Properties

Label 6039.2.a.i.1.4
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 1.4
Root \(1.61181\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.61181 q^{2} +0.597924 q^{4} -4.27930 q^{5} +1.98799 q^{7} +2.25988 q^{8} +O(q^{10})\) \(q-1.61181 q^{2} +0.597924 q^{4} -4.27930 q^{5} +1.98799 q^{7} +2.25988 q^{8} +6.89741 q^{10} -1.00000 q^{11} +0.250746 q^{13} -3.20426 q^{14} -4.83834 q^{16} -3.17083 q^{17} -3.76446 q^{19} -2.55870 q^{20} +1.61181 q^{22} -8.19125 q^{23} +13.3124 q^{25} -0.404154 q^{26} +1.18867 q^{28} +4.02069 q^{29} +2.98063 q^{31} +3.27871 q^{32} +5.11077 q^{34} -8.50721 q^{35} -8.39318 q^{37} +6.06759 q^{38} -9.67069 q^{40} +2.30573 q^{41} +7.83567 q^{43} -0.597924 q^{44} +13.2027 q^{46} -0.852827 q^{47} -3.04789 q^{49} -21.4571 q^{50} +0.149927 q^{52} +7.07558 q^{53} +4.27930 q^{55} +4.49261 q^{56} -6.48058 q^{58} -9.77116 q^{59} -1.00000 q^{61} -4.80420 q^{62} +4.39201 q^{64} -1.07302 q^{65} -13.1758 q^{67} -1.89592 q^{68} +13.7120 q^{70} -2.07157 q^{71} -1.20929 q^{73} +13.5282 q^{74} -2.25086 q^{76} -1.98799 q^{77} +2.92819 q^{79} +20.7047 q^{80} -3.71639 q^{82} -12.1817 q^{83} +13.5689 q^{85} -12.6296 q^{86} -2.25988 q^{88} +9.78144 q^{89} +0.498480 q^{91} -4.89775 q^{92} +1.37459 q^{94} +16.1093 q^{95} +3.01824 q^{97} +4.91261 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + O(q^{10}) \) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + 6q^{10} - 13q^{11} + 13q^{13} - q^{14} + 18q^{16} - 17q^{17} + 14q^{19} + 7q^{20} + 2q^{22} - 7q^{23} + 18q^{25} + 10q^{26} + 19q^{28} + 6q^{29} + 27q^{31} - 5q^{32} + 6q^{34} - 14q^{35} + 10q^{37} - 2q^{38} + 8q^{40} - 3q^{41} + 29q^{43} - 16q^{44} - 24q^{46} - 8q^{47} + 8q^{49} + 27q^{50} + 37q^{52} + 24q^{53} + 3q^{55} - 24q^{56} - 5q^{58} - 13q^{59} - 13q^{61} - 39q^{62} + 47q^{64} + 11q^{65} + 44q^{67} + 8q^{68} - 12q^{70} - 3q^{71} + 48q^{73} + 22q^{74} + 47q^{76} - 11q^{77} - 17q^{79} + 26q^{80} + 56q^{82} - 50q^{83} + 8q^{85} - 18q^{86} + 9q^{88} + 15q^{89} + 47q^{91} - 14q^{92} + 45q^{94} + q^{95} + 27q^{97} - 47q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61181 −1.13972 −0.569860 0.821742i \(-0.693003\pi\)
−0.569860 + 0.821742i \(0.693003\pi\)
\(3\) 0 0
\(4\) 0.597924 0.298962
\(5\) −4.27930 −1.91376 −0.956881 0.290481i \(-0.906185\pi\)
−0.956881 + 0.290481i \(0.906185\pi\)
\(6\) 0 0
\(7\) 1.98799 0.751390 0.375695 0.926743i \(-0.377404\pi\)
0.375695 + 0.926743i \(0.377404\pi\)
\(8\) 2.25988 0.798987
\(9\) 0 0
\(10\) 6.89741 2.18115
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.250746 0.0695444 0.0347722 0.999395i \(-0.488929\pi\)
0.0347722 + 0.999395i \(0.488929\pi\)
\(14\) −3.20426 −0.856374
\(15\) 0 0
\(16\) −4.83834 −1.20958
\(17\) −3.17083 −0.769040 −0.384520 0.923117i \(-0.625633\pi\)
−0.384520 + 0.923117i \(0.625633\pi\)
\(18\) 0 0
\(19\) −3.76446 −0.863627 −0.431814 0.901963i \(-0.642126\pi\)
−0.431814 + 0.901963i \(0.642126\pi\)
\(20\) −2.55870 −0.572142
\(21\) 0 0
\(22\) 1.61181 0.343639
\(23\) −8.19125 −1.70799 −0.853997 0.520278i \(-0.825828\pi\)
−0.853997 + 0.520278i \(0.825828\pi\)
\(24\) 0 0
\(25\) 13.3124 2.66248
\(26\) −0.404154 −0.0792611
\(27\) 0 0
\(28\) 1.18867 0.224637
\(29\) 4.02069 0.746623 0.373312 0.927706i \(-0.378222\pi\)
0.373312 + 0.927706i \(0.378222\pi\)
\(30\) 0 0
\(31\) 2.98063 0.535337 0.267669 0.963511i \(-0.413747\pi\)
0.267669 + 0.963511i \(0.413747\pi\)
\(32\) 3.27871 0.579600
\(33\) 0 0
\(34\) 5.11077 0.876490
\(35\) −8.50721 −1.43798
\(36\) 0 0
\(37\) −8.39318 −1.37983 −0.689916 0.723890i \(-0.742352\pi\)
−0.689916 + 0.723890i \(0.742352\pi\)
\(38\) 6.06759 0.984293
\(39\) 0 0
\(40\) −9.67069 −1.52907
\(41\) 2.30573 0.360094 0.180047 0.983658i \(-0.442375\pi\)
0.180047 + 0.983658i \(0.442375\pi\)
\(42\) 0 0
\(43\) 7.83567 1.19493 0.597464 0.801896i \(-0.296175\pi\)
0.597464 + 0.801896i \(0.296175\pi\)
\(44\) −0.597924 −0.0901405
\(45\) 0 0
\(46\) 13.2027 1.94663
\(47\) −0.852827 −0.124398 −0.0621988 0.998064i \(-0.519811\pi\)
−0.0621988 + 0.998064i \(0.519811\pi\)
\(48\) 0 0
\(49\) −3.04789 −0.435413
\(50\) −21.4571 −3.03449
\(51\) 0 0
\(52\) 0.149927 0.0207911
\(53\) 7.07558 0.971906 0.485953 0.873985i \(-0.338473\pi\)
0.485953 + 0.873985i \(0.338473\pi\)
\(54\) 0 0
\(55\) 4.27930 0.577021
\(56\) 4.49261 0.600351
\(57\) 0 0
\(58\) −6.48058 −0.850942
\(59\) −9.77116 −1.27210 −0.636048 0.771649i \(-0.719432\pi\)
−0.636048 + 0.771649i \(0.719432\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) −4.80420 −0.610135
\(63\) 0 0
\(64\) 4.39201 0.549002
\(65\) −1.07302 −0.133091
\(66\) 0 0
\(67\) −13.1758 −1.60968 −0.804841 0.593490i \(-0.797750\pi\)
−0.804841 + 0.593490i \(0.797750\pi\)
\(68\) −1.89592 −0.229914
\(69\) 0 0
\(70\) 13.7120 1.63890
\(71\) −2.07157 −0.245850 −0.122925 0.992416i \(-0.539227\pi\)
−0.122925 + 0.992416i \(0.539227\pi\)
\(72\) 0 0
\(73\) −1.20929 −0.141537 −0.0707686 0.997493i \(-0.522545\pi\)
−0.0707686 + 0.997493i \(0.522545\pi\)
\(74\) 13.5282 1.57262
\(75\) 0 0
\(76\) −2.25086 −0.258192
\(77\) −1.98799 −0.226553
\(78\) 0 0
\(79\) 2.92819 0.329447 0.164723 0.986340i \(-0.447327\pi\)
0.164723 + 0.986340i \(0.447327\pi\)
\(80\) 20.7047 2.31486
\(81\) 0 0
\(82\) −3.71639 −0.410407
\(83\) −12.1817 −1.33712 −0.668558 0.743660i \(-0.733088\pi\)
−0.668558 + 0.743660i \(0.733088\pi\)
\(84\) 0 0
\(85\) 13.5689 1.47176
\(86\) −12.6296 −1.36188
\(87\) 0 0
\(88\) −2.25988 −0.240904
\(89\) 9.78144 1.03683 0.518415 0.855129i \(-0.326522\pi\)
0.518415 + 0.855129i \(0.326522\pi\)
\(90\) 0 0
\(91\) 0.498480 0.0522549
\(92\) −4.89775 −0.510626
\(93\) 0 0
\(94\) 1.37459 0.141778
\(95\) 16.1093 1.65278
\(96\) 0 0
\(97\) 3.01824 0.306456 0.153228 0.988191i \(-0.451033\pi\)
0.153228 + 0.988191i \(0.451033\pi\)
\(98\) 4.91261 0.496249
\(99\) 0 0
\(100\) 7.95982 0.795982
\(101\) −5.62832 −0.560039 −0.280020 0.959994i \(-0.590341\pi\)
−0.280020 + 0.959994i \(0.590341\pi\)
\(102\) 0 0
\(103\) −16.3398 −1.61001 −0.805003 0.593270i \(-0.797837\pi\)
−0.805003 + 0.593270i \(0.797837\pi\)
\(104\) 0.566654 0.0555650
\(105\) 0 0
\(106\) −11.4045 −1.10770
\(107\) −13.4737 −1.30255 −0.651276 0.758841i \(-0.725766\pi\)
−0.651276 + 0.758841i \(0.725766\pi\)
\(108\) 0 0
\(109\) 13.5673 1.29951 0.649757 0.760142i \(-0.274871\pi\)
0.649757 + 0.760142i \(0.274871\pi\)
\(110\) −6.89741 −0.657642
\(111\) 0 0
\(112\) −9.61857 −0.908869
\(113\) 0.0631308 0.00593884 0.00296942 0.999996i \(-0.499055\pi\)
0.00296942 + 0.999996i \(0.499055\pi\)
\(114\) 0 0
\(115\) 35.0528 3.26869
\(116\) 2.40407 0.223212
\(117\) 0 0
\(118\) 15.7492 1.44983
\(119\) −6.30359 −0.577849
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 1.61181 0.145926
\(123\) 0 0
\(124\) 1.78219 0.160046
\(125\) −35.5713 −3.18160
\(126\) 0 0
\(127\) 5.04839 0.447972 0.223986 0.974592i \(-0.428093\pi\)
0.223986 + 0.974592i \(0.428093\pi\)
\(128\) −13.6365 −1.20531
\(129\) 0 0
\(130\) 1.72950 0.151687
\(131\) −19.1064 −1.66933 −0.834665 0.550757i \(-0.814339\pi\)
−0.834665 + 0.550757i \(0.814339\pi\)
\(132\) 0 0
\(133\) −7.48372 −0.648921
\(134\) 21.2369 1.83459
\(135\) 0 0
\(136\) −7.16569 −0.614453
\(137\) 8.39986 0.717649 0.358824 0.933405i \(-0.383178\pi\)
0.358824 + 0.933405i \(0.383178\pi\)
\(138\) 0 0
\(139\) 7.05747 0.598607 0.299303 0.954158i \(-0.403246\pi\)
0.299303 + 0.954158i \(0.403246\pi\)
\(140\) −5.08667 −0.429902
\(141\) 0 0
\(142\) 3.33897 0.280200
\(143\) −0.250746 −0.0209684
\(144\) 0 0
\(145\) −17.2057 −1.42886
\(146\) 1.94915 0.161313
\(147\) 0 0
\(148\) −5.01849 −0.412517
\(149\) 19.9863 1.63734 0.818672 0.574262i \(-0.194711\pi\)
0.818672 + 0.574262i \(0.194711\pi\)
\(150\) 0 0
\(151\) −15.0434 −1.22421 −0.612107 0.790775i \(-0.709678\pi\)
−0.612107 + 0.790775i \(0.709678\pi\)
\(152\) −8.50722 −0.690027
\(153\) 0 0
\(154\) 3.20426 0.258207
\(155\) −12.7550 −1.02451
\(156\) 0 0
\(157\) −14.2863 −1.14017 −0.570086 0.821585i \(-0.693090\pi\)
−0.570086 + 0.821585i \(0.693090\pi\)
\(158\) −4.71967 −0.375477
\(159\) 0 0
\(160\) −14.0306 −1.10922
\(161\) −16.2841 −1.28337
\(162\) 0 0
\(163\) −12.0009 −0.939983 −0.469991 0.882671i \(-0.655743\pi\)
−0.469991 + 0.882671i \(0.655743\pi\)
\(164\) 1.37865 0.107655
\(165\) 0 0
\(166\) 19.6346 1.52394
\(167\) 17.9733 1.39082 0.695408 0.718615i \(-0.255224\pi\)
0.695408 + 0.718615i \(0.255224\pi\)
\(168\) 0 0
\(169\) −12.9371 −0.995164
\(170\) −21.8705 −1.67739
\(171\) 0 0
\(172\) 4.68514 0.357238
\(173\) 5.97151 0.454006 0.227003 0.973894i \(-0.427107\pi\)
0.227003 + 0.973894i \(0.427107\pi\)
\(174\) 0 0
\(175\) 26.4650 2.00056
\(176\) 4.83834 0.364703
\(177\) 0 0
\(178\) −15.7658 −1.18170
\(179\) −22.5218 −1.68336 −0.841678 0.539980i \(-0.818432\pi\)
−0.841678 + 0.539980i \(0.818432\pi\)
\(180\) 0 0
\(181\) −16.7687 −1.24641 −0.623205 0.782058i \(-0.714170\pi\)
−0.623205 + 0.782058i \(0.714170\pi\)
\(182\) −0.803455 −0.0595560
\(183\) 0 0
\(184\) −18.5112 −1.36466
\(185\) 35.9170 2.64067
\(186\) 0 0
\(187\) 3.17083 0.231874
\(188\) −0.509926 −0.0371902
\(189\) 0 0
\(190\) −25.9651 −1.88370
\(191\) −23.1333 −1.67386 −0.836932 0.547307i \(-0.815653\pi\)
−0.836932 + 0.547307i \(0.815653\pi\)
\(192\) 0 0
\(193\) 1.54885 0.111489 0.0557443 0.998445i \(-0.482247\pi\)
0.0557443 + 0.998445i \(0.482247\pi\)
\(194\) −4.86482 −0.349274
\(195\) 0 0
\(196\) −1.82241 −0.130172
\(197\) 3.90674 0.278344 0.139172 0.990268i \(-0.455556\pi\)
0.139172 + 0.990268i \(0.455556\pi\)
\(198\) 0 0
\(199\) −11.2931 −0.800544 −0.400272 0.916396i \(-0.631084\pi\)
−0.400272 + 0.916396i \(0.631084\pi\)
\(200\) 30.0844 2.12729
\(201\) 0 0
\(202\) 9.07178 0.638288
\(203\) 7.99309 0.561005
\(204\) 0 0
\(205\) −9.86690 −0.689134
\(206\) 26.3366 1.83496
\(207\) 0 0
\(208\) −1.21319 −0.0841197
\(209\) 3.76446 0.260393
\(210\) 0 0
\(211\) 17.2658 1.18863 0.594314 0.804233i \(-0.297424\pi\)
0.594314 + 0.804233i \(0.297424\pi\)
\(212\) 4.23066 0.290563
\(213\) 0 0
\(214\) 21.7170 1.48454
\(215\) −33.5312 −2.28681
\(216\) 0 0
\(217\) 5.92547 0.402247
\(218\) −21.8679 −1.48108
\(219\) 0 0
\(220\) 2.55870 0.172507
\(221\) −0.795073 −0.0534824
\(222\) 0 0
\(223\) −28.3241 −1.89672 −0.948362 0.317191i \(-0.897260\pi\)
−0.948362 + 0.317191i \(0.897260\pi\)
\(224\) 6.51805 0.435506
\(225\) 0 0
\(226\) −0.101755 −0.00676862
\(227\) 14.3403 0.951800 0.475900 0.879500i \(-0.342123\pi\)
0.475900 + 0.879500i \(0.342123\pi\)
\(228\) 0 0
\(229\) −9.21589 −0.609003 −0.304502 0.952512i \(-0.598490\pi\)
−0.304502 + 0.952512i \(0.598490\pi\)
\(230\) −56.4984 −3.72540
\(231\) 0 0
\(232\) 9.08626 0.596542
\(233\) 25.2945 1.65710 0.828550 0.559915i \(-0.189166\pi\)
0.828550 + 0.559915i \(0.189166\pi\)
\(234\) 0 0
\(235\) 3.64950 0.238067
\(236\) −5.84241 −0.380309
\(237\) 0 0
\(238\) 10.1602 0.658586
\(239\) 19.2893 1.24772 0.623860 0.781536i \(-0.285563\pi\)
0.623860 + 0.781536i \(0.285563\pi\)
\(240\) 0 0
\(241\) 12.4108 0.799448 0.399724 0.916636i \(-0.369106\pi\)
0.399724 + 0.916636i \(0.369106\pi\)
\(242\) −1.61181 −0.103611
\(243\) 0 0
\(244\) −0.597924 −0.0382782
\(245\) 13.0428 0.833277
\(246\) 0 0
\(247\) −0.943924 −0.0600604
\(248\) 6.73586 0.427727
\(249\) 0 0
\(250\) 57.3342 3.62613
\(251\) 13.9303 0.879270 0.439635 0.898176i \(-0.355108\pi\)
0.439635 + 0.898176i \(0.355108\pi\)
\(252\) 0 0
\(253\) 8.19125 0.514979
\(254\) −8.13703 −0.510562
\(255\) 0 0
\(256\) 13.1954 0.824713
\(257\) 8.15779 0.508869 0.254434 0.967090i \(-0.418111\pi\)
0.254434 + 0.967090i \(0.418111\pi\)
\(258\) 0 0
\(259\) −16.6856 −1.03679
\(260\) −0.641583 −0.0397893
\(261\) 0 0
\(262\) 30.7958 1.90257
\(263\) 16.6378 1.02593 0.512966 0.858409i \(-0.328547\pi\)
0.512966 + 0.858409i \(0.328547\pi\)
\(264\) 0 0
\(265\) −30.2785 −1.86000
\(266\) 12.0623 0.739588
\(267\) 0 0
\(268\) −7.87815 −0.481234
\(269\) −1.23389 −0.0752315 −0.0376158 0.999292i \(-0.511976\pi\)
−0.0376158 + 0.999292i \(0.511976\pi\)
\(270\) 0 0
\(271\) 4.48114 0.272210 0.136105 0.990694i \(-0.456542\pi\)
0.136105 + 0.990694i \(0.456542\pi\)
\(272\) 15.3415 0.930218
\(273\) 0 0
\(274\) −13.5390 −0.817919
\(275\) −13.3124 −0.802769
\(276\) 0 0
\(277\) 22.3235 1.34129 0.670644 0.741779i \(-0.266018\pi\)
0.670644 + 0.741779i \(0.266018\pi\)
\(278\) −11.3753 −0.682244
\(279\) 0 0
\(280\) −19.2252 −1.14893
\(281\) −1.20667 −0.0719840 −0.0359920 0.999352i \(-0.511459\pi\)
−0.0359920 + 0.999352i \(0.511459\pi\)
\(282\) 0 0
\(283\) −12.8779 −0.765509 −0.382755 0.923850i \(-0.625025\pi\)
−0.382755 + 0.923850i \(0.625025\pi\)
\(284\) −1.23864 −0.0734997
\(285\) 0 0
\(286\) 0.404154 0.0238981
\(287\) 4.58377 0.270571
\(288\) 0 0
\(289\) −6.94583 −0.408578
\(290\) 27.7323 1.62850
\(291\) 0 0
\(292\) −0.723066 −0.0423142
\(293\) −3.55779 −0.207848 −0.103924 0.994585i \(-0.533140\pi\)
−0.103924 + 0.994585i \(0.533140\pi\)
\(294\) 0 0
\(295\) 41.8137 2.43449
\(296\) −18.9676 −1.10247
\(297\) 0 0
\(298\) −32.2141 −1.86611
\(299\) −2.05392 −0.118781
\(300\) 0 0
\(301\) 15.5772 0.897857
\(302\) 24.2471 1.39526
\(303\) 0 0
\(304\) 18.2137 1.04463
\(305\) 4.27930 0.245032
\(306\) 0 0
\(307\) 15.7013 0.896118 0.448059 0.894004i \(-0.352115\pi\)
0.448059 + 0.894004i \(0.352115\pi\)
\(308\) −1.18867 −0.0677307
\(309\) 0 0
\(310\) 20.5586 1.16765
\(311\) 20.3629 1.15467 0.577336 0.816507i \(-0.304092\pi\)
0.577336 + 0.816507i \(0.304092\pi\)
\(312\) 0 0
\(313\) 5.37471 0.303797 0.151898 0.988396i \(-0.451461\pi\)
0.151898 + 0.988396i \(0.451461\pi\)
\(314\) 23.0268 1.29948
\(315\) 0 0
\(316\) 1.75083 0.0984921
\(317\) 3.60000 0.202196 0.101098 0.994876i \(-0.467764\pi\)
0.101098 + 0.994876i \(0.467764\pi\)
\(318\) 0 0
\(319\) −4.02069 −0.225115
\(320\) −18.7948 −1.05066
\(321\) 0 0
\(322\) 26.2469 1.46268
\(323\) 11.9365 0.664164
\(324\) 0 0
\(325\) 3.33803 0.185161
\(326\) 19.3431 1.07132
\(327\) 0 0
\(328\) 5.21066 0.287710
\(329\) −1.69541 −0.0934711
\(330\) 0 0
\(331\) 35.9801 1.97765 0.988823 0.149092i \(-0.0476351\pi\)
0.988823 + 0.149092i \(0.0476351\pi\)
\(332\) −7.28374 −0.399747
\(333\) 0 0
\(334\) −28.9695 −1.58514
\(335\) 56.3833 3.08055
\(336\) 0 0
\(337\) 4.79883 0.261409 0.130704 0.991421i \(-0.458276\pi\)
0.130704 + 0.991421i \(0.458276\pi\)
\(338\) 20.8522 1.13421
\(339\) 0 0
\(340\) 8.11320 0.440000
\(341\) −2.98063 −0.161410
\(342\) 0 0
\(343\) −19.9751 −1.07856
\(344\) 17.7076 0.954732
\(345\) 0 0
\(346\) −9.62493 −0.517439
\(347\) −3.83919 −0.206099 −0.103049 0.994676i \(-0.532860\pi\)
−0.103049 + 0.994676i \(0.532860\pi\)
\(348\) 0 0
\(349\) 0.105346 0.00563902 0.00281951 0.999996i \(-0.499103\pi\)
0.00281951 + 0.999996i \(0.499103\pi\)
\(350\) −42.6564 −2.28008
\(351\) 0 0
\(352\) −3.27871 −0.174756
\(353\) 34.3508 1.82831 0.914155 0.405364i \(-0.132855\pi\)
0.914155 + 0.405364i \(0.132855\pi\)
\(354\) 0 0
\(355\) 8.86485 0.470497
\(356\) 5.84856 0.309973
\(357\) 0 0
\(358\) 36.3007 1.91855
\(359\) 13.8877 0.732965 0.366483 0.930425i \(-0.380562\pi\)
0.366483 + 0.930425i \(0.380562\pi\)
\(360\) 0 0
\(361\) −4.82881 −0.254148
\(362\) 27.0280 1.42056
\(363\) 0 0
\(364\) 0.298054 0.0156223
\(365\) 5.17493 0.270868
\(366\) 0 0
\(367\) −12.2286 −0.638330 −0.319165 0.947699i \(-0.603402\pi\)
−0.319165 + 0.947699i \(0.603402\pi\)
\(368\) 39.6320 2.06596
\(369\) 0 0
\(370\) −57.8912 −3.00962
\(371\) 14.0662 0.730280
\(372\) 0 0
\(373\) 24.3245 1.25948 0.629738 0.776808i \(-0.283162\pi\)
0.629738 + 0.776808i \(0.283162\pi\)
\(374\) −5.11077 −0.264272
\(375\) 0 0
\(376\) −1.92728 −0.0993921
\(377\) 1.00817 0.0519234
\(378\) 0 0
\(379\) 2.33650 0.120018 0.0600090 0.998198i \(-0.480887\pi\)
0.0600090 + 0.998198i \(0.480887\pi\)
\(380\) 9.63213 0.494118
\(381\) 0 0
\(382\) 37.2864 1.90774
\(383\) 29.8315 1.52432 0.762160 0.647389i \(-0.224139\pi\)
0.762160 + 0.647389i \(0.224139\pi\)
\(384\) 0 0
\(385\) 8.50721 0.433568
\(386\) −2.49645 −0.127066
\(387\) 0 0
\(388\) 1.80468 0.0916187
\(389\) −4.92145 −0.249528 −0.124764 0.992186i \(-0.539817\pi\)
−0.124764 + 0.992186i \(0.539817\pi\)
\(390\) 0 0
\(391\) 25.9731 1.31351
\(392\) −6.88786 −0.347889
\(393\) 0 0
\(394\) −6.29692 −0.317234
\(395\) −12.5306 −0.630482
\(396\) 0 0
\(397\) −5.06135 −0.254022 −0.127011 0.991901i \(-0.540538\pi\)
−0.127011 + 0.991901i \(0.540538\pi\)
\(398\) 18.2023 0.912396
\(399\) 0 0
\(400\) −64.4099 −3.22050
\(401\) −1.55341 −0.0775736 −0.0387868 0.999248i \(-0.512349\pi\)
−0.0387868 + 0.999248i \(0.512349\pi\)
\(402\) 0 0
\(403\) 0.747381 0.0372297
\(404\) −3.36531 −0.167431
\(405\) 0 0
\(406\) −12.8833 −0.639389
\(407\) 8.39318 0.416035
\(408\) 0 0
\(409\) −8.42819 −0.416747 −0.208373 0.978049i \(-0.566817\pi\)
−0.208373 + 0.978049i \(0.566817\pi\)
\(410\) 15.9035 0.785420
\(411\) 0 0
\(412\) −9.76995 −0.481331
\(413\) −19.4250 −0.955841
\(414\) 0 0
\(415\) 52.1292 2.55892
\(416\) 0.822124 0.0403079
\(417\) 0 0
\(418\) −6.06759 −0.296776
\(419\) 32.6335 1.59425 0.797125 0.603814i \(-0.206353\pi\)
0.797125 + 0.603814i \(0.206353\pi\)
\(420\) 0 0
\(421\) −5.65215 −0.275469 −0.137735 0.990469i \(-0.543982\pi\)
−0.137735 + 0.990469i \(0.543982\pi\)
\(422\) −27.8292 −1.35470
\(423\) 0 0
\(424\) 15.9899 0.776540
\(425\) −42.2114 −2.04756
\(426\) 0 0
\(427\) −1.98799 −0.0962056
\(428\) −8.05625 −0.389414
\(429\) 0 0
\(430\) 54.0458 2.60632
\(431\) 23.6079 1.13715 0.568576 0.822631i \(-0.307495\pi\)
0.568576 + 0.822631i \(0.307495\pi\)
\(432\) 0 0
\(433\) 11.9983 0.576600 0.288300 0.957540i \(-0.406910\pi\)
0.288300 + 0.957540i \(0.406910\pi\)
\(434\) −9.55072 −0.458449
\(435\) 0 0
\(436\) 8.11223 0.388505
\(437\) 30.8357 1.47507
\(438\) 0 0
\(439\) −9.34054 −0.445799 −0.222900 0.974841i \(-0.571552\pi\)
−0.222900 + 0.974841i \(0.571552\pi\)
\(440\) 9.67069 0.461032
\(441\) 0 0
\(442\) 1.28150 0.0609550
\(443\) −26.2123 −1.24538 −0.622692 0.782467i \(-0.713961\pi\)
−0.622692 + 0.782467i \(0.713961\pi\)
\(444\) 0 0
\(445\) −41.8577 −1.98425
\(446\) 45.6530 2.16173
\(447\) 0 0
\(448\) 8.73129 0.412514
\(449\) 11.4987 0.542658 0.271329 0.962487i \(-0.412537\pi\)
0.271329 + 0.962487i \(0.412537\pi\)
\(450\) 0 0
\(451\) −2.30573 −0.108572
\(452\) 0.0377474 0.00177549
\(453\) 0 0
\(454\) −23.1138 −1.08479
\(455\) −2.13315 −0.100004
\(456\) 0 0
\(457\) 34.8251 1.62905 0.814525 0.580128i \(-0.196997\pi\)
0.814525 + 0.580128i \(0.196997\pi\)
\(458\) 14.8543 0.694094
\(459\) 0 0
\(460\) 20.9589 0.977216
\(461\) −30.9884 −1.44327 −0.721637 0.692272i \(-0.756610\pi\)
−0.721637 + 0.692272i \(0.756610\pi\)
\(462\) 0 0
\(463\) −3.65194 −0.169720 −0.0848599 0.996393i \(-0.527044\pi\)
−0.0848599 + 0.996393i \(0.527044\pi\)
\(464\) −19.4534 −0.903103
\(465\) 0 0
\(466\) −40.7699 −1.88863
\(467\) −33.6723 −1.55817 −0.779084 0.626920i \(-0.784315\pi\)
−0.779084 + 0.626920i \(0.784315\pi\)
\(468\) 0 0
\(469\) −26.1934 −1.20950
\(470\) −5.88230 −0.271330
\(471\) 0 0
\(472\) −22.0816 −1.01639
\(473\) −7.83567 −0.360284
\(474\) 0 0
\(475\) −50.1141 −2.29939
\(476\) −3.76907 −0.172755
\(477\) 0 0
\(478\) −31.0906 −1.42205
\(479\) 0.0546621 0.00249757 0.00124879 0.999999i \(-0.499602\pi\)
0.00124879 + 0.999999i \(0.499602\pi\)
\(480\) 0 0
\(481\) −2.10456 −0.0959595
\(482\) −20.0038 −0.911147
\(483\) 0 0
\(484\) 0.597924 0.0271784
\(485\) −12.9160 −0.586484
\(486\) 0 0
\(487\) 16.1791 0.733145 0.366573 0.930389i \(-0.380531\pi\)
0.366573 + 0.930389i \(0.380531\pi\)
\(488\) −2.25988 −0.102300
\(489\) 0 0
\(490\) −21.0226 −0.949702
\(491\) 0.996030 0.0449502 0.0224751 0.999747i \(-0.492845\pi\)
0.0224751 + 0.999747i \(0.492845\pi\)
\(492\) 0 0
\(493\) −12.7489 −0.574183
\(494\) 1.52142 0.0684521
\(495\) 0 0
\(496\) −14.4213 −0.647535
\(497\) −4.11825 −0.184729
\(498\) 0 0
\(499\) −29.3156 −1.31235 −0.656173 0.754610i \(-0.727826\pi\)
−0.656173 + 0.754610i \(0.727826\pi\)
\(500\) −21.2690 −0.951177
\(501\) 0 0
\(502\) −22.4529 −1.00212
\(503\) −19.9254 −0.888431 −0.444216 0.895920i \(-0.646518\pi\)
−0.444216 + 0.895920i \(0.646518\pi\)
\(504\) 0 0
\(505\) 24.0853 1.07178
\(506\) −13.2027 −0.586933
\(507\) 0 0
\(508\) 3.01855 0.133927
\(509\) 32.0673 1.42136 0.710679 0.703516i \(-0.248388\pi\)
0.710679 + 0.703516i \(0.248388\pi\)
\(510\) 0 0
\(511\) −2.40407 −0.106350
\(512\) 6.00456 0.265367
\(513\) 0 0
\(514\) −13.1488 −0.579968
\(515\) 69.9228 3.08117
\(516\) 0 0
\(517\) 0.852827 0.0375073
\(518\) 26.8939 1.18165
\(519\) 0 0
\(520\) −2.42489 −0.106338
\(521\) −11.7487 −0.514719 −0.257360 0.966316i \(-0.582852\pi\)
−0.257360 + 0.966316i \(0.582852\pi\)
\(522\) 0 0
\(523\) 7.66102 0.334993 0.167497 0.985873i \(-0.446432\pi\)
0.167497 + 0.985873i \(0.446432\pi\)
\(524\) −11.4242 −0.499067
\(525\) 0 0
\(526\) −26.8170 −1.16928
\(527\) −9.45108 −0.411696
\(528\) 0 0
\(529\) 44.0966 1.91724
\(530\) 48.8032 2.11988
\(531\) 0 0
\(532\) −4.47470 −0.194003
\(533\) 0.578151 0.0250425
\(534\) 0 0
\(535\) 57.6580 2.49277
\(536\) −29.7757 −1.28612
\(537\) 0 0
\(538\) 1.98879 0.0857429
\(539\) 3.04789 0.131282
\(540\) 0 0
\(541\) −20.4982 −0.881286 −0.440643 0.897682i \(-0.645250\pi\)
−0.440643 + 0.897682i \(0.645250\pi\)
\(542\) −7.22274 −0.310243
\(543\) 0 0
\(544\) −10.3962 −0.445736
\(545\) −58.0586 −2.48696
\(546\) 0 0
\(547\) −15.0674 −0.644234 −0.322117 0.946700i \(-0.604394\pi\)
−0.322117 + 0.946700i \(0.604394\pi\)
\(548\) 5.02248 0.214550
\(549\) 0 0
\(550\) 21.4571 0.914932
\(551\) −15.1357 −0.644804
\(552\) 0 0
\(553\) 5.82121 0.247543
\(554\) −35.9812 −1.52869
\(555\) 0 0
\(556\) 4.21983 0.178961
\(557\) −22.0552 −0.934508 −0.467254 0.884123i \(-0.654757\pi\)
−0.467254 + 0.884123i \(0.654757\pi\)
\(558\) 0 0
\(559\) 1.96476 0.0831005
\(560\) 41.1607 1.73936
\(561\) 0 0
\(562\) 1.94492 0.0820417
\(563\) 16.2194 0.683566 0.341783 0.939779i \(-0.388969\pi\)
0.341783 + 0.939779i \(0.388969\pi\)
\(564\) 0 0
\(565\) −0.270156 −0.0113655
\(566\) 20.7566 0.872466
\(567\) 0 0
\(568\) −4.68148 −0.196431
\(569\) 35.7962 1.50065 0.750326 0.661068i \(-0.229896\pi\)
0.750326 + 0.661068i \(0.229896\pi\)
\(570\) 0 0
\(571\) −13.2967 −0.556450 −0.278225 0.960516i \(-0.589746\pi\)
−0.278225 + 0.960516i \(0.589746\pi\)
\(572\) −0.149927 −0.00626876
\(573\) 0 0
\(574\) −7.38815 −0.308375
\(575\) −109.045 −4.54751
\(576\) 0 0
\(577\) −19.2277 −0.800462 −0.400231 0.916414i \(-0.631070\pi\)
−0.400231 + 0.916414i \(0.631070\pi\)
\(578\) 11.1953 0.465665
\(579\) 0 0
\(580\) −10.2877 −0.427175
\(581\) −24.2171 −1.00470
\(582\) 0 0
\(583\) −7.07558 −0.293041
\(584\) −2.73285 −0.113086
\(585\) 0 0
\(586\) 5.73447 0.236889
\(587\) −5.61715 −0.231845 −0.115922 0.993258i \(-0.536982\pi\)
−0.115922 + 0.993258i \(0.536982\pi\)
\(588\) 0 0
\(589\) −11.2205 −0.462332
\(590\) −67.3957 −2.77464
\(591\) 0 0
\(592\) 40.6090 1.66902
\(593\) −18.0517 −0.741295 −0.370648 0.928774i \(-0.620864\pi\)
−0.370648 + 0.928774i \(0.620864\pi\)
\(594\) 0 0
\(595\) 26.9749 1.10586
\(596\) 11.9503 0.489504
\(597\) 0 0
\(598\) 3.31053 0.135378
\(599\) −0.631254 −0.0257923 −0.0128962 0.999917i \(-0.504105\pi\)
−0.0128962 + 0.999917i \(0.504105\pi\)
\(600\) 0 0
\(601\) 21.3195 0.869641 0.434820 0.900517i \(-0.356812\pi\)
0.434820 + 0.900517i \(0.356812\pi\)
\(602\) −25.1075 −1.02331
\(603\) 0 0
\(604\) −8.99481 −0.365994
\(605\) −4.27930 −0.173978
\(606\) 0 0
\(607\) 39.8171 1.61613 0.808063 0.589096i \(-0.200516\pi\)
0.808063 + 0.589096i \(0.200516\pi\)
\(608\) −12.3426 −0.500559
\(609\) 0 0
\(610\) −6.89741 −0.279268
\(611\) −0.213843 −0.00865115
\(612\) 0 0
\(613\) 43.2039 1.74499 0.872494 0.488624i \(-0.162501\pi\)
0.872494 + 0.488624i \(0.162501\pi\)
\(614\) −25.3074 −1.02132
\(615\) 0 0
\(616\) −4.49261 −0.181013
\(617\) −3.91242 −0.157508 −0.0787540 0.996894i \(-0.525094\pi\)
−0.0787540 + 0.996894i \(0.525094\pi\)
\(618\) 0 0
\(619\) 24.0600 0.967051 0.483526 0.875330i \(-0.339356\pi\)
0.483526 + 0.875330i \(0.339356\pi\)
\(620\) −7.62654 −0.306289
\(621\) 0 0
\(622\) −32.8210 −1.31600
\(623\) 19.4454 0.779064
\(624\) 0 0
\(625\) 85.6584 3.42633
\(626\) −8.66301 −0.346243
\(627\) 0 0
\(628\) −8.54213 −0.340868
\(629\) 26.6134 1.06114
\(630\) 0 0
\(631\) −35.3657 −1.40789 −0.703943 0.710256i \(-0.748579\pi\)
−0.703943 + 0.710256i \(0.748579\pi\)
\(632\) 6.61734 0.263224
\(633\) 0 0
\(634\) −5.80251 −0.230447
\(635\) −21.6036 −0.857311
\(636\) 0 0
\(637\) −0.764246 −0.0302805
\(638\) 6.48058 0.256569
\(639\) 0 0
\(640\) 58.3547 2.30667
\(641\) 10.9245 0.431493 0.215747 0.976449i \(-0.430781\pi\)
0.215747 + 0.976449i \(0.430781\pi\)
\(642\) 0 0
\(643\) 46.6869 1.84115 0.920576 0.390564i \(-0.127720\pi\)
0.920576 + 0.390564i \(0.127720\pi\)
\(644\) −9.73668 −0.383679
\(645\) 0 0
\(646\) −19.2393 −0.756961
\(647\) −22.9120 −0.900764 −0.450382 0.892836i \(-0.648712\pi\)
−0.450382 + 0.892836i \(0.648712\pi\)
\(648\) 0 0
\(649\) 9.77116 0.383551
\(650\) −5.38027 −0.211031
\(651\) 0 0
\(652\) −7.17563 −0.281019
\(653\) 6.74465 0.263938 0.131969 0.991254i \(-0.457870\pi\)
0.131969 + 0.991254i \(0.457870\pi\)
\(654\) 0 0
\(655\) 81.7619 3.19470
\(656\) −11.1559 −0.435564
\(657\) 0 0
\(658\) 2.73268 0.106531
\(659\) 37.7263 1.46961 0.734805 0.678279i \(-0.237274\pi\)
0.734805 + 0.678279i \(0.237274\pi\)
\(660\) 0 0
\(661\) 40.7549 1.58518 0.792591 0.609754i \(-0.208732\pi\)
0.792591 + 0.609754i \(0.208732\pi\)
\(662\) −57.9930 −2.25396
\(663\) 0 0
\(664\) −27.5291 −1.06834
\(665\) 32.0251 1.24188
\(666\) 0 0
\(667\) −32.9345 −1.27523
\(668\) 10.7467 0.415801
\(669\) 0 0
\(670\) −90.8791 −3.51096
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) −22.5148 −0.867882 −0.433941 0.900941i \(-0.642877\pi\)
−0.433941 + 0.900941i \(0.642877\pi\)
\(674\) −7.73479 −0.297933
\(675\) 0 0
\(676\) −7.73542 −0.297516
\(677\) 0.489159 0.0187999 0.00939995 0.999956i \(-0.497008\pi\)
0.00939995 + 0.999956i \(0.497008\pi\)
\(678\) 0 0
\(679\) 6.00024 0.230268
\(680\) 30.6641 1.17592
\(681\) 0 0
\(682\) 4.80420 0.183963
\(683\) 6.05286 0.231606 0.115803 0.993272i \(-0.463056\pi\)
0.115803 + 0.993272i \(0.463056\pi\)
\(684\) 0 0
\(685\) −35.9455 −1.37341
\(686\) 32.1961 1.22925
\(687\) 0 0
\(688\) −37.9116 −1.44537
\(689\) 1.77417 0.0675906
\(690\) 0 0
\(691\) 44.2515 1.68341 0.841704 0.539939i \(-0.181553\pi\)
0.841704 + 0.539939i \(0.181553\pi\)
\(692\) 3.57051 0.135730
\(693\) 0 0
\(694\) 6.18804 0.234895
\(695\) −30.2010 −1.14559
\(696\) 0 0
\(697\) −7.31107 −0.276927
\(698\) −0.169797 −0.00642691
\(699\) 0 0
\(700\) 15.8240 0.598093
\(701\) 46.3230 1.74960 0.874798 0.484488i \(-0.160994\pi\)
0.874798 + 0.484488i \(0.160994\pi\)
\(702\) 0 0
\(703\) 31.5958 1.19166
\(704\) −4.39201 −0.165530
\(705\) 0 0
\(706\) −55.3670 −2.08376
\(707\) −11.1891 −0.420808
\(708\) 0 0
\(709\) −14.4425 −0.542401 −0.271201 0.962523i \(-0.587421\pi\)
−0.271201 + 0.962523i \(0.587421\pi\)
\(710\) −14.2884 −0.536235
\(711\) 0 0
\(712\) 22.1049 0.828414
\(713\) −24.4151 −0.914353
\(714\) 0 0
\(715\) 1.07302 0.0401286
\(716\) −13.4663 −0.503260
\(717\) 0 0
\(718\) −22.3843 −0.835375
\(719\) 24.6340 0.918693 0.459346 0.888257i \(-0.348084\pi\)
0.459346 + 0.888257i \(0.348084\pi\)
\(720\) 0 0
\(721\) −32.4833 −1.20974
\(722\) 7.78311 0.289658
\(723\) 0 0
\(724\) −10.0264 −0.372630
\(725\) 53.5251 1.98787
\(726\) 0 0
\(727\) 35.6426 1.32191 0.660956 0.750425i \(-0.270151\pi\)
0.660956 + 0.750425i \(0.270151\pi\)
\(728\) 1.12650 0.0417510
\(729\) 0 0
\(730\) −8.34100 −0.308714
\(731\) −24.8456 −0.918947
\(732\) 0 0
\(733\) 14.8831 0.549720 0.274860 0.961484i \(-0.411369\pi\)
0.274860 + 0.961484i \(0.411369\pi\)
\(734\) 19.7102 0.727518
\(735\) 0 0
\(736\) −26.8568 −0.989953
\(737\) 13.1758 0.485338
\(738\) 0 0
\(739\) −14.9543 −0.550104 −0.275052 0.961429i \(-0.588695\pi\)
−0.275052 + 0.961429i \(0.588695\pi\)
\(740\) 21.4756 0.789460
\(741\) 0 0
\(742\) −22.6720 −0.832315
\(743\) 17.4486 0.640126 0.320063 0.947396i \(-0.396296\pi\)
0.320063 + 0.947396i \(0.396296\pi\)
\(744\) 0 0
\(745\) −85.5275 −3.13348
\(746\) −39.2064 −1.43545
\(747\) 0 0
\(748\) 1.89592 0.0693216
\(749\) −26.7856 −0.978724
\(750\) 0 0
\(751\) 21.2302 0.774700 0.387350 0.921933i \(-0.373390\pi\)
0.387350 + 0.921933i \(0.373390\pi\)
\(752\) 4.12626 0.150469
\(753\) 0 0
\(754\) −1.62498 −0.0591782
\(755\) 64.3752 2.34285
\(756\) 0 0
\(757\) 39.8953 1.45002 0.725010 0.688739i \(-0.241835\pi\)
0.725010 + 0.688739i \(0.241835\pi\)
\(758\) −3.76599 −0.136787
\(759\) 0 0
\(760\) 36.4050 1.32055
\(761\) −24.3669 −0.883301 −0.441651 0.897187i \(-0.645607\pi\)
−0.441651 + 0.897187i \(0.645607\pi\)
\(762\) 0 0
\(763\) 26.9717 0.976441
\(764\) −13.8319 −0.500422
\(765\) 0 0
\(766\) −48.0827 −1.73730
\(767\) −2.45008 −0.0884671
\(768\) 0 0
\(769\) 16.9871 0.612572 0.306286 0.951940i \(-0.400914\pi\)
0.306286 + 0.951940i \(0.400914\pi\)
\(770\) −13.7120 −0.494146
\(771\) 0 0
\(772\) 0.926094 0.0333309
\(773\) −50.0956 −1.80181 −0.900907 0.434012i \(-0.857097\pi\)
−0.900907 + 0.434012i \(0.857097\pi\)
\(774\) 0 0
\(775\) 39.6794 1.42533
\(776\) 6.82085 0.244854
\(777\) 0 0
\(778\) 7.93244 0.284392
\(779\) −8.67983 −0.310987
\(780\) 0 0
\(781\) 2.07157 0.0741264
\(782\) −41.8636 −1.49704
\(783\) 0 0
\(784\) 14.7467 0.526669
\(785\) 61.1354 2.18202
\(786\) 0 0
\(787\) −38.7557 −1.38149 −0.690746 0.723098i \(-0.742718\pi\)
−0.690746 + 0.723098i \(0.742718\pi\)
\(788\) 2.33594 0.0832143
\(789\) 0 0
\(790\) 20.1969 0.718573
\(791\) 0.125503 0.00446239
\(792\) 0 0
\(793\) −0.250746 −0.00890424
\(794\) 8.15792 0.289514
\(795\) 0 0
\(796\) −6.75240 −0.239332
\(797\) −44.4888 −1.57588 −0.787938 0.615755i \(-0.788851\pi\)
−0.787938 + 0.615755i \(0.788851\pi\)
\(798\) 0 0
\(799\) 2.70417 0.0956667
\(800\) 43.6476 1.54318
\(801\) 0 0
\(802\) 2.50380 0.0884122
\(803\) 1.20929 0.0426750
\(804\) 0 0
\(805\) 69.6847 2.45606
\(806\) −1.20463 −0.0424314
\(807\) 0 0
\(808\) −12.7193 −0.447464
\(809\) 38.6506 1.35888 0.679442 0.733729i \(-0.262222\pi\)
0.679442 + 0.733729i \(0.262222\pi\)
\(810\) 0 0
\(811\) 19.9320 0.699906 0.349953 0.936767i \(-0.386198\pi\)
0.349953 + 0.936767i \(0.386198\pi\)
\(812\) 4.77927 0.167719
\(813\) 0 0
\(814\) −13.5282 −0.474163
\(815\) 51.3554 1.79890
\(816\) 0 0
\(817\) −29.4971 −1.03197
\(818\) 13.5846 0.474975
\(819\) 0 0
\(820\) −5.89966 −0.206025
\(821\) −17.0881 −0.596379 −0.298189 0.954507i \(-0.596383\pi\)
−0.298189 + 0.954507i \(0.596383\pi\)
\(822\) 0 0
\(823\) −16.4950 −0.574980 −0.287490 0.957784i \(-0.592821\pi\)
−0.287490 + 0.957784i \(0.592821\pi\)
\(824\) −36.9259 −1.28637
\(825\) 0 0
\(826\) 31.3093 1.08939
\(827\) −3.49254 −0.121447 −0.0607237 0.998155i \(-0.519341\pi\)
−0.0607237 + 0.998155i \(0.519341\pi\)
\(828\) 0 0
\(829\) 5.26456 0.182846 0.0914228 0.995812i \(-0.470859\pi\)
0.0914228 + 0.995812i \(0.470859\pi\)
\(830\) −84.0222 −2.91645
\(831\) 0 0
\(832\) 1.10128 0.0381800
\(833\) 9.66435 0.334850
\(834\) 0 0
\(835\) −76.9132 −2.66169
\(836\) 2.25086 0.0778478
\(837\) 0 0
\(838\) −52.5989 −1.81700
\(839\) −22.8286 −0.788130 −0.394065 0.919083i \(-0.628932\pi\)
−0.394065 + 0.919083i \(0.628932\pi\)
\(840\) 0 0
\(841\) −12.8341 −0.442554
\(842\) 9.11018 0.313958
\(843\) 0 0
\(844\) 10.3237 0.355355
\(845\) 55.3619 1.90451
\(846\) 0 0
\(847\) 1.98799 0.0683082
\(848\) −34.2340 −1.17560
\(849\) 0 0
\(850\) 68.0367 2.33364
\(851\) 68.7507 2.35674
\(852\) 0 0
\(853\) 28.5640 0.978014 0.489007 0.872280i \(-0.337359\pi\)
0.489007 + 0.872280i \(0.337359\pi\)
\(854\) 3.20426 0.109648
\(855\) 0 0
\(856\) −30.4489 −1.04072
\(857\) −15.5911 −0.532582 −0.266291 0.963893i \(-0.585798\pi\)
−0.266291 + 0.963893i \(0.585798\pi\)
\(858\) 0 0
\(859\) −4.92436 −0.168017 −0.0840085 0.996465i \(-0.526772\pi\)
−0.0840085 + 0.996465i \(0.526772\pi\)
\(860\) −20.0491 −0.683669
\(861\) 0 0
\(862\) −38.0514 −1.29603
\(863\) −29.5907 −1.00728 −0.503640 0.863914i \(-0.668006\pi\)
−0.503640 + 0.863914i \(0.668006\pi\)
\(864\) 0 0
\(865\) −25.5539 −0.868858
\(866\) −19.3389 −0.657163
\(867\) 0 0
\(868\) 3.54298 0.120257
\(869\) −2.92819 −0.0993319
\(870\) 0 0
\(871\) −3.30378 −0.111944
\(872\) 30.6605 1.03829
\(873\) 0 0
\(874\) −49.7012 −1.68117
\(875\) −70.7155 −2.39062
\(876\) 0 0
\(877\) −57.3331 −1.93600 −0.968001 0.250947i \(-0.919258\pi\)
−0.968001 + 0.250947i \(0.919258\pi\)
\(878\) 15.0551 0.508087
\(879\) 0 0
\(880\) −20.7047 −0.697955
\(881\) 8.30122 0.279675 0.139838 0.990174i \(-0.455342\pi\)
0.139838 + 0.990174i \(0.455342\pi\)
\(882\) 0 0
\(883\) −8.59069 −0.289100 −0.144550 0.989498i \(-0.546173\pi\)
−0.144550 + 0.989498i \(0.546173\pi\)
\(884\) −0.475393 −0.0159892
\(885\) 0 0
\(886\) 42.2492 1.41939
\(887\) −26.4726 −0.888864 −0.444432 0.895813i \(-0.646595\pi\)
−0.444432 + 0.895813i \(0.646595\pi\)
\(888\) 0 0
\(889\) 10.0361 0.336602
\(890\) 67.4666 2.26149
\(891\) 0 0
\(892\) −16.9357 −0.567048
\(893\) 3.21044 0.107433
\(894\) 0 0
\(895\) 96.3774 3.22154
\(896\) −27.1093 −0.905657
\(897\) 0 0
\(898\) −18.5337 −0.618478
\(899\) 11.9842 0.399695
\(900\) 0 0
\(901\) −22.4355 −0.747434
\(902\) 3.71639 0.123742
\(903\) 0 0
\(904\) 0.142668 0.00474506
\(905\) 71.7585 2.38533
\(906\) 0 0
\(907\) 54.7146 1.81677 0.908384 0.418136i \(-0.137316\pi\)
0.908384 + 0.418136i \(0.137316\pi\)
\(908\) 8.57442 0.284552
\(909\) 0 0
\(910\) 3.43822 0.113976
\(911\) 14.0636 0.465947 0.232974 0.972483i \(-0.425154\pi\)
0.232974 + 0.972483i \(0.425154\pi\)
\(912\) 0 0
\(913\) 12.1817 0.403156
\(914\) −56.1314 −1.85666
\(915\) 0 0
\(916\) −5.51041 −0.182069
\(917\) −37.9833 −1.25432
\(918\) 0 0
\(919\) −18.8036 −0.620274 −0.310137 0.950692i \(-0.600375\pi\)
−0.310137 + 0.950692i \(0.600375\pi\)
\(920\) 79.2151 2.61164
\(921\) 0 0
\(922\) 49.9474 1.64493
\(923\) −0.519436 −0.0170975
\(924\) 0 0
\(925\) −111.734 −3.67378
\(926\) 5.88622 0.193433
\(927\) 0 0
\(928\) 13.1827 0.432743
\(929\) −13.3631 −0.438429 −0.219214 0.975677i \(-0.570349\pi\)
−0.219214 + 0.975677i \(0.570349\pi\)
\(930\) 0 0
\(931\) 11.4737 0.376035
\(932\) 15.1242 0.495410
\(933\) 0 0
\(934\) 54.2732 1.77587
\(935\) −13.5689 −0.443752
\(936\) 0 0
\(937\) 7.26525 0.237345 0.118673 0.992933i \(-0.462136\pi\)
0.118673 + 0.992933i \(0.462136\pi\)
\(938\) 42.2188 1.37849
\(939\) 0 0
\(940\) 2.18213 0.0711731
\(941\) −25.5547 −0.833061 −0.416530 0.909122i \(-0.636754\pi\)
−0.416530 + 0.909122i \(0.636754\pi\)
\(942\) 0 0
\(943\) −18.8868 −0.615038
\(944\) 47.2761 1.53871
\(945\) 0 0
\(946\) 12.6296 0.410623
\(947\) 29.0368 0.943570 0.471785 0.881714i \(-0.343610\pi\)
0.471785 + 0.881714i \(0.343610\pi\)
\(948\) 0 0
\(949\) −0.303225 −0.00984311
\(950\) 80.7743 2.62067
\(951\) 0 0
\(952\) −14.2453 −0.461694
\(953\) −41.2250 −1.33541 −0.667705 0.744426i \(-0.732723\pi\)
−0.667705 + 0.744426i \(0.732723\pi\)
\(954\) 0 0
\(955\) 98.9942 3.20338
\(956\) 11.5335 0.373021
\(957\) 0 0
\(958\) −0.0881048 −0.00284654
\(959\) 16.6989 0.539234
\(960\) 0 0
\(961\) −22.1158 −0.713414
\(962\) 3.39214 0.109367
\(963\) 0 0
\(964\) 7.42070 0.239005
\(965\) −6.62799 −0.213362
\(966\) 0 0
\(967\) 13.8906 0.446690 0.223345 0.974739i \(-0.428302\pi\)
0.223345 + 0.974739i \(0.428302\pi\)
\(968\) 2.25988 0.0726352
\(969\) 0 0
\(970\) 20.8180 0.668427
\(971\) 32.4127 1.04017 0.520086 0.854114i \(-0.325900\pi\)
0.520086 + 0.854114i \(0.325900\pi\)
\(972\) 0 0
\(973\) 14.0302 0.449787
\(974\) −26.0776 −0.835580
\(975\) 0 0
\(976\) 4.83834 0.154871
\(977\) −32.1328 −1.02802 −0.514010 0.857784i \(-0.671841\pi\)
−0.514010 + 0.857784i \(0.671841\pi\)
\(978\) 0 0
\(979\) −9.78144 −0.312616
\(980\) 7.79863 0.249118
\(981\) 0 0
\(982\) −1.60541 −0.0512307
\(983\) −3.66030 −0.116745 −0.0583727 0.998295i \(-0.518591\pi\)
−0.0583727 + 0.998295i \(0.518591\pi\)
\(984\) 0 0
\(985\) −16.7181 −0.532684
\(986\) 20.5488 0.654408
\(987\) 0 0
\(988\) −0.564395 −0.0179558
\(989\) −64.1839 −2.04093
\(990\) 0 0
\(991\) −23.8517 −0.757674 −0.378837 0.925463i \(-0.623676\pi\)
−0.378837 + 0.925463i \(0.623676\pi\)
\(992\) 9.77264 0.310282
\(993\) 0 0
\(994\) 6.63783 0.210539
\(995\) 48.3264 1.53205
\(996\) 0 0
\(997\) −54.6646 −1.73124 −0.865622 0.500698i \(-0.833077\pi\)
−0.865622 + 0.500698i \(0.833077\pi\)
\(998\) 47.2511 1.49571
\(999\) 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.1.4 13
3.2 odd 2 2013.2.a.e.1.10 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.e.1.10 13 3.2 odd 2
6039.2.a.i.1.4 13 1.1 even 1 trivial