Properties

Label 2394.2.a.x.1.2
Level $2394$
Weight $2$
Character 2394.1
Self dual yes
Analytic conductor $19.116$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 798)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 2394.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.46410 q^{10} +5.46410 q^{11} -3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -7.46410 q^{17} +1.00000 q^{19} +3.46410 q^{20} +5.46410 q^{22} +5.46410 q^{23} +7.00000 q^{25} -3.46410 q^{26} -1.00000 q^{28} +6.00000 q^{29} +6.92820 q^{31} +1.00000 q^{32} -7.46410 q^{34} -3.46410 q^{35} +10.0000 q^{37} +1.00000 q^{38} +3.46410 q^{40} +2.00000 q^{41} -6.92820 q^{43} +5.46410 q^{44} +5.46410 q^{46} -10.9282 q^{47} +1.00000 q^{49} +7.00000 q^{50} -3.46410 q^{52} -2.00000 q^{53} +18.9282 q^{55} -1.00000 q^{56} +6.00000 q^{58} -4.00000 q^{59} -4.92820 q^{61} +6.92820 q^{62} +1.00000 q^{64} -12.0000 q^{65} +9.46410 q^{67} -7.46410 q^{68} -3.46410 q^{70} -2.92820 q^{71} +10.0000 q^{73} +10.0000 q^{74} +1.00000 q^{76} -5.46410 q^{77} -12.3923 q^{79} +3.46410 q^{80} +2.00000 q^{82} -8.00000 q^{83} -25.8564 q^{85} -6.92820 q^{86} +5.46410 q^{88} +2.00000 q^{89} +3.46410 q^{91} +5.46410 q^{92} -10.9282 q^{94} +3.46410 q^{95} +4.53590 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 4 q^{11} - 2 q^{14} + 2 q^{16} - 8 q^{17} + 2 q^{19} + 4 q^{22} + 4 q^{23} + 14 q^{25} - 2 q^{28} + 12 q^{29} + 2 q^{32} - 8 q^{34} + 20 q^{37} + 2 q^{38} + 4 q^{41} + 4 q^{44} + 4 q^{46} - 8 q^{47} + 2 q^{49} + 14 q^{50} - 4 q^{53} + 24 q^{55} - 2 q^{56} + 12 q^{58} - 8 q^{59} + 4 q^{61} + 2 q^{64} - 24 q^{65} + 12 q^{67} - 8 q^{68} + 8 q^{71} + 20 q^{73} + 20 q^{74} + 2 q^{76} - 4 q^{77} - 4 q^{79} + 4 q^{82} - 16 q^{83} - 24 q^{85} + 4 q^{88} + 4 q^{89} + 4 q^{92} - 8 q^{94} + 16 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 3.46410 1.54919 0.774597 0.632456i \(-0.217953\pi\)
0.774597 + 0.632456i \(0.217953\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 3.46410 1.09545
\(11\) 5.46410 1.64749 0.823744 0.566961i \(-0.191881\pi\)
0.823744 + 0.566961i \(0.191881\pi\)
\(12\) 0 0
\(13\) −3.46410 −0.960769 −0.480384 0.877058i \(-0.659503\pi\)
−0.480384 + 0.877058i \(0.659503\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −7.46410 −1.81031 −0.905155 0.425081i \(-0.860246\pi\)
−0.905155 + 0.425081i \(0.860246\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 3.46410 0.774597
\(21\) 0 0
\(22\) 5.46410 1.16495
\(23\) 5.46410 1.13934 0.569672 0.821872i \(-0.307070\pi\)
0.569672 + 0.821872i \(0.307070\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) −3.46410 −0.679366
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 6.92820 1.24434 0.622171 0.782881i \(-0.286251\pi\)
0.622171 + 0.782881i \(0.286251\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −7.46410 −1.28008
\(35\) −3.46410 −0.585540
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 3.46410 0.547723
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −6.92820 −1.05654 −0.528271 0.849076i \(-0.677159\pi\)
−0.528271 + 0.849076i \(0.677159\pi\)
\(44\) 5.46410 0.823744
\(45\) 0 0
\(46\) 5.46410 0.805638
\(47\) −10.9282 −1.59404 −0.797021 0.603951i \(-0.793592\pi\)
−0.797021 + 0.603951i \(0.793592\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 7.00000 0.989949
\(51\) 0 0
\(52\) −3.46410 −0.480384
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 18.9282 2.55228
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −4.92820 −0.630992 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(62\) 6.92820 0.879883
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) 9.46410 1.15622 0.578112 0.815957i \(-0.303790\pi\)
0.578112 + 0.815957i \(0.303790\pi\)
\(68\) −7.46410 −0.905155
\(69\) 0 0
\(70\) −3.46410 −0.414039
\(71\) −2.92820 −0.347514 −0.173757 0.984789i \(-0.555591\pi\)
−0.173757 + 0.984789i \(0.555591\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −5.46410 −0.622692
\(78\) 0 0
\(79\) −12.3923 −1.39424 −0.697122 0.716953i \(-0.745536\pi\)
−0.697122 + 0.716953i \(0.745536\pi\)
\(80\) 3.46410 0.387298
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) −25.8564 −2.80452
\(86\) −6.92820 −0.747087
\(87\) 0 0
\(88\) 5.46410 0.582475
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 3.46410 0.363137
\(92\) 5.46410 0.569672
\(93\) 0 0
\(94\) −10.9282 −1.12716
\(95\) 3.46410 0.355409
\(96\) 0 0
\(97\) 4.53590 0.460551 0.230275 0.973126i \(-0.426037\pi\)
0.230275 + 0.973126i \(0.426037\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 7.00000 0.700000
\(101\) −4.53590 −0.451339 −0.225669 0.974204i \(-0.572457\pi\)
−0.225669 + 0.974204i \(0.572457\pi\)
\(102\) 0 0
\(103\) −6.92820 −0.682656 −0.341328 0.939944i \(-0.610877\pi\)
−0.341328 + 0.939944i \(0.610877\pi\)
\(104\) −3.46410 −0.339683
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 18.9282 1.80473
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −11.8564 −1.11536 −0.557678 0.830057i \(-0.688308\pi\)
−0.557678 + 0.830057i \(0.688308\pi\)
\(114\) 0 0
\(115\) 18.9282 1.76506
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) 7.46410 0.684233
\(120\) 0 0
\(121\) 18.8564 1.71422
\(122\) −4.92820 −0.446179
\(123\) 0 0
\(124\) 6.92820 0.622171
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) −17.4641 −1.54969 −0.774844 0.632152i \(-0.782172\pi\)
−0.774844 + 0.632152i \(0.782172\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −12.0000 −1.05247
\(131\) 5.07180 0.443125 0.221562 0.975146i \(-0.428884\pi\)
0.221562 + 0.975146i \(0.428884\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 9.46410 0.817574
\(135\) 0 0
\(136\) −7.46410 −0.640041
\(137\) 0.928203 0.0793018 0.0396509 0.999214i \(-0.487375\pi\)
0.0396509 + 0.999214i \(0.487375\pi\)
\(138\) 0 0
\(139\) 9.85641 0.836009 0.418005 0.908445i \(-0.362730\pi\)
0.418005 + 0.908445i \(0.362730\pi\)
\(140\) −3.46410 −0.292770
\(141\) 0 0
\(142\) −2.92820 −0.245729
\(143\) −18.9282 −1.58286
\(144\) 0 0
\(145\) 20.7846 1.72607
\(146\) 10.0000 0.827606
\(147\) 0 0
\(148\) 10.0000 0.821995
\(149\) 3.07180 0.251651 0.125826 0.992052i \(-0.459842\pi\)
0.125826 + 0.992052i \(0.459842\pi\)
\(150\) 0 0
\(151\) 1.46410 0.119147 0.0595734 0.998224i \(-0.481026\pi\)
0.0595734 + 0.998224i \(0.481026\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) −5.46410 −0.440310
\(155\) 24.0000 1.92773
\(156\) 0 0
\(157\) 19.8564 1.58471 0.792357 0.610058i \(-0.208854\pi\)
0.792357 + 0.610058i \(0.208854\pi\)
\(158\) −12.3923 −0.985879
\(159\) 0 0
\(160\) 3.46410 0.273861
\(161\) −5.46410 −0.430632
\(162\) 0 0
\(163\) 1.07180 0.0839496 0.0419748 0.999119i \(-0.486635\pi\)
0.0419748 + 0.999119i \(0.486635\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) 18.9282 1.46471 0.732354 0.680924i \(-0.238422\pi\)
0.732354 + 0.680924i \(0.238422\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) −25.8564 −1.98310
\(171\) 0 0
\(172\) −6.92820 −0.528271
\(173\) 0.928203 0.0705700 0.0352850 0.999377i \(-0.488766\pi\)
0.0352850 + 0.999377i \(0.488766\pi\)
\(174\) 0 0
\(175\) −7.00000 −0.529150
\(176\) 5.46410 0.411872
\(177\) 0 0
\(178\) 2.00000 0.149906
\(179\) 9.85641 0.736702 0.368351 0.929687i \(-0.379922\pi\)
0.368351 + 0.929687i \(0.379922\pi\)
\(180\) 0 0
\(181\) −8.53590 −0.634468 −0.317234 0.948347i \(-0.602754\pi\)
−0.317234 + 0.948347i \(0.602754\pi\)
\(182\) 3.46410 0.256776
\(183\) 0 0
\(184\) 5.46410 0.402819
\(185\) 34.6410 2.54686
\(186\) 0 0
\(187\) −40.7846 −2.98247
\(188\) −10.9282 −0.797021
\(189\) 0 0
\(190\) 3.46410 0.251312
\(191\) 5.46410 0.395369 0.197684 0.980266i \(-0.436658\pi\)
0.197684 + 0.980266i \(0.436658\pi\)
\(192\) 0 0
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 4.53590 0.325659
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −4.92820 −0.351120 −0.175560 0.984469i \(-0.556174\pi\)
−0.175560 + 0.984469i \(0.556174\pi\)
\(198\) 0 0
\(199\) −24.7846 −1.75693 −0.878467 0.477803i \(-0.841433\pi\)
−0.878467 + 0.477803i \(0.841433\pi\)
\(200\) 7.00000 0.494975
\(201\) 0 0
\(202\) −4.53590 −0.319145
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 6.92820 0.483887
\(206\) −6.92820 −0.482711
\(207\) 0 0
\(208\) −3.46410 −0.240192
\(209\) 5.46410 0.377960
\(210\) 0 0
\(211\) 1.46410 0.100793 0.0503965 0.998729i \(-0.483952\pi\)
0.0503965 + 0.998729i \(0.483952\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) −24.0000 −1.63679
\(216\) 0 0
\(217\) −6.92820 −0.470317
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) 18.9282 1.27614
\(221\) 25.8564 1.73929
\(222\) 0 0
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −11.8564 −0.788676
\(227\) −14.9282 −0.990820 −0.495410 0.868659i \(-0.664982\pi\)
−0.495410 + 0.868659i \(0.664982\pi\)
\(228\) 0 0
\(229\) 3.07180 0.202990 0.101495 0.994836i \(-0.467637\pi\)
0.101495 + 0.994836i \(0.467637\pi\)
\(230\) 18.9282 1.24809
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −26.7846 −1.75472 −0.877359 0.479834i \(-0.840697\pi\)
−0.877359 + 0.479834i \(0.840697\pi\)
\(234\) 0 0
\(235\) −37.8564 −2.46948
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 7.46410 0.483826
\(239\) −30.2487 −1.95663 −0.978313 0.207131i \(-0.933587\pi\)
−0.978313 + 0.207131i \(0.933587\pi\)
\(240\) 0 0
\(241\) 21.3205 1.37337 0.686687 0.726953i \(-0.259064\pi\)
0.686687 + 0.726953i \(0.259064\pi\)
\(242\) 18.8564 1.21214
\(243\) 0 0
\(244\) −4.92820 −0.315496
\(245\) 3.46410 0.221313
\(246\) 0 0
\(247\) −3.46410 −0.220416
\(248\) 6.92820 0.439941
\(249\) 0 0
\(250\) 6.92820 0.438178
\(251\) −5.07180 −0.320129 −0.160064 0.987107i \(-0.551170\pi\)
−0.160064 + 0.987107i \(0.551170\pi\)
\(252\) 0 0
\(253\) 29.8564 1.87706
\(254\) −17.4641 −1.09580
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −24.9282 −1.55498 −0.777489 0.628896i \(-0.783507\pi\)
−0.777489 + 0.628896i \(0.783507\pi\)
\(258\) 0 0
\(259\) −10.0000 −0.621370
\(260\) −12.0000 −0.744208
\(261\) 0 0
\(262\) 5.07180 0.313337
\(263\) −11.3205 −0.698052 −0.349026 0.937113i \(-0.613488\pi\)
−0.349026 + 0.937113i \(0.613488\pi\)
\(264\) 0 0
\(265\) −6.92820 −0.425596
\(266\) −1.00000 −0.0613139
\(267\) 0 0
\(268\) 9.46410 0.578112
\(269\) 16.9282 1.03213 0.516065 0.856549i \(-0.327396\pi\)
0.516065 + 0.856549i \(0.327396\pi\)
\(270\) 0 0
\(271\) −29.8564 −1.81365 −0.906824 0.421510i \(-0.861500\pi\)
−0.906824 + 0.421510i \(0.861500\pi\)
\(272\) −7.46410 −0.452578
\(273\) 0 0
\(274\) 0.928203 0.0560748
\(275\) 38.2487 2.30648
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 9.85641 0.591148
\(279\) 0 0
\(280\) −3.46410 −0.207020
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) −2.92820 −0.173757
\(285\) 0 0
\(286\) −18.9282 −1.11925
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) 38.7128 2.27722
\(290\) 20.7846 1.22051
\(291\) 0 0
\(292\) 10.0000 0.585206
\(293\) −7.07180 −0.413139 −0.206569 0.978432i \(-0.566230\pi\)
−0.206569 + 0.978432i \(0.566230\pi\)
\(294\) 0 0
\(295\) −13.8564 −0.806751
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) 3.07180 0.177944
\(299\) −18.9282 −1.09465
\(300\) 0 0
\(301\) 6.92820 0.399335
\(302\) 1.46410 0.0842496
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) −17.0718 −0.977528
\(306\) 0 0
\(307\) 20.7846 1.18624 0.593120 0.805114i \(-0.297896\pi\)
0.593120 + 0.805114i \(0.297896\pi\)
\(308\) −5.46410 −0.311346
\(309\) 0 0
\(310\) 24.0000 1.36311
\(311\) −2.14359 −0.121552 −0.0607760 0.998151i \(-0.519358\pi\)
−0.0607760 + 0.998151i \(0.519358\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 19.8564 1.12056
\(315\) 0 0
\(316\) −12.3923 −0.697122
\(317\) −7.85641 −0.441260 −0.220630 0.975358i \(-0.570811\pi\)
−0.220630 + 0.975358i \(0.570811\pi\)
\(318\) 0 0
\(319\) 32.7846 1.83559
\(320\) 3.46410 0.193649
\(321\) 0 0
\(322\) −5.46410 −0.304502
\(323\) −7.46410 −0.415314
\(324\) 0 0
\(325\) −24.2487 −1.34508
\(326\) 1.07180 0.0593613
\(327\) 0 0
\(328\) 2.00000 0.110432
\(329\) 10.9282 0.602491
\(330\) 0 0
\(331\) 14.5359 0.798965 0.399483 0.916741i \(-0.369190\pi\)
0.399483 + 0.916741i \(0.369190\pi\)
\(332\) −8.00000 −0.439057
\(333\) 0 0
\(334\) 18.9282 1.03571
\(335\) 32.7846 1.79121
\(336\) 0 0
\(337\) 10.7846 0.587475 0.293738 0.955886i \(-0.405101\pi\)
0.293738 + 0.955886i \(0.405101\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) −25.8564 −1.40226
\(341\) 37.8564 2.05004
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −6.92820 −0.373544
\(345\) 0 0
\(346\) 0.928203 0.0499005
\(347\) −27.3205 −1.46664 −0.733321 0.679883i \(-0.762031\pi\)
−0.733321 + 0.679883i \(0.762031\pi\)
\(348\) 0 0
\(349\) 35.8564 1.91935 0.959675 0.281113i \(-0.0907035\pi\)
0.959675 + 0.281113i \(0.0907035\pi\)
\(350\) −7.00000 −0.374166
\(351\) 0 0
\(352\) 5.46410 0.291238
\(353\) −5.32051 −0.283182 −0.141591 0.989925i \(-0.545222\pi\)
−0.141591 + 0.989925i \(0.545222\pi\)
\(354\) 0 0
\(355\) −10.1436 −0.538366
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) 9.85641 0.520927
\(359\) −35.3205 −1.86415 −0.932073 0.362272i \(-0.882001\pi\)
−0.932073 + 0.362272i \(0.882001\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −8.53590 −0.448637
\(363\) 0 0
\(364\) 3.46410 0.181568
\(365\) 34.6410 1.81319
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 5.46410 0.284836
\(369\) 0 0
\(370\) 34.6410 1.80090
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) 10.7846 0.558406 0.279203 0.960232i \(-0.409930\pi\)
0.279203 + 0.960232i \(0.409930\pi\)
\(374\) −40.7846 −2.10892
\(375\) 0 0
\(376\) −10.9282 −0.563579
\(377\) −20.7846 −1.07046
\(378\) 0 0
\(379\) −1.46410 −0.0752058 −0.0376029 0.999293i \(-0.511972\pi\)
−0.0376029 + 0.999293i \(0.511972\pi\)
\(380\) 3.46410 0.177705
\(381\) 0 0
\(382\) 5.46410 0.279568
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) −18.9282 −0.964671
\(386\) −22.0000 −1.11977
\(387\) 0 0
\(388\) 4.53590 0.230275
\(389\) −29.7128 −1.50650 −0.753250 0.657735i \(-0.771515\pi\)
−0.753250 + 0.657735i \(0.771515\pi\)
\(390\) 0 0
\(391\) −40.7846 −2.06257
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −4.92820 −0.248279
\(395\) −42.9282 −2.15995
\(396\) 0 0
\(397\) 19.8564 0.996564 0.498282 0.867015i \(-0.333964\pi\)
0.498282 + 0.867015i \(0.333964\pi\)
\(398\) −24.7846 −1.24234
\(399\) 0 0
\(400\) 7.00000 0.350000
\(401\) −19.8564 −0.991582 −0.495791 0.868442i \(-0.665122\pi\)
−0.495791 + 0.868442i \(0.665122\pi\)
\(402\) 0 0
\(403\) −24.0000 −1.19553
\(404\) −4.53590 −0.225669
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) 54.6410 2.70845
\(408\) 0 0
\(409\) −31.1769 −1.54160 −0.770800 0.637078i \(-0.780143\pi\)
−0.770800 + 0.637078i \(0.780143\pi\)
\(410\) 6.92820 0.342160
\(411\) 0 0
\(412\) −6.92820 −0.341328
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) −27.7128 −1.36037
\(416\) −3.46410 −0.169842
\(417\) 0 0
\(418\) 5.46410 0.267258
\(419\) −18.9282 −0.924703 −0.462352 0.886697i \(-0.652994\pi\)
−0.462352 + 0.886697i \(0.652994\pi\)
\(420\) 0 0
\(421\) 18.7846 0.915506 0.457753 0.889079i \(-0.348654\pi\)
0.457753 + 0.889079i \(0.348654\pi\)
\(422\) 1.46410 0.0712714
\(423\) 0 0
\(424\) −2.00000 −0.0971286
\(425\) −52.2487 −2.53443
\(426\) 0 0
\(427\) 4.92820 0.238492
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) −24.0000 −1.15738
\(431\) 34.9282 1.68243 0.841216 0.540699i \(-0.181840\pi\)
0.841216 + 0.540699i \(0.181840\pi\)
\(432\) 0 0
\(433\) 6.67949 0.320996 0.160498 0.987036i \(-0.448690\pi\)
0.160498 + 0.987036i \(0.448690\pi\)
\(434\) −6.92820 −0.332564
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 5.46410 0.261383
\(438\) 0 0
\(439\) −4.78461 −0.228357 −0.114178 0.993460i \(-0.536424\pi\)
−0.114178 + 0.993460i \(0.536424\pi\)
\(440\) 18.9282 0.902367
\(441\) 0 0
\(442\) 25.8564 1.22986
\(443\) 11.3205 0.537854 0.268927 0.963161i \(-0.413331\pi\)
0.268927 + 0.963161i \(0.413331\pi\)
\(444\) 0 0
\(445\) 6.92820 0.328428
\(446\) −12.0000 −0.568216
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 7.85641 0.370767 0.185383 0.982666i \(-0.440647\pi\)
0.185383 + 0.982666i \(0.440647\pi\)
\(450\) 0 0
\(451\) 10.9282 0.514589
\(452\) −11.8564 −0.557678
\(453\) 0 0
\(454\) −14.9282 −0.700615
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) 12.1436 0.568053 0.284027 0.958816i \(-0.408330\pi\)
0.284027 + 0.958816i \(0.408330\pi\)
\(458\) 3.07180 0.143536
\(459\) 0 0
\(460\) 18.9282 0.882532
\(461\) −38.1051 −1.77473 −0.887366 0.461065i \(-0.847467\pi\)
−0.887366 + 0.461065i \(0.847467\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −26.7846 −1.24077
\(467\) −35.7128 −1.65259 −0.826296 0.563236i \(-0.809556\pi\)
−0.826296 + 0.563236i \(0.809556\pi\)
\(468\) 0 0
\(469\) −9.46410 −0.437012
\(470\) −37.8564 −1.74619
\(471\) 0 0
\(472\) −4.00000 −0.184115
\(473\) −37.8564 −1.74064
\(474\) 0 0
\(475\) 7.00000 0.321182
\(476\) 7.46410 0.342117
\(477\) 0 0
\(478\) −30.2487 −1.38354
\(479\) −27.7128 −1.26623 −0.633115 0.774057i \(-0.718224\pi\)
−0.633115 + 0.774057i \(0.718224\pi\)
\(480\) 0 0
\(481\) −34.6410 −1.57949
\(482\) 21.3205 0.971123
\(483\) 0 0
\(484\) 18.8564 0.857109
\(485\) 15.7128 0.713482
\(486\) 0 0
\(487\) −39.3205 −1.78178 −0.890891 0.454217i \(-0.849919\pi\)
−0.890891 + 0.454217i \(0.849919\pi\)
\(488\) −4.92820 −0.223089
\(489\) 0 0
\(490\) 3.46410 0.156492
\(491\) 40.3923 1.82288 0.911440 0.411434i \(-0.134972\pi\)
0.911440 + 0.411434i \(0.134972\pi\)
\(492\) 0 0
\(493\) −44.7846 −2.01700
\(494\) −3.46410 −0.155857
\(495\) 0 0
\(496\) 6.92820 0.311086
\(497\) 2.92820 0.131348
\(498\) 0 0
\(499\) −22.9282 −1.02641 −0.513204 0.858267i \(-0.671541\pi\)
−0.513204 + 0.858267i \(0.671541\pi\)
\(500\) 6.92820 0.309839
\(501\) 0 0
\(502\) −5.07180 −0.226365
\(503\) 21.8564 0.974529 0.487264 0.873254i \(-0.337995\pi\)
0.487264 + 0.873254i \(0.337995\pi\)
\(504\) 0 0
\(505\) −15.7128 −0.699211
\(506\) 29.8564 1.32728
\(507\) 0 0
\(508\) −17.4641 −0.774844
\(509\) 44.6410 1.97868 0.989339 0.145630i \(-0.0465209\pi\)
0.989339 + 0.145630i \(0.0465209\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −24.9282 −1.09954
\(515\) −24.0000 −1.05757
\(516\) 0 0
\(517\) −59.7128 −2.62617
\(518\) −10.0000 −0.439375
\(519\) 0 0
\(520\) −12.0000 −0.526235
\(521\) −25.7128 −1.12650 −0.563249 0.826287i \(-0.690449\pi\)
−0.563249 + 0.826287i \(0.690449\pi\)
\(522\) 0 0
\(523\) 26.6410 1.16493 0.582465 0.812856i \(-0.302088\pi\)
0.582465 + 0.812856i \(0.302088\pi\)
\(524\) 5.07180 0.221562
\(525\) 0 0
\(526\) −11.3205 −0.493598
\(527\) −51.7128 −2.25265
\(528\) 0 0
\(529\) 6.85641 0.298105
\(530\) −6.92820 −0.300942
\(531\) 0 0
\(532\) −1.00000 −0.0433555
\(533\) −6.92820 −0.300094
\(534\) 0 0
\(535\) 13.8564 0.599065
\(536\) 9.46410 0.408787
\(537\) 0 0
\(538\) 16.9282 0.729827
\(539\) 5.46410 0.235356
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) −29.8564 −1.28244
\(543\) 0 0
\(544\) −7.46410 −0.320021
\(545\) 6.92820 0.296772
\(546\) 0 0
\(547\) −10.2487 −0.438203 −0.219102 0.975702i \(-0.570313\pi\)
−0.219102 + 0.975702i \(0.570313\pi\)
\(548\) 0.928203 0.0396509
\(549\) 0 0
\(550\) 38.2487 1.63093
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 12.3923 0.526974
\(554\) 22.0000 0.934690
\(555\) 0 0
\(556\) 9.85641 0.418005
\(557\) −4.92820 −0.208815 −0.104407 0.994535i \(-0.533295\pi\)
−0.104407 + 0.994535i \(0.533295\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) −3.46410 −0.146385
\(561\) 0 0
\(562\) 26.0000 1.09674
\(563\) −25.8564 −1.08972 −0.544859 0.838528i \(-0.683417\pi\)
−0.544859 + 0.838528i \(0.683417\pi\)
\(564\) 0 0
\(565\) −41.0718 −1.72790
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) −2.92820 −0.122865
\(569\) 45.7128 1.91638 0.958190 0.286131i \(-0.0923694\pi\)
0.958190 + 0.286131i \(0.0923694\pi\)
\(570\) 0 0
\(571\) 37.5692 1.57222 0.786111 0.618085i \(-0.212091\pi\)
0.786111 + 0.618085i \(0.212091\pi\)
\(572\) −18.9282 −0.791428
\(573\) 0 0
\(574\) −2.00000 −0.0834784
\(575\) 38.2487 1.59508
\(576\) 0 0
\(577\) −3.85641 −0.160544 −0.0802722 0.996773i \(-0.525579\pi\)
−0.0802722 + 0.996773i \(0.525579\pi\)
\(578\) 38.7128 1.61024
\(579\) 0 0
\(580\) 20.7846 0.863034
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) −10.9282 −0.452600
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) −7.07180 −0.292133
\(587\) 8.00000 0.330195 0.165098 0.986277i \(-0.447206\pi\)
0.165098 + 0.986277i \(0.447206\pi\)
\(588\) 0 0
\(589\) 6.92820 0.285472
\(590\) −13.8564 −0.570459
\(591\) 0 0
\(592\) 10.0000 0.410997
\(593\) −12.5359 −0.514788 −0.257394 0.966307i \(-0.582864\pi\)
−0.257394 + 0.966307i \(0.582864\pi\)
\(594\) 0 0
\(595\) 25.8564 1.06001
\(596\) 3.07180 0.125826
\(597\) 0 0
\(598\) −18.9282 −0.774032
\(599\) −13.0718 −0.534099 −0.267050 0.963683i \(-0.586049\pi\)
−0.267050 + 0.963683i \(0.586049\pi\)
\(600\) 0 0
\(601\) 6.67949 0.272462 0.136231 0.990677i \(-0.456501\pi\)
0.136231 + 0.990677i \(0.456501\pi\)
\(602\) 6.92820 0.282372
\(603\) 0 0
\(604\) 1.46410 0.0595734
\(605\) 65.3205 2.65566
\(606\) 0 0
\(607\) −17.8564 −0.724769 −0.362385 0.932029i \(-0.618037\pi\)
−0.362385 + 0.932029i \(0.618037\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) −17.0718 −0.691217
\(611\) 37.8564 1.53151
\(612\) 0 0
\(613\) −23.8564 −0.963551 −0.481776 0.876295i \(-0.660008\pi\)
−0.481776 + 0.876295i \(0.660008\pi\)
\(614\) 20.7846 0.838799
\(615\) 0 0
\(616\) −5.46410 −0.220155
\(617\) −12.9282 −0.520470 −0.260235 0.965545i \(-0.583800\pi\)
−0.260235 + 0.965545i \(0.583800\pi\)
\(618\) 0 0
\(619\) 26.6410 1.07079 0.535396 0.844601i \(-0.320162\pi\)
0.535396 + 0.844601i \(0.320162\pi\)
\(620\) 24.0000 0.963863
\(621\) 0 0
\(622\) −2.14359 −0.0859503
\(623\) −2.00000 −0.0801283
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) 19.8564 0.792357
\(629\) −74.6410 −2.97613
\(630\) 0 0
\(631\) 16.7846 0.668185 0.334092 0.942540i \(-0.391570\pi\)
0.334092 + 0.942540i \(0.391570\pi\)
\(632\) −12.3923 −0.492939
\(633\) 0 0
\(634\) −7.85641 −0.312018
\(635\) −60.4974 −2.40077
\(636\) 0 0
\(637\) −3.46410 −0.137253
\(638\) 32.7846 1.29796
\(639\) 0 0
\(640\) 3.46410 0.136931
\(641\) 15.8564 0.626290 0.313145 0.949705i \(-0.398617\pi\)
0.313145 + 0.949705i \(0.398617\pi\)
\(642\) 0 0
\(643\) −14.9282 −0.588711 −0.294355 0.955696i \(-0.595105\pi\)
−0.294355 + 0.955696i \(0.595105\pi\)
\(644\) −5.46410 −0.215316
\(645\) 0 0
\(646\) −7.46410 −0.293671
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −21.8564 −0.857939
\(650\) −24.2487 −0.951113
\(651\) 0 0
\(652\) 1.07180 0.0419748
\(653\) 33.7128 1.31928 0.659642 0.751580i \(-0.270708\pi\)
0.659642 + 0.751580i \(0.270708\pi\)
\(654\) 0 0
\(655\) 17.5692 0.686486
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) 10.9282 0.426026
\(659\) −14.1436 −0.550956 −0.275478 0.961307i \(-0.588836\pi\)
−0.275478 + 0.961307i \(0.588836\pi\)
\(660\) 0 0
\(661\) 45.3205 1.76276 0.881382 0.472405i \(-0.156614\pi\)
0.881382 + 0.472405i \(0.156614\pi\)
\(662\) 14.5359 0.564954
\(663\) 0 0
\(664\) −8.00000 −0.310460
\(665\) −3.46410 −0.134332
\(666\) 0 0
\(667\) 32.7846 1.26943
\(668\) 18.9282 0.732354
\(669\) 0 0
\(670\) 32.7846 1.26658
\(671\) −26.9282 −1.03955
\(672\) 0 0
\(673\) −16.9282 −0.652534 −0.326267 0.945278i \(-0.605791\pi\)
−0.326267 + 0.945278i \(0.605791\pi\)
\(674\) 10.7846 0.415408
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) 19.8564 0.763144 0.381572 0.924339i \(-0.375383\pi\)
0.381572 + 0.924339i \(0.375383\pi\)
\(678\) 0 0
\(679\) −4.53590 −0.174072
\(680\) −25.8564 −0.991548
\(681\) 0 0
\(682\) 37.8564 1.44960
\(683\) −30.9282 −1.18343 −0.591717 0.806145i \(-0.701550\pi\)
−0.591717 + 0.806145i \(0.701550\pi\)
\(684\) 0 0
\(685\) 3.21539 0.122854
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −6.92820 −0.264135
\(689\) 6.92820 0.263944
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 0.928203 0.0352850
\(693\) 0 0
\(694\) −27.3205 −1.03707
\(695\) 34.1436 1.29514
\(696\) 0 0
\(697\) −14.9282 −0.565446
\(698\) 35.8564 1.35718
\(699\) 0 0
\(700\) −7.00000 −0.264575
\(701\) −13.7128 −0.517926 −0.258963 0.965887i \(-0.583381\pi\)
−0.258963 + 0.965887i \(0.583381\pi\)
\(702\) 0 0
\(703\) 10.0000 0.377157
\(704\) 5.46410 0.205936
\(705\) 0 0
\(706\) −5.32051 −0.200240
\(707\) 4.53590 0.170590
\(708\) 0 0
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) −10.1436 −0.380682
\(711\) 0 0
\(712\) 2.00000 0.0749532
\(713\) 37.8564 1.41773
\(714\) 0 0
\(715\) −65.5692 −2.45215
\(716\) 9.85641 0.368351
\(717\) 0 0
\(718\) −35.3205 −1.31815
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 0 0
\(721\) 6.92820 0.258020
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) −8.53590 −0.317234
\(725\) 42.0000 1.55984
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 3.46410 0.128388
\(729\) 0 0
\(730\) 34.6410 1.28212
\(731\) 51.7128 1.91267
\(732\) 0 0
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 5.46410 0.201409
\(737\) 51.7128 1.90487
\(738\) 0 0
\(739\) −9.85641 −0.362574 −0.181287 0.983430i \(-0.558026\pi\)
−0.181287 + 0.983430i \(0.558026\pi\)
\(740\) 34.6410 1.27343
\(741\) 0 0
\(742\) 2.00000 0.0734223
\(743\) −27.7128 −1.01668 −0.508342 0.861155i \(-0.669742\pi\)
−0.508342 + 0.861155i \(0.669742\pi\)
\(744\) 0 0
\(745\) 10.6410 0.389857
\(746\) 10.7846 0.394853
\(747\) 0 0
\(748\) −40.7846 −1.49123
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) 20.3923 0.744126 0.372063 0.928208i \(-0.378651\pi\)
0.372063 + 0.928208i \(0.378651\pi\)
\(752\) −10.9282 −0.398511
\(753\) 0 0
\(754\) −20.7846 −0.756931
\(755\) 5.07180 0.184582
\(756\) 0 0
\(757\) 3.85641 0.140163 0.0700817 0.997541i \(-0.477674\pi\)
0.0700817 + 0.997541i \(0.477674\pi\)
\(758\) −1.46410 −0.0531786
\(759\) 0 0
\(760\) 3.46410 0.125656
\(761\) −4.53590 −0.164426 −0.0822131 0.996615i \(-0.526199\pi\)
−0.0822131 + 0.996615i \(0.526199\pi\)
\(762\) 0 0
\(763\) −2.00000 −0.0724049
\(764\) 5.46410 0.197684
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) 13.8564 0.500326
\(768\) 0 0
\(769\) −19.8564 −0.716040 −0.358020 0.933714i \(-0.616548\pi\)
−0.358020 + 0.933714i \(0.616548\pi\)
\(770\) −18.9282 −0.682125
\(771\) 0 0
\(772\) −22.0000 −0.791797
\(773\) −2.78461 −0.100155 −0.0500777 0.998745i \(-0.515947\pi\)
−0.0500777 + 0.998745i \(0.515947\pi\)
\(774\) 0 0
\(775\) 48.4974 1.74208
\(776\) 4.53590 0.162829
\(777\) 0 0
\(778\) −29.7128 −1.06526
\(779\) 2.00000 0.0716574
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) −40.7846 −1.45845
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 68.7846 2.45503
\(786\) 0 0
\(787\) −11.2154 −0.399785 −0.199893 0.979818i \(-0.564059\pi\)
−0.199893 + 0.979818i \(0.564059\pi\)
\(788\) −4.92820 −0.175560
\(789\) 0 0
\(790\) −42.9282 −1.52732
\(791\) 11.8564 0.421565
\(792\) 0 0
\(793\) 17.0718 0.606237
\(794\) 19.8564 0.704677
\(795\) 0 0
\(796\) −24.7846 −0.878467
\(797\) 44.6410 1.58127 0.790633 0.612290i \(-0.209752\pi\)
0.790633 + 0.612290i \(0.209752\pi\)
\(798\) 0 0
\(799\) 81.5692 2.88571
\(800\) 7.00000 0.247487
\(801\) 0 0
\(802\) −19.8564 −0.701154
\(803\) 54.6410 1.92824
\(804\) 0 0
\(805\) −18.9282 −0.667132
\(806\) −24.0000 −0.845364
\(807\) 0 0
\(808\) −4.53590 −0.159572
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 33.0718 1.16131 0.580654 0.814150i \(-0.302797\pi\)
0.580654 + 0.814150i \(0.302797\pi\)
\(812\) −6.00000 −0.210559
\(813\) 0 0
\(814\) 54.6410 1.91517
\(815\) 3.71281 0.130054
\(816\) 0 0
\(817\) −6.92820 −0.242387
\(818\) −31.1769 −1.09008
\(819\) 0 0
\(820\) 6.92820 0.241943
\(821\) 48.9282 1.70761 0.853803 0.520596i \(-0.174290\pi\)
0.853803 + 0.520596i \(0.174290\pi\)
\(822\) 0 0
\(823\) −21.8564 −0.761866 −0.380933 0.924603i \(-0.624397\pi\)
−0.380933 + 0.924603i \(0.624397\pi\)
\(824\) −6.92820 −0.241355
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) −5.60770 −0.194763 −0.0973817 0.995247i \(-0.531047\pi\)
−0.0973817 + 0.995247i \(0.531047\pi\)
\(830\) −27.7128 −0.961926
\(831\) 0 0
\(832\) −3.46410 −0.120096
\(833\) −7.46410 −0.258616
\(834\) 0 0
\(835\) 65.5692 2.26912
\(836\) 5.46410 0.188980
\(837\) 0 0
\(838\) −18.9282 −0.653864
\(839\) 5.07180 0.175098 0.0875489 0.996160i \(-0.472097\pi\)
0.0875489 + 0.996160i \(0.472097\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 18.7846 0.647360
\(843\) 0 0
\(844\) 1.46410 0.0503965
\(845\) −3.46410 −0.119169
\(846\) 0 0
\(847\) −18.8564 −0.647914
\(848\) −2.00000 −0.0686803
\(849\) 0 0
\(850\) −52.2487 −1.79212
\(851\) 54.6410 1.87307
\(852\) 0 0
\(853\) 35.0718 1.20084 0.600418 0.799687i \(-0.295001\pi\)
0.600418 + 0.799687i \(0.295001\pi\)
\(854\) 4.92820 0.168640
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) −47.5692 −1.62493 −0.812467 0.583007i \(-0.801876\pi\)
−0.812467 + 0.583007i \(0.801876\pi\)
\(858\) 0 0
\(859\) −28.7846 −0.982118 −0.491059 0.871126i \(-0.663390\pi\)
−0.491059 + 0.871126i \(0.663390\pi\)
\(860\) −24.0000 −0.818393
\(861\) 0 0
\(862\) 34.9282 1.18966
\(863\) −18.1436 −0.617615 −0.308808 0.951125i \(-0.599930\pi\)
−0.308808 + 0.951125i \(0.599930\pi\)
\(864\) 0 0
\(865\) 3.21539 0.109327
\(866\) 6.67949 0.226978
\(867\) 0 0
\(868\) −6.92820 −0.235159
\(869\) −67.7128 −2.29700
\(870\) 0 0
\(871\) −32.7846 −1.11086
\(872\) 2.00000 0.0677285
\(873\) 0 0
\(874\) 5.46410 0.184826
\(875\) −6.92820 −0.234216
\(876\) 0 0
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) −4.78461 −0.161473
\(879\) 0 0
\(880\) 18.9282 0.638070
\(881\) −43.1769 −1.45467 −0.727334 0.686284i \(-0.759241\pi\)
−0.727334 + 0.686284i \(0.759241\pi\)
\(882\) 0 0
\(883\) 48.4974 1.63207 0.816034 0.578004i \(-0.196168\pi\)
0.816034 + 0.578004i \(0.196168\pi\)
\(884\) 25.8564 0.869645
\(885\) 0 0
\(886\) 11.3205 0.380320
\(887\) 27.7128 0.930505 0.465253 0.885178i \(-0.345963\pi\)
0.465253 + 0.885178i \(0.345963\pi\)
\(888\) 0 0
\(889\) 17.4641 0.585727
\(890\) 6.92820 0.232234
\(891\) 0 0
\(892\) −12.0000 −0.401790
\(893\) −10.9282 −0.365698
\(894\) 0 0
\(895\) 34.1436 1.14129
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 7.85641 0.262172
\(899\) 41.5692 1.38641
\(900\) 0 0
\(901\) 14.9282 0.497331
\(902\) 10.9282 0.363869
\(903\) 0 0
\(904\) −11.8564 −0.394338
\(905\) −29.5692 −0.982914
\(906\) 0 0
\(907\) −41.4641 −1.37679 −0.688396 0.725335i \(-0.741685\pi\)
−0.688396 + 0.725335i \(0.741685\pi\)
\(908\) −14.9282 −0.495410
\(909\) 0 0
\(910\) 12.0000 0.397796
\(911\) −22.6410 −0.750130 −0.375065 0.926998i \(-0.622380\pi\)
−0.375065 + 0.926998i \(0.622380\pi\)
\(912\) 0 0
\(913\) −43.7128 −1.44668
\(914\) 12.1436 0.401674
\(915\) 0 0
\(916\) 3.07180 0.101495
\(917\) −5.07180 −0.167485
\(918\) 0 0
\(919\) −49.5692 −1.63514 −0.817569 0.575831i \(-0.804679\pi\)
−0.817569 + 0.575831i \(0.804679\pi\)
\(920\) 18.9282 0.624044
\(921\) 0 0
\(922\) −38.1051 −1.25493
\(923\) 10.1436 0.333880
\(924\) 0 0
\(925\) 70.0000 2.30159
\(926\) 24.0000 0.788689
\(927\) 0 0
\(928\) 6.00000 0.196960
\(929\) −25.6077 −0.840161 −0.420081 0.907487i \(-0.637998\pi\)
−0.420081 + 0.907487i \(0.637998\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) −26.7846 −0.877359
\(933\) 0 0
\(934\) −35.7128 −1.16856
\(935\) −141.282 −4.62042
\(936\) 0 0
\(937\) 20.9282 0.683695 0.341847 0.939756i \(-0.388947\pi\)
0.341847 + 0.939756i \(0.388947\pi\)
\(938\) −9.46410 −0.309014
\(939\) 0 0
\(940\) −37.8564 −1.23474
\(941\) 5.21539 0.170017 0.0850084 0.996380i \(-0.472908\pi\)
0.0850084 + 0.996380i \(0.472908\pi\)
\(942\) 0 0
\(943\) 10.9282 0.355871
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −37.8564 −1.23082
\(947\) 28.1051 0.913294 0.456647 0.889648i \(-0.349050\pi\)
0.456647 + 0.889648i \(0.349050\pi\)
\(948\) 0 0
\(949\) −34.6410 −1.12449
\(950\) 7.00000 0.227110
\(951\) 0 0
\(952\) 7.46410 0.241913
\(953\) 31.8564 1.03193 0.515965 0.856610i \(-0.327433\pi\)
0.515965 + 0.856610i \(0.327433\pi\)
\(954\) 0 0
\(955\) 18.9282 0.612502
\(956\) −30.2487 −0.978313
\(957\) 0 0
\(958\) −27.7128 −0.895360
\(959\) −0.928203 −0.0299732
\(960\) 0 0
\(961\) 17.0000 0.548387
\(962\) −34.6410 −1.11687
\(963\) 0 0
\(964\) 21.3205 0.686687
\(965\) −76.2102 −2.45329
\(966\) 0 0
\(967\) −18.9282 −0.608690 −0.304345 0.952562i \(-0.598438\pi\)
−0.304345 + 0.952562i \(0.598438\pi\)
\(968\) 18.8564 0.606068
\(969\) 0 0
\(970\) 15.7128 0.504508
\(971\) 10.6410 0.341486 0.170743 0.985316i \(-0.445383\pi\)
0.170743 + 0.985316i \(0.445383\pi\)
\(972\) 0 0
\(973\) −9.85641 −0.315982
\(974\) −39.3205 −1.25991
\(975\) 0 0
\(976\) −4.92820 −0.157748
\(977\) 7.85641 0.251349 0.125674 0.992072i \(-0.459891\pi\)
0.125674 + 0.992072i \(0.459891\pi\)
\(978\) 0 0
\(979\) 10.9282 0.349267
\(980\) 3.46410 0.110657
\(981\) 0 0
\(982\) 40.3923 1.28897
\(983\) 32.7846 1.04567 0.522833 0.852435i \(-0.324875\pi\)
0.522833 + 0.852435i \(0.324875\pi\)
\(984\) 0 0
\(985\) −17.0718 −0.543953
\(986\) −44.7846 −1.42623
\(987\) 0 0
\(988\) −3.46410 −0.110208
\(989\) −37.8564 −1.20376
\(990\) 0 0
\(991\) 35.6077 1.13112 0.565558 0.824709i \(-0.308661\pi\)
0.565558 + 0.824709i \(0.308661\pi\)
\(992\) 6.92820 0.219971
\(993\) 0 0
\(994\) 2.92820 0.0928770
\(995\) −85.8564 −2.72183
\(996\) 0 0
\(997\) −47.8564 −1.51563 −0.757814 0.652471i \(-0.773732\pi\)
−0.757814 + 0.652471i \(0.773732\pi\)
\(998\) −22.9282 −0.725780
\(999\) 0 0
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 2394.2.a.x.1.2 2
3.2 odd 2 798.2.a.k.1.1 2
12.11 even 2 6384.2.a.br.1.1 2
21.20 even 2 5586.2.a.bd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.a.k.1.1 2 3.2 odd 2
2394.2.a.x.1.2 2 1.1 even 1 trivial
5586.2.a.bd.1.2 2 21.20 even 2
6384.2.a.br.1.1 2 12.11 even 2