Properties

Label 2070.2.a.w.1.1
Level $2070$
Weight $2$
Character 2070.1
Self dual yes
Analytic conductor $16.529$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 2070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.302776 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.302776 q^{7} +1.00000 q^{8} -1.00000 q^{10} +5.30278 q^{11} -0.302776 q^{13} -0.302776 q^{14} +1.00000 q^{16} +3.90833 q^{17} -4.90833 q^{19} -1.00000 q^{20} +5.30278 q^{22} +1.00000 q^{23} +1.00000 q^{25} -0.302776 q^{26} -0.302776 q^{28} -4.60555 q^{29} +2.90833 q^{31} +1.00000 q^{32} +3.90833 q^{34} +0.302776 q^{35} +8.00000 q^{37} -4.90833 q^{38} -1.00000 q^{40} +9.90833 q^{41} +5.21110 q^{43} +5.30278 q^{44} +1.00000 q^{46} -4.60555 q^{47} -6.90833 q^{49} +1.00000 q^{50} -0.302776 q^{52} -3.21110 q^{53} -5.30278 q^{55} -0.302776 q^{56} -4.60555 q^{58} +10.6056 q^{59} -6.51388 q^{61} +2.90833 q^{62} +1.00000 q^{64} +0.302776 q^{65} -4.00000 q^{67} +3.90833 q^{68} +0.302776 q^{70} +12.6972 q^{71} +15.8167 q^{73} +8.00000 q^{74} -4.90833 q^{76} -1.60555 q^{77} +14.4222 q^{79} -1.00000 q^{80} +9.90833 q^{82} +3.21110 q^{83} -3.90833 q^{85} +5.21110 q^{86} +5.30278 q^{88} +0.0916731 q^{91} +1.00000 q^{92} -4.60555 q^{94} +4.90833 q^{95} +2.69722 q^{97} -6.90833 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 3 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 3 q^{7} + 2 q^{8} - 2 q^{10} + 7 q^{11} + 3 q^{13} + 3 q^{14} + 2 q^{16} - 3 q^{17} + q^{19} - 2 q^{20} + 7 q^{22} + 2 q^{23} + 2 q^{25} + 3 q^{26} + 3 q^{28} - 2 q^{29} - 5 q^{31} + 2 q^{32} - 3 q^{34} - 3 q^{35} + 16 q^{37} + q^{38} - 2 q^{40} + 9 q^{41} - 4 q^{43} + 7 q^{44} + 2 q^{46} - 2 q^{47} - 3 q^{49} + 2 q^{50} + 3 q^{52} + 8 q^{53} - 7 q^{55} + 3 q^{56} - 2 q^{58} + 14 q^{59} + 5 q^{61} - 5 q^{62} + 2 q^{64} - 3 q^{65} - 8 q^{67} - 3 q^{68} - 3 q^{70} + 29 q^{71} + 10 q^{73} + 16 q^{74} + q^{76} + 4 q^{77} - 2 q^{80} + 9 q^{82} - 8 q^{83} + 3 q^{85} - 4 q^{86} + 7 q^{88} + 11 q^{91} + 2 q^{92} - 2 q^{94} - q^{95} + 9 q^{97} - 3 q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.302776 −0.114438 −0.0572192 0.998362i \(-0.518223\pi\)
−0.0572192 + 0.998362i \(0.518223\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 5.30278 1.59885 0.799424 0.600768i \(-0.205138\pi\)
0.799424 + 0.600768i \(0.205138\pi\)
\(12\) 0 0
\(13\) −0.302776 −0.0839749 −0.0419874 0.999118i \(-0.513369\pi\)
−0.0419874 + 0.999118i \(0.513369\pi\)
\(14\) −0.302776 −0.0809202
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.90833 0.947909 0.473954 0.880549i \(-0.342826\pi\)
0.473954 + 0.880549i \(0.342826\pi\)
\(18\) 0 0
\(19\) −4.90833 −1.12605 −0.563024 0.826441i \(-0.690362\pi\)
−0.563024 + 0.826441i \(0.690362\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 5.30278 1.13056
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −0.302776 −0.0593792
\(27\) 0 0
\(28\) −0.302776 −0.0572192
\(29\) −4.60555 −0.855229 −0.427615 0.903961i \(-0.640646\pi\)
−0.427615 + 0.903961i \(0.640646\pi\)
\(30\) 0 0
\(31\) 2.90833 0.522351 0.261175 0.965291i \(-0.415890\pi\)
0.261175 + 0.965291i \(0.415890\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.90833 0.670273
\(35\) 0.302776 0.0511784
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −4.90833 −0.796236
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 9.90833 1.54742 0.773710 0.633540i \(-0.218399\pi\)
0.773710 + 0.633540i \(0.218399\pi\)
\(42\) 0 0
\(43\) 5.21110 0.794686 0.397343 0.917670i \(-0.369932\pi\)
0.397343 + 0.917670i \(0.369932\pi\)
\(44\) 5.30278 0.799424
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −4.60555 −0.671789 −0.335894 0.941900i \(-0.609039\pi\)
−0.335894 + 0.941900i \(0.609039\pi\)
\(48\) 0 0
\(49\) −6.90833 −0.986904
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −0.302776 −0.0419874
\(53\) −3.21110 −0.441079 −0.220539 0.975378i \(-0.570782\pi\)
−0.220539 + 0.975378i \(0.570782\pi\)
\(54\) 0 0
\(55\) −5.30278 −0.715026
\(56\) −0.302776 −0.0404601
\(57\) 0 0
\(58\) −4.60555 −0.604739
\(59\) 10.6056 1.38073 0.690363 0.723464i \(-0.257451\pi\)
0.690363 + 0.723464i \(0.257451\pi\)
\(60\) 0 0
\(61\) −6.51388 −0.834017 −0.417008 0.908903i \(-0.636921\pi\)
−0.417008 + 0.908903i \(0.636921\pi\)
\(62\) 2.90833 0.369358
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0.302776 0.0375547
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 3.90833 0.473954
\(69\) 0 0
\(70\) 0.302776 0.0361886
\(71\) 12.6972 1.50688 0.753442 0.657515i \(-0.228392\pi\)
0.753442 + 0.657515i \(0.228392\pi\)
\(72\) 0 0
\(73\) 15.8167 1.85120 0.925600 0.378504i \(-0.123561\pi\)
0.925600 + 0.378504i \(0.123561\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −4.90833 −0.563024
\(77\) −1.60555 −0.182970
\(78\) 0 0
\(79\) 14.4222 1.62262 0.811312 0.584613i \(-0.198754\pi\)
0.811312 + 0.584613i \(0.198754\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 9.90833 1.09419
\(83\) 3.21110 0.352464 0.176232 0.984349i \(-0.443609\pi\)
0.176232 + 0.984349i \(0.443609\pi\)
\(84\) 0 0
\(85\) −3.90833 −0.423918
\(86\) 5.21110 0.561928
\(87\) 0 0
\(88\) 5.30278 0.565278
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0.0916731 0.00960995
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −4.60555 −0.475026
\(95\) 4.90833 0.503584
\(96\) 0 0
\(97\) 2.69722 0.273862 0.136931 0.990581i \(-0.456276\pi\)
0.136931 + 0.990581i \(0.456276\pi\)
\(98\) −6.90833 −0.697846
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 4.60555 0.458269 0.229135 0.973395i \(-0.426410\pi\)
0.229135 + 0.973395i \(0.426410\pi\)
\(102\) 0 0
\(103\) −17.1194 −1.68683 −0.843414 0.537265i \(-0.819458\pi\)
−0.843414 + 0.537265i \(0.819458\pi\)
\(104\) −0.302776 −0.0296896
\(105\) 0 0
\(106\) −3.21110 −0.311890
\(107\) −4.60555 −0.445235 −0.222618 0.974906i \(-0.571460\pi\)
−0.222618 + 0.974906i \(0.571460\pi\)
\(108\) 0 0
\(109\) 19.5139 1.86909 0.934545 0.355844i \(-0.115807\pi\)
0.934545 + 0.355844i \(0.115807\pi\)
\(110\) −5.30278 −0.505600
\(111\) 0 0
\(112\) −0.302776 −0.0286096
\(113\) −12.4222 −1.16858 −0.584291 0.811544i \(-0.698627\pi\)
−0.584291 + 0.811544i \(0.698627\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) −4.60555 −0.427615
\(117\) 0 0
\(118\) 10.6056 0.976320
\(119\) −1.18335 −0.108477
\(120\) 0 0
\(121\) 17.1194 1.55631
\(122\) −6.51388 −0.589739
\(123\) 0 0
\(124\) 2.90833 0.261175
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −11.8167 −1.04856 −0.524279 0.851546i \(-0.675665\pi\)
−0.524279 + 0.851546i \(0.675665\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0.302776 0.0265552
\(131\) −3.21110 −0.280555 −0.140278 0.990112i \(-0.544800\pi\)
−0.140278 + 0.990112i \(0.544800\pi\)
\(132\) 0 0
\(133\) 1.48612 0.128863
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 3.90833 0.335136
\(137\) 6.90833 0.590218 0.295109 0.955464i \(-0.404644\pi\)
0.295109 + 0.955464i \(0.404644\pi\)
\(138\) 0 0
\(139\) −5.39445 −0.457551 −0.228776 0.973479i \(-0.573472\pi\)
−0.228776 + 0.973479i \(0.573472\pi\)
\(140\) 0.302776 0.0255892
\(141\) 0 0
\(142\) 12.6972 1.06553
\(143\) −1.60555 −0.134263
\(144\) 0 0
\(145\) 4.60555 0.382470
\(146\) 15.8167 1.30900
\(147\) 0 0
\(148\) 8.00000 0.657596
\(149\) −9.69722 −0.794428 −0.397214 0.917726i \(-0.630023\pi\)
−0.397214 + 0.917726i \(0.630023\pi\)
\(150\) 0 0
\(151\) −1.90833 −0.155297 −0.0776487 0.996981i \(-0.524741\pi\)
−0.0776487 + 0.996981i \(0.524741\pi\)
\(152\) −4.90833 −0.398118
\(153\) 0 0
\(154\) −1.60555 −0.129379
\(155\) −2.90833 −0.233602
\(156\) 0 0
\(157\) −11.3944 −0.909376 −0.454688 0.890651i \(-0.650249\pi\)
−0.454688 + 0.890651i \(0.650249\pi\)
\(158\) 14.4222 1.14737
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −0.302776 −0.0238621
\(162\) 0 0
\(163\) 5.69722 0.446241 0.223121 0.974791i \(-0.428376\pi\)
0.223121 + 0.974791i \(0.428376\pi\)
\(164\) 9.90833 0.773710
\(165\) 0 0
\(166\) 3.21110 0.249230
\(167\) −21.2111 −1.64136 −0.820682 0.571385i \(-0.806406\pi\)
−0.820682 + 0.571385i \(0.806406\pi\)
\(168\) 0 0
\(169\) −12.9083 −0.992948
\(170\) −3.90833 −0.299755
\(171\) 0 0
\(172\) 5.21110 0.397343
\(173\) −23.3028 −1.77168 −0.885839 0.463993i \(-0.846416\pi\)
−0.885839 + 0.463993i \(0.846416\pi\)
\(174\) 0 0
\(175\) −0.302776 −0.0228877
\(176\) 5.30278 0.399712
\(177\) 0 0
\(178\) 0 0
\(179\) −16.6056 −1.24116 −0.620579 0.784144i \(-0.713102\pi\)
−0.620579 + 0.784144i \(0.713102\pi\)
\(180\) 0 0
\(181\) −8.11943 −0.603512 −0.301756 0.953385i \(-0.597573\pi\)
−0.301756 + 0.953385i \(0.597573\pi\)
\(182\) 0.0916731 0.00679526
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) 20.7250 1.51556
\(188\) −4.60555 −0.335894
\(189\) 0 0
\(190\) 4.90833 0.356087
\(191\) 1.39445 0.100899 0.0504494 0.998727i \(-0.483935\pi\)
0.0504494 + 0.998727i \(0.483935\pi\)
\(192\) 0 0
\(193\) 3.81665 0.274729 0.137364 0.990521i \(-0.456137\pi\)
0.137364 + 0.990521i \(0.456137\pi\)
\(194\) 2.69722 0.193649
\(195\) 0 0
\(196\) −6.90833 −0.493452
\(197\) −0.697224 −0.0496752 −0.0248376 0.999691i \(-0.507907\pi\)
−0.0248376 + 0.999691i \(0.507907\pi\)
\(198\) 0 0
\(199\) 8.42221 0.597034 0.298517 0.954404i \(-0.403508\pi\)
0.298517 + 0.954404i \(0.403508\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 4.60555 0.324045
\(203\) 1.39445 0.0978711
\(204\) 0 0
\(205\) −9.90833 −0.692028
\(206\) −17.1194 −1.19277
\(207\) 0 0
\(208\) −0.302776 −0.0209937
\(209\) −26.0278 −1.80038
\(210\) 0 0
\(211\) −7.21110 −0.496433 −0.248216 0.968705i \(-0.579844\pi\)
−0.248216 + 0.968705i \(0.579844\pi\)
\(212\) −3.21110 −0.220539
\(213\) 0 0
\(214\) −4.60555 −0.314829
\(215\) −5.21110 −0.355394
\(216\) 0 0
\(217\) −0.880571 −0.0597770
\(218\) 19.5139 1.32165
\(219\) 0 0
\(220\) −5.30278 −0.357513
\(221\) −1.18335 −0.0796005
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) −0.302776 −0.0202300
\(225\) 0 0
\(226\) −12.4222 −0.826313
\(227\) 7.39445 0.490787 0.245393 0.969424i \(-0.421083\pi\)
0.245393 + 0.969424i \(0.421083\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) −4.60555 −0.302369
\(233\) −4.18335 −0.274060 −0.137030 0.990567i \(-0.543756\pi\)
−0.137030 + 0.990567i \(0.543756\pi\)
\(234\) 0 0
\(235\) 4.60555 0.300433
\(236\) 10.6056 0.690363
\(237\) 0 0
\(238\) −1.18335 −0.0767049
\(239\) 9.21110 0.595817 0.297908 0.954594i \(-0.403711\pi\)
0.297908 + 0.954594i \(0.403711\pi\)
\(240\) 0 0
\(241\) 14.4222 0.929016 0.464508 0.885569i \(-0.346231\pi\)
0.464508 + 0.885569i \(0.346231\pi\)
\(242\) 17.1194 1.10048
\(243\) 0 0
\(244\) −6.51388 −0.417008
\(245\) 6.90833 0.441357
\(246\) 0 0
\(247\) 1.48612 0.0945597
\(248\) 2.90833 0.184679
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 5.51388 0.348033 0.174016 0.984743i \(-0.444325\pi\)
0.174016 + 0.984743i \(0.444325\pi\)
\(252\) 0 0
\(253\) 5.30278 0.333383
\(254\) −11.8167 −0.741443
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −19.8167 −1.23613 −0.618064 0.786127i \(-0.712083\pi\)
−0.618064 + 0.786127i \(0.712083\pi\)
\(258\) 0 0
\(259\) −2.42221 −0.150509
\(260\) 0.302776 0.0187773
\(261\) 0 0
\(262\) −3.21110 −0.198383
\(263\) 14.5139 0.894964 0.447482 0.894293i \(-0.352321\pi\)
0.447482 + 0.894293i \(0.352321\pi\)
\(264\) 0 0
\(265\) 3.21110 0.197256
\(266\) 1.48612 0.0911200
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −25.8167 −1.57407 −0.787035 0.616909i \(-0.788385\pi\)
−0.787035 + 0.616909i \(0.788385\pi\)
\(270\) 0 0
\(271\) −6.30278 −0.382866 −0.191433 0.981506i \(-0.561314\pi\)
−0.191433 + 0.981506i \(0.561314\pi\)
\(272\) 3.90833 0.236977
\(273\) 0 0
\(274\) 6.90833 0.417347
\(275\) 5.30278 0.319769
\(276\) 0 0
\(277\) −12.7889 −0.768410 −0.384205 0.923248i \(-0.625524\pi\)
−0.384205 + 0.923248i \(0.625524\pi\)
\(278\) −5.39445 −0.323538
\(279\) 0 0
\(280\) 0.302776 0.0180943
\(281\) 19.3944 1.15698 0.578488 0.815691i \(-0.303643\pi\)
0.578488 + 0.815691i \(0.303643\pi\)
\(282\) 0 0
\(283\) 2.00000 0.118888 0.0594438 0.998232i \(-0.481067\pi\)
0.0594438 + 0.998232i \(0.481067\pi\)
\(284\) 12.6972 0.753442
\(285\) 0 0
\(286\) −1.60555 −0.0949382
\(287\) −3.00000 −0.177084
\(288\) 0 0
\(289\) −1.72498 −0.101469
\(290\) 4.60555 0.270447
\(291\) 0 0
\(292\) 15.8167 0.925600
\(293\) −8.78890 −0.513453 −0.256726 0.966484i \(-0.582644\pi\)
−0.256726 + 0.966484i \(0.582644\pi\)
\(294\) 0 0
\(295\) −10.6056 −0.617479
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) −9.69722 −0.561745
\(299\) −0.302776 −0.0175100
\(300\) 0 0
\(301\) −1.57779 −0.0909426
\(302\) −1.90833 −0.109812
\(303\) 0 0
\(304\) −4.90833 −0.281512
\(305\) 6.51388 0.372984
\(306\) 0 0
\(307\) −15.3028 −0.873376 −0.436688 0.899613i \(-0.643849\pi\)
−0.436688 + 0.899613i \(0.643849\pi\)
\(308\) −1.60555 −0.0914848
\(309\) 0 0
\(310\) −2.90833 −0.165182
\(311\) 6.42221 0.364170 0.182085 0.983283i \(-0.441715\pi\)
0.182085 + 0.983283i \(0.441715\pi\)
\(312\) 0 0
\(313\) −12.7250 −0.719258 −0.359629 0.933095i \(-0.617097\pi\)
−0.359629 + 0.933095i \(0.617097\pi\)
\(314\) −11.3944 −0.643026
\(315\) 0 0
\(316\) 14.4222 0.811312
\(317\) 14.7250 0.827037 0.413519 0.910496i \(-0.364300\pi\)
0.413519 + 0.910496i \(0.364300\pi\)
\(318\) 0 0
\(319\) −24.4222 −1.36738
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −0.302776 −0.0168730
\(323\) −19.1833 −1.06739
\(324\) 0 0
\(325\) −0.302776 −0.0167950
\(326\) 5.69722 0.315540
\(327\) 0 0
\(328\) 9.90833 0.547096
\(329\) 1.39445 0.0768784
\(330\) 0 0
\(331\) 9.39445 0.516366 0.258183 0.966096i \(-0.416876\pi\)
0.258183 + 0.966096i \(0.416876\pi\)
\(332\) 3.21110 0.176232
\(333\) 0 0
\(334\) −21.2111 −1.16062
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) −4.48612 −0.244375 −0.122187 0.992507i \(-0.538991\pi\)
−0.122187 + 0.992507i \(0.538991\pi\)
\(338\) −12.9083 −0.702120
\(339\) 0 0
\(340\) −3.90833 −0.211959
\(341\) 15.4222 0.835159
\(342\) 0 0
\(343\) 4.21110 0.227378
\(344\) 5.21110 0.280964
\(345\) 0 0
\(346\) −23.3028 −1.25276
\(347\) 25.5416 1.37115 0.685573 0.728004i \(-0.259552\pi\)
0.685573 + 0.728004i \(0.259552\pi\)
\(348\) 0 0
\(349\) −12.7889 −0.684574 −0.342287 0.939595i \(-0.611202\pi\)
−0.342287 + 0.939595i \(0.611202\pi\)
\(350\) −0.302776 −0.0161840
\(351\) 0 0
\(352\) 5.30278 0.282639
\(353\) −18.4222 −0.980515 −0.490258 0.871578i \(-0.663097\pi\)
−0.490258 + 0.871578i \(0.663097\pi\)
\(354\) 0 0
\(355\) −12.6972 −0.673899
\(356\) 0 0
\(357\) 0 0
\(358\) −16.6056 −0.877631
\(359\) 3.21110 0.169476 0.0847378 0.996403i \(-0.472995\pi\)
0.0847378 + 0.996403i \(0.472995\pi\)
\(360\) 0 0
\(361\) 5.09167 0.267983
\(362\) −8.11943 −0.426748
\(363\) 0 0
\(364\) 0.0916731 0.00480498
\(365\) −15.8167 −0.827881
\(366\) 0 0
\(367\) 29.2111 1.52481 0.762404 0.647102i \(-0.224019\pi\)
0.762404 + 0.647102i \(0.224019\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) −8.00000 −0.415900
\(371\) 0.972244 0.0504764
\(372\) 0 0
\(373\) −2.60555 −0.134910 −0.0674552 0.997722i \(-0.521488\pi\)
−0.0674552 + 0.997722i \(0.521488\pi\)
\(374\) 20.7250 1.07166
\(375\) 0 0
\(376\) −4.60555 −0.237513
\(377\) 1.39445 0.0718178
\(378\) 0 0
\(379\) 4.09167 0.210175 0.105088 0.994463i \(-0.466488\pi\)
0.105088 + 0.994463i \(0.466488\pi\)
\(380\) 4.90833 0.251792
\(381\) 0 0
\(382\) 1.39445 0.0713462
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 1.60555 0.0818265
\(386\) 3.81665 0.194263
\(387\) 0 0
\(388\) 2.69722 0.136931
\(389\) −20.9361 −1.06150 −0.530751 0.847528i \(-0.678090\pi\)
−0.530751 + 0.847528i \(0.678090\pi\)
\(390\) 0 0
\(391\) 3.90833 0.197653
\(392\) −6.90833 −0.348923
\(393\) 0 0
\(394\) −0.697224 −0.0351257
\(395\) −14.4222 −0.725660
\(396\) 0 0
\(397\) −21.7250 −1.09035 −0.545173 0.838324i \(-0.683536\pi\)
−0.545173 + 0.838324i \(0.683536\pi\)
\(398\) 8.42221 0.422167
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 1.39445 0.0696354 0.0348177 0.999394i \(-0.488915\pi\)
0.0348177 + 0.999394i \(0.488915\pi\)
\(402\) 0 0
\(403\) −0.880571 −0.0438643
\(404\) 4.60555 0.229135
\(405\) 0 0
\(406\) 1.39445 0.0692053
\(407\) 42.4222 2.10279
\(408\) 0 0
\(409\) −15.0917 −0.746235 −0.373118 0.927784i \(-0.621711\pi\)
−0.373118 + 0.927784i \(0.621711\pi\)
\(410\) −9.90833 −0.489337
\(411\) 0 0
\(412\) −17.1194 −0.843414
\(413\) −3.21110 −0.158008
\(414\) 0 0
\(415\) −3.21110 −0.157627
\(416\) −0.302776 −0.0148448
\(417\) 0 0
\(418\) −26.0278 −1.27306
\(419\) 39.6333 1.93621 0.968107 0.250538i \(-0.0806073\pi\)
0.968107 + 0.250538i \(0.0806073\pi\)
\(420\) 0 0
\(421\) 34.3028 1.67181 0.835907 0.548870i \(-0.184942\pi\)
0.835907 + 0.548870i \(0.184942\pi\)
\(422\) −7.21110 −0.351031
\(423\) 0 0
\(424\) −3.21110 −0.155945
\(425\) 3.90833 0.189582
\(426\) 0 0
\(427\) 1.97224 0.0954436
\(428\) −4.60555 −0.222618
\(429\) 0 0
\(430\) −5.21110 −0.251302
\(431\) −20.2389 −0.974872 −0.487436 0.873159i \(-0.662068\pi\)
−0.487436 + 0.873159i \(0.662068\pi\)
\(432\) 0 0
\(433\) −34.9083 −1.67759 −0.838794 0.544450i \(-0.816739\pi\)
−0.838794 + 0.544450i \(0.816739\pi\)
\(434\) −0.880571 −0.0422687
\(435\) 0 0
\(436\) 19.5139 0.934545
\(437\) −4.90833 −0.234797
\(438\) 0 0
\(439\) −18.3028 −0.873544 −0.436772 0.899572i \(-0.643878\pi\)
−0.436772 + 0.899572i \(0.643878\pi\)
\(440\) −5.30278 −0.252800
\(441\) 0 0
\(442\) −1.18335 −0.0562860
\(443\) −35.5139 −1.68732 −0.843658 0.536882i \(-0.819602\pi\)
−0.843658 + 0.536882i \(0.819602\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) −0.302776 −0.0143048
\(449\) 12.9083 0.609182 0.304591 0.952483i \(-0.401480\pi\)
0.304591 + 0.952483i \(0.401480\pi\)
\(450\) 0 0
\(451\) 52.5416 2.47409
\(452\) −12.4222 −0.584291
\(453\) 0 0
\(454\) 7.39445 0.347039
\(455\) −0.0916731 −0.00429770
\(456\) 0 0
\(457\) −3.57779 −0.167362 −0.0836811 0.996493i \(-0.526668\pi\)
−0.0836811 + 0.996493i \(0.526668\pi\)
\(458\) 2.00000 0.0934539
\(459\) 0 0
\(460\) −1.00000 −0.0466252
\(461\) −31.8167 −1.48185 −0.740925 0.671588i \(-0.765612\pi\)
−0.740925 + 0.671588i \(0.765612\pi\)
\(462\) 0 0
\(463\) −25.6333 −1.19128 −0.595640 0.803251i \(-0.703102\pi\)
−0.595640 + 0.803251i \(0.703102\pi\)
\(464\) −4.60555 −0.213807
\(465\) 0 0
\(466\) −4.18335 −0.193790
\(467\) −19.8167 −0.917005 −0.458503 0.888693i \(-0.651614\pi\)
−0.458503 + 0.888693i \(0.651614\pi\)
\(468\) 0 0
\(469\) 1.21110 0.0559235
\(470\) 4.60555 0.212438
\(471\) 0 0
\(472\) 10.6056 0.488160
\(473\) 27.6333 1.27058
\(474\) 0 0
\(475\) −4.90833 −0.225209
\(476\) −1.18335 −0.0542386
\(477\) 0 0
\(478\) 9.21110 0.421306
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) −2.42221 −0.110443
\(482\) 14.4222 0.656913
\(483\) 0 0
\(484\) 17.1194 0.778156
\(485\) −2.69722 −0.122475
\(486\) 0 0
\(487\) −11.8167 −0.535464 −0.267732 0.963493i \(-0.586274\pi\)
−0.267732 + 0.963493i \(0.586274\pi\)
\(488\) −6.51388 −0.294869
\(489\) 0 0
\(490\) 6.90833 0.312086
\(491\) 25.8167 1.16509 0.582545 0.812799i \(-0.302057\pi\)
0.582545 + 0.812799i \(0.302057\pi\)
\(492\) 0 0
\(493\) −18.0000 −0.810679
\(494\) 1.48612 0.0668638
\(495\) 0 0
\(496\) 2.90833 0.130588
\(497\) −3.84441 −0.172445
\(498\) 0 0
\(499\) 11.6333 0.520778 0.260389 0.965504i \(-0.416149\pi\)
0.260389 + 0.965504i \(0.416149\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 5.51388 0.246096
\(503\) 2.72498 0.121501 0.0607504 0.998153i \(-0.480651\pi\)
0.0607504 + 0.998153i \(0.480651\pi\)
\(504\) 0 0
\(505\) −4.60555 −0.204944
\(506\) 5.30278 0.235737
\(507\) 0 0
\(508\) −11.8167 −0.524279
\(509\) −29.4500 −1.30535 −0.652673 0.757639i \(-0.726353\pi\)
−0.652673 + 0.757639i \(0.726353\pi\)
\(510\) 0 0
\(511\) −4.78890 −0.211848
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −19.8167 −0.874075
\(515\) 17.1194 0.754372
\(516\) 0 0
\(517\) −24.4222 −1.07409
\(518\) −2.42221 −0.106426
\(519\) 0 0
\(520\) 0.302776 0.0132776
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 8.42221 0.368277 0.184139 0.982900i \(-0.441050\pi\)
0.184139 + 0.982900i \(0.441050\pi\)
\(524\) −3.21110 −0.140278
\(525\) 0 0
\(526\) 14.5139 0.632835
\(527\) 11.3667 0.495141
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 3.21110 0.139481
\(531\) 0 0
\(532\) 1.48612 0.0644316
\(533\) −3.00000 −0.129944
\(534\) 0 0
\(535\) 4.60555 0.199115
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) −25.8167 −1.11303
\(539\) −36.6333 −1.57791
\(540\) 0 0
\(541\) −28.8444 −1.24012 −0.620059 0.784555i \(-0.712891\pi\)
−0.620059 + 0.784555i \(0.712891\pi\)
\(542\) −6.30278 −0.270727
\(543\) 0 0
\(544\) 3.90833 0.167568
\(545\) −19.5139 −0.835883
\(546\) 0 0
\(547\) 7.51388 0.321270 0.160635 0.987014i \(-0.448646\pi\)
0.160635 + 0.987014i \(0.448646\pi\)
\(548\) 6.90833 0.295109
\(549\) 0 0
\(550\) 5.30278 0.226111
\(551\) 22.6056 0.963029
\(552\) 0 0
\(553\) −4.36669 −0.185691
\(554\) −12.7889 −0.543348
\(555\) 0 0
\(556\) −5.39445 −0.228776
\(557\) 6.42221 0.272118 0.136059 0.990701i \(-0.456556\pi\)
0.136059 + 0.990701i \(0.456556\pi\)
\(558\) 0 0
\(559\) −1.57779 −0.0667336
\(560\) 0.302776 0.0127946
\(561\) 0 0
\(562\) 19.3944 0.818105
\(563\) −39.6333 −1.67034 −0.835172 0.549988i \(-0.814632\pi\)
−0.835172 + 0.549988i \(0.814632\pi\)
\(564\) 0 0
\(565\) 12.4222 0.522606
\(566\) 2.00000 0.0840663
\(567\) 0 0
\(568\) 12.6972 0.532764
\(569\) 0.422205 0.0176998 0.00884988 0.999961i \(-0.497183\pi\)
0.00884988 + 0.999961i \(0.497183\pi\)
\(570\) 0 0
\(571\) 9.11943 0.381636 0.190818 0.981625i \(-0.438886\pi\)
0.190818 + 0.981625i \(0.438886\pi\)
\(572\) −1.60555 −0.0671315
\(573\) 0 0
\(574\) −3.00000 −0.125218
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −1.72498 −0.0717497
\(579\) 0 0
\(580\) 4.60555 0.191235
\(581\) −0.972244 −0.0403355
\(582\) 0 0
\(583\) −17.0278 −0.705218
\(584\) 15.8167 0.654498
\(585\) 0 0
\(586\) −8.78890 −0.363066
\(587\) 37.5416 1.54951 0.774755 0.632262i \(-0.217873\pi\)
0.774755 + 0.632262i \(0.217873\pi\)
\(588\) 0 0
\(589\) −14.2750 −0.588192
\(590\) −10.6056 −0.436624
\(591\) 0 0
\(592\) 8.00000 0.328798
\(593\) −19.8167 −0.813772 −0.406886 0.913479i \(-0.633385\pi\)
−0.406886 + 0.913479i \(0.633385\pi\)
\(594\) 0 0
\(595\) 1.18335 0.0485125
\(596\) −9.69722 −0.397214
\(597\) 0 0
\(598\) −0.302776 −0.0123814
\(599\) −4.33053 −0.176941 −0.0884704 0.996079i \(-0.528198\pi\)
−0.0884704 + 0.996079i \(0.528198\pi\)
\(600\) 0 0
\(601\) −3.93608 −0.160556 −0.0802781 0.996773i \(-0.525581\pi\)
−0.0802781 + 0.996773i \(0.525581\pi\)
\(602\) −1.57779 −0.0643061
\(603\) 0 0
\(604\) −1.90833 −0.0776487
\(605\) −17.1194 −0.696004
\(606\) 0 0
\(607\) −26.0555 −1.05756 −0.528780 0.848759i \(-0.677350\pi\)
−0.528780 + 0.848759i \(0.677350\pi\)
\(608\) −4.90833 −0.199059
\(609\) 0 0
\(610\) 6.51388 0.263739
\(611\) 1.39445 0.0564134
\(612\) 0 0
\(613\) 32.4222 1.30952 0.654760 0.755837i \(-0.272770\pi\)
0.654760 + 0.755837i \(0.272770\pi\)
\(614\) −15.3028 −0.617570
\(615\) 0 0
\(616\) −1.60555 −0.0646895
\(617\) −8.09167 −0.325758 −0.162879 0.986646i \(-0.552078\pi\)
−0.162879 + 0.986646i \(0.552078\pi\)
\(618\) 0 0
\(619\) 27.3305 1.09851 0.549253 0.835656i \(-0.314912\pi\)
0.549253 + 0.835656i \(0.314912\pi\)
\(620\) −2.90833 −0.116801
\(621\) 0 0
\(622\) 6.42221 0.257507
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −12.7250 −0.508593
\(627\) 0 0
\(628\) −11.3944 −0.454688
\(629\) 31.2666 1.24668
\(630\) 0 0
\(631\) 30.6056 1.21839 0.609194 0.793021i \(-0.291493\pi\)
0.609194 + 0.793021i \(0.291493\pi\)
\(632\) 14.4222 0.573685
\(633\) 0 0
\(634\) 14.7250 0.584804
\(635\) 11.8167 0.468930
\(636\) 0 0
\(637\) 2.09167 0.0828751
\(638\) −24.4222 −0.966884
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 36.0000 1.42191 0.710957 0.703235i \(-0.248262\pi\)
0.710957 + 0.703235i \(0.248262\pi\)
\(642\) 0 0
\(643\) 16.2389 0.640398 0.320199 0.947350i \(-0.396250\pi\)
0.320199 + 0.947350i \(0.396250\pi\)
\(644\) −0.302776 −0.0119310
\(645\) 0 0
\(646\) −19.1833 −0.754759
\(647\) 30.8444 1.21262 0.606309 0.795229i \(-0.292649\pi\)
0.606309 + 0.795229i \(0.292649\pi\)
\(648\) 0 0
\(649\) 56.2389 2.20757
\(650\) −0.302776 −0.0118758
\(651\) 0 0
\(652\) 5.69722 0.223121
\(653\) −9.27502 −0.362960 −0.181480 0.983395i \(-0.558089\pi\)
−0.181480 + 0.983395i \(0.558089\pi\)
\(654\) 0 0
\(655\) 3.21110 0.125468
\(656\) 9.90833 0.386855
\(657\) 0 0
\(658\) 1.39445 0.0543613
\(659\) 27.6333 1.07644 0.538220 0.842804i \(-0.319097\pi\)
0.538220 + 0.842804i \(0.319097\pi\)
\(660\) 0 0
\(661\) −24.0917 −0.937057 −0.468529 0.883448i \(-0.655216\pi\)
−0.468529 + 0.883448i \(0.655216\pi\)
\(662\) 9.39445 0.365126
\(663\) 0 0
\(664\) 3.21110 0.124615
\(665\) −1.48612 −0.0576293
\(666\) 0 0
\(667\) −4.60555 −0.178328
\(668\) −21.2111 −0.820682
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) −34.5416 −1.33347
\(672\) 0 0
\(673\) 5.63331 0.217148 0.108574 0.994088i \(-0.465372\pi\)
0.108574 + 0.994088i \(0.465372\pi\)
\(674\) −4.48612 −0.172799
\(675\) 0 0
\(676\) −12.9083 −0.496474
\(677\) 12.4222 0.477424 0.238712 0.971090i \(-0.423275\pi\)
0.238712 + 0.971090i \(0.423275\pi\)
\(678\) 0 0
\(679\) −0.816654 −0.0313403
\(680\) −3.90833 −0.149877
\(681\) 0 0
\(682\) 15.4222 0.590547
\(683\) 32.7250 1.25219 0.626093 0.779748i \(-0.284653\pi\)
0.626093 + 0.779748i \(0.284653\pi\)
\(684\) 0 0
\(685\) −6.90833 −0.263954
\(686\) 4.21110 0.160781
\(687\) 0 0
\(688\) 5.21110 0.198671
\(689\) 0.972244 0.0370395
\(690\) 0 0
\(691\) 30.1833 1.14823 0.574114 0.818775i \(-0.305347\pi\)
0.574114 + 0.818775i \(0.305347\pi\)
\(692\) −23.3028 −0.885839
\(693\) 0 0
\(694\) 25.5416 0.969547
\(695\) 5.39445 0.204623
\(696\) 0 0
\(697\) 38.7250 1.46681
\(698\) −12.7889 −0.484067
\(699\) 0 0
\(700\) −0.302776 −0.0114438
\(701\) −42.9083 −1.62063 −0.810313 0.585998i \(-0.800703\pi\)
−0.810313 + 0.585998i \(0.800703\pi\)
\(702\) 0 0
\(703\) −39.2666 −1.48097
\(704\) 5.30278 0.199856
\(705\) 0 0
\(706\) −18.4222 −0.693329
\(707\) −1.39445 −0.0524436
\(708\) 0 0
\(709\) −41.1194 −1.54427 −0.772136 0.635457i \(-0.780812\pi\)
−0.772136 + 0.635457i \(0.780812\pi\)
\(710\) −12.6972 −0.476518
\(711\) 0 0
\(712\) 0 0
\(713\) 2.90833 0.108918
\(714\) 0 0
\(715\) 1.60555 0.0600442
\(716\) −16.6056 −0.620579
\(717\) 0 0
\(718\) 3.21110 0.119837
\(719\) −14.3028 −0.533404 −0.266702 0.963779i \(-0.585934\pi\)
−0.266702 + 0.963779i \(0.585934\pi\)
\(720\) 0 0
\(721\) 5.18335 0.193038
\(722\) 5.09167 0.189492
\(723\) 0 0
\(724\) −8.11943 −0.301756
\(725\) −4.60555 −0.171046
\(726\) 0 0
\(727\) −7.90833 −0.293304 −0.146652 0.989188i \(-0.546850\pi\)
−0.146652 + 0.989188i \(0.546850\pi\)
\(728\) 0.0916731 0.00339763
\(729\) 0 0
\(730\) −15.8167 −0.585401
\(731\) 20.3667 0.753289
\(732\) 0 0
\(733\) −13.6333 −0.503558 −0.251779 0.967785i \(-0.581016\pi\)
−0.251779 + 0.967785i \(0.581016\pi\)
\(734\) 29.2111 1.07820
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −21.2111 −0.781321
\(738\) 0 0
\(739\) −7.63331 −0.280796 −0.140398 0.990095i \(-0.544838\pi\)
−0.140398 + 0.990095i \(0.544838\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) 0.972244 0.0356922
\(743\) −7.33053 −0.268931 −0.134466 0.990918i \(-0.542932\pi\)
−0.134466 + 0.990918i \(0.542932\pi\)
\(744\) 0 0
\(745\) 9.69722 0.355279
\(746\) −2.60555 −0.0953960
\(747\) 0 0
\(748\) 20.7250 0.757780
\(749\) 1.39445 0.0509520
\(750\) 0 0
\(751\) 0.183346 0.00669040 0.00334520 0.999994i \(-0.498935\pi\)
0.00334520 + 0.999994i \(0.498935\pi\)
\(752\) −4.60555 −0.167947
\(753\) 0 0
\(754\) 1.39445 0.0507828
\(755\) 1.90833 0.0694511
\(756\) 0 0
\(757\) −1.21110 −0.0440183 −0.0220091 0.999758i \(-0.507006\pi\)
−0.0220091 + 0.999758i \(0.507006\pi\)
\(758\) 4.09167 0.148616
\(759\) 0 0
\(760\) 4.90833 0.178044
\(761\) −4.54163 −0.164634 −0.0823171 0.996606i \(-0.526232\pi\)
−0.0823171 + 0.996606i \(0.526232\pi\)
\(762\) 0 0
\(763\) −5.90833 −0.213896
\(764\) 1.39445 0.0504494
\(765\) 0 0
\(766\) 0 0
\(767\) −3.21110 −0.115946
\(768\) 0 0
\(769\) −41.2666 −1.48811 −0.744056 0.668117i \(-0.767100\pi\)
−0.744056 + 0.668117i \(0.767100\pi\)
\(770\) 1.60555 0.0578601
\(771\) 0 0
\(772\) 3.81665 0.137364
\(773\) −12.0000 −0.431610 −0.215805 0.976436i \(-0.569238\pi\)
−0.215805 + 0.976436i \(0.569238\pi\)
\(774\) 0 0
\(775\) 2.90833 0.104470
\(776\) 2.69722 0.0968247
\(777\) 0 0
\(778\) −20.9361 −0.750595
\(779\) −48.6333 −1.74247
\(780\) 0 0
\(781\) 67.3305 2.40928
\(782\) 3.90833 0.139761
\(783\) 0 0
\(784\) −6.90833 −0.246726
\(785\) 11.3944 0.406685
\(786\) 0 0
\(787\) −27.4500 −0.978485 −0.489243 0.872148i \(-0.662727\pi\)
−0.489243 + 0.872148i \(0.662727\pi\)
\(788\) −0.697224 −0.0248376
\(789\) 0 0
\(790\) −14.4222 −0.513119
\(791\) 3.76114 0.133731
\(792\) 0 0
\(793\) 1.97224 0.0700364
\(794\) −21.7250 −0.770991
\(795\) 0 0
\(796\) 8.42221 0.298517
\(797\) 19.8167 0.701942 0.350971 0.936386i \(-0.385852\pi\)
0.350971 + 0.936386i \(0.385852\pi\)
\(798\) 0 0
\(799\) −18.0000 −0.636794
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 1.39445 0.0492397
\(803\) 83.8722 2.95978
\(804\) 0 0
\(805\) 0.302776 0.0106714
\(806\) −0.880571 −0.0310168
\(807\) 0 0
\(808\) 4.60555 0.162023
\(809\) −18.2750 −0.642515 −0.321258 0.946992i \(-0.604106\pi\)
−0.321258 + 0.946992i \(0.604106\pi\)
\(810\) 0 0
\(811\) −4.97224 −0.174599 −0.0872995 0.996182i \(-0.527824\pi\)
−0.0872995 + 0.996182i \(0.527824\pi\)
\(812\) 1.39445 0.0489356
\(813\) 0 0
\(814\) 42.4222 1.48690
\(815\) −5.69722 −0.199565
\(816\) 0 0
\(817\) −25.5778 −0.894854
\(818\) −15.0917 −0.527668
\(819\) 0 0
\(820\) −9.90833 −0.346014
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) −0.788897 −0.0274992 −0.0137496 0.999905i \(-0.504377\pi\)
−0.0137496 + 0.999905i \(0.504377\pi\)
\(824\) −17.1194 −0.596384
\(825\) 0 0
\(826\) −3.21110 −0.111729
\(827\) −35.4500 −1.23272 −0.616358 0.787466i \(-0.711393\pi\)
−0.616358 + 0.787466i \(0.711393\pi\)
\(828\) 0 0
\(829\) 16.7889 0.583103 0.291551 0.956555i \(-0.405829\pi\)
0.291551 + 0.956555i \(0.405829\pi\)
\(830\) −3.21110 −0.111459
\(831\) 0 0
\(832\) −0.302776 −0.0104969
\(833\) −27.0000 −0.935495
\(834\) 0 0
\(835\) 21.2111 0.734040
\(836\) −26.0278 −0.900189
\(837\) 0 0
\(838\) 39.6333 1.36911
\(839\) 22.1833 0.765854 0.382927 0.923779i \(-0.374916\pi\)
0.382927 + 0.923779i \(0.374916\pi\)
\(840\) 0 0
\(841\) −7.78890 −0.268583
\(842\) 34.3028 1.18215
\(843\) 0 0
\(844\) −7.21110 −0.248216
\(845\) 12.9083 0.444060
\(846\) 0 0
\(847\) −5.18335 −0.178102
\(848\) −3.21110 −0.110270
\(849\) 0 0
\(850\) 3.90833 0.134055
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) 10.7250 0.367216 0.183608 0.983000i \(-0.441222\pi\)
0.183608 + 0.983000i \(0.441222\pi\)
\(854\) 1.97224 0.0674888
\(855\) 0 0
\(856\) −4.60555 −0.157415
\(857\) 33.6333 1.14889 0.574446 0.818543i \(-0.305218\pi\)
0.574446 + 0.818543i \(0.305218\pi\)
\(858\) 0 0
\(859\) −14.1833 −0.483930 −0.241965 0.970285i \(-0.577792\pi\)
−0.241965 + 0.970285i \(0.577792\pi\)
\(860\) −5.21110 −0.177697
\(861\) 0 0
\(862\) −20.2389 −0.689338
\(863\) −23.4500 −0.798246 −0.399123 0.916897i \(-0.630685\pi\)
−0.399123 + 0.916897i \(0.630685\pi\)
\(864\) 0 0
\(865\) 23.3028 0.792318
\(866\) −34.9083 −1.18623
\(867\) 0 0
\(868\) −0.880571 −0.0298885
\(869\) 76.4777 2.59433
\(870\) 0 0
\(871\) 1.21110 0.0410366
\(872\) 19.5139 0.660823
\(873\) 0 0
\(874\) −4.90833 −0.166027
\(875\) 0.302776 0.0102357
\(876\) 0 0
\(877\) 49.1749 1.66052 0.830260 0.557376i \(-0.188192\pi\)
0.830260 + 0.557376i \(0.188192\pi\)
\(878\) −18.3028 −0.617689
\(879\) 0 0
\(880\) −5.30278 −0.178757
\(881\) 31.2666 1.05340 0.526700 0.850052i \(-0.323429\pi\)
0.526700 + 0.850052i \(0.323429\pi\)
\(882\) 0 0
\(883\) 40.7250 1.37050 0.685252 0.728306i \(-0.259692\pi\)
0.685252 + 0.728306i \(0.259692\pi\)
\(884\) −1.18335 −0.0398002
\(885\) 0 0
\(886\) −35.5139 −1.19311
\(887\) 15.6333 0.524915 0.262458 0.964944i \(-0.415467\pi\)
0.262458 + 0.964944i \(0.415467\pi\)
\(888\) 0 0
\(889\) 3.57779 0.119995
\(890\) 0 0
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) 22.6056 0.756466
\(894\) 0 0
\(895\) 16.6056 0.555062
\(896\) −0.302776 −0.0101150
\(897\) 0 0
\(898\) 12.9083 0.430756
\(899\) −13.3944 −0.446730
\(900\) 0 0
\(901\) −12.5500 −0.418102
\(902\) 52.5416 1.74945
\(903\) 0 0
\(904\) −12.4222 −0.413156
\(905\) 8.11943 0.269899
\(906\) 0 0
\(907\) −30.6611 −1.01808 −0.509042 0.860742i \(-0.670000\pi\)
−0.509042 + 0.860742i \(0.670000\pi\)
\(908\) 7.39445 0.245393
\(909\) 0 0
\(910\) −0.0916731 −0.00303893
\(911\) 25.8167 0.855344 0.427672 0.903934i \(-0.359334\pi\)
0.427672 + 0.903934i \(0.359334\pi\)
\(912\) 0 0
\(913\) 17.0278 0.563536
\(914\) −3.57779 −0.118343
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) 0.972244 0.0321063
\(918\) 0 0
\(919\) 44.0000 1.45143 0.725713 0.687998i \(-0.241510\pi\)
0.725713 + 0.687998i \(0.241510\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 0 0
\(922\) −31.8167 −1.04783
\(923\) −3.84441 −0.126540
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) −25.6333 −0.842363
\(927\) 0 0
\(928\) −4.60555 −0.151185
\(929\) 57.6333 1.89089 0.945444 0.325785i \(-0.105629\pi\)
0.945444 + 0.325785i \(0.105629\pi\)
\(930\) 0 0
\(931\) 33.9083 1.11130
\(932\) −4.18335 −0.137030
\(933\) 0 0
\(934\) −19.8167 −0.648421
\(935\) −20.7250 −0.677779
\(936\) 0 0
\(937\) −44.9638 −1.46890 −0.734452 0.678660i \(-0.762561\pi\)
−0.734452 + 0.678660i \(0.762561\pi\)
\(938\) 1.21110 0.0395439
\(939\) 0 0
\(940\) 4.60555 0.150217
\(941\) 20.9361 0.682497 0.341248 0.939973i \(-0.389150\pi\)
0.341248 + 0.939973i \(0.389150\pi\)
\(942\) 0 0
\(943\) 9.90833 0.322660
\(944\) 10.6056 0.345181
\(945\) 0 0
\(946\) 27.6333 0.898436
\(947\) −41.9361 −1.36274 −0.681370 0.731939i \(-0.738615\pi\)
−0.681370 + 0.731939i \(0.738615\pi\)
\(948\) 0 0
\(949\) −4.78890 −0.155454
\(950\) −4.90833 −0.159247
\(951\) 0 0
\(952\) −1.18335 −0.0383525
\(953\) 1.66947 0.0540794 0.0270397 0.999634i \(-0.491392\pi\)
0.0270397 + 0.999634i \(0.491392\pi\)
\(954\) 0 0
\(955\) −1.39445 −0.0451233
\(956\) 9.21110 0.297908
\(957\) 0 0
\(958\) 30.0000 0.969256
\(959\) −2.09167 −0.0675436
\(960\) 0 0
\(961\) −22.5416 −0.727150
\(962\) −2.42221 −0.0780950
\(963\) 0 0
\(964\) 14.4222 0.464508
\(965\) −3.81665 −0.122862
\(966\) 0 0
\(967\) −5.39445 −0.173474 −0.0867369 0.996231i \(-0.527644\pi\)
−0.0867369 + 0.996231i \(0.527644\pi\)
\(968\) 17.1194 0.550239
\(969\) 0 0
\(970\) −2.69722 −0.0866027
\(971\) 27.9083 0.895621 0.447810 0.894129i \(-0.352204\pi\)
0.447810 + 0.894129i \(0.352204\pi\)
\(972\) 0 0
\(973\) 1.63331 0.0523614
\(974\) −11.8167 −0.378630
\(975\) 0 0
\(976\) −6.51388 −0.208504
\(977\) 11.5139 0.368362 0.184181 0.982892i \(-0.441037\pi\)
0.184181 + 0.982892i \(0.441037\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 6.90833 0.220678
\(981\) 0 0
\(982\) 25.8167 0.823843
\(983\) 19.5416 0.623281 0.311641 0.950200i \(-0.399121\pi\)
0.311641 + 0.950200i \(0.399121\pi\)
\(984\) 0 0
\(985\) 0.697224 0.0222154
\(986\) −18.0000 −0.573237
\(987\) 0 0
\(988\) 1.48612 0.0472798
\(989\) 5.21110 0.165703
\(990\) 0 0
\(991\) 24.3305 0.772885 0.386442 0.922314i \(-0.373704\pi\)
0.386442 + 0.922314i \(0.373704\pi\)
\(992\) 2.90833 0.0923395
\(993\) 0 0
\(994\) −3.84441 −0.121937
\(995\) −8.42221 −0.267002
\(996\) 0 0
\(997\) −31.2111 −0.988466 −0.494233 0.869330i \(-0.664551\pi\)
−0.494233 + 0.869330i \(0.664551\pi\)
\(998\) 11.6333 0.368246
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2070.2.a.w.1.1 2
3.2 odd 2 230.2.a.b.1.2 2
12.11 even 2 1840.2.a.j.1.1 2
15.2 even 4 1150.2.b.f.599.1 4
15.8 even 4 1150.2.b.f.599.4 4
15.14 odd 2 1150.2.a.m.1.1 2
24.5 odd 2 7360.2.a.bc.1.1 2
24.11 even 2 7360.2.a.bu.1.2 2
60.59 even 2 9200.2.a.ca.1.2 2
69.68 even 2 5290.2.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.b.1.2 2 3.2 odd 2
1150.2.a.m.1.1 2 15.14 odd 2
1150.2.b.f.599.1 4 15.2 even 4
1150.2.b.f.599.4 4 15.8 even 4
1840.2.a.j.1.1 2 12.11 even 2
2070.2.a.w.1.1 2 1.1 even 1 trivial
5290.2.a.j.1.2 2 69.68 even 2
7360.2.a.bc.1.1 2 24.5 odd 2
7360.2.a.bu.1.2 2 24.11 even 2
9200.2.a.ca.1.2 2 60.59 even 2