Properties

Label 637.2.a.j.1.1
Level $637$
Weight $2$
Character 637.1
Self dual yes
Analytic conductor $5.086$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) 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 1.1
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.81361 q^{2} +3.10278 q^{3} +1.28917 q^{4} -2.81361 q^{5} -5.62721 q^{6} +1.28917 q^{8} +6.62721 q^{9} +O(q^{10})\) \(q-1.81361 q^{2} +3.10278 q^{3} +1.28917 q^{4} -2.81361 q^{5} -5.62721 q^{6} +1.28917 q^{8} +6.62721 q^{9} +5.10278 q^{10} +3.10278 q^{11} +4.00000 q^{12} -1.00000 q^{13} -8.72999 q^{15} -4.91638 q^{16} +0.524438 q^{17} -12.0192 q^{18} -0.813607 q^{19} -3.62721 q^{20} -5.62721 q^{22} +7.33804 q^{23} +4.00000 q^{24} +2.91638 q^{25} +1.81361 q^{26} +11.2544 q^{27} +8.28917 q^{29} +15.8328 q^{30} -1.39194 q^{31} +6.33804 q^{32} +9.62721 q^{33} -0.951124 q^{34} +8.54359 q^{36} -6.15165 q^{37} +1.47556 q^{38} -3.10278 q^{39} -3.62721 q^{40} +4.20555 q^{41} +6.75971 q^{43} +4.00000 q^{44} -18.6464 q^{45} -13.3083 q^{46} +5.97028 q^{47} -15.2544 q^{48} -5.28917 q^{50} +1.62721 q^{51} -1.28917 q^{52} -2.49472 q^{53} -20.4111 q^{54} -8.72999 q^{55} -2.52444 q^{57} -15.0333 q^{58} +4.47054 q^{59} -11.2544 q^{60} +2.00000 q^{61} +2.52444 q^{62} -1.66196 q^{64} +2.81361 q^{65} -17.4600 q^{66} +10.0383 q^{67} +0.676089 q^{68} +22.7683 q^{69} -8.72999 q^{71} +8.54359 q^{72} +2.34307 q^{73} +11.1567 q^{74} +9.04888 q^{75} -1.04888 q^{76} +5.62721 q^{78} -13.5436 q^{79} +13.8328 q^{80} +15.0383 q^{81} -7.62721 q^{82} -16.4791 q^{83} -1.47556 q^{85} -12.2594 q^{86} +25.7194 q^{87} +4.00000 q^{88} +10.6464 q^{89} +33.8172 q^{90} +9.45998 q^{92} -4.31889 q^{93} -10.8277 q^{94} +2.28917 q^{95} +19.6655 q^{96} +1.18639 q^{97} +20.5628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{5} - 4 q^{6} + 3 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{5} - 4 q^{6} + 3 q^{8} + 7 q^{9} + 8 q^{10} + 2 q^{11} + 12 q^{12} - 3 q^{13} - 6 q^{15} - q^{16} - 4 q^{17} - 15 q^{18} + 4 q^{19} + 2 q^{20} - 4 q^{22} + 10 q^{23} + 12 q^{24} - 5 q^{25} - q^{26} + 8 q^{27} + 24 q^{29} + 20 q^{30} + 4 q^{31} + 7 q^{32} + 16 q^{33} - 14 q^{34} - q^{36} + 10 q^{38} - 2 q^{39} + 2 q^{40} - 2 q^{41} + 10 q^{43} + 12 q^{44} - 22 q^{45} - 18 q^{46} + 8 q^{47} - 20 q^{48} - 15 q^{50} - 8 q^{51} - 3 q^{52} + 8 q^{53} - 32 q^{54} - 6 q^{55} - 2 q^{57} + 12 q^{58} + 4 q^{59} - 8 q^{60} + 6 q^{61} + 2 q^{62} - 17 q^{64} + 2 q^{65} - 12 q^{66} - 12 q^{67} - 22 q^{68} + 6 q^{69} - 6 q^{71} - q^{72} + 10 q^{73} + 30 q^{74} + 16 q^{75} + 8 q^{76} + 4 q^{78} - 14 q^{79} + 14 q^{80} + 3 q^{81} - 10 q^{82} + 12 q^{83} - 10 q^{85} - 26 q^{86} + 26 q^{87} + 12 q^{88} - 2 q^{89} + 28 q^{90} - 12 q^{92} - 22 q^{93} + 10 q^{94} + 6 q^{95} + 4 q^{96} + 10 q^{97} + 14 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.81361 −1.28241 −0.641207 0.767368i \(-0.721566\pi\)
−0.641207 + 0.767368i \(0.721566\pi\)
\(3\) 3.10278 1.79139 0.895694 0.444671i \(-0.146679\pi\)
0.895694 + 0.444671i \(0.146679\pi\)
\(4\) 1.28917 0.644584
\(5\) −2.81361 −1.25828 −0.629142 0.777291i \(-0.716593\pi\)
−0.629142 + 0.777291i \(0.716593\pi\)
\(6\) −5.62721 −2.29730
\(7\) 0 0
\(8\) 1.28917 0.455790
\(9\) 6.62721 2.20907
\(10\) 5.10278 1.61364
\(11\) 3.10278 0.935522 0.467761 0.883855i \(-0.345061\pi\)
0.467761 + 0.883855i \(0.345061\pi\)
\(12\) 4.00000 1.15470
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −8.72999 −2.25407
\(16\) −4.91638 −1.22910
\(17\) 0.524438 0.127195 0.0635974 0.997976i \(-0.479743\pi\)
0.0635974 + 0.997976i \(0.479743\pi\)
\(18\) −12.0192 −2.83294
\(19\) −0.813607 −0.186654 −0.0933271 0.995636i \(-0.529750\pi\)
−0.0933271 + 0.995636i \(0.529750\pi\)
\(20\) −3.62721 −0.811069
\(21\) 0 0
\(22\) −5.62721 −1.19973
\(23\) 7.33804 1.53009 0.765044 0.643978i \(-0.222717\pi\)
0.765044 + 0.643978i \(0.222717\pi\)
\(24\) 4.00000 0.816497
\(25\) 2.91638 0.583276
\(26\) 1.81361 0.355677
\(27\) 11.2544 2.16592
\(28\) 0 0
\(29\) 8.28917 1.53926 0.769630 0.638490i \(-0.220441\pi\)
0.769630 + 0.638490i \(0.220441\pi\)
\(30\) 15.8328 2.89065
\(31\) −1.39194 −0.250000 −0.125000 0.992157i \(-0.539893\pi\)
−0.125000 + 0.992157i \(0.539893\pi\)
\(32\) 6.33804 1.12042
\(33\) 9.62721 1.67588
\(34\) −0.951124 −0.163116
\(35\) 0 0
\(36\) 8.54359 1.42393
\(37\) −6.15165 −1.01133 −0.505663 0.862731i \(-0.668752\pi\)
−0.505663 + 0.862731i \(0.668752\pi\)
\(38\) 1.47556 0.239368
\(39\) −3.10278 −0.496842
\(40\) −3.62721 −0.573513
\(41\) 4.20555 0.656797 0.328398 0.944539i \(-0.393491\pi\)
0.328398 + 0.944539i \(0.393491\pi\)
\(42\) 0 0
\(43\) 6.75971 1.03085 0.515423 0.856936i \(-0.327635\pi\)
0.515423 + 0.856936i \(0.327635\pi\)
\(44\) 4.00000 0.603023
\(45\) −18.6464 −2.77964
\(46\) −13.3083 −1.96221
\(47\) 5.97028 0.870855 0.435427 0.900224i \(-0.356597\pi\)
0.435427 + 0.900224i \(0.356597\pi\)
\(48\) −15.2544 −2.20179
\(49\) 0 0
\(50\) −5.28917 −0.748001
\(51\) 1.62721 0.227855
\(52\) −1.28917 −0.178776
\(53\) −2.49472 −0.342676 −0.171338 0.985212i \(-0.554809\pi\)
−0.171338 + 0.985212i \(0.554809\pi\)
\(54\) −20.4111 −2.77760
\(55\) −8.72999 −1.17715
\(56\) 0 0
\(57\) −2.52444 −0.334370
\(58\) −15.0333 −1.97397
\(59\) 4.47054 0.582015 0.291007 0.956721i \(-0.406010\pi\)
0.291007 + 0.956721i \(0.406010\pi\)
\(60\) −11.2544 −1.45294
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 2.52444 0.320604
\(63\) 0 0
\(64\) −1.66196 −0.207744
\(65\) 2.81361 0.348985
\(66\) −17.4600 −2.14917
\(67\) 10.0383 1.22638 0.613188 0.789937i \(-0.289887\pi\)
0.613188 + 0.789937i \(0.289887\pi\)
\(68\) 0.676089 0.0819878
\(69\) 22.7683 2.74098
\(70\) 0 0
\(71\) −8.72999 −1.03606 −0.518029 0.855363i \(-0.673334\pi\)
−0.518029 + 0.855363i \(0.673334\pi\)
\(72\) 8.54359 1.00687
\(73\) 2.34307 0.274235 0.137118 0.990555i \(-0.456216\pi\)
0.137118 + 0.990555i \(0.456216\pi\)
\(74\) 11.1567 1.29694
\(75\) 9.04888 1.04487
\(76\) −1.04888 −0.120314
\(77\) 0 0
\(78\) 5.62721 0.637156
\(79\) −13.5436 −1.52377 −0.761887 0.647710i \(-0.775727\pi\)
−0.761887 + 0.647710i \(0.775727\pi\)
\(80\) 13.8328 1.54655
\(81\) 15.0383 1.67092
\(82\) −7.62721 −0.842285
\(83\) −16.4791 −1.80882 −0.904410 0.426665i \(-0.859688\pi\)
−0.904410 + 0.426665i \(0.859688\pi\)
\(84\) 0 0
\(85\) −1.47556 −0.160047
\(86\) −12.2594 −1.32197
\(87\) 25.7194 2.75741
\(88\) 4.00000 0.426401
\(89\) 10.6464 1.12851 0.564256 0.825600i \(-0.309163\pi\)
0.564256 + 0.825600i \(0.309163\pi\)
\(90\) 33.8172 3.56464
\(91\) 0 0
\(92\) 9.45998 0.986271
\(93\) −4.31889 −0.447848
\(94\) −10.8277 −1.11680
\(95\) 2.28917 0.234864
\(96\) 19.6655 2.00710
\(97\) 1.18639 0.120460 0.0602300 0.998185i \(-0.480817\pi\)
0.0602300 + 0.998185i \(0.480817\pi\)
\(98\) 0 0
\(99\) 20.5628 2.06663
\(100\) 3.75971 0.375971
\(101\) −13.1028 −1.30377 −0.651887 0.758316i \(-0.726023\pi\)
−0.651887 + 0.758316i \(0.726023\pi\)
\(102\) −2.95112 −0.292205
\(103\) −4.41110 −0.434639 −0.217319 0.976101i \(-0.569731\pi\)
−0.217319 + 0.976101i \(0.569731\pi\)
\(104\) −1.28917 −0.126413
\(105\) 0 0
\(106\) 4.52444 0.439452
\(107\) 0.578337 0.0559100 0.0279550 0.999609i \(-0.491100\pi\)
0.0279550 + 0.999609i \(0.491100\pi\)
\(108\) 14.5089 1.39611
\(109\) 5.57331 0.533827 0.266913 0.963721i \(-0.413996\pi\)
0.266913 + 0.963721i \(0.413996\pi\)
\(110\) 15.8328 1.50959
\(111\) −19.0872 −1.81168
\(112\) 0 0
\(113\) 5.44584 0.512302 0.256151 0.966637i \(-0.417546\pi\)
0.256151 + 0.966637i \(0.417546\pi\)
\(114\) 4.57834 0.428801
\(115\) −20.6464 −1.92528
\(116\) 10.6861 0.992183
\(117\) −6.62721 −0.612686
\(118\) −8.10780 −0.746383
\(119\) 0 0
\(120\) −11.2544 −1.02738
\(121\) −1.37279 −0.124799
\(122\) −3.62721 −0.328392
\(123\) 13.0489 1.17658
\(124\) −1.79445 −0.161146
\(125\) 5.86248 0.524356
\(126\) 0 0
\(127\) −12.8816 −1.14306 −0.571530 0.820581i \(-0.693650\pi\)
−0.571530 + 0.820581i \(0.693650\pi\)
\(128\) −9.66196 −0.854004
\(129\) 20.9739 1.84664
\(130\) −5.10278 −0.447543
\(131\) −9.04888 −0.790604 −0.395302 0.918551i \(-0.629360\pi\)
−0.395302 + 0.918551i \(0.629360\pi\)
\(132\) 12.4111 1.08025
\(133\) 0 0
\(134\) −18.2056 −1.57272
\(135\) −31.6655 −2.72533
\(136\) 0.676089 0.0579741
\(137\) −6.25945 −0.534781 −0.267390 0.963588i \(-0.586161\pi\)
−0.267390 + 0.963588i \(0.586161\pi\)
\(138\) −41.2927 −3.51507
\(139\) 11.5733 0.981636 0.490818 0.871262i \(-0.336698\pi\)
0.490818 + 0.871262i \(0.336698\pi\)
\(140\) 0 0
\(141\) 18.5244 1.56004
\(142\) 15.8328 1.32866
\(143\) −3.10278 −0.259467
\(144\) −32.5819 −2.71516
\(145\) −23.3225 −1.93682
\(146\) −4.24940 −0.351683
\(147\) 0 0
\(148\) −7.93051 −0.651884
\(149\) 8.52444 0.698349 0.349175 0.937058i \(-0.386462\pi\)
0.349175 + 0.937058i \(0.386462\pi\)
\(150\) −16.4111 −1.33996
\(151\) −11.9844 −0.975278 −0.487639 0.873045i \(-0.662142\pi\)
−0.487639 + 0.873045i \(0.662142\pi\)
\(152\) −1.04888 −0.0850751
\(153\) 3.47556 0.280983
\(154\) 0 0
\(155\) 3.91638 0.314571
\(156\) −4.00000 −0.320256
\(157\) 12.8277 1.02377 0.511883 0.859055i \(-0.328948\pi\)
0.511883 + 0.859055i \(0.328948\pi\)
\(158\) 24.5628 1.95411
\(159\) −7.74055 −0.613866
\(160\) −17.8328 −1.40980
\(161\) 0 0
\(162\) −27.2736 −2.14282
\(163\) −13.4600 −1.05427 −0.527133 0.849783i \(-0.676733\pi\)
−0.527133 + 0.849783i \(0.676733\pi\)
\(164\) 5.42166 0.423361
\(165\) −27.0872 −2.10873
\(166\) 29.8867 2.31965
\(167\) 2.02972 0.157064 0.0785322 0.996912i \(-0.474977\pi\)
0.0785322 + 0.996912i \(0.474977\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 2.67609 0.205247
\(171\) −5.39194 −0.412332
\(172\) 8.71440 0.664467
\(173\) −20.2978 −1.54321 −0.771605 0.636102i \(-0.780546\pi\)
−0.771605 + 0.636102i \(0.780546\pi\)
\(174\) −46.6449 −3.53614
\(175\) 0 0
\(176\) −15.2544 −1.14985
\(177\) 13.8711 1.04261
\(178\) −19.3083 −1.44722
\(179\) −11.0036 −0.822445 −0.411223 0.911535i \(-0.634898\pi\)
−0.411223 + 0.911535i \(0.634898\pi\)
\(180\) −24.0383 −1.79171
\(181\) −0.691675 −0.0514118 −0.0257059 0.999670i \(-0.508183\pi\)
−0.0257059 + 0.999670i \(0.508183\pi\)
\(182\) 0 0
\(183\) 6.20555 0.458727
\(184\) 9.45998 0.697399
\(185\) 17.3083 1.27253
\(186\) 7.83276 0.574326
\(187\) 1.62721 0.118994
\(188\) 7.69670 0.561339
\(189\) 0 0
\(190\) −4.15165 −0.301192
\(191\) 7.83276 0.566759 0.283379 0.959008i \(-0.408544\pi\)
0.283379 + 0.959008i \(0.408544\pi\)
\(192\) −5.15667 −0.372151
\(193\) −12.2056 −0.878575 −0.439287 0.898347i \(-0.644769\pi\)
−0.439287 + 0.898347i \(0.644769\pi\)
\(194\) −2.15165 −0.154480
\(195\) 8.72999 0.625167
\(196\) 0 0
\(197\) −18.8222 −1.34103 −0.670513 0.741898i \(-0.733926\pi\)
−0.670513 + 0.741898i \(0.733926\pi\)
\(198\) −37.2927 −2.65028
\(199\) −21.6116 −1.53201 −0.766004 0.642836i \(-0.777758\pi\)
−0.766004 + 0.642836i \(0.777758\pi\)
\(200\) 3.75971 0.265851
\(201\) 31.1466 2.19691
\(202\) 23.7633 1.67198
\(203\) 0 0
\(204\) 2.09775 0.146872
\(205\) −11.8328 −0.826436
\(206\) 8.00000 0.557386
\(207\) 48.6308 3.38007
\(208\) 4.91638 0.340890
\(209\) −2.52444 −0.174619
\(210\) 0 0
\(211\) −17.3764 −1.19624 −0.598119 0.801407i \(-0.704085\pi\)
−0.598119 + 0.801407i \(0.704085\pi\)
\(212\) −3.21611 −0.220884
\(213\) −27.0872 −1.85598
\(214\) −1.04888 −0.0716997
\(215\) −19.0192 −1.29710
\(216\) 14.5089 0.987202
\(217\) 0 0
\(218\) −10.1078 −0.684586
\(219\) 7.27001 0.491262
\(220\) −11.2544 −0.758773
\(221\) −0.524438 −0.0352775
\(222\) 34.6167 2.32332
\(223\) 10.5486 0.706388 0.353194 0.935550i \(-0.385096\pi\)
0.353194 + 0.935550i \(0.385096\pi\)
\(224\) 0 0
\(225\) 19.3275 1.28850
\(226\) −9.87662 −0.656983
\(227\) 6.95112 0.461362 0.230681 0.973029i \(-0.425905\pi\)
0.230681 + 0.973029i \(0.425905\pi\)
\(228\) −3.25443 −0.215530
\(229\) 21.0872 1.39348 0.696740 0.717323i \(-0.254633\pi\)
0.696740 + 0.717323i \(0.254633\pi\)
\(230\) 37.4444 2.46901
\(231\) 0 0
\(232\) 10.6861 0.701579
\(233\) 6.08362 0.398551 0.199276 0.979943i \(-0.436141\pi\)
0.199276 + 0.979943i \(0.436141\pi\)
\(234\) 12.0192 0.785717
\(235\) −16.7980 −1.09578
\(236\) 5.76328 0.375157
\(237\) −42.0227 −2.72967
\(238\) 0 0
\(239\) −14.2056 −0.918881 −0.459440 0.888209i \(-0.651950\pi\)
−0.459440 + 0.888209i \(0.651950\pi\)
\(240\) 42.9200 2.77047
\(241\) −8.44082 −0.543721 −0.271860 0.962337i \(-0.587639\pi\)
−0.271860 + 0.962337i \(0.587639\pi\)
\(242\) 2.48970 0.160044
\(243\) 12.8972 0.827357
\(244\) 2.57834 0.165061
\(245\) 0 0
\(246\) −23.6655 −1.50886
\(247\) 0.813607 0.0517685
\(248\) −1.79445 −0.113948
\(249\) −51.1310 −3.24030
\(250\) −10.6322 −0.672442
\(251\) 23.9844 1.51388 0.756941 0.653483i \(-0.226693\pi\)
0.756941 + 0.653483i \(0.226693\pi\)
\(252\) 0 0
\(253\) 22.7683 1.43143
\(254\) 23.3622 1.46588
\(255\) −4.57834 −0.286707
\(256\) 20.8469 1.30293
\(257\) −15.6116 −0.973827 −0.486913 0.873450i \(-0.661877\pi\)
−0.486913 + 0.873450i \(0.661877\pi\)
\(258\) −38.0383 −2.36816
\(259\) 0 0
\(260\) 3.62721 0.224950
\(261\) 54.9341 3.40033
\(262\) 16.4111 1.01388
\(263\) −15.1708 −0.935472 −0.467736 0.883868i \(-0.654930\pi\)
−0.467736 + 0.883868i \(0.654930\pi\)
\(264\) 12.4111 0.763850
\(265\) 7.01916 0.431183
\(266\) 0 0
\(267\) 33.0333 2.02160
\(268\) 12.9411 0.790502
\(269\) −23.2927 −1.42018 −0.710092 0.704109i \(-0.751347\pi\)
−0.710092 + 0.704109i \(0.751347\pi\)
\(270\) 57.4288 3.49501
\(271\) −12.7456 −0.774238 −0.387119 0.922030i \(-0.626530\pi\)
−0.387119 + 0.922030i \(0.626530\pi\)
\(272\) −2.57834 −0.156335
\(273\) 0 0
\(274\) 11.3522 0.685810
\(275\) 9.04888 0.545668
\(276\) 29.3522 1.76679
\(277\) 8.12193 0.488000 0.244000 0.969775i \(-0.421540\pi\)
0.244000 + 0.969775i \(0.421540\pi\)
\(278\) −20.9894 −1.25886
\(279\) −9.22471 −0.552269
\(280\) 0 0
\(281\) 19.0333 1.13543 0.567715 0.823225i \(-0.307827\pi\)
0.567715 + 0.823225i \(0.307827\pi\)
\(282\) −33.5960 −2.00062
\(283\) −11.1466 −0.662598 −0.331299 0.943526i \(-0.607487\pi\)
−0.331299 + 0.943526i \(0.607487\pi\)
\(284\) −11.2544 −0.667827
\(285\) 7.10278 0.420732
\(286\) 5.62721 0.332744
\(287\) 0 0
\(288\) 42.0036 2.47508
\(289\) −16.7250 −0.983821
\(290\) 42.2978 2.48381
\(291\) 3.68111 0.215791
\(292\) 3.02061 0.176768
\(293\) 14.1758 0.828161 0.414080 0.910240i \(-0.364103\pi\)
0.414080 + 0.910240i \(0.364103\pi\)
\(294\) 0 0
\(295\) −12.5783 −0.732339
\(296\) −7.93051 −0.460952
\(297\) 34.9200 2.02626
\(298\) −15.4600 −0.895572
\(299\) −7.33804 −0.424370
\(300\) 11.6655 0.673509
\(301\) 0 0
\(302\) 21.7350 1.25071
\(303\) −40.6550 −2.33557
\(304\) 4.00000 0.229416
\(305\) −5.62721 −0.322213
\(306\) −6.30330 −0.360336
\(307\) −13.5592 −0.773863 −0.386932 0.922108i \(-0.626465\pi\)
−0.386932 + 0.922108i \(0.626465\pi\)
\(308\) 0 0
\(309\) −13.6867 −0.778606
\(310\) −7.10278 −0.403411
\(311\) −0.426686 −0.0241952 −0.0120976 0.999927i \(-0.503851\pi\)
−0.0120976 + 0.999927i \(0.503851\pi\)
\(312\) −4.00000 −0.226455
\(313\) 18.1517 1.02599 0.512996 0.858391i \(-0.328536\pi\)
0.512996 + 0.858391i \(0.328536\pi\)
\(314\) −23.2645 −1.31289
\(315\) 0 0
\(316\) −17.4600 −0.982200
\(317\) 9.42166 0.529173 0.264587 0.964362i \(-0.414764\pi\)
0.264587 + 0.964362i \(0.414764\pi\)
\(318\) 14.0383 0.787230
\(319\) 25.7194 1.44001
\(320\) 4.67609 0.261401
\(321\) 1.79445 0.100156
\(322\) 0 0
\(323\) −0.426686 −0.0237415
\(324\) 19.3869 1.07705
\(325\) −2.91638 −0.161772
\(326\) 24.4111 1.35201
\(327\) 17.2927 0.956291
\(328\) 5.42166 0.299361
\(329\) 0 0
\(330\) 49.1255 2.70427
\(331\) −17.4005 −0.956420 −0.478210 0.878246i \(-0.658714\pi\)
−0.478210 + 0.878246i \(0.658714\pi\)
\(332\) −21.2444 −1.16594
\(333\) −40.7683 −2.23409
\(334\) −3.68111 −0.201421
\(335\) −28.2439 −1.54313
\(336\) 0 0
\(337\) 22.0524 1.20127 0.600637 0.799522i \(-0.294914\pi\)
0.600637 + 0.799522i \(0.294914\pi\)
\(338\) −1.81361 −0.0986472
\(339\) 16.8972 0.917731
\(340\) −1.90225 −0.103164
\(341\) −4.31889 −0.233881
\(342\) 9.77886 0.528780
\(343\) 0 0
\(344\) 8.71440 0.469849
\(345\) −64.0610 −3.44893
\(346\) 36.8122 1.97903
\(347\) 25.3522 1.36098 0.680488 0.732759i \(-0.261768\pi\)
0.680488 + 0.732759i \(0.261768\pi\)
\(348\) 33.1567 1.77738
\(349\) 5.70529 0.305397 0.152699 0.988273i \(-0.451204\pi\)
0.152699 + 0.988273i \(0.451204\pi\)
\(350\) 0 0
\(351\) −11.2544 −0.600717
\(352\) 19.6655 1.04818
\(353\) 28.6761 1.52627 0.763137 0.646237i \(-0.223658\pi\)
0.763137 + 0.646237i \(0.223658\pi\)
\(354\) −25.1567 −1.33706
\(355\) 24.5628 1.30366
\(356\) 13.7250 0.727422
\(357\) 0 0
\(358\) 19.9561 1.05472
\(359\) 11.0433 0.582845 0.291423 0.956594i \(-0.405871\pi\)
0.291423 + 0.956594i \(0.405871\pi\)
\(360\) −24.0383 −1.26693
\(361\) −18.3380 −0.965160
\(362\) 1.25443 0.0659312
\(363\) −4.25945 −0.223563
\(364\) 0 0
\(365\) −6.59247 −0.345066
\(366\) −11.2544 −0.588278
\(367\) −27.3466 −1.42748 −0.713741 0.700409i \(-0.753001\pi\)
−0.713741 + 0.700409i \(0.753001\pi\)
\(368\) −36.0766 −1.88062
\(369\) 27.8711 1.45091
\(370\) −31.3905 −1.63191
\(371\) 0 0
\(372\) −5.56777 −0.288676
\(373\) 16.1461 0.836014 0.418007 0.908444i \(-0.362729\pi\)
0.418007 + 0.908444i \(0.362729\pi\)
\(374\) −2.95112 −0.152599
\(375\) 18.1900 0.939326
\(376\) 7.69670 0.396927
\(377\) −8.28917 −0.426914
\(378\) 0 0
\(379\) 26.1305 1.34223 0.671117 0.741351i \(-0.265815\pi\)
0.671117 + 0.741351i \(0.265815\pi\)
\(380\) 2.95112 0.151389
\(381\) −39.9688 −2.04767
\(382\) −14.2056 −0.726819
\(383\) 21.0489 1.07555 0.537774 0.843089i \(-0.319265\pi\)
0.537774 + 0.843089i \(0.319265\pi\)
\(384\) −29.9789 −1.52985
\(385\) 0 0
\(386\) 22.1361 1.12670
\(387\) 44.7980 2.27721
\(388\) 1.52946 0.0776466
\(389\) −21.6061 −1.09547 −0.547736 0.836651i \(-0.684510\pi\)
−0.547736 + 0.836651i \(0.684510\pi\)
\(390\) −15.8328 −0.801723
\(391\) 3.84835 0.194619
\(392\) 0 0
\(393\) −28.0766 −1.41628
\(394\) 34.1361 1.71975
\(395\) 38.1063 1.91734
\(396\) 26.5089 1.33212
\(397\) 27.6952 1.38998 0.694992 0.719017i \(-0.255408\pi\)
0.694992 + 0.719017i \(0.255408\pi\)
\(398\) 39.1950 1.96467
\(399\) 0 0
\(400\) −14.3380 −0.716902
\(401\) −2.57834 −0.128756 −0.0643780 0.997926i \(-0.520506\pi\)
−0.0643780 + 0.997926i \(0.520506\pi\)
\(402\) −56.4877 −2.81735
\(403\) 1.39194 0.0693376
\(404\) −16.8917 −0.840393
\(405\) −42.3119 −2.10250
\(406\) 0 0
\(407\) −19.0872 −0.946117
\(408\) 2.09775 0.103854
\(409\) −15.1169 −0.747483 −0.373742 0.927533i \(-0.621925\pi\)
−0.373742 + 0.927533i \(0.621925\pi\)
\(410\) 21.4600 1.05983
\(411\) −19.4217 −0.958000
\(412\) −5.68665 −0.280161
\(413\) 0 0
\(414\) −88.1971 −4.33465
\(415\) 46.3658 2.27601
\(416\) −6.33804 −0.310748
\(417\) 35.9094 1.75849
\(418\) 4.57834 0.223934
\(419\) 9.99446 0.488261 0.244131 0.969742i \(-0.421497\pi\)
0.244131 + 0.969742i \(0.421497\pi\)
\(420\) 0 0
\(421\) −25.9250 −1.26351 −0.631753 0.775170i \(-0.717664\pi\)
−0.631753 + 0.775170i \(0.717664\pi\)
\(422\) 31.5139 1.53407
\(423\) 39.5663 1.92378
\(424\) −3.21611 −0.156188
\(425\) 1.52946 0.0741898
\(426\) 49.1255 2.38014
\(427\) 0 0
\(428\) 0.745574 0.0360387
\(429\) −9.62721 −0.464806
\(430\) 34.4933 1.66341
\(431\) 30.6761 1.47762 0.738808 0.673916i \(-0.235389\pi\)
0.738808 + 0.673916i \(0.235389\pi\)
\(432\) −55.3311 −2.66212
\(433\) 3.51941 0.169132 0.0845661 0.996418i \(-0.473050\pi\)
0.0845661 + 0.996418i \(0.473050\pi\)
\(434\) 0 0
\(435\) −72.3643 −3.46960
\(436\) 7.18494 0.344096
\(437\) −5.97028 −0.285597
\(438\) −13.1849 −0.630001
\(439\) −32.3517 −1.54406 −0.772030 0.635586i \(-0.780759\pi\)
−0.772030 + 0.635586i \(0.780759\pi\)
\(440\) −11.2544 −0.536534
\(441\) 0 0
\(442\) 0.951124 0.0452404
\(443\) 15.4458 0.733854 0.366927 0.930250i \(-0.380410\pi\)
0.366927 + 0.930250i \(0.380410\pi\)
\(444\) −24.6066 −1.16778
\(445\) −29.9547 −1.41999
\(446\) −19.1310 −0.905881
\(447\) 26.4494 1.25101
\(448\) 0 0
\(449\) −14.4705 −0.682907 −0.341453 0.939899i \(-0.610919\pi\)
−0.341453 + 0.939899i \(0.610919\pi\)
\(450\) −35.0524 −1.65239
\(451\) 13.0489 0.614448
\(452\) 7.02061 0.330222
\(453\) −37.1849 −1.74710
\(454\) −12.6066 −0.591657
\(455\) 0 0
\(456\) −3.25443 −0.152402
\(457\) −34.6705 −1.62182 −0.810910 0.585171i \(-0.801027\pi\)
−0.810910 + 0.585171i \(0.801027\pi\)
\(458\) −38.2439 −1.78702
\(459\) 5.90225 0.275493
\(460\) −26.6167 −1.24101
\(461\) −12.5400 −0.584047 −0.292024 0.956411i \(-0.594329\pi\)
−0.292024 + 0.956411i \(0.594329\pi\)
\(462\) 0 0
\(463\) 12.1517 0.564735 0.282368 0.959306i \(-0.408880\pi\)
0.282368 + 0.959306i \(0.408880\pi\)
\(464\) −40.7527 −1.89190
\(465\) 12.1517 0.563519
\(466\) −11.0333 −0.511107
\(467\) 37.0333 1.71370 0.856848 0.515569i \(-0.172419\pi\)
0.856848 + 0.515569i \(0.172419\pi\)
\(468\) −8.54359 −0.394928
\(469\) 0 0
\(470\) 30.4650 1.40525
\(471\) 39.8016 1.83396
\(472\) 5.76328 0.265276
\(473\) 20.9739 0.964379
\(474\) 76.2127 3.50056
\(475\) −2.37279 −0.108871
\(476\) 0 0
\(477\) −16.5330 −0.756996
\(478\) 25.7633 1.17838
\(479\) −12.0086 −0.548687 −0.274343 0.961632i \(-0.588460\pi\)
−0.274343 + 0.961632i \(0.588460\pi\)
\(480\) −55.3311 −2.52551
\(481\) 6.15165 0.280491
\(482\) 15.3083 0.697275
\(483\) 0 0
\(484\) −1.76975 −0.0804434
\(485\) −3.33804 −0.151573
\(486\) −23.3905 −1.06101
\(487\) 11.1184 0.503821 0.251911 0.967751i \(-0.418941\pi\)
0.251911 + 0.967751i \(0.418941\pi\)
\(488\) 2.57834 0.116716
\(489\) −41.7633 −1.88860
\(490\) 0 0
\(491\) 0.0594386 0.00268243 0.00134121 0.999999i \(-0.499573\pi\)
0.00134121 + 0.999999i \(0.499573\pi\)
\(492\) 16.8222 0.758403
\(493\) 4.34715 0.195786
\(494\) −1.47556 −0.0663887
\(495\) −57.8555 −2.60041
\(496\) 6.84333 0.307274
\(497\) 0 0
\(498\) 92.7316 4.15540
\(499\) 10.2978 0.460991 0.230496 0.973073i \(-0.425965\pi\)
0.230496 + 0.973073i \(0.425965\pi\)
\(500\) 7.55773 0.337992
\(501\) 6.29776 0.281363
\(502\) −43.4983 −1.94142
\(503\) 9.32391 0.415733 0.207866 0.978157i \(-0.433348\pi\)
0.207866 + 0.978157i \(0.433348\pi\)
\(504\) 0 0
\(505\) 36.8661 1.64052
\(506\) −41.2927 −1.83569
\(507\) 3.10278 0.137799
\(508\) −16.6066 −0.736799
\(509\) −39.6952 −1.75946 −0.879730 0.475473i \(-0.842277\pi\)
−0.879730 + 0.475473i \(0.842277\pi\)
\(510\) 8.30330 0.367676
\(511\) 0 0
\(512\) −18.4842 −0.816892
\(513\) −9.15667 −0.404277
\(514\) 28.3133 1.24885
\(515\) 12.4111 0.546898
\(516\) 27.0388 1.19032
\(517\) 18.5244 0.814704
\(518\) 0 0
\(519\) −62.9794 −2.76449
\(520\) 3.62721 0.159064
\(521\) −22.3627 −0.979729 −0.489865 0.871798i \(-0.662954\pi\)
−0.489865 + 0.871798i \(0.662954\pi\)
\(522\) −99.6288 −4.36063
\(523\) 20.6550 0.903178 0.451589 0.892226i \(-0.350857\pi\)
0.451589 + 0.892226i \(0.350857\pi\)
\(524\) −11.6655 −0.509611
\(525\) 0 0
\(526\) 27.5139 1.19966
\(527\) −0.729988 −0.0317988
\(528\) −47.3311 −2.05982
\(529\) 30.8469 1.34117
\(530\) −12.7300 −0.552955
\(531\) 29.6272 1.28571
\(532\) 0 0
\(533\) −4.20555 −0.182163
\(534\) −59.9094 −2.59253
\(535\) −1.62721 −0.0703506
\(536\) 12.9411 0.558969
\(537\) −34.1416 −1.47332
\(538\) 42.2439 1.82126
\(539\) 0 0
\(540\) −40.8222 −1.75671
\(541\) 5.62167 0.241695 0.120847 0.992671i \(-0.461439\pi\)
0.120847 + 0.992671i \(0.461439\pi\)
\(542\) 23.1155 0.992894
\(543\) −2.14611 −0.0920985
\(544\) 3.32391 0.142512
\(545\) −15.6811 −0.671705
\(546\) 0 0
\(547\) −10.3970 −0.444542 −0.222271 0.974985i \(-0.571347\pi\)
−0.222271 + 0.974985i \(0.571347\pi\)
\(548\) −8.06949 −0.344711
\(549\) 13.2544 0.565685
\(550\) −16.4111 −0.699772
\(551\) −6.74412 −0.287309
\(552\) 29.3522 1.24931
\(553\) 0 0
\(554\) −14.7300 −0.625817
\(555\) 53.7038 2.27960
\(556\) 14.9200 0.632747
\(557\) 14.6550 0.620951 0.310475 0.950581i \(-0.399512\pi\)
0.310475 + 0.950581i \(0.399512\pi\)
\(558\) 16.7300 0.708237
\(559\) −6.75971 −0.285905
\(560\) 0 0
\(561\) 5.04888 0.213164
\(562\) −34.5189 −1.45609
\(563\) 24.7456 1.04290 0.521451 0.853281i \(-0.325391\pi\)
0.521451 + 0.853281i \(0.325391\pi\)
\(564\) 23.8811 1.00558
\(565\) −15.3225 −0.644621
\(566\) 20.2156 0.849725
\(567\) 0 0
\(568\) −11.2544 −0.472225
\(569\) −20.5330 −0.860789 −0.430395 0.902641i \(-0.641626\pi\)
−0.430395 + 0.902641i \(0.641626\pi\)
\(570\) −12.8816 −0.539552
\(571\) −41.8953 −1.75326 −0.876631 0.481163i \(-0.840214\pi\)
−0.876631 + 0.481163i \(0.840214\pi\)
\(572\) −4.00000 −0.167248
\(573\) 24.3033 1.01529
\(574\) 0 0
\(575\) 21.4005 0.892464
\(576\) −11.0141 −0.458922
\(577\) 20.1744 0.839870 0.419935 0.907554i \(-0.362053\pi\)
0.419935 + 0.907554i \(0.362053\pi\)
\(578\) 30.3325 1.26167
\(579\) −37.8711 −1.57387
\(580\) −30.0666 −1.24845
\(581\) 0 0
\(582\) −6.67609 −0.276733
\(583\) −7.74055 −0.320581
\(584\) 3.02061 0.124994
\(585\) 18.6464 0.770933
\(586\) −25.7094 −1.06204
\(587\) 18.7441 0.773653 0.386826 0.922153i \(-0.373571\pi\)
0.386826 + 0.922153i \(0.373571\pi\)
\(588\) 0 0
\(589\) 1.13249 0.0466636
\(590\) 22.8122 0.939162
\(591\) −58.4011 −2.40230
\(592\) 30.2439 1.24302
\(593\) 2.98084 0.122409 0.0612043 0.998125i \(-0.480506\pi\)
0.0612043 + 0.998125i \(0.480506\pi\)
\(594\) −63.3311 −2.59850
\(595\) 0 0
\(596\) 10.9894 0.450145
\(597\) −67.0560 −2.74442
\(598\) 13.3083 0.544218
\(599\) 7.47411 0.305384 0.152692 0.988274i \(-0.451206\pi\)
0.152692 + 0.988274i \(0.451206\pi\)
\(600\) 11.6655 0.476243
\(601\) −21.4700 −0.875780 −0.437890 0.899028i \(-0.644274\pi\)
−0.437890 + 0.899028i \(0.644274\pi\)
\(602\) 0 0
\(603\) 66.5260 2.70915
\(604\) −15.4499 −0.628649
\(605\) 3.86248 0.157032
\(606\) 73.7321 2.99516
\(607\) 22.9044 0.929660 0.464830 0.885400i \(-0.346116\pi\)
0.464830 + 0.885400i \(0.346116\pi\)
\(608\) −5.15667 −0.209131
\(609\) 0 0
\(610\) 10.2056 0.413211
\(611\) −5.97028 −0.241532
\(612\) 4.48059 0.181117
\(613\) −20.1461 −0.813694 −0.406847 0.913496i \(-0.633372\pi\)
−0.406847 + 0.913496i \(0.633372\pi\)
\(614\) 24.5910 0.992413
\(615\) −36.7144 −1.48047
\(616\) 0 0
\(617\) −13.7844 −0.554939 −0.277470 0.960734i \(-0.589496\pi\)
−0.277470 + 0.960734i \(0.589496\pi\)
\(618\) 24.8222 0.998495
\(619\) 19.6655 0.790424 0.395212 0.918590i \(-0.370671\pi\)
0.395212 + 0.918590i \(0.370671\pi\)
\(620\) 5.04888 0.202768
\(621\) 82.5855 3.31404
\(622\) 0.773841 0.0310282
\(623\) 0 0
\(624\) 15.2544 0.610666
\(625\) −31.0766 −1.24307
\(626\) −32.9200 −1.31575
\(627\) −7.83276 −0.312810
\(628\) 16.5371 0.659903
\(629\) −3.22616 −0.128635
\(630\) 0 0
\(631\) −16.1672 −0.643608 −0.321804 0.946806i \(-0.604289\pi\)
−0.321804 + 0.946806i \(0.604289\pi\)
\(632\) −17.4600 −0.694521
\(633\) −53.9149 −2.14293
\(634\) −17.0872 −0.678619
\(635\) 36.2439 1.43829
\(636\) −9.97887 −0.395688
\(637\) 0 0
\(638\) −46.6449 −1.84669
\(639\) −57.8555 −2.28873
\(640\) 27.1849 1.07458
\(641\) −29.0036 −1.14557 −0.572786 0.819705i \(-0.694137\pi\)
−0.572786 + 0.819705i \(0.694137\pi\)
\(642\) −3.25443 −0.128442
\(643\) 39.2233 1.54681 0.773407 0.633910i \(-0.218551\pi\)
0.773407 + 0.633910i \(0.218551\pi\)
\(644\) 0 0
\(645\) −59.0122 −2.32360
\(646\) 0.773841 0.0304464
\(647\) 11.9844 0.471156 0.235578 0.971855i \(-0.424302\pi\)
0.235578 + 0.971855i \(0.424302\pi\)
\(648\) 19.3869 0.761590
\(649\) 13.8711 0.544487
\(650\) 5.28917 0.207458
\(651\) 0 0
\(652\) −17.3522 −0.679564
\(653\) 45.3311 1.77394 0.886971 0.461826i \(-0.152806\pi\)
0.886971 + 0.461826i \(0.152806\pi\)
\(654\) −31.3622 −1.22636
\(655\) 25.4600 0.994804
\(656\) −20.6761 −0.807266
\(657\) 15.5280 0.605805
\(658\) 0 0
\(659\) 6.12193 0.238477 0.119238 0.992866i \(-0.461955\pi\)
0.119238 + 0.992866i \(0.461955\pi\)
\(660\) −34.9200 −1.35926
\(661\) 27.5280 1.07072 0.535358 0.844625i \(-0.320177\pi\)
0.535358 + 0.844625i \(0.320177\pi\)
\(662\) 31.5577 1.22653
\(663\) −1.62721 −0.0631957
\(664\) −21.2444 −0.824442
\(665\) 0 0
\(666\) 73.9377 2.86503
\(667\) 60.8263 2.35520
\(668\) 2.61665 0.101241
\(669\) 32.7300 1.26541
\(670\) 51.2233 1.97893
\(671\) 6.20555 0.239563
\(672\) 0 0
\(673\) 27.9547 1.07757 0.538787 0.842442i \(-0.318883\pi\)
0.538787 + 0.842442i \(0.318883\pi\)
\(674\) −39.9945 −1.54053
\(675\) 32.8222 1.26333
\(676\) 1.28917 0.0495834
\(677\) 12.6605 0.486583 0.243291 0.969953i \(-0.421773\pi\)
0.243291 + 0.969953i \(0.421773\pi\)
\(678\) −30.6449 −1.17691
\(679\) 0 0
\(680\) −1.90225 −0.0729479
\(681\) 21.5678 0.826479
\(682\) 7.83276 0.299932
\(683\) −28.3033 −1.08300 −0.541498 0.840702i \(-0.682143\pi\)
−0.541498 + 0.840702i \(0.682143\pi\)
\(684\) −6.95112 −0.265783
\(685\) 17.6116 0.672906
\(686\) 0 0
\(687\) 65.4288 2.49626
\(688\) −33.2333 −1.26701
\(689\) 2.49472 0.0950412
\(690\) 116.182 4.42295
\(691\) −12.2353 −0.465452 −0.232726 0.972542i \(-0.574764\pi\)
−0.232726 + 0.972542i \(0.574764\pi\)
\(692\) −26.1672 −0.994729
\(693\) 0 0
\(694\) −45.9789 −1.74533
\(695\) −32.5628 −1.23518
\(696\) 33.1567 1.25680
\(697\) 2.20555 0.0835412
\(698\) −10.3472 −0.391646
\(699\) 18.8761 0.713960
\(700\) 0 0
\(701\) 51.0419 1.92783 0.963913 0.266219i \(-0.0857743\pi\)
0.963913 + 0.266219i \(0.0857743\pi\)
\(702\) 20.4111 0.770367
\(703\) 5.00502 0.188768
\(704\) −5.15667 −0.194349
\(705\) −52.1205 −1.96297
\(706\) −52.0071 −1.95731
\(707\) 0 0
\(708\) 17.8822 0.672053
\(709\) 42.5910 1.59954 0.799770 0.600307i \(-0.204955\pi\)
0.799770 + 0.600307i \(0.204955\pi\)
\(710\) −44.5472 −1.67183
\(711\) −89.7563 −3.36612
\(712\) 13.7250 0.514365
\(713\) −10.2141 −0.382523
\(714\) 0 0
\(715\) 8.72999 0.326483
\(716\) −14.1855 −0.530135
\(717\) −44.0766 −1.64607
\(718\) −20.0283 −0.747448
\(719\) 42.4933 1.58473 0.792366 0.610046i \(-0.208849\pi\)
0.792366 + 0.610046i \(0.208849\pi\)
\(720\) 91.6727 3.41644
\(721\) 0 0
\(722\) 33.2580 1.23773
\(723\) −26.1900 −0.974015
\(724\) −0.891685 −0.0331392
\(725\) 24.1744 0.897814
\(726\) 7.72496 0.286700
\(727\) 3.75614 0.139307 0.0696537 0.997571i \(-0.477811\pi\)
0.0696537 + 0.997571i \(0.477811\pi\)
\(728\) 0 0
\(729\) −5.09775 −0.188806
\(730\) 11.9561 0.442517
\(731\) 3.54505 0.131118
\(732\) 8.00000 0.295689
\(733\) −45.7819 −1.69099 −0.845497 0.533980i \(-0.820696\pi\)
−0.845497 + 0.533980i \(0.820696\pi\)
\(734\) 49.5960 1.83062
\(735\) 0 0
\(736\) 46.5089 1.71434
\(737\) 31.1466 1.14730
\(738\) −50.5472 −1.86067
\(739\) −14.0539 −0.516981 −0.258491 0.966014i \(-0.583225\pi\)
−0.258491 + 0.966014i \(0.583225\pi\)
\(740\) 22.3133 0.820255
\(741\) 2.52444 0.0927375
\(742\) 0 0
\(743\) 4.74557 0.174098 0.0870491 0.996204i \(-0.472256\pi\)
0.0870491 + 0.996204i \(0.472256\pi\)
\(744\) −5.56777 −0.204125
\(745\) −23.9844 −0.878721
\(746\) −29.2827 −1.07212
\(747\) −109.211 −3.99581
\(748\) 2.09775 0.0767014
\(749\) 0 0
\(750\) −32.9894 −1.20460
\(751\) 36.1008 1.31734 0.658669 0.752433i \(-0.271120\pi\)
0.658669 + 0.752433i \(0.271120\pi\)
\(752\) −29.3522 −1.07036
\(753\) 74.4182 2.71195
\(754\) 15.0333 0.547480
\(755\) 33.7194 1.22718
\(756\) 0 0
\(757\) −1.03474 −0.0376084 −0.0188042 0.999823i \(-0.505986\pi\)
−0.0188042 + 0.999823i \(0.505986\pi\)
\(758\) −47.3905 −1.72130
\(759\) 70.6449 2.56425
\(760\) 2.95112 0.107049
\(761\) −29.8414 −1.08175 −0.540874 0.841104i \(-0.681906\pi\)
−0.540874 + 0.841104i \(0.681906\pi\)
\(762\) 72.4877 2.62595
\(763\) 0 0
\(764\) 10.0978 0.365324
\(765\) −9.77886 −0.353556
\(766\) −38.1744 −1.37930
\(767\) −4.47054 −0.161422
\(768\) 64.6832 2.33406
\(769\) −23.6358 −0.852329 −0.426164 0.904646i \(-0.640136\pi\)
−0.426164 + 0.904646i \(0.640136\pi\)
\(770\) 0 0
\(771\) −48.4394 −1.74450
\(772\) −15.7350 −0.566315
\(773\) −13.0278 −0.468576 −0.234288 0.972167i \(-0.575276\pi\)
−0.234288 + 0.972167i \(0.575276\pi\)
\(774\) −81.2460 −2.92033
\(775\) −4.05944 −0.145819
\(776\) 1.52946 0.0549045
\(777\) 0 0
\(778\) 39.1849 1.40485
\(779\) −3.42166 −0.122594
\(780\) 11.2544 0.402973
\(781\) −27.0872 −0.969256
\(782\) −6.97939 −0.249583
\(783\) 93.2898 3.33391
\(784\) 0 0
\(785\) −36.0922 −1.28819
\(786\) 50.9200 1.81625
\(787\) −46.2141 −1.64736 −0.823678 0.567058i \(-0.808082\pi\)
−0.823678 + 0.567058i \(0.808082\pi\)
\(788\) −24.2650 −0.864404
\(789\) −47.0716 −1.67579
\(790\) −69.1099 −2.45882
\(791\) 0 0
\(792\) 26.5089 0.941951
\(793\) −2.00000 −0.0710221
\(794\) −50.2283 −1.78253
\(795\) 21.7789 0.772417
\(796\) −27.8610 −0.987508
\(797\) −53.1155 −1.88145 −0.940723 0.339176i \(-0.889852\pi\)
−0.940723 + 0.339176i \(0.889852\pi\)
\(798\) 0 0
\(799\) 3.13104 0.110768
\(800\) 18.4842 0.653514
\(801\) 70.5558 2.49297
\(802\) 4.67609 0.165118
\(803\) 7.27001 0.256553
\(804\) 40.1533 1.41610
\(805\) 0 0
\(806\) −2.52444 −0.0889195
\(807\) −72.2721 −2.54410
\(808\) −16.8917 −0.594247
\(809\) −54.4635 −1.91484 −0.957418 0.288705i \(-0.906775\pi\)
−0.957418 + 0.288705i \(0.906775\pi\)
\(810\) 76.7371 2.69627
\(811\) −38.0978 −1.33779 −0.668897 0.743356i \(-0.733233\pi\)
−0.668897 + 0.743356i \(0.733233\pi\)
\(812\) 0 0
\(813\) −39.5466 −1.38696
\(814\) 34.6167 1.21331
\(815\) 37.8711 1.32657
\(816\) −8.00000 −0.280056
\(817\) −5.49974 −0.192412
\(818\) 27.4161 0.958582
\(819\) 0 0
\(820\) −15.2544 −0.532708
\(821\) −2.30330 −0.0803858 −0.0401929 0.999192i \(-0.512797\pi\)
−0.0401929 + 0.999192i \(0.512797\pi\)
\(822\) 35.2233 1.22855
\(823\) 23.6172 0.823243 0.411621 0.911355i \(-0.364963\pi\)
0.411621 + 0.911355i \(0.364963\pi\)
\(824\) −5.68665 −0.198104
\(825\) 28.0766 0.977503
\(826\) 0 0
\(827\) −48.1643 −1.67484 −0.837419 0.546562i \(-0.815936\pi\)
−0.837419 + 0.546562i \(0.815936\pi\)
\(828\) 62.6933 2.17874
\(829\) −13.0716 −0.453996 −0.226998 0.973895i \(-0.572891\pi\)
−0.226998 + 0.973895i \(0.572891\pi\)
\(830\) −84.0893 −2.91878
\(831\) 25.2005 0.874197
\(832\) 1.66196 0.0576179
\(833\) 0 0
\(834\) −65.1255 −2.25511
\(835\) −5.71083 −0.197631
\(836\) −3.25443 −0.112557
\(837\) −15.6655 −0.541480
\(838\) −18.1260 −0.626153
\(839\) 17.6756 0.610229 0.305114 0.952316i \(-0.401305\pi\)
0.305114 + 0.952316i \(0.401305\pi\)
\(840\) 0 0
\(841\) 39.7103 1.36932
\(842\) 47.0177 1.62034
\(843\) 59.0560 2.03400
\(844\) −22.4011 −0.771076
\(845\) −2.81361 −0.0967910
\(846\) −71.7577 −2.46708
\(847\) 0 0
\(848\) 12.2650 0.421181
\(849\) −34.5855 −1.18697
\(850\) −2.77384 −0.0951420
\(851\) −45.1411 −1.54742
\(852\) −34.9200 −1.19634
\(853\) 5.48970 0.187964 0.0939818 0.995574i \(-0.470040\pi\)
0.0939818 + 0.995574i \(0.470040\pi\)
\(854\) 0 0
\(855\) 15.1708 0.518831
\(856\) 0.745574 0.0254832
\(857\) 11.0489 0.377422 0.188711 0.982033i \(-0.439569\pi\)
0.188711 + 0.982033i \(0.439569\pi\)
\(858\) 17.4600 0.596074
\(859\) −45.2616 −1.54430 −0.772152 0.635437i \(-0.780820\pi\)
−0.772152 + 0.635437i \(0.780820\pi\)
\(860\) −24.5189 −0.836087
\(861\) 0 0
\(862\) −55.6344 −1.89491
\(863\) −5.90225 −0.200915 −0.100457 0.994941i \(-0.532031\pi\)
−0.100457 + 0.994941i \(0.532031\pi\)
\(864\) 71.3311 2.42673
\(865\) 57.1099 1.94180
\(866\) −6.38283 −0.216898
\(867\) −51.8938 −1.76241
\(868\) 0 0
\(869\) −42.0227 −1.42552
\(870\) 131.240 4.44947
\(871\) −10.0383 −0.340135
\(872\) 7.18494 0.243313
\(873\) 7.86248 0.266105
\(874\) 10.8277 0.366254
\(875\) 0 0
\(876\) 9.37227 0.316660
\(877\) −4.90727 −0.165707 −0.0828534 0.996562i \(-0.526403\pi\)
−0.0828534 + 0.996562i \(0.526403\pi\)
\(878\) 58.6732 1.98012
\(879\) 43.9844 1.48356
\(880\) 42.9200 1.44683
\(881\) −44.2822 −1.49190 −0.745952 0.665999i \(-0.768005\pi\)
−0.745952 + 0.665999i \(0.768005\pi\)
\(882\) 0 0
\(883\) −58.8605 −1.98081 −0.990407 0.138181i \(-0.955874\pi\)
−0.990407 + 0.138181i \(0.955874\pi\)
\(884\) −0.676089 −0.0227393
\(885\) −39.0278 −1.31190
\(886\) −28.0127 −0.941104
\(887\) 10.1289 0.340096 0.170048 0.985436i \(-0.445608\pi\)
0.170048 + 0.985436i \(0.445608\pi\)
\(888\) −24.6066 −0.825744
\(889\) 0 0
\(890\) 54.3260 1.82101
\(891\) 46.6605 1.56319
\(892\) 13.5989 0.455326
\(893\) −4.85746 −0.162549
\(894\) −47.9688 −1.60432
\(895\) 30.9597 1.03487
\(896\) 0 0
\(897\) −22.7683 −0.760211
\(898\) 26.2439 0.875769
\(899\) −11.5381 −0.384816
\(900\) 24.9164 0.830546
\(901\) −1.30833 −0.0435866
\(902\) −23.6655 −0.787976
\(903\) 0 0
\(904\) 7.02061 0.233502
\(905\) 1.94610 0.0646906
\(906\) 67.4389 2.24051
\(907\) 37.9547 1.26026 0.630132 0.776488i \(-0.283001\pi\)
0.630132 + 0.776488i \(0.283001\pi\)
\(908\) 8.96117 0.297387
\(909\) −86.8349 −2.88013
\(910\) 0 0
\(911\) 5.57477 0.184700 0.0923501 0.995727i \(-0.470562\pi\)
0.0923501 + 0.995727i \(0.470562\pi\)
\(912\) 12.4111 0.410973
\(913\) −51.1310 −1.69219
\(914\) 62.8787 2.07984
\(915\) −17.4600 −0.577209
\(916\) 27.1849 0.898216
\(917\) 0 0
\(918\) −10.7044 −0.353296
\(919\) 15.7844 0.520679 0.260340 0.965517i \(-0.416165\pi\)
0.260340 + 0.965517i \(0.416165\pi\)
\(920\) −26.6167 −0.877525
\(921\) −42.0711 −1.38629
\(922\) 22.7427 0.748990
\(923\) 8.72999 0.287351
\(924\) 0 0
\(925\) −17.9406 −0.589882
\(926\) −22.0383 −0.724224
\(927\) −29.2333 −0.960148
\(928\) 52.5371 1.72462
\(929\) 45.2630 1.48503 0.742516 0.669829i \(-0.233632\pi\)
0.742516 + 0.669829i \(0.233632\pi\)
\(930\) −22.0383 −0.722665
\(931\) 0 0
\(932\) 7.84281 0.256900
\(933\) −1.32391 −0.0433429
\(934\) −67.1638 −2.19767
\(935\) −4.57834 −0.149728
\(936\) −8.54359 −0.279256
\(937\) −53.6188 −1.75165 −0.875824 0.482630i \(-0.839682\pi\)
−0.875824 + 0.482630i \(0.839682\pi\)
\(938\) 0 0
\(939\) 56.3205 1.83795
\(940\) −21.6555 −0.706324
\(941\) −20.7753 −0.677255 −0.338628 0.940920i \(-0.609963\pi\)
−0.338628 + 0.940920i \(0.609963\pi\)
\(942\) −72.1844 −2.35190
\(943\) 30.8605 1.00496
\(944\) −21.9789 −0.715351
\(945\) 0 0
\(946\) −38.0383 −1.23673
\(947\) −10.8605 −0.352919 −0.176460 0.984308i \(-0.556465\pi\)
−0.176460 + 0.984308i \(0.556465\pi\)
\(948\) −54.1744 −1.75950
\(949\) −2.34307 −0.0760592
\(950\) 4.30330 0.139618
\(951\) 29.2333 0.947955
\(952\) 0 0
\(953\) −25.7180 −0.833087 −0.416543 0.909116i \(-0.636759\pi\)
−0.416543 + 0.909116i \(0.636759\pi\)
\(954\) 29.9844 0.970781
\(955\) −22.0383 −0.713143
\(956\) −18.3133 −0.592296
\(957\) 79.8016 2.57962
\(958\) 21.7789 0.703643
\(959\) 0 0
\(960\) 14.5089 0.468271
\(961\) −29.0625 −0.937500
\(962\) −11.1567 −0.359706
\(963\) 3.83276 0.123509
\(964\) −10.8816 −0.350474
\(965\) 34.3416 1.10550
\(966\) 0 0
\(967\) 33.5038 1.07741 0.538705 0.842494i \(-0.318914\pi\)
0.538705 + 0.842494i \(0.318914\pi\)
\(968\) −1.76975 −0.0568820
\(969\) −1.32391 −0.0425302
\(970\) 6.05390 0.194379
\(971\) −2.03831 −0.0654126 −0.0327063 0.999465i \(-0.510413\pi\)
−0.0327063 + 0.999465i \(0.510413\pi\)
\(972\) 16.6267 0.533302
\(973\) 0 0
\(974\) −20.1643 −0.646107
\(975\) −9.04888 −0.289796
\(976\) −9.83276 −0.314739
\(977\) 15.1411 0.484406 0.242203 0.970226i \(-0.422130\pi\)
0.242203 + 0.970226i \(0.422130\pi\)
\(978\) 75.7422 2.42197
\(979\) 33.0333 1.05575
\(980\) 0 0
\(981\) 36.9355 1.17926
\(982\) −0.107798 −0.00343998
\(983\) −49.3124 −1.57282 −0.786411 0.617704i \(-0.788063\pi\)
−0.786411 + 0.617704i \(0.788063\pi\)
\(984\) 16.8222 0.536272
\(985\) 52.9583 1.68739
\(986\) −7.88403 −0.251079
\(987\) 0 0
\(988\) 1.04888 0.0333692
\(989\) 49.6030 1.57728
\(990\) 104.927 3.33480
\(991\) 5.43171 0.172544 0.0862720 0.996272i \(-0.472505\pi\)
0.0862720 + 0.996272i \(0.472505\pi\)
\(992\) −8.82220 −0.280105
\(993\) −53.9900 −1.71332
\(994\) 0 0
\(995\) 60.8066 1.92770
\(996\) −65.9165 −2.08865
\(997\) 53.6061 1.69772 0.848861 0.528616i \(-0.177289\pi\)
0.848861 + 0.528616i \(0.177289\pi\)
\(998\) −18.6761 −0.591181
\(999\) −69.2333 −2.19044
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.j.1.1 3
3.2 odd 2 5733.2.a.x.1.3 3
7.2 even 3 637.2.e.i.508.3 6
7.3 odd 6 637.2.e.j.79.3 6
7.4 even 3 637.2.e.i.79.3 6
7.5 odd 6 637.2.e.j.508.3 6
7.6 odd 2 91.2.a.d.1.1 3
13.12 even 2 8281.2.a.bg.1.3 3
21.20 even 2 819.2.a.i.1.3 3
28.27 even 2 1456.2.a.t.1.3 3
35.34 odd 2 2275.2.a.m.1.3 3
56.13 odd 2 5824.2.a.by.1.3 3
56.27 even 2 5824.2.a.bs.1.1 3
91.34 even 4 1183.2.c.f.337.2 6
91.83 even 4 1183.2.c.f.337.5 6
91.90 odd 2 1183.2.a.i.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.1 3 7.6 odd 2
637.2.a.j.1.1 3 1.1 even 1 trivial
637.2.e.i.79.3 6 7.4 even 3
637.2.e.i.508.3 6 7.2 even 3
637.2.e.j.79.3 6 7.3 odd 6
637.2.e.j.508.3 6 7.5 odd 6
819.2.a.i.1.3 3 21.20 even 2
1183.2.a.i.1.3 3 91.90 odd 2
1183.2.c.f.337.2 6 91.34 even 4
1183.2.c.f.337.5 6 91.83 even 4
1456.2.a.t.1.3 3 28.27 even 2
2275.2.a.m.1.3 3 35.34 odd 2
5733.2.a.x.1.3 3 3.2 odd 2
5824.2.a.bs.1.1 3 56.27 even 2
5824.2.a.by.1.3 3 56.13 odd 2
8281.2.a.bg.1.3 3 13.12 even 2