Properties

Label 441.2.c.a.440.1
Level $441$
Weight $2$
Character 441.440
Analytic conductor $3.521$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 440.1
Root \(-1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 441.440
Dual form 441.2.c.a.440.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.44949 q^{5} -2.82843i q^{8} +O(q^{10})\) \(q-1.41421i q^{2} -2.44949 q^{5} -2.82843i q^{8} +3.46410i q^{10} -1.41421i q^{11} -5.19615i q^{13} -4.00000 q^{16} -4.89898 q^{17} +1.73205i q^{19} -2.00000 q^{22} -5.65685i q^{23} +1.00000 q^{25} -7.34847 q^{26} +2.82843i q^{29} -1.73205i q^{31} +6.92820i q^{34} -1.00000 q^{37} +2.44949 q^{38} +6.92820i q^{40} +7.34847 q^{41} -1.00000 q^{43} -8.00000 q^{46} +12.2474 q^{47} -1.41421i q^{50} +2.82843i q^{53} +3.46410i q^{55} +4.00000 q^{58} -4.89898 q^{59} -3.46410i q^{61} -2.44949 q^{62} -8.00000 q^{64} +12.7279i q^{65} +11.0000 q^{67} +7.07107i q^{71} -1.73205i q^{73} +1.41421i q^{74} +5.00000 q^{79} +9.79796 q^{80} -10.3923i q^{82} -7.34847 q^{83} +12.0000 q^{85} +1.41421i q^{86} -4.00000 q^{88} +4.89898 q^{89} -17.3205i q^{94} -4.24264i q^{95} -10.3923i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 16q^{16} - 8q^{22} + 4q^{25} - 4q^{37} - 4q^{43} - 32q^{46} + 16q^{58} - 32q^{64} + 44q^{67} + 20q^{79} + 48q^{85} - 16q^{88} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.41421i − 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(3\) 0 0
\(4\) 0 0
\(5\) −2.44949 −1.09545 −0.547723 0.836660i \(-0.684505\pi\)
−0.547723 + 0.836660i \(0.684505\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) − 2.82843i − 1.00000i
\(9\) 0 0
\(10\) 3.46410i 1.09545i
\(11\) − 1.41421i − 0.426401i −0.977008 0.213201i \(-0.931611\pi\)
0.977008 0.213201i \(-0.0683888\pi\)
\(12\) 0 0
\(13\) − 5.19615i − 1.44115i −0.693375 0.720577i \(-0.743877\pi\)
0.693375 0.720577i \(-0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −4.89898 −1.18818 −0.594089 0.804400i \(-0.702487\pi\)
−0.594089 + 0.804400i \(0.702487\pi\)
\(18\) 0 0
\(19\) 1.73205i 0.397360i 0.980064 + 0.198680i \(0.0636654\pi\)
−0.980064 + 0.198680i \(0.936335\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) − 5.65685i − 1.17954i −0.807573 0.589768i \(-0.799219\pi\)
0.807573 0.589768i \(-0.200781\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −7.34847 −1.44115
\(27\) 0 0
\(28\) 0 0
\(29\) 2.82843i 0.525226i 0.964901 + 0.262613i \(0.0845842\pi\)
−0.964901 + 0.262613i \(0.915416\pi\)
\(30\) 0 0
\(31\) − 1.73205i − 0.311086i −0.987829 0.155543i \(-0.950287\pi\)
0.987829 0.155543i \(-0.0497126\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 6.92820i 1.18818i
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 2.44949 0.397360
\(39\) 0 0
\(40\) 6.92820i 1.09545i
\(41\) 7.34847 1.14764 0.573819 0.818982i \(-0.305461\pi\)
0.573819 + 0.818982i \(0.305461\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 12.2474 1.78647 0.893237 0.449586i \(-0.148429\pi\)
0.893237 + 0.449586i \(0.148429\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) − 1.41421i − 0.200000i
\(51\) 0 0
\(52\) 0 0
\(53\) 2.82843i 0.388514i 0.980951 + 0.194257i \(0.0622296\pi\)
−0.980951 + 0.194257i \(0.937770\pi\)
\(54\) 0 0
\(55\) 3.46410i 0.467099i
\(56\) 0 0
\(57\) 0 0
\(58\) 4.00000 0.525226
\(59\) −4.89898 −0.637793 −0.318896 0.947790i \(-0.603312\pi\)
−0.318896 + 0.947790i \(0.603312\pi\)
\(60\) 0 0
\(61\) − 3.46410i − 0.443533i −0.975100 0.221766i \(-0.928818\pi\)
0.975100 0.221766i \(-0.0711822\pi\)
\(62\) −2.44949 −0.311086
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 12.7279i 1.57870i
\(66\) 0 0
\(67\) 11.0000 1.34386 0.671932 0.740613i \(-0.265465\pi\)
0.671932 + 0.740613i \(0.265465\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.07107i 0.839181i 0.907713 + 0.419591i \(0.137826\pi\)
−0.907713 + 0.419591i \(0.862174\pi\)
\(72\) 0 0
\(73\) − 1.73205i − 0.202721i −0.994850 0.101361i \(-0.967680\pi\)
0.994850 0.101361i \(-0.0323196\pi\)
\(74\) 1.41421i 0.164399i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) 9.79796 1.09545
\(81\) 0 0
\(82\) − 10.3923i − 1.14764i
\(83\) −7.34847 −0.806599 −0.403300 0.915068i \(-0.632137\pi\)
−0.403300 + 0.915068i \(0.632137\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) 1.41421i 0.152499i
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) 4.89898 0.519291 0.259645 0.965704i \(-0.416394\pi\)
0.259645 + 0.965704i \(0.416394\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) − 17.3205i − 1.78647i
\(95\) − 4.24264i − 0.435286i
\(96\) 0 0
\(97\) − 10.3923i − 1.05518i −0.849500 0.527589i \(-0.823096\pi\)
0.849500 0.527589i \(-0.176904\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 17.1464 1.70613 0.853067 0.521802i \(-0.174740\pi\)
0.853067 + 0.521802i \(0.174740\pi\)
\(102\) 0 0
\(103\) − 8.66025i − 0.853320i −0.904412 0.426660i \(-0.859690\pi\)
0.904412 0.426660i \(-0.140310\pi\)
\(104\) −14.6969 −1.44115
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) 2.82843i 0.273434i 0.990610 + 0.136717i \(0.0436552\pi\)
−0.990610 + 0.136717i \(0.956345\pi\)
\(108\) 0 0
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 4.89898 0.467099
\(111\) 0 0
\(112\) 0 0
\(113\) − 1.41421i − 0.133038i −0.997785 0.0665190i \(-0.978811\pi\)
0.997785 0.0665190i \(-0.0211893\pi\)
\(114\) 0 0
\(115\) 13.8564i 1.29212i
\(116\) 0 0
\(117\) 0 0
\(118\) 6.92820i 0.637793i
\(119\) 0 0
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) −4.89898 −0.443533
\(123\) 0 0
\(124\) 0 0
\(125\) 9.79796 0.876356
\(126\) 0 0
\(127\) 11.0000 0.976092 0.488046 0.872818i \(-0.337710\pi\)
0.488046 + 0.872818i \(0.337710\pi\)
\(128\) 11.3137i 1.00000i
\(129\) 0 0
\(130\) 18.0000 1.57870
\(131\) −2.44949 −0.214013 −0.107006 0.994258i \(-0.534127\pi\)
−0.107006 + 0.994258i \(0.534127\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 15.5563i − 1.34386i
\(135\) 0 0
\(136\) 13.8564i 1.18818i
\(137\) 11.3137i 0.966595i 0.875456 + 0.483298i \(0.160561\pi\)
−0.875456 + 0.483298i \(0.839439\pi\)
\(138\) 0 0
\(139\) − 5.19615i − 0.440732i −0.975417 0.220366i \(-0.929275\pi\)
0.975417 0.220366i \(-0.0707252\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 10.0000 0.839181
\(143\) −7.34847 −0.614510
\(144\) 0 0
\(145\) − 6.92820i − 0.575356i
\(146\) −2.44949 −0.202721
\(147\) 0 0
\(148\) 0 0
\(149\) − 5.65685i − 0.463428i −0.972784 0.231714i \(-0.925567\pi\)
0.972784 0.231714i \(-0.0744333\pi\)
\(150\) 0 0
\(151\) −22.0000 −1.79033 −0.895167 0.445730i \(-0.852944\pi\)
−0.895167 + 0.445730i \(0.852944\pi\)
\(152\) 4.89898 0.397360
\(153\) 0 0
\(154\) 0 0
\(155\) 4.24264i 0.340777i
\(156\) 0 0
\(157\) − 17.3205i − 1.38233i −0.722698 0.691164i \(-0.757098\pi\)
0.722698 0.691164i \(-0.242902\pi\)
\(158\) − 7.07107i − 0.562544i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 10.3923i 0.806599i
\(167\) 7.34847 0.568642 0.284321 0.958729i \(-0.408232\pi\)
0.284321 + 0.958729i \(0.408232\pi\)
\(168\) 0 0
\(169\) −14.0000 −1.07692
\(170\) − 16.9706i − 1.30158i
\(171\) 0 0
\(172\) 0 0
\(173\) −9.79796 −0.744925 −0.372463 0.928047i \(-0.621486\pi\)
−0.372463 + 0.928047i \(0.621486\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.65685i 0.426401i
\(177\) 0 0
\(178\) − 6.92820i − 0.519291i
\(179\) − 9.89949i − 0.739923i −0.929047 0.369961i \(-0.879371\pi\)
0.929047 0.369961i \(-0.120629\pi\)
\(180\) 0 0
\(181\) 15.5885i 1.15868i 0.815086 + 0.579340i \(0.196690\pi\)
−0.815086 + 0.579340i \(0.803310\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −16.0000 −1.17954
\(185\) 2.44949 0.180090
\(186\) 0 0
\(187\) 6.92820i 0.506640i
\(188\) 0 0
\(189\) 0 0
\(190\) −6.00000 −0.435286
\(191\) − 1.41421i − 0.102329i −0.998690 0.0511645i \(-0.983707\pi\)
0.998690 0.0511645i \(-0.0162933\pi\)
\(192\) 0 0
\(193\) 11.0000 0.791797 0.395899 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) −14.6969 −1.05518
\(195\) 0 0
\(196\) 0 0
\(197\) 19.7990i 1.41062i 0.708899 + 0.705310i \(0.249192\pi\)
−0.708899 + 0.705310i \(0.750808\pi\)
\(198\) 0 0
\(199\) 13.8564i 0.982255i 0.871088 + 0.491127i \(0.163415\pi\)
−0.871088 + 0.491127i \(0.836585\pi\)
\(200\) − 2.82843i − 0.200000i
\(201\) 0 0
\(202\) − 24.2487i − 1.70613i
\(203\) 0 0
\(204\) 0 0
\(205\) −18.0000 −1.25717
\(206\) −12.2474 −0.853320
\(207\) 0 0
\(208\) 20.7846i 1.44115i
\(209\) 2.44949 0.169435
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) 2.44949 0.167054
\(216\) 0 0
\(217\) 0 0
\(218\) 1.41421i 0.0957826i
\(219\) 0 0
\(220\) 0 0
\(221\) 25.4558i 1.71235i
\(222\) 0 0
\(223\) 20.7846i 1.39184i 0.718119 + 0.695920i \(0.245003\pi\)
−0.718119 + 0.695920i \(0.754997\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) −26.9444 −1.78836 −0.894181 0.447706i \(-0.852241\pi\)
−0.894181 + 0.447706i \(0.852241\pi\)
\(228\) 0 0
\(229\) 22.5167i 1.48794i 0.668211 + 0.743971i \(0.267060\pi\)
−0.668211 + 0.743971i \(0.732940\pi\)
\(230\) 19.5959 1.29212
\(231\) 0 0
\(232\) 8.00000 0.525226
\(233\) − 9.89949i − 0.648537i −0.945965 0.324269i \(-0.894882\pi\)
0.945965 0.324269i \(-0.105118\pi\)
\(234\) 0 0
\(235\) −30.0000 −1.95698
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 26.8701i − 1.73808i −0.494742 0.869040i \(-0.664738\pi\)
0.494742 0.869040i \(-0.335262\pi\)
\(240\) 0 0
\(241\) 13.8564i 0.892570i 0.894891 + 0.446285i \(0.147253\pi\)
−0.894891 + 0.446285i \(0.852747\pi\)
\(242\) − 12.7279i − 0.818182i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 9.00000 0.572656
\(248\) −4.89898 −0.311086
\(249\) 0 0
\(250\) − 13.8564i − 0.876356i
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) − 15.5563i − 0.976092i
\(255\) 0 0
\(256\) 0 0
\(257\) −2.44949 −0.152795 −0.0763975 0.997077i \(-0.524342\pi\)
−0.0763975 + 0.997077i \(0.524342\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 3.46410i 0.214013i
\(263\) − 14.1421i − 0.872041i −0.899937 0.436021i \(-0.856387\pi\)
0.899937 0.436021i \(-0.143613\pi\)
\(264\) 0 0
\(265\) − 6.92820i − 0.425596i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 17.1464 1.04544 0.522718 0.852506i \(-0.324918\pi\)
0.522718 + 0.852506i \(0.324918\pi\)
\(270\) 0 0
\(271\) − 13.8564i − 0.841717i −0.907126 0.420858i \(-0.861729\pi\)
0.907126 0.420858i \(-0.138271\pi\)
\(272\) 19.5959 1.18818
\(273\) 0 0
\(274\) 16.0000 0.966595
\(275\) − 1.41421i − 0.0852803i
\(276\) 0 0
\(277\) 23.0000 1.38194 0.690968 0.722885i \(-0.257185\pi\)
0.690968 + 0.722885i \(0.257185\pi\)
\(278\) −7.34847 −0.440732
\(279\) 0 0
\(280\) 0 0
\(281\) − 22.6274i − 1.34984i −0.737892 0.674919i \(-0.764178\pi\)
0.737892 0.674919i \(-0.235822\pi\)
\(282\) 0 0
\(283\) − 1.73205i − 0.102960i −0.998674 0.0514799i \(-0.983606\pi\)
0.998674 0.0514799i \(-0.0163938\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 10.3923i 0.614510i
\(287\) 0 0
\(288\) 0 0
\(289\) 7.00000 0.411765
\(290\) −9.79796 −0.575356
\(291\) 0 0
\(292\) 0 0
\(293\) −14.6969 −0.858604 −0.429302 0.903161i \(-0.641240\pi\)
−0.429302 + 0.903161i \(0.641240\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 2.82843i 0.164399i
\(297\) 0 0
\(298\) −8.00000 −0.463428
\(299\) −29.3939 −1.69989
\(300\) 0 0
\(301\) 0 0
\(302\) 31.1127i 1.79033i
\(303\) 0 0
\(304\) − 6.92820i − 0.397360i
\(305\) 8.48528i 0.485866i
\(306\) 0 0
\(307\) − 15.5885i − 0.889680i −0.895610 0.444840i \(-0.853260\pi\)
0.895610 0.444840i \(-0.146740\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 6.00000 0.340777
\(311\) 17.1464 0.972285 0.486142 0.873880i \(-0.338404\pi\)
0.486142 + 0.873880i \(0.338404\pi\)
\(312\) 0 0
\(313\) 12.1244i 0.685309i 0.939461 + 0.342655i \(0.111326\pi\)
−0.939461 + 0.342655i \(0.888674\pi\)
\(314\) −24.4949 −1.38233
\(315\) 0 0
\(316\) 0 0
\(317\) − 14.1421i − 0.794301i −0.917753 0.397151i \(-0.869999\pi\)
0.917753 0.397151i \(-0.130001\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 19.5959 1.09545
\(321\) 0 0
\(322\) 0 0
\(323\) − 8.48528i − 0.472134i
\(324\) 0 0
\(325\) − 5.19615i − 0.288231i
\(326\) 14.1421i 0.783260i
\(327\) 0 0
\(328\) − 20.7846i − 1.14764i
\(329\) 0 0
\(330\) 0 0
\(331\) −31.0000 −1.70391 −0.851957 0.523612i \(-0.824584\pi\)
−0.851957 + 0.523612i \(0.824584\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) − 10.3923i − 0.568642i
\(335\) −26.9444 −1.47213
\(336\) 0 0
\(337\) 23.0000 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 19.7990i 1.07692i
\(339\) 0 0
\(340\) 0 0
\(341\) −2.44949 −0.132647
\(342\) 0 0
\(343\) 0 0
\(344\) 2.82843i 0.152499i
\(345\) 0 0
\(346\) 13.8564i 0.744925i
\(347\) − 31.1127i − 1.67022i −0.550085 0.835109i \(-0.685405\pi\)
0.550085 0.835109i \(-0.314595\pi\)
\(348\) 0 0
\(349\) − 10.3923i − 0.556287i −0.960539 0.278144i \(-0.910281\pi\)
0.960539 0.278144i \(-0.0897191\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 17.1464 0.912612 0.456306 0.889823i \(-0.349172\pi\)
0.456306 + 0.889823i \(0.349172\pi\)
\(354\) 0 0
\(355\) − 17.3205i − 0.919277i
\(356\) 0 0
\(357\) 0 0
\(358\) −14.0000 −0.739923
\(359\) 28.2843i 1.49279i 0.665505 + 0.746393i \(0.268216\pi\)
−0.665505 + 0.746393i \(0.731784\pi\)
\(360\) 0 0
\(361\) 16.0000 0.842105
\(362\) 22.0454 1.15868
\(363\) 0 0
\(364\) 0 0
\(365\) 4.24264i 0.222070i
\(366\) 0 0
\(367\) − 1.73205i − 0.0904123i −0.998978 0.0452062i \(-0.985606\pi\)
0.998978 0.0452062i \(-0.0143945\pi\)
\(368\) 22.6274i 1.17954i
\(369\) 0 0
\(370\) − 3.46410i − 0.180090i
\(371\) 0 0
\(372\) 0 0
\(373\) 29.0000 1.50156 0.750782 0.660551i \(-0.229677\pi\)
0.750782 + 0.660551i \(0.229677\pi\)
\(374\) 9.79796 0.506640
\(375\) 0 0
\(376\) − 34.6410i − 1.78647i
\(377\) 14.6969 0.756931
\(378\) 0 0
\(379\) −7.00000 −0.359566 −0.179783 0.983706i \(-0.557540\pi\)
−0.179783 + 0.983706i \(0.557540\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −2.00000 −0.102329
\(383\) 19.5959 1.00130 0.500652 0.865648i \(-0.333094\pi\)
0.500652 + 0.865648i \(0.333094\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 15.5563i − 0.791797i
\(387\) 0 0
\(388\) 0 0
\(389\) − 26.8701i − 1.36237i −0.732113 0.681183i \(-0.761466\pi\)
0.732113 0.681183i \(-0.238534\pi\)
\(390\) 0 0
\(391\) 27.7128i 1.40150i
\(392\) 0 0
\(393\) 0 0
\(394\) 28.0000 1.41062
\(395\) −12.2474 −0.616236
\(396\) 0 0
\(397\) 1.73205i 0.0869291i 0.999055 + 0.0434646i \(0.0138396\pi\)
−0.999055 + 0.0434646i \(0.986160\pi\)
\(398\) 19.5959 0.982255
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 19.7990i 0.988714i 0.869259 + 0.494357i \(0.164597\pi\)
−0.869259 + 0.494357i \(0.835403\pi\)
\(402\) 0 0
\(403\) −9.00000 −0.448322
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.41421i 0.0701000i
\(408\) 0 0
\(409\) − 32.9090i − 1.62724i −0.581394 0.813622i \(-0.697493\pi\)
0.581394 0.813622i \(-0.302507\pi\)
\(410\) 25.4558i 1.25717i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 18.0000 0.883585
\(416\) 0 0
\(417\) 0 0
\(418\) − 3.46410i − 0.169435i
\(419\) −36.7423 −1.79498 −0.897491 0.441034i \(-0.854612\pi\)
−0.897491 + 0.441034i \(0.854612\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) 31.1127i 1.51454i
\(423\) 0 0
\(424\) 8.00000 0.388514
\(425\) −4.89898 −0.237635
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) − 3.46410i − 0.167054i
\(431\) 15.5563i 0.749323i 0.927162 + 0.374661i \(0.122241\pi\)
−0.927162 + 0.374661i \(0.877759\pi\)
\(432\) 0 0
\(433\) 15.5885i 0.749133i 0.927200 + 0.374567i \(0.122209\pi\)
−0.927200 + 0.374567i \(0.877791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 9.79796 0.468700
\(438\) 0 0
\(439\) 27.7128i 1.32266i 0.750095 + 0.661330i \(0.230008\pi\)
−0.750095 + 0.661330i \(0.769992\pi\)
\(440\) 9.79796 0.467099
\(441\) 0 0
\(442\) 36.0000 1.71235
\(443\) − 39.5980i − 1.88136i −0.339300 0.940678i \(-0.610190\pi\)
0.339300 0.940678i \(-0.389810\pi\)
\(444\) 0 0
\(445\) −12.0000 −0.568855
\(446\) 29.3939 1.39184
\(447\) 0 0
\(448\) 0 0
\(449\) 7.07107i 0.333704i 0.985982 + 0.166852i \(0.0533603\pi\)
−0.985982 + 0.166852i \(0.946640\pi\)
\(450\) 0 0
\(451\) − 10.3923i − 0.489355i
\(452\) 0 0
\(453\) 0 0
\(454\) 38.1051i 1.78836i
\(455\) 0 0
\(456\) 0 0
\(457\) 5.00000 0.233890 0.116945 0.993138i \(-0.462690\pi\)
0.116945 + 0.993138i \(0.462690\pi\)
\(458\) 31.8434 1.48794
\(459\) 0 0
\(460\) 0 0
\(461\) 14.6969 0.684505 0.342252 0.939608i \(-0.388810\pi\)
0.342252 + 0.939608i \(0.388810\pi\)
\(462\) 0 0
\(463\) −13.0000 −0.604161 −0.302081 0.953282i \(-0.597681\pi\)
−0.302081 + 0.953282i \(0.597681\pi\)
\(464\) − 11.3137i − 0.525226i
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) 26.9444 1.24684 0.623419 0.781888i \(-0.285743\pi\)
0.623419 + 0.781888i \(0.285743\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 42.4264i 1.95698i
\(471\) 0 0
\(472\) 13.8564i 0.637793i
\(473\) 1.41421i 0.0650256i
\(474\) 0 0
\(475\) 1.73205i 0.0794719i
\(476\) 0 0
\(477\) 0 0
\(478\) −38.0000 −1.73808
\(479\) −4.89898 −0.223840 −0.111920 0.993717i \(-0.535700\pi\)
−0.111920 + 0.993717i \(0.535700\pi\)
\(480\) 0 0
\(481\) 5.19615i 0.236924i
\(482\) 19.5959 0.892570
\(483\) 0 0
\(484\) 0 0
\(485\) 25.4558i 1.15589i
\(486\) 0 0
\(487\) 17.0000 0.770344 0.385172 0.922845i \(-0.374142\pi\)
0.385172 + 0.922845i \(0.374142\pi\)
\(488\) −9.79796 −0.443533
\(489\) 0 0
\(490\) 0 0
\(491\) 11.3137i 0.510581i 0.966864 + 0.255290i \(0.0821710\pi\)
−0.966864 + 0.255290i \(0.917829\pi\)
\(492\) 0 0
\(493\) − 13.8564i − 0.624061i
\(494\) − 12.7279i − 0.572656i
\(495\) 0 0
\(496\) 6.92820i 0.311086i
\(497\) 0 0
\(498\) 0 0
\(499\) −25.0000 −1.11915 −0.559577 0.828778i \(-0.689036\pi\)
−0.559577 + 0.828778i \(0.689036\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −22.0454 −0.982956 −0.491478 0.870890i \(-0.663543\pi\)
−0.491478 + 0.870890i \(0.663543\pi\)
\(504\) 0 0
\(505\) −42.0000 −1.86898
\(506\) 11.3137i 0.502956i
\(507\) 0 0
\(508\) 0 0
\(509\) −2.44949 −0.108572 −0.0542859 0.998525i \(-0.517288\pi\)
−0.0542859 + 0.998525i \(0.517288\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274i 1.00000i
\(513\) 0 0
\(514\) 3.46410i 0.152795i
\(515\) 21.2132i 0.934765i
\(516\) 0 0
\(517\) − 17.3205i − 0.761755i
\(518\) 0 0
\(519\) 0 0
\(520\) 36.0000 1.57870
\(521\) −4.89898 −0.214628 −0.107314 0.994225i \(-0.534225\pi\)
−0.107314 + 0.994225i \(0.534225\pi\)
\(522\) 0 0
\(523\) 1.73205i 0.0757373i 0.999283 + 0.0378686i \(0.0120568\pi\)
−0.999283 + 0.0378686i \(0.987943\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −20.0000 −0.872041
\(527\) 8.48528i 0.369625i
\(528\) 0 0
\(529\) −9.00000 −0.391304
\(530\) −9.79796 −0.425596
\(531\) 0 0
\(532\) 0 0
\(533\) − 38.1838i − 1.65392i
\(534\) 0 0
\(535\) − 6.92820i − 0.299532i
\(536\) − 31.1127i − 1.34386i
\(537\) 0 0
\(538\) − 24.2487i − 1.04544i
\(539\) 0 0
\(540\) 0 0
\(541\) 17.0000 0.730887 0.365444 0.930834i \(-0.380917\pi\)
0.365444 + 0.930834i \(0.380917\pi\)
\(542\) −19.5959 −0.841717
\(543\) 0 0
\(544\) 0 0
\(545\) 2.44949 0.104925
\(546\) 0 0
\(547\) −10.0000 −0.427569 −0.213785 0.976881i \(-0.568579\pi\)
−0.213785 + 0.976881i \(0.568579\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) −4.89898 −0.208704
\(552\) 0 0
\(553\) 0 0
\(554\) − 32.5269i − 1.38194i
\(555\) 0 0
\(556\) 0 0
\(557\) 15.5563i 0.659144i 0.944131 + 0.329572i \(0.106904\pi\)
−0.944131 + 0.329572i \(0.893096\pi\)
\(558\) 0 0
\(559\) 5.19615i 0.219774i
\(560\) 0 0
\(561\) 0 0
\(562\) −32.0000 −1.34984
\(563\) −26.9444 −1.13557 −0.567785 0.823177i \(-0.692200\pi\)
−0.567785 + 0.823177i \(0.692200\pi\)
\(564\) 0 0
\(565\) 3.46410i 0.145736i
\(566\) −2.44949 −0.102960
\(567\) 0 0
\(568\) 20.0000 0.839181
\(569\) − 1.41421i − 0.0592869i −0.999561 0.0296435i \(-0.990563\pi\)
0.999561 0.0296435i \(-0.00943719\pi\)
\(570\) 0 0
\(571\) 11.0000 0.460336 0.230168 0.973151i \(-0.426072\pi\)
0.230168 + 0.973151i \(0.426072\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 5.65685i − 0.235907i
\(576\) 0 0
\(577\) − 1.73205i − 0.0721062i −0.999350 0.0360531i \(-0.988521\pi\)
0.999350 0.0360531i \(-0.0114785\pi\)
\(578\) − 9.89949i − 0.411765i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 4.00000 0.165663
\(584\) −4.89898 −0.202721
\(585\) 0 0
\(586\) 20.7846i 0.858604i
\(587\) 14.6969 0.606608 0.303304 0.952894i \(-0.401910\pi\)
0.303304 + 0.952894i \(0.401910\pi\)
\(588\) 0 0
\(589\) 3.00000 0.123613
\(590\) − 16.9706i − 0.698667i
\(591\) 0 0
\(592\) 4.00000 0.164399
\(593\) −17.1464 −0.704119 −0.352060 0.935978i \(-0.614519\pi\)
−0.352060 + 0.935978i \(0.614519\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 41.5692i 1.69989i
\(599\) − 5.65685i − 0.231133i −0.993300 0.115566i \(-0.963132\pi\)
0.993300 0.115566i \(-0.0368683\pi\)
\(600\) 0 0
\(601\) − 25.9808i − 1.05978i −0.848067 0.529889i \(-0.822234\pi\)
0.848067 0.529889i \(-0.177766\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −22.0454 −0.896273
\(606\) 0 0
\(607\) − 39.8372i − 1.61694i −0.588537 0.808470i \(-0.700296\pi\)
0.588537 0.808470i \(-0.299704\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 12.0000 0.485866
\(611\) − 63.6396i − 2.57458i
\(612\) 0 0
\(613\) 8.00000 0.323117 0.161558 0.986863i \(-0.448348\pi\)
0.161558 + 0.986863i \(0.448348\pi\)
\(614\) −22.0454 −0.889680
\(615\) 0 0
\(616\) 0 0
\(617\) 24.0416i 0.967880i 0.875101 + 0.483940i \(0.160795\pi\)
−0.875101 + 0.483940i \(0.839205\pi\)
\(618\) 0 0
\(619\) 29.4449i 1.18349i 0.806126 + 0.591744i \(0.201561\pi\)
−0.806126 + 0.591744i \(0.798439\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 24.2487i − 0.972285i
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 17.1464 0.685309
\(627\) 0 0
\(628\) 0 0
\(629\) 4.89898 0.195335
\(630\) 0 0
\(631\) 38.0000 1.51276 0.756378 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) − 14.1421i − 0.562544i
\(633\) 0 0
\(634\) −20.0000 −0.794301
\(635\) −26.9444 −1.06926
\(636\) 0 0
\(637\) 0 0
\(638\) − 5.65685i − 0.223957i
\(639\) 0 0
\(640\) − 27.7128i − 1.09545i
\(641\) − 14.1421i − 0.558581i −0.960207 0.279290i \(-0.909901\pi\)
0.960207 0.279290i \(-0.0900992\pi\)
\(642\) 0 0
\(643\) 25.9808i 1.02458i 0.858812 + 0.512291i \(0.171203\pi\)
−0.858812 + 0.512291i \(0.828797\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −12.0000 −0.472134
\(647\) 17.1464 0.674096 0.337048 0.941488i \(-0.390572\pi\)
0.337048 + 0.941488i \(0.390572\pi\)
\(648\) 0 0
\(649\) 6.92820i 0.271956i
\(650\) −7.34847 −0.288231
\(651\) 0 0
\(652\) 0 0
\(653\) 7.07107i 0.276712i 0.990383 + 0.138356i \(0.0441819\pi\)
−0.990383 + 0.138356i \(0.955818\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) −29.3939 −1.14764
\(657\) 0 0
\(658\) 0 0
\(659\) − 22.6274i − 0.881439i −0.897645 0.440720i \(-0.854723\pi\)
0.897645 0.440720i \(-0.145277\pi\)
\(660\) 0 0
\(661\) 29.4449i 1.14527i 0.819810 + 0.572636i \(0.194079\pi\)
−0.819810 + 0.572636i \(0.805921\pi\)
\(662\) 43.8406i 1.70391i
\(663\) 0 0
\(664\) 20.7846i 0.806599i
\(665\) 0 0
\(666\) 0 0
\(667\) 16.0000 0.619522
\(668\) 0 0
\(669\) 0 0
\(670\) 38.1051i 1.47213i
\(671\) −4.89898 −0.189123
\(672\) 0 0
\(673\) 35.0000 1.34915 0.674575 0.738206i \(-0.264327\pi\)
0.674575 + 0.738206i \(0.264327\pi\)
\(674\) − 32.5269i − 1.25289i
\(675\) 0 0
\(676\) 0 0
\(677\) −9.79796 −0.376566 −0.188283 0.982115i \(-0.560292\pi\)
−0.188283 + 0.982115i \(0.560292\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) − 33.9411i − 1.30158i
\(681\) 0 0
\(682\) 3.46410i 0.132647i
\(683\) − 48.0833i − 1.83985i −0.392089 0.919927i \(-0.628247\pi\)
0.392089 0.919927i \(-0.371753\pi\)
\(684\) 0 0
\(685\) − 27.7128i − 1.05885i
\(686\) 0 0
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 14.6969 0.559909
\(690\) 0 0
\(691\) 43.3013i 1.64726i 0.567129 + 0.823629i \(0.308054\pi\)
−0.567129 + 0.823629i \(0.691946\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −44.0000 −1.67022
\(695\) 12.7279i 0.482798i
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) −14.6969 −0.556287
\(699\) 0 0
\(700\) 0 0
\(701\) − 5.65685i − 0.213656i −0.994277 0.106828i \(-0.965931\pi\)
0.994277 0.106828i \(-0.0340695\pi\)
\(702\) 0 0
\(703\) − 1.73205i − 0.0653255i
\(704\) 11.3137i 0.426401i
\(705\) 0 0
\(706\) − 24.2487i − 0.912612i
\(707\) 0 0
\(708\) 0 0
\(709\) −40.0000 −1.50223 −0.751116 0.660171i \(-0.770484\pi\)
−0.751116 + 0.660171i \(0.770484\pi\)
\(710\) −24.4949 −0.919277
\(711\) 0 0
\(712\) − 13.8564i − 0.519291i
\(713\) −9.79796 −0.366936
\(714\) 0 0
\(715\) 18.0000 0.673162
\(716\) 0 0
\(717\) 0 0
\(718\) 40.0000 1.49279
\(719\) 26.9444 1.00486 0.502428 0.864619i \(-0.332440\pi\)
0.502428 + 0.864619i \(0.332440\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 22.6274i − 0.842105i
\(723\) 0 0
\(724\) 0 0
\(725\) 2.82843i 0.105045i
\(726\) 0 0
\(727\) − 25.9808i − 0.963573i −0.876289 0.481787i \(-0.839988\pi\)
0.876289 0.481787i \(-0.160012\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 6.00000 0.222070
\(731\) 4.89898 0.181195
\(732\) 0 0
\(733\) − 39.8372i − 1.47142i −0.677297 0.735710i \(-0.736849\pi\)
0.677297 0.735710i \(-0.263151\pi\)
\(734\) −2.44949 −0.0904123
\(735\) 0 0
\(736\) 0 0
\(737\) − 15.5563i − 0.573025i
\(738\) 0 0
\(739\) −1.00000 −0.0367856 −0.0183928 0.999831i \(-0.505855\pi\)
−0.0183928 + 0.999831i \(0.505855\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.0416i 0.882002i 0.897507 + 0.441001i \(0.145376\pi\)
−0.897507 + 0.441001i \(0.854624\pi\)
\(744\) 0 0
\(745\) 13.8564i 0.507659i
\(746\) − 41.0122i − 1.50156i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 29.0000 1.05823 0.529113 0.848552i \(-0.322525\pi\)
0.529113 + 0.848552i \(0.322525\pi\)
\(752\) −48.9898 −1.78647
\(753\) 0 0
\(754\) − 20.7846i − 0.756931i
\(755\) 53.8888 1.96121
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 9.89949i 0.359566i
\(759\) 0 0
\(760\) −12.0000 −0.435286
\(761\) −24.4949 −0.887939 −0.443970 0.896042i \(-0.646430\pi\)
−0.443970 + 0.896042i \(0.646430\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) − 27.7128i − 1.00130i
\(767\) 25.4558i 0.919157i
\(768\) 0 0
\(769\) 25.9808i 0.936890i 0.883493 + 0.468445i \(0.155186\pi\)
−0.883493 + 0.468445i \(0.844814\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −26.9444 −0.969122 −0.484561 0.874757i \(-0.661021\pi\)
−0.484561 + 0.874757i \(0.661021\pi\)
\(774\) 0 0
\(775\) − 1.73205i − 0.0622171i
\(776\) −29.3939 −1.05518
\(777\) 0 0
\(778\) −38.0000 −1.36237
\(779\) 12.7279i 0.456025i
\(780\) 0 0
\(781\) 10.0000 0.357828
\(782\) 39.1918 1.40150
\(783\) 0 0
\(784\) 0 0
\(785\) 42.4264i 1.51426i
\(786\) 0 0
\(787\) 45.0333i 1.60526i 0.596474 + 0.802632i \(0.296568\pi\)
−0.596474 + 0.802632i \(0.703432\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 17.3205i 0.616236i
\(791\) 0 0
\(792\) 0 0
\(793\) −18.0000 −0.639199
\(794\) 2.44949 0.0869291
\(795\) 0 0
\(796\) 0 0
\(797\) 29.3939 1.04118 0.520592 0.853805i \(-0.325711\pi\)
0.520592 + 0.853805i \(0.325711\pi\)
\(798\) 0 0
\(799\) −60.0000 −2.12265
\(800\) 0 0
\(801\) 0 0
\(802\) 28.0000 0.988714
\(803\) −2.44949 −0.0864406
\(804\) 0 0
\(805\) 0 0
\(806\) 12.7279i 0.448322i
\(807\) 0 0
\(808\) − 48.4974i − 1.70613i
\(809\) 41.0122i 1.44191i 0.692981 + 0.720956i \(0.256297\pi\)
−0.692981 + 0.720956i \(0.743703\pi\)
\(810\) 0 0
\(811\) − 31.1769i − 1.09477i −0.836881 0.547385i \(-0.815623\pi\)
0.836881 0.547385i \(-0.184377\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 2.00000 0.0701000
\(815\) 24.4949 0.858019
\(816\) 0 0
\(817\) − 1.73205i − 0.0605968i
\(818\) −46.5403 −1.62724
\(819\) 0 0
\(820\) 0 0
\(821\) 24.0416i 0.839059i 0.907742 + 0.419529i \(0.137805\pi\)
−0.907742 + 0.419529i \(0.862195\pi\)
\(822\) 0 0
\(823\) −34.0000 −1.18517 −0.592583 0.805510i \(-0.701892\pi\)
−0.592583 + 0.805510i \(0.701892\pi\)
\(824\) −24.4949 −0.853320
\(825\) 0 0
\(826\) 0 0
\(827\) 7.07107i 0.245885i 0.992414 + 0.122943i \(0.0392331\pi\)
−0.992414 + 0.122943i \(0.960767\pi\)
\(828\) 0 0
\(829\) − 1.73205i − 0.0601566i −0.999548 0.0300783i \(-0.990424\pi\)
0.999548 0.0300783i \(-0.00957567\pi\)
\(830\) − 25.4558i − 0.883585i
\(831\) 0 0
\(832\) 41.5692i 1.44115i
\(833\) 0 0
\(834\) 0 0
\(835\) −18.0000 −0.622916
\(836\) 0 0
\(837\) 0 0
\(838\) 51.9615i 1.79498i
\(839\) 14.6969 0.507395 0.253697 0.967284i \(-0.418353\pi\)
0.253697 + 0.967284i \(0.418353\pi\)
\(840\) 0 0
\(841\) 21.0000 0.724138
\(842\) 1.41421i 0.0487370i
\(843\) 0 0
\(844\) 0 0
\(845\) 34.2929 1.17971
\(846\) 0 0
\(847\) 0 0
\(848\) − 11.3137i − 0.388514i
\(849\) 0 0
\(850\) 6.92820i 0.237635i
\(851\) 5.65685i 0.193914i
\(852\) 0 0
\(853\) 36.3731i 1.24539i 0.782465 + 0.622695i \(0.213962\pi\)
−0.782465 + 0.622695i \(0.786038\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) 39.1918 1.33877 0.669384 0.742917i \(-0.266558\pi\)
0.669384 + 0.742917i \(0.266558\pi\)
\(858\) 0 0
\(859\) 38.1051i 1.30013i 0.759879 + 0.650065i \(0.225258\pi\)
−0.759879 + 0.650065i \(0.774742\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 22.0000 0.749323
\(863\) 41.0122i 1.39607i 0.716063 + 0.698036i \(0.245942\pi\)
−0.716063 + 0.698036i \(0.754058\pi\)
\(864\) 0 0
\(865\) 24.0000 0.816024
\(866\) 22.0454 0.749133
\(867\) 0 0
\(868\) 0 0
\(869\) − 7.07107i − 0.239870i
\(870\) 0 0
\(871\) − 57.1577i − 1.93671i
\(872\) 2.82843i 0.0957826i
\(873\) 0 0
\(874\) − 13.8564i − 0.468700i
\(875\) 0 0
\(876\) 0 0
\(877\) 20.0000 0.675352 0.337676 0.941262i \(-0.390359\pi\)
0.337676 + 0.941262i \(0.390359\pi\)
\(878\) 39.1918 1.32266
\(879\) 0 0
\(880\) − 13.8564i − 0.467099i
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 11.0000 0.370179 0.185090 0.982722i \(-0.440742\pi\)
0.185090 + 0.982722i \(0.440742\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −56.0000 −1.88136
\(887\) 41.6413 1.39818 0.699089 0.715034i \(-0.253589\pi\)
0.699089 + 0.715034i \(0.253589\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 16.9706i 0.568855i
\(891\) 0 0
\(892\) 0 0
\(893\) 21.2132i 0.709873i
\(894\) 0 0
\(895\) 24.2487i 0.810545i
\(896\) 0 0
\(897\) 0 0
\(898\) 10.0000 0.333704
\(899\) 4.89898 0.163390
\(900\) 0 0
\(901\) − 13.8564i − 0.461624i
\(902\) −14.6969 −0.489355
\(903\) 0 0
\(904\) −4.00000 −0.133038
\(905\) − 38.1838i − 1.26927i
\(906\) 0 0
\(907\) 5.00000 0.166022 0.0830111 0.996549i \(-0.473546\pi\)
0.0830111 + 0.996549i \(0.473546\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 53.7401i 1.78049i 0.455483 + 0.890245i \(0.349467\pi\)
−0.455483 + 0.890245i \(0.650533\pi\)
\(912\) 0 0
\(913\) 10.3923i 0.343935i
\(914\) − 7.07107i − 0.233890i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 17.0000 0.560778 0.280389 0.959886i \(-0.409536\pi\)
0.280389 + 0.959886i \(0.409536\pi\)
\(920\) 39.1918 1.29212
\(921\) 0 0
\(922\) − 20.7846i − 0.684505i
\(923\) 36.7423 1.20939
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 18.3848i 0.604161i
\(927\) 0 0
\(928\) 0 0
\(929\) 56.3383 1.84840 0.924199 0.381911i \(-0.124734\pi\)
0.924199 + 0.381911i \(0.124734\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) − 38.1051i − 1.24684i
\(935\) − 16.9706i − 0.554997i
\(936\) 0 0
\(937\) 46.7654i 1.52776i 0.645359 + 0.763879i \(0.276708\pi\)
−0.645359 + 0.763879i \(0.723292\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −48.9898 −1.59702 −0.798511 0.601980i \(-0.794379\pi\)
−0.798511 + 0.601980i \(0.794379\pi\)
\(942\) 0 0
\(943\) − 41.5692i − 1.35368i
\(944\) 19.5959 0.637793
\(945\) 0 0
\(946\) 2.00000 0.0650256
\(947\) 49.4975i 1.60845i 0.594324 + 0.804226i \(0.297420\pi\)
−0.594324 + 0.804226i \(0.702580\pi\)
\(948\) 0 0
\(949\) −9.00000 −0.292152
\(950\) 2.44949 0.0794719
\(951\) 0 0
\(952\) 0 0
\(953\) − 5.65685i − 0.183243i −0.995794 0.0916217i \(-0.970795\pi\)
0.995794 0.0916217i \(-0.0292051\pi\)
\(954\) 0 0
\(955\) 3.46410i 0.112096i
\(956\) 0 0
\(957\) 0 0
\(958\) 6.92820i 0.223840i
\(959\) 0 0
\(960\) 0 0
\(961\) 28.0000 0.903226
\(962\) 7.34847 0.236924
\(963\) 0 0
\(964\) 0 0
\(965\) −26.9444 −0.867371
\(966\) 0 0
\(967\) −13.0000 −0.418052 −0.209026 0.977910i \(-0.567029\pi\)
−0.209026 + 0.977910i \(0.567029\pi\)
\(968\) − 25.4558i − 0.818182i
\(969\) 0 0
\(970\) 36.0000 1.15589
\(971\) −39.1918 −1.25773 −0.628863 0.777516i \(-0.716479\pi\)
−0.628863 + 0.777516i \(0.716479\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 24.0416i − 0.770344i
\(975\) 0 0
\(976\) 13.8564i 0.443533i
\(977\) 7.07107i 0.226224i 0.993582 + 0.113112i \(0.0360818\pi\)
−0.993582 + 0.113112i \(0.963918\pi\)
\(978\) 0 0
\(979\) − 6.92820i − 0.221426i
\(980\) 0 0
\(981\) 0 0
\(982\) 16.0000 0.510581
\(983\) −4.89898 −0.156253 −0.0781266 0.996943i \(-0.524894\pi\)
−0.0781266 + 0.996943i \(0.524894\pi\)
\(984\) 0 0
\(985\) − 48.4974i − 1.54526i
\(986\) −19.5959 −0.624061
\(987\) 0 0
\(988\) 0 0
\(989\) 5.65685i 0.179878i
\(990\) 0 0
\(991\) 35.0000 1.11181 0.555906 0.831245i \(-0.312372\pi\)
0.555906 + 0.831245i \(0.312372\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 33.9411i − 1.07601i
\(996\) 0 0
\(997\) 29.4449i 0.932528i 0.884646 + 0.466264i \(0.154400\pi\)
−0.884646 + 0.466264i \(0.845600\pi\)
\(998\) 35.3553i 1.11915i
\(999\) 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.a.440.1 4
3.2 odd 2 inner 441.2.c.a.440.4 4
4.3 odd 2 7056.2.k.b.881.2 4
7.2 even 3 63.2.p.a.17.1 4
7.3 odd 6 63.2.p.a.26.2 yes 4
7.4 even 3 441.2.p.a.215.2 4
7.5 odd 6 441.2.p.a.80.1 4
7.6 odd 2 inner 441.2.c.a.440.2 4
12.11 even 2 7056.2.k.b.881.3 4
21.2 odd 6 63.2.p.a.17.2 yes 4
21.5 even 6 441.2.p.a.80.2 4
21.11 odd 6 441.2.p.a.215.1 4
21.17 even 6 63.2.p.a.26.1 yes 4
21.20 even 2 inner 441.2.c.a.440.3 4
28.3 even 6 1008.2.bt.b.593.1 4
28.23 odd 6 1008.2.bt.b.17.2 4
28.27 even 2 7056.2.k.b.881.4 4
35.2 odd 12 1575.2.bc.a.899.1 8
35.3 even 12 1575.2.bc.a.1349.3 8
35.9 even 6 1575.2.bk.c.1151.2 4
35.17 even 12 1575.2.bc.a.1349.2 8
35.23 odd 12 1575.2.bc.a.899.4 8
35.24 odd 6 1575.2.bk.c.26.1 4
63.2 odd 6 567.2.s.d.458.1 4
63.16 even 3 567.2.s.d.458.2 4
63.23 odd 6 567.2.i.d.269.2 4
63.31 odd 6 567.2.s.d.26.1 4
63.38 even 6 567.2.i.d.215.2 4
63.52 odd 6 567.2.i.d.215.1 4
63.58 even 3 567.2.i.d.269.1 4
63.59 even 6 567.2.s.d.26.2 4
84.23 even 6 1008.2.bt.b.17.1 4
84.59 odd 6 1008.2.bt.b.593.2 4
84.83 odd 2 7056.2.k.b.881.1 4
105.2 even 12 1575.2.bc.a.899.3 8
105.17 odd 12 1575.2.bc.a.1349.4 8
105.23 even 12 1575.2.bc.a.899.2 8
105.38 odd 12 1575.2.bc.a.1349.1 8
105.44 odd 6 1575.2.bk.c.1151.1 4
105.59 even 6 1575.2.bk.c.26.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.p.a.17.1 4 7.2 even 3
63.2.p.a.17.2 yes 4 21.2 odd 6
63.2.p.a.26.1 yes 4 21.17 even 6
63.2.p.a.26.2 yes 4 7.3 odd 6
441.2.c.a.440.1 4 1.1 even 1 trivial
441.2.c.a.440.2 4 7.6 odd 2 inner
441.2.c.a.440.3 4 21.20 even 2 inner
441.2.c.a.440.4 4 3.2 odd 2 inner
441.2.p.a.80.1 4 7.5 odd 6
441.2.p.a.80.2 4 21.5 even 6
441.2.p.a.215.1 4 21.11 odd 6
441.2.p.a.215.2 4 7.4 even 3
567.2.i.d.215.1 4 63.52 odd 6
567.2.i.d.215.2 4 63.38 even 6
567.2.i.d.269.1 4 63.58 even 3
567.2.i.d.269.2 4 63.23 odd 6
567.2.s.d.26.1 4 63.31 odd 6
567.2.s.d.26.2 4 63.59 even 6
567.2.s.d.458.1 4 63.2 odd 6
567.2.s.d.458.2 4 63.16 even 3
1008.2.bt.b.17.1 4 84.23 even 6
1008.2.bt.b.17.2 4 28.23 odd 6
1008.2.bt.b.593.1 4 28.3 even 6
1008.2.bt.b.593.2 4 84.59 odd 6
1575.2.bc.a.899.1 8 35.2 odd 12
1575.2.bc.a.899.2 8 105.23 even 12
1575.2.bc.a.899.3 8 105.2 even 12
1575.2.bc.a.899.4 8 35.23 odd 12
1575.2.bc.a.1349.1 8 105.38 odd 12
1575.2.bc.a.1349.2 8 35.17 even 12
1575.2.bc.a.1349.3 8 35.3 even 12
1575.2.bc.a.1349.4 8 105.17 odd 12
1575.2.bk.c.26.1 4 35.24 odd 6
1575.2.bk.c.26.2 4 105.59 even 6
1575.2.bk.c.1151.1 4 105.44 odd 6
1575.2.bk.c.1151.2 4 35.9 even 6
7056.2.k.b.881.1 4 84.83 odd 2
7056.2.k.b.881.2 4 4.3 odd 2
7056.2.k.b.881.3 4 12.11 even 2
7056.2.k.b.881.4 4 28.27 even 2