LMFDB - Embedded newform 648.4.i.p.217.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,4,Mod(217,648)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("648.217");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$38.2332376837$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-43})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} - 11x + 121 $ x^4 - x^3 - 10x^2 - 11x + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.1
Root			$-2.58945 - 2.07237i$ of defining polynomial
Character	$\chi$	$=$	648.217
Dual form			648.4.i.p.433.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-6.67891 - 11.5682i) q^{5} +(7.17891 - 12.4342i) q^{7} +O(q^{10})$	$q+(-6.67891 - 11.5682i) q^{5} +(7.17891 - 12.4342i) q^{7} +(-19.5367 + 33.8386i) q^{11} +(38.3945 + 66.5013i) q^{13} -62.4313 q^{17} -39.7891 q^{19} +(64.2524 + 111.288i) q^{23} +(-26.7156 + 46.2728i) q^{25} +(-32.4680 + 56.2362i) q^{29} +(4.56873 + 7.91328i) q^{31} -191.789 q^{35} +319.505 q^{37} +(-8.78908 - 15.2231i) q^{41} +(225.473 - 390.530i) q^{43} +(290.799 - 503.678i) q^{47} +(68.4265 + 118.518i) q^{49} +329.450 q^{53} +521.936 q^{55} +(120.927 + 209.451i) q^{59} +(-248.762 + 430.868i) q^{61} +(512.867 - 888.312i) q^{65} +(289.032 + 500.618i) q^{67} -660.927 q^{71} +696.559 q^{73} +(280.505 + 485.848i) q^{77} +(-365.134 + 632.430i) q^{79} +(-545.239 + 944.382i) q^{83} +(416.973 + 722.218i) q^{85} -317.588 q^{89} +1102.52 q^{91} +(265.748 + 460.288i) q^{95} +(742.827 - 1286.61i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} + 6 q^{7}+O(q^{10}) $ 4 * q - 4 * q^5 + 6 * q^7	$ 4 q - 4 q^{5} + 6 q^{7} - 10 q^{11} + 40 q^{13} - 68 q^{17} + 68 q^{19} + 98 q^{23} - 16 q^{25} + 120 q^{29} + 200 q^{31} - 540 q^{35} + 960 q^{37} + 192 q^{41} + 334 q^{43} + 300 q^{47} + 410 q^{49} - 136 q^{53} + 1588 q^{55} + 620 q^{59} - 200 q^{61} + 1370 q^{65} + 1406 q^{67} - 2780 q^{71} + 3604 q^{73} + 804 q^{77} + 334 q^{79} - 500 q^{83} + 1100 q^{85} - 180 q^{89} + 2820 q^{91} + 1222 q^{95} + 200 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 + 6 * q^7 - 10 * q^11 + 40 * q^13 - 68 * q^17 + 68 * q^19 + 98 * q^23 - 16 * q^25 + 120 * q^29 + 200 * q^31 - 540 * q^35 + 960 * q^37 + 192 * q^41 + 334 * q^43 + 300 * q^47 + 410 * q^49 - 136 * q^53 + 1588 * q^55 + 620 * q^59 - 200 * q^61 + 1370 * q^65 + 1406 * q^67 - 2780 * q^71 + 3604 * q^73 + 804 * q^77 + 334 * q^79 - 500 * q^83 + 1100 * q^85 - 180 * q^89 + 2820 * q^91 + 1222 * q^95 + 200 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−6.67891	−	11.5682i	−0.597380	−	1.03469i	−0.993206	−	0.116367i	$-0.962875\pi$
$5$	−6.67891	−	11.5682i	−0.597380	−	1.03469i	0.395827	−	0.918325i	$-0.370458\pi$
$6$		0			0
$7$	7.17891	−	12.4342i	0.387625	−	0.671386i	−0.604505	−	0.796601i	$-0.706629\pi$
$7$	7.17891	−	12.4342i	0.387625	−	0.671386i	0.992130	+	0.125216i	$0.0399624\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−19.5367	+	33.8386i	−0.535504	+	0.927520i	0.463635	+	0.886026i	$0.346545\pi$
$11$	−19.5367	+	33.8386i	−0.535504	+	0.927520i	−0.999139	+	0.0414937i	$0.986788\pi$
$12$		0			0
$13$	38.3945	+	66.5013i	0.819133	+	1.41878i	0.906322	+	0.422588i	$0.138878\pi$
$13$	38.3945	+	66.5013i	0.819133	+	1.41878i	−0.0871887	+	0.996192i	$0.527788\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−62.4313			−0.890694			−0.445347	−	0.895358i	$-0.646920\pi$
$17$	−62.4313			−0.890694			−0.445347	+	0.895358i	$0.646920\pi$
$18$		0			0
$19$	−39.7891			−0.480434			−0.240217	−	0.970719i	$-0.577219\pi$
$19$	−39.7891			−0.480434			−0.240217	+	0.970719i	$0.577219\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	64.2524	+	111.288i	0.582502	+	1.00892i	0.995182	+	0.0980467i	$0.0312595\pi$
$23$	64.2524	+	111.288i	0.582502	+	1.00892i	−0.412680	+	0.910876i	$0.635407\pi$
$24$		0			0
$25$	−26.7156	+	46.2728i	−0.213725	+	0.370183i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−32.4680	+	56.2362i	−0.207902	+	0.360097i	−0.951053	−	0.309027i	$-0.899997\pi$
$29$	−32.4680	+	56.2362i	−0.207902	+	0.360097i	0.743152	+	0.669123i	$0.233330\pi$
$30$		0			0
$31$	4.56873	+	7.91328i	0.0264700	+	0.0458473i	0.878957	−	0.476901i	$-0.158240\pi$
$31$	4.56873	+	7.91328i	0.0264700	+	0.0458473i	−0.852487	+	0.522748i	$0.824907\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−191.789			−0.926236
$36$		0			0
$37$	319.505			1.41963			0.709814	−	0.704389i	$-0.248779\pi$
$37$	319.505			1.41963			0.709814	+	0.704389i	$0.248779\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.78908	−	15.2231i	−0.0334786	−	0.0579867i	0.848801	−	0.528713i	$-0.177325\pi$
$41$	−8.78908	−	15.2231i	−0.0334786	−	0.0579867i	−0.882279	+	0.470726i	$0.843992\pi$
$42$		0			0
$43$	225.473	−	390.530i	0.799634	−	1.38501i	−0.120220	−	0.992747i	$-0.538360\pi$
$43$	225.473	−	390.530i	0.799634	−	1.38501i	0.919855	−	0.392260i	$-0.128307\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	290.799	−	503.678i	0.902496	−	1.56317i	0.0782529	−	0.996934i	$-0.475066\pi$
$47$	290.799	−	503.678i	0.902496	−	1.56317i	0.824243	−	0.566236i	$-0.191601\pi$
$48$		0			0
$49$	68.4265	+	118.518i	0.199494	+	0.345534i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	329.450			0.853839			0.426919	−	0.904290i	$-0.359599\pi$
$53$	329.450			0.853839			0.426919	+	0.904290i	$0.359599\pi$
$54$		0			0
$55$	521.936			1.27960
$56$		0			0
$57$		0			0
$58$		0			0
$59$	120.927	+	209.451i	0.266836	+	0.462173i	0.968043	−	0.250785i	$-0.0806886\pi$
$59$	120.927	+	209.451i	0.266836	+	0.462173i	−0.701207	+	0.712957i	$0.747355\pi$
$60$		0			0
$61$	−248.762	+	430.868i	−0.522142	+	0.904377i	0.477526	+	0.878618i	$0.341534\pi$
$61$	−248.762	+	430.868i	−0.522142	+	0.904377i	−0.999668	+	0.0257594i	$0.991800\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	512.867	−	888.312i	0.978667	−	1.69510i
$66$		0			0
$67$	289.032	+	500.618i	0.527028	+	0.912839i	0.999504	+	0.0314957i	$0.0100271\pi$
$67$	289.032	+	500.618i	0.527028	+	0.912839i	−0.472476	+	0.881344i	$0.656640\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−660.927			−1.10475			−0.552377	−	0.833594i	$-0.686279\pi$
$71$	−660.927			−1.10475			−0.552377	+	0.833594i	$0.686279\pi$
$72$		0			0
$73$	696.559			1.11680			0.558398	−	0.829573i	$-0.311416\pi$
$73$	696.559			1.11680			0.558398	+	0.829573i	$0.311416\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	280.505	+	485.848i	0.415149	+	0.719059i
$78$		0			0
$79$	−365.134	+	632.430i	−0.520010	+	0.900683i	0.479720	+	0.877422i	$0.340738\pi$
$79$	−365.134	+	632.430i	−0.520010	+	0.900683i	−0.999729	+	0.0232613i	$0.992595\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−545.239	+	944.382i	−0.721058	+	1.24891i	0.239519	+	0.970892i	$0.423010\pi$
$83$	−545.239	+	944.382i	−0.721058	+	1.24891i	−0.960576	+	0.278017i	$0.910323\pi$
$84$		0			0
$85$	416.973	+	722.218i	0.532083	+	0.921594i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−317.588			−0.378250			−0.189125	−	0.981953i	$-0.560565\pi$
$89$	−317.588			−0.378250			−0.189125	+	0.981953i	$0.560565\pi$
$90$		0			0
$91$	1102.52			1.27006
$92$		0			0
$93$		0			0
$94$		0			0
$95$	265.748	+	460.288i	0.287001	+	0.497101i
$96$		0			0
$97$	742.827	−	1286.61i	0.777553	−	1.34676i	−0.155795	−	0.987789i	$-0.549794\pi$
$97$	742.827	−	1286.61i	0.777553	−	1.34676i	0.933348	−	0.358972i	$-0.116873\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	501.633	−	868.853i	0.494201	−	0.855982i	−0.505776	−	0.862665i	$-0.668794\pi$
$101$	501.633	−	868.853i	0.494201	−	0.855982i	0.999978	+	0.00668295i	$0.00212727\pi$
$102$		0			0
$103$	858.799	+	1487.48i	0.821553	+	1.42297i	0.904526	+	0.426419i	$0.140225\pi$
$103$	858.799	+	1487.48i	0.821553	+	1.42297i	−0.0829729	+	0.996552i	$0.526441\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1006.27			−0.909161			−0.454581	−	0.890706i	$-0.650211\pi$
$107$	−1006.27			−0.909161			−0.454581	+	0.890706i	$0.650211\pi$
$108$		0			0
$109$	1724.94			1.51577			0.757885	−	0.652388i	$-0.226233\pi$
$109$	1724.94			1.51577			0.757885	+	0.652388i	$0.226233\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	313.509	+	543.014i	0.260995	+	0.452057i	0.966507	−	0.256641i	$-0.0826159\pi$
$113$	313.509	+	543.014i	0.260995	+	0.452057i	−0.705511	+	0.708699i	$0.749283\pi$
$114$		0			0
$115$	858.271	−	1486.57i	0.695950	−	1.20542i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−448.188	+	776.285i	−0.345255	+	0.597999i
$120$		0			0
$121$	−97.8673	−	169.511i	−0.0735291	−	0.127356i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−956.002			−0.684059
$126$		0			0
$127$	−1990.05			−1.39046			−0.695230	−	0.718788i	$-0.744697\pi$
$127$	−1990.05			−1.39046			−0.695230	+	0.718788i	$0.744697\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−78.2072	−	135.459i	−0.0521603	−	0.0903442i	0.838766	−	0.544491i	$-0.183277\pi$
$131$	−78.2072	−	135.459i	−0.0521603	−	0.0903442i	−0.890927	+	0.454147i	$0.849944\pi$
$132$		0			0
$133$	−285.642	+	494.747i	−0.186228	+	0.322556i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1059.66	−	1835.38i	0.660822	−	1.14458i	−0.319579	−	0.947560i	$-0.603541\pi$
$137$	1059.66	−	1835.38i	0.660822	−	1.14458i	0.980400	−	0.197017i	$-0.0631253\pi$
$138$		0			0
$139$	1343.39	+	2326.81i	0.819745	+	1.41984i	0.905870	+	0.423555i	$0.139218\pi$
$139$	1343.39	+	2326.81i	0.819745	+	1.41984i	−0.0861255	+	0.996284i	$0.527449\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3000.41			−1.75460
$144$		0			0
$145$	867.403			0.496785
$146$		0			0
$147$		0			0
$148$		0			0
$149$	775.781	+	1343.69i	0.426540	+	0.738789i	0.996563	−	0.0828401i	$-0.0263991\pi$
$149$	775.781	+	1343.69i	0.426540	+	0.738789i	−0.570023	+	0.821629i	$0.693066\pi$
$150$		0			0
$151$	9.05458	−	15.6830i	0.00487981	−	0.00845208i	−0.863575	−	0.504220i	$-0.831780\pi$
$151$	9.05458	−	15.6830i	0.00487981	−	0.00845208i	0.868455	+	0.495768i	$0.165113\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	61.0283	−	105.704i	0.0316252	−	0.0547765i
$156$		0			0
$157$	−1870.75	−	3240.24i	−0.950971	−	1.64713i	−0.743330	−	0.668925i	$-0.766755\pi$
$157$	−1870.75	−	3240.24i	−0.950971	−	1.64713i	−0.207641	−	0.978205i	$-0.566579\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1845.05			0.903168
$162$		0			0
$163$	2608.90			1.25365			0.626826	−	0.779160i	$-0.284354\pi$
$163$	2608.90			1.25365			0.626826	+	0.779160i	$0.284354\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	781.446	+	1353.50i	0.362097	+	0.627170i	0.988306	−	0.152485i	$-0.0487277\pi$
$167$	781.446	+	1353.50i	0.362097	+	0.627170i	−0.626209	+	0.779655i	$0.715394\pi$
$168$		0			0
$169$	−1849.78	+	3203.92i	−0.841958	+	1.45831i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	751.385	−	1301.44i	0.330212	−	0.571945i	−0.652341	−	0.757926i	$-0.726213\pi$
$173$	751.385	−	1301.44i	0.330212	−	0.571945i	0.982553	+	0.185981i	$0.0595463\pi$
$174$		0			0
$175$	383.578	+	664.377i	0.165690	+	0.286984i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4126.50			1.72307			0.861534	−	0.507700i	$-0.169504\pi$
$179$	4126.50			1.72307			0.861534	+	0.507700i	$0.169504\pi$
$180$		0			0
$181$	1582.33			0.649800			0.324900	−	0.945748i	$-0.394669\pi$
$181$	1582.33			0.649800			0.324900	+	0.945748i	$0.394669\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2133.94	−	3696.10i	−0.848057	−	1.46888i
$186$		0			0
$187$	1219.70	−	2112.59i	0.476970	−	0.826137i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1287.94	+	2230.78i	−0.487916	+	0.845096i	−0.999903	−	0.0138971i	$-0.995576\pi$
$191$	−1287.94	+	2230.78i	−0.487916	+	0.845096i	0.511987	+	0.858993i	$0.328910\pi$
$192$		0			0
$193$	697.649	+	1208.36i	0.260196	+	0.450673i	0.966294	−	0.257441i	$-0.0828794\pi$
$193$	697.649	+	1208.36i	0.260196	+	0.450673i	−0.706098	+	0.708114i	$0.749546\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5047.62			1.82552			0.912761	−	0.408494i	$-0.133946\pi$
$197$	5047.62			1.82552			0.912761	+	0.408494i	$0.133946\pi$
$198$		0			0
$199$	−2441.47			−0.869704			−0.434852	−	0.900502i	$-0.643199\pi$
$199$	−2441.47			−0.869704			−0.434852	+	0.900502i	$0.643199\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	466.169	+	807.429i	0.161176	+	0.279165i
$204$		0			0
$205$	−117.403	+	203.348i	−0.0399989	+	0.0692802i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	777.348	−	1346.41i	0.257274	−	0.445612i
$210$		0			0
$211$	−2133.08	−	3694.61i	−0.695959	−	1.20544i	−0.969856	−	0.243678i	$-0.921646\pi$
$211$	−2133.08	−	3694.61i	−0.695959	−	1.20544i	0.273897	−	0.961759i	$-0.411687\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6023.65			−1.91074
$216$		0			0
$217$	131.194			0.0410416
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2397.02	−	4151.76i	−0.729597	−	1.26370i
$222$		0			0
$223$	−875.574	+	1516.54i	−0.262927	+	0.455404i	−0.967019	−	0.254706i	$-0.918021\pi$
$223$	−875.574	+	1516.54i	−0.262927	+	0.455404i	0.704091	+	0.710110i	$0.251355\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1100.02	+	1905.29i	−0.321635	+	0.557088i	−0.980825	−	0.194888i	$-0.937566\pi$
$227$	−1100.02	+	1905.29i	−0.321635	+	0.557088i	0.659191	+	0.751976i	$0.270899\pi$
$228$		0			0
$229$	595.645	+	1031.69i	0.171884	+	0.297711i	0.939078	−	0.343703i	$-0.111681\pi$
$229$	595.645	+	1031.69i	0.171884	+	0.297711i	−0.767195	+	0.641414i	$0.778348\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2644.52			0.743555			0.371778	−	0.928322i	$-0.378748\pi$
$233$	2644.52			0.743555			0.371778	+	0.928322i	$0.378748\pi$
$234$		0			0
$235$	−7768.87			−2.15653
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−718.761	−	1244.93i	−0.194530	−	0.336937i	0.752216	−	0.658917i	$-0.228985\pi$
$239$	−718.761	−	1244.93i	−0.194530	−	0.336937i	−0.946746	+	0.321980i	$0.895652\pi$
$240$		0			0
$241$	−1633.54	+	2829.37i	−0.436620	+	0.756248i	−0.997426	−	0.0716990i	$-0.977158\pi$
$241$	−1633.54	+	2829.37i	−0.436620	+	0.756248i	0.560806	+	0.827947i	$0.310491\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	914.029	−	1583.15i	0.238348	−	0.412830i
$246$		0			0
$247$	−1527.68	−	2646.03i	−0.393539	−	0.681630i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1871.89			−0.470727			−0.235364	−	0.971907i	$-0.575628\pi$
$251$	−1871.89			−0.470727			−0.235364	+	0.971907i	$0.575628\pi$
$252$		0			0
$253$	−5021.12			−1.24773
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2345.41	+	4062.37i	0.569271	+	0.986007i	0.996638	+	0.0819287i	$0.0261080\pi$
$257$	2345.41	+	4062.37i	0.569271	+	0.986007i	−0.427367	+	0.904078i	$0.640559\pi$
$258$		0			0
$259$	2293.70	−	3972.80i	0.550283	−	0.953118i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2296.69	+	3977.98i	−0.538479	+	0.932673i	0.460507	+	0.887656i	$0.347667\pi$
$263$	−2296.69	+	3977.98i	−0.538479	+	0.932673i	−0.998986	+	0.0450168i	$0.985666\pi$
$264$		0			0
$265$	−2200.37	−	3811.15i	−0.510066	−	0.883460i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	83.0074			0.0188143			0.00940716	−	0.999956i	$-0.497006\pi$
$269$	83.0074			0.0188143			0.00940716	+	0.999956i	$0.497006\pi$
$270$		0			0
$271$	−3464.08			−0.776487			−0.388244	−	0.921557i	$-0.626918\pi$
$271$	−3464.08			−0.776487			−0.388244	+	0.921557i	$0.626918\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1043.87	−	1808.04i	−0.228901	−	0.396469i
$276$		0			0
$277$	635.350	−	1100.46i	0.137814	−	0.238701i	−0.788855	−	0.614580i	$-0.789326\pi$
$277$	635.350	−	1100.46i	0.137814	−	0.238701i	0.926669	+	0.375879i	$0.122659\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3825.90	+	6626.65i	−0.812221	+	1.40681i	0.0990857	+	0.995079i	$0.468408\pi$
$281$	−3825.90	+	6626.65i	−0.812221	+	1.40681i	−0.911306	+	0.411729i	$0.864925\pi$
$282$		0			0
$283$	3.72507	+	6.45201i	0.000782446	+	0.00135524i	0.866416	−	0.499322i	$-0.166418\pi$
$283$	3.72507	+	6.45201i	0.000782446	+	0.00135524i	−0.865634	+	0.500677i	$0.833084\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−252.384			−0.0519086
$288$		0			0
$289$	−1015.34			−0.206663
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1407.01	+	2437.02i	0.280541	+	0.485911i	0.971518	−	0.236965i	$-0.0761528\pi$
$293$	1407.01	+	2437.02i	0.280541	+	0.485911i	−0.690977	+	0.722877i	$0.742819\pi$
$294$		0			0
$295$	1615.31	−	2797.81i	0.318804	−	0.552185i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4933.88	+	8545.73i	−0.954293	+	1.65288i
$300$		0			0
$301$	−3237.30	−	5607.16i	−0.619916	−	1.07373i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6645.83			1.24767
$306$		0			0
$307$	−5096.55			−0.947477			−0.473739	−	0.880666i	$-0.657096\pi$
$307$	−5096.55			−0.947477			−0.473739	+	0.880666i	$0.657096\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3892.41	−	6741.85i	−0.709705	−	1.22924i	−0.964967	−	0.262373i	$-0.915495\pi$
$311$	−3892.41	−	6741.85i	−0.709705	−	1.22924i	0.255262	−	0.966872i	$-0.417838\pi$
$312$		0			0
$313$	123.235	−	213.448i	0.0222544	−	0.0385457i	−0.854684	−	0.519149i	$-0.826249\pi$
$313$	123.235	−	213.448i	0.0222544	−	0.0385457i	0.876938	+	0.480603i	$0.159582\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	29.6600	−	51.3727i	0.00525512	−	0.00910214i	−0.863386	−	0.504544i	$-0.831661\pi$
$317$	29.6600	−	51.3727i	0.00525512	−	0.00910214i	0.868641	+	0.495442i	$0.164994\pi$
$318$		0			0
$319$	−1268.64	−	2197.34i	−0.222665	−	0.385666i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2484.08			0.427920
$324$		0			0
$325$	−4102.94			−0.700277
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4175.23	−	7231.71i	−0.699660	−	1.21185i
$330$		0			0
$331$	−2204.55	+	3818.39i	−0.366082	+	0.634072i	−0.988949	−	0.148255i	$-0.952634\pi$
$331$	−2204.55	+	3818.39i	−0.366082	+	0.634072i	0.622867	+	0.782327i	$0.285968\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3860.84	−	6687.17i	0.629672	−	1.09062i
$336$		0			0
$337$	−3016.15	−	5224.12i	−0.487537	−	0.844440i	0.512360	−	0.858771i	$-0.328771\pi$
$337$	−3016.15	−	5224.12i	−0.487537	−	0.844440i	−0.999897	+	0.0143313i	$0.995438\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−357.032			−0.0566991
$342$		0			0
$343$	6889.64			1.08456
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3120.39	+	5404.68i	0.482742	+	0.836133i	0.999804	−	0.0198147i	$-0.00630762\pi$
$347$	3120.39	+	5404.68i	0.482742	+	0.836133i	−0.517062	+	0.855948i	$0.672974\pi$
$348$		0			0
$349$	−2817.31	+	4879.72i	−0.432111	+	0.748439i	−0.997055	−	0.0766906i	$-0.975565\pi$
$349$	−2817.31	+	4879.72i	−0.432111	+	0.748439i	0.564943	+	0.825130i	$0.308898\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1720.66	−	2980.27i	0.259437	−	0.449359i	−0.706654	−	0.707559i	$-0.749796\pi$
$353$	1720.66	−	2980.27i	0.259437	−	0.449359i	0.966091	+	0.258201i	$0.0831296\pi$
$354$		0			0
$355$	4414.27	+	7645.74i	0.659958	+	1.14308i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	7808.79			1.14800			0.574000	−	0.818855i	$-0.305391\pi$
$359$	7808.79			1.14800			0.574000	+	0.818855i	$0.305391\pi$
$360$		0			0
$361$	−5275.83			−0.769183
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4652.26	−	8057.94i	−0.667151	−	1.15554i
$366$		0			0
$367$	−6804.06	+	11785.0i	−0.967763	+	1.67621i	−0.265763	+	0.964038i	$0.585624\pi$
$367$	−6804.06	+	11785.0i	−0.967763	+	1.67621i	−0.702001	+	0.712176i	$0.747710\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2365.09	−	4096.46i	0.330969	−	0.573255i
$372$		0			0
$373$	1874.08	+	3246.01i	0.260151	+	0.450595i	0.966282	−	0.257486i	$-0.0828942\pi$
$373$	1874.08	+	3246.01i	0.260151	+	0.450595i	−0.706131	+	0.708082i	$0.749561\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4986.37			−0.681197
$378$		0			0
$379$	−10210.3			−1.38382			−0.691909	−	0.721985i	$-0.743230\pi$
$379$	−10210.3			−1.38382			−0.691909	+	0.721985i	$0.743230\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3064.75	−	5308.30i	−0.408880	−	0.708202i	0.585884	−	0.810395i	$-0.300747\pi$
$383$	−3064.75	−	5308.30i	−0.408880	−	0.708202i	−0.994765	+	0.102193i	$0.967414\pi$
$384$		0			0
$385$	3746.93	−	6489.87i	0.496003	−	0.859103i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2432.99	−	4214.05i	0.317114	−	0.549257i	−0.662771	−	0.748822i	$-0.730620\pi$
$389$	2432.99	−	4214.05i	0.317114	−	0.549257i	0.979885	+	0.199565i	$0.0639530\pi$
$390$		0			0
$391$	−4011.36	−	6947.87i	−0.518831	−	0.898642i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9754.78			1.24257
$396$		0			0
$397$	−438.311			−0.0554110			−0.0277055	−	0.999616i	$-0.508820\pi$
$397$	−438.311			−0.0554110			−0.0277055	+	0.999616i	$0.508820\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5876.83	+	10179.0i	0.731857	+	1.26761i	0.956089	+	0.293078i	$0.0946796\pi$
$401$	5876.83	+	10179.0i	0.731857	+	1.26761i	−0.224231	+	0.974536i	$0.571987\pi$
$402$		0			0
$403$	−350.829	+	607.653i	−0.0433648	+	0.0751101i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−6242.08	+	10811.6i	−0.760217	+	1.31673i
$408$		0			0
$409$	−5623.24	−	9739.75i	−0.679833	−	1.17750i	−0.975031	−	0.222069i	$-0.928719\pi$
$409$	−5623.24	−	9739.75i	−0.679833	−	1.17750i	0.295198	−	0.955436i	$-0.404614\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3472.48			0.413728
$414$		0			0
$415$	14566.4			1.72298
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6158.18	+	10666.3i	0.718012	+	1.24363i	0.961786	+	0.273801i	$0.0882810\pi$
$419$	6158.18	+	10666.3i	0.718012	+	1.24363i	−0.243774	+	0.969832i	$0.578386\pi$
$420$		0			0
$421$	2351.30	−	4072.57i	0.272198	−	0.471461i	−0.697226	−	0.716851i	$-0.745583\pi$
$421$	2351.30	−	4072.57i	0.272198	−	0.471461i	0.969424	+	0.245390i	$0.0789160\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1667.89	−	2888.87i	0.190364	−	0.329720i
$426$		0			0
$427$	3571.68	+	6186.32i	0.404790	+	0.701118i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2204.89			−0.246417			−0.123208	−	0.992381i	$-0.539318\pi$
$431$	−2204.89			−0.246417			−0.123208	+	0.992381i	$0.539318\pi$
$432$		0			0
$433$	9426.46			1.04620			0.523102	−	0.852270i	$-0.324775\pi$
$433$	9426.46			1.04620			0.523102	+	0.852270i	$0.324775\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2556.54	−	4428.06i	−0.279854	−	0.484721i
$438$		0			0
$439$	3842.29	−	6655.05i	0.417728	−	0.723527i	−0.577982	−	0.816049i	$-0.696160\pi$
$439$	3842.29	−	6655.05i	0.417728	−	0.723527i	0.995711	+	0.0925227i	$0.0294931\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6197.15	−	10733.8i	0.664640	−	1.15119i	−0.314743	−	0.949177i	$-0.601918\pi$
$443$	6197.15	−	10733.8i	0.664640	−	1.15119i	0.979383	−	0.202013i	$-0.0647483\pi$
$444$		0			0
$445$	2121.14	+	3673.92i	0.225959	+	0.391372i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−14568.7			−1.53127			−0.765635	−	0.643275i	$-0.777575\pi$
$449$	−14568.7			−1.53127			−0.765635	+	0.643275i	$0.777575\pi$
$450$		0			0
$451$	686.840			0.0717118
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−7363.65	−	12754.2i	−0.758711	−	1.31413i
$456$		0			0
$457$	323.024	−	559.493i	0.0330643	−	0.0572691i	−0.849020	−	0.528361i	$-0.822807\pi$
$457$	323.024	−	559.493i	0.0330643	−	0.0572691i	0.882084	+	0.471092i	$0.156140\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9500.49	+	16455.3i	−0.959831	+	1.66248i	−0.236927	+	0.971528i	$0.576140\pi$
$461$	−9500.49	+	16455.3i	−0.959831	+	1.66248i	−0.722904	+	0.690948i	$0.757193\pi$
$462$		0			0
$463$	−6832.86	−	11834.9i	−0.685853	−	1.18793i	−0.973168	−	0.230096i	$-0.926096\pi$
$463$	−6832.86	−	11834.9i	−0.685853	−	1.18793i	0.287315	−	0.957836i	$-0.407237\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3762.83			0.372855			0.186427	−	0.982469i	$-0.440309\pi$
$467$	3762.83			0.372855			0.186427	+	0.982469i	$0.440309\pi$
$468$		0			0
$469$	8299.74			0.817156
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8810.00	+	15259.4i	0.856415	+	1.48335i
$474$		0			0
$475$	1062.99	−	1841.15i	0.102681	−	0.177848i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3381.94	−	5857.68i	0.322598	−	0.558757i	−0.658425	−	0.752646i	$-0.728777\pi$
$479$	3381.94	−	5857.68i	0.322598	−	0.558757i	0.981023	+	0.193890i	$0.0621104\pi$
$480$		0			0
$481$	12267.2	+	21247.5i	1.16286	+	2.01414i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−19845.1			−1.85798
$486$		0			0
$487$	−1447.72			−0.134708			−0.0673538	−	0.997729i	$-0.521456\pi$
$487$	−1447.72			−0.134708			−0.0673538	+	0.997729i	$0.521456\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3919.14	+	6788.15i	0.360220	+	0.623920i	0.987997	−	0.154474i	$-0.0493682\pi$
$491$	3919.14	+	6788.15i	0.360220	+	0.623920i	−0.627777	+	0.778394i	$0.716035\pi$
$492$		0			0
$493$	2027.02	−	3510.90i	0.185177	−	0.320736i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4744.73	+	8218.12i	−0.428230	+	0.741716i
$498$		0			0
$499$	2122.67	+	3676.57i	0.190428	+	0.329832i	0.945392	−	0.325935i	$-0.105679\pi$
$499$	2122.67	+	3676.57i	0.190428	+	0.329832i	−0.754964	+	0.655766i	$0.772346\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−17914.9			−1.58804			−0.794021	−	0.607891i	$-0.792016\pi$
$503$	−17914.9			−1.58804			−0.794021	+	0.607891i	$0.792016\pi$
$504$		0			0
$505$	−13401.4			−1.18090
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7260.14	+	12574.9i	0.632220	+	1.09504i	0.987097	+	0.160124i	$0.0511896\pi$
$509$	7260.14	+	12574.9i	0.632220	+	1.09504i	−0.354877	+	0.934913i	$0.615477\pi$
$510$		0			0
$511$	5000.54	−	8661.18i	0.432898	−	0.749801i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	11471.7	−	19869.5i	0.981558	−	1.70011i
$516$		0			0
$517$	11362.5	+	19680.4i	0.966581	+	1.67417i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3372.49			0.283592			0.141796	−	0.989896i	$-0.454712\pi$
$521$	3372.49			0.283592			0.141796	+	0.989896i	$0.454712\pi$
$522$		0			0
$523$	4339.40			0.362808			0.181404	−	0.983409i	$-0.441936\pi$
$523$	4339.40			0.362808			0.181404	+	0.983409i	$0.441936\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−285.232	−	494.036i	−0.0235766	−	0.0408359i
$528$		0			0
$529$	−2173.23	+	3764.15i	−0.178617	+	0.309373i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	674.906	−	1168.97i	0.0548469	−	0.0949977i
$534$		0			0
$535$	6720.82	+	11640.8i	0.543115	+	0.940702i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5347.32			−0.427320
$540$		0			0
$541$	−3831.98			−0.304528			−0.152264	−	0.988340i	$-0.548656\pi$
$541$	−3831.98			−0.304528			−0.152264	+	0.988340i	$0.548656\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−11520.7	−	19954.4i	−0.905490	−	1.56835i
$546$		0			0
$547$	−6701.34	+	11607.1i	−0.523818	+	0.907280i	0.475797	+	0.879555i	$0.342160\pi$
$547$	−6701.34	+	11607.1i	−0.523818	+	0.907280i	−0.999616	+	0.0277247i	$0.991174\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1291.87	−	2237.59i	0.0998831	−	0.173003i
$552$		0			0
$553$	5242.52	+	9080.32i	0.403137	+	0.698254i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	14810.0			1.12661			0.563304	−	0.826249i	$-0.309530\pi$
$557$	14810.0			1.12661			0.563304	+	0.826249i	$0.309530\pi$
$558$		0			0
$559$	34627.7			2.62003
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6116.61	+	10594.3i	0.457877	+	0.793066i	0.998849	−	0.0479751i	$-0.0152768\pi$
$563$	6116.61	+	10594.3i	0.457877	+	0.793066i	−0.540972	+	0.841041i	$0.681943\pi$
$564$		0			0
$565$	4187.80	−	7253.49i	0.311827	−	0.540100i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4855.84	+	8410.56i	−0.357763	+	0.619664i	−0.987587	−	0.157074i	$-0.949794\pi$
$569$	−4855.84	+	8410.56i	−0.357763	+	0.619664i	0.629823	+	0.776738i	$0.283127\pi$
$570$		0			0
$571$	−4264.81	−	7386.86i	−0.312568	−	0.541384i	0.666349	−	0.745640i	$-0.267856\pi$
$571$	−4264.81	−	7386.86i	−0.312568	−	0.541384i	−0.978918	+	0.204255i	$0.934523\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6866.17			−0.497981
$576$		0			0
$577$	15314.0			1.10491			0.552453	−	0.833544i	$-0.313692\pi$
$577$	15314.0			1.10491			0.552453	+	0.833544i	$0.313692\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7828.44	+	13559.3i	0.558999	+	0.968215i
$582$		0			0
$583$	−6436.38	+	11148.1i	−0.457234	+	0.791953i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9837.53	−	17039.1i	0.691717	−	1.19809i	−0.279557	−	0.960129i	$-0.590188\pi$
$587$	9837.53	−	17039.1i	0.691717	−	1.19809i	0.971275	−	0.237961i	$-0.0764790\pi$
$588$		0			0
$589$	−181.786	−	314.862i	−0.0127171	−	0.0220266i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−15635.9			−1.08278			−0.541391	−	0.840771i	$-0.682102\pi$
$593$	−15635.9			−1.08278			−0.541391	+	0.840771i	$0.682102\pi$
$594$		0			0
$595$	11973.6			0.824994
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−242.594	−	420.186i	−0.0165478	−	0.0286616i	0.857633	−	0.514262i	$-0.171934\pi$
$599$	−242.594	−	420.186i	−0.0165478	−	0.0286616i	−0.874181	+	0.485601i	$0.838601\pi$
$600$		0			0
$601$	7398.12	−	12813.9i	0.502123	−	0.869702i	−0.497874	−	0.867249i	$-0.665886\pi$
$601$	7398.12	−	12813.9i	0.502123	−	0.869702i	0.999997	−	0.00245282i	$-0.000780759\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1307.29	+	2264.30i	−0.0878496	+	0.152160i
$606$		0			0
$607$	−1773.99	−	3072.64i	−0.118623	−	0.205461i	0.800599	−	0.599200i	$-0.204515\pi$
$607$	−1773.99	−	3072.64i	−0.118623	−	0.205461i	−0.919222	+	0.393739i	$0.871181\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	44660.3			2.95706
$612$		0			0
$613$	20450.9			1.34748			0.673740	−	0.738968i	$-0.264687\pi$
$613$	20450.9			1.34748			0.673740	+	0.738968i	$0.264687\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7652.60	−	13254.7i	−0.499323	−	0.864853i	0.500677	−	0.865634i	$-0.333085\pi$
$617$	−7652.60	−	13254.7i	−0.499323	−	0.864853i	−1.00000		0.000781625i	$0.999751\pi$
$618$		0			0
$619$	2075.59	−	3595.03i	0.134774	−	0.233435i	−0.790737	−	0.612156i	$-0.790302\pi$
$619$	2075.59	−	3595.03i	0.134774	−	0.233435i	0.925511	+	0.378720i	$0.123636\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2279.93	+	3948.96i	−0.146619	+	0.253951i
$624$		0			0
$625$	9724.50	+	16843.3i	0.622368	+	1.07797i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−19947.1			−1.26446
$630$		0			0
$631$	25954.4			1.63745			0.818724	−	0.574187i	$-0.194682\pi$
$631$	25954.4			1.63745			0.818724	+	0.574187i	$0.194682\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	13291.4	+	23021.3i	0.830632	+	1.43870i
$636$		0			0
$637$	−5254.41	+	9100.91i	−0.326825	+	0.566077i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8630.66	−	14948.7i	0.531811	−	0.921123i	−0.467500	−	0.883993i	$-0.654845\pi$
$641$	8630.66	−	14948.7i	0.531811	−	0.921123i	0.999310	−	0.0371297i	$-0.0118215\pi$
$642$		0			0
$643$	−1430.18	−	2477.15i	−0.0877152	−	0.151927i	0.818830	−	0.574036i	$-0.194623\pi$
$643$	−1430.18	−	2477.15i	−0.0877152	−	0.151927i	−0.906545	+	0.422109i	$0.861290\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−28384.9			−1.72477			−0.862384	−	0.506255i	$-0.831029\pi$
$647$	−28384.9			−1.72477			−0.862384	+	0.506255i	$0.831029\pi$
$648$		0			0
$649$	−9450.04			−0.571566
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5518.22	+	9557.84i	0.330696	+	0.572783i	0.982649	−	0.185477i	$-0.0593831\pi$
$653$	5518.22	+	9557.84i	0.330696	+	0.572783i	−0.651952	+	0.758260i	$0.726050\pi$
$654$		0			0
$655$	−1044.68	+	1809.43i	−0.0623190	+	0.107940i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	10647.5	−	18441.9i	0.629387	−	1.09013i	−0.358288	−	0.933611i	$-0.616639\pi$
$659$	10647.5	−	18441.9i	0.629387	−	1.09013i	0.987675	−	0.156519i	$-0.0500273\pi$
$660$		0			0
$661$	−6162.37	−	10673.5i	−0.362615	−	0.628067i	0.625776	−	0.780003i	$-0.284783\pi$
$661$	−6162.37	−	10673.5i	−0.362615	−	0.628067i	−0.988390	+	0.151936i	$0.951449\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	7631.11			0.444995
$666$		0			0
$667$	−8344.58			−0.484413
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−9719.98	−	16835.5i	−0.559219	−	0.968595i
$672$		0			0
$673$	−1525.47	+	2642.19i	−0.0873738	+	0.151336i	−0.906400	−	0.422420i	$-0.861181\pi$
$673$	−1525.47	+	2642.19i	−0.0873738	+	0.151336i	0.819027	+	0.573756i	$0.194514\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	842.733	−	1459.66i	0.0478418	−	0.0828643i	−0.841113	−	0.540860i	$-0.818099\pi$
$677$	842.733	−	1459.66i	0.0478418	−	0.0828643i	0.888955	+	0.457995i	$0.151432\pi$
$678$		0			0
$679$	−10665.4	−	18473.0i	−0.602797	−	1.04408i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	34715.5			1.94488			0.972440	−	0.233155i	$-0.0749050\pi$
$683$	34715.5			1.94488			0.972440	+	0.233155i	$0.0749050\pi$
$684$		0			0
$685$	−28309.4			−1.57905
$686$		0			0
$687$		0			0
$688$		0			0
$689$	12649.1	+	21908.9i	0.699408	+	1.21141i
$690$		0			0
$691$	4147.68	−	7184.00i	0.228343	−	0.395502i	−0.728974	−	0.684542i	$-0.760002\pi$
$691$	4147.68	−	7184.00i	0.228343	−	0.395502i	0.957317	+	0.289039i	$0.0933358\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	17944.7	−	31081.1i	0.979398	−	1.69637i
$696$		0			0
$697$	548.714	+	950.400i	0.0298192	+	0.0516484i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−23420.5			−1.26188			−0.630942	−	0.775830i	$-0.717331\pi$
$701$	−23420.5			−1.26188			−0.630942	+	0.775830i	$0.717331\pi$
$702$		0			0
$703$	−12712.8			−0.682037
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7202.35	−	12474.8i	−0.383129	−	0.663599i
$708$		0			0
$709$	2641.33	−	4574.92i	0.139912	−	0.242334i	−0.787551	−	0.616249i	$-0.788651\pi$
$709$	2641.33	−	4574.92i	0.139912	−	0.242334i	0.927463	+	0.373915i	$0.121985\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−587.104	+	1016.89i	−0.0308376	+	0.0534123i
$714$		0			0
$715$	20039.5	+	34709.4i	1.04816	+	1.81547i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−714.975			−0.0370849			−0.0185425	−	0.999828i	$-0.505903\pi$
$719$	−714.975			−0.0370849			−0.0185425	+	0.999828i	$0.505903\pi$
$720$		0			0
$721$	24660.9			1.27382
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1734.81	−	3004.77i	−0.0888677	−	0.153923i
$726$		0			0
$727$	−12897.7	+	22339.5i	−0.657979	+	1.13965i	0.323159	+	0.946345i	$0.395255\pi$
$727$	−12897.7	+	22339.5i	−0.657979	+	1.13965i	−0.981138	+	0.193308i	$0.938078\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−14076.5	+	24381.3i	−0.712230	+	1.23362i
$732$		0			0
$733$	−1815.78	−	3145.02i	−0.0914969	−	0.158477i	0.816644	−	0.577141i	$-0.195832\pi$
$733$	−1815.78	−	3145.02i	−0.0914969	−	0.158477i	−0.908141	+	0.418664i	$0.862498\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−22587.0			−1.12890
$738$		0			0
$739$	−24758.3			−1.23241			−0.616203	−	0.787588i	$-0.711330\pi$
$739$	−24758.3			−1.23241			−0.616203	+	0.787588i	$0.711330\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−4422.66	−	7660.28i	−0.218374	−	0.378235i	0.735937	−	0.677050i	$-0.236742\pi$
$743$	−4422.66	−	7660.28i	−0.218374	−	0.378235i	−0.954311	+	0.298815i	$0.903409\pi$
$744$		0			0
$745$	10362.7	−	17948.8i	0.509612	−	0.882675i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−7223.96	+	12512.3i	−0.352413	+	0.610398i
$750$		0			0
$751$	−830.199	−	1437.95i	−0.0403387	−	0.0698688i	0.845151	−	0.534527i	$-0.179510\pi$
$751$	−830.199	−	1437.95i	−0.0403387	−	0.0698688i	−0.885490	+	0.464659i	$0.846177\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−241.899			−0.0116604
$756$		0			0
$757$	19937.2			0.957239			0.478619	−	0.878022i	$-0.341137\pi$
$757$	19937.2			0.957239			0.478619	+	0.878022i	$0.341137\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4221.87	−	7312.50i	−0.201107	−	0.348328i	0.747778	−	0.663949i	$-0.231121\pi$
$761$	−4221.87	−	7312.50i	−0.201107	−	0.348328i	−0.948886	+	0.315620i	$0.897787\pi$
$762$		0			0
$763$	12383.2	−	21448.3i	0.587550	−	1.01767i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9285.84	+	16083.5i	−0.437148	+	0.757162i
$768$		0			0
$769$	−8553.93	−	14815.8i	−0.401122	−	0.694763i	0.592740	−	0.805394i	$-0.298046\pi$
$769$	−8553.93	−	14815.8i	−0.401122	−	0.694763i	−0.993862	+	0.110631i	$0.964713\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−9821.05			−0.456971			−0.228486	−	0.973547i	$-0.573377\pi$
$773$	−9821.05			−0.456971			−0.228486	+	0.973547i	$0.573377\pi$
$774$		0			0
$775$	−488.226			−0.0226292
$776$		0			0
$777$		0			0
$778$		0			0
$779$	349.710	+	605.715i	0.0160843	+	0.0278588i
$780$		0			0
$781$	12912.3	−	22364.8i	0.591600	−	1.02468i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−24989.2	+	43282.6i	−1.13618	+	1.96792i
$786$		0			0
$787$	2622.78	+	4542.80i	0.118796	+	0.205760i	0.919291	−	0.393579i	$-0.128763\pi$
$787$	2622.78	+	4542.80i	0.118796	+	0.205760i	−0.800495	+	0.599339i	$0.795430\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	9002.62			0.404673
$792$		0			0
$793$	−38204.4			−1.71082
$794$		0			0
$795$		0			0
$796$		0			0
$797$	4609.26	+	7983.47i	0.204854	+	0.354817i	0.950086	−	0.311988i	$-0.100995\pi$
$797$	4609.26	+	7983.47i	0.204854	+	0.354817i	−0.745232	+	0.666805i	$0.767661\pi$
$798$		0			0
$799$	−18154.9	+	31445.2i	−0.803848	+	1.39231i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−13608.5	+	23570.6i	−0.598049	+	1.03585i
$804$		0			0
$805$	−12322.9	−	21343.9i	−0.539534	−	0.934501i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	16397.4			0.712611			0.356305	−	0.934370i	$-0.384036\pi$
$809$	16397.4			0.712611			0.356305	+	0.934370i	$0.384036\pi$
$810$		0			0
$811$	8468.88			0.366686			0.183343	−	0.983049i	$-0.441308\pi$
$811$	8468.88			0.366686			0.183343	+	0.983049i	$0.441308\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−17424.6	−	30180.3i	−0.748906	−	1.29714i
$816$		0			0
$817$	−8971.35	+	15538.8i	−0.384171	+	0.665404i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2616.73	−	4532.31i	0.111236	−	0.192666i	−0.805033	−	0.593230i	$-0.797852\pi$
$821$	2616.73	−	4532.31i	0.111236	−	0.192666i	0.916269	+	0.400564i	$0.131186\pi$
$822$		0			0
$823$	1562.34	+	2706.05i	0.0661722	+	0.114614i	0.897213	−	0.441597i	$-0.145588\pi$
$823$	1562.34	+	2706.05i	0.0661722	+	0.114614i	−0.831041	+	0.556211i	$0.812255\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	11037.6			0.464103			0.232052	−	0.972703i	$-0.425456\pi$
$827$	11037.6			0.464103			0.232052	+	0.972703i	$0.425456\pi$
$828$		0			0
$829$	−23941.9			−1.00306			−0.501529	−	0.865141i	$-0.667229\pi$
$829$	−23941.9			−1.00306			−0.501529	+	0.865141i	$0.667229\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4271.96	−	7399.25i	−0.177688	−	0.307765i
$834$		0			0
$835$	10438.4	−	18079.9i	0.432618	−	0.749317i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−10629.7	+	18411.2i	−0.437399	+	0.757597i	−0.997488	−	0.0708352i	$-0.977434\pi$
$839$	−10629.7	+	18411.2i	−0.437399	+	0.757597i	0.560089	+	0.828432i	$0.310767\pi$
$840$		0			0
$841$	10086.2	+	17469.7i	0.413554	+	0.716296i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	49418.1			2.01187
$846$		0			0
$847$	−2810.32			−0.114007
$848$		0			0
$849$		0			0
$850$		0			0
$851$	20528.9	+	35557.2i	0.826936	+	1.43230i
$852$		0			0
$853$	−4414.37	+	7645.91i	−0.177192	+	0.306906i	−0.940918	−	0.338635i	$-0.890035\pi$
$853$	−4414.37	+	7645.91i	−0.177192	+	0.306906i	0.763725	+	0.645541i	$0.223368\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	19007.4	−	32921.8i	0.757621	−	1.31224i	−0.186440	−	0.982466i	$-0.559695\pi$
$857$	19007.4	−	32921.8i	0.757621	−	1.31224i	0.944061	−	0.329771i	$-0.106972\pi$
$858$		0			0
$859$	−2458.81	−	4258.78i	−0.0976640	−	0.169159i	0.813053	−	0.582189i	$-0.197804\pi$
$859$	−2458.81	−	4258.78i	−0.0976640	−	0.169159i	−0.910717	+	0.413030i	$0.864470\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−20474.1			−0.807585			−0.403792	−	0.914851i	$-0.632308\pi$
$863$	−20474.1			−0.807585			−0.403792	+	0.914851i	$0.632308\pi$
$864$		0			0
$865$	−20073.7			−0.789049
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−14267.0	−	24711.2i	−0.556934	−	0.964639i
$870$		0			0
$871$	−22194.5	+	38442.0i	−0.863412	+	1.49547i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−6863.05	+	11887.2i	−0.265158	+	0.459268i
$876$		0			0
$877$	−7140.36	−	12367.5i	−0.274929	−	0.476191i	0.695188	−	0.718828i	$-0.255321\pi$
$877$	−7140.36	−	12367.5i	−0.274929	−	0.476191i	−0.970117	+	0.242637i	$0.921988\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	12505.2			0.478217			0.239109	−	0.970993i	$-0.423145\pi$
$881$	12505.2			0.478217			0.239109	+	0.970993i	$0.423145\pi$
$882$		0			0
$883$	−34255.9			−1.30555			−0.652776	−	0.757551i	$-0.726396\pi$
$883$	−34255.9			−1.30555			−0.652776	+	0.757551i	$0.726396\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12186.2	+	21107.2i	0.461300	+	0.798995i	0.999026	−	0.0441244i	$-0.0140498\pi$
$887$	12186.2	+	21107.2i	0.461300	+	0.798995i	−0.537726	+	0.843120i	$0.680716\pi$
$888$		0			0
$889$	−14286.4	+	24744.7i	−0.538976	+	0.933534i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−11570.6	+	20040.9i	−0.433590	+	0.750999i
$894$		0			0
$895$	−27560.5	−	47736.2i	−1.02933	−	1.78284i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−593.350			−0.0220126
$900$		0			0
$901$	−20568.0			−0.760510
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10568.2	−	18304.7i	−0.388177	−	0.672343i
$906$		0			0
$907$	21418.0	−	37097.1i	0.784095	−	1.35809i	−0.145443	−	0.989367i	$-0.546461\pi$
$907$	21418.0	−	37097.1i	0.784095	−	1.35809i	0.929538	−	0.368726i	$-0.120206\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	18072.6	−	31302.7i	0.657271	−	1.13843i	−0.324049	−	0.946040i	$-0.605044\pi$
$911$	18072.6	−	31302.7i	0.657271	−	1.13843i	0.981319	−	0.192386i	$-0.0616225\pi$
$912$		0			0
$913$	−21304.4	−	36900.3i	−0.772258	−	1.33759i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2245.77			−0.0808744
$918$		0			0
$919$	−23283.2			−0.835738			−0.417869	−	0.908507i	$-0.637223\pi$
$919$	−23283.2			−0.835738			−0.417869	+	0.908507i	$0.637223\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−25376.0	−	43952.5i	−0.904941	−	1.56740i
$924$		0			0
$925$	−8535.77	+	14784.4i	−0.303410	+	0.525522i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	19595.2	−	33939.9i	0.692033	−	1.19864i	−0.279138	−	0.960251i	$-0.590049\pi$
$929$	19595.2	−	33939.9i	0.692033	−	1.19864i	0.971171	−	0.238385i	$-0.0766180\pi$
$930$		0			0
$931$	−2722.63	−	4715.73i	−0.0958438	−	0.166006i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−32585.1			−1.13973
$936$		0			0
$937$	−36892.8			−1.28627			−0.643135	−	0.765753i	$-0.722366\pi$
$937$	−36892.8			−1.28627			−0.643135	+	0.765753i	$0.722366\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13302.9	−	23041.3i	−0.460853	−	0.798220i	0.538151	−	0.842848i	$-0.319123\pi$
$941$	−13302.9	−	23041.3i	−0.460853	−	0.798220i	−0.999004	+	0.0446282i	$0.985790\pi$
$942$		0			0
$943$	1129.44	−	1956.25i	0.0390027	−	0.0675547i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−6420.67	+	11120.9i	−0.220321	+	0.381607i	−0.954905	−	0.296910i	$-0.904044\pi$
$947$	−6420.67	+	11120.9i	−0.220321	+	0.381607i	0.734585	+	0.678517i	$0.237377\pi$
$948$		0			0
$949$	26744.1	+	46322.1i	0.914804	+	1.58449i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	33977.5			1.15492			0.577460	−	0.816419i	$-0.304044\pi$
$953$	33977.5			1.15492			0.577460	+	0.816419i	$0.304044\pi$
$954$		0			0
$955$	34408.1			1.16589
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−15214.4	−	26352.0i	−0.512301	−	0.887332i
$960$		0			0
$961$	14853.8	−	25727.5i	0.498599	−	0.863598i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	9319.07	−	16141.1i	0.310872	−	0.538446i
$966$		0			0
$967$	−24671.4	−	42732.2i	−0.820455	−	1.42107i	−0.905344	−	0.424679i	$-0.860387\pi$
$967$	−24671.4	−	42732.2i	−0.820455	−	1.42107i	0.0848894	−	0.996390i	$-0.472946\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	43375.7			1.43356			0.716782	−	0.697297i	$-0.245614\pi$
$971$	43375.7			1.43356			0.716782	+	0.697297i	$0.245614\pi$
$972$		0			0
$973$	38576.2			1.27101
$974$		0			0
$975$		0			0
$976$		0			0
$977$	19847.3	+	34376.5i	0.649919	+	1.12569i	0.983142	+	0.182845i	$0.0585305\pi$
$977$	19847.3	+	34376.5i	0.649919	+	1.12569i	−0.333223	+	0.942848i	$0.608136\pi$
$978$		0			0
$979$	6204.62	−	10746.7i	0.202554	−	0.350834i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−4529.45	+	7845.23i	−0.146965	+	0.254551i	−0.930104	−	0.367295i	$-0.880284\pi$
$983$	−4529.45	+	7845.23i	−0.146965	+	0.254551i	0.783139	+	0.621847i	$0.213617\pi$
$984$		0			0
$985$	−33712.6	−	58391.9i	−1.09053	−	1.88885i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	57948.6			1.86315
$990$		0			0
$991$	−33006.3			−1.05800			−0.529001	−	0.848621i	$-0.677433\pi$
$991$	−33006.3			−1.05800			−0.529001	+	0.848621i	$0.677433\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	16306.3	+	28243.4i	0.519544	+	0.899876i
$996$		0			0
$997$	−22162.2	+	38386.1i	−0.703996	+	1.21936i	0.263056	+	0.964781i	$0.415270\pi$
$997$	−22162.2	+	38386.1i	−0.703996	+	1.21936i	−0.967053	+	0.254577i	$0.918064\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.i.p.217.1		4
3.2	odd	2		648.4.i.q.217.2		4
9.2	odd	6		648.4.a.d.1.1	✓	2
9.4	even	3	inner	648.4.i.p.433.1		4
9.5	odd	6		648.4.i.q.433.2		4
9.7	even	3		648.4.a.e.1.2	yes	2
36.7	odd	6		1296.4.a.p.1.2		2
36.11	even	6		1296.4.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.4.a.d.1.1	✓	2	9.2	odd	6
648.4.a.e.1.2	yes	2	9.7	even	3
648.4.i.p.217.1		4	1.1	even	1	trivial
648.4.i.p.433.1		4	9.4	even	3	inner
648.4.i.q.217.2		4	3.2	odd	2
648.4.i.q.433.2		4	9.5	odd	6
1296.4.a.n.1.1		2	36.11	even	6
1296.4.a.p.1.2		2	36.7	odd	6

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-6.67891 - 11.5682i) q^{5} +(7.17891 - 12.4342i) q^{7} +O(q^{10})\)	\(q+(-6.67891 - 11.5682i) q^{5} +(7.17891 - 12.4342i) q^{7} +(-19.5367 + 33.8386i) q^{11} +(38.3945 + 66.5013i) q^{13} -62.4313 q^{17} -39.7891 q^{19} +(64.2524 + 111.288i) q^{23} +(-26.7156 + 46.2728i) q^{25} +(-32.4680 + 56.2362i) q^{29} +(4.56873 + 7.91328i) q^{31} -191.789 q^{35} +319.505 q^{37} +(-8.78908 - 15.2231i) q^{41} +(225.473 - 390.530i) q^{43} +(290.799 - 503.678i) q^{47} +(68.4265 + 118.518i) q^{49} +329.450 q^{53} +521.936 q^{55} +(120.927 + 209.451i) q^{59} +(-248.762 + 430.868i) q^{61} +(512.867 - 888.312i) q^{65} +(289.032 + 500.618i) q^{67} -660.927 q^{71} +696.559 q^{73} +(280.505 + 485.848i) q^{77} +(-365.134 + 632.430i) q^{79} +(-545.239 + 944.382i) q^{83} +(416.973 + 722.218i) q^{85} -317.588 q^{89} +1102.52 q^{91} +(265.748 + 460.288i) q^{95} +(742.827 - 1286.61i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} + 6 q^{7}+O(q^{10}) \) 4 * q - 4 * q^5 + 6 * q^7	\( 4 q - 4 q^{5} + 6 q^{7} - 10 q^{11} + 40 q^{13} - 68 q^{17} + 68 q^{19} + 98 q^{23} - 16 q^{25} + 120 q^{29} + 200 q^{31} - 540 q^{35} + 960 q^{37} + 192 q^{41} + 334 q^{43} + 300 q^{47} + 410 q^{49} - 136 q^{53} + 1588 q^{55} + 620 q^{59} - 200 q^{61} + 1370 q^{65} + 1406 q^{67} - 2780 q^{71} + 3604 q^{73} + 804 q^{77} + 334 q^{79} - 500 q^{83} + 1100 q^{85} - 180 q^{89} + 2820 q^{91} + 1222 q^{95} + 200 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 + 6 * q^7 - 10 * q^11 + 40 * q^13 - 68 * q^17 + 68 * q^19 + 98 * q^23 - 16 * q^25 + 120 * q^29 + 200 * q^31 - 540 * q^35 + 960 * q^37 + 192 * q^41 + 334 * q^43 + 300 * q^47 + 410 * q^49 - 136 * q^53 + 1588 * q^55 + 620 * q^59 - 200 * q^61 + 1370 * q^65 + 1406 * q^67 - 2780 * q^71 + 3604 * q^73 + 804 * q^77 + 334 * q^79 - 500 * q^83 + 1100 * q^85 - 180 * q^89 + 2820 * q^91 + 1222 * q^95 + 200 * q^97

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