76} -8.22878 q^{77} +10.1383 q^{79} +7.73205 q^{80} -10.0302 q^{82} -0.403524 q^{86} -5.21684 q^{88} +2.20489 q^{89} +18.3671 q^{91} -2.26795 q^{92} -14.0526 q^{94} +12.4641 q^{95} -13.1502 q^{97} +14.9516 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4}+O(q^{10}) $ 4 * q + 8 * q^4	$ 4 q + 8 q^{4} + 8 q^{11} - 4 q^{13} - 4 q^{14} - 4 q^{16} - 8 q^{19} + 12 q^{20} - 20 q^{23} - 8 q^{25} + 8 q^{41} - 8 q^{43} + 16 q^{44} + 12 q^{49} + 16 q^{52} + 36 q^{56} + 64 q^{62} - 8 q^{64} + 24 q^{65} + 24 q^{67} + 36 q^{70} + 36 q^{71} - 20 q^{74} + 20 q^{76} + 24 q^{80} - 16 q^{92} + 20 q^{94} + 36 q^{95}+O(q^{100}) $ 4 * q + 8 * q^4 + 8 * q^11 - 4 * q^13 - 4 * q^14 - 4 * q^16 - 8 * q^19 + 12 * q^20 - 20 * q^23 - 8 * q^25 + 8 * q^41 - 8 * q^43 + 16 * q^44 + 12 * q^49 + 16 * q^52 + 36 * q^56 + 64 * q^62 - 8 * q^64 + 24 * q^65 + 24 * q^67 + 36 * q^70 + 36 * q^71 - 20 * q^74 + 20 * q^76 + 24 * q^80 - 16 * q^92 + 20 * q^94 + 36 * q^95

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 $

$2$	1.50597	1.06488	0.532441	−	0.846467i	$-0.321275\pi$
$2$	1.50597	1.06488	0.532441	+	0.846467i	$0.321275\pi$
$3$	0	0
$3$	0	0
$4$	0.267949	0.133975
$5$	−1.73205	−0.774597	−0.387298	−	0.921954i	$-0.626592\pi$
$5$	−1.73205	−0.774597	−0.387298	+	0.921954i	$0.626592\pi$
$6$	0	0
$7$	−4.11439	−1.55509	−0.777547	−	0.628825i	$-0.783536\pi$
$7$	−4.11439	−1.55509	−0.777547	+	0.628825i	$0.783536\pi$
$8$	−2.60842	−0.922215
$9$	0	0
$10$	−2.60842	−0.824854
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−4.46410	−1.23812	−0.619060	−	0.785344i	$-0.712486\pi$
$13$	−4.46410	−1.23812	−0.619060	+	0.785344i	$0.712486\pi$
$14$	−6.19615	−1.65599
$15$	0	0
$16$	−4.46410	−1.11603
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−7.19615	−1.65091	−0.825455	−	0.564467i	$-0.809082\pi$
$19$	−7.19615	−1.65091	−0.825455	+	0.564467i	$0.809082\pi$
$20$	−0.464102	−0.103776
$21$	0	0
$22$	3.01194	0.642148
$23$	−8.46410	−1.76489	−0.882444	−	0.470418i	$-0.844103\pi$
$23$	−8.46410	−1.76489	−0.882444	+	0.470418i	$0.844103\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	−6.72281	−1.31845
$27$	0	0
$28$	−1.10245	−0.208343
$29$	1.73205	0.321634	0.160817	−	0.986984i	$-0.448587\pi$
$29$	1.73205	0.321634	0.160817	+	0.986984i	$0.448587\pi$
$30$	0	0
$31$	6.02388	1.08192	0.540961	−	0.841048i	$-0.318061\pi$
$31$	6.02388	1.08192	0.540961	+	0.841048i	$0.318061\pi$
$32$	−1.50597	−0.266221
$33$	0	0
$34$	0	0
$35$	7.12633	1.20457
$36$	0	0
$37$	9.33123	1.53404	0.767022	−	0.641621i	$-0.221738\pi$
$37$	9.33123	1.53404	0.767022	+	0.641621i	$0.221738\pi$
$38$	−10.8372	−1.75803
$39$	0	0
$40$	4.51791	0.714345
$41$	−6.66025	−1.04016	−0.520078	−	0.854118i	$-0.674097\pi$
$41$	−6.66025	−1.04016	−0.520078	+	0.854118i	$0.674097\pi$
$42$	0	0
$43$	−0.267949	−0.0408619	−0.0204309	−	0.999791i	$-0.506504\pi$
$43$	−0.267949	−0.0408619	−0.0204309	+	0.999791i	$0.506504\pi$
$44$	0.535898	0.0807897
$45$	0	0
$46$	−12.7467	−1.87940
$47$	−9.33123	−1.36110	−0.680550	−	0.732702i	$-0.738259\pi$
$47$	−9.33123	−1.36110	−0.680550	+	0.732702i	$0.738259\pi$
$48$	0	0
$49$	9.92820	1.41831
$50$	−3.01194	−0.425953
$51$	0	0
$52$	−1.19615	−0.165876
$53$	−1.90949	−0.262289	−0.131145	−	0.991363i	$-0.541865\pi$
$53$	−1.90949	−0.262289	−0.131145	+	0.991363i	$0.541865\pi$
$54$	0	0
$55$	−3.46410	−0.467099
$56$	10.7321	1.43413
$57$	0	0
$58$	2.60842	0.342502
$59$	−7.12633	−0.927769	−0.463885	−	0.885896i	$-0.653545\pi$
$59$	−7.12633	−0.927769	−0.463885	+	0.885896i	$0.653545\pi$
$60$	0	0
$61$	−1.10245	−0.141154	−0.0705770	−	0.997506i	$-0.522484\pi$
$61$	−1.10245	−0.141154	−0.0705770	+	0.997506i	$0.522484\pi$
$62$	9.07180	1.15212
$63$	0	0
$64$	6.66025	0.832532
$65$	7.73205	0.959043
$66$	0	0
$67$	4.26795	0.521413	0.260706	−	0.965418i	$-0.416045\pi$
$67$	4.26795	0.521413	0.260706	+	0.965418i	$0.416045\pi$
$68$	0	0
$69$	0	0
$70$	10.7321	1.28273
$71$	5.53590	0.656990	0.328495	−	0.944506i	$-0.393459\pi$
$71$	5.53590	0.656990	0.328495	+	0.944506i	$0.393459\pi$
$72$	0	0
$73$	7.12633	0.834074	0.417037	−	0.908889i	$-0.363069\pi$
$73$	7.12633	0.834074	0.417037	+	0.908889i	$0.363069\pi$
$74$	14.0526	1.63358
$75$	0	0
$76$	−1.92820	−0.221180
$77$	−8.22878	−0.937756
$78$	0	0
$79$	10.1383	1.14064	0.570322	−	0.821421i	$-0.306818\pi$
$79$	10.1383	1.14064	0.570322	+	0.821421i	$0.306818\pi$
$80$	7.73205	0.864470
$81$	0	0
$82$	−10.0302	−1.10764
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	−0.403524	−0.0435131
$87$	0	0
$88$	−5.21684	−0.556117
$89$	2.20489	0.233718	0.116859	−	0.993148i	$-0.462717\pi$
$89$	2.20489	0.233718	0.116859	+	0.993148i	$0.462717\pi$
$90$	0	0
$91$	18.3671	1.92539
$92$	−2.26795	−0.236450
$93$	0	0
$94$	−14.0526	−1.44941
$95$	12.4641	1.27879
$96$	0	0
$97$	−13.1502	−1.33520	−0.667601	−	0.744519i	$-0.732679\pi$
$97$	−13.1502	−1.33520	−0.667601	+	0.744519i	$0.732679\pi$
$98$	14.9516	1.51034
$99$	0	0
$100$	−0.535898	−0.0535898
$101$	−12.3432	−1.22819	−0.614096	−	0.789232i	$-0.710479\pi$
$101$	−12.3432	−1.22819	−0.614096	+	0.789232i	$0.710479\pi$
$102$	0	0
$103$	−6.80385	−0.670403	−0.335202	−	0.942146i	$-0.608804\pi$
$103$	−6.80385	−0.670403	−0.335202	+	0.942146i	$0.608804\pi$
$104$	11.6442	1.14181
$105$	0	0
$106$	−2.87564	−0.279307
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	0	0
$109$	−13.1502	−1.25956	−0.629781	−	0.776773i	$-0.716855\pi$
$109$	−13.1502	−1.25956	−0.629781	+	0.776773i	$0.716855\pi$
$110$	−5.21684	−0.497406
$111$	0	0
$112$	18.3671	1.73552
$113$	11.4641	1.07845	0.539226	−	0.842161i	$-0.318717\pi$
$113$	11.4641	1.07845	0.539226	+	0.842161i	$0.318717\pi$
$114$	0	0
$115$	14.6603	1.36708
$116$	0.464102	0.0430908
$117$	0	0
$118$	−10.7321	−0.987965
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−1.66025	−0.150312
$123$	0	0
$124$	1.61410	0.144950
$125$	12.1244	1.08444
$126$	0	0
$127$	10.9282	0.969721	0.484861	−	0.874591i	$-0.338870\pi$
$127$	10.9282	0.969721	0.484861	+	0.874591i	$0.338870\pi$
$128$	13.0421	1.15277
$129$	0	0
$130$	11.6442	1.02127
$131$	1.00000	0.0873704	0.0436852	−	0.999045i	$-0.486090\pi$
$131$	1.00000	0.0873704	0.0436852	+	0.999045i	$0.486090\pi$
$132$	0	0
$133$	29.6078	2.56732
$134$	6.42741	0.555244
$135$	0	0
$136$	0	0
$137$	2.20489	0.188377	0.0941884	−	0.995554i	$-0.469974\pi$
$137$	2.20489	0.188377	0.0941884	+	0.995554i	$0.469974\pi$
$138$	0	0
$139$	−16.4576	−1.39591	−0.697956	−	0.716141i	$-0.745907\pi$
$139$	−16.4576	−1.39591	−0.697956	+	0.716141i	$0.745907\pi$
$140$	1.90949	0.161382
$141$	0	0
$142$	8.33690	0.699617
$143$	−8.92820	−0.746614
$144$	0	0
$145$	−3.00000	−0.249136
$146$	10.7321	0.888191
$147$	0	0
$148$	2.50029	0.205523
$149$	−1.90949	−0.156432	−0.0782160	−	0.996936i	$-0.524922\pi$
$149$	−1.90949	−0.156432	−0.0782160	+	0.996936i	$0.524922\pi$
$150$	0	0
$151$	1.73205	0.140952	0.0704761	−	0.997513i	$-0.477548\pi$
$151$	1.73205	0.140952	0.0704761	+	0.997513i	$0.477548\pi$
$152$	18.7706	1.52249
$153$	0	0
$154$	−12.3923	−0.998600
$155$	−10.4337	−0.838053
$156$	0	0
$157$	9.07180	0.724008	0.362004	−	0.932177i	$-0.382093\pi$
$157$	9.07180	0.724008	0.362004	+	0.932177i	$0.382093\pi$
$158$	15.2679	1.21465
$159$	0	0
$160$	2.60842	0.206214
$161$	34.8246	2.74456
$162$	0	0
$163$	−6.02388	−0.471827	−0.235914	−	0.971774i	$-0.575808\pi$
$163$	−6.02388	−0.471827	−0.235914	+	0.971774i	$0.575808\pi$
$164$	−1.78461	−0.139355
$165$	0	0
$166$	0	0
$167$	−6.92820	−0.536120	−0.268060	−	0.963402i	$-0.586383\pi$
$167$	−6.92820	−0.536120	−0.268060	+	0.963402i	$0.586383\pi$
$168$	0	0
$169$	6.92820	0.532939
$170$	0	0
$171$	0	0
$172$	−0.0717968	−0.00547445
$173$	14.9282	1.13497	0.567485	−	0.823384i	$-0.307916\pi$
$173$	14.9282	1.13497	0.567485	+	0.823384i	$0.307916\pi$
$174$	0	0
$175$	8.22878	0.622037
$176$	−8.92820	−0.672989
$177$	0	0
$178$	3.32051	0.248883
$179$	6.02388	0.450246	0.225123	−	0.974330i	$-0.427722\pi$
$179$	6.02388	0.450246	0.225123	+	0.974330i	$0.427722\pi$
$180$	0	0
$181$	6.02388	0.447752	0.223876	−	0.974618i	$-0.428129\pi$
$181$	6.02388	0.447752	0.223876	+	0.974618i	$0.428129\pi$
$182$	27.6603	2.05031
$183$	0	0
$184$	22.0779	1.62761
$185$	−16.1622	−1.18827
$186$	0	0
$187$	0	0
$188$	−2.50029	−0.182353
$189$	0	0
$190$	18.7706	1.36176
$191$	22.4814	1.62670	0.813350	−	0.581775i	$-0.197641\pi$
$191$	22.4814	1.62670	0.813350	+	0.581775i	$0.197641\pi$
$192$	0	0
$193$	−10.4337	−0.751032	−0.375516	−	0.926816i	$-0.622535\pi$
$193$	−10.4337	−0.751032	−0.375516	+	0.926816i	$0.622535\pi$
$194$	−19.8038	−1.42183
$195$	0	0
$196$	2.66025	0.190018
$197$	16.6603	1.18699	0.593497	−	0.804836i	$-0.297747\pi$
$197$	16.6603	1.18699	0.593497	+	0.804836i	$0.297747\pi$
$198$	0	0
$199$	−20.2765	−1.43737	−0.718683	−	0.695338i	$-0.755255\pi$
$199$	−20.2765	−1.43737	−0.718683	+	0.695338i	$0.755255\pi$
$200$	5.21684	0.368886
$201$	0	0
$202$	−18.5885	−1.30788
$203$	−7.12633	−0.500170
$204$	0	0
$205$	11.5359	0.805702
$206$	−10.2464	−0.713900
$207$	0	0
$208$	19.9282	1.38177
$209$	−14.3923	−0.995537
$210$	0	0
$211$	0.295400	0.0203362	0.0101681	−	0.999948i	$-0.496763\pi$
$211$	0.295400	0.0203362	0.0101681	+	0.999948i	$0.496763\pi$
$212$	−0.511648	−0.0351401
$213$	0	0
$214$	4.51791	0.308838
$215$	0.464102	0.0316515
$216$	0	0
$217$	−24.7846	−1.68249
$218$	−19.8038	−1.34129
$219$	0	0
$220$	−0.928203	−0.0625794
$221$	0	0
$222$	0	0
$223$	−7.58846	−0.508161	−0.254080	−	0.967183i	$-0.581773\pi$
$223$	−7.58846	−0.508161	−0.254080	+	0.967183i	$0.581773\pi$
$224$	6.19615	0.413998
$225$	0	0
$226$	17.2646	1.14842
$227$	−17.7846	−1.18041	−0.590203	−	0.807255i	$-0.700952\pi$
$227$	−17.7846	−1.18041	−0.590203	+	0.807255i	$0.700952\pi$
$228$	0	0
$229$	−5.39230	−0.356334	−0.178167	−	0.984000i	$-0.557017\pi$
$229$	−5.39230	−0.356334	−0.178167	+	0.984000i	$0.557017\pi$
$230$	22.0779	1.45577
$231$	0	0
$232$	−4.51791	−0.296616
$233$	22.6603	1.48452	0.742261	−	0.670111i	$-0.233753\pi$
$233$	22.6603	1.48452	0.742261	+	0.670111i	$0.233753\pi$
$234$	0	0
$235$	16.1622	1.05430
$236$	−1.90949	−0.124298
$237$	0	0
$238$	0	0
$239$	−27.4029	−1.77255	−0.886273	−	0.463164i	$-0.846714\pi$
$239$	−27.4029	−1.77255	−0.886273	+	0.463164i	$0.846714\pi$
$240$	0	0
$241$	−8.22878	−0.530062	−0.265031	−	0.964240i	$-0.585382\pi$
$241$	−8.22878	−0.530062	−0.265031	+	0.964240i	$0.585382\pi$
$242$	−10.5418	−0.677652
$243$	0	0
$244$	−0.295400	−0.0189110
$245$	−17.1962	−1.09862
$246$	0	0
$247$	32.1244	2.04402
$248$	−15.7128	−0.997765
$249$	0	0
$250$	18.2589	1.15480
$251$	−10.4337	−0.658568	−0.329284	−	0.944231i	$-0.606807\pi$
$251$	−10.4337	−0.658568	−0.329284	+	0.944231i	$0.606807\pi$
$252$	0	0
$253$	−16.9282	−1.06427
$254$	16.4576	1.03264
$255$	0	0
$256$	6.32051	0.395032
$257$	7.93338	0.494871	0.247435	−	0.968904i	$-0.420412\pi$
$257$	7.93338	0.494871	0.247435	+	0.968904i	$0.420412\pi$
$258$	0	0
$259$	−38.3923	−2.38558
$260$	2.07180	0.128487
$261$	0	0
$262$	1.50597	0.0930392
$263$	−14.2527	−0.878857	−0.439428	−	0.898278i	$-0.644819\pi$
$263$	−14.2527	−0.878857	−0.439428	+	0.898278i	$0.644819\pi$
$264$	0	0
$265$	3.30734	0.203168
$266$	44.5885	2.73389
$267$	0	0
$268$	1.14359	0.0698561
$269$	−17.3205	−1.05605	−0.528025	−	0.849229i	$-0.677067\pi$
$269$	−17.3205	−1.05605	−0.528025	+	0.849229i	$0.677067\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	0	0
$273$	0	0
$274$	3.32051	0.200599
$275$	−4.00000	−0.241209
$276$	0	0
$277$	11.5361	0.693138	0.346569	−	0.938024i	$-0.387347\pi$
$277$	11.5361	0.693138	0.346569	+	0.938024i	$0.387347\pi$
$278$	−24.7846	−1.48648
$279$	0	0
$280$	−18.5885	−1.11087
$281$	4.11439	0.245444	0.122722	−	0.992441i	$-0.460838\pi$
$281$	4.11439	0.245444	0.122722	+	0.992441i	$0.460838\pi$
$282$	0	0
$283$	7.93338	0.471590	0.235795	−	0.971803i	$-0.424231\pi$
$283$	7.93338	0.471590	0.235795	+	0.971803i	$0.424231\pi$
$284$	1.48334	0.0880200
$285$	0	0
$286$	−13.4456	−0.795056
$287$	27.4029	1.61754
$288$	0	0
$289$	0	0
$290$	−4.51791	−0.265301
$291$	0	0
$292$	1.90949	0.111745
$293$	28.8007	1.68256	0.841278	−	0.540602i	$-0.181804\pi$
$293$	28.8007	1.68256	0.841278	+	0.540602i	$0.181804\pi$
$294$	0	0
$295$	12.3432	0.718647
$296$	−24.3397	−1.41472
$297$	0	0
$298$	−2.87564	−0.166582
$299$	37.7846	2.18514
$300$	0	0
$301$	1.10245	0.0635440
$302$	2.60842	0.150098
$303$	0	0
$304$	32.1244	1.84246
$305$	1.90949	0.109337
$306$	0	0
$307$	21.5885	1.23212	0.616059	−	0.787700i	$-0.288728\pi$
$307$	21.5885	1.23212	0.616059	+	0.787700i	$0.288728\pi$
$308$	−2.20489	−0.125636
$309$	0	0
$310$	−15.7128	−0.892428
$311$	33.7128	1.91168	0.955839	−	0.293890i	$-0.0949501\pi$
$311$	33.7128	1.91168	0.955839	+	0.293890i	$0.0949501\pi$
$312$	0	0
$313$	−20.2765	−1.14610	−0.573049	−	0.819521i	$-0.694240\pi$
$313$	−20.2765	−1.14610	−0.573049	+	0.819521i	$0.694240\pi$
$314$	13.6619	0.770984
$315$	0	0
$316$	2.71654	0.152817
$317$	11.5885	0.650873	0.325436	−	0.945564i	$-0.394489\pi$
$317$	11.5885	0.650873	0.325436	+	0.945564i	$0.394489\pi$
$318$	0	0
$319$	3.46410	0.193952
$320$	−11.5359	−0.644876
$321$	0	0
$322$	52.4449	2.92264
$323$	0	0
$324$	0	0
$325$	8.92820	0.495248
$326$	−9.07180	−0.502440
$327$	0	0
$328$	17.3727	0.959249
$329$	38.3923	2.11664
$330$	0	0
$331$	−22.1244	−1.21606	−0.608032	−	0.793912i	$-0.708041\pi$
$331$	−22.1244	−1.21606	−0.608032	+	0.793912i	$0.708041\pi$
$332$	0	0
$333$	0	0
$334$	−10.4337	−0.570905
$335$	−7.39230	−0.403885
$336$	0	0
$337$	2.71654	0.147979	0.0739897	−	0.997259i	$-0.476427\pi$
$337$	2.71654	0.147979	0.0739897	+	0.997259i	$0.476427\pi$
$338$	10.4337	0.567517
$339$	0	0
$340$	0	0
$341$	12.0478	0.652423
$342$	0	0
$343$	−12.0478	−0.650518
$344$	0.698924	0.0376834
$345$	0	0
$346$	22.4814	1.20861
$347$	0.856406	0.0459743	0.0229872	−	0.999736i	$-0.492682\pi$
$347$	0.856406	0.0459743	0.0229872	+	0.999736i	$0.492682\pi$
$348$	0	0
$349$	17.2487	0.923302	0.461651	−	0.887062i	$-0.347257\pi$
$349$	17.2487	0.923302	0.461651	+	0.887062i	$0.347257\pi$
$350$	12.3923	0.662397
$351$	0	0
$352$	−3.01194	−0.160537
$353$	14.2527	0.758593	0.379296	−	0.925275i	$-0.376166\pi$
$353$	14.2527	0.758593	0.379296	+	0.925275i	$0.376166\pi$
$354$	0	0
$355$	−9.58846	−0.508902
$356$	0.590800	0.0313123
$357$	0	0
$358$	9.07180	0.479459
$359$	−27.4029	−1.44627	−0.723134	−	0.690707i	$-0.757299\pi$
$359$	−27.4029	−1.44627	−0.723134	+	0.690707i	$0.757299\pi$
$360$	0	0
$361$	32.7846	1.72551
$362$	9.07180	0.476803
$363$	0	0
$364$	4.92144	0.257953
$365$	−12.3432	−0.646071
$366$	0	0
$367$	22.7768	1.18894	0.594471	−	0.804117i	$-0.297362\pi$
$367$	22.7768	1.18894	0.594471	+	0.804117i	$0.297362\pi$
$368$	37.7846	1.96966
$369$	0	0
$370$	−24.3397	−1.26536
$371$	7.85641	0.407884
$372$	0	0
$373$	−19.2487	−0.996660	−0.498330	−	0.866987i	$-0.666053\pi$
$373$	−19.2487	−0.996660	−0.498330	+	0.866987i	$0.666053\pi$
$374$	0	0
$375$	0	0
$376$	24.3397	1.25523
$377$	−7.73205	−0.398221
$378$	0	0
$379$	−34.8246	−1.78882	−0.894410	−	0.447248i	$-0.852404\pi$
$379$	−34.8246	−1.78882	−0.894410	+	0.447248i	$0.852404\pi$
$380$	3.33975	0.171325
$381$	0	0
$382$	33.8564	1.73224
$383$	−20.2765	−1.03608	−0.518042	−	0.855355i	$-0.673339\pi$
$383$	−20.2765	−1.03608	−0.518042	+	0.855355i	$0.673339\pi$
$384$	0	0
$385$	14.2527	0.726383
$386$	−15.7128	−0.799761
$387$	0	0
$388$	−3.52359	−0.178883
$389$	−10.4337	−0.529008	−0.264504	−	0.964385i	$-0.585208\pi$
$389$	−10.4337	−0.529008	−0.264504	+	0.964385i	$0.585208\pi$
$390$	0	0
$391$	0	0
$392$	−25.8969	−1.30799
$393$	0	0
$394$	25.0899	1.26401
$395$	−17.5600	−0.883540
$396$	0	0
$397$	2.20489	0.110660	0.0553302	−	0.998468i	$-0.482379\pi$
$397$	2.20489	0.110660	0.0553302	+	0.998468i	$0.482379\pi$
$398$	−30.5359	−1.53063
$399$	0	0
$400$	8.92820	0.446410
$401$	−13.0526	−0.651814	−0.325907	−	0.945402i	$-0.605670\pi$
$401$	−13.0526	−0.651814	−0.325907	+	0.945402i	$0.605670\pi$
$402$	0	0
$403$	−26.8912	−1.33955
$404$	−3.30734	−0.164546
$405$	0	0
$406$	−10.7321	−0.532623
$407$	18.6625	0.925063
$408$	0	0
$409$	−4.07180	−0.201337	−0.100669	−	0.994920i	$-0.532098\pi$
$409$	−4.07180	−0.201337	−0.100669	+	0.994920i	$0.532098\pi$
$410$	17.3727	0.857978
$411$	0	0
$412$	−1.82309	−0.0898170
$413$	29.3205	1.44277
$414$	0	0
$415$	0	0
$416$	6.72281	0.329613
$417$	0	0
$418$	−21.6744	−1.06013
$419$	−6.07180	−0.296627	−0.148313	−	0.988940i	$-0.547384\pi$
$419$	−6.07180	−0.296627	−0.148313	+	0.988940i	$0.547384\pi$
$420$	0	0
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	0.444864	0.0216556
$423$	0	0
$424$	4.98076	0.241887
$425$	0	0
$426$	0	0
$427$	4.53590	0.219508
$428$	0.803848	0.0388554
$429$	0	0
$430$	0.698924	0.0337051
$431$	27.3923	1.31944	0.659720	−	0.751511i	$-0.270675\pi$
$431$	27.3923	1.31944	0.659720	+	0.751511i	$0.270675\pi$
$432$	0	0
$433$	17.0000	0.816968	0.408484	−	0.912766i	$-0.366058\pi$
$433$	17.0000	0.816968	0.408484	+	0.912766i	$0.366058\pi$
$434$	−37.3249	−1.79165
$435$	0	0
$436$	−3.52359	−0.168749
$437$	60.9090	2.91367
$438$	0	0
$439$	−8.52418	−0.406837	−0.203418	−	0.979092i	$-0.565205\pi$
$439$	−8.52418	−0.406837	−0.203418	+	0.979092i	$0.565205\pi$
$440$	9.03583	0.430766
$441$	0	0
$442$	0	0
$443$	−5.51224	−0.261894	−0.130947	−	0.991389i	$-0.541802\pi$
$443$	−5.51224	−0.261894	−0.130947	+	0.991389i	$0.541802\pi$
$444$	0	0
$445$	−3.81899	−0.181037
$446$	−11.4280	−0.541131
$447$	0	0
$448$	−27.4029	−1.29466
$449$	−13.0718	−0.616896	−0.308448	−	0.951241i	$-0.599810\pi$
$449$	−13.0718	−0.616896	−0.308448	+	0.951241i	$0.599810\pi$
$450$	0	0
$451$	−13.3205	−0.627238
$452$	3.07180	0.144485
$453$	0	0
$454$	−26.7831	−1.25699
$455$	−31.8127	−1.49140
$456$	0	0
$457$	−9.85641	−0.461063	−0.230532	−	0.973065i	$-0.574047\pi$
$457$	−9.85641	−0.461063	−0.230532	+	0.973065i	$0.574047\pi$
$458$	−8.12066	−0.379453
$459$	0	0
$460$	3.92820	0.183153
$461$	18.6625	0.869197	0.434599	−	0.900624i	$-0.356890\pi$
$461$	18.6625	0.869197	0.434599	+	0.900624i	$0.356890\pi$
$462$	0	0
$463$	25.3205	1.17674	0.588372	−	0.808590i	$-0.299769\pi$
$463$	25.3205	1.17674	0.588372	+	0.808590i	$0.299769\pi$
$464$	−7.73205	−0.358951
$465$	0	0
$466$	34.1257	1.58084
$467$	2.71654	0.125707	0.0628533	−	0.998023i	$-0.479980\pi$
$467$	2.71654	0.125707	0.0628533	+	0.998023i	$0.479980\pi$
$468$	0	0
$469$	−17.5600	−0.810846
$470$	24.3397	1.12271
$471$	0	0
$472$	18.5885	0.855603
$473$	−0.535898	−0.0246406
$474$	0	0
$475$	14.3923	0.660364
$476$	0	0
$477$	0	0
$478$	−41.2679	−1.88755
$479$	40.1769	1.83573	0.917865	−	0.396893i	$-0.129911\pi$
$479$	40.1769	1.83573	0.917865	+	0.396893i	$0.129911\pi$
$480$	0	0
$481$	−41.6555	−1.89933
$482$	−12.3923	−0.564454
$483$	0	0
$484$	−1.87564	−0.0852566
$485$	22.7768	1.03424
$486$	0	0
$487$	29.0961	1.31847	0.659236	−	0.751936i	$-0.270880\pi$
$487$	29.0961	1.31847	0.659236	+	0.751936i	$0.270880\pi$
$488$	2.87564	0.130174
$489$	0	0
$490$	−25.8969	−1.16990
$491$	−40.5531	−1.83014	−0.915068	−	0.403300i	$-0.867863\pi$
$491$	−40.5531	−1.83014	−0.915068	+	0.403300i	$0.867863\pi$
$492$	0	0
$493$	0	0
$494$	48.3784	2.17665
$495$	0	0
$496$	−26.8912	−1.20745
$497$	−22.7768	−1.02168
$498$	0	0
$499$	−1.90949	−0.0854807	−0.0427404	−	0.999086i	$-0.513609\pi$
$499$	−1.90949	−0.0854807	−0.0427404	+	0.999086i	$0.513609\pi$
$500$	3.24871	0.145287
$501$	0	0
$502$	−15.7128	−0.701297
$503$	3.67949	0.164060	0.0820302	−	0.996630i	$-0.473860\pi$
$503$	3.67949	0.164060	0.0820302	+	0.996630i	$0.473860\pi$
$504$	0	0
$505$	21.3790	0.951353
$506$	−25.4934	−1.13332
$507$	0	0
$508$	2.92820	0.129918
$509$	−24.6863	−1.09420	−0.547101	−	0.837066i	$-0.684269\pi$
$509$	−24.6863	−1.09420	−0.547101	+	0.837066i	$0.684269\pi$
$510$	0	0
$511$	−29.3205	−1.29706
$512$	−16.5657	−0.732107
$513$	0	0
$514$	11.9474	0.526979
$515$	11.7846	0.519292
$516$	0	0
$517$	−18.6625	−0.820774
$518$	−57.8177	−2.54036
$519$	0	0
$520$	−20.1684	−0.884444
$521$	−34.3923	−1.50675	−0.753377	−	0.657589i	$-0.771576\pi$
$521$	−34.3923	−1.50675	−0.753377	+	0.657589i	$0.771576\pi$
$522$	0	0
$523$	−6.92820	−0.302949	−0.151475	−	0.988461i	$-0.548402\pi$
$523$	−6.92820	−0.302949	−0.151475	+	0.988461i	$0.548402\pi$
$524$	0.267949	0.0117054
$525$	0	0
$526$	−21.4641	−0.935879
$527$	0	0
$528$	0	0
$529$	48.6410	2.11483
$530$	4.98076	0.216350
$531$	0	0
$532$	7.93338	0.343956
$533$	29.7321	1.28784
$534$	0	0
$535$	−5.19615	−0.224649
$536$	−11.1326	−0.480855
$537$	0	0
$538$	−26.0842	−1.12457
$539$	19.8564	0.855276
$540$	0	0
$541$	12.0478	0.517974	0.258987	−	0.965881i	$-0.416611\pi$
$541$	12.0478	0.517974	0.258987	+	0.965881i	$0.416611\pi$
$542$	−18.0717	−0.776244
$543$	0	0
$544$	0	0
$545$	22.7768	0.975653
$546$	0	0
$547$	6.61468	0.282823	0.141412	−	0.989951i	$-0.454836\pi$
$547$	6.61468	0.282823	0.141412	+	0.989951i	$0.454836\pi$
$548$	0.590800	0.0252377
$549$	0	0
$550$	−6.02388	−0.256859
$551$	−12.4641	−0.530989
$552$	0	0
$553$	−41.7128	−1.77381
$554$	17.3731	0.738111
$555$	0	0
$556$	−4.40979	−0.187017
$557$	26.5958	1.12690	0.563451	−	0.826150i	$-0.309473\pi$
$557$	26.5958	1.12690	0.563451	+	0.826150i	$0.309473\pi$
$558$	0	0
$559$	1.19615	0.0505919
$560$	−31.8127	−1.34433
$561$	0	0
$562$	6.19615	0.261369
$563$	29.6078	1.24782	0.623909	−	0.781497i	$-0.285543\pi$
$563$	29.6078	1.24782	0.623909	+	0.781497i	$0.285543\pi$
$564$	0	0
$565$	−19.8564	−0.835365
$566$	11.9474	0.502188
$567$	0	0
$568$	−14.4399	−0.605886
$569$	2.50029	0.104818	0.0524089	−	0.998626i	$-0.483310\pi$
$569$	2.50029	0.104818	0.0524089	+	0.998626i	$0.483310\pi$
$570$	0	0
$571$	−14.5481	−0.608818	−0.304409	−	0.952541i	$-0.598459\pi$
$571$	−14.5481	−0.608818	−0.304409	+	0.952541i	$0.598459\pi$
$572$	−2.39230	−0.100027
$573$	0	0
$574$	41.2679	1.72249
$575$	16.9282	0.705955
$576$	0	0
$577$	−37.8564	−1.57598	−0.787991	−	0.615686i	$-0.788879\pi$
$577$	−37.8564	−1.57598	−0.787991	+	0.615686i	$0.788879\pi$
$578$	0	0
$579$	0	0
$580$	−0.803848	−0.0333780
$581$	0	0
$582$	0	0
$583$	−3.81899	−0.158166
$584$	−18.5885	−0.769196
$585$	0	0
$586$	43.3731	1.79172
$587$	−13.7410	−0.567152	−0.283576	−	0.958950i	$-0.591521\pi$
$587$	−13.7410	−0.567152	−0.283576	+	0.958950i	$0.591521\pi$
$588$	0	0
$589$	−43.3488	−1.78616
$590$	18.5885	0.765275
$591$	0	0
$592$	−41.6555	−1.71203
$593$	26.3004	1.08003	0.540015	−	0.841656i	$-0.318419\pi$
$593$	26.3004	1.08003	0.540015	+	0.841656i	$0.318419\pi$
$594$	0	0
$595$	0	0
$596$	−0.511648	−0.0209579
$597$	0	0
$598$	56.9025	2.32692
$599$	−27.4029	−1.11965	−0.559826	−	0.828610i	$-0.689132\pi$
$599$	−27.4029	−1.11965	−0.559826	+	0.828610i	$0.689132\pi$
$600$	0	0
$601$	19.7649	0.806227	0.403114	−	0.915150i	$-0.367928\pi$
$601$	19.7649	0.806227	0.403114	+	0.915150i	$0.367928\pi$
$602$	1.66025	0.0676669
$603$	0	0
$604$	0.464102	0.0188840
$605$	12.1244	0.492925
$606$	0	0
$607$	30.7102	1.24649	0.623245	−	0.782027i	$-0.285814\pi$
$607$	30.7102	1.24649	0.623245	+	0.782027i	$0.285814\pi$
$608$	10.8372	0.439506
$609$	0	0
$610$	2.87564	0.116431
$611$	41.6555	1.68520
$612$	0	0
$613$	−22.1769	−0.895717	−0.447859	−	0.894104i	$-0.647813\pi$
$613$	−22.1769	−0.895717	−0.447859	+	0.894104i	$0.647813\pi$
$614$	32.5116	1.31206
$615$	0	0
$616$	21.4641	0.864813
$617$	−40.5167	−1.63114	−0.815570	−	0.578659i	$-0.803576\pi$
$617$	−40.5167	−1.63114	−0.815570	+	0.578659i	$0.803576\pi$
$618$	0	0
$619$	6.61468	0.265867	0.132933	−	0.991125i	$-0.457560\pi$
$619$	6.61468	0.265867	0.132933	+	0.991125i	$0.457560\pi$
$620$	−2.79569	−0.112278
$621$	0	0
$622$	50.7705	2.03571
$623$	−9.07180	−0.363454
$624$	0	0
$625$	−11.0000	−0.440000
$626$	−30.5359	−1.22046
$627$	0	0
$628$	2.43078	0.0969987
$629$	0	0
$630$	0	0
$631$	−23.0526	−0.917708	−0.458854	−	0.888512i	$-0.651740\pi$
$631$	−23.0526	−0.917708	−0.458854	+	0.888512i	$0.651740\pi$
$632$	−26.4449	−1.05192
$633$	0	0
$634$	17.4519	0.693103
$635$	−18.9282	−0.751143
$636$	0	0
$637$	−44.3205	−1.75604
$638$	5.21684	0.206537
$639$	0	0
$640$	−22.5896	−0.892931
$641$	8.80385	0.347731	0.173866	−	0.984769i	$-0.444374\pi$
$641$	8.80385	0.347731	0.173866	+	0.984769i	$0.444374\pi$
$642$	0	0
$643$	−32.6197	−1.28640	−0.643198	−	0.765700i	$-0.722393\pi$
$643$	−32.6197	−1.28640	−0.643198	+	0.765700i	$0.722393\pi$
$644$	9.33123	0.367702
$645$	0	0
$646$	0	0
$647$	27.4029	1.07732	0.538659	−	0.842524i	$-0.318931\pi$
$647$	27.4029	1.07732	0.538659	+	0.842524i	$0.318931\pi$
$648$	0	0
$649$	−14.2527	−0.559466
$650$	13.4456	0.527380
$651$	0	0
$652$	−1.61410	−0.0632128
$653$	−4.24871	−0.166265	−0.0831325	−	0.996539i	$-0.526492\pi$
$653$	−4.24871	−0.166265	−0.0831325	+	0.996539i	$0.526492\pi$
$654$	0	0
$655$	−1.73205	−0.0676768
$656$	29.7321	1.16084
$657$	0	0
$658$	57.8177	2.25397
$659$	23.0722	0.898767	0.449384	−	0.893339i	$-0.351644\pi$
$659$	23.0722	0.898767	0.449384	+	0.893339i	$0.351644\pi$
$660$	0	0
$661$	−5.53590	−0.215321	−0.107661	−	0.994188i	$-0.534336\pi$
$661$	−5.53590	−0.215321	−0.107661	+	0.994188i	$0.534336\pi$
$662$	−33.3186	−1.29497
$663$	0	0
$664$	0	0
$665$	−51.2822	−1.98864
$666$	0	0
$667$	−14.6603	−0.567647
$668$	−1.85641	−0.0718265
$669$	0	0
$670$	−11.1326	−0.430090
$671$	−2.20489	−0.0851190
$672$	0	0
$673$	−48.7819	−1.88040	−0.940202	−	0.340618i	$-0.889364\pi$
$673$	−48.7819	−1.88040	−0.940202	+	0.340618i	$0.889364\pi$
$674$	4.09103	0.157581
$675$	0	0
$676$	1.85641	0.0714002
$677$	−10.5167	−0.404188	−0.202094	−	0.979366i	$-0.564775\pi$
$677$	−10.5167	−0.404188	−0.202094	+	0.979366i	$0.564775\pi$
$678$	0	0
$679$	54.1051	2.07636
$680$	0	0
$681$	0	0
$682$	18.1436	0.694754
$683$	−2.78461	−0.106550	−0.0532751	−	0.998580i	$-0.516966\pi$
$683$	−2.78461	−0.106550	−0.0532751	+	0.998580i	$0.516966\pi$
$684$	0	0
$685$	−3.81899	−0.145916
$686$	−18.1436	−0.692726
$687$	0	0
$688$	1.19615	0.0456029
$689$	8.52418	0.324745
$690$	0	0
$691$	−20.2765	−0.771356	−0.385678	−	0.922633i	$-0.626032\pi$
$691$	−20.2765	−0.771356	−0.385678	+	0.922633i	$0.626032\pi$
$692$	4.00000	0.152057
$693$	0	0
$694$	1.28972	0.0489572
$695$	28.5053	1.08127
$696$	0	0
$697$	0	0
$698$	25.9761	0.983208
$699$	0	0
$700$	2.20489	0.0833372
$701$	−41.1439	−1.55398	−0.776992	−	0.629511i	$-0.783255\pi$
$701$	−41.1439	−1.55398	−0.776992	+	0.629511i	$0.783255\pi$
$702$	0	0
$703$	−67.1489	−2.53257
$704$	13.3205	0.502036
$705$	0	0
$706$	21.4641	0.807812
$707$	50.7846	1.90995
$708$	0	0
$709$	34.5292	1.29677	0.648386	−	0.761312i	$-0.275444\pi$
$709$	34.5292	1.29677	0.648386	+	0.761312i	$0.275444\pi$
$710$	−14.4399	−0.541921
$711$	0	0
$712$	−5.75129	−0.215539
$713$	−50.9868	−1.90947
$714$	0	0
$715$	15.4641	0.578325
$716$	1.61410	0.0603216
$717$	0	0
$718$	−41.2679	−1.54011
$719$	−43.3923	−1.61826	−0.809130	−	0.587630i	$-0.800061\pi$
$719$	−43.3923	−1.61826	−0.809130	+	0.587630i	$0.800061\pi$
$720$	0	0
$721$	27.9937	1.04254
$722$	49.3727	1.83746
$723$	0	0
$724$	1.61410	0.0599874
$725$	−3.46410	−0.128654
$726$	0	0
$727$	25.1962	0.934474	0.467237	−	0.884132i	$-0.345250\pi$
$727$	25.1962	0.934474	0.467237	+	0.884132i	$0.345250\pi$
$728$	−47.9090	−1.77562
$729$	0	0
$730$	−18.5885	−0.687990
$731$	0	0
$732$	0	0
$733$	−32.4641	−1.19909	−0.599544	−	0.800341i	$-0.704652\pi$
$733$	−32.4641	−1.19909	−0.599544	+	0.800341i	$0.704652\pi$
$734$	34.3013	1.26608
$735$	0	0
$736$	12.7467	0.469849
$737$	8.53590	0.314424
$738$	0	0
$739$	−7.19615	−0.264715	−0.132357	−	0.991202i	$-0.542255\pi$
$739$	−7.19615	−0.264715	−0.132357	+	0.991202i	$0.542255\pi$
$740$	−4.33064	−0.159197
$741$	0	0
$742$	11.8315	0.434349
$743$	−40.6410	−1.49097	−0.745487	−	0.666520i	$-0.767783\pi$
$743$	−40.6410	−1.49097	−0.745487	+	0.666520i	$0.767783\pi$
$744$	0	0
$745$	3.30734	0.121172
$746$	−28.9880	−1.06133
$747$	0	0
$748$	0	0
$749$	−12.3432	−0.451010
$750$	0	0
$751$	8.22878	0.300272	0.150136	−	0.988665i	$-0.452029\pi$
$751$	8.22878	0.300272	0.150136	+	0.988665i	$0.452029\pi$
$752$	41.6555	1.51902
$753$	0	0
$754$	−11.6442	−0.424058
$755$	−3.00000	−0.109181
$756$	0	0
$757$	24.1769	0.878725	0.439362	−	0.898310i	$-0.355204\pi$
$757$	24.1769	0.878725	0.439362	+	0.898310i	$0.355204\pi$
$758$	−52.4449	−1.90488
$759$	0	0
$760$	−32.5116	−1.17932
$761$	18.9579	0.687222	0.343611	−	0.939112i	$-0.388350\pi$
$761$	18.9579	0.687222	0.343611	+	0.939112i	$0.388350\pi$
$762$	0	0
$763$	54.1051	1.95874
$764$	6.02388	0.217937
$765$	0	0
$766$	−30.5359	−1.10331
$767$	31.8127	1.14869
$768$	0	0
$769$	−37.9282	−1.36773	−0.683863	−	0.729610i	$-0.739701\pi$
$769$	−37.9282	−1.36773	−0.683863	+	0.729610i	$0.739701\pi$
$770$	21.4641	0.773513
$771$	0	0
$772$	−2.79569	−0.100619
$773$	3.81899	0.137360	0.0686798	−	0.997639i	$-0.478121\pi$
$773$	3.81899	0.137360	0.0686798	+	0.997639i	$0.478121\pi$
$774$	0	0
$775$	−12.0478	−0.432769
$776$	34.3013	1.23134
$777$	0	0
$778$	−15.7128	−0.563332
$779$	47.9282	1.71721
$780$	0	0
$781$	11.0718	0.396180
$782$	0	0
$783$	0	0
$784$	−44.3205	−1.58288
$785$	−15.7128	−0.560814
$786$	0	0
$787$	−46.5770	−1.66029	−0.830145	−	0.557547i	$-0.811742\pi$
$787$	−46.5770	−1.66029	−0.830145	+	0.557547i	$0.811742\pi$
$788$	4.46410	0.159027
$789$	0	0
$790$	−26.4449	−0.940866
$791$	−47.1678	−1.67709
$792$	0	0
$793$	4.92144	0.174765
$794$	3.32051	0.117840
$795$	0	0
$796$	−5.43308	−0.192571
$797$	−51.2822	−1.81651	−0.908254	−	0.418420i	$-0.862584\pi$
$797$	−51.2822	−1.81651	−0.908254	+	0.418420i	$0.862584\pi$
$798$	0	0
$799$	0	0
$800$	3.01194	0.106488
$801$	0	0
$802$	−19.6568	−0.694105
$803$	14.2527	0.502966
$804$	0	0
$805$	−60.3180	−2.12593
$806$	−40.4974	−1.42646
$807$	0	0
$808$	32.1962	1.13266
$809$	−26.1244	−0.918483	−0.459242	−	0.888311i	$-0.651879\pi$
$809$	−26.1244	−0.918483	−0.459242	+	0.888311i	$0.651879\pi$
$810$	0	0
$811$	34.8246	1.22286	0.611429	−	0.791299i	$-0.290595\pi$
$811$	34.8246	1.22286	0.611429	+	0.791299i	$0.290595\pi$
$812$	−1.90949	−0.0670101
$813$	0	0
$814$	28.1051	0.985084
$815$	10.4337	0.365476
$816$	0	0
$817$	1.92820	0.0674593
$818$	−6.13201	−0.214401
$819$	0	0
$820$	3.09103	0.107944
$821$	28.5359	0.995910	0.497955	−	0.867203i	$-0.334085\pi$
$821$	28.5359	0.995910	0.497955	+	0.867203i	$0.334085\pi$
$822$	0	0
$823$	4.40979	0.153716	0.0768578	−	0.997042i	$-0.475511\pi$
$823$	4.40979	0.153716	0.0768578	+	0.997042i	$0.475511\pi$
$824$	17.7473	0.618256
$825$	0	0
$826$	44.1558	1.53638
$827$	41.7846	1.45299	0.726497	−	0.687170i	$-0.241147\pi$
$827$	41.7846	1.45299	0.726497	+	0.687170i	$0.241147\pi$
$828$	0	0
$829$	18.3205	0.636298	0.318149	−	0.948041i	$-0.396939\pi$
$829$	18.3205	0.636298	0.318149	+	0.948041i	$0.396939\pi$
$830$	0	0
$831$	0	0
$832$	−29.7321	−1.03077
$833$	0	0
$834$	0	0
$835$	12.0000	0.415277
$836$	−3.85641	−0.133377
$837$	0	0
$838$	−9.14395	−0.315873
$839$	−13.5359	−0.467311	−0.233656	−	0.972319i	$-0.575069\pi$
$839$	−13.5359	−0.467311	−0.233656	+	0.972319i	$0.575069\pi$
$840$	0	0
$841$	−26.0000	−0.896552
$842$	−6.02388	−0.207597
$843$	0	0
$844$	0.0791522	0.00272453
$845$	−12.0000	−0.412813
$846$	0	0
$847$	28.8007	0.989605
$848$	8.52418	0.292722
$849$	0	0
$850$	0	0
$851$	−78.9805	−2.70742
$852$	0	0
$853$	−28.5053	−0.976004	−0.488002	−	0.872843i	$-0.662274\pi$
$853$	−28.5053	−0.976004	−0.488002	+	0.872843i	$0.662274\pi$
$854$	6.83093	0.233750
$855$	0	0
$856$	−7.82526	−0.267462
$857$	24.5359	0.838130	0.419065	−	0.907956i	$-0.362358\pi$
$857$	24.5359	0.838130	0.419065	+	0.907956i	$0.362358\pi$
$858$	0	0
$859$	−30.1436	−1.02849	−0.514243	−	0.857644i	$-0.671927\pi$
$859$	−30.1436	−1.02849	−0.514243	+	0.857644i	$0.671927\pi$
$860$	0.124356	0.00424049
$861$	0	0
$862$	41.2520	1.40505
$863$	10.4337	0.355166	0.177583	−	0.984106i	$-0.443172\pi$
$863$	10.4337	0.355166	0.177583	+	0.984106i	$0.443172\pi$
$864$	0	0
$865$	−25.8564	−0.879144
$866$	25.6015	0.869975
$867$	0	0
$868$	−6.64102	−0.225411
$869$	20.2765	0.687835
$870$	0	0
$871$	−19.0526	−0.645571
$872$	34.3013	1.16159
$873$	0	0
$874$	91.7271	3.10272
$875$	−49.8843	−1.68640
$876$	0	0
$877$	−38.3482	−1.29493	−0.647463	−	0.762097i	$-0.724170\pi$
$877$	−38.3482	−1.29493	−0.647463	+	0.762097i	$0.724170\pi$
$878$	−12.8372	−0.433233
$879$	0	0
$880$	15.4641	0.521295
$881$	43.9808	1.48175	0.740875	−	0.671643i	$-0.234411\pi$
$881$	43.9808	1.48175	0.740875	+	0.671643i	$0.234411\pi$
$882$	0	0
$883$	−48.5167	−1.63272	−0.816358	−	0.577546i	$-0.804010\pi$
$883$	−48.5167	−1.63272	−0.816358	+	0.577546i	$0.804010\pi$
$884$	0	0
$885$	0	0
$886$	−8.30127	−0.278887
$887$	26.0000	0.872995	0.436497	−	0.899706i	$-0.356219\pi$
$887$	26.0000	0.872995	0.436497	+	0.899706i	$0.356219\pi$
$888$	0	0
$889$	−44.9629	−1.50801
$890$	−5.75129	−0.192784
$891$	0	0
$892$	−2.03332	−0.0680806
$893$	67.1489	2.24705
$894$	0	0
$895$	−10.4337	−0.348759
$896$	−53.6603	−1.79266
$897$	0	0
$898$	−19.6857	−0.656922
$899$	10.4337	0.347983
$900$	0	0
$901$	0	0
$902$	−20.0603	−0.667935
$903$	0	0
$904$	−29.9032	−0.994565
$905$	−10.4337	−0.346827
$906$	0	0
$907$	−28.8007	−0.956312	−0.478156	−	0.878275i	$-0.658695\pi$
$907$	−28.8007	−0.956312	−0.478156	+	0.878275i	$0.658695\pi$
$908$	−4.76537	−0.158144
$909$	0	0
$910$	−47.9090	−1.58817
$911$	−7.24871	−0.240161	−0.120080	−	0.992764i	$-0.538315\pi$
$911$	−7.24871	−0.240161	−0.120080	+	0.992764i	$0.538315\pi$
$912$	0	0
$913$	0	0
$914$	−14.8435	−0.490978
$915$	0	0
$916$	−1.44486	−0.0477396
$917$	−4.11439	−0.135869
$918$	0	0
$919$	37.9808	1.25287	0.626435	−	0.779474i	$-0.284513\pi$
$919$	37.9808	1.25287	0.626435	+	0.779474i	$0.284513\pi$
$920$	−38.2401	−1.26074
$921$	0	0
$922$	28.1051	0.925593
$923$	−24.7128	−0.813432
$924$	0	0
$925$	−18.6625	−0.613618
$926$	38.1320	1.25309
$927$	0	0
$928$	−2.60842	−0.0856255
$929$	−18.1244	−0.594641	−0.297320	−	0.954778i	$-0.596093\pi$
$929$	−18.1244	−0.594641	−0.297320	+	0.954778i	$0.596093\pi$
$930$	0	0
$931$	−71.4449	−2.34151
$932$	6.07180	0.198888
$933$	0	0
$934$	4.09103	0.133863
$935$	0	0
$936$	0	0
$937$	−42.8564	−1.40006	−0.700029	−	0.714115i	$-0.746830\pi$
$937$	−42.8564	−1.40006	−0.700029	+	0.714115i	$0.746830\pi$
$938$	−26.4449	−0.863455
$939$	0	0
$940$	4.33064	0.141250
$941$	10.3923	0.338779	0.169390	−	0.985549i	$-0.445820\pi$
$941$	10.3923	0.338779	0.169390	+	0.985549i	$0.445820\pi$
$942$	0	0
$943$	56.3731	1.83576
$944$	31.8127	1.03541
$945$	0	0
$946$	−0.807048	−0.0262394
$947$	14.7846	0.480435	0.240218	−	0.970719i	$-0.422781\pi$
$947$	14.7846	0.480435	0.240218	+	0.970719i	$0.422781\pi$
$948$	0	0
$949$	−31.8127	−1.03268
$950$	21.6744	0.703210
$951$	0	0
$952$	0	0
$953$	38.9390	1.26136	0.630679	−	0.776044i	$-0.282776\pi$
$953$	38.9390	1.26136	0.630679	+	0.776044i	$0.282776\pi$
$954$	0	0
$955$	−38.9390	−1.26004
$956$	−7.34258	−0.237476
$957$	0	0
$958$	60.5053	1.95484
$959$	−9.07180	−0.292944
$960$	0	0
$961$	5.28719	0.170554
$962$	−62.7321	−2.02256
$963$	0	0
$964$	−2.20489	−0.0710149
$965$	18.0717	0.581747
$966$	0	0
$967$	27.5885	0.887185	0.443592	−	0.896229i	$-0.353704\pi$
$967$	27.5885	0.887185	0.443592	+	0.896229i	$0.353704\pi$
$968$	18.2589	0.586864
$969$	0	0
$970$	34.3013	1.10135
$971$	−17.5600	−0.563527	−0.281764	−	0.959484i	$-0.590919\pi$
$971$	−17.5600	−0.563527	−0.281764	+	0.959484i	$0.590919\pi$
$972$	0	0
$973$	67.7128	2.17077
$974$	43.8179	1.40402
$975$	0	0
$976$	4.92144	0.157531
$977$	10.1383	0.324352	0.162176	−	0.986762i	$-0.448149\pi$
$977$	10.1383	0.324352	0.162176	+	0.986762i	$0.448149\pi$
$978$	0	0
$979$	4.40979	0.140937
$980$	−4.60770	−0.147187
$981$	0	0
$982$	−61.0718	−1.94888
$983$	−22.3205	−0.711914	−0.355957	−	0.934502i	$-0.615845\pi$
$983$	−22.3205	−0.711914	−0.355957	+	0.934502i	$0.615845\pi$
$984$	0	0
$985$	−28.8564	−0.919442
$986$	0	0
$987$	0	0
$988$	8.60770	0.273847
$989$	2.26795	0.0721166
$990$	0	0
$991$	−34.2338	−1.08747	−0.543736	−	0.839256i	$-0.682991\pi$
$991$	−34.2338	−1.08747	−0.543736	+	0.839256i	$0.682991\pi$
$992$	−9.07180	−0.288030
$993$	0	0
$994$	−34.3013	−1.08797
$995$	35.1200	1.11338
$996$	0	0
$997$	4.40979	0.139659	0.0698297	−	0.997559i	$-0.477754\pi$
$997$	4.40979	0.139659	0.0698297	+	0.997559i	$0.477754\pi$
$998$	−2.87564	−0.0910269
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7803.2.a.bi.1.3		4
3.2	odd	2		7803.2.a.bh.1.2		4
17.4	even	4		459.2.d.b.271.4	yes	8
17.13	even	4		459.2.d.b.271.3	✓	8
17.16	even	2		7803.2.a.bh.1.3		4
51.38	odd	4		459.2.d.b.271.5	yes	8
51.47	odd	4		459.2.d.b.271.6	yes	8
51.50	odd	2	inner	7803.2.a.bi.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.2.d.b.271.3	✓	8	17.13	even	4
459.2.d.b.271.4	yes	8	17.4	even	4
459.2.d.b.271.5	yes	8	51.38	odd	4
459.2.d.b.271.6	yes	8	51.47	odd	4
7803.2.a.bh.1.2		4	3.2	odd	2
7803.2.a.bh.1.3		4	17.16	even	2
7803.2.a.bi.1.2		4	51.50	odd	2	inner
7803.2.a.bi.1.3		4	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:	\( N \)	\(=\)	\( 7803 = 3^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7803.a (trivial)

\(f(q)\)	\(=\)	\(q+1.50597 q^{2} +0.267949 q^{4} -1.73205 q^{5} -4.11439 q^{7} -2.60842 q^{8} +O(q^{10})\)	\(q+1.50597 q^{2} +0.267949 q^{4} -1.73205 q^{5} -4.11439 q^{7} -2.60842 q^{8} -2.60842 q^{10} +2.00000 q^{11} -4.46410 q^{13} -6.19615 q^{14} -4.46410 q^{16} -7.19615 q^{19} -0.464102 q^{20} +3.01194 q^{22} -8.46410 q^{23} -2.00000 q^{25} -6.72281 q^{26} -1.10245 q^{28} +1.73205 q^{29} +6.02388 q^{31} -1.50597 q^{32} +7.12633 q^{35} +9.33123 q^{37} -10.8372 q^{38} +4.51791 q^{40} -6.66025 q^{41} -0.267949 q^{43} +0.535898 q^{44} -12.7467 q^{46} -9.33123 q^{47} +9.92820 q^{49} -3.01194 q^{50} -1.19615 q^{52} -1.90949 q^{53} -3.46410 q^{55} +10.7321 q^{56} +2.60842 q^{58} -7.12633 q^{59} -1.10245 q^{61} +9.07180 q^{62} +6.66025 q^{64} +7.73205 q^{65} +4.26795 q^{67} +10.7321 q^{70} +5.53590 q^{71} +7.12633 q^{73} +14.0526 q^{74} -1.92820 q^{76} -8.22878 q^{77} +10.1383 q^{79} +7.73205 q^{80} -10.0302 q^{82} -0.403524 q^{86} -5.21684 q^{88} +2.20489 q^{89} +18.3671 q^{91} -2.26795 q^{92} -14.0526 q^{94} +12.4641 q^{95} -13.1502 q^{97} +14.9516 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 4 * q + 8 * q^4	\( 4 q + 8 q^{4} + 8 q^{11} - 4 q^{13} - 4 q^{14} - 4 q^{16} - 8 q^{19} + 12 q^{20} - 20 q^{23} - 8 q^{25} + 8 q^{41} - 8 q^{43} + 16 q^{44} + 12 q^{49} + 16 q^{52} + 36 q^{56} + 64 q^{62} - 8 q^{64} + 24 q^{65} + 24 q^{67} + 36 q^{70} + 36 q^{71} - 20 q^{74} + 20 q^{76} + 24 q^{80} - 16 q^{92} + 20 q^{94} + 36 q^{95}+O(q^{100}) \) 4 * q + 8 * q^4 + 8 * q^11 - 4 * q^13 - 4 * q^14 - 4 * q^16 - 8 * q^19 + 12 * q^20 - 20 * q^23 - 8 * q^25 + 8 * q^41 - 8 * q^43 + 16 * q^44 + 12 * q^49 + 16 * q^52 + 36 * q^56 + 64 * q^62 - 8 * q^64 + 24 * q^65 + 24 * q^67 + 36 * q^70 + 36 * q^71 - 20 * q^74 + 20 * q^76 + 24 * q^80 - 16 * q^92 + 20 * q^94 + 36 * q^95

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