Properties

Label 2205.2.a.v.1.2
Level $2205$
Weight $2$
Character 2205.1
Self dual yes
Analytic conductor $17.607$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.6070136457\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 245)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421 q^{2} +1.00000 q^{5} -2.82843 q^{8} +O(q^{10})\) \(q+1.41421 q^{2} +1.00000 q^{5} -2.82843 q^{8} +1.41421 q^{10} +0.171573 q^{11} -4.41421 q^{13} -4.00000 q^{16} +3.24264 q^{17} -6.00000 q^{19} +0.242641 q^{22} -7.41421 q^{23} +1.00000 q^{25} -6.24264 q^{26} +8.65685 q^{29} -10.2426 q^{31} +4.58579 q^{34} +2.24264 q^{37} -8.48528 q^{38} -2.82843 q^{40} -6.24264 q^{41} +2.00000 q^{43} -10.4853 q^{46} -7.24264 q^{47} +1.41421 q^{50} -4.24264 q^{53} +0.171573 q^{55} +12.2426 q^{58} -2.24264 q^{59} +2.82843 q^{61} -14.4853 q^{62} +8.00000 q^{64} -4.41421 q^{65} -8.24264 q^{67} +3.17157 q^{71} +8.48528 q^{73} +3.17157 q^{74} +1.48528 q^{79} -4.00000 q^{80} -8.82843 q^{82} +3.24264 q^{85} +2.82843 q^{86} -0.485281 q^{88} -8.00000 q^{89} -10.2426 q^{94} -6.00000 q^{95} -13.2426 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + O(q^{10}) \) \( 2q + 2q^{5} + 6q^{11} - 6q^{13} - 8q^{16} - 2q^{17} - 12q^{19} - 8q^{22} - 12q^{23} + 2q^{25} - 4q^{26} + 6q^{29} - 12q^{31} + 12q^{34} - 4q^{37} - 4q^{41} + 4q^{43} - 4q^{46} - 6q^{47} + 6q^{55} + 16q^{58} + 4q^{59} - 12q^{62} + 16q^{64} - 6q^{65} - 8q^{67} + 12q^{71} + 12q^{74} - 14q^{79} - 8q^{80} - 12q^{82} - 2q^{85} + 16q^{88} - 16q^{89} - 12q^{94} - 12q^{95} - 18q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) −2.82843 −1.00000
\(9\) 0 0
\(10\) 1.41421 0.447214
\(11\) 0.171573 0.0517312 0.0258656 0.999665i \(-0.491766\pi\)
0.0258656 + 0.999665i \(0.491766\pi\)
\(12\) 0 0
\(13\) −4.41421 −1.22428 −0.612141 0.790748i \(-0.709692\pi\)
−0.612141 + 0.790748i \(0.709692\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 3.24264 0.786456 0.393228 0.919441i \(-0.371358\pi\)
0.393228 + 0.919441i \(0.371358\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.242641 0.0517312
\(23\) −7.41421 −1.54597 −0.772985 0.634424i \(-0.781237\pi\)
−0.772985 + 0.634424i \(0.781237\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −6.24264 −1.22428
\(27\) 0 0
\(28\) 0 0
\(29\) 8.65685 1.60754 0.803769 0.594942i \(-0.202825\pi\)
0.803769 + 0.594942i \(0.202825\pi\)
\(30\) 0 0
\(31\) −10.2426 −1.83963 −0.919816 0.392349i \(-0.871662\pi\)
−0.919816 + 0.392349i \(0.871662\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 4.58579 0.786456
\(35\) 0 0
\(36\) 0 0
\(37\) 2.24264 0.368688 0.184344 0.982862i \(-0.440984\pi\)
0.184344 + 0.982862i \(0.440984\pi\)
\(38\) −8.48528 −1.37649
\(39\) 0 0
\(40\) −2.82843 −0.447214
\(41\) −6.24264 −0.974937 −0.487468 0.873141i \(-0.662080\pi\)
−0.487468 + 0.873141i \(0.662080\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −10.4853 −1.54597
\(47\) −7.24264 −1.05645 −0.528224 0.849105i \(-0.677142\pi\)
−0.528224 + 0.849105i \(0.677142\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.41421 0.200000
\(51\) 0 0
\(52\) 0 0
\(53\) −4.24264 −0.582772 −0.291386 0.956606i \(-0.594116\pi\)
−0.291386 + 0.956606i \(0.594116\pi\)
\(54\) 0 0
\(55\) 0.171573 0.0231349
\(56\) 0 0
\(57\) 0 0
\(58\) 12.2426 1.60754
\(59\) −2.24264 −0.291967 −0.145983 0.989287i \(-0.546635\pi\)
−0.145983 + 0.989287i \(0.546635\pi\)
\(60\) 0 0
\(61\) 2.82843 0.362143 0.181071 0.983470i \(-0.442043\pi\)
0.181071 + 0.983470i \(0.442043\pi\)
\(62\) −14.4853 −1.83963
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) −4.41421 −0.547516
\(66\) 0 0
\(67\) −8.24264 −1.00700 −0.503499 0.863996i \(-0.667954\pi\)
−0.503499 + 0.863996i \(0.667954\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.17157 0.376396 0.188198 0.982131i \(-0.439735\pi\)
0.188198 + 0.982131i \(0.439735\pi\)
\(72\) 0 0
\(73\) 8.48528 0.993127 0.496564 0.868000i \(-0.334595\pi\)
0.496564 + 0.868000i \(0.334595\pi\)
\(74\) 3.17157 0.368688
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.48528 0.167107 0.0835536 0.996503i \(-0.473373\pi\)
0.0835536 + 0.996503i \(0.473373\pi\)
\(80\) −4.00000 −0.447214
\(81\) 0 0
\(82\) −8.82843 −0.974937
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 3.24264 0.351714
\(86\) 2.82843 0.304997
\(87\) 0 0
\(88\) −0.485281 −0.0517312
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −10.2426 −1.05645
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) −13.2426 −1.34459 −0.672293 0.740285i \(-0.734691\pi\)
−0.672293 + 0.740285i \(0.734691\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.48528 0.247295 0.123647 0.992326i \(-0.460541\pi\)
0.123647 + 0.992326i \(0.460541\pi\)
\(102\) 0 0
\(103\) −19.2426 −1.89603 −0.948017 0.318220i \(-0.896915\pi\)
−0.948017 + 0.318220i \(0.896915\pi\)
\(104\) 12.4853 1.22428
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 2.48528 0.240261 0.120131 0.992758i \(-0.461669\pi\)
0.120131 + 0.992758i \(0.461669\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0.242641 0.0231349
\(111\) 0 0
\(112\) 0 0
\(113\) 13.0711 1.22962 0.614811 0.788674i \(-0.289232\pi\)
0.614811 + 0.788674i \(0.289232\pi\)
\(114\) 0 0
\(115\) −7.41421 −0.691379
\(116\) 0 0
\(117\) 0 0
\(118\) −3.17157 −0.291967
\(119\) 0 0
\(120\) 0 0
\(121\) −10.9706 −0.997324
\(122\) 4.00000 0.362143
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 8.24264 0.731416 0.365708 0.930730i \(-0.380827\pi\)
0.365708 + 0.930730i \(0.380827\pi\)
\(128\) 11.3137 1.00000
\(129\) 0 0
\(130\) −6.24264 −0.547516
\(131\) 12.2426 1.06964 0.534822 0.844965i \(-0.320379\pi\)
0.534822 + 0.844965i \(0.320379\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −11.6569 −1.00700
\(135\) 0 0
\(136\) −9.17157 −0.786456
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −7.75736 −0.657971 −0.328985 0.944335i \(-0.606707\pi\)
−0.328985 + 0.944335i \(0.606707\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4.48528 0.376396
\(143\) −0.757359 −0.0633336
\(144\) 0 0
\(145\) 8.65685 0.718913
\(146\) 12.0000 0.993127
\(147\) 0 0
\(148\) 0 0
\(149\) 9.17157 0.751365 0.375682 0.926749i \(-0.377408\pi\)
0.375682 + 0.926749i \(0.377408\pi\)
\(150\) 0 0
\(151\) −7.48528 −0.609144 −0.304572 0.952489i \(-0.598513\pi\)
−0.304572 + 0.952489i \(0.598513\pi\)
\(152\) 16.9706 1.37649
\(153\) 0 0
\(154\) 0 0
\(155\) −10.2426 −0.822709
\(156\) 0 0
\(157\) 15.1716 1.21082 0.605412 0.795913i \(-0.293008\pi\)
0.605412 + 0.795913i \(0.293008\pi\)
\(158\) 2.10051 0.167107
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −10.2426 −0.802266 −0.401133 0.916020i \(-0.631383\pi\)
−0.401133 + 0.916020i \(0.631383\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.757359 0.0586062 0.0293031 0.999571i \(-0.490671\pi\)
0.0293031 + 0.999571i \(0.490671\pi\)
\(168\) 0 0
\(169\) 6.48528 0.498868
\(170\) 4.58579 0.351714
\(171\) 0 0
\(172\) 0 0
\(173\) −7.24264 −0.550648 −0.275324 0.961352i \(-0.588785\pi\)
−0.275324 + 0.961352i \(0.588785\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −0.686292 −0.0517312
\(177\) 0 0
\(178\) −11.3137 −0.847998
\(179\) 14.4853 1.08268 0.541340 0.840804i \(-0.317917\pi\)
0.541340 + 0.840804i \(0.317917\pi\)
\(180\) 0 0
\(181\) 18.7279 1.39204 0.696018 0.718025i \(-0.254953\pi\)
0.696018 + 0.718025i \(0.254953\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 20.9706 1.54597
\(185\) 2.24264 0.164882
\(186\) 0 0
\(187\) 0.556349 0.0406843
\(188\) 0 0
\(189\) 0 0
\(190\) −8.48528 −0.615587
\(191\) −13.9706 −1.01087 −0.505437 0.862863i \(-0.668669\pi\)
−0.505437 + 0.862863i \(0.668669\pi\)
\(192\) 0 0
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) −18.7279 −1.34459
\(195\) 0 0
\(196\) 0 0
\(197\) −13.4142 −0.955723 −0.477862 0.878435i \(-0.658588\pi\)
−0.477862 + 0.878435i \(0.658588\pi\)
\(198\) 0 0
\(199\) 7.41421 0.525580 0.262790 0.964853i \(-0.415357\pi\)
0.262790 + 0.964853i \(0.415357\pi\)
\(200\) −2.82843 −0.200000
\(201\) 0 0
\(202\) 3.51472 0.247295
\(203\) 0 0
\(204\) 0 0
\(205\) −6.24264 −0.436005
\(206\) −27.2132 −1.89603
\(207\) 0 0
\(208\) 17.6569 1.22428
\(209\) −1.02944 −0.0712077
\(210\) 0 0
\(211\) 9.00000 0.619586 0.309793 0.950804i \(-0.399740\pi\)
0.309793 + 0.950804i \(0.399740\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 3.51472 0.240261
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) 0 0
\(218\) 7.07107 0.478913
\(219\) 0 0
\(220\) 0 0
\(221\) −14.3137 −0.962844
\(222\) 0 0
\(223\) 24.2132 1.62144 0.810718 0.585437i \(-0.199077\pi\)
0.810718 + 0.585437i \(0.199077\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 18.4853 1.22962
\(227\) 15.7279 1.04390 0.521949 0.852976i \(-0.325205\pi\)
0.521949 + 0.852976i \(0.325205\pi\)
\(228\) 0 0
\(229\) −18.0416 −1.19222 −0.596112 0.802901i \(-0.703289\pi\)
−0.596112 + 0.802901i \(0.703289\pi\)
\(230\) −10.4853 −0.691379
\(231\) 0 0
\(232\) −24.4853 −1.60754
\(233\) 9.17157 0.600850 0.300425 0.953805i \(-0.402872\pi\)
0.300425 + 0.953805i \(0.402872\pi\)
\(234\) 0 0
\(235\) −7.24264 −0.472458
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 17.4853 1.13103 0.565514 0.824738i \(-0.308678\pi\)
0.565514 + 0.824738i \(0.308678\pi\)
\(240\) 0 0
\(241\) 0.727922 0.0468896 0.0234448 0.999725i \(-0.492537\pi\)
0.0234448 + 0.999725i \(0.492537\pi\)
\(242\) −15.5147 −0.997324
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 26.4853 1.68522
\(248\) 28.9706 1.83963
\(249\) 0 0
\(250\) 1.41421 0.0894427
\(251\) 17.2132 1.08649 0.543244 0.839575i \(-0.317196\pi\)
0.543244 + 0.839575i \(0.317196\pi\)
\(252\) 0 0
\(253\) −1.27208 −0.0799749
\(254\) 11.6569 0.731416
\(255\) 0 0
\(256\) 0 0
\(257\) −9.51472 −0.593512 −0.296756 0.954953i \(-0.595905\pi\)
−0.296756 + 0.954953i \(0.595905\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 17.3137 1.06964
\(263\) 16.6274 1.02529 0.512645 0.858601i \(-0.328666\pi\)
0.512645 + 0.858601i \(0.328666\pi\)
\(264\) 0 0
\(265\) −4.24264 −0.260623
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −16.2426 −0.990331 −0.495166 0.868799i \(-0.664893\pi\)
−0.495166 + 0.868799i \(0.664893\pi\)
\(270\) 0 0
\(271\) 0.686292 0.0416892 0.0208446 0.999783i \(-0.493364\pi\)
0.0208446 + 0.999783i \(0.493364\pi\)
\(272\) −12.9706 −0.786456
\(273\) 0 0
\(274\) −16.9706 −1.02523
\(275\) 0.171573 0.0103462
\(276\) 0 0
\(277\) 15.2132 0.914073 0.457036 0.889448i \(-0.348911\pi\)
0.457036 + 0.889448i \(0.348911\pi\)
\(278\) −10.9706 −0.657971
\(279\) 0 0
\(280\) 0 0
\(281\) −2.31371 −0.138024 −0.0690121 0.997616i \(-0.521985\pi\)
−0.0690121 + 0.997616i \(0.521985\pi\)
\(282\) 0 0
\(283\) −24.5563 −1.45972 −0.729862 0.683595i \(-0.760416\pi\)
−0.729862 + 0.683595i \(0.760416\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −1.07107 −0.0633336
\(287\) 0 0
\(288\) 0 0
\(289\) −6.48528 −0.381487
\(290\) 12.2426 0.718913
\(291\) 0 0
\(292\) 0 0
\(293\) 25.7279 1.50304 0.751521 0.659710i \(-0.229321\pi\)
0.751521 + 0.659710i \(0.229321\pi\)
\(294\) 0 0
\(295\) −2.24264 −0.130572
\(296\) −6.34315 −0.368688
\(297\) 0 0
\(298\) 12.9706 0.751365
\(299\) 32.7279 1.89270
\(300\) 0 0
\(301\) 0 0
\(302\) −10.5858 −0.609144
\(303\) 0 0
\(304\) 24.0000 1.37649
\(305\) 2.82843 0.161955
\(306\) 0 0
\(307\) −30.8995 −1.76353 −0.881764 0.471691i \(-0.843644\pi\)
−0.881764 + 0.471691i \(0.843644\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −14.4853 −0.822709
\(311\) −10.0000 −0.567048 −0.283524 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) 0 0
\(313\) 18.2132 1.02947 0.514736 0.857349i \(-0.327890\pi\)
0.514736 + 0.857349i \(0.327890\pi\)
\(314\) 21.4558 1.21082
\(315\) 0 0
\(316\) 0 0
\(317\) 0.343146 0.0192730 0.00963649 0.999954i \(-0.496933\pi\)
0.00963649 + 0.999954i \(0.496933\pi\)
\(318\) 0 0
\(319\) 1.48528 0.0831598
\(320\) 8.00000 0.447214
\(321\) 0 0
\(322\) 0 0
\(323\) −19.4558 −1.08255
\(324\) 0 0
\(325\) −4.41421 −0.244857
\(326\) −14.4853 −0.802266
\(327\) 0 0
\(328\) 17.6569 0.974937
\(329\) 0 0
\(330\) 0 0
\(331\) −27.4558 −1.50911 −0.754555 0.656237i \(-0.772147\pi\)
−0.754555 + 0.656237i \(0.772147\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 1.07107 0.0586062
\(335\) −8.24264 −0.450344
\(336\) 0 0
\(337\) 22.2426 1.21163 0.605817 0.795604i \(-0.292846\pi\)
0.605817 + 0.795604i \(0.292846\pi\)
\(338\) 9.17157 0.498868
\(339\) 0 0
\(340\) 0 0
\(341\) −1.75736 −0.0951663
\(342\) 0 0
\(343\) 0 0
\(344\) −5.65685 −0.304997
\(345\) 0 0
\(346\) −10.2426 −0.550648
\(347\) −13.0711 −0.701692 −0.350846 0.936433i \(-0.614106\pi\)
−0.350846 + 0.936433i \(0.614106\pi\)
\(348\) 0 0
\(349\) 10.9706 0.587241 0.293620 0.955922i \(-0.405140\pi\)
0.293620 + 0.955922i \(0.405140\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.21320 −0.330695 −0.165348 0.986235i \(-0.552875\pi\)
−0.165348 + 0.986235i \(0.552875\pi\)
\(354\) 0 0
\(355\) 3.17157 0.168330
\(356\) 0 0
\(357\) 0 0
\(358\) 20.4853 1.08268
\(359\) 11.3137 0.597115 0.298557 0.954392i \(-0.403495\pi\)
0.298557 + 0.954392i \(0.403495\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 26.4853 1.39204
\(363\) 0 0
\(364\) 0 0
\(365\) 8.48528 0.444140
\(366\) 0 0
\(367\) 23.8701 1.24601 0.623003 0.782219i \(-0.285912\pi\)
0.623003 + 0.782219i \(0.285912\pi\)
\(368\) 29.6569 1.54597
\(369\) 0 0
\(370\) 3.17157 0.164882
\(371\) 0 0
\(372\) 0 0
\(373\) −16.4853 −0.853576 −0.426788 0.904352i \(-0.640355\pi\)
−0.426788 + 0.904352i \(0.640355\pi\)
\(374\) 0.786797 0.0406843
\(375\) 0 0
\(376\) 20.4853 1.05645
\(377\) −38.2132 −1.96808
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −19.7574 −1.01087
\(383\) 4.48528 0.229187 0.114594 0.993412i \(-0.463443\pi\)
0.114594 + 0.993412i \(0.463443\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −22.6274 −1.15171
\(387\) 0 0
\(388\) 0 0
\(389\) 6.85786 0.347708 0.173854 0.984771i \(-0.444378\pi\)
0.173854 + 0.984771i \(0.444378\pi\)
\(390\) 0 0
\(391\) −24.0416 −1.21584
\(392\) 0 0
\(393\) 0 0
\(394\) −18.9706 −0.955723
\(395\) 1.48528 0.0747326
\(396\) 0 0
\(397\) −1.58579 −0.0795883 −0.0397942 0.999208i \(-0.512670\pi\)
−0.0397942 + 0.999208i \(0.512670\pi\)
\(398\) 10.4853 0.525580
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −11.8284 −0.590683 −0.295342 0.955392i \(-0.595434\pi\)
−0.295342 + 0.955392i \(0.595434\pi\)
\(402\) 0 0
\(403\) 45.2132 2.25223
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.384776 0.0190727
\(408\) 0 0
\(409\) −2.48528 −0.122889 −0.0614446 0.998110i \(-0.519571\pi\)
−0.0614446 + 0.998110i \(0.519571\pi\)
\(410\) −8.82843 −0.436005
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) −1.45584 −0.0712077
\(419\) 18.7279 0.914919 0.457459 0.889230i \(-0.348759\pi\)
0.457459 + 0.889230i \(0.348759\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 12.7279 0.619586
\(423\) 0 0
\(424\) 12.0000 0.582772
\(425\) 3.24264 0.157291
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 2.82843 0.136399
\(431\) 28.7990 1.38720 0.693599 0.720361i \(-0.256024\pi\)
0.693599 + 0.720361i \(0.256024\pi\)
\(432\) 0 0
\(433\) −10.9706 −0.527212 −0.263606 0.964630i \(-0.584912\pi\)
−0.263606 + 0.964630i \(0.584912\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 44.4853 2.12802
\(438\) 0 0
\(439\) 30.3848 1.45019 0.725093 0.688651i \(-0.241797\pi\)
0.725093 + 0.688651i \(0.241797\pi\)
\(440\) −0.485281 −0.0231349
\(441\) 0 0
\(442\) −20.2426 −0.962844
\(443\) −15.1716 −0.720823 −0.360412 0.932793i \(-0.617364\pi\)
−0.360412 + 0.932793i \(0.617364\pi\)
\(444\) 0 0
\(445\) −8.00000 −0.379236
\(446\) 34.2426 1.62144
\(447\) 0 0
\(448\) 0 0
\(449\) 24.1716 1.14073 0.570364 0.821392i \(-0.306802\pi\)
0.570364 + 0.821392i \(0.306802\pi\)
\(450\) 0 0
\(451\) −1.07107 −0.0504346
\(452\) 0 0
\(453\) 0 0
\(454\) 22.2426 1.04390
\(455\) 0 0
\(456\) 0 0
\(457\) −11.7574 −0.549986 −0.274993 0.961446i \(-0.588676\pi\)
−0.274993 + 0.961446i \(0.588676\pi\)
\(458\) −25.5147 −1.19222
\(459\) 0 0
\(460\) 0 0
\(461\) 3.02944 0.141095 0.0705475 0.997508i \(-0.477525\pi\)
0.0705475 + 0.997508i \(0.477525\pi\)
\(462\) 0 0
\(463\) −21.4558 −0.997138 −0.498569 0.866850i \(-0.666141\pi\)
−0.498569 + 0.866850i \(0.666141\pi\)
\(464\) −34.6274 −1.60754
\(465\) 0 0
\(466\) 12.9706 0.600850
\(467\) −5.72792 −0.265057 −0.132528 0.991179i \(-0.542310\pi\)
−0.132528 + 0.991179i \(0.542310\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −10.2426 −0.472458
\(471\) 0 0
\(472\) 6.34315 0.291967
\(473\) 0.343146 0.0157779
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 0 0
\(477\) 0 0
\(478\) 24.7279 1.13103
\(479\) 11.7574 0.537207 0.268604 0.963251i \(-0.413438\pi\)
0.268604 + 0.963251i \(0.413438\pi\)
\(480\) 0 0
\(481\) −9.89949 −0.451378
\(482\) 1.02944 0.0468896
\(483\) 0 0
\(484\) 0 0
\(485\) −13.2426 −0.601317
\(486\) 0 0
\(487\) 27.6985 1.25514 0.627569 0.778561i \(-0.284050\pi\)
0.627569 + 0.778561i \(0.284050\pi\)
\(488\) −8.00000 −0.362143
\(489\) 0 0
\(490\) 0 0
\(491\) 37.2843 1.68262 0.841308 0.540556i \(-0.181786\pi\)
0.841308 + 0.540556i \(0.181786\pi\)
\(492\) 0 0
\(493\) 28.0711 1.26426
\(494\) 37.4558 1.68522
\(495\) 0 0
\(496\) 40.9706 1.83963
\(497\) 0 0
\(498\) 0 0
\(499\) 3.00000 0.134298 0.0671492 0.997743i \(-0.478610\pi\)
0.0671492 + 0.997743i \(0.478610\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 24.3431 1.08649
\(503\) −41.2426 −1.83892 −0.919459 0.393185i \(-0.871373\pi\)
−0.919459 + 0.393185i \(0.871373\pi\)
\(504\) 0 0
\(505\) 2.48528 0.110594
\(506\) −1.79899 −0.0799749
\(507\) 0 0
\(508\) 0 0
\(509\) −25.2132 −1.11756 −0.558778 0.829317i \(-0.688730\pi\)
−0.558778 + 0.829317i \(0.688730\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −22.6274 −1.00000
\(513\) 0 0
\(514\) −13.4558 −0.593512
\(515\) −19.2426 −0.847932
\(516\) 0 0
\(517\) −1.24264 −0.0546513
\(518\) 0 0
\(519\) 0 0
\(520\) 12.4853 0.547516
\(521\) 14.9706 0.655872 0.327936 0.944700i \(-0.393647\pi\)
0.327936 + 0.944700i \(0.393647\pi\)
\(522\) 0 0
\(523\) −32.4853 −1.42048 −0.710241 0.703959i \(-0.751414\pi\)
−0.710241 + 0.703959i \(0.751414\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 23.5147 1.02529
\(527\) −33.2132 −1.44679
\(528\) 0 0
\(529\) 31.9706 1.39002
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) 0 0
\(533\) 27.5563 1.19360
\(534\) 0 0
\(535\) 2.48528 0.107448
\(536\) 23.3137 1.00700
\(537\) 0 0
\(538\) −22.9706 −0.990331
\(539\) 0 0
\(540\) 0 0
\(541\) −21.9706 −0.944588 −0.472294 0.881441i \(-0.656574\pi\)
−0.472294 + 0.881441i \(0.656574\pi\)
\(542\) 0.970563 0.0416892
\(543\) 0 0
\(544\) 0 0
\(545\) 5.00000 0.214176
\(546\) 0 0
\(547\) −24.4853 −1.04692 −0.523458 0.852052i \(-0.675358\pi\)
−0.523458 + 0.852052i \(0.675358\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0.242641 0.0103462
\(551\) −51.9411 −2.21277
\(552\) 0 0
\(553\) 0 0
\(554\) 21.5147 0.914073
\(555\) 0 0
\(556\) 0 0
\(557\) 7.79899 0.330454 0.165227 0.986256i \(-0.447164\pi\)
0.165227 + 0.986256i \(0.447164\pi\)
\(558\) 0 0
\(559\) −8.82843 −0.373403
\(560\) 0 0
\(561\) 0 0
\(562\) −3.27208 −0.138024
\(563\) −31.9411 −1.34616 −0.673079 0.739571i \(-0.735029\pi\)
−0.673079 + 0.739571i \(0.735029\pi\)
\(564\) 0 0
\(565\) 13.0711 0.549904
\(566\) −34.7279 −1.45972
\(567\) 0 0
\(568\) −8.97056 −0.376396
\(569\) −26.1421 −1.09594 −0.547968 0.836500i \(-0.684598\pi\)
−0.547968 + 0.836500i \(0.684598\pi\)
\(570\) 0 0
\(571\) −17.5147 −0.732968 −0.366484 0.930424i \(-0.619439\pi\)
−0.366484 + 0.930424i \(0.619439\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −7.41421 −0.309194
\(576\) 0 0
\(577\) −15.7279 −0.654762 −0.327381 0.944892i \(-0.606166\pi\)
−0.327381 + 0.944892i \(0.606166\pi\)
\(578\) −9.17157 −0.381487
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −0.727922 −0.0301475
\(584\) −24.0000 −0.993127
\(585\) 0 0
\(586\) 36.3848 1.50304
\(587\) −37.4558 −1.54597 −0.772984 0.634425i \(-0.781237\pi\)
−0.772984 + 0.634425i \(0.781237\pi\)
\(588\) 0 0
\(589\) 61.4558 2.53224
\(590\) −3.17157 −0.130572
\(591\) 0 0
\(592\) −8.97056 −0.368688
\(593\) 19.2426 0.790201 0.395100 0.918638i \(-0.370710\pi\)
0.395100 + 0.918638i \(0.370710\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 46.2843 1.89270
\(599\) 17.8284 0.728450 0.364225 0.931311i \(-0.381334\pi\)
0.364225 + 0.931311i \(0.381334\pi\)
\(600\) 0 0
\(601\) 10.9706 0.447499 0.223749 0.974647i \(-0.428170\pi\)
0.223749 + 0.974647i \(0.428170\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −10.9706 −0.446017
\(606\) 0 0
\(607\) 5.10051 0.207023 0.103512 0.994628i \(-0.466992\pi\)
0.103512 + 0.994628i \(0.466992\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) 31.9706 1.29339
\(612\) 0 0
\(613\) −23.9411 −0.966973 −0.483486 0.875352i \(-0.660630\pi\)
−0.483486 + 0.875352i \(0.660630\pi\)
\(614\) −43.6985 −1.76353
\(615\) 0 0
\(616\) 0 0
\(617\) −4.58579 −0.184617 −0.0923084 0.995730i \(-0.529425\pi\)
−0.0923084 + 0.995730i \(0.529425\pi\)
\(618\) 0 0
\(619\) −16.9289 −0.680431 −0.340216 0.940347i \(-0.610500\pi\)
−0.340216 + 0.940347i \(0.610500\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −14.1421 −0.567048
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 25.7574 1.02947
\(627\) 0 0
\(628\) 0 0
\(629\) 7.27208 0.289957
\(630\) 0 0
\(631\) −42.4558 −1.69014 −0.845070 0.534655i \(-0.820441\pi\)
−0.845070 + 0.534655i \(0.820441\pi\)
\(632\) −4.20101 −0.167107
\(633\) 0 0
\(634\) 0.485281 0.0192730
\(635\) 8.24264 0.327099
\(636\) 0 0
\(637\) 0 0
\(638\) 2.10051 0.0831598
\(639\) 0 0
\(640\) 11.3137 0.447214
\(641\) 23.3137 0.920836 0.460418 0.887702i \(-0.347700\pi\)
0.460418 + 0.887702i \(0.347700\pi\)
\(642\) 0 0
\(643\) −2.27208 −0.0896020 −0.0448010 0.998996i \(-0.514265\pi\)
−0.0448010 + 0.998996i \(0.514265\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −27.5147 −1.08255
\(647\) −11.5147 −0.452690 −0.226345 0.974047i \(-0.572678\pi\)
−0.226345 + 0.974047i \(0.572678\pi\)
\(648\) 0 0
\(649\) −0.384776 −0.0151038
\(650\) −6.24264 −0.244857
\(651\) 0 0
\(652\) 0 0
\(653\) −34.9706 −1.36850 −0.684252 0.729246i \(-0.739871\pi\)
−0.684252 + 0.729246i \(0.739871\pi\)
\(654\) 0 0
\(655\) 12.2426 0.478360
\(656\) 24.9706 0.974937
\(657\) 0 0
\(658\) 0 0
\(659\) −19.9706 −0.777943 −0.388971 0.921250i \(-0.627169\pi\)
−0.388971 + 0.921250i \(0.627169\pi\)
\(660\) 0 0
\(661\) −13.4558 −0.523372 −0.261686 0.965153i \(-0.584278\pi\)
−0.261686 + 0.965153i \(0.584278\pi\)
\(662\) −38.8284 −1.50911
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −64.1838 −2.48521
\(668\) 0 0
\(669\) 0 0
\(670\) −11.6569 −0.450344
\(671\) 0.485281 0.0187341
\(672\) 0 0
\(673\) 3.51472 0.135482 0.0677412 0.997703i \(-0.478421\pi\)
0.0677412 + 0.997703i \(0.478421\pi\)
\(674\) 31.4558 1.21163
\(675\) 0 0
\(676\) 0 0
\(677\) 44.2132 1.69925 0.849626 0.527386i \(-0.176828\pi\)
0.849626 + 0.527386i \(0.176828\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −9.17157 −0.351714
\(681\) 0 0
\(682\) −2.48528 −0.0951663
\(683\) −31.7990 −1.21675 −0.608377 0.793648i \(-0.708179\pi\)
−0.608377 + 0.793648i \(0.708179\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) 18.7279 0.713477
\(690\) 0 0
\(691\) −14.8284 −0.564100 −0.282050 0.959400i \(-0.591014\pi\)
−0.282050 + 0.959400i \(0.591014\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −18.4853 −0.701692
\(695\) −7.75736 −0.294253
\(696\) 0 0
\(697\) −20.2426 −0.766745
\(698\) 15.5147 0.587241
\(699\) 0 0
\(700\) 0 0
\(701\) −46.4558 −1.75461 −0.877307 0.479931i \(-0.840662\pi\)
−0.877307 + 0.479931i \(0.840662\pi\)
\(702\) 0 0
\(703\) −13.4558 −0.507497
\(704\) 1.37258 0.0517312
\(705\) 0 0
\(706\) −8.78680 −0.330695
\(707\) 0 0
\(708\) 0 0
\(709\) 17.0000 0.638448 0.319224 0.947679i \(-0.396578\pi\)
0.319224 + 0.947679i \(0.396578\pi\)
\(710\) 4.48528 0.168330
\(711\) 0 0
\(712\) 22.6274 0.847998
\(713\) 75.9411 2.84402
\(714\) 0 0
\(715\) −0.757359 −0.0283236
\(716\) 0 0
\(717\) 0 0
\(718\) 16.0000 0.597115
\(719\) −25.2132 −0.940294 −0.470147 0.882588i \(-0.655799\pi\)
−0.470147 + 0.882588i \(0.655799\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 24.0416 0.894737
\(723\) 0 0
\(724\) 0 0
\(725\) 8.65685 0.321507
\(726\) 0 0
\(727\) 6.68629 0.247981 0.123990 0.992283i \(-0.460431\pi\)
0.123990 + 0.992283i \(0.460431\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 12.0000 0.444140
\(731\) 6.48528 0.239867
\(732\) 0 0
\(733\) −14.6985 −0.542901 −0.271450 0.962452i \(-0.587503\pi\)
−0.271450 + 0.962452i \(0.587503\pi\)
\(734\) 33.7574 1.24601
\(735\) 0 0
\(736\) 0 0
\(737\) −1.41421 −0.0520932
\(738\) 0 0
\(739\) 29.9706 1.10248 0.551242 0.834345i \(-0.314154\pi\)
0.551242 + 0.834345i \(0.314154\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −11.2721 −0.413532 −0.206766 0.978390i \(-0.566294\pi\)
−0.206766 + 0.978390i \(0.566294\pi\)
\(744\) 0 0
\(745\) 9.17157 0.336020
\(746\) −23.3137 −0.853576
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 29.4853 1.07593 0.537967 0.842966i \(-0.319193\pi\)
0.537967 + 0.842966i \(0.319193\pi\)
\(752\) 28.9706 1.05645
\(753\) 0 0
\(754\) −54.0416 −1.96808
\(755\) −7.48528 −0.272417
\(756\) 0 0
\(757\) 0.485281 0.0176379 0.00881893 0.999961i \(-0.497193\pi\)
0.00881893 + 0.999961i \(0.497193\pi\)
\(758\) 2.82843 0.102733
\(759\) 0 0
\(760\) 16.9706 0.615587
\(761\) −12.7279 −0.461387 −0.230693 0.973026i \(-0.574099\pi\)
−0.230693 + 0.973026i \(0.574099\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 6.34315 0.229187
\(767\) 9.89949 0.357450
\(768\) 0 0
\(769\) 8.82843 0.318361 0.159181 0.987249i \(-0.449115\pi\)
0.159181 + 0.987249i \(0.449115\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −6.21320 −0.223473 −0.111737 0.993738i \(-0.535641\pi\)
−0.111737 + 0.993738i \(0.535641\pi\)
\(774\) 0 0
\(775\) −10.2426 −0.367927
\(776\) 37.4558 1.34459
\(777\) 0 0
\(778\) 9.69848 0.347708
\(779\) 37.4558 1.34199
\(780\) 0 0
\(781\) 0.544156 0.0194714
\(782\) −34.0000 −1.21584
\(783\) 0 0
\(784\) 0 0
\(785\) 15.1716 0.541497
\(786\) 0 0
\(787\) −20.2721 −0.722622 −0.361311 0.932445i \(-0.617671\pi\)
−0.361311 + 0.932445i \(0.617671\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 2.10051 0.0747326
\(791\) 0 0
\(792\) 0 0
\(793\) −12.4853 −0.443365
\(794\) −2.24264 −0.0795883
\(795\) 0 0
\(796\) 0 0
\(797\) −21.1838 −0.750367 −0.375184 0.926950i \(-0.622420\pi\)
−0.375184 + 0.926950i \(0.622420\pi\)
\(798\) 0 0
\(799\) −23.4853 −0.830850
\(800\) 0 0
\(801\) 0 0
\(802\) −16.7279 −0.590683
\(803\) 1.45584 0.0513756
\(804\) 0 0
\(805\) 0 0
\(806\) 63.9411 2.25223
\(807\) 0 0
\(808\) −7.02944 −0.247295
\(809\) 48.5980 1.70861 0.854307 0.519769i \(-0.173982\pi\)
0.854307 + 0.519769i \(0.173982\pi\)
\(810\) 0 0
\(811\) −47.3553 −1.66287 −0.831435 0.555621i \(-0.812480\pi\)
−0.831435 + 0.555621i \(0.812480\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0.544156 0.0190727
\(815\) −10.2426 −0.358784
\(816\) 0 0
\(817\) −12.0000 −0.419827
\(818\) −3.51472 −0.122889
\(819\) 0 0
\(820\) 0 0
\(821\) 23.4853 0.819642 0.409821 0.912166i \(-0.365591\pi\)
0.409821 + 0.912166i \(0.365591\pi\)
\(822\) 0 0
\(823\) 16.7279 0.583099 0.291549 0.956556i \(-0.405829\pi\)
0.291549 + 0.956556i \(0.405829\pi\)
\(824\) 54.4264 1.89603
\(825\) 0 0
\(826\) 0 0
\(827\) −42.0416 −1.46193 −0.730965 0.682415i \(-0.760930\pi\)
−0.730965 + 0.682415i \(0.760930\pi\)
\(828\) 0 0
\(829\) −5.95837 −0.206943 −0.103471 0.994632i \(-0.532995\pi\)
−0.103471 + 0.994632i \(0.532995\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −35.3137 −1.22428
\(833\) 0 0
\(834\) 0 0
\(835\) 0.757359 0.0262095
\(836\) 0 0
\(837\) 0 0
\(838\) 26.4853 0.914919
\(839\) −23.2721 −0.803441 −0.401721 0.915762i \(-0.631588\pi\)
−0.401721 + 0.915762i \(0.631588\pi\)
\(840\) 0 0
\(841\) 45.9411 1.58418
\(842\) −26.8701 −0.926003
\(843\) 0 0
\(844\) 0 0
\(845\) 6.48528 0.223100
\(846\) 0 0
\(847\) 0 0
\(848\) 16.9706 0.582772
\(849\) 0 0
\(850\) 4.58579 0.157291
\(851\) −16.6274 −0.569981
\(852\) 0 0
\(853\) −10.9706 −0.375625 −0.187812 0.982205i \(-0.560140\pi\)
−0.187812 + 0.982205i \(0.560140\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −7.02944 −0.240261
\(857\) −22.4853 −0.768083 −0.384041 0.923316i \(-0.625468\pi\)
−0.384041 + 0.923316i \(0.625468\pi\)
\(858\) 0 0
\(859\) 24.7696 0.845126 0.422563 0.906334i \(-0.361130\pi\)
0.422563 + 0.906334i \(0.361130\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 40.7279 1.38720
\(863\) −18.3848 −0.625825 −0.312913 0.949782i \(-0.601305\pi\)
−0.312913 + 0.949782i \(0.601305\pi\)
\(864\) 0 0
\(865\) −7.24264 −0.246257
\(866\) −15.5147 −0.527212
\(867\) 0 0
\(868\) 0 0
\(869\) 0.254834 0.00864465
\(870\) 0 0
\(871\) 36.3848 1.23285
\(872\) −14.1421 −0.478913
\(873\) 0 0
\(874\) 62.9117 2.12802
\(875\) 0 0
\(876\) 0 0
\(877\) 13.0294 0.439973 0.219986 0.975503i \(-0.429399\pi\)
0.219986 + 0.975503i \(0.429399\pi\)
\(878\) 42.9706 1.45019
\(879\) 0 0
\(880\) −0.686292 −0.0231349
\(881\) −52.9706 −1.78462 −0.892312 0.451420i \(-0.850918\pi\)
−0.892312 + 0.451420i \(0.850918\pi\)
\(882\) 0 0
\(883\) 31.5147 1.06055 0.530277 0.847824i \(-0.322088\pi\)
0.530277 + 0.847824i \(0.322088\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −21.4558 −0.720823
\(887\) 2.97056 0.0997417 0.0498709 0.998756i \(-0.484119\pi\)
0.0498709 + 0.998756i \(0.484119\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −11.3137 −0.379236
\(891\) 0 0
\(892\) 0 0
\(893\) 43.4558 1.45419
\(894\) 0 0
\(895\) 14.4853 0.484190
\(896\) 0 0
\(897\) 0 0
\(898\) 34.1838 1.14073
\(899\) −88.6690 −2.95728
\(900\) 0 0
\(901\) −13.7574 −0.458324
\(902\) −1.51472 −0.0504346
\(903\) 0 0
\(904\) −36.9706 −1.22962
\(905\) 18.7279 0.622537
\(906\) 0 0
\(907\) −44.1838 −1.46710 −0.733549 0.679637i \(-0.762137\pi\)
−0.733549 + 0.679637i \(0.762137\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5.65685 −0.187420 −0.0937100 0.995600i \(-0.529873\pi\)
−0.0937100 + 0.995600i \(0.529873\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −16.6274 −0.549986
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 12.4558 0.410880 0.205440 0.978670i \(-0.434137\pi\)
0.205440 + 0.978670i \(0.434137\pi\)
\(920\) 20.9706 0.691379
\(921\) 0 0
\(922\) 4.28427 0.141095
\(923\) −14.0000 −0.460816
\(924\) 0 0
\(925\) 2.24264 0.0737376
\(926\) −30.3431 −0.997138
\(927\) 0 0
\(928\) 0 0
\(929\) 29.2721 0.960386 0.480193 0.877163i \(-0.340567\pi\)
0.480193 + 0.877163i \(0.340567\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) −8.10051 −0.265057
\(935\) 0.556349 0.0181946
\(936\) 0 0
\(937\) 48.5563 1.58627 0.793133 0.609048i \(-0.208448\pi\)
0.793133 + 0.609048i \(0.208448\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 28.9706 0.944413 0.472207 0.881488i \(-0.343458\pi\)
0.472207 + 0.881488i \(0.343458\pi\)
\(942\) 0 0
\(943\) 46.2843 1.50722
\(944\) 8.97056 0.291967
\(945\) 0 0
\(946\) 0.485281 0.0157779
\(947\) 52.2426 1.69766 0.848829 0.528668i \(-0.177308\pi\)
0.848829 + 0.528668i \(0.177308\pi\)
\(948\) 0 0
\(949\) −37.4558 −1.21587
\(950\) −8.48528 −0.275299
\(951\) 0 0
\(952\) 0 0
\(953\) −53.0122 −1.71723 −0.858617 0.512618i \(-0.828676\pi\)
−0.858617 + 0.512618i \(0.828676\pi\)
\(954\) 0 0
\(955\) −13.9706 −0.452077
\(956\) 0 0
\(957\) 0 0
\(958\) 16.6274 0.537207
\(959\) 0 0
\(960\) 0 0
\(961\) 73.9117 2.38425
\(962\) −14.0000 −0.451378
\(963\) 0 0
\(964\) 0 0
\(965\) −16.0000 −0.515058
\(966\) 0 0
\(967\) −60.4264 −1.94318 −0.971591 0.236666i \(-0.923945\pi\)
−0.971591 + 0.236666i \(0.923945\pi\)
\(968\) 31.0294 0.997324
\(969\) 0 0
\(970\) −18.7279 −0.601317
\(971\) −17.2721 −0.554287 −0.277144 0.960828i \(-0.589388\pi\)
−0.277144 + 0.960828i \(0.589388\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 39.1716 1.25514
\(975\) 0 0
\(976\) −11.3137 −0.362143
\(977\) −42.7696 −1.36832 −0.684160 0.729332i \(-0.739831\pi\)
−0.684160 + 0.729332i \(0.739831\pi\)
\(978\) 0 0
\(979\) −1.37258 −0.0438679
\(980\) 0 0
\(981\) 0 0
\(982\) 52.7279 1.68262
\(983\) 0.213203 0.00680013 0.00340007 0.999994i \(-0.498918\pi\)
0.00340007 + 0.999994i \(0.498918\pi\)
\(984\) 0 0
\(985\) −13.4142 −0.427412
\(986\) 39.6985 1.26426
\(987\) 0 0
\(988\) 0 0
\(989\) −14.8284 −0.471517
\(990\) 0 0
\(991\) 19.9411 0.633451 0.316725 0.948517i \(-0.397417\pi\)
0.316725 + 0.948517i \(0.397417\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 7.41421 0.235046
\(996\) 0 0
\(997\) 21.7279 0.688130 0.344065 0.938946i \(-0.388196\pi\)
0.344065 + 0.938946i \(0.388196\pi\)
\(998\) 4.24264 0.134298
\(999\) 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 2205.2.a.v.1.2 2
3.2 odd 2 245.2.a.e.1.1 2
7.6 odd 2 2205.2.a.t.1.2 2
12.11 even 2 3920.2.a.bw.1.1 2
15.2 even 4 1225.2.b.i.99.2 4
15.8 even 4 1225.2.b.i.99.3 4
15.14 odd 2 1225.2.a.r.1.2 2
21.2 odd 6 245.2.e.g.116.2 4
21.5 even 6 245.2.e.f.116.2 4
21.11 odd 6 245.2.e.g.226.2 4
21.17 even 6 245.2.e.f.226.2 4
21.20 even 2 245.2.a.f.1.1 yes 2
84.83 odd 2 3920.2.a.br.1.2 2
105.62 odd 4 1225.2.b.j.99.1 4
105.83 odd 4 1225.2.b.j.99.4 4
105.104 even 2 1225.2.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.2.a.e.1.1 2 3.2 odd 2
245.2.a.f.1.1 yes 2 21.20 even 2
245.2.e.f.116.2 4 21.5 even 6
245.2.e.f.226.2 4 21.17 even 6
245.2.e.g.116.2 4 21.2 odd 6
245.2.e.g.226.2 4 21.11 odd 6
1225.2.a.p.1.2 2 105.104 even 2
1225.2.a.r.1.2 2 15.14 odd 2
1225.2.b.i.99.2 4 15.2 even 4
1225.2.b.i.99.3 4 15.8 even 4
1225.2.b.j.99.1 4 105.62 odd 4
1225.2.b.j.99.4 4 105.83 odd 4
2205.2.a.t.1.2 2 7.6 odd 2
2205.2.a.v.1.2 2 1.1 even 1 trivial
3920.2.a.br.1.2 2 84.83 odd 2
3920.2.a.bw.1.1 2 12.11 even 2