Properties

Label 3525.2.a.be.1.4
Level $3525$
Weight $2$
Character 3525.1
Self dual yes
Analytic conductor $28.147$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1472667125\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 3 x^{7} - 7 x^{6} + 24 x^{5} + 8 x^{4} - 47 x^{3} + 8 x^{2} + 13 x + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.237165\) of defining polynomial
Character \(\chi\) \(=\) 3525.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.237165 q^{2} -1.00000 q^{3} -1.94375 q^{4} +0.237165 q^{6} -1.64667 q^{7} +0.935320 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.237165 q^{2} -1.00000 q^{3} -1.94375 q^{4} +0.237165 q^{6} -1.64667 q^{7} +0.935320 q^{8} +1.00000 q^{9} +5.84292 q^{11} +1.94375 q^{12} +4.61379 q^{13} +0.390533 q^{14} +3.66568 q^{16} +5.67621 q^{17} -0.237165 q^{18} -6.71365 q^{19} +1.64667 q^{21} -1.38574 q^{22} +6.07976 q^{23} -0.935320 q^{24} -1.09423 q^{26} -1.00000 q^{27} +3.20073 q^{28} +3.23476 q^{29} -6.81968 q^{31} -2.74001 q^{32} -5.84292 q^{33} -1.34620 q^{34} -1.94375 q^{36} -5.84115 q^{37} +1.59224 q^{38} -4.61379 q^{39} -2.60067 q^{41} -0.390533 q^{42} -0.899167 q^{43} -11.3572 q^{44} -1.44191 q^{46} +1.00000 q^{47} -3.66568 q^{48} -4.28847 q^{49} -5.67621 q^{51} -8.96806 q^{52} -4.11309 q^{53} +0.237165 q^{54} -1.54017 q^{56} +6.71365 q^{57} -0.767172 q^{58} +6.93952 q^{59} -6.54592 q^{61} +1.61739 q^{62} -1.64667 q^{63} -6.68153 q^{64} +1.38574 q^{66} +9.16921 q^{67} -11.0332 q^{68} -6.07976 q^{69} +3.98175 q^{71} +0.935320 q^{72} +12.2218 q^{73} +1.38532 q^{74} +13.0497 q^{76} -9.62138 q^{77} +1.09423 q^{78} +15.1032 q^{79} +1.00000 q^{81} +0.616789 q^{82} +14.5712 q^{83} -3.20073 q^{84} +0.213251 q^{86} -3.23476 q^{87} +5.46500 q^{88} -7.03976 q^{89} -7.59740 q^{91} -11.8175 q^{92} +6.81968 q^{93} -0.237165 q^{94} +2.74001 q^{96} -8.59813 q^{97} +1.01707 q^{98} +5.84292 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 3q^{2} - 8q^{3} + 7q^{4} - 3q^{6} + 8q^{7} + 6q^{8} + 8q^{9} + O(q^{10}) \) \( 8q + 3q^{2} - 8q^{3} + 7q^{4} - 3q^{6} + 8q^{7} + 6q^{8} + 8q^{9} - 8q^{11} - 7q^{12} + 10q^{13} + q^{14} + 5q^{16} + 6q^{17} + 3q^{18} - 2q^{19} - 8q^{21} + 10q^{23} - 6q^{24} - 14q^{26} - 8q^{27} + 44q^{28} - 13q^{29} + 10q^{32} + 8q^{33} + 28q^{34} + 7q^{36} + 3q^{37} + 36q^{38} - 10q^{39} - 16q^{41} - q^{42} + 25q^{43} - 17q^{44} - 5q^{46} + 8q^{47} - 5q^{48} + 16q^{49} - 6q^{51} - 17q^{52} + 4q^{53} - 3q^{54} + 37q^{56} + 2q^{57} + 15q^{58} - 8q^{59} + 15q^{61} + 6q^{62} + 8q^{63} - 14q^{64} + 27q^{67} + 14q^{68} - 10q^{69} + 14q^{71} + 6q^{72} + 28q^{73} - 21q^{74} + 6q^{76} + 4q^{77} + 14q^{78} + 7q^{79} + 8q^{81} - 53q^{82} + 60q^{83} - 44q^{84} - 3q^{86} + 13q^{87} + 54q^{88} - 34q^{89} + 23q^{91} - 43q^{92} + 3q^{94} - 10q^{96} + 7q^{97} + 40q^{98} - 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.237165 −0.167701 −0.0838505 0.996478i \(-0.526722\pi\)
−0.0838505 + 0.996478i \(0.526722\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.94375 −0.971876
\(5\) 0 0
\(6\) 0.237165 0.0968222
\(7\) −1.64667 −0.622384 −0.311192 0.950347i \(-0.600728\pi\)
−0.311192 + 0.950347i \(0.600728\pi\)
\(8\) 0.935320 0.330686
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.84292 1.76171 0.880853 0.473389i \(-0.156970\pi\)
0.880853 + 0.473389i \(0.156970\pi\)
\(12\) 1.94375 0.561113
\(13\) 4.61379 1.27963 0.639817 0.768527i \(-0.279010\pi\)
0.639817 + 0.768527i \(0.279010\pi\)
\(14\) 0.390533 0.104374
\(15\) 0 0
\(16\) 3.66568 0.916420
\(17\) 5.67621 1.37668 0.688342 0.725387i \(-0.258339\pi\)
0.688342 + 0.725387i \(0.258339\pi\)
\(18\) −0.237165 −0.0559003
\(19\) −6.71365 −1.54022 −0.770108 0.637913i \(-0.779798\pi\)
−0.770108 + 0.637913i \(0.779798\pi\)
\(20\) 0 0
\(21\) 1.64667 0.359334
\(22\) −1.38574 −0.295440
\(23\) 6.07976 1.26772 0.633859 0.773449i \(-0.281470\pi\)
0.633859 + 0.773449i \(0.281470\pi\)
\(24\) −0.935320 −0.190921
\(25\) 0 0
\(26\) −1.09423 −0.214596
\(27\) −1.00000 −0.192450
\(28\) 3.20073 0.604880
\(29\) 3.23476 0.600680 0.300340 0.953832i \(-0.402900\pi\)
0.300340 + 0.953832i \(0.402900\pi\)
\(30\) 0 0
\(31\) −6.81968 −1.22485 −0.612425 0.790528i \(-0.709806\pi\)
−0.612425 + 0.790528i \(0.709806\pi\)
\(32\) −2.74001 −0.484370
\(33\) −5.84292 −1.01712
\(34\) −1.34620 −0.230871
\(35\) 0 0
\(36\) −1.94375 −0.323959
\(37\) −5.84115 −0.960278 −0.480139 0.877192i \(-0.659414\pi\)
−0.480139 + 0.877192i \(0.659414\pi\)
\(38\) 1.59224 0.258296
\(39\) −4.61379 −0.738797
\(40\) 0 0
\(41\) −2.60067 −0.406157 −0.203078 0.979162i \(-0.565095\pi\)
−0.203078 + 0.979162i \(0.565095\pi\)
\(42\) −0.390533 −0.0602606
\(43\) −0.899167 −0.137122 −0.0685609 0.997647i \(-0.521841\pi\)
−0.0685609 + 0.997647i \(0.521841\pi\)
\(44\) −11.3572 −1.71216
\(45\) 0 0
\(46\) −1.44191 −0.212597
\(47\) 1.00000 0.145865
\(48\) −3.66568 −0.529095
\(49\) −4.28847 −0.612638
\(50\) 0 0
\(51\) −5.67621 −0.794829
\(52\) −8.96806 −1.24365
\(53\) −4.11309 −0.564977 −0.282488 0.959271i \(-0.591160\pi\)
−0.282488 + 0.959271i \(0.591160\pi\)
\(54\) 0.237165 0.0322741
\(55\) 0 0
\(56\) −1.54017 −0.205814
\(57\) 6.71365 0.889244
\(58\) −0.767172 −0.100735
\(59\) 6.93952 0.903449 0.451724 0.892158i \(-0.350809\pi\)
0.451724 + 0.892158i \(0.350809\pi\)
\(60\) 0 0
\(61\) −6.54592 −0.838119 −0.419060 0.907959i \(-0.637640\pi\)
−0.419060 + 0.907959i \(0.637640\pi\)
\(62\) 1.61739 0.205409
\(63\) −1.64667 −0.207461
\(64\) −6.68153 −0.835191
\(65\) 0 0
\(66\) 1.38574 0.170572
\(67\) 9.16921 1.12020 0.560099 0.828426i \(-0.310763\pi\)
0.560099 + 0.828426i \(0.310763\pi\)
\(68\) −11.0332 −1.33797
\(69\) −6.07976 −0.731917
\(70\) 0 0
\(71\) 3.98175 0.472546 0.236273 0.971687i \(-0.424074\pi\)
0.236273 + 0.971687i \(0.424074\pi\)
\(72\) 0.935320 0.110229
\(73\) 12.2218 1.43045 0.715226 0.698893i \(-0.246324\pi\)
0.715226 + 0.698893i \(0.246324\pi\)
\(74\) 1.38532 0.161040
\(75\) 0 0
\(76\) 13.0497 1.49690
\(77\) −9.62138 −1.09646
\(78\) 1.09423 0.123897
\(79\) 15.1032 1.69924 0.849621 0.527394i \(-0.176831\pi\)
0.849621 + 0.527394i \(0.176831\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0.616789 0.0681129
\(83\) 14.5712 1.59940 0.799700 0.600400i \(-0.204992\pi\)
0.799700 + 0.600400i \(0.204992\pi\)
\(84\) −3.20073 −0.349228
\(85\) 0 0
\(86\) 0.213251 0.0229955
\(87\) −3.23476 −0.346803
\(88\) 5.46500 0.582571
\(89\) −7.03976 −0.746213 −0.373106 0.927789i \(-0.621707\pi\)
−0.373106 + 0.927789i \(0.621707\pi\)
\(90\) 0 0
\(91\) −7.59740 −0.796424
\(92\) −11.8175 −1.23206
\(93\) 6.81968 0.707168
\(94\) −0.237165 −0.0244617
\(95\) 0 0
\(96\) 2.74001 0.279651
\(97\) −8.59813 −0.873008 −0.436504 0.899702i \(-0.643783\pi\)
−0.436504 + 0.899702i \(0.643783\pi\)
\(98\) 1.01707 0.102740
\(99\) 5.84292 0.587236
\(100\) 0 0
\(101\) 7.31500 0.727870 0.363935 0.931424i \(-0.381433\pi\)
0.363935 + 0.931424i \(0.381433\pi\)
\(102\) 1.34620 0.133294
\(103\) 0.918971 0.0905489 0.0452744 0.998975i \(-0.485584\pi\)
0.0452744 + 0.998975i \(0.485584\pi\)
\(104\) 4.31537 0.423157
\(105\) 0 0
\(106\) 0.975482 0.0947472
\(107\) 9.87708 0.954853 0.477427 0.878672i \(-0.341570\pi\)
0.477427 + 0.878672i \(0.341570\pi\)
\(108\) 1.94375 0.187038
\(109\) −19.0166 −1.82146 −0.910729 0.413004i \(-0.864479\pi\)
−0.910729 + 0.413004i \(0.864479\pi\)
\(110\) 0 0
\(111\) 5.84115 0.554417
\(112\) −6.03618 −0.570365
\(113\) −4.31060 −0.405507 −0.202753 0.979230i \(-0.564989\pi\)
−0.202753 + 0.979230i \(0.564989\pi\)
\(114\) −1.59224 −0.149127
\(115\) 0 0
\(116\) −6.28758 −0.583787
\(117\) 4.61379 0.426545
\(118\) −1.64581 −0.151509
\(119\) −9.34687 −0.856826
\(120\) 0 0
\(121\) 23.1397 2.10361
\(122\) 1.55246 0.140553
\(123\) 2.60067 0.234495
\(124\) 13.2558 1.19040
\(125\) 0 0
\(126\) 0.390533 0.0347915
\(127\) −1.55525 −0.138006 −0.0690031 0.997616i \(-0.521982\pi\)
−0.0690031 + 0.997616i \(0.521982\pi\)
\(128\) 7.06465 0.624433
\(129\) 0.899167 0.0791673
\(130\) 0 0
\(131\) 15.5392 1.35767 0.678833 0.734292i \(-0.262486\pi\)
0.678833 + 0.734292i \(0.262486\pi\)
\(132\) 11.3572 0.988517
\(133\) 11.0552 0.958606
\(134\) −2.17462 −0.187858
\(135\) 0 0
\(136\) 5.30908 0.455250
\(137\) 12.4815 1.06637 0.533183 0.846000i \(-0.320996\pi\)
0.533183 + 0.846000i \(0.320996\pi\)
\(138\) 1.44191 0.122743
\(139\) 4.67413 0.396454 0.198227 0.980156i \(-0.436482\pi\)
0.198227 + 0.980156i \(0.436482\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) −0.944331 −0.0792465
\(143\) 26.9580 2.25434
\(144\) 3.66568 0.305473
\(145\) 0 0
\(146\) −2.89858 −0.239888
\(147\) 4.28847 0.353707
\(148\) 11.3537 0.933272
\(149\) −20.0538 −1.64287 −0.821435 0.570302i \(-0.806826\pi\)
−0.821435 + 0.570302i \(0.806826\pi\)
\(150\) 0 0
\(151\) −17.3513 −1.41203 −0.706013 0.708198i \(-0.749508\pi\)
−0.706013 + 0.708198i \(0.749508\pi\)
\(152\) −6.27941 −0.509328
\(153\) 5.67621 0.458895
\(154\) 2.28186 0.183877
\(155\) 0 0
\(156\) 8.96806 0.718019
\(157\) 9.02606 0.720358 0.360179 0.932883i \(-0.382716\pi\)
0.360179 + 0.932883i \(0.382716\pi\)
\(158\) −3.58195 −0.284965
\(159\) 4.11309 0.326189
\(160\) 0 0
\(161\) −10.0114 −0.789007
\(162\) −0.237165 −0.0186334
\(163\) 0.183786 0.0143952 0.00719760 0.999974i \(-0.497709\pi\)
0.00719760 + 0.999974i \(0.497709\pi\)
\(164\) 5.05506 0.394734
\(165\) 0 0
\(166\) −3.45579 −0.268221
\(167\) 16.7897 1.29923 0.649615 0.760264i \(-0.274930\pi\)
0.649615 + 0.760264i \(0.274930\pi\)
\(168\) 1.54017 0.118827
\(169\) 8.28702 0.637463
\(170\) 0 0
\(171\) −6.71365 −0.513406
\(172\) 1.74776 0.133265
\(173\) −9.26133 −0.704126 −0.352063 0.935976i \(-0.614520\pi\)
−0.352063 + 0.935976i \(0.614520\pi\)
\(174\) 0.767172 0.0581592
\(175\) 0 0
\(176\) 21.4183 1.61446
\(177\) −6.93952 −0.521606
\(178\) 1.66958 0.125141
\(179\) −0.495946 −0.0370688 −0.0185344 0.999828i \(-0.505900\pi\)
−0.0185344 + 0.999828i \(0.505900\pi\)
\(180\) 0 0
\(181\) −10.8611 −0.807296 −0.403648 0.914914i \(-0.632258\pi\)
−0.403648 + 0.914914i \(0.632258\pi\)
\(182\) 1.80184 0.133561
\(183\) 6.54592 0.483888
\(184\) 5.68652 0.419216
\(185\) 0 0
\(186\) −1.61739 −0.118593
\(187\) 33.1657 2.42531
\(188\) −1.94375 −0.141763
\(189\) 1.64667 0.119778
\(190\) 0 0
\(191\) −7.78821 −0.563535 −0.281767 0.959483i \(-0.590921\pi\)
−0.281767 + 0.959483i \(0.590921\pi\)
\(192\) 6.68153 0.482198
\(193\) −9.91584 −0.713758 −0.356879 0.934151i \(-0.616159\pi\)
−0.356879 + 0.934151i \(0.616159\pi\)
\(194\) 2.03918 0.146404
\(195\) 0 0
\(196\) 8.33572 0.595408
\(197\) 23.4872 1.67339 0.836697 0.547666i \(-0.184484\pi\)
0.836697 + 0.547666i \(0.184484\pi\)
\(198\) −1.38574 −0.0984800
\(199\) 14.0706 0.997438 0.498719 0.866764i \(-0.333804\pi\)
0.498719 + 0.866764i \(0.333804\pi\)
\(200\) 0 0
\(201\) −9.16921 −0.646746
\(202\) −1.73486 −0.122064
\(203\) −5.32660 −0.373854
\(204\) 11.0332 0.772475
\(205\) 0 0
\(206\) −0.217948 −0.0151851
\(207\) 6.07976 0.422572
\(208\) 16.9127 1.17268
\(209\) −39.2273 −2.71341
\(210\) 0 0
\(211\) 4.55801 0.313787 0.156893 0.987616i \(-0.449852\pi\)
0.156893 + 0.987616i \(0.449852\pi\)
\(212\) 7.99483 0.549087
\(213\) −3.98175 −0.272825
\(214\) −2.34250 −0.160130
\(215\) 0 0
\(216\) −0.935320 −0.0636405
\(217\) 11.2298 0.762328
\(218\) 4.51007 0.305460
\(219\) −12.2218 −0.825872
\(220\) 0 0
\(221\) 26.1888 1.76165
\(222\) −1.38532 −0.0929763
\(223\) 8.38549 0.561534 0.280767 0.959776i \(-0.409411\pi\)
0.280767 + 0.959776i \(0.409411\pi\)
\(224\) 4.51191 0.301464
\(225\) 0 0
\(226\) 1.02232 0.0680039
\(227\) −1.27015 −0.0843025 −0.0421513 0.999111i \(-0.513421\pi\)
−0.0421513 + 0.999111i \(0.513421\pi\)
\(228\) −13.0497 −0.864236
\(229\) −5.99457 −0.396133 −0.198066 0.980189i \(-0.563466\pi\)
−0.198066 + 0.980189i \(0.563466\pi\)
\(230\) 0 0
\(231\) 9.62138 0.633041
\(232\) 3.02554 0.198636
\(233\) −9.99821 −0.655005 −0.327502 0.944850i \(-0.606207\pi\)
−0.327502 + 0.944850i \(0.606207\pi\)
\(234\) −1.09423 −0.0715320
\(235\) 0 0
\(236\) −13.4887 −0.878041
\(237\) −15.1032 −0.981058
\(238\) 2.21675 0.143691
\(239\) −7.43603 −0.480997 −0.240498 0.970650i \(-0.577311\pi\)
−0.240498 + 0.970650i \(0.577311\pi\)
\(240\) 0 0
\(241\) −0.976110 −0.0628768 −0.0314384 0.999506i \(-0.510009\pi\)
−0.0314384 + 0.999506i \(0.510009\pi\)
\(242\) −5.48793 −0.352778
\(243\) −1.00000 −0.0641500
\(244\) 12.7237 0.814548
\(245\) 0 0
\(246\) −0.616789 −0.0393250
\(247\) −30.9753 −1.97091
\(248\) −6.37859 −0.405041
\(249\) −14.5712 −0.923414
\(250\) 0 0
\(251\) 22.5387 1.42263 0.711314 0.702874i \(-0.248100\pi\)
0.711314 + 0.702874i \(0.248100\pi\)
\(252\) 3.20073 0.201627
\(253\) 35.5235 2.23335
\(254\) 0.368851 0.0231438
\(255\) 0 0
\(256\) 11.6876 0.730473
\(257\) 7.60490 0.474380 0.237190 0.971463i \(-0.423774\pi\)
0.237190 + 0.971463i \(0.423774\pi\)
\(258\) −0.213251 −0.0132764
\(259\) 9.61846 0.597662
\(260\) 0 0
\(261\) 3.23476 0.200227
\(262\) −3.68536 −0.227682
\(263\) −20.4512 −1.26108 −0.630538 0.776158i \(-0.717166\pi\)
−0.630538 + 0.776158i \(0.717166\pi\)
\(264\) −5.46500 −0.336348
\(265\) 0 0
\(266\) −2.62190 −0.160759
\(267\) 7.03976 0.430826
\(268\) −17.8227 −1.08869
\(269\) −4.79575 −0.292402 −0.146201 0.989255i \(-0.546705\pi\)
−0.146201 + 0.989255i \(0.546705\pi\)
\(270\) 0 0
\(271\) −10.8394 −0.658444 −0.329222 0.944253i \(-0.606786\pi\)
−0.329222 + 0.944253i \(0.606786\pi\)
\(272\) 20.8072 1.26162
\(273\) 7.59740 0.459816
\(274\) −2.96018 −0.178831
\(275\) 0 0
\(276\) 11.8175 0.711333
\(277\) −28.9500 −1.73944 −0.869719 0.493548i \(-0.835700\pi\)
−0.869719 + 0.493548i \(0.835700\pi\)
\(278\) −1.10854 −0.0664858
\(279\) −6.81968 −0.408284
\(280\) 0 0
\(281\) −24.7906 −1.47888 −0.739441 0.673222i \(-0.764910\pi\)
−0.739441 + 0.673222i \(0.764910\pi\)
\(282\) 0.237165 0.0141230
\(283\) 8.99344 0.534604 0.267302 0.963613i \(-0.413868\pi\)
0.267302 + 0.963613i \(0.413868\pi\)
\(284\) −7.73953 −0.459257
\(285\) 0 0
\(286\) −6.39349 −0.378055
\(287\) 4.28246 0.252786
\(288\) −2.74001 −0.161457
\(289\) 15.2194 0.895258
\(290\) 0 0
\(291\) 8.59813 0.504031
\(292\) −23.7561 −1.39022
\(293\) −0.665090 −0.0388550 −0.0194275 0.999811i \(-0.506184\pi\)
−0.0194275 + 0.999811i \(0.506184\pi\)
\(294\) −1.01707 −0.0593170
\(295\) 0 0
\(296\) −5.46334 −0.317550
\(297\) −5.84292 −0.339041
\(298\) 4.75606 0.275511
\(299\) 28.0507 1.62221
\(300\) 0 0
\(301\) 1.48064 0.0853424
\(302\) 4.11512 0.236798
\(303\) −7.31500 −0.420236
\(304\) −24.6101 −1.41149
\(305\) 0 0
\(306\) −1.34620 −0.0769571
\(307\) −0.814634 −0.0464936 −0.0232468 0.999730i \(-0.507400\pi\)
−0.0232468 + 0.999730i \(0.507400\pi\)
\(308\) 18.7016 1.06562
\(309\) −0.918971 −0.0522784
\(310\) 0 0
\(311\) 4.78222 0.271175 0.135587 0.990765i \(-0.456708\pi\)
0.135587 + 0.990765i \(0.456708\pi\)
\(312\) −4.31537 −0.244310
\(313\) −6.26198 −0.353948 −0.176974 0.984216i \(-0.556631\pi\)
−0.176974 + 0.984216i \(0.556631\pi\)
\(314\) −2.14067 −0.120805
\(315\) 0 0
\(316\) −29.3569 −1.65145
\(317\) 15.8462 0.890011 0.445005 0.895528i \(-0.353202\pi\)
0.445005 + 0.895528i \(0.353202\pi\)
\(318\) −0.975482 −0.0547023
\(319\) 18.9005 1.05822
\(320\) 0 0
\(321\) −9.87708 −0.551285
\(322\) 2.37435 0.132317
\(323\) −38.1081 −2.12039
\(324\) −1.94375 −0.107986
\(325\) 0 0
\(326\) −0.0435875 −0.00241409
\(327\) 19.0166 1.05162
\(328\) −2.43246 −0.134310
\(329\) −1.64667 −0.0907841
\(330\) 0 0
\(331\) 31.2797 1.71929 0.859643 0.510895i \(-0.170686\pi\)
0.859643 + 0.510895i \(0.170686\pi\)
\(332\) −28.3229 −1.55442
\(333\) −5.84115 −0.320093
\(334\) −3.98194 −0.217882
\(335\) 0 0
\(336\) 6.03618 0.329301
\(337\) 18.1866 0.990689 0.495344 0.868697i \(-0.335042\pi\)
0.495344 + 0.868697i \(0.335042\pi\)
\(338\) −1.96539 −0.106903
\(339\) 4.31060 0.234120
\(340\) 0 0
\(341\) −39.8468 −2.15783
\(342\) 1.59224 0.0860986
\(343\) 18.5884 1.00368
\(344\) −0.841010 −0.0453442
\(345\) 0 0
\(346\) 2.19646 0.118083
\(347\) −16.2998 −0.875021 −0.437511 0.899213i \(-0.644140\pi\)
−0.437511 + 0.899213i \(0.644140\pi\)
\(348\) 6.28758 0.337049
\(349\) 26.4685 1.41683 0.708413 0.705798i \(-0.249411\pi\)
0.708413 + 0.705798i \(0.249411\pi\)
\(350\) 0 0
\(351\) −4.61379 −0.246266
\(352\) −16.0097 −0.853318
\(353\) 30.7241 1.63528 0.817638 0.575732i \(-0.195283\pi\)
0.817638 + 0.575732i \(0.195283\pi\)
\(354\) 1.64581 0.0874739
\(355\) 0 0
\(356\) 13.6835 0.725226
\(357\) 9.34687 0.494689
\(358\) 0.117621 0.00621647
\(359\) −24.6709 −1.30208 −0.651041 0.759043i \(-0.725667\pi\)
−0.651041 + 0.759043i \(0.725667\pi\)
\(360\) 0 0
\(361\) 26.0731 1.37227
\(362\) 2.57586 0.135384
\(363\) −23.1397 −1.21452
\(364\) 14.7675 0.774026
\(365\) 0 0
\(366\) −1.55246 −0.0811486
\(367\) 0.973102 0.0507955 0.0253977 0.999677i \(-0.491915\pi\)
0.0253977 + 0.999677i \(0.491915\pi\)
\(368\) 22.2864 1.16176
\(369\) −2.60067 −0.135386
\(370\) 0 0
\(371\) 6.77292 0.351633
\(372\) −13.2558 −0.687280
\(373\) −8.61972 −0.446312 −0.223156 0.974783i \(-0.571636\pi\)
−0.223156 + 0.974783i \(0.571636\pi\)
\(374\) −7.86573 −0.406727
\(375\) 0 0
\(376\) 0.935320 0.0482355
\(377\) 14.9245 0.768651
\(378\) −0.390533 −0.0200869
\(379\) 5.44944 0.279919 0.139960 0.990157i \(-0.455303\pi\)
0.139960 + 0.990157i \(0.455303\pi\)
\(380\) 0 0
\(381\) 1.55525 0.0796779
\(382\) 1.84709 0.0945054
\(383\) 6.45312 0.329739 0.164869 0.986315i \(-0.447280\pi\)
0.164869 + 0.986315i \(0.447280\pi\)
\(384\) −7.06465 −0.360516
\(385\) 0 0
\(386\) 2.35169 0.119698
\(387\) −0.899167 −0.0457072
\(388\) 16.7126 0.848456
\(389\) 22.5677 1.14423 0.572113 0.820175i \(-0.306124\pi\)
0.572113 + 0.820175i \(0.306124\pi\)
\(390\) 0 0
\(391\) 34.5100 1.74525
\(392\) −4.01109 −0.202591
\(393\) −15.5392 −0.783849
\(394\) −5.57034 −0.280630
\(395\) 0 0
\(396\) −11.3572 −0.570720
\(397\) 0.566161 0.0284148 0.0142074 0.999899i \(-0.495477\pi\)
0.0142074 + 0.999899i \(0.495477\pi\)
\(398\) −3.33705 −0.167271
\(399\) −11.0552 −0.553452
\(400\) 0 0
\(401\) −23.5606 −1.17656 −0.588280 0.808658i \(-0.700195\pi\)
−0.588280 + 0.808658i \(0.700195\pi\)
\(402\) 2.17462 0.108460
\(403\) −31.4645 −1.56736
\(404\) −14.2185 −0.707399
\(405\) 0 0
\(406\) 1.26328 0.0626957
\(407\) −34.1293 −1.69173
\(408\) −5.30908 −0.262838
\(409\) 37.8278 1.87046 0.935232 0.354036i \(-0.115191\pi\)
0.935232 + 0.354036i \(0.115191\pi\)
\(410\) 0 0
\(411\) −12.4815 −0.615667
\(412\) −1.78625 −0.0880023
\(413\) −11.4271 −0.562292
\(414\) −1.44191 −0.0708658
\(415\) 0 0
\(416\) −12.6418 −0.619817
\(417\) −4.67413 −0.228893
\(418\) 9.30335 0.455042
\(419\) 11.1859 0.546465 0.273233 0.961948i \(-0.411907\pi\)
0.273233 + 0.961948i \(0.411907\pi\)
\(420\) 0 0
\(421\) 39.0593 1.90363 0.951817 0.306668i \(-0.0992141\pi\)
0.951817 + 0.306668i \(0.0992141\pi\)
\(422\) −1.08100 −0.0526223
\(423\) 1.00000 0.0486217
\(424\) −3.84706 −0.186830
\(425\) 0 0
\(426\) 0.944331 0.0457530
\(427\) 10.7790 0.521632
\(428\) −19.1986 −0.928000
\(429\) −26.9580 −1.30154
\(430\) 0 0
\(431\) 32.6413 1.57228 0.786139 0.618049i \(-0.212077\pi\)
0.786139 + 0.618049i \(0.212077\pi\)
\(432\) −3.66568 −0.176365
\(433\) 24.6811 1.18610 0.593050 0.805166i \(-0.297924\pi\)
0.593050 + 0.805166i \(0.297924\pi\)
\(434\) −2.66331 −0.127843
\(435\) 0 0
\(436\) 36.9635 1.77023
\(437\) −40.8174 −1.95256
\(438\) 2.89858 0.138500
\(439\) 30.1029 1.43673 0.718366 0.695665i \(-0.244890\pi\)
0.718366 + 0.695665i \(0.244890\pi\)
\(440\) 0 0
\(441\) −4.28847 −0.204213
\(442\) −6.21107 −0.295431
\(443\) −38.1560 −1.81285 −0.906424 0.422370i \(-0.861198\pi\)
−0.906424 + 0.422370i \(0.861198\pi\)
\(444\) −11.3537 −0.538825
\(445\) 0 0
\(446\) −1.98874 −0.0941698
\(447\) 20.0538 0.948511
\(448\) 11.0023 0.519809
\(449\) 0.588803 0.0277873 0.0138937 0.999903i \(-0.495577\pi\)
0.0138937 + 0.999903i \(0.495577\pi\)
\(450\) 0 0
\(451\) −15.1955 −0.715529
\(452\) 8.37874 0.394103
\(453\) 17.3513 0.815234
\(454\) 0.301234 0.0141376
\(455\) 0 0
\(456\) 6.27941 0.294060
\(457\) 36.8925 1.72576 0.862879 0.505411i \(-0.168659\pi\)
0.862879 + 0.505411i \(0.168659\pi\)
\(458\) 1.42170 0.0664318
\(459\) −5.67621 −0.264943
\(460\) 0 0
\(461\) 3.63431 0.169267 0.0846333 0.996412i \(-0.473028\pi\)
0.0846333 + 0.996412i \(0.473028\pi\)
\(462\) −2.28186 −0.106162
\(463\) 1.66199 0.0772393 0.0386197 0.999254i \(-0.487704\pi\)
0.0386197 + 0.999254i \(0.487704\pi\)
\(464\) 11.8576 0.550475
\(465\) 0 0
\(466\) 2.37123 0.109845
\(467\) 27.7300 1.28319 0.641596 0.767043i \(-0.278273\pi\)
0.641596 + 0.767043i \(0.278273\pi\)
\(468\) −8.96806 −0.414549
\(469\) −15.0987 −0.697193
\(470\) 0 0
\(471\) −9.02606 −0.415899
\(472\) 6.49068 0.298758
\(473\) −5.25376 −0.241568
\(474\) 3.58195 0.164524
\(475\) 0 0
\(476\) 18.1680 0.832729
\(477\) −4.11309 −0.188326
\(478\) 1.76357 0.0806637
\(479\) 31.8914 1.45716 0.728578 0.684962i \(-0.240181\pi\)
0.728578 + 0.684962i \(0.240181\pi\)
\(480\) 0 0
\(481\) −26.9498 −1.22880
\(482\) 0.231499 0.0105445
\(483\) 10.0114 0.455533
\(484\) −44.9779 −2.04445
\(485\) 0 0
\(486\) 0.237165 0.0107580
\(487\) −7.31388 −0.331424 −0.165712 0.986174i \(-0.552992\pi\)
−0.165712 + 0.986174i \(0.552992\pi\)
\(488\) −6.12253 −0.277154
\(489\) −0.183786 −0.00831107
\(490\) 0 0
\(491\) 5.89156 0.265882 0.132941 0.991124i \(-0.457558\pi\)
0.132941 + 0.991124i \(0.457558\pi\)
\(492\) −5.05506 −0.227900
\(493\) 18.3612 0.826946
\(494\) 7.34627 0.330524
\(495\) 0 0
\(496\) −24.9988 −1.12248
\(497\) −6.55664 −0.294105
\(498\) 3.45579 0.154857
\(499\) −23.8545 −1.06787 −0.533936 0.845525i \(-0.679288\pi\)
−0.533936 + 0.845525i \(0.679288\pi\)
\(500\) 0 0
\(501\) −16.7897 −0.750110
\(502\) −5.34539 −0.238576
\(503\) 33.9274 1.51275 0.756373 0.654140i \(-0.226969\pi\)
0.756373 + 0.654140i \(0.226969\pi\)
\(504\) −1.54017 −0.0686045
\(505\) 0 0
\(506\) −8.42494 −0.374534
\(507\) −8.28702 −0.368039
\(508\) 3.02302 0.134125
\(509\) 2.48551 0.110168 0.0550842 0.998482i \(-0.482457\pi\)
0.0550842 + 0.998482i \(0.482457\pi\)
\(510\) 0 0
\(511\) −20.1253 −0.890291
\(512\) −16.9012 −0.746934
\(513\) 6.71365 0.296415
\(514\) −1.80362 −0.0795541
\(515\) 0 0
\(516\) −1.74776 −0.0769408
\(517\) 5.84292 0.256971
\(518\) −2.28116 −0.100229
\(519\) 9.26133 0.406527
\(520\) 0 0
\(521\) −11.2373 −0.492315 −0.246158 0.969230i \(-0.579168\pi\)
−0.246158 + 0.969230i \(0.579168\pi\)
\(522\) −0.767172 −0.0335782
\(523\) −38.2596 −1.67298 −0.836489 0.547984i \(-0.815395\pi\)
−0.836489 + 0.547984i \(0.815395\pi\)
\(524\) −30.2044 −1.31948
\(525\) 0 0
\(526\) 4.85031 0.211484
\(527\) −38.7099 −1.68623
\(528\) −21.4183 −0.932111
\(529\) 13.9634 0.607106
\(530\) 0 0
\(531\) 6.93952 0.301150
\(532\) −21.4886 −0.931647
\(533\) −11.9989 −0.519732
\(534\) −1.66958 −0.0722500
\(535\) 0 0
\(536\) 8.57615 0.370433
\(537\) 0.495946 0.0214017
\(538\) 1.13739 0.0490362
\(539\) −25.0572 −1.07929
\(540\) 0 0
\(541\) 45.3760 1.95087 0.975434 0.220294i \(-0.0707017\pi\)
0.975434 + 0.220294i \(0.0707017\pi\)
\(542\) 2.57072 0.110422
\(543\) 10.8611 0.466093
\(544\) −15.5529 −0.666825
\(545\) 0 0
\(546\) −1.80184 −0.0771115
\(547\) −33.9643 −1.45221 −0.726104 0.687585i \(-0.758671\pi\)
−0.726104 + 0.687585i \(0.758671\pi\)
\(548\) −24.2610 −1.03638
\(549\) −6.54592 −0.279373
\(550\) 0 0
\(551\) −21.7170 −0.925177
\(552\) −5.68652 −0.242034
\(553\) −24.8700 −1.05758
\(554\) 6.86593 0.291705
\(555\) 0 0
\(556\) −9.08535 −0.385305
\(557\) −27.9359 −1.18368 −0.591841 0.806055i \(-0.701599\pi\)
−0.591841 + 0.806055i \(0.701599\pi\)
\(558\) 1.61739 0.0684696
\(559\) −4.14857 −0.175466
\(560\) 0 0
\(561\) −33.1657 −1.40025
\(562\) 5.87946 0.248010
\(563\) 40.1590 1.69250 0.846250 0.532787i \(-0.178855\pi\)
0.846250 + 0.532787i \(0.178855\pi\)
\(564\) 1.94375 0.0818468
\(565\) 0 0
\(566\) −2.13293 −0.0896537
\(567\) −1.64667 −0.0691538
\(568\) 3.72421 0.156264
\(569\) 19.4727 0.816336 0.408168 0.912907i \(-0.366168\pi\)
0.408168 + 0.912907i \(0.366168\pi\)
\(570\) 0 0
\(571\) −10.7006 −0.447807 −0.223904 0.974611i \(-0.571880\pi\)
−0.223904 + 0.974611i \(0.571880\pi\)
\(572\) −52.3997 −2.19094
\(573\) 7.78821 0.325357
\(574\) −1.01565 −0.0423924
\(575\) 0 0
\(576\) −6.68153 −0.278397
\(577\) −5.73302 −0.238669 −0.119334 0.992854i \(-0.538076\pi\)
−0.119334 + 0.992854i \(0.538076\pi\)
\(578\) −3.60951 −0.150136
\(579\) 9.91584 0.412088
\(580\) 0 0
\(581\) −23.9940 −0.995441
\(582\) −2.03918 −0.0845266
\(583\) −24.0325 −0.995323
\(584\) 11.4313 0.473030
\(585\) 0 0
\(586\) 0.157736 0.00651602
\(587\) −16.3376 −0.674323 −0.337162 0.941447i \(-0.609467\pi\)
−0.337162 + 0.941447i \(0.609467\pi\)
\(588\) −8.33572 −0.343759
\(589\) 45.7849 1.88654
\(590\) 0 0
\(591\) −23.4872 −0.966134
\(592\) −21.4118 −0.880018
\(593\) 11.9853 0.492178 0.246089 0.969247i \(-0.420854\pi\)
0.246089 + 0.969247i \(0.420854\pi\)
\(594\) 1.38574 0.0568575
\(595\) 0 0
\(596\) 38.9796 1.59667
\(597\) −14.0706 −0.575871
\(598\) −6.65265 −0.272047
\(599\) 23.8481 0.974408 0.487204 0.873288i \(-0.338017\pi\)
0.487204 + 0.873288i \(0.338017\pi\)
\(600\) 0 0
\(601\) −37.4995 −1.52964 −0.764819 0.644246i \(-0.777171\pi\)
−0.764819 + 0.644246i \(0.777171\pi\)
\(602\) −0.351155 −0.0143120
\(603\) 9.16921 0.373399
\(604\) 33.7266 1.37232
\(605\) 0 0
\(606\) 1.73486 0.0704740
\(607\) 40.9079 1.66040 0.830200 0.557466i \(-0.188226\pi\)
0.830200 + 0.557466i \(0.188226\pi\)
\(608\) 18.3955 0.746035
\(609\) 5.32660 0.215845
\(610\) 0 0
\(611\) 4.61379 0.186654
\(612\) −11.0332 −0.445989
\(613\) 32.7377 1.32226 0.661132 0.750269i \(-0.270076\pi\)
0.661132 + 0.750269i \(0.270076\pi\)
\(614\) 0.193203 0.00779702
\(615\) 0 0
\(616\) −8.99908 −0.362583
\(617\) 43.3902 1.74682 0.873412 0.486983i \(-0.161903\pi\)
0.873412 + 0.486983i \(0.161903\pi\)
\(618\) 0.217948 0.00876714
\(619\) 1.51113 0.0607376 0.0303688 0.999539i \(-0.490332\pi\)
0.0303688 + 0.999539i \(0.490332\pi\)
\(620\) 0 0
\(621\) −6.07976 −0.243972
\(622\) −1.13417 −0.0454763
\(623\) 11.5922 0.464431
\(624\) −16.9127 −0.677048
\(625\) 0 0
\(626\) 1.48512 0.0593575
\(627\) 39.2273 1.56659
\(628\) −17.5444 −0.700099
\(629\) −33.1556 −1.32200
\(630\) 0 0
\(631\) 44.0838 1.75495 0.877475 0.479623i \(-0.159227\pi\)
0.877475 + 0.479623i \(0.159227\pi\)
\(632\) 14.1263 0.561915
\(633\) −4.55801 −0.181165
\(634\) −3.75816 −0.149256
\(635\) 0 0
\(636\) −7.99483 −0.317016
\(637\) −19.7861 −0.783952
\(638\) −4.48253 −0.177465
\(639\) 3.98175 0.157515
\(640\) 0 0
\(641\) −41.9622 −1.65741 −0.828705 0.559686i \(-0.810922\pi\)
−0.828705 + 0.559686i \(0.810922\pi\)
\(642\) 2.34250 0.0924511
\(643\) −38.3467 −1.51225 −0.756124 0.654428i \(-0.772909\pi\)
−0.756124 + 0.654428i \(0.772909\pi\)
\(644\) 19.4596 0.766817
\(645\) 0 0
\(646\) 9.03791 0.355592
\(647\) 20.6208 0.810687 0.405343 0.914164i \(-0.367152\pi\)
0.405343 + 0.914164i \(0.367152\pi\)
\(648\) 0.935320 0.0367429
\(649\) 40.5471 1.59161
\(650\) 0 0
\(651\) −11.2298 −0.440130
\(652\) −0.357234 −0.0139904
\(653\) 2.96171 0.115901 0.0579504 0.998319i \(-0.481543\pi\)
0.0579504 + 0.998319i \(0.481543\pi\)
\(654\) −4.51007 −0.176358
\(655\) 0 0
\(656\) −9.53323 −0.372210
\(657\) 12.2218 0.476818
\(658\) 0.390533 0.0152246
\(659\) −34.5759 −1.34689 −0.673444 0.739238i \(-0.735186\pi\)
−0.673444 + 0.739238i \(0.735186\pi\)
\(660\) 0 0
\(661\) 2.31049 0.0898675 0.0449338 0.998990i \(-0.485692\pi\)
0.0449338 + 0.998990i \(0.485692\pi\)
\(662\) −7.41845 −0.288326
\(663\) −26.1888 −1.01709
\(664\) 13.6288 0.528899
\(665\) 0 0
\(666\) 1.38532 0.0536799
\(667\) 19.6666 0.761492
\(668\) −32.6351 −1.26269
\(669\) −8.38549 −0.324202
\(670\) 0 0
\(671\) −38.2473 −1.47652
\(672\) −4.51191 −0.174051
\(673\) −45.8261 −1.76647 −0.883233 0.468935i \(-0.844638\pi\)
−0.883233 + 0.468935i \(0.844638\pi\)
\(674\) −4.31323 −0.166140
\(675\) 0 0
\(676\) −16.1079 −0.619535
\(677\) −6.48999 −0.249431 −0.124715 0.992193i \(-0.539802\pi\)
−0.124715 + 0.992193i \(0.539802\pi\)
\(678\) −1.02232 −0.0392621
\(679\) 14.1583 0.543346
\(680\) 0 0
\(681\) 1.27015 0.0486721
\(682\) 9.45028 0.361870
\(683\) 12.3351 0.471990 0.235995 0.971754i \(-0.424165\pi\)
0.235995 + 0.971754i \(0.424165\pi\)
\(684\) 13.0497 0.498967
\(685\) 0 0
\(686\) −4.40852 −0.168318
\(687\) 5.99457 0.228707
\(688\) −3.29606 −0.125661
\(689\) −18.9769 −0.722963
\(690\) 0 0
\(691\) 12.5415 0.477100 0.238550 0.971130i \(-0.423328\pi\)
0.238550 + 0.971130i \(0.423328\pi\)
\(692\) 18.0017 0.684323
\(693\) −9.62138 −0.365486
\(694\) 3.86575 0.146742
\(695\) 0 0
\(696\) −3.02554 −0.114683
\(697\) −14.7620 −0.559149
\(698\) −6.27740 −0.237603
\(699\) 9.99821 0.378167
\(700\) 0 0
\(701\) −30.4761 −1.15107 −0.575534 0.817778i \(-0.695206\pi\)
−0.575534 + 0.817778i \(0.695206\pi\)
\(702\) 1.09423 0.0412990
\(703\) 39.2154 1.47904
\(704\) −39.0396 −1.47136
\(705\) 0 0
\(706\) −7.28667 −0.274238
\(707\) −12.0454 −0.453015
\(708\) 13.4887 0.506937
\(709\) 35.8380 1.34593 0.672963 0.739676i \(-0.265021\pi\)
0.672963 + 0.739676i \(0.265021\pi\)
\(710\) 0 0
\(711\) 15.1032 0.566414
\(712\) −6.58443 −0.246762
\(713\) −41.4620 −1.55276
\(714\) −2.21675 −0.0829598
\(715\) 0 0
\(716\) 0.963997 0.0360263
\(717\) 7.43603 0.277704
\(718\) 5.85108 0.218360
\(719\) 33.7554 1.25886 0.629432 0.777056i \(-0.283288\pi\)
0.629432 + 0.777056i \(0.283288\pi\)
\(720\) 0 0
\(721\) −1.51324 −0.0563562
\(722\) −6.18362 −0.230131
\(723\) 0.976110 0.0363019
\(724\) 21.1112 0.784592
\(725\) 0 0
\(726\) 5.48793 0.203676
\(727\) −37.1839 −1.37907 −0.689537 0.724250i \(-0.742186\pi\)
−0.689537 + 0.724250i \(0.742186\pi\)
\(728\) −7.10600 −0.263366
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −5.10386 −0.188773
\(732\) −12.7237 −0.470280
\(733\) −24.2578 −0.895982 −0.447991 0.894038i \(-0.647860\pi\)
−0.447991 + 0.894038i \(0.647860\pi\)
\(734\) −0.230786 −0.00851846
\(735\) 0 0
\(736\) −16.6586 −0.614044
\(737\) 53.5750 1.97346
\(738\) 0.616789 0.0227043
\(739\) −20.9387 −0.770241 −0.385120 0.922866i \(-0.625840\pi\)
−0.385120 + 0.922866i \(0.625840\pi\)
\(740\) 0 0
\(741\) 30.9753 1.13791
\(742\) −1.60630 −0.0589691
\(743\) 6.39891 0.234753 0.117377 0.993087i \(-0.462552\pi\)
0.117377 + 0.993087i \(0.462552\pi\)
\(744\) 6.37859 0.233850
\(745\) 0 0
\(746\) 2.04430 0.0748470
\(747\) 14.5712 0.533133
\(748\) −64.4658 −2.35710
\(749\) −16.2643 −0.594286
\(750\) 0 0
\(751\) −25.7017 −0.937868 −0.468934 0.883233i \(-0.655362\pi\)
−0.468934 + 0.883233i \(0.655362\pi\)
\(752\) 3.66568 0.133674
\(753\) −22.5387 −0.821355
\(754\) −3.53957 −0.128903
\(755\) 0 0
\(756\) −3.20073 −0.116409
\(757\) −37.9646 −1.37985 −0.689923 0.723882i \(-0.742356\pi\)
−0.689923 + 0.723882i \(0.742356\pi\)
\(758\) −1.29242 −0.0469427
\(759\) −35.5235 −1.28942
\(760\) 0 0
\(761\) −25.6250 −0.928906 −0.464453 0.885598i \(-0.653749\pi\)
−0.464453 + 0.885598i \(0.653749\pi\)
\(762\) −0.368851 −0.0133621
\(763\) 31.3141 1.13365
\(764\) 15.1383 0.547686
\(765\) 0 0
\(766\) −1.53045 −0.0552976
\(767\) 32.0175 1.15608
\(768\) −11.6876 −0.421739
\(769\) 13.0484 0.470538 0.235269 0.971930i \(-0.424403\pi\)
0.235269 + 0.971930i \(0.424403\pi\)
\(770\) 0 0
\(771\) −7.60490 −0.273884
\(772\) 19.2739 0.693684
\(773\) −6.21192 −0.223427 −0.111714 0.993740i \(-0.535634\pi\)
−0.111714 + 0.993740i \(0.535634\pi\)
\(774\) 0.213251 0.00766515
\(775\) 0 0
\(776\) −8.04201 −0.288691
\(777\) −9.61846 −0.345060
\(778\) −5.35226 −0.191888
\(779\) 17.4600 0.625569
\(780\) 0 0
\(781\) 23.2650 0.832488
\(782\) −8.18456 −0.292679
\(783\) −3.23476 −0.115601
\(784\) −15.7201 −0.561434
\(785\) 0 0
\(786\) 3.68536 0.131452
\(787\) 42.2577 1.50633 0.753163 0.657834i \(-0.228527\pi\)
0.753163 + 0.657834i \(0.228527\pi\)
\(788\) −45.6533 −1.62633
\(789\) 20.4512 0.728083
\(790\) 0 0
\(791\) 7.09815 0.252381
\(792\) 5.46500 0.194190
\(793\) −30.2015 −1.07249
\(794\) −0.134274 −0.00476520
\(795\) 0 0
\(796\) −27.3498 −0.969386
\(797\) 17.4501 0.618113 0.309056 0.951044i \(-0.399987\pi\)
0.309056 + 0.951044i \(0.399987\pi\)
\(798\) 2.62190 0.0928144
\(799\) 5.67621 0.200810
\(800\) 0 0
\(801\) −7.03976 −0.248738
\(802\) 5.58775 0.197310
\(803\) 71.4110 2.52004
\(804\) 17.8227 0.628557
\(805\) 0 0
\(806\) 7.46229 0.262848
\(807\) 4.79575 0.168818
\(808\) 6.84187 0.240696
\(809\) 7.23858 0.254495 0.127247 0.991871i \(-0.459386\pi\)
0.127247 + 0.991871i \(0.459386\pi\)
\(810\) 0 0
\(811\) −53.0051 −1.86126 −0.930630 0.365960i \(-0.880741\pi\)
−0.930630 + 0.365960i \(0.880741\pi\)
\(812\) 10.3536 0.363340
\(813\) 10.8394 0.380153
\(814\) 8.09429 0.283705
\(815\) 0 0
\(816\) −20.8072 −0.728397
\(817\) 6.03669 0.211197
\(818\) −8.97143 −0.313679
\(819\) −7.59740 −0.265475
\(820\) 0 0
\(821\) −28.7613 −1.00378 −0.501889 0.864932i \(-0.667361\pi\)
−0.501889 + 0.864932i \(0.667361\pi\)
\(822\) 2.96018 0.103248
\(823\) −1.52418 −0.0531296 −0.0265648 0.999647i \(-0.508457\pi\)
−0.0265648 + 0.999647i \(0.508457\pi\)
\(824\) 0.859532 0.0299432
\(825\) 0 0
\(826\) 2.71012 0.0942970
\(827\) 38.7593 1.34779 0.673897 0.738825i \(-0.264619\pi\)
0.673897 + 0.738825i \(0.264619\pi\)
\(828\) −11.8175 −0.410688
\(829\) 30.8699 1.07216 0.536079 0.844168i \(-0.319905\pi\)
0.536079 + 0.844168i \(0.319905\pi\)
\(830\) 0 0
\(831\) 28.9500 1.00426
\(832\) −30.8271 −1.06874
\(833\) −24.3422 −0.843409
\(834\) 1.10854 0.0383856
\(835\) 0 0
\(836\) 76.2482 2.63710
\(837\) 6.81968 0.235723
\(838\) −2.65290 −0.0916428
\(839\) −36.9782 −1.27663 −0.638314 0.769776i \(-0.720368\pi\)
−0.638314 + 0.769776i \(0.720368\pi\)
\(840\) 0 0
\(841\) −18.5363 −0.639183
\(842\) −9.26350 −0.319241
\(843\) 24.7906 0.853833
\(844\) −8.85965 −0.304962
\(845\) 0 0
\(846\) −0.237165 −0.00815390
\(847\) −38.1036 −1.30925
\(848\) −15.0773 −0.517756
\(849\) −8.99344 −0.308654
\(850\) 0 0
\(851\) −35.5127 −1.21736
\(852\) 7.73953 0.265152
\(853\) 2.71422 0.0929333 0.0464666 0.998920i \(-0.485204\pi\)
0.0464666 + 0.998920i \(0.485204\pi\)
\(854\) −2.55640 −0.0874783
\(855\) 0 0
\(856\) 9.23824 0.315756
\(857\) 35.9293 1.22732 0.613660 0.789570i \(-0.289696\pi\)
0.613660 + 0.789570i \(0.289696\pi\)
\(858\) 6.39349 0.218270
\(859\) −11.5320 −0.393465 −0.196733 0.980457i \(-0.563033\pi\)
−0.196733 + 0.980457i \(0.563033\pi\)
\(860\) 0 0
\(861\) −4.28246 −0.145946
\(862\) −7.74139 −0.263673
\(863\) −29.5087 −1.00449 −0.502244 0.864726i \(-0.667492\pi\)
−0.502244 + 0.864726i \(0.667492\pi\)
\(864\) 2.74001 0.0932171
\(865\) 0 0
\(866\) −5.85350 −0.198910
\(867\) −15.2194 −0.516877
\(868\) −21.8279 −0.740888
\(869\) 88.2468 2.99357
\(870\) 0 0
\(871\) 42.3048 1.43344
\(872\) −17.7866 −0.602330
\(873\) −8.59813 −0.291003
\(874\) 9.68045 0.327446
\(875\) 0 0
\(876\) 23.7561 0.802646
\(877\) −10.2327 −0.345535 −0.172767 0.984963i \(-0.555271\pi\)
−0.172767 + 0.984963i \(0.555271\pi\)
\(878\) −7.13935 −0.240941
\(879\) 0.665090 0.0224329
\(880\) 0 0
\(881\) 12.3238 0.415198 0.207599 0.978214i \(-0.433435\pi\)
0.207599 + 0.978214i \(0.433435\pi\)
\(882\) 1.01707 0.0342467
\(883\) −38.2214 −1.28625 −0.643127 0.765760i \(-0.722363\pi\)
−0.643127 + 0.765760i \(0.722363\pi\)
\(884\) −50.9046 −1.71211
\(885\) 0 0
\(886\) 9.04928 0.304016
\(887\) 1.00570 0.0337683 0.0168841 0.999857i \(-0.494625\pi\)
0.0168841 + 0.999857i \(0.494625\pi\)
\(888\) 5.46334 0.183338
\(889\) 2.56099 0.0858928
\(890\) 0 0
\(891\) 5.84292 0.195745
\(892\) −16.2993 −0.545741
\(893\) −6.71365 −0.224664
\(894\) −4.75606 −0.159066
\(895\) 0 0
\(896\) −11.6332 −0.388637
\(897\) −28.0507 −0.936586
\(898\) −0.139644 −0.00465997
\(899\) −22.0600 −0.735743
\(900\) 0 0
\(901\) −23.3468 −0.777794
\(902\) 3.60385 0.119995
\(903\) −1.48064 −0.0492725
\(904\) −4.03179 −0.134095
\(905\) 0 0
\(906\) −4.11512 −0.136716
\(907\) −23.2637 −0.772457 −0.386229 0.922403i \(-0.626222\pi\)
−0.386229 + 0.922403i \(0.626222\pi\)
\(908\) 2.46885 0.0819316
\(909\) 7.31500 0.242623
\(910\) 0 0
\(911\) −47.3133 −1.56756 −0.783779 0.621039i \(-0.786711\pi\)
−0.783779 + 0.621039i \(0.786711\pi\)
\(912\) 24.6101 0.814921
\(913\) 85.1385 2.81767
\(914\) −8.74961 −0.289411
\(915\) 0 0
\(916\) 11.6520 0.384992
\(917\) −25.5880 −0.844990
\(918\) 1.34620 0.0444312
\(919\) −5.25122 −0.173222 −0.0866108 0.996242i \(-0.527604\pi\)
−0.0866108 + 0.996242i \(0.527604\pi\)
\(920\) 0 0
\(921\) 0.814634 0.0268431
\(922\) −0.861931 −0.0283862
\(923\) 18.3709 0.604686
\(924\) −18.7016 −0.615237
\(925\) 0 0
\(926\) −0.394166 −0.0129531
\(927\) 0.918971 0.0301830
\(928\) −8.86328 −0.290952
\(929\) 7.19607 0.236095 0.118048 0.993008i \(-0.462336\pi\)
0.118048 + 0.993008i \(0.462336\pi\)
\(930\) 0 0
\(931\) 28.7913 0.943595
\(932\) 19.4341 0.636584
\(933\) −4.78222 −0.156563
\(934\) −6.57659 −0.215193
\(935\) 0 0
\(936\) 4.31537 0.141052
\(937\) 6.75704 0.220743 0.110371 0.993890i \(-0.464796\pi\)
0.110371 + 0.993890i \(0.464796\pi\)
\(938\) 3.58088 0.116920
\(939\) 6.26198 0.204352
\(940\) 0 0
\(941\) 19.8864 0.648278 0.324139 0.946010i \(-0.394925\pi\)
0.324139 + 0.946010i \(0.394925\pi\)
\(942\) 2.14067 0.0697467
\(943\) −15.8115 −0.514892
\(944\) 25.4381 0.827939
\(945\) 0 0
\(946\) 1.24601 0.0405113
\(947\) 57.2087 1.85903 0.929516 0.368782i \(-0.120225\pi\)
0.929516 + 0.368782i \(0.120225\pi\)
\(948\) 29.3569 0.953467
\(949\) 56.3887 1.83046
\(950\) 0 0
\(951\) −15.8462 −0.513848
\(952\) −8.74232 −0.283340
\(953\) −11.8906 −0.385174 −0.192587 0.981280i \(-0.561688\pi\)
−0.192587 + 0.981280i \(0.561688\pi\)
\(954\) 0.975482 0.0315824
\(955\) 0 0
\(956\) 14.4538 0.467469
\(957\) −18.9005 −0.610965
\(958\) −7.56353 −0.244367
\(959\) −20.5530 −0.663690
\(960\) 0 0
\(961\) 15.5080 0.500259
\(962\) 6.39155 0.206072
\(963\) 9.87708 0.318284
\(964\) 1.89732 0.0611085
\(965\) 0 0
\(966\) −2.37435 −0.0763934
\(967\) −16.0866 −0.517311 −0.258656 0.965970i \(-0.583279\pi\)
−0.258656 + 0.965970i \(0.583279\pi\)
\(968\) 21.6430 0.695634
\(969\) 38.1081 1.22421
\(970\) 0 0
\(971\) 18.8713 0.605608 0.302804 0.953053i \(-0.402077\pi\)
0.302804 + 0.953053i \(0.402077\pi\)
\(972\) 1.94375 0.0623459
\(973\) −7.69676 −0.246747
\(974\) 1.73460 0.0555801
\(975\) 0 0
\(976\) −23.9953 −0.768069
\(977\) −20.3542 −0.651190 −0.325595 0.945509i \(-0.605565\pi\)
−0.325595 + 0.945509i \(0.605565\pi\)
\(978\) 0.0435875 0.00139378
\(979\) −41.1327 −1.31461
\(980\) 0 0
\(981\) −19.0166 −0.607153
\(982\) −1.39727 −0.0445888
\(983\) −19.6706 −0.627396 −0.313698 0.949523i \(-0.601568\pi\)
−0.313698 + 0.949523i \(0.601568\pi\)
\(984\) 2.43246 0.0775441
\(985\) 0 0
\(986\) −4.35463 −0.138680
\(987\) 1.64667 0.0524142
\(988\) 60.2084 1.91548
\(989\) −5.46672 −0.173832
\(990\) 0 0
\(991\) 0.493118 0.0156644 0.00783221 0.999969i \(-0.497507\pi\)
0.00783221 + 0.999969i \(0.497507\pi\)
\(992\) 18.6860 0.593281
\(993\) −31.2797 −0.992630
\(994\) 1.55500 0.0493218
\(995\) 0 0
\(996\) 28.3229 0.897444
\(997\) −23.3345 −0.739010 −0.369505 0.929229i \(-0.620473\pi\)
−0.369505 + 0.929229i \(0.620473\pi\)
\(998\) 5.65744 0.179083
\(999\) 5.84115 0.184806
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 3525.2.a.be.1.4 yes 8
5.4 even 2 3525.2.a.bd.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3525.2.a.bd.1.5 8 5.4 even 2
3525.2.a.be.1.4 yes 8 1.1 even 1 trivial