Properties

Label 3380.2.f.b.3041.2
Level $3380$
Weight $2$
Character 3380.3041
Analytic conductor $26.989$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.9894358832\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3041.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3380.3041
Dual form 3380.2.f.b.3041.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{3} +1.00000i q^{5} +2.00000i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} +1.00000i q^{5} +2.00000i q^{7} +1.00000 q^{9} -2.00000i q^{15} +6.00000 q^{17} +4.00000i q^{19} -4.00000i q^{21} -6.00000 q^{23} -1.00000 q^{25} +4.00000 q^{27} +6.00000 q^{29} +4.00000i q^{31} -2.00000 q^{35} +2.00000i q^{37} -6.00000i q^{41} +10.0000 q^{43} +1.00000i q^{45} -6.00000i q^{47} +3.00000 q^{49} -12.0000 q^{51} -6.00000 q^{53} -8.00000i q^{57} +12.0000i q^{59} +2.00000 q^{61} +2.00000i q^{63} -2.00000i q^{67} +12.0000 q^{69} +12.0000i q^{71} +2.00000i q^{73} +2.00000 q^{75} +8.00000 q^{79} -11.0000 q^{81} -6.00000i q^{83} +6.00000i q^{85} -12.0000 q^{87} -6.00000i q^{89} -8.00000i q^{93} -4.00000 q^{95} -2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 4q^{3} + 2q^{9} + 12q^{17} - 12q^{23} - 2q^{25} + 8q^{27} + 12q^{29} - 4q^{35} + 20q^{43} + 6q^{49} - 24q^{51} - 12q^{53} + 4q^{61} + 24q^{69} + 4q^{75} + 16q^{79} - 22q^{81} - 24q^{87} - 8q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1691\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) − 2.00000i − 0.516398i
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) − 4.00000i − 0.872872i
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 6.00000i − 0.937043i −0.883452 0.468521i \(-0.844787\pi\)
0.883452 0.468521i \(-0.155213\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) − 6.00000i − 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −12.0000 −1.68034
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 8.00000i − 1.05963i
\(58\) 0 0
\(59\) 12.0000i 1.56227i 0.624364 + 0.781133i \(0.285358\pi\)
−0.624364 + 0.781133i \(0.714642\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 2.00000i − 0.244339i −0.992509 0.122169i \(-0.961015\pi\)
0.992509 0.122169i \(-0.0389851\pi\)
\(68\) 0 0
\(69\) 12.0000 1.44463
\(70\) 0 0
\(71\) 12.0000i 1.42414i 0.702109 + 0.712069i \(0.252242\pi\)
−0.702109 + 0.712069i \(0.747758\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 0 0
\(75\) 2.00000 0.230940
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 6.00000i 0.650791i
\(86\) 0 0
\(87\) −12.0000 −1.28654
\(88\) 0 0
\(89\) − 6.00000i − 0.635999i −0.948091 0.317999i \(-0.896989\pi\)
0.948091 0.317999i \(-0.103011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) 4.00000 0.390360
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) − 2.00000i − 0.191565i −0.995402 0.0957826i \(-0.969465\pi\)
0.995402 0.0957826i \(-0.0305354\pi\)
\(110\) 0 0
\(111\) − 4.00000i − 0.379663i
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) − 6.00000i − 0.559503i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 12.0000i 1.10004i
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 12.0000i 1.08200i
\(124\) 0 0
\(125\) − 1.00000i − 0.0894427i
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) −20.0000 −1.76090
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 0 0
\(135\) 4.00000i 0.344265i
\(136\) 0 0
\(137\) 18.0000i 1.53784i 0.639343 + 0.768922i \(0.279207\pi\)
−0.639343 + 0.768922i \(0.720793\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 12.0000i 1.01058i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 6.00000i 0.498273i
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) 0 0
\(149\) 6.00000i 0.491539i 0.969328 + 0.245770i \(0.0790407\pi\)
−0.969328 + 0.245770i \(0.920959\pi\)
\(150\) 0 0
\(151\) 20.0000i 1.62758i 0.581161 + 0.813788i \(0.302599\pi\)
−0.581161 + 0.813788i \(0.697401\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 0 0
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) − 12.0000i − 0.945732i
\(162\) 0 0
\(163\) − 10.0000i − 0.783260i −0.920123 0.391630i \(-0.871911\pi\)
0.920123 0.391630i \(-0.128089\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 18.0000i 1.39288i 0.717614 + 0.696441i \(0.245234\pi\)
−0.717614 + 0.696441i \(0.754766\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 4.00000i 0.305888i
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) − 2.00000i − 0.151186i
\(176\) 0 0
\(177\) − 24.0000i − 1.80395i
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 8.00000i 0.581914i
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 26.0000i 1.87152i 0.352636 + 0.935760i \(0.385285\pi\)
−0.352636 + 0.935760i \(0.614715\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 4.00000i 0.282138i
\(202\) 0 0
\(203\) 12.0000i 0.842235i
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) − 24.0000i − 1.64445i
\(214\) 0 0
\(215\) 10.0000i 0.681994i
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) − 4.00000i − 0.270295i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 10.0000i 0.669650i 0.942280 + 0.334825i \(0.108677\pi\)
−0.942280 + 0.334825i \(0.891323\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) 6.00000i 0.398234i 0.979976 + 0.199117i \(0.0638074\pi\)
−0.979976 + 0.199117i \(0.936193\pi\)
\(228\) 0 0
\(229\) 14.0000i 0.925146i 0.886581 + 0.462573i \(0.153074\pi\)
−0.886581 + 0.462573i \(0.846926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 0 0
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) 24.0000i 1.55243i 0.630468 + 0.776215i \(0.282863\pi\)
−0.630468 + 0.776215i \(0.717137\pi\)
\(240\) 0 0
\(241\) 14.0000i 0.901819i 0.892570 + 0.450910i \(0.148900\pi\)
−0.892570 + 0.450910i \(0.851100\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) 3.00000i 0.191663i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 12.0000i 0.760469i
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) − 12.0000i − 0.751469i
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) − 6.00000i − 0.368577i
\(266\) 0 0
\(267\) 12.0000i 0.734388i
\(268\) 0 0
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 20.0000i 1.21491i 0.794353 + 0.607457i \(0.207810\pi\)
−0.794353 + 0.607457i \(0.792190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 0 0
\(279\) 4.00000i 0.239474i
\(280\) 0 0
\(281\) 6.00000i 0.357930i 0.983855 + 0.178965i \(0.0572749\pi\)
−0.983855 + 0.178965i \(0.942725\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 0 0
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 4.00000i 0.234484i
\(292\) 0 0
\(293\) − 30.0000i − 1.75262i −0.481749 0.876309i \(-0.659998\pi\)
0.481749 0.876309i \(-0.340002\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 20.0000i 1.15278i
\(302\) 0 0
\(303\) 12.0000 0.689382
\(304\) 0 0
\(305\) 2.00000i 0.114520i
\(306\) 0 0
\(307\) 2.00000i 0.114146i 0.998370 + 0.0570730i \(0.0181768\pi\)
−0.998370 + 0.0570730i \(0.981823\pi\)
\(308\) 0 0
\(309\) 28.0000 1.59286
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 24.0000i 1.33540i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 4.00000i 0.221201i
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) − 8.00000i − 0.439720i −0.975531 0.219860i \(-0.929440\pi\)
0.975531 0.219860i \(-0.0705600\pi\)
\(332\) 0 0
\(333\) 2.00000i 0.109599i
\(334\) 0 0
\(335\) 2.00000 0.109272
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 12.0000i 0.646058i
\(346\) 0 0
\(347\) −30.0000 −1.61048 −0.805242 0.592946i \(-0.797965\pi\)
−0.805242 + 0.592946i \(0.797965\pi\)
\(348\) 0 0
\(349\) − 10.0000i − 0.535288i −0.963518 0.267644i \(-0.913755\pi\)
0.963518 0.267644i \(-0.0862451\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 18.0000i − 0.958043i −0.877803 0.479022i \(-0.840992\pi\)
0.877803 0.479022i \(-0.159008\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 0 0
\(357\) − 24.0000i − 1.27021i
\(358\) 0 0
\(359\) 24.0000i 1.26667i 0.773877 + 0.633336i \(0.218315\pi\)
−0.773877 + 0.633336i \(0.781685\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) −22.0000 −1.15470
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) −22.0000 −1.14839 −0.574195 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) 0 0
\(369\) − 6.00000i − 0.312348i
\(370\) 0 0
\(371\) − 12.0000i − 0.623009i
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 2.00000i 0.103280i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 28.0000i 1.43826i 0.694874 + 0.719132i \(0.255460\pi\)
−0.694874 + 0.719132i \(0.744540\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 0 0
\(383\) − 6.00000i − 0.306586i −0.988181 0.153293i \(-0.951012\pi\)
0.988181 0.153293i \(-0.0489878\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 10.0000 0.508329
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.00000i 0.402524i
\(396\) 0 0
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) 0 0
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) − 30.0000i − 1.49813i −0.662497 0.749064i \(-0.730503\pi\)
0.662497 0.749064i \(-0.269497\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 11.0000i − 0.546594i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 34.0000i 1.68119i 0.541663 + 0.840596i \(0.317795\pi\)
−0.541663 + 0.840596i \(0.682205\pi\)
\(410\) 0 0
\(411\) − 36.0000i − 1.77575i
\(412\) 0 0
\(413\) −24.0000 −1.18096
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 0 0
\(417\) 8.00000 0.391762
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) − 26.0000i − 1.26716i −0.773676 0.633581i \(-0.781584\pi\)
0.773676 0.633581i \(-0.218416\pi\)
\(422\) 0 0
\(423\) − 6.00000i − 0.291730i
\(424\) 0 0
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 4.00000i 0.193574i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 36.0000i − 1.73406i −0.498257 0.867029i \(-0.666026\pi\)
0.498257 0.867029i \(-0.333974\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) − 12.0000i − 0.575356i
\(436\) 0 0
\(437\) − 24.0000i − 1.14808i
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 0 0
\(447\) − 12.0000i − 0.567581i
\(448\) 0 0
\(449\) 6.00000i 0.283158i 0.989927 + 0.141579i \(0.0452178\pi\)
−0.989927 + 0.141579i \(0.954782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) − 40.0000i − 1.87936i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 26.0000i − 1.21623i −0.793849 0.608114i \(-0.791926\pi\)
0.793849 0.608114i \(-0.208074\pi\)
\(458\) 0 0
\(459\) 24.0000 1.12022
\(460\) 0 0
\(461\) − 30.0000i − 1.39724i −0.715493 0.698620i \(-0.753798\pi\)
0.715493 0.698620i \(-0.246202\pi\)
\(462\) 0 0
\(463\) 14.0000i 0.650635i 0.945605 + 0.325318i \(0.105471\pi\)
−0.945605 + 0.325318i \(0.894529\pi\)
\(464\) 0 0
\(465\) 8.00000 0.370991
\(466\) 0 0
\(467\) 30.0000 1.38823 0.694117 0.719862i \(-0.255795\pi\)
0.694117 + 0.719862i \(0.255795\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 44.0000 2.02741
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) − 4.00000i − 0.183533i
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) − 24.0000i − 1.09659i −0.836286 0.548294i \(-0.815277\pi\)
0.836286 0.548294i \(-0.184723\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 24.0000i 1.09204i
\(484\) 0 0
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) − 26.0000i − 1.17817i −0.808070 0.589086i \(-0.799488\pi\)
0.808070 0.589086i \(-0.200512\pi\)
\(488\) 0 0
\(489\) 20.0000i 0.904431i
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) 4.00000i 0.179065i 0.995984 + 0.0895323i \(0.0285372\pi\)
−0.995984 + 0.0895323i \(0.971463\pi\)
\(500\) 0 0
\(501\) − 36.0000i − 1.60836i
\(502\) 0 0
\(503\) −18.0000 −0.802580 −0.401290 0.915951i \(-0.631438\pi\)
−0.401290 + 0.915951i \(0.631438\pi\)
\(504\) 0 0
\(505\) − 6.00000i − 0.266996i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 6.00000i − 0.265945i −0.991120 0.132973i \(-0.957548\pi\)
0.991120 0.132973i \(-0.0424523\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) 0 0
\(513\) 16.0000i 0.706417i
\(514\) 0 0
\(515\) − 14.0000i − 0.616914i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 14.0000 0.612177 0.306089 0.952003i \(-0.400980\pi\)
0.306089 + 0.952003i \(0.400980\pi\)
\(524\) 0 0
\(525\) 4.00000i 0.174574i
\(526\) 0 0
\(527\) 24.0000i 1.04546i
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 12.0000i 0.520756i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 6.00000i − 0.259403i
\(536\) 0 0
\(537\) −24.0000 −1.03568
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 14.0000i 0.601907i 0.953639 + 0.300954i \(0.0973049\pi\)
−0.953639 + 0.300954i \(0.902695\pi\)
\(542\) 0 0
\(543\) −20.0000 −0.858282
\(544\) 0 0
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) 26.0000 1.11168 0.555840 0.831289i \(-0.312397\pi\)
0.555840 + 0.831289i \(0.312397\pi\)
\(548\) 0 0
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 24.0000i 1.02243i
\(552\) 0 0
\(553\) 16.0000i 0.680389i
\(554\) 0 0
\(555\) 4.00000 0.169791
\(556\) 0 0
\(557\) − 30.0000i − 1.27114i −0.772043 0.635570i \(-0.780765\pi\)
0.772043 0.635570i \(-0.219235\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) 0 0
\(565\) − 6.00000i − 0.252422i
\(566\) 0 0
\(567\) − 22.0000i − 0.923913i
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) 24.0000 1.00261
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 0 0
\(577\) 22.0000i 0.915872i 0.888985 + 0.457936i \(0.151411\pi\)
−0.888985 + 0.457936i \(0.848589\pi\)
\(578\) 0 0
\(579\) − 52.0000i − 2.16105i
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.00000i 0.247647i 0.992304 + 0.123823i \(0.0395156\pi\)
−0.992304 + 0.123823i \(0.960484\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 36.0000i 1.48084i
\(592\) 0 0
\(593\) 18.0000i 0.739171i 0.929197 + 0.369586i \(0.120500\pi\)
−0.929197 + 0.369586i \(0.879500\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 0 0
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) − 2.00000i − 0.0814463i
\(604\) 0 0
\(605\) 11.0000i 0.447214i
\(606\) 0 0
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 0 0
\(609\) − 24.0000i − 0.972529i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 0 0
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 0 0
\(619\) 20.0000i 0.803868i 0.915669 + 0.401934i \(0.131662\pi\)
−0.915669 + 0.401934i \(0.868338\pi\)
\(620\) 0 0
\(621\) −24.0000 −0.963087
\(622\) 0 0
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12.0000i 0.478471i
\(630\) 0 0
\(631\) − 28.0000i − 1.11466i −0.830290 0.557331i \(-0.811825\pi\)
0.830290 0.557331i \(-0.188175\pi\)
\(632\) 0 0
\(633\) 32.0000 1.27189
\(634\) 0 0
\(635\) − 2.00000i − 0.0793676i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 12.0000i 0.474713i
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) − 14.0000i − 0.552106i −0.961142 0.276053i \(-0.910973\pi\)
0.961142 0.276053i \(-0.0890266\pi\)
\(644\) 0 0
\(645\) − 20.0000i − 0.787499i
\(646\) 0 0
\(647\) −42.0000 −1.65119 −0.825595 0.564263i \(-0.809160\pi\)
−0.825595 + 0.564263i \(0.809160\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 0 0
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.00000i 0.0780274i
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) − 22.0000i − 0.855701i −0.903850 0.427850i \(-0.859271\pi\)
0.903850 0.427850i \(-0.140729\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 8.00000i − 0.310227i
\(666\) 0 0
\(667\) −36.0000 −1.39393
\(668\) 0 0
\(669\) − 20.0000i − 0.773245i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 46.0000 1.77317 0.886585 0.462566i \(-0.153071\pi\)
0.886585 + 0.462566i \(0.153071\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) − 12.0000i − 0.459841i
\(682\) 0 0
\(683\) − 42.0000i − 1.60709i −0.595247 0.803543i \(-0.702946\pi\)
0.595247 0.803543i \(-0.297054\pi\)
\(684\) 0 0
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) − 28.0000i − 1.06827i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 8.00000i − 0.304334i −0.988355 0.152167i \(-0.951375\pi\)
0.988355 0.152167i \(-0.0486252\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 4.00000i − 0.151729i
\(696\) 0 0
\(697\) − 36.0000i − 1.36360i
\(698\) 0 0
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) −12.0000 −0.451946
\(706\) 0 0
\(707\) − 12.0000i − 0.451306i
\(708\) 0 0
\(709\) − 34.0000i − 1.27690i −0.769665 0.638448i \(-0.779577\pi\)
0.769665 0.638448i \(-0.220423\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) − 24.0000i − 0.898807i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 48.0000i − 1.79259i
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) − 28.0000i − 1.04277i
\(722\) 0 0
\(723\) − 28.0000i − 1.04133i
\(724\) 0 0
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 46.0000 1.70605 0.853023 0.521874i \(-0.174767\pi\)
0.853023 + 0.521874i \(0.174767\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 60.0000 2.21918
\(732\) 0 0
\(733\) 22.0000i 0.812589i 0.913742 + 0.406294i \(0.133179\pi\)
−0.913742 + 0.406294i \(0.866821\pi\)
\(734\) 0 0
\(735\) − 6.00000i − 0.221313i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 20.0000i 0.735712i 0.929883 + 0.367856i \(0.119908\pi\)
−0.929883 + 0.367856i \(0.880092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 6.00000i − 0.220119i −0.993925 0.110059i \(-0.964896\pi\)
0.993925 0.110059i \(-0.0351041\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) − 6.00000i − 0.219529i
\(748\) 0 0
\(749\) − 12.0000i − 0.438470i
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −20.0000 −0.727875
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 42.0000i 1.52250i 0.648459 + 0.761249i \(0.275414\pi\)
−0.648459 + 0.761249i \(0.724586\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) 0 0
\(765\) 6.00000i 0.216930i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) − 2.00000i − 0.0721218i −0.999350 0.0360609i \(-0.988519\pi\)
0.999350 0.0360609i \(-0.0114810\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 0 0
\(773\) 30.0000i 1.07903i 0.841978 + 0.539513i \(0.181391\pi\)
−0.841978 + 0.539513i \(0.818609\pi\)
\(774\) 0 0
\(775\) − 4.00000i − 0.143684i
\(776\) 0 0
\(777\) 8.00000 0.286998
\(778\) 0 0
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 24.0000 0.857690
\(784\) 0 0
\(785\) − 22.0000i − 0.785214i
\(786\) 0 0
\(787\) 26.0000i 0.926800i 0.886149 + 0.463400i \(0.153371\pi\)
−0.886149 + 0.463400i \(0.846629\pi\)
\(788\) 0 0
\(789\) 36.0000 1.28163
\(790\) 0 0
\(791\) − 12.0000i − 0.426671i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 12.0000i 0.425596i
\(796\) 0 0
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) − 36.0000i − 1.27359i
\(800\) 0 0
\(801\) − 6.00000i − 0.212000i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 12.0000 0.422944
\(806\) 0 0
\(807\) −36.0000 −1.26726
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 16.0000i 0.561836i 0.959732 + 0.280918i \(0.0906389\pi\)
−0.959732 + 0.280918i \(0.909361\pi\)
\(812\) 0 0
\(813\) − 40.0000i − 1.40286i
\(814\) 0 0
\(815\) 10.0000 0.350285
\(816\) 0 0
\(817\) 40.0000i 1.39942i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 54.0000i 1.88461i 0.334751 + 0.942306i \(0.391348\pi\)
−0.334751 + 0.942306i \(0.608652\pi\)
\(822\) 0 0
\(823\) −38.0000 −1.32460 −0.662298 0.749240i \(-0.730419\pi\)
−0.662298 + 0.749240i \(0.730419\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 30.0000i − 1.04320i −0.853189 0.521601i \(-0.825335\pi\)
0.853189 0.521601i \(-0.174665\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 52.0000 1.80386
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) −18.0000 −0.622916
\(836\) 0 0
\(837\) 16.0000i 0.553041i
\(838\) 0 0
\(839\) − 48.0000i − 1.65714i −0.559883 0.828572i \(-0.689154\pi\)
0.559883 0.828572i \(-0.310846\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) − 12.0000i − 0.413302i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 22.0000i 0.755929i
\(848\) 0 0
\(849\) 28.0000 0.960958
\(850\) 0 0
\(851\) − 12.0000i − 0.411355i
\(852\) 0 0
\(853\) 50.0000i 1.71197i 0.517003 + 0.855984i \(0.327048\pi\)
−0.517003 + 0.855984i \(0.672952\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) 0 0
\(863\) − 6.00000i − 0.204242i −0.994772 0.102121i \(-0.967437\pi\)
0.994772 0.102121i \(-0.0325630\pi\)
\(864\) 0 0
\(865\) 6.00000i 0.204006i
\(866\) 0 0
\(867\) −38.0000 −1.29055
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) − 2.00000i − 0.0676897i
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) − 26.0000i − 0.877958i −0.898497 0.438979i \(-0.855340\pi\)
0.898497 0.438979i \(-0.144660\pi\)
\(878\) 0 0
\(879\) 60.0000i 2.02375i
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) −14.0000 −0.471138 −0.235569 0.971858i \(-0.575695\pi\)
−0.235569 + 0.971858i \(0.575695\pi\)
\(884\) 0 0
\(885\) 24.0000 0.806751
\(886\) 0 0
\(887\) 18.0000 0.604381 0.302190 0.953248i \(-0.402282\pi\)
0.302190 + 0.953248i \(0.402282\pi\)
\(888\) 0 0
\(889\) − 4.00000i − 0.134156i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 24.0000 0.803129
\(894\) 0 0
\(895\) 12.0000i 0.401116i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 24.0000i 0.800445i
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) − 40.0000i − 1.33112i
\(904\) 0 0
\(905\) 10.0000i 0.332411i
\(906\) 0 0
\(907\) 46.0000 1.52740 0.763702 0.645568i \(-0.223379\pi\)
0.763702 + 0.645568i \(0.223379\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) − 4.00000i − 0.132236i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) − 4.00000i − 0.131804i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) − 2.00000i − 0.0657596i
\(926\) 0 0
\(927\) −14.0000 −0.459820
\(928\) 0 0
\(929\) 42.0000i 1.37798i 0.724773 + 0.688988i \(0.241945\pi\)
−0.724773 + 0.688988i \(0.758055\pi\)
\(930\) 0 0
\(931\) 12.0000i 0.393284i
\(932\) 0 0
\(933\) 24.0000 0.785725
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) 44.0000 1.43589
\(940\) 0 0
\(941\) 18.0000i 0.586783i 0.955992 + 0.293392i \(0.0947840\pi\)
−0.955992 + 0.293392i \(0.905216\pi\)
\(942\) 0 0
\(943\) 36.0000i 1.17232i
\(944\) 0 0
\(945\) −8.00000 −0.260240
\(946\) 0 0
\(947\) 18.0000i 0.584921i 0.956278 + 0.292461i \(0.0944741\pi\)
−0.956278 + 0.292461i \(0.905526\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) − 12.0000i − 0.389127i
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) − 12.0000i − 0.388311i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 0 0
\(965\) −26.0000 −0.836970
\(966\) 0 0
\(967\) 22.0000i 0.707472i 0.935345 + 0.353736i \(0.115089\pi\)
−0.935345 + 0.353736i \(0.884911\pi\)
\(968\) 0 0
\(969\) − 48.0000i − 1.54198i
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 0 0
\(973\) − 8.00000i − 0.256468i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) − 2.00000i − 0.0638551i
\(982\) 0 0
\(983\) − 18.0000i − 0.574111i −0.957914 0.287055i \(-0.907324\pi\)
0.957914 0.287055i \(-0.0926764\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 0 0
\(987\) −24.0000 −0.763928
\(988\) 0 0
\(989\) −60.0000 −1.90789
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) 0 0
\(993\) 16.0000i 0.507745i
\(994\) 0 0
\(995\) − 8.00000i − 0.253617i
\(996\) 0 0
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) 0 0
\(999\) 8.00000i 0.253109i
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 3380.2.f.b.3041.2 2
13.5 odd 4 3380.2.a.c.1.1 1
13.8 odd 4 20.2.a.a.1.1 1
13.12 even 2 inner 3380.2.f.b.3041.1 2
39.8 even 4 180.2.a.a.1.1 1
52.47 even 4 80.2.a.b.1.1 1
65.8 even 4 100.2.c.a.49.1 2
65.34 odd 4 100.2.a.a.1.1 1
65.47 even 4 100.2.c.a.49.2 2
91.34 even 4 980.2.a.h.1.1 1
91.47 even 12 980.2.i.c.361.1 2
91.60 odd 12 980.2.i.i.961.1 2
91.73 even 12 980.2.i.c.961.1 2
91.86 odd 12 980.2.i.i.361.1 2
104.21 odd 4 320.2.a.f.1.1 1
104.99 even 4 320.2.a.a.1.1 1
117.34 odd 12 1620.2.i.h.1081.1 2
117.47 even 12 1620.2.i.b.1081.1 2
117.86 even 12 1620.2.i.b.541.1 2
117.112 odd 12 1620.2.i.h.541.1 2
143.21 even 4 2420.2.a.a.1.1 1
156.47 odd 4 720.2.a.h.1.1 1
195.8 odd 4 900.2.d.c.649.1 2
195.47 odd 4 900.2.d.c.649.2 2
195.164 even 4 900.2.a.b.1.1 1
208.21 odd 4 1280.2.d.c.641.2 2
208.99 even 4 1280.2.d.g.641.2 2
208.125 odd 4 1280.2.d.c.641.1 2
208.203 even 4 1280.2.d.g.641.1 2
221.21 odd 4 5780.2.c.a.5201.2 2
221.47 odd 4 5780.2.c.a.5201.1 2
221.203 odd 4 5780.2.a.f.1.1 1
247.151 even 4 7220.2.a.f.1.1 1
260.47 odd 4 400.2.c.b.49.1 2
260.99 even 4 400.2.a.c.1.1 1
260.203 odd 4 400.2.c.b.49.2 2
273.125 odd 4 8820.2.a.g.1.1 1
312.125 even 4 2880.2.a.m.1.1 1
312.203 odd 4 2880.2.a.f.1.1 1
364.307 odd 4 3920.2.a.h.1.1 1
455.34 even 4 4900.2.a.e.1.1 1
455.307 odd 4 4900.2.e.f.2549.1 2
455.398 odd 4 4900.2.e.f.2549.2 2
520.99 even 4 1600.2.a.w.1.1 1
520.203 odd 4 1600.2.c.e.449.1 2
520.229 odd 4 1600.2.a.c.1.1 1
520.307 odd 4 1600.2.c.e.449.2 2
520.333 even 4 1600.2.c.d.449.2 2
520.437 even 4 1600.2.c.d.449.1 2
572.307 odd 4 9680.2.a.ba.1.1 1
780.47 even 4 3600.2.f.j.2449.1 2
780.203 even 4 3600.2.f.j.2449.2 2
780.359 odd 4 3600.2.a.be.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.2.a.a.1.1 1 13.8 odd 4
80.2.a.b.1.1 1 52.47 even 4
100.2.a.a.1.1 1 65.34 odd 4
100.2.c.a.49.1 2 65.8 even 4
100.2.c.a.49.2 2 65.47 even 4
180.2.a.a.1.1 1 39.8 even 4
320.2.a.a.1.1 1 104.99 even 4
320.2.a.f.1.1 1 104.21 odd 4
400.2.a.c.1.1 1 260.99 even 4
400.2.c.b.49.1 2 260.47 odd 4
400.2.c.b.49.2 2 260.203 odd 4
720.2.a.h.1.1 1 156.47 odd 4
900.2.a.b.1.1 1 195.164 even 4
900.2.d.c.649.1 2 195.8 odd 4
900.2.d.c.649.2 2 195.47 odd 4
980.2.a.h.1.1 1 91.34 even 4
980.2.i.c.361.1 2 91.47 even 12
980.2.i.c.961.1 2 91.73 even 12
980.2.i.i.361.1 2 91.86 odd 12
980.2.i.i.961.1 2 91.60 odd 12
1280.2.d.c.641.1 2 208.125 odd 4
1280.2.d.c.641.2 2 208.21 odd 4
1280.2.d.g.641.1 2 208.203 even 4
1280.2.d.g.641.2 2 208.99 even 4
1600.2.a.c.1.1 1 520.229 odd 4
1600.2.a.w.1.1 1 520.99 even 4
1600.2.c.d.449.1 2 520.437 even 4
1600.2.c.d.449.2 2 520.333 even 4
1600.2.c.e.449.1 2 520.203 odd 4
1600.2.c.e.449.2 2 520.307 odd 4
1620.2.i.b.541.1 2 117.86 even 12
1620.2.i.b.1081.1 2 117.47 even 12
1620.2.i.h.541.1 2 117.112 odd 12
1620.2.i.h.1081.1 2 117.34 odd 12
2420.2.a.a.1.1 1 143.21 even 4
2880.2.a.f.1.1 1 312.203 odd 4
2880.2.a.m.1.1 1 312.125 even 4
3380.2.a.c.1.1 1 13.5 odd 4
3380.2.f.b.3041.1 2 13.12 even 2 inner
3380.2.f.b.3041.2 2 1.1 even 1 trivial
3600.2.a.be.1.1 1 780.359 odd 4
3600.2.f.j.2449.1 2 780.47 even 4
3600.2.f.j.2449.2 2 780.203 even 4
3920.2.a.h.1.1 1 364.307 odd 4
4900.2.a.e.1.1 1 455.34 even 4
4900.2.e.f.2549.1 2 455.307 odd 4
4900.2.e.f.2549.2 2 455.398 odd 4
5780.2.a.f.1.1 1 221.203 odd 4
5780.2.c.a.5201.1 2 221.47 odd 4
5780.2.c.a.5201.2 2 221.21 odd 4
7220.2.a.f.1.1 1 247.151 even 4
8820.2.a.g.1.1 1 273.125 odd 4
9680.2.a.ba.1.1 1 572.307 odd 4