Properties

Label 2352.2.h.p.2255.12
Level $2352$
Weight $2$
Character 2352.2255
Analytic conductor $18.781$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 12x^{14} + 97x^{12} - 432x^{10} + 1392x^{8} - 2502x^{6} + 3181x^{4} - 1650x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2255.12
Root \(1.75344 - 1.01235i\) of defining polynomial
Character \(\chi\) \(=\) 2352.2255
Dual form 2352.2.h.p.2255.9

$q$-expansion

\(f(q)\) \(=\) \(q+(0.777403 + 1.54779i) q^{3} +3.40332i q^{5} +(-1.79129 + 2.40651i) q^{9} +O(q^{10})\) \(q+(0.777403 + 1.54779i) q^{3} +3.40332i q^{5} +(-1.79129 + 2.40651i) q^{9} -3.94748 q^{13} +(-5.26761 + 2.64575i) q^{15} -6.09632i q^{17} -4.38774i q^{19} -8.33639 q^{23} -6.58258 q^{25} +(-5.11732 - 0.901703i) q^{27} +3.80848i q^{29} -4.89898i q^{31} -7.58258 q^{37} +(-3.06878 - 6.10985i) q^{39} +0.710314i q^{41} +9.66930i q^{43} +(-8.19012 - 6.09632i) q^{45} +11.7894 q^{47} +(9.43581 - 4.73930i) q^{51} +1.00454i q^{53} +(6.79129 - 3.41105i) q^{57} +4.66442 q^{59} -1.70938 q^{61} -13.4345i q^{65} +3.46410i q^{67} +(-6.48074 - 12.9030i) q^{69} -6.59649 q^{71} -14.3757 q^{73} +(-5.11732 - 10.1884i) q^{75} +6.92820i q^{79} +(-2.58258 - 8.62150i) q^{81} +16.4539 q^{83} +20.7477 q^{85} +(-5.89472 + 2.96073i) q^{87} -6.09632i q^{89} +(7.58258 - 3.80848i) q^{93} +14.9329 q^{95} -9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{9} - 32 q^{25} - 48 q^{37} + 72 q^{57} + 32 q^{81} + 112 q^{85} + 48 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.777403 + 1.54779i 0.448834 + 0.893615i
\(4\) 0 0
\(5\) 3.40332i 1.52201i 0.648746 + 0.761005i \(0.275294\pi\)
−0.648746 + 0.761005i \(0.724706\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.79129 + 2.40651i −0.597096 + 0.802170i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −3.94748 −1.09483 −0.547417 0.836860i \(-0.684389\pi\)
−0.547417 + 0.836860i \(0.684389\pi\)
\(14\) 0 0
\(15\) −5.26761 + 2.64575i −1.36009 + 0.683130i
\(16\) 0 0
\(17\) 6.09632i 1.47858i −0.673390 0.739288i \(-0.735162\pi\)
0.673390 0.739288i \(-0.264838\pi\)
\(18\) 0 0
\(19\) 4.38774i 1.00662i −0.864107 0.503308i \(-0.832116\pi\)
0.864107 0.503308i \(-0.167884\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.33639 −1.73826 −0.869129 0.494585i \(-0.835320\pi\)
−0.869129 + 0.494585i \(0.835320\pi\)
\(24\) 0 0
\(25\) −6.58258 −1.31652
\(26\) 0 0
\(27\) −5.11732 0.901703i −0.984828 0.173533i
\(28\) 0 0
\(29\) 3.80848i 0.707218i 0.935393 + 0.353609i \(0.115046\pi\)
−0.935393 + 0.353609i \(0.884954\pi\)
\(30\) 0 0
\(31\) 4.89898i 0.879883i −0.898027 0.439941i \(-0.854999\pi\)
0.898027 0.439941i \(-0.145001\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.58258 −1.24657 −0.623284 0.781996i \(-0.714202\pi\)
−0.623284 + 0.781996i \(0.714202\pi\)
\(38\) 0 0
\(39\) −3.06878 6.10985i −0.491398 0.978359i
\(40\) 0 0
\(41\) 0.710314i 0.110932i 0.998461 + 0.0554662i \(0.0176645\pi\)
−0.998461 + 0.0554662i \(0.982335\pi\)
\(42\) 0 0
\(43\) 9.66930i 1.47456i 0.675590 + 0.737278i \(0.263889\pi\)
−0.675590 + 0.737278i \(0.736111\pi\)
\(44\) 0 0
\(45\) −8.19012 6.09632i −1.22091 0.908786i
\(46\) 0 0
\(47\) 11.7894 1.71967 0.859833 0.510575i \(-0.170567\pi\)
0.859833 + 0.510575i \(0.170567\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 9.43581 4.73930i 1.32128 0.663635i
\(52\) 0 0
\(53\) 1.00454i 0.137984i 0.997617 + 0.0689918i \(0.0219782\pi\)
−0.997617 + 0.0689918i \(0.978022\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.79129 3.41105i 0.899528 0.451804i
\(58\) 0 0
\(59\) 4.66442 0.607256 0.303628 0.952791i \(-0.401802\pi\)
0.303628 + 0.952791i \(0.401802\pi\)
\(60\) 0 0
\(61\) −1.70938 −0.218863 −0.109432 0.993994i \(-0.534903\pi\)
−0.109432 + 0.993994i \(0.534903\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 13.4345i 1.66635i
\(66\) 0 0
\(67\) 3.46410i 0.423207i 0.977356 + 0.211604i \(0.0678686\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 0 0
\(69\) −6.48074 12.9030i −0.780189 1.55333i
\(70\) 0 0
\(71\) −6.59649 −0.782859 −0.391429 0.920208i \(-0.628019\pi\)
−0.391429 + 0.920208i \(0.628019\pi\)
\(72\) 0 0
\(73\) −14.3757 −1.68255 −0.841274 0.540609i \(-0.818194\pi\)
−0.841274 + 0.540609i \(0.818194\pi\)
\(74\) 0 0
\(75\) −5.11732 10.1884i −0.590897 1.17646i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.92820i 0.779484i 0.920924 + 0.389742i \(0.127436\pi\)
−0.920924 + 0.389742i \(0.872564\pi\)
\(80\) 0 0
\(81\) −2.58258 8.62150i −0.286953 0.957945i
\(82\) 0 0
\(83\) 16.4539 1.80605 0.903023 0.429592i \(-0.141343\pi\)
0.903023 + 0.429592i \(0.141343\pi\)
\(84\) 0 0
\(85\) 20.7477 2.25041
\(86\) 0 0
\(87\) −5.89472 + 2.96073i −0.631980 + 0.317423i
\(88\) 0 0
\(89\) 6.09632i 0.646209i −0.946363 0.323104i \(-0.895273\pi\)
0.946363 0.323104i \(-0.104727\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 7.58258 3.80848i 0.786276 0.394921i
\(94\) 0 0
\(95\) 14.9329 1.53208
\(96\) 0 0
\(97\) −9.89949 −1.00514 −0.502571 0.864536i \(-0.667612\pi\)
−0.502571 + 0.864536i \(0.667612\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 11.6306i 1.15729i 0.815581 + 0.578643i \(0.196418\pi\)
−0.815581 + 0.578643i \(0.803582\pi\)
\(102\) 0 0
\(103\) 18.5734i 1.83010i 0.403345 + 0.915048i \(0.367847\pi\)
−0.403345 + 0.915048i \(0.632153\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.59649 0.637706 0.318853 0.947804i \(-0.396702\pi\)
0.318853 + 0.947804i \(0.396702\pi\)
\(108\) 0 0
\(109\) −4.41742 −0.423113 −0.211556 0.977366i \(-0.567853\pi\)
−0.211556 + 0.977366i \(0.567853\pi\)
\(110\) 0 0
\(111\) −5.89472 11.7362i −0.559502 1.11395i
\(112\) 0 0
\(113\) 12.4300i 1.16931i 0.811280 + 0.584657i \(0.198771\pi\)
−0.811280 + 0.584657i \(0.801229\pi\)
\(114\) 0 0
\(115\) 28.3714i 2.64565i
\(116\) 0 0
\(117\) 7.07107 9.49964i 0.653720 0.878242i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −1.09941 + 0.552200i −0.0991309 + 0.0497902i
\(124\) 0 0
\(125\) 5.38601i 0.481739i
\(126\) 0 0
\(127\) 10.3923i 0.922168i 0.887357 + 0.461084i \(0.152539\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 0 0
\(129\) −14.9660 + 7.51695i −1.31768 + 0.661831i
\(130\) 0 0
\(131\) −7.12502 −0.622516 −0.311258 0.950325i \(-0.600750\pi\)
−0.311258 + 0.950325i \(0.600750\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 3.06878 17.4159i 0.264119 1.49892i
\(136\) 0 0
\(137\) 18.2475i 1.55899i −0.626407 0.779496i \(-0.715475\pi\)
0.626407 0.779496i \(-0.284525\pi\)
\(138\) 0 0
\(139\) 0.511238i 0.0433627i 0.999765 + 0.0216813i \(0.00690192\pi\)
−0.999765 + 0.0216813i \(0.993098\pi\)
\(140\) 0 0
\(141\) 9.16515 + 18.2475i 0.771845 + 1.53672i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −12.9615 −1.07639
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.00454i 0.0822948i 0.999153 + 0.0411474i \(0.0131013\pi\)
−0.999153 + 0.0411474i \(0.986899\pi\)
\(150\) 0 0
\(151\) 15.8745i 1.29185i −0.763401 0.645925i \(-0.776472\pi\)
0.763401 0.645925i \(-0.223528\pi\)
\(152\) 0 0
\(153\) 14.6709 + 10.9203i 1.18607 + 0.882851i
\(154\) 0 0
\(155\) 16.6728 1.33919
\(156\) 0 0
\(157\) 1.70938 0.136423 0.0682116 0.997671i \(-0.478271\pi\)
0.0682116 + 0.997671i \(0.478271\pi\)
\(158\) 0 0
\(159\) −1.55481 + 0.780929i −0.123304 + 0.0619317i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.18710i 0.327959i 0.986464 + 0.163980i \(0.0524331\pi\)
−0.986464 + 0.163980i \(0.947567\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 2.58258 0.198660
\(170\) 0 0
\(171\) 10.5591 + 7.85971i 0.807478 + 0.601047i
\(172\) 0 0
\(173\) 3.40332i 0.258750i 0.991596 + 0.129375i \(0.0412970\pi\)
−0.991596 + 0.129375i \(0.958703\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.62614 + 7.21953i 0.272557 + 0.542653i
\(178\) 0 0
\(179\) −6.59649 −0.493045 −0.246522 0.969137i \(-0.579288\pi\)
−0.246522 + 0.969137i \(0.579288\pi\)
\(180\) 0 0
\(181\) −9.01400 −0.670006 −0.335003 0.942217i \(-0.608737\pi\)
−0.335003 + 0.942217i \(0.608737\pi\)
\(182\) 0 0
\(183\) −1.32888 2.64575i −0.0982333 0.195580i
\(184\) 0 0
\(185\) 25.8059i 1.89729i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.0763 0.729095 0.364548 0.931185i \(-0.381224\pi\)
0.364548 + 0.931185i \(0.381224\pi\)
\(192\) 0 0
\(193\) −18.7477 −1.34949 −0.674745 0.738051i \(-0.735747\pi\)
−0.674745 + 0.738051i \(0.735747\pi\)
\(194\) 0 0
\(195\) 20.7938 10.4440i 1.48907 0.747913i
\(196\) 0 0
\(197\) 1.00454i 0.0715702i −0.999360 0.0357851i \(-0.988607\pi\)
0.999360 0.0357851i \(-0.0113932\pi\)
\(198\) 0 0
\(199\) 9.79796i 0.694559i −0.937762 0.347279i \(-0.887106\pi\)
0.937762 0.347279i \(-0.112894\pi\)
\(200\) 0 0
\(201\) −5.36169 + 2.69300i −0.378185 + 0.189950i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2.41742 −0.168840
\(206\) 0 0
\(207\) 14.9329 20.0616i 1.03791 1.39438i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 15.8745i 1.09285i −0.837509 0.546423i \(-0.815989\pi\)
0.837509 0.546423i \(-0.184011\pi\)
\(212\) 0 0
\(213\) −5.12813 10.2100i −0.351374 0.699575i
\(214\) 0 0
\(215\) −32.9077 −2.24429
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −11.1757 22.2505i −0.755185 1.50355i
\(220\) 0 0
\(221\) 24.0651i 1.61879i
\(222\) 0 0
\(223\) 1.02248i 0.0684701i 0.999414 + 0.0342350i \(0.0108995\pi\)
−0.999414 + 0.0342350i \(0.989101\pi\)
\(224\) 0 0
\(225\) 11.7913 15.8410i 0.786086 1.05607i
\(226\) 0 0
\(227\) 16.4539 1.09208 0.546041 0.837759i \(-0.316134\pi\)
0.546041 + 0.837759i \(0.316134\pi\)
\(228\) 0 0
\(229\) −15.2612 −1.00849 −0.504244 0.863561i \(-0.668229\pi\)
−0.504244 + 0.863561i \(0.668229\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.2475i 1.19544i 0.801706 + 0.597718i \(0.203926\pi\)
−0.801706 + 0.597718i \(0.796074\pi\)
\(234\) 0 0
\(235\) 40.1232i 2.61735i
\(236\) 0 0
\(237\) −10.7234 + 5.38601i −0.696558 + 0.349859i
\(238\) 0 0
\(239\) −8.33639 −0.539236 −0.269618 0.962967i \(-0.586898\pi\)
−0.269618 + 0.962967i \(0.586898\pi\)
\(240\) 0 0
\(241\) 22.8610 1.47260 0.736302 0.676653i \(-0.236570\pi\)
0.736302 + 0.676653i \(0.236570\pi\)
\(242\) 0 0
\(243\) 11.3365 10.6997i 0.727240 0.686384i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 17.3205i 1.10208i
\(248\) 0 0
\(249\) 12.7913 + 25.4671i 0.810615 + 1.61391i
\(250\) 0 0
\(251\) −28.2433 −1.78270 −0.891351 0.453314i \(-0.850241\pi\)
−0.891351 + 0.453314i \(0.850241\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 16.1294 + 32.1131i 1.01006 + 2.01100i
\(256\) 0 0
\(257\) 14.3236i 0.893481i −0.894664 0.446740i \(-0.852585\pi\)
0.894664 0.446740i \(-0.147415\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −9.16515 6.82209i −0.567309 0.422277i
\(262\) 0 0
\(263\) −26.7491 −1.64942 −0.824710 0.565556i \(-0.808661\pi\)
−0.824710 + 0.565556i \(0.808661\pi\)
\(264\) 0 0
\(265\) −3.41875 −0.210012
\(266\) 0 0
\(267\) 9.43581 4.73930i 0.577462 0.290041i
\(268\) 0 0
\(269\) 3.40332i 0.207504i −0.994603 0.103752i \(-0.966915\pi\)
0.994603 0.103752i \(-0.0330848\pi\)
\(270\) 0 0
\(271\) 19.5959i 1.19037i 0.803590 + 0.595184i \(0.202921\pi\)
−0.803590 + 0.595184i \(0.797079\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 11.7894 + 8.77548i 0.705815 + 0.525374i
\(280\) 0 0
\(281\) 10.6306i 0.634167i 0.948398 + 0.317083i \(0.102704\pi\)
−0.948398 + 0.317083i \(0.897296\pi\)
\(282\) 0 0
\(283\) 15.2082i 0.904032i −0.892010 0.452016i \(-0.850705\pi\)
0.892010 0.452016i \(-0.149295\pi\)
\(284\) 0 0
\(285\) 11.6089 + 23.1129i 0.687650 + 1.36909i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −20.1652 −1.18619
\(290\) 0 0
\(291\) −7.69590 15.3223i −0.451142 0.898210i
\(292\) 0 0
\(293\) 1.98269i 0.115830i 0.998322 + 0.0579150i \(0.0184452\pi\)
−0.998322 + 0.0579150i \(0.981555\pi\)
\(294\) 0 0
\(295\) 15.8745i 0.924250i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 32.9077 1.90310
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −18.0017 + 9.04165i −1.03417 + 0.519429i
\(304\) 0 0
\(305\) 5.81755i 0.333112i
\(306\) 0 0
\(307\) 0.511238i 0.0291779i −0.999894 0.0145890i \(-0.995356\pi\)
0.999894 0.0145890i \(-0.00464397\pi\)
\(308\) 0 0
\(309\) −28.7477 + 14.4391i −1.63540 + 0.821409i
\(310\) 0 0
\(311\) −21.1183 −1.19751 −0.598754 0.800933i \(-0.704337\pi\)
−0.598754 + 0.800933i \(0.704337\pi\)
\(312\) 0 0
\(313\) 20.0325 1.13231 0.566153 0.824300i \(-0.308431\pi\)
0.566153 + 0.824300i \(0.308431\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 27.8736i 1.56554i 0.622314 + 0.782768i \(0.286193\pi\)
−0.622314 + 0.782768i \(0.713807\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 5.12813 + 10.2100i 0.286224 + 0.569864i
\(322\) 0 0
\(323\) −26.7491 −1.48836
\(324\) 0 0
\(325\) 25.9846 1.44136
\(326\) 0 0
\(327\) −3.43412 6.83723i −0.189907 0.378100i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.74110i 0.150665i −0.997158 0.0753323i \(-0.975998\pi\)
0.997158 0.0753323i \(-0.0240017\pi\)
\(332\) 0 0
\(333\) 13.5826 18.2475i 0.744321 0.999959i
\(334\) 0 0
\(335\) −11.7894 −0.644126
\(336\) 0 0
\(337\) −28.7477 −1.56599 −0.782994 0.622029i \(-0.786309\pi\)
−0.782994 + 0.622029i \(0.786309\pi\)
\(338\) 0 0
\(339\) −19.2390 + 9.66311i −1.04492 + 0.524828i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 43.9129 22.0560i 2.36419 1.18746i
\(346\) 0 0
\(347\) 3.47981 0.186806 0.0934031 0.995628i \(-0.470225\pi\)
0.0934031 + 0.995628i \(0.470225\pi\)
\(348\) 0 0
\(349\) 14.6709 0.785313 0.392657 0.919685i \(-0.371556\pi\)
0.392657 + 0.919685i \(0.371556\pi\)
\(350\) 0 0
\(351\) 20.2005 + 3.55945i 1.07822 + 0.189989i
\(352\) 0 0
\(353\) 16.8683i 0.897811i 0.893579 + 0.448906i \(0.148186\pi\)
−0.893579 + 0.448906i \(0.851814\pi\)
\(354\) 0 0
\(355\) 22.4499i 1.19152i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.33639 0.439978 0.219989 0.975502i \(-0.429398\pi\)
0.219989 + 0.975502i \(0.429398\pi\)
\(360\) 0 0
\(361\) −0.252273 −0.0132775
\(362\) 0 0
\(363\) −8.55144 17.0257i −0.448834 0.893615i
\(364\) 0 0
\(365\) 48.9251i 2.56086i
\(366\) 0 0
\(367\) 1.02248i 0.0533728i −0.999644 0.0266864i \(-0.991504\pi\)
0.999644 0.0266864i \(-0.00849556\pi\)
\(368\) 0 0
\(369\) −1.70938 1.27238i −0.0889866 0.0662373i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 8.33639 4.18710i 0.430490 0.216221i
\(376\) 0 0
\(377\) 15.0339i 0.774285i
\(378\) 0 0
\(379\) 9.66930i 0.496679i −0.968673 0.248339i \(-0.920115\pi\)
0.968673 0.248339i \(-0.0798848\pi\)
\(380\) 0 0
\(381\) −16.0851 + 8.07901i −0.824063 + 0.413900i
\(382\) 0 0
\(383\) −21.1183 −1.07909 −0.539547 0.841956i \(-0.681404\pi\)
−0.539547 + 0.841956i \(0.681404\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −23.2693 17.3205i −1.18284 0.880451i
\(388\) 0 0
\(389\) 17.4527i 0.884885i −0.896797 0.442443i \(-0.854112\pi\)
0.896797 0.442443i \(-0.145888\pi\)
\(390\) 0 0
\(391\) 50.8213i 2.57015i
\(392\) 0 0
\(393\) −5.53901 11.0280i −0.279406 0.556290i
\(394\) 0 0
\(395\) −23.5789 −1.18638
\(396\) 0 0
\(397\) −5.71846 −0.287001 −0.143501 0.989650i \(-0.545836\pi\)
−0.143501 + 0.989650i \(0.545836\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 25.0696i 1.25192i −0.779856 0.625959i \(-0.784708\pi\)
0.779856 0.625959i \(-0.215292\pi\)
\(402\) 0 0
\(403\) 19.3386i 0.963325i
\(404\) 0 0
\(405\) 29.3417 8.78933i 1.45800 0.436745i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.83297 0.238975 0.119487 0.992836i \(-0.461875\pi\)
0.119487 + 0.992836i \(0.461875\pi\)
\(410\) 0 0
\(411\) 28.2433 14.1857i 1.39314 0.699729i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 55.9977i 2.74882i
\(416\) 0 0
\(417\) −0.791288 + 0.397438i −0.0387495 + 0.0194626i
\(418\) 0 0
\(419\) −7.12502 −0.348080 −0.174040 0.984739i \(-0.555682\pi\)
−0.174040 + 0.984739i \(0.555682\pi\)
\(420\) 0 0
\(421\) −19.4955 −0.950150 −0.475075 0.879945i \(-0.657579\pi\)
−0.475075 + 0.879945i \(0.657579\pi\)
\(422\) 0 0
\(423\) −21.1183 + 28.3714i −1.02681 + 1.37946i
\(424\) 0 0
\(425\) 40.1295i 1.94657i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 38.2022 1.84013 0.920066 0.391762i \(-0.128134\pi\)
0.920066 + 0.391762i \(0.128134\pi\)
\(432\) 0 0
\(433\) 12.1376 0.583296 0.291648 0.956526i \(-0.405796\pi\)
0.291648 + 0.956526i \(0.405796\pi\)
\(434\) 0 0
\(435\) −10.0763 20.0616i −0.483122 0.961881i
\(436\) 0 0
\(437\) 36.5779i 1.74976i
\(438\) 0 0
\(439\) 14.6969i 0.701447i 0.936479 + 0.350723i \(0.114064\pi\)
−0.936479 + 0.350723i \(0.885936\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 23.2693 1.10556 0.552778 0.833328i \(-0.313568\pi\)
0.552778 + 0.833328i \(0.313568\pi\)
\(444\) 0 0
\(445\) 20.7477 0.983537
\(446\) 0 0
\(447\) −1.55481 + 0.780929i −0.0735398 + 0.0369367i
\(448\) 0 0
\(449\) 24.8600i 1.17321i −0.809872 0.586607i \(-0.800463\pi\)
0.809872 0.586607i \(-0.199537\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 24.5704 12.3409i 1.15442 0.579826i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.74773 0.128533 0.0642666 0.997933i \(-0.479529\pi\)
0.0642666 + 0.997933i \(0.479529\pi\)
\(458\) 0 0
\(459\) −5.49707 + 31.1968i −0.256581 + 1.45614i
\(460\) 0 0
\(461\) 8.78933i 0.409360i 0.978829 + 0.204680i \(0.0656153\pi\)
−0.978829 + 0.204680i \(0.934385\pi\)
\(462\) 0 0
\(463\) 5.48220i 0.254780i 0.991853 + 0.127390i \(0.0406599\pi\)
−0.991853 + 0.127390i \(0.959340\pi\)
\(464\) 0 0
\(465\) 12.9615 + 25.8059i 0.601074 + 1.19672i
\(466\) 0 0
\(467\) 25.7827 1.19308 0.596541 0.802583i \(-0.296541\pi\)
0.596541 + 0.802583i \(0.296541\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 1.32888 + 2.64575i 0.0612314 + 0.121910i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 28.8826i 1.32523i
\(476\) 0 0
\(477\) −2.41742 1.79941i −0.110686 0.0823894i
\(478\) 0 0
\(479\) 14.2500 0.651101 0.325550 0.945525i \(-0.394450\pi\)
0.325550 + 0.945525i \(0.394450\pi\)
\(480\) 0 0
\(481\) 29.9320 1.36478
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 33.6911i 1.52984i
\(486\) 0 0
\(487\) 4.91010i 0.222498i 0.993793 + 0.111249i \(0.0354851\pi\)
−0.993793 + 0.111249i \(0.964515\pi\)
\(488\) 0 0
\(489\) −6.48074 + 3.25507i −0.293069 + 0.147199i
\(490\) 0 0
\(491\) −10.0763 −0.454737 −0.227369 0.973809i \(-0.573012\pi\)
−0.227369 + 0.973809i \(0.573012\pi\)
\(492\) 0 0
\(493\) 23.2177 1.04567
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 22.8027i 1.02079i 0.859940 + 0.510395i \(0.170501\pi\)
−0.859940 + 0.510395i \(0.829499\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 14.2500 0.635378 0.317689 0.948195i \(-0.397093\pi\)
0.317689 + 0.948195i \(0.397093\pi\)
\(504\) 0 0
\(505\) −39.5826 −1.76140
\(506\) 0 0
\(507\) 2.00770 + 3.99728i 0.0891652 + 0.177525i
\(508\) 0 0
\(509\) 29.2092i 1.29468i −0.762203 0.647338i \(-0.775882\pi\)
0.762203 0.647338i \(-0.224118\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −3.95644 + 22.4535i −0.174681 + 0.991345i
\(514\) 0 0
\(515\) −63.2113 −2.78542
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −5.26761 + 2.64575i −0.231222 + 0.116136i
\(520\) 0 0
\(521\) 16.8683i 0.739015i 0.929228 + 0.369508i \(0.120474\pi\)
−0.929228 + 0.369508i \(0.879526\pi\)
\(522\) 0 0
\(523\) 32.7591i 1.43246i 0.697866 + 0.716229i \(0.254133\pi\)
−0.697866 + 0.716229i \(0.745867\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −29.8658 −1.30097
\(528\) 0 0
\(529\) 46.4955 2.02154
\(530\) 0 0
\(531\) −8.35532 + 11.2250i −0.362590 + 0.487122i
\(532\) 0 0
\(533\) 2.80395i 0.121452i
\(534\) 0 0
\(535\) 22.4499i 0.970596i
\(536\) 0 0
\(537\) −5.12813 10.2100i −0.221295 0.440592i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 31.4955 1.35410 0.677048 0.735939i \(-0.263259\pi\)
0.677048 + 0.735939i \(0.263259\pi\)
\(542\) 0 0
\(543\) −7.00752 13.9518i −0.300721 0.598727i
\(544\) 0 0
\(545\) 15.0339i 0.643982i
\(546\) 0 0
\(547\) 35.9361i 1.53652i −0.640139 0.768259i \(-0.721123\pi\)
0.640139 0.768259i \(-0.278877\pi\)
\(548\) 0 0
\(549\) 3.06199 4.11363i 0.130682 0.175566i
\(550\) 0 0
\(551\) 16.7106 0.711897
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 39.9421 20.0616i 1.69545 0.851568i
\(556\) 0 0
\(557\) 27.8736i 1.18104i 0.807022 + 0.590521i \(0.201078\pi\)
−0.807022 + 0.590521i \(0.798922\pi\)
\(558\) 0 0
\(559\) 38.1694i 1.61439i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −28.2433 −1.19031 −0.595157 0.803610i \(-0.702910\pi\)
−0.595157 + 0.803610i \(0.702910\pi\)
\(564\) 0 0
\(565\) −42.3032 −1.77971
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 32.6866i 1.37029i 0.728405 + 0.685147i \(0.240262\pi\)
−0.728405 + 0.685147i \(0.759738\pi\)
\(570\) 0 0
\(571\) 16.5975i 0.694584i 0.937757 + 0.347292i \(0.112899\pi\)
−0.937757 + 0.347292i \(0.887101\pi\)
\(572\) 0 0
\(573\) 7.83335 + 15.5960i 0.327243 + 0.651531i
\(574\) 0 0
\(575\) 54.8749 2.28844
\(576\) 0 0
\(577\) 21.0900 0.877988 0.438994 0.898490i \(-0.355335\pi\)
0.438994 + 0.898490i \(0.355335\pi\)
\(578\) 0 0
\(579\) −14.5745 29.0175i −0.605698 1.20593i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 32.3303 + 24.0651i 1.33669 + 0.994969i
\(586\) 0 0
\(587\) −2.20382 −0.0909614 −0.0454807 0.998965i \(-0.514482\pi\)
−0.0454807 + 0.998965i \(0.514482\pi\)
\(588\) 0 0
\(589\) −21.4955 −0.885705
\(590\) 0 0
\(591\) 1.55481 0.780929i 0.0639562 0.0321231i
\(592\) 0 0
\(593\) 15.7442i 0.646537i −0.946307 0.323269i \(-0.895218\pi\)
0.946307 0.323269i \(-0.104782\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 15.1652 7.61697i 0.620668 0.311742i
\(598\) 0 0
\(599\) 26.7491 1.09294 0.546469 0.837479i \(-0.315972\pi\)
0.546469 + 0.837479i \(0.315972\pi\)
\(600\) 0 0
\(601\) −12.1376 −0.495103 −0.247551 0.968875i \(-0.579626\pi\)
−0.247551 + 0.968875i \(0.579626\pi\)
\(602\) 0 0
\(603\) −8.33639 6.20520i −0.339484 0.252695i
\(604\) 0 0
\(605\) 37.4365i 1.52201i
\(606\) 0 0
\(607\) 3.87650i 0.157342i 0.996901 + 0.0786712i \(0.0250677\pi\)
−0.996901 + 0.0786712i \(0.974932\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −46.5385 −1.88275
\(612\) 0 0
\(613\) −44.2432 −1.78697 −0.893483 0.449098i \(-0.851745\pi\)
−0.893483 + 0.449098i \(0.851745\pi\)
\(614\) 0 0
\(615\) −1.87931 3.74166i −0.0757813 0.150878i
\(616\) 0 0
\(617\) 25.0696i 1.00927i 0.863334 + 0.504633i \(0.168372\pi\)
−0.863334 + 0.504633i \(0.831628\pi\)
\(618\) 0 0
\(619\) 19.0847i 0.767078i −0.923525 0.383539i \(-0.874705\pi\)
0.923525 0.383539i \(-0.125295\pi\)
\(620\) 0 0
\(621\) 42.6600 + 7.51695i 1.71189 + 0.301645i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −14.5826 −0.583303
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 46.2258i 1.84314i
\(630\) 0 0
\(631\) 8.37420i 0.333372i 0.986010 + 0.166686i \(0.0533066\pi\)
−0.986010 + 0.166686i \(0.946693\pi\)
\(632\) 0 0
\(633\) 24.5704 12.3409i 0.976584 0.490507i
\(634\) 0 0
\(635\) −35.3683 −1.40355
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 11.8162 15.8745i 0.467442 0.627986i
\(640\) 0 0
\(641\) 32.6866i 1.29104i 0.763742 + 0.645521i \(0.223360\pi\)
−0.763742 + 0.645521i \(0.776640\pi\)
\(642\) 0 0
\(643\) 17.0397i 0.671981i −0.941865 0.335991i \(-0.890929\pi\)
0.941865 0.335991i \(-0.109071\pi\)
\(644\) 0 0
\(645\) −25.5826 50.9341i −1.00731 2.00553i
\(646\) 0 0
\(647\) 2.46060 0.0967361 0.0483681 0.998830i \(-0.484598\pi\)
0.0483681 + 0.998830i \(0.484598\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 11.4255i 0.447112i 0.974691 + 0.223556i \(0.0717666\pi\)
−0.974691 + 0.223556i \(0.928233\pi\)
\(654\) 0 0
\(655\) 24.2487i 0.947476i
\(656\) 0 0
\(657\) 25.7510 34.5952i 1.00464 1.34969i
\(658\) 0 0
\(659\) 46.5385 1.81288 0.906442 0.422330i \(-0.138788\pi\)
0.906442 + 0.422330i \(0.138788\pi\)
\(660\) 0 0
\(661\) 36.1176 1.40481 0.702406 0.711776i \(-0.252109\pi\)
0.702406 + 0.711776i \(0.252109\pi\)
\(662\) 0 0
\(663\) −37.2476 + 18.7083i −1.44658 + 0.726570i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 31.7490i 1.22933i
\(668\) 0 0
\(669\) −1.58258 + 0.794877i −0.0611859 + 0.0307317i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 3.16515 0.122008 0.0610038 0.998138i \(-0.480570\pi\)
0.0610038 + 0.998138i \(0.480570\pi\)
\(674\) 0 0
\(675\) 33.6851 + 5.93553i 1.29654 + 0.228459i
\(676\) 0 0
\(677\) 29.2092i 1.12260i 0.827612 + 0.561301i \(0.189699\pi\)
−0.827612 + 0.561301i \(0.810301\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 12.7913 + 25.4671i 0.490163 + 0.975900i
\(682\) 0 0
\(683\) 10.0763 0.385559 0.192779 0.981242i \(-0.438250\pi\)
0.192779 + 0.981242i \(0.438250\pi\)
\(684\) 0 0
\(685\) 62.1022 2.37280
\(686\) 0 0
\(687\) −11.8641 23.6211i −0.452644 0.901200i
\(688\) 0 0
\(689\) 3.96538i 0.151069i
\(690\) 0 0
\(691\) 23.9837i 0.912381i −0.889882 0.456191i \(-0.849213\pi\)
0.889882 0.456191i \(-0.150787\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.73991 −0.0659984
\(696\) 0 0
\(697\) 4.33030 0.164022
\(698\) 0 0
\(699\) −28.2433 + 14.1857i −1.06826 + 0.536552i
\(700\) 0 0
\(701\) 32.6866i 1.23456i 0.786745 + 0.617278i \(0.211765\pi\)
−0.786745 + 0.617278i \(0.788235\pi\)
\(702\) 0 0
\(703\) 33.2704i 1.25482i
\(704\) 0 0
\(705\) −62.1022 + 31.1919i −2.33890 + 1.17476i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −17.0780 −0.641379 −0.320689 0.947184i \(-0.603915\pi\)
−0.320689 + 0.947184i \(0.603915\pi\)
\(710\) 0 0
\(711\) −16.6728 12.4104i −0.625278 0.465427i
\(712\) 0 0
\(713\) 40.8398i 1.52946i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −6.48074 12.9030i −0.242028 0.481870i
\(718\) 0 0
\(719\) −2.46060 −0.0917649 −0.0458824 0.998947i \(-0.514610\pi\)
−0.0458824 + 0.998947i \(0.514610\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 17.7722 + 35.3839i 0.660955 + 1.31594i
\(724\) 0 0
\(725\) 25.0696i 0.931063i
\(726\) 0 0
\(727\) 43.0683i 1.59732i 0.601785 + 0.798658i \(0.294456\pi\)
−0.601785 + 0.798658i \(0.705544\pi\)
\(728\) 0 0
\(729\) 25.3739 + 9.22860i 0.939773 + 0.341800i
\(730\) 0 0
\(731\) 58.9472 2.18024
\(732\) 0 0
\(733\) −17.4993 −0.646351 −0.323175 0.946339i \(-0.604750\pi\)
−0.323175 + 0.946339i \(0.604750\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 24.9717i 0.918599i 0.888281 + 0.459300i \(0.151900\pi\)
−0.888281 + 0.459300i \(0.848100\pi\)
\(740\) 0 0
\(741\) −26.8085 + 13.4650i −0.984833 + 0.494650i
\(742\) 0 0
\(743\) −21.5294 −0.789836 −0.394918 0.918716i \(-0.629227\pi\)
−0.394918 + 0.918716i \(0.629227\pi\)
\(744\) 0 0
\(745\) −3.41875 −0.125253
\(746\) 0 0
\(747\) −29.4736 + 39.5964i −1.07838 + 1.44876i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 8.94630i 0.326455i −0.986588 0.163228i \(-0.947809\pi\)
0.986588 0.163228i \(-0.0521905\pi\)
\(752\) 0 0
\(753\) −21.9564 43.7146i −0.800137 1.59305i
\(754\) 0 0
\(755\) 54.0260 1.96621
\(756\) 0 0
\(757\) 22.7477 0.826780 0.413390 0.910554i \(-0.364345\pi\)
0.413390 + 0.910554i \(0.364345\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 14.3236i 0.519230i 0.965712 + 0.259615i \(0.0835956\pi\)
−0.965712 + 0.259615i \(0.916404\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −37.1652 + 49.9296i −1.34371 + 1.80521i
\(766\) 0 0
\(767\) −18.4127 −0.664844
\(768\) 0 0
\(769\) −37.0031 −1.33437 −0.667183 0.744894i \(-0.732500\pi\)
−0.667183 + 0.744894i \(0.732500\pi\)
\(770\) 0 0
\(771\) 22.1699 11.1352i 0.798428 0.401025i
\(772\) 0 0
\(773\) 3.40332i 0.122409i 0.998125 + 0.0612044i \(0.0194942\pi\)
−0.998125 + 0.0612044i \(0.980506\pi\)
\(774\) 0 0
\(775\) 32.2479i 1.15838i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.11667 0.111666
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 3.43412 19.4892i 0.122725 0.696488i
\(784\) 0 0
\(785\) 5.81755i 0.207637i
\(786\) 0 0
\(787\) 26.8377i 0.956660i −0.878180 0.478330i \(-0.841242\pi\)
0.878180 0.478330i \(-0.158758\pi\)
\(788\) 0 0
\(789\) −20.7948 41.4019i −0.740316 1.47395i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 6.74773 0.239619
\(794\) 0 0
\(795\) −2.65775 5.29150i −0.0942607 0.187670i
\(796\) 0 0
\(797\) 17.0166i 0.602759i 0.953504 + 0.301379i \(0.0974470\pi\)
−0.953504 + 0.301379i \(0.902553\pi\)
\(798\) 0 0
\(799\) 71.8722i 2.54266i
\(800\) 0 0
\(801\) 14.6709 + 10.9203i 0.518369 + 0.385849i
\(802\) 0 0
\(803\) 0 0