Properties

Label 8470.2.a.bp.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.61803 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.61803 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +1.00000 q^{10} +1.61803 q^{12} -5.23607 q^{13} -1.00000 q^{14} -1.61803 q^{15} +1.00000 q^{16} +2.61803 q^{17} +0.381966 q^{18} +7.85410 q^{19} -1.00000 q^{20} +1.61803 q^{21} -0.472136 q^{23} -1.61803 q^{24} +1.00000 q^{25} +5.23607 q^{26} -5.47214 q^{27} +1.00000 q^{28} +4.47214 q^{29} +1.61803 q^{30} +2.47214 q^{31} -1.00000 q^{32} -2.61803 q^{34} -1.00000 q^{35} -0.381966 q^{36} -11.2361 q^{37} -7.85410 q^{38} -8.47214 q^{39} +1.00000 q^{40} +2.85410 q^{41} -1.61803 q^{42} -2.09017 q^{43} +0.381966 q^{45} +0.472136 q^{46} +4.76393 q^{47} +1.61803 q^{48} +1.00000 q^{49} -1.00000 q^{50} +4.23607 q^{51} -5.23607 q^{52} -2.76393 q^{53} +5.47214 q^{54} -1.00000 q^{56} +12.7082 q^{57} -4.47214 q^{58} +4.14590 q^{59} -1.61803 q^{60} +13.7082 q^{61} -2.47214 q^{62} -0.381966 q^{63} +1.00000 q^{64} +5.23607 q^{65} -3.38197 q^{67} +2.61803 q^{68} -0.763932 q^{69} +1.00000 q^{70} +10.9443 q^{71} +0.381966 q^{72} +2.85410 q^{73} +11.2361 q^{74} +1.61803 q^{75} +7.85410 q^{76} +8.47214 q^{78} +4.00000 q^{79} -1.00000 q^{80} -7.70820 q^{81} -2.85410 q^{82} -8.85410 q^{83} +1.61803 q^{84} -2.61803 q^{85} +2.09017 q^{86} +7.23607 q^{87} -14.8541 q^{89} -0.381966 q^{90} -5.23607 q^{91} -0.472136 q^{92} +4.00000 q^{93} -4.76393 q^{94} -7.85410 q^{95} -1.61803 q^{96} -8.38197 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + q^{3} + 2q^{4} - 2q^{5} - q^{6} + 2q^{7} - 2q^{8} - 3q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + q^{3} + 2q^{4} - 2q^{5} - q^{6} + 2q^{7} - 2q^{8} - 3q^{9} + 2q^{10} + q^{12} - 6q^{13} - 2q^{14} - q^{15} + 2q^{16} + 3q^{17} + 3q^{18} + 9q^{19} - 2q^{20} + q^{21} + 8q^{23} - q^{24} + 2q^{25} + 6q^{26} - 2q^{27} + 2q^{28} + q^{30} - 4q^{31} - 2q^{32} - 3q^{34} - 2q^{35} - 3q^{36} - 18q^{37} - 9q^{38} - 8q^{39} + 2q^{40} - q^{41} - q^{42} + 7q^{43} + 3q^{45} - 8q^{46} + 14q^{47} + q^{48} + 2q^{49} - 2q^{50} + 4q^{51} - 6q^{52} - 10q^{53} + 2q^{54} - 2q^{56} + 12q^{57} + 15q^{59} - q^{60} + 14q^{61} + 4q^{62} - 3q^{63} + 2q^{64} + 6q^{65} - 9q^{67} + 3q^{68} - 6q^{69} + 2q^{70} + 4q^{71} + 3q^{72} - q^{73} + 18q^{74} + q^{75} + 9q^{76} + 8q^{78} + 8q^{79} - 2q^{80} - 2q^{81} + q^{82} - 11q^{83} + q^{84} - 3q^{85} - 7q^{86} + 10q^{87} - 23q^{89} - 3q^{90} - 6q^{91} + 8q^{92} + 8q^{93} - 14q^{94} - 9q^{95} - q^{96} - 19q^{97} - 2q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.61803 0.934172 0.467086 0.884212i \(-0.345304\pi\)
0.467086 + 0.884212i \(0.345304\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.61803 −0.660560
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −0.381966 −0.127322
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 1.61803 0.467086
\(13\) −5.23607 −1.45222 −0.726112 0.687576i \(-0.758675\pi\)
−0.726112 + 0.687576i \(0.758675\pi\)
\(14\) −1.00000 −0.267261
\(15\) −1.61803 −0.417775
\(16\) 1.00000 0.250000
\(17\) 2.61803 0.634967 0.317483 0.948264i \(-0.397162\pi\)
0.317483 + 0.948264i \(0.397162\pi\)
\(18\) 0.381966 0.0900303
\(19\) 7.85410 1.80185 0.900927 0.433970i \(-0.142888\pi\)
0.900927 + 0.433970i \(0.142888\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.61803 0.353084
\(22\) 0 0
\(23\) −0.472136 −0.0984472 −0.0492236 0.998788i \(-0.515675\pi\)
−0.0492236 + 0.998788i \(0.515675\pi\)
\(24\) −1.61803 −0.330280
\(25\) 1.00000 0.200000
\(26\) 5.23607 1.02688
\(27\) −5.47214 −1.05311
\(28\) 1.00000 0.188982
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 1.61803 0.295411
\(31\) 2.47214 0.444009 0.222004 0.975046i \(-0.428740\pi\)
0.222004 + 0.975046i \(0.428740\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.61803 −0.448989
\(35\) −1.00000 −0.169031
\(36\) −0.381966 −0.0636610
\(37\) −11.2361 −1.84720 −0.923599 0.383360i \(-0.874767\pi\)
−0.923599 + 0.383360i \(0.874767\pi\)
\(38\) −7.85410 −1.27410
\(39\) −8.47214 −1.35663
\(40\) 1.00000 0.158114
\(41\) 2.85410 0.445736 0.222868 0.974849i \(-0.428458\pi\)
0.222868 + 0.974849i \(0.428458\pi\)
\(42\) −1.61803 −0.249668
\(43\) −2.09017 −0.318748 −0.159374 0.987218i \(-0.550948\pi\)
−0.159374 + 0.987218i \(0.550948\pi\)
\(44\) 0 0
\(45\) 0.381966 0.0569401
\(46\) 0.472136 0.0696126
\(47\) 4.76393 0.694891 0.347445 0.937700i \(-0.387049\pi\)
0.347445 + 0.937700i \(0.387049\pi\)
\(48\) 1.61803 0.233543
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 4.23607 0.593168
\(52\) −5.23607 −0.726112
\(53\) −2.76393 −0.379655 −0.189828 0.981817i \(-0.560793\pi\)
−0.189828 + 0.981817i \(0.560793\pi\)
\(54\) 5.47214 0.744663
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 12.7082 1.68324
\(58\) −4.47214 −0.587220
\(59\) 4.14590 0.539750 0.269875 0.962895i \(-0.413018\pi\)
0.269875 + 0.962895i \(0.413018\pi\)
\(60\) −1.61803 −0.208887
\(61\) 13.7082 1.75516 0.877578 0.479434i \(-0.159158\pi\)
0.877578 + 0.479434i \(0.159158\pi\)
\(62\) −2.47214 −0.313962
\(63\) −0.381966 −0.0481232
\(64\) 1.00000 0.125000
\(65\) 5.23607 0.649454
\(66\) 0 0
\(67\) −3.38197 −0.413173 −0.206586 0.978428i \(-0.566235\pi\)
−0.206586 + 0.978428i \(0.566235\pi\)
\(68\) 2.61803 0.317483
\(69\) −0.763932 −0.0919666
\(70\) 1.00000 0.119523
\(71\) 10.9443 1.29885 0.649423 0.760427i \(-0.275010\pi\)
0.649423 + 0.760427i \(0.275010\pi\)
\(72\) 0.381966 0.0450151
\(73\) 2.85410 0.334047 0.167024 0.985953i \(-0.446584\pi\)
0.167024 + 0.985953i \(0.446584\pi\)
\(74\) 11.2361 1.30617
\(75\) 1.61803 0.186834
\(76\) 7.85410 0.900927
\(77\) 0 0
\(78\) 8.47214 0.959280
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −1.00000 −0.111803
\(81\) −7.70820 −0.856467
\(82\) −2.85410 −0.315183
\(83\) −8.85410 −0.971864 −0.485932 0.873997i \(-0.661520\pi\)
−0.485932 + 0.873997i \(0.661520\pi\)
\(84\) 1.61803 0.176542
\(85\) −2.61803 −0.283966
\(86\) 2.09017 0.225389
\(87\) 7.23607 0.775788
\(88\) 0 0
\(89\) −14.8541 −1.57453 −0.787266 0.616614i \(-0.788504\pi\)
−0.787266 + 0.616614i \(0.788504\pi\)
\(90\) −0.381966 −0.0402628
\(91\) −5.23607 −0.548889
\(92\) −0.472136 −0.0492236
\(93\) 4.00000 0.414781
\(94\) −4.76393 −0.491362
\(95\) −7.85410 −0.805814
\(96\) −1.61803 −0.165140
\(97\) −8.38197 −0.851060 −0.425530 0.904944i \(-0.639912\pi\)
−0.425530 + 0.904944i \(0.639912\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −14.9443 −1.48701 −0.743505 0.668730i \(-0.766838\pi\)
−0.743505 + 0.668730i \(0.766838\pi\)
\(102\) −4.23607 −0.419433
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 5.23607 0.513439
\(105\) −1.61803 −0.157904
\(106\) 2.76393 0.268457
\(107\) 2.85410 0.275916 0.137958 0.990438i \(-0.455946\pi\)
0.137958 + 0.990438i \(0.455946\pi\)
\(108\) −5.47214 −0.526557
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) −18.1803 −1.72560
\(112\) 1.00000 0.0944911
\(113\) 4.32624 0.406978 0.203489 0.979077i \(-0.434772\pi\)
0.203489 + 0.979077i \(0.434772\pi\)
\(114\) −12.7082 −1.19023
\(115\) 0.472136 0.0440269
\(116\) 4.47214 0.415227
\(117\) 2.00000 0.184900
\(118\) −4.14590 −0.381661
\(119\) 2.61803 0.239995
\(120\) 1.61803 0.147706
\(121\) 0 0
\(122\) −13.7082 −1.24108
\(123\) 4.61803 0.416394
\(124\) 2.47214 0.222004
\(125\) −1.00000 −0.0894427
\(126\) 0.381966 0.0340282
\(127\) 12.4721 1.10672 0.553362 0.832941i \(-0.313345\pi\)
0.553362 + 0.832941i \(0.313345\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −3.38197 −0.297766
\(130\) −5.23607 −0.459234
\(131\) 5.09017 0.444730 0.222365 0.974963i \(-0.428622\pi\)
0.222365 + 0.974963i \(0.428622\pi\)
\(132\) 0 0
\(133\) 7.85410 0.681037
\(134\) 3.38197 0.292157
\(135\) 5.47214 0.470966
\(136\) −2.61803 −0.224495
\(137\) 4.85410 0.414714 0.207357 0.978265i \(-0.433514\pi\)
0.207357 + 0.978265i \(0.433514\pi\)
\(138\) 0.763932 0.0650302
\(139\) −4.94427 −0.419368 −0.209684 0.977769i \(-0.567243\pi\)
−0.209684 + 0.977769i \(0.567243\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 7.70820 0.649148
\(142\) −10.9443 −0.918423
\(143\) 0 0
\(144\) −0.381966 −0.0318305
\(145\) −4.47214 −0.371391
\(146\) −2.85410 −0.236207
\(147\) 1.61803 0.133453
\(148\) −11.2361 −0.923599
\(149\) 11.7082 0.959173 0.479587 0.877494i \(-0.340787\pi\)
0.479587 + 0.877494i \(0.340787\pi\)
\(150\) −1.61803 −0.132112
\(151\) 18.4721 1.50324 0.751621 0.659596i \(-0.229272\pi\)
0.751621 + 0.659596i \(0.229272\pi\)
\(152\) −7.85410 −0.637052
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) −2.47214 −0.198567
\(156\) −8.47214 −0.678314
\(157\) −14.9443 −1.19268 −0.596341 0.802731i \(-0.703380\pi\)
−0.596341 + 0.802731i \(0.703380\pi\)
\(158\) −4.00000 −0.318223
\(159\) −4.47214 −0.354663
\(160\) 1.00000 0.0790569
\(161\) −0.472136 −0.0372095
\(162\) 7.70820 0.605614
\(163\) 22.7984 1.78571 0.892853 0.450348i \(-0.148700\pi\)
0.892853 + 0.450348i \(0.148700\pi\)
\(164\) 2.85410 0.222868
\(165\) 0 0
\(166\) 8.85410 0.687212
\(167\) 8.29180 0.641638 0.320819 0.947140i \(-0.396042\pi\)
0.320819 + 0.947140i \(0.396042\pi\)
\(168\) −1.61803 −0.124834
\(169\) 14.4164 1.10895
\(170\) 2.61803 0.200794
\(171\) −3.00000 −0.229416
\(172\) −2.09017 −0.159374
\(173\) 19.8885 1.51210 0.756049 0.654515i \(-0.227127\pi\)
0.756049 + 0.654515i \(0.227127\pi\)
\(174\) −7.23607 −0.548565
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 6.70820 0.504219
\(178\) 14.8541 1.11336
\(179\) 2.14590 0.160392 0.0801960 0.996779i \(-0.474445\pi\)
0.0801960 + 0.996779i \(0.474445\pi\)
\(180\) 0.381966 0.0284701
\(181\) 20.9443 1.55678 0.778388 0.627784i \(-0.216038\pi\)
0.778388 + 0.627784i \(0.216038\pi\)
\(182\) 5.23607 0.388123
\(183\) 22.1803 1.63962
\(184\) 0.472136 0.0348063
\(185\) 11.2361 0.826092
\(186\) −4.00000 −0.293294
\(187\) 0 0
\(188\) 4.76393 0.347445
\(189\) −5.47214 −0.398039
\(190\) 7.85410 0.569796
\(191\) −1.23607 −0.0894387 −0.0447194 0.999000i \(-0.514239\pi\)
−0.0447194 + 0.999000i \(0.514239\pi\)
\(192\) 1.61803 0.116772
\(193\) −21.4164 −1.54159 −0.770793 0.637085i \(-0.780140\pi\)
−0.770793 + 0.637085i \(0.780140\pi\)
\(194\) 8.38197 0.601790
\(195\) 8.47214 0.606702
\(196\) 1.00000 0.0714286
\(197\) 11.5279 0.821326 0.410663 0.911787i \(-0.365297\pi\)
0.410663 + 0.911787i \(0.365297\pi\)
\(198\) 0 0
\(199\) 5.05573 0.358391 0.179196 0.983813i \(-0.442651\pi\)
0.179196 + 0.983813i \(0.442651\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −5.47214 −0.385975
\(202\) 14.9443 1.05148
\(203\) 4.47214 0.313882
\(204\) 4.23607 0.296584
\(205\) −2.85410 −0.199339
\(206\) 6.00000 0.418040
\(207\) 0.180340 0.0125345
\(208\) −5.23607 −0.363056
\(209\) 0 0
\(210\) 1.61803 0.111655
\(211\) 10.6738 0.734812 0.367406 0.930061i \(-0.380246\pi\)
0.367406 + 0.930061i \(0.380246\pi\)
\(212\) −2.76393 −0.189828
\(213\) 17.7082 1.21335
\(214\) −2.85410 −0.195102
\(215\) 2.09017 0.142548
\(216\) 5.47214 0.372332
\(217\) 2.47214 0.167820
\(218\) −4.00000 −0.270914
\(219\) 4.61803 0.312058
\(220\) 0 0
\(221\) −13.7082 −0.922114
\(222\) 18.1803 1.22018
\(223\) 13.1246 0.878889 0.439445 0.898270i \(-0.355175\pi\)
0.439445 + 0.898270i \(0.355175\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −0.381966 −0.0254644
\(226\) −4.32624 −0.287777
\(227\) 13.6738 0.907559 0.453780 0.891114i \(-0.350075\pi\)
0.453780 + 0.891114i \(0.350075\pi\)
\(228\) 12.7082 0.841621
\(229\) 22.3607 1.47764 0.738818 0.673905i \(-0.235384\pi\)
0.738818 + 0.673905i \(0.235384\pi\)
\(230\) −0.472136 −0.0311317
\(231\) 0 0
\(232\) −4.47214 −0.293610
\(233\) 0.673762 0.0441396 0.0220698 0.999756i \(-0.492974\pi\)
0.0220698 + 0.999756i \(0.492974\pi\)
\(234\) −2.00000 −0.130744
\(235\) −4.76393 −0.310765
\(236\) 4.14590 0.269875
\(237\) 6.47214 0.420410
\(238\) −2.61803 −0.169702
\(239\) 6.29180 0.406982 0.203491 0.979077i \(-0.434771\pi\)
0.203491 + 0.979077i \(0.434771\pi\)
\(240\) −1.61803 −0.104444
\(241\) −30.9787 −1.99551 −0.997757 0.0669372i \(-0.978677\pi\)
−0.997757 + 0.0669372i \(0.978677\pi\)
\(242\) 0 0
\(243\) 3.94427 0.253025
\(244\) 13.7082 0.877578
\(245\) −1.00000 −0.0638877
\(246\) −4.61803 −0.294435
\(247\) −41.1246 −2.61670
\(248\) −2.47214 −0.156981
\(249\) −14.3262 −0.907888
\(250\) 1.00000 0.0632456
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) −0.381966 −0.0240616
\(253\) 0 0
\(254\) −12.4721 −0.782571
\(255\) −4.23607 −0.265273
\(256\) 1.00000 0.0625000
\(257\) 14.8541 0.926573 0.463287 0.886208i \(-0.346670\pi\)
0.463287 + 0.886208i \(0.346670\pi\)
\(258\) 3.38197 0.210552
\(259\) −11.2361 −0.698175
\(260\) 5.23607 0.324727
\(261\) −1.70820 −0.105735
\(262\) −5.09017 −0.314472
\(263\) 26.3607 1.62547 0.812735 0.582634i \(-0.197978\pi\)
0.812735 + 0.582634i \(0.197978\pi\)
\(264\) 0 0
\(265\) 2.76393 0.169787
\(266\) −7.85410 −0.481566
\(267\) −24.0344 −1.47088
\(268\) −3.38197 −0.206586
\(269\) −18.6525 −1.13726 −0.568631 0.822593i \(-0.692527\pi\)
−0.568631 + 0.822593i \(0.692527\pi\)
\(270\) −5.47214 −0.333024
\(271\) −6.18034 −0.375429 −0.187714 0.982224i \(-0.560108\pi\)
−0.187714 + 0.982224i \(0.560108\pi\)
\(272\) 2.61803 0.158742
\(273\) −8.47214 −0.512757
\(274\) −4.85410 −0.293247
\(275\) 0 0
\(276\) −0.763932 −0.0459833
\(277\) −12.0000 −0.721010 −0.360505 0.932757i \(-0.617396\pi\)
−0.360505 + 0.932757i \(0.617396\pi\)
\(278\) 4.94427 0.296538
\(279\) −0.944272 −0.0565321
\(280\) 1.00000 0.0597614
\(281\) 11.3262 0.675667 0.337833 0.941206i \(-0.390306\pi\)
0.337833 + 0.941206i \(0.390306\pi\)
\(282\) −7.70820 −0.459017
\(283\) 16.3607 0.972541 0.486271 0.873808i \(-0.338357\pi\)
0.486271 + 0.873808i \(0.338357\pi\)
\(284\) 10.9443 0.649423
\(285\) −12.7082 −0.752769
\(286\) 0 0
\(287\) 2.85410 0.168472
\(288\) 0.381966 0.0225076
\(289\) −10.1459 −0.596818
\(290\) 4.47214 0.262613
\(291\) −13.5623 −0.795036
\(292\) 2.85410 0.167024
\(293\) 18.4721 1.07915 0.539577 0.841936i \(-0.318584\pi\)
0.539577 + 0.841936i \(0.318584\pi\)
\(294\) −1.61803 −0.0943657
\(295\) −4.14590 −0.241384
\(296\) 11.2361 0.653083
\(297\) 0 0
\(298\) −11.7082 −0.678238
\(299\) 2.47214 0.142967
\(300\) 1.61803 0.0934172
\(301\) −2.09017 −0.120475
\(302\) −18.4721 −1.06295
\(303\) −24.1803 −1.38912
\(304\) 7.85410 0.450464
\(305\) −13.7082 −0.784929
\(306\) 1.00000 0.0571662
\(307\) 18.5623 1.05941 0.529703 0.848183i \(-0.322303\pi\)
0.529703 + 0.848183i \(0.322303\pi\)
\(308\) 0 0
\(309\) −9.70820 −0.552280
\(310\) 2.47214 0.140408
\(311\) −2.18034 −0.123636 −0.0618179 0.998087i \(-0.519690\pi\)
−0.0618179 + 0.998087i \(0.519690\pi\)
\(312\) 8.47214 0.479640
\(313\) −31.2705 −1.76751 −0.883757 0.467946i \(-0.844994\pi\)
−0.883757 + 0.467946i \(0.844994\pi\)
\(314\) 14.9443 0.843354
\(315\) 0.381966 0.0215213
\(316\) 4.00000 0.225018
\(317\) 30.3607 1.70523 0.852613 0.522543i \(-0.175017\pi\)
0.852613 + 0.522543i \(0.175017\pi\)
\(318\) 4.47214 0.250785
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 4.61803 0.257754
\(322\) 0.472136 0.0263111
\(323\) 20.5623 1.14412
\(324\) −7.70820 −0.428234
\(325\) −5.23607 −0.290445
\(326\) −22.7984 −1.26269
\(327\) 6.47214 0.357910
\(328\) −2.85410 −0.157591
\(329\) 4.76393 0.262644
\(330\) 0 0
\(331\) −0.965558 −0.0530719 −0.0265359 0.999648i \(-0.508448\pi\)
−0.0265359 + 0.999648i \(0.508448\pi\)
\(332\) −8.85410 −0.485932
\(333\) 4.29180 0.235189
\(334\) −8.29180 −0.453707
\(335\) 3.38197 0.184777
\(336\) 1.61803 0.0882710
\(337\) −1.20163 −0.0654567 −0.0327284 0.999464i \(-0.510420\pi\)
−0.0327284 + 0.999464i \(0.510420\pi\)
\(338\) −14.4164 −0.784149
\(339\) 7.00000 0.380188
\(340\) −2.61803 −0.141983
\(341\) 0 0
\(342\) 3.00000 0.162221
\(343\) 1.00000 0.0539949
\(344\) 2.09017 0.112694
\(345\) 0.763932 0.0411287
\(346\) −19.8885 −1.06921
\(347\) 15.0902 0.810083 0.405041 0.914298i \(-0.367257\pi\)
0.405041 + 0.914298i \(0.367257\pi\)
\(348\) 7.23607 0.387894
\(349\) −19.4164 −1.03934 −0.519668 0.854368i \(-0.673944\pi\)
−0.519668 + 0.854368i \(0.673944\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 28.6525 1.52936
\(352\) 0 0
\(353\) −11.5623 −0.615399 −0.307700 0.951484i \(-0.599559\pi\)
−0.307700 + 0.951484i \(0.599559\pi\)
\(354\) −6.70820 −0.356537
\(355\) −10.9443 −0.580862
\(356\) −14.8541 −0.787266
\(357\) 4.23607 0.224196
\(358\) −2.14590 −0.113414
\(359\) 29.5967 1.56206 0.781028 0.624496i \(-0.214695\pi\)
0.781028 + 0.624496i \(0.214695\pi\)
\(360\) −0.381966 −0.0201314
\(361\) 42.6869 2.24668
\(362\) −20.9443 −1.10081
\(363\) 0 0
\(364\) −5.23607 −0.274445
\(365\) −2.85410 −0.149391
\(366\) −22.1803 −1.15938
\(367\) 15.8885 0.829375 0.414688 0.909964i \(-0.363891\pi\)
0.414688 + 0.909964i \(0.363891\pi\)
\(368\) −0.472136 −0.0246118
\(369\) −1.09017 −0.0567520
\(370\) −11.2361 −0.584135
\(371\) −2.76393 −0.143496
\(372\) 4.00000 0.207390
\(373\) 14.6525 0.758676 0.379338 0.925258i \(-0.376152\pi\)
0.379338 + 0.925258i \(0.376152\pi\)
\(374\) 0 0
\(375\) −1.61803 −0.0835549
\(376\) −4.76393 −0.245681
\(377\) −23.4164 −1.20601
\(378\) 5.47214 0.281456
\(379\) 15.6180 0.802245 0.401122 0.916025i \(-0.368620\pi\)
0.401122 + 0.916025i \(0.368620\pi\)
\(380\) −7.85410 −0.402907
\(381\) 20.1803 1.03387
\(382\) 1.23607 0.0632427
\(383\) −20.1803 −1.03117 −0.515584 0.856839i \(-0.672425\pi\)
−0.515584 + 0.856839i \(0.672425\pi\)
\(384\) −1.61803 −0.0825700
\(385\) 0 0
\(386\) 21.4164 1.09007
\(387\) 0.798374 0.0405836
\(388\) −8.38197 −0.425530
\(389\) 25.7082 1.30346 0.651729 0.758452i \(-0.274044\pi\)
0.651729 + 0.758452i \(0.274044\pi\)
\(390\) −8.47214 −0.429003
\(391\) −1.23607 −0.0625106
\(392\) −1.00000 −0.0505076
\(393\) 8.23607 0.415455
\(394\) −11.5279 −0.580765
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) 0.111456 0.00559383 0.00279691 0.999996i \(-0.499110\pi\)
0.00279691 + 0.999996i \(0.499110\pi\)
\(398\) −5.05573 −0.253421
\(399\) 12.7082 0.636206
\(400\) 1.00000 0.0500000
\(401\) 2.56231 0.127955 0.0639777 0.997951i \(-0.479621\pi\)
0.0639777 + 0.997951i \(0.479621\pi\)
\(402\) 5.47214 0.272925
\(403\) −12.9443 −0.644800
\(404\) −14.9443 −0.743505
\(405\) 7.70820 0.383024
\(406\) −4.47214 −0.221948
\(407\) 0 0
\(408\) −4.23607 −0.209717
\(409\) −22.9443 −1.13452 −0.567261 0.823538i \(-0.691997\pi\)
−0.567261 + 0.823538i \(0.691997\pi\)
\(410\) 2.85410 0.140954
\(411\) 7.85410 0.387414
\(412\) −6.00000 −0.295599
\(413\) 4.14590 0.204006
\(414\) −0.180340 −0.00886322
\(415\) 8.85410 0.434631
\(416\) 5.23607 0.256719
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) 22.5623 1.10224 0.551120 0.834426i \(-0.314200\pi\)
0.551120 + 0.834426i \(0.314200\pi\)
\(420\) −1.61803 −0.0789520
\(421\) −36.8328 −1.79512 −0.897561 0.440891i \(-0.854663\pi\)
−0.897561 + 0.440891i \(0.854663\pi\)
\(422\) −10.6738 −0.519591
\(423\) −1.81966 −0.0884749
\(424\) 2.76393 0.134228
\(425\) 2.61803 0.126993
\(426\) −17.7082 −0.857965
\(427\) 13.7082 0.663386
\(428\) 2.85410 0.137958
\(429\) 0 0
\(430\) −2.09017 −0.100797
\(431\) −0.652476 −0.0314287 −0.0157143 0.999877i \(-0.505002\pi\)
−0.0157143 + 0.999877i \(0.505002\pi\)
\(432\) −5.47214 −0.263278
\(433\) −8.79837 −0.422823 −0.211411 0.977397i \(-0.567806\pi\)
−0.211411 + 0.977397i \(0.567806\pi\)
\(434\) −2.47214 −0.118666
\(435\) −7.23607 −0.346943
\(436\) 4.00000 0.191565
\(437\) −3.70820 −0.177387
\(438\) −4.61803 −0.220658
\(439\) 30.5410 1.45764 0.728822 0.684704i \(-0.240068\pi\)
0.728822 + 0.684704i \(0.240068\pi\)
\(440\) 0 0
\(441\) −0.381966 −0.0181889
\(442\) 13.7082 0.652033
\(443\) −28.5623 −1.35704 −0.678518 0.734584i \(-0.737377\pi\)
−0.678518 + 0.734584i \(0.737377\pi\)
\(444\) −18.1803 −0.862801
\(445\) 14.8541 0.704152
\(446\) −13.1246 −0.621468
\(447\) 18.9443 0.896033
\(448\) 1.00000 0.0472456
\(449\) −20.0344 −0.945484 −0.472742 0.881201i \(-0.656736\pi\)
−0.472742 + 0.881201i \(0.656736\pi\)
\(450\) 0.381966 0.0180061
\(451\) 0 0
\(452\) 4.32624 0.203489
\(453\) 29.8885 1.40429
\(454\) −13.6738 −0.641741
\(455\) 5.23607 0.245471
\(456\) −12.7082 −0.595116
\(457\) −25.9787 −1.21523 −0.607616 0.794231i \(-0.707874\pi\)
−0.607616 + 0.794231i \(0.707874\pi\)
\(458\) −22.3607 −1.04485
\(459\) −14.3262 −0.668692
\(460\) 0.472136 0.0220135
\(461\) 23.0557 1.07381 0.536906 0.843642i \(-0.319593\pi\)
0.536906 + 0.843642i \(0.319593\pi\)
\(462\) 0 0
\(463\) −10.9443 −0.508623 −0.254312 0.967122i \(-0.581849\pi\)
−0.254312 + 0.967122i \(0.581849\pi\)
\(464\) 4.47214 0.207614
\(465\) −4.00000 −0.185496
\(466\) −0.673762 −0.0312114
\(467\) 12.9443 0.598989 0.299495 0.954098i \(-0.403182\pi\)
0.299495 + 0.954098i \(0.403182\pi\)
\(468\) 2.00000 0.0924500
\(469\) −3.38197 −0.156165
\(470\) 4.76393 0.219744
\(471\) −24.1803 −1.11417
\(472\) −4.14590 −0.190830
\(473\) 0 0
\(474\) −6.47214 −0.297275
\(475\) 7.85410 0.360371
\(476\) 2.61803 0.119997
\(477\) 1.05573 0.0483385
\(478\) −6.29180 −0.287780
\(479\) 22.2918 1.01854 0.509269 0.860607i \(-0.329916\pi\)
0.509269 + 0.860607i \(0.329916\pi\)
\(480\) 1.61803 0.0738528
\(481\) 58.8328 2.68255
\(482\) 30.9787 1.41104
\(483\) −0.763932 −0.0347601
\(484\) 0 0
\(485\) 8.38197 0.380605
\(486\) −3.94427 −0.178916
\(487\) 38.1803 1.73012 0.865058 0.501672i \(-0.167281\pi\)
0.865058 + 0.501672i \(0.167281\pi\)
\(488\) −13.7082 −0.620541
\(489\) 36.8885 1.66816
\(490\) 1.00000 0.0451754
\(491\) −19.2148 −0.867151 −0.433575 0.901117i \(-0.642748\pi\)
−0.433575 + 0.901117i \(0.642748\pi\)
\(492\) 4.61803 0.208197
\(493\) 11.7082 0.527311
\(494\) 41.1246 1.85028
\(495\) 0 0
\(496\) 2.47214 0.111002
\(497\) 10.9443 0.490918
\(498\) 14.3262 0.641974
\(499\) −19.9787 −0.894370 −0.447185 0.894441i \(-0.647573\pi\)
−0.447185 + 0.894441i \(0.647573\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 13.4164 0.599401
\(502\) −20.0000 −0.892644
\(503\) −20.6525 −0.920848 −0.460424 0.887699i \(-0.652303\pi\)
−0.460424 + 0.887699i \(0.652303\pi\)
\(504\) 0.381966 0.0170141
\(505\) 14.9443 0.665011
\(506\) 0 0
\(507\) 23.3262 1.03595
\(508\) 12.4721 0.553362
\(509\) 13.1246 0.581738 0.290869 0.956763i \(-0.406056\pi\)
0.290869 + 0.956763i \(0.406056\pi\)
\(510\) 4.23607 0.187576
\(511\) 2.85410 0.126258
\(512\) −1.00000 −0.0441942
\(513\) −42.9787 −1.89756
\(514\) −14.8541 −0.655186
\(515\) 6.00000 0.264392
\(516\) −3.38197 −0.148883
\(517\) 0 0
\(518\) 11.2361 0.493684
\(519\) 32.1803 1.41256
\(520\) −5.23607 −0.229617
\(521\) −9.67376 −0.423815 −0.211908 0.977290i \(-0.567968\pi\)
−0.211908 + 0.977290i \(0.567968\pi\)
\(522\) 1.70820 0.0747661
\(523\) −20.7984 −0.909449 −0.454725 0.890632i \(-0.650262\pi\)
−0.454725 + 0.890632i \(0.650262\pi\)
\(524\) 5.09017 0.222365
\(525\) 1.61803 0.0706168
\(526\) −26.3607 −1.14938
\(527\) 6.47214 0.281931
\(528\) 0 0
\(529\) −22.7771 −0.990308
\(530\) −2.76393 −0.120058
\(531\) −1.58359 −0.0687220
\(532\) 7.85410 0.340519
\(533\) −14.9443 −0.647308
\(534\) 24.0344 1.04007
\(535\) −2.85410 −0.123394
\(536\) 3.38197 0.146079
\(537\) 3.47214 0.149834
\(538\) 18.6525 0.804165
\(539\) 0 0
\(540\) 5.47214 0.235483
\(541\) −43.0132 −1.84928 −0.924640 0.380842i \(-0.875634\pi\)
−0.924640 + 0.380842i \(0.875634\pi\)
\(542\) 6.18034 0.265468
\(543\) 33.8885 1.45430
\(544\) −2.61803 −0.112247
\(545\) −4.00000 −0.171341
\(546\) 8.47214 0.362574
\(547\) −22.1459 −0.946890 −0.473445 0.880823i \(-0.656990\pi\)
−0.473445 + 0.880823i \(0.656990\pi\)
\(548\) 4.85410 0.207357
\(549\) −5.23607 −0.223470
\(550\) 0 0
\(551\) 35.1246 1.49636
\(552\) 0.763932 0.0325151
\(553\) 4.00000 0.170097
\(554\) 12.0000 0.509831
\(555\) 18.1803 0.771712
\(556\) −4.94427 −0.209684
\(557\) 18.7639 0.795053 0.397527 0.917591i \(-0.369869\pi\)
0.397527 + 0.917591i \(0.369869\pi\)
\(558\) 0.944272 0.0399742
\(559\) 10.9443 0.462893
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −11.3262 −0.477769
\(563\) 29.6180 1.24825 0.624126 0.781324i \(-0.285455\pi\)
0.624126 + 0.781324i \(0.285455\pi\)
\(564\) 7.70820 0.324574
\(565\) −4.32624 −0.182006
\(566\) −16.3607 −0.687691
\(567\) −7.70820 −0.323714
\(568\) −10.9443 −0.459211
\(569\) 44.3262 1.85825 0.929126 0.369763i \(-0.120561\pi\)
0.929126 + 0.369763i \(0.120561\pi\)
\(570\) 12.7082 0.532288
\(571\) −34.8328 −1.45771 −0.728854 0.684669i \(-0.759947\pi\)
−0.728854 + 0.684669i \(0.759947\pi\)
\(572\) 0 0
\(573\) −2.00000 −0.0835512
\(574\) −2.85410 −0.119128
\(575\) −0.472136 −0.0196894
\(576\) −0.381966 −0.0159153
\(577\) −26.3951 −1.09884 −0.549422 0.835545i \(-0.685152\pi\)
−0.549422 + 0.835545i \(0.685152\pi\)
\(578\) 10.1459 0.422014
\(579\) −34.6525 −1.44011
\(580\) −4.47214 −0.185695
\(581\) −8.85410 −0.367330
\(582\) 13.5623 0.562176
\(583\) 0 0
\(584\) −2.85410 −0.118104
\(585\) −2.00000 −0.0826898
\(586\) −18.4721 −0.763077
\(587\) −8.21478 −0.339060 −0.169530 0.985525i \(-0.554225\pi\)
−0.169530 + 0.985525i \(0.554225\pi\)
\(588\) 1.61803 0.0667266
\(589\) 19.4164 0.800039
\(590\) 4.14590 0.170684
\(591\) 18.6525 0.767260
\(592\) −11.2361 −0.461800
\(593\) −34.0344 −1.39763 −0.698814 0.715304i \(-0.746288\pi\)
−0.698814 + 0.715304i \(0.746288\pi\)
\(594\) 0 0
\(595\) −2.61803 −0.107329
\(596\) 11.7082 0.479587
\(597\) 8.18034 0.334799
\(598\) −2.47214 −0.101093
\(599\) −36.0689 −1.47373 −0.736867 0.676037i \(-0.763696\pi\)
−0.736867 + 0.676037i \(0.763696\pi\)
\(600\) −1.61803 −0.0660560
\(601\) −33.3262 −1.35941 −0.679703 0.733488i \(-0.737891\pi\)
−0.679703 + 0.733488i \(0.737891\pi\)
\(602\) 2.09017 0.0851890
\(603\) 1.29180 0.0526060
\(604\) 18.4721 0.751621
\(605\) 0 0
\(606\) 24.1803 0.982259
\(607\) −6.83282 −0.277335 −0.138668 0.990339i \(-0.544282\pi\)
−0.138668 + 0.990339i \(0.544282\pi\)
\(608\) −7.85410 −0.318526
\(609\) 7.23607 0.293220
\(610\) 13.7082 0.555029
\(611\) −24.9443 −1.00914
\(612\) −1.00000 −0.0404226
\(613\) −10.4721 −0.422966 −0.211483 0.977382i \(-0.567829\pi\)
−0.211483 + 0.977382i \(0.567829\pi\)
\(614\) −18.5623 −0.749114
\(615\) −4.61803 −0.186217
\(616\) 0 0
\(617\) −5.56231 −0.223930 −0.111965 0.993712i \(-0.535714\pi\)
−0.111965 + 0.993712i \(0.535714\pi\)
\(618\) 9.70820 0.390521
\(619\) 43.2705 1.73919 0.869594 0.493767i \(-0.164380\pi\)
0.869594 + 0.493767i \(0.164380\pi\)
\(620\) −2.47214 −0.0992834
\(621\) 2.58359 0.103676
\(622\) 2.18034 0.0874237
\(623\) −14.8541 −0.595117
\(624\) −8.47214 −0.339157
\(625\) 1.00000 0.0400000
\(626\) 31.2705 1.24982
\(627\) 0 0
\(628\) −14.9443 −0.596341
\(629\) −29.4164 −1.17291
\(630\) −0.381966 −0.0152179
\(631\) 25.1246 1.00020 0.500098 0.865969i \(-0.333297\pi\)
0.500098 + 0.865969i \(0.333297\pi\)
\(632\) −4.00000 −0.159111
\(633\) 17.2705 0.686441
\(634\) −30.3607 −1.20578
\(635\) −12.4721 −0.494942
\(636\) −4.47214 −0.177332
\(637\) −5.23607 −0.207461
\(638\) 0 0
\(639\) −4.18034 −0.165372
\(640\) 1.00000 0.0395285
\(641\) −40.8541 −1.61364 −0.806820 0.590797i \(-0.798813\pi\)
−0.806820 + 0.590797i \(0.798813\pi\)
\(642\) −4.61803 −0.182259
\(643\) −14.2016 −0.560057 −0.280029 0.959992i \(-0.590344\pi\)
−0.280029 + 0.959992i \(0.590344\pi\)
\(644\) −0.472136 −0.0186048
\(645\) 3.38197 0.133165
\(646\) −20.5623 −0.809013
\(647\) 37.7082 1.48246 0.741231 0.671250i \(-0.234242\pi\)
0.741231 + 0.671250i \(0.234242\pi\)
\(648\) 7.70820 0.302807
\(649\) 0 0
\(650\) 5.23607 0.205375
\(651\) 4.00000 0.156772
\(652\) 22.7984 0.892853
\(653\) 44.6525 1.74739 0.873693 0.486477i \(-0.161718\pi\)
0.873693 + 0.486477i \(0.161718\pi\)
\(654\) −6.47214 −0.253081
\(655\) −5.09017 −0.198889
\(656\) 2.85410 0.111434
\(657\) −1.09017 −0.0425316
\(658\) −4.76393 −0.185717
\(659\) −20.9787 −0.817215 −0.408607 0.912710i \(-0.633985\pi\)
−0.408607 + 0.912710i \(0.633985\pi\)
\(660\) 0 0
\(661\) 1.81966 0.0707766 0.0353883 0.999374i \(-0.488733\pi\)
0.0353883 + 0.999374i \(0.488733\pi\)
\(662\) 0.965558 0.0375275
\(663\) −22.1803 −0.861413
\(664\) 8.85410 0.343606
\(665\) −7.85410 −0.304569
\(666\) −4.29180 −0.166304
\(667\) −2.11146 −0.0817559
\(668\) 8.29180 0.320819
\(669\) 21.2361 0.821034
\(670\) −3.38197 −0.130657
\(671\) 0 0
\(672\) −1.61803 −0.0624170
\(673\) 15.4508 0.595586 0.297793 0.954630i \(-0.403749\pi\)
0.297793 + 0.954630i \(0.403749\pi\)
\(674\) 1.20163 0.0462849
\(675\) −5.47214 −0.210623
\(676\) 14.4164 0.554477
\(677\) 11.4164 0.438768 0.219384 0.975639i \(-0.429595\pi\)
0.219384 + 0.975639i \(0.429595\pi\)
\(678\) −7.00000 −0.268833
\(679\) −8.38197 −0.321670
\(680\) 2.61803 0.100397
\(681\) 22.1246 0.847817
\(682\) 0 0
\(683\) 21.5279 0.823741 0.411870 0.911242i \(-0.364876\pi\)
0.411870 + 0.911242i \(0.364876\pi\)
\(684\) −3.00000 −0.114708
\(685\) −4.85410 −0.185466
\(686\) −1.00000 −0.0381802
\(687\) 36.1803 1.38037
\(688\) −2.09017 −0.0796870
\(689\) 14.4721 0.551344
\(690\) −0.763932 −0.0290824
\(691\) −7.56231 −0.287684 −0.143842 0.989601i \(-0.545946\pi\)
−0.143842 + 0.989601i \(0.545946\pi\)
\(692\) 19.8885 0.756049
\(693\) 0 0
\(694\) −15.0902 −0.572815
\(695\) 4.94427 0.187547
\(696\) −7.23607 −0.274282
\(697\) 7.47214 0.283027
\(698\) 19.4164 0.734922
\(699\) 1.09017 0.0412340
\(700\) 1.00000 0.0377964
\(701\) −34.8328 −1.31562 −0.657809 0.753185i \(-0.728516\pi\)
−0.657809 + 0.753185i \(0.728516\pi\)
\(702\) −28.6525 −1.08142
\(703\) −88.2492 −3.32838
\(704\) 0 0
\(705\) −7.70820 −0.290308
\(706\) 11.5623 0.435153
\(707\) −14.9443 −0.562037
\(708\) 6.70820 0.252110
\(709\) 33.7771 1.26853 0.634263 0.773118i \(-0.281304\pi\)
0.634263 + 0.773118i \(0.281304\pi\)
\(710\) 10.9443 0.410731
\(711\) −1.52786 −0.0572994
\(712\) 14.8541 0.556681
\(713\) −1.16718 −0.0437114
\(714\) −4.23607 −0.158531
\(715\) 0 0
\(716\) 2.14590 0.0801960
\(717\) 10.1803 0.380192
\(718\) −29.5967 −1.10454
\(719\) 24.0689 0.897618 0.448809 0.893628i \(-0.351848\pi\)
0.448809 + 0.893628i \(0.351848\pi\)
\(720\) 0.381966 0.0142350
\(721\) −6.00000 −0.223452
\(722\) −42.6869 −1.58864
\(723\) −50.1246 −1.86415
\(724\) 20.9443 0.778388
\(725\) 4.47214 0.166091
\(726\) 0 0
\(727\) 50.2492 1.86364 0.931820 0.362920i \(-0.118220\pi\)
0.931820 + 0.362920i \(0.118220\pi\)
\(728\) 5.23607 0.194062
\(729\) 29.5066 1.09284
\(730\) 2.85410 0.105635
\(731\) −5.47214 −0.202394
\(732\) 22.1803 0.819809
\(733\) −35.5967 −1.31480 −0.657398 0.753544i \(-0.728343\pi\)
−0.657398 + 0.753544i \(0.728343\pi\)
\(734\) −15.8885 −0.586457
\(735\) −1.61803 −0.0596821
\(736\) 0.472136 0.0174032
\(737\) 0 0
\(738\) 1.09017 0.0401297
\(739\) 10.1459 0.373223 0.186611 0.982434i \(-0.440249\pi\)
0.186611 + 0.982434i \(0.440249\pi\)
\(740\) 11.2361 0.413046
\(741\) −66.5410 −2.44445
\(742\) 2.76393 0.101467
\(743\) −40.2492 −1.47660 −0.738300 0.674472i \(-0.764371\pi\)
−0.738300 + 0.674472i \(0.764371\pi\)
\(744\) −4.00000 −0.146647
\(745\) −11.7082 −0.428955
\(746\) −14.6525 −0.536465
\(747\) 3.38197 0.123740
\(748\) 0 0
\(749\) 2.85410 0.104287
\(750\) 1.61803 0.0590822
\(751\) −45.3050 −1.65320 −0.826601 0.562789i \(-0.809728\pi\)
−0.826601 + 0.562789i \(0.809728\pi\)
\(752\) 4.76393 0.173723
\(753\) 32.3607 1.17929
\(754\) 23.4164 0.852775
\(755\) −18.4721 −0.672270
\(756\) −5.47214 −0.199020
\(757\) 44.6525 1.62292 0.811461 0.584407i \(-0.198673\pi\)
0.811461 + 0.584407i \(0.198673\pi\)
\(758\) −15.6180 −0.567273
\(759\) 0 0
\(760\) 7.85410 0.284898
\(761\) −49.8115 −1.80567 −0.902833 0.429991i \(-0.858517\pi\)
−0.902833 + 0.429991i \(0.858517\pi\)
\(762\) −20.1803 −0.731057
\(763\) 4.00000 0.144810
\(764\) −1.23607 −0.0447194
\(765\) 1.00000 0.0361551
\(766\) 20.1803 0.729145
\(767\) −21.7082 −0.783838
\(768\) 1.61803 0.0583858
\(769\) 31.3050 1.12889 0.564443 0.825472i \(-0.309091\pi\)
0.564443 + 0.825472i \(0.309091\pi\)
\(770\) 0 0
\(771\) 24.0344 0.865579
\(772\) −21.4164 −0.770793
\(773\) 34.7639 1.25037 0.625186 0.780476i \(-0.285023\pi\)
0.625186 + 0.780476i \(0.285023\pi\)
\(774\) −0.798374 −0.0286970
\(775\) 2.47214 0.0888017
\(776\) 8.38197 0.300895
\(777\) −18.1803 −0.652216
\(778\) −25.7082 −0.921684
\(779\) 22.4164 0.803151
\(780\) 8.47214 0.303351
\(781\) 0 0
\(782\) 1.23607 0.0442017
\(783\) −24.4721 −0.874563
\(784\) 1.00000 0.0357143
\(785\) 14.9443 0.533384
\(786\) −8.23607 −0.293771
\(787\) 43.2705 1.54243 0.771214 0.636577i \(-0.219650\pi\)
0.771214 + 0.636577i \(0.219650\pi\)
\(788\) 11.5279 0.410663
\(789\) 42.6525 1.51847
\(790\) 4.00000 0.142314
\(791\) 4.32624 0.153823
\(792\) 0 0
\(793\) −71.7771 −2.54888
\(794\) −0.111456 −0.00395543
\(795\) 4.47214 0.158610
\(796\) 5.05573 0.179196
\(797\) −8.18034 −0.289763 −0.144881 0.989449i \(-0.546280\pi\)
−0.144881 + 0.989449i \(0.546280\pi\)
\(798\) −12.7082 −0.449866
\(799\) 12.4721 0.441232
\(800\) −1.00000 −0.0353553
\(801\) 5.67376 0.200473
\(802\) −2.56231 −0.0904782
\(803\) 0 0
\(804\) −5.47214 −0.192987
\(805\) 0.472136 0.0166406
\(806\) 12.9443 0.455943
\(807\) −30.1803 −1.06240
\(808\) 14.9443 0.525738
\(809\) 27.1591 0.954861 0.477431 0.878669i \(-0.341568\pi\)
0.477431 + 0.878669i \(0.341568\pi\)
\(810\) −7.70820 −0.270839
\(811\) 35.6869 1.25314 0.626569 0.779366i \(-0.284459\pi\)
0.626569 + 0.779366i \(0.284459\pi\)
\(812\) 4.47214 0.156941
\(813\) −10.0000 −0.350715
\(814\) 0 0
\(815\) −22.7984 −0.798592
\(816\) 4.23607 0.148292
\(817\) −16.4164 −0.574337
\(818\) 22.9443 0.802228
\(819\) 2.00000 0.0698857
\(820\) −2.85410 −0.0996696
\(821\) −0.111456 −0.00388985 −0.00194492 0.999998i \(-0.500619\pi\)
−0.00194492 + 0.999998i \(0.500619\pi\)
\(822\) −7.85410 −0.273943
\(823\) −3.81966 −0.133145 −0.0665725 0.997782i \(-0.521206\pi\)
−0.0665725 + 0.997782i \(0.521206\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) −4.14590 −0.144254
\(827\) 52.3262 1.81956 0.909781 0.415089i \(-0.136250\pi\)
0.909781 + 0.415089i \(0.136250\pi\)
\(828\) 0.180340 0.00626724
\(829\) 33.5279 1.16447 0.582235 0.813020i \(-0.302178\pi\)
0.582235 + 0.813020i \(0.302178\pi\)
\(830\) −8.85410 −0.307330
\(831\) −19.4164 −0.673548
\(832\) −5.23607 −0.181528
\(833\) 2.61803 0.0907095
\(834\) 8.00000 0.277017
\(835\) −8.29180 −0.286949
\(836\) 0 0
\(837\) −13.5279 −0.467591
\(838\) −22.5623 −0.779402
\(839\) 36.2918 1.25293 0.626466 0.779449i \(-0.284501\pi\)
0.626466 + 0.779449i \(0.284501\pi\)
\(840\) 1.61803 0.0558275
\(841\) −9.00000 −0.310345
\(842\) 36.8328 1.26934
\(843\) 18.3262 0.631189
\(844\) 10.6738 0.367406
\(845\) −14.4164 −0.495940
\(846\) 1.81966 0.0625612
\(847\) 0 0
\(848\) −2.76393 −0.0949138
\(849\) 26.4721 0.908521
\(850\) −2.61803 −0.0897978
\(851\) 5.30495 0.181851
\(852\) 17.7082 0.606673
\(853\) −13.4164 −0.459369 −0.229685 0.973265i \(-0.573769\pi\)
−0.229685 + 0.973265i \(0.573769\pi\)
\(854\) −13.7082 −0.469085
\(855\) 3.00000 0.102598
\(856\) −2.85410 −0.0975512
\(857\) 24.5623 0.839032 0.419516 0.907748i \(-0.362200\pi\)
0.419516 + 0.907748i \(0.362200\pi\)
\(858\) 0 0
\(859\) 14.3262 0.488805 0.244402 0.969674i \(-0.421408\pi\)
0.244402 + 0.969674i \(0.421408\pi\)
\(860\) 2.09017 0.0712742
\(861\) 4.61803 0.157382
\(862\) 0.652476 0.0222234
\(863\) −14.1115 −0.480360 −0.240180 0.970728i \(-0.577206\pi\)
−0.240180 + 0.970728i \(0.577206\pi\)
\(864\) 5.47214 0.186166
\(865\) −19.8885 −0.676231
\(866\) 8.79837 0.298981
\(867\) −16.4164 −0.557530
\(868\) 2.47214 0.0839098
\(869\) 0 0
\(870\) 7.23607 0.245326
\(871\) 17.7082 0.600020
\(872\) −4.00000 −0.135457
\(873\) 3.20163 0.108359
\(874\) 3.70820 0.125432
\(875\) −1.00000 −0.0338062
\(876\) 4.61803 0.156029
\(877\) −2.76393 −0.0933314 −0.0466657 0.998911i \(-0.514860\pi\)
−0.0466657 + 0.998911i \(0.514860\pi\)
\(878\) −30.5410 −1.03071
\(879\) 29.8885 1.00812
\(880\) 0 0
\(881\) 14.6738 0.494372 0.247186 0.968968i \(-0.420494\pi\)
0.247186 + 0.968968i \(0.420494\pi\)
\(882\) 0.381966 0.0128615
\(883\) −43.2148 −1.45429 −0.727147 0.686482i \(-0.759154\pi\)
−0.727147 + 0.686482i \(0.759154\pi\)
\(884\) −13.7082 −0.461057
\(885\) −6.70820 −0.225494
\(886\) 28.5623 0.959569
\(887\) 26.5410 0.891160 0.445580 0.895242i \(-0.352997\pi\)
0.445580 + 0.895242i \(0.352997\pi\)
\(888\) 18.1803 0.610092
\(889\) 12.4721 0.418302
\(890\) −14.8541 −0.497911
\(891\) 0 0
\(892\) 13.1246 0.439445
\(893\) 37.4164 1.25209
\(894\) −18.9443 −0.633591
\(895\) −2.14590 −0.0717295
\(896\) −1.00000 −0.0334077
\(897\) 4.00000 0.133556
\(898\) 20.0344 0.668558
\(899\) 11.0557 0.368729
\(900\) −0.381966 −0.0127322
\(901\) −7.23607 −0.241068
\(902\) 0 0
\(903\) −3.38197 −0.112545
\(904\) −4.32624 −0.143889
\(905\) −20.9443 −0.696211
\(906\) −29.8885 −0.992980
\(907\) 25.8541 0.858471 0.429236 0.903193i \(-0.358783\pi\)
0.429236 + 0.903193i \(0.358783\pi\)
\(908\) 13.6738 0.453780
\(909\) 5.70820 0.189329
\(910\) −5.23607 −0.173574
\(911\) 9.52786 0.315672 0.157836 0.987465i \(-0.449548\pi\)
0.157836 + 0.987465i \(0.449548\pi\)
\(912\) 12.7082 0.420811
\(913\) 0 0
\(914\) 25.9787 0.859299
\(915\) −22.1803 −0.733259
\(916\) 22.3607 0.738818
\(917\) 5.09017 0.168092
\(918\) 14.3262 0.472836
\(919\) 7.05573 0.232747 0.116373 0.993206i \(-0.462873\pi\)
0.116373 + 0.993206i \(0.462873\pi\)
\(920\) −0.472136 −0.0155659
\(921\) 30.0344 0.989669
\(922\) −23.0557 −0.759300
\(923\) −57.3050 −1.88622
\(924\) 0 0
\(925\) −11.2361 −0.369440
\(926\) 10.9443 0.359651
\(927\) 2.29180 0.0752725
\(928\) −4.47214 −0.146805
\(929\) 10.9098 0.357940 0.178970 0.983855i \(-0.442724\pi\)
0.178970 + 0.983855i \(0.442724\pi\)
\(930\) 4.00000 0.131165
\(931\) 7.85410 0.257408
\(932\) 0.673762 0.0220698
\(933\) −3.52786 −0.115497
\(934\) −12.9443 −0.423550
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −28.5066 −0.931269 −0.465635 0.884977i \(-0.654174\pi\)
−0.465635 + 0.884977i \(0.654174\pi\)
\(938\) 3.38197 0.110425
\(939\) −50.5967 −1.65116
\(940\) −4.76393 −0.155382
\(941\) −24.3607 −0.794135 −0.397068 0.917789i \(-0.629972\pi\)
−0.397068 + 0.917789i \(0.629972\pi\)
\(942\) 24.1803 0.787838
\(943\) −1.34752 −0.0438814
\(944\) 4.14590 0.134937
\(945\) 5.47214 0.178009
\(946\) 0 0
\(947\) −53.6312 −1.74278 −0.871390 0.490591i \(-0.836781\pi\)
−0.871390 + 0.490591i \(0.836781\pi\)
\(948\) 6.47214 0.210205
\(949\) −14.9443 −0.485112
\(950\) −7.85410 −0.254821
\(951\) 49.1246 1.59297
\(952\) −2.61803 −0.0848510
\(953\) 0.854102 0.0276671 0.0138335 0.999904i \(-0.495597\pi\)
0.0138335 + 0.999904i \(0.495597\pi\)
\(954\) −1.05573 −0.0341805
\(955\) 1.23607 0.0399982
\(956\) 6.29180 0.203491
\(957\) 0 0
\(958\) −22.2918 −0.720215
\(959\) 4.85410 0.156747
\(960\) −1.61803 −0.0522218
\(961\) −24.8885 −0.802856
\(962\) −58.8328 −1.89685
\(963\) −1.09017 −0.0351302
\(964\) −30.9787 −0.997757
\(965\) 21.4164 0.689419
\(966\) 0.763932 0.0245791
\(967\) 17.3475 0.557859 0.278929 0.960312i \(-0.410020\pi\)
0.278929 + 0.960312i \(0.410020\pi\)
\(968\) 0 0
\(969\) 33.2705 1.06880
\(970\) −8.38197 −0.269129
\(971\) −34.8328 −1.11784 −0.558919 0.829222i \(-0.688784\pi\)
−0.558919 + 0.829222i \(0.688784\pi\)
\(972\) 3.94427 0.126513
\(973\) −4.94427 −0.158506
\(974\) −38.1803 −1.22338
\(975\) −8.47214 −0.271325
\(976\) 13.7082 0.438789
\(977\) −37.1935 −1.18993 −0.594963 0.803753i \(-0.702833\pi\)
−0.594963 + 0.803753i \(0.702833\pi\)
\(978\) −36.8885 −1.17957
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −1.52786 −0.0487809
\(982\) 19.2148 0.613168
\(983\) −8.94427 −0.285278 −0.142639 0.989775i \(-0.545559\pi\)
−0.142639 + 0.989775i \(0.545559\pi\)
\(984\) −4.61803 −0.147218
\(985\) −11.5279 −0.367308
\(986\) −11.7082 −0.372865
\(987\) 7.70820 0.245355
\(988\) −41.1246 −1.30835
\(989\) 0.986844 0.0313798
\(990\) 0 0
\(991\) 34.3607 1.09150 0.545751 0.837947i \(-0.316244\pi\)
0.545751 + 0.837947i \(0.316244\pi\)
\(992\) −2.47214 −0.0784904
\(993\) −1.56231 −0.0495783
\(994\) −10.9443 −0.347131
\(995\) −5.05573 −0.160277
\(996\) −14.3262 −0.453944
\(997\) −48.8328 −1.54655 −0.773275 0.634070i \(-0.781383\pi\)
−0.773275 + 0.634070i \(0.781383\pi\)
\(998\) 19.9787 0.632415
\(999\) 61.4853 1.94531
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 8470.2.a.bp.1.2 2
11.5 even 5 770.2.n.d.421.1 4
11.9 even 5 770.2.n.d.631.1 yes 4
11.10 odd 2 8470.2.a.cb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.d.421.1 4 11.5 even 5
770.2.n.d.631.1 yes 4 11.9 even 5
8470.2.a.bp.1.2 2 1.1 even 1 trivial
8470.2.a.cb.1.2 2 11.10 odd 2