LMFDB - Embedded newform 8001.2.a.m.1.6

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 = [11,2,0,12,-1] 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:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 15 x^{9} + 25 x^{8} + 88 x^{7} - 112 x^{6} - 247 x^{5} + 215 x^{4} + 313 x^{3} + \cdots + 57 $ x^11 - 2x^10 - 15x^9 + 25x^8 + 88x^7 - 112x^6 - 247x^5 + 215x^4 + 313x^3 - 170x^2 - 121x + 57
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.455441$ 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)

$f(q)$	$=$	\(q+0.455441 q^{2} -1.79257 q^{4} +0.749481 q^{5} -1.00000 q^{7} -1.72729 q^{8} +0.341344 q^{10} +1.95920 q^{11} +2.51893 q^{13} -0.455441 q^{14} +2.79847 q^{16} +1.75736 q^{17} -3.76981 q^{19} -1.34350 q^{20} +0.892299 q^{22} -7.74151 q^{23} -4.43828 q^{25} +1.14723 q^{26} +1.79257 q^{28} -1.80755 q^{29} +7.78196 q^{31} +4.72912 q^{32} +0.800374 q^{34} -0.749481 q^{35} +2.63088 q^{37} -1.71693 q^{38} -1.29457 q^{40} +8.09399 q^{41} -11.7597 q^{43} -3.51201 q^{44} -3.52580 q^{46} +3.25780 q^{47} +1.00000 q^{49} -2.02137 q^{50} -4.51538 q^{52} -0.297989 q^{53} +1.46838 q^{55} +1.72729 q^{56} -0.823230 q^{58} -0.440342 q^{59} +4.23592 q^{61} +3.54422 q^{62} -3.44310 q^{64} +1.88789 q^{65} -13.5402 q^{67} -3.15020 q^{68} -0.341344 q^{70} +1.44289 q^{71} -2.90207 q^{73} +1.19821 q^{74} +6.75767 q^{76} -1.95920 q^{77} +2.11951 q^{79} +2.09740 q^{80} +3.68633 q^{82} +4.47544 q^{83} +1.31711 q^{85} -5.35583 q^{86} -3.38411 q^{88} -2.94756 q^{89} -2.51893 q^{91} +13.8772 q^{92} +1.48374 q^{94} -2.82540 q^{95} -9.80554 q^{97} +0.455441 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8} - 12 q^{10} + 7 q^{11} - 24 q^{13} - 2 q^{14} - 6 q^{16} + 15 q^{17} - 19 q^{19} - 3 q^{20} - 3 q^{22} + 11 q^{23} + 10 q^{25} - 10 q^{26} - 12 q^{28}+ \cdots + 2 q^{98}+O(q^{100}) $ 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8 - 12 * q^10 + 7 * q^11 - 24 * q^13 - 2 * q^14 - 6 * q^16 + 15 * q^17 - 19 * q^19 - 3 * q^20 - 3 * q^22 + 11 * q^23 + 10 * q^25 - 10 * q^26 - 12 * q^28 + 10 * q^29 - 20 * q^31 + 27 * q^32 - 9 * q^34 + q^35 - 22 * q^37 - 8 * q^38 - 29 * q^40 - 9 * q^41 - 17 * q^43 + 9 * q^44 - 18 * q^46 + 7 * q^47 + 11 * q^49 + 47 * q^50 - 66 * q^52 + 28 * q^53 - 24 * q^55 - 15 * q^56 - 39 * q^58 - 35 * q^59 - 6 * q^61 - 18 * q^62 + 11 * q^64 + 43 * q^65 - 22 * q^67 + 12 * q^68 + 12 * q^70 + 22 * q^71 - 29 * q^73 - 14 * q^74 + 10 * q^76 - 7 * q^77 - 20 * q^79 - 66 * q^80 - 24 * q^82 - 17 * q^83 - 50 * q^85 + 12 * q^86 + 2 * q^88 + q^89 + 24 * q^91 + 22 * q^92 + q^94 - 10 * q^95 - 45 * 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 $

$2$	0.455441	0.322045	0.161023	−	0.986951i	$-0.448521\pi$
$2$	0.455441	0.322045	0.161023	+	0.986951i	$0.448521\pi$
$3$	0	0
$3$	0	0
$4$	−1.79257	−0.896287
$5$	0.749481	0.335178	0.167589	−	0.985857i	$-0.446402\pi$
$5$	0.749481	0.335178	0.167589	+	0.985857i	$0.446402\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.72729	−0.610690
$9$	0	0
$10$	0.341344	0.107942
$11$	1.95920	0.590721	0.295361	−	0.955386i	$-0.404560\pi$
$11$	1.95920	0.590721	0.295361	+	0.955386i	$0.404560\pi$
$12$	0	0
$13$	2.51893	0.698627	0.349313	−	0.937006i	$-0.386415\pi$
$13$	2.51893	0.698627	0.349313	+	0.937006i	$0.386415\pi$
$14$	−0.455441	−0.121722
$15$	0	0
$16$	2.79847	0.699617
$17$	1.75736	0.426223	0.213111	−	0.977028i	$-0.431640\pi$
$17$	1.75736	0.426223	0.213111	+	0.977028i	$0.431640\pi$
$18$	0	0
$19$	−3.76981	−0.864854	−0.432427	−	0.901669i	$-0.642343\pi$
$19$	−3.76981	−0.864854	−0.432427	+	0.901669i	$0.642343\pi$
$20$	−1.34350	−0.300416
$21$	0	0
$22$	0.892299	0.190239
$23$	−7.74151	−1.61422	−0.807108	−	0.590404i	$-0.798969\pi$
$23$	−7.74151	−1.61422	−0.807108	+	0.590404i	$0.798969\pi$
$24$	0	0
$25$	−4.43828	−0.887656
$26$	1.14723	0.224989
$27$	0	0
$28$	1.79257	0.338765
$29$	−1.80755	−0.335653	−0.167826	−	0.985817i	$-0.553675\pi$
$29$	−1.80755	−0.335653	−0.167826	+	0.985817i	$0.553675\pi$
$30$	0	0
$31$	7.78196	1.39768	0.698841	−	0.715277i	$-0.253700\pi$
$31$	7.78196	1.39768	0.698841	+	0.715277i	$0.253700\pi$
$32$	4.72912	0.835998
$33$	0	0
$34$	0.800374	0.137263
$35$	−0.749481	−0.126685
$36$	0	0
$37$	2.63088	0.432514	0.216257	−	0.976336i	$-0.430615\pi$
$37$	2.63088	0.432514	0.216257	+	0.976336i	$0.430615\pi$
$38$	−1.71693	−0.278522
$39$	0	0
$40$	−1.29457	−0.204690
$41$	8.09399	1.26407	0.632034	−	0.774940i	$-0.282220\pi$
$41$	8.09399	1.26407	0.632034	+	0.774940i	$0.282220\pi$
$42$	0	0
$43$	−11.7597	−1.79333	−0.896667	−	0.442706i	$-0.854019\pi$
$43$	−11.7597	−1.79333	−0.896667	+	0.442706i	$0.854019\pi$
$44$	−3.51201	−0.529456
$45$	0	0
$46$	−3.52580	−0.519851
$47$	3.25780	0.475199	0.237600	−	0.971363i	$-0.423639\pi$
$47$	3.25780	0.475199	0.237600	+	0.971363i	$0.423639\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.02137	−0.285865
$51$	0	0
$52$	−4.51538	−0.626170
$53$	−0.297989	−0.0409319	−0.0204659	−	0.999791i	$-0.506515\pi$
$53$	−0.297989	−0.0409319	−0.0204659	+	0.999791i	$0.506515\pi$
$54$	0	0
$55$	1.46838	0.197997
$56$	1.72729	0.230819
$57$	0	0
$58$	−0.823230	−0.108095
$59$	−0.440342	−0.0573277	−0.0286638	−	0.999589i	$-0.509125\pi$
$59$	−0.440342	−0.0573277	−0.0286638	+	0.999589i	$0.509125\pi$
$60$	0	0
$61$	4.23592	0.542354	0.271177	−	0.962529i	$-0.412587\pi$
$61$	4.23592	0.542354	0.271177	+	0.962529i	$0.412587\pi$
$62$	3.54422	0.450117
$63$	0	0
$64$	−3.44310	−0.430388
$65$	1.88789	0.234164
$66$	0	0
$67$	−13.5402	−1.65420	−0.827101	−	0.562054i	$-0.810011\pi$
$67$	−13.5402	−1.65420	−0.827101	+	0.562054i	$0.810011\pi$
$68$	−3.15020	−0.382018
$69$	0	0
$70$	−0.341344	−0.0407984
$71$	1.44289	0.171239	0.0856196	−	0.996328i	$-0.472713\pi$
$71$	1.44289	0.171239	0.0856196	+	0.996328i	$0.472713\pi$
$72$	0	0
$73$	−2.90207	−0.339662	−0.169831	−	0.985473i	$-0.554322\pi$
$73$	−2.90207	−0.339662	−0.169831	+	0.985473i	$0.554322\pi$
$74$	1.19821	0.139289
$75$	0	0
$76$	6.75767	0.775158
$77$	−1.95920	−0.223272
$78$	0	0
$79$	2.11951	0.238464	0.119232	−	0.992866i	$-0.461957\pi$
$79$	2.11951	0.238464	0.119232	+	0.992866i	$0.461957\pi$
$80$	2.09740	0.234496
$81$	0	0
$82$	3.68633	0.407087
$83$	4.47544	0.491243	0.245622	−	0.969366i	$-0.421008\pi$
$83$	4.47544	0.491243	0.245622	+	0.969366i	$0.421008\pi$
$84$	0	0
$85$	1.31711	0.142860
$86$	−5.35583	−0.577535
$87$	0	0
$88$	−3.38411	−0.360747
$89$	−2.94756	−0.312441	−0.156220	−	0.987722i	$-0.549931\pi$
$89$	−2.94756	−0.312441	−0.156220	+	0.987722i	$0.549931\pi$
$90$	0	0
$91$	−2.51893	−0.264056
$92$	13.8772	1.44680
$93$	0	0
$94$	1.48374	0.153036
$95$	−2.82540	−0.289880
$96$	0	0
$97$	−9.80554	−0.995602	−0.497801	−	0.867291i	$-0.665859\pi$
$97$	−9.80554	−0.995602	−0.497801	+	0.867291i	$0.665859\pi$
$98$	0.455441	0.0460065
$99$	0	0
$100$	7.95594	0.795594
$101$	17.0317	1.69472	0.847360	−	0.531019i	$-0.178191\pi$
$101$	17.0317	1.69472	0.847360	+	0.531019i	$0.178191\pi$
$102$	0	0
$103$	9.72210	0.957947	0.478974	−	0.877829i	$-0.341009\pi$
$103$	9.72210	0.957947	0.478974	+	0.877829i	$0.341009\pi$
$104$	−4.35094	−0.426644
$105$	0	0
$106$	−0.135716	−0.0131819
$107$	8.65396	0.836610	0.418305	−	0.908307i	$-0.362624\pi$
$107$	8.65396	0.836610	0.418305	+	0.908307i	$0.362624\pi$
$108$	0	0
$109$	−12.5836	−1.20529	−0.602647	−	0.798008i	$-0.705887\pi$
$109$	−12.5836	−1.20529	−0.602647	+	0.798008i	$0.705887\pi$
$110$	0.668761	0.0637639
$111$	0	0
$112$	−2.79847	−0.264430
$113$	−4.54886	−0.427921	−0.213960	−	0.976842i	$-0.568636\pi$
$113$	−4.54886	−0.427921	−0.213960	+	0.976842i	$0.568636\pi$
$114$	0	0
$115$	−5.80211	−0.541050
$116$	3.24016	0.300841
$117$	0	0
$118$	−0.200550	−0.0184621
$119$	−1.75736	−0.161097
$120$	0	0
$121$	−7.16153	−0.651049
$122$	1.92921	0.174663
$123$	0	0
$124$	−13.9497	−1.25272
$125$	−7.07381	−0.632701
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−11.0264	−0.974603
$129$	0	0
$130$	0.859823	0.0754115
$131$	−3.88980	−0.339853	−0.169927	−	0.985457i	$-0.554353\pi$
$131$	−3.88980	−0.339853	−0.169927	+	0.985457i	$0.554353\pi$
$132$	0	0
$133$	3.76981	0.326884
$134$	−6.16677	−0.532727
$135$	0	0
$136$	−3.03548	−0.260290
$137$	−12.5423	−1.07156	−0.535779	−	0.844358i	$-0.679982\pi$
$137$	−12.5423	−1.07156	−0.535779	+	0.844358i	$0.679982\pi$
$138$	0	0
$139$	−21.1389	−1.79298	−0.896488	−	0.443069i	$-0.853890\pi$
$139$	−21.1389	−1.79298	−0.896488	+	0.443069i	$0.853890\pi$
$140$	1.34350	0.113546
$141$	0	0
$142$	0.657149	0.0551468
$143$	4.93510	0.412694
$144$	0	0
$145$	−1.35472	−0.112504
$146$	−1.32172	−0.109386
$147$	0	0
$148$	−4.71605	−0.387657
$149$	18.2486	1.49498	0.747492	−	0.664271i	$-0.231258\pi$
$149$	18.2486	1.49498	0.747492	+	0.664271i	$0.231258\pi$
$150$	0	0
$151$	−4.00291	−0.325752	−0.162876	−	0.986647i	$-0.552077\pi$
$151$	−4.00291	−0.325752	−0.162876	+	0.986647i	$0.552077\pi$
$152$	6.51157	0.528158
$153$	0	0
$154$	−0.892299	−0.0719035
$155$	5.83243	0.468472
$156$	0	0
$157$	−3.26752	−0.260776	−0.130388	−	0.991463i	$-0.541622\pi$
$157$	−3.26752	−0.260776	−0.130388	+	0.991463i	$0.541622\pi$
$158$	0.965313	0.0767961
$159$	0	0
$160$	3.54439	0.280208
$161$	7.74151	0.610116
$162$	0	0
$163$	−5.99201	−0.469331	−0.234665	−	0.972076i	$-0.575399\pi$
$163$	−5.99201	−0.469331	−0.234665	+	0.972076i	$0.575399\pi$
$164$	−14.5091	−1.13297
$165$	0	0
$166$	2.03830	0.158202
$167$	21.2819	1.64684	0.823421	−	0.567431i	$-0.192063\pi$
$167$	21.2819	1.64684	0.823421	+	0.567431i	$0.192063\pi$
$168$	0	0
$169$	−6.65497	−0.511921
$170$	0.599865	0.0460075
$171$	0	0
$172$	21.0801	1.60734
$173$	−2.41503	−0.183612	−0.0918058	−	0.995777i	$-0.529264\pi$
$173$	−2.41503	−0.183612	−0.0918058	+	0.995777i	$0.529264\pi$
$174$	0	0
$175$	4.43828	0.335502
$176$	5.48276	0.413279
$177$	0	0
$178$	−1.34244	−0.100620
$179$	−14.3890	−1.07548	−0.537741	−	0.843110i	$-0.680722\pi$
$179$	−14.3890	−1.07548	−0.537741	+	0.843110i	$0.680722\pi$
$180$	0	0
$181$	6.74044	0.501013	0.250506	−	0.968115i	$-0.419403\pi$
$181$	6.74044	0.501013	0.250506	+	0.968115i	$0.419403\pi$
$182$	−1.14723	−0.0850380
$183$	0	0
$184$	13.3719	0.985786
$185$	1.97179	0.144969
$186$	0	0
$187$	3.44302	0.251779
$188$	−5.83985	−0.425915
$189$	0	0
$190$	−1.28680	−0.0933545
$191$	−17.9119	−1.29606	−0.648031	−	0.761614i	$-0.724407\pi$
$191$	−17.9119	−1.29606	−0.648031	+	0.761614i	$0.724407\pi$
$192$	0	0
$193$	8.61234	0.619930	0.309965	−	0.950748i	$-0.399683\pi$
$193$	8.61234	0.619930	0.309965	+	0.950748i	$0.399683\pi$
$194$	−4.46584	−0.320629
$195$	0	0
$196$	−1.79257	−0.128041
$197$	26.3709	1.87885	0.939423	−	0.342759i	$-0.111361\pi$
$197$	26.3709	1.87885	0.939423	+	0.342759i	$0.111361\pi$
$198$	0	0
$199$	−12.3460	−0.875185	−0.437593	−	0.899173i	$-0.644169\pi$
$199$	−12.3460	−0.875185	−0.437593	+	0.899173i	$0.644169\pi$
$200$	7.66620	0.542082
$201$	0	0
$202$	7.75694	0.545776
$203$	1.80755	0.126865
$204$	0	0
$205$	6.06629	0.423688
$206$	4.42784	0.308502
$207$	0	0
$208$	7.04916	0.488771
$209$	−7.38582	−0.510888
$210$	0	0
$211$	−28.4545	−1.95889	−0.979445	−	0.201711i	$-0.935350\pi$
$211$	−28.4545	−1.95889	−0.979445	+	0.201711i	$0.935350\pi$
$212$	0.534167	0.0366867
$213$	0	0
$214$	3.94136	0.269426
$215$	−8.81365	−0.601086
$216$	0	0
$217$	−7.78196	−0.528274
$218$	−5.73110	−0.388159
$219$	0	0
$220$	−2.63219	−0.177462
$221$	4.42668	0.297771
$222$	0	0
$223$	5.39195	0.361071	0.180536	−	0.983568i	$-0.442217\pi$
$223$	5.39195	0.361071	0.180536	+	0.983568i	$0.442217\pi$
$224$	−4.72912	−0.315978
$225$	0	0
$226$	−2.07174	−0.137810
$227$	−10.8739	−0.721729	−0.360865	−	0.932618i	$-0.617518\pi$
$227$	−10.8739	−0.721729	−0.360865	+	0.932618i	$0.617518\pi$
$228$	0	0
$229$	3.50976	0.231932	0.115966	−	0.993253i	$-0.463004\pi$
$229$	3.50976	0.231932	0.115966	+	0.993253i	$0.463004\pi$
$230$	−2.64252	−0.174243
$231$	0	0
$232$	3.12216	0.204980
$233$	−2.48327	−0.162685	−0.0813424	−	0.996686i	$-0.525921\pi$
$233$	−2.48327	−0.162685	−0.0813424	+	0.996686i	$0.525921\pi$
$234$	0	0
$235$	2.44166	0.159276
$236$	0.789346	0.0513821
$237$	0	0
$238$	−0.800374	−0.0518805
$239$	−21.1743	−1.36965	−0.684826	−	0.728706i	$-0.740122\pi$
$239$	−21.1743	−1.36965	−0.684826	+	0.728706i	$0.740122\pi$
$240$	0	0
$241$	−17.1342	−1.10371	−0.551854	−	0.833941i	$-0.686080\pi$
$241$	−17.1342	−1.10371	−0.551854	+	0.833941i	$0.686080\pi$
$242$	−3.26165	−0.209667
$243$	0	0
$244$	−7.59321	−0.486105
$245$	0.749481	0.0478826
$246$	0	0
$247$	−9.49591	−0.604210
$248$	−13.4417	−0.853550
$249$	0	0
$250$	−3.22170	−0.203758
$251$	2.98515	0.188421	0.0942104	−	0.995552i	$-0.469967\pi$
$251$	2.98515	0.188421	0.0942104	+	0.995552i	$0.469967\pi$
$252$	0	0
$253$	−15.1672	−0.953552
$254$	0.455441	0.0285769
$255$	0	0
$256$	1.86435	0.116522
$257$	14.2653	0.889845	0.444923	−	0.895569i	$-0.353231\pi$
$257$	14.2653	0.889845	0.444923	+	0.895569i	$0.353231\pi$
$258$	0	0
$259$	−2.63088	−0.163475
$260$	−3.38419	−0.209878
$261$	0	0
$262$	−1.77157	−0.109448
$263$	6.23379	0.384392	0.192196	−	0.981357i	$-0.438439\pi$
$263$	6.23379	0.384392	0.192196	+	0.981357i	$0.438439\pi$
$264$	0	0
$265$	−0.223337	−0.0137195
$266$	1.71693	0.105271
$267$	0	0
$268$	24.2718	1.48264
$269$	−9.49493	−0.578916	−0.289458	−	0.957191i	$-0.593475\pi$
$269$	−9.49493	−0.578916	−0.289458	+	0.957191i	$0.593475\pi$
$270$	0	0
$271$	−24.8348	−1.50860	−0.754302	−	0.656528i	$-0.772025\pi$
$271$	−24.8348	−1.50860	−0.754302	+	0.656528i	$0.772025\pi$
$272$	4.91792	0.298193
$273$	0	0
$274$	−5.71225	−0.345090
$275$	−8.69548	−0.524357
$276$	0	0
$277$	16.1475	0.970208	0.485104	−	0.874457i	$-0.338782\pi$
$277$	16.1475	0.970208	0.485104	+	0.874457i	$0.338782\pi$
$278$	−9.62750	−0.577419
$279$	0	0
$280$	1.29457	0.0773655
$281$	9.45191	0.563854	0.281927	−	0.959436i	$-0.409026\pi$
$281$	9.45191	0.563854	0.281927	+	0.959436i	$0.409026\pi$
$282$	0	0
$283$	14.1920	0.843625	0.421812	−	0.906683i	$-0.361394\pi$
$283$	14.1920	0.843625	0.421812	+	0.906683i	$0.361394\pi$
$284$	−2.58648	−0.153479
$285$	0	0
$286$	2.24764	0.132906
$287$	−8.09399	−0.477773
$288$	0	0
$289$	−13.9117	−0.818334
$290$	−0.616995	−0.0362312
$291$	0	0
$292$	5.20218	0.304434
$293$	−18.0263	−1.05311	−0.526554	−	0.850141i	$-0.676516\pi$
$293$	−18.0263	−1.05311	−0.526554	+	0.850141i	$0.676516\pi$
$294$	0	0
$295$	−0.330028	−0.0192150
$296$	−4.54430	−0.264132
$297$	0	0
$298$	8.31115	0.481452
$299$	−19.5004	−1.12773
$300$	0	0
$301$	11.7597	0.677817
$302$	−1.82309	−0.104907
$303$	0	0
$304$	−10.5497	−0.605067
$305$	3.17474	0.181785
$306$	0	0
$307$	16.0943	0.918552	0.459276	−	0.888294i	$-0.348109\pi$
$307$	16.0943	0.918552	0.459276	+	0.888294i	$0.348109\pi$
$308$	3.51201	0.200115
$309$	0	0
$310$	2.65633	0.150869
$311$	−19.3077	−1.09484	−0.547421	−	0.836857i	$-0.684390\pi$
$311$	−19.3077	−1.09484	−0.547421	+	0.836857i	$0.684390\pi$
$312$	0	0
$313$	−30.8613	−1.74438	−0.872191	−	0.489166i	$-0.837301\pi$
$313$	−30.8613	−1.74438	−0.872191	+	0.489166i	$0.837301\pi$
$314$	−1.48816	−0.0839817
$315$	0	0
$316$	−3.79938	−0.213732
$317$	−30.6756	−1.72291	−0.861457	−	0.507831i	$-0.830447\pi$
$317$	−30.6756	−1.72291	−0.861457	+	0.507831i	$0.830447\pi$
$318$	0	0
$319$	−3.54135	−0.198277
$320$	−2.58054	−0.144257
$321$	0	0
$322$	3.52580	0.196485
$323$	−6.62492	−0.368621
$324$	0	0
$325$	−11.1797	−0.620140
$326$	−2.72901	−0.151146
$327$	0	0
$328$	−13.9807	−0.771954
$329$	−3.25780	−0.179608
$330$	0	0
$331$	−10.2450	−0.563114	−0.281557	−	0.959544i	$-0.590851\pi$
$331$	−10.2450	−0.563114	−0.281557	+	0.959544i	$0.590851\pi$
$332$	−8.02255	−0.440295
$333$	0	0
$334$	9.69263	0.530357
$335$	−10.1481	−0.554452
$336$	0	0
$337$	−1.30395	−0.0710307	−0.0355154	−	0.999369i	$-0.511307\pi$
$337$	−1.30395	−0.0710307	−0.0355154	+	0.999369i	$0.511307\pi$
$338$	−3.03094	−0.164862
$339$	0	0
$340$	−2.36101	−0.128044
$341$	15.2464	0.825640
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	20.3124	1.09517
$345$	0	0
$346$	−1.09990	−0.0591312
$347$	−18.6437	−1.00085	−0.500424	−	0.865780i	$-0.666823\pi$
$347$	−18.6437	−1.00085	−0.500424	+	0.865780i	$0.666823\pi$
$348$	0	0
$349$	7.47368	0.400057	0.200029	−	0.979790i	$-0.435897\pi$
$349$	7.47368	0.400057	0.200029	+	0.979790i	$0.435897\pi$
$350$	2.02137	0.108047
$351$	0	0
$352$	9.26529	0.493842
$353$	10.4557	0.556502	0.278251	−	0.960508i	$-0.410245\pi$
$353$	10.4557	0.556502	0.278251	+	0.960508i	$0.410245\pi$
$354$	0	0
$355$	1.08142	0.0573956
$356$	5.28372	0.280037
$357$	0	0
$358$	−6.55332	−0.346354
$359$	−35.8946	−1.89444	−0.947222	−	0.320579i	$-0.896123\pi$
$359$	−35.8946	−1.89444	−0.947222	+	0.320579i	$0.896123\pi$
$360$	0	0
$361$	−4.78851	−0.252027
$362$	3.06987	0.161349
$363$	0	0
$364$	4.51538	0.236670
$365$	−2.17505	−0.113847
$366$	0	0
$367$	3.15632	0.164758	0.0823792	−	0.996601i	$-0.473748\pi$
$367$	3.15632	0.164758	0.0823792	+	0.996601i	$0.473748\pi$
$368$	−21.6644	−1.12933
$369$	0	0
$370$	0.898035	0.0466866
$371$	0.297989	0.0154708
$372$	0	0
$373$	−4.08938	−0.211740	−0.105870	−	0.994380i	$-0.533763\pi$
$373$	−4.08938	−0.211740	−0.105870	+	0.994380i	$0.533763\pi$
$374$	1.56809	0.0810841
$375$	0	0
$376$	−5.62718	−0.290200
$377$	−4.55309	−0.234496
$378$	0	0
$379$	−34.1819	−1.75581	−0.877903	−	0.478839i	$-0.841058\pi$
$379$	−34.1819	−1.75581	−0.877903	+	0.478839i	$0.841058\pi$
$380$	5.06474	0.259816
$381$	0	0
$382$	−8.15782	−0.417391
$383$	−35.9813	−1.83856	−0.919279	−	0.393606i	$-0.871228\pi$
$383$	−35.9813	−1.83856	−0.919279	+	0.393606i	$0.871228\pi$
$384$	0	0
$385$	−1.46838	−0.0748357
$386$	3.92241	0.199645
$387$	0	0
$388$	17.5772	0.892345
$389$	10.0249	0.508281	0.254141	−	0.967167i	$-0.418207\pi$
$389$	10.0249	0.508281	0.254141	+	0.967167i	$0.418207\pi$
$390$	0	0
$391$	−13.6046	−0.688016
$392$	−1.72729	−0.0872414
$393$	0	0
$394$	12.0104	0.605074
$395$	1.58853	0.0799279
$396$	0	0
$397$	−33.3894	−1.67576	−0.837882	−	0.545852i	$-0.816206\pi$
$397$	−33.3894	−1.67576	−0.837882	+	0.545852i	$0.816206\pi$
$398$	−5.62287	−0.281849
$399$	0	0
$400$	−12.4204	−0.621019
$401$	−38.9495	−1.94505	−0.972524	−	0.232804i	$-0.925210\pi$
$401$	−38.9495	−1.94505	−0.972524	+	0.232804i	$0.925210\pi$
$402$	0	0
$403$	19.6023	0.976458
$404$	−30.5306	−1.51896
$405$	0	0
$406$	0.823230	0.0408562
$407$	5.15442	0.255495
$408$	0	0
$409$	−27.7656	−1.37292	−0.686461	−	0.727167i	$-0.740837\pi$
$409$	−27.7656	−1.37292	−0.686461	+	0.727167i	$0.740837\pi$
$410$	2.76283	0.136447
$411$	0	0
$412$	−17.4276	−0.858596
$413$	0.440342	0.0216678
$414$	0	0
$415$	3.35425	0.164654
$416$	11.9123	0.584051
$417$	0	0
$418$	−3.36380	−0.164529
$419$	−31.9442	−1.56058	−0.780288	−	0.625420i	$-0.784928\pi$
$419$	−31.9442	−1.56058	−0.780288	+	0.625420i	$0.784928\pi$
$420$	0	0
$421$	16.0201	0.780771	0.390386	−	0.920651i	$-0.372342\pi$
$421$	16.0201	0.780771	0.390386	+	0.920651i	$0.372342\pi$
$422$	−12.9593	−0.630851
$423$	0	0
$424$	0.514713	0.0249967
$425$	−7.79966	−0.378339
$426$	0	0
$427$	−4.23592	−0.204991
$428$	−15.5129	−0.749842
$429$	0	0
$430$	−4.01410	−0.193577
$431$	30.1897	1.45419	0.727094	−	0.686537i	$-0.240870\pi$
$431$	30.1897	1.45419	0.727094	+	0.686537i	$0.240870\pi$
$432$	0	0
$433$	22.6091	1.08652	0.543262	−	0.839563i	$-0.317189\pi$
$433$	22.6091	1.08652	0.543262	+	0.839563i	$0.317189\pi$
$434$	−3.54422	−0.170128
$435$	0	0
$436$	22.5571	1.08029
$437$	29.1840	1.39606
$438$	0	0
$439$	30.2295	1.44278	0.721389	−	0.692530i	$-0.243504\pi$
$439$	30.2295	1.44278	0.721389	+	0.692530i	$0.243504\pi$
$440$	−2.53633	−0.120915
$441$	0	0
$442$	2.01609	0.0958956
$443$	4.67658	0.222191	0.111096	−	0.993810i	$-0.464564\pi$
$443$	4.67658	0.222191	0.111096	+	0.993810i	$0.464564\pi$
$444$	0	0
$445$	−2.20914	−0.104723
$446$	2.45571	0.116281
$447$	0	0
$448$	3.44310	0.162671
$449$	1.91605	0.0904240	0.0452120	−	0.998977i	$-0.485604\pi$
$449$	1.91605	0.0904240	0.0452120	+	0.998977i	$0.485604\pi$
$450$	0	0
$451$	15.8577	0.746712
$452$	8.15417	0.383540
$453$	0	0
$454$	−4.95244	−0.232429
$455$	−1.88789	−0.0885058
$456$	0	0
$457$	−8.96919	−0.419561	−0.209780	−	0.977749i	$-0.567275\pi$
$457$	−8.96919	−0.419561	−0.209780	+	0.977749i	$0.567275\pi$
$458$	1.59849	0.0746924
$459$	0	0
$460$	10.4007	0.484936
$461$	40.4148	1.88231	0.941153	−	0.337979i	$-0.109743\pi$
$461$	40.4148	1.88231	0.941153	+	0.337979i	$0.109743\pi$
$462$	0	0
$463$	18.4440	0.857165	0.428582	−	0.903503i	$-0.359013\pi$
$463$	18.4440	0.857165	0.428582	+	0.903503i	$0.359013\pi$
$464$	−5.05836	−0.234829
$465$	0	0
$466$	−1.13098	−0.0523918
$467$	−1.77695	−0.0822273	−0.0411136	−	0.999154i	$-0.513091\pi$
$467$	−1.77695	−0.0822273	−0.0411136	+	0.999154i	$0.513091\pi$
$468$	0	0
$469$	13.5402	0.625229
$470$	1.11203	0.0512942
$471$	0	0
$472$	0.760600	0.0350094
$473$	−23.0396	−1.05936
$474$	0	0
$475$	16.7315	0.767693
$476$	3.15020	0.144389
$477$	0	0
$478$	−9.64364	−0.441090
$479$	5.47638	0.250222	0.125111	−	0.992143i	$-0.460071\pi$
$479$	5.47638	0.250222	0.125111	+	0.992143i	$0.460071\pi$
$480$	0	0
$481$	6.62702	0.302166
$482$	−7.80359	−0.355444
$483$	0	0
$484$	12.8376	0.583526
$485$	−7.34907	−0.333704
$486$	0	0
$487$	−34.6833	−1.57165	−0.785826	−	0.618448i	$-0.787762\pi$
$487$	−34.6833	−1.57165	−0.785826	+	0.618448i	$0.787762\pi$
$488$	−7.31668	−0.331210
$489$	0	0
$490$	0.341344	0.0154204
$491$	−5.81719	−0.262526	−0.131263	−	0.991348i	$-0.541903\pi$
$491$	−5.81719	−0.262526	−0.131263	+	0.991348i	$0.541903\pi$
$492$	0	0
$493$	−3.17651	−0.143063
$494$	−4.32482	−0.194583
$495$	0	0
$496$	21.7776	0.977842
$497$	−1.44289	−0.0647223
$498$	0	0
$499$	−2.83170	−0.126764	−0.0633822	−	0.997989i	$-0.520189\pi$
$499$	−2.83170	−0.126764	−0.0633822	+	0.997989i	$0.520189\pi$
$500$	12.6803	0.567081
$501$	0	0
$502$	1.35956	0.0606800
$503$	11.1295	0.496242	0.248121	−	0.968729i	$-0.420187\pi$
$503$	11.1295	0.496242	0.248121	+	0.968729i	$0.420187\pi$
$504$	0	0
$505$	12.7650	0.568033
$506$	−6.90774	−0.307087
$507$	0	0
$508$	−1.79257	−0.0795326
$509$	−0.0947118	−0.00419803	−0.00209901	−	0.999998i	$-0.500668\pi$
$509$	−0.0947118	−0.00419803	−0.00209901	+	0.999998i	$0.500668\pi$
$510$	0	0
$511$	2.90207	0.128380
$512$	22.9018	1.01213
$513$	0	0
$514$	6.49700	0.286570
$515$	7.28653	0.321083
$516$	0	0
$517$	6.38269	0.280710
$518$	−1.19821	−0.0526463
$519$	0	0
$520$	−3.26094	−0.143002
$521$	−17.2116	−0.754054	−0.377027	−	0.926202i	$-0.623054\pi$
$521$	−17.2116	−0.754054	−0.377027	+	0.926202i	$0.623054\pi$
$522$	0	0
$523$	−30.7334	−1.34388	−0.671938	−	0.740607i	$-0.734538\pi$
$523$	−30.7334	−1.34388	−0.671938	+	0.740607i	$0.734538\pi$
$524$	6.97275	0.304606
$525$	0	0
$526$	2.83912	0.123791
$527$	13.6757	0.595724
$528$	0	0
$529$	36.9310	1.60569
$530$	−0.101717	−0.00441829
$531$	0	0
$532$	−6.75767	−0.292982
$533$	20.3882	0.883112
$534$	0	0
$535$	6.48598	0.280413
$536$	23.3879	1.01020
$537$	0	0
$538$	−4.32438	−0.186437
$539$	1.95920	0.0843887
$540$	0	0
$541$	−45.4903	−1.95578	−0.977891	−	0.209117i	$-0.932941\pi$
$541$	−45.4903	−1.95578	−0.977891	+	0.209117i	$0.932941\pi$
$542$	−11.3108	−0.485839
$543$	0	0
$544$	8.31077	0.356321
$545$	−9.43119	−0.403988
$546$	0	0
$547$	−2.40242	−0.102720	−0.0513599	−	0.998680i	$-0.516356\pi$
$547$	−2.40242	−0.102720	−0.0513599	+	0.998680i	$0.516356\pi$
$548$	22.4829	0.960423
$549$	0	0
$550$	−3.96027	−0.168867
$551$	6.81411	0.290291
$552$	0	0
$553$	−2.11951	−0.0901309
$554$	7.35422	0.312451
$555$	0	0
$556$	37.8930	1.60702
$557$	14.6511	0.620788	0.310394	−	0.950608i	$-0.399539\pi$
$557$	14.6511	0.620788	0.310394	+	0.950608i	$0.399539\pi$
$558$	0	0
$559$	−29.6219	−1.25287
$560$	−2.09740	−0.0886313
$561$	0	0
$562$	4.30478	0.181586
$563$	37.5930	1.58436	0.792178	−	0.610291i	$-0.208947\pi$
$563$	37.5930	1.58436	0.792178	+	0.610291i	$0.208947\pi$
$564$	0	0
$565$	−3.40929	−0.143430
$566$	6.46360	0.271685
$567$	0	0
$568$	−2.49229	−0.104574
$569$	−3.82876	−0.160510	−0.0802550	−	0.996774i	$-0.525573\pi$
$569$	−3.82876	−0.160510	−0.0802550	+	0.996774i	$0.525573\pi$
$570$	0	0
$571$	17.2172	0.720519	0.360260	−	0.932852i	$-0.382688\pi$
$571$	17.2172	0.720519	0.360260	+	0.932852i	$0.382688\pi$
$572$	−8.84653	−0.369892
$573$	0	0
$574$	−3.68633	−0.153864
$575$	34.3590	1.43287
$576$	0	0
$577$	17.9929	0.749055	0.374527	−	0.927216i	$-0.377805\pi$
$577$	17.9929	0.749055	0.374527	+	0.927216i	$0.377805\pi$
$578$	−6.33595	−0.263541
$579$	0	0
$580$	2.42844	0.100835
$581$	−4.47544	−0.185672
$582$	0	0
$583$	−0.583819	−0.0241793
$584$	5.01273	0.207428
$585$	0	0
$586$	−8.20992	−0.339149
$587$	15.1685	0.626070	0.313035	−	0.949742i	$-0.398654\pi$
$587$	15.1685	0.626070	0.313035	+	0.949742i	$0.398654\pi$
$588$	0	0
$589$	−29.3365	−1.20879
$590$	−0.150308	−0.00618809
$591$	0	0
$592$	7.36244	0.302594
$593$	−29.4625	−1.20988	−0.604940	−	0.796271i	$-0.706803\pi$
$593$	−29.4625	−1.20988	−0.604940	+	0.796271i	$0.706803\pi$
$594$	0	0
$595$	−1.31711	−0.0539962
$596$	−32.7120	−1.33993
$597$	0	0
$598$	−8.88126	−0.363182
$599$	−6.27754	−0.256493	−0.128247	−	0.991742i	$-0.540935\pi$
$599$	−6.27754	−0.256493	−0.128247	+	0.991742i	$0.540935\pi$
$600$	0	0
$601$	21.5091	0.877373	0.438687	−	0.898640i	$-0.355444\pi$
$601$	21.5091	0.877373	0.438687	+	0.898640i	$0.355444\pi$
$602$	5.35583	0.218288
$603$	0	0
$604$	7.17551	0.291967
$605$	−5.36743	−0.218217
$606$	0	0
$607$	36.2458	1.47117	0.735586	−	0.677431i	$-0.236907\pi$
$607$	36.2458	1.47117	0.735586	+	0.677431i	$0.236907\pi$
$608$	−17.8279	−0.723017
$609$	0	0
$610$	1.44591	0.0585431
$611$	8.20619	0.331987
$612$	0	0
$613$	4.88666	0.197370	0.0986851	−	0.995119i	$-0.468536\pi$
$613$	4.88666	0.197370	0.0986851	+	0.995119i	$0.468536\pi$
$614$	7.33001	0.295815
$615$	0	0
$616$	3.38411	0.136350
$617$	−4.27798	−0.172225	−0.0861126	−	0.996285i	$-0.527444\pi$
$617$	−4.27798	−0.172225	−0.0861126	+	0.996285i	$0.527444\pi$
$618$	0	0
$619$	32.3159	1.29888	0.649442	−	0.760411i	$-0.275003\pi$
$619$	32.3159	1.29888	0.649442	+	0.760411i	$0.275003\pi$
$620$	−10.4551	−0.419886
$621$	0	0
$622$	−8.79353	−0.352589
$623$	2.94756	0.118092
$624$	0	0
$625$	16.8897	0.675588
$626$	−14.0555	−0.561770
$627$	0	0
$628$	5.85726	0.233730
$629$	4.62341	0.184347
$630$	0	0
$631$	−0.860434	−0.0342533	−0.0171267	−	0.999853i	$-0.505452\pi$
$631$	−0.860434	−0.0342533	−0.0171267	+	0.999853i	$0.505452\pi$
$632$	−3.66102	−0.145628
$633$	0	0
$634$	−13.9709	−0.554856
$635$	0.749481	0.0297422
$636$	0	0
$637$	2.51893	0.0998038
$638$	−1.61287	−0.0638542
$639$	0	0
$640$	−8.26405	−0.326665
$641$	−1.41230	−0.0557827	−0.0278913	−	0.999611i	$-0.508879\pi$
$641$	−1.41230	−0.0557827	−0.0278913	+	0.999611i	$0.508879\pi$
$642$	0	0
$643$	−11.5508	−0.455518	−0.227759	−	0.973718i	$-0.573140\pi$
$643$	−11.5508	−0.455518	−0.227759	+	0.973718i	$0.573140\pi$
$644$	−13.8772	−0.546839
$645$	0	0
$646$	−3.01726	−0.118712
$647$	−1.61666	−0.0635574	−0.0317787	−	0.999495i	$-0.510117\pi$
$647$	−1.61666	−0.0635574	−0.0317787	+	0.999495i	$0.510117\pi$
$648$	0	0
$649$	−0.862719	−0.0338647
$650$	−5.09170	−0.199713
$651$	0	0
$652$	10.7411	0.420655
$653$	14.8742	0.582071	0.291035	−	0.956712i	$-0.406000\pi$
$653$	14.8742	0.582071	0.291035	+	0.956712i	$0.406000\pi$
$654$	0	0
$655$	−2.91533	−0.113911
$656$	22.6508	0.884364
$657$	0	0
$658$	−1.48374	−0.0578420
$659$	−19.5325	−0.760878	−0.380439	−	0.924806i	$-0.624227\pi$
$659$	−19.5325	−0.760878	−0.380439	+	0.924806i	$0.624227\pi$
$660$	0	0
$661$	36.3289	1.41303	0.706516	−	0.707697i	$-0.250266\pi$
$661$	36.3289	1.41303	0.706516	+	0.707697i	$0.250266\pi$
$662$	−4.66597	−0.181348
$663$	0	0
$664$	−7.73039	−0.299997
$665$	2.82540	0.109564
$666$	0	0
$667$	13.9931	0.541817
$668$	−38.1493	−1.47604
$669$	0	0
$670$	−4.62187	−0.178559
$671$	8.29902	0.320380
$672$	0	0
$673$	6.49459	0.250348	0.125174	−	0.992135i	$-0.460051\pi$
$673$	6.49459	0.250348	0.125174	+	0.992135i	$0.460051\pi$
$674$	−0.593872	−0.0228751
$675$	0	0
$676$	11.9295	0.458828
$677$	−47.4029	−1.82184	−0.910921	−	0.412582i	$-0.864627\pi$
$677$	−47.4029	−1.82184	−0.910921	+	0.412582i	$0.864627\pi$
$678$	0	0
$679$	9.80554	0.376302
$680$	−2.27503	−0.0872435
$681$	0	0
$682$	6.94384	0.265893
$683$	32.5349	1.24491	0.622456	−	0.782655i	$-0.286135\pi$
$683$	32.5349	1.24491	0.622456	+	0.782655i	$0.286135\pi$
$684$	0	0
$685$	−9.40018	−0.359163
$686$	−0.455441	−0.0173888
$687$	0	0
$688$	−32.9091	−1.25465
$689$	−0.750614	−0.0285961
$690$	0	0
$691$	−1.09394	−0.0416155	−0.0208078	−	0.999783i	$-0.506624\pi$
$691$	−1.09394	−0.0416155	−0.0208078	+	0.999783i	$0.506624\pi$
$692$	4.32912	0.164569
$693$	0	0
$694$	−8.49112	−0.322318
$695$	−15.8432	−0.600966
$696$	0	0
$697$	14.2241	0.538775
$698$	3.40382	0.128836
$699$	0	0
$700$	−7.95594	−0.300706
$701$	−37.3987	−1.41253	−0.706264	−	0.707949i	$-0.749621\pi$
$701$	−37.3987	−1.41253	−0.706264	+	0.707949i	$0.749621\pi$
$702$	0	0
$703$	−9.91793	−0.374062
$704$	−6.74573	−0.254239
$705$	0	0
$706$	4.76196	0.179219
$707$	−17.0317	−0.640544
$708$	0	0
$709$	27.4538	1.03105	0.515524	−	0.856875i	$-0.327597\pi$
$709$	27.4538	1.03105	0.515524	+	0.856875i	$0.327597\pi$
$710$	0.492521	0.0184840
$711$	0	0
$712$	5.09130	0.190804
$713$	−60.2441	−2.25616
$714$	0	0
$715$	3.69876	0.138326
$716$	25.7933	0.963941
$717$	0	0
$718$	−16.3479	−0.610096
$719$	−14.8902	−0.555312	−0.277656	−	0.960681i	$-0.589557\pi$
$719$	−14.8902	−0.555312	−0.277656	+	0.960681i	$0.589557\pi$
$720$	0	0
$721$	−9.72210	−0.362070
$722$	−2.18088	−0.0811641
$723$	0	0
$724$	−12.0827	−0.449051
$725$	8.02239	0.297944
$726$	0	0
$727$	8.84554	0.328063	0.164032	−	0.986455i	$-0.447550\pi$
$727$	8.84554	0.328063	0.164032	+	0.986455i	$0.447550\pi$
$728$	4.35094	0.161256
$729$	0	0
$730$	−0.990605	−0.0366639
$731$	−20.6660	−0.764360
$732$	0	0
$733$	51.0985	1.88737	0.943683	−	0.330851i	$-0.107336\pi$
$733$	51.0985	1.88737	0.943683	+	0.330851i	$0.107336\pi$
$734$	1.43752	0.0530597
$735$	0	0
$736$	−36.6105	−1.34948
$737$	−26.5280	−0.977171
$738$	0	0
$739$	1.57429	0.0579112	0.0289556	−	0.999581i	$-0.490782\pi$
$739$	1.57429	0.0579112	0.0289556	+	0.999581i	$0.490782\pi$
$740$	−3.53459	−0.129934
$741$	0	0
$742$	0.135716	0.00498230
$743$	−39.0343	−1.43203	−0.716015	−	0.698085i	$-0.754036\pi$
$743$	−39.0343	−1.43203	−0.716015	+	0.698085i	$0.754036\pi$
$744$	0	0
$745$	13.6770	0.501086
$746$	−1.86247	−0.0681899
$747$	0	0
$748$	−6.17187	−0.225666
$749$	−8.65396	−0.316209
$750$	0	0
$751$	9.74428	0.355574	0.177787	−	0.984069i	$-0.443106\pi$
$751$	9.74428	0.355574	0.177787	+	0.984069i	$0.443106\pi$
$752$	9.11686	0.332458
$753$	0	0
$754$	−2.07366	−0.0755183
$755$	−3.00010	−0.109185
$756$	0	0
$757$	−17.7563	−0.645363	−0.322682	−	0.946508i	$-0.604584\pi$
$757$	−17.7563	−0.645363	−0.322682	+	0.946508i	$0.604584\pi$
$758$	−15.5678	−0.565449
$759$	0	0
$760$	4.88030	0.177027
$761$	14.0688	0.509994	0.254997	−	0.966942i	$-0.417925\pi$
$761$	14.0688	0.509994	0.254997	+	0.966942i	$0.417925\pi$
$762$	0	0
$763$	12.5836	0.455558
$764$	32.1085	1.16164
$765$	0	0
$766$	−16.3873	−0.592099
$767$	−1.10919	−0.0400507
$768$	0	0
$769$	−2.66067	−0.0959461	−0.0479730	−	0.998849i	$-0.515276\pi$
$769$	−2.66067	−0.0959461	−0.0479730	+	0.998849i	$0.515276\pi$
$770$	−0.668761	−0.0241005
$771$	0	0
$772$	−15.4383	−0.555635
$773$	3.52993	0.126963	0.0634814	−	0.997983i	$-0.479780\pi$
$773$	3.52993	0.126963	0.0634814	+	0.997983i	$0.479780\pi$
$774$	0	0
$775$	−34.5385	−1.24066
$776$	16.9370	0.608004
$777$	0	0
$778$	4.56573	0.163690
$779$	−30.5128	−1.09324
$780$	0	0
$781$	2.82690	0.101155
$782$	−6.19610	−0.221572
$783$	0	0
$784$	2.79847	0.0999453
$785$	−2.44894	−0.0874065
$786$	0	0
$787$	9.63057	0.343293	0.171646	−	0.985159i	$-0.445091\pi$
$787$	9.63057	0.343293	0.171646	+	0.985159i	$0.445091\pi$
$788$	−47.2717	−1.68399
$789$	0	0
$790$	0.723483	0.0257404
$791$	4.54886	0.161739
$792$	0	0
$793$	10.6700	0.378903
$794$	−15.2069	−0.539672
$795$	0	0
$796$	22.1311	0.784417
$797$	5.94571	0.210608	0.105304	−	0.994440i	$-0.466418\pi$
$797$	5.94571	0.210608	0.105304	+	0.994440i	$0.466418\pi$
$798$	0	0
$799$	5.72514	0.202541
$800$	−20.9892	−0.742079
$801$	0	0
$802$	−17.7392	−0.626393
$803$	−5.68574	−0.200645
$804$	0	0
$805$	5.80211	0.204498
$806$	8.92766	0.314464
$807$	0	0
$808$	−29.4188	−1.03495
$809$	38.1282	1.34051	0.670257	−	0.742129i	$-0.266184\pi$
$809$	38.1282	1.34051	0.670257	+	0.742129i	$0.266184\pi$
$810$	0	0
$811$	21.9571	0.771018	0.385509	−	0.922704i	$-0.374026\pi$
$811$	21.9571	0.771018	0.385509	+	0.922704i	$0.374026\pi$
$812$	−3.24016	−0.113707
$813$	0	0
$814$	2.34753	0.0822810
$815$	−4.49090	−0.157309
$816$	0	0
$817$	44.3318	1.55097
$818$	−12.6456	−0.442143
$819$	0	0
$820$	−10.8743	−0.379746
$821$	−31.2927	−1.09212	−0.546061	−	0.837746i	$-0.683873\pi$
$821$	−31.2927	−1.09212	−0.546061	+	0.837746i	$0.683873\pi$
$822$	0	0
$823$	18.8063	0.655547	0.327773	−	0.944756i	$-0.393702\pi$
$823$	18.8063	0.655547	0.327773	+	0.944756i	$0.393702\pi$
$824$	−16.7929	−0.585009
$825$	0	0
$826$	0.200550	0.00697802
$827$	23.1897	0.806386	0.403193	−	0.915115i	$-0.367900\pi$
$827$	23.1897	0.806386	0.403193	+	0.915115i	$0.367900\pi$
$828$	0	0
$829$	−56.7692	−1.97168	−0.985838	−	0.167700i	$-0.946366\pi$
$829$	−56.7692	−1.97168	−0.985838	+	0.167700i	$0.946366\pi$
$830$	1.52766	0.0530260
$831$	0	0
$832$	−8.67295	−0.300681
$833$	1.75736	0.0608890
$834$	0	0
$835$	15.9504	0.551985
$836$	13.2396	0.457902
$837$	0	0
$838$	−14.5487	−0.502576
$839$	28.5757	0.986542	0.493271	−	0.869876i	$-0.335801\pi$
$839$	28.5757	0.986542	0.493271	+	0.869876i	$0.335801\pi$
$840$	0	0
$841$	−25.7328	−0.887337
$842$	7.29620	0.251444
$843$	0	0
$844$	51.0068	1.75573
$845$	−4.98777	−0.171585
$846$	0	0
$847$	7.16153	0.246073
$848$	−0.833912	−0.0286366
$849$	0	0
$850$	−3.55228	−0.121842
$851$	−20.3670	−0.698171
$852$	0	0
$853$	17.4300	0.596792	0.298396	−	0.954442i	$-0.403548\pi$
$853$	17.4300	0.596792	0.298396	+	0.954442i	$0.403548\pi$
$854$	−1.92921	−0.0660163
$855$	0	0
$856$	−14.9479	−0.510909
$857$	31.4574	1.07456	0.537282	−	0.843403i	$-0.319451\pi$
$857$	31.4574	1.07456	0.537282	+	0.843403i	$0.319451\pi$
$858$	0	0
$859$	−17.7552	−0.605799	−0.302900	−	0.953022i	$-0.597955\pi$
$859$	−17.7552	−0.605799	−0.302900	+	0.953022i	$0.597955\pi$
$860$	15.7991	0.538746
$861$	0	0
$862$	13.7496	0.468315
$863$	−16.9150	−0.575793	−0.287897	−	0.957661i	$-0.592956\pi$
$863$	−16.9150	−0.575793	−0.287897	+	0.957661i	$0.592956\pi$
$864$	0	0
$865$	−1.81002	−0.0615426
$866$	10.2971	0.349910
$867$	0	0
$868$	13.9497	0.473485
$869$	4.15255	0.140866
$870$	0	0
$871$	−34.1069	−1.15567
$872$	21.7356	0.736060
$873$	0	0
$874$	13.2916	0.449595
$875$	7.07381	0.239138
$876$	0	0
$877$	−13.3300	−0.450121	−0.225061	−	0.974345i	$-0.572258\pi$
$877$	−13.3300	−0.450121	−0.225061	+	0.974345i	$0.572258\pi$
$878$	13.7678	0.464639
$879$	0	0
$880$	4.10922	0.138522
$881$	31.1651	1.04998	0.524990	−	0.851109i	$-0.324069\pi$
$881$	31.1651	1.04998	0.524990	+	0.851109i	$0.324069\pi$
$882$	0	0
$883$	20.0110	0.673422	0.336711	−	0.941608i	$-0.390685\pi$
$883$	20.0110	0.673422	0.336711	+	0.941608i	$0.390685\pi$
$884$	−7.93515	−0.266888
$885$	0	0
$886$	2.12991	0.0715556
$887$	10.0796	0.338441	0.169221	−	0.985578i	$-0.445875\pi$
$887$	10.0796	0.338441	0.169221	+	0.985578i	$0.445875\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−1.00613	−0.0337256
$891$	0	0
$892$	−9.66546	−0.323624
$893$	−12.2813	−0.410978
$894$	0	0
$895$	−10.7843	−0.360478
$896$	11.0264	0.368365
$897$	0	0
$898$	0.872647	0.0291206
$899$	−14.0663	−0.469136
$900$	0	0
$901$	−0.523674	−0.0174461
$902$	7.22226	0.240475
$903$	0	0
$904$	7.85721	0.261327
$905$	5.05183	0.167928
$906$	0	0
$907$	32.5788	1.08176	0.540880	−	0.841100i	$-0.318091\pi$
$907$	32.5788	1.08176	0.540880	+	0.841100i	$0.318091\pi$
$908$	19.4924	0.646876
$909$	0	0
$910$	−0.859823	−0.0285029
$911$	34.9407	1.15764	0.578819	−	0.815456i	$-0.303514\pi$
$911$	34.9407	1.15764	0.578819	+	0.815456i	$0.303514\pi$
$912$	0	0
$913$	8.76828	0.290188
$914$	−4.08493	−0.135118
$915$	0	0
$916$	−6.29151	−0.207877
$917$	3.88980	0.128453
$918$	0	0
$919$	−36.5952	−1.20716	−0.603582	−	0.797301i	$-0.706260\pi$
$919$	−36.5952	−1.20716	−0.603582	+	0.797301i	$0.706260\pi$
$920$	10.0219	0.330414
$921$	0	0
$922$	18.4066	0.606188
$923$	3.63454	0.119632
$924$	0	0
$925$	−11.6766	−0.383924
$926$	8.40014	0.276046
$927$	0	0
$928$	−8.54811	−0.280605
$929$	−4.37077	−0.143400	−0.0717002	−	0.997426i	$-0.522842\pi$
$929$	−4.37077	−0.143400	−0.0717002	+	0.997426i	$0.522842\pi$
$930$	0	0
$931$	−3.76981	−0.123551
$932$	4.45145	0.145812
$933$	0	0
$934$	−0.809294	−0.0264809
$935$	2.58048	0.0843907
$936$	0	0
$937$	36.0803	1.17869	0.589346	−	0.807881i	$-0.299386\pi$
$937$	36.0803	1.17869	0.589346	+	0.807881i	$0.299386\pi$
$938$	6.16677	0.201352
$939$	0	0
$940$	−4.37686	−0.142757
$941$	−17.2953	−0.563810	−0.281905	−	0.959442i	$-0.590966\pi$
$941$	−17.2953	−0.563810	−0.281905	+	0.959442i	$0.590966\pi$
$942$	0	0
$943$	−62.6597	−2.04048
$944$	−1.23228	−0.0401074
$945$	0	0
$946$	−10.4932	−0.341162
$947$	−28.8863	−0.938679	−0.469340	−	0.883018i	$-0.655508\pi$
$947$	−28.8863	−0.938679	−0.469340	+	0.883018i	$0.655508\pi$
$948$	0	0
$949$	−7.31013	−0.237297
$950$	7.62020	0.247232
$951$	0	0
$952$	3.03548	0.0983804
$953$	45.3794	1.46998	0.734992	−	0.678076i	$-0.237186\pi$
$953$	45.3794	1.46998	0.734992	+	0.678076i	$0.237186\pi$
$954$	0	0
$955$	−13.4247	−0.434412
$956$	37.9565	1.22760
$957$	0	0
$958$	2.49417	0.0805828
$959$	12.5423	0.405011
$960$	0	0
$961$	29.5589	0.953514
$962$	3.01821	0.0973111
$963$	0	0
$964$	30.7142	0.989239
$965$	6.45479	0.207787
$966$	0	0
$967$	−9.53966	−0.306775	−0.153387	−	0.988166i	$-0.549018\pi$
$967$	−9.53966	−0.306775	−0.153387	+	0.988166i	$0.549018\pi$
$968$	12.3701	0.397589
$969$	0	0
$970$	−3.34706	−0.107468
$971$	−48.0293	−1.54133	−0.770667	−	0.637239i	$-0.780077\pi$
$971$	−48.0293	−1.54133	−0.770667	+	0.637239i	$0.780077\pi$
$972$	0	0
$973$	21.1389	0.677681
$974$	−15.7962	−0.506143
$975$	0	0
$976$	11.8541	0.379440
$977$	10.7980	0.345460	0.172730	−	0.984969i	$-0.444741\pi$
$977$	10.7980	0.345460	0.172730	+	0.984969i	$0.444741\pi$
$978$	0	0
$979$	−5.77486	−0.184565
$980$	−1.34350	−0.0429165
$981$	0	0
$982$	−2.64939	−0.0845453
$983$	3.04790	0.0972130	0.0486065	−	0.998818i	$-0.484522\pi$
$983$	3.04790	0.0972130	0.0486065	+	0.998818i	$0.484522\pi$
$984$	0	0
$985$	19.7645	0.629748
$986$	−1.44671	−0.0460727
$987$	0	0
$988$	17.0221	0.541546
$989$	91.0377	2.89483
$990$	0	0
$991$	−44.1141	−1.40133	−0.700665	−	0.713491i	$-0.747113\pi$
$991$	−44.1141	−1.40133	−0.700665	+	0.713491i	$0.747113\pi$
$992$	36.8018	1.16846
$993$	0	0
$994$	−0.657149	−0.0208435
$995$	−9.25310	−0.293343
$996$	0	0
$997$	−60.8537	−1.92726	−0.963629	−	0.267245i	$-0.913887\pi$
$997$	−60.8537	−1.92726	−0.963629	+	0.267245i	$0.913887\pi$
$998$	−1.28967	−0.0408239
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.m.1.6		11
3.2	odd	2		2667.2.a.k.1.6	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.k.1.6	✓	11	3.2	odd	2
8001.2.a.m.1.6		11	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)

\(f(q)\)	\(=\)	\(q+0.455441 q^{2} -1.79257 q^{4} +0.749481 q^{5} -1.00000 q^{7} -1.72729 q^{8} +0.341344 q^{10} +1.95920 q^{11} +2.51893 q^{13} -0.455441 q^{14} +2.79847 q^{16} +1.75736 q^{17} -3.76981 q^{19} -1.34350 q^{20} +0.892299 q^{22} -7.74151 q^{23} -4.43828 q^{25} +1.14723 q^{26} +1.79257 q^{28} -1.80755 q^{29} +7.78196 q^{31} +4.72912 q^{32} +0.800374 q^{34} -0.749481 q^{35} +2.63088 q^{37} -1.71693 q^{38} -1.29457 q^{40} +8.09399 q^{41} -11.7597 q^{43} -3.51201 q^{44} -3.52580 q^{46} +3.25780 q^{47} +1.00000 q^{49} -2.02137 q^{50} -4.51538 q^{52} -0.297989 q^{53} +1.46838 q^{55} +1.72729 q^{56} -0.823230 q^{58} -0.440342 q^{59} +4.23592 q^{61} +3.54422 q^{62} -3.44310 q^{64} +1.88789 q^{65} -13.5402 q^{67} -3.15020 q^{68} -0.341344 q^{70} +1.44289 q^{71} -2.90207 q^{73} +1.19821 q^{74} +6.75767 q^{76} -1.95920 q^{77} +2.11951 q^{79} +2.09740 q^{80} +3.68633 q^{82} +4.47544 q^{83} +1.31711 q^{85} -5.35583 q^{86} -3.38411 q^{88} -2.94756 q^{89} -2.51893 q^{91} +13.8772 q^{92} +1.48374 q^{94} -2.82540 q^{95} -9.80554 q^{97} +0.455441 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8} - 12 q^{10} + 7 q^{11} - 24 q^{13} - 2 q^{14} - 6 q^{16} + 15 q^{17} - 19 q^{19} - 3 q^{20} - 3 q^{22} + 11 q^{23} + 10 q^{25} - 10 q^{26} - 12 q^{28}+ \cdots + 2 q^{98}+O(q^{100}) \) 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8 - 12 * q^10 + 7 * q^11 - 24 * q^13 - 2 * q^14 - 6 * q^16 + 15 * q^17 - 19 * q^19 - 3 * q^20 - 3 * q^22 + 11 * q^23 + 10 * q^25 - 10 * q^26 - 12 * q^28 + 10 * q^29 - 20 * q^31 + 27 * q^32 - 9 * q^34 + q^35 - 22 * q^37 - 8 * q^38 - 29 * q^40 - 9 * q^41 - 17 * q^43 + 9 * q^44 - 18 * q^46 + 7 * q^47 + 11 * q^49 + 47 * q^50 - 66 * q^52 + 28 * q^53 - 24 * q^55 - 15 * q^56 - 39 * q^58 - 35 * q^59 - 6 * q^61 - 18 * q^62 + 11 * q^64 + 43 * q^65 - 22 * q^67 + 12 * q^68 + 12 * q^70 + 22 * q^71 - 29 * q^73 - 14 * q^74 + 10 * q^76 - 7 * q^77 - 20 * q^79 - 66 * q^80 - 24 * q^82 - 17 * q^83 - 50 * q^85 + 12 * q^86 + 2 * q^88 + q^89 + 24 * q^91 + 22 * q^92 + q^94 - 10 * q^95 - 45 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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