Properties

Label 2541.2.a.v.1.1
Level 2541
Weight 2
Character 2541.1
Self dual yes
Analytic conductor 20.290
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2541.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.2899871536\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\)
Character \(\chi\) = 2541.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -0.618034 q^{5} +2.23607 q^{6} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -0.618034 q^{5} +2.23607 q^{6} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +1.38197 q^{10} -3.00000 q^{12} -3.23607 q^{13} +2.23607 q^{14} +0.618034 q^{15} -1.00000 q^{16} -4.85410 q^{17} -2.23607 q^{18} -2.85410 q^{19} -1.85410 q^{20} +1.00000 q^{21} +4.38197 q^{23} +2.23607 q^{24} -4.61803 q^{25} +7.23607 q^{26} -1.00000 q^{27} -3.00000 q^{28} +6.00000 q^{29} -1.38197 q^{30} -3.09017 q^{31} +6.70820 q^{32} +10.8541 q^{34} +0.618034 q^{35} +3.00000 q^{36} +4.61803 q^{37} +6.38197 q^{38} +3.23607 q^{39} +1.38197 q^{40} -7.38197 q^{41} -2.23607 q^{42} -9.70820 q^{43} -0.618034 q^{45} -9.79837 q^{46} +4.47214 q^{47} +1.00000 q^{48} +1.00000 q^{49} +10.3262 q^{50} +4.85410 q^{51} -9.70820 q^{52} +6.76393 q^{53} +2.23607 q^{54} +2.23607 q^{56} +2.85410 q^{57} -13.4164 q^{58} +3.23607 q^{59} +1.85410 q^{60} +6.90983 q^{62} -1.00000 q^{63} -13.0000 q^{64} +2.00000 q^{65} -14.5623 q^{68} -4.38197 q^{69} -1.38197 q^{70} -4.94427 q^{71} -2.23607 q^{72} -13.7082 q^{73} -10.3262 q^{74} +4.61803 q^{75} -8.56231 q^{76} -7.23607 q^{78} +10.0000 q^{79} +0.618034 q^{80} +1.00000 q^{81} +16.5066 q^{82} -16.4721 q^{83} +3.00000 q^{84} +3.00000 q^{85} +21.7082 q^{86} -6.00000 q^{87} +5.61803 q^{89} +1.38197 q^{90} +3.23607 q^{91} +13.1459 q^{92} +3.09017 q^{93} -10.0000 q^{94} +1.76393 q^{95} -6.70820 q^{96} -6.00000 q^{97} -2.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 6q^{4} + q^{5} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 6q^{4} + q^{5} - 2q^{7} + 2q^{9} + 5q^{10} - 6q^{12} - 2q^{13} - q^{15} - 2q^{16} - 3q^{17} + q^{19} + 3q^{20} + 2q^{21} + 11q^{23} - 7q^{25} + 10q^{26} - 2q^{27} - 6q^{28} + 12q^{29} - 5q^{30} + 5q^{31} + 15q^{34} - q^{35} + 6q^{36} + 7q^{37} + 15q^{38} + 2q^{39} + 5q^{40} - 17q^{41} - 6q^{43} + q^{45} + 5q^{46} + 2q^{48} + 2q^{49} + 5q^{50} + 3q^{51} - 6q^{52} + 18q^{53} - q^{57} + 2q^{59} - 3q^{60} + 25q^{62} - 2q^{63} - 26q^{64} + 4q^{65} - 9q^{68} - 11q^{69} - 5q^{70} + 8q^{71} - 14q^{73} - 5q^{74} + 7q^{75} + 3q^{76} - 10q^{78} + 20q^{79} - q^{80} + 2q^{81} - 5q^{82} - 24q^{83} + 6q^{84} + 6q^{85} + 30q^{86} - 12q^{87} + 9q^{89} + 5q^{90} + 2q^{91} + 33q^{92} - 5q^{93} - 20q^{94} + 8q^{95} - 12q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.23607 −1.58114 −0.790569 0.612372i \(-0.790215\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.00000 1.50000
\(5\) −0.618034 −0.276393 −0.138197 0.990405i \(-0.544131\pi\)
−0.138197 + 0.990405i \(0.544131\pi\)
\(6\) 2.23607 0.912871
\(7\) −1.00000 −0.377964
\(8\) −2.23607 −0.790569
\(9\) 1.00000 0.333333
\(10\) 1.38197 0.437016
\(11\) 0 0
\(12\) −3.00000 −0.866025
\(13\) −3.23607 −0.897524 −0.448762 0.893651i \(-0.648135\pi\)
−0.448762 + 0.893651i \(0.648135\pi\)
\(14\) 2.23607 0.597614
\(15\) 0.618034 0.159576
\(16\) −1.00000 −0.250000
\(17\) −4.85410 −1.17729 −0.588646 0.808391i \(-0.700339\pi\)
−0.588646 + 0.808391i \(0.700339\pi\)
\(18\) −2.23607 −0.527046
\(19\) −2.85410 −0.654776 −0.327388 0.944890i \(-0.606168\pi\)
−0.327388 + 0.944890i \(0.606168\pi\)
\(20\) −1.85410 −0.414590
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 4.38197 0.913703 0.456852 0.889543i \(-0.348977\pi\)
0.456852 + 0.889543i \(0.348977\pi\)
\(24\) 2.23607 0.456435
\(25\) −4.61803 −0.923607
\(26\) 7.23607 1.41911
\(27\) −1.00000 −0.192450
\(28\) −3.00000 −0.566947
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −1.38197 −0.252311
\(31\) −3.09017 −0.555011 −0.277505 0.960724i \(-0.589508\pi\)
−0.277505 + 0.960724i \(0.589508\pi\)
\(32\) 6.70820 1.18585
\(33\) 0 0
\(34\) 10.8541 1.86146
\(35\) 0.618034 0.104467
\(36\) 3.00000 0.500000
\(37\) 4.61803 0.759200 0.379600 0.925151i \(-0.376062\pi\)
0.379600 + 0.925151i \(0.376062\pi\)
\(38\) 6.38197 1.03529
\(39\) 3.23607 0.518186
\(40\) 1.38197 0.218508
\(41\) −7.38197 −1.15287 −0.576435 0.817143i \(-0.695556\pi\)
−0.576435 + 0.817143i \(0.695556\pi\)
\(42\) −2.23607 −0.345033
\(43\) −9.70820 −1.48049 −0.740244 0.672339i \(-0.765290\pi\)
−0.740244 + 0.672339i \(0.765290\pi\)
\(44\) 0 0
\(45\) −0.618034 −0.0921311
\(46\) −9.79837 −1.44469
\(47\) 4.47214 0.652328 0.326164 0.945313i \(-0.394244\pi\)
0.326164 + 0.945313i \(0.394244\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 10.3262 1.46035
\(51\) 4.85410 0.679710
\(52\) −9.70820 −1.34629
\(53\) 6.76393 0.929098 0.464549 0.885548i \(-0.346217\pi\)
0.464549 + 0.885548i \(0.346217\pi\)
\(54\) 2.23607 0.304290
\(55\) 0 0
\(56\) 2.23607 0.298807
\(57\) 2.85410 0.378035
\(58\) −13.4164 −1.76166
\(59\) 3.23607 0.421300 0.210650 0.977562i \(-0.432442\pi\)
0.210650 + 0.977562i \(0.432442\pi\)
\(60\) 1.85410 0.239364
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 6.90983 0.877549
\(63\) −1.00000 −0.125988
\(64\) −13.0000 −1.62500
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −14.5623 −1.76594
\(69\) −4.38197 −0.527527
\(70\) −1.38197 −0.165177
\(71\) −4.94427 −0.586777 −0.293389 0.955993i \(-0.594783\pi\)
−0.293389 + 0.955993i \(0.594783\pi\)
\(72\) −2.23607 −0.263523
\(73\) −13.7082 −1.60442 −0.802212 0.597039i \(-0.796344\pi\)
−0.802212 + 0.597039i \(0.796344\pi\)
\(74\) −10.3262 −1.20040
\(75\) 4.61803 0.533245
\(76\) −8.56231 −0.982164
\(77\) 0 0
\(78\) −7.23607 −0.819323
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0.618034 0.0690983
\(81\) 1.00000 0.111111
\(82\) 16.5066 1.82285
\(83\) −16.4721 −1.80805 −0.904026 0.427478i \(-0.859402\pi\)
−0.904026 + 0.427478i \(0.859402\pi\)
\(84\) 3.00000 0.327327
\(85\) 3.00000 0.325396
\(86\) 21.7082 2.34086
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 5.61803 0.595510 0.297755 0.954642i \(-0.403762\pi\)
0.297755 + 0.954642i \(0.403762\pi\)
\(90\) 1.38197 0.145672
\(91\) 3.23607 0.339232
\(92\) 13.1459 1.37055
\(93\) 3.09017 0.320436
\(94\) −10.0000 −1.03142
\(95\) 1.76393 0.180976
\(96\) −6.70820 −0.684653
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) −2.23607 −0.225877
\(99\) 0 0
\(100\) −13.8541 −1.38541
\(101\) 6.79837 0.676463 0.338232 0.941063i \(-0.390171\pi\)
0.338232 + 0.941063i \(0.390171\pi\)
\(102\) −10.8541 −1.07472
\(103\) 1.32624 0.130678 0.0653391 0.997863i \(-0.479187\pi\)
0.0653391 + 0.997863i \(0.479187\pi\)
\(104\) 7.23607 0.709555
\(105\) −0.618034 −0.0603139
\(106\) −15.1246 −1.46903
\(107\) −15.7984 −1.52729 −0.763643 0.645638i \(-0.776591\pi\)
−0.763643 + 0.645638i \(0.776591\pi\)
\(108\) −3.00000 −0.288675
\(109\) 17.5623 1.68216 0.841082 0.540908i \(-0.181919\pi\)
0.841082 + 0.540908i \(0.181919\pi\)
\(110\) 0 0
\(111\) −4.61803 −0.438324
\(112\) 1.00000 0.0944911
\(113\) −5.23607 −0.492568 −0.246284 0.969198i \(-0.579210\pi\)
−0.246284 + 0.969198i \(0.579210\pi\)
\(114\) −6.38197 −0.597726
\(115\) −2.70820 −0.252541
\(116\) 18.0000 1.67126
\(117\) −3.23607 −0.299175
\(118\) −7.23607 −0.666134
\(119\) 4.85410 0.444975
\(120\) −1.38197 −0.126156
\(121\) 0 0
\(122\) 0 0
\(123\) 7.38197 0.665609
\(124\) −9.27051 −0.832516
\(125\) 5.94427 0.531672
\(126\) 2.23607 0.199205
\(127\) −0.291796 −0.0258927 −0.0129464 0.999916i \(-0.504121\pi\)
−0.0129464 + 0.999916i \(0.504121\pi\)
\(128\) 15.6525 1.38350
\(129\) 9.70820 0.854760
\(130\) −4.47214 −0.392232
\(131\) 16.6525 1.45493 0.727467 0.686143i \(-0.240698\pi\)
0.727467 + 0.686143i \(0.240698\pi\)
\(132\) 0 0
\(133\) 2.85410 0.247482
\(134\) 0 0
\(135\) 0.618034 0.0531919
\(136\) 10.8541 0.930732
\(137\) 8.18034 0.698894 0.349447 0.936956i \(-0.386370\pi\)
0.349447 + 0.936956i \(0.386370\pi\)
\(138\) 9.79837 0.834093
\(139\) −13.1459 −1.11502 −0.557510 0.830170i \(-0.688243\pi\)
−0.557510 + 0.830170i \(0.688243\pi\)
\(140\) 1.85410 0.156700
\(141\) −4.47214 −0.376622
\(142\) 11.0557 0.927776
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) −3.70820 −0.307950
\(146\) 30.6525 2.53682
\(147\) −1.00000 −0.0824786
\(148\) 13.8541 1.13880
\(149\) 4.94427 0.405051 0.202525 0.979277i \(-0.435085\pi\)
0.202525 + 0.979277i \(0.435085\pi\)
\(150\) −10.3262 −0.843134
\(151\) 15.5279 1.26364 0.631820 0.775115i \(-0.282308\pi\)
0.631820 + 0.775115i \(0.282308\pi\)
\(152\) 6.38197 0.517646
\(153\) −4.85410 −0.392431
\(154\) 0 0
\(155\) 1.90983 0.153401
\(156\) 9.70820 0.777278
\(157\) −23.1246 −1.84554 −0.922772 0.385345i \(-0.874082\pi\)
−0.922772 + 0.385345i \(0.874082\pi\)
\(158\) −22.3607 −1.77892
\(159\) −6.76393 −0.536415
\(160\) −4.14590 −0.327762
\(161\) −4.38197 −0.345347
\(162\) −2.23607 −0.175682
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) −22.1459 −1.72930
\(165\) 0 0
\(166\) 36.8328 2.85878
\(167\) 21.2361 1.64330 0.821648 0.569995i \(-0.193055\pi\)
0.821648 + 0.569995i \(0.193055\pi\)
\(168\) −2.23607 −0.172516
\(169\) −2.52786 −0.194451
\(170\) −6.70820 −0.514496
\(171\) −2.85410 −0.218259
\(172\) −29.1246 −2.22073
\(173\) 13.6180 1.03536 0.517680 0.855574i \(-0.326796\pi\)
0.517680 + 0.855574i \(0.326796\pi\)
\(174\) 13.4164 1.01710
\(175\) 4.61803 0.349091
\(176\) 0 0
\(177\) −3.23607 −0.243238
\(178\) −12.5623 −0.941585
\(179\) −14.7984 −1.10608 −0.553041 0.833154i \(-0.686533\pi\)
−0.553041 + 0.833154i \(0.686533\pi\)
\(180\) −1.85410 −0.138197
\(181\) −19.8885 −1.47830 −0.739152 0.673539i \(-0.764773\pi\)
−0.739152 + 0.673539i \(0.764773\pi\)
\(182\) −7.23607 −0.536373
\(183\) 0 0
\(184\) −9.79837 −0.722346
\(185\) −2.85410 −0.209838
\(186\) −6.90983 −0.506653
\(187\) 0 0
\(188\) 13.4164 0.978492
\(189\) 1.00000 0.0727393
\(190\) −3.94427 −0.286148
\(191\) 4.90983 0.355263 0.177631 0.984097i \(-0.443157\pi\)
0.177631 + 0.984097i \(0.443157\pi\)
\(192\) 13.0000 0.938194
\(193\) 21.8541 1.57309 0.786546 0.617531i \(-0.211867\pi\)
0.786546 + 0.617531i \(0.211867\pi\)
\(194\) 13.4164 0.963242
\(195\) −2.00000 −0.143223
\(196\) 3.00000 0.214286
\(197\) −22.4721 −1.60107 −0.800537 0.599284i \(-0.795452\pi\)
−0.800537 + 0.599284i \(0.795452\pi\)
\(198\) 0 0
\(199\) 13.1459 0.931888 0.465944 0.884814i \(-0.345715\pi\)
0.465944 + 0.884814i \(0.345715\pi\)
\(200\) 10.3262 0.730175
\(201\) 0 0
\(202\) −15.2016 −1.06958
\(203\) −6.00000 −0.421117
\(204\) 14.5623 1.01957
\(205\) 4.56231 0.318645
\(206\) −2.96556 −0.206620
\(207\) 4.38197 0.304568
\(208\) 3.23607 0.224381
\(209\) 0 0
\(210\) 1.38197 0.0953647
\(211\) 18.3607 1.26400 0.632001 0.774968i \(-0.282234\pi\)
0.632001 + 0.774968i \(0.282234\pi\)
\(212\) 20.2918 1.39365
\(213\) 4.94427 0.338776
\(214\) 35.3262 2.41485
\(215\) 6.00000 0.409197
\(216\) 2.23607 0.152145
\(217\) 3.09017 0.209774
\(218\) −39.2705 −2.65973
\(219\) 13.7082 0.926315
\(220\) 0 0
\(221\) 15.7082 1.05665
\(222\) 10.3262 0.693052
\(223\) 19.6180 1.31372 0.656860 0.754012i \(-0.271884\pi\)
0.656860 + 0.754012i \(0.271884\pi\)
\(224\) −6.70820 −0.448211
\(225\) −4.61803 −0.307869
\(226\) 11.7082 0.778818
\(227\) −12.4721 −0.827805 −0.413902 0.910321i \(-0.635835\pi\)
−0.413902 + 0.910321i \(0.635835\pi\)
\(228\) 8.56231 0.567053
\(229\) 16.1803 1.06923 0.534613 0.845097i \(-0.320457\pi\)
0.534613 + 0.845097i \(0.320457\pi\)
\(230\) 6.05573 0.399303
\(231\) 0 0
\(232\) −13.4164 −0.880830
\(233\) 15.7082 1.02908 0.514539 0.857467i \(-0.327963\pi\)
0.514539 + 0.857467i \(0.327963\pi\)
\(234\) 7.23607 0.473037
\(235\) −2.76393 −0.180299
\(236\) 9.70820 0.631950
\(237\) −10.0000 −0.649570
\(238\) −10.8541 −0.703567
\(239\) 19.0344 1.23124 0.615618 0.788045i \(-0.288907\pi\)
0.615618 + 0.788045i \(0.288907\pi\)
\(240\) −0.618034 −0.0398939
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −0.618034 −0.0394847
\(246\) −16.5066 −1.05242
\(247\) 9.23607 0.587677
\(248\) 6.90983 0.438775
\(249\) 16.4721 1.04388
\(250\) −13.2918 −0.840647
\(251\) 27.7082 1.74893 0.874463 0.485092i \(-0.161214\pi\)
0.874463 + 0.485092i \(0.161214\pi\)
\(252\) −3.00000 −0.188982
\(253\) 0 0
\(254\) 0.652476 0.0409400
\(255\) −3.00000 −0.187867
\(256\) −9.00000 −0.562500
\(257\) 24.4508 1.52520 0.762601 0.646869i \(-0.223922\pi\)
0.762601 + 0.646869i \(0.223922\pi\)
\(258\) −21.7082 −1.35149
\(259\) −4.61803 −0.286951
\(260\) 6.00000 0.372104
\(261\) 6.00000 0.371391
\(262\) −37.2361 −2.30045
\(263\) −10.8541 −0.669293 −0.334646 0.942344i \(-0.608617\pi\)
−0.334646 + 0.942344i \(0.608617\pi\)
\(264\) 0 0
\(265\) −4.18034 −0.256796
\(266\) −6.38197 −0.391303
\(267\) −5.61803 −0.343818
\(268\) 0 0
\(269\) −13.4164 −0.818013 −0.409006 0.912532i \(-0.634125\pi\)
−0.409006 + 0.912532i \(0.634125\pi\)
\(270\) −1.38197 −0.0841038
\(271\) 22.5066 1.36718 0.683589 0.729868i \(-0.260418\pi\)
0.683589 + 0.729868i \(0.260418\pi\)
\(272\) 4.85410 0.294323
\(273\) −3.23607 −0.195856
\(274\) −18.2918 −1.10505
\(275\) 0 0
\(276\) −13.1459 −0.791290
\(277\) 0.0901699 0.00541779 0.00270889 0.999996i \(-0.499138\pi\)
0.00270889 + 0.999996i \(0.499138\pi\)
\(278\) 29.3951 1.76300
\(279\) −3.09017 −0.185004
\(280\) −1.38197 −0.0825883
\(281\) 26.9443 1.60736 0.803680 0.595061i \(-0.202872\pi\)
0.803680 + 0.595061i \(0.202872\pi\)
\(282\) 10.0000 0.595491
\(283\) 17.2705 1.02663 0.513313 0.858202i \(-0.328418\pi\)
0.513313 + 0.858202i \(0.328418\pi\)
\(284\) −14.8328 −0.880166
\(285\) −1.76393 −0.104486
\(286\) 0 0
\(287\) 7.38197 0.435744
\(288\) 6.70820 0.395285
\(289\) 6.56231 0.386018
\(290\) 8.29180 0.486911
\(291\) 6.00000 0.351726
\(292\) −41.1246 −2.40664
\(293\) 18.2148 1.06412 0.532059 0.846707i \(-0.321418\pi\)
0.532059 + 0.846707i \(0.321418\pi\)
\(294\) 2.23607 0.130410
\(295\) −2.00000 −0.116445
\(296\) −10.3262 −0.600200
\(297\) 0 0
\(298\) −11.0557 −0.640441
\(299\) −14.1803 −0.820070
\(300\) 13.8541 0.799867
\(301\) 9.70820 0.559572
\(302\) −34.7214 −1.99799
\(303\) −6.79837 −0.390556
\(304\) 2.85410 0.163694
\(305\) 0 0
\(306\) 10.8541 0.620488
\(307\) 17.5623 1.00233 0.501167 0.865351i \(-0.332904\pi\)
0.501167 + 0.865351i \(0.332904\pi\)
\(308\) 0 0
\(309\) −1.32624 −0.0754470
\(310\) −4.27051 −0.242549
\(311\) 20.6525 1.17109 0.585547 0.810638i \(-0.300880\pi\)
0.585547 + 0.810638i \(0.300880\pi\)
\(312\) −7.23607 −0.409662
\(313\) −7.81966 −0.441993 −0.220997 0.975275i \(-0.570931\pi\)
−0.220997 + 0.975275i \(0.570931\pi\)
\(314\) 51.7082 2.91806
\(315\) 0.618034 0.0348223
\(316\) 30.0000 1.68763
\(317\) 6.76393 0.379900 0.189950 0.981794i \(-0.439167\pi\)
0.189950 + 0.981794i \(0.439167\pi\)
\(318\) 15.1246 0.848146
\(319\) 0 0
\(320\) 8.03444 0.449139
\(321\) 15.7984 0.881779
\(322\) 9.79837 0.546042
\(323\) 13.8541 0.770863
\(324\) 3.00000 0.166667
\(325\) 14.9443 0.828959
\(326\) 22.3607 1.23844
\(327\) −17.5623 −0.971198
\(328\) 16.5066 0.911423
\(329\) −4.47214 −0.246557
\(330\) 0 0
\(331\) 17.4164 0.957292 0.478646 0.878008i \(-0.341128\pi\)
0.478646 + 0.878008i \(0.341128\pi\)
\(332\) −49.4164 −2.71208
\(333\) 4.61803 0.253067
\(334\) −47.4853 −2.59828
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) −27.0902 −1.47570 −0.737848 0.674967i \(-0.764158\pi\)
−0.737848 + 0.674967i \(0.764158\pi\)
\(338\) 5.65248 0.307454
\(339\) 5.23607 0.284384
\(340\) 9.00000 0.488094
\(341\) 0 0
\(342\) 6.38197 0.345097
\(343\) −1.00000 −0.0539949
\(344\) 21.7082 1.17043
\(345\) 2.70820 0.145805
\(346\) −30.4508 −1.63705
\(347\) 14.5066 0.778754 0.389377 0.921078i \(-0.372690\pi\)
0.389377 + 0.921078i \(0.372690\pi\)
\(348\) −18.0000 −0.964901
\(349\) 21.7082 1.16201 0.581007 0.813899i \(-0.302659\pi\)
0.581007 + 0.813899i \(0.302659\pi\)
\(350\) −10.3262 −0.551961
\(351\) 3.23607 0.172729
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 7.23607 0.384593
\(355\) 3.05573 0.162181
\(356\) 16.8541 0.893266
\(357\) −4.85410 −0.256906
\(358\) 33.0902 1.74887
\(359\) −6.43769 −0.339768 −0.169884 0.985464i \(-0.554339\pi\)
−0.169884 + 0.985464i \(0.554339\pi\)
\(360\) 1.38197 0.0728360
\(361\) −10.8541 −0.571269
\(362\) 44.4721 2.33740
\(363\) 0 0
\(364\) 9.70820 0.508848
\(365\) 8.47214 0.443452
\(366\) 0 0
\(367\) 2.72949 0.142478 0.0712391 0.997459i \(-0.477305\pi\)
0.0712391 + 0.997459i \(0.477305\pi\)
\(368\) −4.38197 −0.228426
\(369\) −7.38197 −0.384290
\(370\) 6.38197 0.331783
\(371\) −6.76393 −0.351166
\(372\) 9.27051 0.480654
\(373\) −16.3262 −0.845341 −0.422670 0.906284i \(-0.638907\pi\)
−0.422670 + 0.906284i \(0.638907\pi\)
\(374\) 0 0
\(375\) −5.94427 −0.306961
\(376\) −10.0000 −0.515711
\(377\) −19.4164 −0.999996
\(378\) −2.23607 −0.115011
\(379\) 21.5279 1.10581 0.552906 0.833244i \(-0.313519\pi\)
0.552906 + 0.833244i \(0.313519\pi\)
\(380\) 5.29180 0.271463
\(381\) 0.291796 0.0149492
\(382\) −10.9787 −0.561720
\(383\) −20.8328 −1.06451 −0.532254 0.846585i \(-0.678655\pi\)
−0.532254 + 0.846585i \(0.678655\pi\)
\(384\) −15.6525 −0.798762
\(385\) 0 0
\(386\) −48.8673 −2.48728
\(387\) −9.70820 −0.493496
\(388\) −18.0000 −0.913812
\(389\) 15.2361 0.772499 0.386250 0.922394i \(-0.373770\pi\)
0.386250 + 0.922394i \(0.373770\pi\)
\(390\) 4.47214 0.226455
\(391\) −21.2705 −1.07570
\(392\) −2.23607 −0.112938
\(393\) −16.6525 −0.840006
\(394\) 50.2492 2.53152
\(395\) −6.18034 −0.310967
\(396\) 0 0
\(397\) 30.5410 1.53281 0.766405 0.642358i \(-0.222044\pi\)
0.766405 + 0.642358i \(0.222044\pi\)
\(398\) −29.3951 −1.47344
\(399\) −2.85410 −0.142884
\(400\) 4.61803 0.230902
\(401\) 7.41641 0.370358 0.185179 0.982705i \(-0.440714\pi\)
0.185179 + 0.982705i \(0.440714\pi\)
\(402\) 0 0
\(403\) 10.0000 0.498135
\(404\) 20.3951 1.01470
\(405\) −0.618034 −0.0307104
\(406\) 13.4164 0.665845
\(407\) 0 0
\(408\) −10.8541 −0.537358
\(409\) 22.1803 1.09675 0.548374 0.836233i \(-0.315247\pi\)
0.548374 + 0.836233i \(0.315247\pi\)
\(410\) −10.2016 −0.503822
\(411\) −8.18034 −0.403506
\(412\) 3.97871 0.196017
\(413\) −3.23607 −0.159236
\(414\) −9.79837 −0.481564
\(415\) 10.1803 0.499733
\(416\) −21.7082 −1.06433
\(417\) 13.1459 0.643757
\(418\) 0 0
\(419\) −3.23607 −0.158092 −0.0790461 0.996871i \(-0.525187\pi\)
−0.0790461 + 0.996871i \(0.525187\pi\)
\(420\) −1.85410 −0.0904709
\(421\) 9.79837 0.477544 0.238772 0.971076i \(-0.423255\pi\)
0.238772 + 0.971076i \(0.423255\pi\)
\(422\) −41.0557 −1.99856
\(423\) 4.47214 0.217443
\(424\) −15.1246 −0.734516
\(425\) 22.4164 1.08736
\(426\) −11.0557 −0.535652
\(427\) 0 0
\(428\) −47.3951 −2.29093
\(429\) 0 0
\(430\) −13.4164 −0.646997
\(431\) −15.0902 −0.726868 −0.363434 0.931620i \(-0.618396\pi\)
−0.363434 + 0.931620i \(0.618396\pi\)
\(432\) 1.00000 0.0481125
\(433\) −31.2361 −1.50111 −0.750555 0.660808i \(-0.770214\pi\)
−0.750555 + 0.660808i \(0.770214\pi\)
\(434\) −6.90983 −0.331682
\(435\) 3.70820 0.177795
\(436\) 52.6869 2.52325
\(437\) −12.5066 −0.598271
\(438\) −30.6525 −1.46463
\(439\) −1.85410 −0.0884915 −0.0442457 0.999021i \(-0.514088\pi\)
−0.0442457 + 0.999021i \(0.514088\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −35.1246 −1.67071
\(443\) 20.3820 0.968376 0.484188 0.874964i \(-0.339115\pi\)
0.484188 + 0.874964i \(0.339115\pi\)
\(444\) −13.8541 −0.657487
\(445\) −3.47214 −0.164595
\(446\) −43.8673 −2.07717
\(447\) −4.94427 −0.233856
\(448\) 13.0000 0.614192
\(449\) −41.8885 −1.97684 −0.988421 0.151734i \(-0.951514\pi\)
−0.988421 + 0.151734i \(0.951514\pi\)
\(450\) 10.3262 0.486784
\(451\) 0 0
\(452\) −15.7082 −0.738852
\(453\) −15.5279 −0.729563
\(454\) 27.8885 1.30887
\(455\) −2.00000 −0.0937614
\(456\) −6.38197 −0.298863
\(457\) −21.4164 −1.00182 −0.500909 0.865500i \(-0.667001\pi\)
−0.500909 + 0.865500i \(0.667001\pi\)
\(458\) −36.1803 −1.69060
\(459\) 4.85410 0.226570
\(460\) −8.12461 −0.378812
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) −23.4164 −1.08825 −0.544126 0.839003i \(-0.683139\pi\)
−0.544126 + 0.839003i \(0.683139\pi\)
\(464\) −6.00000 −0.278543
\(465\) −1.90983 −0.0885662
\(466\) −35.1246 −1.62712
\(467\) 26.1803 1.21148 0.605741 0.795662i \(-0.292877\pi\)
0.605741 + 0.795662i \(0.292877\pi\)
\(468\) −9.70820 −0.448762
\(469\) 0 0
\(470\) 6.18034 0.285078
\(471\) 23.1246 1.06553
\(472\) −7.23607 −0.333067
\(473\) 0 0
\(474\) 22.3607 1.02706
\(475\) 13.1803 0.604755
\(476\) 14.5623 0.667462
\(477\) 6.76393 0.309699
\(478\) −42.5623 −1.94675
\(479\) 0.944272 0.0431449 0.0215724 0.999767i \(-0.493133\pi\)
0.0215724 + 0.999767i \(0.493133\pi\)
\(480\) 4.14590 0.189233
\(481\) −14.9443 −0.681400
\(482\) 26.8328 1.22220
\(483\) 4.38197 0.199386
\(484\) 0 0
\(485\) 3.70820 0.168381
\(486\) 2.23607 0.101430
\(487\) 27.7082 1.25558 0.627789 0.778383i \(-0.283960\pi\)
0.627789 + 0.778383i \(0.283960\pi\)
\(488\) 0 0
\(489\) 10.0000 0.452216
\(490\) 1.38197 0.0624309
\(491\) −1.50658 −0.0679909 −0.0339955 0.999422i \(-0.510823\pi\)
−0.0339955 + 0.999422i \(0.510823\pi\)
\(492\) 22.1459 0.998414
\(493\) −29.1246 −1.31171
\(494\) −20.6525 −0.929199
\(495\) 0 0
\(496\) 3.09017 0.138753
\(497\) 4.94427 0.221781
\(498\) −36.8328 −1.65052
\(499\) 21.1246 0.945668 0.472834 0.881152i \(-0.343231\pi\)
0.472834 + 0.881152i \(0.343231\pi\)
\(500\) 17.8328 0.797508
\(501\) −21.2361 −0.948758
\(502\) −61.9574 −2.76530
\(503\) 14.4721 0.645281 0.322640 0.946522i \(-0.395430\pi\)
0.322640 + 0.946522i \(0.395430\pi\)
\(504\) 2.23607 0.0996024
\(505\) −4.20163 −0.186970
\(506\) 0 0
\(507\) 2.52786 0.112266
\(508\) −0.875388 −0.0388391
\(509\) −27.0902 −1.20075 −0.600375 0.799718i \(-0.704982\pi\)
−0.600375 + 0.799718i \(0.704982\pi\)
\(510\) 6.70820 0.297044
\(511\) 13.7082 0.606415
\(512\) −11.1803 −0.494106
\(513\) 2.85410 0.126012
\(514\) −54.6738 −2.41156
\(515\) −0.819660 −0.0361185
\(516\) 29.1246 1.28214
\(517\) 0 0
\(518\) 10.3262 0.453709
\(519\) −13.6180 −0.597765
\(520\) −4.47214 −0.196116
\(521\) −12.3262 −0.540022 −0.270011 0.962857i \(-0.587027\pi\)
−0.270011 + 0.962857i \(0.587027\pi\)
\(522\) −13.4164 −0.587220
\(523\) −13.1459 −0.574830 −0.287415 0.957806i \(-0.592796\pi\)
−0.287415 + 0.957806i \(0.592796\pi\)
\(524\) 49.9574 2.18240
\(525\) −4.61803 −0.201548
\(526\) 24.2705 1.05824
\(527\) 15.0000 0.653410
\(528\) 0 0
\(529\) −3.79837 −0.165147
\(530\) 9.34752 0.406031
\(531\) 3.23607 0.140433
\(532\) 8.56231 0.371223
\(533\) 23.8885 1.03473
\(534\) 12.5623 0.543624
\(535\) 9.76393 0.422132
\(536\) 0 0
\(537\) 14.7984 0.638597
\(538\) 30.0000 1.29339
\(539\) 0 0
\(540\) 1.85410 0.0797878
\(541\) 16.3262 0.701920 0.350960 0.936390i \(-0.385855\pi\)
0.350960 + 0.936390i \(0.385855\pi\)
\(542\) −50.3262 −2.16170
\(543\) 19.8885 0.853499
\(544\) −32.5623 −1.39610
\(545\) −10.8541 −0.464939
\(546\) 7.23607 0.309675
\(547\) 36.4721 1.55944 0.779718 0.626131i \(-0.215362\pi\)
0.779718 + 0.626131i \(0.215362\pi\)
\(548\) 24.5410 1.04834
\(549\) 0 0
\(550\) 0 0
\(551\) −17.1246 −0.729533
\(552\) 9.79837 0.417046
\(553\) −10.0000 −0.425243
\(554\) −0.201626 −0.00856627
\(555\) 2.85410 0.121150
\(556\) −39.4377 −1.67253
\(557\) −22.3607 −0.947452 −0.473726 0.880672i \(-0.657091\pi\)
−0.473726 + 0.880672i \(0.657091\pi\)
\(558\) 6.90983 0.292516
\(559\) 31.4164 1.32877
\(560\) −0.618034 −0.0261167
\(561\) 0 0
\(562\) −60.2492 −2.54146
\(563\) 9.52786 0.401552 0.200776 0.979637i \(-0.435654\pi\)
0.200776 + 0.979637i \(0.435654\pi\)
\(564\) −13.4164 −0.564933
\(565\) 3.23607 0.136142
\(566\) −38.6180 −1.62324
\(567\) −1.00000 −0.0419961
\(568\) 11.0557 0.463888
\(569\) −11.5967 −0.486161 −0.243080 0.970006i \(-0.578158\pi\)
−0.243080 + 0.970006i \(0.578158\pi\)
\(570\) 3.94427 0.165207
\(571\) −15.3475 −0.642274 −0.321137 0.947033i \(-0.604065\pi\)
−0.321137 + 0.947033i \(0.604065\pi\)
\(572\) 0 0
\(573\) −4.90983 −0.205111
\(574\) −16.5066 −0.688971
\(575\) −20.2361 −0.843902
\(576\) −13.0000 −0.541667
\(577\) −35.7082 −1.48655 −0.743276 0.668985i \(-0.766729\pi\)
−0.743276 + 0.668985i \(0.766729\pi\)
\(578\) −14.6738 −0.610348
\(579\) −21.8541 −0.908225
\(580\) −11.1246 −0.461924
\(581\) 16.4721 0.683379
\(582\) −13.4164 −0.556128
\(583\) 0 0
\(584\) 30.6525 1.26841
\(585\) 2.00000 0.0826898
\(586\) −40.7295 −1.68252
\(587\) −1.41641 −0.0584614 −0.0292307 0.999573i \(-0.509306\pi\)
−0.0292307 + 0.999573i \(0.509306\pi\)
\(588\) −3.00000 −0.123718
\(589\) 8.81966 0.363408
\(590\) 4.47214 0.184115
\(591\) 22.4721 0.924380
\(592\) −4.61803 −0.189800
\(593\) −22.5066 −0.924234 −0.462117 0.886819i \(-0.652910\pi\)
−0.462117 + 0.886819i \(0.652910\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) 14.8328 0.607576
\(597\) −13.1459 −0.538026
\(598\) 31.7082 1.29664
\(599\) −23.6180 −0.965007 −0.482503 0.875894i \(-0.660272\pi\)
−0.482503 + 0.875894i \(0.660272\pi\)
\(600\) −10.3262 −0.421567
\(601\) −3.52786 −0.143905 −0.0719523 0.997408i \(-0.522923\pi\)
−0.0719523 + 0.997408i \(0.522923\pi\)
\(602\) −21.7082 −0.884760
\(603\) 0 0
\(604\) 46.5836 1.89546
\(605\) 0 0
\(606\) 15.2016 0.617524
\(607\) 11.9787 0.486201 0.243100 0.970001i \(-0.421836\pi\)
0.243100 + 0.970001i \(0.421836\pi\)
\(608\) −19.1459 −0.776469
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) −14.4721 −0.585480
\(612\) −14.5623 −0.588646
\(613\) 21.9787 0.887712 0.443856 0.896098i \(-0.353610\pi\)
0.443856 + 0.896098i \(0.353610\pi\)
\(614\) −39.2705 −1.58483
\(615\) −4.56231 −0.183970
\(616\) 0 0
\(617\) −14.0689 −0.566392 −0.283196 0.959062i \(-0.591395\pi\)
−0.283196 + 0.959062i \(0.591395\pi\)
\(618\) 2.96556 0.119292
\(619\) −36.6869 −1.47457 −0.737286 0.675581i \(-0.763893\pi\)
−0.737286 + 0.675581i \(0.763893\pi\)
\(620\) 5.72949 0.230102
\(621\) −4.38197 −0.175842
\(622\) −46.1803 −1.85166
\(623\) −5.61803 −0.225082
\(624\) −3.23607 −0.129546
\(625\) 19.4164 0.776656
\(626\) 17.4853 0.698853
\(627\) 0 0
\(628\) −69.3738 −2.76832
\(629\) −22.4164 −0.893801
\(630\) −1.38197 −0.0550588
\(631\) −45.3050 −1.80356 −0.901781 0.432194i \(-0.857740\pi\)
−0.901781 + 0.432194i \(0.857740\pi\)
\(632\) −22.3607 −0.889460
\(633\) −18.3607 −0.729772
\(634\) −15.1246 −0.600675
\(635\) 0.180340 0.00715657
\(636\) −20.2918 −0.804622
\(637\) −3.23607 −0.128218
\(638\) 0 0
\(639\) −4.94427 −0.195592
\(640\) −9.67376 −0.382389
\(641\) 12.7639 0.504145 0.252073 0.967708i \(-0.418888\pi\)
0.252073 + 0.967708i \(0.418888\pi\)
\(642\) −35.3262 −1.39422
\(643\) 10.2705 0.405029 0.202515 0.979279i \(-0.435089\pi\)
0.202515 + 0.979279i \(0.435089\pi\)
\(644\) −13.1459 −0.518021
\(645\) −6.00000 −0.236250
\(646\) −30.9787 −1.21884
\(647\) 36.4721 1.43387 0.716934 0.697141i \(-0.245545\pi\)
0.716934 + 0.697141i \(0.245545\pi\)
\(648\) −2.23607 −0.0878410
\(649\) 0 0
\(650\) −33.4164 −1.31070
\(651\) −3.09017 −0.121113
\(652\) −30.0000 −1.17489
\(653\) 16.3607 0.640243 0.320121 0.947377i \(-0.396276\pi\)
0.320121 + 0.947377i \(0.396276\pi\)
\(654\) 39.2705 1.53560
\(655\) −10.2918 −0.402134
\(656\) 7.38197 0.288217
\(657\) −13.7082 −0.534808
\(658\) 10.0000 0.389841
\(659\) 25.8541 1.00713 0.503566 0.863957i \(-0.332021\pi\)
0.503566 + 0.863957i \(0.332021\pi\)
\(660\) 0 0
\(661\) −32.5410 −1.26570 −0.632849 0.774275i \(-0.718115\pi\)
−0.632849 + 0.774275i \(0.718115\pi\)
\(662\) −38.9443 −1.51361
\(663\) −15.7082 −0.610056
\(664\) 36.8328 1.42939
\(665\) −1.76393 −0.0684023
\(666\) −10.3262 −0.400134
\(667\) 26.2918 1.01802
\(668\) 63.7082 2.46494
\(669\) −19.6180 −0.758477
\(670\) 0 0
\(671\) 0 0
\(672\) 6.70820 0.258775
\(673\) 44.8328 1.72818 0.864089 0.503339i \(-0.167895\pi\)
0.864089 + 0.503339i \(0.167895\pi\)
\(674\) 60.5755 2.33328
\(675\) 4.61803 0.177748
\(676\) −7.58359 −0.291677
\(677\) −34.3607 −1.32059 −0.660294 0.751007i \(-0.729568\pi\)
−0.660294 + 0.751007i \(0.729568\pi\)
\(678\) −11.7082 −0.449651
\(679\) 6.00000 0.230259
\(680\) −6.70820 −0.257248
\(681\) 12.4721 0.477933
\(682\) 0 0
\(683\) 26.7984 1.02541 0.512706 0.858564i \(-0.328643\pi\)
0.512706 + 0.858564i \(0.328643\pi\)
\(684\) −8.56231 −0.327388
\(685\) −5.05573 −0.193169
\(686\) 2.23607 0.0853735
\(687\) −16.1803 −0.617318
\(688\) 9.70820 0.370122
\(689\) −21.8885 −0.833887
\(690\) −6.05573 −0.230538
\(691\) 48.7426 1.85426 0.927129 0.374743i \(-0.122269\pi\)
0.927129 + 0.374743i \(0.122269\pi\)
\(692\) 40.8541 1.55304
\(693\) 0 0
\(694\) −32.4377 −1.23132
\(695\) 8.12461 0.308184
\(696\) 13.4164 0.508548
\(697\) 35.8328 1.35726
\(698\) −48.5410 −1.83730
\(699\) −15.7082 −0.594139
\(700\) 13.8541 0.523636
\(701\) −39.5967 −1.49555 −0.747774 0.663953i \(-0.768877\pi\)
−0.747774 + 0.663953i \(0.768877\pi\)
\(702\) −7.23607 −0.273108
\(703\) −13.1803 −0.497106
\(704\) 0 0
\(705\) 2.76393 0.104096
\(706\) 40.2492 1.51480
\(707\) −6.79837 −0.255679
\(708\) −9.70820 −0.364857
\(709\) −8.03444 −0.301740 −0.150870 0.988554i \(-0.548207\pi\)
−0.150870 + 0.988554i \(0.548207\pi\)
\(710\) −6.83282 −0.256431
\(711\) 10.0000 0.375029
\(712\) −12.5623 −0.470792
\(713\) −13.5410 −0.507115
\(714\) 10.8541 0.406205
\(715\) 0 0
\(716\) −44.3951 −1.65912
\(717\) −19.0344 −0.710854
\(718\) 14.3951 0.537221
\(719\) 39.0132 1.45495 0.727473 0.686137i \(-0.240695\pi\)
0.727473 + 0.686137i \(0.240695\pi\)
\(720\) 0.618034 0.0230328
\(721\) −1.32624 −0.0493917
\(722\) 24.2705 0.903255
\(723\) 12.0000 0.446285
\(724\) −59.6656 −2.21746
\(725\) −27.7082 −1.02906
\(726\) 0 0
\(727\) −18.8541 −0.699260 −0.349630 0.936888i \(-0.613693\pi\)
−0.349630 + 0.936888i \(0.613693\pi\)
\(728\) −7.23607 −0.268187
\(729\) 1.00000 0.0370370
\(730\) −18.9443 −0.701159
\(731\) 47.1246 1.74297
\(732\) 0 0
\(733\) 18.0689 0.667389 0.333695 0.942681i \(-0.391705\pi\)
0.333695 + 0.942681i \(0.391705\pi\)
\(734\) −6.10333 −0.225278
\(735\) 0.618034 0.0227965
\(736\) 29.3951 1.08352
\(737\) 0 0
\(738\) 16.5066 0.607616
\(739\) −41.1246 −1.51279 −0.756397 0.654113i \(-0.773042\pi\)
−0.756397 + 0.654113i \(0.773042\pi\)
\(740\) −8.56231 −0.314757
\(741\) −9.23607 −0.339295
\(742\) 15.1246 0.555242
\(743\) 16.3262 0.598952 0.299476 0.954104i \(-0.403188\pi\)
0.299476 + 0.954104i \(0.403188\pi\)
\(744\) −6.90983 −0.253327
\(745\) −3.05573 −0.111953
\(746\) 36.5066 1.33660
\(747\) −16.4721 −0.602684
\(748\) 0 0
\(749\) 15.7984 0.577260
\(750\) 13.2918 0.485348
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) −4.47214 −0.163082
\(753\) −27.7082 −1.00974
\(754\) 43.4164 1.58113
\(755\) −9.59675 −0.349261
\(756\) 3.00000 0.109109
\(757\) 10.5623 0.383894 0.191947 0.981405i \(-0.438520\pi\)
0.191947 + 0.981405i \(0.438520\pi\)
\(758\) −48.1378 −1.74844
\(759\) 0 0
\(760\) −3.94427 −0.143074
\(761\) −1.41641 −0.0513447 −0.0256724 0.999670i \(-0.508173\pi\)
−0.0256724 + 0.999670i \(0.508173\pi\)
\(762\) −0.652476 −0.0236367
\(763\) −17.5623 −0.635798
\(764\) 14.7295 0.532894
\(765\) 3.00000 0.108465
\(766\) 46.5836 1.68313
\(767\) −10.4721 −0.378127
\(768\) 9.00000 0.324760
\(769\) −1.05573 −0.0380705 −0.0190353 0.999819i \(-0.506059\pi\)
−0.0190353 + 0.999819i \(0.506059\pi\)
\(770\) 0 0
\(771\) −24.4508 −0.880576
\(772\) 65.5623 2.35964
\(773\) 8.47214 0.304722 0.152361 0.988325i \(-0.451312\pi\)
0.152361 + 0.988325i \(0.451312\pi\)
\(774\) 21.7082 0.780285
\(775\) 14.2705 0.512612
\(776\) 13.4164 0.481621
\(777\) 4.61803 0.165671
\(778\) −34.0689 −1.22143
\(779\) 21.0689 0.754871
\(780\) −6.00000 −0.214834
\(781\) 0 0
\(782\) 47.5623 1.70082
\(783\) −6.00000 −0.214423
\(784\) −1.00000 −0.0357143
\(785\) 14.2918 0.510096
\(786\) 37.2361 1.32817
\(787\) 25.8541 0.921599 0.460800 0.887504i \(-0.347563\pi\)
0.460800 + 0.887504i \(0.347563\pi\)
\(788\) −67.4164 −2.40161
\(789\) 10.8541 0.386416
\(790\) 13.8197 0.491681
\(791\) 5.23607 0.186173
\(792\) 0 0
\(793\) 0 0
\(794\) −68.2918 −2.42359
\(795\) 4.18034 0.148261
\(796\) 39.4377 1.39783
\(797\) 41.5623 1.47221 0.736106 0.676866i \(-0.236662\pi\)
0.736106 + 0.676866i \(0.236662\pi\)
\(798\) 6.38197 0.225919
\(799\) −21.7082 −0.767981
\(800\) −30.9787 −1.09526
\(801\) 5.61803 0.198503
\(802\) −16.5836 −0.585587
\(803\) 0 0
\(804\) 0 0
\(805\) 2.70820 0.0954516
\(806\) −22.3607 −0.787621
\(807\) 13.4164 0.472280
\(808\) −15.2016 −0.534791
\(809\) −7.23607 −0.254407 −0.127203 0.991877i \(-0.540600\pi\)
−0.127203 + 0.991877i \(0.540600\pi\)
\(810\) 1.38197 0.0485573
\(811\) −8.58359 −0.301411 −0.150705 0.988579i \(-0.548154\pi\)
−0.150705 + 0.988579i \(0.548154\pi\)
\(812\) −18.0000 −0.631676
\(813\) −22.5066 −0.789340
\(814\) 0 0
\(815\) 6.18034 0.216488
\(816\) −4.85410 −0.169928
\(817\) 27.7082 0.969387
\(818\) −49.5967 −1.73411
\(819\) 3.23607 0.113077
\(820\) 13.6869 0.477968
\(821\) 10.9443 0.381958 0.190979 0.981594i \(-0.438834\pi\)
0.190979 + 0.981594i \(0.438834\pi\)
\(822\) 18.2918 0.638000
\(823\) 4.83282 0.168461 0.0842307 0.996446i \(-0.473157\pi\)
0.0842307 + 0.996446i \(0.473157\pi\)
\(824\) −2.96556 −0.103310
\(825\) 0 0
\(826\) 7.23607 0.251775
\(827\) −29.8673 −1.03859 −0.519293 0.854596i \(-0.673805\pi\)
−0.519293 + 0.854596i \(0.673805\pi\)
\(828\) 13.1459 0.456852
\(829\) −30.2492 −1.05060 −0.525299 0.850917i \(-0.676047\pi\)
−0.525299 + 0.850917i \(0.676047\pi\)
\(830\) −22.7639 −0.790148
\(831\) −0.0901699 −0.00312796
\(832\) 42.0689 1.45848
\(833\) −4.85410 −0.168185
\(834\) −29.3951 −1.01787
\(835\) −13.1246 −0.454196
\(836\) 0 0
\(837\) 3.09017 0.106812
\(838\) 7.23607 0.249966
\(839\) −40.2492 −1.38956 −0.694779 0.719224i \(-0.744498\pi\)
−0.694779 + 0.719224i \(0.744498\pi\)
\(840\) 1.38197 0.0476824
\(841\) 7.00000 0.241379
\(842\) −21.9098 −0.755063
\(843\) −26.9443 −0.928010
\(844\) 55.0820 1.89600
\(845\) 1.56231 0.0537450
\(846\) −10.0000 −0.343807
\(847\) 0 0
\(848\) −6.76393 −0.232274
\(849\) −17.2705 −0.592722
\(850\) −50.1246 −1.71926
\(851\) 20.2361 0.693683
\(852\) 14.8328 0.508164
\(853\) 41.7082 1.42806 0.714031 0.700114i \(-0.246868\pi\)
0.714031 + 0.700114i \(0.246868\pi\)
\(854\) 0 0
\(855\) 1.76393 0.0603252
\(856\) 35.3262 1.20743
\(857\) 34.3607 1.17374 0.586869 0.809682i \(-0.300360\pi\)
0.586869 + 0.809682i \(0.300360\pi\)
\(858\) 0 0
\(859\) 12.0000 0.409435 0.204717 0.978821i \(-0.434372\pi\)
0.204717 + 0.978821i \(0.434372\pi\)
\(860\) 18.0000 0.613795
\(861\) −7.38197 −0.251577
\(862\) 33.7426 1.14928
\(863\) −12.3262 −0.419590 −0.209795 0.977745i \(-0.567280\pi\)
−0.209795 + 0.977745i \(0.567280\pi\)
\(864\) −6.70820 −0.228218
\(865\) −8.41641 −0.286166
\(866\) 69.8460 2.37346
\(867\) −6.56231 −0.222868
\(868\) 9.27051 0.314662
\(869\) 0 0
\(870\) −8.29180 −0.281118
\(871\) 0 0
\(872\) −39.2705 −1.32987
\(873\) −6.00000 −0.203069
\(874\) 27.9656 0.945949
\(875\) −5.94427 −0.200953
\(876\) 41.1246 1.38947
\(877\) −12.4721 −0.421154 −0.210577 0.977577i \(-0.567534\pi\)
−0.210577 + 0.977577i \(0.567534\pi\)
\(878\) 4.14590 0.139917
\(879\) −18.2148 −0.614369
\(880\) 0 0
\(881\) 11.3262 0.381591 0.190795 0.981630i \(-0.438893\pi\)
0.190795 + 0.981630i \(0.438893\pi\)
\(882\) −2.23607 −0.0752923
\(883\) 19.5967 0.659483 0.329742 0.944071i \(-0.393038\pi\)
0.329742 + 0.944071i \(0.393038\pi\)
\(884\) 47.1246 1.58497
\(885\) 2.00000 0.0672293
\(886\) −45.5755 −1.53114
\(887\) −27.5967 −0.926608 −0.463304 0.886199i \(-0.653336\pi\)
−0.463304 + 0.886199i \(0.653336\pi\)
\(888\) 10.3262 0.346526
\(889\) 0.291796 0.00978653
\(890\) 7.76393 0.260248
\(891\) 0 0
\(892\) 58.8541 1.97058
\(893\) −12.7639 −0.427129
\(894\) 11.0557 0.369759
\(895\) 9.14590 0.305714
\(896\) −15.6525 −0.522913
\(897\) 14.1803 0.473468
\(898\) 93.6656 3.12566
\(899\) −18.5410 −0.618378
\(900\) −13.8541 −0.461803
\(901\) −32.8328 −1.09382
\(902\) 0 0
\(903\) −9.70820 −0.323069
\(904\) 11.7082 0.389409
\(905\) 12.2918 0.408593
\(906\) 34.7214 1.15354
\(907\) 16.2918 0.540960 0.270480 0.962726i \(-0.412818\pi\)
0.270480 + 0.962726i \(0.412818\pi\)
\(908\) −37.4164 −1.24171
\(909\) 6.79837 0.225488
\(910\) 4.47214 0.148250
\(911\) −0.944272 −0.0312851 −0.0156426 0.999878i \(-0.504979\pi\)
−0.0156426 + 0.999878i \(0.504979\pi\)
\(912\) −2.85410 −0.0945088
\(913\) 0 0
\(914\) 47.8885 1.58401
\(915\) 0 0
\(916\) 48.5410 1.60384
\(917\) −16.6525 −0.549913
\(918\) −10.8541 −0.358239
\(919\) 30.4721 1.00518 0.502592 0.864524i \(-0.332380\pi\)
0.502592 + 0.864524i \(0.332380\pi\)
\(920\) 6.05573 0.199651
\(921\) −17.5623 −0.578698
\(922\) −13.4164 −0.441846
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) −21.3262 −0.701202
\(926\) 52.3607 1.72068
\(927\) 1.32624 0.0435594
\(928\) 40.2492 1.32125
\(929\) 23.7295 0.778539 0.389270 0.921124i \(-0.372727\pi\)
0.389270 + 0.921124i \(0.372727\pi\)
\(930\) 4.27051 0.140036
\(931\) −2.85410 −0.0935394
\(932\) 47.1246 1.54362
\(933\) −20.6525 −0.676132
\(934\) −58.5410 −1.91552
\(935\) 0 0
\(936\) 7.23607 0.236518
\(937\) −34.0689 −1.11298 −0.556491 0.830854i \(-0.687853\pi\)
−0.556491 + 0.830854i \(0.687853\pi\)
\(938\) 0 0
\(939\) 7.81966 0.255185
\(940\) −8.29180 −0.270449
\(941\) 0.493422 0.0160851 0.00804255 0.999968i \(-0.497440\pi\)
0.00804255 + 0.999968i \(0.497440\pi\)
\(942\) −51.7082 −1.68474
\(943\) −32.3475 −1.05338
\(944\) −3.23607 −0.105325
\(945\) −0.618034 −0.0201046
\(946\) 0 0
\(947\) 13.8541 0.450198 0.225099 0.974336i \(-0.427729\pi\)
0.225099 + 0.974336i \(0.427729\pi\)
\(948\) −30.0000 −0.974355
\(949\) 44.3607 1.44001
\(950\) −29.4721 −0.956202
\(951\) −6.76393 −0.219336
\(952\) −10.8541 −0.351783
\(953\) −11.8885 −0.385108 −0.192554 0.981286i \(-0.561677\pi\)
−0.192554 + 0.981286i \(0.561677\pi\)
\(954\) −15.1246 −0.489677
\(955\) −3.03444 −0.0981922
\(956\) 57.1033 1.84685
\(957\) 0 0
\(958\) −2.11146 −0.0682181
\(959\) −8.18034 −0.264157
\(960\) −8.03444 −0.259310
\(961\) −21.4508 −0.691963
\(962\) 33.4164 1.07739
\(963\) −15.7984 −0.509095
\(964\) −36.0000 −1.15948
\(965\) −13.5066 −0.434792
\(966\) −9.79837 −0.315258
\(967\) 3.63932 0.117033 0.0585163 0.998286i \(-0.481363\pi\)
0.0585163 + 0.998286i \(0.481363\pi\)
\(968\) 0 0
\(969\) −13.8541 −0.445058
\(970\) −8.29180 −0.266234
\(971\) 54.5410 1.75030 0.875152 0.483848i \(-0.160761\pi\)
0.875152 + 0.483848i \(0.160761\pi\)
\(972\) −3.00000 −0.0962250
\(973\) 13.1459 0.421438
\(974\) −61.9574 −1.98524
\(975\) −14.9443 −0.478600
\(976\) 0 0
\(977\) −13.6393 −0.436361 −0.218180 0.975908i \(-0.570012\pi\)
−0.218180 + 0.975908i \(0.570012\pi\)
\(978\) −22.3607 −0.715016
\(979\) 0 0
\(980\) −1.85410 −0.0592271
\(981\) 17.5623 0.560721
\(982\) 3.36881 0.107503
\(983\) −10.9443 −0.349068 −0.174534 0.984651i \(-0.555842\pi\)
−0.174534 + 0.984651i \(0.555842\pi\)
\(984\) −16.5066 −0.526210
\(985\) 13.8885 0.442526
\(986\) 65.1246 2.07399
\(987\) 4.47214 0.142350
\(988\) 27.7082 0.881515
\(989\) −42.5410 −1.35273
\(990\) 0 0
\(991\) −33.2361 −1.05578 −0.527889 0.849313i \(-0.677016\pi\)
−0.527889 + 0.849313i \(0.677016\pi\)
\(992\) −20.7295 −0.658162
\(993\) −17.4164 −0.552693
\(994\) −11.0557 −0.350666
\(995\) −8.12461 −0.257568
\(996\) 49.4164 1.56582
\(997\) 44.4296 1.40710 0.703549 0.710647i \(-0.251597\pi\)
0.703549 + 0.710647i \(0.251597\pi\)
\(998\) −47.2361 −1.49523
\(999\) −4.61803 −0.146108
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 2541.2.a.v.1.1 2
3.2 odd 2 7623.2.a.bj.1.2 2
11.7 odd 10 231.2.j.e.148.1 yes 4
11.8 odd 10 231.2.j.e.64.1 4
11.10 odd 2 2541.2.a.w.1.2 2
33.8 even 10 693.2.m.a.64.1 4
33.29 even 10 693.2.m.a.379.1 4
33.32 even 2 7623.2.a.bk.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.j.e.64.1 4 11.8 odd 10
231.2.j.e.148.1 yes 4 11.7 odd 10
693.2.m.a.64.1 4 33.8 even 10
693.2.m.a.379.1 4 33.29 even 10
2541.2.a.v.1.1 2 1.1 even 1 trivial
2541.2.a.w.1.2 2 11.10 odd 2
7623.2.a.bj.1.2 2 3.2 odd 2
7623.2.a.bk.1.1 2 33.32 even 2