Properties

Label 5070.2.b.l.1351.2
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.l.1351.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -5.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -5.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.00000i q^{11} -1.00000 q^{12} +5.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} +8.00000 q^{17} +1.00000i q^{18} +5.00000i q^{19} -1.00000i q^{20} -5.00000i q^{21} +3.00000 q^{22} +4.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +5.00000i q^{28} -4.00000 q^{29} -1.00000 q^{30} +2.00000i q^{31} +1.00000i q^{32} -3.00000i q^{33} +8.00000i q^{34} +5.00000 q^{35} -1.00000 q^{36} -7.00000i q^{37} -5.00000 q^{38} +1.00000 q^{40} -6.00000i q^{41} +5.00000 q^{42} -6.00000 q^{43} +3.00000i q^{44} +1.00000i q^{45} +4.00000i q^{46} -3.00000i q^{47} +1.00000 q^{48} -18.0000 q^{49} -1.00000i q^{50} +8.00000 q^{51} +1.00000 q^{53} +1.00000i q^{54} +3.00000 q^{55} -5.00000 q^{56} +5.00000i q^{57} -4.00000i q^{58} +12.0000i q^{59} -1.00000i q^{60} +2.00000 q^{61} -2.00000 q^{62} -5.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} -8.00000i q^{67} -8.00000 q^{68} +4.00000 q^{69} +5.00000i q^{70} -2.00000i q^{71} -1.00000i q^{72} +7.00000 q^{74} -1.00000 q^{75} -5.00000i q^{76} -15.0000 q^{77} -2.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +6.00000 q^{82} -8.00000i q^{83} +5.00000i q^{84} +8.00000i q^{85} -6.00000i q^{86} -4.00000 q^{87} -3.00000 q^{88} -11.0000i q^{89} -1.00000 q^{90} -4.00000 q^{92} +2.00000i q^{93} +3.00000 q^{94} -5.00000 q^{95} +1.00000i q^{96} -18.0000i q^{98} -3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{4} + 2q^{9} - 2q^{10} - 2q^{12} + 10q^{14} + 2q^{16} + 16q^{17} + 6q^{22} + 8q^{23} - 2q^{25} + 2q^{27} - 8q^{29} - 2q^{30} + 10q^{35} - 2q^{36} - 10q^{38} + 2q^{40} + 10q^{42} - 12q^{43} + 2q^{48} - 36q^{49} + 16q^{51} + 2q^{53} + 6q^{55} - 10q^{56} + 4q^{61} - 4q^{62} - 2q^{64} + 6q^{66} - 16q^{68} + 8q^{69} + 14q^{74} - 2q^{75} - 30q^{77} - 4q^{79} + 2q^{81} + 12q^{82} - 8q^{87} - 6q^{88} - 2q^{90} - 8q^{92} + 6q^{94} - 10q^{95} + O(q^{100}) \)

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
\(7\) − 5.00000i − 1.88982i −0.327327 0.944911i \(-0.606148\pi\)
0.327327 0.944911i \(-0.393852\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) − 3.00000i − 0.904534i −0.891883 0.452267i \(-0.850615\pi\)
0.891883 0.452267i \(-0.149385\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 5.00000 1.33631
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) 8.00000 1.94029 0.970143 0.242536i \(-0.0779791\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 5.00000i 1.14708i 0.819178 + 0.573539i \(0.194430\pi\)
−0.819178 + 0.573539i \(0.805570\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) − 5.00000i − 1.09109i
\(22\) 3.00000 0.639602
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 5.00000i 0.944911i
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) −1.00000 −0.182574
\(31\) 2.00000i 0.359211i 0.983739 + 0.179605i \(0.0574821\pi\)
−0.983739 + 0.179605i \(0.942518\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 3.00000i − 0.522233i
\(34\) 8.00000i 1.37199i
\(35\) 5.00000 0.845154
\(36\) −1.00000 −0.166667
\(37\) − 7.00000i − 1.15079i −0.817875 0.575396i \(-0.804848\pi\)
0.817875 0.575396i \(-0.195152\pi\)
\(38\) −5.00000 −0.811107
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) − 6.00000i − 0.937043i −0.883452 0.468521i \(-0.844787\pi\)
0.883452 0.468521i \(-0.155213\pi\)
\(42\) 5.00000 0.771517
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 3.00000i 0.452267i
\(45\) 1.00000i 0.149071i
\(46\) 4.00000i 0.589768i
\(47\) − 3.00000i − 0.437595i −0.975770 0.218797i \(-0.929787\pi\)
0.975770 0.218797i \(-0.0702134\pi\)
\(48\) 1.00000 0.144338
\(49\) −18.0000 −2.57143
\(50\) − 1.00000i − 0.141421i
\(51\) 8.00000 1.12022
\(52\) 0 0
\(53\) 1.00000 0.137361 0.0686803 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 3.00000 0.404520
\(56\) −5.00000 −0.668153
\(57\) 5.00000i 0.662266i
\(58\) − 4.00000i − 0.525226i
\(59\) 12.0000i 1.56227i 0.624364 + 0.781133i \(0.285358\pi\)
−0.624364 + 0.781133i \(0.714642\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −2.00000 −0.254000
\(63\) − 5.00000i − 0.629941i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) −8.00000 −0.970143
\(69\) 4.00000 0.481543
\(70\) 5.00000i 0.597614i
\(71\) − 2.00000i − 0.237356i −0.992933 0.118678i \(-0.962134\pi\)
0.992933 0.118678i \(-0.0378657\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 7.00000 0.813733
\(75\) −1.00000 −0.115470
\(76\) − 5.00000i − 0.573539i
\(77\) −15.0000 −1.70941
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) − 8.00000i − 0.878114i −0.898459 0.439057i \(-0.855313\pi\)
0.898459 0.439057i \(-0.144687\pi\)
\(84\) 5.00000i 0.545545i
\(85\) 8.00000i 0.867722i
\(86\) − 6.00000i − 0.646997i
\(87\) −4.00000 −0.428845
\(88\) −3.00000 −0.319801
\(89\) − 11.0000i − 1.16600i −0.812473 0.582999i \(-0.801879\pi\)
0.812473 0.582999i \(-0.198121\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 2.00000i 0.207390i
\(94\) 3.00000 0.309426
\(95\) −5.00000 −0.512989
\(96\) 1.00000i 0.102062i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) − 18.0000i − 1.81827i
\(99\) − 3.00000i − 0.301511i
\(100\) 1.00000 0.100000
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 8.00000i 0.792118i
\(103\) 7.00000 0.689730 0.344865 0.938652i \(-0.387925\pi\)
0.344865 + 0.938652i \(0.387925\pi\)
\(104\) 0 0
\(105\) 5.00000 0.487950
\(106\) 1.00000i 0.0971286i
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −1.00000 −0.0962250
\(109\) − 14.0000i − 1.34096i −0.741929 0.670478i \(-0.766089\pi\)
0.741929 0.670478i \(-0.233911\pi\)
\(110\) 3.00000i 0.286039i
\(111\) − 7.00000i − 0.664411i
\(112\) − 5.00000i − 0.472456i
\(113\) 8.00000 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) −5.00000 −0.468293
\(115\) 4.00000i 0.373002i
\(116\) 4.00000 0.371391
\(117\) 0 0
\(118\) −12.0000 −1.10469
\(119\) − 40.0000i − 3.66679i
\(120\) 1.00000 0.0912871
\(121\) 2.00000 0.181818
\(122\) 2.00000i 0.181071i
\(123\) − 6.00000i − 0.541002i
\(124\) − 2.00000i − 0.179605i
\(125\) − 1.00000i − 0.0894427i
\(126\) 5.00000 0.445435
\(127\) 21.0000 1.86345 0.931724 0.363166i \(-0.118304\pi\)
0.931724 + 0.363166i \(0.118304\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) −19.0000 −1.66004 −0.830019 0.557735i \(-0.811670\pi\)
−0.830019 + 0.557735i \(0.811670\pi\)
\(132\) 3.00000i 0.261116i
\(133\) 25.0000 2.16777
\(134\) 8.00000 0.691095
\(135\) 1.00000i 0.0860663i
\(136\) − 8.00000i − 0.685994i
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 4.00000i 0.340503i
\(139\) 7.00000 0.593732 0.296866 0.954919i \(-0.404058\pi\)
0.296866 + 0.954919i \(0.404058\pi\)
\(140\) −5.00000 −0.422577
\(141\) − 3.00000i − 0.252646i
\(142\) 2.00000 0.167836
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 4.00000i − 0.332182i
\(146\) 0 0
\(147\) −18.0000 −1.48461
\(148\) 7.00000i 0.575396i
\(149\) 2.00000i 0.163846i 0.996639 + 0.0819232i \(0.0261062\pi\)
−0.996639 + 0.0819232i \(0.973894\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) 22.0000i 1.79033i 0.445730 + 0.895167i \(0.352944\pi\)
−0.445730 + 0.895167i \(0.647056\pi\)
\(152\) 5.00000 0.405554
\(153\) 8.00000 0.646762
\(154\) − 15.0000i − 1.20873i
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) 15.0000 1.19713 0.598565 0.801074i \(-0.295738\pi\)
0.598565 + 0.801074i \(0.295738\pi\)
\(158\) − 2.00000i − 0.159111i
\(159\) 1.00000 0.0793052
\(160\) −1.00000 −0.0790569
\(161\) − 20.0000i − 1.57622i
\(162\) 1.00000i 0.0785674i
\(163\) − 20.0000i − 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) 6.00000i 0.468521i
\(165\) 3.00000 0.233550
\(166\) 8.00000 0.620920
\(167\) − 23.0000i − 1.77979i −0.456162 0.889897i \(-0.650776\pi\)
0.456162 0.889897i \(-0.349224\pi\)
\(168\) −5.00000 −0.385758
\(169\) 0 0
\(170\) −8.00000 −0.613572
\(171\) 5.00000i 0.382360i
\(172\) 6.00000 0.457496
\(173\) −5.00000 −0.380143 −0.190071 0.981770i \(-0.560872\pi\)
−0.190071 + 0.981770i \(0.560872\pi\)
\(174\) − 4.00000i − 0.303239i
\(175\) 5.00000i 0.377964i
\(176\) − 3.00000i − 0.226134i
\(177\) 12.0000i 0.901975i
\(178\) 11.0000 0.824485
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) − 4.00000i − 0.294884i
\(185\) 7.00000 0.514650
\(186\) −2.00000 −0.146647
\(187\) − 24.0000i − 1.75505i
\(188\) 3.00000i 0.218797i
\(189\) − 5.00000i − 0.363696i
\(190\) − 5.00000i − 0.362738i
\(191\) 2.00000 0.144715 0.0723575 0.997379i \(-0.476948\pi\)
0.0723575 + 0.997379i \(0.476948\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 24.0000i 1.72756i 0.503871 + 0.863779i \(0.331909\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 18.0000 1.28571
\(197\) − 3.00000i − 0.213741i −0.994273 0.106871i \(-0.965917\pi\)
0.994273 0.106871i \(-0.0340831\pi\)
\(198\) 3.00000 0.213201
\(199\) −22.0000 −1.55954 −0.779769 0.626067i \(-0.784664\pi\)
−0.779769 + 0.626067i \(0.784664\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) − 8.00000i − 0.564276i
\(202\) 8.00000i 0.562878i
\(203\) 20.0000i 1.40372i
\(204\) −8.00000 −0.560112
\(205\) 6.00000 0.419058
\(206\) 7.00000i 0.487713i
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 15.0000 1.03757
\(210\) 5.00000i 0.345033i
\(211\) −15.0000 −1.03264 −0.516321 0.856395i \(-0.672699\pi\)
−0.516321 + 0.856395i \(0.672699\pi\)
\(212\) −1.00000 −0.0686803
\(213\) − 2.00000i − 0.137038i
\(214\) − 6.00000i − 0.410152i
\(215\) − 6.00000i − 0.409197i
\(216\) − 1.00000i − 0.0680414i
\(217\) 10.0000 0.678844
\(218\) 14.0000 0.948200
\(219\) 0 0
\(220\) −3.00000 −0.202260
\(221\) 0 0
\(222\) 7.00000 0.469809
\(223\) 3.00000i 0.200895i 0.994942 + 0.100447i \(0.0320274\pi\)
−0.994942 + 0.100447i \(0.967973\pi\)
\(224\) 5.00000 0.334077
\(225\) −1.00000 −0.0666667
\(226\) 8.00000i 0.532152i
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) − 5.00000i − 0.331133i
\(229\) 14.0000i 0.925146i 0.886581 + 0.462573i \(0.153074\pi\)
−0.886581 + 0.462573i \(0.846926\pi\)
\(230\) −4.00000 −0.263752
\(231\) −15.0000 −0.986928
\(232\) 4.00000i 0.262613i
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 0 0
\(235\) 3.00000 0.195698
\(236\) − 12.0000i − 0.781133i
\(237\) −2.00000 −0.129914
\(238\) 40.0000 2.59281
\(239\) 18.0000i 1.16432i 0.813073 + 0.582162i \(0.197793\pi\)
−0.813073 + 0.582162i \(0.802207\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) − 25.0000i − 1.61039i −0.593009 0.805196i \(-0.702060\pi\)
0.593009 0.805196i \(-0.297940\pi\)
\(242\) 2.00000i 0.128565i
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) − 18.0000i − 1.14998i
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) 2.00000 0.127000
\(249\) − 8.00000i − 0.506979i
\(250\) 1.00000 0.0632456
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 5.00000i 0.314970i
\(253\) − 12.0000i − 0.754434i
\(254\) 21.0000i 1.31766i
\(255\) 8.00000i 0.500979i
\(256\) 1.00000 0.0625000
\(257\) −28.0000 −1.74659 −0.873296 0.487190i \(-0.838022\pi\)
−0.873296 + 0.487190i \(0.838022\pi\)
\(258\) − 6.00000i − 0.373544i
\(259\) −35.0000 −2.17479
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) − 19.0000i − 1.17382i
\(263\) −15.0000 −0.924940 −0.462470 0.886635i \(-0.653037\pi\)
−0.462470 + 0.886635i \(0.653037\pi\)
\(264\) −3.00000 −0.184637
\(265\) 1.00000i 0.0614295i
\(266\) 25.0000i 1.53285i
\(267\) − 11.0000i − 0.673189i
\(268\) 8.00000i 0.488678i
\(269\) 4.00000 0.243884 0.121942 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 4.00000i 0.242983i 0.992592 + 0.121491i \(0.0387677\pi\)
−0.992592 + 0.121491i \(0.961232\pi\)
\(272\) 8.00000 0.485071
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 3.00000i 0.180907i
\(276\) −4.00000 −0.240772
\(277\) 15.0000 0.901263 0.450631 0.892710i \(-0.351199\pi\)
0.450631 + 0.892710i \(0.351199\pi\)
\(278\) 7.00000i 0.419832i
\(279\) 2.00000i 0.119737i
\(280\) − 5.00000i − 0.298807i
\(281\) 18.0000i 1.07379i 0.843649 + 0.536895i \(0.180403\pi\)
−0.843649 + 0.536895i \(0.819597\pi\)
\(282\) 3.00000 0.178647
\(283\) 10.0000 0.594438 0.297219 0.954809i \(-0.403941\pi\)
0.297219 + 0.954809i \(0.403941\pi\)
\(284\) 2.00000i 0.118678i
\(285\) −5.00000 −0.296174
\(286\) 0 0
\(287\) −30.0000 −1.77084
\(288\) 1.00000i 0.0589256i
\(289\) 47.0000 2.76471
\(290\) 4.00000 0.234888
\(291\) 0 0
\(292\) 0 0
\(293\) − 9.00000i − 0.525786i −0.964825 0.262893i \(-0.915323\pi\)
0.964825 0.262893i \(-0.0846766\pi\)
\(294\) − 18.0000i − 1.04978i
\(295\) −12.0000 −0.698667
\(296\) −7.00000 −0.406867
\(297\) − 3.00000i − 0.174078i
\(298\) −2.00000 −0.115857
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 30.0000i 1.72917i
\(302\) −22.0000 −1.26596
\(303\) 8.00000 0.459588
\(304\) 5.00000i 0.286770i
\(305\) 2.00000i 0.114520i
\(306\) 8.00000i 0.457330i
\(307\) − 6.00000i − 0.342438i −0.985233 0.171219i \(-0.945229\pi\)
0.985233 0.171219i \(-0.0547706\pi\)
\(308\) 15.0000 0.854704
\(309\) 7.00000 0.398216
\(310\) − 2.00000i − 0.113592i
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 15.0000i 0.846499i
\(315\) 5.00000 0.281718
\(316\) 2.00000 0.112509
\(317\) 23.0000i 1.29181i 0.763418 + 0.645904i \(0.223520\pi\)
−0.763418 + 0.645904i \(0.776480\pi\)
\(318\) 1.00000i 0.0560772i
\(319\) 12.0000i 0.671871i
\(320\) − 1.00000i − 0.0559017i
\(321\) −6.00000 −0.334887
\(322\) 20.0000 1.11456
\(323\) 40.0000i 2.22566i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 20.0000 1.10770
\(327\) − 14.0000i − 0.774202i
\(328\) −6.00000 −0.331295
\(329\) −15.0000 −0.826977
\(330\) 3.00000i 0.165145i
\(331\) − 4.00000i − 0.219860i −0.993939 0.109930i \(-0.964937\pi\)
0.993939 0.109930i \(-0.0350627\pi\)
\(332\) 8.00000i 0.439057i
\(333\) − 7.00000i − 0.383598i
\(334\) 23.0000 1.25850
\(335\) 8.00000 0.437087
\(336\) − 5.00000i − 0.272772i
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 8.00000 0.434500
\(340\) − 8.00000i − 0.433861i
\(341\) 6.00000 0.324918
\(342\) −5.00000 −0.270369
\(343\) 55.0000i 2.96972i
\(344\) 6.00000i 0.323498i
\(345\) 4.00000i 0.215353i
\(346\) − 5.00000i − 0.268802i
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) 4.00000 0.214423
\(349\) 8.00000i 0.428230i 0.976808 + 0.214115i \(0.0686868\pi\)
−0.976808 + 0.214115i \(0.931313\pi\)
\(350\) −5.00000 −0.267261
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) − 16.0000i − 0.851594i −0.904819 0.425797i \(-0.859994\pi\)
0.904819 0.425797i \(-0.140006\pi\)
\(354\) −12.0000 −0.637793
\(355\) 2.00000 0.106149
\(356\) 11.0000i 0.582999i
\(357\) − 40.0000i − 2.11702i
\(358\) − 4.00000i − 0.211407i
\(359\) − 18.0000i − 0.950004i −0.879985 0.475002i \(-0.842447\pi\)
0.879985 0.475002i \(-0.157553\pi\)
\(360\) 1.00000 0.0527046
\(361\) −6.00000 −0.315789
\(362\) 2.00000i 0.105118i
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) 0 0
\(366\) 2.00000i 0.104542i
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 4.00000 0.208514
\(369\) − 6.00000i − 0.312348i
\(370\) 7.00000i 0.363913i
\(371\) − 5.00000i − 0.259587i
\(372\) − 2.00000i − 0.103695i
\(373\) 38.0000 1.96757 0.983783 0.179364i \(-0.0574041\pi\)
0.983783 + 0.179364i \(0.0574041\pi\)
\(374\) 24.0000 1.24101
\(375\) − 1.00000i − 0.0516398i
\(376\) −3.00000 −0.154713
\(377\) 0 0
\(378\) 5.00000 0.257172
\(379\) 25.0000i 1.28416i 0.766636 + 0.642082i \(0.221929\pi\)
−0.766636 + 0.642082i \(0.778071\pi\)
\(380\) 5.00000 0.256495
\(381\) 21.0000 1.07586
\(382\) 2.00000i 0.102329i
\(383\) 28.0000i 1.43073i 0.698749 + 0.715367i \(0.253740\pi\)
−0.698749 + 0.715367i \(0.746260\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) − 15.0000i − 0.764471i
\(386\) −24.0000 −1.22157
\(387\) −6.00000 −0.304997
\(388\) 0 0
\(389\) 32.0000 1.62246 0.811232 0.584724i \(-0.198797\pi\)
0.811232 + 0.584724i \(0.198797\pi\)
\(390\) 0 0
\(391\) 32.0000 1.61831
\(392\) 18.0000i 0.909137i
\(393\) −19.0000 −0.958423
\(394\) 3.00000 0.151138
\(395\) − 2.00000i − 0.100631i
\(396\) 3.00000i 0.150756i
\(397\) 25.0000i 1.25471i 0.778732 + 0.627357i \(0.215863\pi\)
−0.778732 + 0.627357i \(0.784137\pi\)
\(398\) − 22.0000i − 1.10276i
\(399\) 25.0000 1.25157
\(400\) −1.00000 −0.0500000
\(401\) − 19.0000i − 0.948815i −0.880305 0.474407i \(-0.842662\pi\)
0.880305 0.474407i \(-0.157338\pi\)
\(402\) 8.00000 0.399004
\(403\) 0 0
\(404\) −8.00000 −0.398015
\(405\) 1.00000i 0.0496904i
\(406\) −20.0000 −0.992583
\(407\) −21.0000 −1.04093
\(408\) − 8.00000i − 0.396059i
\(409\) − 25.0000i − 1.23617i −0.786111 0.618085i \(-0.787909\pi\)
0.786111 0.618085i \(-0.212091\pi\)
\(410\) 6.00000i 0.296319i
\(411\) − 12.0000i − 0.591916i
\(412\) −7.00000 −0.344865
\(413\) 60.0000 2.95241
\(414\) 4.00000i 0.196589i
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) 7.00000 0.342791
\(418\) 15.0000i 0.733674i
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) −5.00000 −0.243975
\(421\) − 12.0000i − 0.584844i −0.956289 0.292422i \(-0.905539\pi\)
0.956289 0.292422i \(-0.0944612\pi\)
\(422\) − 15.0000i − 0.730189i
\(423\) − 3.00000i − 0.145865i
\(424\) − 1.00000i − 0.0485643i
\(425\) −8.00000 −0.388057
\(426\) 2.00000 0.0969003
\(427\) − 10.0000i − 0.483934i
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) 6.00000 0.289346
\(431\) − 12.0000i − 0.578020i −0.957326 0.289010i \(-0.906674\pi\)
0.957326 0.289010i \(-0.0933260\pi\)
\(432\) 1.00000 0.0481125
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 10.0000i 0.480015i
\(435\) − 4.00000i − 0.191785i
\(436\) 14.0000i 0.670478i
\(437\) 20.0000i 0.956730i
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) − 3.00000i − 0.143019i
\(441\) −18.0000 −0.857143
\(442\) 0 0
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 7.00000i 0.332205i
\(445\) 11.0000 0.521450
\(446\) −3.00000 −0.142054
\(447\) 2.00000i 0.0945968i
\(448\) 5.00000i 0.236228i
\(449\) − 27.0000i − 1.27421i −0.770778 0.637104i \(-0.780132\pi\)
0.770778 0.637104i \(-0.219868\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) −18.0000 −0.847587
\(452\) −8.00000 −0.376288
\(453\) 22.0000i 1.03365i
\(454\) 0 0
\(455\) 0 0
\(456\) 5.00000 0.234146
\(457\) − 30.0000i − 1.40334i −0.712502 0.701670i \(-0.752438\pi\)
0.712502 0.701670i \(-0.247562\pi\)
\(458\) −14.0000 −0.654177
\(459\) 8.00000 0.373408
\(460\) − 4.00000i − 0.186501i
\(461\) − 8.00000i − 0.372597i −0.982493 0.186299i \(-0.940351\pi\)
0.982493 0.186299i \(-0.0596492\pi\)
\(462\) − 15.0000i − 0.697863i
\(463\) 8.00000i 0.371792i 0.982569 + 0.185896i \(0.0595187\pi\)
−0.982569 + 0.185896i \(0.940481\pi\)
\(464\) −4.00000 −0.185695
\(465\) −2.00000 −0.0927478
\(466\) 14.0000i 0.648537i
\(467\) 24.0000 1.11059 0.555294 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) 0 0
\(469\) −40.0000 −1.84703
\(470\) 3.00000i 0.138380i
\(471\) 15.0000 0.691164
\(472\) 12.0000 0.552345
\(473\) 18.0000i 0.827641i
\(474\) − 2.00000i − 0.0918630i
\(475\) − 5.00000i − 0.229416i
\(476\) 40.0000i 1.83340i
\(477\) 1.00000 0.0457869
\(478\) −18.0000 −0.823301
\(479\) − 28.0000i − 1.27935i −0.768644 0.639676i \(-0.779068\pi\)
0.768644 0.639676i \(-0.220932\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 25.0000 1.13872
\(483\) − 20.0000i − 0.910032i
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 1.00000i 0.0453609i
\(487\) − 37.0000i − 1.67663i −0.545186 0.838315i \(-0.683541\pi\)
0.545186 0.838315i \(-0.316459\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) − 20.0000i − 0.904431i
\(490\) 18.0000 0.813157
\(491\) −21.0000 −0.947717 −0.473858 0.880601i \(-0.657139\pi\)
−0.473858 + 0.880601i \(0.657139\pi\)
\(492\) 6.00000i 0.270501i
\(493\) −32.0000 −1.44121
\(494\) 0 0
\(495\) 3.00000 0.134840
\(496\) 2.00000i 0.0898027i
\(497\) −10.0000 −0.448561
\(498\) 8.00000 0.358489
\(499\) − 4.00000i − 0.179065i −0.995984 0.0895323i \(-0.971463\pi\)
0.995984 0.0895323i \(-0.0285372\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) − 23.0000i − 1.02756i
\(502\) − 15.0000i − 0.669483i
\(503\) −11.0000 −0.490466 −0.245233 0.969464i \(-0.578864\pi\)
−0.245233 + 0.969464i \(0.578864\pi\)
\(504\) −5.00000 −0.222718
\(505\) 8.00000i 0.355995i
\(506\) 12.0000 0.533465
\(507\) 0 0
\(508\) −21.0000 −0.931724
\(509\) − 10.0000i − 0.443242i −0.975133 0.221621i \(-0.928865\pi\)
0.975133 0.221621i \(-0.0711348\pi\)
\(510\) −8.00000 −0.354246
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) 5.00000i 0.220755i
\(514\) − 28.0000i − 1.23503i
\(515\) 7.00000i 0.308457i
\(516\) 6.00000 0.264135
\(517\) −9.00000 −0.395820
\(518\) − 35.0000i − 1.53781i
\(519\) −5.00000 −0.219476
\(520\) 0 0
\(521\) 5.00000 0.219054 0.109527 0.993984i \(-0.465066\pi\)
0.109527 + 0.993984i \(0.465066\pi\)
\(522\) − 4.00000i − 0.175075i
\(523\) −10.0000 −0.437269 −0.218635 0.975807i \(-0.570160\pi\)
−0.218635 + 0.975807i \(0.570160\pi\)
\(524\) 19.0000 0.830019
\(525\) 5.00000i 0.218218i
\(526\) − 15.0000i − 0.654031i
\(527\) 16.0000i 0.696971i
\(528\) − 3.00000i − 0.130558i
\(529\) −7.00000 −0.304348
\(530\) −1.00000 −0.0434372
\(531\) 12.0000i 0.520756i
\(532\) −25.0000 −1.08389
\(533\) 0 0
\(534\) 11.0000 0.476017
\(535\) − 6.00000i − 0.259403i
\(536\) −8.00000 −0.345547
\(537\) −4.00000 −0.172613
\(538\) 4.00000i 0.172452i
\(539\) 54.0000i 2.32594i
\(540\) − 1.00000i − 0.0430331i
\(541\) 34.0000i 1.46177i 0.682498 + 0.730887i \(0.260893\pi\)
−0.682498 + 0.730887i \(0.739107\pi\)
\(542\) −4.00000 −0.171815
\(543\) 2.00000 0.0858282
\(544\) 8.00000i 0.342997i
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) 6.00000 0.256541 0.128271 0.991739i \(-0.459057\pi\)
0.128271 + 0.991739i \(0.459057\pi\)
\(548\) 12.0000i 0.512615i
\(549\) 2.00000 0.0853579
\(550\) −3.00000 −0.127920
\(551\) − 20.0000i − 0.852029i
\(552\) − 4.00000i − 0.170251i
\(553\) 10.0000i 0.425243i
\(554\) 15.0000i 0.637289i
\(555\) 7.00000 0.297133
\(556\) −7.00000 −0.296866
\(557\) 15.0000i 0.635570i 0.948163 + 0.317785i \(0.102939\pi\)
−0.948163 + 0.317785i \(0.897061\pi\)
\(558\) −2.00000 −0.0846668
\(559\) 0 0
\(560\) 5.00000 0.211289
\(561\) − 24.0000i − 1.01328i
\(562\) −18.0000 −0.759284
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 3.00000i 0.126323i
\(565\) 8.00000i 0.336563i
\(566\) 10.0000i 0.420331i
\(567\) − 5.00000i − 0.209980i
\(568\) −2.00000 −0.0839181
\(569\) 15.0000 0.628833 0.314416 0.949285i \(-0.398191\pi\)
0.314416 + 0.949285i \(0.398191\pi\)
\(570\) − 5.00000i − 0.209427i
\(571\) 33.0000 1.38101 0.690504 0.723329i \(-0.257389\pi\)
0.690504 + 0.723329i \(0.257389\pi\)
\(572\) 0 0
\(573\) 2.00000 0.0835512
\(574\) − 30.0000i − 1.25218i
\(575\) −4.00000 −0.166812
\(576\) −1.00000 −0.0416667
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 47.0000i 1.95494i
\(579\) 24.0000i 0.997406i
\(580\) 4.00000i 0.166091i
\(581\) −40.0000 −1.65948
\(582\) 0 0
\(583\) − 3.00000i − 0.124247i
\(584\) 0 0
\(585\) 0 0
\(586\) 9.00000 0.371787
\(587\) − 18.0000i − 0.742940i −0.928445 0.371470i \(-0.878854\pi\)
0.928445 0.371470i \(-0.121146\pi\)
\(588\) 18.0000 0.742307
\(589\) −10.0000 −0.412043
\(590\) − 12.0000i − 0.494032i
\(591\) − 3.00000i − 0.123404i
\(592\) − 7.00000i − 0.287698i
\(593\) 20.0000i 0.821302i 0.911793 + 0.410651i \(0.134698\pi\)
−0.911793 + 0.410651i \(0.865302\pi\)
\(594\) 3.00000 0.123091
\(595\) 40.0000 1.63984
\(596\) − 2.00000i − 0.0819232i
\(597\) −22.0000 −0.900400
\(598\) 0 0
\(599\) 34.0000 1.38920 0.694601 0.719395i \(-0.255581\pi\)
0.694601 + 0.719395i \(0.255581\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) 37.0000 1.50926 0.754631 0.656150i \(-0.227816\pi\)
0.754631 + 0.656150i \(0.227816\pi\)
\(602\) −30.0000 −1.22271
\(603\) − 8.00000i − 0.325785i
\(604\) − 22.0000i − 0.895167i
\(605\) 2.00000i 0.0813116i
\(606\) 8.00000i 0.324978i
\(607\) −29.0000 −1.17707 −0.588537 0.808470i \(-0.700296\pi\)
−0.588537 + 0.808470i \(0.700296\pi\)
\(608\) −5.00000 −0.202777
\(609\) 20.0000i 0.810441i
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) −8.00000 −0.323381
\(613\) − 25.0000i − 1.00974i −0.863195 0.504870i \(-0.831540\pi\)
0.863195 0.504870i \(-0.168460\pi\)
\(614\) 6.00000 0.242140
\(615\) 6.00000 0.241943
\(616\) 15.0000i 0.604367i
\(617\) 14.0000i 0.563619i 0.959470 + 0.281809i \(0.0909346\pi\)
−0.959470 + 0.281809i \(0.909065\pi\)
\(618\) 7.00000i 0.281581i
\(619\) 17.0000i 0.683288i 0.939829 + 0.341644i \(0.110984\pi\)
−0.939829 + 0.341644i \(0.889016\pi\)
\(620\) 2.00000 0.0803219
\(621\) 4.00000 0.160514
\(622\) 12.0000i 0.481156i
\(623\) −55.0000 −2.20353
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 6.00000i − 0.239808i
\(627\) 15.0000 0.599042
\(628\) −15.0000 −0.598565
\(629\) − 56.0000i − 2.23287i
\(630\) 5.00000i 0.199205i
\(631\) 12.0000i 0.477712i 0.971055 + 0.238856i \(0.0767725\pi\)
−0.971055 + 0.238856i \(0.923228\pi\)
\(632\) 2.00000i 0.0795557i
\(633\) −15.0000 −0.596196
\(634\) −23.0000 −0.913447
\(635\) 21.0000i 0.833360i
\(636\) −1.00000 −0.0396526
\(637\) 0 0
\(638\) −12.0000 −0.475085
\(639\) − 2.00000i − 0.0791188i
\(640\) 1.00000 0.0395285
\(641\) −27.0000 −1.06644 −0.533218 0.845978i \(-0.679017\pi\)
−0.533218 + 0.845978i \(0.679017\pi\)
\(642\) − 6.00000i − 0.236801i
\(643\) 44.0000i 1.73519i 0.497271 + 0.867595i \(0.334335\pi\)
−0.497271 + 0.867595i \(0.665665\pi\)
\(644\) 20.0000i 0.788110i
\(645\) − 6.00000i − 0.236250i
\(646\) −40.0000 −1.57378
\(647\) −3.00000 −0.117942 −0.0589711 0.998260i \(-0.518782\pi\)
−0.0589711 + 0.998260i \(0.518782\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) 20.0000i 0.783260i
\(653\) 27.0000 1.05659 0.528296 0.849060i \(-0.322831\pi\)
0.528296 + 0.849060i \(0.322831\pi\)
\(654\) 14.0000 0.547443
\(655\) − 19.0000i − 0.742391i
\(656\) − 6.00000i − 0.234261i
\(657\) 0 0
\(658\) − 15.0000i − 0.584761i
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) −3.00000 −0.116775
\(661\) − 10.0000i − 0.388955i −0.980907 0.194477i \(-0.937699\pi\)
0.980907 0.194477i \(-0.0623011\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) −8.00000 −0.310460
\(665\) 25.0000i 0.969458i
\(666\) 7.00000 0.271244
\(667\) −16.0000 −0.619522
\(668\) 23.0000i 0.889897i
\(669\) 3.00000i 0.115987i
\(670\) 8.00000i 0.309067i
\(671\) − 6.00000i − 0.231627i
\(672\) 5.00000 0.192879
\(673\) 32.0000 1.23351 0.616755 0.787155i \(-0.288447\pi\)
0.616755 + 0.787155i \(0.288447\pi\)
\(674\) − 14.0000i − 0.539260i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) 8.00000i 0.307238i
\(679\) 0 0
\(680\) 8.00000 0.306786
\(681\) 0 0
\(682\) 6.00000i 0.229752i
\(683\) 30.0000i 1.14792i 0.818884 + 0.573959i \(0.194593\pi\)
−0.818884 + 0.573959i \(0.805407\pi\)
\(684\) − 5.00000i − 0.191180i
\(685\) 12.0000 0.458496
\(686\) −55.0000 −2.09991
\(687\) 14.0000i 0.534133i
\(688\) −6.00000 −0.228748
\(689\) 0 0
\(690\) −4.00000 −0.152277
\(691\) − 17.0000i − 0.646710i −0.946278 0.323355i \(-0.895189\pi\)
0.946278 0.323355i \(-0.104811\pi\)
\(692\) 5.00000 0.190071
\(693\) −15.0000 −0.569803
\(694\) − 16.0000i − 0.607352i
\(695\) 7.00000i 0.265525i
\(696\) 4.00000i 0.151620i
\(697\) − 48.0000i − 1.81813i
\(698\) −8.00000 −0.302804
\(699\) 14.0000 0.529529
\(700\) − 5.00000i − 0.188982i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 35.0000 1.32005
\(704\) 3.00000i 0.113067i
\(705\) 3.00000 0.112987
\(706\) 16.0000 0.602168
\(707\) − 40.0000i − 1.50435i
\(708\) − 12.0000i − 0.450988i
\(709\) 32.0000i 1.20179i 0.799330 + 0.600893i \(0.205188\pi\)
−0.799330 + 0.600893i \(0.794812\pi\)
\(710\) 2.00000i 0.0750587i
\(711\) −2.00000 −0.0750059
\(712\) −11.0000 −0.412242
\(713\) 8.00000i 0.299602i
\(714\) 40.0000 1.49696
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 18.0000i 0.672222i
\(718\) 18.0000 0.671754
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) − 35.0000i − 1.30347i
\(722\) − 6.00000i − 0.223297i
\(723\) − 25.0000i − 0.929760i
\(724\) −2.00000 −0.0743294
\(725\) 4.00000 0.148556
\(726\) 2.00000i 0.0742270i
\(727\) 11.0000 0.407967 0.203984 0.978974i \(-0.434611\pi\)
0.203984 + 0.978974i \(0.434611\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −48.0000 −1.77534
\(732\) −2.00000 −0.0739221
\(733\) 43.0000i 1.58824i 0.607760 + 0.794121i \(0.292068\pi\)
−0.607760 + 0.794121i \(0.707932\pi\)
\(734\) − 8.00000i − 0.295285i
\(735\) − 18.0000i − 0.663940i
\(736\) 4.00000i 0.147442i
\(737\) −24.0000 −0.884051
\(738\) 6.00000 0.220863
\(739\) 19.0000i 0.698926i 0.936950 + 0.349463i \(0.113636\pi\)
−0.936950 + 0.349463i \(0.886364\pi\)
\(740\) −7.00000 −0.257325
\(741\) 0 0
\(742\) 5.00000 0.183556
\(743\) 16.0000i 0.586983i 0.955962 + 0.293492i \(0.0948173\pi\)
−0.955962 + 0.293492i \(0.905183\pi\)
\(744\) 2.00000 0.0733236
\(745\) −2.00000 −0.0732743
\(746\) 38.0000i 1.39128i
\(747\) − 8.00000i − 0.292705i
\(748\) 24.0000i 0.877527i
\(749\) 30.0000i 1.09618i
\(750\) 1.00000 0.0365148
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) − 3.00000i − 0.109399i
\(753\) −15.0000 −0.546630
\(754\) 0 0
\(755\) −22.0000 −0.800662
\(756\) 5.00000i 0.181848i
\(757\) 17.0000 0.617876 0.308938 0.951082i \(-0.400027\pi\)
0.308938 + 0.951082i \(0.400027\pi\)
\(758\) −25.0000 −0.908041
\(759\) − 12.0000i − 0.435572i
\(760\) 5.00000i 0.181369i
\(761\) 9.00000i 0.326250i 0.986605 + 0.163125i \(0.0521573\pi\)
−0.986605 + 0.163125i \(0.947843\pi\)
\(762\) 21.0000i 0.760750i
\(763\) −70.0000 −2.53417
\(764\) −2.00000 −0.0723575
\(765\) 8.00000i 0.289241i
\(766\) −28.0000 −1.01168
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 34.0000i 1.22607i 0.790055 + 0.613036i \(0.210052\pi\)
−0.790055 + 0.613036i \(0.789948\pi\)
\(770\) 15.0000 0.540562
\(771\) −28.0000 −1.00840
\(772\) − 24.0000i − 0.863779i
\(773\) − 1.00000i − 0.0359675i −0.999838 0.0179838i \(-0.994275\pi\)
0.999838 0.0179838i \(-0.00572471\pi\)
\(774\) − 6.00000i − 0.215666i
\(775\) − 2.00000i − 0.0718421i
\(776\) 0 0
\(777\) −35.0000 −1.25562
\(778\) 32.0000i 1.14726i
\(779\) 30.0000 1.07486
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) 32.0000i 1.14432i
\(783\) −4.00000 −0.142948
\(784\) −18.0000 −0.642857
\(785\) 15.0000i 0.535373i
\(786\) − 19.0000i − 0.677708i
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) 3.00000i 0.106871i
\(789\) −15.0000 −0.534014
\(790\) 2.00000 0.0711568
\(791\) − 40.0000i − 1.42224i
\(792\) −3.00000 −0.106600
\(793\) 0 0
\(794\) −25.0000 −0.887217
\(795\) 1.00000i 0.0354663i
\(796\) 22.0000 0.779769
\(797\) 10.0000 0.354218 0.177109 0.984191i \(-0.443325\pi\)
0.177109 + 0.984191i \(0.443325\pi\)
\(798\) 25.0000i 0.884990i
\(799\) − 24.0000i − 0.849059i
\(800\) − 1.00000i − 0.0353553i
\(801\) − 11.0000i − 0.388666i
\(802\) 19.0000 0.670913
\(803\) 0 0
\(804\) 8.00000i 0.282138i
\(805\) 20.0000 0.704907
\(806\) 0 0
\(807\) 4.00000 0.140807
\(808\) − 8.00000i − 0.281439i
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) −1.00000 −0.0351364
\(811\) − 25.0000i − 0.877869i −0.898519 0.438934i \(-0.855356\pi\)
0.898519 0.438934i \(-0.144644\pi\)
\(812\) − 20.0000i − 0.701862i
\(813\) 4.00000i 0.140286i
\(814\) − 21.0000i − 0.736050i
\(815\) 20.0000 0.700569
\(816\) 8.00000 0.280056
\(817\) − 30.0000i − 1.04957i
\(818\) 25.0000 0.874105
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) 22.0000i 0.767805i 0.923374 + 0.383903i \(0.125420\pi\)
−0.923374 + 0.383903i \(0.874580\pi\)
\(822\) 12.0000 0.418548
\(823\) 35.0000 1.22002 0.610012 0.792392i \(-0.291165\pi\)
0.610012 + 0.792392i \(0.291165\pi\)
\(824\) − 7.00000i − 0.243857i
\(825\) 3.00000i 0.104447i
\(826\) 60.0000i 2.08767i
\(827\) 6.00000i 0.208640i 0.994544 + 0.104320i \(0.0332667\pi\)
−0.994544 + 0.104320i \(0.966733\pi\)
\(828\) −4.00000 −0.139010
\(829\) 16.0000 0.555703 0.277851 0.960624i \(-0.410378\pi\)
0.277851 + 0.960624i \(0.410378\pi\)
\(830\) 8.00000i 0.277684i
\(831\) 15.0000 0.520344
\(832\) 0 0
\(833\) −144.000 −4.98930
\(834\) 7.00000i 0.242390i
\(835\) 23.0000 0.795948
\(836\) −15.0000 −0.518786
\(837\) 2.00000i 0.0691301i
\(838\) − 12.0000i − 0.414533i
\(839\) − 14.0000i − 0.483334i −0.970359 0.241667i \(-0.922306\pi\)
0.970359 0.241667i \(-0.0776941\pi\)
\(840\) − 5.00000i − 0.172516i
\(841\) −13.0000 −0.448276
\(842\) 12.0000 0.413547
\(843\) 18.0000i 0.619953i
\(844\) 15.0000 0.516321
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) − 10.0000i − 0.343604i
\(848\) 1.00000 0.0343401
\(849\) 10.0000 0.343199
\(850\) − 8.00000i − 0.274398i
\(851\) − 28.0000i − 0.959828i
\(852\) 2.00000i 0.0685189i
\(853\) − 26.0000i − 0.890223i −0.895475 0.445112i \(-0.853164\pi\)
0.895475 0.445112i \(-0.146836\pi\)
\(854\) 10.0000 0.342193
\(855\) −5.00000 −0.170996
\(856\) 6.00000i 0.205076i
\(857\) −38.0000 −1.29806 −0.649028 0.760765i \(-0.724824\pi\)
−0.649028 + 0.760765i \(0.724824\pi\)
\(858\) 0 0
\(859\) 3.00000 0.102359 0.0511793 0.998689i \(-0.483702\pi\)
0.0511793 + 0.998689i \(0.483702\pi\)
\(860\) 6.00000i 0.204598i
\(861\) −30.0000 −1.02240
\(862\) 12.0000 0.408722
\(863\) 24.0000i 0.816970i 0.912765 + 0.408485i \(0.133943\pi\)
−0.912765 + 0.408485i \(0.866057\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) − 5.00000i − 0.170005i
\(866\) − 16.0000i − 0.543702i
\(867\) 47.0000 1.59620
\(868\) −10.0000 −0.339422
\(869\) 6.00000i 0.203536i
\(870\) 4.00000 0.135613
\(871\) 0 0
\(872\) −14.0000 −0.474100
\(873\) 0 0
\(874\) −20.0000 −0.676510
\(875\) −5.00000 −0.169031
\(876\) 0 0
\(877\) 54.0000i 1.82345i 0.410801 + 0.911725i \(0.365249\pi\)
−0.410801 + 0.911725i \(0.634751\pi\)
\(878\) − 10.0000i − 0.337484i
\(879\) − 9.00000i − 0.303562i
\(880\) 3.00000 0.101130
\(881\) 15.0000 0.505363 0.252681 0.967550i \(-0.418688\pi\)
0.252681 + 0.967550i \(0.418688\pi\)
\(882\) − 18.0000i − 0.606092i
\(883\) −30.0000 −1.00958 −0.504790 0.863242i \(-0.668430\pi\)
−0.504790 + 0.863242i \(0.668430\pi\)
\(884\) 0 0
\(885\) −12.0000 −0.403376
\(886\) 6.00000i 0.201574i
\(887\) 41.0000 1.37665 0.688323 0.725405i \(-0.258347\pi\)
0.688323 + 0.725405i \(0.258347\pi\)
\(888\) −7.00000 −0.234905
\(889\) − 105.000i − 3.52159i
\(890\) 11.0000i 0.368721i
\(891\) − 3.00000i − 0.100504i
\(892\) − 3.00000i − 0.100447i
\(893\) 15.0000 0.501956
\(894\) −2.00000 −0.0668900
\(895\) − 4.00000i − 0.133705i
\(896\) −5.00000 −0.167038
\(897\) 0 0
\(898\) 27.0000 0.901002
\(899\) − 8.00000i − 0.266815i
\(900\) 1.00000 0.0333333
\(901\) 8.00000 0.266519
\(902\) − 18.0000i − 0.599334i
\(903\) 30.0000i 0.998337i
\(904\) − 8.00000i − 0.266076i
\(905\) 2.00000i 0.0664822i
\(906\) −22.0000 −0.730901
\(907\) −10.0000 −0.332045 −0.166022 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) 0 0
\(909\) 8.00000 0.265343
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 5.00000i 0.165567i
\(913\) −24.0000 −0.794284
\(914\) 30.0000 0.992312
\(915\) 2.00000i 0.0661180i
\(916\) − 14.0000i − 0.462573i
\(917\) 95.0000i 3.13718i
\(918\) 8.00000i 0.264039i
\(919\) −2.00000 −0.0659739 −0.0329870 0.999456i \(-0.510502\pi\)
−0.0329870 + 0.999456i \(0.510502\pi\)
\(920\) 4.00000 0.131876
\(921\) − 6.00000i − 0.197707i
\(922\) 8.00000 0.263466
\(923\) 0 0
\(924\) 15.0000 0.493464
\(925\) 7.00000i 0.230159i
\(926\) −8.00000 −0.262896
\(927\) 7.00000 0.229910
\(928\) − 4.00000i − 0.131306i
\(929\) − 2.00000i − 0.0656179i −0.999462 0.0328089i \(-0.989555\pi\)
0.999462 0.0328089i \(-0.0104453\pi\)
\(930\) − 2.00000i − 0.0655826i
\(931\) − 90.0000i − 2.94963i
\(932\) −14.0000 −0.458585
\(933\) 12.0000 0.392862
\(934\) 24.0000i 0.785304i
\(935\) 24.0000 0.784884
\(936\) 0 0
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) − 40.0000i − 1.30605i
\(939\) −6.00000 −0.195803
\(940\) −3.00000 −0.0978492
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 15.0000i 0.488726i
\(943\) − 24.0000i − 0.781548i
\(944\) 12.0000i 0.390567i
\(945\) 5.00000 0.162650
\(946\) −18.0000 −0.585230
\(947\) 58.0000i 1.88475i 0.334563 + 0.942373i \(0.391411\pi\)
−0.334563 + 0.942373i \(0.608589\pi\)
\(948\) 2.00000 0.0649570
\(949\) 0 0
\(950\) 5.00000 0.162221
\(951\) 23.0000i 0.745826i
\(952\) −40.0000 −1.29641
\(953\) 54.0000 1.74923 0.874616 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 1.00000i 0.0323762i
\(955\) 2.00000i 0.0647185i
\(956\) − 18.0000i − 0.582162i
\(957\) 12.0000i 0.387905i
\(958\) 28.0000 0.904639
\(959\) −60.0000 −1.93750
\(960\) − 1.00000i − 0.0322749i
\(961\) 27.0000 0.870968
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 25.0000i 0.805196i
\(965\) −24.0000 −0.772587
\(966\) 20.0000 0.643489
\(967\) 31.0000i 0.996893i 0.866921 + 0.498446i \(0.166096\pi\)
−0.866921 + 0.498446i \(0.833904\pi\)
\(968\) − 2.00000i − 0.0642824i
\(969\) 40.0000i 1.28499i
\(970\) 0 0
\(971\) 15.0000 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) −1.00000 −0.0320750
\(973\) − 35.0000i − 1.12205i
\(974\) 37.0000 1.18556
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) − 20.0000i − 0.639857i −0.947442 0.319928i \(-0.896341\pi\)
0.947442 0.319928i \(-0.103659\pi\)
\(978\) 20.0000 0.639529
\(979\) −33.0000 −1.05468
\(980\) 18.0000i 0.574989i
\(981\) − 14.0000i − 0.446986i
\(982\) − 21.0000i − 0.670137i
\(983\) 23.0000i 0.733586i 0.930303 + 0.366793i \(0.119544\pi\)
−0.930303 + 0.366793i \(0.880456\pi\)
\(984\) −6.00000 −0.191273
\(985\) 3.00000 0.0955879
\(986\) − 32.0000i − 1.01909i
\(987\) −15.0000 −0.477455
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) 3.00000i 0.0953463i
\(991\) 10.0000 0.317660 0.158830 0.987306i \(-0.449228\pi\)
0.158830 + 0.987306i \(0.449228\pi\)
\(992\) −2.00000 −0.0635001
\(993\) − 4.00000i − 0.126936i
\(994\) − 10.0000i − 0.317181i
\(995\) − 22.0000i − 0.697447i
\(996\) 8.00000i 0.253490i
\(997\) −1.00000 −0.0316703 −0.0158352 0.999875i \(-0.505041\pi\)
−0.0158352 + 0.999875i \(0.505041\pi\)
\(998\) 4.00000 0.126618
\(999\) − 7.00000i − 0.221470i
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.l.1351.2 2
13.5 odd 4 5070.2.a.x.1.1 1
13.7 odd 12 390.2.i.d.211.1 yes 2
13.8 odd 4 5070.2.a.i.1.1 1
13.11 odd 12 390.2.i.d.61.1 2
13.12 even 2 inner 5070.2.b.l.1351.1 2
39.11 even 12 1170.2.i.g.451.1 2
39.20 even 12 1170.2.i.g.991.1 2
65.7 even 12 1950.2.z.j.1849.1 4
65.24 odd 12 1950.2.i.i.451.1 2
65.33 even 12 1950.2.z.j.1849.2 4
65.37 even 12 1950.2.z.j.1699.2 4
65.59 odd 12 1950.2.i.i.601.1 2
65.63 even 12 1950.2.z.j.1699.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.d.61.1 2 13.11 odd 12
390.2.i.d.211.1 yes 2 13.7 odd 12
1170.2.i.g.451.1 2 39.11 even 12
1170.2.i.g.991.1 2 39.20 even 12
1950.2.i.i.451.1 2 65.24 odd 12
1950.2.i.i.601.1 2 65.59 odd 12
1950.2.z.j.1699.1 4 65.63 even 12
1950.2.z.j.1699.2 4 65.37 even 12
1950.2.z.j.1849.1 4 65.7 even 12
1950.2.z.j.1849.2 4 65.33 even 12
5070.2.a.i.1.1 1 13.8 odd 4
5070.2.a.x.1.1 1 13.5 odd 4
5070.2.b.l.1351.1 2 13.12 even 2 inner
5070.2.b.l.1351.2 2 1.1 even 1 trivial