Properties

Label 1400.2.g.d.449.1
Level $1400$
Weight $2$
Character 1400.449
Analytic conductor $11.179$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1400.449
Dual form 1400.2.g.d.449.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-2.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} +5.00000 q^{11} +8.00000i q^{17} +2.00000 q^{19} +2.00000 q^{21} +7.00000i q^{23} -4.00000i q^{27} +3.00000 q^{29} +4.00000 q^{31} -10.0000i q^{33} -1.00000i q^{37} -2.00000 q^{41} -3.00000i q^{43} +6.00000i q^{47} -1.00000 q^{49} +16.0000 q^{51} -10.0000i q^{53} -4.00000i q^{57} +4.00000 q^{59} -6.00000 q^{61} -1.00000i q^{63} +13.0000i q^{67} +14.0000 q^{69} +5.00000 q^{71} -6.00000i q^{73} +5.00000i q^{77} +13.0000 q^{79} -11.0000 q^{81} -16.0000i q^{83} -6.00000i q^{87} -8.00000i q^{93} -12.0000i q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 10q^{11} + 4q^{19} + 4q^{21} + 6q^{29} + 8q^{31} - 4q^{41} - 2q^{49} + 32q^{51} + 8q^{59} - 12q^{61} + 28q^{69} + 10q^{71} + 26q^{79} - 22q^{81} - 10q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 2.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.00000i 1.94029i 0.242536 + 0.970143i \(0.422021\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 7.00000i 1.45960i 0.683660 + 0.729800i \(0.260387\pi\)
−0.683660 + 0.729800i \(0.739613\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 4.00000i − 0.769800i
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) − 10.0000i − 1.74078i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 1.00000i − 0.164399i −0.996616 0.0821995i \(-0.973806\pi\)
0.996616 0.0821995i \(-0.0261945\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) − 3.00000i − 0.457496i −0.973486 0.228748i \(-0.926537\pi\)
0.973486 0.228748i \(-0.0734631\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 16.0000 2.24045
\(52\) 0 0
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.00000i − 0.529813i
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) − 1.00000i − 0.125988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 13.0000i 1.58820i 0.607785 + 0.794101i \(0.292058\pi\)
−0.607785 + 0.794101i \(0.707942\pi\)
\(68\) 0 0
\(69\) 14.0000 1.68540
\(70\) 0 0
\(71\) 5.00000 0.593391 0.296695 0.954972i \(-0.404115\pi\)
0.296695 + 0.954972i \(0.404115\pi\)
\(72\) 0 0
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.00000i 0.569803i
\(78\) 0 0
\(79\) 13.0000 1.46261 0.731307 0.682048i \(-0.238911\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) − 16.0000i − 1.75623i −0.478451 0.878114i \(-0.658802\pi\)
0.478451 0.878114i \(-0.341198\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 6.00000i − 0.643268i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 12.0000i − 1.21842i −0.793011 0.609208i \(-0.791488\pi\)
0.793011 0.609208i \(-0.208512\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) 2.00000i 0.197066i 0.995134 + 0.0985329i \(0.0314150\pi\)
−0.995134 + 0.0985329i \(0.968585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) − 1.00000i − 0.0940721i −0.998893 0.0470360i \(-0.985022\pi\)
0.998893 0.0470360i \(-0.0149776\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 4.00000i 0.360668i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 13.0000i 1.15356i 0.816898 + 0.576782i \(0.195692\pi\)
−0.816898 + 0.576782i \(0.804308\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 2.00000i 0.173422i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 18.0000i − 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.00000i 0.164957i
\(148\) 0 0
\(149\) 17.0000 1.39269 0.696347 0.717705i \(-0.254807\pi\)
0.696347 + 0.717705i \(0.254807\pi\)
\(150\) 0 0
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) − 8.00000i − 0.646762i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) 0 0
\(159\) −20.0000 −1.58610
\(160\) 0 0
\(161\) −7.00000 −0.551677
\(162\) 0 0
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) 22.0000i 1.67263i 0.548250 + 0.836315i \(0.315294\pi\)
−0.548250 + 0.836315i \(0.684706\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 8.00000i − 0.601317i
\(178\) 0 0
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) 12.0000i 0.887066i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 40.0000i 2.92509i
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) − 17.0000i − 1.22369i −0.790979 0.611843i \(-0.790428\pi\)
0.790979 0.611843i \(-0.209572\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 3.00000i − 0.213741i −0.994273 0.106871i \(-0.965917\pi\)
0.994273 0.106871i \(-0.0340831\pi\)
\(198\) 0 0
\(199\) −26.0000 −1.84309 −0.921546 0.388270i \(-0.873073\pi\)
−0.921546 + 0.388270i \(0.873073\pi\)
\(200\) 0 0
\(201\) 26.0000 1.83390
\(202\) 0 0
\(203\) 3.00000i 0.210559i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 7.00000i − 0.486534i
\(208\) 0 0
\(209\) 10.0000 0.691714
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) − 10.0000i − 0.685189i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.00000i 0.271538i
\(218\) 0 0
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 14.0000i 0.937509i 0.883328 + 0.468755i \(0.155297\pi\)
−0.883328 + 0.468755i \(0.844703\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.0000i 0.929213i 0.885517 + 0.464606i \(0.153804\pi\)
−0.885517 + 0.464606i \(0.846196\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 10.0000 0.657952
\(232\) 0 0
\(233\) 29.0000i 1.89985i 0.312473 + 0.949927i \(0.398843\pi\)
−0.312473 + 0.949927i \(0.601157\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 26.0000i − 1.68888i
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) 10.0000i 0.641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −32.0000 −2.02792
\(250\) 0 0
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) 35.0000i 2.20043i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 12.0000i − 0.748539i −0.927320 0.374270i \(-0.877893\pi\)
0.927320 0.374270i \(-0.122107\pi\)
\(258\) 0 0
\(259\) 1.00000 0.0621370
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) 0 0
\(263\) 5.00000i 0.308313i 0.988046 + 0.154157i \(0.0492660\pi\)
−0.988046 + 0.154157i \(0.950734\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000i 1.32185i 0.750451 + 0.660926i \(0.229836\pi\)
−0.750451 + 0.660926i \(0.770164\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 9.00000 0.536895 0.268447 0.963294i \(-0.413489\pi\)
0.268447 + 0.963294i \(0.413489\pi\)
\(282\) 0 0
\(283\) − 10.0000i − 0.594438i −0.954809 0.297219i \(-0.903941\pi\)
0.954809 0.297219i \(-0.0960592\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 2.00000i − 0.118056i
\(288\) 0 0
\(289\) −47.0000 −2.76471
\(290\) 0 0
\(291\) −24.0000 −1.40690
\(292\) 0 0
\(293\) 12.0000i 0.701047i 0.936554 + 0.350524i \(0.113996\pi\)
−0.936554 + 0.350524i \(0.886004\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 20.0000i − 1.16052i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 3.00000 0.172917
\(302\) 0 0
\(303\) 36.0000i 2.06815i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) − 14.0000i − 0.791327i −0.918396 0.395663i \(-0.870515\pi\)
0.918396 0.395663i \(-0.129485\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.00000i 0.0561656i 0.999606 + 0.0280828i \(0.00894021\pi\)
−0.999606 + 0.0280828i \(0.991060\pi\)
\(318\) 0 0
\(319\) 15.0000 0.839839
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) 16.0000i 0.890264i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 10.0000i 0.553001i
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 3.00000 0.164895 0.0824475 0.996595i \(-0.473726\pi\)
0.0824475 + 0.996595i \(0.473726\pi\)
\(332\) 0 0
\(333\) 1.00000i 0.0547997i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) 0 0
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 20.0000 1.08306
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 29.0000i − 1.55680i −0.627768 0.778401i \(-0.716031\pi\)
0.627768 0.778401i \(-0.283969\pi\)
\(348\) 0 0
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 20.0000i 1.06449i 0.846590 + 0.532246i \(0.178652\pi\)
−0.846590 + 0.532246i \(0.821348\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 16.0000i 0.846810i
\(358\) 0 0
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) − 28.0000i − 1.46962i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 22.0000i − 1.14839i −0.818718 0.574195i \(-0.805315\pi\)
0.818718 0.574195i \(-0.194685\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 10.0000 0.519174
\(372\) 0 0
\(373\) − 1.00000i − 0.0517780i −0.999665 0.0258890i \(-0.991758\pi\)
0.999665 0.0258890i \(-0.00824165\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −15.0000 −0.770498 −0.385249 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) 0 0
\(381\) 26.0000 1.33202
\(382\) 0 0
\(383\) − 12.0000i − 0.613171i −0.951843 0.306586i \(-0.900813\pi\)
0.951843 0.306586i \(-0.0991866\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.00000i 0.152499i
\(388\) 0 0
\(389\) −5.00000 −0.253510 −0.126755 0.991934i \(-0.540456\pi\)
−0.126755 + 0.991934i \(0.540456\pi\)
\(390\) 0 0
\(391\) −56.0000 −2.83204
\(392\) 0 0
\(393\) − 8.00000i − 0.403547i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 18.0000i − 0.903394i −0.892171 0.451697i \(-0.850819\pi\)
0.892171 0.451697i \(-0.149181\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) 25.0000 1.24844 0.624220 0.781248i \(-0.285417\pi\)
0.624220 + 0.781248i \(0.285417\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 5.00000i − 0.247841i
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) −36.0000 −1.77575
\(412\) 0 0
\(413\) 4.00000i 0.196827i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 4.00000i − 0.195881i
\(418\) 0 0
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) −33.0000 −1.60832 −0.804161 0.594412i \(-0.797385\pi\)
−0.804161 + 0.594412i \(0.797385\pi\)
\(422\) 0 0
\(423\) − 6.00000i − 0.291730i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 6.00000i − 0.290360i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 20.0000 0.963366 0.481683 0.876346i \(-0.340026\pi\)
0.481683 + 0.876346i \(0.340026\pi\)
\(432\) 0 0
\(433\) − 24.0000i − 1.15337i −0.816968 0.576683i \(-0.804347\pi\)
0.816968 0.576683i \(-0.195653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 14.0000i 0.669711i
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 4.00000i 0.190046i 0.995475 + 0.0950229i \(0.0302924\pi\)
−0.995475 + 0.0950229i \(0.969708\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 34.0000i − 1.60814i
\(448\) 0 0
\(449\) −31.0000 −1.46298 −0.731490 0.681852i \(-0.761175\pi\)
−0.731490 + 0.681852i \(0.761175\pi\)
\(450\) 0 0
\(451\) −10.0000 −0.470882
\(452\) 0 0
\(453\) 38.0000i 1.78540i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 37.0000i 1.73079i 0.501093 + 0.865393i \(0.332931\pi\)
−0.501093 + 0.865393i \(0.667069\pi\)
\(458\) 0 0
\(459\) 32.0000 1.49363
\(460\) 0 0
\(461\) −40.0000 −1.86299 −0.931493 0.363760i \(-0.881493\pi\)
−0.931493 + 0.363760i \(0.881493\pi\)
\(462\) 0 0
\(463\) − 24.0000i − 1.11537i −0.830051 0.557687i \(-0.811689\pi\)
0.830051 0.557687i \(-0.188311\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 14.0000i − 0.647843i −0.946084 0.323921i \(-0.894999\pi\)
0.946084 0.323921i \(-0.105001\pi\)
\(468\) 0 0
\(469\) −13.0000 −0.600284
\(470\) 0 0
\(471\) 20.0000 0.921551
\(472\) 0 0
\(473\) − 15.0000i − 0.689701i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 10.0000i 0.457869i
\(478\) 0 0
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 14.0000i 0.637022i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 13.0000i − 0.589086i −0.955638 0.294543i \(-0.904833\pi\)
0.955638 0.294543i \(-0.0951675\pi\)
\(488\) 0 0
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) −11.0000 −0.496423 −0.248212 0.968706i \(-0.579843\pi\)
−0.248212 + 0.968706i \(0.579843\pi\)
\(492\) 0 0
\(493\) 24.0000i 1.08091i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.00000i 0.224281i
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) 0 0
\(503\) − 38.0000i − 1.69434i −0.531325 0.847168i \(-0.678306\pi\)
0.531325 0.847168i \(-0.321694\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 26.0000i − 1.15470i
\(508\) 0 0
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) − 8.00000i − 0.353209i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 30.0000i 1.31940i
\(518\) 0 0
\(519\) 44.0000 1.93139
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 32.0000i 1.39394i
\(528\) 0 0
\(529\) −26.0000 −1.13043
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 48.0000i − 2.07135i
\(538\) 0 0
\(539\) −5.00000 −0.215365
\(540\) 0 0
\(541\) −13.0000 −0.558914 −0.279457 0.960158i \(-0.590154\pi\)
−0.279457 + 0.960158i \(0.590154\pi\)
\(542\) 0 0
\(543\) 44.0000i 1.88822i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 17.0000i 0.726868i 0.931620 + 0.363434i \(0.118396\pi\)
−0.931620 + 0.363434i \(0.881604\pi\)
\(548\) 0 0
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 13.0000i 0.552816i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.00000i 0.381342i 0.981654 + 0.190671i \(0.0610664\pi\)
−0.981654 + 0.190671i \(0.938934\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 80.0000 3.37760
\(562\) 0 0
\(563\) 6.00000i 0.252870i 0.991975 + 0.126435i \(0.0403535\pi\)
−0.991975 + 0.126435i \(0.959647\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 11.0000i − 0.461957i
\(568\) 0 0
\(569\) −13.0000 −0.544988 −0.272494 0.962157i \(-0.587849\pi\)
−0.272494 + 0.962157i \(0.587849\pi\)
\(570\) 0 0
\(571\) 19.0000 0.795125 0.397563 0.917575i \(-0.369856\pi\)
0.397563 + 0.917575i \(0.369856\pi\)
\(572\) 0 0
\(573\) − 24.0000i − 1.00261i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 0 0
\(579\) −34.0000 −1.41299
\(580\) 0 0
\(581\) 16.0000 0.663792
\(582\) 0 0
\(583\) − 50.0000i − 2.07079i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 30.0000i − 1.23823i −0.785299 0.619116i \(-0.787491\pi\)
0.785299 0.619116i \(-0.212509\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) − 12.0000i − 0.492781i −0.969171 0.246390i \(-0.920755\pi\)
0.969171 0.246390i \(-0.0792446\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 52.0000i 2.12822i
\(598\) 0 0
\(599\) 5.00000 0.204294 0.102147 0.994769i \(-0.467429\pi\)
0.102147 + 0.994769i \(0.467429\pi\)
\(600\) 0 0
\(601\) 40.0000 1.63163 0.815817 0.578310i \(-0.196288\pi\)
0.815817 + 0.578310i \(0.196288\pi\)
\(602\) 0 0
\(603\) − 13.0000i − 0.529401i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 48.0000i 1.94826i 0.225989 + 0.974130i \(0.427439\pi\)
−0.225989 + 0.974130i \(0.572561\pi\)
\(608\) 0 0
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 35.0000i − 1.41364i −0.707395 0.706818i \(-0.750130\pi\)
0.707395 0.706818i \(-0.249870\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 9.00000i − 0.362326i −0.983453 0.181163i \(-0.942014\pi\)
0.983453 0.181163i \(-0.0579862\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 28.0000 1.12360
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 20.0000i − 0.798723i
\(628\) 0 0
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) 0 0
\(633\) − 40.0000i − 1.58986i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −5.00000 −0.197797
\(640\) 0 0
\(641\) −7.00000 −0.276483 −0.138242 0.990399i \(-0.544145\pi\)
−0.138242 + 0.990399i \(0.544145\pi\)
\(642\) 0 0
\(643\) − 16.0000i − 0.630978i −0.948929 0.315489i \(-0.897831\pi\)
0.948929 0.315489i \(-0.102169\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 26.0000i − 1.02217i −0.859532 0.511083i \(-0.829245\pi\)
0.859532 0.511083i \(-0.170755\pi\)
\(648\) 0 0
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) 0 0
\(653\) − 18.0000i − 0.704394i −0.935926 0.352197i \(-0.885435\pi\)
0.935926 0.352197i \(-0.114565\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 30.0000 1.16686 0.583432 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 21.0000i 0.813123i
\(668\) 0 0
\(669\) 28.0000 1.08254
\(670\) 0 0
\(671\) −30.0000 −1.15814
\(672\) 0 0
\(673\) 30.0000i 1.15642i 0.815890 + 0.578208i \(0.196248\pi\)
−0.815890 + 0.578208i \(0.803752\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.00000i 0.230599i 0.993331 + 0.115299i \(0.0367827\pi\)
−0.993331 + 0.115299i \(0.963217\pi\)
\(678\) 0 0
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) 28.0000 1.07296
\(682\) 0 0
\(683\) − 27.0000i − 1.03313i −0.856249 0.516563i \(-0.827211\pi\)
0.856249 0.516563i \(-0.172789\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 28.0000i 1.06827i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 14.0000 0.532585 0.266293 0.963892i \(-0.414201\pi\)
0.266293 + 0.963892i \(0.414201\pi\)
\(692\) 0 0
\(693\) − 5.00000i − 0.189934i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 16.0000i − 0.606043i
\(698\) 0 0
\(699\) 58.0000 2.19376
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) − 2.00000i − 0.0754314i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 18.0000i − 0.676960i
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) −13.0000 −0.487538
\(712\) 0 0
\(713\) 28.0000i 1.04861i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 48.0000i 1.79259i
\(718\) 0 0
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) −2.00000 −0.0744839
\(722\) 0 0
\(723\) 44.0000i 1.63638i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 40.0000i 1.48352i 0.670667 + 0.741759i \(0.266008\pi\)
−0.670667 + 0.741759i \(0.733992\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 28.0000i 1.03420i 0.855924 + 0.517102i \(0.172989\pi\)
−0.855924 + 0.517102i \(0.827011\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 65.0000i 2.39431i
\(738\) 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
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 16.0000i 0.585409i
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 12.0000 0.437886 0.218943 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) 0 0
\(753\) 8.00000i 0.291536i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 15.0000i 0.545184i 0.962130 + 0.272592i \(0.0878810\pi\)
−0.962130 + 0.272592i \(0.912119\pi\)
\(758\) 0 0
\(759\) 70.0000 2.54084
\(760\) 0 0
\(761\) 48.0000 1.74000 0.869999 0.493053i \(-0.164119\pi\)
0.869999 + 0.493053i \(0.164119\pi\)
\(762\) 0 0
\(763\) − 5.00000i − 0.181012i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) −24.0000 −0.864339
\(772\) 0 0
\(773\) 36.0000i 1.29483i 0.762138 + 0.647415i \(0.224150\pi\)
−0.762138 + 0.647415i \(0.775850\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 2.00000i − 0.0717496i
\(778\) 0 0
\(779\) −4.00000 −0.143315
\(780\) 0 0
\(781\) 25.0000 0.894570
\(782\) 0 0
\(783\) − 12.0000i − 0.428845i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 18.0000i − 0.641631i −0.947142 0.320815i \(-0.896043\pi\)
0.947142 0.320815i \(-0.103957\pi\)
\(788\) 0 0
\(789\) 10.0000 0.356009
\(790\) 0 0
\(791\) 1.00000 0.0355559
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 44.0000i − 1.55856i −0.626676 0.779280i \(-0.715585\pi\)
0.626676 0.779280i \(-0.284415\pi\)
\(798\) 0 0
\(799\) −48.0000 −1.69812
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 30.0000i − 1.05868i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 48.0000i − 1.68968i
\(808\) 0 0
\(809\) 29.0000 1.01959 0.509793 0.860297i \(-0.329722\pi\)
0.509793 + 0.860297i \(0.329722\pi\)
\(810\) 0 0
\(811\) −50.0000 −1.75574 −0.877869 0.478901i \(-0.841035\pi\)
−0.877869 + 0.478901i \(0.841035\pi\)
\(812\) 0 0
\(813\) 16.0000i 0.561144i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 6.00000i − 0.209913i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −54.0000 −1.88461 −0.942306 0.334751i \(-0.891348\pi\)
−0.942306 + 0.334751i \(0.891348\pi\)
\(822\) 0 0
\(823\) − 53.0000i − 1.84746i −0.383040 0.923732i \(-0.625123\pi\)
0.383040 0.923732i \(-0.374877\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 11.0000i 0.382507i 0.981541 + 0.191254i \(0.0612553\pi\)
−0.981541 + 0.191254i \(0.938745\pi\)
\(828\) 0 0
\(829\) −20.0000 −0.694629 −0.347314 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) 0 0
\(831\) 44.0000 1.52634
\(832\) 0 0
\(833\) − 8.00000i − 0.277184i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 16.0000i − 0.553041i
\(838\) 0 0
\(839\) −46.0000 −1.58810 −0.794048 0.607855i \(-0.792030\pi\)
−0.794048 + 0.607855i \(0.792030\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) − 18.0000i − 0.619953i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 14.0000i 0.481046i
\(848\) 0 0
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 7.00000 0.239957
\(852\) 0 0
\(853\) − 28.0000i − 0.958702i −0.877623 0.479351i \(-0.840872\pi\)
0.877623 0.479351i \(-0.159128\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 54.0000i − 1.84460i −0.386469 0.922302i \(-0.626305\pi\)
0.386469 0.922302i \(-0.373695\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) 0 0
\(863\) − 43.0000i − 1.46374i −0.681446 0.731869i \(-0.738649\pi\)
0.681446 0.731869i \(-0.261351\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 94.0000i 3.19241i
\(868\) 0 0
\(869\) 65.0000 2.20497
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 12.0000i 0.406138i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 18.0000i − 0.607817i −0.952701 0.303908i \(-0.901708\pi\)
0.952701 0.303908i \(-0.0982917\pi\)
\(878\) 0 0
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) 20.0000 0.673817 0.336909 0.941537i \(-0.390619\pi\)
0.336909 + 0.941537i \(0.390619\pi\)
\(882\) 0 0
\(883\) 35.0000i 1.17784i 0.808190 + 0.588922i \(0.200447\pi\)
−0.808190 + 0.588922i \(0.799553\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 6.00000i 0.201460i 0.994914 + 0.100730i \(0.0321179\pi\)
−0.994914 + 0.100730i \(0.967882\pi\)
\(888\) 0 0
\(889\) −13.0000 −0.436006
\(890\) 0 0
\(891\) −55.0000 −1.84257
\(892\) 0 0
\(893\) 12.0000i 0.401565i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) 80.0000 2.66519
\(902\) 0 0
\(903\) − 6.00000i − 0.199667i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 28.0000i − 0.929725i −0.885383 0.464862i \(-0.846104\pi\)
0.885383 0.464862i \(-0.153896\pi\)
\(908\) 0 0
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 27.0000 0.894550 0.447275 0.894397i \(-0.352395\pi\)
0.447275 + 0.894397i \(0.352395\pi\)
\(912\) 0 0
\(913\) − 80.0000i − 2.64761i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.00000i 0.132092i
\(918\) 0 0
\(919\) −15.0000 −0.494804 −0.247402 0.968913i \(-0.579577\pi\)
−0.247402 + 0.968913i \(0.579577\pi\)
\(920\) 0 0
\(921\) −24.0000 −0.790827
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 2.00000i − 0.0656886i
\(928\) 0 0
\(929\) 22.0000 0.721797 0.360898 0.932605i \(-0.382470\pi\)
0.360898 + 0.932605i \(0.382470\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 0 0
\(933\) 8.00000i 0.261908i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 22.0000i − 0.718709i −0.933201 0.359354i \(-0.882997\pi\)
0.933201 0.359354i \(-0.117003\pi\)
\(938\) 0 0
\(939\) −28.0000 −0.913745
\(940\) 0 0
\(941\) −12.0000 −0.391189 −0.195594 0.980685i \(-0.562664\pi\)
−0.195594 + 0.980685i \(0.562664\pi\)
\(942\) 0 0
\(943\) − 14.0000i − 0.455903i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 20.0000i − 0.649913i −0.945729 0.324956i \(-0.894650\pi\)
0.945729 0.324956i \(-0.105350\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) 0 0
\(953\) − 5.00000i − 0.161966i −0.996715 0.0809829i \(-0.974194\pi\)
0.996715 0.0809829i \(-0.0258059\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 30.0000i − 0.969762i
\(958\) 0 0
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 12.0000i 0.386695i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 56.0000i 1.80084i 0.435023 + 0.900419i \(0.356740\pi\)
−0.435023 + 0.900419i \(0.643260\pi\)
\(968\) 0 0
\(969\) 32.0000 1.02799
\(970\) 0 0
\(971\) −26.0000 −0.834380 −0.417190 0.908819i \(-0.636985\pi\)
−0.417190 + 0.908819i \(0.636985\pi\)
\(972\) 0 0
\(973\) 2.00000i 0.0641171i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 51.0000i 1.63163i 0.578310 + 0.815817i \(0.303713\pi\)
−0.578310 + 0.815817i \(0.696287\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 5.00000 0.159638
\(982\) 0 0
\(983\) 12.0000i 0.382741i 0.981518 + 0.191370i \(0.0612931\pi\)
−0.981518 + 0.191370i \(0.938707\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 12.0000i 0.381964i
\(988\) 0 0
\(989\) 21.0000 0.667761
\(990\) 0 0
\(991\) −13.0000 −0.412959 −0.206479 0.978451i \(-0.566201\pi\)
−0.206479 + 0.978451i \(0.566201\pi\)
\(992\) 0 0
\(993\) − 6.00000i − 0.190404i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 2.00000i 0.0633406i 0.999498 + 0.0316703i \(0.0100827\pi\)
−0.999498 + 0.0316703i \(0.989917\pi\)
\(998\) 0 0
\(999\) −4.00000 −0.126554
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 1400.2.g.d.449.1 2
4.3 odd 2 2800.2.g.d.449.2 2
5.2 odd 4 1400.2.a.b.1.1 1
5.3 odd 4 1400.2.a.m.1.1 yes 1
5.4 even 2 inner 1400.2.g.d.449.2 2
20.3 even 4 2800.2.a.d.1.1 1
20.7 even 4 2800.2.a.bc.1.1 1
20.19 odd 2 2800.2.g.d.449.1 2
35.13 even 4 9800.2.a.l.1.1 1
35.27 even 4 9800.2.a.bo.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1400.2.a.b.1.1 1 5.2 odd 4
1400.2.a.m.1.1 yes 1 5.3 odd 4
1400.2.g.d.449.1 2 1.1 even 1 trivial
1400.2.g.d.449.2 2 5.4 even 2 inner
2800.2.a.d.1.1 1 20.3 even 4
2800.2.a.bc.1.1 1 20.7 even 4
2800.2.g.d.449.1 2 20.19 odd 2
2800.2.g.d.449.2 2 4.3 odd 2
9800.2.a.l.1.1 1 35.13 even 4
9800.2.a.bo.1.1 1 35.27 even 4