LMFDB - Embedded newform 2000.2.c.i.1249.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2000,2,Mod(1249,2000)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2000.1249"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2000 = 2^{4} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2000.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-22,0,-10,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$15.9700804043$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.25728160000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 23x^{6} + 181x^{4} + 540x^{2} + 400 $ x^8 + 23x^6 + 181x^4 + 540*x^2 + 400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1000)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.7
Root			$2.65792i$ of defining polynomial
Character	$\chi$	$=$	2000.1249
Dual form			2000.2.c.i.1249.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.65792i q^{3} -4.68257i q^{7} -4.06454 q^{9} +0.0398855 q^{11} +1.02465i q^{13} +4.95852i q^{17} -8.21924 q^{19} +12.4459 q^{21} -4.15471i q^{23} -2.82945i q^{27} +0.878753 q^{29} -5.45690 q^{31} +0.106013i q^{33} -9.51202i q^{37} -2.72344 q^{39} -5.28378 q^{41} -8.95852i q^{43} -9.09017i q^{47} -14.9265 q^{49} -13.1794 q^{51} -1.18095i q^{53} -21.8461i q^{57} +9.34049 q^{59} +2.57815 q^{61} +19.0325i q^{63} +8.46015i q^{67} +11.0429 q^{69} -15.0383 q^{71} -3.58597i q^{73} -0.186767i q^{77} -10.2439 q^{79} -4.67315 q^{81} -9.65890i q^{83} +2.33565i q^{87} -0.759096 q^{89} +4.79800 q^{91} -14.5040i q^{93} -6.67217i q^{97} -0.162116 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 22 q^{9} - 10 q^{11} + 6 q^{21} - 20 q^{29} - 18 q^{31} + 42 q^{39} + 34 q^{41} - 34 q^{49} - 50 q^{51} + 36 q^{59} + 22 q^{61} + 16 q^{69} - 52 q^{71} - 16 q^{79} - 8 q^{81} - 10 q^{89} + 66 q^{91}+ \cdots + 16 q^{99}+O(q^{100}) $ 8 * q - 22 * q^9 - 10 * q^11 + 6 * q^21 - 20 * q^29 - 18 * q^31 + 42 * q^39 + 34 * q^41 - 34 * q^49 - 50 * q^51 + 36 * q^59 + 22 * q^61 + 16 * q^69 - 52 * q^71 - 16 * q^79 - 8 * q^81 - 10 * q^89 + 66 * q^91 + 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2000\mathbb{Z}\right)^\times$.

$n$	$501$	$751$	$1377$
$\chi(n)$	$1$	$1$	$-1$

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	0
$2$	0	0
$3$	2.65792i	1.53455i	0.641318	+	0.767275i	$0.278388\pi$
$3$	2.65792i	1.53455i	−0.641318	+	0.767275i	$0.721612\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.68257i	− 1.76985i	−0.465738	−	0.884923i	$-0.654211\pi$
$7$	− 4.68257i	− 1.76985i	0.465738	−	0.884923i	$-0.345789\pi$
$8$	0	0
$9$	−4.06454	−1.35485
$10$	0	0
$11$	0.0398855	0.0120259	0.00601297	−	0.999982i	$-0.498086\pi$
$11$	0.0398855	0.0120259	0.00601297	+	0.999982i	$0.498086\pi$
$12$	0	0
$13$	1.02465i	0.284187i	0.989853	+	0.142093i	$0.0453834\pi$
$13$	1.02465i	0.284187i	−0.989853	+	0.142093i	$0.954617\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.95852i	1.20262i	0.799016	+	0.601309i	$0.205354\pi$
$17$	4.95852i	1.20262i	−0.799016	+	0.601309i	$0.794646\pi$
$18$	0	0
$19$	−8.21924	−1.88562	−0.942812	−	0.333326i	$-0.891829\pi$
$19$	−8.21924	−1.88562	−0.942812	+	0.333326i	$0.891829\pi$
$20$	0	0
$21$	12.4459	2.71592
$22$	0	0
$23$	− 4.15471i	− 0.866316i	−0.901318	−	0.433158i	$-0.857399\pi$
$23$	− 4.15471i	− 0.866316i	0.901318	−	0.433158i	$-0.142601\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2.82945i	− 0.544528i
$28$	0	0
$29$	0.878753	0.163180	0.0815901	−	0.996666i	$-0.474000\pi$
$29$	0.878753	0.163180	0.0815901	+	0.996666i	$0.474000\pi$
$30$	0	0
$31$	−5.45690	−0.980088	−0.490044	−	0.871698i	$-0.663019\pi$
$31$	−5.45690	−0.980088	−0.490044	+	0.871698i	$0.663019\pi$
$32$	0	0
$33$	0.106013i	0.0184544i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 9.51202i	− 1.56377i	−0.623425	−	0.781883i	$-0.714259\pi$
$37$	− 9.51202i	− 1.56377i	0.623425	−	0.781883i	$-0.285741\pi$
$38$	0	0
$39$	−2.72344	−0.436099
$40$	0	0
$41$	−5.28378	−0.825188	−0.412594	−	0.910915i	$-0.635377\pi$
$41$	−5.28378	−0.825188	−0.412594	+	0.910915i	$0.635377\pi$
$42$	0	0
$43$	− 8.95852i	− 1.36616i	−0.730343	−	0.683081i	$-0.760640\pi$
$43$	− 8.95852i	− 1.36616i	0.730343	−	0.683081i	$-0.239360\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 9.09017i	− 1.32594i	−0.748647	−	0.662969i	$-0.769296\pi$
$47$	− 9.09017i	− 1.32594i	0.748647	−	0.662969i	$-0.230704\pi$
$48$	0	0
$49$	−14.9265	−2.13235
$50$	0	0
$51$	−13.1794	−1.84548
$52$	0	0
$53$	− 1.18095i	− 0.162216i	−0.996705	−	0.0811078i	$-0.974154\pi$
$53$	− 1.18095i	− 0.162216i	0.996705	−	0.0811078i	$-0.0258458\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 21.8461i	− 2.89358i
$58$	0	0
$59$	9.34049	1.21603	0.608014	−	0.793926i	$-0.291966\pi$
$59$	9.34049	1.21603	0.608014	+	0.793926i	$0.291966\pi$
$60$	0	0
$61$	2.57815	0.330098	0.165049	−	0.986285i	$-0.447222\pi$
$61$	2.57815	0.330098	0.165049	+	0.986285i	$0.447222\pi$
$62$	0	0
$63$	19.0325i	2.39787i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.46015i	1.03357i	0.856115	+	0.516786i	$0.172872\pi$
$67$	8.46015i	1.03357i	−0.856115	+	0.516786i	$0.827128\pi$
$68$	0	0
$69$	11.0429	1.32941
$70$	0	0
$71$	−15.0383	−1.78472	−0.892359	−	0.451327i	$-0.850951\pi$
$71$	−15.0383	−1.78472	−0.892359	+	0.451327i	$0.850951\pi$
$72$	0	0
$73$	− 3.58597i	− 0.419706i	−0.977733	−	0.209853i	$-0.932701\pi$
$73$	− 3.58597i	− 0.419706i	0.977733	−	0.209853i	$-0.0672986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.186767i	− 0.0212840i
$78$	0	0
$79$	−10.2439	−1.15253	−0.576264	−	0.817264i	$-0.695490\pi$
$79$	−10.2439	−1.15253	−0.576264	+	0.817264i	$0.695490\pi$
$80$	0	0
$81$	−4.67315	−0.519239
$82$	0	0
$83$	− 9.65890i	− 1.06020i	−0.847934	−	0.530101i	$-0.822154\pi$
$83$	− 9.65890i	− 1.06020i	0.847934	−	0.530101i	$-0.177846\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.33565i	0.250408i
$88$	0	0
$89$	−0.759096	−0.0804640	−0.0402320	−	0.999190i	$-0.512810\pi$
$89$	−0.759096	−0.0804640	−0.0402320	+	0.999190i	$0.512810\pi$
$90$	0	0
$91$	4.79800	0.502967
$92$	0	0
$93$	− 14.5040i	− 1.50400i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 6.67217i	− 0.677456i	−0.940884	−	0.338728i	$-0.890003\pi$
$97$	− 6.67217i	− 0.677456i	0.940884	−	0.338728i	$-0.109997\pi$
$98$	0	0
$99$	−0.162116	−0.0162933
$100$	0	0
$101$	11.5597	1.15024	0.575118	−	0.818070i	$-0.304956\pi$
$101$	11.5597	1.15024	0.575118	+	0.818070i	$0.304956\pi$
$102$	0	0
$103$	10.3829i	1.02306i	0.859265	+	0.511531i	$0.170922\pi$
$103$	10.3829i	1.02306i	−0.859265	+	0.511531i	$0.829078\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.94329i	− 0.574559i	−0.957847	−	0.287280i	$-0.907249\pi$
$107$	− 5.94329i	− 0.574559i	0.957847	−	0.287280i	$-0.0927509\pi$
$108$	0	0
$109$	9.17936	0.879223	0.439611	−	0.898188i	$-0.355116\pi$
$109$	9.17936	0.879223	0.439611	+	0.898188i	$0.355116\pi$
$110$	0	0
$111$	25.2822	2.39968
$112$	0	0
$113$	14.9249i	1.40401i	0.712170	+	0.702007i	$0.247712\pi$
$113$	14.9249i	1.40401i	−0.712170	+	0.702007i	$0.752288\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.16473i	− 0.385029i
$118$	0	0
$119$	23.2186	2.12845
$120$	0	0
$121$	−10.9984	−0.999855
$122$	0	0
$123$	− 14.0439i	− 1.26629i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.68416i	− 0.593123i	−0.955014	−	0.296562i	$-0.904160\pi$
$127$	− 6.68416i	− 0.593123i	0.955014	−	0.296562i	$-0.0958400\pi$
$128$	0	0
$129$	23.8110	2.09644
$130$	0	0
$131$	−0.552515	−0.0482734	−0.0241367	−	0.999709i	$-0.507684\pi$
$131$	−0.552515	−0.0482734	−0.0241367	+	0.999709i	$0.507684\pi$
$132$	0	0
$133$	38.4872i	3.33726i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.95168i	− 0.593922i	−0.954890	−	0.296961i	$-0.904027\pi$
$137$	− 6.95168i	− 0.593922i	0.954890	−	0.296961i	$-0.0959732\pi$
$138$	0	0
$139$	7.19300	0.610102	0.305051	−	0.952336i	$-0.401326\pi$
$139$	7.19300	0.610102	0.305051	+	0.952336i	$0.401326\pi$
$140$	0	0
$141$	24.1609	2.03472
$142$	0	0
$143$	0.0408687i	0.00341761i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 39.6733i	− 3.27220i
$148$	0	0
$149$	−9.22182	−0.755481	−0.377740	−	0.925912i	$-0.623299\pi$
$149$	−9.22182	−0.755481	−0.377740	+	0.925912i	$0.623299\pi$
$150$	0	0
$151$	−3.09501	−0.251868	−0.125934	−	0.992039i	$-0.540193\pi$
$151$	−3.09501	−0.251868	−0.125934	+	0.992039i	$0.540193\pi$
$152$	0	0
$153$	− 20.1541i	− 1.62936i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.39463i	0.191112i	0.995424	+	0.0955560i	$0.0304629\pi$
$157$	2.39463i	0.191112i	−0.995424	+	0.0955560i	$0.969537\pi$
$158$	0	0
$159$	3.13886	0.248928
$160$	0	0
$161$	−19.4547	−1.53325
$162$	0	0
$163$	6.59240i	0.516357i	0.966097	+	0.258178i	$0.0831222\pi$
$163$	6.59240i	0.516357i	−0.966097	+	0.258178i	$0.916878\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 2.24548i	− 0.173761i	−0.996219	−	0.0868804i	$-0.972310\pi$
$167$	− 2.24548i	− 0.173761i	0.996219	−	0.0868804i	$-0.0276898\pi$
$168$	0	0
$169$	11.9501	0.919238
$170$	0	0
$171$	33.4074	2.55473
$172$	0	0
$173$	− 10.0782i	− 0.766230i	−0.923701	−	0.383115i	$-0.874851\pi$
$173$	− 10.0782i	− 0.766230i	0.923701	−	0.383115i	$-0.125149\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	24.8263i	1.86606i
$178$	0	0
$179$	11.0256	0.824095	0.412047	−	0.911162i	$-0.364814\pi$
$179$	11.0256	0.824095	0.412047	+	0.911162i	$0.364814\pi$
$180$	0	0
$181$	−12.8950	−0.958476	−0.479238	−	0.877685i	$-0.659087\pi$
$181$	−12.8950	−0.958476	−0.479238	+	0.877685i	$0.659087\pi$
$182$	0	0
$183$	6.85251i	0.506552i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.197773i	0.0144626i
$188$	0	0
$189$	−13.2491	−0.963731
$190$	0	0
$191$	16.6032	1.20136	0.600682	−	0.799488i	$-0.294896\pi$
$191$	16.6032	1.20136	0.600682	+	0.799488i	$0.294896\pi$
$192$	0	0
$193$	4.98477i	0.358811i	0.983775	+	0.179406i	$0.0574174\pi$
$193$	4.98477i	0.358811i	−0.983775	+	0.179406i	$0.942583\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.95528i	0.566790i	0.959003	+	0.283395i	$0.0914608\pi$
$197$	7.95528i	0.566790i	−0.959003	+	0.283395i	$0.908539\pi$
$198$	0	0
$199$	−4.26329	−0.302217	−0.151108	−	0.988517i	$-0.548284\pi$
$199$	−4.26329	−0.302217	−0.151108	+	0.988517i	$0.548284\pi$
$200$	0	0
$201$	−22.4864	−1.58607
$202$	0	0
$203$	− 4.11482i	− 0.288804i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	16.8870i	1.17372i
$208$	0	0
$209$	−0.327829	−0.0226764
$210$	0	0
$211$	−20.6315	−1.42033	−0.710165	−	0.704035i	$-0.751380\pi$
$211$	−20.6315	−1.42033	−0.710165	+	0.704035i	$0.751380\pi$
$212$	0	0
$213$	− 39.9706i	− 2.73874i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	25.5523i	1.73460i
$218$	0	0
$219$	9.53123	0.644061
$220$	0	0
$221$	−5.08075	−0.341769
$222$	0	0
$223$	− 5.79482i	− 0.388050i	−0.980997	−	0.194025i	$-0.937846\pi$
$223$	− 5.79482i	− 0.388050i	0.980997	−	0.194025i	$-0.0621542\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.95754i	0.660905i	0.943823	+	0.330453i	$0.107201\pi$
$227$	9.95754i	0.660905i	−0.943823	+	0.330453i	$0.892799\pi$
$228$	0	0
$229$	1.91142	0.126310	0.0631551	−	0.998004i	$-0.479884\pi$
$229$	1.91142	0.126310	0.0631551	+	0.998004i	$0.479884\pi$
$230$	0	0
$231$	0.496411	0.0326614
$232$	0	0
$233$	− 2.74650i	− 0.179929i	−0.995945	−	0.0899646i	$-0.971325\pi$
$233$	− 2.74650i	− 0.179929i	0.995945	−	0.0899646i	$-0.0286754\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 27.2274i	− 1.76861i
$238$	0	0
$239$	−16.8189	−1.08792	−0.543961	−	0.839111i	$-0.683076\pi$
$239$	−16.8189	−1.08792	−0.543961	+	0.839111i	$0.683076\pi$
$240$	0	0
$241$	1.69879	0.109429	0.0547143	−	0.998502i	$-0.482575\pi$
$241$	1.69879	0.109429	0.0547143	+	0.998502i	$0.482575\pi$
$242$	0	0
$243$	− 20.9092i	− 1.34133i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 8.42185i	− 0.535870i
$248$	0	0
$249$	25.6726	1.62693
$250$	0	0
$251$	8.25270	0.520906	0.260453	−	0.965487i	$-0.416128\pi$
$251$	8.25270	0.520906	0.260453	+	0.965487i	$0.416128\pi$
$252$	0	0
$253$	− 0.165713i	− 0.0104183i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 24.6824i	− 1.53964i	−0.638258	−	0.769822i	$-0.720345\pi$
$257$	− 24.6824i	− 1.53964i	0.638258	−	0.769822i	$-0.279655\pi$
$258$	0	0
$259$	−44.5407	−2.76762
$260$	0	0
$261$	−3.57172	−0.221084
$262$	0	0
$263$	12.0759i	0.744633i	0.928106	+	0.372317i	$0.121436\pi$
$263$	12.0759i	0.744633i	−0.928106	+	0.372317i	$0.878564\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2.01762i	− 0.123476i
$268$	0	0
$269$	−17.3577	−1.05832	−0.529160	−	0.848522i	$-0.677493\pi$
$269$	−17.3577	−1.05832	−0.529160	+	0.848522i	$0.677493\pi$
$270$	0	0
$271$	7.40302	0.449701	0.224851	−	0.974393i	$-0.427811\pi$
$271$	7.40302	0.449701	0.224851	+	0.974393i	$0.427811\pi$
$272$	0	0
$273$	12.7527i	0.771828i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.3208i	1.10079i	0.834904	+	0.550395i	$0.185523\pi$
$277$	18.3208i	1.10079i	−0.834904	+	0.550395i	$0.814477\pi$
$278$	0	0
$279$	22.1798	1.32787
$280$	0	0
$281$	−20.3539	−1.21421	−0.607105	−	0.794621i	$-0.707669\pi$
$281$	−20.3539	−1.21421	−0.607105	+	0.794621i	$0.707669\pi$
$282$	0	0
$283$	3.33149i	0.198036i	0.995086	+	0.0990182i	$0.0315702\pi$
$283$	3.33149i	0.198036i	−0.995086	+	0.0990182i	$0.968430\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	24.7417i	1.46045i
$288$	0	0
$289$	−7.58696	−0.446292
$290$	0	0
$291$	17.7341	1.03959
$292$	0	0
$293$	25.3525i	1.48111i	0.671995	+	0.740556i	$0.265438\pi$
$293$	25.3525i	1.48111i	−0.671995	+	0.740556i	$0.734562\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 0.112854i	− 0.00654846i
$298$	0	0
$299$	4.25712	0.246196
$300$	0	0
$301$	−41.9489	−2.41790
$302$	0	0
$303$	30.7248i	1.76510i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.2611i	1.09929i	0.835399	+	0.549644i	$0.185237\pi$
$307$	19.2611i	1.09929i	−0.835399	+	0.549644i	$0.814763\pi$
$308$	0	0
$309$	−27.5970	−1.56994
$310$	0	0
$311$	−29.6626	−1.68201	−0.841005	−	0.541028i	$-0.818035\pi$
$311$	−29.6626	−1.68201	−0.841005	+	0.541028i	$0.818035\pi$
$312$	0	0
$313$	15.5759i	0.880405i	0.897899	+	0.440202i	$0.145093\pi$
$313$	15.5759i	0.880405i	−0.897899	+	0.440202i	$0.854907\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 32.2994i	− 1.81412i	−0.421005	−	0.907058i	$-0.638323\pi$
$317$	− 32.2994i	− 1.81412i	0.421005	−	0.907058i	$-0.361677\pi$
$318$	0	0
$319$	0.0350495	0.00196240
$320$	0	0
$321$	15.7968	0.881690
$322$	0	0
$323$	− 40.7553i	− 2.26769i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	24.3980i	1.34921i
$328$	0	0
$329$	−42.5654	−2.34670
$330$	0	0
$331$	−20.3804	−1.12021	−0.560103	−	0.828423i	$-0.689239\pi$
$331$	−20.3804	−1.12021	−0.560103	+	0.828423i	$0.689239\pi$
$332$	0	0
$333$	38.6620i	2.11866i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 4.85795i	− 0.264630i	−0.991208	−	0.132315i	$-0.957759\pi$
$337$	− 4.85795i	− 0.264630i	0.991208	−	0.132315i	$-0.0422410\pi$
$338$	0	0
$339$	−39.6691	−2.15453
$340$	0	0
$341$	−0.217651	−0.0117865
$342$	0	0
$343$	37.1162i	2.00409i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.5630i	1.42598i	0.701177	+	0.712988i	$0.252658\pi$
$347$	26.5630i	1.42598i	−0.701177	+	0.712988i	$0.747342\pi$
$348$	0	0
$349$	−4.01622	−0.214983	−0.107492	−	0.994206i	$-0.534282\pi$
$349$	−4.01622	−0.214983	−0.107492	+	0.994206i	$0.534282\pi$
$350$	0	0
$351$	2.89920	0.154748
$352$	0	0
$353$	− 23.1038i	− 1.22969i	−0.788647	−	0.614846i	$-0.789218\pi$
$353$	− 23.1038i	− 1.22969i	0.788647	−	0.614846i	$-0.210782\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	61.7133i	3.26621i
$358$	0	0
$359$	23.2611	1.22767	0.613837	−	0.789433i	$-0.289625\pi$
$359$	23.2611	1.22767	0.613837	+	0.789433i	$0.289625\pi$
$360$	0	0
$361$	48.5559	2.55558
$362$	0	0
$363$	− 29.2329i	− 1.53433i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0.305184i	0.0159305i	0.999968	+	0.00796523i	$0.00253544\pi$
$367$	0.305184i	0.0159305i	−0.999968	+	0.00796523i	$0.997465\pi$
$368$	0	0
$369$	21.4761	1.11800
$370$	0	0
$371$	−5.52987	−0.287097
$372$	0	0
$373$	− 6.97859i	− 0.361338i	−0.983544	−	0.180669i	$-0.942174\pi$
$373$	− 6.97859i	− 0.361338i	0.983544	−	0.180669i	$-0.0578263\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.900414i	0.0463737i
$378$	0	0
$379$	27.8509	1.43061	0.715303	−	0.698815i	$-0.246289\pi$
$379$	27.8509	1.43061	0.715303	+	0.698815i	$0.246289\pi$
$380$	0	0
$381$	17.7660	0.910178
$382$	0	0
$383$	7.54707i	0.385637i	0.981234	+	0.192819i	$0.0617629\pi$
$383$	7.54707i	0.385637i	−0.981234	+	0.192819i	$0.938237\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	36.4122i	1.85094i
$388$	0	0
$389$	−11.4831	−0.582218	−0.291109	−	0.956690i	$-0.594024\pi$
$389$	−11.4831	−0.582218	−0.291109	+	0.956690i	$0.594024\pi$
$390$	0	0
$391$	20.6012	1.04185
$392$	0	0
$393$	− 1.46854i	− 0.0740780i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.39077i	0.220367i	0.993911	+	0.110183i	$0.0351438\pi$
$397$	4.39077i	0.220367i	−0.993911	+	0.110183i	$0.964856\pi$
$398$	0	0
$399$	−102.296	−5.12120
$400$	0	0
$401$	−9.31883	−0.465360	−0.232680	−	0.972553i	$-0.574749\pi$
$401$	−9.31883	−0.465360	−0.232680	+	0.972553i	$0.574749\pi$
$402$	0	0
$403$	− 5.59142i	− 0.278528i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 0.379392i	− 0.0188058i
$408$	0	0
$409$	22.7657	1.12569	0.562846	−	0.826562i	$-0.309706\pi$
$409$	22.7657	1.12569	0.562846	+	0.826562i	$0.309706\pi$
$410$	0	0
$411$	18.4770	0.911404
$412$	0	0
$413$	− 43.7375i	− 2.15218i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	19.1184i	0.936233i
$418$	0	0
$419$	18.1849	0.888391	0.444196	−	0.895930i	$-0.353489\pi$
$419$	18.1849	0.888391	0.444196	+	0.895930i	$0.353489\pi$
$420$	0	0
$421$	−14.6177	−0.712424	−0.356212	−	0.934405i	$-0.615932\pi$
$421$	−14.6177	−0.712424	−0.356212	+	0.934405i	$0.615932\pi$
$422$	0	0
$423$	36.9473i	1.79644i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 12.0724i	− 0.584222i
$428$	0	0
$429$	−0.108626	−0.00524450
$430$	0	0
$431$	−9.73641	−0.468986	−0.234493	−	0.972118i	$-0.575343\pi$
$431$	−9.73641	−0.468986	−0.234493	+	0.972118i	$0.575343\pi$
$432$	0	0
$433$	− 20.1544i	− 0.968558i	−0.874914	−	0.484279i	$-0.839082\pi$
$433$	− 20.1544i	− 0.968558i	0.874914	−	0.484279i	$-0.160918\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	34.1485i	1.63355i
$438$	0	0
$439$	8.93773	0.426574	0.213287	−	0.976990i	$-0.431583\pi$
$439$	8.93773	0.426574	0.213287	+	0.976990i	$0.431583\pi$
$440$	0	0
$441$	60.6691	2.88901
$442$	0	0
$443$	4.43390i	0.210661i	0.994437	+	0.105331i	$0.0335901\pi$
$443$	4.43390i	0.210661i	−0.994437	+	0.105331i	$0.966410\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 24.5108i	− 1.15932i
$448$	0	0
$449$	−38.3645	−1.81053	−0.905267	−	0.424843i	$-0.860329\pi$
$449$	−38.3645	−1.81053	−0.905267	+	0.424843i	$0.860329\pi$
$450$	0	0
$451$	−0.210746	−0.00992365
$452$	0	0
$453$	− 8.22628i	− 0.386504i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 31.8529i	− 1.49002i	−0.667055	−	0.745009i	$-0.732445\pi$
$457$	− 31.8529i	− 1.49002i	0.667055	−	0.745009i	$-0.267555\pi$
$458$	0	0
$459$	14.0299	0.654860
$460$	0	0
$461$	35.4680	1.65191	0.825954	−	0.563737i	$-0.190637\pi$
$461$	35.4680	1.65191	0.825954	+	0.563737i	$0.190637\pi$
$462$	0	0
$463$	− 0.845101i	− 0.0392752i	−0.999807	−	0.0196376i	$-0.993749\pi$
$463$	− 0.845101i	− 0.0392752i	0.999807	−	0.0196376i	$-0.00625124\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 13.7848i	− 0.637884i	−0.947774	−	0.318942i	$-0.896672\pi$
$467$	− 13.7848i	− 0.637884i	0.947774	−	0.318942i	$-0.103328\pi$
$468$	0	0
$469$	39.6152	1.82926
$470$	0	0
$471$	−6.36473	−0.293271
$472$	0	0
$473$	− 0.357315i	− 0.0164294i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.80000i	0.219777i
$478$	0	0
$479$	7.08075	0.323528	0.161764	−	0.986829i	$-0.448282\pi$
$479$	7.08075	0.323528	0.161764	+	0.986829i	$0.448282\pi$
$480$	0	0
$481$	9.74650	0.444402
$482$	0	0
$483$	− 51.7090i	− 2.35284i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 25.3059i	− 1.14672i	−0.819304	−	0.573359i	$-0.805640\pi$
$487$	− 25.3059i	− 1.14672i	0.819304	−	0.573359i	$-0.194360\pi$
$488$	0	0
$489$	−17.5221	−0.792375
$490$	0	0
$491$	−24.4890	−1.10517	−0.552587	−	0.833455i	$-0.686359\pi$
$491$	−24.4890	−1.10517	−0.552587	+	0.833455i	$0.686359\pi$
$492$	0	0
$493$	4.35732i	0.196244i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	70.4179i	3.15867i
$498$	0	0
$499$	3.14230	0.140669	0.0703344	−	0.997523i	$-0.477593\pi$
$499$	3.14230	0.140669	0.0703344	+	0.997523i	$0.477593\pi$
$500$	0	0
$501$	5.96831	0.266645
$502$	0	0
$503$	− 20.7790i	− 0.926489i	−0.886231	−	0.463244i	$-0.846685\pi$
$503$	− 20.7790i	− 0.926489i	0.886231	−	0.463244i	$-0.153315\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	31.7624i	1.41062i
$508$	0	0
$509$	27.9139	1.23726	0.618630	−	0.785682i	$-0.287688\pi$
$509$	27.9139	1.23726	0.618630	+	0.785682i	$0.287688\pi$
$510$	0	0
$511$	−16.7916	−0.742815
$512$	0	0
$513$	23.2559i	1.02678i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 0.362566i	− 0.0159456i
$518$	0	0
$519$	26.7870	1.17582
$520$	0	0
$521$	24.0675	1.05442	0.527209	−	0.849736i	$-0.323239\pi$
$521$	24.0675	1.05442	0.527209	+	0.849736i	$0.323239\pi$
$522$	0	0
$523$	40.1150i	1.75411i	0.480393	+	0.877053i	$0.340494\pi$
$523$	40.1150i	1.75411i	−0.480393	+	0.877053i	$0.659506\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 27.0582i	− 1.17867i
$528$	0	0
$529$	5.73842	0.249496
$530$	0	0
$531$	−37.9648	−1.64753
$532$	0	0
$533$	− 5.41403i	− 0.234508i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	29.3052i	1.26461i
$538$	0	0
$539$	−0.595350	−0.0256435
$540$	0	0
$541$	−28.9359	−1.24405	−0.622027	−	0.782996i	$-0.713690\pi$
$541$	−28.9359	−1.24405	−0.622027	+	0.782996i	$0.713690\pi$
$542$	0	0
$543$	− 34.2738i	− 1.47083i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	25.9290i	1.10865i	0.832302	+	0.554323i	$0.187023\pi$
$547$	25.9290i	1.10865i	−0.832302	+	0.554323i	$0.812977\pi$
$548$	0	0
$549$	−10.4790	−0.447232
$550$	0	0
$551$	−7.22268	−0.307697
$552$	0	0
$553$	47.9677i	2.03980i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.7002i	0.580496i	0.956952	+	0.290248i	$0.0937377\pi$
$557$	13.7002i	0.580496i	−0.956952	+	0.290248i	$0.906262\pi$
$558$	0	0
$559$	9.17936	0.388245
$560$	0	0
$561$	−0.525666	−0.0221936
$562$	0	0
$563$	24.8866i	1.04885i	0.851458	+	0.524423i	$0.175719\pi$
$563$	24.8866i	1.04885i	−0.851458	+	0.524423i	$0.824281\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	21.8824i	0.918973i
$568$	0	0
$569$	−28.2649	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$569$	−28.2649	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$570$	0	0
$571$	7.04643	0.294884	0.147442	−	0.989071i	$-0.452896\pi$
$571$	7.04643	0.294884	0.147442	+	0.989071i	$0.452896\pi$
$572$	0	0
$573$	44.1299i	1.84355i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1.86295i	− 0.0775556i	−0.999248	−	0.0387778i	$-0.987654\pi$
$577$	− 1.86295i	− 0.0775556i	0.999248	−	0.0387778i	$-0.0123465\pi$
$578$	0	0
$579$	−13.2491	−0.550614
$580$	0	0
$581$	−45.2285	−1.87639
$582$	0	0
$583$	− 0.0471027i	− 0.00195079i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 17.9871i	− 0.742409i	−0.928551	−	0.371204i	$-0.878945\pi$
$587$	− 17.9871i	− 0.742409i	0.928551	−	0.371204i	$-0.121055\pi$
$588$	0	0
$589$	44.8516	1.84808
$590$	0	0
$591$	−21.1445	−0.869768
$592$	0	0
$593$	− 29.1161i	− 1.19565i	−0.801625	−	0.597827i	$-0.796031\pi$
$593$	− 29.1161i	− 1.19565i	0.801625	−	0.597827i	$-0.203969\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 11.3315i	− 0.463767i
$598$	0	0
$599$	29.0913	1.18864	0.594318	−	0.804230i	$-0.297422\pi$
$599$	29.0913	1.18864	0.594318	+	0.804230i	$0.297422\pi$
$600$	0	0
$601$	32.3954	1.32144	0.660718	−	0.750634i	$-0.270252\pi$
$601$	32.3954	1.32144	0.660718	+	0.750634i	$0.270252\pi$
$602$	0	0
$603$	− 34.3866i	− 1.40033i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	19.0239i	0.772154i	0.922467	+	0.386077i	$0.126170\pi$
$607$	19.0239i	0.772154i	−0.922467	+	0.386077i	$0.873830\pi$
$608$	0	0
$609$	10.9369	0.443184
$610$	0	0
$611$	9.31425	0.376814
$612$	0	0
$613$	− 2.03474i	− 0.0821823i	−0.999155	−	0.0410911i	$-0.986917\pi$
$613$	− 2.03474i	− 0.0821823i	0.999155	−	0.0410911i	$-0.0130834\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 9.73728i	− 0.392008i	−0.980603	−	0.196004i	$-0.937203\pi$
$617$	− 9.73728i	− 0.392008i	0.980603	−	0.196004i	$-0.0627966\pi$
$618$	0	0
$619$	31.1593	1.25240	0.626200	−	0.779662i	$-0.284609\pi$
$619$	31.1593	1.25240	0.626200	+	0.779662i	$0.284609\pi$
$620$	0	0
$621$	−11.7555	−0.471734
$622$	0	0
$623$	3.55452i	0.142409i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 0.871343i	− 0.0347981i
$628$	0	0
$629$	47.1656	1.88061
$630$	0	0
$631$	−28.6829	−1.14185	−0.570925	−	0.821002i	$-0.693415\pi$
$631$	−28.6829	−1.14185	−0.570925	+	0.821002i	$0.693415\pi$
$632$	0	0
$633$	− 54.8368i	− 2.17957i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 15.2944i	− 0.605987i
$638$	0	0
$639$	61.1237	2.41802
$640$	0	0
$641$	22.3152	0.881399	0.440699	−	0.897655i	$-0.354731\pi$
$641$	22.3152	0.881399	0.440699	+	0.897655i	$0.354731\pi$
$642$	0	0
$643$	36.8046i	1.45143i	0.687994	+	0.725716i	$0.258491\pi$
$643$	36.8046i	1.45143i	−0.687994	+	0.725716i	$0.741509\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	49.8385i	1.95935i	0.200584	+	0.979676i	$0.435716\pi$
$647$	49.8385i	1.95935i	−0.200584	+	0.979676i	$0.564284\pi$
$648$	0	0
$649$	0.372550	0.0146239
$650$	0	0
$651$	−67.9160	−2.66184
$652$	0	0
$653$	− 15.8293i	− 0.619447i	−0.950827	−	0.309723i	$-0.899764\pi$
$653$	− 15.8293i	− 0.619447i	0.950827	−	0.309723i	$-0.100236\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14.5753i	0.568637i
$658$	0	0
$659$	−10.1311	−0.394651	−0.197325	−	0.980338i	$-0.563226\pi$
$659$	−10.1311	−0.394651	−0.197325	+	0.980338i	$0.563226\pi$
$660$	0	0
$661$	−28.7912	−1.11985	−0.559924	−	0.828544i	$-0.689170\pi$
$661$	−28.7912	−1.11985	−0.559924	+	0.828544i	$0.689170\pi$
$662$	0	0
$663$	− 13.5042i	− 0.524461i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.65096i	− 0.141366i
$668$	0	0
$669$	15.4022	0.595482
$670$	0	0
$671$	0.102831	0.00396974
$672$	0	0
$673$	20.6035i	0.794205i	0.917774	+	0.397103i	$0.129984\pi$
$673$	20.6035i	0.794205i	−0.917774	+	0.397103i	$0.870016\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 29.5265i	− 1.13479i	−0.823444	−	0.567397i	$-0.807951\pi$
$677$	− 29.5265i	− 1.13479i	0.823444	−	0.567397i	$-0.192049\pi$
$678$	0	0
$679$	−31.2429	−1.19899
$680$	0	0
$681$	−26.4663	−1.01419
$682$	0	0
$683$	− 3.62067i	− 0.138541i	−0.997598	−	0.0692706i	$-0.977933\pi$
$683$	− 3.62067i	− 0.138541i	0.997598	−	0.0692706i	$-0.0220672\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.08040i	0.193829i
$688$	0	0
$689$	1.21006	0.0460996
$690$	0	0
$691$	−4.34306	−0.165218	−0.0826090	−	0.996582i	$-0.526325\pi$
$691$	−4.34306	−0.165218	−0.0826090	+	0.996582i	$0.526325\pi$
$692$	0	0
$693$	0.759120i	0.0288366i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 26.1997i	− 0.992386i
$698$	0	0
$699$	7.29997	0.276110
$700$	0	0
$701$	−15.6469	−0.590976	−0.295488	−	0.955346i	$-0.595482\pi$
$701$	−15.6469	−0.590976	−0.295488	+	0.955346i	$0.595482\pi$
$702$	0	0
$703$	78.1816i	2.94868i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 54.1293i	− 2.03574i
$708$	0	0
$709$	10.6054	0.398296	0.199148	−	0.979969i	$-0.436183\pi$
$709$	10.6054	0.398296	0.199148	+	0.979969i	$0.436183\pi$
$710$	0	0
$711$	41.6367	1.56150
$712$	0	0
$713$	22.6718i	0.849066i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 44.7032i	− 1.66947i
$718$	0	0
$719$	−28.6068	−1.06685	−0.533427	−	0.845846i	$-0.679096\pi$
$719$	−28.6068	−1.06685	−0.533427	+	0.845846i	$0.679096\pi$
$720$	0	0
$721$	48.6189	1.81066
$722$	0	0
$723$	4.51524i	0.167924i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 0.410174i	− 0.0152125i	−0.999971	−	0.00760625i	$-0.997579\pi$
$727$	− 0.410174i	− 0.0152125i	0.999971	−	0.00760625i	$-0.00242117\pi$
$728$	0	0
$729$	41.5556	1.53910
$730$	0	0
$731$	44.4211	1.64297
$732$	0	0
$733$	− 7.18559i	− 0.265406i	−0.991156	−	0.132703i	$-0.957634\pi$
$733$	− 7.18559i	− 0.265406i	0.991156	−	0.132703i	$-0.0423656\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.337437i	0.0124297i
$738$	0	0
$739$	−45.5555	−1.67579	−0.837894	−	0.545834i	$-0.816213\pi$
$739$	−45.5555	−1.67579	−0.837894	+	0.545834i	$0.816213\pi$
$740$	0	0
$741$	22.3846	0.822319
$742$	0	0
$743$	11.8535i	0.434864i	0.976075	+	0.217432i	$0.0697680\pi$
$743$	11.8535i	0.434864i	−0.976075	+	0.217432i	$0.930232\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	39.2590i	1.43641i
$748$	0	0
$749$	−27.8299	−1.01688
$750$	0	0
$751$	−0.674433	−0.0246104	−0.0123052	−	0.999924i	$-0.503917\pi$
$751$	−0.674433	−0.0246104	−0.0123052	+	0.999924i	$0.503917\pi$
$752$	0	0
$753$	21.9350i	0.799356i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 3.41666i	− 0.124181i	−0.998071	−	0.0620904i	$-0.980223\pi$
$757$	− 3.41666i	− 0.124181i	0.998071	−	0.0620904i	$-0.0197767\pi$
$758$	0	0
$759$	0.440451	0.0159874
$760$	0	0
$761$	−26.1930	−0.949496	−0.474748	−	0.880122i	$-0.657461\pi$
$761$	−26.1930	−0.949496	−0.474748	+	0.880122i	$0.657461\pi$
$762$	0	0
$763$	− 42.9830i	− 1.55609i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.57074i	0.345579i
$768$	0	0
$769$	−15.1742	−0.547195	−0.273597	−	0.961844i	$-0.588214\pi$
$769$	−15.1742	−0.547195	−0.273597	+	0.961844i	$0.588214\pi$
$770$	0	0
$771$	65.6038	2.36266
$772$	0	0
$773$	− 29.0032i	− 1.04317i	−0.853198	−	0.521587i	$-0.825340\pi$
$773$	− 29.0032i	− 1.04317i	0.853198	−	0.521587i	$-0.174660\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 118.386i	− 4.24706i
$778$	0	0
$779$	43.4287	1.55599
$780$	0	0
$781$	−0.599810	−0.0214629
$782$	0	0
$783$	− 2.48639i	− 0.0888562i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	19.2660i	0.686758i	0.939197	+	0.343379i	$0.111571\pi$
$787$	19.2660i	0.686758i	−0.939197	+	0.343379i	$0.888429\pi$
$788$	0	0
$789$	−32.0968	−1.14268
$790$	0	0
$791$	69.8868	2.48489
$792$	0	0
$793$	2.64170i	0.0938096i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.54279i	0.0546483i	0.999627	+	0.0273242i	$0.00869863\pi$
$797$	1.54279i	0.0546483i	−0.999627	+	0.0273242i	$0.991301\pi$
$798$	0	0
$799$	45.0738	1.59460
$800$	0	0
$801$	3.08537	0.109016
$802$	0	0
$803$	− 0.143028i	− 0.00504736i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 46.1354i	− 1.62404i
$808$	0	0
$809$	−36.7800	−1.29312	−0.646558	−	0.762865i	$-0.723792\pi$
$809$	−36.7800	−1.29312	−0.646558	+	0.762865i	$0.723792\pi$
$810$	0	0
$811$	17.4212	0.611743	0.305871	−	0.952073i	$-0.401052\pi$
$811$	17.4212	0.611743	0.305871	+	0.952073i	$0.401052\pi$
$812$	0	0
$813$	19.6766i	0.690090i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	73.6323i	2.57607i
$818$	0	0
$819$	−19.5016	−0.681442
$820$	0	0
$821$	18.6670	0.651484	0.325742	−	0.945459i	$-0.394386\pi$
$821$	18.6670	0.651484	0.325742	+	0.945459i	$0.394386\pi$
$822$	0	0
$823$	39.4544i	1.37530i	0.726044	+	0.687648i	$0.241357\pi$
$823$	39.4544i	1.37530i	−0.726044	+	0.687648i	$0.758643\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 26.1204i	− 0.908297i	−0.890926	−	0.454148i	$-0.849944\pi$
$827$	− 26.1204i	− 0.908297i	0.890926	−	0.454148i	$-0.150056\pi$
$828$	0	0
$829$	9.81880	0.341021	0.170510	−	0.985356i	$-0.445458\pi$
$829$	9.81880	0.341021	0.170510	+	0.985356i	$0.445458\pi$
$830$	0	0
$831$	−48.6952	−1.68922
$832$	0	0
$833$	− 74.0132i	− 2.56441i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	15.4400i	0.533686i
$838$	0	0
$839$	13.9336	0.481042	0.240521	−	0.970644i	$-0.422682\pi$
$839$	13.9336	0.481042	0.240521	+	0.970644i	$0.422682\pi$
$840$	0	0
$841$	−28.2278	−0.973372
$842$	0	0
$843$	− 54.0990i	− 1.86327i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	51.5008i	1.76959i
$848$	0	0
$849$	−8.85483	−0.303897
$850$	0	0
$851$	−39.5197	−1.35472
$852$	0	0
$853$	18.1138i	0.620205i	0.950703	+	0.310102i	$0.100363\pi$
$853$	18.1138i	0.620205i	−0.950703	+	0.310102i	$0.899637\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 9.72704i	− 0.332269i	−0.986103	−	0.166135i	$-0.946871\pi$
$857$	− 9.72704i	− 0.332269i	0.986103	−	0.166135i	$-0.0531286\pi$
$858$	0	0
$859$	16.7910	0.572902	0.286451	−	0.958095i	$-0.407524\pi$
$859$	16.7910	0.572902	0.286451	+	0.958095i	$0.407524\pi$
$860$	0	0
$861$	−65.7613	−2.24114
$862$	0	0
$863$	− 3.06190i	− 0.104228i	−0.998641	−	0.0521141i	$-0.983404\pi$
$863$	− 3.06190i	− 0.104228i	0.998641	−	0.0521141i	$-0.0165960\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 20.1655i	− 0.684857i
$868$	0	0
$869$	−0.408583	−0.0138602
$870$	0	0
$871$	−8.66869	−0.293727
$872$	0	0
$873$	27.1193i	0.917849i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	21.7836i	0.735580i	0.929909	+	0.367790i	$0.119886\pi$
$877$	21.7836i	0.735580i	−0.929909	+	0.367790i	$0.880114\pi$
$878$	0	0
$879$	−67.3850	−2.27284
$880$	0	0
$881$	47.4065	1.59717	0.798583	−	0.601885i	$-0.205584\pi$
$881$	47.4065	1.59717	0.798583	+	0.601885i	$0.205584\pi$
$882$	0	0
$883$	− 16.8303i	− 0.566384i	−0.959063	−	0.283192i	$-0.908607\pi$
$883$	− 16.8303i	− 0.566384i	0.959063	−	0.283192i	$-0.0913934\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.25584i	− 0.0421671i	−0.999778	−	0.0210835i	$-0.993288\pi$
$887$	− 1.25584i	− 0.0421671i	0.999778	−	0.0210835i	$-0.00671160\pi$
$888$	0	0
$889$	−31.2991	−1.04974
$890$	0	0
$891$	−0.186391	−0.00624434
$892$	0	0
$893$	74.7143i	2.50022i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	11.3151i	0.377800i
$898$	0	0
$899$	−4.79527	−0.159931
$900$	0	0
$901$	5.85576	0.195084
$902$	0	0
$903$	− 111.497i	− 3.71038i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 28.0199i	− 0.930385i	−0.885210	−	0.465192i	$-0.845985\pi$
$907$	− 28.0199i	− 0.930385i	0.885210	−	0.465192i	$-0.154015\pi$
$908$	0	0
$909$	−46.9849	−1.55839
$910$	0	0
$911$	20.7419	0.687210	0.343605	−	0.939114i	$-0.388352\pi$
$911$	20.7419	0.687210	0.343605	+	0.939114i	$0.388352\pi$
$912$	0	0
$913$	− 0.385250i	− 0.0127499i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.58719i	0.0854365i
$918$	0	0
$919$	−0.224493	−0.00740534	−0.00370267	−	0.999993i	$-0.501179\pi$
$919$	−0.224493	−0.00740534	−0.00370267	+	0.999993i	$0.501179\pi$
$920$	0	0
$921$	−51.1944	−1.68691
$922$	0	0
$923$	− 15.4090i	− 0.507193i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 42.2019i	− 1.38609i
$928$	0	0
$929$	33.2747	1.09171	0.545854	−	0.837880i	$-0.316205\pi$
$929$	33.2747	1.09171	0.545854	+	0.837880i	$0.316205\pi$
$930$	0	0
$931$	122.684	4.02081
$932$	0	0
$933$	− 78.8407i	− 2.58113i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 18.4776i	− 0.603636i	−0.953366	−	0.301818i	$-0.902406\pi$
$937$	− 18.4776i	− 0.603636i	0.953366	−	0.301818i	$-0.0975935\pi$
$938$	0	0
$939$	−41.3996	−1.35103
$940$	0	0
$941$	−13.3752	−0.436018	−0.218009	−	0.975947i	$-0.569956\pi$
$941$	−13.3752	−0.436018	−0.218009	+	0.975947i	$0.569956\pi$
$942$	0	0
$943$	21.9525i	0.714873i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 37.0656i	− 1.20447i	−0.798319	−	0.602235i	$-0.794277\pi$
$947$	− 37.0656i	− 1.20447i	0.798319	−	0.602235i	$-0.205723\pi$
$948$	0	0
$949$	3.67437	0.119275
$950$	0	0
$951$	85.8493	2.78385
$952$	0	0
$953$	27.8429i	0.901920i	0.892544	+	0.450960i	$0.148918\pi$
$953$	27.8429i	0.901920i	−0.892544	+	0.450960i	$0.851082\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.0931588i	0.00301140i
$958$	0	0
$959$	−32.5517	−1.05115
$960$	0	0
$961$	−1.22223	−0.0394268
$962$	0	0
$963$	24.1567i	0.778439i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 54.0629i	− 1.73855i	−0.494331	−	0.869274i	$-0.664587\pi$
$967$	− 54.0629i	− 1.73855i	0.494331	−	0.869274i	$-0.335413\pi$
$968$	0	0
$969$	108.324	3.47988
$970$	0	0
$971$	−53.4131	−1.71411	−0.857054	−	0.515227i	$-0.827708\pi$
$971$	−53.4131	−1.71411	−0.857054	+	0.515227i	$0.827708\pi$
$972$	0	0
$973$	− 33.6817i	− 1.07979i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 40.7391i	− 1.30336i	−0.758494	−	0.651679i	$-0.774065\pi$
$977$	− 40.7391i	− 1.30336i	0.758494	−	0.651679i	$-0.225935\pi$
$978$	0	0
$979$	−0.0302769	−0.000967655 0
$980$	0	0
$981$	−37.3098	−1.19121
$982$	0	0
$983$	6.22306i	0.198485i	0.995063	+	0.0992423i	$0.0316419\pi$
$983$	6.22306i	0.198485i	−0.995063	+	0.0992423i	$0.968358\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 113.135i	− 3.60114i
$988$	0	0
$989$	−37.2200	−1.18353
$990$	0	0
$991$	27.7894	0.882759	0.441379	−	0.897321i	$-0.354489\pi$
$991$	27.7894	0.882759	0.441379	+	0.897321i	$0.354489\pi$
$992$	0	0
$993$	− 54.1694i	− 1.71901i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 8.56549i	− 0.271272i	−0.990759	−	0.135636i	$-0.956692\pi$
$997$	− 8.56549i	− 0.271272i	0.990759	−	0.135636i	$-0.0433077\pi$
$998$	0	0
$999$	−26.9138	−0.851515

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2000.2.c.i.1249.7		8
4.3	odd	2		1000.2.c.c.249.2		8
5.2	odd	4		2000.2.a.q.1.4		4
5.3	odd	4		2000.2.a.n.1.1		4
5.4	even	2	inner	2000.2.c.i.1249.2		8
20.3	even	4		1000.2.a.g.1.4	yes	4
20.7	even	4		1000.2.a.f.1.1	✓	4
20.19	odd	2		1000.2.c.c.249.7		8
40.3	even	4		8000.2.a.bd.1.1		4
40.13	odd	4		8000.2.a.bo.1.4		4
40.27	even	4		8000.2.a.bn.1.4		4
40.37	odd	4		8000.2.a.be.1.1		4
60.23	odd	4		9000.2.a.bb.1.4		4
60.47	odd	4		9000.2.a.q.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.a.f.1.1	✓	4	20.7	even	4
1000.2.a.g.1.4	yes	4	20.3	even	4
1000.2.c.c.249.2		8	4.3	odd	2
1000.2.c.c.249.7		8	20.19	odd	2
2000.2.a.n.1.1		4	5.3	odd	4
2000.2.a.q.1.4		4	5.2	odd	4
2000.2.c.i.1249.2		8	5.4	even	2	inner
2000.2.c.i.1249.7		8	1.1	even	1	trivial
8000.2.a.bd.1.1		4	40.3	even	4
8000.2.a.be.1.1		4	40.37	odd	4
8000.2.a.bn.1.4		4	40.27	even	4
8000.2.a.bo.1.4		4	40.13	odd	4
9000.2.a.q.1.1		4	60.47	odd	4
9000.2.a.bb.1.4		4	60.23	odd	4

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:	\( N \)	\(=\)	\( 2000 = 2^{4} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2000.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.65792i q^{3} -4.68257i q^{7} -4.06454 q^{9} +0.0398855 q^{11} +1.02465i q^{13} +4.95852i q^{17} -8.21924 q^{19} +12.4459 q^{21} -4.15471i q^{23} -2.82945i q^{27} +0.878753 q^{29} -5.45690 q^{31} +0.106013i q^{33} -9.51202i q^{37} -2.72344 q^{39} -5.28378 q^{41} -8.95852i q^{43} -9.09017i q^{47} -14.9265 q^{49} -13.1794 q^{51} -1.18095i q^{53} -21.8461i q^{57} +9.34049 q^{59} +2.57815 q^{61} +19.0325i q^{63} +8.46015i q^{67} +11.0429 q^{69} -15.0383 q^{71} -3.58597i q^{73} -0.186767i q^{77} -10.2439 q^{79} -4.67315 q^{81} -9.65890i q^{83} +2.33565i q^{87} -0.759096 q^{89} +4.79800 q^{91} -14.5040i q^{93} -6.67217i q^{97} -0.162116 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 22 q^{9} - 10 q^{11} + 6 q^{21} - 20 q^{29} - 18 q^{31} + 42 q^{39} + 34 q^{41} - 34 q^{49} - 50 q^{51} + 36 q^{59} + 22 q^{61} + 16 q^{69} - 52 q^{71} - 16 q^{79} - 8 q^{81} - 10 q^{89} + 66 q^{91}+ \cdots + 16 q^{99}+O(q^{100}) \) 8 * q - 22 * q^9 - 10 * q^11 + 6 * q^21 - 20 * q^29 - 18 * q^31 + 42 * q^39 + 34 * q^41 - 34 * q^49 - 50 * q^51 + 36 * q^59 + 22 * q^61 + 16 * q^69 - 52 * q^71 - 16 * q^79 - 8 * q^81 - 10 * q^89 + 66 * q^91 + 16 * q^99

Introduction

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