LMFDB - Embedded newform 1287.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1287,2,Mod(1,1287)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1287, 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("1287.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1287 = 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1287.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:	$10.2767467401$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 429)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.571993$ of defining polynomial
Character	$\chi$	$=$	1287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.571993 q^{2} -1.67282 q^{4} +4.24482 q^{5} -2.67282 q^{7} -2.10083 q^{8} +O(q^{10})$	\(q+0.571993 q^{2} -1.67282 q^{4} +4.24482 q^{5} -2.67282 q^{7} -2.10083 q^{8} +2.42801 q^{10} -1.00000 q^{11} -1.00000 q^{13} -1.52884 q^{14} +2.14399 q^{16} +0.428007 q^{17} +6.67282 q^{19} -7.10083 q^{20} -0.571993 q^{22} +7.81681 q^{23} +13.0185 q^{25} -0.571993 q^{26} +4.47116 q^{28} +2.91764 q^{29} +1.75518 q^{31} +5.42801 q^{32} +0.244817 q^{34} -11.3456 q^{35} +7.63362 q^{37} +3.81681 q^{38} -8.91764 q^{40} +2.38485 q^{41} -11.7737 q^{43} +1.67282 q^{44} +4.47116 q^{46} -6.48963 q^{47} +0.143987 q^{49} +7.44648 q^{50} +1.67282 q^{52} +2.85601 q^{53} -4.24482 q^{55} +5.61515 q^{56} +1.66887 q^{58} -12.4896 q^{59} +3.14399 q^{61} +1.00395 q^{62} -1.18319 q^{64} -4.24482 q^{65} +13.3888 q^{67} -0.715980 q^{68} -6.48963 q^{70} +7.34565 q^{71} +11.7305 q^{73} +4.36638 q^{74} -11.1625 q^{76} +2.67282 q^{77} +6.42801 q^{79} +9.10083 q^{80} +1.36412 q^{82} -1.79834 q^{83} +1.81681 q^{85} -6.73445 q^{86} +2.10083 q^{88} -3.59046 q^{89} +2.67282 q^{91} -13.0761 q^{92} -3.71203 q^{94} +28.3249 q^{95} -11.6336 q^{97} +0.0823593 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 5 q^{4} + 2 q^{5} + 2 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + q^2 + 5 * q^4 + 2 * q^5 + 2 * q^7 + 3 * q^8	$ 3 q + q^{2} + 5 q^{4} + 2 q^{5} + 2 q^{7} + 3 q^{8} + 8 q^{10} - 3 q^{11} - 3 q^{13} + 4 q^{14} + 5 q^{16} + 2 q^{17} + 10 q^{19} - 12 q^{20} - q^{22} + 12 q^{23} + 9 q^{25} - q^{26} + 22 q^{28} - 12 q^{29} + 16 q^{31} + 17 q^{32} - 10 q^{34} - 14 q^{35} - 6 q^{40} - 16 q^{43} - 5 q^{44} + 22 q^{46} + 2 q^{47} - q^{49} - 7 q^{50} - 5 q^{52} + 10 q^{53} - 2 q^{55} + 24 q^{56} - 16 q^{59} + 8 q^{61} - 2 q^{62} - 15 q^{64} - 2 q^{65} + 28 q^{67} + 2 q^{70} + 2 q^{71} + 8 q^{73} + 36 q^{74} - 2 q^{76} - 2 q^{77} + 20 q^{79} + 18 q^{80} - 46 q^{82} - 24 q^{83} - 6 q^{85} + 12 q^{86} - 3 q^{88} + 20 q^{89} - 2 q^{91} + 8 q^{92} - 14 q^{94} + 22 q^{95} - 12 q^{97} + 21 q^{98}+O(q^{100}) $ 3 * q + q^2 + 5 * q^4 + 2 * q^5 + 2 * q^7 + 3 * q^8 + 8 * q^10 - 3 * q^11 - 3 * q^13 + 4 * q^14 + 5 * q^16 + 2 * q^17 + 10 * q^19 - 12 * q^20 - q^22 + 12 * q^23 + 9 * q^25 - q^26 + 22 * q^28 - 12 * q^29 + 16 * q^31 + 17 * q^32 - 10 * q^34 - 14 * q^35 - 6 * q^40 - 16 * q^43 - 5 * q^44 + 22 * q^46 + 2 * q^47 - q^49 - 7 * q^50 - 5 * q^52 + 10 * q^53 - 2 * q^55 + 24 * q^56 - 16 * q^59 + 8 * q^61 - 2 * q^62 - 15 * q^64 - 2 * q^65 + 28 * q^67 + 2 * q^70 + 2 * q^71 + 8 * q^73 + 36 * q^74 - 2 * q^76 - 2 * q^77 + 20 * q^79 + 18 * q^80 - 46 * q^82 - 24 * q^83 - 6 * q^85 + 12 * q^86 - 3 * q^88 + 20 * q^89 - 2 * q^91 + 8 * q^92 - 14 * q^94 + 22 * q^95 - 12 * q^97 + 21 * q^98

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 $

$2$	0.571993	0.404460	0.202230	−	0.979338i	$-0.435181\pi$
$2$	0.571993	0.404460	0.202230	+	0.979338i	$0.435181\pi$
$3$	0	0
$3$	0	0
$4$	−1.67282	−0.836412
$5$	4.24482	1.89834	0.949170	−	0.314764i	$-0.101925\pi$
$5$	4.24482	1.89834	0.949170	+	0.314764i	$0.101925\pi$
$6$	0	0
$7$	−2.67282	−1.01023	−0.505116	−	0.863051i	$-0.668550\pi$
$7$	−2.67282	−1.01023	−0.505116	+	0.863051i	$0.668550\pi$
$8$	−2.10083	−0.742756
$9$	0	0
$10$	2.42801	0.767803
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.52884	−0.408599
$15$	0	0
$16$	2.14399	0.535997
$17$	0.428007	0.103807	0.0519034	−	0.998652i	$-0.483471\pi$
$17$	0.428007	0.103807	0.0519034	+	0.998652i	$0.483471\pi$
$18$	0	0
$19$	6.67282	1.53085	0.765425	−	0.643525i	$-0.222529\pi$
$19$	6.67282	1.53085	0.765425	+	0.643525i	$0.222529\pi$
$20$	−7.10083	−1.58779
$21$	0	0
$22$	−0.571993	−0.121949
$23$	7.81681	1.62992	0.814959	−	0.579519i	$-0.196760\pi$
$23$	7.81681	1.62992	0.814959	+	0.579519i	$0.196760\pi$
$24$	0	0
$25$	13.0185	2.60369
$26$	−0.571993	−0.112177
$27$	0	0
$28$	4.47116	0.844970
$29$	2.91764	0.541792	0.270896	−	0.962609i	$-0.412680\pi$
$29$	2.91764	0.541792	0.270896	+	0.962609i	$0.412680\pi$
$30$	0	0
$31$	1.75518	0.315240	0.157620	−	0.987500i	$-0.449618\pi$
$31$	1.75518	0.315240	0.157620	+	0.987500i	$0.449618\pi$
$32$	5.42801	0.959545
$33$	0	0
$34$	0.244817	0.0419858
$35$	−11.3456	−1.91776
$36$	0	0
$37$	7.63362	1.25496	0.627480	−	0.778633i	$-0.284087\pi$
$37$	7.63362	1.25496	0.627480	+	0.778633i	$0.284087\pi$
$38$	3.81681	0.619168
$39$	0	0
$40$	−8.91764	−1.41000
$41$	2.38485	0.372451	0.186226	−	0.982507i	$-0.440374\pi$
$41$	2.38485	0.372451	0.186226	+	0.982507i	$0.440374\pi$
$42$	0	0
$43$	−11.7737	−1.79547	−0.897733	−	0.440541i	$-0.854787\pi$
$43$	−11.7737	−1.79547	−0.897733	+	0.440541i	$0.854787\pi$
$44$	1.67282	0.252188
$45$	0	0
$46$	4.47116	0.659237
$47$	−6.48963	−0.946610	−0.473305	−	0.880899i	$-0.656939\pi$
$47$	−6.48963	−0.946610	−0.473305	+	0.880899i	$0.656939\pi$
$48$	0	0
$49$	0.143987	0.0205695
$50$	7.44648	1.05309
$51$	0	0
$52$	1.67282	0.231979
$53$	2.85601	0.392304	0.196152	−	0.980574i	$-0.437155\pi$
$53$	2.85601	0.392304	0.196152	+	0.980574i	$0.437155\pi$
$54$	0	0
$55$	−4.24482	−0.572371
$56$	5.61515	0.750356
$57$	0	0
$58$	1.66887	0.219133
$59$	−12.4896	−1.62601	−0.813006	−	0.582255i	$-0.802170\pi$
$59$	−12.4896	−1.62601	−0.813006	+	0.582255i	$0.802170\pi$
$60$	0	0
$61$	3.14399	0.402546	0.201273	−	0.979535i	$-0.435492\pi$
$61$	3.14399	0.402546	0.201273	+	0.979535i	$0.435492\pi$
$62$	1.00395	0.127502
$63$	0	0
$64$	−1.18319	−0.147899
$65$	−4.24482	−0.526505
$66$	0	0
$67$	13.3888	1.63570	0.817851	−	0.575430i	$-0.195165\pi$
$67$	13.3888	1.63570	0.817851	+	0.575430i	$0.195165\pi$
$68$	−0.715980	−0.0868253
$69$	0	0
$70$	−6.48963	−0.775660
$71$	7.34565	0.871768	0.435884	−	0.900003i	$-0.356436\pi$
$71$	7.34565	0.871768	0.435884	+	0.900003i	$0.356436\pi$
$72$	0	0
$73$	11.7305	1.37295	0.686475	−	0.727153i	$-0.259157\pi$
$73$	11.7305	1.37295	0.686475	+	0.727153i	$0.259157\pi$
$74$	4.36638	0.507581
$75$	0	0
$76$	−11.1625	−1.28042
$77$	2.67282	0.304597
$78$	0	0
$79$	6.42801	0.723207	0.361604	−	0.932332i	$-0.382229\pi$
$79$	6.42801	0.723207	0.361604	+	0.932332i	$0.382229\pi$
$80$	9.10083	1.01750
$81$	0	0
$82$	1.36412	0.150642
$83$	−1.79834	−0.197393	−0.0986967	−	0.995118i	$-0.531467\pi$
$83$	−1.79834	−0.197393	−0.0986967	+	0.995118i	$0.531467\pi$
$84$	0	0
$85$	1.81681	0.197061
$86$	−6.73445	−0.726195
$87$	0	0
$88$	2.10083	0.223949
$89$	−3.59046	−0.380588	−0.190294	−	0.981727i	$-0.560944\pi$
$89$	−3.59046	−0.380588	−0.190294	+	0.981727i	$0.560944\pi$
$90$	0	0
$91$	2.67282	0.280188
$92$	−13.0761	−1.36328
$93$	0	0
$94$	−3.71203	−0.382866
$95$	28.3249	2.90607
$96$	0	0
$97$	−11.6336	−1.18122	−0.590608	−	0.806959i	$-0.701112\pi$
$97$	−11.6336	−1.18122	−0.590608	+	0.806959i	$0.701112\pi$
$98$	0.0823593	0.00831955
$99$	0	0
$100$	−21.7776	−2.17776
$101$	−7.48568	−0.744853	−0.372427	−	0.928062i	$-0.621474\pi$
$101$	−7.48568	−0.744853	−0.372427	+	0.928062i	$0.621474\pi$
$102$	0	0
$103$	1.71203	0.168691	0.0843455	−	0.996437i	$-0.473120\pi$
$103$	1.71203	0.168691	0.0843455	+	0.996437i	$0.473120\pi$
$104$	2.10083	0.206003
$105$	0	0
$106$	1.63362	0.158671
$107$	−4.48963	−0.434029	−0.217015	−	0.976168i	$-0.569632\pi$
$107$	−4.48963	−0.434029	−0.217015	+	0.976168i	$0.569632\pi$
$108$	0	0
$109$	−10.0185	−0.959595	−0.479798	−	0.877379i	$-0.659290\pi$
$109$	−10.0185	−0.959595	−0.479798	+	0.877379i	$0.659290\pi$
$110$	−2.42801	−0.231501
$111$	0	0
$112$	−5.73050	−0.541481
$113$	10.4896	0.986782	0.493391	−	0.869808i	$-0.335757\pi$
$113$	10.4896	0.986782	0.493391	+	0.869808i	$0.335757\pi$
$114$	0	0
$115$	33.1809	3.09414
$116$	−4.88070	−0.453161
$117$	0	0
$118$	−7.14399	−0.657657
$119$	−1.14399	−0.104869
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.79834	0.162814
$123$	0	0
$124$	−2.93611	−0.263671
$125$	34.0369	3.04436
$126$	0	0
$127$	10.6297	0.943230	0.471615	−	0.881804i	$-0.343671\pi$
$127$	10.6297	0.943230	0.471615	+	0.881804i	$0.343671\pi$
$128$	−11.5328	−1.01936
$129$	0	0
$130$	−2.42801	−0.212950
$131$	−19.6336	−1.71540	−0.857699	−	0.514153i	$-0.828106\pi$
$131$	−19.6336	−1.71540	−0.857699	+	0.514153i	$0.828106\pi$
$132$	0	0
$133$	−17.8353	−1.54652
$134$	7.65831	0.661577
$135$	0	0
$136$	−0.899170	−0.0771032
$137$	8.81286	0.752933	0.376467	−	0.926430i	$-0.377139\pi$
$137$	8.81286	0.752933	0.376467	+	0.926430i	$0.377139\pi$
$138$	0	0
$139$	−9.40727	−0.797915	−0.398957	−	0.916970i	$-0.630628\pi$
$139$	−9.40727	−0.797915	−0.398957	+	0.916970i	$0.630628\pi$
$140$	18.9793	1.60404
$141$	0	0
$142$	4.20166	0.352596
$143$	1.00000	0.0836242
$144$	0	0
$145$	12.3849	1.02851
$146$	6.70977	0.555304
$147$	0	0
$148$	−12.7697	−1.04966
$149$	4.67282	0.382813	0.191406	−	0.981511i	$-0.438695\pi$
$149$	4.67282	0.382813	0.191406	+	0.981511i	$0.438695\pi$
$150$	0	0
$151$	−5.73050	−0.466341	−0.233171	−	0.972436i	$-0.574910\pi$
$151$	−5.73050	−0.466341	−0.233171	+	0.972436i	$0.574910\pi$
$152$	−14.0185	−1.13705
$153$	0	0
$154$	1.52884	0.123197
$155$	7.45043	0.598433
$156$	0	0
$157$	8.38485	0.669184	0.334592	−	0.942363i	$-0.391402\pi$
$157$	8.38485	0.669184	0.334592	+	0.942363i	$0.391402\pi$
$158$	3.67678	0.292509
$159$	0	0
$160$	23.0409	1.82154
$161$	−20.8930	−1.64660
$162$	0	0
$163$	−4.82076	−0.377591	−0.188796	−	0.982016i	$-0.560458\pi$
$163$	−4.82076	−0.377591	−0.188796	+	0.982016i	$0.560458\pi$
$164$	−3.98943	−0.311523
$165$	0	0
$166$	−1.02864	−0.0798378
$167$	−19.6336	−1.51930	−0.759648	−	0.650335i	$-0.774629\pi$
$167$	−19.6336	−1.51930	−0.759648	+	0.650335i	$0.774629\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	1.03920	0.0797032
$171$	0	0
$172$	19.6952	1.50175
$173$	−4.42801	−0.336655	−0.168328	−	0.985731i	$-0.553837\pi$
$173$	−4.42801	−0.336655	−0.168328	+	0.985731i	$0.553837\pi$
$174$	0	0
$175$	−34.7961	−2.63034
$176$	−2.14399	−0.161609
$177$	0	0
$178$	−2.05372	−0.153933
$179$	−13.2488	−0.990260	−0.495130	−	0.868819i	$-0.664879\pi$
$179$	−13.2488	−0.990260	−0.495130	+	0.868819i	$0.664879\pi$
$180$	0	0
$181$	11.4425	0.850516	0.425258	−	0.905072i	$-0.360183\pi$
$181$	11.4425	0.850516	0.425258	+	0.905072i	$0.360183\pi$
$182$	1.52884	0.113325
$183$	0	0
$184$	−16.4218	−1.21063
$185$	32.4033	2.38234
$186$	0	0
$187$	−0.428007	−0.0312990
$188$	10.8560	0.791756
$189$	0	0
$190$	16.2017	1.17539
$191$	−0.489634	−0.0354287	−0.0177143	−	0.999843i	$-0.505639\pi$
$191$	−0.489634	−0.0354287	−0.0177143	+	0.999843i	$0.505639\pi$
$192$	0	0
$193$	−1.03920	−0.0748035	−0.0374017	−	0.999300i	$-0.511908\pi$
$193$	−1.03920	−0.0748035	−0.0374017	+	0.999300i	$0.511908\pi$
$194$	−6.65435	−0.477755
$195$	0	0
$196$	−0.240864	−0.0172046
$197$	−9.16246	−0.652798	−0.326399	−	0.945232i	$-0.605835\pi$
$197$	−9.16246	−0.652798	−0.326399	+	0.945232i	$0.605835\pi$
$198$	0	0
$199$	−8.40332	−0.595696	−0.297848	−	0.954613i	$-0.596269\pi$
$199$	−8.40332	−0.595696	−0.297848	+	0.954613i	$0.596269\pi$
$200$	−27.3496	−1.93391
$201$	0	0
$202$	−4.28176	−0.301264
$203$	−7.79834	−0.547336
$204$	0	0
$205$	10.1233	0.707039
$206$	0.979268	0.0682288
$207$	0	0
$208$	−2.14399	−0.148659
$209$	−6.67282	−0.461569
$210$	0	0
$211$	14.4649	0.995808	0.497904	−	0.867232i	$-0.334103\pi$
$211$	14.4649	0.995808	0.497904	+	0.867232i	$0.334103\pi$
$212$	−4.77761	−0.328127
$213$	0	0
$214$	−2.56804	−0.175548
$215$	−49.9770	−3.40840
$216$	0	0
$217$	−4.69129	−0.318466
$218$	−5.73050	−0.388118
$219$	0	0
$220$	7.10083	0.478738
$221$	−0.428007	−0.0287908
$222$	0	0
$223$	−2.73445	−0.183112	−0.0915562	−	0.995800i	$-0.529184\pi$
$223$	−2.73445	−0.183112	−0.0915562	+	0.995800i	$0.529184\pi$
$224$	−14.5081	−0.969364
$225$	0	0
$226$	6.00000	0.399114
$227$	−18.7961	−1.24754	−0.623770	−	0.781608i	$-0.714400\pi$
$227$	−18.7961	−1.24754	−0.623770	+	0.781608i	$0.714400\pi$
$228$	0	0
$229$	4.48963	0.296683	0.148342	−	0.988936i	$-0.452606\pi$
$229$	4.48963	0.296683	0.148342	+	0.988936i	$0.452606\pi$
$230$	18.9793	1.25146
$231$	0	0
$232$	−6.12947	−0.402419
$233$	11.0040	0.720893	0.360446	−	0.932780i	$-0.382624\pi$
$233$	11.0040	0.720893	0.360446	+	0.932780i	$0.382624\pi$
$234$	0	0
$235$	−27.5473	−1.79699
$236$	20.8930	1.36002
$237$	0	0
$238$	−0.654353	−0.0424154
$239$	−25.2778	−1.63509	−0.817543	−	0.575868i	$-0.804664\pi$
$239$	−25.2778	−1.63509	−0.817543	+	0.575868i	$0.804664\pi$
$240$	0	0
$241$	15.2593	0.982940	0.491470	−	0.870894i	$-0.336460\pi$
$241$	15.2593	0.982940	0.491470	+	0.870894i	$0.336460\pi$
$242$	0.571993	0.0367691
$243$	0	0
$244$	−5.25934	−0.336694
$245$	0.611196	0.0390479
$246$	0	0
$247$	−6.67282	−0.424582
$248$	−3.68734	−0.234146
$249$	0	0
$250$	19.4689	1.23132
$251$	−7.91369	−0.499508	−0.249754	−	0.968309i	$-0.580350\pi$
$251$	−7.91369	−0.499508	−0.249754	+	0.968309i	$0.580350\pi$
$252$	0	0
$253$	−7.81681	−0.491439
$254$	6.08010	0.381499
$255$	0	0
$256$	−4.23030	−0.264394
$257$	15.3456	0.957235	0.478618	−	0.878023i	$-0.341138\pi$
$257$	15.3456	0.957235	0.478618	+	0.878023i	$0.341138\pi$
$258$	0	0
$259$	−20.4033	−1.26780
$260$	7.10083	0.440375
$261$	0	0
$262$	−11.2303	−0.693810
$263$	26.7776	1.65118	0.825589	−	0.564272i	$-0.190843\pi$
$263$	26.7776	1.65118	0.825589	+	0.564272i	$0.190843\pi$
$264$	0	0
$265$	12.1233	0.744726
$266$	−10.2017	−0.625504
$267$	0	0
$268$	−22.3971	−1.36812
$269$	−6.85601	−0.418019	−0.209009	−	0.977914i	$-0.567024\pi$
$269$	−6.85601	−0.418019	−0.209009	+	0.977914i	$0.567024\pi$
$270$	0	0
$271$	−18.1832	−1.10455	−0.552275	−	0.833662i	$-0.686240\pi$
$271$	−18.1832	−1.10455	−0.552275	+	0.833662i	$0.686240\pi$
$272$	0.917641	0.0556401
$273$	0	0
$274$	5.04090	0.304532
$275$	−13.0185	−0.785043
$276$	0	0
$277$	−23.7490	−1.42694	−0.713469	−	0.700687i	$-0.752877\pi$
$277$	−23.7490	−1.42694	−0.713469	+	0.700687i	$0.752877\pi$
$278$	−5.38090	−0.322725
$279$	0	0
$280$	23.8353	1.42443
$281$	−23.6442	−1.41049	−0.705247	−	0.708962i	$-0.749164\pi$
$281$	−23.6442	−1.41049	−0.705247	+	0.708962i	$0.749164\pi$
$282$	0	0
$283$	28.0616	1.66809	0.834045	−	0.551696i	$-0.186019\pi$
$283$	28.0616	1.66809	0.834045	+	0.551696i	$0.186019\pi$
$284$	−12.2880	−0.729157
$285$	0	0
$286$	0.571993	0.0338227
$287$	−6.37429	−0.376262
$288$	0	0
$289$	−16.8168	−0.989224
$290$	7.08405	0.415990
$291$	0	0
$292$	−19.6231	−1.14835
$293$	27.4874	1.60583	0.802915	−	0.596094i	$-0.203281\pi$
$293$	27.4874	1.60583	0.802915	+	0.596094i	$0.203281\pi$
$294$	0	0
$295$	−53.0162	−3.08672
$296$	−16.0369	−0.932128
$297$	0	0
$298$	2.67282	0.154833
$299$	−7.81681	−0.452058
$300$	0	0
$301$	31.4689	1.81384
$302$	−3.27781	−0.188617
$303$	0	0
$304$	14.3064	0.820531
$305$	13.3456	0.764170
$306$	0	0
$307$	9.15455	0.522478	0.261239	−	0.965274i	$-0.415869\pi$
$307$	9.15455	0.522478	0.261239	+	0.965274i	$0.415869\pi$
$308$	−4.47116	−0.254768
$309$	0	0
$310$	4.26160	0.242042
$311$	−15.2409	−0.864230	−0.432115	−	0.901818i	$-0.642233\pi$
$311$	−15.2409	−0.864230	−0.432115	+	0.901818i	$0.642233\pi$
$312$	0	0
$313$	−8.87448	−0.501616	−0.250808	−	0.968037i	$-0.580696\pi$
$313$	−8.87448	−0.501616	−0.250808	+	0.968037i	$0.580696\pi$
$314$	4.79608	0.270658
$315$	0	0
$316$	−10.7529	−0.604899
$317$	−7.01452	−0.393975	−0.196987	−	0.980406i	$-0.563116\pi$
$317$	−7.01452	−0.393975	−0.196987	+	0.980406i	$0.563116\pi$
$318$	0	0
$319$	−2.91764	−0.163357
$320$	−5.02242	−0.280762
$321$	0	0
$322$	−11.9506	−0.665983
$323$	2.85601	0.158913
$324$	0	0
$325$	−13.0185	−0.722135
$326$	−2.75744	−0.152721
$327$	0	0
$328$	−5.01017	−0.276640
$329$	17.3456	0.956296
$330$	0	0
$331$	24.3681	1.33939	0.669695	−	0.742636i	$-0.266425\pi$
$331$	24.3681	1.33939	0.669695	+	0.742636i	$0.266425\pi$
$332$	3.00830	0.165102
$333$	0	0
$334$	−11.2303	−0.614495
$335$	56.8330	3.10512
$336$	0	0
$337$	−28.6050	−1.55821	−0.779106	−	0.626892i	$-0.784327\pi$
$337$	−28.6050	−1.55821	−0.779106	+	0.626892i	$0.784327\pi$
$338$	0.571993	0.0311123
$339$	0	0
$340$	−3.03920	−0.164824
$341$	−1.75518	−0.0950485
$342$	0	0
$343$	18.3249	0.989452
$344$	24.7345	1.33359
$345$	0	0
$346$	−2.53279	−0.136164
$347$	1.42405	0.0764472	0.0382236	−	0.999269i	$-0.487830\pi$
$347$	1.42405	0.0764472	0.0382236	+	0.999269i	$0.487830\pi$
$348$	0	0
$349$	18.1233	0.970116	0.485058	−	0.874482i	$-0.338799\pi$
$349$	18.1233	0.970116	0.485058	+	0.874482i	$0.338799\pi$
$350$	−19.9031	−1.06387
$351$	0	0
$352$	−5.42801	−0.289314
$353$	9.46721	0.503889	0.251944	−	0.967742i	$-0.418930\pi$
$353$	9.46721	0.503889	0.251944	+	0.967742i	$0.418930\pi$
$354$	0	0
$355$	31.1809	1.65491
$356$	6.00621	0.318329
$357$	0	0
$358$	−7.57821	−0.400521
$359$	5.04711	0.266376	0.133188	−	0.991091i	$-0.457479\pi$
$359$	5.04711	0.266376	0.133188	+	0.991091i	$0.457479\pi$
$360$	0	0
$361$	25.5266	1.34350
$362$	6.54505	0.344000
$363$	0	0
$364$	−4.47116	−0.234353
$365$	49.7938	2.60633
$366$	0	0
$367$	16.6992	0.871691	0.435846	−	0.900021i	$-0.356449\pi$
$367$	16.6992	0.871691	0.435846	+	0.900021i	$0.356449\pi$
$368$	16.7591	0.873630
$369$	0	0
$370$	18.5345	0.963562
$371$	−7.63362	−0.396318
$372$	0	0
$373$	28.6050	1.48111	0.740555	−	0.671996i	$-0.234563\pi$
$373$	28.6050	1.48111	0.740555	+	0.671996i	$0.234563\pi$
$374$	−0.244817	−0.0126592
$375$	0	0
$376$	13.6336	0.703100
$377$	−2.91764	−0.150266
$378$	0	0
$379$	−31.8291	−1.63495	−0.817475	−	0.575965i	$-0.804627\pi$
$379$	−31.8291	−1.63495	−0.817475	+	0.575965i	$0.804627\pi$
$380$	−47.3826	−2.43068
$381$	0	0
$382$	−0.280067	−0.0143295
$383$	−1.83528	−0.0937785	−0.0468892	−	0.998900i	$-0.514931\pi$
$383$	−1.83528	−0.0937785	−0.0468892	+	0.998900i	$0.514931\pi$
$384$	0	0
$385$	11.3456	0.578228
$386$	−0.594417	−0.0302550
$387$	0	0
$388$	19.4610	0.987982
$389$	−19.4689	−0.987113	−0.493556	−	0.869714i	$-0.664303\pi$
$389$	−19.4689	−0.987113	−0.493556	+	0.869714i	$0.664303\pi$
$390$	0	0
$391$	3.34565	0.169197
$392$	−0.302491	−0.0152781
$393$	0	0
$394$	−5.24086	−0.264031
$395$	27.2857	1.37289
$396$	0	0
$397$	−21.0162	−1.05477	−0.527387	−	0.849625i	$-0.676828\pi$
$397$	−21.0162	−1.05477	−0.527387	+	0.849625i	$0.676828\pi$
$398$	−4.80664	−0.240935
$399$	0	0
$400$	27.9114	1.39557
$401$	35.2162	1.75861	0.879306	−	0.476257i	$-0.158007\pi$
$401$	35.2162	1.75861	0.879306	+	0.476257i	$0.158007\pi$
$402$	0	0
$403$	−1.75518	−0.0874319
$404$	12.5222	0.623004
$405$	0	0
$406$	−4.46060	−0.221376
$407$	−7.63362	−0.378385
$408$	0	0
$409$	−6.96080	−0.344189	−0.172095	−	0.985080i	$-0.555053\pi$
$409$	−6.96080	−0.344189	−0.172095	+	0.985080i	$0.555053\pi$
$410$	5.79043	0.285969
$411$	0	0
$412$	−2.86392	−0.141095
$413$	33.3826	1.64265
$414$	0	0
$415$	−7.63362	−0.374720
$416$	−5.42801	−0.266130
$417$	0	0
$418$	−3.81681	−0.186686
$419$	18.5944	0.908397	0.454198	−	0.890901i	$-0.349926\pi$
$419$	18.5944	0.908397	0.454198	+	0.890901i	$0.349926\pi$
$420$	0	0
$421$	−30.6129	−1.49198	−0.745990	−	0.665957i	$-0.768024\pi$
$421$	−30.6129	−1.49198	−0.745990	+	0.665957i	$0.768024\pi$
$422$	8.27385	0.402765
$423$	0	0
$424$	−6.00000	−0.291386
$425$	5.57199	0.270281
$426$	0	0
$427$	−8.40332	−0.406665
$428$	7.51037	0.363027
$429$	0	0
$430$	−28.5865	−1.37856
$431$	8.20957	0.395441	0.197720	−	0.980258i	$-0.436646\pi$
$431$	8.20957	0.395441	0.197720	+	0.980258i	$0.436646\pi$
$432$	0	0
$433$	−28.1153	−1.35114	−0.675569	−	0.737297i	$-0.736102\pi$
$433$	−28.1153	−1.35114	−0.675569	+	0.737297i	$0.736102\pi$
$434$	−2.68339	−0.128807
$435$	0	0
$436$	16.7591	0.802617
$437$	52.1602	2.49516
$438$	0	0
$439$	−16.7529	−0.799573	−0.399787	−	0.916608i	$-0.630916\pi$
$439$	−16.7529	−0.799573	−0.399787	+	0.916608i	$0.630916\pi$
$440$	8.91764	0.425132
$441$	0	0
$442$	−0.244817	−0.0116448
$443$	−12.9793	−0.616664	−0.308332	−	0.951279i	$-0.599771\pi$
$443$	−12.9793	−0.616664	−0.308332	+	0.951279i	$0.599771\pi$
$444$	0	0
$445$	−15.2409	−0.722486
$446$	−1.56409	−0.0740617
$447$	0	0
$448$	3.16246	0.149412
$449$	−24.1954	−1.14185	−0.570927	−	0.821001i	$-0.693416\pi$
$449$	−24.1954	−1.14185	−0.570927	+	0.821001i	$0.693416\pi$
$450$	0	0
$451$	−2.38485	−0.112298
$452$	−17.5473	−0.825356
$453$	0	0
$454$	−10.7512	−0.504580
$455$	11.3456	0.531892
$456$	0	0
$457$	−4.32492	−0.202311	−0.101156	−	0.994871i	$-0.532254\pi$
$457$	−4.32492	−0.202311	−0.101156	+	0.994871i	$0.532254\pi$
$458$	2.56804	0.119997
$459$	0	0
$460$	−55.5058	−2.58797
$461$	6.38485	0.297372	0.148686	−	0.988884i	$-0.452496\pi$
$461$	6.38485	0.297372	0.148686	+	0.988884i	$0.452496\pi$
$462$	0	0
$463$	30.6560	1.42471	0.712354	−	0.701821i	$-0.247629\pi$
$463$	30.6560	1.42471	0.712354	+	0.701821i	$0.247629\pi$
$464$	6.25538	0.290399
$465$	0	0
$466$	6.29419	0.291573
$467$	−31.8538	−1.47402	−0.737008	−	0.675884i	$-0.763762\pi$
$467$	−31.8538	−1.47402	−0.737008	+	0.675884i	$0.763762\pi$
$468$	0	0
$469$	−35.7859	−1.65244
$470$	−15.7569	−0.726810
$471$	0	0
$472$	26.2386	1.20773
$473$	11.7737	0.541353
$474$	0	0
$475$	86.8700	3.98587
$476$	1.91369	0.0877137
$477$	0	0
$478$	−14.4587	−0.661327
$479$	−29.8168	−1.36236	−0.681182	−	0.732114i	$-0.738534\pi$
$479$	−29.8168	−1.36236	−0.681182	+	0.732114i	$0.738534\pi$
$480$	0	0
$481$	−7.63362	−0.348063
$482$	8.72824	0.397560
$483$	0	0
$484$	−1.67282	−0.0760374
$485$	−49.3826	−2.24235
$486$	0	0
$487$	−19.4257	−0.880265	−0.440132	−	0.897933i	$-0.645068\pi$
$487$	−19.4257	−0.880265	−0.440132	+	0.897933i	$0.645068\pi$
$488$	−6.60498	−0.298994
$489$	0	0
$490$	0.349600	0.0157933
$491$	−16.7282	−0.754935	−0.377467	−	0.926023i	$-0.623205\pi$
$491$	−16.7282	−0.754935	−0.377467	+	0.926023i	$0.623205\pi$
$492$	0	0
$493$	1.24877	0.0562418
$494$	−3.81681	−0.171726
$495$	0	0
$496$	3.76309	0.168968
$497$	−19.6336	−0.880688
$498$	0	0
$499$	18.0017	0.805866	0.402933	−	0.915229i	$-0.367991\pi$
$499$	18.0017	0.805866	0.402933	+	0.915229i	$0.367991\pi$
$500$	−56.9378	−2.54634
$501$	0	0
$502$	−4.52658	−0.202031
$503$	−30.0448	−1.33963	−0.669817	−	0.742526i	$-0.733627\pi$
$503$	−30.0448	−1.33963	−0.669817	+	0.742526i	$0.733627\pi$
$504$	0	0
$505$	−31.7753	−1.41398
$506$	−4.47116	−0.198767
$507$	0	0
$508$	−17.7816	−0.788929
$509$	−1.79213	−0.0794346	−0.0397173	−	0.999211i	$-0.512646\pi$
$509$	−1.79213	−0.0794346	−0.0397173	+	0.999211i	$0.512646\pi$
$510$	0	0
$511$	−31.3536	−1.38700
$512$	20.6459	0.912428
$513$	0	0
$514$	8.77761	0.387164
$515$	7.26724	0.320233
$516$	0	0
$517$	6.48963	0.285414
$518$	−11.6706	−0.512775
$519$	0	0
$520$	8.91764	0.391064
$521$	29.0946	1.27466	0.637329	−	0.770592i	$-0.280039\pi$
$521$	29.0946	1.27466	0.637329	+	0.770592i	$0.280039\pi$
$522$	0	0
$523$	−20.0695	−0.877579	−0.438790	−	0.898590i	$-0.644593\pi$
$523$	−20.0695	−0.877579	−0.438790	+	0.898590i	$0.644593\pi$
$524$	32.8436	1.43478
$525$	0	0
$526$	15.3166	0.667836
$527$	0.751230	0.0327241
$528$	0	0
$529$	38.1025	1.65663
$530$	6.93442	0.301212
$531$	0	0
$532$	29.8353	1.29352
$533$	−2.38485	−0.103299
$534$	0	0
$535$	−19.0577	−0.823935
$536$	−28.1276	−1.21493
$537$	0	0
$538$	−3.92159	−0.169072
$539$	−0.143987	−0.00620194
$540$	0	0
$541$	24.1153	1.03680	0.518400	−	0.855138i	$-0.326528\pi$
$541$	24.1153	1.03680	0.518400	+	0.855138i	$0.326528\pi$
$542$	−10.4007	−0.446747
$543$	0	0
$544$	2.32322	0.0996074
$545$	−42.5266	−1.82164
$546$	0	0
$547$	32.5019	1.38968	0.694840	−	0.719164i	$-0.255475\pi$
$547$	32.5019	1.38968	0.694840	+	0.719164i	$0.255475\pi$
$548$	−14.7424	−0.629762
$549$	0	0
$550$	−7.44648	−0.317519
$551$	19.4689	0.829403
$552$	0	0
$553$	−17.1809	−0.730607
$554$	−13.5843	−0.577139
$555$	0	0
$556$	15.7367	0.667385
$557$	−41.6890	−1.76642	−0.883211	−	0.468977i	$-0.844623\pi$
$557$	−41.6890	−1.76642	−0.883211	+	0.468977i	$0.844623\pi$
$558$	0	0
$559$	11.7737	0.497973
$560$	−24.3249	−1.02792
$561$	0	0
$562$	−13.5243	−0.570489
$563$	5.74897	0.242290	0.121145	−	0.992635i	$-0.461343\pi$
$563$	5.74897	0.242290	0.121145	+	0.992635i	$0.461343\pi$
$564$	0	0
$565$	44.5266	1.87325
$566$	16.0511	0.674676
$567$	0	0
$568$	−15.4320	−0.647511
$569$	−15.8521	−0.664553	−0.332276	−	0.943182i	$-0.607817\pi$
$569$	−15.8521	−0.664553	−0.332276	+	0.943182i	$0.607817\pi$
$570$	0	0
$571$	−5.57199	−0.233181	−0.116590	−	0.993180i	$-0.537196\pi$
$571$	−5.57199	−0.233181	−0.116590	+	0.993180i	$0.537196\pi$
$572$	−1.67282	−0.0699443
$573$	0	0
$574$	−3.64605	−0.152183
$575$	101.763	4.24381
$576$	0	0
$577$	−2.20957	−0.0919855	−0.0459927	−	0.998942i	$-0.514645\pi$
$577$	−2.20957	−0.0919855	−0.0459927	+	0.998942i	$0.514645\pi$
$578$	−9.61910	−0.400102
$579$	0	0
$580$	−20.7177	−0.860254
$581$	4.80664	0.199413
$582$	0	0
$583$	−2.85601	−0.118284
$584$	−24.6438	−1.01977
$585$	0	0
$586$	15.7226	0.649494
$587$	15.1809	0.626584	0.313292	−	0.949657i	$-0.398568\pi$
$587$	15.1809	0.626584	0.313292	+	0.949657i	$0.398568\pi$
$588$	0	0
$589$	11.7120	0.482586
$590$	−30.3249	−1.24846
$591$	0	0
$592$	16.3664	0.672654
$593$	7.81681	0.320998	0.160499	−	0.987036i	$-0.448690\pi$
$593$	7.81681	0.320998	0.160499	+	0.987036i	$0.448690\pi$
$594$	0	0
$595$	−4.85601	−0.199077
$596$	−7.81681	−0.320189
$597$	0	0
$598$	−4.47116	−0.182839
$599$	15.5104	0.633736	0.316868	−	0.948470i	$-0.397369\pi$
$599$	15.5104	0.633736	0.316868	+	0.948470i	$0.397369\pi$
$600$	0	0
$601$	−3.38259	−0.137979	−0.0689894	−	0.997617i	$-0.521977\pi$
$601$	−3.38259	−0.137979	−0.0689894	+	0.997617i	$0.521977\pi$
$602$	18.0000	0.733625
$603$	0	0
$604$	9.58611	0.390053
$605$	4.24482	0.172576
$606$	0	0
$607$	−14.9097	−0.605167	−0.302584	−	0.953123i	$-0.597849\pi$
$607$	−14.9097	−0.605167	−0.302584	+	0.953123i	$0.597849\pi$
$608$	36.2201	1.46892
$609$	0	0
$610$	7.63362	0.309076
$611$	6.48963	0.262542
$612$	0	0
$613$	−5.82472	−0.235258	−0.117629	−	0.993058i	$-0.537529\pi$
$613$	−5.82472	−0.235258	−0.117629	+	0.993058i	$0.537529\pi$
$614$	5.23634	0.211322
$615$	0	0
$616$	−5.61515	−0.226241
$617$	−35.6353	−1.43462	−0.717312	−	0.696752i	$-0.754628\pi$
$617$	−35.6353	−1.43462	−0.717312	+	0.696752i	$0.754628\pi$
$618$	0	0
$619$	26.5697	1.06793	0.533964	−	0.845507i	$-0.320702\pi$
$619$	26.5697	1.06793	0.533964	+	0.845507i	$0.320702\pi$
$620$	−12.4633	−0.500536
$621$	0	0
$622$	−8.71767	−0.349547
$623$	9.59668	0.384483
$624$	0	0
$625$	79.3882	3.17553
$626$	−5.07615	−0.202884
$627$	0	0
$628$	−14.0264	−0.559713
$629$	3.26724	0.130273
$630$	0	0
$631$	14.4834	0.576576	0.288288	−	0.957544i	$-0.406914\pi$
$631$	14.4834	0.576576	0.288288	+	0.957544i	$0.406914\pi$
$632$	−13.5042	−0.537166
$633$	0	0
$634$	−4.01226	−0.159347
$635$	45.1210	1.79057
$636$	0	0
$637$	−0.143987	−0.00570495
$638$	−1.66887	−0.0660712
$639$	0	0
$640$	−48.9546	−1.93510
$641$	−19.0162	−0.751095	−0.375548	−	0.926803i	$-0.622545\pi$
$641$	−19.0162	−0.751095	−0.375548	+	0.926803i	$0.622545\pi$
$642$	0	0
$643$	11.5905	0.457083	0.228542	−	0.973534i	$-0.426604\pi$
$643$	11.5905	0.457083	0.228542	+	0.973534i	$0.426604\pi$
$644$	34.9502	1.37723
$645$	0	0
$646$	1.63362	0.0642739
$647$	−9.22239	−0.362570	−0.181285	−	0.983431i	$-0.558026\pi$
$647$	−9.22239	−0.362570	−0.181285	+	0.983431i	$0.558026\pi$
$648$	0	0
$649$	12.4896	0.490261
$650$	−7.44648	−0.292075
$651$	0	0
$652$	8.06429	0.315822
$653$	−35.0162	−1.37029	−0.685145	−	0.728407i	$-0.740261\pi$
$653$	−35.0162	−1.37029	−0.685145	+	0.728407i	$0.740261\pi$
$654$	0	0
$655$	−83.3411	−3.25641
$656$	5.11309	0.199633
$657$	0	0
$658$	9.92159	0.386784
$659$	4.81455	0.187548	0.0937741	−	0.995593i	$-0.470107\pi$
$659$	4.81455	0.187548	0.0937741	+	0.995593i	$0.470107\pi$
$660$	0	0
$661$	7.43196	0.289070	0.144535	−	0.989500i	$-0.453831\pi$
$661$	7.43196	0.289070	0.144535	+	0.989500i	$0.453831\pi$
$662$	13.9384	0.541730
$663$	0	0
$664$	3.77801	0.146615
$665$	−75.7075	−2.93581
$666$	0	0
$667$	22.8066	0.883077
$668$	32.8436	1.27076
$669$	0	0
$670$	32.5081	1.25590
$671$	−3.14399	−0.121372
$672$	0	0
$673$	−9.13608	−0.352170	−0.176085	−	0.984375i	$-0.556343\pi$
$673$	−9.13608	−0.352170	−0.176085	+	0.984375i	$0.556343\pi$
$674$	−16.3619	−0.630235
$675$	0	0
$676$	−1.67282	−0.0643394
$677$	8.50641	0.326928	0.163464	−	0.986549i	$-0.447733\pi$
$677$	8.50641	0.326928	0.163464	+	0.986549i	$0.447733\pi$
$678$	0	0
$679$	31.0946	1.19330
$680$	−3.81681	−0.146368
$681$	0	0
$682$	−1.00395	−0.0384433
$683$	25.7075	0.983670	0.491835	−	0.870688i	$-0.336326\pi$
$683$	25.7075	0.983670	0.491835	+	0.870688i	$0.336326\pi$
$684$	0	0
$685$	37.4090	1.42932
$686$	10.4817	0.400194
$687$	0	0
$688$	−25.2426	−0.962363
$689$	−2.85601	−0.108805
$690$	0	0
$691$	−14.7424	−0.560826	−0.280413	−	0.959879i	$-0.590471\pi$
$691$	−14.7424	−0.560826	−0.280413	+	0.959879i	$0.590471\pi$
$692$	7.40727	0.281582
$693$	0	0
$694$	0.814549	0.0309199
$695$	−39.9322	−1.51471
$696$	0	0
$697$	1.02073	0.0386630
$698$	10.3664	0.392373
$699$	0	0
$700$	58.2077	2.20004
$701$	33.3210	1.25852	0.629258	−	0.777197i	$-0.283359\pi$
$701$	33.3210	1.25852	0.629258	+	0.777197i	$0.283359\pi$
$702$	0	0
$703$	50.9378	1.92116
$704$	1.18319	0.0445931
$705$	0	0
$706$	5.41518	0.203803
$707$	20.0079	0.752475
$708$	0	0
$709$	1.25934	0.0472953	0.0236477	−	0.999720i	$-0.492472\pi$
$709$	1.25934	0.0472953	0.0236477	+	0.999720i	$0.492472\pi$
$710$	17.8353	0.669346
$711$	0	0
$712$	7.54296	0.282684
$713$	13.7199	0.513816
$714$	0	0
$715$	4.24482	0.158747
$716$	22.1629	0.828265
$717$	0	0
$718$	2.88691	0.107739
$719$	15.2672	0.569372	0.284686	−	0.958621i	$-0.408111\pi$
$719$	15.2672	0.569372	0.284686	+	0.958621i	$0.408111\pi$
$720$	0	0
$721$	−4.57595	−0.170417
$722$	14.6010	0.543394
$723$	0	0
$724$	−19.1413	−0.711382
$725$	37.9832	1.41066
$726$	0	0
$727$	−24.1233	−0.894682	−0.447341	−	0.894363i	$-0.647629\pi$
$727$	−24.1233	−0.894682	−0.447341	+	0.894363i	$0.647629\pi$
$728$	−5.61515	−0.208111
$729$	0	0
$730$	28.4817	1.05416
$731$	−5.03920	−0.186382
$732$	0	0
$733$	47.5737	1.75717	0.878587	−	0.477582i	$-0.158487\pi$
$733$	47.5737	1.75717	0.878587	+	0.477582i	$0.158487\pi$
$734$	9.55183	0.352564
$735$	0	0
$736$	42.4297	1.56398
$737$	−13.3888	−0.493183
$738$	0	0
$739$	−17.7305	−0.652227	−0.326113	−	0.945331i	$-0.605739\pi$
$739$	−17.7305	−0.652227	−0.326113	+	0.945331i	$0.605739\pi$
$740$	−54.2050	−1.99262
$741$	0	0
$742$	−4.36638	−0.160295
$743$	−5.99209	−0.219829	−0.109914	−	0.993941i	$-0.535058\pi$
$743$	−5.99209	−0.219829	−0.109914	+	0.993941i	$0.535058\pi$
$744$	0	0
$745$	19.8353	0.726708
$746$	16.3619	0.599050
$747$	0	0
$748$	0.715980	0.0261788
$749$	12.0000	0.438470
$750$	0	0
$751$	45.5058	1.66053	0.830266	−	0.557367i	$-0.188189\pi$
$751$	45.5058	1.66053	0.830266	+	0.557367i	$0.188189\pi$
$752$	−13.9137	−0.507380
$753$	0	0
$754$	−1.66887	−0.0607767
$755$	−24.3249	−0.885274
$756$	0	0
$757$	46.1708	1.67810	0.839052	−	0.544051i	$-0.183110\pi$
$757$	46.1708	1.67810	0.839052	+	0.544051i	$0.183110\pi$
$758$	−18.2060	−0.661272
$759$	0	0
$760$	−59.5058	−2.15850
$761$	4.01057	0.145383	0.0726914	−	0.997354i	$-0.476841\pi$
$761$	4.01057	0.145383	0.0726914	+	0.997354i	$0.476841\pi$
$762$	0	0
$763$	26.7776	0.969414
$764$	0.819071	0.0296330
$765$	0	0
$766$	−1.04977	−0.0379297
$767$	12.4896	0.450975
$768$	0	0
$769$	41.8432	1.50890	0.754452	−	0.656355i	$-0.227903\pi$
$769$	41.8432	1.50890	0.754452	+	0.656355i	$0.227903\pi$
$770$	6.48963	0.233870
$771$	0	0
$772$	1.73840	0.0625665
$773$	26.9440	0.969109	0.484554	−	0.874761i	$-0.338982\pi$
$773$	26.9440	0.969109	0.484554	+	0.874761i	$0.338982\pi$
$774$	0	0
$775$	22.8498	0.820789
$776$	24.4403	0.877354
$777$	0	0
$778$	−11.1361	−0.399248
$779$	15.9137	0.570167
$780$	0	0
$781$	−7.34565	−0.262848
$782$	1.91369	0.0684333
$783$	0	0
$784$	0.308705	0.0110252
$785$	35.5922	1.27034
$786$	0	0
$787$	5.53674	0.197364	0.0986818	−	0.995119i	$-0.468537\pi$
$787$	5.53674	0.197364	0.0986818	+	0.995119i	$0.468537\pi$
$788$	15.3272	0.546008
$789$	0	0
$790$	15.6072	0.555281
$791$	−28.0369	−0.996879
$792$	0	0
$793$	−3.14399	−0.111646
$794$	−12.0211	−0.426614
$795$	0	0
$796$	14.0573	0.498247
$797$	−14.2465	−0.504637	−0.252319	−	0.967644i	$-0.581193\pi$
$797$	−14.2465	−0.504637	−0.252319	+	0.967644i	$0.581193\pi$
$798$	0	0
$799$	−2.77761	−0.0982647
$800$	70.6643	2.49836
$801$	0	0
$802$	20.1434	0.711289
$803$	−11.7305	−0.413960
$804$	0	0
$805$	−88.6868	−3.12580
$806$	−1.00395	−0.0353627
$807$	0	0
$808$	15.7261	0.553244
$809$	−40.6745	−1.43004	−0.715020	−	0.699104i	$-0.753582\pi$
$809$	−40.6745	−1.43004	−0.715020	+	0.699104i	$0.753582\pi$
$810$	0	0
$811$	21.9031	0.769123	0.384561	−	0.923099i	$-0.374353\pi$
$811$	21.9031	0.769123	0.384561	+	0.923099i	$0.374353\pi$
$812$	13.0452	0.457798
$813$	0	0
$814$	−4.36638	−0.153042
$815$	−20.4633	−0.716797
$816$	0	0
$817$	−78.5635	−2.74859
$818$	−3.98153	−0.139211
$819$	0	0
$820$	−16.9344	−0.591376
$821$	−16.2201	−0.566087	−0.283043	−	0.959107i	$-0.591344\pi$
$821$	−16.2201	−0.566087	−0.283043	+	0.959107i	$0.591344\pi$
$822$	0	0
$823$	−47.5473	−1.65739	−0.828697	−	0.559697i	$-0.810918\pi$
$823$	−47.5473	−1.65739	−0.828697	+	0.559697i	$0.810918\pi$
$824$	−3.59668	−0.125296
$825$	0	0
$826$	19.0946	0.664387
$827$	27.4090	0.953103	0.476552	−	0.879147i	$-0.341887\pi$
$827$	27.4090	0.953103	0.476552	+	0.879147i	$0.341887\pi$
$828$	0	0
$829$	6.97927	0.242400	0.121200	−	0.992628i	$-0.461326\pi$
$829$	6.97927	0.242400	0.121200	+	0.992628i	$0.461326\pi$
$830$	−4.36638	−0.151559
$831$	0	0
$832$	1.18319	0.0410197
$833$	0.0616272	0.00213526
$834$	0	0
$835$	−83.3411	−2.88414
$836$	11.1625	0.386062
$837$	0	0
$838$	10.6359	0.367410
$839$	−33.1888	−1.14581	−0.572903	−	0.819623i	$-0.694183\pi$
$839$	−33.1888	−1.14581	−0.572903	+	0.819623i	$0.694183\pi$
$840$	0	0
$841$	−20.4874	−0.706461
$842$	−17.5104	−0.603447
$843$	0	0
$844$	−24.1973	−0.832906
$845$	4.24482	0.146026
$846$	0	0
$847$	−2.67282	−0.0918393
$848$	6.12325	0.210273
$849$	0	0
$850$	3.18714	0.109318
$851$	59.6706	2.04548
$852$	0	0
$853$	6.61289	0.226421	0.113210	−	0.993571i	$-0.463887\pi$
$853$	6.61289	0.226421	0.113210	+	0.993571i	$0.463887\pi$
$854$	−4.80664	−0.164480
$855$	0	0
$856$	9.43196	0.322378
$857$	−11.0409	−0.377150	−0.188575	−	0.982059i	$-0.560387\pi$
$857$	−11.0409	−0.377150	−0.188575	+	0.982059i	$0.560387\pi$
$858$	0	0
$859$	−8.52658	−0.290923	−0.145462	−	0.989364i	$-0.546467\pi$
$859$	−8.52658	−0.290923	−0.145462	+	0.989364i	$0.546467\pi$
$860$	83.6027	2.85083
$861$	0	0
$862$	4.69582	0.159940
$863$	17.2593	0.587515	0.293757	−	0.955880i	$-0.405094\pi$
$863$	17.2593	0.587515	0.293757	+	0.955880i	$0.405094\pi$
$864$	0	0
$865$	−18.7961	−0.639086
$866$	−16.0818	−0.546481
$867$	0	0
$868$	7.84771	0.266369
$869$	−6.42801	−0.218055
$870$	0	0
$871$	−13.3888	−0.453662
$872$	21.0471	0.712745
$873$	0	0
$874$	29.8353	1.00919
$875$	−90.9747	−3.07551
$876$	0	0
$877$	12.1647	0.410773	0.205387	−	0.978681i	$-0.434155\pi$
$877$	12.1647	0.410773	0.205387	+	0.978681i	$0.434155\pi$
$878$	−9.58256	−0.323396
$879$	0	0
$880$	−9.10083	−0.306789
$881$	−1.09462	−0.0368786	−0.0184393	−	0.999830i	$-0.505870\pi$
$881$	−1.09462	−0.0368786	−0.0184393	+	0.999830i	$0.505870\pi$
$882$	0	0
$883$	−40.9793	−1.37906	−0.689531	−	0.724256i	$-0.742183\pi$
$883$	−40.9793	−1.37906	−0.689531	+	0.724256i	$0.742183\pi$
$884$	0.715980	0.0240810
$885$	0	0
$886$	−7.42405	−0.249416
$887$	−32.0818	−1.07720	−0.538601	−	0.842561i	$-0.681047\pi$
$887$	−32.0818	−1.07720	−0.538601	+	0.842561i	$0.681047\pi$
$888$	0	0
$889$	−28.4112	−0.952882
$890$	−8.71767	−0.292217
$891$	0	0
$892$	4.57425	0.153157
$893$	−43.3042	−1.44912
$894$	0	0
$895$	−56.2386	−1.87985
$896$	30.8251	1.02979
$897$	0	0
$898$	−13.8396	−0.461835
$899$	5.12099	0.170795
$900$	0	0
$901$	1.22239	0.0407238
$902$	−1.36412	−0.0454202
$903$	0	0
$904$	−22.0369	−0.732938
$905$	48.5714	1.61457
$906$	0	0
$907$	−2.81455	−0.0934556	−0.0467278	−	0.998908i	$-0.514879\pi$
$907$	−2.81455	−0.0934556	−0.0467278	+	0.998908i	$0.514879\pi$
$908$	31.4425	1.04346
$909$	0	0
$910$	6.48963	0.215129
$911$	−7.18093	−0.237915	−0.118957	−	0.992899i	$-0.537955\pi$
$911$	−7.18093	−0.237915	−0.118957	+	0.992899i	$0.537955\pi$
$912$	0	0
$913$	1.79834	0.0595163
$914$	−2.47382	−0.0818268
$915$	0	0
$916$	−7.51037	−0.248149
$917$	52.4772	1.73295
$918$	0	0
$919$	24.0123	0.792091	0.396046	−	0.918231i	$-0.370382\pi$
$919$	24.0123	0.792091	0.396046	+	0.918231i	$0.370382\pi$
$920$	−69.7075	−2.29819
$921$	0	0
$922$	3.65209	0.120275
$923$	−7.34565	−0.241785
$924$	0	0
$925$	99.3781	3.26753
$926$	17.5351	0.576238
$927$	0	0
$928$	15.8370	0.519874
$929$	22.0017	0.721852	0.360926	−	0.932594i	$-0.382461\pi$
$929$	22.0017	0.721852	0.360926	+	0.932594i	$0.382461\pi$
$930$	0	0
$931$	0.960797	0.0314888
$932$	−18.4077	−0.602963
$933$	0	0
$934$	−18.2201	−0.596181
$935$	−1.81681	−0.0594160
$936$	0	0
$937$	−18.4975	−0.604288	−0.302144	−	0.953262i	$-0.597702\pi$
$937$	−18.4975	−0.604288	−0.302144	+	0.953262i	$0.597702\pi$
$938$	−20.4693	−0.668346
$939$	0	0
$940$	46.0818	1.50302
$941$	−34.4218	−1.12212	−0.561059	−	0.827776i	$-0.689606\pi$
$941$	−34.4218	−1.12212	−0.561059	+	0.827776i	$0.689606\pi$
$942$	0	0
$943$	18.6419	0.607065
$944$	−26.7776	−0.871537
$945$	0	0
$946$	6.73445	0.218956
$947$	41.5473	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$947$	41.5473	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$948$	0	0
$949$	−11.7305	−0.380788
$950$	49.6890	1.61213
$951$	0	0
$952$	2.40332	0.0778921
$953$	3.53053	0.114365	0.0571825	−	0.998364i	$-0.481788\pi$
$953$	3.53053	0.114365	0.0571825	+	0.998364i	$0.481788\pi$
$954$	0	0
$955$	−2.07841	−0.0672557
$956$	42.2853	1.36760
$957$	0	0
$958$	−17.0550	−0.551023
$959$	−23.5552	−0.760638
$960$	0	0
$961$	−27.9193	−0.900624
$962$	−4.36638	−0.140778
$963$	0	0
$964$	−25.5262	−0.822143
$965$	−4.41123	−0.142002
$966$	0	0
$967$	44.5450	1.43247	0.716236	−	0.697858i	$-0.245863\pi$
$967$	44.5450	1.43247	0.716236	+	0.697858i	$0.245863\pi$
$968$	−2.10083	−0.0675232
$969$	0	0
$970$	−28.2465	−0.906941
$971$	14.5450	0.466773	0.233386	−	0.972384i	$-0.425019\pi$
$971$	14.5450	0.466773	0.233386	+	0.972384i	$0.425019\pi$
$972$	0	0
$973$	25.1440	0.806079
$974$	−11.1114	−0.356032
$975$	0	0
$976$	6.74066	0.215763
$977$	−38.3681	−1.22750	−0.613752	−	0.789499i	$-0.710340\pi$
$977$	−38.3681	−1.22750	−0.613752	+	0.789499i	$0.710340\pi$
$978$	0	0
$979$	3.59046	0.114752
$980$	−1.02242	−0.0326601
$981$	0	0
$982$	−9.56844	−0.305341
$983$	34.6544	1.10530	0.552651	−	0.833413i	$-0.313616\pi$
$983$	34.6544	1.10530	0.552651	+	0.833413i	$0.313616\pi$
$984$	0	0
$985$	−38.8930	−1.23923
$986$	0.714288	0.0227476
$987$	0	0
$988$	11.1625	0.355125
$989$	−92.0324	−2.92646
$990$	0	0
$991$	−1.06558	−0.0338493	−0.0169246	−	0.999857i	$-0.505388\pi$
$991$	−1.06558	−0.0338493	−0.0169246	+	0.999857i	$0.505388\pi$
$992$	9.52715	0.302487
$993$	0	0
$994$	−11.2303	−0.356203
$995$	−35.6706	−1.13083
$996$	0	0
$997$	−8.53110	−0.270183	−0.135091	−	0.990833i	$-0.543133\pi$
$997$	−8.53110	−0.270183	−0.135091	+	0.990833i	$0.543133\pi$
$998$	10.2968	0.325941
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.a.j.1.2		3
3.2	odd	2		429.2.a.e.1.2	✓	3
12.11	even	2		6864.2.a.bu.1.1		3
33.32	even	2		4719.2.a.u.1.2		3
39.38	odd	2		5577.2.a.l.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.e.1.2	✓	3	3.2	odd	2
1287.2.a.j.1.2		3	1.1	even	1	trivial
4719.2.a.u.1.2		3	33.32	even	2
5577.2.a.l.1.2		3	39.38	odd	2
6864.2.a.bu.1.1		3	12.11	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:	\( N \)	\(=\)	\( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1287.a (trivial)

\(f(q)\)	\(=\)	\(q+0.571993 q^{2} -1.67282 q^{4} +4.24482 q^{5} -2.67282 q^{7} -2.10083 q^{8} +O(q^{10})\)	\(q+0.571993 q^{2} -1.67282 q^{4} +4.24482 q^{5} -2.67282 q^{7} -2.10083 q^{8} +2.42801 q^{10} -1.00000 q^{11} -1.00000 q^{13} -1.52884 q^{14} +2.14399 q^{16} +0.428007 q^{17} +6.67282 q^{19} -7.10083 q^{20} -0.571993 q^{22} +7.81681 q^{23} +13.0185 q^{25} -0.571993 q^{26} +4.47116 q^{28} +2.91764 q^{29} +1.75518 q^{31} +5.42801 q^{32} +0.244817 q^{34} -11.3456 q^{35} +7.63362 q^{37} +3.81681 q^{38} -8.91764 q^{40} +2.38485 q^{41} -11.7737 q^{43} +1.67282 q^{44} +4.47116 q^{46} -6.48963 q^{47} +0.143987 q^{49} +7.44648 q^{50} +1.67282 q^{52} +2.85601 q^{53} -4.24482 q^{55} +5.61515 q^{56} +1.66887 q^{58} -12.4896 q^{59} +3.14399 q^{61} +1.00395 q^{62} -1.18319 q^{64} -4.24482 q^{65} +13.3888 q^{67} -0.715980 q^{68} -6.48963 q^{70} +7.34565 q^{71} +11.7305 q^{73} +4.36638 q^{74} -11.1625 q^{76} +2.67282 q^{77} +6.42801 q^{79} +9.10083 q^{80} +1.36412 q^{82} -1.79834 q^{83} +1.81681 q^{85} -6.73445 q^{86} +2.10083 q^{88} -3.59046 q^{89} +2.67282 q^{91} -13.0761 q^{92} -3.71203 q^{94} +28.3249 q^{95} -11.6336 q^{97} +0.0823593 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 5 q^{4} + 2 q^{5} + 2 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + q^2 + 5 * q^4 + 2 * q^5 + 2 * q^7 + 3 * q^8	\( 3 q + q^{2} + 5 q^{4} + 2 q^{5} + 2 q^{7} + 3 q^{8} + 8 q^{10} - 3 q^{11} - 3 q^{13} + 4 q^{14} + 5 q^{16} + 2 q^{17} + 10 q^{19} - 12 q^{20} - q^{22} + 12 q^{23} + 9 q^{25} - q^{26} + 22 q^{28} - 12 q^{29} + 16 q^{31} + 17 q^{32} - 10 q^{34} - 14 q^{35} - 6 q^{40} - 16 q^{43} - 5 q^{44} + 22 q^{46} + 2 q^{47} - q^{49} - 7 q^{50} - 5 q^{52} + 10 q^{53} - 2 q^{55} + 24 q^{56} - 16 q^{59} + 8 q^{61} - 2 q^{62} - 15 q^{64} - 2 q^{65} + 28 q^{67} + 2 q^{70} + 2 q^{71} + 8 q^{73} + 36 q^{74} - 2 q^{76} - 2 q^{77} + 20 q^{79} + 18 q^{80} - 46 q^{82} - 24 q^{83} - 6 q^{85} + 12 q^{86} - 3 q^{88} + 20 q^{89} - 2 q^{91} + 8 q^{92} - 14 q^{94} + 22 q^{95} - 12 q^{97} + 21 q^{98}+O(q^{100}) \) 3 * q + q^2 + 5 * q^4 + 2 * q^5 + 2 * q^7 + 3 * q^8 + 8 * q^10 - 3 * q^11 - 3 * q^13 + 4 * q^14 + 5 * q^16 + 2 * q^17 + 10 * q^19 - 12 * q^20 - q^22 + 12 * q^23 + 9 * q^25 - q^26 + 22 * q^28 - 12 * q^29 + 16 * q^31 + 17 * q^32 - 10 * q^34 - 14 * q^35 - 6 * q^40 - 16 * q^43 - 5 * q^44 + 22 * q^46 + 2 * q^47 - q^49 - 7 * q^50 - 5 * q^52 + 10 * q^53 - 2 * q^55 + 24 * q^56 - 16 * q^59 + 8 * q^61 - 2 * q^62 - 15 * q^64 - 2 * q^65 + 28 * q^67 + 2 * q^70 + 2 * q^71 + 8 * q^73 + 36 * q^74 - 2 * q^76 - 2 * q^77 + 20 * q^79 + 18 * q^80 - 46 * q^82 - 24 * q^83 - 6 * q^85 + 12 * q^86 - 3 * q^88 + 20 * q^89 - 2 * q^91 + 8 * q^92 - 14 * q^94 + 22 * q^95 - 12 * q^97 + 21 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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