Properties

Label 3969.2.a.y.1.2
Level $3969$
Weight $2$
Character 3969.1
Self dual yes
Analytic conductor $31.693$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3969.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.6926245622\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial: \( x^{4} - 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.28825\) of defining polynomial
Character \(\chi\) \(=\) 3969.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.618034 q^{2} -1.61803 q^{4} +2.28825 q^{5} +2.23607 q^{8} +O(q^{10})\) \(q-0.618034 q^{2} -1.61803 q^{4} +2.28825 q^{5} +2.23607 q^{8} -1.41421 q^{10} +1.00000 q^{11} +4.57649 q^{13} +1.85410 q^{16} -3.16228 q^{17} -4.03631 q^{19} -3.70246 q^{20} -0.618034 q^{22} -7.23607 q^{23} +0.236068 q^{25} -2.82843 q^{26} +1.23607 q^{29} -5.99070 q^{31} -5.61803 q^{32} +1.95440 q^{34} -10.7082 q^{37} +2.49458 q^{38} +5.11667 q^{40} -5.78437 q^{41} +5.94427 q^{43} -1.61803 q^{44} +4.47214 q^{46} -3.03476 q^{47} -0.145898 q^{50} -7.40492 q^{52} +10.7082 q^{53} +2.28825 q^{55} -0.763932 q^{58} -14.4760 q^{59} +8.94665 q^{61} +3.70246 q^{62} -0.236068 q^{64} +10.4721 q^{65} +1.47214 q^{67} +5.11667 q^{68} +5.47214 q^{71} -6.73722 q^{73} +6.61803 q^{74} +6.53089 q^{76} -7.76393 q^{79} +4.24264 q^{80} +3.57494 q^{82} -8.61280 q^{83} -7.23607 q^{85} -3.67376 q^{86} +2.23607 q^{88} +6.32456 q^{89} +11.7082 q^{92} +1.87558 q^{94} -9.23607 q^{95} +6.19704 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} + 4 q^{11} - 6 q^{16} + 2 q^{22} - 20 q^{23} - 8 q^{25} - 4 q^{29} - 18 q^{32} - 16 q^{37} - 12 q^{43} - 2 q^{44} - 14 q^{50} + 16 q^{53} - 12 q^{58} + 8 q^{64} + 24 q^{65} - 12 q^{67} + 4 q^{71} + 22 q^{74} - 40 q^{79} - 20 q^{85} - 46 q^{86} + 20 q^{92} - 28 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.618034 −0.437016 −0.218508 0.975835i \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) 0 0
\(4\) −1.61803 −0.809017
\(5\) 2.28825 1.02333 0.511667 0.859184i \(-0.329028\pi\)
0.511667 + 0.859184i \(0.329028\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 2.23607 0.790569
\(9\) 0 0
\(10\) −1.41421 −0.447214
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 4.57649 1.26929 0.634645 0.772804i \(-0.281146\pi\)
0.634645 + 0.772804i \(0.281146\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) −3.16228 −0.766965 −0.383482 0.923548i \(-0.625275\pi\)
−0.383482 + 0.923548i \(0.625275\pi\)
\(18\) 0 0
\(19\) −4.03631 −0.925993 −0.462996 0.886360i \(-0.653226\pi\)
−0.462996 + 0.886360i \(0.653226\pi\)
\(20\) −3.70246 −0.827895
\(21\) 0 0
\(22\) −0.618034 −0.131765
\(23\) −7.23607 −1.50882 −0.754412 0.656401i \(-0.772078\pi\)
−0.754412 + 0.656401i \(0.772078\pi\)
\(24\) 0 0
\(25\) 0.236068 0.0472136
\(26\) −2.82843 −0.554700
\(27\) 0 0
\(28\) 0 0
\(29\) 1.23607 0.229532 0.114766 0.993393i \(-0.463388\pi\)
0.114766 + 0.993393i \(0.463388\pi\)
\(30\) 0 0
\(31\) −5.99070 −1.07596 −0.537981 0.842957i \(-0.680813\pi\)
−0.537981 + 0.842957i \(0.680813\pi\)
\(32\) −5.61803 −0.993137
\(33\) 0 0
\(34\) 1.95440 0.335176
\(35\) 0 0
\(36\) 0 0
\(37\) −10.7082 −1.76042 −0.880209 0.474586i \(-0.842598\pi\)
−0.880209 + 0.474586i \(0.842598\pi\)
\(38\) 2.49458 0.404674
\(39\) 0 0
\(40\) 5.11667 0.809017
\(41\) −5.78437 −0.903367 −0.451684 0.892178i \(-0.649176\pi\)
−0.451684 + 0.892178i \(0.649176\pi\)
\(42\) 0 0
\(43\) 5.94427 0.906493 0.453246 0.891385i \(-0.350266\pi\)
0.453246 + 0.891385i \(0.350266\pi\)
\(44\) −1.61803 −0.243928
\(45\) 0 0
\(46\) 4.47214 0.659380
\(47\) −3.03476 −0.442665 −0.221332 0.975198i \(-0.571041\pi\)
−0.221332 + 0.975198i \(0.571041\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −0.145898 −0.0206331
\(51\) 0 0
\(52\) −7.40492 −1.02688
\(53\) 10.7082 1.47088 0.735442 0.677587i \(-0.236974\pi\)
0.735442 + 0.677587i \(0.236974\pi\)
\(54\) 0 0
\(55\) 2.28825 0.308547
\(56\) 0 0
\(57\) 0 0
\(58\) −0.763932 −0.100309
\(59\) −14.4760 −1.88461 −0.942306 0.334752i \(-0.891348\pi\)
−0.942306 + 0.334752i \(0.891348\pi\)
\(60\) 0 0
\(61\) 8.94665 1.14550 0.572751 0.819730i \(-0.305876\pi\)
0.572751 + 0.819730i \(0.305876\pi\)
\(62\) 3.70246 0.470213
\(63\) 0 0
\(64\) −0.236068 −0.0295085
\(65\) 10.4721 1.29891
\(66\) 0 0
\(67\) 1.47214 0.179850 0.0899250 0.995949i \(-0.471337\pi\)
0.0899250 + 0.995949i \(0.471337\pi\)
\(68\) 5.11667 0.620488
\(69\) 0 0
\(70\) 0 0
\(71\) 5.47214 0.649423 0.324712 0.945813i \(-0.394733\pi\)
0.324712 + 0.945813i \(0.394733\pi\)
\(72\) 0 0
\(73\) −6.73722 −0.788532 −0.394266 0.918996i \(-0.629001\pi\)
−0.394266 + 0.918996i \(0.629001\pi\)
\(74\) 6.61803 0.769331
\(75\) 0 0
\(76\) 6.53089 0.749144
\(77\) 0 0
\(78\) 0 0
\(79\) −7.76393 −0.873511 −0.436755 0.899580i \(-0.643872\pi\)
−0.436755 + 0.899580i \(0.643872\pi\)
\(80\) 4.24264 0.474342
\(81\) 0 0
\(82\) 3.57494 0.394786
\(83\) −8.61280 −0.945378 −0.472689 0.881229i \(-0.656717\pi\)
−0.472689 + 0.881229i \(0.656717\pi\)
\(84\) 0 0
\(85\) −7.23607 −0.784862
\(86\) −3.67376 −0.396152
\(87\) 0 0
\(88\) 2.23607 0.238366
\(89\) 6.32456 0.670402 0.335201 0.942147i \(-0.391196\pi\)
0.335201 + 0.942147i \(0.391196\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 11.7082 1.22066
\(93\) 0 0
\(94\) 1.87558 0.193452
\(95\) −9.23607 −0.947601
\(96\) 0 0
\(97\) 6.19704 0.629214 0.314607 0.949222i \(-0.398127\pi\)
0.314607 + 0.949222i \(0.398127\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −0.381966 −0.0381966
\(101\) 18.7186 1.86257 0.931286 0.364288i \(-0.118687\pi\)
0.931286 + 0.364288i \(0.118687\pi\)
\(102\) 0 0
\(103\) −5.78437 −0.569951 −0.284976 0.958535i \(-0.591986\pi\)
−0.284976 + 0.958535i \(0.591986\pi\)
\(104\) 10.2333 1.00346
\(105\) 0 0
\(106\) −6.61803 −0.642800
\(107\) 11.9443 1.15470 0.577348 0.816498i \(-0.304088\pi\)
0.577348 + 0.816498i \(0.304088\pi\)
\(108\) 0 0
\(109\) −14.9443 −1.43140 −0.715701 0.698407i \(-0.753893\pi\)
−0.715701 + 0.698407i \(0.753893\pi\)
\(110\) −1.41421 −0.134840
\(111\) 0 0
\(112\) 0 0
\(113\) −3.76393 −0.354081 −0.177040 0.984204i \(-0.556652\pi\)
−0.177040 + 0.984204i \(0.556652\pi\)
\(114\) 0 0
\(115\) −16.5579 −1.54403
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) 8.94665 0.823606
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −5.52933 −0.500602
\(123\) 0 0
\(124\) 9.69316 0.870472
\(125\) −10.9010 −0.975019
\(126\) 0 0
\(127\) −17.6525 −1.56640 −0.783202 0.621767i \(-0.786415\pi\)
−0.783202 + 0.621767i \(0.786415\pi\)
\(128\) 11.3820 1.00603
\(129\) 0 0
\(130\) −6.47214 −0.567644
\(131\) 20.2604 1.77016 0.885078 0.465443i \(-0.154105\pi\)
0.885078 + 0.465443i \(0.154105\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −0.909830 −0.0785973
\(135\) 0 0
\(136\) −7.07107 −0.606339
\(137\) 8.23607 0.703655 0.351827 0.936065i \(-0.385560\pi\)
0.351827 + 0.936065i \(0.385560\pi\)
\(138\) 0 0
\(139\) −9.69316 −0.822163 −0.411082 0.911598i \(-0.634849\pi\)
−0.411082 + 0.911598i \(0.634849\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −3.38197 −0.283808
\(143\) 4.57649 0.382705
\(144\) 0 0
\(145\) 2.82843 0.234888
\(146\) 4.16383 0.344601
\(147\) 0 0
\(148\) 17.3262 1.42421
\(149\) −10.5279 −0.862476 −0.431238 0.902238i \(-0.641923\pi\)
−0.431238 + 0.902238i \(0.641923\pi\)
\(150\) 0 0
\(151\) −18.7082 −1.52245 −0.761226 0.648487i \(-0.775402\pi\)
−0.761226 + 0.648487i \(0.775402\pi\)
\(152\) −9.02546 −0.732062
\(153\) 0 0
\(154\) 0 0
\(155\) −13.7082 −1.10107
\(156\) 0 0
\(157\) −14.8098 −1.18195 −0.590977 0.806689i \(-0.701258\pi\)
−0.590977 + 0.806689i \(0.701258\pi\)
\(158\) 4.79837 0.381738
\(159\) 0 0
\(160\) −12.8554 −1.01631
\(161\) 0 0
\(162\) 0 0
\(163\) −3.76393 −0.294814 −0.147407 0.989076i \(-0.547093\pi\)
−0.147407 + 0.989076i \(0.547093\pi\)
\(164\) 9.35931 0.730840
\(165\) 0 0
\(166\) 5.32300 0.413145
\(167\) 18.1784 1.40669 0.703345 0.710848i \(-0.251689\pi\)
0.703345 + 0.710848i \(0.251689\pi\)
\(168\) 0 0
\(169\) 7.94427 0.611098
\(170\) 4.47214 0.342997
\(171\) 0 0
\(172\) −9.61803 −0.733368
\(173\) −16.0965 −1.22380 −0.611898 0.790936i \(-0.709594\pi\)
−0.611898 + 0.790936i \(0.709594\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.85410 0.139758
\(177\) 0 0
\(178\) −3.90879 −0.292976
\(179\) −24.3607 −1.82080 −0.910401 0.413726i \(-0.864227\pi\)
−0.910401 + 0.413726i \(0.864227\pi\)
\(180\) 0 0
\(181\) 16.6367 1.23660 0.618299 0.785943i \(-0.287822\pi\)
0.618299 + 0.785943i \(0.287822\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −16.1803 −1.19283
\(185\) −24.5030 −1.80150
\(186\) 0 0
\(187\) −3.16228 −0.231249
\(188\) 4.91034 0.358123
\(189\) 0 0
\(190\) 5.70820 0.414117
\(191\) 22.4164 1.62199 0.810997 0.585050i \(-0.198925\pi\)
0.810997 + 0.585050i \(0.198925\pi\)
\(192\) 0 0
\(193\) −7.52786 −0.541868 −0.270934 0.962598i \(-0.587332\pi\)
−0.270934 + 0.962598i \(0.587332\pi\)
\(194\) −3.82998 −0.274976
\(195\) 0 0
\(196\) 0 0
\(197\) −8.41641 −0.599644 −0.299822 0.953995i \(-0.596927\pi\)
−0.299822 + 0.953995i \(0.596927\pi\)
\(198\) 0 0
\(199\) 4.37016 0.309792 0.154896 0.987931i \(-0.450496\pi\)
0.154896 + 0.987931i \(0.450496\pi\)
\(200\) 0.527864 0.0373256
\(201\) 0 0
\(202\) −11.5687 −0.813974
\(203\) 0 0
\(204\) 0 0
\(205\) −13.2361 −0.924447
\(206\) 3.57494 0.249078
\(207\) 0 0
\(208\) 8.48528 0.588348
\(209\) −4.03631 −0.279197
\(210\) 0 0
\(211\) −16.7082 −1.15024 −0.575120 0.818069i \(-0.695045\pi\)
−0.575120 + 0.818069i \(0.695045\pi\)
\(212\) −17.3262 −1.18997
\(213\) 0 0
\(214\) −7.38197 −0.504621
\(215\) 13.6020 0.927646
\(216\) 0 0
\(217\) 0 0
\(218\) 9.23607 0.625545
\(219\) 0 0
\(220\) −3.70246 −0.249620
\(221\) −14.4721 −0.973501
\(222\) 0 0
\(223\) 5.32300 0.356455 0.178227 0.983989i \(-0.442964\pi\)
0.178227 + 0.983989i \(0.442964\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2.32624 0.154739
\(227\) 0.952843 0.0632424 0.0316212 0.999500i \(-0.489933\pi\)
0.0316212 + 0.999500i \(0.489933\pi\)
\(228\) 0 0
\(229\) 5.78437 0.382242 0.191121 0.981566i \(-0.438788\pi\)
0.191121 + 0.981566i \(0.438788\pi\)
\(230\) 10.2333 0.674767
\(231\) 0 0
\(232\) 2.76393 0.181461
\(233\) 5.05573 0.331212 0.165606 0.986192i \(-0.447042\pi\)
0.165606 + 0.986192i \(0.447042\pi\)
\(234\) 0 0
\(235\) −6.94427 −0.452994
\(236\) 23.4226 1.52468
\(237\) 0 0
\(238\) 0 0
\(239\) −3.94427 −0.255134 −0.127567 0.991830i \(-0.540717\pi\)
−0.127567 + 0.991830i \(0.540717\pi\)
\(240\) 0 0
\(241\) 16.3516 1.05330 0.526649 0.850083i \(-0.323448\pi\)
0.526649 + 0.850083i \(0.323448\pi\)
\(242\) 6.18034 0.397287
\(243\) 0 0
\(244\) −14.4760 −0.926730
\(245\) 0 0
\(246\) 0 0
\(247\) −18.4721 −1.17535
\(248\) −13.3956 −0.850623
\(249\) 0 0
\(250\) 6.73722 0.426099
\(251\) −18.5123 −1.16849 −0.584243 0.811579i \(-0.698608\pi\)
−0.584243 + 0.811579i \(0.698608\pi\)
\(252\) 0 0
\(253\) −7.23607 −0.454928
\(254\) 10.9098 0.684544
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) 9.56564 0.596689 0.298344 0.954458i \(-0.403566\pi\)
0.298344 + 0.954458i \(0.403566\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −16.9443 −1.05084
\(261\) 0 0
\(262\) −12.5216 −0.773586
\(263\) 9.76393 0.602070 0.301035 0.953613i \(-0.402668\pi\)
0.301035 + 0.953613i \(0.402668\pi\)
\(264\) 0 0
\(265\) 24.5030 1.50521
\(266\) 0 0
\(267\) 0 0
\(268\) −2.38197 −0.145502
\(269\) 4.16383 0.253873 0.126937 0.991911i \(-0.459486\pi\)
0.126937 + 0.991911i \(0.459486\pi\)
\(270\) 0 0
\(271\) 26.1723 1.58985 0.794926 0.606707i \(-0.207510\pi\)
0.794926 + 0.606707i \(0.207510\pi\)
\(272\) −5.86319 −0.355508
\(273\) 0 0
\(274\) −5.09017 −0.307508
\(275\) 0.236068 0.0142354
\(276\) 0 0
\(277\) 10.2361 0.615026 0.307513 0.951544i \(-0.400503\pi\)
0.307513 + 0.951544i \(0.400503\pi\)
\(278\) 5.99070 0.359299
\(279\) 0 0
\(280\) 0 0
\(281\) −5.00000 −0.298275 −0.149137 0.988816i \(-0.547650\pi\)
−0.149137 + 0.988816i \(0.547650\pi\)
\(282\) 0 0
\(283\) −16.3516 −0.972000 −0.486000 0.873959i \(-0.661544\pi\)
−0.486000 + 0.873959i \(0.661544\pi\)
\(284\) −8.85410 −0.525394
\(285\) 0 0
\(286\) −2.82843 −0.167248
\(287\) 0 0
\(288\) 0 0
\(289\) −7.00000 −0.411765
\(290\) −1.74806 −0.102650
\(291\) 0 0
\(292\) 10.9010 0.637935
\(293\) 13.5231 0.790030 0.395015 0.918675i \(-0.370739\pi\)
0.395015 + 0.918675i \(0.370739\pi\)
\(294\) 0 0
\(295\) −33.1246 −1.92859
\(296\) −23.9443 −1.39173
\(297\) 0 0
\(298\) 6.50658 0.376916
\(299\) −33.1158 −1.91514
\(300\) 0 0
\(301\) 0 0
\(302\) 11.5623 0.665336
\(303\) 0 0
\(304\) −7.48373 −0.429221
\(305\) 20.4721 1.17223
\(306\) 0 0
\(307\) −12.9830 −0.740977 −0.370488 0.928837i \(-0.620810\pi\)
−0.370488 + 0.928837i \(0.620810\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 8.47214 0.481185
\(311\) −12.6004 −0.714503 −0.357252 0.934008i \(-0.616286\pi\)
−0.357252 + 0.934008i \(0.616286\pi\)
\(312\) 0 0
\(313\) 0.746512 0.0421954 0.0210977 0.999777i \(-0.493284\pi\)
0.0210977 + 0.999777i \(0.493284\pi\)
\(314\) 9.15298 0.516533
\(315\) 0 0
\(316\) 12.5623 0.706685
\(317\) −34.8328 −1.95641 −0.978203 0.207651i \(-0.933418\pi\)
−0.978203 + 0.207651i \(0.933418\pi\)
\(318\) 0 0
\(319\) 1.23607 0.0692065
\(320\) −0.540182 −0.0301971
\(321\) 0 0
\(322\) 0 0
\(323\) 12.7639 0.710204
\(324\) 0 0
\(325\) 1.08036 0.0599278
\(326\) 2.32624 0.128838
\(327\) 0 0
\(328\) −12.9343 −0.714175
\(329\) 0 0
\(330\) 0 0
\(331\) 2.00000 0.109930 0.0549650 0.998488i \(-0.482495\pi\)
0.0549650 + 0.998488i \(0.482495\pi\)
\(332\) 13.9358 0.764827
\(333\) 0 0
\(334\) −11.2349 −0.614746
\(335\) 3.36861 0.184047
\(336\) 0 0
\(337\) 5.18034 0.282191 0.141096 0.989996i \(-0.454938\pi\)
0.141096 + 0.989996i \(0.454938\pi\)
\(338\) −4.90983 −0.267060
\(339\) 0 0
\(340\) 11.7082 0.634967
\(341\) −5.99070 −0.324415
\(342\) 0 0
\(343\) 0 0
\(344\) 13.2918 0.716646
\(345\) 0 0
\(346\) 9.94820 0.534819
\(347\) 4.05573 0.217723 0.108861 0.994057i \(-0.465280\pi\)
0.108861 + 0.994057i \(0.465280\pi\)
\(348\) 0 0
\(349\) 18.7673 1.00459 0.502296 0.864696i \(-0.332489\pi\)
0.502296 + 0.864696i \(0.332489\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −5.61803 −0.299442
\(353\) 33.2433 1.76936 0.884682 0.466195i \(-0.154376\pi\)
0.884682 + 0.466195i \(0.154376\pi\)
\(354\) 0 0
\(355\) 12.5216 0.664577
\(356\) −10.2333 −0.542366
\(357\) 0 0
\(358\) 15.0557 0.795720
\(359\) −5.18034 −0.273408 −0.136704 0.990612i \(-0.543651\pi\)
−0.136704 + 0.990612i \(0.543651\pi\)
\(360\) 0 0
\(361\) −2.70820 −0.142537
\(362\) −10.2821 −0.540413
\(363\) 0 0
\(364\) 0 0
\(365\) −15.4164 −0.806932
\(366\) 0 0
\(367\) −4.57649 −0.238891 −0.119445 0.992841i \(-0.538112\pi\)
−0.119445 + 0.992841i \(0.538112\pi\)
\(368\) −13.4164 −0.699379
\(369\) 0 0
\(370\) 15.1437 0.787283
\(371\) 0 0
\(372\) 0 0
\(373\) 13.1803 0.682452 0.341226 0.939981i \(-0.389158\pi\)
0.341226 + 0.939981i \(0.389158\pi\)
\(374\) 1.95440 0.101059
\(375\) 0 0
\(376\) −6.78593 −0.349957
\(377\) 5.65685 0.291343
\(378\) 0 0
\(379\) −0.347524 −0.0178511 −0.00892556 0.999960i \(-0.502841\pi\)
−0.00892556 + 0.999960i \(0.502841\pi\)
\(380\) 14.9443 0.766625
\(381\) 0 0
\(382\) −13.8541 −0.708838
\(383\) −24.6305 −1.25856 −0.629280 0.777178i \(-0.716650\pi\)
−0.629280 + 0.777178i \(0.716650\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.65248 0.236805
\(387\) 0 0
\(388\) −10.0270 −0.509045
\(389\) −6.94427 −0.352089 −0.176044 0.984382i \(-0.556330\pi\)
−0.176044 + 0.984382i \(0.556330\pi\)
\(390\) 0 0
\(391\) 22.8825 1.15722
\(392\) 0 0
\(393\) 0 0
\(394\) 5.20163 0.262054
\(395\) −17.7658 −0.893894
\(396\) 0 0
\(397\) −31.9867 −1.60537 −0.802684 0.596405i \(-0.796595\pi\)
−0.802684 + 0.596405i \(0.796595\pi\)
\(398\) −2.70091 −0.135384
\(399\) 0 0
\(400\) 0.437694 0.0218847
\(401\) −21.0689 −1.05213 −0.526065 0.850444i \(-0.676333\pi\)
−0.526065 + 0.850444i \(0.676333\pi\)
\(402\) 0 0
\(403\) −27.4164 −1.36571
\(404\) −30.2874 −1.50685
\(405\) 0 0
\(406\) 0 0
\(407\) −10.7082 −0.530786
\(408\) 0 0
\(409\) 19.9265 0.985302 0.492651 0.870227i \(-0.336028\pi\)
0.492651 + 0.870227i \(0.336028\pi\)
\(410\) 8.18034 0.403998
\(411\) 0 0
\(412\) 9.35931 0.461100
\(413\) 0 0
\(414\) 0 0
\(415\) −19.7082 −0.967438
\(416\) −25.7109 −1.26058
\(417\) 0 0
\(418\) 2.49458 0.122014
\(419\) −23.0100 −1.12411 −0.562055 0.827100i \(-0.689989\pi\)
−0.562055 + 0.827100i \(0.689989\pi\)
\(420\) 0 0
\(421\) −20.8885 −1.01805 −0.509023 0.860753i \(-0.669993\pi\)
−0.509023 + 0.860753i \(0.669993\pi\)
\(422\) 10.3262 0.502673
\(423\) 0 0
\(424\) 23.9443 1.16284
\(425\) −0.746512 −0.0362112
\(426\) 0 0
\(427\) 0 0
\(428\) −19.3262 −0.934169
\(429\) 0 0
\(430\) −8.40647 −0.405396
\(431\) 3.41641 0.164563 0.0822813 0.996609i \(-0.473779\pi\)
0.0822813 + 0.996609i \(0.473779\pi\)
\(432\) 0 0
\(433\) −33.2433 −1.59757 −0.798786 0.601615i \(-0.794524\pi\)
−0.798786 + 0.601615i \(0.794524\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 24.1803 1.15803
\(437\) 29.2070 1.39716
\(438\) 0 0
\(439\) −35.7379 −1.70568 −0.852838 0.522175i \(-0.825121\pi\)
−0.852838 + 0.522175i \(0.825121\pi\)
\(440\) 5.11667 0.243928
\(441\) 0 0
\(442\) 8.94427 0.425436
\(443\) −15.7082 −0.746319 −0.373160 0.927767i \(-0.621726\pi\)
−0.373160 + 0.927767i \(0.621726\pi\)
\(444\) 0 0
\(445\) 14.4721 0.686045
\(446\) −3.28980 −0.155776
\(447\) 0 0
\(448\) 0 0
\(449\) 22.5279 1.06316 0.531578 0.847009i \(-0.321599\pi\)
0.531578 + 0.847009i \(0.321599\pi\)
\(450\) 0 0
\(451\) −5.78437 −0.272376
\(452\) 6.09017 0.286457
\(453\) 0 0
\(454\) −0.588890 −0.0276380
\(455\) 0 0
\(456\) 0 0
\(457\) −29.8328 −1.39552 −0.697760 0.716331i \(-0.745820\pi\)
−0.697760 + 0.716331i \(0.745820\pi\)
\(458\) −3.57494 −0.167046
\(459\) 0 0
\(460\) 26.7912 1.24915
\(461\) 8.69161 0.404809 0.202404 0.979302i \(-0.435124\pi\)
0.202404 + 0.979302i \(0.435124\pi\)
\(462\) 0 0
\(463\) −1.11146 −0.0516537 −0.0258269 0.999666i \(-0.508222\pi\)
−0.0258269 + 0.999666i \(0.508222\pi\)
\(464\) 2.29180 0.106394
\(465\) 0 0
\(466\) −3.12461 −0.144745
\(467\) −13.4443 −0.622129 −0.311065 0.950389i \(-0.600686\pi\)
−0.311065 + 0.950389i \(0.600686\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 4.29180 0.197966
\(471\) 0 0
\(472\) −32.3693 −1.48992
\(473\) 5.94427 0.273318
\(474\) 0 0
\(475\) −0.952843 −0.0437195
\(476\) 0 0
\(477\) 0 0
\(478\) 2.43769 0.111498
\(479\) 14.0633 0.642570 0.321285 0.946983i \(-0.395885\pi\)
0.321285 + 0.946983i \(0.395885\pi\)
\(480\) 0 0
\(481\) −49.0060 −2.23448
\(482\) −10.1058 −0.460308
\(483\) 0 0
\(484\) 16.1803 0.735470
\(485\) 14.1803 0.643896
\(486\) 0 0
\(487\) 3.94427 0.178732 0.0893660 0.995999i \(-0.471516\pi\)
0.0893660 + 0.995999i \(0.471516\pi\)
\(488\) 20.0053 0.905598
\(489\) 0 0
\(490\) 0 0
\(491\) 22.0000 0.992846 0.496423 0.868081i \(-0.334646\pi\)
0.496423 + 0.868081i \(0.334646\pi\)
\(492\) 0 0
\(493\) −3.90879 −0.176043
\(494\) 11.4164 0.513648
\(495\) 0 0
\(496\) −11.1074 −0.498736
\(497\) 0 0
\(498\) 0 0
\(499\) 21.8885 0.979866 0.489933 0.871760i \(-0.337021\pi\)
0.489933 + 0.871760i \(0.337021\pi\)
\(500\) 17.6383 0.788807
\(501\) 0 0
\(502\) 11.4412 0.510647
\(503\) −6.48218 −0.289026 −0.144513 0.989503i \(-0.546162\pi\)
−0.144513 + 0.989503i \(0.546162\pi\)
\(504\) 0 0
\(505\) 42.8328 1.90604
\(506\) 4.47214 0.198811
\(507\) 0 0
\(508\) 28.5623 1.26725
\(509\) 21.4682 0.951563 0.475782 0.879563i \(-0.342165\pi\)
0.475782 + 0.879563i \(0.342165\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −18.7082 −0.826794
\(513\) 0 0
\(514\) −5.91189 −0.260762
\(515\) −13.2361 −0.583251
\(516\) 0 0
\(517\) −3.03476 −0.133469
\(518\) 0 0
\(519\) 0 0
\(520\) 23.4164 1.02688
\(521\) −9.23179 −0.404452 −0.202226 0.979339i \(-0.564818\pi\)
−0.202226 + 0.979339i \(0.564818\pi\)
\(522\) 0 0
\(523\) −1.28669 −0.0562632 −0.0281316 0.999604i \(-0.508956\pi\)
−0.0281316 + 0.999604i \(0.508956\pi\)
\(524\) −32.7820 −1.43209
\(525\) 0 0
\(526\) −6.03444 −0.263114
\(527\) 18.9443 0.825225
\(528\) 0 0
\(529\) 29.3607 1.27655
\(530\) −15.1437 −0.657800
\(531\) 0 0
\(532\) 0 0
\(533\) −26.4721 −1.14664
\(534\) 0 0
\(535\) 27.3314 1.18164
\(536\) 3.29180 0.142184
\(537\) 0 0
\(538\) −2.57339 −0.110947
\(539\) 0 0
\(540\) 0 0
\(541\) 30.5967 1.31546 0.657728 0.753255i \(-0.271517\pi\)
0.657728 + 0.753255i \(0.271517\pi\)
\(542\) −16.1753 −0.694790
\(543\) 0 0
\(544\) 17.7658 0.761702
\(545\) −34.1962 −1.46480
\(546\) 0 0
\(547\) 10.7082 0.457850 0.228925 0.973444i \(-0.426479\pi\)
0.228925 + 0.973444i \(0.426479\pi\)
\(548\) −13.3262 −0.569269
\(549\) 0 0
\(550\) −0.145898 −0.00622111
\(551\) −4.98915 −0.212545
\(552\) 0 0
\(553\) 0 0
\(554\) −6.32624 −0.268776
\(555\) 0 0
\(556\) 15.6839 0.665144
\(557\) 7.29180 0.308963 0.154482 0.987996i \(-0.450629\pi\)
0.154482 + 0.987996i \(0.450629\pi\)
\(558\) 0 0
\(559\) 27.2039 1.15060
\(560\) 0 0
\(561\) 0 0
\(562\) 3.09017 0.130351
\(563\) 27.9504 1.17797 0.588985 0.808144i \(-0.299528\pi\)
0.588985 + 0.808144i \(0.299528\pi\)
\(564\) 0 0
\(565\) −8.61280 −0.362343
\(566\) 10.1058 0.424780
\(567\) 0 0
\(568\) 12.2361 0.513414
\(569\) −15.5836 −0.653298 −0.326649 0.945146i \(-0.605920\pi\)
−0.326649 + 0.945146i \(0.605920\pi\)
\(570\) 0 0
\(571\) −17.8197 −0.745730 −0.372865 0.927886i \(-0.621624\pi\)
−0.372865 + 0.927886i \(0.621624\pi\)
\(572\) −7.40492 −0.309615
\(573\) 0 0
\(574\) 0 0
\(575\) −1.70820 −0.0712370
\(576\) 0 0
\(577\) 3.03476 0.126339 0.0631693 0.998003i \(-0.479879\pi\)
0.0631693 + 0.998003i \(0.479879\pi\)
\(578\) 4.32624 0.179948
\(579\) 0 0
\(580\) −4.57649 −0.190028
\(581\) 0 0
\(582\) 0 0
\(583\) 10.7082 0.443488
\(584\) −15.0649 −0.623389
\(585\) 0 0
\(586\) −8.35776 −0.345256
\(587\) 28.2055 1.16416 0.582082 0.813130i \(-0.302238\pi\)
0.582082 + 0.813130i \(0.302238\pi\)
\(588\) 0 0
\(589\) 24.1803 0.996334
\(590\) 20.4721 0.842824
\(591\) 0 0
\(592\) −19.8541 −0.815999
\(593\) 34.2750 1.40750 0.703752 0.710445i \(-0.251506\pi\)
0.703752 + 0.710445i \(0.251506\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 17.0344 0.697758
\(597\) 0 0
\(598\) 20.4667 0.836945
\(599\) −13.8328 −0.565194 −0.282597 0.959239i \(-0.591196\pi\)
−0.282597 + 0.959239i \(0.591196\pi\)
\(600\) 0 0
\(601\) 28.8245 1.17577 0.587887 0.808943i \(-0.299960\pi\)
0.587887 + 0.808943i \(0.299960\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 30.2705 1.23169
\(605\) −22.8825 −0.930304
\(606\) 0 0
\(607\) 2.20943 0.0896782 0.0448391 0.998994i \(-0.485722\pi\)
0.0448391 + 0.998994i \(0.485722\pi\)
\(608\) 22.6761 0.919638
\(609\) 0 0
\(610\) −12.6525 −0.512284
\(611\) −13.8885 −0.561870
\(612\) 0 0
\(613\) 16.8197 0.679340 0.339670 0.940545i \(-0.389685\pi\)
0.339670 + 0.940545i \(0.389685\pi\)
\(614\) 8.02391 0.323819
\(615\) 0 0
\(616\) 0 0
\(617\) −5.41641 −0.218056 −0.109028 0.994039i \(-0.534774\pi\)
−0.109028 + 0.994039i \(0.534774\pi\)
\(618\) 0 0
\(619\) 45.0485 1.81065 0.905326 0.424717i \(-0.139626\pi\)
0.905326 + 0.424717i \(0.139626\pi\)
\(620\) 22.1803 0.890784
\(621\) 0 0
\(622\) 7.78748 0.312249
\(623\) 0 0
\(624\) 0 0
\(625\) −26.1246 −1.04498
\(626\) −0.461370 −0.0184401
\(627\) 0 0
\(628\) 23.9628 0.956221
\(629\) 33.8623 1.35018
\(630\) 0 0
\(631\) −42.6525 −1.69797 −0.848984 0.528418i \(-0.822785\pi\)
−0.848984 + 0.528418i \(0.822785\pi\)
\(632\) −17.3607 −0.690571
\(633\) 0 0
\(634\) 21.5279 0.854981
\(635\) −40.3932 −1.60296
\(636\) 0 0
\(637\) 0 0
\(638\) −0.763932 −0.0302444
\(639\) 0 0
\(640\) 26.0447 1.02951
\(641\) −21.4721 −0.848098 −0.424049 0.905639i \(-0.639392\pi\)
−0.424049 + 0.905639i \(0.639392\pi\)
\(642\) 0 0
\(643\) 3.08347 0.121600 0.0608000 0.998150i \(-0.480635\pi\)
0.0608000 + 0.998150i \(0.480635\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −7.88854 −0.310371
\(647\) 36.7394 1.44438 0.722188 0.691696i \(-0.243136\pi\)
0.722188 + 0.691696i \(0.243136\pi\)
\(648\) 0 0
\(649\) −14.4760 −0.568232
\(650\) −0.667701 −0.0261894
\(651\) 0 0
\(652\) 6.09017 0.238509
\(653\) 16.7082 0.653843 0.326921 0.945052i \(-0.393989\pi\)
0.326921 + 0.945052i \(0.393989\pi\)
\(654\) 0 0
\(655\) 46.3607 1.81146
\(656\) −10.7248 −0.418734
\(657\) 0 0
\(658\) 0 0
\(659\) −16.2361 −0.632467 −0.316234 0.948681i \(-0.602418\pi\)
−0.316234 + 0.948681i \(0.602418\pi\)
\(660\) 0 0
\(661\) 4.98915 0.194056 0.0970278 0.995282i \(-0.469066\pi\)
0.0970278 + 0.995282i \(0.469066\pi\)
\(662\) −1.23607 −0.0480411
\(663\) 0 0
\(664\) −19.2588 −0.747387
\(665\) 0 0
\(666\) 0 0
\(667\) −8.94427 −0.346324
\(668\) −29.4133 −1.13804
\(669\) 0 0
\(670\) −2.08191 −0.0804314
\(671\) 8.94665 0.345382
\(672\) 0 0
\(673\) −21.6525 −0.834642 −0.417321 0.908759i \(-0.637031\pi\)
−0.417321 + 0.908759i \(0.637031\pi\)
\(674\) −3.20163 −0.123322
\(675\) 0 0
\(676\) −12.8541 −0.494389
\(677\) 7.07107 0.271763 0.135882 0.990725i \(-0.456613\pi\)
0.135882 + 0.990725i \(0.456613\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −16.1803 −0.620488
\(681\) 0 0
\(682\) 3.70246 0.141774
\(683\) −22.2361 −0.850839 −0.425420 0.904996i \(-0.639874\pi\)
−0.425420 + 0.904996i \(0.639874\pi\)
\(684\) 0 0
\(685\) 18.8461 0.720074
\(686\) 0 0
\(687\) 0 0
\(688\) 11.0213 0.420183
\(689\) 49.0060 1.86698
\(690\) 0 0
\(691\) −37.7711 −1.43688 −0.718440 0.695589i \(-0.755144\pi\)
−0.718440 + 0.695589i \(0.755144\pi\)
\(692\) 26.0447 0.990072
\(693\) 0 0
\(694\) −2.50658 −0.0951484
\(695\) −22.1803 −0.841348
\(696\) 0 0
\(697\) 18.2918 0.692851
\(698\) −11.5989 −0.439023
\(699\) 0 0
\(700\) 0 0
\(701\) 9.18034 0.346737 0.173368 0.984857i \(-0.444535\pi\)
0.173368 + 0.984857i \(0.444535\pi\)
\(702\) 0 0
\(703\) 43.2216 1.63013
\(704\) −0.236068 −0.00889715
\(705\) 0 0
\(706\) −20.5455 −0.773240
\(707\) 0 0
\(708\) 0 0
\(709\) 25.6525 0.963399 0.481699 0.876336i \(-0.340020\pi\)
0.481699 + 0.876336i \(0.340020\pi\)
\(710\) −7.73877 −0.290431
\(711\) 0 0
\(712\) 14.1421 0.529999
\(713\) 43.3491 1.62344
\(714\) 0 0
\(715\) 10.4721 0.391636
\(716\) 39.4164 1.47306
\(717\) 0 0
\(718\) 3.20163 0.119484
\(719\) 51.7556 1.93016 0.965079 0.261958i \(-0.0843680\pi\)
0.965079 + 0.261958i \(0.0843680\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.67376 0.0622910
\(723\) 0 0
\(724\) −26.9188 −1.00043
\(725\) 0.291796 0.0108370
\(726\) 0 0
\(727\) −48.1320 −1.78512 −0.892558 0.450933i \(-0.851091\pi\)
−0.892558 + 0.450933i \(0.851091\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 9.52786 0.352642
\(731\) −18.7974 −0.695248
\(732\) 0 0
\(733\) −5.78437 −0.213651 −0.106825 0.994278i \(-0.534069\pi\)
−0.106825 + 0.994278i \(0.534069\pi\)
\(734\) 2.82843 0.104399
\(735\) 0 0
\(736\) 40.6525 1.49847
\(737\) 1.47214 0.0542268
\(738\) 0 0
\(739\) 11.3607 0.417909 0.208955 0.977925i \(-0.432994\pi\)
0.208955 + 0.977925i \(0.432994\pi\)
\(740\) 39.6467 1.45744
\(741\) 0 0
\(742\) 0 0
\(743\) 39.2492 1.43991 0.719957 0.694018i \(-0.244161\pi\)
0.719957 + 0.694018i \(0.244161\pi\)
\(744\) 0 0
\(745\) −24.0903 −0.882602
\(746\) −8.14590 −0.298243
\(747\) 0 0
\(748\) 5.11667 0.187084
\(749\) 0 0
\(750\) 0 0
\(751\) −32.3050 −1.17882 −0.589412 0.807832i \(-0.700641\pi\)
−0.589412 + 0.807832i \(0.700641\pi\)
\(752\) −5.62675 −0.205186
\(753\) 0 0
\(754\) −3.49613 −0.127321
\(755\) −42.8090 −1.55798
\(756\) 0 0
\(757\) −42.0689 −1.52902 −0.764510 0.644612i \(-0.777019\pi\)
−0.764510 + 0.644612i \(0.777019\pi\)
\(758\) 0.214782 0.00780122
\(759\) 0 0
\(760\) −20.6525 −0.749144
\(761\) 20.2604 0.734437 0.367219 0.930135i \(-0.380310\pi\)
0.367219 + 0.930135i \(0.380310\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −36.2705 −1.31222
\(765\) 0 0
\(766\) 15.2225 0.550011
\(767\) −66.2492 −2.39212
\(768\) 0 0
\(769\) −14.1120 −0.508893 −0.254446 0.967087i \(-0.581893\pi\)
−0.254446 + 0.967087i \(0.581893\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 12.1803 0.438380
\(773\) −42.6026 −1.53231 −0.766155 0.642656i \(-0.777833\pi\)
−0.766155 + 0.642656i \(0.777833\pi\)
\(774\) 0 0
\(775\) −1.41421 −0.0508001
\(776\) 13.8570 0.497437
\(777\) 0 0
\(778\) 4.29180 0.153868
\(779\) 23.3475 0.836512
\(780\) 0 0
\(781\) 5.47214 0.195808
\(782\) −14.1421 −0.505722
\(783\) 0 0
\(784\) 0 0
\(785\) −33.8885 −1.20953
\(786\) 0 0
\(787\) 43.6343 1.55539 0.777697 0.628639i \(-0.216388\pi\)
0.777697 + 0.628639i \(0.216388\pi\)
\(788\) 13.6180 0.485122
\(789\) 0 0
\(790\) 10.9799 0.390646
\(791\) 0 0
\(792\) 0 0
\(793\) 40.9443 1.45397
\(794\) 19.7689 0.701572
\(795\) 0 0
\(796\) −7.07107 −0.250627
\(797\) −20.8794 −0.739585 −0.369792 0.929114i \(-0.620571\pi\)
−0.369792 + 0.929114i \(0.620571\pi\)
\(798\) 0 0
\(799\) 9.59675 0.339509
\(800\) −1.32624 −0.0468896
\(801\) 0 0
\(802\) 13.0213 0.459798
\(803\) −6.73722 −0.237751
\(804\) 0 0
\(805\) 0 0
\(806\) 16.9443 0.596837
\(807\) 0 0
\(808\) 41.8561 1.47249
\(809\) −23.5410 −0.827658 −0.413829 0.910355i \(-0.635809\pi\)
−0.413829 + 0.910355i \(0.635809\pi\)
\(810\) 0 0
\(811\) −50.4990 −1.77326 −0.886630 0.462479i \(-0.846960\pi\)
−0.886630 + 0.462479i \(0.846960\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 6.61803 0.231962
\(815\) −8.61280 −0.301693
\(816\) 0 0
\(817\) −23.9929 −0.839406
\(818\) −12.3153 −0.430593
\(819\) 0 0
\(820\) 21.4164 0.747893
\(821\) −10.8197 −0.377609 −0.188804 0.982015i \(-0.560461\pi\)
−0.188804 + 0.982015i \(0.560461\pi\)
\(822\) 0 0
\(823\) 17.4164 0.607098 0.303549 0.952816i \(-0.401828\pi\)
0.303549 + 0.952816i \(0.401828\pi\)
\(824\) −12.9343 −0.450586
\(825\) 0 0
\(826\) 0 0
\(827\) 9.65248 0.335649 0.167825 0.985817i \(-0.446326\pi\)
0.167825 + 0.985817i \(0.446326\pi\)
\(828\) 0 0
\(829\) −3.24109 −0.112568 −0.0562838 0.998415i \(-0.517925\pi\)
−0.0562838 + 0.998415i \(0.517925\pi\)
\(830\) 12.1803 0.422786
\(831\) 0 0
\(832\) −1.08036 −0.0374548
\(833\) 0 0
\(834\) 0 0
\(835\) 41.5967 1.43951
\(836\) 6.53089 0.225875
\(837\) 0 0
\(838\) 14.2209 0.491254
\(839\) 19.0038 0.656083 0.328041 0.944663i \(-0.393611\pi\)
0.328041 + 0.944663i \(0.393611\pi\)
\(840\) 0 0
\(841\) −27.4721 −0.947315
\(842\) 12.9098 0.444902
\(843\) 0 0
\(844\) 27.0344 0.930564
\(845\) 18.1784 0.625358
\(846\) 0 0
\(847\) 0 0
\(848\) 19.8541 0.681793
\(849\) 0 0
\(850\) 0.461370 0.0158249
\(851\) 77.4853 2.65616
\(852\) 0 0
\(853\) −37.7410 −1.29223 −0.646114 0.763241i \(-0.723607\pi\)
−0.646114 + 0.763241i \(0.723607\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 26.7082 0.912868
\(857\) −55.2517 −1.88736 −0.943682 0.330854i \(-0.892663\pi\)
−0.943682 + 0.330854i \(0.892663\pi\)
\(858\) 0 0
\(859\) −41.3460 −1.41071 −0.705354 0.708855i \(-0.749212\pi\)
−0.705354 + 0.708855i \(0.749212\pi\)
\(860\) −22.0084 −0.750481
\(861\) 0 0
\(862\) −2.11146 −0.0719165
\(863\) 40.3050 1.37200 0.685998 0.727603i \(-0.259366\pi\)
0.685998 + 0.727603i \(0.259366\pi\)
\(864\) 0 0
\(865\) −36.8328 −1.25235
\(866\) 20.5455 0.698165
\(867\) 0 0
\(868\) 0 0
\(869\) −7.76393 −0.263373
\(870\) 0 0
\(871\) 6.73722 0.228282
\(872\) −33.4164 −1.13162
\(873\) 0 0
\(874\) −18.0509 −0.610582
\(875\) 0 0
\(876\) 0 0
\(877\) 14.2361 0.480718 0.240359 0.970684i \(-0.422735\pi\)
0.240359 + 0.970684i \(0.422735\pi\)
\(878\) 22.0872 0.745408
\(879\) 0 0
\(880\) 4.24264 0.143019
\(881\) 24.9157 0.839430 0.419715 0.907656i \(-0.362130\pi\)
0.419715 + 0.907656i \(0.362130\pi\)
\(882\) 0 0
\(883\) 10.3475 0.348222 0.174111 0.984726i \(-0.444295\pi\)
0.174111 + 0.984726i \(0.444295\pi\)
\(884\) 23.4164 0.787579
\(885\) 0 0
\(886\) 9.70820 0.326153
\(887\) −14.1421 −0.474846 −0.237423 0.971406i \(-0.576303\pi\)
−0.237423 + 0.971406i \(0.576303\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −8.94427 −0.299813
\(891\) 0 0
\(892\) −8.61280 −0.288378
\(893\) 12.2492 0.409905
\(894\) 0 0
\(895\) −55.7432 −1.86329
\(896\) 0 0
\(897\) 0 0
\(898\) −13.9230 −0.464616
\(899\) −7.40492 −0.246968
\(900\) 0 0
\(901\) −33.8623 −1.12812
\(902\) 3.57494 0.119032
\(903\) 0 0
\(904\) −8.41641 −0.279926
\(905\) 38.0689 1.26545
\(906\) 0 0
\(907\) −55.5410 −1.84421 −0.922105 0.386941i \(-0.873532\pi\)
−0.922105 + 0.386941i \(0.873532\pi\)
\(908\) −1.54173 −0.0511642
\(909\) 0 0
\(910\) 0 0
\(911\) 40.2492 1.33352 0.666758 0.745274i \(-0.267681\pi\)
0.666758 + 0.745274i \(0.267681\pi\)
\(912\) 0 0
\(913\) −8.61280 −0.285042
\(914\) 18.4377 0.609865
\(915\) 0 0
\(916\) −9.35931 −0.309240
\(917\) 0 0
\(918\) 0 0
\(919\) −3.18034 −0.104910 −0.0524549 0.998623i \(-0.516705\pi\)
−0.0524549 + 0.998623i \(0.516705\pi\)
\(920\) −37.0246 −1.22066
\(921\) 0 0
\(922\) −5.37171 −0.176908
\(923\) 25.0432 0.824306
\(924\) 0 0
\(925\) −2.52786 −0.0831157
\(926\) 0.686918 0.0225735
\(927\) 0 0
\(928\) −6.94427 −0.227957
\(929\) 32.0841 1.05265 0.526323 0.850284i \(-0.323570\pi\)
0.526323 + 0.850284i \(0.323570\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −8.18034 −0.267956
\(933\) 0 0
\(934\) 8.30905 0.271881
\(935\) −7.23607 −0.236645
\(936\) 0 0
\(937\) 51.1667 1.67154 0.835772 0.549077i \(-0.185020\pi\)
0.835772 + 0.549077i \(0.185020\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 11.2361 0.366480
\(941\) 21.2920 0.694100 0.347050 0.937847i \(-0.387183\pi\)
0.347050 + 0.937847i \(0.387183\pi\)
\(942\) 0 0
\(943\) 41.8561 1.36302
\(944\) −26.8400 −0.873566
\(945\) 0 0
\(946\) −3.67376 −0.119444
\(947\) −42.7639 −1.38964 −0.694821 0.719183i \(-0.744516\pi\)
−0.694821 + 0.719183i \(0.744516\pi\)
\(948\) 0 0
\(949\) −30.8328 −1.00088
\(950\) 0.588890 0.0191061
\(951\) 0 0
\(952\) 0 0
\(953\) 57.4164 1.85990 0.929950 0.367686i \(-0.119850\pi\)
0.929950 + 0.367686i \(0.119850\pi\)
\(954\) 0 0
\(955\) 51.2942 1.65984
\(956\) 6.38197 0.206408
\(957\) 0 0
\(958\) −8.69161 −0.280813
\(959\) 0 0
\(960\) 0 0
\(961\) 4.88854 0.157695
\(962\) 30.2874 0.976504
\(963\) 0 0
\(964\) −26.4574 −0.852135
\(965\) −17.2256 −0.554512
\(966\) 0 0
\(967\) −0.472136 −0.0151829 −0.00759143 0.999971i \(-0.502416\pi\)
−0.00759143 + 0.999971i \(0.502416\pi\)
\(968\) −22.3607 −0.718699
\(969\) 0 0
\(970\) −8.76393 −0.281393
\(971\) −5.99070 −0.192251 −0.0961254 0.995369i \(-0.530645\pi\)
−0.0961254 + 0.995369i \(0.530645\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −2.43769 −0.0781088
\(975\) 0 0
\(976\) 16.5880 0.530969
\(977\) 31.3050 1.00153 0.500767 0.865582i \(-0.333051\pi\)
0.500767 + 0.865582i \(0.333051\pi\)
\(978\) 0 0
\(979\) 6.32456 0.202134
\(980\) 0 0
\(981\) 0 0
\(982\) −13.5967 −0.433890
\(983\) 2.36706 0.0754974 0.0377487 0.999287i \(-0.487981\pi\)
0.0377487 + 0.999287i \(0.487981\pi\)
\(984\) 0 0
\(985\) −19.2588 −0.613637
\(986\) 2.41577 0.0769336
\(987\) 0 0
\(988\) 29.8885 0.950881
\(989\) −43.0132 −1.36774
\(990\) 0 0
\(991\) 20.4164 0.648549 0.324274 0.945963i \(-0.394880\pi\)
0.324274 + 0.945963i \(0.394880\pi\)
\(992\) 33.6560 1.06858
\(993\) 0 0
\(994\) 0 0
\(995\) 10.0000 0.317021
\(996\) 0 0
\(997\) 19.8778 0.629536 0.314768 0.949169i \(-0.398073\pi\)
0.314768 + 0.949169i \(0.398073\pi\)
\(998\) −13.5279 −0.428217
\(999\) 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 3969.2.a.y.1.2 yes 4
3.2 odd 2 3969.2.a.r.1.3 4
7.6 odd 2 inner 3969.2.a.y.1.1 yes 4
21.20 even 2 3969.2.a.r.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3969.2.a.r.1.3 4 3.2 odd 2
3969.2.a.r.1.4 yes 4 21.20 even 2
3969.2.a.y.1.1 yes 4 7.6 odd 2 inner
3969.2.a.y.1.2 yes 4 1.1 even 1 trivial