Properties

Label 4410.2.a.bt.1.2
Level $4410$
Weight $2$
Character 4410.1
Self dual yes
Analytic conductor $35.214$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.2140272914\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 490)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4410.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} -1.00000 q^{10} +0.828427 q^{11} -4.82843 q^{13} +1.00000 q^{16} -2.58579 q^{17} +0.585786 q^{19} -1.00000 q^{20} +0.828427 q^{22} +1.17157 q^{23} +1.00000 q^{25} -4.82843 q^{26} +4.82843 q^{29} -2.82843 q^{31} +1.00000 q^{32} -2.58579 q^{34} -7.65685 q^{37} +0.585786 q^{38} -1.00000 q^{40} +3.07107 q^{41} -8.82843 q^{43} +0.828427 q^{44} +1.17157 q^{46} -5.17157 q^{47} +1.00000 q^{50} -4.82843 q^{52} -6.48528 q^{53} -0.828427 q^{55} +4.82843 q^{58} -8.58579 q^{59} +9.31371 q^{61} -2.82843 q^{62} +1.00000 q^{64} +4.82843 q^{65} +1.65685 q^{67} -2.58579 q^{68} +4.48528 q^{71} -9.41421 q^{73} -7.65685 q^{74} +0.585786 q^{76} -6.82843 q^{79} -1.00000 q^{80} +3.07107 q^{82} +2.24264 q^{83} +2.58579 q^{85} -8.82843 q^{86} +0.828427 q^{88} -12.7279 q^{89} +1.17157 q^{92} -5.17157 q^{94} -0.585786 q^{95} -7.75736 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} + 2q^{8} - 2q^{10} - 4q^{11} - 4q^{13} + 2q^{16} - 8q^{17} + 4q^{19} - 2q^{20} - 4q^{22} + 8q^{23} + 2q^{25} - 4q^{26} + 4q^{29} + 2q^{32} - 8q^{34} - 4q^{37} + 4q^{38} - 2q^{40} - 8q^{41} - 12q^{43} - 4q^{44} + 8q^{46} - 16q^{47} + 2q^{50} - 4q^{52} + 4q^{53} + 4q^{55} + 4q^{58} - 20q^{59} - 4q^{61} + 2q^{64} + 4q^{65} - 8q^{67} - 8q^{68} - 8q^{71} - 16q^{73} - 4q^{74} + 4q^{76} - 8q^{79} - 2q^{80} - 8q^{82} - 4q^{83} + 8q^{85} - 12q^{86} - 4q^{88} + 8q^{92} - 16q^{94} - 4q^{95} - 24q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 0.828427 0.249780 0.124890 0.992171i \(-0.460142\pi\)
0.124890 + 0.992171i \(0.460142\pi\)
\(12\) 0 0
\(13\) −4.82843 −1.33916 −0.669582 0.742738i \(-0.733527\pi\)
−0.669582 + 0.742738i \(0.733527\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.58579 −0.627145 −0.313573 0.949564i \(-0.601526\pi\)
−0.313573 + 0.949564i \(0.601526\pi\)
\(18\) 0 0
\(19\) 0.585786 0.134389 0.0671943 0.997740i \(-0.478595\pi\)
0.0671943 + 0.997740i \(0.478595\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 0.828427 0.176621
\(23\) 1.17157 0.244290 0.122145 0.992512i \(-0.461023\pi\)
0.122145 + 0.992512i \(0.461023\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −4.82843 −0.946932
\(27\) 0 0
\(28\) 0 0
\(29\) 4.82843 0.896616 0.448308 0.893879i \(-0.352027\pi\)
0.448308 + 0.893879i \(0.352027\pi\)
\(30\) 0 0
\(31\) −2.82843 −0.508001 −0.254000 0.967204i \(-0.581746\pi\)
−0.254000 + 0.967204i \(0.581746\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.58579 −0.443459
\(35\) 0 0
\(36\) 0 0
\(37\) −7.65685 −1.25878 −0.629390 0.777090i \(-0.716695\pi\)
−0.629390 + 0.777090i \(0.716695\pi\)
\(38\) 0.585786 0.0950271
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 3.07107 0.479620 0.239810 0.970820i \(-0.422915\pi\)
0.239810 + 0.970820i \(0.422915\pi\)
\(42\) 0 0
\(43\) −8.82843 −1.34632 −0.673161 0.739496i \(-0.735064\pi\)
−0.673161 + 0.739496i \(0.735064\pi\)
\(44\) 0.828427 0.124890
\(45\) 0 0
\(46\) 1.17157 0.172739
\(47\) −5.17157 −0.754351 −0.377176 0.926142i \(-0.623105\pi\)
−0.377176 + 0.926142i \(0.623105\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −4.82843 −0.669582
\(53\) −6.48528 −0.890822 −0.445411 0.895326i \(-0.646942\pi\)
−0.445411 + 0.895326i \(0.646942\pi\)
\(54\) 0 0
\(55\) −0.828427 −0.111705
\(56\) 0 0
\(57\) 0 0
\(58\) 4.82843 0.634004
\(59\) −8.58579 −1.11777 −0.558887 0.829244i \(-0.688771\pi\)
−0.558887 + 0.829244i \(0.688771\pi\)
\(60\) 0 0
\(61\) 9.31371 1.19250 0.596249 0.802799i \(-0.296657\pi\)
0.596249 + 0.802799i \(0.296657\pi\)
\(62\) −2.82843 −0.359211
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.82843 0.598893
\(66\) 0 0
\(67\) 1.65685 0.202417 0.101208 0.994865i \(-0.467729\pi\)
0.101208 + 0.994865i \(0.467729\pi\)
\(68\) −2.58579 −0.313573
\(69\) 0 0
\(70\) 0 0
\(71\) 4.48528 0.532305 0.266152 0.963931i \(-0.414248\pi\)
0.266152 + 0.963931i \(0.414248\pi\)
\(72\) 0 0
\(73\) −9.41421 −1.10185 −0.550925 0.834555i \(-0.685725\pi\)
−0.550925 + 0.834555i \(0.685725\pi\)
\(74\) −7.65685 −0.890091
\(75\) 0 0
\(76\) 0.585786 0.0671943
\(77\) 0 0
\(78\) 0 0
\(79\) −6.82843 −0.768258 −0.384129 0.923279i \(-0.625498\pi\)
−0.384129 + 0.923279i \(0.625498\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 3.07107 0.339143
\(83\) 2.24264 0.246162 0.123081 0.992397i \(-0.460723\pi\)
0.123081 + 0.992397i \(0.460723\pi\)
\(84\) 0 0
\(85\) 2.58579 0.280468
\(86\) −8.82843 −0.951994
\(87\) 0 0
\(88\) 0.828427 0.0883106
\(89\) −12.7279 −1.34916 −0.674579 0.738203i \(-0.735675\pi\)
−0.674579 + 0.738203i \(0.735675\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.17157 0.122145
\(93\) 0 0
\(94\) −5.17157 −0.533407
\(95\) −0.585786 −0.0601004
\(96\) 0 0
\(97\) −7.75736 −0.787641 −0.393820 0.919187i \(-0.628847\pi\)
−0.393820 + 0.919187i \(0.628847\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −13.3137 −1.32476 −0.662382 0.749166i \(-0.730454\pi\)
−0.662382 + 0.749166i \(0.730454\pi\)
\(102\) 0 0
\(103\) −14.8284 −1.46109 −0.730544 0.682865i \(-0.760734\pi\)
−0.730544 + 0.682865i \(0.760734\pi\)
\(104\) −4.82843 −0.473466
\(105\) 0 0
\(106\) −6.48528 −0.629906
\(107\) −9.65685 −0.933563 −0.466782 0.884373i \(-0.654587\pi\)
−0.466782 + 0.884373i \(0.654587\pi\)
\(108\) 0 0
\(109\) 2.48528 0.238047 0.119023 0.992891i \(-0.462024\pi\)
0.119023 + 0.992891i \(0.462024\pi\)
\(110\) −0.828427 −0.0789874
\(111\) 0 0
\(112\) 0 0
\(113\) 15.3137 1.44059 0.720296 0.693667i \(-0.244006\pi\)
0.720296 + 0.693667i \(0.244006\pi\)
\(114\) 0 0
\(115\) −1.17157 −0.109250
\(116\) 4.82843 0.448308
\(117\) 0 0
\(118\) −8.58579 −0.790386
\(119\) 0 0
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) 9.31371 0.843224
\(123\) 0 0
\(124\) −2.82843 −0.254000
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.82843 0.250982 0.125491 0.992095i \(-0.459949\pi\)
0.125491 + 0.992095i \(0.459949\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 4.82843 0.423481
\(131\) 6.24264 0.545422 0.272711 0.962096i \(-0.412080\pi\)
0.272711 + 0.962096i \(0.412080\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.65685 0.143130
\(135\) 0 0
\(136\) −2.58579 −0.221729
\(137\) 16.0000 1.36697 0.683486 0.729964i \(-0.260463\pi\)
0.683486 + 0.729964i \(0.260463\pi\)
\(138\) 0 0
\(139\) 19.8995 1.68785 0.843927 0.536459i \(-0.180238\pi\)
0.843927 + 0.536459i \(0.180238\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4.48528 0.376396
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) −4.82843 −0.400979
\(146\) −9.41421 −0.779126
\(147\) 0 0
\(148\) −7.65685 −0.629390
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −11.3137 −0.920697 −0.460348 0.887738i \(-0.652275\pi\)
−0.460348 + 0.887738i \(0.652275\pi\)
\(152\) 0.585786 0.0475136
\(153\) 0 0
\(154\) 0 0
\(155\) 2.82843 0.227185
\(156\) 0 0
\(157\) 6.48528 0.517582 0.258791 0.965933i \(-0.416676\pi\)
0.258791 + 0.965933i \(0.416676\pi\)
\(158\) −6.82843 −0.543240
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) −20.1421 −1.57765 −0.788827 0.614615i \(-0.789311\pi\)
−0.788827 + 0.614615i \(0.789311\pi\)
\(164\) 3.07107 0.239810
\(165\) 0 0
\(166\) 2.24264 0.174063
\(167\) −15.7990 −1.22256 −0.611281 0.791413i \(-0.709346\pi\)
−0.611281 + 0.791413i \(0.709346\pi\)
\(168\) 0 0
\(169\) 10.3137 0.793362
\(170\) 2.58579 0.198321
\(171\) 0 0
\(172\) −8.82843 −0.673161
\(173\) 8.82843 0.671213 0.335606 0.942002i \(-0.391059\pi\)
0.335606 + 0.942002i \(0.391059\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.828427 0.0624450
\(177\) 0 0
\(178\) −12.7279 −0.953998
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −2.48528 −0.184730 −0.0923648 0.995725i \(-0.529443\pi\)
−0.0923648 + 0.995725i \(0.529443\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1.17157 0.0863695
\(185\) 7.65685 0.562943
\(186\) 0 0
\(187\) −2.14214 −0.156648
\(188\) −5.17157 −0.377176
\(189\) 0 0
\(190\) −0.585786 −0.0424974
\(191\) −10.1421 −0.733859 −0.366930 0.930249i \(-0.619591\pi\)
−0.366930 + 0.930249i \(0.619591\pi\)
\(192\) 0 0
\(193\) 5.65685 0.407189 0.203595 0.979055i \(-0.434738\pi\)
0.203595 + 0.979055i \(0.434738\pi\)
\(194\) −7.75736 −0.556946
\(195\) 0 0
\(196\) 0 0
\(197\) 25.7990 1.83810 0.919051 0.394139i \(-0.128957\pi\)
0.919051 + 0.394139i \(0.128957\pi\)
\(198\) 0 0
\(199\) 16.4853 1.16861 0.584305 0.811534i \(-0.301367\pi\)
0.584305 + 0.811534i \(0.301367\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −13.3137 −0.936749
\(203\) 0 0
\(204\) 0 0
\(205\) −3.07107 −0.214493
\(206\) −14.8284 −1.03315
\(207\) 0 0
\(208\) −4.82843 −0.334791
\(209\) 0.485281 0.0335676
\(210\) 0 0
\(211\) 18.6274 1.28236 0.641182 0.767389i \(-0.278444\pi\)
0.641182 + 0.767389i \(0.278444\pi\)
\(212\) −6.48528 −0.445411
\(213\) 0 0
\(214\) −9.65685 −0.660129
\(215\) 8.82843 0.602094
\(216\) 0 0
\(217\) 0 0
\(218\) 2.48528 0.168324
\(219\) 0 0
\(220\) −0.828427 −0.0558525
\(221\) 12.4853 0.839851
\(222\) 0 0
\(223\) −7.31371 −0.489762 −0.244881 0.969553i \(-0.578749\pi\)
−0.244881 + 0.969553i \(0.578749\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 15.3137 1.01865
\(227\) −18.2426 −1.21081 −0.605403 0.795919i \(-0.706988\pi\)
−0.605403 + 0.795919i \(0.706988\pi\)
\(228\) 0 0
\(229\) −16.1421 −1.06670 −0.533351 0.845894i \(-0.679068\pi\)
−0.533351 + 0.845894i \(0.679068\pi\)
\(230\) −1.17157 −0.0772512
\(231\) 0 0
\(232\) 4.82843 0.317002
\(233\) −23.3137 −1.52733 −0.763666 0.645612i \(-0.776603\pi\)
−0.763666 + 0.645612i \(0.776603\pi\)
\(234\) 0 0
\(235\) 5.17157 0.337356
\(236\) −8.58579 −0.558887
\(237\) 0 0
\(238\) 0 0
\(239\) −1.65685 −0.107173 −0.0535865 0.998563i \(-0.517065\pi\)
−0.0535865 + 0.998563i \(0.517065\pi\)
\(240\) 0 0
\(241\) −13.4142 −0.864085 −0.432043 0.901853i \(-0.642207\pi\)
−0.432043 + 0.901853i \(0.642207\pi\)
\(242\) −10.3137 −0.662990
\(243\) 0 0
\(244\) 9.31371 0.596249
\(245\) 0 0
\(246\) 0 0
\(247\) −2.82843 −0.179969
\(248\) −2.82843 −0.179605
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −0.585786 −0.0369745 −0.0184873 0.999829i \(-0.505885\pi\)
−0.0184873 + 0.999829i \(0.505885\pi\)
\(252\) 0 0
\(253\) 0.970563 0.0610188
\(254\) 2.82843 0.177471
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 9.89949 0.617514 0.308757 0.951141i \(-0.400087\pi\)
0.308757 + 0.951141i \(0.400087\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 4.82843 0.299446
\(261\) 0 0
\(262\) 6.24264 0.385672
\(263\) −28.0000 −1.72655 −0.863277 0.504730i \(-0.831592\pi\)
−0.863277 + 0.504730i \(0.831592\pi\)
\(264\) 0 0
\(265\) 6.48528 0.398388
\(266\) 0 0
\(267\) 0 0
\(268\) 1.65685 0.101208
\(269\) −18.4853 −1.12707 −0.563534 0.826093i \(-0.690559\pi\)
−0.563534 + 0.826093i \(0.690559\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) −2.58579 −0.156786
\(273\) 0 0
\(274\) 16.0000 0.966595
\(275\) 0.828427 0.0499560
\(276\) 0 0
\(277\) −8.14214 −0.489214 −0.244607 0.969622i \(-0.578659\pi\)
−0.244607 + 0.969622i \(0.578659\pi\)
\(278\) 19.8995 1.19349
\(279\) 0 0
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) −2.24264 −0.133311 −0.0666556 0.997776i \(-0.521233\pi\)
−0.0666556 + 0.997776i \(0.521233\pi\)
\(284\) 4.48528 0.266152
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 0 0
\(289\) −10.3137 −0.606689
\(290\) −4.82843 −0.283535
\(291\) 0 0
\(292\) −9.41421 −0.550925
\(293\) 8.34315 0.487412 0.243706 0.969849i \(-0.421637\pi\)
0.243706 + 0.969849i \(0.421637\pi\)
\(294\) 0 0
\(295\) 8.58579 0.499884
\(296\) −7.65685 −0.445046
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) −5.65685 −0.327144
\(300\) 0 0
\(301\) 0 0
\(302\) −11.3137 −0.651031
\(303\) 0 0
\(304\) 0.585786 0.0335972
\(305\) −9.31371 −0.533301
\(306\) 0 0
\(307\) 14.9289 0.852039 0.426020 0.904714i \(-0.359915\pi\)
0.426020 + 0.904714i \(0.359915\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 2.82843 0.160644
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) 14.3848 0.813076 0.406538 0.913634i \(-0.366736\pi\)
0.406538 + 0.913634i \(0.366736\pi\)
\(314\) 6.48528 0.365986
\(315\) 0 0
\(316\) −6.82843 −0.384129
\(317\) −10.4853 −0.588912 −0.294456 0.955665i \(-0.595138\pi\)
−0.294456 + 0.955665i \(0.595138\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) −1.51472 −0.0842812
\(324\) 0 0
\(325\) −4.82843 −0.267833
\(326\) −20.1421 −1.11557
\(327\) 0 0
\(328\) 3.07107 0.169571
\(329\) 0 0
\(330\) 0 0
\(331\) 33.7990 1.85776 0.928880 0.370380i \(-0.120773\pi\)
0.928880 + 0.370380i \(0.120773\pi\)
\(332\) 2.24264 0.123081
\(333\) 0 0
\(334\) −15.7990 −0.864482
\(335\) −1.65685 −0.0905236
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 10.3137 0.560992
\(339\) 0 0
\(340\) 2.58579 0.140234
\(341\) −2.34315 −0.126888
\(342\) 0 0
\(343\) 0 0
\(344\) −8.82843 −0.475997
\(345\) 0 0
\(346\) 8.82843 0.474619
\(347\) 3.17157 0.170259 0.0851295 0.996370i \(-0.472870\pi\)
0.0851295 + 0.996370i \(0.472870\pi\)
\(348\) 0 0
\(349\) 2.48528 0.133034 0.0665170 0.997785i \(-0.478811\pi\)
0.0665170 + 0.997785i \(0.478811\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.828427 0.0441553
\(353\) 2.38478 0.126929 0.0634644 0.997984i \(-0.479785\pi\)
0.0634644 + 0.997984i \(0.479785\pi\)
\(354\) 0 0
\(355\) −4.48528 −0.238054
\(356\) −12.7279 −0.674579
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) −28.2843 −1.49279 −0.746393 0.665505i \(-0.768216\pi\)
−0.746393 + 0.665505i \(0.768216\pi\)
\(360\) 0 0
\(361\) −18.6569 −0.981940
\(362\) −2.48528 −0.130623
\(363\) 0 0
\(364\) 0 0
\(365\) 9.41421 0.492762
\(366\) 0 0
\(367\) 24.9706 1.30345 0.651726 0.758454i \(-0.274045\pi\)
0.651726 + 0.758454i \(0.274045\pi\)
\(368\) 1.17157 0.0610725
\(369\) 0 0
\(370\) 7.65685 0.398061
\(371\) 0 0
\(372\) 0 0
\(373\) −30.4853 −1.57847 −0.789234 0.614093i \(-0.789522\pi\)
−0.789234 + 0.614093i \(0.789522\pi\)
\(374\) −2.14214 −0.110767
\(375\) 0 0
\(376\) −5.17157 −0.266704
\(377\) −23.3137 −1.20072
\(378\) 0 0
\(379\) 34.4853 1.77139 0.885695 0.464268i \(-0.153682\pi\)
0.885695 + 0.464268i \(0.153682\pi\)
\(380\) −0.585786 −0.0300502
\(381\) 0 0
\(382\) −10.1421 −0.518917
\(383\) 32.4853 1.65992 0.829960 0.557823i \(-0.188363\pi\)
0.829960 + 0.557823i \(0.188363\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5.65685 0.287926
\(387\) 0 0
\(388\) −7.75736 −0.393820
\(389\) −28.1421 −1.42686 −0.713431 0.700725i \(-0.752860\pi\)
−0.713431 + 0.700725i \(0.752860\pi\)
\(390\) 0 0
\(391\) −3.02944 −0.153205
\(392\) 0 0
\(393\) 0 0
\(394\) 25.7990 1.29973
\(395\) 6.82843 0.343575
\(396\) 0 0
\(397\) 33.7990 1.69632 0.848161 0.529738i \(-0.177710\pi\)
0.848161 + 0.529738i \(0.177710\pi\)
\(398\) 16.4853 0.826332
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) 13.6569 0.680296
\(404\) −13.3137 −0.662382
\(405\) 0 0
\(406\) 0 0
\(407\) −6.34315 −0.314418
\(408\) 0 0
\(409\) −10.5858 −0.523433 −0.261717 0.965145i \(-0.584289\pi\)
−0.261717 + 0.965145i \(0.584289\pi\)
\(410\) −3.07107 −0.151669
\(411\) 0 0
\(412\) −14.8284 −0.730544
\(413\) 0 0
\(414\) 0 0
\(415\) −2.24264 −0.110087
\(416\) −4.82843 −0.236733
\(417\) 0 0
\(418\) 0.485281 0.0237359
\(419\) 20.8701 1.01957 0.509785 0.860302i \(-0.329725\pi\)
0.509785 + 0.860302i \(0.329725\pi\)
\(420\) 0 0
\(421\) 17.3137 0.843819 0.421909 0.906638i \(-0.361360\pi\)
0.421909 + 0.906638i \(0.361360\pi\)
\(422\) 18.6274 0.906768
\(423\) 0 0
\(424\) −6.48528 −0.314953
\(425\) −2.58579 −0.125429
\(426\) 0 0
\(427\) 0 0
\(428\) −9.65685 −0.466782
\(429\) 0 0
\(430\) 8.82843 0.425745
\(431\) 22.3431 1.07623 0.538116 0.842871i \(-0.319136\pi\)
0.538116 + 0.842871i \(0.319136\pi\)
\(432\) 0 0
\(433\) 10.5858 0.508720 0.254360 0.967110i \(-0.418135\pi\)
0.254360 + 0.967110i \(0.418135\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.48528 0.119023
\(437\) 0.686292 0.0328298
\(438\) 0 0
\(439\) −24.9706 −1.19178 −0.595890 0.803066i \(-0.703201\pi\)
−0.595890 + 0.803066i \(0.703201\pi\)
\(440\) −0.828427 −0.0394937
\(441\) 0 0
\(442\) 12.4853 0.593864
\(443\) −3.02944 −0.143933 −0.0719665 0.997407i \(-0.522927\pi\)
−0.0719665 + 0.997407i \(0.522927\pi\)
\(444\) 0 0
\(445\) 12.7279 0.603361
\(446\) −7.31371 −0.346314
\(447\) 0 0
\(448\) 0 0
\(449\) 16.6274 0.784696 0.392348 0.919817i \(-0.371663\pi\)
0.392348 + 0.919817i \(0.371663\pi\)
\(450\) 0 0
\(451\) 2.54416 0.119800
\(452\) 15.3137 0.720296
\(453\) 0 0
\(454\) −18.2426 −0.856170
\(455\) 0 0
\(456\) 0 0
\(457\) 21.6569 1.01306 0.506532 0.862221i \(-0.330927\pi\)
0.506532 + 0.862221i \(0.330927\pi\)
\(458\) −16.1421 −0.754272
\(459\) 0 0
\(460\) −1.17157 −0.0546249
\(461\) 12.8284 0.597479 0.298740 0.954335i \(-0.403434\pi\)
0.298740 + 0.954335i \(0.403434\pi\)
\(462\) 0 0
\(463\) −16.9706 −0.788689 −0.394344 0.918963i \(-0.629028\pi\)
−0.394344 + 0.918963i \(0.629028\pi\)
\(464\) 4.82843 0.224154
\(465\) 0 0
\(466\) −23.3137 −1.07999
\(467\) 15.8995 0.735741 0.367870 0.929877i \(-0.380087\pi\)
0.367870 + 0.929877i \(0.380087\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 5.17157 0.238547
\(471\) 0 0
\(472\) −8.58579 −0.395193
\(473\) −7.31371 −0.336285
\(474\) 0 0
\(475\) 0.585786 0.0268777
\(476\) 0 0
\(477\) 0 0
\(478\) −1.65685 −0.0757827
\(479\) 17.1716 0.784589 0.392295 0.919840i \(-0.371681\pi\)
0.392295 + 0.919840i \(0.371681\pi\)
\(480\) 0 0
\(481\) 36.9706 1.68571
\(482\) −13.4142 −0.611001
\(483\) 0 0
\(484\) −10.3137 −0.468805
\(485\) 7.75736 0.352244
\(486\) 0 0
\(487\) 31.7990 1.44095 0.720475 0.693481i \(-0.243924\pi\)
0.720475 + 0.693481i \(0.243924\pi\)
\(488\) 9.31371 0.421612
\(489\) 0 0
\(490\) 0 0
\(491\) −32.2843 −1.45697 −0.728484 0.685062i \(-0.759775\pi\)
−0.728484 + 0.685062i \(0.759775\pi\)
\(492\) 0 0
\(493\) −12.4853 −0.562309
\(494\) −2.82843 −0.127257
\(495\) 0 0
\(496\) −2.82843 −0.127000
\(497\) 0 0
\(498\) 0 0
\(499\) 30.3431 1.35835 0.679173 0.733978i \(-0.262339\pi\)
0.679173 + 0.733978i \(0.262339\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −0.585786 −0.0261449
\(503\) −17.6569 −0.787280 −0.393640 0.919265i \(-0.628784\pi\)
−0.393640 + 0.919265i \(0.628784\pi\)
\(504\) 0 0
\(505\) 13.3137 0.592452
\(506\) 0.970563 0.0431468
\(507\) 0 0
\(508\) 2.82843 0.125491
\(509\) −5.79899 −0.257036 −0.128518 0.991707i \(-0.541022\pi\)
−0.128518 + 0.991707i \(0.541022\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 9.89949 0.436648
\(515\) 14.8284 0.653419
\(516\) 0 0
\(517\) −4.28427 −0.188422
\(518\) 0 0
\(519\) 0 0
\(520\) 4.82843 0.211741
\(521\) −19.0711 −0.835519 −0.417759 0.908558i \(-0.637184\pi\)
−0.417759 + 0.908558i \(0.637184\pi\)
\(522\) 0 0
\(523\) 23.8995 1.04505 0.522526 0.852623i \(-0.324990\pi\)
0.522526 + 0.852623i \(0.324990\pi\)
\(524\) 6.24264 0.272711
\(525\) 0 0
\(526\) −28.0000 −1.22086
\(527\) 7.31371 0.318590
\(528\) 0 0
\(529\) −21.6274 −0.940322
\(530\) 6.48528 0.281703
\(531\) 0 0
\(532\) 0 0
\(533\) −14.8284 −0.642290
\(534\) 0 0
\(535\) 9.65685 0.417502
\(536\) 1.65685 0.0715652
\(537\) 0 0
\(538\) −18.4853 −0.796957
\(539\) 0 0
\(540\) 0 0
\(541\) −14.9706 −0.643635 −0.321817 0.946802i \(-0.604294\pi\)
−0.321817 + 0.946802i \(0.604294\pi\)
\(542\) 12.0000 0.515444
\(543\) 0 0
\(544\) −2.58579 −0.110865
\(545\) −2.48528 −0.106458
\(546\) 0 0
\(547\) −10.4853 −0.448318 −0.224159 0.974553i \(-0.571964\pi\)
−0.224159 + 0.974553i \(0.571964\pi\)
\(548\) 16.0000 0.683486
\(549\) 0 0
\(550\) 0.828427 0.0353243
\(551\) 2.82843 0.120495
\(552\) 0 0
\(553\) 0 0
\(554\) −8.14214 −0.345926
\(555\) 0 0
\(556\) 19.8995 0.843927
\(557\) −15.1716 −0.642840 −0.321420 0.946937i \(-0.604160\pi\)
−0.321420 + 0.946937i \(0.604160\pi\)
\(558\) 0 0
\(559\) 42.6274 1.80295
\(560\) 0 0
\(561\) 0 0
\(562\) −8.00000 −0.337460
\(563\) 36.5858 1.54191 0.770954 0.636891i \(-0.219780\pi\)
0.770954 + 0.636891i \(0.219780\pi\)
\(564\) 0 0
\(565\) −15.3137 −0.644253
\(566\) −2.24264 −0.0942652
\(567\) 0 0
\(568\) 4.48528 0.188198
\(569\) 29.3137 1.22889 0.614447 0.788958i \(-0.289379\pi\)
0.614447 + 0.788958i \(0.289379\pi\)
\(570\) 0 0
\(571\) −2.20101 −0.0921094 −0.0460547 0.998939i \(-0.514665\pi\)
−0.0460547 + 0.998939i \(0.514665\pi\)
\(572\) −4.00000 −0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) 1.17157 0.0488580
\(576\) 0 0
\(577\) 6.10051 0.253967 0.126984 0.991905i \(-0.459470\pi\)
0.126984 + 0.991905i \(0.459470\pi\)
\(578\) −10.3137 −0.428994
\(579\) 0 0
\(580\) −4.82843 −0.200490
\(581\) 0 0
\(582\) 0 0
\(583\) −5.37258 −0.222510
\(584\) −9.41421 −0.389563
\(585\) 0 0
\(586\) 8.34315 0.344652
\(587\) −17.0711 −0.704598 −0.352299 0.935887i \(-0.614600\pi\)
−0.352299 + 0.935887i \(0.614600\pi\)
\(588\) 0 0
\(589\) −1.65685 −0.0682695
\(590\) 8.58579 0.353471
\(591\) 0 0
\(592\) −7.65685 −0.314695
\(593\) −3.27208 −0.134368 −0.0671841 0.997741i \(-0.521401\pi\)
−0.0671841 + 0.997741i \(0.521401\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) −5.65685 −0.231326
\(599\) 10.8284 0.442438 0.221219 0.975224i \(-0.428997\pi\)
0.221219 + 0.975224i \(0.428997\pi\)
\(600\) 0 0
\(601\) 6.58579 0.268640 0.134320 0.990938i \(-0.457115\pi\)
0.134320 + 0.990938i \(0.457115\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −11.3137 −0.460348
\(605\) 10.3137 0.419312
\(606\) 0 0
\(607\) 16.2843 0.660958 0.330479 0.943813i \(-0.392790\pi\)
0.330479 + 0.943813i \(0.392790\pi\)
\(608\) 0.585786 0.0237568
\(609\) 0 0
\(610\) −9.31371 −0.377101
\(611\) 24.9706 1.01020
\(612\) 0 0
\(613\) −12.3431 −0.498535 −0.249267 0.968435i \(-0.580190\pi\)
−0.249267 + 0.968435i \(0.580190\pi\)
\(614\) 14.9289 0.602483
\(615\) 0 0
\(616\) 0 0
\(617\) 33.3137 1.34116 0.670580 0.741837i \(-0.266045\pi\)
0.670580 + 0.741837i \(0.266045\pi\)
\(618\) 0 0
\(619\) 29.0711 1.16846 0.584232 0.811586i \(-0.301396\pi\)
0.584232 + 0.811586i \(0.301396\pi\)
\(620\) 2.82843 0.113592
\(621\) 0 0
\(622\) −4.00000 −0.160385
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 14.3848 0.574931
\(627\) 0 0
\(628\) 6.48528 0.258791
\(629\) 19.7990 0.789437
\(630\) 0 0
\(631\) −12.4853 −0.497031 −0.248516 0.968628i \(-0.579943\pi\)
−0.248516 + 0.968628i \(0.579943\pi\)
\(632\) −6.82843 −0.271620
\(633\) 0 0
\(634\) −10.4853 −0.416424
\(635\) −2.82843 −0.112243
\(636\) 0 0
\(637\) 0 0
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 24.6274 0.972724 0.486362 0.873757i \(-0.338324\pi\)
0.486362 + 0.873757i \(0.338324\pi\)
\(642\) 0 0
\(643\) 4.78680 0.188773 0.0943864 0.995536i \(-0.469911\pi\)
0.0943864 + 0.995536i \(0.469911\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −1.51472 −0.0595958
\(647\) 23.1127 0.908654 0.454327 0.890835i \(-0.349880\pi\)
0.454327 + 0.890835i \(0.349880\pi\)
\(648\) 0 0
\(649\) −7.11270 −0.279198
\(650\) −4.82843 −0.189386
\(651\) 0 0
\(652\) −20.1421 −0.788827
\(653\) −4.34315 −0.169960 −0.0849802 0.996383i \(-0.527083\pi\)
−0.0849802 + 0.996383i \(0.527083\pi\)
\(654\) 0 0
\(655\) −6.24264 −0.243920
\(656\) 3.07107 0.119905
\(657\) 0 0
\(658\) 0 0
\(659\) 27.1716 1.05845 0.529227 0.848480i \(-0.322482\pi\)
0.529227 + 0.848480i \(0.322482\pi\)
\(660\) 0 0
\(661\) −38.2843 −1.48909 −0.744543 0.667575i \(-0.767332\pi\)
−0.744543 + 0.667575i \(0.767332\pi\)
\(662\) 33.7990 1.31364
\(663\) 0 0
\(664\) 2.24264 0.0870313
\(665\) 0 0
\(666\) 0 0
\(667\) 5.65685 0.219034
\(668\) −15.7990 −0.611281
\(669\) 0 0
\(670\) −1.65685 −0.0640099
\(671\) 7.71573 0.297862
\(672\) 0 0
\(673\) −48.0000 −1.85026 −0.925132 0.379646i \(-0.876046\pi\)
−0.925132 + 0.379646i \(0.876046\pi\)
\(674\) 6.00000 0.231111
\(675\) 0 0
\(676\) 10.3137 0.396681
\(677\) −39.4558 −1.51641 −0.758206 0.652015i \(-0.773924\pi\)
−0.758206 + 0.652015i \(0.773924\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 2.58579 0.0991604
\(681\) 0 0
\(682\) −2.34315 −0.0897237
\(683\) −33.6569 −1.28784 −0.643922 0.765091i \(-0.722694\pi\)
−0.643922 + 0.765091i \(0.722694\pi\)
\(684\) 0 0
\(685\) −16.0000 −0.611329
\(686\) 0 0
\(687\) 0 0
\(688\) −8.82843 −0.336581
\(689\) 31.3137 1.19296
\(690\) 0 0
\(691\) 1.75736 0.0668531 0.0334265 0.999441i \(-0.489358\pi\)
0.0334265 + 0.999441i \(0.489358\pi\)
\(692\) 8.82843 0.335606
\(693\) 0 0
\(694\) 3.17157 0.120391
\(695\) −19.8995 −0.754831
\(696\) 0 0
\(697\) −7.94113 −0.300792
\(698\) 2.48528 0.0940693
\(699\) 0 0
\(700\) 0 0
\(701\) −2.48528 −0.0938678 −0.0469339 0.998898i \(-0.514945\pi\)
−0.0469339 + 0.998898i \(0.514945\pi\)
\(702\) 0 0
\(703\) −4.48528 −0.169166
\(704\) 0.828427 0.0312225
\(705\) 0 0
\(706\) 2.38478 0.0897522
\(707\) 0 0
\(708\) 0 0
\(709\) 45.1127 1.69424 0.847121 0.531399i \(-0.178334\pi\)
0.847121 + 0.531399i \(0.178334\pi\)
\(710\) −4.48528 −0.168330
\(711\) 0 0
\(712\) −12.7279 −0.476999
\(713\) −3.31371 −0.124099
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) −28.2843 −1.05556
\(719\) −41.4558 −1.54604 −0.773021 0.634380i \(-0.781255\pi\)
−0.773021 + 0.634380i \(0.781255\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −18.6569 −0.694336
\(723\) 0 0
\(724\) −2.48528 −0.0923648
\(725\) 4.82843 0.179323
\(726\) 0 0
\(727\) −3.51472 −0.130354 −0.0651768 0.997874i \(-0.520761\pi\)
−0.0651768 + 0.997874i \(0.520761\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 9.41421 0.348436
\(731\) 22.8284 0.844340
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 24.9706 0.921680
\(735\) 0 0
\(736\) 1.17157 0.0431847
\(737\) 1.37258 0.0505597
\(738\) 0 0
\(739\) 3.17157 0.116668 0.0583341 0.998297i \(-0.481421\pi\)
0.0583341 + 0.998297i \(0.481421\pi\)
\(740\) 7.65685 0.281472
\(741\) 0 0
\(742\) 0 0
\(743\) −51.7990 −1.90032 −0.950160 0.311762i \(-0.899081\pi\)
−0.950160 + 0.311762i \(0.899081\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) −30.4853 −1.11615
\(747\) 0 0
\(748\) −2.14214 −0.0783242
\(749\) 0 0
\(750\) 0 0
\(751\) −39.3137 −1.43458 −0.717289 0.696776i \(-0.754617\pi\)
−0.717289 + 0.696776i \(0.754617\pi\)
\(752\) −5.17157 −0.188588
\(753\) 0 0
\(754\) −23.3137 −0.849035
\(755\) 11.3137 0.411748
\(756\) 0 0
\(757\) 3.65685 0.132911 0.0664553 0.997789i \(-0.478831\pi\)
0.0664553 + 0.997789i \(0.478831\pi\)
\(758\) 34.4853 1.25256
\(759\) 0 0
\(760\) −0.585786 −0.0212487
\(761\) −22.3848 −0.811448 −0.405724 0.913996i \(-0.632980\pi\)
−0.405724 + 0.913996i \(0.632980\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −10.1421 −0.366930
\(765\) 0 0
\(766\) 32.4853 1.17374
\(767\) 41.4558 1.49688
\(768\) 0 0
\(769\) 19.5563 0.705220 0.352610 0.935770i \(-0.385294\pi\)
0.352610 + 0.935770i \(0.385294\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5.65685 0.203595
\(773\) 2.00000 0.0719350 0.0359675 0.999353i \(-0.488549\pi\)
0.0359675 + 0.999353i \(0.488549\pi\)
\(774\) 0 0
\(775\) −2.82843 −0.101600
\(776\) −7.75736 −0.278473
\(777\) 0 0
\(778\) −28.1421 −1.00894
\(779\) 1.79899 0.0644555
\(780\) 0 0
\(781\) 3.71573 0.132959
\(782\) −3.02944 −0.108332
\(783\) 0 0
\(784\) 0 0
\(785\) −6.48528 −0.231470
\(786\) 0 0
\(787\) −1.27208 −0.0453447 −0.0226723 0.999743i \(-0.507217\pi\)
−0.0226723 + 0.999743i \(0.507217\pi\)
\(788\) 25.7990 0.919051
\(789\) 0 0
\(790\) 6.82843 0.242945
\(791\) 0 0
\(792\) 0 0
\(793\) −44.9706 −1.59695
\(794\) 33.7990 1.19948
\(795\) 0 0
\(796\) 16.4853 0.584305
\(797\) 41.7990 1.48060 0.740298 0.672279i \(-0.234684\pi\)
0.740298 + 0.672279i \(0.234684\pi\)
\(798\) 0 0
\(799\) 13.3726 0.473088
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 6.00000 0.211867
\(803\) −7.79899 −0.275220
\(804\) 0 0
\(805\) 0 0
\(806\) 13.6569 0.481042
\(807\) 0 0
\(808\) −13.3137 −0.468375
\(809\) −3.02944 −0.106509 −0.0532547 0.998581i \(-0.516960\pi\)
−0.0532547 + 0.998581i \(0.516960\pi\)
\(810\) 0 0
\(811\) 32.5858 1.14424 0.572121 0.820169i \(-0.306121\pi\)
0.572121 + 0.820169i \(0.306121\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −6.34315 −0.222327
\(815\) 20.1421 0.705548
\(816\) 0 0
\(817\) −5.17157 −0.180930
\(818\) −10.5858 −0.370123
\(819\) 0 0
\(820\) −3.07107 −0.107246
\(821\) −17.3137 −0.604253 −0.302126 0.953268i \(-0.597696\pi\)
−0.302126 + 0.953268i \(0.597696\pi\)
\(822\) 0 0
\(823\) 20.2843 0.707065 0.353533 0.935422i \(-0.384980\pi\)
0.353533 + 0.935422i \(0.384980\pi\)
\(824\) −14.8284 −0.516573
\(825\) 0 0
\(826\) 0 0
\(827\) 5.37258 0.186823 0.0934115 0.995628i \(-0.470223\pi\)
0.0934115 + 0.995628i \(0.470223\pi\)
\(828\) 0 0
\(829\) −5.02944 −0.174680 −0.0873398 0.996179i \(-0.527837\pi\)
−0.0873398 + 0.996179i \(0.527837\pi\)
\(830\) −2.24264 −0.0778432
\(831\) 0 0
\(832\) −4.82843 −0.167396
\(833\) 0 0
\(834\) 0 0
\(835\) 15.7990 0.546747
\(836\) 0.485281 0.0167838
\(837\) 0 0
\(838\) 20.8701 0.720944
\(839\) 42.1421 1.45491 0.727454 0.686156i \(-0.240703\pi\)
0.727454 + 0.686156i \(0.240703\pi\)
\(840\) 0 0
\(841\) −5.68629 −0.196079
\(842\) 17.3137 0.596670
\(843\) 0 0
\(844\) 18.6274 0.641182
\(845\) −10.3137 −0.354802
\(846\) 0 0
\(847\) 0 0
\(848\) −6.48528 −0.222705
\(849\) 0 0
\(850\) −2.58579 −0.0886917
\(851\) −8.97056 −0.307507
\(852\) 0 0
\(853\) −43.1716 −1.47817 −0.739083 0.673614i \(-0.764741\pi\)
−0.739083 + 0.673614i \(0.764741\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −9.65685 −0.330064
\(857\) 4.92893 0.168369 0.0841846 0.996450i \(-0.473171\pi\)
0.0841846 + 0.996450i \(0.473171\pi\)
\(858\) 0 0
\(859\) −7.21320 −0.246111 −0.123056 0.992400i \(-0.539269\pi\)
−0.123056 + 0.992400i \(0.539269\pi\)
\(860\) 8.82843 0.301047
\(861\) 0 0
\(862\) 22.3431 0.761011
\(863\) 4.97056 0.169200 0.0846000 0.996415i \(-0.473039\pi\)
0.0846000 + 0.996415i \(0.473039\pi\)
\(864\) 0 0
\(865\) −8.82843 −0.300176
\(866\) 10.5858 0.359720
\(867\) 0 0
\(868\) 0 0
\(869\) −5.65685 −0.191896
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 2.48528 0.0841622
\(873\) 0 0
\(874\) 0.686292 0.0232142
\(875\) 0 0
\(876\) 0 0
\(877\) −30.2843 −1.02263 −0.511314 0.859394i \(-0.670841\pi\)
−0.511314 + 0.859394i \(0.670841\pi\)
\(878\) −24.9706 −0.842716
\(879\) 0 0
\(880\) −0.828427 −0.0279263
\(881\) 2.38478 0.0803452 0.0401726 0.999193i \(-0.487209\pi\)
0.0401726 + 0.999193i \(0.487209\pi\)
\(882\) 0 0
\(883\) −41.6569 −1.40186 −0.700932 0.713228i \(-0.747233\pi\)
−0.700932 + 0.713228i \(0.747233\pi\)
\(884\) 12.4853 0.419925
\(885\) 0 0
\(886\) −3.02944 −0.101776
\(887\) 55.1127 1.85050 0.925252 0.379354i \(-0.123854\pi\)
0.925252 + 0.379354i \(0.123854\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 12.7279 0.426641
\(891\) 0 0
\(892\) −7.31371 −0.244881
\(893\) −3.02944 −0.101376
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) 0 0
\(897\) 0 0
\(898\) 16.6274 0.554864
\(899\) −13.6569 −0.455482
\(900\) 0 0
\(901\) 16.7696 0.558675
\(902\) 2.54416 0.0847111
\(903\) 0 0
\(904\) 15.3137 0.509326
\(905\) 2.48528 0.0826135
\(906\) 0 0
\(907\) 0.284271 0.00943907 0.00471954 0.999989i \(-0.498498\pi\)
0.00471954 + 0.999989i \(0.498498\pi\)
\(908\) −18.2426 −0.605403
\(909\) 0 0
\(910\) 0 0
\(911\) −36.2843 −1.20215 −0.601076 0.799192i \(-0.705261\pi\)
−0.601076 + 0.799192i \(0.705261\pi\)
\(912\) 0 0
\(913\) 1.85786 0.0614863
\(914\) 21.6569 0.716345
\(915\) 0 0
\(916\) −16.1421 −0.533351
\(917\) 0 0
\(918\) 0 0
\(919\) 15.5147 0.511783 0.255892 0.966705i \(-0.417631\pi\)
0.255892 + 0.966705i \(0.417631\pi\)
\(920\) −1.17157 −0.0386256
\(921\) 0 0
\(922\) 12.8284 0.422482
\(923\) −21.6569 −0.712844
\(924\) 0 0
\(925\) −7.65685 −0.251756
\(926\) −16.9706 −0.557687
\(927\) 0 0
\(928\) 4.82843 0.158501
\(929\) 17.2132 0.564747 0.282373 0.959305i \(-0.408878\pi\)
0.282373 + 0.959305i \(0.408878\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −23.3137 −0.763666
\(933\) 0 0
\(934\) 15.8995 0.520247
\(935\) 2.14214 0.0700553
\(936\) 0 0
\(937\) 20.2426 0.661298 0.330649 0.943754i \(-0.392732\pi\)
0.330649 + 0.943754i \(0.392732\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 5.17157 0.168678
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) 0 0
\(943\) 3.59798 0.117166
\(944\) −8.58579 −0.279444
\(945\) 0 0
\(946\) −7.31371 −0.237789
\(947\) 4.82843 0.156903 0.0784514 0.996918i \(-0.475002\pi\)
0.0784514 + 0.996918i \(0.475002\pi\)
\(948\) 0 0
\(949\) 45.4558 1.47556
\(950\) 0.585786 0.0190054
\(951\) 0 0
\(952\) 0 0
\(953\) −0.343146 −0.0111156 −0.00555779 0.999985i \(-0.501769\pi\)
−0.00555779 + 0.999985i \(0.501769\pi\)
\(954\) 0 0
\(955\) 10.1421 0.328192
\(956\) −1.65685 −0.0535865
\(957\) 0 0
\(958\) 17.1716 0.554788
\(959\) 0 0
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 36.9706 1.19198
\(963\) 0 0
\(964\) −13.4142 −0.432043
\(965\) −5.65685 −0.182101
\(966\) 0 0
\(967\) −37.4558 −1.20450 −0.602249 0.798308i \(-0.705729\pi\)
−0.602249 + 0.798308i \(0.705729\pi\)
\(968\) −10.3137 −0.331495
\(969\) 0 0
\(970\) 7.75736 0.249074
\(971\) 33.3553 1.07042 0.535212 0.844718i \(-0.320232\pi\)
0.535212 + 0.844718i \(0.320232\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 31.7990 1.01891
\(975\) 0 0
\(976\) 9.31371 0.298125
\(977\) 12.6863 0.405870 0.202935 0.979192i \(-0.434952\pi\)
0.202935 + 0.979192i \(0.434952\pi\)
\(978\) 0 0
\(979\) −10.5442 −0.336993
\(980\) 0 0
\(981\) 0 0
\(982\) −32.2843 −1.03023
\(983\) 12.2010 0.389152 0.194576 0.980887i \(-0.437667\pi\)
0.194576 + 0.980887i \(0.437667\pi\)
\(984\) 0 0
\(985\) −25.7990 −0.822024
\(986\) −12.4853 −0.397612
\(987\) 0 0
\(988\) −2.82843 −0.0899843
\(989\) −10.3431 −0.328893
\(990\) 0 0
\(991\) 44.7696 1.42215 0.711076 0.703115i \(-0.248208\pi\)
0.711076 + 0.703115i \(0.248208\pi\)
\(992\) −2.82843 −0.0898027
\(993\) 0 0
\(994\) 0 0
\(995\) −16.4853 −0.522619
\(996\) 0 0
\(997\) −18.2843 −0.579069 −0.289534 0.957168i \(-0.593500\pi\)
−0.289534 + 0.957168i \(0.593500\pi\)
\(998\) 30.3431 0.960495
\(999\) 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 4410.2.a.bt.1.2 2
3.2 odd 2 490.2.a.m.1.2 yes 2
7.6 odd 2 4410.2.a.by.1.2 2
12.11 even 2 3920.2.a.bm.1.1 2
15.2 even 4 2450.2.c.t.99.1 4
15.8 even 4 2450.2.c.t.99.4 4
15.14 odd 2 2450.2.a.bn.1.1 2
21.2 odd 6 490.2.e.i.361.1 4
21.5 even 6 490.2.e.j.361.2 4
21.11 odd 6 490.2.e.i.471.1 4
21.17 even 6 490.2.e.j.471.2 4
21.20 even 2 490.2.a.l.1.1 2
84.83 odd 2 3920.2.a.ca.1.2 2
105.62 odd 4 2450.2.c.w.99.2 4
105.83 odd 4 2450.2.c.w.99.3 4
105.104 even 2 2450.2.a.bs.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
490.2.a.l.1.1 2 21.20 even 2
490.2.a.m.1.2 yes 2 3.2 odd 2
490.2.e.i.361.1 4 21.2 odd 6
490.2.e.i.471.1 4 21.11 odd 6
490.2.e.j.361.2 4 21.5 even 6
490.2.e.j.471.2 4 21.17 even 6
2450.2.a.bn.1.1 2 15.14 odd 2
2450.2.a.bs.1.2 2 105.104 even 2
2450.2.c.t.99.1 4 15.2 even 4
2450.2.c.t.99.4 4 15.8 even 4
2450.2.c.w.99.2 4 105.62 odd 4
2450.2.c.w.99.3 4 105.83 odd 4
3920.2.a.bm.1.1 2 12.11 even 2
3920.2.a.ca.1.2 2 84.83 odd 2
4410.2.a.bt.1.2 2 1.1 even 1 trivial
4410.2.a.by.1.2 2 7.6 odd 2