Properties

Label 1183.2.c.c.337.2
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 337.2
Root \(0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 1183.337
Dual form 1183.2.c.c.337.3

$q$-expansion

\(f(q)\) \(=\) \(q-0.381966i q^{2} -0.381966 q^{3} +1.85410 q^{4} -0.381966i q^{5} +0.145898i q^{6} +1.00000i q^{7} -1.47214i q^{8} -2.85410 q^{9} +O(q^{10})\) \(q-0.381966i q^{2} -0.381966 q^{3} +1.85410 q^{4} -0.381966i q^{5} +0.145898i q^{6} +1.00000i q^{7} -1.47214i q^{8} -2.85410 q^{9} -0.145898 q^{10} -4.85410i q^{11} -0.708204 q^{12} +0.381966 q^{14} +0.145898i q^{15} +3.14590 q^{16} -7.47214 q^{17} +1.09017i q^{18} -4.85410i q^{19} -0.708204i q^{20} -0.381966i q^{21} -1.85410 q^{22} -4.47214 q^{23} +0.562306i q^{24} +4.85410 q^{25} +2.23607 q^{27} +1.85410i q^{28} -4.09017 q^{29} +0.0557281 q^{30} -8.70820i q^{31} -4.14590i q^{32} +1.85410i q^{33} +2.85410i q^{34} +0.381966 q^{35} -5.29180 q^{36} +4.00000i q^{37} -1.85410 q^{38} -0.562306 q^{40} -5.23607i q^{41} -0.145898 q^{42} +7.56231 q^{43} -9.00000i q^{44} +1.09017i q^{45} +1.70820i q^{46} +2.23607i q^{47} -1.20163 q^{48} -1.00000 q^{49} -1.85410i q^{50} +2.85410 q^{51} +8.23607 q^{53} -0.854102i q^{54} -1.85410 q^{55} +1.47214 q^{56} +1.85410i q^{57} +1.56231i q^{58} +2.23607i q^{59} +0.270510i q^{60} -6.00000 q^{61} -3.32624 q^{62} -2.85410i q^{63} +4.70820 q^{64} +0.708204 q^{66} -0.708204i q^{67} -13.8541 q^{68} +1.70820 q^{69} -0.145898i q^{70} -8.18034i q^{71} +4.20163i q^{72} -2.00000i q^{73} +1.52786 q^{74} -1.85410 q^{75} -9.00000i q^{76} +4.85410 q^{77} +4.00000 q^{79} -1.20163i q^{80} +7.70820 q^{81} -2.00000 q^{82} +6.70820i q^{83} -0.708204i q^{84} +2.85410i q^{85} -2.88854i q^{86} +1.56231 q^{87} -7.14590 q^{88} +16.0902i q^{89} +0.416408 q^{90} -8.29180 q^{92} +3.32624i q^{93} +0.854102 q^{94} -1.85410 q^{95} +1.58359i q^{96} -12.1459i q^{97} +0.381966i q^{98} +13.8541i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{3} - 6q^{4} + 2q^{9} + O(q^{10}) \) \( 4q - 6q^{3} - 6q^{4} + 2q^{9} - 14q^{10} + 24q^{12} + 6q^{14} + 26q^{16} - 12q^{17} + 6q^{22} + 6q^{25} + 6q^{29} + 36q^{30} + 6q^{35} - 48q^{36} + 6q^{38} + 38q^{40} - 14q^{42} - 10q^{43} - 54q^{48} - 4q^{49} - 2q^{51} + 24q^{53} + 6q^{55} - 12q^{56} - 24q^{61} + 18q^{62} - 8q^{64} - 24q^{66} - 42q^{68} - 20q^{69} + 24q^{74} + 6q^{75} + 6q^{77} + 16q^{79} + 4q^{81} - 8q^{82} - 34q^{87} - 42q^{88} - 52q^{90} - 60q^{92} - 10q^{94} + 6q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.381966i − 0.270091i −0.990839 0.135045i \(-0.956882\pi\)
0.990839 0.135045i \(-0.0431180\pi\)
\(3\) −0.381966 −0.220528 −0.110264 0.993902i \(-0.535170\pi\)
−0.110264 + 0.993902i \(0.535170\pi\)
\(4\) 1.85410 0.927051
\(5\) − 0.381966i − 0.170820i −0.996346 0.0854102i \(-0.972780\pi\)
0.996346 0.0854102i \(-0.0272201\pi\)
\(6\) 0.145898i 0.0595626i
\(7\) 1.00000i 0.377964i
\(8\) − 1.47214i − 0.520479i
\(9\) −2.85410 −0.951367
\(10\) −0.145898 −0.0461370
\(11\) − 4.85410i − 1.46357i −0.681537 0.731783i \(-0.738688\pi\)
0.681537 0.731783i \(-0.261312\pi\)
\(12\) −0.708204 −0.204441
\(13\) 0 0
\(14\) 0.381966 0.102085
\(15\) 0.145898i 0.0376707i
\(16\) 3.14590 0.786475
\(17\) −7.47214 −1.81226 −0.906130 0.423000i \(-0.860977\pi\)
−0.906130 + 0.423000i \(0.860977\pi\)
\(18\) 1.09017i 0.256956i
\(19\) − 4.85410i − 1.11361i −0.830644 0.556804i \(-0.812028\pi\)
0.830644 0.556804i \(-0.187972\pi\)
\(20\) − 0.708204i − 0.158359i
\(21\) − 0.381966i − 0.0833518i
\(22\) −1.85410 −0.395296
\(23\) −4.47214 −0.932505 −0.466252 0.884652i \(-0.654396\pi\)
−0.466252 + 0.884652i \(0.654396\pi\)
\(24\) 0.562306i 0.114780i
\(25\) 4.85410 0.970820
\(26\) 0 0
\(27\) 2.23607 0.430331
\(28\) 1.85410i 0.350392i
\(29\) −4.09017 −0.759525 −0.379763 0.925084i \(-0.623994\pi\)
−0.379763 + 0.925084i \(0.623994\pi\)
\(30\) 0.0557281 0.0101745
\(31\) − 8.70820i − 1.56404i −0.623254 0.782020i \(-0.714190\pi\)
0.623254 0.782020i \(-0.285810\pi\)
\(32\) − 4.14590i − 0.732898i
\(33\) 1.85410i 0.322758i
\(34\) 2.85410i 0.489474i
\(35\) 0.381966 0.0645640
\(36\) −5.29180 −0.881966
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) −1.85410 −0.300775
\(39\) 0 0
\(40\) −0.562306 −0.0889084
\(41\) − 5.23607i − 0.817736i −0.912593 0.408868i \(-0.865924\pi\)
0.912593 0.408868i \(-0.134076\pi\)
\(42\) −0.145898 −0.0225126
\(43\) 7.56231 1.15324 0.576620 0.817012i \(-0.304371\pi\)
0.576620 + 0.817012i \(0.304371\pi\)
\(44\) − 9.00000i − 1.35680i
\(45\) 1.09017i 0.162513i
\(46\) 1.70820i 0.251861i
\(47\) 2.23607i 0.326164i 0.986613 + 0.163082i \(0.0521435\pi\)
−0.986613 + 0.163082i \(0.947856\pi\)
\(48\) −1.20163 −0.173440
\(49\) −1.00000 −0.142857
\(50\) − 1.85410i − 0.262210i
\(51\) 2.85410 0.399654
\(52\) 0 0
\(53\) 8.23607 1.13131 0.565655 0.824642i \(-0.308623\pi\)
0.565655 + 0.824642i \(0.308623\pi\)
\(54\) − 0.854102i − 0.116229i
\(55\) −1.85410 −0.250007
\(56\) 1.47214 0.196722
\(57\) 1.85410i 0.245582i
\(58\) 1.56231i 0.205141i
\(59\) 2.23607i 0.291111i 0.989350 + 0.145556i \(0.0464970\pi\)
−0.989350 + 0.145556i \(0.953503\pi\)
\(60\) 0.270510i 0.0349227i
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −3.32624 −0.422433
\(63\) − 2.85410i − 0.359583i
\(64\) 4.70820 0.588525
\(65\) 0 0
\(66\) 0.708204 0.0871739
\(67\) − 0.708204i − 0.0865209i −0.999064 0.0432604i \(-0.986225\pi\)
0.999064 0.0432604i \(-0.0137745\pi\)
\(68\) −13.8541 −1.68006
\(69\) 1.70820 0.205644
\(70\) − 0.145898i − 0.0174382i
\(71\) − 8.18034i − 0.970828i −0.874284 0.485414i \(-0.838669\pi\)
0.874284 0.485414i \(-0.161331\pi\)
\(72\) 4.20163i 0.495166i
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 1.52786 0.177611
\(75\) −1.85410 −0.214093
\(76\) − 9.00000i − 1.03237i
\(77\) 4.85410 0.553176
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) − 1.20163i − 0.134346i
\(81\) 7.70820 0.856467
\(82\) −2.00000 −0.220863
\(83\) 6.70820i 0.736321i 0.929762 + 0.368161i \(0.120012\pi\)
−0.929762 + 0.368161i \(0.879988\pi\)
\(84\) − 0.708204i − 0.0772714i
\(85\) 2.85410i 0.309571i
\(86\) − 2.88854i − 0.311480i
\(87\) 1.56231 0.167497
\(88\) −7.14590 −0.761755
\(89\) 16.0902i 1.70555i 0.522275 + 0.852777i \(0.325084\pi\)
−0.522275 + 0.852777i \(0.674916\pi\)
\(90\) 0.416408 0.0438932
\(91\) 0 0
\(92\) −8.29180 −0.864479
\(93\) 3.32624i 0.344915i
\(94\) 0.854102 0.0880939
\(95\) −1.85410 −0.190227
\(96\) 1.58359i 0.161625i
\(97\) − 12.1459i − 1.23323i −0.787265 0.616615i \(-0.788504\pi\)
0.787265 0.616615i \(-0.211496\pi\)
\(98\) 0.381966i 0.0385844i
\(99\) 13.8541i 1.39239i
\(100\) 9.00000 0.900000
\(101\) −8.56231 −0.851981 −0.425991 0.904728i \(-0.640074\pi\)
−0.425991 + 0.904728i \(0.640074\pi\)
\(102\) − 1.09017i − 0.107943i
\(103\) −4.70820 −0.463913 −0.231957 0.972726i \(-0.574513\pi\)
−0.231957 + 0.972726i \(0.574513\pi\)
\(104\) 0 0
\(105\) −0.145898 −0.0142382
\(106\) − 3.14590i − 0.305557i
\(107\) −5.61803 −0.543116 −0.271558 0.962422i \(-0.587539\pi\)
−0.271558 + 0.962422i \(0.587539\pi\)
\(108\) 4.14590 0.398939
\(109\) − 10.7082i − 1.02566i −0.858490 0.512830i \(-0.828597\pi\)
0.858490 0.512830i \(-0.171403\pi\)
\(110\) 0.708204i 0.0675246i
\(111\) − 1.52786i − 0.145018i
\(112\) 3.14590i 0.297259i
\(113\) −7.47214 −0.702919 −0.351460 0.936203i \(-0.614315\pi\)
−0.351460 + 0.936203i \(0.614315\pi\)
\(114\) 0.708204 0.0663294
\(115\) 1.70820i 0.159291i
\(116\) −7.58359 −0.704119
\(117\) 0 0
\(118\) 0.854102 0.0786265
\(119\) − 7.47214i − 0.684970i
\(120\) 0.214782 0.0196068
\(121\) −12.5623 −1.14203
\(122\) 2.29180i 0.207489i
\(123\) 2.00000i 0.180334i
\(124\) − 16.1459i − 1.44994i
\(125\) − 3.76393i − 0.336656i
\(126\) −1.09017 −0.0971201
\(127\) 14.1459 1.25525 0.627623 0.778518i \(-0.284028\pi\)
0.627623 + 0.778518i \(0.284028\pi\)
\(128\) − 10.0902i − 0.891853i
\(129\) −2.88854 −0.254322
\(130\) 0 0
\(131\) 0.326238 0.0285035 0.0142518 0.999898i \(-0.495463\pi\)
0.0142518 + 0.999898i \(0.495463\pi\)
\(132\) 3.43769i 0.299213i
\(133\) 4.85410 0.420904
\(134\) −0.270510 −0.0233685
\(135\) − 0.854102i − 0.0735094i
\(136\) 11.0000i 0.943242i
\(137\) − 0.381966i − 0.0326336i −0.999867 0.0163168i \(-0.994806\pi\)
0.999867 0.0163168i \(-0.00519402\pi\)
\(138\) − 0.652476i − 0.0555424i
\(139\) −15.5623 −1.31998 −0.659989 0.751275i \(-0.729439\pi\)
−0.659989 + 0.751275i \(0.729439\pi\)
\(140\) 0.708204 0.0598542
\(141\) − 0.854102i − 0.0719284i
\(142\) −3.12461 −0.262212
\(143\) 0 0
\(144\) −8.97871 −0.748226
\(145\) 1.56231i 0.129742i
\(146\) −0.763932 −0.0632235
\(147\) 0.381966 0.0315040
\(148\) 7.41641i 0.609625i
\(149\) 4.85410i 0.397664i 0.980034 + 0.198832i \(0.0637147\pi\)
−0.980034 + 0.198832i \(0.936285\pi\)
\(150\) 0.708204i 0.0578246i
\(151\) − 14.7082i − 1.19694i −0.801146 0.598468i \(-0.795776\pi\)
0.801146 0.598468i \(-0.204224\pi\)
\(152\) −7.14590 −0.579609
\(153\) 21.3262 1.72412
\(154\) − 1.85410i − 0.149408i
\(155\) −3.32624 −0.267170
\(156\) 0 0
\(157\) 8.14590 0.650113 0.325057 0.945695i \(-0.394617\pi\)
0.325057 + 0.945695i \(0.394617\pi\)
\(158\) − 1.52786i − 0.121550i
\(159\) −3.14590 −0.249486
\(160\) −1.58359 −0.125194
\(161\) − 4.47214i − 0.352454i
\(162\) − 2.94427i − 0.231324i
\(163\) 9.70820i 0.760405i 0.924903 + 0.380203i \(0.124146\pi\)
−0.924903 + 0.380203i \(0.875854\pi\)
\(164\) − 9.70820i − 0.758083i
\(165\) 0.708204 0.0551336
\(166\) 2.56231 0.198874
\(167\) 9.76393i 0.755556i 0.925896 + 0.377778i \(0.123312\pi\)
−0.925896 + 0.377778i \(0.876688\pi\)
\(168\) −0.562306 −0.0433828
\(169\) 0 0
\(170\) 1.09017 0.0836122
\(171\) 13.8541i 1.05945i
\(172\) 14.0213 1.06911
\(173\) 9.00000 0.684257 0.342129 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) − 0.596748i − 0.0452393i
\(175\) 4.85410i 0.366936i
\(176\) − 15.2705i − 1.15106i
\(177\) − 0.854102i − 0.0641982i
\(178\) 6.14590 0.460655
\(179\) 9.00000 0.672692 0.336346 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 2.02129i 0.150658i
\(181\) 3.70820 0.275629 0.137814 0.990458i \(-0.455992\pi\)
0.137814 + 0.990458i \(0.455992\pi\)
\(182\) 0 0
\(183\) 2.29180 0.169414
\(184\) 6.58359i 0.485349i
\(185\) 1.52786 0.112331
\(186\) 1.27051 0.0931583
\(187\) 36.2705i 2.65236i
\(188\) 4.14590i 0.302371i
\(189\) 2.23607i 0.162650i
\(190\) 0.708204i 0.0513785i
\(191\) 23.6180 1.70894 0.854470 0.519500i \(-0.173882\pi\)
0.854470 + 0.519500i \(0.173882\pi\)
\(192\) −1.79837 −0.129786
\(193\) − 6.00000i − 0.431889i −0.976406 0.215945i \(-0.930717\pi\)
0.976406 0.215945i \(-0.0692831\pi\)
\(194\) −4.63932 −0.333084
\(195\) 0 0
\(196\) −1.85410 −0.132436
\(197\) − 7.79837i − 0.555611i −0.960637 0.277806i \(-0.910393\pi\)
0.960637 0.277806i \(-0.0896071\pi\)
\(198\) 5.29180 0.376072
\(199\) −2.41641 −0.171295 −0.0856473 0.996326i \(-0.527296\pi\)
−0.0856473 + 0.996326i \(0.527296\pi\)
\(200\) − 7.14590i − 0.505291i
\(201\) 0.270510i 0.0190803i
\(202\) 3.27051i 0.230112i
\(203\) − 4.09017i − 0.287074i
\(204\) 5.29180 0.370500
\(205\) −2.00000 −0.139686
\(206\) 1.79837i 0.125299i
\(207\) 12.7639 0.887155
\(208\) 0 0
\(209\) −23.5623 −1.62984
\(210\) 0.0557281i 0.00384560i
\(211\) −8.70820 −0.599497 −0.299749 0.954018i \(-0.596903\pi\)
−0.299749 + 0.954018i \(0.596903\pi\)
\(212\) 15.2705 1.04878
\(213\) 3.12461i 0.214095i
\(214\) 2.14590i 0.146691i
\(215\) − 2.88854i − 0.196997i
\(216\) − 3.29180i − 0.223978i
\(217\) 8.70820 0.591151
\(218\) −4.09017 −0.277021
\(219\) 0.763932i 0.0516217i
\(220\) −3.43769 −0.231769
\(221\) 0 0
\(222\) −0.583592 −0.0391681
\(223\) 13.2705i 0.888659i 0.895863 + 0.444330i \(0.146558\pi\)
−0.895863 + 0.444330i \(0.853442\pi\)
\(224\) 4.14590 0.277009
\(225\) −13.8541 −0.923607
\(226\) 2.85410i 0.189852i
\(227\) 7.47214i 0.495943i 0.968767 + 0.247972i \(0.0797639\pi\)
−0.968767 + 0.247972i \(0.920236\pi\)
\(228\) 3.43769i 0.227667i
\(229\) 27.1246i 1.79244i 0.443605 + 0.896222i \(0.353699\pi\)
−0.443605 + 0.896222i \(0.646301\pi\)
\(230\) 0.652476 0.0430230
\(231\) −1.85410 −0.121991
\(232\) 6.02129i 0.395317i
\(233\) −0.381966 −0.0250234 −0.0125117 0.999922i \(-0.503983\pi\)
−0.0125117 + 0.999922i \(0.503983\pi\)
\(234\) 0 0
\(235\) 0.854102 0.0557155
\(236\) 4.14590i 0.269875i
\(237\) −1.52786 −0.0992454
\(238\) −2.85410 −0.185004
\(239\) 11.2918i 0.730406i 0.930928 + 0.365203i \(0.119000\pi\)
−0.930928 + 0.365203i \(0.881000\pi\)
\(240\) 0.458980i 0.0296271i
\(241\) 4.43769i 0.285857i 0.989733 + 0.142929i \(0.0456519\pi\)
−0.989733 + 0.142929i \(0.954348\pi\)
\(242\) 4.79837i 0.308451i
\(243\) −9.65248 −0.619207
\(244\) −11.1246 −0.712180
\(245\) 0.381966i 0.0244029i
\(246\) 0.763932 0.0487065
\(247\) 0 0
\(248\) −12.8197 −0.814049
\(249\) − 2.56231i − 0.162380i
\(250\) −1.43769 −0.0909278
\(251\) −5.23607 −0.330498 −0.165249 0.986252i \(-0.552843\pi\)
−0.165249 + 0.986252i \(0.552843\pi\)
\(252\) − 5.29180i − 0.333352i
\(253\) 21.7082i 1.36478i
\(254\) − 5.40325i − 0.339030i
\(255\) − 1.09017i − 0.0682691i
\(256\) 5.56231 0.347644
\(257\) 25.7426 1.60578 0.802891 0.596126i \(-0.203294\pi\)
0.802891 + 0.596126i \(0.203294\pi\)
\(258\) 1.10333i 0.0686900i
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 11.6738 0.722588
\(262\) − 0.124612i − 0.00769854i
\(263\) 9.00000 0.554964 0.277482 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) 2.72949 0.167989
\(265\) − 3.14590i − 0.193251i
\(266\) − 1.85410i − 0.113682i
\(267\) − 6.14590i − 0.376123i
\(268\) − 1.31308i − 0.0802093i
\(269\) 13.7426 0.837904 0.418952 0.908008i \(-0.362398\pi\)
0.418952 + 0.908008i \(0.362398\pi\)
\(270\) −0.326238 −0.0198542
\(271\) 18.4164i 1.11872i 0.828926 + 0.559359i \(0.188952\pi\)
−0.828926 + 0.559359i \(0.811048\pi\)
\(272\) −23.5066 −1.42530
\(273\) 0 0
\(274\) −0.145898 −0.00881402
\(275\) − 23.5623i − 1.42086i
\(276\) 3.16718 0.190642
\(277\) 5.00000 0.300421 0.150210 0.988654i \(-0.452005\pi\)
0.150210 + 0.988654i \(0.452005\pi\)
\(278\) 5.94427i 0.356514i
\(279\) 24.8541i 1.48798i
\(280\) − 0.562306i − 0.0336042i
\(281\) − 2.18034i − 0.130068i −0.997883 0.0650341i \(-0.979284\pi\)
0.997883 0.0650341i \(-0.0207156\pi\)
\(282\) −0.326238 −0.0194272
\(283\) 13.4164 0.797523 0.398761 0.917055i \(-0.369440\pi\)
0.398761 + 0.917055i \(0.369440\pi\)
\(284\) − 15.1672i − 0.900007i
\(285\) 0.708204 0.0419504
\(286\) 0 0
\(287\) 5.23607 0.309075
\(288\) 11.8328i 0.697255i
\(289\) 38.8328 2.28428
\(290\) 0.596748 0.0350422
\(291\) 4.63932i 0.271962i
\(292\) − 3.70820i − 0.217006i
\(293\) − 11.2361i − 0.656418i −0.944605 0.328209i \(-0.893555\pi\)
0.944605 0.328209i \(-0.106445\pi\)
\(294\) − 0.145898i − 0.00850895i
\(295\) 0.854102 0.0497277
\(296\) 5.88854 0.342265
\(297\) − 10.8541i − 0.629819i
\(298\) 1.85410 0.107405
\(299\) 0 0
\(300\) −3.43769 −0.198475
\(301\) 7.56231i 0.435884i
\(302\) −5.61803 −0.323282
\(303\) 3.27051 0.187886
\(304\) − 15.2705i − 0.875824i
\(305\) 2.29180i 0.131228i
\(306\) − 8.14590i − 0.465670i
\(307\) 1.85410i 0.105819i 0.998599 + 0.0529096i \(0.0168495\pi\)
−0.998599 + 0.0529096i \(0.983150\pi\)
\(308\) 9.00000 0.512823
\(309\) 1.79837 0.102306
\(310\) 1.27051i 0.0721601i
\(311\) 12.3262 0.698957 0.349478 0.936944i \(-0.386359\pi\)
0.349478 + 0.936944i \(0.386359\pi\)
\(312\) 0 0
\(313\) −15.1246 −0.854894 −0.427447 0.904041i \(-0.640587\pi\)
−0.427447 + 0.904041i \(0.640587\pi\)
\(314\) − 3.11146i − 0.175590i
\(315\) −1.09017 −0.0614241
\(316\) 7.41641 0.417206
\(317\) − 21.7639i − 1.22238i −0.791482 0.611192i \(-0.790690\pi\)
0.791482 0.611192i \(-0.209310\pi\)
\(318\) 1.20163i 0.0673838i
\(319\) 19.8541i 1.11162i
\(320\) − 1.79837i − 0.100532i
\(321\) 2.14590 0.119772
\(322\) −1.70820 −0.0951945
\(323\) 36.2705i 2.01815i
\(324\) 14.2918 0.793989
\(325\) 0 0
\(326\) 3.70820 0.205378
\(327\) 4.09017i 0.226187i
\(328\) −7.70820 −0.425614
\(329\) −2.23607 −0.123278
\(330\) − 0.270510i − 0.0148911i
\(331\) 16.8541i 0.926385i 0.886258 + 0.463193i \(0.153296\pi\)
−0.886258 + 0.463193i \(0.846704\pi\)
\(332\) 12.4377i 0.682607i
\(333\) − 11.4164i − 0.625615i
\(334\) 3.72949 0.204069
\(335\) −0.270510 −0.0147795
\(336\) − 1.20163i − 0.0655541i
\(337\) −8.56231 −0.466419 −0.233209 0.972427i \(-0.574923\pi\)
−0.233209 + 0.972427i \(0.574923\pi\)
\(338\) 0 0
\(339\) 2.85410 0.155014
\(340\) 5.29180i 0.286988i
\(341\) −42.2705 −2.28908
\(342\) 5.29180 0.286148
\(343\) − 1.00000i − 0.0539949i
\(344\) − 11.1327i − 0.600237i
\(345\) − 0.652476i − 0.0351281i
\(346\) − 3.43769i − 0.184812i
\(347\) 35.2361 1.89157 0.945786 0.324792i \(-0.105294\pi\)
0.945786 + 0.324792i \(0.105294\pi\)
\(348\) 2.89667 0.155278
\(349\) 7.29180i 0.390321i 0.980771 + 0.195160i \(0.0625228\pi\)
−0.980771 + 0.195160i \(0.937477\pi\)
\(350\) 1.85410 0.0991059
\(351\) 0 0
\(352\) −20.1246 −1.07265
\(353\) − 28.8541i − 1.53575i −0.640600 0.767874i \(-0.721314\pi\)
0.640600 0.767874i \(-0.278686\pi\)
\(354\) −0.326238 −0.0173393
\(355\) −3.12461 −0.165837
\(356\) 29.8328i 1.58114i
\(357\) 2.85410i 0.151055i
\(358\) − 3.43769i − 0.181688i
\(359\) − 10.9098i − 0.575799i −0.957661 0.287899i \(-0.907043\pi\)
0.957661 0.287899i \(-0.0929569\pi\)
\(360\) 1.60488 0.0845845
\(361\) −4.56231 −0.240121
\(362\) − 1.41641i − 0.0744447i
\(363\) 4.79837 0.251849
\(364\) 0 0
\(365\) −0.763932 −0.0399860
\(366\) − 0.875388i − 0.0457573i
\(367\) 25.4164 1.32673 0.663363 0.748298i \(-0.269129\pi\)
0.663363 + 0.748298i \(0.269129\pi\)
\(368\) −14.0689 −0.733391
\(369\) 14.9443i 0.777968i
\(370\) − 0.583592i − 0.0303395i
\(371\) 8.23607i 0.427595i
\(372\) 6.16718i 0.319754i
\(373\) −0.437694 −0.0226629 −0.0113315 0.999936i \(-0.503607\pi\)
−0.0113315 + 0.999936i \(0.503607\pi\)
\(374\) 13.8541 0.716379
\(375\) 1.43769i 0.0742422i
\(376\) 3.29180 0.169761
\(377\) 0 0
\(378\) 0.854102 0.0439303
\(379\) 12.8541i 0.660271i 0.943934 + 0.330135i \(0.107094\pi\)
−0.943934 + 0.330135i \(0.892906\pi\)
\(380\) −3.43769 −0.176350
\(381\) −5.40325 −0.276817
\(382\) − 9.02129i − 0.461569i
\(383\) − 24.9787i − 1.27635i −0.769890 0.638176i \(-0.779689\pi\)
0.769890 0.638176i \(-0.220311\pi\)
\(384\) 3.85410i 0.196679i
\(385\) − 1.85410i − 0.0944938i
\(386\) −2.29180 −0.116649
\(387\) −21.5836 −1.09716
\(388\) − 22.5197i − 1.14327i
\(389\) −23.8885 −1.21120 −0.605599 0.795770i \(-0.707066\pi\)
−0.605599 + 0.795770i \(0.707066\pi\)
\(390\) 0 0
\(391\) 33.4164 1.68994
\(392\) 1.47214i 0.0743541i
\(393\) −0.124612 −0.00628583
\(394\) −2.97871 −0.150065
\(395\) − 1.52786i − 0.0768752i
\(396\) 25.6869i 1.29082i
\(397\) − 25.4164i − 1.27561i −0.770197 0.637806i \(-0.779842\pi\)
0.770197 0.637806i \(-0.220158\pi\)
\(398\) 0.922986i 0.0462651i
\(399\) −1.85410 −0.0928212
\(400\) 15.2705 0.763525
\(401\) − 20.4508i − 1.02127i −0.859799 0.510633i \(-0.829411\pi\)
0.859799 0.510633i \(-0.170589\pi\)
\(402\) 0.103326 0.00515341
\(403\) 0 0
\(404\) −15.8754 −0.789830
\(405\) − 2.94427i − 0.146302i
\(406\) −1.56231 −0.0775359
\(407\) 19.4164 0.962436
\(408\) − 4.20163i − 0.208011i
\(409\) − 34.5623i − 1.70900i −0.519455 0.854498i \(-0.673865\pi\)
0.519455 0.854498i \(-0.326135\pi\)
\(410\) 0.763932i 0.0377279i
\(411\) 0.145898i 0.00719662i
\(412\) −8.72949 −0.430071
\(413\) −2.23607 −0.110030
\(414\) − 4.87539i − 0.239612i
\(415\) 2.56231 0.125779
\(416\) 0 0
\(417\) 5.94427 0.291092
\(418\) 9.00000i 0.440204i
\(419\) 5.94427 0.290397 0.145198 0.989403i \(-0.453618\pi\)
0.145198 + 0.989403i \(0.453618\pi\)
\(420\) −0.270510 −0.0131995
\(421\) 25.4164i 1.23872i 0.785107 + 0.619360i \(0.212608\pi\)
−0.785107 + 0.619360i \(0.787392\pi\)
\(422\) 3.32624i 0.161919i
\(423\) − 6.38197i − 0.310302i
\(424\) − 12.1246i − 0.588823i
\(425\) −36.2705 −1.75938
\(426\) 1.19350 0.0578250
\(427\) − 6.00000i − 0.290360i
\(428\) −10.4164 −0.503496
\(429\) 0 0
\(430\) −1.10333 −0.0532071
\(431\) 16.7984i 0.809149i 0.914505 + 0.404575i \(0.132580\pi\)
−0.914505 + 0.404575i \(0.867420\pi\)
\(432\) 7.03444 0.338445
\(433\) −1.00000 −0.0480569 −0.0240285 0.999711i \(-0.507649\pi\)
−0.0240285 + 0.999711i \(0.507649\pi\)
\(434\) − 3.32624i − 0.159665i
\(435\) − 0.596748i − 0.0286119i
\(436\) − 19.8541i − 0.950839i
\(437\) 21.7082i 1.03844i
\(438\) 0.291796 0.0139426
\(439\) −8.14590 −0.388783 −0.194391 0.980924i \(-0.562273\pi\)
−0.194391 + 0.980924i \(0.562273\pi\)
\(440\) 2.72949i 0.130123i
\(441\) 2.85410 0.135910
\(442\) 0 0
\(443\) −0.763932 −0.0362955 −0.0181478 0.999835i \(-0.505777\pi\)
−0.0181478 + 0.999835i \(0.505777\pi\)
\(444\) − 2.83282i − 0.134439i
\(445\) 6.14590 0.291344
\(446\) 5.06888 0.240019
\(447\) − 1.85410i − 0.0876960i
\(448\) 4.70820i 0.222442i
\(449\) 28.4721i 1.34368i 0.740695 + 0.671842i \(0.234496\pi\)
−0.740695 + 0.671842i \(0.765504\pi\)
\(450\) 5.29180i 0.249458i
\(451\) −25.4164 −1.19681
\(452\) −13.8541 −0.651642
\(453\) 5.61803i 0.263958i
\(454\) 2.85410 0.133950
\(455\) 0 0
\(456\) 2.72949 0.127820
\(457\) 11.4164i 0.534037i 0.963691 + 0.267019i \(0.0860385\pi\)
−0.963691 + 0.267019i \(0.913962\pi\)
\(458\) 10.3607 0.484123
\(459\) −16.7082 −0.779872
\(460\) 3.16718i 0.147671i
\(461\) − 39.2148i − 1.82641i −0.407495 0.913207i \(-0.633598\pi\)
0.407495 0.913207i \(-0.366402\pi\)
\(462\) 0.708204i 0.0329486i
\(463\) − 6.70820i − 0.311757i −0.987776 0.155878i \(-0.950179\pi\)
0.987776 0.155878i \(-0.0498208\pi\)
\(464\) −12.8673 −0.597347
\(465\) 1.27051 0.0589185
\(466\) 0.145898i 0.00675860i
\(467\) −33.6525 −1.55725 −0.778625 0.627489i \(-0.784083\pi\)
−0.778625 + 0.627489i \(0.784083\pi\)
\(468\) 0 0
\(469\) 0.708204 0.0327018
\(470\) − 0.326238i − 0.0150482i
\(471\) −3.11146 −0.143368
\(472\) 3.29180 0.151517
\(473\) − 36.7082i − 1.68785i
\(474\) 0.583592i 0.0268053i
\(475\) − 23.5623i − 1.08111i
\(476\) − 13.8541i − 0.635002i
\(477\) −23.5066 −1.07629
\(478\) 4.31308 0.197276
\(479\) − 21.9787i − 1.00423i −0.864800 0.502117i \(-0.832555\pi\)
0.864800 0.502117i \(-0.167445\pi\)
\(480\) 0.604878 0.0276088
\(481\) 0 0
\(482\) 1.69505 0.0772073
\(483\) 1.70820i 0.0777260i
\(484\) −23.2918 −1.05872
\(485\) −4.63932 −0.210661
\(486\) 3.68692i 0.167242i
\(487\) 16.9787i 0.769379i 0.923046 + 0.384689i \(0.125691\pi\)
−0.923046 + 0.384689i \(0.874309\pi\)
\(488\) 8.83282i 0.399843i
\(489\) − 3.70820i − 0.167691i
\(490\) 0.145898 0.00659100
\(491\) −14.6180 −0.659703 −0.329851 0.944033i \(-0.606999\pi\)
−0.329851 + 0.944033i \(0.606999\pi\)
\(492\) 3.70820i 0.167179i
\(493\) 30.5623 1.37646
\(494\) 0 0
\(495\) 5.29180 0.237849
\(496\) − 27.3951i − 1.23008i
\(497\) 8.18034 0.366938
\(498\) −0.978714 −0.0438572
\(499\) − 8.14590i − 0.364660i −0.983237 0.182330i \(-0.941636\pi\)
0.983237 0.182330i \(-0.0583640\pi\)
\(500\) − 6.97871i − 0.312098i
\(501\) − 3.72949i − 0.166621i
\(502\) 2.00000i 0.0892644i
\(503\) 24.3820 1.08714 0.543569 0.839364i \(-0.317073\pi\)
0.543569 + 0.839364i \(0.317073\pi\)
\(504\) −4.20163 −0.187155
\(505\) 3.27051i 0.145536i
\(506\) 8.29180 0.368615
\(507\) 0 0
\(508\) 26.2279 1.16368
\(509\) − 30.5967i − 1.35618i −0.734980 0.678089i \(-0.762809\pi\)
0.734980 0.678089i \(-0.237191\pi\)
\(510\) −0.416408 −0.0184389
\(511\) 2.00000 0.0884748
\(512\) − 22.3050i − 0.985749i
\(513\) − 10.8541i − 0.479220i
\(514\) − 9.83282i − 0.433707i
\(515\) 1.79837i 0.0792458i
\(516\) −5.35565 −0.235770
\(517\) 10.8541 0.477363
\(518\) 1.52786i 0.0671305i
\(519\) −3.43769 −0.150898
\(520\) 0 0
\(521\) −12.6525 −0.554315 −0.277158 0.960824i \(-0.589392\pi\)
−0.277158 + 0.960824i \(0.589392\pi\)
\(522\) − 4.45898i − 0.195164i
\(523\) −39.1246 −1.71080 −0.855400 0.517968i \(-0.826689\pi\)
−0.855400 + 0.517968i \(0.826689\pi\)
\(524\) 0.604878 0.0264242
\(525\) − 1.85410i − 0.0809196i
\(526\) − 3.43769i − 0.149891i
\(527\) 65.0689i 2.83445i
\(528\) 5.83282i 0.253841i
\(529\) −3.00000 −0.130435
\(530\) −1.20163 −0.0521953
\(531\) − 6.38197i − 0.276954i
\(532\) 9.00000 0.390199
\(533\) 0 0
\(534\) −2.34752 −0.101587
\(535\) 2.14590i 0.0927753i
\(536\) −1.04257 −0.0450323
\(537\) −3.43769 −0.148347
\(538\) − 5.24922i − 0.226310i
\(539\) 4.85410i 0.209081i
\(540\) − 1.58359i − 0.0681470i
\(541\) 1.72949i 0.0743566i 0.999309 + 0.0371783i \(0.0118369\pi\)
−0.999309 + 0.0371783i \(0.988163\pi\)
\(542\) 7.03444 0.302155
\(543\) −1.41641 −0.0607839
\(544\) 30.9787i 1.32820i
\(545\) −4.09017 −0.175204
\(546\) 0 0
\(547\) −3.00000 −0.128271 −0.0641354 0.997941i \(-0.520429\pi\)
−0.0641354 + 0.997941i \(0.520429\pi\)
\(548\) − 0.708204i − 0.0302530i
\(549\) 17.1246 0.730861
\(550\) −9.00000 −0.383761
\(551\) 19.8541i 0.845813i
\(552\) − 2.51471i − 0.107033i
\(553\) 4.00000i 0.170097i
\(554\) − 1.90983i − 0.0811409i
\(555\) −0.583592 −0.0247721
\(556\) −28.8541 −1.22369
\(557\) 18.9787i 0.804154i 0.915606 + 0.402077i \(0.131712\pi\)
−0.915606 + 0.402077i \(0.868288\pi\)
\(558\) 9.49342 0.401889
\(559\) 0 0
\(560\) 1.20163 0.0507780
\(561\) − 13.8541i − 0.584921i
\(562\) −0.832816 −0.0351302
\(563\) 38.9443 1.64131 0.820653 0.571427i \(-0.193610\pi\)
0.820653 + 0.571427i \(0.193610\pi\)
\(564\) − 1.58359i − 0.0666813i
\(565\) 2.85410i 0.120073i
\(566\) − 5.12461i − 0.215404i
\(567\) 7.70820i 0.323714i
\(568\) −12.0426 −0.505295
\(569\) 2.94427 0.123430 0.0617151 0.998094i \(-0.480343\pi\)
0.0617151 + 0.998094i \(0.480343\pi\)
\(570\) − 0.270510i − 0.0113304i
\(571\) 35.6869 1.49345 0.746726 0.665132i \(-0.231625\pi\)
0.746726 + 0.665132i \(0.231625\pi\)
\(572\) 0 0
\(573\) −9.02129 −0.376870
\(574\) − 2.00000i − 0.0834784i
\(575\) −21.7082 −0.905295
\(576\) −13.4377 −0.559904
\(577\) − 9.83282i − 0.409345i −0.978830 0.204673i \(-0.934387\pi\)
0.978830 0.204673i \(-0.0656130\pi\)
\(578\) − 14.8328i − 0.616964i
\(579\) 2.29180i 0.0952438i
\(580\) 2.89667i 0.120278i
\(581\) −6.70820 −0.278303
\(582\) 1.77206 0.0734544
\(583\) − 39.9787i − 1.65575i
\(584\) −2.94427 −0.121835
\(585\) 0 0
\(586\) −4.29180 −0.177292
\(587\) 31.0902i 1.28323i 0.767027 + 0.641614i \(0.221735\pi\)
−0.767027 + 0.641614i \(0.778265\pi\)
\(588\) 0.708204 0.0292058
\(589\) −42.2705 −1.74173
\(590\) − 0.326238i − 0.0134310i
\(591\) 2.97871i 0.122528i
\(592\) 12.5836i 0.517182i
\(593\) 19.2016i 0.788516i 0.919000 + 0.394258i \(0.128998\pi\)
−0.919000 + 0.394258i \(0.871002\pi\)
\(594\) −4.14590 −0.170108
\(595\) −2.85410 −0.117007
\(596\) 9.00000i 0.368654i
\(597\) 0.922986 0.0377753
\(598\) 0 0
\(599\) −8.50658 −0.347569 −0.173785 0.984784i \(-0.555600\pi\)
−0.173785 + 0.984784i \(0.555600\pi\)
\(600\) 2.72949i 0.111431i
\(601\) −33.3951 −1.36222 −0.681108 0.732183i \(-0.738501\pi\)
−0.681108 + 0.732183i \(0.738501\pi\)
\(602\) 2.88854 0.117728
\(603\) 2.02129i 0.0823131i
\(604\) − 27.2705i − 1.10962i
\(605\) 4.79837i 0.195082i
\(606\) − 1.24922i − 0.0507462i
\(607\) −23.0000 −0.933541 −0.466771 0.884378i \(-0.654583\pi\)
−0.466771 + 0.884378i \(0.654583\pi\)
\(608\) −20.1246 −0.816161
\(609\) 1.56231i 0.0633078i
\(610\) 0.875388 0.0354434
\(611\) 0 0
\(612\) 39.5410 1.59835
\(613\) − 14.4377i − 0.583133i −0.956551 0.291566i \(-0.905824\pi\)
0.956551 0.291566i \(-0.0941765\pi\)
\(614\) 0.708204 0.0285808
\(615\) 0.763932 0.0308047
\(616\) − 7.14590i − 0.287916i
\(617\) − 17.9443i − 0.722409i −0.932487 0.361205i \(-0.882366\pi\)
0.932487 0.361205i \(-0.117634\pi\)
\(618\) − 0.686918i − 0.0276319i
\(619\) − 17.4164i − 0.700025i −0.936745 0.350012i \(-0.886177\pi\)
0.936745 0.350012i \(-0.113823\pi\)
\(620\) −6.16718 −0.247680
\(621\) −10.0000 −0.401286
\(622\) − 4.70820i − 0.188782i
\(623\) −16.0902 −0.644639
\(624\) 0 0
\(625\) 22.8328 0.913313
\(626\) 5.77709i 0.230899i
\(627\) 9.00000 0.359425
\(628\) 15.1033 0.602688
\(629\) − 29.8885i − 1.19173i
\(630\) 0.416408i 0.0165901i
\(631\) − 39.3951i − 1.56830i −0.620574 0.784148i \(-0.713101\pi\)
0.620574 0.784148i \(-0.286899\pi\)
\(632\) − 5.88854i − 0.234234i
\(633\) 3.32624 0.132206
\(634\) −8.31308 −0.330155
\(635\) − 5.40325i − 0.214422i
\(636\) −5.83282 −0.231286
\(637\) 0 0
\(638\) 7.58359 0.300237
\(639\) 23.3475i 0.923614i
\(640\) −3.85410 −0.152347
\(641\) 9.49342 0.374968 0.187484 0.982268i \(-0.439967\pi\)
0.187484 + 0.982268i \(0.439967\pi\)
\(642\) − 0.819660i − 0.0323494i
\(643\) 7.00000i 0.276053i 0.990429 + 0.138027i \(0.0440759\pi\)
−0.990429 + 0.138027i \(0.955924\pi\)
\(644\) − 8.29180i − 0.326743i
\(645\) 1.10333i 0.0434434i
\(646\) 13.8541 0.545082
\(647\) −29.2361 −1.14939 −0.574694 0.818368i \(-0.694879\pi\)
−0.574694 + 0.818368i \(0.694879\pi\)
\(648\) − 11.3475i − 0.445773i
\(649\) 10.8541 0.426061
\(650\) 0 0
\(651\) −3.32624 −0.130366
\(652\) 18.0000i 0.704934i
\(653\) −2.61803 −0.102452 −0.0512258 0.998687i \(-0.516313\pi\)
−0.0512258 + 0.998687i \(0.516313\pi\)
\(654\) 1.56231 0.0610910
\(655\) − 0.124612i − 0.00486899i
\(656\) − 16.4721i − 0.643129i
\(657\) 5.70820i 0.222698i
\(658\) 0.854102i 0.0332964i
\(659\) 11.8885 0.463112 0.231556 0.972822i \(-0.425618\pi\)
0.231556 + 0.972822i \(0.425618\pi\)
\(660\) 1.31308 0.0511117
\(661\) 18.5410i 0.721162i 0.932728 + 0.360581i \(0.117422\pi\)
−0.932728 + 0.360581i \(0.882578\pi\)
\(662\) 6.43769 0.250208
\(663\) 0 0
\(664\) 9.87539 0.383239
\(665\) − 1.85410i − 0.0718990i
\(666\) −4.36068 −0.168973
\(667\) 18.2918 0.708261
\(668\) 18.1033i 0.700439i
\(669\) − 5.06888i − 0.195974i
\(670\) 0.103326i 0.00399181i
\(671\) 29.1246i 1.12434i
\(672\) −1.58359 −0.0610884
\(673\) −41.2492 −1.59004 −0.795020 0.606583i \(-0.792540\pi\)
−0.795020 + 0.606583i \(0.792540\pi\)
\(674\) 3.27051i 0.125975i
\(675\) 10.8541 0.417775
\(676\) 0 0
\(677\) 1.25735 0.0483240 0.0241620 0.999708i \(-0.492308\pi\)
0.0241620 + 0.999708i \(0.492308\pi\)
\(678\) − 1.09017i − 0.0418677i
\(679\) 12.1459 0.466117
\(680\) 4.20163 0.161125
\(681\) − 2.85410i − 0.109369i
\(682\) 16.1459i 0.618258i
\(683\) − 7.47214i − 0.285913i −0.989729 0.142957i \(-0.954339\pi\)
0.989729 0.142957i \(-0.0456610\pi\)
\(684\) 25.6869i 0.982164i
\(685\) −0.145898 −0.00557448
\(686\) −0.381966 −0.0145835
\(687\) − 10.3607i − 0.395285i
\(688\) 23.7902 0.906995
\(689\) 0 0
\(690\) −0.249224 −0.00948778
\(691\) − 0.854102i − 0.0324916i −0.999868 0.0162458i \(-0.994829\pi\)
0.999868 0.0162458i \(-0.00517142\pi\)
\(692\) 16.6869 0.634341
\(693\) −13.8541 −0.526274
\(694\) − 13.4590i − 0.510896i
\(695\) 5.94427i 0.225479i
\(696\) − 2.29993i − 0.0871785i
\(697\) 39.1246i 1.48195i
\(698\) 2.78522 0.105422
\(699\) 0.145898 0.00551837
\(700\) 9.00000i 0.340168i
\(701\) −6.76393 −0.255470 −0.127735 0.991808i \(-0.540771\pi\)
−0.127735 + 0.991808i \(0.540771\pi\)
\(702\) 0 0
\(703\) 19.4164 0.732304
\(704\) − 22.8541i − 0.861346i
\(705\) −0.326238 −0.0122868
\(706\) −11.0213 −0.414792
\(707\) − 8.56231i − 0.322019i
\(708\) − 1.58359i − 0.0595150i
\(709\) 3.43769i 0.129105i 0.997914 + 0.0645527i \(0.0205620\pi\)
−0.997914 + 0.0645527i \(0.979438\pi\)
\(710\) 1.19350i 0.0447911i
\(711\) −11.4164 −0.428149
\(712\) 23.6869 0.887705
\(713\) 38.9443i 1.45847i
\(714\) 1.09017 0.0407986
\(715\) 0 0
\(716\) 16.6869 0.623619
\(717\) − 4.31308i − 0.161075i
\(718\) −4.16718 −0.155518
\(719\) 32.1246 1.19805 0.599023 0.800732i \(-0.295556\pi\)
0.599023 + 0.800732i \(0.295556\pi\)
\(720\) 3.42956i 0.127812i
\(721\) − 4.70820i − 0.175343i
\(722\) 1.74265i 0.0648546i
\(723\) − 1.69505i − 0.0630395i
\(724\) 6.87539 0.255522
\(725\) −19.8541 −0.737363
\(726\) − 1.83282i − 0.0680222i
\(727\) 17.2918 0.641317 0.320659 0.947195i \(-0.396096\pi\)
0.320659 + 0.947195i \(0.396096\pi\)
\(728\) 0 0
\(729\) −19.4377 −0.719915
\(730\) 0.291796i 0.0107999i
\(731\) −56.5066 −2.08997
\(732\) 4.24922 0.157056
\(733\) − 1.27051i − 0.0469274i −0.999725 0.0234637i \(-0.992531\pi\)
0.999725 0.0234637i \(-0.00746941\pi\)
\(734\) − 9.70820i − 0.358336i
\(735\) − 0.145898i − 0.00538153i
\(736\) 18.5410i 0.683431i
\(737\) −3.43769 −0.126629
\(738\) 5.70820 0.210122
\(739\) − 47.1246i − 1.73351i −0.498737 0.866753i \(-0.666203\pi\)
0.498737 0.866753i \(-0.333797\pi\)
\(740\) 2.83282 0.104136
\(741\) 0 0
\(742\) 3.14590 0.115490
\(743\) − 23.6738i − 0.868506i −0.900791 0.434253i \(-0.857012\pi\)
0.900791 0.434253i \(-0.142988\pi\)
\(744\) 4.89667 0.179521
\(745\) 1.85410 0.0679290
\(746\) 0.167184i 0.00612105i
\(747\) − 19.1459i − 0.700512i
\(748\) 67.2492i 2.45888i
\(749\) − 5.61803i − 0.205278i
\(750\) 0.549150 0.0200521
\(751\) −9.29180 −0.339062 −0.169531 0.985525i \(-0.554225\pi\)
−0.169531 + 0.985525i \(0.554225\pi\)
\(752\) 7.03444i 0.256520i
\(753\) 2.00000 0.0728841
\(754\) 0 0
\(755\) −5.61803 −0.204461
\(756\) 4.14590i 0.150785i
\(757\) 28.0000 1.01768 0.508839 0.860862i \(-0.330075\pi\)
0.508839 + 0.860862i \(0.330075\pi\)
\(758\) 4.90983 0.178333
\(759\) − 8.29180i − 0.300973i
\(760\) 2.72949i 0.0990090i
\(761\) − 22.1459i − 0.802788i −0.915905 0.401394i \(-0.868526\pi\)
0.915905 0.401394i \(-0.131474\pi\)
\(762\) 2.06386i 0.0747657i
\(763\) 10.7082 0.387663
\(764\) 43.7902 1.58428
\(765\) − 8.14590i − 0.294516i
\(766\) −9.54102 −0.344731
\(767\) 0 0
\(768\) −2.12461 −0.0766653
\(769\) − 8.41641i − 0.303503i −0.988419 0.151752i \(-0.951509\pi\)
0.988419 0.151752i \(-0.0484914\pi\)
\(770\) −0.708204 −0.0255219
\(771\) −9.83282 −0.354120
\(772\) − 11.1246i − 0.400384i
\(773\) 19.3607i 0.696355i 0.937429 + 0.348178i \(0.113199\pi\)
−0.937429 + 0.348178i \(0.886801\pi\)
\(774\) 8.24420i 0.296332i
\(775\) − 42.2705i − 1.51840i
\(776\) −17.8804 −0.641869
\(777\) 1.52786 0.0548118
\(778\) 9.12461i 0.327133i
\(779\) −25.4164 −0.910637
\(780\) 0 0
\(781\) −39.7082 −1.42087
\(782\) − 12.7639i − 0.456437i
\(783\) −9.14590 −0.326848
\(784\) −3.14590 −0.112354
\(785\) − 3.11146i − 0.111053i
\(786\) 0.0475975i 0.00169775i
\(787\) 29.4164i 1.04858i 0.851539 + 0.524291i \(0.175670\pi\)
−0.851539 + 0.524291i \(0.824330\pi\)
\(788\) − 14.4590i − 0.515080i
\(789\) −3.43769 −0.122385
\(790\) −0.583592 −0.0207633
\(791\) − 7.47214i − 0.265679i
\(792\) 20.3951 0.724709
\(793\) 0 0
\(794\) −9.70820 −0.344531
\(795\) 1.20163i 0.0426173i
\(796\) −4.48027 −0.158799
\(797\) −14.1803 −0.502293 −0.251147 0.967949i \(-0.580808\pi\)
−0.251147 + 0.967949i \(0.580808\pi\)
\(798\) 0.708204i 0.0250701i
\(799\) − 16.7082i − 0.591094i
\(800\) − 20.1246i − 0.711512i
\(801\) − 45.9230i − 1.62261i
\(802\) −7.81153 −0.275835
\(803\) −9.70820 −0.342595
\(804\) 0.501553i 0.0176884i
\(805\) −1.70820 −0.0602063
\(806\) 0 0
\(807\) −5.24922 −0.184781
\(808\) 12.6049i 0.443438i
\(809\) 22.4164 0.788119 0.394059 0.919085i \(-0.371070\pi\)
0.394059 + 0.919085i \(0.371070\pi\)
\(810\) −1.12461 −0.0395148
\(811\) − 5.72949i − 0.201190i −0.994927 0.100595i \(-0.967925\pi\)
0.994927 0.100595i \(-0.0320746\pi\)
\(812\) − 7.58359i − 0.266132i
\(813\) − 7.03444i − 0.246709i
\(814\) − 7.41641i − 0.259945i
\(815\) 3.70820 0.129893
\(816\) 8.97871 0.314318
\(817\) − 36.7082i − 1.28426i
\(818\) −13.2016 −0.461584
\(819\) 0 0
\(820\) −3.70820 −0.129496
\(821\) − 37.3607i − 1.30390i −0.758263 0.651948i \(-0.773952\pi\)
0.758263 0.651948i \(-0.226048\pi\)
\(822\) 0.0557281 0.00194374
\(823\) 11.5836 0.403779 0.201889 0.979408i \(-0.435292\pi\)
0.201889 + 0.979408i \(0.435292\pi\)
\(824\) 6.93112i 0.241457i
\(825\) 9.00000i 0.313340i
\(826\) 0.854102i 0.0297180i
\(827\) − 30.9787i − 1.07724i −0.842550 0.538618i \(-0.818947\pi\)
0.842550 0.538618i \(-0.181053\pi\)
\(828\) 23.6656 0.822438
\(829\) 12.5623 0.436307 0.218153 0.975914i \(-0.429997\pi\)
0.218153 + 0.975914i \(0.429997\pi\)
\(830\) − 0.978714i − 0.0339717i
\(831\) −1.90983 −0.0662513
\(832\) 0 0
\(833\) 7.47214 0.258894
\(834\) − 2.27051i − 0.0786213i
\(835\) 3.72949 0.129064
\(836\) −43.6869 −1.51094
\(837\) − 19.4721i − 0.673055i
\(838\) − 2.27051i − 0.0784335i
\(839\) − 28.7426i − 0.992306i −0.868235 0.496153i \(-0.834745\pi\)
0.868235 0.496153i \(-0.165255\pi\)
\(840\) 0.214782i 0.00741067i
\(841\) −12.2705 −0.423121
\(842\) 9.70820 0.334567
\(843\) 0.832816i 0.0286837i
\(844\) −16.1459 −0.555765
\(845\) 0 0
\(846\) −2.43769 −0.0838096
\(847\) − 12.5623i − 0.431646i
\(848\) 25.9098 0.889747
\(849\) −5.12461 −0.175876
\(850\) 13.8541i 0.475192i
\(851\) − 17.8885i − 0.613211i
\(852\) 5.79335i 0.198477i
\(853\) − 26.1246i − 0.894490i −0.894412 0.447245i \(-0.852405\pi\)
0.894412 0.447245i \(-0.147595\pi\)
\(854\) −2.29180 −0.0784236
\(855\) 5.29180 0.180976
\(856\) 8.27051i 0.282680i
\(857\) −29.4508 −1.00602 −0.503011 0.864280i \(-0.667774\pi\)
−0.503011 + 0.864280i \(0.667774\pi\)
\(858\) 0 0
\(859\) 36.2492 1.23681 0.618404 0.785861i \(-0.287780\pi\)
0.618404 + 0.785861i \(0.287780\pi\)
\(860\) − 5.35565i − 0.182626i
\(861\) −2.00000 −0.0681598
\(862\) 6.41641 0.218544
\(863\) 23.8885i 0.813175i 0.913612 + 0.406588i \(0.133281\pi\)
−0.913612 + 0.406588i \(0.866719\pi\)
\(864\) − 9.27051i − 0.315389i
\(865\) − 3.43769i − 0.116885i
\(866\) 0.381966i 0.0129797i
\(867\) −14.8328 −0.503749
\(868\) 16.1459 0.548027
\(869\) − 19.4164i − 0.658657i
\(870\) −0.227937 −0.00772780
\(871\) 0 0
\(872\) −15.7639 −0.533834
\(873\) 34.6656i 1.17325i
\(874\) 8.29180 0.280474
\(875\) 3.76393 0.127244
\(876\) 1.41641i 0.0478560i
\(877\) 12.7082i 0.429126i 0.976710 + 0.214563i \(0.0688327\pi\)
−0.976710 + 0.214563i \(0.931167\pi\)
\(878\) 3.11146i 0.105007i
\(879\) 4.29180i 0.144759i
\(880\) −5.83282 −0.196624
\(881\) −12.5967 −0.424395 −0.212198 0.977227i \(-0.568062\pi\)
−0.212198 + 0.977227i \(0.568062\pi\)
\(882\) − 1.09017i − 0.0367079i
\(883\) −29.0000 −0.975928 −0.487964 0.872864i \(-0.662260\pi\)
−0.487964 + 0.872864i \(0.662260\pi\)
\(884\) 0 0
\(885\) −0.326238 −0.0109664
\(886\) 0.291796i 0.00980308i
\(887\) 23.3475 0.783933 0.391967 0.919979i \(-0.371795\pi\)
0.391967 + 0.919979i \(0.371795\pi\)
\(888\) −2.24922 −0.0754790
\(889\) 14.1459i 0.474438i
\(890\) − 2.34752i − 0.0786892i
\(891\) − 37.4164i − 1.25350i
\(892\) 24.6049i 0.823832i
\(893\) 10.8541 0.363219
\(894\) −0.708204 −0.0236859
\(895\) − 3.43769i − 0.114909i
\(896\) 10.0902 0.337089
\(897\) 0 0
\(898\) 10.8754 0.362916
\(899\) 35.6180i 1.18793i
\(900\) −25.6869 −0.856231
\(901\) −61.5410 −2.05023
\(902\) 9.70820i 0.323248i
\(903\) − 2.88854i − 0.0961247i
\(904\) 11.0000i 0.365855i
\(905\) − 1.41641i − 0.0470830i
\(906\) 2.14590 0.0712927
\(907\) 24.0000 0.796907 0.398453 0.917189i \(-0.369547\pi\)
0.398453 + 0.917189i \(0.369547\pi\)
\(908\) 13.8541i 0.459765i
\(909\) 24.4377 0.810547
\(910\) 0 0
\(911\) 37.6869 1.24862 0.624312 0.781175i \(-0.285380\pi\)
0.624312 + 0.781175i \(0.285380\pi\)
\(912\) 5.83282i 0.193144i
\(913\) 32.5623 1.07766
\(914\) 4.36068 0.144238
\(915\) − 0.875388i − 0.0289394i
\(916\) 50.2918i 1.66169i
\(917\) 0.326238i 0.0107733i
\(918\) 6.38197i 0.210636i
\(919\) −30.0000 −0.989609 −0.494804 0.869004i \(-0.664760\pi\)
−0.494804 + 0.869004i \(0.664760\pi\)
\(920\) 2.51471 0.0829075
\(921\) − 0.708204i − 0.0233361i
\(922\) −14.9787 −0.493298
\(923\) 0 0
\(924\) −3.43769 −0.113092
\(925\) 19.4164i 0.638408i
\(926\) −2.56231 −0.0842026
\(927\) 13.4377 0.441352
\(928\) 16.9574i 0.556655i
\(929\) 47.0689i 1.54428i 0.635452 + 0.772140i \(0.280814\pi\)
−0.635452 + 0.772140i \(0.719186\pi\)
\(930\) − 0.485292i − 0.0159133i
\(931\) 4.85410i 0.159087i
\(932\) −0.708204 −0.0231980
\(933\) −4.70820 −0.154140
\(934\) 12.8541i 0.420599i
\(935\) 13.8541 0.453078
\(936\) 0 0
\(937\) −56.1246 −1.83351 −0.916756 0.399449i \(-0.869202\pi\)
−0.916756 + 0.399449i \(0.869202\pi\)
\(938\) − 0.270510i − 0.00883246i
\(939\) 5.77709 0.188528
\(940\) 1.58359 0.0516511
\(941\) − 51.6525i − 1.68382i −0.539616 0.841911i \(-0.681431\pi\)
0.539616 0.841911i \(-0.318569\pi\)
\(942\) 1.18847i 0.0387225i
\(943\) 23.4164i 0.762543i
\(944\) 7.03444i 0.228952i
\(945\) 0.854102 0.0277839
\(946\) −14.0213 −0.455871
\(947\) 45.8673i 1.49049i 0.666793 + 0.745243i \(0.267666\pi\)
−0.666793 + 0.745243i \(0.732334\pi\)
\(948\) −2.83282 −0.0920056
\(949\) 0 0
\(950\) −9.00000 −0.291999
\(951\) 8.31308i 0.269570i
\(952\) −11.0000 −0.356512
\(953\) 44.7771 1.45047 0.725236 0.688500i \(-0.241731\pi\)
0.725236 + 0.688500i \(0.241731\pi\)
\(954\) 8.97871i 0.290697i
\(955\) − 9.02129i − 0.291922i
\(956\) 20.9361i 0.677123i
\(957\) − 7.58359i − 0.245143i
\(958\) −8.39512 −0.271234
\(959\) 0.381966 0.0123343
\(960\) 0.686918i 0.0221702i
\(961\) −44.8328 −1.44622
\(962\) 0 0
\(963\) 16.0344 0.516703
\(964\) 8.22794i 0.265004i
\(965\) −2.29180 −0.0737755
\(966\) 0.652476 0.0209931
\(967\) 39.0000i 1.25416i 0.778957 + 0.627078i \(0.215749\pi\)
−0.778957 + 0.627078i \(0.784251\pi\)
\(968\) 18.4934i 0.594401i
\(969\) − 13.8541i − 0.445058i
\(970\) 1.77206i 0.0568975i
\(971\) 58.4164 1.87467 0.937336 0.348427i \(-0.113284\pi\)
0.937336 + 0.348427i \(0.113284\pi\)
\(972\) −17.8967 −0.574036
\(973\) − 15.5623i − 0.498905i
\(974\) 6.48529 0.207802
\(975\) 0 0
\(976\) −18.8754 −0.604186
\(977\) 31.4721i 1.00688i 0.864029 + 0.503441i \(0.167933\pi\)
−0.864029 + 0.503441i \(0.832067\pi\)
\(978\) −1.41641 −0.0452917
\(979\) 78.1033 2.49619
\(980\) 0.708204i 0.0226227i
\(981\) 30.5623i 0.975779i
\(982\) 5.58359i 0.178180i
\(983\) 20.6180i 0.657613i 0.944397 + 0.328807i \(0.106646\pi\)
−0.944397 + 0.328807i \(0.893354\pi\)
\(984\) 2.94427 0.0938600
\(985\) −2.97871 −0.0949097
\(986\) − 11.6738i − 0.371768i
\(987\) 0.854102 0.0271864
\(988\) 0 0
\(989\) −33.8197 −1.07540
\(990\) − 2.02129i − 0.0642407i
\(991\) −22.8541 −0.725984 −0.362992 0.931792i \(-0.618245\pi\)
−0.362992 + 0.931792i \(0.618245\pi\)
\(992\) −36.1033 −1.14628
\(993\) − 6.43769i − 0.204294i
\(994\) − 3.12461i − 0.0991067i
\(995\) 0.922986i 0.0292606i
\(996\) − 4.75078i − 0.150534i
\(997\) 49.0000 1.55185 0.775923 0.630828i \(-0.217285\pi\)
0.775923 + 0.630828i \(0.217285\pi\)
\(998\) −3.11146 −0.0984914
\(999\) 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.337.2 4
13.5 odd 4 1183.2.a.c.1.2 2
13.7 odd 12 91.2.f.a.29.2 yes 4
13.8 odd 4 1183.2.a.g.1.1 2
13.11 odd 12 91.2.f.a.22.2 4
13.12 even 2 inner 1183.2.c.c.337.3 4
39.11 even 12 819.2.o.c.568.1 4
39.20 even 12 819.2.o.c.757.1 4
52.7 even 12 1456.2.s.h.1121.2 4
52.11 even 12 1456.2.s.h.113.2 4
91.11 odd 12 637.2.g.b.373.2 4
91.20 even 12 637.2.f.c.393.2 4
91.24 even 12 637.2.g.c.373.2 4
91.33 even 12 637.2.g.c.263.2 4
91.34 even 4 8281.2.a.bb.1.1 2
91.37 odd 12 637.2.h.g.165.1 4
91.46 odd 12 637.2.h.g.471.1 4
91.59 even 12 637.2.h.f.471.1 4
91.72 odd 12 637.2.g.b.263.2 4
91.76 even 12 637.2.f.c.295.2 4
91.83 even 4 8281.2.a.n.1.2 2
91.89 even 12 637.2.h.f.165.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.a.22.2 4 13.11 odd 12
91.2.f.a.29.2 yes 4 13.7 odd 12
637.2.f.c.295.2 4 91.76 even 12
637.2.f.c.393.2 4 91.20 even 12
637.2.g.b.263.2 4 91.72 odd 12
637.2.g.b.373.2 4 91.11 odd 12
637.2.g.c.263.2 4 91.33 even 12
637.2.g.c.373.2 4 91.24 even 12
637.2.h.f.165.1 4 91.89 even 12
637.2.h.f.471.1 4 91.59 even 12
637.2.h.g.165.1 4 91.37 odd 12
637.2.h.g.471.1 4 91.46 odd 12
819.2.o.c.568.1 4 39.11 even 12
819.2.o.c.757.1 4 39.20 even 12
1183.2.a.c.1.2 2 13.5 odd 4
1183.2.a.g.1.1 2 13.8 odd 4
1183.2.c.c.337.2 4 1.1 even 1 trivial
1183.2.c.c.337.3 4 13.12 even 2 inner
1456.2.s.h.113.2 4 52.11 even 12
1456.2.s.h.1121.2 4 52.7 even 12
8281.2.a.n.1.2 2 91.83 even 4
8281.2.a.bb.1.1 2 91.34 even 4