Properties

Label 2352.2.b.l.1567.2
Level $2352$
Weight $2$
Character 2352.1567
Analytic conductor $18.781$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.339738624.1
Defining polynomial: \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.2
Root \(-1.60021 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1567
Dual form 2352.2.b.l.1567.7

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.21486i q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -3.21486i q^{5} +1.00000 q^{9} -1.36710i q^{11} +2.93015i q^{13} -3.21486i q^{15} -6.91037i q^{17} -7.35449 q^{19} -3.62774i q^{23} -5.33530 q^{25} +1.00000 q^{27} -1.11185 q^{29} -8.70319 q^{31} -1.36710i q^{33} +7.63792 q^{37} +2.93015i q^{39} +0.833147i q^{41} +4.82362i q^{43} -3.21486i q^{45} -2.95367 q^{47} -6.91037i q^{51} -4.57794 q^{53} -4.39502 q^{55} -7.35449 q^{57} +14.0985 q^{59} -11.0618i q^{61} +9.42002 q^{65} -12.1545i q^{67} -3.62774i q^{69} +12.6249i q^{71} +6.49011i q^{73} -5.33530 q^{75} +7.79367i q^{79} +1.00000 q^{81} -8.87502 q^{83} -22.2159 q^{85} -1.11185 q^{87} -12.6146i q^{89} -8.70319 q^{93} +23.6436i q^{95} -13.8811i q^{97} -1.36710i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{3} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{3} + 8q^{9} - 24q^{25} + 8q^{27} + 16q^{29} - 32q^{31} - 16q^{47} + 16q^{53} - 64q^{55} + 48q^{59} - 16q^{65} - 24q^{75} + 8q^{81} - 64q^{85} + 16q^{87} - 32q^{93} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) − 3.21486i − 1.43773i −0.695151 0.718864i \(-0.744662\pi\)
0.695151 0.718864i \(-0.255338\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) − 1.36710i − 0.412195i −0.978531 0.206098i \(-0.933924\pi\)
0.978531 0.206098i \(-0.0660765\pi\)
\(12\) 0 0
\(13\) 2.93015i 0.812678i 0.913722 + 0.406339i \(0.133195\pi\)
−0.913722 + 0.406339i \(0.866805\pi\)
\(14\) 0 0
\(15\) − 3.21486i − 0.830072i
\(16\) 0 0
\(17\) − 6.91037i − 1.67601i −0.545661 0.838006i \(-0.683721\pi\)
0.545661 0.838006i \(-0.316279\pi\)
\(18\) 0 0
\(19\) −7.35449 −1.68724 −0.843618 0.536943i \(-0.819579\pi\)
−0.843618 + 0.536943i \(0.819579\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 3.62774i − 0.756436i −0.925717 0.378218i \(-0.876537\pi\)
0.925717 0.378218i \(-0.123463\pi\)
\(24\) 0 0
\(25\) −5.33530 −1.06706
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.11185 −0.206466 −0.103233 0.994657i \(-0.532919\pi\)
−0.103233 + 0.994657i \(0.532919\pi\)
\(30\) 0 0
\(31\) −8.70319 −1.56314 −0.781569 0.623819i \(-0.785580\pi\)
−0.781569 + 0.623819i \(0.785580\pi\)
\(32\) 0 0
\(33\) − 1.36710i − 0.237981i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.63792 1.25567 0.627833 0.778348i \(-0.283942\pi\)
0.627833 + 0.778348i \(0.283942\pi\)
\(38\) 0 0
\(39\) 2.93015i 0.469200i
\(40\) 0 0
\(41\) 0.833147i 0.130116i 0.997881 + 0.0650579i \(0.0207232\pi\)
−0.997881 + 0.0650579i \(0.979277\pi\)
\(42\) 0 0
\(43\) 4.82362i 0.735595i 0.929906 + 0.367797i \(0.119888\pi\)
−0.929906 + 0.367797i \(0.880112\pi\)
\(44\) 0 0
\(45\) − 3.21486i − 0.479243i
\(46\) 0 0
\(47\) −2.95367 −0.430837 −0.215418 0.976522i \(-0.569112\pi\)
−0.215418 + 0.976522i \(0.569112\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) − 6.91037i − 0.967646i
\(52\) 0 0
\(53\) −4.57794 −0.628829 −0.314414 0.949286i \(-0.601808\pi\)
−0.314414 + 0.949286i \(0.601808\pi\)
\(54\) 0 0
\(55\) −4.39502 −0.592625
\(56\) 0 0
\(57\) −7.35449 −0.974127
\(58\) 0 0
\(59\) 14.0985 1.83546 0.917732 0.397200i \(-0.130018\pi\)
0.917732 + 0.397200i \(0.130018\pi\)
\(60\) 0 0
\(61\) − 11.0618i − 1.41632i −0.706052 0.708160i \(-0.749525\pi\)
0.706052 0.708160i \(-0.250475\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.42002 1.16841
\(66\) 0 0
\(67\) − 12.1545i − 1.48490i −0.669900 0.742452i \(-0.733663\pi\)
0.669900 0.742452i \(-0.266337\pi\)
\(68\) 0 0
\(69\) − 3.62774i − 0.436728i
\(70\) 0 0
\(71\) 12.6249i 1.49830i 0.662402 + 0.749148i \(0.269537\pi\)
−0.662402 + 0.749148i \(0.730463\pi\)
\(72\) 0 0
\(73\) 6.49011i 0.759610i 0.925067 + 0.379805i \(0.124009\pi\)
−0.925067 + 0.379805i \(0.875991\pi\)
\(74\) 0 0
\(75\) −5.33530 −0.616068
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 7.79367i 0.876856i 0.898766 + 0.438428i \(0.144465\pi\)
−0.898766 + 0.438428i \(0.855535\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −8.87502 −0.974159 −0.487080 0.873358i \(-0.661938\pi\)
−0.487080 + 0.873358i \(0.661938\pi\)
\(84\) 0 0
\(85\) −22.2159 −2.40965
\(86\) 0 0
\(87\) −1.11185 −0.119203
\(88\) 0 0
\(89\) − 12.6146i − 1.33715i −0.743646 0.668574i \(-0.766905\pi\)
0.743646 0.668574i \(-0.233095\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −8.70319 −0.902478
\(94\) 0 0
\(95\) 23.6436i 2.42579i
\(96\) 0 0
\(97\) − 13.8811i − 1.40942i −0.709497 0.704708i \(-0.751078\pi\)
0.709497 0.704708i \(-0.248922\pi\)
\(98\) 0 0
\(99\) − 1.36710i − 0.137398i
\(100\) 0 0
\(101\) 1.65891i 0.165068i 0.996588 + 0.0825338i \(0.0263013\pi\)
−0.996588 + 0.0825338i \(0.973699\pi\)
\(102\) 0 0
\(103\) −1.30791 −0.128872 −0.0644359 0.997922i \(-0.520525\pi\)
−0.0644359 + 0.997922i \(0.520525\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 7.15204i − 0.691414i −0.938343 0.345707i \(-0.887639\pi\)
0.938343 0.345707i \(-0.112361\pi\)
\(108\) 0 0
\(109\) −15.0443 −1.44098 −0.720491 0.693464i \(-0.756083\pi\)
−0.720491 + 0.693464i \(0.756083\pi\)
\(110\) 0 0
\(111\) 7.63792 0.724959
\(112\) 0 0
\(113\) −15.4530 −1.45369 −0.726846 0.686800i \(-0.759015\pi\)
−0.726846 + 0.686800i \(0.759015\pi\)
\(114\) 0 0
\(115\) −11.6627 −1.08755
\(116\) 0 0
\(117\) 2.93015i 0.270893i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 9.13104 0.830095
\(122\) 0 0
\(123\) 0.833147i 0.0751224i
\(124\) 0 0
\(125\) 1.07795i 0.0964149i
\(126\) 0 0
\(127\) 2.16478i 0.192094i 0.995377 + 0.0960468i \(0.0306198\pi\)
−0.995377 + 0.0960468i \(0.969380\pi\)
\(128\) 0 0
\(129\) 4.82362i 0.424696i
\(130\) 0 0
\(131\) 4.97877 0.434997 0.217499 0.976061i \(-0.430210\pi\)
0.217499 + 0.976061i \(0.430210\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) − 3.21486i − 0.276691i
\(136\) 0 0
\(137\) −3.68980 −0.315241 −0.157620 0.987500i \(-0.550382\pi\)
−0.157620 + 0.987500i \(0.550382\pi\)
\(138\) 0 0
\(139\) 12.0577 1.02272 0.511360 0.859367i \(-0.329142\pi\)
0.511360 + 0.859367i \(0.329142\pi\)
\(140\) 0 0
\(141\) −2.95367 −0.248744
\(142\) 0 0
\(143\) 4.00580 0.334982
\(144\) 0 0
\(145\) 3.57445i 0.296842i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.30262 0.270561 0.135280 0.990807i \(-0.456806\pi\)
0.135280 + 0.990807i \(0.456806\pi\)
\(150\) 0 0
\(151\) 18.9511i 1.54222i 0.636701 + 0.771111i \(0.280299\pi\)
−0.636701 + 0.771111i \(0.719701\pi\)
\(152\) 0 0
\(153\) − 6.91037i − 0.558671i
\(154\) 0 0
\(155\) 27.9795i 2.24737i
\(156\) 0 0
\(157\) − 0.886436i − 0.0707453i −0.999374 0.0353726i \(-0.988738\pi\)
0.999374 0.0353726i \(-0.0112618\pi\)
\(158\) 0 0
\(159\) −4.57794 −0.363055
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 19.3538i − 1.51591i −0.652310 0.757953i \(-0.726200\pi\)
0.652310 0.757953i \(-0.273800\pi\)
\(164\) 0 0
\(165\) −4.39502 −0.342152
\(166\) 0 0
\(167\) −25.4855 −1.97213 −0.986065 0.166360i \(-0.946799\pi\)
−0.986065 + 0.166360i \(0.946799\pi\)
\(168\) 0 0
\(169\) 4.41421 0.339555
\(170\) 0 0
\(171\) −7.35449 −0.562412
\(172\) 0 0
\(173\) − 21.6924i − 1.64925i −0.565683 0.824623i \(-0.691387\pi\)
0.565683 0.824623i \(-0.308613\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 14.0985 1.05971
\(178\) 0 0
\(179\) − 12.2824i − 0.918032i −0.888428 0.459016i \(-0.848202\pi\)
0.888428 0.459016i \(-0.151798\pi\)
\(180\) 0 0
\(181\) − 2.74444i − 0.203993i −0.994785 0.101996i \(-0.967477\pi\)
0.994785 0.101996i \(-0.0325230\pi\)
\(182\) 0 0
\(183\) − 11.0618i − 0.817713i
\(184\) 0 0
\(185\) − 24.5548i − 1.80531i
\(186\) 0 0
\(187\) −9.44716 −0.690844
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 22.8857i 1.65595i 0.560766 + 0.827974i \(0.310507\pi\)
−0.560766 + 0.827974i \(0.689493\pi\)
\(192\) 0 0
\(193\) 14.6163 1.05211 0.526053 0.850452i \(-0.323671\pi\)
0.526053 + 0.850452i \(0.323671\pi\)
\(194\) 0 0
\(195\) 9.42002 0.674581
\(196\) 0 0
\(197\) 24.5430 1.74861 0.874307 0.485374i \(-0.161316\pi\)
0.874307 + 0.485374i \(0.161316\pi\)
\(198\) 0 0
\(199\) −6.04659 −0.428631 −0.214316 0.976764i \(-0.568752\pi\)
−0.214316 + 0.976764i \(0.568752\pi\)
\(200\) 0 0
\(201\) − 12.1545i − 0.857309i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2.67845 0.187071
\(206\) 0 0
\(207\) − 3.62774i − 0.252145i
\(208\) 0 0
\(209\) 10.0543i 0.695471i
\(210\) 0 0
\(211\) − 9.33513i − 0.642657i −0.946968 0.321328i \(-0.895871\pi\)
0.946968 0.321328i \(-0.104129\pi\)
\(212\) 0 0
\(213\) 12.6249i 0.865042i
\(214\) 0 0
\(215\) 15.5072 1.05758
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 6.49011i 0.438561i
\(220\) 0 0
\(221\) 20.2484 1.36206
\(222\) 0 0
\(223\) 5.48794 0.367500 0.183750 0.982973i \(-0.441176\pi\)
0.183750 + 0.982973i \(0.441176\pi\)
\(224\) 0 0
\(225\) −5.33530 −0.355687
\(226\) 0 0
\(227\) −1.03813 −0.0689028 −0.0344514 0.999406i \(-0.510968\pi\)
−0.0344514 + 0.999406i \(0.510968\pi\)
\(228\) 0 0
\(229\) − 15.6039i − 1.03113i −0.856850 0.515566i \(-0.827582\pi\)
0.856850 0.515566i \(-0.172418\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.21561 0.603734 0.301867 0.953350i \(-0.402390\pi\)
0.301867 + 0.953350i \(0.402390\pi\)
\(234\) 0 0
\(235\) 9.49562i 0.619426i
\(236\) 0 0
\(237\) 7.79367i 0.506253i
\(238\) 0 0
\(239\) − 1.54809i − 0.100137i −0.998746 0.0500687i \(-0.984056\pi\)
0.998746 0.0500687i \(-0.0159440\pi\)
\(240\) 0 0
\(241\) − 3.68527i − 0.237389i −0.992931 0.118695i \(-0.962129\pi\)
0.992931 0.118695i \(-0.0378709\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 21.5498i − 1.37118i
\(248\) 0 0
\(249\) −8.87502 −0.562431
\(250\) 0 0
\(251\) 26.1940 1.65335 0.826676 0.562679i \(-0.190229\pi\)
0.826676 + 0.562679i \(0.190229\pi\)
\(252\) 0 0
\(253\) −4.95947 −0.311799
\(254\) 0 0
\(255\) −22.2159 −1.39121
\(256\) 0 0
\(257\) − 0.833147i − 0.0519703i −0.999662 0.0259851i \(-0.991728\pi\)
0.999662 0.0259851i \(-0.00827226\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.11185 −0.0688220
\(262\) 0 0
\(263\) − 5.04212i − 0.310911i −0.987843 0.155455i \(-0.950316\pi\)
0.987843 0.155455i \(-0.0496845\pi\)
\(264\) 0 0
\(265\) 14.7174i 0.904085i
\(266\) 0 0
\(267\) − 12.6146i − 0.772002i
\(268\) 0 0
\(269\) − 4.85594i − 0.296072i −0.988982 0.148036i \(-0.952705\pi\)
0.988982 0.148036i \(-0.0472951\pi\)
\(270\) 0 0
\(271\) −22.2488 −1.35152 −0.675759 0.737122i \(-0.736184\pi\)
−0.675759 + 0.737122i \(0.736184\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.29388i 0.439837i
\(276\) 0 0
\(277\) 0.670606 0.0402928 0.0201464 0.999797i \(-0.493587\pi\)
0.0201464 + 0.999797i \(0.493587\pi\)
\(278\) 0 0
\(279\) −8.70319 −0.521046
\(280\) 0 0
\(281\) 25.5971 1.52700 0.763499 0.645810i \(-0.223480\pi\)
0.763499 + 0.645810i \(0.223480\pi\)
\(282\) 0 0
\(283\) 11.1115 0.660510 0.330255 0.943892i \(-0.392865\pi\)
0.330255 + 0.943892i \(0.392865\pi\)
\(284\) 0 0
\(285\) 23.6436i 1.40053i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −30.7533 −1.80902
\(290\) 0 0
\(291\) − 13.8811i − 0.813727i
\(292\) 0 0
\(293\) 19.2818i 1.12645i 0.826303 + 0.563226i \(0.190440\pi\)
−0.826303 + 0.563226i \(0.809560\pi\)
\(294\) 0 0
\(295\) − 45.3245i − 2.63890i
\(296\) 0 0
\(297\) − 1.36710i − 0.0793270i
\(298\) 0 0
\(299\) 10.6298 0.614738
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 1.65891i 0.0953019i
\(304\) 0 0
\(305\) −35.5622 −2.03628
\(306\) 0 0
\(307\) 26.0058 1.48423 0.742115 0.670273i \(-0.233823\pi\)
0.742115 + 0.670273i \(0.233823\pi\)
\(308\) 0 0
\(309\) −1.30791 −0.0744042
\(310\) 0 0
\(311\) 19.0772 1.08177 0.540885 0.841096i \(-0.318089\pi\)
0.540885 + 0.841096i \(0.318089\pi\)
\(312\) 0 0
\(313\) 21.3583i 1.20724i 0.797272 + 0.603620i \(0.206276\pi\)
−0.797272 + 0.603620i \(0.793724\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.74131 0.434795 0.217398 0.976083i \(-0.430243\pi\)
0.217398 + 0.976083i \(0.430243\pi\)
\(318\) 0 0
\(319\) 1.52001i 0.0851043i
\(320\) 0 0
\(321\) − 7.15204i − 0.399188i
\(322\) 0 0
\(323\) 50.8223i 2.82783i
\(324\) 0 0
\(325\) − 15.6332i − 0.867176i
\(326\) 0 0
\(327\) −15.0443 −0.831951
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.675988i 0.0371557i 0.999827 + 0.0185778i \(0.00591385\pi\)
−0.999827 + 0.0185778i \(0.994086\pi\)
\(332\) 0 0
\(333\) 7.63792 0.418555
\(334\) 0 0
\(335\) −39.0748 −2.13489
\(336\) 0 0
\(337\) −0.176755 −0.00962847 −0.00481423 0.999988i \(-0.501532\pi\)
−0.00481423 + 0.999988i \(0.501532\pi\)
\(338\) 0 0
\(339\) −15.4530 −0.839290
\(340\) 0 0
\(341\) 11.8981i 0.644318i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −11.6627 −0.627896
\(346\) 0 0
\(347\) 6.22637i 0.334249i 0.985936 + 0.167125i \(0.0534482\pi\)
−0.985936 + 0.167125i \(0.946552\pi\)
\(348\) 0 0
\(349\) − 17.4414i − 0.933616i −0.884359 0.466808i \(-0.845404\pi\)
0.884359 0.466808i \(-0.154596\pi\)
\(350\) 0 0
\(351\) 2.93015i 0.156400i
\(352\) 0 0
\(353\) 8.70218i 0.463170i 0.972815 + 0.231585i \(0.0743912\pi\)
−0.972815 + 0.231585i \(0.925609\pi\)
\(354\) 0 0
\(355\) 40.5871 2.15414
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 0.764263i − 0.0403362i −0.999797 0.0201681i \(-0.993580\pi\)
0.999797 0.0201681i \(-0.00642015\pi\)
\(360\) 0 0
\(361\) 35.0886 1.84677
\(362\) 0 0
\(363\) 9.13104 0.479256
\(364\) 0 0
\(365\) 20.8648 1.09211
\(366\) 0 0
\(367\) 26.5162 1.38413 0.692067 0.721833i \(-0.256700\pi\)
0.692067 + 0.721833i \(0.256700\pi\)
\(368\) 0 0
\(369\) 0.833147i 0.0433719i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −20.1310 −1.04235 −0.521173 0.853451i \(-0.674505\pi\)
−0.521173 + 0.853451i \(0.674505\pi\)
\(374\) 0 0
\(375\) 1.07795i 0.0556652i
\(376\) 0 0
\(377\) − 3.25790i − 0.167790i
\(378\) 0 0
\(379\) − 10.2608i − 0.527061i −0.964651 0.263531i \(-0.915113\pi\)
0.964651 0.263531i \(-0.0848870\pi\)
\(380\) 0 0
\(381\) 2.16478i 0.110905i
\(382\) 0 0
\(383\) −2.34315 −0.119729 −0.0598646 0.998207i \(-0.519067\pi\)
−0.0598646 + 0.998207i \(0.519067\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.82362i 0.245198i
\(388\) 0 0
\(389\) 24.0440 1.21908 0.609540 0.792755i \(-0.291354\pi\)
0.609540 + 0.792755i \(0.291354\pi\)
\(390\) 0 0
\(391\) −25.0690 −1.26780
\(392\) 0 0
\(393\) 4.97877 0.251146
\(394\) 0 0
\(395\) 25.0555 1.26068
\(396\) 0 0
\(397\) 18.3526i 0.921088i 0.887637 + 0.460544i \(0.152346\pi\)
−0.887637 + 0.460544i \(0.847654\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.96972 −0.148301 −0.0741503 0.997247i \(-0.523624\pi\)
−0.0741503 + 0.997247i \(0.523624\pi\)
\(402\) 0 0
\(403\) − 25.5016i − 1.27033i
\(404\) 0 0
\(405\) − 3.21486i − 0.159748i
\(406\) 0 0
\(407\) − 10.4418i − 0.517580i
\(408\) 0 0
\(409\) 23.9502i 1.18426i 0.805842 + 0.592131i \(0.201713\pi\)
−0.805842 + 0.592131i \(0.798287\pi\)
\(410\) 0 0
\(411\) −3.68980 −0.182004
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 28.5319i 1.40058i
\(416\) 0 0
\(417\) 12.0577 0.590467
\(418\) 0 0
\(419\) −36.3065 −1.77369 −0.886844 0.462069i \(-0.847107\pi\)
−0.886844 + 0.462069i \(0.847107\pi\)
\(420\) 0 0
\(421\) −22.6274 −1.10279 −0.551396 0.834243i \(-0.685905\pi\)
−0.551396 + 0.834243i \(0.685905\pi\)
\(422\) 0 0
\(423\) −2.95367 −0.143612
\(424\) 0 0
\(425\) 36.8689i 1.78841i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 4.00580 0.193402
\(430\) 0 0
\(431\) − 30.6040i − 1.47414i −0.675816 0.737071i \(-0.736208\pi\)
0.675816 0.737071i \(-0.263792\pi\)
\(432\) 0 0
\(433\) − 14.4650i − 0.695146i −0.937653 0.347573i \(-0.887006\pi\)
0.937653 0.347573i \(-0.112994\pi\)
\(434\) 0 0
\(435\) 3.57445i 0.171382i
\(436\) 0 0
\(437\) 26.6802i 1.27629i
\(438\) 0 0
\(439\) −9.10981 −0.434788 −0.217394 0.976084i \(-0.569756\pi\)
−0.217394 + 0.976084i \(0.569756\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.83319i 0.134609i 0.997732 + 0.0673045i \(0.0214399\pi\)
−0.997732 + 0.0673045i \(0.978560\pi\)
\(444\) 0 0
\(445\) −40.5542 −1.92245
\(446\) 0 0
\(447\) 3.30262 0.156208
\(448\) 0 0
\(449\) 10.3630 0.489058 0.244529 0.969642i \(-0.421367\pi\)
0.244529 + 0.969642i \(0.421367\pi\)
\(450\) 0 0
\(451\) 1.13899 0.0536331
\(452\) 0 0
\(453\) 18.9511i 0.890402i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −6.65951 −0.311519 −0.155759 0.987795i \(-0.549782\pi\)
−0.155759 + 0.987795i \(0.549782\pi\)
\(458\) 0 0
\(459\) − 6.91037i − 0.322549i
\(460\) 0 0
\(461\) − 1.71311i − 0.0797873i −0.999204 0.0398936i \(-0.987298\pi\)
0.999204 0.0398936i \(-0.0127019\pi\)
\(462\) 0 0
\(463\) − 24.1733i − 1.12343i −0.827331 0.561715i \(-0.810142\pi\)
0.827331 0.561715i \(-0.189858\pi\)
\(464\) 0 0
\(465\) 27.9795i 1.29752i
\(466\) 0 0
\(467\) 37.2654 1.72444 0.862220 0.506535i \(-0.169074\pi\)
0.862220 + 0.506535i \(0.169074\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) − 0.886436i − 0.0408448i
\(472\) 0 0
\(473\) 6.59436 0.303209
\(474\) 0 0
\(475\) 39.2385 1.80038
\(476\) 0 0
\(477\) −4.57794 −0.209610
\(478\) 0 0
\(479\) 21.8596 0.998790 0.499395 0.866374i \(-0.333556\pi\)
0.499395 + 0.866374i \(0.333556\pi\)
\(480\) 0 0
\(481\) 22.3803i 1.02045i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −44.6259 −2.02636
\(486\) 0 0
\(487\) − 3.37269i − 0.152831i −0.997076 0.0764155i \(-0.975652\pi\)
0.997076 0.0764155i \(-0.0243475\pi\)
\(488\) 0 0
\(489\) − 19.3538i − 0.875208i
\(490\) 0 0
\(491\) − 9.68824i − 0.437224i −0.975812 0.218612i \(-0.929847\pi\)
0.975812 0.218612i \(-0.0701529\pi\)
\(492\) 0 0
\(493\) 7.68332i 0.346039i
\(494\) 0 0
\(495\) −4.39502 −0.197542
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 13.7401i − 0.615089i −0.951534 0.307545i \(-0.900493\pi\)
0.951534 0.307545i \(-0.0995074\pi\)
\(500\) 0 0
\(501\) −25.4855 −1.13861
\(502\) 0 0
\(503\) −1.92715 −0.0859273 −0.0429637 0.999077i \(-0.513680\pi\)
−0.0429637 + 0.999077i \(0.513680\pi\)
\(504\) 0 0
\(505\) 5.33316 0.237322
\(506\) 0 0
\(507\) 4.41421 0.196042
\(508\) 0 0
\(509\) 14.1621i 0.627723i 0.949469 + 0.313862i \(0.101623\pi\)
−0.949469 + 0.313862i \(0.898377\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −7.35449 −0.324709
\(514\) 0 0
\(515\) 4.20473i 0.185283i
\(516\) 0 0
\(517\) 4.03795i 0.177589i
\(518\) 0 0
\(519\) − 21.6924i − 0.952193i
\(520\) 0 0
\(521\) − 21.6026i − 0.946427i −0.880948 0.473213i \(-0.843094\pi\)
0.880948 0.473213i \(-0.156906\pi\)
\(522\) 0 0
\(523\) 13.9650 0.610648 0.305324 0.952249i \(-0.401235\pi\)
0.305324 + 0.952249i \(0.401235\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 60.1423i 2.61984i
\(528\) 0 0
\(529\) 9.83952 0.427805
\(530\) 0 0
\(531\) 14.0985 0.611821
\(532\) 0 0
\(533\) −2.44125 −0.105742
\(534\) 0 0
\(535\) −22.9928 −0.994064
\(536\) 0 0
\(537\) − 12.2824i − 0.530026i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 3.88005 0.166816 0.0834081 0.996515i \(-0.473419\pi\)
0.0834081 + 0.996515i \(0.473419\pi\)
\(542\) 0 0
\(543\) − 2.74444i − 0.117775i
\(544\) 0 0
\(545\) 48.3652i 2.07174i
\(546\) 0 0
\(547\) 35.1640i 1.50351i 0.659445 + 0.751753i \(0.270791\pi\)
−0.659445 + 0.751753i \(0.729209\pi\)
\(548\) 0 0
\(549\) − 11.0618i − 0.472107i
\(550\) 0 0
\(551\) 8.17712 0.348357
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) − 24.5548i − 1.04229i
\(556\) 0 0
\(557\) 42.4346 1.79801 0.899006 0.437936i \(-0.144290\pi\)
0.899006 + 0.437936i \(0.144290\pi\)
\(558\) 0 0
\(559\) −14.1339 −0.597802
\(560\) 0 0
\(561\) −9.44716 −0.398859
\(562\) 0 0
\(563\) −16.5488 −0.697447 −0.348724 0.937226i \(-0.613385\pi\)
−0.348724 + 0.937226i \(0.613385\pi\)
\(564\) 0 0
\(565\) 49.6790i 2.09001i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −24.2566 −1.01689 −0.508446 0.861094i \(-0.669780\pi\)
−0.508446 + 0.861094i \(0.669780\pi\)
\(570\) 0 0
\(571\) − 4.62563i − 0.193576i −0.995305 0.0967882i \(-0.969143\pi\)
0.995305 0.0967882i \(-0.0308569\pi\)
\(572\) 0 0
\(573\) 22.8857i 0.956062i
\(574\) 0 0
\(575\) 19.3551i 0.807163i
\(576\) 0 0
\(577\) 26.7907i 1.11531i 0.830072 + 0.557656i \(0.188299\pi\)
−0.830072 + 0.557656i \(0.811701\pi\)
\(578\) 0 0
\(579\) 14.6163 0.607434
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 6.25850i 0.259200i
\(584\) 0 0
\(585\) 9.42002 0.389470
\(586\) 0 0
\(587\) 27.8591 1.14987 0.574933 0.818200i \(-0.305028\pi\)
0.574933 + 0.818200i \(0.305028\pi\)
\(588\) 0 0
\(589\) 64.0075 2.63738
\(590\) 0 0
\(591\) 24.5430 1.00956
\(592\) 0 0
\(593\) − 1.56906i − 0.0644334i −0.999481 0.0322167i \(-0.989743\pi\)
0.999481 0.0322167i \(-0.0102567\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −6.04659 −0.247470
\(598\) 0 0
\(599\) 40.1156i 1.63908i 0.573021 + 0.819540i \(0.305771\pi\)
−0.573021 + 0.819540i \(0.694229\pi\)
\(600\) 0 0
\(601\) 24.7827i 1.01091i 0.862854 + 0.505453i \(0.168675\pi\)
−0.862854 + 0.505453i \(0.831325\pi\)
\(602\) 0 0
\(603\) − 12.1545i − 0.494968i
\(604\) 0 0
\(605\) − 29.3550i − 1.19345i
\(606\) 0 0
\(607\) −4.66240 −0.189241 −0.0946205 0.995513i \(-0.530164\pi\)
−0.0946205 + 0.995513i \(0.530164\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 8.65470i − 0.350132i
\(612\) 0 0
\(613\) 28.5817 1.15440 0.577202 0.816601i \(-0.304144\pi\)
0.577202 + 0.816601i \(0.304144\pi\)
\(614\) 0 0
\(615\) 2.67845 0.108005
\(616\) 0 0
\(617\) 10.1524 0.408720 0.204360 0.978896i \(-0.434489\pi\)
0.204360 + 0.978896i \(0.434489\pi\)
\(618\) 0 0
\(619\) −8.64022 −0.347280 −0.173640 0.984809i \(-0.555553\pi\)
−0.173640 + 0.984809i \(0.555553\pi\)
\(620\) 0 0
\(621\) − 3.62774i − 0.145576i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −23.2111 −0.928442
\(626\) 0 0
\(627\) 10.0543i 0.401530i
\(628\) 0 0
\(629\) − 52.7809i − 2.10451i
\(630\) 0 0
\(631\) − 5.46211i − 0.217443i −0.994072 0.108722i \(-0.965324\pi\)
0.994072 0.108722i \(-0.0346757\pi\)
\(632\) 0 0
\(633\) − 9.33513i − 0.371038i
\(634\) 0 0
\(635\) 6.95947 0.276178
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 12.6249i 0.499432i
\(640\) 0 0
\(641\) 1.42822 0.0564114 0.0282057 0.999602i \(-0.491021\pi\)
0.0282057 + 0.999602i \(0.491021\pi\)
\(642\) 0 0
\(643\) −24.6036 −0.970270 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(644\) 0 0
\(645\) 15.5072 0.610597
\(646\) 0 0
\(647\) 20.2785 0.797229 0.398614 0.917119i \(-0.369491\pi\)
0.398614 + 0.917119i \(0.369491\pi\)
\(648\) 0 0
\(649\) − 19.2740i − 0.756570i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 42.8841 1.67818 0.839092 0.543990i \(-0.183087\pi\)
0.839092 + 0.543990i \(0.183087\pi\)
\(654\) 0 0
\(655\) − 16.0060i − 0.625407i
\(656\) 0 0
\(657\) 6.49011i 0.253203i
\(658\) 0 0
\(659\) − 31.4717i − 1.22596i −0.790098 0.612981i \(-0.789970\pi\)
0.790098 0.612981i \(-0.210030\pi\)
\(660\) 0 0
\(661\) 12.3154i 0.479015i 0.970895 + 0.239507i \(0.0769859\pi\)
−0.970895 + 0.239507i \(0.923014\pi\)
\(662\) 0 0
\(663\) 20.2484 0.786384
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.03351i 0.156178i
\(668\) 0 0
\(669\) 5.48794 0.212176
\(670\) 0 0
\(671\) −15.1226 −0.583801
\(672\) 0 0
\(673\) 37.1864 1.43343 0.716716 0.697365i \(-0.245645\pi\)
0.716716 + 0.697365i \(0.245645\pi\)
\(674\) 0 0
\(675\) −5.33530 −0.205356
\(676\) 0 0
\(677\) − 35.6046i − 1.36840i −0.729296 0.684198i \(-0.760152\pi\)
0.729296 0.684198i \(-0.239848\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −1.03813 −0.0397810
\(682\) 0 0
\(683\) − 32.2056i − 1.23231i −0.787623 0.616157i \(-0.788689\pi\)
0.787623 0.616157i \(-0.211311\pi\)
\(684\) 0 0
\(685\) 11.8622i 0.453230i
\(686\) 0 0
\(687\) − 15.6039i − 0.595325i
\(688\) 0 0
\(689\) − 13.4141i − 0.511035i
\(690\) 0 0
\(691\) −38.6168 −1.46905 −0.734527 0.678580i \(-0.762596\pi\)
−0.734527 + 0.678580i \(0.762596\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 38.7637i − 1.47039i
\(696\) 0 0
\(697\) 5.75736 0.218076
\(698\) 0 0
\(699\) 9.21561 0.348566
\(700\) 0 0
\(701\) 30.8725 1.16604 0.583018 0.812459i \(-0.301872\pi\)
0.583018 + 0.812459i \(0.301872\pi\)
\(702\) 0 0
\(703\) −56.1730 −2.11861
\(704\) 0 0
\(705\) 9.49562i 0.357626i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 39.8452 1.49642 0.748209 0.663463i \(-0.230914\pi\)
0.748209 + 0.663463i \(0.230914\pi\)
\(710\) 0 0
\(711\) 7.79367i 0.292285i
\(712\) 0 0
\(713\) 31.5729i 1.18241i
\(714\) 0 0
\(715\) − 12.8781i − 0.481613i
\(716\) 0 0
\(717\) − 1.54809i − 0.0578144i
\(718\) 0 0
\(719\) 2.09266 0.0780431 0.0390216 0.999238i \(-0.487576\pi\)
0.0390216 + 0.999238i \(0.487576\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 3.68527i − 0.137057i
\(724\) 0 0
\(725\) 5.93207 0.220312
\(726\) 0 0
\(727\) 21.5164 0.798001 0.399000 0.916951i \(-0.369357\pi\)
0.399000 + 0.916951i \(0.369357\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 33.3330 1.23287
\(732\) 0 0
\(733\) − 7.72856i − 0.285461i −0.989762 0.142730i \(-0.954412\pi\)
0.989762 0.142730i \(-0.0455882\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −16.6163 −0.612070
\(738\) 0 0
\(739\) − 16.1063i − 0.592481i −0.955113 0.296240i \(-0.904267\pi\)
0.955113 0.296240i \(-0.0957329\pi\)
\(740\) 0 0
\(741\) − 21.5498i − 0.791651i
\(742\) 0 0
\(743\) − 29.3296i − 1.07600i −0.842945 0.537999i \(-0.819180\pi\)
0.842945 0.537999i \(-0.180820\pi\)
\(744\) 0 0
\(745\) − 10.6174i − 0.388993i
\(746\) 0 0
\(747\) −8.87502 −0.324720
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 13.5991i 0.496238i 0.968730 + 0.248119i \(0.0798125\pi\)
−0.968730 + 0.248119i \(0.920188\pi\)
\(752\) 0 0
\(753\) 26.1940 0.954563
\(754\) 0 0
\(755\) 60.9252 2.21730
\(756\) 0 0
\(757\) −2.59688 −0.0943851 −0.0471926 0.998886i \(-0.515027\pi\)
−0.0471926 + 0.998886i \(0.515027\pi\)
\(758\) 0 0
\(759\) −4.95947 −0.180017
\(760\) 0 0
\(761\) 17.2920i 0.626833i 0.949616 + 0.313416i \(0.101474\pi\)
−0.949616 + 0.313416i \(0.898526\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −22.2159 −0.803216
\(766\) 0 0
\(767\) 41.3106i 1.49164i
\(768\) 0 0
\(769\) − 41.4025i − 1.49301i −0.665378 0.746507i \(-0.731730\pi\)
0.665378 0.746507i \(-0.268270\pi\)
\(770\) 0 0
\(771\) − 0.833147i − 0.0300051i
\(772\) 0 0
\(773\) 29.4669i 1.05985i 0.848044 + 0.529925i \(0.177780\pi\)
−0.848044 + 0.529925i \(0.822220\pi\)
\(774\) 0 0
\(775\) 46.4341 1.66796
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 6.12738i − 0.219536i
\(780\) 0 0
\(781\) 17.2594 0.617591
\(782\) 0 0
\(783\) −1.11185 −0.0397344
\(784\) 0 0
\(785\) −2.84976 −0.101712
\(786\) 0 0
\(787\) −54.1841 −1.93146 −0.965728 0.259557i \(-0.916423\pi\)
−0.965728 + 0.259557i \(0.916423\pi\)
\(788\) 0 0
\(789\) − 5.04212i − 0.179504i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 32.4128 1.15101
\(794\) 0 0
\(795\) 14.7174i 0.521974i
\(796\) 0 0
\(797\) 0.512549i 0.0181554i 0.999959 + 0.00907771i \(0.00288956\pi\)
−0.999959 + 0.00907771i \(0.997110\pi\)
\(798\) 0 0
\(799\) 20.4110i 0.722088i
\(800\) 0 0
\(801\) − 12.6146i − 0.445716i
\(802\) 0 0
\(803\) 8.87261 0.313108
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 4.85594i − 0.170937i
\(808\) 0 0
\(809\) −53.1407 −1.86833 −0.934164 0.356844i \(-0.883853\pi\)
−0.934164 + 0.356844i \(0.883853\pi\)
\(810\) 0 0
\(811\) 10.8127 0.379687 0.189843 0.981814i \(-0.439202\pi\)
0.189843 + 0.981814i \(0.439202\pi\)
\(812\) 0 0
\(813\) −22.2488 −0.780300
\(814\) 0 0
\(815\) −62.2196 −2.17946
\(816\) 0 0
\(817\) − 35.4753i − 1.24112i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.70078 −0.0942579 −0.0471290 0.998889i \(-0.515007\pi\)
−0.0471290 + 0.998889i \(0.515007\pi\)
\(822\) 0 0
\(823\) 12.6132i 0.439670i 0.975537 + 0.219835i \(0.0705519\pi\)
−0.975537 + 0.219835i \(0.929448\pi\)
\(824\) 0 0
\(825\) 7.29388i 0.253940i
\(826\) 0 0
\(827\) 2.95803i 0.102861i 0.998677 + 0.0514304i \(0.0163780\pi\)
−0.998677 + 0.0514304i \(0.983622\pi\)
\(828\) 0 0
\(829\) − 15.8101i − 0.549106i −0.961572 0.274553i \(-0.911470\pi\)
0.961572 0.274553i \(-0.0885300\pi\)
\(830\) 0 0
\(831\) 0.670606 0.0232631
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 81.9324i 2.83539i
\(836\) 0 0
\(837\) −8.70319 −0.300826
\(838\) 0 0
\(839\) 5.13130 0.177152 0.0885761 0.996069i \(-0.471768\pi\)
0.0885761 + 0.996069i \(0.471768\pi\)
\(840\) 0 0
\(841\) −27.7638 −0.957372
\(842\) 0 0
\(843\) 25.5971 0.881612
\(844\) 0 0
\(845\) − 14.1911i − 0.488187i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 11.1115 0.381345
\(850\) 0 0
\(851\) − 27.7084i − 0.949831i
\(852\) 0 0
\(853\) 45.1627i 1.54634i 0.634198 + 0.773170i \(0.281330\pi\)
−0.634198 + 0.773170i \(0.718670\pi\)
\(854\) 0 0
\(855\) 23.6436i 0.808596i
\(856\) 0 0
\(857\) − 1.47725i − 0.0504618i −0.999682 0.0252309i \(-0.991968\pi\)
0.999682 0.0252309i \(-0.00803210\pi\)
\(858\) 0 0
\(859\) 3.08183 0.105151 0.0525753 0.998617i \(-0.483257\pi\)
0.0525753 + 0.998617i \(0.483257\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 46.4000i − 1.57948i −0.613445 0.789738i \(-0.710217\pi\)
0.613445 0.789738i \(-0.289783\pi\)
\(864\) 0 0
\(865\) −69.7381 −2.37117
\(866\) 0 0
\(867\) −30.7533 −1.04444
\(868\) 0 0
\(869\) 10.6547 0.361436
\(870\) 0 0
\(871\) 35.6144 1.20675
\(872\) 0 0
\(873\) − 13.8811i − 0.469806i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −50.4774 −1.70450 −0.852251 0.523133i \(-0.824763\pi\)
−0.852251 + 0.523133i \(0.824763\pi\)
\(878\) 0 0
\(879\) 19.2818i 0.650357i
\(880\) 0 0
\(881\) − 3.66920i − 0.123619i −0.998088 0.0618093i \(-0.980313\pi\)
0.998088 0.0618093i \(-0.0196870\pi\)
\(882\) 0 0
\(883\) 18.0393i 0.607071i 0.952820 + 0.303535i \(0.0981671\pi\)
−0.952820 + 0.303535i \(0.901833\pi\)
\(884\) 0 0
\(885\) − 45.3245i − 1.52357i
\(886\) 0 0
\(887\) −12.8996 −0.433126 −0.216563 0.976269i \(-0.569485\pi\)
−0.216563 + 0.976269i \(0.569485\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 1.36710i − 0.0457995i
\(892\) 0 0
\(893\) 21.7227 0.726924
\(894\) 0 0
\(895\) −39.4863 −1.31988
\(896\) 0 0
\(897\) 10.6298 0.354919
\(898\) 0 0
\(899\) 9.67667 0.322735
\(900\) 0 0
\(901\) 31.6353i 1.05392i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8.82299 −0.293286
\(906\) 0 0
\(907\) 0.765170i 0.0254071i 0.999919 + 0.0127035i \(0.00404377\pi\)
−0.999919 + 0.0127035i \(0.995956\pi\)
\(908\) 0 0
\(909\) 1.65891i 0.0550226i
\(910\) 0 0
\(911\) 18.7415i 0.620934i 0.950584 + 0.310467i \(0.100485\pi\)
−0.950584 + 0.310467i \(0.899515\pi\)
\(912\) 0 0
\(913\) 12.1330i 0.401544i
\(914\) 0 0
\(915\) −35.5622 −1.17565
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 21.0460i 0.694242i 0.937820 + 0.347121i \(0.112841\pi\)
−0.937820 + 0.347121i \(0.887159\pi\)
\(920\) 0 0
\(921\) 26.0058 0.856920
\(922\) 0 0
\(923\) −36.9928 −1.21763
\(924\) 0 0
\(925\) −40.7506 −1.33987
\(926\) 0 0
\(927\) −1.30791 −0.0429573
\(928\) 0 0
\(929\) 32.7286i 1.07379i 0.843649 + 0.536895i \(0.180403\pi\)
−0.843649 + 0.536895i \(0.819597\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 19.0772 0.624561
\(934\) 0 0
\(935\) 30.3713i 0.993246i
\(936\) 0 0
\(937\) 2.53824i 0.0829207i 0.999140 + 0.0414603i \(0.0132010\pi\)
−0.999140 + 0.0414603i \(0.986799\pi\)
\(938\) 0 0
\(939\) 21.3583i 0.697001i
\(940\) 0 0
\(941\) 46.1741i 1.50523i 0.658459 + 0.752617i \(0.271209\pi\)
−0.658459 + 0.752617i \(0.728791\pi\)
\(942\) 0 0
\(943\) 3.02244 0.0984242
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1.51338i − 0.0491782i −0.999698 0.0245891i \(-0.992172\pi\)
0.999698 0.0245891i \(-0.00782775\pi\)
\(948\) 0 0
\(949\) −19.0170 −0.617318
\(950\) 0 0
\(951\) 7.74131 0.251029
\(952\) 0 0
\(953\) −43.1132 −1.39657 −0.698287 0.715818i \(-0.746054\pi\)
−0.698287 + 0.715818i \(0.746054\pi\)
\(954\) 0 0
\(955\) 73.5741 2.38080
\(956\) 0 0
\(957\) 1.52001i 0.0491350i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 44.7454 1.44340
\(962\) 0 0
\(963\) − 7.15204i − 0.230471i
\(964\) 0 0
\(965\) − 46.9894i − 1.51264i
\(966\) 0 0
\(967\) 22.5218i 0.724253i 0.932129 + 0.362127i \(0.117949\pi\)
−0.932129 + 0.362127i \(0.882051\pi\)
\(968\) 0 0
\(969\) 50.8223i 1.63265i
\(970\) 0 0
\(971\) −24.4894 −0.785902 −0.392951 0.919559i \(-0.628546\pi\)
−0.392951 + 0.919559i \(0.628546\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) − 15.6332i − 0.500665i
\(976\) 0 0
\(977\) 16.6382 0.532302 0.266151 0.963931i \(-0.414248\pi\)
0.266151 + 0.963931i \(0.414248\pi\)
\(978\) 0 0
\(979\) −17.2454 −0.551166
\(980\) 0 0
\(981\) −15.0443 −0.480327
\(982\) 0 0
\(983\) 5.35617 0.170835 0.0854177 0.996345i \(-0.472778\pi\)
0.0854177 + 0.996345i \(0.472778\pi\)
\(984\) 0 0
\(985\) − 78.9021i − 2.51403i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 17.4988 0.556430
\(990\) 0 0
\(991\) 4.67605i 0.148540i 0.997238 + 0.0742698i \(0.0236626\pi\)
−0.997238 + 0.0742698i \(0.976337\pi\)
\(992\) 0 0
\(993\) 0.675988i 0.0214518i
\(994\) 0 0
\(995\) 19.4389i 0.616255i
\(996\) 0 0
\(997\) 0.906930i 0.0287228i 0.999897 + 0.0143614i \(0.00457153\pi\)
−0.999897 + 0.0143614i \(0.995428\pi\)
\(998\) 0 0
\(999\) 7.63792 0.241653
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 2352.2.b.l.1567.2 yes 8
3.2 odd 2 7056.2.b.w.1567.7 8
4.3 odd 2 2352.2.b.k.1567.2 8
7.2 even 3 2352.2.bl.o.31.1 8
7.3 odd 6 2352.2.bl.t.607.1 8
7.4 even 3 2352.2.bl.q.607.4 8
7.5 odd 6 2352.2.bl.r.31.4 8
7.6 odd 2 2352.2.b.k.1567.7 yes 8
12.11 even 2 7056.2.b.x.1567.7 8
21.20 even 2 7056.2.b.x.1567.2 8
28.3 even 6 2352.2.bl.o.607.1 8
28.11 odd 6 2352.2.bl.r.607.4 8
28.19 even 6 2352.2.bl.q.31.4 8
28.23 odd 6 2352.2.bl.t.31.1 8
28.27 even 2 inner 2352.2.b.l.1567.7 yes 8
84.83 odd 2 7056.2.b.w.1567.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2352.2.b.k.1567.2 8 4.3 odd 2
2352.2.b.k.1567.7 yes 8 7.6 odd 2
2352.2.b.l.1567.2 yes 8 1.1 even 1 trivial
2352.2.b.l.1567.7 yes 8 28.27 even 2 inner
2352.2.bl.o.31.1 8 7.2 even 3
2352.2.bl.o.607.1 8 28.3 even 6
2352.2.bl.q.31.4 8 28.19 even 6
2352.2.bl.q.607.4 8 7.4 even 3
2352.2.bl.r.31.4 8 7.5 odd 6
2352.2.bl.r.607.4 8 28.11 odd 6
2352.2.bl.t.31.1 8 28.23 odd 6
2352.2.bl.t.607.1 8 7.3 odd 6
7056.2.b.w.1567.2 8 84.83 odd 2
7056.2.b.w.1567.7 8 3.2 odd 2
7056.2.b.x.1567.2 8 21.20 even 2
7056.2.b.x.1567.7 8 12.11 even 2