Properties

Label 3822.2.a.bu.1.2
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3822.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.41421 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.41421 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.41421 q^{10} +2.41421 q^{11} +1.00000 q^{12} -1.00000 q^{13} +3.41421 q^{15} +1.00000 q^{16} -0.414214 q^{17} +1.00000 q^{18} +7.82843 q^{19} +3.41421 q^{20} +2.41421 q^{22} -1.41421 q^{23} +1.00000 q^{24} +6.65685 q^{25} -1.00000 q^{26} +1.00000 q^{27} +3.82843 q^{29} +3.41421 q^{30} -8.48528 q^{31} +1.00000 q^{32} +2.41421 q^{33} -0.414214 q^{34} +1.00000 q^{36} -1.41421 q^{37} +7.82843 q^{38} -1.00000 q^{39} +3.41421 q^{40} -9.89949 q^{41} -10.4853 q^{43} +2.41421 q^{44} +3.41421 q^{45} -1.41421 q^{46} -1.00000 q^{47} +1.00000 q^{48} +6.65685 q^{50} -0.414214 q^{51} -1.00000 q^{52} -7.48528 q^{53} +1.00000 q^{54} +8.24264 q^{55} +7.82843 q^{57} +3.82843 q^{58} +12.0711 q^{59} +3.41421 q^{60} +1.58579 q^{61} -8.48528 q^{62} +1.00000 q^{64} -3.41421 q^{65} +2.41421 q^{66} -3.82843 q^{67} -0.414214 q^{68} -1.41421 q^{69} -5.00000 q^{71} +1.00000 q^{72} +1.41421 q^{73} -1.41421 q^{74} +6.65685 q^{75} +7.82843 q^{76} -1.00000 q^{78} -0.343146 q^{79} +3.41421 q^{80} +1.00000 q^{81} -9.89949 q^{82} +3.65685 q^{83} -1.41421 q^{85} -10.4853 q^{86} +3.82843 q^{87} +2.41421 q^{88} -5.41421 q^{89} +3.41421 q^{90} -1.41421 q^{92} -8.48528 q^{93} -1.00000 q^{94} +26.7279 q^{95} +1.00000 q^{96} +15.0711 q^{97} +2.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 4 q^{10} + 2 q^{11} + 2 q^{12} - 2 q^{13} + 4 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} + 10 q^{19} + 4 q^{20} + 2 q^{22} + 2 q^{24} + 2 q^{25} - 2 q^{26} + 2 q^{27} + 2 q^{29} + 4 q^{30} + 2 q^{32} + 2 q^{33} + 2 q^{34} + 2 q^{36} + 10 q^{38} - 2 q^{39} + 4 q^{40} - 4 q^{43} + 2 q^{44} + 4 q^{45} - 2 q^{47} + 2 q^{48} + 2 q^{50} + 2 q^{51} - 2 q^{52} + 2 q^{53} + 2 q^{54} + 8 q^{55} + 10 q^{57} + 2 q^{58} + 10 q^{59} + 4 q^{60} + 6 q^{61} + 2 q^{64} - 4 q^{65} + 2 q^{66} - 2 q^{67} + 2 q^{68} - 10 q^{71} + 2 q^{72} + 2 q^{75} + 10 q^{76} - 2 q^{78} - 12 q^{79} + 4 q^{80} + 2 q^{81} - 4 q^{83} - 4 q^{86} + 2 q^{87} + 2 q^{88} - 8 q^{89} + 4 q^{90} - 2 q^{94} + 28 q^{95} + 2 q^{96} + 16 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 3.41421 1.52688 0.763441 0.645877i \(-0.223508\pi\)
0.763441 + 0.645877i \(0.223508\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.41421 1.07967
\(11\) 2.41421 0.727913 0.363956 0.931416i \(-0.381426\pi\)
0.363956 + 0.931416i \(0.381426\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 3.41421 0.881546
\(16\) 1.00000 0.250000
\(17\) −0.414214 −0.100462 −0.0502308 0.998738i \(-0.515996\pi\)
−0.0502308 + 0.998738i \(0.515996\pi\)
\(18\) 1.00000 0.235702
\(19\) 7.82843 1.79596 0.897982 0.440032i \(-0.145033\pi\)
0.897982 + 0.440032i \(0.145033\pi\)
\(20\) 3.41421 0.763441
\(21\) 0 0
\(22\) 2.41421 0.514712
\(23\) −1.41421 −0.294884 −0.147442 0.989071i \(-0.547104\pi\)
−0.147442 + 0.989071i \(0.547104\pi\)
\(24\) 1.00000 0.204124
\(25\) 6.65685 1.33137
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.82843 0.710921 0.355461 0.934691i \(-0.384324\pi\)
0.355461 + 0.934691i \(0.384324\pi\)
\(30\) 3.41421 0.623347
\(31\) −8.48528 −1.52400 −0.762001 0.647576i \(-0.775783\pi\)
−0.762001 + 0.647576i \(0.775783\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.41421 0.420261
\(34\) −0.414214 −0.0710370
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.41421 −0.232495 −0.116248 0.993220i \(-0.537087\pi\)
−0.116248 + 0.993220i \(0.537087\pi\)
\(38\) 7.82843 1.26994
\(39\) −1.00000 −0.160128
\(40\) 3.41421 0.539835
\(41\) −9.89949 −1.54604 −0.773021 0.634381i \(-0.781255\pi\)
−0.773021 + 0.634381i \(0.781255\pi\)
\(42\) 0 0
\(43\) −10.4853 −1.59899 −0.799495 0.600672i \(-0.794900\pi\)
−0.799495 + 0.600672i \(0.794900\pi\)
\(44\) 2.41421 0.363956
\(45\) 3.41421 0.508961
\(46\) −1.41421 −0.208514
\(47\) −1.00000 −0.145865 −0.0729325 0.997337i \(-0.523236\pi\)
−0.0729325 + 0.997337i \(0.523236\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 6.65685 0.941421
\(51\) −0.414214 −0.0580015
\(52\) −1.00000 −0.138675
\(53\) −7.48528 −1.02818 −0.514091 0.857736i \(-0.671871\pi\)
−0.514091 + 0.857736i \(0.671871\pi\)
\(54\) 1.00000 0.136083
\(55\) 8.24264 1.11144
\(56\) 0 0
\(57\) 7.82843 1.03690
\(58\) 3.82843 0.502697
\(59\) 12.0711 1.57152 0.785760 0.618532i \(-0.212272\pi\)
0.785760 + 0.618532i \(0.212272\pi\)
\(60\) 3.41421 0.440773
\(61\) 1.58579 0.203039 0.101520 0.994834i \(-0.467630\pi\)
0.101520 + 0.994834i \(0.467630\pi\)
\(62\) −8.48528 −1.07763
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.41421 −0.423481
\(66\) 2.41421 0.297169
\(67\) −3.82843 −0.467717 −0.233858 0.972271i \(-0.575135\pi\)
−0.233858 + 0.972271i \(0.575135\pi\)
\(68\) −0.414214 −0.0502308
\(69\) −1.41421 −0.170251
\(70\) 0 0
\(71\) −5.00000 −0.593391 −0.296695 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) 1.00000 0.117851
\(73\) 1.41421 0.165521 0.0827606 0.996569i \(-0.473626\pi\)
0.0827606 + 0.996569i \(0.473626\pi\)
\(74\) −1.41421 −0.164399
\(75\) 6.65685 0.768667
\(76\) 7.82843 0.897982
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) −0.343146 −0.0386069 −0.0193035 0.999814i \(-0.506145\pi\)
−0.0193035 + 0.999814i \(0.506145\pi\)
\(80\) 3.41421 0.381721
\(81\) 1.00000 0.111111
\(82\) −9.89949 −1.09322
\(83\) 3.65685 0.401392 0.200696 0.979654i \(-0.435680\pi\)
0.200696 + 0.979654i \(0.435680\pi\)
\(84\) 0 0
\(85\) −1.41421 −0.153393
\(86\) −10.4853 −1.13066
\(87\) 3.82843 0.410450
\(88\) 2.41421 0.257356
\(89\) −5.41421 −0.573905 −0.286953 0.957945i \(-0.592642\pi\)
−0.286953 + 0.957945i \(0.592642\pi\)
\(90\) 3.41421 0.359890
\(91\) 0 0
\(92\) −1.41421 −0.147442
\(93\) −8.48528 −0.879883
\(94\) −1.00000 −0.103142
\(95\) 26.7279 2.74223
\(96\) 1.00000 0.102062
\(97\) 15.0711 1.53024 0.765118 0.643891i \(-0.222681\pi\)
0.765118 + 0.643891i \(0.222681\pi\)
\(98\) 0 0
\(99\) 2.41421 0.242638
\(100\) 6.65685 0.665685
\(101\) 9.17157 0.912606 0.456303 0.889825i \(-0.349173\pi\)
0.456303 + 0.889825i \(0.349173\pi\)
\(102\) −0.414214 −0.0410133
\(103\) 11.1716 1.10077 0.550384 0.834912i \(-0.314481\pi\)
0.550384 + 0.834912i \(0.314481\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −7.48528 −0.727035
\(107\) 3.31371 0.320348 0.160174 0.987089i \(-0.448794\pi\)
0.160174 + 0.987089i \(0.448794\pi\)
\(108\) 1.00000 0.0962250
\(109\) 1.65685 0.158698 0.0793489 0.996847i \(-0.474716\pi\)
0.0793489 + 0.996847i \(0.474716\pi\)
\(110\) 8.24264 0.785905
\(111\) −1.41421 −0.134231
\(112\) 0 0
\(113\) −14.8995 −1.40163 −0.700813 0.713345i \(-0.747179\pi\)
−0.700813 + 0.713345i \(0.747179\pi\)
\(114\) 7.82843 0.733199
\(115\) −4.82843 −0.450253
\(116\) 3.82843 0.355461
\(117\) −1.00000 −0.0924500
\(118\) 12.0711 1.11123
\(119\) 0 0
\(120\) 3.41421 0.311674
\(121\) −5.17157 −0.470143
\(122\) 1.58579 0.143570
\(123\) −9.89949 −0.892607
\(124\) −8.48528 −0.762001
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) 11.8995 1.05591 0.527955 0.849273i \(-0.322959\pi\)
0.527955 + 0.849273i \(0.322959\pi\)
\(128\) 1.00000 0.0883883
\(129\) −10.4853 −0.923178
\(130\) −3.41421 −0.299446
\(131\) −21.5563 −1.88339 −0.941693 0.336472i \(-0.890766\pi\)
−0.941693 + 0.336472i \(0.890766\pi\)
\(132\) 2.41421 0.210130
\(133\) 0 0
\(134\) −3.82843 −0.330726
\(135\) 3.41421 0.293849
\(136\) −0.414214 −0.0355185
\(137\) −3.07107 −0.262379 −0.131190 0.991357i \(-0.541880\pi\)
−0.131190 + 0.991357i \(0.541880\pi\)
\(138\) −1.41421 −0.120386
\(139\) −1.07107 −0.0908468 −0.0454234 0.998968i \(-0.514464\pi\)
−0.0454234 + 0.998968i \(0.514464\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) −5.00000 −0.419591
\(143\) −2.41421 −0.201887
\(144\) 1.00000 0.0833333
\(145\) 13.0711 1.08549
\(146\) 1.41421 0.117041
\(147\) 0 0
\(148\) −1.41421 −0.116248
\(149\) −19.0711 −1.56236 −0.781181 0.624304i \(-0.785383\pi\)
−0.781181 + 0.624304i \(0.785383\pi\)
\(150\) 6.65685 0.543530
\(151\) −10.0711 −0.819572 −0.409786 0.912182i \(-0.634397\pi\)
−0.409786 + 0.912182i \(0.634397\pi\)
\(152\) 7.82843 0.634969
\(153\) −0.414214 −0.0334872
\(154\) 0 0
\(155\) −28.9706 −2.32697
\(156\) −1.00000 −0.0800641
\(157\) 15.7279 1.25522 0.627612 0.778526i \(-0.284032\pi\)
0.627612 + 0.778526i \(0.284032\pi\)
\(158\) −0.343146 −0.0272992
\(159\) −7.48528 −0.593621
\(160\) 3.41421 0.269917
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −7.34315 −0.575160 −0.287580 0.957757i \(-0.592851\pi\)
−0.287580 + 0.957757i \(0.592851\pi\)
\(164\) −9.89949 −0.773021
\(165\) 8.24264 0.641689
\(166\) 3.65685 0.283827
\(167\) −10.6569 −0.824652 −0.412326 0.911036i \(-0.635284\pi\)
−0.412326 + 0.911036i \(0.635284\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −1.41421 −0.108465
\(171\) 7.82843 0.598655
\(172\) −10.4853 −0.799495
\(173\) 11.4853 0.873210 0.436605 0.899653i \(-0.356181\pi\)
0.436605 + 0.899653i \(0.356181\pi\)
\(174\) 3.82843 0.290232
\(175\) 0 0
\(176\) 2.41421 0.181978
\(177\) 12.0711 0.907317
\(178\) −5.41421 −0.405812
\(179\) 8.34315 0.623596 0.311798 0.950148i \(-0.399069\pi\)
0.311798 + 0.950148i \(0.399069\pi\)
\(180\) 3.41421 0.254480
\(181\) 16.8995 1.25613 0.628065 0.778161i \(-0.283847\pi\)
0.628065 + 0.778161i \(0.283847\pi\)
\(182\) 0 0
\(183\) 1.58579 0.117225
\(184\) −1.41421 −0.104257
\(185\) −4.82843 −0.354993
\(186\) −8.48528 −0.622171
\(187\) −1.00000 −0.0731272
\(188\) −1.00000 −0.0729325
\(189\) 0 0
\(190\) 26.7279 1.93905
\(191\) −21.6569 −1.56703 −0.783517 0.621370i \(-0.786577\pi\)
−0.783517 + 0.621370i \(0.786577\pi\)
\(192\) 1.00000 0.0721688
\(193\) −17.0711 −1.22880 −0.614401 0.788994i \(-0.710602\pi\)
−0.614401 + 0.788994i \(0.710602\pi\)
\(194\) 15.0711 1.08204
\(195\) −3.41421 −0.244497
\(196\) 0 0
\(197\) 15.5563 1.10834 0.554172 0.832402i \(-0.313035\pi\)
0.554172 + 0.832402i \(0.313035\pi\)
\(198\) 2.41421 0.171571
\(199\) 22.7279 1.61114 0.805570 0.592501i \(-0.201859\pi\)
0.805570 + 0.592501i \(0.201859\pi\)
\(200\) 6.65685 0.470711
\(201\) −3.82843 −0.270036
\(202\) 9.17157 0.645310
\(203\) 0 0
\(204\) −0.414214 −0.0290008
\(205\) −33.7990 −2.36062
\(206\) 11.1716 0.778360
\(207\) −1.41421 −0.0982946
\(208\) −1.00000 −0.0693375
\(209\) 18.8995 1.30731
\(210\) 0 0
\(211\) −27.0711 −1.86365 −0.931825 0.362909i \(-0.881784\pi\)
−0.931825 + 0.362909i \(0.881784\pi\)
\(212\) −7.48528 −0.514091
\(213\) −5.00000 −0.342594
\(214\) 3.31371 0.226520
\(215\) −35.7990 −2.44147
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 1.65685 0.112216
\(219\) 1.41421 0.0955637
\(220\) 8.24264 0.555719
\(221\) 0.414214 0.0278630
\(222\) −1.41421 −0.0949158
\(223\) 22.0711 1.47799 0.738994 0.673712i \(-0.235301\pi\)
0.738994 + 0.673712i \(0.235301\pi\)
\(224\) 0 0
\(225\) 6.65685 0.443790
\(226\) −14.8995 −0.991100
\(227\) 26.8284 1.78067 0.890333 0.455311i \(-0.150472\pi\)
0.890333 + 0.455311i \(0.150472\pi\)
\(228\) 7.82843 0.518450
\(229\) −9.51472 −0.628750 −0.314375 0.949299i \(-0.601795\pi\)
−0.314375 + 0.949299i \(0.601795\pi\)
\(230\) −4.82843 −0.318377
\(231\) 0 0
\(232\) 3.82843 0.251349
\(233\) 7.72792 0.506273 0.253137 0.967431i \(-0.418538\pi\)
0.253137 + 0.967431i \(0.418538\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −3.41421 −0.222719
\(236\) 12.0711 0.785760
\(237\) −0.343146 −0.0222897
\(238\) 0 0
\(239\) 21.4853 1.38977 0.694884 0.719122i \(-0.255456\pi\)
0.694884 + 0.719122i \(0.255456\pi\)
\(240\) 3.41421 0.220387
\(241\) −26.1421 −1.68396 −0.841981 0.539506i \(-0.818611\pi\)
−0.841981 + 0.539506i \(0.818611\pi\)
\(242\) −5.17157 −0.332441
\(243\) 1.00000 0.0641500
\(244\) 1.58579 0.101520
\(245\) 0 0
\(246\) −9.89949 −0.631169
\(247\) −7.82843 −0.498111
\(248\) −8.48528 −0.538816
\(249\) 3.65685 0.231744
\(250\) 5.65685 0.357771
\(251\) −19.1716 −1.21010 −0.605049 0.796188i \(-0.706847\pi\)
−0.605049 + 0.796188i \(0.706847\pi\)
\(252\) 0 0
\(253\) −3.41421 −0.214650
\(254\) 11.8995 0.746641
\(255\) −1.41421 −0.0885615
\(256\) 1.00000 0.0625000
\(257\) −9.51472 −0.593512 −0.296756 0.954953i \(-0.595905\pi\)
−0.296756 + 0.954953i \(0.595905\pi\)
\(258\) −10.4853 −0.652785
\(259\) 0 0
\(260\) −3.41421 −0.211741
\(261\) 3.82843 0.236974
\(262\) −21.5563 −1.33176
\(263\) 23.4142 1.44378 0.721891 0.692007i \(-0.243273\pi\)
0.721891 + 0.692007i \(0.243273\pi\)
\(264\) 2.41421 0.148585
\(265\) −25.5563 −1.56991
\(266\) 0 0
\(267\) −5.41421 −0.331344
\(268\) −3.82843 −0.233858
\(269\) −19.1421 −1.16712 −0.583558 0.812071i \(-0.698340\pi\)
−0.583558 + 0.812071i \(0.698340\pi\)
\(270\) 3.41421 0.207782
\(271\) 16.2132 0.984882 0.492441 0.870346i \(-0.336105\pi\)
0.492441 + 0.870346i \(0.336105\pi\)
\(272\) −0.414214 −0.0251154
\(273\) 0 0
\(274\) −3.07107 −0.185530
\(275\) 16.0711 0.969122
\(276\) −1.41421 −0.0851257
\(277\) 3.92893 0.236067 0.118033 0.993010i \(-0.462341\pi\)
0.118033 + 0.993010i \(0.462341\pi\)
\(278\) −1.07107 −0.0642384
\(279\) −8.48528 −0.508001
\(280\) 0 0
\(281\) 23.3137 1.39078 0.695390 0.718633i \(-0.255232\pi\)
0.695390 + 0.718633i \(0.255232\pi\)
\(282\) −1.00000 −0.0595491
\(283\) 24.2426 1.44108 0.720538 0.693416i \(-0.243895\pi\)
0.720538 + 0.693416i \(0.243895\pi\)
\(284\) −5.00000 −0.296695
\(285\) 26.7279 1.58323
\(286\) −2.41421 −0.142755
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −16.8284 −0.989907
\(290\) 13.0711 0.767560
\(291\) 15.0711 0.883482
\(292\) 1.41421 0.0827606
\(293\) −9.89949 −0.578335 −0.289167 0.957279i \(-0.593378\pi\)
−0.289167 + 0.957279i \(0.593378\pi\)
\(294\) 0 0
\(295\) 41.2132 2.39953
\(296\) −1.41421 −0.0821995
\(297\) 2.41421 0.140087
\(298\) −19.0711 −1.10476
\(299\) 1.41421 0.0817861
\(300\) 6.65685 0.384334
\(301\) 0 0
\(302\) −10.0711 −0.579525
\(303\) 9.17157 0.526893
\(304\) 7.82843 0.448991
\(305\) 5.41421 0.310017
\(306\) −0.414214 −0.0236790
\(307\) −34.1127 −1.94691 −0.973457 0.228869i \(-0.926497\pi\)
−0.973457 + 0.228869i \(0.926497\pi\)
\(308\) 0 0
\(309\) 11.1716 0.635529
\(310\) −28.9706 −1.64542
\(311\) 27.8995 1.58204 0.791018 0.611793i \(-0.209552\pi\)
0.791018 + 0.611793i \(0.209552\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −20.4853 −1.15790 −0.578948 0.815364i \(-0.696537\pi\)
−0.578948 + 0.815364i \(0.696537\pi\)
\(314\) 15.7279 0.887578
\(315\) 0 0
\(316\) −0.343146 −0.0193035
\(317\) −9.79899 −0.550366 −0.275183 0.961392i \(-0.588738\pi\)
−0.275183 + 0.961392i \(0.588738\pi\)
\(318\) −7.48528 −0.419754
\(319\) 9.24264 0.517489
\(320\) 3.41421 0.190860
\(321\) 3.31371 0.184953
\(322\) 0 0
\(323\) −3.24264 −0.180425
\(324\) 1.00000 0.0555556
\(325\) −6.65685 −0.369256
\(326\) −7.34315 −0.406699
\(327\) 1.65685 0.0916242
\(328\) −9.89949 −0.546608
\(329\) 0 0
\(330\) 8.24264 0.453742
\(331\) −5.51472 −0.303116 −0.151558 0.988448i \(-0.548429\pi\)
−0.151558 + 0.988448i \(0.548429\pi\)
\(332\) 3.65685 0.200696
\(333\) −1.41421 −0.0774984
\(334\) −10.6569 −0.583117
\(335\) −13.0711 −0.714149
\(336\) 0 0
\(337\) 2.31371 0.126036 0.0630179 0.998012i \(-0.479927\pi\)
0.0630179 + 0.998012i \(0.479927\pi\)
\(338\) 1.00000 0.0543928
\(339\) −14.8995 −0.809229
\(340\) −1.41421 −0.0766965
\(341\) −20.4853 −1.10934
\(342\) 7.82843 0.423313
\(343\) 0 0
\(344\) −10.4853 −0.565328
\(345\) −4.82843 −0.259954
\(346\) 11.4853 0.617453
\(347\) 16.7279 0.898002 0.449001 0.893531i \(-0.351780\pi\)
0.449001 + 0.893531i \(0.351780\pi\)
\(348\) 3.82843 0.205225
\(349\) −18.7279 −1.00248 −0.501241 0.865308i \(-0.667123\pi\)
−0.501241 + 0.865308i \(0.667123\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 2.41421 0.128678
\(353\) 32.6274 1.73658 0.868291 0.496055i \(-0.165219\pi\)
0.868291 + 0.496055i \(0.165219\pi\)
\(354\) 12.0711 0.641570
\(355\) −17.0711 −0.906038
\(356\) −5.41421 −0.286953
\(357\) 0 0
\(358\) 8.34315 0.440949
\(359\) 4.82843 0.254835 0.127417 0.991849i \(-0.459331\pi\)
0.127417 + 0.991849i \(0.459331\pi\)
\(360\) 3.41421 0.179945
\(361\) 42.2843 2.22549
\(362\) 16.8995 0.888218
\(363\) −5.17157 −0.271437
\(364\) 0 0
\(365\) 4.82843 0.252731
\(366\) 1.58579 0.0828904
\(367\) 32.6274 1.70314 0.851569 0.524243i \(-0.175652\pi\)
0.851569 + 0.524243i \(0.175652\pi\)
\(368\) −1.41421 −0.0737210
\(369\) −9.89949 −0.515347
\(370\) −4.82843 −0.251018
\(371\) 0 0
\(372\) −8.48528 −0.439941
\(373\) −8.41421 −0.435671 −0.217836 0.975985i \(-0.569900\pi\)
−0.217836 + 0.975985i \(0.569900\pi\)
\(374\) −1.00000 −0.0517088
\(375\) 5.65685 0.292119
\(376\) −1.00000 −0.0515711
\(377\) −3.82843 −0.197174
\(378\) 0 0
\(379\) −26.6274 −1.36776 −0.683879 0.729595i \(-0.739709\pi\)
−0.683879 + 0.729595i \(0.739709\pi\)
\(380\) 26.7279 1.37111
\(381\) 11.8995 0.609630
\(382\) −21.6569 −1.10806
\(383\) −16.8284 −0.859892 −0.429946 0.902854i \(-0.641467\pi\)
−0.429946 + 0.902854i \(0.641467\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −17.0711 −0.868894
\(387\) −10.4853 −0.532997
\(388\) 15.0711 0.765118
\(389\) −21.4853 −1.08935 −0.544674 0.838648i \(-0.683346\pi\)
−0.544674 + 0.838648i \(0.683346\pi\)
\(390\) −3.41421 −0.172885
\(391\) 0.585786 0.0296245
\(392\) 0 0
\(393\) −21.5563 −1.08737
\(394\) 15.5563 0.783718
\(395\) −1.17157 −0.0589482
\(396\) 2.41421 0.121319
\(397\) −21.5563 −1.08188 −0.540941 0.841060i \(-0.681932\pi\)
−0.540941 + 0.841060i \(0.681932\pi\)
\(398\) 22.7279 1.13925
\(399\) 0 0
\(400\) 6.65685 0.332843
\(401\) 0.100505 0.00501898 0.00250949 0.999997i \(-0.499201\pi\)
0.00250949 + 0.999997i \(0.499201\pi\)
\(402\) −3.82843 −0.190945
\(403\) 8.48528 0.422682
\(404\) 9.17157 0.456303
\(405\) 3.41421 0.169654
\(406\) 0 0
\(407\) −3.41421 −0.169236
\(408\) −0.414214 −0.0205066
\(409\) −35.4142 −1.75112 −0.875560 0.483109i \(-0.839507\pi\)
−0.875560 + 0.483109i \(0.839507\pi\)
\(410\) −33.7990 −1.66921
\(411\) −3.07107 −0.151485
\(412\) 11.1716 0.550384
\(413\) 0 0
\(414\) −1.41421 −0.0695048
\(415\) 12.4853 0.612878
\(416\) −1.00000 −0.0490290
\(417\) −1.07107 −0.0524504
\(418\) 18.8995 0.924405
\(419\) 23.4558 1.14589 0.572946 0.819593i \(-0.305800\pi\)
0.572946 + 0.819593i \(0.305800\pi\)
\(420\) 0 0
\(421\) −13.0711 −0.637045 −0.318522 0.947915i \(-0.603187\pi\)
−0.318522 + 0.947915i \(0.603187\pi\)
\(422\) −27.0711 −1.31780
\(423\) −1.00000 −0.0486217
\(424\) −7.48528 −0.363517
\(425\) −2.75736 −0.133752
\(426\) −5.00000 −0.242251
\(427\) 0 0
\(428\) 3.31371 0.160174
\(429\) −2.41421 −0.116559
\(430\) −35.7990 −1.72638
\(431\) 13.5147 0.650981 0.325491 0.945545i \(-0.394471\pi\)
0.325491 + 0.945545i \(0.394471\pi\)
\(432\) 1.00000 0.0481125
\(433\) 3.97056 0.190813 0.0954065 0.995438i \(-0.469585\pi\)
0.0954065 + 0.995438i \(0.469585\pi\)
\(434\) 0 0
\(435\) 13.0711 0.626710
\(436\) 1.65685 0.0793489
\(437\) −11.0711 −0.529601
\(438\) 1.41421 0.0675737
\(439\) −19.4142 −0.926590 −0.463295 0.886204i \(-0.653333\pi\)
−0.463295 + 0.886204i \(0.653333\pi\)
\(440\) 8.24264 0.392952
\(441\) 0 0
\(442\) 0.414214 0.0197021
\(443\) 25.6985 1.22097 0.610486 0.792027i \(-0.290974\pi\)
0.610486 + 0.792027i \(0.290974\pi\)
\(444\) −1.41421 −0.0671156
\(445\) −18.4853 −0.876286
\(446\) 22.0711 1.04510
\(447\) −19.0711 −0.902031
\(448\) 0 0
\(449\) −8.97056 −0.423347 −0.211674 0.977340i \(-0.567891\pi\)
−0.211674 + 0.977340i \(0.567891\pi\)
\(450\) 6.65685 0.313807
\(451\) −23.8995 −1.12538
\(452\) −14.8995 −0.700813
\(453\) −10.0711 −0.473180
\(454\) 26.8284 1.25912
\(455\) 0 0
\(456\) 7.82843 0.366600
\(457\) 11.0711 0.517883 0.258941 0.965893i \(-0.416626\pi\)
0.258941 + 0.965893i \(0.416626\pi\)
\(458\) −9.51472 −0.444594
\(459\) −0.414214 −0.0193338
\(460\) −4.82843 −0.225127
\(461\) −12.4853 −0.581498 −0.290749 0.956799i \(-0.593904\pi\)
−0.290749 + 0.956799i \(0.593904\pi\)
\(462\) 0 0
\(463\) 12.9706 0.602793 0.301397 0.953499i \(-0.402547\pi\)
0.301397 + 0.953499i \(0.402547\pi\)
\(464\) 3.82843 0.177730
\(465\) −28.9706 −1.34348
\(466\) 7.72792 0.357989
\(467\) 8.58579 0.397303 0.198651 0.980070i \(-0.436344\pi\)
0.198651 + 0.980070i \(0.436344\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) −3.41421 −0.157486
\(471\) 15.7279 0.724704
\(472\) 12.0711 0.555616
\(473\) −25.3137 −1.16393
\(474\) −0.343146 −0.0157612
\(475\) 52.1127 2.39109
\(476\) 0 0
\(477\) −7.48528 −0.342727
\(478\) 21.4853 0.982714
\(479\) −30.1127 −1.37588 −0.687942 0.725766i \(-0.741486\pi\)
−0.687942 + 0.725766i \(0.741486\pi\)
\(480\) 3.41421 0.155837
\(481\) 1.41421 0.0644826
\(482\) −26.1421 −1.19074
\(483\) 0 0
\(484\) −5.17157 −0.235071
\(485\) 51.4558 2.33649
\(486\) 1.00000 0.0453609
\(487\) −26.2132 −1.18783 −0.593917 0.804526i \(-0.702419\pi\)
−0.593917 + 0.804526i \(0.702419\pi\)
\(488\) 1.58579 0.0717852
\(489\) −7.34315 −0.332069
\(490\) 0 0
\(491\) 12.3431 0.557038 0.278519 0.960431i \(-0.410156\pi\)
0.278519 + 0.960431i \(0.410156\pi\)
\(492\) −9.89949 −0.446304
\(493\) −1.58579 −0.0714202
\(494\) −7.82843 −0.352218
\(495\) 8.24264 0.370479
\(496\) −8.48528 −0.381000
\(497\) 0 0
\(498\) 3.65685 0.163868
\(499\) −7.65685 −0.342768 −0.171384 0.985204i \(-0.554824\pi\)
−0.171384 + 0.985204i \(0.554824\pi\)
\(500\) 5.65685 0.252982
\(501\) −10.6569 −0.476113
\(502\) −19.1716 −0.855669
\(503\) 2.14214 0.0955131 0.0477566 0.998859i \(-0.484793\pi\)
0.0477566 + 0.998859i \(0.484793\pi\)
\(504\) 0 0
\(505\) 31.3137 1.39344
\(506\) −3.41421 −0.151780
\(507\) 1.00000 0.0444116
\(508\) 11.8995 0.527955
\(509\) 34.6274 1.53483 0.767417 0.641149i \(-0.221542\pi\)
0.767417 + 0.641149i \(0.221542\pi\)
\(510\) −1.41421 −0.0626224
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 7.82843 0.345634
\(514\) −9.51472 −0.419676
\(515\) 38.1421 1.68074
\(516\) −10.4853 −0.461589
\(517\) −2.41421 −0.106177
\(518\) 0 0
\(519\) 11.4853 0.504148
\(520\) −3.41421 −0.149723
\(521\) 4.62742 0.202731 0.101365 0.994849i \(-0.467679\pi\)
0.101365 + 0.994849i \(0.467679\pi\)
\(522\) 3.82843 0.167566
\(523\) 11.1716 0.488499 0.244249 0.969712i \(-0.421458\pi\)
0.244249 + 0.969712i \(0.421458\pi\)
\(524\) −21.5563 −0.941693
\(525\) 0 0
\(526\) 23.4142 1.02091
\(527\) 3.51472 0.153104
\(528\) 2.41421 0.105065
\(529\) −21.0000 −0.913043
\(530\) −25.5563 −1.11010
\(531\) 12.0711 0.523840
\(532\) 0 0
\(533\) 9.89949 0.428795
\(534\) −5.41421 −0.234296
\(535\) 11.3137 0.489134
\(536\) −3.82843 −0.165363
\(537\) 8.34315 0.360033
\(538\) −19.1421 −0.825276
\(539\) 0 0
\(540\) 3.41421 0.146924
\(541\) 8.92893 0.383885 0.191942 0.981406i \(-0.438521\pi\)
0.191942 + 0.981406i \(0.438521\pi\)
\(542\) 16.2132 0.696417
\(543\) 16.8995 0.725227
\(544\) −0.414214 −0.0177593
\(545\) 5.65685 0.242313
\(546\) 0 0
\(547\) 22.8701 0.977853 0.488927 0.872325i \(-0.337389\pi\)
0.488927 + 0.872325i \(0.337389\pi\)
\(548\) −3.07107 −0.131190
\(549\) 1.58579 0.0676797
\(550\) 16.0711 0.685273
\(551\) 29.9706 1.27679
\(552\) −1.41421 −0.0601929
\(553\) 0 0
\(554\) 3.92893 0.166924
\(555\) −4.82843 −0.204955
\(556\) −1.07107 −0.0454234
\(557\) 12.6274 0.535041 0.267520 0.963552i \(-0.413796\pi\)
0.267520 + 0.963552i \(0.413796\pi\)
\(558\) −8.48528 −0.359211
\(559\) 10.4853 0.443480
\(560\) 0 0
\(561\) −1.00000 −0.0422200
\(562\) 23.3137 0.983429
\(563\) −40.7696 −1.71823 −0.859116 0.511781i \(-0.828986\pi\)
−0.859116 + 0.511781i \(0.828986\pi\)
\(564\) −1.00000 −0.0421076
\(565\) −50.8701 −2.14012
\(566\) 24.2426 1.01899
\(567\) 0 0
\(568\) −5.00000 −0.209795
\(569\) −1.72792 −0.0724383 −0.0362191 0.999344i \(-0.511531\pi\)
−0.0362191 + 0.999344i \(0.511531\pi\)
\(570\) 26.7279 1.11951
\(571\) −8.68629 −0.363510 −0.181755 0.983344i \(-0.558178\pi\)
−0.181755 + 0.983344i \(0.558178\pi\)
\(572\) −2.41421 −0.100943
\(573\) −21.6569 −0.904728
\(574\) 0 0
\(575\) −9.41421 −0.392600
\(576\) 1.00000 0.0416667
\(577\) −27.6569 −1.15137 −0.575685 0.817672i \(-0.695265\pi\)
−0.575685 + 0.817672i \(0.695265\pi\)
\(578\) −16.8284 −0.699970
\(579\) −17.0711 −0.709449
\(580\) 13.0711 0.542747
\(581\) 0 0
\(582\) 15.0711 0.624716
\(583\) −18.0711 −0.748427
\(584\) 1.41421 0.0585206
\(585\) −3.41421 −0.141160
\(586\) −9.89949 −0.408944
\(587\) 4.41421 0.182194 0.0910970 0.995842i \(-0.470963\pi\)
0.0910970 + 0.995842i \(0.470963\pi\)
\(588\) 0 0
\(589\) −66.4264 −2.73705
\(590\) 41.2132 1.69672
\(591\) 15.5563 0.639903
\(592\) −1.41421 −0.0581238
\(593\) −2.38478 −0.0979310 −0.0489655 0.998800i \(-0.515592\pi\)
−0.0489655 + 0.998800i \(0.515592\pi\)
\(594\) 2.41421 0.0990564
\(595\) 0 0
\(596\) −19.0711 −0.781181
\(597\) 22.7279 0.930192
\(598\) 1.41421 0.0578315
\(599\) −29.5563 −1.20764 −0.603820 0.797121i \(-0.706355\pi\)
−0.603820 + 0.797121i \(0.706355\pi\)
\(600\) 6.65685 0.271765
\(601\) −40.7990 −1.66423 −0.832113 0.554606i \(-0.812869\pi\)
−0.832113 + 0.554606i \(0.812869\pi\)
\(602\) 0 0
\(603\) −3.82843 −0.155906
\(604\) −10.0711 −0.409786
\(605\) −17.6569 −0.717853
\(606\) 9.17157 0.372570
\(607\) 30.5269 1.23905 0.619525 0.784977i \(-0.287325\pi\)
0.619525 + 0.784977i \(0.287325\pi\)
\(608\) 7.82843 0.317485
\(609\) 0 0
\(610\) 5.41421 0.219215
\(611\) 1.00000 0.0404557
\(612\) −0.414214 −0.0167436
\(613\) 31.7990 1.28435 0.642175 0.766558i \(-0.278032\pi\)
0.642175 + 0.766558i \(0.278032\pi\)
\(614\) −34.1127 −1.37668
\(615\) −33.7990 −1.36291
\(616\) 0 0
\(617\) 4.14214 0.166756 0.0833781 0.996518i \(-0.473429\pi\)
0.0833781 + 0.996518i \(0.473429\pi\)
\(618\) 11.1716 0.449387
\(619\) −15.1716 −0.609797 −0.304898 0.952385i \(-0.598623\pi\)
−0.304898 + 0.952385i \(0.598623\pi\)
\(620\) −28.9706 −1.16349
\(621\) −1.41421 −0.0567504
\(622\) 27.8995 1.11867
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) −13.9706 −0.558823
\(626\) −20.4853 −0.818757
\(627\) 18.8995 0.754773
\(628\) 15.7279 0.627612
\(629\) 0.585786 0.0233568
\(630\) 0 0
\(631\) −12.6274 −0.502690 −0.251345 0.967898i \(-0.580873\pi\)
−0.251345 + 0.967898i \(0.580873\pi\)
\(632\) −0.343146 −0.0136496
\(633\) −27.0711 −1.07598
\(634\) −9.79899 −0.389168
\(635\) 40.6274 1.61225
\(636\) −7.48528 −0.296811
\(637\) 0 0
\(638\) 9.24264 0.365920
\(639\) −5.00000 −0.197797
\(640\) 3.41421 0.134959
\(641\) −7.79899 −0.308042 −0.154021 0.988068i \(-0.549222\pi\)
−0.154021 + 0.988068i \(0.549222\pi\)
\(642\) 3.31371 0.130782
\(643\) 19.4853 0.768424 0.384212 0.923245i \(-0.374473\pi\)
0.384212 + 0.923245i \(0.374473\pi\)
\(644\) 0 0
\(645\) −35.7990 −1.40958
\(646\) −3.24264 −0.127580
\(647\) 42.9706 1.68935 0.844674 0.535282i \(-0.179795\pi\)
0.844674 + 0.535282i \(0.179795\pi\)
\(648\) 1.00000 0.0392837
\(649\) 29.1421 1.14393
\(650\) −6.65685 −0.261103
\(651\) 0 0
\(652\) −7.34315 −0.287580
\(653\) −6.14214 −0.240360 −0.120180 0.992752i \(-0.538347\pi\)
−0.120180 + 0.992752i \(0.538347\pi\)
\(654\) 1.65685 0.0647881
\(655\) −73.5980 −2.87571
\(656\) −9.89949 −0.386510
\(657\) 1.41421 0.0551737
\(658\) 0 0
\(659\) 13.4142 0.522544 0.261272 0.965265i \(-0.415858\pi\)
0.261272 + 0.965265i \(0.415858\pi\)
\(660\) 8.24264 0.320844
\(661\) −31.7574 −1.23522 −0.617609 0.786485i \(-0.711899\pi\)
−0.617609 + 0.786485i \(0.711899\pi\)
\(662\) −5.51472 −0.214336
\(663\) 0.414214 0.0160867
\(664\) 3.65685 0.141913
\(665\) 0 0
\(666\) −1.41421 −0.0547997
\(667\) −5.41421 −0.209639
\(668\) −10.6569 −0.412326
\(669\) 22.0711 0.853317
\(670\) −13.0711 −0.504979
\(671\) 3.82843 0.147795
\(672\) 0 0
\(673\) 13.8579 0.534181 0.267091 0.963671i \(-0.413938\pi\)
0.267091 + 0.963671i \(0.413938\pi\)
\(674\) 2.31371 0.0891207
\(675\) 6.65685 0.256222
\(676\) 1.00000 0.0384615
\(677\) −15.0000 −0.576497 −0.288248 0.957556i \(-0.593073\pi\)
−0.288248 + 0.957556i \(0.593073\pi\)
\(678\) −14.8995 −0.572212
\(679\) 0 0
\(680\) −1.41421 −0.0542326
\(681\) 26.8284 1.02807
\(682\) −20.4853 −0.784422
\(683\) 14.1421 0.541134 0.270567 0.962701i \(-0.412789\pi\)
0.270567 + 0.962701i \(0.412789\pi\)
\(684\) 7.82843 0.299327
\(685\) −10.4853 −0.400622
\(686\) 0 0
\(687\) −9.51472 −0.363009
\(688\) −10.4853 −0.399748
\(689\) 7.48528 0.285167
\(690\) −4.82843 −0.183815
\(691\) 9.34315 0.355430 0.177715 0.984082i \(-0.443129\pi\)
0.177715 + 0.984082i \(0.443129\pi\)
\(692\) 11.4853 0.436605
\(693\) 0 0
\(694\) 16.7279 0.634983
\(695\) −3.65685 −0.138712
\(696\) 3.82843 0.145116
\(697\) 4.10051 0.155318
\(698\) −18.7279 −0.708862
\(699\) 7.72792 0.292297
\(700\) 0 0
\(701\) −34.8284 −1.31545 −0.657726 0.753257i \(-0.728481\pi\)
−0.657726 + 0.753257i \(0.728481\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −11.0711 −0.417553
\(704\) 2.41421 0.0909891
\(705\) −3.41421 −0.128587
\(706\) 32.6274 1.22795
\(707\) 0 0
\(708\) 12.0711 0.453659
\(709\) 16.8701 0.633568 0.316784 0.948498i \(-0.397397\pi\)
0.316784 + 0.948498i \(0.397397\pi\)
\(710\) −17.0711 −0.640666
\(711\) −0.343146 −0.0128690
\(712\) −5.41421 −0.202906
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) −8.24264 −0.308257
\(716\) 8.34315 0.311798
\(717\) 21.4853 0.802383
\(718\) 4.82843 0.180195
\(719\) 13.3137 0.496518 0.248259 0.968694i \(-0.420142\pi\)
0.248259 + 0.968694i \(0.420142\pi\)
\(720\) 3.41421 0.127240
\(721\) 0 0
\(722\) 42.2843 1.57366
\(723\) −26.1421 −0.972236
\(724\) 16.8995 0.628065
\(725\) 25.4853 0.946500
\(726\) −5.17157 −0.191935
\(727\) −34.4264 −1.27680 −0.638402 0.769703i \(-0.720404\pi\)
−0.638402 + 0.769703i \(0.720404\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 4.82843 0.178708
\(731\) 4.34315 0.160637
\(732\) 1.58579 0.0586124
\(733\) 31.4142 1.16031 0.580155 0.814506i \(-0.302992\pi\)
0.580155 + 0.814506i \(0.302992\pi\)
\(734\) 32.6274 1.20430
\(735\) 0 0
\(736\) −1.41421 −0.0521286
\(737\) −9.24264 −0.340457
\(738\) −9.89949 −0.364405
\(739\) −30.0000 −1.10357 −0.551784 0.833987i \(-0.686053\pi\)
−0.551784 + 0.833987i \(0.686053\pi\)
\(740\) −4.82843 −0.177497
\(741\) −7.82843 −0.287584
\(742\) 0 0
\(743\) 17.8284 0.654062 0.327031 0.945014i \(-0.393952\pi\)
0.327031 + 0.945014i \(0.393952\pi\)
\(744\) −8.48528 −0.311086
\(745\) −65.1127 −2.38554
\(746\) −8.41421 −0.308066
\(747\) 3.65685 0.133797
\(748\) −1.00000 −0.0365636
\(749\) 0 0
\(750\) 5.65685 0.206559
\(751\) 3.31371 0.120919 0.0604595 0.998171i \(-0.480743\pi\)
0.0604595 + 0.998171i \(0.480743\pi\)
\(752\) −1.00000 −0.0364662
\(753\) −19.1716 −0.698651
\(754\) −3.82843 −0.139423
\(755\) −34.3848 −1.25139
\(756\) 0 0
\(757\) −39.2426 −1.42630 −0.713149 0.701012i \(-0.752732\pi\)
−0.713149 + 0.701012i \(0.752732\pi\)
\(758\) −26.6274 −0.967151
\(759\) −3.41421 −0.123928
\(760\) 26.7279 0.969524
\(761\) 19.1127 0.692835 0.346417 0.938080i \(-0.387398\pi\)
0.346417 + 0.938080i \(0.387398\pi\)
\(762\) 11.8995 0.431073
\(763\) 0 0
\(764\) −21.6569 −0.783517
\(765\) −1.41421 −0.0511310
\(766\) −16.8284 −0.608036
\(767\) −12.0711 −0.435861
\(768\) 1.00000 0.0360844
\(769\) 9.89949 0.356985 0.178492 0.983941i \(-0.442878\pi\)
0.178492 + 0.983941i \(0.442878\pi\)
\(770\) 0 0
\(771\) −9.51472 −0.342664
\(772\) −17.0711 −0.614401
\(773\) 14.9706 0.538454 0.269227 0.963077i \(-0.413232\pi\)
0.269227 + 0.963077i \(0.413232\pi\)
\(774\) −10.4853 −0.376886
\(775\) −56.4853 −2.02901
\(776\) 15.0711 0.541020
\(777\) 0 0
\(778\) −21.4853 −0.770285
\(779\) −77.4975 −2.77664
\(780\) −3.41421 −0.122248
\(781\) −12.0711 −0.431937
\(782\) 0.585786 0.0209477
\(783\) 3.82843 0.136817
\(784\) 0 0
\(785\) 53.6985 1.91658
\(786\) −21.5563 −0.768890
\(787\) 3.68629 0.131402 0.0657011 0.997839i \(-0.479072\pi\)
0.0657011 + 0.997839i \(0.479072\pi\)
\(788\) 15.5563 0.554172
\(789\) 23.4142 0.833568
\(790\) −1.17157 −0.0416827
\(791\) 0 0
\(792\) 2.41421 0.0857853
\(793\) −1.58579 −0.0563129
\(794\) −21.5563 −0.765006
\(795\) −25.5563 −0.906390
\(796\) 22.7279 0.805570
\(797\) −45.6569 −1.61725 −0.808624 0.588325i \(-0.799787\pi\)
−0.808624 + 0.588325i \(0.799787\pi\)
\(798\) 0 0
\(799\) 0.414214 0.0146538
\(800\) 6.65685 0.235355
\(801\) −5.41421 −0.191302
\(802\) 0.100505 0.00354896
\(803\) 3.41421 0.120485
\(804\) −3.82843 −0.135018
\(805\) 0 0
\(806\) 8.48528 0.298881
\(807\) −19.1421 −0.673835
\(808\) 9.17157 0.322655
\(809\) −20.4142 −0.717726 −0.358863 0.933390i \(-0.616835\pi\)
−0.358863 + 0.933390i \(0.616835\pi\)
\(810\) 3.41421 0.119963
\(811\) −4.54416 −0.159567 −0.0797834 0.996812i \(-0.525423\pi\)
−0.0797834 + 0.996812i \(0.525423\pi\)
\(812\) 0 0
\(813\) 16.2132 0.568622
\(814\) −3.41421 −0.119668
\(815\) −25.0711 −0.878201
\(816\) −0.414214 −0.0145004
\(817\) −82.0833 −2.87173
\(818\) −35.4142 −1.23823
\(819\) 0 0
\(820\) −33.7990 −1.18031
\(821\) 28.1005 0.980714 0.490357 0.871522i \(-0.336866\pi\)
0.490357 + 0.871522i \(0.336866\pi\)
\(822\) −3.07107 −0.107116
\(823\) 43.8406 1.52819 0.764094 0.645105i \(-0.223186\pi\)
0.764094 + 0.645105i \(0.223186\pi\)
\(824\) 11.1716 0.389180
\(825\) 16.0711 0.559523
\(826\) 0 0
\(827\) −24.3553 −0.846918 −0.423459 0.905915i \(-0.639184\pi\)
−0.423459 + 0.905915i \(0.639184\pi\)
\(828\) −1.41421 −0.0491473
\(829\) −32.6985 −1.13567 −0.567833 0.823144i \(-0.692218\pi\)
−0.567833 + 0.823144i \(0.692218\pi\)
\(830\) 12.4853 0.433370
\(831\) 3.92893 0.136293
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) −1.07107 −0.0370880
\(835\) −36.3848 −1.25915
\(836\) 18.8995 0.653653
\(837\) −8.48528 −0.293294
\(838\) 23.4558 0.810269
\(839\) 26.1716 0.903543 0.451772 0.892134i \(-0.350792\pi\)
0.451772 + 0.892134i \(0.350792\pi\)
\(840\) 0 0
\(841\) −14.3431 −0.494591
\(842\) −13.0711 −0.450459
\(843\) 23.3137 0.802967
\(844\) −27.0711 −0.931825
\(845\) 3.41421 0.117453
\(846\) −1.00000 −0.0343807
\(847\) 0 0
\(848\) −7.48528 −0.257046
\(849\) 24.2426 0.832005
\(850\) −2.75736 −0.0945766
\(851\) 2.00000 0.0685591
\(852\) −5.00000 −0.171297
\(853\) 18.7279 0.641232 0.320616 0.947209i \(-0.396110\pi\)
0.320616 + 0.947209i \(0.396110\pi\)
\(854\) 0 0
\(855\) 26.7279 0.914076
\(856\) 3.31371 0.113260
\(857\) 10.5563 0.360598 0.180299 0.983612i \(-0.442293\pi\)
0.180299 + 0.983612i \(0.442293\pi\)
\(858\) −2.41421 −0.0824199
\(859\) −27.7990 −0.948489 −0.474245 0.880393i \(-0.657279\pi\)
−0.474245 + 0.880393i \(0.657279\pi\)
\(860\) −35.7990 −1.22074
\(861\) 0 0
\(862\) 13.5147 0.460313
\(863\) 16.6863 0.568008 0.284004 0.958823i \(-0.408337\pi\)
0.284004 + 0.958823i \(0.408337\pi\)
\(864\) 1.00000 0.0340207
\(865\) 39.2132 1.33329
\(866\) 3.97056 0.134925
\(867\) −16.8284 −0.571523
\(868\) 0 0
\(869\) −0.828427 −0.0281025
\(870\) 13.0711 0.443151
\(871\) 3.82843 0.129721
\(872\) 1.65685 0.0561082
\(873\) 15.0711 0.510078
\(874\) −11.0711 −0.374484
\(875\) 0 0
\(876\) 1.41421 0.0477818
\(877\) 8.04163 0.271547 0.135773 0.990740i \(-0.456648\pi\)
0.135773 + 0.990740i \(0.456648\pi\)
\(878\) −19.4142 −0.655198
\(879\) −9.89949 −0.333902
\(880\) 8.24264 0.277859
\(881\) −1.51472 −0.0510322 −0.0255161 0.999674i \(-0.508123\pi\)
−0.0255161 + 0.999674i \(0.508123\pi\)
\(882\) 0 0
\(883\) −18.2426 −0.613914 −0.306957 0.951723i \(-0.599311\pi\)
−0.306957 + 0.951723i \(0.599311\pi\)
\(884\) 0.414214 0.0139315
\(885\) 41.2132 1.38537
\(886\) 25.6985 0.863357
\(887\) 52.9117 1.77660 0.888300 0.459263i \(-0.151886\pi\)
0.888300 + 0.459263i \(0.151886\pi\)
\(888\) −1.41421 −0.0474579
\(889\) 0 0
\(890\) −18.4853 −0.619628
\(891\) 2.41421 0.0808792
\(892\) 22.0711 0.738994
\(893\) −7.82843 −0.261968
\(894\) −19.0711 −0.637832
\(895\) 28.4853 0.952158
\(896\) 0 0
\(897\) 1.41421 0.0472192
\(898\) −8.97056 −0.299352
\(899\) −32.4853 −1.08344
\(900\) 6.65685 0.221895
\(901\) 3.10051 0.103293
\(902\) −23.8995 −0.795766
\(903\) 0 0
\(904\) −14.8995 −0.495550
\(905\) 57.6985 1.91796
\(906\) −10.0711 −0.334589
\(907\) 17.6152 0.584904 0.292452 0.956280i \(-0.405529\pi\)
0.292452 + 0.956280i \(0.405529\pi\)
\(908\) 26.8284 0.890333
\(909\) 9.17157 0.304202
\(910\) 0 0
\(911\) −15.6569 −0.518735 −0.259367 0.965779i \(-0.583514\pi\)
−0.259367 + 0.965779i \(0.583514\pi\)
\(912\) 7.82843 0.259225
\(913\) 8.82843 0.292178
\(914\) 11.0711 0.366198
\(915\) 5.41421 0.178988
\(916\) −9.51472 −0.314375
\(917\) 0 0
\(918\) −0.414214 −0.0136711
\(919\) −37.7990 −1.24687 −0.623437 0.781874i \(-0.714264\pi\)
−0.623437 + 0.781874i \(0.714264\pi\)
\(920\) −4.82843 −0.159189
\(921\) −34.1127 −1.12405
\(922\) −12.4853 −0.411181
\(923\) 5.00000 0.164577
\(924\) 0 0
\(925\) −9.41421 −0.309537
\(926\) 12.9706 0.426239
\(927\) 11.1716 0.366923
\(928\) 3.82843 0.125674
\(929\) −57.3137 −1.88040 −0.940201 0.340620i \(-0.889363\pi\)
−0.940201 + 0.340620i \(0.889363\pi\)
\(930\) −28.9706 −0.949982
\(931\) 0 0
\(932\) 7.72792 0.253137
\(933\) 27.8995 0.913388
\(934\) 8.58579 0.280936
\(935\) −3.41421 −0.111657
\(936\) −1.00000 −0.0326860
\(937\) −31.3431 −1.02394 −0.511968 0.859005i \(-0.671083\pi\)
−0.511968 + 0.859005i \(0.671083\pi\)
\(938\) 0 0
\(939\) −20.4853 −0.668512
\(940\) −3.41421 −0.111359
\(941\) −20.9706 −0.683621 −0.341810 0.939769i \(-0.611040\pi\)
−0.341810 + 0.939769i \(0.611040\pi\)
\(942\) 15.7279 0.512443
\(943\) 14.0000 0.455903
\(944\) 12.0711 0.392880
\(945\) 0 0
\(946\) −25.3137 −0.823020
\(947\) 5.58579 0.181514 0.0907568 0.995873i \(-0.471071\pi\)
0.0907568 + 0.995873i \(0.471071\pi\)
\(948\) −0.343146 −0.0111449
\(949\) −1.41421 −0.0459073
\(950\) 52.1127 1.69076
\(951\) −9.79899 −0.317754
\(952\) 0 0
\(953\) −44.4142 −1.43872 −0.719359 0.694639i \(-0.755564\pi\)
−0.719359 + 0.694639i \(0.755564\pi\)
\(954\) −7.48528 −0.242345
\(955\) −73.9411 −2.39268
\(956\) 21.4853 0.694884
\(957\) 9.24264 0.298772
\(958\) −30.1127 −0.972897
\(959\) 0 0
\(960\) 3.41421 0.110193
\(961\) 41.0000 1.32258
\(962\) 1.41421 0.0455961
\(963\) 3.31371 0.106783
\(964\) −26.1421 −0.841981
\(965\) −58.2843 −1.87624
\(966\) 0 0
\(967\) −16.4142 −0.527846 −0.263923 0.964544i \(-0.585016\pi\)
−0.263923 + 0.964544i \(0.585016\pi\)
\(968\) −5.17157 −0.166221
\(969\) −3.24264 −0.104169
\(970\) 51.4558 1.65215
\(971\) −32.5269 −1.04384 −0.521919 0.852995i \(-0.674784\pi\)
−0.521919 + 0.852995i \(0.674784\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −26.2132 −0.839925
\(975\) −6.65685 −0.213190
\(976\) 1.58579 0.0507598
\(977\) 48.8284 1.56216 0.781080 0.624431i \(-0.214669\pi\)
0.781080 + 0.624431i \(0.214669\pi\)
\(978\) −7.34315 −0.234808
\(979\) −13.0711 −0.417753
\(980\) 0 0
\(981\) 1.65685 0.0528993
\(982\) 12.3431 0.393886
\(983\) 12.5147 0.399158 0.199579 0.979882i \(-0.436043\pi\)
0.199579 + 0.979882i \(0.436043\pi\)
\(984\) −9.89949 −0.315584
\(985\) 53.1127 1.69231
\(986\) −1.58579 −0.0505017
\(987\) 0 0
\(988\) −7.82843 −0.249055
\(989\) 14.8284 0.471517
\(990\) 8.24264 0.261968
\(991\) −23.3137 −0.740584 −0.370292 0.928915i \(-0.620742\pi\)
−0.370292 + 0.928915i \(0.620742\pi\)
\(992\) −8.48528 −0.269408
\(993\) −5.51472 −0.175004
\(994\) 0 0
\(995\) 77.5980 2.46002
\(996\) 3.65685 0.115872
\(997\) 46.4975 1.47259 0.736295 0.676661i \(-0.236574\pi\)
0.736295 + 0.676661i \(0.236574\pi\)
\(998\) −7.65685 −0.242373
\(999\) −1.41421 −0.0447437
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 3822.2.a.bu.1.2 2
7.3 odd 6 546.2.i.i.79.2 4
7.5 odd 6 546.2.i.i.235.2 yes 4
7.6 odd 2 3822.2.a.bn.1.1 2
21.5 even 6 1638.2.j.m.235.1 4
21.17 even 6 1638.2.j.m.1171.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.i.79.2 4 7.3 odd 6
546.2.i.i.235.2 yes 4 7.5 odd 6
1638.2.j.m.235.1 4 21.5 even 6
1638.2.j.m.1171.1 4 21.17 even 6
3822.2.a.bn.1.1 2 7.6 odd 2
3822.2.a.bu.1.2 2 1.1 even 1 trivial