Properties

Label 684.3.e.a.305.5
Level $684$
Weight $3$
Character 684.305
Analytic conductor $18.638$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 156 x^{10} + 8721 x^{8} + 208784 x^{6} + 2024760 x^{4} + 7117056 x^{2} + 6533136\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 305.5
Root \(-2.16068i\) of defining polynomial
Character \(\chi\) \(=\) 684.305
Dual form 684.3.e.a.305.8

$q$-expansion

\(f(q)\) \(=\) \(q-2.16068i q^{5} -9.11429 q^{7} +O(q^{10})\) \(q-2.16068i q^{5} -9.11429 q^{7} -1.10046i q^{11} +2.28570 q^{13} +19.2684i q^{17} +4.35890 q^{19} +1.28620i q^{23} +20.3315 q^{25} +37.0190i q^{29} +31.3785 q^{31} +19.6930i q^{35} +34.9511 q^{37} +43.5088i q^{41} +0.553740 q^{43} -9.35139i q^{47} +34.0703 q^{49} +75.0713i q^{53} -2.37773 q^{55} -38.1174i q^{59} -62.4816 q^{61} -4.93866i q^{65} +85.6623 q^{67} -15.9252i q^{71} +67.1233 q^{73} +10.0299i q^{77} -58.5499 q^{79} +115.564i q^{83} +41.6328 q^{85} -6.59543i q^{89} -20.8325 q^{91} -9.41818i q^{95} -14.0228 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{7} + O(q^{10}) \) \( 12 q + 16 q^{7} - 16 q^{13} - 12 q^{25} - 40 q^{31} - 32 q^{37} + 92 q^{43} - 84 q^{55} - 48 q^{61} - 88 q^{67} + 148 q^{73} - 56 q^{79} + 228 q^{85} - 8 q^{91} + 72 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) − 2.16068i − 0.432136i −0.976378 0.216068i \(-0.930677\pi\)
0.976378 0.216068i \(-0.0693232\pi\)
\(6\) 0 0
\(7\) −9.11429 −1.30204 −0.651021 0.759060i \(-0.725659\pi\)
−0.651021 + 0.759060i \(0.725659\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.10046i − 0.100041i −0.998748 0.0500207i \(-0.984071\pi\)
0.998748 0.0500207i \(-0.0159287\pi\)
\(12\) 0 0
\(13\) 2.28570 0.175823 0.0879116 0.996128i \(-0.471981\pi\)
0.0879116 + 0.996128i \(0.471981\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 19.2684i 1.13343i 0.823913 + 0.566717i \(0.191787\pi\)
−0.823913 + 0.566717i \(0.808213\pi\)
\(18\) 0 0
\(19\) 4.35890 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.28620i 0.0559216i 0.999609 + 0.0279608i \(0.00890136\pi\)
−0.999609 + 0.0279608i \(0.991099\pi\)
\(24\) 0 0
\(25\) 20.3315 0.813259
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 37.0190i 1.27652i 0.769821 + 0.638259i \(0.220345\pi\)
−0.769821 + 0.638259i \(0.779655\pi\)
\(30\) 0 0
\(31\) 31.3785 1.01221 0.506104 0.862472i \(-0.331085\pi\)
0.506104 + 0.862472i \(0.331085\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 19.6930i 0.562659i
\(36\) 0 0
\(37\) 34.9511 0.944625 0.472312 0.881431i \(-0.343419\pi\)
0.472312 + 0.881431i \(0.343419\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 43.5088i 1.06119i 0.847626 + 0.530595i \(0.178031\pi\)
−0.847626 + 0.530595i \(0.821969\pi\)
\(42\) 0 0
\(43\) 0.553740 0.0128777 0.00643884 0.999979i \(-0.497950\pi\)
0.00643884 + 0.999979i \(0.497950\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 9.35139i − 0.198966i −0.995039 0.0994828i \(-0.968281\pi\)
0.995039 0.0994828i \(-0.0317188\pi\)
\(48\) 0 0
\(49\) 34.0703 0.695312
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 75.0713i 1.41644i 0.705992 + 0.708220i \(0.250501\pi\)
−0.705992 + 0.708220i \(0.749499\pi\)
\(54\) 0 0
\(55\) −2.37773 −0.0432315
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 38.1174i − 0.646057i −0.946389 0.323028i \(-0.895299\pi\)
0.946389 0.323028i \(-0.104701\pi\)
\(60\) 0 0
\(61\) −62.4816 −1.02429 −0.512145 0.858899i \(-0.671149\pi\)
−0.512145 + 0.858899i \(0.671149\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 4.93866i − 0.0759794i
\(66\) 0 0
\(67\) 85.6623 1.27854 0.639271 0.768982i \(-0.279236\pi\)
0.639271 + 0.768982i \(0.279236\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 15.9252i − 0.224298i −0.993691 0.112149i \(-0.964227\pi\)
0.993691 0.112149i \(-0.0357734\pi\)
\(72\) 0 0
\(73\) 67.1233 0.919497 0.459748 0.888049i \(-0.347940\pi\)
0.459748 + 0.888049i \(0.347940\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.0299i 0.130258i
\(78\) 0 0
\(79\) −58.5499 −0.741138 −0.370569 0.928805i \(-0.620837\pi\)
−0.370569 + 0.928805i \(0.620837\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 115.564i 1.39233i 0.717881 + 0.696166i \(0.245112\pi\)
−0.717881 + 0.696166i \(0.754888\pi\)
\(84\) 0 0
\(85\) 41.6328 0.489797
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 6.59543i − 0.0741060i −0.999313 0.0370530i \(-0.988203\pi\)
0.999313 0.0370530i \(-0.0117970\pi\)
\(90\) 0 0
\(91\) −20.8325 −0.228929
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 9.41818i − 0.0991387i
\(96\) 0 0
\(97\) −14.0228 −0.144565 −0.0722825 0.997384i \(-0.523028\pi\)
−0.0722825 + 0.997384i \(0.523028\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 99.4086i 0.984243i 0.870526 + 0.492122i \(0.163779\pi\)
−0.870526 + 0.492122i \(0.836221\pi\)
\(102\) 0 0
\(103\) 10.8374 0.105217 0.0526087 0.998615i \(-0.483246\pi\)
0.0526087 + 0.998615i \(0.483246\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 133.136i 1.24426i 0.782913 + 0.622131i \(0.213733\pi\)
−0.782913 + 0.622131i \(0.786267\pi\)
\(108\) 0 0
\(109\) 8.56068 0.0785384 0.0392692 0.999229i \(-0.487497\pi\)
0.0392692 + 0.999229i \(0.487497\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 190.600i 1.68672i 0.537345 + 0.843362i \(0.319427\pi\)
−0.537345 + 0.843362i \(0.680573\pi\)
\(114\) 0 0
\(115\) 2.77906 0.0241657
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 175.618i − 1.47578i
\(120\) 0 0
\(121\) 119.789 0.989992
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 97.9467i − 0.783574i
\(126\) 0 0
\(127\) −66.8407 −0.526305 −0.263153 0.964754i \(-0.584762\pi\)
−0.263153 + 0.964754i \(0.584762\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 75.5187i 0.576479i 0.957558 + 0.288239i \(0.0930699\pi\)
−0.957558 + 0.288239i \(0.906930\pi\)
\(132\) 0 0
\(133\) −39.7283 −0.298709
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 169.943i − 1.24046i −0.784420 0.620230i \(-0.787039\pi\)
0.784420 0.620230i \(-0.212961\pi\)
\(138\) 0 0
\(139\) 130.111 0.936052 0.468026 0.883715i \(-0.344965\pi\)
0.468026 + 0.883715i \(0.344965\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 2.51531i − 0.0175896i
\(144\) 0 0
\(145\) 79.9863 0.551629
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 63.5185i − 0.426298i −0.977020 0.213149i \(-0.931628\pi\)
0.977020 0.213149i \(-0.0683720\pi\)
\(150\) 0 0
\(151\) −5.44525 −0.0360613 −0.0180306 0.999837i \(-0.505740\pi\)
−0.0180306 + 0.999837i \(0.505740\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 67.7988i − 0.437412i
\(156\) 0 0
\(157\) −111.650 −0.711143 −0.355572 0.934649i \(-0.615714\pi\)
−0.355572 + 0.934649i \(0.615714\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 11.7228i − 0.0728122i
\(162\) 0 0
\(163\) −234.008 −1.43563 −0.717817 0.696232i \(-0.754858\pi\)
−0.717817 + 0.696232i \(0.754858\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 146.835i − 0.879253i −0.898181 0.439627i \(-0.855111\pi\)
0.898181 0.439627i \(-0.144889\pi\)
\(168\) 0 0
\(169\) −163.776 −0.969086
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 41.9823i − 0.242672i −0.992611 0.121336i \(-0.961282\pi\)
0.992611 0.121336i \(-0.0387179\pi\)
\(174\) 0 0
\(175\) −185.307 −1.05890
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 234.866i − 1.31210i −0.754717 0.656051i \(-0.772226\pi\)
0.754717 0.656051i \(-0.227774\pi\)
\(180\) 0 0
\(181\) 95.8804 0.529726 0.264863 0.964286i \(-0.414673\pi\)
0.264863 + 0.964286i \(0.414673\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 75.5181i − 0.408206i
\(186\) 0 0
\(187\) 21.2040 0.113390
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 164.583i − 0.861691i −0.902426 0.430845i \(-0.858215\pi\)
0.902426 0.430845i \(-0.141785\pi\)
\(192\) 0 0
\(193\) 185.715 0.962255 0.481128 0.876651i \(-0.340227\pi\)
0.481128 + 0.876651i \(0.340227\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 159.624i 0.810275i 0.914256 + 0.405137i \(0.132776\pi\)
−0.914256 + 0.405137i \(0.867224\pi\)
\(198\) 0 0
\(199\) −291.623 −1.46544 −0.732721 0.680529i \(-0.761750\pi\)
−0.732721 + 0.680529i \(0.761750\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 337.402i − 1.66208i
\(204\) 0 0
\(205\) 94.0084 0.458578
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 4.79678i − 0.0229511i
\(210\) 0 0
\(211\) −139.147 −0.659464 −0.329732 0.944075i \(-0.606958\pi\)
−0.329732 + 0.944075i \(0.606958\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 1.19645i − 0.00556490i
\(216\) 0 0
\(217\) −285.993 −1.31794
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 44.0417i 0.199284i
\(222\) 0 0
\(223\) 98.6258 0.442268 0.221134 0.975243i \(-0.429024\pi\)
0.221134 + 0.975243i \(0.429024\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 103.162i 0.454460i 0.973841 + 0.227230i \(0.0729669\pi\)
−0.973841 + 0.227230i \(0.927033\pi\)
\(228\) 0 0
\(229\) −8.27807 −0.0361488 −0.0180744 0.999837i \(-0.505754\pi\)
−0.0180744 + 0.999837i \(0.505754\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 164.236i 0.704874i 0.935835 + 0.352437i \(0.114647\pi\)
−0.935835 + 0.352437i \(0.885353\pi\)
\(234\) 0 0
\(235\) −20.2053 −0.0859802
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 306.024i − 1.28044i −0.768194 0.640218i \(-0.778844\pi\)
0.768194 0.640218i \(-0.221156\pi\)
\(240\) 0 0
\(241\) 159.742 0.662829 0.331414 0.943485i \(-0.392474\pi\)
0.331414 + 0.943485i \(0.392474\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 73.6149i − 0.300469i
\(246\) 0 0
\(247\) 9.96314 0.0403366
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 311.540i − 1.24120i −0.784129 0.620598i \(-0.786890\pi\)
0.784129 0.620598i \(-0.213110\pi\)
\(252\) 0 0
\(253\) 1.41540 0.00559448
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 299.046i 1.16360i 0.813331 + 0.581802i \(0.197652\pi\)
−0.813331 + 0.581802i \(0.802348\pi\)
\(258\) 0 0
\(259\) −318.555 −1.22994
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 256.742i 0.976207i 0.872786 + 0.488104i \(0.162311\pi\)
−0.872786 + 0.488104i \(0.837689\pi\)
\(264\) 0 0
\(265\) 162.205 0.612094
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 101.739i − 0.378212i −0.981957 0.189106i \(-0.939441\pi\)
0.981957 0.189106i \(-0.0605589\pi\)
\(270\) 0 0
\(271\) −413.450 −1.52565 −0.762823 0.646608i \(-0.776187\pi\)
−0.762823 + 0.646608i \(0.776187\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 22.3739i − 0.0813596i
\(276\) 0 0
\(277\) −52.8385 −0.190753 −0.0953764 0.995441i \(-0.530405\pi\)
−0.0953764 + 0.995441i \(0.530405\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 249.276i 0.887102i 0.896249 + 0.443551i \(0.146281\pi\)
−0.896249 + 0.443551i \(0.853719\pi\)
\(282\) 0 0
\(283\) 264.801 0.935691 0.467846 0.883810i \(-0.345030\pi\)
0.467846 + 0.883810i \(0.345030\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 396.551i − 1.38171i
\(288\) 0 0
\(289\) −82.2702 −0.284672
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 140.716i − 0.480261i −0.970741 0.240130i \(-0.922810\pi\)
0.970741 0.240130i \(-0.0771902\pi\)
\(294\) 0 0
\(295\) −82.3594 −0.279184
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.93986i 0.00983231i
\(300\) 0 0
\(301\) −5.04695 −0.0167673
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 135.003i 0.442632i
\(306\) 0 0
\(307\) −224.226 −0.730379 −0.365189 0.930933i \(-0.618996\pi\)
−0.365189 + 0.930933i \(0.618996\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 190.561i − 0.612737i −0.951913 0.306369i \(-0.900886\pi\)
0.951913 0.306369i \(-0.0991140\pi\)
\(312\) 0 0
\(313\) 346.633 1.10745 0.553727 0.832698i \(-0.313205\pi\)
0.553727 + 0.832698i \(0.313205\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 394.319i − 1.24391i −0.783054 0.621954i \(-0.786339\pi\)
0.783054 0.621954i \(-0.213661\pi\)
\(318\) 0 0
\(319\) 40.7378 0.127705
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 83.9889i 0.260028i
\(324\) 0 0
\(325\) 46.4716 0.142990
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 85.2313i 0.259062i
\(330\) 0 0
\(331\) −207.167 −0.625882 −0.312941 0.949773i \(-0.601314\pi\)
−0.312941 + 0.949773i \(0.601314\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 185.089i − 0.552504i
\(336\) 0 0
\(337\) 288.916 0.857319 0.428659 0.903466i \(-0.358986\pi\)
0.428659 + 0.903466i \(0.358986\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 34.5306i − 0.101263i
\(342\) 0 0
\(343\) 136.074 0.396717
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 111.771i − 0.322106i −0.986946 0.161053i \(-0.948511\pi\)
0.986946 0.161053i \(-0.0514891\pi\)
\(348\) 0 0
\(349\) −364.709 −1.04501 −0.522505 0.852636i \(-0.675003\pi\)
−0.522505 + 0.852636i \(0.675003\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 227.805i 0.645341i 0.946511 + 0.322671i \(0.104581\pi\)
−0.946511 + 0.322671i \(0.895419\pi\)
\(354\) 0 0
\(355\) −34.4091 −0.0969272
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 125.832i 0.350506i 0.984523 + 0.175253i \(0.0560744\pi\)
−0.984523 + 0.175253i \(0.943926\pi\)
\(360\) 0 0
\(361\) 19.0000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 145.032i − 0.397347i
\(366\) 0 0
\(367\) 372.047 1.01375 0.506876 0.862019i \(-0.330800\pi\)
0.506876 + 0.862019i \(0.330800\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 684.221i − 1.84426i
\(372\) 0 0
\(373\) 122.215 0.327655 0.163828 0.986489i \(-0.447616\pi\)
0.163828 + 0.986489i \(0.447616\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 84.6144i 0.224442i
\(378\) 0 0
\(379\) −76.5257 −0.201915 −0.100957 0.994891i \(-0.532191\pi\)
−0.100957 + 0.994891i \(0.532191\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 494.820i 1.29196i 0.763355 + 0.645979i \(0.223551\pi\)
−0.763355 + 0.645979i \(0.776449\pi\)
\(384\) 0 0
\(385\) 21.6713 0.0562892
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 685.225i 1.76150i 0.473578 + 0.880752i \(0.342962\pi\)
−0.473578 + 0.880752i \(0.657038\pi\)
\(390\) 0 0
\(391\) −24.7829 −0.0633834
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 126.508i 0.320272i
\(396\) 0 0
\(397\) −171.558 −0.432136 −0.216068 0.976378i \(-0.569323\pi\)
−0.216068 + 0.976378i \(0.569323\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 704.126i 1.75593i 0.478729 + 0.877963i \(0.341098\pi\)
−0.478729 + 0.877963i \(0.658902\pi\)
\(402\) 0 0
\(403\) 71.7218 0.177970
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 38.4622i − 0.0945016i
\(408\) 0 0
\(409\) 223.677 0.546889 0.273444 0.961888i \(-0.411837\pi\)
0.273444 + 0.961888i \(0.411837\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 347.413i 0.841193i
\(414\) 0 0
\(415\) 249.696 0.601676
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 527.562i 1.25910i 0.776961 + 0.629549i \(0.216760\pi\)
−0.776961 + 0.629549i \(0.783240\pi\)
\(420\) 0 0
\(421\) 239.059 0.567836 0.283918 0.958849i \(-0.408366\pi\)
0.283918 + 0.958849i \(0.408366\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 391.754i 0.921775i
\(426\) 0 0
\(427\) 569.476 1.33367
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 165.894i 0.384905i 0.981306 + 0.192452i \(0.0616441\pi\)
−0.981306 + 0.192452i \(0.938356\pi\)
\(432\) 0 0
\(433\) 266.348 0.615123 0.307561 0.951528i \(-0.400487\pi\)
0.307561 + 0.951528i \(0.400487\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5.60640i 0.0128293i
\(438\) 0 0
\(439\) 216.344 0.492811 0.246406 0.969167i \(-0.420750\pi\)
0.246406 + 0.969167i \(0.420750\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 307.303i 0.693686i 0.937923 + 0.346843i \(0.112746\pi\)
−0.937923 + 0.346843i \(0.887254\pi\)
\(444\) 0 0
\(445\) −14.2506 −0.0320238
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 652.614i − 1.45348i −0.686911 0.726741i \(-0.741034\pi\)
0.686911 0.726741i \(-0.258966\pi\)
\(450\) 0 0
\(451\) 47.8795 0.106163
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 45.0124i 0.0989284i
\(456\) 0 0
\(457\) 141.214 0.309002 0.154501 0.987993i \(-0.450623\pi\)
0.154501 + 0.987993i \(0.450623\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 43.3889i − 0.0941190i −0.998892 0.0470595i \(-0.985015\pi\)
0.998892 0.0470595i \(-0.0149850\pi\)
\(462\) 0 0
\(463\) 640.051 1.38240 0.691199 0.722664i \(-0.257083\pi\)
0.691199 + 0.722664i \(0.257083\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 80.7873i − 0.172992i −0.996252 0.0864961i \(-0.972433\pi\)
0.996252 0.0864961i \(-0.0275670\pi\)
\(468\) 0 0
\(469\) −780.751 −1.66471
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 0.609367i − 0.00128830i
\(474\) 0 0
\(475\) 88.6228 0.186574
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 585.566i 1.22248i 0.791447 + 0.611238i \(0.209328\pi\)
−0.791447 + 0.611238i \(0.790672\pi\)
\(480\) 0 0
\(481\) 79.8878 0.166087
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 30.2988i 0.0624717i
\(486\) 0 0
\(487\) −314.449 −0.645686 −0.322843 0.946453i \(-0.604639\pi\)
−0.322843 + 0.946453i \(0.604639\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 422.460i 0.860408i 0.902732 + 0.430204i \(0.141558\pi\)
−0.902732 + 0.430204i \(0.858442\pi\)
\(492\) 0 0
\(493\) −713.297 −1.44685
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 145.147i 0.292045i
\(498\) 0 0
\(499\) 975.125 1.95416 0.977079 0.212877i \(-0.0682833\pi\)
0.977079 + 0.212877i \(0.0682833\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 675.102i 1.34215i 0.741389 + 0.671076i \(0.234168\pi\)
−0.741389 + 0.671076i \(0.765832\pi\)
\(504\) 0 0
\(505\) 214.790 0.425327
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 3.87656i − 0.00761603i −0.999993 0.00380801i \(-0.998788\pi\)
0.999993 0.00380801i \(-0.00121213\pi\)
\(510\) 0 0
\(511\) −611.781 −1.19722
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 23.4161i − 0.0454682i
\(516\) 0 0
\(517\) −10.2908 −0.0199048
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 890.215i − 1.70867i −0.519726 0.854333i \(-0.673966\pi\)
0.519726 0.854333i \(-0.326034\pi\)
\(522\) 0 0
\(523\) −786.449 −1.50373 −0.751864 0.659319i \(-0.770845\pi\)
−0.751864 + 0.659319i \(0.770845\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 604.612i 1.14727i
\(528\) 0 0
\(529\) 527.346 0.996873
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 99.4480i 0.186582i
\(534\) 0 0
\(535\) 287.664 0.537690
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 37.4928i − 0.0695600i
\(540\) 0 0
\(541\) 462.603 0.855088 0.427544 0.903995i \(-0.359379\pi\)
0.427544 + 0.903995i \(0.359379\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 18.4969i − 0.0339392i
\(546\) 0 0
\(547\) −980.369 −1.79227 −0.896133 0.443786i \(-0.853635\pi\)
−0.896133 + 0.443786i \(0.853635\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 161.362i 0.292853i
\(552\) 0 0
\(553\) 533.641 0.964993
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 22.4835i − 0.0403654i −0.999796 0.0201827i \(-0.993575\pi\)
0.999796 0.0201827i \(-0.00642479\pi\)
\(558\) 0 0
\(559\) 1.26568 0.00226419
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 620.228i − 1.10165i −0.834621 0.550824i \(-0.814313\pi\)
0.834621 0.550824i \(-0.185687\pi\)
\(564\) 0 0
\(565\) 411.825 0.728894
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 160.563i − 0.282184i −0.989997 0.141092i \(-0.954939\pi\)
0.989997 0.141092i \(-0.0450613\pi\)
\(570\) 0 0
\(571\) 541.216 0.947840 0.473920 0.880568i \(-0.342839\pi\)
0.473920 + 0.880568i \(0.342839\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 26.1503i 0.0454787i
\(576\) 0 0
\(577\) −43.5123 −0.0754112 −0.0377056 0.999289i \(-0.512005\pi\)
−0.0377056 + 0.999289i \(0.512005\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 1053.28i − 1.81287i
\(582\) 0 0
\(583\) 82.6126 0.141703
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 64.7751i − 0.110349i −0.998477 0.0551747i \(-0.982428\pi\)
0.998477 0.0551747i \(-0.0175716\pi\)
\(588\) 0 0
\(589\) 136.776 0.232217
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 702.719i 1.18502i 0.805562 + 0.592512i \(0.201864\pi\)
−0.805562 + 0.592512i \(0.798136\pi\)
\(594\) 0 0
\(595\) −379.453 −0.637736
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 858.099i − 1.43255i −0.697817 0.716276i \(-0.745845\pi\)
0.697817 0.716276i \(-0.254155\pi\)
\(600\) 0 0
\(601\) −1116.90 −1.85840 −0.929200 0.369577i \(-0.879502\pi\)
−0.929200 + 0.369577i \(0.879502\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 258.825i − 0.427811i
\(606\) 0 0
\(607\) 1135.10 1.87002 0.935012 0.354616i \(-0.115388\pi\)
0.935012 + 0.354616i \(0.115388\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 21.3745i − 0.0349828i
\(612\) 0 0
\(613\) 228.738 0.373145 0.186573 0.982441i \(-0.440262\pi\)
0.186573 + 0.982441i \(0.440262\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1084.17i 1.75716i 0.477591 + 0.878582i \(0.341510\pi\)
−0.477591 + 0.878582i \(0.658490\pi\)
\(618\) 0 0
\(619\) −479.659 −0.774893 −0.387446 0.921892i \(-0.626643\pi\)
−0.387446 + 0.921892i \(0.626643\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 60.1127i 0.0964890i
\(624\) 0 0
\(625\) 296.655 0.474649
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 673.451i 1.07067i
\(630\) 0 0
\(631\) −39.9974 −0.0633873 −0.0316937 0.999498i \(-0.510090\pi\)
−0.0316937 + 0.999498i \(0.510090\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 144.421i 0.227435i
\(636\) 0 0
\(637\) 77.8745 0.122252
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1121.58i 1.74973i 0.484364 + 0.874866i \(0.339051\pi\)
−0.484364 + 0.874866i \(0.660949\pi\)
\(642\) 0 0
\(643\) 653.314 1.01604 0.508020 0.861345i \(-0.330378\pi\)
0.508020 + 0.861345i \(0.330378\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 475.997i 0.735698i 0.929885 + 0.367849i \(0.119906\pi\)
−0.929885 + 0.367849i \(0.880094\pi\)
\(648\) 0 0
\(649\) −41.9465 −0.0646325
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 949.403i − 1.45391i −0.686685 0.726955i \(-0.740935\pi\)
0.686685 0.726955i \(-0.259065\pi\)
\(654\) 0 0
\(655\) 163.172 0.249117
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 462.803i − 0.702281i −0.936323 0.351140i \(-0.885794\pi\)
0.936323 0.351140i \(-0.114206\pi\)
\(660\) 0 0
\(661\) 395.105 0.597738 0.298869 0.954294i \(-0.403391\pi\)
0.298869 + 0.954294i \(0.403391\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 85.8400i 0.129083i
\(666\) 0 0
\(667\) −47.6138 −0.0713849
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 68.7583i 0.102471i
\(672\) 0 0
\(673\) −718.411 −1.06748 −0.533738 0.845650i \(-0.679213\pi\)
−0.533738 + 0.845650i \(0.679213\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 1162.70i − 1.71743i −0.512457 0.858713i \(-0.671265\pi\)
0.512457 0.858713i \(-0.328735\pi\)
\(678\) 0 0
\(679\) 127.808 0.188230
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 419.297i 0.613905i 0.951725 + 0.306952i \(0.0993092\pi\)
−0.951725 + 0.306952i \(0.900691\pi\)
\(684\) 0 0
\(685\) −367.192 −0.536047
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 171.590i 0.249043i
\(690\) 0 0
\(691\) −767.513 −1.11073 −0.555364 0.831607i \(-0.687421\pi\)
−0.555364 + 0.831607i \(0.687421\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 281.129i − 0.404502i
\(696\) 0 0
\(697\) −838.343 −1.20279
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 1227.38i − 1.75089i −0.483314 0.875447i \(-0.660567\pi\)
0.483314 0.875447i \(-0.339433\pi\)
\(702\) 0 0
\(703\) 152.348 0.216712
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 906.039i − 1.28153i
\(708\) 0 0
\(709\) 302.852 0.427154 0.213577 0.976926i \(-0.431489\pi\)
0.213577 + 0.976926i \(0.431489\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 40.3589i 0.0566043i
\(714\) 0 0
\(715\) −5.43478 −0.00760109
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 834.235i 1.16027i 0.814520 + 0.580136i \(0.197001\pi\)
−0.814520 + 0.580136i \(0.802999\pi\)
\(720\) 0 0
\(721\) −98.7751 −0.136997
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 752.652i 1.03814i
\(726\) 0 0
\(727\) −760.433 −1.04599 −0.522994 0.852336i \(-0.675185\pi\)
−0.522994 + 0.852336i \(0.675185\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 10.6697i 0.0145960i
\(732\) 0 0
\(733\) −486.420 −0.663602 −0.331801 0.943349i \(-0.607656\pi\)
−0.331801 + 0.943349i \(0.607656\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 94.2676i − 0.127907i
\(738\) 0 0
\(739\) 542.728 0.734408 0.367204 0.930140i \(-0.380315\pi\)
0.367204 + 0.930140i \(0.380315\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 250.265i 0.336830i 0.985716 + 0.168415i \(0.0538649\pi\)
−0.985716 + 0.168415i \(0.946135\pi\)
\(744\) 0 0
\(745\) −137.243 −0.184219
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 1213.44i − 1.62008i
\(750\) 0 0
\(751\) −1194.76 −1.59090 −0.795448 0.606022i \(-0.792764\pi\)
−0.795448 + 0.606022i \(0.792764\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 11.7654i 0.0155834i
\(756\) 0 0
\(757\) −1179.94 −1.55871 −0.779355 0.626583i \(-0.784453\pi\)
−0.779355 + 0.626583i \(0.784453\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 523.434i − 0.687823i −0.939002 0.343912i \(-0.888248\pi\)
0.939002 0.343912i \(-0.111752\pi\)
\(762\) 0 0
\(763\) −78.0246 −0.102260
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 87.1249i − 0.113592i
\(768\) 0 0
\(769\) 1050.26 1.36574 0.682872 0.730538i \(-0.260731\pi\)
0.682872 + 0.730538i \(0.260731\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 1231.05i − 1.59256i −0.604928 0.796280i \(-0.706798\pi\)
0.604928 0.796280i \(-0.293202\pi\)
\(774\) 0 0
\(775\) 637.970 0.823188
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 189.650i 0.243453i
\(780\) 0 0
\(781\) −17.5249 −0.0224391
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 241.239i 0.307310i
\(786\) 0 0
\(787\) −151.612 −0.192646 −0.0963228 0.995350i \(-0.530708\pi\)
−0.0963228 + 0.995350i \(0.530708\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 1737.18i − 2.19619i
\(792\) 0 0
\(793\) −142.814 −0.180094
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1199.06i − 1.50446i −0.658899 0.752232i \(-0.728977\pi\)
0.658899 0.752232i \(-0.271023\pi\)
\(798\) 0 0
\(799\) 180.186 0.225514
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 73.8662i − 0.0919878i
\(804\) 0 0
\(805\) −25.3291 −0.0314648
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 21.7239i − 0.0268528i −0.999910 0.0134264i \(-0.995726\pi\)
0.999910 0.0134264i \(-0.00427389\pi\)
\(810\) 0 0
\(811\) −1050.05 −1.29476 −0.647381 0.762166i \(-0.724136\pi\)
−0.647381 + 0.762166i \(0.724136\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 505.617i 0.620388i
\(816\) 0 0
\(817\) 2.41370 0.00295434
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 1316.52i − 1.60355i −0.597624 0.801776i \(-0.703889\pi\)
0.597624 0.801776i \(-0.296111\pi\)
\(822\) 0 0
\(823\) −4.71401 −0.00572784 −0.00286392 0.999996i \(-0.500912\pi\)
−0.00286392 + 0.999996i \(0.500912\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 107.497i − 0.129985i −0.997886 0.0649924i \(-0.979298\pi\)
0.997886 0.0649924i \(-0.0207023\pi\)
\(828\) 0 0
\(829\) −1234.04 −1.48859 −0.744293 0.667853i \(-0.767213\pi\)
−0.744293 + 0.667853i \(0.767213\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 656.479i 0.788090i
\(834\) 0 0
\(835\) −317.264 −0.379957
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 212.712i − 0.253530i −0.991933 0.126765i \(-0.959541\pi\)
0.991933 0.126765i \(-0.0404595\pi\)
\(840\) 0 0
\(841\) −529.410 −0.629500
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 353.866i 0.418777i
\(846\) 0 0
\(847\) −1091.79 −1.28901
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 44.9540i 0.0528249i
\(852\) 0 0
\(853\) −581.654 −0.681892 −0.340946 0.940083i \(-0.610747\pi\)
−0.340946 + 0.940083i \(0.610747\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 452.387i − 0.527873i −0.964540 0.263937i \(-0.914979\pi\)
0.964540 0.263937i \(-0.0850209\pi\)
\(858\) 0 0
\(859\) 880.415 1.02493 0.512465 0.858708i \(-0.328732\pi\)
0.512465 + 0.858708i \(0.328732\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 470.242i − 0.544892i −0.962171 0.272446i \(-0.912167\pi\)
0.962171 0.272446i \(-0.0878326\pi\)
\(864\) 0 0
\(865\) −90.7103 −0.104867
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 64.4316i 0.0741445i
\(870\) 0 0
\(871\) 195.798 0.224797
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 892.715i 1.02025i
\(876\) 0 0
\(877\) 64.6856 0.0737579 0.0368789 0.999320i \(-0.488258\pi\)
0.0368789 + 0.999320i \(0.488258\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 804.020i 0.912622i 0.889820 + 0.456311i \(0.150829\pi\)
−0.889820 + 0.456311i \(0.849171\pi\)
\(882\) 0 0
\(883\) 894.396 1.01291 0.506453 0.862267i \(-0.330956\pi\)
0.506453 + 0.862267i \(0.330956\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 1450.55i − 1.63535i −0.575682 0.817674i \(-0.695263\pi\)
0.575682 0.817674i \(-0.304737\pi\)
\(888\) 0 0
\(889\) 609.206 0.685271
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 40.7618i − 0.0456459i
\(894\) 0 0
\(895\) −507.470 −0.567006
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1161.60i 1.29210i
\(900\) 0 0
\(901\) −1446.50 −1.60544
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 207.167i − 0.228914i
\(906\) 0 0
\(907\) 1663.35 1.83391 0.916954 0.398993i \(-0.130640\pi\)
0.916954 + 0.398993i \(0.130640\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 150.701i 0.165424i 0.996574 + 0.0827118i \(0.0263581\pi\)
−0.996574 + 0.0827118i \(0.973642\pi\)
\(912\) 0 0
\(913\) 127.173 0.139291
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 688.300i − 0.750600i
\(918\) 0 0
\(919\) 826.616 0.899473 0.449736 0.893161i \(-0.351518\pi\)
0.449736 + 0.893161i \(0.351518\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 36.4001i − 0.0394368i
\(924\) 0 0
\(925\) 710.607 0.768224
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 1130.62i − 1.21703i −0.793541 0.608517i \(-0.791765\pi\)
0.793541 0.608517i \(-0.208235\pi\)
\(930\) 0 0
\(931\) 148.509 0.159515
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 45.8150i − 0.0490000i
\(936\) 0 0
\(937\) 34.1269 0.0364214 0.0182107 0.999834i \(-0.494203\pi\)
0.0182107 + 0.999834i \(0.494203\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 424.451i − 0.451063i −0.974236 0.225532i \(-0.927588\pi\)
0.974236 0.225532i \(-0.0724119\pi\)
\(942\) 0 0
\(943\) −55.9608 −0.0593434
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1653.85i 1.74641i 0.487356 + 0.873203i \(0.337961\pi\)
−0.487356 + 0.873203i \(0.662039\pi\)
\(948\) 0 0
\(949\) 153.424 0.161669
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 458.801i − 0.481428i −0.970596 0.240714i \(-0.922618\pi\)
0.970596 0.240714i \(-0.0773816\pi\)
\(954\) 0 0
\(955\) −355.611 −0.372367
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1548.91i 1.61513i
\(960\) 0 0
\(961\) 23.6087 0.0245668
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 401.271i − 0.415825i
\(966\) 0 0
\(967\) 953.711 0.986257 0.493129 0.869956i \(-0.335853\pi\)
0.493129 + 0.869956i \(0.335853\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 668.931i − 0.688909i −0.938803 0.344454i \(-0.888064\pi\)
0.938803 0.344454i \(-0.111936\pi\)
\(972\) 0 0
\(973\) −1185.87 −1.21878
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 1236.97i − 1.26609i −0.774116 0.633044i \(-0.781805\pi\)
0.774116 0.633044i \(-0.218195\pi\)
\(978\) 0 0
\(979\) −7.25798 −0.00741367
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 608.856i 0.619385i 0.950837 + 0.309693i \(0.100226\pi\)
−0.950837 + 0.309693i \(0.899774\pi\)
\(984\) 0 0
\(985\) 344.896 0.350149
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0.712219i 0 0.000720140i
\(990\) 0 0
\(991\) 1705.01 1.72049 0.860246 0.509878i \(-0.170310\pi\)
0.860246 + 0.509878i \(0.170310\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 630.104i 0.633270i
\(996\) 0 0
\(997\) 74.8450 0.0750702 0.0375351 0.999295i \(-0.488049\pi\)
0.0375351 + 0.999295i \(0.488049\pi\)
\(998\) 0 0
\(999\) 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 684.3.e.a.305.5 12
3.2 odd 2 inner 684.3.e.a.305.8 yes 12
4.3 odd 2 2736.3.h.c.305.5 12
12.11 even 2 2736.3.h.c.305.8 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
684.3.e.a.305.5 12 1.1 even 1 trivial
684.3.e.a.305.8 yes 12 3.2 odd 2 inner
2736.3.h.c.305.5 12 4.3 odd 2
2736.3.h.c.305.8 12 12.11 even 2