Properties

Label 1950.2.e.o.1249.2
Level $1950$
Weight $2$
Character 1950.1249
Analytic conductor $15.571$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.5708283941\)
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 1249.2
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1950.1249
Dual form 1950.2.e.o.1249.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.82843i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.82843i q^{7} +1.00000i q^{8} -1.00000 q^{9} +5.65685 q^{11} -1.00000i q^{12} -1.00000i q^{13} +2.82843 q^{14} +1.00000 q^{16} -0.828427i q^{17} +1.00000i q^{18} -2.82843 q^{19} -2.82843 q^{21} -5.65685i q^{22} -8.48528i q^{23} -1.00000 q^{24} -1.00000 q^{26} -1.00000i q^{27} -2.82843i q^{28} +8.82843 q^{29} +4.00000 q^{31} -1.00000i q^{32} +5.65685i q^{33} -0.828427 q^{34} +1.00000 q^{36} +11.6569i q^{37} +2.82843i q^{38} +1.00000 q^{39} -7.65685 q^{41} +2.82843i q^{42} +9.65685i q^{43} -5.65685 q^{44} -8.48528 q^{46} +8.00000i q^{47} +1.00000i q^{48} -1.00000 q^{49} +0.828427 q^{51} +1.00000i q^{52} +13.3137i q^{53} -1.00000 q^{54} -2.82843 q^{56} -2.82843i q^{57} -8.82843i q^{58} +2.34315 q^{59} +6.00000 q^{61} -4.00000i q^{62} -2.82843i q^{63} -1.00000 q^{64} +5.65685 q^{66} -5.65685i q^{67} +0.828427i q^{68} +8.48528 q^{69} -5.65685 q^{71} -1.00000i q^{72} -14.4853i q^{73} +11.6569 q^{74} +2.82843 q^{76} +16.0000i q^{77} -1.00000i q^{78} -2.34315 q^{79} +1.00000 q^{81} +7.65685i q^{82} +6.34315i q^{83} +2.82843 q^{84} +9.65685 q^{86} +8.82843i q^{87} +5.65685i q^{88} +15.6569 q^{89} +2.82843 q^{91} +8.48528i q^{92} +4.00000i q^{93} +8.00000 q^{94} +1.00000 q^{96} +3.17157i q^{97} +1.00000i q^{98} -5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} + 4q^{16} - 4q^{24} - 4q^{26} + 24q^{29} + 16q^{31} + 8q^{34} + 4q^{36} + 4q^{39} - 8q^{41} - 4q^{49} - 8q^{51} - 4q^{54} + 32q^{59} + 24q^{61} - 4q^{64} + 24q^{74} - 32q^{79} + 4q^{81} + 16q^{86} + 40q^{89} + 32q^{94} + 4q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 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
\(10\) 0 0
\(11\) 5.65685 1.70561 0.852803 0.522233i \(-0.174901\pi\)
0.852803 + 0.522233i \(0.174901\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 1.00000i − 0.277350i
\(14\) 2.82843 0.755929
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 0.828427i − 0.200923i −0.994941 0.100462i \(-0.967968\pi\)
0.994941 0.100462i \(-0.0320319\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) −2.82843 −0.617213
\(22\) − 5.65685i − 1.20605i
\(23\) − 8.48528i − 1.76930i −0.466252 0.884652i \(-0.654396\pi\)
0.466252 0.884652i \(-0.345604\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) − 1.00000i − 0.192450i
\(28\) − 2.82843i − 0.534522i
\(29\) 8.82843 1.63940 0.819699 0.572795i \(-0.194141\pi\)
0.819699 + 0.572795i \(0.194141\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 5.65685i 0.984732i
\(34\) −0.828427 −0.142074
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 11.6569i 1.91638i 0.286141 + 0.958188i \(0.407627\pi\)
−0.286141 + 0.958188i \(0.592373\pi\)
\(38\) 2.82843i 0.458831i
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −7.65685 −1.19580 −0.597900 0.801571i \(-0.703998\pi\)
−0.597900 + 0.801571i \(0.703998\pi\)
\(42\) 2.82843i 0.436436i
\(43\) 9.65685i 1.47266i 0.676625 + 0.736328i \(0.263442\pi\)
−0.676625 + 0.736328i \(0.736558\pi\)
\(44\) −5.65685 −0.852803
\(45\) 0 0
\(46\) −8.48528 −1.25109
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0.828427 0.116003
\(52\) 1.00000i 0.138675i
\(53\) 13.3137i 1.82878i 0.404836 + 0.914389i \(0.367329\pi\)
−0.404836 + 0.914389i \(0.632671\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −2.82843 −0.377964
\(57\) − 2.82843i − 0.374634i
\(58\) − 8.82843i − 1.15923i
\(59\) 2.34315 0.305052 0.152526 0.988299i \(-0.451259\pi\)
0.152526 + 0.988299i \(0.451259\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) − 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 \(-0.887693\pi\)
0.938401 0.345547i \(-0.112307\pi\)
\(68\) 0.828427i 0.100462i
\(69\) 8.48528 1.02151
\(70\) 0 0
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 14.4853i − 1.69537i −0.530497 0.847687i \(-0.677995\pi\)
0.530497 0.847687i \(-0.322005\pi\)
\(74\) 11.6569 1.35508
\(75\) 0 0
\(76\) 2.82843 0.324443
\(77\) 16.0000i 1.82337i
\(78\) − 1.00000i − 0.113228i
\(79\) −2.34315 −0.263624 −0.131812 0.991275i \(-0.542080\pi\)
−0.131812 + 0.991275i \(0.542080\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 7.65685i 0.845558i
\(83\) 6.34315i 0.696251i 0.937448 + 0.348125i \(0.113182\pi\)
−0.937448 + 0.348125i \(0.886818\pi\)
\(84\) 2.82843 0.308607
\(85\) 0 0
\(86\) 9.65685 1.04133
\(87\) 8.82843i 0.946507i
\(88\) 5.65685i 0.603023i
\(89\) 15.6569 1.65962 0.829812 0.558044i \(-0.188448\pi\)
0.829812 + 0.558044i \(0.188448\pi\)
\(90\) 0 0
\(91\) 2.82843 0.296500
\(92\) 8.48528i 0.884652i
\(93\) 4.00000i 0.414781i
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 3.17157i 0.322024i 0.986952 + 0.161012i \(0.0514759\pi\)
−0.986952 + 0.161012i \(0.948524\pi\)
\(98\) 1.00000i 0.101015i
\(99\) −5.65685 −0.568535
\(100\) 0 0
\(101\) 16.1421 1.60620 0.803101 0.595843i \(-0.203182\pi\)
0.803101 + 0.595843i \(0.203182\pi\)
\(102\) − 0.828427i − 0.0820265i
\(103\) − 1.65685i − 0.163255i −0.996663 0.0816274i \(-0.973988\pi\)
0.996663 0.0816274i \(-0.0260117\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 13.3137 1.29314
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −8.82843 −0.845610 −0.422805 0.906221i \(-0.638954\pi\)
−0.422805 + 0.906221i \(0.638954\pi\)
\(110\) 0 0
\(111\) −11.6569 −1.10642
\(112\) 2.82843i 0.267261i
\(113\) 6.48528i 0.610084i 0.952339 + 0.305042i \(0.0986705\pi\)
−0.952339 + 0.305042i \(0.901330\pi\)
\(114\) −2.82843 −0.264906
\(115\) 0 0
\(116\) −8.82843 −0.819699
\(117\) 1.00000i 0.0924500i
\(118\) − 2.34315i − 0.215704i
\(119\) 2.34315 0.214796
\(120\) 0 0
\(121\) 21.0000 1.90909
\(122\) − 6.00000i − 0.543214i
\(123\) − 7.65685i − 0.690395i
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) −2.82843 −0.251976
\(127\) − 9.65685i − 0.856907i −0.903564 0.428454i \(-0.859059\pi\)
0.903564 0.428454i \(-0.140941\pi\)
\(128\) 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
\(133\) − 8.00000i − 0.693688i
\(134\) −5.65685 −0.488678
\(135\) 0 0
\(136\) 0.828427 0.0710370
\(137\) 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 \(-0.586696\pi\)
−0.269009 + 0.963138i \(0.586696\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 5.65685i 0.474713i
\(143\) − 5.65685i − 0.473050i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −14.4853 −1.19881
\(147\) − 1.00000i − 0.0824786i
\(148\) − 11.6569i − 0.958188i
\(149\) −3.65685 −0.299581 −0.149791 0.988718i \(-0.547860\pi\)
−0.149791 + 0.988718i \(0.547860\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) − 2.82843i − 0.229416i
\(153\) 0.828427i 0.0669744i
\(154\) 16.0000 1.28932
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) − 5.31371i − 0.424080i −0.977261 0.212040i \(-0.931989\pi\)
0.977261 0.212040i \(-0.0680107\pi\)
\(158\) 2.34315i 0.186411i
\(159\) −13.3137 −1.05585
\(160\) 0 0
\(161\) 24.0000 1.89146
\(162\) − 1.00000i − 0.0785674i
\(163\) 11.3137i 0.886158i 0.896483 + 0.443079i \(0.146114\pi\)
−0.896483 + 0.443079i \(0.853886\pi\)
\(164\) 7.65685 0.597900
\(165\) 0 0
\(166\) 6.34315 0.492324
\(167\) − 8.97056i − 0.694163i −0.937835 0.347081i \(-0.887173\pi\)
0.937835 0.347081i \(-0.112827\pi\)
\(168\) − 2.82843i − 0.218218i
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 2.82843 0.216295
\(172\) − 9.65685i − 0.736328i
\(173\) − 9.31371i − 0.708108i −0.935225 0.354054i \(-0.884803\pi\)
0.935225 0.354054i \(-0.115197\pi\)
\(174\) 8.82843 0.669281
\(175\) 0 0
\(176\) 5.65685 0.426401
\(177\) 2.34315i 0.176122i
\(178\) − 15.6569i − 1.17353i
\(179\) −7.51472 −0.561676 −0.280838 0.959755i \(-0.590612\pi\)
−0.280838 + 0.959755i \(0.590612\pi\)
\(180\) 0 0
\(181\) −7.65685 −0.569129 −0.284565 0.958657i \(-0.591849\pi\)
−0.284565 + 0.958657i \(0.591849\pi\)
\(182\) − 2.82843i − 0.209657i
\(183\) 6.00000i 0.443533i
\(184\) 8.48528 0.625543
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) − 4.68629i − 0.342696i
\(188\) − 8.00000i − 0.583460i
\(189\) 2.82843 0.205738
\(190\) 0 0
\(191\) 11.3137 0.818631 0.409316 0.912393i \(-0.365768\pi\)
0.409316 + 0.912393i \(0.365768\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 2.48528i 0.178894i 0.995992 + 0.0894472i \(0.0285100\pi\)
−0.995992 + 0.0894472i \(0.971490\pi\)
\(194\) 3.17157 0.227706
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 13.3137i 0.948562i 0.880373 + 0.474281i \(0.157292\pi\)
−0.880373 + 0.474281i \(0.842708\pi\)
\(198\) 5.65685i 0.402015i
\(199\) −10.3431 −0.733206 −0.366603 0.930377i \(-0.619479\pi\)
−0.366603 + 0.930377i \(0.619479\pi\)
\(200\) 0 0
\(201\) 5.65685 0.399004
\(202\) − 16.1421i − 1.13576i
\(203\) 24.9706i 1.75259i
\(204\) −0.828427 −0.0580015
\(205\) 0 0
\(206\) −1.65685 −0.115439
\(207\) 8.48528i 0.589768i
\(208\) − 1.00000i − 0.0693375i
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 0.686292 0.0472463 0.0236231 0.999721i \(-0.492480\pi\)
0.0236231 + 0.999721i \(0.492480\pi\)
\(212\) − 13.3137i − 0.914389i
\(213\) − 5.65685i − 0.387601i
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 11.3137i 0.768025i
\(218\) 8.82843i 0.597937i
\(219\) 14.4853 0.978825
\(220\) 0 0
\(221\) −0.828427 −0.0557260
\(222\) 11.6569i 0.782357i
\(223\) 10.8284i 0.725125i 0.931959 + 0.362563i \(0.118098\pi\)
−0.931959 + 0.362563i \(0.881902\pi\)
\(224\) 2.82843 0.188982
\(225\) 0 0
\(226\) 6.48528 0.431394
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) 2.82843i 0.187317i
\(229\) −4.14214 −0.273720 −0.136860 0.990590i \(-0.543701\pi\)
−0.136860 + 0.990590i \(0.543701\pi\)
\(230\) 0 0
\(231\) −16.0000 −1.05272
\(232\) 8.82843i 0.579615i
\(233\) 5.51472i 0.361281i 0.983549 + 0.180641i \(0.0578171\pi\)
−0.983549 + 0.180641i \(0.942183\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) −2.34315 −0.152526
\(237\) − 2.34315i − 0.152204i
\(238\) − 2.34315i − 0.151884i
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 5.31371 0.342286 0.171143 0.985246i \(-0.445254\pi\)
0.171143 + 0.985246i \(0.445254\pi\)
\(242\) − 21.0000i − 1.34993i
\(243\) 1.00000i 0.0641500i
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) −7.65685 −0.488183
\(247\) 2.82843i 0.179969i
\(248\) 4.00000i 0.254000i
\(249\) −6.34315 −0.401981
\(250\) 0 0
\(251\) 10.8284 0.683484 0.341742 0.939794i \(-0.388983\pi\)
0.341742 + 0.939794i \(0.388983\pi\)
\(252\) 2.82843i 0.178174i
\(253\) − 48.0000i − 3.01773i
\(254\) −9.65685 −0.605925
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 4.82843i 0.301189i 0.988596 + 0.150595i \(0.0481188\pi\)
−0.988596 + 0.150595i \(0.951881\pi\)
\(258\) 9.65685i 0.601209i
\(259\) −32.9706 −2.04869
\(260\) 0 0
\(261\) −8.82843 −0.546466
\(262\) 6.14214i 0.379462i
\(263\) 16.4853i 1.01653i 0.861202 + 0.508263i \(0.169712\pi\)
−0.861202 + 0.508263i \(0.830288\pi\)
\(264\) −5.65685 −0.348155
\(265\) 0 0
\(266\) −8.00000 −0.490511
\(267\) 15.6569i 0.958184i
\(268\) 5.65685i 0.345547i
\(269\) 14.4853 0.883183 0.441592 0.897216i \(-0.354414\pi\)
0.441592 + 0.897216i \(0.354414\pi\)
\(270\) 0 0
\(271\) 7.31371 0.444276 0.222138 0.975015i \(-0.428696\pi\)
0.222138 + 0.975015i \(0.428696\pi\)
\(272\) − 0.828427i − 0.0502308i
\(273\) 2.82843i 0.171184i
\(274\) 17.3137 1.04596
\(275\) 0 0
\(276\) −8.48528 −0.510754
\(277\) − 26.0000i − 1.56219i −0.624413 0.781094i \(-0.714662\pi\)
0.624413 0.781094i \(-0.285338\pi\)
\(278\) 6.34315i 0.380437i
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 8.34315 0.497710 0.248855 0.968541i \(-0.419946\pi\)
0.248855 + 0.968541i \(0.419946\pi\)
\(282\) 8.00000i 0.476393i
\(283\) − 17.6569i − 1.04959i −0.851228 0.524796i \(-0.824142\pi\)
0.851228 0.524796i \(-0.175858\pi\)
\(284\) 5.65685 0.335673
\(285\) 0 0
\(286\) −5.65685 −0.334497
\(287\) − 21.6569i − 1.27836i
\(288\) 1.00000i 0.0589256i
\(289\) 16.3137 0.959630
\(290\) 0 0
\(291\) −3.17157 −0.185921
\(292\) 14.4853i 0.847687i
\(293\) − 16.6274i − 0.971384i −0.874130 0.485692i \(-0.838568\pi\)
0.874130 0.485692i \(-0.161432\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) −11.6569 −0.677541
\(297\) − 5.65685i − 0.328244i
\(298\) 3.65685i 0.211836i
\(299\) −8.48528 −0.490716
\(300\) 0 0
\(301\) −27.3137 −1.57434
\(302\) − 12.0000i − 0.690522i
\(303\) 16.1421i 0.927341i
\(304\) −2.82843 −0.162221
\(305\) 0 0
\(306\) 0.828427 0.0473580
\(307\) 21.6569i 1.23602i 0.786169 + 0.618011i \(0.212061\pi\)
−0.786169 + 0.618011i \(0.787939\pi\)
\(308\) − 16.0000i − 0.911685i
\(309\) 1.65685 0.0942551
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 1.00000i 0.0566139i
\(313\) − 30.9706i − 1.75056i −0.483617 0.875280i \(-0.660677\pi\)
0.483617 0.875280i \(-0.339323\pi\)
\(314\) −5.31371 −0.299870
\(315\) 0 0
\(316\) 2.34315 0.131812
\(317\) − 25.3137i − 1.42176i −0.703314 0.710880i \(-0.748297\pi\)
0.703314 0.710880i \(-0.251703\pi\)
\(318\) 13.3137i 0.746596i
\(319\) 49.9411 2.79617
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) − 24.0000i − 1.33747i
\(323\) 2.34315i 0.130376i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 11.3137 0.626608
\(327\) − 8.82843i − 0.488213i
\(328\) − 7.65685i − 0.422779i
\(329\) −22.6274 −1.24749
\(330\) 0 0
\(331\) −8.48528 −0.466393 −0.233197 0.972430i \(-0.574919\pi\)
−0.233197 + 0.972430i \(0.574919\pi\)
\(332\) − 6.34315i − 0.348125i
\(333\) − 11.6569i − 0.638792i
\(334\) −8.97056 −0.490847
\(335\) 0 0
\(336\) −2.82843 −0.154303
\(337\) − 10.9706i − 0.597605i −0.954315 0.298802i \(-0.903413\pi\)
0.954315 0.298802i \(-0.0965871\pi\)
\(338\) 1.00000i 0.0543928i
\(339\) −6.48528 −0.352232
\(340\) 0 0
\(341\) 22.6274 1.22534
\(342\) − 2.82843i − 0.152944i
\(343\) 16.9706i 0.916324i
\(344\) −9.65685 −0.520663
\(345\) 0 0
\(346\) −9.31371 −0.500708
\(347\) 9.65685i 0.518407i 0.965823 + 0.259204i \(0.0834600\pi\)
−0.965823 + 0.259204i \(0.916540\pi\)
\(348\) − 8.82843i − 0.473253i
\(349\) −12.1421 −0.649954 −0.324977 0.945722i \(-0.605356\pi\)
−0.324977 + 0.945722i \(0.605356\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) − 5.65685i − 0.301511i
\(353\) 5.31371i 0.282820i 0.989951 + 0.141410i \(0.0451636\pi\)
−0.989951 + 0.141410i \(0.954836\pi\)
\(354\) 2.34315 0.124537
\(355\) 0 0
\(356\) −15.6569 −0.829812
\(357\) 2.34315i 0.124012i
\(358\) 7.51472i 0.397165i
\(359\) 28.2843 1.49279 0.746393 0.665505i \(-0.231784\pi\)
0.746393 + 0.665505i \(0.231784\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 7.65685i 0.402435i
\(363\) 21.0000i 1.10221i
\(364\) −2.82843 −0.148250
\(365\) 0 0
\(366\) 6.00000 0.313625
\(367\) − 25.6569i − 1.33928i −0.742687 0.669638i \(-0.766449\pi\)
0.742687 0.669638i \(-0.233551\pi\)
\(368\) − 8.48528i − 0.442326i
\(369\) 7.65685 0.398600
\(370\) 0 0
\(371\) −37.6569 −1.95505
\(372\) − 4.00000i − 0.207390i
\(373\) − 2.68629i − 0.139091i −0.997579 0.0695455i \(-0.977845\pi\)
0.997579 0.0695455i \(-0.0221549\pi\)
\(374\) −4.68629 −0.242322
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) − 8.82843i − 0.454687i
\(378\) − 2.82843i − 0.145479i
\(379\) 7.51472 0.386005 0.193003 0.981198i \(-0.438177\pi\)
0.193003 + 0.981198i \(0.438177\pi\)
\(380\) 0 0
\(381\) 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 \(-0.273676\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 2.48528 0.126497
\(387\) − 9.65685i − 0.490885i
\(388\) − 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
\(393\) − 6.14214i − 0.309830i
\(394\) 13.3137 0.670735
\(395\) 0 0
\(396\) 5.65685 0.284268
\(397\) − 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
\(400\) 0 0
\(401\) −26.9706 −1.34685 −0.673423 0.739258i \(-0.735177\pi\)
−0.673423 + 0.739258i \(0.735177\pi\)
\(402\) − 5.65685i − 0.282138i
\(403\) − 4.00000i − 0.199254i
\(404\) −16.1421 −0.803101
\(405\) 0 0
\(406\) 24.9706 1.23927
\(407\) 65.9411i 3.26858i
\(408\) 0.828427i 0.0410133i
\(409\) 3.65685 0.180820 0.0904099 0.995905i \(-0.471182\pi\)
0.0904099 + 0.995905i \(0.471182\pi\)
\(410\) 0 0
\(411\) −17.3137 −0.854022
\(412\) 1.65685i 0.0816274i
\(413\) 6.62742i 0.326114i
\(414\) 8.48528 0.417029
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) − 6.34315i − 0.310625i
\(418\) 16.0000i 0.782586i
\(419\) 10.8284 0.529003 0.264502 0.964385i \(-0.414793\pi\)
0.264502 + 0.964385i \(0.414793\pi\)
\(420\) 0 0
\(421\) −24.1421 −1.17662 −0.588308 0.808637i \(-0.700206\pi\)
−0.588308 + 0.808637i \(0.700206\pi\)
\(422\) − 0.686292i − 0.0334081i
\(423\) − 8.00000i − 0.388973i
\(424\) −13.3137 −0.646571
\(425\) 0 0
\(426\) −5.65685 −0.274075
\(427\) 16.9706i 0.821263i
\(428\) − 4.00000i − 0.193347i
\(429\) 5.65685 0.273115
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 22.9706i − 1.10389i −0.833879 0.551947i \(-0.813885\pi\)
0.833879 0.551947i \(-0.186115\pi\)
\(434\) 11.3137 0.543075
\(435\) 0 0
\(436\) 8.82843 0.422805
\(437\) 24.0000i 1.14808i
\(438\) − 14.4853i − 0.692134i
\(439\) −22.6274 −1.07995 −0.539974 0.841682i \(-0.681566\pi\)
−0.539974 + 0.841682i \(0.681566\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0.828427i 0.0394043i
\(443\) 30.3431i 1.44165i 0.693119 + 0.720823i \(0.256236\pi\)
−0.693119 + 0.720823i \(0.743764\pi\)
\(444\) 11.6569 0.553210
\(445\) 0 0
\(446\) 10.8284 0.512741
\(447\) − 3.65685i − 0.172963i
\(448\) − 2.82843i − 0.133631i
\(449\) −26.2843 −1.24043 −0.620216 0.784431i \(-0.712955\pi\)
−0.620216 + 0.784431i \(0.712955\pi\)
\(450\) 0 0
\(451\) −43.3137 −2.03956
\(452\) − 6.48528i − 0.305042i
\(453\) 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 \(-0.838034\pi\)
0.873315 0.487156i \(-0.161966\pi\)
\(458\) 4.14214i 0.193549i
\(459\) −0.828427 −0.0386677
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 16.0000i 0.744387i
\(463\) − 3.79899i − 0.176554i −0.996096 0.0882770i \(-0.971864\pi\)
0.996096 0.0882770i \(-0.0281361\pi\)
\(464\) 8.82843 0.409849
\(465\) 0 0
\(466\) 5.51472 0.255464
\(467\) − 7.31371i − 0.338438i −0.985578 0.169219i \(-0.945875\pi\)
0.985578 0.169219i \(-0.0541245\pi\)
\(468\) − 1.00000i − 0.0462250i
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 5.31371 0.244843
\(472\) 2.34315i 0.107852i
\(473\) 54.6274i 2.51177i
\(474\) −2.34315 −0.107624
\(475\) 0 0
\(476\) −2.34315 −0.107398
\(477\) − 13.3137i − 0.609593i
\(478\) 16.0000i 0.731823i
\(479\) 11.3137 0.516937 0.258468 0.966020i \(-0.416782\pi\)
0.258468 + 0.966020i \(0.416782\pi\)
\(480\) 0 0
\(481\) 11.6569 0.531507
\(482\) − 5.31371i − 0.242033i
\(483\) 24.0000i 1.09204i
\(484\) −21.0000 −0.954545
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 16.4853i − 0.747019i −0.927626 0.373510i \(-0.878154\pi\)
0.927626 0.373510i \(-0.121846\pi\)
\(488\) 6.00000i 0.271607i
\(489\) −11.3137 −0.511624
\(490\) 0 0
\(491\) −38.1421 −1.72133 −0.860665 0.509171i \(-0.829952\pi\)
−0.860665 + 0.509171i \(0.829952\pi\)
\(492\) 7.65685i 0.345198i
\(493\) − 7.31371i − 0.329393i
\(494\) 2.82843 0.127257
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) − 16.0000i − 0.717698i
\(498\) 6.34315i 0.284243i
\(499\) −0.485281 −0.0217242 −0.0108621 0.999941i \(-0.503458\pi\)
−0.0108621 + 0.999941i \(0.503458\pi\)
\(500\) 0 0
\(501\) 8.97056 0.400775
\(502\) − 10.8284i − 0.483296i
\(503\) − 23.5147i − 1.04847i −0.851574 0.524235i \(-0.824351\pi\)
0.851574 0.524235i \(-0.175649\pi\)
\(504\) 2.82843 0.125988
\(505\) 0 0
\(506\) −48.0000 −2.13386
\(507\) − 1.00000i − 0.0444116i
\(508\) 9.65685i 0.428454i
\(509\) 37.3137 1.65390 0.826951 0.562275i \(-0.190074\pi\)
0.826951 + 0.562275i \(0.190074\pi\)
\(510\) 0 0
\(511\) 40.9706 1.81243
\(512\) − 1.00000i − 0.0441942i
\(513\) 2.82843i 0.124878i
\(514\) 4.82843 0.212973
\(515\) 0 0
\(516\) 9.65685 0.425119
\(517\) 45.2548i 1.99031i
\(518\) 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
\(523\) − 10.6274i − 0.464704i −0.972632 0.232352i \(-0.925358\pi\)
0.972632 0.232352i \(-0.0746422\pi\)
\(524\) 6.14214 0.268320
\(525\) 0 0
\(526\) 16.4853 0.718792
\(527\) − 3.31371i − 0.144347i
\(528\) 5.65685i 0.246183i
\(529\) −49.0000 −2.13043
\(530\) 0 0
\(531\) −2.34315 −0.101684
\(532\) 8.00000i 0.346844i
\(533\) 7.65685i 0.331655i
\(534\) 15.6569 0.677538
\(535\) 0 0
\(536\) 5.65685 0.244339
\(537\) − 7.51472i − 0.324284i
\(538\) − 14.4853i − 0.624505i
\(539\) −5.65685 −0.243658
\(540\) 0 0
\(541\) 14.4853 0.622771 0.311385 0.950284i \(-0.399207\pi\)
0.311385 + 0.950284i \(0.399207\pi\)
\(542\) − 7.31371i − 0.314151i
\(543\) − 7.65685i − 0.328587i
\(544\) −0.828427 −0.0355185
\(545\) 0 0
\(546\) 2.82843 0.121046
\(547\) − 0.686292i − 0.0293437i −0.999892 0.0146719i \(-0.995330\pi\)
0.999892 0.0146719i \(-0.00467036\pi\)
\(548\) − 17.3137i − 0.739605i
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −24.9706 −1.06378
\(552\) 8.48528i 0.361158i
\(553\) − 6.62742i − 0.281826i
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) 6.34315 0.269009
\(557\) − 10.6863i − 0.452793i −0.974035 0.226396i \(-0.927306\pi\)
0.974035 0.226396i \(-0.0726945\pi\)
\(558\) 4.00000i 0.169334i
\(559\) 9.65685 0.408441
\(560\) 0 0
\(561\) 4.68629 0.197855
\(562\) − 8.34315i − 0.351934i
\(563\) − 30.3431i − 1.27881i −0.768870 0.639406i \(-0.779181\pi\)
0.768870 0.639406i \(-0.220819\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −17.6569 −0.742173
\(567\) 2.82843i 0.118783i
\(568\) − 5.65685i − 0.237356i
\(569\) −31.6569 −1.32712 −0.663562 0.748121i \(-0.730956\pi\)
−0.663562 + 0.748121i \(0.730956\pi\)
\(570\) 0 0
\(571\) 20.9706 0.877591 0.438795 0.898587i \(-0.355405\pi\)
0.438795 + 0.898587i \(0.355405\pi\)
\(572\) 5.65685i 0.236525i
\(573\) 11.3137i 0.472637i
\(574\) −21.6569 −0.903940
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 23.4558i 0.976480i 0.872710 + 0.488240i \(0.162361\pi\)
−0.872710 + 0.488240i \(0.837639\pi\)
\(578\) − 16.3137i − 0.678561i
\(579\) −2.48528 −0.103285
\(580\) 0 0
\(581\) −17.9411 −0.744323
\(582\) 3.17157i 0.131466i
\(583\) 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 \(-0.982732\pi\)
0.998529 0.0542226i \(-0.0172680\pi\)
\(588\) 1.00000i 0.0412393i
\(589\) −11.3137 −0.466173
\(590\) 0 0
\(591\) −13.3137 −0.547653
\(592\) 11.6569i 0.479094i
\(593\) − 0.343146i − 0.0140913i −0.999975 0.00704565i \(-0.997757\pi\)
0.999975 0.00704565i \(-0.00224272\pi\)
\(594\) −5.65685 −0.232104
\(595\) 0 0
\(596\) 3.65685 0.149791
\(597\) − 10.3431i − 0.423317i
\(598\) 8.48528i 0.346989i
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 0 0
\(601\) 29.3137 1.19573 0.597866 0.801596i \(-0.296016\pi\)
0.597866 + 0.801596i \(0.296016\pi\)
\(602\) 27.3137i 1.11322i
\(603\) 5.65685i 0.230365i
\(604\) −12.0000 −0.488273
\(605\) 0 0
\(606\) 16.1421 0.655729
\(607\) − 28.9706i − 1.17588i −0.808905 0.587939i \(-0.799939\pi\)
0.808905 0.587939i \(-0.200061\pi\)
\(608\) 2.82843i 0.114708i
\(609\) −24.9706 −1.01186
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) − 0.828427i − 0.0334872i
\(613\) 22.2843i 0.900053i 0.893015 + 0.450027i \(0.148586\pi\)
−0.893015 + 0.450027i \(0.851414\pi\)
\(614\) 21.6569 0.874000
\(615\) 0 0
\(616\) −16.0000 −0.644658
\(617\) − 2.00000i − 0.0805170i −0.999189 0.0402585i \(-0.987182\pi\)
0.999189 0.0402585i \(-0.0128181\pi\)
\(618\) − 1.65685i − 0.0666485i
\(619\) 34.8284 1.39987 0.699936 0.714205i \(-0.253212\pi\)
0.699936 + 0.714205i \(0.253212\pi\)
\(620\) 0 0
\(621\) −8.48528 −0.340503
\(622\) 24.0000i 0.962312i
\(623\) 44.2843i 1.77421i
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) −30.9706 −1.23783
\(627\) − 16.0000i − 0.638978i
\(628\) 5.31371i 0.212040i
\(629\) 9.65685 0.385044
\(630\) 0 0
\(631\) −33.6569 −1.33986 −0.669929 0.742425i \(-0.733676\pi\)
−0.669929 + 0.742425i \(0.733676\pi\)
\(632\) − 2.34315i − 0.0932053i
\(633\) 0.686292i 0.0272776i
\(634\) −25.3137 −1.00534
\(635\) 0 0
\(636\) 13.3137 0.527923
\(637\) 1.00000i 0.0396214i
\(638\) − 49.9411i − 1.97719i
\(639\) 5.65685 0.223782
\(640\) 0 0
\(641\) −4.62742 −0.182772 −0.0913860 0.995816i \(-0.529130\pi\)
−0.0913860 + 0.995816i \(0.529130\pi\)
\(642\) 4.00000i 0.157867i
\(643\) 39.5980i 1.56159i 0.624786 + 0.780796i \(0.285186\pi\)
−0.624786 + 0.780796i \(0.714814\pi\)
\(644\) −24.0000 −0.945732
\(645\) 0 0
\(646\) 2.34315 0.0921898
\(647\) − 8.48528i − 0.333591i −0.985992 0.166795i \(-0.946658\pi\)
0.985992 0.166795i \(-0.0533419\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 13.2548 0.520298
\(650\) 0 0
\(651\) −11.3137 −0.443419
\(652\) − 11.3137i − 0.443079i
\(653\) − 42.2843i − 1.65471i −0.561678 0.827356i \(-0.689844\pi\)
0.561678 0.827356i \(-0.310156\pi\)
\(654\) −8.82843 −0.345219
\(655\) 0 0
\(656\) −7.65685 −0.298950
\(657\) 14.4853i 0.565125i
\(658\) 22.6274i 0.882109i
\(659\) 7.51472 0.292732 0.146366 0.989231i \(-0.453242\pi\)
0.146366 + 0.989231i \(0.453242\pi\)
\(660\) 0 0
\(661\) −8.14214 −0.316692 −0.158346 0.987384i \(-0.550616\pi\)
−0.158346 + 0.987384i \(0.550616\pi\)
\(662\) 8.48528i 0.329790i
\(663\) − 0.828427i − 0.0321734i
\(664\) −6.34315 −0.246162
\(665\) 0 0
\(666\) −11.6569 −0.451694
\(667\) − 74.9117i − 2.90059i
\(668\) 8.97056i 0.347081i
\(669\) −10.8284 −0.418651
\(670\) 0 0
\(671\) 33.9411 1.31028
\(672\) 2.82843i 0.109109i
\(673\) 32.6274i 1.25769i 0.777529 + 0.628847i \(0.216473\pi\)
−0.777529 + 0.628847i \(0.783527\pi\)
\(674\) −10.9706 −0.422570
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) − 12.3431i − 0.474386i −0.971463 0.237193i \(-0.923773\pi\)
0.971463 0.237193i \(-0.0762273\pi\)
\(678\) 6.48528i 0.249066i
\(679\) −8.97056 −0.344259
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) − 22.6274i − 0.866449i
\(683\) 33.6569i 1.28784i 0.765091 + 0.643922i \(0.222694\pi\)
−0.765091 + 0.643922i \(0.777306\pi\)
\(684\) −2.82843 −0.108148
\(685\) 0 0
\(686\) 16.9706 0.647939
\(687\) − 4.14214i − 0.158032i
\(688\) 9.65685i 0.368164i
\(689\) 13.3137 0.507212
\(690\) 0 0
\(691\) −27.7990 −1.05752 −0.528762 0.848770i \(-0.677343\pi\)
−0.528762 + 0.848770i \(0.677343\pi\)
\(692\) 9.31371i 0.354054i
\(693\) − 16.0000i − 0.607790i
\(694\) 9.65685 0.366569
\(695\) 0 0
\(696\) −8.82843 −0.334641
\(697\) 6.34315i 0.240264i
\(698\) 12.1421i 0.459587i
\(699\) −5.51472 −0.208586
\(700\) 0 0
\(701\) 0.142136 0.00536839 0.00268419 0.999996i \(-0.499146\pi\)
0.00268419 + 0.999996i \(0.499146\pi\)
\(702\) 1.00000i 0.0377426i
\(703\) − 32.9706i − 1.24351i
\(704\) −5.65685 −0.213201
\(705\) 0 0
\(706\) 5.31371 0.199984
\(707\) 45.6569i 1.71710i
\(708\) − 2.34315i − 0.0880608i
\(709\) 7.17157 0.269334 0.134667 0.990891i \(-0.457004\pi\)
0.134667 + 0.990891i \(0.457004\pi\)
\(710\) 0 0
\(711\) 2.34315 0.0878748
\(712\) 15.6569i 0.586765i
\(713\) − 33.9411i − 1.27111i
\(714\) 2.34315 0.0876900
\(715\) 0 0
\(716\) 7.51472 0.280838
\(717\) − 16.0000i − 0.597531i
\(718\) − 28.2843i − 1.05556i
\(719\) 29.6569 1.10601 0.553007 0.833177i \(-0.313480\pi\)
0.553007 + 0.833177i \(0.313480\pi\)
\(720\) 0 0
\(721\) 4.68629 0.174527
\(722\) 11.0000i 0.409378i
\(723\) 5.31371i 0.197619i
\(724\) 7.65685 0.284565
\(725\) 0 0
\(726\) 21.0000 0.779383
\(727\) 45.9411i 1.70386i 0.523654 + 0.851931i \(0.324568\pi\)
−0.523654 + 0.851931i \(0.675432\pi\)
\(728\) 2.82843i 0.104828i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) − 6.00000i − 0.221766i
\(733\) − 0.343146i − 0.0126744i −0.999980 0.00633719i \(-0.997983\pi\)
0.999980 0.00633719i \(-0.00201720\pi\)
\(734\) −25.6569 −0.947012
\(735\) 0 0
\(736\) −8.48528 −0.312772
\(737\) − 32.0000i − 1.17874i
\(738\) − 7.65685i − 0.281853i
\(739\) 14.1421 0.520227 0.260113 0.965578i \(-0.416240\pi\)
0.260113 + 0.965578i \(0.416240\pi\)
\(740\) 0 0
\(741\) −2.82843 −0.103905
\(742\) 37.6569i 1.38243i
\(743\) − 36.2843i − 1.33114i −0.746335 0.665570i \(-0.768188\pi\)
0.746335 0.665570i \(-0.231812\pi\)
\(744\) −4.00000 −0.146647
\(745\) 0 0
\(746\) −2.68629 −0.0983521
\(747\) − 6.34315i − 0.232084i
\(748\) 4.68629i 0.171348i
\(749\) −11.3137 −0.413394
\(750\) 0 0
\(751\) 11.3137 0.412843 0.206422 0.978463i \(-0.433818\pi\)
0.206422 + 0.978463i \(0.433818\pi\)
\(752\) 8.00000i 0.291730i
\(753\) 10.8284i 0.394610i
\(754\) −8.82843 −0.321512
\(755\) 0 0
\(756\) −2.82843 −0.102869
\(757\) − 19.9411i − 0.724773i −0.932028 0.362386i \(-0.881962\pi\)
0.932028 0.362386i \(-0.118038\pi\)
\(758\) − 7.51472i − 0.272947i
\(759\) 48.0000 1.74229
\(760\) 0 0
\(761\) 27.6569 1.00256 0.501280 0.865285i \(-0.332863\pi\)
0.501280 + 0.865285i \(0.332863\pi\)
\(762\) − 9.65685i − 0.349831i
\(763\) − 24.9706i − 0.903995i
\(764\) −11.3137 −0.409316
\(765\) 0 0
\(766\) −29.6569 −1.07155
\(767\) − 2.34315i − 0.0846061i
\(768\) 1.00000i 0.0360844i
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −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 \(-0.408289\pi\)
\(774\) −9.65685 −0.347108
\(775\) 0 0
\(776\) −3.17157 −0.113853
\(777\) − 32.9706i − 1.18281i
\(778\) − 6.48528i − 0.232509i
\(779\) 21.6569 0.775937
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 7.02944i 0.251372i
\(783\) − 8.82843i − 0.315502i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) −6.14214 −0.219083
\(787\) 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
\(790\) 0 0
\(791\) −18.3431 −0.652207
\(792\) − 5.65685i − 0.201008i
\(793\) − 6.00000i − 0.213066i
\(794\) −30.2843 −1.07475
\(795\) 0 0
\(796\) 10.3431 0.366603
\(797\) − 16.6274i − 0.588973i −0.955656 0.294487i \(-0.904851\pi\)
0.955656 0.294487i \(-0.0951487\pi\)
\(798\) − 8.00000i − 0.283197i
\(799\) 6.62742 0.234461
\(800\) 0 0
\(801\) −15.6569 −0.553208
\(802\) 26.9706i 0.952364i
\(803\) − 81.9411i − 2.89164i
\(804\) −5.65685 −0.199502
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 14.4853i 0.509906i
\(808\) 16.1421i 0.567878i
\(809\) −13.3137 −0.468085 −0.234043 0.972226i \(-0.575196\pi\)
−0.234043 + 0.972226i \(0.575196\pi\)
\(810\) 0 0
\(811\) −1.85786 −0.0652384 −0.0326192 0.999468i \(-0.510385\pi\)
−0.0326192 + 0.999468i \(0.510385\pi\)
\(812\) − 24.9706i − 0.876295i
\(813\) 7.31371i 0.256503i
\(814\) 65.9411 2.31124
\(815\) 0 0
\(816\) 0.828427 0.0290008
\(817\) − 27.3137i − 0.955586i
\(818\) − 3.65685i − 0.127859i
\(819\) −2.82843 −0.0988332
\(820\) 0 0
\(821\) 34.2843 1.19653 0.598265 0.801299i \(-0.295857\pi\)
0.598265 + 0.801299i \(0.295857\pi\)
\(822\) 17.3137i 0.603885i
\(823\) − 52.9706i − 1.84644i −0.384275 0.923219i \(-0.625548\pi\)
0.384275 0.923219i \(-0.374452\pi\)
\(824\) 1.65685 0.0577193
\(825\) 0 0
\(826\) 6.62742 0.230597
\(827\) − 9.65685i − 0.335802i −0.985804 0.167901i \(-0.946301\pi\)
0.985804 0.167901i \(-0.0536989\pi\)
\(828\) − 8.48528i − 0.294884i
\(829\) 53.3137 1.85166 0.925831 0.377938i \(-0.123367\pi\)
0.925831 + 0.377938i \(0.123367\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) 1.00000i 0.0346688i
\(833\) 0.828427i 0.0287033i
\(834\) −6.34315 −0.219645
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) − 4.00000i − 0.138260i
\(838\) − 10.8284i − 0.374062i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 48.9411 1.68763
\(842\) 24.1421i 0.831993i
\(843\) 8.34315i 0.287353i
\(844\) −0.686292 −0.0236231
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) 59.3970i 2.04090i
\(848\) 13.3137i 0.457195i
\(849\) 17.6569 0.605982
\(850\) 0 0
\(851\) 98.9117 3.39065
\(852\) 5.65685i 0.193801i
\(853\) 38.2843i 1.31083i 0.755270 + 0.655414i \(0.227506\pi\)
−0.755270 + 0.655414i \(0.772494\pi\)
\(854\) 16.9706 0.580721
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 20.8284i 0.711486i 0.934584 + 0.355743i \(0.115772\pi\)
−0.934584 + 0.355743i \(0.884228\pi\)
\(858\) − 5.65685i − 0.193122i
\(859\) −37.9411 −1.29453 −0.647267 0.762263i \(-0.724088\pi\)
−0.647267 + 0.762263i \(0.724088\pi\)
\(860\) 0 0
\(861\) 21.6569 0.738064
\(862\) 16.0000i 0.544962i
\(863\) − 28.2843i − 0.962808i −0.876499 0.481404i \(-0.840127\pi\)
0.876499 0.481404i \(-0.159873\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −22.9706 −0.780571
\(867\) 16.3137i 0.554043i
\(868\) − 11.3137i − 0.384012i
\(869\) −13.2548 −0.449639
\(870\) 0 0
\(871\) −5.65685 −0.191675
\(872\) − 8.82843i − 0.298968i
\(873\) − 3.17157i − 0.107341i
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) −14.4853 −0.489412
\(877\) 51.2548i 1.73075i 0.501122 + 0.865376i \(0.332921\pi\)
−0.501122 + 0.865376i \(0.667079\pi\)
\(878\) 22.6274i 0.763638i
\(879\) 16.6274 0.560829
\(880\) 0 0
\(881\) −10.2843 −0.346486 −0.173243 0.984879i \(-0.555425\pi\)
−0.173243 + 0.984879i \(0.555425\pi\)
\(882\) − 1.00000i − 0.0336718i
\(883\) 31.3137i 1.05379i 0.849930 + 0.526895i \(0.176644\pi\)
−0.849930 + 0.526895i \(0.823356\pi\)
\(884\) 0.828427 0.0278630
\(885\) 0 0
\(886\) 30.3431 1.01940
\(887\) 40.4853i 1.35936i 0.733508 + 0.679681i \(0.237882\pi\)
−0.733508 + 0.679681i \(0.762118\pi\)
\(888\) − 11.6569i − 0.391178i
\(889\) 27.3137 0.916072
\(890\) 0 0
\(891\) 5.65685 0.189512
\(892\) − 10.8284i − 0.362563i
\(893\) − 22.6274i − 0.757198i
\(894\) −3.65685 −0.122304
\(895\) 0 0
\(896\) −2.82843 −0.0944911
\(897\) − 8.48528i − 0.283315i
\(898\) 26.2843i 0.877117i
\(899\) 35.3137 1.17778
\(900\) 0 0
\(901\) 11.0294 0.367444
\(902\) 43.3137i 1.44219i
\(903\) − 27.3137i − 0.908943i
\(904\) −6.48528 −0.215697
\(905\) 0 0
\(906\) 12.0000 0.398673
\(907\) − 8.28427i − 0.275075i −0.990497 0.137537i \(-0.956081\pi\)
0.990497 0.137537i \(-0.0439187\pi\)
\(908\) 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
\(913\) 35.8823i 1.18753i
\(914\) −20.8284 −0.688943
\(915\) 0 0
\(916\) 4.14214 0.136860
\(917\) − 17.3726i − 0.573693i
\(918\) 0.828427i 0.0273422i
\(919\) −41.9411 −1.38351 −0.691755 0.722132i \(-0.743162\pi\)
−0.691755 + 0.722132i \(0.743162\pi\)
\(920\) 0 0
\(921\) −21.6569 −0.713618
\(922\) − 14.0000i − 0.461065i
\(923\) 5.65685i 0.186198i
\(924\) 16.0000 0.526361
\(925\) 0 0
\(926\) −3.79899 −0.124843
\(927\) 1.65685i 0.0544182i
\(928\) − 8.82843i − 0.289807i
\(929\) 33.5980 1.10231 0.551157 0.834402i \(-0.314187\pi\)
0.551157 + 0.834402i \(0.314187\pi\)
\(930\) 0 0
\(931\) 2.82843 0.0926980
\(932\) − 5.51472i − 0.180641i
\(933\) − 24.0000i − 0.785725i
\(934\) −7.31371 −0.239312
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) − 16.6274i − 0.543194i −0.962411 0.271597i \(-0.912448\pi\)
0.962411 0.271597i \(-0.0875518\pi\)
\(938\) − 16.0000i − 0.522419i
\(939\) 30.9706 1.01069
\(940\) 0 0
\(941\) 54.9706 1.79199 0.895995 0.444065i \(-0.146464\pi\)
0.895995 + 0.444065i \(0.146464\pi\)
\(942\) − 5.31371i − 0.173130i
\(943\) 64.9706i 2.11573i
\(944\) 2.34315 0.0762629
\(945\) 0 0
\(946\) 54.6274 1.77609
\(947\) − 30.3431i − 0.986020i −0.870024 0.493010i \(-0.835897\pi\)
0.870024 0.493010i \(-0.164103\pi\)
\(948\) 2.34315i 0.0761018i
\(949\) −14.4853 −0.470212
\(950\) 0 0
\(951\) 25.3137 0.820853
\(952\) 2.34315i 0.0759418i
\(953\) − 27.8579i − 0.902405i −0.892422 0.451202i \(-0.850995\pi\)
0.892422 0.451202i \(-0.149005\pi\)
\(954\) −13.3137 −0.431047
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 49.9411i 1.61437i
\(958\) − 11.3137i − 0.365529i
\(959\) −48.9706 −1.58134
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) − 11.6569i − 0.375832i
\(963\) − 4.00000i − 0.128898i
\(964\) −5.31371 −0.171143
\(965\) 0 0
\(966\) 24.0000 0.772187
\(967\) 7.51472i 0.241657i 0.992673 + 0.120829i \(0.0385551\pi\)
−0.992673 + 0.120829i \(0.961445\pi\)
\(968\) 21.0000i 0.674966i
\(969\) −2.34315 −0.0752727
\(970\) 0 0
\(971\) 15.5147 0.497891 0.248946 0.968517i \(-0.419916\pi\)
0.248946 + 0.968517i \(0.419916\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) − 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 \(0.0426089\pi\)
−0.991054 + 0.133460i \(0.957391\pi\)
\(978\) 11.3137i 0.361773i
\(979\) 88.5685 2.83066
\(980\) 0 0
\(981\) 8.82843 0.281870
\(982\) 38.1421i 1.21716i
\(983\) − 2.34315i − 0.0747347i −0.999302 0.0373674i \(-0.988103\pi\)
0.999302 0.0373674i \(-0.0118972\pi\)
\(984\) 7.65685 0.244092
\(985\) 0 0
\(986\) −7.31371 −0.232916
\(987\) − 22.6274i − 0.720239i
\(988\) − 2.82843i − 0.0899843i
\(989\) 81.9411 2.60558
\(990\) 0 0
\(991\) −42.9117 −1.36313 −0.681567 0.731755i \(-0.738701\pi\)
−0.681567 + 0.731755i \(0.738701\pi\)
\(992\) − 4.00000i − 0.127000i
\(993\) − 8.48528i − 0.269272i
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) 6.34315 0.200990
\(997\) − 61.3137i − 1.94182i −0.239435 0.970912i \(-0.576962\pi\)
0.239435 0.970912i \(-0.423038\pi\)
\(998\) 0.485281i 0.0153613i
\(999\) 11.6569 0.368807
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 1950.2.e.o.1249.2 4
3.2 odd 2 5850.2.e.bk.5149.4 4
5.2 odd 4 390.2.a.h.1.1 2
5.3 odd 4 1950.2.a.bd.1.2 2
5.4 even 2 inner 1950.2.e.o.1249.3 4
15.2 even 4 1170.2.a.o.1.1 2
15.8 even 4 5850.2.a.cl.1.2 2
15.14 odd 2 5850.2.e.bk.5149.1 4
20.7 even 4 3120.2.a.bc.1.2 2
60.47 odd 4 9360.2.a.ch.1.2 2
65.12 odd 4 5070.2.a.bc.1.2 2
65.47 even 4 5070.2.b.q.1351.4 4
65.57 even 4 5070.2.b.q.1351.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.h.1.1 2 5.2 odd 4
1170.2.a.o.1.1 2 15.2 even 4
1950.2.a.bd.1.2 2 5.3 odd 4
1950.2.e.o.1249.2 4 1.1 even 1 trivial
1950.2.e.o.1249.3 4 5.4 even 2 inner
3120.2.a.bc.1.2 2 20.7 even 4
5070.2.a.bc.1.2 2 65.12 odd 4
5070.2.b.q.1351.1 4 65.57 even 4
5070.2.b.q.1351.4 4 65.47 even 4
5850.2.a.cl.1.2 2 15.8 even 4
5850.2.e.bk.5149.1 4 15.14 odd 2
5850.2.e.bk.5149.4 4 3.2 odd 2
9360.2.a.ch.1.2 2 60.47 odd 4