Properties

Label 441.2.c.b.440.2
Level $441$
Weight $2$
Character 441.440
Analytic conductor $3.521$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 440.2
Root \(0.382683 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 441.440
Dual form 441.2.c.b.440.8

$q$-expansion

\(f(q)\) \(=\) \(q-2.41421i q^{2} -3.82843 q^{4} +3.37849 q^{5} +4.41421i q^{8} +O(q^{10})\) \(q-2.41421i q^{2} -3.82843 q^{4} +3.37849 q^{5} +4.41421i q^{8} -8.15640i q^{10} -0.828427i q^{11} -3.37849i q^{13} +3.00000 q^{16} +1.39942 q^{17} -6.75699i q^{19} -12.9343 q^{20} -2.00000 q^{22} -2.00000i q^{23} +6.41421 q^{25} -8.15640 q^{26} +4.82843i q^{29} +6.75699i q^{31} +1.58579i q^{32} -3.37849i q^{34} -2.58579 q^{37} -16.3128 q^{38} +14.9134i q^{40} +8.15640 q^{41} -12.4853 q^{43} +3.17157i q^{44} -4.82843 q^{46} +6.75699 q^{47} -15.4853i q^{50} +12.9343i q^{52} +7.07107i q^{53} -2.79884i q^{55} +11.6569 q^{58} +6.75699 q^{59} +8.15640i q^{61} +16.3128 q^{62} +9.82843 q^{64} -11.4142i q^{65} +8.48528 q^{67} -5.35757 q^{68} -4.82843i q^{71} +1.39942i q^{73} +6.24264i q^{74} +25.8686i q^{76} -9.65685 q^{79} +10.1355 q^{80} -19.6913i q^{82} -13.5140 q^{83} +4.72792 q^{85} +30.1421i q^{86} +3.65685 q^{88} +6.17733 q^{89} +7.65685i q^{92} -16.3128i q^{94} -22.8284i q^{95} -1.39942i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + O(q^{10}) \) \( 8q - 8q^{4} + 24q^{16} - 16q^{22} + 40q^{25} - 32q^{37} - 32q^{43} - 16q^{46} + 48q^{58} + 56q^{64} - 32q^{79} - 64q^{85} - 16q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.41421i − 1.70711i −0.521005 0.853553i \(-0.674443\pi\)
0.521005 0.853553i \(-0.325557\pi\)
\(3\) 0 0
\(4\) −3.82843 −1.91421
\(5\) 3.37849 1.51091 0.755454 0.655202i \(-0.227416\pi\)
0.755454 + 0.655202i \(0.227416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 4.41421i 1.56066i
\(9\) 0 0
\(10\) − 8.15640i − 2.57928i
\(11\) − 0.828427i − 0.249780i −0.992171 0.124890i \(-0.960142\pi\)
0.992171 0.124890i \(-0.0398578\pi\)
\(12\) 0 0
\(13\) − 3.37849i − 0.937025i −0.883457 0.468513i \(-0.844790\pi\)
0.883457 0.468513i \(-0.155210\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 1.39942 0.339409 0.169704 0.985495i \(-0.445719\pi\)
0.169704 + 0.985495i \(0.445719\pi\)
\(18\) 0 0
\(19\) − 6.75699i − 1.55016i −0.631864 0.775079i \(-0.717710\pi\)
0.631864 0.775079i \(-0.282290\pi\)
\(20\) −12.9343 −2.89220
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) − 2.00000i − 0.417029i −0.978019 0.208514i \(-0.933137\pi\)
0.978019 0.208514i \(-0.0668628\pi\)
\(24\) 0 0
\(25\) 6.41421 1.28284
\(26\) −8.15640 −1.59960
\(27\) 0 0
\(28\) 0 0
\(29\) 4.82843i 0.896616i 0.893879 + 0.448308i \(0.147973\pi\)
−0.893879 + 0.448308i \(0.852027\pi\)
\(30\) 0 0
\(31\) 6.75699i 1.21359i 0.794858 + 0.606795i \(0.207545\pi\)
−0.794858 + 0.606795i \(0.792455\pi\)
\(32\) 1.58579i 0.280330i
\(33\) 0 0
\(34\) − 3.37849i − 0.579407i
\(35\) 0 0
\(36\) 0 0
\(37\) −2.58579 −0.425101 −0.212550 0.977150i \(-0.568177\pi\)
−0.212550 + 0.977150i \(0.568177\pi\)
\(38\) −16.3128 −2.64629
\(39\) 0 0
\(40\) 14.9134i 2.35801i
\(41\) 8.15640 1.27382 0.636908 0.770940i \(-0.280213\pi\)
0.636908 + 0.770940i \(0.280213\pi\)
\(42\) 0 0
\(43\) −12.4853 −1.90399 −0.951994 0.306117i \(-0.900970\pi\)
−0.951994 + 0.306117i \(0.900970\pi\)
\(44\) 3.17157i 0.478133i
\(45\) 0 0
\(46\) −4.82843 −0.711913
\(47\) 6.75699 0.985608 0.492804 0.870140i \(-0.335972\pi\)
0.492804 + 0.870140i \(0.335972\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) − 15.4853i − 2.18995i
\(51\) 0 0
\(52\) 12.9343i 1.79367i
\(53\) 7.07107i 0.971286i 0.874157 + 0.485643i \(0.161414\pi\)
−0.874157 + 0.485643i \(0.838586\pi\)
\(54\) 0 0
\(55\) − 2.79884i − 0.377395i
\(56\) 0 0
\(57\) 0 0
\(58\) 11.6569 1.53062
\(59\) 6.75699 0.879685 0.439842 0.898075i \(-0.355034\pi\)
0.439842 + 0.898075i \(0.355034\pi\)
\(60\) 0 0
\(61\) 8.15640i 1.04432i 0.852847 + 0.522160i \(0.174874\pi\)
−0.852847 + 0.522160i \(0.825126\pi\)
\(62\) 16.3128 2.07173
\(63\) 0 0
\(64\) 9.82843 1.22855
\(65\) − 11.4142i − 1.41576i
\(66\) 0 0
\(67\) 8.48528 1.03664 0.518321 0.855186i \(-0.326557\pi\)
0.518321 + 0.855186i \(0.326557\pi\)
\(68\) −5.35757 −0.649701
\(69\) 0 0
\(70\) 0 0
\(71\) − 4.82843i − 0.573029i −0.958076 0.286514i \(-0.907503\pi\)
0.958076 0.286514i \(-0.0924966\pi\)
\(72\) 0 0
\(73\) 1.39942i 0.163789i 0.996641 + 0.0818947i \(0.0260971\pi\)
−0.996641 + 0.0818947i \(0.973903\pi\)
\(74\) 6.24264i 0.725692i
\(75\) 0 0
\(76\) 25.8686i 2.96734i
\(77\) 0 0
\(78\) 0 0
\(79\) −9.65685 −1.08648 −0.543240 0.839577i \(-0.682803\pi\)
−0.543240 + 0.839577i \(0.682803\pi\)
\(80\) 10.1355 1.13318
\(81\) 0 0
\(82\) − 19.6913i − 2.17454i
\(83\) −13.5140 −1.48335 −0.741676 0.670759i \(-0.765969\pi\)
−0.741676 + 0.670759i \(0.765969\pi\)
\(84\) 0 0
\(85\) 4.72792 0.512815
\(86\) 30.1421i 3.25031i
\(87\) 0 0
\(88\) 3.65685 0.389822
\(89\) 6.17733 0.654795 0.327398 0.944887i \(-0.393828\pi\)
0.327398 + 0.944887i \(0.393828\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 7.65685i 0.798282i
\(93\) 0 0
\(94\) − 16.3128i − 1.68254i
\(95\) − 22.8284i − 2.34215i
\(96\) 0 0
\(97\) − 1.39942i − 0.142089i −0.997473 0.0710447i \(-0.977367\pi\)
0.997473 0.0710447i \(-0.0226333\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −24.5563 −2.45563
\(101\) 12.9343 1.28701 0.643506 0.765441i \(-0.277479\pi\)
0.643506 + 0.765441i \(0.277479\pi\)
\(102\) 0 0
\(103\) − 2.79884i − 0.275777i −0.990448 0.137889i \(-0.955968\pi\)
0.990448 0.137889i \(-0.0440316\pi\)
\(104\) 14.9134 1.46238
\(105\) 0 0
\(106\) 17.0711 1.65809
\(107\) 9.31371i 0.900390i 0.892930 + 0.450195i \(0.148646\pi\)
−0.892930 + 0.450195i \(0.851354\pi\)
\(108\) 0 0
\(109\) −2.58579 −0.247673 −0.123837 0.992303i \(-0.539520\pi\)
−0.123837 + 0.992303i \(0.539520\pi\)
\(110\) −6.75699 −0.644253
\(111\) 0 0
\(112\) 0 0
\(113\) 5.89949i 0.554978i 0.960729 + 0.277489i \(0.0895022\pi\)
−0.960729 + 0.277489i \(0.910498\pi\)
\(114\) 0 0
\(115\) − 6.75699i − 0.630092i
\(116\) − 18.4853i − 1.71632i
\(117\) 0 0
\(118\) − 16.3128i − 1.50172i
\(119\) 0 0
\(120\) 0 0
\(121\) 10.3137 0.937610
\(122\) 19.6913 1.78277
\(123\) 0 0
\(124\) − 25.8686i − 2.32307i
\(125\) 4.77791 0.427349
\(126\) 0 0
\(127\) −0.485281 −0.0430618 −0.0215309 0.999768i \(-0.506854\pi\)
−0.0215309 + 0.999768i \(0.506854\pi\)
\(128\) − 20.5563i − 1.81694i
\(129\) 0 0
\(130\) −27.5563 −2.41685
\(131\) −9.55582 −0.834896 −0.417448 0.908701i \(-0.637075\pi\)
−0.417448 + 0.908701i \(0.637075\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 20.4853i − 1.76966i
\(135\) 0 0
\(136\) 6.17733i 0.529701i
\(137\) 0.828427i 0.0707773i 0.999374 + 0.0353887i \(0.0112669\pi\)
−0.999374 + 0.0353887i \(0.988733\pi\)
\(138\) 0 0
\(139\) 9.55582i 0.810514i 0.914203 + 0.405257i \(0.132818\pi\)
−0.914203 + 0.405257i \(0.867182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −11.6569 −0.978221
\(143\) −2.79884 −0.234050
\(144\) 0 0
\(145\) 16.3128i 1.35470i
\(146\) 3.37849 0.279606
\(147\) 0 0
\(148\) 9.89949 0.813733
\(149\) − 4.24264i − 0.347571i −0.984784 0.173785i \(-0.944400\pi\)
0.984784 0.173785i \(-0.0555999\pi\)
\(150\) 0 0
\(151\) −12.4853 −1.01604 −0.508019 0.861346i \(-0.669622\pi\)
−0.508019 + 0.861346i \(0.669622\pi\)
\(152\) 29.8268 2.41927
\(153\) 0 0
\(154\) 0 0
\(155\) 22.8284i 1.83362i
\(156\) 0 0
\(157\) 7.33664i 0.585528i 0.956185 + 0.292764i \(0.0945750\pi\)
−0.956185 + 0.292764i \(0.905425\pi\)
\(158\) 23.3137i 1.85474i
\(159\) 0 0
\(160\) 5.35757i 0.423553i
\(161\) 0 0
\(162\) 0 0
\(163\) 14.8284 1.16145 0.580726 0.814099i \(-0.302769\pi\)
0.580726 + 0.814099i \(0.302769\pi\)
\(164\) −31.2262 −2.43836
\(165\) 0 0
\(166\) 32.6256i 2.53224i
\(167\) 2.79884 0.216580 0.108290 0.994119i \(-0.465462\pi\)
0.108290 + 0.994119i \(0.465462\pi\)
\(168\) 0 0
\(169\) 1.58579 0.121984
\(170\) − 11.4142i − 0.875430i
\(171\) 0 0
\(172\) 47.7990 3.64464
\(173\) 8.15640 0.620120 0.310060 0.950717i \(-0.399651\pi\)
0.310060 + 0.950717i \(0.399651\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) − 2.48528i − 0.187335i
\(177\) 0 0
\(178\) − 14.9134i − 1.11781i
\(179\) 9.51472i 0.711163i 0.934645 + 0.355582i \(0.115717\pi\)
−0.934645 + 0.355582i \(0.884283\pi\)
\(180\) 0 0
\(181\) 17.7122i 1.31654i 0.752782 + 0.658270i \(0.228711\pi\)
−0.752782 + 0.658270i \(0.771289\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 8.82843 0.650840
\(185\) −8.73606 −0.642288
\(186\) 0 0
\(187\) − 1.15932i − 0.0847775i
\(188\) −25.8686 −1.88666
\(189\) 0 0
\(190\) −55.1127 −3.99830
\(191\) 20.8284i 1.50709i 0.657395 + 0.753546i \(0.271658\pi\)
−0.657395 + 0.753546i \(0.728342\pi\)
\(192\) 0 0
\(193\) −17.6569 −1.27097 −0.635484 0.772114i \(-0.719199\pi\)
−0.635484 + 0.772114i \(0.719199\pi\)
\(194\) −3.37849 −0.242562
\(195\) 0 0
\(196\) 0 0
\(197\) − 16.2426i − 1.15724i −0.815597 0.578620i \(-0.803591\pi\)
0.815597 0.578620i \(-0.196409\pi\)
\(198\) 0 0
\(199\) 19.1116i 1.35479i 0.735620 + 0.677394i \(0.236891\pi\)
−0.735620 + 0.677394i \(0.763109\pi\)
\(200\) 28.3137i 2.00208i
\(201\) 0 0
\(202\) − 31.2262i − 2.19707i
\(203\) 0 0
\(204\) 0 0
\(205\) 27.5563 1.92462
\(206\) −6.75699 −0.470781
\(207\) 0 0
\(208\) − 10.1355i − 0.702769i
\(209\) −5.59767 −0.387199
\(210\) 0 0
\(211\) 7.31371 0.503496 0.251748 0.967793i \(-0.418995\pi\)
0.251748 + 0.967793i \(0.418995\pi\)
\(212\) − 27.0711i − 1.85925i
\(213\) 0 0
\(214\) 22.4853 1.53706
\(215\) −42.1814 −2.87675
\(216\) 0 0
\(217\) 0 0
\(218\) 6.24264i 0.422805i
\(219\) 0 0
\(220\) 10.7151i 0.722414i
\(221\) − 4.72792i − 0.318034i
\(222\) 0 0
\(223\) − 23.0698i − 1.54487i −0.635095 0.772434i \(-0.719039\pi\)
0.635095 0.772434i \(-0.280961\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 14.2426 0.947406
\(227\) 2.79884 0.185765 0.0928826 0.995677i \(-0.470392\pi\)
0.0928826 + 0.995677i \(0.470392\pi\)
\(228\) 0 0
\(229\) 12.1146i 0.800552i 0.916395 + 0.400276i \(0.131086\pi\)
−0.916395 + 0.400276i \(0.868914\pi\)
\(230\) −16.3128 −1.07563
\(231\) 0 0
\(232\) −21.3137 −1.39931
\(233\) 10.8284i 0.709394i 0.934981 + 0.354697i \(0.115416\pi\)
−0.934981 + 0.354697i \(0.884584\pi\)
\(234\) 0 0
\(235\) 22.8284 1.48916
\(236\) −25.8686 −1.68390
\(237\) 0 0
\(238\) 0 0
\(239\) 0.343146i 0.0221963i 0.999938 + 0.0110981i \(0.00353272\pi\)
−0.999938 + 0.0110981i \(0.996467\pi\)
\(240\) 0 0
\(241\) − 8.97616i − 0.578205i −0.957298 0.289103i \(-0.906643\pi\)
0.957298 0.289103i \(-0.0933569\pi\)
\(242\) − 24.8995i − 1.60060i
\(243\) 0 0
\(244\) − 31.2262i − 1.99905i
\(245\) 0 0
\(246\) 0 0
\(247\) −22.8284 −1.45254
\(248\) −29.8268 −1.89400
\(249\) 0 0
\(250\) − 11.5349i − 0.729531i
\(251\) −25.8686 −1.63281 −0.816407 0.577477i \(-0.804037\pi\)
−0.816407 + 0.577477i \(0.804037\pi\)
\(252\) 0 0
\(253\) −1.65685 −0.104166
\(254\) 1.17157i 0.0735110i
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) 21.6704 1.35176 0.675880 0.737011i \(-0.263764\pi\)
0.675880 + 0.737011i \(0.263764\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 43.6985i 2.71006i
\(261\) 0 0
\(262\) 23.0698i 1.42526i
\(263\) − 24.1421i − 1.48867i −0.667808 0.744334i \(-0.732767\pi\)
0.667808 0.744334i \(-0.267233\pi\)
\(264\) 0 0
\(265\) 23.8896i 1.46752i
\(266\) 0 0
\(267\) 0 0
\(268\) −32.4853 −1.98435
\(269\) −31.2262 −1.90389 −0.951947 0.306262i \(-0.900922\pi\)
−0.951947 + 0.306262i \(0.900922\pi\)
\(270\) 0 0
\(271\) 6.75699i 0.410458i 0.978714 + 0.205229i \(0.0657939\pi\)
−0.978714 + 0.205229i \(0.934206\pi\)
\(272\) 4.19825 0.254556
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) − 5.31371i − 0.320429i
\(276\) 0 0
\(277\) −20.9706 −1.26000 −0.630000 0.776596i \(-0.716945\pi\)
−0.630000 + 0.776596i \(0.716945\pi\)
\(278\) 23.0698 1.38363
\(279\) 0 0
\(280\) 0 0
\(281\) − 18.8284i − 1.12321i −0.827406 0.561605i \(-0.810184\pi\)
0.827406 0.561605i \(-0.189816\pi\)
\(282\) 0 0
\(283\) − 29.8268i − 1.77302i −0.462711 0.886509i \(-0.653123\pi\)
0.462711 0.886509i \(-0.346877\pi\)
\(284\) 18.4853i 1.09690i
\(285\) 0 0
\(286\) 6.75699i 0.399549i
\(287\) 0 0
\(288\) 0 0
\(289\) −15.0416 −0.884802
\(290\) 39.3826 2.31263
\(291\) 0 0
\(292\) − 5.35757i − 0.313528i
\(293\) 7.33664 0.428611 0.214306 0.976767i \(-0.431251\pi\)
0.214306 + 0.976767i \(0.431251\pi\)
\(294\) 0 0
\(295\) 22.8284 1.32912
\(296\) − 11.4142i − 0.663438i
\(297\) 0 0
\(298\) −10.2426 −0.593340
\(299\) −6.75699 −0.390767
\(300\) 0 0
\(301\) 0 0
\(302\) 30.1421i 1.73448i
\(303\) 0 0
\(304\) − 20.2710i − 1.16262i
\(305\) 27.5563i 1.57787i
\(306\) 0 0
\(307\) − 12.3547i − 0.705117i −0.935790 0.352559i \(-0.885312\pi\)
0.935790 0.352559i \(-0.114688\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 55.1127 3.13019
\(311\) 10.7151 0.607600 0.303800 0.952736i \(-0.401745\pi\)
0.303800 + 0.952736i \(0.401745\pi\)
\(312\) 0 0
\(313\) − 12.9343i − 0.731091i −0.930794 0.365545i \(-0.880883\pi\)
0.930794 0.365545i \(-0.119117\pi\)
\(314\) 17.7122 0.999559
\(315\) 0 0
\(316\) 36.9706 2.07976
\(317\) − 23.7574i − 1.33435i −0.744903 0.667173i \(-0.767504\pi\)
0.744903 0.667173i \(-0.232496\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 33.2053 1.85623
\(321\) 0 0
\(322\) 0 0
\(323\) − 9.45584i − 0.526137i
\(324\) 0 0
\(325\) − 21.6704i − 1.20206i
\(326\) − 35.7990i − 1.98272i
\(327\) 0 0
\(328\) 36.0041i 1.98799i
\(329\) 0 0
\(330\) 0 0
\(331\) 10.3431 0.568511 0.284255 0.958749i \(-0.408254\pi\)
0.284255 + 0.958749i \(0.408254\pi\)
\(332\) 51.7373 2.83945
\(333\) 0 0
\(334\) − 6.75699i − 0.369726i
\(335\) 28.6675 1.56627
\(336\) 0 0
\(337\) −12.9289 −0.704284 −0.352142 0.935947i \(-0.614547\pi\)
−0.352142 + 0.935947i \(0.614547\pi\)
\(338\) − 3.82843i − 0.208239i
\(339\) 0 0
\(340\) −18.1005 −0.981638
\(341\) 5.59767 0.303131
\(342\) 0 0
\(343\) 0 0
\(344\) − 55.1127i − 2.97148i
\(345\) 0 0
\(346\) − 19.6913i − 1.05861i
\(347\) 18.9706i 1.01839i 0.860650 + 0.509197i \(0.170057\pi\)
−0.860650 + 0.509197i \(0.829943\pi\)
\(348\) 0 0
\(349\) 6.17733i 0.330665i 0.986238 + 0.165332i \(0.0528697\pi\)
−0.986238 + 0.165332i \(0.947130\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.31371 0.0700209
\(353\) 23.6494 1.25873 0.629367 0.777109i \(-0.283314\pi\)
0.629367 + 0.777109i \(0.283314\pi\)
\(354\) 0 0
\(355\) − 16.3128i − 0.865794i
\(356\) −23.6494 −1.25342
\(357\) 0 0
\(358\) 22.9706 1.21403
\(359\) 1.02944i 0.0543316i 0.999631 + 0.0271658i \(0.00864821\pi\)
−0.999631 + 0.0271658i \(0.991352\pi\)
\(360\) 0 0
\(361\) −26.6569 −1.40299
\(362\) 42.7611 2.24747
\(363\) 0 0
\(364\) 0 0
\(365\) 4.72792i 0.247471i
\(366\) 0 0
\(367\) − 3.95815i − 0.206614i −0.994650 0.103307i \(-0.967058\pi\)
0.994650 0.103307i \(-0.0329424\pi\)
\(368\) − 6.00000i − 0.312772i
\(369\) 0 0
\(370\) 21.0907i 1.09645i
\(371\) 0 0
\(372\) 0 0
\(373\) −22.6274 −1.17160 −0.585802 0.810454i \(-0.699220\pi\)
−0.585802 + 0.810454i \(0.699220\pi\)
\(374\) −2.79884 −0.144724
\(375\) 0 0
\(376\) 29.8268i 1.53820i
\(377\) 16.3128 0.840152
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 87.3970i 4.48337i
\(381\) 0 0
\(382\) 50.2843 2.57277
\(383\) −27.0279 −1.38106 −0.690532 0.723302i \(-0.742623\pi\)
−0.690532 + 0.723302i \(0.742623\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 42.6274i 2.16968i
\(387\) 0 0
\(388\) 5.35757i 0.271989i
\(389\) 13.7990i 0.699637i 0.936818 + 0.349818i \(0.113757\pi\)
−0.936818 + 0.349818i \(0.886243\pi\)
\(390\) 0 0
\(391\) − 2.79884i − 0.141543i
\(392\) 0 0
\(393\) 0 0
\(394\) −39.2132 −1.97553
\(395\) −32.6256 −1.64157
\(396\) 0 0
\(397\) − 27.2680i − 1.36854i −0.729227 0.684272i \(-0.760120\pi\)
0.729227 0.684272i \(-0.239880\pi\)
\(398\) 46.1396 2.31277
\(399\) 0 0
\(400\) 19.2426 0.962132
\(401\) − 10.8284i − 0.540746i −0.962756 0.270373i \(-0.912853\pi\)
0.962756 0.270373i \(-0.0871470\pi\)
\(402\) 0 0
\(403\) 22.8284 1.13716
\(404\) −49.5181 −2.46362
\(405\) 0 0
\(406\) 0 0
\(407\) 2.14214i 0.106182i
\(408\) 0 0
\(409\) 17.7122i 0.875813i 0.899020 + 0.437907i \(0.144280\pi\)
−0.899020 + 0.437907i \(0.855720\pi\)
\(410\) − 66.5269i − 3.28553i
\(411\) 0 0
\(412\) 10.7151i 0.527897i
\(413\) 0 0
\(414\) 0 0
\(415\) −45.6569 −2.24121
\(416\) 5.35757 0.262676
\(417\) 0 0
\(418\) 13.5140i 0.660990i
\(419\) −35.4244 −1.73060 −0.865299 0.501256i \(-0.832871\pi\)
−0.865299 + 0.501256i \(0.832871\pi\)
\(420\) 0 0
\(421\) −23.3137 −1.13624 −0.568120 0.822946i \(-0.692329\pi\)
−0.568120 + 0.822946i \(0.692329\pi\)
\(422\) − 17.6569i − 0.859522i
\(423\) 0 0
\(424\) −31.2132 −1.51585
\(425\) 8.97616 0.435408
\(426\) 0 0
\(427\) 0 0
\(428\) − 35.6569i − 1.72354i
\(429\) 0 0
\(430\) 101.835i 4.91092i
\(431\) 15.4558i 0.744482i 0.928136 + 0.372241i \(0.121410\pi\)
−0.928136 + 0.372241i \(0.878590\pi\)
\(432\) 0 0
\(433\) − 8.15640i − 0.391972i −0.980607 0.195986i \(-0.937209\pi\)
0.980607 0.195986i \(-0.0627907\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 9.89949 0.474100
\(437\) −13.5140 −0.646461
\(438\) 0 0
\(439\) − 28.6675i − 1.36822i −0.729377 0.684112i \(-0.760190\pi\)
0.729377 0.684112i \(-0.239810\pi\)
\(440\) 12.3547 0.588985
\(441\) 0 0
\(442\) −11.4142 −0.542919
\(443\) 24.8284i 1.17963i 0.807537 + 0.589817i \(0.200800\pi\)
−0.807537 + 0.589817i \(0.799200\pi\)
\(444\) 0 0
\(445\) 20.8701 0.989336
\(446\) −55.6954 −2.63725
\(447\) 0 0
\(448\) 0 0
\(449\) − 24.2426i − 1.14408i −0.820225 0.572040i \(-0.806152\pi\)
0.820225 0.572040i \(-0.193848\pi\)
\(450\) 0 0
\(451\) − 6.75699i − 0.318174i
\(452\) − 22.5858i − 1.06235i
\(453\) 0 0
\(454\) − 6.75699i − 0.317121i
\(455\) 0 0
\(456\) 0 0
\(457\) 28.6274 1.33913 0.669567 0.742752i \(-0.266480\pi\)
0.669567 + 0.742752i \(0.266480\pi\)
\(458\) 29.2471 1.36663
\(459\) 0 0
\(460\) 25.8686i 1.20613i
\(461\) 21.6704 1.00929 0.504645 0.863327i \(-0.331623\pi\)
0.504645 + 0.863327i \(0.331623\pi\)
\(462\) 0 0
\(463\) 3.51472 0.163343 0.0816714 0.996659i \(-0.473974\pi\)
0.0816714 + 0.996659i \(0.473974\pi\)
\(464\) 14.4853i 0.672462i
\(465\) 0 0
\(466\) 26.1421 1.21101
\(467\) −2.79884 −0.129515 −0.0647573 0.997901i \(-0.520627\pi\)
−0.0647573 + 0.997901i \(0.520627\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) − 55.1127i − 2.54216i
\(471\) 0 0
\(472\) 29.8268i 1.37289i
\(473\) 10.3431i 0.475578i
\(474\) 0 0
\(475\) − 43.3407i − 1.98861i
\(476\) 0 0
\(477\) 0 0
\(478\) 0.828427 0.0378914
\(479\) −8.39651 −0.383646 −0.191823 0.981430i \(-0.561440\pi\)
−0.191823 + 0.981430i \(0.561440\pi\)
\(480\) 0 0
\(481\) 8.73606i 0.398330i
\(482\) −21.6704 −0.987059
\(483\) 0 0
\(484\) −39.4853 −1.79479
\(485\) − 4.72792i − 0.214684i
\(486\) 0 0
\(487\) 9.45584 0.428485 0.214243 0.976780i \(-0.431272\pi\)
0.214243 + 0.976780i \(0.431272\pi\)
\(488\) −36.0041 −1.62983
\(489\) 0 0
\(490\) 0 0
\(491\) 2.68629i 0.121231i 0.998161 + 0.0606153i \(0.0193063\pi\)
−0.998161 + 0.0606153i \(0.980694\pi\)
\(492\) 0 0
\(493\) 6.75699i 0.304319i
\(494\) 55.1127i 2.47964i
\(495\) 0 0
\(496\) 20.2710i 0.910193i
\(497\) 0 0
\(498\) 0 0
\(499\) −2.14214 −0.0958952 −0.0479476 0.998850i \(-0.515268\pi\)
−0.0479476 + 0.998850i \(0.515268\pi\)
\(500\) −18.2919 −0.818038
\(501\) 0 0
\(502\) 62.4524i 2.78739i
\(503\) 32.6256 1.45470 0.727352 0.686264i \(-0.240751\pi\)
0.727352 + 0.686264i \(0.240751\pi\)
\(504\) 0 0
\(505\) 43.6985 1.94456
\(506\) 4.00000i 0.177822i
\(507\) 0 0
\(508\) 1.85786 0.0824294
\(509\) 36.8239 1.63219 0.816095 0.577918i \(-0.196135\pi\)
0.816095 + 0.577918i \(0.196135\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 31.2426i 1.38074i
\(513\) 0 0
\(514\) − 52.3169i − 2.30760i
\(515\) − 9.45584i − 0.416674i
\(516\) 0 0
\(517\) − 5.59767i − 0.246185i
\(518\) 0 0
\(519\) 0 0
\(520\) 50.3848 2.20952
\(521\) −42.7611 −1.87340 −0.936699 0.350136i \(-0.886135\pi\)
−0.936699 + 0.350136i \(0.886135\pi\)
\(522\) 0 0
\(523\) − 17.4721i − 0.764003i −0.924162 0.382001i \(-0.875235\pi\)
0.924162 0.382001i \(-0.124765\pi\)
\(524\) 36.5838 1.59817
\(525\) 0 0
\(526\) −58.2843 −2.54131
\(527\) 9.45584i 0.411903i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 57.6745 2.50522
\(531\) 0 0
\(532\) 0 0
\(533\) − 27.5563i − 1.19360i
\(534\) 0 0
\(535\) 31.4663i 1.36041i
\(536\) 37.4558i 1.61785i
\(537\) 0 0
\(538\) 75.3867i 3.25015i
\(539\) 0 0
\(540\) 0 0
\(541\) 18.9706 0.815608 0.407804 0.913069i \(-0.366295\pi\)
0.407804 + 0.913069i \(0.366295\pi\)
\(542\) 16.3128 0.700695
\(543\) 0 0
\(544\) 2.21918i 0.0951464i
\(545\) −8.73606 −0.374212
\(546\) 0 0
\(547\) −26.6274 −1.13851 −0.569253 0.822162i \(-0.692768\pi\)
−0.569253 + 0.822162i \(0.692768\pi\)
\(548\) − 3.17157i − 0.135483i
\(549\) 0 0
\(550\) −12.8284 −0.547006
\(551\) 32.6256 1.38990
\(552\) 0 0
\(553\) 0 0
\(554\) 50.6274i 2.15095i
\(555\) 0 0
\(556\) − 36.5838i − 1.55150i
\(557\) 10.5858i 0.448534i 0.974528 + 0.224267i \(0.0719988\pi\)
−0.974528 + 0.224267i \(0.928001\pi\)
\(558\) 0 0
\(559\) 42.1814i 1.78408i
\(560\) 0 0
\(561\) 0 0
\(562\) −45.4558 −1.91744
\(563\) 2.79884 0.117957 0.0589784 0.998259i \(-0.481216\pi\)
0.0589784 + 0.998259i \(0.481216\pi\)
\(564\) 0 0
\(565\) 19.9314i 0.838520i
\(566\) −72.0082 −3.02673
\(567\) 0 0
\(568\) 21.3137 0.894303
\(569\) − 25.1127i − 1.05278i −0.850244 0.526390i \(-0.823545\pi\)
0.850244 0.526390i \(-0.176455\pi\)
\(570\) 0 0
\(571\) −11.3137 −0.473464 −0.236732 0.971575i \(-0.576076\pi\)
−0.236732 + 0.971575i \(0.576076\pi\)
\(572\) 10.7151 0.448022
\(573\) 0 0
\(574\) 0 0
\(575\) − 12.8284i − 0.534982i
\(576\) 0 0
\(577\) − 13.7541i − 0.572590i −0.958142 0.286295i \(-0.907576\pi\)
0.958142 0.286295i \(-0.0924237\pi\)
\(578\) 36.3137i 1.51045i
\(579\) 0 0
\(580\) − 62.4524i − 2.59319i
\(581\) 0 0
\(582\) 0 0
\(583\) 5.85786 0.242608
\(584\) −6.17733 −0.255620
\(585\) 0 0
\(586\) − 17.7122i − 0.731685i
\(587\) −20.2710 −0.836672 −0.418336 0.908292i \(-0.637387\pi\)
−0.418336 + 0.908292i \(0.637387\pi\)
\(588\) 0 0
\(589\) 45.6569 1.88126
\(590\) − 55.1127i − 2.26895i
\(591\) 0 0
\(592\) −7.75736 −0.318826
\(593\) −20.5111 −0.842288 −0.421144 0.906994i \(-0.638371\pi\)
−0.421144 + 0.906994i \(0.638371\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 16.2426i 0.665324i
\(597\) 0 0
\(598\) 16.3128i 0.667080i
\(599\) − 6.48528i − 0.264981i −0.991184 0.132491i \(-0.957703\pi\)
0.991184 0.132491i \(-0.0422975\pi\)
\(600\) 0 0
\(601\) − 27.6076i − 1.12614i −0.826410 0.563069i \(-0.809621\pi\)
0.826410 0.563069i \(-0.190379\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 47.7990 1.94491
\(605\) 34.8448 1.41664
\(606\) 0 0
\(607\) − 23.0698i − 0.936374i −0.883629 0.468187i \(-0.844907\pi\)
0.883629 0.468187i \(-0.155093\pi\)
\(608\) 10.7151 0.434556
\(609\) 0 0
\(610\) 66.5269 2.69360
\(611\) − 22.8284i − 0.923539i
\(612\) 0 0
\(613\) −10.8701 −0.439037 −0.219519 0.975608i \(-0.570449\pi\)
−0.219519 + 0.975608i \(0.570449\pi\)
\(614\) −29.8268 −1.20371
\(615\) 0 0
\(616\) 0 0
\(617\) 30.4853i 1.22729i 0.789582 + 0.613646i \(0.210298\pi\)
−0.789582 + 0.613646i \(0.789702\pi\)
\(618\) 0 0
\(619\) − 24.7093i − 0.993151i −0.867994 0.496576i \(-0.834591\pi\)
0.867994 0.496576i \(-0.165409\pi\)
\(620\) − 87.3970i − 3.50995i
\(621\) 0 0
\(622\) − 25.8686i − 1.03724i
\(623\) 0 0
\(624\) 0 0
\(625\) −15.9289 −0.637157
\(626\) −31.2262 −1.24805
\(627\) 0 0
\(628\) − 28.0878i − 1.12083i
\(629\) −3.61859 −0.144283
\(630\) 0 0
\(631\) −21.1716 −0.842827 −0.421414 0.906869i \(-0.638466\pi\)
−0.421414 + 0.906869i \(0.638466\pi\)
\(632\) − 42.6274i − 1.69563i
\(633\) 0 0
\(634\) −57.3553 −2.27787
\(635\) −1.63952 −0.0650624
\(636\) 0 0
\(637\) 0 0
\(638\) − 9.65685i − 0.382319i
\(639\) 0 0
\(640\) − 69.4495i − 2.74523i
\(641\) − 7.51472i − 0.296814i −0.988926 0.148407i \(-0.952586\pi\)
0.988926 0.148407i \(-0.0474145\pi\)
\(642\) 0 0
\(643\) − 6.75699i − 0.266469i −0.991085 0.133235i \(-0.957464\pi\)
0.991085 0.133235i \(-0.0425364\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −22.8284 −0.898172
\(647\) 25.8686 1.01700 0.508500 0.861062i \(-0.330200\pi\)
0.508500 + 0.861062i \(0.330200\pi\)
\(648\) 0 0
\(649\) − 5.59767i − 0.219728i
\(650\) −52.3169 −2.05204
\(651\) 0 0
\(652\) −56.7696 −2.22327
\(653\) 38.4853i 1.50605i 0.657995 + 0.753023i \(0.271405\pi\)
−0.657995 + 0.753023i \(0.728595\pi\)
\(654\) 0 0
\(655\) −32.2843 −1.26145
\(656\) 24.4692 0.955362
\(657\) 0 0
\(658\) 0 0
\(659\) 47.4558i 1.84862i 0.381646 + 0.924309i \(0.375357\pi\)
−0.381646 + 0.924309i \(0.624643\pi\)
\(660\) 0 0
\(661\) − 7.33664i − 0.285362i −0.989769 0.142681i \(-0.954428\pi\)
0.989769 0.142681i \(-0.0455724\pi\)
\(662\) − 24.9706i − 0.970508i
\(663\) 0 0
\(664\) − 59.6536i − 2.31501i
\(665\) 0 0
\(666\) 0 0
\(667\) 9.65685 0.373915
\(668\) −10.7151 −0.414581
\(669\) 0 0
\(670\) − 69.2094i − 2.67379i
\(671\) 6.75699 0.260851
\(672\) 0 0
\(673\) −37.8995 −1.46092 −0.730459 0.682956i \(-0.760694\pi\)
−0.730459 + 0.682956i \(0.760694\pi\)
\(674\) 31.2132i 1.20229i
\(675\) 0 0
\(676\) −6.07107 −0.233503
\(677\) −32.8657 −1.26313 −0.631566 0.775322i \(-0.717588\pi\)
−0.631566 + 0.775322i \(0.717588\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 20.8701i 0.800330i
\(681\) 0 0
\(682\) − 13.5140i − 0.517477i
\(683\) − 38.7696i − 1.48348i −0.670690 0.741738i \(-0.734002\pi\)
0.670690 0.741738i \(-0.265998\pi\)
\(684\) 0 0
\(685\) 2.79884i 0.106938i
\(686\) 0 0
\(687\) 0 0
\(688\) −37.4558 −1.42799
\(689\) 23.8896 0.910119
\(690\) 0 0
\(691\) 9.55582i 0.363521i 0.983343 + 0.181760i \(0.0581795\pi\)
−0.983343 + 0.181760i \(0.941821\pi\)
\(692\) −31.2262 −1.18704
\(693\) 0 0
\(694\) 45.7990 1.73851
\(695\) 32.2843i 1.22461i
\(696\) 0 0
\(697\) 11.4142 0.432344
\(698\) 14.9134 0.564480
\(699\) 0 0
\(700\) 0 0
\(701\) 40.7696i 1.53984i 0.638138 + 0.769922i \(0.279705\pi\)
−0.638138 + 0.769922i \(0.720295\pi\)
\(702\) 0 0
\(703\) 17.4721i 0.658974i
\(704\) − 8.14214i − 0.306868i
\(705\) 0 0
\(706\) − 57.0948i − 2.14879i
\(707\) 0 0
\(708\) 0 0
\(709\) −32.7279 −1.22912 −0.614561 0.788869i \(-0.710667\pi\)
−0.614561 + 0.788869i \(0.710667\pi\)
\(710\) −39.3826 −1.47800
\(711\) 0 0
\(712\) 27.2680i 1.02191i
\(713\) 13.5140 0.506102
\(714\) 0 0
\(715\) −9.45584 −0.353629
\(716\) − 36.4264i − 1.36132i
\(717\) 0 0
\(718\) 2.48528 0.0927499
\(719\) −13.5140 −0.503986 −0.251993 0.967729i \(-0.581086\pi\)
−0.251993 + 0.967729i \(0.581086\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 64.3553i 2.39506i
\(723\) 0 0
\(724\) − 67.8100i − 2.52014i
\(725\) 30.9706i 1.15022i
\(726\) 0 0
\(727\) 43.3407i 1.60742i 0.595022 + 0.803710i \(0.297143\pi\)
−0.595022 + 0.803710i \(0.702857\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 11.4142 0.422459
\(731\) −17.4721 −0.646230
\(732\) 0 0
\(733\) 23.3099i 0.860971i 0.902598 + 0.430485i \(0.141658\pi\)
−0.902598 + 0.430485i \(0.858342\pi\)
\(734\) −9.55582 −0.352712
\(735\) 0 0
\(736\) 3.17157 0.116906
\(737\) − 7.02944i − 0.258933i
\(738\) 0 0
\(739\) −18.8284 −0.692615 −0.346307 0.938121i \(-0.612565\pi\)
−0.346307 + 0.938121i \(0.612565\pi\)
\(740\) 33.4454 1.22948
\(741\) 0 0
\(742\) 0 0
\(743\) − 52.4264i − 1.92334i −0.274212 0.961669i \(-0.588417\pi\)
0.274212 0.961669i \(-0.411583\pi\)
\(744\) 0 0
\(745\) − 14.3337i − 0.525147i
\(746\) 54.6274i 2.00005i
\(747\) 0 0
\(748\) 4.43835i 0.162282i
\(749\) 0 0
\(750\) 0 0
\(751\) 20.6863 0.754853 0.377427 0.926039i \(-0.376809\pi\)
0.377427 + 0.926039i \(0.376809\pi\)
\(752\) 20.2710 0.739206
\(753\) 0 0
\(754\) − 39.3826i − 1.43423i
\(755\) −42.1814 −1.53514
\(756\) 0 0
\(757\) 6.87006 0.249696 0.124848 0.992176i \(-0.460156\pi\)
0.124848 + 0.992176i \(0.460156\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 100.770 3.65530
\(761\) −0.579658 −0.0210126 −0.0105063 0.999945i \(-0.503344\pi\)
−0.0105063 + 0.999945i \(0.503344\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) − 79.7401i − 2.88490i
\(765\) 0 0
\(766\) 65.2512i 2.35762i
\(767\) − 22.8284i − 0.824287i
\(768\) 0 0
\(769\) 45.8995i 1.65518i 0.561335 + 0.827589i \(0.310288\pi\)
−0.561335 + 0.827589i \(0.689712\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 67.5980 2.43290
\(773\) 18.8715 0.678762 0.339381 0.940649i \(-0.389782\pi\)
0.339381 + 0.940649i \(0.389782\pi\)
\(774\) 0 0
\(775\) 43.3407i 1.55685i
\(776\) 6.17733 0.221753
\(777\) 0 0
\(778\) 33.3137 1.19435
\(779\) − 55.1127i − 1.97462i
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) −6.75699 −0.241629
\(783\) 0 0
\(784\) 0 0
\(785\) 24.7868i 0.884679i
\(786\) 0 0
\(787\) 13.5140i 0.481721i 0.970560 + 0.240861i \(0.0774296\pi\)
−0.970560 + 0.240861i \(0.922570\pi\)
\(788\) 62.1838i 2.21520i
\(789\) 0 0
\(790\) 78.7652i 2.80234i
\(791\) 0 0
\(792\) 0 0
\(793\) 27.5563 0.978555
\(794\) −65.8309 −2.33625
\(795\) 0 0
\(796\) − 73.1675i − 2.59335i
\(797\) −1.73897 −0.0615976 −0.0307988 0.999526i \(-0.509805\pi\)
−0.0307988 + 0.999526i \(0.509805\pi\)
\(798\) 0 0
\(799\) 9.45584 0.334524
\(800\) 10.1716i 0.359619i
\(801\) 0 0
\(802\) −26.1421 −0.923111
\(803\) 1.15932 0.0409114
\(804\) 0 0
\(805\) 0 0
\(806\) − 55.1127i − 1.94126i
\(807\) 0 0
\(808\) 57.0948i 2.00859i
\(809\) − 32.0416i − 1.12652i −0.826278 0.563262i \(-0.809546\pi\)
0.826278 0.563262i \(-0.190454\pi\)
\(810\) 0 0
\(811\) − 9.55582i − 0.335550i −0.985825 0.167775i \(-0.946342\pi\)
0.985825 0.167775i \(-0.0536583\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 5.17157 0.181264
\(815\) 50.0977 1.75485
\(816\) 0 0
\(817\) 84.3629i 2.95148i
\(818\) 42.7611 1.49511
\(819\) 0 0
\(820\) −105.497 −3.68413
\(821\) − 32.2426i − 1.12528i −0.826703 0.562638i \(-0.809787\pi\)
0.826703 0.562638i \(-0.190213\pi\)
\(822\) 0 0
\(823\) 25.9411 0.904251 0.452125 0.891954i \(-0.350666\pi\)
0.452125 + 0.891954i \(0.350666\pi\)
\(824\) 12.3547 0.430395
\(825\) 0 0
\(826\) 0 0
\(827\) − 28.3431i − 0.985588i −0.870146 0.492794i \(-0.835976\pi\)
0.870146 0.492794i \(-0.164024\pi\)
\(828\) 0 0
\(829\) 34.8448i 1.21021i 0.796146 + 0.605105i \(0.206869\pi\)
−0.796146 + 0.605105i \(0.793131\pi\)
\(830\) 110.225i 3.82598i
\(831\) 0 0
\(832\) − 33.2053i − 1.15119i
\(833\) 0 0
\(834\) 0 0
\(835\) 9.45584 0.327233
\(836\) 21.4303 0.741181
\(837\) 0 0
\(838\) 85.5222i 2.95431i
\(839\) −20.2710 −0.699831 −0.349916 0.936781i \(-0.613790\pi\)
−0.349916 + 0.936781i \(0.613790\pi\)
\(840\) 0 0
\(841\) 5.68629 0.196079
\(842\) 56.2843i 1.93968i
\(843\) 0 0
\(844\) −28.0000 −0.963800
\(845\) 5.35757 0.184306
\(846\) 0 0
\(847\) 0 0
\(848\) 21.2132i 0.728464i
\(849\) 0 0
\(850\) − 21.6704i − 0.743288i
\(851\) 5.17157i 0.177279i
\(852\) 0 0
\(853\) − 38.8029i − 1.32859i −0.747472 0.664294i \(-0.768732\pi\)
0.747472 0.664294i \(-0.231268\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −41.1127 −1.40520
\(857\) −19.6913 −0.672642 −0.336321 0.941747i \(-0.609183\pi\)
−0.336321 + 0.941747i \(0.609183\pi\)
\(858\) 0 0
\(859\) 29.8268i 1.01768i 0.860862 + 0.508838i \(0.169925\pi\)
−0.860862 + 0.508838i \(0.830075\pi\)
\(860\) 161.489 5.50671
\(861\) 0 0
\(862\) 37.3137 1.27091
\(863\) 33.3137i 1.13401i 0.823714 + 0.567006i \(0.191898\pi\)
−0.823714 + 0.567006i \(0.808102\pi\)
\(864\) 0 0
\(865\) 27.5563 0.936944
\(866\) −19.6913 −0.669138
\(867\) 0 0
\(868\) 0 0
\(869\) 8.00000i 0.271381i
\(870\) 0 0
\(871\) − 28.6675i − 0.971360i
\(872\) − 11.4142i − 0.386534i
\(873\) 0 0
\(874\) 32.6256i 1.10358i
\(875\) 0 0
\(876\) 0 0
\(877\) 32.5269 1.09836 0.549178 0.835705i \(-0.314941\pi\)
0.549178 + 0.835705i \(0.314941\pi\)
\(878\) −69.2094 −2.33570
\(879\) 0 0
\(880\) − 8.39651i − 0.283046i
\(881\) 20.5111 0.691035 0.345518 0.938412i \(-0.387703\pi\)
0.345518 + 0.938412i \(0.387703\pi\)
\(882\) 0 0
\(883\) −37.4558 −1.26049 −0.630245 0.776397i \(-0.717045\pi\)
−0.630245 + 0.776397i \(0.717045\pi\)
\(884\) 18.1005i 0.608786i
\(885\) 0 0
\(886\) 59.9411 2.01376
\(887\) 17.9523 0.602780 0.301390 0.953501i \(-0.402549\pi\)
0.301390 + 0.953501i \(0.402549\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) − 50.3848i − 1.68890i
\(891\) 0 0
\(892\) 88.3210i 2.95721i
\(893\) − 45.6569i − 1.52785i
\(894\) 0 0
\(895\) 32.1454i 1.07450i
\(896\) 0 0
\(897\) 0 0
\(898\) −58.5269 −1.95307
\(899\) −32.6256 −1.08813
\(900\) 0 0
\(901\) 9.89538i 0.329663i
\(902\) −16.3128 −0.543157
\(903\) 0 0
\(904\) −26.0416 −0.866132
\(905\) 59.8406i 1.98917i
\(906\) 0 0
\(907\) −44.7696 −1.48655 −0.743274 0.668987i \(-0.766728\pi\)
−0.743274 + 0.668987i \(0.766728\pi\)
\(908\) −10.7151 −0.355594
\(909\) 0 0
\(910\) 0 0
\(911\) − 25.5147i − 0.845340i −0.906284 0.422670i \(-0.861093\pi\)
0.906284 0.422670i \(-0.138907\pi\)
\(912\) 0 0
\(913\) 11.1953i 0.370512i
\(914\) − 69.1127i − 2.28604i
\(915\) 0 0
\(916\) − 46.3797i − 1.53243i
\(917\) 0 0
\(918\) 0 0
\(919\) 9.45584 0.311920 0.155960 0.987763i \(-0.450153\pi\)
0.155960 + 0.987763i \(0.450153\pi\)
\(920\) 29.8268 0.983360
\(921\) 0 0
\(922\) − 52.3169i − 1.72297i
\(923\) −16.3128 −0.536943
\(924\) 0 0
\(925\) −16.5858 −0.545337
\(926\) − 8.48528i − 0.278844i
\(927\) 0 0
\(928\) −7.65685 −0.251349
\(929\) −19.6913 −0.646051 −0.323025 0.946390i \(-0.604700\pi\)
−0.323025 + 0.946390i \(0.604700\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 41.4558i − 1.35793i
\(933\) 0 0
\(934\) 6.75699i 0.221095i
\(935\) − 3.91674i − 0.128091i
\(936\) 0 0
\(937\) − 1.73897i − 0.0568098i −0.999596 0.0284049i \(-0.990957\pi\)
0.999596 0.0284049i \(-0.00904277\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −87.3970 −2.85057
\(941\) 14.9134 0.486163 0.243081 0.970006i \(-0.421842\pi\)
0.243081 + 0.970006i \(0.421842\pi\)
\(942\) 0 0
\(943\) − 16.3128i − 0.531218i
\(944\) 20.2710 0.659763
\(945\) 0 0
\(946\) 24.9706 0.811863
\(947\) − 22.9706i − 0.746443i −0.927742 0.373221i \(-0.878253\pi\)
0.927742 0.373221i \(-0.121747\pi\)
\(948\) 0 0
\(949\) 4.72792 0.153475
\(950\) −104.634 −3.39477
\(951\) 0 0
\(952\) 0 0
\(953\) 26.3848i 0.854687i 0.904089 + 0.427343i \(0.140550\pi\)
−0.904089 + 0.427343i \(0.859450\pi\)
\(954\) 0 0
\(955\) 70.3687i 2.27708i
\(956\) − 1.31371i − 0.0424884i
\(957\) 0 0
\(958\) 20.2710i 0.654925i
\(959\) 0 0
\(960\) 0 0
\(961\) −14.6569 −0.472802
\(962\) 21.0907 0.679992
\(963\) 0 0
\(964\) 34.3646i 1.10681i
\(965\) −59.6536 −1.92032
\(966\) 0 0
\(967\) 43.1127 1.38641 0.693205 0.720740i \(-0.256198\pi\)
0.693205 + 0.720740i \(0.256198\pi\)
\(968\) 45.5269i 1.46329i
\(969\) 0 0
\(970\) −11.4142 −0.366488
\(971\) 32.6256 1.04701 0.523503 0.852024i \(-0.324625\pi\)
0.523503 + 0.852024i \(0.324625\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 22.8284i − 0.731470i
\(975\) 0 0
\(976\) 24.4692i 0.783240i
\(977\) − 39.1716i − 1.25321i −0.779337 0.626605i \(-0.784444\pi\)
0.779337 0.626605i \(-0.215556\pi\)
\(978\) 0 0
\(979\) − 5.11747i − 0.163555i
\(980\) 0 0
\(981\) 0 0
\(982\) 6.48528 0.206954
\(983\) 17.4721 0.557274 0.278637 0.960396i \(-0.410117\pi\)
0.278637 + 0.960396i \(0.410117\pi\)
\(984\) 0 0
\(985\) − 54.8756i − 1.74848i
\(986\) 16.3128 0.519506
\(987\) 0 0
\(988\) 87.3970 2.78047
\(989\) 24.9706i 0.794018i
\(990\) 0 0
\(991\) 59.3970 1.88681 0.943403 0.331647i \(-0.107604\pi\)
0.943403 + 0.331647i \(0.107604\pi\)
\(992\) −10.7151 −0.340206
\(993\) 0 0
\(994\) 0 0
\(995\) 64.5685i 2.04696i
\(996\) 0 0
\(997\) − 11.7750i − 0.372918i −0.982463 0.186459i \(-0.940299\pi\)
0.982463 0.186459i \(-0.0597011\pi\)
\(998\) 5.17157i 0.163703i
\(999\) 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 441.2.c.b.440.2 yes 8
3.2 odd 2 inner 441.2.c.b.440.7 yes 8
4.3 odd 2 7056.2.k.g.881.8 8
7.2 even 3 441.2.p.c.80.1 16
7.3 odd 6 441.2.p.c.215.8 16
7.4 even 3 441.2.p.c.215.7 16
7.5 odd 6 441.2.p.c.80.2 16
7.6 odd 2 inner 441.2.c.b.440.1 8
12.11 even 2 7056.2.k.g.881.1 8
21.2 odd 6 441.2.p.c.80.8 16
21.5 even 6 441.2.p.c.80.7 16
21.11 odd 6 441.2.p.c.215.2 16
21.17 even 6 441.2.p.c.215.1 16
21.20 even 2 inner 441.2.c.b.440.8 yes 8
28.27 even 2 7056.2.k.g.881.2 8
84.83 odd 2 7056.2.k.g.881.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.2.c.b.440.1 8 7.6 odd 2 inner
441.2.c.b.440.2 yes 8 1.1 even 1 trivial
441.2.c.b.440.7 yes 8 3.2 odd 2 inner
441.2.c.b.440.8 yes 8 21.20 even 2 inner
441.2.p.c.80.1 16 7.2 even 3
441.2.p.c.80.2 16 7.5 odd 6
441.2.p.c.80.7 16 21.5 even 6
441.2.p.c.80.8 16 21.2 odd 6
441.2.p.c.215.1 16 21.17 even 6
441.2.p.c.215.2 16 21.11 odd 6
441.2.p.c.215.7 16 7.4 even 3
441.2.p.c.215.8 16 7.3 odd 6
7056.2.k.g.881.1 8 12.11 even 2
7056.2.k.g.881.2 8 28.27 even 2
7056.2.k.g.881.7 8 84.83 odd 2
7056.2.k.g.881.8 8 4.3 odd 2