Properties

Label 7623.2.a.cc.1.3
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2541)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.34292\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.34292 q^{2} +3.48929 q^{4} +0.146365 q^{5} -1.00000 q^{7} +3.48929 q^{8} +O(q^{10})\) \(q+2.34292 q^{2} +3.48929 q^{4} +0.146365 q^{5} -1.00000 q^{7} +3.48929 q^{8} +0.342923 q^{10} -4.34292 q^{13} -2.34292 q^{14} +1.19656 q^{16} -0.146365 q^{17} +1.83221 q^{19} +0.510711 q^{20} -8.81079 q^{23} -4.97858 q^{25} -10.1751 q^{26} -3.48929 q^{28} +4.34292 q^{29} +0.292731 q^{31} -4.17513 q^{32} -0.342923 q^{34} -0.146365 q^{35} -3.48929 q^{37} +4.29273 q^{38} +0.510711 q^{40} +2.80344 q^{41} +7.86098 q^{43} -20.6430 q^{46} +0.949808 q^{47} +1.00000 q^{49} -11.6644 q^{50} -15.1537 q^{52} +4.51071 q^{53} -3.48929 q^{56} +10.1751 q^{58} -8.02877 q^{59} -5.43910 q^{61} +0.685846 q^{62} -12.1751 q^{64} -0.635654 q^{65} -7.76060 q^{67} -0.510711 q^{68} -0.342923 q^{70} +3.53948 q^{71} -16.1825 q^{73} -8.17513 q^{74} +6.39312 q^{76} -13.0790 q^{79} +0.175135 q^{80} +6.56825 q^{82} -12.9070 q^{83} -0.0214229 q^{85} +18.4177 q^{86} +9.81079 q^{89} +4.34292 q^{91} -30.7434 q^{92} +2.22533 q^{94} +0.268173 q^{95} -17.4219 q^{97} +2.34292 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} + 3q^{4} - q^{5} - 3q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + q^{2} + 3q^{4} - q^{5} - 3q^{7} + 3q^{8} - 5q^{10} - 7q^{13} - q^{14} - q^{16} + q^{17} - 8q^{19} + 9q^{20} + 2q^{23} - 11q^{26} - 3q^{28} + 7q^{29} - 2q^{31} + 7q^{32} + 5q^{34} + q^{35} - 3q^{37} + 10q^{38} + 9q^{40} + 13q^{41} - 8q^{43} - 20q^{46} + 6q^{47} + 3q^{49} - 8q^{50} - 11q^{52} + 21q^{53} - 3q^{56} + 11q^{58} - 6q^{59} - 12q^{61} - 10q^{62} - 17q^{64} + 7q^{65} + 2q^{67} - 9q^{68} + 5q^{70} + 4q^{73} - 5q^{74} + 10q^{76} - 18q^{79} - 19q^{80} - 9q^{82} - 12q^{83} - 15q^{85} + 36q^{86} + q^{89} + 7q^{91} - 44q^{92} - 16q^{94} + 8q^{95} - 25q^{97} + q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.34292 1.65670 0.828348 0.560213i \(-0.189281\pi\)
0.828348 + 0.560213i \(0.189281\pi\)
\(3\) 0 0
\(4\) 3.48929 1.74464
\(5\) 0.146365 0.0654566 0.0327283 0.999464i \(-0.489580\pi\)
0.0327283 + 0.999464i \(0.489580\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 3.48929 1.23365
\(9\) 0 0
\(10\) 0.342923 0.108442
\(11\) 0 0
\(12\) 0 0
\(13\) −4.34292 −1.20451 −0.602255 0.798304i \(-0.705731\pi\)
−0.602255 + 0.798304i \(0.705731\pi\)
\(14\) −2.34292 −0.626173
\(15\) 0 0
\(16\) 1.19656 0.299139
\(17\) −0.146365 −0.0354988 −0.0177494 0.999842i \(-0.505650\pi\)
−0.0177494 + 0.999842i \(0.505650\pi\)
\(18\) 0 0
\(19\) 1.83221 0.420338 0.210169 0.977665i \(-0.432599\pi\)
0.210169 + 0.977665i \(0.432599\pi\)
\(20\) 0.510711 0.114199
\(21\) 0 0
\(22\) 0 0
\(23\) −8.81079 −1.83718 −0.918588 0.395216i \(-0.870670\pi\)
−0.918588 + 0.395216i \(0.870670\pi\)
\(24\) 0 0
\(25\) −4.97858 −0.995715
\(26\) −10.1751 −1.99551
\(27\) 0 0
\(28\) −3.48929 −0.659414
\(29\) 4.34292 0.806461 0.403230 0.915099i \(-0.367887\pi\)
0.403230 + 0.915099i \(0.367887\pi\)
\(30\) 0 0
\(31\) 0.292731 0.0525760 0.0262880 0.999654i \(-0.491631\pi\)
0.0262880 + 0.999654i \(0.491631\pi\)
\(32\) −4.17513 −0.738067
\(33\) 0 0
\(34\) −0.342923 −0.0588108
\(35\) −0.146365 −0.0247403
\(36\) 0 0
\(37\) −3.48929 −0.573636 −0.286818 0.957985i \(-0.592597\pi\)
−0.286818 + 0.957985i \(0.592597\pi\)
\(38\) 4.29273 0.696373
\(39\) 0 0
\(40\) 0.510711 0.0807506
\(41\) 2.80344 0.437824 0.218912 0.975745i \(-0.429749\pi\)
0.218912 + 0.975745i \(0.429749\pi\)
\(42\) 0 0
\(43\) 7.86098 1.19879 0.599394 0.800454i \(-0.295408\pi\)
0.599394 + 0.800454i \(0.295408\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −20.6430 −3.04364
\(47\) 0.949808 0.138544 0.0692719 0.997598i \(-0.477932\pi\)
0.0692719 + 0.997598i \(0.477932\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −11.6644 −1.64960
\(51\) 0 0
\(52\) −15.1537 −2.10144
\(53\) 4.51071 0.619594 0.309797 0.950803i \(-0.399739\pi\)
0.309797 + 0.950803i \(0.399739\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.48929 −0.466276
\(57\) 0 0
\(58\) 10.1751 1.33606
\(59\) −8.02877 −1.04526 −0.522628 0.852561i \(-0.675048\pi\)
−0.522628 + 0.852561i \(0.675048\pi\)
\(60\) 0 0
\(61\) −5.43910 −0.696405 −0.348202 0.937419i \(-0.613208\pi\)
−0.348202 + 0.937419i \(0.613208\pi\)
\(62\) 0.685846 0.0871026
\(63\) 0 0
\(64\) −12.1751 −1.52189
\(65\) −0.635654 −0.0788432
\(66\) 0 0
\(67\) −7.76060 −0.948108 −0.474054 0.880496i \(-0.657210\pi\)
−0.474054 + 0.880496i \(0.657210\pi\)
\(68\) −0.510711 −0.0619329
\(69\) 0 0
\(70\) −0.342923 −0.0409871
\(71\) 3.53948 0.420059 0.210030 0.977695i \(-0.432644\pi\)
0.210030 + 0.977695i \(0.432644\pi\)
\(72\) 0 0
\(73\) −16.1825 −1.89402 −0.947008 0.321210i \(-0.895911\pi\)
−0.947008 + 0.321210i \(0.895911\pi\)
\(74\) −8.17513 −0.950340
\(75\) 0 0
\(76\) 6.39312 0.733341
\(77\) 0 0
\(78\) 0 0
\(79\) −13.0790 −1.47150 −0.735749 0.677254i \(-0.763170\pi\)
−0.735749 + 0.677254i \(0.763170\pi\)
\(80\) 0.175135 0.0195807
\(81\) 0 0
\(82\) 6.56825 0.725342
\(83\) −12.9070 −1.41672 −0.708362 0.705850i \(-0.750565\pi\)
−0.708362 + 0.705850i \(0.750565\pi\)
\(84\) 0 0
\(85\) −0.0214229 −0.00232364
\(86\) 18.4177 1.98603
\(87\) 0 0
\(88\) 0 0
\(89\) 9.81079 1.03994 0.519971 0.854184i \(-0.325943\pi\)
0.519971 + 0.854184i \(0.325943\pi\)
\(90\) 0 0
\(91\) 4.34292 0.455262
\(92\) −30.7434 −3.20522
\(93\) 0 0
\(94\) 2.22533 0.229525
\(95\) 0.268173 0.0275139
\(96\) 0 0
\(97\) −17.4219 −1.76892 −0.884462 0.466612i \(-0.845474\pi\)
−0.884462 + 0.466612i \(0.845474\pi\)
\(98\) 2.34292 0.236671
\(99\) 0 0
\(100\) −17.3717 −1.73717
\(101\) −15.6357 −1.55581 −0.777903 0.628385i \(-0.783716\pi\)
−0.777903 + 0.628385i \(0.783716\pi\)
\(102\) 0 0
\(103\) 4.16779 0.410664 0.205332 0.978692i \(-0.434173\pi\)
0.205332 + 0.978692i \(0.434173\pi\)
\(104\) −15.1537 −1.48594
\(105\) 0 0
\(106\) 10.5682 1.02648
\(107\) −5.49663 −0.531380 −0.265690 0.964059i \(-0.585600\pi\)
−0.265690 + 0.964059i \(0.585600\pi\)
\(108\) 0 0
\(109\) −5.87819 −0.563029 −0.281514 0.959557i \(-0.590837\pi\)
−0.281514 + 0.959557i \(0.590837\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.19656 −0.113064
\(113\) 3.88240 0.365226 0.182613 0.983185i \(-0.441544\pi\)
0.182613 + 0.983185i \(0.441544\pi\)
\(114\) 0 0
\(115\) −1.28960 −0.120255
\(116\) 15.1537 1.40699
\(117\) 0 0
\(118\) −18.8108 −1.73167
\(119\) 0.146365 0.0134173
\(120\) 0 0
\(121\) 0 0
\(122\) −12.7434 −1.15373
\(123\) 0 0
\(124\) 1.02142 0.0917265
\(125\) −1.46052 −0.130633
\(126\) 0 0
\(127\) 17.6686 1.56784 0.783919 0.620863i \(-0.213218\pi\)
0.783919 + 0.620863i \(0.213218\pi\)
\(128\) −20.1751 −1.78325
\(129\) 0 0
\(130\) −1.48929 −0.130619
\(131\) 18.0288 1.57518 0.787590 0.616199i \(-0.211328\pi\)
0.787590 + 0.616199i \(0.211328\pi\)
\(132\) 0 0
\(133\) −1.83221 −0.158873
\(134\) −18.1825 −1.57073
\(135\) 0 0
\(136\) −0.510711 −0.0437931
\(137\) 21.5542 1.84150 0.920749 0.390156i \(-0.127579\pi\)
0.920749 + 0.390156i \(0.127579\pi\)
\(138\) 0 0
\(139\) −13.5395 −1.14840 −0.574202 0.818714i \(-0.694688\pi\)
−0.574202 + 0.818714i \(0.694688\pi\)
\(140\) −0.510711 −0.0431630
\(141\) 0 0
\(142\) 8.29273 0.695911
\(143\) 0 0
\(144\) 0 0
\(145\) 0.635654 0.0527882
\(146\) −37.9143 −3.13781
\(147\) 0 0
\(148\) −12.1751 −1.00079
\(149\) 10.3001 0.843815 0.421908 0.906639i \(-0.361361\pi\)
0.421908 + 0.906639i \(0.361361\pi\)
\(150\) 0 0
\(151\) −1.31836 −0.107287 −0.0536435 0.998560i \(-0.517083\pi\)
−0.0536435 + 0.998560i \(0.517083\pi\)
\(152\) 6.39312 0.518550
\(153\) 0 0
\(154\) 0 0
\(155\) 0.0428457 0.00344145
\(156\) 0 0
\(157\) 9.85677 0.786656 0.393328 0.919398i \(-0.371324\pi\)
0.393328 + 0.919398i \(0.371324\pi\)
\(158\) −30.6430 −2.43783
\(159\) 0 0
\(160\) −0.611096 −0.0483114
\(161\) 8.81079 0.694387
\(162\) 0 0
\(163\) −0.292731 −0.0229285 −0.0114642 0.999934i \(-0.503649\pi\)
−0.0114642 + 0.999934i \(0.503649\pi\)
\(164\) 9.78202 0.763847
\(165\) 0 0
\(166\) −30.2400 −2.34708
\(167\) −15.7360 −1.21769 −0.608846 0.793289i \(-0.708367\pi\)
−0.608846 + 0.793289i \(0.708367\pi\)
\(168\) 0 0
\(169\) 5.86098 0.450845
\(170\) −0.0501921 −0.00384956
\(171\) 0 0
\(172\) 27.4292 2.09146
\(173\) −17.7360 −1.34845 −0.674223 0.738528i \(-0.735521\pi\)
−0.674223 + 0.738528i \(0.735521\pi\)
\(174\) 0 0
\(175\) 4.97858 0.376345
\(176\) 0 0
\(177\) 0 0
\(178\) 22.9859 1.72287
\(179\) 23.2285 1.73618 0.868088 0.496410i \(-0.165349\pi\)
0.868088 + 0.496410i \(0.165349\pi\)
\(180\) 0 0
\(181\) 25.1109 1.86648 0.933238 0.359259i \(-0.116970\pi\)
0.933238 + 0.359259i \(0.116970\pi\)
\(182\) 10.1751 0.754231
\(183\) 0 0
\(184\) −30.7434 −2.26643
\(185\) −0.510711 −0.0375483
\(186\) 0 0
\(187\) 0 0
\(188\) 3.31415 0.241710
\(189\) 0 0
\(190\) 0.628308 0.0455822
\(191\) 17.8898 1.29446 0.647228 0.762296i \(-0.275928\pi\)
0.647228 + 0.762296i \(0.275928\pi\)
\(192\) 0 0
\(193\) 10.3288 0.743487 0.371743 0.928336i \(-0.378760\pi\)
0.371743 + 0.928336i \(0.378760\pi\)
\(194\) −40.8181 −2.93057
\(195\) 0 0
\(196\) 3.48929 0.249235
\(197\) 14.8610 1.05880 0.529401 0.848372i \(-0.322417\pi\)
0.529401 + 0.848372i \(0.322417\pi\)
\(198\) 0 0
\(199\) 0.0674041 0.00477815 0.00238908 0.999997i \(-0.499240\pi\)
0.00238908 + 0.999997i \(0.499240\pi\)
\(200\) −17.3717 −1.22836
\(201\) 0 0
\(202\) −36.6331 −2.57750
\(203\) −4.34292 −0.304813
\(204\) 0 0
\(205\) 0.410327 0.0286585
\(206\) 9.76481 0.680346
\(207\) 0 0
\(208\) −5.19656 −0.360316
\(209\) 0 0
\(210\) 0 0
\(211\) 13.1323 0.904064 0.452032 0.892002i \(-0.350699\pi\)
0.452032 + 0.892002i \(0.350699\pi\)
\(212\) 15.7392 1.08097
\(213\) 0 0
\(214\) −12.8782 −0.880335
\(215\) 1.15058 0.0784687
\(216\) 0 0
\(217\) −0.292731 −0.0198719
\(218\) −13.7722 −0.932768
\(219\) 0 0
\(220\) 0 0
\(221\) 0.635654 0.0427587
\(222\) 0 0
\(223\) 5.37169 0.359715 0.179858 0.983693i \(-0.442436\pi\)
0.179858 + 0.983693i \(0.442436\pi\)
\(224\) 4.17513 0.278963
\(225\) 0 0
\(226\) 9.09617 0.605068
\(227\) −6.85785 −0.455171 −0.227586 0.973758i \(-0.573083\pi\)
−0.227586 + 0.973758i \(0.573083\pi\)
\(228\) 0 0
\(229\) −8.76060 −0.578917 −0.289458 0.957191i \(-0.593475\pi\)
−0.289458 + 0.957191i \(0.593475\pi\)
\(230\) −3.02142 −0.199227
\(231\) 0 0
\(232\) 15.1537 0.994890
\(233\) 1.36435 0.0893813 0.0446906 0.999001i \(-0.485770\pi\)
0.0446906 + 0.999001i \(0.485770\pi\)
\(234\) 0 0
\(235\) 0.139019 0.00906861
\(236\) −28.0147 −1.82360
\(237\) 0 0
\(238\) 0.342923 0.0222284
\(239\) 4.97858 0.322037 0.161019 0.986951i \(-0.448522\pi\)
0.161019 + 0.986951i \(0.448522\pi\)
\(240\) 0 0
\(241\) −14.1678 −0.912627 −0.456314 0.889819i \(-0.650831\pi\)
−0.456314 + 0.889819i \(0.650831\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −18.9786 −1.21498
\(245\) 0.146365 0.00935095
\(246\) 0 0
\(247\) −7.95715 −0.506302
\(248\) 1.02142 0.0648604
\(249\) 0 0
\(250\) −3.42188 −0.216419
\(251\) 2.82908 0.178570 0.0892848 0.996006i \(-0.471542\pi\)
0.0892848 + 0.996006i \(0.471542\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 41.3963 2.59743
\(255\) 0 0
\(256\) −22.9185 −1.43241
\(257\) 22.9901 1.43409 0.717043 0.697029i \(-0.245495\pi\)
0.717043 + 0.697029i \(0.245495\pi\)
\(258\) 0 0
\(259\) 3.48929 0.216814
\(260\) −2.21798 −0.137553
\(261\) 0 0
\(262\) 42.2400 2.60960
\(263\) 3.63986 0.224444 0.112222 0.993683i \(-0.464203\pi\)
0.112222 + 0.993683i \(0.464203\pi\)
\(264\) 0 0
\(265\) 0.660212 0.0405565
\(266\) −4.29273 −0.263204
\(267\) 0 0
\(268\) −27.0790 −1.65411
\(269\) 7.54683 0.460138 0.230069 0.973174i \(-0.426105\pi\)
0.230069 + 0.973174i \(0.426105\pi\)
\(270\) 0 0
\(271\) −23.6644 −1.43751 −0.718756 0.695263i \(-0.755288\pi\)
−0.718756 + 0.695263i \(0.755288\pi\)
\(272\) −0.175135 −0.0106191
\(273\) 0 0
\(274\) 50.4998 3.05080
\(275\) 0 0
\(276\) 0 0
\(277\) −31.8929 −1.91626 −0.958129 0.286337i \(-0.907562\pi\)
−0.958129 + 0.286337i \(0.907562\pi\)
\(278\) −31.7220 −1.90256
\(279\) 0 0
\(280\) −0.510711 −0.0305208
\(281\) −30.5082 −1.81997 −0.909983 0.414645i \(-0.863906\pi\)
−0.909983 + 0.414645i \(0.863906\pi\)
\(282\) 0 0
\(283\) −22.6430 −1.34599 −0.672993 0.739649i \(-0.734992\pi\)
−0.672993 + 0.739649i \(0.734992\pi\)
\(284\) 12.3503 0.732854
\(285\) 0 0
\(286\) 0 0
\(287\) −2.80344 −0.165482
\(288\) 0 0
\(289\) −16.9786 −0.998740
\(290\) 1.48929 0.0874540
\(291\) 0 0
\(292\) −56.4653 −3.30438
\(293\) −1.81079 −0.105787 −0.0528937 0.998600i \(-0.516844\pi\)
−0.0528937 + 0.998600i \(0.516844\pi\)
\(294\) 0 0
\(295\) −1.17513 −0.0684190
\(296\) −12.1751 −0.707665
\(297\) 0 0
\(298\) 24.1323 1.39795
\(299\) 38.2646 2.21290
\(300\) 0 0
\(301\) −7.86098 −0.453099
\(302\) −3.08883 −0.177742
\(303\) 0 0
\(304\) 2.19235 0.125740
\(305\) −0.796096 −0.0455843
\(306\) 0 0
\(307\) −29.8469 −1.70345 −0.851726 0.523987i \(-0.824444\pi\)
−0.851726 + 0.523987i \(0.824444\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0.100384 0.00570144
\(311\) 5.09304 0.288800 0.144400 0.989519i \(-0.453875\pi\)
0.144400 + 0.989519i \(0.453875\pi\)
\(312\) 0 0
\(313\) −22.0319 −1.24532 −0.622658 0.782494i \(-0.713947\pi\)
−0.622658 + 0.782494i \(0.713947\pi\)
\(314\) 23.0937 1.30325
\(315\) 0 0
\(316\) −45.6363 −2.56724
\(317\) 7.08883 0.398148 0.199074 0.979984i \(-0.436207\pi\)
0.199074 + 0.979984i \(0.436207\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.78202 −0.0996179
\(321\) 0 0
\(322\) 20.6430 1.15039
\(323\) −0.268173 −0.0149215
\(324\) 0 0
\(325\) 21.6216 1.19935
\(326\) −0.685846 −0.0379855
\(327\) 0 0
\(328\) 9.78202 0.540122
\(329\) −0.949808 −0.0523646
\(330\) 0 0
\(331\) −34.6044 −1.90203 −0.951014 0.309148i \(-0.899956\pi\)
−0.951014 + 0.309148i \(0.899956\pi\)
\(332\) −45.0361 −2.47168
\(333\) 0 0
\(334\) −36.8683 −2.01735
\(335\) −1.13588 −0.0620599
\(336\) 0 0
\(337\) 12.5254 0.682302 0.341151 0.940008i \(-0.389183\pi\)
0.341151 + 0.940008i \(0.389183\pi\)
\(338\) 13.7318 0.746913
\(339\) 0 0
\(340\) −0.0747505 −0.00405392
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 27.4292 1.47889
\(345\) 0 0
\(346\) −41.5542 −2.23397
\(347\) 27.6461 1.48412 0.742061 0.670332i \(-0.233848\pi\)
0.742061 + 0.670332i \(0.233848\pi\)
\(348\) 0 0
\(349\) 11.2969 0.604711 0.302356 0.953195i \(-0.402227\pi\)
0.302356 + 0.953195i \(0.402227\pi\)
\(350\) 11.6644 0.623490
\(351\) 0 0
\(352\) 0 0
\(353\) −16.8034 −0.894357 −0.447178 0.894445i \(-0.647571\pi\)
−0.447178 + 0.894445i \(0.647571\pi\)
\(354\) 0 0
\(355\) 0.518058 0.0274957
\(356\) 34.2327 1.81433
\(357\) 0 0
\(358\) 54.4225 2.87632
\(359\) 29.0607 1.53376 0.766882 0.641788i \(-0.221807\pi\)
0.766882 + 0.641788i \(0.221807\pi\)
\(360\) 0 0
\(361\) −15.6430 −0.823316
\(362\) 58.8328 3.09218
\(363\) 0 0
\(364\) 15.1537 0.794270
\(365\) −2.36856 −0.123976
\(366\) 0 0
\(367\) 17.6974 0.923797 0.461898 0.886933i \(-0.347168\pi\)
0.461898 + 0.886933i \(0.347168\pi\)
\(368\) −10.5426 −0.549572
\(369\) 0 0
\(370\) −1.19656 −0.0622061
\(371\) −4.51071 −0.234184
\(372\) 0 0
\(373\) 12.7820 0.661828 0.330914 0.943661i \(-0.392643\pi\)
0.330914 + 0.943661i \(0.392643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 3.31415 0.170914
\(377\) −18.8610 −0.971390
\(378\) 0 0
\(379\) −19.6258 −1.00811 −0.504055 0.863672i \(-0.668159\pi\)
−0.504055 + 0.863672i \(0.668159\pi\)
\(380\) 0.935731 0.0480020
\(381\) 0 0
\(382\) 41.9143 2.14452
\(383\) −29.5500 −1.50993 −0.754966 0.655764i \(-0.772347\pi\)
−0.754966 + 0.655764i \(0.772347\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 24.1997 1.23173
\(387\) 0 0
\(388\) −60.7900 −3.08614
\(389\) 13.0962 0.664002 0.332001 0.943279i \(-0.392276\pi\)
0.332001 + 0.943279i \(0.392276\pi\)
\(390\) 0 0
\(391\) 1.28960 0.0652176
\(392\) 3.48929 0.176236
\(393\) 0 0
\(394\) 34.8181 1.75411
\(395\) −1.91431 −0.0963193
\(396\) 0 0
\(397\) 24.4679 1.22801 0.614003 0.789303i \(-0.289558\pi\)
0.614003 + 0.789303i \(0.289558\pi\)
\(398\) 0.157923 0.00791595
\(399\) 0 0
\(400\) −5.95715 −0.297858
\(401\) −24.1751 −1.20725 −0.603624 0.797269i \(-0.706277\pi\)
−0.603624 + 0.797269i \(0.706277\pi\)
\(402\) 0 0
\(403\) −1.27131 −0.0633284
\(404\) −54.5573 −2.71433
\(405\) 0 0
\(406\) −10.1751 −0.504983
\(407\) 0 0
\(408\) 0 0
\(409\) −5.22112 −0.258168 −0.129084 0.991634i \(-0.541204\pi\)
−0.129084 + 0.991634i \(0.541204\pi\)
\(410\) 0.961365 0.0474784
\(411\) 0 0
\(412\) 14.5426 0.716463
\(413\) 8.02877 0.395070
\(414\) 0 0
\(415\) −1.88913 −0.0927339
\(416\) 18.1323 0.889009
\(417\) 0 0
\(418\) 0 0
\(419\) 22.6718 1.10759 0.553794 0.832654i \(-0.313179\pi\)
0.553794 + 0.832654i \(0.313179\pi\)
\(420\) 0 0
\(421\) −2.41454 −0.117677 −0.0588387 0.998268i \(-0.518740\pi\)
−0.0588387 + 0.998268i \(0.518740\pi\)
\(422\) 30.7679 1.49776
\(423\) 0 0
\(424\) 15.7392 0.764362
\(425\) 0.728692 0.0353467
\(426\) 0 0
\(427\) 5.43910 0.263216
\(428\) −19.1793 −0.927069
\(429\) 0 0
\(430\) 2.69571 0.129999
\(431\) −1.35700 −0.0653644 −0.0326822 0.999466i \(-0.510405\pi\)
−0.0326822 + 0.999466i \(0.510405\pi\)
\(432\) 0 0
\(433\) 20.6503 0.992392 0.496196 0.868210i \(-0.334730\pi\)
0.496196 + 0.868210i \(0.334730\pi\)
\(434\) −0.685846 −0.0329217
\(435\) 0 0
\(436\) −20.5107 −0.982285
\(437\) −16.1432 −0.772235
\(438\) 0 0
\(439\) 20.3074 0.969220 0.484610 0.874730i \(-0.338961\pi\)
0.484610 + 0.874730i \(0.338961\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.48929 0.0708382
\(443\) 9.81752 0.466444 0.233222 0.972423i \(-0.425073\pi\)
0.233222 + 0.972423i \(0.425073\pi\)
\(444\) 0 0
\(445\) 1.43596 0.0680711
\(446\) 12.5855 0.595939
\(447\) 0 0
\(448\) 12.1751 0.575221
\(449\) −10.0748 −0.475457 −0.237728 0.971332i \(-0.576403\pi\)
−0.237728 + 0.971332i \(0.576403\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 13.5468 0.637189
\(453\) 0 0
\(454\) −16.0674 −0.754081
\(455\) 0.635654 0.0297999
\(456\) 0 0
\(457\) −14.6791 −0.686660 −0.343330 0.939215i \(-0.611555\pi\)
−0.343330 + 0.939215i \(0.611555\pi\)
\(458\) −20.5254 −0.959089
\(459\) 0 0
\(460\) −4.49977 −0.209803
\(461\) 27.1396 1.26402 0.632009 0.774961i \(-0.282230\pi\)
0.632009 + 0.774961i \(0.282230\pi\)
\(462\) 0 0
\(463\) 23.2369 1.07991 0.539955 0.841694i \(-0.318441\pi\)
0.539955 + 0.841694i \(0.318441\pi\)
\(464\) 5.19656 0.241244
\(465\) 0 0
\(466\) 3.19656 0.148078
\(467\) −23.8568 −1.10396 −0.551980 0.833857i \(-0.686127\pi\)
−0.551980 + 0.833857i \(0.686127\pi\)
\(468\) 0 0
\(469\) 7.76060 0.358351
\(470\) 0.325711 0.0150239
\(471\) 0 0
\(472\) −28.0147 −1.28948
\(473\) 0 0
\(474\) 0 0
\(475\) −9.12181 −0.418537
\(476\) 0.510711 0.0234084
\(477\) 0 0
\(478\) 11.6644 0.533518
\(479\) −17.2081 −0.786259 −0.393129 0.919483i \(-0.628608\pi\)
−0.393129 + 0.919483i \(0.628608\pi\)
\(480\) 0 0
\(481\) 15.1537 0.690950
\(482\) −33.1940 −1.51195
\(483\) 0 0
\(484\) 0 0
\(485\) −2.54996 −0.115788
\(486\) 0 0
\(487\) −3.16044 −0.143213 −0.0716066 0.997433i \(-0.522813\pi\)
−0.0716066 + 0.997433i \(0.522813\pi\)
\(488\) −18.9786 −0.859120
\(489\) 0 0
\(490\) 0.342923 0.0154917
\(491\) 4.22533 0.190686 0.0953432 0.995444i \(-0.469605\pi\)
0.0953432 + 0.995444i \(0.469605\pi\)
\(492\) 0 0
\(493\) −0.635654 −0.0286284
\(494\) −18.6430 −0.838788
\(495\) 0 0
\(496\) 0.350269 0.0157276
\(497\) −3.53948 −0.158767
\(498\) 0 0
\(499\) 30.5615 1.36812 0.684061 0.729425i \(-0.260212\pi\)
0.684061 + 0.729425i \(0.260212\pi\)
\(500\) −5.09617 −0.227908
\(501\) 0 0
\(502\) 6.62831 0.295836
\(503\) −0.513847 −0.0229113 −0.0114557 0.999934i \(-0.503647\pi\)
−0.0114557 + 0.999934i \(0.503647\pi\)
\(504\) 0 0
\(505\) −2.28852 −0.101838
\(506\) 0 0
\(507\) 0 0
\(508\) 61.6510 2.73532
\(509\) −24.9217 −1.10463 −0.552316 0.833635i \(-0.686256\pi\)
−0.552316 + 0.833635i \(0.686256\pi\)
\(510\) 0 0
\(511\) 16.1825 0.715871
\(512\) −13.3461 −0.589818
\(513\) 0 0
\(514\) 53.8641 2.37584
\(515\) 0.610020 0.0268807
\(516\) 0 0
\(517\) 0 0
\(518\) 8.17513 0.359195
\(519\) 0 0
\(520\) −2.21798 −0.0972649
\(521\) 31.8855 1.39693 0.698465 0.715644i \(-0.253867\pi\)
0.698465 + 0.715644i \(0.253867\pi\)
\(522\) 0 0
\(523\) −5.34713 −0.233814 −0.116907 0.993143i \(-0.537298\pi\)
−0.116907 + 0.993143i \(0.537298\pi\)
\(524\) 62.9076 2.74813
\(525\) 0 0
\(526\) 8.52792 0.371835
\(527\) −0.0428457 −0.00186639
\(528\) 0 0
\(529\) 54.6300 2.37522
\(530\) 1.54683 0.0671899
\(531\) 0 0
\(532\) −6.39312 −0.277177
\(533\) −12.1751 −0.527364
\(534\) 0 0
\(535\) −0.804518 −0.0347823
\(536\) −27.0790 −1.16963
\(537\) 0 0
\(538\) 17.6816 0.762309
\(539\) 0 0
\(540\) 0 0
\(541\) −27.8610 −1.19784 −0.598919 0.800810i \(-0.704403\pi\)
−0.598919 + 0.800810i \(0.704403\pi\)
\(542\) −55.4439 −2.38152
\(543\) 0 0
\(544\) 0.611096 0.0262005
\(545\) −0.860365 −0.0368540
\(546\) 0 0
\(547\) 30.7005 1.31266 0.656330 0.754474i \(-0.272108\pi\)
0.656330 + 0.754474i \(0.272108\pi\)
\(548\) 75.2087 3.21276
\(549\) 0 0
\(550\) 0 0
\(551\) 7.95715 0.338986
\(552\) 0 0
\(553\) 13.0790 0.556174
\(554\) −74.7226 −3.17466
\(555\) 0 0
\(556\) −47.2432 −2.00356
\(557\) 16.7434 0.709440 0.354720 0.934973i \(-0.384576\pi\)
0.354720 + 0.934973i \(0.384576\pi\)
\(558\) 0 0
\(559\) −34.1396 −1.44395
\(560\) −0.175135 −0.00740079
\(561\) 0 0
\(562\) −71.4783 −3.01513
\(563\) −27.6932 −1.16713 −0.583564 0.812067i \(-0.698342\pi\)
−0.583564 + 0.812067i \(0.698342\pi\)
\(564\) 0 0
\(565\) 0.568250 0.0239065
\(566\) −53.0508 −2.22989
\(567\) 0 0
\(568\) 12.3503 0.518206
\(569\) −11.6827 −0.489765 −0.244882 0.969553i \(-0.578749\pi\)
−0.244882 + 0.969553i \(0.578749\pi\)
\(570\) 0 0
\(571\) 24.1151 1.00918 0.504592 0.863358i \(-0.331643\pi\)
0.504592 + 0.863358i \(0.331643\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −6.56825 −0.274153
\(575\) 43.8652 1.82930
\(576\) 0 0
\(577\) 44.2572 1.84245 0.921226 0.389027i \(-0.127189\pi\)
0.921226 + 0.389027i \(0.127189\pi\)
\(578\) −39.7795 −1.65461
\(579\) 0 0
\(580\) 2.21798 0.0920966
\(581\) 12.9070 0.535471
\(582\) 0 0
\(583\) 0 0
\(584\) −56.4653 −2.33655
\(585\) 0 0
\(586\) −4.24254 −0.175258
\(587\) −8.42188 −0.347608 −0.173804 0.984780i \(-0.555606\pi\)
−0.173804 + 0.984780i \(0.555606\pi\)
\(588\) 0 0
\(589\) 0.536345 0.0220997
\(590\) −2.75325 −0.113350
\(591\) 0 0
\(592\) −4.17513 −0.171597
\(593\) 1.94560 0.0798961 0.0399480 0.999202i \(-0.487281\pi\)
0.0399480 + 0.999202i \(0.487281\pi\)
\(594\) 0 0
\(595\) 0.0214229 0.000878251 0
\(596\) 35.9399 1.47216
\(597\) 0 0
\(598\) 89.6510 3.66610
\(599\) −31.0361 −1.26810 −0.634051 0.773292i \(-0.718609\pi\)
−0.634051 + 0.773292i \(0.718609\pi\)
\(600\) 0 0
\(601\) 5.80031 0.236599 0.118300 0.992978i \(-0.462256\pi\)
0.118300 + 0.992978i \(0.462256\pi\)
\(602\) −18.4177 −0.750648
\(603\) 0 0
\(604\) −4.60015 −0.187178
\(605\) 0 0
\(606\) 0 0
\(607\) −2.57560 −0.104540 −0.0522701 0.998633i \(-0.516646\pi\)
−0.0522701 + 0.998633i \(0.516646\pi\)
\(608\) −7.64973 −0.310238
\(609\) 0 0
\(610\) −1.86519 −0.0755194
\(611\) −4.12494 −0.166877
\(612\) 0 0
\(613\) −15.8353 −0.639584 −0.319792 0.947488i \(-0.603613\pi\)
−0.319792 + 0.947488i \(0.603613\pi\)
\(614\) −69.9290 −2.82210
\(615\) 0 0
\(616\) 0 0
\(617\) 1.72448 0.0694250 0.0347125 0.999397i \(-0.488948\pi\)
0.0347125 + 0.999397i \(0.488948\pi\)
\(618\) 0 0
\(619\) −17.3288 −0.696505 −0.348253 0.937401i \(-0.613225\pi\)
−0.348253 + 0.937401i \(0.613225\pi\)
\(620\) 0.149501 0.00600411
\(621\) 0 0
\(622\) 11.9326 0.478454
\(623\) −9.81079 −0.393061
\(624\) 0 0
\(625\) 24.6791 0.987165
\(626\) −51.6191 −2.06311
\(627\) 0 0
\(628\) 34.3931 1.37243
\(629\) 0.510711 0.0203634
\(630\) 0 0
\(631\) 6.54683 0.260625 0.130313 0.991473i \(-0.458402\pi\)
0.130313 + 0.991473i \(0.458402\pi\)
\(632\) −45.6363 −1.81531
\(633\) 0 0
\(634\) 16.6086 0.659611
\(635\) 2.58608 0.102625
\(636\) 0 0
\(637\) −4.34292 −0.172073
\(638\) 0 0
\(639\) 0 0
\(640\) −2.95294 −0.116725
\(641\) −16.0930 −0.635637 −0.317818 0.948152i \(-0.602950\pi\)
−0.317818 + 0.948152i \(0.602950\pi\)
\(642\) 0 0
\(643\) −3.21377 −0.126739 −0.0633694 0.997990i \(-0.520185\pi\)
−0.0633694 + 0.997990i \(0.520185\pi\)
\(644\) 30.7434 1.21146
\(645\) 0 0
\(646\) −0.628308 −0.0247204
\(647\) 37.8083 1.48640 0.743198 0.669071i \(-0.233308\pi\)
0.743198 + 0.669071i \(0.233308\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 50.6577 1.98696
\(651\) 0 0
\(652\) −1.02142 −0.0400020
\(653\) 13.9754 0.546901 0.273451 0.961886i \(-0.411835\pi\)
0.273451 + 0.961886i \(0.411835\pi\)
\(654\) 0 0
\(655\) 2.63879 0.103106
\(656\) 3.35448 0.130970
\(657\) 0 0
\(658\) −2.22533 −0.0867523
\(659\) 8.31729 0.323996 0.161998 0.986791i \(-0.448206\pi\)
0.161998 + 0.986791i \(0.448206\pi\)
\(660\) 0 0
\(661\) −38.2474 −1.48765 −0.743825 0.668374i \(-0.766990\pi\)
−0.743825 + 0.668374i \(0.766990\pi\)
\(662\) −81.0754 −3.15108
\(663\) 0 0
\(664\) −45.0361 −1.74774
\(665\) −0.268173 −0.0103993
\(666\) 0 0
\(667\) −38.2646 −1.48161
\(668\) −54.9076 −2.12444
\(669\) 0 0
\(670\) −2.66129 −0.102815
\(671\) 0 0
\(672\) 0 0
\(673\) 7.86098 0.303019 0.151509 0.988456i \(-0.451587\pi\)
0.151509 + 0.988456i \(0.451587\pi\)
\(674\) 29.3461 1.13037
\(675\) 0 0
\(676\) 20.4507 0.786564
\(677\) −31.9603 −1.22833 −0.614167 0.789176i \(-0.710508\pi\)
−0.614167 + 0.789176i \(0.710508\pi\)
\(678\) 0 0
\(679\) 17.4219 0.668591
\(680\) −0.0747505 −0.00286655
\(681\) 0 0
\(682\) 0 0
\(683\) −35.1856 −1.34634 −0.673170 0.739488i \(-0.735068\pi\)
−0.673170 + 0.739488i \(0.735068\pi\)
\(684\) 0 0
\(685\) 3.15479 0.120538
\(686\) −2.34292 −0.0894532
\(687\) 0 0
\(688\) 9.40612 0.358605
\(689\) −19.5897 −0.746307
\(690\) 0 0
\(691\) 2.05754 0.0782725 0.0391362 0.999234i \(-0.487539\pi\)
0.0391362 + 0.999234i \(0.487539\pi\)
\(692\) −61.8862 −2.35256
\(693\) 0 0
\(694\) 64.7728 2.45874
\(695\) −1.98171 −0.0751706
\(696\) 0 0
\(697\) −0.410327 −0.0155423
\(698\) 26.4679 1.00182
\(699\) 0 0
\(700\) 17.3717 0.656588
\(701\) −11.1207 −0.420024 −0.210012 0.977699i \(-0.567350\pi\)
−0.210012 + 0.977699i \(0.567350\pi\)
\(702\) 0 0
\(703\) −6.39312 −0.241121
\(704\) 0 0
\(705\) 0 0
\(706\) −39.3692 −1.48168
\(707\) 15.6357 0.588039
\(708\) 0 0
\(709\) 26.6901 1.00237 0.501183 0.865341i \(-0.332898\pi\)
0.501183 + 0.865341i \(0.332898\pi\)
\(710\) 1.21377 0.0455520
\(711\) 0 0
\(712\) 34.2327 1.28292
\(713\) −2.57919 −0.0965915
\(714\) 0 0
\(715\) 0 0
\(716\) 81.0508 3.02901
\(717\) 0 0
\(718\) 68.0869 2.54098
\(719\) −19.8996 −0.742130 −0.371065 0.928607i \(-0.621007\pi\)
−0.371065 + 0.928607i \(0.621007\pi\)
\(720\) 0 0
\(721\) −4.16779 −0.155217
\(722\) −36.6503 −1.36398
\(723\) 0 0
\(724\) 87.6191 3.25634
\(725\) −21.6216 −0.803005
\(726\) 0 0
\(727\) 17.3618 0.643915 0.321957 0.946754i \(-0.395659\pi\)
0.321957 + 0.946754i \(0.395659\pi\)
\(728\) 15.1537 0.561634
\(729\) 0 0
\(730\) −5.54935 −0.205391
\(731\) −1.15058 −0.0425556
\(732\) 0 0
\(733\) −33.3790 −1.23288 −0.616441 0.787401i \(-0.711426\pi\)
−0.616441 + 0.787401i \(0.711426\pi\)
\(734\) 41.4637 1.53045
\(735\) 0 0
\(736\) 36.7862 1.35596
\(737\) 0 0
\(738\) 0 0
\(739\) −15.2369 −0.560498 −0.280249 0.959927i \(-0.590417\pi\)
−0.280249 + 0.959927i \(0.590417\pi\)
\(740\) −1.78202 −0.0655083
\(741\) 0 0
\(742\) −10.5682 −0.387973
\(743\) −23.3387 −0.856214 −0.428107 0.903728i \(-0.640819\pi\)
−0.428107 + 0.903728i \(0.640819\pi\)
\(744\) 0 0
\(745\) 1.50758 0.0552333
\(746\) 29.9473 1.09645
\(747\) 0 0
\(748\) 0 0
\(749\) 5.49663 0.200843
\(750\) 0 0
\(751\) 29.6174 1.08075 0.540377 0.841423i \(-0.318282\pi\)
0.540377 + 0.841423i \(0.318282\pi\)
\(752\) 1.13650 0.0414439
\(753\) 0 0
\(754\) −44.1898 −1.60930
\(755\) −0.192963 −0.00702265
\(756\) 0 0
\(757\) −25.4360 −0.924486 −0.462243 0.886753i \(-0.652955\pi\)
−0.462243 + 0.886753i \(0.652955\pi\)
\(758\) −45.9817 −1.67013
\(759\) 0 0
\(760\) 0.935731 0.0339425
\(761\) 44.9473 1.62934 0.814669 0.579926i \(-0.196919\pi\)
0.814669 + 0.579926i \(0.196919\pi\)
\(762\) 0 0
\(763\) 5.87819 0.212805
\(764\) 62.4225 2.25837
\(765\) 0 0
\(766\) −69.2333 −2.50150
\(767\) 34.8683 1.25902
\(768\) 0 0
\(769\) 32.6749 1.17829 0.589144 0.808028i \(-0.299465\pi\)
0.589144 + 0.808028i \(0.299465\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 36.0403 1.29712
\(773\) 49.6222 1.78479 0.892393 0.451259i \(-0.149025\pi\)
0.892393 + 0.451259i \(0.149025\pi\)
\(774\) 0 0
\(775\) −1.45738 −0.0523508
\(776\) −60.7900 −2.18223
\(777\) 0 0
\(778\) 30.6833 1.10005
\(779\) 5.13650 0.184034
\(780\) 0 0
\(781\) 0 0
\(782\) 3.02142 0.108046
\(783\) 0 0
\(784\) 1.19656 0.0427342
\(785\) 1.44269 0.0514918
\(786\) 0 0
\(787\) −40.2583 −1.43505 −0.717527 0.696531i \(-0.754726\pi\)
−0.717527 + 0.696531i \(0.754726\pi\)
\(788\) 51.8543 1.84723
\(789\) 0 0
\(790\) −4.48508 −0.159572
\(791\) −3.88240 −0.138042
\(792\) 0 0
\(793\) 23.6216 0.838827
\(794\) 57.3263 2.03444
\(795\) 0 0
\(796\) 0.235192 0.00833618
\(797\) −17.4862 −0.619391 −0.309696 0.950836i \(-0.600227\pi\)
−0.309696 + 0.950836i \(0.600227\pi\)
\(798\) 0 0
\(799\) −0.139019 −0.00491814
\(800\) 20.7862 0.734904
\(801\) 0 0
\(802\) −56.6405 −2.00004
\(803\) 0 0
\(804\) 0 0
\(805\) 1.28960 0.0454523
\(806\) −2.97858 −0.104916
\(807\) 0 0
\(808\) −54.5573 −1.91932
\(809\) 25.9473 0.912258 0.456129 0.889914i \(-0.349236\pi\)
0.456129 + 0.889914i \(0.349236\pi\)
\(810\) 0 0
\(811\) 18.2829 0.641998 0.320999 0.947079i \(-0.395981\pi\)
0.320999 + 0.947079i \(0.395981\pi\)
\(812\) −15.1537 −0.531791
\(813\) 0 0
\(814\) 0 0
\(815\) −0.0428457 −0.00150082
\(816\) 0 0
\(817\) 14.4030 0.503897
\(818\) −12.2327 −0.427705
\(819\) 0 0
\(820\) 1.43175 0.0499989
\(821\) −25.3288 −0.883983 −0.441991 0.897019i \(-0.645728\pi\)
−0.441991 + 0.897019i \(0.645728\pi\)
\(822\) 0 0
\(823\) 39.2713 1.36891 0.684456 0.729054i \(-0.260040\pi\)
0.684456 + 0.729054i \(0.260040\pi\)
\(824\) 14.5426 0.506616
\(825\) 0 0
\(826\) 18.8108 0.654511
\(827\) −27.4047 −0.952954 −0.476477 0.879187i \(-0.658086\pi\)
−0.476477 + 0.879187i \(0.658086\pi\)
\(828\) 0 0
\(829\) 38.5155 1.33770 0.668850 0.743397i \(-0.266787\pi\)
0.668850 + 0.743397i \(0.266787\pi\)
\(830\) −4.42610 −0.153632
\(831\) 0 0
\(832\) 52.8757 1.83313
\(833\) −0.146365 −0.00507126
\(834\) 0 0
\(835\) −2.30321 −0.0797060
\(836\) 0 0
\(837\) 0 0
\(838\) 53.1182 1.83494
\(839\) 17.7276 0.612025 0.306013 0.952027i \(-0.401005\pi\)
0.306013 + 0.952027i \(0.401005\pi\)
\(840\) 0 0
\(841\) −10.1390 −0.349621
\(842\) −5.65708 −0.194956
\(843\) 0 0
\(844\) 45.8223 1.57727
\(845\) 0.857845 0.0295108
\(846\) 0 0
\(847\) 0 0
\(848\) 5.39733 0.185345
\(849\) 0 0
\(850\) 1.70727 0.0585588
\(851\) 30.7434 1.05387
\(852\) 0 0
\(853\) −15.3032 −0.523972 −0.261986 0.965072i \(-0.584377\pi\)
−0.261986 + 0.965072i \(0.584377\pi\)
\(854\) 12.7434 0.436070
\(855\) 0 0
\(856\) −19.1793 −0.655537
\(857\) 10.7722 0.367970 0.183985 0.982929i \(-0.441100\pi\)
0.183985 + 0.982929i \(0.441100\pi\)
\(858\) 0 0
\(859\) 47.2516 1.61220 0.806101 0.591777i \(-0.201574\pi\)
0.806101 + 0.591777i \(0.201574\pi\)
\(860\) 4.01469 0.136900
\(861\) 0 0
\(862\) −3.17935 −0.108289
\(863\) 34.0294 1.15837 0.579187 0.815195i \(-0.303370\pi\)
0.579187 + 0.815195i \(0.303370\pi\)
\(864\) 0 0
\(865\) −2.59594 −0.0882647
\(866\) 48.3822 1.64409
\(867\) 0 0
\(868\) −1.02142 −0.0346694
\(869\) 0 0
\(870\) 0 0
\(871\) 33.7037 1.14201
\(872\) −20.5107 −0.694580
\(873\) 0 0
\(874\) −37.8223 −1.27936
\(875\) 1.46052 0.0493746
\(876\) 0 0
\(877\) 3.25410 0.109883 0.0549415 0.998490i \(-0.482503\pi\)
0.0549415 + 0.998490i \(0.482503\pi\)
\(878\) 47.5787 1.60570
\(879\) 0 0
\(880\) 0 0
\(881\) 13.3545 0.449924 0.224962 0.974368i \(-0.427774\pi\)
0.224962 + 0.974368i \(0.427774\pi\)
\(882\) 0 0
\(883\) −25.4741 −0.857273 −0.428636 0.903477i \(-0.641006\pi\)
−0.428636 + 0.903477i \(0.641006\pi\)
\(884\) 2.21798 0.0745988
\(885\) 0 0
\(886\) 23.0017 0.772757
\(887\) 21.6932 0.728386 0.364193 0.931323i \(-0.381345\pi\)
0.364193 + 0.931323i \(0.381345\pi\)
\(888\) 0 0
\(889\) −17.6686 −0.592587
\(890\) 3.36435 0.112773
\(891\) 0 0
\(892\) 18.7434 0.627575
\(893\) 1.74025 0.0582352
\(894\) 0 0
\(895\) 3.39985 0.113644
\(896\) 20.1751 0.674004
\(897\) 0 0
\(898\) −23.6044 −0.787688
\(899\) 1.27131 0.0424005
\(900\) 0 0
\(901\) −0.660212 −0.0219949
\(902\) 0 0
\(903\) 0 0
\(904\) 13.5468 0.450561
\(905\) 3.67536 0.122173
\(906\) 0 0
\(907\) 1.21377 0.0403026 0.0201513 0.999797i \(-0.493585\pi\)
0.0201513 + 0.999797i \(0.493585\pi\)
\(908\) −23.9290 −0.794112
\(909\) 0 0
\(910\) 1.48929 0.0493694
\(911\) −57.0080 −1.88876 −0.944379 0.328859i \(-0.893336\pi\)
−0.944379 + 0.328859i \(0.893336\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −34.3920 −1.13759
\(915\) 0 0
\(916\) −30.5682 −1.01000
\(917\) −18.0288 −0.595362
\(918\) 0 0
\(919\) −6.45065 −0.212787 −0.106394 0.994324i \(-0.533930\pi\)
−0.106394 + 0.994324i \(0.533930\pi\)
\(920\) −4.49977 −0.148353
\(921\) 0 0
\(922\) 63.5861 2.09410
\(923\) −15.3717 −0.505965
\(924\) 0 0
\(925\) 17.3717 0.571178
\(926\) 54.4422 1.78908
\(927\) 0 0
\(928\) −18.1323 −0.595222
\(929\) −2.74652 −0.0901104 −0.0450552 0.998984i \(-0.514346\pi\)
−0.0450552 + 0.998984i \(0.514346\pi\)
\(930\) 0 0
\(931\) 1.83221 0.0600483
\(932\) 4.76060 0.155939
\(933\) 0 0
\(934\) −55.8946 −1.82893
\(935\) 0 0
\(936\) 0 0
\(937\) −25.8923 −0.845864 −0.422932 0.906162i \(-0.638999\pi\)
−0.422932 + 0.906162i \(0.638999\pi\)
\(938\) 18.1825 0.593679
\(939\) 0 0
\(940\) 0.485078 0.0158215
\(941\) −9.01890 −0.294008 −0.147004 0.989136i \(-0.546963\pi\)
−0.147004 + 0.989136i \(0.546963\pi\)
\(942\) 0 0
\(943\) −24.7005 −0.804360
\(944\) −9.60688 −0.312677
\(945\) 0 0
\(946\) 0 0
\(947\) −39.2003 −1.27384 −0.636919 0.770930i \(-0.719792\pi\)
−0.636919 + 0.770930i \(0.719792\pi\)
\(948\) 0 0
\(949\) 70.2793 2.28136
\(950\) −21.3717 −0.693389
\(951\) 0 0
\(952\) 0.510711 0.0165523
\(953\) −55.4464 −1.79609 −0.898043 0.439907i \(-0.855011\pi\)
−0.898043 + 0.439907i \(0.855011\pi\)
\(954\) 0 0
\(955\) 2.61844 0.0847308
\(956\) 17.3717 0.561841
\(957\) 0 0
\(958\) −40.3173 −1.30259
\(959\) −21.5542 −0.696021
\(960\) 0 0
\(961\) −30.9143 −0.997236
\(962\) 35.5040 1.14469
\(963\) 0 0
\(964\) −49.4355 −1.59221
\(965\) 1.51179 0.0486661
\(966\) 0 0
\(967\) −23.9467 −0.770073 −0.385037 0.922901i \(-0.625811\pi\)
−0.385037 + 0.922901i \(0.625811\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −5.97437 −0.191825
\(971\) 6.42188 0.206088 0.103044 0.994677i \(-0.467142\pi\)
0.103044 + 0.994677i \(0.467142\pi\)
\(972\) 0 0
\(973\) 13.5395 0.434056
\(974\) −7.40467 −0.237261
\(975\) 0 0
\(976\) −6.50819 −0.208322
\(977\) −38.5584 −1.23359 −0.616796 0.787123i \(-0.711570\pi\)
−0.616796 + 0.787123i \(0.711570\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.510711 0.0163141
\(981\) 0 0
\(982\) 9.89962 0.315909
\(983\) −21.6069 −0.689153 −0.344576 0.938758i \(-0.611977\pi\)
−0.344576 + 0.938758i \(0.611977\pi\)
\(984\) 0 0
\(985\) 2.17513 0.0693056
\(986\) −1.48929 −0.0474286
\(987\) 0 0
\(988\) −27.7648 −0.883316
\(989\) −69.2614 −2.20239
\(990\) 0 0
\(991\) 34.4507 1.09436 0.547181 0.837015i \(-0.315701\pi\)
0.547181 + 0.837015i \(0.315701\pi\)
\(992\) −1.22219 −0.0388046
\(993\) 0 0
\(994\) −8.29273 −0.263029
\(995\) 0.00986564 0.000312762 0
\(996\) 0 0
\(997\) 7.76733 0.245994 0.122997 0.992407i \(-0.460749\pi\)
0.122997 + 0.992407i \(0.460749\pi\)
\(998\) 71.6033 2.26656
\(999\) 0 0
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 7623.2.a.cc.1.3 3
3.2 odd 2 2541.2.a.bh.1.1 3
11.10 odd 2 7623.2.a.ca.1.1 3
33.32 even 2 2541.2.a.bj.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.bh.1.1 3 3.2 odd 2
2541.2.a.bj.1.3 yes 3 33.32 even 2
7623.2.a.ca.1.1 3 11.10 odd 2
7623.2.a.cc.1.3 3 1.1 even 1 trivial