Properties

Label 637.2.a.k.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 $1$

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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: \( x^{5} - x^{4} - 7x^{3} + 8x + 2 \) Copy content Toggle raw display
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

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

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 \(-0.225967\pi\)
0.758430 + 0.651754i \(0.225967\pi\)
\(4\) −0.398235 −0.199118
\(5\) 2.90260 1.29808 0.649041 0.760753i \(-0.275170\pi\)
0.649041 + 0.760753i \(0.275170\pi\)
\(6\) 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 \(-0.661147\pi\)
−0.484909 + 0.874565i \(0.661147\pi\)
\(18\) 4.93916 1.16417
\(19\) −6.96210 −1.59722 −0.798608 0.601852i \(-0.794430\pi\)
−0.798608 + 0.601852i \(0.794430\pi\)
\(20\) −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 \(-0.113714\pi\)
0.936864 + 0.349693i \(0.113714\pi\)
\(32\) 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 \(-0.487043\pi\)
0.0406950 + 0.999172i \(0.487043\pi\)
\(42\) 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 \(-0.779307\pi\)
−0.769123 + 0.639101i \(0.779307\pi\)
\(48\) −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 \(-0.542323\pi\)
−0.132569 + 0.991174i \(0.542323\pi\)
\(60\) −3.03692 −0.392065
\(61\) −2.40081 −0.307393 −0.153696 0.988118i \(-0.549118\pi\)
−0.153696 + 0.988118i \(0.549118\pi\)
\(62\) 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 \(-0.555715\pi\)
−0.174141 + 0.984721i \(0.555715\pi\)
\(74\) −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 \(-0.748024\pi\)
−0.702703 + 0.711483i \(0.748024\pi\)
\(84\) 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 \(-0.454617\pi\)
0.142092 + 0.989853i \(0.454617\pi\)
\(90\) 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 \(-0.462275\pi\)
0.118238 + 0.992985i \(0.462275\pi\)
\(98\) 0 0
\(99\) 7.94789 0.798793
\(100\) −1.36399 −0.136399
\(101\) −1.45324 −0.144603 −0.0723014 0.997383i \(-0.523034\pi\)
−0.0723014 + 0.997383i \(0.523034\pi\)
\(102\) −13.2960 −1.31650
\(103\) 11.6353 1.14646 0.573230 0.819394i \(-0.305690\pi\)
0.573230 + 0.819394i \(0.305690\pi\)
\(104\) 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 \(-0.652728\pi\)
−0.461609 + 0.887084i \(0.652728\pi\)
\(132\) −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 \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 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 \(-0.340013\pi\)
0.481717 + 0.876327i \(0.340013\pi\)
\(158\) −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 \(-0.562681\pi\)
−0.195647 + 0.980674i \(0.562681\pi\)
\(168\) 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 \(-0.492792\pi\)
0.0226419 + 0.999744i \(0.492792\pi\)
\(174\) 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 \(-0.522415\pi\)
−0.0703618 + 0.997522i \(0.522415\pi\)
\(182\) 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 \(-0.881224\pi\)
−0.931185 + 0.364547i \(0.881224\pi\)
\(200\) −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 \(-0.699225\pi\)
−0.585813 + 0.810446i \(0.699225\pi\)
\(224\) 0 0
\(225\) 13.3667 0.891116
\(226\) −1.38719 −0.0922743
\(227\) 9.51630 0.631619 0.315810 0.948823i \(-0.397724\pi\)
0.315810 + 0.948823i \(0.397724\pi\)
\(228\) 7.28427 0.482413
\(229\) −21.1170 −1.39545 −0.697725 0.716366i \(-0.745804\pi\)
−0.697725 + 0.716366i \(0.745804\pi\)
\(230\) −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 \(-0.354662\pi\)
0.440893 + 0.897560i \(0.354662\pi\)
\(242\) −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 \(-0.346890\pi\)
0.462674 + 0.886528i \(0.346890\pi\)
\(252\) 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 \(-0.482590\pi\)
0.0546675 + 0.998505i \(0.482590\pi\)
\(258\) 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 \(-0.265096\pi\)
0.672789 + 0.739835i \(0.265096\pi\)
\(270\) 8.71141 0.530159
\(271\) 8.96210 0.544409 0.272204 0.962239i \(-0.412247\pi\)
0.272204 + 0.962239i \(0.412247\pi\)
\(272\) 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 \(-0.502852\pi\)
−0.00895970 + 0.999960i \(0.502852\pi\)
\(284\) −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 \(-0.309950\pi\)
0.562214 + 0.826992i \(0.309950\pi\)
\(294\) 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 \(-0.467445\pi\)
0.102098 + 0.994774i \(0.467445\pi\)
\(308\) 0 0
\(309\) 30.5692 1.73902
\(310\) 38.3243 2.17667
\(311\) 23.8306 1.35131 0.675655 0.737218i \(-0.263861\pi\)
0.675655 + 0.737218i \(0.263861\pi\)
\(312\) 7.97439 0.451461
\(313\) 18.0814 1.02202 0.511009 0.859575i \(-0.329272\pi\)
0.511009 + 0.859575i \(0.329272\pi\)
\(314\) 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 \(-0.679159\pi\)
−0.533595 + 0.845740i \(0.679159\pi\)
\(350\) 0 0
\(351\) −2.37138 −0.126575
\(352\) 4.51456 0.240627
\(353\) −22.9152 −1.21965 −0.609825 0.792536i \(-0.708760\pi\)
−0.609825 + 0.792536i \(0.708760\pi\)
\(354\) −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 \(-0.408554\pi\)
0.283351 + 0.959016i \(0.408554\pi\)
\(368\) 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 \(-0.475041\pi\)
0.0783313 + 0.996927i \(0.475041\pi\)
\(384\) 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 \(-0.642335\pi\)
−0.432404 + 0.901680i \(0.642335\pi\)
\(398\) −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 \(-0.390563\pi\)
0.337073 + 0.941478i \(0.390563\pi\)
\(410\) 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 \(-0.414617\pi\)
0.265033 + 0.964239i \(0.414617\pi\)
\(420\) 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 \(-0.457314\pi\)
0.133700 + 0.991022i \(0.457314\pi\)
\(434\) 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 \(-0.344047\pi\)
0.470573 + 0.882361i \(0.344047\pi\)
\(440\) −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 \(-0.737509\pi\)
−0.678821 + 0.734304i \(0.737509\pi\)
\(462\) 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 \(-0.592803\pi\)
−0.287437 + 0.957800i \(0.592803\pi\)
\(468\) 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 \(-0.808103\pi\)
−0.823716 + 0.567003i \(0.808103\pi\)
\(480\) 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 \(-0.624559\pi\)
−0.381403 + 0.924409i \(0.624559\pi\)
\(504\) 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 \(-0.523179\pi\)
−0.0727547 + 0.997350i \(0.523179\pi\)
\(510\) −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 \(-0.466675\pi\)
0.104502 + 0.994525i \(0.466675\pi\)
\(522\) 5.41363 0.236948
\(523\) −25.5124 −1.11558 −0.557789 0.829983i \(-0.688350\pi\)
−0.557789 + 0.829983i \(0.688350\pi\)
\(524\) 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 \(-0.174149\pi\)
0.854034 + 0.520218i \(0.174149\pi\)
\(564\) 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 \(-0.757308\pi\)
−0.723154 + 0.690687i \(0.757308\pi\)
\(578\) −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 \(-0.343719\pi\)
0.471483 + 0.881875i \(0.343719\pi\)
\(588\) 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 \(-0.617646\pi\)
−0.361240 + 0.932473i \(0.617646\pi\)
\(594\) 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 \(-0.490666\pi\)
0.0293193 + 0.999570i \(0.490666\pi\)
\(602\) 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 \(-0.266268\pi\)
0.670061 + 0.742306i \(0.266268\pi\)
\(608\) −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 \(-0.774425\pi\)
−0.759231 + 0.650822i \(0.774425\pi\)
\(620\) −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 \(-0.238190\pi\)
0.732849 + 0.680391i \(0.238190\pi\)
\(644\) 0 0
\(645\) 2.50912 0.0987965
\(646\) 35.2334 1.38624
\(647\) −18.8319 −0.740357 −0.370178 0.928961i \(-0.620703\pi\)
−0.370178 + 0.928961i \(0.620703\pi\)
\(648\) 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 \(-0.539088\pi\)
−0.122490 + 0.992470i \(0.539088\pi\)
\(662\) −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 \(-0.654923\pi\)
−0.467715 + 0.883879i \(0.654923\pi\)
\(678\) −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 \(-0.507132\pi\)
−0.0224037 + 0.999749i \(0.507132\pi\)
\(692\) −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 \(-0.433383\pi\)
0.207759 + 0.978180i \(0.433383\pi\)
\(720\) −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 \(-0.463041\pi\)
0.115851 + 0.993267i \(0.463041\pi\)
\(728\) 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 \(-0.693791\pi\)
−0.571894 + 0.820327i \(0.693791\pi\)
\(734\) 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 \(-0.577656\pi\)
−0.241552 + 0.970388i \(0.577656\pi\)
\(762\) 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 \(-0.553308\pi\)
−0.166689 + 0.986010i \(0.553308\pi\)
\(770\) 0 0
\(771\) 4.60503 0.165846
\(772\) −5.41101 −0.194746
\(773\) 10.1419 0.364780 0.182390 0.983226i \(-0.441617\pi\)
0.182390 + 0.983226i \(0.441617\pi\)
\(774\) 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 \(-0.798948\pi\)
−0.807070 + 0.590456i \(0.798948\pi\)
\(788\) 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 \(-0.659212\pi\)
−0.479583 + 0.877497i \(0.659212\pi\)
\(798\) 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 \(-0.350468\pi\)
0.452681 + 0.891673i \(0.350468\pi\)
\(812\) 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 \(-0.756761\pi\)
−0.721966 + 0.691928i \(0.756761\pi\)
\(830\) −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 \(-0.790660\pi\)
−0.791425 + 0.611266i \(0.790660\pi\)
\(840\) 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 \(-0.740281\pi\)
−0.685191 + 0.728364i \(0.740281\pi\)
\(854\) 0 0
\(855\) −78.8645 −2.69711
\(856\) −59.5692 −2.03603
\(857\) −32.8702 −1.12282 −0.561412 0.827536i \(-0.689742\pi\)
−0.561412 + 0.827536i \(0.689742\pi\)
\(858\) −6.77179 −0.231185
\(859\) 34.0503 1.16178 0.580891 0.813981i \(-0.302704\pi\)
0.580891 + 0.813981i \(0.302704\pi\)
\(860\) −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 \(-0.399571\pi\)
0.310298 + 0.950639i \(0.399571\pi\)
\(882\) 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 \(-0.520721\pi\)
−0.0650498 + 0.997882i \(0.520721\pi\)
\(888\) 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 \(-0.455684\pi\)
0.138773 + 0.990324i \(0.455684\pi\)
\(930\) 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 \(-0.316566\pi\)
0.544905 + 0.838498i \(0.316566\pi\)
\(938\) 0 0
\(939\) 47.5048 1.55026
\(940\) 12.1899 0.397592
\(941\) 13.4037 0.436949 0.218475 0.975843i \(-0.429892\pi\)
0.218475 + 0.975843i \(0.429892\pi\)
\(942\) 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 \(-0.478519\pi\)
0.0674337 + 0.997724i \(0.478519\pi\)
\(972\) 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 \(-0.330562\pi\)
0.507520 + 0.861640i \(0.330562\pi\)
\(984\) −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 \(-0.613003\pi\)
−0.347599 + 0.937643i \(0.613003\pi\)
\(998\) 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.k.1.3 5
3.2 odd 2 5733.2.a.bm.1.3 5
7.2 even 3 637.2.e.m.508.3 10
7.3 odd 6 91.2.e.c.79.3 yes 10
7.4 even 3 637.2.e.m.79.3 10
7.5 odd 6 91.2.e.c.53.3 10
7.6 odd 2 637.2.a.l.1.3 5
13.12 even 2 8281.2.a.bx.1.3 5
21.5 even 6 819.2.j.h.235.3 10
21.17 even 6 819.2.j.h.352.3 10
21.20 even 2 5733.2.a.bl.1.3 5
28.3 even 6 1456.2.r.p.625.1 10
28.19 even 6 1456.2.r.p.417.1 10
91.12 odd 6 1183.2.e.f.508.3 10
91.38 odd 6 1183.2.e.f.170.3 10
91.90 odd 2 8281.2.a.bw.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.c.53.3 10 7.5 odd 6
91.2.e.c.79.3 yes 10 7.3 odd 6
637.2.a.k.1.3 5 1.1 even 1 trivial
637.2.a.l.1.3 5 7.6 odd 2
637.2.e.m.79.3 10 7.4 even 3
637.2.e.m.508.3 10 7.2 even 3
819.2.j.h.235.3 10 21.5 even 6
819.2.j.h.352.3 10 21.17 even 6
1183.2.e.f.170.3 10 91.38 odd 6
1183.2.e.f.508.3 10 91.12 odd 6
1456.2.r.p.417.1 10 28.19 even 6
1456.2.r.p.625.1 10 28.3 even 6
5733.2.a.bl.1.3 5 21.20 even 2
5733.2.a.bm.1.3 5 3.2 odd 2
8281.2.a.bw.1.3 5 91.90 odd 2
8281.2.a.bx.1.3 5 13.12 even 2