LMFDB - Embedded newform 4000.2.d.c.2001.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4000,2,Mod(2001,4000)] 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("4000.2001"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4000.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [40,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 1000)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2001.13
Character	$\chi$	$=$	4000.2001
Dual form			4000.2.d.c.2001.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.62662i q^{3} -0.269237 q^{7} -3.89913 q^{9} -4.58163i q^{11} -1.55366i q^{13} +0.609109 q^{17} -6.69309i q^{19} +0.707183i q^{21} -3.52675 q^{23} +2.36168i q^{27} +7.73304i q^{29} +3.34096 q^{31} -12.0342 q^{33} -5.32768i q^{37} -4.08087 q^{39} +4.38291 q^{41} -12.2644i q^{43} +9.54651 q^{47} -6.92751 q^{49} -1.59990i q^{51} -10.6574i q^{53} -17.5802 q^{57} +10.2256i q^{59} +13.2628i q^{61} +1.04979 q^{63} +3.93398i q^{67} +9.26343i q^{69} -6.95638 q^{71} +5.93156 q^{73} +1.23354i q^{77} -10.1382 q^{79} -5.49416 q^{81} +6.52313i q^{83} +20.3118 q^{87} +6.69564 q^{89} +0.418302i q^{91} -8.77544i q^{93} -3.99182 q^{97} +17.8644i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 24 q^{9} - 48 q^{31} - 8 q^{39} + 44 q^{41} + 12 q^{49} - 96 q^{71} - 96 q^{79} - 56 q^{81} - 44 q^{89}+O(q^{100}) $ 40 * q - 24 * q^9 - 48 * q^31 - 8 * q^39 + 44 * q^41 + 12 * q^49 - 96 * q^71 - 96 * q^79 - 56 * q^81 - 44 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$.

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 2.62662i	− 1.51648i	−0.651976	−	0.758240i	$-0.726060\pi$
$3$	− 2.62662i	− 1.51648i	0.651976	−	0.758240i	$-0.273940\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.269237	−0.101762	−0.0508810	−	0.998705i	$-0.516203\pi$
$7$	−0.269237	−0.101762	−0.0508810	+	0.998705i	$0.516203\pi$
$8$	0	0
$9$	−3.89913	−1.29971
$10$	0	0
$11$	− 4.58163i	− 1.38141i	−0.723135	−	0.690706i	$-0.757300\pi$
$11$	− 4.58163i	− 1.38141i	0.723135	−	0.690706i	$-0.242700\pi$
$12$	0	0
$13$	− 1.55366i	− 0.430908i	−0.976514	−	0.215454i	$-0.930877\pi$
$13$	− 1.55366i	− 0.430908i	0.976514	−	0.215454i	$-0.0691231\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.609109	0.147731	0.0738653	−	0.997268i	$-0.476467\pi$
$17$	0.609109	0.147731	0.0738653	+	0.997268i	$0.476467\pi$
$18$	0	0
$19$	− 6.69309i	− 1.53550i	−0.640749	−	0.767750i	$-0.721376\pi$
$19$	− 6.69309i	− 1.53550i	0.640749	−	0.767750i	$-0.278624\pi$
$20$	0	0
$21$	0.707183i	0.154320i
$22$	0	0
$23$	−3.52675	−0.735378	−0.367689	−	0.929949i	$-0.619851\pi$
$23$	−3.52675	−0.735378	−0.367689	+	0.929949i	$0.619851\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.36168i	0.454506i
$28$	0	0
$29$	7.73304i	1.43599i	0.696048	+	0.717995i	$0.254940\pi$
$29$	7.73304i	1.43599i	−0.696048	+	0.717995i	$0.745060\pi$
$30$	0	0
$31$	3.34096	0.600054	0.300027	−	0.953931i	$-0.403004\pi$
$31$	3.34096	0.600054	0.300027	+	0.953931i	$0.403004\pi$
$32$	0	0
$33$	−12.0342	−2.09488
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 5.32768i	− 0.875865i	−0.899008	−	0.437933i	$-0.855711\pi$
$37$	− 5.32768i	− 0.875865i	0.899008	−	0.437933i	$-0.144289\pi$
$38$	0	0
$39$	−4.08087	−0.653463
$40$	0	0
$41$	4.38291	0.684496	0.342248	−	0.939610i	$-0.388812\pi$
$41$	4.38291	0.684496	0.342248	+	0.939610i	$0.388812\pi$
$42$	0	0
$43$	− 12.2644i	− 1.87031i	−0.354238	−	0.935155i	$-0.615260\pi$
$43$	− 12.2644i	− 1.87031i	0.354238	−	0.935155i	$-0.384740\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.54651	1.39250	0.696251	−	0.717798i	$-0.254850\pi$
$47$	9.54651	1.39250	0.696251	+	0.717798i	$0.254850\pi$
$48$	0	0
$49$	−6.92751	−0.989645
$50$	0	0
$51$	− 1.59990i	− 0.224030i
$52$	0	0
$53$	− 10.6574i	− 1.46391i	−0.681353	−	0.731955i	$-0.738608\pi$
$53$	− 10.6574i	− 1.46391i	0.681353	−	0.731955i	$-0.261392\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−17.5802	−2.32855
$58$	0	0
$59$	10.2256i	1.33126i	0.746283	+	0.665629i	$0.231837\pi$
$59$	10.2256i	1.33126i	−0.746283	+	0.665629i	$0.768163\pi$
$60$	0	0
$61$	13.2628i	1.69812i	0.528294	+	0.849062i	$0.322832\pi$
$61$	13.2628i	1.69812i	−0.528294	+	0.849062i	$0.677168\pi$
$62$	0	0
$63$	1.04979	0.132261
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.93398i	0.480613i	0.970697	+	0.240306i	$0.0772479\pi$
$67$	3.93398i	0.480613i	−0.970697	+	0.240306i	$0.922752\pi$
$68$	0	0
$69$	9.26343i	1.11519i
$70$	0	0
$71$	−6.95638	−0.825571	−0.412785	−	0.910828i	$-0.635444\pi$
$71$	−6.95638	−0.825571	−0.412785	+	0.910828i	$0.635444\pi$
$72$	0	0
$73$	5.93156	0.694236	0.347118	−	0.937821i	$-0.387160\pi$
$73$	5.93156	0.694236	0.347118	+	0.937821i	$0.387160\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.23354i	0.140575i
$78$	0	0
$79$	−10.1382	−1.14064	−0.570321	−	0.821422i	$-0.693181\pi$
$79$	−10.1382	−1.14064	−0.570321	+	0.821422i	$0.693181\pi$
$80$	0	0
$81$	−5.49416	−0.610462
$82$	0	0
$83$	6.52313i	0.716007i	0.933720	+	0.358003i	$0.116542\pi$
$83$	6.52313i	0.716007i	−0.933720	+	0.358003i	$0.883458\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	20.3118	2.17765
$88$	0	0
$89$	6.69564	0.709736	0.354868	−	0.934916i	$-0.384526\pi$
$89$	6.69564	0.709736	0.354868	+	0.934916i	$0.384526\pi$
$90$	0	0
$91$	0.418302i	0.0438500i
$92$	0	0
$93$	− 8.77544i	− 0.909970i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.99182	−0.405307	−0.202654	−	0.979250i	$-0.564957\pi$
$97$	−3.99182	−0.405307	−0.202654	+	0.979250i	$0.564957\pi$
$98$	0	0
$99$	17.8644i	1.79544i
$100$	0	0
$101$	0.361133i	0.0359340i	0.999839	+	0.0179670i	$0.00571939\pi$
$101$	0.361133i	0.0359340i	−0.999839	+	0.0179670i	$0.994281\pi$
$102$	0	0
$103$	−19.0514	−1.87719	−0.938596	−	0.345019i	$-0.887872\pi$
$103$	−19.0514	−1.87719	−0.938596	+	0.345019i	$0.887872\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.68019i	− 0.742472i	−0.928539	−	0.371236i	$-0.878934\pi$
$107$	− 7.68019i	− 0.742472i	0.928539	−	0.371236i	$-0.121066\pi$
$108$	0	0
$109$	1.81184i	0.173543i	0.996228	+	0.0867716i	$0.0276550\pi$
$109$	1.81184i	0.173543i	−0.996228	+	0.0867716i	$0.972345\pi$
$110$	0	0
$111$	−13.9938	−1.32823
$112$	0	0
$113$	1.11278	0.104681	0.0523406	−	0.998629i	$-0.483332\pi$
$113$	1.11278	0.104681	0.0523406	+	0.998629i	$0.483332\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.05792i	0.560055i
$118$	0	0
$119$	−0.163994	−0.0150333
$120$	0	0
$121$	−9.99130	−0.908300
$122$	0	0
$123$	− 11.5122i	− 1.03802i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−9.81903	−0.871298	−0.435649	−	0.900117i	$-0.643481\pi$
$127$	−9.81903	−0.871298	−0.435649	+	0.900117i	$0.643481\pi$
$128$	0	0
$129$	−32.2140	−2.83629
$130$	0	0
$131$	− 5.00551i	− 0.437333i	−0.975800	−	0.218667i	$-0.929829\pi$
$131$	− 5.00551i	− 0.437333i	0.975800	−	0.218667i	$-0.0701708\pi$
$132$	0	0
$133$	1.80203i	0.156255i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.1561	−1.03856	−0.519282	−	0.854603i	$-0.673800\pi$
$137$	−12.1561	−1.03856	−0.519282	+	0.854603i	$0.673800\pi$
$138$	0	0
$139$	7.81838i	0.663147i	0.943429	+	0.331573i	$0.107579\pi$
$139$	7.81838i	0.663147i	−0.943429	+	0.331573i	$0.892421\pi$
$140$	0	0
$141$	− 25.0751i	− 2.11170i
$142$	0	0
$143$	−7.11829	−0.595261
$144$	0	0
$145$	0	0
$146$	0	0
$147$	18.1959i	1.50078i
$148$	0	0
$149$	− 17.0047i	− 1.39308i	−0.717520	−	0.696538i	$-0.754723\pi$
$149$	− 17.0047i	− 1.39308i	0.717520	−	0.696538i	$-0.245277\pi$
$150$	0	0
$151$	13.2804	1.08075	0.540373	−	0.841425i	$-0.318283\pi$
$151$	13.2804	1.08075	0.540373	+	0.841425i	$0.318283\pi$
$152$	0	0
$153$	−2.37500	−0.192007
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 19.0554i	− 1.52078i	−0.649465	−	0.760391i	$-0.725007\pi$
$157$	− 19.0554i	− 1.52078i	0.649465	−	0.760391i	$-0.274993\pi$
$158$	0	0
$159$	−27.9930	−2.21999
$160$	0	0
$161$	0.949531	0.0748335
$162$	0	0
$163$	16.7152i	1.30923i	0.755961	+	0.654617i	$0.227170\pi$
$163$	16.7152i	1.30923i	−0.755961	+	0.654617i	$0.772830\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.2045	−0.944409	−0.472204	−	0.881489i	$-0.656542\pi$
$167$	−12.2045	−0.944409	−0.472204	+	0.881489i	$0.656542\pi$
$168$	0	0
$169$	10.5861	0.814319
$170$	0	0
$171$	26.0972i	1.99571i
$172$	0	0
$173$	2.55690i	0.194397i	0.995265	+	0.0971986i	$0.0309882\pi$
$173$	2.55690i	0.194397i	−0.995265	+	0.0971986i	$0.969012\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	26.8587	2.01883
$178$	0	0
$179$	18.3566i	1.37204i	0.727584	+	0.686019i	$0.240643\pi$
$179$	18.3566i	1.37204i	−0.727584	+	0.686019i	$0.759357\pi$
$180$	0	0
$181$	8.74009i	0.649646i	0.945775	+	0.324823i	$0.105305\pi$
$181$	8.74009i	0.649646i	−0.945775	+	0.324823i	$0.894695\pi$
$182$	0	0
$183$	34.8362	2.57517
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.79071i	− 0.204077i
$188$	0	0
$189$	− 0.635851i	− 0.0462514i
$190$	0	0
$191$	21.5766	1.56123	0.780614	−	0.625013i	$-0.214906\pi$
$191$	21.5766	1.56123	0.780614	+	0.625013i	$0.214906\pi$
$192$	0	0
$193$	9.90462	0.712950	0.356475	−	0.934305i	$-0.383978\pi$
$193$	9.90462	0.712950	0.356475	+	0.934305i	$0.383978\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 9.37856i	− 0.668195i	−0.942539	−	0.334097i	$-0.891569\pi$
$197$	− 9.37856i	− 0.668195i	0.942539	−	0.334097i	$-0.108431\pi$
$198$	0	0
$199$	−16.8855	−1.19698	−0.598491	−	0.801130i	$-0.704233\pi$
$199$	−16.8855	−1.19698	−0.598491	+	0.801130i	$0.704233\pi$
$200$	0	0
$201$	10.3331	0.728839
$202$	0	0
$203$	− 2.08202i	− 0.146129i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	13.7513	0.955779
$208$	0	0
$209$	−30.6652	−2.12116
$210$	0	0
$211$	14.0265i	0.965623i	0.875724	+	0.482811i	$0.160384\pi$
$211$	14.0265i	0.965623i	−0.875724	+	0.482811i	$0.839616\pi$
$212$	0	0
$213$	18.2718i	1.25196i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.899510	−0.0610627
$218$	0	0
$219$	− 15.5799i	− 1.05280i
$220$	0	0
$221$	− 0.946347i	− 0.0636582i
$222$	0	0
$223$	2.06137	0.138039	0.0690197	−	0.997615i	$-0.478013\pi$
$223$	2.06137	0.138039	0.0690197	+	0.997615i	$0.478013\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.13275i	0.207928i	0.994581	+	0.103964i	$0.0331526\pi$
$227$	3.13275i	0.207928i	−0.994581	+	0.103964i	$0.966847\pi$
$228$	0	0
$229$	17.0738i	1.12827i	0.825682	+	0.564136i	$0.190791\pi$
$229$	17.0738i	1.12827i	−0.825682	+	0.564136i	$0.809209\pi$
$230$	0	0
$231$	3.24005	0.213179
$232$	0	0
$233$	−7.63474	−0.500169	−0.250084	−	0.968224i	$-0.580458\pi$
$233$	−7.63474	−0.500169	−0.250084	+	0.968224i	$0.580458\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	26.6293i	1.72976i
$238$	0	0
$239$	−4.92104	−0.318315	−0.159158	−	0.987253i	$-0.550878\pi$
$239$	−4.92104	−0.318315	−0.159158	+	0.987253i	$0.550878\pi$
$240$	0	0
$241$	19.1214	1.23172	0.615860	−	0.787855i	$-0.288809\pi$
$241$	19.1214	1.23172	0.615860	+	0.787855i	$0.288809\pi$
$242$	0	0
$243$	21.5161i	1.38026i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10.3988	−0.661659
$248$	0	0
$249$	17.1338	1.08581
$250$	0	0
$251$	− 29.7004i	− 1.87467i	−0.348423	−	0.937337i	$-0.613283\pi$
$251$	− 29.7004i	− 1.87467i	0.348423	−	0.937337i	$-0.386717\pi$
$252$	0	0
$253$	16.1583i	1.01586i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.1697	−0.821501	−0.410750	−	0.911748i	$-0.634733\pi$
$257$	−13.1697	−0.821501	−0.410750	+	0.911748i	$0.634733\pi$
$258$	0	0
$259$	1.43441i	0.0891297i
$260$	0	0
$261$	− 30.1522i	− 1.86637i
$262$	0	0
$263$	17.7204	1.09269	0.546345	−	0.837560i	$-0.316019\pi$
$263$	17.7204	1.09269	0.546345	+	0.837560i	$0.316019\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 17.5869i	− 1.07630i
$268$	0	0
$269$	− 1.52445i	− 0.0929474i	−0.998920	−	0.0464737i	$-0.985202\pi$
$269$	− 1.52445i	− 0.0929474i	0.998920	−	0.0464737i	$-0.0147984\pi$
$270$	0	0
$271$	−16.1497	−0.981024	−0.490512	−	0.871435i	$-0.663190\pi$
$271$	−16.1497	−0.981024	−0.490512	+	0.871435i	$0.663190\pi$
$272$	0	0
$273$	1.09872	0.0664976
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.3707i	0.743283i	0.928376	+	0.371642i	$0.121205\pi$
$277$	12.3707i	0.743283i	−0.928376	+	0.371642i	$0.878795\pi$
$278$	0	0
$279$	−13.0269	−0.779897
$280$	0	0
$281$	−4.03820	−0.240899	−0.120450	−	0.992719i	$-0.538434\pi$
$281$	−4.03820	−0.240899	−0.120450	+	0.992719i	$0.538434\pi$
$282$	0	0
$283$	4.13989i	0.246091i	0.992401	+	0.123046i	$0.0392661\pi$
$283$	4.13989i	0.246091i	−0.992401	+	0.123046i	$0.960734\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.18004	−0.0696557
$288$	0	0
$289$	−16.6290	−0.978176
$290$	0	0
$291$	10.4850i	0.614641i
$292$	0	0
$293$	25.2638i	1.47593i	0.674840	+	0.737964i	$0.264213\pi$
$293$	25.2638i	1.47593i	−0.674840	+	0.737964i	$0.735787\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	10.8203	0.627860
$298$	0	0
$299$	5.47937i	0.316880i
$300$	0	0
$301$	3.30204i	0.190326i
$302$	0	0
$303$	0.948558	0.0544933
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 10.6692i	− 0.608922i	−0.952525	−	0.304461i	$-0.901524\pi$
$307$	− 10.6692i	− 0.608922i	0.952525	−	0.304461i	$-0.0984764\pi$
$308$	0	0
$309$	50.0408i	2.84672i
$310$	0	0
$311$	−1.78111	−0.100998	−0.0504988	−	0.998724i	$-0.516081\pi$
$311$	−1.78111	−0.100998	−0.0504988	+	0.998724i	$0.516081\pi$
$312$	0	0
$313$	26.6250	1.50494	0.752468	−	0.658629i	$-0.228863\pi$
$313$	26.6250	1.50494	0.752468	+	0.658629i	$0.228863\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 9.16369i	− 0.514684i	−0.966320	−	0.257342i	$-0.917153\pi$
$317$	− 9.16369i	− 0.514684i	0.966320	−	0.257342i	$-0.0828467\pi$
$318$	0	0
$319$	35.4299	1.98369
$320$	0	0
$321$	−20.1729	−1.12594
$322$	0	0
$323$	− 4.07682i	− 0.226840i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.75903	0.263175
$328$	0	0
$329$	−2.57027	−0.141704
$330$	0	0
$331$	− 7.83419i	− 0.430606i	−0.976547	−	0.215303i	$-0.930926\pi$
$331$	− 7.83419i	− 0.430606i	0.976547	−	0.215303i	$-0.0690739\pi$
$332$	0	0
$333$	20.7733i	1.13837i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−0.636306	−0.0346618	−0.0173309	−	0.999850i	$-0.505517\pi$
$337$	−0.636306	−0.0346618	−0.0173309	+	0.999850i	$0.505517\pi$
$338$	0	0
$339$	− 2.92284i	− 0.158747i
$340$	0	0
$341$	− 15.3070i	− 0.828923i
$342$	0	0
$343$	3.74980	0.202470
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 23.5999i	− 1.26691i	−0.773780	−	0.633455i	$-0.781636\pi$
$347$	− 23.5999i	− 1.26691i	0.773780	−	0.633455i	$-0.218364\pi$
$348$	0	0
$349$	19.1542i	1.02530i	0.858597	+	0.512651i	$0.171336\pi$
$349$	19.1542i	1.02530i	−0.858597	+	0.512651i	$0.828664\pi$
$350$	0	0
$351$	3.66925	0.195850
$352$	0	0
$353$	21.9765	1.16969	0.584845	−	0.811145i	$-0.301155\pi$
$353$	21.9765	1.16969	0.584845	+	0.811145i	$0.301155\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.430751i	0.0227978i
$358$	0	0
$359$	5.63031	0.297156	0.148578	−	0.988901i	$-0.452530\pi$
$359$	5.63031	0.297156	0.148578	+	0.988901i	$0.452530\pi$
$360$	0	0
$361$	−25.7974	−1.35776
$362$	0	0
$363$	26.2434i	1.37742i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.21463	−0.272201	−0.136101	−	0.990695i	$-0.543457\pi$
$367$	−5.21463	−0.272201	−0.136101	+	0.990695i	$0.543457\pi$
$368$	0	0
$369$	−17.0896	−0.889647
$370$	0	0
$371$	2.86937i	0.148970i
$372$	0	0
$373$	− 12.6000i	− 0.652405i	−0.945300	−	0.326202i	$-0.894231\pi$
$373$	− 12.6000i	− 0.652405i	0.945300	−	0.326202i	$-0.105769\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0145	0.618779
$378$	0	0
$379$	− 1.28403i	− 0.0659562i	−0.999456	−	0.0329781i	$-0.989501\pi$
$379$	− 1.28403i	− 0.0659562i	0.999456	−	0.0329781i	$-0.0104992\pi$
$380$	0	0
$381$	25.7909i	1.32131i
$382$	0	0
$383$	18.3642	0.938369	0.469185	−	0.883100i	$-0.344548\pi$
$383$	18.3642	0.938369	0.469185	+	0.883100i	$0.344548\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	47.8207i	2.43086i
$388$	0	0
$389$	− 20.5311i	− 1.04097i	−0.853871	−	0.520485i	$-0.825751\pi$
$389$	− 20.5311i	− 1.04097i	0.853871	−	0.520485i	$-0.174249\pi$
$390$	0	0
$391$	−2.14817	−0.108638
$392$	0	0
$393$	−13.1476	−0.663207
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 17.0191i	− 0.854164i	−0.904213	−	0.427082i	$-0.859542\pi$
$397$	− 17.0191i	− 0.854164i	0.904213	−	0.427082i	$-0.140458\pi$
$398$	0	0
$399$	4.73324	0.236958
$400$	0	0
$401$	7.62575	0.380812	0.190406	−	0.981705i	$-0.439020\pi$
$401$	7.62575	0.380812	0.190406	+	0.981705i	$0.439020\pi$
$402$	0	0
$403$	− 5.19072i	− 0.258568i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−24.4094	−1.20993
$408$	0	0
$409$	7.37530	0.364685	0.182342	−	0.983235i	$-0.441632\pi$
$409$	7.37530	0.364685	0.182342	+	0.983235i	$0.441632\pi$
$410$	0	0
$411$	31.9294i	1.57496i
$412$	0	0
$413$	− 2.75310i	− 0.135471i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	20.5359	1.00565
$418$	0	0
$419$	− 22.0835i	− 1.07885i	−0.842034	−	0.539425i	$-0.818642\pi$
$419$	− 22.0835i	− 1.07885i	0.842034	−	0.539425i	$-0.181358\pi$
$420$	0	0
$421$	29.2485i	1.42549i	0.701425	+	0.712743i	$0.252547\pi$
$421$	29.2485i	1.42549i	−0.701425	+	0.712743i	$0.747453\pi$
$422$	0	0
$423$	−37.2231	−1.80985
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 3.57083i	− 0.172804i
$428$	0	0
$429$	18.6970i	0.902701i
$430$	0	0
$431$	6.99018	0.336705	0.168353	−	0.985727i	$-0.446155\pi$
$431$	6.99018	0.336705	0.168353	+	0.985727i	$0.446155\pi$
$432$	0	0
$433$	19.1288	0.919271	0.459636	−	0.888108i	$-0.347980\pi$
$433$	19.1288	0.919271	0.459636	+	0.888108i	$0.347980\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	23.6049i	1.12917i
$438$	0	0
$439$	2.78856	0.133091	0.0665453	−	0.997783i	$-0.478802\pi$
$439$	2.78856	0.133091	0.0665453	+	0.997783i	$0.478802\pi$
$440$	0	0
$441$	27.0113	1.28625
$442$	0	0
$443$	16.8836i	0.802164i	0.916042	+	0.401082i	$0.131366\pi$
$443$	16.8836i	0.802164i	−0.916042	+	0.401082i	$0.868634\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−44.6648	−2.11257
$448$	0	0
$449$	−13.1404	−0.620135	−0.310068	−	0.950714i	$-0.600352\pi$
$449$	−13.1404	−0.620135	−0.310068	+	0.950714i	$0.600352\pi$
$450$	0	0
$451$	− 20.0809i	− 0.945571i
$452$	0	0
$453$	− 34.8827i	− 1.63893i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	33.4760	1.56594	0.782971	−	0.622058i	$-0.213703\pi$
$457$	33.4760	1.56594	0.782971	+	0.622058i	$0.213703\pi$
$458$	0	0
$459$	1.43852i	0.0671444i
$460$	0	0
$461$	27.6037i	1.28563i	0.766021	+	0.642816i	$0.222234\pi$
$461$	27.6037i	1.28563i	−0.766021	+	0.642816i	$0.777766\pi$
$462$	0	0
$463$	−14.9227	−0.693517	−0.346758	−	0.937955i	$-0.612718\pi$
$463$	−14.9227	−0.693517	−0.346758	+	0.937955i	$0.612718\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 11.8685i	− 0.549210i	−0.961557	−	0.274605i	$-0.911453\pi$
$467$	− 11.8685i	− 0.549210i	0.961557	−	0.274605i	$-0.0885472\pi$
$468$	0	0
$469$	− 1.05917i	− 0.0489081i
$470$	0	0
$471$	−50.0512	−2.30624
$472$	0	0
$473$	−56.1911	−2.58367
$474$	0	0
$475$	0	0
$476$	0	0
$477$	41.5547i	1.90266i
$478$	0	0
$479$	−15.4425	−0.705586	−0.352793	−	0.935701i	$-0.614768\pi$
$479$	−15.4425	−0.705586	−0.352793	+	0.935701i	$0.614768\pi$
$480$	0	0
$481$	−8.27740	−0.377417
$482$	0	0
$483$	− 2.49406i	− 0.113484i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7.58171	−0.343560	−0.171780	−	0.985135i	$-0.554952\pi$
$487$	−7.58171	−0.343560	−0.171780	+	0.985135i	$0.554952\pi$
$488$	0	0
$489$	43.9044	1.98543
$490$	0	0
$491$	− 5.55509i	− 0.250698i	−0.992113	−	0.125349i	$-0.959995\pi$
$491$	− 5.55509i	− 0.250698i	0.992113	−	0.125349i	$-0.0400050\pi$
$492$	0	0
$493$	4.71026i	0.212140i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.87291	0.0840117
$498$	0	0
$499$	− 14.1201i	− 0.632102i	−0.948742	−	0.316051i	$-0.897643\pi$
$499$	− 14.1201i	− 0.632102i	0.948742	−	0.316051i	$-0.102357\pi$
$500$	0	0
$501$	32.0565i	1.43218i
$502$	0	0
$503$	14.3503	0.639847	0.319924	−	0.947443i	$-0.396343\pi$
$503$	14.3503	0.639847	0.319924	+	0.947443i	$0.396343\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 27.8058i	− 1.23490i
$508$	0	0
$509$	− 42.9057i	− 1.90176i	−0.309557	−	0.950881i	$-0.600181\pi$
$509$	− 42.9057i	− 1.90176i	0.309557	−	0.950881i	$-0.399819\pi$
$510$	0	0
$511$	−1.59699	−0.0706468
$512$	0	0
$513$	15.8069	0.697893
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 43.7386i	− 1.92362i
$518$	0	0
$519$	6.71600	0.294800
$520$	0	0
$521$	11.4542	0.501817	0.250909	−	0.968011i	$-0.419271\pi$
$521$	11.4542	0.501817	0.250909	+	0.968011i	$0.419271\pi$
$522$	0	0
$523$	− 21.7522i	− 0.951156i	−0.879674	−	0.475578i	$-0.842239\pi$
$523$	− 21.7522i	− 0.951156i	0.879674	−	0.475578i	$-0.157761\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.03501	0.0886464
$528$	0	0
$529$	−10.5620	−0.459219
$530$	0	0
$531$	− 39.8709i	− 1.73025i
$532$	0	0
$533$	− 6.80955i	− 0.294955i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	48.2158	2.08067
$538$	0	0
$539$	31.7393i	1.36711i
$540$	0	0
$541$	− 36.5960i	− 1.57339i	−0.617344	−	0.786693i	$-0.711791\pi$
$541$	− 36.5960i	− 1.57339i	0.617344	−	0.786693i	$-0.288209\pi$
$542$	0	0
$543$	22.9569	0.985175
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 15.8078i	− 0.675893i	−0.941165	−	0.337946i	$-0.890268\pi$
$547$	− 15.8078i	− 0.675893i	0.941165	−	0.337946i	$-0.109732\pi$
$548$	0	0
$549$	− 51.7133i	− 2.20707i
$550$	0	0
$551$	51.7580	2.20496
$552$	0	0
$553$	2.72959	0.116074
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 12.7121i	− 0.538629i	−0.963052	−	0.269315i	$-0.913203\pi$
$557$	− 12.7121i	− 0.538629i	0.963052	−	0.269315i	$-0.0867971\pi$
$558$	0	0
$559$	−19.0548	−0.805931
$560$	0	0
$561$	−7.33013	−0.309478
$562$	0	0
$563$	− 14.3700i	− 0.605624i	−0.953050	−	0.302812i	$-0.902075\pi$
$563$	− 14.3700i	− 0.605624i	0.953050	−	0.302812i	$-0.0979255\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.47923	0.0621219
$568$	0	0
$569$	32.5495	1.36455	0.682274	−	0.731097i	$-0.260991\pi$
$569$	32.5495	1.36455	0.682274	+	0.731097i	$0.260991\pi$
$570$	0	0
$571$	− 16.2274i	− 0.679097i	−0.940589	−	0.339548i	$-0.889726\pi$
$571$	− 16.2274i	− 0.679097i	0.940589	−	0.339548i	$-0.110274\pi$
$572$	0	0
$573$	− 56.6736i	− 2.36757i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−30.1137	−1.25365	−0.626825	−	0.779160i	$-0.715646\pi$
$577$	−30.1137	−1.25365	−0.626825	+	0.779160i	$0.715646\pi$
$578$	0	0
$579$	− 26.0157i	− 1.08117i
$580$	0	0
$581$	− 1.75627i	− 0.0728622i
$582$	0	0
$583$	−48.8283	−2.02226
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 14.1963i	− 0.585943i	−0.956121	−	0.292971i	$-0.905356\pi$
$587$	− 14.1963i	− 0.585943i	0.956121	−	0.292971i	$-0.0946440\pi$
$588$	0	0
$589$	− 22.3614i	− 0.921384i
$590$	0	0
$591$	−24.6339	−1.01330
$592$	0	0
$593$	−12.2025	−0.501098	−0.250549	−	0.968104i	$-0.580611\pi$
$593$	−12.2025	−0.501098	−0.250549	+	0.968104i	$0.580611\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	44.3518i	1.81520i
$598$	0	0
$599$	2.44106	0.0997391	0.0498695	−	0.998756i	$-0.484119\pi$
$599$	2.44106	0.0997391	0.0498695	+	0.998756i	$0.484119\pi$
$600$	0	0
$601$	−13.7234	−0.559789	−0.279895	−	0.960031i	$-0.590300\pi$
$601$	−13.7234	−0.559789	−0.279895	+	0.960031i	$0.590300\pi$
$602$	0	0
$603$	− 15.3391i	− 0.624657i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−25.0502	−1.01676	−0.508378	−	0.861134i	$-0.669755\pi$
$607$	−25.0502	−1.01676	−0.508378	+	0.861134i	$0.669755\pi$
$608$	0	0
$609$	−5.46868	−0.221602
$610$	0	0
$611$	− 14.8320i	− 0.600040i
$612$	0	0
$613$	26.4690i	1.06907i	0.845146	+	0.534536i	$0.179514\pi$
$613$	26.4690i	1.06907i	−0.845146	+	0.534536i	$0.820486\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20.2771	−0.816324	−0.408162	−	0.912909i	$-0.633830\pi$
$617$	−20.2771	−0.816324	−0.408162	+	0.912909i	$0.633830\pi$
$618$	0	0
$619$	− 13.1584i	− 0.528880i	−0.964402	−	0.264440i	$-0.914813\pi$
$619$	− 13.1584i	− 0.528880i	0.964402	−	0.264440i	$-0.0851871\pi$
$620$	0	0
$621$	− 8.32906i	− 0.334234i
$622$	0	0
$623$	−1.80271	−0.0722241
$624$	0	0
$625$	0	0
$626$	0	0
$627$	80.5459i	3.21669i
$628$	0	0
$629$	− 3.24514i	− 0.129392i
$630$	0	0
$631$	−30.4366	−1.21166	−0.605830	−	0.795594i	$-0.707159\pi$
$631$	−30.4366	−1.21166	−0.605830	+	0.795594i	$0.707159\pi$
$632$	0	0
$633$	36.8422	1.46435
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10.7630i	0.426445i
$638$	0	0
$639$	27.1239	1.07300
$640$	0	0
$641$	22.4237	0.885683	0.442841	−	0.896600i	$-0.353971\pi$
$641$	22.4237	0.885683	0.442841	+	0.896600i	$0.353971\pi$
$642$	0	0
$643$	− 36.0640i	− 1.42222i	−0.703078	−	0.711112i	$-0.748192\pi$
$643$	− 36.0640i	− 1.42222i	0.703078	−	0.711112i	$-0.251808\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.35270	−0.0924942	−0.0462471	−	0.998930i	$-0.514726\pi$
$647$	−2.35270	−0.0924942	−0.0462471	+	0.998930i	$0.514726\pi$
$648$	0	0
$649$	46.8498	1.83902
$650$	0	0
$651$	2.36267i	0.0926004i
$652$	0	0
$653$	− 13.9239i	− 0.544886i	−0.962172	−	0.272443i	$-0.912168\pi$
$653$	− 13.9239i	− 0.544886i	0.962172	−	0.272443i	$-0.0878317\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−23.1279	−0.902306
$658$	0	0
$659$	20.8197i	0.811022i	0.914090	+	0.405511i	$0.132906\pi$
$659$	20.8197i	0.811022i	−0.914090	+	0.405511i	$0.867094\pi$
$660$	0	0
$661$	− 9.83003i	− 0.382344i	−0.981557	−	0.191172i	$-0.938771\pi$
$661$	− 9.83003i	− 0.382344i	0.981557	−	0.191172i	$-0.0612288\pi$
$662$	0	0
$663$	−2.48569	−0.0965364
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 27.2725i	− 1.05600i
$668$	0	0
$669$	− 5.41443i	− 0.209334i
$670$	0	0
$671$	60.7650	2.34581
$672$	0	0
$673$	−45.7820	−1.76476	−0.882382	−	0.470533i	$-0.844062\pi$
$673$	−45.7820	−1.76476	−0.882382	+	0.470533i	$0.844062\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 49.9162i	− 1.91844i	−0.282669	−	0.959218i	$-0.591220\pi$
$677$	− 49.9162i	− 1.91844i	0.282669	−	0.959218i	$-0.408780\pi$
$678$	0	0
$679$	1.07474	0.0412449
$680$	0	0
$681$	8.22853	0.315318
$682$	0	0
$683$	14.4414i	0.552584i	0.961074	+	0.276292i	$0.0891056\pi$
$683$	14.4414i	0.552584i	−0.961074	+	0.276292i	$0.910894\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	44.8465	1.71100
$688$	0	0
$689$	−16.5580	−0.630810
$690$	0	0
$691$	20.6371i	0.785072i	0.919737	+	0.392536i	$0.128402\pi$
$691$	20.6371i	0.785072i	−0.919737	+	0.392536i	$0.871598\pi$
$692$	0	0
$693$	− 4.80975i	− 0.182707i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.66967	0.101121
$698$	0	0
$699$	20.0536i	0.758496i
$700$	0	0
$701$	− 30.0040i	− 1.13323i	−0.823981	−	0.566617i	$-0.808252\pi$
$701$	− 30.0040i	− 1.13323i	0.823981	−	0.566617i	$-0.191748\pi$
$702$	0	0
$703$	−35.6586	−1.34489
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 0.0972302i	− 0.00365672i
$708$	0	0
$709$	− 8.10493i	− 0.304387i	−0.988351	−	0.152194i	$-0.951366\pi$
$709$	− 8.10493i	− 0.304387i	0.988351	−	0.152194i	$-0.0486337\pi$
$710$	0	0
$711$	39.5303	1.48250
$712$	0	0
$713$	−11.7827	−0.441267
$714$	0	0
$715$	0	0
$716$	0	0
$717$	12.9257i	0.482719i
$718$	0	0
$719$	38.0224	1.41800	0.708998	−	0.705210i	$-0.249148\pi$
$719$	38.0224	1.41800	0.708998	+	0.705210i	$0.249148\pi$
$720$	0	0
$721$	5.12934	0.191027
$722$	0	0
$723$	− 50.2248i	− 1.86788i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	33.5323	1.24364	0.621821	−	0.783159i	$-0.286393\pi$
$727$	33.5323	1.24364	0.621821	+	0.783159i	$0.286393\pi$
$728$	0	0
$729$	40.0322	1.48267
$730$	0	0
$731$	− 7.47038i	− 0.276302i
$732$	0	0
$733$	8.77824i	0.324232i	0.986772	+	0.162116i	$0.0518318\pi$
$733$	8.77824i	0.324232i	−0.986772	+	0.162116i	$0.948168\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18.0240	0.663924
$738$	0	0
$739$	10.4760i	0.385365i	0.981261	+	0.192683i	$0.0617187\pi$
$739$	10.4760i	0.385365i	−0.981261	+	0.192683i	$0.938281\pi$
$740$	0	0
$741$	27.3136i	1.00339i
$742$	0	0
$743$	−4.29242	−0.157474	−0.0787368	−	0.996895i	$-0.525089\pi$
$743$	−4.29242	−0.157474	−0.0787368	+	0.996895i	$0.525089\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 25.4346i	− 0.930602i
$748$	0	0
$749$	2.06779i	0.0755554i
$750$	0	0
$751$	−54.2895	−1.98105	−0.990526	−	0.137324i	$-0.956150\pi$
$751$	−54.2895	−1.98105	−0.990526	+	0.137324i	$0.956150\pi$
$752$	0	0
$753$	−78.0118	−2.84291
$754$	0	0
$755$	0	0
$756$	0	0
$757$	15.6468i	0.568691i	0.958722	+	0.284345i	$0.0917763\pi$
$757$	15.6468i	0.568691i	−0.958722	+	0.284345i	$0.908224\pi$
$758$	0	0
$759$	42.4416	1.54053
$760$	0	0
$761$	33.5439	1.21597	0.607983	−	0.793950i	$-0.291979\pi$
$761$	33.5439	1.21597	0.607983	+	0.793950i	$0.291979\pi$
$762$	0	0
$763$	− 0.487815i	− 0.0176601i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	15.8871	0.573649
$768$	0	0
$769$	−12.1137	−0.436832	−0.218416	−	0.975856i	$-0.570089\pi$
$769$	−12.1137	−0.436832	−0.218416	+	0.975856i	$0.570089\pi$
$770$	0	0
$771$	34.5917i	1.24579i
$772$	0	0
$773$	− 29.0451i	− 1.04468i	−0.852737	−	0.522340i	$-0.825059\pi$
$773$	− 29.0451i	− 1.04468i	0.852737	−	0.522340i	$-0.174941\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.76764	0.135163
$778$	0	0
$779$	− 29.3352i	− 1.05104i
$780$	0	0
$781$	31.8716i	1.14045i
$782$	0	0
$783$	−18.2630	−0.652665
$784$	0	0
$785$	0	0
$786$	0	0
$787$	29.0437i	1.03530i	0.855593	+	0.517649i	$0.173193\pi$
$787$	29.0437i	1.03530i	−0.855593	+	0.517649i	$0.826807\pi$
$788$	0	0
$789$	− 46.5449i	− 1.65704i
$790$	0	0
$791$	−0.299601	−0.0106526
$792$	0	0
$793$	20.6058	0.731734
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 40.5313i	− 1.43569i	−0.696201	−	0.717847i	$-0.745128\pi$
$797$	− 40.5313i	− 1.43569i	0.696201	−	0.717847i	$-0.254872\pi$
$798$	0	0
$799$	5.81486	0.205715
$800$	0	0
$801$	−26.1072	−0.922452
$802$	0	0
$803$	− 27.1762i	− 0.959026i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.00415	−0.140953
$808$	0	0
$809$	52.0674	1.83059	0.915297	−	0.402781i	$-0.131956\pi$
$809$	52.0674	1.83059	0.915297	+	0.402781i	$0.131956\pi$
$810$	0	0
$811$	− 27.5707i	− 0.968138i	−0.875030	−	0.484069i	$-0.839158\pi$
$811$	− 27.5707i	− 0.968138i	0.875030	−	0.484069i	$-0.160842\pi$
$812$	0	0
$813$	42.4191i	1.48770i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−82.0870	−2.87186
$818$	0	0
$819$	− 1.63102i	− 0.0569923i
$820$	0	0
$821$	− 51.8915i	− 1.81103i	−0.424318	−	0.905513i	$-0.639486\pi$
$821$	− 51.8915i	− 1.81103i	0.424318	−	0.905513i	$-0.360514\pi$
$822$	0	0
$823$	42.8229	1.49271	0.746356	−	0.665547i	$-0.231802\pi$
$823$	42.8229	1.49271	0.746356	+	0.665547i	$0.231802\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 24.7187i	− 0.859554i	−0.902935	−	0.429777i	$-0.858592\pi$
$827$	− 24.7187i	− 0.859554i	0.902935	−	0.429777i	$-0.141408\pi$
$828$	0	0
$829$	9.70912i	0.337212i	0.985684	+	0.168606i	$0.0539265\pi$
$829$	9.70912i	0.337212i	−0.985684	+	0.168606i	$0.946073\pi$
$830$	0	0
$831$	32.4931	1.12717
$832$	0	0
$833$	−4.21961	−0.146201
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7.89028i	0.272728i
$838$	0	0
$839$	15.1090	0.521620	0.260810	−	0.965390i	$-0.416010\pi$
$839$	15.1090	0.521620	0.260810	+	0.965390i	$0.416010\pi$
$840$	0	0
$841$	−30.8000	−1.06207
$842$	0	0
$843$	10.6068i	0.365319i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.69003	0.0924304
$848$	0	0
$849$	10.8739	0.373192
$850$	0	0
$851$	18.7894i	0.644092i
$852$	0	0
$853$	− 15.5755i	− 0.533295i	−0.963794	−	0.266647i	$-0.914084\pi$
$853$	− 15.5755i	− 0.533295i	0.963794	−	0.266647i	$-0.0859159\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24.5547	−0.838773	−0.419386	−	0.907808i	$-0.637755\pi$
$857$	−24.5547	−0.838773	−0.419386	+	0.907808i	$0.637755\pi$
$858$	0	0
$859$	− 38.2985i	− 1.30673i	−0.757044	−	0.653363i	$-0.773357\pi$
$859$	− 38.2985i	− 1.30673i	0.757044	−	0.653363i	$-0.226643\pi$
$860$	0	0
$861$	3.09952i	0.105631i
$862$	0	0
$863$	45.7916	1.55876	0.779382	−	0.626549i	$-0.215533\pi$
$863$	45.7916	1.55876	0.779382	+	0.626549i	$0.215533\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	43.6780i	1.48338i
$868$	0	0
$869$	46.4496i	1.57570i
$870$	0	0
$871$	6.11207	0.207100
$872$	0	0
$873$	15.5646	0.526783
$874$	0	0
$875$	0	0
$876$	0	0
$877$	42.8419i	1.44667i	0.690499	+	0.723334i	$0.257391\pi$
$877$	42.8419i	1.44667i	−0.690499	+	0.723334i	$0.742609\pi$
$878$	0	0
$879$	66.3585	2.23822
$880$	0	0
$881$	26.0361	0.877180	0.438590	−	0.898687i	$-0.355478\pi$
$881$	26.0361	0.877180	0.438590	+	0.898687i	$0.355478\pi$
$882$	0	0
$883$	− 19.3468i	− 0.651072i	−0.945530	−	0.325536i	$-0.894455\pi$
$883$	− 19.3468i	− 0.651072i	0.945530	−	0.325536i	$-0.105545\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−26.1028	−0.876445	−0.438222	−	0.898867i	$-0.644392\pi$
$887$	−26.1028	−0.876445	−0.438222	+	0.898867i	$0.644392\pi$
$888$	0	0
$889$	2.64365	0.0886650
$890$	0	0
$891$	25.1722i	0.843300i
$892$	0	0
$893$	− 63.8957i	− 2.13819i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	14.3922	0.480542
$898$	0	0
$899$	25.8358i	0.861672i
$900$	0	0
$901$	− 6.49153i	− 0.216264i
$902$	0	0
$903$	8.67321	0.288626
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 47.0568i	− 1.56249i	−0.624222	−	0.781247i	$-0.714584\pi$
$907$	− 47.0568i	− 1.56249i	0.624222	−	0.781247i	$-0.285416\pi$
$908$	0	0
$909$	− 1.40810i	− 0.0467039i
$910$	0	0
$911$	−57.6189	−1.90900	−0.954499	−	0.298214i	$-0.903609\pi$
$911$	−57.6189	−1.90900	−0.954499	+	0.298214i	$0.903609\pi$
$912$	0	0
$913$	29.8866	0.989101
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.34767i	0.0445039i
$918$	0	0
$919$	−22.6047	−0.745661	−0.372830	−	0.927900i	$-0.621613\pi$
$919$	−22.6047	−0.745661	−0.372830	+	0.927900i	$0.621613\pi$
$920$	0	0
$921$	−28.0239	−0.923418
$922$	0	0
$923$	10.8079i	0.355745i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	74.2840	2.43981
$928$	0	0
$929$	22.2533	0.730108	0.365054	−	0.930986i	$-0.381050\pi$
$929$	22.2533	0.730108	0.365054	+	0.930986i	$0.381050\pi$
$930$	0	0
$931$	46.3665i	1.51960i
$932$	0	0
$933$	4.67831i	0.153161i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.8873	0.682358	0.341179	−	0.939998i	$-0.389174\pi$
$937$	20.8873	0.682358	0.341179	+	0.939998i	$0.389174\pi$
$938$	0	0
$939$	− 69.9339i	− 2.28221i
$940$	0	0
$941$	13.7530i	0.448335i	0.974551	+	0.224168i	$0.0719664\pi$
$941$	13.7530i	0.448335i	−0.974551	+	0.224168i	$0.928034\pi$
$942$	0	0
$943$	−15.4574	−0.503364
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.2715i	1.01619i	0.861302	+	0.508094i	$0.169650\pi$
$947$	31.2715i	1.01619i	−0.861302	+	0.508094i	$0.830350\pi$
$948$	0	0
$949$	− 9.21562i	− 0.299152i
$950$	0	0
$951$	−24.0695	−0.780508
$952$	0	0
$953$	−26.2447	−0.850150	−0.425075	−	0.905158i	$-0.639752\pi$
$953$	−26.2447	−0.850150	−0.425075	+	0.905158i	$0.639752\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 93.0609i	− 3.00823i
$958$	0	0
$959$	3.27286	0.105686
$960$	0	0
$961$	−19.8380	−0.639935
$962$	0	0
$963$	29.9461i	0.964998i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	18.5089	0.595208	0.297604	−	0.954689i	$-0.403813\pi$
$967$	18.5089	0.595208	0.297604	+	0.954689i	$0.403813\pi$
$968$	0	0
$969$	−10.7083	−0.343999
$970$	0	0
$971$	16.7762i	0.538373i	0.963088	+	0.269187i	$0.0867548\pi$
$971$	16.7762i	0.538373i	−0.963088	+	0.269187i	$0.913245\pi$
$972$	0	0
$973$	− 2.10500i	− 0.0674831i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11.2815	0.360926	0.180463	−	0.983582i	$-0.442240\pi$
$977$	11.2815	0.360926	0.180463	+	0.983582i	$0.442240\pi$
$978$	0	0
$979$	− 30.6769i	− 0.980438i
$980$	0	0
$981$	− 7.06462i	− 0.225556i
$982$	0	0
$983$	−28.6404	−0.913488	−0.456744	−	0.889598i	$-0.650984\pi$
$983$	−28.6404	−0.913488	−0.456744	+	0.889598i	$0.650984\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6.75113i	0.214891i
$988$	0	0
$989$	43.2537i	1.37539i
$990$	0	0
$991$	17.4805	0.555287	0.277643	−	0.960684i	$-0.410447\pi$
$991$	17.4805	0.555287	0.277643	+	0.960684i	$0.410447\pi$
$992$	0	0
$993$	−20.5774	−0.653005
$994$	0	0
$995$	0	0
$996$	0	0
$997$	17.6990i	0.560533i	0.959922	+	0.280266i	$0.0904228\pi$
$997$	17.6990i	0.560533i	−0.959922	+	0.280266i	$0.909577\pi$
$998$	0	0
$999$	12.5823	0.398086

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.d.c.2001.13		40
4.3	odd	2		1000.2.d.c.501.29	yes	40
5.2	odd	4		4000.2.f.d.3249.1		20
5.3	odd	4		4000.2.f.c.3249.20		20
5.4	even	2	inner	4000.2.d.c.2001.28		40
8.3	odd	2		1000.2.d.c.501.30	yes	40
8.5	even	2	inner	4000.2.d.c.2001.14		40
20.3	even	4		1000.2.f.c.749.5		20
20.7	even	4		1000.2.f.d.749.16		20
20.19	odd	2		1000.2.d.c.501.12	yes	40
40.3	even	4		1000.2.f.d.749.15		20
40.13	odd	4		4000.2.f.d.3249.2		20
40.19	odd	2		1000.2.d.c.501.11	✓	40
40.27	even	4		1000.2.f.c.749.6		20
40.29	even	2	inner	4000.2.d.c.2001.27		40
40.37	odd	4		4000.2.f.c.3249.19		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.d.c.501.11	✓	40	40.19	odd	2
1000.2.d.c.501.12	yes	40	20.19	odd	2
1000.2.d.c.501.29	yes	40	4.3	odd	2
1000.2.d.c.501.30	yes	40	8.3	odd	2
1000.2.f.c.749.5		20	20.3	even	4
1000.2.f.c.749.6		20	40.27	even	4
1000.2.f.d.749.15		20	40.3	even	4
1000.2.f.d.749.16		20	20.7	even	4
4000.2.d.c.2001.13		40	1.1	even	1	trivial
4000.2.d.c.2001.14		40	8.5	even	2	inner
4000.2.d.c.2001.27		40	40.29	even	2	inner
4000.2.d.c.2001.28		40	5.4	even	2	inner
4000.2.f.c.3249.19		20	40.37	odd	4
4000.2.f.c.3249.20		20	5.3	odd	4
4000.2.f.d.3249.1		20	5.2	odd	4
4000.2.f.d.3249.2		20	40.13	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:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.62662i q^{3} -0.269237 q^{7} -3.89913 q^{9} -4.58163i q^{11} -1.55366i q^{13} +0.609109 q^{17} -6.69309i q^{19} +0.707183i q^{21} -3.52675 q^{23} +2.36168i q^{27} +7.73304i q^{29} +3.34096 q^{31} -12.0342 q^{33} -5.32768i q^{37} -4.08087 q^{39} +4.38291 q^{41} -12.2644i q^{43} +9.54651 q^{47} -6.92751 q^{49} -1.59990i q^{51} -10.6574i q^{53} -17.5802 q^{57} +10.2256i q^{59} +13.2628i q^{61} +1.04979 q^{63} +3.93398i q^{67} +9.26343i q^{69} -6.95638 q^{71} +5.93156 q^{73} +1.23354i q^{77} -10.1382 q^{79} -5.49416 q^{81} +6.52313i q^{83} +20.3118 q^{87} +6.69564 q^{89} +0.418302i q^{91} -8.77544i q^{93} -3.99182 q^{97} +17.8644i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 24 q^{9} - 48 q^{31} - 8 q^{39} + 44 q^{41} + 12 q^{49} - 96 q^{71} - 96 q^{79} - 56 q^{81} - 44 q^{89}+O(q^{100}) \) 40 * q - 24 * q^9 - 48 * q^31 - 8 * q^39 + 44 * q^41 + 12 * q^49 - 96 * q^71 - 96 * q^79 - 56 * q^81 - 44 * q^89

Introduction

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