LMFDB - Embedded newform 8001.2.a.w.1.9

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 = [20,-8,0,24,-3] 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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} + \cdots + 31 $ x^20 - 8x^19 + 152x^17 - 274x^16 - 1061x^15 + 3125x^14 + 2977x^13 - 15474x^12 - 56x^11 + 39579x^10 - 17664x^9 - 52271x^8 + 35701x^7 + 32493x^6 - 25504x^5 - 8607x^4 + 6812x^3 + 609x^2 - 425x + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.831296$ 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)

$f(q)$	$=$	\(q-0.831296 q^{2} -1.30895 q^{4} +1.71517 q^{5} +1.00000 q^{7} +2.75071 q^{8} -1.42582 q^{10} -6.30225 q^{11} +2.82193 q^{13} -0.831296 q^{14} +0.331239 q^{16} -3.67069 q^{17} +4.30307 q^{19} -2.24507 q^{20} +5.23903 q^{22} +1.07254 q^{23} -2.05818 q^{25} -2.34586 q^{26} -1.30895 q^{28} -6.99923 q^{29} +4.64341 q^{31} -5.77678 q^{32} +3.05143 q^{34} +1.71517 q^{35} +4.42007 q^{37} -3.57712 q^{38} +4.71795 q^{40} +0.227674 q^{41} +0.781819 q^{43} +8.24932 q^{44} -0.891595 q^{46} +6.94665 q^{47} +1.00000 q^{49} +1.71095 q^{50} -3.69376 q^{52} -0.313469 q^{53} -10.8095 q^{55} +2.75071 q^{56} +5.81843 q^{58} +2.11774 q^{59} -1.64281 q^{61} -3.86004 q^{62} +4.13974 q^{64} +4.84011 q^{65} -5.95221 q^{67} +4.80474 q^{68} -1.42582 q^{70} -6.51071 q^{71} +0.998203 q^{73} -3.67439 q^{74} -5.63250 q^{76} -6.30225 q^{77} -3.86107 q^{79} +0.568133 q^{80} -0.189264 q^{82} +2.10845 q^{83} -6.29587 q^{85} -0.649923 q^{86} -17.3357 q^{88} -0.415935 q^{89} +2.82193 q^{91} -1.40389 q^{92} -5.77472 q^{94} +7.38052 q^{95} -9.24065 q^{97} -0.831296 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28}+ \cdots - 8 q^{98}+O(q^{100}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * 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 $

$2$	−0.831296	−0.587815	−0.293907	−	0.955834i	$-0.594956\pi$
$2$	−0.831296	−0.587815	−0.293907	+	0.955834i	$0.594956\pi$
$3$	0	0
$3$	0	0
$4$	−1.30895	−0.654474
$5$	1.71517	0.767049	0.383525	−	0.923531i	$-0.374710\pi$
$5$	1.71517	0.767049	0.383525	+	0.923531i	$0.374710\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.75071	0.972524
$9$	0	0
$10$	−1.42582	−0.450883
$11$	−6.30225	−1.90020	−0.950100	−	0.311945i	$-0.899020\pi$
$11$	−6.30225	−1.90020	−0.950100	+	0.311945i	$0.899020\pi$
$12$	0	0
$13$	2.82193	0.782664	0.391332	−	0.920250i	$-0.372014\pi$
$13$	2.82193	0.782664	0.391332	+	0.920250i	$0.372014\pi$
$14$	−0.831296	−0.222173
$15$	0	0
$16$	0.331239	0.0828099
$17$	−3.67069	−0.890272	−0.445136	−	0.895463i	$-0.646845\pi$
$17$	−3.67069	−0.890272	−0.445136	+	0.895463i	$0.646845\pi$
$18$	0	0
$19$	4.30307	0.987193	0.493596	−	0.869691i	$-0.335682\pi$
$19$	4.30307	0.987193	0.493596	+	0.869691i	$0.335682\pi$
$20$	−2.24507	−0.502014
$21$	0	0
$22$	5.23903	1.11697
$23$	1.07254	0.223639	0.111820	−	0.993729i	$-0.464332\pi$
$23$	1.07254	0.223639	0.111820	+	0.993729i	$0.464332\pi$
$24$	0	0
$25$	−2.05818	−0.411636
$26$	−2.34586	−0.460061
$27$	0	0
$28$	−1.30895	−0.247368
$29$	−6.99923	−1.29972	−0.649862	−	0.760052i	$-0.725174\pi$
$29$	−6.99923	−1.29972	−0.649862	+	0.760052i	$0.725174\pi$
$30$	0	0
$31$	4.64341	0.833981	0.416990	−	0.908911i	$-0.363085\pi$
$31$	4.64341	0.833981	0.416990	+	0.908911i	$0.363085\pi$
$32$	−5.77678	−1.02120
$33$	0	0
$34$	3.05143	0.523315
$35$	1.71517	0.289917
$36$	0	0
$37$	4.42007	0.726655	0.363328	−	0.931661i	$-0.381641\pi$
$37$	4.42007	0.726655	0.363328	+	0.931661i	$0.381641\pi$
$38$	−3.57712	−0.580286
$39$	0	0
$40$	4.71795	0.745974
$41$	0.227674	0.0355567	0.0177783	−	0.999842i	$-0.494341\pi$
$41$	0.227674	0.0355567	0.0177783	+	0.999842i	$0.494341\pi$
$42$	0	0
$43$	0.781819	0.119226	0.0596131	−	0.998222i	$-0.481013\pi$
$43$	0.781819	0.119226	0.0596131	+	0.998222i	$0.481013\pi$
$44$	8.24932	1.24363
$45$	0	0
$46$	−0.891595	−0.131459
$47$	6.94665	1.01327	0.506636	−	0.862160i	$-0.330889\pi$
$47$	6.94665	1.01327	0.506636	+	0.862160i	$0.330889\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.71095	0.241966
$51$	0	0
$52$	−3.69376	−0.512233
$53$	−0.313469	−0.0430582	−0.0215291	−	0.999768i	$-0.506853\pi$
$53$	−0.313469	−0.0430582	−0.0215291	+	0.999768i	$0.506853\pi$
$54$	0	0
$55$	−10.8095	−1.45755
$56$	2.75071	0.367580
$57$	0	0
$58$	5.81843	0.763997
$59$	2.11774	0.275706	0.137853	−	0.990453i	$-0.455980\pi$
$59$	2.11774	0.275706	0.137853	+	0.990453i	$0.455980\pi$
$60$	0	0
$61$	−1.64281	−0.210340	−0.105170	−	0.994454i	$-0.533539\pi$
$61$	−1.64281	−0.210340	−0.105170	+	0.994454i	$0.533539\pi$
$62$	−3.86004	−0.490226
$63$	0	0
$64$	4.13974	0.517467
$65$	4.84011	0.600341
$66$	0	0
$67$	−5.95221	−0.727178	−0.363589	−	0.931560i	$-0.618449\pi$
$67$	−5.95221	−0.727178	−0.363589	+	0.931560i	$0.618449\pi$
$68$	4.80474	0.582660
$69$	0	0
$70$	−1.42582	−0.170418
$71$	−6.51071	−0.772679	−0.386339	−	0.922357i	$-0.626261\pi$
$71$	−6.51071	−0.772679	−0.386339	+	0.922357i	$0.626261\pi$
$72$	0	0
$73$	0.998203	0.116831	0.0584154	−	0.998292i	$-0.481395\pi$
$73$	0.998203	0.116831	0.0584154	+	0.998292i	$0.481395\pi$
$74$	−3.67439	−0.427139
$75$	0	0
$76$	−5.63250	−0.646092
$77$	−6.30225	−0.718208
$78$	0	0
$79$	−3.86107	−0.434404	−0.217202	−	0.976127i	$-0.569693\pi$
$79$	−3.86107	−0.434404	−0.217202	+	0.976127i	$0.569693\pi$
$80$	0.568133	0.0635192
$81$	0	0
$82$	−0.189264	−0.0209007
$83$	2.10845	0.231432	0.115716	−	0.993282i	$-0.463084\pi$
$83$	2.10845	0.231432	0.115716	+	0.993282i	$0.463084\pi$
$84$	0	0
$85$	−6.29587	−0.682883
$86$	−0.649923	−0.0700830
$87$	0	0
$88$	−17.3357	−1.84799
$89$	−0.415935	−0.0440890	−0.0220445	−	0.999757i	$-0.507018\pi$
$89$	−0.415935	−0.0440890	−0.0220445	+	0.999757i	$0.507018\pi$
$90$	0	0
$91$	2.82193	0.295819
$92$	−1.40389	−0.146366
$93$	0	0
$94$	−5.77472	−0.595616
$95$	7.38052	0.757225
$96$	0	0
$97$	−9.24065	−0.938246	−0.469123	−	0.883133i	$-0.655430\pi$
$97$	−9.24065	−0.938246	−0.469123	+	0.883133i	$0.655430\pi$
$98$	−0.831296	−0.0839735
$99$	0	0
$100$	2.69405	0.269405
$101$	−13.8607	−1.37919	−0.689597	−	0.724193i	$-0.742213\pi$
$101$	−13.8607	−1.37919	−0.689597	+	0.724193i	$0.742213\pi$
$102$	0	0
$103$	−15.2610	−1.50371	−0.751856	−	0.659327i	$-0.770841\pi$
$103$	−15.2610	−1.50371	−0.751856	+	0.659327i	$0.770841\pi$
$104$	7.76233	0.761159
$105$	0	0
$106$	0.260585	0.0253103
$107$	−12.1395	−1.17357	−0.586786	−	0.809742i	$-0.699607\pi$
$107$	−12.1395	−1.17357	−0.586786	+	0.809742i	$0.699607\pi$
$108$	0	0
$109$	8.07430	0.773377	0.386689	−	0.922210i	$-0.373619\pi$
$109$	8.07430	0.773377	0.386689	+	0.922210i	$0.373619\pi$
$110$	8.98586	0.856768
$111$	0	0
$112$	0.331239	0.0312992
$113$	18.4989	1.74023	0.870115	−	0.492848i	$-0.164044\pi$
$113$	18.4989	1.74023	0.870115	+	0.492848i	$0.164044\pi$
$114$	0	0
$115$	1.83959	0.171542
$116$	9.16163	0.850636
$117$	0	0
$118$	−1.76047	−0.162064
$119$	−3.67069	−0.336491
$120$	0	0
$121$	28.7184	2.61076
$122$	1.36566	0.123641
$123$	0	0
$124$	−6.07798	−0.545819
$125$	−12.1060	−1.08279
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	8.11223	0.717026
$129$	0	0
$130$	−4.02356	−0.352890
$131$	−4.26528	−0.372659	−0.186330	−	0.982487i	$-0.559659\pi$
$131$	−4.26528	−0.372659	−0.186330	+	0.982487i	$0.559659\pi$
$132$	0	0
$133$	4.30307	0.373124
$134$	4.94804	0.427446
$135$	0	0
$136$	−10.0970	−0.865811
$137$	−8.66409	−0.740223	−0.370112	−	0.928987i	$-0.620681\pi$
$137$	−8.66409	−0.740223	−0.370112	+	0.928987i	$0.620681\pi$
$138$	0	0
$139$	−20.6701	−1.75322	−0.876609	−	0.481203i	$-0.840200\pi$
$139$	−20.6701	−1.75322	−0.876609	+	0.481203i	$0.840200\pi$
$140$	−2.24507	−0.189743
$141$	0	0
$142$	5.41232	0.454192
$143$	−17.7845	−1.48722
$144$	0	0
$145$	−12.0049	−0.996953
$146$	−0.829802	−0.0686749
$147$	0	0
$148$	−5.78564	−0.475577
$149$	8.86934	0.726605	0.363302	−	0.931671i	$-0.381649\pi$
$149$	8.86934	0.726605	0.363302	+	0.931671i	$0.381649\pi$
$150$	0	0
$151$	1.72988	0.140776	0.0703879	−	0.997520i	$-0.477576\pi$
$151$	1.72988	0.140776	0.0703879	+	0.997520i	$0.477576\pi$
$152$	11.8365	0.960068
$153$	0	0
$154$	5.23903	0.422173
$155$	7.96425	0.639704
$156$	0	0
$157$	15.8721	1.26673	0.633364	−	0.773854i	$-0.281674\pi$
$157$	15.8721	1.26673	0.633364	+	0.773854i	$0.281674\pi$
$158$	3.20969	0.255349
$159$	0	0
$160$	−9.90819	−0.783311
$161$	1.07254	0.0845277
$162$	0	0
$163$	23.4618	1.83767	0.918835	−	0.394642i	$-0.129131\pi$
$163$	23.4618	1.83767	0.918835	+	0.394642i	$0.129131\pi$
$164$	−0.298013	−0.0232709
$165$	0	0
$166$	−1.75274	−0.136039
$167$	21.8922	1.69407	0.847036	−	0.531535i	$-0.178385\pi$
$167$	21.8922	1.69407	0.847036	+	0.531535i	$0.178385\pi$
$168$	0	0
$169$	−5.03669	−0.387438
$170$	5.23373	0.401408
$171$	0	0
$172$	−1.02336	−0.0780305
$173$	2.68178	0.203892	0.101946	−	0.994790i	$-0.467493\pi$
$173$	2.68178	0.203892	0.101946	+	0.994790i	$0.467493\pi$
$174$	0	0
$175$	−2.05818	−0.155584
$176$	−2.08755	−0.157355
$177$	0	0
$178$	0.345765	0.0259162
$179$	2.96401	0.221540	0.110770	−	0.993846i	$-0.464668\pi$
$179$	2.96401	0.221540	0.110770	+	0.993846i	$0.464668\pi$
$180$	0	0
$181$	−3.65550	−0.271711	−0.135856	−	0.990729i	$-0.543378\pi$
$181$	−3.65550	−0.271711	−0.135856	+	0.990729i	$0.543378\pi$
$182$	−2.34586	−0.173887
$183$	0	0
$184$	2.95024	0.217495
$185$	7.58119	0.557380
$186$	0	0
$187$	23.1336	1.69170
$188$	−9.09279	−0.663160
$189$	0	0
$190$	−6.13539	−0.445108
$191$	−20.0960	−1.45409	−0.727047	−	0.686587i	$-0.759108\pi$
$191$	−20.0960	−1.45409	−0.727047	+	0.686587i	$0.759108\pi$
$192$	0	0
$193$	−19.2213	−1.38358	−0.691791	−	0.722098i	$-0.743178\pi$
$193$	−19.2213	−1.38358	−0.691791	+	0.722098i	$0.743178\pi$
$194$	7.68171	0.551515
$195$	0	0
$196$	−1.30895	−0.0934963
$197$	−22.2096	−1.58237	−0.791183	−	0.611579i	$-0.790535\pi$
$197$	−22.2096	−1.58237	−0.791183	+	0.611579i	$0.790535\pi$
$198$	0	0
$199$	12.3056	0.872323	0.436162	−	0.899868i	$-0.356338\pi$
$199$	12.3056	0.872323	0.436162	+	0.899868i	$0.356338\pi$
$200$	−5.66146	−0.400326
$201$	0	0
$202$	11.5224	0.810711
$203$	−6.99923	−0.491250
$204$	0	0
$205$	0.390500	0.0272737
$206$	12.6864	0.883904
$207$	0	0
$208$	0.934736	0.0648123
$209$	−27.1190	−1.87586
$210$	0	0
$211$	−3.25081	−0.223795	−0.111897	−	0.993720i	$-0.535693\pi$
$211$	−3.25081	−0.223795	−0.111897	+	0.993720i	$0.535693\pi$
$212$	0.410314	0.0281805
$213$	0	0
$214$	10.0915	0.689842
$215$	1.34096	0.0914524
$216$	0	0
$217$	4.64341	0.315215
$218$	−6.71213	−0.454603
$219$	0	0
$220$	14.1490	0.953926
$221$	−10.3584	−0.696784
$222$	0	0
$223$	−14.5106	−0.971705	−0.485852	−	0.874041i	$-0.661491\pi$
$223$	−14.5106	−0.971705	−0.485852	+	0.874041i	$0.661491\pi$
$224$	−5.77678	−0.385978
$225$	0	0
$226$	−15.3781	−1.02293
$227$	−12.5720	−0.834435	−0.417218	−	0.908807i	$-0.636995\pi$
$227$	−12.5720	−0.834435	−0.417218	+	0.908807i	$0.636995\pi$
$228$	0	0
$229$	−9.51965	−0.629076	−0.314538	−	0.949245i	$-0.601850\pi$
$229$	−9.51965	−0.629076	−0.314538	+	0.949245i	$0.601850\pi$
$230$	−1.52924	−0.100835
$231$	0	0
$232$	−19.2529	−1.26401
$233$	2.10048	0.137607	0.0688036	−	0.997630i	$-0.478082\pi$
$233$	2.10048	0.137607	0.0688036	+	0.997630i	$0.478082\pi$
$234$	0	0
$235$	11.9147	0.777230
$236$	−2.77201	−0.180442
$237$	0	0
$238$	3.05143	0.197795
$239$	3.60174	0.232977	0.116489	−	0.993192i	$-0.462836\pi$
$239$	3.60174	0.232977	0.116489	+	0.993192i	$0.462836\pi$
$240$	0	0
$241$	−19.2665	−1.24106	−0.620532	−	0.784181i	$-0.713083\pi$
$241$	−19.2665	−1.24106	−0.620532	+	0.784181i	$0.713083\pi$
$242$	−23.8735	−1.53464
$243$	0	0
$244$	2.15035	0.137662
$245$	1.71517	0.109578
$246$	0	0
$247$	12.1430	0.772640
$248$	12.7727	0.811066
$249$	0	0
$250$	10.0637	0.636482
$251$	18.2768	1.15362	0.576811	−	0.816877i	$-0.304297\pi$
$251$	18.2768	1.15362	0.576811	+	0.816877i	$0.304297\pi$
$252$	0	0
$253$	−6.75940	−0.424960
$254$	0.831296	0.0521601
$255$	0	0
$256$	−15.0231	−0.938946
$257$	15.6453	0.975927	0.487963	−	0.872864i	$-0.337740\pi$
$257$	15.6453	0.975927	0.487963	+	0.872864i	$0.337740\pi$
$258$	0	0
$259$	4.42007	0.274650
$260$	−6.33545	−0.392908
$261$	0	0
$262$	3.54571	0.219055
$263$	−3.88781	−0.239733	−0.119866	−	0.992790i	$-0.538247\pi$
$263$	−3.88781	−0.239733	−0.119866	+	0.992790i	$0.538247\pi$
$264$	0	0
$265$	−0.537653	−0.0330278
$266$	−3.57712	−0.219328
$267$	0	0
$268$	7.79113	0.475919
$269$	26.5902	1.62123	0.810615	−	0.585579i	$-0.199133\pi$
$269$	26.5902	1.62123	0.810615	+	0.585579i	$0.199133\pi$
$270$	0	0
$271$	−12.0907	−0.734460	−0.367230	−	0.930130i	$-0.619694\pi$
$271$	−12.0907	−0.734460	−0.367230	+	0.930130i	$0.619694\pi$
$272$	−1.21588	−0.0737233
$273$	0	0
$274$	7.20242	0.435114
$275$	12.9712	0.782190
$276$	0	0
$277$	16.1812	0.972237	0.486118	−	0.873893i	$-0.338412\pi$
$277$	16.1812	0.972237	0.486118	+	0.873893i	$0.338412\pi$
$278$	17.1830	1.03057
$279$	0	0
$280$	4.71795	0.281952
$281$	−18.9746	−1.13193	−0.565964	−	0.824430i	$-0.691496\pi$
$281$	−18.9746	−1.13193	−0.565964	+	0.824430i	$0.691496\pi$
$282$	0	0
$283$	−14.0476	−0.835045	−0.417522	−	0.908667i	$-0.637101\pi$
$283$	−14.0476	−0.835045	−0.417522	+	0.908667i	$0.637101\pi$
$284$	8.52217	0.505698
$285$	0	0
$286$	14.7842	0.874209
$287$	0.227674	0.0134392
$288$	0	0
$289$	−3.52606	−0.207415
$290$	9.97962	0.586024
$291$	0	0
$292$	−1.30660	−0.0764627
$293$	−5.69187	−0.332523	−0.166261	−	0.986082i	$-0.553170\pi$
$293$	−5.69187	−0.332523	−0.166261	+	0.986082i	$0.553170\pi$
$294$	0	0
$295$	3.63229	0.211480
$296$	12.1583	0.706690
$297$	0	0
$298$	−7.37305	−0.427109
$299$	3.02663	0.175034
$300$	0	0
$301$	0.781819	0.0450633
$302$	−1.43804	−0.0827501
$303$	0	0
$304$	1.42535	0.0817493
$305$	−2.81770	−0.161341
$306$	0	0
$307$	−28.1042	−1.60399	−0.801995	−	0.597331i	$-0.796228\pi$
$307$	−28.1042	−1.60399	−0.801995	+	0.597331i	$0.796228\pi$
$308$	8.24932	0.470049
$309$	0	0
$310$	−6.62065	−0.376028
$311$	−10.5342	−0.597341	−0.298671	−	0.954356i	$-0.596543\pi$
$311$	−10.5342	−0.597341	−0.298671	+	0.954356i	$0.596543\pi$
$312$	0	0
$313$	−10.0212	−0.566434	−0.283217	−	0.959056i	$-0.591402\pi$
$313$	−10.0212	−0.566434	−0.283217	+	0.959056i	$0.591402\pi$
$314$	−13.1944	−0.744601
$315$	0	0
$316$	5.05394	0.284306
$317$	−7.70845	−0.432950	−0.216475	−	0.976288i	$-0.569456\pi$
$317$	−7.70845	−0.432950	−0.216475	+	0.976288i	$0.569456\pi$
$318$	0	0
$319$	44.1109	2.46974
$320$	7.10037	0.396923
$321$	0	0
$322$	−0.891595	−0.0496866
$323$	−15.7952	−0.878870
$324$	0	0
$325$	−5.80804	−0.322172
$326$	−19.5037	−1.08021
$327$	0	0
$328$	0.626265	0.0345797
$329$	6.94665	0.382981
$330$	0	0
$331$	12.1205	0.666201	0.333100	−	0.942891i	$-0.391905\pi$
$331$	12.1205	0.666201	0.333100	+	0.942891i	$0.391905\pi$
$332$	−2.75985	−0.151466
$333$	0	0
$334$	−18.1989	−0.995801
$335$	−10.2091	−0.557781
$336$	0	0
$337$	−24.2692	−1.32203	−0.661013	−	0.750374i	$-0.729873\pi$
$337$	−24.2692	−1.32203	−0.661013	+	0.750374i	$0.729873\pi$
$338$	4.18698	0.227742
$339$	0	0
$340$	8.24096	0.446929
$341$	−29.2639	−1.58473
$342$	0	0
$343$	1.00000	0.0539949
$344$	2.15056	0.115950
$345$	0	0
$346$	−2.22935	−0.119851
$347$	−13.9889	−0.750962	−0.375481	−	0.926830i	$-0.622522\pi$
$347$	−13.9889	−0.750962	−0.375481	+	0.926830i	$0.622522\pi$
$348$	0	0
$349$	−9.95992	−0.533142	−0.266571	−	0.963815i	$-0.585891\pi$
$349$	−9.95992	−0.533142	−0.266571	+	0.963815i	$0.585891\pi$
$350$	1.71095	0.0914544
$351$	0	0
$352$	36.4068	1.94049
$353$	31.6257	1.68327	0.841633	−	0.540050i	$-0.181595\pi$
$353$	31.6257	1.68327	0.841633	+	0.540050i	$0.181595\pi$
$354$	0	0
$355$	−11.1670	−0.592682
$356$	0.544437	0.0288551
$357$	0	0
$358$	−2.46396	−0.130225
$359$	22.5955	1.19255	0.596274	−	0.802781i	$-0.296647\pi$
$359$	22.5955	1.19255	0.596274	+	0.802781i	$0.296647\pi$
$360$	0	0
$361$	−0.483568	−0.0254510
$362$	3.03880	0.159716
$363$	0	0
$364$	−3.69376	−0.193606
$365$	1.71209	0.0896150
$366$	0	0
$367$	5.28894	0.276081	0.138040	−	0.990427i	$-0.455920\pi$
$367$	5.28894	0.276081	0.138040	+	0.990427i	$0.455920\pi$
$368$	0.355266	0.0185195
$369$	0	0
$370$	−6.30221	−0.327636
$371$	−0.313469	−0.0162745
$372$	0	0
$373$	20.0442	1.03785	0.518925	−	0.854820i	$-0.326333\pi$
$373$	20.0442	1.03785	0.518925	+	0.854820i	$0.326333\pi$
$374$	−19.2309	−0.994404
$375$	0	0
$376$	19.1082	0.985432
$377$	−19.7514	−1.01725
$378$	0	0
$379$	14.6363	0.751817	0.375909	−	0.926657i	$-0.377331\pi$
$379$	14.6363	0.751817	0.375909	+	0.926657i	$0.377331\pi$
$380$	−9.66071	−0.495584
$381$	0	0
$382$	16.7057	0.854738
$383$	−37.2043	−1.90105	−0.950525	−	0.310649i	$-0.899454\pi$
$383$	−37.2043	−1.90105	−0.950525	+	0.310649i	$0.899454\pi$
$384$	0	0
$385$	−10.8095	−0.550901
$386$	15.9786	0.813290
$387$	0	0
$388$	12.0955	0.614057
$389$	−28.9147	−1.46603	−0.733016	−	0.680212i	$-0.761888\pi$
$389$	−28.9147	−1.46603	−0.733016	+	0.680212i	$0.761888\pi$
$390$	0	0
$391$	−3.93695	−0.199100
$392$	2.75071	0.138932
$393$	0	0
$394$	18.4627	0.930138
$395$	−6.62240	−0.333209
$396$	0	0
$397$	−35.2847	−1.77089	−0.885445	−	0.464744i	$-0.846146\pi$
$397$	−35.2847	−1.77089	−0.885445	+	0.464744i	$0.846146\pi$
$398$	−10.2296	−0.512765
$399$	0	0
$400$	−0.681750	−0.0340875
$401$	20.6780	1.03261	0.516305	−	0.856405i	$-0.327307\pi$
$401$	20.6780	1.03261	0.516305	+	0.856405i	$0.327307\pi$
$402$	0	0
$403$	13.1034	0.652726
$404$	18.1430	0.902647
$405$	0	0
$406$	5.81843	0.288764
$407$	−27.8564	−1.38079
$408$	0	0
$409$	25.3476	1.25336	0.626679	−	0.779278i	$-0.284414\pi$
$409$	25.3476	1.25336	0.626679	+	0.779278i	$0.284414\pi$
$410$	−0.324621	−0.0160319
$411$	0	0
$412$	19.9759	0.984141
$413$	2.11774	0.104207
$414$	0	0
$415$	3.61636	0.177520
$416$	−16.3017	−0.799257
$417$	0	0
$418$	22.5439	1.10266
$419$	2.32255	0.113464	0.0567320	−	0.998389i	$-0.481932\pi$
$419$	2.32255	0.113464	0.0567320	+	0.998389i	$0.481932\pi$
$420$	0	0
$421$	36.8840	1.79762	0.898809	−	0.438341i	$-0.144434\pi$
$421$	36.8840	1.79762	0.898809	+	0.438341i	$0.144434\pi$
$422$	2.70238	0.131550
$423$	0	0
$424$	−0.862262	−0.0418751
$425$	7.55493	0.366468
$426$	0	0
$427$	−1.64281	−0.0795011
$428$	15.8900	0.768072
$429$	0	0
$430$	−1.11473	−0.0537571
$431$	18.3720	0.884946	0.442473	−	0.896782i	$-0.354101\pi$
$431$	18.3720	0.884946	0.442473	+	0.896782i	$0.354101\pi$
$432$	0	0
$433$	12.1349	0.583164	0.291582	−	0.956546i	$-0.405818\pi$
$433$	12.1349	0.583164	0.291582	+	0.956546i	$0.405818\pi$
$434$	−3.86004	−0.185288
$435$	0	0
$436$	−10.5688	−0.506155
$437$	4.61520	0.220775
$438$	0	0
$439$	−5.10805	−0.243794	−0.121897	−	0.992543i	$-0.538898\pi$
$439$	−5.10805	−0.243794	−0.121897	+	0.992543i	$0.538898\pi$
$440$	−29.7337	−1.41750
$441$	0	0
$442$	8.61092	0.409580
$443$	−2.48478	−0.118055	−0.0590277	−	0.998256i	$-0.518800\pi$
$443$	−2.48478	−0.118055	−0.0590277	+	0.998256i	$0.518800\pi$
$444$	0	0
$445$	−0.713400	−0.0338184
$446$	12.0626	0.571182
$447$	0	0
$448$	4.13974	0.195584
$449$	38.9469	1.83802	0.919010	−	0.394234i	$-0.128990\pi$
$449$	38.9469	1.83802	0.919010	+	0.394234i	$0.128990\pi$
$450$	0	0
$451$	−1.43486	−0.0675648
$452$	−24.2141	−1.13894
$453$	0	0
$454$	10.4511	0.490493
$455$	4.84011	0.226908
$456$	0	0
$457$	−13.5198	−0.632431	−0.316216	−	0.948687i	$-0.602412\pi$
$457$	−13.5198	−0.632431	−0.316216	+	0.948687i	$0.602412\pi$
$458$	7.91364	0.369780
$459$	0	0
$460$	−2.40792	−0.112270
$461$	29.9460	1.39472	0.697362	−	0.716719i	$-0.254357\pi$
$461$	29.9460	1.39472	0.697362	+	0.716719i	$0.254357\pi$
$462$	0	0
$463$	−1.16296	−0.0540472	−0.0270236	−	0.999635i	$-0.508603\pi$
$463$	−1.16296	−0.0540472	−0.0270236	+	0.999635i	$0.508603\pi$
$464$	−2.31842	−0.107630
$465$	0	0
$466$	−1.74612	−0.0808875
$467$	−27.9120	−1.29161	−0.645806	−	0.763502i	$-0.723478\pi$
$467$	−27.9120	−1.29161	−0.645806	+	0.763502i	$0.723478\pi$
$468$	0	0
$469$	−5.95221	−0.274847
$470$	−9.90464	−0.456867
$471$	0	0
$472$	5.82529	0.268131
$473$	−4.92722	−0.226554
$474$	0	0
$475$	−8.85649	−0.406364
$476$	4.80474	0.220225
$477$	0	0
$478$	−2.99411	−0.136947
$479$	−28.8892	−1.31998	−0.659990	−	0.751275i	$-0.729439\pi$
$479$	−28.8892	−1.31998	−0.659990	+	0.751275i	$0.729439\pi$
$480$	0	0
$481$	12.4731	0.568727
$482$	16.0161	0.729515
$483$	0	0
$484$	−37.5909	−1.70868
$485$	−15.8493	−0.719681
$486$	0	0
$487$	−24.6580	−1.11736	−0.558681	−	0.829383i	$-0.688692\pi$
$487$	−24.6580	−1.11736	−0.558681	+	0.829383i	$0.688692\pi$
$488$	−4.51890	−0.204561
$489$	0	0
$490$	−1.42582	−0.0644118
$491$	6.48797	0.292798	0.146399	−	0.989226i	$-0.453232\pi$
$491$	6.48797	0.292798	0.146399	+	0.989226i	$0.453232\pi$
$492$	0	0
$493$	25.6920	1.15711
$494$	−10.0944	−0.454169
$495$	0	0
$496$	1.53808	0.0690618
$497$	−6.51071	−0.292045
$498$	0	0
$499$	4.08922	0.183059	0.0915293	−	0.995802i	$-0.470824\pi$
$499$	4.08922	0.183059	0.0915293	+	0.995802i	$0.470824\pi$
$500$	15.8461	0.708660
$501$	0	0
$502$	−15.1934	−0.678117
$503$	−31.7289	−1.41472	−0.707361	−	0.706852i	$-0.750114\pi$
$503$	−31.7289	−1.41472	−0.707361	+	0.706852i	$0.750114\pi$
$504$	0	0
$505$	−23.7736	−1.05791
$506$	5.61906	0.249798
$507$	0	0
$508$	1.30895	0.0580752
$509$	−3.38619	−0.150090	−0.0750450	−	0.997180i	$-0.523910\pi$
$509$	−3.38619	−0.150090	−0.0750450	+	0.997180i	$0.523910\pi$
$510$	0	0
$511$	0.998203	0.0441579
$512$	−3.73579	−0.165100
$513$	0	0
$514$	−13.0059	−0.573664
$515$	−26.1753	−1.15342
$516$	0	0
$517$	−43.7795	−1.92542
$518$	−3.67439	−0.161443
$519$	0	0
$520$	13.3137	0.583846
$521$	−42.3641	−1.85601	−0.928004	−	0.372571i	$-0.878476\pi$
$521$	−42.3641	−1.85601	−0.928004	+	0.372571i	$0.878476\pi$
$522$	0	0
$523$	−35.2029	−1.53932	−0.769658	−	0.638456i	$-0.779573\pi$
$523$	−35.2029	−1.53932	−0.769658	+	0.638456i	$0.779573\pi$
$524$	5.58303	0.243896
$525$	0	0
$526$	3.23192	0.140918
$527$	−17.0445	−0.742470
$528$	0	0
$529$	−21.8497	−0.949985
$530$	0.446949	0.0194142
$531$	0	0
$532$	−5.63250	−0.244200
$533$	0.642480	0.0278289
$534$	0	0
$535$	−20.8214	−0.900187
$536$	−16.3728	−0.707198
$537$	0	0
$538$	−22.1043	−0.952983
$539$	−6.30225	−0.271457
$540$	0	0
$541$	−24.2460	−1.04241	−0.521207	−	0.853430i	$-0.674518\pi$
$541$	−24.2460	−1.04241	−0.521207	+	0.853430i	$0.674518\pi$
$542$	10.0510	0.431727
$543$	0	0
$544$	21.2048	0.909147
$545$	13.8488	0.593218
$546$	0	0
$547$	13.6074	0.581809	0.290905	−	0.956752i	$-0.406044\pi$
$547$	13.6074	0.581809	0.290905	+	0.956752i	$0.406044\pi$
$548$	11.3408	0.484457
$549$	0	0
$550$	−10.7829	−0.459783
$551$	−30.1182	−1.28308
$552$	0	0
$553$	−3.86107	−0.164189
$554$	−13.4514	−0.571495
$555$	0	0
$556$	27.0561	1.14744
$557$	10.8155	0.458266	0.229133	−	0.973395i	$-0.426411\pi$
$557$	10.8155	0.458266	0.229133	+	0.973395i	$0.426411\pi$
$558$	0	0
$559$	2.20624	0.0933141
$560$	0.568133	0.0240080
$561$	0	0
$562$	15.7735	0.665364
$563$	−32.0424	−1.35043	−0.675213	−	0.737623i	$-0.735948\pi$
$563$	−32.0424	−1.35043	−0.675213	+	0.737623i	$0.735948\pi$
$564$	0	0
$565$	31.7288	1.33484
$566$	11.6777	0.490852
$567$	0	0
$568$	−17.9091	−0.751449
$569$	−29.8595	−1.25178	−0.625888	−	0.779913i	$-0.715263\pi$
$569$	−29.8595	−1.25178	−0.625888	+	0.779913i	$0.715263\pi$
$570$	0	0
$571$	−41.1937	−1.72390	−0.861951	−	0.506992i	$-0.830757\pi$
$571$	−41.1937	−1.72390	−0.861951	+	0.506992i	$0.830757\pi$
$572$	23.2790	0.973345
$573$	0	0
$574$	−0.189264	−0.00789973
$575$	−2.20747	−0.0920579
$576$	0	0
$577$	−4.05530	−0.168824	−0.0844121	−	0.996431i	$-0.526901\pi$
$577$	−4.05530	−0.168824	−0.0844121	+	0.996431i	$0.526901\pi$
$578$	2.93120	0.121922
$579$	0	0
$580$	15.7138	0.652480
$581$	2.10845	0.0874732
$582$	0	0
$583$	1.97556	0.0818192
$584$	2.74577	0.113621
$585$	0	0
$586$	4.73163	0.195462
$587$	2.36252	0.0975118	0.0487559	−	0.998811i	$-0.484474\pi$
$587$	2.36252	0.0975118	0.0487559	+	0.998811i	$0.484474\pi$
$588$	0	0
$589$	19.9809	0.823300
$590$	−3.01950	−0.124311
$591$	0	0
$592$	1.46410	0.0601742
$593$	−3.17018	−0.130184	−0.0650918	−	0.997879i	$-0.520734\pi$
$593$	−3.17018	−0.130184	−0.0650918	+	0.997879i	$0.520734\pi$
$594$	0	0
$595$	−6.29587	−0.258105
$596$	−11.6095	−0.475544
$597$	0	0
$598$	−2.51602	−0.102888
$599$	−13.0217	−0.532050	−0.266025	−	0.963966i	$-0.585710\pi$
$599$	−13.0217	−0.532050	−0.266025	+	0.963966i	$0.585710\pi$
$600$	0	0
$601$	10.0547	0.410139	0.205069	−	0.978747i	$-0.434258\pi$
$601$	10.0547	0.410139	0.205069	+	0.978747i	$0.434258\pi$
$602$	−0.649923	−0.0264889
$603$	0	0
$604$	−2.26433	−0.0921341
$605$	49.2570	2.00258
$606$	0	0
$607$	−34.1582	−1.38644	−0.693219	−	0.720727i	$-0.743808\pi$
$607$	−34.1582	−1.38644	−0.693219	+	0.720727i	$0.743808\pi$
$608$	−24.8579	−1.00812
$609$	0	0
$610$	2.34235	0.0948388
$611$	19.6030	0.793051
$612$	0	0
$613$	−11.3482	−0.458349	−0.229175	−	0.973385i	$-0.573603\pi$
$613$	−11.3482	−0.458349	−0.229175	+	0.973385i	$0.573603\pi$
$614$	23.3629	0.942849
$615$	0	0
$616$	−17.3357	−0.698475
$617$	−37.2169	−1.49830	−0.749149	−	0.662402i	$-0.769537\pi$
$617$	−37.2169	−1.49830	−0.749149	+	0.662402i	$0.769537\pi$
$618$	0	0
$619$	−1.53605	−0.0617392	−0.0308696	−	0.999523i	$-0.509828\pi$
$619$	−1.53605	−0.0617392	−0.0308696	+	0.999523i	$0.509828\pi$
$620$	−10.4248	−0.418670
$621$	0	0
$622$	8.75706	0.351126
$623$	−0.415935	−0.0166641
$624$	0	0
$625$	−10.4730	−0.418920
$626$	8.33061	0.332958
$627$	0	0
$628$	−20.7757	−0.829040
$629$	−16.2247	−0.646921
$630$	0	0
$631$	19.8102	0.788633	0.394316	−	0.918975i	$-0.370981\pi$
$631$	19.8102	0.788633	0.394316	+	0.918975i	$0.370981\pi$
$632$	−10.6207	−0.422468
$633$	0	0
$634$	6.40800	0.254494
$635$	−1.71517	−0.0680646
$636$	0	0
$637$	2.82193	0.111809
$638$	−36.6692	−1.45175
$639$	0	0
$640$	13.9139	0.549994
$641$	12.7864	0.505032	0.252516	−	0.967593i	$-0.418742\pi$
$641$	12.7864	0.505032	0.252516	+	0.967593i	$0.418742\pi$
$642$	0	0
$643$	−9.72685	−0.383589	−0.191795	−	0.981435i	$-0.561431\pi$
$643$	−9.72685	−0.383589	−0.191795	+	0.981435i	$0.561431\pi$
$644$	−1.40389	−0.0553212
$645$	0	0
$646$	13.1305	0.516613
$647$	−43.0173	−1.69118	−0.845592	−	0.533829i	$-0.820753\pi$
$647$	−43.0173	−1.69118	−0.845592	+	0.533829i	$0.820753\pi$
$648$	0	0
$649$	−13.3465	−0.523896
$650$	4.82820	0.189378
$651$	0	0
$652$	−30.7103	−1.20271
$653$	−6.94865	−0.271922	−0.135961	−	0.990714i	$-0.543412\pi$
$653$	−6.94865	−0.271922	−0.135961	+	0.990714i	$0.543412\pi$
$654$	0	0
$655$	−7.31570	−0.285848
$656$	0.0754145	0.00294444
$657$	0	0
$658$	−5.77472	−0.225122
$659$	29.5240	1.15009	0.575046	−	0.818121i	$-0.304984\pi$
$659$	29.5240	1.15009	0.575046	+	0.818121i	$0.304984\pi$
$660$	0	0
$661$	−17.4148	−0.677355	−0.338678	−	0.940902i	$-0.609980\pi$
$661$	−17.4148	−0.677355	−0.338678	+	0.940902i	$0.609980\pi$
$662$	−10.0757	−0.391603
$663$	0	0
$664$	5.79974	0.225074
$665$	7.38052	0.286204
$666$	0	0
$667$	−7.50694	−0.290670
$668$	−28.6558	−1.10873
$669$	0	0
$670$	8.48675	0.327872
$671$	10.3534	0.399689
$672$	0	0
$673$	−39.6379	−1.52793	−0.763965	−	0.645258i	$-0.776750\pi$
$673$	−39.6379	−1.52793	−0.763965	+	0.645258i	$0.776750\pi$
$674$	20.1749	0.777107
$675$	0	0
$676$	6.59276	0.253568
$677$	25.3837	0.975574	0.487787	−	0.872963i	$-0.337804\pi$
$677$	25.3837	0.975574	0.487787	+	0.872963i	$0.337804\pi$
$678$	0	0
$679$	−9.24065	−0.354624
$680$	−17.3181	−0.664120
$681$	0	0
$682$	24.3270	0.931528
$683$	−8.40979	−0.321792	−0.160896	−	0.986971i	$-0.551438\pi$
$683$	−8.40979	−0.321792	−0.160896	+	0.986971i	$0.551438\pi$
$684$	0	0
$685$	−14.8604	−0.567788
$686$	−0.831296	−0.0317390
$687$	0	0
$688$	0.258969	0.00987311
$689$	−0.884587	−0.0337001
$690$	0	0
$691$	−22.9450	−0.872868	−0.436434	−	0.899736i	$-0.643759\pi$
$691$	−22.9450	−0.872868	−0.436434	+	0.899736i	$0.643759\pi$
$692$	−3.51031	−0.133442
$693$	0	0
$694$	11.6289	0.441427
$695$	−35.4529	−1.34480
$696$	0	0
$697$	−0.835719	−0.0316551
$698$	8.27964	0.313389
$699$	0	0
$700$	2.69405	0.101825
$701$	−0.256451	−0.00968603	−0.00484301	−	0.999988i	$-0.501542\pi$
$701$	−0.256451	−0.00968603	−0.00484301	+	0.999988i	$0.501542\pi$
$702$	0	0
$703$	19.0199	0.717349
$704$	−26.0897	−0.983291
$705$	0	0
$706$	−26.2903	−0.989448
$707$	−13.8607	−0.521287
$708$	0	0
$709$	−19.5392	−0.733811	−0.366906	−	0.930258i	$-0.619583\pi$
$709$	−19.5392	−0.733811	−0.366906	+	0.930258i	$0.619583\pi$
$710$	9.28307	0.348387
$711$	0	0
$712$	−1.14412	−0.0428776
$713$	4.98023	0.186511
$714$	0	0
$715$	−30.5036	−1.14077
$716$	−3.87973	−0.144992
$717$	0	0
$718$	−18.7836	−0.700997
$719$	−10.9444	−0.408156	−0.204078	−	0.978955i	$-0.565420\pi$
$719$	−10.9444	−0.408156	−0.204078	+	0.978955i	$0.565420\pi$
$720$	0	0
$721$	−15.2610	−0.568350
$722$	0.401988	0.0149604
$723$	0	0
$724$	4.78486	0.177828
$725$	14.4057	0.535013
$726$	0	0
$727$	9.91038	0.367556	0.183778	−	0.982968i	$-0.441167\pi$
$727$	9.91038	0.367556	0.183778	+	0.982968i	$0.441167\pi$
$728$	7.76233	0.287691
$729$	0	0
$730$	−1.42325	−0.0526770
$731$	−2.86981	−0.106144
$732$	0	0
$733$	−16.0686	−0.593509	−0.296754	−	0.954954i	$-0.595904\pi$
$733$	−16.0686	−0.593509	−0.296754	+	0.954954i	$0.595904\pi$
$734$	−4.39667	−0.162284
$735$	0	0
$736$	−6.19581	−0.228381
$737$	37.5123	1.38178
$738$	0	0
$739$	−3.90553	−0.143667	−0.0718336	−	0.997417i	$-0.522885\pi$
$739$	−3.90553	−0.143667	−0.0718336	+	0.997417i	$0.522885\pi$
$740$	−9.92338	−0.364791
$741$	0	0
$742$	0.260585	0.00956638
$743$	0.181845	0.00667125	0.00333562	−	0.999994i	$-0.498938\pi$
$743$	0.181845	0.00667125	0.00333562	+	0.999994i	$0.498938\pi$
$744$	0	0
$745$	15.2125	0.557342
$746$	−16.6627	−0.610063
$747$	0	0
$748$	−30.2807	−1.10717
$749$	−12.1395	−0.443568
$750$	0	0
$751$	−44.8201	−1.63551	−0.817755	−	0.575567i	$-0.804782\pi$
$751$	−44.8201	−1.63551	−0.817755	+	0.575567i	$0.804782\pi$
$752$	2.30100	0.0839089
$753$	0	0
$754$	16.4192	0.597953
$755$	2.96705	0.107982
$756$	0	0
$757$	34.7956	1.26467	0.632334	−	0.774696i	$-0.282097\pi$
$757$	34.7956	1.26467	0.632334	+	0.774696i	$0.282097\pi$
$758$	−12.1671	−0.441929
$759$	0	0
$760$	20.3017	0.736420
$761$	48.7895	1.76862	0.884310	−	0.466900i	$-0.154629\pi$
$761$	48.7895	1.76862	0.884310	+	0.466900i	$0.154629\pi$
$762$	0	0
$763$	8.07430	0.292309
$764$	26.3046	0.951667
$765$	0	0
$766$	30.9277	1.11746
$767$	5.97611	0.215785
$768$	0	0
$769$	26.8710	0.968993	0.484497	−	0.874793i	$-0.339003\pi$
$769$	26.8710	0.968993	0.484497	+	0.874793i	$0.339003\pi$
$770$	8.98586	0.323828
$771$	0	0
$772$	25.1597	0.905518
$773$	9.79444	0.352282	0.176141	−	0.984365i	$-0.443639\pi$
$773$	9.79444	0.352282	0.176141	+	0.984365i	$0.443639\pi$
$774$	0	0
$775$	−9.55696	−0.343296
$776$	−25.4184	−0.912467
$777$	0	0
$778$	24.0366	0.861755
$779$	0.979696	0.0351013
$780$	0	0
$781$	41.0321	1.46824
$782$	3.27277	0.117034
$783$	0	0
$784$	0.331239	0.0118300
$785$	27.2233	0.971642
$786$	0	0
$787$	−35.8458	−1.27777	−0.638883	−	0.769304i	$-0.720603\pi$
$787$	−35.8458	−1.27777	−0.638883	+	0.769304i	$0.720603\pi$
$788$	29.0712	1.03562
$789$	0	0
$790$	5.50517	0.195865
$791$	18.4989	0.657745
$792$	0	0
$793$	−4.63590	−0.164626
$794$	29.3320	1.04096
$795$	0	0
$796$	−16.1074	−0.570913
$797$	35.2094	1.24718	0.623590	−	0.781752i	$-0.285673\pi$
$797$	35.2094	1.24718	0.623590	+	0.781752i	$0.285673\pi$
$798$	0	0
$799$	−25.4990	−0.902088
$800$	11.8897	0.420363
$801$	0	0
$802$	−17.1895	−0.606984
$803$	−6.29093	−0.222002
$804$	0	0
$805$	1.83959	0.0648369
$806$	−10.8928	−0.383682
$807$	0	0
$808$	−38.1269	−1.34130
$809$	−31.3378	−1.10178	−0.550889	−	0.834579i	$-0.685711\pi$
$809$	−31.3378	−1.10178	−0.550889	+	0.834579i	$0.685711\pi$
$810$	0	0
$811$	8.40960	0.295301	0.147650	−	0.989040i	$-0.452829\pi$
$811$	8.40960	0.295301	0.147650	+	0.989040i	$0.452829\pi$
$812$	9.16163	0.321510
$813$	0	0
$814$	23.1569	0.811649
$815$	40.2411	1.40958
$816$	0	0
$817$	3.36422	0.117699
$818$	−21.0713	−0.736742
$819$	0	0
$820$	−0.511144	−0.0178499
$821$	37.1830	1.29769	0.648847	−	0.760919i	$-0.275251\pi$
$821$	37.1830	1.29769	0.648847	+	0.760919i	$0.275251\pi$
$822$	0	0
$823$	38.4266	1.33947	0.669733	−	0.742602i	$-0.266409\pi$
$823$	38.4266	1.33947	0.669733	+	0.742602i	$0.266409\pi$
$824$	−41.9787	−1.46240
$825$	0	0
$826$	−1.76047	−0.0612544
$827$	−19.1887	−0.667255	−0.333627	−	0.942705i	$-0.608273\pi$
$827$	−19.1887	−0.667255	−0.333627	+	0.942705i	$0.608273\pi$
$828$	0	0
$829$	−16.9870	−0.589984	−0.294992	−	0.955500i	$-0.595317\pi$
$829$	−16.9870	−0.589984	−0.294992	+	0.955500i	$0.595317\pi$
$830$	−3.00626	−0.104349
$831$	0	0
$832$	11.6821	0.405003
$833$	−3.67069	−0.127182
$834$	0	0
$835$	37.5490	1.29944
$836$	35.4974	1.22770
$837$	0	0
$838$	−1.93073	−0.0666958
$839$	50.1822	1.73248	0.866241	−	0.499626i	$-0.166529\pi$
$839$	50.1822	1.73248	0.866241	+	0.499626i	$0.166529\pi$
$840$	0	0
$841$	19.9893	0.689285
$842$	−30.6615	−1.05667
$843$	0	0
$844$	4.25513	0.146468
$845$	−8.63880	−0.297184
$846$	0	0
$847$	28.7184	0.986775
$848$	−0.103833	−0.00356564
$849$	0	0
$850$	−6.28038	−0.215415
$851$	4.74069	0.162509
$852$	0	0
$853$	−13.6028	−0.465750	−0.232875	−	0.972507i	$-0.574813\pi$
$853$	−13.6028	−0.465750	−0.232875	+	0.972507i	$0.574813\pi$
$854$	1.36566	0.0467319
$855$	0	0
$856$	−33.3923	−1.14133
$857$	7.98670	0.272820	0.136410	−	0.990652i	$-0.456443\pi$
$857$	7.98670	0.272820	0.136410	+	0.990652i	$0.456443\pi$
$858$	0	0
$859$	−56.8903	−1.94107	−0.970537	−	0.240954i	$-0.922540\pi$
$859$	−56.8903	−1.94107	−0.970537	+	0.240954i	$0.922540\pi$
$860$	−1.75524	−0.0598532
$861$	0	0
$862$	−15.2725	−0.520185
$863$	−24.5638	−0.836160	−0.418080	−	0.908410i	$-0.637297\pi$
$863$	−24.5638	−0.836160	−0.418080	+	0.908410i	$0.637297\pi$
$864$	0	0
$865$	4.59972	0.156395
$866$	−10.0877	−0.342792
$867$	0	0
$868$	−6.07798	−0.206300
$869$	24.3334	0.825455
$870$	0	0
$871$	−16.7967	−0.569135
$872$	22.2101	0.752128
$873$	0	0
$874$	−3.83660	−0.129775
$875$	−12.1060	−0.409258
$876$	0	0
$877$	−10.2102	−0.344775	−0.172387	−	0.985029i	$-0.555148\pi$
$877$	−10.2102	−0.344775	−0.172387	+	0.985029i	$0.555148\pi$
$878$	4.24630	0.143306
$879$	0	0
$880$	−3.58052	−0.120699
$881$	−55.4646	−1.86865	−0.934325	−	0.356423i	$-0.883996\pi$
$881$	−55.4646	−1.86865	−0.934325	+	0.356423i	$0.883996\pi$
$882$	0	0
$883$	−44.3327	−1.49191	−0.745957	−	0.665994i	$-0.768008\pi$
$883$	−44.3327	−1.49191	−0.745957	+	0.665994i	$0.768008\pi$
$884$	13.5586	0.456027
$885$	0	0
$886$	2.06558	0.0693947
$887$	−1.91326	−0.0642410	−0.0321205	−	0.999484i	$-0.510226\pi$
$887$	−1.91326	−0.0642410	−0.0321205	+	0.999484i	$0.510226\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	0.593047	0.0198790
$891$	0	0
$892$	18.9937	0.635955
$893$	29.8919	1.00029
$894$	0	0
$895$	5.08378	0.169932
$896$	8.11223	0.271010
$897$	0	0
$898$	−32.3764	−1.08042
$899$	−32.5003	−1.08395
$900$	0	0
$901$	1.15064	0.0383335
$902$	1.19279	0.0397156
$903$	0	0
$904$	50.8852	1.69242
$905$	−6.26982	−0.208416
$906$	0	0
$907$	22.8392	0.758362	0.379181	−	0.925322i	$-0.376206\pi$
$907$	22.8392	0.758362	0.379181	+	0.925322i	$0.376206\pi$
$908$	16.4561	0.546116
$909$	0	0
$910$	−4.02356	−0.133380
$911$	−41.8099	−1.38522	−0.692612	−	0.721310i	$-0.743540\pi$
$911$	−41.8099	−1.38522	−0.692612	+	0.721310i	$0.743540\pi$
$912$	0	0
$913$	−13.2880	−0.439768
$914$	11.2390	0.371753
$915$	0	0
$916$	12.4607	0.411714
$917$	−4.26528	−0.140852
$918$	0	0
$919$	24.2427	0.799692	0.399846	−	0.916582i	$-0.369064\pi$
$919$	24.2427	0.799692	0.399846	+	0.916582i	$0.369064\pi$
$920$	5.06018	0.166829
$921$	0	0
$922$	−24.8940	−0.819839
$923$	−18.3728	−0.604747
$924$	0	0
$925$	−9.09729	−0.299117
$926$	0.966761	0.0317697
$927$	0	0
$928$	40.4331	1.32728
$929$	−46.8694	−1.53774	−0.768868	−	0.639407i	$-0.779180\pi$
$929$	−46.8694	−1.53774	−0.768868	+	0.639407i	$0.779180\pi$
$930$	0	0
$931$	4.30307	0.141028
$932$	−2.74942	−0.0900603
$933$	0	0
$934$	23.2031	0.759228
$935$	39.6781	1.29761
$936$	0	0
$937$	−16.9803	−0.554722	−0.277361	−	0.960766i	$-0.589460\pi$
$937$	−16.9803	−0.554722	−0.277361	+	0.960766i	$0.589460\pi$
$938$	4.94804	0.161559
$939$	0	0
$940$	−15.5957	−0.508676
$941$	5.11076	0.166606	0.0833030	−	0.996524i	$-0.473453\pi$
$941$	5.11076	0.166606	0.0833030	+	0.996524i	$0.473453\pi$
$942$	0	0
$943$	0.244188	0.00795187
$944$	0.701478	0.0228312
$945$	0	0
$946$	4.09598	0.133172
$947$	43.6068	1.41703	0.708516	−	0.705695i	$-0.249365\pi$
$947$	43.6068	1.41703	0.708516	+	0.705695i	$0.249365\pi$
$948$	0	0
$949$	2.81686	0.0914393
$950$	7.36236	0.238867
$951$	0	0
$952$	−10.0970	−0.327246
$953$	12.1825	0.394628	0.197314	−	0.980340i	$-0.436778\pi$
$953$	12.1825	0.394628	0.197314	+	0.980340i	$0.436778\pi$
$954$	0	0
$955$	−34.4681	−1.11536
$956$	−4.71449	−0.152477
$957$	0	0
$958$	24.0154	0.775903
$959$	−8.66409	−0.279778
$960$	0	0
$961$	−9.43876	−0.304476
$962$	−10.3689	−0.334306
$963$	0	0
$964$	25.2188	0.812244
$965$	−32.9679	−1.06128
$966$	0	0
$967$	−37.5746	−1.20832	−0.604159	−	0.796864i	$-0.706491\pi$
$967$	−37.5746	−1.20832	−0.604159	+	0.796864i	$0.706491\pi$
$968$	78.9961	2.53903
$969$	0	0
$970$	13.1755	0.423039
$971$	−12.7002	−0.407568	−0.203784	−	0.979016i	$-0.565324\pi$
$971$	−12.7002	−0.407568	−0.203784	+	0.979016i	$0.565324\pi$
$972$	0	0
$973$	−20.6701	−0.662654
$974$	20.4981	0.656801
$975$	0	0
$976$	−0.544163	−0.0174182
$977$	26.7450	0.855649	0.427825	−	0.903862i	$-0.359280\pi$
$977$	26.7450	0.855649	0.427825	+	0.903862i	$0.359280\pi$
$978$	0	0
$979$	2.62133	0.0837779
$980$	−2.24507	−0.0717162
$981$	0	0
$982$	−5.39342	−0.172111
$983$	33.7781	1.07735	0.538677	−	0.842512i	$-0.318924\pi$
$983$	33.7781	1.07735	0.538677	+	0.842512i	$0.318924\pi$
$984$	0	0
$985$	−38.0933	−1.21375
$986$	−21.3576	−0.680166
$987$	0	0
$988$	−15.8945	−0.505672
$989$	0.838530	0.0266637
$990$	0	0
$991$	−59.6306	−1.89423	−0.947114	−	0.320898i	$-0.896015\pi$
$991$	−59.6306	−1.89423	−0.947114	+	0.320898i	$0.896015\pi$
$992$	−26.8240	−0.851662
$993$	0	0
$994$	5.41232	0.171668
$995$	21.1063	0.669115
$996$	0	0
$997$	−24.9160	−0.789099	−0.394550	−	0.918875i	$-0.629099\pi$
$997$	−24.9160	−0.789099	−0.394550	+	0.918875i	$0.629099\pi$
$998$	−3.39935	−0.107605
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.w.1.9		20
3.2	odd	2		889.2.a.d.1.12	✓	20
21.20	even	2		6223.2.a.l.1.12		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.d.1.12	✓	20	3.2	odd	2
6223.2.a.l.1.12		20	21.20	even	2
8001.2.a.w.1.9		20	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)

\(f(q)\)	\(=\)	\(q-0.831296 q^{2} -1.30895 q^{4} +1.71517 q^{5} +1.00000 q^{7} +2.75071 q^{8} -1.42582 q^{10} -6.30225 q^{11} +2.82193 q^{13} -0.831296 q^{14} +0.331239 q^{16} -3.67069 q^{17} +4.30307 q^{19} -2.24507 q^{20} +5.23903 q^{22} +1.07254 q^{23} -2.05818 q^{25} -2.34586 q^{26} -1.30895 q^{28} -6.99923 q^{29} +4.64341 q^{31} -5.77678 q^{32} +3.05143 q^{34} +1.71517 q^{35} +4.42007 q^{37} -3.57712 q^{38} +4.71795 q^{40} +0.227674 q^{41} +0.781819 q^{43} +8.24932 q^{44} -0.891595 q^{46} +6.94665 q^{47} +1.00000 q^{49} +1.71095 q^{50} -3.69376 q^{52} -0.313469 q^{53} -10.8095 q^{55} +2.75071 q^{56} +5.81843 q^{58} +2.11774 q^{59} -1.64281 q^{61} -3.86004 q^{62} +4.13974 q^{64} +4.84011 q^{65} -5.95221 q^{67} +4.80474 q^{68} -1.42582 q^{70} -6.51071 q^{71} +0.998203 q^{73} -3.67439 q^{74} -5.63250 q^{76} -6.30225 q^{77} -3.86107 q^{79} +0.568133 q^{80} -0.189264 q^{82} +2.10845 q^{83} -6.29587 q^{85} -0.649923 q^{86} -17.3357 q^{88} -0.415935 q^{89} +2.82193 q^{91} -1.40389 q^{92} -5.77472 q^{94} +7.38052 q^{95} -9.24065 q^{97} -0.831296 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28}+ \cdots - 8 q^{98}+O(q^{100}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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