Properties

Label 637.2.a.l.1.3
Level $637$
Weight $2$
Character 637.1
Self dual yes
Analytic conductor $5.086$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.746052.1
Defining polynomial: \(x^{5} - x^{4} - 7 x^{3} + 8 x + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.265608\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.26561 q^{2} -2.62728 q^{3} -0.398235 q^{4} -2.90260 q^{5} -3.32511 q^{6} -3.03523 q^{8} +3.90260 q^{9} +O(q^{10})\) \(q+1.26561 q^{2} -2.62728 q^{3} -0.398235 q^{4} -2.90260 q^{5} -3.32511 q^{6} -3.03523 q^{8} +3.90260 q^{9} -3.67356 q^{10} +2.03656 q^{11} +1.04628 q^{12} +1.00000 q^{13} +7.62594 q^{15} -3.04494 q^{16} +3.99866 q^{17} +4.93916 q^{18} +6.96210 q^{19} +1.15592 q^{20} +2.57749 q^{22} -0.627280 q^{23} +7.97439 q^{24} +3.42509 q^{25} +1.26561 q^{26} -2.37138 q^{27} +1.09606 q^{29} +9.65146 q^{30} -10.4325 q^{31} +2.21675 q^{32} -5.35062 q^{33} +5.06074 q^{34} -1.55415 q^{36} -3.08537 q^{37} +8.81129 q^{38} -2.62728 q^{39} +8.81005 q^{40} -0.521150 q^{41} +0.329024 q^{43} -0.811031 q^{44} -11.3277 q^{45} -0.793891 q^{46} +10.5457 q^{47} +7.99991 q^{48} +4.33482 q^{50} -10.5056 q^{51} -0.398235 q^{52} +7.11900 q^{53} -3.00124 q^{54} -5.91133 q^{55} -18.2914 q^{57} +1.38719 q^{58} +2.03656 q^{59} -3.03692 q^{60} +2.40081 q^{61} -13.2034 q^{62} +8.89542 q^{64} -2.90260 q^{65} -6.77179 q^{66} +14.6942 q^{67} -1.59241 q^{68} +1.64804 q^{69} +3.60141 q^{71} -11.8453 q^{72} +2.97573 q^{73} -3.90487 q^{74} -8.99866 q^{75} -2.77255 q^{76} -3.32511 q^{78} -8.76150 q^{79} +8.83824 q^{80} -5.47751 q^{81} -0.659572 q^{82} +12.8039 q^{83} -11.6065 q^{85} +0.416416 q^{86} -2.87966 q^{87} -6.18143 q^{88} -2.68098 q^{89} -14.3364 q^{90} +0.249805 q^{92} +27.4090 q^{93} +13.3467 q^{94} -20.2082 q^{95} -5.82403 q^{96} -2.32902 q^{97} +7.94789 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 4q^{2} + 8q^{4} + 2q^{5} - 5q^{6} + 9q^{8} + 3q^{9} + O(q^{10}) \) \( 5q + 4q^{2} + 8q^{4} + 2q^{5} - 5q^{6} + 9q^{8} + 3q^{9} - 5q^{10} + 11q^{11} + 5q^{12} + 5q^{13} + 10q^{16} - 5q^{17} + 9q^{18} + 9q^{19} + q^{20} + 8q^{22} + 10q^{23} + 9q^{25} + 4q^{26} - 3q^{29} - 13q^{30} - 6q^{31} + 22q^{32} + 8q^{33} - 22q^{34} + 7q^{36} + 4q^{37} - 10q^{38} + 28q^{40} + 14q^{41} + 2q^{43} - 32q^{45} + 3q^{46} + q^{47} - 23q^{48} + 9q^{50} - 8q^{51} + 8q^{52} + 17q^{53} + 23q^{54} - 16q^{57} - 27q^{58} + 11q^{59} - 29q^{60} - 11q^{61} - 23q^{62} + 9q^{64} + 2q^{65} + 21q^{66} + 13q^{67} - 32q^{68} + 18q^{69} + 15q^{71} - 19q^{72} - 33q^{74} - 20q^{75} + 8q^{76} - 5q^{78} + 2q^{79} + 55q^{80} - 19q^{81} + 34q^{82} + 6q^{83} - 22q^{85} + 28q^{86} - 8q^{87} - 3q^{88} - 4q^{89} - 34q^{90} + 21q^{92} + 18q^{93} + 20q^{94} - 12q^{95} - 37q^{96} - 12q^{97} + 11q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.26561 0.894920 0.447460 0.894304i \(-0.352329\pi\)
0.447460 + 0.894304i \(0.352329\pi\)
\(3\) −2.62728 −1.51686 −0.758430 0.651754i \(-0.774033\pi\)
−0.758430 + 0.651754i \(0.774033\pi\)
\(4\) −0.398235 −0.199118
\(5\) −2.90260 −1.29808 −0.649041 0.760753i \(-0.724830\pi\)
−0.649041 + 0.760753i \(0.724830\pi\)
\(6\) −3.32511 −1.35747
\(7\) 0 0
\(8\) −3.03523 −1.07311
\(9\) 3.90260 1.30087
\(10\) −3.67356 −1.16168
\(11\) 2.03656 0.614047 0.307024 0.951702i \(-0.400667\pi\)
0.307024 + 0.951702i \(0.400667\pi\)
\(12\) 1.04628 0.302034
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 7.62594 1.96901
\(16\) −3.04494 −0.761235
\(17\) 3.99866 0.969818 0.484909 0.874565i \(-0.338853\pi\)
0.484909 + 0.874565i \(0.338853\pi\)
\(18\) 4.93916 1.16417
\(19\) 6.96210 1.59722 0.798608 0.601852i \(-0.205570\pi\)
0.798608 + 0.601852i \(0.205570\pi\)
\(20\) 1.15592 0.258471
\(21\) 0 0
\(22\) 2.57749 0.549523
\(23\) −0.627280 −0.130797 −0.0653985 0.997859i \(-0.520832\pi\)
−0.0653985 + 0.997859i \(0.520832\pi\)
\(24\) 7.97439 1.62777
\(25\) 3.42509 0.685017
\(26\) 1.26561 0.248206
\(27\) −2.37138 −0.456373
\(28\) 0 0
\(29\) 1.09606 0.203534 0.101767 0.994808i \(-0.467550\pi\)
0.101767 + 0.994808i \(0.467550\pi\)
\(30\) 9.65146 1.76211
\(31\) −10.4325 −1.87373 −0.936864 0.349693i \(-0.886286\pi\)
−0.936864 + 0.349693i \(0.886286\pi\)
\(32\) 2.21675 0.391870
\(33\) −5.35062 −0.931424
\(34\) 5.06074 0.867910
\(35\) 0 0
\(36\) −1.55415 −0.259025
\(37\) −3.08537 −0.507232 −0.253616 0.967305i \(-0.581620\pi\)
−0.253616 + 0.967305i \(0.581620\pi\)
\(38\) 8.81129 1.42938
\(39\) −2.62728 −0.420701
\(40\) 8.81005 1.39299
\(41\) −0.521150 −0.0813900 −0.0406950 0.999172i \(-0.512957\pi\)
−0.0406950 + 0.999172i \(0.512957\pi\)
\(42\) 0 0
\(43\) 0.329024 0.0501757 0.0250879 0.999685i \(-0.492013\pi\)
0.0250879 + 0.999685i \(0.492013\pi\)
\(44\) −0.811031 −0.122268
\(45\) −11.3277 −1.68863
\(46\) −0.793891 −0.117053
\(47\) 10.5457 1.53825 0.769123 0.639101i \(-0.220693\pi\)
0.769123 + 0.639101i \(0.220693\pi\)
\(48\) 7.99991 1.15469
\(49\) 0 0
\(50\) 4.33482 0.613036
\(51\) −10.5056 −1.47108
\(52\) −0.398235 −0.0552253
\(53\) 7.11900 0.977870 0.488935 0.872320i \(-0.337386\pi\)
0.488935 + 0.872320i \(0.337386\pi\)
\(54\) −3.00124 −0.408417
\(55\) −5.91133 −0.797084
\(56\) 0 0
\(57\) −18.2914 −2.42275
\(58\) 1.38719 0.182147
\(59\) 2.03656 0.265138 0.132569 0.991174i \(-0.457677\pi\)
0.132569 + 0.991174i \(0.457677\pi\)
\(60\) −3.03692 −0.392065
\(61\) 2.40081 0.307393 0.153696 0.988118i \(-0.450882\pi\)
0.153696 + 0.988118i \(0.450882\pi\)
\(62\) −13.2034 −1.67684
\(63\) 0 0
\(64\) 8.89542 1.11193
\(65\) −2.90260 −0.360023
\(66\) −6.77179 −0.833550
\(67\) 14.6942 1.79518 0.897589 0.440832i \(-0.145317\pi\)
0.897589 + 0.440832i \(0.145317\pi\)
\(68\) −1.59241 −0.193108
\(69\) 1.64804 0.198401
\(70\) 0 0
\(71\) 3.60141 0.427409 0.213704 0.976898i \(-0.431447\pi\)
0.213704 + 0.976898i \(0.431447\pi\)
\(72\) −11.8453 −1.39598
\(73\) 2.97573 0.348283 0.174141 0.984721i \(-0.444285\pi\)
0.174141 + 0.984721i \(0.444285\pi\)
\(74\) −3.90487 −0.453932
\(75\) −8.99866 −1.03908
\(76\) −2.77255 −0.318034
\(77\) 0 0
\(78\) −3.32511 −0.376494
\(79\) −8.76150 −0.985746 −0.492873 0.870101i \(-0.664053\pi\)
−0.492873 + 0.870101i \(0.664053\pi\)
\(80\) 8.83824 0.988145
\(81\) −5.47751 −0.608613
\(82\) −0.659572 −0.0728376
\(83\) 12.8039 1.40541 0.702703 0.711483i \(-0.251976\pi\)
0.702703 + 0.711483i \(0.251976\pi\)
\(84\) 0 0
\(85\) −11.6065 −1.25890
\(86\) 0.416416 0.0449033
\(87\) −2.87966 −0.308732
\(88\) −6.18143 −0.658943
\(89\) −2.68098 −0.284184 −0.142092 0.989853i \(-0.545383\pi\)
−0.142092 + 0.989853i \(0.545383\pi\)
\(90\) −14.3364 −1.51119
\(91\) 0 0
\(92\) 0.249805 0.0260440
\(93\) 27.4090 2.84219
\(94\) 13.3467 1.37661
\(95\) −20.2082 −2.07332
\(96\) −5.82403 −0.594413
\(97\) −2.32902 −0.236477 −0.118238 0.992985i \(-0.537725\pi\)
−0.118238 + 0.992985i \(0.537725\pi\)
\(98\) 0 0
\(99\) 7.94789 0.798793
\(100\) −1.36399 −0.136399
\(101\) 1.45324 0.144603 0.0723014 0.997383i \(-0.476966\pi\)
0.0723014 + 0.997383i \(0.476966\pi\)
\(102\) −13.2960 −1.31650
\(103\) −11.6353 −1.14646 −0.573230 0.819394i \(-0.694310\pi\)
−0.573230 + 0.819394i \(0.694310\pi\)
\(104\) −3.03523 −0.297628
\(105\) 0 0
\(106\) 9.00987 0.875115
\(107\) 19.6259 1.89731 0.948656 0.316310i \(-0.102444\pi\)
0.948656 + 0.316310i \(0.102444\pi\)
\(108\) 0.944368 0.0908719
\(109\) −1.10676 −0.106008 −0.0530040 0.998594i \(-0.516880\pi\)
−0.0530040 + 0.998594i \(0.516880\pi\)
\(110\) −7.48143 −0.713326
\(111\) 8.10613 0.769400
\(112\) 0 0
\(113\) −1.09606 −0.103109 −0.0515545 0.998670i \(-0.516418\pi\)
−0.0515545 + 0.998670i \(0.516418\pi\)
\(114\) −23.1497 −2.16817
\(115\) 1.82074 0.169785
\(116\) −0.436491 −0.0405272
\(117\) 3.90260 0.360796
\(118\) 2.57749 0.237277
\(119\) 0 0
\(120\) −23.1465 −2.11297
\(121\) −6.85241 −0.622946
\(122\) 3.03849 0.275092
\(123\) 1.36921 0.123457
\(124\) 4.15458 0.373092
\(125\) 4.57134 0.408873
\(126\) 0 0
\(127\) 5.18143 0.459778 0.229889 0.973217i \(-0.426164\pi\)
0.229889 + 0.973217i \(0.426164\pi\)
\(128\) 6.82461 0.603216
\(129\) −0.864439 −0.0761096
\(130\) −3.67356 −0.322192
\(131\) 10.5667 0.923217 0.461609 0.887084i \(-0.347272\pi\)
0.461609 + 0.887084i \(0.347272\pi\)
\(132\) 2.13081 0.185463
\(133\) 0 0
\(134\) 18.5971 1.60654
\(135\) 6.88318 0.592410
\(136\) −12.1368 −1.04073
\(137\) −5.87177 −0.501659 −0.250830 0.968031i \(-0.580703\pi\)
−0.250830 + 0.968031i \(0.580703\pi\)
\(138\) 2.08577 0.177553
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −27.7065 −2.33331
\(142\) 4.55797 0.382497
\(143\) 2.03656 0.170306
\(144\) −11.8832 −0.990265
\(145\) −3.18143 −0.264204
\(146\) 3.76611 0.311685
\(147\) 0 0
\(148\) 1.22870 0.100999
\(149\) −10.1054 −0.827868 −0.413934 0.910307i \(-0.635846\pi\)
−0.413934 + 0.910307i \(0.635846\pi\)
\(150\) −11.3888 −0.929890
\(151\) −0.187726 −0.0152769 −0.00763847 0.999971i \(-0.502431\pi\)
−0.00763847 + 0.999971i \(0.502431\pi\)
\(152\) −21.1316 −1.71400
\(153\) 15.6052 1.26160
\(154\) 0 0
\(155\) 30.2813 2.43225
\(156\) 1.04628 0.0837691
\(157\) −12.0718 −0.963434 −0.481717 0.876327i \(-0.659987\pi\)
−0.481717 + 0.876327i \(0.659987\pi\)
\(158\) −11.0886 −0.882164
\(159\) −18.7036 −1.48329
\(160\) −6.43435 −0.508680
\(161\) 0 0
\(162\) −6.93239 −0.544660
\(163\) 14.9136 1.16812 0.584060 0.811711i \(-0.301463\pi\)
0.584060 + 0.811711i \(0.301463\pi\)
\(164\) 0.207540 0.0162062
\(165\) 15.5307 1.20906
\(166\) 16.2047 1.25773
\(167\) 5.05664 0.391294 0.195647 0.980674i \(-0.437319\pi\)
0.195647 + 0.980674i \(0.437319\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −14.6893 −1.12662
\(171\) 27.1703 2.07776
\(172\) −0.131029 −0.00999087
\(173\) −0.595615 −0.0452837 −0.0226419 0.999744i \(-0.507208\pi\)
−0.0226419 + 0.999744i \(0.507208\pi\)
\(174\) −3.64453 −0.276291
\(175\) 0 0
\(176\) −6.20121 −0.467434
\(177\) −5.35062 −0.402177
\(178\) −3.39308 −0.254322
\(179\) 8.07664 0.603676 0.301838 0.953359i \(-0.402400\pi\)
0.301838 + 0.953359i \(0.402400\pi\)
\(180\) 4.51108 0.336236
\(181\) 1.89324 0.140724 0.0703618 0.997522i \(-0.477585\pi\)
0.0703618 + 0.997522i \(0.477585\pi\)
\(182\) 0 0
\(183\) −6.30761 −0.466272
\(184\) 1.90394 0.140360
\(185\) 8.95559 0.658428
\(186\) 34.6891 2.54353
\(187\) 8.14353 0.595514
\(188\) −4.19966 −0.306292
\(189\) 0 0
\(190\) −25.5757 −1.85545
\(191\) 3.70174 0.267849 0.133924 0.990992i \(-0.457242\pi\)
0.133924 + 0.990992i \(0.457242\pi\)
\(192\) −23.3708 −1.68664
\(193\) 13.5875 0.978047 0.489024 0.872271i \(-0.337353\pi\)
0.489024 + 0.872271i \(0.337353\pi\)
\(194\) −2.94763 −0.211628
\(195\) 7.62594 0.546105
\(196\) 0 0
\(197\) −9.70258 −0.691280 −0.345640 0.938367i \(-0.612338\pi\)
−0.345640 + 0.938367i \(0.612338\pi\)
\(198\) 10.0589 0.714856
\(199\) 26.2720 1.86237 0.931185 0.364547i \(-0.118776\pi\)
0.931185 + 0.364547i \(0.118776\pi\)
\(200\) −10.3959 −0.735102
\(201\) −38.6057 −2.72304
\(202\) 1.83923 0.129408
\(203\) 0 0
\(204\) 4.18370 0.292918
\(205\) 1.51269 0.105651
\(206\) −14.7257 −1.02599
\(207\) −2.44802 −0.170149
\(208\) −3.04494 −0.211128
\(209\) 14.1788 0.980765
\(210\) 0 0
\(211\) 10.0338 0.690758 0.345379 0.938463i \(-0.387750\pi\)
0.345379 + 0.938463i \(0.387750\pi\)
\(212\) −2.83504 −0.194711
\(213\) −9.46191 −0.648320
\(214\) 24.8388 1.69794
\(215\) −0.955026 −0.0651322
\(216\) 7.19769 0.489740
\(217\) 0 0
\(218\) −1.40072 −0.0948688
\(219\) −7.81807 −0.528296
\(220\) 2.35410 0.158713
\(221\) 3.99866 0.268979
\(222\) 10.2592 0.688552
\(223\) 17.4961 1.17163 0.585813 0.810446i \(-0.300775\pi\)
0.585813 + 0.810446i \(0.300775\pi\)
\(224\) 0 0
\(225\) 13.3667 0.891116
\(226\) −1.38719 −0.0922743
\(227\) −9.51630 −0.631619 −0.315810 0.948823i \(-0.602276\pi\)
−0.315810 + 0.948823i \(0.602276\pi\)
\(228\) 7.28427 0.482413
\(229\) 21.1170 1.39545 0.697725 0.716366i \(-0.254196\pi\)
0.697725 + 0.716366i \(0.254196\pi\)
\(230\) 2.30435 0.151944
\(231\) 0 0
\(232\) −3.32680 −0.218415
\(233\) 14.1788 0.928881 0.464441 0.885604i \(-0.346255\pi\)
0.464441 + 0.885604i \(0.346255\pi\)
\(234\) 4.93916 0.322883
\(235\) −30.6099 −1.99677
\(236\) −0.811031 −0.0527936
\(237\) 23.0189 1.49524
\(238\) 0 0
\(239\) −16.5275 −1.06907 −0.534536 0.845145i \(-0.679514\pi\)
−0.534536 + 0.845145i \(0.679514\pi\)
\(240\) −23.2205 −1.49888
\(241\) −13.6890 −0.881786 −0.440893 0.897560i \(-0.645338\pi\)
−0.440893 + 0.897560i \(0.645338\pi\)
\(242\) −8.67247 −0.557487
\(243\) 21.5051 1.37955
\(244\) −0.956089 −0.0612073
\(245\) 0 0
\(246\) 1.73288 0.110484
\(247\) 6.96210 0.442988
\(248\) 31.6649 2.01073
\(249\) −33.6393 −2.13181
\(250\) 5.78553 0.365909
\(251\) −14.6603 −0.925349 −0.462674 0.886528i \(-0.653110\pi\)
−0.462674 + 0.886528i \(0.653110\pi\)
\(252\) 0 0
\(253\) −1.27750 −0.0803155
\(254\) 6.55767 0.411464
\(255\) 30.4936 1.90958
\(256\) −9.15355 −0.572097
\(257\) −1.75277 −0.109335 −0.0546675 0.998505i \(-0.517410\pi\)
−0.0546675 + 0.998505i \(0.517410\pi\)
\(258\) −1.09404 −0.0681120
\(259\) 0 0
\(260\) 1.15592 0.0716870
\(261\) 4.27750 0.264770
\(262\) 13.3733 0.826206
\(263\) 26.9416 1.66129 0.830645 0.556802i \(-0.187972\pi\)
0.830645 + 0.556802i \(0.187972\pi\)
\(264\) 16.2404 0.999525
\(265\) −20.6636 −1.26936
\(266\) 0 0
\(267\) 7.04370 0.431067
\(268\) −5.85174 −0.357452
\(269\) −22.0691 −1.34558 −0.672789 0.739835i \(-0.734904\pi\)
−0.672789 + 0.739835i \(0.734904\pi\)
\(270\) 8.71141 0.530159
\(271\) −8.96210 −0.544409 −0.272204 0.962239i \(-0.587753\pi\)
−0.272204 + 0.962239i \(0.587753\pi\)
\(272\) −12.1757 −0.738259
\(273\) 0 0
\(274\) −7.43137 −0.448945
\(275\) 6.97541 0.420633
\(276\) −0.656308 −0.0395051
\(277\) −7.52925 −0.452389 −0.226194 0.974082i \(-0.572628\pi\)
−0.226194 + 0.974082i \(0.572628\pi\)
\(278\) −5.06243 −0.303625
\(279\) −40.7138 −2.43747
\(280\) 0 0
\(281\) 29.7762 1.77630 0.888151 0.459553i \(-0.151990\pi\)
0.888151 + 0.459553i \(0.151990\pi\)
\(282\) −35.0655 −2.08812
\(283\) 0.301451 0.0179194 0.00895970 0.999960i \(-0.497148\pi\)
0.00895970 + 0.999960i \(0.497148\pi\)
\(284\) −1.43421 −0.0851046
\(285\) 53.0926 3.14493
\(286\) 2.57749 0.152410
\(287\) 0 0
\(288\) 8.65110 0.509771
\(289\) −1.01069 −0.0594526
\(290\) −4.02645 −0.236441
\(291\) 6.11900 0.358702
\(292\) −1.18504 −0.0693492
\(293\) −19.2471 −1.12443 −0.562214 0.826992i \(-0.690050\pi\)
−0.562214 + 0.826992i \(0.690050\pi\)
\(294\) 0 0
\(295\) −5.91133 −0.344171
\(296\) 9.36480 0.544318
\(297\) −4.82947 −0.280234
\(298\) −12.7895 −0.740876
\(299\) −0.627280 −0.0362765
\(300\) 3.58358 0.206898
\(301\) 0 0
\(302\) −0.237588 −0.0136716
\(303\) −3.81807 −0.219342
\(304\) −21.1992 −1.21586
\(305\) −6.96860 −0.399021
\(306\) 19.7501 1.12904
\(307\) −3.57779 −0.204195 −0.102098 0.994774i \(-0.532555\pi\)
−0.102098 + 0.994774i \(0.532555\pi\)
\(308\) 0 0
\(309\) 30.5692 1.73902
\(310\) 38.3243 2.17667
\(311\) −23.8306 −1.35131 −0.675655 0.737218i \(-0.736139\pi\)
−0.675655 + 0.737218i \(0.736139\pi\)
\(312\) 7.97439 0.451461
\(313\) −18.0814 −1.02202 −0.511009 0.859575i \(-0.670728\pi\)
−0.511009 + 0.859575i \(0.670728\pi\)
\(314\) −15.2782 −0.862196
\(315\) 0 0
\(316\) 3.48914 0.196279
\(317\) −27.5482 −1.54726 −0.773630 0.633638i \(-0.781561\pi\)
−0.773630 + 0.633638i \(0.781561\pi\)
\(318\) −23.6714 −1.32743
\(319\) 2.23220 0.124979
\(320\) −25.8198 −1.44337
\(321\) −51.5628 −2.87796
\(322\) 0 0
\(323\) 27.8391 1.54901
\(324\) 2.18134 0.121185
\(325\) 3.42509 0.189990
\(326\) 18.8747 1.04537
\(327\) 2.90776 0.160799
\(328\) 1.58181 0.0873408
\(329\) 0 0
\(330\) 19.6558 1.08202
\(331\) −18.1814 −0.999339 −0.499669 0.866216i \(-0.666545\pi\)
−0.499669 + 0.866216i \(0.666545\pi\)
\(332\) −5.09895 −0.279841
\(333\) −12.0410 −0.659841
\(334\) 6.39972 0.350177
\(335\) −42.6513 −2.33029
\(336\) 0 0
\(337\) −17.1381 −0.933572 −0.466786 0.884370i \(-0.654588\pi\)
−0.466786 + 0.884370i \(0.654588\pi\)
\(338\) 1.26561 0.0688400
\(339\) 2.87966 0.156402
\(340\) 4.62212 0.250670
\(341\) −21.2464 −1.15056
\(342\) 34.3869 1.85943
\(343\) 0 0
\(344\) −0.998663 −0.0538443
\(345\) −4.78360 −0.257540
\(346\) −0.753815 −0.0405253
\(347\) 22.2688 1.19545 0.597725 0.801701i \(-0.296071\pi\)
0.597725 + 0.801701i \(0.296071\pi\)
\(348\) 1.14678 0.0614741
\(349\) 19.9368 1.06719 0.533595 0.845740i \(-0.320841\pi\)
0.533595 + 0.845740i \(0.320841\pi\)
\(350\) 0 0
\(351\) −2.37138 −0.126575
\(352\) 4.51456 0.240627
\(353\) 22.9152 1.21965 0.609825 0.792536i \(-0.291240\pi\)
0.609825 + 0.792536i \(0.291240\pi\)
\(354\) −6.77179 −0.359917
\(355\) −10.4535 −0.554812
\(356\) 1.06766 0.0565860
\(357\) 0 0
\(358\) 10.2219 0.540242
\(359\) 27.2314 1.43722 0.718610 0.695413i \(-0.244779\pi\)
0.718610 + 0.695413i \(0.244779\pi\)
\(360\) 34.3821 1.81210
\(361\) 29.4708 1.55110
\(362\) 2.39611 0.125936
\(363\) 18.0032 0.944923
\(364\) 0 0
\(365\) −8.63735 −0.452099
\(366\) −7.98297 −0.417276
\(367\) −10.8564 −0.566702 −0.283351 0.959016i \(-0.591446\pi\)
−0.283351 + 0.959016i \(0.591446\pi\)
\(368\) 1.91003 0.0995671
\(369\) −2.03384 −0.105878
\(370\) 11.3343 0.589241
\(371\) 0 0
\(372\) −10.9152 −0.565929
\(373\) −2.37144 −0.122789 −0.0613943 0.998114i \(-0.519555\pi\)
−0.0613943 + 0.998114i \(0.519555\pi\)
\(374\) 10.3065 0.532938
\(375\) −12.0102 −0.620204
\(376\) −32.0085 −1.65071
\(377\) 1.09606 0.0564501
\(378\) 0 0
\(379\) −29.2197 −1.50092 −0.750458 0.660918i \(-0.770167\pi\)
−0.750458 + 0.660918i \(0.770167\pi\)
\(380\) 8.04761 0.412834
\(381\) −13.6131 −0.697419
\(382\) 4.68496 0.239703
\(383\) −3.06595 −0.156663 −0.0783313 0.996927i \(-0.524959\pi\)
−0.0783313 + 0.996927i \(0.524959\pi\)
\(384\) −17.9302 −0.914995
\(385\) 0 0
\(386\) 17.1964 0.875274
\(387\) 1.28405 0.0652719
\(388\) 0.927499 0.0470866
\(389\) 27.7410 1.40652 0.703261 0.710932i \(-0.251726\pi\)
0.703261 + 0.710932i \(0.251726\pi\)
\(390\) 9.65146 0.488721
\(391\) −2.50828 −0.126849
\(392\) 0 0
\(393\) −27.7617 −1.40039
\(394\) −12.2797 −0.618641
\(395\) 25.4311 1.27958
\(396\) −3.16513 −0.159054
\(397\) 17.2312 0.864808 0.432404 0.901680i \(-0.357665\pi\)
0.432404 + 0.901680i \(0.357665\pi\)
\(398\) 33.2500 1.66667
\(399\) 0 0
\(400\) −10.4292 −0.521459
\(401\) 16.6440 0.831163 0.415582 0.909556i \(-0.363578\pi\)
0.415582 + 0.909556i \(0.363578\pi\)
\(402\) −48.8597 −2.43690
\(403\) −10.4325 −0.519679
\(404\) −0.578731 −0.0287930
\(405\) 15.8990 0.790029
\(406\) 0 0
\(407\) −6.28355 −0.311464
\(408\) 31.8869 1.57864
\(409\) −13.6338 −0.674147 −0.337073 0.941478i \(-0.609437\pi\)
−0.337073 + 0.941478i \(0.609437\pi\)
\(410\) 1.91447 0.0945491
\(411\) 15.4268 0.760948
\(412\) 4.63359 0.228280
\(413\) 0 0
\(414\) −3.09824 −0.152270
\(415\) −37.1645 −1.82433
\(416\) 2.21675 0.108685
\(417\) 10.5091 0.514634
\(418\) 17.9448 0.877707
\(419\) −10.8502 −0.530066 −0.265033 0.964239i \(-0.585383\pi\)
−0.265033 + 0.964239i \(0.585383\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 12.6989 0.618173
\(423\) 41.1556 2.00105
\(424\) −21.6078 −1.04937
\(425\) 13.6958 0.664342
\(426\) −11.9751 −0.580194
\(427\) 0 0
\(428\) −7.81574 −0.377788
\(429\) −5.35062 −0.258331
\(430\) −1.20869 −0.0582881
\(431\) 1.20953 0.0582609 0.0291304 0.999576i \(-0.490726\pi\)
0.0291304 + 0.999576i \(0.490726\pi\)
\(432\) 7.22072 0.347407
\(433\) −5.56422 −0.267399 −0.133700 0.991022i \(-0.542686\pi\)
−0.133700 + 0.991022i \(0.542686\pi\)
\(434\) 0 0
\(435\) 8.35851 0.400760
\(436\) 0.440749 0.0211081
\(437\) −4.36719 −0.208911
\(438\) −9.89461 −0.472783
\(439\) −19.7192 −0.941146 −0.470573 0.882361i \(-0.655953\pi\)
−0.470573 + 0.882361i \(0.655953\pi\)
\(440\) 17.9422 0.855362
\(441\) 0 0
\(442\) 5.06074 0.240715
\(443\) 22.2310 1.05623 0.528113 0.849174i \(-0.322900\pi\)
0.528113 + 0.849174i \(0.322900\pi\)
\(444\) −3.22815 −0.153201
\(445\) 7.78182 0.368894
\(446\) 22.1432 1.04851
\(447\) 26.5498 1.25576
\(448\) 0 0
\(449\) 18.4579 0.871082 0.435541 0.900169i \(-0.356557\pi\)
0.435541 + 0.900169i \(0.356557\pi\)
\(450\) 16.9171 0.797478
\(451\) −1.06136 −0.0499773
\(452\) 0.436491 0.0205308
\(453\) 0.493209 0.0231730
\(454\) −12.0439 −0.565249
\(455\) 0 0
\(456\) 55.5185 2.59989
\(457\) −29.9819 −1.40250 −0.701248 0.712917i \(-0.747373\pi\)
−0.701248 + 0.712917i \(0.747373\pi\)
\(458\) 26.7258 1.24882
\(459\) −9.48236 −0.442599
\(460\) −0.725084 −0.0338072
\(461\) 29.1498 1.35764 0.678821 0.734304i \(-0.262491\pi\)
0.678821 + 0.734304i \(0.262491\pi\)
\(462\) 0 0
\(463\) 1.55900 0.0724530 0.0362265 0.999344i \(-0.488466\pi\)
0.0362265 + 0.999344i \(0.488466\pi\)
\(464\) −3.33744 −0.154937
\(465\) −79.5575 −3.68939
\(466\) 17.9448 0.831275
\(467\) 12.4231 0.574874 0.287437 0.957800i \(-0.407197\pi\)
0.287437 + 0.957800i \(0.407197\pi\)
\(468\) −1.55415 −0.0718407
\(469\) 0 0
\(470\) −38.7402 −1.78695
\(471\) 31.7160 1.46139
\(472\) −6.18143 −0.284523
\(473\) 0.670079 0.0308103
\(474\) 29.1329 1.33812
\(475\) 23.8458 1.09412
\(476\) 0 0
\(477\) 27.7826 1.27208
\(478\) −20.9173 −0.956735
\(479\) 36.0558 1.64743 0.823716 0.567003i \(-0.191897\pi\)
0.823716 + 0.567003i \(0.191897\pi\)
\(480\) 16.9048 0.771597
\(481\) −3.08537 −0.140681
\(482\) −17.3249 −0.789128
\(483\) 0 0
\(484\) 2.72887 0.124040
\(485\) 6.76023 0.306966
\(486\) 27.2170 1.23459
\(487\) 7.30004 0.330796 0.165398 0.986227i \(-0.447109\pi\)
0.165398 + 0.986227i \(0.447109\pi\)
\(488\) −7.28702 −0.329868
\(489\) −39.1821 −1.77188
\(490\) 0 0
\(491\) 4.49178 0.202711 0.101356 0.994850i \(-0.467682\pi\)
0.101356 + 0.994850i \(0.467682\pi\)
\(492\) −0.545267 −0.0245825
\(493\) 4.38279 0.197391
\(494\) 8.81129 0.396439
\(495\) −23.0696 −1.03690
\(496\) 31.7663 1.42635
\(497\) 0 0
\(498\) −42.5742 −1.90780
\(499\) 11.3674 0.508873 0.254437 0.967089i \(-0.418110\pi\)
0.254437 + 0.967089i \(0.418110\pi\)
\(500\) −1.82047 −0.0814139
\(501\) −13.2852 −0.593539
\(502\) −18.5542 −0.828113
\(503\) 17.1080 0.762806 0.381403 0.924409i \(-0.375441\pi\)
0.381403 + 0.924409i \(0.375441\pi\)
\(504\) 0 0
\(505\) −4.21818 −0.187706
\(506\) −1.61681 −0.0718759
\(507\) −2.62728 −0.116682
\(508\) −2.06343 −0.0915498
\(509\) 3.28284 0.145509 0.0727547 0.997350i \(-0.476821\pi\)
0.0727547 + 0.997350i \(0.476821\pi\)
\(510\) 38.5929 1.70892
\(511\) 0 0
\(512\) −25.2340 −1.11520
\(513\) −16.5098 −0.728926
\(514\) −2.21833 −0.0978462
\(515\) 33.7726 1.48820
\(516\) 0.344250 0.0151548
\(517\) 21.4770 0.944555
\(518\) 0 0
\(519\) 1.56485 0.0686891
\(520\) 8.81005 0.386346
\(521\) −4.77061 −0.209004 −0.104502 0.994525i \(-0.533325\pi\)
−0.104502 + 0.994525i \(0.533325\pi\)
\(522\) 5.41363 0.236948
\(523\) 25.5124 1.11558 0.557789 0.829983i \(-0.311650\pi\)
0.557789 + 0.829983i \(0.311650\pi\)
\(524\) −4.20803 −0.183829
\(525\) 0 0
\(526\) 34.0975 1.48672
\(527\) −41.7160 −1.81718
\(528\) 16.2923 0.709032
\(529\) −22.6065 −0.982892
\(530\) −26.1520 −1.13597
\(531\) 7.94789 0.344909
\(532\) 0 0
\(533\) −0.521150 −0.0225735
\(534\) 8.91456 0.385771
\(535\) −56.9663 −2.46287
\(536\) −44.6001 −1.92643
\(537\) −21.2196 −0.915693
\(538\) −27.9309 −1.20418
\(539\) 0 0
\(540\) −2.74112 −0.117959
\(541\) −16.5157 −0.710064 −0.355032 0.934854i \(-0.615530\pi\)
−0.355032 + 0.934854i \(0.615530\pi\)
\(542\) −11.3425 −0.487202
\(543\) −4.97408 −0.213458
\(544\) 8.86405 0.380043
\(545\) 3.21247 0.137607
\(546\) 0 0
\(547\) 23.3317 0.997591 0.498796 0.866720i \(-0.333776\pi\)
0.498796 + 0.866720i \(0.333776\pi\)
\(548\) 2.33835 0.0998892
\(549\) 9.36942 0.399877
\(550\) 8.82814 0.376433
\(551\) 7.63090 0.325087
\(552\) −5.00218 −0.212907
\(553\) 0 0
\(554\) −9.52909 −0.404852
\(555\) −23.5289 −0.998744
\(556\) 1.59294 0.0675557
\(557\) −20.0471 −0.849422 −0.424711 0.905329i \(-0.639624\pi\)
−0.424711 + 0.905329i \(0.639624\pi\)
\(558\) −51.5277 −2.18134
\(559\) 0.329024 0.0139162
\(560\) 0 0
\(561\) −21.3953 −0.903312
\(562\) 37.6851 1.58965
\(563\) −40.5284 −1.70807 −0.854034 0.520218i \(-0.825851\pi\)
−0.854034 + 0.520218i \(0.825851\pi\)
\(564\) 11.0337 0.464602
\(565\) 3.18143 0.133844
\(566\) 0.381519 0.0160364
\(567\) 0 0
\(568\) −10.9311 −0.458659
\(569\) −21.4504 −0.899246 −0.449623 0.893219i \(-0.648442\pi\)
−0.449623 + 0.893219i \(0.648442\pi\)
\(570\) 67.1944 2.81446
\(571\) −10.9559 −0.458489 −0.229244 0.973369i \(-0.573625\pi\)
−0.229244 + 0.973369i \(0.573625\pi\)
\(572\) −0.811031 −0.0339109
\(573\) −9.72552 −0.406289
\(574\) 0 0
\(575\) −2.14849 −0.0895982
\(576\) 34.7153 1.44647
\(577\) 34.7415 1.44631 0.723154 0.690687i \(-0.242692\pi\)
0.723154 + 0.690687i \(0.242692\pi\)
\(578\) −1.27914 −0.0532053
\(579\) −35.6981 −1.48356
\(580\) 1.26696 0.0526076
\(581\) 0 0
\(582\) 7.74426 0.321010
\(583\) 14.4983 0.600458
\(584\) −9.03201 −0.373747
\(585\) −11.3277 −0.468342
\(586\) −24.3593 −1.00627
\(587\) −22.8463 −0.942967 −0.471483 0.881875i \(-0.656281\pi\)
−0.471483 + 0.881875i \(0.656281\pi\)
\(588\) 0 0
\(589\) −72.6320 −2.99275
\(590\) −7.48143 −0.308006
\(591\) 25.4914 1.04858
\(592\) 9.39476 0.386122
\(593\) 17.5935 0.722480 0.361240 0.932473i \(-0.382354\pi\)
0.361240 + 0.932473i \(0.382354\pi\)
\(594\) −6.11222 −0.250787
\(595\) 0 0
\(596\) 4.02433 0.164843
\(597\) −69.0238 −2.82496
\(598\) −0.793891 −0.0324646
\(599\) 31.0073 1.26692 0.633461 0.773774i \(-0.281634\pi\)
0.633461 + 0.773774i \(0.281634\pi\)
\(600\) 27.3130 1.11505
\(601\) −1.43754 −0.0586385 −0.0293193 0.999570i \(-0.509334\pi\)
−0.0293193 + 0.999570i \(0.509334\pi\)
\(602\) 0 0
\(603\) 57.3455 2.33529
\(604\) 0.0747591 0.00304191
\(605\) 19.8898 0.808635
\(606\) −4.83218 −0.196294
\(607\) −33.0171 −1.34012 −0.670061 0.742306i \(-0.733732\pi\)
−0.670061 + 0.742306i \(0.733732\pi\)
\(608\) 15.4333 0.625901
\(609\) 0 0
\(610\) −8.81953 −0.357092
\(611\) 10.5457 0.426633
\(612\) −6.21453 −0.251208
\(613\) −43.1657 −1.74345 −0.871723 0.489999i \(-0.836997\pi\)
−0.871723 + 0.489999i \(0.836997\pi\)
\(614\) −4.52808 −0.182738
\(615\) −3.97426 −0.160258
\(616\) 0 0
\(617\) 2.45772 0.0989441 0.0494721 0.998776i \(-0.484246\pi\)
0.0494721 + 0.998776i \(0.484246\pi\)
\(618\) 38.6886 1.55628
\(619\) 37.7789 1.51846 0.759231 0.650822i \(-0.225575\pi\)
0.759231 + 0.650822i \(0.225575\pi\)
\(620\) −12.0591 −0.484305
\(621\) 1.48752 0.0596922
\(622\) −30.1602 −1.20932
\(623\) 0 0
\(624\) 7.99991 0.320253
\(625\) −30.3942 −1.21577
\(626\) −22.8839 −0.914625
\(627\) −37.2516 −1.48768
\(628\) 4.80741 0.191837
\(629\) −12.3374 −0.491922
\(630\) 0 0
\(631\) −28.4828 −1.13388 −0.566942 0.823758i \(-0.691874\pi\)
−0.566942 + 0.823758i \(0.691874\pi\)
\(632\) 26.5932 1.05782
\(633\) −26.3617 −1.04778
\(634\) −34.8652 −1.38467
\(635\) −15.0396 −0.596829
\(636\) 7.44843 0.295350
\(637\) 0 0
\(638\) 2.82509 0.111847
\(639\) 14.0549 0.556002
\(640\) −19.8091 −0.783024
\(641\) −27.1922 −1.07403 −0.537014 0.843573i \(-0.680448\pi\)
−0.537014 + 0.843573i \(0.680448\pi\)
\(642\) −65.2584 −2.57554
\(643\) −37.1664 −1.46570 −0.732849 0.680391i \(-0.761810\pi\)
−0.732849 + 0.680391i \(0.761810\pi\)
\(644\) 0 0
\(645\) 2.50912 0.0987965
\(646\) 35.2334 1.38624
\(647\) 18.8319 0.740357 0.370178 0.928961i \(-0.379297\pi\)
0.370178 + 0.928961i \(0.379297\pi\)
\(648\) 16.6255 0.653111
\(649\) 4.14759 0.162807
\(650\) 4.33482 0.170026
\(651\) 0 0
\(652\) −5.93910 −0.232593
\(653\) −26.0185 −1.01818 −0.509090 0.860713i \(-0.670018\pi\)
−0.509090 + 0.860713i \(0.670018\pi\)
\(654\) 3.68008 0.143903
\(655\) −30.6709 −1.19841
\(656\) 1.58687 0.0619569
\(657\) 11.6131 0.453069
\(658\) 0 0
\(659\) −33.3339 −1.29851 −0.649253 0.760573i \(-0.724918\pi\)
−0.649253 + 0.760573i \(0.724918\pi\)
\(660\) −6.18488 −0.240746
\(661\) 6.29841 0.244980 0.122490 0.992470i \(-0.460912\pi\)
0.122490 + 0.992470i \(0.460912\pi\)
\(662\) −23.0105 −0.894329
\(663\) −10.5056 −0.408004
\(664\) −38.8626 −1.50816
\(665\) 0 0
\(666\) −15.2391 −0.590505
\(667\) −0.687538 −0.0266216
\(668\) −2.01373 −0.0779136
\(669\) −45.9672 −1.77719
\(670\) −53.9799 −2.08542
\(671\) 4.88941 0.188754
\(672\) 0 0
\(673\) 18.3188 0.706137 0.353068 0.935598i \(-0.385138\pi\)
0.353068 + 0.935598i \(0.385138\pi\)
\(674\) −21.6901 −0.835473
\(675\) −8.12219 −0.312623
\(676\) −0.398235 −0.0153167
\(677\) 24.3392 0.935430 0.467715 0.883879i \(-0.345077\pi\)
0.467715 + 0.883879i \(0.345077\pi\)
\(678\) 3.64453 0.139967
\(679\) 0 0
\(680\) 35.2284 1.35095
\(681\) 25.0020 0.958078
\(682\) −26.8896 −1.02966
\(683\) 11.7682 0.450297 0.225149 0.974324i \(-0.427713\pi\)
0.225149 + 0.974324i \(0.427713\pi\)
\(684\) −10.8202 −0.413719
\(685\) 17.0434 0.651195
\(686\) 0 0
\(687\) −55.4802 −2.11670
\(688\) −1.00186 −0.0381955
\(689\) 7.11900 0.271212
\(690\) −6.05417 −0.230478
\(691\) 1.17785 0.0448074 0.0224037 0.999749i \(-0.492868\pi\)
0.0224037 + 0.999749i \(0.492868\pi\)
\(692\) 0.237195 0.00901679
\(693\) 0 0
\(694\) 28.1835 1.06983
\(695\) 11.6104 0.440408
\(696\) 8.74043 0.331305
\(697\) −2.08390 −0.0789335
\(698\) 25.2321 0.955050
\(699\) −37.2516 −1.40898
\(700\) 0 0
\(701\) −31.2867 −1.18168 −0.590841 0.806788i \(-0.701204\pi\)
−0.590841 + 0.806788i \(0.701204\pi\)
\(702\) −3.00124 −0.113275
\(703\) −21.4806 −0.810158
\(704\) 18.1161 0.682776
\(705\) 80.4208 3.02882
\(706\) 29.0016 1.09149
\(707\) 0 0
\(708\) 2.13081 0.0800806
\(709\) 15.3748 0.577411 0.288706 0.957418i \(-0.406775\pi\)
0.288706 + 0.957418i \(0.406775\pi\)
\(710\) −13.2300 −0.496512
\(711\) −34.1926 −1.28232
\(712\) 8.13739 0.304962
\(713\) 6.54409 0.245078
\(714\) 0 0
\(715\) −5.91133 −0.221071
\(716\) −3.21640 −0.120203
\(717\) 43.4223 1.62163
\(718\) 34.4643 1.28620
\(719\) −11.1417 −0.415517 −0.207759 0.978180i \(-0.566617\pi\)
−0.207759 + 0.978180i \(0.566617\pi\)
\(720\) 34.4921 1.28545
\(721\) 0 0
\(722\) 37.2985 1.38811
\(723\) 35.9648 1.33755
\(724\) −0.753956 −0.0280206
\(725\) 3.75411 0.139424
\(726\) 22.7850 0.845631
\(727\) −6.24735 −0.231702 −0.115851 0.993267i \(-0.536959\pi\)
−0.115851 + 0.993267i \(0.536959\pi\)
\(728\) 0 0
\(729\) −40.0674 −1.48398
\(730\) −10.9315 −0.404593
\(731\) 1.31566 0.0486613
\(732\) 2.51191 0.0928430
\(733\) 30.9669 1.14379 0.571894 0.820327i \(-0.306209\pi\)
0.571894 + 0.820327i \(0.306209\pi\)
\(734\) −13.7400 −0.507153
\(735\) 0 0
\(736\) −1.39053 −0.0512554
\(737\) 29.9256 1.10232
\(738\) −2.57405 −0.0947520
\(739\) 2.33744 0.0859843 0.0429921 0.999075i \(-0.486311\pi\)
0.0429921 + 0.999075i \(0.486311\pi\)
\(740\) −3.56643 −0.131105
\(741\) −18.2914 −0.671951
\(742\) 0 0
\(743\) 24.3612 0.893726 0.446863 0.894603i \(-0.352541\pi\)
0.446863 + 0.894603i \(0.352541\pi\)
\(744\) −83.1927 −3.04999
\(745\) 29.3320 1.07464
\(746\) −3.00132 −0.109886
\(747\) 49.9684 1.82825
\(748\) −3.24304 −0.118577
\(749\) 0 0
\(750\) −15.2002 −0.555033
\(751\) −12.0253 −0.438810 −0.219405 0.975634i \(-0.570412\pi\)
−0.219405 + 0.975634i \(0.570412\pi\)
\(752\) −32.1110 −1.17097
\(753\) 38.5167 1.40363
\(754\) 1.38719 0.0505184
\(755\) 0.544894 0.0198307
\(756\) 0 0
\(757\) 25.9905 0.944641 0.472321 0.881427i \(-0.343416\pi\)
0.472321 + 0.881427i \(0.343416\pi\)
\(758\) −36.9807 −1.34320
\(759\) 3.35634 0.121827
\(760\) 61.3364 2.22491
\(761\) 13.3270 0.483103 0.241552 0.970388i \(-0.422344\pi\)
0.241552 + 0.970388i \(0.422344\pi\)
\(762\) −17.2288 −0.624134
\(763\) 0 0
\(764\) −1.47416 −0.0533334
\(765\) −45.2956 −1.63767
\(766\) −3.88029 −0.140200
\(767\) 2.03656 0.0735361
\(768\) 24.0489 0.867792
\(769\) 9.24486 0.333378 0.166689 0.986010i \(-0.446692\pi\)
0.166689 + 0.986010i \(0.446692\pi\)
\(770\) 0 0
\(771\) 4.60503 0.165846
\(772\) −5.41101 −0.194746
\(773\) −10.1419 −0.364780 −0.182390 0.983226i \(-0.558383\pi\)
−0.182390 + 0.983226i \(0.558383\pi\)
\(774\) 1.62510 0.0584132
\(775\) −35.7322 −1.28354
\(776\) 7.06912 0.253766
\(777\) 0 0
\(778\) 35.1092 1.25873
\(779\) −3.62830 −0.129997
\(780\) −3.03692 −0.108739
\(781\) 7.33450 0.262449
\(782\) −3.17450 −0.113520
\(783\) −2.59919 −0.0928873
\(784\) 0 0
\(785\) 35.0396 1.25062
\(786\) −35.1354 −1.25324
\(787\) 45.2823 1.61414 0.807070 0.590456i \(-0.201052\pi\)
0.807070 + 0.590456i \(0.201052\pi\)
\(788\) 3.86391 0.137646
\(789\) −70.7831 −2.51995
\(790\) 32.1859 1.14512
\(791\) 0 0
\(792\) −24.1237 −0.857197
\(793\) 2.40081 0.0852554
\(794\) 21.8079 0.773935
\(795\) 54.2891 1.92544
\(796\) −10.4624 −0.370831
\(797\) 27.0784 0.959165 0.479583 0.877497i \(-0.340788\pi\)
0.479583 + 0.877497i \(0.340788\pi\)
\(798\) 0 0
\(799\) 42.1686 1.49182
\(800\) 7.59257 0.268438
\(801\) −10.4628 −0.369685
\(802\) 21.0648 0.743825
\(803\) 6.06026 0.213862
\(804\) 15.3741 0.542204
\(805\) 0 0
\(806\) −13.2034 −0.465071
\(807\) 57.9817 2.04105
\(808\) −4.41091 −0.155175
\(809\) −25.7798 −0.906370 −0.453185 0.891417i \(-0.649712\pi\)
−0.453185 + 0.891417i \(0.649712\pi\)
\(810\) 20.1219 0.707013
\(811\) −25.7829 −0.905362 −0.452681 0.891673i \(-0.649532\pi\)
−0.452681 + 0.891673i \(0.649532\pi\)
\(812\) 0 0
\(813\) 23.5459 0.825792
\(814\) −7.95252 −0.278736
\(815\) −43.2881 −1.51632
\(816\) 31.9889 1.11984
\(817\) 2.29070 0.0801414
\(818\) −17.2550 −0.603308
\(819\) 0 0
\(820\) −0.602407 −0.0210370
\(821\) −1.71073 −0.0597050 −0.0298525 0.999554i \(-0.509504\pi\)
−0.0298525 + 0.999554i \(0.509504\pi\)
\(822\) 19.5243 0.680987
\(823\) −40.3773 −1.40747 −0.703733 0.710465i \(-0.748485\pi\)
−0.703733 + 0.710465i \(0.748485\pi\)
\(824\) 35.3158 1.23028
\(825\) −18.3264 −0.638042
\(826\) 0 0
\(827\) 19.5698 0.680509 0.340254 0.940333i \(-0.389487\pi\)
0.340254 + 0.940333i \(0.389487\pi\)
\(828\) 0.974889 0.0338797
\(829\) 41.5742 1.44393 0.721966 0.691928i \(-0.243239\pi\)
0.721966 + 0.691928i \(0.243239\pi\)
\(830\) −47.0357 −1.63263
\(831\) 19.7815 0.686211
\(832\) 8.89542 0.308393
\(833\) 0 0
\(834\) 13.3004 0.460556
\(835\) −14.6774 −0.507932
\(836\) −5.64648 −0.195288
\(837\) 24.7394 0.855119
\(838\) −13.7321 −0.474367
\(839\) 45.8480 1.58285 0.791425 0.611266i \(-0.209340\pi\)
0.791425 + 0.611266i \(0.209340\pi\)
\(840\) 0 0
\(841\) −27.7986 −0.958574
\(842\) −12.6561 −0.436157
\(843\) −78.2305 −2.69440
\(844\) −3.99583 −0.137542
\(845\) −2.90260 −0.0998525
\(846\) 52.0869 1.79078
\(847\) 0 0
\(848\) −21.6769 −0.744388
\(849\) −0.791997 −0.0271812
\(850\) 17.3335 0.594534
\(851\) 1.93539 0.0663443
\(852\) 3.76807 0.129092
\(853\) 40.0236 1.37038 0.685191 0.728364i \(-0.259719\pi\)
0.685191 + 0.728364i \(0.259719\pi\)
\(854\) 0 0
\(855\) −78.8645 −2.69711
\(856\) −59.5692 −2.03603
\(857\) 32.8702 1.12282 0.561412 0.827536i \(-0.310258\pi\)
0.561412 + 0.827536i \(0.310258\pi\)
\(858\) −6.77179 −0.231185
\(859\) −34.0503 −1.16178 −0.580891 0.813981i \(-0.697296\pi\)
−0.580891 + 0.813981i \(0.697296\pi\)
\(860\) 0.380325 0.0129690
\(861\) 0 0
\(862\) 1.53079 0.0521388
\(863\) −14.0642 −0.478749 −0.239375 0.970927i \(-0.576942\pi\)
−0.239375 + 0.970927i \(0.576942\pi\)
\(864\) −5.25677 −0.178839
\(865\) 1.72883 0.0587820
\(866\) −7.04212 −0.239301
\(867\) 2.65537 0.0901813
\(868\) 0 0
\(869\) −17.8434 −0.605295
\(870\) 10.5786 0.358648
\(871\) 14.6942 0.497893
\(872\) 3.35926 0.113759
\(873\) −9.08925 −0.307625
\(874\) −5.52715 −0.186959
\(875\) 0 0
\(876\) 3.11343 0.105193
\(877\) 51.0669 1.72441 0.862204 0.506562i \(-0.169084\pi\)
0.862204 + 0.506562i \(0.169084\pi\)
\(878\) −24.9568 −0.842251
\(879\) 50.5675 1.70560
\(880\) 17.9996 0.606768
\(881\) −18.4203 −0.620597 −0.310298 0.950639i \(-0.600429\pi\)
−0.310298 + 0.950639i \(0.600429\pi\)
\(882\) 0 0
\(883\) 0.126678 0.00426305 0.00213153 0.999998i \(-0.499322\pi\)
0.00213153 + 0.999998i \(0.499322\pi\)
\(884\) −1.59241 −0.0535585
\(885\) 15.5307 0.522059
\(886\) 28.1357 0.945239
\(887\) 3.87470 0.130100 0.0650498 0.997882i \(-0.479279\pi\)
0.0650498 + 0.997882i \(0.479279\pi\)
\(888\) −24.6039 −0.825654
\(889\) 0 0
\(890\) 9.84874 0.330131
\(891\) −11.1553 −0.373717
\(892\) −6.96757 −0.233291
\(893\) 73.4201 2.45691
\(894\) 33.6016 1.12381
\(895\) −23.4433 −0.783622
\(896\) 0 0
\(897\) 1.64804 0.0550265
\(898\) 23.3605 0.779549
\(899\) −11.4347 −0.381367
\(900\) −5.32311 −0.177437
\(901\) 28.4665 0.948356
\(902\) −1.34326 −0.0447257
\(903\) 0 0
\(904\) 3.32680 0.110648
\(905\) −5.49533 −0.182671
\(906\) 0.624210 0.0207380
\(907\) −46.7741 −1.55311 −0.776555 0.630050i \(-0.783035\pi\)
−0.776555 + 0.630050i \(0.783035\pi\)
\(908\) 3.78973 0.125766
\(909\) 5.67142 0.188109
\(910\) 0 0
\(911\) 5.93675 0.196693 0.0983467 0.995152i \(-0.468645\pi\)
0.0983467 + 0.995152i \(0.468645\pi\)
\(912\) 55.6961 1.84428
\(913\) 26.0759 0.862986
\(914\) −37.9454 −1.25512
\(915\) 18.3085 0.605260
\(916\) −8.40952 −0.277858
\(917\) 0 0
\(918\) −12.0010 −0.396091
\(919\) −8.58701 −0.283259 −0.141630 0.989920i \(-0.545234\pi\)
−0.141630 + 0.989920i \(0.545234\pi\)
\(920\) −5.52637 −0.182199
\(921\) 9.39985 0.309736
\(922\) 36.8922 1.21498
\(923\) 3.60141 0.118542
\(924\) 0 0
\(925\) −10.5677 −0.347463
\(926\) 1.97309 0.0648396
\(927\) −45.4079 −1.49139
\(928\) 2.42970 0.0797589
\(929\) −8.45945 −0.277546 −0.138773 0.990324i \(-0.544316\pi\)
−0.138773 + 0.990324i \(0.544316\pi\)
\(930\) −100.689 −3.30171
\(931\) 0 0
\(932\) −5.64648 −0.184957
\(933\) 62.6097 2.04975
\(934\) 15.7228 0.514466
\(935\) −23.6374 −0.773026
\(936\) −11.8453 −0.387175
\(937\) −33.3596 −1.08981 −0.544905 0.838498i \(-0.683434\pi\)
−0.544905 + 0.838498i \(0.683434\pi\)
\(938\) 0 0
\(939\) 47.5048 1.55026
\(940\) 12.1899 0.397592
\(941\) −13.4037 −0.436949 −0.218475 0.975843i \(-0.570108\pi\)
−0.218475 + 0.975843i \(0.570108\pi\)
\(942\) 40.1400 1.30783
\(943\) 0.326907 0.0106456
\(944\) −6.20121 −0.201832
\(945\) 0 0
\(946\) 0.848057 0.0275727
\(947\) 43.0794 1.39989 0.699946 0.714196i \(-0.253207\pi\)
0.699946 + 0.714196i \(0.253207\pi\)
\(948\) −9.16695 −0.297729
\(949\) 2.97573 0.0965962
\(950\) 30.1794 0.979150
\(951\) 72.3768 2.34698
\(952\) 0 0
\(953\) 16.7332 0.542040 0.271020 0.962574i \(-0.412639\pi\)
0.271020 + 0.962574i \(0.412639\pi\)
\(954\) 35.1619 1.13841
\(955\) −10.7447 −0.347690
\(956\) 6.58182 0.212871
\(957\) −5.86462 −0.189576
\(958\) 45.6325 1.47432
\(959\) 0 0
\(960\) 67.8360 2.18940
\(961\) 77.8366 2.51086
\(962\) −3.90487 −0.125898
\(963\) 76.5922 2.46815
\(964\) 5.45144 0.175579
\(965\) −39.4390 −1.26959
\(966\) 0 0
\(967\) 44.7594 1.43937 0.719683 0.694303i \(-0.244287\pi\)
0.719683 + 0.694303i \(0.244287\pi\)
\(968\) 20.7986 0.668493
\(969\) −73.1411 −2.34963
\(970\) 8.55580 0.274710
\(971\) −4.20259 −0.134867 −0.0674337 0.997724i \(-0.521481\pi\)
−0.0674337 + 0.997724i \(0.521481\pi\)
\(972\) −8.56409 −0.274693
\(973\) 0 0
\(974\) 9.23899 0.296036
\(975\) −8.99866 −0.288188
\(976\) −7.31033 −0.233998
\(977\) −25.6899 −0.821892 −0.410946 0.911660i \(-0.634801\pi\)
−0.410946 + 0.911660i \(0.634801\pi\)
\(978\) −49.5892 −1.58569
\(979\) −5.45999 −0.174502
\(980\) 0 0
\(981\) −4.31923 −0.137902
\(982\) 5.68483 0.181410
\(983\) −31.8244 −1.01504 −0.507520 0.861640i \(-0.669438\pi\)
−0.507520 + 0.861640i \(0.669438\pi\)
\(984\) −4.15586 −0.132484
\(985\) 28.1627 0.897339
\(986\) 5.54689 0.176649
\(987\) 0 0
\(988\) −2.77255 −0.0882067
\(989\) −0.206390 −0.00656283
\(990\) −29.1970 −0.927942
\(991\) 9.47478 0.300976 0.150488 0.988612i \(-0.451915\pi\)
0.150488 + 0.988612i \(0.451915\pi\)
\(992\) −23.1262 −0.734259
\(993\) 47.7676 1.51586
\(994\) 0 0
\(995\) −76.2570 −2.41751
\(996\) 13.3964 0.424480
\(997\) 21.9511 0.695198 0.347599 0.937643i \(-0.386997\pi\)
0.347599 + 0.937643i \(0.386997\pi\)
\(998\) 14.3866 0.455401
\(999\) 7.31659 0.231487
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 637.2.a.l.1.3 5
3.2 odd 2 5733.2.a.bl.1.3 5
7.2 even 3 91.2.e.c.53.3 10
7.3 odd 6 637.2.e.m.79.3 10
7.4 even 3 91.2.e.c.79.3 yes 10
7.5 odd 6 637.2.e.m.508.3 10
7.6 odd 2 637.2.a.k.1.3 5
13.12 even 2 8281.2.a.bw.1.3 5
21.2 odd 6 819.2.j.h.235.3 10
21.11 odd 6 819.2.j.h.352.3 10
21.20 even 2 5733.2.a.bm.1.3 5
28.11 odd 6 1456.2.r.p.625.1 10
28.23 odd 6 1456.2.r.p.417.1 10
91.25 even 6 1183.2.e.f.170.3 10
91.51 even 6 1183.2.e.f.508.3 10
91.90 odd 2 8281.2.a.bx.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.c.53.3 10 7.2 even 3
91.2.e.c.79.3 yes 10 7.4 even 3
637.2.a.k.1.3 5 7.6 odd 2
637.2.a.l.1.3 5 1.1 even 1 trivial
637.2.e.m.79.3 10 7.3 odd 6
637.2.e.m.508.3 10 7.5 odd 6
819.2.j.h.235.3 10 21.2 odd 6
819.2.j.h.352.3 10 21.11 odd 6
1183.2.e.f.170.3 10 91.25 even 6
1183.2.e.f.508.3 10 91.51 even 6
1456.2.r.p.417.1 10 28.23 odd 6
1456.2.r.p.625.1 10 28.11 odd 6
5733.2.a.bl.1.3 5 3.2 odd 2
5733.2.a.bm.1.3 5 21.20 even 2
8281.2.a.bw.1.3 5 13.12 even 2
8281.2.a.bx.1.3 5 91.90 odd 2