Properties

Label 4560.2.a.bo.1.2
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +3.64575 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +3.64575 q^{7} +1.00000 q^{9} -5.64575 q^{11} -5.64575 q^{13} +1.00000 q^{15} -4.00000 q^{17} +1.00000 q^{19} +3.64575 q^{21} +1.29150 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.93725 q^{29} -6.00000 q^{31} -5.64575 q^{33} +3.64575 q^{35} -1.64575 q^{37} -5.64575 q^{39} -4.35425 q^{41} -0.354249 q^{43} +1.00000 q^{45} -9.29150 q^{47} +6.29150 q^{49} -4.00000 q^{51} +0.708497 q^{53} -5.64575 q^{55} +1.00000 q^{57} -0.708497 q^{59} -0.708497 q^{61} +3.64575 q^{63} -5.64575 q^{65} -14.5830 q^{67} +1.29150 q^{69} +3.29150 q^{71} +10.0000 q^{73} +1.00000 q^{75} -20.5830 q^{77} +14.5830 q^{79} +1.00000 q^{81} -6.00000 q^{83} -4.00000 q^{85} -6.93725 q^{87} -1.06275 q^{89} -20.5830 q^{91} -6.00000 q^{93} +1.00000 q^{95} +12.9373 q^{97} -5.64575 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 6 q^{11} - 6 q^{13} + 2 q^{15} - 8 q^{17} + 2 q^{19} + 2 q^{21} - 8 q^{23} + 2 q^{25} + 2 q^{27} + 2 q^{29} - 12 q^{31} - 6 q^{33} + 2 q^{35} + 2 q^{37} - 6 q^{39} - 14 q^{41} - 6 q^{43} + 2 q^{45} - 8 q^{47} + 2 q^{49} - 8 q^{51} + 12 q^{53} - 6 q^{55} + 2 q^{57} - 12 q^{59} - 12 q^{61} + 2 q^{63} - 6 q^{65} - 8 q^{67} - 8 q^{69} - 4 q^{71} + 20 q^{73} + 2 q^{75} - 20 q^{77} + 8 q^{79} + 2 q^{81} - 12 q^{83} - 8 q^{85} + 2 q^{87} - 18 q^{89} - 20 q^{91} - 12 q^{93} + 2 q^{95} + 10 q^{97} - 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.64575 1.37796 0.688982 0.724778i \(-0.258058\pi\)
0.688982 + 0.724778i \(0.258058\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.64575 −1.70226 −0.851129 0.524957i \(-0.824082\pi\)
−0.851129 + 0.524957i \(0.824082\pi\)
\(12\) 0 0
\(13\) −5.64575 −1.56585 −0.782925 0.622116i \(-0.786273\pi\)
−0.782925 + 0.622116i \(0.786273\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 3.64575 0.795568
\(22\) 0 0
\(23\) 1.29150 0.269297 0.134648 0.990893i \(-0.457009\pi\)
0.134648 + 0.990893i \(0.457009\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.93725 −1.28822 −0.644108 0.764935i \(-0.722771\pi\)
−0.644108 + 0.764935i \(0.722771\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 0 0
\(33\) −5.64575 −0.982799
\(34\) 0 0
\(35\) 3.64575 0.616244
\(36\) 0 0
\(37\) −1.64575 −0.270560 −0.135280 0.990807i \(-0.543193\pi\)
−0.135280 + 0.990807i \(0.543193\pi\)
\(38\) 0 0
\(39\) −5.64575 −0.904044
\(40\) 0 0
\(41\) −4.35425 −0.680019 −0.340010 0.940422i \(-0.610430\pi\)
−0.340010 + 0.940422i \(0.610430\pi\)
\(42\) 0 0
\(43\) −0.354249 −0.0540224 −0.0270112 0.999635i \(-0.508599\pi\)
−0.0270112 + 0.999635i \(0.508599\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −9.29150 −1.35530 −0.677652 0.735382i \(-0.737003\pi\)
−0.677652 + 0.735382i \(0.737003\pi\)
\(48\) 0 0
\(49\) 6.29150 0.898786
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) 0.708497 0.0973196 0.0486598 0.998815i \(-0.484505\pi\)
0.0486598 + 0.998815i \(0.484505\pi\)
\(54\) 0 0
\(55\) −5.64575 −0.761273
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −0.708497 −0.0922385 −0.0461193 0.998936i \(-0.514685\pi\)
−0.0461193 + 0.998936i \(0.514685\pi\)
\(60\) 0 0
\(61\) −0.708497 −0.0907138 −0.0453569 0.998971i \(-0.514443\pi\)
−0.0453569 + 0.998971i \(0.514443\pi\)
\(62\) 0 0
\(63\) 3.64575 0.459321
\(64\) 0 0
\(65\) −5.64575 −0.700269
\(66\) 0 0
\(67\) −14.5830 −1.78160 −0.890799 0.454398i \(-0.849854\pi\)
−0.890799 + 0.454398i \(0.849854\pi\)
\(68\) 0 0
\(69\) 1.29150 0.155479
\(70\) 0 0
\(71\) 3.29150 0.390629 0.195315 0.980741i \(-0.437427\pi\)
0.195315 + 0.980741i \(0.437427\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −20.5830 −2.34565
\(78\) 0 0
\(79\) 14.5830 1.64072 0.820358 0.571850i \(-0.193774\pi\)
0.820358 + 0.571850i \(0.193774\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) −6.93725 −0.743752
\(88\) 0 0
\(89\) −1.06275 −0.112651 −0.0563254 0.998412i \(-0.517938\pi\)
−0.0563254 + 0.998412i \(0.517938\pi\)
\(90\) 0 0
\(91\) −20.5830 −2.15769
\(92\) 0 0
\(93\) −6.00000 −0.622171
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 12.9373 1.31358 0.656790 0.754074i \(-0.271914\pi\)
0.656790 + 0.754074i \(0.271914\pi\)
\(98\) 0 0
\(99\) −5.64575 −0.567419
\(100\) 0 0
\(101\) −1.29150 −0.128509 −0.0642547 0.997934i \(-0.520467\pi\)
−0.0642547 + 0.997934i \(0.520467\pi\)
\(102\) 0 0
\(103\) −10.5830 −1.04277 −0.521387 0.853320i \(-0.674585\pi\)
−0.521387 + 0.853320i \(0.674585\pi\)
\(104\) 0 0
\(105\) 3.64575 0.355789
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −5.29150 −0.506834 −0.253417 0.967357i \(-0.581554\pi\)
−0.253417 + 0.967357i \(0.581554\pi\)
\(110\) 0 0
\(111\) −1.64575 −0.156208
\(112\) 0 0
\(113\) −4.00000 −0.376288 −0.188144 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 0 0
\(115\) 1.29150 0.120433
\(116\) 0 0
\(117\) −5.64575 −0.521950
\(118\) 0 0
\(119\) −14.5830 −1.33682
\(120\) 0 0
\(121\) 20.8745 1.89768
\(122\) 0 0
\(123\) −4.35425 −0.392609
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 0 0
\(129\) −0.354249 −0.0311899
\(130\) 0 0
\(131\) −12.2288 −1.06843 −0.534216 0.845348i \(-0.679393\pi\)
−0.534216 + 0.845348i \(0.679393\pi\)
\(132\) 0 0
\(133\) 3.64575 0.316127
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 13.8745 1.17682 0.588410 0.808563i \(-0.299754\pi\)
0.588410 + 0.808563i \(0.299754\pi\)
\(140\) 0 0
\(141\) −9.29150 −0.782486
\(142\) 0 0
\(143\) 31.8745 2.66548
\(144\) 0 0
\(145\) −6.93725 −0.576108
\(146\) 0 0
\(147\) 6.29150 0.518914
\(148\) 0 0
\(149\) 0.583005 0.0477617 0.0238808 0.999715i \(-0.492398\pi\)
0.0238808 + 0.999715i \(0.492398\pi\)
\(150\) 0 0
\(151\) −12.5830 −1.02399 −0.511995 0.858988i \(-0.671093\pi\)
−0.511995 + 0.858988i \(0.671093\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) 0 0
\(157\) −2.70850 −0.216162 −0.108081 0.994142i \(-0.534471\pi\)
−0.108081 + 0.994142i \(0.534471\pi\)
\(158\) 0 0
\(159\) 0.708497 0.0561875
\(160\) 0 0
\(161\) 4.70850 0.371082
\(162\) 0 0
\(163\) 7.64575 0.598861 0.299431 0.954118i \(-0.403203\pi\)
0.299431 + 0.954118i \(0.403203\pi\)
\(164\) 0 0
\(165\) −5.64575 −0.439521
\(166\) 0 0
\(167\) 10.7085 0.828648 0.414324 0.910129i \(-0.364018\pi\)
0.414324 + 0.910129i \(0.364018\pi\)
\(168\) 0 0
\(169\) 18.8745 1.45189
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 3.64575 0.275593
\(176\) 0 0
\(177\) −0.708497 −0.0532539
\(178\) 0 0
\(179\) −3.29150 −0.246018 −0.123009 0.992406i \(-0.539254\pi\)
−0.123009 + 0.992406i \(0.539254\pi\)
\(180\) 0 0
\(181\) 6.70850 0.498639 0.249319 0.968421i \(-0.419793\pi\)
0.249319 + 0.968421i \(0.419793\pi\)
\(182\) 0 0
\(183\) −0.708497 −0.0523736
\(184\) 0 0
\(185\) −1.64575 −0.120998
\(186\) 0 0
\(187\) 22.5830 1.65143
\(188\) 0 0
\(189\) 3.64575 0.265189
\(190\) 0 0
\(191\) 20.2288 1.46370 0.731851 0.681465i \(-0.238657\pi\)
0.731851 + 0.681465i \(0.238657\pi\)
\(192\) 0 0
\(193\) −6.35425 −0.457389 −0.228694 0.973498i \(-0.573446\pi\)
−0.228694 + 0.973498i \(0.573446\pi\)
\(194\) 0 0
\(195\) −5.64575 −0.404301
\(196\) 0 0
\(197\) 17.1660 1.22303 0.611514 0.791234i \(-0.290561\pi\)
0.611514 + 0.791234i \(0.290561\pi\)
\(198\) 0 0
\(199\) 3.29150 0.233328 0.116664 0.993171i \(-0.462780\pi\)
0.116664 + 0.993171i \(0.462780\pi\)
\(200\) 0 0
\(201\) −14.5830 −1.02861
\(202\) 0 0
\(203\) −25.2915 −1.77512
\(204\) 0 0
\(205\) −4.35425 −0.304114
\(206\) 0 0
\(207\) 1.29150 0.0897656
\(208\) 0 0
\(209\) −5.64575 −0.390525
\(210\) 0 0
\(211\) 18.5830 1.27931 0.639653 0.768663i \(-0.279078\pi\)
0.639653 + 0.768663i \(0.279078\pi\)
\(212\) 0 0
\(213\) 3.29150 0.225530
\(214\) 0 0
\(215\) −0.354249 −0.0241596
\(216\) 0 0
\(217\) −21.8745 −1.48494
\(218\) 0 0
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 22.5830 1.51910
\(222\) 0 0
\(223\) 18.5830 1.24441 0.622205 0.782854i \(-0.286237\pi\)
0.622205 + 0.782854i \(0.286237\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 21.2915 1.41317 0.706583 0.707630i \(-0.250236\pi\)
0.706583 + 0.707630i \(0.250236\pi\)
\(228\) 0 0
\(229\) 19.2915 1.27482 0.637409 0.770525i \(-0.280006\pi\)
0.637409 + 0.770525i \(0.280006\pi\)
\(230\) 0 0
\(231\) −20.5830 −1.35426
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) −9.29150 −0.606111
\(236\) 0 0
\(237\) 14.5830 0.947268
\(238\) 0 0
\(239\) −10.3542 −0.669761 −0.334880 0.942261i \(-0.608696\pi\)
−0.334880 + 0.942261i \(0.608696\pi\)
\(240\) 0 0
\(241\) −4.58301 −0.295217 −0.147609 0.989046i \(-0.547158\pi\)
−0.147609 + 0.989046i \(0.547158\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 6.29150 0.401949
\(246\) 0 0
\(247\) −5.64575 −0.359231
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 21.6458 1.36627 0.683134 0.730293i \(-0.260617\pi\)
0.683134 + 0.730293i \(0.260617\pi\)
\(252\) 0 0
\(253\) −7.29150 −0.458413
\(254\) 0 0
\(255\) −4.00000 −0.250490
\(256\) 0 0
\(257\) −26.5830 −1.65820 −0.829101 0.559099i \(-0.811147\pi\)
−0.829101 + 0.559099i \(0.811147\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) −6.93725 −0.429405
\(262\) 0 0
\(263\) 16.5830 1.02255 0.511276 0.859417i \(-0.329173\pi\)
0.511276 + 0.859417i \(0.329173\pi\)
\(264\) 0 0
\(265\) 0.708497 0.0435226
\(266\) 0 0
\(267\) −1.06275 −0.0650390
\(268\) 0 0
\(269\) −22.2288 −1.35531 −0.677656 0.735379i \(-0.737004\pi\)
−0.677656 + 0.735379i \(0.737004\pi\)
\(270\) 0 0
\(271\) 15.2915 0.928893 0.464446 0.885601i \(-0.346253\pi\)
0.464446 + 0.885601i \(0.346253\pi\)
\(272\) 0 0
\(273\) −20.5830 −1.24574
\(274\) 0 0
\(275\) −5.64575 −0.340452
\(276\) 0 0
\(277\) −20.5830 −1.23671 −0.618356 0.785898i \(-0.712201\pi\)
−0.618356 + 0.785898i \(0.712201\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −5.77124 −0.344284 −0.172142 0.985072i \(-0.555069\pi\)
−0.172142 + 0.985072i \(0.555069\pi\)
\(282\) 0 0
\(283\) −25.5203 −1.51702 −0.758511 0.651660i \(-0.774073\pi\)
−0.758511 + 0.651660i \(0.774073\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) −15.8745 −0.937043
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 12.9373 0.758395
\(292\) 0 0
\(293\) 26.5830 1.55300 0.776498 0.630120i \(-0.216994\pi\)
0.776498 + 0.630120i \(0.216994\pi\)
\(294\) 0 0
\(295\) −0.708497 −0.0412503
\(296\) 0 0
\(297\) −5.64575 −0.327600
\(298\) 0 0
\(299\) −7.29150 −0.421678
\(300\) 0 0
\(301\) −1.29150 −0.0744410
\(302\) 0 0
\(303\) −1.29150 −0.0741949
\(304\) 0 0
\(305\) −0.708497 −0.0405684
\(306\) 0 0
\(307\) −28.4575 −1.62416 −0.812078 0.583549i \(-0.801664\pi\)
−0.812078 + 0.583549i \(0.801664\pi\)
\(308\) 0 0
\(309\) −10.5830 −0.602046
\(310\) 0 0
\(311\) −7.06275 −0.400492 −0.200246 0.979746i \(-0.564174\pi\)
−0.200246 + 0.979746i \(0.564174\pi\)
\(312\) 0 0
\(313\) 6.70850 0.379187 0.189593 0.981863i \(-0.439283\pi\)
0.189593 + 0.981863i \(0.439283\pi\)
\(314\) 0 0
\(315\) 3.64575 0.205415
\(316\) 0 0
\(317\) −32.4575 −1.82300 −0.911498 0.411305i \(-0.865073\pi\)
−0.911498 + 0.411305i \(0.865073\pi\)
\(318\) 0 0
\(319\) 39.1660 2.19288
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) −5.64575 −0.313170
\(326\) 0 0
\(327\) −5.29150 −0.292621
\(328\) 0 0
\(329\) −33.8745 −1.86756
\(330\) 0 0
\(331\) −32.5830 −1.79092 −0.895462 0.445138i \(-0.853155\pi\)
−0.895462 + 0.445138i \(0.853155\pi\)
\(332\) 0 0
\(333\) −1.64575 −0.0901866
\(334\) 0 0
\(335\) −14.5830 −0.796755
\(336\) 0 0
\(337\) −0.937254 −0.0510555 −0.0255277 0.999674i \(-0.508127\pi\)
−0.0255277 + 0.999674i \(0.508127\pi\)
\(338\) 0 0
\(339\) −4.00000 −0.217250
\(340\) 0 0
\(341\) 33.8745 1.83441
\(342\) 0 0
\(343\) −2.58301 −0.139469
\(344\) 0 0
\(345\) 1.29150 0.0695322
\(346\) 0 0
\(347\) −26.0000 −1.39575 −0.697877 0.716218i \(-0.745872\pi\)
−0.697877 + 0.716218i \(0.745872\pi\)
\(348\) 0 0
\(349\) −19.1660 −1.02593 −0.512967 0.858409i \(-0.671454\pi\)
−0.512967 + 0.858409i \(0.671454\pi\)
\(350\) 0 0
\(351\) −5.64575 −0.301348
\(352\) 0 0
\(353\) −20.5830 −1.09552 −0.547761 0.836635i \(-0.684520\pi\)
−0.547761 + 0.836635i \(0.684520\pi\)
\(354\) 0 0
\(355\) 3.29150 0.174695
\(356\) 0 0
\(357\) −14.5830 −0.771814
\(358\) 0 0
\(359\) −22.1033 −1.16657 −0.583283 0.812269i \(-0.698232\pi\)
−0.583283 + 0.812269i \(0.698232\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 20.8745 1.09563
\(364\) 0 0
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) 3.64575 0.190307 0.0951533 0.995463i \(-0.469666\pi\)
0.0951533 + 0.995463i \(0.469666\pi\)
\(368\) 0 0
\(369\) −4.35425 −0.226673
\(370\) 0 0
\(371\) 2.58301 0.134103
\(372\) 0 0
\(373\) 20.9373 1.08409 0.542045 0.840349i \(-0.317650\pi\)
0.542045 + 0.840349i \(0.317650\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 39.1660 2.01715
\(378\) 0 0
\(379\) −10.0000 −0.513665 −0.256833 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) −2.58301 −0.131985 −0.0659927 0.997820i \(-0.521021\pi\)
−0.0659927 + 0.997820i \(0.521021\pi\)
\(384\) 0 0
\(385\) −20.5830 −1.04901
\(386\) 0 0
\(387\) −0.354249 −0.0180075
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −5.16601 −0.261256
\(392\) 0 0
\(393\) −12.2288 −0.616859
\(394\) 0 0
\(395\) 14.5830 0.733751
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 3.64575 0.182516
\(400\) 0 0
\(401\) 10.9373 0.546180 0.273090 0.961988i \(-0.411954\pi\)
0.273090 + 0.961988i \(0.411954\pi\)
\(402\) 0 0
\(403\) 33.8745 1.68741
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 9.29150 0.460563
\(408\) 0 0
\(409\) −17.2915 −0.855010 −0.427505 0.904013i \(-0.640607\pi\)
−0.427505 + 0.904013i \(0.640607\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 0 0
\(413\) −2.58301 −0.127101
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) 13.8745 0.679438
\(418\) 0 0
\(419\) −23.0627 −1.12669 −0.563344 0.826222i \(-0.690486\pi\)
−0.563344 + 0.826222i \(0.690486\pi\)
\(420\) 0 0
\(421\) 7.41699 0.361482 0.180741 0.983531i \(-0.442150\pi\)
0.180741 + 0.983531i \(0.442150\pi\)
\(422\) 0 0
\(423\) −9.29150 −0.451768
\(424\) 0 0
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) −2.58301 −0.125000
\(428\) 0 0
\(429\) 31.8745 1.53892
\(430\) 0 0
\(431\) 7.29150 0.351219 0.175610 0.984460i \(-0.443810\pi\)
0.175610 + 0.984460i \(0.443810\pi\)
\(432\) 0 0
\(433\) −22.3542 −1.07428 −0.537138 0.843494i \(-0.680495\pi\)
−0.537138 + 0.843494i \(0.680495\pi\)
\(434\) 0 0
\(435\) −6.93725 −0.332616
\(436\) 0 0
\(437\) 1.29150 0.0617809
\(438\) 0 0
\(439\) 22.5830 1.07783 0.538914 0.842361i \(-0.318835\pi\)
0.538914 + 0.842361i \(0.318835\pi\)
\(440\) 0 0
\(441\) 6.29150 0.299595
\(442\) 0 0
\(443\) −37.2915 −1.77177 −0.885886 0.463902i \(-0.846449\pi\)
−0.885886 + 0.463902i \(0.846449\pi\)
\(444\) 0 0
\(445\) −1.06275 −0.0503790
\(446\) 0 0
\(447\) 0.583005 0.0275752
\(448\) 0 0
\(449\) −9.77124 −0.461133 −0.230567 0.973057i \(-0.574058\pi\)
−0.230567 + 0.973057i \(0.574058\pi\)
\(450\) 0 0
\(451\) 24.5830 1.15757
\(452\) 0 0
\(453\) −12.5830 −0.591201
\(454\) 0 0
\(455\) −20.5830 −0.964946
\(456\) 0 0
\(457\) 31.8745 1.49103 0.745513 0.666491i \(-0.232204\pi\)
0.745513 + 0.666491i \(0.232204\pi\)
\(458\) 0 0
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) 25.7490 1.19925 0.599626 0.800281i \(-0.295316\pi\)
0.599626 + 0.800281i \(0.295316\pi\)
\(462\) 0 0
\(463\) −1.52026 −0.0706524 −0.0353262 0.999376i \(-0.511247\pi\)
−0.0353262 + 0.999376i \(0.511247\pi\)
\(464\) 0 0
\(465\) −6.00000 −0.278243
\(466\) 0 0
\(467\) 11.8745 0.549487 0.274743 0.961518i \(-0.411407\pi\)
0.274743 + 0.961518i \(0.411407\pi\)
\(468\) 0 0
\(469\) −53.1660 −2.45498
\(470\) 0 0
\(471\) −2.70850 −0.124801
\(472\) 0 0
\(473\) 2.00000 0.0919601
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0.708497 0.0324399
\(478\) 0 0
\(479\) −9.64575 −0.440726 −0.220363 0.975418i \(-0.570724\pi\)
−0.220363 + 0.975418i \(0.570724\pi\)
\(480\) 0 0
\(481\) 9.29150 0.423656
\(482\) 0 0
\(483\) 4.70850 0.214244
\(484\) 0 0
\(485\) 12.9373 0.587450
\(486\) 0 0
\(487\) 13.8745 0.628714 0.314357 0.949305i \(-0.398211\pi\)
0.314357 + 0.949305i \(0.398211\pi\)
\(488\) 0 0
\(489\) 7.64575 0.345753
\(490\) 0 0
\(491\) 32.2288 1.45446 0.727232 0.686392i \(-0.240807\pi\)
0.727232 + 0.686392i \(0.240807\pi\)
\(492\) 0 0
\(493\) 27.7490 1.24975
\(494\) 0 0
\(495\) −5.64575 −0.253758
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) −43.0405 −1.92676 −0.963379 0.268143i \(-0.913590\pi\)
−0.963379 + 0.268143i \(0.913590\pi\)
\(500\) 0 0
\(501\) 10.7085 0.478420
\(502\) 0 0
\(503\) 2.00000 0.0891756 0.0445878 0.999005i \(-0.485803\pi\)
0.0445878 + 0.999005i \(0.485803\pi\)
\(504\) 0 0
\(505\) −1.29150 −0.0574711
\(506\) 0 0
\(507\) 18.8745 0.838246
\(508\) 0 0
\(509\) 24.1033 1.06836 0.534179 0.845371i \(-0.320621\pi\)
0.534179 + 0.845371i \(0.320621\pi\)
\(510\) 0 0
\(511\) 36.4575 1.61279
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) −10.5830 −0.466343
\(516\) 0 0
\(517\) 52.4575 2.30708
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −36.1033 −1.58171 −0.790856 0.612002i \(-0.790365\pi\)
−0.790856 + 0.612002i \(0.790365\pi\)
\(522\) 0 0
\(523\) 12.7085 0.555704 0.277852 0.960624i \(-0.410378\pi\)
0.277852 + 0.960624i \(0.410378\pi\)
\(524\) 0 0
\(525\) 3.64575 0.159114
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) −21.3320 −0.927479
\(530\) 0 0
\(531\) −0.708497 −0.0307462
\(532\) 0 0
\(533\) 24.5830 1.06481
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −3.29150 −0.142039
\(538\) 0 0
\(539\) −35.5203 −1.52997
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 0 0
\(543\) 6.70850 0.287889
\(544\) 0 0
\(545\) −5.29150 −0.226663
\(546\) 0 0
\(547\) 29.8745 1.27734 0.638671 0.769480i \(-0.279485\pi\)
0.638671 + 0.769480i \(0.279485\pi\)
\(548\) 0 0
\(549\) −0.708497 −0.0302379
\(550\) 0 0
\(551\) −6.93725 −0.295537
\(552\) 0 0
\(553\) 53.1660 2.26085
\(554\) 0 0
\(555\) −1.64575 −0.0698583
\(556\) 0 0
\(557\) −32.5830 −1.38059 −0.690293 0.723530i \(-0.742518\pi\)
−0.690293 + 0.723530i \(0.742518\pi\)
\(558\) 0 0
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 22.5830 0.953455
\(562\) 0 0
\(563\) −39.8745 −1.68051 −0.840255 0.542191i \(-0.817595\pi\)
−0.840255 + 0.542191i \(0.817595\pi\)
\(564\) 0 0
\(565\) −4.00000 −0.168281
\(566\) 0 0
\(567\) 3.64575 0.153107
\(568\) 0 0
\(569\) −22.9373 −0.961580 −0.480790 0.876836i \(-0.659650\pi\)
−0.480790 + 0.876836i \(0.659650\pi\)
\(570\) 0 0
\(571\) −27.2915 −1.14211 −0.571057 0.820910i \(-0.693466\pi\)
−0.571057 + 0.820910i \(0.693466\pi\)
\(572\) 0 0
\(573\) 20.2288 0.845068
\(574\) 0 0
\(575\) 1.29150 0.0538594
\(576\) 0 0
\(577\) −2.70850 −0.112756 −0.0563781 0.998409i \(-0.517955\pi\)
−0.0563781 + 0.998409i \(0.517955\pi\)
\(578\) 0 0
\(579\) −6.35425 −0.264074
\(580\) 0 0
\(581\) −21.8745 −0.907508
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) 0 0
\(585\) −5.64575 −0.233423
\(586\) 0 0
\(587\) 22.7085 0.937280 0.468640 0.883389i \(-0.344744\pi\)
0.468640 + 0.883389i \(0.344744\pi\)
\(588\) 0 0
\(589\) −6.00000 −0.247226
\(590\) 0 0
\(591\) 17.1660 0.706115
\(592\) 0 0
\(593\) 24.5830 1.00950 0.504752 0.863265i \(-0.331584\pi\)
0.504752 + 0.863265i \(0.331584\pi\)
\(594\) 0 0
\(595\) −14.5830 −0.597845
\(596\) 0 0
\(597\) 3.29150 0.134712
\(598\) 0 0
\(599\) −30.5830 −1.24959 −0.624794 0.780790i \(-0.714817\pi\)
−0.624794 + 0.780790i \(0.714817\pi\)
\(600\) 0 0
\(601\) 33.2915 1.35799 0.678994 0.734143i \(-0.262416\pi\)
0.678994 + 0.734143i \(0.262416\pi\)
\(602\) 0 0
\(603\) −14.5830 −0.593866
\(604\) 0 0
\(605\) 20.8745 0.848669
\(606\) 0 0
\(607\) −31.0405 −1.25990 −0.629948 0.776637i \(-0.716924\pi\)
−0.629948 + 0.776637i \(0.716924\pi\)
\(608\) 0 0
\(609\) −25.2915 −1.02486
\(610\) 0 0
\(611\) 52.4575 2.12220
\(612\) 0 0
\(613\) 22.4575 0.907050 0.453525 0.891243i \(-0.350166\pi\)
0.453525 + 0.891243i \(0.350166\pi\)
\(614\) 0 0
\(615\) −4.35425 −0.175580
\(616\) 0 0
\(617\) 12.0000 0.483102 0.241551 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 0 0
\(619\) −12.7085 −0.510798 −0.255399 0.966836i \(-0.582207\pi\)
−0.255399 + 0.966836i \(0.582207\pi\)
\(620\) 0 0
\(621\) 1.29150 0.0518262
\(622\) 0 0
\(623\) −3.87451 −0.155229
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −5.64575 −0.225470
\(628\) 0 0
\(629\) 6.58301 0.262482
\(630\) 0 0
\(631\) −6.58301 −0.262065 −0.131033 0.991378i \(-0.541829\pi\)
−0.131033 + 0.991378i \(0.541829\pi\)
\(632\) 0 0
\(633\) 18.5830 0.738608
\(634\) 0 0
\(635\) −4.00000 −0.158735
\(636\) 0 0
\(637\) −35.5203 −1.40736
\(638\) 0 0
\(639\) 3.29150 0.130210
\(640\) 0 0
\(641\) −1.06275 −0.0419759 −0.0209880 0.999780i \(-0.506681\pi\)
−0.0209880 + 0.999780i \(0.506681\pi\)
\(642\) 0 0
\(643\) 8.35425 0.329459 0.164730 0.986339i \(-0.447325\pi\)
0.164730 + 0.986339i \(0.447325\pi\)
\(644\) 0 0
\(645\) −0.354249 −0.0139485
\(646\) 0 0
\(647\) 24.5830 0.966458 0.483229 0.875494i \(-0.339464\pi\)
0.483229 + 0.875494i \(0.339464\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) −21.8745 −0.857330
\(652\) 0 0
\(653\) 30.5830 1.19681 0.598403 0.801195i \(-0.295802\pi\)
0.598403 + 0.801195i \(0.295802\pi\)
\(654\) 0 0
\(655\) −12.2288 −0.477817
\(656\) 0 0
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) −46.5830 −1.81462 −0.907308 0.420466i \(-0.861866\pi\)
−0.907308 + 0.420466i \(0.861866\pi\)
\(660\) 0 0
\(661\) −9.29150 −0.361398 −0.180699 0.983538i \(-0.557836\pi\)
−0.180699 + 0.983538i \(0.557836\pi\)
\(662\) 0 0
\(663\) 22.5830 0.877051
\(664\) 0 0
\(665\) 3.64575 0.141376
\(666\) 0 0
\(667\) −8.95948 −0.346913
\(668\) 0 0
\(669\) 18.5830 0.718460
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) −28.9373 −1.11545 −0.557725 0.830026i \(-0.688325\pi\)
−0.557725 + 0.830026i \(0.688325\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −12.4575 −0.478781 −0.239391 0.970923i \(-0.576948\pi\)
−0.239391 + 0.970923i \(0.576948\pi\)
\(678\) 0 0
\(679\) 47.1660 1.81007
\(680\) 0 0
\(681\) 21.2915 0.815892
\(682\) 0 0
\(683\) 43.7490 1.67401 0.837005 0.547196i \(-0.184305\pi\)
0.837005 + 0.547196i \(0.184305\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) 0 0
\(687\) 19.2915 0.736017
\(688\) 0 0
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) −31.0405 −1.18084 −0.590418 0.807097i \(-0.701037\pi\)
−0.590418 + 0.807097i \(0.701037\pi\)
\(692\) 0 0
\(693\) −20.5830 −0.781884
\(694\) 0 0
\(695\) 13.8745 0.526290
\(696\) 0 0
\(697\) 17.4170 0.659716
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −4.58301 −0.173098 −0.0865489 0.996248i \(-0.527584\pi\)
−0.0865489 + 0.996248i \(0.527584\pi\)
\(702\) 0 0
\(703\) −1.64575 −0.0620707
\(704\) 0 0
\(705\) −9.29150 −0.349938
\(706\) 0 0
\(707\) −4.70850 −0.177081
\(708\) 0 0
\(709\) −15.2915 −0.574284 −0.287142 0.957888i \(-0.592705\pi\)
−0.287142 + 0.957888i \(0.592705\pi\)
\(710\) 0 0
\(711\) 14.5830 0.546905
\(712\) 0 0
\(713\) −7.74902 −0.290203
\(714\) 0 0
\(715\) 31.8745 1.19204
\(716\) 0 0
\(717\) −10.3542 −0.386687
\(718\) 0 0
\(719\) 14.8118 0.552386 0.276193 0.961102i \(-0.410927\pi\)
0.276193 + 0.961102i \(0.410927\pi\)
\(720\) 0 0
\(721\) −38.5830 −1.43691
\(722\) 0 0
\(723\) −4.58301 −0.170444
\(724\) 0 0
\(725\) −6.93725 −0.257643
\(726\) 0 0
\(727\) 44.1033 1.63570 0.817850 0.575432i \(-0.195166\pi\)
0.817850 + 0.575432i \(0.195166\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 1.41699 0.0524094
\(732\) 0 0
\(733\) 28.5830 1.05574 0.527869 0.849326i \(-0.322991\pi\)
0.527869 + 0.849326i \(0.322991\pi\)
\(734\) 0 0
\(735\) 6.29150 0.232066
\(736\) 0 0
\(737\) 82.3320 3.03274
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) −5.64575 −0.207402
\(742\) 0 0
\(743\) −10.5830 −0.388253 −0.194126 0.980977i \(-0.562187\pi\)
−0.194126 + 0.980977i \(0.562187\pi\)
\(744\) 0 0
\(745\) 0.583005 0.0213597
\(746\) 0 0
\(747\) −6.00000 −0.219529
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 23.4170 0.854498 0.427249 0.904134i \(-0.359483\pi\)
0.427249 + 0.904134i \(0.359483\pi\)
\(752\) 0 0
\(753\) 21.6458 0.788815
\(754\) 0 0
\(755\) −12.5830 −0.457942
\(756\) 0 0
\(757\) 7.87451 0.286204 0.143102 0.989708i \(-0.454292\pi\)
0.143102 + 0.989708i \(0.454292\pi\)
\(758\) 0 0
\(759\) −7.29150 −0.264665
\(760\) 0 0
\(761\) −37.7490 −1.36840 −0.684200 0.729294i \(-0.739849\pi\)
−0.684200 + 0.729294i \(0.739849\pi\)
\(762\) 0 0
\(763\) −19.2915 −0.698399
\(764\) 0 0
\(765\) −4.00000 −0.144620
\(766\) 0 0
\(767\) 4.00000 0.144432
\(768\) 0 0
\(769\) 45.7490 1.64975 0.824876 0.565314i \(-0.191245\pi\)
0.824876 + 0.565314i \(0.191245\pi\)
\(770\) 0 0
\(771\) −26.5830 −0.957364
\(772\) 0 0
\(773\) −19.2915 −0.693867 −0.346934 0.937890i \(-0.612777\pi\)
−0.346934 + 0.937890i \(0.612777\pi\)
\(774\) 0 0
\(775\) −6.00000 −0.215526
\(776\) 0 0
\(777\) −6.00000 −0.215249
\(778\) 0 0
\(779\) −4.35425 −0.156007
\(780\) 0 0
\(781\) −18.5830 −0.664952
\(782\) 0 0
\(783\) −6.93725 −0.247917
\(784\) 0 0
\(785\) −2.70850 −0.0966704
\(786\) 0 0
\(787\) −6.12549 −0.218350 −0.109175 0.994023i \(-0.534821\pi\)
−0.109175 + 0.994023i \(0.534821\pi\)
\(788\) 0 0
\(789\) 16.5830 0.590371
\(790\) 0 0
\(791\) −14.5830 −0.518512
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 0 0
\(795\) 0.708497 0.0251278
\(796\) 0 0
\(797\) 40.0000 1.41687 0.708436 0.705775i \(-0.249401\pi\)
0.708436 + 0.705775i \(0.249401\pi\)
\(798\) 0 0
\(799\) 37.1660 1.31484
\(800\) 0 0
\(801\) −1.06275 −0.0375503
\(802\) 0 0
\(803\) −56.4575 −1.99234
\(804\) 0 0
\(805\) 4.70850 0.165953
\(806\) 0 0
\(807\) −22.2288 −0.782489
\(808\) 0 0
\(809\) −40.5830 −1.42682 −0.713411 0.700746i \(-0.752851\pi\)
−0.713411 + 0.700746i \(0.752851\pi\)
\(810\) 0 0
\(811\) 46.3320 1.62694 0.813469 0.581609i \(-0.197577\pi\)
0.813469 + 0.581609i \(0.197577\pi\)
\(812\) 0 0
\(813\) 15.2915 0.536296
\(814\) 0 0
\(815\) 7.64575 0.267819
\(816\) 0 0
\(817\) −0.354249 −0.0123936
\(818\) 0 0
\(819\) −20.5830 −0.719228
\(820\) 0 0
\(821\) −47.6235 −1.66207 −0.831036 0.556218i \(-0.812252\pi\)
−0.831036 + 0.556218i \(0.812252\pi\)
\(822\) 0 0
\(823\) 22.2288 0.774846 0.387423 0.921902i \(-0.373365\pi\)
0.387423 + 0.921902i \(0.373365\pi\)
\(824\) 0 0
\(825\) −5.64575 −0.196560
\(826\) 0 0
\(827\) 22.4575 0.780924 0.390462 0.920619i \(-0.372315\pi\)
0.390462 + 0.920619i \(0.372315\pi\)
\(828\) 0 0
\(829\) 13.2915 0.461633 0.230816 0.972997i \(-0.425860\pi\)
0.230816 + 0.972997i \(0.425860\pi\)
\(830\) 0 0
\(831\) −20.5830 −0.714017
\(832\) 0 0
\(833\) −25.1660 −0.871951
\(834\) 0 0
\(835\) 10.7085 0.370583
\(836\) 0 0
\(837\) −6.00000 −0.207390
\(838\) 0 0
\(839\) −40.4575 −1.39675 −0.698374 0.715733i \(-0.746093\pi\)
−0.698374 + 0.715733i \(0.746093\pi\)
\(840\) 0 0
\(841\) 19.1255 0.659500
\(842\) 0 0
\(843\) −5.77124 −0.198772
\(844\) 0 0
\(845\) 18.8745 0.649303
\(846\) 0 0
\(847\) 76.1033 2.61494
\(848\) 0 0
\(849\) −25.5203 −0.875853
\(850\) 0 0
\(851\) −2.12549 −0.0728609
\(852\) 0 0
\(853\) 25.2915 0.865965 0.432982 0.901402i \(-0.357461\pi\)
0.432982 + 0.901402i \(0.357461\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) −43.0405 −1.47024 −0.735118 0.677939i \(-0.762873\pi\)
−0.735118 + 0.677939i \(0.762873\pi\)
\(858\) 0 0
\(859\) −50.3320 −1.71731 −0.858653 0.512557i \(-0.828698\pi\)
−0.858653 + 0.512557i \(0.828698\pi\)
\(860\) 0 0
\(861\) −15.8745 −0.541002
\(862\) 0 0
\(863\) 20.0000 0.680808 0.340404 0.940279i \(-0.389436\pi\)
0.340404 + 0.940279i \(0.389436\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) −82.3320 −2.79292
\(870\) 0 0
\(871\) 82.3320 2.78971
\(872\) 0 0
\(873\) 12.9373 0.437860
\(874\) 0 0
\(875\) 3.64575 0.123249
\(876\) 0 0
\(877\) −40.9373 −1.38235 −0.691176 0.722686i \(-0.742907\pi\)
−0.691176 + 0.722686i \(0.742907\pi\)
\(878\) 0 0
\(879\) 26.5830 0.896623
\(880\) 0 0
\(881\) −31.8745 −1.07388 −0.536940 0.843621i \(-0.680420\pi\)
−0.536940 + 0.843621i \(0.680420\pi\)
\(882\) 0 0
\(883\) 36.8118 1.23881 0.619407 0.785070i \(-0.287373\pi\)
0.619407 + 0.785070i \(0.287373\pi\)
\(884\) 0 0
\(885\) −0.708497 −0.0238159
\(886\) 0 0
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) −14.5830 −0.489098
\(890\) 0 0
\(891\) −5.64575 −0.189140
\(892\) 0 0
\(893\) −9.29150 −0.310928
\(894\) 0 0
\(895\) −3.29150 −0.110023
\(896\) 0 0
\(897\) −7.29150 −0.243456
\(898\) 0 0
\(899\) 41.6235 1.38822
\(900\) 0 0
\(901\) −2.83399 −0.0944139
\(902\) 0 0
\(903\) −1.29150 −0.0429785
\(904\) 0 0
\(905\) 6.70850 0.222998
\(906\) 0 0
\(907\) −27.0405 −0.897866 −0.448933 0.893566i \(-0.648196\pi\)
−0.448933 + 0.893566i \(0.648196\pi\)
\(908\) 0 0
\(909\) −1.29150 −0.0428364
\(910\) 0 0
\(911\) −5.41699 −0.179473 −0.0897365 0.995966i \(-0.528602\pi\)
−0.0897365 + 0.995966i \(0.528602\pi\)
\(912\) 0 0
\(913\) 33.8745 1.12108
\(914\) 0 0
\(915\) −0.708497 −0.0234222
\(916\) 0 0
\(917\) −44.5830 −1.47226
\(918\) 0 0
\(919\) −57.1660 −1.88573 −0.942866 0.333171i \(-0.891881\pi\)
−0.942866 + 0.333171i \(0.891881\pi\)
\(920\) 0 0
\(921\) −28.4575 −0.937707
\(922\) 0 0
\(923\) −18.5830 −0.611667
\(924\) 0 0
\(925\) −1.64575 −0.0541120
\(926\) 0 0
\(927\) −10.5830 −0.347591
\(928\) 0 0
\(929\) 19.8745 0.652061 0.326031 0.945359i \(-0.394289\pi\)
0.326031 + 0.945359i \(0.394289\pi\)
\(930\) 0 0
\(931\) 6.29150 0.206196
\(932\) 0 0
\(933\) −7.06275 −0.231224
\(934\) 0 0
\(935\) 22.5830 0.738543
\(936\) 0 0
\(937\) 51.8745 1.69467 0.847333 0.531062i \(-0.178207\pi\)
0.847333 + 0.531062i \(0.178207\pi\)
\(938\) 0 0
\(939\) 6.70850 0.218924
\(940\) 0 0
\(941\) −38.2288 −1.24622 −0.623111 0.782133i \(-0.714131\pi\)
−0.623111 + 0.782133i \(0.714131\pi\)
\(942\) 0 0
\(943\) −5.62352 −0.183127
\(944\) 0 0
\(945\) 3.64575 0.118596
\(946\) 0 0
\(947\) −32.5830 −1.05881 −0.529403 0.848371i \(-0.677584\pi\)
−0.529403 + 0.848371i \(0.677584\pi\)
\(948\) 0 0
\(949\) −56.4575 −1.83269
\(950\) 0 0
\(951\) −32.4575 −1.05251
\(952\) 0 0
\(953\) 10.5830 0.342817 0.171409 0.985200i \(-0.445168\pi\)
0.171409 + 0.985200i \(0.445168\pi\)
\(954\) 0 0
\(955\) 20.2288 0.654587
\(956\) 0 0
\(957\) 39.1660 1.26606
\(958\) 0 0
\(959\) 21.8745 0.706365
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −6.35425 −0.204551
\(966\) 0 0
\(967\) 36.3542 1.16907 0.584537 0.811367i \(-0.301276\pi\)
0.584537 + 0.811367i \(0.301276\pi\)
\(968\) 0 0
\(969\) −4.00000 −0.128499
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 50.5830 1.62162
\(974\) 0 0
\(975\) −5.64575 −0.180809
\(976\) 0 0
\(977\) 23.0405 0.737131 0.368566 0.929602i \(-0.379849\pi\)
0.368566 + 0.929602i \(0.379849\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −5.29150 −0.168945
\(982\) 0 0
\(983\) 18.4575 0.588703 0.294352 0.955697i \(-0.404896\pi\)
0.294352 + 0.955697i \(0.404896\pi\)
\(984\) 0 0
\(985\) 17.1660 0.546955
\(986\) 0 0
\(987\) −33.8745 −1.07824
\(988\) 0 0
\(989\) −0.457513 −0.0145481
\(990\) 0 0
\(991\) −27.7490 −0.881477 −0.440738 0.897636i \(-0.645283\pi\)
−0.440738 + 0.897636i \(0.645283\pi\)
\(992\) 0 0
\(993\) −32.5830 −1.03399
\(994\) 0 0
\(995\) 3.29150 0.104348
\(996\) 0 0
\(997\) 37.2915 1.18103 0.590517 0.807025i \(-0.298924\pi\)
0.590517 + 0.807025i \(0.298924\pi\)
\(998\) 0 0
\(999\) −1.64575 −0.0520693
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 4560.2.a.bo.1.2 2
4.3 odd 2 285.2.a.d.1.2 2
12.11 even 2 855.2.a.g.1.1 2
20.3 even 4 1425.2.c.i.799.1 4
20.7 even 4 1425.2.c.i.799.4 4
20.19 odd 2 1425.2.a.p.1.1 2
60.59 even 2 4275.2.a.u.1.2 2
76.75 even 2 5415.2.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.d.1.2 2 4.3 odd 2
855.2.a.g.1.1 2 12.11 even 2
1425.2.a.p.1.1 2 20.19 odd 2
1425.2.c.i.799.1 4 20.3 even 4
1425.2.c.i.799.4 4 20.7 even 4
4275.2.a.u.1.2 2 60.59 even 2
4560.2.a.bo.1.2 2 1.1 even 1 trivial
5415.2.a.s.1.1 2 76.75 even 2