Properties

Label 4410.2.a.by.1.2
Level $4410$
Weight $2$
Character 4410.1
Self dual yes
Analytic conductor $35.214$
Analytic rank $0$
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: \(0\)
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 \(-0.266473\pi\)
0.669582 + 0.742738i \(0.266473\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.58579 0.627145 0.313573 0.949564i \(-0.398474\pi\)
0.313573 + 0.949564i \(0.398474\pi\)
\(18\) 0 0
\(19\) −0.585786 −0.134389 −0.0671943 0.997740i \(-0.521405\pi\)
−0.0671943 + 0.997740i \(0.521405\pi\)
\(20\) 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 \(-0.418254\pi\)
0.254000 + 0.967204i \(0.418254\pi\)
\(32\) 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 \(-0.577085\pi\)
−0.239810 + 0.970820i \(0.577085\pi\)
\(42\) 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 \(-0.376895\pi\)
0.377176 + 0.926142i \(0.376895\pi\)
\(48\) 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 \(-0.311229\pi\)
0.558887 + 0.829244i \(0.311229\pi\)
\(60\) 0 0
\(61\) −9.31371 −1.19250 −0.596249 0.802799i \(-0.703343\pi\)
−0.596249 + 0.802799i \(0.703343\pi\)
\(62\) 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 \(-0.314275\pi\)
0.550925 + 0.834555i \(0.314275\pi\)
\(74\) −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 \(-0.539277\pi\)
−0.123081 + 0.992397i \(0.539277\pi\)
\(84\) 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 \(-0.264325\pi\)
0.674579 + 0.738203i \(0.264325\pi\)
\(90\) 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 \(-0.371153\pi\)
0.393820 + 0.919187i \(0.371153\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 13.3137 1.32476 0.662382 0.749166i \(-0.269546\pi\)
0.662382 + 0.749166i \(0.269546\pi\)
\(102\) 0 0
\(103\) 14.8284 1.46109 0.730544 0.682865i \(-0.239266\pi\)
0.730544 + 0.682865i \(0.239266\pi\)
\(104\) 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 \(-0.587920\pi\)
−0.272711 + 0.962096i \(0.587920\pi\)
\(132\) 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 \(-0.819762\pi\)
−0.843927 + 0.536459i \(0.819762\pi\)
\(140\) 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 \(-0.583324\pi\)
−0.258791 + 0.965933i \(0.583324\pi\)
\(158\) −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 \(-0.290654\pi\)
0.611281 + 0.791413i \(0.290654\pi\)
\(168\) 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 \(-0.608941\pi\)
−0.335606 + 0.942002i \(0.608941\pi\)
\(174\) 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 \(-0.470557\pi\)
0.0923648 + 0.995725i \(0.470557\pi\)
\(182\) 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 \(-0.698633\pi\)
−0.584305 + 0.811534i \(0.698633\pi\)
\(200\) 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 \(-0.421251\pi\)
0.244881 + 0.969553i \(0.421251\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 15.3137 1.01865
\(227\) 18.2426 1.21081 0.605403 0.795919i \(-0.293012\pi\)
0.605403 + 0.795919i \(0.293012\pi\)
\(228\) 0 0
\(229\) 16.1421 1.06670 0.533351 0.845894i \(-0.320932\pi\)
0.533351 + 0.845894i \(0.320932\pi\)
\(230\) 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 \(-0.357793\pi\)
0.432043 + 0.901853i \(0.357793\pi\)
\(242\) −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 \(-0.494115\pi\)
0.0184873 + 0.999829i \(0.494115\pi\)
\(252\) 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 \(-0.599913\pi\)
−0.308757 + 0.951141i \(0.599913\pi\)
\(258\) 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 \(-0.309441\pi\)
0.563534 + 0.826093i \(0.309441\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 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 \(-0.478767\pi\)
0.0666556 + 0.997776i \(0.478767\pi\)
\(284\) 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 \(-0.578363\pi\)
−0.243706 + 0.969849i \(0.578363\pi\)
\(294\) 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 \(-0.640085\pi\)
−0.426020 + 0.904714i \(0.640085\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 2.82843 0.160644
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) −14.3848 −0.813076 −0.406538 0.913634i \(-0.633264\pi\)
−0.406538 + 0.913634i \(0.633264\pi\)
\(314\) −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 \(-0.521189\pi\)
−0.0665170 + 0.997785i \(0.521189\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.828427 0.0441553
\(353\) −2.38478 −0.126929 −0.0634644 0.997984i \(-0.520215\pi\)
−0.0634644 + 0.997984i \(0.520215\pi\)
\(354\) 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 \(-0.725955\pi\)
−0.651726 + 0.758454i \(0.725955\pi\)
\(368\) 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 \(-0.811637\pi\)
−0.829960 + 0.557823i \(0.811637\pi\)
\(384\) 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 \(-0.822290\pi\)
−0.848161 + 0.529738i \(0.822290\pi\)
\(398\) −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 \(-0.415711\pi\)
0.261717 + 0.965145i \(0.415711\pi\)
\(410\) −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 \(-0.670275\pi\)
−0.509785 + 0.860302i \(0.670275\pi\)
\(420\) 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 \(-0.581865\pi\)
−0.254360 + 0.967110i \(0.581865\pi\)
\(434\) 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 \(-0.296799\pi\)
0.595890 + 0.803066i \(0.296799\pi\)
\(440\) 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 \(-0.596566\pi\)
−0.298740 + 0.954335i \(0.596566\pi\)
\(462\) 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 \(-0.619913\pi\)
−0.367870 + 0.929877i \(0.619913\pi\)
\(468\) 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 \(-0.628319\pi\)
−0.392295 + 0.919840i \(0.628319\pi\)
\(480\) 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 \(-0.371216\pi\)
0.393640 + 0.919265i \(0.371216\pi\)
\(504\) 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 \(-0.458978\pi\)
0.128518 + 0.991707i \(0.458978\pi\)
\(510\) 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 \(-0.362816\pi\)
0.417759 + 0.908558i \(0.362816\pi\)
\(522\) 0 0
\(523\) −23.8995 −1.04505 −0.522526 0.852623i \(-0.675010\pi\)
−0.522526 + 0.852623i \(0.675010\pi\)
\(524\) −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 \(-0.780220\pi\)
−0.770954 + 0.636891i \(0.780220\pi\)
\(564\) 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 \(-0.540530\pi\)
−0.126984 + 0.991905i \(0.540530\pi\)
\(578\) −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 \(-0.385400\pi\)
0.352299 + 0.935887i \(0.385400\pi\)
\(588\) 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 \(-0.478599\pi\)
0.0671841 + 0.997741i \(0.478599\pi\)
\(594\) 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 \(-0.542885\pi\)
−0.134320 + 0.990938i \(0.542885\pi\)
\(602\) 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 \(-0.607210\pi\)
−0.330479 + 0.943813i \(0.607210\pi\)
\(608\) −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 \(-0.698604\pi\)
−0.584232 + 0.811586i \(0.698604\pi\)
\(620\) 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 \(-0.530089\pi\)
−0.0943864 + 0.995536i \(0.530089\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −1.51472 −0.0595958
\(647\) −23.1127 −0.908654 −0.454327 0.890835i \(-0.650120\pi\)
−0.454327 + 0.890835i \(0.650120\pi\)
\(648\) 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 \(-0.232668\pi\)
0.744543 + 0.667575i \(0.232668\pi\)
\(662\) 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 \(-0.226076\pi\)
0.758206 + 0.652015i \(0.226076\pi\)
\(678\) 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 \(-0.510642\pi\)
−0.0334265 + 0.999441i \(0.510642\pi\)
\(692\) −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 \(-0.218745\pi\)
0.773021 + 0.634380i \(0.218745\pi\)
\(720\) 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 \(-0.479239\pi\)
0.0651768 + 0.997874i \(0.479239\pi\)
\(728\) 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 \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) −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 \(-0.367020\pi\)
0.405724 + 0.913996i \(0.367020\pi\)
\(762\) 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 \(-0.614706\pi\)
−0.352610 + 0.935770i \(0.614706\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5.65685 0.203595
\(773\) −2.00000 −0.0719350 −0.0359675 0.999353i \(-0.511451\pi\)
−0.0359675 + 0.999353i \(0.511451\pi\)
\(774\) 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 \(-0.492783\pi\)
0.0226723 + 0.999743i \(0.492783\pi\)
\(788\) 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 \(-0.765316\pi\)
−0.740298 + 0.672279i \(0.765316\pi\)
\(798\) 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 \(-0.693879\pi\)
−0.572121 + 0.820169i \(0.693879\pi\)
\(812\) 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 \(-0.472163\pi\)
0.0873398 + 0.996179i \(0.472163\pi\)
\(830\) −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 \(-0.759297\pi\)
−0.727454 + 0.686156i \(0.759297\pi\)
\(840\) 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 \(-0.235259\pi\)
0.739083 + 0.673614i \(0.235259\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −9.65685 −0.330064
\(857\) −4.92893 −0.168369 −0.0841846 0.996450i \(-0.526829\pi\)
−0.0841846 + 0.996450i \(0.526829\pi\)
\(858\) 0 0
\(859\) 7.21320 0.246111 0.123056 0.992400i \(-0.460731\pi\)
0.123056 + 0.992400i \(0.460731\pi\)
\(860\) −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 \(-0.512791\pi\)
−0.0401726 + 0.999193i \(0.512791\pi\)
\(882\) 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 \(-0.876146\pi\)
−0.925252 + 0.379354i \(0.876146\pi\)
\(888\) 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 \(-0.591122\pi\)
−0.282373 + 0.959305i \(0.591122\pi\)
\(930\) 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 \(-0.607268\pi\)
−0.330649 + 0.943754i \(0.607268\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 5.17157 0.168678
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 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 \(-0.679768\pi\)
−0.535212 + 0.844718i \(0.679768\pi\)
\(972\) 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 \(-0.562333\pi\)
−0.194576 + 0.980887i \(0.562333\pi\)
\(984\) 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 \(-0.406500\pi\)
0.289534 + 0.957168i \(0.406500\pi\)
\(998\) 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.by.1.2 2
3.2 odd 2 490.2.a.l.1.1 2
7.6 odd 2 4410.2.a.bt.1.2 2
12.11 even 2 3920.2.a.ca.1.2 2
15.2 even 4 2450.2.c.w.99.2 4
15.8 even 4 2450.2.c.w.99.3 4
15.14 odd 2 2450.2.a.bs.1.2 2
21.2 odd 6 490.2.e.j.361.2 4
21.5 even 6 490.2.e.i.361.1 4
21.11 odd 6 490.2.e.j.471.2 4
21.17 even 6 490.2.e.i.471.1 4
21.20 even 2 490.2.a.m.1.2 yes 2
84.83 odd 2 3920.2.a.bm.1.1 2
105.62 odd 4 2450.2.c.t.99.1 4
105.83 odd 4 2450.2.c.t.99.4 4
105.104 even 2 2450.2.a.bn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
490.2.a.l.1.1 2 3.2 odd 2
490.2.a.m.1.2 yes 2 21.20 even 2
490.2.e.i.361.1 4 21.5 even 6
490.2.e.i.471.1 4 21.17 even 6
490.2.e.j.361.2 4 21.2 odd 6
490.2.e.j.471.2 4 21.11 odd 6
2450.2.a.bn.1.1 2 105.104 even 2
2450.2.a.bs.1.2 2 15.14 odd 2
2450.2.c.t.99.1 4 105.62 odd 4
2450.2.c.t.99.4 4 105.83 odd 4
2450.2.c.w.99.2 4 15.2 even 4
2450.2.c.w.99.3 4 15.8 even 4
3920.2.a.bm.1.1 2 84.83 odd 2
3920.2.a.ca.1.2 2 12.11 even 2
4410.2.a.bt.1.2 2 7.6 odd 2
4410.2.a.by.1.2 2 1.1 even 1 trivial