Properties

Label 1148.2.d.a.1065.19
Level $1148$
Weight $2$
Character 1148.1065
Analytic conductor $9.167$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \(x^{20} + 39 x^{18} + 531 x^{16} + 3488 x^{14} + 12661 x^{12} + 27027 x^{10} + 34540 x^{8} + 25909 x^{6} + 10677 x^{4} + 2092 x^{2} + 144\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1065.19
Root \(-1.52399i\) of defining polynomial
Character \(\chi\) \(=\) 1148.1065
Dual form 1148.2.d.a.1065.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.64112i q^{3} +0.508610 q^{5} +1.00000i q^{7} -3.97553 q^{9} +O(q^{10})\) \(q+2.64112i q^{3} +0.508610 q^{5} +1.00000i q^{7} -3.97553 q^{9} -1.38158i q^{11} +2.62949i q^{13} +1.34330i q^{15} +3.63814i q^{17} +0.0724145i q^{19} -2.64112 q^{21} -3.11469 q^{23} -4.74132 q^{25} -2.57650i q^{27} +5.90463i q^{29} -2.48761 q^{31} +3.64891 q^{33} +0.508610i q^{35} -0.477276 q^{37} -6.94482 q^{39} +(-0.803934 + 6.35246i) q^{41} -0.169419 q^{43} -2.02199 q^{45} -7.87970i q^{47} -1.00000 q^{49} -9.60879 q^{51} -1.97267i q^{53} -0.702683i q^{55} -0.191256 q^{57} -4.22422 q^{59} +10.4481 q^{61} -3.97553i q^{63} +1.33739i q^{65} -8.77120i q^{67} -8.22627i q^{69} +0.853928i q^{71} +6.17800 q^{73} -12.5224i q^{75} +1.38158 q^{77} -1.30234i q^{79} -5.12175 q^{81} +5.00782 q^{83} +1.85040i q^{85} -15.5949 q^{87} +6.89431i q^{89} -2.62949 q^{91} -6.57009i q^{93} +0.0368307i q^{95} -9.36862i q^{97} +5.49250i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{5} - 20q^{9} + O(q^{10}) \) \( 20q + 4q^{5} - 20q^{9} + 4q^{21} + 8q^{31} + 20q^{37} + 4q^{39} - 16q^{41} + 20q^{43} - 4q^{45} - 20q^{49} + 52q^{51} - 36q^{57} + 20q^{59} - 4q^{61} - 12q^{73} + 8q^{77} + 20q^{81} - 48q^{83} + 44q^{87} - 4q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 2.64112i 1.52485i 0.647075 + 0.762427i \(0.275992\pi\)
−0.647075 + 0.762427i \(0.724008\pi\)
\(4\) 0 0
\(5\) 0.508610 0.227457 0.113729 0.993512i \(-0.463721\pi\)
0.113729 + 0.993512i \(0.463721\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −3.97553 −1.32518
\(10\) 0 0
\(11\) 1.38158i 0.416561i −0.978069 0.208280i \(-0.933213\pi\)
0.978069 0.208280i \(-0.0667867\pi\)
\(12\) 0 0
\(13\) 2.62949i 0.729291i 0.931146 + 0.364645i \(0.118810\pi\)
−0.931146 + 0.364645i \(0.881190\pi\)
\(14\) 0 0
\(15\) 1.34330i 0.346839i
\(16\) 0 0
\(17\) 3.63814i 0.882380i 0.897414 + 0.441190i \(0.145443\pi\)
−0.897414 + 0.441190i \(0.854557\pi\)
\(18\) 0 0
\(19\) 0.0724145i 0.0166130i 0.999966 + 0.00830652i \(0.00264408\pi\)
−0.999966 + 0.00830652i \(0.997356\pi\)
\(20\) 0 0
\(21\) −2.64112 −0.576340
\(22\) 0 0
\(23\) −3.11469 −0.649457 −0.324729 0.945807i \(-0.605273\pi\)
−0.324729 + 0.945807i \(0.605273\pi\)
\(24\) 0 0
\(25\) −4.74132 −0.948263
\(26\) 0 0
\(27\) 2.57650i 0.495847i
\(28\) 0 0
\(29\) 5.90463i 1.09646i 0.836327 + 0.548231i \(0.184699\pi\)
−0.836327 + 0.548231i \(0.815301\pi\)
\(30\) 0 0
\(31\) −2.48761 −0.446789 −0.223394 0.974728i \(-0.571714\pi\)
−0.223394 + 0.974728i \(0.571714\pi\)
\(32\) 0 0
\(33\) 3.64891 0.635194
\(34\) 0 0
\(35\) 0.508610i 0.0859708i
\(36\) 0 0
\(37\) −0.477276 −0.0784637 −0.0392319 0.999230i \(-0.512491\pi\)
−0.0392319 + 0.999230i \(0.512491\pi\)
\(38\) 0 0
\(39\) −6.94482 −1.11206
\(40\) 0 0
\(41\) −0.803934 + 6.35246i −0.125553 + 0.992087i
\(42\) 0 0
\(43\) −0.169419 −0.0258362 −0.0129181 0.999917i \(-0.504112\pi\)
−0.0129181 + 0.999917i \(0.504112\pi\)
\(44\) 0 0
\(45\) −2.02199 −0.301421
\(46\) 0 0
\(47\) 7.87970i 1.14937i −0.818374 0.574686i \(-0.805124\pi\)
0.818374 0.574686i \(-0.194876\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −9.60879 −1.34550
\(52\) 0 0
\(53\) 1.97267i 0.270966i −0.990780 0.135483i \(-0.956741\pi\)
0.990780 0.135483i \(-0.0432587\pi\)
\(54\) 0 0
\(55\) 0.702683i 0.0947498i
\(56\) 0 0
\(57\) −0.191256 −0.0253324
\(58\) 0 0
\(59\) −4.22422 −0.549947 −0.274974 0.961452i \(-0.588669\pi\)
−0.274974 + 0.961452i \(0.588669\pi\)
\(60\) 0 0
\(61\) 10.4481 1.33775 0.668873 0.743377i \(-0.266777\pi\)
0.668873 + 0.743377i \(0.266777\pi\)
\(62\) 0 0
\(63\) 3.97553i 0.500870i
\(64\) 0 0
\(65\) 1.33739i 0.165882i
\(66\) 0 0
\(67\) 8.77120i 1.07157i −0.844354 0.535786i \(-0.820015\pi\)
0.844354 0.535786i \(-0.179985\pi\)
\(68\) 0 0
\(69\) 8.22627i 0.990327i
\(70\) 0 0
\(71\) 0.853928i 0.101343i 0.998715 + 0.0506713i \(0.0161361\pi\)
−0.998715 + 0.0506713i \(0.983864\pi\)
\(72\) 0 0
\(73\) 6.17800 0.723081 0.361540 0.932356i \(-0.382251\pi\)
0.361540 + 0.932356i \(0.382251\pi\)
\(74\) 0 0
\(75\) 12.5224i 1.44596i
\(76\) 0 0
\(77\) 1.38158 0.157445
\(78\) 0 0
\(79\) 1.30234i 0.146525i −0.997313 0.0732624i \(-0.976659\pi\)
0.997313 0.0732624i \(-0.0233411\pi\)
\(80\) 0 0
\(81\) −5.12175 −0.569083
\(82\) 0 0
\(83\) 5.00782 0.549680 0.274840 0.961490i \(-0.411375\pi\)
0.274840 + 0.961490i \(0.411375\pi\)
\(84\) 0 0
\(85\) 1.85040i 0.200704i
\(86\) 0 0
\(87\) −15.5949 −1.67194
\(88\) 0 0
\(89\) 6.89431i 0.730796i 0.930851 + 0.365398i \(0.119067\pi\)
−0.930851 + 0.365398i \(0.880933\pi\)
\(90\) 0 0
\(91\) −2.62949 −0.275646
\(92\) 0 0
\(93\) 6.57009i 0.681287i
\(94\) 0 0
\(95\) 0.0368307i 0.00377876i
\(96\) 0 0
\(97\) 9.36862i 0.951239i −0.879651 0.475620i \(-0.842224\pi\)
0.879651 0.475620i \(-0.157776\pi\)
\(98\) 0 0
\(99\) 5.49250i 0.552017i
\(100\) 0 0
\(101\) 9.89885i 0.984972i 0.870320 + 0.492486i \(0.163912\pi\)
−0.870320 + 0.492486i \(0.836088\pi\)
\(102\) 0 0
\(103\) −0.792482 −0.0780856 −0.0390428 0.999238i \(-0.512431\pi\)
−0.0390428 + 0.999238i \(0.512431\pi\)
\(104\) 0 0
\(105\) −1.34330 −0.131093
\(106\) 0 0
\(107\) −14.0102 −1.35442 −0.677209 0.735791i \(-0.736811\pi\)
−0.677209 + 0.735791i \(0.736811\pi\)
\(108\) 0 0
\(109\) 11.0323i 1.05670i 0.849026 + 0.528351i \(0.177190\pi\)
−0.849026 + 0.528351i \(0.822810\pi\)
\(110\) 0 0
\(111\) 1.26055i 0.119646i
\(112\) 0 0
\(113\) −1.22227 −0.114982 −0.0574909 0.998346i \(-0.518310\pi\)
−0.0574909 + 0.998346i \(0.518310\pi\)
\(114\) 0 0
\(115\) −1.58416 −0.147724
\(116\) 0 0
\(117\) 10.4536i 0.966439i
\(118\) 0 0
\(119\) −3.63814 −0.333508
\(120\) 0 0
\(121\) 9.09125 0.826477
\(122\) 0 0
\(123\) −16.7776 2.12329i −1.51279 0.191451i
\(124\) 0 0
\(125\) −4.95453 −0.443147
\(126\) 0 0
\(127\) 4.17280 0.370276 0.185138 0.982713i \(-0.440727\pi\)
0.185138 + 0.982713i \(0.440727\pi\)
\(128\) 0 0
\(129\) 0.447458i 0.0393965i
\(130\) 0 0
\(131\) 10.5869 0.924980 0.462490 0.886624i \(-0.346956\pi\)
0.462490 + 0.886624i \(0.346956\pi\)
\(132\) 0 0
\(133\) −0.0724145 −0.00627914
\(134\) 0 0
\(135\) 1.31043i 0.112784i
\(136\) 0 0
\(137\) 9.05390i 0.773527i 0.922179 + 0.386764i \(0.126407\pi\)
−0.922179 + 0.386764i \(0.873593\pi\)
\(138\) 0 0
\(139\) −4.99820 −0.423942 −0.211971 0.977276i \(-0.567988\pi\)
−0.211971 + 0.977276i \(0.567988\pi\)
\(140\) 0 0
\(141\) 20.8113 1.75262
\(142\) 0 0
\(143\) 3.63285 0.303794
\(144\) 0 0
\(145\) 3.00315i 0.249398i
\(146\) 0 0
\(147\) 2.64112i 0.217836i
\(148\) 0 0
\(149\) 1.83394i 0.150242i −0.997174 0.0751210i \(-0.976066\pi\)
0.997174 0.0751210i \(-0.0239343\pi\)
\(150\) 0 0
\(151\) 16.2271i 1.32054i 0.751026 + 0.660272i \(0.229559\pi\)
−0.751026 + 0.660272i \(0.770441\pi\)
\(152\) 0 0
\(153\) 14.4636i 1.16931i
\(154\) 0 0
\(155\) −1.26522 −0.101625
\(156\) 0 0
\(157\) 2.55114i 0.203603i 0.994805 + 0.101801i \(0.0324606\pi\)
−0.994805 + 0.101801i \(0.967539\pi\)
\(158\) 0 0
\(159\) 5.21005 0.413184
\(160\) 0 0
\(161\) 3.11469i 0.245472i
\(162\) 0 0
\(163\) 8.52310 0.667581 0.333790 0.942647i \(-0.391672\pi\)
0.333790 + 0.942647i \(0.391672\pi\)
\(164\) 0 0
\(165\) 1.85587 0.144480
\(166\) 0 0
\(167\) 6.36055i 0.492194i −0.969245 0.246097i \(-0.920852\pi\)
0.969245 0.246097i \(-0.0791482\pi\)
\(168\) 0 0
\(169\) 6.08576 0.468135
\(170\) 0 0
\(171\) 0.287886i 0.0220152i
\(172\) 0 0
\(173\) 11.0590 0.840804 0.420402 0.907338i \(-0.361889\pi\)
0.420402 + 0.907338i \(0.361889\pi\)
\(174\) 0 0
\(175\) 4.74132i 0.358410i
\(176\) 0 0
\(177\) 11.1567i 0.838589i
\(178\) 0 0
\(179\) 2.28487i 0.170779i −0.996348 0.0853895i \(-0.972787\pi\)
0.996348 0.0853895i \(-0.0272135\pi\)
\(180\) 0 0
\(181\) 2.46071i 0.182903i 0.995810 + 0.0914515i \(0.0291506\pi\)
−0.995810 + 0.0914515i \(0.970849\pi\)
\(182\) 0 0
\(183\) 27.5948i 2.03987i
\(184\) 0 0
\(185\) −0.242747 −0.0178471
\(186\) 0 0
\(187\) 5.02637 0.367565
\(188\) 0 0
\(189\) 2.57650 0.187413
\(190\) 0 0
\(191\) 6.76820i 0.489730i 0.969557 + 0.244865i \(0.0787437\pi\)
−0.969557 + 0.244865i \(0.921256\pi\)
\(192\) 0 0
\(193\) 12.6140i 0.907979i 0.891007 + 0.453989i \(0.150000\pi\)
−0.891007 + 0.453989i \(0.850000\pi\)
\(194\) 0 0
\(195\) −3.53220 −0.252946
\(196\) 0 0
\(197\) 10.9937 0.783265 0.391633 0.920122i \(-0.371910\pi\)
0.391633 + 0.920122i \(0.371910\pi\)
\(198\) 0 0
\(199\) 22.0421i 1.56252i 0.624206 + 0.781260i \(0.285423\pi\)
−0.624206 + 0.781260i \(0.714577\pi\)
\(200\) 0 0
\(201\) 23.1658 1.63399
\(202\) 0 0
\(203\) −5.90463 −0.414424
\(204\) 0 0
\(205\) −0.408889 + 3.23092i −0.0285580 + 0.225657i
\(206\) 0 0
\(207\) 12.3825 0.860646
\(208\) 0 0
\(209\) 0.100046 0.00692034
\(210\) 0 0
\(211\) 1.59811i 0.110018i −0.998486 0.0550092i \(-0.982481\pi\)
0.998486 0.0550092i \(-0.0175188\pi\)
\(212\) 0 0
\(213\) −2.25533 −0.154533
\(214\) 0 0
\(215\) −0.0861684 −0.00587664
\(216\) 0 0
\(217\) 2.48761i 0.168870i
\(218\) 0 0
\(219\) 16.3169i 1.10259i
\(220\) 0 0
\(221\) −9.56648 −0.643511
\(222\) 0 0
\(223\) 28.1669 1.88620 0.943098 0.332515i \(-0.107897\pi\)
0.943098 + 0.332515i \(0.107897\pi\)
\(224\) 0 0
\(225\) 18.8492 1.25662
\(226\) 0 0
\(227\) 11.3800i 0.755320i −0.925944 0.377660i \(-0.876729\pi\)
0.925944 0.377660i \(-0.123271\pi\)
\(228\) 0 0
\(229\) 7.55923i 0.499528i 0.968307 + 0.249764i \(0.0803530\pi\)
−0.968307 + 0.249764i \(0.919647\pi\)
\(230\) 0 0
\(231\) 3.64891i 0.240081i
\(232\) 0 0
\(233\) 4.36747i 0.286122i −0.989714 0.143061i \(-0.954305\pi\)
0.989714 0.143061i \(-0.0456945\pi\)
\(234\) 0 0
\(235\) 4.00769i 0.261433i
\(236\) 0 0
\(237\) 3.43964 0.223429
\(238\) 0 0
\(239\) 25.4430i 1.64577i 0.568208 + 0.822885i \(0.307637\pi\)
−0.568208 + 0.822885i \(0.692363\pi\)
\(240\) 0 0
\(241\) 19.8105 1.27611 0.638053 0.769993i \(-0.279740\pi\)
0.638053 + 0.769993i \(0.279740\pi\)
\(242\) 0 0
\(243\) 21.2567i 1.36361i
\(244\) 0 0
\(245\) −0.508610 −0.0324939
\(246\) 0 0
\(247\) −0.190414 −0.0121157
\(248\) 0 0
\(249\) 13.2263i 0.838181i
\(250\) 0 0
\(251\) −0.0327888 −0.00206961 −0.00103480 0.999999i \(-0.500329\pi\)
−0.00103480 + 0.999999i \(0.500329\pi\)
\(252\) 0 0
\(253\) 4.30318i 0.270538i
\(254\) 0 0
\(255\) −4.88712 −0.306044
\(256\) 0 0
\(257\) 12.6386i 0.788371i −0.919031 0.394186i \(-0.871027\pi\)
0.919031 0.394186i \(-0.128973\pi\)
\(258\) 0 0
\(259\) 0.477276i 0.0296565i
\(260\) 0 0
\(261\) 23.4740i 1.45301i
\(262\) 0 0
\(263\) 13.3942i 0.825922i 0.910749 + 0.412961i \(0.135505\pi\)
−0.910749 + 0.412961i \(0.864495\pi\)
\(264\) 0 0
\(265\) 1.00332i 0.0616333i
\(266\) 0 0
\(267\) −18.2087 −1.11436
\(268\) 0 0
\(269\) −27.8300 −1.69683 −0.848413 0.529335i \(-0.822442\pi\)
−0.848413 + 0.529335i \(0.822442\pi\)
\(270\) 0 0
\(271\) −0.756271 −0.0459402 −0.0229701 0.999736i \(-0.507312\pi\)
−0.0229701 + 0.999736i \(0.507312\pi\)
\(272\) 0 0
\(273\) 6.94482i 0.420320i
\(274\) 0 0
\(275\) 6.55049i 0.395009i
\(276\) 0 0
\(277\) −8.35529 −0.502021 −0.251010 0.967984i \(-0.580763\pi\)
−0.251010 + 0.967984i \(0.580763\pi\)
\(278\) 0 0
\(279\) 9.88958 0.592074
\(280\) 0 0
\(281\) 17.9049i 1.06812i 0.845447 + 0.534060i \(0.179334\pi\)
−0.845447 + 0.534060i \(0.820666\pi\)
\(282\) 0 0
\(283\) 27.2783 1.62153 0.810763 0.585374i \(-0.199052\pi\)
0.810763 + 0.585374i \(0.199052\pi\)
\(284\) 0 0
\(285\) −0.0972745 −0.00576205
\(286\) 0 0
\(287\) −6.35246 0.803934i −0.374974 0.0474547i
\(288\) 0 0
\(289\) 3.76391 0.221406
\(290\) 0 0
\(291\) 24.7437 1.45050
\(292\) 0 0
\(293\) 19.1720i 1.12004i −0.828479 0.560021i \(-0.810793\pi\)
0.828479 0.560021i \(-0.189207\pi\)
\(294\) 0 0
\(295\) −2.14848 −0.125089
\(296\) 0 0
\(297\) −3.55963 −0.206551
\(298\) 0 0
\(299\) 8.19005i 0.473643i
\(300\) 0 0
\(301\) 0.169419i 0.00976518i
\(302\) 0 0
\(303\) −26.1441 −1.50194
\(304\) 0 0
\(305\) 5.31402 0.304280
\(306\) 0 0
\(307\) 20.2272 1.15443 0.577213 0.816594i \(-0.304140\pi\)
0.577213 + 0.816594i \(0.304140\pi\)
\(308\) 0 0
\(309\) 2.09304i 0.119069i
\(310\) 0 0
\(311\) 3.34846i 0.189874i 0.995483 + 0.0949368i \(0.0302649\pi\)
−0.995483 + 0.0949368i \(0.969735\pi\)
\(312\) 0 0
\(313\) 2.29250i 0.129580i 0.997899 + 0.0647899i \(0.0206377\pi\)
−0.997899 + 0.0647899i \(0.979362\pi\)
\(314\) 0 0
\(315\) 2.02199i 0.113926i
\(316\) 0 0
\(317\) 1.09588i 0.0615507i 0.999526 + 0.0307754i \(0.00979765\pi\)
−0.999526 + 0.0307754i \(0.990202\pi\)
\(318\) 0 0
\(319\) 8.15770 0.456743
\(320\) 0 0
\(321\) 37.0027i 2.06529i
\(322\) 0 0
\(323\) −0.263454 −0.0146590
\(324\) 0 0
\(325\) 12.4673i 0.691560i
\(326\) 0 0
\(327\) −29.1377 −1.61132
\(328\) 0 0
\(329\) 7.87970 0.434422
\(330\) 0 0
\(331\) 29.3385i 1.61259i −0.591515 0.806294i \(-0.701470\pi\)
0.591515 0.806294i \(-0.298530\pi\)
\(332\) 0 0
\(333\) 1.89743 0.103978
\(334\) 0 0
\(335\) 4.46112i 0.243737i
\(336\) 0 0
\(337\) 0.155119 0.00844988 0.00422494 0.999991i \(-0.498655\pi\)
0.00422494 + 0.999991i \(0.498655\pi\)
\(338\) 0 0
\(339\) 3.22817i 0.175330i
\(340\) 0 0
\(341\) 3.43683i 0.186115i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 4.18396i 0.225257i
\(346\) 0 0
\(347\) 16.6077i 0.891550i −0.895145 0.445775i \(-0.852928\pi\)
0.895145 0.445775i \(-0.147072\pi\)
\(348\) 0 0
\(349\) −23.3856 −1.25180 −0.625900 0.779903i \(-0.715268\pi\)
−0.625900 + 0.779903i \(0.715268\pi\)
\(350\) 0 0
\(351\) 6.77489 0.361617
\(352\) 0 0
\(353\) 15.1859 0.808262 0.404131 0.914701i \(-0.367574\pi\)
0.404131 + 0.914701i \(0.367574\pi\)
\(354\) 0 0
\(355\) 0.434316i 0.0230511i
\(356\) 0 0
\(357\) 9.60879i 0.508551i
\(358\) 0 0
\(359\) −6.88726 −0.363495 −0.181748 0.983345i \(-0.558175\pi\)
−0.181748 + 0.983345i \(0.558175\pi\)
\(360\) 0 0
\(361\) 18.9948 0.999724
\(362\) 0 0
\(363\) 24.0111i 1.26026i
\(364\) 0 0
\(365\) 3.14219 0.164470
\(366\) 0 0
\(367\) 10.5256 0.549430 0.274715 0.961526i \(-0.411416\pi\)
0.274715 + 0.961526i \(0.411416\pi\)
\(368\) 0 0
\(369\) 3.19606 25.2544i 0.166380 1.31469i
\(370\) 0 0
\(371\) 1.97267 0.102416
\(372\) 0 0
\(373\) 20.8894 1.08161 0.540807 0.841147i \(-0.318119\pi\)
0.540807 + 0.841147i \(0.318119\pi\)
\(374\) 0 0
\(375\) 13.0855i 0.675734i
\(376\) 0 0
\(377\) −15.5262 −0.799640
\(378\) 0 0
\(379\) −13.4170 −0.689183 −0.344592 0.938753i \(-0.611983\pi\)
−0.344592 + 0.938753i \(0.611983\pi\)
\(380\) 0 0
\(381\) 11.0209i 0.564617i
\(382\) 0 0
\(383\) 4.10325i 0.209666i −0.994490 0.104833i \(-0.966569\pi\)
0.994490 0.104833i \(-0.0334309\pi\)
\(384\) 0 0
\(385\) 0.702683 0.0358121
\(386\) 0 0
\(387\) 0.673532 0.0342376
\(388\) 0 0
\(389\) −2.82685 −0.143327 −0.0716636 0.997429i \(-0.522831\pi\)
−0.0716636 + 0.997429i \(0.522831\pi\)
\(390\) 0 0
\(391\) 11.3317i 0.573068i
\(392\) 0 0
\(393\) 27.9613i 1.41046i
\(394\) 0 0
\(395\) 0.662383i 0.0333281i
\(396\) 0 0
\(397\) 13.6818i 0.686670i −0.939213 0.343335i \(-0.888443\pi\)
0.939213 0.343335i \(-0.111557\pi\)
\(398\) 0 0
\(399\) 0.191256i 0.00957476i
\(400\) 0 0
\(401\) 29.0564 1.45101 0.725503 0.688219i \(-0.241607\pi\)
0.725503 + 0.688219i \(0.241607\pi\)
\(402\) 0 0
\(403\) 6.54117i 0.325839i
\(404\) 0 0
\(405\) −2.60497 −0.129442
\(406\) 0 0
\(407\) 0.659393i 0.0326849i
\(408\) 0 0
\(409\) 26.6631 1.31840 0.659202 0.751966i \(-0.270894\pi\)
0.659202 + 0.751966i \(0.270894\pi\)
\(410\) 0 0
\(411\) −23.9125 −1.17952
\(412\) 0 0
\(413\) 4.22422i 0.207860i
\(414\) 0 0
\(415\) 2.54703 0.125029
\(416\) 0 0
\(417\) 13.2009i 0.646449i
\(418\) 0 0
\(419\) −8.54721 −0.417559 −0.208779 0.977963i \(-0.566949\pi\)
−0.208779 + 0.977963i \(0.566949\pi\)
\(420\) 0 0
\(421\) 5.69195i 0.277409i 0.990334 + 0.138704i \(0.0442938\pi\)
−0.990334 + 0.138704i \(0.955706\pi\)
\(422\) 0 0
\(423\) 31.3260i 1.52312i
\(424\) 0 0
\(425\) 17.2496i 0.836728i
\(426\) 0 0
\(427\) 10.4481i 0.505620i
\(428\) 0 0
\(429\) 9.59480i 0.463241i
\(430\) 0 0
\(431\) −17.8902 −0.861740 −0.430870 0.902414i \(-0.641793\pi\)
−0.430870 + 0.902414i \(0.641793\pi\)
\(432\) 0 0
\(433\) −12.3887 −0.595364 −0.297682 0.954665i \(-0.596214\pi\)
−0.297682 + 0.954665i \(0.596214\pi\)
\(434\) 0 0
\(435\) −7.93170 −0.380296
\(436\) 0 0
\(437\) 0.225549i 0.0107895i
\(438\) 0 0
\(439\) 14.4192i 0.688192i −0.938934 0.344096i \(-0.888185\pi\)
0.938934 0.344096i \(-0.111815\pi\)
\(440\) 0 0
\(441\) 3.97553 0.189311
\(442\) 0 0
\(443\) −6.41166 −0.304627 −0.152314 0.988332i \(-0.548672\pi\)
−0.152314 + 0.988332i \(0.548672\pi\)
\(444\) 0 0
\(445\) 3.50652i 0.166225i
\(446\) 0 0
\(447\) 4.84366 0.229097
\(448\) 0 0
\(449\) 2.48174 0.117120 0.0585602 0.998284i \(-0.481349\pi\)
0.0585602 + 0.998284i \(0.481349\pi\)
\(450\) 0 0
\(451\) 8.77640 + 1.11070i 0.413265 + 0.0523006i
\(452\) 0 0
\(453\) −42.8578 −2.01364
\(454\) 0 0
\(455\) −1.33739 −0.0626977
\(456\) 0 0
\(457\) 26.4456i 1.23707i −0.785756 0.618537i \(-0.787726\pi\)
0.785756 0.618537i \(-0.212274\pi\)
\(458\) 0 0
\(459\) 9.37367 0.437525
\(460\) 0 0
\(461\) −24.4793 −1.14012 −0.570058 0.821605i \(-0.693079\pi\)
−0.570058 + 0.821605i \(0.693079\pi\)
\(462\) 0 0
\(463\) 29.5271i 1.37224i 0.727489 + 0.686119i \(0.240687\pi\)
−0.727489 + 0.686119i \(0.759313\pi\)
\(464\) 0 0
\(465\) 3.34161i 0.154964i
\(466\) 0 0
\(467\) −38.5304 −1.78297 −0.891487 0.453047i \(-0.850337\pi\)
−0.891487 + 0.453047i \(0.850337\pi\)
\(468\) 0 0
\(469\) 8.77120 0.405016
\(470\) 0 0
\(471\) −6.73787 −0.310465
\(472\) 0 0
\(473\) 0.234066i 0.0107624i
\(474\) 0 0
\(475\) 0.343340i 0.0157535i
\(476\) 0 0
\(477\) 7.84239i 0.359079i
\(478\) 0 0
\(479\) 20.8626i 0.953235i 0.879111 + 0.476618i \(0.158137\pi\)
−0.879111 + 0.476618i \(0.841863\pi\)
\(480\) 0 0
\(481\) 1.25500i 0.0572229i
\(482\) 0 0
\(483\) 8.22627 0.374308
\(484\) 0 0
\(485\) 4.76497i 0.216366i
\(486\) 0 0
\(487\) 28.7296 1.30186 0.650931 0.759137i \(-0.274379\pi\)
0.650931 + 0.759137i \(0.274379\pi\)
\(488\) 0 0
\(489\) 22.5106i 1.01796i
\(490\) 0 0
\(491\) −32.6652 −1.47416 −0.737079 0.675806i \(-0.763796\pi\)
−0.737079 + 0.675806i \(0.763796\pi\)
\(492\) 0 0
\(493\) −21.4819 −0.967496
\(494\) 0 0
\(495\) 2.79354i 0.125560i
\(496\) 0 0
\(497\) −0.853928 −0.0383039
\(498\) 0 0
\(499\) 14.7265i 0.659250i −0.944112 0.329625i \(-0.893078\pi\)
0.944112 0.329625i \(-0.106922\pi\)
\(500\) 0 0
\(501\) 16.7990 0.750524
\(502\) 0 0
\(503\) 14.5002i 0.646532i 0.946308 + 0.323266i \(0.104781\pi\)
−0.946308 + 0.323266i \(0.895219\pi\)
\(504\) 0 0
\(505\) 5.03465i 0.224039i
\(506\) 0 0
\(507\) 16.0732i 0.713837i
\(508\) 0 0
\(509\) 18.2537i 0.809080i 0.914520 + 0.404540i \(0.132568\pi\)
−0.914520 + 0.404540i \(0.867432\pi\)
\(510\) 0 0
\(511\) 6.17800i 0.273299i
\(512\) 0 0
\(513\) 0.186576 0.00823753
\(514\) 0 0
\(515\) −0.403064 −0.0177611
\(516\) 0 0
\(517\) −10.8864 −0.478784
\(518\) 0 0
\(519\) 29.2083i 1.28210i
\(520\) 0 0
\(521\) 0.597365i 0.0261710i 0.999914 + 0.0130855i \(0.00416537\pi\)
−0.999914 + 0.0130855i \(0.995835\pi\)
\(522\) 0 0
\(523\) −23.8032 −1.04084 −0.520421 0.853910i \(-0.674225\pi\)
−0.520421 + 0.853910i \(0.674225\pi\)
\(524\) 0 0
\(525\) 12.5224 0.546522
\(526\) 0 0
\(527\) 9.05030i 0.394237i
\(528\) 0 0
\(529\) −13.2987 −0.578206
\(530\) 0 0
\(531\) 16.7935 0.728777
\(532\) 0 0
\(533\) −16.7037 2.11394i −0.723520 0.0915649i
\(534\) 0 0
\(535\) −7.12573 −0.308072
\(536\) 0 0
\(537\) 6.03462 0.260413
\(538\) 0 0
\(539\) 1.38158i 0.0595087i
\(540\) 0 0
\(541\) 12.9049 0.554823 0.277411 0.960751i \(-0.410523\pi\)
0.277411 + 0.960751i \(0.410523\pi\)
\(542\) 0 0
\(543\) −6.49903 −0.278900
\(544\) 0 0
\(545\) 5.61114i 0.240355i
\(546\) 0 0
\(547\) 17.0350i 0.728366i 0.931327 + 0.364183i \(0.118652\pi\)
−0.931327 + 0.364183i \(0.881348\pi\)
\(548\) 0 0
\(549\) −41.5369 −1.77275
\(550\) 0 0
\(551\) −0.427581 −0.0182156
\(552\) 0 0
\(553\) 1.30234 0.0553812
\(554\) 0 0
\(555\) 0.641126i 0.0272143i
\(556\) 0 0
\(557\) 8.89705i 0.376980i 0.982075 + 0.188490i \(0.0603593\pi\)
−0.982075 + 0.188490i \(0.939641\pi\)
\(558\) 0 0
\(559\) 0.445488i 0.0188421i
\(560\) 0 0
\(561\) 13.2753i 0.560482i
\(562\) 0 0
\(563\) 23.0993i 0.973521i −0.873536 0.486760i \(-0.838178\pi\)
0.873536 0.486760i \(-0.161822\pi\)
\(564\) 0 0
\(565\) −0.621660 −0.0261534
\(566\) 0 0
\(567\) 5.12175i 0.215093i
\(568\) 0 0
\(569\) 42.3708 1.77628 0.888139 0.459575i \(-0.151998\pi\)
0.888139 + 0.459575i \(0.151998\pi\)
\(570\) 0 0
\(571\) 22.8718i 0.957156i −0.878045 0.478578i \(-0.841152\pi\)
0.878045 0.478578i \(-0.158848\pi\)
\(572\) 0 0
\(573\) −17.8757 −0.746766
\(574\) 0 0
\(575\) 14.7677 0.615856
\(576\) 0 0
\(577\) 12.4374i 0.517775i 0.965908 + 0.258887i \(0.0833558\pi\)
−0.965908 + 0.258887i \(0.916644\pi\)
\(578\) 0 0
\(579\) −33.3152 −1.38453
\(580\) 0 0
\(581\) 5.00782i 0.207760i
\(582\) 0 0
\(583\) −2.72539 −0.112874
\(584\) 0 0
\(585\) 5.31682i 0.219824i
\(586\) 0 0
\(587\) 2.01267i 0.0830716i 0.999137 + 0.0415358i \(0.0132251\pi\)
−0.999137 + 0.0415358i \(0.986775\pi\)
\(588\) 0 0
\(589\) 0.180139i 0.00742251i
\(590\) 0 0
\(591\) 29.0356i 1.19436i
\(592\) 0 0
\(593\) 16.6460i 0.683568i 0.939778 + 0.341784i \(0.111031\pi\)
−0.939778 + 0.341784i \(0.888969\pi\)
\(594\) 0 0
\(595\) −1.85040 −0.0758588
\(596\) 0 0
\(597\) −58.2158 −2.38261
\(598\) 0 0
\(599\) −17.2512 −0.704866 −0.352433 0.935837i \(-0.614646\pi\)
−0.352433 + 0.935837i \(0.614646\pi\)
\(600\) 0 0
\(601\) 1.49145i 0.0608376i −0.999537 0.0304188i \(-0.990316\pi\)
0.999537 0.0304188i \(-0.00968409\pi\)
\(602\) 0 0
\(603\) 34.8702i 1.42002i
\(604\) 0 0
\(605\) 4.62390 0.187988
\(606\) 0 0
\(607\) 3.38778 0.137506 0.0687529 0.997634i \(-0.478098\pi\)
0.0687529 + 0.997634i \(0.478098\pi\)
\(608\) 0 0
\(609\) 15.5949i 0.631936i
\(610\) 0 0
\(611\) 20.7196 0.838227
\(612\) 0 0
\(613\) 16.3851 0.661789 0.330894 0.943668i \(-0.392650\pi\)
0.330894 + 0.943668i \(0.392650\pi\)
\(614\) 0 0
\(615\) −8.53326 1.07993i −0.344094 0.0435468i
\(616\) 0 0
\(617\) −32.3742 −1.30334 −0.651668 0.758505i \(-0.725930\pi\)
−0.651668 + 0.758505i \(0.725930\pi\)
\(618\) 0 0
\(619\) 36.7849 1.47851 0.739255 0.673426i \(-0.235178\pi\)
0.739255 + 0.673426i \(0.235178\pi\)
\(620\) 0 0
\(621\) 8.02498i 0.322031i
\(622\) 0 0
\(623\) −6.89431 −0.276215
\(624\) 0 0
\(625\) 21.1867 0.847466
\(626\) 0 0
\(627\) 0.264234i 0.0105525i
\(628\) 0 0
\(629\) 1.73640i 0.0692348i
\(630\) 0 0
\(631\) −24.1438 −0.961149 −0.480575 0.876954i \(-0.659572\pi\)
−0.480575 + 0.876954i \(0.659572\pi\)
\(632\) 0 0
\(633\) 4.22080 0.167762
\(634\) 0 0
\(635\) 2.12233 0.0842220
\(636\) 0 0
\(637\) 2.62949i 0.104184i
\(638\) 0 0
\(639\) 3.39482i 0.134297i
\(640\) 0 0
\(641\) 23.9449i 0.945765i 0.881125 + 0.472883i \(0.156787\pi\)
−0.881125 + 0.472883i \(0.843213\pi\)
\(642\) 0 0
\(643\) 30.1202i 1.18783i −0.804529 0.593913i \(-0.797582\pi\)
0.804529 0.593913i \(-0.202418\pi\)
\(644\) 0 0
\(645\) 0.227581i 0.00896101i
\(646\) 0 0
\(647\) 25.5113 1.00295 0.501477 0.865171i \(-0.332790\pi\)
0.501477 + 0.865171i \(0.332790\pi\)
\(648\) 0 0
\(649\) 5.83609i 0.229086i
\(650\) 0 0
\(651\) 6.57009 0.257502
\(652\) 0 0
\(653\) 48.7156i 1.90639i 0.302356 + 0.953195i \(0.402227\pi\)
−0.302356 + 0.953195i \(0.597773\pi\)
\(654\) 0 0
\(655\) 5.38459 0.210394
\(656\) 0 0
\(657\) −24.5609 −0.958210
\(658\) 0 0
\(659\) 25.5196i 0.994102i −0.867721 0.497051i \(-0.834416\pi\)
0.867721 0.497051i \(-0.165584\pi\)
\(660\) 0 0
\(661\) 8.94107 0.347767 0.173884 0.984766i \(-0.444368\pi\)
0.173884 + 0.984766i \(0.444368\pi\)
\(662\) 0 0
\(663\) 25.2663i 0.981260i
\(664\) 0 0
\(665\) −0.0368307 −0.00142824
\(666\) 0 0
\(667\) 18.3911i 0.712105i
\(668\) 0 0
\(669\) 74.3923i 2.87617i
\(670\) 0 0
\(671\) 14.4349i 0.557253i
\(672\) 0 0
\(673\) 38.1905i 1.47214i −0.676907 0.736068i \(-0.736680\pi\)
0.676907 0.736068i \(-0.263320\pi\)
\(674\) 0 0
\(675\) 12.2160i 0.470194i
\(676\) 0 0
\(677\) −25.4119 −0.976658 −0.488329 0.872660i \(-0.662393\pi\)
−0.488329 + 0.872660i \(0.662393\pi\)
\(678\) 0 0
\(679\) 9.36862 0.359535
\(680\) 0 0
\(681\) 30.0561 1.15175
\(682\) 0 0
\(683\) 14.5602i 0.557132i −0.960417 0.278566i \(-0.910141\pi\)
0.960417 0.278566i \(-0.0898591\pi\)
\(684\) 0 0
\(685\) 4.60491i 0.175944i
\(686\) 0 0
\(687\) −19.9648 −0.761706
\(688\) 0 0
\(689\) 5.18711 0.197613
\(690\) 0 0
\(691\) 13.1979i 0.502072i 0.967978 + 0.251036i \(0.0807712\pi\)
−0.967978 + 0.251036i \(0.919229\pi\)
\(692\) 0 0
\(693\) −5.49250 −0.208643
\(694\) 0 0
\(695\) −2.54214 −0.0964287
\(696\) 0 0
\(697\) −23.1111 2.92483i −0.875397 0.110786i
\(698\) 0 0
\(699\) 11.5350 0.436294
\(700\) 0 0
\(701\) −20.5136 −0.774787 −0.387393 0.921914i \(-0.626624\pi\)
−0.387393 + 0.921914i \(0.626624\pi\)
\(702\) 0 0
\(703\) 0.0345617i 0.00130352i
\(704\) 0 0
\(705\) 10.5848 0.398647
\(706\) 0 0
\(707\) −9.89885 −0.372284
\(708\) 0 0
\(709\) 10.5794i 0.397319i 0.980069 + 0.198659i \(0.0636588\pi\)
−0.980069 + 0.198659i \(0.936341\pi\)
\(710\) 0 0
\(711\) 5.17750i 0.194171i
\(712\) 0 0
\(713\) 7.74814 0.290170
\(714\) 0 0
\(715\) 1.84770 0.0691001
\(716\) 0 0
\(717\) −67.1980 −2.50956
\(718\) 0 0
\(719\) 12.9033i 0.481211i −0.970623 0.240605i \(-0.922654\pi\)
0.970623 0.240605i \(-0.0773460\pi\)
\(720\) 0 0
\(721\) 0.792482i 0.0295136i
\(722\) 0 0
\(723\) 52.3219i 1.94587i
\(724\) 0 0
\(725\) 27.9957i 1.03974i
\(726\) 0 0
\(727\) 12.9443i 0.480078i −0.970763 0.240039i \(-0.922840\pi\)
0.970763 0.240039i \(-0.0771603\pi\)
\(728\) 0 0
\(729\) 40.7762 1.51023
\(730\) 0 0
\(731\) 0.616372i 0.0227974i
\(732\) 0 0
\(733\) −15.2751 −0.564197 −0.282099 0.959385i \(-0.591031\pi\)
−0.282099 + 0.959385i \(0.591031\pi\)
\(734\) 0 0
\(735\) 1.34330i 0.0495484i
\(736\) 0 0
\(737\) −12.1181 −0.446375
\(738\) 0 0
\(739\) 14.9475 0.549852 0.274926 0.961465i \(-0.411347\pi\)
0.274926 + 0.961465i \(0.411347\pi\)
\(740\) 0 0
\(741\) 0.502906i 0.0184747i
\(742\) 0 0
\(743\) −20.8721 −0.765723 −0.382861 0.923806i \(-0.625061\pi\)
−0.382861 + 0.923806i \(0.625061\pi\)
\(744\) 0 0
\(745\) 0.932759i 0.0341736i
\(746\) 0 0
\(747\) −19.9088 −0.728423
\(748\) 0 0
\(749\) 14.0102i 0.511922i
\(750\) 0 0
\(751\) 46.1750i 1.68495i −0.538736 0.842475i \(-0.681098\pi\)
0.538736 0.842475i \(-0.318902\pi\)
\(752\) 0 0
\(753\) 0.0865991i 0.00315585i
\(754\) 0 0
\(755\) 8.25328i 0.300367i
\(756\) 0 0
\(757\) 5.19901i 0.188961i 0.995527 + 0.0944806i \(0.0301190\pi\)
−0.995527 + 0.0944806i \(0.969881\pi\)
\(758\) 0 0
\(759\) −11.3652 −0.412531
\(760\) 0 0
\(761\) −24.4753 −0.887227 −0.443614 0.896218i \(-0.646304\pi\)
−0.443614 + 0.896218i \(0.646304\pi\)
\(762\) 0 0
\(763\) −11.0323 −0.399396
\(764\) 0 0
\(765\) 7.35631i 0.265968i
\(766\) 0 0
\(767\) 11.1076i 0.401071i
\(768\) 0 0
\(769\) −42.7224 −1.54061 −0.770305 0.637676i \(-0.779896\pi\)
−0.770305 + 0.637676i \(0.779896\pi\)
\(770\) 0 0
\(771\) 33.3800 1.20215
\(772\) 0 0
\(773\) 51.4835i 1.85173i −0.377852 0.925866i \(-0.623337\pi\)
0.377852 0.925866i \(-0.376663\pi\)
\(774\) 0 0
\(775\) 11.7946 0.423673
\(776\) 0 0
\(777\) 1.26055 0.0452218
\(778\) 0 0
\(779\) −0.460010 0.0582165i −0.0164816 0.00208582i
\(780\) 0 0
\(781\) 1.17977 0.0422154
\(782\) 0 0
\(783\) 15.2133 0.543678
\(784\) 0 0
\(785\) 1.29753i 0.0463110i
\(786\) 0 0
\(787\) 29.6859 1.05819 0.529094 0.848563i \(-0.322532\pi\)
0.529094 + 0.848563i \(0.322532\pi\)
\(788\) 0 0
\(789\) −35.3757 −1.25941
\(790\) 0 0
\(791\) 1.22227i 0.0434590i
\(792\) 0 0
\(793\) 27.4733i 0.975606i
\(794\) 0 0
\(795\) 2.64988 0.0939817
\(796\) 0 0
\(797\) −5.71982 −0.202606 −0.101303 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(798\) 0 0
\(799\) 28.6675 1.01418
\(800\) 0 0
\(801\) 27.4086i 0.968434i
\(802\) 0 0
\(803\) 8.53538i 0.301207i
\(804\) 0 0
\(805\) 1.58416i 0.0558343i