LMFDB - Embedded newform 8001.2.a.z.1.25

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:	$1$
Dimension:	$32$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.25
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)

$f(q)$	$=$	$q+1.63487 q^{2} +0.672794 q^{4} -0.107826 q^{5} -1.00000 q^{7} -2.16981 q^{8} +O(q^{10})$	\(q+1.63487 q^{2} +0.672794 q^{4} -0.107826 q^{5} -1.00000 q^{7} -2.16981 q^{8} -0.176281 q^{10} +1.23406 q^{11} -2.67752 q^{13} -1.63487 q^{14} -4.89294 q^{16} +6.14025 q^{17} +1.57630 q^{19} -0.0725448 q^{20} +2.01753 q^{22} +5.97787 q^{23} -4.98837 q^{25} -4.37740 q^{26} -0.672794 q^{28} -9.60657 q^{29} +1.81504 q^{31} -3.65969 q^{32} +10.0385 q^{34} +0.107826 q^{35} +7.63656 q^{37} +2.57704 q^{38} +0.233962 q^{40} -4.73470 q^{41} +8.65406 q^{43} +0.830270 q^{44} +9.77303 q^{46} -11.0930 q^{47} +1.00000 q^{49} -8.15533 q^{50} -1.80142 q^{52} -13.7545 q^{53} -0.133064 q^{55} +2.16981 q^{56} -15.7055 q^{58} +5.68940 q^{59} -8.36686 q^{61} +2.96735 q^{62} +3.80276 q^{64} +0.288707 q^{65} -4.40536 q^{67} +4.13113 q^{68} +0.176281 q^{70} -13.9036 q^{71} -8.13243 q^{73} +12.4848 q^{74} +1.06052 q^{76} -1.23406 q^{77} +10.6927 q^{79} +0.527586 q^{80} -7.74061 q^{82} +10.3228 q^{83} -0.662079 q^{85} +14.1482 q^{86} -2.67768 q^{88} +15.7224 q^{89} +2.67752 q^{91} +4.02188 q^{92} -18.1355 q^{94} -0.169966 q^{95} -12.2148 q^{97} +1.63487 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) $ 32 * q + 30 * q^4 - 32 * q^7	$ 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) $ 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * 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 $

$2$	1.63487	1.15603	0.578013	−	0.816027i	$-0.303828\pi$
$2$	1.63487	1.15603	0.578013	+	0.816027i	$0.303828\pi$
$3$	0	0
$3$	0	0
$4$	0.672794	0.336397
$5$	−0.107826	−0.0482213	−0.0241106	−	0.999709i	$-0.507675\pi$
$5$	−0.107826	−0.0482213	−0.0241106	+	0.999709i	$0.507675\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.16981	−0.767142
$9$	0	0
$10$	−0.176281	−0.0557451
$11$	1.23406	0.372084	0.186042	−	0.982542i	$-0.440434\pi$
$11$	1.23406	0.372084	0.186042	+	0.982542i	$0.440434\pi$
$12$	0	0
$13$	−2.67752	−0.742611	−0.371306	−	0.928511i	$-0.621090\pi$
$13$	−2.67752	−0.742611	−0.371306	+	0.928511i	$0.621090\pi$
$14$	−1.63487	−0.436937
$15$	0	0
$16$	−4.89294	−1.22323
$17$	6.14025	1.48923	0.744615	−	0.667494i	$-0.232633\pi$
$17$	6.14025	1.48923	0.744615	+	0.667494i	$0.232633\pi$
$18$	0	0
$19$	1.57630	0.361628	0.180814	−	0.983517i	$-0.442127\pi$
$19$	1.57630	0.361628	0.180814	+	0.983517i	$0.442127\pi$
$20$	−0.0725448	−0.0162215
$21$	0	0
$22$	2.01753	0.430139
$23$	5.97787	1.24647	0.623236	−	0.782034i	$-0.285818\pi$
$23$	5.97787	1.24647	0.623236	+	0.782034i	$0.285818\pi$
$24$	0	0
$25$	−4.98837	−0.997675
$26$	−4.37740	−0.858478
$27$	0	0
$28$	−0.672794	−0.127146
$29$	−9.60657	−1.78389	−0.891947	−	0.452139i	$-0.850661\pi$
$29$	−9.60657	−1.78389	−0.891947	+	0.452139i	$0.850661\pi$
$30$	0	0
$31$	1.81504	0.325990	0.162995	−	0.986627i	$-0.447885\pi$
$31$	1.81504	0.325990	0.162995	+	0.986627i	$0.447885\pi$
$32$	−3.65969	−0.646949
$33$	0	0
$34$	10.0385	1.72159
$35$	0.107826	0.0182259
$36$	0	0
$37$	7.63656	1.25544	0.627721	−	0.778438i	$-0.283988\pi$
$37$	7.63656	1.25544	0.627721	+	0.778438i	$0.283988\pi$
$38$	2.57704	0.418051
$39$	0	0
$40$	0.233962	0.0369926
$41$	−4.73470	−0.739436	−0.369718	−	0.929144i	$-0.620546\pi$
$41$	−4.73470	−0.739436	−0.369718	+	0.929144i	$0.620546\pi$
$42$	0	0
$43$	8.65406	1.31973	0.659866	−	0.751384i	$-0.270613\pi$
$43$	8.65406	1.31973	0.659866	+	0.751384i	$0.270613\pi$
$44$	0.830270	0.125168
$45$	0	0
$46$	9.77303	1.44095
$47$	−11.0930	−1.61807	−0.809037	−	0.587758i	$-0.800011\pi$
$47$	−11.0930	−1.61807	−0.809037	+	0.587758i	$0.800011\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−8.15533	−1.15334
$51$	0	0
$52$	−1.80142	−0.249812
$53$	−13.7545	−1.88933	−0.944665	−	0.328038i	$-0.893613\pi$
$53$	−13.7545	−1.88933	−0.944665	+	0.328038i	$0.893613\pi$
$54$	0	0
$55$	−0.133064	−0.0179424
$56$	2.16981	0.289953
$57$	0	0
$58$	−15.7055	−2.06223
$59$	5.68940	0.740697	0.370348	−	0.928893i	$-0.379238\pi$
$59$	5.68940	0.740697	0.370348	+	0.928893i	$0.379238\pi$
$60$	0	0
$61$	−8.36686	−1.07127	−0.535634	−	0.844450i	$-0.679927\pi$
$61$	−8.36686	−1.07127	−0.535634	+	0.844450i	$0.679927\pi$
$62$	2.96735	0.376853
$63$	0	0
$64$	3.80276	0.475344
$65$	0.288707	0.0358097
$66$	0	0
$67$	−4.40536	−0.538200	−0.269100	−	0.963112i	$-0.586726\pi$
$67$	−4.40536	−0.538200	−0.269100	+	0.963112i	$0.586726\pi$
$68$	4.13113	0.500973
$69$	0	0
$70$	0.176281	0.0210697
$71$	−13.9036	−1.65006	−0.825030	−	0.565089i	$-0.808842\pi$
$71$	−13.9036	−1.65006	−0.825030	+	0.565089i	$0.808842\pi$
$72$	0	0
$73$	−8.13243	−0.951829	−0.475914	−	0.879492i	$-0.657883\pi$
$73$	−8.13243	−0.951829	−0.475914	+	0.879492i	$0.657883\pi$
$74$	12.4848	1.45133
$75$	0	0
$76$	1.06052	0.121651
$77$	−1.23406	−0.140634
$78$	0	0
$79$	10.6927	1.20302	0.601510	−	0.798865i	$-0.294566\pi$
$79$	10.6927	1.20302	0.601510	+	0.798865i	$0.294566\pi$
$80$	0.527586	0.0589859
$81$	0	0
$82$	−7.74061	−0.854807
$83$	10.3228	1.13307	0.566535	−	0.824038i	$-0.308284\pi$
$83$	10.3228	1.13307	0.566535	+	0.824038i	$0.308284\pi$
$84$	0	0
$85$	−0.662079	−0.0718126
$86$	14.1482	1.52564
$87$	0	0
$88$	−2.67768	−0.285441
$89$	15.7224	1.66658	0.833288	−	0.552839i	$-0.186456\pi$
$89$	15.7224	1.66658	0.833288	+	0.552839i	$0.186456\pi$
$90$	0	0
$91$	2.67752	0.280681
$92$	4.02188	0.419310
$93$	0	0
$94$	−18.1355	−1.87054
$95$	−0.169966	−0.0174382
$96$	0	0
$97$	−12.2148	−1.24022	−0.620111	−	0.784514i	$-0.712912\pi$
$97$	−12.2148	−1.24022	−0.620111	+	0.784514i	$0.712912\pi$
$98$	1.63487	0.165147
$99$	0	0
$100$	−3.35615	−0.335615
$101$	−4.73958	−0.471606	−0.235803	−	0.971801i	$-0.575772\pi$
$101$	−4.73958	−0.471606	−0.235803	+	0.971801i	$0.575772\pi$
$102$	0	0
$103$	4.33072	0.426718	0.213359	−	0.976974i	$-0.431560\pi$
$103$	4.33072	0.426718	0.213359	+	0.976974i	$0.431560\pi$
$104$	5.80971	0.569689
$105$	0	0
$106$	−22.4868	−2.18411
$107$	3.48445	0.336854	0.168427	−	0.985714i	$-0.446131\pi$
$107$	3.48445	0.336854	0.168427	+	0.985714i	$0.446131\pi$
$108$	0	0
$109$	−15.1221	−1.44843	−0.724217	−	0.689572i	$-0.757799\pi$
$109$	−15.1221	−1.44843	−0.724217	+	0.689572i	$0.757799\pi$
$110$	−0.217542	−0.0207419
$111$	0	0
$112$	4.89294	0.462339
$113$	−11.6269	−1.09377	−0.546885	−	0.837208i	$-0.684187\pi$
$113$	−11.6269	−1.09377	−0.546885	+	0.837208i	$0.684187\pi$
$114$	0	0
$115$	−0.644570	−0.0601065
$116$	−6.46324	−0.600097
$117$	0	0
$118$	9.30142	0.856265
$119$	−6.14025	−0.562876
$120$	0	0
$121$	−9.47709	−0.861554
$122$	−13.6787	−1.23841
$123$	0	0
$124$	1.22115	0.109662
$125$	1.07701	0.0963305
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	13.5364	1.19646
$129$	0	0
$130$	0.471998	0.0413969
$131$	−2.66536	−0.232873	−0.116437	−	0.993198i	$-0.537147\pi$
$131$	−2.66536	−0.232873	−0.116437	+	0.993198i	$0.537147\pi$
$132$	0	0
$133$	−1.57630	−0.136682
$134$	−7.20218	−0.622174
$135$	0	0
$136$	−13.3232	−1.14245
$137$	−18.7227	−1.59959	−0.799795	−	0.600274i	$-0.795058\pi$
$137$	−18.7227	−1.59959	−0.799795	+	0.600274i	$0.795058\pi$
$138$	0	0
$139$	−13.7227	−1.16395	−0.581973	−	0.813208i	$-0.697719\pi$
$139$	−13.7227	−1.16395	−0.581973	+	0.813208i	$0.697719\pi$
$140$	0.0725448	0.00613115
$141$	0	0
$142$	−22.7306	−1.90751
$143$	−3.30423	−0.276314
$144$	0	0
$145$	1.03584	0.0860217
$146$	−13.2954	−1.10034
$147$	0	0
$148$	5.13784	0.422327
$149$	0.137822	0.0112908	0.00564541	−	0.999984i	$-0.498203\pi$
$149$	0.137822	0.0112908	0.00564541	+	0.999984i	$0.498203\pi$
$150$	0	0
$151$	−5.95504	−0.484614	−0.242307	−	0.970200i	$-0.577904\pi$
$151$	−5.95504	−0.484614	−0.242307	+	0.970200i	$0.577904\pi$
$152$	−3.42026	−0.277420
$153$	0	0
$154$	−2.01753	−0.162577
$155$	−0.195708	−0.0157197
$156$	0	0
$157$	−11.1340	−0.888588	−0.444294	−	0.895881i	$-0.646545\pi$
$157$	−11.1340	−0.888588	−0.444294	+	0.895881i	$0.646545\pi$
$158$	17.4811	1.39072
$159$	0	0
$160$	0.394611	0.0311967
$161$	−5.97787	−0.471122
$162$	0	0
$163$	11.5262	0.902799	0.451400	−	0.892322i	$-0.350925\pi$
$163$	11.5262	0.902799	0.451400	+	0.892322i	$0.350925\pi$
$164$	−3.18548	−0.248744
$165$	0	0
$166$	16.8763	1.30986
$167$	−19.8842	−1.53869	−0.769344	−	0.638835i	$-0.779417\pi$
$167$	−19.8842	−1.53869	−0.769344	+	0.638835i	$0.779417\pi$
$168$	0	0
$169$	−5.83087	−0.448529
$170$	−1.08241	−0.0830173
$171$	0	0
$172$	5.82240	0.443954
$173$	−22.1209	−1.68182	−0.840909	−	0.541177i	$-0.817979\pi$
$173$	−22.1209	−1.68182	−0.840909	+	0.541177i	$0.817979\pi$
$174$	0	0
$175$	4.98837	0.377086
$176$	−6.03819	−0.455146
$177$	0	0
$178$	25.7041	1.92661
$179$	8.04366	0.601211	0.300606	−	0.953749i	$-0.402811\pi$
$179$	8.04366	0.601211	0.300606	+	0.953749i	$0.402811\pi$
$180$	0	0
$181$	8.83086	0.656392	0.328196	−	0.944610i	$-0.393559\pi$
$181$	8.83086	0.656392	0.328196	+	0.944610i	$0.393559\pi$
$182$	4.37740	0.324474
$183$	0	0
$184$	−12.9708	−0.956221
$185$	−0.823421	−0.0605391
$186$	0	0
$187$	7.57745	0.554118
$188$	−7.46328	−0.544315
$189$	0	0
$190$	−0.277872	−0.0201590
$191$	11.0056	0.796337	0.398169	−	0.917312i	$-0.369646\pi$
$191$	11.0056	0.796337	0.398169	+	0.917312i	$0.369646\pi$
$192$	0	0
$193$	6.67906	0.480769	0.240385	−	0.970678i	$-0.422726\pi$
$193$	6.67906	0.480769	0.240385	+	0.970678i	$0.422726\pi$
$194$	−19.9695	−1.43373
$195$	0	0
$196$	0.672794	0.0480567
$197$	14.2032	1.01194	0.505970	−	0.862551i	$-0.331135\pi$
$197$	14.2032	1.01194	0.505970	+	0.862551i	$0.331135\pi$
$198$	0	0
$199$	−8.78561	−0.622795	−0.311398	−	0.950280i	$-0.600797\pi$
$199$	−8.78561	−0.622795	−0.311398	+	0.950280i	$0.600797\pi$
$200$	10.8238	0.765359
$201$	0	0
$202$	−7.74859	−0.545189
$203$	9.60657	0.674249
$204$	0	0
$205$	0.510524	0.0356565
$206$	7.08015	0.493298
$207$	0	0
$208$	13.1009	0.908387
$209$	1.94525	0.134556
$210$	0	0
$211$	−17.0024	−1.17049	−0.585247	−	0.810855i	$-0.699002\pi$
$211$	−17.0024	−1.17049	−0.585247	+	0.810855i	$0.699002\pi$
$212$	−9.25397	−0.635565
$213$	0	0
$214$	5.69662	0.389413
$215$	−0.933133	−0.0636392
$216$	0	0
$217$	−1.81504	−0.123213
$218$	−24.7226	−1.67443
$219$	0	0
$220$	−0.0895248	−0.00603576
$221$	−16.4407	−1.10592
$222$	0	0
$223$	−1.34976	−0.0903867	−0.0451934	−	0.998978i	$-0.514390\pi$
$223$	−1.34976	−0.0903867	−0.0451934	+	0.998978i	$0.514390\pi$
$224$	3.65969	0.244524
$225$	0	0
$226$	−19.0085	−1.26443
$227$	−12.2240	−0.811338	−0.405669	−	0.914020i	$-0.632961\pi$
$227$	−12.2240	−0.811338	−0.405669	+	0.914020i	$0.632961\pi$
$228$	0	0
$229$	−6.61263	−0.436975	−0.218488	−	0.975840i	$-0.570112\pi$
$229$	−6.61263	−0.436975	−0.218488	+	0.975840i	$0.570112\pi$
$230$	−1.05379	−0.0694847
$231$	0	0
$232$	20.8444	1.36850
$233$	−7.28567	−0.477300	−0.238650	−	0.971106i	$-0.576705\pi$
$233$	−7.28567	−0.477300	−0.238650	+	0.971106i	$0.576705\pi$
$234$	0	0
$235$	1.19611	0.0780256
$236$	3.82780	0.249168
$237$	0	0
$238$	−10.0385	−0.650699
$239$	−10.5610	−0.683135	−0.341567	−	0.939857i	$-0.610958\pi$
$239$	−10.5610	−0.683135	−0.341567	+	0.939857i	$0.610958\pi$
$240$	0	0
$241$	21.1530	1.36258	0.681292	−	0.732012i	$-0.261418\pi$
$241$	21.1530	1.36258	0.681292	+	0.732012i	$0.261418\pi$
$242$	−15.4938	−0.995979
$243$	0	0
$244$	−5.62918	−0.360371
$245$	−0.107826	−0.00688876
$246$	0	0
$247$	−4.22057	−0.268549
$248$	−3.93828	−0.250081
$249$	0	0
$250$	1.76077	0.111361
$251$	4.58972	0.289700	0.144850	−	0.989454i	$-0.453730\pi$
$251$	4.58972	0.289700	0.144850	+	0.989454i	$0.453730\pi$
$252$	0	0
$253$	7.37707	0.463792
$254$	−1.63487	−0.102581
$255$	0	0
$256$	14.5247	0.907794
$257$	22.0168	1.37337	0.686687	−	0.726954i	$-0.259064\pi$
$257$	22.0168	1.37337	0.686687	+	0.726954i	$0.259064\pi$
$258$	0	0
$259$	−7.63656	−0.474513
$260$	0.194240	0.0120463
$261$	0	0
$262$	−4.35750	−0.269208
$263$	−1.29988	−0.0801540	−0.0400770	−	0.999197i	$-0.512760\pi$
$263$	−1.29988	−0.0801540	−0.0400770	+	0.999197i	$0.512760\pi$
$264$	0	0
$265$	1.48310	0.0911059
$266$	−2.57704	−0.158008
$267$	0	0
$268$	−2.96390	−0.181049
$269$	5.22689	0.318689	0.159345	−	0.987223i	$-0.449062\pi$
$269$	5.22689	0.318689	0.159345	+	0.987223i	$0.449062\pi$
$270$	0	0
$271$	−30.8828	−1.87600	−0.937999	−	0.346637i	$-0.887324\pi$
$271$	−30.8828	−1.87600	−0.937999	+	0.346637i	$0.887324\pi$
$272$	−30.0439	−1.82168
$273$	0	0
$274$	−30.6092	−1.84917
$275$	−6.15597	−0.371219
$276$	0	0
$277$	1.95927	0.117721	0.0588606	−	0.998266i	$-0.481253\pi$
$277$	1.95927	0.117721	0.0588606	+	0.998266i	$0.481253\pi$
$278$	−22.4349	−1.34555
$279$	0	0
$280$	−0.233962	−0.0139819
$281$	24.0387	1.43403	0.717015	−	0.697058i	$-0.245508\pi$
$281$	24.0387	1.43403	0.717015	+	0.697058i	$0.245508\pi$
$282$	0	0
$283$	−19.8765	−1.18154	−0.590768	−	0.806841i	$-0.701175\pi$
$283$	−19.8765	−1.18154	−0.590768	+	0.806841i	$0.701175\pi$
$284$	−9.35430	−0.555075
$285$	0	0
$286$	−5.40198	−0.319426
$287$	4.73470	0.279480
$288$	0	0
$289$	20.7027	1.21781
$290$	1.69346	0.0994434
$291$	0	0
$292$	−5.47145	−0.320192
$293$	−12.0901	−0.706313	−0.353156	−	0.935564i	$-0.614892\pi$
$293$	−12.0901	−0.706313	−0.353156	+	0.935564i	$0.614892\pi$
$294$	0	0
$295$	−0.613466	−0.0357174
$296$	−16.5699	−0.963103
$297$	0	0
$298$	0.225321	0.0130525
$299$	−16.0059	−0.925644
$300$	0	0
$301$	−8.65406	−0.498812
$302$	−9.73571	−0.560227
$303$	0	0
$304$	−7.71273	−0.442355
$305$	0.902166	0.0516579
$306$	0	0
$307$	−27.6351	−1.57722	−0.788608	−	0.614896i	$-0.789198\pi$
$307$	−27.6351	−1.57722	−0.788608	+	0.614896i	$0.789198\pi$
$308$	−0.830270	−0.0473090
$309$	0	0
$310$	−0.319957	−0.0181724
$311$	23.9836	1.35999	0.679994	−	0.733218i	$-0.261983\pi$
$311$	23.9836	1.35999	0.679994	+	0.733218i	$0.261983\pi$
$312$	0	0
$313$	23.6714	1.33799	0.668993	−	0.743269i	$-0.266726\pi$
$313$	23.6714	1.33799	0.668993	+	0.743269i	$0.266726\pi$
$314$	−18.2026	−1.02723
$315$	0	0
$316$	7.19397	0.404693
$317$	−21.3847	−1.20108	−0.600542	−	0.799593i	$-0.705049\pi$
$317$	−21.3847	−1.20108	−0.600542	+	0.799593i	$0.705049\pi$
$318$	0	0
$319$	−11.8551	−0.663758
$320$	−0.410036	−0.0229217
$321$	0	0
$322$	−9.77303	−0.544630
$323$	9.67887	0.538547
$324$	0	0
$325$	13.3565	0.740884
$326$	18.8438	1.04366
$327$	0	0
$328$	10.2734	0.567252
$329$	11.0930	0.611574
$330$	0	0
$331$	2.40761	0.132334	0.0661670	−	0.997809i	$-0.478923\pi$
$331$	2.40761	0.132334	0.0661670	+	0.997809i	$0.478923\pi$
$332$	6.94509	0.381161
$333$	0	0
$334$	−32.5081	−1.77876
$335$	0.475013	0.0259527
$336$	0	0
$337$	−27.4338	−1.49442	−0.747208	−	0.664590i	$-0.768606\pi$
$337$	−27.4338	−1.49442	−0.747208	+	0.664590i	$0.768606\pi$
$338$	−9.53271	−0.518511
$339$	0	0
$340$	−0.445443	−0.0241576
$341$	2.23987	0.121296
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−18.7776	−1.01242
$345$	0	0
$346$	−36.1647	−1.94423
$347$	24.7643	1.32942	0.664709	−	0.747103i	$-0.268556\pi$
$347$	24.7643	1.32942	0.664709	+	0.747103i	$0.268556\pi$
$348$	0	0
$349$	−33.4203	−1.78895	−0.894473	−	0.447123i	$-0.852449\pi$
$349$	−33.4203	−1.78895	−0.894473	+	0.447123i	$0.852449\pi$
$350$	8.15533	0.435921
$351$	0	0
$352$	−4.51629	−0.240719
$353$	9.29165	0.494545	0.247272	−	0.968946i	$-0.420466\pi$
$353$	9.29165	0.494545	0.247272	+	0.968946i	$0.420466\pi$
$354$	0	0
$355$	1.49918	0.0795680
$356$	10.5780	0.560631
$357$	0	0
$358$	13.1503	0.695016
$359$	2.66632	0.140723	0.0703614	−	0.997522i	$-0.477585\pi$
$359$	2.66632	0.140723	0.0703614	+	0.997522i	$0.477585\pi$
$360$	0	0
$361$	−16.5153	−0.869225
$362$	14.4373	0.758807
$363$	0	0
$364$	1.80142	0.0944202
$365$	0.876888	0.0458984
$366$	0	0
$367$	5.53961	0.289165	0.144583	−	0.989493i	$-0.453816\pi$
$367$	5.53961	0.289165	0.144583	+	0.989493i	$0.453816\pi$
$368$	−29.2493	−1.52473
$369$	0	0
$370$	−1.34618	−0.0699848
$371$	13.7545	0.714099
$372$	0	0
$373$	7.13583	0.369479	0.184740	−	0.982788i	$-0.440856\pi$
$373$	7.13583	0.369479	0.184740	+	0.982788i	$0.440856\pi$
$374$	12.3881	0.640576
$375$	0	0
$376$	24.0696	1.24129
$377$	25.7218	1.32474
$378$	0	0
$379$	22.1188	1.13617	0.568083	−	0.822971i	$-0.307685\pi$
$379$	22.1188	1.13617	0.568083	+	0.822971i	$0.307685\pi$
$380$	−0.114352	−0.00586615
$381$	0	0
$382$	17.9927	0.920587
$383$	5.74924	0.293773	0.146886	−	0.989153i	$-0.453075\pi$
$383$	5.74924	0.293773	0.146886	+	0.989153i	$0.453075\pi$
$384$	0	0
$385$	0.133064	0.00678158
$386$	10.9194	0.555782
$387$	0	0
$388$	−8.21803	−0.417207
$389$	−18.0335	−0.914335	−0.457167	−	0.889381i	$-0.651136\pi$
$389$	−18.0335	−0.914335	−0.457167	+	0.889381i	$0.651136\pi$
$390$	0	0
$391$	36.7056	1.85628
$392$	−2.16981	−0.109592
$393$	0	0
$394$	23.2204	1.16983
$395$	−1.15295	−0.0580112
$396$	0	0
$397$	22.5113	1.12981	0.564906	−	0.825156i	$-0.308913\pi$
$397$	22.5113	1.12981	0.564906	+	0.825156i	$0.308913\pi$
$398$	−14.3633	−0.719968
$399$	0	0
$400$	24.4078	1.22039
$401$	33.0927	1.65257	0.826284	−	0.563253i	$-0.190450\pi$
$401$	33.0927	1.65257	0.826284	+	0.563253i	$0.190450\pi$
$402$	0	0
$403$	−4.85980	−0.242084
$404$	−3.18876	−0.158647
$405$	0	0
$406$	15.7055	0.779449
$407$	9.42399	0.467130
$408$	0	0
$409$	−10.6802	−0.528103	−0.264051	−	0.964509i	$-0.585059\pi$
$409$	−10.6802	−0.528103	−0.264051	+	0.964509i	$0.585059\pi$
$410$	0.834640	0.0412199
$411$	0	0
$412$	2.91368	0.143547
$413$	−5.68940	−0.279957
$414$	0	0
$415$	−1.11306	−0.0546381
$416$	9.79891	0.480431
$417$	0	0
$418$	3.18023	0.155550
$419$	−8.09996	−0.395709	−0.197855	−	0.980231i	$-0.563397\pi$
$419$	−8.09996	−0.395709	−0.197855	+	0.980231i	$0.563397\pi$
$420$	0	0
$421$	22.9457	1.11830	0.559152	−	0.829065i	$-0.311127\pi$
$421$	22.9457	1.11830	0.559152	+	0.829065i	$0.311127\pi$
$422$	−27.7967	−1.35312
$423$	0	0
$424$	29.8447	1.44938
$425$	−30.6299	−1.48577
$426$	0	0
$427$	8.36686	0.404901
$428$	2.34432	0.113317
$429$	0	0
$430$	−1.52555	−0.0735686
$431$	18.5620	0.894099	0.447049	−	0.894509i	$-0.352475\pi$
$431$	18.5620	0.894099	0.447049	+	0.894509i	$0.352475\pi$
$432$	0	0
$433$	−9.97229	−0.479238	−0.239619	−	0.970867i	$-0.577023\pi$
$433$	−9.97229	−0.479238	−0.239619	+	0.970867i	$0.577023\pi$
$434$	−2.96735	−0.142437
$435$	0	0
$436$	−10.1741	−0.487249
$437$	9.42290	0.450759
$438$	0	0
$439$	−13.8980	−0.663316	−0.331658	−	0.943400i	$-0.607608\pi$
$439$	−13.8980	−0.663316	−0.331658	+	0.943400i	$0.607608\pi$
$440$	0.288723	0.0137644
$441$	0	0
$442$	−26.8783	−1.27847
$443$	26.4427	1.25633	0.628165	−	0.778080i	$-0.283806\pi$
$443$	26.4427	1.25633	0.628165	+	0.778080i	$0.283806\pi$
$444$	0	0
$445$	−1.69529	−0.0803644
$446$	−2.20668	−0.104489
$447$	0	0
$448$	−3.80276	−0.179663
$449$	12.1557	0.573664	0.286832	−	0.957981i	$-0.407398\pi$
$449$	12.1557	0.573664	0.286832	+	0.957981i	$0.407398\pi$
$450$	0	0
$451$	−5.84291	−0.275132
$452$	−7.82254	−0.367941
$453$	0	0
$454$	−19.9847	−0.937928
$455$	−0.288707	−0.0135348
$456$	0	0
$457$	−13.8183	−0.646395	−0.323197	−	0.946332i	$-0.604758\pi$
$457$	−13.8183	−0.646395	−0.323197	+	0.946332i	$0.604758\pi$
$458$	−10.8108	−0.505155
$459$	0	0
$460$	−0.433663	−0.0202197
$461$	−11.8257	−0.550779	−0.275389	−	0.961333i	$-0.588807\pi$
$461$	−11.8257	−0.550779	−0.275389	+	0.961333i	$0.588807\pi$
$462$	0	0
$463$	23.5068	1.09245	0.546227	−	0.837637i	$-0.316064\pi$
$463$	23.5068	1.09245	0.546227	+	0.837637i	$0.316064\pi$
$464$	47.0043	2.18212
$465$	0	0
$466$	−11.9111	−0.551772
$467$	31.0128	1.43510	0.717551	−	0.696506i	$-0.245263\pi$
$467$	31.0128	1.43510	0.717551	+	0.696506i	$0.245263\pi$
$468$	0	0
$469$	4.40536	0.203421
$470$	1.95548	0.0901997
$471$	0	0
$472$	−12.3449	−0.568220
$473$	10.6797	0.491051
$474$	0	0
$475$	−7.86316	−0.360787
$476$	−4.13113	−0.189350
$477$	0	0
$478$	−17.2659	−0.789722
$479$	−21.5723	−0.985664	−0.492832	−	0.870125i	$-0.664038\pi$
$479$	−21.5723	−0.985664	−0.492832	+	0.870125i	$0.664038\pi$
$480$	0	0
$481$	−20.4471	−0.932306
$482$	34.5824	1.57518
$483$	0	0
$484$	−6.37613	−0.289824
$485$	1.31707	0.0598051
$486$	0	0
$487$	−16.3925	−0.742813	−0.371407	−	0.928470i	$-0.621124\pi$
$487$	−16.3925	−0.742813	−0.371407	+	0.928470i	$0.621124\pi$
$488$	18.1545	0.821815
$489$	0	0
$490$	−0.176281	−0.00796359
$491$	5.03723	0.227327	0.113663	−	0.993519i	$-0.463741\pi$
$491$	5.03723	0.227327	0.113663	+	0.993519i	$0.463741\pi$
$492$	0	0
$493$	−58.9867	−2.65663
$494$	−6.90008	−0.310449
$495$	0	0
$496$	−8.88086	−0.398762
$497$	13.9036	0.623664
$498$	0	0
$499$	10.7980	0.483387	0.241693	−	0.970353i	$-0.422297\pi$
$499$	10.7980	0.483387	0.241693	+	0.970353i	$0.422297\pi$
$500$	0.724605	0.0324053
$501$	0	0
$502$	7.50359	0.334901
$503$	−7.50427	−0.334599	−0.167299	−	0.985906i	$-0.553505\pi$
$503$	−7.50427	−0.334599	−0.167299	+	0.985906i	$0.553505\pi$
$504$	0	0
$505$	0.511050	0.0227414
$506$	12.0605	0.536156
$507$	0	0
$508$	−0.672794	−0.0298504
$509$	23.3270	1.03395	0.516976	−	0.856000i	$-0.327058\pi$
$509$	23.3270	1.03395	0.516976	+	0.856000i	$0.327058\pi$
$510$	0	0
$511$	8.13243	0.359757
$512$	−3.32680	−0.147025
$513$	0	0
$514$	35.9946	1.58766
$515$	−0.466964	−0.0205769
$516$	0	0
$517$	−13.6894	−0.602059
$518$	−12.4848	−0.548549
$519$	0	0
$520$	−0.626438	−0.0274711
$521$	25.1871	1.10347	0.551734	−	0.834020i	$-0.313966\pi$
$521$	25.1871	1.10347	0.551734	+	0.834020i	$0.313966\pi$
$522$	0	0
$523$	−5.75769	−0.251766	−0.125883	−	0.992045i	$-0.540176\pi$
$523$	−5.75769	−0.251766	−0.125883	+	0.992045i	$0.540176\pi$
$524$	−1.79324	−0.0783379
$525$	0	0
$526$	−2.12513	−0.0926601
$527$	11.1448	0.485474
$528$	0	0
$529$	12.7349	0.553692
$530$	2.42467	0.105321
$531$	0	0
$532$	−1.06052	−0.0459796
$533$	12.6773	0.549113
$534$	0	0
$535$	−0.375715	−0.0162436
$536$	9.55877	0.412876
$537$	0	0
$538$	8.54528	0.368413
$539$	1.23406	0.0531548
$540$	0	0
$541$	0.385713	0.0165831	0.00829154	−	0.999966i	$-0.497361\pi$
$541$	0.385713	0.0165831	0.00829154	+	0.999966i	$0.497361\pi$
$542$	−50.4894	−2.16870
$543$	0	0
$544$	−22.4714	−0.963455
$545$	1.63056	0.0698454
$546$	0	0
$547$	−18.7454	−0.801495	−0.400747	−	0.916189i	$-0.631249\pi$
$547$	−18.7454	−0.801495	−0.400747	+	0.916189i	$0.631249\pi$
$548$	−12.5965	−0.538097
$549$	0	0
$550$	−10.0642	−0.429139
$551$	−15.1428	−0.645106
$552$	0	0
$553$	−10.6927	−0.454699
$554$	3.20315	0.136089
$555$	0	0
$556$	−9.23258	−0.391548
$557$	10.1358	0.429467	0.214734	−	0.976673i	$-0.431112\pi$
$557$	10.1358	0.429467	0.214734	+	0.976673i	$0.431112\pi$
$558$	0	0
$559$	−23.1714	−0.980047
$560$	−0.527586	−0.0222946
$561$	0	0
$562$	39.3002	1.65778
$563$	−0.675398	−0.0284646	−0.0142323	−	0.999899i	$-0.504530\pi$
$563$	−0.675398	−0.0284646	−0.0142323	+	0.999899i	$0.504530\pi$
$564$	0	0
$565$	1.25369	0.0527430
$566$	−32.4955	−1.36589
$567$	0	0
$568$	30.1682	1.26583
$569$	−16.3639	−0.686009	−0.343004	−	0.939334i	$-0.611445\pi$
$569$	−16.3639	−0.686009	−0.343004	+	0.939334i	$0.611445\pi$
$570$	0	0
$571$	−7.26607	−0.304075	−0.152038	−	0.988375i	$-0.548584\pi$
$571$	−7.26607	−0.304075	−0.152038	+	0.988375i	$0.548584\pi$
$572$	−2.22307	−0.0929511
$573$	0	0
$574$	7.74061	0.323087
$575$	−29.8198	−1.24357
$576$	0	0
$577$	−25.9010	−1.07827	−0.539137	−	0.842218i	$-0.681250\pi$
$577$	−25.9010	−1.07827	−0.539137	+	0.842218i	$0.681250\pi$
$578$	33.8462	1.40782
$579$	0	0
$580$	0.696906	0.0289375
$581$	−10.3228	−0.428260
$582$	0	0
$583$	−16.9739	−0.702989
$584$	17.6458	0.730188
$585$	0	0
$586$	−19.7658	−0.816516
$587$	9.99126	0.412383	0.206192	−	0.978512i	$-0.433893\pi$
$587$	9.99126	0.412383	0.206192	+	0.978512i	$0.433893\pi$
$588$	0	0
$589$	2.86104	0.117887
$590$	−1.00294	−0.0412902
$591$	0	0
$592$	−37.3652	−1.53570
$593$	−32.8401	−1.34858	−0.674291	−	0.738466i	$-0.735551\pi$
$593$	−32.8401	−1.34858	−0.674291	+	0.738466i	$0.735551\pi$
$594$	0	0
$595$	0.662079	0.0271426
$596$	0.0927258	0.00379820
$597$	0	0
$598$	−26.1675	−1.07007
$599$	33.2490	1.35852	0.679258	−	0.733900i	$-0.262302\pi$
$599$	33.2490	1.35852	0.679258	+	0.733900i	$0.262302\pi$
$600$	0	0
$601$	9.22830	0.376430	0.188215	−	0.982128i	$-0.439730\pi$
$601$	9.22830	0.376430	0.188215	+	0.982128i	$0.439730\pi$
$602$	−14.1482	−0.576639
$603$	0	0
$604$	−4.00652	−0.163023
$605$	1.02188	0.0415452
$606$	0	0
$607$	43.9163	1.78251	0.891253	−	0.453506i	$-0.149827\pi$
$607$	43.9163	1.78251	0.891253	+	0.453506i	$0.149827\pi$
$608$	−5.76877	−0.233954
$609$	0	0
$610$	1.47492	0.0597179
$611$	29.7016	1.20160
$612$	0	0
$613$	41.6037	1.68036	0.840179	−	0.542309i	$-0.182450\pi$
$613$	41.6037	1.68036	0.840179	+	0.542309i	$0.182450\pi$
$614$	−45.1797	−1.82330
$615$	0	0
$616$	2.67768	0.107887
$617$	−17.5822	−0.707833	−0.353916	−	0.935277i	$-0.615150\pi$
$617$	−17.5822	−0.707833	−0.353916	+	0.935277i	$0.615150\pi$
$618$	0	0
$619$	−19.9761	−0.802909	−0.401455	−	0.915879i	$-0.631495\pi$
$619$	−19.9761	−0.802909	−0.401455	+	0.915879i	$0.631495\pi$
$620$	−0.131672	−0.00528806
$621$	0	0
$622$	39.2101	1.57218
$623$	−15.7224	−0.629906
$624$	0	0
$625$	24.8257	0.993030
$626$	38.6996	1.54675
$627$	0	0
$628$	−7.49088	−0.298919
$629$	46.8904	1.86964
$630$	0	0
$631$	23.2542	0.925733	0.462867	−	0.886428i	$-0.346821\pi$
$631$	23.2542	0.925733	0.462867	+	0.886428i	$0.346821\pi$
$632$	−23.2010	−0.922888
$633$	0	0
$634$	−34.9612	−1.38849
$635$	0.107826	0.00427895
$636$	0	0
$637$	−2.67752	−0.106087
$638$	−19.3815	−0.767322
$639$	0	0
$640$	−1.45958	−0.0576948
$641$	−9.68321	−0.382464	−0.191232	−	0.981545i	$-0.561248\pi$
$641$	−9.68321	−0.382464	−0.191232	+	0.981545i	$0.561248\pi$
$642$	0	0
$643$	−24.4462	−0.964063	−0.482032	−	0.876154i	$-0.660101\pi$
$643$	−24.4462	−0.964063	−0.482032	+	0.876154i	$0.660101\pi$
$644$	−4.02188	−0.158484
$645$	0	0
$646$	15.8237	0.622574
$647$	−48.3444	−1.90062	−0.950308	−	0.311312i	$-0.899231\pi$
$647$	−48.3444	−1.90062	−0.950308	+	0.311312i	$0.899231\pi$
$648$	0	0
$649$	7.02107	0.275601
$650$	21.8361	0.856482
$651$	0	0
$652$	7.75474	0.303699
$653$	2.27517	0.0890342	0.0445171	−	0.999009i	$-0.485825\pi$
$653$	2.27517	0.0890342	0.0445171	+	0.999009i	$0.485825\pi$
$654$	0	0
$655$	0.287395	0.0112294
$656$	23.1666	0.904503
$657$	0	0
$658$	18.1355	0.706996
$659$	21.4031	0.833745	0.416873	−	0.908965i	$-0.363126\pi$
$659$	21.4031	0.833745	0.416873	+	0.908965i	$0.363126\pi$
$660$	0	0
$661$	−17.6134	−0.685083	−0.342541	−	0.939503i	$-0.611288\pi$
$661$	−17.6134	−0.685083	−0.342541	+	0.939503i	$0.611288\pi$
$662$	3.93612	0.152982
$663$	0	0
$664$	−22.3984	−0.869225
$665$	0.169966	0.00659100
$666$	0	0
$667$	−57.4268	−2.22357
$668$	−13.3780	−0.517610
$669$	0	0
$670$	0.776583	0.0300020
$671$	−10.3252	−0.398601
$672$	0	0
$673$	18.3102	0.705805	0.352903	−	0.935660i	$-0.385195\pi$
$673$	18.3102	0.705805	0.352903	+	0.935660i	$0.385195\pi$
$674$	−44.8507	−1.72758
$675$	0	0
$676$	−3.92298	−0.150884
$677$	−27.3081	−1.04954	−0.524769	−	0.851245i	$-0.675848\pi$
$677$	−27.3081	−1.04954	−0.524769	+	0.851245i	$0.675848\pi$
$678$	0	0
$679$	12.2148	0.468760
$680$	1.43658	0.0550905
$681$	0	0
$682$	3.66189	0.140221
$683$	−14.9870	−0.573461	−0.286731	−	0.958011i	$-0.592568\pi$
$683$	−14.9870	−0.573461	−0.286731	+	0.958011i	$0.592568\pi$
$684$	0	0
$685$	2.01880	0.0771343
$686$	−1.63487	−0.0624196
$687$	0	0
$688$	−42.3438	−1.61434
$689$	36.8281	1.40304
$690$	0	0
$691$	−42.1601	−1.60384	−0.801922	−	0.597428i	$-0.796189\pi$
$691$	−42.1601	−1.60384	−0.801922	+	0.597428i	$0.796189\pi$
$692$	−14.8828	−0.565759
$693$	0	0
$694$	40.4864	1.53684
$695$	1.47967	0.0561270
$696$	0	0
$697$	−29.0722	−1.10119
$698$	−54.6377	−2.06807
$699$	0	0
$700$	3.35615	0.126851
$701$	32.7665	1.23757	0.618786	−	0.785559i	$-0.287625\pi$
$701$	32.7665	1.23757	0.618786	+	0.785559i	$0.287625\pi$
$702$	0	0
$703$	12.0375	0.454003
$704$	4.69284	0.176868
$705$	0	0
$706$	15.1906	0.571707
$707$	4.73958	0.178250
$708$	0	0
$709$	25.0222	0.939728	0.469864	−	0.882739i	$-0.344303\pi$
$709$	25.0222	0.939728	0.469864	+	0.882739i	$0.344303\pi$
$710$	2.45096	0.0919827
$711$	0	0
$712$	−34.1147	−1.27850
$713$	10.8501	0.406338
$714$	0	0
$715$	0.356282	0.0133242
$716$	5.41173	0.202246
$717$	0	0
$718$	4.35908	0.162679
$719$	−41.1989	−1.53646	−0.768230	−	0.640174i	$-0.778862\pi$
$719$	−41.1989	−1.53646	−0.768230	+	0.640174i	$0.778862\pi$
$720$	0	0
$721$	−4.33072	−0.161284
$722$	−27.0003	−1.00485
$723$	0	0
$724$	5.94135	0.220809
$725$	47.9211	1.77975
$726$	0	0
$727$	−2.12027	−0.0786366	−0.0393183	−	0.999227i	$-0.512519\pi$
$727$	−2.12027	−0.0786366	−0.0393183	+	0.999227i	$0.512519\pi$
$728$	−5.80971	−0.215322
$729$	0	0
$730$	1.43360	0.0530598
$731$	53.1381	1.96538
$732$	0	0
$733$	−26.4888	−0.978385	−0.489192	−	0.872176i	$-0.662708\pi$
$733$	−26.4888	−0.978385	−0.489192	+	0.872176i	$0.662708\pi$
$734$	9.05653	0.334282
$735$	0	0
$736$	−21.8772	−0.806403
$737$	−5.43649	−0.200256
$738$	0	0
$739$	36.6556	1.34840	0.674199	−	0.738549i	$-0.264489\pi$
$739$	36.6556	1.34840	0.674199	+	0.738549i	$0.264489\pi$
$740$	−0.553993	−0.0203652
$741$	0	0
$742$	22.4868	0.825518
$743$	−16.4701	−0.604231	−0.302115	−	0.953271i	$-0.597693\pi$
$743$	−16.4701	−0.604231	−0.302115	+	0.953271i	$0.597693\pi$
$744$	0	0
$745$	−0.0148608	−0.000544458 0
$746$	11.6661	0.427128
$747$	0	0
$748$	5.09807	0.186404
$749$	−3.48445	−0.127319
$750$	0	0
$751$	15.7106	0.573288	0.286644	−	0.958037i	$-0.407460\pi$
$751$	15.7106	0.573288	0.286644	+	0.958037i	$0.407460\pi$
$752$	54.2771	1.97928
$753$	0	0
$754$	42.0518	1.53143
$755$	0.642109	0.0233687
$756$	0	0
$757$	−14.4985	−0.526957	−0.263478	−	0.964665i	$-0.584870\pi$
$757$	−14.4985	−0.526957	−0.263478	+	0.964665i	$0.584870\pi$
$758$	36.1613	1.31344
$759$	0	0
$760$	0.368794	0.0133775
$761$	28.2418	1.02376	0.511882	−	0.859056i	$-0.328949\pi$
$761$	28.2418	1.02376	0.511882	+	0.859056i	$0.328949\pi$
$762$	0	0
$763$	15.1221	0.547457
$764$	7.40451	0.267886
$765$	0	0
$766$	9.39926	0.339609
$767$	−15.2335	−0.550050
$768$	0	0
$769$	14.1456	0.510103	0.255051	−	0.966928i	$-0.417908\pi$
$769$	14.1456	0.510103	0.255051	+	0.966928i	$0.417908\pi$
$770$	0.217542	0.00783968
$771$	0	0
$772$	4.49364	0.161729
$773$	13.2490	0.476534	0.238267	−	0.971200i	$-0.423421\pi$
$773$	13.2490	0.476534	0.238267	+	0.971200i	$0.423421\pi$
$774$	0	0
$775$	−9.05408	−0.325232
$776$	26.5037	0.951427
$777$	0	0
$778$	−29.4824	−1.05700
$779$	−7.46330	−0.267400
$780$	0	0
$781$	−17.1580	−0.613961
$782$	60.0089	2.14591
$783$	0	0
$784$	−4.89294	−0.174748
$785$	1.20053	0.0428489
$786$	0	0
$787$	51.2817	1.82800	0.913998	−	0.405718i	$-0.132979\pi$
$787$	51.2817	1.82800	0.913998	+	0.405718i	$0.132979\pi$
$788$	9.55586	0.340414
$789$	0	0
$790$	−1.88492	−0.0670625
$791$	11.6269	0.413406
$792$	0	0
$793$	22.4025	0.795535
$794$	36.8031	1.30609
$795$	0	0
$796$	−5.91091	−0.209507
$797$	49.0876	1.73877	0.869385	−	0.494135i	$-0.164515\pi$
$797$	49.0876	1.73877	0.869385	+	0.494135i	$0.164515\pi$
$798$	0	0
$799$	−68.1135	−2.40968
$800$	18.2559	0.645444
$801$	0	0
$802$	54.1021	1.91041
$803$	−10.0359	−0.354160
$804$	0	0
$805$	0.644570	0.0227181
$806$	−7.94514	−0.279856
$807$	0	0
$808$	10.2840	0.361789
$809$	8.25607	0.290268	0.145134	−	0.989412i	$-0.453639\pi$
$809$	8.25607	0.290268	0.145134	+	0.989412i	$0.453639\pi$
$810$	0	0
$811$	10.1645	0.356923	0.178461	−	0.983947i	$-0.442888\pi$
$811$	10.1645	0.356923	0.178461	+	0.983947i	$0.442888\pi$
$812$	6.46324	0.226815
$813$	0	0
$814$	15.4070	0.540015
$815$	−1.24282	−0.0435342
$816$	0	0
$817$	13.6414	0.477251
$818$	−17.4608	−0.610501
$819$	0	0
$820$	0.343478	0.0119948
$821$	11.6755	0.407478	0.203739	−	0.979025i	$-0.434691\pi$
$821$	11.6755	0.407478	0.203739	+	0.979025i	$0.434691\pi$
$822$	0	0
$823$	−46.4146	−1.61791	−0.808956	−	0.587870i	$-0.799967\pi$
$823$	−46.4146	−1.61791	−0.808956	+	0.587870i	$0.799967\pi$
$824$	−9.39682	−0.327354
$825$	0	0
$826$	−9.30142	−0.323638
$827$	31.2992	1.08838	0.544190	−	0.838962i	$-0.316837\pi$
$827$	31.2992	1.08838	0.544190	+	0.838962i	$0.316837\pi$
$828$	0	0
$829$	13.8062	0.479510	0.239755	−	0.970833i	$-0.422933\pi$
$829$	13.8062	0.479510	0.239755	+	0.970833i	$0.422933\pi$
$830$	−1.81971	−0.0631630
$831$	0	0
$832$	−10.1820	−0.352996
$833$	6.14025	0.212747
$834$	0	0
$835$	2.14404	0.0741975
$836$	1.30875	0.0452642
$837$	0	0
$838$	−13.2424	−0.457450
$839$	−29.4290	−1.01600	−0.508000	−	0.861357i	$-0.669615\pi$
$839$	−29.4290	−1.01600	−0.508000	+	0.861357i	$0.669615\pi$
$840$	0	0
$841$	63.2861	2.18228
$842$	37.5132	1.29279
$843$	0	0
$844$	−11.4391	−0.393751
$845$	0.628720	0.0216286
$846$	0	0
$847$	9.47709	0.325637
$848$	67.3000	2.31109
$849$	0	0
$850$	−50.0758	−1.71759
$851$	45.6504	1.56487
$852$	0	0
$853$	17.7046	0.606195	0.303097	−	0.952960i	$-0.401979\pi$
$853$	17.7046	0.606195	0.303097	+	0.952960i	$0.401979\pi$
$854$	13.6787	0.468076
$855$	0	0
$856$	−7.56058	−0.258415
$857$	0.809325	0.0276460	0.0138230	−	0.999904i	$-0.495600\pi$
$857$	0.809325	0.0276460	0.0138230	+	0.999904i	$0.495600\pi$
$858$	0	0
$859$	−3.42756	−0.116947	−0.0584734	−	0.998289i	$-0.518623\pi$
$859$	−3.42756	−0.116947	−0.0584734	+	0.998289i	$0.518623\pi$
$860$	−0.627807	−0.0214080
$861$	0	0
$862$	30.3464	1.03360
$863$	13.9962	0.476436	0.238218	−	0.971212i	$-0.423437\pi$
$863$	13.9962	0.476436	0.238218	+	0.971212i	$0.423437\pi$
$864$	0	0
$865$	2.38521	0.0810994
$866$	−16.3034	−0.554012
$867$	0	0
$868$	−1.22115	−0.0414484
$869$	13.1954	0.447624
$870$	0	0
$871$	11.7954	0.399673
$872$	32.8120	1.11116
$873$	0	0
$874$	15.4052	0.521089
$875$	−1.07701	−0.0364095
$876$	0	0
$877$	−39.9269	−1.34824	−0.674118	−	0.738624i	$-0.735476\pi$
$877$	−39.9269	−1.34824	−0.674118	+	0.738624i	$0.735476\pi$
$878$	−22.7214	−0.766810
$879$	0	0
$880$	0.651075	0.0219477
$881$	−26.7623	−0.901646	−0.450823	−	0.892613i	$-0.648869\pi$
$881$	−26.7623	−0.901646	−0.450823	+	0.892613i	$0.648869\pi$
$882$	0	0
$883$	−26.3077	−0.885326	−0.442663	−	0.896688i	$-0.645966\pi$
$883$	−26.3077	−0.885326	−0.442663	+	0.896688i	$0.645966\pi$
$884$	−11.0612	−0.372028
$885$	0	0
$886$	43.2303	1.45235
$887$	−47.3136	−1.58864	−0.794318	−	0.607502i	$-0.792172\pi$
$887$	−47.3136	−1.58864	−0.794318	+	0.607502i	$0.792172\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−2.77158	−0.0929034
$891$	0	0
$892$	−0.908112	−0.0304058
$893$	−17.4858	−0.585140
$894$	0	0
$895$	−0.867316	−0.0289912
$896$	−13.5364	−0.452219
$897$	0	0
$898$	19.8730	0.663171
$899$	−17.4363	−0.581532
$900$	0	0
$901$	−84.4562	−2.81365
$902$	−9.55240	−0.318060
$903$	0	0
$904$	25.2282	0.839078
$905$	−0.952197	−0.0316521
$906$	0	0
$907$	18.1057	0.601191	0.300596	−	0.953752i	$-0.402815\pi$
$907$	18.1057	0.601191	0.300596	+	0.953752i	$0.402815\pi$
$908$	−8.22426	−0.272932
$909$	0	0
$910$	−0.471998	−0.0156466
$911$	−10.1254	−0.335471	−0.167735	−	0.985832i	$-0.553645\pi$
$911$	−10.1254	−0.335471	−0.167735	+	0.985832i	$0.553645\pi$
$912$	0	0
$913$	12.7389	0.421597
$914$	−22.5912	−0.747249
$915$	0	0
$916$	−4.44894	−0.146997
$917$	2.66536	0.0880178
$918$	0	0
$919$	25.0065	0.824889	0.412444	−	0.910983i	$-0.364675\pi$
$919$	25.0065	0.824889	0.412444	+	0.910983i	$0.364675\pi$
$920$	1.39859	0.0461102
$921$	0	0
$922$	−19.3335	−0.636715
$923$	37.2273	1.22535
$924$	0	0
$925$	−38.0940	−1.25252
$926$	38.4305	1.26291
$927$	0	0
$928$	35.1571	1.15409
$929$	−12.6245	−0.414196	−0.207098	−	0.978320i	$-0.566402\pi$
$929$	−12.6245	−0.414196	−0.207098	+	0.978320i	$0.566402\pi$
$930$	0	0
$931$	1.57630	0.0516611
$932$	−4.90176	−0.160563
$933$	0	0
$934$	50.7019	1.65902
$935$	−0.817047	−0.0267203
$936$	0	0
$937$	4.17602	0.136425	0.0682123	−	0.997671i	$-0.478270\pi$
$937$	4.17602	0.136425	0.0682123	+	0.997671i	$0.478270\pi$
$938$	7.20218	0.235160
$939$	0	0
$940$	0.804736	0.0262476
$941$	−4.50287	−0.146789	−0.0733946	−	0.997303i	$-0.523383\pi$
$941$	−4.50287	−0.146789	−0.0733946	+	0.997303i	$0.523383\pi$
$942$	0	0
$943$	−28.3034	−0.921686
$944$	−27.8379	−0.906045
$945$	0	0
$946$	17.4598	0.567668
$947$	−10.8130	−0.351377	−0.175688	−	0.984446i	$-0.556215\pi$
$947$	−10.8130	−0.351377	−0.175688	+	0.984446i	$0.556215\pi$
$948$	0	0
$949$	21.7748	0.706839
$950$	−12.8552	−0.417079
$951$	0	0
$952$	13.3232	0.431806
$953$	1.02312	0.0331420	0.0165710	−	0.999863i	$-0.494725\pi$
$953$	1.02312	0.0331420	0.0165710	+	0.999863i	$0.494725\pi$
$954$	0	0
$955$	−1.18669	−0.0384004
$956$	−7.10539	−0.229805
$957$	0	0
$958$	−35.2679	−1.13945
$959$	18.7227	0.604588
$960$	0	0
$961$	−27.7056	−0.893730
$962$	−33.4283	−1.07777
$963$	0	0
$964$	14.2316	0.458369
$965$	−0.720177	−0.0231833
$966$	0	0
$967$	−33.4538	−1.07580	−0.537902	−	0.843008i	$-0.680783\pi$
$967$	−33.4538	−1.07580	−0.537902	+	0.843008i	$0.680783\pi$
$968$	20.5634	0.660934
$969$	0	0
$970$	2.15324	0.0691363
$971$	16.1758	0.519106	0.259553	−	0.965729i	$-0.416425\pi$
$971$	16.1758	0.519106	0.259553	+	0.965729i	$0.416425\pi$
$972$	0	0
$973$	13.7227	0.439931
$974$	−26.7995	−0.858712
$975$	0	0
$976$	40.9385	1.31041
$977$	24.4857	0.783368	0.391684	−	0.920100i	$-0.371893\pi$
$977$	24.4857	0.783368	0.391684	+	0.920100i	$0.371893\pi$
$978$	0	0
$979$	19.4025	0.620106
$980$	−0.0725448	−0.00231736
$981$	0	0
$982$	8.23520	0.262796
$983$	−27.0424	−0.862518	−0.431259	−	0.902228i	$-0.641931\pi$
$983$	−27.0424	−0.862518	−0.431259	+	0.902228i	$0.641931\pi$
$984$	0	0
$985$	−1.53148	−0.0487970
$986$	−96.4355	−3.07113
$987$	0	0
$988$	−2.83958	−0.0903390
$989$	51.7328	1.64501
$990$	0	0
$991$	−34.0897	−1.08290	−0.541448	−	0.840734i	$-0.682124\pi$
$991$	−34.0897	−1.08290	−0.541448	+	0.840734i	$0.682124\pi$
$992$	−6.64248	−0.210899
$993$	0	0
$994$	22.7306	0.720972
$995$	0.947318	0.0300320
$996$	0	0
$997$	−18.0695	−0.572266	−0.286133	−	0.958190i	$-0.592370\pi$
$997$	−18.0695	−0.572266	−0.286133	+	0.958190i	$0.592370\pi$
$998$	17.6534	0.558808
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.z.1.25	yes	32
3.2	odd	2	inner	8001.2.a.z.1.8	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.z.1.8	✓	32	3.2	odd	2	inner
8001.2.a.z.1.25	yes	32	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)

\(f(q)\)	\(=\)	\(q+1.63487 q^{2} +0.672794 q^{4} -0.107826 q^{5} -1.00000 q^{7} -2.16981 q^{8} +O(q^{10})\)	\(q+1.63487 q^{2} +0.672794 q^{4} -0.107826 q^{5} -1.00000 q^{7} -2.16981 q^{8} -0.176281 q^{10} +1.23406 q^{11} -2.67752 q^{13} -1.63487 q^{14} -4.89294 q^{16} +6.14025 q^{17} +1.57630 q^{19} -0.0725448 q^{20} +2.01753 q^{22} +5.97787 q^{23} -4.98837 q^{25} -4.37740 q^{26} -0.672794 q^{28} -9.60657 q^{29} +1.81504 q^{31} -3.65969 q^{32} +10.0385 q^{34} +0.107826 q^{35} +7.63656 q^{37} +2.57704 q^{38} +0.233962 q^{40} -4.73470 q^{41} +8.65406 q^{43} +0.830270 q^{44} +9.77303 q^{46} -11.0930 q^{47} +1.00000 q^{49} -8.15533 q^{50} -1.80142 q^{52} -13.7545 q^{53} -0.133064 q^{55} +2.16981 q^{56} -15.7055 q^{58} +5.68940 q^{59} -8.36686 q^{61} +2.96735 q^{62} +3.80276 q^{64} +0.288707 q^{65} -4.40536 q^{67} +4.13113 q^{68} +0.176281 q^{70} -13.9036 q^{71} -8.13243 q^{73} +12.4848 q^{74} +1.06052 q^{76} -1.23406 q^{77} +10.6927 q^{79} +0.527586 q^{80} -7.74061 q^{82} +10.3228 q^{83} -0.662079 q^{85} +14.1482 q^{86} -2.67768 q^{88} +15.7224 q^{89} +2.67752 q^{91} +4.02188 q^{92} -18.1355 q^{94} -0.169966 q^{95} -12.2148 q^{97} +1.63487 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) \) 32 * q + 30 * q^4 - 32 * q^7	\( 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) \) 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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