Properties

Label 462.4.a.f.1.1
Level $462$
Weight $4$
Character 462.1
Self dual yes
Analytic conductor $27.259$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 462.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.2588824227\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 462.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +1.00000 q^{5} -6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +1.00000 q^{5} -6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +2.00000 q^{10} -11.0000 q^{11} -12.0000 q^{12} -43.0000 q^{13} -14.0000 q^{14} -3.00000 q^{15} +16.0000 q^{16} +100.000 q^{17} +18.0000 q^{18} -87.0000 q^{19} +4.00000 q^{20} +21.0000 q^{21} -22.0000 q^{22} -58.0000 q^{23} -24.0000 q^{24} -124.000 q^{25} -86.0000 q^{26} -27.0000 q^{27} -28.0000 q^{28} -223.000 q^{29} -6.00000 q^{30} +88.0000 q^{31} +32.0000 q^{32} +33.0000 q^{33} +200.000 q^{34} -7.00000 q^{35} +36.0000 q^{36} +37.0000 q^{37} -174.000 q^{38} +129.000 q^{39} +8.00000 q^{40} +128.000 q^{41} +42.0000 q^{42} -458.000 q^{43} -44.0000 q^{44} +9.00000 q^{45} -116.000 q^{46} -341.000 q^{47} -48.0000 q^{48} +49.0000 q^{49} -248.000 q^{50} -300.000 q^{51} -172.000 q^{52} -342.000 q^{53} -54.0000 q^{54} -11.0000 q^{55} -56.0000 q^{56} +261.000 q^{57} -446.000 q^{58} -105.000 q^{59} -12.0000 q^{60} +190.000 q^{61} +176.000 q^{62} -63.0000 q^{63} +64.0000 q^{64} -43.0000 q^{65} +66.0000 q^{66} -579.000 q^{67} +400.000 q^{68} +174.000 q^{69} -14.0000 q^{70} +128.000 q^{71} +72.0000 q^{72} -161.000 q^{73} +74.0000 q^{74} +372.000 q^{75} -348.000 q^{76} +77.0000 q^{77} +258.000 q^{78} -396.000 q^{79} +16.0000 q^{80} +81.0000 q^{81} +256.000 q^{82} -420.000 q^{83} +84.0000 q^{84} +100.000 q^{85} -916.000 q^{86} +669.000 q^{87} -88.0000 q^{88} -798.000 q^{89} +18.0000 q^{90} +301.000 q^{91} -232.000 q^{92} -264.000 q^{93} -682.000 q^{94} -87.0000 q^{95} -96.0000 q^{96} +1414.00 q^{97} +98.0000 q^{98} -99.0000 q^{99} +O(q^{100})\)

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 \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) 1.00000 0.0894427 0.0447214 0.998999i \(-0.485760\pi\)
0.0447214 + 0.998999i \(0.485760\pi\)
\(6\) −6.00000 −0.408248
\(7\) −7.00000 −0.377964
\(8\) 8.00000 0.353553
\(9\) 9.00000 0.333333
\(10\) 2.00000 0.0632456
\(11\) −11.0000 −0.301511
\(12\) −12.0000 −0.288675
\(13\) −43.0000 −0.917389 −0.458694 0.888594i \(-0.651683\pi\)
−0.458694 + 0.888594i \(0.651683\pi\)
\(14\) −14.0000 −0.267261
\(15\) −3.00000 −0.0516398
\(16\) 16.0000 0.250000
\(17\) 100.000 1.42668 0.713340 0.700818i \(-0.247182\pi\)
0.713340 + 0.700818i \(0.247182\pi\)
\(18\) 18.0000 0.235702
\(19\) −87.0000 −1.05048 −0.525241 0.850953i \(-0.676025\pi\)
−0.525241 + 0.850953i \(0.676025\pi\)
\(20\) 4.00000 0.0447214
\(21\) 21.0000 0.218218
\(22\) −22.0000 −0.213201
\(23\) −58.0000 −0.525819 −0.262909 0.964821i \(-0.584682\pi\)
−0.262909 + 0.964821i \(0.584682\pi\)
\(24\) −24.0000 −0.204124
\(25\) −124.000 −0.992000
\(26\) −86.0000 −0.648692
\(27\) −27.0000 −0.192450
\(28\) −28.0000 −0.188982
\(29\) −223.000 −1.42793 −0.713967 0.700180i \(-0.753103\pi\)
−0.713967 + 0.700180i \(0.753103\pi\)
\(30\) −6.00000 −0.0365148
\(31\) 88.0000 0.509847 0.254924 0.966961i \(-0.417950\pi\)
0.254924 + 0.966961i \(0.417950\pi\)
\(32\) 32.0000 0.176777
\(33\) 33.0000 0.174078
\(34\) 200.000 1.00882
\(35\) −7.00000 −0.0338062
\(36\) 36.0000 0.166667
\(37\) 37.0000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) −174.000 −0.742803
\(39\) 129.000 0.529655
\(40\) 8.00000 0.0316228
\(41\) 128.000 0.487567 0.243783 0.969830i \(-0.421611\pi\)
0.243783 + 0.969830i \(0.421611\pi\)
\(42\) 42.0000 0.154303
\(43\) −458.000 −1.62429 −0.812144 0.583458i \(-0.801699\pi\)
−0.812144 + 0.583458i \(0.801699\pi\)
\(44\) −44.0000 −0.150756
\(45\) 9.00000 0.0298142
\(46\) −116.000 −0.371810
\(47\) −341.000 −1.05830 −0.529149 0.848529i \(-0.677489\pi\)
−0.529149 + 0.848529i \(0.677489\pi\)
\(48\) −48.0000 −0.144338
\(49\) 49.0000 0.142857
\(50\) −248.000 −0.701450
\(51\) −300.000 −0.823694
\(52\) −172.000 −0.458694
\(53\) −342.000 −0.886364 −0.443182 0.896432i \(-0.646151\pi\)
−0.443182 + 0.896432i \(0.646151\pi\)
\(54\) −54.0000 −0.136083
\(55\) −11.0000 −0.0269680
\(56\) −56.0000 −0.133631
\(57\) 261.000 0.606496
\(58\) −446.000 −1.00970
\(59\) −105.000 −0.231692 −0.115846 0.993267i \(-0.536958\pi\)
−0.115846 + 0.993267i \(0.536958\pi\)
\(60\) −12.0000 −0.0258199
\(61\) 190.000 0.398803 0.199402 0.979918i \(-0.436100\pi\)
0.199402 + 0.979918i \(0.436100\pi\)
\(62\) 176.000 0.360516
\(63\) −63.0000 −0.125988
\(64\) 64.0000 0.125000
\(65\) −43.0000 −0.0820537
\(66\) 66.0000 0.123091
\(67\) −579.000 −1.05576 −0.527881 0.849318i \(-0.677013\pi\)
−0.527881 + 0.849318i \(0.677013\pi\)
\(68\) 400.000 0.713340
\(69\) 174.000 0.303582
\(70\) −14.0000 −0.0239046
\(71\) 128.000 0.213955 0.106978 0.994261i \(-0.465883\pi\)
0.106978 + 0.994261i \(0.465883\pi\)
\(72\) 72.0000 0.117851
\(73\) −161.000 −0.258132 −0.129066 0.991636i \(-0.541198\pi\)
−0.129066 + 0.991636i \(0.541198\pi\)
\(74\) 74.0000 0.116248
\(75\) 372.000 0.572731
\(76\) −348.000 −0.525241
\(77\) 77.0000 0.113961
\(78\) 258.000 0.374522
\(79\) −396.000 −0.563968 −0.281984 0.959419i \(-0.590993\pi\)
−0.281984 + 0.959419i \(0.590993\pi\)
\(80\) 16.0000 0.0223607
\(81\) 81.0000 0.111111
\(82\) 256.000 0.344762
\(83\) −420.000 −0.555434 −0.277717 0.960663i \(-0.589578\pi\)
−0.277717 + 0.960663i \(0.589578\pi\)
\(84\) 84.0000 0.109109
\(85\) 100.000 0.127606
\(86\) −916.000 −1.14854
\(87\) 669.000 0.824418
\(88\) −88.0000 −0.106600
\(89\) −798.000 −0.950425 −0.475213 0.879871i \(-0.657629\pi\)
−0.475213 + 0.879871i \(0.657629\pi\)
\(90\) 18.0000 0.0210819
\(91\) 301.000 0.346740
\(92\) −232.000 −0.262909
\(93\) −264.000 −0.294360
\(94\) −682.000 −0.748329
\(95\) −87.0000 −0.0939580
\(96\) −96.0000 −0.102062
\(97\) 1414.00 1.48010 0.740051 0.672550i \(-0.234801\pi\)
0.740051 + 0.672550i \(0.234801\pi\)
\(98\) 98.0000 0.101015
\(99\) −99.0000 −0.100504
\(100\) −496.000 −0.496000
\(101\) −1764.00 −1.73787 −0.868933 0.494929i \(-0.835194\pi\)
−0.868933 + 0.494929i \(0.835194\pi\)
\(102\) −600.000 −0.582440
\(103\) 978.000 0.935584 0.467792 0.883838i \(-0.345050\pi\)
0.467792 + 0.883838i \(0.345050\pi\)
\(104\) −344.000 −0.324346
\(105\) 21.0000 0.0195180
\(106\) −684.000 −0.626754
\(107\) 481.000 0.434580 0.217290 0.976107i \(-0.430278\pi\)
0.217290 + 0.976107i \(0.430278\pi\)
\(108\) −108.000 −0.0962250
\(109\) −284.000 −0.249562 −0.124781 0.992184i \(-0.539823\pi\)
−0.124781 + 0.992184i \(0.539823\pi\)
\(110\) −22.0000 −0.0190693
\(111\) −111.000 −0.0949158
\(112\) −112.000 −0.0944911
\(113\) 1498.00 1.24708 0.623540 0.781792i \(-0.285694\pi\)
0.623540 + 0.781792i \(0.285694\pi\)
\(114\) 522.000 0.428858
\(115\) −58.0000 −0.0470307
\(116\) −892.000 −0.713967
\(117\) −387.000 −0.305796
\(118\) −210.000 −0.163831
\(119\) −700.000 −0.539234
\(120\) −24.0000 −0.0182574
\(121\) 121.000 0.0909091
\(122\) 380.000 0.281997
\(123\) −384.000 −0.281497
\(124\) 352.000 0.254924
\(125\) −249.000 −0.178170
\(126\) −126.000 −0.0890871
\(127\) −616.000 −0.430403 −0.215201 0.976570i \(-0.569041\pi\)
−0.215201 + 0.976570i \(0.569041\pi\)
\(128\) 128.000 0.0883883
\(129\) 1374.00 0.937783
\(130\) −86.0000 −0.0580208
\(131\) 2748.00 1.83278 0.916389 0.400289i \(-0.131090\pi\)
0.916389 + 0.400289i \(0.131090\pi\)
\(132\) 132.000 0.0870388
\(133\) 609.000 0.397045
\(134\) −1158.00 −0.746537
\(135\) −27.0000 −0.0172133
\(136\) 800.000 0.504408
\(137\) 2184.00 1.36198 0.680992 0.732291i \(-0.261549\pi\)
0.680992 + 0.732291i \(0.261549\pi\)
\(138\) 348.000 0.214665
\(139\) 908.000 0.554069 0.277034 0.960860i \(-0.410648\pi\)
0.277034 + 0.960860i \(0.410648\pi\)
\(140\) −28.0000 −0.0169031
\(141\) 1023.00 0.611008
\(142\) 256.000 0.151289
\(143\) 473.000 0.276603
\(144\) 144.000 0.0833333
\(145\) −223.000 −0.127718
\(146\) −322.000 −0.182527
\(147\) −147.000 −0.0824786
\(148\) 148.000 0.0821995
\(149\) 421.000 0.231474 0.115737 0.993280i \(-0.463077\pi\)
0.115737 + 0.993280i \(0.463077\pi\)
\(150\) 744.000 0.404982
\(151\) 948.000 0.510908 0.255454 0.966821i \(-0.417775\pi\)
0.255454 + 0.966821i \(0.417775\pi\)
\(152\) −696.000 −0.371402
\(153\) 900.000 0.475560
\(154\) 154.000 0.0805823
\(155\) 88.0000 0.0456021
\(156\) 516.000 0.264827
\(157\) 2368.00 1.20374 0.601869 0.798595i \(-0.294423\pi\)
0.601869 + 0.798595i \(0.294423\pi\)
\(158\) −792.000 −0.398786
\(159\) 1026.00 0.511743
\(160\) 32.0000 0.0158114
\(161\) 406.000 0.198741
\(162\) 162.000 0.0785674
\(163\) −2603.00 −1.25081 −0.625407 0.780299i \(-0.715067\pi\)
−0.625407 + 0.780299i \(0.715067\pi\)
\(164\) 512.000 0.243783
\(165\) 33.0000 0.0155700
\(166\) −840.000 −0.392751
\(167\) −1402.00 −0.649641 −0.324820 0.945776i \(-0.605304\pi\)
−0.324820 + 0.945776i \(0.605304\pi\)
\(168\) 168.000 0.0771517
\(169\) −348.000 −0.158398
\(170\) 200.000 0.0902312
\(171\) −783.000 −0.350161
\(172\) −1832.00 −0.812144
\(173\) 836.000 0.367398 0.183699 0.982983i \(-0.441193\pi\)
0.183699 + 0.982983i \(0.441193\pi\)
\(174\) 1338.00 0.582951
\(175\) 868.000 0.374941
\(176\) −176.000 −0.0753778
\(177\) 315.000 0.133768
\(178\) −1596.00 −0.672052
\(179\) −3292.00 −1.37461 −0.687306 0.726368i \(-0.741207\pi\)
−0.687306 + 0.726368i \(0.741207\pi\)
\(180\) 36.0000 0.0149071
\(181\) 4838.00 1.98677 0.993386 0.114823i \(-0.0366302\pi\)
0.993386 + 0.114823i \(0.0366302\pi\)
\(182\) 602.000 0.245182
\(183\) −570.000 −0.230249
\(184\) −464.000 −0.185905
\(185\) 37.0000 0.0147043
\(186\) −528.000 −0.208144
\(187\) −1100.00 −0.430160
\(188\) −1364.00 −0.529149
\(189\) 189.000 0.0727393
\(190\) −174.000 −0.0664384
\(191\) −1232.00 −0.466725 −0.233362 0.972390i \(-0.574973\pi\)
−0.233362 + 0.972390i \(0.574973\pi\)
\(192\) −192.000 −0.0721688
\(193\) 2366.00 0.882427 0.441213 0.897402i \(-0.354548\pi\)
0.441213 + 0.897402i \(0.354548\pi\)
\(194\) 2828.00 1.04659
\(195\) 129.000 0.0473738
\(196\) 196.000 0.0714286
\(197\) 318.000 0.115008 0.0575040 0.998345i \(-0.481686\pi\)
0.0575040 + 0.998345i \(0.481686\pi\)
\(198\) −198.000 −0.0710669
\(199\) −806.000 −0.287115 −0.143557 0.989642i \(-0.545854\pi\)
−0.143557 + 0.989642i \(0.545854\pi\)
\(200\) −992.000 −0.350725
\(201\) 1737.00 0.609545
\(202\) −3528.00 −1.22886
\(203\) 1561.00 0.539708
\(204\) −1200.00 −0.411847
\(205\) 128.000 0.0436093
\(206\) 1956.00 0.661558
\(207\) −522.000 −0.175273
\(208\) −688.000 −0.229347
\(209\) 957.000 0.316732
\(210\) 42.0000 0.0138013
\(211\) 4462.00 1.45581 0.727907 0.685676i \(-0.240493\pi\)
0.727907 + 0.685676i \(0.240493\pi\)
\(212\) −1368.00 −0.443182
\(213\) −384.000 −0.123527
\(214\) 962.000 0.307294
\(215\) −458.000 −0.145281
\(216\) −216.000 −0.0680414
\(217\) −616.000 −0.192704
\(218\) −568.000 −0.176467
\(219\) 483.000 0.149032
\(220\) −44.0000 −0.0134840
\(221\) −4300.00 −1.30882
\(222\) −222.000 −0.0671156
\(223\) 3802.00 1.14171 0.570854 0.821052i \(-0.306612\pi\)
0.570854 + 0.821052i \(0.306612\pi\)
\(224\) −224.000 −0.0668153
\(225\) −1116.00 −0.330667
\(226\) 2996.00 0.881818
\(227\) −3370.00 −0.985351 −0.492676 0.870213i \(-0.663981\pi\)
−0.492676 + 0.870213i \(0.663981\pi\)
\(228\) 1044.00 0.303248
\(229\) −898.000 −0.259133 −0.129567 0.991571i \(-0.541359\pi\)
−0.129567 + 0.991571i \(0.541359\pi\)
\(230\) −116.000 −0.0332557
\(231\) −231.000 −0.0657952
\(232\) −1784.00 −0.504851
\(233\) −462.000 −0.129900 −0.0649498 0.997889i \(-0.520689\pi\)
−0.0649498 + 0.997889i \(0.520689\pi\)
\(234\) −774.000 −0.216231
\(235\) −341.000 −0.0946570
\(236\) −420.000 −0.115846
\(237\) 1188.00 0.325607
\(238\) −1400.00 −0.381296
\(239\) −5665.00 −1.53322 −0.766608 0.642116i \(-0.778057\pi\)
−0.766608 + 0.642116i \(0.778057\pi\)
\(240\) −48.0000 −0.0129099
\(241\) 1593.00 0.425785 0.212892 0.977076i \(-0.431712\pi\)
0.212892 + 0.977076i \(0.431712\pi\)
\(242\) 242.000 0.0642824
\(243\) −243.000 −0.0641500
\(244\) 760.000 0.199402
\(245\) 49.0000 0.0127775
\(246\) −768.000 −0.199048
\(247\) 3741.00 0.963701
\(248\) 704.000 0.180258
\(249\) 1260.00 0.320680
\(250\) −498.000 −0.125985
\(251\) 4545.00 1.14294 0.571470 0.820623i \(-0.306373\pi\)
0.571470 + 0.820623i \(0.306373\pi\)
\(252\) −252.000 −0.0629941
\(253\) 638.000 0.158540
\(254\) −1232.00 −0.304341
\(255\) −300.000 −0.0736734
\(256\) 256.000 0.0625000
\(257\) −2859.00 −0.693928 −0.346964 0.937878i \(-0.612787\pi\)
−0.346964 + 0.937878i \(0.612787\pi\)
\(258\) 2748.00 0.663112
\(259\) −259.000 −0.0621370
\(260\) −172.000 −0.0410269
\(261\) −2007.00 −0.475978
\(262\) 5496.00 1.29597
\(263\) −6279.00 −1.47217 −0.736083 0.676891i \(-0.763327\pi\)
−0.736083 + 0.676891i \(0.763327\pi\)
\(264\) 264.000 0.0615457
\(265\) −342.000 −0.0792788
\(266\) 1218.00 0.280753
\(267\) 2394.00 0.548728
\(268\) −2316.00 −0.527881
\(269\) −6270.00 −1.42115 −0.710574 0.703623i \(-0.751565\pi\)
−0.710574 + 0.703623i \(0.751565\pi\)
\(270\) −54.0000 −0.0121716
\(271\) 6721.00 1.50654 0.753269 0.657713i \(-0.228476\pi\)
0.753269 + 0.657713i \(0.228476\pi\)
\(272\) 1600.00 0.356670
\(273\) −903.000 −0.200191
\(274\) 4368.00 0.963068
\(275\) 1364.00 0.299099
\(276\) 696.000 0.151791
\(277\) −5388.00 −1.16871 −0.584357 0.811497i \(-0.698653\pi\)
−0.584357 + 0.811497i \(0.698653\pi\)
\(278\) 1816.00 0.391786
\(279\) 792.000 0.169949
\(280\) −56.0000 −0.0119523
\(281\) 4765.00 1.01159 0.505794 0.862654i \(-0.331200\pi\)
0.505794 + 0.862654i \(0.331200\pi\)
\(282\) 2046.00 0.432048
\(283\) −903.000 −0.189674 −0.0948371 0.995493i \(-0.530233\pi\)
−0.0948371 + 0.995493i \(0.530233\pi\)
\(284\) 512.000 0.106978
\(285\) 261.000 0.0542467
\(286\) 946.000 0.195588
\(287\) −896.000 −0.184283
\(288\) 288.000 0.0589256
\(289\) 5087.00 1.03542
\(290\) −446.000 −0.0903104
\(291\) −4242.00 −0.854538
\(292\) −644.000 −0.129066
\(293\) 672.000 0.133989 0.0669943 0.997753i \(-0.478659\pi\)
0.0669943 + 0.997753i \(0.478659\pi\)
\(294\) −294.000 −0.0583212
\(295\) −105.000 −0.0207232
\(296\) 296.000 0.0581238
\(297\) 297.000 0.0580259
\(298\) 842.000 0.163677
\(299\) 2494.00 0.482380
\(300\) 1488.00 0.286366
\(301\) 3206.00 0.613923
\(302\) 1896.00 0.361267
\(303\) 5292.00 1.00336
\(304\) −1392.00 −0.262621
\(305\) 190.000 0.0356701
\(306\) 1800.00 0.336272
\(307\) −1876.00 −0.348759 −0.174379 0.984679i \(-0.555792\pi\)
−0.174379 + 0.984679i \(0.555792\pi\)
\(308\) 308.000 0.0569803
\(309\) −2934.00 −0.540160
\(310\) 176.000 0.0322456
\(311\) −4688.00 −0.854766 −0.427383 0.904071i \(-0.640564\pi\)
−0.427383 + 0.904071i \(0.640564\pi\)
\(312\) 1032.00 0.187261
\(313\) 4912.00 0.887037 0.443519 0.896265i \(-0.353730\pi\)
0.443519 + 0.896265i \(0.353730\pi\)
\(314\) 4736.00 0.851172
\(315\) −63.0000 −0.0112687
\(316\) −1584.00 −0.281984
\(317\) −558.000 −0.0988656 −0.0494328 0.998777i \(-0.515741\pi\)
−0.0494328 + 0.998777i \(0.515741\pi\)
\(318\) 2052.00 0.361857
\(319\) 2453.00 0.430538
\(320\) 64.0000 0.0111803
\(321\) −1443.00 −0.250905
\(322\) 812.000 0.140531
\(323\) −8700.00 −1.49870
\(324\) 324.000 0.0555556
\(325\) 5332.00 0.910050
\(326\) −5206.00 −0.884459
\(327\) 852.000 0.144085
\(328\) 1024.00 0.172381
\(329\) 2387.00 0.399999
\(330\) 66.0000 0.0110096
\(331\) 4660.00 0.773827 0.386914 0.922116i \(-0.373541\pi\)
0.386914 + 0.922116i \(0.373541\pi\)
\(332\) −1680.00 −0.277717
\(333\) 333.000 0.0547997
\(334\) −2804.00 −0.459365
\(335\) −579.000 −0.0944303
\(336\) 336.000 0.0545545
\(337\) −9906.00 −1.60123 −0.800615 0.599180i \(-0.795493\pi\)
−0.800615 + 0.599180i \(0.795493\pi\)
\(338\) −696.000 −0.112004
\(339\) −4494.00 −0.720002
\(340\) 400.000 0.0638031
\(341\) −968.000 −0.153725
\(342\) −1566.00 −0.247601
\(343\) −343.000 −0.0539949
\(344\) −3664.00 −0.574272
\(345\) 174.000 0.0271532
\(346\) 1672.00 0.259790
\(347\) −10664.0 −1.64978 −0.824890 0.565294i \(-0.808763\pi\)
−0.824890 + 0.565294i \(0.808763\pi\)
\(348\) 2676.00 0.412209
\(349\) 5489.00 0.841889 0.420945 0.907086i \(-0.361699\pi\)
0.420945 + 0.907086i \(0.361699\pi\)
\(350\) 1736.00 0.265123
\(351\) 1161.00 0.176552
\(352\) −352.000 −0.0533002
\(353\) −1845.00 −0.278185 −0.139093 0.990279i \(-0.544419\pi\)
−0.139093 + 0.990279i \(0.544419\pi\)
\(354\) 630.000 0.0945879
\(355\) 128.000 0.0191367
\(356\) −3192.00 −0.475213
\(357\) 2100.00 0.311327
\(358\) −6584.00 −0.971998
\(359\) −848.000 −0.124668 −0.0623339 0.998055i \(-0.519854\pi\)
−0.0623339 + 0.998055i \(0.519854\pi\)
\(360\) 72.0000 0.0105409
\(361\) 710.000 0.103514
\(362\) 9676.00 1.40486
\(363\) −363.000 −0.0524864
\(364\) 1204.00 0.173370
\(365\) −161.000 −0.0230880
\(366\) −1140.00 −0.162811
\(367\) −8474.00 −1.20528 −0.602642 0.798012i \(-0.705885\pi\)
−0.602642 + 0.798012i \(0.705885\pi\)
\(368\) −928.000 −0.131455
\(369\) 1152.00 0.162522
\(370\) 74.0000 0.0103975
\(371\) 2394.00 0.335014
\(372\) −1056.00 −0.147180
\(373\) 11396.0 1.58194 0.790969 0.611857i \(-0.209577\pi\)
0.790969 + 0.611857i \(0.209577\pi\)
\(374\) −2200.00 −0.304169
\(375\) 747.000 0.102866
\(376\) −2728.00 −0.374165
\(377\) 9589.00 1.30997
\(378\) 378.000 0.0514344
\(379\) 2155.00 0.292071 0.146036 0.989279i \(-0.453349\pi\)
0.146036 + 0.989279i \(0.453349\pi\)
\(380\) −348.000 −0.0469790
\(381\) 1848.00 0.248493
\(382\) −2464.00 −0.330024
\(383\) −5864.00 −0.782340 −0.391170 0.920318i \(-0.627930\pi\)
−0.391170 + 0.920318i \(0.627930\pi\)
\(384\) −384.000 −0.0510310
\(385\) 77.0000 0.0101929
\(386\) 4732.00 0.623970
\(387\) −4122.00 −0.541429
\(388\) 5656.00 0.740051
\(389\) 10104.0 1.31695 0.658474 0.752603i \(-0.271202\pi\)
0.658474 + 0.752603i \(0.271202\pi\)
\(390\) 258.000 0.0334983
\(391\) −5800.00 −0.750175
\(392\) 392.000 0.0505076
\(393\) −8244.00 −1.05815
\(394\) 636.000 0.0813229
\(395\) −396.000 −0.0504428
\(396\) −396.000 −0.0502519
\(397\) −5868.00 −0.741830 −0.370915 0.928667i \(-0.620956\pi\)
−0.370915 + 0.928667i \(0.620956\pi\)
\(398\) −1612.00 −0.203021
\(399\) −1827.00 −0.229234
\(400\) −1984.00 −0.248000
\(401\) −2418.00 −0.301120 −0.150560 0.988601i \(-0.548108\pi\)
−0.150560 + 0.988601i \(0.548108\pi\)
\(402\) 3474.00 0.431013
\(403\) −3784.00 −0.467728
\(404\) −7056.00 −0.868933
\(405\) 81.0000 0.00993808
\(406\) 3122.00 0.381631
\(407\) −407.000 −0.0495682
\(408\) −2400.00 −0.291220
\(409\) −2354.00 −0.284591 −0.142296 0.989824i \(-0.545448\pi\)
−0.142296 + 0.989824i \(0.545448\pi\)
\(410\) 256.000 0.0308364
\(411\) −6552.00 −0.786341
\(412\) 3912.00 0.467792
\(413\) 735.000 0.0875714
\(414\) −1044.00 −0.123937
\(415\) −420.000 −0.0496795
\(416\) −1376.00 −0.162173
\(417\) −2724.00 −0.319892
\(418\) 1914.00 0.223964
\(419\) −11443.0 −1.33419 −0.667097 0.744971i \(-0.732463\pi\)
−0.667097 + 0.744971i \(0.732463\pi\)
\(420\) 84.0000 0.00975900
\(421\) −13325.0 −1.54257 −0.771284 0.636492i \(-0.780385\pi\)
−0.771284 + 0.636492i \(0.780385\pi\)
\(422\) 8924.00 1.02942
\(423\) −3069.00 −0.352766
\(424\) −2736.00 −0.313377
\(425\) −12400.0 −1.41527
\(426\) −768.000 −0.0873468
\(427\) −1330.00 −0.150734
\(428\) 1924.00 0.217290
\(429\) −1419.00 −0.159697
\(430\) −916.000 −0.102729
\(431\) −8503.00 −0.950290 −0.475145 0.879907i \(-0.657604\pi\)
−0.475145 + 0.879907i \(0.657604\pi\)
\(432\) −432.000 −0.0481125
\(433\) 4496.00 0.498993 0.249497 0.968376i \(-0.419735\pi\)
0.249497 + 0.968376i \(0.419735\pi\)
\(434\) −1232.00 −0.136262
\(435\) 669.000 0.0737381
\(436\) −1136.00 −0.124781
\(437\) 5046.00 0.552364
\(438\) 966.000 0.105382
\(439\) −7247.00 −0.787883 −0.393941 0.919136i \(-0.628889\pi\)
−0.393941 + 0.919136i \(0.628889\pi\)
\(440\) −88.0000 −0.00953463
\(441\) 441.000 0.0476190
\(442\) −8600.00 −0.925476
\(443\) 8568.00 0.918912 0.459456 0.888201i \(-0.348044\pi\)
0.459456 + 0.888201i \(0.348044\pi\)
\(444\) −444.000 −0.0474579
\(445\) −798.000 −0.0850086
\(446\) 7604.00 0.807309
\(447\) −1263.00 −0.133642
\(448\) −448.000 −0.0472456
\(449\) −7812.00 −0.821094 −0.410547 0.911840i \(-0.634662\pi\)
−0.410547 + 0.911840i \(0.634662\pi\)
\(450\) −2232.00 −0.233817
\(451\) −1408.00 −0.147007
\(452\) 5992.00 0.623540
\(453\) −2844.00 −0.294973
\(454\) −6740.00 −0.696749
\(455\) 301.000 0.0310134
\(456\) 2088.00 0.214429
\(457\) 4262.00 0.436254 0.218127 0.975920i \(-0.430005\pi\)
0.218127 + 0.975920i \(0.430005\pi\)
\(458\) −1796.00 −0.183235
\(459\) −2700.00 −0.274565
\(460\) −232.000 −0.0235153
\(461\) 8316.00 0.840162 0.420081 0.907487i \(-0.362002\pi\)
0.420081 + 0.907487i \(0.362002\pi\)
\(462\) −462.000 −0.0465242
\(463\) −16103.0 −1.61635 −0.808175 0.588943i \(-0.799544\pi\)
−0.808175 + 0.588943i \(0.799544\pi\)
\(464\) −3568.00 −0.356983
\(465\) −264.000 −0.0263284
\(466\) −924.000 −0.0918529
\(467\) −8913.00 −0.883179 −0.441589 0.897217i \(-0.645585\pi\)
−0.441589 + 0.897217i \(0.645585\pi\)
\(468\) −1548.00 −0.152898
\(469\) 4053.00 0.399041
\(470\) −682.000 −0.0669326
\(471\) −7104.00 −0.694979
\(472\) −840.000 −0.0819155
\(473\) 5038.00 0.489741
\(474\) 2376.00 0.230239
\(475\) 10788.0 1.04208
\(476\) −2800.00 −0.269617
\(477\) −3078.00 −0.295455
\(478\) −11330.0 −1.08415
\(479\) −4100.00 −0.391093 −0.195547 0.980694i \(-0.562648\pi\)
−0.195547 + 0.980694i \(0.562648\pi\)
\(480\) −96.0000 −0.00912871
\(481\) −1591.00 −0.150818
\(482\) 3186.00 0.301075
\(483\) −1218.00 −0.114743
\(484\) 484.000 0.0454545
\(485\) 1414.00 0.132384
\(486\) −486.000 −0.0453609
\(487\) 10496.0 0.976631 0.488315 0.872667i \(-0.337612\pi\)
0.488315 + 0.872667i \(0.337612\pi\)
\(488\) 1520.00 0.140998
\(489\) 7809.00 0.722158
\(490\) 98.0000 0.00903508
\(491\) 2625.00 0.241272 0.120636 0.992697i \(-0.461507\pi\)
0.120636 + 0.992697i \(0.461507\pi\)
\(492\) −1536.00 −0.140748
\(493\) −22300.0 −2.03720
\(494\) 7482.00 0.681439
\(495\) −99.0000 −0.00898933
\(496\) 1408.00 0.127462
\(497\) −896.000 −0.0808674
\(498\) 2520.00 0.226755
\(499\) −12397.0 −1.11216 −0.556078 0.831130i \(-0.687694\pi\)
−0.556078 + 0.831130i \(0.687694\pi\)
\(500\) −996.000 −0.0890849
\(501\) 4206.00 0.375070
\(502\) 9090.00 0.808180
\(503\) −1854.00 −0.164345 −0.0821727 0.996618i \(-0.526186\pi\)
−0.0821727 + 0.996618i \(0.526186\pi\)
\(504\) −504.000 −0.0445435
\(505\) −1764.00 −0.155440
\(506\) 1276.00 0.112105
\(507\) 1044.00 0.0914510
\(508\) −2464.00 −0.215201
\(509\) 17530.0 1.52653 0.763265 0.646086i \(-0.223595\pi\)
0.763265 + 0.646086i \(0.223595\pi\)
\(510\) −600.000 −0.0520950
\(511\) 1127.00 0.0975647
\(512\) 512.000 0.0441942
\(513\) 2349.00 0.202165
\(514\) −5718.00 −0.490681
\(515\) 978.000 0.0836812
\(516\) 5496.00 0.468891
\(517\) 3751.00 0.319089
\(518\) −518.000 −0.0439375
\(519\) −2508.00 −0.212117
\(520\) −344.000 −0.0290104
\(521\) −22255.0 −1.87142 −0.935709 0.352772i \(-0.885239\pi\)
−0.935709 + 0.352772i \(0.885239\pi\)
\(522\) −4014.00 −0.336567
\(523\) −7789.00 −0.651222 −0.325611 0.945504i \(-0.605570\pi\)
−0.325611 + 0.945504i \(0.605570\pi\)
\(524\) 10992.0 0.916389
\(525\) −2604.00 −0.216472
\(526\) −12558.0 −1.04098
\(527\) 8800.00 0.727389
\(528\) 528.000 0.0435194
\(529\) −8803.00 −0.723514
\(530\) −684.000 −0.0560586
\(531\) −945.000 −0.0772307
\(532\) 2436.00 0.198523
\(533\) −5504.00 −0.447288
\(534\) 4788.00 0.388009
\(535\) 481.000 0.0388700
\(536\) −4632.00 −0.373269
\(537\) 9876.00 0.793633
\(538\) −12540.0 −1.00490
\(539\) −539.000 −0.0430730
\(540\) −108.000 −0.00860663
\(541\) 8288.00 0.658649 0.329324 0.944217i \(-0.393179\pi\)
0.329324 + 0.944217i \(0.393179\pi\)
\(542\) 13442.0 1.06528
\(543\) −14514.0 −1.14706
\(544\) 3200.00 0.252204
\(545\) −284.000 −0.0223215
\(546\) −1806.00 −0.141556
\(547\) −6432.00 −0.502765 −0.251383 0.967888i \(-0.580885\pi\)
−0.251383 + 0.967888i \(0.580885\pi\)
\(548\) 8736.00 0.680992
\(549\) 1710.00 0.132934
\(550\) 2728.00 0.211495
\(551\) 19401.0 1.50002
\(552\) 1392.00 0.107332
\(553\) 2772.00 0.213160
\(554\) −10776.0 −0.826405
\(555\) −111.000 −0.00848953
\(556\) 3632.00 0.277034
\(557\) −9815.00 −0.746634 −0.373317 0.927704i \(-0.621780\pi\)
−0.373317 + 0.927704i \(0.621780\pi\)
\(558\) 1584.00 0.120172
\(559\) 19694.0 1.49010
\(560\) −112.000 −0.00845154
\(561\) 3300.00 0.248353
\(562\) 9530.00 0.715300
\(563\) −20042.0 −1.50030 −0.750151 0.661267i \(-0.770019\pi\)
−0.750151 + 0.661267i \(0.770019\pi\)
\(564\) 4092.00 0.305504
\(565\) 1498.00 0.111542
\(566\) −1806.00 −0.134120
\(567\) −567.000 −0.0419961
\(568\) 1024.00 0.0756445
\(569\) −8562.00 −0.630822 −0.315411 0.948955i \(-0.602142\pi\)
−0.315411 + 0.948955i \(0.602142\pi\)
\(570\) 522.000 0.0383582
\(571\) 27004.0 1.97913 0.989564 0.144093i \(-0.0460265\pi\)
0.989564 + 0.144093i \(0.0460265\pi\)
\(572\) 1892.00 0.138302
\(573\) 3696.00 0.269464
\(574\) −1792.00 −0.130308
\(575\) 7192.00 0.521612
\(576\) 576.000 0.0416667
\(577\) 14360.0 1.03607 0.518037 0.855358i \(-0.326663\pi\)
0.518037 + 0.855358i \(0.326663\pi\)
\(578\) 10174.0 0.732150
\(579\) −7098.00 −0.509469
\(580\) −892.000 −0.0638591
\(581\) 2940.00 0.209934
\(582\) −8484.00 −0.604249
\(583\) 3762.00 0.267249
\(584\) −1288.00 −0.0912634
\(585\) −387.000 −0.0273512
\(586\) 1344.00 0.0947442
\(587\) 15453.0 1.08656 0.543282 0.839550i \(-0.317181\pi\)
0.543282 + 0.839550i \(0.317181\pi\)
\(588\) −588.000 −0.0412393
\(589\) −7656.00 −0.535586
\(590\) −210.000 −0.0146535
\(591\) −954.000 −0.0663999
\(592\) 592.000 0.0410997
\(593\) 18000.0 1.24649 0.623247 0.782025i \(-0.285813\pi\)
0.623247 + 0.782025i \(0.285813\pi\)
\(594\) 594.000 0.0410305
\(595\) −700.000 −0.0482306
\(596\) 1684.00 0.115737
\(597\) 2418.00 0.165766
\(598\) 4988.00 0.341094
\(599\) 27180.0 1.85400 0.926999 0.375064i \(-0.122379\pi\)
0.926999 + 0.375064i \(0.122379\pi\)
\(600\) 2976.00 0.202491
\(601\) −16153.0 −1.09633 −0.548165 0.836370i \(-0.684674\pi\)
−0.548165 + 0.836370i \(0.684674\pi\)
\(602\) 6412.00 0.434109
\(603\) −5211.00 −0.351921
\(604\) 3792.00 0.255454
\(605\) 121.000 0.00813116
\(606\) 10584.0 0.709481
\(607\) −11443.0 −0.765168 −0.382584 0.923921i \(-0.624966\pi\)
−0.382584 + 0.923921i \(0.624966\pi\)
\(608\) −2784.00 −0.185701
\(609\) −4683.00 −0.311601
\(610\) 380.000 0.0252225
\(611\) 14663.0 0.970870
\(612\) 3600.00 0.237780
\(613\) −9488.00 −0.625150 −0.312575 0.949893i \(-0.601191\pi\)
−0.312575 + 0.949893i \(0.601191\pi\)
\(614\) −3752.00 −0.246610
\(615\) −384.000 −0.0251778
\(616\) 616.000 0.0402911
\(617\) 22902.0 1.49433 0.747164 0.664640i \(-0.231415\pi\)
0.747164 + 0.664640i \(0.231415\pi\)
\(618\) −5868.00 −0.381951
\(619\) 19448.0 1.26281 0.631406 0.775452i \(-0.282478\pi\)
0.631406 + 0.775452i \(0.282478\pi\)
\(620\) 352.000 0.0228011
\(621\) 1566.00 0.101194
\(622\) −9376.00 −0.604411
\(623\) 5586.00 0.359227
\(624\) 2064.00 0.132414
\(625\) 15251.0 0.976064
\(626\) 9824.00 0.627230
\(627\) −2871.00 −0.182866
\(628\) 9472.00 0.601869
\(629\) 3700.00 0.234545
\(630\) −126.000 −0.00796819
\(631\) −13608.0 −0.858520 −0.429260 0.903181i \(-0.641226\pi\)
−0.429260 + 0.903181i \(0.641226\pi\)
\(632\) −3168.00 −0.199393
\(633\) −13386.0 −0.840515
\(634\) −1116.00 −0.0699086
\(635\) −616.000 −0.0384964
\(636\) 4104.00 0.255871
\(637\) −2107.00 −0.131056
\(638\) 4906.00 0.304436
\(639\) 1152.00 0.0713183
\(640\) 128.000 0.00790569
\(641\) 17628.0 1.08622 0.543108 0.839663i \(-0.317248\pi\)
0.543108 + 0.839663i \(0.317248\pi\)
\(642\) −2886.00 −0.177416
\(643\) −21074.0 −1.29250 −0.646250 0.763126i \(-0.723664\pi\)
−0.646250 + 0.763126i \(0.723664\pi\)
\(644\) 1624.00 0.0993704
\(645\) 1374.00 0.0838778
\(646\) −17400.0 −1.05974
\(647\) −4899.00 −0.297681 −0.148840 0.988861i \(-0.547554\pi\)
−0.148840 + 0.988861i \(0.547554\pi\)
\(648\) 648.000 0.0392837
\(649\) 1155.00 0.0698578
\(650\) 10664.0 0.643502
\(651\) 1848.00 0.111258
\(652\) −10412.0 −0.625407
\(653\) 7760.00 0.465042 0.232521 0.972591i \(-0.425303\pi\)
0.232521 + 0.972591i \(0.425303\pi\)
\(654\) 1704.00 0.101883
\(655\) 2748.00 0.163929
\(656\) 2048.00 0.121892
\(657\) −1449.00 −0.0860439
\(658\) 4774.00 0.282842
\(659\) 18365.0 1.08558 0.542791 0.839868i \(-0.317367\pi\)
0.542791 + 0.839868i \(0.317367\pi\)
\(660\) 132.000 0.00778499
\(661\) 13286.0 0.781794 0.390897 0.920435i \(-0.372165\pi\)
0.390897 + 0.920435i \(0.372165\pi\)
\(662\) 9320.00 0.547178
\(663\) 12900.0 0.755648
\(664\) −3360.00 −0.196375
\(665\) 609.000 0.0355128
\(666\) 666.000 0.0387492
\(667\) 12934.0 0.750834
\(668\) −5608.00 −0.324820
\(669\) −11406.0 −0.659165
\(670\) −1158.00 −0.0667723
\(671\) −2090.00 −0.120244
\(672\) 672.000 0.0385758
\(673\) −15242.0 −0.873010 −0.436505 0.899702i \(-0.643784\pi\)
−0.436505 + 0.899702i \(0.643784\pi\)
\(674\) −19812.0 −1.13224
\(675\) 3348.00 0.190910
\(676\) −1392.00 −0.0791989
\(677\) 14170.0 0.804427 0.402214 0.915546i \(-0.368241\pi\)
0.402214 + 0.915546i \(0.368241\pi\)
\(678\) −8988.00 −0.509118
\(679\) −9898.00 −0.559426
\(680\) 800.000 0.0451156
\(681\) 10110.0 0.568893
\(682\) −1936.00 −0.108700
\(683\) −7116.00 −0.398662 −0.199331 0.979932i \(-0.563877\pi\)
−0.199331 + 0.979932i \(0.563877\pi\)
\(684\) −3132.00 −0.175080
\(685\) 2184.00 0.121819
\(686\) −686.000 −0.0381802
\(687\) 2694.00 0.149611
\(688\) −7328.00 −0.406072
\(689\) 14706.0 0.813141
\(690\) 348.000 0.0192002
\(691\) 2010.00 0.110657 0.0553285 0.998468i \(-0.482379\pi\)
0.0553285 + 0.998468i \(0.482379\pi\)
\(692\) 3344.00 0.183699
\(693\) 693.000 0.0379869
\(694\) −21328.0 −1.16657
\(695\) 908.000 0.0495574
\(696\) 5352.00 0.291476
\(697\) 12800.0 0.695602
\(698\) 10978.0 0.595306
\(699\) 1386.00 0.0749976
\(700\) 3472.00 0.187470
\(701\) 7794.00 0.419936 0.209968 0.977708i \(-0.432664\pi\)
0.209968 + 0.977708i \(0.432664\pi\)
\(702\) 2322.00 0.124841
\(703\) −3219.00 −0.172698
\(704\) −704.000 −0.0376889
\(705\) 1023.00 0.0546502
\(706\) −3690.00 −0.196707
\(707\) 12348.0 0.656852
\(708\) 1260.00 0.0668838
\(709\) 29991.0 1.58863 0.794313 0.607509i \(-0.207831\pi\)
0.794313 + 0.607509i \(0.207831\pi\)
\(710\) 256.000 0.0135317
\(711\) −3564.00 −0.187989
\(712\) −6384.00 −0.336026
\(713\) −5104.00 −0.268087
\(714\) 4200.00 0.220142
\(715\) 473.000 0.0247401
\(716\) −13168.0 −0.687306
\(717\) 16995.0 0.885202
\(718\) −1696.00 −0.0881534
\(719\) 19613.0 1.01730 0.508652 0.860972i \(-0.330144\pi\)
0.508652 + 0.860972i \(0.330144\pi\)
\(720\) 144.000 0.00745356
\(721\) −6846.00 −0.353618
\(722\) 1420.00 0.0731952
\(723\) −4779.00 −0.245827
\(724\) 19352.0 0.993386
\(725\) 27652.0 1.41651
\(726\) −726.000 −0.0371135
\(727\) 16646.0 0.849197 0.424598 0.905382i \(-0.360415\pi\)
0.424598 + 0.905382i \(0.360415\pi\)
\(728\) 2408.00 0.122591
\(729\) 729.000 0.0370370
\(730\) −322.000 −0.0163257
\(731\) −45800.0 −2.31734
\(732\) −2280.00 −0.115125
\(733\) −36346.0 −1.83147 −0.915737 0.401779i \(-0.868392\pi\)
−0.915737 + 0.401779i \(0.868392\pi\)
\(734\) −16948.0 −0.852264
\(735\) −147.000 −0.00737711
\(736\) −1856.00 −0.0929525
\(737\) 6369.00 0.318324
\(738\) 2304.00 0.114921
\(739\) −25698.0 −1.27918 −0.639591 0.768715i \(-0.720896\pi\)
−0.639591 + 0.768715i \(0.720896\pi\)
\(740\) 148.000 0.00735215
\(741\) −11223.0 −0.556393
\(742\) 4788.00 0.236891
\(743\) −27543.0 −1.35997 −0.679983 0.733228i \(-0.738013\pi\)
−0.679983 + 0.733228i \(0.738013\pi\)
\(744\) −2112.00 −0.104072
\(745\) 421.000 0.0207037
\(746\) 22792.0 1.11860
\(747\) −3780.00 −0.185145
\(748\) −4400.00 −0.215080
\(749\) −3367.00 −0.164256
\(750\) 1494.00 0.0727376
\(751\) 26005.0 1.26356 0.631782 0.775146i \(-0.282324\pi\)
0.631782 + 0.775146i \(0.282324\pi\)
\(752\) −5456.00 −0.264574
\(753\) −13635.0 −0.659877
\(754\) 19178.0 0.926289
\(755\) 948.000 0.0456970
\(756\) 756.000 0.0363696
\(757\) 17621.0 0.846032 0.423016 0.906122i \(-0.360971\pi\)
0.423016 + 0.906122i \(0.360971\pi\)
\(758\) 4310.00 0.206525
\(759\) −1914.00 −0.0915333
\(760\) −696.000 −0.0332192
\(761\) 22968.0 1.09407 0.547036 0.837109i \(-0.315756\pi\)
0.547036 + 0.837109i \(0.315756\pi\)
\(762\) 3696.00 0.175711
\(763\) 1988.00 0.0943256
\(764\) −4928.00 −0.233362
\(765\) 900.000 0.0425354
\(766\) −11728.0 −0.553198
\(767\) 4515.00 0.212552
\(768\) −768.000 −0.0360844
\(769\) 28273.0 1.32581 0.662907 0.748702i \(-0.269323\pi\)
0.662907 + 0.748702i \(0.269323\pi\)
\(770\) 154.000 0.00720750
\(771\) 8577.00 0.400640
\(772\) 9464.00 0.441213
\(773\) −13239.0 −0.616007 −0.308004 0.951385i \(-0.599661\pi\)
−0.308004 + 0.951385i \(0.599661\pi\)
\(774\) −8244.00 −0.382848
\(775\) −10912.0 −0.505769
\(776\) 11312.0 0.523295
\(777\) 777.000 0.0358748
\(778\) 20208.0 0.931224
\(779\) −11136.0 −0.512180
\(780\) 516.000 0.0236869
\(781\) −1408.00 −0.0645099
\(782\) −11600.0 −0.530454
\(783\) 6021.00 0.274806
\(784\) 784.000 0.0357143
\(785\) 2368.00 0.107666
\(786\) −16488.0 −0.748228
\(787\) 1775.00 0.0803963 0.0401982 0.999192i \(-0.487201\pi\)
0.0401982 + 0.999192i \(0.487201\pi\)
\(788\) 1272.00 0.0575040
\(789\) 18837.0 0.849956
\(790\) −792.000 −0.0356685
\(791\) −10486.0 −0.471352
\(792\) −792.000 −0.0355335
\(793\) −8170.00 −0.365858
\(794\) −11736.0 −0.524553
\(795\) 1026.00 0.0457717
\(796\) −3224.00 −0.143557
\(797\) −16481.0 −0.732481 −0.366240 0.930520i \(-0.619355\pi\)
−0.366240 + 0.930520i \(0.619355\pi\)
\(798\) −3654.00 −0.162093
\(799\) −34100.0 −1.50985
\(800\) −3968.00 −0.175362
\(801\) −7182.00 −0.316808
\(802\) −4836.00 −0.212924
\(803\) 1771.00 0.0778297
\(804\) 6948.00 0.304772
\(805\) 406.000 0.0177759
\(806\) −7568.00 −0.330734
\(807\) 18810.0 0.820500
\(808\) −14112.0 −0.614429
\(809\) −29079.0 −1.26374 −0.631868 0.775076i \(-0.717712\pi\)
−0.631868 + 0.775076i \(0.717712\pi\)
\(810\) 162.000 0.00702728
\(811\) −40381.0 −1.74842 −0.874210 0.485548i \(-0.838620\pi\)
−0.874210 + 0.485548i \(0.838620\pi\)
\(812\) 6244.00 0.269854
\(813\) −20163.0 −0.869800
\(814\) −814.000 −0.0350500
\(815\) −2603.00 −0.111876
\(816\) −4800.00 −0.205924
\(817\) 39846.0 1.70629
\(818\) −4708.00 −0.201236
\(819\) 2709.00 0.115580
\(820\) 512.000 0.0218047
\(821\) −6345.00 −0.269722 −0.134861 0.990865i \(-0.543059\pi\)
−0.134861 + 0.990865i \(0.543059\pi\)
\(822\) −13104.0 −0.556027
\(823\) 7757.00 0.328544 0.164272 0.986415i \(-0.447472\pi\)
0.164272 + 0.986415i \(0.447472\pi\)
\(824\) 7824.00 0.330779
\(825\) −4092.00 −0.172685
\(826\) 1470.00 0.0619223
\(827\) −15659.0 −0.658424 −0.329212 0.944256i \(-0.606783\pi\)
−0.329212 + 0.944256i \(0.606783\pi\)
\(828\) −2088.00 −0.0876365
\(829\) −5384.00 −0.225566 −0.112783 0.993620i \(-0.535976\pi\)
−0.112783 + 0.993620i \(0.535976\pi\)
\(830\) −840.000 −0.0351287
\(831\) 16164.0 0.674757
\(832\) −2752.00 −0.114674
\(833\) 4900.00 0.203811
\(834\) −5448.00 −0.226198
\(835\) −1402.00 −0.0581056
\(836\) 3828.00 0.158366
\(837\) −2376.00 −0.0981202
\(838\) −22886.0 −0.943417
\(839\) 40395.0 1.66221 0.831103 0.556119i \(-0.187710\pi\)
0.831103 + 0.556119i \(0.187710\pi\)
\(840\) 168.000 0.00690066
\(841\) 25340.0 1.03899
\(842\) −26650.0 −1.09076
\(843\) −14295.0 −0.584040
\(844\) 17848.0 0.727907
\(845\) −348.000 −0.0141675
\(846\) −6138.00 −0.249443
\(847\) −847.000 −0.0343604
\(848\) −5472.00 −0.221591
\(849\) 2709.00 0.109508
\(850\) −24800.0 −1.00074
\(851\) −2146.00 −0.0864441
\(852\) −1536.00 −0.0617635
\(853\) −33202.0 −1.33273 −0.666363 0.745628i \(-0.732150\pi\)
−0.666363 + 0.745628i \(0.732150\pi\)
\(854\) −2660.00 −0.106585
\(855\) −783.000 −0.0313193
\(856\) 3848.00 0.153647
\(857\) 21034.0 0.838399 0.419199 0.907894i \(-0.362311\pi\)
0.419199 + 0.907894i \(0.362311\pi\)
\(858\) −2838.00 −0.112923
\(859\) 20920.0 0.830944 0.415472 0.909606i \(-0.363616\pi\)
0.415472 + 0.909606i \(0.363616\pi\)
\(860\) −1832.00 −0.0726403
\(861\) 2688.00 0.106396
\(862\) −17006.0 −0.671957
\(863\) 8354.00 0.329517 0.164759 0.986334i \(-0.447315\pi\)
0.164759 + 0.986334i \(0.447315\pi\)
\(864\) −864.000 −0.0340207
\(865\) 836.000 0.0328611
\(866\) 8992.00 0.352841
\(867\) −15261.0 −0.597798
\(868\) −2464.00 −0.0963521
\(869\) 4356.00 0.170043
\(870\) 1338.00 0.0521407
\(871\) 24897.0 0.968545
\(872\) −2272.00 −0.0882335
\(873\) 12726.0 0.493368
\(874\) 10092.0 0.390580
\(875\) 1743.00 0.0673419
\(876\) 1932.00 0.0745162
\(877\) 19564.0 0.753283 0.376642 0.926359i \(-0.377079\pi\)
0.376642 + 0.926359i \(0.377079\pi\)
\(878\) −14494.0 −0.557117
\(879\) −2016.00 −0.0773584
\(880\) −176.000 −0.00674200
\(881\) 32061.0 1.22606 0.613032 0.790058i \(-0.289950\pi\)
0.613032 + 0.790058i \(0.289950\pi\)
\(882\) 882.000 0.0336718
\(883\) 46711.0 1.78024 0.890119 0.455728i \(-0.150621\pi\)
0.890119 + 0.455728i \(0.150621\pi\)
\(884\) −17200.0 −0.654410
\(885\) 315.000 0.0119645
\(886\) 17136.0 0.649769
\(887\) −7120.00 −0.269522 −0.134761 0.990878i \(-0.543027\pi\)
−0.134761 + 0.990878i \(0.543027\pi\)
\(888\) −888.000 −0.0335578
\(889\) 4312.00 0.162677
\(890\) −1596.00 −0.0601102
\(891\) −891.000 −0.0335013
\(892\) 15208.0 0.570854
\(893\) 29667.0 1.11172
\(894\) −2526.00 −0.0944990
\(895\) −3292.00 −0.122949
\(896\) −896.000 −0.0334077
\(897\) −7482.00 −0.278502
\(898\) −15624.0 −0.580601
\(899\) −19624.0 −0.728028
\(900\) −4464.00 −0.165333
\(901\) −34200.0 −1.26456
\(902\) −2816.00 −0.103950
\(903\) −9618.00 −0.354449
\(904\) 11984.0 0.440909
\(905\) 4838.00 0.177702
\(906\) −5688.00 −0.208577
\(907\) −21940.0 −0.803204 −0.401602 0.915814i \(-0.631546\pi\)
−0.401602 + 0.915814i \(0.631546\pi\)
\(908\) −13480.0 −0.492676
\(909\) −15876.0 −0.579289
\(910\) 602.000 0.0219298
\(911\) 66.0000 0.00240030 0.00120015 0.999999i \(-0.499618\pi\)
0.00120015 + 0.999999i \(0.499618\pi\)
\(912\) 4176.00 0.151624
\(913\) 4620.00 0.167470
\(914\) 8524.00 0.308478
\(915\) −570.000 −0.0205941
\(916\) −3592.00 −0.129567
\(917\) −19236.0 −0.692725
\(918\) −5400.00 −0.194147
\(919\) −42594.0 −1.52889 −0.764443 0.644691i \(-0.776986\pi\)
−0.764443 + 0.644691i \(0.776986\pi\)
\(920\) −464.000 −0.0166279
\(921\) 5628.00 0.201356
\(922\) 16632.0 0.594084
\(923\) −5504.00 −0.196280
\(924\) −924.000 −0.0328976
\(925\) −4588.00 −0.163084
\(926\) −32206.0 −1.14293
\(927\) 8802.00 0.311861
\(928\) −7136.00 −0.252425
\(929\) −6609.00 −0.233406 −0.116703 0.993167i \(-0.537233\pi\)
−0.116703 + 0.993167i \(0.537233\pi\)
\(930\) −528.000 −0.0186170
\(931\) −4263.00 −0.150069
\(932\) −1848.00 −0.0649498
\(933\) 14064.0 0.493499
\(934\) −17826.0 −0.624502
\(935\) −1100.00 −0.0384747
\(936\) −3096.00 −0.108115
\(937\) 10098.0 0.352068 0.176034 0.984384i \(-0.443673\pi\)
0.176034 + 0.984384i \(0.443673\pi\)
\(938\) 8106.00 0.282164
\(939\) −14736.0 −0.512131
\(940\) −1364.00 −0.0473285
\(941\) −41824.0 −1.44891 −0.724455 0.689323i \(-0.757908\pi\)
−0.724455 + 0.689323i \(0.757908\pi\)
\(942\) −14208.0 −0.491424
\(943\) −7424.00 −0.256372
\(944\) −1680.00 −0.0579230
\(945\) 189.000 0.00650600
\(946\) 10076.0 0.346299
\(947\) −28838.0 −0.989556 −0.494778 0.869020i \(-0.664751\pi\)
−0.494778 + 0.869020i \(0.664751\pi\)
\(948\) 4752.00 0.162804
\(949\) 6923.00 0.236807
\(950\) 21576.0 0.736861
\(951\) 1674.00 0.0570801
\(952\) −5600.00 −0.190648
\(953\) 16165.0 0.549460 0.274730 0.961521i \(-0.411411\pi\)
0.274730 + 0.961521i \(0.411411\pi\)
\(954\) −6156.00 −0.208918
\(955\) −1232.00 −0.0417451
\(956\) −22660.0 −0.766608
\(957\) −7359.00 −0.248571
\(958\) −8200.00 −0.276545
\(959\) −15288.0 −0.514781
\(960\) −192.000 −0.00645497
\(961\) −22047.0 −0.740056
\(962\) −3182.00 −0.106644
\(963\) 4329.00 0.144860
\(964\) 6372.00 0.212892
\(965\) 2366.00 0.0789267
\(966\) −2436.00 −0.0811356
\(967\) −40114.0 −1.33400 −0.667001 0.745057i \(-0.732422\pi\)
−0.667001 + 0.745057i \(0.732422\pi\)
\(968\) 968.000 0.0321412
\(969\) 26100.0 0.865276
\(970\) 2828.00 0.0936099
\(971\) −6515.00 −0.215321 −0.107660 0.994188i \(-0.534336\pi\)
−0.107660 + 0.994188i \(0.534336\pi\)
\(972\) −972.000 −0.0320750
\(973\) −6356.00 −0.209418
\(974\) 20992.0 0.690582
\(975\) −15996.0 −0.525417
\(976\) 3040.00 0.0997008
\(977\) 12790.0 0.418821 0.209411 0.977828i \(-0.432845\pi\)
0.209411 + 0.977828i \(0.432845\pi\)
\(978\) 15618.0 0.510643
\(979\) 8778.00 0.286564
\(980\) 196.000 0.00638877
\(981\) −2556.00 −0.0831874
\(982\) 5250.00 0.170605
\(983\) 4788.00 0.155355 0.0776773 0.996979i \(-0.475250\pi\)
0.0776773 + 0.996979i \(0.475250\pi\)
\(984\) −3072.00 −0.0995242
\(985\) 318.000 0.0102866
\(986\) −44600.0 −1.44052
\(987\) −7161.00 −0.230939
\(988\) 14964.0 0.481850
\(989\) 26564.0 0.854081
\(990\) −198.000 −0.00635642
\(991\) 26989.0 0.865120 0.432560 0.901605i \(-0.357610\pi\)
0.432560 + 0.901605i \(0.357610\pi\)
\(992\) 2816.00 0.0901291
\(993\) −13980.0 −0.446769
\(994\) −1792.00 −0.0571819
\(995\) −806.000 −0.0256803
\(996\) 5040.00 0.160340
\(997\) −55518.0 −1.76356 −0.881782 0.471658i \(-0.843656\pi\)
−0.881782 + 0.471658i \(0.843656\pi\)
\(998\) −24794.0 −0.786413
\(999\) −999.000 −0.0316386
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 462.4.a.f.1.1 1
3.2 odd 2 1386.4.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.4.a.f.1.1 1 1.1 even 1 trivial
1386.4.a.d.1.1 1 3.2 odd 2