Properties

Label 637.2.a.e.1.2
Level $637$
Weight $2$
Character 637.1
Self dual yes
Analytic conductor $5.086$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
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 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.381966 q^{2} -2.23607 q^{3} -1.85410 q^{4} +2.23607 q^{5} +0.854102 q^{6} +1.47214 q^{8} +2.00000 q^{9} +O(q^{10})\) \(q-0.381966 q^{2} -2.23607 q^{3} -1.85410 q^{4} +2.23607 q^{5} +0.854102 q^{6} +1.47214 q^{8} +2.00000 q^{9} -0.854102 q^{10} -3.00000 q^{11} +4.14590 q^{12} +1.00000 q^{13} -5.00000 q^{15} +3.14590 q^{16} +7.47214 q^{17} -0.763932 q^{18} -3.00000 q^{19} -4.14590 q^{20} +1.14590 q^{22} -3.76393 q^{23} -3.29180 q^{24} -0.381966 q^{26} +2.23607 q^{27} -4.47214 q^{29} +1.90983 q^{30} -5.00000 q^{31} -4.14590 q^{32} +6.70820 q^{33} -2.85410 q^{34} -3.70820 q^{36} -8.70820 q^{37} +1.14590 q^{38} -2.23607 q^{39} +3.29180 q^{40} -4.47214 q^{41} -8.00000 q^{43} +5.56231 q^{44} +4.47214 q^{45} +1.43769 q^{46} -1.47214 q^{47} -7.03444 q^{48} -16.7082 q^{51} -1.85410 q^{52} +1.47214 q^{53} -0.854102 q^{54} -6.70820 q^{55} +6.70820 q^{57} +1.70820 q^{58} -7.47214 q^{59} +9.27051 q^{60} -3.00000 q^{61} +1.90983 q^{62} -4.70820 q^{64} +2.23607 q^{65} -2.56231 q^{66} -3.00000 q^{67} -13.8541 q^{68} +8.41641 q^{69} +8.94427 q^{71} +2.94427 q^{72} -10.7082 q^{73} +3.32624 q^{74} +5.56231 q^{76} +0.854102 q^{78} +10.7082 q^{79} +7.03444 q^{80} -11.0000 q^{81} +1.70820 q^{82} +16.7082 q^{85} +3.05573 q^{86} +10.0000 q^{87} -4.41641 q^{88} +2.23607 q^{89} -1.70820 q^{90} +6.97871 q^{92} +11.1803 q^{93} +0.562306 q^{94} -6.70820 q^{95} +9.27051 q^{96} +17.4164 q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} - 6 q^{8} + 4 q^{9} + O(q^{10}) \) \( 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} - 6 q^{8} + 4 q^{9} + 5 q^{10} - 6 q^{11} + 15 q^{12} + 2 q^{13} - 10 q^{15} + 13 q^{16} + 6 q^{17} - 6 q^{18} - 6 q^{19} - 15 q^{20} + 9 q^{22} - 12 q^{23} - 20 q^{24} - 3 q^{26} + 15 q^{30} - 10 q^{31} - 15 q^{32} + q^{34} + 6 q^{36} - 4 q^{37} + 9 q^{38} + 20 q^{40} - 16 q^{43} - 9 q^{44} + 23 q^{46} + 6 q^{47} + 15 q^{48} - 20 q^{51} + 3 q^{52} - 6 q^{53} + 5 q^{54} - 10 q^{58} - 6 q^{59} - 15 q^{60} - 6 q^{61} + 15 q^{62} + 4 q^{64} + 15 q^{66} - 6 q^{67} - 21 q^{68} - 10 q^{69} - 12 q^{72} - 8 q^{73} - 9 q^{74} - 9 q^{76} - 5 q^{78} + 8 q^{79} - 15 q^{80} - 22 q^{81} - 10 q^{82} + 20 q^{85} + 24 q^{86} + 20 q^{87} + 18 q^{88} + 10 q^{90} - 33 q^{92} - 19 q^{94} - 15 q^{96} + 8 q^{97} - 12 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.381966 −0.270091 −0.135045 0.990839i \(-0.543118\pi\)
−0.135045 + 0.990839i \(0.543118\pi\)
\(3\) −2.23607 −1.29099 −0.645497 0.763763i \(-0.723350\pi\)
−0.645497 + 0.763763i \(0.723350\pi\)
\(4\) −1.85410 −0.927051
\(5\) 2.23607 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0.854102 0.348686
\(7\) 0 0
\(8\) 1.47214 0.520479
\(9\) 2.00000 0.666667
\(10\) −0.854102 −0.270091
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 4.14590 1.19682
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −5.00000 −1.29099
\(16\) 3.14590 0.786475
\(17\) 7.47214 1.81226 0.906130 0.423000i \(-0.139023\pi\)
0.906130 + 0.423000i \(0.139023\pi\)
\(18\) −0.763932 −0.180061
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) −4.14590 −0.927051
\(21\) 0 0
\(22\) 1.14590 0.244306
\(23\) −3.76393 −0.784834 −0.392417 0.919787i \(-0.628361\pi\)
−0.392417 + 0.919787i \(0.628361\pi\)
\(24\) −3.29180 −0.671935
\(25\) 0 0
\(26\) −0.381966 −0.0749097
\(27\) 2.23607 0.430331
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 1.90983 0.348686
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) −4.14590 −0.732898
\(33\) 6.70820 1.16775
\(34\) −2.85410 −0.489474
\(35\) 0 0
\(36\) −3.70820 −0.618034
\(37\) −8.70820 −1.43162 −0.715810 0.698295i \(-0.753942\pi\)
−0.715810 + 0.698295i \(0.753942\pi\)
\(38\) 1.14590 0.185889
\(39\) −2.23607 −0.358057
\(40\) 3.29180 0.520479
\(41\) −4.47214 −0.698430 −0.349215 0.937043i \(-0.613552\pi\)
−0.349215 + 0.937043i \(0.613552\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 5.56231 0.838549
\(45\) 4.47214 0.666667
\(46\) 1.43769 0.211976
\(47\) −1.47214 −0.214733 −0.107367 0.994220i \(-0.534242\pi\)
−0.107367 + 0.994220i \(0.534242\pi\)
\(48\) −7.03444 −1.01533
\(49\) 0 0
\(50\) 0 0
\(51\) −16.7082 −2.33962
\(52\) −1.85410 −0.257118
\(53\) 1.47214 0.202213 0.101107 0.994876i \(-0.467762\pi\)
0.101107 + 0.994876i \(0.467762\pi\)
\(54\) −0.854102 −0.116229
\(55\) −6.70820 −0.904534
\(56\) 0 0
\(57\) 6.70820 0.888523
\(58\) 1.70820 0.224298
\(59\) −7.47214 −0.972789 −0.486395 0.873739i \(-0.661688\pi\)
−0.486395 + 0.873739i \(0.661688\pi\)
\(60\) 9.27051 1.19682
\(61\) −3.00000 −0.384111 −0.192055 0.981384i \(-0.561515\pi\)
−0.192055 + 0.981384i \(0.561515\pi\)
\(62\) 1.90983 0.242549
\(63\) 0 0
\(64\) −4.70820 −0.588525
\(65\) 2.23607 0.277350
\(66\) −2.56231 −0.315398
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) −13.8541 −1.68006
\(69\) 8.41641 1.01322
\(70\) 0 0
\(71\) 8.94427 1.06149 0.530745 0.847532i \(-0.321912\pi\)
0.530745 + 0.847532i \(0.321912\pi\)
\(72\) 2.94427 0.346986
\(73\) −10.7082 −1.25330 −0.626650 0.779301i \(-0.715575\pi\)
−0.626650 + 0.779301i \(0.715575\pi\)
\(74\) 3.32624 0.386667
\(75\) 0 0
\(76\) 5.56231 0.638040
\(77\) 0 0
\(78\) 0.854102 0.0967080
\(79\) 10.7082 1.20477 0.602384 0.798207i \(-0.294218\pi\)
0.602384 + 0.798207i \(0.294218\pi\)
\(80\) 7.03444 0.786475
\(81\) −11.0000 −1.22222
\(82\) 1.70820 0.188640
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 16.7082 1.81226
\(86\) 3.05573 0.329508
\(87\) 10.0000 1.07211
\(88\) −4.41641 −0.470791
\(89\) 2.23607 0.237023 0.118511 0.992953i \(-0.462188\pi\)
0.118511 + 0.992953i \(0.462188\pi\)
\(90\) −1.70820 −0.180061
\(91\) 0 0
\(92\) 6.97871 0.727581
\(93\) 11.1803 1.15935
\(94\) 0.562306 0.0579974
\(95\) −6.70820 −0.688247
\(96\) 9.27051 0.946167
\(97\) 17.4164 1.76837 0.884184 0.467139i \(-0.154715\pi\)
0.884184 + 0.467139i \(0.154715\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 9.00000 0.895533 0.447767 0.894150i \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) 6.38197 0.631909
\(103\) −10.7082 −1.05511 −0.527555 0.849521i \(-0.676891\pi\)
−0.527555 + 0.849521i \(0.676891\pi\)
\(104\) 1.47214 0.144355
\(105\) 0 0
\(106\) −0.562306 −0.0546160
\(107\) −14.2361 −1.37625 −0.688126 0.725591i \(-0.741567\pi\)
−0.688126 + 0.725591i \(0.741567\pi\)
\(108\) −4.14590 −0.398939
\(109\) 10.7082 1.02566 0.512830 0.858490i \(-0.328597\pi\)
0.512830 + 0.858490i \(0.328597\pi\)
\(110\) 2.56231 0.244306
\(111\) 19.4721 1.84821
\(112\) 0 0
\(113\) −14.9443 −1.40584 −0.702919 0.711269i \(-0.748121\pi\)
−0.702919 + 0.711269i \(0.748121\pi\)
\(114\) −2.56231 −0.239982
\(115\) −8.41641 −0.784834
\(116\) 8.29180 0.769874
\(117\) 2.00000 0.184900
\(118\) 2.85410 0.262741
\(119\) 0 0
\(120\) −7.36068 −0.671935
\(121\) −2.00000 −0.181818
\(122\) 1.14590 0.103745
\(123\) 10.0000 0.901670
\(124\) 9.27051 0.832516
\(125\) −11.1803 −1.00000
\(126\) 0 0
\(127\) 15.4164 1.36798 0.683992 0.729489i \(-0.260242\pi\)
0.683992 + 0.729489i \(0.260242\pi\)
\(128\) 10.0902 0.891853
\(129\) 17.8885 1.57500
\(130\) −0.854102 −0.0749097
\(131\) 3.76393 0.328856 0.164428 0.986389i \(-0.447422\pi\)
0.164428 + 0.986389i \(0.447422\pi\)
\(132\) −12.4377 −1.08256
\(133\) 0 0
\(134\) 1.14590 0.0989905
\(135\) 5.00000 0.430331
\(136\) 11.0000 0.943242
\(137\) −3.76393 −0.321574 −0.160787 0.986989i \(-0.551403\pi\)
−0.160787 + 0.986989i \(0.551403\pi\)
\(138\) −3.21478 −0.273660
\(139\) −3.41641 −0.289776 −0.144888 0.989448i \(-0.546282\pi\)
−0.144888 + 0.989448i \(0.546282\pi\)
\(140\) 0 0
\(141\) 3.29180 0.277219
\(142\) −3.41641 −0.286699
\(143\) −3.00000 −0.250873
\(144\) 6.29180 0.524316
\(145\) −10.0000 −0.830455
\(146\) 4.09017 0.338505
\(147\) 0 0
\(148\) 16.1459 1.32718
\(149\) −12.7082 −1.04110 −0.520548 0.853832i \(-0.674272\pi\)
−0.520548 + 0.853832i \(0.674272\pi\)
\(150\) 0 0
\(151\) −6.41641 −0.522160 −0.261080 0.965317i \(-0.584079\pi\)
−0.261080 + 0.965317i \(0.584079\pi\)
\(152\) −4.41641 −0.358218
\(153\) 14.9443 1.20817
\(154\) 0 0
\(155\) −11.1803 −0.898027
\(156\) 4.14590 0.331937
\(157\) −7.00000 −0.558661 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) −4.09017 −0.325396
\(159\) −3.29180 −0.261056
\(160\) −9.27051 −0.732898
\(161\) 0 0
\(162\) 4.20163 0.330111
\(163\) 10.4164 0.815876 0.407938 0.913010i \(-0.366248\pi\)
0.407938 + 0.913010i \(0.366248\pi\)
\(164\) 8.29180 0.647480
\(165\) 15.0000 1.16775
\(166\) 0 0
\(167\) −13.5279 −1.04682 −0.523409 0.852082i \(-0.675340\pi\)
−0.523409 + 0.852082i \(0.675340\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −6.38197 −0.489474
\(171\) −6.00000 −0.458831
\(172\) 14.8328 1.13099
\(173\) −10.4164 −0.791945 −0.395972 0.918262i \(-0.629592\pi\)
−0.395972 + 0.918262i \(0.629592\pi\)
\(174\) −3.81966 −0.289568
\(175\) 0 0
\(176\) −9.43769 −0.711393
\(177\) 16.7082 1.25587
\(178\) −0.854102 −0.0640176
\(179\) 20.1246 1.50418 0.752092 0.659058i \(-0.229045\pi\)
0.752092 + 0.659058i \(0.229045\pi\)
\(180\) −8.29180 −0.618034
\(181\) −1.41641 −0.105281 −0.0526404 0.998614i \(-0.516764\pi\)
−0.0526404 + 0.998614i \(0.516764\pi\)
\(182\) 0 0
\(183\) 6.70820 0.495885
\(184\) −5.54102 −0.408489
\(185\) −19.4721 −1.43162
\(186\) −4.27051 −0.313129
\(187\) −22.4164 −1.63925
\(188\) 2.72949 0.199069
\(189\) 0 0
\(190\) 2.56231 0.185889
\(191\) −11.1803 −0.808981 −0.404491 0.914542i \(-0.632551\pi\)
−0.404491 + 0.914542i \(0.632551\pi\)
\(192\) 10.5279 0.759783
\(193\) −12.7082 −0.914757 −0.457378 0.889272i \(-0.651211\pi\)
−0.457378 + 0.889272i \(0.651211\pi\)
\(194\) −6.65248 −0.477620
\(195\) −5.00000 −0.358057
\(196\) 0 0
\(197\) −26.9443 −1.91970 −0.959850 0.280514i \(-0.909495\pi\)
−0.959850 + 0.280514i \(0.909495\pi\)
\(198\) 2.29180 0.162871
\(199\) 7.29180 0.516902 0.258451 0.966024i \(-0.416788\pi\)
0.258451 + 0.966024i \(0.416788\pi\)
\(200\) 0 0
\(201\) 6.70820 0.473160
\(202\) −3.43769 −0.241875
\(203\) 0 0
\(204\) 30.9787 2.16894
\(205\) −10.0000 −0.698430
\(206\) 4.09017 0.284976
\(207\) −7.52786 −0.523223
\(208\) 3.14590 0.218129
\(209\) 9.00000 0.622543
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −2.72949 −0.187462
\(213\) −20.0000 −1.37038
\(214\) 5.43769 0.371713
\(215\) −17.8885 −1.21999
\(216\) 3.29180 0.223978
\(217\) 0 0
\(218\) −4.09017 −0.277021
\(219\) 23.9443 1.61800
\(220\) 12.4377 0.838549
\(221\) 7.47214 0.502630
\(222\) −7.43769 −0.499185
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 5.70820 0.379704
\(227\) 11.9443 0.792769 0.396385 0.918085i \(-0.370265\pi\)
0.396385 + 0.918085i \(0.370265\pi\)
\(228\) −12.4377 −0.823706
\(229\) 16.1246 1.06554 0.532772 0.846259i \(-0.321150\pi\)
0.532772 + 0.846259i \(0.321150\pi\)
\(230\) 3.21478 0.211976
\(231\) 0 0
\(232\) −6.58359 −0.432234
\(233\) −5.94427 −0.389422 −0.194711 0.980861i \(-0.562377\pi\)
−0.194711 + 0.980861i \(0.562377\pi\)
\(234\) −0.763932 −0.0499398
\(235\) −3.29180 −0.214733
\(236\) 13.8541 0.901825
\(237\) −23.9443 −1.55535
\(238\) 0 0
\(239\) −7.41641 −0.479728 −0.239864 0.970807i \(-0.577103\pi\)
−0.239864 + 0.970807i \(0.577103\pi\)
\(240\) −15.7295 −1.01533
\(241\) 8.70820 0.560945 0.280472 0.959862i \(-0.409509\pi\)
0.280472 + 0.959862i \(0.409509\pi\)
\(242\) 0.763932 0.0491074
\(243\) 17.8885 1.14755
\(244\) 5.56231 0.356090
\(245\) 0 0
\(246\) −3.81966 −0.243533
\(247\) −3.00000 −0.190885
\(248\) −7.36068 −0.467404
\(249\) 0 0
\(250\) 4.27051 0.270091
\(251\) −10.4721 −0.660995 −0.330498 0.943807i \(-0.607217\pi\)
−0.330498 + 0.943807i \(0.607217\pi\)
\(252\) 0 0
\(253\) 11.2918 0.709909
\(254\) −5.88854 −0.369480
\(255\) −37.3607 −2.33962
\(256\) 5.56231 0.347644
\(257\) 17.9443 1.11933 0.559666 0.828718i \(-0.310929\pi\)
0.559666 + 0.828718i \(0.310929\pi\)
\(258\) −6.83282 −0.425393
\(259\) 0 0
\(260\) −4.14590 −0.257118
\(261\) −8.94427 −0.553637
\(262\) −1.43769 −0.0888210
\(263\) −14.1246 −0.870961 −0.435480 0.900198i \(-0.643421\pi\)
−0.435480 + 0.900198i \(0.643421\pi\)
\(264\) 9.87539 0.607788
\(265\) 3.29180 0.202213
\(266\) 0 0
\(267\) −5.00000 −0.305995
\(268\) 5.56231 0.339772
\(269\) −4.52786 −0.276069 −0.138034 0.990427i \(-0.544078\pi\)
−0.138034 + 0.990427i \(0.544078\pi\)
\(270\) −1.90983 −0.116229
\(271\) −6.41641 −0.389769 −0.194885 0.980826i \(-0.562433\pi\)
−0.194885 + 0.980826i \(0.562433\pi\)
\(272\) 23.5066 1.42530
\(273\) 0 0
\(274\) 1.43769 0.0868543
\(275\) 0 0
\(276\) −15.6049 −0.939303
\(277\) 26.4164 1.58721 0.793604 0.608435i \(-0.208202\pi\)
0.793604 + 0.608435i \(0.208202\pi\)
\(278\) 1.30495 0.0782658
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) 9.05573 0.540219 0.270110 0.962830i \(-0.412940\pi\)
0.270110 + 0.962830i \(0.412940\pi\)
\(282\) −1.25735 −0.0748744
\(283\) 14.1246 0.839621 0.419811 0.907612i \(-0.362097\pi\)
0.419811 + 0.907612i \(0.362097\pi\)
\(284\) −16.5836 −0.984055
\(285\) 15.0000 0.888523
\(286\) 1.14590 0.0677584
\(287\) 0 0
\(288\) −8.29180 −0.488599
\(289\) 38.8328 2.28428
\(290\) 3.81966 0.224298
\(291\) −38.9443 −2.28295
\(292\) 19.8541 1.16187
\(293\) −2.94427 −0.172006 −0.0860031 0.996295i \(-0.527409\pi\)
−0.0860031 + 0.996295i \(0.527409\pi\)
\(294\) 0 0
\(295\) −16.7082 −0.972789
\(296\) −12.8197 −0.745128
\(297\) −6.70820 −0.389249
\(298\) 4.85410 0.281191
\(299\) −3.76393 −0.217674
\(300\) 0 0
\(301\) 0 0
\(302\) 2.45085 0.141031
\(303\) −20.1246 −1.15613
\(304\) −9.43769 −0.541289
\(305\) −6.70820 −0.384111
\(306\) −5.70820 −0.326316
\(307\) 7.41641 0.423277 0.211638 0.977348i \(-0.432120\pi\)
0.211638 + 0.977348i \(0.432120\pi\)
\(308\) 0 0
\(309\) 23.9443 1.36214
\(310\) 4.27051 0.242549
\(311\) 32.2361 1.82794 0.913970 0.405782i \(-0.133001\pi\)
0.913970 + 0.405782i \(0.133001\pi\)
\(312\) −3.29180 −0.186361
\(313\) 32.4164 1.83228 0.916142 0.400854i \(-0.131287\pi\)
0.916142 + 0.400854i \(0.131287\pi\)
\(314\) 2.67376 0.150889
\(315\) 0 0
\(316\) −19.8541 −1.11688
\(317\) −3.76393 −0.211403 −0.105702 0.994398i \(-0.533709\pi\)
−0.105702 + 0.994398i \(0.533709\pi\)
\(318\) 1.25735 0.0705089
\(319\) 13.4164 0.751175
\(320\) −10.5279 −0.588525
\(321\) 31.8328 1.77673
\(322\) 0 0
\(323\) −22.4164 −1.24728
\(324\) 20.3951 1.13306
\(325\) 0 0
\(326\) −3.97871 −0.220361
\(327\) −23.9443 −1.32412
\(328\) −6.58359 −0.363518
\(329\) 0 0
\(330\) −5.72949 −0.315398
\(331\) −28.4164 −1.56191 −0.780954 0.624589i \(-0.785266\pi\)
−0.780954 + 0.624589i \(0.785266\pi\)
\(332\) 0 0
\(333\) −17.4164 −0.954413
\(334\) 5.16718 0.282736
\(335\) −6.70820 −0.366508
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) −0.381966 −0.0207762
\(339\) 33.4164 1.81493
\(340\) −30.9787 −1.68006
\(341\) 15.0000 0.812296
\(342\) 2.29180 0.123926
\(343\) 0 0
\(344\) −11.7771 −0.634978
\(345\) 18.8197 1.01322
\(346\) 3.97871 0.213897
\(347\) −35.0689 −1.88260 −0.941298 0.337576i \(-0.890393\pi\)
−0.941298 + 0.337576i \(0.890393\pi\)
\(348\) −18.5410 −0.993903
\(349\) 2.58359 0.138297 0.0691483 0.997606i \(-0.477972\pi\)
0.0691483 + 0.997606i \(0.477972\pi\)
\(350\) 0 0
\(351\) 2.23607 0.119352
\(352\) 12.4377 0.662931
\(353\) −30.7082 −1.63443 −0.817216 0.576331i \(-0.804484\pi\)
−0.817216 + 0.576331i \(0.804484\pi\)
\(354\) −6.38197 −0.339198
\(355\) 20.0000 1.06149
\(356\) −4.14590 −0.219732
\(357\) 0 0
\(358\) −7.68692 −0.406266
\(359\) −5.94427 −0.313727 −0.156863 0.987620i \(-0.550138\pi\)
−0.156863 + 0.987620i \(0.550138\pi\)
\(360\) 6.58359 0.346986
\(361\) −10.0000 −0.526316
\(362\) 0.541020 0.0284354
\(363\) 4.47214 0.234726
\(364\) 0 0
\(365\) −23.9443 −1.25330
\(366\) −2.56231 −0.133934
\(367\) −0.708204 −0.0369679 −0.0184840 0.999829i \(-0.505884\pi\)
−0.0184840 + 0.999829i \(0.505884\pi\)
\(368\) −11.8409 −0.617252
\(369\) −8.94427 −0.465620
\(370\) 7.43769 0.386667
\(371\) 0 0
\(372\) −20.7295 −1.07477
\(373\) −28.4164 −1.47135 −0.735673 0.677337i \(-0.763134\pi\)
−0.735673 + 0.677337i \(0.763134\pi\)
\(374\) 8.56231 0.442746
\(375\) 25.0000 1.29099
\(376\) −2.16718 −0.111764
\(377\) −4.47214 −0.230327
\(378\) 0 0
\(379\) 11.4164 0.586421 0.293211 0.956048i \(-0.405276\pi\)
0.293211 + 0.956048i \(0.405276\pi\)
\(380\) 12.4377 0.638040
\(381\) −34.4721 −1.76606
\(382\) 4.27051 0.218498
\(383\) −15.0000 −0.766464 −0.383232 0.923652i \(-0.625189\pi\)
−0.383232 + 0.923652i \(0.625189\pi\)
\(384\) −22.5623 −1.15138
\(385\) 0 0
\(386\) 4.85410 0.247067
\(387\) −16.0000 −0.813326
\(388\) −32.2918 −1.63937
\(389\) −7.47214 −0.378852 −0.189426 0.981895i \(-0.560663\pi\)
−0.189426 + 0.981895i \(0.560663\pi\)
\(390\) 1.90983 0.0967080
\(391\) −28.1246 −1.42232
\(392\) 0 0
\(393\) −8.41641 −0.424552
\(394\) 10.2918 0.518493
\(395\) 23.9443 1.20477
\(396\) 11.1246 0.559033
\(397\) −14.1246 −0.708894 −0.354447 0.935076i \(-0.615331\pi\)
−0.354447 + 0.935076i \(0.615331\pi\)
\(398\) −2.78522 −0.139610
\(399\) 0 0
\(400\) 0 0
\(401\) 9.76393 0.487587 0.243794 0.969827i \(-0.421608\pi\)
0.243794 + 0.969827i \(0.421608\pi\)
\(402\) −2.56231 −0.127796
\(403\) −5.00000 −0.249068
\(404\) −16.6869 −0.830205
\(405\) −24.5967 −1.22222
\(406\) 0 0
\(407\) 26.1246 1.29495
\(408\) −24.5967 −1.21772
\(409\) 4.70820 0.232806 0.116403 0.993202i \(-0.462864\pi\)
0.116403 + 0.993202i \(0.462864\pi\)
\(410\) 3.81966 0.188640
\(411\) 8.41641 0.415151
\(412\) 19.8541 0.978141
\(413\) 0 0
\(414\) 2.87539 0.141318
\(415\) 0 0
\(416\) −4.14590 −0.203269
\(417\) 7.63932 0.374099
\(418\) −3.43769 −0.168143
\(419\) −15.0557 −0.735520 −0.367760 0.929921i \(-0.619875\pi\)
−0.367760 + 0.929921i \(0.619875\pi\)
\(420\) 0 0
\(421\) −13.4164 −0.653876 −0.326938 0.945046i \(-0.606017\pi\)
−0.326938 + 0.945046i \(0.606017\pi\)
\(422\) −1.52786 −0.0743753
\(423\) −2.94427 −0.143155
\(424\) 2.16718 0.105248
\(425\) 0 0
\(426\) 7.63932 0.370126
\(427\) 0 0
\(428\) 26.3951 1.27586
\(429\) 6.70820 0.323875
\(430\) 6.83282 0.329508
\(431\) 13.3607 0.643561 0.321781 0.946814i \(-0.395719\pi\)
0.321781 + 0.946814i \(0.395719\pi\)
\(432\) 7.03444 0.338445
\(433\) −2.58359 −0.124160 −0.0620798 0.998071i \(-0.519773\pi\)
−0.0620798 + 0.998071i \(0.519773\pi\)
\(434\) 0 0
\(435\) 22.3607 1.07211
\(436\) −19.8541 −0.950839
\(437\) 11.2918 0.540160
\(438\) −9.14590 −0.437008
\(439\) 16.1246 0.769586 0.384793 0.923003i \(-0.374273\pi\)
0.384793 + 0.923003i \(0.374273\pi\)
\(440\) −9.87539 −0.470791
\(441\) 0 0
\(442\) −2.85410 −0.135756
\(443\) −2.23607 −0.106239 −0.0531194 0.998588i \(-0.516916\pi\)
−0.0531194 + 0.998588i \(0.516916\pi\)
\(444\) −36.1033 −1.71339
\(445\) 5.00000 0.237023
\(446\) −1.52786 −0.0723465
\(447\) 28.4164 1.34405
\(448\) 0 0
\(449\) 10.3607 0.488951 0.244475 0.969656i \(-0.421384\pi\)
0.244475 + 0.969656i \(0.421384\pi\)
\(450\) 0 0
\(451\) 13.4164 0.631754
\(452\) 27.7082 1.30328
\(453\) 14.3475 0.674105
\(454\) −4.56231 −0.214120
\(455\) 0 0
\(456\) 9.87539 0.462457
\(457\) 34.1246 1.59628 0.798141 0.602471i \(-0.205817\pi\)
0.798141 + 0.602471i \(0.205817\pi\)
\(458\) −6.15905 −0.287794
\(459\) 16.7082 0.779872
\(460\) 15.6049 0.727581
\(461\) 10.3607 0.482545 0.241272 0.970457i \(-0.422435\pi\)
0.241272 + 0.970457i \(0.422435\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −14.0689 −0.653132
\(465\) 25.0000 1.15935
\(466\) 2.27051 0.105179
\(467\) 21.6525 1.00196 0.500979 0.865460i \(-0.332974\pi\)
0.500979 + 0.865460i \(0.332974\pi\)
\(468\) −3.70820 −0.171412
\(469\) 0 0
\(470\) 1.25735 0.0579974
\(471\) 15.6525 0.721228
\(472\) −11.0000 −0.506316
\(473\) 24.0000 1.10352
\(474\) 9.14590 0.420085
\(475\) 0 0
\(476\) 0 0
\(477\) 2.94427 0.134809
\(478\) 2.83282 0.129570
\(479\) 29.8328 1.36310 0.681548 0.731773i \(-0.261307\pi\)
0.681548 + 0.731773i \(0.261307\pi\)
\(480\) 20.7295 0.946167
\(481\) −8.70820 −0.397060
\(482\) −3.32624 −0.151506
\(483\) 0 0
\(484\) 3.70820 0.168555
\(485\) 38.9443 1.76837
\(486\) −6.83282 −0.309943
\(487\) 31.8328 1.44248 0.721241 0.692684i \(-0.243572\pi\)
0.721241 + 0.692684i \(0.243572\pi\)
\(488\) −4.41641 −0.199921
\(489\) −23.2918 −1.05329
\(490\) 0 0
\(491\) −34.4721 −1.55571 −0.777853 0.628446i \(-0.783691\pi\)
−0.777853 + 0.628446i \(0.783691\pi\)
\(492\) −18.5410 −0.835894
\(493\) −33.4164 −1.50500
\(494\) 1.14590 0.0515564
\(495\) −13.4164 −0.603023
\(496\) −15.7295 −0.706275
\(497\) 0 0
\(498\) 0 0
\(499\) −0.416408 −0.0186410 −0.00932049 0.999957i \(-0.502967\pi\)
−0.00932049 + 0.999957i \(0.502967\pi\)
\(500\) 20.7295 0.927051
\(501\) 30.2492 1.35144
\(502\) 4.00000 0.178529
\(503\) −3.05573 −0.136248 −0.0681241 0.997677i \(-0.521701\pi\)
−0.0681241 + 0.997677i \(0.521701\pi\)
\(504\) 0 0
\(505\) 20.1246 0.895533
\(506\) −4.31308 −0.191740
\(507\) −2.23607 −0.0993073
\(508\) −28.5836 −1.26819
\(509\) 15.7639 0.698724 0.349362 0.936988i \(-0.386398\pi\)
0.349362 + 0.936988i \(0.386398\pi\)
\(510\) 14.2705 0.631909
\(511\) 0 0
\(512\) −22.3050 −0.985749
\(513\) −6.70820 −0.296174
\(514\) −6.85410 −0.302321
\(515\) −23.9443 −1.05511
\(516\) −33.1672 −1.46010
\(517\) 4.41641 0.194233
\(518\) 0 0
\(519\) 23.2918 1.02240
\(520\) 3.29180 0.144355
\(521\) 0.0557281 0.00244149 0.00122075 0.999999i \(-0.499611\pi\)
0.00122075 + 0.999999i \(0.499611\pi\)
\(522\) 3.41641 0.149532
\(523\) −19.2918 −0.843571 −0.421786 0.906696i \(-0.638597\pi\)
−0.421786 + 0.906696i \(0.638597\pi\)
\(524\) −6.97871 −0.304867
\(525\) 0 0
\(526\) 5.39512 0.235238
\(527\) −37.3607 −1.62746
\(528\) 21.1033 0.918404
\(529\) −8.83282 −0.384035
\(530\) −1.25735 −0.0546160
\(531\) −14.9443 −0.648526
\(532\) 0 0
\(533\) −4.47214 −0.193710
\(534\) 1.90983 0.0826464
\(535\) −31.8328 −1.37625
\(536\) −4.41641 −0.190760
\(537\) −45.0000 −1.94189
\(538\) 1.72949 0.0745636
\(539\) 0 0
\(540\) −9.27051 −0.398939
\(541\) 14.7082 0.632355 0.316178 0.948700i \(-0.397600\pi\)
0.316178 + 0.948700i \(0.397600\pi\)
\(542\) 2.45085 0.105273
\(543\) 3.16718 0.135917
\(544\) −30.9787 −1.32820
\(545\) 23.9443 1.02566
\(546\) 0 0
\(547\) −31.4164 −1.34327 −0.671634 0.740883i \(-0.734407\pi\)
−0.671634 + 0.740883i \(0.734407\pi\)
\(548\) 6.97871 0.298116
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) 13.4164 0.571558
\(552\) 12.3901 0.527358
\(553\) 0 0
\(554\) −10.0902 −0.428690
\(555\) 43.5410 1.84821
\(556\) 6.33437 0.268637
\(557\) −5.29180 −0.224221 −0.112110 0.993696i \(-0.535761\pi\)
−0.112110 + 0.993696i \(0.535761\pi\)
\(558\) 3.81966 0.161699
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 50.1246 2.11626
\(562\) −3.45898 −0.145908
\(563\) 36.5967 1.54237 0.771185 0.636612i \(-0.219665\pi\)
0.771185 + 0.636612i \(0.219665\pi\)
\(564\) −6.10333 −0.256996
\(565\) −33.4164 −1.40584
\(566\) −5.39512 −0.226774
\(567\) 0 0
\(568\) 13.1672 0.552483
\(569\) 16.5279 0.692884 0.346442 0.938071i \(-0.387390\pi\)
0.346442 + 0.938071i \(0.387390\pi\)
\(570\) −5.72949 −0.239982
\(571\) 4.12461 0.172610 0.0863048 0.996269i \(-0.472494\pi\)
0.0863048 + 0.996269i \(0.472494\pi\)
\(572\) 5.56231 0.232572
\(573\) 25.0000 1.04439
\(574\) 0 0
\(575\) 0 0
\(576\) −9.41641 −0.392350
\(577\) 32.7082 1.36166 0.680830 0.732441i \(-0.261619\pi\)
0.680830 + 0.732441i \(0.261619\pi\)
\(578\) −14.8328 −0.616964
\(579\) 28.4164 1.18095
\(580\) 18.5410 0.769874
\(581\) 0 0
\(582\) 14.8754 0.616605
\(583\) −4.41641 −0.182909
\(584\) −15.7639 −0.652316
\(585\) 4.47214 0.184900
\(586\) 1.12461 0.0464573
\(587\) 41.8885 1.72893 0.864463 0.502697i \(-0.167659\pi\)
0.864463 + 0.502697i \(0.167659\pi\)
\(588\) 0 0
\(589\) 15.0000 0.618064
\(590\) 6.38197 0.262741
\(591\) 60.2492 2.47832
\(592\) −27.3951 −1.12593
\(593\) −32.2361 −1.32378 −0.661888 0.749602i \(-0.730245\pi\)
−0.661888 + 0.749602i \(0.730245\pi\)
\(594\) 2.56231 0.105133
\(595\) 0 0
\(596\) 23.5623 0.965150
\(597\) −16.3050 −0.667317
\(598\) 1.43769 0.0587917
\(599\) 41.0689 1.67803 0.839015 0.544109i \(-0.183132\pi\)
0.839015 + 0.544109i \(0.183132\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) −6.00000 −0.244339
\(604\) 11.8967 0.484069
\(605\) −4.47214 −0.181818
\(606\) 7.68692 0.312260
\(607\) 16.1246 0.654478 0.327239 0.944942i \(-0.393882\pi\)
0.327239 + 0.944942i \(0.393882\pi\)
\(608\) 12.4377 0.504415
\(609\) 0 0
\(610\) 2.56231 0.103745
\(611\) −1.47214 −0.0595562
\(612\) −27.7082 −1.12004
\(613\) 22.1246 0.893605 0.446802 0.894633i \(-0.352563\pi\)
0.446802 + 0.894633i \(0.352563\pi\)
\(614\) −2.83282 −0.114323
\(615\) 22.3607 0.901670
\(616\) 0 0
\(617\) 4.47214 0.180041 0.0900207 0.995940i \(-0.471307\pi\)
0.0900207 + 0.995940i \(0.471307\pi\)
\(618\) −9.14590 −0.367902
\(619\) −17.0000 −0.683288 −0.341644 0.939829i \(-0.610984\pi\)
−0.341644 + 0.939829i \(0.610984\pi\)
\(620\) 20.7295 0.832516
\(621\) −8.41641 −0.337739
\(622\) −12.3131 −0.493710
\(623\) 0 0
\(624\) −7.03444 −0.281603
\(625\) −25.0000 −1.00000
\(626\) −12.3820 −0.494883
\(627\) −20.1246 −0.803700
\(628\) 12.9787 0.517907
\(629\) −65.0689 −2.59447
\(630\) 0 0
\(631\) −30.8328 −1.22744 −0.613718 0.789526i \(-0.710327\pi\)
−0.613718 + 0.789526i \(0.710327\pi\)
\(632\) 15.7639 0.627056
\(633\) −8.94427 −0.355503
\(634\) 1.43769 0.0570981
\(635\) 34.4721 1.36798
\(636\) 6.10333 0.242013
\(637\) 0 0
\(638\) −5.12461 −0.202885
\(639\) 17.8885 0.707660
\(640\) 22.5623 0.891853
\(641\) −5.94427 −0.234785 −0.117392 0.993086i \(-0.537453\pi\)
−0.117392 + 0.993086i \(0.537453\pi\)
\(642\) −12.1591 −0.479880
\(643\) 18.8328 0.742694 0.371347 0.928494i \(-0.378896\pi\)
0.371347 + 0.928494i \(0.378896\pi\)
\(644\) 0 0
\(645\) 40.0000 1.57500
\(646\) 8.56231 0.336879
\(647\) −15.7639 −0.619744 −0.309872 0.950778i \(-0.600286\pi\)
−0.309872 + 0.950778i \(0.600286\pi\)
\(648\) −16.1935 −0.636141
\(649\) 22.4164 0.879921
\(650\) 0 0
\(651\) 0 0
\(652\) −19.3131 −0.756359
\(653\) 13.4721 0.527205 0.263603 0.964631i \(-0.415089\pi\)
0.263603 + 0.964631i \(0.415089\pi\)
\(654\) 9.14590 0.357633
\(655\) 8.41641 0.328856
\(656\) −14.0689 −0.549298
\(657\) −21.4164 −0.835534
\(658\) 0 0
\(659\) 8.94427 0.348419 0.174210 0.984709i \(-0.444263\pi\)
0.174210 + 0.984709i \(0.444263\pi\)
\(660\) −27.8115 −1.08256
\(661\) −6.70820 −0.260919 −0.130459 0.991454i \(-0.541645\pi\)
−0.130459 + 0.991454i \(0.541645\pi\)
\(662\) 10.8541 0.421857
\(663\) −16.7082 −0.648893
\(664\) 0 0
\(665\) 0 0
\(666\) 6.65248 0.257778
\(667\) 16.8328 0.651769
\(668\) 25.0820 0.970453
\(669\) −8.94427 −0.345806
\(670\) 2.56231 0.0989905
\(671\) 9.00000 0.347441
\(672\) 0 0
\(673\) 17.4164 0.671353 0.335677 0.941977i \(-0.391035\pi\)
0.335677 + 0.941977i \(0.391035\pi\)
\(674\) −6.87539 −0.264830
\(675\) 0 0
\(676\) −1.85410 −0.0713116
\(677\) −32.8885 −1.26401 −0.632005 0.774965i \(-0.717768\pi\)
−0.632005 + 0.774965i \(0.717768\pi\)
\(678\) −12.7639 −0.490196
\(679\) 0 0
\(680\) 24.5967 0.943242
\(681\) −26.7082 −1.02346
\(682\) −5.72949 −0.219394
\(683\) −4.52786 −0.173254 −0.0866270 0.996241i \(-0.527609\pi\)
−0.0866270 + 0.996241i \(0.527609\pi\)
\(684\) 11.1246 0.425360
\(685\) −8.41641 −0.321574
\(686\) 0 0
\(687\) −36.0557 −1.37561
\(688\) −25.1672 −0.959490
\(689\) 1.47214 0.0560839
\(690\) −7.18847 −0.273660
\(691\) −1.83282 −0.0697236 −0.0348618 0.999392i \(-0.511099\pi\)
−0.0348618 + 0.999392i \(0.511099\pi\)
\(692\) 19.3131 0.734173
\(693\) 0 0
\(694\) 13.3951 0.508472
\(695\) −7.63932 −0.289776
\(696\) 14.7214 0.558012
\(697\) −33.4164 −1.26574
\(698\) −0.986844 −0.0373526
\(699\) 13.2918 0.502742
\(700\) 0 0
\(701\) 22.3607 0.844551 0.422276 0.906467i \(-0.361231\pi\)
0.422276 + 0.906467i \(0.361231\pi\)
\(702\) −0.854102 −0.0322360
\(703\) 26.1246 0.985308
\(704\) 14.1246 0.532341
\(705\) 7.36068 0.277219
\(706\) 11.7295 0.441445
\(707\) 0 0
\(708\) −30.9787 −1.16425
\(709\) −9.87539 −0.370878 −0.185439 0.982656i \(-0.559371\pi\)
−0.185439 + 0.982656i \(0.559371\pi\)
\(710\) −7.63932 −0.286699
\(711\) 21.4164 0.803178
\(712\) 3.29180 0.123365
\(713\) 18.8197 0.704802
\(714\) 0 0
\(715\) −6.70820 −0.250873
\(716\) −37.3131 −1.39446
\(717\) 16.5836 0.619326
\(718\) 2.27051 0.0847347
\(719\) −11.2918 −0.421113 −0.210556 0.977582i \(-0.567528\pi\)
−0.210556 + 0.977582i \(0.567528\pi\)
\(720\) 14.0689 0.524316
\(721\) 0 0
\(722\) 3.81966 0.142153
\(723\) −19.4721 −0.724177
\(724\) 2.62616 0.0976006
\(725\) 0 0
\(726\) −1.70820 −0.0633974
\(727\) −14.8328 −0.550119 −0.275059 0.961427i \(-0.588698\pi\)
−0.275059 + 0.961427i \(0.588698\pi\)
\(728\) 0 0
\(729\) −7.00000 −0.259259
\(730\) 9.14590 0.338505
\(731\) −59.7771 −2.21094
\(732\) −12.4377 −0.459710
\(733\) −15.2918 −0.564815 −0.282408 0.959294i \(-0.591133\pi\)
−0.282408 + 0.959294i \(0.591133\pi\)
\(734\) 0.270510 0.00998470
\(735\) 0 0
\(736\) 15.6049 0.575203
\(737\) 9.00000 0.331519
\(738\) 3.41641 0.125760
\(739\) 35.8328 1.31813 0.659066 0.752085i \(-0.270952\pi\)
0.659066 + 0.752085i \(0.270952\pi\)
\(740\) 36.1033 1.32718
\(741\) 6.70820 0.246432
\(742\) 0 0
\(743\) 15.0557 0.552341 0.276171 0.961109i \(-0.410935\pi\)
0.276171 + 0.961109i \(0.410935\pi\)
\(744\) 16.4590 0.603415
\(745\) −28.4164 −1.04110
\(746\) 10.8541 0.397397
\(747\) 0 0
\(748\) 41.5623 1.51967
\(749\) 0 0
\(750\) −9.54915 −0.348686
\(751\) 30.1246 1.09926 0.549631 0.835407i \(-0.314768\pi\)
0.549631 + 0.835407i \(0.314768\pi\)
\(752\) −4.63119 −0.168882
\(753\) 23.4164 0.853341
\(754\) 1.70820 0.0622091
\(755\) −14.3475 −0.522160
\(756\) 0 0
\(757\) 0.832816 0.0302692 0.0151346 0.999885i \(-0.495182\pi\)
0.0151346 + 0.999885i \(0.495182\pi\)
\(758\) −4.36068 −0.158387
\(759\) −25.2492 −0.916489
\(760\) −9.87539 −0.358218
\(761\) −33.5410 −1.21586 −0.607931 0.793990i \(-0.708000\pi\)
−0.607931 + 0.793990i \(0.708000\pi\)
\(762\) 13.1672 0.476997
\(763\) 0 0
\(764\) 20.7295 0.749967
\(765\) 33.4164 1.20817
\(766\) 5.72949 0.207015
\(767\) −7.47214 −0.269803
\(768\) −12.4377 −0.448807
\(769\) −46.0000 −1.65880 −0.829401 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 0 0
\(771\) −40.1246 −1.44505
\(772\) 23.5623 0.848026
\(773\) −11.0689 −0.398120 −0.199060 0.979987i \(-0.563789\pi\)
−0.199060 + 0.979987i \(0.563789\pi\)
\(774\) 6.11146 0.219672
\(775\) 0 0
\(776\) 25.6393 0.920398
\(777\) 0 0
\(778\) 2.85410 0.102325
\(779\) 13.4164 0.480693
\(780\) 9.27051 0.331937
\(781\) −26.8328 −0.960154
\(782\) 10.7426 0.384156
\(783\) −10.0000 −0.357371
\(784\) 0 0
\(785\) −15.6525 −0.558661
\(786\) 3.21478 0.114667
\(787\) 6.41641 0.228720 0.114360 0.993439i \(-0.463518\pi\)
0.114360 + 0.993439i \(0.463518\pi\)
\(788\) 49.9574 1.77966
\(789\) 31.5836 1.12441
\(790\) −9.14590 −0.325396
\(791\) 0 0
\(792\) −8.83282 −0.313860
\(793\) −3.00000 −0.106533
\(794\) 5.39512 0.191466
\(795\) −7.36068 −0.261056
\(796\) −13.5197 −0.479194
\(797\) −26.9443 −0.954415 −0.477208 0.878791i \(-0.658351\pi\)
−0.477208 + 0.878791i \(0.658351\pi\)
\(798\) 0 0
\(799\) −11.0000 −0.389152
\(800\) 0 0
\(801\) 4.47214 0.158015
\(802\) −3.72949 −0.131693
\(803\) 32.1246 1.13365
\(804\) −12.4377 −0.438644
\(805\) 0 0
\(806\) 1.90983 0.0672709
\(807\) 10.1246 0.356403
\(808\) 13.2492 0.466106
\(809\) −4.41641 −0.155273 −0.0776363 0.996982i \(-0.524737\pi\)
−0.0776363 + 0.996982i \(0.524737\pi\)
\(810\) 9.39512 0.330111
\(811\) −38.8328 −1.36360 −0.681802 0.731536i \(-0.738804\pi\)
−0.681802 + 0.731536i \(0.738804\pi\)
\(812\) 0 0
\(813\) 14.3475 0.503190
\(814\) −9.97871 −0.349754
\(815\) 23.2918 0.815876
\(816\) −52.5623 −1.84005
\(817\) 24.0000 0.839654
\(818\) −1.79837 −0.0628787
\(819\) 0 0
\(820\) 18.5410 0.647480
\(821\) 33.7639 1.17837 0.589185 0.807998i \(-0.299449\pi\)
0.589185 + 0.807998i \(0.299449\pi\)
\(822\) −3.21478 −0.112128
\(823\) −6.12461 −0.213491 −0.106745 0.994286i \(-0.534043\pi\)
−0.106745 + 0.994286i \(0.534043\pi\)
\(824\) −15.7639 −0.549163
\(825\) 0 0
\(826\) 0 0
\(827\) −26.8328 −0.933068 −0.466534 0.884503i \(-0.654498\pi\)
−0.466534 + 0.884503i \(0.654498\pi\)
\(828\) 13.9574 0.485054
\(829\) 25.0000 0.868286 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) 0 0
\(831\) −59.0689 −2.04908
\(832\) −4.70820 −0.163228
\(833\) 0 0
\(834\) −2.91796 −0.101041
\(835\) −30.2492 −1.04682
\(836\) −16.6869 −0.577129
\(837\) −11.1803 −0.386449
\(838\) 5.75078 0.198657
\(839\) −29.8885 −1.03187 −0.515934 0.856629i \(-0.672555\pi\)
−0.515934 + 0.856629i \(0.672555\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 5.12461 0.176606
\(843\) −20.2492 −0.697420
\(844\) −7.41641 −0.255283
\(845\) 2.23607 0.0769231
\(846\) 1.12461 0.0386650
\(847\) 0 0
\(848\) 4.63119 0.159036
\(849\) −31.5836 −1.08395
\(850\) 0 0
\(851\) 32.7771 1.12358
\(852\) 37.0820 1.27041
\(853\) −28.2492 −0.967235 −0.483617 0.875279i \(-0.660677\pi\)
−0.483617 + 0.875279i \(0.660677\pi\)
\(854\) 0 0
\(855\) −13.4164 −0.458831
\(856\) −20.9574 −0.716310
\(857\) −43.3607 −1.48117 −0.740586 0.671961i \(-0.765452\pi\)
−0.740586 + 0.671961i \(0.765452\pi\)
\(858\) −2.56231 −0.0874757
\(859\) −7.29180 −0.248793 −0.124396 0.992233i \(-0.539699\pi\)
−0.124396 + 0.992233i \(0.539699\pi\)
\(860\) 33.1672 1.13099
\(861\) 0 0
\(862\) −5.10333 −0.173820
\(863\) 6.05573 0.206139 0.103070 0.994674i \(-0.467134\pi\)
0.103070 + 0.994674i \(0.467134\pi\)
\(864\) −9.27051 −0.315389
\(865\) −23.2918 −0.791945
\(866\) 0.986844 0.0335343
\(867\) −86.8328 −2.94900
\(868\) 0 0
\(869\) −32.1246 −1.08975
\(870\) −8.54102 −0.289568
\(871\) −3.00000 −0.101651
\(872\) 15.7639 0.533834
\(873\) 34.8328 1.17891
\(874\) −4.31308 −0.145892
\(875\) 0 0
\(876\) −44.3951 −1.49997
\(877\) −8.12461 −0.274349 −0.137174 0.990547i \(-0.543802\pi\)
−0.137174 + 0.990547i \(0.543802\pi\)
\(878\) −6.15905 −0.207858
\(879\) 6.58359 0.222059
\(880\) −21.1033 −0.711393
\(881\) 19.3050 0.650400 0.325200 0.945645i \(-0.394568\pi\)
0.325200 + 0.945645i \(0.394568\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −13.8541 −0.465964
\(885\) 37.3607 1.25587
\(886\) 0.854102 0.0286941
\(887\) 15.7639 0.529301 0.264651 0.964344i \(-0.414743\pi\)
0.264651 + 0.964344i \(0.414743\pi\)
\(888\) 28.6656 0.961956
\(889\) 0 0
\(890\) −1.90983 −0.0640176
\(891\) 33.0000 1.10554
\(892\) −7.41641 −0.248320
\(893\) 4.41641 0.147789
\(894\) −10.8541 −0.363015
\(895\) 45.0000 1.50418
\(896\) 0 0
\(897\) 8.41641 0.281016
\(898\) −3.95743 −0.132061
\(899\) 22.3607 0.745770
\(900\) 0 0
\(901\) 11.0000 0.366463
\(902\) −5.12461 −0.170631
\(903\) 0 0
\(904\) −22.0000 −0.731709
\(905\) −3.16718 −0.105281
\(906\) −5.48027 −0.182070
\(907\) 5.29180 0.175711 0.0878556 0.996133i \(-0.471999\pi\)
0.0878556 + 0.996133i \(0.471999\pi\)
\(908\) −22.1459 −0.734937
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 46.2492 1.53231 0.766153 0.642659i \(-0.222169\pi\)
0.766153 + 0.642659i \(0.222169\pi\)
\(912\) 21.1033 0.698801
\(913\) 0 0
\(914\) −13.0344 −0.431141
\(915\) 15.0000 0.495885
\(916\) −29.8967 −0.987814
\(917\) 0 0
\(918\) −6.38197 −0.210636
\(919\) −20.1246 −0.663850 −0.331925 0.943306i \(-0.607698\pi\)
−0.331925 + 0.943306i \(0.607698\pi\)
\(920\) −12.3901 −0.408489
\(921\) −16.5836 −0.546448
\(922\) −3.95743 −0.130331
\(923\) 8.94427 0.294404
\(924\) 0 0
\(925\) 0 0
\(926\) 9.16718 0.301252
\(927\) −21.4164 −0.703407
\(928\) 18.5410 0.608639
\(929\) 47.1803 1.54794 0.773968 0.633224i \(-0.218269\pi\)
0.773968 + 0.633224i \(0.218269\pi\)
\(930\) −9.54915 −0.313129
\(931\) 0 0
\(932\) 11.0213 0.361014
\(933\) −72.0820 −2.35986
\(934\) −8.27051 −0.270619
\(935\) −50.1246 −1.63925
\(936\) 2.94427 0.0962365
\(937\) 1.41641 0.0462720 0.0231360 0.999732i \(-0.492635\pi\)
0.0231360 + 0.999732i \(0.492635\pi\)
\(938\) 0 0
\(939\) −72.4853 −2.36547
\(940\) 6.10333 0.199069
\(941\) −45.7639 −1.49186 −0.745931 0.666023i \(-0.767995\pi\)
−0.745931 + 0.666023i \(0.767995\pi\)
\(942\) −5.97871 −0.194797
\(943\) 16.8328 0.548152
\(944\) −23.5066 −0.765074
\(945\) 0 0
\(946\) −9.16718 −0.298051
\(947\) −31.4721 −1.02271 −0.511354 0.859370i \(-0.670856\pi\)
−0.511354 + 0.859370i \(0.670856\pi\)
\(948\) 44.3951 1.44189
\(949\) −10.7082 −0.347603
\(950\) 0 0
\(951\) 8.41641 0.272921
\(952\) 0 0
\(953\) 29.7771 0.964574 0.482287 0.876013i \(-0.339806\pi\)
0.482287 + 0.876013i \(0.339806\pi\)
\(954\) −1.12461 −0.0364107
\(955\) −25.0000 −0.808981
\(956\) 13.7508 0.444732
\(957\) −30.0000 −0.969762
\(958\) −11.3951 −0.368160
\(959\) 0 0
\(960\) 23.5410 0.759783
\(961\) −6.00000 −0.193548
\(962\) 3.32624 0.107242
\(963\) −28.4721 −0.917502
\(964\) −16.1459 −0.520024
\(965\) −28.4164 −0.914757
\(966\) 0 0
\(967\) 43.4164 1.39618 0.698089 0.716011i \(-0.254034\pi\)
0.698089 + 0.716011i \(0.254034\pi\)
\(968\) −2.94427 −0.0946325
\(969\) 50.1246 1.61023
\(970\) −14.8754 −0.477620
\(971\) 24.7082 0.792924 0.396462 0.918051i \(-0.370238\pi\)
0.396462 + 0.918051i \(0.370238\pi\)
\(972\) −33.1672 −1.06384
\(973\) 0 0
\(974\) −12.1591 −0.389601
\(975\) 0 0
\(976\) −9.43769 −0.302093
\(977\) −33.6525 −1.07664 −0.538319 0.842741i \(-0.680940\pi\)
−0.538319 + 0.842741i \(0.680940\pi\)
\(978\) 8.89667 0.284484
\(979\) −6.70820 −0.214395
\(980\) 0 0
\(981\) 21.4164 0.683773
\(982\) 13.1672 0.420182
\(983\) 1.47214 0.0469538 0.0234769 0.999724i \(-0.492526\pi\)
0.0234769 + 0.999724i \(0.492526\pi\)
\(984\) 14.7214 0.469300
\(985\) −60.2492 −1.91970
\(986\) 12.7639 0.406486
\(987\) 0 0
\(988\) 5.56231 0.176961
\(989\) 30.1115 0.957489
\(990\) 5.12461 0.162871
\(991\) 17.2918 0.549292 0.274646 0.961545i \(-0.411439\pi\)
0.274646 + 0.961545i \(0.411439\pi\)
\(992\) 20.7295 0.658162
\(993\) 63.5410 2.01641
\(994\) 0 0
\(995\) 16.3050 0.516902
\(996\) 0 0
\(997\) 0.416408 0.0131878 0.00659388 0.999978i \(-0.497901\pi\)
0.00659388 + 0.999978i \(0.497901\pi\)
\(998\) 0.159054 0.00503476
\(999\) −19.4721 −0.616071
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.e.1.2 2
3.2 odd 2 5733.2.a.w.1.1 2
7.2 even 3 637.2.e.h.508.1 4
7.3 odd 6 91.2.e.b.79.1 yes 4
7.4 even 3 637.2.e.h.79.1 4
7.5 odd 6 91.2.e.b.53.1 4
7.6 odd 2 637.2.a.f.1.2 2
13.12 even 2 8281.2.a.ba.1.1 2
21.5 even 6 819.2.j.c.235.2 4
21.17 even 6 819.2.j.c.352.2 4
21.20 even 2 5733.2.a.v.1.1 2
28.3 even 6 1456.2.r.j.625.2 4
28.19 even 6 1456.2.r.j.417.2 4
91.12 odd 6 1183.2.e.d.508.2 4
91.38 odd 6 1183.2.e.d.170.2 4
91.90 odd 2 8281.2.a.z.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.b.53.1 4 7.5 odd 6
91.2.e.b.79.1 yes 4 7.3 odd 6
637.2.a.e.1.2 2 1.1 even 1 trivial
637.2.a.f.1.2 2 7.6 odd 2
637.2.e.h.79.1 4 7.4 even 3
637.2.e.h.508.1 4 7.2 even 3
819.2.j.c.235.2 4 21.5 even 6
819.2.j.c.352.2 4 21.17 even 6
1183.2.e.d.170.2 4 91.38 odd 6
1183.2.e.d.508.2 4 91.12 odd 6
1456.2.r.j.417.2 4 28.19 even 6
1456.2.r.j.625.2 4 28.3 even 6
5733.2.a.v.1.1 2 21.20 even 2
5733.2.a.w.1.1 2 3.2 odd 2
8281.2.a.z.1.1 2 91.90 odd 2
8281.2.a.ba.1.1 2 13.12 even 2