LMFDB - Embedded newform 507.4.a.j.1.3

Newspace parameters

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$29.9139683729$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5054412.1
Defining polynomial:	$ x^{4} - 29x^{2} + 48 $ x^4 - 29*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.32750$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

$f(q)$	$=$	$q+1.32750 q^{2} -3.00000 q^{3} -6.23774 q^{4} +15.4241 q^{5} -3.98251 q^{6} -7.96501 q^{7} -18.9006 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+1.32750 q^{2} -3.00000 q^{3} -6.23774 q^{4} +15.4241 q^{5} -3.98251 q^{6} -7.96501 q^{7} -18.9006 q^{8} +9.00000 q^{9} +20.4755 q^{10} +12.7691 q^{11} +18.7132 q^{12} -10.5736 q^{14} -46.2722 q^{15} +24.8113 q^{16} -54.0000 q^{17} +11.9475 q^{18} -84.5794 q^{19} -96.2113 q^{20} +23.8950 q^{21} +16.9510 q^{22} +122.853 q^{23} +56.7019 q^{24} +112.902 q^{25} -27.0000 q^{27} +49.6837 q^{28} +140.853 q^{29} -61.4264 q^{30} -116.439 q^{31} +184.142 q^{32} -38.3072 q^{33} -71.6851 q^{34} -122.853 q^{35} -56.1397 q^{36} -433.898 q^{37} -112.279 q^{38} -291.525 q^{40} -205.823 q^{41} +31.7207 q^{42} -418.853 q^{43} -79.6501 q^{44} +138.817 q^{45} +163.087 q^{46} +485.861 q^{47} -74.4339 q^{48} -279.559 q^{49} +149.877 q^{50} +162.000 q^{51} -674.559 q^{53} -35.8425 q^{54} +196.951 q^{55} +150.544 q^{56} +253.738 q^{57} +186.982 q^{58} +186.226 q^{59} +288.634 q^{60} -671.902 q^{61} -154.574 q^{62} -71.6851 q^{63} +45.9584 q^{64} -50.8529 q^{66} +14.0364 q^{67} +336.838 q^{68} -368.559 q^{69} -163.087 q^{70} -346.789 q^{71} -170.106 q^{72} -832.900 q^{73} -576.000 q^{74} -338.706 q^{75} +527.584 q^{76} -101.706 q^{77} -335.608 q^{79} +382.691 q^{80} +81.0000 q^{81} -273.230 q^{82} +568.797 q^{83} -149.051 q^{84} -832.900 q^{85} -556.028 q^{86} -422.559 q^{87} -241.343 q^{88} -236.671 q^{89} +184.279 q^{90} -766.324 q^{92} +349.318 q^{93} +644.981 q^{94} -1304.56 q^{95} -552.426 q^{96} -1278.94 q^{97} -371.115 q^{98} +114.922 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 26 q^{4} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 + 26 * q^4 + 36 * q^9	$ 4 q - 12 q^{3} + 26 q^{4} + 36 q^{9} - 20 q^{10} - 78 q^{12} - 348 q^{14} + 354 q^{16} - 216 q^{17} - 136 q^{22} - 120 q^{23} + 44 q^{25} - 108 q^{27} - 48 q^{29} + 60 q^{30} + 120 q^{35} + 234 q^{36} + 468 q^{38} - 1268 q^{40} + 1044 q^{42} - 1064 q^{43} - 1062 q^{48} + 716 q^{49} + 648 q^{51} - 864 q^{53} + 584 q^{55} - 3372 q^{56} - 2280 q^{61} - 924 q^{62} + 1050 q^{64} + 408 q^{66} - 1404 q^{68} + 360 q^{69} - 2304 q^{74} - 132 q^{75} + 816 q^{77} + 288 q^{79} + 324 q^{81} + 28 q^{82} + 144 q^{87} - 2392 q^{88} - 180 q^{90} - 8568 q^{92} + 6656 q^{94} - 3384 q^{95}+O(q^{100}) $ 4 * q - 12 * q^3 + 26 * q^4 + 36 * q^9 - 20 * q^10 - 78 * q^12 - 348 * q^14 + 354 * q^16 - 216 * q^17 - 136 * q^22 - 120 * q^23 + 44 * q^25 - 108 * q^27 - 48 * q^29 + 60 * q^30 + 120 * q^35 + 234 * q^36 + 468 * q^38 - 1268 * q^40 + 1044 * q^42 - 1064 * q^43 - 1062 * q^48 + 716 * q^49 + 648 * q^51 - 864 * q^53 + 584 * q^55 - 3372 * q^56 - 2280 * q^61 - 924 * q^62 + 1050 * q^64 + 408 * q^66 - 1404 * q^68 + 360 * q^69 - 2304 * q^74 - 132 * q^75 + 816 * q^77 + 288 * q^79 + 324 * q^81 + 28 * q^82 + 144 * q^87 - 2392 * q^88 - 180 * q^90 - 8568 * q^92 + 6656 * q^94 - 3384 * q^95

Display 100 coefficients 10 coefficients

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 $

$2$	1.32750	0.469343	0.234671	−	0.972075i	$-0.424599\pi$
$2$	1.32750	0.469343	0.234671	+	0.972075i	$0.424599\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−6.23774	−0.779717
$5$	15.4241	1.37957	0.689785	−	0.724014i	$-0.257705\pi$
$5$	15.4241	1.37957	0.689785	+	0.724014i	$0.257705\pi$
$6$	−3.98251	−0.270975
$7$	−7.96501	−0.430070	−0.215035	−	0.976606i	$-0.568987\pi$
$7$	−7.96501	−0.430070	−0.215035	+	0.976606i	$0.568987\pi$
$8$	−18.9006	−0.835297
$9$	9.00000	0.333333
$10$	20.4755	0.647491
$11$	12.7691	0.350002	0.175001	−	0.984568i	$-0.444007\pi$
$11$	12.7691	0.350002	0.175001	+	0.984568i	$0.444007\pi$
$12$	18.7132	0.450170
$13$	0	0
$13$	0	0
$14$	−10.5736	−0.201850
$15$	−46.2722	−0.796496
$16$	24.8113	0.387677
$17$	−54.0000	−0.770407	−0.385204	−	0.922832i	$-0.625869\pi$
$17$	−54.0000	−0.770407	−0.385204	+	0.922832i	$0.625869\pi$
$18$	11.9475	0.156448
$19$	−84.5794	−1.02126	−0.510628	−	0.859802i	$-0.670587\pi$
$19$	−84.5794	−1.02126	−0.510628	+	0.859802i	$0.670587\pi$
$20$	−96.2113	−1.07568
$21$	23.8950	0.248301
$22$	16.9510	0.164271
$23$	122.853	1.11376	0.556882	−	0.830591i	$-0.311997\pi$
$23$	122.853	1.11376	0.556882	+	0.830591i	$0.311997\pi$
$24$	56.7019	0.482259
$25$	112.902	0.903215
$26$	0	0
$27$	−27.0000	−0.192450
$28$	49.6837	0.335333
$29$	140.853	0.901921	0.450961	−	0.892544i	$-0.351081\pi$
$29$	140.853	0.901921	0.450961	+	0.892544i	$0.351081\pi$
$30$	−61.4264	−0.373829
$31$	−116.439	−0.674617	−0.337309	−	0.941394i	$-0.609517\pi$
$31$	−116.439	−0.674617	−0.337309	+	0.941394i	$0.609517\pi$
$32$	184.142	1.01725
$33$	−38.3072	−0.202074
$34$	−71.6851	−0.361585
$35$	−122.853	−0.593312
$36$	−56.1397	−0.259906
$37$	−433.898	−1.92790	−0.963951	−	0.266081i	$-0.914271\pi$
$37$	−433.898	−1.92790	−0.963951	+	0.266081i	$0.914271\pi$
$38$	−112.279	−0.479319
$39$	0	0
$40$	−291.525	−1.15235
$41$	−205.823	−0.784003	−0.392002	−	0.919965i	$-0.628217\pi$
$41$	−205.823	−0.784003	−0.392002	+	0.919965i	$0.628217\pi$
$42$	31.7207	0.116538
$43$	−418.853	−1.48545	−0.742726	−	0.669595i	$-0.766468\pi$
$43$	−418.853	−1.48545	−0.742726	+	0.669595i	$0.766468\pi$
$44$	−79.6501	−0.272902
$45$	138.817	0.459857
$46$	163.087	0.522737
$47$	485.861	1.50787	0.753937	−	0.656947i	$-0.228152\pi$
$47$	485.861	1.50787	0.753937	+	0.656947i	$0.228152\pi$
$48$	−74.4339	−0.223825
$49$	−279.559	−0.815040
$50$	149.877	0.423918
$51$	162.000	0.444795
$52$	0	0
$53$	−674.559	−1.74826	−0.874130	−	0.485693i	$-0.838567\pi$
$53$	−674.559	−1.74826	−0.874130	+	0.485693i	$0.838567\pi$
$54$	−35.8425	−0.0903251
$55$	196.951	0.482852
$56$	150.544	0.359236
$57$	253.738	0.589622
$58$	186.982	0.423310
$59$	186.226	0.410925	0.205462	−	0.978665i	$-0.434130\pi$
$59$	186.226	0.410925	0.205462	+	0.978665i	$0.434130\pi$
$60$	288.634	0.621041
$61$	−671.902	−1.41030	−0.705149	−	0.709059i	$-0.749120\pi$
$61$	−671.902	−1.41030	−0.705149	+	0.709059i	$0.749120\pi$
$62$	−154.574	−0.316627
$63$	−71.6851	−0.143357
$64$	45.9584	0.0897626
$65$	0	0
$66$	−50.8529	−0.0948418
$67$	14.0364	0.0255944	0.0127972	−	0.999918i	$-0.495926\pi$
$67$	14.0364	0.0255944	0.0127972	+	0.999918i	$0.495926\pi$
$68$	336.838	0.600700
$69$	−368.559	−0.643033
$70$	−163.087	−0.278467
$71$	−346.789	−0.579665	−0.289833	−	0.957077i	$-0.593600\pi$
$71$	−346.789	−0.579665	−0.289833	+	0.957077i	$0.593600\pi$
$72$	−170.106	−0.278432
$73$	−832.900	−1.33539	−0.667695	−	0.744435i	$-0.732719\pi$
$73$	−832.900	−1.33539	−0.667695	+	0.744435i	$0.732719\pi$
$74$	−576.000	−0.904846
$75$	−338.706	−0.521472
$76$	527.584	0.796290
$77$	−101.706	−0.150525
$78$	0	0
$79$	−335.608	−0.477960	−0.238980	−	0.971025i	$-0.576813\pi$
$79$	−335.608	−0.477960	−0.238980	+	0.971025i	$0.576813\pi$
$80$	382.691	0.534827
$81$	81.0000	0.111111
$82$	−273.230	−0.367966
$83$	568.797	0.752212	0.376106	−	0.926577i	$-0.377263\pi$
$83$	568.797	0.752212	0.376106	+	0.926577i	$0.377263\pi$
$84$	−149.051	−0.193605
$85$	−832.900	−1.06283
$86$	−556.028	−0.697186
$87$	−422.559	−0.520725
$88$	−241.343	−0.292355
$89$	−236.671	−0.281877	−0.140939	−	0.990018i	$-0.545012\pi$
$89$	−236.671	−0.281877	−0.140939	+	0.990018i	$0.545012\pi$
$90$	184.279	0.215830
$91$	0	0
$92$	−766.324	−0.868422
$93$	349.318	0.389491
$94$	644.981	0.707710
$95$	−1304.56	−1.40889
$96$	−552.426	−0.587310
$97$	−1278.94	−1.33873	−0.669365	−	0.742934i	$-0.733434\pi$
$97$	−1278.94	−1.33873	−0.669365	+	0.742934i	$0.733434\pi$
$98$	−371.115	−0.382533
$99$	114.922	0.116667
$100$	−704.253	−0.704253
$101$	632.264	0.622898	0.311449	−	0.950263i	$-0.399186\pi$
$101$	632.264	0.622898	0.311449	+	0.950263i	$0.399186\pi$
$102$	215.055	0.208761
$103$	1506.26	1.44094	0.720469	−	0.693487i	$-0.243927\pi$
$103$	1506.26	1.44094	0.720469	+	0.693487i	$0.243927\pi$
$104$	0	0
$105$	368.559	0.342549
$106$	−895.478	−0.820533
$107$	−1268.56	−1.14613	−0.573066	−	0.819509i	$-0.694246\pi$
$107$	−1268.56	−1.14613	−0.573066	+	0.819509i	$0.694246\pi$
$108$	168.419	0.150057
$109$	347.425	0.305296	0.152648	−	0.988281i	$-0.451220\pi$
$109$	347.425	0.305296	0.152648	+	0.988281i	$0.451220\pi$
$110$	261.453	0.226623
$111$	1301.69	1.11307
$112$	−197.622	−0.166728
$113$	659.706	0.549203	0.274601	−	0.961558i	$-0.411454\pi$
$113$	659.706	0.549203	0.274601	+	0.961558i	$0.411454\pi$
$114$	336.838	0.276735
$115$	1894.89	1.53652
$116$	−878.603	−0.703244
$117$	0	0
$118$	247.215	0.192865
$119$	430.111	0.331329
$120$	874.574	0.665311
$121$	−1167.95	−0.877499
$122$	−891.951	−0.661913
$123$	617.469	0.452645
$124$	726.319	0.526011
$125$	−186.602	−0.133521
$126$	−95.1621	−0.0672834
$127$	275.019	0.192157	0.0960787	−	0.995374i	$-0.469370\pi$
$127$	275.019	0.192157	0.0960787	+	0.995374i	$0.469370\pi$
$128$	−1412.13	−0.975121
$129$	1256.56	0.857626
$130$	0	0
$131$	−1183.97	−0.789648	−0.394824	−	0.918757i	$-0.629195\pi$
$131$	−1183.97	−0.789648	−0.394824	+	0.918757i	$0.629195\pi$
$132$	238.950	0.157560
$133$	673.676	0.439211
$134$	18.6334	0.0120125
$135$	−416.450	−0.265499
$136$	1020.63	0.643519
$137$	2557.36	1.59482	0.797410	−	0.603438i	$-0.206203\pi$
$137$	2557.36	1.59482	0.797410	+	0.603438i	$0.206203\pi$
$138$	−489.262	−0.301803
$139$	545.736	0.333012	0.166506	−	0.986040i	$-0.446751\pi$
$139$	545.736	0.333012	0.166506	+	0.986040i	$0.446751\pi$
$140$	766.324	0.462616
$141$	−1457.58	−0.870571
$142$	−460.362	−0.272062
$143$	0	0
$144$	223.302	0.129226
$145$	2172.52	1.24426
$146$	−1105.68	−0.626756
$147$	838.676	0.470563
$148$	2706.54	1.50322
$149$	1376.78	0.756981	0.378491	−	0.925605i	$-0.376443\pi$
$149$	1376.78	0.756981	0.378491	+	0.925605i	$0.376443\pi$
$150$	−449.632	−0.244749
$151$	2733.47	1.47316	0.736579	−	0.676352i	$-0.236440\pi$
$151$	2733.47	1.47316	0.736579	+	0.676352i	$0.236440\pi$
$152$	1598.60	0.853052
$153$	−486.000	−0.256802
$154$	−135.015	−0.0706479
$155$	−1795.97	−0.930683
$156$	0	0
$157$	1029.97	0.523570	0.261785	−	0.965126i	$-0.415689\pi$
$157$	1029.97	0.523570	0.261785	+	0.965126i	$0.415689\pi$
$158$	−445.520	−0.224327
$159$	2023.68	1.00936
$160$	2840.22	1.40337
$161$	−978.524	−0.478997
$162$	107.528	0.0521492
$163$	−2882.91	−1.38532	−0.692660	−	0.721264i	$-0.743561\pi$
$163$	−2882.91	−1.38532	−0.692660	+	0.721264i	$0.743561\pi$
$164$	1283.87	0.611301
$165$	−590.853	−0.278775
$166$	755.079	0.353045
$167$	1153.90	0.534679	0.267340	−	0.963602i	$-0.413855\pi$
$167$	1153.90	0.534679	0.267340	+	0.963602i	$0.413855\pi$
$168$	−451.631	−0.207405
$169$	0	0
$170$	−1105.68	−0.498832
$171$	−761.215	−0.340418
$172$	2612.69	1.15823
$173$	−1688.85	−0.742203	−0.371101	−	0.928592i	$-0.621020\pi$
$173$	−1688.85	−0.742203	−0.371101	+	0.928592i	$0.621020\pi$
$174$	−560.947	−0.244398
$175$	−899.265	−0.388446
$176$	316.817	0.135687
$177$	−558.678	−0.237247
$178$	−314.181	−0.132297
$179$	−942.793	−0.393674	−0.196837	−	0.980436i	$-0.563067\pi$
$179$	−942.793	−0.393674	−0.196837	+	0.980436i	$0.563067\pi$
$180$	−865.902	−0.358558
$181$	482.030	0.197950	0.0989751	−	0.995090i	$-0.468444\pi$
$181$	482.030	0.197950	0.0989751	+	0.995090i	$0.468444\pi$
$182$	0	0
$183$	2015.71	0.814236
$184$	−2322.00	−0.930325
$185$	−6692.47	−2.65968
$186$	463.721	0.182805
$187$	−689.530	−0.269644
$188$	−3030.67	−1.17572
$189$	215.055	0.0827670
$190$	−1731.80	−0.661254
$191$	4223.32	1.59994	0.799971	−	0.600039i	$-0.204848\pi$
$191$	4223.32	1.59994	0.799971	+	0.600039i	$0.204848\pi$
$192$	−137.875	−0.0518244
$193$	−229.092	−0.0854424	−0.0427212	−	0.999087i	$-0.513603\pi$
$193$	−229.092	−0.0854424	−0.0427212	+	0.999087i	$0.513603\pi$
$194$	−1697.80	−0.628323
$195$	0	0
$196$	1743.81	0.635501
$197$	−228.335	−0.0825798	−0.0412899	−	0.999147i	$-0.513147\pi$
$197$	−228.335	−0.0825798	−0.0412899	+	0.999147i	$0.513147\pi$
$198$	152.559	0.0547569
$199$	2939.02	1.04694	0.523471	−	0.852043i	$-0.324637\pi$
$199$	2939.02	1.04694	0.523471	+	0.852043i	$0.324637\pi$
$200$	−2133.92	−0.754453
$201$	−42.1093	−0.0147769
$202$	839.332	0.292352
$203$	−1121.89	−0.387889
$204$	−1010.51	−0.346814
$205$	−3174.63	−1.08159
$206$	1999.57	0.676294
$207$	1105.68	0.371255
$208$	0	0
$209$	−1080.00	−0.357441
$210$	489.262	0.160773
$211$	−1607.02	−0.524321	−0.262161	−	0.965024i	$-0.584435\pi$
$211$	−1607.02	−0.524321	−0.262161	+	0.965024i	$0.584435\pi$
$212$	4207.72	1.36315
$213$	1040.37	0.334670
$214$	−1684.01	−0.537929
$215$	−6460.42	−2.04929
$216$	510.317	0.160753
$217$	927.441	0.290133
$218$	461.207	0.143288
$219$	2498.70	0.770988
$220$	−1228.53	−0.376488
$221$	0	0
$222$	1728.00	0.522413
$223$	130.867	0.0392981	0.0196490	−	0.999807i	$-0.493745\pi$
$223$	130.867	0.0392981	0.0196490	+	0.999807i	$0.493745\pi$
$224$	−1466.69	−0.437489
$225$	1016.12	0.301072
$226$	875.761	0.257764
$227$	4325.19	1.26464	0.632319	−	0.774708i	$-0.282103\pi$
$227$	4325.19	1.26464	0.632319	+	0.774708i	$0.282103\pi$
$228$	−1582.75	−0.459738
$229$	−2621.57	−0.756499	−0.378250	−	0.925704i	$-0.623474\pi$
$229$	−2621.57	−0.756499	−0.378250	+	0.925704i	$0.623474\pi$
$230$	2515.47	0.721153
$231$	305.117	0.0869058
$232$	−2662.21	−0.753373
$233$	−4643.12	−1.30550	−0.652748	−	0.757575i	$-0.726384\pi$
$233$	−4643.12	−1.30550	−0.652748	+	0.757575i	$0.726384\pi$
$234$	0	0
$235$	7493.95	2.08022
$236$	−1161.63	−0.320405
$237$	1006.82	0.275950
$238$	570.972	0.155507
$239$	6696.69	1.81244	0.906219	−	0.422809i	$-0.138956\pi$
$239$	6696.69	1.81244	0.906219	+	0.422809i	$0.138956\pi$
$240$	−1148.07	−0.308783
$241$	2301.47	0.615148	0.307574	−	0.951524i	$-0.400483\pi$
$241$	2301.47	0.615148	0.307574	+	0.951524i	$0.400483\pi$
$242$	−1550.46	−0.411848
$243$	−243.000	−0.0641500
$244$	4191.15	1.09963
$245$	−4311.93	−1.12440
$246$	819.691	0.212445
$247$	0	0
$248$	2200.78	0.563506
$249$	−1706.39	−0.434290
$250$	−247.714	−0.0626673
$251$	828.000	0.208219	0.104109	−	0.994566i	$-0.466801\pi$
$251$	828.000	0.208219	0.104109	+	0.994566i	$0.466801\pi$
$252$	447.153	0.111778
$253$	1568.72	0.389820
$254$	365.088	0.0901877
$255$	2498.70	0.613626
$256$	−2242.27	−0.547429
$257$	884.763	0.214747	0.107374	−	0.994219i	$-0.465756\pi$
$257$	884.763	0.214747	0.107374	+	0.994219i	$0.465756\pi$
$258$	1668.08	0.402521
$259$	3456.00	0.829133
$260$	0	0
$261$	1267.68	0.300640
$262$	−1571.72	−0.370616
$263$	−8343.94	−1.95631	−0.978155	−	0.207878i	$-0.933344\pi$
$263$	−8343.94	−1.95631	−0.978155	+	0.207878i	$0.933344\pi$
$264$	724.030	0.168792
$265$	−10404.4	−2.41185
$266$	894.306	0.206141
$267$	710.013	0.162742
$268$	−87.5556	−0.0199564
$269$	2762.56	0.626157	0.313078	−	0.949727i	$-0.398640\pi$
$269$	2762.56	0.626157	0.313078	+	0.949727i	$0.398640\pi$
$270$	−552.838	−0.124610
$271$	−3116.54	−0.698585	−0.349293	−	0.937014i	$-0.613578\pi$
$271$	−3116.54	−0.698585	−0.349293	+	0.937014i	$0.613578\pi$
$272$	−1339.81	−0.298669
$273$	0	0
$274$	3394.91	0.748517
$275$	1441.65	0.316127
$276$	2298.97	0.501384
$277$	−502.060	−0.108902	−0.0544510	−	0.998516i	$-0.517341\pi$
$277$	−502.060	−0.108902	−0.0544510	+	0.998516i	$0.517341\pi$
$278$	724.465	0.156297
$279$	−1047.96	−0.224872
$280$	2322.00	0.495592
$281$	6607.56	1.40275	0.701377	−	0.712791i	$-0.252569\pi$
$281$	6607.56	1.40275	0.701377	+	0.712791i	$0.252569\pi$
$282$	−1934.94	−0.408596
$283$	−4368.98	−0.917699	−0.458850	−	0.888514i	$-0.651738\pi$
$283$	−4368.98	−0.917699	−0.458850	+	0.888514i	$0.651738\pi$
$284$	2163.18	0.451975
$285$	3913.68	0.813425
$286$	0	0
$287$	1639.38	0.337176
$288$	1657.28	0.339084
$289$	−1997.00	−0.406473
$290$	2884.03	0.583986
$291$	3836.82	0.772916
$292$	5195.41	1.04123
$293$	5348.12	1.06635	0.533175	−	0.846005i	$-0.320999\pi$
$293$	5348.12	1.06635	0.533175	+	0.846005i	$0.320999\pi$
$294$	1113.34	0.220856
$295$	2872.36	0.566900
$296$	8200.94	1.61037
$297$	−344.765	−0.0673579
$298$	1827.68	0.355284
$299$	0	0
$300$	2112.76	0.406600
$301$	3336.17	0.638849
$302$	3628.69	0.691416
$303$	−1896.79	−0.359630
$304$	−2098.53	−0.395917
$305$	−10363.5	−1.94561
$306$	−645.166	−0.120528
$307$	4502.46	0.837032	0.418516	−	0.908210i	$-0.362550\pi$
$307$	4502.46	0.837032	0.418516	+	0.908210i	$0.362550\pi$
$308$	634.414	0.117367
$309$	−4518.79	−0.831926
$310$	−2384.15	−0.436809
$311$	−7447.20	−1.35785	−0.678926	−	0.734207i	$-0.737554\pi$
$311$	−7447.20	−1.35785	−0.678926	+	0.734207i	$0.737554\pi$
$312$	0	0
$313$	−6508.93	−1.17542	−0.587710	−	0.809072i	$-0.699970\pi$
$313$	−6508.93	−1.17542	−0.587710	+	0.809072i	$0.699970\pi$
$314$	1367.29	0.245734
$315$	−1105.68	−0.197771
$316$	2093.43	0.372673
$317$	−2465.57	−0.436846	−0.218423	−	0.975854i	$-0.570091\pi$
$317$	−2465.57	−0.436846	−0.218423	+	0.975854i	$0.570091\pi$
$318$	2686.43	0.473735
$319$	1798.56	0.315674
$320$	708.866	0.123834
$321$	3805.68	0.661720
$322$	−1298.99	−0.224814
$323$	4567.29	0.786782
$324$	−505.257	−0.0866353
$325$	0	0
$326$	−3827.07	−0.650190
$327$	−1042.27	−0.176263
$328$	3890.18	0.654876
$329$	−3869.89	−0.648491
$330$	−784.358	−0.130841
$331$	4114.84	0.683300	0.341650	−	0.939827i	$-0.389014\pi$
$331$	4114.84	0.683300	0.341650	+	0.939827i	$0.389014\pi$
$332$	−3548.01	−0.586513
$333$	−3905.08	−0.642634
$334$	1531.80	0.250948
$335$	216.499	0.0353092
$336$	592.867	0.0962605
$337$	4798.05	0.775568	0.387784	−	0.921750i	$-0.373241\pi$
$337$	4798.05	0.775568	0.387784	+	0.921750i	$0.373241\pi$
$338$	0	0
$339$	−1979.12	−0.317082
$340$	5195.41	0.828708
$341$	−1486.82	−0.236117
$342$	−1010.51	−0.159773
$343$	4958.69	0.780594
$344$	7916.58	1.24079
$345$	−5684.67	−0.887109
$346$	−2241.96	−0.348348
$347$	−3314.76	−0.512812	−0.256406	−	0.966569i	$-0.582538\pi$
$347$	−3314.76	−0.512812	−0.256406	+	0.966569i	$0.582538\pi$
$348$	2635.81	0.406018
$349$	371.740	0.0570166	0.0285083	−	0.999594i	$-0.490924\pi$
$349$	371.740	0.0570166	0.0285083	+	0.999594i	$0.490924\pi$
$350$	−1193.78	−0.182314
$351$	0	0
$352$	2351.32	0.356039
$353$	−7539.10	−1.13673	−0.568365	−	0.822776i	$-0.692424\pi$
$353$	−7539.10	−1.13673	−0.568365	+	0.822776i	$0.692424\pi$
$354$	−741.646	−0.111350
$355$	−5348.89	−0.799689
$356$	1476.29	0.219785
$357$	−1290.33	−0.191293
$358$	−1251.56	−0.184768
$359$	−12741.5	−1.87317	−0.936586	−	0.350437i	$-0.886033\pi$
$359$	−12741.5	−1.87317	−0.936586	+	0.350437i	$0.886033\pi$
$360$	−2623.72	−0.384117
$361$	294.676	0.0429619
$362$	639.896	0.0929065
$363$	3503.85	0.506624
$364$	0	0
$365$	−12846.7	−1.84227
$366$	2675.85	0.382156
$367$	7187.26	1.02227	0.511134	−	0.859501i	$-0.329226\pi$
$367$	7187.26	1.02227	0.511134	+	0.859501i	$0.329226\pi$
$368$	3048.14	0.431781
$369$	−1852.41	−0.261334
$370$	−8884.26	−1.24830
$371$	5372.87	0.751874
$372$	−2178.96	−0.303693
$373$	−2087.99	−0.289845	−0.144922	−	0.989443i	$-0.546293\pi$
$373$	−2087.99	−0.289845	−0.144922	+	0.989443i	$0.546293\pi$
$374$	−915.352	−0.126555
$375$	559.805	0.0770886
$376$	−9183.07	−1.25952
$377$	0	0
$378$	285.486	0.0388461
$379$	−3982.08	−0.539699	−0.269850	−	0.962902i	$-0.586974\pi$
$379$	−3982.08	−0.539699	−0.269850	+	0.962902i	$0.586974\pi$
$380$	8137.50	1.09854
$381$	−825.057	−0.110942
$382$	5606.47	0.750921
$383$	−8638.43	−1.15249	−0.576244	−	0.817278i	$-0.695482\pi$
$383$	−8638.43	−1.15249	−0.576244	+	0.817278i	$0.695482\pi$
$384$	4236.38	0.562987
$385$	−1568.72	−0.207660
$386$	−304.120	−0.0401018
$387$	−3769.68	−0.495151
$388$	7977.70	1.04383
$389$	−1275.74	−0.166279	−0.0831393	−	0.996538i	$-0.526495\pi$
$389$	−1275.74	−0.166279	−0.0831393	+	0.996538i	$0.526495\pi$
$390$	0	0
$391$	−6634.06	−0.858053
$392$	5283.83	0.680801
$393$	3551.91	0.455904
$394$	−303.115	−0.0387582
$395$	−5176.44	−0.659379
$396$	−716.851	−0.0909675
$397$	4622.65	0.584394	0.292197	−	0.956358i	$-0.405614\pi$
$397$	4622.65	0.584394	0.292197	+	0.956358i	$0.405614\pi$
$398$	3901.55	0.491375
$399$	−2021.03	−0.253579
$400$	2801.24	0.350155
$401$	138.075	0.0171949	0.00859743	−	0.999963i	$-0.497263\pi$
$401$	138.075	0.0171949	0.00859743	+	0.999963i	$0.497263\pi$
$402$	−55.9001	−0.00693544
$403$	0	0
$404$	−3943.90	−0.485684
$405$	1249.35	0.153286
$406$	−1489.32	−0.182053
$407$	−5540.47	−0.674769
$408$	−3061.90	−0.371536
$409$	1204.64	0.145637	0.0728186	−	0.997345i	$-0.476801\pi$
$409$	1204.64	0.145637	0.0728186	+	0.997345i	$0.476801\pi$
$410$	−4214.32	−0.507635
$411$	−7672.09	−0.920770
$412$	−9395.68	−1.12352
$413$	−1483.29	−0.176726
$414$	1467.79	0.174246
$415$	8773.16	1.03773
$416$	0	0
$417$	−1637.21	−0.192265
$418$	−1433.70	−0.167762
$419$	5199.85	0.606275	0.303138	−	0.952947i	$-0.401966\pi$
$419$	5199.85	0.606275	0.303138	+	0.952947i	$0.401966\pi$
$420$	−2298.97	−0.267091
$421$	14136.5	1.63651	0.818256	−	0.574854i	$-0.194941\pi$
$421$	14136.5	1.63651	0.818256	+	0.574854i	$0.194941\pi$
$422$	−2133.32	−0.246086
$423$	4372.75	0.502625
$424$	12749.6	1.46032
$425$	−6096.70	−0.695844
$426$	1381.09	0.157075
$427$	5351.71	0.606527
$428$	7912.94	0.893660
$429$	0	0
$430$	−8576.21	−0.961818
$431$	2279.83	0.254793	0.127396	−	0.991852i	$-0.459338\pi$
$431$	2279.83	0.254793	0.127396	+	0.991852i	$0.459338\pi$
$432$	−669.905	−0.0746084
$433$	−13298.7	−1.47597	−0.737984	−	0.674819i	$-0.764222\pi$
$433$	−13298.7	−1.47597	−0.737984	+	0.674819i	$0.764222\pi$
$434$	1231.18	0.136172
$435$	−6517.57	−0.718376
$436$	−2167.14	−0.238045
$437$	−10390.8	−1.13744
$438$	3317.03	0.361858
$439$	−10452.3	−1.13635	−0.568177	−	0.822907i	$-0.692351\pi$
$439$	−10452.3	−1.13635	−0.568177	+	0.822907i	$0.692351\pi$
$440$	−3722.50	−0.403325
$441$	−2516.03	−0.271680
$442$	0	0
$443$	5363.50	0.575232	0.287616	−	0.957746i	$-0.407137\pi$
$443$	5363.50	0.575232	0.287616	+	0.957746i	$0.407137\pi$
$444$	−8119.62	−0.867883
$445$	−3650.43	−0.388870
$446$	173.726	0.0184443
$447$	−4130.34	−0.437043
$448$	−366.059	−0.0386042
$449$	−9681.73	−1.01762	−0.508808	−	0.860880i	$-0.669914\pi$
$449$	−9681.73	−1.01762	−0.508808	+	0.860880i	$0.669914\pi$
$450$	1348.90	0.141306
$451$	−2628.17	−0.274402
$452$	−4115.07	−0.428223
$453$	−8200.42	−0.850528
$454$	5741.69	0.593548
$455$	0	0
$456$	−4795.81	−0.492510
$457$	3537.94	0.362139	0.181070	−	0.983470i	$-0.442044\pi$
$457$	3537.94	0.362139	0.181070	+	0.983470i	$0.442044\pi$
$458$	−3480.14	−0.355057
$459$	1458.00	0.148265
$460$	−11819.8	−1.19805
$461$	15074.9	1.52302	0.761508	−	0.648156i	$-0.224459\pi$
$461$	15074.9	1.52302	0.761508	+	0.648156i	$0.224459\pi$
$462$	405.044	0.0407886
$463$	11070.0	1.11116	0.555580	−	0.831463i	$-0.312496\pi$
$463$	11070.0	1.11116	0.555580	+	0.831463i	$0.312496\pi$
$464$	3494.74	0.349654
$465$	5387.91	0.537330
$466$	−6163.75	−0.612725
$467$	13252.8	1.31320	0.656600	−	0.754239i	$-0.271994\pi$
$467$	13252.8	1.31320	0.656600	+	0.754239i	$0.271994\pi$
$468$	0	0
$469$	−111.800	−0.0110074
$470$	9948.23	0.976335
$471$	−3089.91	−0.302284
$472$	−3519.79	−0.343244
$473$	−5348.36	−0.519911
$474$	1336.56	0.129515
$475$	−9549.18	−0.922413
$476$	−2682.92	−0.258343
$477$	−6071.03	−0.582753
$478$	8889.86	0.850654
$479$	−12241.4	−1.16769	−0.583843	−	0.811866i	$-0.698452\pi$
$479$	−12241.4	−1.16769	−0.583843	+	0.811866i	$0.698452\pi$
$480$	−8520.66	−0.810236
$481$	0	0
$482$	3055.20	0.288715
$483$	2935.57	0.276549
$484$	7285.37	0.684201
$485$	−19726.5	−1.84687
$486$	−322.583	−0.0301084
$487$	13413.3	1.24808	0.624041	−	0.781392i	$-0.285490\pi$
$487$	13413.3	1.24808	0.624041	+	0.781392i	$0.285490\pi$
$488$	12699.4	1.17802
$489$	8648.74	0.799815
$490$	−5724.10	−0.527731
$491$	−737.885	−0.0678214	−0.0339107	−	0.999425i	$-0.510796\pi$
$491$	−737.885	−0.0678214	−0.0339107	+	0.999425i	$0.510796\pi$
$492$	−3851.61	−0.352935
$493$	−7606.06	−0.694847
$494$	0	0
$495$	1772.56	0.160951
$496$	−2889.01	−0.261533
$497$	2762.17	0.249297
$498$	−2265.24	−0.203831
$499$	−1865.65	−0.167370	−0.0836852	−	0.996492i	$-0.526669\pi$
$499$	−1865.65	−0.167370	−0.0836852	+	0.996492i	$0.526669\pi$
$500$	1163.97	0.104109
$501$	−3461.70	−0.308697
$502$	1099.17	0.0977259
$503$	−16632.0	−1.47432	−0.737161	−	0.675717i	$-0.763834\pi$
$503$	−16632.0	−1.47432	−0.737161	+	0.675717i	$0.763834\pi$
$504$	1354.89	0.119745
$505$	9752.09	0.859331
$506$	2082.47	0.182959
$507$	0	0
$508$	−1715.50	−0.149829
$509$	−4128.91	−0.359549	−0.179775	−	0.983708i	$-0.557537\pi$
$509$	−4128.91	−0.359549	−0.179775	+	0.983708i	$0.557537\pi$
$510$	3317.03	0.288001
$511$	6634.06	0.574312
$512$	8320.40	0.718190
$513$	2283.64	0.196541
$514$	1174.52	0.100790
$515$	23232.7	1.98788
$516$	−7838.08	−0.668706
$517$	6203.99	0.527758
$518$	4587.85	0.389147
$519$	5066.56	0.428511
$520$	0	0
$521$	−988.234	−0.0831005	−0.0415502	−	0.999136i	$-0.513230\pi$
$521$	−988.234	−0.0831005	−0.0415502	+	0.999136i	$0.513230\pi$
$522$	1682.84	0.141103
$523$	−9441.62	−0.789394	−0.394697	−	0.918811i	$-0.629150\pi$
$523$	−9441.62	−0.789394	−0.394697	+	0.918811i	$0.629150\pi$
$524$	7385.30	0.615703
$525$	2697.79	0.224269
$526$	−11076.6	−0.918180
$527$	6287.73	0.519730
$528$	−950.452	−0.0783392
$529$	2925.83	0.240472
$530$	−13811.9	−1.13198
$531$	1676.03	0.136975
$532$	−4202.21	−0.342461
$533$	0	0
$534$	942.544	0.0763817
$535$	−19566.3	−1.58117
$536$	−265.297	−0.0213789
$537$	2828.38	0.227288
$538$	3667.30	0.293882
$539$	−3569.70	−0.285265
$540$	2597.71	0.207014
$541$	−14001.5	−1.11270	−0.556351	−	0.830948i	$-0.687799\pi$
$541$	−14001.5	−1.11270	−0.556351	+	0.830948i	$0.687799\pi$
$542$	−4137.22	−0.327876
$543$	−1446.09	−0.114287
$544$	−9943.67	−0.783697
$545$	5358.70	0.421177
$546$	0	0
$547$	−4244.85	−0.331804	−0.165902	−	0.986142i	$-0.553053\pi$
$547$	−4244.85	−0.331804	−0.165902	+	0.986142i	$0.553053\pi$
$548$	−15952.2	−1.24351
$549$	−6047.12	−0.470100
$550$	1913.80	0.148372
$551$	−11913.3	−0.921092
$552$	6965.99	0.537123
$553$	2673.12	0.205556
$554$	−666.485	−0.0511124
$555$	20077.4	1.53556
$556$	−3404.16	−0.259655
$557$	11732.2	0.892473	0.446236	−	0.894915i	$-0.352764\pi$
$557$	11732.2	0.892473	0.446236	+	0.894915i	$0.352764\pi$
$558$	−1391.16	−0.105542
$559$	0	0
$560$	−3048.14	−0.230013
$561$	2068.59	0.155679
$562$	8771.54	0.658372
$563$	−9941.80	−0.744222	−0.372111	−	0.928188i	$-0.621366\pi$
$563$	−9941.80	−0.744222	−0.372111	+	0.928188i	$0.621366\pi$
$564$	9092.02	0.678800
$565$	10175.3	0.757664
$566$	−5799.83	−0.430716
$567$	−645.166	−0.0477856
$568$	6554.52	0.484193
$569$	3690.77	0.271925	0.135962	−	0.990714i	$-0.456587\pi$
$569$	3690.77	0.271925	0.135962	+	0.990714i	$0.456587\pi$
$570$	5195.41	0.381775
$571$	−5685.09	−0.416661	−0.208331	−	0.978058i	$-0.566803\pi$
$571$	−5685.09	−0.416661	−0.208331	+	0.978058i	$0.566803\pi$
$572$	0	0
$573$	−12670.0	−0.923727
$574$	2176.28	0.158251
$575$	13870.3	1.00597
$576$	413.626	0.0299209
$577$	−7746.50	−0.558910	−0.279455	−	0.960159i	$-0.590154\pi$
$577$	−7746.50	−0.558910	−0.279455	+	0.960159i	$0.590154\pi$
$578$	−2651.02	−0.190775
$579$	687.275	0.0493302
$580$	−13551.6	−0.970175
$581$	−4530.47	−0.323504
$582$	5093.39	0.362762
$583$	−8613.48	−0.611894
$584$	15742.3	1.11545
$585$	0	0
$586$	7099.64	0.500484
$587$	2766.54	0.194527	0.0972635	−	0.995259i	$-0.468991\pi$
$587$	2766.54	0.194527	0.0972635	+	0.995259i	$0.468991\pi$
$588$	−5231.44	−0.366906
$589$	9848.38	0.688957
$590$	3813.07	0.266070
$591$	685.006	0.0476774
$592$	−10765.6	−0.747402
$593$	1440.79	0.0997743	0.0498871	−	0.998755i	$-0.484114\pi$
$593$	1440.79	0.0997743	0.0498871	+	0.998755i	$0.484114\pi$
$594$	−457.676	−0.0316139
$595$	6634.06	0.457092
$596$	−8587.99	−0.590231
$597$	−8817.06	−0.604452
$598$	0	0
$599$	23837.5	1.62600	0.813001	−	0.582263i	$-0.197832\pi$
$599$	23837.5	1.62600	0.813001	+	0.582263i	$0.197832\pi$
$600$	6401.75	0.435584
$601$	−6694.23	−0.454348	−0.227174	−	0.973854i	$-0.572949\pi$
$601$	−6694.23	−0.454348	−0.227174	+	0.973854i	$0.572949\pi$
$602$	4428.77	0.299839
$603$	126.328	0.00853145
$604$	−17050.7	−1.14865
$605$	−18014.6	−1.21057
$606$	−2518.00	−0.168790
$607$	3330.50	0.222703	0.111352	−	0.993781i	$-0.464482\pi$
$607$	3330.50	0.222703	0.111352	+	0.993781i	$0.464482\pi$
$608$	−15574.6	−1.03887
$609$	3365.68	0.223948
$610$	−13757.5	−0.913156
$611$	0	0
$612$	3031.54	0.200233
$613$	13490.3	0.888857	0.444428	−	0.895814i	$-0.353407\pi$
$613$	13490.3	0.888857	0.444428	+	0.895814i	$0.353407\pi$
$614$	5977.02	0.392855
$615$	9523.88	0.624455
$616$	1922.30	0.125733
$617$	−7470.76	−0.487458	−0.243729	−	0.969843i	$-0.578371\pi$
$617$	−7470.76	−0.487458	−0.243729	+	0.969843i	$0.578371\pi$
$618$	−5998.71	−0.390458
$619$	−24806.9	−1.61078	−0.805389	−	0.592746i	$-0.798044\pi$
$619$	−24806.9	−1.61078	−0.805389	+	0.592746i	$0.798044\pi$
$620$	11202.8	0.725669
$621$	−3317.03	−0.214344
$622$	−9886.17	−0.637298
$623$	1885.09	0.121227
$624$	0	0
$625$	−16990.9	−1.08742
$626$	−8640.61	−0.551675
$627$	3240.00	0.206369
$628$	−6424.68	−0.408237
$629$	23430.5	1.48527
$630$	−1467.79	−0.0928222
$631$	−314.333	−0.0198311	−0.00991554	−	0.999951i	$-0.503156\pi$
$631$	−314.333	−0.0198311	−0.00991554	+	0.999951i	$0.503156\pi$
$632$	6343.19	0.399238
$633$	4821.06	0.302717
$634$	−3273.05	−0.205031
$635$	4241.91	0.265095
$636$	−12623.2	−0.787014
$637$	0	0
$638$	2387.59	0.148159
$639$	−3121.10	−0.193222
$640$	−21780.7	−1.34525
$641$	5550.41	0.342009	0.171005	−	0.985270i	$-0.445299\pi$
$641$	5550.41	0.342009	0.171005	+	0.985270i	$0.445299\pi$
$642$	5052.04	0.310573
$643$	−5479.48	−0.336065	−0.168032	−	0.985781i	$-0.553741\pi$
$643$	−5479.48	−0.336065	−0.168032	+	0.985781i	$0.553741\pi$
$644$	6103.78	0.373482
$645$	19381.2	1.18316
$646$	6063.08	0.369271
$647$	4724.83	0.287098	0.143549	−	0.989643i	$-0.454149\pi$
$647$	4724.83	0.287098	0.143549	+	0.989643i	$0.454149\pi$
$648$	−1530.95	−0.0928108
$649$	2377.93	0.143824
$650$	0	0
$651$	−2782.32	−0.167508
$652$	17982.9	1.08016
$653$	3463.91	0.207585	0.103793	−	0.994599i	$-0.466902\pi$
$653$	3463.91	0.207585	0.103793	+	0.994599i	$0.466902\pi$
$654$	−1383.62	−0.0827276
$655$	−18261.6	−1.08938
$656$	−5106.73	−0.303940
$657$	−7496.10	−0.445130
$658$	−5137.28	−0.304365
$659$	−2606.35	−0.154065	−0.0770327	−	0.997029i	$-0.524545\pi$
$659$	−2606.35	−0.154065	−0.0770327	+	0.997029i	$0.524545\pi$
$660$	3685.59	0.217366
$661$	22436.7	1.32025	0.660127	−	0.751154i	$-0.270502\pi$
$661$	22436.7	1.32025	0.660127	+	0.751154i	$0.270502\pi$
$662$	5462.46	0.320702
$663$	0	0
$664$	−10750.6	−0.628321
$665$	10390.8	0.605923
$666$	−5184.00	−0.301615
$667$	17304.2	1.00453
$668$	−7197.73	−0.416899
$669$	−392.600	−0.0226888
$670$	287.403	0.0165721
$671$	−8579.56	−0.493607
$672$	4400.08	0.252584
$673$	−633.970	−0.0363117	−0.0181558	−	0.999835i	$-0.505779\pi$
$673$	−633.970	−0.0363117	−0.0181558	+	0.999835i	$0.505779\pi$
$674$	6369.42	0.364007
$675$	−3048.35	−0.173824
$676$	0	0
$677$	24457.4	1.38844	0.694221	−	0.719762i	$-0.255749\pi$
$677$	24457.4	1.38844	0.694221	+	0.719762i	$0.255749\pi$
$678$	−2627.28	−0.148820
$679$	10186.8	0.575747
$680$	15742.3	0.887780
$681$	−12975.6	−0.730139
$682$	−1973.76	−0.110820
$683$	12367.6	0.692875	0.346437	−	0.938073i	$-0.387391\pi$
$683$	12367.6	0.692875	0.346437	+	0.938073i	$0.387391\pi$
$684$	4748.26	0.265430
$685$	39445.0	2.20017
$686$	6582.66	0.366366
$687$	7864.71	0.436765
$688$	−10392.3	−0.575875
$689$	0	0
$690$	−7546.41	−0.416358
$691$	−1050.99	−0.0578605	−0.0289302	−	0.999581i	$-0.509210\pi$
$691$	−1050.99	−0.0578605	−0.0289302	+	0.999581i	$0.509210\pi$
$692$	10534.6	0.578709
$693$	−915.352	−0.0501751
$694$	−4400.35	−0.240685
$695$	8417.46	0.459414
$696$	7986.62	0.434960
$697$	11114.4	0.604002
$698$	493.486	0.0267603
$699$	13929.4	0.753729
$700$	5609.38	0.302878
$701$	−24294.1	−1.30895	−0.654476	−	0.756083i	$-0.727111\pi$
$701$	−24294.1	−1.30895	−0.654476	+	0.756083i	$0.727111\pi$
$702$	0	0
$703$	36698.8	1.96888
$704$	586.846	0.0314170
$705$	−22481.8	−1.20101
$706$	−10008.2	−0.533516
$707$	−5035.99	−0.267890
$708$	3484.89	0.184986
$709$	27465.9	1.45487	0.727436	−	0.686176i	$-0.240712\pi$
$709$	27465.9	1.45487	0.727436	+	0.686176i	$0.240712\pi$
$710$	−7100.66	−0.375328
$711$	−3020.47	−0.159320
$712$	4473.23	0.235451
$713$	−14304.9	−0.751365
$714$	−1712.92	−0.0897820
$715$	0	0
$716$	5880.90	0.306955
$717$	−20090.1	−1.04641
$718$	−16914.3	−0.879160
$719$	36433.5	1.88976	0.944882	−	0.327411i	$-0.106176\pi$
$719$	36433.5	1.88976	0.944882	+	0.327411i	$0.106176\pi$
$720$	3444.22	0.178276
$721$	−11997.4	−0.619704
$722$	391.183	0.0201639
$723$	−6904.40	−0.355156
$724$	−3006.78	−0.154345
$725$	15902.6	0.814629
$726$	4651.37	0.237780
$727$	551.608	0.0281403	0.0140701	−	0.999901i	$-0.495521\pi$
$727$	551.608	0.0281403	0.0140701	+	0.999901i	$0.495521\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−17054.0	−0.864654
$731$	22618.1	1.14440
$732$	−12573.4	−0.634874
$733$	20317.2	1.02379	0.511893	−	0.859049i	$-0.328945\pi$
$733$	20317.2	1.02379	0.511893	+	0.859049i	$0.328945\pi$
$734$	9541.10	0.479794
$735$	12935.8	0.649175
$736$	22622.4	1.13298
$737$	179.232	0.00895807
$738$	−2459.07	−0.122655
$739$	−29931.6	−1.48992	−0.744962	−	0.667107i	$-0.767532\pi$
$739$	−29931.6	−1.48992	−0.744962	+	0.667107i	$0.767532\pi$
$740$	41745.9	2.07380
$741$	0	0
$742$	7132.49	0.352887
$743$	24376.4	1.20361	0.601805	−	0.798643i	$-0.294448\pi$
$743$	24376.4	1.20361	0.601805	+	0.798643i	$0.294448\pi$
$744$	−6602.33	−0.325340
$745$	21235.5	1.04431
$746$	−2771.81	−0.136037
$747$	5119.17	0.250737
$748$	4301.11	0.210246
$749$	10104.1	0.492917
$750$	743.142	0.0361810
$751$	22692.2	1.10260	0.551298	−	0.834308i	$-0.314133\pi$
$751$	22692.2	1.10260	0.551298	+	0.834308i	$0.314133\pi$
$752$	12054.8	0.584567
$753$	−2484.00	−0.120215
$754$	0	0
$755$	42161.3	2.03232
$756$	−1341.46	−0.0645349
$757$	−33063.5	−1.58747	−0.793734	−	0.608265i	$-0.791866\pi$
$757$	−33063.5	−1.58747	−0.793734	+	0.608265i	$0.791866\pi$
$758$	−5286.22	−0.253304
$759$	−4706.15	−0.225062
$760$	24657.0	1.17685
$761$	−216.324	−0.0103045	−0.00515226	−	0.999987i	$-0.501640\pi$
$761$	−216.324	−0.0103045	−0.00515226	+	0.999987i	$0.501640\pi$
$762$	−1095.27	−0.0520699
$763$	−2767.24	−0.131299
$764$	−26344.0	−1.24750
$765$	−7496.10	−0.354277
$766$	−11467.5	−0.540912
$767$	0	0
$768$	6726.80	0.316058
$769$	19214.4	0.901025	0.450512	−	0.892770i	$-0.351241\pi$
$769$	19214.4	0.901025	0.450512	+	0.892770i	$0.351241\pi$
$770$	−2082.47	−0.0974638
$771$	−2654.29	−0.123984
$772$	1429.01	0.0666209
$773$	33175.6	1.54365	0.771826	−	0.635834i	$-0.219344\pi$
$773$	33175.6	1.54365	0.771826	+	0.635834i	$0.219344\pi$
$774$	−5004.25	−0.232395
$775$	−13146.2	−0.609325
$776$	24172.8	1.11824
$777$	−10368.0	−0.478700
$778$	−1693.54	−0.0780416
$779$	17408.4	0.800667
$780$	0	0
$781$	−4428.17	−0.202884
$782$	−8806.72	−0.402721
$783$	−3803.03	−0.173575
$784$	−6936.21	−0.315972
$785$	15886.3	0.722302
$786$	4715.17	0.213975
$787$	−14887.9	−0.674330	−0.337165	−	0.941446i	$-0.609468\pi$
$787$	−14887.9	−0.674330	−0.337165	+	0.941446i	$0.609468\pi$
$788$	1424.30	0.0643889
$789$	25031.8	1.12948
$790$	−6871.73	−0.309475
$791$	−5254.56	−0.236196
$792$	−2172.09	−0.0974518
$793$	0	0
$794$	6136.58	0.274281
$795$	31213.3	1.39248
$796$	−18332.8	−0.816319
$797$	12954.5	0.575751	0.287876	−	0.957668i	$-0.407051\pi$
$797$	12954.5	0.575751	0.287876	+	0.957668i	$0.407051\pi$
$798$	−2682.92	−0.119015
$799$	−26236.5	−1.16168
$800$	20790.0	0.918796
$801$	−2130.04	−0.0939591
$802$	183.295	0.00807028
$803$	−10635.4	−0.467389
$804$	262.667	0.0115218
$805$	−15092.8	−0.660810
$806$	0	0
$807$	−8287.68	−0.361512
$808$	−11950.2	−0.520305
$809$	8275.59	0.359647	0.179823	−	0.983699i	$-0.442447\pi$
$809$	8275.59	0.359647	0.179823	+	0.983699i	$0.442447\pi$
$810$	1658.51	0.0719435
$811$	26327.1	1.13991	0.569956	−	0.821675i	$-0.306960\pi$
$811$	26327.1	1.13991	0.569956	+	0.821675i	$0.306960\pi$
$812$	6998.09	0.302444
$813$	9349.63	0.403328
$814$	−7354.98	−0.316698
$815$	−44466.3	−1.91115
$816$	4019.43	0.172437
$817$	35426.3	1.51703
$818$	1599.16	0.0683538
$819$	0	0
$820$	19802.5	0.843333
$821$	−34439.0	−1.46398	−0.731992	−	0.681314i	$-0.761409\pi$
$821$	−34439.0	−1.46398	−0.731992	+	0.681314i	$0.761409\pi$
$822$	−10184.7	−0.432157
$823$	−13870.5	−0.587479	−0.293739	−	0.955886i	$-0.594900\pi$
$823$	−13870.5	−0.587479	−0.293739	+	0.955886i	$0.594900\pi$
$824$	−28469.3	−1.20361
$825$	−4324.96	−0.182516
$826$	−1969.07	−0.0829453
$827$	2132.30	0.0896583	0.0448292	−	0.998995i	$-0.485726\pi$
$827$	2132.30	0.0896583	0.0448292	+	0.998995i	$0.485726\pi$
$828$	−6896.92	−0.289474
$829$	−6212.39	−0.260272	−0.130136	−	0.991496i	$-0.541541\pi$
$829$	−6212.39	−0.260272	−0.130136	+	0.991496i	$0.541541\pi$
$830$	11646.4	0.487051
$831$	1506.18	0.0628746
$832$	0	0
$833$	15096.2	0.627913
$834$	−2173.40	−0.0902380
$835$	17797.8	0.737628
$836$	6736.76	0.278703
$837$	3143.87	0.129830
$838$	6902.81	0.284551
$839$	−4550.52	−0.187248	−0.0936242	−	0.995608i	$-0.529845\pi$
$839$	−4550.52	−0.187248	−0.0936242	+	0.995608i	$0.529845\pi$
$840$	−6965.99	−0.286130
$841$	−4549.47	−0.186538
$842$	18766.2	0.768085
$843$	−19822.7	−0.809880
$844$	10024.2	0.408822
$845$	0	0
$846$	5804.83	0.235903
$847$	9302.74	0.377386
$848$	−16736.7	−0.677759
$849$	13106.9	0.529834
$850$	−8093.38	−0.326589
$851$	−53305.6	−2.14723
$852$	−6489.53	−0.260948
$853$	12262.8	0.492228	0.246114	−	0.969241i	$-0.420846\pi$
$853$	12262.8	0.492228	0.246114	+	0.969241i	$0.420846\pi$
$854$	7104.40	0.284669
$855$	−11741.0	−0.469631
$856$	23976.5	0.957362
$857$	−34949.1	−1.39304	−0.696521	−	0.717536i	$-0.745270\pi$
$857$	−34949.1	−1.39304	−0.696521	+	0.717536i	$0.745270\pi$
$858$	0	0
$859$	−21762.1	−0.864394	−0.432197	−	0.901779i	$-0.642261\pi$
$859$	−21762.1	−0.864394	−0.432197	+	0.901779i	$0.642261\pi$
$860$	40298.4	1.59786
$861$	−4918.14	−0.194669
$862$	3026.48	0.119585
$863$	−19811.5	−0.781450	−0.390725	−	0.920507i	$-0.627776\pi$
$863$	−19811.5	−0.781450	−0.390725	+	0.920507i	$0.627776\pi$
$864$	−4971.83	−0.195770
$865$	−26049.0	−1.02392
$866$	−17654.0	−0.692735
$867$	5991.00	0.234677
$868$	−5785.14	−0.226222
$869$	−4285.40	−0.167287
$870$	−8652.09	−0.337165
$871$	0	0
$872$	−6566.54	−0.255013
$873$	−11510.5	−0.446243
$874$	−13793.8	−0.533848
$875$	1486.28	0.0574235
$876$	−15586.2	−0.601153
$877$	−25716.8	−0.990186	−0.495093	−	0.868840i	$-0.664866\pi$
$877$	−25716.8	−0.990186	−0.495093	+	0.868840i	$0.664866\pi$
$878$	−13875.4	−0.533339
$879$	−16044.4	−0.615658
$880$	4886.61	0.187190
$881$	34709.6	1.32735	0.663676	−	0.748020i	$-0.268995\pi$
$881$	34709.6	1.32735	0.663676	+	0.748020i	$0.268995\pi$
$882$	−3340.03	−0.127511
$883$	−3848.68	−0.146680	−0.0733400	−	0.997307i	$-0.523366\pi$
$883$	−3848.68	−0.146680	−0.0733400	+	0.997307i	$0.523366\pi$
$884$	0	0
$885$	−8617.09	−0.327300
$886$	7120.06	0.269981
$887$	32804.8	1.24180	0.620900	−	0.783890i	$-0.286767\pi$
$887$	32804.8	1.24180	0.620900	+	0.783890i	$0.286767\pi$
$888$	−24602.8	−0.929748
$889$	−2190.53	−0.0826412
$890$	−4845.95	−0.182513
$891$	1034.29	0.0388891
$892$	−816.311	−0.0306414
$893$	−41093.8	−1.53992
$894$	−5483.03	−0.205123
$895$	−14541.7	−0.543101
$896$	11247.6	0.419371
$897$	0	0
$898$	−12852.5	−0.477610
$899$	−16400.8	−0.608452
$900$	−6338.27	−0.234751
$901$	36426.2	1.34687
$902$	−3488.90	−0.128789
$903$	−10008.5	−0.368840
$904$	−12468.8	−0.458748
$905$	7434.86	0.273086
$906$	−10886.1	−0.399189
$907$	−22262.2	−0.814999	−0.407500	−	0.913205i	$-0.633599\pi$
$907$	−22262.2	−0.814999	−0.407500	+	0.913205i	$0.633599\pi$
$908$	−26979.4	−0.986060
$909$	5690.38	0.207633
$910$	0	0
$911$	−13515.3	−0.491528	−0.245764	−	0.969330i	$-0.579039\pi$
$911$	−13515.3	−0.491528	−0.245764	+	0.969330i	$0.579039\pi$
$912$	6295.58	0.228583
$913$	7263.01	0.263275
$914$	4696.62	0.169968
$915$	31090.4	1.12330
$916$	16352.7	0.589855
$917$	9430.33	0.339604
$918$	1935.50	0.0695871
$919$	26600.6	0.954811	0.477405	−	0.878683i	$-0.341577\pi$
$919$	26600.6	0.954811	0.477405	+	0.878683i	$0.341577\pi$
$920$	−35814.6	−1.28345
$921$	−13507.4	−0.483260
$922$	20012.0	0.714816
$923$	0	0
$924$	−1903.24	−0.0677620
$925$	−48987.9	−1.74131
$926$	14695.5	0.521515
$927$	13556.4	0.480313
$928$	25936.9	0.917480
$929$	−32887.7	−1.16148	−0.580738	−	0.814090i	$-0.697236\pi$
$929$	−32887.7	−1.16148	−0.580738	+	0.814090i	$0.697236\pi$
$930$	7152.46	0.252192
$931$	23644.9	0.832363
$932$	28962.6	1.01792
$933$	22341.6	0.783956
$934$	17593.1	0.616341
$935$	−10635.4	−0.371993
$936$	0	0
$937$	−9261.78	−0.322913	−0.161456	−	0.986880i	$-0.551619\pi$
$937$	−9261.78	−0.322913	−0.161456	+	0.986880i	$0.551619\pi$
$938$	−148.415	−0.00516623
$939$	19526.8	0.678629
$940$	−46745.3	−1.62198
$941$	12054.9	0.417619	0.208810	−	0.977956i	$-0.433041\pi$
$941$	12054.9	0.417619	0.208810	+	0.977956i	$0.433041\pi$
$942$	−4101.86	−0.141875
$943$	−25285.9	−0.873195
$944$	4620.51	0.159306
$945$	3317.03	0.114183
$946$	−7099.96	−0.244016
$947$	20221.4	0.693885	0.346942	−	0.937886i	$-0.387220\pi$
$947$	20221.4	0.693885	0.346942	+	0.937886i	$0.387220\pi$
$948$	−6280.30	−0.215163
$949$	0	0
$950$	−12676.5	−0.432928
$951$	7396.72	0.252213
$952$	−8129.36	−0.276758
$953$	−20331.5	−0.691083	−0.345542	−	0.938403i	$-0.612305\pi$
$953$	−20331.5	−0.691083	−0.345542	+	0.938403i	$0.612305\pi$
$954$	−8059.30	−0.273511
$955$	65140.8	2.20723
$956$	−41772.2	−1.41319
$957$	−5395.68	−0.182254
$958$	−16250.4	−0.548045
$959$	−20369.4	−0.685885
$960$	−2126.60	−0.0714955
$961$	−16232.9	−0.544891
$962$	0	0
$963$	−11417.0	−0.382044
$964$	−14356.0	−0.479641
$965$	−3533.53	−0.117874
$966$	3896.98	0.129796
$967$	−11082.2	−0.368541	−0.184270	−	0.982876i	$-0.558992\pi$
$967$	−11082.2	−0.368541	−0.184270	+	0.982876i	$0.558992\pi$
$968$	22075.0	0.732973
$969$	−13701.9	−0.454249
$970$	−26186.9	−0.866816
$971$	−36694.1	−1.21274	−0.606369	−	0.795184i	$-0.707374\pi$
$971$	−36694.1	−1.21274	−0.606369	+	0.795184i	$0.707374\pi$
$972$	1515.77	0.0500189
$973$	−4346.79	−0.143219
$974$	17806.2	0.585778
$975$	0	0
$976$	−16670.8	−0.546740
$977$	17155.3	0.561767	0.280883	−	0.959742i	$-0.409373\pi$
$977$	17155.3	0.561767	0.280883	+	0.959742i	$0.409373\pi$
$978$	11481.2	0.375387
$979$	−3022.07	−0.0986575
$980$	26896.7	0.876718
$981$	3126.82	0.101765
$982$	−979.544	−0.0318315
$983$	−38419.7	−1.24659	−0.623296	−	0.781986i	$-0.714207\pi$
$983$	−38419.7	−1.24659	−0.623296	+	0.781986i	$0.714207\pi$
$984$	−11670.5	−0.378093
$985$	−3521.86	−0.113925
$986$	−10097.1	−0.326121
$987$	11609.7	0.374407
$988$	0	0
$989$	−51457.3	−1.65445
$990$	2353.07	0.0755410
$991$	51728.9	1.65815	0.829073	−	0.559140i	$-0.188869\pi$
$991$	51728.9	1.65815	0.829073	+	0.559140i	$0.188869\pi$
$992$	−21441.4	−0.686255
$993$	−12344.5	−0.394503
$994$	3666.79	0.117006
$995$	45331.6	1.44433
$996$	10644.0	0.338623
$997$	−26846.8	−0.852805	−0.426403	−	0.904533i	$-0.640219\pi$
$997$	−26846.8	−0.852805	−0.426403	+	0.904533i	$0.640219\pi$
$998$	−2476.65	−0.0785541
$999$	11715.2	0.371025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.j.1.3		4
3.2	odd	2		1521.4.a.x.1.2		4
13.5	odd	4		39.4.b.a.25.2	✓	4
13.8	odd	4		39.4.b.a.25.3	yes	4
13.12	even	2	inner	507.4.a.j.1.2		4
39.5	even	4		117.4.b.d.64.3		4
39.8	even	4		117.4.b.d.64.2		4
39.38	odd	2		1521.4.a.x.1.3		4
52.31	even	4		624.4.c.e.337.1		4
52.47	even	4		624.4.c.e.337.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.b.a.25.2	✓	4	13.5	odd	4
39.4.b.a.25.3	yes	4	13.8	odd	4
117.4.b.d.64.2		4	39.8	even	4
117.4.b.d.64.3		4	39.5	even	4
507.4.a.j.1.2		4	13.12	even	2	inner
507.4.a.j.1.3		4	1.1	even	1	trivial
624.4.c.e.337.1		4	52.31	even	4
624.4.c.e.337.4		4	52.47	even	4
1521.4.a.x.1.2		4	3.2	odd	2
1521.4.a.x.1.3		4	39.38	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

\(f(q)\)	\(=\)	\(q+1.32750 q^{2} -3.00000 q^{3} -6.23774 q^{4} +15.4241 q^{5} -3.98251 q^{6} -7.96501 q^{7} -18.9006 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+1.32750 q^{2} -3.00000 q^{3} -6.23774 q^{4} +15.4241 q^{5} -3.98251 q^{6} -7.96501 q^{7} -18.9006 q^{8} +9.00000 q^{9} +20.4755 q^{10} +12.7691 q^{11} +18.7132 q^{12} -10.5736 q^{14} -46.2722 q^{15} +24.8113 q^{16} -54.0000 q^{17} +11.9475 q^{18} -84.5794 q^{19} -96.2113 q^{20} +23.8950 q^{21} +16.9510 q^{22} +122.853 q^{23} +56.7019 q^{24} +112.902 q^{25} -27.0000 q^{27} +49.6837 q^{28} +140.853 q^{29} -61.4264 q^{30} -116.439 q^{31} +184.142 q^{32} -38.3072 q^{33} -71.6851 q^{34} -122.853 q^{35} -56.1397 q^{36} -433.898 q^{37} -112.279 q^{38} -291.525 q^{40} -205.823 q^{41} +31.7207 q^{42} -418.853 q^{43} -79.6501 q^{44} +138.817 q^{45} +163.087 q^{46} +485.861 q^{47} -74.4339 q^{48} -279.559 q^{49} +149.877 q^{50} +162.000 q^{51} -674.559 q^{53} -35.8425 q^{54} +196.951 q^{55} +150.544 q^{56} +253.738 q^{57} +186.982 q^{58} +186.226 q^{59} +288.634 q^{60} -671.902 q^{61} -154.574 q^{62} -71.6851 q^{63} +45.9584 q^{64} -50.8529 q^{66} +14.0364 q^{67} +336.838 q^{68} -368.559 q^{69} -163.087 q^{70} -346.789 q^{71} -170.106 q^{72} -832.900 q^{73} -576.000 q^{74} -338.706 q^{75} +527.584 q^{76} -101.706 q^{77} -335.608 q^{79} +382.691 q^{80} +81.0000 q^{81} -273.230 q^{82} +568.797 q^{83} -149.051 q^{84} -832.900 q^{85} -556.028 q^{86} -422.559 q^{87} -241.343 q^{88} -236.671 q^{89} +184.279 q^{90} -766.324 q^{92} +349.318 q^{93} +644.981 q^{94} -1304.56 q^{95} -552.426 q^{96} -1278.94 q^{97} -371.115 q^{98} +114.922 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 26 q^{4} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 + 26 * q^4 + 36 * q^9	\( 4 q - 12 q^{3} + 26 q^{4} + 36 q^{9} - 20 q^{10} - 78 q^{12} - 348 q^{14} + 354 q^{16} - 216 q^{17} - 136 q^{22} - 120 q^{23} + 44 q^{25} - 108 q^{27} - 48 q^{29} + 60 q^{30} + 120 q^{35} + 234 q^{36} + 468 q^{38} - 1268 q^{40} + 1044 q^{42} - 1064 q^{43} - 1062 q^{48} + 716 q^{49} + 648 q^{51} - 864 q^{53} + 584 q^{55} - 3372 q^{56} - 2280 q^{61} - 924 q^{62} + 1050 q^{64} + 408 q^{66} - 1404 q^{68} + 360 q^{69} - 2304 q^{74} - 132 q^{75} + 816 q^{77} + 288 q^{79} + 324 q^{81} + 28 q^{82} + 144 q^{87} - 2392 q^{88} - 180 q^{90} - 8568 q^{92} + 6656 q^{94} - 3384 q^{95}+O(q^{100}) \) 4 * q - 12 * q^3 + 26 * q^4 + 36 * q^9 - 20 * q^10 - 78 * q^12 - 348 * q^14 + 354 * q^16 - 216 * q^17 - 136 * q^22 - 120 * q^23 + 44 * q^25 - 108 * q^27 - 48 * q^29 + 60 * q^30 + 120 * q^35 + 234 * q^36 + 468 * q^38 - 1268 * q^40 + 1044 * q^42 - 1064 * q^43 - 1062 * q^48 + 716 * q^49 + 648 * q^51 - 864 * q^53 + 584 * q^55 - 3372 * q^56 - 2280 * q^61 - 924 * q^62 + 1050 * q^64 + 408 * q^66 - 1404 * q^68 + 360 * q^69 - 2304 * q^74 - 132 * q^75 + 816 * q^77 + 288 * q^79 + 324 * q^81 + 28 * q^82 + 144 * q^87 - 2392 * q^88 - 180 * q^90 - 8568 * q^92 + 6656 * q^94 - 3384 * q^95

Introduction

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