Properties

Label 1400.2.g.k.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: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
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 \(1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 1400.449
Dual form 1400.2.g.k.449.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.56155i q^{3} +1.00000i q^{7} +0.561553 q^{9} +O(q^{10})\) \(q+1.56155i q^{3} +1.00000i q^{7} +0.561553 q^{9} +1.56155 q^{11} -6.68466i q^{13} +7.56155i q^{17} +7.12311 q^{19} -1.56155 q^{21} -3.12311i q^{23} +5.56155i q^{27} -0.438447 q^{29} +6.24621 q^{31} +2.43845i q^{33} -8.24621i q^{37} +10.4384 q^{39} -1.12311 q^{41} +7.12311i q^{43} +2.43845i q^{47} -1.00000 q^{49} -11.8078 q^{51} +13.1231i q^{53} +11.1231i q^{57} +4.00000 q^{59} -6.87689 q^{61} +0.561553i q^{63} +2.24621i q^{67} +4.87689 q^{69} +4.24621i q^{73} +1.56155i q^{77} -0.684658 q^{79} -7.00000 q^{81} -12.0000i q^{83} -0.684658i q^{87} -5.12311 q^{89} +6.68466 q^{91} +9.75379i q^{93} +1.31534i q^{97} +0.876894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{9} + O(q^{10}) \) \( 4q - 6q^{9} - 2q^{11} + 12q^{19} + 2q^{21} - 10q^{29} - 8q^{31} + 50q^{39} + 12q^{41} - 4q^{49} - 6q^{51} + 16q^{59} - 44q^{61} + 36q^{69} + 22q^{79} - 28q^{81} - 4q^{89} + 2q^{91} + 20q^{99} + O(q^{100}) \)

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
\(3\) 1.56155i 0.901563i 0.892634 + 0.450781i \(0.148855\pi\)
−0.892634 + 0.450781i \(0.851145\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0.561553 0.187184
\(10\) 0 0
\(11\) 1.56155 0.470826 0.235413 0.971895i \(-0.424356\pi\)
0.235413 + 0.971895i \(0.424356\pi\)
\(12\) 0 0
\(13\) − 6.68466i − 1.85399i −0.375073 0.926995i \(-0.622382\pi\)
0.375073 0.926995i \(-0.377618\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.56155i 1.83395i 0.398949 + 0.916973i \(0.369375\pi\)
−0.398949 + 0.916973i \(0.630625\pi\)
\(18\) 0 0
\(19\) 7.12311 1.63415 0.817076 0.576530i \(-0.195593\pi\)
0.817076 + 0.576530i \(0.195593\pi\)
\(20\) 0 0
\(21\) −1.56155 −0.340759
\(22\) 0 0
\(23\) − 3.12311i − 0.651213i −0.945505 0.325606i \(-0.894432\pi\)
0.945505 0.325606i \(-0.105568\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.56155i 1.07032i
\(28\) 0 0
\(29\) −0.438447 −0.0814176 −0.0407088 0.999171i \(-0.512962\pi\)
−0.0407088 + 0.999171i \(0.512962\pi\)
\(30\) 0 0
\(31\) 6.24621 1.12185 0.560926 0.827866i \(-0.310445\pi\)
0.560926 + 0.827866i \(0.310445\pi\)
\(32\) 0 0
\(33\) 2.43845i 0.424479i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 8.24621i − 1.35567i −0.735215 0.677834i \(-0.762919\pi\)
0.735215 0.677834i \(-0.237081\pi\)
\(38\) 0 0
\(39\) 10.4384 1.67149
\(40\) 0 0
\(41\) −1.12311 −0.175400 −0.0876998 0.996147i \(-0.527952\pi\)
−0.0876998 + 0.996147i \(0.527952\pi\)
\(42\) 0 0
\(43\) 7.12311i 1.08626i 0.839648 + 0.543132i \(0.182762\pi\)
−0.839648 + 0.543132i \(0.817238\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.43845i 0.355684i 0.984059 + 0.177842i \(0.0569116\pi\)
−0.984059 + 0.177842i \(0.943088\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −11.8078 −1.65342
\(52\) 0 0
\(53\) 13.1231i 1.80260i 0.433198 + 0.901299i \(0.357385\pi\)
−0.433198 + 0.901299i \(0.642615\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 11.1231i 1.47329i
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −6.87689 −0.880496 −0.440248 0.897876i \(-0.645109\pi\)
−0.440248 + 0.897876i \(0.645109\pi\)
\(62\) 0 0
\(63\) 0.561553i 0.0707490i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.24621i 0.274418i 0.990542 + 0.137209i \(0.0438133\pi\)
−0.990542 + 0.137209i \(0.956187\pi\)
\(68\) 0 0
\(69\) 4.87689 0.587109
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 4.24621i 0.496981i 0.968634 + 0.248491i \(0.0799345\pi\)
−0.968634 + 0.248491i \(0.920065\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.56155i 0.177955i
\(78\) 0 0
\(79\) −0.684658 −0.0770301 −0.0385150 0.999258i \(-0.512263\pi\)
−0.0385150 + 0.999258i \(0.512263\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 0.684658i − 0.0734031i
\(88\) 0 0
\(89\) −5.12311 −0.543048 −0.271524 0.962432i \(-0.587528\pi\)
−0.271524 + 0.962432i \(0.587528\pi\)
\(90\) 0 0
\(91\) 6.68466 0.700743
\(92\) 0 0
\(93\) 9.75379i 1.01142i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.31534i 0.133553i 0.997768 + 0.0667764i \(0.0212714\pi\)
−0.997768 + 0.0667764i \(0.978729\pi\)
\(98\) 0 0
\(99\) 0.876894 0.0881312
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 11.8078i 1.16345i 0.813384 + 0.581727i \(0.197623\pi\)
−0.813384 + 0.581727i \(0.802377\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.1231i 1.46201i 0.682374 + 0.731003i \(0.260947\pi\)
−0.682374 + 0.731003i \(0.739053\pi\)
\(108\) 0 0
\(109\) 4.43845 0.425126 0.212563 0.977147i \(-0.431819\pi\)
0.212563 + 0.977147i \(0.431819\pi\)
\(110\) 0 0
\(111\) 12.8769 1.22222
\(112\) 0 0
\(113\) − 8.24621i − 0.775738i −0.921714 0.387869i \(-0.873211\pi\)
0.921714 0.387869i \(-0.126789\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 3.75379i − 0.347038i
\(118\) 0 0
\(119\) −7.56155 −0.693166
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) 0 0
\(123\) − 1.75379i − 0.158134i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 6.24621i − 0.554262i −0.960832 0.277131i \(-0.910616\pi\)
0.960832 0.277131i \(-0.0893835\pi\)
\(128\) 0 0
\(129\) −11.1231 −0.979335
\(130\) 0 0
\(131\) 15.1231 1.32131 0.660656 0.750689i \(-0.270278\pi\)
0.660656 + 0.750689i \(0.270278\pi\)
\(132\) 0 0
\(133\) 7.12311i 0.617652i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 7.36932i − 0.629603i −0.949157 0.314802i \(-0.898062\pi\)
0.949157 0.314802i \(-0.101938\pi\)
\(138\) 0 0
\(139\) 21.3693 1.81252 0.906261 0.422719i \(-0.138924\pi\)
0.906261 + 0.422719i \(0.138924\pi\)
\(140\) 0 0
\(141\) −3.80776 −0.320672
\(142\) 0 0
\(143\) − 10.4384i − 0.872907i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 1.56155i − 0.128795i
\(148\) 0 0
\(149\) 0.246211 0.0201704 0.0100852 0.999949i \(-0.496790\pi\)
0.0100852 + 0.999949i \(0.496790\pi\)
\(150\) 0 0
\(151\) −19.8078 −1.61193 −0.805966 0.591961i \(-0.798354\pi\)
−0.805966 + 0.591961i \(0.798354\pi\)
\(152\) 0 0
\(153\) 4.24621i 0.343286i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.24621i 0.338885i 0.985540 + 0.169442i \(0.0541966\pi\)
−0.985540 + 0.169442i \(0.945803\pi\)
\(158\) 0 0
\(159\) −20.4924 −1.62515
\(160\) 0 0
\(161\) 3.12311 0.246135
\(162\) 0 0
\(163\) − 19.6155i − 1.53641i −0.640206 0.768203i \(-0.721151\pi\)
0.640206 0.768203i \(-0.278849\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.19224i 0.324405i 0.986757 + 0.162202i \(0.0518598\pi\)
−0.986757 + 0.162202i \(0.948140\pi\)
\(168\) 0 0
\(169\) −31.6847 −2.43728
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) 23.1771i 1.76212i 0.473004 + 0.881060i \(0.343170\pi\)
−0.473004 + 0.881060i \(0.656830\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.24621i 0.469494i
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −5.12311 −0.380797 −0.190399 0.981707i \(-0.560978\pi\)
−0.190399 + 0.981707i \(0.560978\pi\)
\(182\) 0 0
\(183\) − 10.7386i − 0.793823i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 11.8078i 0.863469i
\(188\) 0 0
\(189\) −5.56155 −0.404543
\(190\) 0 0
\(191\) −0.684658 −0.0495401 −0.0247701 0.999693i \(-0.507885\pi\)
−0.0247701 + 0.999693i \(0.507885\pi\)
\(192\) 0 0
\(193\) − 13.1231i − 0.944622i −0.881432 0.472311i \(-0.843420\pi\)
0.881432 0.472311i \(-0.156580\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 13.1231i − 0.934983i −0.883998 0.467491i \(-0.845158\pi\)
0.883998 0.467491i \(-0.154842\pi\)
\(198\) 0 0
\(199\) 14.2462 1.00989 0.504944 0.863152i \(-0.331513\pi\)
0.504944 + 0.863152i \(0.331513\pi\)
\(200\) 0 0
\(201\) −3.50758 −0.247405
\(202\) 0 0
\(203\) − 0.438447i − 0.0307730i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 1.75379i − 0.121897i
\(208\) 0 0
\(209\) 11.1231 0.769401
\(210\) 0 0
\(211\) −17.5616 −1.20899 −0.604494 0.796610i \(-0.706624\pi\)
−0.604494 + 0.796610i \(0.706624\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6.24621i 0.424020i
\(218\) 0 0
\(219\) −6.63068 −0.448060
\(220\) 0 0
\(221\) 50.5464 3.40012
\(222\) 0 0
\(223\) − 24.6847i − 1.65301i −0.562932 0.826503i \(-0.690327\pi\)
0.562932 0.826503i \(-0.309673\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 11.3153i − 0.751026i −0.926817 0.375513i \(-0.877467\pi\)
0.926817 0.375513i \(-0.122533\pi\)
\(228\) 0 0
\(229\) 11.3693 0.751306 0.375653 0.926760i \(-0.377419\pi\)
0.375653 + 0.926760i \(0.377419\pi\)
\(230\) 0 0
\(231\) −2.43845 −0.160438
\(232\) 0 0
\(233\) 10.8769i 0.712569i 0.934378 + 0.356285i \(0.115957\pi\)
−0.934378 + 0.356285i \(0.884043\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 1.06913i − 0.0694475i
\(238\) 0 0
\(239\) 18.0540 1.16781 0.583907 0.811820i \(-0.301523\pi\)
0.583907 + 0.811820i \(0.301523\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 5.75379i 0.369106i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 47.6155i − 3.02970i
\(248\) 0 0
\(249\) 18.7386 1.18751
\(250\) 0 0
\(251\) 13.3693 0.843864 0.421932 0.906628i \(-0.361352\pi\)
0.421932 + 0.906628i \(0.361352\pi\)
\(252\) 0 0
\(253\) − 4.87689i − 0.306608i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 18.4924i − 1.15353i −0.816912 0.576763i \(-0.804316\pi\)
0.816912 0.576763i \(-0.195684\pi\)
\(258\) 0 0
\(259\) 8.24621 0.512395
\(260\) 0 0
\(261\) −0.246211 −0.0152401
\(262\) 0 0
\(263\) − 9.36932i − 0.577737i −0.957369 0.288868i \(-0.906721\pi\)
0.957369 0.288868i \(-0.0932790\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 8.00000i − 0.489592i
\(268\) 0 0
\(269\) −4.24621 −0.258896 −0.129448 0.991586i \(-0.541321\pi\)
−0.129448 + 0.991586i \(0.541321\pi\)
\(270\) 0 0
\(271\) −6.24621 −0.379430 −0.189715 0.981839i \(-0.560756\pi\)
−0.189715 + 0.981839i \(0.560756\pi\)
\(272\) 0 0
\(273\) 10.4384i 0.631764i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 8.24621i − 0.495467i −0.968828 0.247733i \(-0.920314\pi\)
0.968828 0.247733i \(-0.0796857\pi\)
\(278\) 0 0
\(279\) 3.50758 0.209993
\(280\) 0 0
\(281\) −19.5616 −1.16694 −0.583472 0.812133i \(-0.698306\pi\)
−0.583472 + 0.812133i \(0.698306\pi\)
\(282\) 0 0
\(283\) 4.68466i 0.278474i 0.990259 + 0.139237i \(0.0444650\pi\)
−0.990259 + 0.139237i \(0.955535\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 1.12311i − 0.0662948i
\(288\) 0 0
\(289\) −40.1771 −2.36336
\(290\) 0 0
\(291\) −2.05398 −0.120406
\(292\) 0 0
\(293\) − 32.0540i − 1.87261i −0.351184 0.936307i \(-0.614221\pi\)
0.351184 0.936307i \(-0.385779\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 8.68466i 0.503935i
\(298\) 0 0
\(299\) −20.8769 −1.20734
\(300\) 0 0
\(301\) −7.12311 −0.410569
\(302\) 0 0
\(303\) 9.36932i 0.538253i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 28.6847i − 1.63712i −0.574421 0.818560i \(-0.694773\pi\)
0.574421 0.818560i \(-0.305227\pi\)
\(308\) 0 0
\(309\) −18.4384 −1.04893
\(310\) 0 0
\(311\) −12.8769 −0.730182 −0.365091 0.930972i \(-0.618962\pi\)
−0.365091 + 0.930972i \(0.618962\pi\)
\(312\) 0 0
\(313\) − 26.6847i − 1.50831i −0.656699 0.754153i \(-0.728048\pi\)
0.656699 0.754153i \(-0.271952\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 10.0000i − 0.561656i −0.959758 0.280828i \(-0.909391\pi\)
0.959758 0.280828i \(-0.0906090\pi\)
\(318\) 0 0
\(319\) −0.684658 −0.0383335
\(320\) 0 0
\(321\) −23.6155 −1.31809
\(322\) 0 0
\(323\) 53.8617i 2.99695i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 6.93087i 0.383278i
\(328\) 0 0
\(329\) −2.43845 −0.134436
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) − 4.63068i − 0.253760i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000i 0.108947i 0.998515 + 0.0544735i \(0.0173480\pi\)
−0.998515 + 0.0544735i \(0.982652\pi\)
\(338\) 0 0
\(339\) 12.8769 0.699377
\(340\) 0 0
\(341\) 9.75379 0.528197
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.1231i 0.811851i 0.913906 + 0.405925i \(0.133051\pi\)
−0.913906 + 0.405925i \(0.866949\pi\)
\(348\) 0 0
\(349\) 11.7538 0.629166 0.314583 0.949230i \(-0.398135\pi\)
0.314583 + 0.949230i \(0.398135\pi\)
\(350\) 0 0
\(351\) 37.1771 1.98437
\(352\) 0 0
\(353\) 2.19224i 0.116681i 0.998297 + 0.0583405i \(0.0185809\pi\)
−0.998297 + 0.0583405i \(0.981419\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 11.8078i − 0.624933i
\(358\) 0 0
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) 31.7386 1.67045
\(362\) 0 0
\(363\) − 13.3693i − 0.701707i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 14.9309i 0.779385i 0.920945 + 0.389693i \(0.127419\pi\)
−0.920945 + 0.389693i \(0.872581\pi\)
\(368\) 0 0
\(369\) −0.630683 −0.0328321
\(370\) 0 0
\(371\) −13.1231 −0.681318
\(372\) 0 0
\(373\) − 15.3693i − 0.795793i −0.917430 0.397897i \(-0.869740\pi\)
0.917430 0.397897i \(-0.130260\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.93087i 0.150947i
\(378\) 0 0
\(379\) −32.4924 −1.66902 −0.834512 0.550990i \(-0.814250\pi\)
−0.834512 + 0.550990i \(0.814250\pi\)
\(380\) 0 0
\(381\) 9.75379 0.499702
\(382\) 0 0
\(383\) 9.75379i 0.498395i 0.968453 + 0.249198i \(0.0801669\pi\)
−0.968453 + 0.249198i \(0.919833\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.00000i 0.203331i
\(388\) 0 0
\(389\) −22.3002 −1.13066 −0.565332 0.824863i \(-0.691252\pi\)
−0.565332 + 0.824863i \(0.691252\pi\)
\(390\) 0 0
\(391\) 23.6155 1.19429
\(392\) 0 0
\(393\) 23.6155i 1.19125i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 23.1771i − 1.16322i −0.813466 0.581612i \(-0.802422\pi\)
0.813466 0.581612i \(-0.197578\pi\)
\(398\) 0 0
\(399\) −11.1231 −0.556852
\(400\) 0 0
\(401\) −12.9309 −0.645737 −0.322868 0.946444i \(-0.604647\pi\)
−0.322868 + 0.946444i \(0.604647\pi\)
\(402\) 0 0
\(403\) − 41.7538i − 2.07990i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 12.8769i − 0.638284i
\(408\) 0 0
\(409\) 18.4924 0.914391 0.457196 0.889366i \(-0.348854\pi\)
0.457196 + 0.889366i \(0.348854\pi\)
\(410\) 0 0
\(411\) 11.5076 0.567627
\(412\) 0 0
\(413\) 4.00000i 0.196827i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 33.3693i 1.63410i
\(418\) 0 0
\(419\) 18.2462 0.891386 0.445693 0.895186i \(-0.352957\pi\)
0.445693 + 0.895186i \(0.352957\pi\)
\(420\) 0 0
\(421\) 3.56155 0.173579 0.0867897 0.996227i \(-0.472339\pi\)
0.0867897 + 0.996227i \(0.472339\pi\)
\(422\) 0 0
\(423\) 1.36932i 0.0665785i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 6.87689i − 0.332796i
\(428\) 0 0
\(429\) 16.3002 0.786980
\(430\) 0 0
\(431\) −22.9309 −1.10454 −0.552271 0.833665i \(-0.686238\pi\)
−0.552271 + 0.833665i \(0.686238\pi\)
\(432\) 0 0
\(433\) − 19.7538i − 0.949307i −0.880173 0.474653i \(-0.842573\pi\)
0.880173 0.474653i \(-0.157427\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 22.2462i − 1.06418i
\(438\) 0 0
\(439\) −19.1231 −0.912696 −0.456348 0.889801i \(-0.650843\pi\)
−0.456348 + 0.889801i \(0.650843\pi\)
\(440\) 0 0
\(441\) −0.561553 −0.0267406
\(442\) 0 0
\(443\) 19.6155i 0.931962i 0.884795 + 0.465981i \(0.154298\pi\)
−0.884795 + 0.465981i \(0.845702\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0.384472i 0.0181849i
\(448\) 0 0
\(449\) 21.3153 1.00593 0.502967 0.864306i \(-0.332242\pi\)
0.502967 + 0.864306i \(0.332242\pi\)
\(450\) 0 0
\(451\) −1.75379 −0.0825827
\(452\) 0 0
\(453\) − 30.9309i − 1.45326i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.63068i 0.403726i 0.979414 + 0.201863i \(0.0646996\pi\)
−0.979414 + 0.201863i \(0.935300\pi\)
\(458\) 0 0
\(459\) −42.0540 −1.96291
\(460\) 0 0
\(461\) 18.8769 0.879185 0.439592 0.898197i \(-0.355123\pi\)
0.439592 + 0.898197i \(0.355123\pi\)
\(462\) 0 0
\(463\) 6.24621i 0.290286i 0.989411 + 0.145143i \(0.0463642\pi\)
−0.989411 + 0.145143i \(0.953636\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 25.5616i 1.18285i 0.806361 + 0.591424i \(0.201434\pi\)
−0.806361 + 0.591424i \(0.798566\pi\)
\(468\) 0 0
\(469\) −2.24621 −0.103720
\(470\) 0 0
\(471\) −6.63068 −0.305526
\(472\) 0 0
\(473\) 11.1231i 0.511441i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 7.36932i 0.337418i
\(478\) 0 0
\(479\) −17.3693 −0.793624 −0.396812 0.917900i \(-0.629884\pi\)
−0.396812 + 0.917900i \(0.629884\pi\)
\(480\) 0 0
\(481\) −55.1231 −2.51340
\(482\) 0 0
\(483\) 4.87689i 0.221906i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 3.12311i 0.141521i 0.997493 + 0.0707607i \(0.0225427\pi\)
−0.997493 + 0.0707607i \(0.977457\pi\)
\(488\) 0 0
\(489\) 30.6307 1.38517
\(490\) 0 0
\(491\) −3.31534 −0.149619 −0.0748096 0.997198i \(-0.523835\pi\)
−0.0748096 + 0.997198i \(0.523835\pi\)
\(492\) 0 0
\(493\) − 3.31534i − 0.149315i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0.192236 0.00860566 0.00430283 0.999991i \(-0.498630\pi\)
0.00430283 + 0.999991i \(0.498630\pi\)
\(500\) 0 0
\(501\) −6.54640 −0.292471
\(502\) 0 0
\(503\) 29.1771i 1.30094i 0.759531 + 0.650471i \(0.225428\pi\)
−0.759531 + 0.650471i \(0.774572\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 49.4773i − 2.19736i
\(508\) 0 0
\(509\) −32.7386 −1.45111 −0.725557 0.688162i \(-0.758418\pi\)
−0.725557 + 0.688162i \(0.758418\pi\)
\(510\) 0 0
\(511\) −4.24621 −0.187841
\(512\) 0 0
\(513\) 39.6155i 1.74907i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 3.80776i 0.167465i
\(518\) 0 0
\(519\) −36.1922 −1.58866
\(520\) 0 0
\(521\) −12.2462 −0.536516 −0.268258 0.963347i \(-0.586448\pi\)
−0.268258 + 0.963347i \(0.586448\pi\)
\(522\) 0 0
\(523\) 12.0000i 0.524723i 0.964970 + 0.262362i \(0.0845013\pi\)
−0.964970 + 0.262362i \(0.915499\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 47.2311i 2.05742i
\(528\) 0 0
\(529\) 13.2462 0.575922
\(530\) 0 0
\(531\) 2.24621 0.0974773
\(532\) 0 0
\(533\) 7.50758i 0.325189i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 6.24621i 0.269544i
\(538\) 0 0
\(539\) −1.56155 −0.0672608
\(540\) 0 0
\(541\) 41.4233 1.78093 0.890463 0.455055i \(-0.150380\pi\)
0.890463 + 0.455055i \(0.150380\pi\)
\(542\) 0 0
\(543\) − 8.00000i − 0.343313i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 12.0000i 0.513083i 0.966533 + 0.256541i \(0.0825830\pi\)
−0.966533 + 0.256541i \(0.917417\pi\)
\(548\) 0 0
\(549\) −3.86174 −0.164815
\(550\) 0 0
\(551\) −3.12311 −0.133049
\(552\) 0 0
\(553\) − 0.684658i − 0.0291146i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 17.6155i − 0.746394i −0.927752 0.373197i \(-0.878262\pi\)
0.927752 0.373197i \(-0.121738\pi\)
\(558\) 0 0
\(559\) 47.6155 2.01392
\(560\) 0 0
\(561\) −18.4384 −0.778472
\(562\) 0 0
\(563\) 7.50758i 0.316407i 0.987407 + 0.158203i \(0.0505701\pi\)
−0.987407 + 0.158203i \(0.949430\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 7.00000i − 0.293972i
\(568\) 0 0
\(569\) −18.9848 −0.795886 −0.397943 0.917410i \(-0.630276\pi\)
−0.397943 + 0.917410i \(0.630276\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) − 1.06913i − 0.0446636i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 3.56155i − 0.148269i −0.997248 0.0741347i \(-0.976381\pi\)
0.997248 0.0741347i \(-0.0236195\pi\)
\(578\) 0 0
\(579\) 20.4924 0.851636
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 20.4924i 0.848709i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10.2462i 0.422906i 0.977388 + 0.211453i \(0.0678196\pi\)
−0.977388 + 0.211453i \(0.932180\pi\)
\(588\) 0 0
\(589\) 44.4924 1.83328
\(590\) 0 0
\(591\) 20.4924 0.842946
\(592\) 0 0
\(593\) − 37.4233i − 1.53679i −0.639976 0.768395i \(-0.721056\pi\)
0.639976 0.768395i \(-0.278944\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 22.2462i 0.910477i
\(598\) 0 0
\(599\) 46.9309 1.91754 0.958772 0.284178i \(-0.0917205\pi\)
0.958772 + 0.284178i \(0.0917205\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 1.26137i 0.0513668i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 8.68466i − 0.352499i −0.984345 0.176250i \(-0.943603\pi\)
0.984345 0.176250i \(-0.0563966\pi\)
\(608\) 0 0
\(609\) 0.684658 0.0277438
\(610\) 0 0
\(611\) 16.3002 0.659435
\(612\) 0 0
\(613\) − 16.7386i − 0.676067i −0.941134 0.338034i \(-0.890238\pi\)
0.941134 0.338034i \(-0.109762\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 15.7538i − 0.634224i −0.948388 0.317112i \(-0.897287\pi\)
0.948388 0.317112i \(-0.102713\pi\)
\(618\) 0 0
\(619\) −10.6307 −0.427283 −0.213642 0.976912i \(-0.568532\pi\)
−0.213642 + 0.976912i \(0.568532\pi\)
\(620\) 0 0
\(621\) 17.3693 0.697007
\(622\) 0 0
\(623\) − 5.12311i − 0.205253i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 17.3693i 0.693664i
\(628\) 0 0
\(629\) 62.3542 2.48622
\(630\) 0 0
\(631\) −27.4233 −1.09170 −0.545852 0.837882i \(-0.683794\pi\)
−0.545852 + 0.837882i \(0.683794\pi\)
\(632\) 0 0
\(633\) − 27.4233i − 1.08998i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6.68466i 0.264856i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −22.9848 −0.907847 −0.453923 0.891041i \(-0.649976\pi\)
−0.453923 + 0.891041i \(0.649976\pi\)
\(642\) 0 0
\(643\) 30.0540i 1.18521i 0.805492 + 0.592607i \(0.201901\pi\)
−0.805492 + 0.592607i \(0.798099\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.00000i 0.314512i 0.987558 + 0.157256i \(0.0502649\pi\)
−0.987558 + 0.157256i \(0.949735\pi\)
\(648\) 0 0
\(649\) 6.24621 0.245185
\(650\) 0 0
\(651\) −9.75379 −0.382281
\(652\) 0 0
\(653\) 38.4924i 1.50632i 0.657835 + 0.753162i \(0.271473\pi\)
−0.657835 + 0.753162i \(0.728527\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.38447i 0.0930271i
\(658\) 0 0
\(659\) 0.192236 0.00748845 0.00374422 0.999993i \(-0.498808\pi\)
0.00374422 + 0.999993i \(0.498808\pi\)
\(660\) 0 0
\(661\) −17.6155 −0.685165 −0.342582 0.939488i \(-0.611302\pi\)
−0.342582 + 0.939488i \(0.611302\pi\)
\(662\) 0 0
\(663\) 78.9309i 3.06542i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.36932i 0.0530202i
\(668\) 0 0
\(669\) 38.5464 1.49029
\(670\) 0 0
\(671\) −10.7386 −0.414560
\(672\) 0 0
\(673\) − 41.6155i − 1.60416i −0.597216 0.802080i \(-0.703727\pi\)
0.597216 0.802080i \(-0.296273\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 1.31534i − 0.0505527i −0.999681 0.0252763i \(-0.991953\pi\)
0.999681 0.0252763i \(-0.00804657\pi\)
\(678\) 0 0
\(679\) −1.31534 −0.0504782
\(680\) 0 0
\(681\) 17.6695 0.677097
\(682\) 0 0
\(683\) − 44.9848i − 1.72130i −0.509199 0.860649i \(-0.670058\pi\)
0.509199 0.860649i \(-0.329942\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 17.7538i 0.677349i
\(688\) 0 0
\(689\) 87.7235 3.34200
\(690\) 0 0
\(691\) 16.4924 0.627401 0.313701 0.949522i \(-0.398431\pi\)
0.313701 + 0.949522i \(0.398431\pi\)
\(692\) 0 0
\(693\) 0.876894i 0.0333105i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 8.49242i − 0.321673i
\(698\) 0 0
\(699\) −16.9848 −0.642426
\(700\) 0 0
\(701\) −9.31534 −0.351836 −0.175918 0.984405i \(-0.556289\pi\)
−0.175918 + 0.984405i \(0.556289\pi\)
\(702\) 0 0
\(703\) − 58.7386i − 2.21537i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.00000i 0.225653i
\(708\) 0 0
\(709\) 4.05398 0.152250 0.0761251 0.997098i \(-0.475745\pi\)
0.0761251 + 0.997098i \(0.475745\pi\)
\(710\) 0 0
\(711\) −0.384472 −0.0144188
\(712\) 0 0
\(713\) − 19.5076i − 0.730565i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 28.1922i 1.05286i
\(718\) 0 0
\(719\) 23.6155 0.880711 0.440355 0.897824i \(-0.354852\pi\)
0.440355 + 0.897824i \(0.354852\pi\)
\(720\) 0 0
\(721\) −11.8078 −0.439744
\(722\) 0 0
\(723\) 3.12311i 0.116150i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 48.9848i 1.81675i 0.418159 + 0.908374i \(0.362675\pi\)
−0.418159 + 0.908374i \(0.637325\pi\)
\(728\) 0 0
\(729\) −29.9848 −1.11055
\(730\) 0 0
\(731\) −53.8617 −1.99215
\(732\) 0 0
\(733\) − 16.4384i − 0.607168i −0.952805 0.303584i \(-0.901817\pi\)
0.952805 0.303584i \(-0.0981833\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.50758i 0.129203i
\(738\) 0 0
\(739\) 6.43845 0.236842 0.118421 0.992963i \(-0.462217\pi\)
0.118421 + 0.992963i \(0.462217\pi\)
\(740\) 0 0
\(741\) 74.3542 2.73147
\(742\) 0 0
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 6.73863i − 0.246554i
\(748\) 0 0
\(749\) −15.1231 −0.552586
\(750\) 0 0
\(751\) −31.3153 −1.14271 −0.571357 0.820702i \(-0.693583\pi\)
−0.571357 + 0.820702i \(0.693583\pi\)
\(752\) 0 0
\(753\) 20.8769i 0.760796i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 15.3693i 0.558607i 0.960203 + 0.279304i \(0.0901036\pi\)
−0.960203 + 0.279304i \(0.909896\pi\)
\(758\) 0 0
\(759\) 7.61553 0.276426
\(760\) 0 0
\(761\) 26.0000 0.942499 0.471250 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) 0 0
\(763\) 4.43845i 0.160683i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 26.7386i − 0.965476i
\(768\) 0 0
\(769\) −42.9848 −1.55007 −0.775037 0.631916i \(-0.782269\pi\)
−0.775037 + 0.631916i \(0.782269\pi\)
\(770\) 0 0
\(771\) 28.8769 1.03998
\(772\) 0 0
\(773\) 29.8078i 1.07211i 0.844183 + 0.536055i \(0.180086\pi\)
−0.844183 + 0.536055i \(0.819914\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 12.8769i 0.461956i
\(778\) 0 0
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 2.43845i − 0.0871430i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 33.1771i 1.18264i 0.806439 + 0.591318i \(0.201392\pi\)
−0.806439 + 0.591318i \(0.798608\pi\)
\(788\) 0 0
\(789\) 14.6307 0.520866
\(790\) 0 0
\(791\) 8.24621 0.293202
\(792\) 0 0
\(793\) 45.9697i 1.63243i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 17.8078i 0.630783i 0.948962 + 0.315392i \(0.102136\pi\)
−0.948962 + 0.315392i \(0.897864\pi\)
\(798\) 0 0
\(799\) −18.4384 −0.652305
\(800\) 0 0
\(801\) −2.87689 −0.101650
\(802\) 0 0
\(803\) 6.63068i 0.233992i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 6.63068i − 0.233411i
\(808\) 0 0
\(809\) −4.05398 −0.142530 −0.0712651 0.997457i \(-0.522704\pi\)
−0.0712651 + 0.997457i \(0.522704\pi\)
\(810\) 0 0
\(811\) −27.6155 −0.969712 −0.484856 0.874594i \(-0.661128\pi\)
−0.484856 + 0.874594i \(0.661128\pi\)
\(812\) 0 0
\(813\) − 9.75379i − 0.342080i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 50.7386i 1.77512i
\(818\) 0 0
\(819\) 3.75379 0.131168
\(820\) 0 0
\(821\) 54.3002 1.89509 0.947545 0.319623i \(-0.103556\pi\)
0.947545 + 0.319623i \(0.103556\pi\)
\(822\) 0 0
\(823\) 17.7538i 0.618858i 0.950923 + 0.309429i \(0.100138\pi\)
−0.950923 + 0.309429i \(0.899862\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 24.8769i 0.865054i 0.901621 + 0.432527i \(0.142378\pi\)
−0.901621 + 0.432527i \(0.857622\pi\)
\(828\) 0 0
\(829\) −5.61553 −0.195035 −0.0975177 0.995234i \(-0.531090\pi\)
−0.0975177 + 0.995234i \(0.531090\pi\)
\(830\) 0 0
\(831\) 12.8769 0.446695
\(832\) 0 0
\(833\) − 7.56155i − 0.261992i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 34.7386i 1.20074i
\(838\) 0 0
\(839\) −19.1231 −0.660203 −0.330101 0.943945i \(-0.607083\pi\)
−0.330101 + 0.943945i \(0.607083\pi\)
\(840\) 0 0
\(841\) −28.8078 −0.993371
\(842\) 0 0
\(843\) − 30.5464i − 1.05207i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 8.56155i − 0.294178i
\(848\) 0 0
\(849\) −7.31534 −0.251062
\(850\) 0 0
\(851\) −25.7538 −0.882829
\(852\) 0 0
\(853\) − 15.7538i − 0.539399i −0.962944 0.269700i \(-0.913076\pi\)
0.962944 0.269700i \(-0.0869244\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 52.7386i 1.80152i 0.434320 + 0.900759i \(0.356989\pi\)
−0.434320 + 0.900759i \(0.643011\pi\)
\(858\) 0 0
\(859\) −36.9848 −1.26191 −0.630953 0.775821i \(-0.717336\pi\)
−0.630953 + 0.775821i \(0.717336\pi\)
\(860\) 0 0
\(861\) 1.75379 0.0597690
\(862\) 0 0
\(863\) − 28.4924i − 0.969893i −0.874544 0.484947i \(-0.838839\pi\)
0.874544 0.484947i \(-0.161161\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 62.7386i − 2.13072i
\(868\) 0 0
\(869\) −1.06913 −0.0362678
\(870\) 0 0
\(871\) 15.0152 0.508769
\(872\) 0 0
\(873\) 0.738634i 0.0249990i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 13.5076i − 0.456118i −0.973647 0.228059i \(-0.926762\pi\)
0.973647 0.228059i \(-0.0732380\pi\)
\(878\) 0 0
\(879\) 50.0540 1.68828
\(880\) 0 0
\(881\) 54.1080 1.82294 0.911472 0.411363i \(-0.134947\pi\)
0.911472 + 0.411363i \(0.134947\pi\)
\(882\) 0 0
\(883\) − 21.7538i − 0.732073i −0.930600 0.366037i \(-0.880714\pi\)
0.930600 0.366037i \(-0.119286\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 36.4924i − 1.22530i −0.790356 0.612648i \(-0.790104\pi\)
0.790356 0.612648i \(-0.209896\pi\)
\(888\) 0 0
\(889\) 6.24621 0.209491
\(890\) 0 0
\(891\) −10.9309 −0.366198
\(892\) 0 0
\(893\) 17.3693i 0.581242i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 32.6004i − 1.08849i
\(898\) 0 0
\(899\) −2.73863 −0.0913385
\(900\) 0 0
\(901\) −99.2311 −3.30587
\(902\) 0 0
\(903\) − 11.1231i − 0.370154i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 48.1080i − 1.59740i −0.601731 0.798699i \(-0.705522\pi\)
0.601731 0.798699i \(-0.294478\pi\)
\(908\) 0 0
\(909\) 3.36932 0.111753
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) − 18.7386i − 0.620158i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 15.1231i 0.499409i
\(918\) 0 0
\(919\) −2.43845 −0.0804370 −0.0402185 0.999191i \(-0.512805\pi\)
−0.0402185 + 0.999191i \(0.512805\pi\)
\(920\) 0 0
\(921\) 44.7926 1.47597
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 6.63068i 0.217780i
\(928\) 0 0
\(929\) 2.87689 0.0943878 0.0471939 0.998886i \(-0.484972\pi\)
0.0471939 + 0.998886i \(0.484972\pi\)
\(930\) 0 0
\(931\) −7.12311 −0.233450
\(932\) 0 0
\(933\) − 20.1080i − 0.658305i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 37.8078i 1.23513i 0.786521 + 0.617563i \(0.211880\pi\)
−0.786521 + 0.617563i \(0.788120\pi\)
\(938\) 0 0
\(939\) 41.6695 1.35983
\(940\) 0 0
\(941\) 28.6307 0.933334 0.466667 0.884433i \(-0.345455\pi\)
0.466667 + 0.884433i \(0.345455\pi\)
\(942\) 0 0
\(943\) 3.50758i 0.114222i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 35.2311i − 1.14486i −0.819955 0.572428i \(-0.806002\pi\)
0.819955 0.572428i \(-0.193998\pi\)
\(948\) 0 0
\(949\) 28.3845 0.921399
\(950\) 0 0
\(951\) 15.6155 0.506368
\(952\) 0 0
\(953\) 51.8617i 1.67997i 0.542612 + 0.839983i \(0.317435\pi\)
−0.542612 + 0.839983i \(0.682565\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 1.06913i − 0.0345601i
\(958\) 0 0
\(959\) 7.36932 0.237968
\(960\) 0 0
\(961\) 8.01515 0.258553
\(962\) 0 0
\(963\) 8.49242i 0.273664i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 44.1080i 1.41842i 0.704999 + 0.709208i \(0.250947\pi\)
−0.704999 + 0.709208i \(0.749053\pi\)
\(968\) 0 0
\(969\) −84.1080 −2.70194
\(970\) 0 0
\(971\) −22.7386 −0.729717 −0.364859 0.931063i \(-0.618883\pi\)
−0.364859 + 0.931063i \(0.618883\pi\)
\(972\) 0 0
\(973\) 21.3693i 0.685069i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 10.9848i 0.351436i 0.984441 + 0.175718i \(0.0562247\pi\)
−0.984441 + 0.175718i \(0.943775\pi\)
\(978\) 0 0
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) 2.49242 0.0795769
\(982\) 0 0
\(983\) 29.1771i 0.930604i 0.885152 + 0.465302i \(0.154054\pi\)
−0.885152 + 0.465302i \(0.845946\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 3.80776i − 0.121202i
\(988\) 0 0
\(989\) 22.2462 0.707388
\(990\) 0 0
\(991\) −20.4924 −0.650963 −0.325482 0.945548i \(-0.605526\pi\)
−0.325482 + 0.945548i \(0.605526\pi\)
\(992\) 0 0
\(993\) − 18.7386i − 0.594653i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 56.9309i − 1.80302i −0.432760 0.901509i \(-0.642460\pi\)
0.432760 0.901509i \(-0.357540\pi\)
\(998\) 0 0
\(999\) 45.8617 1.45100
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.k.449.3 4
4.3 odd 2 2800.2.g.u.449.2 4
5.2 odd 4 1400.2.a.p.1.2 2
5.3 odd 4 280.2.a.d.1.1 2
5.4 even 2 inner 1400.2.g.k.449.2 4
15.8 even 4 2520.2.a.w.1.1 2
20.3 even 4 560.2.a.g.1.2 2
20.7 even 4 2800.2.a.bn.1.1 2
20.19 odd 2 2800.2.g.u.449.3 4
35.3 even 12 1960.2.q.u.961.1 4
35.13 even 4 1960.2.a.r.1.2 2
35.18 odd 12 1960.2.q.s.961.2 4
35.23 odd 12 1960.2.q.s.361.2 4
35.27 even 4 9800.2.a.by.1.1 2
35.33 even 12 1960.2.q.u.361.1 4
40.3 even 4 2240.2.a.bi.1.1 2
40.13 odd 4 2240.2.a.be.1.2 2
60.23 odd 4 5040.2.a.bq.1.2 2
140.83 odd 4 3920.2.a.bu.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.d.1.1 2 5.3 odd 4
560.2.a.g.1.2 2 20.3 even 4
1400.2.a.p.1.2 2 5.2 odd 4
1400.2.g.k.449.2 4 5.4 even 2 inner
1400.2.g.k.449.3 4 1.1 even 1 trivial
1960.2.a.r.1.2 2 35.13 even 4
1960.2.q.s.361.2 4 35.23 odd 12
1960.2.q.s.961.2 4 35.18 odd 12
1960.2.q.u.361.1 4 35.33 even 12
1960.2.q.u.961.1 4 35.3 even 12
2240.2.a.be.1.2 2 40.13 odd 4
2240.2.a.bi.1.1 2 40.3 even 4
2520.2.a.w.1.1 2 15.8 even 4
2800.2.a.bn.1.1 2 20.7 even 4
2800.2.g.u.449.2 4 4.3 odd 2
2800.2.g.u.449.3 4 20.19 odd 2
3920.2.a.bu.1.1 2 140.83 odd 4
5040.2.a.bq.1.2 2 60.23 odd 4
9800.2.a.by.1.1 2 35.27 even 4