Properties

Label 425.2.b.b.324.2
Level $425$
Weight $2$
Character 425.324
Analytic conductor $3.394$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 324.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 425.324
Dual form 425.2.b.b.324.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{4} -4.00000i q^{7} +3.00000i q^{8} +3.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{4} -4.00000i q^{7} +3.00000i q^{8} +3.00000 q^{9} -2.00000i q^{13} +4.00000 q^{14} -1.00000 q^{16} -1.00000i q^{17} +3.00000i q^{18} +4.00000 q^{19} +4.00000i q^{23} +2.00000 q^{26} -4.00000i q^{28} -6.00000 q^{29} +4.00000 q^{31} +5.00000i q^{32} +1.00000 q^{34} +3.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} -6.00000 q^{41} +4.00000i q^{43} -4.00000 q^{46} -9.00000 q^{49} -2.00000i q^{52} +6.00000i q^{53} +12.0000 q^{56} -6.00000i q^{58} +12.0000 q^{59} -10.0000 q^{61} +4.00000i q^{62} -12.0000i q^{63} -7.00000 q^{64} -4.00000i q^{67} -1.00000i q^{68} -4.00000 q^{71} +9.00000i q^{72} -6.00000i q^{73} -2.00000 q^{74} +4.00000 q^{76} -12.0000 q^{79} +9.00000 q^{81} -6.00000i q^{82} -4.00000i q^{83} -4.00000 q^{86} -10.0000 q^{89} -8.00000 q^{91} +4.00000i q^{92} -2.00000i q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 6 q^{9} + 8 q^{14} - 2 q^{16} + 8 q^{19} + 4 q^{26} - 12 q^{29} + 8 q^{31} + 2 q^{34} + 6 q^{36} - 12 q^{41} - 8 q^{46} - 18 q^{49} + 24 q^{56} + 24 q^{59} - 20 q^{61} - 14 q^{64} - 8 q^{71} - 4 q^{74} + 8 q^{76} - 24 q^{79} + 18 q^{81} - 8 q^{86} - 20 q^{89} - 16 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) 3.00000i 1.06066i
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) − 1.00000i − 0.242536i
\(18\) 3.00000i 0.707107i
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) − 4.00000i − 0.755929i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 5.00000i 0.883883i
\(33\) 0 0
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) − 2.00000i − 0.277350i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 12.0000 1.60357
\(57\) 0 0
\(58\) − 6.00000i − 0.787839i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 4.00000i 0.508001i
\(63\) − 12.0000i − 1.51186i
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) − 1.00000i − 0.121268i
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 9.00000i 1.06066i
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) − 6.00000i − 0.662589i
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 4.00000i 0.417029i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) − 9.00000i − 0.909137i
\(99\) 0 0
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) − 8.00000i − 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.00000i 0.377964i
\(113\) − 14.0000i − 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) − 6.00000i − 0.554700i
\(118\) 12.0000i 1.10469i
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) − 10.0000i − 0.905357i
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 12.0000 1.06904
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) 3.00000i 0.265165i
\(129\) 0 0
\(130\) 0 0
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 0 0
\(133\) − 16.0000i − 1.38738i
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 4.00000i − 0.335673i
\(143\) 0 0
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) 2.00000i 0.164399i
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 12.0000i 0.973329i
\(153\) − 3.00000i − 0.242536i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) − 12.0000i − 0.954669i
\(159\) 0 0
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 9.00000i 0.707107i
\(163\) 24.0000i 1.87983i 0.341415 + 0.939913i \(0.389094\pi\)
−0.341415 + 0.939913i \(0.610906\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 4.00000i 0.309529i 0.987951 + 0.154765i \(0.0494619\pi\)
−0.987951 + 0.154765i \(0.950538\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 12.0000 0.917663
\(172\) 4.00000i 0.304997i
\(173\) 22.0000i 1.67263i 0.548250 + 0.836315i \(0.315294\pi\)
−0.548250 + 0.836315i \(0.684706\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) − 10.0000i − 0.749532i
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) − 8.00000i − 0.592999i
\(183\) 0 0
\(184\) −12.0000 −0.884652
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 10.0000i − 0.703598i
\(203\) 24.0000i 1.68447i
\(204\) 0 0
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 12.0000i 0.834058i
\(208\) 2.00000i 0.138675i
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) 0 0
\(217\) − 16.0000i − 1.08615i
\(218\) − 6.00000i − 0.406371i
\(219\) 0 0
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) 24.0000i 1.60716i 0.595198 + 0.803579i \(0.297074\pi\)
−0.595198 + 0.803579i \(0.702926\pi\)
\(224\) 20.0000 1.33631
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) 24.0000i 1.59294i 0.604681 + 0.796468i \(0.293301\pi\)
−0.604681 + 0.796468i \(0.706699\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 18.0000i − 1.18176i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) − 4.00000i − 0.259281i
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) − 11.0000i − 0.707107i
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) − 8.00000i − 0.509028i
\(248\) 12.0000i 0.762001i
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) − 12.0000i − 0.755929i
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −18.0000 −1.11417
\(262\) 16.0000i 0.988483i
\(263\) − 16.0000i − 0.986602i −0.869859 0.493301i \(-0.835790\pi\)
0.869859 0.493301i \(-0.164210\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 16.0000 0.981023
\(267\) 0 0
\(268\) − 4.00000i − 0.244339i
\(269\) −22.0000 −1.34136 −0.670682 0.741745i \(-0.733998\pi\)
−0.670682 + 0.741745i \(0.733998\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 1.00000i 0.0606339i
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) − 14.0000i − 0.841178i −0.907251 0.420589i \(-0.861823\pi\)
0.907251 0.420589i \(-0.138177\pi\)
\(278\) 8.00000i 0.479808i
\(279\) 12.0000 0.718421
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) − 16.0000i − 0.951101i −0.879688 0.475551i \(-0.842249\pi\)
0.879688 0.475551i \(-0.157751\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) 0 0
\(287\) 24.0000i 1.41668i
\(288\) 15.0000i 0.883883i
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) − 6.00000i − 0.351123i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) 10.0000i 0.579284i
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) − 16.0000i − 0.920697i
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 3.00000 0.171499
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 28.0000 1.58773 0.793867 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) 0 0
\(313\) − 22.0000i − 1.24351i −0.783210 0.621757i \(-0.786419\pi\)
0.783210 0.621757i \(-0.213581\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) 10.0000i 0.561656i 0.959758 + 0.280828i \(0.0906090\pi\)
−0.959758 + 0.280828i \(0.909391\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 16.0000i 0.891645i
\(323\) − 4.00000i − 0.222566i
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) −24.0000 −1.32924
\(327\) 0 0
\(328\) − 18.0000i − 0.993884i
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) − 4.00000i − 0.219529i
\(333\) 6.00000i 0.328798i
\(334\) −4.00000 −0.218870
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 12.0000i 0.648886i
\(343\) 8.00000i 0.431959i
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −22.0000 −1.18273
\(347\) − 32.0000i − 1.71785i −0.512101 0.858925i \(-0.671133\pi\)
0.512101 0.858925i \(-0.328867\pi\)
\(348\) 0 0
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 30.0000i − 1.59674i −0.602168 0.798369i \(-0.705696\pi\)
0.602168 0.798369i \(-0.294304\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) − 12.0000i − 0.634220i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 2.00000i − 0.105118i
\(363\) 0 0
\(364\) −8.00000 −0.419314
\(365\) 0 0
\(366\) 0 0
\(367\) − 28.0000i − 1.46159i −0.682598 0.730794i \(-0.739150\pi\)
0.682598 0.730794i \(-0.260850\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) 6.00000i 0.310668i 0.987862 + 0.155334i \(0.0496454\pi\)
−0.987862 + 0.155334i \(0.950355\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 16.0000i − 0.818631i
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 12.0000i 0.609994i
\(388\) − 2.00000i − 0.101535i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) − 27.0000i − 1.36371i
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) − 6.00000i − 0.301131i −0.988600 0.150566i \(-0.951890\pi\)
0.988600 0.150566i \(-0.0481095\pi\)
\(398\) 20.0000i 1.00251i
\(399\) 0 0
\(400\) 0 0
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 0 0
\(403\) − 8.00000i − 0.398508i
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) −24.0000 −1.19110
\(407\) 0 0
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8.00000i 0.394132i
\(413\) − 48.0000i − 2.36193i
\(414\) −12.0000 −0.589768
\(415\) 0 0
\(416\) 10.0000 0.490290
\(417\) 0 0
\(418\) 0 0
\(419\) −8.00000 −0.390826 −0.195413 0.980721i \(-0.562605\pi\)
−0.195413 + 0.980721i \(0.562605\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 8.00000i 0.389434i
\(423\) 0 0
\(424\) −18.0000 −0.874157
\(425\) 0 0
\(426\) 0 0
\(427\) 40.0000i 1.93574i
\(428\) − 8.00000i − 0.386695i
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) 2.00000i 0.0961139i 0.998845 + 0.0480569i \(0.0153029\pi\)
−0.998845 + 0.0480569i \(0.984697\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) 16.0000i 0.765384i
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) −27.0000 −1.28571
\(442\) − 2.00000i − 0.0951303i
\(443\) 28.0000i 1.33032i 0.746701 + 0.665160i \(0.231637\pi\)
−0.746701 + 0.665160i \(0.768363\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −24.0000 −1.13643
\(447\) 0 0
\(448\) 28.0000i 1.32288i
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 14.0000i − 0.658505i
\(453\) 0 0
\(454\) −24.0000 −1.12638
\(455\) 0 0
\(456\) 0 0
\(457\) 6.00000i 0.280668i 0.990104 + 0.140334i \(0.0448177\pi\)
−0.990104 + 0.140334i \(0.955182\pi\)
\(458\) − 6.00000i − 0.280362i
\(459\) 0 0
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 32.0000i 1.48717i 0.668644 + 0.743583i \(0.266875\pi\)
−0.668644 + 0.743583i \(0.733125\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) − 12.0000i − 0.555294i −0.960683 0.277647i \(-0.910445\pi\)
0.960683 0.277647i \(-0.0895545\pi\)
\(468\) − 6.00000i − 0.277350i
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 0 0
\(472\) 36.0000i 1.65703i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −4.00000 −0.183340
\(477\) 18.0000i 0.824163i
\(478\) 16.0000i 0.731823i
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 18.0000i 0.819878i
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) 0 0
\(487\) − 20.0000i − 0.906287i −0.891438 0.453143i \(-0.850303\pi\)
0.891438 0.453143i \(-0.149697\pi\)
\(488\) − 30.0000i − 1.35804i
\(489\) 0 0
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) 6.00000i 0.270226i
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 16.0000i 0.717698i
\(498\) 0 0
\(499\) 40.0000 1.79065 0.895323 0.445418i \(-0.146945\pi\)
0.895323 + 0.445418i \(0.146945\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 12.0000i 0.535586i
\(503\) − 12.0000i − 0.535054i −0.963550 0.267527i \(-0.913794\pi\)
0.963550 0.267527i \(-0.0862064\pi\)
\(504\) 36.0000 1.60357
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) − 8.00000i − 0.354943i
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) −24.0000 −1.06170
\(512\) − 11.0000i − 0.486136i
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 8.00000i 0.351500i
\(519\) 0 0
\(520\) 0 0
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) − 18.0000i − 0.787839i
\(523\) − 36.0000i − 1.57417i −0.616844 0.787085i \(-0.711589\pi\)
0.616844 0.787085i \(-0.288411\pi\)
\(524\) 16.0000 0.698963
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) − 4.00000i − 0.174243i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 36.0000 1.56227
\(532\) − 16.0000i − 0.693688i
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) − 22.0000i − 0.948487i
\(539\) 0 0
\(540\) 0 0
\(541\) 6.00000 0.257960 0.128980 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) − 16.0000i − 0.687259i
\(543\) 0 0
\(544\) 5.00000 0.214373
\(545\) 0 0
\(546\) 0 0
\(547\) 32.0000i 1.36822i 0.729378 + 0.684111i \(0.239809\pi\)
−0.729378 + 0.684111i \(0.760191\pi\)
\(548\) 6.00000i 0.256307i
\(549\) −30.0000 −1.28037
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 48.0000i 2.04117i
\(554\) 14.0000 0.594803
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) − 30.0000i − 1.27114i −0.772043 0.635570i \(-0.780765\pi\)
0.772043 0.635570i \(-0.219235\pi\)
\(558\) 12.0000i 0.508001i
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) − 6.00000i − 0.253095i
\(563\) − 4.00000i − 0.168580i −0.996441 0.0842900i \(-0.973138\pi\)
0.996441 0.0842900i \(-0.0268622\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) − 36.0000i − 1.51186i
\(568\) − 12.0000i − 0.503509i
\(569\) 38.0000 1.59304 0.796521 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) −21.0000 −0.875000
\(577\) 14.0000i 0.582828i 0.956597 + 0.291414i \(0.0941257\pi\)
−0.956597 + 0.291414i \(0.905874\pi\)
\(578\) − 1.00000i − 0.0415945i
\(579\) 0 0
\(580\) 0 0
\(581\) −16.0000 −0.663792
\(582\) 0 0
\(583\) 0 0
\(584\) 18.0000 0.744845
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) − 4.00000i − 0.165098i −0.996587 0.0825488i \(-0.973694\pi\)
0.996587 0.0825488i \(-0.0263060\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 0 0
\(592\) − 2.00000i − 0.0821995i
\(593\) 18.0000i 0.739171i 0.929197 + 0.369586i \(0.120500\pi\)
−0.929197 + 0.369586i \(0.879500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) 8.00000i 0.327144i
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 16.0000i 0.652111i
\(603\) − 12.0000i − 0.488678i
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) 0 0
\(607\) − 20.0000i − 0.811775i −0.913923 0.405887i \(-0.866962\pi\)
0.913923 0.405887i \(-0.133038\pi\)
\(608\) 20.0000i 0.811107i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) − 3.00000i − 0.121268i
\(613\) − 26.0000i − 1.05013i −0.851062 0.525065i \(-0.824041\pi\)
0.851062 0.525065i \(-0.175959\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 0 0
\(619\) 48.0000 1.92928 0.964641 0.263566i \(-0.0848986\pi\)
0.964641 + 0.263566i \(0.0848986\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 28.0000i 1.12270i
\(623\) 40.0000i 1.60257i
\(624\) 0 0
\(625\) 0 0
\(626\) 22.0000 0.879297
\(627\) 0 0
\(628\) 2.00000i 0.0798087i
\(629\) 2.00000 0.0797452
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) − 36.0000i − 1.43200i
\(633\) 0 0
\(634\) −10.0000 −0.397151
\(635\) 0 0
\(636\) 0 0
\(637\) 18.0000i 0.713186i
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) 32.0000i 1.26196i 0.775800 + 0.630978i \(0.217346\pi\)
−0.775800 + 0.630978i \(0.782654\pi\)
\(644\) 16.0000 0.630488
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) − 8.00000i − 0.314512i −0.987558 0.157256i \(-0.949735\pi\)
0.987558 0.157256i \(-0.0502649\pi\)
\(648\) 27.0000i 1.06066i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 24.0000i 0.939913i
\(653\) 6.00000i 0.234798i 0.993085 + 0.117399i \(0.0374557\pi\)
−0.993085 + 0.117399i \(0.962544\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) − 18.0000i − 0.702247i
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 4.00000i 0.155464i
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) − 24.0000i − 0.929284i
\(668\) 4.00000i 0.154765i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 2.00000i 0.0770943i 0.999257 + 0.0385472i \(0.0122730\pi\)
−0.999257 + 0.0385472i \(0.987727\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) − 30.0000i − 1.15299i −0.817099 0.576497i \(-0.804419\pi\)
0.817099 0.576497i \(-0.195581\pi\)
\(678\) 0 0
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 40.0000i − 1.53056i −0.643699 0.765279i \(-0.722601\pi\)
0.643699 0.765279i \(-0.277399\pi\)
\(684\) 12.0000 0.458831
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) 0 0
\(688\) − 4.00000i − 0.152499i
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 22.0000i 0.836315i
\(693\) 0 0
\(694\) 32.0000 1.21470
\(695\) 0 0
\(696\) 0 0
\(697\) 6.00000i 0.227266i
\(698\) 18.0000i 0.681310i
\(699\) 0 0
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 8.00000i 0.301726i
\(704\) 0 0
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 40.0000i 1.50435i
\(708\) 0 0
\(709\) 34.0000 1.27690 0.638448 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) 0 0
\(711\) −36.0000 −1.35011
\(712\) − 30.0000i − 1.12430i
\(713\) 16.0000i 0.599205i
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) −4.00000 −0.149175 −0.0745874 0.997214i \(-0.523764\pi\)
−0.0745874 + 0.997214i \(0.523764\pi\)
\(720\) 0 0
\(721\) 32.0000 1.19174
\(722\) − 3.00000i − 0.111648i
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 0 0
\(727\) − 40.0000i − 1.48352i −0.670667 0.741759i \(-0.733992\pi\)
0.670667 0.741759i \(-0.266008\pi\)
\(728\) − 24.0000i − 0.889499i
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 4.00000 0.147945
\(732\) 0 0
\(733\) − 50.0000i − 1.84679i −0.383849 0.923396i \(-0.625402\pi\)
0.383849 0.923396i \(-0.374598\pi\)
\(734\) 28.0000 1.03350
\(735\) 0 0
\(736\) −20.0000 −0.737210
\(737\) 0 0
\(738\) − 18.0000i − 0.662589i
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 24.0000i 0.881068i
\(743\) 12.0000i 0.440237i 0.975473 + 0.220119i \(0.0706445\pi\)
−0.975473 + 0.220119i \(0.929356\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) − 12.0000i − 0.439057i
\(748\) 0 0
\(749\) −32.0000 −1.16925
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) − 22.0000i − 0.799604i −0.916602 0.399802i \(-0.869079\pi\)
0.916602 0.399802i \(-0.130921\pi\)
\(758\) 8.00000i 0.290573i
\(759\) 0 0
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 0 0
\(763\) 24.0000i 0.868858i
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) − 24.0000i − 0.866590i
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.00000i 0.0719816i
\(773\) − 26.0000i − 0.935155i −0.883952 0.467578i \(-0.845127\pi\)
0.883952 0.467578i \(-0.154873\pi\)
\(774\) −12.0000 −0.431331
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) − 6.00000i − 0.215110i
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 4.00000i 0.143040i
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 0 0
\(787\) 32.0000i 1.14068i 0.821410 + 0.570338i \(0.193188\pi\)
−0.821410 + 0.570338i \(0.806812\pi\)
\(788\) 18.0000i 0.641223i
\(789\) 0 0
\(790\) 0 0
\(791\) −56.0000 −1.99113
\(792\) 0 0
\(793\) 20.0000i 0.710221i
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 50.0000i 1.77109i 0.464553 + 0.885545i \(0.346215\pi\)
−0.464553 + 0.885545i \(0.653785\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) − 14.0000i − 0.494357i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 0 0
\(808\) − 30.0000i − 1.05540i
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) 40.0000 1.40459 0.702295 0.711886i \(-0.252159\pi\)
0.702295 + 0.711886i \(0.252159\pi\)
\(812\) 24.0000i 0.842235i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 16.0000i 0.559769i
\(818\) − 26.0000i − 0.909069i
\(819\) −24.0000 −0.838628
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) 20.0000i 0.697156i 0.937280 + 0.348578i \(0.113335\pi\)
−0.937280 + 0.348578i \(0.886665\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) 48.0000 1.67013
\(827\) 48.0000i 1.66912i 0.550914 + 0.834562i \(0.314279\pi\)
−0.550914 + 0.834562i \(0.685721\pi\)
\(828\) 12.0000i 0.417029i
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 14.0000i 0.485363i
\(833\) 9.00000i 0.311832i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) − 8.00000i − 0.276355i
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 22.0000i 0.758170i
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) 0 0
\(847\) 44.0000i 1.51186i
\(848\) − 6.00000i − 0.206041i
\(849\) 0 0
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) 14.0000i 0.479351i 0.970853 + 0.239675i \(0.0770410\pi\)
−0.970853 + 0.239675i \(0.922959\pi\)
\(854\) −40.0000 −1.36877
\(855\) 0 0
\(856\) 24.0000 0.820303
\(857\) − 10.0000i − 0.341593i −0.985306 0.170797i \(-0.945366\pi\)
0.985306 0.170797i \(-0.0546341\pi\)
\(858\) 0 0
\(859\) −52.0000 −1.77422 −0.887109 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 12.0000i 0.408722i
\(863\) 16.0000i 0.544646i 0.962206 + 0.272323i \(0.0877920\pi\)
−0.962206 + 0.272323i \(0.912208\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) 0 0
\(868\) − 16.0000i − 0.543075i
\(869\) 0 0
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) − 18.0000i − 0.609557i
\(873\) − 6.00000i − 0.203069i
\(874\) −16.0000 −0.541208
\(875\) 0 0
\(876\) 0 0
\(877\) − 6.00000i − 0.202606i −0.994856 0.101303i \(-0.967699\pi\)
0.994856 0.101303i \(-0.0323011\pi\)
\(878\) 20.0000i 0.674967i
\(879\) 0 0
\(880\) 0 0
\(881\) −46.0000 −1.54978 −0.774890 0.632096i \(-0.782195\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(882\) − 27.0000i − 0.909137i
\(883\) − 12.0000i − 0.403832i −0.979403 0.201916i \(-0.935283\pi\)
0.979403 0.201916i \(-0.0647168\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) −28.0000 −0.940678
\(887\) − 12.0000i − 0.402921i −0.979497 0.201460i \(-0.935431\pi\)
0.979497 0.201460i \(-0.0645687\pi\)
\(888\) 0 0
\(889\) −32.0000 −1.07325
\(890\) 0 0
\(891\) 0 0
\(892\) 24.0000i 0.803579i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 12.0000 0.400892
\(897\) 0 0
\(898\) − 34.0000i − 1.13459i
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 6.00000 0.199889
\(902\) 0 0
\(903\) 0 0
\(904\) 42.0000 1.39690
\(905\) 0 0
\(906\) 0 0
\(907\) − 32.0000i − 1.06254i −0.847202 0.531271i \(-0.821714\pi\)
0.847202 0.531271i \(-0.178286\pi\)
\(908\) 24.0000i 0.796468i
\(909\) −30.0000 −0.995037
\(910\) 0 0
\(911\) −4.00000 −0.132526 −0.0662630 0.997802i \(-0.521108\pi\)
−0.0662630 + 0.997802i \(0.521108\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −6.00000 −0.198462
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) − 64.0000i − 2.11347i
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 2.00000i − 0.0658665i
\(923\) 8.00000i 0.263323i
\(924\) 0 0
\(925\) 0 0
\(926\) −32.0000 −1.05159
\(927\) 24.0000i 0.788263i
\(928\) − 30.0000i − 0.984798i
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) − 6.00000i − 0.196537i
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 18.0000 0.588348
\(937\) − 10.0000i − 0.326686i −0.986569 0.163343i \(-0.947772\pi\)
0.986569 0.163343i \(-0.0522277\pi\)
\(938\) − 16.0000i − 0.522419i
\(939\) 0 0
\(940\) 0 0
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 0 0
\(943\) − 24.0000i − 0.781548i
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) − 32.0000i − 1.03986i −0.854209 0.519930i \(-0.825958\pi\)
0.854209 0.519930i \(-0.174042\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) − 12.0000i − 0.388922i
\(953\) − 22.0000i − 0.712650i −0.934362 0.356325i \(-0.884030\pi\)
0.934362 0.356325i \(-0.115970\pi\)
\(954\) −18.0000 −0.582772
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 0 0
\(958\) − 36.0000i − 1.16311i
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 4.00000i 0.128965i
\(963\) − 24.0000i − 0.773389i
\(964\) 18.0000 0.579741
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) − 33.0000i − 1.06066i
\(969\) 0 0
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) − 32.0000i − 1.02587i
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) − 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
\(981\) −18.0000 −0.574696
\(982\) 20.0000i 0.638226i
\(983\) 12.0000i 0.382741i 0.981518 + 0.191370i \(0.0612931\pi\)
−0.981518 + 0.191370i \(0.938707\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −6.00000 −0.191079
\(987\) 0 0
\(988\) − 8.00000i − 0.254514i
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) −12.0000 −0.381193 −0.190596 0.981669i \(-0.561042\pi\)
−0.190596 + 0.981669i \(0.561042\pi\)
\(992\) 20.0000i 0.635001i
\(993\) 0 0
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) 0 0
\(997\) − 46.0000i − 1.45683i −0.685134 0.728417i \(-0.740256\pi\)
0.685134 0.728417i \(-0.259744\pi\)
\(998\) 40.0000i 1.26618i
\(999\) 0 0
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 425.2.b.b.324.2 2
5.2 odd 4 17.2.a.a.1.1 1
5.3 odd 4 425.2.a.d.1.1 1
5.4 even 2 inner 425.2.b.b.324.1 2
15.2 even 4 153.2.a.c.1.1 1
15.8 even 4 3825.2.a.d.1.1 1
20.3 even 4 6800.2.a.n.1.1 1
20.7 even 4 272.2.a.b.1.1 1
35.2 odd 12 833.2.e.b.18.1 2
35.12 even 12 833.2.e.a.18.1 2
35.17 even 12 833.2.e.a.324.1 2
35.27 even 4 833.2.a.a.1.1 1
35.32 odd 12 833.2.e.b.324.1 2
40.27 even 4 1088.2.a.h.1.1 1
40.37 odd 4 1088.2.a.i.1.1 1
55.32 even 4 2057.2.a.e.1.1 1
60.47 odd 4 2448.2.a.o.1.1 1
65.12 odd 4 2873.2.a.c.1.1 1
85.2 odd 8 289.2.c.a.38.2 4
85.7 even 16 289.2.d.d.134.2 8
85.12 even 16 289.2.d.d.110.1 8
85.22 even 16 289.2.d.d.110.2 8
85.27 even 16 289.2.d.d.134.1 8
85.32 odd 8 289.2.c.a.38.1 4
85.33 odd 4 7225.2.a.g.1.1 1
85.37 even 16 289.2.d.d.179.2 8
85.42 odd 8 289.2.c.a.251.1 4
85.47 odd 4 289.2.b.a.288.1 2
85.57 even 16 289.2.d.d.155.2 8
85.62 even 16 289.2.d.d.155.1 8
85.67 odd 4 289.2.a.a.1.1 1
85.72 odd 4 289.2.b.a.288.2 2
85.77 odd 8 289.2.c.a.251.2 4
85.82 even 16 289.2.d.d.179.1 8
95.37 even 4 6137.2.a.b.1.1 1
105.62 odd 4 7497.2.a.l.1.1 1
115.22 even 4 8993.2.a.a.1.1 1
120.77 even 4 9792.2.a.n.1.1 1
120.107 odd 4 9792.2.a.i.1.1 1
255.152 even 4 2601.2.a.g.1.1 1
340.67 even 4 4624.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.2.a.a.1.1 1 5.2 odd 4
153.2.a.c.1.1 1 15.2 even 4
272.2.a.b.1.1 1 20.7 even 4
289.2.a.a.1.1 1 85.67 odd 4
289.2.b.a.288.1 2 85.47 odd 4
289.2.b.a.288.2 2 85.72 odd 4
289.2.c.a.38.1 4 85.32 odd 8
289.2.c.a.38.2 4 85.2 odd 8
289.2.c.a.251.1 4 85.42 odd 8
289.2.c.a.251.2 4 85.77 odd 8
289.2.d.d.110.1 8 85.12 even 16
289.2.d.d.110.2 8 85.22 even 16
289.2.d.d.134.1 8 85.27 even 16
289.2.d.d.134.2 8 85.7 even 16
289.2.d.d.155.1 8 85.62 even 16
289.2.d.d.155.2 8 85.57 even 16
289.2.d.d.179.1 8 85.82 even 16
289.2.d.d.179.2 8 85.37 even 16
425.2.a.d.1.1 1 5.3 odd 4
425.2.b.b.324.1 2 5.4 even 2 inner
425.2.b.b.324.2 2 1.1 even 1 trivial
833.2.a.a.1.1 1 35.27 even 4
833.2.e.a.18.1 2 35.12 even 12
833.2.e.a.324.1 2 35.17 even 12
833.2.e.b.18.1 2 35.2 odd 12
833.2.e.b.324.1 2 35.32 odd 12
1088.2.a.h.1.1 1 40.27 even 4
1088.2.a.i.1.1 1 40.37 odd 4
2057.2.a.e.1.1 1 55.32 even 4
2448.2.a.o.1.1 1 60.47 odd 4
2601.2.a.g.1.1 1 255.152 even 4
2873.2.a.c.1.1 1 65.12 odd 4
3825.2.a.d.1.1 1 15.8 even 4
4624.2.a.d.1.1 1 340.67 even 4
6137.2.a.b.1.1 1 95.37 even 4
6800.2.a.n.1.1 1 20.3 even 4
7225.2.a.g.1.1 1 85.33 odd 4
7497.2.a.l.1.1 1 105.62 odd 4
8993.2.a.a.1.1 1 115.22 even 4
9792.2.a.i.1.1 1 120.107 odd 4
9792.2.a.n.1.1 1 120.77 even 4