Properties

Label 2070.2.a.w.1.2
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 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 2070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.30278 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.30278 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.69722 q^{11} +3.30278 q^{13} +3.30278 q^{14} +1.00000 q^{16} -6.90833 q^{17} +5.90833 q^{19} -1.00000 q^{20} +1.69722 q^{22} +1.00000 q^{23} +1.00000 q^{25} +3.30278 q^{26} +3.30278 q^{28} +2.60555 q^{29} -7.90833 q^{31} +1.00000 q^{32} -6.90833 q^{34} -3.30278 q^{35} +8.00000 q^{37} +5.90833 q^{38} -1.00000 q^{40} -0.908327 q^{41} -9.21110 q^{43} +1.69722 q^{44} +1.00000 q^{46} +2.60555 q^{47} +3.90833 q^{49} +1.00000 q^{50} +3.30278 q^{52} +11.2111 q^{53} -1.69722 q^{55} +3.30278 q^{56} +2.60555 q^{58} +3.39445 q^{59} +11.5139 q^{61} -7.90833 q^{62} +1.00000 q^{64} -3.30278 q^{65} -4.00000 q^{67} -6.90833 q^{68} -3.30278 q^{70} +16.3028 q^{71} -5.81665 q^{73} +8.00000 q^{74} +5.90833 q^{76} +5.60555 q^{77} -14.4222 q^{79} -1.00000 q^{80} -0.908327 q^{82} -11.2111 q^{83} +6.90833 q^{85} -9.21110 q^{86} +1.69722 q^{88} +10.9083 q^{91} +1.00000 q^{92} +2.60555 q^{94} -5.90833 q^{95} +6.30278 q^{97} +3.90833 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} + 3q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} + 3q^{7} + 2q^{8} - 2q^{10} + 7q^{11} + 3q^{13} + 3q^{14} + 2q^{16} - 3q^{17} + q^{19} - 2q^{20} + 7q^{22} + 2q^{23} + 2q^{25} + 3q^{26} + 3q^{28} - 2q^{29} - 5q^{31} + 2q^{32} - 3q^{34} - 3q^{35} + 16q^{37} + q^{38} - 2q^{40} + 9q^{41} - 4q^{43} + 7q^{44} + 2q^{46} - 2q^{47} - 3q^{49} + 2q^{50} + 3q^{52} + 8q^{53} - 7q^{55} + 3q^{56} - 2q^{58} + 14q^{59} + 5q^{61} - 5q^{62} + 2q^{64} - 3q^{65} - 8q^{67} - 3q^{68} - 3q^{70} + 29q^{71} + 10q^{73} + 16q^{74} + q^{76} + 4q^{77} - 2q^{80} + 9q^{82} - 8q^{83} + 3q^{85} - 4q^{86} + 7q^{88} + 11q^{91} + 2q^{92} - 2q^{94} - q^{95} + 9q^{97} - 3q^{98} + O(q^{100}) \)

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
\(7\) 3.30278 1.24833 0.624166 0.781292i \(-0.285439\pi\)
0.624166 + 0.781292i \(0.285439\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 1.69722 0.511732 0.255866 0.966712i \(-0.417639\pi\)
0.255866 + 0.966712i \(0.417639\pi\)
\(12\) 0 0
\(13\) 3.30278 0.916025 0.458013 0.888946i \(-0.348561\pi\)
0.458013 + 0.888946i \(0.348561\pi\)
\(14\) 3.30278 0.882704
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.90833 −1.67552 −0.837758 0.546042i \(-0.816134\pi\)
−0.837758 + 0.546042i \(0.816134\pi\)
\(18\) 0 0
\(19\) 5.90833 1.35546 0.677732 0.735309i \(-0.262963\pi\)
0.677732 + 0.735309i \(0.262963\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 1.69722 0.361849
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 3.30278 0.647728
\(27\) 0 0
\(28\) 3.30278 0.624166
\(29\) 2.60555 0.483839 0.241919 0.970296i \(-0.422223\pi\)
0.241919 + 0.970296i \(0.422223\pi\)
\(30\) 0 0
\(31\) −7.90833 −1.42038 −0.710189 0.704011i \(-0.751390\pi\)
−0.710189 + 0.704011i \(0.751390\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −6.90833 −1.18477
\(35\) −3.30278 −0.558271
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 5.90833 0.958457
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −0.908327 −0.141857 −0.0709284 0.997481i \(-0.522596\pi\)
−0.0709284 + 0.997481i \(0.522596\pi\)
\(42\) 0 0
\(43\) −9.21110 −1.40468 −0.702340 0.711842i \(-0.747861\pi\)
−0.702340 + 0.711842i \(0.747861\pi\)
\(44\) 1.69722 0.255866
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 2.60555 0.380059 0.190029 0.981778i \(-0.439142\pi\)
0.190029 + 0.981778i \(0.439142\pi\)
\(48\) 0 0
\(49\) 3.90833 0.558332
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 3.30278 0.458013
\(53\) 11.2111 1.53996 0.769982 0.638066i \(-0.220265\pi\)
0.769982 + 0.638066i \(0.220265\pi\)
\(54\) 0 0
\(55\) −1.69722 −0.228854
\(56\) 3.30278 0.441352
\(57\) 0 0
\(58\) 2.60555 0.342126
\(59\) 3.39445 0.441920 0.220960 0.975283i \(-0.429081\pi\)
0.220960 + 0.975283i \(0.429081\pi\)
\(60\) 0 0
\(61\) 11.5139 1.47420 0.737101 0.675783i \(-0.236194\pi\)
0.737101 + 0.675783i \(0.236194\pi\)
\(62\) −7.90833 −1.00436
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.30278 −0.409659
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −6.90833 −0.837758
\(69\) 0 0
\(70\) −3.30278 −0.394757
\(71\) 16.3028 1.93478 0.967392 0.253285i \(-0.0815110\pi\)
0.967392 + 0.253285i \(0.0815110\pi\)
\(72\) 0 0
\(73\) −5.81665 −0.680788 −0.340394 0.940283i \(-0.610560\pi\)
−0.340394 + 0.940283i \(0.610560\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 5.90833 0.677732
\(77\) 5.60555 0.638812
\(78\) 0 0
\(79\) −14.4222 −1.62262 −0.811312 0.584613i \(-0.801246\pi\)
−0.811312 + 0.584613i \(0.801246\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −0.908327 −0.100308
\(83\) −11.2111 −1.23058 −0.615289 0.788301i \(-0.710961\pi\)
−0.615289 + 0.788301i \(0.710961\pi\)
\(84\) 0 0
\(85\) 6.90833 0.749313
\(86\) −9.21110 −0.993259
\(87\) 0 0
\(88\) 1.69722 0.180925
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 10.9083 1.14350
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 2.60555 0.268742
\(95\) −5.90833 −0.606182
\(96\) 0 0
\(97\) 6.30278 0.639950 0.319975 0.947426i \(-0.396325\pi\)
0.319975 + 0.947426i \(0.396325\pi\)
\(98\) 3.90833 0.394801
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −2.60555 −0.259262 −0.129631 0.991562i \(-0.541379\pi\)
−0.129631 + 0.991562i \(0.541379\pi\)
\(102\) 0 0
\(103\) 8.11943 0.800031 0.400016 0.916508i \(-0.369005\pi\)
0.400016 + 0.916508i \(0.369005\pi\)
\(104\) 3.30278 0.323864
\(105\) 0 0
\(106\) 11.2111 1.08892
\(107\) 2.60555 0.251888 0.125944 0.992037i \(-0.459804\pi\)
0.125944 + 0.992037i \(0.459804\pi\)
\(108\) 0 0
\(109\) 1.48612 0.142345 0.0711723 0.997464i \(-0.477326\pi\)
0.0711723 + 0.997464i \(0.477326\pi\)
\(110\) −1.69722 −0.161824
\(111\) 0 0
\(112\) 3.30278 0.312083
\(113\) 16.4222 1.54487 0.772436 0.635093i \(-0.219038\pi\)
0.772436 + 0.635093i \(0.219038\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 2.60555 0.241919
\(117\) 0 0
\(118\) 3.39445 0.312484
\(119\) −22.8167 −2.09160
\(120\) 0 0
\(121\) −8.11943 −0.738130
\(122\) 11.5139 1.04242
\(123\) 0 0
\(124\) −7.90833 −0.710189
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 9.81665 0.871087 0.435544 0.900168i \(-0.356556\pi\)
0.435544 + 0.900168i \(0.356556\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −3.30278 −0.289673
\(131\) 11.2111 0.979519 0.489759 0.871858i \(-0.337085\pi\)
0.489759 + 0.871858i \(0.337085\pi\)
\(132\) 0 0
\(133\) 19.5139 1.69207
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −6.90833 −0.592384
\(137\) −3.90833 −0.333911 −0.166955 0.985964i \(-0.553394\pi\)
−0.166955 + 0.985964i \(0.553394\pi\)
\(138\) 0 0
\(139\) −12.6056 −1.06919 −0.534594 0.845109i \(-0.679536\pi\)
−0.534594 + 0.845109i \(0.679536\pi\)
\(140\) −3.30278 −0.279135
\(141\) 0 0
\(142\) 16.3028 1.36810
\(143\) 5.60555 0.468760
\(144\) 0 0
\(145\) −2.60555 −0.216379
\(146\) −5.81665 −0.481390
\(147\) 0 0
\(148\) 8.00000 0.657596
\(149\) −13.3028 −1.08981 −0.544903 0.838499i \(-0.683433\pi\)
−0.544903 + 0.838499i \(0.683433\pi\)
\(150\) 0 0
\(151\) 8.90833 0.724949 0.362475 0.931994i \(-0.381932\pi\)
0.362475 + 0.931994i \(0.381932\pi\)
\(152\) 5.90833 0.479229
\(153\) 0 0
\(154\) 5.60555 0.451708
\(155\) 7.90833 0.635212
\(156\) 0 0
\(157\) −18.6056 −1.48488 −0.742442 0.669910i \(-0.766333\pi\)
−0.742442 + 0.669910i \(0.766333\pi\)
\(158\) −14.4222 −1.14737
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 3.30278 0.260295
\(162\) 0 0
\(163\) 9.30278 0.728650 0.364325 0.931272i \(-0.381300\pi\)
0.364325 + 0.931272i \(0.381300\pi\)
\(164\) −0.908327 −0.0709284
\(165\) 0 0
\(166\) −11.2111 −0.870150
\(167\) −6.78890 −0.525341 −0.262670 0.964886i \(-0.584603\pi\)
−0.262670 + 0.964886i \(0.584603\pi\)
\(168\) 0 0
\(169\) −2.09167 −0.160898
\(170\) 6.90833 0.529844
\(171\) 0 0
\(172\) −9.21110 −0.702340
\(173\) −19.6972 −1.49755 −0.748776 0.662823i \(-0.769358\pi\)
−0.748776 + 0.662823i \(0.769358\pi\)
\(174\) 0 0
\(175\) 3.30278 0.249666
\(176\) 1.69722 0.127933
\(177\) 0 0
\(178\) 0 0
\(179\) −9.39445 −0.702174 −0.351087 0.936343i \(-0.614188\pi\)
−0.351087 + 0.936343i \(0.614188\pi\)
\(180\) 0 0
\(181\) 17.1194 1.27248 0.636239 0.771492i \(-0.280489\pi\)
0.636239 + 0.771492i \(0.280489\pi\)
\(182\) 10.9083 0.808579
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) −11.7250 −0.857416
\(188\) 2.60555 0.190029
\(189\) 0 0
\(190\) −5.90833 −0.428635
\(191\) 8.60555 0.622676 0.311338 0.950299i \(-0.399223\pi\)
0.311338 + 0.950299i \(0.399223\pi\)
\(192\) 0 0
\(193\) −17.8167 −1.28247 −0.641235 0.767344i \(-0.721578\pi\)
−0.641235 + 0.767344i \(0.721578\pi\)
\(194\) 6.30278 0.452513
\(195\) 0 0
\(196\) 3.90833 0.279166
\(197\) −4.30278 −0.306560 −0.153280 0.988183i \(-0.548984\pi\)
−0.153280 + 0.988183i \(0.548984\pi\)
\(198\) 0 0
\(199\) −20.4222 −1.44769 −0.723846 0.689962i \(-0.757627\pi\)
−0.723846 + 0.689962i \(0.757627\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −2.60555 −0.183326
\(203\) 8.60555 0.603991
\(204\) 0 0
\(205\) 0.908327 0.0634403
\(206\) 8.11943 0.565707
\(207\) 0 0
\(208\) 3.30278 0.229006
\(209\) 10.0278 0.693634
\(210\) 0 0
\(211\) 7.21110 0.496433 0.248216 0.968705i \(-0.420156\pi\)
0.248216 + 0.968705i \(0.420156\pi\)
\(212\) 11.2111 0.769982
\(213\) 0 0
\(214\) 2.60555 0.178112
\(215\) 9.21110 0.628192
\(216\) 0 0
\(217\) −26.1194 −1.77310
\(218\) 1.48612 0.100653
\(219\) 0 0
\(220\) −1.69722 −0.114427
\(221\) −22.8167 −1.53481
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 3.30278 0.220676
\(225\) 0 0
\(226\) 16.4222 1.09239
\(227\) 14.6056 0.969404 0.484702 0.874679i \(-0.338928\pi\)
0.484702 + 0.874679i \(0.338928\pi\)
\(228\) 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
\(232\) 2.60555 0.171063
\(233\) −25.8167 −1.69131 −0.845653 0.533734i \(-0.820788\pi\)
−0.845653 + 0.533734i \(0.820788\pi\)
\(234\) 0 0
\(235\) −2.60555 −0.169967
\(236\) 3.39445 0.220960
\(237\) 0 0
\(238\) −22.8167 −1.47898
\(239\) −5.21110 −0.337078 −0.168539 0.985695i \(-0.553905\pi\)
−0.168539 + 0.985695i \(0.553905\pi\)
\(240\) 0 0
\(241\) −14.4222 −0.929016 −0.464508 0.885569i \(-0.653769\pi\)
−0.464508 + 0.885569i \(0.653769\pi\)
\(242\) −8.11943 −0.521937
\(243\) 0 0
\(244\) 11.5139 0.737101
\(245\) −3.90833 −0.249694
\(246\) 0 0
\(247\) 19.5139 1.24164
\(248\) −7.90833 −0.502179
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −12.5139 −0.789869 −0.394934 0.918709i \(-0.629233\pi\)
−0.394934 + 0.918709i \(0.629233\pi\)
\(252\) 0 0
\(253\) 1.69722 0.106704
\(254\) 9.81665 0.615952
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 1.81665 0.113320 0.0566599 0.998394i \(-0.481955\pi\)
0.0566599 + 0.998394i \(0.481955\pi\)
\(258\) 0 0
\(259\) 26.4222 1.64180
\(260\) −3.30278 −0.204829
\(261\) 0 0
\(262\) 11.2111 0.692624
\(263\) −3.51388 −0.216675 −0.108338 0.994114i \(-0.534553\pi\)
−0.108338 + 0.994114i \(0.534553\pi\)
\(264\) 0 0
\(265\) −11.2111 −0.688693
\(266\) 19.5139 1.19647
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −4.18335 −0.255063 −0.127532 0.991835i \(-0.540705\pi\)
−0.127532 + 0.991835i \(0.540705\pi\)
\(270\) 0 0
\(271\) −2.69722 −0.163845 −0.0819224 0.996639i \(-0.526106\pi\)
−0.0819224 + 0.996639i \(0.526106\pi\)
\(272\) −6.90833 −0.418879
\(273\) 0 0
\(274\) −3.90833 −0.236111
\(275\) 1.69722 0.102346
\(276\) 0 0
\(277\) −27.2111 −1.63496 −0.817478 0.575959i \(-0.804629\pi\)
−0.817478 + 0.575959i \(0.804629\pi\)
\(278\) −12.6056 −0.756031
\(279\) 0 0
\(280\) −3.30278 −0.197379
\(281\) 26.6056 1.58715 0.793577 0.608470i \(-0.208216\pi\)
0.793577 + 0.608470i \(0.208216\pi\)
\(282\) 0 0
\(283\) 2.00000 0.118888 0.0594438 0.998232i \(-0.481067\pi\)
0.0594438 + 0.998232i \(0.481067\pi\)
\(284\) 16.3028 0.967392
\(285\) 0 0
\(286\) 5.60555 0.331463
\(287\) −3.00000 −0.177084
\(288\) 0 0
\(289\) 30.7250 1.80735
\(290\) −2.60555 −0.153003
\(291\) 0 0
\(292\) −5.81665 −0.340394
\(293\) −23.2111 −1.35601 −0.678004 0.735059i \(-0.737155\pi\)
−0.678004 + 0.735059i \(0.737155\pi\)
\(294\) 0 0
\(295\) −3.39445 −0.197632
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) −13.3028 −0.770609
\(299\) 3.30278 0.191004
\(300\) 0 0
\(301\) −30.4222 −1.75351
\(302\) 8.90833 0.512617
\(303\) 0 0
\(304\) 5.90833 0.338866
\(305\) −11.5139 −0.659283
\(306\) 0 0
\(307\) −11.6972 −0.667596 −0.333798 0.942645i \(-0.608330\pi\)
−0.333798 + 0.942645i \(0.608330\pi\)
\(308\) 5.60555 0.319406
\(309\) 0 0
\(310\) 7.90833 0.449163
\(311\) −22.4222 −1.27145 −0.635723 0.771917i \(-0.719298\pi\)
−0.635723 + 0.771917i \(0.719298\pi\)
\(312\) 0 0
\(313\) 19.7250 1.11492 0.557461 0.830203i \(-0.311776\pi\)
0.557461 + 0.830203i \(0.311776\pi\)
\(314\) −18.6056 −1.04997
\(315\) 0 0
\(316\) −14.4222 −0.811312
\(317\) −17.7250 −0.995534 −0.497767 0.867311i \(-0.665847\pi\)
−0.497767 + 0.867311i \(0.665847\pi\)
\(318\) 0 0
\(319\) 4.42221 0.247596
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 3.30278 0.184056
\(323\) −40.8167 −2.27110
\(324\) 0 0
\(325\) 3.30278 0.183205
\(326\) 9.30278 0.515233
\(327\) 0 0
\(328\) −0.908327 −0.0501540
\(329\) 8.60555 0.474439
\(330\) 0 0
\(331\) 16.6056 0.912724 0.456362 0.889794i \(-0.349152\pi\)
0.456362 + 0.889794i \(0.349152\pi\)
\(332\) −11.2111 −0.615289
\(333\) 0 0
\(334\) −6.78890 −0.371472
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) −22.5139 −1.22641 −0.613205 0.789924i \(-0.710120\pi\)
−0.613205 + 0.789924i \(0.710120\pi\)
\(338\) −2.09167 −0.113772
\(339\) 0 0
\(340\) 6.90833 0.374657
\(341\) −13.4222 −0.726853
\(342\) 0 0
\(343\) −10.2111 −0.551348
\(344\) −9.21110 −0.496629
\(345\) 0 0
\(346\) −19.6972 −1.05893
\(347\) −28.5416 −1.53220 −0.766098 0.642724i \(-0.777804\pi\)
−0.766098 + 0.642724i \(0.777804\pi\)
\(348\) 0 0
\(349\) −27.2111 −1.45658 −0.728288 0.685271i \(-0.759684\pi\)
−0.728288 + 0.685271i \(0.759684\pi\)
\(350\) 3.30278 0.176541
\(351\) 0 0
\(352\) 1.69722 0.0904624
\(353\) 10.4222 0.554718 0.277359 0.960766i \(-0.410541\pi\)
0.277359 + 0.960766i \(0.410541\pi\)
\(354\) 0 0
\(355\) −16.3028 −0.865261
\(356\) 0 0
\(357\) 0 0
\(358\) −9.39445 −0.496512
\(359\) −11.2111 −0.591699 −0.295850 0.955235i \(-0.595603\pi\)
−0.295850 + 0.955235i \(0.595603\pi\)
\(360\) 0 0
\(361\) 15.9083 0.837280
\(362\) 17.1194 0.899777
\(363\) 0 0
\(364\) 10.9083 0.571752
\(365\) 5.81665 0.304458
\(366\) 0 0
\(367\) 14.7889 0.771974 0.385987 0.922504i \(-0.373861\pi\)
0.385987 + 0.922504i \(0.373861\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) −8.00000 −0.415900
\(371\) 37.0278 1.92239
\(372\) 0 0
\(373\) 4.60555 0.238466 0.119233 0.992866i \(-0.461956\pi\)
0.119233 + 0.992866i \(0.461956\pi\)
\(374\) −11.7250 −0.606284
\(375\) 0 0
\(376\) 2.60555 0.134371
\(377\) 8.60555 0.443208
\(378\) 0 0
\(379\) 14.9083 0.765789 0.382895 0.923792i \(-0.374927\pi\)
0.382895 + 0.923792i \(0.374927\pi\)
\(380\) −5.90833 −0.303091
\(381\) 0 0
\(382\) 8.60555 0.440298
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −5.60555 −0.285685
\(386\) −17.8167 −0.906844
\(387\) 0 0
\(388\) 6.30278 0.319975
\(389\) 25.9361 1.31501 0.657506 0.753449i \(-0.271612\pi\)
0.657506 + 0.753449i \(0.271612\pi\)
\(390\) 0 0
\(391\) −6.90833 −0.349369
\(392\) 3.90833 0.197400
\(393\) 0 0
\(394\) −4.30278 −0.216771
\(395\) 14.4222 0.725660
\(396\) 0 0
\(397\) 10.7250 0.538271 0.269136 0.963102i \(-0.413262\pi\)
0.269136 + 0.963102i \(0.413262\pi\)
\(398\) −20.4222 −1.02367
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 8.60555 0.429741 0.214870 0.976643i \(-0.431067\pi\)
0.214870 + 0.976643i \(0.431067\pi\)
\(402\) 0 0
\(403\) −26.1194 −1.30110
\(404\) −2.60555 −0.129631
\(405\) 0 0
\(406\) 8.60555 0.427086
\(407\) 13.5778 0.673026
\(408\) 0 0
\(409\) −25.9083 −1.28108 −0.640542 0.767923i \(-0.721290\pi\)
−0.640542 + 0.767923i \(0.721290\pi\)
\(410\) 0.908327 0.0448591
\(411\) 0 0
\(412\) 8.11943 0.400016
\(413\) 11.2111 0.551662
\(414\) 0 0
\(415\) 11.2111 0.550331
\(416\) 3.30278 0.161932
\(417\) 0 0
\(418\) 10.0278 0.490474
\(419\) −3.63331 −0.177499 −0.0887493 0.996054i \(-0.528287\pi\)
−0.0887493 + 0.996054i \(0.528287\pi\)
\(420\) 0 0
\(421\) 30.6972 1.49609 0.748046 0.663647i \(-0.230992\pi\)
0.748046 + 0.663647i \(0.230992\pi\)
\(422\) 7.21110 0.351031
\(423\) 0 0
\(424\) 11.2111 0.544459
\(425\) −6.90833 −0.335103
\(426\) 0 0
\(427\) 38.0278 1.84029
\(428\) 2.60555 0.125944
\(429\) 0 0
\(430\) 9.21110 0.444199
\(431\) 30.2389 1.45655 0.728277 0.685283i \(-0.240321\pi\)
0.728277 + 0.685283i \(0.240321\pi\)
\(432\) 0 0
\(433\) −24.0917 −1.15777 −0.578886 0.815409i \(-0.696512\pi\)
−0.578886 + 0.815409i \(0.696512\pi\)
\(434\) −26.1194 −1.25377
\(435\) 0 0
\(436\) 1.48612 0.0711723
\(437\) 5.90833 0.282634
\(438\) 0 0
\(439\) −14.6972 −0.701460 −0.350730 0.936477i \(-0.614067\pi\)
−0.350730 + 0.936477i \(0.614067\pi\)
\(440\) −1.69722 −0.0809120
\(441\) 0 0
\(442\) −22.8167 −1.08528
\(443\) −17.4861 −0.830791 −0.415395 0.909641i \(-0.636357\pi\)
−0.415395 + 0.909641i \(0.636357\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) 3.30278 0.156041
\(449\) 2.09167 0.0987122 0.0493561 0.998781i \(-0.484283\pi\)
0.0493561 + 0.998781i \(0.484283\pi\)
\(450\) 0 0
\(451\) −1.54163 −0.0725927
\(452\) 16.4222 0.772436
\(453\) 0 0
\(454\) 14.6056 0.685472
\(455\) −10.9083 −0.511390
\(456\) 0 0
\(457\) −32.4222 −1.51665 −0.758323 0.651879i \(-0.773981\pi\)
−0.758323 + 0.651879i \(0.773981\pi\)
\(458\) 2.00000 0.0934539
\(459\) 0 0
\(460\) −1.00000 −0.0466252
\(461\) −10.1833 −0.474286 −0.237143 0.971475i \(-0.576211\pi\)
−0.237143 + 0.971475i \(0.576211\pi\)
\(462\) 0 0
\(463\) 17.6333 0.819489 0.409745 0.912200i \(-0.365618\pi\)
0.409745 + 0.912200i \(0.365618\pi\)
\(464\) 2.60555 0.120960
\(465\) 0 0
\(466\) −25.8167 −1.19593
\(467\) 1.81665 0.0840647 0.0420324 0.999116i \(-0.486617\pi\)
0.0420324 + 0.999116i \(0.486617\pi\)
\(468\) 0 0
\(469\) −13.2111 −0.610032
\(470\) −2.60555 −0.120185
\(471\) 0 0
\(472\) 3.39445 0.156242
\(473\) −15.6333 −0.718820
\(474\) 0 0
\(475\) 5.90833 0.271093
\(476\) −22.8167 −1.04580
\(477\) 0 0
\(478\) −5.21110 −0.238350
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) 26.4222 1.20475
\(482\) −14.4222 −0.656913
\(483\) 0 0
\(484\) −8.11943 −0.369065
\(485\) −6.30278 −0.286194
\(486\) 0 0
\(487\) 9.81665 0.444835 0.222418 0.974952i \(-0.428605\pi\)
0.222418 + 0.974952i \(0.428605\pi\)
\(488\) 11.5139 0.521209
\(489\) 0 0
\(490\) −3.90833 −0.176560
\(491\) 4.18335 0.188792 0.0943959 0.995535i \(-0.469908\pi\)
0.0943959 + 0.995535i \(0.469908\pi\)
\(492\) 0 0
\(493\) −18.0000 −0.810679
\(494\) 19.5139 0.877971
\(495\) 0 0
\(496\) −7.90833 −0.355094
\(497\) 53.8444 2.41525
\(498\) 0 0
\(499\) −31.6333 −1.41610 −0.708051 0.706162i \(-0.750425\pi\)
−0.708051 + 0.706162i \(0.750425\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −12.5139 −0.558522
\(503\) −29.7250 −1.32537 −0.662686 0.748898i \(-0.730583\pi\)
−0.662686 + 0.748898i \(0.730583\pi\)
\(504\) 0 0
\(505\) 2.60555 0.115946
\(506\) 1.69722 0.0754508
\(507\) 0 0
\(508\) 9.81665 0.435544
\(509\) 35.4500 1.57129 0.785646 0.618676i \(-0.212331\pi\)
0.785646 + 0.618676i \(0.212331\pi\)
\(510\) 0 0
\(511\) −19.2111 −0.849849
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 1.81665 0.0801292
\(515\) −8.11943 −0.357785
\(516\) 0 0
\(517\) 4.42221 0.194488
\(518\) 26.4222 1.16093
\(519\) 0 0
\(520\) −3.30278 −0.144836
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) −20.4222 −0.893001 −0.446500 0.894783i \(-0.647330\pi\)
−0.446500 + 0.894783i \(0.647330\pi\)
\(524\) 11.2111 0.489759
\(525\) 0 0
\(526\) −3.51388 −0.153212
\(527\) 54.6333 2.37986
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −11.2111 −0.486979
\(531\) 0 0
\(532\) 19.5139 0.846034
\(533\) −3.00000 −0.129944
\(534\) 0 0
\(535\) −2.60555 −0.112648
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) −4.18335 −0.180357
\(539\) 6.63331 0.285717
\(540\) 0 0
\(541\) 28.8444 1.24012 0.620059 0.784555i \(-0.287109\pi\)
0.620059 + 0.784555i \(0.287109\pi\)
\(542\) −2.69722 −0.115856
\(543\) 0 0
\(544\) −6.90833 −0.296192
\(545\) −1.48612 −0.0636585
\(546\) 0 0
\(547\) −10.5139 −0.449541 −0.224770 0.974412i \(-0.572163\pi\)
−0.224770 + 0.974412i \(0.572163\pi\)
\(548\) −3.90833 −0.166955
\(549\) 0 0
\(550\) 1.69722 0.0723699
\(551\) 15.3944 0.655826
\(552\) 0 0
\(553\) −47.6333 −2.02557
\(554\) −27.2111 −1.15609
\(555\) 0 0
\(556\) −12.6056 −0.534594
\(557\) −22.4222 −0.950059 −0.475030 0.879970i \(-0.657563\pi\)
−0.475030 + 0.879970i \(0.657563\pi\)
\(558\) 0 0
\(559\) −30.4222 −1.28672
\(560\) −3.30278 −0.139568
\(561\) 0 0
\(562\) 26.6056 1.12229
\(563\) 3.63331 0.153126 0.0765628 0.997065i \(-0.475605\pi\)
0.0765628 + 0.997065i \(0.475605\pi\)
\(564\) 0 0
\(565\) −16.4222 −0.690887
\(566\) 2.00000 0.0840663
\(567\) 0 0
\(568\) 16.3028 0.684049
\(569\) −28.4222 −1.19152 −0.595760 0.803162i \(-0.703149\pi\)
−0.595760 + 0.803162i \(0.703149\pi\)
\(570\) 0 0
\(571\) −16.1194 −0.674577 −0.337289 0.941401i \(-0.609510\pi\)
−0.337289 + 0.941401i \(0.609510\pi\)
\(572\) 5.60555 0.234380
\(573\) 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 \(0.486745\pi\)
\(578\) 30.7250 1.27799
\(579\) 0 0
\(580\) −2.60555 −0.108190
\(581\) −37.0278 −1.53617
\(582\) 0 0
\(583\) 19.0278 0.788049
\(584\) −5.81665 −0.240695
\(585\) 0 0
\(586\) −23.2111 −0.958842
\(587\) −16.5416 −0.682746 −0.341373 0.939928i \(-0.610892\pi\)
−0.341373 + 0.939928i \(0.610892\pi\)
\(588\) 0 0
\(589\) −46.7250 −1.92527
\(590\) −3.39445 −0.139747
\(591\) 0 0
\(592\) 8.00000 0.328798
\(593\) 1.81665 0.0746010 0.0373005 0.999304i \(-0.488124\pi\)
0.0373005 + 0.999304i \(0.488124\pi\)
\(594\) 0 0
\(595\) 22.8167 0.935392
\(596\) −13.3028 −0.544903
\(597\) 0 0
\(598\) 3.30278 0.135061
\(599\) 35.3305 1.44357 0.721783 0.692119i \(-0.243323\pi\)
0.721783 + 0.692119i \(0.243323\pi\)
\(600\) 0 0
\(601\) 42.9361 1.75140 0.875700 0.482856i \(-0.160401\pi\)
0.875700 + 0.482856i \(0.160401\pi\)
\(602\) −30.4222 −1.23992
\(603\) 0 0
\(604\) 8.90833 0.362475
\(605\) 8.11943 0.330102
\(606\) 0 0
\(607\) 46.0555 1.86934 0.934668 0.355522i \(-0.115697\pi\)
0.934668 + 0.355522i \(0.115697\pi\)
\(608\) 5.90833 0.239614
\(609\) 0 0
\(610\) −11.5139 −0.466183
\(611\) 8.60555 0.348143
\(612\) 0 0
\(613\) 3.57779 0.144506 0.0722529 0.997386i \(-0.476981\pi\)
0.0722529 + 0.997386i \(0.476981\pi\)
\(614\) −11.6972 −0.472062
\(615\) 0 0
\(616\) 5.60555 0.225854
\(617\) −18.9083 −0.761221 −0.380610 0.924736i \(-0.624286\pi\)
−0.380610 + 0.924736i \(0.624286\pi\)
\(618\) 0 0
\(619\) −12.3305 −0.495606 −0.247803 0.968810i \(-0.579709\pi\)
−0.247803 + 0.968810i \(0.579709\pi\)
\(620\) 7.90833 0.317606
\(621\) 0 0
\(622\) −22.4222 −0.899049
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 19.7250 0.788369
\(627\) 0 0
\(628\) −18.6056 −0.742442
\(629\) −55.2666 −2.20362
\(630\) 0 0
\(631\) 23.3944 0.931318 0.465659 0.884964i \(-0.345817\pi\)
0.465659 + 0.884964i \(0.345817\pi\)
\(632\) −14.4222 −0.573685
\(633\) 0 0
\(634\) −17.7250 −0.703949
\(635\) −9.81665 −0.389562
\(636\) 0 0
\(637\) 12.9083 0.511447
\(638\) 4.42221 0.175077
\(639\) 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
\(643\) −34.2389 −1.35025 −0.675124 0.737704i \(-0.735910\pi\)
−0.675124 + 0.737704i \(0.735910\pi\)
\(644\) 3.30278 0.130148
\(645\) 0 0
\(646\) −40.8167 −1.60591
\(647\) −26.8444 −1.05536 −0.527681 0.849442i \(-0.676938\pi\)
−0.527681 + 0.849442i \(0.676938\pi\)
\(648\) 0 0
\(649\) 5.76114 0.226145
\(650\) 3.30278 0.129546
\(651\) 0 0
\(652\) 9.30278 0.364325
\(653\) −41.7250 −1.63282 −0.816412 0.577469i \(-0.804040\pi\)
−0.816412 + 0.577469i \(0.804040\pi\)
\(654\) 0 0
\(655\) −11.2111 −0.438054
\(656\) −0.908327 −0.0354642
\(657\) 0 0
\(658\) 8.60555 0.335479
\(659\) −15.6333 −0.608987 −0.304494 0.952514i \(-0.598487\pi\)
−0.304494 + 0.952514i \(0.598487\pi\)
\(660\) 0 0
\(661\) −34.9083 −1.35778 −0.678888 0.734242i \(-0.737538\pi\)
−0.678888 + 0.734242i \(0.737538\pi\)
\(662\) 16.6056 0.645393
\(663\) 0 0
\(664\) −11.2111 −0.435075
\(665\) −19.5139 −0.756716
\(666\) 0 0
\(667\) 2.60555 0.100887
\(668\) −6.78890 −0.262670
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) 19.5416 0.754396
\(672\) 0 0
\(673\) −37.6333 −1.45066 −0.725329 0.688403i \(-0.758312\pi\)
−0.725329 + 0.688403i \(0.758312\pi\)
\(674\) −22.5139 −0.867202
\(675\) 0 0
\(676\) −2.09167 −0.0804490
\(677\) −16.4222 −0.631157 −0.315578 0.948900i \(-0.602198\pi\)
−0.315578 + 0.948900i \(0.602198\pi\)
\(678\) 0 0
\(679\) 20.8167 0.798870
\(680\) 6.90833 0.264922
\(681\) 0 0
\(682\) −13.4222 −0.513963
\(683\) 0.275019 0.0105233 0.00526166 0.999986i \(-0.498325\pi\)
0.00526166 + 0.999986i \(0.498325\pi\)
\(684\) 0 0
\(685\) 3.90833 0.149329
\(686\) −10.2111 −0.389862
\(687\) 0 0
\(688\) −9.21110 −0.351170
\(689\) 37.0278 1.41065
\(690\) 0 0
\(691\) 51.8167 1.97120 0.985599 0.169098i \(-0.0540855\pi\)
0.985599 + 0.169098i \(0.0540855\pi\)
\(692\) −19.6972 −0.748776
\(693\) 0 0
\(694\) −28.5416 −1.08343
\(695\) 12.6056 0.478156
\(696\) 0 0
\(697\) 6.27502 0.237683
\(698\) −27.2111 −1.02996
\(699\) 0 0
\(700\) 3.30278 0.124833
\(701\) −32.0917 −1.21209 −0.606043 0.795432i \(-0.707244\pi\)
−0.606043 + 0.795432i \(0.707244\pi\)
\(702\) 0 0
\(703\) 47.2666 1.78269
\(704\) 1.69722 0.0639665
\(705\) 0 0
\(706\) 10.4222 0.392245
\(707\) −8.60555 −0.323645
\(708\) 0 0
\(709\) −15.8806 −0.596407 −0.298204 0.954502i \(-0.596387\pi\)
−0.298204 + 0.954502i \(0.596387\pi\)
\(710\) −16.3028 −0.611832
\(711\) 0 0
\(712\) 0 0
\(713\) −7.90833 −0.296169
\(714\) 0 0
\(715\) −5.60555 −0.209636
\(716\) −9.39445 −0.351087
\(717\) 0 0
\(718\) −11.2111 −0.418395
\(719\) −10.6972 −0.398939 −0.199470 0.979904i \(-0.563922\pi\)
−0.199470 + 0.979904i \(0.563922\pi\)
\(720\) 0 0
\(721\) 26.8167 0.998704
\(722\) 15.9083 0.592047
\(723\) 0 0
\(724\) 17.1194 0.636239
\(725\) 2.60555 0.0967677
\(726\) 0 0
\(727\) 2.90833 0.107864 0.0539319 0.998545i \(-0.482825\pi\)
0.0539319 + 0.998545i \(0.482825\pi\)
\(728\) 10.9083 0.404289
\(729\) 0 0
\(730\) 5.81665 0.215284
\(731\) 63.6333 2.35356
\(732\) 0 0
\(733\) 29.6333 1.09453 0.547266 0.836959i \(-0.315669\pi\)
0.547266 + 0.836959i \(0.315669\pi\)
\(734\) 14.7889 0.545868
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −6.78890 −0.250072
\(738\) 0 0
\(739\) 35.6333 1.31079 0.655396 0.755285i \(-0.272502\pi\)
0.655396 + 0.755285i \(0.272502\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) 37.0278 1.35933
\(743\) 32.3305 1.18609 0.593046 0.805169i \(-0.297925\pi\)
0.593046 + 0.805169i \(0.297925\pi\)
\(744\) 0 0
\(745\) 13.3028 0.487376
\(746\) 4.60555 0.168621
\(747\) 0 0
\(748\) −11.7250 −0.428708
\(749\) 8.60555 0.314440
\(750\) 0 0
\(751\) 21.8167 0.796101 0.398051 0.917364i \(-0.369687\pi\)
0.398051 + 0.917364i \(0.369687\pi\)
\(752\) 2.60555 0.0950147
\(753\) 0 0
\(754\) 8.60555 0.313396
\(755\) −8.90833 −0.324207
\(756\) 0 0
\(757\) 13.2111 0.480166 0.240083 0.970752i \(-0.422825\pi\)
0.240083 + 0.970752i \(0.422825\pi\)
\(758\) 14.9083 0.541495
\(759\) 0 0
\(760\) −5.90833 −0.214318
\(761\) 49.5416 1.79588 0.897941 0.440115i \(-0.145062\pi\)
0.897941 + 0.440115i \(0.145062\pi\)
\(762\) 0 0
\(763\) 4.90833 0.177693
\(764\) 8.60555 0.311338
\(765\) 0 0
\(766\) 0 0
\(767\) 11.2111 0.404809
\(768\) 0 0
\(769\) 45.2666 1.63236 0.816178 0.577801i \(-0.196089\pi\)
0.816178 + 0.577801i \(0.196089\pi\)
\(770\) −5.60555 −0.202010
\(771\) 0 0
\(772\) −17.8167 −0.641235
\(773\) −12.0000 −0.431610 −0.215805 0.976436i \(-0.569238\pi\)
−0.215805 + 0.976436i \(0.569238\pi\)
\(774\) 0 0
\(775\) −7.90833 −0.284075
\(776\) 6.30278 0.226256
\(777\) 0 0
\(778\) 25.9361 0.929854
\(779\) −5.36669 −0.192282
\(780\) 0 0
\(781\) 27.6695 0.990091
\(782\) −6.90833 −0.247041
\(783\) 0 0
\(784\) 3.90833 0.139583
\(785\) 18.6056 0.664061
\(786\) 0 0
\(787\) 37.4500 1.33495 0.667473 0.744634i \(-0.267376\pi\)
0.667473 + 0.744634i \(0.267376\pi\)
\(788\) −4.30278 −0.153280
\(789\) 0 0
\(790\) 14.4222 0.513119
\(791\) 54.2389 1.92851
\(792\) 0 0
\(793\) 38.0278 1.35041
\(794\) 10.7250 0.380615
\(795\) 0 0
\(796\) −20.4222 −0.723846
\(797\) −1.81665 −0.0643492 −0.0321746 0.999482i \(-0.510243\pi\)
−0.0321746 + 0.999482i \(0.510243\pi\)
\(798\) 0 0
\(799\) −18.0000 −0.636794
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 8.60555 0.303873
\(803\) −9.87217 −0.348381
\(804\) 0 0
\(805\) −3.30278 −0.116408
\(806\) −26.1194 −0.920018
\(807\) 0 0
\(808\) −2.60555 −0.0916630
\(809\) −50.7250 −1.78340 −0.891698 0.452631i \(-0.850485\pi\)
−0.891698 + 0.452631i \(0.850485\pi\)
\(810\) 0 0
\(811\) −41.0278 −1.44068 −0.720340 0.693621i \(-0.756014\pi\)
−0.720340 + 0.693621i \(0.756014\pi\)
\(812\) 8.60555 0.301996
\(813\) 0 0
\(814\) 13.5778 0.475901
\(815\) −9.30278 −0.325862
\(816\) 0 0
\(817\) −54.4222 −1.90399
\(818\) −25.9083 −0.905863
\(819\) 0 0
\(820\) 0.908327 0.0317202
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) −15.2111 −0.530226 −0.265113 0.964217i \(-0.585409\pi\)
−0.265113 + 0.964217i \(0.585409\pi\)
\(824\) 8.11943 0.282854
\(825\) 0 0
\(826\) 11.2111 0.390084
\(827\) 29.4500 1.02408 0.512038 0.858963i \(-0.328891\pi\)
0.512038 + 0.858963i \(0.328891\pi\)
\(828\) 0 0
\(829\) 31.2111 1.08401 0.542003 0.840376i \(-0.317666\pi\)
0.542003 + 0.840376i \(0.317666\pi\)
\(830\) 11.2111 0.389143
\(831\) 0 0
\(832\) 3.30278 0.114503
\(833\) −27.0000 −0.935495
\(834\) 0 0
\(835\) 6.78890 0.234939
\(836\) 10.0278 0.346817
\(837\) 0 0
\(838\) −3.63331 −0.125511
\(839\) 43.8167 1.51272 0.756359 0.654156i \(-0.226976\pi\)
0.756359 + 0.654156i \(0.226976\pi\)
\(840\) 0 0
\(841\) −22.2111 −0.765900
\(842\) 30.6972 1.05790
\(843\) 0 0
\(844\) 7.21110 0.248216
\(845\) 2.09167 0.0719557
\(846\) 0 0
\(847\) −26.8167 −0.921431
\(848\) 11.2111 0.384991
\(849\) 0 0
\(850\) −6.90833 −0.236954
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) −21.7250 −0.743849 −0.371925 0.928263i \(-0.621302\pi\)
−0.371925 + 0.928263i \(0.621302\pi\)
\(854\) 38.0278 1.30128
\(855\) 0 0
\(856\) 2.60555 0.0890559
\(857\) −9.63331 −0.329068 −0.164534 0.986371i \(-0.552612\pi\)
−0.164534 + 0.986371i \(0.552612\pi\)
\(858\) 0 0
\(859\) −35.8167 −1.22205 −0.611024 0.791612i \(-0.709242\pi\)
−0.611024 + 0.791612i \(0.709242\pi\)
\(860\) 9.21110 0.314096
\(861\) 0 0
\(862\) 30.2389 1.02994
\(863\) 41.4500 1.41097 0.705487 0.708723i \(-0.250729\pi\)
0.705487 + 0.708723i \(0.250729\pi\)
\(864\) 0 0
\(865\) 19.6972 0.669726
\(866\) −24.0917 −0.818668
\(867\) 0 0
\(868\) −26.1194 −0.886551
\(869\) −24.4777 −0.830350
\(870\) 0 0
\(871\) −13.2111 −0.447641
\(872\) 1.48612 0.0503264
\(873\) 0 0
\(874\) 5.90833 0.199852
\(875\) −3.30278 −0.111654
\(876\) 0 0
\(877\) −48.1749 −1.62675 −0.813376 0.581738i \(-0.802373\pi\)
−0.813376 + 0.581738i \(0.802373\pi\)
\(878\) −14.6972 −0.496007
\(879\) 0 0
\(880\) −1.69722 −0.0572134
\(881\) −55.2666 −1.86198 −0.930990 0.365045i \(-0.881054\pi\)
−0.930990 + 0.365045i \(0.881054\pi\)
\(882\) 0 0
\(883\) 8.27502 0.278477 0.139238 0.990259i \(-0.455535\pi\)
0.139238 + 0.990259i \(0.455535\pi\)
\(884\) −22.8167 −0.767407
\(885\) 0 0
\(886\) −17.4861 −0.587458
\(887\) −27.6333 −0.927836 −0.463918 0.885878i \(-0.653557\pi\)
−0.463918 + 0.885878i \(0.653557\pi\)
\(888\) 0 0
\(889\) 32.4222 1.08741
\(890\) 0 0
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) 15.3944 0.515156
\(894\) 0 0
\(895\) 9.39445 0.314022
\(896\) 3.30278 0.110338
\(897\) 0 0
\(898\) 2.09167 0.0698000
\(899\) −20.6056 −0.687234
\(900\) 0 0
\(901\) −77.4500 −2.58023
\(902\) −1.54163 −0.0513308
\(903\) 0 0
\(904\) 16.4222 0.546194
\(905\) −17.1194 −0.569069
\(906\) 0 0
\(907\) 48.6611 1.61576 0.807882 0.589344i \(-0.200614\pi\)
0.807882 + 0.589344i \(0.200614\pi\)
\(908\) 14.6056 0.484702
\(909\) 0 0
\(910\) −10.9083 −0.361608
\(911\) 4.18335 0.138600 0.0693002 0.997596i \(-0.477923\pi\)
0.0693002 + 0.997596i \(0.477923\pi\)
\(912\) 0 0
\(913\) −19.0278 −0.629727
\(914\) −32.4222 −1.07243
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) 37.0278 1.22276
\(918\) 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
\(922\) −10.1833 −0.335371
\(923\) 53.8444 1.77231
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 17.6333 0.579466
\(927\) 0 0
\(928\) 2.60555 0.0855314
\(929\) 14.3667 0.471356 0.235678 0.971831i \(-0.424269\pi\)
0.235678 + 0.971831i \(0.424269\pi\)
\(930\) 0 0
\(931\) 23.0917 0.756799
\(932\) −25.8167 −0.845653
\(933\) 0 0
\(934\) 1.81665 0.0594427
\(935\) 11.7250 0.383448
\(936\) 0 0
\(937\) 37.9638 1.24022 0.620112 0.784513i \(-0.287087\pi\)
0.620112 + 0.784513i \(0.287087\pi\)
\(938\) −13.2111 −0.431358
\(939\) 0 0
\(940\) −2.60555 −0.0849837
\(941\) −25.9361 −0.845492 −0.422746 0.906248i \(-0.638934\pi\)
−0.422746 + 0.906248i \(0.638934\pi\)
\(942\) 0 0
\(943\) −0.908327 −0.0295792
\(944\) 3.39445 0.110480
\(945\) 0 0
\(946\) −15.6333 −0.508283
\(947\) 4.93608 0.160401 0.0802006 0.996779i \(-0.474444\pi\)
0.0802006 + 0.996779i \(0.474444\pi\)
\(948\) 0 0
\(949\) −19.2111 −0.623619
\(950\) 5.90833 0.191691
\(951\) 0 0
\(952\) −22.8167 −0.739492
\(953\) 41.3305 1.33883 0.669414 0.742890i \(-0.266545\pi\)
0.669414 + 0.742890i \(0.266545\pi\)
\(954\) 0 0
\(955\) −8.60555 −0.278469
\(956\) −5.21110 −0.168539
\(957\) 0 0
\(958\) 30.0000 0.969256
\(959\) −12.9083 −0.416832
\(960\) 0 0
\(961\) 31.5416 1.01747
\(962\) 26.4222 0.851886
\(963\) 0 0
\(964\) −14.4222 −0.464508
\(965\) 17.8167 0.573538
\(966\) 0 0
\(967\) −12.6056 −0.405367 −0.202684 0.979244i \(-0.564966\pi\)
−0.202684 + 0.979244i \(0.564966\pi\)
\(968\) −8.11943 −0.260968
\(969\) 0 0
\(970\) −6.30278 −0.202370
\(971\) 17.0917 0.548498 0.274249 0.961659i \(-0.411571\pi\)
0.274249 + 0.961659i \(0.411571\pi\)
\(972\) 0 0
\(973\) −41.6333 −1.33470
\(974\) 9.81665 0.314546
\(975\) 0 0
\(976\) 11.5139 0.368550
\(977\) −6.51388 −0.208397 −0.104199 0.994556i \(-0.533228\pi\)
−0.104199 + 0.994556i \(0.533228\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −3.90833 −0.124847
\(981\) 0 0
\(982\) 4.18335 0.133496
\(983\) −34.5416 −1.10171 −0.550854 0.834602i \(-0.685698\pi\)
−0.550854 + 0.834602i \(0.685698\pi\)
\(984\) 0 0
\(985\) 4.30278 0.137098
\(986\) −18.0000 −0.573237
\(987\) 0 0
\(988\) 19.5139 0.620819
\(989\) −9.21110 −0.292896
\(990\) 0 0
\(991\) −15.3305 −0.486990 −0.243495 0.969902i \(-0.578294\pi\)
−0.243495 + 0.969902i \(0.578294\pi\)
\(992\) −7.90833 −0.251090
\(993\) 0 0
\(994\) 53.8444 1.70784
\(995\) 20.4222 0.647427
\(996\) 0 0
\(997\) −16.7889 −0.531710 −0.265855 0.964013i \(-0.585654\pi\)
−0.265855 + 0.964013i \(0.585654\pi\)
\(998\) −31.6333 −1.00133
\(999\) 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.1.2 2
3.2 odd 2 230.2.a.b.1.1 2
12.11 even 2 1840.2.a.j.1.2 2
15.2 even 4 1150.2.b.f.599.2 4
15.8 even 4 1150.2.b.f.599.3 4
15.14 odd 2 1150.2.a.m.1.2 2
24.5 odd 2 7360.2.a.bc.1.2 2
24.11 even 2 7360.2.a.bu.1.1 2
60.59 even 2 9200.2.a.ca.1.1 2
69.68 even 2 5290.2.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.b.1.1 2 3.2 odd 2
1150.2.a.m.1.2 2 15.14 odd 2
1150.2.b.f.599.2 4 15.2 even 4
1150.2.b.f.599.3 4 15.8 even 4
1840.2.a.j.1.2 2 12.11 even 2
2070.2.a.w.1.2 2 1.1 even 1 trivial
5290.2.a.j.1.1 2 69.68 even 2
7360.2.a.bc.1.2 2 24.5 odd 2
7360.2.a.bu.1.1 2 24.11 even 2
9200.2.a.ca.1.1 2 60.59 even 2