Properties

Label 3234.2.a.bf.1.3
Level $3234$
Weight $2$
Character 3234.1
Self dual yes
Analytic conductor $25.824$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.8236200137\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.2700.1
Defining polynomial: \(x^{3} - 15 x - 20\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.41883\) of defining polynomial
Character \(\chi\) \(=\) 3234.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.41883 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.41883 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +4.41883 q^{10} -1.00000 q^{11} -1.00000 q^{12} -4.41883 q^{15} +1.00000 q^{16} +2.37683 q^{17} +1.00000 q^{18} -3.68842 q^{19} +4.41883 q^{20} -1.00000 q^{22} +4.73042 q^{23} -1.00000 q^{24} +14.5261 q^{25} -1.00000 q^{27} -6.52608 q^{29} -4.41883 q^{30} -3.37683 q^{31} +1.00000 q^{32} +1.00000 q^{33} +2.37683 q^{34} +1.00000 q^{36} +7.68842 q^{37} -3.68842 q^{38} +4.41883 q^{40} +12.1492 q^{41} +3.06525 q^{43} -1.00000 q^{44} +4.41883 q^{45} +4.73042 q^{46} -4.10725 q^{47} -1.00000 q^{48} +14.5261 q^{50} -2.37683 q^{51} +8.00000 q^{53} -1.00000 q^{54} -4.41883 q^{55} +3.68842 q^{57} -6.52608 q^{58} -12.5261 q^{59} -4.41883 q^{60} +3.79567 q^{61} -3.37683 q^{62} +1.00000 q^{64} +1.00000 q^{66} +10.1492 q^{67} +2.37683 q^{68} -4.73042 q^{69} +0.934749 q^{71} +1.00000 q^{72} -7.46083 q^{73} +7.68842 q^{74} -14.5261 q^{75} -3.68842 q^{76} -0.418833 q^{79} +4.41883 q^{80} +1.00000 q^{81} +12.1492 q^{82} -2.14925 q^{83} +10.5028 q^{85} +3.06525 q^{86} +6.52608 q^{87} -1.00000 q^{88} -12.8377 q^{89} +4.41883 q^{90} +4.73042 q^{92} +3.37683 q^{93} -4.10725 q^{94} -16.2985 q^{95} -1.00000 q^{96} +7.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} - 3 q^{11} - 3 q^{12} + 3 q^{16} + 3 q^{17} + 3 q^{18} - 9 q^{19} - 3 q^{22} + 3 q^{23} - 3 q^{24} + 15 q^{25} - 3 q^{27} + 9 q^{29} - 6 q^{31} + 3 q^{32} + 3 q^{33} + 3 q^{34} + 3 q^{36} + 21 q^{37} - 9 q^{38} + 12 q^{41} + 3 q^{43} - 3 q^{44} + 3 q^{46} + 3 q^{47} - 3 q^{48} + 15 q^{50} - 3 q^{51} + 24 q^{53} - 3 q^{54} + 9 q^{57} + 9 q^{58} - 9 q^{59} - 6 q^{61} - 6 q^{62} + 3 q^{64} + 3 q^{66} + 6 q^{67} + 3 q^{68} - 3 q^{69} + 9 q^{71} + 3 q^{72} + 21 q^{74} - 15 q^{75} - 9 q^{76} + 12 q^{79} + 3 q^{81} + 12 q^{82} + 18 q^{83} + 3 q^{86} - 9 q^{87} - 3 q^{88} - 12 q^{89} + 3 q^{92} + 6 q^{93} + 3 q^{94} - 3 q^{96} + 21 q^{97} - 3 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 4.41883 1.97616 0.988081 0.153935i \(-0.0491945\pi\)
0.988081 + 0.153935i \(0.0491945\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 4.41883 1.39736
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −4.41883 −1.14094
\(16\) 1.00000 0.250000
\(17\) 2.37683 0.576467 0.288233 0.957560i \(-0.406932\pi\)
0.288233 + 0.957560i \(0.406932\pi\)
\(18\) 1.00000 0.235702
\(19\) −3.68842 −0.846181 −0.423090 0.906087i \(-0.639055\pi\)
−0.423090 + 0.906087i \(0.639055\pi\)
\(20\) 4.41883 0.988081
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 4.73042 0.986360 0.493180 0.869927i \(-0.335834\pi\)
0.493180 + 0.869927i \(0.335834\pi\)
\(24\) −1.00000 −0.204124
\(25\) 14.5261 2.90522
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.52608 −1.21186 −0.605932 0.795517i \(-0.707199\pi\)
−0.605932 + 0.795517i \(0.707199\pi\)
\(30\) −4.41883 −0.806765
\(31\) −3.37683 −0.606497 −0.303249 0.952911i \(-0.598071\pi\)
−0.303249 + 0.952911i \(0.598071\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) 2.37683 0.407624
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 7.68842 1.26397 0.631984 0.774981i \(-0.282241\pi\)
0.631984 + 0.774981i \(0.282241\pi\)
\(38\) −3.68842 −0.598340
\(39\) 0 0
\(40\) 4.41883 0.698679
\(41\) 12.1492 1.89739 0.948697 0.316187i \(-0.102403\pi\)
0.948697 + 0.316187i \(0.102403\pi\)
\(42\) 0 0
\(43\) 3.06525 0.467446 0.233723 0.972303i \(-0.424909\pi\)
0.233723 + 0.972303i \(0.424909\pi\)
\(44\) −1.00000 −0.150756
\(45\) 4.41883 0.658721
\(46\) 4.73042 0.697462
\(47\) −4.10725 −0.599104 −0.299552 0.954080i \(-0.596837\pi\)
−0.299552 + 0.954080i \(0.596837\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 14.5261 2.05430
\(51\) −2.37683 −0.332823
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.41883 −0.595835
\(56\) 0 0
\(57\) 3.68842 0.488543
\(58\) −6.52608 −0.856917
\(59\) −12.5261 −1.63076 −0.815379 0.578928i \(-0.803471\pi\)
−0.815379 + 0.578928i \(0.803471\pi\)
\(60\) −4.41883 −0.570469
\(61\) 3.79567 0.485985 0.242993 0.970028i \(-0.421871\pi\)
0.242993 + 0.970028i \(0.421871\pi\)
\(62\) −3.37683 −0.428858
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) 10.1492 1.23993 0.619964 0.784630i \(-0.287147\pi\)
0.619964 + 0.784630i \(0.287147\pi\)
\(68\) 2.37683 0.288233
\(69\) −4.73042 −0.569475
\(70\) 0 0
\(71\) 0.934749 0.110934 0.0554672 0.998461i \(-0.482335\pi\)
0.0554672 + 0.998461i \(0.482335\pi\)
\(72\) 1.00000 0.117851
\(73\) −7.46083 −0.873224 −0.436612 0.899650i \(-0.643822\pi\)
−0.436612 + 0.899650i \(0.643822\pi\)
\(74\) 7.68842 0.893760
\(75\) −14.5261 −1.67733
\(76\) −3.68842 −0.423090
\(77\) 0 0
\(78\) 0 0
\(79\) −0.418833 −0.0471224 −0.0235612 0.999722i \(-0.507500\pi\)
−0.0235612 + 0.999722i \(0.507500\pi\)
\(80\) 4.41883 0.494041
\(81\) 1.00000 0.111111
\(82\) 12.1492 1.34166
\(83\) −2.14925 −0.235911 −0.117955 0.993019i \(-0.537634\pi\)
−0.117955 + 0.993019i \(0.537634\pi\)
\(84\) 0 0
\(85\) 10.5028 1.13919
\(86\) 3.06525 0.330534
\(87\) 6.52608 0.699669
\(88\) −1.00000 −0.106600
\(89\) −12.8377 −1.36079 −0.680395 0.732846i \(-0.738192\pi\)
−0.680395 + 0.732846i \(0.738192\pi\)
\(90\) 4.41883 0.465786
\(91\) 0 0
\(92\) 4.73042 0.493180
\(93\) 3.37683 0.350161
\(94\) −4.10725 −0.423630
\(95\) −16.2985 −1.67219
\(96\) −1.00000 −0.102062
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 14.5261 1.45261
\(101\) −15.3637 −1.52875 −0.764375 0.644772i \(-0.776952\pi\)
−0.764375 + 0.644772i \(0.776952\pi\)
\(102\) −2.37683 −0.235342
\(103\) −2.62317 −0.258468 −0.129234 0.991614i \(-0.541252\pi\)
−0.129234 + 0.991614i \(0.541252\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 8.00000 0.777029
\(107\) 10.9029 1.05402 0.527012 0.849858i \(-0.323312\pi\)
0.527012 + 0.849858i \(0.323312\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −11.7957 −1.12982 −0.564910 0.825153i \(-0.691089\pi\)
−0.564910 + 0.825153i \(0.691089\pi\)
\(110\) −4.41883 −0.421319
\(111\) −7.68842 −0.729752
\(112\) 0 0
\(113\) 5.37683 0.505810 0.252905 0.967491i \(-0.418614\pi\)
0.252905 + 0.967491i \(0.418614\pi\)
\(114\) 3.68842 0.345452
\(115\) 20.9029 1.94921
\(116\) −6.52608 −0.605932
\(117\) 0 0
\(118\) −12.5261 −1.15312
\(119\) 0 0
\(120\) −4.41883 −0.403382
\(121\) 1.00000 0.0909091
\(122\) 3.79567 0.343643
\(123\) −12.1492 −1.09546
\(124\) −3.37683 −0.303249
\(125\) 42.0942 3.76502
\(126\) 0 0
\(127\) 0.730416 0.0648139 0.0324070 0.999475i \(-0.489683\pi\)
0.0324070 + 0.999475i \(0.489683\pi\)
\(128\) 1.00000 0.0883883
\(129\) −3.06525 −0.269880
\(130\) 0 0
\(131\) 7.37683 0.644517 0.322258 0.946652i \(-0.395558\pi\)
0.322258 + 0.946652i \(0.395558\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) 10.1492 0.876762
\(135\) −4.41883 −0.380313
\(136\) 2.37683 0.203812
\(137\) −8.83767 −0.755053 −0.377526 0.925999i \(-0.623225\pi\)
−0.377526 + 0.925999i \(0.623225\pi\)
\(138\) −4.73042 −0.402680
\(139\) 1.47392 0.125016 0.0625080 0.998044i \(-0.480090\pi\)
0.0625080 + 0.998044i \(0.480090\pi\)
\(140\) 0 0
\(141\) 4.10725 0.345893
\(142\) 0.934749 0.0784424
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −28.8377 −2.39484
\(146\) −7.46083 −0.617463
\(147\) 0 0
\(148\) 7.68842 0.631984
\(149\) −9.36375 −0.767108 −0.383554 0.923518i \(-0.625300\pi\)
−0.383554 + 0.923518i \(0.625300\pi\)
\(150\) −14.5261 −1.18605
\(151\) −13.5681 −1.10415 −0.552077 0.833793i \(-0.686165\pi\)
−0.552077 + 0.833793i \(0.686165\pi\)
\(152\) −3.68842 −0.299170
\(153\) 2.37683 0.192156
\(154\) 0 0
\(155\) −14.9217 −1.19854
\(156\) 0 0
\(157\) −18.7406 −1.49566 −0.747831 0.663890i \(-0.768904\pi\)
−0.747831 + 0.663890i \(0.768904\pi\)
\(158\) −0.418833 −0.0333205
\(159\) −8.00000 −0.634441
\(160\) 4.41883 0.349339
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 18.7724 1.47037 0.735185 0.677867i \(-0.237096\pi\)
0.735185 + 0.677867i \(0.237096\pi\)
\(164\) 12.1492 0.948697
\(165\) 4.41883 0.344006
\(166\) −2.14925 −0.166814
\(167\) −23.8898 −1.84865 −0.924325 0.381606i \(-0.875371\pi\)
−0.924325 + 0.381606i \(0.875371\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 10.5028 0.805530
\(171\) −3.68842 −0.282060
\(172\) 3.06525 0.233723
\(173\) 11.3768 0.864965 0.432482 0.901642i \(-0.357638\pi\)
0.432482 + 0.901642i \(0.357638\pi\)
\(174\) 6.52608 0.494741
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 12.5261 0.941518
\(178\) −12.8377 −0.962224
\(179\) 24.6101 1.83944 0.919722 0.392571i \(-0.128414\pi\)
0.919722 + 0.392571i \(0.128414\pi\)
\(180\) 4.41883 0.329360
\(181\) 17.4608 1.29785 0.648927 0.760851i \(-0.275218\pi\)
0.648927 + 0.760851i \(0.275218\pi\)
\(182\) 0 0
\(183\) −3.79567 −0.280584
\(184\) 4.73042 0.348731
\(185\) 33.9738 2.49781
\(186\) 3.37683 0.247601
\(187\) −2.37683 −0.173811
\(188\) −4.10725 −0.299552
\(189\) 0 0
\(190\) −16.2985 −1.18242
\(191\) −16.8377 −1.21833 −0.609165 0.793043i \(-0.708495\pi\)
−0.609165 + 0.793043i \(0.708495\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 5.46083 0.393079 0.196540 0.980496i \(-0.437030\pi\)
0.196540 + 0.980496i \(0.437030\pi\)
\(194\) 7.00000 0.502571
\(195\) 0 0
\(196\) 0 0
\(197\) −9.68842 −0.690271 −0.345136 0.938553i \(-0.612167\pi\)
−0.345136 + 0.938553i \(0.612167\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 20.2985 1.43892 0.719461 0.694533i \(-0.244389\pi\)
0.719461 + 0.694533i \(0.244389\pi\)
\(200\) 14.5261 1.02715
\(201\) −10.1492 −0.715873
\(202\) −15.3637 −1.08099
\(203\) 0 0
\(204\) −2.37683 −0.166412
\(205\) 53.6855 3.74956
\(206\) −2.62317 −0.182765
\(207\) 4.73042 0.328787
\(208\) 0 0
\(209\) 3.68842 0.255133
\(210\) 0 0
\(211\) 22.5130 1.54986 0.774929 0.632048i \(-0.217785\pi\)
0.774929 + 0.632048i \(0.217785\pi\)
\(212\) 8.00000 0.549442
\(213\) −0.934749 −0.0640480
\(214\) 10.9029 0.745308
\(215\) 13.5448 0.923750
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −11.7957 −0.798903
\(219\) 7.46083 0.504156
\(220\) −4.41883 −0.297918
\(221\) 0 0
\(222\) −7.68842 −0.516013
\(223\) −7.46083 −0.499614 −0.249807 0.968296i \(-0.580367\pi\)
−0.249807 + 0.968296i \(0.580367\pi\)
\(224\) 0 0
\(225\) 14.5261 0.968405
\(226\) 5.37683 0.357662
\(227\) −2.77241 −0.184012 −0.0920058 0.995758i \(-0.529328\pi\)
−0.0920058 + 0.995758i \(0.529328\pi\)
\(228\) 3.68842 0.244271
\(229\) −2.53917 −0.167793 −0.0838965 0.996474i \(-0.526737\pi\)
−0.0838965 + 0.996474i \(0.526737\pi\)
\(230\) 20.9029 1.37830
\(231\) 0 0
\(232\) −6.52608 −0.428458
\(233\) −24.0522 −1.57571 −0.787855 0.615861i \(-0.788808\pi\)
−0.787855 + 0.615861i \(0.788808\pi\)
\(234\) 0 0
\(235\) −18.1492 −1.18393
\(236\) −12.5261 −0.815379
\(237\) 0.418833 0.0272061
\(238\) 0 0
\(239\) 7.59133 0.491042 0.245521 0.969391i \(-0.421041\pi\)
0.245521 + 0.969391i \(0.421041\pi\)
\(240\) −4.41883 −0.285234
\(241\) 16.2985 1.04988 0.524939 0.851140i \(-0.324088\pi\)
0.524939 + 0.851140i \(0.324088\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 3.79567 0.242993
\(245\) 0 0
\(246\) −12.1492 −0.774608
\(247\) 0 0
\(248\) −3.37683 −0.214429
\(249\) 2.14925 0.136203
\(250\) 42.0942 2.66227
\(251\) −8.61008 −0.543463 −0.271732 0.962373i \(-0.587596\pi\)
−0.271732 + 0.962373i \(0.587596\pi\)
\(252\) 0 0
\(253\) −4.73042 −0.297399
\(254\) 0.730416 0.0458304
\(255\) −10.5028 −0.657713
\(256\) 1.00000 0.0625000
\(257\) 0.837665 0.0522521 0.0261261 0.999659i \(-0.491683\pi\)
0.0261261 + 0.999659i \(0.491683\pi\)
\(258\) −3.06525 −0.190834
\(259\) 0 0
\(260\) 0 0
\(261\) −6.52608 −0.403954
\(262\) 7.37683 0.455742
\(263\) 10.8377 0.668279 0.334140 0.942524i \(-0.391554\pi\)
0.334140 + 0.942524i \(0.391554\pi\)
\(264\) 1.00000 0.0615457
\(265\) 35.3507 2.17157
\(266\) 0 0
\(267\) 12.8377 0.785652
\(268\) 10.1492 0.619964
\(269\) 17.8797 1.09014 0.545071 0.838390i \(-0.316503\pi\)
0.545071 + 0.838390i \(0.316503\pi\)
\(270\) −4.41883 −0.268922
\(271\) −8.83767 −0.536850 −0.268425 0.963301i \(-0.586503\pi\)
−0.268425 + 0.963301i \(0.586503\pi\)
\(272\) 2.37683 0.144117
\(273\) 0 0
\(274\) −8.83767 −0.533903
\(275\) −14.5261 −0.875956
\(276\) −4.73042 −0.284738
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) 1.47392 0.0883997
\(279\) −3.37683 −0.202166
\(280\) 0 0
\(281\) −15.9217 −0.949807 −0.474903 0.880038i \(-0.657517\pi\)
−0.474903 + 0.880038i \(0.657517\pi\)
\(282\) 4.10725 0.244583
\(283\) −18.0840 −1.07498 −0.537491 0.843269i \(-0.680628\pi\)
−0.537491 + 0.843269i \(0.680628\pi\)
\(284\) 0.934749 0.0554672
\(285\) 16.2985 0.965440
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −11.3507 −0.667686
\(290\) −28.8377 −1.69341
\(291\) −7.00000 −0.410347
\(292\) −7.46083 −0.436612
\(293\) 25.1492 1.46923 0.734617 0.678482i \(-0.237362\pi\)
0.734617 + 0.678482i \(0.237362\pi\)
\(294\) 0 0
\(295\) −55.3507 −3.22264
\(296\) 7.68842 0.446880
\(297\) 1.00000 0.0580259
\(298\) −9.36375 −0.542427
\(299\) 0 0
\(300\) −14.5261 −0.838664
\(301\) 0 0
\(302\) −13.5681 −0.780755
\(303\) 15.3637 0.882624
\(304\) −3.68842 −0.211545
\(305\) 16.7724 0.960386
\(306\) 2.37683 0.135875
\(307\) −3.86950 −0.220844 −0.110422 0.993885i \(-0.535220\pi\)
−0.110422 + 0.993885i \(0.535220\pi\)
\(308\) 0 0
\(309\) 2.62317 0.149227
\(310\) −14.9217 −0.847494
\(311\) −13.5681 −0.769375 −0.384688 0.923047i \(-0.625691\pi\)
−0.384688 + 0.923047i \(0.625691\pi\)
\(312\) 0 0
\(313\) 2.52608 0.142783 0.0713913 0.997448i \(-0.477256\pi\)
0.0713913 + 0.997448i \(0.477256\pi\)
\(314\) −18.7406 −1.05759
\(315\) 0 0
\(316\) −0.418833 −0.0235612
\(317\) −20.0102 −1.12388 −0.561941 0.827177i \(-0.689945\pi\)
−0.561941 + 0.827177i \(0.689945\pi\)
\(318\) −8.00000 −0.448618
\(319\) 6.52608 0.365390
\(320\) 4.41883 0.247020
\(321\) −10.9029 −0.608541
\(322\) 0 0
\(323\) −8.76675 −0.487795
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 18.7724 1.03971
\(327\) 11.7957 0.652302
\(328\) 12.1492 0.670830
\(329\) 0 0
\(330\) 4.41883 0.243249
\(331\) −16.1492 −0.887643 −0.443821 0.896115i \(-0.646378\pi\)
−0.443821 + 0.896115i \(0.646378\pi\)
\(332\) −2.14925 −0.117955
\(333\) 7.68842 0.421323
\(334\) −23.8898 −1.30719
\(335\) 44.8478 2.45030
\(336\) 0 0
\(337\) 23.1362 1.26031 0.630154 0.776471i \(-0.282992\pi\)
0.630154 + 0.776471i \(0.282992\pi\)
\(338\) −13.0000 −0.707107
\(339\) −5.37683 −0.292030
\(340\) 10.5028 0.569596
\(341\) 3.37683 0.182866
\(342\) −3.68842 −0.199447
\(343\) 0 0
\(344\) 3.06525 0.165267
\(345\) −20.9029 −1.12538
\(346\) 11.3768 0.611622
\(347\) 15.6566 0.840489 0.420245 0.907411i \(-0.361944\pi\)
0.420245 + 0.907411i \(0.361944\pi\)
\(348\) 6.52608 0.349835
\(349\) −20.0102 −1.07112 −0.535560 0.844497i \(-0.679899\pi\)
−0.535560 + 0.844497i \(0.679899\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 7.46083 0.397100 0.198550 0.980091i \(-0.436377\pi\)
0.198550 + 0.980091i \(0.436377\pi\)
\(354\) 12.5261 0.665754
\(355\) 4.13050 0.219224
\(356\) −12.8377 −0.680395
\(357\) 0 0
\(358\) 24.6101 1.30068
\(359\) −11.2463 −0.593559 −0.296779 0.954946i \(-0.595913\pi\)
−0.296779 + 0.954946i \(0.595913\pi\)
\(360\) 4.41883 0.232893
\(361\) −5.39558 −0.283978
\(362\) 17.4608 0.917721
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −32.9682 −1.72563
\(366\) −3.79567 −0.198403
\(367\) −5.91600 −0.308813 −0.154406 0.988007i \(-0.549347\pi\)
−0.154406 + 0.988007i \(0.549347\pi\)
\(368\) 4.73042 0.246590
\(369\) 12.1492 0.632465
\(370\) 33.9738 1.76622
\(371\) 0 0
\(372\) 3.37683 0.175081
\(373\) −23.1260 −1.19742 −0.598709 0.800966i \(-0.704320\pi\)
−0.598709 + 0.800966i \(0.704320\pi\)
\(374\) −2.37683 −0.122903
\(375\) −42.0942 −2.17373
\(376\) −4.10725 −0.211815
\(377\) 0 0
\(378\) 0 0
\(379\) −6.77241 −0.347876 −0.173938 0.984757i \(-0.555649\pi\)
−0.173938 + 0.984757i \(0.555649\pi\)
\(380\) −16.2985 −0.836095
\(381\) −0.730416 −0.0374203
\(382\) −16.8377 −0.861490
\(383\) 25.3637 1.29603 0.648013 0.761629i \(-0.275600\pi\)
0.648013 + 0.761629i \(0.275600\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 5.46083 0.277949
\(387\) 3.06525 0.155815
\(388\) 7.00000 0.355371
\(389\) 25.2565 1.28056 0.640278 0.768144i \(-0.278819\pi\)
0.640278 + 0.768144i \(0.278819\pi\)
\(390\) 0 0
\(391\) 11.2434 0.568604
\(392\) 0 0
\(393\) −7.37683 −0.372112
\(394\) −9.68842 −0.488095
\(395\) −1.85075 −0.0931214
\(396\) −1.00000 −0.0502519
\(397\) −13.4739 −0.676237 −0.338118 0.941104i \(-0.609790\pi\)
−0.338118 + 0.941104i \(0.609790\pi\)
\(398\) 20.2985 1.01747
\(399\) 0 0
\(400\) 14.5261 0.726304
\(401\) 33.8898 1.69238 0.846189 0.532883i \(-0.178892\pi\)
0.846189 + 0.532883i \(0.178892\pi\)
\(402\) −10.1492 −0.506199
\(403\) 0 0
\(404\) −15.3637 −0.764375
\(405\) 4.41883 0.219574
\(406\) 0 0
\(407\) −7.68842 −0.381101
\(408\) −2.37683 −0.117671
\(409\) −4.21450 −0.208394 −0.104197 0.994557i \(-0.533227\pi\)
−0.104197 + 0.994557i \(0.533227\pi\)
\(410\) 53.6855 2.65134
\(411\) 8.83767 0.435930
\(412\) −2.62317 −0.129234
\(413\) 0 0
\(414\) 4.73042 0.232487
\(415\) −9.49717 −0.466198
\(416\) 0 0
\(417\) −1.47392 −0.0721781
\(418\) 3.68842 0.180406
\(419\) −22.3116 −1.08999 −0.544996 0.838439i \(-0.683469\pi\)
−0.544996 + 0.838439i \(0.683469\pi\)
\(420\) 0 0
\(421\) 18.3116 0.892452 0.446226 0.894920i \(-0.352768\pi\)
0.446226 + 0.894920i \(0.352768\pi\)
\(422\) 22.5130 1.09592
\(423\) −4.10725 −0.199701
\(424\) 8.00000 0.388514
\(425\) 34.5261 1.67476
\(426\) −0.934749 −0.0452888
\(427\) 0 0
\(428\) 10.9029 0.527012
\(429\) 0 0
\(430\) 13.5448 0.653190
\(431\) −28.8377 −1.38906 −0.694531 0.719463i \(-0.744388\pi\)
−0.694531 + 0.719463i \(0.744388\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 7.62317 0.366346 0.183173 0.983081i \(-0.441363\pi\)
0.183173 + 0.983081i \(0.441363\pi\)
\(434\) 0 0
\(435\) 28.8377 1.38266
\(436\) −11.7957 −0.564910
\(437\) −17.4477 −0.834639
\(438\) 7.46083 0.356492
\(439\) −33.3739 −1.59285 −0.796425 0.604737i \(-0.793278\pi\)
−0.796425 + 0.604737i \(0.793278\pi\)
\(440\) −4.41883 −0.210660
\(441\) 0 0
\(442\) 0 0
\(443\) −4.85075 −0.230466 −0.115233 0.993338i \(-0.536761\pi\)
−0.115233 + 0.993338i \(0.536761\pi\)
\(444\) −7.68842 −0.364876
\(445\) −56.7275 −2.68914
\(446\) −7.46083 −0.353281
\(447\) 9.36375 0.442890
\(448\) 0 0
\(449\) 14.3450 0.676982 0.338491 0.940970i \(-0.390083\pi\)
0.338491 + 0.940970i \(0.390083\pi\)
\(450\) 14.5261 0.684766
\(451\) −12.1492 −0.572086
\(452\) 5.37683 0.252905
\(453\) 13.5681 0.637484
\(454\) −2.77241 −0.130116
\(455\) 0 0
\(456\) 3.68842 0.172726
\(457\) −25.1827 −1.17800 −0.588998 0.808135i \(-0.700477\pi\)
−0.588998 + 0.808135i \(0.700477\pi\)
\(458\) −2.53917 −0.118648
\(459\) −2.37683 −0.110941
\(460\) 20.9029 0.974603
\(461\) 13.0187 0.606344 0.303172 0.952936i \(-0.401954\pi\)
0.303172 + 0.952936i \(0.401954\pi\)
\(462\) 0 0
\(463\) −14.2145 −0.660604 −0.330302 0.943875i \(-0.607151\pi\)
−0.330302 + 0.943875i \(0.607151\pi\)
\(464\) −6.52608 −0.302966
\(465\) 14.9217 0.691976
\(466\) −24.0522 −1.11420
\(467\) −2.39558 −0.110854 −0.0554271 0.998463i \(-0.517652\pi\)
−0.0554271 + 0.998463i \(0.517652\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −18.1492 −0.837162
\(471\) 18.7406 0.863520
\(472\) −12.5261 −0.576560
\(473\) −3.06525 −0.140940
\(474\) 0.418833 0.0192376
\(475\) −53.5782 −2.45834
\(476\) 0 0
\(477\) 8.00000 0.366295
\(478\) 7.59133 0.347219
\(479\) 10.8377 0.495186 0.247593 0.968864i \(-0.420360\pi\)
0.247593 + 0.968864i \(0.420360\pi\)
\(480\) −4.41883 −0.201691
\(481\) 0 0
\(482\) 16.2985 0.742376
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 30.9318 1.40454
\(486\) −1.00000 −0.0453609
\(487\) −34.6435 −1.56985 −0.784923 0.619593i \(-0.787298\pi\)
−0.784923 + 0.619593i \(0.787298\pi\)
\(488\) 3.79567 0.171822
\(489\) −18.7724 −0.848918
\(490\) 0 0
\(491\) 38.1492 1.72165 0.860826 0.508900i \(-0.169948\pi\)
0.860826 + 0.508900i \(0.169948\pi\)
\(492\) −12.1492 −0.547730
\(493\) −15.5114 −0.698599
\(494\) 0 0
\(495\) −4.41883 −0.198612
\(496\) −3.37683 −0.151624
\(497\) 0 0
\(498\) 2.14925 0.0963101
\(499\) −0.623166 −0.0278968 −0.0139484 0.999903i \(-0.504440\pi\)
−0.0139484 + 0.999903i \(0.504440\pi\)
\(500\) 42.0942 1.88251
\(501\) 23.8898 1.06732
\(502\) −8.61008 −0.384287
\(503\) 31.1362 1.38829 0.694146 0.719834i \(-0.255782\pi\)
0.694146 + 0.719834i \(0.255782\pi\)
\(504\) 0 0
\(505\) −67.8898 −3.02106
\(506\) −4.73042 −0.210293
\(507\) 13.0000 0.577350
\(508\) 0.730416 0.0324070
\(509\) −42.1043 −1.86624 −0.933121 0.359563i \(-0.882926\pi\)
−0.933121 + 0.359563i \(0.882926\pi\)
\(510\) −10.5028 −0.465073
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 3.68842 0.162848
\(514\) 0.837665 0.0369478
\(515\) −11.5913 −0.510775
\(516\) −3.06525 −0.134940
\(517\) 4.10725 0.180637
\(518\) 0 0
\(519\) −11.3768 −0.499388
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) −6.52608 −0.285639
\(523\) −25.2667 −1.10483 −0.552417 0.833568i \(-0.686294\pi\)
−0.552417 + 0.833568i \(0.686294\pi\)
\(524\) 7.37683 0.322258
\(525\) 0 0
\(526\) 10.8377 0.472545
\(527\) −8.02617 −0.349626
\(528\) 1.00000 0.0435194
\(529\) −0.623166 −0.0270942
\(530\) 35.3507 1.53553
\(531\) −12.5261 −0.543586
\(532\) 0 0
\(533\) 0 0
\(534\) 12.8377 0.555540
\(535\) 48.1782 2.08292
\(536\) 10.1492 0.438381
\(537\) −24.6101 −1.06200
\(538\) 17.8797 0.770847
\(539\) 0 0
\(540\) −4.41883 −0.190156
\(541\) −4.41883 −0.189980 −0.0949902 0.995478i \(-0.530282\pi\)
−0.0949902 + 0.995478i \(0.530282\pi\)
\(542\) −8.83767 −0.379610
\(543\) −17.4608 −0.749316
\(544\) 2.37683 0.101906
\(545\) −52.1231 −2.23271
\(546\) 0 0
\(547\) −27.6622 −1.18275 −0.591376 0.806396i \(-0.701415\pi\)
−0.591376 + 0.806396i \(0.701415\pi\)
\(548\) −8.83767 −0.377526
\(549\) 3.79567 0.161995
\(550\) −14.5261 −0.619394
\(551\) 24.0709 1.02546
\(552\) −4.73042 −0.201340
\(553\) 0 0
\(554\) −16.0000 −0.679775
\(555\) −33.9738 −1.44211
\(556\) 1.47392 0.0625080
\(557\) −7.98691 −0.338416 −0.169208 0.985580i \(-0.554121\pi\)
−0.169208 + 0.985580i \(0.554121\pi\)
\(558\) −3.37683 −0.142953
\(559\) 0 0
\(560\) 0 0
\(561\) 2.37683 0.100350
\(562\) −15.9217 −0.671615
\(563\) −27.3507 −1.15269 −0.576346 0.817205i \(-0.695522\pi\)
−0.576346 + 0.817205i \(0.695522\pi\)
\(564\) 4.10725 0.172946
\(565\) 23.7593 0.999562
\(566\) −18.0840 −0.760127
\(567\) 0 0
\(568\) 0.934749 0.0392212
\(569\) −29.4477 −1.23451 −0.617257 0.786762i \(-0.711756\pi\)
−0.617257 + 0.786762i \(0.711756\pi\)
\(570\) 16.2985 0.682669
\(571\) −14.5261 −0.607898 −0.303949 0.952688i \(-0.598305\pi\)
−0.303949 + 0.952688i \(0.598305\pi\)
\(572\) 0 0
\(573\) 16.8377 0.703404
\(574\) 0 0
\(575\) 68.7144 2.86559
\(576\) 1.00000 0.0416667
\(577\) 13.3956 0.557665 0.278833 0.960340i \(-0.410053\pi\)
0.278833 + 0.960340i \(0.410053\pi\)
\(578\) −11.3507 −0.472125
\(579\) −5.46083 −0.226944
\(580\) −28.8377 −1.19742
\(581\) 0 0
\(582\) −7.00000 −0.290159
\(583\) −8.00000 −0.331326
\(584\) −7.46083 −0.308731
\(585\) 0 0
\(586\) 25.1492 1.03891
\(587\) −11.2928 −0.466105 −0.233053 0.972464i \(-0.574871\pi\)
−0.233053 + 0.972464i \(0.574871\pi\)
\(588\) 0 0
\(589\) 12.4552 0.513206
\(590\) −55.3507 −2.27875
\(591\) 9.68842 0.398528
\(592\) 7.68842 0.315992
\(593\) −19.5782 −0.803982 −0.401991 0.915644i \(-0.631682\pi\)
−0.401991 + 0.915644i \(0.631682\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) −9.36375 −0.383554
\(597\) −20.2985 −0.830762
\(598\) 0 0
\(599\) 10.9318 0.446662 0.223331 0.974743i \(-0.428307\pi\)
0.223331 + 0.974743i \(0.428307\pi\)
\(600\) −14.5261 −0.593025
\(601\) −41.2202 −1.68141 −0.840703 0.541497i \(-0.817858\pi\)
−0.840703 + 0.541497i \(0.817858\pi\)
\(602\) 0 0
\(603\) 10.1492 0.413309
\(604\) −13.5681 −0.552077
\(605\) 4.41883 0.179651
\(606\) 15.3637 0.624110
\(607\) −31.3405 −1.27207 −0.636036 0.771660i \(-0.719427\pi\)
−0.636036 + 0.771660i \(0.719427\pi\)
\(608\) −3.68842 −0.149585
\(609\) 0 0
\(610\) 16.7724 0.679095
\(611\) 0 0
\(612\) 2.37683 0.0960778
\(613\) 17.0885 0.690198 0.345099 0.938566i \(-0.387845\pi\)
0.345099 + 0.938566i \(0.387845\pi\)
\(614\) −3.86950 −0.156160
\(615\) −53.6855 −2.16481
\(616\) 0 0
\(617\) −6.96817 −0.280528 −0.140264 0.990114i \(-0.544795\pi\)
−0.140264 + 0.990114i \(0.544795\pi\)
\(618\) 2.62317 0.105519
\(619\) −0.0187473 −0.000753517 0 −0.000376759 1.00000i \(-0.500120\pi\)
−0.000376759 1.00000i \(0.500120\pi\)
\(620\) −14.9217 −0.599268
\(621\) −4.73042 −0.189825
\(622\) −13.5681 −0.544030
\(623\) 0 0
\(624\) 0 0
\(625\) 113.377 4.53507
\(626\) 2.52608 0.100963
\(627\) −3.68842 −0.147301
\(628\) −18.7406 −0.747831
\(629\) 18.2741 0.728636
\(630\) 0 0
\(631\) −5.78550 −0.230317 −0.115159 0.993347i \(-0.536738\pi\)
−0.115159 + 0.993347i \(0.536738\pi\)
\(632\) −0.418833 −0.0166603
\(633\) −22.5130 −0.894811
\(634\) −20.0102 −0.794705
\(635\) 3.22759 0.128083
\(636\) −8.00000 −0.317221
\(637\) 0 0
\(638\) 6.52608 0.258370
\(639\) 0.934749 0.0369781
\(640\) 4.41883 0.174670
\(641\) 15.3768 0.607348 0.303674 0.952776i \(-0.401787\pi\)
0.303674 + 0.952776i \(0.401787\pi\)
\(642\) −10.9029 −0.430304
\(643\) −35.8058 −1.41204 −0.706022 0.708190i \(-0.749512\pi\)
−0.706022 + 0.708190i \(0.749512\pi\)
\(644\) 0 0
\(645\) −13.5448 −0.533327
\(646\) −8.76675 −0.344923
\(647\) 9.25650 0.363910 0.181955 0.983307i \(-0.441757\pi\)
0.181955 + 0.983307i \(0.441757\pi\)
\(648\) 1.00000 0.0392837
\(649\) 12.5261 0.491692
\(650\) 0 0
\(651\) 0 0
\(652\) 18.7724 0.735185
\(653\) 24.8478 0.972371 0.486185 0.873856i \(-0.338388\pi\)
0.486185 + 0.873856i \(0.338388\pi\)
\(654\) 11.7957 0.461247
\(655\) 32.5970 1.27367
\(656\) 12.1492 0.474348
\(657\) −7.46083 −0.291075
\(658\) 0 0
\(659\) −6.60442 −0.257272 −0.128636 0.991692i \(-0.541060\pi\)
−0.128636 + 0.991692i \(0.541060\pi\)
\(660\) 4.41883 0.172003
\(661\) −13.2332 −0.514714 −0.257357 0.966316i \(-0.582852\pi\)
−0.257357 + 0.966316i \(0.582852\pi\)
\(662\) −16.1492 −0.627658
\(663\) 0 0
\(664\) −2.14925 −0.0834070
\(665\) 0 0
\(666\) 7.68842 0.297920
\(667\) −30.8711 −1.19533
\(668\) −23.8898 −0.924325
\(669\) 7.46083 0.288452
\(670\) 44.8478 1.73262
\(671\) −3.79567 −0.146530
\(672\) 0 0
\(673\) 0.130501 0.00503045 0.00251523 0.999997i \(-0.499199\pi\)
0.00251523 + 0.999997i \(0.499199\pi\)
\(674\) 23.1362 0.891172
\(675\) −14.5261 −0.559109
\(676\) −13.0000 −0.500000
\(677\) 28.1174 1.08064 0.540320 0.841460i \(-0.318303\pi\)
0.540320 + 0.841460i \(0.318303\pi\)
\(678\) −5.37683 −0.206496
\(679\) 0 0
\(680\) 10.5028 0.402765
\(681\) 2.77241 0.106239
\(682\) 3.37683 0.129306
\(683\) 15.9029 0.608508 0.304254 0.952591i \(-0.401593\pi\)
0.304254 + 0.952591i \(0.401593\pi\)
\(684\) −3.68842 −0.141030
\(685\) −39.0522 −1.49211
\(686\) 0 0
\(687\) 2.53917 0.0968753
\(688\) 3.06525 0.116862
\(689\) 0 0
\(690\) −20.9029 −0.795760
\(691\) −15.9813 −0.607956 −0.303978 0.952679i \(-0.598315\pi\)
−0.303978 + 0.952679i \(0.598315\pi\)
\(692\) 11.3768 0.432482
\(693\) 0 0
\(694\) 15.6566 0.594316
\(695\) 6.51300 0.247052
\(696\) 6.52608 0.247371
\(697\) 28.8767 1.09378
\(698\) −20.0102 −0.757396
\(699\) 24.0522 0.909736
\(700\) 0 0
\(701\) 13.9869 0.528278 0.264139 0.964485i \(-0.414912\pi\)
0.264139 + 0.964485i \(0.414912\pi\)
\(702\) 0 0
\(703\) −28.3581 −1.06955
\(704\) −1.00000 −0.0376889
\(705\) 18.1492 0.683540
\(706\) 7.46083 0.280792
\(707\) 0 0
\(708\) 12.5261 0.470759
\(709\) 7.64191 0.286998 0.143499 0.989650i \(-0.454165\pi\)
0.143499 + 0.989650i \(0.454165\pi\)
\(710\) 4.13050 0.155015
\(711\) −0.418833 −0.0157075
\(712\) −12.8377 −0.481112
\(713\) −15.9738 −0.598225
\(714\) 0 0
\(715\) 0 0
\(716\) 24.6101 0.919722
\(717\) −7.59133 −0.283504
\(718\) −11.2463 −0.419709
\(719\) −19.4376 −0.724899 −0.362450 0.932003i \(-0.618060\pi\)
−0.362450 + 0.932003i \(0.618060\pi\)
\(720\) 4.41883 0.164680
\(721\) 0 0
\(722\) −5.39558 −0.200803
\(723\) −16.2985 −0.606148
\(724\) 17.4608 0.648927
\(725\) −94.7984 −3.52072
\(726\) −1.00000 −0.0371135
\(727\) −44.9217 −1.66605 −0.833026 0.553234i \(-0.813394\pi\)
−0.833026 + 0.553234i \(0.813394\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −32.9682 −1.22021
\(731\) 7.28559 0.269467
\(732\) −3.79567 −0.140292
\(733\) 25.0420 0.924947 0.462474 0.886633i \(-0.346962\pi\)
0.462474 + 0.886633i \(0.346962\pi\)
\(734\) −5.91600 −0.218364
\(735\) 0 0
\(736\) 4.73042 0.174365
\(737\) −10.1492 −0.373852
\(738\) 12.1492 0.447220
\(739\) −6.62317 −0.243637 −0.121819 0.992552i \(-0.538873\pi\)
−0.121819 + 0.992552i \(0.538873\pi\)
\(740\) 33.9738 1.24890
\(741\) 0 0
\(742\) 0 0
\(743\) −39.3972 −1.44534 −0.722671 0.691192i \(-0.757086\pi\)
−0.722671 + 0.691192i \(0.757086\pi\)
\(744\) 3.37683 0.123801
\(745\) −41.3768 −1.51593
\(746\) −23.1260 −0.846703
\(747\) −2.14925 −0.0786369
\(748\) −2.37683 −0.0869056
\(749\) 0 0
\(750\) −42.0942 −1.53706
\(751\) −10.7072 −0.390710 −0.195355 0.980733i \(-0.562586\pi\)
−0.195355 + 0.980733i \(0.562586\pi\)
\(752\) −4.10725 −0.149776
\(753\) 8.61008 0.313769
\(754\) 0 0
\(755\) −59.9551 −2.18199
\(756\) 0 0
\(757\) −17.2797 −0.628043 −0.314022 0.949416i \(-0.601676\pi\)
−0.314022 + 0.949416i \(0.601676\pi\)
\(758\) −6.77241 −0.245985
\(759\) 4.73042 0.171703
\(760\) −16.2985 −0.591209
\(761\) −32.7087 −1.18569 −0.592846 0.805316i \(-0.701996\pi\)
−0.592846 + 0.805316i \(0.701996\pi\)
\(762\) −0.730416 −0.0264602
\(763\) 0 0
\(764\) −16.8377 −0.609165
\(765\) 10.5028 0.379731
\(766\) 25.3637 0.916429
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 12.2145 0.440466 0.220233 0.975447i \(-0.429318\pi\)
0.220233 + 0.975447i \(0.429318\pi\)
\(770\) 0 0
\(771\) −0.837665 −0.0301678
\(772\) 5.46083 0.196540
\(773\) 0.204334 0.00734937 0.00367468 0.999993i \(-0.498830\pi\)
0.00367468 + 0.999993i \(0.498830\pi\)
\(774\) 3.06525 0.110178
\(775\) −49.0522 −1.76201
\(776\) 7.00000 0.251285
\(777\) 0 0
\(778\) 25.2565 0.905489
\(779\) −44.8115 −1.60554
\(780\) 0 0
\(781\) −0.934749 −0.0334480
\(782\) 11.2434 0.402064
\(783\) 6.52608 0.233223
\(784\) 0 0
\(785\) −82.8115 −2.95567
\(786\) −7.37683 −0.263123
\(787\) 25.5579 0.911041 0.455521 0.890225i \(-0.349453\pi\)
0.455521 + 0.890225i \(0.349453\pi\)
\(788\) −9.68842 −0.345136
\(789\) −10.8377 −0.385831
\(790\) −1.85075 −0.0658468
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) 0 0
\(794\) −13.4739 −0.478171
\(795\) −35.3507 −1.25376
\(796\) 20.2985 0.719461
\(797\) 30.0477 1.06434 0.532171 0.846637i \(-0.321376\pi\)
0.532171 + 0.846637i \(0.321376\pi\)
\(798\) 0 0
\(799\) −9.76225 −0.345364
\(800\) 14.5261 0.513575
\(801\) −12.8377 −0.453597
\(802\) 33.8898 1.19669
\(803\) 7.46083 0.263287
\(804\) −10.1492 −0.357936
\(805\) 0 0
\(806\) 0 0
\(807\) −17.8797 −0.629394
\(808\) −15.3637 −0.540495
\(809\) 2.34342 0.0823901 0.0411951 0.999151i \(-0.486883\pi\)
0.0411951 + 0.999151i \(0.486883\pi\)
\(810\) 4.41883 0.155262
\(811\) −11.8695 −0.416794 −0.208397 0.978044i \(-0.566825\pi\)
−0.208397 + 0.978044i \(0.566825\pi\)
\(812\) 0 0
\(813\) 8.83767 0.309950
\(814\) −7.68842 −0.269479
\(815\) 82.9522 2.90569
\(816\) −2.37683 −0.0832058
\(817\) −11.3059 −0.395544
\(818\) −4.21450 −0.147357
\(819\) 0 0
\(820\) 53.6855 1.87478
\(821\) −37.4812 −1.30810 −0.654051 0.756451i \(-0.726932\pi\)
−0.654051 + 0.756451i \(0.726932\pi\)
\(822\) 8.83767 0.308249
\(823\) 1.29284 0.0450654 0.0225327 0.999746i \(-0.492827\pi\)
0.0225327 + 0.999746i \(0.492827\pi\)
\(824\) −2.62317 −0.0913823
\(825\) 14.5261 0.505733
\(826\) 0 0
\(827\) 43.3319 1.50680 0.753399 0.657563i \(-0.228413\pi\)
0.753399 + 0.657563i \(0.228413\pi\)
\(828\) 4.73042 0.164393
\(829\) 40.1174 1.39334 0.696668 0.717394i \(-0.254665\pi\)
0.696668 + 0.717394i \(0.254665\pi\)
\(830\) −9.49717 −0.329652
\(831\) 16.0000 0.555034
\(832\) 0 0
\(833\) 0 0
\(834\) −1.47392 −0.0510376
\(835\) −105.565 −3.65323
\(836\) 3.68842 0.127567
\(837\) 3.37683 0.116720
\(838\) −22.3116 −0.770741
\(839\) 54.6912 1.88815 0.944074 0.329733i \(-0.106959\pi\)
0.944074 + 0.329733i \(0.106959\pi\)
\(840\) 0 0
\(841\) 13.5897 0.468612
\(842\) 18.3116 0.631059
\(843\) 15.9217 0.548371
\(844\) 22.5130 0.774929
\(845\) −57.4448 −1.97616
\(846\) −4.10725 −0.141210
\(847\) 0 0
\(848\) 8.00000 0.274721
\(849\) 18.0840 0.620641
\(850\) 34.5261 1.18423
\(851\) 36.3694 1.24673
\(852\) −0.934749 −0.0320240
\(853\) −20.8275 −0.713120 −0.356560 0.934272i \(-0.616050\pi\)
−0.356560 + 0.934272i \(0.616050\pi\)
\(854\) 0 0
\(855\) −16.2985 −0.557397
\(856\) 10.9029 0.372654
\(857\) −36.8433 −1.25854 −0.629272 0.777185i \(-0.716647\pi\)
−0.629272 + 0.777185i \(0.716647\pi\)
\(858\) 0 0
\(859\) 38.5782 1.31627 0.658136 0.752899i \(-0.271345\pi\)
0.658136 + 0.752899i \(0.271345\pi\)
\(860\) 13.5448 0.461875
\(861\) 0 0
\(862\) −28.8377 −0.982215
\(863\) −17.2565 −0.587418 −0.293709 0.955895i \(-0.594890\pi\)
−0.293709 + 0.955895i \(0.594890\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 50.2723 1.70931
\(866\) 7.62317 0.259046
\(867\) 11.3507 0.385489
\(868\) 0 0
\(869\) 0.418833 0.0142079
\(870\) 28.8377 0.977688
\(871\) 0 0
\(872\) −11.7957 −0.399452
\(873\) 7.00000 0.236914
\(874\) −17.4477 −0.590179
\(875\) 0 0
\(876\) 7.46083 0.252078
\(877\) 0.250837 0.00847016 0.00423508 0.999991i \(-0.498652\pi\)
0.00423508 + 0.999991i \(0.498652\pi\)
\(878\) −33.3739 −1.12632
\(879\) −25.1492 −0.848263
\(880\) −4.41883 −0.148959
\(881\) 16.2985 0.549110 0.274555 0.961571i \(-0.411469\pi\)
0.274555 + 0.961571i \(0.411469\pi\)
\(882\) 0 0
\(883\) −33.9551 −1.14268 −0.571340 0.820714i \(-0.693576\pi\)
−0.571340 + 0.820714i \(0.693576\pi\)
\(884\) 0 0
\(885\) 55.3507 1.86059
\(886\) −4.85075 −0.162964
\(887\) 19.4608 0.653431 0.326715 0.945123i \(-0.394058\pi\)
0.326715 + 0.945123i \(0.394058\pi\)
\(888\) −7.68842 −0.258006
\(889\) 0 0
\(890\) −56.7275 −1.90151
\(891\) −1.00000 −0.0335013
\(892\) −7.46083 −0.249807
\(893\) 15.1492 0.506950
\(894\) 9.36375 0.313171
\(895\) 108.748 3.63504
\(896\) 0 0
\(897\) 0 0
\(898\) 14.3450 0.478699
\(899\) 22.0375 0.734992
\(900\) 14.5261 0.484203
\(901\) 19.0147 0.633470
\(902\) −12.1492 −0.404526
\(903\) 0 0
\(904\) 5.37683 0.178831
\(905\) 77.1565 2.56477
\(906\) 13.5681 0.450769
\(907\) 0.772415 0.0256476 0.0128238 0.999918i \(-0.495918\pi\)
0.0128238 + 0.999918i \(0.495918\pi\)
\(908\) −2.77241 −0.0920058
\(909\) −15.3637 −0.509583
\(910\) 0 0
\(911\) −8.94491 −0.296358 −0.148179 0.988961i \(-0.547341\pi\)
−0.148179 + 0.988961i \(0.547341\pi\)
\(912\) 3.68842 0.122136
\(913\) 2.14925 0.0711297
\(914\) −25.1827 −0.832969
\(915\) −16.7724 −0.554479
\(916\) −2.53917 −0.0838965
\(917\) 0 0
\(918\) −2.37683 −0.0784472
\(919\) 58.8812 1.94231 0.971157 0.238443i \(-0.0766369\pi\)
0.971157 + 0.238443i \(0.0766369\pi\)
\(920\) 20.9029 0.689149
\(921\) 3.86950 0.127504
\(922\) 13.0187 0.428750
\(923\) 0 0
\(924\) 0 0
\(925\) 111.683 3.67210
\(926\) −14.2145 −0.467117
\(927\) −2.62317 −0.0861561
\(928\) −6.52608 −0.214229
\(929\) −22.5392 −0.739486 −0.369743 0.929134i \(-0.620554\pi\)
−0.369743 + 0.929134i \(0.620554\pi\)
\(930\) 14.9217 0.489301
\(931\) 0 0
\(932\) −24.0522 −0.787855
\(933\) 13.5681 0.444199
\(934\) −2.39558 −0.0783858
\(935\) −10.5028 −0.343479
\(936\) 0 0
\(937\) −12.9478 −0.422987 −0.211494 0.977379i \(-0.567833\pi\)
−0.211494 + 0.977379i \(0.567833\pi\)
\(938\) 0 0
\(939\) −2.52608 −0.0824356
\(940\) −18.1492 −0.591963
\(941\) 48.8246 1.59164 0.795818 0.605536i \(-0.207041\pi\)
0.795818 + 0.605536i \(0.207041\pi\)
\(942\) 18.7406 0.610601
\(943\) 57.4710 1.87151
\(944\) −12.5261 −0.407689
\(945\) 0 0
\(946\) −3.06525 −0.0996599
\(947\) −31.1231 −1.01136 −0.505682 0.862720i \(-0.668759\pi\)
−0.505682 + 0.862720i \(0.668759\pi\)
\(948\) 0.418833 0.0136031
\(949\) 0 0
\(950\) −53.5782 −1.73831
\(951\) 20.0102 0.648874
\(952\) 0 0
\(953\) 16.2797 0.527353 0.263676 0.964611i \(-0.415065\pi\)
0.263676 + 0.964611i \(0.415065\pi\)
\(954\) 8.00000 0.259010
\(955\) −74.4028 −2.40762
\(956\) 7.59133 0.245521
\(957\) −6.52608 −0.210958
\(958\) 10.8377 0.350149
\(959\) 0 0
\(960\) −4.41883 −0.142617
\(961\) −19.5970 −0.632161
\(962\) 0 0
\(963\) 10.9029 0.351342
\(964\) 16.2985 0.524939
\(965\) 24.1305 0.776788
\(966\) 0 0
\(967\) 3.91308 0.125836 0.0629181 0.998019i \(-0.479959\pi\)
0.0629181 + 0.998019i \(0.479959\pi\)
\(968\) 1.00000 0.0321412
\(969\) 8.76675 0.281629
\(970\) 30.9318 0.993161
\(971\) 39.3132 1.26162 0.630810 0.775938i \(-0.282723\pi\)
0.630810 + 0.775938i \(0.282723\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −34.6435 −1.11005
\(975\) 0 0
\(976\) 3.79567 0.121496
\(977\) −23.3507 −0.747054 −0.373527 0.927619i \(-0.621852\pi\)
−0.373527 + 0.927619i \(0.621852\pi\)
\(978\) −18.7724 −0.600276
\(979\) 12.8377 0.410294
\(980\) 0 0
\(981\) −11.7957 −0.376607
\(982\) 38.1492 1.21739
\(983\) −4.51592 −0.144035 −0.0720177 0.997403i \(-0.522944\pi\)
−0.0720177 + 0.997403i \(0.522944\pi\)
\(984\) −12.1492 −0.387304
\(985\) −42.8115 −1.36409
\(986\) −15.5114 −0.493984
\(987\) 0 0
\(988\) 0 0
\(989\) 14.4999 0.461070
\(990\) −4.41883 −0.140440
\(991\) 59.9273 1.90365 0.951827 0.306635i \(-0.0992032\pi\)
0.951827 + 0.306635i \(0.0992032\pi\)
\(992\) −3.37683 −0.107215
\(993\) 16.1492 0.512481
\(994\) 0 0
\(995\) 89.6957 2.84354
\(996\) 2.14925 0.0681015
\(997\) −21.5073 −0.681144 −0.340572 0.940218i \(-0.610621\pi\)
−0.340572 + 0.940218i \(0.610621\pi\)
\(998\) −0.623166 −0.0197260
\(999\) −7.68842 −0.243251
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 3234.2.a.bf.1.3 3
3.2 odd 2 9702.2.a.dv.1.1 3
7.2 even 3 462.2.i.g.67.1 6
7.4 even 3 462.2.i.g.331.1 yes 6
7.6 odd 2 3234.2.a.bh.1.1 3
21.2 odd 6 1386.2.k.v.991.3 6
21.11 odd 6 1386.2.k.v.793.3 6
21.20 even 2 9702.2.a.dw.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.g.67.1 6 7.2 even 3
462.2.i.g.331.1 yes 6 7.4 even 3
1386.2.k.v.793.3 6 21.11 odd 6
1386.2.k.v.991.3 6 21.2 odd 6
3234.2.a.bf.1.3 3 1.1 even 1 trivial
3234.2.a.bh.1.1 3 7.6 odd 2
9702.2.a.dv.1.1 3 3.2 odd 2
9702.2.a.dw.1.3 3 21.20 even 2