Properties

Label 1183.2.c.c.337.1
Level $1183$
Weight $2$
Character 1183.337
Analytic conductor $9.446$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 1183.337
Dual form 1183.2.c.c.337.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.61803i q^{2} -2.61803 q^{3} -4.85410 q^{4} -2.61803i q^{5} +6.85410i q^{6} +1.00000i q^{7} +7.47214i q^{8} +3.85410 q^{9} +O(q^{10})\) \(q-2.61803i q^{2} -2.61803 q^{3} -4.85410 q^{4} -2.61803i q^{5} +6.85410i q^{6} +1.00000i q^{7} +7.47214i q^{8} +3.85410 q^{9} -6.85410 q^{10} +1.85410i q^{11} +12.7082 q^{12} +2.61803 q^{14} +6.85410i q^{15} +9.85410 q^{16} +1.47214 q^{17} -10.0902i q^{18} +1.85410i q^{19} +12.7082i q^{20} -2.61803i q^{21} +4.85410 q^{22} +4.47214 q^{23} -19.5623i q^{24} -1.85410 q^{25} -2.23607 q^{27} -4.85410i q^{28} +7.09017 q^{29} +17.9443 q^{30} +4.70820i q^{31} -10.8541i q^{32} -4.85410i q^{33} -3.85410i q^{34} +2.61803 q^{35} -18.7082 q^{36} +4.00000i q^{37} +4.85410 q^{38} +19.5623 q^{40} -0.763932i q^{41} -6.85410 q^{42} -12.5623 q^{43} -9.00000i q^{44} -10.0902i q^{45} -11.7082i q^{46} -2.23607i q^{47} -25.7984 q^{48} -1.00000 q^{49} +4.85410i q^{50} -3.85410 q^{51} +3.76393 q^{53} +5.85410i q^{54} +4.85410 q^{55} -7.47214 q^{56} -4.85410i q^{57} -18.5623i q^{58} -2.23607i q^{59} -33.2705i q^{60} -6.00000 q^{61} +12.3262 q^{62} +3.85410i q^{63} -8.70820 q^{64} -12.7082 q^{66} +12.7082i q^{67} -7.14590 q^{68} -11.7082 q^{69} -6.85410i q^{70} +14.1803i q^{71} +28.7984i q^{72} -2.00000i q^{73} +10.4721 q^{74} +4.85410 q^{75} -9.00000i q^{76} -1.85410 q^{77} +4.00000 q^{79} -25.7984i q^{80} -5.70820 q^{81} -2.00000 q^{82} -6.70820i q^{83} +12.7082i q^{84} -3.85410i q^{85} +32.8885i q^{86} -18.5623 q^{87} -13.8541 q^{88} +4.90983i q^{89} -26.4164 q^{90} -21.7082 q^{92} -12.3262i q^{93} -5.85410 q^{94} +4.85410 q^{95} +28.4164i q^{96} -18.8541i q^{97} +2.61803i q^{98} +7.14590i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{3} - 6q^{4} + 2q^{9} + O(q^{10}) \) \( 4q - 6q^{3} - 6q^{4} + 2q^{9} - 14q^{10} + 24q^{12} + 6q^{14} + 26q^{16} - 12q^{17} + 6q^{22} + 6q^{25} + 6q^{29} + 36q^{30} + 6q^{35} - 48q^{36} + 6q^{38} + 38q^{40} - 14q^{42} - 10q^{43} - 54q^{48} - 4q^{49} - 2q^{51} + 24q^{53} + 6q^{55} - 12q^{56} - 24q^{61} + 18q^{62} - 8q^{64} - 24q^{66} - 42q^{68} - 20q^{69} + 24q^{74} + 6q^{75} + 6q^{77} + 16q^{79} + 4q^{81} - 8q^{82} - 34q^{87} - 42q^{88} - 52q^{90} - 60q^{92} - 10q^{94} + 6q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.61803i − 1.85123i −0.378467 0.925615i \(-0.623549\pi\)
0.378467 0.925615i \(-0.376451\pi\)
\(3\) −2.61803 −1.51152 −0.755761 0.654847i \(-0.772733\pi\)
−0.755761 + 0.654847i \(0.772733\pi\)
\(4\) −4.85410 −2.42705
\(5\) − 2.61803i − 1.17082i −0.810737 0.585410i \(-0.800933\pi\)
0.810737 0.585410i \(-0.199067\pi\)
\(6\) 6.85410i 2.79818i
\(7\) 1.00000i 0.377964i
\(8\) 7.47214i 2.64180i
\(9\) 3.85410 1.28470
\(10\) −6.85410 −2.16746
\(11\) 1.85410i 0.559033i 0.960141 + 0.279516i \(0.0901741\pi\)
−0.960141 + 0.279516i \(0.909826\pi\)
\(12\) 12.7082 3.66854
\(13\) 0 0
\(14\) 2.61803 0.699699
\(15\) 6.85410i 1.76972i
\(16\) 9.85410 2.46353
\(17\) 1.47214 0.357045 0.178523 0.983936i \(-0.442868\pi\)
0.178523 + 0.983936i \(0.442868\pi\)
\(18\) − 10.0902i − 2.37828i
\(19\) 1.85410i 0.425360i 0.977122 + 0.212680i \(0.0682192\pi\)
−0.977122 + 0.212680i \(0.931781\pi\)
\(20\) 12.7082i 2.84164i
\(21\) − 2.61803i − 0.571302i
\(22\) 4.85410 1.03490
\(23\) 4.47214 0.932505 0.466252 0.884652i \(-0.345604\pi\)
0.466252 + 0.884652i \(0.345604\pi\)
\(24\) − 19.5623i − 3.99314i
\(25\) −1.85410 −0.370820
\(26\) 0 0
\(27\) −2.23607 −0.430331
\(28\) − 4.85410i − 0.917339i
\(29\) 7.09017 1.31661 0.658306 0.752751i \(-0.271273\pi\)
0.658306 + 0.752751i \(0.271273\pi\)
\(30\) 17.9443 3.27616
\(31\) 4.70820i 0.845618i 0.906219 + 0.422809i \(0.138956\pi\)
−0.906219 + 0.422809i \(0.861044\pi\)
\(32\) − 10.8541i − 1.91875i
\(33\) − 4.85410i − 0.844991i
\(34\) − 3.85410i − 0.660973i
\(35\) 2.61803 0.442529
\(36\) −18.7082 −3.11803
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 4.85410 0.787439
\(39\) 0 0
\(40\) 19.5623 3.09307
\(41\) − 0.763932i − 0.119306i −0.998219 0.0596531i \(-0.981001\pi\)
0.998219 0.0596531i \(-0.0189994\pi\)
\(42\) −6.85410 −1.05761
\(43\) −12.5623 −1.91573 −0.957867 0.287213i \(-0.907271\pi\)
−0.957867 + 0.287213i \(0.907271\pi\)
\(44\) − 9.00000i − 1.35680i
\(45\) − 10.0902i − 1.50415i
\(46\) − 11.7082i − 1.72628i
\(47\) − 2.23607i − 0.326164i −0.986613 0.163082i \(-0.947856\pi\)
0.986613 0.163082i \(-0.0521435\pi\)
\(48\) −25.7984 −3.72367
\(49\) −1.00000 −0.142857
\(50\) 4.85410i 0.686474i
\(51\) −3.85410 −0.539682
\(52\) 0 0
\(53\) 3.76393 0.517016 0.258508 0.966009i \(-0.416769\pi\)
0.258508 + 0.966009i \(0.416769\pi\)
\(54\) 5.85410i 0.796642i
\(55\) 4.85410 0.654527
\(56\) −7.47214 −0.998506
\(57\) − 4.85410i − 0.642942i
\(58\) − 18.5623i − 2.43735i
\(59\) − 2.23607i − 0.291111i −0.989350 0.145556i \(-0.953503\pi\)
0.989350 0.145556i \(-0.0464970\pi\)
\(60\) − 33.2705i − 4.29520i
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 12.3262 1.56543
\(63\) 3.85410i 0.485571i
\(64\) −8.70820 −1.08853
\(65\) 0 0
\(66\) −12.7082 −1.56427
\(67\) 12.7082i 1.55255i 0.630392 + 0.776277i \(0.282894\pi\)
−0.630392 + 0.776277i \(0.717106\pi\)
\(68\) −7.14590 −0.866567
\(69\) −11.7082 −1.40950
\(70\) − 6.85410i − 0.819222i
\(71\) 14.1803i 1.68290i 0.540338 + 0.841448i \(0.318297\pi\)
−0.540338 + 0.841448i \(0.681703\pi\)
\(72\) 28.7984i 3.39392i
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 10.4721 1.21736
\(75\) 4.85410 0.560503
\(76\) − 9.00000i − 1.03237i
\(77\) −1.85410 −0.211295
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) − 25.7984i − 2.88435i
\(81\) −5.70820 −0.634245
\(82\) −2.00000 −0.220863
\(83\) − 6.70820i − 0.736321i −0.929762 0.368161i \(-0.879988\pi\)
0.929762 0.368161i \(-0.120012\pi\)
\(84\) 12.7082i 1.38658i
\(85\) − 3.85410i − 0.418036i
\(86\) 32.8885i 3.54646i
\(87\) −18.5623 −1.99009
\(88\) −13.8541 −1.47685
\(89\) 4.90983i 0.520441i 0.965549 + 0.260220i \(0.0837952\pi\)
−0.965549 + 0.260220i \(0.916205\pi\)
\(90\) −26.4164 −2.78453
\(91\) 0 0
\(92\) −21.7082 −2.26324
\(93\) − 12.3262i − 1.27817i
\(94\) −5.85410 −0.603805
\(95\) 4.85410 0.498020
\(96\) 28.4164i 2.90024i
\(97\) − 18.8541i − 1.91434i −0.289520 0.957172i \(-0.593496\pi\)
0.289520 0.957172i \(-0.406504\pi\)
\(98\) 2.61803i 0.264461i
\(99\) 7.14590i 0.718190i
\(100\) 9.00000 0.900000
\(101\) 11.5623 1.15049 0.575246 0.817980i \(-0.304906\pi\)
0.575246 + 0.817980i \(0.304906\pi\)
\(102\) 10.0902i 0.999076i
\(103\) 8.70820 0.858045 0.429022 0.903294i \(-0.358858\pi\)
0.429022 + 0.903294i \(0.358858\pi\)
\(104\) 0 0
\(105\) −6.85410 −0.668892
\(106\) − 9.85410i − 0.957115i
\(107\) −3.38197 −0.326947 −0.163473 0.986548i \(-0.552270\pi\)
−0.163473 + 0.986548i \(0.552270\pi\)
\(108\) 10.8541 1.04444
\(109\) 2.70820i 0.259399i 0.991553 + 0.129699i \(0.0414012\pi\)
−0.991553 + 0.129699i \(0.958599\pi\)
\(110\) − 12.7082i − 1.21168i
\(111\) − 10.4721i − 0.993971i
\(112\) 9.85410i 0.931125i
\(113\) 1.47214 0.138487 0.0692435 0.997600i \(-0.477941\pi\)
0.0692435 + 0.997600i \(0.477941\pi\)
\(114\) −12.7082 −1.19023
\(115\) − 11.7082i − 1.09180i
\(116\) −34.4164 −3.19548
\(117\) 0 0
\(118\) −5.85410 −0.538914
\(119\) 1.47214i 0.134950i
\(120\) −51.2148 −4.67525
\(121\) 7.56231 0.687482
\(122\) 15.7082i 1.42215i
\(123\) 2.00000i 0.180334i
\(124\) − 22.8541i − 2.05236i
\(125\) − 8.23607i − 0.736656i
\(126\) 10.0902 0.898904
\(127\) 20.8541 1.85050 0.925251 0.379355i \(-0.123854\pi\)
0.925251 + 0.379355i \(0.123854\pi\)
\(128\) 1.09017i 0.0963583i
\(129\) 32.8885 2.89567
\(130\) 0 0
\(131\) −15.3262 −1.33906 −0.669530 0.742785i \(-0.733504\pi\)
−0.669530 + 0.742785i \(0.733504\pi\)
\(132\) 23.5623i 2.05084i
\(133\) −1.85410 −0.160771
\(134\) 33.2705 2.87413
\(135\) 5.85410i 0.503841i
\(136\) 11.0000i 0.943242i
\(137\) − 2.61803i − 0.223674i −0.993727 0.111837i \(-0.964327\pi\)
0.993727 0.111837i \(-0.0356734\pi\)
\(138\) 30.6525i 2.60931i
\(139\) 4.56231 0.386970 0.193485 0.981103i \(-0.438021\pi\)
0.193485 + 0.981103i \(0.438021\pi\)
\(140\) −12.7082 −1.07404
\(141\) 5.85410i 0.493004i
\(142\) 37.1246 3.11543
\(143\) 0 0
\(144\) 37.9787 3.16489
\(145\) − 18.5623i − 1.54152i
\(146\) −5.23607 −0.433340
\(147\) 2.61803 0.215932
\(148\) − 19.4164i − 1.59602i
\(149\) − 1.85410i − 0.151894i −0.997112 0.0759470i \(-0.975802\pi\)
0.997112 0.0759470i \(-0.0241980\pi\)
\(150\) − 12.7082i − 1.03762i
\(151\) − 1.29180i − 0.105125i −0.998618 0.0525624i \(-0.983261\pi\)
0.998618 0.0525624i \(-0.0167389\pi\)
\(152\) −13.8541 −1.12372
\(153\) 5.67376 0.458696
\(154\) 4.85410i 0.391155i
\(155\) 12.3262 0.990067
\(156\) 0 0
\(157\) 14.8541 1.18549 0.592743 0.805392i \(-0.298045\pi\)
0.592743 + 0.805392i \(0.298045\pi\)
\(158\) − 10.4721i − 0.833118i
\(159\) −9.85410 −0.781481
\(160\) −28.4164 −2.24651
\(161\) 4.47214i 0.352454i
\(162\) 14.9443i 1.17413i
\(163\) − 3.70820i − 0.290449i −0.989399 0.145224i \(-0.953610\pi\)
0.989399 0.145224i \(-0.0463904\pi\)
\(164\) 3.70820i 0.289562i
\(165\) −12.7082 −0.989332
\(166\) −17.5623 −1.36310
\(167\) 14.2361i 1.10162i 0.834631 + 0.550810i \(0.185681\pi\)
−0.834631 + 0.550810i \(0.814319\pi\)
\(168\) 19.5623 1.50926
\(169\) 0 0
\(170\) −10.0902 −0.773881
\(171\) 7.14590i 0.546460i
\(172\) 60.9787 4.64958
\(173\) 9.00000 0.684257 0.342129 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) 48.5967i 3.68411i
\(175\) − 1.85410i − 0.140157i
\(176\) 18.2705i 1.37719i
\(177\) 5.85410i 0.440021i
\(178\) 12.8541 0.963456
\(179\) 9.00000 0.672692 0.336346 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 48.9787i 3.65066i
\(181\) −9.70820 −0.721605 −0.360803 0.932642i \(-0.617497\pi\)
−0.360803 + 0.932642i \(0.617497\pi\)
\(182\) 0 0
\(183\) 15.7082 1.16118
\(184\) 33.4164i 2.46349i
\(185\) 10.4721 0.769927
\(186\) −32.2705 −2.36619
\(187\) 2.72949i 0.199600i
\(188\) 10.8541i 0.791617i
\(189\) − 2.23607i − 0.162650i
\(190\) − 12.7082i − 0.921950i
\(191\) 21.3820 1.54714 0.773572 0.633708i \(-0.218468\pi\)
0.773572 + 0.633708i \(0.218468\pi\)
\(192\) 22.7984 1.64533
\(193\) − 6.00000i − 0.431889i −0.976406 0.215945i \(-0.930717\pi\)
0.976406 0.215945i \(-0.0692831\pi\)
\(194\) −49.3607 −3.54389
\(195\) 0 0
\(196\) 4.85410 0.346722
\(197\) 16.7984i 1.19683i 0.801185 + 0.598417i \(0.204203\pi\)
−0.801185 + 0.598417i \(0.795797\pi\)
\(198\) 18.7082 1.32953
\(199\) 24.4164 1.73083 0.865417 0.501053i \(-0.167054\pi\)
0.865417 + 0.501053i \(0.167054\pi\)
\(200\) − 13.8541i − 0.979633i
\(201\) − 33.2705i − 2.34672i
\(202\) − 30.2705i − 2.12983i
\(203\) 7.09017i 0.497632i
\(204\) 18.7082 1.30984
\(205\) −2.00000 −0.139686
\(206\) − 22.7984i − 1.58844i
\(207\) 17.2361 1.19799
\(208\) 0 0
\(209\) −3.43769 −0.237790
\(210\) 17.9443i 1.23827i
\(211\) 4.70820 0.324126 0.162063 0.986780i \(-0.448185\pi\)
0.162063 + 0.986780i \(0.448185\pi\)
\(212\) −18.2705 −1.25482
\(213\) − 37.1246i − 2.54374i
\(214\) 8.85410i 0.605254i
\(215\) 32.8885i 2.24298i
\(216\) − 16.7082i − 1.13685i
\(217\) −4.70820 −0.319614
\(218\) 7.09017 0.480207
\(219\) 5.23607i 0.353821i
\(220\) −23.5623 −1.58857
\(221\) 0 0
\(222\) −27.4164 −1.84007
\(223\) − 20.2705i − 1.35741i −0.734409 0.678707i \(-0.762541\pi\)
0.734409 0.678707i \(-0.237459\pi\)
\(224\) 10.8541 0.725220
\(225\) −7.14590 −0.476393
\(226\) − 3.85410i − 0.256371i
\(227\) − 1.47214i − 0.0977091i −0.998806 0.0488545i \(-0.984443\pi\)
0.998806 0.0488545i \(-0.0155571\pi\)
\(228\) 23.5623i 1.56045i
\(229\) − 13.1246i − 0.867299i −0.901082 0.433649i \(-0.857226\pi\)
0.901082 0.433649i \(-0.142774\pi\)
\(230\) −30.6525 −2.02116
\(231\) 4.85410 0.319376
\(232\) 52.9787i 3.47822i
\(233\) −2.61803 −0.171513 −0.0857566 0.996316i \(-0.527331\pi\)
−0.0857566 + 0.996316i \(0.527331\pi\)
\(234\) 0 0
\(235\) −5.85410 −0.381880
\(236\) 10.8541i 0.706542i
\(237\) −10.4721 −0.680238
\(238\) 3.85410 0.249824
\(239\) 24.7082i 1.59824i 0.601171 + 0.799120i \(0.294701\pi\)
−0.601171 + 0.799120i \(0.705299\pi\)
\(240\) 67.5410i 4.35975i
\(241\) 24.5623i 1.58220i 0.611689 + 0.791099i \(0.290491\pi\)
−0.611689 + 0.791099i \(0.709509\pi\)
\(242\) − 19.7984i − 1.27269i
\(243\) 21.6525 1.38901
\(244\) 29.1246 1.86451
\(245\) 2.61803i 0.167260i
\(246\) 5.23607 0.333840
\(247\) 0 0
\(248\) −35.1803 −2.23395
\(249\) 17.5623i 1.11297i
\(250\) −21.5623 −1.36372
\(251\) −0.763932 −0.0482190 −0.0241095 0.999709i \(-0.507675\pi\)
−0.0241095 + 0.999709i \(0.507675\pi\)
\(252\) − 18.7082i − 1.17851i
\(253\) 8.29180i 0.521301i
\(254\) − 54.5967i − 3.42570i
\(255\) 10.0902i 0.631871i
\(256\) −14.5623 −0.910144
\(257\) −16.7426 −1.04438 −0.522189 0.852830i \(-0.674884\pi\)
−0.522189 + 0.852830i \(0.674884\pi\)
\(258\) − 86.1033i − 5.36056i
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 27.3262 1.69145
\(262\) 40.1246i 2.47891i
\(263\) 9.00000 0.554964 0.277482 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) 36.2705 2.23230
\(265\) − 9.85410i − 0.605333i
\(266\) 4.85410i 0.297624i
\(267\) − 12.8541i − 0.786658i
\(268\) − 61.6869i − 3.76813i
\(269\) −28.7426 −1.75247 −0.876235 0.481884i \(-0.839953\pi\)
−0.876235 + 0.481884i \(0.839953\pi\)
\(270\) 15.3262 0.932725
\(271\) − 8.41641i − 0.511260i −0.966775 0.255630i \(-0.917717\pi\)
0.966775 0.255630i \(-0.0822829\pi\)
\(272\) 14.5066 0.879590
\(273\) 0 0
\(274\) −6.85410 −0.414071
\(275\) − 3.43769i − 0.207301i
\(276\) 56.8328 3.42093
\(277\) 5.00000 0.300421 0.150210 0.988654i \(-0.452005\pi\)
0.150210 + 0.988654i \(0.452005\pi\)
\(278\) − 11.9443i − 0.716370i
\(279\) 18.1459i 1.08637i
\(280\) 19.5623i 1.16907i
\(281\) 20.1803i 1.20386i 0.798550 + 0.601929i \(0.205601\pi\)
−0.798550 + 0.601929i \(0.794399\pi\)
\(282\) 15.3262 0.912664
\(283\) −13.4164 −0.797523 −0.398761 0.917055i \(-0.630560\pi\)
−0.398761 + 0.917055i \(0.630560\pi\)
\(284\) − 68.8328i − 4.08448i
\(285\) −12.7082 −0.752769
\(286\) 0 0
\(287\) 0.763932 0.0450935
\(288\) − 41.8328i − 2.46502i
\(289\) −14.8328 −0.872519
\(290\) −48.5967 −2.85370
\(291\) 49.3607i 2.89357i
\(292\) 9.70820i 0.568130i
\(293\) − 6.76393i − 0.395153i −0.980287 0.197577i \(-0.936693\pi\)
0.980287 0.197577i \(-0.0633071\pi\)
\(294\) − 6.85410i − 0.399739i
\(295\) −5.85410 −0.340839
\(296\) −29.8885 −1.73724
\(297\) − 4.14590i − 0.240569i
\(298\) −4.85410 −0.281191
\(299\) 0 0
\(300\) −23.5623 −1.36037
\(301\) − 12.5623i − 0.724079i
\(302\) −3.38197 −0.194610
\(303\) −30.2705 −1.73900
\(304\) 18.2705i 1.04789i
\(305\) 15.7082i 0.899449i
\(306\) − 14.8541i − 0.849152i
\(307\) − 4.85410i − 0.277038i −0.990360 0.138519i \(-0.955766\pi\)
0.990360 0.138519i \(-0.0442342\pi\)
\(308\) 9.00000 0.512823
\(309\) −22.7984 −1.29695
\(310\) − 32.2705i − 1.83284i
\(311\) −3.32624 −0.188614 −0.0943068 0.995543i \(-0.530063\pi\)
−0.0943068 + 0.995543i \(0.530063\pi\)
\(312\) 0 0
\(313\) 25.1246 1.42013 0.710064 0.704138i \(-0.248666\pi\)
0.710064 + 0.704138i \(0.248666\pi\)
\(314\) − 38.8885i − 2.19461i
\(315\) 10.0902 0.568517
\(316\) −19.4164 −1.09226
\(317\) − 26.2361i − 1.47356i −0.676130 0.736782i \(-0.736344\pi\)
0.676130 0.736782i \(-0.263656\pi\)
\(318\) 25.7984i 1.44670i
\(319\) 13.1459i 0.736029i
\(320\) 22.7984i 1.27447i
\(321\) 8.85410 0.494188
\(322\) 11.7082 0.652473
\(323\) 2.72949i 0.151873i
\(324\) 27.7082 1.53934
\(325\) 0 0
\(326\) −9.70820 −0.537688
\(327\) − 7.09017i − 0.392087i
\(328\) 5.70820 0.315183
\(329\) 2.23607 0.123278
\(330\) 33.2705i 1.83148i
\(331\) 10.1459i 0.557669i 0.960339 + 0.278834i \(0.0899481\pi\)
−0.960339 + 0.278834i \(0.910052\pi\)
\(332\) 32.5623i 1.78709i
\(333\) 15.4164i 0.844814i
\(334\) 37.2705 2.03935
\(335\) 33.2705 1.81776
\(336\) − 25.7984i − 1.40742i
\(337\) 11.5623 0.629839 0.314919 0.949118i \(-0.398022\pi\)
0.314919 + 0.949118i \(0.398022\pi\)
\(338\) 0 0
\(339\) −3.85410 −0.209326
\(340\) 18.7082i 1.01459i
\(341\) −8.72949 −0.472728
\(342\) 18.7082 1.01162
\(343\) − 1.00000i − 0.0539949i
\(344\) − 93.8673i − 5.06098i
\(345\) 30.6525i 1.65027i
\(346\) − 23.5623i − 1.26672i
\(347\) 30.7639 1.65149 0.825747 0.564040i \(-0.190754\pi\)
0.825747 + 0.564040i \(0.190754\pi\)
\(348\) 90.1033 4.83005
\(349\) 20.7082i 1.10848i 0.832355 + 0.554242i \(0.186992\pi\)
−0.832355 + 0.554242i \(0.813008\pi\)
\(350\) −4.85410 −0.259463
\(351\) 0 0
\(352\) 20.1246 1.07265
\(353\) − 22.1459i − 1.17871i −0.807875 0.589354i \(-0.799382\pi\)
0.807875 0.589354i \(-0.200618\pi\)
\(354\) 15.3262 0.814580
\(355\) 37.1246 1.97037
\(356\) − 23.8328i − 1.26314i
\(357\) − 3.85410i − 0.203981i
\(358\) − 23.5623i − 1.24531i
\(359\) − 22.0902i − 1.16587i −0.812517 0.582937i \(-0.801903\pi\)
0.812517 0.582937i \(-0.198097\pi\)
\(360\) 75.3951 3.97367
\(361\) 15.5623 0.819069
\(362\) 25.4164i 1.33586i
\(363\) −19.7984 −1.03915
\(364\) 0 0
\(365\) −5.23607 −0.274068
\(366\) − 41.1246i − 2.14962i
\(367\) −1.41641 −0.0739359 −0.0369679 0.999316i \(-0.511770\pi\)
−0.0369679 + 0.999316i \(0.511770\pi\)
\(368\) 44.0689 2.29725
\(369\) − 2.94427i − 0.153273i
\(370\) − 27.4164i − 1.42531i
\(371\) 3.76393i 0.195414i
\(372\) 59.8328i 3.10219i
\(373\) −20.5623 −1.06468 −0.532338 0.846532i \(-0.678686\pi\)
−0.532338 + 0.846532i \(0.678686\pi\)
\(374\) 7.14590 0.369506
\(375\) 21.5623i 1.11347i
\(376\) 16.7082 0.861660
\(377\) 0 0
\(378\) −5.85410 −0.301103
\(379\) 6.14590i 0.315694i 0.987464 + 0.157847i \(0.0504552\pi\)
−0.987464 + 0.157847i \(0.949545\pi\)
\(380\) −23.5623 −1.20872
\(381\) −54.5967 −2.79708
\(382\) − 55.9787i − 2.86412i
\(383\) 21.9787i 1.12306i 0.827456 + 0.561530i \(0.189787\pi\)
−0.827456 + 0.561530i \(0.810213\pi\)
\(384\) − 2.85410i − 0.145648i
\(385\) 4.85410i 0.247388i
\(386\) −15.7082 −0.799527
\(387\) −48.4164 −2.46114
\(388\) 91.5197i 4.64621i
\(389\) 11.8885 0.602773 0.301387 0.953502i \(-0.402551\pi\)
0.301387 + 0.953502i \(0.402551\pi\)
\(390\) 0 0
\(391\) 6.58359 0.332947
\(392\) − 7.47214i − 0.377400i
\(393\) 40.1246 2.02402
\(394\) 43.9787 2.21562
\(395\) − 10.4721i − 0.526910i
\(396\) − 34.6869i − 1.74308i
\(397\) 1.41641i 0.0710875i 0.999368 + 0.0355437i \(0.0113163\pi\)
−0.999368 + 0.0355437i \(0.988684\pi\)
\(398\) − 63.9230i − 3.20417i
\(399\) 4.85410 0.243009
\(400\) −18.2705 −0.913525
\(401\) 35.4508i 1.77033i 0.465276 + 0.885165i \(0.345955\pi\)
−0.465276 + 0.885165i \(0.654045\pi\)
\(402\) −87.1033 −4.34432
\(403\) 0 0
\(404\) −56.1246 −2.79230
\(405\) 14.9443i 0.742587i
\(406\) 18.5623 0.921232
\(407\) −7.41641 −0.367618
\(408\) − 28.7984i − 1.42573i
\(409\) − 14.4377i − 0.713898i −0.934124 0.356949i \(-0.883817\pi\)
0.934124 0.356949i \(-0.116183\pi\)
\(410\) 5.23607i 0.258591i
\(411\) 6.85410i 0.338088i
\(412\) −42.2705 −2.08252
\(413\) 2.23607 0.110030
\(414\) − 45.1246i − 2.21775i
\(415\) −17.5623 −0.862100
\(416\) 0 0
\(417\) −11.9443 −0.584914
\(418\) 9.00000i 0.440204i
\(419\) −11.9443 −0.583516 −0.291758 0.956492i \(-0.594240\pi\)
−0.291758 + 0.956492i \(0.594240\pi\)
\(420\) 33.2705 1.62343
\(421\) − 1.41641i − 0.0690315i −0.999404 0.0345157i \(-0.989011\pi\)
0.999404 0.0345157i \(-0.0109889\pi\)
\(422\) − 12.3262i − 0.600032i
\(423\) − 8.61803i − 0.419023i
\(424\) 28.1246i 1.36585i
\(425\) −2.72949 −0.132400
\(426\) −97.1935 −4.70904
\(427\) − 6.00000i − 0.290360i
\(428\) 16.4164 0.793517
\(429\) 0 0
\(430\) 86.1033 4.15227
\(431\) − 7.79837i − 0.375634i −0.982204 0.187817i \(-0.939859\pi\)
0.982204 0.187817i \(-0.0601413\pi\)
\(432\) −22.0344 −1.06013
\(433\) −1.00000 −0.0480569 −0.0240285 0.999711i \(-0.507649\pi\)
−0.0240285 + 0.999711i \(0.507649\pi\)
\(434\) 12.3262i 0.591678i
\(435\) 48.5967i 2.33004i
\(436\) − 13.1459i − 0.629574i
\(437\) 8.29180i 0.396650i
\(438\) 13.7082 0.655003
\(439\) −14.8541 −0.708948 −0.354474 0.935066i \(-0.615340\pi\)
−0.354474 + 0.935066i \(0.615340\pi\)
\(440\) 36.2705i 1.72913i
\(441\) −3.85410 −0.183529
\(442\) 0 0
\(443\) −5.23607 −0.248773 −0.124387 0.992234i \(-0.539696\pi\)
−0.124387 + 0.992234i \(0.539696\pi\)
\(444\) 50.8328i 2.41242i
\(445\) 12.8541 0.609343
\(446\) −53.0689 −2.51288
\(447\) 4.85410i 0.229591i
\(448\) − 8.70820i − 0.411424i
\(449\) 19.5279i 0.921577i 0.887510 + 0.460788i \(0.152433\pi\)
−0.887510 + 0.460788i \(0.847567\pi\)
\(450\) 18.7082i 0.881913i
\(451\) 1.41641 0.0666960
\(452\) −7.14590 −0.336115
\(453\) 3.38197i 0.158899i
\(454\) −3.85410 −0.180882
\(455\) 0 0
\(456\) 36.2705 1.69852
\(457\) − 15.4164i − 0.721149i −0.932730 0.360575i \(-0.882581\pi\)
0.932730 0.360575i \(-0.117419\pi\)
\(458\) −34.3607 −1.60557
\(459\) −3.29180 −0.153648
\(460\) 56.8328i 2.64984i
\(461\) 12.2148i 0.568899i 0.958691 + 0.284450i \(0.0918108\pi\)
−0.958691 + 0.284450i \(0.908189\pi\)
\(462\) − 12.7082i − 0.591239i
\(463\) 6.70820i 0.311757i 0.987776 + 0.155878i \(0.0498208\pi\)
−0.987776 + 0.155878i \(0.950179\pi\)
\(464\) 69.8673 3.24351
\(465\) −32.2705 −1.49651
\(466\) 6.85410i 0.317510i
\(467\) −2.34752 −0.108630 −0.0543152 0.998524i \(-0.517298\pi\)
−0.0543152 + 0.998524i \(0.517298\pi\)
\(468\) 0 0
\(469\) −12.7082 −0.586810
\(470\) 15.3262i 0.706947i
\(471\) −38.8885 −1.79189
\(472\) 16.7082 0.769057
\(473\) − 23.2918i − 1.07096i
\(474\) 27.4164i 1.25928i
\(475\) − 3.43769i − 0.157732i
\(476\) − 7.14590i − 0.327532i
\(477\) 14.5066 0.664211
\(478\) 64.6869 2.95871
\(479\) 24.9787i 1.14131i 0.821191 + 0.570653i \(0.193310\pi\)
−0.821191 + 0.570653i \(0.806690\pi\)
\(480\) 74.3951 3.39566
\(481\) 0 0
\(482\) 64.3050 2.92901
\(483\) − 11.7082i − 0.532742i
\(484\) −36.7082 −1.66855
\(485\) −49.3607 −2.24135
\(486\) − 56.6869i − 2.57137i
\(487\) − 29.9787i − 1.35847i −0.733923 0.679233i \(-0.762313\pi\)
0.733923 0.679233i \(-0.237687\pi\)
\(488\) − 44.8328i − 2.02949i
\(489\) 9.70820i 0.439020i
\(490\) 6.85410 0.309637
\(491\) −12.3820 −0.558790 −0.279395 0.960176i \(-0.590134\pi\)
−0.279395 + 0.960176i \(0.590134\pi\)
\(492\) − 9.70820i − 0.437680i
\(493\) 10.4377 0.470090
\(494\) 0 0
\(495\) 18.7082 0.840871
\(496\) 46.3951i 2.08320i
\(497\) −14.1803 −0.636075
\(498\) 45.9787 2.06036
\(499\) − 14.8541i − 0.664961i −0.943110 0.332480i \(-0.892114\pi\)
0.943110 0.332480i \(-0.107886\pi\)
\(500\) 39.9787i 1.78790i
\(501\) − 37.2705i − 1.66512i
\(502\) 2.00000i 0.0892644i
\(503\) 26.6180 1.18684 0.593420 0.804893i \(-0.297777\pi\)
0.593420 + 0.804893i \(0.297777\pi\)
\(504\) −28.7984 −1.28278
\(505\) − 30.2705i − 1.34702i
\(506\) 21.7082 0.965047
\(507\) 0 0
\(508\) −101.228 −4.49126
\(509\) 18.5967i 0.824286i 0.911119 + 0.412143i \(0.135220\pi\)
−0.911119 + 0.412143i \(0.864780\pi\)
\(510\) 26.4164 1.16974
\(511\) 2.00000 0.0884748
\(512\) 40.3050i 1.78124i
\(513\) − 4.14590i − 0.183046i
\(514\) 43.8328i 1.93338i
\(515\) − 22.7984i − 1.00462i
\(516\) −159.644 −7.02795
\(517\) 4.14590 0.182336
\(518\) 10.4721i 0.460119i
\(519\) −23.5623 −1.03427
\(520\) 0 0
\(521\) 18.6525 0.817180 0.408590 0.912718i \(-0.366021\pi\)
0.408590 + 0.912718i \(0.366021\pi\)
\(522\) − 71.5410i − 3.13127i
\(523\) 1.12461 0.0491758 0.0245879 0.999698i \(-0.492173\pi\)
0.0245879 + 0.999698i \(0.492173\pi\)
\(524\) 74.3951 3.24997
\(525\) 4.85410i 0.211850i
\(526\) − 23.5623i − 1.02737i
\(527\) 6.93112i 0.301924i
\(528\) − 47.8328i − 2.08166i
\(529\) −3.00000 −0.130435
\(530\) −25.7984 −1.12061
\(531\) − 8.61803i − 0.373991i
\(532\) 9.00000 0.390199
\(533\) 0 0
\(534\) −33.6525 −1.45629
\(535\) 8.85410i 0.382796i
\(536\) −94.9574 −4.10154
\(537\) −23.5623 −1.01679
\(538\) 75.2492i 3.24422i
\(539\) − 1.85410i − 0.0798618i
\(540\) − 28.4164i − 1.22285i
\(541\) 35.2705i 1.51640i 0.652023 + 0.758199i \(0.273920\pi\)
−0.652023 + 0.758199i \(0.726080\pi\)
\(542\) −22.0344 −0.946460
\(543\) 25.4164 1.09072
\(544\) − 15.9787i − 0.685082i
\(545\) 7.09017 0.303710
\(546\) 0 0
\(547\) −3.00000 −0.128271 −0.0641354 0.997941i \(-0.520429\pi\)
−0.0641354 + 0.997941i \(0.520429\pi\)
\(548\) 12.7082i 0.542868i
\(549\) −23.1246 −0.986934
\(550\) −9.00000 −0.383761
\(551\) 13.1459i 0.560034i
\(552\) − 87.4853i − 3.72362i
\(553\) 4.00000i 0.170097i
\(554\) − 13.0902i − 0.556148i
\(555\) −27.4164 −1.16376
\(556\) −22.1459 −0.939195
\(557\) − 27.9787i − 1.18550i −0.805388 0.592748i \(-0.798043\pi\)
0.805388 0.592748i \(-0.201957\pi\)
\(558\) 47.5066 2.01111
\(559\) 0 0
\(560\) 25.7984 1.09018
\(561\) − 7.14590i − 0.301700i
\(562\) 52.8328 2.22862
\(563\) 21.0557 0.887393 0.443697 0.896177i \(-0.353667\pi\)
0.443697 + 0.896177i \(0.353667\pi\)
\(564\) − 28.4164i − 1.19655i
\(565\) − 3.85410i − 0.162143i
\(566\) 35.1246i 1.47640i
\(567\) − 5.70820i − 0.239722i
\(568\) −105.957 −4.44587
\(569\) −14.9443 −0.626496 −0.313248 0.949671i \(-0.601417\pi\)
−0.313248 + 0.949671i \(0.601417\pi\)
\(570\) 33.2705i 1.39355i
\(571\) −24.6869 −1.03312 −0.516558 0.856252i \(-0.672787\pi\)
−0.516558 + 0.856252i \(0.672787\pi\)
\(572\) 0 0
\(573\) −55.9787 −2.33854
\(574\) − 2.00000i − 0.0834784i
\(575\) −8.29180 −0.345792
\(576\) −33.5623 −1.39843
\(577\) 43.8328i 1.82478i 0.409318 + 0.912392i \(0.365767\pi\)
−0.409318 + 0.912392i \(0.634233\pi\)
\(578\) 38.8328i 1.61523i
\(579\) 15.7082i 0.652811i
\(580\) 90.1033i 3.74134i
\(581\) 6.70820 0.278303
\(582\) 129.228 5.35667
\(583\) 6.97871i 0.289029i
\(584\) 14.9443 0.618398
\(585\) 0 0
\(586\) −17.7082 −0.731519
\(587\) 19.9098i 0.821767i 0.911688 + 0.410883i \(0.134780\pi\)
−0.911688 + 0.410883i \(0.865220\pi\)
\(588\) −12.7082 −0.524077
\(589\) −8.72949 −0.359692
\(590\) 15.3262i 0.630971i
\(591\) − 43.9787i − 1.80904i
\(592\) 39.4164i 1.62000i
\(593\) 43.7984i 1.79858i 0.437349 + 0.899292i \(0.355917\pi\)
−0.437349 + 0.899292i \(0.644083\pi\)
\(594\) −10.8541 −0.445349
\(595\) 3.85410 0.158003
\(596\) 9.00000i 0.368654i
\(597\) −63.9230 −2.61619
\(598\) 0 0
\(599\) 29.5066 1.20561 0.602803 0.797890i \(-0.294050\pi\)
0.602803 + 0.797890i \(0.294050\pi\)
\(600\) 36.2705i 1.48074i
\(601\) 40.3951 1.64775 0.823876 0.566771i \(-0.191807\pi\)
0.823876 + 0.566771i \(0.191807\pi\)
\(602\) −32.8885 −1.34044
\(603\) 48.9787i 1.99457i
\(604\) 6.27051i 0.255143i
\(605\) − 19.7984i − 0.804918i
\(606\) 79.2492i 3.21928i
\(607\) −23.0000 −0.933541 −0.466771 0.884378i \(-0.654583\pi\)
−0.466771 + 0.884378i \(0.654583\pi\)
\(608\) 20.1246 0.816161
\(609\) − 18.5623i − 0.752183i
\(610\) 41.1246 1.66509
\(611\) 0 0
\(612\) −27.5410 −1.11328
\(613\) − 34.5623i − 1.39596i −0.716118 0.697979i \(-0.754083\pi\)
0.716118 0.697979i \(-0.245917\pi\)
\(614\) −12.7082 −0.512861
\(615\) 5.23607 0.211139
\(616\) − 13.8541i − 0.558198i
\(617\) − 0.0557281i − 0.00224353i −0.999999 0.00112176i \(-0.999643\pi\)
0.999999 0.00112176i \(-0.000357069\pi\)
\(618\) 59.6869i 2.40096i
\(619\) 9.41641i 0.378477i 0.981931 + 0.189239i \(0.0606020\pi\)
−0.981931 + 0.189239i \(0.939398\pi\)
\(620\) −59.8328 −2.40294
\(621\) −10.0000 −0.401286
\(622\) 8.70820i 0.349167i
\(623\) −4.90983 −0.196708
\(624\) 0 0
\(625\) −30.8328 −1.23331
\(626\) − 65.7771i − 2.62898i
\(627\) 9.00000 0.359425
\(628\) −72.1033 −2.87724
\(629\) 5.88854i 0.234792i
\(630\) − 26.4164i − 1.05245i
\(631\) 34.3951i 1.36925i 0.728896 + 0.684624i \(0.240034\pi\)
−0.728896 + 0.684624i \(0.759966\pi\)
\(632\) 29.8885i 1.18890i
\(633\) −12.3262 −0.489924
\(634\) −68.6869 −2.72791
\(635\) − 54.5967i − 2.16661i
\(636\) 47.8328 1.89669
\(637\) 0 0
\(638\) 34.4164 1.36256
\(639\) 54.6525i 2.16202i
\(640\) 2.85410 0.112818
\(641\) 47.5066 1.87640 0.938199 0.346098i \(-0.112493\pi\)
0.938199 + 0.346098i \(0.112493\pi\)
\(642\) − 23.1803i − 0.914855i
\(643\) 7.00000i 0.276053i 0.990429 + 0.138027i \(0.0440759\pi\)
−0.990429 + 0.138027i \(0.955924\pi\)
\(644\) − 21.7082i − 0.855423i
\(645\) − 86.1033i − 3.39032i
\(646\) 7.14590 0.281152
\(647\) −24.7639 −0.973571 −0.486785 0.873522i \(-0.661831\pi\)
−0.486785 + 0.873522i \(0.661831\pi\)
\(648\) − 42.6525i − 1.67555i
\(649\) 4.14590 0.162741
\(650\) 0 0
\(651\) 12.3262 0.483103
\(652\) 18.0000i 0.704934i
\(653\) −0.381966 −0.0149475 −0.00747374 0.999972i \(-0.502379\pi\)
−0.00747374 + 0.999972i \(0.502379\pi\)
\(654\) −18.5623 −0.725844
\(655\) 40.1246i 1.56780i
\(656\) − 7.52786i − 0.293914i
\(657\) − 7.70820i − 0.300726i
\(658\) − 5.85410i − 0.228217i
\(659\) −23.8885 −0.930566 −0.465283 0.885162i \(-0.654047\pi\)
−0.465283 + 0.885162i \(0.654047\pi\)
\(660\) 61.6869 2.40116
\(661\) − 48.5410i − 1.88803i −0.329907 0.944013i \(-0.607017\pi\)
0.329907 0.944013i \(-0.392983\pi\)
\(662\) 26.5623 1.03237
\(663\) 0 0
\(664\) 50.1246 1.94521
\(665\) 4.85410i 0.188234i
\(666\) 40.3607 1.56394
\(667\) 31.7082 1.22775
\(668\) − 69.1033i − 2.67369i
\(669\) 53.0689i 2.05176i
\(670\) − 87.1033i − 3.36510i
\(671\) − 11.1246i − 0.429461i
\(672\) −28.4164 −1.09619
\(673\) 39.2492 1.51295 0.756473 0.654025i \(-0.226921\pi\)
0.756473 + 0.654025i \(0.226921\pi\)
\(674\) − 30.2705i − 1.16598i
\(675\) 4.14590 0.159576
\(676\) 0 0
\(677\) 43.7426 1.68117 0.840583 0.541682i \(-0.182212\pi\)
0.840583 + 0.541682i \(0.182212\pi\)
\(678\) 10.0902i 0.387511i
\(679\) 18.8541 0.723554
\(680\) 28.7984 1.10437
\(681\) 3.85410i 0.147690i
\(682\) 22.8541i 0.875129i
\(683\) 1.47214i 0.0563297i 0.999603 + 0.0281649i \(0.00896634\pi\)
−0.999603 + 0.0281649i \(0.991034\pi\)
\(684\) − 34.6869i − 1.32629i
\(685\) −6.85410 −0.261882
\(686\) −2.61803 −0.0999570
\(687\) 34.3607i 1.31094i
\(688\) −123.790 −4.71946
\(689\) 0 0
\(690\) 80.2492 3.05504
\(691\) 5.85410i 0.222701i 0.993781 + 0.111350i \(0.0355175\pi\)
−0.993781 + 0.111350i \(0.964482\pi\)
\(692\) −43.6869 −1.66073
\(693\) −7.14590 −0.271450
\(694\) − 80.5410i − 3.05730i
\(695\) − 11.9443i − 0.453072i
\(696\) − 138.700i − 5.25741i
\(697\) − 1.12461i − 0.0425977i
\(698\) 54.2148 2.05206
\(699\) 6.85410 0.259246
\(700\) 9.00000i 0.340168i
\(701\) −11.2361 −0.424380 −0.212190 0.977228i \(-0.568060\pi\)
−0.212190 + 0.977228i \(0.568060\pi\)
\(702\) 0 0
\(703\) −7.41641 −0.279715
\(704\) − 16.1459i − 0.608521i
\(705\) 15.3262 0.577220
\(706\) −57.9787 −2.18206
\(707\) 11.5623i 0.434845i
\(708\) − 28.4164i − 1.06795i
\(709\) 23.5623i 0.884901i 0.896793 + 0.442450i \(0.145891\pi\)
−0.896793 + 0.442450i \(0.854109\pi\)
\(710\) − 97.1935i − 3.64761i
\(711\) 15.4164 0.578160
\(712\) −36.6869 −1.37490
\(713\) 21.0557i 0.788543i
\(714\) −10.0902 −0.377615
\(715\) 0 0
\(716\) −43.6869 −1.63266
\(717\) − 64.6869i − 2.41578i
\(718\) −57.8328 −2.15830
\(719\) −8.12461 −0.302997 −0.151498 0.988457i \(-0.548410\pi\)
−0.151498 + 0.988457i \(0.548410\pi\)
\(720\) − 99.4296i − 3.70552i
\(721\) 8.70820i 0.324310i
\(722\) − 40.7426i − 1.51628i
\(723\) − 64.3050i − 2.39153i
\(724\) 47.1246 1.75137
\(725\) −13.1459 −0.488226
\(726\) 51.8328i 1.92370i
\(727\) 30.7082 1.13890 0.569452 0.822025i \(-0.307155\pi\)
0.569452 + 0.822025i \(0.307155\pi\)
\(728\) 0 0
\(729\) −39.5623 −1.46527
\(730\) 13.7082i 0.507363i
\(731\) −18.4934 −0.684004
\(732\) −76.2492 −2.81825
\(733\) 32.2705i 1.19194i 0.803007 + 0.595969i \(0.203232\pi\)
−0.803007 + 0.595969i \(0.796768\pi\)
\(734\) 3.70820i 0.136872i
\(735\) − 6.85410i − 0.252817i
\(736\) − 48.5410i − 1.78925i
\(737\) −23.5623 −0.867929
\(738\) −7.70820 −0.283743
\(739\) − 6.87539i − 0.252915i −0.991972 0.126458i \(-0.959639\pi\)
0.991972 0.126458i \(-0.0403608\pi\)
\(740\) −50.8328 −1.86865
\(741\) 0 0
\(742\) 9.85410 0.361755
\(743\) − 39.3262i − 1.44274i −0.692550 0.721370i \(-0.743513\pi\)
0.692550 0.721370i \(-0.256487\pi\)
\(744\) 92.1033 3.37667
\(745\) −4.85410 −0.177841
\(746\) 53.8328i 1.97096i
\(747\) − 25.8541i − 0.945952i
\(748\) − 13.2492i − 0.484440i
\(749\) − 3.38197i − 0.123574i
\(750\) 56.4508 2.06129
\(751\) −22.7082 −0.828634 −0.414317 0.910133i \(-0.635980\pi\)
−0.414317 + 0.910133i \(0.635980\pi\)
\(752\) − 22.0344i − 0.803513i
\(753\) 2.00000 0.0728841
\(754\) 0 0
\(755\) −3.38197 −0.123082
\(756\) 10.8541i 0.394760i
\(757\) 28.0000 1.01768 0.508839 0.860862i \(-0.330075\pi\)
0.508839 + 0.860862i \(0.330075\pi\)
\(758\) 16.0902 0.584421
\(759\) − 21.7082i − 0.787958i
\(760\) 36.2705i 1.31567i
\(761\) − 28.8541i − 1.04596i −0.852345 0.522980i \(-0.824820\pi\)
0.852345 0.522980i \(-0.175180\pi\)
\(762\) 142.936i 5.17803i
\(763\) −2.70820 −0.0980436
\(764\) −103.790 −3.75500
\(765\) − 14.8541i − 0.537051i
\(766\) 57.5410 2.07904
\(767\) 0 0
\(768\) 38.1246 1.37570
\(769\) 18.4164i 0.664113i 0.943260 + 0.332056i \(0.107742\pi\)
−0.943260 + 0.332056i \(0.892258\pi\)
\(770\) 12.7082 0.457972
\(771\) 43.8328 1.57860
\(772\) 29.1246i 1.04822i
\(773\) − 25.3607i − 0.912160i −0.889939 0.456080i \(-0.849253\pi\)
0.889939 0.456080i \(-0.150747\pi\)
\(774\) 126.756i 4.55614i
\(775\) − 8.72949i − 0.313573i
\(776\) 140.880 5.05731
\(777\) 10.4721 0.375686
\(778\) − 31.1246i − 1.11587i
\(779\) 1.41641 0.0507481
\(780\) 0 0
\(781\) −26.2918 −0.940794
\(782\) − 17.2361i − 0.616361i
\(783\) −15.8541 −0.566579
\(784\) −9.85410 −0.351932
\(785\) − 38.8885i − 1.38799i
\(786\) − 105.048i − 3.74692i
\(787\) 2.58359i 0.0920951i 0.998939 + 0.0460476i \(0.0146626\pi\)
−0.998939 + 0.0460476i \(0.985337\pi\)
\(788\) − 81.5410i − 2.90478i
\(789\) −23.5623 −0.838840
\(790\) −27.4164 −0.975432
\(791\) 1.47214i 0.0523431i
\(792\) −53.3951 −1.89731
\(793\) 0 0
\(794\) 3.70820 0.131599
\(795\) 25.7984i 0.914974i
\(796\) −118.520 −4.20082
\(797\) 8.18034 0.289763 0.144881 0.989449i \(-0.453720\pi\)
0.144881 + 0.989449i \(0.453720\pi\)
\(798\) − 12.7082i − 0.449866i
\(799\) − 3.29180i − 0.116455i
\(800\) 20.1246i 0.711512i
\(801\) 18.9230i 0.668611i
\(802\) 92.8115 3.27729
\(803\) 3.70820 0.130860
\(804\) 161.498i 5.69561i
\(805\) 11.7082 0.412660
\(806\) 0 0
\(807\) 75.2492 2.64890
\(808\) 86.3951i 3.03937i
\(809\) −4.41641 −0.155273 −0.0776363 0.996982i \(-0.524737\pi\)
−0.0776363 + 0.996982i \(0.524737\pi\)
\(810\) 39.1246 1.37470
\(811\) − 39.2705i − 1.37897i −0.724298 0.689487i \(-0.757836\pi\)
0.724298 0.689487i \(-0.242164\pi\)
\(812\) − 34.4164i − 1.20778i
\(813\) 22.0344i 0.772782i
\(814\) 19.4164i 0.680545i
\(815\) −9.70820 −0.340064
\(816\) −37.9787 −1.32952
\(817\) − 23.2918i − 0.814877i
\(818\) −37.7984 −1.32159
\(819\) 0 0
\(820\) 9.70820 0.339025
\(821\) 7.36068i 0.256889i 0.991717 + 0.128445i \(0.0409985\pi\)
−0.991717 + 0.128445i \(0.959002\pi\)
\(822\) 17.9443 0.625878
\(823\) 38.4164 1.33911 0.669556 0.742762i \(-0.266484\pi\)
0.669556 + 0.742762i \(0.266484\pi\)
\(824\) 65.0689i 2.26678i
\(825\) 9.00000i 0.313340i
\(826\) − 5.85410i − 0.203690i
\(827\) 15.9787i 0.555634i 0.960634 + 0.277817i \(0.0896109\pi\)
−0.960634 + 0.277817i \(0.910389\pi\)
\(828\) −83.6656 −2.90758
\(829\) −7.56231 −0.262650 −0.131325 0.991339i \(-0.541923\pi\)
−0.131325 + 0.991339i \(0.541923\pi\)
\(830\) 45.9787i 1.59594i
\(831\) −13.0902 −0.454093
\(832\) 0 0
\(833\) −1.47214 −0.0510065
\(834\) 31.2705i 1.08281i
\(835\) 37.2705 1.28980
\(836\) 16.6869 0.577129
\(837\) − 10.5279i − 0.363896i
\(838\) 31.2705i 1.08022i
\(839\) 13.7426i 0.474449i 0.971455 + 0.237224i \(0.0762377\pi\)
−0.971455 + 0.237224i \(0.923762\pi\)
\(840\) − 51.2148i − 1.76708i
\(841\) 21.2705 0.733466
\(842\) −3.70820 −0.127793
\(843\) − 52.8328i − 1.81966i
\(844\) −22.8541 −0.786671
\(845\) 0 0
\(846\) −22.5623 −0.775708
\(847\) 7.56231i 0.259844i
\(848\) 37.0902 1.27368
\(849\) 35.1246 1.20547
\(850\) 7.14590i 0.245102i
\(851\) 17.8885i 0.613211i
\(852\) 180.207i 6.17378i
\(853\) 14.1246i 0.483617i 0.970324 + 0.241809i \(0.0777407\pi\)
−0.970324 + 0.241809i \(0.922259\pi\)
\(854\) −15.7082 −0.537524
\(855\) 18.7082 0.639807
\(856\) − 25.2705i − 0.863728i
\(857\) 26.4508 0.903544 0.451772 0.892133i \(-0.350792\pi\)
0.451772 + 0.892133i \(0.350792\pi\)
\(858\) 0 0
\(859\) −44.2492 −1.50976 −0.754882 0.655861i \(-0.772306\pi\)
−0.754882 + 0.655861i \(0.772306\pi\)
\(860\) − 159.644i − 5.44383i
\(861\) −2.00000 −0.0681598
\(862\) −20.4164 −0.695386
\(863\) − 11.8885i − 0.404691i −0.979314 0.202345i \(-0.935144\pi\)
0.979314 0.202345i \(-0.0648564\pi\)
\(864\) 24.2705i 0.825700i
\(865\) − 23.5623i − 0.801142i
\(866\) 2.61803i 0.0889644i
\(867\) 38.8328 1.31883
\(868\) 22.8541 0.775719
\(869\) 7.41641i 0.251584i
\(870\) 127.228 4.31343
\(871\) 0 0
\(872\) −20.2361 −0.685280
\(873\) − 72.6656i − 2.45936i
\(874\) 21.7082 0.734291
\(875\) 8.23607 0.278430
\(876\) − 25.4164i − 0.858741i
\(877\) − 0.708204i − 0.0239143i −0.999929 0.0119572i \(-0.996194\pi\)
0.999929 0.0119572i \(-0.00380618\pi\)
\(878\) 38.8885i 1.31242i
\(879\) 17.7082i 0.597283i
\(880\) 47.8328 1.61244
\(881\) 36.5967 1.23298 0.616488 0.787364i \(-0.288555\pi\)
0.616488 + 0.787364i \(0.288555\pi\)
\(882\) 10.0902i 0.339754i
\(883\) −29.0000 −0.975928 −0.487964 0.872864i \(-0.662260\pi\)
−0.487964 + 0.872864i \(0.662260\pi\)
\(884\) 0 0
\(885\) 15.3262 0.515186
\(886\) 13.7082i 0.460536i
\(887\) 54.6525 1.83505 0.917525 0.397677i \(-0.130184\pi\)
0.917525 + 0.397677i \(0.130184\pi\)
\(888\) 78.2492 2.62587
\(889\) 20.8541i 0.699424i
\(890\) − 33.6525i − 1.12803i
\(891\) − 10.5836i − 0.354564i
\(892\) 98.3951i 3.29451i
\(893\) 4.14590 0.138737
\(894\) 12.7082 0.425026
\(895\) − 23.5623i − 0.787601i
\(896\) −1.09017 −0.0364200
\(897\) 0 0
\(898\) 51.1246 1.70605
\(899\) 33.3820i 1.11335i
\(900\) 34.6869 1.15623
\(901\) 5.54102 0.184598
\(902\) − 3.70820i − 0.123470i
\(903\) 32.8885i 1.09446i
\(904\) 11.0000i 0.365855i
\(905\) 25.4164i 0.844870i
\(906\) 8.85410 0.294158
\(907\) 24.0000 0.796907 0.398453 0.917189i \(-0.369547\pi\)
0.398453 + 0.917189i \(0.369547\pi\)
\(908\) 7.14590i 0.237145i
\(909\) 44.5623 1.47804
\(910\) 0 0
\(911\) −22.6869 −0.751651 −0.375826 0.926690i \(-0.622641\pi\)
−0.375826 + 0.926690i \(0.622641\pi\)
\(912\) − 47.8328i − 1.58390i
\(913\) 12.4377 0.411628
\(914\) −40.3607 −1.33501
\(915\) − 41.1246i − 1.35954i
\(916\) 63.7082i 2.10498i
\(917\) − 15.3262i − 0.506117i
\(918\) 8.61803i 0.284438i
\(919\) −30.0000 −0.989609 −0.494804 0.869004i \(-0.664760\pi\)
−0.494804 + 0.869004i \(0.664760\pi\)
\(920\) 87.4853 2.88430
\(921\) 12.7082i 0.418750i
\(922\) 31.9787 1.05316
\(923\) 0 0
\(924\) −23.5623 −0.775143
\(925\) − 7.41641i − 0.243850i
\(926\) 17.5623 0.577133
\(927\) 33.5623 1.10233
\(928\) − 76.9574i − 2.52625i
\(929\) − 11.0689i − 0.363158i −0.983376 0.181579i \(-0.941879\pi\)
0.983376 0.181579i \(-0.0581209\pi\)
\(930\) 84.4853i 2.77038i
\(931\) − 1.85410i − 0.0607657i
\(932\) 12.7082 0.416271
\(933\) 8.70820 0.285094
\(934\) 6.14590i 0.201100i
\(935\) 7.14590 0.233696
\(936\) 0 0
\(937\) −15.8754 −0.518626 −0.259313 0.965793i \(-0.583496\pi\)
−0.259313 + 0.965793i \(0.583496\pi\)
\(938\) 33.2705i 1.08632i
\(939\) −65.7771 −2.14655
\(940\) 28.4164 0.926841
\(941\) − 20.3475i − 0.663310i −0.943401 0.331655i \(-0.892393\pi\)
0.943401 0.331655i \(-0.107607\pi\)
\(942\) 101.812i 3.31720i
\(943\) − 3.41641i − 0.111254i
\(944\) − 22.0344i − 0.717160i
\(945\) −5.85410 −0.190434
\(946\) −60.9787 −1.98259
\(947\) − 36.8673i − 1.19802i −0.800740 0.599012i \(-0.795560\pi\)
0.800740 0.599012i \(-0.204440\pi\)
\(948\) 50.8328 1.65097
\(949\) 0 0
\(950\) −9.00000 −0.291999
\(951\) 68.6869i 2.22733i
\(952\) −11.0000 −0.356512
\(953\) −26.7771 −0.867395 −0.433697 0.901059i \(-0.642791\pi\)
−0.433697 + 0.901059i \(0.642791\pi\)
\(954\) − 37.9787i − 1.22961i
\(955\) − 55.9787i − 1.81143i
\(956\) − 119.936i − 3.87901i
\(957\) − 34.4164i − 1.11252i
\(958\) 65.3951 2.11282
\(959\) 2.61803 0.0845407
\(960\) − 59.6869i − 1.92639i
\(961\) 8.83282 0.284930
\(962\) 0 0
\(963\) −13.0344 −0.420029
\(964\) − 119.228i − 3.84007i
\(965\) −15.7082 −0.505665
\(966\) −30.6525 −0.986227
\(967\) 39.0000i 1.25416i 0.778957 + 0.627078i \(0.215749\pi\)
−0.778957 + 0.627078i \(0.784251\pi\)
\(968\) 56.5066i 1.81619i
\(969\) − 7.14590i − 0.229559i
\(970\) 129.228i 4.14926i
\(971\) 31.5836 1.01357 0.506783 0.862074i \(-0.330835\pi\)
0.506783 + 0.862074i \(0.330835\pi\)
\(972\) −105.103 −3.37119
\(973\) 4.56231i 0.146261i
\(974\) −78.4853 −2.51483
\(975\) 0 0
\(976\) −59.1246 −1.89253
\(977\) 22.5279i 0.720730i 0.932811 + 0.360365i \(0.117348\pi\)
−0.932811 + 0.360365i \(0.882652\pi\)
\(978\) 25.4164 0.812727
\(979\) −9.10333 −0.290944
\(980\) − 12.7082i − 0.405949i
\(981\) 10.4377i 0.333250i
\(982\) 32.4164i 1.03445i
\(983\) 18.3820i 0.586294i 0.956067 + 0.293147i \(0.0947025\pi\)
−0.956067 + 0.293147i \(0.905298\pi\)
\(984\) −14.9443 −0.476406
\(985\) 43.9787 1.40128
\(986\) − 27.3262i − 0.870245i
\(987\) −5.85410 −0.186338
\(988\) 0 0
\(989\) −56.1803 −1.78643
\(990\) − 48.9787i − 1.55665i
\(991\) −16.1459 −0.512891 −0.256446 0.966559i \(-0.582551\pi\)
−0.256446 + 0.966559i \(0.582551\pi\)
\(992\) 51.1033 1.62253
\(993\) − 26.5623i − 0.842929i
\(994\) 37.1246i 1.17752i
\(995\) − 63.9230i − 2.02649i
\(996\) − 85.2492i − 2.70123i
\(997\) 49.0000 1.55185 0.775923 0.630828i \(-0.217285\pi\)
0.775923 + 0.630828i \(0.217285\pi\)
\(998\) −38.8885 −1.23100
\(999\) − 8.94427i − 0.282984i
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 1183.2.c.c.337.1 4
13.5 odd 4 1183.2.a.c.1.1 2
13.7 odd 12 91.2.f.a.29.1 yes 4
13.8 odd 4 1183.2.a.g.1.2 2
13.11 odd 12 91.2.f.a.22.1 4
13.12 even 2 inner 1183.2.c.c.337.4 4
39.11 even 12 819.2.o.c.568.2 4
39.20 even 12 819.2.o.c.757.2 4
52.7 even 12 1456.2.s.h.1121.1 4
52.11 even 12 1456.2.s.h.113.1 4
91.11 odd 12 637.2.g.b.373.1 4
91.20 even 12 637.2.f.c.393.1 4
91.24 even 12 637.2.g.c.373.1 4
91.33 even 12 637.2.g.c.263.1 4
91.34 even 4 8281.2.a.bb.1.2 2
91.37 odd 12 637.2.h.g.165.2 4
91.46 odd 12 637.2.h.g.471.2 4
91.59 even 12 637.2.h.f.471.2 4
91.72 odd 12 637.2.g.b.263.1 4
91.76 even 12 637.2.f.c.295.1 4
91.83 even 4 8281.2.a.n.1.1 2
91.89 even 12 637.2.h.f.165.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.a.22.1 4 13.11 odd 12
91.2.f.a.29.1 yes 4 13.7 odd 12
637.2.f.c.295.1 4 91.76 even 12
637.2.f.c.393.1 4 91.20 even 12
637.2.g.b.263.1 4 91.72 odd 12
637.2.g.b.373.1 4 91.11 odd 12
637.2.g.c.263.1 4 91.33 even 12
637.2.g.c.373.1 4 91.24 even 12
637.2.h.f.165.2 4 91.89 even 12
637.2.h.f.471.2 4 91.59 even 12
637.2.h.g.165.2 4 91.37 odd 12
637.2.h.g.471.2 4 91.46 odd 12
819.2.o.c.568.2 4 39.11 even 12
819.2.o.c.757.2 4 39.20 even 12
1183.2.a.c.1.1 2 13.5 odd 4
1183.2.a.g.1.2 2 13.8 odd 4
1183.2.c.c.337.1 4 1.1 even 1 trivial
1183.2.c.c.337.4 4 13.12 even 2 inner
1456.2.s.h.113.1 4 52.11 even 12
1456.2.s.h.1121.1 4 52.7 even 12
8281.2.a.n.1.1 2 91.83 even 4
8281.2.a.bb.1.2 2 91.34 even 4