Properties

Label 8034.2.a.w.1.10
Level 8034
Weight 2
Character 8034.1
Self dual yes
Analytic conductor 64.152
Analytic rank 1
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 6 x^{11} - 8 x^{10} + 94 x^{9} - 7 x^{8} - 580 x^{7} + 180 x^{6} + 1787 x^{5} - 308 x^{4} - 2790 x^{3} - 352 x^{2} + 1768 x + 768\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-2.20360\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.09722 q^{5} -1.00000 q^{6} -0.0373018 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.09722 q^{5} -1.00000 q^{6} -0.0373018 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.09722 q^{10} -3.17762 q^{11} +1.00000 q^{12} +1.00000 q^{13} +0.0373018 q^{14} +2.09722 q^{15} +1.00000 q^{16} -2.38289 q^{17} -1.00000 q^{18} +2.91209 q^{19} +2.09722 q^{20} -0.0373018 q^{21} +3.17762 q^{22} +3.99261 q^{23} -1.00000 q^{24} -0.601670 q^{25} -1.00000 q^{26} +1.00000 q^{27} -0.0373018 q^{28} -2.46598 q^{29} -2.09722 q^{30} +0.760072 q^{31} -1.00000 q^{32} -3.17762 q^{33} +2.38289 q^{34} -0.0782300 q^{35} +1.00000 q^{36} -8.84213 q^{37} -2.91209 q^{38} +1.00000 q^{39} -2.09722 q^{40} -9.70721 q^{41} +0.0373018 q^{42} -10.8256 q^{43} -3.17762 q^{44} +2.09722 q^{45} -3.99261 q^{46} -5.29982 q^{47} +1.00000 q^{48} -6.99861 q^{49} +0.601670 q^{50} -2.38289 q^{51} +1.00000 q^{52} -10.7397 q^{53} -1.00000 q^{54} -6.66417 q^{55} +0.0373018 q^{56} +2.91209 q^{57} +2.46598 q^{58} +4.58591 q^{59} +2.09722 q^{60} -1.68161 q^{61} -0.760072 q^{62} -0.0373018 q^{63} +1.00000 q^{64} +2.09722 q^{65} +3.17762 q^{66} +7.20145 q^{67} -2.38289 q^{68} +3.99261 q^{69} +0.0782300 q^{70} -14.3856 q^{71} -1.00000 q^{72} +15.2846 q^{73} +8.84213 q^{74} -0.601670 q^{75} +2.91209 q^{76} +0.118531 q^{77} -1.00000 q^{78} +6.39400 q^{79} +2.09722 q^{80} +1.00000 q^{81} +9.70721 q^{82} -5.88671 q^{83} -0.0373018 q^{84} -4.99743 q^{85} +10.8256 q^{86} -2.46598 q^{87} +3.17762 q^{88} +2.63980 q^{89} -2.09722 q^{90} -0.0373018 q^{91} +3.99261 q^{92} +0.760072 q^{93} +5.29982 q^{94} +6.10728 q^{95} -1.00000 q^{96} -2.73678 q^{97} +6.99861 q^{98} -3.17762 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + 4q^{10} - 9q^{11} + 12q^{12} + 12q^{13} - 4q^{15} + 12q^{16} - 20q^{17} - 12q^{18} + 4q^{19} - 4q^{20} + 9q^{22} - 30q^{23} - 12q^{24} + 14q^{25} - 12q^{26} + 12q^{27} - 29q^{29} + 4q^{30} + 6q^{31} - 12q^{32} - 9q^{33} + 20q^{34} - 22q^{35} + 12q^{36} + 7q^{37} - 4q^{38} + 12q^{39} + 4q^{40} - 8q^{41} - 8q^{43} - 9q^{44} - 4q^{45} + 30q^{46} - 16q^{47} + 12q^{48} + 10q^{49} - 14q^{50} - 20q^{51} + 12q^{52} - 9q^{53} - 12q^{54} - 20q^{55} + 4q^{57} + 29q^{58} - 29q^{59} - 4q^{60} - 26q^{61} - 6q^{62} + 12q^{64} - 4q^{65} + 9q^{66} + 12q^{67} - 20q^{68} - 30q^{69} + 22q^{70} - 35q^{71} - 12q^{72} + 18q^{73} - 7q^{74} + 14q^{75} + 4q^{76} - 25q^{77} - 12q^{78} - 37q^{79} - 4q^{80} + 12q^{81} + 8q^{82} - 24q^{83} - 17q^{85} + 8q^{86} - 29q^{87} + 9q^{88} + 15q^{89} + 4q^{90} - 30q^{92} + 6q^{93} + 16q^{94} - 54q^{95} - 12q^{96} - 11q^{97} - 10q^{98} - 9q^{99} + O(q^{100}) \)

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
\(5\) 2.09722 0.937905 0.468953 0.883223i \(-0.344632\pi\)
0.468953 + 0.883223i \(0.344632\pi\)
\(6\) −1.00000 −0.408248
\(7\) −0.0373018 −0.0140987 −0.00704937 0.999975i \(-0.502244\pi\)
−0.00704937 + 0.999975i \(0.502244\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.09722 −0.663199
\(11\) −3.17762 −0.958090 −0.479045 0.877790i \(-0.659017\pi\)
−0.479045 + 0.877790i \(0.659017\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 0.0373018 0.00996931
\(15\) 2.09722 0.541500
\(16\) 1.00000 0.250000
\(17\) −2.38289 −0.577935 −0.288967 0.957339i \(-0.593312\pi\)
−0.288967 + 0.957339i \(0.593312\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.91209 0.668078 0.334039 0.942559i \(-0.391588\pi\)
0.334039 + 0.942559i \(0.391588\pi\)
\(20\) 2.09722 0.468953
\(21\) −0.0373018 −0.00813991
\(22\) 3.17762 0.677472
\(23\) 3.99261 0.832518 0.416259 0.909246i \(-0.363341\pi\)
0.416259 + 0.909246i \(0.363341\pi\)
\(24\) −1.00000 −0.204124
\(25\) −0.601670 −0.120334
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) −0.0373018 −0.00704937
\(29\) −2.46598 −0.457921 −0.228960 0.973436i \(-0.573533\pi\)
−0.228960 + 0.973436i \(0.573533\pi\)
\(30\) −2.09722 −0.382898
\(31\) 0.760072 0.136513 0.0682564 0.997668i \(-0.478256\pi\)
0.0682564 + 0.997668i \(0.478256\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.17762 −0.553153
\(34\) 2.38289 0.408661
\(35\) −0.0782300 −0.0132233
\(36\) 1.00000 0.166667
\(37\) −8.84213 −1.45364 −0.726818 0.686830i \(-0.759002\pi\)
−0.726818 + 0.686830i \(0.759002\pi\)
\(38\) −2.91209 −0.472403
\(39\) 1.00000 0.160128
\(40\) −2.09722 −0.331600
\(41\) −9.70721 −1.51601 −0.758006 0.652248i \(-0.773826\pi\)
−0.758006 + 0.652248i \(0.773826\pi\)
\(42\) 0.0373018 0.00575579
\(43\) −10.8256 −1.65089 −0.825447 0.564480i \(-0.809077\pi\)
−0.825447 + 0.564480i \(0.809077\pi\)
\(44\) −3.17762 −0.479045
\(45\) 2.09722 0.312635
\(46\) −3.99261 −0.588679
\(47\) −5.29982 −0.773058 −0.386529 0.922277i \(-0.626326\pi\)
−0.386529 + 0.922277i \(0.626326\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.99861 −0.999801
\(50\) 0.601670 0.0850890
\(51\) −2.38289 −0.333671
\(52\) 1.00000 0.138675
\(53\) −10.7397 −1.47521 −0.737604 0.675234i \(-0.764043\pi\)
−0.737604 + 0.675234i \(0.764043\pi\)
\(54\) −1.00000 −0.136083
\(55\) −6.66417 −0.898597
\(56\) 0.0373018 0.00498466
\(57\) 2.91209 0.385715
\(58\) 2.46598 0.323799
\(59\) 4.58591 0.597034 0.298517 0.954404i \(-0.403508\pi\)
0.298517 + 0.954404i \(0.403508\pi\)
\(60\) 2.09722 0.270750
\(61\) −1.68161 −0.215308 −0.107654 0.994188i \(-0.534334\pi\)
−0.107654 + 0.994188i \(0.534334\pi\)
\(62\) −0.760072 −0.0965292
\(63\) −0.0373018 −0.00469958
\(64\) 1.00000 0.125000
\(65\) 2.09722 0.260128
\(66\) 3.17762 0.391138
\(67\) 7.20145 0.879797 0.439899 0.898047i \(-0.355014\pi\)
0.439899 + 0.898047i \(0.355014\pi\)
\(68\) −2.38289 −0.288967
\(69\) 3.99261 0.480654
\(70\) 0.0782300 0.00935027
\(71\) −14.3856 −1.70726 −0.853628 0.520882i \(-0.825603\pi\)
−0.853628 + 0.520882i \(0.825603\pi\)
\(72\) −1.00000 −0.117851
\(73\) 15.2846 1.78893 0.894463 0.447141i \(-0.147558\pi\)
0.894463 + 0.447141i \(0.147558\pi\)
\(74\) 8.84213 1.02788
\(75\) −0.601670 −0.0694749
\(76\) 2.91209 0.334039
\(77\) 0.118531 0.0135079
\(78\) −1.00000 −0.113228
\(79\) 6.39400 0.719381 0.359691 0.933072i \(-0.382882\pi\)
0.359691 + 0.933072i \(0.382882\pi\)
\(80\) 2.09722 0.234476
\(81\) 1.00000 0.111111
\(82\) 9.70721 1.07198
\(83\) −5.88671 −0.646151 −0.323075 0.946373i \(-0.604717\pi\)
−0.323075 + 0.946373i \(0.604717\pi\)
\(84\) −0.0373018 −0.00406996
\(85\) −4.99743 −0.542048
\(86\) 10.8256 1.16736
\(87\) −2.46598 −0.264381
\(88\) 3.17762 0.338736
\(89\) 2.63980 0.279818 0.139909 0.990164i \(-0.455319\pi\)
0.139909 + 0.990164i \(0.455319\pi\)
\(90\) −2.09722 −0.221066
\(91\) −0.0373018 −0.00391029
\(92\) 3.99261 0.416259
\(93\) 0.760072 0.0788158
\(94\) 5.29982 0.546635
\(95\) 6.10728 0.626594
\(96\) −1.00000 −0.102062
\(97\) −2.73678 −0.277878 −0.138939 0.990301i \(-0.544369\pi\)
−0.138939 + 0.990301i \(0.544369\pi\)
\(98\) 6.99861 0.706966
\(99\) −3.17762 −0.319363
\(100\) −0.601670 −0.0601670
\(101\) 17.7613 1.76731 0.883656 0.468137i \(-0.155075\pi\)
0.883656 + 0.468137i \(0.155075\pi\)
\(102\) 2.38289 0.235941
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −0.0782300 −0.00763446
\(106\) 10.7397 1.04313
\(107\) 7.44643 0.719873 0.359937 0.932977i \(-0.382798\pi\)
0.359937 + 0.932977i \(0.382798\pi\)
\(108\) 1.00000 0.0962250
\(109\) −3.30101 −0.316180 −0.158090 0.987425i \(-0.550534\pi\)
−0.158090 + 0.987425i \(0.550534\pi\)
\(110\) 6.66417 0.635404
\(111\) −8.84213 −0.839257
\(112\) −0.0373018 −0.00352468
\(113\) −6.21481 −0.584640 −0.292320 0.956321i \(-0.594427\pi\)
−0.292320 + 0.956321i \(0.594427\pi\)
\(114\) −2.91209 −0.272742
\(115\) 8.37339 0.780823
\(116\) −2.46598 −0.228960
\(117\) 1.00000 0.0924500
\(118\) −4.58591 −0.422167
\(119\) 0.0888858 0.00814815
\(120\) −2.09722 −0.191449
\(121\) −0.902708 −0.0820644
\(122\) 1.68161 0.152246
\(123\) −9.70721 −0.875270
\(124\) 0.760072 0.0682564
\(125\) −11.7479 −1.05077
\(126\) 0.0373018 0.00332310
\(127\) 4.59734 0.407948 0.203974 0.978976i \(-0.434614\pi\)
0.203974 + 0.978976i \(0.434614\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.8256 −0.953144
\(130\) −2.09722 −0.183938
\(131\) −17.3267 −1.51385 −0.756923 0.653505i \(-0.773298\pi\)
−0.756923 + 0.653505i \(0.773298\pi\)
\(132\) −3.17762 −0.276577
\(133\) −0.108626 −0.00941906
\(134\) −7.20145 −0.622111
\(135\) 2.09722 0.180500
\(136\) 2.38289 0.204331
\(137\) −14.5720 −1.24497 −0.622485 0.782632i \(-0.713877\pi\)
−0.622485 + 0.782632i \(0.713877\pi\)
\(138\) −3.99261 −0.339874
\(139\) −15.9341 −1.35152 −0.675758 0.737124i \(-0.736183\pi\)
−0.675758 + 0.737124i \(0.736183\pi\)
\(140\) −0.0782300 −0.00661164
\(141\) −5.29982 −0.446325
\(142\) 14.3856 1.20721
\(143\) −3.17762 −0.265726
\(144\) 1.00000 0.0833333
\(145\) −5.17170 −0.429486
\(146\) −15.2846 −1.26496
\(147\) −6.99861 −0.577236
\(148\) −8.84213 −0.726818
\(149\) −12.4885 −1.02310 −0.511548 0.859255i \(-0.670928\pi\)
−0.511548 + 0.859255i \(0.670928\pi\)
\(150\) 0.601670 0.0491262
\(151\) 11.6943 0.951669 0.475834 0.879535i \(-0.342146\pi\)
0.475834 + 0.879535i \(0.342146\pi\)
\(152\) −2.91209 −0.236201
\(153\) −2.38289 −0.192645
\(154\) −0.118531 −0.00955150
\(155\) 1.59404 0.128036
\(156\) 1.00000 0.0800641
\(157\) −11.0964 −0.885592 −0.442796 0.896622i \(-0.646013\pi\)
−0.442796 + 0.896622i \(0.646013\pi\)
\(158\) −6.39400 −0.508680
\(159\) −10.7397 −0.851712
\(160\) −2.09722 −0.165800
\(161\) −0.148932 −0.0117374
\(162\) −1.00000 −0.0785674
\(163\) 5.69850 0.446341 0.223170 0.974779i \(-0.428359\pi\)
0.223170 + 0.974779i \(0.428359\pi\)
\(164\) −9.70721 −0.758006
\(165\) −6.66417 −0.518805
\(166\) 5.88671 0.456898
\(167\) 19.9987 1.54755 0.773773 0.633464i \(-0.218367\pi\)
0.773773 + 0.633464i \(0.218367\pi\)
\(168\) 0.0373018 0.00287789
\(169\) 1.00000 0.0769231
\(170\) 4.99743 0.383286
\(171\) 2.91209 0.222693
\(172\) −10.8256 −0.825447
\(173\) 7.43256 0.565087 0.282544 0.959254i \(-0.408822\pi\)
0.282544 + 0.959254i \(0.408822\pi\)
\(174\) 2.46598 0.186945
\(175\) 0.0224434 0.00169656
\(176\) −3.17762 −0.239522
\(177\) 4.58591 0.344698
\(178\) −2.63980 −0.197861
\(179\) 24.5685 1.83634 0.918170 0.396188i \(-0.129667\pi\)
0.918170 + 0.396188i \(0.129667\pi\)
\(180\) 2.09722 0.156318
\(181\) −6.64229 −0.493718 −0.246859 0.969051i \(-0.579398\pi\)
−0.246859 + 0.969051i \(0.579398\pi\)
\(182\) 0.0373018 0.00276499
\(183\) −1.68161 −0.124308
\(184\) −3.99261 −0.294339
\(185\) −18.5439 −1.36337
\(186\) −0.760072 −0.0557312
\(187\) 7.57191 0.553713
\(188\) −5.29982 −0.386529
\(189\) −0.0373018 −0.00271330
\(190\) −6.10728 −0.443069
\(191\) −12.5288 −0.906550 −0.453275 0.891371i \(-0.649745\pi\)
−0.453275 + 0.891371i \(0.649745\pi\)
\(192\) 1.00000 0.0721688
\(193\) 4.59720 0.330914 0.165457 0.986217i \(-0.447090\pi\)
0.165457 + 0.986217i \(0.447090\pi\)
\(194\) 2.73678 0.196489
\(195\) 2.09722 0.150185
\(196\) −6.99861 −0.499901
\(197\) 15.5509 1.10796 0.553978 0.832532i \(-0.313109\pi\)
0.553978 + 0.832532i \(0.313109\pi\)
\(198\) 3.17762 0.225824
\(199\) −9.66523 −0.685150 −0.342575 0.939491i \(-0.611299\pi\)
−0.342575 + 0.939491i \(0.611299\pi\)
\(200\) 0.601670 0.0425445
\(201\) 7.20145 0.507951
\(202\) −17.7613 −1.24968
\(203\) 0.0919853 0.00645610
\(204\) −2.38289 −0.166835
\(205\) −20.3582 −1.42188
\(206\) 1.00000 0.0696733
\(207\) 3.99261 0.277506
\(208\) 1.00000 0.0693375
\(209\) −9.25351 −0.640079
\(210\) 0.0782300 0.00539838
\(211\) −3.17494 −0.218572 −0.109286 0.994010i \(-0.534856\pi\)
−0.109286 + 0.994010i \(0.534856\pi\)
\(212\) −10.7397 −0.737604
\(213\) −14.3856 −0.985685
\(214\) −7.44643 −0.509027
\(215\) −22.7037 −1.54838
\(216\) −1.00000 −0.0680414
\(217\) −0.0283520 −0.00192466
\(218\) 3.30101 0.223573
\(219\) 15.2846 1.03284
\(220\) −6.66417 −0.449299
\(221\) −2.38289 −0.160290
\(222\) 8.84213 0.593445
\(223\) −1.36542 −0.0914350 −0.0457175 0.998954i \(-0.514557\pi\)
−0.0457175 + 0.998954i \(0.514557\pi\)
\(224\) 0.0373018 0.00249233
\(225\) −0.601670 −0.0401114
\(226\) 6.21481 0.413403
\(227\) −10.1984 −0.676892 −0.338446 0.940986i \(-0.609901\pi\)
−0.338446 + 0.940986i \(0.609901\pi\)
\(228\) 2.91209 0.192858
\(229\) 20.1236 1.32980 0.664902 0.746931i \(-0.268473\pi\)
0.664902 + 0.746931i \(0.268473\pi\)
\(230\) −8.37339 −0.552125
\(231\) 0.118531 0.00779876
\(232\) 2.46598 0.161899
\(233\) −20.9544 −1.37277 −0.686383 0.727240i \(-0.740803\pi\)
−0.686383 + 0.727240i \(0.740803\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −11.1149 −0.725055
\(236\) 4.58591 0.298517
\(237\) 6.39400 0.415335
\(238\) −0.0888858 −0.00576161
\(239\) −22.3195 −1.44373 −0.721865 0.692034i \(-0.756715\pi\)
−0.721865 + 0.692034i \(0.756715\pi\)
\(240\) 2.09722 0.135375
\(241\) 26.3016 1.69424 0.847118 0.531405i \(-0.178336\pi\)
0.847118 + 0.531405i \(0.178336\pi\)
\(242\) 0.902708 0.0580283
\(243\) 1.00000 0.0641500
\(244\) −1.68161 −0.107654
\(245\) −14.6776 −0.937719
\(246\) 9.70721 0.618909
\(247\) 2.91209 0.185292
\(248\) −0.760072 −0.0482646
\(249\) −5.88671 −0.373055
\(250\) 11.7479 0.743004
\(251\) 14.0004 0.883696 0.441848 0.897090i \(-0.354323\pi\)
0.441848 + 0.897090i \(0.354323\pi\)
\(252\) −0.0373018 −0.00234979
\(253\) −12.6870 −0.797627
\(254\) −4.59734 −0.288463
\(255\) −4.99743 −0.312951
\(256\) 1.00000 0.0625000
\(257\) 11.7882 0.735330 0.367665 0.929958i \(-0.380157\pi\)
0.367665 + 0.929958i \(0.380157\pi\)
\(258\) 10.8256 0.673974
\(259\) 0.329827 0.0204944
\(260\) 2.09722 0.130064
\(261\) −2.46598 −0.152640
\(262\) 17.3267 1.07045
\(263\) −3.34403 −0.206201 −0.103101 0.994671i \(-0.532876\pi\)
−0.103101 + 0.994671i \(0.532876\pi\)
\(264\) 3.17762 0.195569
\(265\) −22.5235 −1.38361
\(266\) 0.108626 0.00666028
\(267\) 2.63980 0.161553
\(268\) 7.20145 0.439899
\(269\) 16.1066 0.982037 0.491019 0.871149i \(-0.336625\pi\)
0.491019 + 0.871149i \(0.336625\pi\)
\(270\) −2.09722 −0.127633
\(271\) −9.09091 −0.552234 −0.276117 0.961124i \(-0.589048\pi\)
−0.276117 + 0.961124i \(0.589048\pi\)
\(272\) −2.38289 −0.144484
\(273\) −0.0373018 −0.00225760
\(274\) 14.5720 0.880327
\(275\) 1.91188 0.115291
\(276\) 3.99261 0.240327
\(277\) 20.2183 1.21480 0.607401 0.794396i \(-0.292212\pi\)
0.607401 + 0.794396i \(0.292212\pi\)
\(278\) 15.9341 0.955666
\(279\) 0.760072 0.0455043
\(280\) 0.0782300 0.00467513
\(281\) 10.4908 0.625827 0.312913 0.949782i \(-0.398695\pi\)
0.312913 + 0.949782i \(0.398695\pi\)
\(282\) 5.29982 0.315600
\(283\) −20.5499 −1.22157 −0.610783 0.791798i \(-0.709145\pi\)
−0.610783 + 0.791798i \(0.709145\pi\)
\(284\) −14.3856 −0.853628
\(285\) 6.10728 0.361764
\(286\) 3.17762 0.187897
\(287\) 0.362096 0.0213739
\(288\) −1.00000 −0.0589256
\(289\) −11.3219 −0.665992
\(290\) 5.17170 0.303693
\(291\) −2.73678 −0.160433
\(292\) 15.2846 0.894463
\(293\) −2.35918 −0.137825 −0.0689124 0.997623i \(-0.521953\pi\)
−0.0689124 + 0.997623i \(0.521953\pi\)
\(294\) 6.99861 0.408167
\(295\) 9.61765 0.559961
\(296\) 8.84213 0.513938
\(297\) −3.17762 −0.184384
\(298\) 12.4885 0.723438
\(299\) 3.99261 0.230899
\(300\) −0.601670 −0.0347375
\(301\) 0.403815 0.0232755
\(302\) −11.6943 −0.672931
\(303\) 17.7613 1.02036
\(304\) 2.91209 0.167020
\(305\) −3.52671 −0.201939
\(306\) 2.38289 0.136220
\(307\) −23.9915 −1.36927 −0.684633 0.728888i \(-0.740038\pi\)
−0.684633 + 0.728888i \(0.740038\pi\)
\(308\) 0.118531 0.00675393
\(309\) −1.00000 −0.0568880
\(310\) −1.59404 −0.0905352
\(311\) −9.06740 −0.514165 −0.257083 0.966389i \(-0.582761\pi\)
−0.257083 + 0.966389i \(0.582761\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 27.0568 1.52934 0.764670 0.644422i \(-0.222902\pi\)
0.764670 + 0.644422i \(0.222902\pi\)
\(314\) 11.0964 0.626208
\(315\) −0.0782300 −0.00440776
\(316\) 6.39400 0.359691
\(317\) 25.3558 1.42412 0.712062 0.702116i \(-0.247761\pi\)
0.712062 + 0.702116i \(0.247761\pi\)
\(318\) 10.7397 0.602251
\(319\) 7.83595 0.438729
\(320\) 2.09722 0.117238
\(321\) 7.44643 0.415619
\(322\) 0.148932 0.00829963
\(323\) −6.93916 −0.386105
\(324\) 1.00000 0.0555556
\(325\) −0.601670 −0.0333747
\(326\) −5.69850 −0.315610
\(327\) −3.30101 −0.182546
\(328\) 9.70721 0.535991
\(329\) 0.197693 0.0108991
\(330\) 6.66417 0.366851
\(331\) −13.8421 −0.760829 −0.380414 0.924816i \(-0.624219\pi\)
−0.380414 + 0.924816i \(0.624219\pi\)
\(332\) −5.88671 −0.323075
\(333\) −8.84213 −0.484545
\(334\) −19.9987 −1.09428
\(335\) 15.1030 0.825166
\(336\) −0.0373018 −0.00203498
\(337\) −0.107575 −0.00585998 −0.00292999 0.999996i \(-0.500933\pi\)
−0.00292999 + 0.999996i \(0.500933\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −6.21481 −0.337542
\(340\) −4.99743 −0.271024
\(341\) −2.41522 −0.130792
\(342\) −2.91209 −0.157468
\(343\) 0.522173 0.0281947
\(344\) 10.8256 0.583679
\(345\) 8.37339 0.450808
\(346\) −7.43256 −0.399577
\(347\) −33.6802 −1.80805 −0.904025 0.427481i \(-0.859401\pi\)
−0.904025 + 0.427481i \(0.859401\pi\)
\(348\) −2.46598 −0.132190
\(349\) −14.2268 −0.761543 −0.380771 0.924669i \(-0.624342\pi\)
−0.380771 + 0.924669i \(0.624342\pi\)
\(350\) −0.0224434 −0.00119965
\(351\) 1.00000 0.0533761
\(352\) 3.17762 0.169368
\(353\) 2.62388 0.139655 0.0698274 0.997559i \(-0.477755\pi\)
0.0698274 + 0.997559i \(0.477755\pi\)
\(354\) −4.58591 −0.243738
\(355\) −30.1698 −1.60124
\(356\) 2.63980 0.139909
\(357\) 0.0888858 0.00470434
\(358\) −24.5685 −1.29849
\(359\) −15.8065 −0.834234 −0.417117 0.908853i \(-0.636959\pi\)
−0.417117 + 0.908853i \(0.636959\pi\)
\(360\) −2.09722 −0.110533
\(361\) −10.5198 −0.553672
\(362\) 6.64229 0.349111
\(363\) −0.902708 −0.0473799
\(364\) −0.0373018 −0.00195514
\(365\) 32.0552 1.67784
\(366\) 1.68161 0.0878993
\(367\) 11.4273 0.596500 0.298250 0.954488i \(-0.403597\pi\)
0.298250 + 0.954488i \(0.403597\pi\)
\(368\) 3.99261 0.208129
\(369\) −9.70721 −0.505337
\(370\) 18.5439 0.964050
\(371\) 0.400609 0.0207986
\(372\) 0.760072 0.0394079
\(373\) −5.05968 −0.261980 −0.130990 0.991384i \(-0.541816\pi\)
−0.130990 + 0.991384i \(0.541816\pi\)
\(374\) −7.57191 −0.391534
\(375\) −11.7479 −0.606661
\(376\) 5.29982 0.273317
\(377\) −2.46598 −0.127004
\(378\) 0.0373018 0.00191860
\(379\) 12.8369 0.659389 0.329695 0.944088i \(-0.393054\pi\)
0.329695 + 0.944088i \(0.393054\pi\)
\(380\) 6.10728 0.313297
\(381\) 4.59734 0.235529
\(382\) 12.5288 0.641028
\(383\) −19.1538 −0.978715 −0.489357 0.872083i \(-0.662769\pi\)
−0.489357 + 0.872083i \(0.662769\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0.248585 0.0126691
\(386\) −4.59720 −0.233991
\(387\) −10.8256 −0.550298
\(388\) −2.73678 −0.138939
\(389\) 11.7164 0.594045 0.297022 0.954870i \(-0.404006\pi\)
0.297022 + 0.954870i \(0.404006\pi\)
\(390\) −2.09722 −0.106197
\(391\) −9.51394 −0.481141
\(392\) 6.99861 0.353483
\(393\) −17.3267 −0.874019
\(394\) −15.5509 −0.783443
\(395\) 13.4096 0.674712
\(396\) −3.17762 −0.159682
\(397\) 23.4133 1.17508 0.587540 0.809195i \(-0.300097\pi\)
0.587540 + 0.809195i \(0.300097\pi\)
\(398\) 9.66523 0.484474
\(399\) −0.108626 −0.00543810
\(400\) −0.601670 −0.0300835
\(401\) 28.4310 1.41978 0.709889 0.704314i \(-0.248745\pi\)
0.709889 + 0.704314i \(0.248745\pi\)
\(402\) −7.20145 −0.359176
\(403\) 0.760072 0.0378619
\(404\) 17.7613 0.883656
\(405\) 2.09722 0.104212
\(406\) −0.0919853 −0.00456515
\(407\) 28.0969 1.39271
\(408\) 2.38289 0.117970
\(409\) −5.14846 −0.254575 −0.127287 0.991866i \(-0.540627\pi\)
−0.127287 + 0.991866i \(0.540627\pi\)
\(410\) 20.3582 1.00542
\(411\) −14.5720 −0.718784
\(412\) −1.00000 −0.0492665
\(413\) −0.171062 −0.00841743
\(414\) −3.99261 −0.196226
\(415\) −12.3457 −0.606028
\(416\) −1.00000 −0.0490290
\(417\) −15.9341 −0.780298
\(418\) 9.25351 0.452604
\(419\) −15.5597 −0.760143 −0.380072 0.924957i \(-0.624101\pi\)
−0.380072 + 0.924957i \(0.624101\pi\)
\(420\) −0.0782300 −0.00381723
\(421\) 19.2163 0.936547 0.468274 0.883584i \(-0.344876\pi\)
0.468274 + 0.883584i \(0.344876\pi\)
\(422\) 3.17494 0.154554
\(423\) −5.29982 −0.257686
\(424\) 10.7397 0.521565
\(425\) 1.43371 0.0695452
\(426\) 14.3856 0.696985
\(427\) 0.0627271 0.00303558
\(428\) 7.44643 0.359937
\(429\) −3.17762 −0.153417
\(430\) 22.7037 1.09487
\(431\) −24.8965 −1.19922 −0.599611 0.800292i \(-0.704678\pi\)
−0.599611 + 0.800292i \(0.704678\pi\)
\(432\) 1.00000 0.0481125
\(433\) 29.0280 1.39500 0.697498 0.716586i \(-0.254296\pi\)
0.697498 + 0.716586i \(0.254296\pi\)
\(434\) 0.0283520 0.00136094
\(435\) −5.17170 −0.247964
\(436\) −3.30101 −0.158090
\(437\) 11.6268 0.556187
\(438\) −15.2846 −0.730326
\(439\) −12.1657 −0.580636 −0.290318 0.956930i \(-0.593761\pi\)
−0.290318 + 0.956930i \(0.593761\pi\)
\(440\) 6.66417 0.317702
\(441\) −6.99861 −0.333267
\(442\) 2.38289 0.113342
\(443\) −26.0682 −1.23854 −0.619269 0.785179i \(-0.712571\pi\)
−0.619269 + 0.785179i \(0.712571\pi\)
\(444\) −8.84213 −0.419629
\(445\) 5.53623 0.262443
\(446\) 1.36542 0.0646543
\(447\) −12.4885 −0.590685
\(448\) −0.0373018 −0.00176234
\(449\) −17.1631 −0.809977 −0.404988 0.914322i \(-0.632724\pi\)
−0.404988 + 0.914322i \(0.632724\pi\)
\(450\) 0.601670 0.0283630
\(451\) 30.8459 1.45248
\(452\) −6.21481 −0.292320
\(453\) 11.6943 0.549446
\(454\) 10.1984 0.478635
\(455\) −0.0782300 −0.00366748
\(456\) −2.91209 −0.136371
\(457\) 1.04913 0.0490760 0.0245380 0.999699i \(-0.492189\pi\)
0.0245380 + 0.999699i \(0.492189\pi\)
\(458\) −20.1236 −0.940313
\(459\) −2.38289 −0.111224
\(460\) 8.37339 0.390411
\(461\) 1.36203 0.0634362 0.0317181 0.999497i \(-0.489902\pi\)
0.0317181 + 0.999497i \(0.489902\pi\)
\(462\) −0.118531 −0.00551456
\(463\) −17.9663 −0.834963 −0.417482 0.908685i \(-0.637087\pi\)
−0.417482 + 0.908685i \(0.637087\pi\)
\(464\) −2.46598 −0.114480
\(465\) 1.59404 0.0739217
\(466\) 20.9544 0.970692
\(467\) −12.2355 −0.566194 −0.283097 0.959091i \(-0.591362\pi\)
−0.283097 + 0.959091i \(0.591362\pi\)
\(468\) 1.00000 0.0462250
\(469\) −0.268627 −0.0124040
\(470\) 11.1149 0.512691
\(471\) −11.0964 −0.511297
\(472\) −4.58591 −0.211083
\(473\) 34.3998 1.58170
\(474\) −6.39400 −0.293686
\(475\) −1.75212 −0.0803926
\(476\) 0.0888858 0.00407407
\(477\) −10.7397 −0.491736
\(478\) 22.3195 1.02087
\(479\) 18.4397 0.842531 0.421265 0.906937i \(-0.361586\pi\)
0.421265 + 0.906937i \(0.361586\pi\)
\(480\) −2.09722 −0.0957245
\(481\) −8.84213 −0.403166
\(482\) −26.3016 −1.19801
\(483\) −0.148932 −0.00677662
\(484\) −0.902708 −0.0410322
\(485\) −5.73963 −0.260623
\(486\) −1.00000 −0.0453609
\(487\) 24.9829 1.13208 0.566041 0.824377i \(-0.308474\pi\)
0.566041 + 0.824377i \(0.308474\pi\)
\(488\) 1.68161 0.0761230
\(489\) 5.69850 0.257695
\(490\) 14.6776 0.663067
\(491\) −21.6861 −0.978679 −0.489339 0.872093i \(-0.662762\pi\)
−0.489339 + 0.872093i \(0.662762\pi\)
\(492\) −9.70721 −0.437635
\(493\) 5.87614 0.264648
\(494\) −2.91209 −0.131021
\(495\) −6.66417 −0.299532
\(496\) 0.760072 0.0341282
\(497\) 0.536608 0.0240702
\(498\) 5.88671 0.263790
\(499\) 28.2202 1.26331 0.631654 0.775250i \(-0.282376\pi\)
0.631654 + 0.775250i \(0.282376\pi\)
\(500\) −11.7479 −0.525384
\(501\) 19.9987 0.893476
\(502\) −14.0004 −0.624868
\(503\) −15.1995 −0.677713 −0.338856 0.940838i \(-0.610040\pi\)
−0.338856 + 0.940838i \(0.610040\pi\)
\(504\) 0.0373018 0.00166155
\(505\) 37.2493 1.65757
\(506\) 12.6870 0.564007
\(507\) 1.00000 0.0444116
\(508\) 4.59734 0.203974
\(509\) −27.6342 −1.22487 −0.612433 0.790522i \(-0.709809\pi\)
−0.612433 + 0.790522i \(0.709809\pi\)
\(510\) 4.99743 0.221290
\(511\) −0.570142 −0.0252216
\(512\) −1.00000 −0.0441942
\(513\) 2.91209 0.128572
\(514\) −11.7882 −0.519957
\(515\) −2.09722 −0.0924145
\(516\) −10.8256 −0.476572
\(517\) 16.8408 0.740659
\(518\) −0.329827 −0.0144918
\(519\) 7.43256 0.326253
\(520\) −2.09722 −0.0919692
\(521\) −40.7700 −1.78616 −0.893082 0.449893i \(-0.851462\pi\)
−0.893082 + 0.449893i \(0.851462\pi\)
\(522\) 2.46598 0.107933
\(523\) 9.57148 0.418532 0.209266 0.977859i \(-0.432893\pi\)
0.209266 + 0.977859i \(0.432893\pi\)
\(524\) −17.3267 −0.756923
\(525\) 0.0224434 0.000979509 0
\(526\) 3.34403 0.145806
\(527\) −1.81116 −0.0788955
\(528\) −3.17762 −0.138288
\(529\) −7.05903 −0.306914
\(530\) 22.5235 0.978357
\(531\) 4.58591 0.199011
\(532\) −0.108626 −0.00470953
\(533\) −9.70721 −0.420466
\(534\) −2.63980 −0.114235
\(535\) 15.6168 0.675173
\(536\) −7.20145 −0.311055
\(537\) 24.5685 1.06021
\(538\) −16.1066 −0.694405
\(539\) 22.2389 0.957899
\(540\) 2.09722 0.0902500
\(541\) 41.1381 1.76867 0.884333 0.466857i \(-0.154614\pi\)
0.884333 + 0.466857i \(0.154614\pi\)
\(542\) 9.09091 0.390488
\(543\) −6.64229 −0.285048
\(544\) 2.38289 0.102165
\(545\) −6.92295 −0.296547
\(546\) 0.0373018 0.00159637
\(547\) 18.1876 0.777644 0.388822 0.921313i \(-0.372882\pi\)
0.388822 + 0.921313i \(0.372882\pi\)
\(548\) −14.5720 −0.622485
\(549\) −1.68161 −0.0717695
\(550\) −1.91188 −0.0815229
\(551\) −7.18114 −0.305927
\(552\) −3.99261 −0.169937
\(553\) −0.238508 −0.0101424
\(554\) −20.2183 −0.858994
\(555\) −18.5439 −0.787144
\(556\) −15.9341 −0.675758
\(557\) 6.12099 0.259355 0.129677 0.991556i \(-0.458606\pi\)
0.129677 + 0.991556i \(0.458606\pi\)
\(558\) −0.760072 −0.0321764
\(559\) −10.8256 −0.457875
\(560\) −0.0782300 −0.00330582
\(561\) 7.57191 0.319686
\(562\) −10.4908 −0.442526
\(563\) −46.7197 −1.96900 −0.984501 0.175382i \(-0.943884\pi\)
−0.984501 + 0.175382i \(0.943884\pi\)
\(564\) −5.29982 −0.223163
\(565\) −13.0338 −0.548337
\(566\) 20.5499 0.863778
\(567\) −0.0373018 −0.00156653
\(568\) 14.3856 0.603606
\(569\) −13.9598 −0.585224 −0.292612 0.956231i \(-0.594524\pi\)
−0.292612 + 0.956231i \(0.594524\pi\)
\(570\) −6.10728 −0.255806
\(571\) 1.97131 0.0824968 0.0412484 0.999149i \(-0.486866\pi\)
0.0412484 + 0.999149i \(0.486866\pi\)
\(572\) −3.17762 −0.132863
\(573\) −12.5288 −0.523397
\(574\) −0.362096 −0.0151136
\(575\) −2.40224 −0.100180
\(576\) 1.00000 0.0416667
\(577\) 20.8428 0.867697 0.433849 0.900986i \(-0.357155\pi\)
0.433849 + 0.900986i \(0.357155\pi\)
\(578\) 11.3219 0.470927
\(579\) 4.59720 0.191053
\(580\) −5.17170 −0.214743
\(581\) 0.219585 0.00910991
\(582\) 2.73678 0.113443
\(583\) 34.1267 1.41338
\(584\) −15.2846 −0.632481
\(585\) 2.09722 0.0867094
\(586\) 2.35918 0.0974569
\(587\) −34.8380 −1.43792 −0.718958 0.695053i \(-0.755381\pi\)
−0.718958 + 0.695053i \(0.755381\pi\)
\(588\) −6.99861 −0.288618
\(589\) 2.21339 0.0912013
\(590\) −9.61765 −0.395953
\(591\) 15.5509 0.639678
\(592\) −8.84213 −0.363409
\(593\) −2.11731 −0.0869477 −0.0434738 0.999055i \(-0.513843\pi\)
−0.0434738 + 0.999055i \(0.513843\pi\)
\(594\) 3.17762 0.130379
\(595\) 0.186413 0.00764219
\(596\) −12.4885 −0.511548
\(597\) −9.66523 −0.395571
\(598\) −3.99261 −0.163270
\(599\) 35.5037 1.45064 0.725320 0.688411i \(-0.241692\pi\)
0.725320 + 0.688411i \(0.241692\pi\)
\(600\) 0.601670 0.0245631
\(601\) 16.5908 0.676752 0.338376 0.941011i \(-0.390123\pi\)
0.338376 + 0.941011i \(0.390123\pi\)
\(602\) −0.403815 −0.0164583
\(603\) 7.20145 0.293266
\(604\) 11.6943 0.475834
\(605\) −1.89318 −0.0769686
\(606\) −17.7613 −0.721502
\(607\) −40.3761 −1.63882 −0.819408 0.573210i \(-0.805698\pi\)
−0.819408 + 0.573210i \(0.805698\pi\)
\(608\) −2.91209 −0.118101
\(609\) 0.0919853 0.00372743
\(610\) 3.52671 0.142792
\(611\) −5.29982 −0.214408
\(612\) −2.38289 −0.0963224
\(613\) −14.6447 −0.591493 −0.295747 0.955266i \(-0.595568\pi\)
−0.295747 + 0.955266i \(0.595568\pi\)
\(614\) 23.9915 0.968218
\(615\) −20.3582 −0.820920
\(616\) −0.118531 −0.00477575
\(617\) 14.9516 0.601927 0.300963 0.953636i \(-0.402692\pi\)
0.300963 + 0.953636i \(0.402692\pi\)
\(618\) 1.00000 0.0402259
\(619\) 18.1481 0.729435 0.364717 0.931118i \(-0.381166\pi\)
0.364717 + 0.931118i \(0.381166\pi\)
\(620\) 1.59404 0.0640181
\(621\) 3.99261 0.160218
\(622\) 9.06740 0.363570
\(623\) −0.0984690 −0.00394508
\(624\) 1.00000 0.0400320
\(625\) −21.6296 −0.865186
\(626\) −27.0568 −1.08141
\(627\) −9.25351 −0.369550
\(628\) −11.0964 −0.442796
\(629\) 21.0698 0.840107
\(630\) 0.0782300 0.00311676
\(631\) 12.3163 0.490305 0.245153 0.969485i \(-0.421162\pi\)
0.245153 + 0.969485i \(0.421162\pi\)
\(632\) −6.39400 −0.254340
\(633\) −3.17494 −0.126193
\(634\) −25.3558 −1.00701
\(635\) 9.64162 0.382616
\(636\) −10.7397 −0.425856
\(637\) −6.99861 −0.277295
\(638\) −7.83595 −0.310228
\(639\) −14.3856 −0.569086
\(640\) −2.09722 −0.0828999
\(641\) 9.98898 0.394541 0.197270 0.980349i \(-0.436792\pi\)
0.197270 + 0.980349i \(0.436792\pi\)
\(642\) −7.44643 −0.293887
\(643\) 6.16171 0.242994 0.121497 0.992592i \(-0.461231\pi\)
0.121497 + 0.992592i \(0.461231\pi\)
\(644\) −0.148932 −0.00586872
\(645\) −22.7037 −0.893958
\(646\) 6.93916 0.273018
\(647\) 27.2248 1.07032 0.535159 0.844751i \(-0.320252\pi\)
0.535159 + 0.844751i \(0.320252\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −14.5723 −0.572012
\(650\) 0.601670 0.0235995
\(651\) −0.0283520 −0.00111120
\(652\) 5.69850 0.223170
\(653\) 27.1892 1.06400 0.531998 0.846745i \(-0.321441\pi\)
0.531998 + 0.846745i \(0.321441\pi\)
\(654\) 3.30101 0.129080
\(655\) −36.3380 −1.41984
\(656\) −9.70721 −0.379003
\(657\) 15.2846 0.596309
\(658\) −0.197693 −0.00770686
\(659\) 13.5823 0.529093 0.264546 0.964373i \(-0.414778\pi\)
0.264546 + 0.964373i \(0.414778\pi\)
\(660\) −6.66417 −0.259403
\(661\) −31.9323 −1.24202 −0.621010 0.783802i \(-0.713278\pi\)
−0.621010 + 0.783802i \(0.713278\pi\)
\(662\) 13.8421 0.537987
\(663\) −2.38289 −0.0925436
\(664\) 5.88671 0.228449
\(665\) −0.227812 −0.00883418
\(666\) 8.84213 0.342625
\(667\) −9.84570 −0.381227
\(668\) 19.9987 0.773773
\(669\) −1.36542 −0.0527900
\(670\) −15.1030 −0.583481
\(671\) 5.34353 0.206285
\(672\) 0.0373018 0.00143895
\(673\) 23.8672 0.920012 0.460006 0.887916i \(-0.347847\pi\)
0.460006 + 0.887916i \(0.347847\pi\)
\(674\) 0.107575 0.00414363
\(675\) −0.601670 −0.0231583
\(676\) 1.00000 0.0384615
\(677\) −51.5593 −1.98158 −0.990792 0.135390i \(-0.956771\pi\)
−0.990792 + 0.135390i \(0.956771\pi\)
\(678\) 6.21481 0.238678
\(679\) 0.102087 0.00391773
\(680\) 4.99743 0.191643
\(681\) −10.1984 −0.390804
\(682\) 2.41522 0.0924836
\(683\) 11.2269 0.429586 0.214793 0.976660i \(-0.431092\pi\)
0.214793 + 0.976660i \(0.431092\pi\)
\(684\) 2.91209 0.111346
\(685\) −30.5607 −1.16766
\(686\) −0.522173 −0.0199366
\(687\) 20.1236 0.767763
\(688\) −10.8256 −0.412723
\(689\) −10.7397 −0.409149
\(690\) −8.37339 −0.318769
\(691\) −34.2987 −1.30478 −0.652392 0.757881i \(-0.726235\pi\)
−0.652392 + 0.757881i \(0.726235\pi\)
\(692\) 7.43256 0.282544
\(693\) 0.118531 0.00450262
\(694\) 33.6802 1.27848
\(695\) −33.4174 −1.26759
\(696\) 2.46598 0.0934727
\(697\) 23.1312 0.876156
\(698\) 14.2268 0.538492
\(699\) −20.9544 −0.792567
\(700\) 0.0224434 0.000848279 0
\(701\) −8.77197 −0.331313 −0.165656 0.986184i \(-0.552974\pi\)
−0.165656 + 0.986184i \(0.552974\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −25.7490 −0.971143
\(704\) −3.17762 −0.119761
\(705\) −11.1149 −0.418611
\(706\) −2.62388 −0.0987509
\(707\) −0.662526 −0.0249169
\(708\) 4.58591 0.172349
\(709\) 25.0019 0.938967 0.469484 0.882941i \(-0.344440\pi\)
0.469484 + 0.882941i \(0.344440\pi\)
\(710\) 30.1698 1.13225
\(711\) 6.39400 0.239794
\(712\) −2.63980 −0.0989305
\(713\) 3.03467 0.113649
\(714\) −0.0888858 −0.00332647
\(715\) −6.66417 −0.249226
\(716\) 24.5685 0.918170
\(717\) −22.3195 −0.833538
\(718\) 15.8065 0.589892
\(719\) −39.3676 −1.46816 −0.734081 0.679061i \(-0.762387\pi\)
−0.734081 + 0.679061i \(0.762387\pi\)
\(720\) 2.09722 0.0781588
\(721\) 0.0373018 0.00138919
\(722\) 10.5198 0.391505
\(723\) 26.3016 0.978168
\(724\) −6.64229 −0.246859
\(725\) 1.48371 0.0551035
\(726\) 0.902708 0.0335026
\(727\) −5.79125 −0.214786 −0.107393 0.994217i \(-0.534250\pi\)
−0.107393 + 0.994217i \(0.534250\pi\)
\(728\) 0.0373018 0.00138250
\(729\) 1.00000 0.0370370
\(730\) −32.0552 −1.18641
\(731\) 25.7962 0.954108
\(732\) −1.68161 −0.0621542
\(733\) 24.9648 0.922098 0.461049 0.887375i \(-0.347473\pi\)
0.461049 + 0.887375i \(0.347473\pi\)
\(734\) −11.4273 −0.421789
\(735\) −14.6776 −0.541392
\(736\) −3.99261 −0.147170
\(737\) −22.8835 −0.842925
\(738\) 9.70721 0.357327
\(739\) 16.6930 0.614061 0.307031 0.951700i \(-0.400665\pi\)
0.307031 + 0.951700i \(0.400665\pi\)
\(740\) −18.5439 −0.681687
\(741\) 2.91209 0.106978
\(742\) −0.400609 −0.0147068
\(743\) −37.9447 −1.39206 −0.696028 0.718014i \(-0.745051\pi\)
−0.696028 + 0.718014i \(0.745051\pi\)
\(744\) −0.760072 −0.0278656
\(745\) −26.1911 −0.959567
\(746\) 5.05968 0.185248
\(747\) −5.88671 −0.215384
\(748\) 7.57191 0.276857
\(749\) −0.277765 −0.0101493
\(750\) 11.7479 0.428974
\(751\) 29.4524 1.07474 0.537368 0.843348i \(-0.319419\pi\)
0.537368 + 0.843348i \(0.319419\pi\)
\(752\) −5.29982 −0.193264
\(753\) 14.0004 0.510202
\(754\) 2.46598 0.0898056
\(755\) 24.5255 0.892575
\(756\) −0.0373018 −0.00135665
\(757\) 8.22118 0.298804 0.149402 0.988777i \(-0.452265\pi\)
0.149402 + 0.988777i \(0.452265\pi\)
\(758\) −12.8369 −0.466259
\(759\) −12.6870 −0.460510
\(760\) −6.10728 −0.221534
\(761\) −28.5972 −1.03665 −0.518324 0.855185i \(-0.673444\pi\)
−0.518324 + 0.855185i \(0.673444\pi\)
\(762\) −4.59734 −0.166544
\(763\) 0.123134 0.00445773
\(764\) −12.5288 −0.453275
\(765\) −4.99743 −0.180683
\(766\) 19.1538 0.692056
\(767\) 4.58591 0.165588
\(768\) 1.00000 0.0360844
\(769\) 38.8323 1.40033 0.700165 0.713981i \(-0.253110\pi\)
0.700165 + 0.713981i \(0.253110\pi\)
\(770\) −0.248585 −0.00895840
\(771\) 11.7882 0.424543
\(772\) 4.59720 0.165457
\(773\) −36.2728 −1.30464 −0.652320 0.757943i \(-0.726204\pi\)
−0.652320 + 0.757943i \(0.726204\pi\)
\(774\) 10.8256 0.389119
\(775\) −0.457313 −0.0164272
\(776\) 2.73678 0.0982447
\(777\) 0.329827 0.0118325
\(778\) −11.7164 −0.420053
\(779\) −28.2682 −1.01281
\(780\) 2.09722 0.0750925
\(781\) 45.7120 1.63570
\(782\) 9.51394 0.340218
\(783\) −2.46598 −0.0881269
\(784\) −6.99861 −0.249950
\(785\) −23.2717 −0.830601
\(786\) 17.3267 0.618025
\(787\) −44.7173 −1.59400 −0.797000 0.603979i \(-0.793581\pi\)
−0.797000 + 0.603979i \(0.793581\pi\)
\(788\) 15.5509 0.553978
\(789\) −3.34403 −0.119050
\(790\) −13.4096 −0.477093
\(791\) 0.231823 0.00824269
\(792\) 3.17762 0.112912
\(793\) −1.68161 −0.0597158
\(794\) −23.4133 −0.830907
\(795\) −22.5235 −0.798825
\(796\) −9.66523 −0.342575
\(797\) 49.2580 1.74481 0.872403 0.488787i \(-0.162560\pi\)
0.872403 + 0.488787i \(0.162560\pi\)
\(798\) 0.108626 0.00384531
\(799\) 12.6289 0.446777
\(800\) 0.601670 0.0212723
\(801\) 2.63980 0.0932726
\(802\) −28.4310 −1.00393
\(803\) −48.5687 −1.71395
\(804\) 7.20145 0.253976
\(805\) −0.312342 −0.0110086
\(806\) −0.760072 −0.0267724
\(807\) 16.1066 0.566979
\(808\) −17.7613 −0.624839
\(809\) −33.0157 −1.16077 −0.580384 0.814343i \(-0.697098\pi\)
−0.580384 + 0.814343i \(0.697098\pi\)
\(810\) −2.09722 −0.0736888
\(811\) 11.8002 0.414360 0.207180 0.978303i \(-0.433571\pi\)
0.207180 + 0.978303i \(0.433571\pi\)
\(812\) 0.0919853 0.00322805
\(813\) −9.09091 −0.318832
\(814\) −28.0969 −0.984797
\(815\) 11.9510 0.418625
\(816\) −2.38289 −0.0834177
\(817\) −31.5252 −1.10293
\(818\) 5.14846 0.180012
\(819\) −0.0373018 −0.00130343
\(820\) −20.3582 −0.710938
\(821\) −16.4095 −0.572695 −0.286348 0.958126i \(-0.592441\pi\)
−0.286348 + 0.958126i \(0.592441\pi\)
\(822\) 14.5720 0.508257
\(823\) −12.2205 −0.425979 −0.212989 0.977055i \(-0.568320\pi\)
−0.212989 + 0.977055i \(0.568320\pi\)
\(824\) 1.00000 0.0348367
\(825\) 1.91188 0.0665632
\(826\) 0.171062 0.00595202
\(827\) −38.8199 −1.34990 −0.674951 0.737863i \(-0.735835\pi\)
−0.674951 + 0.737863i \(0.735835\pi\)
\(828\) 3.99261 0.138753
\(829\) 41.0684 1.42637 0.713183 0.700978i \(-0.247253\pi\)
0.713183 + 0.700978i \(0.247253\pi\)
\(830\) 12.3457 0.428527
\(831\) 20.2183 0.701366
\(832\) 1.00000 0.0346688
\(833\) 16.6769 0.577820
\(834\) 15.9341 0.551754
\(835\) 41.9416 1.45145
\(836\) −9.25351 −0.320039
\(837\) 0.760072 0.0262719
\(838\) 15.5597 0.537502
\(839\) 38.0471 1.31353 0.656766 0.754094i \(-0.271924\pi\)
0.656766 + 0.754094i \(0.271924\pi\)
\(840\) 0.0782300 0.00269919
\(841\) −22.9190 −0.790309
\(842\) −19.2163 −0.662239
\(843\) 10.4908 0.361321
\(844\) −3.17494 −0.109286
\(845\) 2.09722 0.0721465
\(846\) 5.29982 0.182212
\(847\) 0.0336726 0.00115700
\(848\) −10.7397 −0.368802
\(849\) −20.5499 −0.705272
\(850\) −1.43371 −0.0491759
\(851\) −35.3032 −1.21018
\(852\) −14.3856 −0.492843
\(853\) −3.25026 −0.111287 −0.0556434 0.998451i \(-0.517721\pi\)
−0.0556434 + 0.998451i \(0.517721\pi\)
\(854\) −0.0627271 −0.00214648
\(855\) 6.10728 0.208865
\(856\) −7.44643 −0.254514
\(857\) 5.18334 0.177059 0.0885297 0.996074i \(-0.471783\pi\)
0.0885297 + 0.996074i \(0.471783\pi\)
\(858\) 3.17762 0.108482
\(859\) −12.7539 −0.435159 −0.217580 0.976043i \(-0.569816\pi\)
−0.217580 + 0.976043i \(0.569816\pi\)
\(860\) −22.7037 −0.774191
\(861\) 0.362096 0.0123402
\(862\) 24.8965 0.847977
\(863\) 23.8338 0.811312 0.405656 0.914026i \(-0.367043\pi\)
0.405656 + 0.914026i \(0.367043\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 15.5877 0.529998
\(866\) −29.0280 −0.986412
\(867\) −11.3219 −0.384510
\(868\) −0.0283520 −0.000962330 0
\(869\) −20.3177 −0.689232
\(870\) 5.17170 0.175337
\(871\) 7.20145 0.244012
\(872\) 3.30101 0.111786
\(873\) −2.73678 −0.0926260
\(874\) −11.6268 −0.393284
\(875\) 0.438219 0.0148145
\(876\) 15.2846 0.516419
\(877\) −38.2283 −1.29088 −0.645438 0.763812i \(-0.723325\pi\)
−0.645438 + 0.763812i \(0.723325\pi\)
\(878\) 12.1657 0.410571
\(879\) −2.35918 −0.0795732
\(880\) −6.66417 −0.224649
\(881\) 8.71342 0.293562 0.146781 0.989169i \(-0.453109\pi\)
0.146781 + 0.989169i \(0.453109\pi\)
\(882\) 6.99861 0.235655
\(883\) 31.4124 1.05711 0.528555 0.848899i \(-0.322734\pi\)
0.528555 + 0.848899i \(0.322734\pi\)
\(884\) −2.38289 −0.0801451
\(885\) 9.61765 0.323294
\(886\) 26.0682 0.875779
\(887\) −33.5550 −1.12667 −0.563334 0.826229i \(-0.690482\pi\)
−0.563334 + 0.826229i \(0.690482\pi\)
\(888\) 8.84213 0.296722
\(889\) −0.171489 −0.00575155
\(890\) −5.53623 −0.185575
\(891\) −3.17762 −0.106454
\(892\) −1.36542 −0.0457175
\(893\) −15.4335 −0.516463
\(894\) 12.4885 0.417677
\(895\) 51.5256 1.72231
\(896\) 0.0373018 0.00124616
\(897\) 3.99261 0.133310
\(898\) 17.1631 0.572740
\(899\) −1.87432 −0.0625121
\(900\) −0.601670 −0.0200557
\(901\) 25.5914 0.852574
\(902\) −30.8459 −1.02705
\(903\) 0.403815 0.0134381
\(904\) 6.21481 0.206701
\(905\) −13.9303 −0.463060
\(906\) −11.6943 −0.388517
\(907\) −8.26058 −0.274288 −0.137144 0.990551i \(-0.543792\pi\)
−0.137144 + 0.990551i \(0.543792\pi\)
\(908\) −10.1984 −0.338446
\(909\) 17.7613 0.589104
\(910\) 0.0782300 0.00259330
\(911\) 18.2362 0.604191 0.302095 0.953278i \(-0.402314\pi\)
0.302095 + 0.953278i \(0.402314\pi\)
\(912\) 2.91209 0.0964288
\(913\) 18.7058 0.619070
\(914\) −1.04913 −0.0347020
\(915\) −3.52671 −0.116589
\(916\) 20.1236 0.664902
\(917\) 0.646318 0.0213433
\(918\) 2.38289 0.0786469
\(919\) −50.2704 −1.65827 −0.829133 0.559051i \(-0.811165\pi\)
−0.829133 + 0.559051i \(0.811165\pi\)
\(920\) −8.37339 −0.276062
\(921\) −23.9915 −0.790547
\(922\) −1.36203 −0.0448562
\(923\) −14.3856 −0.473508
\(924\) 0.118531 0.00389938
\(925\) 5.32004 0.174922
\(926\) 17.9663 0.590408
\(927\) −1.00000 −0.0328443
\(928\) 2.46598 0.0809497
\(929\) −26.0641 −0.855137 −0.427568 0.903983i \(-0.640630\pi\)
−0.427568 + 0.903983i \(0.640630\pi\)
\(930\) −1.59404 −0.0522705
\(931\) −20.3805 −0.667945
\(932\) −20.9544 −0.686383
\(933\) −9.06740 −0.296853
\(934\) 12.2355 0.400359
\(935\) 15.8800 0.519330
\(936\) −1.00000 −0.0326860
\(937\) 50.1578 1.63858 0.819292 0.573376i \(-0.194367\pi\)
0.819292 + 0.573376i \(0.194367\pi\)
\(938\) 0.268627 0.00877097
\(939\) 27.0568 0.882965
\(940\) −11.1149 −0.362528
\(941\) 52.6830 1.71742 0.858708 0.512466i \(-0.171268\pi\)
0.858708 + 0.512466i \(0.171268\pi\)
\(942\) 11.0964 0.361542
\(943\) −38.7572 −1.26211
\(944\) 4.58591 0.149259
\(945\) −0.0782300 −0.00254482
\(946\) −34.3998 −1.11843
\(947\) 14.4382 0.469180 0.234590 0.972094i \(-0.424625\pi\)
0.234590 + 0.972094i \(0.424625\pi\)
\(948\) 6.39400 0.207668
\(949\) 15.2846 0.496159
\(950\) 1.75212 0.0568461
\(951\) 25.3558 0.822219
\(952\) −0.0888858 −0.00288081
\(953\) −54.7392 −1.77318 −0.886588 0.462561i \(-0.846931\pi\)
−0.886588 + 0.462561i \(0.846931\pi\)
\(954\) 10.7397 0.347710
\(955\) −26.2756 −0.850258
\(956\) −22.3195 −0.721865
\(957\) 7.83595 0.253300
\(958\) −18.4397 −0.595759
\(959\) 0.543561 0.0175525
\(960\) 2.09722 0.0676875
\(961\) −30.4223 −0.981364
\(962\) 8.84213 0.285082
\(963\) 7.44643 0.239958
\(964\) 26.3016 0.847118
\(965\) 9.64134 0.310366
\(966\) 0.148932 0.00479179
\(967\) 1.08880 0.0350135 0.0175068 0.999847i \(-0.494427\pi\)
0.0175068 + 0.999847i \(0.494427\pi\)
\(968\) 0.902708 0.0290141
\(969\) −6.93916 −0.222918
\(970\) 5.73963 0.184288
\(971\) −32.3608 −1.03851 −0.519254 0.854620i \(-0.673790\pi\)
−0.519254 + 0.854620i \(0.673790\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0.594371 0.0190547
\(974\) −24.9829 −0.800504
\(975\) −0.601670 −0.0192689
\(976\) −1.68161 −0.0538271
\(977\) 32.4148 1.03704 0.518521 0.855065i \(-0.326483\pi\)
0.518521 + 0.855065i \(0.326483\pi\)
\(978\) −5.69850 −0.182218
\(979\) −8.38828 −0.268091
\(980\) −14.6776 −0.468859
\(981\) −3.30101 −0.105393
\(982\) 21.6861 0.692030
\(983\) 52.2186 1.66552 0.832758 0.553638i \(-0.186761\pi\)
0.832758 + 0.553638i \(0.186761\pi\)
\(984\) 9.70721 0.309455
\(985\) 32.6136 1.03916
\(986\) −5.87614 −0.187135
\(987\) 0.197693 0.00629262
\(988\) 2.91209 0.0926458
\(989\) −43.2226 −1.37440
\(990\) 6.66417 0.211801
\(991\) −26.0290 −0.826838 −0.413419 0.910541i \(-0.635665\pi\)
−0.413419 + 0.910541i \(0.635665\pi\)
\(992\) −0.760072 −0.0241323
\(993\) −13.8421 −0.439265
\(994\) −0.536608 −0.0170202
\(995\) −20.2701 −0.642606
\(996\) −5.88671 −0.186528
\(997\) 26.7927 0.848534 0.424267 0.905537i \(-0.360532\pi\)
0.424267 + 0.905537i \(0.360532\pi\)
\(998\) −28.2202 −0.893294
\(999\) −8.84213 −0.279752
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 8034.2.a.w.1.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.w.1.10 12 1.1 even 1 trivial