Properties

Label 1800.4.f.t.649.2
Level $1800$
Weight $4$
Character 1800.649
Analytic conductor $106.203$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(106.203438010\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1800.649
Dual form 1800.4.f.t.649.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{7} +O(q^{10})\) \(q+2.00000i q^{7} +34.0000 q^{11} +68.0000i q^{13} +38.0000i q^{17} -4.00000 q^{19} +152.000i q^{23} -46.0000 q^{29} -260.000 q^{31} -312.000i q^{37} -48.0000 q^{41} +200.000i q^{43} -104.000i q^{47} +339.000 q^{49} -414.000i q^{53} -2.00000 q^{59} -38.0000 q^{61} -244.000i q^{67} -708.000 q^{71} +378.000i q^{73} +68.0000i q^{77} +852.000 q^{79} +844.000i q^{83} -1380.00 q^{89} -136.000 q^{91} +514.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 68 q^{11} - 8 q^{19} - 92 q^{29} - 520 q^{31} - 96 q^{41} + 678 q^{49} - 4 q^{59} - 76 q^{61} - 1416 q^{71} + 1704 q^{79} - 2760 q^{89} - 272 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.107990i 0.998541 + 0.0539949i \(0.0171955\pi\)
−0.998541 + 0.0539949i \(0.982805\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 34.0000 0.931944 0.465972 0.884799i \(-0.345705\pi\)
0.465972 + 0.884799i \(0.345705\pi\)
\(12\) 0 0
\(13\) 68.0000i 1.45075i 0.688352 + 0.725377i \(0.258335\pi\)
−0.688352 + 0.725377i \(0.741665\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 38.0000i 0.542138i 0.962560 + 0.271069i \(0.0873772\pi\)
−0.962560 + 0.271069i \(0.912623\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.0482980 −0.0241490 0.999708i \(-0.507688\pi\)
−0.0241490 + 0.999708i \(0.507688\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 152.000i 1.37801i 0.724757 + 0.689004i \(0.241952\pi\)
−0.724757 + 0.689004i \(0.758048\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −46.0000 −0.294551 −0.147276 0.989095i \(-0.547050\pi\)
−0.147276 + 0.989095i \(0.547050\pi\)
\(30\) 0 0
\(31\) −260.000 −1.50637 −0.753184 0.657810i \(-0.771483\pi\)
−0.753184 + 0.657810i \(0.771483\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 312.000i − 1.38628i −0.720801 0.693142i \(-0.756226\pi\)
0.720801 0.693142i \(-0.243774\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −48.0000 −0.182838 −0.0914188 0.995813i \(-0.529140\pi\)
−0.0914188 + 0.995813i \(0.529140\pi\)
\(42\) 0 0
\(43\) 200.000i 0.709296i 0.935000 + 0.354648i \(0.115399\pi\)
−0.935000 + 0.354648i \(0.884601\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 104.000i − 0.322765i −0.986892 0.161383i \(-0.948405\pi\)
0.986892 0.161383i \(-0.0515953\pi\)
\(48\) 0 0
\(49\) 339.000 0.988338
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 414.000i − 1.07297i −0.843911 0.536484i \(-0.819752\pi\)
0.843911 0.536484i \(-0.180248\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.00000 −0.00441318 −0.00220659 0.999998i \(-0.500702\pi\)
−0.00220659 + 0.999998i \(0.500702\pi\)
\(60\) 0 0
\(61\) −38.0000 −0.0797607 −0.0398803 0.999204i \(-0.512698\pi\)
−0.0398803 + 0.999204i \(0.512698\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 244.000i − 0.444916i −0.974942 0.222458i \(-0.928592\pi\)
0.974942 0.222458i \(-0.0714080\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −708.000 −1.18344 −0.591719 0.806144i \(-0.701551\pi\)
−0.591719 + 0.806144i \(0.701551\pi\)
\(72\) 0 0
\(73\) 378.000i 0.606049i 0.952983 + 0.303024i \(0.0979963\pi\)
−0.952983 + 0.303024i \(0.902004\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 68.0000i 0.100641i
\(78\) 0 0
\(79\) 852.000 1.21339 0.606693 0.794936i \(-0.292496\pi\)
0.606693 + 0.794936i \(0.292496\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 844.000i 1.11616i 0.829788 + 0.558079i \(0.188461\pi\)
−0.829788 + 0.558079i \(0.811539\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1380.00 −1.64359 −0.821796 0.569782i \(-0.807028\pi\)
−0.821796 + 0.569782i \(0.807028\pi\)
\(90\) 0 0
\(91\) −136.000 −0.156667
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 514.000i 0.538029i 0.963136 + 0.269014i \(0.0866979\pi\)
−0.963136 + 0.269014i \(0.913302\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 702.000 0.691600 0.345800 0.938308i \(-0.387608\pi\)
0.345800 + 0.938308i \(0.387608\pi\)
\(102\) 0 0
\(103\) − 898.000i − 0.859054i −0.903054 0.429527i \(-0.858680\pi\)
0.903054 0.429527i \(-0.141320\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 876.000i − 0.791459i −0.918367 0.395730i \(-0.870492\pi\)
0.918367 0.395730i \(-0.129508\pi\)
\(108\) 0 0
\(109\) −602.000 −0.529001 −0.264501 0.964386i \(-0.585207\pi\)
−0.264501 + 0.964386i \(0.585207\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1350.00i 1.12387i 0.827181 + 0.561935i \(0.189943\pi\)
−0.827181 + 0.561935i \(0.810057\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −76.0000 −0.0585455
\(120\) 0 0
\(121\) −175.000 −0.131480
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 366.000i − 0.255726i −0.991792 0.127863i \(-0.959188\pi\)
0.991792 0.127863i \(-0.0408118\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −498.000 −0.332141 −0.166070 0.986114i \(-0.553108\pi\)
−0.166070 + 0.986114i \(0.553108\pi\)
\(132\) 0 0
\(133\) − 8.00000i − 0.00521570i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2026.00i 1.26345i 0.775192 + 0.631726i \(0.217653\pi\)
−0.775192 + 0.631726i \(0.782347\pi\)
\(138\) 0 0
\(139\) −2460.00 −1.50111 −0.750556 0.660807i \(-0.770214\pi\)
−0.750556 + 0.660807i \(0.770214\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2312.00i 1.35202i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3362.00 −1.84850 −0.924248 0.381794i \(-0.875306\pi\)
−0.924248 + 0.381794i \(0.875306\pi\)
\(150\) 0 0
\(151\) 2096.00 1.12960 0.564802 0.825227i \(-0.308953\pi\)
0.564802 + 0.825227i \(0.308953\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2092.00i 1.06344i 0.846921 + 0.531719i \(0.178454\pi\)
−0.846921 + 0.531719i \(0.821546\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −304.000 −0.148811
\(162\) 0 0
\(163\) − 244.000i − 0.117249i −0.998280 0.0586244i \(-0.981329\pi\)
0.998280 0.0586244i \(-0.0186714\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 2064.00i − 0.956390i −0.878254 0.478195i \(-0.841291\pi\)
0.878254 0.478195i \(-0.158709\pi\)
\(168\) 0 0
\(169\) −2427.00 −1.10469
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1258.00i 0.552855i 0.961035 + 0.276428i \(0.0891506\pi\)
−0.961035 + 0.276428i \(0.910849\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3986.00 −1.66440 −0.832200 0.554475i \(-0.812919\pi\)
−0.832200 + 0.554475i \(0.812919\pi\)
\(180\) 0 0
\(181\) 2570.00 1.05540 0.527698 0.849432i \(-0.323055\pi\)
0.527698 + 0.849432i \(0.323055\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1292.00i 0.505243i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4684.00 −1.77446 −0.887231 0.461325i \(-0.847374\pi\)
−0.887231 + 0.461325i \(0.847374\pi\)
\(192\) 0 0
\(193\) − 214.000i − 0.0798138i −0.999203 0.0399069i \(-0.987294\pi\)
0.999203 0.0399069i \(-0.0127061\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 3014.00i − 1.09004i −0.838422 0.545022i \(-0.816521\pi\)
0.838422 0.545022i \(-0.183479\pi\)
\(198\) 0 0
\(199\) 1792.00 0.638349 0.319175 0.947696i \(-0.396594\pi\)
0.319175 + 0.947696i \(0.396594\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 92.0000i − 0.0318085i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −136.000 −0.0450111
\(210\) 0 0
\(211\) −4540.00 −1.48126 −0.740631 0.671911i \(-0.765474\pi\)
−0.740631 + 0.671911i \(0.765474\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 520.000i − 0.162672i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2584.00 −0.786510
\(222\) 0 0
\(223\) − 6506.00i − 1.95369i −0.213937 0.976847i \(-0.568629\pi\)
0.213937 0.976847i \(-0.431371\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3696.00i 1.08067i 0.841450 + 0.540335i \(0.181703\pi\)
−0.841450 + 0.540335i \(0.818297\pi\)
\(228\) 0 0
\(229\) 3386.00 0.977088 0.488544 0.872539i \(-0.337528\pi\)
0.488544 + 0.872539i \(0.337528\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3306.00i 0.929542i 0.885431 + 0.464771i \(0.153863\pi\)
−0.885431 + 0.464771i \(0.846137\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4188.00 −1.13347 −0.566735 0.823900i \(-0.691794\pi\)
−0.566735 + 0.823900i \(0.691794\pi\)
\(240\) 0 0
\(241\) 5462.00 1.45991 0.729955 0.683495i \(-0.239541\pi\)
0.729955 + 0.683495i \(0.239541\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 272.000i − 0.0700686i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3366.00 0.846454 0.423227 0.906024i \(-0.360897\pi\)
0.423227 + 0.906024i \(0.360897\pi\)
\(252\) 0 0
\(253\) 5168.00i 1.28423i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 1158.00i − 0.281066i −0.990076 0.140533i \(-0.955118\pi\)
0.990076 0.140533i \(-0.0448817\pi\)
\(258\) 0 0
\(259\) 624.000 0.149705
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8304.00i 1.94695i 0.228804 + 0.973473i \(0.426519\pi\)
−0.228804 + 0.973473i \(0.573481\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7478.00 −1.69495 −0.847475 0.530835i \(-0.821878\pi\)
−0.847475 + 0.530835i \(0.821878\pi\)
\(270\) 0 0
\(271\) −6792.00 −1.52245 −0.761226 0.648486i \(-0.775402\pi\)
−0.761226 + 0.648486i \(0.775402\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 2296.00i − 0.498026i −0.968500 0.249013i \(-0.919894\pi\)
0.968500 0.249013i \(-0.0801062\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3980.00 −0.844936 −0.422468 0.906378i \(-0.638836\pi\)
−0.422468 + 0.906378i \(0.638836\pi\)
\(282\) 0 0
\(283\) 1972.00i 0.414216i 0.978318 + 0.207108i \(0.0664052\pi\)
−0.978318 + 0.207108i \(0.933595\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 96.0000i − 0.0197446i
\(288\) 0 0
\(289\) 3469.00 0.706086
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 9254.00i 1.84513i 0.385836 + 0.922567i \(0.373913\pi\)
−0.385836 + 0.922567i \(0.626087\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −10336.0 −1.99915
\(300\) 0 0
\(301\) −400.000 −0.0765967
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 5888.00i − 1.09461i −0.836933 0.547306i \(-0.815653\pi\)
0.836933 0.547306i \(-0.184347\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4604.00 0.839450 0.419725 0.907651i \(-0.362127\pi\)
0.419725 + 0.907651i \(0.362127\pi\)
\(312\) 0 0
\(313\) 8026.00i 1.44938i 0.689074 + 0.724691i \(0.258017\pi\)
−0.689074 + 0.724691i \(0.741983\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 2838.00i − 0.502833i −0.967879 0.251416i \(-0.919104\pi\)
0.967879 0.251416i \(-0.0808963\pi\)
\(318\) 0 0
\(319\) −1564.00 −0.274505
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 152.000i − 0.0261842i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 208.000 0.0348554
\(330\) 0 0
\(331\) −1020.00 −0.169378 −0.0846892 0.996407i \(-0.526990\pi\)
−0.0846892 + 0.996407i \(0.526990\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 814.000i 0.131577i 0.997834 + 0.0657884i \(0.0209562\pi\)
−0.997834 + 0.0657884i \(0.979044\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −8840.00 −1.40385
\(342\) 0 0
\(343\) 1364.00i 0.214720i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4544.00i 0.702982i 0.936191 + 0.351491i \(0.114325\pi\)
−0.936191 + 0.351491i \(0.885675\pi\)
\(348\) 0 0
\(349\) −6978.00 −1.07027 −0.535134 0.844767i \(-0.679739\pi\)
−0.535134 + 0.844767i \(0.679739\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2818.00i 0.424892i 0.977173 + 0.212446i \(0.0681430\pi\)
−0.977173 + 0.212446i \(0.931857\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 744.000 0.109378 0.0546892 0.998503i \(-0.482583\pi\)
0.0546892 + 0.998503i \(0.482583\pi\)
\(360\) 0 0
\(361\) −6843.00 −0.997667
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 6454.00i − 0.917973i −0.888443 0.458986i \(-0.848213\pi\)
0.888443 0.458986i \(-0.151787\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 828.000 0.115870
\(372\) 0 0
\(373\) 5900.00i 0.819009i 0.912308 + 0.409505i \(0.134298\pi\)
−0.912308 + 0.409505i \(0.865702\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 3128.00i − 0.427321i
\(378\) 0 0
\(379\) 11876.0 1.60958 0.804788 0.593563i \(-0.202279\pi\)
0.804788 + 0.593563i \(0.202279\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 552.000i − 0.0736446i −0.999322 0.0368223i \(-0.988276\pi\)
0.999322 0.0368223i \(-0.0117236\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1722.00 −0.224444 −0.112222 0.993683i \(-0.535797\pi\)
−0.112222 + 0.993683i \(0.535797\pi\)
\(390\) 0 0
\(391\) −5776.00 −0.747071
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 4576.00i 0.578496i 0.957254 + 0.289248i \(0.0934052\pi\)
−0.957254 + 0.289248i \(0.906595\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2892.00 −0.360149 −0.180074 0.983653i \(-0.557634\pi\)
−0.180074 + 0.983653i \(0.557634\pi\)
\(402\) 0 0
\(403\) − 17680.0i − 2.18537i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 10608.0i − 1.29194i
\(408\) 0 0
\(409\) 230.000 0.0278063 0.0139031 0.999903i \(-0.495574\pi\)
0.0139031 + 0.999903i \(0.495574\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 4.00000i 0 0.000476579i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −15438.0 −1.79999 −0.899995 0.435901i \(-0.856430\pi\)
−0.899995 + 0.435901i \(0.856430\pi\)
\(420\) 0 0
\(421\) 12294.0 1.42321 0.711607 0.702578i \(-0.247968\pi\)
0.711607 + 0.702578i \(0.247968\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 76.0000i − 0.00861334i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −17488.0 −1.95445 −0.977224 0.212209i \(-0.931934\pi\)
−0.977224 + 0.212209i \(0.931934\pi\)
\(432\) 0 0
\(433\) 8698.00i 0.965356i 0.875798 + 0.482678i \(0.160336\pi\)
−0.875798 + 0.482678i \(0.839664\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 608.000i − 0.0665551i
\(438\) 0 0
\(439\) −8536.00 −0.928021 −0.464010 0.885830i \(-0.653590\pi\)
−0.464010 + 0.885830i \(0.653590\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 8712.00i − 0.934356i −0.884163 0.467178i \(-0.845271\pi\)
0.884163 0.467178i \(-0.154729\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −5484.00 −0.576405 −0.288203 0.957569i \(-0.593058\pi\)
−0.288203 + 0.957569i \(0.593058\pi\)
\(450\) 0 0
\(451\) −1632.00 −0.170394
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 19402.0i 1.98597i 0.118250 + 0.992984i \(0.462272\pi\)
−0.118250 + 0.992984i \(0.537728\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 13578.0 1.37178 0.685890 0.727705i \(-0.259413\pi\)
0.685890 + 0.727705i \(0.259413\pi\)
\(462\) 0 0
\(463\) − 6222.00i − 0.624537i −0.949994 0.312269i \(-0.898911\pi\)
0.949994 0.312269i \(-0.101089\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 15260.0i 1.51210i 0.654516 + 0.756048i \(0.272872\pi\)
−0.654516 + 0.756048i \(0.727128\pi\)
\(468\) 0 0
\(469\) 488.000 0.0480464
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6800.00i 0.661024i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9812.00 0.935953 0.467977 0.883741i \(-0.344983\pi\)
0.467977 + 0.883741i \(0.344983\pi\)
\(480\) 0 0
\(481\) 21216.0 2.01116
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 7226.00i − 0.672364i −0.941797 0.336182i \(-0.890864\pi\)
0.941797 0.336182i \(-0.109136\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 6750.00 0.620414 0.310207 0.950669i \(-0.399602\pi\)
0.310207 + 0.950669i \(0.399602\pi\)
\(492\) 0 0
\(493\) − 1748.00i − 0.159688i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 1416.00i − 0.127799i
\(498\) 0 0
\(499\) 4156.00 0.372842 0.186421 0.982470i \(-0.440311\pi\)
0.186421 + 0.982470i \(0.440311\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 14088.0i 1.24881i 0.781100 + 0.624406i \(0.214659\pi\)
−0.781100 + 0.624406i \(0.785341\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 16970.0 1.47776 0.738882 0.673835i \(-0.235354\pi\)
0.738882 + 0.673835i \(0.235354\pi\)
\(510\) 0 0
\(511\) −756.000 −0.0654471
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 3536.00i − 0.300799i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 8500.00 0.714763 0.357382 0.933958i \(-0.383670\pi\)
0.357382 + 0.933958i \(0.383670\pi\)
\(522\) 0 0
\(523\) − 20620.0i − 1.72400i −0.506912 0.861998i \(-0.669213\pi\)
0.506912 0.861998i \(-0.330787\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 9880.00i − 0.816660i
\(528\) 0 0
\(529\) −10937.0 −0.898907
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 3264.00i − 0.265252i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 11526.0 0.921076
\(540\) 0 0
\(541\) 5314.00 0.422304 0.211152 0.977453i \(-0.432278\pi\)
0.211152 + 0.977453i \(0.432278\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 24104.0i 1.88412i 0.335447 + 0.942059i \(0.391113\pi\)
−0.335447 + 0.942059i \(0.608887\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 184.000 0.0142262
\(552\) 0 0
\(553\) 1704.00i 0.131033i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 23582.0i 1.79390i 0.442134 + 0.896949i \(0.354222\pi\)
−0.442134 + 0.896949i \(0.645778\pi\)
\(558\) 0 0
\(559\) −13600.0 −1.02901
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 2680.00i − 0.200619i −0.994956 0.100310i \(-0.968017\pi\)
0.994956 0.100310i \(-0.0319833\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −25004.0 −1.84222 −0.921109 0.389304i \(-0.872715\pi\)
−0.921109 + 0.389304i \(0.872715\pi\)
\(570\) 0 0
\(571\) −11180.0 −0.819384 −0.409692 0.912224i \(-0.634364\pi\)
−0.409692 + 0.912224i \(0.634364\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 15862.0i − 1.14444i −0.820099 0.572222i \(-0.806082\pi\)
0.820099 0.572222i \(-0.193918\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1688.00 −0.120534
\(582\) 0 0
\(583\) − 14076.0i − 0.999946i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 15036.0i − 1.05724i −0.848857 0.528622i \(-0.822709\pi\)
0.848857 0.528622i \(-0.177291\pi\)
\(588\) 0 0
\(589\) 1040.00 0.0727546
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 12786.0i − 0.885427i −0.896663 0.442713i \(-0.854016\pi\)
0.896663 0.442713i \(-0.145984\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 13464.0 0.918404 0.459202 0.888332i \(-0.348135\pi\)
0.459202 + 0.888332i \(0.348135\pi\)
\(600\) 0 0
\(601\) 8518.00 0.578131 0.289065 0.957309i \(-0.406656\pi\)
0.289065 + 0.957309i \(0.406656\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 11082.0i − 0.741029i −0.928827 0.370514i \(-0.879181\pi\)
0.928827 0.370514i \(-0.120819\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7072.00 0.468253
\(612\) 0 0
\(613\) 26568.0i 1.75052i 0.483649 + 0.875262i \(0.339311\pi\)
−0.483649 + 0.875262i \(0.660689\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 3282.00i − 0.214146i −0.994251 0.107073i \(-0.965852\pi\)
0.994251 0.107073i \(-0.0341479\pi\)
\(618\) 0 0
\(619\) 2308.00 0.149865 0.0749324 0.997189i \(-0.476126\pi\)
0.0749324 + 0.997189i \(0.476126\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 2760.00i − 0.177491i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 11856.0 0.751558
\(630\) 0 0
\(631\) −24572.0 −1.55023 −0.775116 0.631819i \(-0.782308\pi\)
−0.775116 + 0.631819i \(0.782308\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 23052.0i 1.43384i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2136.00 −0.131618 −0.0658088 0.997832i \(-0.520963\pi\)
−0.0658088 + 0.997832i \(0.520963\pi\)
\(642\) 0 0
\(643\) 5508.00i 0.337814i 0.985632 + 0.168907i \(0.0540237\pi\)
−0.985632 + 0.168907i \(0.945976\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4536.00i 0.275624i 0.990458 + 0.137812i \(0.0440069\pi\)
−0.990458 + 0.137812i \(0.955993\pi\)
\(648\) 0 0
\(649\) −68.0000 −0.00411284
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 27914.0i − 1.67283i −0.548095 0.836416i \(-0.684647\pi\)
0.548095 0.836416i \(-0.315353\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 22842.0 1.35022 0.675112 0.737715i \(-0.264095\pi\)
0.675112 + 0.737715i \(0.264095\pi\)
\(660\) 0 0
\(661\) 16458.0 0.968445 0.484222 0.874945i \(-0.339103\pi\)
0.484222 + 0.874945i \(0.339103\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 6992.00i − 0.405894i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1292.00 −0.0743325
\(672\) 0 0
\(673\) 16050.0i 0.919290i 0.888103 + 0.459645i \(0.152023\pi\)
−0.888103 + 0.459645i \(0.847977\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5314.00i 0.301674i 0.988559 + 0.150837i \(0.0481969\pi\)
−0.988559 + 0.150837i \(0.951803\pi\)
\(678\) 0 0
\(679\) −1028.00 −0.0581016
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 15876.0i − 0.889426i −0.895673 0.444713i \(-0.853306\pi\)
0.895673 0.444713i \(-0.146694\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 28152.0 1.55661
\(690\) 0 0
\(691\) −13372.0 −0.736172 −0.368086 0.929792i \(-0.619987\pi\)
−0.368086 + 0.929792i \(0.619987\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 1824.00i − 0.0991233i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 3230.00 0.174031 0.0870153 0.996207i \(-0.472267\pi\)
0.0870153 + 0.996207i \(0.472267\pi\)
\(702\) 0 0
\(703\) 1248.00i 0.0669548i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1404.00i 0.0746858i
\(708\) 0 0
\(709\) 6154.00 0.325978 0.162989 0.986628i \(-0.447887\pi\)
0.162989 + 0.986628i \(0.447887\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 39520.0i − 2.07579i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 20264.0 1.05107 0.525535 0.850772i \(-0.323865\pi\)
0.525535 + 0.850772i \(0.323865\pi\)
\(720\) 0 0
\(721\) 1796.00 0.0927691
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 25354.0i − 1.29344i −0.762729 0.646718i \(-0.776141\pi\)
0.762729 0.646718i \(-0.223859\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −7600.00 −0.384536
\(732\) 0 0
\(733\) 13344.0i 0.672404i 0.941790 + 0.336202i \(0.109142\pi\)
−0.941790 + 0.336202i \(0.890858\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 8296.00i − 0.414636i
\(738\) 0 0
\(739\) 28452.0 1.41627 0.708135 0.706077i \(-0.249537\pi\)
0.708135 + 0.706077i \(0.249537\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 5784.00i − 0.285591i −0.989752 0.142796i \(-0.954391\pi\)
0.989752 0.142796i \(-0.0456092\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1752.00 0.0854695
\(750\) 0 0
\(751\) 852.000 0.0413980 0.0206990 0.999786i \(-0.493411\pi\)
0.0206990 + 0.999786i \(0.493411\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 5704.00i − 0.273864i −0.990580 0.136932i \(-0.956276\pi\)
0.990580 0.136932i \(-0.0437243\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 24828.0 1.18267 0.591337 0.806425i \(-0.298600\pi\)
0.591337 + 0.806425i \(0.298600\pi\)
\(762\) 0 0
\(763\) − 1204.00i − 0.0571268i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 136.000i − 0.00640245i
\(768\) 0 0
\(769\) 13298.0 0.623587 0.311793 0.950150i \(-0.399070\pi\)
0.311793 + 0.950150i \(0.399070\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 642.000i − 0.0298721i −0.999888 0.0149361i \(-0.995246\pi\)
0.999888 0.0149361i \(-0.00475447\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 192.000 0.00883070
\(780\) 0 0
\(781\) −24072.0 −1.10290
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 20236.0i − 0.916564i −0.888807 0.458282i \(-0.848465\pi\)
0.888807 0.458282i \(-0.151535\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2700.00 −0.121367
\(792\) 0 0
\(793\) − 2584.00i − 0.115713i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 11562.0i − 0.513861i −0.966430 0.256930i \(-0.917289\pi\)
0.966430 0.256930i \(-0.0827111\pi\)
\(798\) 0 0
\(799\) 3952.00 0.174983
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 12852.0i 0.564804i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18984.0 0.825021 0.412510 0.910953i \(-0.364652\pi\)
0.412510 + 0.910953i \(0.364652\pi\)
\(810\) 0 0
\(811\) −2332.00 −0.100971 −0.0504856 0.998725i \(-0.516077\pi\)
−0.0504856 + 0.998725i \(0.516077\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 800.000i − 0.0342576i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 19126.0 0.813035 0.406518 0.913643i \(-0.366743\pi\)
0.406518 + 0.913643i \(0.366743\pi\)
\(822\) 0 0
\(823\) − 37102.0i − 1.57144i −0.618583 0.785720i \(-0.712293\pi\)
0.618583 0.785720i \(-0.287707\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 11304.0i 0.475307i 0.971350 + 0.237653i \(0.0763782\pi\)
−0.971350 + 0.237653i \(0.923622\pi\)
\(828\) 0 0
\(829\) 974.000 0.0408063 0.0204031 0.999792i \(-0.493505\pi\)
0.0204031 + 0.999792i \(0.493505\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 12882.0i 0.535816i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 16480.0 0.678132 0.339066 0.940763i \(-0.389889\pi\)
0.339066 + 0.940763i \(0.389889\pi\)
\(840\) 0 0
\(841\) −22273.0 −0.913240
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 350.000i − 0.0141985i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 47424.0 1.91031
\(852\) 0 0
\(853\) − 11192.0i − 0.449246i −0.974446 0.224623i \(-0.927885\pi\)
0.974446 0.224623i \(-0.0721150\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 34278.0i 1.36629i 0.730281 + 0.683147i \(0.239389\pi\)
−0.730281 + 0.683147i \(0.760611\pi\)
\(858\) 0 0
\(859\) 14020.0 0.556876 0.278438 0.960454i \(-0.410183\pi\)
0.278438 + 0.960454i \(0.410183\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 30528.0i − 1.20415i −0.798438 0.602077i \(-0.794340\pi\)
0.798438 0.602077i \(-0.205660\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 28968.0 1.13081
\(870\) 0 0
\(871\) 16592.0 0.645463
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2932.00i 0.112892i 0.998406 + 0.0564462i \(0.0179769\pi\)
−0.998406 + 0.0564462i \(0.982023\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −7116.00 −0.272127 −0.136064 0.990700i \(-0.543445\pi\)
−0.136064 + 0.990700i \(0.543445\pi\)
\(882\) 0 0
\(883\) 35140.0i 1.33925i 0.742701 + 0.669624i \(0.233545\pi\)
−0.742701 + 0.669624i \(0.766455\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 20296.0i 0.768290i 0.923273 + 0.384145i \(0.125504\pi\)
−0.923273 + 0.384145i \(0.874496\pi\)
\(888\) 0 0
\(889\) 732.000 0.0276159
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 416.000i 0.0155889i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 11960.0 0.443702
\(900\) 0 0
\(901\) 15732.0 0.581697
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 19512.0i 0.714317i 0.934044 + 0.357158i \(0.116254\pi\)
−0.934044 + 0.357158i \(0.883746\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 16720.0 0.608077 0.304039 0.952660i \(-0.401665\pi\)
0.304039 + 0.952660i \(0.401665\pi\)
\(912\) 0 0
\(913\) 28696.0i 1.04020i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 996.000i − 0.0358678i
\(918\) 0 0
\(919\) −7340.00 −0.263465 −0.131732 0.991285i \(-0.542054\pi\)
−0.131732 + 0.991285i \(0.542054\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 48144.0i − 1.71688i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 48932.0 1.72810 0.864051 0.503404i \(-0.167919\pi\)
0.864051 + 0.503404i \(0.167919\pi\)
\(930\) 0 0
\(931\) −1356.00 −0.0477348
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 30298.0i 1.05634i 0.849138 + 0.528171i \(0.177122\pi\)
−0.849138 + 0.528171i \(0.822878\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −8414.00 −0.291486 −0.145743 0.989322i \(-0.546557\pi\)
−0.145743 + 0.989322i \(0.546557\pi\)
\(942\) 0 0
\(943\) − 7296.00i − 0.251952i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 23912.0i − 0.820523i −0.911968 0.410262i \(-0.865437\pi\)
0.911968 0.410262i \(-0.134563\pi\)
\(948\) 0 0
\(949\) −25704.0 −0.879228
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 22866.0i − 0.777232i −0.921400 0.388616i \(-0.872953\pi\)
0.921400 0.388616i \(-0.127047\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4052.00 −0.136440
\(960\) 0 0
\(961\) 37809.0 1.26914
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 738.000i 0.0245424i 0.999925 + 0.0122712i \(0.00390614\pi\)
−0.999925 + 0.0122712i \(0.996094\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −44098.0 −1.45744 −0.728719 0.684813i \(-0.759884\pi\)
−0.728719 + 0.684813i \(0.759884\pi\)
\(972\) 0 0
\(973\) − 4920.00i − 0.162105i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 34426.0i − 1.12731i −0.826009 0.563657i \(-0.809394\pi\)
0.826009 0.563657i \(-0.190606\pi\)
\(978\) 0 0
\(979\) −46920.0 −1.53174
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 30216.0i − 0.980408i −0.871608 0.490204i \(-0.836922\pi\)
0.871608 0.490204i \(-0.163078\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −30400.0 −0.977415
\(990\) 0 0
\(991\) 4592.00 0.147194 0.0735972 0.997288i \(-0.476552\pi\)
0.0735972 + 0.997288i \(0.476552\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 24276.0i − 0.771142i −0.922678 0.385571i \(-0.874004\pi\)
0.922678 0.385571i \(-0.125996\pi\)
\(998\) 0 0
\(999\) 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 1800.4.f.t.649.2 2
3.2 odd 2 1800.4.f.f.649.2 2
5.2 odd 4 1800.4.a.r.1.1 1
5.3 odd 4 360.4.a.d.1.1 1
5.4 even 2 inner 1800.4.f.t.649.1 2
15.2 even 4 1800.4.a.q.1.1 1
15.8 even 4 360.4.a.k.1.1 yes 1
15.14 odd 2 1800.4.f.f.649.1 2
20.3 even 4 720.4.a.g.1.1 1
60.23 odd 4 720.4.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.4.a.d.1.1 1 5.3 odd 4
360.4.a.k.1.1 yes 1 15.8 even 4
720.4.a.g.1.1 1 20.3 even 4
720.4.a.x.1.1 1 60.23 odd 4
1800.4.a.q.1.1 1 15.2 even 4
1800.4.a.r.1.1 1 5.2 odd 4
1800.4.f.f.649.1 2 15.14 odd 2
1800.4.f.f.649.2 2 3.2 odd 2
1800.4.f.t.649.1 2 5.4 even 2 inner
1800.4.f.t.649.2 2 1.1 even 1 trivial