Properties

Label 280.2.a.d.1.1
Level $280$
Weight $2$
Character 280.1
Self dual yes
Analytic conductor $2.236$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.23581125660\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 280.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.56155 q^{3} +1.00000 q^{5} +1.00000 q^{7} -0.561553 q^{9} +O(q^{10})\) \(q-1.56155 q^{3} +1.00000 q^{5} +1.00000 q^{7} -0.561553 q^{9} +1.56155 q^{11} +6.68466 q^{13} -1.56155 q^{15} +7.56155 q^{17} -7.12311 q^{19} -1.56155 q^{21} +3.12311 q^{23} +1.00000 q^{25} +5.56155 q^{27} +0.438447 q^{29} +6.24621 q^{31} -2.43845 q^{33} +1.00000 q^{35} -8.24621 q^{37} -10.4384 q^{39} -1.12311 q^{41} -7.12311 q^{43} -0.561553 q^{45} +2.43845 q^{47} +1.00000 q^{49} -11.8078 q^{51} -13.1231 q^{53} +1.56155 q^{55} +11.1231 q^{57} -4.00000 q^{59} -6.87689 q^{61} -0.561553 q^{63} +6.68466 q^{65} +2.24621 q^{67} -4.87689 q^{69} -4.24621 q^{73} -1.56155 q^{75} +1.56155 q^{77} +0.684658 q^{79} -7.00000 q^{81} +12.0000 q^{83} +7.56155 q^{85} -0.684658 q^{87} +5.12311 q^{89} +6.68466 q^{91} -9.75379 q^{93} -7.12311 q^{95} +1.31534 q^{97} -0.876894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9} + O(q^{10}) \) \( 2 q + q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9} - q^{11} + q^{13} + q^{15} + 11 q^{17} - 6 q^{19} + q^{21} - 2 q^{23} + 2 q^{25} + 7 q^{27} + 5 q^{29} - 4 q^{31} - 9 q^{33} + 2 q^{35} - 25 q^{39} + 6 q^{41} - 6 q^{43} + 3 q^{45} + 9 q^{47} + 2 q^{49} - 3 q^{51} - 18 q^{53} - q^{55} + 14 q^{57} - 8 q^{59} - 22 q^{61} + 3 q^{63} + q^{65} - 12 q^{67} - 18 q^{69} + 8 q^{73} + q^{75} - q^{77} - 11 q^{79} - 14 q^{81} + 24 q^{83} + 11 q^{85} + 11 q^{87} + 2 q^{89} + q^{91} - 36 q^{93} - 6 q^{95} + 15 q^{97} - 10 q^{99} + O(q^{100}) \)

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
\(3\) −1.56155 −0.901563 −0.450781 0.892634i \(-0.648855\pi\)
−0.450781 + 0.892634i \(0.648855\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) 1.56155 0.470826 0.235413 0.971895i \(-0.424356\pi\)
0.235413 + 0.971895i \(0.424356\pi\)
\(12\) 0 0
\(13\) 6.68466 1.85399 0.926995 0.375073i \(-0.122382\pi\)
0.926995 + 0.375073i \(0.122382\pi\)
\(14\) 0 0
\(15\) −1.56155 −0.403191
\(16\) 0 0
\(17\) 7.56155 1.83395 0.916973 0.398949i \(-0.130625\pi\)
0.916973 + 0.398949i \(0.130625\pi\)
\(18\) 0 0
\(19\) −7.12311 −1.63415 −0.817076 0.576530i \(-0.804407\pi\)
−0.817076 + 0.576530i \(0.804407\pi\)
\(20\) 0 0
\(21\) −1.56155 −0.340759
\(22\) 0 0
\(23\) 3.12311 0.651213 0.325606 0.945505i \(-0.394432\pi\)
0.325606 + 0.945505i \(0.394432\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.56155 1.07032
\(28\) 0 0
\(29\) 0.438447 0.0814176 0.0407088 0.999171i \(-0.487038\pi\)
0.0407088 + 0.999171i \(0.487038\pi\)
\(30\) 0 0
\(31\) 6.24621 1.12185 0.560926 0.827866i \(-0.310445\pi\)
0.560926 + 0.827866i \(0.310445\pi\)
\(32\) 0 0
\(33\) −2.43845 −0.424479
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −8.24621 −1.35567 −0.677834 0.735215i \(-0.737081\pi\)
−0.677834 + 0.735215i \(0.737081\pi\)
\(38\) 0 0
\(39\) −10.4384 −1.67149
\(40\) 0 0
\(41\) −1.12311 −0.175400 −0.0876998 0.996147i \(-0.527952\pi\)
−0.0876998 + 0.996147i \(0.527952\pi\)
\(42\) 0 0
\(43\) −7.12311 −1.08626 −0.543132 0.839648i \(-0.682762\pi\)
−0.543132 + 0.839648i \(0.682762\pi\)
\(44\) 0 0
\(45\) −0.561553 −0.0837114
\(46\) 0 0
\(47\) 2.43845 0.355684 0.177842 0.984059i \(-0.443088\pi\)
0.177842 + 0.984059i \(0.443088\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −11.8078 −1.65342
\(52\) 0 0
\(53\) −13.1231 −1.80260 −0.901299 0.433198i \(-0.857385\pi\)
−0.901299 + 0.433198i \(0.857385\pi\)
\(54\) 0 0
\(55\) 1.56155 0.210560
\(56\) 0 0
\(57\) 11.1231 1.47329
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −6.87689 −0.880496 −0.440248 0.897876i \(-0.645109\pi\)
−0.440248 + 0.897876i \(0.645109\pi\)
\(62\) 0 0
\(63\) −0.561553 −0.0707490
\(64\) 0 0
\(65\) 6.68466 0.829130
\(66\) 0 0
\(67\) 2.24621 0.274418 0.137209 0.990542i \(-0.456187\pi\)
0.137209 + 0.990542i \(0.456187\pi\)
\(68\) 0 0
\(69\) −4.87689 −0.587109
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −4.24621 −0.496981 −0.248491 0.968634i \(-0.579935\pi\)
−0.248491 + 0.968634i \(0.579935\pi\)
\(74\) 0 0
\(75\) −1.56155 −0.180313
\(76\) 0 0
\(77\) 1.56155 0.177955
\(78\) 0 0
\(79\) 0.684658 0.0770301 0.0385150 0.999258i \(-0.487737\pi\)
0.0385150 + 0.999258i \(0.487737\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 7.56155 0.820166
\(86\) 0 0
\(87\) −0.684658 −0.0734031
\(88\) 0 0
\(89\) 5.12311 0.543048 0.271524 0.962432i \(-0.412472\pi\)
0.271524 + 0.962432i \(0.412472\pi\)
\(90\) 0 0
\(91\) 6.68466 0.700743
\(92\) 0 0
\(93\) −9.75379 −1.01142
\(94\) 0 0
\(95\) −7.12311 −0.730815
\(96\) 0 0
\(97\) 1.31534 0.133553 0.0667764 0.997768i \(-0.478729\pi\)
0.0667764 + 0.997768i \(0.478729\pi\)
\(98\) 0 0
\(99\) −0.876894 −0.0881312
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −11.8078 −1.16345 −0.581727 0.813384i \(-0.697623\pi\)
−0.581727 + 0.813384i \(0.697623\pi\)
\(104\) 0 0
\(105\) −1.56155 −0.152392
\(106\) 0 0
\(107\) 15.1231 1.46201 0.731003 0.682374i \(-0.239053\pi\)
0.731003 + 0.682374i \(0.239053\pi\)
\(108\) 0 0
\(109\) −4.43845 −0.425126 −0.212563 0.977147i \(-0.568181\pi\)
−0.212563 + 0.977147i \(0.568181\pi\)
\(110\) 0 0
\(111\) 12.8769 1.22222
\(112\) 0 0
\(113\) 8.24621 0.775738 0.387869 0.921714i \(-0.373211\pi\)
0.387869 + 0.921714i \(0.373211\pi\)
\(114\) 0 0
\(115\) 3.12311 0.291231
\(116\) 0 0
\(117\) −3.75379 −0.347038
\(118\) 0 0
\(119\) 7.56155 0.693166
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) 0 0
\(123\) 1.75379 0.158134
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −6.24621 −0.554262 −0.277131 0.960832i \(-0.589384\pi\)
−0.277131 + 0.960832i \(0.589384\pi\)
\(128\) 0 0
\(129\) 11.1231 0.979335
\(130\) 0 0
\(131\) 15.1231 1.32131 0.660656 0.750689i \(-0.270278\pi\)
0.660656 + 0.750689i \(0.270278\pi\)
\(132\) 0 0
\(133\) −7.12311 −0.617652
\(134\) 0 0
\(135\) 5.56155 0.478662
\(136\) 0 0
\(137\) −7.36932 −0.629603 −0.314802 0.949157i \(-0.601938\pi\)
−0.314802 + 0.949157i \(0.601938\pi\)
\(138\) 0 0
\(139\) −21.3693 −1.81252 −0.906261 0.422719i \(-0.861076\pi\)
−0.906261 + 0.422719i \(0.861076\pi\)
\(140\) 0 0
\(141\) −3.80776 −0.320672
\(142\) 0 0
\(143\) 10.4384 0.872907
\(144\) 0 0
\(145\) 0.438447 0.0364111
\(146\) 0 0
\(147\) −1.56155 −0.128795
\(148\) 0 0
\(149\) −0.246211 −0.0201704 −0.0100852 0.999949i \(-0.503210\pi\)
−0.0100852 + 0.999949i \(0.503210\pi\)
\(150\) 0 0
\(151\) −19.8078 −1.61193 −0.805966 0.591961i \(-0.798354\pi\)
−0.805966 + 0.591961i \(0.798354\pi\)
\(152\) 0 0
\(153\) −4.24621 −0.343286
\(154\) 0 0
\(155\) 6.24621 0.501708
\(156\) 0 0
\(157\) 4.24621 0.338885 0.169442 0.985540i \(-0.445803\pi\)
0.169442 + 0.985540i \(0.445803\pi\)
\(158\) 0 0
\(159\) 20.4924 1.62515
\(160\) 0 0
\(161\) 3.12311 0.246135
\(162\) 0 0
\(163\) 19.6155 1.53641 0.768203 0.640206i \(-0.221151\pi\)
0.768203 + 0.640206i \(0.221151\pi\)
\(164\) 0 0
\(165\) −2.43845 −0.189833
\(166\) 0 0
\(167\) 4.19224 0.324405 0.162202 0.986757i \(-0.448140\pi\)
0.162202 + 0.986757i \(0.448140\pi\)
\(168\) 0 0
\(169\) 31.6847 2.43728
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) −23.1771 −1.76212 −0.881060 0.473004i \(-0.843170\pi\)
−0.881060 + 0.473004i \(0.843170\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 6.24621 0.469494
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −5.12311 −0.380797 −0.190399 0.981707i \(-0.560978\pi\)
−0.190399 + 0.981707i \(0.560978\pi\)
\(182\) 0 0
\(183\) 10.7386 0.793823
\(184\) 0 0
\(185\) −8.24621 −0.606274
\(186\) 0 0
\(187\) 11.8078 0.863469
\(188\) 0 0
\(189\) 5.56155 0.404543
\(190\) 0 0
\(191\) −0.684658 −0.0495401 −0.0247701 0.999693i \(-0.507885\pi\)
−0.0247701 + 0.999693i \(0.507885\pi\)
\(192\) 0 0
\(193\) 13.1231 0.944622 0.472311 0.881432i \(-0.343420\pi\)
0.472311 + 0.881432i \(0.343420\pi\)
\(194\) 0 0
\(195\) −10.4384 −0.747513
\(196\) 0 0
\(197\) −13.1231 −0.934983 −0.467491 0.883998i \(-0.654842\pi\)
−0.467491 + 0.883998i \(0.654842\pi\)
\(198\) 0 0
\(199\) −14.2462 −1.00989 −0.504944 0.863152i \(-0.668487\pi\)
−0.504944 + 0.863152i \(0.668487\pi\)
\(200\) 0 0
\(201\) −3.50758 −0.247405
\(202\) 0 0
\(203\) 0.438447 0.0307730
\(204\) 0 0
\(205\) −1.12311 −0.0784411
\(206\) 0 0
\(207\) −1.75379 −0.121897
\(208\) 0 0
\(209\) −11.1231 −0.769401
\(210\) 0 0
\(211\) −17.5616 −1.20899 −0.604494 0.796610i \(-0.706624\pi\)
−0.604494 + 0.796610i \(0.706624\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −7.12311 −0.485792
\(216\) 0 0
\(217\) 6.24621 0.424020
\(218\) 0 0
\(219\) 6.63068 0.448060
\(220\) 0 0
\(221\) 50.5464 3.40012
\(222\) 0 0
\(223\) 24.6847 1.65301 0.826503 0.562932i \(-0.190327\pi\)
0.826503 + 0.562932i \(0.190327\pi\)
\(224\) 0 0
\(225\) −0.561553 −0.0374369
\(226\) 0 0
\(227\) −11.3153 −0.751026 −0.375513 0.926817i \(-0.622533\pi\)
−0.375513 + 0.926817i \(0.622533\pi\)
\(228\) 0 0
\(229\) −11.3693 −0.751306 −0.375653 0.926760i \(-0.622581\pi\)
−0.375653 + 0.926760i \(0.622581\pi\)
\(230\) 0 0
\(231\) −2.43845 −0.160438
\(232\) 0 0
\(233\) −10.8769 −0.712569 −0.356285 0.934378i \(-0.615957\pi\)
−0.356285 + 0.934378i \(0.615957\pi\)
\(234\) 0 0
\(235\) 2.43845 0.159067
\(236\) 0 0
\(237\) −1.06913 −0.0694475
\(238\) 0 0
\(239\) −18.0540 −1.16781 −0.583907 0.811820i \(-0.698477\pi\)
−0.583907 + 0.811820i \(0.698477\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) −5.75379 −0.369106
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −47.6155 −3.02970
\(248\) 0 0
\(249\) −18.7386 −1.18751
\(250\) 0 0
\(251\) 13.3693 0.843864 0.421932 0.906628i \(-0.361352\pi\)
0.421932 + 0.906628i \(0.361352\pi\)
\(252\) 0 0
\(253\) 4.87689 0.306608
\(254\) 0 0
\(255\) −11.8078 −0.739431
\(256\) 0 0
\(257\) −18.4924 −1.15353 −0.576763 0.816912i \(-0.695684\pi\)
−0.576763 + 0.816912i \(0.695684\pi\)
\(258\) 0 0
\(259\) −8.24621 −0.512395
\(260\) 0 0
\(261\) −0.246211 −0.0152401
\(262\) 0 0
\(263\) 9.36932 0.577737 0.288868 0.957369i \(-0.406721\pi\)
0.288868 + 0.957369i \(0.406721\pi\)
\(264\) 0 0
\(265\) −13.1231 −0.806146
\(266\) 0 0
\(267\) −8.00000 −0.489592
\(268\) 0 0
\(269\) 4.24621 0.258896 0.129448 0.991586i \(-0.458679\pi\)
0.129448 + 0.991586i \(0.458679\pi\)
\(270\) 0 0
\(271\) −6.24621 −0.379430 −0.189715 0.981839i \(-0.560756\pi\)
−0.189715 + 0.981839i \(0.560756\pi\)
\(272\) 0 0
\(273\) −10.4384 −0.631764
\(274\) 0 0
\(275\) 1.56155 0.0941652
\(276\) 0 0
\(277\) −8.24621 −0.495467 −0.247733 0.968828i \(-0.579686\pi\)
−0.247733 + 0.968828i \(0.579686\pi\)
\(278\) 0 0
\(279\) −3.50758 −0.209993
\(280\) 0 0
\(281\) −19.5616 −1.16694 −0.583472 0.812133i \(-0.698306\pi\)
−0.583472 + 0.812133i \(0.698306\pi\)
\(282\) 0 0
\(283\) −4.68466 −0.278474 −0.139237 0.990259i \(-0.544465\pi\)
−0.139237 + 0.990259i \(0.544465\pi\)
\(284\) 0 0
\(285\) 11.1231 0.658876
\(286\) 0 0
\(287\) −1.12311 −0.0662948
\(288\) 0 0
\(289\) 40.1771 2.36336
\(290\) 0 0
\(291\) −2.05398 −0.120406
\(292\) 0 0
\(293\) 32.0540 1.87261 0.936307 0.351184i \(-0.114221\pi\)
0.936307 + 0.351184i \(0.114221\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) 8.68466 0.503935
\(298\) 0 0
\(299\) 20.8769 1.20734
\(300\) 0 0
\(301\) −7.12311 −0.410569
\(302\) 0 0
\(303\) −9.36932 −0.538253
\(304\) 0 0
\(305\) −6.87689 −0.393770
\(306\) 0 0
\(307\) −28.6847 −1.63712 −0.818560 0.574421i \(-0.805227\pi\)
−0.818560 + 0.574421i \(0.805227\pi\)
\(308\) 0 0
\(309\) 18.4384 1.04893
\(310\) 0 0
\(311\) −12.8769 −0.730182 −0.365091 0.930972i \(-0.618962\pi\)
−0.365091 + 0.930972i \(0.618962\pi\)
\(312\) 0 0
\(313\) 26.6847 1.50831 0.754153 0.656699i \(-0.228048\pi\)
0.754153 + 0.656699i \(0.228048\pi\)
\(314\) 0 0
\(315\) −0.561553 −0.0316399
\(316\) 0 0
\(317\) −10.0000 −0.561656 −0.280828 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 0 0
\(319\) 0.684658 0.0383335
\(320\) 0 0
\(321\) −23.6155 −1.31809
\(322\) 0 0
\(323\) −53.8617 −2.99695
\(324\) 0 0
\(325\) 6.68466 0.370798
\(326\) 0 0
\(327\) 6.93087 0.383278
\(328\) 0 0
\(329\) 2.43845 0.134436
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) 4.63068 0.253760
\(334\) 0 0
\(335\) 2.24621 0.122724
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) −12.8769 −0.699377
\(340\) 0 0
\(341\) 9.75379 0.528197
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −4.87689 −0.262563
\(346\) 0 0
\(347\) 15.1231 0.811851 0.405925 0.913906i \(-0.366949\pi\)
0.405925 + 0.913906i \(0.366949\pi\)
\(348\) 0 0
\(349\) −11.7538 −0.629166 −0.314583 0.949230i \(-0.601865\pi\)
−0.314583 + 0.949230i \(0.601865\pi\)
\(350\) 0 0
\(351\) 37.1771 1.98437
\(352\) 0 0
\(353\) −2.19224 −0.116681 −0.0583405 0.998297i \(-0.518581\pi\)
−0.0583405 + 0.998297i \(0.518581\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −11.8078 −0.624933
\(358\) 0 0
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) 31.7386 1.67045
\(362\) 0 0
\(363\) 13.3693 0.701707
\(364\) 0 0
\(365\) −4.24621 −0.222257
\(366\) 0 0
\(367\) 14.9309 0.779385 0.389693 0.920945i \(-0.372581\pi\)
0.389693 + 0.920945i \(0.372581\pi\)
\(368\) 0 0
\(369\) 0.630683 0.0328321
\(370\) 0 0
\(371\) −13.1231 −0.681318
\(372\) 0 0
\(373\) 15.3693 0.795793 0.397897 0.917430i \(-0.369740\pi\)
0.397897 + 0.917430i \(0.369740\pi\)
\(374\) 0 0
\(375\) −1.56155 −0.0806382
\(376\) 0 0
\(377\) 2.93087 0.150947
\(378\) 0 0
\(379\) 32.4924 1.66902 0.834512 0.550990i \(-0.185750\pi\)
0.834512 + 0.550990i \(0.185750\pi\)
\(380\) 0 0
\(381\) 9.75379 0.499702
\(382\) 0 0
\(383\) −9.75379 −0.498395 −0.249198 0.968453i \(-0.580167\pi\)
−0.249198 + 0.968453i \(0.580167\pi\)
\(384\) 0 0
\(385\) 1.56155 0.0795841
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 0 0
\(389\) 22.3002 1.13066 0.565332 0.824863i \(-0.308748\pi\)
0.565332 + 0.824863i \(0.308748\pi\)
\(390\) 0 0
\(391\) 23.6155 1.19429
\(392\) 0 0
\(393\) −23.6155 −1.19125
\(394\) 0 0
\(395\) 0.684658 0.0344489
\(396\) 0 0
\(397\) −23.1771 −1.16322 −0.581612 0.813466i \(-0.697578\pi\)
−0.581612 + 0.813466i \(0.697578\pi\)
\(398\) 0 0
\(399\) 11.1231 0.556852
\(400\) 0 0
\(401\) −12.9309 −0.645737 −0.322868 0.946444i \(-0.604647\pi\)
−0.322868 + 0.946444i \(0.604647\pi\)
\(402\) 0 0
\(403\) 41.7538 2.07990
\(404\) 0 0
\(405\) −7.00000 −0.347833
\(406\) 0 0
\(407\) −12.8769 −0.638284
\(408\) 0 0
\(409\) −18.4924 −0.914391 −0.457196 0.889366i \(-0.651146\pi\)
−0.457196 + 0.889366i \(0.651146\pi\)
\(410\) 0 0
\(411\) 11.5076 0.567627
\(412\) 0 0
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) 33.3693 1.63410
\(418\) 0 0
\(419\) −18.2462 −0.891386 −0.445693 0.895186i \(-0.647043\pi\)
−0.445693 + 0.895186i \(0.647043\pi\)
\(420\) 0 0
\(421\) 3.56155 0.173579 0.0867897 0.996227i \(-0.472339\pi\)
0.0867897 + 0.996227i \(0.472339\pi\)
\(422\) 0 0
\(423\) −1.36932 −0.0665785
\(424\) 0 0
\(425\) 7.56155 0.366789
\(426\) 0 0
\(427\) −6.87689 −0.332796
\(428\) 0 0
\(429\) −16.3002 −0.786980
\(430\) 0 0
\(431\) −22.9309 −1.10454 −0.552271 0.833665i \(-0.686238\pi\)
−0.552271 + 0.833665i \(0.686238\pi\)
\(432\) 0 0
\(433\) 19.7538 0.949307 0.474653 0.880173i \(-0.342573\pi\)
0.474653 + 0.880173i \(0.342573\pi\)
\(434\) 0 0
\(435\) −0.684658 −0.0328269
\(436\) 0 0
\(437\) −22.2462 −1.06418
\(438\) 0 0
\(439\) 19.1231 0.912696 0.456348 0.889801i \(-0.349157\pi\)
0.456348 + 0.889801i \(0.349157\pi\)
\(440\) 0 0
\(441\) −0.561553 −0.0267406
\(442\) 0 0
\(443\) −19.6155 −0.931962 −0.465981 0.884795i \(-0.654298\pi\)
−0.465981 + 0.884795i \(0.654298\pi\)
\(444\) 0 0
\(445\) 5.12311 0.242858
\(446\) 0 0
\(447\) 0.384472 0.0181849
\(448\) 0 0
\(449\) −21.3153 −1.00593 −0.502967 0.864306i \(-0.667758\pi\)
−0.502967 + 0.864306i \(0.667758\pi\)
\(450\) 0 0
\(451\) −1.75379 −0.0825827
\(452\) 0 0
\(453\) 30.9309 1.45326
\(454\) 0 0
\(455\) 6.68466 0.313382
\(456\) 0 0
\(457\) 8.63068 0.403726 0.201863 0.979414i \(-0.435300\pi\)
0.201863 + 0.979414i \(0.435300\pi\)
\(458\) 0 0
\(459\) 42.0540 1.96291
\(460\) 0 0
\(461\) 18.8769 0.879185 0.439592 0.898197i \(-0.355123\pi\)
0.439592 + 0.898197i \(0.355123\pi\)
\(462\) 0 0
\(463\) −6.24621 −0.290286 −0.145143 0.989411i \(-0.546364\pi\)
−0.145143 + 0.989411i \(0.546364\pi\)
\(464\) 0 0
\(465\) −9.75379 −0.452321
\(466\) 0 0
\(467\) 25.5616 1.18285 0.591424 0.806361i \(-0.298566\pi\)
0.591424 + 0.806361i \(0.298566\pi\)
\(468\) 0 0
\(469\) 2.24621 0.103720
\(470\) 0 0
\(471\) −6.63068 −0.305526
\(472\) 0 0
\(473\) −11.1231 −0.511441
\(474\) 0 0
\(475\) −7.12311 −0.326831
\(476\) 0 0
\(477\) 7.36932 0.337418
\(478\) 0 0
\(479\) 17.3693 0.793624 0.396812 0.917900i \(-0.370116\pi\)
0.396812 + 0.917900i \(0.370116\pi\)
\(480\) 0 0
\(481\) −55.1231 −2.51340
\(482\) 0 0
\(483\) −4.87689 −0.221906
\(484\) 0 0
\(485\) 1.31534 0.0597266
\(486\) 0 0
\(487\) 3.12311 0.141521 0.0707607 0.997493i \(-0.477457\pi\)
0.0707607 + 0.997493i \(0.477457\pi\)
\(488\) 0 0
\(489\) −30.6307 −1.38517
\(490\) 0 0
\(491\) −3.31534 −0.149619 −0.0748096 0.997198i \(-0.523835\pi\)
−0.0748096 + 0.997198i \(0.523835\pi\)
\(492\) 0 0
\(493\) 3.31534 0.149315
\(494\) 0 0
\(495\) −0.876894 −0.0394135
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −0.192236 −0.00860566 −0.00430283 0.999991i \(-0.501370\pi\)
−0.00430283 + 0.999991i \(0.501370\pi\)
\(500\) 0 0
\(501\) −6.54640 −0.292471
\(502\) 0 0
\(503\) −29.1771 −1.30094 −0.650471 0.759531i \(-0.725428\pi\)
−0.650471 + 0.759531i \(0.725428\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) −49.4773 −2.19736
\(508\) 0 0
\(509\) 32.7386 1.45111 0.725557 0.688162i \(-0.241582\pi\)
0.725557 + 0.688162i \(0.241582\pi\)
\(510\) 0 0
\(511\) −4.24621 −0.187841
\(512\) 0 0
\(513\) −39.6155 −1.74907
\(514\) 0 0
\(515\) −11.8078 −0.520312
\(516\) 0 0
\(517\) 3.80776 0.167465
\(518\) 0 0
\(519\) 36.1922 1.58866
\(520\) 0 0
\(521\) −12.2462 −0.536516 −0.268258 0.963347i \(-0.586448\pi\)
−0.268258 + 0.963347i \(0.586448\pi\)
\(522\) 0 0
\(523\) −12.0000 −0.524723 −0.262362 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) 0 0
\(525\) −1.56155 −0.0681518
\(526\) 0 0
\(527\) 47.2311 2.05742
\(528\) 0 0
\(529\) −13.2462 −0.575922
\(530\) 0 0
\(531\) 2.24621 0.0974773
\(532\) 0 0
\(533\) −7.50758 −0.325189
\(534\) 0 0
\(535\) 15.1231 0.653829
\(536\) 0 0
\(537\) 6.24621 0.269544
\(538\) 0 0
\(539\) 1.56155 0.0672608
\(540\) 0 0
\(541\) 41.4233 1.78093 0.890463 0.455055i \(-0.150380\pi\)
0.890463 + 0.455055i \(0.150380\pi\)
\(542\) 0 0
\(543\) 8.00000 0.343313
\(544\) 0 0
\(545\) −4.43845 −0.190122
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 0 0
\(549\) 3.86174 0.164815
\(550\) 0 0
\(551\) −3.12311 −0.133049
\(552\) 0 0
\(553\) 0.684658 0.0291146
\(554\) 0 0
\(555\) 12.8769 0.546594
\(556\) 0 0
\(557\) −17.6155 −0.746394 −0.373197 0.927752i \(-0.621738\pi\)
−0.373197 + 0.927752i \(0.621738\pi\)
\(558\) 0 0
\(559\) −47.6155 −2.01392
\(560\) 0 0
\(561\) −18.4384 −0.778472
\(562\) 0 0
\(563\) −7.50758 −0.316407 −0.158203 0.987407i \(-0.550570\pi\)
−0.158203 + 0.987407i \(0.550570\pi\)
\(564\) 0 0
\(565\) 8.24621 0.346921
\(566\) 0 0
\(567\) −7.00000 −0.293972
\(568\) 0 0
\(569\) 18.9848 0.795886 0.397943 0.917410i \(-0.369724\pi\)
0.397943 + 0.917410i \(0.369724\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 1.06913 0.0446636
\(574\) 0 0
\(575\) 3.12311 0.130243
\(576\) 0 0
\(577\) −3.56155 −0.148269 −0.0741347 0.997248i \(-0.523619\pi\)
−0.0741347 + 0.997248i \(0.523619\pi\)
\(578\) 0 0
\(579\) −20.4924 −0.851636
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) −20.4924 −0.848709
\(584\) 0 0
\(585\) −3.75379 −0.155200
\(586\) 0 0
\(587\) 10.2462 0.422906 0.211453 0.977388i \(-0.432180\pi\)
0.211453 + 0.977388i \(0.432180\pi\)
\(588\) 0 0
\(589\) −44.4924 −1.83328
\(590\) 0 0
\(591\) 20.4924 0.842946
\(592\) 0 0
\(593\) 37.4233 1.53679 0.768395 0.639976i \(-0.221056\pi\)
0.768395 + 0.639976i \(0.221056\pi\)
\(594\) 0 0
\(595\) 7.56155 0.309993
\(596\) 0 0
\(597\) 22.2462 0.910477
\(598\) 0 0
\(599\) −46.9309 −1.91754 −0.958772 0.284178i \(-0.908279\pi\)
−0.958772 + 0.284178i \(0.908279\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) −1.26137 −0.0513668
\(604\) 0 0
\(605\) −8.56155 −0.348077
\(606\) 0 0
\(607\) −8.68466 −0.352499 −0.176250 0.984345i \(-0.556397\pi\)
−0.176250 + 0.984345i \(0.556397\pi\)
\(608\) 0 0
\(609\) −0.684658 −0.0277438
\(610\) 0 0
\(611\) 16.3002 0.659435
\(612\) 0 0
\(613\) 16.7386 0.676067 0.338034 0.941134i \(-0.390238\pi\)
0.338034 + 0.941134i \(0.390238\pi\)
\(614\) 0 0
\(615\) 1.75379 0.0707196
\(616\) 0 0
\(617\) −15.7538 −0.634224 −0.317112 0.948388i \(-0.602713\pi\)
−0.317112 + 0.948388i \(0.602713\pi\)
\(618\) 0 0
\(619\) 10.6307 0.427283 0.213642 0.976912i \(-0.431468\pi\)
0.213642 + 0.976912i \(0.431468\pi\)
\(620\) 0 0
\(621\) 17.3693 0.697007
\(622\) 0 0
\(623\) 5.12311 0.205253
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 17.3693 0.693664
\(628\) 0 0
\(629\) −62.3542 −2.48622
\(630\) 0 0
\(631\) −27.4233 −1.09170 −0.545852 0.837882i \(-0.683794\pi\)
−0.545852 + 0.837882i \(0.683794\pi\)
\(632\) 0 0
\(633\) 27.4233 1.08998
\(634\) 0 0
\(635\) −6.24621 −0.247873
\(636\) 0 0
\(637\) 6.68466 0.264856
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −22.9848 −0.907847 −0.453923 0.891041i \(-0.649976\pi\)
−0.453923 + 0.891041i \(0.649976\pi\)
\(642\) 0 0
\(643\) −30.0540 −1.18521 −0.592607 0.805492i \(-0.701901\pi\)
−0.592607 + 0.805492i \(0.701901\pi\)
\(644\) 0 0
\(645\) 11.1231 0.437972
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 0 0
\(649\) −6.24621 −0.245185
\(650\) 0 0
\(651\) −9.75379 −0.382281
\(652\) 0 0
\(653\) −38.4924 −1.50632 −0.753162 0.657835i \(-0.771473\pi\)
−0.753162 + 0.657835i \(0.771473\pi\)
\(654\) 0 0
\(655\) 15.1231 0.590909
\(656\) 0 0
\(657\) 2.38447 0.0930271
\(658\) 0 0
\(659\) −0.192236 −0.00748845 −0.00374422 0.999993i \(-0.501192\pi\)
−0.00374422 + 0.999993i \(0.501192\pi\)
\(660\) 0 0
\(661\) −17.6155 −0.685165 −0.342582 0.939488i \(-0.611302\pi\)
−0.342582 + 0.939488i \(0.611302\pi\)
\(662\) 0 0
\(663\) −78.9309 −3.06542
\(664\) 0 0
\(665\) −7.12311 −0.276222
\(666\) 0 0
\(667\) 1.36932 0.0530202
\(668\) 0 0
\(669\) −38.5464 −1.49029
\(670\) 0 0
\(671\) −10.7386 −0.414560
\(672\) 0 0
\(673\) 41.6155 1.60416 0.802080 0.597216i \(-0.203727\pi\)
0.802080 + 0.597216i \(0.203727\pi\)
\(674\) 0 0
\(675\) 5.56155 0.214064
\(676\) 0 0
\(677\) −1.31534 −0.0505527 −0.0252763 0.999681i \(-0.508047\pi\)
−0.0252763 + 0.999681i \(0.508047\pi\)
\(678\) 0 0
\(679\) 1.31534 0.0504782
\(680\) 0 0
\(681\) 17.6695 0.677097
\(682\) 0 0
\(683\) 44.9848 1.72130 0.860649 0.509199i \(-0.170058\pi\)
0.860649 + 0.509199i \(0.170058\pi\)
\(684\) 0 0
\(685\) −7.36932 −0.281567
\(686\) 0 0
\(687\) 17.7538 0.677349
\(688\) 0 0
\(689\) −87.7235 −3.34200
\(690\) 0 0
\(691\) 16.4924 0.627401 0.313701 0.949522i \(-0.398431\pi\)
0.313701 + 0.949522i \(0.398431\pi\)
\(692\) 0 0
\(693\) −0.876894 −0.0333105
\(694\) 0 0
\(695\) −21.3693 −0.810584
\(696\) 0 0
\(697\) −8.49242 −0.321673
\(698\) 0 0
\(699\) 16.9848 0.642426
\(700\) 0 0
\(701\) −9.31534 −0.351836 −0.175918 0.984405i \(-0.556289\pi\)
−0.175918 + 0.984405i \(0.556289\pi\)
\(702\) 0 0
\(703\) 58.7386 2.21537
\(704\) 0 0
\(705\) −3.80776 −0.143409
\(706\) 0 0
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) −4.05398 −0.152250 −0.0761251 0.997098i \(-0.524255\pi\)
−0.0761251 + 0.997098i \(0.524255\pi\)
\(710\) 0 0
\(711\) −0.384472 −0.0144188
\(712\) 0 0
\(713\) 19.5076 0.730565
\(714\) 0 0
\(715\) 10.4384 0.390376
\(716\) 0 0
\(717\) 28.1922 1.05286
\(718\) 0 0
\(719\) −23.6155 −0.880711 −0.440355 0.897824i \(-0.645148\pi\)
−0.440355 + 0.897824i \(0.645148\pi\)
\(720\) 0 0
\(721\) −11.8078 −0.439744
\(722\) 0 0
\(723\) −3.12311 −0.116150
\(724\) 0 0
\(725\) 0.438447 0.0162835
\(726\) 0 0
\(727\) 48.9848 1.81675 0.908374 0.418159i \(-0.137325\pi\)
0.908374 + 0.418159i \(0.137325\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) −53.8617 −1.99215
\(732\) 0 0
\(733\) 16.4384 0.607168 0.303584 0.952805i \(-0.401817\pi\)
0.303584 + 0.952805i \(0.401817\pi\)
\(734\) 0 0
\(735\) −1.56155 −0.0575987
\(736\) 0 0
\(737\) 3.50758 0.129203
\(738\) 0 0
\(739\) −6.43845 −0.236842 −0.118421 0.992963i \(-0.537783\pi\)
−0.118421 + 0.992963i \(0.537783\pi\)
\(740\) 0 0
\(741\) 74.3542 2.73147
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −0.246211 −0.00902048
\(746\) 0 0
\(747\) −6.73863 −0.246554
\(748\) 0 0
\(749\) 15.1231 0.552586
\(750\) 0 0
\(751\) −31.3153 −1.14271 −0.571357 0.820702i \(-0.693583\pi\)
−0.571357 + 0.820702i \(0.693583\pi\)
\(752\) 0 0
\(753\) −20.8769 −0.760796
\(754\) 0 0
\(755\) −19.8078 −0.720878
\(756\) 0 0
\(757\) 15.3693 0.558607 0.279304 0.960203i \(-0.409896\pi\)
0.279304 + 0.960203i \(0.409896\pi\)
\(758\) 0 0
\(759\) −7.61553 −0.276426
\(760\) 0 0
\(761\) 26.0000 0.942499 0.471250 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) 0 0
\(763\) −4.43845 −0.160683
\(764\) 0 0
\(765\) −4.24621 −0.153522
\(766\) 0 0
\(767\) −26.7386 −0.965476
\(768\) 0 0
\(769\) 42.9848 1.55007 0.775037 0.631916i \(-0.217731\pi\)
0.775037 + 0.631916i \(0.217731\pi\)
\(770\) 0 0
\(771\) 28.8769 1.03998
\(772\) 0 0
\(773\) −29.8078 −1.07211 −0.536055 0.844183i \(-0.680086\pi\)
−0.536055 + 0.844183i \(0.680086\pi\)
\(774\) 0 0
\(775\) 6.24621 0.224371
\(776\) 0 0
\(777\) 12.8769 0.461956
\(778\) 0 0
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 2.43845 0.0871430
\(784\) 0 0
\(785\) 4.24621 0.151554
\(786\) 0 0
\(787\) 33.1771 1.18264 0.591318 0.806439i \(-0.298608\pi\)
0.591318 + 0.806439i \(0.298608\pi\)
\(788\) 0 0
\(789\) −14.6307 −0.520866
\(790\) 0 0
\(791\) 8.24621 0.293202
\(792\) 0 0
\(793\) −45.9697 −1.63243
\(794\) 0 0
\(795\) 20.4924 0.726791
\(796\) 0 0
\(797\) 17.8078 0.630783 0.315392 0.948962i \(-0.397864\pi\)
0.315392 + 0.948962i \(0.397864\pi\)
\(798\) 0 0
\(799\) 18.4384 0.652305
\(800\) 0 0
\(801\) −2.87689 −0.101650
\(802\) 0 0
\(803\) −6.63068 −0.233992
\(804\) 0 0
\(805\) 3.12311 0.110075
\(806\) 0 0
\(807\) −6.63068 −0.233411
\(808\) 0 0
\(809\) 4.05398 0.142530 0.0712651 0.997457i \(-0.477296\pi\)
0.0712651 + 0.997457i \(0.477296\pi\)
\(810\) 0 0
\(811\) −27.6155 −0.969712 −0.484856 0.874594i \(-0.661128\pi\)
−0.484856 + 0.874594i \(0.661128\pi\)
\(812\) 0 0
\(813\) 9.75379 0.342080
\(814\) 0 0
\(815\) 19.6155 0.687102
\(816\) 0 0
\(817\) 50.7386 1.77512
\(818\) 0 0
\(819\) −3.75379 −0.131168
\(820\) 0 0
\(821\) 54.3002 1.89509 0.947545 0.319623i \(-0.103556\pi\)
0.947545 + 0.319623i \(0.103556\pi\)
\(822\) 0 0
\(823\) −17.7538 −0.618858 −0.309429 0.950923i \(-0.600138\pi\)
−0.309429 + 0.950923i \(0.600138\pi\)
\(824\) 0 0
\(825\) −2.43845 −0.0848958
\(826\) 0 0
\(827\) 24.8769 0.865054 0.432527 0.901621i \(-0.357622\pi\)
0.432527 + 0.901621i \(0.357622\pi\)
\(828\) 0 0
\(829\) 5.61553 0.195035 0.0975177 0.995234i \(-0.468910\pi\)
0.0975177 + 0.995234i \(0.468910\pi\)
\(830\) 0 0
\(831\) 12.8769 0.446695
\(832\) 0 0
\(833\) 7.56155 0.261992
\(834\) 0 0
\(835\) 4.19224 0.145078
\(836\) 0 0
\(837\) 34.7386 1.20074
\(838\) 0 0
\(839\) 19.1231 0.660203 0.330101 0.943945i \(-0.392917\pi\)
0.330101 + 0.943945i \(0.392917\pi\)
\(840\) 0 0
\(841\) −28.8078 −0.993371
\(842\) 0 0
\(843\) 30.5464 1.05207
\(844\) 0 0
\(845\) 31.6847 1.08999
\(846\) 0 0
\(847\) −8.56155 −0.294178
\(848\) 0 0
\(849\) 7.31534 0.251062
\(850\) 0 0
\(851\) −25.7538 −0.882829
\(852\) 0 0
\(853\) 15.7538 0.539399 0.269700 0.962944i \(-0.413076\pi\)
0.269700 + 0.962944i \(0.413076\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) 52.7386 1.80152 0.900759 0.434320i \(-0.143011\pi\)
0.900759 + 0.434320i \(0.143011\pi\)
\(858\) 0 0
\(859\) 36.9848 1.26191 0.630953 0.775821i \(-0.282664\pi\)
0.630953 + 0.775821i \(0.282664\pi\)
\(860\) 0 0
\(861\) 1.75379 0.0597690
\(862\) 0 0
\(863\) 28.4924 0.969893 0.484947 0.874544i \(-0.338839\pi\)
0.484947 + 0.874544i \(0.338839\pi\)
\(864\) 0 0
\(865\) −23.1771 −0.788044
\(866\) 0 0
\(867\) −62.7386 −2.13072
\(868\) 0 0
\(869\) 1.06913 0.0362678
\(870\) 0 0
\(871\) 15.0152 0.508769
\(872\) 0 0
\(873\) −0.738634 −0.0249990
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −13.5076 −0.456118 −0.228059 0.973647i \(-0.573238\pi\)
−0.228059 + 0.973647i \(0.573238\pi\)
\(878\) 0 0
\(879\) −50.0540 −1.68828
\(880\) 0 0
\(881\) 54.1080 1.82294 0.911472 0.411363i \(-0.134947\pi\)
0.911472 + 0.411363i \(0.134947\pi\)
\(882\) 0 0
\(883\) 21.7538 0.732073 0.366037 0.930600i \(-0.380714\pi\)
0.366037 + 0.930600i \(0.380714\pi\)
\(884\) 0 0
\(885\) 6.24621 0.209964
\(886\) 0 0
\(887\) −36.4924 −1.22530 −0.612648 0.790356i \(-0.709896\pi\)
−0.612648 + 0.790356i \(0.709896\pi\)
\(888\) 0 0
\(889\) −6.24621 −0.209491
\(890\) 0 0
\(891\) −10.9309 −0.366198
\(892\) 0 0
\(893\) −17.3693 −0.581242
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) 0 0
\(897\) −32.6004 −1.08849
\(898\) 0 0
\(899\) 2.73863 0.0913385
\(900\) 0 0
\(901\) −99.2311 −3.30587
\(902\) 0 0
\(903\) 11.1231 0.370154
\(904\) 0 0
\(905\) −5.12311 −0.170298
\(906\) 0 0
\(907\) −48.1080 −1.59740 −0.798699 0.601731i \(-0.794478\pi\)
−0.798699 + 0.601731i \(0.794478\pi\)
\(908\) 0 0
\(909\) −3.36932 −0.111753
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) 18.7386 0.620158
\(914\) 0 0
\(915\) 10.7386 0.355008
\(916\) 0 0
\(917\) 15.1231 0.499409
\(918\) 0 0
\(919\) 2.43845 0.0804370 0.0402185 0.999191i \(-0.487195\pi\)
0.0402185 + 0.999191i \(0.487195\pi\)
\(920\) 0 0
\(921\) 44.7926 1.47597
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −8.24621 −0.271134
\(926\) 0 0
\(927\) 6.63068 0.217780
\(928\) 0 0
\(929\) −2.87689 −0.0943878 −0.0471939 0.998886i \(-0.515028\pi\)
−0.0471939 + 0.998886i \(0.515028\pi\)
\(930\) 0 0
\(931\) −7.12311 −0.233450
\(932\) 0 0
\(933\) 20.1080 0.658305
\(934\) 0 0
\(935\) 11.8078 0.386155
\(936\) 0 0
\(937\) 37.8078 1.23513 0.617563 0.786521i \(-0.288120\pi\)
0.617563 + 0.786521i \(0.288120\pi\)
\(938\) 0 0
\(939\) −41.6695 −1.35983
\(940\) 0 0
\(941\) 28.6307 0.933334 0.466667 0.884433i \(-0.345455\pi\)
0.466667 + 0.884433i \(0.345455\pi\)
\(942\) 0 0
\(943\) −3.50758 −0.114222
\(944\) 0 0
\(945\) 5.56155 0.180917
\(946\) 0 0
\(947\) −35.2311 −1.14486 −0.572428 0.819955i \(-0.693998\pi\)
−0.572428 + 0.819955i \(0.693998\pi\)
\(948\) 0 0
\(949\) −28.3845 −0.921399
\(950\) 0 0
\(951\) 15.6155 0.506368
\(952\) 0 0
\(953\) −51.8617 −1.67997 −0.839983 0.542612i \(-0.817435\pi\)
−0.839983 + 0.542612i \(0.817435\pi\)
\(954\) 0 0
\(955\) −0.684658 −0.0221550
\(956\) 0 0
\(957\) −1.06913 −0.0345601
\(958\) 0 0
\(959\) −7.36932 −0.237968
\(960\) 0 0
\(961\) 8.01515 0.258553
\(962\) 0 0
\(963\) −8.49242 −0.273664
\(964\) 0 0
\(965\) 13.1231 0.422448
\(966\) 0 0
\(967\) 44.1080 1.41842 0.709208 0.704999i \(-0.249053\pi\)
0.709208 + 0.704999i \(0.249053\pi\)
\(968\) 0 0
\(969\) 84.1080 2.70194
\(970\) 0 0
\(971\) −22.7386 −0.729717 −0.364859 0.931063i \(-0.618883\pi\)
−0.364859 + 0.931063i \(0.618883\pi\)
\(972\) 0 0
\(973\) −21.3693 −0.685069
\(974\) 0 0
\(975\) −10.4384 −0.334298
\(976\) 0 0
\(977\) 10.9848 0.351436 0.175718 0.984441i \(-0.443775\pi\)
0.175718 + 0.984441i \(0.443775\pi\)
\(978\) 0 0
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) 2.49242 0.0795769
\(982\) 0 0
\(983\) −29.1771 −0.930604 −0.465302 0.885152i \(-0.654054\pi\)
−0.465302 + 0.885152i \(0.654054\pi\)
\(984\) 0 0
\(985\) −13.1231 −0.418137
\(986\) 0 0
\(987\) −3.80776 −0.121202
\(988\) 0 0
\(989\) −22.2462 −0.707388
\(990\) 0 0
\(991\) −20.4924 −0.650963 −0.325482 0.945548i \(-0.605526\pi\)
−0.325482 + 0.945548i \(0.605526\pi\)
\(992\) 0 0
\(993\) 18.7386 0.594653
\(994\) 0 0
\(995\) −14.2462 −0.451635
\(996\) 0 0
\(997\) −56.9309 −1.80302 −0.901509 0.432760i \(-0.857540\pi\)
−0.901509 + 0.432760i \(0.857540\pi\)
\(998\) 0 0
\(999\) −45.8617 −1.45100
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 280.2.a.d.1.1 2
3.2 odd 2 2520.2.a.w.1.1 2
4.3 odd 2 560.2.a.g.1.2 2
5.2 odd 4 1400.2.g.k.449.3 4
5.3 odd 4 1400.2.g.k.449.2 4
5.4 even 2 1400.2.a.p.1.2 2
7.2 even 3 1960.2.q.s.361.2 4
7.3 odd 6 1960.2.q.u.961.1 4
7.4 even 3 1960.2.q.s.961.2 4
7.5 odd 6 1960.2.q.u.361.1 4
7.6 odd 2 1960.2.a.r.1.2 2
8.3 odd 2 2240.2.a.bi.1.1 2
8.5 even 2 2240.2.a.be.1.2 2
12.11 even 2 5040.2.a.bq.1.2 2
20.3 even 4 2800.2.g.u.449.3 4
20.7 even 4 2800.2.g.u.449.2 4
20.19 odd 2 2800.2.a.bn.1.1 2
28.27 even 2 3920.2.a.bu.1.1 2
35.34 odd 2 9800.2.a.by.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.d.1.1 2 1.1 even 1 trivial
560.2.a.g.1.2 2 4.3 odd 2
1400.2.a.p.1.2 2 5.4 even 2
1400.2.g.k.449.2 4 5.3 odd 4
1400.2.g.k.449.3 4 5.2 odd 4
1960.2.a.r.1.2 2 7.6 odd 2
1960.2.q.s.361.2 4 7.2 even 3
1960.2.q.s.961.2 4 7.4 even 3
1960.2.q.u.361.1 4 7.5 odd 6
1960.2.q.u.961.1 4 7.3 odd 6
2240.2.a.be.1.2 2 8.5 even 2
2240.2.a.bi.1.1 2 8.3 odd 2
2520.2.a.w.1.1 2 3.2 odd 2
2800.2.a.bn.1.1 2 20.19 odd 2
2800.2.g.u.449.2 4 20.7 even 4
2800.2.g.u.449.3 4 20.3 even 4
3920.2.a.bu.1.1 2 28.27 even 2
5040.2.a.bq.1.2 2 12.11 even 2
9800.2.a.by.1.1 2 35.34 odd 2