Properties

Label 3850.2.a.bd.1.1
Level $3850$
Weight $2$
Character 3850.1
Self dual yes
Analytic conductor $30.742$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.7424047782\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
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 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} +2.73205 q^{6} -1.00000 q^{7} -1.00000 q^{8} +4.46410 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} +2.73205 q^{6} -1.00000 q^{7} -1.00000 q^{8} +4.46410 q^{9} +1.00000 q^{11} -2.73205 q^{12} +1.46410 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.46410 q^{17} -4.46410 q^{18} +6.73205 q^{19} +2.73205 q^{21} -1.00000 q^{22} +8.19615 q^{23} +2.73205 q^{24} -1.46410 q^{26} -4.00000 q^{27} -1.00000 q^{28} -4.73205 q^{29} +2.00000 q^{31} -1.00000 q^{32} -2.73205 q^{33} -3.46410 q^{34} +4.46410 q^{36} -0.732051 q^{37} -6.73205 q^{38} -4.00000 q^{39} -2.19615 q^{41} -2.73205 q^{42} -2.00000 q^{43} +1.00000 q^{44} -8.19615 q^{46} -6.92820 q^{47} -2.73205 q^{48} +1.00000 q^{49} -9.46410 q^{51} +1.46410 q^{52} +7.26795 q^{53} +4.00000 q^{54} +1.00000 q^{56} -18.3923 q^{57} +4.73205 q^{58} +6.92820 q^{59} -4.92820 q^{61} -2.00000 q^{62} -4.46410 q^{63} +1.00000 q^{64} +2.73205 q^{66} +4.00000 q^{67} +3.46410 q^{68} -22.3923 q^{69} +9.46410 q^{71} -4.46410 q^{72} +14.3923 q^{73} +0.732051 q^{74} +6.73205 q^{76} -1.00000 q^{77} +4.00000 q^{78} -12.1962 q^{79} -2.46410 q^{81} +2.19615 q^{82} +16.3923 q^{83} +2.73205 q^{84} +2.00000 q^{86} +12.9282 q^{87} -1.00000 q^{88} +3.46410 q^{89} -1.46410 q^{91} +8.19615 q^{92} -5.46410 q^{93} +6.92820 q^{94} +2.73205 q^{96} -14.5885 q^{97} -1.00000 q^{98} +4.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + 2q^{11} - 2q^{12} - 4q^{13} + 2q^{14} + 2q^{16} - 2q^{18} + 10q^{19} + 2q^{21} - 2q^{22} + 6q^{23} + 2q^{24} + 4q^{26} - 8q^{27} - 2q^{28} - 6q^{29} + 4q^{31} - 2q^{32} - 2q^{33} + 2q^{36} + 2q^{37} - 10q^{38} - 8q^{39} + 6q^{41} - 2q^{42} - 4q^{43} + 2q^{44} - 6q^{46} - 2q^{48} + 2q^{49} - 12q^{51} - 4q^{52} + 18q^{53} + 8q^{54} + 2q^{56} - 16q^{57} + 6q^{58} + 4q^{61} - 4q^{62} - 2q^{63} + 2q^{64} + 2q^{66} + 8q^{67} - 24q^{69} + 12q^{71} - 2q^{72} + 8q^{73} - 2q^{74} + 10q^{76} - 2q^{77} + 8q^{78} - 14q^{79} + 2q^{81} - 6q^{82} + 12q^{83} + 2q^{84} + 4q^{86} + 12q^{87} - 2q^{88} + 4q^{91} + 6q^{92} - 4q^{93} + 2q^{96} + 2q^{97} - 2q^{98} + 2q^{99} + O(q^{100}) \)

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
\(3\) −2.73205 −1.57735 −0.788675 0.614810i \(-0.789233\pi\)
−0.788675 + 0.614810i \(0.789233\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.73205 1.11536
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 4.46410 1.48803
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −2.73205 −0.788675
\(13\) 1.46410 0.406069 0.203034 0.979172i \(-0.434920\pi\)
0.203034 + 0.979172i \(0.434920\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) −4.46410 −1.05220
\(19\) 6.73205 1.54444 0.772219 0.635356i \(-0.219147\pi\)
0.772219 + 0.635356i \(0.219147\pi\)
\(20\) 0 0
\(21\) 2.73205 0.596182
\(22\) −1.00000 −0.213201
\(23\) 8.19615 1.70902 0.854508 0.519438i \(-0.173859\pi\)
0.854508 + 0.519438i \(0.173859\pi\)
\(24\) 2.73205 0.557678
\(25\) 0 0
\(26\) −1.46410 −0.287134
\(27\) −4.00000 −0.769800
\(28\) −1.00000 −0.188982
\(29\) −4.73205 −0.878720 −0.439360 0.898311i \(-0.644795\pi\)
−0.439360 + 0.898311i \(0.644795\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.73205 −0.475589
\(34\) −3.46410 −0.594089
\(35\) 0 0
\(36\) 4.46410 0.744017
\(37\) −0.732051 −0.120348 −0.0601742 0.998188i \(-0.519166\pi\)
−0.0601742 + 0.998188i \(0.519166\pi\)
\(38\) −6.73205 −1.09208
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −2.19615 −0.342981 −0.171491 0.985186i \(-0.554858\pi\)
−0.171491 + 0.985186i \(0.554858\pi\)
\(42\) −2.73205 −0.421565
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −8.19615 −1.20846
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) −2.73205 −0.394338
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −9.46410 −1.32524
\(52\) 1.46410 0.203034
\(53\) 7.26795 0.998330 0.499165 0.866507i \(-0.333640\pi\)
0.499165 + 0.866507i \(0.333640\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −18.3923 −2.43612
\(58\) 4.73205 0.621349
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0 0
\(61\) −4.92820 −0.630992 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(62\) −2.00000 −0.254000
\(63\) −4.46410 −0.562424
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.73205 0.336292
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 3.46410 0.420084
\(69\) −22.3923 −2.69572
\(70\) 0 0
\(71\) 9.46410 1.12318 0.561591 0.827415i \(-0.310189\pi\)
0.561591 + 0.827415i \(0.310189\pi\)
\(72\) −4.46410 −0.526099
\(73\) 14.3923 1.68449 0.842246 0.539093i \(-0.181233\pi\)
0.842246 + 0.539093i \(0.181233\pi\)
\(74\) 0.732051 0.0850992
\(75\) 0 0
\(76\) 6.73205 0.772219
\(77\) −1.00000 −0.113961
\(78\) 4.00000 0.452911
\(79\) −12.1962 −1.37217 −0.686087 0.727519i \(-0.740673\pi\)
−0.686087 + 0.727519i \(0.740673\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 2.19615 0.242524
\(83\) 16.3923 1.79929 0.899645 0.436623i \(-0.143826\pi\)
0.899645 + 0.436623i \(0.143826\pi\)
\(84\) 2.73205 0.298091
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) 12.9282 1.38605
\(88\) −1.00000 −0.106600
\(89\) 3.46410 0.367194 0.183597 0.983002i \(-0.441226\pi\)
0.183597 + 0.983002i \(0.441226\pi\)
\(90\) 0 0
\(91\) −1.46410 −0.153480
\(92\) 8.19615 0.854508
\(93\) −5.46410 −0.566601
\(94\) 6.92820 0.714590
\(95\) 0 0
\(96\) 2.73205 0.278839
\(97\) −14.5885 −1.48123 −0.740617 0.671928i \(-0.765467\pi\)
−0.740617 + 0.671928i \(0.765467\pi\)
\(98\) −1.00000 −0.101015
\(99\) 4.46410 0.448659
\(100\) 0 0
\(101\) −7.85641 −0.781742 −0.390871 0.920446i \(-0.627826\pi\)
−0.390871 + 0.920446i \(0.627826\pi\)
\(102\) 9.46410 0.937086
\(103\) −12.3923 −1.22105 −0.610525 0.791997i \(-0.709042\pi\)
−0.610525 + 0.791997i \(0.709042\pi\)
\(104\) −1.46410 −0.143567
\(105\) 0 0
\(106\) −7.26795 −0.705926
\(107\) −7.85641 −0.759507 −0.379754 0.925088i \(-0.623991\pi\)
−0.379754 + 0.925088i \(0.623991\pi\)
\(108\) −4.00000 −0.384900
\(109\) −15.6603 −1.49998 −0.749990 0.661449i \(-0.769942\pi\)
−0.749990 + 0.661449i \(0.769942\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −1.00000 −0.0944911
\(113\) −7.85641 −0.739069 −0.369534 0.929217i \(-0.620483\pi\)
−0.369534 + 0.929217i \(0.620483\pi\)
\(114\) 18.3923 1.72260
\(115\) 0 0
\(116\) −4.73205 −0.439360
\(117\) 6.53590 0.604244
\(118\) −6.92820 −0.637793
\(119\) −3.46410 −0.317554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 4.92820 0.446179
\(123\) 6.00000 0.541002
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) 4.46410 0.397694
\(127\) −1.07180 −0.0951066 −0.0475533 0.998869i \(-0.515142\pi\)
−0.0475533 + 0.998869i \(0.515142\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 5.46410 0.481087
\(130\) 0 0
\(131\) 5.66025 0.494539 0.247269 0.968947i \(-0.420467\pi\)
0.247269 + 0.968947i \(0.420467\pi\)
\(132\) −2.73205 −0.237795
\(133\) −6.73205 −0.583743
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −3.46410 −0.297044
\(137\) 0.928203 0.0793018 0.0396509 0.999214i \(-0.487375\pi\)
0.0396509 + 0.999214i \(0.487375\pi\)
\(138\) 22.3923 1.90616
\(139\) 13.6603 1.15865 0.579324 0.815097i \(-0.303317\pi\)
0.579324 + 0.815097i \(0.303317\pi\)
\(140\) 0 0
\(141\) 18.9282 1.59404
\(142\) −9.46410 −0.794210
\(143\) 1.46410 0.122434
\(144\) 4.46410 0.372008
\(145\) 0 0
\(146\) −14.3923 −1.19112
\(147\) −2.73205 −0.225336
\(148\) −0.732051 −0.0601742
\(149\) −7.26795 −0.595414 −0.297707 0.954657i \(-0.596222\pi\)
−0.297707 + 0.954657i \(0.596222\pi\)
\(150\) 0 0
\(151\) 11.1244 0.905287 0.452644 0.891692i \(-0.350481\pi\)
0.452644 + 0.891692i \(0.350481\pi\)
\(152\) −6.73205 −0.546041
\(153\) 15.4641 1.25020
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) −4.53590 −0.362004 −0.181002 0.983483i \(-0.557934\pi\)
−0.181002 + 0.983483i \(0.557934\pi\)
\(158\) 12.1962 0.970274
\(159\) −19.8564 −1.57472
\(160\) 0 0
\(161\) −8.19615 −0.645947
\(162\) 2.46410 0.193598
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −2.19615 −0.171491
\(165\) 0 0
\(166\) −16.3923 −1.27229
\(167\) −13.8564 −1.07224 −0.536120 0.844141i \(-0.680111\pi\)
−0.536120 + 0.844141i \(0.680111\pi\)
\(168\) −2.73205 −0.210782
\(169\) −10.8564 −0.835108
\(170\) 0 0
\(171\) 30.0526 2.29818
\(172\) −2.00000 −0.152499
\(173\) −0.928203 −0.0705700 −0.0352850 0.999377i \(-0.511234\pi\)
−0.0352850 + 0.999377i \(0.511234\pi\)
\(174\) −12.9282 −0.980085
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −18.9282 −1.42273
\(178\) −3.46410 −0.259645
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 1.46410 0.108526
\(183\) 13.4641 0.995295
\(184\) −8.19615 −0.604228
\(185\) 0 0
\(186\) 5.46410 0.400647
\(187\) 3.46410 0.253320
\(188\) −6.92820 −0.505291
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) −2.73205 −0.197169
\(193\) −12.3923 −0.892018 −0.446009 0.895029i \(-0.647155\pi\)
−0.446009 + 0.895029i \(0.647155\pi\)
\(194\) 14.5885 1.04739
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 24.2487 1.72765 0.863825 0.503793i \(-0.168062\pi\)
0.863825 + 0.503793i \(0.168062\pi\)
\(198\) −4.46410 −0.317250
\(199\) 2.92820 0.207575 0.103787 0.994600i \(-0.466904\pi\)
0.103787 + 0.994600i \(0.466904\pi\)
\(200\) 0 0
\(201\) −10.9282 −0.770816
\(202\) 7.85641 0.552775
\(203\) 4.73205 0.332125
\(204\) −9.46410 −0.662620
\(205\) 0 0
\(206\) 12.3923 0.863413
\(207\) 36.5885 2.54307
\(208\) 1.46410 0.101517
\(209\) 6.73205 0.465666
\(210\) 0 0
\(211\) 26.9282 1.85381 0.926907 0.375291i \(-0.122457\pi\)
0.926907 + 0.375291i \(0.122457\pi\)
\(212\) 7.26795 0.499165
\(213\) −25.8564 −1.77165
\(214\) 7.85641 0.537053
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) −2.00000 −0.135769
\(218\) 15.6603 1.06065
\(219\) −39.3205 −2.65703
\(220\) 0 0
\(221\) 5.07180 0.341166
\(222\) −2.00000 −0.134231
\(223\) 25.4641 1.70520 0.852601 0.522562i \(-0.175024\pi\)
0.852601 + 0.522562i \(0.175024\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 7.85641 0.522600
\(227\) −6.92820 −0.459841 −0.229920 0.973209i \(-0.573847\pi\)
−0.229920 + 0.973209i \(0.573847\pi\)
\(228\) −18.3923 −1.21806
\(229\) 24.3923 1.61189 0.805944 0.591991i \(-0.201658\pi\)
0.805944 + 0.591991i \(0.201658\pi\)
\(230\) 0 0
\(231\) 2.73205 0.179756
\(232\) 4.73205 0.310674
\(233\) 7.85641 0.514690 0.257345 0.966320i \(-0.417152\pi\)
0.257345 + 0.966320i \(0.417152\pi\)
\(234\) −6.53590 −0.427265
\(235\) 0 0
\(236\) 6.92820 0.450988
\(237\) 33.3205 2.16440
\(238\) 3.46410 0.224544
\(239\) 1.26795 0.0820168 0.0410084 0.999159i \(-0.486943\pi\)
0.0410084 + 0.999159i \(0.486943\pi\)
\(240\) 0 0
\(241\) 3.26795 0.210507 0.105254 0.994445i \(-0.466435\pi\)
0.105254 + 0.994445i \(0.466435\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 18.7321 1.20166
\(244\) −4.92820 −0.315496
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 9.85641 0.627148
\(248\) −2.00000 −0.127000
\(249\) −44.7846 −2.83811
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −4.46410 −0.281212
\(253\) 8.19615 0.515288
\(254\) 1.07180 0.0672505
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 23.6603 1.47589 0.737943 0.674863i \(-0.235797\pi\)
0.737943 + 0.674863i \(0.235797\pi\)
\(258\) −5.46410 −0.340180
\(259\) 0.732051 0.0454874
\(260\) 0 0
\(261\) −21.1244 −1.30756
\(262\) −5.66025 −0.349692
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 2.73205 0.168146
\(265\) 0 0
\(266\) 6.73205 0.412769
\(267\) −9.46410 −0.579194
\(268\) 4.00000 0.244339
\(269\) 28.3923 1.73111 0.865555 0.500814i \(-0.166966\pi\)
0.865555 + 0.500814i \(0.166966\pi\)
\(270\) 0 0
\(271\) 0.392305 0.0238308 0.0119154 0.999929i \(-0.496207\pi\)
0.0119154 + 0.999929i \(0.496207\pi\)
\(272\) 3.46410 0.210042
\(273\) 4.00000 0.242091
\(274\) −0.928203 −0.0560748
\(275\) 0 0
\(276\) −22.3923 −1.34786
\(277\) −7.07180 −0.424903 −0.212452 0.977172i \(-0.568145\pi\)
−0.212452 + 0.977172i \(0.568145\pi\)
\(278\) −13.6603 −0.819288
\(279\) 8.92820 0.534518
\(280\) 0 0
\(281\) 22.3923 1.33581 0.667906 0.744245i \(-0.267191\pi\)
0.667906 + 0.744245i \(0.267191\pi\)
\(282\) −18.9282 −1.12716
\(283\) 31.7128 1.88513 0.942566 0.334021i \(-0.108406\pi\)
0.942566 + 0.334021i \(0.108406\pi\)
\(284\) 9.46410 0.561591
\(285\) 0 0
\(286\) −1.46410 −0.0865741
\(287\) 2.19615 0.129635
\(288\) −4.46410 −0.263050
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 39.8564 2.33642
\(292\) 14.3923 0.842246
\(293\) −21.4641 −1.25395 −0.626973 0.779041i \(-0.715706\pi\)
−0.626973 + 0.779041i \(0.715706\pi\)
\(294\) 2.73205 0.159336
\(295\) 0 0
\(296\) 0.732051 0.0425496
\(297\) −4.00000 −0.232104
\(298\) 7.26795 0.421021
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) −11.1244 −0.640135
\(303\) 21.4641 1.23308
\(304\) 6.73205 0.386110
\(305\) 0 0
\(306\) −15.4641 −0.884024
\(307\) −24.3923 −1.39214 −0.696071 0.717973i \(-0.745070\pi\)
−0.696071 + 0.717973i \(0.745070\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 33.8564 1.92602
\(310\) 0 0
\(311\) 24.9282 1.41355 0.706774 0.707439i \(-0.250150\pi\)
0.706774 + 0.707439i \(0.250150\pi\)
\(312\) 4.00000 0.226455
\(313\) −22.1962 −1.25460 −0.627300 0.778777i \(-0.715840\pi\)
−0.627300 + 0.778777i \(0.715840\pi\)
\(314\) 4.53590 0.255976
\(315\) 0 0
\(316\) −12.1962 −0.686087
\(317\) −30.5885 −1.71802 −0.859009 0.511960i \(-0.828920\pi\)
−0.859009 + 0.511960i \(0.828920\pi\)
\(318\) 19.8564 1.11349
\(319\) −4.73205 −0.264944
\(320\) 0 0
\(321\) 21.4641 1.19801
\(322\) 8.19615 0.456754
\(323\) 23.3205 1.29759
\(324\) −2.46410 −0.136895
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 42.7846 2.36599
\(328\) 2.19615 0.121262
\(329\) 6.92820 0.381964
\(330\) 0 0
\(331\) −18.7846 −1.03250 −0.516248 0.856439i \(-0.672672\pi\)
−0.516248 + 0.856439i \(0.672672\pi\)
\(332\) 16.3923 0.899645
\(333\) −3.26795 −0.179083
\(334\) 13.8564 0.758189
\(335\) 0 0
\(336\) 2.73205 0.149046
\(337\) −22.7846 −1.24116 −0.620578 0.784144i \(-0.713102\pi\)
−0.620578 + 0.784144i \(0.713102\pi\)
\(338\) 10.8564 0.590511
\(339\) 21.4641 1.16577
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) −30.0526 −1.62506
\(343\) −1.00000 −0.0539949
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) 0.928203 0.0499005
\(347\) 23.0718 1.23856 0.619279 0.785171i \(-0.287425\pi\)
0.619279 + 0.785171i \(0.287425\pi\)
\(348\) 12.9282 0.693024
\(349\) −5.60770 −0.300173 −0.150087 0.988673i \(-0.547955\pi\)
−0.150087 + 0.988673i \(0.547955\pi\)
\(350\) 0 0
\(351\) −5.85641 −0.312592
\(352\) −1.00000 −0.0533002
\(353\) −14.1962 −0.755585 −0.377792 0.925890i \(-0.623317\pi\)
−0.377792 + 0.925890i \(0.623317\pi\)
\(354\) 18.9282 1.00602
\(355\) 0 0
\(356\) 3.46410 0.183597
\(357\) 9.46410 0.500893
\(358\) 6.00000 0.317110
\(359\) −34.0526 −1.79723 −0.898613 0.438743i \(-0.855424\pi\)
−0.898613 + 0.438743i \(0.855424\pi\)
\(360\) 0 0
\(361\) 26.3205 1.38529
\(362\) −14.0000 −0.735824
\(363\) −2.73205 −0.143395
\(364\) −1.46410 −0.0767398
\(365\) 0 0
\(366\) −13.4641 −0.703780
\(367\) −12.3923 −0.646873 −0.323437 0.946250i \(-0.604838\pi\)
−0.323437 + 0.946250i \(0.604838\pi\)
\(368\) 8.19615 0.427254
\(369\) −9.80385 −0.510368
\(370\) 0 0
\(371\) −7.26795 −0.377333
\(372\) −5.46410 −0.283300
\(373\) −18.3923 −0.952317 −0.476159 0.879359i \(-0.657971\pi\)
−0.476159 + 0.879359i \(0.657971\pi\)
\(374\) −3.46410 −0.179124
\(375\) 0 0
\(376\) 6.92820 0.357295
\(377\) −6.92820 −0.356821
\(378\) −4.00000 −0.205738
\(379\) 6.14359 0.315575 0.157788 0.987473i \(-0.449564\pi\)
0.157788 + 0.987473i \(0.449564\pi\)
\(380\) 0 0
\(381\) 2.92820 0.150016
\(382\) 12.0000 0.613973
\(383\) 33.4641 1.70994 0.854968 0.518681i \(-0.173577\pi\)
0.854968 + 0.518681i \(0.173577\pi\)
\(384\) 2.73205 0.139419
\(385\) 0 0
\(386\) 12.3923 0.630752
\(387\) −8.92820 −0.453846
\(388\) −14.5885 −0.740617
\(389\) −1.60770 −0.0815134 −0.0407567 0.999169i \(-0.512977\pi\)
−0.0407567 + 0.999169i \(0.512977\pi\)
\(390\) 0 0
\(391\) 28.3923 1.43586
\(392\) −1.00000 −0.0505076
\(393\) −15.4641 −0.780061
\(394\) −24.2487 −1.22163
\(395\) 0 0
\(396\) 4.46410 0.224330
\(397\) −30.3923 −1.52535 −0.762673 0.646784i \(-0.776113\pi\)
−0.762673 + 0.646784i \(0.776113\pi\)
\(398\) −2.92820 −0.146778
\(399\) 18.3923 0.920767
\(400\) 0 0
\(401\) 2.53590 0.126637 0.0633184 0.997993i \(-0.479832\pi\)
0.0633184 + 0.997993i \(0.479832\pi\)
\(402\) 10.9282 0.545049
\(403\) 2.92820 0.145864
\(404\) −7.85641 −0.390871
\(405\) 0 0
\(406\) −4.73205 −0.234848
\(407\) −0.732051 −0.0362864
\(408\) 9.46410 0.468543
\(409\) 10.8756 0.537766 0.268883 0.963173i \(-0.413346\pi\)
0.268883 + 0.963173i \(0.413346\pi\)
\(410\) 0 0
\(411\) −2.53590 −0.125087
\(412\) −12.3923 −0.610525
\(413\) −6.92820 −0.340915
\(414\) −36.5885 −1.79822
\(415\) 0 0
\(416\) −1.46410 −0.0717835
\(417\) −37.3205 −1.82759
\(418\) −6.73205 −0.329275
\(419\) 30.9282 1.51094 0.755471 0.655182i \(-0.227408\pi\)
0.755471 + 0.655182i \(0.227408\pi\)
\(420\) 0 0
\(421\) −35.8564 −1.74753 −0.873767 0.486344i \(-0.838330\pi\)
−0.873767 + 0.486344i \(0.838330\pi\)
\(422\) −26.9282 −1.31084
\(423\) −30.9282 −1.50378
\(424\) −7.26795 −0.352963
\(425\) 0 0
\(426\) 25.8564 1.25275
\(427\) 4.92820 0.238492
\(428\) −7.85641 −0.379754
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 3.12436 0.150495 0.0752475 0.997165i \(-0.476025\pi\)
0.0752475 + 0.997165i \(0.476025\pi\)
\(432\) −4.00000 −0.192450
\(433\) 18.1962 0.874451 0.437226 0.899352i \(-0.355961\pi\)
0.437226 + 0.899352i \(0.355961\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0 0
\(436\) −15.6603 −0.749990
\(437\) 55.1769 2.63947
\(438\) 39.3205 1.87881
\(439\) 26.2487 1.25278 0.626391 0.779509i \(-0.284531\pi\)
0.626391 + 0.779509i \(0.284531\pi\)
\(440\) 0 0
\(441\) 4.46410 0.212576
\(442\) −5.07180 −0.241241
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) −25.4641 −1.20576
\(447\) 19.8564 0.939176
\(448\) −1.00000 −0.0472456
\(449\) 40.3923 1.90623 0.953115 0.302607i \(-0.0978570\pi\)
0.953115 + 0.302607i \(0.0978570\pi\)
\(450\) 0 0
\(451\) −2.19615 −0.103413
\(452\) −7.85641 −0.369534
\(453\) −30.3923 −1.42796
\(454\) 6.92820 0.325157
\(455\) 0 0
\(456\) 18.3923 0.861299
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) −24.3923 −1.13978
\(459\) −13.8564 −0.646762
\(460\) 0 0
\(461\) 22.3923 1.04291 0.521457 0.853278i \(-0.325389\pi\)
0.521457 + 0.853278i \(0.325389\pi\)
\(462\) −2.73205 −0.127107
\(463\) −27.5167 −1.27881 −0.639404 0.768871i \(-0.720819\pi\)
−0.639404 + 0.768871i \(0.720819\pi\)
\(464\) −4.73205 −0.219680
\(465\) 0 0
\(466\) −7.85641 −0.363941
\(467\) 34.0526 1.57576 0.787882 0.615826i \(-0.211178\pi\)
0.787882 + 0.615826i \(0.211178\pi\)
\(468\) 6.53590 0.302122
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 12.3923 0.571007
\(472\) −6.92820 −0.318896
\(473\) −2.00000 −0.0919601
\(474\) −33.3205 −1.53046
\(475\) 0 0
\(476\) −3.46410 −0.158777
\(477\) 32.4449 1.48555
\(478\) −1.26795 −0.0579946
\(479\) −32.7846 −1.49797 −0.748984 0.662589i \(-0.769458\pi\)
−0.748984 + 0.662589i \(0.769458\pi\)
\(480\) 0 0
\(481\) −1.07180 −0.0488697
\(482\) −3.26795 −0.148851
\(483\) 22.3923 1.01889
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −18.7321 −0.849703
\(487\) −4.19615 −0.190146 −0.0950729 0.995470i \(-0.530308\pi\)
−0.0950729 + 0.995470i \(0.530308\pi\)
\(488\) 4.92820 0.223089
\(489\) −10.9282 −0.494190
\(490\) 0 0
\(491\) −27.7128 −1.25066 −0.625331 0.780360i \(-0.715036\pi\)
−0.625331 + 0.780360i \(0.715036\pi\)
\(492\) 6.00000 0.270501
\(493\) −16.3923 −0.738272
\(494\) −9.85641 −0.443461
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −9.46410 −0.424523
\(498\) 44.7846 2.00685
\(499\) 12.1436 0.543622 0.271811 0.962351i \(-0.412377\pi\)
0.271811 + 0.962351i \(0.412377\pi\)
\(500\) 0 0
\(501\) 37.8564 1.69130
\(502\) −12.0000 −0.535586
\(503\) −8.78461 −0.391686 −0.195843 0.980635i \(-0.562744\pi\)
−0.195843 + 0.980635i \(0.562744\pi\)
\(504\) 4.46410 0.198847
\(505\) 0 0
\(506\) −8.19615 −0.364363
\(507\) 29.6603 1.31726
\(508\) −1.07180 −0.0475533
\(509\) 11.0718 0.490749 0.245374 0.969428i \(-0.421089\pi\)
0.245374 + 0.969428i \(0.421089\pi\)
\(510\) 0 0
\(511\) −14.3923 −0.636678
\(512\) −1.00000 −0.0441942
\(513\) −26.9282 −1.18891
\(514\) −23.6603 −1.04361
\(515\) 0 0
\(516\) 5.46410 0.240544
\(517\) −6.92820 −0.304702
\(518\) −0.732051 −0.0321645
\(519\) 2.53590 0.111314
\(520\) 0 0
\(521\) 10.3923 0.455295 0.227648 0.973744i \(-0.426897\pi\)
0.227648 + 0.973744i \(0.426897\pi\)
\(522\) 21.1244 0.924588
\(523\) 29.1769 1.27582 0.637909 0.770112i \(-0.279800\pi\)
0.637909 + 0.770112i \(0.279800\pi\)
\(524\) 5.66025 0.247269
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 6.92820 0.301797
\(528\) −2.73205 −0.118897
\(529\) 44.1769 1.92074
\(530\) 0 0
\(531\) 30.9282 1.34217
\(532\) −6.73205 −0.291871
\(533\) −3.21539 −0.139274
\(534\) 9.46410 0.409552
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 16.3923 0.707380
\(538\) −28.3923 −1.22408
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −20.7321 −0.891340 −0.445670 0.895197i \(-0.647035\pi\)
−0.445670 + 0.895197i \(0.647035\pi\)
\(542\) −0.392305 −0.0168509
\(543\) −38.2487 −1.64141
\(544\) −3.46410 −0.148522
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) −28.7846 −1.23074 −0.615371 0.788238i \(-0.710994\pi\)
−0.615371 + 0.788238i \(0.710994\pi\)
\(548\) 0.928203 0.0396509
\(549\) −22.0000 −0.938937
\(550\) 0 0
\(551\) −31.8564 −1.35713
\(552\) 22.3923 0.953080
\(553\) 12.1962 0.518633
\(554\) 7.07180 0.300452
\(555\) 0 0
\(556\) 13.6603 0.579324
\(557\) 25.6077 1.08503 0.542516 0.840045i \(-0.317472\pi\)
0.542516 + 0.840045i \(0.317472\pi\)
\(558\) −8.92820 −0.377961
\(559\) −2.92820 −0.123850
\(560\) 0 0
\(561\) −9.46410 −0.399575
\(562\) −22.3923 −0.944562
\(563\) −5.07180 −0.213751 −0.106875 0.994272i \(-0.534085\pi\)
−0.106875 + 0.994272i \(0.534085\pi\)
\(564\) 18.9282 0.797021
\(565\) 0 0
\(566\) −31.7128 −1.33299
\(567\) 2.46410 0.103483
\(568\) −9.46410 −0.397105
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 3.60770 0.150977 0.0754887 0.997147i \(-0.475948\pi\)
0.0754887 + 0.997147i \(0.475948\pi\)
\(572\) 1.46410 0.0612172
\(573\) 32.7846 1.36960
\(574\) −2.19615 −0.0916656
\(575\) 0 0
\(576\) 4.46410 0.186004
\(577\) 34.5885 1.43994 0.719968 0.694007i \(-0.244156\pi\)
0.719968 + 0.694007i \(0.244156\pi\)
\(578\) 5.00000 0.207973
\(579\) 33.8564 1.40702
\(580\) 0 0
\(581\) −16.3923 −0.680067
\(582\) −39.8564 −1.65210
\(583\) 7.26795 0.301008
\(584\) −14.3923 −0.595558
\(585\) 0 0
\(586\) 21.4641 0.886674
\(587\) 11.4115 0.471005 0.235502 0.971874i \(-0.424326\pi\)
0.235502 + 0.971874i \(0.424326\pi\)
\(588\) −2.73205 −0.112668
\(589\) 13.4641 0.554779
\(590\) 0 0
\(591\) −66.2487 −2.72511
\(592\) −0.732051 −0.0300871
\(593\) −0.248711 −0.0102133 −0.00510667 0.999987i \(-0.501626\pi\)
−0.00510667 + 0.999987i \(0.501626\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −7.26795 −0.297707
\(597\) −8.00000 −0.327418
\(598\) −12.0000 −0.490716
\(599\) 37.1769 1.51901 0.759504 0.650503i \(-0.225442\pi\)
0.759504 + 0.650503i \(0.225442\pi\)
\(600\) 0 0
\(601\) −5.51666 −0.225029 −0.112515 0.993650i \(-0.535891\pi\)
−0.112515 + 0.993650i \(0.535891\pi\)
\(602\) −2.00000 −0.0815139
\(603\) 17.8564 0.727169
\(604\) 11.1244 0.452644
\(605\) 0 0
\(606\) −21.4641 −0.871920
\(607\) −20.9282 −0.849450 −0.424725 0.905323i \(-0.639629\pi\)
−0.424725 + 0.905323i \(0.639629\pi\)
\(608\) −6.73205 −0.273021
\(609\) −12.9282 −0.523877
\(610\) 0 0
\(611\) −10.1436 −0.410366
\(612\) 15.4641 0.625099
\(613\) 35.1769 1.42078 0.710391 0.703807i \(-0.248518\pi\)
0.710391 + 0.703807i \(0.248518\pi\)
\(614\) 24.3923 0.984393
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) −17.3205 −0.697297 −0.348649 0.937253i \(-0.613359\pi\)
−0.348649 + 0.937253i \(0.613359\pi\)
\(618\) −33.8564 −1.36190
\(619\) 28.7846 1.15695 0.578476 0.815700i \(-0.303648\pi\)
0.578476 + 0.815700i \(0.303648\pi\)
\(620\) 0 0
\(621\) −32.7846 −1.31560
\(622\) −24.9282 −0.999530
\(623\) −3.46410 −0.138786
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) 22.1962 0.887137
\(627\) −18.3923 −0.734518
\(628\) −4.53590 −0.181002
\(629\) −2.53590 −0.101113
\(630\) 0 0
\(631\) −46.9282 −1.86818 −0.934091 0.357035i \(-0.883788\pi\)
−0.934091 + 0.357035i \(0.883788\pi\)
\(632\) 12.1962 0.485137
\(633\) −73.5692 −2.92411
\(634\) 30.5885 1.21482
\(635\) 0 0
\(636\) −19.8564 −0.787358
\(637\) 1.46410 0.0580098
\(638\) 4.73205 0.187344
\(639\) 42.2487 1.67133
\(640\) 0 0
\(641\) −35.5692 −1.40490 −0.702450 0.711733i \(-0.747910\pi\)
−0.702450 + 0.711733i \(0.747910\pi\)
\(642\) −21.4641 −0.847121
\(643\) −33.2679 −1.31196 −0.655980 0.754778i \(-0.727744\pi\)
−0.655980 + 0.754778i \(0.727744\pi\)
\(644\) −8.19615 −0.322974
\(645\) 0 0
\(646\) −23.3205 −0.917533
\(647\) 26.5359 1.04323 0.521617 0.853180i \(-0.325329\pi\)
0.521617 + 0.853180i \(0.325329\pi\)
\(648\) 2.46410 0.0967991
\(649\) 6.92820 0.271956
\(650\) 0 0
\(651\) 5.46410 0.214155
\(652\) 4.00000 0.156652
\(653\) −11.6603 −0.456301 −0.228151 0.973626i \(-0.573268\pi\)
−0.228151 + 0.973626i \(0.573268\pi\)
\(654\) −42.7846 −1.67301
\(655\) 0 0
\(656\) −2.19615 −0.0857453
\(657\) 64.2487 2.50658
\(658\) −6.92820 −0.270089
\(659\) 30.2487 1.17832 0.589161 0.808015i \(-0.299458\pi\)
0.589161 + 0.808015i \(0.299458\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) 18.7846 0.730085
\(663\) −13.8564 −0.538138
\(664\) −16.3923 −0.636145
\(665\) 0 0
\(666\) 3.26795 0.126630
\(667\) −38.7846 −1.50175
\(668\) −13.8564 −0.536120
\(669\) −69.5692 −2.68970
\(670\) 0 0
\(671\) −4.92820 −0.190251
\(672\) −2.73205 −0.105391
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 22.7846 0.877630
\(675\) 0 0
\(676\) −10.8564 −0.417554
\(677\) 4.14359 0.159251 0.0796256 0.996825i \(-0.474628\pi\)
0.0796256 + 0.996825i \(0.474628\pi\)
\(678\) −21.4641 −0.824324
\(679\) 14.5885 0.559854
\(680\) 0 0
\(681\) 18.9282 0.725330
\(682\) −2.00000 −0.0765840
\(683\) −6.24871 −0.239100 −0.119550 0.992828i \(-0.538145\pi\)
−0.119550 + 0.992828i \(0.538145\pi\)
\(684\) 30.0526 1.14909
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) −66.6410 −2.54251
\(688\) −2.00000 −0.0762493
\(689\) 10.6410 0.405390
\(690\) 0 0
\(691\) −13.4641 −0.512199 −0.256099 0.966650i \(-0.582437\pi\)
−0.256099 + 0.966650i \(0.582437\pi\)
\(692\) −0.928203 −0.0352850
\(693\) −4.46410 −0.169577
\(694\) −23.0718 −0.875793
\(695\) 0 0
\(696\) −12.9282 −0.490042
\(697\) −7.60770 −0.288162
\(698\) 5.60770 0.212254
\(699\) −21.4641 −0.811847
\(700\) 0 0
\(701\) 41.9090 1.58288 0.791440 0.611247i \(-0.209332\pi\)
0.791440 + 0.611247i \(0.209332\pi\)
\(702\) 5.85641 0.221036
\(703\) −4.92820 −0.185871
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 14.1962 0.534279
\(707\) 7.85641 0.295471
\(708\) −18.9282 −0.711365
\(709\) 25.3205 0.950932 0.475466 0.879734i \(-0.342280\pi\)
0.475466 + 0.879734i \(0.342280\pi\)
\(710\) 0 0
\(711\) −54.4449 −2.04184
\(712\) −3.46410 −0.129823
\(713\) 16.3923 0.613897
\(714\) −9.46410 −0.354185
\(715\) 0 0
\(716\) −6.00000 −0.224231
\(717\) −3.46410 −0.129369
\(718\) 34.0526 1.27083
\(719\) −27.7128 −1.03351 −0.516757 0.856132i \(-0.672861\pi\)
−0.516757 + 0.856132i \(0.672861\pi\)
\(720\) 0 0
\(721\) 12.3923 0.461514
\(722\) −26.3205 −0.979548
\(723\) −8.92820 −0.332043
\(724\) 14.0000 0.520306
\(725\) 0 0
\(726\) 2.73205 0.101396
\(727\) 4.00000 0.148352 0.0741759 0.997245i \(-0.476367\pi\)
0.0741759 + 0.997245i \(0.476367\pi\)
\(728\) 1.46410 0.0542632
\(729\) −43.7846 −1.62165
\(730\) 0 0
\(731\) −6.92820 −0.256249
\(732\) 13.4641 0.497648
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) 12.3923 0.457408
\(735\) 0 0
\(736\) −8.19615 −0.302114
\(737\) 4.00000 0.147342
\(738\) 9.80385 0.360885
\(739\) 11.7128 0.430863 0.215431 0.976519i \(-0.430884\pi\)
0.215431 + 0.976519i \(0.430884\pi\)
\(740\) 0 0
\(741\) −26.9282 −0.989232
\(742\) 7.26795 0.266815
\(743\) 32.7846 1.20275 0.601375 0.798967i \(-0.294620\pi\)
0.601375 + 0.798967i \(0.294620\pi\)
\(744\) 5.46410 0.200324
\(745\) 0 0
\(746\) 18.3923 0.673390
\(747\) 73.1769 2.67740
\(748\) 3.46410 0.126660
\(749\) 7.85641 0.287067
\(750\) 0 0
\(751\) −11.6077 −0.423571 −0.211785 0.977316i \(-0.567928\pi\)
−0.211785 + 0.977316i \(0.567928\pi\)
\(752\) −6.92820 −0.252646
\(753\) −32.7846 −1.19474
\(754\) 6.92820 0.252310
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 43.3731 1.57642 0.788210 0.615406i \(-0.211008\pi\)
0.788210 + 0.615406i \(0.211008\pi\)
\(758\) −6.14359 −0.223145
\(759\) −22.3923 −0.812789
\(760\) 0 0
\(761\) −12.3397 −0.447315 −0.223658 0.974668i \(-0.571800\pi\)
−0.223658 + 0.974668i \(0.571800\pi\)
\(762\) −2.92820 −0.106078
\(763\) 15.6603 0.566939
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −33.4641 −1.20911
\(767\) 10.1436 0.366264
\(768\) −2.73205 −0.0985844
\(769\) −18.1962 −0.656170 −0.328085 0.944648i \(-0.606403\pi\)
−0.328085 + 0.944648i \(0.606403\pi\)
\(770\) 0 0
\(771\) −64.6410 −2.32799
\(772\) −12.3923 −0.446009
\(773\) 0.928203 0.0333851 0.0166926 0.999861i \(-0.494686\pi\)
0.0166926 + 0.999861i \(0.494686\pi\)
\(774\) 8.92820 0.320918
\(775\) 0 0
\(776\) 14.5885 0.523695
\(777\) −2.00000 −0.0717496
\(778\) 1.60770 0.0576387
\(779\) −14.7846 −0.529714
\(780\) 0 0
\(781\) 9.46410 0.338652
\(782\) −28.3923 −1.01531
\(783\) 18.9282 0.676439
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 15.4641 0.551586
\(787\) −18.1436 −0.646749 −0.323375 0.946271i \(-0.604817\pi\)
−0.323375 + 0.946271i \(0.604817\pi\)
\(788\) 24.2487 0.863825
\(789\) −65.5692 −2.33433
\(790\) 0 0
\(791\) 7.85641 0.279342
\(792\) −4.46410 −0.158625
\(793\) −7.21539 −0.256226
\(794\) 30.3923 1.07858
\(795\) 0 0
\(796\) 2.92820 0.103787
\(797\) 25.6077 0.907071 0.453536 0.891238i \(-0.350163\pi\)
0.453536 + 0.891238i \(0.350163\pi\)
\(798\) −18.3923 −0.651081
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 15.4641 0.546397
\(802\) −2.53590 −0.0895457
\(803\) 14.3923 0.507893
\(804\) −10.9282 −0.385408
\(805\) 0 0
\(806\) −2.92820 −0.103142
\(807\) −77.5692 −2.73057
\(808\) 7.85641 0.276387
\(809\) −31.1769 −1.09612 −0.548061 0.836438i \(-0.684634\pi\)
−0.548061 + 0.836438i \(0.684634\pi\)
\(810\) 0 0
\(811\) −43.1244 −1.51430 −0.757150 0.653241i \(-0.773409\pi\)
−0.757150 + 0.653241i \(0.773409\pi\)
\(812\) 4.73205 0.166062
\(813\) −1.07180 −0.0375896
\(814\) 0.732051 0.0256584
\(815\) 0 0
\(816\) −9.46410 −0.331310
\(817\) −13.4641 −0.471049
\(818\) −10.8756 −0.380258
\(819\) −6.53590 −0.228383
\(820\) 0 0
\(821\) 17.9090 0.625027 0.312514 0.949913i \(-0.398829\pi\)
0.312514 + 0.949913i \(0.398829\pi\)
\(822\) 2.53590 0.0884496
\(823\) 0.875644 0.0305230 0.0152615 0.999884i \(-0.495142\pi\)
0.0152615 + 0.999884i \(0.495142\pi\)
\(824\) 12.3923 0.431706
\(825\) 0 0
\(826\) 6.92820 0.241063
\(827\) 25.8564 0.899115 0.449558 0.893251i \(-0.351582\pi\)
0.449558 + 0.893251i \(0.351582\pi\)
\(828\) 36.5885 1.27154
\(829\) 2.24871 0.0781010 0.0390505 0.999237i \(-0.487567\pi\)
0.0390505 + 0.999237i \(0.487567\pi\)
\(830\) 0 0
\(831\) 19.3205 0.670221
\(832\) 1.46410 0.0507586
\(833\) 3.46410 0.120024
\(834\) 37.3205 1.29230
\(835\) 0 0
\(836\) 6.73205 0.232833
\(837\) −8.00000 −0.276520
\(838\) −30.9282 −1.06840
\(839\) −31.8564 −1.09981 −0.549903 0.835229i \(-0.685335\pi\)
−0.549903 + 0.835229i \(0.685335\pi\)
\(840\) 0 0
\(841\) −6.60770 −0.227852
\(842\) 35.8564 1.23569
\(843\) −61.1769 −2.10704
\(844\) 26.9282 0.926907
\(845\) 0 0
\(846\) 30.9282 1.06333
\(847\) −1.00000 −0.0343604
\(848\) 7.26795 0.249582
\(849\) −86.6410 −2.97351
\(850\) 0 0
\(851\) −6.00000 −0.205677
\(852\) −25.8564 −0.885826
\(853\) 54.7846 1.87579 0.937895 0.346920i \(-0.112773\pi\)
0.937895 + 0.346920i \(0.112773\pi\)
\(854\) −4.92820 −0.168640
\(855\) 0 0
\(856\) 7.85641 0.268526
\(857\) 27.4641 0.938156 0.469078 0.883157i \(-0.344586\pi\)
0.469078 + 0.883157i \(0.344586\pi\)
\(858\) 4.00000 0.136558
\(859\) −12.7846 −0.436205 −0.218103 0.975926i \(-0.569987\pi\)
−0.218103 + 0.975926i \(0.569987\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) −3.12436 −0.106416
\(863\) 7.51666 0.255870 0.127935 0.991783i \(-0.459165\pi\)
0.127935 + 0.991783i \(0.459165\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) −18.1962 −0.618330
\(867\) 13.6603 0.463927
\(868\) −2.00000 −0.0678844
\(869\) −12.1962 −0.413726
\(870\) 0 0
\(871\) 5.85641 0.198437
\(872\) 15.6603 0.530323
\(873\) −65.1244 −2.20413
\(874\) −55.1769 −1.86639
\(875\) 0 0
\(876\) −39.3205 −1.32852
\(877\) 21.3205 0.719942 0.359971 0.932963i \(-0.382787\pi\)
0.359971 + 0.932963i \(0.382787\pi\)
\(878\) −26.2487 −0.885851
\(879\) 58.6410 1.97791
\(880\) 0 0
\(881\) 11.0718 0.373018 0.186509 0.982453i \(-0.440283\pi\)
0.186509 + 0.982453i \(0.440283\pi\)
\(882\) −4.46410 −0.150314
\(883\) 35.6077 1.19829 0.599147 0.800639i \(-0.295506\pi\)
0.599147 + 0.800639i \(0.295506\pi\)
\(884\) 5.07180 0.170583
\(885\) 0 0
\(886\) 0 0
\(887\) −13.8564 −0.465253 −0.232626 0.972566i \(-0.574732\pi\)
−0.232626 + 0.972566i \(0.574732\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 1.07180 0.0359469
\(890\) 0 0
\(891\) −2.46410 −0.0825505
\(892\) 25.4641 0.852601
\(893\) −46.6410 −1.56078
\(894\) −19.8564 −0.664098
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) −32.7846 −1.09465
\(898\) −40.3923 −1.34791
\(899\) −9.46410 −0.315645
\(900\) 0 0
\(901\) 25.1769 0.838765
\(902\) 2.19615 0.0731239
\(903\) −5.46410 −0.181834
\(904\) 7.85641 0.261300
\(905\) 0 0
\(906\) 30.3923 1.00972
\(907\) 31.0333 1.03044 0.515222 0.857057i \(-0.327709\pi\)
0.515222 + 0.857057i \(0.327709\pi\)
\(908\) −6.92820 −0.229920
\(909\) −35.0718 −1.16326
\(910\) 0 0
\(911\) 9.46410 0.313560 0.156780 0.987634i \(-0.449889\pi\)
0.156780 + 0.987634i \(0.449889\pi\)
\(912\) −18.3923 −0.609030
\(913\) 16.3923 0.542506
\(914\) 2.00000 0.0661541
\(915\) 0 0
\(916\) 24.3923 0.805944
\(917\) −5.66025 −0.186918
\(918\) 13.8564 0.457330
\(919\) −25.3731 −0.836980 −0.418490 0.908221i \(-0.637441\pi\)
−0.418490 + 0.908221i \(0.637441\pi\)
\(920\) 0 0
\(921\) 66.6410 2.19590
\(922\) −22.3923 −0.737451
\(923\) 13.8564 0.456089
\(924\) 2.73205 0.0898779
\(925\) 0 0
\(926\) 27.5167 0.904254
\(927\) −55.3205 −1.81696
\(928\) 4.73205 0.155337
\(929\) −25.6077 −0.840161 −0.420081 0.907487i \(-0.637998\pi\)
−0.420081 + 0.907487i \(0.637998\pi\)
\(930\) 0 0
\(931\) 6.73205 0.220634
\(932\) 7.85641 0.257345
\(933\) −68.1051 −2.22966
\(934\) −34.0526 −1.11423
\(935\) 0 0
\(936\) −6.53590 −0.213633
\(937\) 1.21539 0.0397051 0.0198525 0.999803i \(-0.493680\pi\)
0.0198525 + 0.999803i \(0.493680\pi\)
\(938\) 4.00000 0.130605
\(939\) 60.6410 1.97894
\(940\) 0 0
\(941\) −14.7846 −0.481965 −0.240982 0.970530i \(-0.577470\pi\)
−0.240982 + 0.970530i \(0.577470\pi\)
\(942\) −12.3923 −0.403763
\(943\) −18.0000 −0.586161
\(944\) 6.92820 0.225494
\(945\) 0 0
\(946\) 2.00000 0.0650256
\(947\) −47.3205 −1.53771 −0.768855 0.639423i \(-0.779173\pi\)
−0.768855 + 0.639423i \(0.779173\pi\)
\(948\) 33.3205 1.08220
\(949\) 21.0718 0.684019
\(950\) 0 0
\(951\) 83.5692 2.70992
\(952\) 3.46410 0.112272
\(953\) −26.5359 −0.859582 −0.429791 0.902928i \(-0.641413\pi\)
−0.429791 + 0.902928i \(0.641413\pi\)
\(954\) −32.4449 −1.05044
\(955\) 0 0
\(956\) 1.26795 0.0410084
\(957\) 12.9282 0.417909
\(958\) 32.7846 1.05922
\(959\) −0.928203 −0.0299732
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 1.07180 0.0345561
\(963\) −35.0718 −1.13017
\(964\) 3.26795 0.105254
\(965\) 0 0
\(966\) −22.3923 −0.720461
\(967\) −26.9282 −0.865953 −0.432976 0.901405i \(-0.642537\pi\)
−0.432976 + 0.901405i \(0.642537\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −63.7128 −2.04675
\(970\) 0 0
\(971\) 42.9282 1.37763 0.688816 0.724936i \(-0.258131\pi\)
0.688816 + 0.724936i \(0.258131\pi\)
\(972\) 18.7321 0.600831
\(973\) −13.6603 −0.437928
\(974\) 4.19615 0.134453
\(975\) 0 0
\(976\) −4.92820 −0.157748
\(977\) 33.7128 1.07857 0.539284 0.842124i \(-0.318695\pi\)
0.539284 + 0.842124i \(0.318695\pi\)
\(978\) 10.9282 0.349445
\(979\) 3.46410 0.110713
\(980\) 0 0
\(981\) −69.9090 −2.23202
\(982\) 27.7128 0.884351
\(983\) 37.1769 1.18576 0.592880 0.805291i \(-0.297991\pi\)
0.592880 + 0.805291i \(0.297991\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 16.3923 0.522037
\(987\) −18.9282 −0.602491
\(988\) 9.85641 0.313574
\(989\) −16.3923 −0.521245
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 51.3205 1.62861
\(994\) 9.46410 0.300183
\(995\) 0 0
\(996\) −44.7846 −1.41905
\(997\) −17.7128 −0.560970 −0.280485 0.959858i \(-0.590495\pi\)
−0.280485 + 0.959858i \(0.590495\pi\)
\(998\) −12.1436 −0.384399
\(999\) 2.92820 0.0926443
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 3850.2.a.bd.1.1 2
5.2 odd 4 3850.2.c.x.1849.2 4
5.3 odd 4 3850.2.c.x.1849.3 4
5.4 even 2 770.2.a.j.1.2 2
15.14 odd 2 6930.2.a.bv.1.1 2
20.19 odd 2 6160.2.a.t.1.1 2
35.34 odd 2 5390.2.a.bs.1.1 2
55.54 odd 2 8470.2.a.br.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.j.1.2 2 5.4 even 2
3850.2.a.bd.1.1 2 1.1 even 1 trivial
3850.2.c.x.1849.2 4 5.2 odd 4
3850.2.c.x.1849.3 4 5.3 odd 4
5390.2.a.bs.1.1 2 35.34 odd 2
6160.2.a.t.1.1 2 20.19 odd 2
6930.2.a.bv.1.1 2 15.14 odd 2
8470.2.a.br.1.2 2 55.54 odd 2