LMFDB - Embedded newform 2023.4.a.u.1.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2023,4,Mod(1,2023)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2023, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2023.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 2023 = 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2023.a (trivial)

Newform invariants

comment:select newform

sage:traces = [56,8,-24,240,-80,-68] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$119.360863942$
Analytic rank:	$1$
Dimension:	$56$
Twist minimal:	no (minimal twist has level 119)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	2023.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.64206 q^{2} +6.07912 q^{3} +5.26462 q^{4} +0.735620 q^{5} -22.1405 q^{6} -7.00000 q^{7} +9.96243 q^{8} +9.95570 q^{9} -2.67917 q^{10} +32.2063 q^{11} +32.0042 q^{12} +62.7330 q^{13} +25.4944 q^{14} +4.47192 q^{15} -78.4007 q^{16} -36.2593 q^{18} -33.0331 q^{19} +3.87276 q^{20} -42.5538 q^{21} -117.297 q^{22} +185.724 q^{23} +60.5628 q^{24} -124.459 q^{25} -228.478 q^{26} -103.614 q^{27} -36.8523 q^{28} -188.550 q^{29} -16.2870 q^{30} -69.3292 q^{31} +205.841 q^{32} +195.786 q^{33} -5.14934 q^{35} +52.4129 q^{36} -416.317 q^{37} +120.309 q^{38} +381.362 q^{39} +7.32856 q^{40} -308.248 q^{41} +154.984 q^{42} -376.954 q^{43} +169.554 q^{44} +7.32361 q^{45} -676.417 q^{46} +193.539 q^{47} -476.607 q^{48} +49.0000 q^{49} +453.287 q^{50} +330.265 q^{52} -234.320 q^{53} +377.370 q^{54} +23.6916 q^{55} -69.7370 q^{56} -200.812 q^{57} +686.711 q^{58} +162.432 q^{59} +23.5430 q^{60} -264.554 q^{61} +252.501 q^{62} -69.6899 q^{63} -122.480 q^{64} +46.1477 q^{65} -713.064 q^{66} -273.786 q^{67} +1129.04 q^{69} +18.7542 q^{70} +21.0261 q^{71} +99.1829 q^{72} +1105.25 q^{73} +1516.25 q^{74} -756.600 q^{75} -173.907 q^{76} -225.444 q^{77} -1388.94 q^{78} +434.602 q^{79} -57.6732 q^{80} -898.688 q^{81} +1122.66 q^{82} +598.442 q^{83} -224.030 q^{84} +1372.89 q^{86} -1146.22 q^{87} +320.853 q^{88} +895.951 q^{89} -26.6730 q^{90} -439.131 q^{91} +977.764 q^{92} -421.461 q^{93} -704.880 q^{94} -24.2998 q^{95} +1251.33 q^{96} -324.024 q^{97} -178.461 q^{98} +320.636 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 8 q^{2} - 24 q^{3} + 240 q^{4} - 80 q^{5} - 68 q^{6} - 392 q^{7} + 96 q^{8} + 576 q^{9} - 80 q^{10} - 176 q^{11} - 288 q^{12} - 96 q^{13} - 56 q^{14} + 192 q^{15} + 1088 q^{16} + 216 q^{18} + 48 q^{19}+ \cdots - 6576 q^{99}+O(q^{100}) $ 56 * q + 8 * q^2 - 24 * q^3 + 240 * q^4 - 80 * q^5 - 68 * q^6 - 392 * q^7 + 96 * q^8 + 576 * q^9 - 80 * q^10 - 176 * q^11 - 288 * q^12 - 96 * q^13 - 56 * q^14 + 192 * q^15 + 1088 * q^16 + 216 * q^18 + 48 * q^19 - 1660 * q^20 + 168 * q^21 - 352 * q^22 - 608 * q^23 - 736 * q^24 + 1792 * q^25 - 864 * q^27 - 1680 * q^28 - 928 * q^29 - 100 * q^30 - 496 * q^31 + 436 * q^32 + 384 * q^33 + 560 * q^35 + 2804 * q^36 - 1392 * q^37 - 624 * q^39 + 1020 * q^40 - 2088 * q^41 + 476 * q^42 + 1312 * q^43 - 2112 * q^44 - 4840 * q^45 - 736 * q^46 - 5228 * q^48 + 2744 * q^49 + 2020 * q^50 - 1536 * q^52 + 1272 * q^53 - 2268 * q^54 - 336 * q^55 - 672 * q^56 - 2448 * q^57 - 1376 * q^58 + 2372 * q^60 - 976 * q^61 - 3776 * q^62 - 4032 * q^63 + 5376 * q^64 - 6080 * q^65 + 976 * q^67 + 1248 * q^69 + 560 * q^70 - 3216 * q^71 + 4532 * q^72 - 4088 * q^73 + 2456 * q^74 + 760 * q^75 - 2640 * q^76 + 1232 * q^77 - 3744 * q^78 - 1792 * q^79 - 13920 * q^80 + 4184 * q^81 - 8204 * q^82 - 2832 * q^83 + 2016 * q^84 + 4908 * q^86 + 832 * q^87 - 2424 * q^88 - 2768 * q^89 - 8660 * q^90 + 672 * q^91 - 6784 * q^92 + 784 * q^93 - 928 * q^94 - 4560 * q^95 - 6656 * q^96 - 9464 * q^97 + 392 * q^98 - 6576 * q^99

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 $

$2$	−3.64206	−1.28766	−0.643832	−	0.765167i	$-0.722656\pi$
$2$	−3.64206	−1.28766	−0.643832	+	0.765167i	$0.722656\pi$
$3$	6.07912	1.16993	0.584964	−	0.811060i	$-0.301109\pi$
$3$	6.07912	1.16993	0.584964	+	0.811060i	$0.301109\pi$
$4$	5.26462	0.658077
$5$	0.735620	0.0657959	0.0328979	−	0.999459i	$-0.489526\pi$
$5$	0.735620	0.0657959	0.0328979	+	0.999459i	$0.489526\pi$
$6$	−22.1405	−1.50647
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	9.96243	0.440281
$9$	9.95570	0.368729
$10$	−2.67917	−0.0847229
$11$	32.2063	0.882778	0.441389	−	0.897316i	$-0.354486\pi$
$11$	32.2063	0.882778	0.441389	+	0.897316i	$0.354486\pi$
$12$	32.0042	0.769902
$13$	62.7330	1.33839	0.669193	−	0.743089i	$-0.266640\pi$
$13$	62.7330	1.33839	0.669193	+	0.743089i	$0.266640\pi$
$14$	25.4944	0.486691
$15$	4.47192	0.0769764
$16$	−78.4007	−1.22501
$17$	0	0
$17$	0	0
$18$	−36.2593	−0.474799
$19$	−33.0331	−0.398859	−0.199429	−	0.979912i	$-0.563909\pi$
$19$	−33.0331	−0.398859	−0.199429	+	0.979912i	$0.563909\pi$
$20$	3.87276	0.0432988
$21$	−42.5538	−0.442191
$22$	−117.297	−1.13672
$23$	185.724	1.68374	0.841871	−	0.539679i	$-0.181454\pi$
$23$	185.724	1.68374	0.841871	+	0.539679i	$0.181454\pi$
$24$	60.5628	0.515097
$25$	−124.459	−0.995671
$26$	−228.478	−1.72339
$27$	−103.614	−0.738541
$28$	−36.8523	−0.248730
$29$	−188.550	−1.20734	−0.603670	−	0.797234i	$-0.706296\pi$
$29$	−188.550	−1.20734	−0.603670	+	0.797234i	$0.706296\pi$
$30$	−16.2870	−0.0991196
$31$	−69.3292	−0.401674	−0.200837	−	0.979625i	$-0.564366\pi$
$31$	−69.3292	−0.401674	−0.200837	+	0.979625i	$0.564366\pi$
$32$	205.841	1.13712
$33$	195.786	1.03279
$34$	0	0
$35$	−5.14934	−0.0248685
$36$	52.4129	0.242652
$37$	−416.317	−1.84979	−0.924893	−	0.380228i	$-0.875845\pi$
$37$	−416.317	−1.84979	−0.924893	+	0.380228i	$0.875845\pi$
$38$	120.309	0.513596
$39$	381.362	1.56581
$40$	7.32856	0.0289687
$41$	−308.248	−1.17415	−0.587076	−	0.809532i	$-0.699721\pi$
$41$	−308.248	−1.17415	−0.587076	+	0.809532i	$0.699721\pi$
$42$	154.984	0.569393
$43$	−376.954	−1.33686	−0.668430	−	0.743775i	$-0.733033\pi$
$43$	−376.954	−1.33686	−0.668430	+	0.743775i	$0.733033\pi$
$44$	169.554	0.580936
$45$	7.32361	0.0242609
$46$	−676.417	−2.16809
$47$	193.539	0.600649	0.300325	−	0.953837i	$-0.402905\pi$
$47$	193.539	0.600649	0.300325	+	0.953837i	$0.402905\pi$
$48$	−476.607	−1.43317
$49$	49.0000	0.142857
$50$	453.287	1.28209
$51$	0	0
$52$	330.265	0.880761
$53$	−234.320	−0.607290	−0.303645	−	0.952785i	$-0.598204\pi$
$53$	−234.320	−0.607290	−0.303645	+	0.952785i	$0.598204\pi$
$54$	377.370	0.950992
$55$	23.6916	0.0580831
$56$	−69.7370	−0.166411
$57$	−200.812	−0.466635
$58$	686.711	1.55465
$59$	162.432	0.358421	0.179211	−	0.983811i	$-0.442646\pi$
$59$	162.432	0.358421	0.179211	+	0.983811i	$0.442646\pi$
$60$	23.5430	0.0506564
$61$	−264.554	−0.555289	−0.277644	−	0.960684i	$-0.589554\pi$
$61$	−264.554	−0.555289	−0.277644	+	0.960684i	$0.589554\pi$
$62$	252.501	0.517221
$63$	−69.6899	−0.139367
$64$	−122.480	−0.239218
$65$	46.1477	0.0880602
$66$	−713.064	−1.32988
$67$	−273.786	−0.499228	−0.249614	−	0.968345i	$-0.580304\pi$
$67$	−273.786	−0.499228	−0.249614	+	0.968345i	$0.580304\pi$
$68$	0	0
$69$	1129.04	1.96986
$70$	18.7542	0.0320223
$71$	21.0261	0.0351456	0.0175728	−	0.999846i	$-0.494406\pi$
$71$	21.0261	0.0351456	0.0175728	+	0.999846i	$0.494406\pi$
$72$	99.1829	0.162345
$73$	1105.25	1.77206	0.886028	−	0.463632i	$-0.153454\pi$
$73$	1105.25	1.77206	0.886028	+	0.463632i	$0.153454\pi$
$74$	1516.25	2.38190
$75$	−756.600	−1.16486
$76$	−173.907	−0.262480
$77$	−225.444	−0.333659
$78$	−1388.94	−2.01624
$79$	434.602	0.618944	0.309472	−	0.950909i	$-0.399848\pi$
$79$	434.602	0.618944	0.309472	+	0.950909i	$0.399848\pi$
$80$	−57.6732	−0.0806007
$81$	−898.688	−1.23277
$82$	1122.66	1.51191
$83$	598.442	0.791416	0.395708	−	0.918376i	$-0.370499\pi$
$83$	598.442	0.791416	0.395708	+	0.918376i	$0.370499\pi$
$84$	−224.030	−0.290996
$85$	0	0
$86$	1372.89	1.72142
$87$	−1146.22	−1.41250
$88$	320.853	0.388671
$89$	895.951	1.06709	0.533543	−	0.845773i	$-0.320860\pi$
$89$	895.951	1.06709	0.533543	+	0.845773i	$0.320860\pi$
$90$	−26.6730	−0.0312398
$91$	−439.131	−0.505862
$92$	977.764	1.10803
$93$	−421.461	−0.469929
$94$	−704.880	−0.773434
$95$	−24.2998	−0.0262432
$96$	1251.33	1.33035
$97$	−324.024	−0.339172	−0.169586	−	0.985515i	$-0.554243\pi$
$97$	−324.024	−0.339172	−0.169586	+	0.985515i	$0.554243\pi$
$98$	−178.461	−0.183952
$99$	320.636	0.325506
$100$	−655.228	−0.655228
$101$	−172.147	−0.169597	−0.0847983	−	0.996398i	$-0.527025\pi$
$101$	−172.147	−0.169597	−0.0847983	+	0.996398i	$0.527025\pi$
$102$	0	0
$103$	−816.148	−0.780752	−0.390376	−	0.920655i	$-0.627655\pi$
$103$	−816.148	−0.780752	−0.390376	+	0.920655i	$0.627655\pi$
$104$	624.974	0.589266
$105$	−31.3035	−0.0290943
$106$	853.409	0.781985
$107$	−1220.20	−1.10244	−0.551220	−	0.834360i	$-0.685837\pi$
$107$	−1220.20	−1.10244	−0.551220	+	0.834360i	$0.685837\pi$
$108$	−545.490	−0.486017
$109$	137.554	0.120875	0.0604373	−	0.998172i	$-0.480750\pi$
$109$	137.554	0.120875	0.0604373	+	0.998172i	$0.480750\pi$
$110$	−86.2862	−0.0747915
$111$	−2530.84	−2.16411
$112$	548.805	0.463011
$113$	332.678	0.276954	0.138477	−	0.990366i	$-0.455779\pi$
$113$	332.678	0.276954	0.138477	+	0.990366i	$0.455779\pi$
$114$	731.370	0.600869
$115$	136.622	0.110783
$116$	−992.645	−0.794524
$117$	624.551	0.493502
$118$	−591.588	−0.461526
$119$	0	0
$120$	44.5512	0.0338913
$121$	−293.757	−0.220704
$122$	963.521	0.715025
$123$	−1873.88	−1.37367
$124$	−364.992	−0.264333
$125$	−183.507	−0.131307
$126$	253.815	0.179457
$127$	−1990.76	−1.39096	−0.695479	−	0.718546i	$-0.744808\pi$
$127$	−1990.76	−1.39096	−0.695479	+	0.718546i	$0.744808\pi$
$128$	−1200.65	−0.829089
$129$	−2291.55	−1.56403
$130$	−168.073	−0.113392
$131$	−136.261	−0.0908792	−0.0454396	−	0.998967i	$-0.514469\pi$
$131$	−136.261	−0.0908792	−0.0454396	+	0.998967i	$0.514469\pi$
$132$	1030.74	0.679653
$133$	231.232	0.150754
$134$	997.145	0.642837
$135$	−76.2208	−0.0485929
$136$	0	0
$137$	2431.06	1.51606	0.758028	−	0.652222i	$-0.226163\pi$
$137$	2431.06	1.51606	0.758028	+	0.652222i	$0.226163\pi$
$138$	−4112.02	−2.53651
$139$	−1468.74	−0.896235	−0.448118	−	0.893975i	$-0.647905\pi$
$139$	−1468.74	−0.896235	−0.448118	+	0.893975i	$0.647905\pi$
$140$	−27.1093	−0.0163654
$141$	1176.55	0.702716
$142$	−76.5782	−0.0452557
$143$	2020.40	1.18150
$144$	−780.534	−0.451698
$145$	−138.701	−0.0794380
$146$	−4025.40	−2.28181
$147$	297.877	0.167132
$148$	−2191.75	−1.21730
$149$	−2354.30	−1.29444	−0.647222	−	0.762302i	$-0.724069\pi$
$149$	−2354.30	−1.29444	−0.647222	+	0.762302i	$0.724069\pi$
$150$	2755.59	1.49995
$151$	−2958.11	−1.59422	−0.797110	−	0.603834i	$-0.793639\pi$
$151$	−2958.11	−1.59422	−0.797110	+	0.603834i	$0.793639\pi$
$152$	−329.090	−0.175610
$153$	0	0
$154$	821.081	0.429640
$155$	−51.0000	−0.0264285
$156$	2007.72	1.03043
$157$	1622.53	0.824791	0.412396	−	0.911005i	$-0.364692\pi$
$157$	1622.53	0.824791	0.412396	+	0.911005i	$0.364692\pi$
$158$	−1582.85	−0.796992
$159$	−1424.46	−0.710485
$160$	151.421	0.0748179
$161$	−1300.07	−0.636395
$162$	3273.08	1.58739
$163$	473.822	0.227685	0.113842	−	0.993499i	$-0.463684\pi$
$163$	473.822	0.227685	0.113842	+	0.993499i	$0.463684\pi$
$164$	−1622.81	−0.772683
$165$	144.024	0.0679530
$166$	−2179.56	−1.01908
$167$	3947.69	1.82923	0.914615	−	0.404327i	$-0.132494\pi$
$167$	3947.69	1.82923	0.914615	+	0.404327i	$0.132494\pi$
$168$	−423.940	−0.194688
$169$	1738.43	0.791276
$170$	0	0
$171$	−328.867	−0.147071
$172$	−1984.52	−0.879757
$173$	3644.85	1.60181	0.800904	−	0.598793i	$-0.204353\pi$
$173$	3644.85	1.60181	0.800904	+	0.598793i	$0.204353\pi$
$174$	4174.60	1.81883
$175$	871.212	0.376328
$176$	−2524.99	−1.08141
$177$	987.444	0.419327
$178$	−3263.11	−1.37405
$179$	−2600.97	−1.08606	−0.543032	−	0.839712i	$-0.682724\pi$
$179$	−2600.97	−1.08606	−0.543032	+	0.839712i	$0.682724\pi$
$180$	38.5560	0.0159655
$181$	−1409.92	−0.578996	−0.289498	−	0.957179i	$-0.593488\pi$
$181$	−1409.92	−0.578996	−0.289498	+	0.957179i	$0.593488\pi$
$182$	1599.34	0.651380
$183$	−1608.25	−0.649648
$184$	1850.26	0.741320
$185$	−306.251	−0.121708
$186$	1534.99	0.605111
$187$	0	0
$188$	1018.91	0.395274
$189$	725.301	0.279142
$190$	88.5014	0.0337925
$191$	−4734.73	−1.79368	−0.896841	−	0.442352i	$-0.854144\pi$
$191$	−4734.73	−1.79368	−0.896841	+	0.442352i	$0.854144\pi$
$192$	−744.568	−0.279868
$193$	1126.36	0.420089	0.210044	−	0.977692i	$-0.432639\pi$
$193$	1126.36	0.420089	0.210044	+	0.977692i	$0.432639\pi$
$194$	1180.12	0.436740
$195$	280.537	0.103024
$196$	257.966	0.0940110
$197$	−2589.54	−0.936532	−0.468266	−	0.883587i	$-0.655121\pi$
$197$	−2589.54	−0.936532	−0.468266	+	0.883587i	$0.655121\pi$
$198$	−1167.78	−0.419142
$199$	418.193	0.148969	0.0744846	−	0.997222i	$-0.476269\pi$
$199$	418.193	0.148969	0.0744846	+	0.997222i	$0.476269\pi$
$200$	−1239.91	−0.438375
$201$	−1664.38	−0.584060
$202$	626.970	0.218383
$203$	1319.85	0.456332
$204$	0	0
$205$	−226.753	−0.0772543
$206$	2972.46	1.00535
$207$	1849.01	0.620845
$208$	−4918.32	−1.63954
$209$	−1063.87	−0.352103
$210$	114.009	0.0374637
$211$	2437.68	0.795340	0.397670	−	0.917528i	$-0.369819\pi$
$211$	2437.68	0.795340	0.397670	+	0.917528i	$0.369819\pi$
$212$	−1233.61	−0.399644
$213$	127.820	0.0411177
$214$	4444.04	1.41957
$215$	−277.295	−0.0879598
$216$	−1032.25	−0.325166
$217$	485.305	0.151819
$218$	−500.982	−0.155646
$219$	6718.97	2.07318
$220$	124.727	0.0382232
$221$	0	0
$222$	9217.48	2.78665
$223$	−3027.09	−0.909008	−0.454504	−	0.890745i	$-0.650183\pi$
$223$	−3027.09	−0.909008	−0.454504	+	0.890745i	$0.650183\pi$
$224$	−1440.89	−0.429791
$225$	−1239.07	−0.367133
$226$	−1211.64	−0.356623
$227$	−3437.57	−1.00511	−0.502554	−	0.864546i	$-0.667606\pi$
$227$	−3437.57	−1.00511	−0.502554	+	0.864546i	$0.667606\pi$
$228$	−1057.20	−0.307082
$229$	2002.54	0.577868	0.288934	−	0.957349i	$-0.406699\pi$
$229$	2002.54	0.577868	0.288934	+	0.957349i	$0.406699\pi$
$230$	−497.586	−0.142652
$231$	−1370.50	−0.390356
$232$	−1878.42	−0.531570
$233$	−225.210	−0.0633218	−0.0316609	−	0.999499i	$-0.510080\pi$
$233$	−225.210	−0.0633218	−0.0316609	+	0.999499i	$0.510080\pi$
$234$	−2274.65	−0.635465
$235$	142.371	0.0395202
$236$	855.143	0.235869
$237$	2642.00	0.724120
$238$	0	0
$239$	−2740.22	−0.741631	−0.370816	−	0.928707i	$-0.620922\pi$
$239$	−2740.22	−0.741631	−0.370816	+	0.928707i	$0.620922\pi$
$240$	−350.602	−0.0942969
$241$	−6477.47	−1.73133	−0.865665	−	0.500623i	$-0.833104\pi$
$241$	−6477.47	−1.73133	−0.865665	+	0.500623i	$0.833104\pi$
$242$	1069.88	0.284192
$243$	−2665.64	−0.703708
$244$	−1392.77	−0.365423
$245$	36.0454	0.00939941
$246$	6824.77	1.76883
$247$	−2072.27	−0.533827
$248$	−690.688	−0.176850
$249$	3638.00	0.925899
$250$	668.344	0.169079
$251$	−659.198	−0.165770	−0.0828849	−	0.996559i	$-0.526413\pi$
$251$	−659.198	−0.165770	−0.0828849	+	0.996559i	$0.526413\pi$
$252$	−366.891	−0.0917140
$253$	5981.47	1.48637
$254$	7250.49	1.79109
$255$	0	0
$256$	5352.68	1.30681
$257$	4277.39	1.03820	0.519098	−	0.854715i	$-0.326268\pi$
$257$	4277.39	1.03820	0.519098	+	0.854715i	$0.326268\pi$
$258$	8345.96	2.01394
$259$	2914.22	0.699153
$260$	242.950	0.0579504
$261$	−1877.15	−0.445182
$262$	496.271	0.117022
$263$	2658.57	0.623324	0.311662	−	0.950193i	$-0.399114\pi$
$263$	2658.57	0.623324	0.311662	+	0.950193i	$0.399114\pi$
$264$	1950.50	0.454716
$265$	−172.371	−0.0399572
$266$	−842.160	−0.194121
$267$	5446.59	1.24841
$268$	−1441.38	−0.328530
$269$	−7118.60	−1.61349	−0.806744	−	0.590901i	$-0.798773\pi$
$269$	−7118.60	−1.61349	−0.806744	+	0.590901i	$0.798773\pi$
$270$	277.601	0.0625713
$271$	8364.15	1.87486	0.937428	−	0.348179i	$-0.113200\pi$
$271$	8364.15	1.87486	0.937428	+	0.348179i	$0.113200\pi$
$272$	0	0
$273$	−2669.53	−0.591822
$274$	−8854.08	−1.95217
$275$	−4008.35	−0.878956
$276$	5943.95	1.29632
$277$	7457.60	1.61763	0.808815	−	0.588063i	$-0.200109\pi$
$277$	7457.60	1.61763	0.808815	+	0.588063i	$0.200109\pi$
$278$	5349.23	1.15405
$279$	−690.221	−0.148109
$280$	−51.2999	−0.0109491
$281$	−1554.39	−0.329989	−0.164995	−	0.986294i	$-0.552761\pi$
$281$	−1554.39	−0.329989	−0.164995	+	0.986294i	$0.552761\pi$
$282$	−4285.05	−0.904862
$283$	1866.67	0.392092	0.196046	−	0.980595i	$-0.437190\pi$
$283$	1866.67	0.392092	0.196046	+	0.980595i	$0.437190\pi$
$284$	110.694	0.0231285
$285$	−147.721	−0.0307027
$286$	−7358.41	−1.52137
$287$	2157.74	0.443788
$288$	2049.29	0.419290
$289$	0	0
$290$	505.159	0.102289
$291$	−1969.78	−0.396807
$292$	5818.74	1.16615
$293$	2675.22	0.533406	0.266703	−	0.963779i	$-0.414066\pi$
$293$	2675.22	0.533406	0.266703	+	0.963779i	$0.414066\pi$
$294$	−1084.89	−0.215210
$295$	119.488	0.0235826
$296$	−4147.53	−0.814426
$297$	−3337.03	−0.651967
$298$	8574.52	1.66681
$299$	11651.0	2.25350
$300$	−3983.21	−0.766569
$301$	2638.68	0.505285
$302$	10773.6	2.05282
$303$	−1046.50	−0.198416
$304$	2589.82	0.488606
$305$	−194.611	−0.0365357
$306$	0	0
$307$	−3232.31	−0.600904	−0.300452	−	0.953797i	$-0.597137\pi$
$307$	−3232.31	−0.600904	−0.300452	+	0.953797i	$0.597137\pi$
$308$	−1186.88	−0.219573
$309$	−4961.46	−0.913423
$310$	185.745	0.0340310
$311$	−8395.94	−1.53084	−0.765418	−	0.643534i	$-0.777468\pi$
$311$	−8395.94	−1.53084	−0.765418	+	0.643534i	$0.777468\pi$
$312$	3799.29	0.689399
$313$	5792.15	1.04598	0.522990	−	0.852339i	$-0.324817\pi$
$313$	5792.15	1.04598	0.522990	+	0.852339i	$0.324817\pi$
$314$	−5909.37	−1.06205
$315$	−51.2653	−0.00916975
$316$	2288.02	0.407313
$317$	−8772.96	−1.55438	−0.777190	−	0.629266i	$-0.783356\pi$
$317$	−8772.96	−1.55438	−0.777190	+	0.629266i	$0.783356\pi$
$318$	5187.98	0.914865
$319$	−6072.50	−1.06581
$320$	−90.0984	−0.0157395
$321$	−7417.73	−1.28977
$322$	4734.92	0.819462
$323$	0	0
$324$	−4731.25	−0.811257
$325$	−7807.68	−1.33259
$326$	−1725.69	−0.293181
$327$	836.210	0.141414
$328$	−3070.90	−0.516957
$329$	−1354.77	−0.227024
$330$	−524.544	−0.0875006
$331$	1897.29	0.315059	0.157529	−	0.987514i	$-0.449647\pi$
$331$	1897.29	0.315059	0.157529	+	0.987514i	$0.449647\pi$
$332$	3150.57	0.520813
$333$	−4144.72	−0.682070
$334$	−14377.7	−2.35543
$335$	−201.402	−0.0328471
$336$	3336.25	0.541689
$337$	−2295.11	−0.370988	−0.185494	−	0.982645i	$-0.559388\pi$
$337$	−2295.11	−0.370988	−0.185494	+	0.982645i	$0.559388\pi$
$338$	−6331.48	−1.01890
$339$	2022.39	0.324015
$340$	0	0
$341$	−2232.84	−0.354589
$342$	1197.76	0.189378
$343$	−343.000	−0.0539949
$344$	−3755.38	−0.588594
$345$	830.542	0.129608
$346$	−13274.8	−2.06259
$347$	1088.46	0.168391	0.0841954	−	0.996449i	$-0.473168\pi$
$347$	1088.46	0.168391	0.0841954	+	0.996449i	$0.473168\pi$
$348$	−6034.41	−0.929535
$349$	−8916.91	−1.36765	−0.683827	−	0.729644i	$-0.739686\pi$
$349$	−8916.91	−1.36765	−0.683827	+	0.729644i	$0.739686\pi$
$350$	−3173.01	−0.484584
$351$	−6500.04	−0.988452
$352$	6629.37	1.00383
$353$	−352.870	−0.0532051	−0.0266025	−	0.999646i	$-0.508469\pi$
$353$	−352.870	−0.0532051	−0.0266025	+	0.999646i	$0.508469\pi$
$354$	−3596.33	−0.539952
$355$	15.4672	0.00231243
$356$	4716.84	0.702224
$357$	0	0
$358$	9472.89	1.39849
$359$	2905.65	0.427171	0.213585	−	0.976924i	$-0.431486\pi$
$359$	2905.65	0.427171	0.213585	+	0.976924i	$0.431486\pi$
$360$	72.9610	0.0106816
$361$	−5767.81	−0.840912
$362$	5135.00	0.745552
$363$	−1785.78	−0.258207
$364$	−2311.86	−0.332896
$365$	813.046	0.116594
$366$	5857.36	0.836527
$367$	−12008.6	−1.70802	−0.854012	−	0.520253i	$-0.825837\pi$
$367$	−12008.6	−1.70802	−0.854012	+	0.520253i	$0.825837\pi$
$368$	−14560.9	−2.06260
$369$	−3068.82	−0.432945
$370$	1115.39	0.156719
$371$	1640.24	0.229534
$372$	−2218.83	−0.309250
$373$	5972.30	0.829046	0.414523	−	0.910039i	$-0.363948\pi$
$373$	5972.30	0.829046	0.414523	+	0.910039i	$0.363948\pi$
$374$	0	0
$375$	−1115.56	−0.153619
$376$	1928.12	0.264455
$377$	−11828.3	−1.61589
$378$	−2641.59	−0.359441
$379$	−11731.1	−1.58993	−0.794966	−	0.606655i	$-0.792511\pi$
$379$	−11731.1	−1.58993	−0.794966	+	0.606655i	$0.792511\pi$
$380$	−127.929	−0.0172701
$381$	−12102.1	−1.62732
$382$	17244.2	2.30966
$383$	2568.97	0.342737	0.171369	−	0.985207i	$-0.445181\pi$
$383$	2568.97	0.342737	0.171369	+	0.985207i	$0.445181\pi$
$384$	−7298.89	−0.969974
$385$	−165.841	−0.0219534
$386$	−4102.27	−0.540933
$387$	−3752.84	−0.492939
$388$	−1705.86	−0.223201
$389$	3199.38	0.417006	0.208503	−	0.978022i	$-0.433141\pi$
$389$	3199.38	0.417006	0.208503	+	0.978022i	$0.433141\pi$
$390$	−1021.73	−0.132660
$391$	0	0
$392$	488.159	0.0628973
$393$	−828.346	−0.106322
$394$	9431.26	1.20594
$395$	319.702	0.0407240
$396$	1688.02	0.214208
$397$	9940.93	1.25673	0.628364	−	0.777920i	$-0.283725\pi$
$397$	9940.93	1.25673	0.628364	+	0.777920i	$0.283725\pi$
$398$	−1523.08	−0.191822
$399$	1405.69	0.176372
$400$	9757.67	1.21971
$401$	14580.6	1.81576	0.907880	−	0.419230i	$-0.137700\pi$
$401$	14580.6	1.81576	0.907880	+	0.419230i	$0.137700\pi$
$402$	6061.76	0.752073
$403$	−4349.23	−0.537595
$404$	−906.288	−0.111608
$405$	−661.093	−0.0811110
$406$	−4806.98	−0.587602
$407$	−13408.0	−1.63295
$408$	0	0
$409$	−9924.06	−1.19979	−0.599894	−	0.800080i	$-0.704791\pi$
$409$	−9924.06	−1.19979	−0.599894	+	0.800080i	$0.704791\pi$
$410$	825.850	0.0994776
$411$	14778.7	1.77368
$412$	−4296.71	−0.513795
$413$	−1137.02	−0.135470
$414$	−6734.20	−0.799440
$415$	440.226	0.0520719
$416$	12913.0	1.52191
$417$	−8928.63	−1.04853
$418$	3874.69	0.453391
$419$	−646.685	−0.0754000	−0.0377000	−	0.999289i	$-0.512003\pi$
$419$	−646.685	−0.0754000	−0.0377000	+	0.999289i	$0.512003\pi$
$420$	−164.801	−0.0191463
$421$	3364.64	0.389508	0.194754	−	0.980852i	$-0.437609\pi$
$421$	3364.64	0.389508	0.194754	+	0.980852i	$0.437609\pi$
$422$	−8878.18	−1.02413
$423$	1926.81	0.221477
$424$	−2334.40	−0.267378
$425$	0	0
$426$	−465.528	−0.0529458
$427$	1851.88	0.209879
$428$	−6423.88	−0.725490
$429$	12282.2	1.38227
$430$	1009.93	0.113263
$431$	−2829.51	−0.316224	−0.158112	−	0.987421i	$-0.550541\pi$
$431$	−2829.51	−0.316224	−0.158112	+	0.987421i	$0.550541\pi$
$432$	8123.44	0.904721
$433$	−11912.4	−1.32211	−0.661053	−	0.750340i	$-0.729890\pi$
$433$	−11912.4	−1.32211	−0.661053	+	0.750340i	$0.729890\pi$
$434$	−1767.51	−0.195491
$435$	−843.182	−0.0929367
$436$	724.171	0.0795448
$437$	−6135.03	−0.671575
$438$	−24470.9	−2.66955
$439$	−10454.9	−1.13664	−0.568320	−	0.822807i	$-0.692407\pi$
$439$	−10454.9	−1.13664	−0.568320	+	0.822807i	$0.692407\pi$
$440$	236.026	0.0255729
$441$	487.829	0.0526756
$442$	0	0
$443$	3285.55	0.352373	0.176186	−	0.984357i	$-0.443624\pi$
$443$	3285.55	0.352373	0.176186	+	0.984357i	$0.443624\pi$
$444$	−13323.9	−1.42415
$445$	659.079	0.0702098
$446$	11024.8	1.17050
$447$	−14312.1	−1.51440
$448$	857.357	0.0904159
$449$	−5781.14	−0.607637	−0.303818	−	0.952730i	$-0.598262\pi$
$449$	−5781.14	−0.607637	−0.303818	+	0.952730i	$0.598262\pi$
$450$	4512.79	0.472744
$451$	−9927.51	−1.03652
$452$	1751.42	0.182257
$453$	−17982.7	−1.86512
$454$	12519.8	1.29424
$455$	−323.034	−0.0332836
$456$	−2000.58	−0.205451
$457$	−11445.4	−1.17154	−0.585770	−	0.810478i	$-0.699208\pi$
$457$	−11445.4	−1.17154	−0.585770	+	0.810478i	$0.699208\pi$
$458$	−7293.39	−0.744099
$459$	0	0
$460$	719.263	0.0729039
$461$	−1438.84	−0.145365	−0.0726827	−	0.997355i	$-0.523156\pi$
$461$	−1438.84	−0.145365	−0.0726827	+	0.997355i	$0.523156\pi$
$462$	4991.45	0.502647
$463$	9137.07	0.917139	0.458570	−	0.888658i	$-0.348362\pi$
$463$	9137.07	0.917139	0.458570	+	0.888658i	$0.348362\pi$
$464$	14782.5	1.47901
$465$	−310.035	−0.0309194
$466$	820.228	0.0815372
$467$	3003.70	0.297633	0.148817	−	0.988865i	$-0.452454\pi$
$467$	3003.70	0.297633	0.148817	+	0.988865i	$0.452454\pi$
$468$	3288.02	0.324763
$469$	1916.50	0.188690
$470$	−518.524	−0.0508888
$471$	9863.57	0.964946
$472$	1618.22	0.157806
$473$	−12140.3	−1.18015
$474$	−9622.33	−0.932422
$475$	4111.26	0.397132
$476$	0	0
$477$	−2332.82	−0.223926
$478$	9980.04	0.954971
$479$	4784.77	0.456412	0.228206	−	0.973613i	$-0.426714\pi$
$479$	4784.77	0.456412	0.228206	+	0.973613i	$0.426714\pi$
$480$	920.505	0.0875314
$481$	−26116.8	−2.47573
$482$	23591.4	2.22937
$483$	−7903.25	−0.744535
$484$	−1546.52	−0.145240
$485$	−238.359	−0.0223161
$486$	9708.44	0.906139
$487$	867.346	0.0807048	0.0403524	−	0.999186i	$-0.487152\pi$
$487$	867.346	0.0807048	0.0403524	+	0.999186i	$0.487152\pi$
$488$	−2635.60	−0.244483
$489$	2880.42	0.266374
$490$	−131.280	−0.0121033
$491$	5793.56	0.532504	0.266252	−	0.963903i	$-0.414215\pi$
$491$	5793.56	0.532504	0.266252	+	0.963903i	$0.414215\pi$
$492$	−9865.24	−0.903983
$493$	0	0
$494$	7547.32	0.687389
$495$	235.866	0.0214170
$496$	5435.46	0.492055
$497$	−147.182	−0.0132838
$498$	−13249.8	−1.19225
$499$	−8742.03	−0.784262	−0.392131	−	0.919909i	$-0.628262\pi$
$499$	−8742.03	−0.784262	−0.392131	+	0.919909i	$0.628262\pi$
$500$	−966.094	−0.0864101
$501$	23998.5	2.14006
$502$	2400.84	0.213456
$503$	−15513.4	−1.37516	−0.687581	−	0.726108i	$-0.741327\pi$
$503$	−15513.4	−1.37516	−0.687581	+	0.726108i	$0.741327\pi$
$504$	−694.281	−0.0613605
$505$	−126.635	−0.0111588
$506$	−21784.9	−1.91394
$507$	10568.1	0.925735
$508$	−10480.6	−0.915358
$509$	−11863.0	−1.03304	−0.516522	−	0.856274i	$-0.672774\pi$
$509$	−11863.0	−1.03304	−0.516522	+	0.856274i	$0.672774\pi$
$510$	0	0
$511$	−7736.77	−0.669774
$512$	−9889.59	−0.853637
$513$	3422.70	0.294573
$514$	−15578.5	−1.33685
$515$	−600.375	−0.0513703
$516$	−12064.1	−1.02925
$517$	6233.16	0.530240
$518$	−10613.8	−0.900274
$519$	22157.5	1.87400
$520$	459.743	0.0387713
$521$	−10291.0	−0.865371	−0.432685	−	0.901545i	$-0.642434\pi$
$521$	−10291.0	−0.865371	−0.432685	+	0.901545i	$0.642434\pi$
$522$	6836.69	0.573245
$523$	13001.8	1.08705	0.543525	−	0.839393i	$-0.317089\pi$
$523$	13001.8	1.08705	0.543525	+	0.839393i	$0.317089\pi$
$524$	−717.362	−0.0598055
$525$	5296.20	0.440277
$526$	−9682.67	−0.802632
$527$	0	0
$528$	−15349.7	−1.26517
$529$	22326.3	1.83499
$530$	627.785	0.0514514
$531$	1617.12	0.132160
$532$	1217.35	0.0992080
$533$	−19337.3	−1.57147
$534$	−19836.8	−1.60753
$535$	−897.602	−0.0725359
$536$	−2727.57	−0.219801
$537$	−15811.6	−1.27062
$538$	25926.4	2.07763
$539$	1578.11	0.126111
$540$	−401.273	−0.0319779
$541$	−794.987	−0.0631777	−0.0315889	−	0.999501i	$-0.510057\pi$
$541$	−794.987	−0.0631777	−0.0315889	+	0.999501i	$0.510057\pi$
$542$	−30462.8	−2.41418
$543$	−8571.05	−0.677383
$544$	0	0
$545$	101.188	0.00795304
$546$	9722.60	0.762067
$547$	−17667.8	−1.38102	−0.690511	−	0.723322i	$-0.742614\pi$
$547$	−17667.8	−1.38102	−0.690511	+	0.723322i	$0.742614\pi$
$548$	12798.6	0.997682
$549$	−2633.82	−0.204751
$550$	14598.7	1.13180
$551$	6228.40	0.481558
$552$	11247.9	0.867291
$553$	−3042.22	−0.233939
$554$	−27161.0	−2.08296
$555$	−1861.74	−0.142390
$556$	−7732.34	−0.589792
$557$	−13613.8	−1.03561	−0.517804	−	0.855499i	$-0.673250\pi$
$557$	−13613.8	−1.03561	−0.517804	+	0.855499i	$0.673250\pi$
$558$	2513.83	0.190715
$559$	−23647.5	−1.78923
$560$	403.712	0.0304642
$561$	0	0
$562$	5661.18	0.424915
$563$	−11486.5	−0.859858	−0.429929	−	0.902863i	$-0.641461\pi$
$563$	−11486.5	−0.859858	−0.429929	+	0.902863i	$0.641461\pi$
$564$	6194.06	0.462442
$565$	244.725	0.0182224
$566$	−6798.52	−0.504882
$567$	6290.82	0.465943
$568$	209.471	0.0154739
$569$	−13577.4	−1.00034	−0.500171	−	0.865926i	$-0.666730\pi$
$569$	−13577.4	−1.00034	−0.500171	+	0.865926i	$0.666730\pi$
$570$	538.011	0.0395347
$571$	−3590.99	−0.263184	−0.131592	−	0.991304i	$-0.542009\pi$
$571$	−3590.99	−0.263184	−0.131592	+	0.991304i	$0.542009\pi$
$572$	10636.6	0.777516
$573$	−28783.0	−2.09848
$574$	−7858.61	−0.571449
$575$	−23115.0	−1.67645
$576$	−1219.37	−0.0882067
$577$	12933.1	0.933125	0.466562	−	0.884488i	$-0.345492\pi$
$577$	12933.1	0.933125	0.466562	+	0.884488i	$0.345492\pi$
$578$	0	0
$579$	6847.27	0.491473
$580$	−730.209	−0.0522764
$581$	−4189.09	−0.299127
$582$	7174.07	0.510953
$583$	−7546.58	−0.536102
$584$	11011.0	0.780203
$585$	459.432	0.0324704
$586$	−9743.32	−0.686848
$587$	10241.5	0.720124	0.360062	−	0.932928i	$-0.382756\pi$
$587$	10241.5	0.720124	0.360062	+	0.932928i	$0.382756\pi$
$588$	1568.21	0.109986
$589$	2290.16	0.160211
$590$	−435.184	−0.0303665
$591$	−15742.1	−1.09567
$592$	32639.5	2.26601
$593$	21175.2	1.46638	0.733190	−	0.680024i	$-0.238031\pi$
$593$	21175.2	1.46638	0.733190	+	0.680024i	$0.238031\pi$
$594$	12153.7	0.839514
$595$	0	0
$596$	−12394.5	−0.851844
$597$	2542.24	0.174283
$598$	−42433.7	−2.90174
$599$	−14663.1	−1.00020	−0.500099	−	0.865968i	$-0.666703\pi$
$599$	−14663.1	−1.00020	−0.500099	+	0.865968i	$0.666703\pi$
$600$	−7537.58	−0.512867
$601$	3880.89	0.263402	0.131701	−	0.991289i	$-0.457956\pi$
$601$	3880.89	0.263402	0.131701	+	0.991289i	$0.457956\pi$
$602$	−9610.23	−0.650637
$603$	−2725.73	−0.184080
$604$	−15573.3	−1.04912
$605$	−216.093	−0.0145214
$606$	3811.43	0.255493
$607$	570.656	0.0381585	0.0190793	−	0.999818i	$-0.493927\pi$
$607$	570.656	0.0381585	0.0190793	+	0.999818i	$0.493927\pi$
$608$	−6799.56	−0.453551
$609$	8023.53	0.533875
$610$	708.785	0.0470457
$611$	12141.3	0.803901
$612$	0	0
$613$	4023.93	0.265130	0.132565	−	0.991174i	$-0.457679\pi$
$613$	4023.93	0.265130	0.132565	+	0.991174i	$0.457679\pi$
$614$	11772.3	0.773762
$615$	−1378.46	−0.0903820
$616$	−2245.97	−0.146904
$617$	1888.25	0.123206	0.0616030	−	0.998101i	$-0.480379\pi$
$617$	1888.25	0.123206	0.0616030	+	0.998101i	$0.480379\pi$
$618$	18070.0	1.17618
$619$	27267.7	1.77057	0.885284	−	0.465051i	$-0.153964\pi$
$619$	27267.7	1.77057	0.885284	+	0.465051i	$0.153964\pi$
$620$	−268.495	−0.0173920
$621$	−19243.6	−1.24351
$622$	30578.5	1.97120
$623$	−6271.65	−0.403320
$624$	−29899.0	−1.91814
$625$	15422.4	0.987031
$626$	−21095.4	−1.34687
$627$	−6467.41	−0.411935
$628$	8542.02	0.542776
$629$	0	0
$630$	186.711	0.0118075
$631$	−13639.8	−0.860523	−0.430262	−	0.902704i	$-0.641579\pi$
$631$	−13639.8	−0.860523	−0.430262	+	0.902704i	$0.641579\pi$
$632$	4329.70	0.272510
$633$	14818.9	0.930490
$634$	31951.7	2.00152
$635$	−1464.45	−0.0915193
$636$	−7499.24	−0.467554
$637$	3073.92	0.191198
$638$	22116.4	1.37241
$639$	209.329	0.0129592
$640$	−883.222	−0.0545506
$641$	−8582.79	−0.528861	−0.264430	−	0.964405i	$-0.585184\pi$
$641$	−8582.79	−0.528861	−0.264430	+	0.964405i	$0.585184\pi$
$642$	27015.8	1.66079
$643$	12537.9	0.768966	0.384483	−	0.923132i	$-0.374380\pi$
$643$	12537.9	0.768966	0.384483	+	0.923132i	$0.374380\pi$
$644$	−6844.35	−0.418797
$645$	−1685.71	−0.102907
$646$	0	0
$647$	−5026.97	−0.305457	−0.152728	−	0.988268i	$-0.548806\pi$
$647$	−5026.97	−0.305457	−0.152728	+	0.988268i	$0.548806\pi$
$648$	−8953.12	−0.542765
$649$	5231.33	0.316406
$650$	28436.1	1.71593
$651$	2950.22	0.177617
$652$	2494.49	0.149834
$653$	3193.86	0.191402	0.0957010	−	0.995410i	$-0.469491\pi$
$653$	3193.86	0.191402	0.0957010	+	0.995410i	$0.469491\pi$
$654$	−3045.53	−0.182094
$655$	−100.236	−0.00597947
$656$	24166.9	1.43835
$657$	11003.6	0.653409
$658$	4934.16	0.292331
$659$	27591.3	1.63096	0.815481	−	0.578784i	$-0.196473\pi$
$659$	27591.3	1.63096	0.815481	+	0.578784i	$0.196473\pi$
$660$	758.231	0.0447183
$661$	28203.5	1.65959	0.829794	−	0.558070i	$-0.188458\pi$
$661$	28203.5	1.65959	0.829794	+	0.558070i	$0.188458\pi$
$662$	−6910.04	−0.405689
$663$	0	0
$664$	5961.94	0.348446
$665$	170.099	0.00991901
$666$	15095.3	0.878277
$667$	−35018.2	−2.03285
$668$	20783.1	1.20377
$669$	−18402.0	−1.06347
$670$	733.520	0.0422960
$671$	−8520.28	−0.490197
$672$	−8759.32	−0.502825
$673$	−1286.62	−0.0736934	−0.0368467	−	0.999321i	$-0.511731\pi$
$673$	−1286.62	−0.0736934	−0.0368467	+	0.999321i	$0.511731\pi$
$674$	8358.95	0.477707
$675$	12895.7	0.735343
$676$	9152.19	0.520721
$677$	−5662.54	−0.321461	−0.160730	−	0.986998i	$-0.551385\pi$
$677$	−5662.54	−0.321461	−0.160730	+	0.986998i	$0.551385\pi$
$678$	−7365.68	−0.417223
$679$	2268.17	0.128195
$680$	0	0
$681$	−20897.4	−1.17590
$682$	8132.13	0.456591
$683$	−21299.4	−1.19326	−0.596631	−	0.802516i	$-0.703494\pi$
$683$	−21299.4	−1.19326	−0.596631	+	0.802516i	$0.703494\pi$
$684$	−1731.36	−0.0967840
$685$	1788.34	0.0997502
$686$	1249.23	0.0695273
$687$	12173.7	0.676063
$688$	29553.5	1.63767
$689$	−14699.6	−0.812788
$690$	−3024.88	−0.166892
$691$	27147.7	1.49457	0.747286	−	0.664503i	$-0.231357\pi$
$691$	27147.7	1.49457	0.747286	+	0.664503i	$0.231357\pi$
$692$	19188.7	1.05411
$693$	−2244.45	−0.123030
$694$	−3964.24	−0.216831
$695$	−1080.43	−0.0589686
$696$	−11419.1	−0.621898
$697$	0	0
$698$	32475.9	1.76108
$699$	−1369.08	−0.0740819
$700$	4586.60	0.247653
$701$	−21492.8	−1.15802	−0.579010	−	0.815320i	$-0.696561\pi$
$701$	−21492.8	−1.15802	−0.579010	+	0.815320i	$0.696561\pi$
$702$	23673.6	1.27279
$703$	13752.2	0.737803
$704$	−3944.61	−0.211176
$705$	865.490	0.0462358
$706$	1285.18	0.0685103
$707$	1205.03	0.0641015
$708$	5198.51	0.275949
$709$	13610.8	0.720966	0.360483	−	0.932766i	$-0.382612\pi$
$709$	13610.8	0.720966	0.360483	+	0.932766i	$0.382612\pi$
$710$	−56.3325	−0.00297763
$711$	4326.77	0.228223
$712$	8925.85	0.469818
$713$	−12876.1	−0.676315
$714$	0	0
$715$	1486.24	0.0777376
$716$	−13693.1	−0.714714
$717$	−16658.1	−0.867654
$718$	−10582.6	−0.550052
$719$	−15101.1	−0.783277	−0.391638	−	0.920119i	$-0.628092\pi$
$719$	−15101.1	−0.783277	−0.391638	+	0.920119i	$0.628092\pi$
$720$	−574.176	−0.0297198
$721$	5713.04	0.295097
$722$	21006.7	1.08281
$723$	−39377.3	−2.02553
$724$	−7422.67	−0.381024
$725$	23466.7	1.20211
$726$	6503.93	0.332484
$727$	2541.87	0.129674	0.0648369	−	0.997896i	$-0.479347\pi$
$727$	2541.87	0.129674	0.0648369	+	0.997896i	$0.479347\pi$
$728$	−4374.81	−0.222722
$729$	8059.81	0.409481
$730$	−2961.17	−0.150134
$731$	0	0
$732$	−8466.84	−0.427518
$733$	−6613.18	−0.333238	−0.166619	−	0.986021i	$-0.553285\pi$
$733$	−6613.18	−0.333238	−0.166619	+	0.986021i	$0.553285\pi$
$734$	43736.1	2.19936
$735$	219.124	0.0109966
$736$	38229.5	1.91462
$737$	−8817.62	−0.440707
$738$	11176.8	0.557487
$739$	−14151.4	−0.704422	−0.352211	−	0.935921i	$-0.614570\pi$
$739$	−14151.4	−0.704422	−0.352211	+	0.935921i	$0.614570\pi$
$740$	−1612.29	−0.0800934
$741$	−12597.6	−0.624538
$742$	−5973.86	−0.295563
$743$	−31300.8	−1.54551	−0.772755	−	0.634704i	$-0.781122\pi$
$743$	−31300.8	−1.54551	−0.772755	+	0.634704i	$0.781122\pi$
$744$	−4198.77	−0.206901
$745$	−1731.87	−0.0851690
$746$	−21751.5	−1.06753
$747$	5957.91	0.291819
$748$	0	0
$749$	8541.39	0.416683
$750$	4062.94	0.197810
$751$	−21798.7	−1.05918	−0.529592	−	0.848253i	$-0.677655\pi$
$751$	−21798.7	−1.05918	−0.529592	+	0.848253i	$0.677655\pi$
$752$	−15173.6	−0.735803
$753$	−4007.34	−0.193939
$754$	43079.5	2.08072
$755$	−2176.04	−0.104893
$756$	3818.43	0.183697
$757$	31100.5	1.49322	0.746610	−	0.665262i	$-0.231680\pi$
$757$	31100.5	1.49322	0.746610	+	0.665262i	$0.231680\pi$
$758$	42725.2	2.04730
$759$	36362.0	1.73894
$760$	−242.085	−0.0115544
$761$	17649.7	0.840736	0.420368	−	0.907354i	$-0.361901\pi$
$761$	17649.7	0.840736	0.420368	+	0.907354i	$0.361901\pi$
$762$	44076.6	2.09544
$763$	−962.881	−0.0456863
$764$	−24926.6	−1.18038
$765$	0	0
$766$	−9356.36	−0.441330
$767$	10189.9	0.479706
$768$	32539.6	1.52887
$769$	−33607.0	−1.57594	−0.787970	−	0.615714i	$-0.788868\pi$
$769$	−33607.0	−1.57594	−0.787970	+	0.615714i	$0.788868\pi$
$770$	604.003	0.0282685
$771$	26002.8	1.21461
$772$	5929.85	0.276451
$773$	18841.8	0.876707	0.438353	−	0.898803i	$-0.355562\pi$
$773$	18841.8	0.876707	0.438353	+	0.898803i	$0.355562\pi$
$774$	13668.1	0.634740
$775$	8628.64	0.399935
$776$	−3228.07	−0.149331
$777$	17715.9	0.817958
$778$	−11652.4	−0.536963
$779$	10182.4	0.468321
$780$	1476.92	0.0677978
$781$	677.171	0.0310257
$782$	0	0
$783$	19536.5	0.891670
$784$	−3841.64	−0.175002
$785$	1193.57	0.0542678
$786$	3016.89	0.136907
$787$	4863.69	0.220295	0.110147	−	0.993915i	$-0.464868\pi$
$787$	4863.69	0.220295	0.110147	+	0.993915i	$0.464868\pi$
$788$	−13632.9	−0.616311
$789$	16161.7	0.729244
$790$	−1164.38	−0.0524388
$791$	−2328.75	−0.104679
$792$	3194.31	0.143314
$793$	−16596.3	−0.743191
$794$	−36205.5	−1.61824
$795$	−1047.86	−0.0467470
$796$	2201.62	0.0980333
$797$	35226.2	1.56559	0.782795	−	0.622279i	$-0.213793\pi$
$797$	35226.2	1.56559	0.782795	+	0.622279i	$0.213793\pi$
$798$	−5119.59	−0.227107
$799$	0	0
$800$	−25618.7	−1.13220
$801$	8919.81	0.393466
$802$	−53103.4	−2.33809
$803$	35596.1	1.56433
$804$	−8762.31	−0.384357
$805$	−956.354	−0.0418721
$806$	15840.2	0.692241
$807$	−43274.8	−1.88766
$808$	−1715.00	−0.0746703
$809$	−22023.2	−0.957100	−0.478550	−	0.878060i	$-0.658837\pi$
$809$	−22023.2	−0.957100	−0.478550	+	0.878060i	$0.658837\pi$
$810$	2407.74	0.104444
$811$	1882.97	0.0815290	0.0407645	−	0.999169i	$-0.487021\pi$
$811$	1882.97	0.0815290	0.0407645	+	0.999169i	$0.487021\pi$
$812$	6948.51	0.300302
$813$	50846.7	2.19344
$814$	48832.8	2.10269
$815$	348.553	0.0149807
$816$	0	0
$817$	12452.0	0.533218
$818$	36144.0	1.54492
$819$	−4371.86	−0.186526
$820$	−1193.77	−0.0508393
$821$	23579.7	1.00236	0.501180	−	0.865343i	$-0.332899\pi$
$821$	23579.7	1.00236	0.501180	+	0.865343i	$0.332899\pi$
$822$	−53825.0	−2.28390
$823$	−23585.3	−0.998943	−0.499472	−	0.866330i	$-0.666473\pi$
$823$	−23585.3	−0.998943	−0.499472	+	0.866330i	$0.666473\pi$
$824$	−8130.82	−0.343751
$825$	−24367.3	−1.02831
$826$	4141.11	0.174440
$827$	−11879.3	−0.499497	−0.249749	−	0.968311i	$-0.580348\pi$
$827$	−11879.3	−0.499497	−0.249749	+	0.968311i	$0.580348\pi$
$828$	9734.32	0.408564
$829$	−30188.9	−1.26478	−0.632391	−	0.774650i	$-0.717926\pi$
$829$	−30188.9	−1.26478	−0.632391	+	0.774650i	$0.717926\pi$
$830$	−1603.33	−0.0670511
$831$	45335.6	1.89251
$832$	−7683.52	−0.320166
$833$	0	0
$834$	32518.6	1.35015
$835$	2904.00	0.120356
$836$	−5600.88	−0.231711
$837$	7183.50	0.296653
$838$	2355.27	0.0970899
$839$	31302.3	1.28805	0.644025	−	0.765004i	$-0.277263\pi$
$839$	31302.3	1.28805	0.644025	+	0.765004i	$0.277263\pi$
$840$	−311.859	−0.0128097
$841$	11162.2	0.457672
$842$	−12254.2	−0.501555
$843$	−9449.31	−0.386063
$844$	12833.5	0.523395
$845$	1278.83	0.0520627
$846$	−7017.57	−0.285188
$847$	2056.30	0.0834181
$848$	18370.9	0.743937
$849$	11347.7	0.458719
$850$	0	0
$851$	−77319.9	−3.11456
$852$	672.923	0.0270587
$853$	30088.3	1.20774	0.603870	−	0.797083i	$-0.293625\pi$
$853$	30088.3	1.20774	0.603870	+	0.797083i	$0.293625\pi$
$854$	−6744.65	−0.270254
$855$	−241.922	−0.00967666
$856$	−12156.1	−0.485384
$857$	−23044.9	−0.918552	−0.459276	−	0.888294i	$-0.651891\pi$
$857$	−23044.9	−0.918552	−0.459276	+	0.888294i	$0.651891\pi$
$858$	−44732.7	−1.77989
$859$	49288.0	1.95772	0.978862	−	0.204524i	$-0.0655647\pi$
$859$	49288.0	1.95772	0.978862	+	0.204524i	$0.0655647\pi$
$860$	−1459.85	−0.0578843
$861$	13117.1	0.519199
$862$	10305.2	0.407190
$863$	27315.2	1.07743	0.538714	−	0.842488i	$-0.318910\pi$
$863$	27315.2	1.07743	0.538714	+	0.842488i	$0.318910\pi$
$864$	−21328.1	−0.839810
$865$	2681.22	0.105392
$866$	43385.6	1.70243
$867$	0	0
$868$	2554.94	0.0999083
$869$	13996.9	0.546390
$870$	3070.92	0.119671
$871$	−17175.4	−0.668159
$872$	1370.38	0.0532188
$873$	−3225.89	−0.125063
$874$	22344.2	0.864762
$875$	1284.55	0.0496293
$876$	35372.8	1.36431
$877$	−25724.5	−0.990485	−0.495243	−	0.868755i	$-0.664921\pi$
$877$	−25724.5	−0.990485	−0.495243	+	0.868755i	$0.664921\pi$
$878$	38077.4	1.46361
$879$	16263.0	0.624046
$880$	−1857.44	−0.0711525
$881$	−14406.5	−0.550929	−0.275465	−	0.961311i	$-0.588832\pi$
$881$	−14406.5	−0.550929	−0.275465	+	0.961311i	$0.588832\pi$
$882$	−1776.70	−0.0678285
$883$	11914.3	0.454075	0.227037	−	0.973886i	$-0.427096\pi$
$883$	11914.3	0.454075	0.227037	+	0.973886i	$0.427096\pi$
$884$	0	0
$885$	726.383	0.0275900
$886$	−11966.2	−0.453737
$887$	39206.4	1.48413	0.742065	−	0.670328i	$-0.233847\pi$
$887$	39206.4	1.48413	0.742065	+	0.670328i	$0.233847\pi$
$888$	−25213.3	−0.952819
$889$	13935.4	0.525733
$890$	−2400.41	−0.0904066
$891$	−28943.4	−1.08826
$892$	−15936.5	−0.598197
$893$	−6393.18	−0.239574
$894$	52125.6	1.95004
$895$	−1913.32	−0.0714585
$896$	8404.54	0.313366
$897$	70827.9	2.63643
$898$	21055.3	0.782432
$899$	13072.0	0.484958
$900$	−6523.25	−0.241602
$901$	0	0
$902$	36156.6	1.33468
$903$	16040.8	0.591147
$904$	3314.29	0.121937
$905$	−1037.16	−0.0380955
$906$	65494.0	2.40165
$907$	26831.7	0.982283	0.491142	−	0.871080i	$-0.336580\pi$
$907$	26831.7	0.982283	0.491142	+	0.871080i	$0.336580\pi$
$908$	−18097.5	−0.661439
$909$	−1713.84	−0.0625353
$910$	1176.51	0.0428581
$911$	−4405.66	−0.160226	−0.0801131	−	0.996786i	$-0.525528\pi$
$911$	−4405.66	−0.160226	−0.0801131	+	0.996786i	$0.525528\pi$
$912$	15743.8	0.571634
$913$	19273.6	0.698645
$914$	41684.9	1.50855
$915$	−1183.06	−0.0427441
$916$	10542.6	0.380282
$917$	953.826	0.0343491
$918$	0	0
$919$	−6547.68	−0.235025	−0.117513	−	0.993071i	$-0.537492\pi$
$919$	−6547.68	−0.235025	−0.117513	+	0.993071i	$0.537492\pi$
$920$	1361.09	0.0487758
$921$	−19649.6	−0.703013
$922$	5240.35	0.187182
$923$	1319.03	0.0470383
$924$	−7215.16	−0.256885
$925$	51814.3	1.84178
$926$	−33277.8	−1.18097
$927$	−8125.33	−0.287886
$928$	−38811.3	−1.37289
$929$	12420.1	0.438632	0.219316	−	0.975654i	$-0.429617\pi$
$929$	12420.1	0.438632	0.219316	+	0.975654i	$0.429617\pi$
$930$	1129.17	0.0398138
$931$	−1618.62	−0.0569798
$932$	−1185.64	−0.0416707
$933$	−51039.9	−1.79097
$934$	−10939.7	−0.383251
$935$	0	0
$936$	6222.05	0.217280
$937$	18965.1	0.661218	0.330609	−	0.943768i	$-0.392746\pi$
$937$	18965.1	0.661218	0.330609	+	0.943768i	$0.392746\pi$
$938$	−6980.01	−0.242970
$939$	35211.2	1.22372
$940$	749.529	0.0260074
$941$	−37572.2	−1.30162	−0.650808	−	0.759243i	$-0.725570\pi$
$941$	−37572.2	−1.30162	−0.650808	+	0.759243i	$0.725570\pi$
$942$	−35923.7	−1.24253
$943$	−57248.9	−1.97697
$944$	−12734.8	−0.439070
$945$	533.546	0.0183664
$946$	44215.7	1.51964
$947$	−29832.7	−1.02369	−0.511844	−	0.859078i	$-0.671038\pi$
$947$	−29832.7	−1.02369	−0.511844	+	0.859078i	$0.671038\pi$
$948$	13909.1	0.476527
$949$	69335.9	2.37169
$950$	−14973.5	−0.511372
$951$	−53331.9	−1.81851
$952$	0	0
$953$	10086.1	0.342834	0.171417	−	0.985199i	$-0.445165\pi$
$953$	10086.1	0.342834	0.171417	+	0.985199i	$0.445165\pi$
$954$	8496.28	0.288341
$955$	−3482.97	−0.118017
$956$	−14426.2	−0.488051
$957$	−36915.4	−1.24692
$958$	−17426.4	−0.587706
$959$	−17017.4	−0.573015
$960$	−547.719	−0.0184141
$961$	−24984.5	−0.838658
$962$	95119.1	3.18790
$963$	−12147.9	−0.406502
$964$	−34101.4	−1.13935
$965$	828.572	0.0276401
$966$	28784.1	0.958711
$967$	18781.5	0.624583	0.312291	−	0.949986i	$-0.398903\pi$
$967$	18781.5	0.624583	0.312291	+	0.949986i	$0.398903\pi$
$968$	−2926.53	−0.0971717
$969$	0	0
$970$	868.118	0.0287357
$971$	−14106.2	−0.466209	−0.233105	−	0.972452i	$-0.574888\pi$
$971$	−14106.2	−0.466209	−0.233105	+	0.972452i	$0.574888\pi$
$972$	−14033.6	−0.463094
$973$	10281.2	0.338745
$974$	−3158.93	−0.103921
$975$	−47463.8	−1.55904
$976$	20741.2	0.680235
$977$	−49111.2	−1.60819	−0.804097	−	0.594497i	$-0.797351\pi$
$977$	−49111.2	−1.60819	−0.804097	+	0.594497i	$0.797351\pi$
$978$	−10490.7	−0.343001
$979$	28855.2	0.941999
$980$	189.765	0.00618554
$981$	1369.45	0.0445700
$982$	−21100.5	−0.685686
$983$	20614.4	0.668867	0.334433	−	0.942419i	$-0.391455\pi$
$983$	20614.4	0.668867	0.334433	+	0.942419i	$0.391455\pi$
$984$	−18668.4	−0.604802
$985$	−1904.92	−0.0616199
$986$	0	0
$987$	−8235.82	−0.265602
$988$	−10909.7	−0.351299
$989$	−70009.3	−2.25093
$990$	−859.039	−0.0275778
$991$	−5123.38	−0.164228	−0.0821138	−	0.996623i	$-0.526167\pi$
$991$	−5123.38	−0.164228	−0.0821138	+	0.996623i	$0.526167\pi$
$992$	−14270.8	−0.456752
$993$	11533.8	0.368596
$994$	536.048	0.0171050
$995$	307.631	0.00980156
$996$	19152.7	0.609313
$997$	−31441.4	−0.998755	−0.499377	−	0.866385i	$-0.666438\pi$
$997$	−31441.4	−0.998755	−0.499377	+	0.866385i	$0.666438\pi$
$998$	31839.0	1.00987
$999$	43136.4	1.36614

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2023.4.a.u.1.12		56
17.5	odd	16		119.4.k.a.8.23	✓	112
17.7	odd	16		119.4.k.a.15.23	yes	112
17.16	even	2		2023.4.a.v.1.12		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
119.4.k.a.8.23	✓	112	17.5	odd	16
119.4.k.a.15.23	yes	112	17.7	odd	16
2023.4.a.u.1.12		56	1.1	even	1	trivial
2023.4.a.v.1.12		56	17.16	even	2

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:	\( N \)	\(=\)	\( 2023 = 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2023.a (trivial)

\(f(q)\)	\(=\)	\(q-3.64206 q^{2} +6.07912 q^{3} +5.26462 q^{4} +0.735620 q^{5} -22.1405 q^{6} -7.00000 q^{7} +9.96243 q^{8} +9.95570 q^{9} -2.67917 q^{10} +32.2063 q^{11} +32.0042 q^{12} +62.7330 q^{13} +25.4944 q^{14} +4.47192 q^{15} -78.4007 q^{16} -36.2593 q^{18} -33.0331 q^{19} +3.87276 q^{20} -42.5538 q^{21} -117.297 q^{22} +185.724 q^{23} +60.5628 q^{24} -124.459 q^{25} -228.478 q^{26} -103.614 q^{27} -36.8523 q^{28} -188.550 q^{29} -16.2870 q^{30} -69.3292 q^{31} +205.841 q^{32} +195.786 q^{33} -5.14934 q^{35} +52.4129 q^{36} -416.317 q^{37} +120.309 q^{38} +381.362 q^{39} +7.32856 q^{40} -308.248 q^{41} +154.984 q^{42} -376.954 q^{43} +169.554 q^{44} +7.32361 q^{45} -676.417 q^{46} +193.539 q^{47} -476.607 q^{48} +49.0000 q^{49} +453.287 q^{50} +330.265 q^{52} -234.320 q^{53} +377.370 q^{54} +23.6916 q^{55} -69.7370 q^{56} -200.812 q^{57} +686.711 q^{58} +162.432 q^{59} +23.5430 q^{60} -264.554 q^{61} +252.501 q^{62} -69.6899 q^{63} -122.480 q^{64} +46.1477 q^{65} -713.064 q^{66} -273.786 q^{67} +1129.04 q^{69} +18.7542 q^{70} +21.0261 q^{71} +99.1829 q^{72} +1105.25 q^{73} +1516.25 q^{74} -756.600 q^{75} -173.907 q^{76} -225.444 q^{77} -1388.94 q^{78} +434.602 q^{79} -57.6732 q^{80} -898.688 q^{81} +1122.66 q^{82} +598.442 q^{83} -224.030 q^{84} +1372.89 q^{86} -1146.22 q^{87} +320.853 q^{88} +895.951 q^{89} -26.6730 q^{90} -439.131 q^{91} +977.764 q^{92} -421.461 q^{93} -704.880 q^{94} -24.2998 q^{95} +1251.33 q^{96} -324.024 q^{97} -178.461 q^{98} +320.636 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 8 q^{2} - 24 q^{3} + 240 q^{4} - 80 q^{5} - 68 q^{6} - 392 q^{7} + 96 q^{8} + 576 q^{9} - 80 q^{10} - 176 q^{11} - 288 q^{12} - 96 q^{13} - 56 q^{14} + 192 q^{15} + 1088 q^{16} + 216 q^{18} + 48 q^{19}+ \cdots - 6576 q^{99}+O(q^{100}) \) 56 * q + 8 * q^2 - 24 * q^3 + 240 * q^4 - 80 * q^5 - 68 * q^6 - 392 * q^7 + 96 * q^8 + 576 * q^9 - 80 * q^10 - 176 * q^11 - 288 * q^12 - 96 * q^13 - 56 * q^14 + 192 * q^15 + 1088 * q^16 + 216 * q^18 + 48 * q^19 - 1660 * q^20 + 168 * q^21 - 352 * q^22 - 608 * q^23 - 736 * q^24 + 1792 * q^25 - 864 * q^27 - 1680 * q^28 - 928 * q^29 - 100 * q^30 - 496 * q^31 + 436 * q^32 + 384 * q^33 + 560 * q^35 + 2804 * q^36 - 1392 * q^37 - 624 * q^39 + 1020 * q^40 - 2088 * q^41 + 476 * q^42 + 1312 * q^43 - 2112 * q^44 - 4840 * q^45 - 736 * q^46 - 5228 * q^48 + 2744 * q^49 + 2020 * q^50 - 1536 * q^52 + 1272 * q^53 - 2268 * q^54 - 336 * q^55 - 672 * q^56 - 2448 * q^57 - 1376 * q^58 + 2372 * q^60 - 976 * q^61 - 3776 * q^62 - 4032 * q^63 + 5376 * q^64 - 6080 * q^65 + 976 * q^67 + 1248 * q^69 + 560 * q^70 - 3216 * q^71 + 4532 * q^72 - 4088 * q^73 + 2456 * q^74 + 760 * q^75 - 2640 * q^76 + 1232 * q^77 - 3744 * q^78 - 1792 * q^79 - 13920 * q^80 + 4184 * q^81 - 8204 * q^82 - 2832 * q^83 + 2016 * q^84 + 4908 * q^86 + 832 * q^87 - 2424 * q^88 - 2768 * q^89 - 8660 * q^90 + 672 * q^91 - 6784 * q^92 + 784 * q^93 - 928 * q^94 - 4560 * q^95 - 6656 * q^96 - 9464 * q^97 + 392 * q^98 - 6576 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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