Properties

Label 3420.2.a.i.1.1
Level $3420$
Weight $2$
Character 3420.1
Self dual yes
Analytic conductor $27.309$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.3088374913\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 1140)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 3420.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} -2.60555 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} -2.60555 q^{7} +4.60555 q^{11} +4.60555 q^{13} -2.00000 q^{17} -1.00000 q^{19} +2.00000 q^{23} +1.00000 q^{25} -2.60555 q^{29} +4.00000 q^{31} -2.60555 q^{35} +3.39445 q^{37} -6.60555 q^{41} +10.6056 q^{43} -6.00000 q^{47} -0.211103 q^{49} +4.60555 q^{55} -5.21110 q^{59} -7.21110 q^{61} +4.60555 q^{65} +4.00000 q^{67} +9.21110 q^{71} +6.00000 q^{73} -12.0000 q^{77} +8.00000 q^{79} +11.2111 q^{83} -2.00000 q^{85} -6.60555 q^{89} -12.0000 q^{91} -1.00000 q^{95} +16.6056 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 2 q^{7} + O(q^{10}) \) \( 2 q + 2 q^{5} + 2 q^{7} + 2 q^{11} + 2 q^{13} - 4 q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 2 q^{29} + 8 q^{31} + 2 q^{35} + 14 q^{37} - 6 q^{41} + 14 q^{43} - 12 q^{47} + 14 q^{49} + 2 q^{55} + 4 q^{59} + 2 q^{65} + 8 q^{67} + 4 q^{71} + 12 q^{73} - 24 q^{77} + 16 q^{79} + 8 q^{83} - 4 q^{85} - 6 q^{89} - 24 q^{91} - 2 q^{95} + 26 q^{97} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.60555 −0.984806 −0.492403 0.870367i \(-0.663881\pi\)
−0.492403 + 0.870367i \(0.663881\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.60555 1.38863 0.694313 0.719673i \(-0.255708\pi\)
0.694313 + 0.719673i \(0.255708\pi\)
\(12\) 0 0
\(13\) 4.60555 1.27735 0.638675 0.769477i \(-0.279483\pi\)
0.638675 + 0.769477i \(0.279483\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.60555 −0.483839 −0.241919 0.970296i \(-0.577777\pi\)
−0.241919 + 0.970296i \(0.577777\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.60555 −0.440419
\(36\) 0 0
\(37\) 3.39445 0.558044 0.279022 0.960285i \(-0.409990\pi\)
0.279022 + 0.960285i \(0.409990\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.60555 −1.03161 −0.515807 0.856705i \(-0.672508\pi\)
−0.515807 + 0.856705i \(0.672508\pi\)
\(42\) 0 0
\(43\) 10.6056 1.61733 0.808666 0.588268i \(-0.200190\pi\)
0.808666 + 0.588268i \(0.200190\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −0.211103 −0.0301575
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 4.60555 0.621012
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.21110 −0.678428 −0.339214 0.940709i \(-0.610161\pi\)
−0.339214 + 0.940709i \(0.610161\pi\)
\(60\) 0 0
\(61\) −7.21110 −0.923287 −0.461644 0.887066i \(-0.652740\pi\)
−0.461644 + 0.887066i \(0.652740\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.60555 0.571248
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.21110 1.09316 0.546578 0.837408i \(-0.315930\pi\)
0.546578 + 0.837408i \(0.315930\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 11.2111 1.23058 0.615289 0.788301i \(-0.289039\pi\)
0.615289 + 0.788301i \(0.289039\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.60555 −0.700187 −0.350094 0.936715i \(-0.613850\pi\)
−0.350094 + 0.936715i \(0.613850\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 16.6056 1.68604 0.843019 0.537884i \(-0.180776\pi\)
0.843019 + 0.537884i \(0.180776\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.21110 0.717532 0.358766 0.933428i \(-0.383198\pi\)
0.358766 + 0.933428i \(0.383198\pi\)
\(102\) 0 0
\(103\) 18.4222 1.81519 0.907597 0.419843i \(-0.137915\pi\)
0.907597 + 0.419843i \(0.137915\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −18.4222 −1.78094 −0.890471 0.455040i \(-0.849625\pi\)
−0.890471 + 0.455040i \(0.849625\pi\)
\(108\) 0 0
\(109\) −11.2111 −1.07383 −0.536914 0.843637i \(-0.680410\pi\)
−0.536914 + 0.843637i \(0.680410\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.21110 0.113931 0.0569655 0.998376i \(-0.481858\pi\)
0.0569655 + 0.998376i \(0.481858\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.21110 0.477701
\(120\) 0 0
\(121\) 10.2111 0.928282
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 18.4222 1.63471 0.817353 0.576137i \(-0.195441\pi\)
0.817353 + 0.576137i \(0.195441\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 15.3944 1.34502 0.672510 0.740088i \(-0.265216\pi\)
0.672510 + 0.740088i \(0.265216\pi\)
\(132\) 0 0
\(133\) 2.60555 0.225930
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.4222 1.06130 0.530650 0.847591i \(-0.321948\pi\)
0.530650 + 0.847591i \(0.321948\pi\)
\(138\) 0 0
\(139\) 9.21110 0.781276 0.390638 0.920544i \(-0.372255\pi\)
0.390638 + 0.920544i \(0.372255\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 21.2111 1.77376
\(144\) 0 0
\(145\) −2.60555 −0.216379
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −16.4222 −1.34536 −0.672680 0.739934i \(-0.734857\pi\)
−0.672680 + 0.739934i \(0.734857\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 15.2111 1.21398 0.606989 0.794710i \(-0.292377\pi\)
0.606989 + 0.794710i \(0.292377\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.21110 −0.410692
\(162\) 0 0
\(163\) −17.0278 −1.33372 −0.666858 0.745184i \(-0.732361\pi\)
−0.666858 + 0.745184i \(0.732361\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.78890 0.525341 0.262670 0.964886i \(-0.415397\pi\)
0.262670 + 0.964886i \(0.415397\pi\)
\(168\) 0 0
\(169\) 8.21110 0.631623
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.21110 0.700307 0.350154 0.936692i \(-0.386129\pi\)
0.350154 + 0.936692i \(0.386129\pi\)
\(174\) 0 0
\(175\) −2.60555 −0.196961
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 21.2111 1.58539 0.792696 0.609617i \(-0.208677\pi\)
0.792696 + 0.609617i \(0.208677\pi\)
\(180\) 0 0
\(181\) −0.788897 −0.0586383 −0.0293191 0.999570i \(-0.509334\pi\)
−0.0293191 + 0.999570i \(0.509334\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.39445 0.249565
\(186\) 0 0
\(187\) −9.21110 −0.673583
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5.81665 0.420878 0.210439 0.977607i \(-0.432511\pi\)
0.210439 + 0.977607i \(0.432511\pi\)
\(192\) 0 0
\(193\) −0.605551 −0.0435885 −0.0217943 0.999762i \(-0.506938\pi\)
−0.0217943 + 0.999762i \(0.506938\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −14.0000 −0.997459 −0.498729 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) 0 0
\(199\) 17.2111 1.22006 0.610031 0.792377i \(-0.291157\pi\)
0.610031 + 0.792377i \(0.291157\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.78890 0.476487
\(204\) 0 0
\(205\) −6.60555 −0.461352
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.60555 −0.318573
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.6056 0.723293
\(216\) 0 0
\(217\) −10.4222 −0.707505
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −9.21110 −0.619606
\(222\) 0 0
\(223\) −10.4222 −0.697922 −0.348961 0.937137i \(-0.613466\pi\)
−0.348961 + 0.937137i \(0.613466\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 13.2111 0.876852 0.438426 0.898767i \(-0.355536\pi\)
0.438426 + 0.898767i \(0.355536\pi\)
\(228\) 0 0
\(229\) −3.21110 −0.212196 −0.106098 0.994356i \(-0.533836\pi\)
−0.106098 + 0.994356i \(0.533836\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.4222 1.07585 0.537927 0.842991i \(-0.319208\pi\)
0.537927 + 0.842991i \(0.319208\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −25.8167 −1.66994 −0.834970 0.550295i \(-0.814515\pi\)
−0.834970 + 0.550295i \(0.814515\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.211103 −0.0134868
\(246\) 0 0
\(247\) −4.60555 −0.293044
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.18335 0.390289 0.195145 0.980774i \(-0.437482\pi\)
0.195145 + 0.980774i \(0.437482\pi\)
\(252\) 0 0
\(253\) 9.21110 0.579097
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.78890 −0.173967 −0.0869833 0.996210i \(-0.527723\pi\)
−0.0869833 + 0.996210i \(0.527723\pi\)
\(258\) 0 0
\(259\) −8.84441 −0.549565
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7.21110 −0.444656 −0.222328 0.974972i \(-0.571366\pi\)
−0.222328 + 0.974972i \(0.571366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 17.3944 1.06056 0.530279 0.847823i \(-0.322087\pi\)
0.530279 + 0.847823i \(0.322087\pi\)
\(270\) 0 0
\(271\) 11.6333 0.706673 0.353337 0.935496i \(-0.385047\pi\)
0.353337 + 0.935496i \(0.385047\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.60555 0.277725
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −18.2389 −1.08804 −0.544020 0.839073i \(-0.683098\pi\)
−0.544020 + 0.839073i \(0.683098\pi\)
\(282\) 0 0
\(283\) −18.6056 −1.10599 −0.552993 0.833186i \(-0.686514\pi\)
−0.552993 + 0.833186i \(0.686514\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 17.2111 1.01594
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −19.6333 −1.14699 −0.573495 0.819209i \(-0.694413\pi\)
−0.573495 + 0.819209i \(0.694413\pi\)
\(294\) 0 0
\(295\) −5.21110 −0.303402
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 9.21110 0.532692
\(300\) 0 0
\(301\) −27.6333 −1.59276
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7.21110 −0.412907
\(306\) 0 0
\(307\) −11.6333 −0.663948 −0.331974 0.943289i \(-0.607715\pi\)
−0.331974 + 0.943289i \(0.607715\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −16.6056 −0.941614 −0.470807 0.882236i \(-0.656037\pi\)
−0.470807 + 0.882236i \(0.656037\pi\)
\(312\) 0 0
\(313\) 0.788897 0.0445911 0.0222956 0.999751i \(-0.492903\pi\)
0.0222956 + 0.999751i \(0.492903\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.42221 0.360707 0.180353 0.983602i \(-0.442276\pi\)
0.180353 + 0.983602i \(0.442276\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.00000 0.111283
\(324\) 0 0
\(325\) 4.60555 0.255470
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 15.6333 0.861892
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 3.02776 0.164932 0.0824662 0.996594i \(-0.473720\pi\)
0.0824662 + 0.996594i \(0.473720\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 18.4222 0.997618
\(342\) 0 0
\(343\) 18.7889 1.01451
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −21.6333 −1.16134 −0.580668 0.814140i \(-0.697209\pi\)
−0.580668 + 0.814140i \(0.697209\pi\)
\(348\) 0 0
\(349\) 34.8444 1.86518 0.932589 0.360939i \(-0.117544\pi\)
0.932589 + 0.360939i \(0.117544\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −0.422205 −0.0224717 −0.0112359 0.999937i \(-0.503577\pi\)
−0.0112359 + 0.999937i \(0.503577\pi\)
\(354\) 0 0
\(355\) 9.21110 0.488875
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 13.8167 0.729215 0.364608 0.931161i \(-0.381203\pi\)
0.364608 + 0.931161i \(0.381203\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 25.0278 1.30644 0.653219 0.757169i \(-0.273418\pi\)
0.653219 + 0.757169i \(0.273418\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 16.6056 0.859803 0.429901 0.902876i \(-0.358548\pi\)
0.429901 + 0.902876i \(0.358548\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 26.4222 1.35722 0.678609 0.734500i \(-0.262583\pi\)
0.678609 + 0.734500i \(0.262583\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −36.4222 −1.84668 −0.923340 0.383984i \(-0.874552\pi\)
−0.923340 + 0.383984i \(0.874552\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −7.57779 −0.380319 −0.190159 0.981753i \(-0.560900\pi\)
−0.190159 + 0.981753i \(0.560900\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −25.3944 −1.26814 −0.634069 0.773276i \(-0.718617\pi\)
−0.634069 + 0.773276i \(0.718617\pi\)
\(402\) 0 0
\(403\) 18.4222 0.917675
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 15.6333 0.774914
\(408\) 0 0
\(409\) −12.7889 −0.632370 −0.316185 0.948698i \(-0.602402\pi\)
−0.316185 + 0.948698i \(0.602402\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 13.5778 0.668120
\(414\) 0 0
\(415\) 11.2111 0.550331
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 7.02776 0.343328 0.171664 0.985156i \(-0.445086\pi\)
0.171664 + 0.985156i \(0.445086\pi\)
\(420\) 0 0
\(421\) −8.42221 −0.410473 −0.205237 0.978712i \(-0.565796\pi\)
−0.205237 + 0.978712i \(0.565796\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 18.7889 0.909258
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 9.21110 0.443683 0.221842 0.975083i \(-0.428793\pi\)
0.221842 + 0.975083i \(0.428793\pi\)
\(432\) 0 0
\(433\) −16.6056 −0.798012 −0.399006 0.916948i \(-0.630645\pi\)
−0.399006 + 0.916948i \(0.630645\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.00000 −0.0956730
\(438\) 0 0
\(439\) 34.4222 1.64288 0.821441 0.570293i \(-0.193170\pi\)
0.821441 + 0.570293i \(0.193170\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −18.8444 −0.895325 −0.447662 0.894203i \(-0.647743\pi\)
−0.447662 + 0.894203i \(0.647743\pi\)
\(444\) 0 0
\(445\) −6.60555 −0.313133
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.2389 0.860745 0.430372 0.902651i \(-0.358382\pi\)
0.430372 + 0.902651i \(0.358382\pi\)
\(450\) 0 0
\(451\) −30.4222 −1.43253
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) 32.0555 1.49949 0.749747 0.661725i \(-0.230175\pi\)
0.749747 + 0.661725i \(0.230175\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 22.8444 1.06397 0.531985 0.846754i \(-0.321446\pi\)
0.531985 + 0.846754i \(0.321446\pi\)
\(462\) 0 0
\(463\) 33.0278 1.53493 0.767465 0.641091i \(-0.221518\pi\)
0.767465 + 0.641091i \(0.221518\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −22.8444 −1.05711 −0.528557 0.848898i \(-0.677267\pi\)
−0.528557 + 0.848898i \(0.677267\pi\)
\(468\) 0 0
\(469\) −10.4222 −0.481253
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 48.8444 2.24587
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −28.6056 −1.30702 −0.653510 0.756917i \(-0.726704\pi\)
−0.653510 + 0.756917i \(0.726704\pi\)
\(480\) 0 0
\(481\) 15.6333 0.712817
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 16.6056 0.754019
\(486\) 0 0
\(487\) 0.366692 0.0166164 0.00830821 0.999965i \(-0.497355\pi\)
0.00830821 + 0.999965i \(0.497355\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 12.2389 0.552332 0.276166 0.961110i \(-0.410936\pi\)
0.276166 + 0.961110i \(0.410936\pi\)
\(492\) 0 0
\(493\) 5.21110 0.234696
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) −27.6333 −1.23704 −0.618518 0.785770i \(-0.712267\pi\)
−0.618518 + 0.785770i \(0.712267\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12.7889 −0.570229 −0.285114 0.958494i \(-0.592032\pi\)
−0.285114 + 0.958494i \(0.592032\pi\)
\(504\) 0 0
\(505\) 7.21110 0.320890
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 35.4500 1.57129 0.785646 0.618676i \(-0.212331\pi\)
0.785646 + 0.618676i \(0.212331\pi\)
\(510\) 0 0
\(511\) −15.6333 −0.691577
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 18.4222 0.811779
\(516\) 0 0
\(517\) −27.6333 −1.21531
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.18335 0.183276 0.0916379 0.995792i \(-0.470790\pi\)
0.0916379 + 0.995792i \(0.470790\pi\)
\(522\) 0 0
\(523\) −25.2111 −1.10240 −0.551202 0.834372i \(-0.685831\pi\)
−0.551202 + 0.834372i \(0.685831\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −30.4222 −1.31773
\(534\) 0 0
\(535\) −18.4222 −0.796461
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.972244 −0.0418775
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −11.2111 −0.480231
\(546\) 0 0
\(547\) −22.7889 −0.974383 −0.487191 0.873295i \(-0.661979\pi\)
−0.487191 + 0.873295i \(0.661979\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.60555 0.111000
\(552\) 0 0
\(553\) −20.8444 −0.886394
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.42221 0.187375 0.0936874 0.995602i \(-0.470135\pi\)
0.0936874 + 0.995602i \(0.470135\pi\)
\(558\) 0 0
\(559\) 48.8444 2.06590
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −31.6333 −1.33318 −0.666592 0.745422i \(-0.732248\pi\)
−0.666592 + 0.745422i \(0.732248\pi\)
\(564\) 0 0
\(565\) 1.21110 0.0509515
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −39.4500 −1.65383 −0.826914 0.562328i \(-0.809906\pi\)
−0.826914 + 0.562328i \(0.809906\pi\)
\(570\) 0 0
\(571\) 7.63331 0.319444 0.159722 0.987162i \(-0.448940\pi\)
0.159722 + 0.987162i \(0.448940\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.00000 0.0834058
\(576\) 0 0
\(577\) −23.2111 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −29.2111 −1.21188
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.4222 −0.512719 −0.256360 0.966581i \(-0.582523\pi\)
−0.256360 + 0.966581i \(0.582523\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.57779 0.146922 0.0734612 0.997298i \(-0.476595\pi\)
0.0734612 + 0.997298i \(0.476595\pi\)
\(594\) 0 0
\(595\) 5.21110 0.213634
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −36.8444 −1.50542 −0.752711 0.658351i \(-0.771254\pi\)
−0.752711 + 0.658351i \(0.771254\pi\)
\(600\) 0 0
\(601\) 4.78890 0.195343 0.0976716 0.995219i \(-0.468861\pi\)
0.0976716 + 0.995219i \(0.468861\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10.2111 0.415140
\(606\) 0 0
\(607\) −23.6333 −0.959246 −0.479623 0.877475i \(-0.659227\pi\)
−0.479623 + 0.877475i \(0.659227\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −27.6333 −1.11792
\(612\) 0 0
\(613\) 4.78890 0.193422 0.0967109 0.995313i \(-0.469168\pi\)
0.0967109 + 0.995313i \(0.469168\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −34.0000 −1.36879 −0.684394 0.729112i \(-0.739933\pi\)
−0.684394 + 0.729112i \(0.739933\pi\)
\(618\) 0 0
\(619\) −4.36669 −0.175512 −0.0877561 0.996142i \(-0.527970\pi\)
−0.0877561 + 0.996142i \(0.527970\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 17.2111 0.689548
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −6.78890 −0.270691
\(630\) 0 0
\(631\) 36.8444 1.46675 0.733376 0.679823i \(-0.237943\pi\)
0.733376 + 0.679823i \(0.237943\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 18.4222 0.731063
\(636\) 0 0
\(637\) −0.972244 −0.0385217
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −35.4500 −1.40019 −0.700095 0.714050i \(-0.746859\pi\)
−0.700095 + 0.714050i \(0.746859\pi\)
\(642\) 0 0
\(643\) 5.39445 0.212736 0.106368 0.994327i \(-0.466078\pi\)
0.106368 + 0.994327i \(0.466078\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.36669 0.0930443 0.0465221 0.998917i \(-0.485186\pi\)
0.0465221 + 0.998917i \(0.485186\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −42.0000 −1.64359 −0.821794 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) 0 0
\(655\) 15.3944 0.601511
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −31.2111 −1.21397 −0.606986 0.794713i \(-0.707621\pi\)
−0.606986 + 0.794713i \(0.707621\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.60555 0.101039
\(666\) 0 0
\(667\) −5.21110 −0.201775
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −33.2111 −1.28210
\(672\) 0 0
\(673\) 36.6056 1.41104 0.705520 0.708690i \(-0.250713\pi\)
0.705520 + 0.708690i \(0.250713\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −10.4222 −0.400558 −0.200279 0.979739i \(-0.564185\pi\)
−0.200279 + 0.979739i \(0.564185\pi\)
\(678\) 0 0
\(679\) −43.2666 −1.66042
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 48.0000 1.83667 0.918334 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) 0 0
\(685\) 12.4222 0.474628
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −42.0555 −1.59987 −0.799934 0.600089i \(-0.795132\pi\)
−0.799934 + 0.600089i \(0.795132\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.21110 0.349397
\(696\) 0 0
\(697\) 13.2111 0.500406
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) −3.39445 −0.128024
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −18.7889 −0.706629
\(708\) 0 0
\(709\) −25.6333 −0.962679 −0.481340 0.876534i \(-0.659850\pi\)
−0.481340 + 0.876534i \(0.659850\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 21.2111 0.793250
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −16.2389 −0.605607 −0.302804 0.953053i \(-0.597923\pi\)
−0.302804 + 0.953053i \(0.597923\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2.60555 −0.0967677
\(726\) 0 0
\(727\) −17.3944 −0.645124 −0.322562 0.946548i \(-0.604544\pi\)
−0.322562 + 0.946548i \(0.604544\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −21.2111 −0.784521
\(732\) 0 0
\(733\) −24.4222 −0.902055 −0.451027 0.892510i \(-0.648942\pi\)
−0.451027 + 0.892510i \(0.648942\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 18.4222 0.678591
\(738\) 0 0
\(739\) 38.4222 1.41338 0.706692 0.707521i \(-0.250187\pi\)
0.706692 + 0.707521i \(0.250187\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 20.0000 0.733729 0.366864 0.930274i \(-0.380431\pi\)
0.366864 + 0.930274i \(0.380431\pi\)
\(744\) 0 0
\(745\) −16.4222 −0.601663
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) 40.8444 1.49043 0.745217 0.666822i \(-0.232346\pi\)
0.745217 + 0.666822i \(0.232346\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) −36.0555 −1.31046 −0.655230 0.755429i \(-0.727428\pi\)
−0.655230 + 0.755429i \(0.727428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 29.2111 1.05751
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) −20.4222 −0.736444 −0.368222 0.929738i \(-0.620033\pi\)
−0.368222 + 0.929738i \(0.620033\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 43.2666 1.55619 0.778096 0.628145i \(-0.216186\pi\)
0.778096 + 0.628145i \(0.216186\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6.60555 0.236668
\(780\) 0 0
\(781\) 42.4222 1.51799
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 15.2111 0.542908
\(786\) 0 0
\(787\) −19.6333 −0.699852 −0.349926 0.936777i \(-0.613793\pi\)
−0.349926 + 0.936777i \(0.613793\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.15559 −0.112200
\(792\) 0 0
\(793\) −33.2111 −1.17936
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −17.2111 −0.609649 −0.304824 0.952409i \(-0.598598\pi\)
−0.304824 + 0.952409i \(0.598598\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 27.6333 0.975158
\(804\) 0 0
\(805\) −5.21110 −0.183667
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 48.4222 1.70243 0.851217 0.524814i \(-0.175865\pi\)
0.851217 + 0.524814i \(0.175865\pi\)
\(810\) 0 0
\(811\) −40.8444 −1.43424 −0.717121 0.696949i \(-0.754540\pi\)
−0.717121 + 0.696949i \(0.754540\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −17.0278 −0.596456
\(816\) 0 0
\(817\) −10.6056 −0.371041
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −43.2111 −1.50808 −0.754039 0.656830i \(-0.771897\pi\)
−0.754039 + 0.656830i \(0.771897\pi\)
\(822\) 0 0
\(823\) −47.8167 −1.66678 −0.833392 0.552683i \(-0.813604\pi\)
−0.833392 + 0.552683i \(0.813604\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 27.6333 0.960904 0.480452 0.877021i \(-0.340473\pi\)
0.480452 + 0.877021i \(0.340473\pi\)
\(828\) 0 0
\(829\) 15.2111 0.528303 0.264152 0.964481i \(-0.414908\pi\)
0.264152 + 0.964481i \(0.414908\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.422205 0.0146285
\(834\) 0 0
\(835\) 6.78890 0.234939
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −27.6333 −0.954008 −0.477004 0.878901i \(-0.658277\pi\)
−0.477004 + 0.878901i \(0.658277\pi\)
\(840\) 0 0
\(841\) −22.2111 −0.765900
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 8.21110 0.282471
\(846\) 0 0
\(847\) −26.6056 −0.914178
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 6.78890 0.232720
\(852\) 0 0
\(853\) 29.6333 1.01463 0.507313 0.861762i \(-0.330639\pi\)
0.507313 + 0.861762i \(0.330639\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −6.42221 −0.219378 −0.109689 0.993966i \(-0.534986\pi\)
−0.109689 + 0.993966i \(0.534986\pi\)
\(858\) 0 0
\(859\) 8.84441 0.301767 0.150884 0.988552i \(-0.451788\pi\)
0.150884 + 0.988552i \(0.451788\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.42221 −0.218614 −0.109307 0.994008i \(-0.534863\pi\)
−0.109307 + 0.994008i \(0.534863\pi\)
\(864\) 0 0
\(865\) 9.21110 0.313187
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 36.8444 1.24986
\(870\) 0 0
\(871\) 18.4222 0.624213
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.60555 −0.0880837
\(876\) 0 0
\(877\) 27.0278 0.912662 0.456331 0.889810i \(-0.349163\pi\)
0.456331 + 0.889810i \(0.349163\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −59.2111 −1.99487 −0.997436 0.0715590i \(-0.977203\pi\)
−0.997436 + 0.0715590i \(0.977203\pi\)
\(882\) 0 0
\(883\) 27.8167 0.936105 0.468052 0.883701i \(-0.344956\pi\)
0.468052 + 0.883701i \(0.344956\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 20.8444 0.699887 0.349943 0.936771i \(-0.386201\pi\)
0.349943 + 0.936771i \(0.386201\pi\)
\(888\) 0 0
\(889\) −48.0000 −1.60987
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) 21.2111 0.709009
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −10.4222 −0.347600
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −0.788897 −0.0262238
\(906\) 0 0
\(907\) 30.0555 0.997977 0.498988 0.866609i \(-0.333705\pi\)
0.498988 + 0.866609i \(0.333705\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −10.4222 −0.345303 −0.172652 0.984983i \(-0.555233\pi\)
−0.172652 + 0.984983i \(0.555233\pi\)
\(912\) 0 0
\(913\) 51.6333 1.70881
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −40.1110 −1.32458
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 42.4222 1.39634
\(924\) 0 0
\(925\) 3.39445 0.111609
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 45.6333 1.49718 0.748590 0.663033i \(-0.230731\pi\)
0.748590 + 0.663033i \(0.230731\pi\)
\(930\) 0 0
\(931\) 0.211103 0.00691861
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −9.21110 −0.301235
\(936\) 0 0
\(937\) −19.2111 −0.627599 −0.313800 0.949489i \(-0.601602\pi\)
−0.313800 + 0.949489i \(0.601602\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 30.2389 0.985759 0.492879 0.870098i \(-0.335944\pi\)
0.492879 + 0.870098i \(0.335944\pi\)
\(942\) 0 0
\(943\) −13.2111 −0.430213
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 7.21110 0.234329 0.117165 0.993113i \(-0.462619\pi\)
0.117165 + 0.993113i \(0.462619\pi\)
\(948\) 0 0
\(949\) 27.6333 0.897015
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −5.21110 −0.168804 −0.0844021 0.996432i \(-0.526898\pi\)
−0.0844021 + 0.996432i \(0.526898\pi\)
\(954\) 0 0
\(955\) 5.81665 0.188222
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −32.3667 −1.04518
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −0.605551 −0.0194934
\(966\) 0 0
\(967\) −26.2389 −0.843785 −0.421892 0.906646i \(-0.638634\pi\)
−0.421892 + 0.906646i \(0.638634\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) −24.0000 −0.769405
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −6.42221 −0.205465 −0.102732 0.994709i \(-0.532758\pi\)
−0.102732 + 0.994709i \(0.532758\pi\)
\(978\) 0 0
\(979\) −30.4222 −0.972298
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 9.21110 0.293789 0.146894 0.989152i \(-0.453072\pi\)
0.146894 + 0.989152i \(0.453072\pi\)
\(984\) 0 0
\(985\) −14.0000 −0.446077
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 21.2111 0.674474
\(990\) 0 0
\(991\) 50.4222 1.60171 0.800857 0.598856i \(-0.204378\pi\)
0.800857 + 0.598856i \(0.204378\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 17.2111 0.545629
\(996\) 0 0
\(997\) −61.6333 −1.95195 −0.975973 0.217891i \(-0.930082\pi\)
−0.975973 + 0.217891i \(0.930082\pi\)
\(998\) 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 3420.2.a.i.1.1 2
3.2 odd 2 1140.2.a.e.1.1 2
12.11 even 2 4560.2.a.bl.1.2 2
15.2 even 4 5700.2.f.n.3649.3 4
15.8 even 4 5700.2.f.n.3649.2 4
15.14 odd 2 5700.2.a.u.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.a.e.1.1 2 3.2 odd 2
3420.2.a.i.1.1 2 1.1 even 1 trivial
4560.2.a.bl.1.2 2 12.11 even 2
5700.2.a.u.1.2 2 15.14 odd 2
5700.2.f.n.3649.2 4 15.8 even 4
5700.2.f.n.3649.3 4 15.2 even 4