Properties

Label 8004.2.a.g.1.9
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} - 1156 x^{3} - 616 x^{2} + 136 x + 16\)
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 1.9
Root \(-1.42583\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.42583 q^{5} -1.49015 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.42583 q^{5} -1.49015 q^{7} +1.00000 q^{9} +4.45964 q^{11} -0.159001 q^{13} -1.42583 q^{15} +3.47517 q^{17} +1.29574 q^{19} +1.49015 q^{21} +1.00000 q^{23} -2.96701 q^{25} -1.00000 q^{27} -1.00000 q^{29} -8.87791 q^{31} -4.45964 q^{33} -2.12470 q^{35} -9.07562 q^{37} +0.159001 q^{39} -6.40770 q^{41} -12.0404 q^{43} +1.42583 q^{45} -10.2522 q^{47} -4.77945 q^{49} -3.47517 q^{51} +8.00324 q^{53} +6.35868 q^{55} -1.29574 q^{57} +12.2266 q^{59} -10.3000 q^{61} -1.49015 q^{63} -0.226708 q^{65} -0.485962 q^{67} -1.00000 q^{69} +7.64325 q^{71} -3.71841 q^{73} +2.96701 q^{75} -6.64554 q^{77} +12.4035 q^{79} +1.00000 q^{81} +1.30451 q^{83} +4.95499 q^{85} +1.00000 q^{87} -18.2043 q^{89} +0.236935 q^{91} +8.87791 q^{93} +1.84750 q^{95} +2.78330 q^{97} +4.45964 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} - 5q^{11} - 6q^{13} + 3q^{15} - 7q^{17} - 3q^{19} - 4q^{21} + 12q^{23} + 11q^{25} - 12q^{27} - 12q^{29} + 2q^{31} + 5q^{33} - 9q^{35} - 20q^{37} + 6q^{39} - 3q^{41} + 5q^{43} - 3q^{45} - 2q^{49} + 7q^{51} - 3q^{53} + 19q^{55} + 3q^{57} - 20q^{59} - 17q^{61} + 4q^{63} - 4q^{65} - 9q^{67} - 12q^{69} + 7q^{71} - 9q^{73} - 11q^{75} - 34q^{77} + 14q^{79} + 12q^{81} + 5q^{83} - 12q^{85} + 12q^{87} - 22q^{89} - 3q^{91} - 2q^{93} - 27q^{95} + 17q^{97} - 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.42583 0.637650 0.318825 0.947814i \(-0.396712\pi\)
0.318825 + 0.947814i \(0.396712\pi\)
\(6\) 0 0
\(7\) −1.49015 −0.563224 −0.281612 0.959528i \(-0.590869\pi\)
−0.281612 + 0.959528i \(0.590869\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.45964 1.34463 0.672316 0.740264i \(-0.265299\pi\)
0.672316 + 0.740264i \(0.265299\pi\)
\(12\) 0 0
\(13\) −0.159001 −0.0440988 −0.0220494 0.999757i \(-0.507019\pi\)
−0.0220494 + 0.999757i \(0.507019\pi\)
\(14\) 0 0
\(15\) −1.42583 −0.368147
\(16\) 0 0
\(17\) 3.47517 0.842852 0.421426 0.906863i \(-0.361530\pi\)
0.421426 + 0.906863i \(0.361530\pi\)
\(18\) 0 0
\(19\) 1.29574 0.297262 0.148631 0.988893i \(-0.452513\pi\)
0.148631 + 0.988893i \(0.452513\pi\)
\(20\) 0 0
\(21\) 1.49015 0.325177
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −2.96701 −0.593403
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −8.87791 −1.59452 −0.797260 0.603636i \(-0.793718\pi\)
−0.797260 + 0.603636i \(0.793718\pi\)
\(32\) 0 0
\(33\) −4.45964 −0.776324
\(34\) 0 0
\(35\) −2.12470 −0.359140
\(36\) 0 0
\(37\) −9.07562 −1.49202 −0.746012 0.665933i \(-0.768034\pi\)
−0.746012 + 0.665933i \(0.768034\pi\)
\(38\) 0 0
\(39\) 0.159001 0.0254605
\(40\) 0 0
\(41\) −6.40770 −1.00071 −0.500357 0.865819i \(-0.666798\pi\)
−0.500357 + 0.865819i \(0.666798\pi\)
\(42\) 0 0
\(43\) −12.0404 −1.83614 −0.918072 0.396414i \(-0.870255\pi\)
−0.918072 + 0.396414i \(0.870255\pi\)
\(44\) 0 0
\(45\) 1.42583 0.212550
\(46\) 0 0
\(47\) −10.2522 −1.49543 −0.747716 0.664019i \(-0.768850\pi\)
−0.747716 + 0.664019i \(0.768850\pi\)
\(48\) 0 0
\(49\) −4.77945 −0.682779
\(50\) 0 0
\(51\) −3.47517 −0.486621
\(52\) 0 0
\(53\) 8.00324 1.09933 0.549665 0.835385i \(-0.314755\pi\)
0.549665 + 0.835385i \(0.314755\pi\)
\(54\) 0 0
\(55\) 6.35868 0.857405
\(56\) 0 0
\(57\) −1.29574 −0.171625
\(58\) 0 0
\(59\) 12.2266 1.59177 0.795885 0.605447i \(-0.207006\pi\)
0.795885 + 0.605447i \(0.207006\pi\)
\(60\) 0 0
\(61\) −10.3000 −1.31878 −0.659388 0.751803i \(-0.729184\pi\)
−0.659388 + 0.751803i \(0.729184\pi\)
\(62\) 0 0
\(63\) −1.49015 −0.187741
\(64\) 0 0
\(65\) −0.226708 −0.0281196
\(66\) 0 0
\(67\) −0.485962 −0.0593697 −0.0296848 0.999559i \(-0.509450\pi\)
−0.0296848 + 0.999559i \(0.509450\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 7.64325 0.907087 0.453544 0.891234i \(-0.350160\pi\)
0.453544 + 0.891234i \(0.350160\pi\)
\(72\) 0 0
\(73\) −3.71841 −0.435207 −0.217603 0.976037i \(-0.569824\pi\)
−0.217603 + 0.976037i \(0.569824\pi\)
\(74\) 0 0
\(75\) 2.96701 0.342601
\(76\) 0 0
\(77\) −6.64554 −0.757329
\(78\) 0 0
\(79\) 12.4035 1.39550 0.697751 0.716341i \(-0.254184\pi\)
0.697751 + 0.716341i \(0.254184\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.30451 0.143188 0.0715941 0.997434i \(-0.477191\pi\)
0.0715941 + 0.997434i \(0.477191\pi\)
\(84\) 0 0
\(85\) 4.95499 0.537444
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) −18.2043 −1.92965 −0.964827 0.262884i \(-0.915326\pi\)
−0.964827 + 0.262884i \(0.915326\pi\)
\(90\) 0 0
\(91\) 0.236935 0.0248375
\(92\) 0 0
\(93\) 8.87791 0.920597
\(94\) 0 0
\(95\) 1.84750 0.189549
\(96\) 0 0
\(97\) 2.78330 0.282601 0.141301 0.989967i \(-0.454872\pi\)
0.141301 + 0.989967i \(0.454872\pi\)
\(98\) 0 0
\(99\) 4.45964 0.448211
\(100\) 0 0
\(101\) 8.42981 0.838797 0.419399 0.907802i \(-0.362241\pi\)
0.419399 + 0.907802i \(0.362241\pi\)
\(102\) 0 0
\(103\) 14.8072 1.45900 0.729499 0.683982i \(-0.239753\pi\)
0.729499 + 0.683982i \(0.239753\pi\)
\(104\) 0 0
\(105\) 2.12470 0.207349
\(106\) 0 0
\(107\) 13.5258 1.30759 0.653797 0.756670i \(-0.273175\pi\)
0.653797 + 0.756670i \(0.273175\pi\)
\(108\) 0 0
\(109\) −12.8502 −1.23083 −0.615415 0.788203i \(-0.711012\pi\)
−0.615415 + 0.788203i \(0.711012\pi\)
\(110\) 0 0
\(111\) 9.07562 0.861420
\(112\) 0 0
\(113\) 0.391242 0.0368050 0.0184025 0.999831i \(-0.494142\pi\)
0.0184025 + 0.999831i \(0.494142\pi\)
\(114\) 0 0
\(115\) 1.42583 0.132959
\(116\) 0 0
\(117\) −0.159001 −0.0146996
\(118\) 0 0
\(119\) −5.17852 −0.474714
\(120\) 0 0
\(121\) 8.88842 0.808038
\(122\) 0 0
\(123\) 6.40770 0.577763
\(124\) 0 0
\(125\) −11.3596 −1.01603
\(126\) 0 0
\(127\) −11.4109 −1.01256 −0.506278 0.862370i \(-0.668979\pi\)
−0.506278 + 0.862370i \(0.668979\pi\)
\(128\) 0 0
\(129\) 12.0404 1.06010
\(130\) 0 0
\(131\) 7.75502 0.677559 0.338779 0.940866i \(-0.389986\pi\)
0.338779 + 0.940866i \(0.389986\pi\)
\(132\) 0 0
\(133\) −1.93084 −0.167425
\(134\) 0 0
\(135\) −1.42583 −0.122716
\(136\) 0 0
\(137\) 8.38978 0.716787 0.358394 0.933571i \(-0.383325\pi\)
0.358394 + 0.933571i \(0.383325\pi\)
\(138\) 0 0
\(139\) −13.9105 −1.17988 −0.589939 0.807448i \(-0.700848\pi\)
−0.589939 + 0.807448i \(0.700848\pi\)
\(140\) 0 0
\(141\) 10.2522 0.863388
\(142\) 0 0
\(143\) −0.709086 −0.0592968
\(144\) 0 0
\(145\) −1.42583 −0.118409
\(146\) 0 0
\(147\) 4.77945 0.394203
\(148\) 0 0
\(149\) −10.7393 −0.879797 −0.439898 0.898048i \(-0.644986\pi\)
−0.439898 + 0.898048i \(0.644986\pi\)
\(150\) 0 0
\(151\) 9.86355 0.802685 0.401342 0.915928i \(-0.368544\pi\)
0.401342 + 0.915928i \(0.368544\pi\)
\(152\) 0 0
\(153\) 3.47517 0.280951
\(154\) 0 0
\(155\) −12.6584 −1.01675
\(156\) 0 0
\(157\) 4.75666 0.379623 0.189812 0.981821i \(-0.439212\pi\)
0.189812 + 0.981821i \(0.439212\pi\)
\(158\) 0 0
\(159\) −8.00324 −0.634698
\(160\) 0 0
\(161\) −1.49015 −0.117440
\(162\) 0 0
\(163\) 5.52008 0.432366 0.216183 0.976353i \(-0.430639\pi\)
0.216183 + 0.976353i \(0.430639\pi\)
\(164\) 0 0
\(165\) −6.35868 −0.495023
\(166\) 0 0
\(167\) −3.79049 −0.293317 −0.146659 0.989187i \(-0.546852\pi\)
−0.146659 + 0.989187i \(0.546852\pi\)
\(168\) 0 0
\(169\) −12.9747 −0.998055
\(170\) 0 0
\(171\) 1.29574 0.0990875
\(172\) 0 0
\(173\) 23.1910 1.76318 0.881590 0.472016i \(-0.156474\pi\)
0.881590 + 0.472016i \(0.156474\pi\)
\(174\) 0 0
\(175\) 4.42130 0.334219
\(176\) 0 0
\(177\) −12.2266 −0.919009
\(178\) 0 0
\(179\) −23.2446 −1.73738 −0.868691 0.495355i \(-0.835038\pi\)
−0.868691 + 0.495355i \(0.835038\pi\)
\(180\) 0 0
\(181\) −4.88281 −0.362936 −0.181468 0.983397i \(-0.558085\pi\)
−0.181468 + 0.983397i \(0.558085\pi\)
\(182\) 0 0
\(183\) 10.3000 0.761395
\(184\) 0 0
\(185\) −12.9403 −0.951388
\(186\) 0 0
\(187\) 15.4980 1.13333
\(188\) 0 0
\(189\) 1.49015 0.108392
\(190\) 0 0
\(191\) −21.9079 −1.58520 −0.792599 0.609743i \(-0.791273\pi\)
−0.792599 + 0.609743i \(0.791273\pi\)
\(192\) 0 0
\(193\) −8.29118 −0.596812 −0.298406 0.954439i \(-0.596455\pi\)
−0.298406 + 0.954439i \(0.596455\pi\)
\(194\) 0 0
\(195\) 0.226708 0.0162349
\(196\) 0 0
\(197\) 4.20469 0.299572 0.149786 0.988718i \(-0.452142\pi\)
0.149786 + 0.988718i \(0.452142\pi\)
\(198\) 0 0
\(199\) −12.9265 −0.916335 −0.458167 0.888866i \(-0.651494\pi\)
−0.458167 + 0.888866i \(0.651494\pi\)
\(200\) 0 0
\(201\) 0.485962 0.0342771
\(202\) 0 0
\(203\) 1.49015 0.104588
\(204\) 0 0
\(205\) −9.13628 −0.638105
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 5.77852 0.399709
\(210\) 0 0
\(211\) −24.3532 −1.67654 −0.838271 0.545254i \(-0.816433\pi\)
−0.838271 + 0.545254i \(0.816433\pi\)
\(212\) 0 0
\(213\) −7.64325 −0.523707
\(214\) 0 0
\(215\) −17.1675 −1.17082
\(216\) 0 0
\(217\) 13.2294 0.898072
\(218\) 0 0
\(219\) 3.71841 0.251267
\(220\) 0 0
\(221\) −0.552554 −0.0371688
\(222\) 0 0
\(223\) −8.47197 −0.567325 −0.283662 0.958924i \(-0.591550\pi\)
−0.283662 + 0.958924i \(0.591550\pi\)
\(224\) 0 0
\(225\) −2.96701 −0.197801
\(226\) 0 0
\(227\) 17.6827 1.17364 0.586821 0.809717i \(-0.300379\pi\)
0.586821 + 0.809717i \(0.300379\pi\)
\(228\) 0 0
\(229\) −4.93562 −0.326155 −0.163077 0.986613i \(-0.552142\pi\)
−0.163077 + 0.986613i \(0.552142\pi\)
\(230\) 0 0
\(231\) 6.64554 0.437244
\(232\) 0 0
\(233\) −26.5583 −1.73989 −0.869947 0.493145i \(-0.835847\pi\)
−0.869947 + 0.493145i \(0.835847\pi\)
\(234\) 0 0
\(235\) −14.6178 −0.953562
\(236\) 0 0
\(237\) −12.4035 −0.805693
\(238\) 0 0
\(239\) −0.0342729 −0.00221693 −0.00110846 0.999999i \(-0.500353\pi\)
−0.00110846 + 0.999999i \(0.500353\pi\)
\(240\) 0 0
\(241\) −15.3244 −0.987134 −0.493567 0.869708i \(-0.664307\pi\)
−0.493567 + 0.869708i \(0.664307\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −6.81468 −0.435374
\(246\) 0 0
\(247\) −0.206023 −0.0131089
\(248\) 0 0
\(249\) −1.30451 −0.0826698
\(250\) 0 0
\(251\) 5.44938 0.343962 0.171981 0.985100i \(-0.444983\pi\)
0.171981 + 0.985100i \(0.444983\pi\)
\(252\) 0 0
\(253\) 4.45964 0.280375
\(254\) 0 0
\(255\) −4.95499 −0.310294
\(256\) 0 0
\(257\) −1.38441 −0.0863574 −0.0431787 0.999067i \(-0.513748\pi\)
−0.0431787 + 0.999067i \(0.513748\pi\)
\(258\) 0 0
\(259\) 13.5240 0.840343
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) 6.66705 0.411108 0.205554 0.978646i \(-0.434100\pi\)
0.205554 + 0.978646i \(0.434100\pi\)
\(264\) 0 0
\(265\) 11.4112 0.700987
\(266\) 0 0
\(267\) 18.2043 1.11409
\(268\) 0 0
\(269\) 30.4318 1.85546 0.927730 0.373251i \(-0.121757\pi\)
0.927730 + 0.373251i \(0.121757\pi\)
\(270\) 0 0
\(271\) −18.2818 −1.11054 −0.555270 0.831670i \(-0.687385\pi\)
−0.555270 + 0.831670i \(0.687385\pi\)
\(272\) 0 0
\(273\) −0.236935 −0.0143400
\(274\) 0 0
\(275\) −13.2318 −0.797909
\(276\) 0 0
\(277\) 4.36334 0.262168 0.131084 0.991371i \(-0.458154\pi\)
0.131084 + 0.991371i \(0.458154\pi\)
\(278\) 0 0
\(279\) −8.87791 −0.531507
\(280\) 0 0
\(281\) −14.1143 −0.841990 −0.420995 0.907063i \(-0.638319\pi\)
−0.420995 + 0.907063i \(0.638319\pi\)
\(282\) 0 0
\(283\) 2.84332 0.169018 0.0845090 0.996423i \(-0.473068\pi\)
0.0845090 + 0.996423i \(0.473068\pi\)
\(284\) 0 0
\(285\) −1.84750 −0.109436
\(286\) 0 0
\(287\) 9.54843 0.563626
\(288\) 0 0
\(289\) −4.92322 −0.289601
\(290\) 0 0
\(291\) −2.78330 −0.163160
\(292\) 0 0
\(293\) −9.50985 −0.555571 −0.277786 0.960643i \(-0.589600\pi\)
−0.277786 + 0.960643i \(0.589600\pi\)
\(294\) 0 0
\(295\) 17.4331 1.01499
\(296\) 0 0
\(297\) −4.45964 −0.258775
\(298\) 0 0
\(299\) −0.159001 −0.00919524
\(300\) 0 0
\(301\) 17.9420 1.03416
\(302\) 0 0
\(303\) −8.42981 −0.484280
\(304\) 0 0
\(305\) −14.6860 −0.840917
\(306\) 0 0
\(307\) −5.05037 −0.288240 −0.144120 0.989560i \(-0.546035\pi\)
−0.144120 + 0.989560i \(0.546035\pi\)
\(308\) 0 0
\(309\) −14.8072 −0.842353
\(310\) 0 0
\(311\) 32.6238 1.84993 0.924964 0.380056i \(-0.124095\pi\)
0.924964 + 0.380056i \(0.124095\pi\)
\(312\) 0 0
\(313\) −6.08094 −0.343715 −0.171858 0.985122i \(-0.554977\pi\)
−0.171858 + 0.985122i \(0.554977\pi\)
\(314\) 0 0
\(315\) −2.12470 −0.119713
\(316\) 0 0
\(317\) 26.3432 1.47958 0.739791 0.672837i \(-0.234925\pi\)
0.739791 + 0.672837i \(0.234925\pi\)
\(318\) 0 0
\(319\) −4.45964 −0.249692
\(320\) 0 0
\(321\) −13.5258 −0.754939
\(322\) 0 0
\(323\) 4.50290 0.250548
\(324\) 0 0
\(325\) 0.471757 0.0261684
\(326\) 0 0
\(327\) 12.8502 0.710620
\(328\) 0 0
\(329\) 15.2773 0.842263
\(330\) 0 0
\(331\) −2.49309 −0.137032 −0.0685162 0.997650i \(-0.521826\pi\)
−0.0685162 + 0.997650i \(0.521826\pi\)
\(332\) 0 0
\(333\) −9.07562 −0.497341
\(334\) 0 0
\(335\) −0.692898 −0.0378571
\(336\) 0 0
\(337\) −10.6144 −0.578201 −0.289101 0.957299i \(-0.593356\pi\)
−0.289101 + 0.957299i \(0.593356\pi\)
\(338\) 0 0
\(339\) −0.391242 −0.0212494
\(340\) 0 0
\(341\) −39.5923 −2.14404
\(342\) 0 0
\(343\) 17.5532 0.947781
\(344\) 0 0
\(345\) −1.42583 −0.0767640
\(346\) 0 0
\(347\) 16.7762 0.900592 0.450296 0.892879i \(-0.351318\pi\)
0.450296 + 0.892879i \(0.351318\pi\)
\(348\) 0 0
\(349\) 9.12257 0.488320 0.244160 0.969735i \(-0.421488\pi\)
0.244160 + 0.969735i \(0.421488\pi\)
\(350\) 0 0
\(351\) 0.159001 0.00848683
\(352\) 0 0
\(353\) −25.6428 −1.36483 −0.682415 0.730965i \(-0.739070\pi\)
−0.682415 + 0.730965i \(0.739070\pi\)
\(354\) 0 0
\(355\) 10.8980 0.578404
\(356\) 0 0
\(357\) 5.17852 0.274076
\(358\) 0 0
\(359\) −21.9300 −1.15742 −0.578710 0.815534i \(-0.696444\pi\)
−0.578710 + 0.815534i \(0.696444\pi\)
\(360\) 0 0
\(361\) −17.3211 −0.911635
\(362\) 0 0
\(363\) −8.88842 −0.466521
\(364\) 0 0
\(365\) −5.30181 −0.277510
\(366\) 0 0
\(367\) 35.9231 1.87517 0.937586 0.347755i \(-0.113056\pi\)
0.937586 + 0.347755i \(0.113056\pi\)
\(368\) 0 0
\(369\) −6.40770 −0.333571
\(370\) 0 0
\(371\) −11.9260 −0.619169
\(372\) 0 0
\(373\) −14.6467 −0.758377 −0.379188 0.925320i \(-0.623797\pi\)
−0.379188 + 0.925320i \(0.623797\pi\)
\(374\) 0 0
\(375\) 11.3596 0.586607
\(376\) 0 0
\(377\) 0.159001 0.00818895
\(378\) 0 0
\(379\) 0.639273 0.0328372 0.0164186 0.999865i \(-0.494774\pi\)
0.0164186 + 0.999865i \(0.494774\pi\)
\(380\) 0 0
\(381\) 11.4109 0.584600
\(382\) 0 0
\(383\) 1.78915 0.0914212 0.0457106 0.998955i \(-0.485445\pi\)
0.0457106 + 0.998955i \(0.485445\pi\)
\(384\) 0 0
\(385\) −9.47540 −0.482911
\(386\) 0 0
\(387\) −12.0404 −0.612048
\(388\) 0 0
\(389\) −17.9578 −0.910498 −0.455249 0.890364i \(-0.650450\pi\)
−0.455249 + 0.890364i \(0.650450\pi\)
\(390\) 0 0
\(391\) 3.47517 0.175747
\(392\) 0 0
\(393\) −7.75502 −0.391189
\(394\) 0 0
\(395\) 17.6852 0.889841
\(396\) 0 0
\(397\) 9.23664 0.463574 0.231787 0.972767i \(-0.425543\pi\)
0.231787 + 0.972767i \(0.425543\pi\)
\(398\) 0 0
\(399\) 1.93084 0.0966630
\(400\) 0 0
\(401\) −10.3628 −0.517493 −0.258746 0.965945i \(-0.583309\pi\)
−0.258746 + 0.965945i \(0.583309\pi\)
\(402\) 0 0
\(403\) 1.41159 0.0703165
\(404\) 0 0
\(405\) 1.42583 0.0708500
\(406\) 0 0
\(407\) −40.4740 −2.00622
\(408\) 0 0
\(409\) −14.8047 −0.732044 −0.366022 0.930606i \(-0.619280\pi\)
−0.366022 + 0.930606i \(0.619280\pi\)
\(410\) 0 0
\(411\) −8.38978 −0.413837
\(412\) 0 0
\(413\) −18.2195 −0.896523
\(414\) 0 0
\(415\) 1.86000 0.0913039
\(416\) 0 0
\(417\) 13.9105 0.681202
\(418\) 0 0
\(419\) −34.8281 −1.70146 −0.850731 0.525601i \(-0.823841\pi\)
−0.850731 + 0.525601i \(0.823841\pi\)
\(420\) 0 0
\(421\) 6.17148 0.300779 0.150390 0.988627i \(-0.451947\pi\)
0.150390 + 0.988627i \(0.451947\pi\)
\(422\) 0 0
\(423\) −10.2522 −0.498477
\(424\) 0 0
\(425\) −10.3109 −0.500151
\(426\) 0 0
\(427\) 15.3485 0.742766
\(428\) 0 0
\(429\) 0.709086 0.0342350
\(430\) 0 0
\(431\) 1.31883 0.0635258 0.0317629 0.999495i \(-0.489888\pi\)
0.0317629 + 0.999495i \(0.489888\pi\)
\(432\) 0 0
\(433\) −16.3422 −0.785357 −0.392678 0.919676i \(-0.628451\pi\)
−0.392678 + 0.919676i \(0.628451\pi\)
\(434\) 0 0
\(435\) 1.42583 0.0683632
\(436\) 0 0
\(437\) 1.29574 0.0619835
\(438\) 0 0
\(439\) 29.0443 1.38621 0.693103 0.720838i \(-0.256243\pi\)
0.693103 + 0.720838i \(0.256243\pi\)
\(440\) 0 0
\(441\) −4.77945 −0.227593
\(442\) 0 0
\(443\) 8.80448 0.418313 0.209157 0.977882i \(-0.432928\pi\)
0.209157 + 0.977882i \(0.432928\pi\)
\(444\) 0 0
\(445\) −25.9562 −1.23044
\(446\) 0 0
\(447\) 10.7393 0.507951
\(448\) 0 0
\(449\) 6.87568 0.324483 0.162242 0.986751i \(-0.448128\pi\)
0.162242 + 0.986751i \(0.448128\pi\)
\(450\) 0 0
\(451\) −28.5760 −1.34559
\(452\) 0 0
\(453\) −9.86355 −0.463430
\(454\) 0 0
\(455\) 0.337828 0.0158376
\(456\) 0 0
\(457\) 23.2243 1.08638 0.543192 0.839608i \(-0.317215\pi\)
0.543192 + 0.839608i \(0.317215\pi\)
\(458\) 0 0
\(459\) −3.47517 −0.162207
\(460\) 0 0
\(461\) 31.7004 1.47643 0.738217 0.674564i \(-0.235668\pi\)
0.738217 + 0.674564i \(0.235668\pi\)
\(462\) 0 0
\(463\) 29.6451 1.37772 0.688862 0.724892i \(-0.258110\pi\)
0.688862 + 0.724892i \(0.258110\pi\)
\(464\) 0 0
\(465\) 12.6584 0.587018
\(466\) 0 0
\(467\) −33.5566 −1.55281 −0.776407 0.630232i \(-0.782960\pi\)
−0.776407 + 0.630232i \(0.782960\pi\)
\(468\) 0 0
\(469\) 0.724156 0.0334384
\(470\) 0 0
\(471\) −4.75666 −0.219176
\(472\) 0 0
\(473\) −53.6959 −2.46894
\(474\) 0 0
\(475\) −3.84447 −0.176396
\(476\) 0 0
\(477\) 8.00324 0.366443
\(478\) 0 0
\(479\) −15.5699 −0.711409 −0.355704 0.934599i \(-0.615759\pi\)
−0.355704 + 0.934599i \(0.615759\pi\)
\(480\) 0 0
\(481\) 1.44303 0.0657965
\(482\) 0 0
\(483\) 1.49015 0.0678042
\(484\) 0 0
\(485\) 3.96851 0.180201
\(486\) 0 0
\(487\) −6.07824 −0.275431 −0.137716 0.990472i \(-0.543976\pi\)
−0.137716 + 0.990472i \(0.543976\pi\)
\(488\) 0 0
\(489\) −5.52008 −0.249627
\(490\) 0 0
\(491\) −1.40028 −0.0631935 −0.0315968 0.999501i \(-0.510059\pi\)
−0.0315968 + 0.999501i \(0.510059\pi\)
\(492\) 0 0
\(493\) −3.47517 −0.156514
\(494\) 0 0
\(495\) 6.35868 0.285802
\(496\) 0 0
\(497\) −11.3896 −0.510893
\(498\) 0 0
\(499\) 29.9133 1.33910 0.669551 0.742766i \(-0.266487\pi\)
0.669551 + 0.742766i \(0.266487\pi\)
\(500\) 0 0
\(501\) 3.79049 0.169347
\(502\) 0 0
\(503\) 11.9620 0.533359 0.266680 0.963785i \(-0.414073\pi\)
0.266680 + 0.963785i \(0.414073\pi\)
\(504\) 0 0
\(505\) 12.0195 0.534859
\(506\) 0 0
\(507\) 12.9747 0.576227
\(508\) 0 0
\(509\) 9.60495 0.425732 0.212866 0.977081i \(-0.431720\pi\)
0.212866 + 0.977081i \(0.431720\pi\)
\(510\) 0 0
\(511\) 5.54099 0.245119
\(512\) 0 0
\(513\) −1.29574 −0.0572082
\(514\) 0 0
\(515\) 21.1125 0.930330
\(516\) 0 0
\(517\) −45.7210 −2.01081
\(518\) 0 0
\(519\) −23.1910 −1.01797
\(520\) 0 0
\(521\) 24.4955 1.07317 0.536584 0.843847i \(-0.319715\pi\)
0.536584 + 0.843847i \(0.319715\pi\)
\(522\) 0 0
\(523\) −16.4556 −0.719552 −0.359776 0.933039i \(-0.617147\pi\)
−0.359776 + 0.933039i \(0.617147\pi\)
\(524\) 0 0
\(525\) −4.42130 −0.192961
\(526\) 0 0
\(527\) −30.8522 −1.34394
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 12.2266 0.530590
\(532\) 0 0
\(533\) 1.01883 0.0441303
\(534\) 0 0
\(535\) 19.2855 0.833786
\(536\) 0 0
\(537\) 23.2446 1.00308
\(538\) 0 0
\(539\) −21.3146 −0.918087
\(540\) 0 0
\(541\) −25.3934 −1.09175 −0.545873 0.837868i \(-0.683802\pi\)
−0.545873 + 0.837868i \(0.683802\pi\)
\(542\) 0 0
\(543\) 4.88281 0.209541
\(544\) 0 0
\(545\) −18.3222 −0.784839
\(546\) 0 0
\(547\) −10.0145 −0.428190 −0.214095 0.976813i \(-0.568680\pi\)
−0.214095 + 0.976813i \(0.568680\pi\)
\(548\) 0 0
\(549\) −10.3000 −0.439592
\(550\) 0 0
\(551\) −1.29574 −0.0552002
\(552\) 0 0
\(553\) −18.4831 −0.785980
\(554\) 0 0
\(555\) 12.9403 0.549284
\(556\) 0 0
\(557\) −7.79216 −0.330164 −0.165082 0.986280i \(-0.552789\pi\)
−0.165082 + 0.986280i \(0.552789\pi\)
\(558\) 0 0
\(559\) 1.91443 0.0809718
\(560\) 0 0
\(561\) −15.4980 −0.654326
\(562\) 0 0
\(563\) 9.81478 0.413644 0.206822 0.978379i \(-0.433688\pi\)
0.206822 + 0.978379i \(0.433688\pi\)
\(564\) 0 0
\(565\) 0.557844 0.0234687
\(566\) 0 0
\(567\) −1.49015 −0.0625804
\(568\) 0 0
\(569\) 4.39753 0.184354 0.0921770 0.995743i \(-0.470617\pi\)
0.0921770 + 0.995743i \(0.470617\pi\)
\(570\) 0 0
\(571\) −1.66013 −0.0694743 −0.0347372 0.999396i \(-0.511059\pi\)
−0.0347372 + 0.999396i \(0.511059\pi\)
\(572\) 0 0
\(573\) 21.9079 0.915215
\(574\) 0 0
\(575\) −2.96701 −0.123733
\(576\) 0 0
\(577\) −29.3118 −1.22027 −0.610133 0.792299i \(-0.708884\pi\)
−0.610133 + 0.792299i \(0.708884\pi\)
\(578\) 0 0
\(579\) 8.29118 0.344570
\(580\) 0 0
\(581\) −1.94391 −0.0806470
\(582\) 0 0
\(583\) 35.6916 1.47819
\(584\) 0 0
\(585\) −0.226708 −0.00937320
\(586\) 0 0
\(587\) −35.0457 −1.44649 −0.723246 0.690590i \(-0.757351\pi\)
−0.723246 + 0.690590i \(0.757351\pi\)
\(588\) 0 0
\(589\) −11.5034 −0.473991
\(590\) 0 0
\(591\) −4.20469 −0.172958
\(592\) 0 0
\(593\) 39.1421 1.60737 0.803686 0.595053i \(-0.202869\pi\)
0.803686 + 0.595053i \(0.202869\pi\)
\(594\) 0 0
\(595\) −7.38368 −0.302701
\(596\) 0 0
\(597\) 12.9265 0.529046
\(598\) 0 0
\(599\) 18.8311 0.769417 0.384708 0.923038i \(-0.374302\pi\)
0.384708 + 0.923038i \(0.374302\pi\)
\(600\) 0 0
\(601\) −23.4667 −0.957228 −0.478614 0.878025i \(-0.658861\pi\)
−0.478614 + 0.878025i \(0.658861\pi\)
\(602\) 0 0
\(603\) −0.485962 −0.0197899
\(604\) 0 0
\(605\) 12.6734 0.515245
\(606\) 0 0
\(607\) −27.8701 −1.13121 −0.565607 0.824675i \(-0.691358\pi\)
−0.565607 + 0.824675i \(0.691358\pi\)
\(608\) 0 0
\(609\) −1.49015 −0.0603839
\(610\) 0 0
\(611\) 1.63010 0.0659468
\(612\) 0 0
\(613\) 29.5067 1.19176 0.595881 0.803072i \(-0.296803\pi\)
0.595881 + 0.803072i \(0.296803\pi\)
\(614\) 0 0
\(615\) 9.13628 0.368410
\(616\) 0 0
\(617\) −31.9522 −1.28635 −0.643175 0.765720i \(-0.722383\pi\)
−0.643175 + 0.765720i \(0.722383\pi\)
\(618\) 0 0
\(619\) −11.5996 −0.466227 −0.233114 0.972450i \(-0.574891\pi\)
−0.233114 + 0.972450i \(0.574891\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 27.1272 1.08683
\(624\) 0 0
\(625\) −1.36176 −0.0544703
\(626\) 0 0
\(627\) −5.77852 −0.230772
\(628\) 0 0
\(629\) −31.5393 −1.25755
\(630\) 0 0
\(631\) −23.1670 −0.922262 −0.461131 0.887332i \(-0.652556\pi\)
−0.461131 + 0.887332i \(0.652556\pi\)
\(632\) 0 0
\(633\) 24.3532 0.967952
\(634\) 0 0
\(635\) −16.2700 −0.645657
\(636\) 0 0
\(637\) 0.759936 0.0301098
\(638\) 0 0
\(639\) 7.64325 0.302362
\(640\) 0 0
\(641\) −12.5953 −0.497486 −0.248743 0.968569i \(-0.580017\pi\)
−0.248743 + 0.968569i \(0.580017\pi\)
\(642\) 0 0
\(643\) 20.8087 0.820617 0.410308 0.911947i \(-0.365421\pi\)
0.410308 + 0.911947i \(0.365421\pi\)
\(644\) 0 0
\(645\) 17.1675 0.675971
\(646\) 0 0
\(647\) 27.5547 1.08329 0.541643 0.840608i \(-0.317802\pi\)
0.541643 + 0.840608i \(0.317802\pi\)
\(648\) 0 0
\(649\) 54.5264 2.14035
\(650\) 0 0
\(651\) −13.2294 −0.518502
\(652\) 0 0
\(653\) −35.4803 −1.38845 −0.694227 0.719756i \(-0.744253\pi\)
−0.694227 + 0.719756i \(0.744253\pi\)
\(654\) 0 0
\(655\) 11.0573 0.432045
\(656\) 0 0
\(657\) −3.71841 −0.145069
\(658\) 0 0
\(659\) −18.5600 −0.722995 −0.361498 0.932373i \(-0.617734\pi\)
−0.361498 + 0.932373i \(0.617734\pi\)
\(660\) 0 0
\(661\) 31.9109 1.24119 0.620595 0.784131i \(-0.286891\pi\)
0.620595 + 0.784131i \(0.286891\pi\)
\(662\) 0 0
\(663\) 0.552554 0.0214594
\(664\) 0 0
\(665\) −2.75305 −0.106759
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) 8.47197 0.327545
\(670\) 0 0
\(671\) −45.9342 −1.77327
\(672\) 0 0
\(673\) −7.56551 −0.291629 −0.145814 0.989312i \(-0.546580\pi\)
−0.145814 + 0.989312i \(0.546580\pi\)
\(674\) 0 0
\(675\) 2.96701 0.114200
\(676\) 0 0
\(677\) −17.0944 −0.656989 −0.328495 0.944506i \(-0.606541\pi\)
−0.328495 + 0.944506i \(0.606541\pi\)
\(678\) 0 0
\(679\) −4.14753 −0.159168
\(680\) 0 0
\(681\) −17.6827 −0.677603
\(682\) 0 0
\(683\) 3.29592 0.126115 0.0630575 0.998010i \(-0.479915\pi\)
0.0630575 + 0.998010i \(0.479915\pi\)
\(684\) 0 0
\(685\) 11.9624 0.457059
\(686\) 0 0
\(687\) 4.93562 0.188306
\(688\) 0 0
\(689\) −1.27252 −0.0484792
\(690\) 0 0
\(691\) −4.51318 −0.171690 −0.0858448 0.996309i \(-0.527359\pi\)
−0.0858448 + 0.996309i \(0.527359\pi\)
\(692\) 0 0
\(693\) −6.64554 −0.252443
\(694\) 0 0
\(695\) −19.8340 −0.752348
\(696\) 0 0
\(697\) −22.2678 −0.843454
\(698\) 0 0
\(699\) 26.5583 1.00453
\(700\) 0 0
\(701\) −5.03168 −0.190044 −0.0950220 0.995475i \(-0.530292\pi\)
−0.0950220 + 0.995475i \(0.530292\pi\)
\(702\) 0 0
\(703\) −11.7596 −0.443522
\(704\) 0 0
\(705\) 14.6178 0.550539
\(706\) 0 0
\(707\) −12.5617 −0.472431
\(708\) 0 0
\(709\) −3.07674 −0.115549 −0.0577747 0.998330i \(-0.518400\pi\)
−0.0577747 + 0.998330i \(0.518400\pi\)
\(710\) 0 0
\(711\) 12.4035 0.465167
\(712\) 0 0
\(713\) −8.87791 −0.332480
\(714\) 0 0
\(715\) −1.01103 −0.0378106
\(716\) 0 0
\(717\) 0.0342729 0.00127994
\(718\) 0 0
\(719\) 22.0964 0.824057 0.412028 0.911171i \(-0.364820\pi\)
0.412028 + 0.911171i \(0.364820\pi\)
\(720\) 0 0
\(721\) −22.0650 −0.821743
\(722\) 0 0
\(723\) 15.3244 0.569922
\(724\) 0 0
\(725\) 2.96701 0.110192
\(726\) 0 0
\(727\) 18.9934 0.704425 0.352212 0.935920i \(-0.385430\pi\)
0.352212 + 0.935920i \(0.385430\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −41.8424 −1.54760
\(732\) 0 0
\(733\) −10.8286 −0.399964 −0.199982 0.979800i \(-0.564088\pi\)
−0.199982 + 0.979800i \(0.564088\pi\)
\(734\) 0 0
\(735\) 6.81468 0.251363
\(736\) 0 0
\(737\) −2.16722 −0.0798304
\(738\) 0 0
\(739\) 52.1079 1.91682 0.958410 0.285395i \(-0.0921247\pi\)
0.958410 + 0.285395i \(0.0921247\pi\)
\(740\) 0 0
\(741\) 0.206023 0.00756844
\(742\) 0 0
\(743\) 23.1036 0.847589 0.423795 0.905758i \(-0.360698\pi\)
0.423795 + 0.905758i \(0.360698\pi\)
\(744\) 0 0
\(745\) −15.3124 −0.561002
\(746\) 0 0
\(747\) 1.30451 0.0477294
\(748\) 0 0
\(749\) −20.1555 −0.736468
\(750\) 0 0
\(751\) −31.5778 −1.15229 −0.576146 0.817347i \(-0.695444\pi\)
−0.576146 + 0.817347i \(0.695444\pi\)
\(752\) 0 0
\(753\) −5.44938 −0.198586
\(754\) 0 0
\(755\) 14.0637 0.511832
\(756\) 0 0
\(757\) −4.93030 −0.179195 −0.0895974 0.995978i \(-0.528558\pi\)
−0.0895974 + 0.995978i \(0.528558\pi\)
\(758\) 0 0
\(759\) −4.45964 −0.161875
\(760\) 0 0
\(761\) 15.0272 0.544735 0.272368 0.962193i \(-0.412193\pi\)
0.272368 + 0.962193i \(0.412193\pi\)
\(762\) 0 0
\(763\) 19.1488 0.693233
\(764\) 0 0
\(765\) 4.95499 0.179148
\(766\) 0 0
\(767\) −1.94404 −0.0701952
\(768\) 0 0
\(769\) 49.6405 1.79008 0.895041 0.445984i \(-0.147146\pi\)
0.895041 + 0.445984i \(0.147146\pi\)
\(770\) 0 0
\(771\) 1.38441 0.0498585
\(772\) 0 0
\(773\) −4.88562 −0.175724 −0.0878618 0.996133i \(-0.528003\pi\)
−0.0878618 + 0.996133i \(0.528003\pi\)
\(774\) 0 0
\(775\) 26.3409 0.946193
\(776\) 0 0
\(777\) −13.5240 −0.485172
\(778\) 0 0
\(779\) −8.30269 −0.297475
\(780\) 0 0
\(781\) 34.0862 1.21970
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) 6.78219 0.242067
\(786\) 0 0
\(787\) −3.31990 −0.118342 −0.0591708 0.998248i \(-0.518846\pi\)
−0.0591708 + 0.998248i \(0.518846\pi\)
\(788\) 0 0
\(789\) −6.66705 −0.237353
\(790\) 0 0
\(791\) −0.583010 −0.0207294
\(792\) 0 0
\(793\) 1.63770 0.0581565
\(794\) 0 0
\(795\) −11.4112 −0.404715
\(796\) 0 0
\(797\) −28.8324 −1.02129 −0.510647 0.859790i \(-0.670594\pi\)
−0.510647 + 0.859790i \(0.670594\pi\)
\(798\) 0 0
\(799\) −35.6280 −1.26043
\(800\) 0 0
\(801\) −18.2043 −0.643218
\(802\) 0 0
\(803\) −16.5828 −0.585194
\(804\) 0 0
\(805\) −2.12470 −0.0748858
\(806\) 0 0
\(807\) −30.4318 −1.07125
\(808\) 0 0
\(809\) −44.7935 −1.57486 −0.787428 0.616407i \(-0.788588\pi\)
−0.787428 + 0.616407i \(0.788588\pi\)
\(810\) 0 0
\(811\) −17.1891 −0.603591 −0.301795 0.953373i \(-0.597586\pi\)
−0.301795 + 0.953373i \(0.597586\pi\)
\(812\) 0 0
\(813\) 18.2818 0.641170
\(814\) 0 0
\(815\) 7.87069 0.275698
\(816\) 0 0
\(817\) −15.6012 −0.545816
\(818\) 0 0
\(819\) 0.236935 0.00827917
\(820\) 0 0
\(821\) −5.07521 −0.177126 −0.0885630 0.996071i \(-0.528227\pi\)
−0.0885630 + 0.996071i \(0.528227\pi\)
\(822\) 0 0
\(823\) 34.8400 1.21444 0.607222 0.794532i \(-0.292284\pi\)
0.607222 + 0.794532i \(0.292284\pi\)
\(824\) 0 0
\(825\) 13.2318 0.460673
\(826\) 0 0
\(827\) −6.00246 −0.208726 −0.104363 0.994539i \(-0.533280\pi\)
−0.104363 + 0.994539i \(0.533280\pi\)
\(828\) 0 0
\(829\) −6.25611 −0.217284 −0.108642 0.994081i \(-0.534650\pi\)
−0.108642 + 0.994081i \(0.534650\pi\)
\(830\) 0 0
\(831\) −4.36334 −0.151363
\(832\) 0 0
\(833\) −16.6094 −0.575481
\(834\) 0 0
\(835\) −5.40459 −0.187034
\(836\) 0 0
\(837\) 8.87791 0.306866
\(838\) 0 0
\(839\) −0.807807 −0.0278886 −0.0139443 0.999903i \(-0.504439\pi\)
−0.0139443 + 0.999903i \(0.504439\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 14.1143 0.486123
\(844\) 0 0
\(845\) −18.4997 −0.636410
\(846\) 0 0
\(847\) −13.2451 −0.455106
\(848\) 0 0
\(849\) −2.84332 −0.0975826
\(850\) 0 0
\(851\) −9.07562 −0.311108
\(852\) 0 0
\(853\) 17.6367 0.603868 0.301934 0.953329i \(-0.402368\pi\)
0.301934 + 0.953329i \(0.402368\pi\)
\(854\) 0 0
\(855\) 1.84750 0.0631831
\(856\) 0 0
\(857\) −26.1242 −0.892387 −0.446193 0.894937i \(-0.647221\pi\)
−0.446193 + 0.894937i \(0.647221\pi\)
\(858\) 0 0
\(859\) 51.9607 1.77288 0.886438 0.462848i \(-0.153172\pi\)
0.886438 + 0.462848i \(0.153172\pi\)
\(860\) 0 0
\(861\) −9.54843 −0.325410
\(862\) 0 0
\(863\) 37.1896 1.26595 0.632975 0.774172i \(-0.281834\pi\)
0.632975 + 0.774172i \(0.281834\pi\)
\(864\) 0 0
\(865\) 33.0664 1.12429
\(866\) 0 0
\(867\) 4.92322 0.167201
\(868\) 0 0
\(869\) 55.3151 1.87644
\(870\) 0 0
\(871\) 0.0772682 0.00261813
\(872\) 0 0
\(873\) 2.78330 0.0942004
\(874\) 0 0
\(875\) 16.9275 0.572254
\(876\) 0 0
\(877\) 2.89445 0.0977385 0.0488692 0.998805i \(-0.484438\pi\)
0.0488692 + 0.998805i \(0.484438\pi\)
\(878\) 0 0
\(879\) 9.50985 0.320759
\(880\) 0 0
\(881\) −7.22220 −0.243322 −0.121661 0.992572i \(-0.538822\pi\)
−0.121661 + 0.992572i \(0.538822\pi\)
\(882\) 0 0
\(883\) 19.5749 0.658747 0.329374 0.944200i \(-0.393162\pi\)
0.329374 + 0.944200i \(0.393162\pi\)
\(884\) 0 0
\(885\) −17.4331 −0.586006
\(886\) 0 0
\(887\) 14.0585 0.472038 0.236019 0.971748i \(-0.424157\pi\)
0.236019 + 0.971748i \(0.424157\pi\)
\(888\) 0 0
\(889\) 17.0040 0.570296
\(890\) 0 0
\(891\) 4.45964 0.149404
\(892\) 0 0
\(893\) −13.2841 −0.444536
\(894\) 0 0
\(895\) −33.1428 −1.10784
\(896\) 0 0
\(897\) 0.159001 0.00530888
\(898\) 0 0
\(899\) 8.87791 0.296095
\(900\) 0 0
\(901\) 27.8126 0.926572
\(902\) 0 0
\(903\) −17.9420 −0.597073
\(904\) 0 0
\(905\) −6.96205 −0.231426
\(906\) 0 0
\(907\) 53.7323 1.78415 0.892076 0.451886i \(-0.149249\pi\)
0.892076 + 0.451886i \(0.149249\pi\)
\(908\) 0 0
\(909\) 8.42981 0.279599
\(910\) 0 0
\(911\) 20.9457 0.693962 0.346981 0.937872i \(-0.387207\pi\)
0.346981 + 0.937872i \(0.387207\pi\)
\(912\) 0 0
\(913\) 5.81763 0.192536
\(914\) 0 0
\(915\) 14.6860 0.485504
\(916\) 0 0
\(917\) −11.5561 −0.381617
\(918\) 0 0
\(919\) 27.6524 0.912169 0.456085 0.889936i \(-0.349251\pi\)
0.456085 + 0.889936i \(0.349251\pi\)
\(920\) 0 0
\(921\) 5.05037 0.166415
\(922\) 0 0
\(923\) −1.21528 −0.0400015
\(924\) 0 0
\(925\) 26.9275 0.885371
\(926\) 0 0
\(927\) 14.8072 0.486333
\(928\) 0 0
\(929\) 16.7689 0.550168 0.275084 0.961420i \(-0.411294\pi\)
0.275084 + 0.961420i \(0.411294\pi\)
\(930\) 0 0
\(931\) −6.19291 −0.202964
\(932\) 0 0
\(933\) −32.6238 −1.06806
\(934\) 0 0
\(935\) 22.0975 0.722665
\(936\) 0 0
\(937\) −38.0173 −1.24197 −0.620985 0.783822i \(-0.713267\pi\)
−0.620985 + 0.783822i \(0.713267\pi\)
\(938\) 0 0
\(939\) 6.08094 0.198444
\(940\) 0 0
\(941\) −0.763245 −0.0248811 −0.0124405 0.999923i \(-0.503960\pi\)
−0.0124405 + 0.999923i \(0.503960\pi\)
\(942\) 0 0
\(943\) −6.40770 −0.208663
\(944\) 0 0
\(945\) 2.12470 0.0691164
\(946\) 0 0
\(947\) −27.1708 −0.882932 −0.441466 0.897278i \(-0.645541\pi\)
−0.441466 + 0.897278i \(0.645541\pi\)
\(948\) 0 0
\(949\) 0.591229 0.0191921
\(950\) 0 0
\(951\) −26.3432 −0.854237
\(952\) 0 0
\(953\) 24.2213 0.784605 0.392302 0.919836i \(-0.371679\pi\)
0.392302 + 0.919836i \(0.371679\pi\)
\(954\) 0 0
\(955\) −31.2369 −1.01080
\(956\) 0 0
\(957\) 4.45964 0.144160
\(958\) 0 0
\(959\) −12.5020 −0.403712
\(960\) 0 0
\(961\) 47.8173 1.54249
\(962\) 0 0
\(963\) 13.5258 0.435864
\(964\) 0 0
\(965\) −11.8218 −0.380557
\(966\) 0 0
\(967\) 29.2987 0.942184 0.471092 0.882084i \(-0.343860\pi\)
0.471092 + 0.882084i \(0.343860\pi\)
\(968\) 0 0
\(969\) −4.50290 −0.144654
\(970\) 0 0
\(971\) −41.1612 −1.32092 −0.660462 0.750859i \(-0.729640\pi\)
−0.660462 + 0.750859i \(0.729640\pi\)
\(972\) 0 0
\(973\) 20.7288 0.664535
\(974\) 0 0
\(975\) −0.471757 −0.0151083
\(976\) 0 0
\(977\) 12.6314 0.404115 0.202057 0.979374i \(-0.435237\pi\)
0.202057 + 0.979374i \(0.435237\pi\)
\(978\) 0 0
\(979\) −81.1848 −2.59468
\(980\) 0 0
\(981\) −12.8502 −0.410277
\(982\) 0 0
\(983\) −9.56242 −0.304994 −0.152497 0.988304i \(-0.548731\pi\)
−0.152497 + 0.988304i \(0.548731\pi\)
\(984\) 0 0
\(985\) 5.99516 0.191022
\(986\) 0 0
\(987\) −15.2773 −0.486281
\(988\) 0 0
\(989\) −12.0404 −0.382862
\(990\) 0 0
\(991\) 18.8307 0.598177 0.299089 0.954225i \(-0.403317\pi\)
0.299089 + 0.954225i \(0.403317\pi\)
\(992\) 0 0
\(993\) 2.49309 0.0791157
\(994\) 0 0
\(995\) −18.4310 −0.584300
\(996\) 0 0
\(997\) −39.5586 −1.25283 −0.626416 0.779489i \(-0.715479\pi\)
−0.626416 + 0.779489i \(0.715479\pi\)
\(998\) 0 0
\(999\) 9.07562 0.287140
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 8004.2.a.g.1.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.g.1.9 12 1.1 even 1 trivial