Properties

Label 1400.2.g.i.449.3
Level $1400$
Weight $2$
Character 1400.449
Analytic conductor $11.179$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{33})\)
Defining polynomial: \(x^{4} + 17 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.3
Root \(2.37228i\) of defining polynomial
Character \(\chi\) \(=\) 1400.449
Dual form 1400.2.g.i.449.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.37228i q^{3} +1.00000i q^{7} -2.62772 q^{9} +O(q^{10})\) \(q+2.37228i q^{3} +1.00000i q^{7} -2.62772 q^{9} +6.37228 q^{11} +4.37228i q^{13} +0.372281i q^{17} +4.74456 q^{19} -2.37228 q^{21} -4.74456i q^{23} +0.883156i q^{27} +4.37228 q^{29} -8.00000 q^{31} +15.1168i q^{33} +2.00000i q^{37} -10.3723 q^{39} +6.74456 q^{41} -8.74456i q^{43} +7.11684i q^{47} -1.00000 q^{49} -0.883156 q^{51} +10.7446i q^{53} +11.2554i q^{57} -8.00000 q^{59} -2.74456 q^{61} -2.62772i q^{63} +4.00000i q^{67} +11.2554 q^{69} +8.00000 q^{71} -6.00000i q^{73} +6.37228i q^{77} -15.1168 q^{79} -9.97825 q^{81} -9.48913i q^{83} +10.3723i q^{87} -14.7446 q^{89} -4.37228 q^{91} -18.9783i q^{93} +9.86141i q^{97} -16.7446 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 22q^{9} + O(q^{10}) \) \( 4q - 22q^{9} + 14q^{11} - 4q^{19} + 2q^{21} + 6q^{29} - 32q^{31} - 30q^{39} + 4q^{41} - 4q^{49} - 38q^{51} - 32q^{59} + 12q^{61} + 68q^{69} + 32q^{71} - 26q^{79} + 52q^{81} - 36q^{89} - 6q^{91} - 44q^{99} + O(q^{100}) \)

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
\(3\) 2.37228i 1.36964i 0.728714 + 0.684819i \(0.240119\pi\)
−0.728714 + 0.684819i \(0.759881\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −2.62772 −0.875906
\(10\) 0 0
\(11\) 6.37228 1.92132 0.960658 0.277736i \(-0.0895839\pi\)
0.960658 + 0.277736i \(0.0895839\pi\)
\(12\) 0 0
\(13\) 4.37228i 1.21265i 0.795216 + 0.606326i \(0.207357\pi\)
−0.795216 + 0.606326i \(0.792643\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.372281i 0.0902915i 0.998980 + 0.0451457i \(0.0143752\pi\)
−0.998980 + 0.0451457i \(0.985625\pi\)
\(18\) 0 0
\(19\) 4.74456 1.08848 0.544239 0.838930i \(-0.316819\pi\)
0.544239 + 0.838930i \(0.316819\pi\)
\(20\) 0 0
\(21\) −2.37228 −0.517674
\(22\) 0 0
\(23\) − 4.74456i − 0.989310i −0.869090 0.494655i \(-0.835294\pi\)
0.869090 0.494655i \(-0.164706\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0.883156i 0.169963i
\(28\) 0 0
\(29\) 4.37228 0.811912 0.405956 0.913893i \(-0.366939\pi\)
0.405956 + 0.913893i \(0.366939\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 15.1168i 2.63150i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) −10.3723 −1.66089
\(40\) 0 0
\(41\) 6.74456 1.05332 0.526662 0.850075i \(-0.323443\pi\)
0.526662 + 0.850075i \(0.323443\pi\)
\(42\) 0 0
\(43\) − 8.74456i − 1.33353i −0.745267 0.666767i \(-0.767678\pi\)
0.745267 0.666767i \(-0.232322\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.11684i 1.03810i 0.854744 + 0.519049i \(0.173714\pi\)
−0.854744 + 0.519049i \(0.826286\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −0.883156 −0.123667
\(52\) 0 0
\(53\) 10.7446i 1.47588i 0.674867 + 0.737940i \(0.264201\pi\)
−0.674867 + 0.737940i \(0.735799\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 11.2554i 1.49082i
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −2.74456 −0.351405 −0.175703 0.984443i \(-0.556220\pi\)
−0.175703 + 0.984443i \(0.556220\pi\)
\(62\) 0 0
\(63\) − 2.62772i − 0.331061i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 11.2554 1.35500
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 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
\(77\) 6.37228i 0.726189i
\(78\) 0 0
\(79\) −15.1168 −1.70078 −0.850389 0.526155i \(-0.823633\pi\)
−0.850389 + 0.526155i \(0.823633\pi\)
\(80\) 0 0
\(81\) −9.97825 −1.10869
\(82\) 0 0
\(83\) − 9.48913i − 1.04157i −0.853689 0.520783i \(-0.825640\pi\)
0.853689 0.520783i \(-0.174360\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 10.3723i 1.11203i
\(88\) 0 0
\(89\) −14.7446 −1.56292 −0.781460 0.623955i \(-0.785525\pi\)
−0.781460 + 0.623955i \(0.785525\pi\)
\(90\) 0 0
\(91\) −4.37228 −0.458340
\(92\) 0 0
\(93\) − 18.9783i − 1.96795i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.86141i 1.00127i 0.865657 + 0.500637i \(0.166901\pi\)
−0.865657 + 0.500637i \(0.833099\pi\)
\(98\) 0 0
\(99\) −16.7446 −1.68289
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) − 5.62772i − 0.554516i −0.960796 0.277258i \(-0.910574\pi\)
0.960796 0.277258i \(-0.0894256\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 8.74456i − 0.845369i −0.906277 0.422684i \(-0.861088\pi\)
0.906277 0.422684i \(-0.138912\pi\)
\(108\) 0 0
\(109\) −0.372281 −0.0356581 −0.0178290 0.999841i \(-0.505675\pi\)
−0.0178290 + 0.999841i \(0.505675\pi\)
\(110\) 0 0
\(111\) −4.74456 −0.450334
\(112\) 0 0
\(113\) 2.00000i 0.188144i 0.995565 + 0.0940721i \(0.0299884\pi\)
−0.995565 + 0.0940721i \(0.970012\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 11.4891i − 1.06217i
\(118\) 0 0
\(119\) −0.372281 −0.0341270
\(120\) 0 0
\(121\) 29.6060 2.69145
\(122\) 0 0
\(123\) 16.0000i 1.44267i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) 0 0
\(129\) 20.7446 1.82646
\(130\) 0 0
\(131\) −4.74456 −0.414534 −0.207267 0.978284i \(-0.566457\pi\)
−0.207267 + 0.978284i \(0.566457\pi\)
\(132\) 0 0
\(133\) 4.74456i 0.411406i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 14.7446i − 1.25971i −0.776712 0.629857i \(-0.783114\pi\)
0.776712 0.629857i \(-0.216886\pi\)
\(138\) 0 0
\(139\) 4.74456 0.402429 0.201214 0.979547i \(-0.435511\pi\)
0.201214 + 0.979547i \(0.435511\pi\)
\(140\) 0 0
\(141\) −16.8832 −1.42182
\(142\) 0 0
\(143\) 27.8614i 2.32989i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 2.37228i − 0.195662i
\(148\) 0 0
\(149\) −15.4891 −1.26892 −0.634459 0.772956i \(-0.718777\pi\)
−0.634459 + 0.772956i \(0.718777\pi\)
\(150\) 0 0
\(151\) 15.1168 1.23019 0.615096 0.788452i \(-0.289117\pi\)
0.615096 + 0.788452i \(0.289117\pi\)
\(152\) 0 0
\(153\) − 0.978251i − 0.0790869i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 15.4891i 1.23617i 0.786113 + 0.618083i \(0.212091\pi\)
−0.786113 + 0.618083i \(0.787909\pi\)
\(158\) 0 0
\(159\) −25.4891 −2.02142
\(160\) 0 0
\(161\) 4.74456 0.373924
\(162\) 0 0
\(163\) − 16.7446i − 1.31154i −0.754963 0.655768i \(-0.772345\pi\)
0.754963 0.655768i \(-0.227655\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.62772i 0.435486i 0.976006 + 0.217743i \(0.0698695\pi\)
−0.976006 + 0.217743i \(0.930131\pi\)
\(168\) 0 0
\(169\) −6.11684 −0.470526
\(170\) 0 0
\(171\) −12.4674 −0.953404
\(172\) 0 0
\(173\) − 0.372281i − 0.0283040i −0.999900 0.0141520i \(-0.995495\pi\)
0.999900 0.0141520i \(-0.00450488\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 18.9783i − 1.42649i
\(178\) 0 0
\(179\) 22.9783 1.71748 0.858738 0.512416i \(-0.171249\pi\)
0.858738 + 0.512416i \(0.171249\pi\)
\(180\) 0 0
\(181\) 16.2337 1.20664 0.603320 0.797499i \(-0.293844\pi\)
0.603320 + 0.797499i \(0.293844\pi\)
\(182\) 0 0
\(183\) − 6.51087i − 0.481298i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2.37228i 0.173478i
\(188\) 0 0
\(189\) −0.883156 −0.0642401
\(190\) 0 0
\(191\) 3.86141 0.279402 0.139701 0.990194i \(-0.455386\pi\)
0.139701 + 0.990194i \(0.455386\pi\)
\(192\) 0 0
\(193\) 6.74456i 0.485484i 0.970091 + 0.242742i \(0.0780469\pi\)
−0.970091 + 0.242742i \(0.921953\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.23369i 0.586626i 0.956016 + 0.293313i \(0.0947578\pi\)
−0.956016 + 0.293313i \(0.905242\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −9.48913 −0.669311
\(202\) 0 0
\(203\) 4.37228i 0.306874i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 12.4674i 0.866543i
\(208\) 0 0
\(209\) 30.2337 2.09131
\(210\) 0 0
\(211\) −14.3723 −0.989429 −0.494714 0.869056i \(-0.664727\pi\)
−0.494714 + 0.869056i \(0.664727\pi\)
\(212\) 0 0
\(213\) 18.9783i 1.30037i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 8.00000i − 0.543075i
\(218\) 0 0
\(219\) 14.2337 0.961823
\(220\) 0 0
\(221\) −1.62772 −0.109492
\(222\) 0 0
\(223\) − 5.62772i − 0.376860i −0.982087 0.188430i \(-0.939660\pi\)
0.982087 0.188430i \(-0.0603398\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 19.8614i − 1.31825i −0.752034 0.659124i \(-0.770927\pi\)
0.752034 0.659124i \(-0.229073\pi\)
\(228\) 0 0
\(229\) 12.2337 0.808425 0.404212 0.914665i \(-0.367546\pi\)
0.404212 + 0.914665i \(0.367546\pi\)
\(230\) 0 0
\(231\) −15.1168 −0.994615
\(232\) 0 0
\(233\) − 1.25544i − 0.0822464i −0.999154 0.0411232i \(-0.986906\pi\)
0.999154 0.0411232i \(-0.0130936\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 35.8614i − 2.32945i
\(238\) 0 0
\(239\) −13.6277 −0.881504 −0.440752 0.897629i \(-0.645288\pi\)
−0.440752 + 0.897629i \(0.645288\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 0 0
\(243\) − 21.0217i − 1.34855i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 20.7446i 1.31994i
\(248\) 0 0
\(249\) 22.5109 1.42657
\(250\) 0 0
\(251\) 4.74456 0.299474 0.149737 0.988726i \(-0.452157\pi\)
0.149737 + 0.988726i \(0.452157\pi\)
\(252\) 0 0
\(253\) − 30.2337i − 1.90078i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 23.4891i 1.46521i 0.680653 + 0.732606i \(0.261696\pi\)
−0.680653 + 0.732606i \(0.738304\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) −11.4891 −0.711159
\(262\) 0 0
\(263\) 22.2337i 1.37099i 0.728078 + 0.685494i \(0.240414\pi\)
−0.728078 + 0.685494i \(0.759586\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 34.9783i − 2.14063i
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 9.48913 0.576423 0.288212 0.957567i \(-0.406939\pi\)
0.288212 + 0.957567i \(0.406939\pi\)
\(272\) 0 0
\(273\) − 10.3723i − 0.627759i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 24.9783i − 1.50080i −0.660985 0.750399i \(-0.729861\pi\)
0.660985 0.750399i \(-0.270139\pi\)
\(278\) 0 0
\(279\) 21.0217 1.25854
\(280\) 0 0
\(281\) 18.6060 1.10994 0.554970 0.831871i \(-0.312730\pi\)
0.554970 + 0.831871i \(0.312730\pi\)
\(282\) 0 0
\(283\) − 8.88316i − 0.528049i −0.964516 0.264024i \(-0.914950\pi\)
0.964516 0.264024i \(-0.0850500\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.74456i 0.398119i
\(288\) 0 0
\(289\) 16.8614 0.991847
\(290\) 0 0
\(291\) −23.3940 −1.37138
\(292\) 0 0
\(293\) 25.1168i 1.46734i 0.679505 + 0.733671i \(0.262195\pi\)
−0.679505 + 0.733671i \(0.737805\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.62772i 0.326553i
\(298\) 0 0
\(299\) 20.7446 1.19969
\(300\) 0 0
\(301\) 8.74456 0.504028
\(302\) 0 0
\(303\) − 14.2337i − 0.817704i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 31.1168i − 1.77593i −0.459909 0.887966i \(-0.652118\pi\)
0.459909 0.887966i \(-0.347882\pi\)
\(308\) 0 0
\(309\) 13.3505 0.759485
\(310\) 0 0
\(311\) −12.7446 −0.722678 −0.361339 0.932435i \(-0.617680\pi\)
−0.361339 + 0.932435i \(0.617680\pi\)
\(312\) 0 0
\(313\) 2.88316i 0.162966i 0.996675 + 0.0814828i \(0.0259656\pi\)
−0.996675 + 0.0814828i \(0.974034\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 14.0000i − 0.786318i −0.919470 0.393159i \(-0.871382\pi\)
0.919470 0.393159i \(-0.128618\pi\)
\(318\) 0 0
\(319\) 27.8614 1.55994
\(320\) 0 0
\(321\) 20.7446 1.15785
\(322\) 0 0
\(323\) 1.76631i 0.0982802i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 0.883156i − 0.0488386i
\(328\) 0 0
\(329\) −7.11684 −0.392364
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 0 0
\(333\) − 5.25544i − 0.287996i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 7.48913i 0.407959i 0.978975 + 0.203979i \(0.0653875\pi\)
−0.978975 + 0.203979i \(0.934612\pi\)
\(338\) 0 0
\(339\) −4.74456 −0.257689
\(340\) 0 0
\(341\) −50.9783 −2.76063
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.7446i 1.32836i 0.747574 + 0.664179i \(0.231219\pi\)
−0.747574 + 0.664179i \(0.768781\pi\)
\(348\) 0 0
\(349\) −19.4891 −1.04323 −0.521614 0.853181i \(-0.674670\pi\)
−0.521614 + 0.853181i \(0.674670\pi\)
\(350\) 0 0
\(351\) −3.86141 −0.206107
\(352\) 0 0
\(353\) − 1.86141i − 0.0990727i −0.998772 0.0495363i \(-0.984226\pi\)
0.998772 0.0495363i \(-0.0157744\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 0.883156i − 0.0467416i
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 3.51087 0.184783
\(362\) 0 0
\(363\) 70.2337i 3.68631i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 19.8614i − 1.03676i −0.855151 0.518378i \(-0.826536\pi\)
0.855151 0.518378i \(-0.173464\pi\)
\(368\) 0 0
\(369\) −17.7228 −0.922613
\(370\) 0 0
\(371\) −10.7446 −0.557830
\(372\) 0 0
\(373\) − 30.7446i − 1.59189i −0.605367 0.795947i \(-0.706974\pi\)
0.605367 0.795947i \(-0.293026\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 19.1168i 0.984568i
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) 18.9783 0.972285
\(382\) 0 0
\(383\) − 17.4891i − 0.893653i −0.894621 0.446826i \(-0.852554\pi\)
0.894621 0.446826i \(-0.147446\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 22.9783i 1.16805i
\(388\) 0 0
\(389\) −17.8614 −0.905609 −0.452805 0.891610i \(-0.649576\pi\)
−0.452805 + 0.891610i \(0.649576\pi\)
\(390\) 0 0
\(391\) 1.76631 0.0893262
\(392\) 0 0
\(393\) − 11.2554i − 0.567762i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 31.6277i − 1.58735i −0.608342 0.793675i \(-0.708165\pi\)
0.608342 0.793675i \(-0.291835\pi\)
\(398\) 0 0
\(399\) −11.2554 −0.563477
\(400\) 0 0
\(401\) −38.6060 −1.92789 −0.963945 0.266101i \(-0.914264\pi\)
−0.963945 + 0.266101i \(0.914264\pi\)
\(402\) 0 0
\(403\) − 34.9783i − 1.74239i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.7446i 0.631725i
\(408\) 0 0
\(409\) −11.4891 −0.568101 −0.284050 0.958809i \(-0.591678\pi\)
−0.284050 + 0.958809i \(0.591678\pi\)
\(410\) 0 0
\(411\) 34.9783 1.72535
\(412\) 0 0
\(413\) − 8.00000i − 0.393654i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 11.2554i 0.551181i
\(418\) 0 0
\(419\) 14.5109 0.708903 0.354451 0.935074i \(-0.384668\pi\)
0.354451 + 0.935074i \(0.384668\pi\)
\(420\) 0 0
\(421\) −18.6060 −0.906799 −0.453400 0.891307i \(-0.649789\pi\)
−0.453400 + 0.891307i \(0.649789\pi\)
\(422\) 0 0
\(423\) − 18.7011i − 0.909277i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 2.74456i − 0.132819i
\(428\) 0 0
\(429\) −66.0951 −3.19110
\(430\) 0 0
\(431\) 18.3723 0.884962 0.442481 0.896778i \(-0.354098\pi\)
0.442481 + 0.896778i \(0.354098\pi\)
\(432\) 0 0
\(433\) 28.9783i 1.39261i 0.717748 + 0.696303i \(0.245173\pi\)
−0.717748 + 0.696303i \(0.754827\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 22.5109i − 1.07684i
\(438\) 0 0
\(439\) 6.23369 0.297518 0.148759 0.988874i \(-0.452472\pi\)
0.148759 + 0.988874i \(0.452472\pi\)
\(440\) 0 0
\(441\) 2.62772 0.125129
\(442\) 0 0
\(443\) − 16.7446i − 0.795558i −0.917481 0.397779i \(-0.869781\pi\)
0.917481 0.397779i \(-0.130219\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 36.7446i − 1.73796i
\(448\) 0 0
\(449\) −17.1168 −0.807794 −0.403897 0.914805i \(-0.632345\pi\)
−0.403897 + 0.914805i \(0.632345\pi\)
\(450\) 0 0
\(451\) 42.9783 2.02377
\(452\) 0 0
\(453\) 35.8614i 1.68492i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 5.25544i − 0.245839i −0.992417 0.122919i \(-0.960774\pi\)
0.992417 0.122919i \(-0.0392257\pi\)
\(458\) 0 0
\(459\) −0.328782 −0.0153463
\(460\) 0 0
\(461\) −1.25544 −0.0584715 −0.0292358 0.999573i \(-0.509307\pi\)
−0.0292358 + 0.999573i \(0.509307\pi\)
\(462\) 0 0
\(463\) − 6.51087i − 0.302586i −0.988489 0.151293i \(-0.951656\pi\)
0.988489 0.151293i \(-0.0483437\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 8.60597i − 0.398237i −0.979975 0.199118i \(-0.936192\pi\)
0.979975 0.199118i \(-0.0638078\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −36.7446 −1.69310
\(472\) 0 0
\(473\) − 55.7228i − 2.56214i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 28.2337i − 1.29273i
\(478\) 0 0
\(479\) −22.2337 −1.01588 −0.507942 0.861392i \(-0.669593\pi\)
−0.507942 + 0.861392i \(0.669593\pi\)
\(480\) 0 0
\(481\) −8.74456 −0.398718
\(482\) 0 0
\(483\) 11.2554i 0.512140i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 30.2337i − 1.37002i −0.728534 0.685010i \(-0.759798\pi\)
0.728534 0.685010i \(-0.240202\pi\)
\(488\) 0 0
\(489\) 39.7228 1.79633
\(490\) 0 0
\(491\) 35.1168 1.58480 0.792400 0.610001i \(-0.208831\pi\)
0.792400 + 0.610001i \(0.208831\pi\)
\(492\) 0 0
\(493\) 1.62772i 0.0733088i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000i 0.358849i
\(498\) 0 0
\(499\) −31.8614 −1.42631 −0.713156 0.701005i \(-0.752735\pi\)
−0.713156 + 0.701005i \(0.752735\pi\)
\(500\) 0 0
\(501\) −13.3505 −0.596458
\(502\) 0 0
\(503\) − 18.3723i − 0.819180i −0.912270 0.409590i \(-0.865672\pi\)
0.912270 0.409590i \(-0.134328\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 14.5109i − 0.644451i
\(508\) 0 0
\(509\) 40.9783 1.81633 0.908165 0.418613i \(-0.137484\pi\)
0.908165 + 0.418613i \(0.137484\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) 4.19019i 0.185001i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 45.3505i 1.99451i
\(518\) 0 0
\(519\) 0.883156 0.0387662
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) − 36.4674i − 1.59461i −0.603579 0.797304i \(-0.706259\pi\)
0.603579 0.797304i \(-0.293741\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 2.97825i − 0.129735i
\(528\) 0 0
\(529\) 0.489125 0.0212663
\(530\) 0 0
\(531\) 21.0217 0.912266
\(532\) 0 0
\(533\) 29.4891i 1.27732i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 54.5109i 2.35232i
\(538\) 0 0
\(539\) −6.37228 −0.274474
\(540\) 0 0
\(541\) −31.3505 −1.34786 −0.673932 0.738793i \(-0.735396\pi\)
−0.673932 + 0.738793i \(0.735396\pi\)
\(542\) 0 0
\(543\) 38.5109i 1.65266i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 30.9783i 1.32453i 0.749268 + 0.662267i \(0.230406\pi\)
−0.749268 + 0.662267i \(0.769594\pi\)
\(548\) 0 0
\(549\) 7.21194 0.307798
\(550\) 0 0
\(551\) 20.7446 0.883748
\(552\) 0 0
\(553\) − 15.1168i − 0.642834i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.76631i 0.159584i 0.996812 + 0.0797919i \(0.0254256\pi\)
−0.996812 + 0.0797919i \(0.974574\pi\)
\(558\) 0 0
\(559\) 38.2337 1.61711
\(560\) 0 0
\(561\) −5.62772 −0.237602
\(562\) 0 0
\(563\) 17.4891i 0.737079i 0.929612 + 0.368539i \(0.120142\pi\)
−0.929612 + 0.368539i \(0.879858\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 9.97825i − 0.419047i
\(568\) 0 0
\(569\) 24.9783 1.04714 0.523571 0.851982i \(-0.324599\pi\)
0.523571 + 0.851982i \(0.324599\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 9.16034i 0.382679i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 22.6060i 0.941099i 0.882374 + 0.470549i \(0.155944\pi\)
−0.882374 + 0.470549i \(0.844056\pi\)
\(578\) 0 0
\(579\) −16.0000 −0.664937
\(580\) 0 0
\(581\) 9.48913 0.393675
\(582\) 0 0
\(583\) 68.4674i 2.83563i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 34.9783i 1.44371i 0.692046 + 0.721853i \(0.256710\pi\)
−0.692046 + 0.721853i \(0.743290\pi\)
\(588\) 0 0
\(589\) −37.9565 −1.56397
\(590\) 0 0
\(591\) −19.5326 −0.803465
\(592\) 0 0
\(593\) − 19.6277i − 0.806014i −0.915197 0.403007i \(-0.867965\pi\)
0.915197 0.403007i \(-0.132035\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 37.9565i 1.55346i
\(598\) 0 0
\(599\) 32.6060 1.33224 0.666122 0.745843i \(-0.267953\pi\)
0.666122 + 0.745843i \(0.267953\pi\)
\(600\) 0 0
\(601\) 16.5109 0.673493 0.336746 0.941595i \(-0.390674\pi\)
0.336746 + 0.941595i \(0.390674\pi\)
\(602\) 0 0
\(603\) − 10.5109i − 0.428036i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 24.6060i − 0.998725i −0.866393 0.499363i \(-0.833568\pi\)
0.866393 0.499363i \(-0.166432\pi\)
\(608\) 0 0
\(609\) −10.3723 −0.420306
\(610\) 0 0
\(611\) −31.1168 −1.25885
\(612\) 0 0
\(613\) − 8.51087i − 0.343751i −0.985119 0.171875i \(-0.945017\pi\)
0.985119 0.171875i \(-0.0549827\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 0 0
\(619\) 12.7446 0.512247 0.256124 0.966644i \(-0.417555\pi\)
0.256124 + 0.966644i \(0.417555\pi\)
\(620\) 0 0
\(621\) 4.19019 0.168146
\(622\) 0 0
\(623\) − 14.7446i − 0.590728i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 71.7228i 2.86433i
\(628\) 0 0
\(629\) −0.744563 −0.0296877
\(630\) 0 0
\(631\) 2.37228 0.0944390 0.0472195 0.998885i \(-0.484964\pi\)
0.0472195 + 0.998885i \(0.484964\pi\)
\(632\) 0 0
\(633\) − 34.0951i − 1.35516i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 4.37228i − 0.173236i
\(638\) 0 0
\(639\) −21.0217 −0.831608
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 18.3723i 0.724532i 0.932075 + 0.362266i \(0.117997\pi\)
−0.932075 + 0.362266i \(0.882003\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.0000i 0.629025i 0.949253 + 0.314512i \(0.101841\pi\)
−0.949253 + 0.314512i \(0.898159\pi\)
\(648\) 0 0
\(649\) −50.9783 −2.00107
\(650\) 0 0
\(651\) 18.9783 0.743816
\(652\) 0 0
\(653\) 6.00000i 0.234798i 0.993085 + 0.117399i \(0.0374557\pi\)
−0.993085 + 0.117399i \(0.962544\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 15.7663i 0.615102i
\(658\) 0 0
\(659\) 11.1168 0.433051 0.216525 0.976277i \(-0.430528\pi\)
0.216525 + 0.976277i \(0.430528\pi\)
\(660\) 0 0
\(661\) 14.7446 0.573497 0.286749 0.958006i \(-0.407426\pi\)
0.286749 + 0.958006i \(0.407426\pi\)
\(662\) 0 0
\(663\) − 3.86141i − 0.149965i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 20.7446i − 0.803233i
\(668\) 0 0
\(669\) 13.3505 0.516161
\(670\) 0 0
\(671\) −17.4891 −0.675160
\(672\) 0 0
\(673\) 25.7228i 0.991542i 0.868453 + 0.495771i \(0.165114\pi\)
−0.868453 + 0.495771i \(0.834886\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 15.3505i − 0.589969i −0.955502 0.294984i \(-0.904686\pi\)
0.955502 0.294984i \(-0.0953145\pi\)
\(678\) 0 0
\(679\) −9.86141 −0.378446
\(680\) 0 0
\(681\) 47.1168 1.80552
\(682\) 0 0
\(683\) 36.0000i 1.37750i 0.724998 + 0.688751i \(0.241841\pi\)
−0.724998 + 0.688751i \(0.758159\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 29.0217i 1.10725i
\(688\) 0 0
\(689\) −46.9783 −1.78973
\(690\) 0 0
\(691\) −9.48913 −0.360983 −0.180492 0.983577i \(-0.557769\pi\)
−0.180492 + 0.983577i \(0.557769\pi\)
\(692\) 0 0
\(693\) − 16.7446i − 0.636073i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.51087i 0.0951062i
\(698\) 0 0
\(699\) 2.97825 0.112648
\(700\) 0 0
\(701\) 2.13859 0.0807736 0.0403868 0.999184i \(-0.487141\pi\)
0.0403868 + 0.999184i \(0.487141\pi\)
\(702\) 0 0
\(703\) 9.48913i 0.357889i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 6.00000i − 0.225653i
\(708\) 0 0
\(709\) −6.60597 −0.248092 −0.124046 0.992276i \(-0.539587\pi\)
−0.124046 + 0.992276i \(0.539587\pi\)
\(710\) 0 0
\(711\) 39.7228 1.48972
\(712\) 0 0
\(713\) 37.9565i 1.42148i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 32.3288i − 1.20734i
\(718\) 0 0
\(719\) −3.25544 −0.121407 −0.0607037 0.998156i \(-0.519334\pi\)
−0.0607037 + 0.998156i \(0.519334\pi\)
\(720\) 0 0
\(721\) 5.62772 0.209587
\(722\) 0 0
\(723\) 61.6793i 2.29388i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 32.0000i − 1.18681i −0.804902 0.593407i \(-0.797782\pi\)
0.804902 0.593407i \(-0.202218\pi\)
\(728\) 0 0
\(729\) 19.9348 0.738324
\(730\) 0 0
\(731\) 3.25544 0.120407
\(732\) 0 0
\(733\) − 10.1386i − 0.374477i −0.982314 0.187239i \(-0.940046\pi\)
0.982314 0.187239i \(-0.0599538\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 25.4891i 0.938904i
\(738\) 0 0
\(739\) 20.6060 0.758003 0.379001 0.925396i \(-0.376268\pi\)
0.379001 + 0.925396i \(0.376268\pi\)
\(740\) 0 0
\(741\) −49.2119 −1.80785
\(742\) 0 0
\(743\) − 6.51087i − 0.238861i −0.992843 0.119430i \(-0.961893\pi\)
0.992843 0.119430i \(-0.0381068\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 24.9348i 0.912315i
\(748\) 0 0
\(749\) 8.74456 0.319519
\(750\) 0 0
\(751\) 20.1386 0.734868 0.367434 0.930050i \(-0.380236\pi\)
0.367434 + 0.930050i \(0.380236\pi\)
\(752\) 0 0
\(753\) 11.2554i 0.410171i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 3.76631i 0.136889i 0.997655 + 0.0684445i \(0.0218036\pi\)
−0.997655 + 0.0684445i \(0.978196\pi\)
\(758\) 0 0
\(759\) 71.7228 2.60337
\(760\) 0 0
\(761\) 4.97825 0.180461 0.0902307 0.995921i \(-0.471240\pi\)
0.0902307 + 0.995921i \(0.471240\pi\)
\(762\) 0 0
\(763\) − 0.372281i − 0.0134775i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 34.9783i − 1.26299i
\(768\) 0 0
\(769\) −3.48913 −0.125821 −0.0629105 0.998019i \(-0.520038\pi\)
−0.0629105 + 0.998019i \(0.520038\pi\)
\(770\) 0 0
\(771\) −55.7228 −2.00681
\(772\) 0 0
\(773\) 4.37228i 0.157260i 0.996904 + 0.0786300i \(0.0250546\pi\)
−0.996904 + 0.0786300i \(0.974945\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 4.74456i − 0.170210i
\(778\) 0 0
\(779\) 32.0000 1.14652
\(780\) 0 0
\(781\) 50.9783 1.82415
\(782\) 0 0
\(783\) 3.86141i 0.137995i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 31.1168i 1.10920i 0.832119 + 0.554598i \(0.187128\pi\)
−0.832119 + 0.554598i \(0.812872\pi\)
\(788\) 0 0
\(789\) −52.7446 −1.87776
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) − 12.0000i − 0.426132i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 15.6277i − 0.553562i −0.960933 0.276781i \(-0.910732\pi\)
0.960933 0.276781i \(-0.0892677\pi\)
\(798\) 0 0
\(799\) −2.64947 −0.0937314
\(800\) 0 0
\(801\) 38.7446 1.36897
\(802\) 0 0
\(803\) − 38.2337i − 1.34924i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 14.2337i 0.501050i
\(808\) 0 0
\(809\) −20.3723 −0.716251 −0.358126 0.933673i \(-0.616584\pi\)
−0.358126 + 0.933673i \(0.616584\pi\)
\(810\) 0 0
\(811\) 12.7446 0.447522 0.223761 0.974644i \(-0.428166\pi\)
0.223761 + 0.974644i \(0.428166\pi\)
\(812\) 0 0
\(813\) 22.5109i 0.789491i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 41.4891i − 1.45152i
\(818\) 0 0
\(819\) 11.4891 0.401463
\(820\) 0 0
\(821\) −25.1168 −0.876584 −0.438292 0.898833i \(-0.644416\pi\)
−0.438292 + 0.898833i \(0.644416\pi\)
\(822\) 0 0
\(823\) − 8.00000i − 0.278862i −0.990232 0.139431i \(-0.955473\pi\)
0.990232 0.139431i \(-0.0445274\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 24.7446i − 0.860453i −0.902721 0.430226i \(-0.858434\pi\)
0.902721 0.430226i \(-0.141566\pi\)
\(828\) 0 0
\(829\) 12.2337 0.424894 0.212447 0.977173i \(-0.431857\pi\)
0.212447 + 0.977173i \(0.431857\pi\)
\(830\) 0 0
\(831\) 59.2554 2.05555
\(832\) 0 0
\(833\) − 0.372281i − 0.0128988i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 7.06525i − 0.244211i
\(838\) 0 0
\(839\) 11.2554 0.388581 0.194290 0.980944i \(-0.437760\pi\)
0.194290 + 0.980944i \(0.437760\pi\)
\(840\) 0 0
\(841\) −9.88316 −0.340798
\(842\) 0 0
\(843\) 44.1386i 1.52021i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 29.6060i 1.01727i
\(848\) 0 0
\(849\) 21.0733 0.723235
\(850\) 0 0
\(851\) 9.48913 0.325283
\(852\) 0 0
\(853\) − 39.4891i − 1.35208i −0.736864 0.676041i \(-0.763694\pi\)
0.736864 0.676041i \(-0.236306\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 18.0000i − 0.614868i −0.951569 0.307434i \(-0.900530\pi\)
0.951569 0.307434i \(-0.0994704\pi\)
\(858\) 0 0
\(859\) −44.4674 −1.51721 −0.758604 0.651552i \(-0.774118\pi\)
−0.758604 + 0.651552i \(0.774118\pi\)
\(860\) 0 0
\(861\) −16.0000 −0.545279
\(862\) 0 0
\(863\) 25.4891i 0.867660i 0.900995 + 0.433830i \(0.142838\pi\)
−0.900995 + 0.433830i \(0.857162\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 40.0000i 1.35847i
\(868\) 0 0
\(869\) −96.3288 −3.26773
\(870\) 0 0
\(871\) −17.4891 −0.592596
\(872\) 0 0
\(873\) − 25.9130i − 0.877022i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7.02175i 0.237108i 0.992948 + 0.118554i \(0.0378258\pi\)
−0.992948 + 0.118554i \(0.962174\pi\)
\(878\) 0 0
\(879\) −59.5842 −2.00973
\(880\) 0 0
\(881\) −25.2554 −0.850877 −0.425439 0.904987i \(-0.639880\pi\)
−0.425439 + 0.904987i \(0.639880\pi\)
\(882\) 0 0
\(883\) − 10.5109i − 0.353719i −0.984236 0.176860i \(-0.943406\pi\)
0.984236 0.176860i \(-0.0565938\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 13.0217i 0.437228i 0.975811 + 0.218614i \(0.0701535\pi\)
−0.975811 + 0.218614i \(0.929847\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −63.5842 −2.13015
\(892\) 0 0
\(893\) 33.7663i 1.12995i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 49.2119i 1.64314i
\(898\) 0 0
\(899\) −34.9783 −1.16659
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 0 0
\(903\) 20.7446i 0.690336i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 11.7228i − 0.389250i −0.980878 0.194625i \(-0.937651\pi\)
0.980878 0.194625i \(-0.0623489\pi\)
\(908\) 0 0
\(909\) 15.7663 0.522936
\(910\) 0 0
\(911\) −45.9565 −1.52261 −0.761303 0.648396i \(-0.775440\pi\)
−0.761303 + 0.648396i \(0.775440\pi\)
\(912\) 0 0
\(913\) − 60.4674i − 2.00118i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 4.74456i − 0.156679i
\(918\) 0 0
\(919\) −13.6277 −0.449537 −0.224768 0.974412i \(-0.572163\pi\)
−0.224768 + 0.974412i \(0.572163\pi\)
\(920\) 0 0
\(921\) 73.8179 2.43238
\(922\) 0 0
\(923\) 34.9783i 1.15132i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 14.7881i 0.485704i
\(928\) 0 0
\(929\) 7.76631 0.254804 0.127402 0.991851i \(-0.459336\pi\)
0.127402 + 0.991851i \(0.459336\pi\)
\(930\) 0 0
\(931\) −4.74456 −0.155497
\(932\) 0 0
\(933\) − 30.2337i − 0.989807i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 28.0951i − 0.917827i −0.888481 0.458913i \(-0.848239\pi\)
0.888481 0.458913i \(-0.151761\pi\)
\(938\) 0 0
\(939\) −6.83966 −0.223204
\(940\) 0 0
\(941\) 32.2337 1.05079 0.525394 0.850859i \(-0.323918\pi\)
0.525394 + 0.850859i \(0.323918\pi\)
\(942\) 0 0
\(943\) − 32.0000i − 1.04206i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28.0000i 0.909878i 0.890523 + 0.454939i \(0.150339\pi\)
−0.890523 + 0.454939i \(0.849661\pi\)
\(948\) 0 0
\(949\) 26.2337 0.851582
\(950\) 0 0
\(951\) 33.2119 1.07697
\(952\) 0 0
\(953\) 37.2554i 1.20682i 0.797430 + 0.603411i \(0.206192\pi\)
−0.797430 + 0.603411i \(0.793808\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 66.0951i 2.13655i
\(958\) 0 0
\(959\) 14.7446 0.476127
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 22.9783i 0.740464i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 1.76631i − 0.0568008i −0.999597 0.0284004i \(-0.990959\pi\)
0.999597 0.0284004i \(-0.00904134\pi\)
\(968\) 0 0
\(969\) −4.19019 −0.134608
\(970\) 0 0
\(971\) 33.4891 1.07472 0.537359 0.843354i \(-0.319422\pi\)
0.537359 + 0.843354i \(0.319422\pi\)
\(972\) 0 0
\(973\) 4.74456i 0.152104i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 52.9783i − 1.69492i −0.530856 0.847462i \(-0.678129\pi\)
0.530856 0.847462i \(-0.321871\pi\)
\(978\) 0 0
\(979\) −93.9565 −3.00286
\(980\) 0 0
\(981\) 0.978251 0.0312331
\(982\) 0 0
\(983\) − 10.3723i − 0.330824i −0.986225 0.165412i \(-0.947105\pi\)
0.986225 0.165412i \(-0.0528954\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 16.8832i − 0.537397i
\(988\) 0 0
\(989\) −41.4891 −1.31928
\(990\) 0 0
\(991\) −37.9565 −1.20573 −0.602864 0.797844i \(-0.705974\pi\)
−0.602864 + 0.797844i \(0.705974\pi\)
\(992\) 0 0
\(993\) 28.4674i 0.903385i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 6.88316i 0.217992i 0.994042 + 0.108996i \(0.0347635\pi\)
−0.994042 + 0.108996i \(0.965236\pi\)
\(998\) 0 0
\(999\) −1.76631 −0.0558836
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.i.449.3 4
4.3 odd 2 2800.2.g.r.449.2 4
5.2 odd 4 280.2.a.c.1.2 2
5.3 odd 4 1400.2.a.r.1.1 2
5.4 even 2 inner 1400.2.g.i.449.2 4
15.2 even 4 2520.2.a.x.1.1 2
20.3 even 4 2800.2.a.bk.1.2 2
20.7 even 4 560.2.a.h.1.1 2
20.19 odd 2 2800.2.g.r.449.3 4
35.2 odd 12 1960.2.q.t.361.1 4
35.12 even 12 1960.2.q.r.361.2 4
35.13 even 4 9800.2.a.bu.1.2 2
35.17 even 12 1960.2.q.r.961.2 4
35.27 even 4 1960.2.a.s.1.1 2
35.32 odd 12 1960.2.q.t.961.1 4
40.27 even 4 2240.2.a.bg.1.2 2
40.37 odd 4 2240.2.a.bk.1.1 2
60.47 odd 4 5040.2.a.by.1.2 2
140.27 odd 4 3920.2.a.bt.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.c.1.2 2 5.2 odd 4
560.2.a.h.1.1 2 20.7 even 4
1400.2.a.r.1.1 2 5.3 odd 4
1400.2.g.i.449.2 4 5.4 even 2 inner
1400.2.g.i.449.3 4 1.1 even 1 trivial
1960.2.a.s.1.1 2 35.27 even 4
1960.2.q.r.361.2 4 35.12 even 12
1960.2.q.r.961.2 4 35.17 even 12
1960.2.q.t.361.1 4 35.2 odd 12
1960.2.q.t.961.1 4 35.32 odd 12
2240.2.a.bg.1.2 2 40.27 even 4
2240.2.a.bk.1.1 2 40.37 odd 4
2520.2.a.x.1.1 2 15.2 even 4
2800.2.a.bk.1.2 2 20.3 even 4
2800.2.g.r.449.2 4 4.3 odd 2
2800.2.g.r.449.3 4 20.19 odd 2
3920.2.a.bt.1.2 2 140.27 odd 4
5040.2.a.by.1.2 2 60.47 odd 4
9800.2.a.bu.1.2 2 35.13 even 4