Properties

Label 6039.2.a.i.1.2
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2216577807\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2013)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.53773\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.53773 q^{2} +4.44007 q^{4} -0.994065 q^{5} +4.88646 q^{7} -6.19224 q^{8} +O(q^{10})\) \(q-2.53773 q^{2} +4.44007 q^{4} -0.994065 q^{5} +4.88646 q^{7} -6.19224 q^{8} +2.52267 q^{10} -1.00000 q^{11} +4.29008 q^{13} -12.4005 q^{14} +6.83409 q^{16} +3.82389 q^{17} -2.93110 q^{19} -4.41372 q^{20} +2.53773 q^{22} +5.41371 q^{23} -4.01184 q^{25} -10.8871 q^{26} +21.6962 q^{28} +2.31016 q^{29} -6.92787 q^{31} -4.95859 q^{32} -9.70401 q^{34} -4.85746 q^{35} +0.949760 q^{37} +7.43834 q^{38} +6.15549 q^{40} +4.66091 q^{41} +4.23337 q^{43} -4.44007 q^{44} -13.7385 q^{46} -8.27575 q^{47} +16.8775 q^{49} +10.1810 q^{50} +19.0483 q^{52} +11.5324 q^{53} +0.994065 q^{55} -30.2582 q^{56} -5.86257 q^{58} -5.12033 q^{59} -1.00000 q^{61} +17.5811 q^{62} -1.08461 q^{64} -4.26462 q^{65} -6.83412 q^{67} +16.9784 q^{68} +12.3269 q^{70} -4.72943 q^{71} +15.9491 q^{73} -2.41023 q^{74} -13.0143 q^{76} -4.88646 q^{77} -6.64878 q^{79} -6.79353 q^{80} -11.8281 q^{82} +8.78161 q^{83} -3.80120 q^{85} -10.7431 q^{86} +6.19224 q^{88} -0.0660596 q^{89} +20.9633 q^{91} +24.0372 q^{92} +21.0016 q^{94} +2.91370 q^{95} +11.3016 q^{97} -42.8306 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + O(q^{10}) \) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + 6q^{10} - 13q^{11} + 13q^{13} - q^{14} + 18q^{16} - 17q^{17} + 14q^{19} + 7q^{20} + 2q^{22} - 7q^{23} + 18q^{25} + 10q^{26} + 19q^{28} + 6q^{29} + 27q^{31} - 5q^{32} + 6q^{34} - 14q^{35} + 10q^{37} - 2q^{38} + 8q^{40} - 3q^{41} + 29q^{43} - 16q^{44} - 24q^{46} - 8q^{47} + 8q^{49} + 27q^{50} + 37q^{52} + 24q^{53} + 3q^{55} - 24q^{56} - 5q^{58} - 13q^{59} - 13q^{61} - 39q^{62} + 47q^{64} + 11q^{65} + 44q^{67} + 8q^{68} - 12q^{70} - 3q^{71} + 48q^{73} + 22q^{74} + 47q^{76} - 11q^{77} - 17q^{79} + 26q^{80} + 56q^{82} - 50q^{83} + 8q^{85} - 18q^{86} + 9q^{88} + 15q^{89} + 47q^{91} - 14q^{92} + 45q^{94} + q^{95} + 27q^{97} - 47q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.53773 −1.79445 −0.897223 0.441578i \(-0.854419\pi\)
−0.897223 + 0.441578i \(0.854419\pi\)
\(3\) 0 0
\(4\) 4.44007 2.22004
\(5\) −0.994065 −0.444559 −0.222280 0.974983i \(-0.571350\pi\)
−0.222280 + 0.974983i \(0.571350\pi\)
\(6\) 0 0
\(7\) 4.88646 1.84691 0.923455 0.383707i \(-0.125353\pi\)
0.923455 + 0.383707i \(0.125353\pi\)
\(8\) −6.19224 −2.18929
\(9\) 0 0
\(10\) 2.52267 0.797737
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 4.29008 1.18985 0.594927 0.803779i \(-0.297181\pi\)
0.594927 + 0.803779i \(0.297181\pi\)
\(14\) −12.4005 −3.31418
\(15\) 0 0
\(16\) 6.83409 1.70852
\(17\) 3.82389 0.927431 0.463715 0.885984i \(-0.346516\pi\)
0.463715 + 0.885984i \(0.346516\pi\)
\(18\) 0 0
\(19\) −2.93110 −0.672441 −0.336220 0.941783i \(-0.609149\pi\)
−0.336220 + 0.941783i \(0.609149\pi\)
\(20\) −4.41372 −0.986937
\(21\) 0 0
\(22\) 2.53773 0.541046
\(23\) 5.41371 1.12884 0.564418 0.825489i \(-0.309101\pi\)
0.564418 + 0.825489i \(0.309101\pi\)
\(24\) 0 0
\(25\) −4.01184 −0.802367
\(26\) −10.8871 −2.13513
\(27\) 0 0
\(28\) 21.6962 4.10021
\(29\) 2.31016 0.428986 0.214493 0.976725i \(-0.431190\pi\)
0.214493 + 0.976725i \(0.431190\pi\)
\(30\) 0 0
\(31\) −6.92787 −1.24428 −0.622141 0.782905i \(-0.713737\pi\)
−0.622141 + 0.782905i \(0.713737\pi\)
\(32\) −4.95859 −0.876564
\(33\) 0 0
\(34\) −9.70401 −1.66422
\(35\) −4.85746 −0.821061
\(36\) 0 0
\(37\) 0.949760 0.156140 0.0780698 0.996948i \(-0.475124\pi\)
0.0780698 + 0.996948i \(0.475124\pi\)
\(38\) 7.43834 1.20666
\(39\) 0 0
\(40\) 6.15549 0.973268
\(41\) 4.66091 0.727912 0.363956 0.931416i \(-0.381426\pi\)
0.363956 + 0.931416i \(0.381426\pi\)
\(42\) 0 0
\(43\) 4.23337 0.645583 0.322791 0.946470i \(-0.395379\pi\)
0.322791 + 0.946470i \(0.395379\pi\)
\(44\) −4.44007 −0.669366
\(45\) 0 0
\(46\) −13.7385 −2.02563
\(47\) −8.27575 −1.20714 −0.603571 0.797309i \(-0.706256\pi\)
−0.603571 + 0.797309i \(0.706256\pi\)
\(48\) 0 0
\(49\) 16.8775 2.41108
\(50\) 10.1810 1.43980
\(51\) 0 0
\(52\) 19.0483 2.64152
\(53\) 11.5324 1.58410 0.792051 0.610455i \(-0.209013\pi\)
0.792051 + 0.610455i \(0.209013\pi\)
\(54\) 0 0
\(55\) 0.994065 0.134040
\(56\) −30.2582 −4.04342
\(57\) 0 0
\(58\) −5.86257 −0.769793
\(59\) −5.12033 −0.666610 −0.333305 0.942819i \(-0.608164\pi\)
−0.333305 + 0.942819i \(0.608164\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) 17.5811 2.23280
\(63\) 0 0
\(64\) −1.08461 −0.135577
\(65\) −4.26462 −0.528961
\(66\) 0 0
\(67\) −6.83412 −0.834920 −0.417460 0.908695i \(-0.637080\pi\)
−0.417460 + 0.908695i \(0.637080\pi\)
\(68\) 16.9784 2.05893
\(69\) 0 0
\(70\) 12.3269 1.47335
\(71\) −4.72943 −0.561280 −0.280640 0.959813i \(-0.590547\pi\)
−0.280640 + 0.959813i \(0.590547\pi\)
\(72\) 0 0
\(73\) 15.9491 1.86670 0.933348 0.358972i \(-0.116873\pi\)
0.933348 + 0.358972i \(0.116873\pi\)
\(74\) −2.41023 −0.280184
\(75\) 0 0
\(76\) −13.0143 −1.49284
\(77\) −4.88646 −0.556864
\(78\) 0 0
\(79\) −6.64878 −0.748046 −0.374023 0.927419i \(-0.622022\pi\)
−0.374023 + 0.927419i \(0.622022\pi\)
\(80\) −6.79353 −0.759540
\(81\) 0 0
\(82\) −11.8281 −1.30620
\(83\) 8.78161 0.963906 0.481953 0.876197i \(-0.339928\pi\)
0.481953 + 0.876197i \(0.339928\pi\)
\(84\) 0 0
\(85\) −3.80120 −0.412298
\(86\) −10.7431 −1.15846
\(87\) 0 0
\(88\) 6.19224 0.660095
\(89\) −0.0660596 −0.00700230 −0.00350115 0.999994i \(-0.501114\pi\)
−0.00350115 + 0.999994i \(0.501114\pi\)
\(90\) 0 0
\(91\) 20.9633 2.19755
\(92\) 24.0372 2.50606
\(93\) 0 0
\(94\) 21.0016 2.16615
\(95\) 2.91370 0.298940
\(96\) 0 0
\(97\) 11.3016 1.14751 0.573753 0.819028i \(-0.305487\pi\)
0.573753 + 0.819028i \(0.305487\pi\)
\(98\) −42.8306 −4.32654
\(99\) 0 0
\(100\) −17.8128 −1.78128
\(101\) 6.79683 0.676310 0.338155 0.941091i \(-0.390197\pi\)
0.338155 + 0.941091i \(0.390197\pi\)
\(102\) 0 0
\(103\) 12.1574 1.19790 0.598952 0.800785i \(-0.295584\pi\)
0.598952 + 0.800785i \(0.295584\pi\)
\(104\) −26.5652 −2.60493
\(105\) 0 0
\(106\) −29.2662 −2.84258
\(107\) −3.63314 −0.351229 −0.175614 0.984459i \(-0.556191\pi\)
−0.175614 + 0.984459i \(0.556191\pi\)
\(108\) 0 0
\(109\) 4.81079 0.460790 0.230395 0.973097i \(-0.425998\pi\)
0.230395 + 0.973097i \(0.425998\pi\)
\(110\) −2.52267 −0.240527
\(111\) 0 0
\(112\) 33.3945 3.15549
\(113\) 4.35919 0.410078 0.205039 0.978754i \(-0.434268\pi\)
0.205039 + 0.978754i \(0.434268\pi\)
\(114\) 0 0
\(115\) −5.38157 −0.501834
\(116\) 10.2573 0.952365
\(117\) 0 0
\(118\) 12.9940 1.19620
\(119\) 18.6853 1.71288
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.53773 0.229755
\(123\) 0 0
\(124\) −30.7602 −2.76235
\(125\) 8.95835 0.801259
\(126\) 0 0
\(127\) −20.5780 −1.82600 −0.913001 0.407957i \(-0.866241\pi\)
−0.913001 + 0.407957i \(0.866241\pi\)
\(128\) 12.6696 1.11985
\(129\) 0 0
\(130\) 10.8225 0.949192
\(131\) −8.91370 −0.778794 −0.389397 0.921070i \(-0.627317\pi\)
−0.389397 + 0.921070i \(0.627317\pi\)
\(132\) 0 0
\(133\) −14.3227 −1.24194
\(134\) 17.3431 1.49822
\(135\) 0 0
\(136\) −23.6785 −2.03041
\(137\) 11.8906 1.01589 0.507943 0.861391i \(-0.330406\pi\)
0.507943 + 0.861391i \(0.330406\pi\)
\(138\) 0 0
\(139\) 3.99386 0.338755 0.169377 0.985551i \(-0.445824\pi\)
0.169377 + 0.985551i \(0.445824\pi\)
\(140\) −21.5675 −1.82278
\(141\) 0 0
\(142\) 12.0020 1.00719
\(143\) −4.29008 −0.358755
\(144\) 0 0
\(145\) −2.29645 −0.190710
\(146\) −40.4744 −3.34969
\(147\) 0 0
\(148\) 4.21700 0.346636
\(149\) −18.2887 −1.49827 −0.749134 0.662418i \(-0.769530\pi\)
−0.749134 + 0.662418i \(0.769530\pi\)
\(150\) 0 0
\(151\) 11.2751 0.917554 0.458777 0.888551i \(-0.348288\pi\)
0.458777 + 0.888551i \(0.348288\pi\)
\(152\) 18.1501 1.47217
\(153\) 0 0
\(154\) 12.4005 0.999263
\(155\) 6.88675 0.553157
\(156\) 0 0
\(157\) −16.6823 −1.33139 −0.665695 0.746224i \(-0.731865\pi\)
−0.665695 + 0.746224i \(0.731865\pi\)
\(158\) 16.8728 1.34233
\(159\) 0 0
\(160\) 4.92916 0.389685
\(161\) 26.4539 2.08486
\(162\) 0 0
\(163\) 10.4676 0.819882 0.409941 0.912112i \(-0.365549\pi\)
0.409941 + 0.912112i \(0.365549\pi\)
\(164\) 20.6948 1.61599
\(165\) 0 0
\(166\) −22.2853 −1.72968
\(167\) 21.0144 1.62614 0.813070 0.582166i \(-0.197795\pi\)
0.813070 + 0.582166i \(0.197795\pi\)
\(168\) 0 0
\(169\) 5.40481 0.415754
\(170\) 9.64641 0.739846
\(171\) 0 0
\(172\) 18.7965 1.43322
\(173\) −3.21998 −0.244811 −0.122405 0.992480i \(-0.539061\pi\)
−0.122405 + 0.992480i \(0.539061\pi\)
\(174\) 0 0
\(175\) −19.6037 −1.48190
\(176\) −6.83409 −0.515139
\(177\) 0 0
\(178\) 0.167641 0.0125652
\(179\) 7.27099 0.543460 0.271730 0.962374i \(-0.412404\pi\)
0.271730 + 0.962374i \(0.412404\pi\)
\(180\) 0 0
\(181\) 13.4180 0.997355 0.498678 0.866788i \(-0.333819\pi\)
0.498678 + 0.866788i \(0.333819\pi\)
\(182\) −53.1993 −3.94339
\(183\) 0 0
\(184\) −33.5230 −2.47135
\(185\) −0.944123 −0.0694133
\(186\) 0 0
\(187\) −3.82389 −0.279631
\(188\) −36.7449 −2.67990
\(189\) 0 0
\(190\) −7.39419 −0.536431
\(191\) −16.2153 −1.17329 −0.586647 0.809843i \(-0.699552\pi\)
−0.586647 + 0.809843i \(0.699552\pi\)
\(192\) 0 0
\(193\) 15.4786 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(194\) −28.6805 −2.05914
\(195\) 0 0
\(196\) 74.9374 5.35267
\(197\) 17.7360 1.26364 0.631818 0.775117i \(-0.282309\pi\)
0.631818 + 0.775117i \(0.282309\pi\)
\(198\) 0 0
\(199\) −2.42918 −0.172200 −0.0861000 0.996286i \(-0.527440\pi\)
−0.0861000 + 0.996286i \(0.527440\pi\)
\(200\) 24.8423 1.75661
\(201\) 0 0
\(202\) −17.2485 −1.21360
\(203\) 11.2885 0.792299
\(204\) 0 0
\(205\) −4.63325 −0.323600
\(206\) −30.8522 −2.14957
\(207\) 0 0
\(208\) 29.3188 2.03289
\(209\) 2.93110 0.202748
\(210\) 0 0
\(211\) −23.4818 −1.61655 −0.808276 0.588804i \(-0.799599\pi\)
−0.808276 + 0.588804i \(0.799599\pi\)
\(212\) 51.2048 3.51676
\(213\) 0 0
\(214\) 9.21992 0.630261
\(215\) −4.20824 −0.287000
\(216\) 0 0
\(217\) −33.8528 −2.29808
\(218\) −12.2085 −0.826863
\(219\) 0 0
\(220\) 4.41372 0.297573
\(221\) 16.4048 1.10351
\(222\) 0 0
\(223\) 19.8001 1.32591 0.662957 0.748657i \(-0.269301\pi\)
0.662957 + 0.748657i \(0.269301\pi\)
\(224\) −24.2300 −1.61893
\(225\) 0 0
\(226\) −11.0624 −0.735862
\(227\) −22.2706 −1.47815 −0.739077 0.673621i \(-0.764738\pi\)
−0.739077 + 0.673621i \(0.764738\pi\)
\(228\) 0 0
\(229\) 4.54201 0.300144 0.150072 0.988675i \(-0.452049\pi\)
0.150072 + 0.988675i \(0.452049\pi\)
\(230\) 13.6570 0.900515
\(231\) 0 0
\(232\) −14.3051 −0.939175
\(233\) 9.10518 0.596500 0.298250 0.954488i \(-0.403597\pi\)
0.298250 + 0.954488i \(0.403597\pi\)
\(234\) 0 0
\(235\) 8.22663 0.536646
\(236\) −22.7346 −1.47990
\(237\) 0 0
\(238\) −47.4183 −3.07367
\(239\) 7.66236 0.495637 0.247818 0.968807i \(-0.420286\pi\)
0.247818 + 0.968807i \(0.420286\pi\)
\(240\) 0 0
\(241\) −3.33162 −0.214608 −0.107304 0.994226i \(-0.534222\pi\)
−0.107304 + 0.994226i \(0.534222\pi\)
\(242\) −2.53773 −0.163131
\(243\) 0 0
\(244\) −4.44007 −0.284246
\(245\) −16.7774 −1.07187
\(246\) 0 0
\(247\) −12.5747 −0.800107
\(248\) 42.8991 2.72409
\(249\) 0 0
\(250\) −22.7339 −1.43782
\(251\) −29.0747 −1.83518 −0.917589 0.397529i \(-0.869868\pi\)
−0.917589 + 0.397529i \(0.869868\pi\)
\(252\) 0 0
\(253\) −5.41371 −0.340357
\(254\) 52.2214 3.27666
\(255\) 0 0
\(256\) −29.9829 −1.87393
\(257\) 20.2898 1.26564 0.632822 0.774297i \(-0.281896\pi\)
0.632822 + 0.774297i \(0.281896\pi\)
\(258\) 0 0
\(259\) 4.64097 0.288376
\(260\) −18.9352 −1.17431
\(261\) 0 0
\(262\) 22.6206 1.39750
\(263\) −32.3400 −1.99417 −0.997085 0.0762942i \(-0.975691\pi\)
−0.997085 + 0.0762942i \(0.975691\pi\)
\(264\) 0 0
\(265\) −11.4640 −0.704227
\(266\) 36.3472 2.22859
\(267\) 0 0
\(268\) −30.3440 −1.85355
\(269\) −17.3768 −1.05948 −0.529742 0.848159i \(-0.677711\pi\)
−0.529742 + 0.848159i \(0.677711\pi\)
\(270\) 0 0
\(271\) −11.5350 −0.700701 −0.350350 0.936619i \(-0.613937\pi\)
−0.350350 + 0.936619i \(0.613937\pi\)
\(272\) 26.1328 1.58454
\(273\) 0 0
\(274\) −30.1752 −1.82295
\(275\) 4.01184 0.241923
\(276\) 0 0
\(277\) −11.7386 −0.705307 −0.352653 0.935754i \(-0.614721\pi\)
−0.352653 + 0.935754i \(0.614721\pi\)
\(278\) −10.1353 −0.607877
\(279\) 0 0
\(280\) 30.0786 1.79754
\(281\) −7.78483 −0.464404 −0.232202 0.972668i \(-0.574593\pi\)
−0.232202 + 0.972668i \(0.574593\pi\)
\(282\) 0 0
\(283\) −24.6554 −1.46561 −0.732806 0.680438i \(-0.761790\pi\)
−0.732806 + 0.680438i \(0.761790\pi\)
\(284\) −20.9990 −1.24606
\(285\) 0 0
\(286\) 10.8871 0.643766
\(287\) 22.7754 1.34439
\(288\) 0 0
\(289\) −2.37783 −0.139873
\(290\) 5.82777 0.342219
\(291\) 0 0
\(292\) 70.8150 4.14413
\(293\) 13.4592 0.786297 0.393149 0.919475i \(-0.371386\pi\)
0.393149 + 0.919475i \(0.371386\pi\)
\(294\) 0 0
\(295\) 5.08994 0.296348
\(296\) −5.88114 −0.341835
\(297\) 0 0
\(298\) 46.4118 2.68856
\(299\) 23.2252 1.34315
\(300\) 0 0
\(301\) 20.6862 1.19233
\(302\) −28.6131 −1.64650
\(303\) 0 0
\(304\) −20.0314 −1.14888
\(305\) 0.994065 0.0569200
\(306\) 0 0
\(307\) 15.7969 0.901579 0.450789 0.892630i \(-0.351143\pi\)
0.450789 + 0.892630i \(0.351143\pi\)
\(308\) −21.6962 −1.23626
\(309\) 0 0
\(310\) −17.4767 −0.992611
\(311\) −20.8528 −1.18246 −0.591228 0.806504i \(-0.701357\pi\)
−0.591228 + 0.806504i \(0.701357\pi\)
\(312\) 0 0
\(313\) 7.74049 0.437518 0.218759 0.975779i \(-0.429799\pi\)
0.218759 + 0.975779i \(0.429799\pi\)
\(314\) 42.3351 2.38911
\(315\) 0 0
\(316\) −29.5211 −1.66069
\(317\) −3.37084 −0.189325 −0.0946627 0.995509i \(-0.530177\pi\)
−0.0946627 + 0.995509i \(0.530177\pi\)
\(318\) 0 0
\(319\) −2.31016 −0.129344
\(320\) 1.07818 0.0602719
\(321\) 0 0
\(322\) −67.1328 −3.74116
\(323\) −11.2082 −0.623642
\(324\) 0 0
\(325\) −17.2111 −0.954700
\(326\) −26.5638 −1.47123
\(327\) 0 0
\(328\) −28.8615 −1.59361
\(329\) −40.4391 −2.22948
\(330\) 0 0
\(331\) 11.3276 0.622622 0.311311 0.950308i \(-0.399232\pi\)
0.311311 + 0.950308i \(0.399232\pi\)
\(332\) 38.9910 2.13991
\(333\) 0 0
\(334\) −53.3288 −2.91802
\(335\) 6.79355 0.371172
\(336\) 0 0
\(337\) 27.5129 1.49872 0.749362 0.662161i \(-0.230360\pi\)
0.749362 + 0.662161i \(0.230360\pi\)
\(338\) −13.7159 −0.746049
\(339\) 0 0
\(340\) −16.8776 −0.915316
\(341\) 6.92787 0.375165
\(342\) 0 0
\(343\) 48.2662 2.60613
\(344\) −26.2140 −1.41337
\(345\) 0 0
\(346\) 8.17145 0.439300
\(347\) 7.46395 0.400686 0.200343 0.979726i \(-0.435794\pi\)
0.200343 + 0.979726i \(0.435794\pi\)
\(348\) 0 0
\(349\) −8.64667 −0.462845 −0.231423 0.972853i \(-0.574338\pi\)
−0.231423 + 0.972853i \(0.574338\pi\)
\(350\) 49.7489 2.65919
\(351\) 0 0
\(352\) 4.95859 0.264294
\(353\) −33.8131 −1.79969 −0.899845 0.436210i \(-0.856320\pi\)
−0.899845 + 0.436210i \(0.856320\pi\)
\(354\) 0 0
\(355\) 4.70136 0.249522
\(356\) −0.293309 −0.0155454
\(357\) 0 0
\(358\) −18.4518 −0.975209
\(359\) −4.96562 −0.262076 −0.131038 0.991377i \(-0.541831\pi\)
−0.131038 + 0.991377i \(0.541831\pi\)
\(360\) 0 0
\(361\) −10.4087 −0.547824
\(362\) −34.0514 −1.78970
\(363\) 0 0
\(364\) 93.0787 4.87865
\(365\) −15.8544 −0.829857
\(366\) 0 0
\(367\) 28.2934 1.47690 0.738451 0.674307i \(-0.235558\pi\)
0.738451 + 0.674307i \(0.235558\pi\)
\(368\) 36.9978 1.92864
\(369\) 0 0
\(370\) 2.39593 0.124558
\(371\) 56.3528 2.92569
\(372\) 0 0
\(373\) −18.5022 −0.958007 −0.479003 0.877813i \(-0.659002\pi\)
−0.479003 + 0.877813i \(0.659002\pi\)
\(374\) 9.70401 0.501782
\(375\) 0 0
\(376\) 51.2454 2.64278
\(377\) 9.91079 0.510432
\(378\) 0 0
\(379\) 10.1763 0.522721 0.261361 0.965241i \(-0.415829\pi\)
0.261361 + 0.965241i \(0.415829\pi\)
\(380\) 12.9371 0.663657
\(381\) 0 0
\(382\) 41.1499 2.10541
\(383\) 18.0563 0.922634 0.461317 0.887235i \(-0.347377\pi\)
0.461317 + 0.887235i \(0.347377\pi\)
\(384\) 0 0
\(385\) 4.85746 0.247559
\(386\) −39.2804 −1.99932
\(387\) 0 0
\(388\) 50.1801 2.54751
\(389\) −12.9240 −0.655271 −0.327636 0.944804i \(-0.606252\pi\)
−0.327636 + 0.944804i \(0.606252\pi\)
\(390\) 0 0
\(391\) 20.7014 1.04692
\(392\) −104.510 −5.27854
\(393\) 0 0
\(394\) −45.0091 −2.26753
\(395\) 6.60932 0.332551
\(396\) 0 0
\(397\) 25.2642 1.26798 0.633988 0.773343i \(-0.281417\pi\)
0.633988 + 0.773343i \(0.281417\pi\)
\(398\) 6.16460 0.309004
\(399\) 0 0
\(400\) −27.4173 −1.37086
\(401\) 33.4265 1.66924 0.834620 0.550826i \(-0.185687\pi\)
0.834620 + 0.550826i \(0.185687\pi\)
\(402\) 0 0
\(403\) −29.7211 −1.48052
\(404\) 30.1784 1.50143
\(405\) 0 0
\(406\) −28.6472 −1.42174
\(407\) −0.949760 −0.0470779
\(408\) 0 0
\(409\) −3.25363 −0.160882 −0.0804408 0.996759i \(-0.525633\pi\)
−0.0804408 + 0.996759i \(0.525633\pi\)
\(410\) 11.7579 0.580683
\(411\) 0 0
\(412\) 53.9797 2.65939
\(413\) −25.0203 −1.23117
\(414\) 0 0
\(415\) −8.72948 −0.428513
\(416\) −21.2728 −1.04298
\(417\) 0 0
\(418\) −7.43834 −0.363821
\(419\) 35.2785 1.72347 0.861735 0.507359i \(-0.169378\pi\)
0.861735 + 0.507359i \(0.169378\pi\)
\(420\) 0 0
\(421\) 7.26689 0.354167 0.177083 0.984196i \(-0.443334\pi\)
0.177083 + 0.984196i \(0.443334\pi\)
\(422\) 59.5904 2.90081
\(423\) 0 0
\(424\) −71.4116 −3.46806
\(425\) −15.3408 −0.744140
\(426\) 0 0
\(427\) −4.88646 −0.236473
\(428\) −16.1314 −0.779740
\(429\) 0 0
\(430\) 10.6794 0.515005
\(431\) 7.77524 0.374520 0.187260 0.982310i \(-0.440039\pi\)
0.187260 + 0.982310i \(0.440039\pi\)
\(432\) 0 0
\(433\) −35.7180 −1.71650 −0.858250 0.513232i \(-0.828448\pi\)
−0.858250 + 0.513232i \(0.828448\pi\)
\(434\) 85.9092 4.12378
\(435\) 0 0
\(436\) 21.3603 1.02297
\(437\) −15.8681 −0.759075
\(438\) 0 0
\(439\) 15.8024 0.754205 0.377103 0.926172i \(-0.376920\pi\)
0.377103 + 0.926172i \(0.376920\pi\)
\(440\) −6.15549 −0.293451
\(441\) 0 0
\(442\) −41.6310 −1.98018
\(443\) 2.20489 0.104757 0.0523787 0.998627i \(-0.483320\pi\)
0.0523787 + 0.998627i \(0.483320\pi\)
\(444\) 0 0
\(445\) 0.0656675 0.00311294
\(446\) −50.2473 −2.37928
\(447\) 0 0
\(448\) −5.29993 −0.250398
\(449\) 11.0478 0.521379 0.260690 0.965423i \(-0.416050\pi\)
0.260690 + 0.965423i \(0.416050\pi\)
\(450\) 0 0
\(451\) −4.66091 −0.219474
\(452\) 19.3551 0.910387
\(453\) 0 0
\(454\) 56.5169 2.65247
\(455\) −20.8389 −0.976943
\(456\) 0 0
\(457\) −32.3367 −1.51265 −0.756324 0.654197i \(-0.773007\pi\)
−0.756324 + 0.654197i \(0.773007\pi\)
\(458\) −11.5264 −0.538593
\(459\) 0 0
\(460\) −23.8946 −1.11409
\(461\) 15.6990 0.731175 0.365587 0.930777i \(-0.380868\pi\)
0.365587 + 0.930777i \(0.380868\pi\)
\(462\) 0 0
\(463\) −5.21953 −0.242572 −0.121286 0.992618i \(-0.538702\pi\)
−0.121286 + 0.992618i \(0.538702\pi\)
\(464\) 15.7879 0.732933
\(465\) 0 0
\(466\) −23.1065 −1.07039
\(467\) −38.1044 −1.76326 −0.881630 0.471941i \(-0.843554\pi\)
−0.881630 + 0.471941i \(0.843554\pi\)
\(468\) 0 0
\(469\) −33.3947 −1.54202
\(470\) −20.8770 −0.962982
\(471\) 0 0
\(472\) 31.7063 1.45940
\(473\) −4.23337 −0.194650
\(474\) 0 0
\(475\) 11.7591 0.539544
\(476\) 82.9642 3.80266
\(477\) 0 0
\(478\) −19.4450 −0.889393
\(479\) 13.9545 0.637597 0.318799 0.947822i \(-0.396721\pi\)
0.318799 + 0.947822i \(0.396721\pi\)
\(480\) 0 0
\(481\) 4.07455 0.185783
\(482\) 8.45475 0.385103
\(483\) 0 0
\(484\) 4.44007 0.201821
\(485\) −11.2346 −0.510135
\(486\) 0 0
\(487\) −28.5606 −1.29421 −0.647103 0.762403i \(-0.724020\pi\)
−0.647103 + 0.762403i \(0.724020\pi\)
\(488\) 6.19224 0.280310
\(489\) 0 0
\(490\) 42.5764 1.92341
\(491\) −3.97165 −0.179238 −0.0896189 0.995976i \(-0.528565\pi\)
−0.0896189 + 0.995976i \(0.528565\pi\)
\(492\) 0 0
\(493\) 8.83382 0.397855
\(494\) 31.9111 1.43575
\(495\) 0 0
\(496\) −47.3457 −2.12589
\(497\) −23.1102 −1.03663
\(498\) 0 0
\(499\) −24.1339 −1.08038 −0.540192 0.841542i \(-0.681648\pi\)
−0.540192 + 0.841542i \(0.681648\pi\)
\(500\) 39.7757 1.77882
\(501\) 0 0
\(502\) 73.7837 3.29313
\(503\) 13.5112 0.602433 0.301217 0.953556i \(-0.402607\pi\)
0.301217 + 0.953556i \(0.402607\pi\)
\(504\) 0 0
\(505\) −6.75649 −0.300660
\(506\) 13.7385 0.610752
\(507\) 0 0
\(508\) −91.3678 −4.05379
\(509\) 25.6211 1.13563 0.567817 0.823155i \(-0.307788\pi\)
0.567817 + 0.823155i \(0.307788\pi\)
\(510\) 0 0
\(511\) 77.9345 3.44762
\(512\) 50.7492 2.24282
\(513\) 0 0
\(514\) −51.4901 −2.27113
\(515\) −12.0852 −0.532539
\(516\) 0 0
\(517\) 8.27575 0.363967
\(518\) −11.7775 −0.517475
\(519\) 0 0
\(520\) 26.4076 1.15805
\(521\) −1.69899 −0.0744343 −0.0372172 0.999307i \(-0.511849\pi\)
−0.0372172 + 0.999307i \(0.511849\pi\)
\(522\) 0 0
\(523\) −6.21946 −0.271958 −0.135979 0.990712i \(-0.543418\pi\)
−0.135979 + 0.990712i \(0.543418\pi\)
\(524\) −39.5775 −1.72895
\(525\) 0 0
\(526\) 82.0702 3.57843
\(527\) −26.4914 −1.15399
\(528\) 0 0
\(529\) 6.30822 0.274271
\(530\) 29.0925 1.26370
\(531\) 0 0
\(532\) −63.5939 −2.75714
\(533\) 19.9957 0.866109
\(534\) 0 0
\(535\) 3.61157 0.156142
\(536\) 42.3185 1.82788
\(537\) 0 0
\(538\) 44.0977 1.90119
\(539\) −16.8775 −0.726967
\(540\) 0 0
\(541\) 29.7950 1.28099 0.640493 0.767964i \(-0.278730\pi\)
0.640493 + 0.767964i \(0.278730\pi\)
\(542\) 29.2727 1.25737
\(543\) 0 0
\(544\) −18.9611 −0.812952
\(545\) −4.78224 −0.204849
\(546\) 0 0
\(547\) −7.33226 −0.313505 −0.156752 0.987638i \(-0.550102\pi\)
−0.156752 + 0.987638i \(0.550102\pi\)
\(548\) 52.7953 2.25530
\(549\) 0 0
\(550\) −10.1810 −0.434117
\(551\) −6.77132 −0.288468
\(552\) 0 0
\(553\) −32.4890 −1.38157
\(554\) 29.7895 1.26563
\(555\) 0 0
\(556\) 17.7330 0.752048
\(557\) −28.4514 −1.20552 −0.602762 0.797921i \(-0.705933\pi\)
−0.602762 + 0.797921i \(0.705933\pi\)
\(558\) 0 0
\(559\) 18.1615 0.768149
\(560\) −33.1963 −1.40280
\(561\) 0 0
\(562\) 19.7558 0.833348
\(563\) 13.4623 0.567369 0.283684 0.958918i \(-0.408443\pi\)
0.283684 + 0.958918i \(0.408443\pi\)
\(564\) 0 0
\(565\) −4.33331 −0.182304
\(566\) 62.5687 2.62996
\(567\) 0 0
\(568\) 29.2858 1.22880
\(569\) 9.09616 0.381331 0.190666 0.981655i \(-0.438935\pi\)
0.190666 + 0.981655i \(0.438935\pi\)
\(570\) 0 0
\(571\) 46.0280 1.92621 0.963107 0.269120i \(-0.0867329\pi\)
0.963107 + 0.269120i \(0.0867329\pi\)
\(572\) −19.0483 −0.796448
\(573\) 0 0
\(574\) −57.7977 −2.41243
\(575\) −21.7189 −0.905741
\(576\) 0 0
\(577\) −7.77431 −0.323649 −0.161824 0.986820i \(-0.551738\pi\)
−0.161824 + 0.986820i \(0.551738\pi\)
\(578\) 6.03430 0.250994
\(579\) 0 0
\(580\) −10.1964 −0.423383
\(581\) 42.9110 1.78025
\(582\) 0 0
\(583\) −11.5324 −0.477625
\(584\) −98.7605 −4.08674
\(585\) 0 0
\(586\) −34.1559 −1.41097
\(587\) −37.6111 −1.55238 −0.776188 0.630501i \(-0.782849\pi\)
−0.776188 + 0.630501i \(0.782849\pi\)
\(588\) 0 0
\(589\) 20.3063 0.836706
\(590\) −12.9169 −0.531780
\(591\) 0 0
\(592\) 6.49075 0.266768
\(593\) −24.6250 −1.01123 −0.505614 0.862760i \(-0.668734\pi\)
−0.505614 + 0.862760i \(0.668734\pi\)
\(594\) 0 0
\(595\) −18.5744 −0.761477
\(596\) −81.2031 −3.32621
\(597\) 0 0
\(598\) −58.9394 −2.41021
\(599\) 22.6718 0.926343 0.463171 0.886269i \(-0.346711\pi\)
0.463171 + 0.886269i \(0.346711\pi\)
\(600\) 0 0
\(601\) −29.1465 −1.18891 −0.594455 0.804129i \(-0.702632\pi\)
−0.594455 + 0.804129i \(0.702632\pi\)
\(602\) −52.4960 −2.13958
\(603\) 0 0
\(604\) 50.0622 2.03700
\(605\) −0.994065 −0.0404145
\(606\) 0 0
\(607\) 8.63031 0.350293 0.175147 0.984542i \(-0.443960\pi\)
0.175147 + 0.984542i \(0.443960\pi\)
\(608\) 14.5341 0.589437
\(609\) 0 0
\(610\) −2.52267 −0.102140
\(611\) −35.5036 −1.43632
\(612\) 0 0
\(613\) −17.1543 −0.692854 −0.346427 0.938077i \(-0.612605\pi\)
−0.346427 + 0.938077i \(0.612605\pi\)
\(614\) −40.0884 −1.61783
\(615\) 0 0
\(616\) 30.2582 1.21914
\(617\) 43.1267 1.73622 0.868108 0.496376i \(-0.165336\pi\)
0.868108 + 0.496376i \(0.165336\pi\)
\(618\) 0 0
\(619\) −16.8162 −0.675902 −0.337951 0.941164i \(-0.609734\pi\)
−0.337951 + 0.941164i \(0.609734\pi\)
\(620\) 30.5777 1.22803
\(621\) 0 0
\(622\) 52.9189 2.12185
\(623\) −0.322798 −0.0129326
\(624\) 0 0
\(625\) 11.1540 0.446160
\(626\) −19.6433 −0.785103
\(627\) 0 0
\(628\) −74.0705 −2.95574
\(629\) 3.63178 0.144809
\(630\) 0 0
\(631\) 37.2478 1.48281 0.741406 0.671057i \(-0.234160\pi\)
0.741406 + 0.671057i \(0.234160\pi\)
\(632\) 41.1709 1.63769
\(633\) 0 0
\(634\) 8.55429 0.339734
\(635\) 20.4559 0.811766
\(636\) 0 0
\(637\) 72.4060 2.86883
\(638\) 5.86257 0.232101
\(639\) 0 0
\(640\) −12.5944 −0.497839
\(641\) 15.9557 0.630213 0.315107 0.949056i \(-0.397960\pi\)
0.315107 + 0.949056i \(0.397960\pi\)
\(642\) 0 0
\(643\) 35.2795 1.39129 0.695644 0.718387i \(-0.255119\pi\)
0.695644 + 0.718387i \(0.255119\pi\)
\(644\) 117.457 4.62846
\(645\) 0 0
\(646\) 28.4434 1.11909
\(647\) 25.5200 1.00329 0.501647 0.865072i \(-0.332727\pi\)
0.501647 + 0.865072i \(0.332727\pi\)
\(648\) 0 0
\(649\) 5.12033 0.200990
\(650\) 43.6771 1.71316
\(651\) 0 0
\(652\) 46.4767 1.82017
\(653\) 3.73660 0.146225 0.0731123 0.997324i \(-0.476707\pi\)
0.0731123 + 0.997324i \(0.476707\pi\)
\(654\) 0 0
\(655\) 8.86080 0.346220
\(656\) 31.8531 1.24365
\(657\) 0 0
\(658\) 102.624 4.00069
\(659\) 13.9968 0.545237 0.272618 0.962122i \(-0.412110\pi\)
0.272618 + 0.962122i \(0.412110\pi\)
\(660\) 0 0
\(661\) 16.3657 0.636551 0.318275 0.947998i \(-0.396896\pi\)
0.318275 + 0.947998i \(0.396896\pi\)
\(662\) −28.7464 −1.11726
\(663\) 0 0
\(664\) −54.3778 −2.11027
\(665\) 14.2377 0.552115
\(666\) 0 0
\(667\) 12.5065 0.484255
\(668\) 93.3053 3.61009
\(669\) 0 0
\(670\) −17.2402 −0.666047
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) 27.6477 1.06574 0.532871 0.846196i \(-0.321113\pi\)
0.532871 + 0.846196i \(0.321113\pi\)
\(674\) −69.8203 −2.68938
\(675\) 0 0
\(676\) 23.9977 0.922989
\(677\) −29.5915 −1.13729 −0.568647 0.822581i \(-0.692533\pi\)
−0.568647 + 0.822581i \(0.692533\pi\)
\(678\) 0 0
\(679\) 55.2250 2.11934
\(680\) 23.5379 0.902639
\(681\) 0 0
\(682\) −17.5811 −0.673214
\(683\) 4.11331 0.157391 0.0786957 0.996899i \(-0.474924\pi\)
0.0786957 + 0.996899i \(0.474924\pi\)
\(684\) 0 0
\(685\) −11.8201 −0.451622
\(686\) −122.487 −4.67656
\(687\) 0 0
\(688\) 28.9312 1.10299
\(689\) 49.4751 1.88485
\(690\) 0 0
\(691\) 34.3281 1.30590 0.652951 0.757400i \(-0.273531\pi\)
0.652951 + 0.757400i \(0.273531\pi\)
\(692\) −14.2970 −0.543489
\(693\) 0 0
\(694\) −18.9415 −0.719009
\(695\) −3.97016 −0.150597
\(696\) 0 0
\(697\) 17.8228 0.675088
\(698\) 21.9429 0.830551
\(699\) 0 0
\(700\) −87.0418 −3.28987
\(701\) 3.38074 0.127689 0.0638443 0.997960i \(-0.479664\pi\)
0.0638443 + 0.997960i \(0.479664\pi\)
\(702\) 0 0
\(703\) −2.78384 −0.104995
\(704\) 1.08461 0.0408779
\(705\) 0 0
\(706\) 85.8085 3.22945
\(707\) 33.2124 1.24908
\(708\) 0 0
\(709\) 28.0573 1.05371 0.526856 0.849954i \(-0.323371\pi\)
0.526856 + 0.849954i \(0.323371\pi\)
\(710\) −11.9308 −0.447754
\(711\) 0 0
\(712\) 0.409057 0.0153301
\(713\) −37.5055 −1.40459
\(714\) 0 0
\(715\) 4.26462 0.159488
\(716\) 32.2837 1.20650
\(717\) 0 0
\(718\) 12.6014 0.470280
\(719\) −0.0861753 −0.00321380 −0.00160690 0.999999i \(-0.500511\pi\)
−0.00160690 + 0.999999i \(0.500511\pi\)
\(720\) 0 0
\(721\) 59.4067 2.21242
\(722\) 26.4143 0.983040
\(723\) 0 0
\(724\) 59.5771 2.21416
\(725\) −9.26799 −0.344205
\(726\) 0 0
\(727\) −12.6578 −0.469453 −0.234727 0.972061i \(-0.575419\pi\)
−0.234727 + 0.972061i \(0.575419\pi\)
\(728\) −129.810 −4.81108
\(729\) 0 0
\(730\) 40.2342 1.48913
\(731\) 16.1880 0.598733
\(732\) 0 0
\(733\) 25.0980 0.927016 0.463508 0.886093i \(-0.346591\pi\)
0.463508 + 0.886093i \(0.346591\pi\)
\(734\) −71.8009 −2.65022
\(735\) 0 0
\(736\) −26.8444 −0.989497
\(737\) 6.83412 0.251738
\(738\) 0 0
\(739\) 31.4481 1.15684 0.578419 0.815740i \(-0.303670\pi\)
0.578419 + 0.815740i \(0.303670\pi\)
\(740\) −4.19197 −0.154100
\(741\) 0 0
\(742\) −143.008 −5.25000
\(743\) −2.21453 −0.0812431 −0.0406215 0.999175i \(-0.512934\pi\)
−0.0406215 + 0.999175i \(0.512934\pi\)
\(744\) 0 0
\(745\) 18.1802 0.666069
\(746\) 46.9535 1.71909
\(747\) 0 0
\(748\) −16.9784 −0.620790
\(749\) −17.7532 −0.648688
\(750\) 0 0
\(751\) 34.7061 1.26644 0.633222 0.773970i \(-0.281732\pi\)
0.633222 + 0.773970i \(0.281732\pi\)
\(752\) −56.5572 −2.06243
\(753\) 0 0
\(754\) −25.1509 −0.915942
\(755\) −11.2082 −0.407907
\(756\) 0 0
\(757\) −23.4680 −0.852960 −0.426480 0.904497i \(-0.640247\pi\)
−0.426480 + 0.904497i \(0.640247\pi\)
\(758\) −25.8247 −0.937995
\(759\) 0 0
\(760\) −18.0424 −0.654465
\(761\) 5.93914 0.215294 0.107647 0.994189i \(-0.465668\pi\)
0.107647 + 0.994189i \(0.465668\pi\)
\(762\) 0 0
\(763\) 23.5078 0.851038
\(764\) −71.9969 −2.60476
\(765\) 0 0
\(766\) −45.8220 −1.65562
\(767\) −21.9666 −0.793169
\(768\) 0 0
\(769\) −4.29907 −0.155028 −0.0775142 0.996991i \(-0.524698\pi\)
−0.0775142 + 0.996991i \(0.524698\pi\)
\(770\) −12.3269 −0.444231
\(771\) 0 0
\(772\) 68.7260 2.47350
\(773\) 19.6190 0.705646 0.352823 0.935690i \(-0.385222\pi\)
0.352823 + 0.935690i \(0.385222\pi\)
\(774\) 0 0
\(775\) 27.7935 0.998371
\(776\) −69.9824 −2.51222
\(777\) 0 0
\(778\) 32.7975 1.17585
\(779\) −13.6616 −0.489477
\(780\) 0 0
\(781\) 4.72943 0.169232
\(782\) −52.5347 −1.87864
\(783\) 0 0
\(784\) 115.343 4.11938
\(785\) 16.5833 0.591882
\(786\) 0 0
\(787\) −19.9085 −0.709662 −0.354831 0.934931i \(-0.615462\pi\)
−0.354831 + 0.934931i \(0.615462\pi\)
\(788\) 78.7490 2.80532
\(789\) 0 0
\(790\) −16.7727 −0.596744
\(791\) 21.3010 0.757376
\(792\) 0 0
\(793\) −4.29008 −0.152345
\(794\) −64.1138 −2.27531
\(795\) 0 0
\(796\) −10.7857 −0.382290
\(797\) 31.1383 1.10298 0.551488 0.834183i \(-0.314060\pi\)
0.551488 + 0.834183i \(0.314060\pi\)
\(798\) 0 0
\(799\) −31.6456 −1.11954
\(800\) 19.8931 0.703326
\(801\) 0 0
\(802\) −84.8274 −2.99536
\(803\) −15.9491 −0.562830
\(804\) 0 0
\(805\) −26.2969 −0.926843
\(806\) 75.4242 2.65670
\(807\) 0 0
\(808\) −42.0876 −1.48064
\(809\) −28.9916 −1.01929 −0.509645 0.860385i \(-0.670223\pi\)
−0.509645 + 0.860385i \(0.670223\pi\)
\(810\) 0 0
\(811\) 52.5821 1.84641 0.923204 0.384309i \(-0.125560\pi\)
0.923204 + 0.384309i \(0.125560\pi\)
\(812\) 50.1219 1.75893
\(813\) 0 0
\(814\) 2.41023 0.0844787
\(815\) −10.4054 −0.364486
\(816\) 0 0
\(817\) −12.4084 −0.434116
\(818\) 8.25684 0.288693
\(819\) 0 0
\(820\) −20.5719 −0.718403
\(821\) 3.62268 0.126432 0.0632162 0.998000i \(-0.479864\pi\)
0.0632162 + 0.998000i \(0.479864\pi\)
\(822\) 0 0
\(823\) 52.2870 1.82261 0.911306 0.411730i \(-0.135075\pi\)
0.911306 + 0.411730i \(0.135075\pi\)
\(824\) −75.2815 −2.62256
\(825\) 0 0
\(826\) 63.4947 2.20926
\(827\) 21.6441 0.752637 0.376319 0.926490i \(-0.377190\pi\)
0.376319 + 0.926490i \(0.377190\pi\)
\(828\) 0 0
\(829\) 23.1504 0.804048 0.402024 0.915629i \(-0.368307\pi\)
0.402024 + 0.915629i \(0.368307\pi\)
\(830\) 22.1531 0.768944
\(831\) 0 0
\(832\) −4.65308 −0.161317
\(833\) 64.5379 2.23611
\(834\) 0 0
\(835\) −20.8896 −0.722916
\(836\) 13.0143 0.450109
\(837\) 0 0
\(838\) −89.5274 −3.09267
\(839\) −45.0573 −1.55555 −0.777776 0.628541i \(-0.783652\pi\)
−0.777776 + 0.628541i \(0.783652\pi\)
\(840\) 0 0
\(841\) −23.6631 −0.815971
\(842\) −18.4414 −0.635533
\(843\) 0 0
\(844\) −104.261 −3.58880
\(845\) −5.37273 −0.184827
\(846\) 0 0
\(847\) 4.88646 0.167901
\(848\) 78.8137 2.70647
\(849\) 0 0
\(850\) 38.9309 1.33532
\(851\) 5.14172 0.176256
\(852\) 0 0
\(853\) −25.4763 −0.872291 −0.436146 0.899876i \(-0.643657\pi\)
−0.436146 + 0.899876i \(0.643657\pi\)
\(854\) 12.4005 0.424337
\(855\) 0 0
\(856\) 22.4973 0.768941
\(857\) 19.0519 0.650802 0.325401 0.945576i \(-0.394501\pi\)
0.325401 + 0.945576i \(0.394501\pi\)
\(858\) 0 0
\(859\) −0.566950 −0.0193441 −0.00967204 0.999953i \(-0.503079\pi\)
−0.00967204 + 0.999953i \(0.503079\pi\)
\(860\) −18.6849 −0.637150
\(861\) 0 0
\(862\) −19.7315 −0.672056
\(863\) −5.04043 −0.171578 −0.0857891 0.996313i \(-0.527341\pi\)
−0.0857891 + 0.996313i \(0.527341\pi\)
\(864\) 0 0
\(865\) 3.20087 0.108833
\(866\) 90.6427 3.08017
\(867\) 0 0
\(868\) −150.309 −5.10181
\(869\) 6.64878 0.225544
\(870\) 0 0
\(871\) −29.3189 −0.993434
\(872\) −29.7896 −1.00880
\(873\) 0 0
\(874\) 40.2690 1.36212
\(875\) 43.7746 1.47985
\(876\) 0 0
\(877\) 21.1269 0.713406 0.356703 0.934218i \(-0.383901\pi\)
0.356703 + 0.934218i \(0.383901\pi\)
\(878\) −40.1021 −1.35338
\(879\) 0 0
\(880\) 6.79353 0.229010
\(881\) 43.4917 1.46527 0.732635 0.680621i \(-0.238290\pi\)
0.732635 + 0.680621i \(0.238290\pi\)
\(882\) 0 0
\(883\) −11.2005 −0.376925 −0.188463 0.982080i \(-0.560350\pi\)
−0.188463 + 0.982080i \(0.560350\pi\)
\(884\) 72.8386 2.44983
\(885\) 0 0
\(886\) −5.59541 −0.187981
\(887\) 36.9615 1.24105 0.620523 0.784188i \(-0.286920\pi\)
0.620523 + 0.784188i \(0.286920\pi\)
\(888\) 0 0
\(889\) −100.554 −3.37246
\(890\) −0.166646 −0.00558600
\(891\) 0 0
\(892\) 87.9139 2.94358
\(893\) 24.2571 0.811731
\(894\) 0 0
\(895\) −7.22784 −0.241600
\(896\) 61.9098 2.06826
\(897\) 0 0
\(898\) −28.0364 −0.935587
\(899\) −16.0045 −0.533780
\(900\) 0 0
\(901\) 44.0988 1.46914
\(902\) 11.8281 0.393834
\(903\) 0 0
\(904\) −26.9931 −0.897778
\(905\) −13.3384 −0.443384
\(906\) 0 0
\(907\) 52.8559 1.75505 0.877526 0.479529i \(-0.159193\pi\)
0.877526 + 0.479529i \(0.159193\pi\)
\(908\) −98.8832 −3.28155
\(909\) 0 0
\(910\) 52.8835 1.75307
\(911\) 6.97550 0.231109 0.115554 0.993301i \(-0.463136\pi\)
0.115554 + 0.993301i \(0.463136\pi\)
\(912\) 0 0
\(913\) −8.78161 −0.290629
\(914\) 82.0619 2.71437
\(915\) 0 0
\(916\) 20.1668 0.666331
\(917\) −43.5565 −1.43836
\(918\) 0 0
\(919\) −33.6287 −1.10931 −0.554655 0.832081i \(-0.687150\pi\)
−0.554655 + 0.832081i \(0.687150\pi\)
\(920\) 33.3240 1.09866
\(921\) 0 0
\(922\) −39.8398 −1.31205
\(923\) −20.2897 −0.667842
\(924\) 0 0
\(925\) −3.81028 −0.125281
\(926\) 13.2458 0.435282
\(927\) 0 0
\(928\) −11.4552 −0.376034
\(929\) 6.54384 0.214696 0.107348 0.994221i \(-0.465764\pi\)
0.107348 + 0.994221i \(0.465764\pi\)
\(930\) 0 0
\(931\) −49.4697 −1.62131
\(932\) 40.4276 1.32425
\(933\) 0 0
\(934\) 96.6986 3.16408
\(935\) 3.80120 0.124312
\(936\) 0 0
\(937\) −35.2517 −1.15162 −0.575811 0.817583i \(-0.695314\pi\)
−0.575811 + 0.817583i \(0.695314\pi\)
\(938\) 84.7466 2.76708
\(939\) 0 0
\(940\) 36.5268 1.19137
\(941\) −2.92372 −0.0953107 −0.0476553 0.998864i \(-0.515175\pi\)
−0.0476553 + 0.998864i \(0.515175\pi\)
\(942\) 0 0
\(943\) 25.2328 0.821693
\(944\) −34.9928 −1.13892
\(945\) 0 0
\(946\) 10.7431 0.349290
\(947\) 23.8069 0.773621 0.386811 0.922159i \(-0.373577\pi\)
0.386811 + 0.922159i \(0.373577\pi\)
\(948\) 0 0
\(949\) 68.4228 2.22110
\(950\) −29.8414 −0.968183
\(951\) 0 0
\(952\) −115.704 −3.74999
\(953\) 43.2272 1.40027 0.700133 0.714012i \(-0.253124\pi\)
0.700133 + 0.714012i \(0.253124\pi\)
\(954\) 0 0
\(955\) 16.1190 0.521599
\(956\) 34.0214 1.10033
\(957\) 0 0
\(958\) −35.4127 −1.14413
\(959\) 58.1032 1.87625
\(960\) 0 0
\(961\) 16.9954 0.548239
\(962\) −10.3401 −0.333378
\(963\) 0 0
\(964\) −14.7926 −0.476438
\(965\) −15.3867 −0.495315
\(966\) 0 0
\(967\) −4.95474 −0.159334 −0.0796669 0.996822i \(-0.525386\pi\)
−0.0796669 + 0.996822i \(0.525386\pi\)
\(968\) −6.19224 −0.199026
\(969\) 0 0
\(970\) 28.5103 0.915409
\(971\) −59.7469 −1.91737 −0.958685 0.284471i \(-0.908182\pi\)
−0.958685 + 0.284471i \(0.908182\pi\)
\(972\) 0 0
\(973\) 19.5159 0.625650
\(974\) 72.4791 2.32238
\(975\) 0 0
\(976\) −6.83409 −0.218754
\(977\) 34.4068 1.10077 0.550386 0.834911i \(-0.314481\pi\)
0.550386 + 0.834911i \(0.314481\pi\)
\(978\) 0 0
\(979\) 0.0660596 0.00211127
\(980\) −74.4927 −2.37958
\(981\) 0 0
\(982\) 10.0790 0.321633
\(983\) 25.9260 0.826912 0.413456 0.910524i \(-0.364321\pi\)
0.413456 + 0.910524i \(0.364321\pi\)
\(984\) 0 0
\(985\) −17.6307 −0.561761
\(986\) −22.4178 −0.713929
\(987\) 0 0
\(988\) −55.8324 −1.77627
\(989\) 22.9182 0.728757
\(990\) 0 0
\(991\) −43.2573 −1.37411 −0.687057 0.726604i \(-0.741098\pi\)
−0.687057 + 0.726604i \(0.741098\pi\)
\(992\) 34.3525 1.09069
\(993\) 0 0
\(994\) 58.6474 1.86018
\(995\) 2.41476 0.0765531
\(996\) 0 0
\(997\) 4.53605 0.143658 0.0718291 0.997417i \(-0.477116\pi\)
0.0718291 + 0.997417i \(0.477116\pi\)
\(998\) 61.2454 1.93869
\(999\) 0 0
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 6039.2.a.i.1.2 13
3.2 odd 2 2013.2.a.e.1.12 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.e.1.12 13 3.2 odd 2
6039.2.a.i.1.2 13 1.1 even 1 trivial