Properties

Label 5070.2.b.q.1351.4
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.4
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.q.1351.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.82843i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.82843i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -5.65685i q^{11} -1.00000 q^{12} -2.82843 q^{14} +1.00000i q^{15} +1.00000 q^{16} -0.828427 q^{17} +1.00000i q^{18} +2.82843i q^{19} -1.00000i q^{20} +2.82843i q^{21} +5.65685 q^{22} +8.48528 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -2.82843i q^{28} -8.82843 q^{29} -1.00000 q^{30} +4.00000i q^{31} +1.00000i q^{32} -5.65685i q^{33} -0.828427i q^{34} -2.82843 q^{35} -1.00000 q^{36} +11.6569i q^{37} -2.82843 q^{38} +1.00000 q^{40} -7.65685i q^{41} -2.82843 q^{42} -9.65685 q^{43} +5.65685i q^{44} +1.00000i q^{45} +8.48528i q^{46} +8.00000i q^{47} +1.00000 q^{48} -1.00000 q^{49} -1.00000i q^{50} -0.828427 q^{51} +13.3137 q^{53} +1.00000i q^{54} +5.65685 q^{55} +2.82843 q^{56} +2.82843i q^{57} -8.82843i q^{58} +2.34315i q^{59} -1.00000i q^{60} +6.00000 q^{61} -4.00000 q^{62} +2.82843i q^{63} -1.00000 q^{64} +5.65685 q^{66} +5.65685i q^{67} +0.828427 q^{68} +8.48528 q^{69} -2.82843i q^{70} -5.65685i q^{71} -1.00000i q^{72} +14.4853i q^{73} -11.6569 q^{74} -1.00000 q^{75} -2.82843i q^{76} +16.0000 q^{77} +2.34315 q^{79} +1.00000i q^{80} +1.00000 q^{81} +7.65685 q^{82} +6.34315i q^{83} -2.82843i q^{84} -0.828427i q^{85} -9.65685i q^{86} -8.82843 q^{87} -5.65685 q^{88} +15.6569i q^{89} -1.00000 q^{90} -8.48528 q^{92} +4.00000i q^{93} -8.00000 q^{94} -2.82843 q^{95} +1.00000i q^{96} -3.17157i q^{97} -1.00000i q^{98} -5.65685i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 4q^{4} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 4q^{4} + 4q^{9} - 4q^{10} - 4q^{12} + 4q^{16} + 8q^{17} - 4q^{25} + 4q^{27} - 24q^{29} - 4q^{30} - 4q^{36} + 4q^{40} - 16q^{43} + 4q^{48} - 4q^{49} + 8q^{51} + 8q^{53} + 24q^{61} - 16q^{62} - 4q^{64} - 8q^{68} - 24q^{74} - 4q^{75} + 64q^{77} + 32q^{79} + 4q^{81} + 8q^{82} - 24q^{87} - 4q^{90} - 32q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 1.00000i 0.447214i
\(6\) 1.00000i 0.408248i
\(7\) 2.82843i 1.06904i 0.845154 + 0.534522i \(0.179509\pi\)
−0.845154 + 0.534522i \(0.820491\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) − 5.65685i − 1.70561i −0.522233 0.852803i \(-0.674901\pi\)
0.522233 0.852803i \(-0.325099\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −2.82843 −0.755929
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) −0.828427 −0.200923 −0.100462 0.994941i \(-0.532032\pi\)
−0.100462 + 0.994941i \(0.532032\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 2.82843i 0.648886i 0.945905 + 0.324443i \(0.105177\pi\)
−0.945905 + 0.324443i \(0.894823\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) 2.82843i 0.617213i
\(22\) 5.65685 1.20605
\(23\) 8.48528 1.76930 0.884652 0.466252i \(-0.154396\pi\)
0.884652 + 0.466252i \(0.154396\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) − 2.82843i − 0.534522i
\(29\) −8.82843 −1.63940 −0.819699 0.572795i \(-0.805859\pi\)
−0.819699 + 0.572795i \(0.805859\pi\)
\(30\) −1.00000 −0.182574
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 5.65685i − 0.984732i
\(34\) − 0.828427i − 0.142074i
\(35\) −2.82843 −0.478091
\(36\) −1.00000 −0.166667
\(37\) 11.6569i 1.91638i 0.286141 + 0.958188i \(0.407627\pi\)
−0.286141 + 0.958188i \(0.592373\pi\)
\(38\) −2.82843 −0.458831
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) − 7.65685i − 1.19580i −0.801571 0.597900i \(-0.796002\pi\)
0.801571 0.597900i \(-0.203998\pi\)
\(42\) −2.82843 −0.436436
\(43\) −9.65685 −1.47266 −0.736328 0.676625i \(-0.763442\pi\)
−0.736328 + 0.676625i \(0.763442\pi\)
\(44\) 5.65685i 0.852803i
\(45\) 1.00000i 0.149071i
\(46\) 8.48528i 1.25109i
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 1.00000 0.144338
\(49\) −1.00000 −0.142857
\(50\) − 1.00000i − 0.141421i
\(51\) −0.828427 −0.116003
\(52\) 0 0
\(53\) 13.3137 1.82878 0.914389 0.404836i \(-0.132671\pi\)
0.914389 + 0.404836i \(0.132671\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 5.65685 0.762770
\(56\) 2.82843 0.377964
\(57\) 2.82843i 0.374634i
\(58\) − 8.82843i − 1.15923i
\(59\) 2.34315i 0.305052i 0.988299 + 0.152526i \(0.0487407\pi\)
−0.988299 + 0.152526i \(0.951259\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −4.00000 −0.508001
\(63\) 2.82843i 0.356348i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 5.65685 0.696311
\(67\) 5.65685i 0.691095i 0.938401 + 0.345547i \(0.112307\pi\)
−0.938401 + 0.345547i \(0.887693\pi\)
\(68\) 0.828427 0.100462
\(69\) 8.48528 1.02151
\(70\) − 2.82843i − 0.338062i
\(71\) − 5.65685i − 0.671345i −0.941979 0.335673i \(-0.891036\pi\)
0.941979 0.335673i \(-0.108964\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 14.4853i 1.69537i 0.530497 + 0.847687i \(0.322005\pi\)
−0.530497 + 0.847687i \(0.677995\pi\)
\(74\) −11.6569 −1.35508
\(75\) −1.00000 −0.115470
\(76\) − 2.82843i − 0.324443i
\(77\) 16.0000 1.82337
\(78\) 0 0
\(79\) 2.34315 0.263624 0.131812 0.991275i \(-0.457920\pi\)
0.131812 + 0.991275i \(0.457920\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) 7.65685 0.845558
\(83\) 6.34315i 0.696251i 0.937448 + 0.348125i \(0.113182\pi\)
−0.937448 + 0.348125i \(0.886818\pi\)
\(84\) − 2.82843i − 0.308607i
\(85\) − 0.828427i − 0.0898555i
\(86\) − 9.65685i − 1.04133i
\(87\) −8.82843 −0.946507
\(88\) −5.65685 −0.603023
\(89\) 15.6569i 1.65962i 0.558044 + 0.829812i \(0.311552\pi\)
−0.558044 + 0.829812i \(0.688448\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −8.48528 −0.884652
\(93\) 4.00000i 0.414781i
\(94\) −8.00000 −0.825137
\(95\) −2.82843 −0.290191
\(96\) 1.00000i 0.102062i
\(97\) − 3.17157i − 0.322024i −0.986952 0.161012i \(-0.948524\pi\)
0.986952 0.161012i \(-0.0514759\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) − 5.65685i − 0.568535i
\(100\) 1.00000 0.100000
\(101\) −16.1421 −1.60620 −0.803101 0.595843i \(-0.796818\pi\)
−0.803101 + 0.595843i \(0.796818\pi\)
\(102\) − 0.828427i − 0.0820265i
\(103\) 1.65685 0.163255 0.0816274 0.996663i \(-0.473988\pi\)
0.0816274 + 0.996663i \(0.473988\pi\)
\(104\) 0 0
\(105\) −2.82843 −0.276026
\(106\) 13.3137i 1.29314i
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 8.82843i 0.845610i 0.906221 + 0.422805i \(0.138954\pi\)
−0.906221 + 0.422805i \(0.861046\pi\)
\(110\) 5.65685i 0.539360i
\(111\) 11.6569i 1.10642i
\(112\) 2.82843i 0.267261i
\(113\) 6.48528 0.610084 0.305042 0.952339i \(-0.401330\pi\)
0.305042 + 0.952339i \(0.401330\pi\)
\(114\) −2.82843 −0.264906
\(115\) 8.48528i 0.791257i
\(116\) 8.82843 0.819699
\(117\) 0 0
\(118\) −2.34315 −0.215704
\(119\) − 2.34315i − 0.214796i
\(120\) 1.00000 0.0912871
\(121\) −21.0000 −1.90909
\(122\) 6.00000i 0.543214i
\(123\) − 7.65685i − 0.690395i
\(124\) − 4.00000i − 0.359211i
\(125\) − 1.00000i − 0.0894427i
\(126\) −2.82843 −0.251976
\(127\) −9.65685 −0.856907 −0.428454 0.903564i \(-0.640941\pi\)
−0.428454 + 0.903564i \(0.640941\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −9.65685 −0.850239
\(130\) 0 0
\(131\) −6.14214 −0.536641 −0.268320 0.963330i \(-0.586469\pi\)
−0.268320 + 0.963330i \(0.586469\pi\)
\(132\) 5.65685i 0.492366i
\(133\) −8.00000 −0.693688
\(134\) −5.65685 −0.488678
\(135\) 1.00000i 0.0860663i
\(136\) 0.828427i 0.0710370i
\(137\) 17.3137i 1.47921i 0.673041 + 0.739605i \(0.264988\pi\)
−0.673041 + 0.739605i \(0.735012\pi\)
\(138\) 8.48528i 0.722315i
\(139\) 6.34315 0.538019 0.269009 0.963138i \(-0.413304\pi\)
0.269009 + 0.963138i \(0.413304\pi\)
\(140\) 2.82843 0.239046
\(141\) 8.00000i 0.673722i
\(142\) 5.65685 0.474713
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 8.82843i − 0.733161i
\(146\) −14.4853 −1.19881
\(147\) −1.00000 −0.0824786
\(148\) − 11.6569i − 0.958188i
\(149\) 3.65685i 0.299581i 0.988718 + 0.149791i \(0.0478599\pi\)
−0.988718 + 0.149791i \(0.952140\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) − 12.0000i − 0.976546i −0.872691 0.488273i \(-0.837627\pi\)
0.872691 0.488273i \(-0.162373\pi\)
\(152\) 2.82843 0.229416
\(153\) −0.828427 −0.0669744
\(154\) 16.0000i 1.28932i
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 5.31371 0.424080 0.212040 0.977261i \(-0.431989\pi\)
0.212040 + 0.977261i \(0.431989\pi\)
\(158\) 2.34315i 0.186411i
\(159\) 13.3137 1.05585
\(160\) −1.00000 −0.0790569
\(161\) 24.0000i 1.89146i
\(162\) 1.00000i 0.0785674i
\(163\) − 11.3137i − 0.886158i −0.896483 0.443079i \(-0.853886\pi\)
0.896483 0.443079i \(-0.146114\pi\)
\(164\) 7.65685i 0.597900i
\(165\) 5.65685 0.440386
\(166\) −6.34315 −0.492324
\(167\) − 8.97056i − 0.694163i −0.937835 0.347081i \(-0.887173\pi\)
0.937835 0.347081i \(-0.112827\pi\)
\(168\) 2.82843 0.218218
\(169\) 0 0
\(170\) 0.828427 0.0635375
\(171\) 2.82843i 0.216295i
\(172\) 9.65685 0.736328
\(173\) 9.31371 0.708108 0.354054 0.935225i \(-0.384803\pi\)
0.354054 + 0.935225i \(0.384803\pi\)
\(174\) − 8.82843i − 0.669281i
\(175\) − 2.82843i − 0.213809i
\(176\) − 5.65685i − 0.426401i
\(177\) 2.34315i 0.176122i
\(178\) −15.6569 −1.17353
\(179\) −7.51472 −0.561676 −0.280838 0.959755i \(-0.590612\pi\)
−0.280838 + 0.959755i \(0.590612\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) 7.65685 0.569129 0.284565 0.958657i \(-0.408151\pi\)
0.284565 + 0.958657i \(0.408151\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) − 8.48528i − 0.625543i
\(185\) −11.6569 −0.857029
\(186\) −4.00000 −0.293294
\(187\) 4.68629i 0.342696i
\(188\) − 8.00000i − 0.583460i
\(189\) 2.82843i 0.205738i
\(190\) − 2.82843i − 0.205196i
\(191\) 11.3137 0.818631 0.409316 0.912393i \(-0.365768\pi\)
0.409316 + 0.912393i \(0.365768\pi\)
\(192\) −1.00000 −0.0721688
\(193\) − 2.48528i − 0.178894i −0.995992 0.0894472i \(-0.971490\pi\)
0.995992 0.0894472i \(-0.0285100\pi\)
\(194\) 3.17157 0.227706
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 13.3137i − 0.948562i −0.880373 0.474281i \(-0.842708\pi\)
0.880373 0.474281i \(-0.157292\pi\)
\(198\) 5.65685 0.402015
\(199\) −10.3431 −0.733206 −0.366603 0.930377i \(-0.619479\pi\)
−0.366603 + 0.930377i \(0.619479\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 5.65685i 0.399004i
\(202\) − 16.1421i − 1.13576i
\(203\) − 24.9706i − 1.75259i
\(204\) 0.828427 0.0580015
\(205\) 7.65685 0.534778
\(206\) 1.65685i 0.115439i
\(207\) 8.48528 0.589768
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) − 2.82843i − 0.195180i
\(211\) 0.686292 0.0472463 0.0236231 0.999721i \(-0.492480\pi\)
0.0236231 + 0.999721i \(0.492480\pi\)
\(212\) −13.3137 −0.914389
\(213\) − 5.65685i − 0.387601i
\(214\) − 4.00000i − 0.273434i
\(215\) − 9.65685i − 0.658592i
\(216\) − 1.00000i − 0.0680414i
\(217\) −11.3137 −0.768025
\(218\) −8.82843 −0.597937
\(219\) 14.4853i 0.978825i
\(220\) −5.65685 −0.381385
\(221\) 0 0
\(222\) −11.6569 −0.782357
\(223\) 10.8284i 0.725125i 0.931959 + 0.362563i \(0.118098\pi\)
−0.931959 + 0.362563i \(0.881902\pi\)
\(224\) −2.82843 −0.188982
\(225\) −1.00000 −0.0666667
\(226\) 6.48528i 0.431394i
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) − 2.82843i − 0.187317i
\(229\) − 4.14214i − 0.273720i −0.990590 0.136860i \(-0.956299\pi\)
0.990590 0.136860i \(-0.0437011\pi\)
\(230\) −8.48528 −0.559503
\(231\) 16.0000 1.05272
\(232\) 8.82843i 0.579615i
\(233\) −5.51472 −0.361281 −0.180641 0.983549i \(-0.557817\pi\)
−0.180641 + 0.983549i \(0.557817\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) − 2.34315i − 0.152526i
\(237\) 2.34315 0.152204
\(238\) 2.34315 0.151884
\(239\) 16.0000i 1.03495i 0.855697 + 0.517477i \(0.173129\pi\)
−0.855697 + 0.517477i \(0.826871\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) − 5.31371i − 0.342286i −0.985246 0.171143i \(-0.945254\pi\)
0.985246 0.171143i \(-0.0547460\pi\)
\(242\) − 21.0000i − 1.34993i
\(243\) 1.00000 0.0641500
\(244\) −6.00000 −0.384111
\(245\) − 1.00000i − 0.0638877i
\(246\) 7.65685 0.488183
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) 6.34315i 0.401981i
\(250\) 1.00000 0.0632456
\(251\) −10.8284 −0.683484 −0.341742 0.939794i \(-0.611017\pi\)
−0.341742 + 0.939794i \(0.611017\pi\)
\(252\) − 2.82843i − 0.178174i
\(253\) − 48.0000i − 3.01773i
\(254\) − 9.65685i − 0.605925i
\(255\) − 0.828427i − 0.0518781i
\(256\) 1.00000 0.0625000
\(257\) 4.82843 0.301189 0.150595 0.988596i \(-0.451881\pi\)
0.150595 + 0.988596i \(0.451881\pi\)
\(258\) − 9.65685i − 0.601209i
\(259\) −32.9706 −2.04869
\(260\) 0 0
\(261\) −8.82843 −0.546466
\(262\) − 6.14214i − 0.379462i
\(263\) 16.4853 1.01653 0.508263 0.861202i \(-0.330288\pi\)
0.508263 + 0.861202i \(0.330288\pi\)
\(264\) −5.65685 −0.348155
\(265\) 13.3137i 0.817855i
\(266\) − 8.00000i − 0.490511i
\(267\) 15.6569i 0.958184i
\(268\) − 5.65685i − 0.345547i
\(269\) −14.4853 −0.883183 −0.441592 0.897216i \(-0.645586\pi\)
−0.441592 + 0.897216i \(0.645586\pi\)
\(270\) −1.00000 −0.0608581
\(271\) − 7.31371i − 0.444276i −0.975015 0.222138i \(-0.928696\pi\)
0.975015 0.222138i \(-0.0713036\pi\)
\(272\) −0.828427 −0.0502308
\(273\) 0 0
\(274\) −17.3137 −1.04596
\(275\) 5.65685i 0.341121i
\(276\) −8.48528 −0.510754
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 6.34315i 0.380437i
\(279\) 4.00000i 0.239474i
\(280\) 2.82843i 0.169031i
\(281\) − 8.34315i − 0.497710i −0.968541 0.248855i \(-0.919946\pi\)
0.968541 0.248855i \(-0.0800543\pi\)
\(282\) −8.00000 −0.476393
\(283\) 17.6569 1.04959 0.524796 0.851228i \(-0.324142\pi\)
0.524796 + 0.851228i \(0.324142\pi\)
\(284\) 5.65685i 0.335673i
\(285\) −2.82843 −0.167542
\(286\) 0 0
\(287\) 21.6569 1.27836
\(288\) 1.00000i 0.0589256i
\(289\) −16.3137 −0.959630
\(290\) 8.82843 0.518423
\(291\) − 3.17157i − 0.185921i
\(292\) − 14.4853i − 0.847687i
\(293\) 16.6274i 0.971384i 0.874130 + 0.485692i \(0.161432\pi\)
−0.874130 + 0.485692i \(0.838568\pi\)
\(294\) − 1.00000i − 0.0583212i
\(295\) −2.34315 −0.136423
\(296\) 11.6569 0.677541
\(297\) − 5.65685i − 0.328244i
\(298\) −3.65685 −0.211836
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) − 27.3137i − 1.57434i
\(302\) 12.0000 0.690522
\(303\) −16.1421 −0.927341
\(304\) 2.82843i 0.162221i
\(305\) 6.00000i 0.343559i
\(306\) − 0.828427i − 0.0473580i
\(307\) 21.6569i 1.23602i 0.786169 + 0.618011i \(0.212061\pi\)
−0.786169 + 0.618011i \(0.787939\pi\)
\(308\) −16.0000 −0.911685
\(309\) 1.65685 0.0942551
\(310\) − 4.00000i − 0.227185i
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −30.9706 −1.75056 −0.875280 0.483617i \(-0.839323\pi\)
−0.875280 + 0.483617i \(0.839323\pi\)
\(314\) 5.31371i 0.299870i
\(315\) −2.82843 −0.159364
\(316\) −2.34315 −0.131812
\(317\) 25.3137i 1.42176i 0.703314 + 0.710880i \(0.251703\pi\)
−0.703314 + 0.710880i \(0.748297\pi\)
\(318\) 13.3137i 0.746596i
\(319\) 49.9411i 2.79617i
\(320\) − 1.00000i − 0.0559017i
\(321\) −4.00000 −0.223258
\(322\) −24.0000 −1.33747
\(323\) − 2.34315i − 0.130376i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 11.3137 0.626608
\(327\) 8.82843i 0.488213i
\(328\) −7.65685 −0.422779
\(329\) −22.6274 −1.24749
\(330\) 5.65685i 0.311400i
\(331\) − 8.48528i − 0.466393i −0.972430 0.233197i \(-0.925081\pi\)
0.972430 0.233197i \(-0.0749186\pi\)
\(332\) − 6.34315i − 0.348125i
\(333\) 11.6569i 0.638792i
\(334\) 8.97056 0.490847
\(335\) −5.65685 −0.309067
\(336\) 2.82843i 0.154303i
\(337\) −10.9706 −0.597605 −0.298802 0.954315i \(-0.596587\pi\)
−0.298802 + 0.954315i \(0.596587\pi\)
\(338\) 0 0
\(339\) 6.48528 0.352232
\(340\) 0.828427i 0.0449278i
\(341\) 22.6274 1.22534
\(342\) −2.82843 −0.152944
\(343\) 16.9706i 0.916324i
\(344\) 9.65685i 0.520663i
\(345\) 8.48528i 0.456832i
\(346\) 9.31371i 0.500708i
\(347\) −9.65685 −0.518407 −0.259204 0.965823i \(-0.583460\pi\)
−0.259204 + 0.965823i \(0.583460\pi\)
\(348\) 8.82843 0.473253
\(349\) − 12.1421i − 0.649954i −0.945722 0.324977i \(-0.894644\pi\)
0.945722 0.324977i \(-0.105356\pi\)
\(350\) 2.82843 0.151186
\(351\) 0 0
\(352\) 5.65685 0.301511
\(353\) 5.31371i 0.282820i 0.989951 + 0.141410i \(0.0451636\pi\)
−0.989951 + 0.141410i \(0.954836\pi\)
\(354\) −2.34315 −0.124537
\(355\) 5.65685 0.300235
\(356\) − 15.6569i − 0.829812i
\(357\) − 2.34315i − 0.124012i
\(358\) − 7.51472i − 0.397165i
\(359\) 28.2843i 1.49279i 0.665505 + 0.746393i \(0.268216\pi\)
−0.665505 + 0.746393i \(0.731784\pi\)
\(360\) 1.00000 0.0527046
\(361\) 11.0000 0.578947
\(362\) 7.65685i 0.402435i
\(363\) −21.0000 −1.10221
\(364\) 0 0
\(365\) −14.4853 −0.758194
\(366\) 6.00000i 0.313625i
\(367\) 25.6569 1.33928 0.669638 0.742687i \(-0.266449\pi\)
0.669638 + 0.742687i \(0.266449\pi\)
\(368\) 8.48528 0.442326
\(369\) − 7.65685i − 0.398600i
\(370\) − 11.6569i − 0.606011i
\(371\) 37.6569i 1.95505i
\(372\) − 4.00000i − 0.207390i
\(373\) −2.68629 −0.139091 −0.0695455 0.997579i \(-0.522155\pi\)
−0.0695455 + 0.997579i \(0.522155\pi\)
\(374\) −4.68629 −0.242322
\(375\) − 1.00000i − 0.0516398i
\(376\) 8.00000 0.412568
\(377\) 0 0
\(378\) −2.82843 −0.145479
\(379\) − 7.51472i − 0.386005i −0.981198 0.193003i \(-0.938177\pi\)
0.981198 0.193003i \(-0.0618226\pi\)
\(380\) 2.82843 0.145095
\(381\) −9.65685 −0.494736
\(382\) 11.3137i 0.578860i
\(383\) − 29.6569i − 1.51539i −0.652606 0.757697i \(-0.726324\pi\)
0.652606 0.757697i \(-0.273676\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) 16.0000i 0.815436i
\(386\) 2.48528 0.126497
\(387\) −9.65685 −0.490885
\(388\) 3.17157i 0.161012i
\(389\) 6.48528 0.328817 0.164408 0.986392i \(-0.447429\pi\)
0.164408 + 0.986392i \(0.447429\pi\)
\(390\) 0 0
\(391\) −7.02944 −0.355494
\(392\) 1.00000i 0.0505076i
\(393\) −6.14214 −0.309830
\(394\) 13.3137 0.670735
\(395\) 2.34315i 0.117896i
\(396\) 5.65685i 0.284268i
\(397\) − 30.2843i − 1.51992i −0.649968 0.759962i \(-0.725218\pi\)
0.649968 0.759962i \(-0.274782\pi\)
\(398\) − 10.3431i − 0.518455i
\(399\) −8.00000 −0.400501
\(400\) −1.00000 −0.0500000
\(401\) 26.9706i 1.34685i 0.739258 + 0.673423i \(0.235177\pi\)
−0.739258 + 0.673423i \(0.764823\pi\)
\(402\) −5.65685 −0.282138
\(403\) 0 0
\(404\) 16.1421 0.803101
\(405\) 1.00000i 0.0496904i
\(406\) 24.9706 1.23927
\(407\) 65.9411 3.26858
\(408\) 0.828427i 0.0410133i
\(409\) − 3.65685i − 0.180820i −0.995905 0.0904099i \(-0.971182\pi\)
0.995905 0.0904099i \(-0.0288177\pi\)
\(410\) 7.65685i 0.378145i
\(411\) 17.3137i 0.854022i
\(412\) −1.65685 −0.0816274
\(413\) −6.62742 −0.326114
\(414\) 8.48528i 0.417029i
\(415\) −6.34315 −0.311373
\(416\) 0 0
\(417\) 6.34315 0.310625
\(418\) 16.0000i 0.782586i
\(419\) −10.8284 −0.529003 −0.264502 0.964385i \(-0.585207\pi\)
−0.264502 + 0.964385i \(0.585207\pi\)
\(420\) 2.82843 0.138013
\(421\) − 24.1421i − 1.17662i −0.808637 0.588308i \(-0.799794\pi\)
0.808637 0.588308i \(-0.200206\pi\)
\(422\) 0.686292i 0.0334081i
\(423\) 8.00000i 0.388973i
\(424\) − 13.3137i − 0.646571i
\(425\) 0.828427 0.0401846
\(426\) 5.65685 0.274075
\(427\) 16.9706i 0.821263i
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) 9.65685 0.465695
\(431\) − 16.0000i − 0.770693i −0.922772 0.385346i \(-0.874082\pi\)
0.922772 0.385346i \(-0.125918\pi\)
\(432\) 1.00000 0.0481125
\(433\) 22.9706 1.10389 0.551947 0.833879i \(-0.313885\pi\)
0.551947 + 0.833879i \(0.313885\pi\)
\(434\) − 11.3137i − 0.543075i
\(435\) − 8.82843i − 0.423291i
\(436\) − 8.82843i − 0.422805i
\(437\) 24.0000i 1.14808i
\(438\) −14.4853 −0.692134
\(439\) −22.6274 −1.07995 −0.539974 0.841682i \(-0.681566\pi\)
−0.539974 + 0.841682i \(0.681566\pi\)
\(440\) − 5.65685i − 0.269680i
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) 30.3431 1.44165 0.720823 0.693119i \(-0.243764\pi\)
0.720823 + 0.693119i \(0.243764\pi\)
\(444\) − 11.6569i − 0.553210i
\(445\) −15.6569 −0.742206
\(446\) −10.8284 −0.512741
\(447\) 3.65685i 0.172963i
\(448\) − 2.82843i − 0.133631i
\(449\) − 26.2843i − 1.24043i −0.784431 0.620216i \(-0.787045\pi\)
0.784431 0.620216i \(-0.212955\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) −43.3137 −2.03956
\(452\) −6.48528 −0.305042
\(453\) − 12.0000i − 0.563809i
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 2.82843 0.132453
\(457\) 20.8284i 0.974313i 0.873315 + 0.487156i \(0.161966\pi\)
−0.873315 + 0.487156i \(0.838034\pi\)
\(458\) 4.14214 0.193549
\(459\) −0.828427 −0.0386677
\(460\) − 8.48528i − 0.395628i
\(461\) 14.0000i 0.652045i 0.945362 + 0.326023i \(0.105709\pi\)
−0.945362 + 0.326023i \(0.894291\pi\)
\(462\) 16.0000i 0.744387i
\(463\) 3.79899i 0.176554i 0.996096 + 0.0882770i \(0.0281361\pi\)
−0.996096 + 0.0882770i \(0.971864\pi\)
\(464\) −8.82843 −0.409849
\(465\) −4.00000 −0.185496
\(466\) − 5.51472i − 0.255464i
\(467\) −7.31371 −0.338438 −0.169219 0.985578i \(-0.554125\pi\)
−0.169219 + 0.985578i \(0.554125\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) − 8.00000i − 0.369012i
\(471\) 5.31371 0.244843
\(472\) 2.34315 0.107852
\(473\) 54.6274i 2.51177i
\(474\) 2.34315i 0.107624i
\(475\) − 2.82843i − 0.129777i
\(476\) 2.34315i 0.107398i
\(477\) 13.3137 0.609593
\(478\) −16.0000 −0.731823
\(479\) 11.3137i 0.516937i 0.966020 + 0.258468i \(0.0832177\pi\)
−0.966020 + 0.258468i \(0.916782\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 5.31371 0.242033
\(483\) 24.0000i 1.09204i
\(484\) 21.0000 0.954545
\(485\) 3.17157 0.144014
\(486\) 1.00000i 0.0453609i
\(487\) 16.4853i 0.747019i 0.927626 + 0.373510i \(0.121846\pi\)
−0.927626 + 0.373510i \(0.878154\pi\)
\(488\) − 6.00000i − 0.271607i
\(489\) − 11.3137i − 0.511624i
\(490\) 1.00000 0.0451754
\(491\) 38.1421 1.72133 0.860665 0.509171i \(-0.170048\pi\)
0.860665 + 0.509171i \(0.170048\pi\)
\(492\) 7.65685i 0.345198i
\(493\) 7.31371 0.329393
\(494\) 0 0
\(495\) 5.65685 0.254257
\(496\) 4.00000i 0.179605i
\(497\) 16.0000 0.717698
\(498\) −6.34315 −0.284243
\(499\) 0.485281i 0.0217242i 0.999941 + 0.0108621i \(0.00345758\pi\)
−0.999941 + 0.0108621i \(0.996542\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) − 8.97056i − 0.400775i
\(502\) − 10.8284i − 0.483296i
\(503\) −23.5147 −1.04847 −0.524235 0.851574i \(-0.675649\pi\)
−0.524235 + 0.851574i \(0.675649\pi\)
\(504\) 2.82843 0.125988
\(505\) − 16.1421i − 0.718316i
\(506\) 48.0000 2.13386
\(507\) 0 0
\(508\) 9.65685 0.428454
\(509\) − 37.3137i − 1.65390i −0.562275 0.826951i \(-0.690074\pi\)
0.562275 0.826951i \(-0.309926\pi\)
\(510\) 0.828427 0.0366834
\(511\) −40.9706 −1.81243
\(512\) 1.00000i 0.0441942i
\(513\) 2.82843i 0.124878i
\(514\) 4.82843i 0.212973i
\(515\) 1.65685i 0.0730097i
\(516\) 9.65685 0.425119
\(517\) 45.2548 1.99031
\(518\) − 32.9706i − 1.44864i
\(519\) 9.31371 0.408826
\(520\) 0 0
\(521\) 26.9706 1.18160 0.590801 0.806817i \(-0.298812\pi\)
0.590801 + 0.806817i \(0.298812\pi\)
\(522\) − 8.82843i − 0.386410i
\(523\) −10.6274 −0.464704 −0.232352 0.972632i \(-0.574642\pi\)
−0.232352 + 0.972632i \(0.574642\pi\)
\(524\) 6.14214 0.268320
\(525\) − 2.82843i − 0.123443i
\(526\) 16.4853i 0.718792i
\(527\) − 3.31371i − 0.144347i
\(528\) − 5.65685i − 0.246183i
\(529\) 49.0000 2.13043
\(530\) −13.3137 −0.578311
\(531\) 2.34315i 0.101684i
\(532\) 8.00000 0.346844
\(533\) 0 0
\(534\) −15.6569 −0.677538
\(535\) − 4.00000i − 0.172935i
\(536\) 5.65685 0.244339
\(537\) −7.51472 −0.324284
\(538\) − 14.4853i − 0.624505i
\(539\) 5.65685i 0.243658i
\(540\) − 1.00000i − 0.0430331i
\(541\) − 14.4853i − 0.622771i −0.950284 0.311385i \(-0.899207\pi\)
0.950284 0.311385i \(-0.100793\pi\)
\(542\) 7.31371 0.314151
\(543\) 7.65685 0.328587
\(544\) − 0.828427i − 0.0355185i
\(545\) −8.82843 −0.378168
\(546\) 0 0
\(547\) 0.686292 0.0293437 0.0146719 0.999892i \(-0.495330\pi\)
0.0146719 + 0.999892i \(0.495330\pi\)
\(548\) − 17.3137i − 0.739605i
\(549\) 6.00000 0.256074
\(550\) −5.65685 −0.241209
\(551\) − 24.9706i − 1.06378i
\(552\) − 8.48528i − 0.361158i
\(553\) 6.62742i 0.281826i
\(554\) − 26.0000i − 1.10463i
\(555\) −11.6569 −0.494806
\(556\) −6.34315 −0.269009
\(557\) − 10.6863i − 0.452793i −0.974035 0.226396i \(-0.927306\pi\)
0.974035 0.226396i \(-0.0726945\pi\)
\(558\) −4.00000 −0.169334
\(559\) 0 0
\(560\) −2.82843 −0.119523
\(561\) 4.68629i 0.197855i
\(562\) 8.34315 0.351934
\(563\) 30.3431 1.27881 0.639406 0.768870i \(-0.279181\pi\)
0.639406 + 0.768870i \(0.279181\pi\)
\(564\) − 8.00000i − 0.336861i
\(565\) 6.48528i 0.272838i
\(566\) 17.6569i 0.742173i
\(567\) 2.82843i 0.118783i
\(568\) −5.65685 −0.237356
\(569\) −31.6569 −1.32712 −0.663562 0.748121i \(-0.730956\pi\)
−0.663562 + 0.748121i \(0.730956\pi\)
\(570\) − 2.82843i − 0.118470i
\(571\) −20.9706 −0.877591 −0.438795 0.898587i \(-0.644595\pi\)
−0.438795 + 0.898587i \(0.644595\pi\)
\(572\) 0 0
\(573\) 11.3137 0.472637
\(574\) 21.6569i 0.903940i
\(575\) −8.48528 −0.353861
\(576\) −1.00000 −0.0416667
\(577\) − 23.4558i − 0.976480i −0.872710 0.488240i \(-0.837639\pi\)
0.872710 0.488240i \(-0.162361\pi\)
\(578\) − 16.3137i − 0.678561i
\(579\) − 2.48528i − 0.103285i
\(580\) 8.82843i 0.366580i
\(581\) −17.9411 −0.744323
\(582\) 3.17157 0.131466
\(583\) − 75.3137i − 3.11918i
\(584\) 14.4853 0.599405
\(585\) 0 0
\(586\) −16.6274 −0.686872
\(587\) 2.62742i 0.108445i 0.998529 + 0.0542226i \(0.0172680\pi\)
−0.998529 + 0.0542226i \(0.982732\pi\)
\(588\) 1.00000 0.0412393
\(589\) −11.3137 −0.466173
\(590\) − 2.34315i − 0.0964658i
\(591\) − 13.3137i − 0.547653i
\(592\) 11.6569i 0.479094i
\(593\) 0.343146i 0.0140913i 0.999975 + 0.00704565i \(0.00224272\pi\)
−0.999975 + 0.00704565i \(0.997757\pi\)
\(594\) 5.65685 0.232104
\(595\) 2.34315 0.0960596
\(596\) − 3.65685i − 0.149791i
\(597\) −10.3431 −0.423317
\(598\) 0 0
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) 29.3137 1.19573 0.597866 0.801596i \(-0.296016\pi\)
0.597866 + 0.801596i \(0.296016\pi\)
\(602\) 27.3137 1.11322
\(603\) 5.65685i 0.230365i
\(604\) 12.0000i 0.488273i
\(605\) − 21.0000i − 0.853771i
\(606\) − 16.1421i − 0.655729i
\(607\) 28.9706 1.17588 0.587939 0.808905i \(-0.299939\pi\)
0.587939 + 0.808905i \(0.299939\pi\)
\(608\) −2.82843 −0.114708
\(609\) − 24.9706i − 1.01186i
\(610\) −6.00000 −0.242933
\(611\) 0 0
\(612\) 0.828427 0.0334872
\(613\) 22.2843i 0.900053i 0.893015 + 0.450027i \(0.148586\pi\)
−0.893015 + 0.450027i \(0.851414\pi\)
\(614\) −21.6569 −0.874000
\(615\) 7.65685 0.308754
\(616\) − 16.0000i − 0.644658i
\(617\) 2.00000i 0.0805170i 0.999189 + 0.0402585i \(0.0128181\pi\)
−0.999189 + 0.0402585i \(0.987182\pi\)
\(618\) 1.65685i 0.0666485i
\(619\) 34.8284i 1.39987i 0.714205 + 0.699936i \(0.246788\pi\)
−0.714205 + 0.699936i \(0.753212\pi\)
\(620\) 4.00000 0.160644
\(621\) 8.48528 0.340503
\(622\) 24.0000i 0.962312i
\(623\) −44.2843 −1.77421
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 30.9706i − 1.23783i
\(627\) 16.0000 0.638978
\(628\) −5.31371 −0.212040
\(629\) − 9.65685i − 0.385044i
\(630\) − 2.82843i − 0.112687i
\(631\) 33.6569i 1.33986i 0.742425 + 0.669929i \(0.233676\pi\)
−0.742425 + 0.669929i \(0.766324\pi\)
\(632\) − 2.34315i − 0.0932053i
\(633\) 0.686292 0.0272776
\(634\) −25.3137 −1.00534
\(635\) − 9.65685i − 0.383221i
\(636\) −13.3137 −0.527923
\(637\) 0 0
\(638\) −49.9411 −1.97719
\(639\) − 5.65685i − 0.223782i
\(640\) 1.00000 0.0395285
\(641\) 4.62742 0.182772 0.0913860 0.995816i \(-0.470870\pi\)
0.0913860 + 0.995816i \(0.470870\pi\)
\(642\) − 4.00000i − 0.157867i
\(643\) 39.5980i 1.56159i 0.624786 + 0.780796i \(0.285186\pi\)
−0.624786 + 0.780796i \(0.714814\pi\)
\(644\) − 24.0000i − 0.945732i
\(645\) − 9.65685i − 0.380238i
\(646\) 2.34315 0.0921898
\(647\) −8.48528 −0.333591 −0.166795 0.985992i \(-0.553342\pi\)
−0.166795 + 0.985992i \(0.553342\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 13.2548 0.520298
\(650\) 0 0
\(651\) −11.3137 −0.443419
\(652\) 11.3137i 0.443079i
\(653\) −42.2843 −1.65471 −0.827356 0.561678i \(-0.810156\pi\)
−0.827356 + 0.561678i \(0.810156\pi\)
\(654\) −8.82843 −0.345219
\(655\) − 6.14214i − 0.239993i
\(656\) − 7.65685i − 0.298950i
\(657\) 14.4853i 0.565125i
\(658\) − 22.6274i − 0.882109i
\(659\) −7.51472 −0.292732 −0.146366 0.989231i \(-0.546758\pi\)
−0.146366 + 0.989231i \(0.546758\pi\)
\(660\) −5.65685 −0.220193
\(661\) 8.14214i 0.316692i 0.987384 + 0.158346i \(0.0506162\pi\)
−0.987384 + 0.158346i \(0.949384\pi\)
\(662\) 8.48528 0.329790
\(663\) 0 0
\(664\) 6.34315 0.246162
\(665\) − 8.00000i − 0.310227i
\(666\) −11.6569 −0.451694
\(667\) −74.9117 −2.90059
\(668\) 8.97056i 0.347081i
\(669\) 10.8284i 0.418651i
\(670\) − 5.65685i − 0.218543i
\(671\) − 33.9411i − 1.31028i
\(672\) −2.82843 −0.109109
\(673\) −32.6274 −1.25769 −0.628847 0.777529i \(-0.716473\pi\)
−0.628847 + 0.777529i \(0.716473\pi\)
\(674\) − 10.9706i − 0.422570i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 12.3431 0.474386 0.237193 0.971463i \(-0.423773\pi\)
0.237193 + 0.971463i \(0.423773\pi\)
\(678\) 6.48528i 0.249066i
\(679\) 8.97056 0.344259
\(680\) −0.828427 −0.0317687
\(681\) 4.00000i 0.153280i
\(682\) 22.6274i 0.866449i
\(683\) − 33.6569i − 1.28784i −0.765091 0.643922i \(-0.777306\pi\)
0.765091 0.643922i \(-0.222694\pi\)
\(684\) − 2.82843i − 0.108148i
\(685\) −17.3137 −0.661523
\(686\) −16.9706 −0.647939
\(687\) − 4.14214i − 0.158032i
\(688\) −9.65685 −0.368164
\(689\) 0 0
\(690\) −8.48528 −0.323029
\(691\) − 27.7990i − 1.05752i −0.848770 0.528762i \(-0.822657\pi\)
0.848770 0.528762i \(-0.177343\pi\)
\(692\) −9.31371 −0.354054
\(693\) 16.0000 0.607790
\(694\) − 9.65685i − 0.366569i
\(695\) 6.34315i 0.240609i
\(696\) 8.82843i 0.334641i
\(697\) 6.34315i 0.240264i
\(698\) 12.1421 0.459587
\(699\) −5.51472 −0.208586
\(700\) 2.82843i 0.106904i
\(701\) −0.142136 −0.00536839 −0.00268419 0.999996i \(-0.500854\pi\)
−0.00268419 + 0.999996i \(0.500854\pi\)
\(702\) 0 0
\(703\) −32.9706 −1.24351
\(704\) 5.65685i 0.213201i
\(705\) −8.00000 −0.301297
\(706\) −5.31371 −0.199984
\(707\) − 45.6569i − 1.71710i
\(708\) − 2.34315i − 0.0880608i
\(709\) 7.17157i 0.269334i 0.990891 + 0.134667i \(0.0429965\pi\)
−0.990891 + 0.134667i \(0.957004\pi\)
\(710\) 5.65685i 0.212298i
\(711\) 2.34315 0.0878748
\(712\) 15.6569 0.586765
\(713\) 33.9411i 1.27111i
\(714\) 2.34315 0.0876900
\(715\) 0 0
\(716\) 7.51472 0.280838
\(717\) 16.0000i 0.597531i
\(718\) −28.2843 −1.05556
\(719\) 29.6569 1.10601 0.553007 0.833177i \(-0.313480\pi\)
0.553007 + 0.833177i \(0.313480\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) 4.68629i 0.174527i
\(722\) 11.0000i 0.409378i
\(723\) − 5.31371i − 0.197619i
\(724\) −7.65685 −0.284565
\(725\) 8.82843 0.327880
\(726\) − 21.0000i − 0.779383i
\(727\) 45.9411 1.70386 0.851931 0.523654i \(-0.175432\pi\)
0.851931 + 0.523654i \(0.175432\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 14.4853i − 0.536124i
\(731\) 8.00000 0.295891
\(732\) −6.00000 −0.221766
\(733\) − 0.343146i − 0.0126744i −0.999980 0.00633719i \(-0.997983\pi\)
0.999980 0.00633719i \(-0.00201720\pi\)
\(734\) 25.6569i 0.947012i
\(735\) − 1.00000i − 0.0368856i
\(736\) 8.48528i 0.312772i
\(737\) 32.0000 1.17874
\(738\) 7.65685 0.281853
\(739\) 14.1421i 0.520227i 0.965578 + 0.260113i \(0.0837600\pi\)
−0.965578 + 0.260113i \(0.916240\pi\)
\(740\) 11.6569 0.428514
\(741\) 0 0
\(742\) −37.6569 −1.38243
\(743\) − 36.2843i − 1.33114i −0.746335 0.665570i \(-0.768188\pi\)
0.746335 0.665570i \(-0.231812\pi\)
\(744\) 4.00000 0.146647
\(745\) −3.65685 −0.133977
\(746\) − 2.68629i − 0.0983521i
\(747\) 6.34315i 0.232084i
\(748\) − 4.68629i − 0.171348i
\(749\) − 11.3137i − 0.413394i
\(750\) 1.00000 0.0365148
\(751\) −11.3137 −0.412843 −0.206422 0.978463i \(-0.566182\pi\)
−0.206422 + 0.978463i \(0.566182\pi\)
\(752\) 8.00000i 0.291730i
\(753\) −10.8284 −0.394610
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) − 2.82843i − 0.102869i
\(757\) 19.9411 0.724773 0.362386 0.932028i \(-0.381962\pi\)
0.362386 + 0.932028i \(0.381962\pi\)
\(758\) 7.51472 0.272947
\(759\) − 48.0000i − 1.74229i
\(760\) 2.82843i 0.102598i
\(761\) − 27.6569i − 1.00256i −0.865285 0.501280i \(-0.832863\pi\)
0.865285 0.501280i \(-0.167137\pi\)
\(762\) − 9.65685i − 0.349831i
\(763\) −24.9706 −0.903995
\(764\) −11.3137 −0.409316
\(765\) − 0.828427i − 0.0299518i
\(766\) 29.6569 1.07155
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) − 14.0000i − 0.504853i −0.967616 0.252426i \(-0.918771\pi\)
0.967616 0.252426i \(-0.0812286\pi\)
\(770\) −16.0000 −0.576600
\(771\) 4.82843 0.173892
\(772\) 2.48528i 0.0894472i
\(773\) − 53.3137i − 1.91756i −0.284148 0.958780i \(-0.591711\pi\)
0.284148 0.958780i \(-0.408289\pi\)
\(774\) − 9.65685i − 0.347108i
\(775\) − 4.00000i − 0.143684i
\(776\) −3.17157 −0.113853
\(777\) −32.9706 −1.18281
\(778\) 6.48528i 0.232509i
\(779\) 21.6569 0.775937
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) − 7.02944i − 0.251372i
\(783\) −8.82843 −0.315502
\(784\) −1.00000 −0.0357143
\(785\) 5.31371i 0.189654i
\(786\) − 6.14214i − 0.219083i
\(787\) 24.0000i 0.855508i 0.903895 + 0.427754i \(0.140695\pi\)
−0.903895 + 0.427754i \(0.859305\pi\)
\(788\) 13.3137i 0.474281i
\(789\) 16.4853 0.586892
\(790\) −2.34315 −0.0833654
\(791\) 18.3431i 0.652207i
\(792\) −5.65685 −0.201008
\(793\) 0 0
\(794\) 30.2843 1.07475
\(795\) 13.3137i 0.472189i
\(796\) 10.3431 0.366603
\(797\) −16.6274 −0.588973 −0.294487 0.955656i \(-0.595149\pi\)
−0.294487 + 0.955656i \(0.595149\pi\)
\(798\) − 8.00000i − 0.283197i
\(799\) − 6.62742i − 0.234461i
\(800\) − 1.00000i − 0.0353553i
\(801\) 15.6569i 0.553208i
\(802\) −26.9706 −0.952364
\(803\) 81.9411 2.89164
\(804\) − 5.65685i − 0.199502i
\(805\) −24.0000 −0.845889
\(806\) 0 0
\(807\) −14.4853 −0.509906
\(808\) 16.1421i 0.567878i
\(809\) 13.3137 0.468085 0.234043 0.972226i \(-0.424804\pi\)
0.234043 + 0.972226i \(0.424804\pi\)
\(810\) −1.00000 −0.0351364
\(811\) − 1.85786i − 0.0652384i −0.999468 0.0326192i \(-0.989615\pi\)
0.999468 0.0326192i \(-0.0103849\pi\)
\(812\) 24.9706i 0.876295i
\(813\) − 7.31371i − 0.256503i
\(814\) 65.9411i 2.31124i
\(815\) 11.3137 0.396302
\(816\) −0.828427 −0.0290008
\(817\) − 27.3137i − 0.955586i
\(818\) 3.65685 0.127859
\(819\) 0 0
\(820\) −7.65685 −0.267389
\(821\) 34.2843i 1.19653i 0.801299 + 0.598265i \(0.204143\pi\)
−0.801299 + 0.598265i \(0.795857\pi\)
\(822\) −17.3137 −0.603885
\(823\) 52.9706 1.84644 0.923219 0.384275i \(-0.125548\pi\)
0.923219 + 0.384275i \(0.125548\pi\)
\(824\) − 1.65685i − 0.0577193i
\(825\) 5.65685i 0.196946i
\(826\) − 6.62742i − 0.230597i
\(827\) − 9.65685i − 0.335802i −0.985804 0.167901i \(-0.946301\pi\)
0.985804 0.167901i \(-0.0536989\pi\)
\(828\) −8.48528 −0.294884
\(829\) 53.3137 1.85166 0.925831 0.377938i \(-0.123367\pi\)
0.925831 + 0.377938i \(0.123367\pi\)
\(830\) − 6.34315i − 0.220174i
\(831\) −26.0000 −0.901930
\(832\) 0 0
\(833\) 0.828427 0.0287033
\(834\) 6.34315i 0.219645i
\(835\) 8.97056 0.310439
\(836\) −16.0000 −0.553372
\(837\) 4.00000i 0.138260i
\(838\) − 10.8284i − 0.374062i
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 2.82843i 0.0975900i
\(841\) 48.9411 1.68763
\(842\) 24.1421 0.831993
\(843\) − 8.34315i − 0.287353i
\(844\) −0.686292 −0.0236231
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) − 59.3970i − 2.04090i
\(848\) 13.3137 0.457195
\(849\) 17.6569 0.605982
\(850\) 0.828427i 0.0284148i
\(851\) 98.9117i 3.39065i
\(852\) 5.65685i 0.193801i
\(853\) − 38.2843i − 1.31083i −0.755270 0.655414i \(-0.772494\pi\)
0.755270 0.655414i \(-0.227506\pi\)
\(854\) −16.9706 −0.580721
\(855\) −2.82843 −0.0967302
\(856\) 4.00000i 0.136717i
\(857\) 20.8284 0.711486 0.355743 0.934584i \(-0.384228\pi\)
0.355743 + 0.934584i \(0.384228\pi\)
\(858\) 0 0
\(859\) 37.9411 1.29453 0.647267 0.762263i \(-0.275912\pi\)
0.647267 + 0.762263i \(0.275912\pi\)
\(860\) 9.65685i 0.329296i
\(861\) 21.6569 0.738064
\(862\) 16.0000 0.544962
\(863\) − 28.2843i − 0.962808i −0.876499 0.481404i \(-0.840127\pi\)
0.876499 0.481404i \(-0.159873\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) 9.31371i 0.316676i
\(866\) 22.9706i 0.780571i
\(867\) −16.3137 −0.554043
\(868\) 11.3137 0.384012
\(869\) − 13.2548i − 0.449639i
\(870\) 8.82843 0.299312
\(871\) 0 0
\(872\) 8.82843 0.298968
\(873\) − 3.17157i − 0.107341i
\(874\) −24.0000 −0.811812
\(875\) 2.82843 0.0956183
\(876\) − 14.4853i − 0.489412i
\(877\) − 51.2548i − 1.73075i −0.501122 0.865376i \(-0.667079\pi\)
0.501122 0.865376i \(-0.332921\pi\)
\(878\) − 22.6274i − 0.763638i
\(879\) 16.6274i 0.560829i
\(880\) 5.65685 0.190693
\(881\) 10.2843 0.346486 0.173243 0.984879i \(-0.444575\pi\)
0.173243 + 0.984879i \(0.444575\pi\)
\(882\) − 1.00000i − 0.0336718i
\(883\) −31.3137 −1.05379 −0.526895 0.849930i \(-0.676644\pi\)
−0.526895 + 0.849930i \(0.676644\pi\)
\(884\) 0 0
\(885\) −2.34315 −0.0787640
\(886\) 30.3431i 1.01940i
\(887\) −40.4853 −1.35936 −0.679681 0.733508i \(-0.737882\pi\)
−0.679681 + 0.733508i \(0.737882\pi\)
\(888\) 11.6569 0.391178
\(889\) − 27.3137i − 0.916072i
\(890\) − 15.6569i − 0.524819i
\(891\) − 5.65685i − 0.189512i
\(892\) − 10.8284i − 0.362563i
\(893\) −22.6274 −0.757198
\(894\) −3.65685 −0.122304
\(895\) − 7.51472i − 0.251189i
\(896\) 2.82843 0.0944911
\(897\) 0 0
\(898\) 26.2843 0.877117
\(899\) − 35.3137i − 1.17778i
\(900\) 1.00000 0.0333333
\(901\) −11.0294 −0.367444
\(902\) − 43.3137i − 1.44219i
\(903\) − 27.3137i − 0.908943i
\(904\) − 6.48528i − 0.215697i
\(905\) 7.65685i 0.254522i
\(906\) 12.0000 0.398673
\(907\) −8.28427 −0.275075 −0.137537 0.990497i \(-0.543919\pi\)
−0.137537 + 0.990497i \(0.543919\pi\)
\(908\) − 4.00000i − 0.132745i
\(909\) −16.1421 −0.535401
\(910\) 0 0
\(911\) 24.9706 0.827312 0.413656 0.910433i \(-0.364252\pi\)
0.413656 + 0.910433i \(0.364252\pi\)
\(912\) 2.82843i 0.0936586i
\(913\) 35.8823 1.18753
\(914\) −20.8284 −0.688943
\(915\) 6.00000i 0.198354i
\(916\) 4.14214i 0.136860i
\(917\) − 17.3726i − 0.573693i
\(918\) − 0.828427i − 0.0273422i
\(919\) 41.9411 1.38351 0.691755 0.722132i \(-0.256838\pi\)
0.691755 + 0.722132i \(0.256838\pi\)
\(920\) 8.48528 0.279751
\(921\) 21.6569i 0.713618i
\(922\) −14.0000 −0.461065
\(923\) 0 0
\(924\) −16.0000 −0.526361
\(925\) − 11.6569i − 0.383275i
\(926\) −3.79899 −0.124843
\(927\) 1.65685 0.0544182
\(928\) − 8.82843i − 0.289807i
\(929\) − 33.5980i − 1.10231i −0.834402 0.551157i \(-0.814187\pi\)
0.834402 0.551157i \(-0.185813\pi\)
\(930\) − 4.00000i − 0.131165i
\(931\) − 2.82843i − 0.0926980i
\(932\) 5.51472 0.180641
\(933\) 24.0000 0.785725
\(934\) − 7.31371i − 0.239312i
\(935\) −4.68629 −0.153258
\(936\) 0 0
\(937\) 16.6274 0.543194 0.271597 0.962411i \(-0.412448\pi\)
0.271597 + 0.962411i \(0.412448\pi\)
\(938\) − 16.0000i − 0.522419i
\(939\) −30.9706 −1.01069
\(940\) 8.00000 0.260931
\(941\) 54.9706i 1.79199i 0.444065 + 0.895995i \(0.353536\pi\)
−0.444065 + 0.895995i \(0.646464\pi\)
\(942\) 5.31371i 0.173130i
\(943\) − 64.9706i − 2.11573i
\(944\) 2.34315i 0.0762629i
\(945\) −2.82843 −0.0920087
\(946\) −54.6274 −1.77609
\(947\) − 30.3431i − 0.986020i −0.870024 0.493010i \(-0.835897\pi\)
0.870024 0.493010i \(-0.164103\pi\)
\(948\) −2.34315 −0.0761018
\(949\) 0 0
\(950\) 2.82843 0.0917663
\(951\) 25.3137i 0.820853i
\(952\) −2.34315 −0.0759418
\(953\) 27.8579 0.902405 0.451202 0.892422i \(-0.350995\pi\)
0.451202 + 0.892422i \(0.350995\pi\)
\(954\) 13.3137i 0.431047i
\(955\) 11.3137i 0.366103i
\(956\) − 16.0000i − 0.517477i
\(957\) 49.9411i 1.61437i
\(958\) −11.3137 −0.365529
\(959\) −48.9706 −1.58134
\(960\) − 1.00000i − 0.0322749i
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) −4.00000 −0.128898
\(964\) 5.31371i 0.171143i
\(965\) 2.48528 0.0800040
\(966\) −24.0000 −0.772187
\(967\) − 7.51472i − 0.241657i −0.992673 0.120829i \(-0.961445\pi\)
0.992673 0.120829i \(-0.0385551\pi\)
\(968\) 21.0000i 0.674966i
\(969\) − 2.34315i − 0.0752727i
\(970\) 3.17157i 0.101833i
\(971\) 15.5147 0.497891 0.248946 0.968517i \(-0.419916\pi\)
0.248946 + 0.968517i \(0.419916\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 17.9411i 0.575166i
\(974\) −16.4853 −0.528222
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) − 8.34315i − 0.266921i −0.991054 0.133460i \(-0.957391\pi\)
0.991054 0.133460i \(-0.0426089\pi\)
\(978\) 11.3137 0.361773
\(979\) 88.5685 2.83066
\(980\) 1.00000i 0.0319438i
\(981\) 8.82843i 0.281870i
\(982\) 38.1421i 1.21716i
\(983\) 2.34315i 0.0747347i 0.999302 + 0.0373674i \(0.0118972\pi\)
−0.999302 + 0.0373674i \(0.988103\pi\)
\(984\) −7.65685 −0.244092
\(985\) 13.3137 0.424210
\(986\) 7.31371i 0.232916i
\(987\) −22.6274 −0.720239
\(988\) 0 0
\(989\) −81.9411 −2.60558
\(990\) 5.65685i 0.179787i
\(991\) −42.9117 −1.36313 −0.681567 0.731755i \(-0.738701\pi\)
−0.681567 + 0.731755i \(0.738701\pi\)
\(992\) −4.00000 −0.127000
\(993\) − 8.48528i − 0.269272i
\(994\) 16.0000i 0.507489i
\(995\) − 10.3431i − 0.327900i
\(996\) − 6.34315i − 0.200990i
\(997\) 61.3137 1.94182 0.970912 0.239435i \(-0.0769623\pi\)
0.970912 + 0.239435i \(0.0769623\pi\)
\(998\) −0.485281 −0.0153613
\(999\) 11.6569i 0.368807i
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 5070.2.b.q.1351.4 4
13.5 odd 4 390.2.a.h.1.1 2
13.8 odd 4 5070.2.a.bc.1.2 2
13.12 even 2 inner 5070.2.b.q.1351.1 4
39.5 even 4 1170.2.a.o.1.1 2
52.31 even 4 3120.2.a.bc.1.2 2
65.18 even 4 1950.2.e.o.1249.2 4
65.44 odd 4 1950.2.a.bd.1.2 2
65.57 even 4 1950.2.e.o.1249.3 4
156.83 odd 4 9360.2.a.ch.1.2 2
195.44 even 4 5850.2.a.cl.1.2 2
195.83 odd 4 5850.2.e.bk.5149.4 4
195.122 odd 4 5850.2.e.bk.5149.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.h.1.1 2 13.5 odd 4
1170.2.a.o.1.1 2 39.5 even 4
1950.2.a.bd.1.2 2 65.44 odd 4
1950.2.e.o.1249.2 4 65.18 even 4
1950.2.e.o.1249.3 4 65.57 even 4
3120.2.a.bc.1.2 2 52.31 even 4
5070.2.a.bc.1.2 2 13.8 odd 4
5070.2.b.q.1351.1 4 13.12 even 2 inner
5070.2.b.q.1351.4 4 1.1 even 1 trivial
5850.2.a.cl.1.2 2 195.44 even 4
5850.2.e.bk.5149.1 4 195.122 odd 4
5850.2.e.bk.5149.4 4 195.83 odd 4
9360.2.a.ch.1.2 2 156.83 odd 4