LMFDB - Embedded newform 8001.2.a.t.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8001,2,Mod(1,8001)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [16,2,0,12,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} + \cdots + 1 $ x^16 - 2x^15 - 20x^14 + 38x^13 + 155x^12 - 275x^11 - 593x^10 + 957x^9 + 1177x^8 - 1655x^7 - 1150x^6 + 1279x^5 + 474x^4 - 280x^3 - 83x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.170601$ of defining polynomial
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)

$f(q)$	$=$	\(q-0.170601 q^{2} -1.97090 q^{4} +0.206353 q^{5} -1.00000 q^{7} +0.677439 q^{8} -0.0352041 q^{10} -2.95191 q^{11} -6.02043 q^{13} +0.170601 q^{14} +3.82622 q^{16} +6.47770 q^{17} -5.49983 q^{19} -0.406701 q^{20} +0.503600 q^{22} -7.87219 q^{23} -4.95742 q^{25} +1.02709 q^{26} +1.97090 q^{28} -3.67353 q^{29} +0.182599 q^{31} -2.00764 q^{32} -1.10510 q^{34} -0.206353 q^{35} -9.06492 q^{37} +0.938277 q^{38} +0.139792 q^{40} +0.717792 q^{41} +4.41197 q^{43} +5.81791 q^{44} +1.34301 q^{46} +3.34219 q^{47} +1.00000 q^{49} +0.845741 q^{50} +11.8656 q^{52} +6.62197 q^{53} -0.609137 q^{55} -0.677439 q^{56} +0.626708 q^{58} -7.35511 q^{59} -15.3403 q^{61} -0.0311515 q^{62} -7.30993 q^{64} -1.24234 q^{65} +6.35295 q^{67} -12.7669 q^{68} +0.0352041 q^{70} +7.06485 q^{71} +8.10741 q^{73} +1.54649 q^{74} +10.8396 q^{76} +2.95191 q^{77} +5.73229 q^{79} +0.789553 q^{80} -0.122456 q^{82} -9.79819 q^{83} +1.33670 q^{85} -0.752687 q^{86} -1.99974 q^{88} +14.8905 q^{89} +6.02043 q^{91} +15.5153 q^{92} -0.570182 q^{94} -1.13491 q^{95} -14.8333 q^{97} -0.170601 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28}+ \cdots + 2 q^{98}+O(q^{100}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * q^98

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 $

$2$	−0.170601	−0.120633	−0.0603166	−	0.998179i	$-0.519211\pi$
$2$	−0.170601	−0.120633	−0.0603166	+	0.998179i	$0.519211\pi$
$3$	0	0
$3$	0	0
$4$	−1.97090	−0.985448
$5$	0.206353	0.0922840	0.0461420	−	0.998935i	$-0.485307\pi$
$5$	0.206353	0.0922840	0.0461420	+	0.998935i	$0.485307\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.677439	0.239511
$9$	0	0
$10$	−0.0352041	−0.0111325
$11$	−2.95191	−0.890035	−0.445018	−	0.895522i	$-0.646803\pi$
$11$	−2.95191	−0.890035	−0.445018	+	0.895522i	$0.646803\pi$
$12$	0	0
$13$	−6.02043	−1.66977	−0.834883	−	0.550427i	$-0.814465\pi$
$13$	−6.02043	−1.66977	−0.834883	+	0.550427i	$0.814465\pi$
$14$	0.170601	0.0455951
$15$	0	0
$16$	3.82622	0.956555
$17$	6.47770	1.57107	0.785537	−	0.618815i	$-0.212387\pi$
$17$	6.47770	1.57107	0.785537	+	0.618815i	$0.212387\pi$
$18$	0	0
$19$	−5.49983	−1.26175	−0.630874	−	0.775886i	$-0.717303\pi$
$19$	−5.49983	−1.26175	−0.630874	+	0.775886i	$0.717303\pi$
$20$	−0.406701	−0.0909410
$21$	0	0
$22$	0.503600	0.107368
$23$	−7.87219	−1.64147	−0.820733	−	0.571312i	$-0.806435\pi$
$23$	−7.87219	−1.64147	−0.820733	+	0.571312i	$0.806435\pi$
$24$	0	0
$25$	−4.95742	−0.991484
$26$	1.02709	0.201429
$27$	0	0
$28$	1.97090	0.372464
$29$	−3.67353	−0.682157	−0.341079	−	0.940035i	$-0.610792\pi$
$29$	−3.67353	−0.682157	−0.341079	+	0.940035i	$0.610792\pi$
$30$	0	0
$31$	0.182599	0.0327957	0.0163978	−	0.999866i	$-0.494780\pi$
$31$	0.182599	0.0327957	0.0163978	+	0.999866i	$0.494780\pi$
$32$	−2.00764	−0.354903
$33$	0	0
$34$	−1.10510	−0.189524
$35$	−0.206353	−0.0348801
$36$	0	0
$37$	−9.06492	−1.49026	−0.745132	−	0.666917i	$-0.767614\pi$
$37$	−9.06492	−1.49026	−0.745132	+	0.666917i	$0.767614\pi$
$38$	0.938277	0.152209
$39$	0	0
$40$	0.139792	0.0221030
$41$	0.717792	0.112100	0.0560501	−	0.998428i	$-0.482149\pi$
$41$	0.717792	0.112100	0.0560501	+	0.998428i	$0.482149\pi$
$42$	0	0
$43$	4.41197	0.672819	0.336410	−	0.941716i	$-0.390787\pi$
$43$	4.41197	0.672819	0.336410	+	0.941716i	$0.390787\pi$
$44$	5.81791	0.877083
$45$	0	0
$46$	1.34301	0.198015
$47$	3.34219	0.487509	0.243754	−	0.969837i	$-0.421621\pi$
$47$	3.34219	0.487509	0.243754	+	0.969837i	$0.421621\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.845741	0.119606
$51$	0	0
$52$	11.8656	1.64547
$53$	6.62197	0.909597	0.454799	−	0.890594i	$-0.349711\pi$
$53$	6.62197	0.909597	0.454799	+	0.890594i	$0.349711\pi$
$54$	0	0
$55$	−0.609137	−0.0821360
$56$	−0.677439	−0.0905266
$57$	0	0
$58$	0.626708	0.0822909
$59$	−7.35511	−0.957553	−0.478777	−	0.877937i	$-0.658920\pi$
$59$	−7.35511	−0.957553	−0.478777	+	0.877937i	$0.658920\pi$
$60$	0	0
$61$	−15.3403	−1.96413	−0.982064	−	0.188549i	$-0.939621\pi$
$61$	−15.3403	−1.96413	−0.982064	+	0.188549i	$0.939621\pi$
$62$	−0.0311515	−0.00395625
$63$	0	0
$64$	−7.30993	−0.913742
$65$	−1.24234	−0.154093
$66$	0	0
$67$	6.35295	0.776137	0.388068	−	0.921631i	$-0.373142\pi$
$67$	6.35295	0.776137	0.388068	+	0.921631i	$0.373142\pi$
$68$	−12.7669	−1.54821
$69$	0	0
$70$	0.0352041	0.00420770
$71$	7.06485	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$71$	7.06485	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$72$	0	0
$73$	8.10741	0.948900	0.474450	−	0.880282i	$-0.342647\pi$
$73$	8.10741	0.948900	0.474450	+	0.880282i	$0.342647\pi$
$74$	1.54649	0.179775
$75$	0	0
$76$	10.8396	1.24339
$77$	2.95191	0.336402
$78$	0	0
$79$	5.73229	0.644933	0.322467	−	0.946581i	$-0.395488\pi$
$79$	5.73229	0.644933	0.322467	+	0.946581i	$0.395488\pi$
$80$	0.789553	0.0882747
$81$	0	0
$82$	−0.122456	−0.0135230
$83$	−9.79819	−1.07549	−0.537746	−	0.843107i	$-0.680724\pi$
$83$	−9.79819	−1.07549	−0.537746	+	0.843107i	$0.680724\pi$
$84$	0	0
$85$	1.33670	0.144985
$86$	−0.752687	−0.0811644
$87$	0	0
$88$	−1.99974	−0.213173
$89$	14.8905	1.57839	0.789193	−	0.614145i	$-0.210499\pi$
$89$	14.8905	1.57839	0.789193	+	0.614145i	$0.210499\pi$
$90$	0	0
$91$	6.02043	0.631113
$92$	15.5153	1.61758
$93$	0	0
$94$	−0.570182	−0.0588098
$95$	−1.13491	−0.116439
$96$	0	0
$97$	−14.8333	−1.50609	−0.753046	−	0.657967i	$-0.771416\pi$
$97$	−14.8333	−1.50609	−0.753046	+	0.657967i	$0.771416\pi$
$98$	−0.170601	−0.0172333
$99$	0	0
$100$	9.77055	0.977055
$101$	−15.4154	−1.53389	−0.766943	−	0.641716i	$-0.778223\pi$
$101$	−15.4154	−1.53389	−0.766943	+	0.641716i	$0.778223\pi$
$102$	0	0
$103$	12.6158	1.24307	0.621534	−	0.783387i	$-0.286510\pi$
$103$	12.6158	1.24307	0.621534	+	0.783387i	$0.286510\pi$
$104$	−4.07848	−0.399927
$105$	0	0
$106$	−1.12972	−0.109728
$107$	−18.0545	−1.74539	−0.872697	−	0.488262i	$-0.837631\pi$
$107$	−18.0545	−1.74539	−0.872697	+	0.488262i	$0.837631\pi$
$108$	0	0
$109$	5.67574	0.543638	0.271819	−	0.962348i	$-0.412375\pi$
$109$	5.67574	0.543638	0.271819	+	0.962348i	$0.412375\pi$
$110$	0.103919	0.00990833
$111$	0	0
$112$	−3.82622	−0.361544
$113$	0.331384	0.0311740	0.0155870	−	0.999879i	$-0.495038\pi$
$113$	0.331384	0.0311740	0.0155870	+	0.999879i	$0.495038\pi$
$114$	0	0
$115$	−1.62445	−0.151481
$116$	7.24014	0.672230
$117$	0	0
$118$	1.25479	0.115513
$119$	−6.47770	−0.593810
$120$	0	0
$121$	−2.28621	−0.207838
$122$	2.61708	0.236939
$123$	0	0
$124$	−0.359883	−0.0323184
$125$	−2.05475	−0.183782
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	5.26235	0.465131
$129$	0	0
$130$	0.211944	0.0185887
$131$	9.55770	0.835061	0.417530	−	0.908663i	$-0.362896\pi$
$131$	9.55770	0.835061	0.417530	+	0.908663i	$0.362896\pi$
$132$	0	0
$133$	5.49983	0.476896
$134$	−1.08382	−0.0936279
$135$	0	0
$136$	4.38825	0.376289
$137$	−19.1217	−1.63367	−0.816837	−	0.576868i	$-0.804275\pi$
$137$	−19.1217	−1.63367	−0.816837	+	0.576868i	$0.804275\pi$
$138$	0	0
$139$	−19.5559	−1.65871	−0.829353	−	0.558725i	$-0.811291\pi$
$139$	−19.5559	−1.65871	−0.829353	+	0.558725i	$0.811291\pi$
$140$	0.406701	0.0343725
$141$	0	0
$142$	−1.20527	−0.101144
$143$	17.7718	1.48615
$144$	0	0
$145$	−0.758045	−0.0629522
$146$	−1.38313	−0.114469
$147$	0	0
$148$	17.8660	1.46858
$149$	8.65860	0.709340	0.354670	−	0.934992i	$-0.384593\pi$
$149$	8.65860	0.709340	0.354670	+	0.934992i	$0.384593\pi$
$150$	0	0
$151$	−14.9600	−1.21743	−0.608715	−	0.793389i	$-0.708315\pi$
$151$	−14.9600	−1.21743	−0.608715	+	0.793389i	$0.708315\pi$
$152$	−3.72580	−0.302202
$153$	0	0
$154$	−0.503600	−0.0405812
$155$	0.0376798	0.00302652
$156$	0	0
$157$	13.5556	1.08185	0.540926	−	0.841070i	$-0.318074\pi$
$157$	13.5556	1.08185	0.540926	+	0.841070i	$0.318074\pi$
$158$	−0.977936	−0.0778004
$159$	0	0
$160$	−0.414282	−0.0327519
$161$	7.87219	0.620416
$162$	0	0
$163$	13.3149	1.04291	0.521453	−	0.853280i	$-0.325390\pi$
$163$	13.3149	1.04291	0.521453	+	0.853280i	$0.325390\pi$
$164$	−1.41469	−0.110469
$165$	0	0
$166$	1.67158	0.129740
$167$	−5.40055	−0.417907	−0.208953	−	0.977926i	$-0.567006\pi$
$167$	−5.40055	−0.417907	−0.208953	+	0.977926i	$0.567006\pi$
$168$	0	0
$169$	23.2456	1.78812
$170$	−0.228042	−0.0174900
$171$	0	0
$172$	−8.69553	−0.663028
$173$	−0.466383	−0.0354584	−0.0177292	−	0.999843i	$-0.505644\pi$
$173$	−0.466383	−0.0354584	−0.0177292	+	0.999843i	$0.505644\pi$
$174$	0	0
$175$	4.95742	0.374746
$176$	−11.2947	−0.851367
$177$	0	0
$178$	−2.54033	−0.190406
$179$	−10.0960	−0.754611	−0.377306	−	0.926089i	$-0.623149\pi$
$179$	−10.0960	−0.754611	−0.377306	+	0.926089i	$0.623149\pi$
$180$	0	0
$181$	−22.2609	−1.65464	−0.827319	−	0.561732i	$-0.810135\pi$
$181$	−22.2609	−1.65464	−0.827319	+	0.561732i	$0.810135\pi$
$182$	−1.02709	−0.0761331
$183$	0	0
$184$	−5.33293	−0.393149
$185$	−1.87058	−0.137528
$186$	0	0
$187$	−19.1216	−1.39831
$188$	−6.58711	−0.480414
$189$	0	0
$190$	0.193617	0.0140464
$191$	8.20390	0.593613	0.296807	−	0.954938i	$-0.404078\pi$
$191$	8.20390	0.593613	0.296807	+	0.954938i	$0.404078\pi$
$192$	0	0
$193$	7.11018	0.511802	0.255901	−	0.966703i	$-0.417628\pi$
$193$	7.11018	0.511802	0.255901	+	0.966703i	$0.417628\pi$
$194$	2.53058	0.181685
$195$	0	0
$196$	−1.97090	−0.140778
$197$	12.9689	0.923994	0.461997	−	0.886882i	$-0.347133\pi$
$197$	12.9689	0.923994	0.461997	+	0.886882i	$0.347133\pi$
$198$	0	0
$199$	20.6661	1.46498	0.732491	−	0.680777i	$-0.238358\pi$
$199$	20.6661	1.46498	0.732491	+	0.680777i	$0.238358\pi$
$200$	−3.35835	−0.237471
$201$	0	0
$202$	2.62988	0.185038
$203$	3.67353	0.257831
$204$	0	0
$205$	0.148119	0.0103451
$206$	−2.15226	−0.149955
$207$	0	0
$208$	−23.0355	−1.59722
$209$	16.2350	1.12300
$210$	0	0
$211$	2.16808	0.149257	0.0746283	−	0.997211i	$-0.476223\pi$
$211$	2.16808	0.149257	0.0746283	+	0.997211i	$0.476223\pi$
$212$	−13.0512	−0.896360
$213$	0	0
$214$	3.08012	0.210552
$215$	0.910425	0.0620905
$216$	0	0
$217$	−0.182599	−0.0123956
$218$	−0.968289	−0.0655808
$219$	0	0
$220$	1.20054	0.0809407
$221$	−38.9986	−2.62333
$222$	0	0
$223$	−1.10332	−0.0738836	−0.0369418	−	0.999317i	$-0.511762\pi$
$223$	−1.10332	−0.0738836	−0.0369418	+	0.999317i	$0.511762\pi$
$224$	2.00764	0.134141
$225$	0	0
$226$	−0.0565346	−0.00376062
$227$	8.00286	0.531169	0.265584	−	0.964088i	$-0.414435\pi$
$227$	8.00286	0.531169	0.265584	+	0.964088i	$0.414435\pi$
$228$	0	0
$229$	0.363896	0.0240469	0.0120235	−	0.999928i	$-0.496173\pi$
$229$	0.363896	0.0240469	0.0120235	+	0.999928i	$0.496173\pi$
$230$	0.277134	0.0182736
$231$	0	0
$232$	−2.48859	−0.163384
$233$	28.6092	1.87425	0.937125	−	0.348994i	$-0.113477\pi$
$233$	28.6092	1.87425	0.937125	+	0.348994i	$0.113477\pi$
$234$	0	0
$235$	0.689672	0.0449893
$236$	14.4961	0.943619
$237$	0	0
$238$	1.10510	0.0716332
$239$	28.4902	1.84288	0.921438	−	0.388525i	$-0.127015\pi$
$239$	28.4902	1.84288	0.921438	+	0.388525i	$0.127015\pi$
$240$	0	0
$241$	−2.12643	−0.136975	−0.0684877	−	0.997652i	$-0.521817\pi$
$241$	−2.12643	−0.136975	−0.0684877	+	0.997652i	$0.521817\pi$
$242$	0.390031	0.0250721
$243$	0	0
$244$	30.2342	1.93554
$245$	0.206353	0.0131834
$246$	0	0
$247$	33.1113	2.10682
$248$	0.123700	0.00785493
$249$	0	0
$250$	0.350542	0.0221702
$251$	−12.3182	−0.777519	−0.388759	−	0.921339i	$-0.627096\pi$
$251$	−12.3182	−0.777519	−0.388759	+	0.921339i	$0.627096\pi$
$252$	0	0
$253$	23.2380	1.46096
$254$	0.170601	0.0107045
$255$	0	0
$256$	13.7221	0.857631
$257$	4.35584	0.271710	0.135855	−	0.990729i	$-0.456622\pi$
$257$	4.35584	0.271710	0.135855	+	0.990729i	$0.456622\pi$
$258$	0	0
$259$	9.06492	0.563267
$260$	2.44851	0.151850
$261$	0	0
$262$	−1.63056	−0.100736
$263$	7.24987	0.447046	0.223523	−	0.974699i	$-0.428244\pi$
$263$	7.24987	0.447046	0.223523	+	0.974699i	$0.428244\pi$
$264$	0	0
$265$	1.36646	0.0839413
$266$	−0.938277	−0.0575295
$267$	0	0
$268$	−12.5210	−0.764842
$269$	1.81003	0.110360	0.0551798	−	0.998476i	$-0.482427\pi$
$269$	1.81003	0.110360	0.0551798	+	0.998476i	$0.482427\pi$
$270$	0	0
$271$	15.9554	0.969219	0.484609	−	0.874731i	$-0.338962\pi$
$271$	15.9554	0.969219	0.484609	+	0.874731i	$0.338962\pi$
$272$	24.7851	1.50282
$273$	0	0
$274$	3.26218	0.197075
$275$	14.6339	0.882455
$276$	0	0
$277$	−10.5622	−0.634621	−0.317310	−	0.948322i	$-0.602780\pi$
$277$	−10.5622	−0.634621	−0.317310	+	0.948322i	$0.602780\pi$
$278$	3.33625	0.200095
$279$	0	0
$280$	−0.139792	−0.00835416
$281$	−8.87376	−0.529364	−0.264682	−	0.964336i	$-0.585267\pi$
$281$	−8.87376	−0.529364	−0.264682	+	0.964336i	$0.585267\pi$
$282$	0	0
$283$	−13.8510	−0.823359	−0.411680	−	0.911329i	$-0.635058\pi$
$283$	−13.8510	−0.823359	−0.411680	+	0.911329i	$0.635058\pi$
$284$	−13.9241	−0.826243
$285$	0	0
$286$	−3.03189	−0.179279
$287$	−0.717792	−0.0423699
$288$	0	0
$289$	24.9606	1.46827
$290$	0.129323	0.00759413
$291$	0	0
$292$	−15.9788	−0.935091
$293$	9.28962	0.542705	0.271353	−	0.962480i	$-0.412529\pi$
$293$	9.28962	0.542705	0.271353	+	0.962480i	$0.412529\pi$
$294$	0	0
$295$	−1.51775	−0.0883669
$296$	−6.14094	−0.356935
$297$	0	0
$298$	−1.47717	−0.0855700
$299$	47.3940	2.74087
$300$	0	0
$301$	−4.41197	−0.254302
$302$	2.55220	0.146863
$303$	0	0
$304$	−21.0435	−1.20693
$305$	−3.16553	−0.181258
$306$	0	0
$307$	28.5119	1.62726	0.813631	−	0.581382i	$-0.197488\pi$
$307$	28.5119	1.62726	0.813631	+	0.581382i	$0.197488\pi$
$308$	−5.81791	−0.331506
$309$	0	0
$310$	−0.00642822	−0.000365099 0
$311$	−7.51913	−0.426371	−0.213185	−	0.977012i	$-0.568384\pi$
$311$	−7.51913	−0.426371	−0.213185	+	0.977012i	$0.568384\pi$
$312$	0	0
$313$	15.1262	0.854983	0.427492	−	0.904019i	$-0.359397\pi$
$313$	15.1262	0.854983	0.427492	+	0.904019i	$0.359397\pi$
$314$	−2.31259	−0.130507
$315$	0	0
$316$	−11.2977	−0.635548
$317$	10.3384	0.580661	0.290331	−	0.956926i	$-0.406235\pi$
$317$	10.3384	0.580661	0.290331	+	0.956926i	$0.406235\pi$
$318$	0	0
$319$	10.8439	0.607144
$320$	−1.50843	−0.0843237
$321$	0	0
$322$	−1.34301	−0.0748428
$323$	−35.6263	−1.98230
$324$	0	0
$325$	29.8458	1.65555
$326$	−2.27154	−0.125809
$327$	0	0
$328$	0.486260	0.0268492
$329$	−3.34219	−0.184261
$330$	0	0
$331$	1.33558	0.0734100	0.0367050	−	0.999326i	$-0.488314\pi$
$331$	1.33558	0.0734100	0.0367050	+	0.999326i	$0.488314\pi$
$332$	19.3112	1.05984
$333$	0	0
$334$	0.921339	0.0504135
$335$	1.31095	0.0716250
$336$	0	0
$337$	−12.6188	−0.687388	−0.343694	−	0.939082i	$-0.611678\pi$
$337$	−12.6188	−0.687388	−0.343694	+	0.939082i	$0.611678\pi$
$338$	−3.96572	−0.215707
$339$	0	0
$340$	−2.63449	−0.142875
$341$	−0.539015	−0.0291893
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	2.98884	0.161148
$345$	0	0
$346$	0.0795655	0.00427747
$347$	2.34794	0.126044	0.0630221	−	0.998012i	$-0.479926\pi$
$347$	2.34794	0.126044	0.0630221	+	0.998012i	$0.479926\pi$
$348$	0	0
$349$	−14.3453	−0.767885	−0.383943	−	0.923357i	$-0.625434\pi$
$349$	−14.3453	−0.767885	−0.383943	+	0.923357i	$0.625434\pi$
$350$	−0.845741	−0.0452068
$351$	0	0
$352$	5.92637	0.315876
$353$	−18.7887	−1.00002	−0.500011	−	0.866019i	$-0.666671\pi$
$353$	−18.7887	−1.00002	−0.500011	+	0.866019i	$0.666671\pi$
$354$	0	0
$355$	1.45786	0.0773750
$356$	−29.3476	−1.55542
$357$	0	0
$358$	1.72239	0.0910312
$359$	11.5788	0.611107	0.305553	−	0.952175i	$-0.401159\pi$
$359$	11.5788	0.611107	0.305553	+	0.952175i	$0.401159\pi$
$360$	0	0
$361$	11.2481	0.592006
$362$	3.79773	0.199604
$363$	0	0
$364$	−11.8656	−0.621928
$365$	1.67299	0.0875683
$366$	0	0
$367$	5.31371	0.277373	0.138687	−	0.990336i	$-0.455712\pi$
$367$	5.31371	0.277373	0.138687	+	0.990336i	$0.455712\pi$
$368$	−30.1207	−1.57015
$369$	0	0
$370$	0.319123	0.0165904
$371$	−6.62197	−0.343795
$372$	0	0
$373$	−36.9922	−1.91539	−0.957693	−	0.287792i	$-0.907079\pi$
$373$	−36.9922	−1.91539	−0.957693	+	0.287792i	$0.907079\pi$
$374$	3.26217	0.168683
$375$	0	0
$376$	2.26413	0.116764
$377$	22.1162	1.13904
$378$	0	0
$379$	−14.5568	−0.747730	−0.373865	−	0.927483i	$-0.621968\pi$
$379$	−14.5568	−0.747730	−0.373865	+	0.927483i	$0.621968\pi$
$380$	2.23678	0.114745
$381$	0	0
$382$	−1.39959	−0.0716095
$383$	16.5337	0.844833	0.422416	−	0.906402i	$-0.361182\pi$
$383$	16.5337	0.844833	0.422416	+	0.906402i	$0.361182\pi$
$384$	0	0
$385$	0.609137	0.0310445
$386$	−1.21301	−0.0617403
$387$	0	0
$388$	29.2349	1.48418
$389$	−0.903542	−0.0458114	−0.0229057	−	0.999738i	$-0.507292\pi$
$389$	−0.903542	−0.0458114	−0.0229057	+	0.999738i	$0.507292\pi$
$390$	0	0
$391$	−50.9937	−2.57886
$392$	0.677439	0.0342159
$393$	0	0
$394$	−2.21250	−0.111464
$395$	1.18288	0.0595170
$396$	0	0
$397$	−19.6722	−0.987318	−0.493659	−	0.869656i	$-0.664341\pi$
$397$	−19.6722	−0.987318	−0.493659	+	0.869656i	$0.664341\pi$
$398$	−3.52566	−0.176725
$399$	0	0
$400$	−18.9682	−0.948408
$401$	−19.5957	−0.978562	−0.489281	−	0.872126i	$-0.662741\pi$
$401$	−19.5957	−0.978562	−0.489281	+	0.872126i	$0.662741\pi$
$402$	0	0
$403$	−1.09932	−0.0547612
$404$	30.3821	1.51156
$405$	0	0
$406$	−0.626708	−0.0311030
$407$	26.7589	1.32639
$408$	0	0
$409$	−16.0598	−0.794106	−0.397053	−	0.917796i	$-0.629967\pi$
$409$	−16.0598	−0.794106	−0.397053	+	0.917796i	$0.629967\pi$
$410$	−0.0252692	−0.00124796
$411$	0	0
$412$	−24.8643	−1.22498
$413$	7.35511	0.361921
$414$	0	0
$415$	−2.02189	−0.0992506
$416$	12.0868	0.592606
$417$	0	0
$418$	−2.76971	−0.135471
$419$	12.9330	0.631818	0.315909	−	0.948790i	$-0.397691\pi$
$419$	12.9330	0.631818	0.315909	+	0.948790i	$0.397691\pi$
$420$	0	0
$421$	−24.9490	−1.21594	−0.607970	−	0.793960i	$-0.708016\pi$
$421$	−24.9490	−1.21594	−0.607970	+	0.793960i	$0.708016\pi$
$422$	−0.369876	−0.0180053
$423$	0	0
$424$	4.48598	0.217859
$425$	−32.1127	−1.55769
$426$	0	0
$427$	15.3403	0.742370
$428$	35.5835	1.71999
$429$	0	0
$430$	−0.155320	−0.00749017
$431$	7.50132	0.361326	0.180663	−	0.983545i	$-0.442176\pi$
$431$	7.50132	0.361326	0.180663	+	0.983545i	$0.442176\pi$
$432$	0	0
$433$	−14.1428	−0.679661	−0.339831	−	0.940487i	$-0.610370\pi$
$433$	−14.1428	−0.679661	−0.339831	+	0.940487i	$0.610370\pi$
$434$	0.0311515	0.00149532
$435$	0	0
$436$	−11.1863	−0.535726
$437$	43.2957	2.07112
$438$	0	0
$439$	9.05446	0.432146	0.216073	−	0.976377i	$-0.430675\pi$
$439$	9.05446	0.432146	0.216073	+	0.976377i	$0.430675\pi$
$440$	−0.412653	−0.0196725
$441$	0	0
$442$	6.65320	0.316460
$443$	29.2026	1.38746	0.693729	−	0.720236i	$-0.255967\pi$
$443$	29.2026	1.38746	0.693729	+	0.720236i	$0.255967\pi$
$444$	0	0
$445$	3.07270	0.145660
$446$	0.188227	0.00891282
$447$	0	0
$448$	7.30993	0.345362
$449$	−1.14072	−0.0538341	−0.0269170	−	0.999638i	$-0.508569\pi$
$449$	−1.14072	−0.0538341	−0.0269170	+	0.999638i	$0.508569\pi$
$450$	0	0
$451$	−2.11886	−0.0997732
$452$	−0.653124	−0.0307204
$453$	0	0
$454$	−1.36530	−0.0640766
$455$	1.24234	0.0582416
$456$	0	0
$457$	−21.9756	−1.02798	−0.513988	−	0.857797i	$-0.671832\pi$
$457$	−21.9756	−1.02798	−0.513988	+	0.857797i	$0.671832\pi$
$458$	−0.0620810	−0.00290086
$459$	0	0
$460$	3.20163	0.149277
$461$	22.1987	1.03390	0.516948	−	0.856017i	$-0.327068\pi$
$461$	22.1987	1.03390	0.516948	+	0.856017i	$0.327068\pi$
$462$	0	0
$463$	3.57602	0.166192	0.0830958	−	0.996542i	$-0.473519\pi$
$463$	3.57602	0.166192	0.0830958	+	0.996542i	$0.473519\pi$
$464$	−14.0557	−0.652521
$465$	0	0
$466$	−4.88076	−0.226097
$467$	5.02468	0.232514	0.116257	−	0.993219i	$-0.462910\pi$
$467$	5.02468	0.232514	0.116257	+	0.993219i	$0.462910\pi$
$468$	0	0
$469$	−6.35295	−0.293352
$470$	−0.117659	−0.00542720
$471$	0	0
$472$	−4.98264	−0.229345
$473$	−13.0238	−0.598833
$474$	0	0
$475$	27.2649	1.25100
$476$	12.7669	0.585169
$477$	0	0
$478$	−4.86046	−0.222312
$479$	39.0051	1.78219	0.891094	−	0.453818i	$-0.149938\pi$
$479$	39.0051	1.78219	0.891094	+	0.453818i	$0.149938\pi$
$480$	0	0
$481$	54.5747	2.48839
$482$	0.362771	0.0165238
$483$	0	0
$484$	4.50589	0.204813
$485$	−3.06090	−0.138988
$486$	0	0
$487$	−1.09342	−0.0495476	−0.0247738	−	0.999693i	$-0.507887\pi$
$487$	−1.09342	−0.0495476	−0.0247738	+	0.999693i	$0.507887\pi$
$488$	−10.3921	−0.470430
$489$	0	0
$490$	−0.0352041	−0.00159036
$491$	26.2655	1.18535	0.592673	−	0.805443i	$-0.298073\pi$
$491$	26.2655	1.18535	0.592673	+	0.805443i	$0.298073\pi$
$492$	0	0
$493$	−23.7960	−1.07172
$494$	−5.64883	−0.254153
$495$	0	0
$496$	0.698663	0.0313709
$497$	−7.06485	−0.316902
$498$	0	0
$499$	−29.5452	−1.32262	−0.661311	−	0.750111i	$-0.730000\pi$
$499$	−29.5452	−1.32262	−0.661311	+	0.750111i	$0.730000\pi$
$500$	4.04969	0.181108
$501$	0	0
$502$	2.10150	0.0937946
$503$	42.8080	1.90871	0.954357	−	0.298670i	$-0.0965429\pi$
$503$	42.8080	1.90871	0.954357	+	0.298670i	$0.0965429\pi$
$504$	0	0
$505$	−3.18101	−0.141553
$506$	−3.96443	−0.176241
$507$	0	0
$508$	1.97090	0.0874443
$509$	36.8649	1.63401	0.817004	−	0.576632i	$-0.195633\pi$
$509$	36.8649	1.63401	0.817004	+	0.576632i	$0.195633\pi$
$510$	0	0
$511$	−8.10741	−0.358651
$512$	−12.8657	−0.568590
$513$	0	0
$514$	−0.743112	−0.0327773
$515$	2.60330	0.114715
$516$	0	0
$517$	−9.86586	−0.433900
$518$	−1.54649	−0.0679487
$519$	0	0
$520$	−0.841607	−0.0369069
$521$	2.49859	0.109465	0.0547327	−	0.998501i	$-0.482569\pi$
$521$	2.49859	0.109465	0.0547327	+	0.998501i	$0.482569\pi$
$522$	0	0
$523$	17.7123	0.774506	0.387253	−	0.921974i	$-0.373424\pi$
$523$	17.7123	0.774506	0.387253	+	0.921974i	$0.373424\pi$
$524$	−18.8372	−0.822908
$525$	0	0
$526$	−1.23684	−0.0539286
$527$	1.18282	0.0515245
$528$	0	0
$529$	38.9714	1.69441
$530$	−0.233120	−0.0101261
$531$	0	0
$532$	−10.8396	−0.469956
$533$	−4.32142	−0.187181
$534$	0	0
$535$	−3.72560	−0.161072
$536$	4.30374	0.185893
$537$	0	0
$538$	−0.308794	−0.0133130
$539$	−2.95191	−0.127148
$540$	0	0
$541$	36.2962	1.56050	0.780248	−	0.625470i	$-0.215093\pi$
$541$	36.2962	1.56050	0.780248	+	0.625470i	$0.215093\pi$
$542$	−2.72200	−0.116920
$543$	0	0
$544$	−13.0049	−0.557579
$545$	1.17121	0.0501691
$546$	0	0
$547$	1.05534	0.0451231	0.0225615	−	0.999745i	$-0.492818\pi$
$547$	1.05534	0.0451231	0.0225615	+	0.999745i	$0.492818\pi$
$548$	37.6868	1.60990
$549$	0	0
$550$	−2.49655	−0.106453
$551$	20.2038	0.860710
$552$	0	0
$553$	−5.73229	−0.243762
$554$	1.80192	0.0765564
$555$	0	0
$556$	38.5426	1.63457
$557$	−4.73085	−0.200453	−0.100226	−	0.994965i	$-0.531957\pi$
$557$	−4.73085	−0.200453	−0.100226	+	0.994965i	$0.531957\pi$
$558$	0	0
$559$	−26.5620	−1.12345
$560$	−0.789553	−0.0333647
$561$	0	0
$562$	1.51387	0.0638589
$563$	−29.8718	−1.25894	−0.629472	−	0.777023i	$-0.716729\pi$
$563$	−29.8718	−1.25894	−0.629472	+	0.777023i	$0.716729\pi$
$564$	0	0
$565$	0.0683823	0.00287686
$566$	2.36300	0.0993245
$567$	0	0
$568$	4.78601	0.200817
$569$	40.9556	1.71695	0.858473	−	0.512859i	$-0.171413\pi$
$569$	40.9556	1.71695	0.858473	+	0.512859i	$0.171413\pi$
$570$	0	0
$571$	28.0854	1.17534	0.587668	−	0.809102i	$-0.300046\pi$
$571$	28.0854	1.17534	0.587668	+	0.809102i	$0.300046\pi$
$572$	−35.0263	−1.46452
$573$	0	0
$574$	0.122456	0.00511122
$575$	39.0258	1.62749
$576$	0	0
$577$	−24.4957	−1.01977	−0.509884	−	0.860243i	$-0.670312\pi$
$577$	−24.4957	−1.01977	−0.509884	+	0.860243i	$0.670312\pi$
$578$	−4.25832	−0.177123
$579$	0	0
$580$	1.49403	0.0620361
$581$	9.79819	0.406497
$582$	0	0
$583$	−19.5475	−0.809573
$584$	5.49228	0.227272
$585$	0	0
$586$	−1.58482	−0.0654683
$587$	−26.5944	−1.09767	−0.548834	−	0.835931i	$-0.684928\pi$
$587$	−26.5944	−1.09767	−0.548834	+	0.835931i	$0.684928\pi$
$588$	0	0
$589$	−1.00426	−0.0413799
$590$	0.258930	0.0106600
$591$	0	0
$592$	−34.6844	−1.42552
$593$	−1.49143	−0.0612459	−0.0306229	−	0.999531i	$-0.509749\pi$
$593$	−1.49143	−0.0612459	−0.0306229	+	0.999531i	$0.509749\pi$
$594$	0	0
$595$	−1.33670	−0.0547992
$596$	−17.0652	−0.699017
$597$	0	0
$598$	−8.08547	−0.330639
$599$	28.5590	1.16689	0.583445	−	0.812153i	$-0.301704\pi$
$599$	28.5590	1.16689	0.583445	+	0.812153i	$0.301704\pi$
$600$	0	0
$601$	−7.45416	−0.304062	−0.152031	−	0.988376i	$-0.548581\pi$
$601$	−7.45416	−0.304062	−0.152031	+	0.988376i	$0.548581\pi$
$602$	0.752687	0.0306773
$603$	0	0
$604$	29.4847	1.19971
$605$	−0.471768	−0.0191801
$606$	0	0
$607$	6.18969	0.251232	0.125616	−	0.992079i	$-0.459909\pi$
$607$	6.18969	0.251232	0.125616	+	0.992079i	$0.459909\pi$
$608$	11.0417	0.447798
$609$	0	0
$610$	0.540042	0.0218657
$611$	−20.1214	−0.814026
$612$	0	0
$613$	37.3292	1.50771	0.753857	−	0.657039i	$-0.228191\pi$
$613$	37.3292	1.50771	0.753857	+	0.657039i	$0.228191\pi$
$614$	−4.86417	−0.196302
$615$	0	0
$616$	1.99974	0.0805719
$617$	−45.8234	−1.84478	−0.922391	−	0.386257i	$-0.873768\pi$
$617$	−45.8234	−1.84478	−0.922391	+	0.386257i	$0.873768\pi$
$618$	0	0
$619$	−19.5672	−0.786473	−0.393236	−	0.919437i	$-0.628645\pi$
$619$	−19.5672	−0.786473	−0.393236	+	0.919437i	$0.628645\pi$
$620$	−0.0742630	−0.00298247
$621$	0	0
$622$	1.28277	0.0514345
$623$	−14.8905	−0.596574
$624$	0	0
$625$	24.3631	0.974524
$626$	−2.58055	−0.103139
$627$	0	0
$628$	−26.7166	−1.06611
$629$	−58.7199	−2.34132
$630$	0	0
$631$	−2.98989	−0.119026	−0.0595129	−	0.998228i	$-0.518955\pi$
$631$	−2.98989	−0.119026	−0.0595129	+	0.998228i	$0.518955\pi$
$632$	3.88328	0.154469
$633$	0	0
$634$	−1.76374	−0.0700470
$635$	−0.206353	−0.00818888
$636$	0	0
$637$	−6.02043	−0.238538
$638$	−1.84999	−0.0732417
$639$	0	0
$640$	1.08590	0.0429241
$641$	−45.0880	−1.78087	−0.890434	−	0.455112i	$-0.849599\pi$
$641$	−45.0880	−1.78087	−0.890434	+	0.455112i	$0.849599\pi$
$642$	0	0
$643$	43.4923	1.71517	0.857585	−	0.514342i	$-0.171964\pi$
$643$	43.4923	1.71517	0.857585	+	0.514342i	$0.171964\pi$
$644$	−15.5153	−0.611387
$645$	0	0
$646$	6.07788	0.239131
$647$	11.9165	0.468485	0.234242	−	0.972178i	$-0.424739\pi$
$647$	11.9165	0.468485	0.234242	+	0.972178i	$0.424739\pi$
$648$	0	0
$649$	21.7116	0.852256
$650$	−5.09173	−0.199714
$651$	0	0
$652$	−26.2424	−1.02773
$653$	0.522585	0.0204503	0.0102252	−	0.999948i	$-0.496745\pi$
$653$	0.522585	0.0204503	0.0102252	+	0.999948i	$0.496745\pi$
$654$	0	0
$655$	1.97226	0.0770627
$656$	2.74643	0.107230
$657$	0	0
$658$	0.570182	0.0222280
$659$	−18.0786	−0.704243	−0.352121	−	0.935954i	$-0.614540\pi$
$659$	−18.0786	−0.704243	−0.352121	+	0.935954i	$0.614540\pi$
$660$	0	0
$661$	−10.6760	−0.415249	−0.207625	−	0.978209i	$-0.566573\pi$
$661$	−10.6760	−0.415249	−0.207625	+	0.978209i	$0.566573\pi$
$662$	−0.227851	−0.00885569
$663$	0	0
$664$	−6.63768	−0.257592
$665$	1.13491	0.0440098
$666$	0	0
$667$	28.9187	1.11974
$668$	10.6439	0.411825
$669$	0	0
$670$	−0.223650	−0.00864036
$671$	45.2833	1.74814
$672$	0	0
$673$	44.6274	1.72026	0.860130	−	0.510074i	$-0.170382\pi$
$673$	44.6274	1.72026	0.860130	+	0.510074i	$0.170382\pi$
$674$	2.15278	0.0829219
$675$	0	0
$676$	−45.8146	−1.76210
$677$	45.2122	1.73765	0.868823	−	0.495123i	$-0.164877\pi$
$677$	45.2122	1.73765	0.868823	+	0.495123i	$0.164877\pi$
$678$	0	0
$679$	14.8333	0.569250
$680$	0.905530	0.0347255
$681$	0	0
$682$	0.0919566	0.00352120
$683$	−6.72334	−0.257261	−0.128631	−	0.991693i	$-0.541058\pi$
$683$	−6.72334	−0.257261	−0.128631	+	0.991693i	$0.541058\pi$
$684$	0	0
$685$	−3.94582	−0.150762
$686$	0.170601	0.00651358
$687$	0	0
$688$	16.8812	0.643589
$689$	−39.8671	−1.51882
$690$	0	0
$691$	−9.59035	−0.364834	−0.182417	−	0.983221i	$-0.558392\pi$
$691$	−9.59035	−0.364834	−0.182417	+	0.983221i	$0.558392\pi$
$692$	0.919192	0.0349424
$693$	0	0
$694$	−0.400562	−0.0152051
$695$	−4.03542	−0.153072
$696$	0	0
$697$	4.64964	0.176118
$698$	2.44732	0.0926325
$699$	0	0
$700$	−9.77055	−0.369292
$701$	−18.0322	−0.681066	−0.340533	−	0.940233i	$-0.610607\pi$
$701$	−18.0322	−0.681066	−0.340533	+	0.940233i	$0.610607\pi$
$702$	0	0
$703$	49.8555	1.88034
$704$	21.5783	0.813262
$705$	0	0
$706$	3.20538	0.120636
$707$	15.4154	0.579754
$708$	0	0
$709$	29.8573	1.12131	0.560657	−	0.828048i	$-0.310549\pi$
$709$	29.8573	1.12131	0.560657	+	0.828048i	$0.310549\pi$
$710$	−0.248712	−0.00933399
$711$	0	0
$712$	10.0874	0.378041
$713$	−1.43745	−0.0538330
$714$	0	0
$715$	3.66726	0.137148
$716$	19.8982	0.743630
$717$	0	0
$718$	−1.97536	−0.0737198
$719$	−25.0473	−0.934108	−0.467054	−	0.884229i	$-0.654685\pi$
$719$	−25.0473	−0.934108	−0.467054	+	0.884229i	$0.654685\pi$
$720$	0	0
$721$	−12.6158	−0.469836
$722$	−1.91894	−0.0714156
$723$	0	0
$724$	43.8739	1.63056
$725$	18.2112	0.676348
$726$	0	0
$727$	43.1090	1.59882	0.799412	−	0.600784i	$-0.205145\pi$
$727$	43.1090	1.59882	0.799412	+	0.600784i	$0.205145\pi$
$728$	4.07848	0.151158
$729$	0	0
$730$	−0.285414	−0.0105636
$731$	28.5794	1.05705
$732$	0	0
$733$	5.25092	0.193947	0.0969736	−	0.995287i	$-0.469084\pi$
$733$	5.25092	0.193947	0.0969736	+	0.995287i	$0.469084\pi$
$734$	−0.906525	−0.0334605
$735$	0	0
$736$	15.8045	0.582562
$737$	−18.7534	−0.690789
$738$	0	0
$739$	5.24635	0.192990	0.0964950	−	0.995333i	$-0.469237\pi$
$739$	5.24635	0.192990	0.0964950	+	0.995333i	$0.469237\pi$
$740$	3.68671	0.135526
$741$	0	0
$742$	1.12972	0.0414732
$743$	−11.7455	−0.430899	−0.215449	−	0.976515i	$-0.569122\pi$
$743$	−11.7455	−0.430899	−0.215449	+	0.976515i	$0.569122\pi$
$744$	0	0
$745$	1.78673	0.0654607
$746$	6.31092	0.231059
$747$	0	0
$748$	37.6867	1.37796
$749$	18.0545	0.659697
$750$	0	0
$751$	24.2930	0.886463	0.443231	−	0.896407i	$-0.353832\pi$
$751$	24.2930	0.886463	0.443231	+	0.896407i	$0.353832\pi$
$752$	12.7880	0.466329
$753$	0	0
$754$	−3.77305	−0.137407
$755$	−3.08705	−0.112349
$756$	0	0
$757$	−28.3351	−1.02986	−0.514929	−	0.857233i	$-0.672182\pi$
$757$	−28.3351	−1.02986	−0.514929	+	0.857233i	$0.672182\pi$
$758$	2.48340	0.0902011
$759$	0	0
$760$	−0.768831	−0.0278884
$761$	0.521485	0.0189038	0.00945190	−	0.999955i	$-0.496991\pi$
$761$	0.521485	0.0189038	0.00945190	+	0.999955i	$0.496991\pi$
$762$	0	0
$763$	−5.67574	−0.205476
$764$	−16.1690	−0.584975
$765$	0	0
$766$	−2.82067	−0.101915
$767$	44.2809	1.59889
$768$	0	0
$769$	−42.0950	−1.51798	−0.758992	−	0.651100i	$-0.774308\pi$
$769$	−42.0950	−1.51798	−0.758992	+	0.651100i	$0.774308\pi$
$770$	−0.103919	−0.00374500
$771$	0	0
$772$	−14.0134	−0.504354
$773$	−44.2658	−1.59213	−0.796066	−	0.605210i	$-0.793089\pi$
$773$	−44.2658	−1.59213	−0.796066	+	0.605210i	$0.793089\pi$
$774$	0	0
$775$	−0.905218	−0.0325164
$776$	−10.0487	−0.360726
$777$	0	0
$778$	0.154145	0.00552638
$779$	−3.94773	−0.141442
$780$	0	0
$781$	−20.8548	−0.746244
$782$	8.69959	0.311097
$783$	0	0
$784$	3.82622	0.136651
$785$	2.79723	0.0998376
$786$	0	0
$787$	−30.4455	−1.08526	−0.542632	−	0.839971i	$-0.682572\pi$
$787$	−30.4455	−1.08526	−0.542632	+	0.839971i	$0.682572\pi$
$788$	−25.5603	−0.910547
$789$	0	0
$790$	−0.201800	−0.00717973
$791$	−0.331384	−0.0117827
$792$	0	0
$793$	92.3554	3.27963
$794$	3.35610	0.119103
$795$	0	0
$796$	−40.7307	−1.44366
$797$	31.4005	1.11226	0.556131	−	0.831095i	$-0.312285\pi$
$797$	31.4005	1.11226	0.556131	+	0.831095i	$0.312285\pi$
$798$	0	0
$799$	21.6497	0.765912
$800$	9.95269	0.351881
$801$	0	0
$802$	3.34305	0.118047
$803$	−23.9324	−0.844554
$804$	0	0
$805$	1.62445	0.0572545
$806$	0.187546	0.00660602
$807$	0	0
$808$	−10.4430	−0.367382
$809$	35.4110	1.24499	0.622493	−	0.782626i	$-0.286120\pi$
$809$	35.4110	1.24499	0.622493	+	0.782626i	$0.286120\pi$
$810$	0	0
$811$	−33.1853	−1.16529	−0.582647	−	0.812725i	$-0.697983\pi$
$811$	−33.1853	−1.16529	−0.582647	+	0.812725i	$0.697983\pi$
$812$	−7.24014	−0.254079
$813$	0	0
$814$	−4.56509	−0.160006
$815$	2.74758	0.0962436
$816$	0	0
$817$	−24.2651	−0.848928
$818$	2.73982	0.0957956
$819$	0	0
$820$	−0.291926	−0.0101945
$821$	40.6379	1.41827	0.709136	−	0.705072i	$-0.249085\pi$
$821$	40.6379	1.41827	0.709136	+	0.705072i	$0.249085\pi$
$822$	0	0
$823$	−47.6335	−1.66040	−0.830200	−	0.557465i	$-0.811774\pi$
$823$	−47.6335	−1.66040	−0.830200	+	0.557465i	$0.811774\pi$
$824$	8.54641	0.297728
$825$	0	0
$826$	−1.25479	−0.0436597
$827$	−33.7637	−1.17408	−0.587039	−	0.809559i	$-0.699706\pi$
$827$	−33.7637	−1.17408	−0.587039	+	0.809559i	$0.699706\pi$
$828$	0	0
$829$	27.3714	0.950649	0.475325	−	0.879811i	$-0.342331\pi$
$829$	27.3714	0.950649	0.475325	+	0.879811i	$0.342331\pi$
$830$	0.344937	0.0119729
$831$	0	0
$832$	44.0089	1.52574
$833$	6.47770	0.224439
$834$	0	0
$835$	−1.11442	−0.0385661
$836$	−31.9975	−1.10666
$837$	0	0
$838$	−2.20638	−0.0762182
$839$	−19.6027	−0.676761	−0.338380	−	0.941009i	$-0.609879\pi$
$839$	−19.6027	−0.676761	−0.338380	+	0.941009i	$0.609879\pi$
$840$	0	0
$841$	−15.5052	−0.534661
$842$	4.25633	0.146683
$843$	0	0
$844$	−4.27305	−0.147085
$845$	4.79680	0.165015
$846$	0	0
$847$	2.28621	0.0785553
$848$	25.3371	0.870080
$849$	0	0
$850$	5.47846	0.187910
$851$	71.3609	2.44622
$852$	0	0
$853$	16.1746	0.553809	0.276905	−	0.960897i	$-0.410691\pi$
$853$	16.1746	0.553809	0.276905	+	0.960897i	$0.410691\pi$
$854$	−2.61708	−0.0895545
$855$	0	0
$856$	−12.2308	−0.418041
$857$	−21.3510	−0.729335	−0.364668	−	0.931138i	$-0.618817\pi$
$857$	−21.3510	−0.729335	−0.364668	+	0.931138i	$0.618817\pi$
$858$	0	0
$859$	0.615958	0.0210162	0.0105081	−	0.999945i	$-0.496655\pi$
$859$	0.615958	0.0210162	0.0105081	+	0.999945i	$0.496655\pi$
$860$	−1.79435	−0.0611869
$861$	0	0
$862$	−1.27973	−0.0435879
$863$	50.4704	1.71803	0.859016	−	0.511949i	$-0.171076\pi$
$863$	50.4704	1.71803	0.859016	+	0.511949i	$0.171076\pi$
$864$	0	0
$865$	−0.0962396	−0.00327225
$866$	2.41278	0.0819897
$867$	0	0
$868$	0.359883	0.0122152
$869$	−16.9212	−0.574013
$870$	0	0
$871$	−38.2475	−1.29597
$872$	3.84497	0.130207
$873$	0	0
$874$	−7.38630	−0.249845
$875$	2.05475	0.0694631
$876$	0	0
$877$	40.6889	1.37397	0.686983	−	0.726673i	$-0.258935\pi$
$877$	40.6889	1.37397	0.686983	+	0.726673i	$0.258935\pi$
$878$	−1.54470	−0.0521312
$879$	0	0
$880$	−2.33069	−0.0785676
$881$	−34.1963	−1.15210	−0.576052	−	0.817413i	$-0.695407\pi$
$881$	−34.1963	−1.15210	−0.576052	+	0.817413i	$0.695407\pi$
$882$	0	0
$883$	−20.2286	−0.680745	−0.340373	−	0.940291i	$-0.610553\pi$
$883$	−20.2286	−0.680745	−0.340373	+	0.940291i	$0.610553\pi$
$884$	76.8621	2.58515
$885$	0	0
$886$	−4.98200	−0.167374
$887$	−43.6034	−1.46406	−0.732030	−	0.681273i	$-0.761427\pi$
$887$	−43.6034	−1.46406	−0.732030	+	0.681273i	$0.761427\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−0.524206	−0.0175714
$891$	0	0
$892$	2.17452	0.0728084
$893$	−18.3815	−0.615113
$894$	0	0
$895$	−2.08334	−0.0696385
$896$	−5.26235	−0.175803
$897$	0	0
$898$	0.194609	0.00649418
$899$	−0.670782	−0.0223718
$900$	0	0
$901$	42.8952	1.42904
$902$	0.361480	0.0120360
$903$	0	0
$904$	0.224493	0.00746652
$905$	−4.59361	−0.152697
$906$	0	0
$907$	24.4877	0.813102	0.406551	−	0.913628i	$-0.366731\pi$
$907$	24.4877	0.813102	0.406551	+	0.913628i	$0.366731\pi$
$908$	−15.7728	−0.523439
$909$	0	0
$910$	−0.211944	−0.00702587
$911$	−30.1286	−0.998206	−0.499103	−	0.866543i	$-0.666337\pi$
$911$	−30.1286	−0.998206	−0.499103	+	0.866543i	$0.666337\pi$
$912$	0	0
$913$	28.9234	0.957225
$914$	3.74906	0.124008
$915$	0	0
$916$	−0.717200	−0.0236970
$917$	−9.55770	−0.315623
$918$	0	0
$919$	−37.4612	−1.23573	−0.617865	−	0.786284i	$-0.712002\pi$
$919$	−37.4612	−1.23573	−0.617865	+	0.786284i	$0.712002\pi$
$920$	−1.10047	−0.0362814
$921$	0	0
$922$	−3.78713	−0.124722
$923$	−42.5335	−1.40001
$924$	0	0
$925$	44.9386	1.47757
$926$	−0.610073	−0.0200482
$927$	0	0
$928$	7.37511	0.242100
$929$	10.3163	0.338468	0.169234	−	0.985576i	$-0.445871\pi$
$929$	10.3163	0.338468	0.169234	+	0.985576i	$0.445871\pi$
$930$	0	0
$931$	−5.49983	−0.180250
$932$	−56.3857	−1.84698
$933$	0	0
$934$	−0.857216	−0.0280490
$935$	−3.94581	−0.129042
$936$	0	0
$937$	24.4343	0.798235	0.399117	−	0.916900i	$-0.369317\pi$
$937$	24.4343	0.798235	0.399117	+	0.916900i	$0.369317\pi$
$938$	1.08382	0.0353880
$939$	0	0
$940$	−1.35927	−0.0443346
$941$	−10.3783	−0.338324	−0.169162	−	0.985588i	$-0.554106\pi$
$941$	−10.3783	−0.338324	−0.169162	+	0.985588i	$0.554106\pi$
$942$	0	0
$943$	−5.65060	−0.184009
$944$	−28.1422	−0.915952
$945$	0	0
$946$	2.22187	0.0722391
$947$	−20.3460	−0.661157	−0.330579	−	0.943778i	$-0.607244\pi$
$947$	−20.3460	−0.661157	−0.330579	+	0.943778i	$0.607244\pi$
$948$	0	0
$949$	−48.8101	−1.58444
$950$	−4.65143	−0.150912
$951$	0	0
$952$	−4.38825	−0.142224
$953$	−25.1505	−0.814704	−0.407352	−	0.913271i	$-0.633548\pi$
$953$	−25.1505	−0.814704	−0.407352	+	0.913271i	$0.633548\pi$
$954$	0	0
$955$	1.69290	0.0547810
$956$	−56.1511	−1.81606
$957$	0	0
$958$	−6.65432	−0.214991
$959$	19.1217	0.617471
$960$	0	0
$961$	−30.9667	−0.998924
$962$	−9.31051	−0.300183
$963$	0	0
$964$	4.19097	0.134982
$965$	1.46721	0.0472311
$966$	0	0
$967$	−27.8536	−0.895711	−0.447855	−	0.894106i	$-0.647812\pi$
$967$	−27.8536	−0.895711	−0.447855	+	0.894106i	$0.647812\pi$
$968$	−1.54877	−0.0497794
$969$	0	0
$970$	0.522193	0.0167666
$971$	2.52631	0.0810732	0.0405366	−	0.999178i	$-0.487093\pi$
$971$	2.52631	0.0810732	0.0405366	+	0.999178i	$0.487093\pi$
$972$	0	0
$973$	19.5559	0.626932
$974$	0.186539	0.00597708
$975$	0	0
$976$	−58.6954	−1.87880
$977$	−52.7486	−1.68758	−0.843789	−	0.536674i	$-0.819680\pi$
$977$	−52.7486	−1.68758	−0.843789	+	0.536674i	$0.819680\pi$
$978$	0	0
$979$	−43.9554	−1.40482
$980$	−0.406701	−0.0129916
$981$	0	0
$982$	−4.48092	−0.142992
$983$	7.36372	0.234866	0.117433	−	0.993081i	$-0.462533\pi$
$983$	7.36372	0.234866	0.117433	+	0.993081i	$0.462533\pi$
$984$	0	0
$985$	2.67617	0.0852698
$986$	4.05963	0.129285
$987$	0	0
$988$	−65.2590	−2.07616
$989$	−34.7319	−1.10441
$990$	0	0
$991$	31.3967	0.997350	0.498675	−	0.866789i	$-0.333820\pi$
$991$	31.3967	0.997350	0.498675	+	0.866789i	$0.333820\pi$
$992$	−0.366592	−0.0116393
$993$	0	0
$994$	1.20527	0.0382289
$995$	4.26452	0.135194
$996$	0	0
$997$	−16.1969	−0.512960	−0.256480	−	0.966550i	$-0.582563\pi$
$997$	−16.1969	−0.512960	−0.256480	+	0.966550i	$0.582563\pi$
$998$	5.04044	0.159552
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.t.1.8		16
3.2	odd	2		889.2.a.c.1.9	✓	16
21.20	even	2		6223.2.a.k.1.9		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.9	✓	16	3.2	odd	2
6223.2.a.k.1.9		16	21.20	even	2
8001.2.a.t.1.8		16	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-0.170601 q^{2} -1.97090 q^{4} +0.206353 q^{5} -1.00000 q^{7} +0.677439 q^{8} -0.0352041 q^{10} -2.95191 q^{11} -6.02043 q^{13} +0.170601 q^{14} +3.82622 q^{16} +6.47770 q^{17} -5.49983 q^{19} -0.406701 q^{20} +0.503600 q^{22} -7.87219 q^{23} -4.95742 q^{25} +1.02709 q^{26} +1.97090 q^{28} -3.67353 q^{29} +0.182599 q^{31} -2.00764 q^{32} -1.10510 q^{34} -0.206353 q^{35} -9.06492 q^{37} +0.938277 q^{38} +0.139792 q^{40} +0.717792 q^{41} +4.41197 q^{43} +5.81791 q^{44} +1.34301 q^{46} +3.34219 q^{47} +1.00000 q^{49} +0.845741 q^{50} +11.8656 q^{52} +6.62197 q^{53} -0.609137 q^{55} -0.677439 q^{56} +0.626708 q^{58} -7.35511 q^{59} -15.3403 q^{61} -0.0311515 q^{62} -7.30993 q^{64} -1.24234 q^{65} +6.35295 q^{67} -12.7669 q^{68} +0.0352041 q^{70} +7.06485 q^{71} +8.10741 q^{73} +1.54649 q^{74} +10.8396 q^{76} +2.95191 q^{77} +5.73229 q^{79} +0.789553 q^{80} -0.122456 q^{82} -9.79819 q^{83} +1.33670 q^{85} -0.752687 q^{86} -1.99974 q^{88} +14.8905 q^{89} +6.02043 q^{91} +15.5153 q^{92} -0.570182 q^{94} -1.13491 q^{95} -14.8333 q^{97} -0.170601 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28}+ \cdots + 2 q^{98}+O(q^{100}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

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