LMFDB - Embedded newform 8001.2.a.y.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.24740 q^{2} -0.443996 q^{4} -1.34151 q^{5} -1.00000 q^{7} -3.04864 q^{8} +O(q^{10})$	\(q+1.24740 q^{2} -0.443996 q^{4} -1.34151 q^{5} -1.00000 q^{7} -3.04864 q^{8} -1.67339 q^{10} +5.34830 q^{11} -4.35012 q^{13} -1.24740 q^{14} -2.91487 q^{16} +3.15816 q^{17} +2.16622 q^{19} +0.595623 q^{20} +6.67146 q^{22} -6.10623 q^{23} -3.20036 q^{25} -5.42633 q^{26} +0.443996 q^{28} +1.86915 q^{29} +0.389776 q^{31} +2.46126 q^{32} +3.93948 q^{34} +1.34151 q^{35} +1.82278 q^{37} +2.70214 q^{38} +4.08977 q^{40} -4.04302 q^{41} -12.1420 q^{43} -2.37462 q^{44} -7.61690 q^{46} +2.55141 q^{47} +1.00000 q^{49} -3.99213 q^{50} +1.93144 q^{52} +0.687516 q^{53} -7.17477 q^{55} +3.04864 q^{56} +2.33157 q^{58} +7.83644 q^{59} -8.37683 q^{61} +0.486206 q^{62} +8.89993 q^{64} +5.83571 q^{65} +2.01012 q^{67} -1.40221 q^{68} +1.67339 q^{70} +16.1489 q^{71} +3.64873 q^{73} +2.27374 q^{74} -0.961794 q^{76} -5.34830 q^{77} +7.25997 q^{79} +3.91032 q^{80} -5.04326 q^{82} -1.44730 q^{83} -4.23668 q^{85} -15.1460 q^{86} -16.3050 q^{88} -5.15405 q^{89} +4.35012 q^{91} +2.71114 q^{92} +3.18263 q^{94} -2.90600 q^{95} +1.05155 q^{97} +1.24740 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) $ 28 * q + 30 * q^4 - 28 * q^7	$ 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) $ 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * q^94 - 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.24740	0.882044	0.441022	−	0.897496i	$-0.354616\pi$
$2$	1.24740	0.882044	0.441022	+	0.897496i	$0.354616\pi$
$3$	0	0
$3$	0	0
$4$	−0.443996	−0.221998
$5$	−1.34151	−0.599940	−0.299970	−	0.953949i	$-0.596977\pi$
$5$	−1.34151	−0.599940	−0.299970	+	0.953949i	$0.596977\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.04864	−1.07786
$9$	0	0
$10$	−1.67339	−0.529173
$11$	5.34830	1.61257	0.806287	−	0.591525i	$-0.201474\pi$
$11$	5.34830	1.61257	0.806287	+	0.591525i	$0.201474\pi$
$12$	0	0
$13$	−4.35012	−1.20651	−0.603253	−	0.797550i	$-0.706129\pi$
$13$	−4.35012	−1.20651	−0.603253	+	0.797550i	$0.706129\pi$
$14$	−1.24740	−0.333381
$15$	0	0
$16$	−2.91487	−0.728719
$17$	3.15816	0.765965	0.382983	−	0.923756i	$-0.374897\pi$
$17$	3.15816	0.765965	0.382983	+	0.923756i	$0.374897\pi$
$18$	0	0
$19$	2.16622	0.496965	0.248483	−	0.968636i	$-0.420068\pi$
$19$	2.16622	0.496965	0.248483	+	0.968636i	$0.420068\pi$
$20$	0.595623	0.133185
$21$	0	0
$22$	6.67146	1.42236
$23$	−6.10623	−1.27324	−0.636618	−	0.771179i	$-0.719667\pi$
$23$	−6.10623	−1.27324	−0.636618	+	0.771179i	$0.719667\pi$
$24$	0	0
$25$	−3.20036	−0.640073
$26$	−5.42633	−1.06419
$27$	0	0
$28$	0.443996	0.0839074
$29$	1.86915	0.347092	0.173546	−	0.984826i	$-0.444477\pi$
$29$	1.86915	0.347092	0.173546	+	0.984826i	$0.444477\pi$
$30$	0	0
$31$	0.389776	0.0700059	0.0350029	−	0.999387i	$-0.488856\pi$
$31$	0.389776	0.0700059	0.0350029	+	0.999387i	$0.488856\pi$
$32$	2.46126	0.435094
$33$	0	0
$34$	3.93948	0.675615
$35$	1.34151	0.226756
$36$	0	0
$37$	1.82278	0.299664	0.149832	−	0.988711i	$-0.452127\pi$
$37$	1.82278	0.299664	0.149832	+	0.988711i	$0.452127\pi$
$38$	2.70214	0.438345
$39$	0	0
$40$	4.08977	0.646649
$41$	−4.04302	−0.631414	−0.315707	−	0.948857i	$-0.602242\pi$
$41$	−4.04302	−0.631414	−0.315707	+	0.948857i	$0.602242\pi$
$42$	0	0
$43$	−12.1420	−1.85164	−0.925821	−	0.377962i	$-0.876625\pi$
$43$	−12.1420	−1.85164	−0.925821	+	0.377962i	$0.876625\pi$
$44$	−2.37462	−0.357988
$45$	0	0
$46$	−7.61690	−1.12305
$47$	2.55141	0.372162	0.186081	−	0.982534i	$-0.440421\pi$
$47$	2.55141	0.372162	0.186081	+	0.982534i	$0.440421\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.99213	−0.564572
$51$	0	0
$52$	1.93144	0.267842
$53$	0.687516	0.0944376	0.0472188	−	0.998885i	$-0.484964\pi$
$53$	0.687516	0.0944376	0.0472188	+	0.998885i	$0.484964\pi$
$54$	0	0
$55$	−7.17477	−0.967446
$56$	3.04864	0.407391
$57$	0	0
$58$	2.33157	0.306151
$59$	7.83644	1.02022	0.510109	−	0.860110i	$-0.329605\pi$
$59$	7.83644	1.02022	0.510109	+	0.860110i	$0.329605\pi$
$60$	0	0
$61$	−8.37683	−1.07254	−0.536272	−	0.844045i	$-0.680168\pi$
$61$	−8.37683	−1.07254	−0.536272	+	0.844045i	$0.680168\pi$
$62$	0.486206	0.0617483
$63$	0	0
$64$	8.89993	1.11249
$65$	5.83571	0.723831
$66$	0	0
$67$	2.01012	0.245576	0.122788	−	0.992433i	$-0.460817\pi$
$67$	2.01012	0.245576	0.122788	+	0.992433i	$0.460817\pi$
$68$	−1.40221	−0.170043
$69$	0	0
$70$	1.67339	0.200009
$71$	16.1489	1.91652	0.958258	−	0.285904i	$-0.0922938\pi$
$71$	16.1489	1.91652	0.958258	+	0.285904i	$0.0922938\pi$
$72$	0	0
$73$	3.64873	0.427051	0.213526	−	0.976937i	$-0.431505\pi$
$73$	3.64873	0.427051	0.213526	+	0.976937i	$0.431505\pi$
$74$	2.27374	0.264317
$75$	0	0
$76$	−0.961794	−0.110325
$77$	−5.34830	−0.609495
$78$	0	0
$79$	7.25997	0.816810	0.408405	−	0.912801i	$-0.366085\pi$
$79$	7.25997	0.816810	0.408405	+	0.912801i	$0.366085\pi$
$80$	3.91032	0.437187
$81$	0	0
$82$	−5.04326	−0.556935
$83$	−1.44730	−0.158861	−0.0794307	−	0.996840i	$-0.525310\pi$
$83$	−1.44730	−0.158861	−0.0794307	+	0.996840i	$0.525310\pi$
$84$	0	0
$85$	−4.23668	−0.459533
$86$	−15.1460	−1.63323
$87$	0	0
$88$	−16.3050	−1.73812
$89$	−5.15405	−0.546328	−0.273164	−	0.961967i	$-0.588070\pi$
$89$	−5.15405	−0.546328	−0.273164	+	0.961967i	$0.588070\pi$
$90$	0	0
$91$	4.35012	0.456016
$92$	2.71114	0.282656
$93$	0	0
$94$	3.18263	0.328263
$95$	−2.90600	−0.298149
$96$	0	0
$97$	1.05155	0.106769	0.0533843	−	0.998574i	$-0.482999\pi$
$97$	1.05155	0.106769	0.0533843	+	0.998574i	$0.482999\pi$
$98$	1.24740	0.126006
$99$	0	0
$100$	1.42095	0.142095
$101$	11.1364	1.10811	0.554055	−	0.832480i	$-0.313080\pi$
$101$	11.1364	1.10811	0.554055	+	0.832480i	$0.313080\pi$
$102$	0	0
$103$	2.94345	0.290027	0.145013	−	0.989430i	$-0.453677\pi$
$103$	2.94345	0.290027	0.145013	+	0.989430i	$0.453677\pi$
$104$	13.2619	1.30044
$105$	0	0
$106$	0.857607	0.0832981
$107$	−17.8510	−1.72572	−0.862858	−	0.505446i	$-0.831328\pi$
$107$	−17.8510	−1.72572	−0.862858	+	0.505446i	$0.831328\pi$
$108$	0	0
$109$	6.89841	0.660748	0.330374	−	0.943850i	$-0.392825\pi$
$109$	6.89841	0.660748	0.330374	+	0.943850i	$0.392825\pi$
$110$	−8.94980	−0.853330
$111$	0	0
$112$	2.91487	0.275430
$113$	−7.27356	−0.684239	−0.342119	−	0.939656i	$-0.611145\pi$
$113$	−7.27356	−0.684239	−0.342119	+	0.939656i	$0.611145\pi$
$114$	0	0
$115$	8.19154	0.763865
$116$	−0.829895	−0.0770538
$117$	0	0
$118$	9.77516	0.899877
$119$	−3.15816	−0.289508
$120$	0	0
$121$	17.6043	1.60039
$122$	−10.4492	−0.946030
$123$	0	0
$124$	−0.173059	−0.0155412
$125$	11.0008	0.983944
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	6.17923	0.546172
$129$	0	0
$130$	7.27946	0.638451
$131$	−0.921159	−0.0804820	−0.0402410	−	0.999190i	$-0.512813\pi$
$131$	−0.921159	−0.0804820	−0.0402410	+	0.999190i	$0.512813\pi$
$132$	0	0
$133$	−2.16622	−0.187835
$134$	2.50743	0.216609
$135$	0	0
$136$	−9.62807	−0.825600
$137$	8.98921	0.768000	0.384000	−	0.923333i	$-0.374546\pi$
$137$	8.98921	0.768000	0.384000	+	0.923333i	$0.374546\pi$
$138$	0	0
$139$	−3.44202	−0.291948	−0.145974	−	0.989288i	$-0.546632\pi$
$139$	−3.44202	−0.291948	−0.145974	+	0.989288i	$0.546632\pi$
$140$	−0.595623	−0.0503394
$141$	0	0
$142$	20.1441	1.69045
$143$	−23.2657	−1.94558
$144$	0	0
$145$	−2.50747	−0.208234
$146$	4.55142	0.376678
$147$	0	0
$148$	−0.809309	−0.0665248
$149$	1.20512	0.0987276	0.0493638	−	0.998781i	$-0.484281\pi$
$149$	1.20512	0.0987276	0.0493638	+	0.998781i	$0.484281\pi$
$150$	0	0
$151$	13.6943	1.11443	0.557214	−	0.830369i	$-0.311870\pi$
$151$	13.6943	1.11443	0.557214	+	0.830369i	$0.311870\pi$
$152$	−6.60403	−0.535657
$153$	0	0
$154$	−6.67146	−0.537602
$155$	−0.522887	−0.0419993
$156$	0	0
$157$	19.2880	1.53935	0.769674	−	0.638437i	$-0.220419\pi$
$157$	19.2880	1.53935	0.769674	+	0.638437i	$0.220419\pi$
$158$	9.05607	0.720463
$159$	0	0
$160$	−3.30180	−0.261030
$161$	6.10623	0.481238
$162$	0	0
$163$	−5.43179	−0.425451	−0.212725	−	0.977112i	$-0.568234\pi$
$163$	−5.43179	−0.425451	−0.212725	+	0.977112i	$0.568234\pi$
$164$	1.79509	0.140173
$165$	0	0
$166$	−1.80536	−0.140123
$167$	7.20217	0.557321	0.278660	−	0.960390i	$-0.410110\pi$
$167$	7.20217	0.557321	0.278660	+	0.960390i	$0.410110\pi$
$168$	0	0
$169$	5.92354	0.455657
$170$	−5.28483	−0.405328
$171$	0	0
$172$	5.39102	0.411061
$173$	19.7557	1.50200	0.751000	−	0.660302i	$-0.229572\pi$
$173$	19.7557	1.50200	0.751000	+	0.660302i	$0.229572\pi$
$174$	0	0
$175$	3.20036	0.241925
$176$	−15.5896	−1.17511
$177$	0	0
$178$	−6.42915	−0.481885
$179$	17.8014	1.33054	0.665269	−	0.746604i	$-0.268317\pi$
$179$	17.8014	1.33054	0.665269	+	0.746604i	$0.268317\pi$
$180$	0	0
$181$	23.4288	1.74145	0.870723	−	0.491773i	$-0.163651\pi$
$181$	23.4288	1.74145	0.870723	+	0.491773i	$0.163651\pi$
$182$	5.42633	0.402227
$183$	0	0
$184$	18.6157	1.37237
$185$	−2.44528	−0.179780
$186$	0	0
$187$	16.8908	1.23517
$188$	−1.13282	−0.0826192
$189$	0	0
$190$	−3.62494	−0.262981
$191$	−16.1706	−1.17006	−0.585030	−	0.811011i	$-0.698917\pi$
$191$	−16.1706	−1.17006	−0.585030	+	0.811011i	$0.698917\pi$
$192$	0	0
$193$	−10.0893	−0.726243	−0.363121	−	0.931742i	$-0.618289\pi$
$193$	−10.0893	−0.726243	−0.363121	+	0.931742i	$0.618289\pi$
$194$	1.31170	0.0941745
$195$	0	0
$196$	−0.443996	−0.0317140
$197$	4.50647	0.321073	0.160536	−	0.987030i	$-0.448678\pi$
$197$	4.50647	0.321073	0.160536	+	0.987030i	$0.448678\pi$
$198$	0	0
$199$	22.4241	1.58960	0.794800	−	0.606871i	$-0.207576\pi$
$199$	22.4241	1.58960	0.794800	+	0.606871i	$0.207576\pi$
$200$	9.75675	0.689906
$201$	0	0
$202$	13.8915	0.977402
$203$	−1.86915	−0.131189
$204$	0	0
$205$	5.42374	0.378810
$206$	3.67166	0.255817
$207$	0	0
$208$	12.6801	0.879204
$209$	11.5856	0.801393
$210$	0	0
$211$	0.809347	0.0557178	0.0278589	−	0.999612i	$-0.491131\pi$
$211$	0.809347	0.0557178	0.0278589	+	0.999612i	$0.491131\pi$
$212$	−0.305255	−0.0209650
$213$	0	0
$214$	−22.2673	−1.52216
$215$	16.2886	1.11087
$216$	0	0
$217$	−0.389776	−0.0264597
$218$	8.60506	0.582809
$219$	0	0
$220$	3.18557	0.214771
$221$	−13.7384	−0.924142
$222$	0	0
$223$	6.93385	0.464325	0.232162	−	0.972677i	$-0.425420\pi$
$223$	6.93385	0.464325	0.232162	+	0.972677i	$0.425420\pi$
$224$	−2.46126	−0.164450
$225$	0	0
$226$	−9.07303	−0.603529
$227$	−10.5103	−0.697590	−0.348795	−	0.937199i	$-0.613409\pi$
$227$	−10.5103	−0.697590	−0.348795	+	0.937199i	$0.613409\pi$
$228$	0	0
$229$	−15.7823	−1.04293	−0.521463	−	0.853274i	$-0.674614\pi$
$229$	−15.7823	−1.04293	−0.521463	+	0.853274i	$0.674614\pi$
$230$	10.2181	0.673763
$231$	0	0
$232$	−5.69836	−0.374116
$233$	−2.53665	−0.166182	−0.0830908	−	0.996542i	$-0.526479\pi$
$233$	−2.53665	−0.166182	−0.0830908	+	0.996542i	$0.526479\pi$
$234$	0	0
$235$	−3.42273	−0.223274
$236$	−3.47935	−0.226486
$237$	0	0
$238$	−3.93948	−0.255359
$239$	−11.0886	−0.717263	−0.358631	−	0.933479i	$-0.616756\pi$
$239$	−11.0886	−0.717263	−0.358631	+	0.933479i	$0.616756\pi$
$240$	0	0
$241$	26.3087	1.69469	0.847345	−	0.531042i	$-0.178200\pi$
$241$	26.3087	1.69469	0.847345	+	0.531042i	$0.178200\pi$
$242$	21.9596	1.41162
$243$	0	0
$244$	3.71928	0.238103
$245$	−1.34151	−0.0857057
$246$	0	0
$247$	−9.42333	−0.599592
$248$	−1.18829	−0.0754563
$249$	0	0
$250$	13.7224	0.867882
$251$	12.5462	0.791911	0.395956	−	0.918270i	$-0.370413\pi$
$251$	12.5462	0.791911	0.395956	+	0.918270i	$0.370413\pi$
$252$	0	0
$253$	−32.6579	−2.05319
$254$	1.24740	0.0782688
$255$	0	0
$256$	−10.0919	−0.630743
$257$	5.65644	0.352839	0.176420	−	0.984315i	$-0.443548\pi$
$257$	5.65644	0.352839	0.176420	+	0.984315i	$0.443548\pi$
$258$	0	0
$259$	−1.82278	−0.113262
$260$	−2.59103	−0.160689
$261$	0	0
$262$	−1.14905	−0.0709887
$263$	17.2959	1.06651	0.533256	−	0.845954i	$-0.320968\pi$
$263$	17.2959	1.06651	0.533256	+	0.845954i	$0.320968\pi$
$264$	0	0
$265$	−0.922307	−0.0566569
$266$	−2.70214	−0.165679
$267$	0	0
$268$	−0.892487	−0.0545173
$269$	27.3003	1.66453	0.832265	−	0.554378i	$-0.187044\pi$
$269$	27.3003	1.66453	0.832265	+	0.554378i	$0.187044\pi$
$270$	0	0
$271$	−0.959296	−0.0582731	−0.0291365	−	0.999575i	$-0.509276\pi$
$271$	−0.959296	−0.0582731	−0.0291365	+	0.999575i	$0.509276\pi$
$272$	−9.20563	−0.558173
$273$	0	0
$274$	11.2131	0.677410
$275$	−17.1165	−1.03216
$276$	0	0
$277$	11.6931	0.702568	0.351284	−	0.936269i	$-0.385745\pi$
$277$	11.6931	0.702568	0.351284	+	0.936269i	$0.385745\pi$
$278$	−4.29357	−0.257511
$279$	0	0
$280$	−4.08977	−0.244410
$281$	5.71153	0.340721	0.170361	−	0.985382i	$-0.445507\pi$
$281$	5.71153	0.340721	0.170361	+	0.985382i	$0.445507\pi$
$282$	0	0
$283$	−7.87883	−0.468348	−0.234174	−	0.972195i	$-0.575239\pi$
$283$	−7.87883	−0.468348	−0.234174	+	0.972195i	$0.575239\pi$
$284$	−7.17003	−0.425463
$285$	0	0
$286$	−29.0217	−1.71609
$287$	4.04302	0.238652
$288$	0	0
$289$	−7.02605	−0.413297
$290$	−3.12782	−0.183672
$291$	0	0
$292$	−1.62002	−0.0948046
$293$	−22.9436	−1.34038	−0.670189	−	0.742190i	$-0.733787\pi$
$293$	−22.9436	−1.34038	−0.670189	+	0.742190i	$0.733787\pi$
$294$	0	0
$295$	−10.5126	−0.612069
$296$	−5.55701	−0.322995
$297$	0	0
$298$	1.50327	0.0870821
$299$	26.5628	1.53617
$300$	0	0
$301$	12.1420	0.699855
$302$	17.0823	0.982975
$303$	0	0
$304$	−6.31427	−0.362148
$305$	11.2376	0.643461
$306$	0	0
$307$	9.06672	0.517465	0.258732	−	0.965949i	$-0.416695\pi$
$307$	9.06672	0.517465	0.258732	+	0.965949i	$0.416695\pi$
$308$	2.37462	0.135307
$309$	0	0
$310$	−0.652249	−0.0370452
$311$	12.3150	0.698319	0.349160	−	0.937063i	$-0.386467\pi$
$311$	12.3150	0.698319	0.349160	+	0.937063i	$0.386467\pi$
$312$	0	0
$313$	0.643671	0.0363824	0.0181912	−	0.999835i	$-0.494209\pi$
$313$	0.643671	0.0363824	0.0181912	+	0.999835i	$0.494209\pi$
$314$	24.0598	1.35777
$315$	0	0
$316$	−3.22340	−0.181330
$317$	21.7594	1.22213	0.611066	−	0.791580i	$-0.290741\pi$
$317$	21.7594	1.22213	0.611066	+	0.791580i	$0.290741\pi$
$318$	0	0
$319$	9.99677	0.559712
$320$	−11.9393	−0.667427
$321$	0	0
$322$	7.61690	0.424473
$323$	6.84127	0.380658
$324$	0	0
$325$	13.9220	0.772251
$326$	−6.77561	−0.375267
$327$	0	0
$328$	12.3257	0.680573
$329$	−2.55141	−0.140664
$330$	0	0
$331$	22.5245	1.23806	0.619029	−	0.785368i	$-0.287526\pi$
$331$	22.5245	1.23806	0.619029	+	0.785368i	$0.287526\pi$
$332$	0.642594	0.0352669
$333$	0	0
$334$	8.98398	0.491582
$335$	−2.69659	−0.147331
$336$	0	0
$337$	17.9305	0.976735	0.488368	−	0.872638i	$-0.337593\pi$
$337$	17.9305	0.976735	0.488368	+	0.872638i	$0.337593\pi$
$338$	7.38902	0.401910
$339$	0	0
$340$	1.88107	0.102015
$341$	2.08464	0.112890
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	37.0167	1.99580
$345$	0	0
$346$	24.6433	1.32483
$347$	10.9499	0.587820	0.293910	−	0.955833i	$-0.405043\pi$
$347$	10.9499	0.587820	0.293910	+	0.955833i	$0.405043\pi$
$348$	0	0
$349$	13.0008	0.695917	0.347959	−	0.937510i	$-0.386875\pi$
$349$	13.0008	0.695917	0.347959	+	0.937510i	$0.386875\pi$
$350$	3.99213	0.213388
$351$	0	0
$352$	13.1636	0.701621
$353$	−6.83064	−0.363558	−0.181779	−	0.983339i	$-0.558186\pi$
$353$	−6.83064	−0.363558	−0.181779	+	0.983339i	$0.558186\pi$
$354$	0	0
$355$	−21.6638	−1.14979
$356$	2.28838	0.121284
$357$	0	0
$358$	22.2054	1.17359
$359$	−15.5017	−0.818147	−0.409074	−	0.912501i	$-0.634148\pi$
$359$	−15.5017	−0.818147	−0.409074	+	0.912501i	$0.634148\pi$
$360$	0	0
$361$	−14.3075	−0.753025
$362$	29.2250	1.53603
$363$	0	0
$364$	−1.93144	−0.101235
$365$	−4.89479	−0.256205
$366$	0	0
$367$	−9.57294	−0.499704	−0.249852	−	0.968284i	$-0.580382\pi$
$367$	−9.57294	−0.499704	−0.249852	+	0.968284i	$0.580382\pi$
$368$	17.7989	0.927831
$369$	0	0
$370$	−3.05023	−0.158574
$371$	−0.687516	−0.0356941
$372$	0	0
$373$	−7.99725	−0.414082	−0.207041	−	0.978332i	$-0.566383\pi$
$373$	−7.99725	−0.414082	−0.207041	+	0.978332i	$0.566383\pi$
$374$	21.0695	1.08948
$375$	0	0
$376$	−7.77833	−0.401137
$377$	−8.13102	−0.418769
$378$	0	0
$379$	−17.0692	−0.876785	−0.438393	−	0.898784i	$-0.644452\pi$
$379$	−17.0692	−0.876785	−0.438393	+	0.898784i	$0.644452\pi$
$380$	1.29025	0.0661886
$381$	0	0
$382$	−20.1711	−1.03205
$383$	6.50697	0.332491	0.166245	−	0.986084i	$-0.446836\pi$
$383$	6.50697	0.332491	0.166245	+	0.986084i	$0.446836\pi$
$384$	0	0
$385$	7.17477	0.365660
$386$	−12.5854	−0.640578
$387$	0	0
$388$	−0.466883	−0.0237024
$389$	−7.96414	−0.403798	−0.201899	−	0.979406i	$-0.564711\pi$
$389$	−7.96414	−0.403798	−0.201899	+	0.979406i	$0.564711\pi$
$390$	0	0
$391$	−19.2844	−0.975255
$392$	−3.04864	−0.153979
$393$	0	0
$394$	5.62137	0.283200
$395$	−9.73929	−0.490037
$396$	0	0
$397$	5.65801	0.283967	0.141984	−	0.989869i	$-0.454652\pi$
$397$	5.65801	0.283967	0.141984	+	0.989869i	$0.454652\pi$
$398$	27.9718	1.40210
$399$	0	0
$400$	9.32866	0.466433
$401$	−8.34238	−0.416598	−0.208299	−	0.978065i	$-0.566793\pi$
$401$	−8.34238	−0.416598	−0.208299	+	0.978065i	$0.566793\pi$
$402$	0	0
$403$	−1.69557	−0.0844625
$404$	−4.94451	−0.245998
$405$	0	0
$406$	−2.33157	−0.115714
$407$	9.74880	0.483230
$408$	0	0
$409$	35.1591	1.73850	0.869252	−	0.494370i	$-0.164601\pi$
$409$	35.1591	1.73850	0.869252	+	0.494370i	$0.164601\pi$
$410$	6.76556	0.334127
$411$	0	0
$412$	−1.30688	−0.0643854
$413$	−7.83644	−0.385606
$414$	0	0
$415$	1.94156	0.0953072
$416$	−10.7068	−0.524944
$417$	0	0
$418$	14.4519	0.706864
$419$	−5.32538	−0.260162	−0.130081	−	0.991503i	$-0.541524\pi$
$419$	−5.32538	−0.260162	−0.130081	+	0.991503i	$0.541524\pi$
$420$	0	0
$421$	8.82064	0.429892	0.214946	−	0.976626i	$-0.431043\pi$
$421$	8.82064	0.429892	0.214946	+	0.976626i	$0.431043\pi$
$422$	1.00958	0.0491455
$423$	0	0
$424$	−2.09599	−0.101790
$425$	−10.1072	−0.490273
$426$	0	0
$427$	8.37683	0.405383
$428$	7.92575	0.383106
$429$	0	0
$430$	20.3184	0.979839
$431$	−19.9202	−0.959524	−0.479762	−	0.877399i	$-0.659277\pi$
$431$	−19.9202	−0.959524	−0.479762	+	0.877399i	$0.659277\pi$
$432$	0	0
$433$	−20.1919	−0.970360	−0.485180	−	0.874414i	$-0.661246\pi$
$433$	−20.1919	−0.970360	−0.485180	+	0.874414i	$0.661246\pi$
$434$	−0.486206	−0.0233387
$435$	0	0
$436$	−3.06287	−0.146685
$437$	−13.2274	−0.632754
$438$	0	0
$439$	26.8589	1.28190	0.640952	−	0.767581i	$-0.278540\pi$
$439$	26.8589	1.28190	0.640952	+	0.767581i	$0.278540\pi$
$440$	21.8733	1.04277
$441$	0	0
$442$	−17.1372	−0.815134
$443$	−8.75427	−0.415928	−0.207964	−	0.978137i	$-0.566684\pi$
$443$	−8.75427	−0.415928	−0.207964	+	0.978137i	$0.566684\pi$
$444$	0	0
$445$	6.91419	0.327764
$446$	8.64927	0.409555
$447$	0	0
$448$	−8.89993	−0.420482
$449$	8.94965	0.422360	0.211180	−	0.977447i	$-0.432269\pi$
$449$	8.94965	0.422360	0.211180	+	0.977447i	$0.432269\pi$
$450$	0	0
$451$	−21.6233	−1.01820
$452$	3.22943	0.151900
$453$	0	0
$454$	−13.1105	−0.615305
$455$	−5.83571	−0.273582
$456$	0	0
$457$	−3.94645	−0.184607	−0.0923035	−	0.995731i	$-0.529423\pi$
$457$	−3.94645	−0.184607	−0.0923035	+	0.995731i	$0.529423\pi$
$458$	−19.6869	−0.919907
$459$	0	0
$460$	−3.63701	−0.169577
$461$	−17.6827	−0.823566	−0.411783	−	0.911282i	$-0.635094\pi$
$461$	−17.6827	−0.823566	−0.411783	+	0.911282i	$0.635094\pi$
$462$	0	0
$463$	19.1389	0.889460	0.444730	−	0.895665i	$-0.353300\pi$
$463$	19.1389	0.889460	0.444730	+	0.895665i	$0.353300\pi$
$464$	−5.44834	−0.252933
$465$	0	0
$466$	−3.16422	−0.146580
$467$	−21.0237	−0.972861	−0.486430	−	0.873719i	$-0.661701\pi$
$467$	−21.0237	−0.972861	−0.486430	+	0.873719i	$0.661701\pi$
$468$	0	0
$469$	−2.01012	−0.0928189
$470$	−4.26951	−0.196938
$471$	0	0
$472$	−23.8905	−1.09965
$473$	−64.9392	−2.98591
$474$	0	0
$475$	−6.93270	−0.318094
$476$	1.40221	0.0642701
$477$	0	0
$478$	−13.8319	−0.632657
$479$	−2.35198	−0.107465	−0.0537324	−	0.998555i	$-0.517112\pi$
$479$	−2.35198	−0.107465	−0.0537324	+	0.998555i	$0.517112\pi$
$480$	0	0
$481$	−7.92933	−0.361546
$482$	32.8174	1.49479
$483$	0	0
$484$	−7.81625	−0.355284
$485$	−1.41066	−0.0640546
$486$	0	0
$487$	27.2298	1.23390	0.616949	−	0.787003i	$-0.288368\pi$
$487$	27.2298	1.23390	0.616949	+	0.787003i	$0.288368\pi$
$488$	25.5379	1.15605
$489$	0	0
$490$	−1.67339	−0.0755962
$491$	−10.4846	−0.473165	−0.236583	−	0.971611i	$-0.576027\pi$
$491$	−10.4846	−0.473165	−0.236583	+	0.971611i	$0.576027\pi$
$492$	0	0
$493$	5.90306	0.265861
$494$	−11.7546	−0.528867
$495$	0	0
$496$	−1.13615	−0.0510146
$497$	−16.1489	−0.724375
$498$	0	0
$499$	9.97856	0.446702	0.223351	−	0.974738i	$-0.428300\pi$
$499$	9.97856	0.446702	0.223351	+	0.974738i	$0.428300\pi$
$500$	−4.88433	−0.218434
$501$	0	0
$502$	15.6502	0.698501
$503$	−11.4444	−0.510283	−0.255141	−	0.966904i	$-0.582122\pi$
$503$	−11.4444	−0.510283	−0.255141	+	0.966904i	$0.582122\pi$
$504$	0	0
$505$	−14.9395	−0.664799
$506$	−40.7375	−1.81100
$507$	0	0
$508$	−0.443996	−0.0196991
$509$	2.37848	0.105424	0.0527121	−	0.998610i	$-0.483213\pi$
$509$	2.37848	0.105424	0.0527121	+	0.998610i	$0.483213\pi$
$510$	0	0
$511$	−3.64873	−0.161410
$512$	−24.9471	−1.10252
$513$	0	0
$514$	7.05584	0.311220
$515$	−3.94866	−0.173999
$516$	0	0
$517$	13.6457	0.600138
$518$	−2.27374	−0.0999024
$519$	0	0
$520$	−17.7910	−0.780186
$521$	−17.4353	−0.763856	−0.381928	−	0.924192i	$-0.624740\pi$
$521$	−17.4353	−0.763856	−0.381928	+	0.924192i	$0.624740\pi$
$522$	0	0
$523$	32.1999	1.40800	0.704001	−	0.710199i	$-0.251395\pi$
$523$	32.1999	1.40800	0.704001	+	0.710199i	$0.251395\pi$
$524$	0.408991	0.0178669
$525$	0	0
$526$	21.5749	0.940711
$527$	1.23097	0.0536221
$528$	0	0
$529$	14.2860	0.621131
$530$	−1.15048	−0.0499739
$531$	0	0
$532$	0.961794	0.0416991
$533$	17.5876	0.761805
$534$	0	0
$535$	23.9472	1.03533
$536$	−6.12814	−0.264695
$537$	0	0
$538$	34.0544	1.46819
$539$	5.34830	0.230368
$540$	0	0
$541$	−10.6946	−0.459795	−0.229897	−	0.973215i	$-0.573839\pi$
$541$	−10.6946	−0.459795	−0.229897	+	0.973215i	$0.573839\pi$
$542$	−1.19662	−0.0513994
$543$	0	0
$544$	7.77306	0.333267
$545$	−9.25425	−0.396409
$546$	0	0
$547$	22.0976	0.944825	0.472413	−	0.881377i	$-0.343383\pi$
$547$	22.0976	0.944825	0.472413	+	0.881377i	$0.343383\pi$
$548$	−3.99118	−0.170495
$549$	0	0
$550$	−21.3511	−0.910414
$551$	4.04899	0.172493
$552$	0	0
$553$	−7.25997	−0.308725
$554$	14.5859	0.619696
$555$	0	0
$556$	1.52824	0.0648120
$557$	−29.3270	−1.24262	−0.621312	−	0.783563i	$-0.713400\pi$
$557$	−29.3270	−1.24262	−0.621312	+	0.783563i	$0.713400\pi$
$558$	0	0
$559$	52.8193	2.23402
$560$	−3.91032	−0.165241
$561$	0	0
$562$	7.12456	0.300531
$563$	−34.8575	−1.46907	−0.734533	−	0.678572i	$-0.762599\pi$
$563$	−34.8575	−1.46907	−0.734533	+	0.678572i	$0.762599\pi$
$564$	0	0
$565$	9.75752	0.410502
$566$	−9.82805	−0.413104
$567$	0	0
$568$	−49.2320	−2.06573
$569$	−10.5395	−0.441840	−0.220920	−	0.975292i	$-0.570906\pi$
$569$	−10.5395	−0.441840	−0.220920	+	0.975292i	$0.570906\pi$
$570$	0	0
$571$	−8.46093	−0.354079	−0.177040	−	0.984204i	$-0.556652\pi$
$571$	−8.46093	−0.354079	−0.177040	+	0.984204i	$0.556652\pi$
$572$	10.3299	0.431915
$573$	0	0
$574$	5.04326	0.210502
$575$	19.5421	0.814964
$576$	0	0
$577$	17.0207	0.708579	0.354290	−	0.935136i	$-0.384723\pi$
$577$	17.0207	0.708579	0.354290	+	0.935136i	$0.384723\pi$
$578$	−8.76429	−0.364546
$579$	0	0
$580$	1.11331	0.0462276
$581$	1.44730	0.0600440
$582$	0	0
$583$	3.67704	0.152288
$584$	−11.1236	−0.460300
$585$	0	0
$586$	−28.6198	−1.18227
$587$	39.3515	1.62421	0.812104	−	0.583512i	$-0.198322\pi$
$587$	39.3515	1.62421	0.812104	+	0.583512i	$0.198322\pi$
$588$	0	0
$589$	0.844342	0.0347905
$590$	−13.1134	−0.539872
$591$	0	0
$592$	−5.31319	−0.218371
$593$	36.0916	1.48211	0.741053	−	0.671447i	$-0.234327\pi$
$593$	36.0916	1.48211	0.741053	+	0.671447i	$0.234327\pi$
$594$	0	0
$595$	4.23668	0.173687
$596$	−0.535070	−0.0219173
$597$	0	0
$598$	33.1344	1.35497
$599$	−16.7381	−0.683899	−0.341950	−	0.939718i	$-0.611087\pi$
$599$	−16.7381	−0.683899	−0.341950	+	0.939718i	$0.611087\pi$
$600$	0	0
$601$	34.8782	1.42271	0.711357	−	0.702831i	$-0.248081\pi$
$601$	34.8782	1.42271	0.711357	+	0.702831i	$0.248081\pi$
$602$	15.1460	0.617303
$603$	0	0
$604$	−6.08023	−0.247401
$605$	−23.6163	−0.960138
$606$	0	0
$607$	−17.1367	−0.695555	−0.347778	−	0.937577i	$-0.613064\pi$
$607$	−17.1367	−0.695555	−0.347778	+	0.937577i	$0.613064\pi$
$608$	5.33165	0.216227
$609$	0	0
$610$	14.0177	0.567561
$611$	−11.0989	−0.449015
$612$	0	0
$613$	19.2408	0.777130	0.388565	−	0.921421i	$-0.372971\pi$
$613$	19.2408	0.777130	0.388565	+	0.921421i	$0.372971\pi$
$614$	11.3098	0.456427
$615$	0	0
$616$	16.3050	0.656948
$617$	−30.2743	−1.21880	−0.609398	−	0.792864i	$-0.708589\pi$
$617$	−30.2743	−1.21880	−0.609398	+	0.792864i	$0.708589\pi$
$618$	0	0
$619$	−34.7250	−1.39571	−0.697857	−	0.716237i	$-0.745863\pi$
$619$	−34.7250	−1.39571	−0.697857	+	0.716237i	$0.745863\pi$
$620$	0.232160	0.00932376
$621$	0	0
$622$	15.3617	0.615948
$623$	5.15405	0.206493
$624$	0	0
$625$	1.24413	0.0497654
$626$	0.802914	0.0320909
$627$	0	0
$628$	−8.56379	−0.341732
$629$	5.75664	0.229532
$630$	0	0
$631$	−3.79750	−0.151176	−0.0755881	−	0.997139i	$-0.524083\pi$
$631$	−3.79750	−0.151176	−0.0755881	+	0.997139i	$0.524083\pi$
$632$	−22.1330	−0.880404
$633$	0	0
$634$	27.1427	1.07797
$635$	−1.34151	−0.0532360
$636$	0	0
$637$	−4.35012	−0.172358
$638$	12.4700	0.493690
$639$	0	0
$640$	−8.28947	−0.327670
$641$	−3.57080	−0.141038	−0.0705190	−	0.997510i	$-0.522466\pi$
$641$	−3.57080	−0.141038	−0.0705190	+	0.997510i	$0.522466\pi$
$642$	0	0
$643$	−24.5350	−0.967565	−0.483782	−	0.875188i	$-0.660737\pi$
$643$	−24.5350	−0.967565	−0.483782	+	0.875188i	$0.660737\pi$
$644$	−2.71114	−0.106834
$645$	0	0
$646$	8.53379	0.335757
$647$	3.17518	0.124829	0.0624147	−	0.998050i	$-0.480120\pi$
$647$	3.17518	0.124829	0.0624147	+	0.998050i	$0.480120\pi$
$648$	0	0
$649$	41.9116	1.64517
$650$	17.3662	0.681160
$651$	0	0
$652$	2.41170	0.0944493
$653$	−17.8809	−0.699733	−0.349866	−	0.936800i	$-0.613773\pi$
$653$	−17.8809	−0.699733	−0.349866	+	0.936800i	$0.613773\pi$
$654$	0	0
$655$	1.23574	0.0482843
$656$	11.7849	0.460123
$657$	0	0
$658$	−3.18263	−0.124072
$659$	−18.7474	−0.730295	−0.365148	−	0.930950i	$-0.618982\pi$
$659$	−18.7474	−0.730295	−0.365148	+	0.930950i	$0.618982\pi$
$660$	0	0
$661$	−47.9298	−1.86425	−0.932127	−	0.362131i	$-0.882049\pi$
$661$	−47.9298	−1.86425	−0.932127	+	0.362131i	$0.882049\pi$
$662$	28.0970	1.09202
$663$	0	0
$664$	4.41228	0.171230
$665$	2.90600	0.112690
$666$	0	0
$667$	−11.4134	−0.441930
$668$	−3.19774	−0.123724
$669$	0	0
$670$	−3.36373	−0.129952
$671$	−44.8018	−1.72955
$672$	0	0
$673$	36.8538	1.42061	0.710305	−	0.703894i	$-0.248557\pi$
$673$	36.8538	1.42061	0.710305	+	0.703894i	$0.248557\pi$
$674$	22.3665	0.861523
$675$	0	0
$676$	−2.63003	−0.101155
$677$	5.44042	0.209092	0.104546	−	0.994520i	$-0.466661\pi$
$677$	5.44042	0.209092	0.104546	+	0.994520i	$0.466661\pi$
$678$	0	0
$679$	−1.05155	−0.0403547
$680$	12.9161	0.495310
$681$	0	0
$682$	2.60038	0.0995736
$683$	−33.7346	−1.29082	−0.645408	−	0.763838i	$-0.723313\pi$
$683$	−33.7346	−1.29082	−0.645408	+	0.763838i	$0.723313\pi$
$684$	0	0
$685$	−12.0591	−0.460754
$686$	−1.24740	−0.0476259
$687$	0	0
$688$	35.3925	1.34933
$689$	−2.99078	−0.113940
$690$	0	0
$691$	0.00717620	0.000272995 0	0.000136498	−	1.00000i	$-0.499957\pi$
$691$	0.00717620	0.000272995 0	0.000136498		1.00000i	$0.499957\pi$
$692$	−8.77147	−0.333441
$693$	0	0
$694$	13.6589	0.518484
$695$	4.61749	0.175151
$696$	0	0
$697$	−12.7685	−0.483641
$698$	16.2172	0.613830
$699$	0	0
$700$	−1.42095	−0.0537068
$701$	40.5873	1.53296	0.766480	−	0.642268i	$-0.222006\pi$
$701$	40.5873	1.53296	0.766480	+	0.642268i	$0.222006\pi$
$702$	0	0
$703$	3.94856	0.148923
$704$	47.5995	1.79397
$705$	0	0
$706$	−8.52053	−0.320675
$707$	−11.1364	−0.418826
$708$	0	0
$709$	−25.8397	−0.970430	−0.485215	−	0.874395i	$-0.661259\pi$
$709$	−25.8397	−0.970430	−0.485215	+	0.874395i	$0.661259\pi$
$710$	−27.0234	−1.01417
$711$	0	0
$712$	15.7128	0.588863
$713$	−2.38006	−0.0891340
$714$	0	0
$715$	31.2111	1.16723
$716$	−7.90375	−0.295377
$717$	0	0
$718$	−19.3368	−0.721642
$719$	−11.3043	−0.421581	−0.210791	−	0.977531i	$-0.567604\pi$
$719$	−11.3043	−0.421581	−0.210791	+	0.977531i	$0.567604\pi$
$720$	0	0
$721$	−2.94345	−0.109620
$722$	−17.8471	−0.664202
$723$	0	0
$724$	−10.4023	−0.386598
$725$	−5.98195	−0.222164
$726$	0	0
$727$	−36.2507	−1.34446	−0.672232	−	0.740340i	$-0.734664\pi$
$727$	−36.2507	−1.34446	−0.672232	+	0.740340i	$0.734664\pi$
$728$	−13.2619	−0.491520
$729$	0	0
$730$	−6.10575	−0.225984
$731$	−38.3464	−1.41829
$732$	0	0
$733$	41.7267	1.54121	0.770605	−	0.637313i	$-0.219954\pi$
$733$	41.7267	1.54121	0.770605	+	0.637313i	$0.219954\pi$
$734$	−11.9413	−0.440761
$735$	0	0
$736$	−15.0290	−0.553978
$737$	10.7507	0.396009
$738$	0	0
$739$	35.4262	1.30317	0.651587	−	0.758574i	$-0.274104\pi$
$739$	35.4262	1.30317	0.651587	+	0.758574i	$0.274104\pi$
$740$	1.08569	0.0399109
$741$	0	0
$742$	−0.857607	−0.0314837
$743$	−33.0960	−1.21417	−0.607087	−	0.794635i	$-0.707662\pi$
$743$	−33.0960	−1.21417	−0.607087	+	0.794635i	$0.707662\pi$
$744$	0	0
$745$	−1.61668	−0.0592306
$746$	−9.97577	−0.365239
$747$	0	0
$748$	−7.49943	−0.274206
$749$	17.8510	0.652260
$750$	0	0
$751$	−0.221433	−0.00808019	−0.00404010	−	0.999992i	$-0.501286\pi$
$751$	−0.221433	−0.00808019	−0.00404010	+	0.999992i	$0.501286\pi$
$752$	−7.43704	−0.271201
$753$	0	0
$754$	−10.1426	−0.369373
$755$	−18.3710	−0.668589
$756$	0	0
$757$	9.55021	0.347108	0.173554	−	0.984824i	$-0.444475\pi$
$757$	9.55021	0.347108	0.173554	+	0.984824i	$0.444475\pi$
$758$	−21.2921	−0.773363
$759$	0	0
$760$	8.85934	0.321362
$761$	29.0094	1.05159	0.525794	−	0.850612i	$-0.323768\pi$
$761$	29.0094	1.05159	0.525794	+	0.850612i	$0.323768\pi$
$762$	0	0
$763$	−6.89841	−0.249739
$764$	7.17967	0.259751
$765$	0	0
$766$	8.11679	0.293271
$767$	−34.0894	−1.23090
$768$	0	0
$769$	−46.1448	−1.66402	−0.832012	−	0.554758i	$-0.812811\pi$
$769$	−46.1448	−1.66402	−0.832012	+	0.554758i	$0.812811\pi$
$770$	8.94980	0.322529
$771$	0	0
$772$	4.47961	0.161225
$773$	9.10578	0.327512	0.163756	−	0.986501i	$-0.447639\pi$
$773$	9.10578	0.327512	0.163756	+	0.986501i	$0.447639\pi$
$774$	0	0
$775$	−1.24743	−0.0448088
$776$	−3.20579	−0.115081
$777$	0	0
$778$	−9.93446	−0.356168
$779$	−8.75808	−0.313791
$780$	0	0
$781$	86.3689	3.09052
$782$	−24.0554	−0.860218
$783$	0	0
$784$	−2.91487	−0.104103
$785$	−25.8749	−0.923516
$786$	0	0
$787$	−16.1514	−0.575734	−0.287867	−	0.957670i	$-0.592946\pi$
$787$	−16.1514	−0.575734	−0.287867	+	0.957670i	$0.592946\pi$
$788$	−2.00086	−0.0712776
$789$	0	0
$790$	−12.1488	−0.432234
$791$	7.27356	0.258618
$792$	0	0
$793$	36.4402	1.29403
$794$	7.05780	0.250472
$795$	0	0
$796$	−9.95620	−0.352888
$797$	−42.5067	−1.50566	−0.752832	−	0.658213i	$-0.771313\pi$
$797$	−42.5067	−1.50566	−0.752832	+	0.658213i	$0.771313\pi$
$798$	0	0
$799$	8.05775	0.285063
$800$	−7.87694	−0.278492
$801$	0	0
$802$	−10.4063	−0.367458
$803$	19.5145	0.688651
$804$	0	0
$805$	−8.19154	−0.288714
$806$	−2.11506	−0.0744997
$807$	0	0
$808$	−33.9508	−1.19438
$809$	15.8564	0.557482	0.278741	−	0.960366i	$-0.410083\pi$
$809$	15.8564	0.557482	0.278741	+	0.960366i	$0.410083\pi$
$810$	0	0
$811$	−34.5190	−1.21213	−0.606063	−	0.795417i	$-0.707252\pi$
$811$	−34.5190	−1.21213	−0.606063	+	0.795417i	$0.707252\pi$
$812$	0.829895	0.0291236
$813$	0	0
$814$	12.1606	0.426230
$815$	7.28678	0.255245
$816$	0	0
$817$	−26.3023	−0.920202
$818$	43.8574	1.53344
$819$	0	0
$820$	−2.40812	−0.0840951
$821$	30.4097	1.06130	0.530652	−	0.847590i	$-0.321947\pi$
$821$	30.4097	1.06130	0.530652	+	0.847590i	$0.321947\pi$
$822$	0	0
$823$	−49.9569	−1.74139	−0.870694	−	0.491826i	$-0.836330\pi$
$823$	−49.9569	−1.74139	−0.870694	+	0.491826i	$0.836330\pi$
$824$	−8.97352	−0.312607
$825$	0	0
$826$	−9.77516	−0.340121
$827$	38.7472	1.34737	0.673685	−	0.739018i	$-0.264710\pi$
$827$	38.7472	1.34737	0.673685	+	0.739018i	$0.264710\pi$
$828$	0	0
$829$	−13.4751	−0.468009	−0.234004	−	0.972236i	$-0.575183\pi$
$829$	−13.4751	−0.468009	−0.234004	+	0.972236i	$0.575183\pi$
$830$	2.42189	0.0840652
$831$	0	0
$832$	−38.7158	−1.34223
$833$	3.15816	0.109424
$834$	0	0
$835$	−9.66176	−0.334359
$836$	−5.14396	−0.177908
$837$	0	0
$838$	−6.64288	−0.229474
$839$	12.9736	0.447899	0.223949	−	0.974601i	$-0.428105\pi$
$839$	12.9736	0.447899	0.223949	+	0.974601i	$0.428105\pi$
$840$	0	0
$841$	−25.5063	−0.879527
$842$	11.0029	0.379183
$843$	0	0
$844$	−0.359347	−0.0123692
$845$	−7.94647	−0.273367
$846$	0	0
$847$	−17.6043	−0.604891
$848$	−2.00402	−0.0688185
$849$	0	0
$850$	−12.6078	−0.432443
$851$	−11.1303	−0.381543
$852$	0	0
$853$	25.5643	0.875305	0.437653	−	0.899144i	$-0.355810\pi$
$853$	25.5643	0.875305	0.437653	+	0.899144i	$0.355810\pi$
$854$	10.4492	0.357566
$855$	0	0
$856$	54.4211	1.86007
$857$	33.0517	1.12903	0.564513	−	0.825424i	$-0.309064\pi$
$857$	33.0517	1.12903	0.564513	+	0.825424i	$0.309064\pi$
$858$	0	0
$859$	44.4442	1.51641	0.758207	−	0.652013i	$-0.226075\pi$
$859$	44.4442	1.51641	0.758207	+	0.652013i	$0.226075\pi$
$860$	−7.23208	−0.246612
$861$	0	0
$862$	−24.8485	−0.846342
$863$	23.4382	0.797845	0.398923	−	0.916985i	$-0.369384\pi$
$863$	23.4382	0.797845	0.398923	+	0.916985i	$0.369384\pi$
$864$	0	0
$865$	−26.5024	−0.901109
$866$	−25.1873	−0.855901
$867$	0	0
$868$	0.173059	0.00587401
$869$	38.8285	1.31717
$870$	0	0
$871$	−8.74428	−0.296289
$872$	−21.0307	−0.712191
$873$	0	0
$874$	−16.4999	−0.558117
$875$	−11.0008	−0.371896
$876$	0	0
$877$	−17.3000	−0.584178	−0.292089	−	0.956391i	$-0.594350\pi$
$877$	−17.3000	−0.584178	−0.292089	+	0.956391i	$0.594350\pi$
$878$	33.5037	1.13070
$879$	0	0
$880$	20.9136	0.704996
$881$	−39.8133	−1.34135	−0.670673	−	0.741753i	$-0.733995\pi$
$881$	−39.8133	−1.34135	−0.670673	+	0.741753i	$0.733995\pi$
$882$	0	0
$883$	−11.6937	−0.393525	−0.196763	−	0.980451i	$-0.563043\pi$
$883$	−11.6937	−0.393525	−0.196763	+	0.980451i	$0.563043\pi$
$884$	6.09978	0.205158
$885$	0	0
$886$	−10.9201	−0.366867
$887$	15.0546	0.505484	0.252742	−	0.967534i	$-0.418668\pi$
$887$	15.0546	0.505484	0.252742	+	0.967534i	$0.418668\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	8.62475	0.289102
$891$	0	0
$892$	−3.07860	−0.103079
$893$	5.52692	0.184951
$894$	0	0
$895$	−23.8807	−0.798242
$896$	−6.17923	−0.206434
$897$	0	0
$898$	11.1638	0.372540
$899$	0.728550	0.0242985
$900$	0	0
$901$	2.17128	0.0723359
$902$	−26.9729	−0.898098
$903$	0	0
$904$	22.1744	0.737511
$905$	−31.4298	−1.04476
$906$	0	0
$907$	−16.3901	−0.544225	−0.272112	−	0.962265i	$-0.587722\pi$
$907$	−16.3901	−0.544225	−0.272112	+	0.962265i	$0.587722\pi$
$908$	4.66651	0.154864
$909$	0	0
$910$	−7.27946	−0.241312
$911$	−37.9073	−1.25592	−0.627962	−	0.778244i	$-0.716111\pi$
$911$	−37.9073	−1.25592	−0.627962	+	0.778244i	$0.716111\pi$
$912$	0	0
$913$	−7.74057	−0.256176
$914$	−4.92279	−0.162832
$915$	0	0
$916$	7.00730	0.231528
$917$	0.921159	0.0304193
$918$	0	0
$919$	6.73720	0.222240	0.111120	−	0.993807i	$-0.464556\pi$
$919$	6.73720	0.222240	0.111120	+	0.993807i	$0.464556\pi$
$920$	−24.9730	−0.823337
$921$	0	0
$922$	−22.0574	−0.726422
$923$	−70.2495	−2.31229
$924$	0	0
$925$	−5.83357	−0.191807
$926$	23.8738	0.784543
$927$	0	0
$928$	4.60047	0.151018
$929$	−1.91596	−0.0628605	−0.0314302	−	0.999506i	$-0.510006\pi$
$929$	−1.91596	−0.0628605	−0.0314302	+	0.999506i	$0.510006\pi$
$930$	0	0
$931$	2.16622	0.0709951
$932$	1.12626	0.0368920
$933$	0	0
$934$	−26.2249	−0.858106
$935$	−22.6591	−0.741030
$936$	0	0
$937$	23.3384	0.762433	0.381216	−	0.924486i	$-0.375505\pi$
$937$	23.3384	0.762433	0.381216	+	0.924486i	$0.375505\pi$
$938$	−2.50743	−0.0818704
$939$	0	0
$940$	1.51968	0.0495665
$941$	38.3365	1.24973	0.624867	−	0.780731i	$-0.285153\pi$
$941$	38.3365	1.24973	0.624867	+	0.780731i	$0.285153\pi$
$942$	0	0
$943$	24.6876	0.803939
$944$	−22.8422	−0.743451
$945$	0	0
$946$	−81.0051	−2.63370
$947$	−59.7658	−1.94213	−0.971064	−	0.238821i	$-0.923239\pi$
$947$	−59.7658	−1.94213	−0.971064	+	0.238821i	$0.923239\pi$
$948$	0	0
$949$	−15.8724	−0.515240
$950$	−8.64784	−0.280573
$951$	0	0
$952$	9.62807	0.312048
$953$	−52.0956	−1.68754	−0.843771	−	0.536703i	$-0.819670\pi$
$953$	−52.0956	−1.68754	−0.843771	+	0.536703i	$0.819670\pi$
$954$	0	0
$955$	21.6929	0.701966
$956$	4.92330	0.159231
$957$	0	0
$958$	−2.93386	−0.0947886
$959$	−8.98921	−0.290277
$960$	0	0
$961$	−30.8481	−0.995099
$962$	−9.89104	−0.318900
$963$	0	0
$964$	−11.6809	−0.376218
$965$	13.5348	0.435702
$966$	0	0
$967$	17.9971	0.578748	0.289374	−	0.957216i	$-0.406553\pi$
$967$	17.9971	0.578748	0.289374	+	0.957216i	$0.406553\pi$
$968$	−53.6692	−1.72499
$969$	0	0
$970$	−1.75965	−0.0564990
$971$	−21.2618	−0.682324	−0.341162	−	0.940005i	$-0.610820\pi$
$971$	−21.2618	−0.682324	−0.341162	+	0.940005i	$0.610820\pi$
$972$	0	0
$973$	3.44202	0.110346
$974$	33.9664	1.08835
$975$	0	0
$976$	24.4174	0.781582
$977$	−41.7439	−1.33551	−0.667753	−	0.744383i	$-0.732744\pi$
$977$	−41.7439	−1.33551	−0.667753	+	0.744383i	$0.732744\pi$
$978$	0	0
$979$	−27.5654	−0.880994
$980$	0.595623	0.0190265
$981$	0	0
$982$	−13.0785	−0.417353
$983$	−24.8258	−0.791821	−0.395910	−	0.918289i	$-0.629571\pi$
$983$	−24.8258	−0.791821	−0.395910	+	0.918289i	$0.629571\pi$
$984$	0	0
$985$	−6.04546	−0.192624
$986$	7.36347	0.234501
$987$	0	0
$988$	4.18392	0.133108
$989$	74.1420	2.35758
$990$	0	0
$991$	36.9314	1.17317	0.586583	−	0.809889i	$-0.300473\pi$
$991$	36.9314	1.17317	0.586583	+	0.809889i	$0.300473\pi$
$992$	0.959342	0.0304591
$993$	0	0
$994$	−20.1441	−0.638931
$995$	−30.0820	−0.953664
$996$	0	0
$997$	42.3243	1.34042	0.670212	−	0.742169i	$-0.266203\pi$
$997$	42.3243	1.34042	0.670212	+	0.742169i	$0.266203\pi$
$998$	12.4472	0.394011
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.y.1.19	yes	28
3.2	odd	2	inner	8001.2.a.y.1.10	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.y.1.10	✓	28	3.2	odd	2	inner
8001.2.a.y.1.19	yes	28	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+1.24740 q^{2} -0.443996 q^{4} -1.34151 q^{5} -1.00000 q^{7} -3.04864 q^{8} +O(q^{10})\)	\(q+1.24740 q^{2} -0.443996 q^{4} -1.34151 q^{5} -1.00000 q^{7} -3.04864 q^{8} -1.67339 q^{10} +5.34830 q^{11} -4.35012 q^{13} -1.24740 q^{14} -2.91487 q^{16} +3.15816 q^{17} +2.16622 q^{19} +0.595623 q^{20} +6.67146 q^{22} -6.10623 q^{23} -3.20036 q^{25} -5.42633 q^{26} +0.443996 q^{28} +1.86915 q^{29} +0.389776 q^{31} +2.46126 q^{32} +3.93948 q^{34} +1.34151 q^{35} +1.82278 q^{37} +2.70214 q^{38} +4.08977 q^{40} -4.04302 q^{41} -12.1420 q^{43} -2.37462 q^{44} -7.61690 q^{46} +2.55141 q^{47} +1.00000 q^{49} -3.99213 q^{50} +1.93144 q^{52} +0.687516 q^{53} -7.17477 q^{55} +3.04864 q^{56} +2.33157 q^{58} +7.83644 q^{59} -8.37683 q^{61} +0.486206 q^{62} +8.89993 q^{64} +5.83571 q^{65} +2.01012 q^{67} -1.40221 q^{68} +1.67339 q^{70} +16.1489 q^{71} +3.64873 q^{73} +2.27374 q^{74} -0.961794 q^{76} -5.34830 q^{77} +7.25997 q^{79} +3.91032 q^{80} -5.04326 q^{82} -1.44730 q^{83} -4.23668 q^{85} -15.1460 q^{86} -16.3050 q^{88} -5.15405 q^{89} +4.35012 q^{91} +2.71114 q^{92} +3.18263 q^{94} -2.90600 q^{95} +1.05155 q^{97} +1.24740 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) \) 28 * q + 30 * q^4 - 28 * q^7	\( 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) \) 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * q^94 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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