Properties

Label 8470.2.a.da.1.4
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.38498000.1
Defining polynomial: \(x^{6} - x^{5} - 12 x^{4} + 5 x^{3} + 38 x^{2} - x - 19\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.752765\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.247235 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.247235 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.93888 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.247235 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.247235 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.93888 q^{9} -1.00000 q^{10} -0.247235 q^{12} +2.83439 q^{13} +1.00000 q^{14} +0.247235 q^{15} +1.00000 q^{16} -5.76549 q^{17} -2.93888 q^{18} +3.66941 q^{19} -1.00000 q^{20} -0.247235 q^{21} -7.17432 q^{23} -0.247235 q^{24} +1.00000 q^{25} +2.83439 q^{26} +1.46830 q^{27} +1.00000 q^{28} +1.41528 q^{29} +0.247235 q^{30} -0.356435 q^{31} +1.00000 q^{32} -5.76549 q^{34} -1.00000 q^{35} -2.93888 q^{36} +2.94752 q^{37} +3.66941 q^{38} -0.700760 q^{39} -1.00000 q^{40} +2.22578 q^{41} -0.247235 q^{42} +9.64018 q^{43} +2.93888 q^{45} -7.17432 q^{46} +3.13615 q^{47} -0.247235 q^{48} +1.00000 q^{49} +1.00000 q^{50} +1.42543 q^{51} +2.83439 q^{52} +0.874146 q^{53} +1.46830 q^{54} +1.00000 q^{56} -0.907206 q^{57} +1.41528 q^{58} -10.4962 q^{59} +0.247235 q^{60} -9.20135 q^{61} -0.356435 q^{62} -2.93888 q^{63} +1.00000 q^{64} -2.83439 q^{65} -13.5466 q^{67} -5.76549 q^{68} +1.77374 q^{69} -1.00000 q^{70} -12.5139 q^{71} -2.93888 q^{72} -7.19836 q^{73} +2.94752 q^{74} -0.247235 q^{75} +3.66941 q^{76} -0.700760 q^{78} +6.14508 q^{79} -1.00000 q^{80} +8.45361 q^{81} +2.22578 q^{82} +3.08217 q^{83} -0.247235 q^{84} +5.76549 q^{85} +9.64018 q^{86} -0.349905 q^{87} -5.75443 q^{89} +2.93888 q^{90} +2.83439 q^{91} -7.17432 q^{92} +0.0881230 q^{93} +3.13615 q^{94} -3.66941 q^{95} -0.247235 q^{96} +12.5278 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} - 5q^{3} + 6q^{4} - 6q^{5} - 5q^{6} + 6q^{7} + 6q^{8} + 11q^{9} + O(q^{10}) \) \( 6q + 6q^{2} - 5q^{3} + 6q^{4} - 6q^{5} - 5q^{6} + 6q^{7} + 6q^{8} + 11q^{9} - 6q^{10} - 5q^{12} + 6q^{14} + 5q^{15} + 6q^{16} - 15q^{17} + 11q^{18} - 13q^{19} - 6q^{20} - 5q^{21} - 2q^{23} - 5q^{24} + 6q^{25} - 26q^{27} + 6q^{28} - 4q^{29} + 5q^{30} - 2q^{31} + 6q^{32} - 15q^{34} - 6q^{35} + 11q^{36} - 13q^{38} - 30q^{39} - 6q^{40} - 17q^{41} - 5q^{42} + 15q^{43} - 11q^{45} - 2q^{46} - 18q^{47} - 5q^{48} + 6q^{49} + 6q^{50} + 6q^{51} + 22q^{53} - 26q^{54} + 6q^{56} - 2q^{57} - 4q^{58} - 7q^{59} + 5q^{60} - 20q^{61} - 2q^{62} + 11q^{63} + 6q^{64} - 29q^{67} - 15q^{68} + 12q^{69} - 6q^{70} - 2q^{71} + 11q^{72} + q^{73} - 5q^{75} - 13q^{76} - 30q^{78} - 12q^{79} - 6q^{80} + 22q^{81} - 17q^{82} - 35q^{83} - 5q^{84} + 15q^{85} + 15q^{86} - 29q^{89} - 11q^{90} - 2q^{92} + 8q^{93} - 18q^{94} + 13q^{95} - 5q^{96} - 19q^{97} + 6q^{98} + O(q^{100}) \)

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
\(3\) −0.247235 −0.142741 −0.0713705 0.997450i \(-0.522737\pi\)
−0.0713705 + 0.997450i \(0.522737\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −0.247235 −0.100933
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −2.93888 −0.979625
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −0.247235 −0.0713705
\(13\) 2.83439 0.786119 0.393060 0.919513i \(-0.371417\pi\)
0.393060 + 0.919513i \(0.371417\pi\)
\(14\) 1.00000 0.267261
\(15\) 0.247235 0.0638357
\(16\) 1.00000 0.250000
\(17\) −5.76549 −1.39834 −0.699168 0.714957i \(-0.746446\pi\)
−0.699168 + 0.714957i \(0.746446\pi\)
\(18\) −2.93888 −0.692699
\(19\) 3.66941 0.841821 0.420910 0.907102i \(-0.361711\pi\)
0.420910 + 0.907102i \(0.361711\pi\)
\(20\) −1.00000 −0.223607
\(21\) −0.247235 −0.0539510
\(22\) 0 0
\(23\) −7.17432 −1.49595 −0.747974 0.663728i \(-0.768973\pi\)
−0.747974 + 0.663728i \(0.768973\pi\)
\(24\) −0.247235 −0.0504666
\(25\) 1.00000 0.200000
\(26\) 2.83439 0.555870
\(27\) 1.46830 0.282574
\(28\) 1.00000 0.188982
\(29\) 1.41528 0.262810 0.131405 0.991329i \(-0.458051\pi\)
0.131405 + 0.991329i \(0.458051\pi\)
\(30\) 0.247235 0.0451387
\(31\) −0.356435 −0.0640176 −0.0320088 0.999488i \(-0.510190\pi\)
−0.0320088 + 0.999488i \(0.510190\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −5.76549 −0.988773
\(35\) −1.00000 −0.169031
\(36\) −2.93888 −0.489813
\(37\) 2.94752 0.484570 0.242285 0.970205i \(-0.422103\pi\)
0.242285 + 0.970205i \(0.422103\pi\)
\(38\) 3.66941 0.595257
\(39\) −0.700760 −0.112211
\(40\) −1.00000 −0.158114
\(41\) 2.22578 0.347608 0.173804 0.984780i \(-0.444394\pi\)
0.173804 + 0.984780i \(0.444394\pi\)
\(42\) −0.247235 −0.0381491
\(43\) 9.64018 1.47011 0.735057 0.678006i \(-0.237156\pi\)
0.735057 + 0.678006i \(0.237156\pi\)
\(44\) 0 0
\(45\) 2.93888 0.438102
\(46\) −7.17432 −1.05780
\(47\) 3.13615 0.457455 0.228727 0.973491i \(-0.426544\pi\)
0.228727 + 0.973491i \(0.426544\pi\)
\(48\) −0.247235 −0.0356852
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 1.42543 0.199600
\(52\) 2.83439 0.393060
\(53\) 0.874146 0.120073 0.0600366 0.998196i \(-0.480878\pi\)
0.0600366 + 0.998196i \(0.480878\pi\)
\(54\) 1.46830 0.199810
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −0.907206 −0.120162
\(58\) 1.41528 0.185835
\(59\) −10.4962 −1.36649 −0.683243 0.730191i \(-0.739431\pi\)
−0.683243 + 0.730191i \(0.739431\pi\)
\(60\) 0.247235 0.0319179
\(61\) −9.20135 −1.17811 −0.589056 0.808092i \(-0.700500\pi\)
−0.589056 + 0.808092i \(0.700500\pi\)
\(62\) −0.356435 −0.0452673
\(63\) −2.93888 −0.370263
\(64\) 1.00000 0.125000
\(65\) −2.83439 −0.351563
\(66\) 0 0
\(67\) −13.5466 −1.65498 −0.827492 0.561478i \(-0.810233\pi\)
−0.827492 + 0.561478i \(0.810233\pi\)
\(68\) −5.76549 −0.699168
\(69\) 1.77374 0.213533
\(70\) −1.00000 −0.119523
\(71\) −12.5139 −1.48512 −0.742561 0.669779i \(-0.766389\pi\)
−0.742561 + 0.669779i \(0.766389\pi\)
\(72\) −2.93888 −0.346350
\(73\) −7.19836 −0.842504 −0.421252 0.906944i \(-0.638409\pi\)
−0.421252 + 0.906944i \(0.638409\pi\)
\(74\) 2.94752 0.342643
\(75\) −0.247235 −0.0285482
\(76\) 3.66941 0.420910
\(77\) 0 0
\(78\) −0.700760 −0.0793455
\(79\) 6.14508 0.691376 0.345688 0.938350i \(-0.387646\pi\)
0.345688 + 0.938350i \(0.387646\pi\)
\(80\) −1.00000 −0.111803
\(81\) 8.45361 0.939290
\(82\) 2.22578 0.245796
\(83\) 3.08217 0.338312 0.169156 0.985589i \(-0.445896\pi\)
0.169156 + 0.985589i \(0.445896\pi\)
\(84\) −0.247235 −0.0269755
\(85\) 5.76549 0.625355
\(86\) 9.64018 1.03953
\(87\) −0.349905 −0.0375138
\(88\) 0 0
\(89\) −5.75443 −0.609968 −0.304984 0.952357i \(-0.598651\pi\)
−0.304984 + 0.952357i \(0.598651\pi\)
\(90\) 2.93888 0.309785
\(91\) 2.83439 0.297125
\(92\) −7.17432 −0.747974
\(93\) 0.0881230 0.00913793
\(94\) 3.13615 0.323469
\(95\) −3.66941 −0.376474
\(96\) −0.247235 −0.0252333
\(97\) 12.5278 1.27200 0.636001 0.771689i \(-0.280588\pi\)
0.636001 + 0.771689i \(0.280588\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −5.90123 −0.587195 −0.293597 0.955929i \(-0.594852\pi\)
−0.293597 + 0.955929i \(0.594852\pi\)
\(102\) 1.42543 0.141138
\(103\) −11.9671 −1.17915 −0.589576 0.807713i \(-0.700705\pi\)
−0.589576 + 0.807713i \(0.700705\pi\)
\(104\) 2.83439 0.277935
\(105\) 0.247235 0.0241276
\(106\) 0.874146 0.0849046
\(107\) −6.28895 −0.607976 −0.303988 0.952676i \(-0.598318\pi\)
−0.303988 + 0.952676i \(0.598318\pi\)
\(108\) 1.46830 0.141287
\(109\) −3.42125 −0.327696 −0.163848 0.986486i \(-0.552391\pi\)
−0.163848 + 0.986486i \(0.552391\pi\)
\(110\) 0 0
\(111\) −0.728730 −0.0691680
\(112\) 1.00000 0.0944911
\(113\) −16.8223 −1.58251 −0.791253 0.611489i \(-0.790571\pi\)
−0.791253 + 0.611489i \(0.790571\pi\)
\(114\) −0.907206 −0.0849676
\(115\) 7.17432 0.669008
\(116\) 1.41528 0.131405
\(117\) −8.32993 −0.770102
\(118\) −10.4962 −0.966251
\(119\) −5.76549 −0.528521
\(120\) 0.247235 0.0225693
\(121\) 0 0
\(122\) −9.20135 −0.833051
\(123\) −0.550290 −0.0496180
\(124\) −0.356435 −0.0320088
\(125\) −1.00000 −0.0894427
\(126\) −2.93888 −0.261816
\(127\) 0.175760 0.0155961 0.00779807 0.999970i \(-0.497518\pi\)
0.00779807 + 0.999970i \(0.497518\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.38339 −0.209845
\(130\) −2.83439 −0.248593
\(131\) −11.4391 −0.999438 −0.499719 0.866187i \(-0.666563\pi\)
−0.499719 + 0.866187i \(0.666563\pi\)
\(132\) 0 0
\(133\) 3.66941 0.318178
\(134\) −13.5466 −1.17025
\(135\) −1.46830 −0.126371
\(136\) −5.76549 −0.494387
\(137\) −9.95341 −0.850377 −0.425189 0.905105i \(-0.639792\pi\)
−0.425189 + 0.905105i \(0.639792\pi\)
\(138\) 1.77374 0.150991
\(139\) −0.627224 −0.0532005 −0.0266002 0.999646i \(-0.508468\pi\)
−0.0266002 + 0.999646i \(0.508468\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −0.775365 −0.0652975
\(142\) −12.5139 −1.05014
\(143\) 0 0
\(144\) −2.93888 −0.244906
\(145\) −1.41528 −0.117532
\(146\) −7.19836 −0.595740
\(147\) −0.247235 −0.0203916
\(148\) 2.94752 0.242285
\(149\) −15.0315 −1.23143 −0.615713 0.787970i \(-0.711132\pi\)
−0.615713 + 0.787970i \(0.711132\pi\)
\(150\) −0.247235 −0.0201866
\(151\) 0.516467 0.0420295 0.0210148 0.999779i \(-0.493310\pi\)
0.0210148 + 0.999779i \(0.493310\pi\)
\(152\) 3.66941 0.297629
\(153\) 16.9441 1.36985
\(154\) 0 0
\(155\) 0.356435 0.0286295
\(156\) −0.700760 −0.0561057
\(157\) −22.7782 −1.81790 −0.908949 0.416907i \(-0.863114\pi\)
−0.908949 + 0.416907i \(0.863114\pi\)
\(158\) 6.14508 0.488876
\(159\) −0.216119 −0.0171394
\(160\) −1.00000 −0.0790569
\(161\) −7.17432 −0.565415
\(162\) 8.45361 0.664178
\(163\) 3.72353 0.291649 0.145825 0.989310i \(-0.453416\pi\)
0.145825 + 0.989310i \(0.453416\pi\)
\(164\) 2.22578 0.173804
\(165\) 0 0
\(166\) 3.08217 0.239223
\(167\) 15.5825 1.20581 0.602906 0.797812i \(-0.294009\pi\)
0.602906 + 0.797812i \(0.294009\pi\)
\(168\) −0.247235 −0.0190746
\(169\) −4.96622 −0.382017
\(170\) 5.76549 0.442193
\(171\) −10.7839 −0.824669
\(172\) 9.64018 0.735057
\(173\) 5.89952 0.448532 0.224266 0.974528i \(-0.428002\pi\)
0.224266 + 0.974528i \(0.428002\pi\)
\(174\) −0.349905 −0.0265263
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 2.59502 0.195053
\(178\) −5.75443 −0.431313
\(179\) 16.3323 1.22073 0.610367 0.792119i \(-0.291022\pi\)
0.610367 + 0.792119i \(0.291022\pi\)
\(180\) 2.93888 0.219051
\(181\) −3.48610 −0.259120 −0.129560 0.991572i \(-0.541356\pi\)
−0.129560 + 0.991572i \(0.541356\pi\)
\(182\) 2.83439 0.210099
\(183\) 2.27489 0.168165
\(184\) −7.17432 −0.528898
\(185\) −2.94752 −0.216706
\(186\) 0.0881230 0.00646149
\(187\) 0 0
\(188\) 3.13615 0.228727
\(189\) 1.46830 0.106803
\(190\) −3.66941 −0.266207
\(191\) 25.9703 1.87914 0.939572 0.342352i \(-0.111223\pi\)
0.939572 + 0.342352i \(0.111223\pi\)
\(192\) −0.247235 −0.0178426
\(193\) −0.677502 −0.0487677 −0.0243838 0.999703i \(-0.507762\pi\)
−0.0243838 + 0.999703i \(0.507762\pi\)
\(194\) 12.5278 0.899441
\(195\) 0.700760 0.0501825
\(196\) 1.00000 0.0714286
\(197\) 5.80812 0.413811 0.206906 0.978361i \(-0.433661\pi\)
0.206906 + 0.978361i \(0.433661\pi\)
\(198\) 0 0
\(199\) 10.9475 0.776046 0.388023 0.921650i \(-0.373158\pi\)
0.388023 + 0.921650i \(0.373158\pi\)
\(200\) 1.00000 0.0707107
\(201\) 3.34919 0.236234
\(202\) −5.90123 −0.415209
\(203\) 1.41528 0.0993329
\(204\) 1.42543 0.0998000
\(205\) −2.22578 −0.155455
\(206\) −11.9671 −0.833787
\(207\) 21.0844 1.46547
\(208\) 2.83439 0.196530
\(209\) 0 0
\(210\) 0.247235 0.0170608
\(211\) −6.42786 −0.442512 −0.221256 0.975216i \(-0.571016\pi\)
−0.221256 + 0.975216i \(0.571016\pi\)
\(212\) 0.874146 0.0600366
\(213\) 3.09386 0.211988
\(214\) −6.28895 −0.429904
\(215\) −9.64018 −0.657455
\(216\) 1.46830 0.0999049
\(217\) −0.356435 −0.0241964
\(218\) −3.42125 −0.231716
\(219\) 1.77968 0.120260
\(220\) 0 0
\(221\) −16.3417 −1.09926
\(222\) −0.728730 −0.0489091
\(223\) 23.2048 1.55391 0.776955 0.629557i \(-0.216763\pi\)
0.776955 + 0.629557i \(0.216763\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.93888 −0.195925
\(226\) −16.8223 −1.11900
\(227\) −25.5715 −1.69724 −0.848621 0.529001i \(-0.822567\pi\)
−0.848621 + 0.529001i \(0.822567\pi\)
\(228\) −0.907206 −0.0600812
\(229\) 4.02501 0.265980 0.132990 0.991117i \(-0.457542\pi\)
0.132990 + 0.991117i \(0.457542\pi\)
\(230\) 7.17432 0.473060
\(231\) 0 0
\(232\) 1.41528 0.0929174
\(233\) −3.67721 −0.240902 −0.120451 0.992719i \(-0.538434\pi\)
−0.120451 + 0.992719i \(0.538434\pi\)
\(234\) −8.32993 −0.544544
\(235\) −3.13615 −0.204580
\(236\) −10.4962 −0.683243
\(237\) −1.51928 −0.0986877
\(238\) −5.76549 −0.373721
\(239\) −1.09444 −0.0707931 −0.0353966 0.999373i \(-0.511269\pi\)
−0.0353966 + 0.999373i \(0.511269\pi\)
\(240\) 0.247235 0.0159589
\(241\) 6.24902 0.402535 0.201267 0.979536i \(-0.435494\pi\)
0.201267 + 0.979536i \(0.435494\pi\)
\(242\) 0 0
\(243\) −6.49491 −0.416649
\(244\) −9.20135 −0.589056
\(245\) −1.00000 −0.0638877
\(246\) −0.550290 −0.0350852
\(247\) 10.4006 0.661772
\(248\) −0.356435 −0.0226336
\(249\) −0.762019 −0.0482910
\(250\) −1.00000 −0.0632456
\(251\) −19.2916 −1.21767 −0.608836 0.793296i \(-0.708363\pi\)
−0.608836 + 0.793296i \(0.708363\pi\)
\(252\) −2.93888 −0.185132
\(253\) 0 0
\(254\) 0.175760 0.0110281
\(255\) −1.42543 −0.0892638
\(256\) 1.00000 0.0625000
\(257\) 25.2663 1.57607 0.788035 0.615630i \(-0.211099\pi\)
0.788035 + 0.615630i \(0.211099\pi\)
\(258\) −2.38339 −0.148383
\(259\) 2.94752 0.183150
\(260\) −2.83439 −0.175782
\(261\) −4.15932 −0.257455
\(262\) −11.4391 −0.706710
\(263\) 2.99352 0.184589 0.0922943 0.995732i \(-0.470580\pi\)
0.0922943 + 0.995732i \(0.470580\pi\)
\(264\) 0 0
\(265\) −0.874146 −0.0536984
\(266\) 3.66941 0.224986
\(267\) 1.42269 0.0870675
\(268\) −13.5466 −0.827492
\(269\) −22.5013 −1.37193 −0.685964 0.727635i \(-0.740619\pi\)
−0.685964 + 0.727635i \(0.740619\pi\)
\(270\) −1.46830 −0.0893576
\(271\) −13.6286 −0.827881 −0.413940 0.910304i \(-0.635848\pi\)
−0.413940 + 0.910304i \(0.635848\pi\)
\(272\) −5.76549 −0.349584
\(273\) −0.700760 −0.0424119
\(274\) −9.95341 −0.601307
\(275\) 0 0
\(276\) 1.77374 0.106767
\(277\) −23.5465 −1.41477 −0.707387 0.706827i \(-0.750126\pi\)
−0.707387 + 0.706827i \(0.750126\pi\)
\(278\) −0.627224 −0.0376184
\(279\) 1.04752 0.0627132
\(280\) −1.00000 −0.0597614
\(281\) −19.1837 −1.14440 −0.572201 0.820113i \(-0.693910\pi\)
−0.572201 + 0.820113i \(0.693910\pi\)
\(282\) −0.775365 −0.0461723
\(283\) 17.2722 1.02673 0.513363 0.858172i \(-0.328399\pi\)
0.513363 + 0.858172i \(0.328399\pi\)
\(284\) −12.5139 −0.742561
\(285\) 0.907206 0.0537382
\(286\) 0 0
\(287\) 2.22578 0.131384
\(288\) −2.93888 −0.173175
\(289\) 16.2409 0.955345
\(290\) −1.41528 −0.0831079
\(291\) −3.09730 −0.181567
\(292\) −7.19836 −0.421252
\(293\) −32.9318 −1.92389 −0.961947 0.273235i \(-0.911906\pi\)
−0.961947 + 0.273235i \(0.911906\pi\)
\(294\) −0.247235 −0.0144190
\(295\) 10.4962 0.611111
\(296\) 2.94752 0.171321
\(297\) 0 0
\(298\) −15.0315 −0.870750
\(299\) −20.3348 −1.17599
\(300\) −0.247235 −0.0142741
\(301\) 9.64018 0.555651
\(302\) 0.516467 0.0297194
\(303\) 1.45899 0.0838168
\(304\) 3.66941 0.210455
\(305\) 9.20135 0.526868
\(306\) 16.9441 0.968627
\(307\) −7.97195 −0.454983 −0.227492 0.973780i \(-0.573052\pi\)
−0.227492 + 0.973780i \(0.573052\pi\)
\(308\) 0 0
\(309\) 2.95868 0.168313
\(310\) 0.356435 0.0202441
\(311\) −4.00836 −0.227293 −0.113647 0.993521i \(-0.536253\pi\)
−0.113647 + 0.993521i \(0.536253\pi\)
\(312\) −0.700760 −0.0396727
\(313\) 5.27260 0.298025 0.149013 0.988835i \(-0.452390\pi\)
0.149013 + 0.988835i \(0.452390\pi\)
\(314\) −22.7782 −1.28545
\(315\) 2.93888 0.165587
\(316\) 6.14508 0.345688
\(317\) −4.65978 −0.261720 −0.130860 0.991401i \(-0.541774\pi\)
−0.130860 + 0.991401i \(0.541774\pi\)
\(318\) −0.216119 −0.0121194
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 1.55485 0.0867831
\(322\) −7.17432 −0.399809
\(323\) −21.1560 −1.17715
\(324\) 8.45361 0.469645
\(325\) 2.83439 0.157224
\(326\) 3.72353 0.206227
\(327\) 0.845851 0.0467757
\(328\) 2.22578 0.122898
\(329\) 3.13615 0.172902
\(330\) 0 0
\(331\) −25.0199 −1.37522 −0.687608 0.726082i \(-0.741339\pi\)
−0.687608 + 0.726082i \(0.741339\pi\)
\(332\) 3.08217 0.169156
\(333\) −8.66240 −0.474697
\(334\) 15.5825 0.852638
\(335\) 13.5466 0.740131
\(336\) −0.247235 −0.0134878
\(337\) 15.9367 0.868130 0.434065 0.900882i \(-0.357079\pi\)
0.434065 + 0.900882i \(0.357079\pi\)
\(338\) −4.96622 −0.270127
\(339\) 4.15905 0.225889
\(340\) 5.76549 0.312678
\(341\) 0 0
\(342\) −10.7839 −0.583129
\(343\) 1.00000 0.0539949
\(344\) 9.64018 0.519764
\(345\) −1.77374 −0.0954949
\(346\) 5.89952 0.317160
\(347\) −12.4275 −0.667142 −0.333571 0.942725i \(-0.608254\pi\)
−0.333571 + 0.942725i \(0.608254\pi\)
\(348\) −0.349905 −0.0187569
\(349\) −16.2129 −0.867854 −0.433927 0.900948i \(-0.642872\pi\)
−0.433927 + 0.900948i \(0.642872\pi\)
\(350\) 1.00000 0.0534522
\(351\) 4.16173 0.222137
\(352\) 0 0
\(353\) −21.8320 −1.16200 −0.580999 0.813904i \(-0.697338\pi\)
−0.580999 + 0.813904i \(0.697338\pi\)
\(354\) 2.59502 0.137924
\(355\) 12.5139 0.664166
\(356\) −5.75443 −0.304984
\(357\) 1.42543 0.0754417
\(358\) 16.3323 0.863189
\(359\) −18.2106 −0.961116 −0.480558 0.876963i \(-0.659566\pi\)
−0.480558 + 0.876963i \(0.659566\pi\)
\(360\) 2.93888 0.154892
\(361\) −5.53541 −0.291338
\(362\) −3.48610 −0.183226
\(363\) 0 0
\(364\) 2.83439 0.148563
\(365\) 7.19836 0.376779
\(366\) 2.27489 0.118911
\(367\) −5.27857 −0.275539 −0.137770 0.990464i \(-0.543993\pi\)
−0.137770 + 0.990464i \(0.543993\pi\)
\(368\) −7.17432 −0.373987
\(369\) −6.54129 −0.340526
\(370\) −2.94752 −0.153234
\(371\) 0.874146 0.0453834
\(372\) 0.0881230 0.00456897
\(373\) 18.5799 0.962031 0.481016 0.876712i \(-0.340268\pi\)
0.481016 + 0.876712i \(0.340268\pi\)
\(374\) 0 0
\(375\) 0.247235 0.0127671
\(376\) 3.13615 0.161735
\(377\) 4.01145 0.206600
\(378\) 1.46830 0.0755210
\(379\) 22.8406 1.17324 0.586621 0.809862i \(-0.300458\pi\)
0.586621 + 0.809862i \(0.300458\pi\)
\(380\) −3.66941 −0.188237
\(381\) −0.0434539 −0.00222621
\(382\) 25.9703 1.32876
\(383\) −22.4735 −1.14834 −0.574171 0.818736i \(-0.694675\pi\)
−0.574171 + 0.818736i \(0.694675\pi\)
\(384\) −0.247235 −0.0126166
\(385\) 0 0
\(386\) −0.677502 −0.0344840
\(387\) −28.3313 −1.44016
\(388\) 12.5278 0.636001
\(389\) 36.2498 1.83794 0.918968 0.394333i \(-0.129024\pi\)
0.918968 + 0.394333i \(0.129024\pi\)
\(390\) 0.700760 0.0354844
\(391\) 41.3634 2.09184
\(392\) 1.00000 0.0505076
\(393\) 2.82814 0.142661
\(394\) 5.80812 0.292609
\(395\) −6.14508 −0.309193
\(396\) 0 0
\(397\) −8.02992 −0.403010 −0.201505 0.979487i \(-0.564583\pi\)
−0.201505 + 0.979487i \(0.564583\pi\)
\(398\) 10.9475 0.548748
\(399\) −0.907206 −0.0454171
\(400\) 1.00000 0.0500000
\(401\) 0.672421 0.0335791 0.0167896 0.999859i \(-0.494655\pi\)
0.0167896 + 0.999859i \(0.494655\pi\)
\(402\) 3.34919 0.167043
\(403\) −1.01028 −0.0503254
\(404\) −5.90123 −0.293597
\(405\) −8.45361 −0.420063
\(406\) 1.41528 0.0702390
\(407\) 0 0
\(408\) 1.42543 0.0705692
\(409\) −21.7385 −1.07490 −0.537450 0.843296i \(-0.680612\pi\)
−0.537450 + 0.843296i \(0.680612\pi\)
\(410\) −2.22578 −0.109923
\(411\) 2.46083 0.121384
\(412\) −11.9671 −0.589576
\(413\) −10.4962 −0.516483
\(414\) 21.0844 1.03624
\(415\) −3.08217 −0.151298
\(416\) 2.83439 0.138968
\(417\) 0.155072 0.00759389
\(418\) 0 0
\(419\) −7.38003 −0.360538 −0.180269 0.983617i \(-0.557697\pi\)
−0.180269 + 0.983617i \(0.557697\pi\)
\(420\) 0.247235 0.0120638
\(421\) 29.4497 1.43529 0.717646 0.696408i \(-0.245220\pi\)
0.717646 + 0.696408i \(0.245220\pi\)
\(422\) −6.42786 −0.312903
\(423\) −9.21676 −0.448134
\(424\) 0.874146 0.0424523
\(425\) −5.76549 −0.279667
\(426\) 3.09386 0.149898
\(427\) −9.20135 −0.445285
\(428\) −6.28895 −0.303988
\(429\) 0 0
\(430\) −9.64018 −0.464891
\(431\) −19.6701 −0.947477 −0.473739 0.880665i \(-0.657096\pi\)
−0.473739 + 0.880665i \(0.657096\pi\)
\(432\) 1.46830 0.0706434
\(433\) −17.0504 −0.819390 −0.409695 0.912222i \(-0.634365\pi\)
−0.409695 + 0.912222i \(0.634365\pi\)
\(434\) −0.356435 −0.0171094
\(435\) 0.349905 0.0167767
\(436\) −3.42125 −0.163848
\(437\) −26.3255 −1.25932
\(438\) 1.77968 0.0850365
\(439\) 29.9589 1.42986 0.714929 0.699197i \(-0.246459\pi\)
0.714929 + 0.699197i \(0.246459\pi\)
\(440\) 0 0
\(441\) −2.93888 −0.139946
\(442\) −16.3417 −0.777293
\(443\) 41.2954 1.96201 0.981003 0.193994i \(-0.0621442\pi\)
0.981003 + 0.193994i \(0.0621442\pi\)
\(444\) −0.728730 −0.0345840
\(445\) 5.75443 0.272786
\(446\) 23.2048 1.09878
\(447\) 3.71630 0.175775
\(448\) 1.00000 0.0472456
\(449\) 33.9263 1.60108 0.800541 0.599278i \(-0.204546\pi\)
0.800541 + 0.599278i \(0.204546\pi\)
\(450\) −2.93888 −0.138540
\(451\) 0 0
\(452\) −16.8223 −0.791253
\(453\) −0.127689 −0.00599934
\(454\) −25.5715 −1.20013
\(455\) −2.83439 −0.132878
\(456\) −0.907206 −0.0424838
\(457\) 19.9928 0.935225 0.467613 0.883934i \(-0.345114\pi\)
0.467613 + 0.883934i \(0.345114\pi\)
\(458\) 4.02501 0.188076
\(459\) −8.46544 −0.395133
\(460\) 7.17432 0.334504
\(461\) −17.4739 −0.813840 −0.406920 0.913464i \(-0.633397\pi\)
−0.406920 + 0.913464i \(0.633397\pi\)
\(462\) 0 0
\(463\) −41.7730 −1.94135 −0.970677 0.240386i \(-0.922726\pi\)
−0.970677 + 0.240386i \(0.922726\pi\)
\(464\) 1.41528 0.0657026
\(465\) −0.0881230 −0.00408661
\(466\) −3.67721 −0.170343
\(467\) −13.7476 −0.636164 −0.318082 0.948063i \(-0.603039\pi\)
−0.318082 + 0.948063i \(0.603039\pi\)
\(468\) −8.32993 −0.385051
\(469\) −13.5466 −0.625525
\(470\) −3.13615 −0.144660
\(471\) 5.63156 0.259489
\(472\) −10.4962 −0.483125
\(473\) 0 0
\(474\) −1.51928 −0.0697827
\(475\) 3.66941 0.168364
\(476\) −5.76549 −0.264261
\(477\) −2.56901 −0.117627
\(478\) −1.09444 −0.0500583
\(479\) −20.6636 −0.944145 −0.472072 0.881560i \(-0.656494\pi\)
−0.472072 + 0.881560i \(0.656494\pi\)
\(480\) 0.247235 0.0112847
\(481\) 8.35444 0.380930
\(482\) 6.24902 0.284635
\(483\) 1.77374 0.0807079
\(484\) 0 0
\(485\) −12.5278 −0.568856
\(486\) −6.49491 −0.294615
\(487\) −22.1049 −1.00167 −0.500835 0.865543i \(-0.666974\pi\)
−0.500835 + 0.865543i \(0.666974\pi\)
\(488\) −9.20135 −0.416526
\(489\) −0.920586 −0.0416303
\(490\) −1.00000 −0.0451754
\(491\) 10.6974 0.482768 0.241384 0.970430i \(-0.422399\pi\)
0.241384 + 0.970430i \(0.422399\pi\)
\(492\) −0.550290 −0.0248090
\(493\) −8.15976 −0.367497
\(494\) 10.4006 0.467943
\(495\) 0 0
\(496\) −0.356435 −0.0160044
\(497\) −12.5139 −0.561323
\(498\) −0.762019 −0.0341469
\(499\) 1.85809 0.0831794 0.0415897 0.999135i \(-0.486758\pi\)
0.0415897 + 0.999135i \(0.486758\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −3.85254 −0.172119
\(502\) −19.2916 −0.861024
\(503\) −13.5487 −0.604108 −0.302054 0.953291i \(-0.597672\pi\)
−0.302054 + 0.953291i \(0.597672\pi\)
\(504\) −2.93888 −0.130908
\(505\) 5.90123 0.262601
\(506\) 0 0
\(507\) 1.22782 0.0545295
\(508\) 0.175760 0.00779807
\(509\) −6.36772 −0.282244 −0.141122 0.989992i \(-0.545071\pi\)
−0.141122 + 0.989992i \(0.545071\pi\)
\(510\) −1.42543 −0.0631190
\(511\) −7.19836 −0.318437
\(512\) 1.00000 0.0441942
\(513\) 5.38778 0.237876
\(514\) 25.2663 1.11445
\(515\) 11.9671 0.527333
\(516\) −2.38339 −0.104923
\(517\) 0 0
\(518\) 2.94752 0.129507
\(519\) −1.45857 −0.0640240
\(520\) −2.83439 −0.124296
\(521\) −24.1693 −1.05888 −0.529439 0.848348i \(-0.677598\pi\)
−0.529439 + 0.848348i \(0.677598\pi\)
\(522\) −4.15932 −0.182048
\(523\) −30.5849 −1.33738 −0.668692 0.743540i \(-0.733145\pi\)
−0.668692 + 0.743540i \(0.733145\pi\)
\(524\) −11.4391 −0.499719
\(525\) −0.247235 −0.0107902
\(526\) 2.99352 0.130524
\(527\) 2.05502 0.0895181
\(528\) 0 0
\(529\) 28.4708 1.23786
\(530\) −0.874146 −0.0379705
\(531\) 30.8469 1.33864
\(532\) 3.66941 0.159089
\(533\) 6.30873 0.273261
\(534\) 1.42269 0.0615660
\(535\) 6.28895 0.271895
\(536\) −13.5466 −0.585125
\(537\) −4.03791 −0.174249
\(538\) −22.5013 −0.970100
\(539\) 0 0
\(540\) −1.46830 −0.0631854
\(541\) 22.1914 0.954082 0.477041 0.878881i \(-0.341709\pi\)
0.477041 + 0.878881i \(0.341709\pi\)
\(542\) −13.6286 −0.585400
\(543\) 0.861886 0.0369871
\(544\) −5.76549 −0.247193
\(545\) 3.42125 0.146550
\(546\) −0.700760 −0.0299898
\(547\) 17.1879 0.734903 0.367452 0.930043i \(-0.380230\pi\)
0.367452 + 0.930043i \(0.380230\pi\)
\(548\) −9.95341 −0.425189
\(549\) 27.0416 1.15411
\(550\) 0 0
\(551\) 5.19323 0.221239
\(552\) 1.77374 0.0754954
\(553\) 6.14508 0.261315
\(554\) −23.5465 −1.00040
\(555\) 0.728730 0.0309329
\(556\) −0.627224 −0.0266002
\(557\) −6.02990 −0.255495 −0.127747 0.991807i \(-0.540775\pi\)
−0.127747 + 0.991807i \(0.540775\pi\)
\(558\) 1.04752 0.0443449
\(559\) 27.3240 1.15568
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −19.1837 −0.809215
\(563\) 27.1986 1.14628 0.573142 0.819456i \(-0.305724\pi\)
0.573142 + 0.819456i \(0.305724\pi\)
\(564\) −0.775365 −0.0326488
\(565\) 16.8223 0.707718
\(566\) 17.2722 0.726005
\(567\) 8.45361 0.355018
\(568\) −12.5139 −0.525070
\(569\) 31.8782 1.33641 0.668203 0.743979i \(-0.267064\pi\)
0.668203 + 0.743979i \(0.267064\pi\)
\(570\) 0.907206 0.0379987
\(571\) −26.4539 −1.10706 −0.553531 0.832829i \(-0.686720\pi\)
−0.553531 + 0.832829i \(0.686720\pi\)
\(572\) 0 0
\(573\) −6.42075 −0.268231
\(574\) 2.22578 0.0929022
\(575\) −7.17432 −0.299190
\(576\) −2.93888 −0.122453
\(577\) −30.0526 −1.25111 −0.625554 0.780181i \(-0.715127\pi\)
−0.625554 + 0.780181i \(0.715127\pi\)
\(578\) 16.2409 0.675531
\(579\) 0.167502 0.00696115
\(580\) −1.41528 −0.0587661
\(581\) 3.08217 0.127870
\(582\) −3.09730 −0.128387
\(583\) 0 0
\(584\) −7.19836 −0.297870
\(585\) 8.32993 0.344400
\(586\) −32.9318 −1.36040
\(587\) 26.7574 1.10440 0.552198 0.833713i \(-0.313789\pi\)
0.552198 + 0.833713i \(0.313789\pi\)
\(588\) −0.247235 −0.0101958
\(589\) −1.30791 −0.0538913
\(590\) 10.4962 0.432121
\(591\) −1.43597 −0.0590679
\(592\) 2.94752 0.121142
\(593\) −3.33504 −0.136954 −0.0684768 0.997653i \(-0.521814\pi\)
−0.0684768 + 0.997653i \(0.521814\pi\)
\(594\) 0 0
\(595\) 5.76549 0.236362
\(596\) −15.0315 −0.615713
\(597\) −2.70660 −0.110774
\(598\) −20.3348 −0.831553
\(599\) −3.33155 −0.136123 −0.0680617 0.997681i \(-0.521681\pi\)
−0.0680617 + 0.997681i \(0.521681\pi\)
\(600\) −0.247235 −0.0100933
\(601\) −41.5433 −1.69459 −0.847293 0.531126i \(-0.821769\pi\)
−0.847293 + 0.531126i \(0.821769\pi\)
\(602\) 9.64018 0.392904
\(603\) 39.8118 1.62126
\(604\) 0.516467 0.0210148
\(605\) 0 0
\(606\) 1.45899 0.0592674
\(607\) 20.9676 0.851049 0.425525 0.904947i \(-0.360090\pi\)
0.425525 + 0.904947i \(0.360090\pi\)
\(608\) 3.66941 0.148814
\(609\) −0.349905 −0.0141789
\(610\) 9.20135 0.372552
\(611\) 8.88908 0.359614
\(612\) 16.9441 0.684923
\(613\) −2.77191 −0.111956 −0.0559782 0.998432i \(-0.517828\pi\)
−0.0559782 + 0.998432i \(0.517828\pi\)
\(614\) −7.97195 −0.321722
\(615\) 0.550290 0.0221898
\(616\) 0 0
\(617\) 48.3567 1.94677 0.973384 0.229180i \(-0.0736043\pi\)
0.973384 + 0.229180i \(0.0736043\pi\)
\(618\) 2.95868 0.119016
\(619\) 46.8795 1.88425 0.942123 0.335268i \(-0.108827\pi\)
0.942123 + 0.335268i \(0.108827\pi\)
\(620\) 0.356435 0.0143148
\(621\) −10.5340 −0.422716
\(622\) −4.00836 −0.160721
\(623\) −5.75443 −0.230546
\(624\) −0.700760 −0.0280529
\(625\) 1.00000 0.0400000
\(626\) 5.27260 0.210736
\(627\) 0 0
\(628\) −22.7782 −0.908949
\(629\) −16.9939 −0.677592
\(630\) 2.93888 0.117088
\(631\) −21.5815 −0.859145 −0.429573 0.903032i \(-0.641336\pi\)
−0.429573 + 0.903032i \(0.641336\pi\)
\(632\) 6.14508 0.244438
\(633\) 1.58919 0.0631647
\(634\) −4.65978 −0.185064
\(635\) −0.175760 −0.00697481
\(636\) −0.216119 −0.00856969
\(637\) 2.83439 0.112303
\(638\) 0 0
\(639\) 36.7766 1.45486
\(640\) −1.00000 −0.0395285
\(641\) 22.9849 0.907850 0.453925 0.891040i \(-0.350023\pi\)
0.453925 + 0.891040i \(0.350023\pi\)
\(642\) 1.55485 0.0613649
\(643\) −32.9082 −1.29777 −0.648886 0.760886i \(-0.724765\pi\)
−0.648886 + 0.760886i \(0.724765\pi\)
\(644\) −7.17432 −0.282708
\(645\) 2.38339 0.0938457
\(646\) −21.1560 −0.832370
\(647\) 19.9201 0.783140 0.391570 0.920148i \(-0.371932\pi\)
0.391570 + 0.920148i \(0.371932\pi\)
\(648\) 8.45361 0.332089
\(649\) 0 0
\(650\) 2.83439 0.111174
\(651\) 0.0881230 0.00345381
\(652\) 3.72353 0.145825
\(653\) 34.0859 1.33389 0.666943 0.745109i \(-0.267602\pi\)
0.666943 + 0.745109i \(0.267602\pi\)
\(654\) 0.845851 0.0330754
\(655\) 11.4391 0.446962
\(656\) 2.22578 0.0869021
\(657\) 21.1551 0.825338
\(658\) 3.13615 0.122260
\(659\) 7.54952 0.294088 0.147044 0.989130i \(-0.453024\pi\)
0.147044 + 0.989130i \(0.453024\pi\)
\(660\) 0 0
\(661\) 41.0094 1.59508 0.797541 0.603265i \(-0.206134\pi\)
0.797541 + 0.603265i \(0.206134\pi\)
\(662\) −25.0199 −0.972424
\(663\) 4.04022 0.156909
\(664\) 3.08217 0.119611
\(665\) −3.66941 −0.142294
\(666\) −8.66240 −0.335661
\(667\) −10.1536 −0.393150
\(668\) 15.5825 0.602906
\(669\) −5.73703 −0.221807
\(670\) 13.5466 0.523352
\(671\) 0 0
\(672\) −0.247235 −0.00953728
\(673\) 48.3741 1.86469 0.932343 0.361576i \(-0.117761\pi\)
0.932343 + 0.361576i \(0.117761\pi\)
\(674\) 15.9367 0.613860
\(675\) 1.46830 0.0565147
\(676\) −4.96622 −0.191008
\(677\) 5.38363 0.206910 0.103455 0.994634i \(-0.467010\pi\)
0.103455 + 0.994634i \(0.467010\pi\)
\(678\) 4.15905 0.159727
\(679\) 12.5278 0.480771
\(680\) 5.76549 0.221096
\(681\) 6.32217 0.242266
\(682\) 0 0
\(683\) 12.5734 0.481106 0.240553 0.970636i \(-0.422671\pi\)
0.240553 + 0.970636i \(0.422671\pi\)
\(684\) −10.7839 −0.412334
\(685\) 9.95341 0.380300
\(686\) 1.00000 0.0381802
\(687\) −0.995121 −0.0379662
\(688\) 9.64018 0.367528
\(689\) 2.47767 0.0943919
\(690\) −1.77374 −0.0675251
\(691\) −26.0427 −0.990712 −0.495356 0.868690i \(-0.664962\pi\)
−0.495356 + 0.868690i \(0.664962\pi\)
\(692\) 5.89952 0.224266
\(693\) 0 0
\(694\) −12.4275 −0.471741
\(695\) 0.627224 0.0237920
\(696\) −0.349905 −0.0132631
\(697\) −12.8327 −0.486073
\(698\) −16.2129 −0.613666
\(699\) 0.909134 0.0343866
\(700\) 1.00000 0.0377964
\(701\) −46.1549 −1.74325 −0.871624 0.490176i \(-0.836933\pi\)
−0.871624 + 0.490176i \(0.836933\pi\)
\(702\) 4.16173 0.157074
\(703\) 10.8157 0.407921
\(704\) 0 0
\(705\) 0.775365 0.0292019
\(706\) −21.8320 −0.821656
\(707\) −5.90123 −0.221939
\(708\) 2.59502 0.0975267
\(709\) 44.5132 1.67173 0.835865 0.548935i \(-0.184967\pi\)
0.835865 + 0.548935i \(0.184967\pi\)
\(710\) 12.5139 0.469637
\(711\) −18.0596 −0.677289
\(712\) −5.75443 −0.215656
\(713\) 2.55718 0.0957670
\(714\) 1.42543 0.0533453
\(715\) 0 0
\(716\) 16.3323 0.610367
\(717\) 0.270582 0.0101051
\(718\) −18.2106 −0.679612
\(719\) 24.2008 0.902536 0.451268 0.892389i \(-0.350972\pi\)
0.451268 + 0.892389i \(0.350972\pi\)
\(720\) 2.93888 0.109525
\(721\) −11.9671 −0.445678
\(722\) −5.53541 −0.206007
\(723\) −1.54498 −0.0574582
\(724\) −3.48610 −0.129560
\(725\) 1.41528 0.0525620
\(726\) 0 0
\(727\) −5.25083 −0.194742 −0.0973712 0.995248i \(-0.531043\pi\)
−0.0973712 + 0.995248i \(0.531043\pi\)
\(728\) 2.83439 0.105050
\(729\) −23.7551 −0.879817
\(730\) 7.19836 0.266423
\(731\) −55.5803 −2.05571
\(732\) 2.27489 0.0840825
\(733\) −47.6047 −1.75832 −0.879160 0.476526i \(-0.841896\pi\)
−0.879160 + 0.476526i \(0.841896\pi\)
\(734\) −5.27857 −0.194836
\(735\) 0.247235 0.00911939
\(736\) −7.17432 −0.264449
\(737\) 0 0
\(738\) −6.54129 −0.240788
\(739\) 27.2113 1.00098 0.500491 0.865742i \(-0.333153\pi\)
0.500491 + 0.865742i \(0.333153\pi\)
\(740\) −2.94752 −0.108353
\(741\) −2.57138 −0.0944619
\(742\) 0.874146 0.0320909
\(743\) −5.29833 −0.194377 −0.0971885 0.995266i \(-0.530985\pi\)
−0.0971885 + 0.995266i \(0.530985\pi\)
\(744\) 0.0881230 0.00323075
\(745\) 15.0315 0.550710
\(746\) 18.5799 0.680259
\(747\) −9.05811 −0.331419
\(748\) 0 0
\(749\) −6.28895 −0.229793
\(750\) 0.247235 0.00902773
\(751\) −20.0625 −0.732089 −0.366045 0.930597i \(-0.619288\pi\)
−0.366045 + 0.930597i \(0.619288\pi\)
\(752\) 3.13615 0.114364
\(753\) 4.76954 0.173812
\(754\) 4.01145 0.146088
\(755\) −0.516467 −0.0187962
\(756\) 1.46830 0.0534014
\(757\) −4.36865 −0.158781 −0.0793907 0.996844i \(-0.525297\pi\)
−0.0793907 + 0.996844i \(0.525297\pi\)
\(758\) 22.8406 0.829607
\(759\) 0 0
\(760\) −3.66941 −0.133104
\(761\) 2.16435 0.0784576 0.0392288 0.999230i \(-0.487510\pi\)
0.0392288 + 0.999230i \(0.487510\pi\)
\(762\) −0.0434539 −0.00157417
\(763\) −3.42125 −0.123857
\(764\) 25.9703 0.939572
\(765\) −16.9441 −0.612613
\(766\) −22.4735 −0.812000
\(767\) −29.7503 −1.07422
\(768\) −0.247235 −0.00892131
\(769\) −32.7883 −1.18237 −0.591187 0.806534i \(-0.701341\pi\)
−0.591187 + 0.806534i \(0.701341\pi\)
\(770\) 0 0
\(771\) −6.24671 −0.224970
\(772\) −0.677502 −0.0243838
\(773\) −7.29845 −0.262507 −0.131253 0.991349i \(-0.541900\pi\)
−0.131253 + 0.991349i \(0.541900\pi\)
\(774\) −28.3313 −1.01835
\(775\) −0.356435 −0.0128035
\(776\) 12.5278 0.449720
\(777\) −0.728730 −0.0261430
\(778\) 36.2498 1.29962
\(779\) 8.16730 0.292624
\(780\) 0.700760 0.0250912
\(781\) 0 0
\(782\) 41.3634 1.47915
\(783\) 2.07804 0.0742632
\(784\) 1.00000 0.0357143
\(785\) 22.7782 0.812989
\(786\) 2.82814 0.100876
\(787\) −0.836360 −0.0298130 −0.0149065 0.999889i \(-0.504745\pi\)
−0.0149065 + 0.999889i \(0.504745\pi\)
\(788\) 5.80812 0.206906
\(789\) −0.740103 −0.0263484
\(790\) −6.14508 −0.218632
\(791\) −16.8223 −0.598131
\(792\) 0 0
\(793\) −26.0802 −0.926137
\(794\) −8.02992 −0.284971
\(795\) 0.216119 0.00766496
\(796\) 10.9475 0.388023
\(797\) −2.03251 −0.0719952 −0.0359976 0.999352i \(-0.511461\pi\)
−0.0359976 + 0.999352i \(0.511461\pi\)
\(798\) −0.907206 −0.0321147
\(799\) −18.0814 −0.639676
\(800\) 1.00000 0.0353553
\(801\) 16.9115 0.597540
\(802\) 0.672421 0.0237440
\(803\) 0 0
\(804\) 3.34919 0.118117
\(805\) 7.17432 0.252861
\(806\) −1.01028 −0.0355855
\(807\) 5.56310 0.195830
\(808\) −5.90123 −0.207605
\(809\) −42.1998 −1.48367 −0.741833 0.670585i \(-0.766043\pi\)
−0.741833 + 0.670585i \(0.766043\pi\)
\(810\) −8.45361 −0.297030
\(811\) −3.03390 −0.106535 −0.0532673 0.998580i \(-0.516964\pi\)
−0.0532673 + 0.998580i \(0.516964\pi\)
\(812\) 1.41528 0.0496665
\(813\) 3.36947 0.118173
\(814\) 0 0
\(815\) −3.72353 −0.130430
\(816\) 1.42543 0.0499000
\(817\) 35.3738 1.23757
\(818\) −21.7385 −0.760069
\(819\) −8.32993 −0.291071
\(820\) −2.22578 −0.0777276
\(821\) 34.5900 1.20720 0.603599 0.797288i \(-0.293733\pi\)
0.603599 + 0.797288i \(0.293733\pi\)
\(822\) 2.46083 0.0858312
\(823\) 50.7373 1.76859 0.884295 0.466928i \(-0.154639\pi\)
0.884295 + 0.466928i \(0.154639\pi\)
\(824\) −11.9671 −0.416894
\(825\) 0 0
\(826\) −10.4962 −0.365209
\(827\) −12.9190 −0.449239 −0.224619 0.974447i \(-0.572114\pi\)
−0.224619 + 0.974447i \(0.572114\pi\)
\(828\) 21.0844 0.732734
\(829\) 12.8251 0.445434 0.222717 0.974883i \(-0.428507\pi\)
0.222717 + 0.974883i \(0.428507\pi\)
\(830\) −3.08217 −0.106984
\(831\) 5.82151 0.201946
\(832\) 2.83439 0.0982649
\(833\) −5.76549 −0.199762
\(834\) 0.155072 0.00536969
\(835\) −15.5825 −0.539255
\(836\) 0 0
\(837\) −0.523352 −0.0180897
\(838\) −7.38003 −0.254939
\(839\) −8.02823 −0.277165 −0.138583 0.990351i \(-0.544255\pi\)
−0.138583 + 0.990351i \(0.544255\pi\)
\(840\) 0.247235 0.00853041
\(841\) −26.9970 −0.930931
\(842\) 29.4497 1.01490
\(843\) 4.74287 0.163353
\(844\) −6.42786 −0.221256
\(845\) 4.96622 0.170843
\(846\) −9.21676 −0.316879
\(847\) 0 0
\(848\) 0.874146 0.0300183
\(849\) −4.27029 −0.146556
\(850\) −5.76549 −0.197755
\(851\) −21.1465 −0.724891
\(852\) 3.09386 0.105994
\(853\) −30.1638 −1.03279 −0.516395 0.856351i \(-0.672726\pi\)
−0.516395 + 0.856351i \(0.672726\pi\)
\(854\) −9.20135 −0.314864
\(855\) 10.7839 0.368803
\(856\) −6.28895 −0.214952
\(857\) 32.8279 1.12138 0.560689 0.828026i \(-0.310536\pi\)
0.560689 + 0.828026i \(0.310536\pi\)
\(858\) 0 0
\(859\) 40.7882 1.39168 0.695838 0.718199i \(-0.255033\pi\)
0.695838 + 0.718199i \(0.255033\pi\)
\(860\) −9.64018 −0.328727
\(861\) −0.550290 −0.0187538
\(862\) −19.6701 −0.669968
\(863\) 42.0371 1.43096 0.715479 0.698634i \(-0.246208\pi\)
0.715479 + 0.698634i \(0.246208\pi\)
\(864\) 1.46830 0.0499524
\(865\) −5.89952 −0.200590
\(866\) −17.0504 −0.579397
\(867\) −4.01530 −0.136367
\(868\) −0.356435 −0.0120982
\(869\) 0 0
\(870\) 0.349905 0.0118629
\(871\) −38.3964 −1.30101
\(872\) −3.42125 −0.115858
\(873\) −36.8175 −1.24608
\(874\) −26.3255 −0.890474
\(875\) −1.00000 −0.0338062
\(876\) 1.77968 0.0601299
\(877\) −2.16379 −0.0730659 −0.0365330 0.999332i \(-0.511631\pi\)
−0.0365330 + 0.999332i \(0.511631\pi\)
\(878\) 29.9589 1.01106
\(879\) 8.14187 0.274619
\(880\) 0 0
\(881\) 9.74157 0.328202 0.164101 0.986444i \(-0.447528\pi\)
0.164101 + 0.986444i \(0.447528\pi\)
\(882\) −2.93888 −0.0989571
\(883\) 25.2808 0.850768 0.425384 0.905013i \(-0.360139\pi\)
0.425384 + 0.905013i \(0.360139\pi\)
\(884\) −16.3417 −0.549629
\(885\) −2.59502 −0.0872306
\(886\) 41.2954 1.38735
\(887\) −45.8358 −1.53901 −0.769507 0.638638i \(-0.779498\pi\)
−0.769507 + 0.638638i \(0.779498\pi\)
\(888\) −0.728730 −0.0244546
\(889\) 0.175760 0.00589479
\(890\) 5.75443 0.192889
\(891\) 0 0
\(892\) 23.2048 0.776955
\(893\) 11.5078 0.385095
\(894\) 3.71630 0.124292
\(895\) −16.3323 −0.545929
\(896\) 1.00000 0.0334077
\(897\) 5.02747 0.167862
\(898\) 33.9263 1.13214
\(899\) −0.504454 −0.0168245
\(900\) −2.93888 −0.0979625
\(901\) −5.03988 −0.167903
\(902\) 0 0
\(903\) −2.38339 −0.0793141
\(904\) −16.8223 −0.559500
\(905\) 3.48610 0.115882
\(906\) −0.127689 −0.00424217
\(907\) −15.1299 −0.502381 −0.251190 0.967938i \(-0.580822\pi\)
−0.251190 + 0.967938i \(0.580822\pi\)
\(908\) −25.5715 −0.848621
\(909\) 17.3430 0.575231
\(910\) −2.83439 −0.0939592
\(911\) 14.7067 0.487255 0.243627 0.969869i \(-0.421663\pi\)
0.243627 + 0.969869i \(0.421663\pi\)
\(912\) −0.907206 −0.0300406
\(913\) 0 0
\(914\) 19.9928 0.661304
\(915\) −2.27489 −0.0752056
\(916\) 4.02501 0.132990
\(917\) −11.4391 −0.377752
\(918\) −8.46544 −0.279401
\(919\) 1.78904 0.0590151 0.0295075 0.999565i \(-0.490606\pi\)
0.0295075 + 0.999565i \(0.490606\pi\)
\(920\) 7.17432 0.236530
\(921\) 1.97094 0.0649448
\(922\) −17.4739 −0.575472
\(923\) −35.4692 −1.16748
\(924\) 0 0
\(925\) 2.94752 0.0969140
\(926\) −41.7730 −1.37275
\(927\) 35.1698 1.15513
\(928\) 1.41528 0.0464587
\(929\) 50.8053 1.66687 0.833433 0.552620i \(-0.186372\pi\)
0.833433 + 0.552620i \(0.186372\pi\)
\(930\) −0.0881230 −0.00288967
\(931\) 3.66941 0.120260
\(932\) −3.67721 −0.120451
\(933\) 0.991006 0.0324441
\(934\) −13.7476 −0.449836
\(935\) 0 0
\(936\) −8.32993 −0.272272
\(937\) 56.0500 1.83107 0.915537 0.402233i \(-0.131766\pi\)
0.915537 + 0.402233i \(0.131766\pi\)
\(938\) −13.5466 −0.442313
\(939\) −1.30357 −0.0425404
\(940\) −3.13615 −0.102290
\(941\) 31.8530 1.03838 0.519188 0.854660i \(-0.326234\pi\)
0.519188 + 0.854660i \(0.326234\pi\)
\(942\) 5.63156 0.183486
\(943\) −15.9684 −0.520004
\(944\) −10.4962 −0.341621
\(945\) −1.46830 −0.0477637
\(946\) 0 0
\(947\) 16.7087 0.542959 0.271479 0.962444i \(-0.412487\pi\)
0.271479 + 0.962444i \(0.412487\pi\)
\(948\) −1.51928 −0.0493438
\(949\) −20.4030 −0.662308
\(950\) 3.66941 0.119051
\(951\) 1.15206 0.0373581
\(952\) −5.76549 −0.186861
\(953\) 2.96987 0.0962036 0.0481018 0.998842i \(-0.484683\pi\)
0.0481018 + 0.998842i \(0.484683\pi\)
\(954\) −2.56901 −0.0831747
\(955\) −25.9703 −0.840379
\(956\) −1.09444 −0.0353966
\(957\) 0 0
\(958\) −20.6636 −0.667611
\(959\) −9.95341 −0.321412
\(960\) 0.247235 0.00797946
\(961\) −30.8730 −0.995902
\(962\) 8.35444 0.269358
\(963\) 18.4824 0.595588
\(964\) 6.24902 0.201267
\(965\) 0.677502 0.0218096
\(966\) 1.77374 0.0570691
\(967\) −20.4576 −0.657873 −0.328936 0.944352i \(-0.606690\pi\)
−0.328936 + 0.944352i \(0.606690\pi\)
\(968\) 0 0
\(969\) 5.23049 0.168027
\(970\) −12.5278 −0.402242
\(971\) −60.0213 −1.92618 −0.963088 0.269187i \(-0.913245\pi\)
−0.963088 + 0.269187i \(0.913245\pi\)
\(972\) −6.49491 −0.208324
\(973\) −0.627224 −0.0201079
\(974\) −22.1049 −0.708287
\(975\) −0.700760 −0.0224423
\(976\) −9.20135 −0.294528
\(977\) 16.7790 0.536808 0.268404 0.963306i \(-0.413504\pi\)
0.268404 + 0.963306i \(0.413504\pi\)
\(978\) −0.920586 −0.0294371
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 10.0546 0.321019
\(982\) 10.6974 0.341369
\(983\) 48.6058 1.55028 0.775141 0.631788i \(-0.217679\pi\)
0.775141 + 0.631788i \(0.217679\pi\)
\(984\) −0.550290 −0.0175426
\(985\) −5.80812 −0.185062
\(986\) −8.15976 −0.259860
\(987\) −0.775365 −0.0246802
\(988\) 10.4006 0.330886
\(989\) −69.1617 −2.19921
\(990\) 0 0
\(991\) 55.7217 1.77006 0.885028 0.465537i \(-0.154139\pi\)
0.885028 + 0.465537i \(0.154139\pi\)
\(992\) −0.356435 −0.0113168
\(993\) 6.18578 0.196300
\(994\) −12.5139 −0.396915
\(995\) −10.9475 −0.347058
\(996\) −0.762019 −0.0241455
\(997\) 38.8069 1.22903 0.614513 0.788907i \(-0.289353\pi\)
0.614513 + 0.788907i \(0.289353\pi\)
\(998\) 1.85809 0.0588167
\(999\) 4.32784 0.136927
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 8470.2.a.da.1.4 6
11.3 even 5 770.2.n.h.141.2 yes 12
11.4 even 5 770.2.n.h.71.2 12
11.10 odd 2 8470.2.a.cu.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.h.71.2 12 11.4 even 5
770.2.n.h.141.2 yes 12 11.3 even 5
8470.2.a.cu.1.4 6 11.10 odd 2
8470.2.a.da.1.4 6 1.1 even 1 trivial