Properties

Label 3450.2.a.bq.1.3
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.363328\) of defining polynomial
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.50466 q^{11} +1.00000 q^{12} +2.72666 q^{13} -2.00000 q^{14} +1.00000 q^{16} -0.726656 q^{17} -1.00000 q^{18} -7.78734 q^{19} +2.00000 q^{21} -3.50466 q^{22} +1.00000 q^{23} -1.00000 q^{24} -2.72666 q^{26} +1.00000 q^{27} +2.00000 q^{28} -2.00000 q^{29} +8.28267 q^{31} -1.00000 q^{32} +3.50466 q^{33} +0.726656 q^{34} +1.00000 q^{36} +9.50466 q^{37} +7.78734 q^{38} +2.72666 q^{39} +0.726656 q^{41} -2.00000 q^{42} +5.50466 q^{43} +3.50466 q^{44} -1.00000 q^{46} +2.72666 q^{47} +1.00000 q^{48} -3.00000 q^{49} -0.726656 q^{51} +2.72666 q^{52} -0.231321 q^{53} -1.00000 q^{54} -2.00000 q^{56} -7.78734 q^{57} +2.00000 q^{58} +9.29200 q^{59} +4.05135 q^{61} -8.28267 q^{62} +2.00000 q^{63} +1.00000 q^{64} -3.50466 q^{66} +4.05135 q^{67} -0.726656 q^{68} +1.00000 q^{69} -11.7360 q^{71} -1.00000 q^{72} +3.71733 q^{73} -9.50466 q^{74} -7.78734 q^{76} +7.00933 q^{77} -2.72666 q^{78} -9.73599 q^{79} +1.00000 q^{81} -0.726656 q^{82} -13.5233 q^{83} +2.00000 q^{84} -5.50466 q^{86} -2.00000 q^{87} -3.50466 q^{88} -9.55602 q^{89} +5.45331 q^{91} +1.00000 q^{92} +8.28267 q^{93} -2.72666 q^{94} -1.00000 q^{96} +4.54669 q^{97} +3.00000 q^{98} +3.50466 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{6} + 6q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{6} + 6q^{7} - 3q^{8} + 3q^{9} + 3q^{12} + 4q^{13} - 6q^{14} + 3q^{16} + 2q^{17} - 3q^{18} + 4q^{19} + 6q^{21} + 3q^{23} - 3q^{24} - 4q^{26} + 3q^{27} + 6q^{28} - 6q^{29} + 8q^{31} - 3q^{32} - 2q^{34} + 3q^{36} + 18q^{37} - 4q^{38} + 4q^{39} - 2q^{41} - 6q^{42} + 6q^{43} - 3q^{46} + 4q^{47} + 3q^{48} - 9q^{49} + 2q^{51} + 4q^{52} + 14q^{53} - 3q^{54} - 6q^{56} + 4q^{57} + 6q^{58} - 10q^{59} + 10q^{61} - 8q^{62} + 6q^{63} + 3q^{64} + 10q^{67} + 2q^{68} + 3q^{69} - 10q^{71} - 3q^{72} + 28q^{73} - 18q^{74} + 4q^{76} - 4q^{78} - 4q^{79} + 3q^{81} + 2q^{82} + 12q^{83} + 6q^{84} - 6q^{86} - 6q^{87} - 16q^{89} + 8q^{91} + 3q^{92} + 8q^{93} - 4q^{94} - 3q^{96} + 22q^{97} + 9q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.50466 1.05670 0.528348 0.849028i \(-0.322812\pi\)
0.528348 + 0.849028i \(0.322812\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.72666 0.756238 0.378119 0.925757i \(-0.376571\pi\)
0.378119 + 0.925757i \(0.376571\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.726656 −0.176240 −0.0881200 0.996110i \(-0.528086\pi\)
−0.0881200 + 0.996110i \(0.528086\pi\)
\(18\) −1.00000 −0.235702
\(19\) −7.78734 −1.78654 −0.893269 0.449523i \(-0.851594\pi\)
−0.893269 + 0.449523i \(0.851594\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) −3.50466 −0.747197
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −2.72666 −0.534741
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 8.28267 1.48761 0.743806 0.668396i \(-0.233019\pi\)
0.743806 + 0.668396i \(0.233019\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.50466 0.610084
\(34\) 0.726656 0.124621
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 9.50466 1.56256 0.781279 0.624182i \(-0.214568\pi\)
0.781279 + 0.624182i \(0.214568\pi\)
\(38\) 7.78734 1.26327
\(39\) 2.72666 0.436614
\(40\) 0 0
\(41\) 0.726656 0.113485 0.0567423 0.998389i \(-0.481929\pi\)
0.0567423 + 0.998389i \(0.481929\pi\)
\(42\) −2.00000 −0.308607
\(43\) 5.50466 0.839453 0.419727 0.907651i \(-0.362126\pi\)
0.419727 + 0.907651i \(0.362126\pi\)
\(44\) 3.50466 0.528348
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 2.72666 0.397724 0.198862 0.980028i \(-0.436275\pi\)
0.198862 + 0.980028i \(0.436275\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −0.726656 −0.101752
\(52\) 2.72666 0.378119
\(53\) −0.231321 −0.0317744 −0.0158872 0.999874i \(-0.505057\pi\)
−0.0158872 + 0.999874i \(0.505057\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) −7.78734 −1.03146
\(58\) 2.00000 0.262613
\(59\) 9.29200 1.20972 0.604858 0.796334i \(-0.293230\pi\)
0.604858 + 0.796334i \(0.293230\pi\)
\(60\) 0 0
\(61\) 4.05135 0.518722 0.259361 0.965780i \(-0.416488\pi\)
0.259361 + 0.965780i \(0.416488\pi\)
\(62\) −8.28267 −1.05190
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.50466 −0.431394
\(67\) 4.05135 0.494951 0.247476 0.968894i \(-0.420399\pi\)
0.247476 + 0.968894i \(0.420399\pi\)
\(68\) −0.726656 −0.0881200
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −11.7360 −1.39281 −0.696403 0.717651i \(-0.745217\pi\)
−0.696403 + 0.717651i \(0.745217\pi\)
\(72\) −1.00000 −0.117851
\(73\) 3.71733 0.435080 0.217540 0.976051i \(-0.430197\pi\)
0.217540 + 0.976051i \(0.430197\pi\)
\(74\) −9.50466 −1.10489
\(75\) 0 0
\(76\) −7.78734 −0.893269
\(77\) 7.00933 0.798787
\(78\) −2.72666 −0.308733
\(79\) −9.73599 −1.09538 −0.547692 0.836680i \(-0.684493\pi\)
−0.547692 + 0.836680i \(0.684493\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −0.726656 −0.0802458
\(83\) −13.5233 −1.48438 −0.742189 0.670191i \(-0.766212\pi\)
−0.742189 + 0.670191i \(0.766212\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −5.50466 −0.593583
\(87\) −2.00000 −0.214423
\(88\) −3.50466 −0.373598
\(89\) −9.55602 −1.01294 −0.506468 0.862259i \(-0.669049\pi\)
−0.506468 + 0.862259i \(0.669049\pi\)
\(90\) 0 0
\(91\) 5.45331 0.571663
\(92\) 1.00000 0.104257
\(93\) 8.28267 0.858873
\(94\) −2.72666 −0.281233
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 4.54669 0.461646 0.230823 0.972996i \(-0.425858\pi\)
0.230823 + 0.972996i \(0.425858\pi\)
\(98\) 3.00000 0.303046
\(99\) 3.50466 0.352232
\(100\) 0 0
\(101\) 3.45331 0.343617 0.171809 0.985130i \(-0.445039\pi\)
0.171809 + 0.985130i \(0.445039\pi\)
\(102\) 0.726656 0.0719497
\(103\) 14.4626 1.42505 0.712523 0.701649i \(-0.247552\pi\)
0.712523 + 0.701649i \(0.247552\pi\)
\(104\) −2.72666 −0.267371
\(105\) 0 0
\(106\) 0.231321 0.0224679
\(107\) −13.0607 −1.26262 −0.631312 0.775529i \(-0.717483\pi\)
−0.631312 + 0.775529i \(0.717483\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.9580 −1.04958 −0.524792 0.851231i \(-0.675857\pi\)
−0.524792 + 0.851231i \(0.675857\pi\)
\(110\) 0 0
\(111\) 9.50466 0.902143
\(112\) 2.00000 0.188982
\(113\) 11.7360 1.10403 0.552014 0.833835i \(-0.313859\pi\)
0.552014 + 0.833835i \(0.313859\pi\)
\(114\) 7.78734 0.729351
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 2.72666 0.252079
\(118\) −9.29200 −0.855398
\(119\) −1.45331 −0.133225
\(120\) 0 0
\(121\) 1.28267 0.116607
\(122\) −4.05135 −0.366792
\(123\) 0.726656 0.0655204
\(124\) 8.28267 0.743806
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 12.2827 1.08991 0.544955 0.838465i \(-0.316547\pi\)
0.544955 + 0.838465i \(0.316547\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 5.50466 0.484659
\(130\) 0 0
\(131\) −14.8480 −1.29728 −0.648639 0.761097i \(-0.724661\pi\)
−0.648639 + 0.761097i \(0.724661\pi\)
\(132\) 3.50466 0.305042
\(133\) −15.5747 −1.35050
\(134\) −4.05135 −0.349983
\(135\) 0 0
\(136\) 0.726656 0.0623103
\(137\) 3.27334 0.279661 0.139830 0.990175i \(-0.455344\pi\)
0.139830 + 0.990175i \(0.455344\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 14.0187 1.18905 0.594524 0.804078i \(-0.297341\pi\)
0.594524 + 0.804078i \(0.297341\pi\)
\(140\) 0 0
\(141\) 2.72666 0.229626
\(142\) 11.7360 0.984862
\(143\) 9.55602 0.799114
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −3.71733 −0.307648
\(147\) −3.00000 −0.247436
\(148\) 9.50466 0.781279
\(149\) −4.49534 −0.368272 −0.184136 0.982901i \(-0.558949\pi\)
−0.184136 + 0.982901i \(0.558949\pi\)
\(150\) 0 0
\(151\) 22.3013 1.81486 0.907428 0.420207i \(-0.138043\pi\)
0.907428 + 0.420207i \(0.138043\pi\)
\(152\) 7.78734 0.631636
\(153\) −0.726656 −0.0587467
\(154\) −7.00933 −0.564828
\(155\) 0 0
\(156\) 2.72666 0.218307
\(157\) 9.60737 0.766751 0.383376 0.923592i \(-0.374761\pi\)
0.383376 + 0.923592i \(0.374761\pi\)
\(158\) 9.73599 0.774553
\(159\) −0.231321 −0.0183449
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) −1.00000 −0.0785674
\(163\) 14.1214 1.10607 0.553035 0.833158i \(-0.313470\pi\)
0.553035 + 0.833158i \(0.313470\pi\)
\(164\) 0.726656 0.0567423
\(165\) 0 0
\(166\) 13.5233 1.04961
\(167\) 8.74531 0.676733 0.338366 0.941014i \(-0.390126\pi\)
0.338366 + 0.941014i \(0.390126\pi\)
\(168\) −2.00000 −0.154303
\(169\) −5.56534 −0.428103
\(170\) 0 0
\(171\) −7.78734 −0.595513
\(172\) 5.50466 0.419727
\(173\) 5.89730 0.448363 0.224182 0.974547i \(-0.428029\pi\)
0.224182 + 0.974547i \(0.428029\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 3.50466 0.264174
\(177\) 9.29200 0.698430
\(178\) 9.55602 0.716254
\(179\) −0.623954 −0.0466365 −0.0233182 0.999728i \(-0.507423\pi\)
−0.0233182 + 0.999728i \(0.507423\pi\)
\(180\) 0 0
\(181\) −14.0700 −1.04582 −0.522908 0.852389i \(-0.675153\pi\)
−0.522908 + 0.852389i \(0.675153\pi\)
\(182\) −5.45331 −0.404226
\(183\) 4.05135 0.299485
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −8.28267 −0.607315
\(187\) −2.54669 −0.186232
\(188\) 2.72666 0.198862
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) −19.5747 −1.41637 −0.708187 0.706025i \(-0.750487\pi\)
−0.708187 + 0.706025i \(0.750487\pi\)
\(192\) 1.00000 0.0721688
\(193\) −10.0187 −0.721159 −0.360579 0.932729i \(-0.617421\pi\)
−0.360579 + 0.932729i \(0.617421\pi\)
\(194\) −4.54669 −0.326433
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −3.50466 −0.249066
\(199\) 23.1893 1.64385 0.821923 0.569599i \(-0.192901\pi\)
0.821923 + 0.569599i \(0.192901\pi\)
\(200\) 0 0
\(201\) 4.05135 0.285760
\(202\) −3.45331 −0.242974
\(203\) −4.00000 −0.280745
\(204\) −0.726656 −0.0508761
\(205\) 0 0
\(206\) −14.4626 −1.00766
\(207\) 1.00000 0.0695048
\(208\) 2.72666 0.189060
\(209\) −27.2920 −1.88783
\(210\) 0 0
\(211\) −7.57467 −0.521462 −0.260731 0.965411i \(-0.583964\pi\)
−0.260731 + 0.965411i \(0.583964\pi\)
\(212\) −0.231321 −0.0158872
\(213\) −11.7360 −0.804136
\(214\) 13.0607 0.892810
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 16.5653 1.12453
\(218\) 10.9580 0.742168
\(219\) 3.71733 0.251194
\(220\) 0 0
\(221\) −1.98134 −0.133280
\(222\) −9.50466 −0.637911
\(223\) −26.5840 −1.78020 −0.890098 0.455769i \(-0.849364\pi\)
−0.890098 + 0.455769i \(0.849364\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) −11.7360 −0.780666
\(227\) −11.9673 −0.794298 −0.397149 0.917754i \(-0.630000\pi\)
−0.397149 + 0.917754i \(0.630000\pi\)
\(228\) −7.78734 −0.515729
\(229\) 26.5327 1.75333 0.876663 0.481104i \(-0.159764\pi\)
0.876663 + 0.481104i \(0.159764\pi\)
\(230\) 0 0
\(231\) 7.00933 0.461180
\(232\) 2.00000 0.131306
\(233\) 24.4040 1.59876 0.799381 0.600825i \(-0.205161\pi\)
0.799381 + 0.600825i \(0.205161\pi\)
\(234\) −2.72666 −0.178247
\(235\) 0 0
\(236\) 9.29200 0.604858
\(237\) −9.73599 −0.632420
\(238\) 1.45331 0.0942043
\(239\) −3.55602 −0.230020 −0.115010 0.993364i \(-0.536690\pi\)
−0.115010 + 0.993364i \(0.536690\pi\)
\(240\) 0 0
\(241\) 3.55602 0.229063 0.114532 0.993420i \(-0.463463\pi\)
0.114532 + 0.993420i \(0.463463\pi\)
\(242\) −1.28267 −0.0824533
\(243\) 1.00000 0.0641500
\(244\) 4.05135 0.259361
\(245\) 0 0
\(246\) −0.726656 −0.0463299
\(247\) −21.2334 −1.35105
\(248\) −8.28267 −0.525950
\(249\) −13.5233 −0.857006
\(250\) 0 0
\(251\) 6.05135 0.381958 0.190979 0.981594i \(-0.438834\pi\)
0.190979 + 0.981594i \(0.438834\pi\)
\(252\) 2.00000 0.125988
\(253\) 3.50466 0.220336
\(254\) −12.2827 −0.770683
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 16.9066 1.05461 0.527303 0.849677i \(-0.323203\pi\)
0.527303 + 0.849677i \(0.323203\pi\)
\(258\) −5.50466 −0.342705
\(259\) 19.0093 1.18118
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 14.8480 0.917314
\(263\) 0.887968 0.0547545 0.0273772 0.999625i \(-0.491284\pi\)
0.0273772 + 0.999625i \(0.491284\pi\)
\(264\) −3.50466 −0.215697
\(265\) 0 0
\(266\) 15.5747 0.954944
\(267\) −9.55602 −0.584819
\(268\) 4.05135 0.247476
\(269\) 20.1214 1.22682 0.613410 0.789764i \(-0.289797\pi\)
0.613410 + 0.789764i \(0.289797\pi\)
\(270\) 0 0
\(271\) −30.5840 −1.85785 −0.928923 0.370273i \(-0.879264\pi\)
−0.928923 + 0.370273i \(0.879264\pi\)
\(272\) −0.726656 −0.0440600
\(273\) 5.45331 0.330050
\(274\) −3.27334 −0.197750
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) −4.38538 −0.263492 −0.131746 0.991284i \(-0.542058\pi\)
−0.131746 + 0.991284i \(0.542058\pi\)
\(278\) −14.0187 −0.840783
\(279\) 8.28267 0.495871
\(280\) 0 0
\(281\) 10.1214 0.603790 0.301895 0.953341i \(-0.402381\pi\)
0.301895 + 0.953341i \(0.402381\pi\)
\(282\) −2.72666 −0.162370
\(283\) 25.1820 1.49692 0.748458 0.663182i \(-0.230794\pi\)
0.748458 + 0.663182i \(0.230794\pi\)
\(284\) −11.7360 −0.696403
\(285\) 0 0
\(286\) −9.55602 −0.565059
\(287\) 1.45331 0.0857864
\(288\) −1.00000 −0.0589256
\(289\) −16.4720 −0.968939
\(290\) 0 0
\(291\) 4.54669 0.266532
\(292\) 3.71733 0.217540
\(293\) 0.759350 0.0443617 0.0221809 0.999754i \(-0.492939\pi\)
0.0221809 + 0.999754i \(0.492939\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) −9.50466 −0.552447
\(297\) 3.50466 0.203361
\(298\) 4.49534 0.260408
\(299\) 2.72666 0.157687
\(300\) 0 0
\(301\) 11.0093 0.634567
\(302\) −22.3013 −1.28330
\(303\) 3.45331 0.198388
\(304\) −7.78734 −0.446634
\(305\) 0 0
\(306\) 0.726656 0.0415402
\(307\) −27.5747 −1.57377 −0.786885 0.617100i \(-0.788307\pi\)
−0.786885 + 0.617100i \(0.788307\pi\)
\(308\) 7.00933 0.399394
\(309\) 14.4626 0.822751
\(310\) 0 0
\(311\) 24.1214 1.36780 0.683898 0.729577i \(-0.260283\pi\)
0.683898 + 0.729577i \(0.260283\pi\)
\(312\) −2.72666 −0.154367
\(313\) 8.58400 0.485196 0.242598 0.970127i \(-0.422000\pi\)
0.242598 + 0.970127i \(0.422000\pi\)
\(314\) −9.60737 −0.542175
\(315\) 0 0
\(316\) −9.73599 −0.547692
\(317\) 24.1214 1.35479 0.677395 0.735619i \(-0.263109\pi\)
0.677395 + 0.735619i \(0.263109\pi\)
\(318\) 0.231321 0.0129718
\(319\) −7.00933 −0.392447
\(320\) 0 0
\(321\) −13.0607 −0.728976
\(322\) −2.00000 −0.111456
\(323\) 5.65872 0.314860
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −14.1214 −0.782110
\(327\) −10.9580 −0.605978
\(328\) −0.726656 −0.0401229
\(329\) 5.45331 0.300651
\(330\) 0 0
\(331\) 8.10270 0.445365 0.222682 0.974891i \(-0.428519\pi\)
0.222682 + 0.974891i \(0.428519\pi\)
\(332\) −13.5233 −0.742189
\(333\) 9.50466 0.520852
\(334\) −8.74531 −0.478522
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) −24.1214 −1.31397 −0.656987 0.753902i \(-0.728169\pi\)
−0.656987 + 0.753902i \(0.728169\pi\)
\(338\) 5.56534 0.302715
\(339\) 11.7360 0.637411
\(340\) 0 0
\(341\) 29.0280 1.57195
\(342\) 7.78734 0.421091
\(343\) −20.0000 −1.07990
\(344\) −5.50466 −0.296792
\(345\) 0 0
\(346\) −5.89730 −0.317041
\(347\) −21.9160 −1.17651 −0.588255 0.808675i \(-0.700185\pi\)
−0.588255 + 0.808675i \(0.700185\pi\)
\(348\) −2.00000 −0.107211
\(349\) 26.5653 1.42201 0.711005 0.703187i \(-0.248240\pi\)
0.711005 + 0.703187i \(0.248240\pi\)
\(350\) 0 0
\(351\) 2.72666 0.145538
\(352\) −3.50466 −0.186799
\(353\) −22.9507 −1.22154 −0.610772 0.791807i \(-0.709141\pi\)
−0.610772 + 0.791807i \(0.709141\pi\)
\(354\) −9.29200 −0.493864
\(355\) 0 0
\(356\) −9.55602 −0.506468
\(357\) −1.45331 −0.0769175
\(358\) 0.623954 0.0329770
\(359\) 5.02799 0.265367 0.132683 0.991158i \(-0.457641\pi\)
0.132683 + 0.991158i \(0.457641\pi\)
\(360\) 0 0
\(361\) 41.6426 2.19172
\(362\) 14.0700 0.739503
\(363\) 1.28267 0.0673228
\(364\) 5.45331 0.285831
\(365\) 0 0
\(366\) −4.05135 −0.211768
\(367\) 7.45331 0.389060 0.194530 0.980897i \(-0.437682\pi\)
0.194530 + 0.980897i \(0.437682\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0.726656 0.0378282
\(370\) 0 0
\(371\) −0.462642 −0.0240192
\(372\) 8.28267 0.429437
\(373\) 14.4953 0.750540 0.375270 0.926916i \(-0.377550\pi\)
0.375270 + 0.926916i \(0.377550\pi\)
\(374\) 2.54669 0.131686
\(375\) 0 0
\(376\) −2.72666 −0.140617
\(377\) −5.45331 −0.280860
\(378\) −2.00000 −0.102869
\(379\) 17.1379 0.880317 0.440159 0.897920i \(-0.354922\pi\)
0.440159 + 0.897920i \(0.354922\pi\)
\(380\) 0 0
\(381\) 12.2827 0.629260
\(382\) 19.5747 1.00153
\(383\) 0.565344 0.0288878 0.0144439 0.999896i \(-0.495402\pi\)
0.0144439 + 0.999896i \(0.495402\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 10.0187 0.509936
\(387\) 5.50466 0.279818
\(388\) 4.54669 0.230823
\(389\) −29.5233 −1.49689 −0.748446 0.663196i \(-0.769200\pi\)
−0.748446 + 0.663196i \(0.769200\pi\)
\(390\) 0 0
\(391\) −0.726656 −0.0367486
\(392\) 3.00000 0.151523
\(393\) −14.8480 −0.748983
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) 3.50466 0.176116
\(397\) −37.7360 −1.89391 −0.946957 0.321359i \(-0.895860\pi\)
−0.946957 + 0.321359i \(0.895860\pi\)
\(398\) −23.1893 −1.16237
\(399\) −15.5747 −0.779709
\(400\) 0 0
\(401\) −6.90663 −0.344900 −0.172450 0.985018i \(-0.555168\pi\)
−0.172450 + 0.985018i \(0.555168\pi\)
\(402\) −4.05135 −0.202063
\(403\) 22.5840 1.12499
\(404\) 3.45331 0.171809
\(405\) 0 0
\(406\) 4.00000 0.198517
\(407\) 33.3107 1.65115
\(408\) 0.726656 0.0359749
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) 3.27334 0.161462
\(412\) 14.4626 0.712523
\(413\) 18.5840 0.914459
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) −2.72666 −0.133685
\(417\) 14.0187 0.686497
\(418\) 27.2920 1.33490
\(419\) 27.9673 1.36629 0.683146 0.730282i \(-0.260611\pi\)
0.683146 + 0.730282i \(0.260611\pi\)
\(420\) 0 0
\(421\) 2.59804 0.126621 0.0633103 0.997994i \(-0.479834\pi\)
0.0633103 + 0.997994i \(0.479834\pi\)
\(422\) 7.57467 0.368729
\(423\) 2.72666 0.132575
\(424\) 0.231321 0.0112339
\(425\) 0 0
\(426\) 11.7360 0.568610
\(427\) 8.10270 0.392117
\(428\) −13.0607 −0.631312
\(429\) 9.55602 0.461369
\(430\) 0 0
\(431\) −15.2147 −0.732868 −0.366434 0.930444i \(-0.619421\pi\)
−0.366434 + 0.930444i \(0.619421\pi\)
\(432\) 1.00000 0.0481125
\(433\) −27.1307 −1.30382 −0.651909 0.758297i \(-0.726032\pi\)
−0.651909 + 0.758297i \(0.726032\pi\)
\(434\) −16.5653 −0.795162
\(435\) 0 0
\(436\) −10.9580 −0.524792
\(437\) −7.78734 −0.372519
\(438\) −3.71733 −0.177621
\(439\) −1.65872 −0.0791663 −0.0395832 0.999216i \(-0.512603\pi\)
−0.0395832 + 0.999216i \(0.512603\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 1.98134 0.0942429
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 9.50466 0.451071
\(445\) 0 0
\(446\) 26.5840 1.25879
\(447\) −4.49534 −0.212622
\(448\) 2.00000 0.0944911
\(449\) −14.9507 −0.705568 −0.352784 0.935705i \(-0.614765\pi\)
−0.352784 + 0.935705i \(0.614765\pi\)
\(450\) 0 0
\(451\) 2.54669 0.119919
\(452\) 11.7360 0.552014
\(453\) 22.3013 1.04781
\(454\) 11.9673 0.561654
\(455\) 0 0
\(456\) 7.78734 0.364675
\(457\) 1.43466 0.0671104 0.0335552 0.999437i \(-0.489317\pi\)
0.0335552 + 0.999437i \(0.489317\pi\)
\(458\) −26.5327 −1.23979
\(459\) −0.726656 −0.0339174
\(460\) 0 0
\(461\) −13.5747 −0.632236 −0.316118 0.948720i \(-0.602379\pi\)
−0.316118 + 0.948720i \(0.602379\pi\)
\(462\) −7.00933 −0.326103
\(463\) 1.73599 0.0806781 0.0403390 0.999186i \(-0.487156\pi\)
0.0403390 + 0.999186i \(0.487156\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −24.4040 −1.13049
\(467\) −28.1727 −1.30368 −0.651839 0.758358i \(-0.726002\pi\)
−0.651839 + 0.758358i \(0.726002\pi\)
\(468\) 2.72666 0.126040
\(469\) 8.10270 0.374148
\(470\) 0 0
\(471\) 9.60737 0.442684
\(472\) −9.29200 −0.427699
\(473\) 19.2920 0.887047
\(474\) 9.73599 0.447189
\(475\) 0 0
\(476\) −1.45331 −0.0666125
\(477\) −0.231321 −0.0105915
\(478\) 3.55602 0.162648
\(479\) −26.3786 −1.20527 −0.602634 0.798017i \(-0.705882\pi\)
−0.602634 + 0.798017i \(0.705882\pi\)
\(480\) 0 0
\(481\) 25.9160 1.18167
\(482\) −3.55602 −0.161972
\(483\) 2.00000 0.0910032
\(484\) 1.28267 0.0583033
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 20.2827 0.919096 0.459548 0.888153i \(-0.348011\pi\)
0.459548 + 0.888153i \(0.348011\pi\)
\(488\) −4.05135 −0.183396
\(489\) 14.1214 0.638590
\(490\) 0 0
\(491\) 18.6426 0.841329 0.420665 0.907216i \(-0.361797\pi\)
0.420665 + 0.907216i \(0.361797\pi\)
\(492\) 0.726656 0.0327602
\(493\) 1.45331 0.0654539
\(494\) 21.2334 0.955335
\(495\) 0 0
\(496\) 8.28267 0.371903
\(497\) −23.4720 −1.05286
\(498\) 13.5233 0.605995
\(499\) 20.9907 0.939671 0.469836 0.882754i \(-0.344313\pi\)
0.469836 + 0.882754i \(0.344313\pi\)
\(500\) 0 0
\(501\) 8.74531 0.390712
\(502\) −6.05135 −0.270085
\(503\) −13.5560 −0.604433 −0.302216 0.953239i \(-0.597727\pi\)
−0.302216 + 0.953239i \(0.597727\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) −3.50466 −0.155801
\(507\) −5.56534 −0.247166
\(508\) 12.2827 0.544955
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) 0 0
\(511\) 7.43466 0.328890
\(512\) −1.00000 −0.0441942
\(513\) −7.78734 −0.343819
\(514\) −16.9066 −0.745719
\(515\) 0 0
\(516\) 5.50466 0.242329
\(517\) 9.55602 0.420273
\(518\) −19.0093 −0.835222
\(519\) 5.89730 0.258863
\(520\) 0 0
\(521\) −44.6027 −1.95408 −0.977039 0.213061i \(-0.931657\pi\)
−0.977039 + 0.213061i \(0.931657\pi\)
\(522\) 2.00000 0.0875376
\(523\) 19.7287 0.862677 0.431339 0.902190i \(-0.358041\pi\)
0.431339 + 0.902190i \(0.358041\pi\)
\(524\) −14.8480 −0.648639
\(525\) 0 0
\(526\) −0.887968 −0.0387173
\(527\) −6.01866 −0.262177
\(528\) 3.50466 0.152521
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 9.29200 0.403238
\(532\) −15.5747 −0.675248
\(533\) 1.98134 0.0858215
\(534\) 9.55602 0.413529
\(535\) 0 0
\(536\) −4.05135 −0.174992
\(537\) −0.623954 −0.0269256
\(538\) −20.1214 −0.867493
\(539\) −10.5140 −0.452870
\(540\) 0 0
\(541\) −13.4720 −0.579205 −0.289603 0.957147i \(-0.593523\pi\)
−0.289603 + 0.957147i \(0.593523\pi\)
\(542\) 30.5840 1.31370
\(543\) −14.0700 −0.603802
\(544\) 0.726656 0.0311551
\(545\) 0 0
\(546\) −5.45331 −0.233380
\(547\) 23.0466 0.985403 0.492702 0.870198i \(-0.336009\pi\)
0.492702 + 0.870198i \(0.336009\pi\)
\(548\) 3.27334 0.139830
\(549\) 4.05135 0.172907
\(550\) 0 0
\(551\) 15.5747 0.663503
\(552\) −1.00000 −0.0425628
\(553\) −19.4720 −0.828032
\(554\) 4.38538 0.186317
\(555\) 0 0
\(556\) 14.0187 0.594524
\(557\) 37.2593 1.57873 0.789364 0.613926i \(-0.210411\pi\)
0.789364 + 0.613926i \(0.210411\pi\)
\(558\) −8.28267 −0.350633
\(559\) 15.0093 0.634827
\(560\) 0 0
\(561\) −2.54669 −0.107521
\(562\) −10.1214 −0.426944
\(563\) −46.5513 −1.96190 −0.980952 0.194251i \(-0.937772\pi\)
−0.980952 + 0.194251i \(0.937772\pi\)
\(564\) 2.72666 0.114813
\(565\) 0 0
\(566\) −25.1820 −1.05848
\(567\) 2.00000 0.0839921
\(568\) 11.7360 0.492431
\(569\) −20.0000 −0.838444 −0.419222 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 0 0
\(571\) −13.6660 −0.571903 −0.285952 0.958244i \(-0.592310\pi\)
−0.285952 + 0.958244i \(0.592310\pi\)
\(572\) 9.55602 0.399557
\(573\) −19.5747 −0.817744
\(574\) −1.45331 −0.0606601
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 6.30133 0.262328 0.131164 0.991361i \(-0.458129\pi\)
0.131164 + 0.991361i \(0.458129\pi\)
\(578\) 16.4720 0.685144
\(579\) −10.0187 −0.416361
\(580\) 0 0
\(581\) −27.0466 −1.12208
\(582\) −4.54669 −0.188466
\(583\) −0.810702 −0.0335758
\(584\) −3.71733 −0.153824
\(585\) 0 0
\(586\) −0.759350 −0.0313685
\(587\) 35.5747 1.46832 0.734162 0.678974i \(-0.237575\pi\)
0.734162 + 0.678974i \(0.237575\pi\)
\(588\) −3.00000 −0.123718
\(589\) −64.5000 −2.65767
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) 9.50466 0.390639
\(593\) −4.54669 −0.186710 −0.0933550 0.995633i \(-0.529759\pi\)
−0.0933550 + 0.995633i \(0.529759\pi\)
\(594\) −3.50466 −0.143798
\(595\) 0 0
\(596\) −4.49534 −0.184136
\(597\) 23.1893 0.949075
\(598\) −2.72666 −0.111501
\(599\) 16.1214 0.658701 0.329350 0.944208i \(-0.393170\pi\)
0.329350 + 0.944208i \(0.393170\pi\)
\(600\) 0 0
\(601\) −9.39470 −0.383218 −0.191609 0.981471i \(-0.561371\pi\)
−0.191609 + 0.981471i \(0.561371\pi\)
\(602\) −11.0093 −0.448707
\(603\) 4.05135 0.164984
\(604\) 22.3013 0.907428
\(605\) 0 0
\(606\) −3.45331 −0.140281
\(607\) −10.9066 −0.442686 −0.221343 0.975196i \(-0.571044\pi\)
−0.221343 + 0.975196i \(0.571044\pi\)
\(608\) 7.78734 0.315818
\(609\) −4.00000 −0.162088
\(610\) 0 0
\(611\) 7.43466 0.300774
\(612\) −0.726656 −0.0293733
\(613\) −43.4206 −1.75374 −0.876871 0.480725i \(-0.840373\pi\)
−0.876871 + 0.480725i \(0.840373\pi\)
\(614\) 27.5747 1.11282
\(615\) 0 0
\(616\) −7.00933 −0.282414
\(617\) −20.0959 −0.809031 −0.404516 0.914531i \(-0.632560\pi\)
−0.404516 + 0.914531i \(0.632560\pi\)
\(618\) −14.4626 −0.581773
\(619\) 27.3247 1.09827 0.549136 0.835733i \(-0.314957\pi\)
0.549136 + 0.835733i \(0.314957\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) −24.1214 −0.967178
\(623\) −19.1120 −0.765707
\(624\) 2.72666 0.109154
\(625\) 0 0
\(626\) −8.58400 −0.343086
\(627\) −27.2920 −1.08994
\(628\) 9.60737 0.383376
\(629\) −6.90663 −0.275385
\(630\) 0 0
\(631\) 39.8947 1.58818 0.794091 0.607799i \(-0.207947\pi\)
0.794091 + 0.607799i \(0.207947\pi\)
\(632\) 9.73599 0.387277
\(633\) −7.57467 −0.301066
\(634\) −24.1214 −0.957982
\(635\) 0 0
\(636\) −0.231321 −0.00917247
\(637\) −8.17997 −0.324102
\(638\) 7.00933 0.277502
\(639\) −11.7360 −0.464268
\(640\) 0 0
\(641\) 49.1680 1.94202 0.971010 0.239040i \(-0.0768327\pi\)
0.971010 + 0.239040i \(0.0768327\pi\)
\(642\) 13.0607 0.515464
\(643\) −20.9766 −0.827238 −0.413619 0.910450i \(-0.635735\pi\)
−0.413619 + 0.910450i \(0.635735\pi\)
\(644\) 2.00000 0.0788110
\(645\) 0 0
\(646\) −5.65872 −0.222639
\(647\) −25.9813 −1.02143 −0.510716 0.859749i \(-0.670620\pi\)
−0.510716 + 0.859749i \(0.670620\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 32.5653 1.27830
\(650\) 0 0
\(651\) 16.5653 0.649247
\(652\) 14.1214 0.553035
\(653\) −0.0840454 −0.00328895 −0.00164448 0.999999i \(-0.500523\pi\)
−0.00164448 + 0.999999i \(0.500523\pi\)
\(654\) 10.9580 0.428491
\(655\) 0 0
\(656\) 0.726656 0.0283712
\(657\) 3.71733 0.145027
\(658\) −5.45331 −0.212592
\(659\) 10.6167 0.413568 0.206784 0.978387i \(-0.433700\pi\)
0.206784 + 0.978387i \(0.433700\pi\)
\(660\) 0 0
\(661\) −39.9860 −1.55527 −0.777637 0.628714i \(-0.783582\pi\)
−0.777637 + 0.628714i \(0.783582\pi\)
\(662\) −8.10270 −0.314920
\(663\) −1.98134 −0.0769490
\(664\) 13.5233 0.524807
\(665\) 0 0
\(666\) −9.50466 −0.368298
\(667\) −2.00000 −0.0774403
\(668\) 8.74531 0.338366
\(669\) −26.5840 −1.02780
\(670\) 0 0
\(671\) 14.1986 0.548132
\(672\) −2.00000 −0.0771517
\(673\) 2.86667 0.110502 0.0552511 0.998472i \(-0.482404\pi\)
0.0552511 + 0.998472i \(0.482404\pi\)
\(674\) 24.1214 0.929120
\(675\) 0 0
\(676\) −5.56534 −0.214052
\(677\) −3.76868 −0.144842 −0.0724211 0.997374i \(-0.523073\pi\)
−0.0724211 + 0.997374i \(0.523073\pi\)
\(678\) −11.7360 −0.450718
\(679\) 9.09337 0.348972
\(680\) 0 0
\(681\) −11.9673 −0.458588
\(682\) −29.0280 −1.11154
\(683\) −39.6120 −1.51571 −0.757855 0.652423i \(-0.773753\pi\)
−0.757855 + 0.652423i \(0.773753\pi\)
\(684\) −7.78734 −0.297756
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 26.5327 1.01228
\(688\) 5.50466 0.209863
\(689\) −0.630732 −0.0240290
\(690\) 0 0
\(691\) −49.0653 −1.86653 −0.933266 0.359186i \(-0.883054\pi\)
−0.933266 + 0.359186i \(0.883054\pi\)
\(692\) 5.89730 0.224182
\(693\) 7.00933 0.266262
\(694\) 21.9160 0.831918
\(695\) 0 0
\(696\) 2.00000 0.0758098
\(697\) −0.528030 −0.0200005
\(698\) −26.5653 −1.00551
\(699\) 24.4040 0.923045
\(700\) 0 0
\(701\) −10.5140 −0.397108 −0.198554 0.980090i \(-0.563625\pi\)
−0.198554 + 0.980090i \(0.563625\pi\)
\(702\) −2.72666 −0.102911
\(703\) −74.0160 −2.79157
\(704\) 3.50466 0.132087
\(705\) 0 0
\(706\) 22.9507 0.863762
\(707\) 6.90663 0.259750
\(708\) 9.29200 0.349215
\(709\) −17.1447 −0.643884 −0.321942 0.946759i \(-0.604336\pi\)
−0.321942 + 0.946759i \(0.604336\pi\)
\(710\) 0 0
\(711\) −9.73599 −0.365128
\(712\) 9.55602 0.358127
\(713\) 8.28267 0.310189
\(714\) 1.45331 0.0543889
\(715\) 0 0
\(716\) −0.623954 −0.0233182
\(717\) −3.55602 −0.132802
\(718\) −5.02799 −0.187643
\(719\) 29.0093 1.08187 0.540933 0.841066i \(-0.318071\pi\)
0.540933 + 0.841066i \(0.318071\pi\)
\(720\) 0 0
\(721\) 28.9253 1.07723
\(722\) −41.6426 −1.54978
\(723\) 3.55602 0.132250
\(724\) −14.0700 −0.522908
\(725\) 0 0
\(726\) −1.28267 −0.0476044
\(727\) −38.9253 −1.44366 −0.721829 0.692071i \(-0.756698\pi\)
−0.721829 + 0.692071i \(0.756698\pi\)
\(728\) −5.45331 −0.202113
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.00000 −0.147945
\(732\) 4.05135 0.149742
\(733\) −5.50466 −0.203319 −0.101660 0.994819i \(-0.532415\pi\)
−0.101660 + 0.994819i \(0.532415\pi\)
\(734\) −7.45331 −0.275107
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 14.1986 0.523013
\(738\) −0.726656 −0.0267486
\(739\) 28.6027 1.05217 0.526083 0.850433i \(-0.323660\pi\)
0.526083 + 0.850433i \(0.323660\pi\)
\(740\) 0 0
\(741\) −21.2334 −0.780028
\(742\) 0.462642 0.0169841
\(743\) −9.98134 −0.366180 −0.183090 0.983096i \(-0.558610\pi\)
−0.183090 + 0.983096i \(0.558610\pi\)
\(744\) −8.28267 −0.303657
\(745\) 0 0
\(746\) −14.4953 −0.530712
\(747\) −13.5233 −0.494792
\(748\) −2.54669 −0.0931161
\(749\) −26.1214 −0.954454
\(750\) 0 0
\(751\) −21.8387 −0.796905 −0.398453 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(752\) 2.72666 0.0994309
\(753\) 6.05135 0.220524
\(754\) 5.45331 0.198598
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 16.5513 0.601568 0.300784 0.953692i \(-0.402752\pi\)
0.300784 + 0.953692i \(0.402752\pi\)
\(758\) −17.1379 −0.622478
\(759\) 3.50466 0.127211
\(760\) 0 0
\(761\) 20.5840 0.746169 0.373085 0.927797i \(-0.378300\pi\)
0.373085 + 0.927797i \(0.378300\pi\)
\(762\) −12.2827 −0.444954
\(763\) −21.9160 −0.793411
\(764\) −19.5747 −0.708187
\(765\) 0 0
\(766\) −0.565344 −0.0204267
\(767\) 25.3361 0.914833
\(768\) 1.00000 0.0360844
\(769\) 7.76142 0.279884 0.139942 0.990160i \(-0.455308\pi\)
0.139942 + 0.990160i \(0.455308\pi\)
\(770\) 0 0
\(771\) 16.9066 0.608877
\(772\) −10.0187 −0.360579
\(773\) 21.8247 0.784978 0.392489 0.919757i \(-0.371614\pi\)
0.392489 + 0.919757i \(0.371614\pi\)
\(774\) −5.50466 −0.197861
\(775\) 0 0
\(776\) −4.54669 −0.163217
\(777\) 19.0093 0.681956
\(778\) 29.5233 1.05846
\(779\) −5.65872 −0.202745
\(780\) 0 0
\(781\) −41.1307 −1.47177
\(782\) 0.726656 0.0259852
\(783\) −2.00000 −0.0714742
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 14.8480 0.529611
\(787\) 46.4299 1.65505 0.827524 0.561430i \(-0.189748\pi\)
0.827524 + 0.561430i \(0.189748\pi\)
\(788\) 2.00000 0.0712470
\(789\) 0.887968 0.0316125
\(790\) 0 0
\(791\) 23.4720 0.834567
\(792\) −3.50466 −0.124533
\(793\) 11.0466 0.392278
\(794\) 37.7360 1.33920
\(795\) 0 0
\(796\) 23.1893 0.821923
\(797\) −44.3340 −1.57039 −0.785196 0.619248i \(-0.787438\pi\)
−0.785196 + 0.619248i \(0.787438\pi\)
\(798\) 15.5747 0.551337
\(799\) −1.98134 −0.0700949
\(800\) 0 0
\(801\) −9.55602 −0.337645
\(802\) 6.90663 0.243881
\(803\) 13.0280 0.459748
\(804\) 4.05135 0.142880
\(805\) 0 0
\(806\) −22.5840 −0.795488
\(807\) 20.1214 0.708305
\(808\) −3.45331 −0.121487
\(809\) −1.96269 −0.0690043 −0.0345022 0.999405i \(-0.510985\pi\)
−0.0345022 + 0.999405i \(0.510985\pi\)
\(810\) 0 0
\(811\) 31.8973 1.12007 0.560033 0.828470i \(-0.310789\pi\)
0.560033 + 0.828470i \(0.310789\pi\)
\(812\) −4.00000 −0.140372
\(813\) −30.5840 −1.07263
\(814\) −33.3107 −1.16754
\(815\) 0 0
\(816\) −0.726656 −0.0254381
\(817\) −42.8667 −1.49972
\(818\) −2.00000 −0.0699284
\(819\) 5.45331 0.190554
\(820\) 0 0
\(821\) 11.4906 0.401026 0.200513 0.979691i \(-0.435739\pi\)
0.200513 + 0.979691i \(0.435739\pi\)
\(822\) −3.27334 −0.114171
\(823\) −37.1307 −1.29429 −0.647147 0.762365i \(-0.724038\pi\)
−0.647147 + 0.762365i \(0.724038\pi\)
\(824\) −14.4626 −0.503830
\(825\) 0 0
\(826\) −18.5840 −0.646620
\(827\) −40.2100 −1.39824 −0.699120 0.715005i \(-0.746425\pi\)
−0.699120 + 0.715005i \(0.746425\pi\)
\(828\) 1.00000 0.0347524
\(829\) −6.88797 −0.239229 −0.119615 0.992820i \(-0.538166\pi\)
−0.119615 + 0.992820i \(0.538166\pi\)
\(830\) 0 0
\(831\) −4.38538 −0.152127
\(832\) 2.72666 0.0945298
\(833\) 2.17997 0.0755315
\(834\) −14.0187 −0.485426
\(835\) 0 0
\(836\) −27.2920 −0.943914
\(837\) 8.28267 0.286291
\(838\) −27.9673 −0.966115
\(839\) −26.3786 −0.910690 −0.455345 0.890315i \(-0.650484\pi\)
−0.455345 + 0.890315i \(0.650484\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −2.59804 −0.0895343
\(843\) 10.1214 0.348598
\(844\) −7.57467 −0.260731
\(845\) 0 0
\(846\) −2.72666 −0.0937444
\(847\) 2.56534 0.0881463
\(848\) −0.231321 −0.00794359
\(849\) 25.1820 0.864245
\(850\) 0 0
\(851\) 9.50466 0.325816
\(852\) −11.7360 −0.402068
\(853\) 11.7546 0.402471 0.201236 0.979543i \(-0.435504\pi\)
0.201236 + 0.979543i \(0.435504\pi\)
\(854\) −8.10270 −0.277269
\(855\) 0 0
\(856\) 13.0607 0.446405
\(857\) −53.5347 −1.82871 −0.914356 0.404912i \(-0.867302\pi\)
−0.914356 + 0.404912i \(0.867302\pi\)
\(858\) −9.55602 −0.326237
\(859\) −7.11203 −0.242659 −0.121330 0.992612i \(-0.538716\pi\)
−0.121330 + 0.992612i \(0.538716\pi\)
\(860\) 0 0
\(861\) 1.45331 0.0495288
\(862\) 15.2147 0.518216
\(863\) 28.0373 0.954401 0.477201 0.878794i \(-0.341651\pi\)
0.477201 + 0.878794i \(0.341651\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 27.1307 0.921938
\(867\) −16.4720 −0.559417
\(868\) 16.5653 0.562264
\(869\) −34.1214 −1.15749
\(870\) 0 0
\(871\) 11.0466 0.374301
\(872\) 10.9580 0.371084
\(873\) 4.54669 0.153882
\(874\) 7.78734 0.263411
\(875\) 0 0
\(876\) 3.71733 0.125597
\(877\) −46.4040 −1.56695 −0.783476 0.621422i \(-0.786555\pi\)
−0.783476 + 0.621422i \(0.786555\pi\)
\(878\) 1.65872 0.0559790
\(879\) 0.759350 0.0256123
\(880\) 0 0
\(881\) 14.6494 0.493550 0.246775 0.969073i \(-0.420629\pi\)
0.246775 + 0.969073i \(0.420629\pi\)
\(882\) 3.00000 0.101015
\(883\) −13.7614 −0.463109 −0.231554 0.972822i \(-0.574381\pi\)
−0.231554 + 0.972822i \(0.574381\pi\)
\(884\) −1.98134 −0.0666398
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 35.8573 1.20397 0.601986 0.798507i \(-0.294376\pi\)
0.601986 + 0.798507i \(0.294376\pi\)
\(888\) −9.50466 −0.318956
\(889\) 24.5653 0.823895
\(890\) 0 0
\(891\) 3.50466 0.117411
\(892\) −26.5840 −0.890098
\(893\) −21.2334 −0.710548
\(894\) 4.49534 0.150347
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 2.72666 0.0910404
\(898\) 14.9507 0.498912
\(899\) −16.5653 −0.552485
\(900\) 0 0
\(901\) 0.168091 0.00559992
\(902\) −2.54669 −0.0847954
\(903\) 11.0093 0.366368
\(904\) −11.7360 −0.390333
\(905\) 0 0
\(906\) −22.3013 −0.740912
\(907\) 1.60737 0.0533718 0.0266859 0.999644i \(-0.491505\pi\)
0.0266859 + 0.999644i \(0.491505\pi\)
\(908\) −11.9673 −0.397149
\(909\) 3.45331 0.114539
\(910\) 0 0
\(911\) −15.8973 −0.526701 −0.263350 0.964700i \(-0.584828\pi\)
−0.263350 + 0.964700i \(0.584828\pi\)
\(912\) −7.78734 −0.257864
\(913\) −47.3947 −1.56854
\(914\) −1.43466 −0.0474542
\(915\) 0 0
\(916\) 26.5327 0.876663
\(917\) −29.6960 −0.980649
\(918\) 0.726656 0.0239832
\(919\) −42.3013 −1.39539 −0.697696 0.716394i \(-0.745791\pi\)
−0.697696 + 0.716394i \(0.745791\pi\)
\(920\) 0 0
\(921\) −27.5747 −0.908616
\(922\) 13.5747 0.447058
\(923\) −32.0000 −1.05329
\(924\) 7.00933 0.230590
\(925\) 0 0
\(926\) −1.73599 −0.0570480
\(927\) 14.4626 0.475015
\(928\) 2.00000 0.0656532
\(929\) −42.0628 −1.38003 −0.690017 0.723793i \(-0.742397\pi\)
−0.690017 + 0.723793i \(0.742397\pi\)
\(930\) 0 0
\(931\) 23.3620 0.765659
\(932\) 24.4040 0.799381
\(933\) 24.1214 0.789698
\(934\) 28.1727 0.921839
\(935\) 0 0
\(936\) −2.72666 −0.0891236
\(937\) 8.44398 0.275853 0.137926 0.990442i \(-0.455956\pi\)
0.137926 + 0.990442i \(0.455956\pi\)
\(938\) −8.10270 −0.264563
\(939\) 8.58400 0.280128
\(940\) 0 0
\(941\) −19.2993 −0.629138 −0.314569 0.949235i \(-0.601860\pi\)
−0.314569 + 0.949235i \(0.601860\pi\)
\(942\) −9.60737 −0.313025
\(943\) 0.726656 0.0236632
\(944\) 9.29200 0.302429
\(945\) 0 0
\(946\) −19.2920 −0.627237
\(947\) −3.36927 −0.109486 −0.0547432 0.998500i \(-0.517434\pi\)
−0.0547432 + 0.998500i \(0.517434\pi\)
\(948\) −9.73599 −0.316210
\(949\) 10.1359 0.329024
\(950\) 0 0
\(951\) 24.1214 0.782189
\(952\) 1.45331 0.0471021
\(953\) 24.0441 0.778865 0.389432 0.921055i \(-0.372671\pi\)
0.389432 + 0.921055i \(0.372671\pi\)
\(954\) 0.231321 0.00748929
\(955\) 0 0
\(956\) −3.55602 −0.115010
\(957\) −7.00933 −0.226579
\(958\) 26.3786 0.852254
\(959\) 6.54669 0.211404
\(960\) 0 0
\(961\) 37.6027 1.21299
\(962\) −25.9160 −0.835564
\(963\) −13.0607 −0.420875
\(964\) 3.55602 0.114532
\(965\) 0 0
\(966\) −2.00000 −0.0643489
\(967\) −26.9439 −0.866459 −0.433229 0.901284i \(-0.642626\pi\)
−0.433229 + 0.901284i \(0.642626\pi\)
\(968\) −1.28267 −0.0412266
\(969\) 5.65872 0.181784
\(970\) 0 0
\(971\) −5.84595 −0.187605 −0.0938027 0.995591i \(-0.529902\pi\)
−0.0938027 + 0.995591i \(0.529902\pi\)
\(972\) 1.00000 0.0320750
\(973\) 28.0373 0.898835
\(974\) −20.2827 −0.649899
\(975\) 0 0
\(976\) 4.05135 0.129681
\(977\) 23.8387 0.762667 0.381334 0.924437i \(-0.375465\pi\)
0.381334 + 0.924437i \(0.375465\pi\)
\(978\) −14.1214 −0.451551
\(979\) −33.4906 −1.07037
\(980\) 0 0
\(981\) −10.9580 −0.349861
\(982\) −18.6426 −0.594910
\(983\) 44.6680 1.42469 0.712345 0.701830i \(-0.247633\pi\)
0.712345 + 0.701830i \(0.247633\pi\)
\(984\) −0.726656 −0.0231650
\(985\) 0 0
\(986\) −1.45331 −0.0462829
\(987\) 5.45331 0.173581
\(988\) −21.2334 −0.675524
\(989\) 5.50466 0.175038
\(990\) 0 0
\(991\) −10.0586 −0.319522 −0.159761 0.987156i \(-0.551072\pi\)
−0.159761 + 0.987156i \(0.551072\pi\)
\(992\) −8.28267 −0.262975
\(993\) 8.10270 0.257132
\(994\) 23.4720 0.744486
\(995\) 0 0
\(996\) −13.5233 −0.428503
\(997\) −38.2640 −1.21183 −0.605917 0.795528i \(-0.707194\pi\)
−0.605917 + 0.795528i \(0.707194\pi\)
\(998\) −20.9907 −0.664448
\(999\) 9.50466 0.300714
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 3450.2.a.bq.1.3 3
5.2 odd 4 690.2.d.d.139.1 6
5.3 odd 4 690.2.d.d.139.4 yes 6
5.4 even 2 3450.2.a.br.1.3 3
15.2 even 4 2070.2.d.d.829.6 6
15.8 even 4 2070.2.d.d.829.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.d.d.139.1 6 5.2 odd 4
690.2.d.d.139.4 yes 6 5.3 odd 4
2070.2.d.d.829.3 6 15.8 even 4
2070.2.d.d.829.6 6 15.2 even 4
3450.2.a.bq.1.3 3 1.1 even 1 trivial
3450.2.a.br.1.3 3 5.4 even 2