Properties

Label 1216.2.c.g.609.3
Level $1216$
Weight $2$
Character 1216.609
Analytic conductor $9.710$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 609.3
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1216.609
Dual form 1216.2.c.g.609.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} -3.46410i q^{5} +1.73205 q^{7} +2.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -3.46410i q^{5} +1.73205 q^{7} +2.00000 q^{9} -5.19615i q^{13} +3.46410 q^{15} -3.00000 q^{17} -1.00000i q^{19} +1.73205i q^{21} -1.73205 q^{23} -7.00000 q^{25} +5.00000i q^{27} -1.73205i q^{29} -3.46410 q^{31} -6.00000i q^{35} +5.19615 q^{39} -10.0000i q^{43} -6.92820i q^{45} +10.3923 q^{47} -4.00000 q^{49} -3.00000i q^{51} +5.19615i q^{53} +1.00000 q^{57} +9.00000i q^{59} -10.3923i q^{61} +3.46410 q^{63} -18.0000 q^{65} -13.0000i q^{67} -1.73205i q^{69} -3.46410 q^{71} -1.00000 q^{73} -7.00000i q^{75} +10.3923 q^{79} +1.00000 q^{81} +10.3923i q^{85} +1.73205 q^{87} +18.0000 q^{89} -9.00000i q^{91} -3.46410i q^{93} -3.46410 q^{95} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{9} + O(q^{10}) \) \( 4q + 8q^{9} - 12q^{17} - 28q^{25} - 16q^{49} + 4q^{57} - 72q^{65} - 4q^{73} + 4q^{81} + 72q^{89} - 56q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) 0 0
\(5\) − 3.46410i − 1.54919i −0.632456 0.774597i \(-0.717953\pi\)
0.632456 0.774597i \(-0.282047\pi\)
\(6\) 0 0
\(7\) 1.73205 0.654654 0.327327 0.944911i \(-0.393852\pi\)
0.327327 + 0.944911i \(0.393852\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) − 5.19615i − 1.44115i −0.693375 0.720577i \(-0.743877\pi\)
0.693375 0.720577i \(-0.256123\pi\)
\(14\) 0 0
\(15\) 3.46410 0.894427
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) − 1.00000i − 0.229416i
\(20\) 0 0
\(21\) 1.73205i 0.377964i
\(22\) 0 0
\(23\) −1.73205 −0.361158 −0.180579 0.983561i \(-0.557797\pi\)
−0.180579 + 0.983561i \(0.557797\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) 5.00000i 0.962250i
\(28\) 0 0
\(29\) − 1.73205i − 0.321634i −0.986984 0.160817i \(-0.948587\pi\)
0.986984 0.160817i \(-0.0514129\pi\)
\(30\) 0 0
\(31\) −3.46410 −0.622171 −0.311086 0.950382i \(-0.600693\pi\)
−0.311086 + 0.950382i \(0.600693\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 6.00000i − 1.01419i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 5.19615 0.832050
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) − 10.0000i − 1.52499i −0.646997 0.762493i \(-0.723975\pi\)
0.646997 0.762493i \(-0.276025\pi\)
\(44\) 0 0
\(45\) − 6.92820i − 1.03280i
\(46\) 0 0
\(47\) 10.3923 1.51587 0.757937 0.652328i \(-0.226208\pi\)
0.757937 + 0.652328i \(0.226208\pi\)
\(48\) 0 0
\(49\) −4.00000 −0.571429
\(50\) 0 0
\(51\) − 3.00000i − 0.420084i
\(52\) 0 0
\(53\) 5.19615i 0.713746i 0.934153 + 0.356873i \(0.116157\pi\)
−0.934153 + 0.356873i \(0.883843\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 9.00000i 1.17170i 0.810419 + 0.585850i \(0.199239\pi\)
−0.810419 + 0.585850i \(0.800761\pi\)
\(60\) 0 0
\(61\) − 10.3923i − 1.33060i −0.746577 0.665299i \(-0.768304\pi\)
0.746577 0.665299i \(-0.231696\pi\)
\(62\) 0 0
\(63\) 3.46410 0.436436
\(64\) 0 0
\(65\) −18.0000 −2.23263
\(66\) 0 0
\(67\) − 13.0000i − 1.58820i −0.607785 0.794101i \(-0.707942\pi\)
0.607785 0.794101i \(-0.292058\pi\)
\(68\) 0 0
\(69\) − 1.73205i − 0.208514i
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 0 0
\(75\) − 7.00000i − 0.808290i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.3923 1.16923 0.584613 0.811312i \(-0.301246\pi\)
0.584613 + 0.811312i \(0.301246\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 10.3923i 1.12720i
\(86\) 0 0
\(87\) 1.73205 0.185695
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) − 9.00000i − 0.943456i
\(92\) 0 0
\(93\) − 3.46410i − 0.359211i
\(94\) 0 0
\(95\) −3.46410 −0.355409
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 13.8564 1.36531 0.682656 0.730740i \(-0.260825\pi\)
0.682656 + 0.730740i \(0.260825\pi\)
\(104\) 0 0
\(105\) 6.00000 0.585540
\(106\) 0 0
\(107\) 3.00000i 0.290021i 0.989430 + 0.145010i \(0.0463216\pi\)
−0.989430 + 0.145010i \(0.953678\pi\)
\(108\) 0 0
\(109\) − 5.19615i − 0.497701i −0.968542 0.248851i \(-0.919947\pi\)
0.968542 0.248851i \(-0.0800528\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 6.00000i 0.559503i
\(116\) 0 0
\(117\) − 10.3923i − 0.960769i
\(118\) 0 0
\(119\) −5.19615 −0.476331
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.92820i 0.619677i
\(126\) 0 0
\(127\) −10.3923 −0.922168 −0.461084 0.887357i \(-0.652539\pi\)
−0.461084 + 0.887357i \(0.652539\pi\)
\(128\) 0 0
\(129\) 10.0000 0.880451
\(130\) 0 0
\(131\) 18.0000i 1.57267i 0.617802 + 0.786334i \(0.288023\pi\)
−0.617802 + 0.786334i \(0.711977\pi\)
\(132\) 0 0
\(133\) − 1.73205i − 0.150188i
\(134\) 0 0
\(135\) 17.3205 1.49071
\(136\) 0 0
\(137\) 21.0000 1.79415 0.897076 0.441877i \(-0.145687\pi\)
0.897076 + 0.441877i \(0.145687\pi\)
\(138\) 0 0
\(139\) 22.0000i 1.86602i 0.359856 + 0.933008i \(0.382826\pi\)
−0.359856 + 0.933008i \(0.617174\pi\)
\(140\) 0 0
\(141\) 10.3923i 0.875190i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) − 4.00000i − 0.329914i
\(148\) 0 0
\(149\) − 20.7846i − 1.70274i −0.524564 0.851371i \(-0.675772\pi\)
0.524564 0.851371i \(-0.324228\pi\)
\(150\) 0 0
\(151\) −13.8564 −1.12762 −0.563809 0.825905i \(-0.690665\pi\)
−0.563809 + 0.825905i \(0.690665\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 12.0000i 0.963863i
\(156\) 0 0
\(157\) 3.46410i 0.276465i 0.990400 + 0.138233i \(0.0441422\pi\)
−0.990400 + 0.138233i \(0.955858\pi\)
\(158\) 0 0
\(159\) −5.19615 −0.412082
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) − 14.0000i − 1.09656i −0.836293 0.548282i \(-0.815282\pi\)
0.836293 0.548282i \(-0.184718\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.8564 1.07224 0.536120 0.844141i \(-0.319889\pi\)
0.536120 + 0.844141i \(0.319889\pi\)
\(168\) 0 0
\(169\) −14.0000 −1.07692
\(170\) 0 0
\(171\) − 2.00000i − 0.152944i
\(172\) 0 0
\(173\) 20.7846i 1.58022i 0.612962 + 0.790112i \(0.289978\pi\)
−0.612962 + 0.790112i \(0.710022\pi\)
\(174\) 0 0
\(175\) −12.1244 −0.916515
\(176\) 0 0
\(177\) −9.00000 −0.676481
\(178\) 0 0
\(179\) 12.0000i 0.896922i 0.893802 + 0.448461i \(0.148028\pi\)
−0.893802 + 0.448461i \(0.851972\pi\)
\(180\) 0 0
\(181\) 13.8564i 1.02994i 0.857209 + 0.514969i \(0.172197\pi\)
−0.857209 + 0.514969i \(0.827803\pi\)
\(182\) 0 0
\(183\) 10.3923 0.768221
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 8.66025i 0.629941i
\(190\) 0 0
\(191\) −1.73205 −0.125327 −0.0626634 0.998035i \(-0.519959\pi\)
−0.0626634 + 0.998035i \(0.519959\pi\)
\(192\) 0 0
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) 0 0
\(195\) − 18.0000i − 1.28901i
\(196\) 0 0
\(197\) − 13.8564i − 0.987228i −0.869681 0.493614i \(-0.835676\pi\)
0.869681 0.493614i \(-0.164324\pi\)
\(198\) 0 0
\(199\) −22.5167 −1.59616 −0.798082 0.602549i \(-0.794152\pi\)
−0.798082 + 0.602549i \(0.794152\pi\)
\(200\) 0 0
\(201\) 13.0000 0.916949
\(202\) 0 0
\(203\) − 3.00000i − 0.210559i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −3.46410 −0.240772
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 5.00000i 0.344214i 0.985078 + 0.172107i \(0.0550575\pi\)
−0.985078 + 0.172107i \(0.944942\pi\)
\(212\) 0 0
\(213\) − 3.46410i − 0.237356i
\(214\) 0 0
\(215\) −34.6410 −2.36250
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) 0 0
\(219\) − 1.00000i − 0.0675737i
\(220\) 0 0
\(221\) 15.5885i 1.04859i
\(222\) 0 0
\(223\) −6.92820 −0.463947 −0.231973 0.972722i \(-0.574518\pi\)
−0.231973 + 0.972722i \(0.574518\pi\)
\(224\) 0 0
\(225\) −14.0000 −0.933333
\(226\) 0 0
\(227\) 15.0000i 0.995585i 0.867296 + 0.497792i \(0.165856\pi\)
−0.867296 + 0.497792i \(0.834144\pi\)
\(228\) 0 0
\(229\) − 3.46410i − 0.228914i −0.993428 0.114457i \(-0.963487\pi\)
0.993428 0.114457i \(-0.0365129\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) − 36.0000i − 2.34838i
\(236\) 0 0
\(237\) 10.3923i 0.675053i
\(238\) 0 0
\(239\) −19.0526 −1.23241 −0.616204 0.787587i \(-0.711330\pi\)
−0.616204 + 0.787587i \(0.711330\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 16.0000i 1.02640i
\(244\) 0 0
\(245\) 13.8564i 0.885253i
\(246\) 0 0
\(247\) −5.19615 −0.330623
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 18.0000i − 1.13615i −0.822977 0.568075i \(-0.807688\pi\)
0.822977 0.568075i \(-0.192312\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −10.3923 −0.650791
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) − 3.46410i − 0.214423i
\(262\) 0 0
\(263\) −24.2487 −1.49524 −0.747620 0.664127i \(-0.768803\pi\)
−0.747620 + 0.664127i \(0.768803\pi\)
\(264\) 0 0
\(265\) 18.0000 1.10573
\(266\) 0 0
\(267\) 18.0000i 1.10158i
\(268\) 0 0
\(269\) 27.7128i 1.68968i 0.535019 + 0.844840i \(0.320304\pi\)
−0.535019 + 0.844840i \(0.679696\pi\)
\(270\) 0 0
\(271\) 22.5167 1.36779 0.683895 0.729581i \(-0.260285\pi\)
0.683895 + 0.729581i \(0.260285\pi\)
\(272\) 0 0
\(273\) 9.00000 0.544705
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 10.3923i − 0.624413i −0.950014 0.312207i \(-0.898932\pi\)
0.950014 0.312207i \(-0.101068\pi\)
\(278\) 0 0
\(279\) −6.92820 −0.414781
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) 28.0000i 1.66443i 0.554455 + 0.832214i \(0.312927\pi\)
−0.554455 + 0.832214i \(0.687073\pi\)
\(284\) 0 0
\(285\) − 3.46410i − 0.205196i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) − 14.0000i − 0.820695i
\(292\) 0 0
\(293\) − 1.73205i − 0.101187i −0.998719 0.0505937i \(-0.983889\pi\)
0.998719 0.0505937i \(-0.0161114\pi\)
\(294\) 0 0
\(295\) 31.1769 1.81519
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 9.00000i 0.520483i
\(300\) 0 0
\(301\) − 17.3205i − 0.998337i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −36.0000 −2.06135
\(306\) 0 0
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) 0 0
\(309\) 13.8564i 0.788263i
\(310\) 0 0
\(311\) −1.73205 −0.0982156 −0.0491078 0.998793i \(-0.515638\pi\)
−0.0491078 + 0.998793i \(0.515638\pi\)
\(312\) 0 0
\(313\) 19.0000 1.07394 0.536972 0.843600i \(-0.319568\pi\)
0.536972 + 0.843600i \(0.319568\pi\)
\(314\) 0 0
\(315\) − 12.0000i − 0.676123i
\(316\) 0 0
\(317\) 8.66025i 0.486408i 0.969975 + 0.243204i \(0.0781985\pi\)
−0.969975 + 0.243204i \(0.921801\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) 3.00000i 0.166924i
\(324\) 0 0
\(325\) 36.3731i 2.01761i
\(326\) 0 0
\(327\) 5.19615 0.287348
\(328\) 0 0
\(329\) 18.0000 0.992372
\(330\) 0 0
\(331\) 7.00000i 0.384755i 0.981321 + 0.192377i \(0.0616198\pi\)
−0.981321 + 0.192377i \(0.938380\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −45.0333 −2.46043
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) 6.00000i 0.325875i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −19.0526 −1.02874
\(344\) 0 0
\(345\) −6.00000 −0.323029
\(346\) 0 0
\(347\) 6.00000i 0.322097i 0.986947 + 0.161048i \(0.0514875\pi\)
−0.986947 + 0.161048i \(0.948512\pi\)
\(348\) 0 0
\(349\) − 10.3923i − 0.556287i −0.960539 0.278144i \(-0.910281\pi\)
0.960539 0.278144i \(-0.0897191\pi\)
\(350\) 0 0
\(351\) 25.9808 1.38675
\(352\) 0 0
\(353\) 27.0000 1.43706 0.718532 0.695493i \(-0.244814\pi\)
0.718532 + 0.695493i \(0.244814\pi\)
\(354\) 0 0
\(355\) 12.0000i 0.636894i
\(356\) 0 0
\(357\) − 5.19615i − 0.275010i
\(358\) 0 0
\(359\) −8.66025 −0.457071 −0.228535 0.973536i \(-0.573394\pi\)
−0.228535 + 0.973536i \(0.573394\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 11.0000i 0.577350i
\(364\) 0 0
\(365\) 3.46410i 0.181319i
\(366\) 0 0
\(367\) 31.1769 1.62742 0.813711 0.581270i \(-0.197444\pi\)
0.813711 + 0.581270i \(0.197444\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 9.00000i 0.467257i
\(372\) 0 0
\(373\) 12.1244i 0.627775i 0.949460 + 0.313888i \(0.101632\pi\)
−0.949460 + 0.313888i \(0.898368\pi\)
\(374\) 0 0
\(375\) −6.92820 −0.357771
\(376\) 0 0
\(377\) −9.00000 −0.463524
\(378\) 0 0
\(379\) − 1.00000i − 0.0513665i −0.999670 0.0256833i \(-0.991824\pi\)
0.999670 0.0256833i \(-0.00817614\pi\)
\(380\) 0 0
\(381\) − 10.3923i − 0.532414i
\(382\) 0 0
\(383\) 34.6410 1.77007 0.885037 0.465521i \(-0.154133\pi\)
0.885037 + 0.465521i \(0.154133\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 20.0000i − 1.01666i
\(388\) 0 0
\(389\) 27.7128i 1.40510i 0.711637 + 0.702548i \(0.247954\pi\)
−0.711637 + 0.702548i \(0.752046\pi\)
\(390\) 0 0
\(391\) 5.19615 0.262781
\(392\) 0 0
\(393\) −18.0000 −0.907980
\(394\) 0 0
\(395\) − 36.0000i − 1.81136i
\(396\) 0 0
\(397\) − 17.3205i − 0.869291i −0.900602 0.434646i \(-0.856874\pi\)
0.900602 0.434646i \(-0.143126\pi\)
\(398\) 0 0
\(399\) 1.73205 0.0867110
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) 18.0000i 0.896644i
\(404\) 0 0
\(405\) − 3.46410i − 0.172133i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) 21.0000i 1.03585i
\(412\) 0 0
\(413\) 15.5885i 0.767058i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −22.0000 −1.07734
\(418\) 0 0
\(419\) 6.00000i 0.293119i 0.989202 + 0.146560i \(0.0468200\pi\)
−0.989202 + 0.146560i \(0.953180\pi\)
\(420\) 0 0
\(421\) − 8.66025i − 0.422075i −0.977478 0.211037i \(-0.932316\pi\)
0.977478 0.211037i \(-0.0676842\pi\)
\(422\) 0 0
\(423\) 20.7846 1.01058
\(424\) 0 0
\(425\) 21.0000 1.01865
\(426\) 0 0
\(427\) − 18.0000i − 0.871081i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −24.2487 −1.16802 −0.584010 0.811747i \(-0.698517\pi\)
−0.584010 + 0.811747i \(0.698517\pi\)
\(432\) 0 0
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) − 6.00000i − 0.287678i
\(436\) 0 0
\(437\) 1.73205i 0.0828552i
\(438\) 0 0
\(439\) 34.6410 1.65333 0.826663 0.562698i \(-0.190236\pi\)
0.826663 + 0.562698i \(0.190236\pi\)
\(440\) 0 0
\(441\) −8.00000 −0.380952
\(442\) 0 0
\(443\) − 12.0000i − 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) 0 0
\(445\) − 62.3538i − 2.95585i
\(446\) 0 0
\(447\) 20.7846 0.983078
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) − 13.8564i − 0.651031i
\(454\) 0 0
\(455\) −31.1769 −1.46160
\(456\) 0 0
\(457\) −1.00000 −0.0467780 −0.0233890 0.999726i \(-0.507446\pi\)
−0.0233890 + 0.999726i \(0.507446\pi\)
\(458\) 0 0
\(459\) − 15.0000i − 0.700140i
\(460\) 0 0
\(461\) − 13.8564i − 0.645357i −0.946509 0.322679i \(-0.895417\pi\)
0.946509 0.322679i \(-0.104583\pi\)
\(462\) 0 0
\(463\) 10.3923 0.482971 0.241486 0.970404i \(-0.422365\pi\)
0.241486 + 0.970404i \(0.422365\pi\)
\(464\) 0 0
\(465\) −12.0000 −0.556487
\(466\) 0 0
\(467\) 6.00000i 0.277647i 0.990317 + 0.138823i \(0.0443321\pi\)
−0.990317 + 0.138823i \(0.955668\pi\)
\(468\) 0 0
\(469\) − 22.5167i − 1.03972i
\(470\) 0 0
\(471\) −3.46410 −0.159617
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 7.00000i 0.321182i
\(476\) 0 0
\(477\) 10.3923i 0.475831i
\(478\) 0 0
\(479\) −31.1769 −1.42451 −0.712255 0.701921i \(-0.752326\pi\)
−0.712255 + 0.701921i \(0.752326\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) − 3.00000i − 0.136505i
\(484\) 0 0
\(485\) 48.4974i 2.20215i
\(486\) 0 0
\(487\) 34.6410 1.56973 0.784867 0.619664i \(-0.212731\pi\)
0.784867 + 0.619664i \(0.212731\pi\)
\(488\) 0 0
\(489\) 14.0000 0.633102
\(490\) 0 0
\(491\) 12.0000i 0.541552i 0.962642 + 0.270776i \(0.0872803\pi\)
−0.962642 + 0.270776i \(0.912720\pi\)
\(492\) 0 0
\(493\) 5.19615i 0.234023i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) 2.00000i 0.0895323i 0.998997 + 0.0447661i \(0.0142543\pi\)
−0.998997 + 0.0447661i \(0.985746\pi\)
\(500\) 0 0
\(501\) 13.8564i 0.619059i
\(502\) 0 0
\(503\) −5.19615 −0.231685 −0.115842 0.993268i \(-0.536957\pi\)
−0.115842 + 0.993268i \(0.536957\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 14.0000i − 0.621762i
\(508\) 0 0
\(509\) 27.7128i 1.22835i 0.789170 + 0.614174i \(0.210511\pi\)
−0.789170 + 0.614174i \(0.789489\pi\)
\(510\) 0 0
\(511\) −1.73205 −0.0766214
\(512\) 0 0
\(513\) 5.00000 0.220755
\(514\) 0 0
\(515\) − 48.0000i − 2.11513i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −20.7846 −0.912343
\(520\) 0 0
\(521\) −36.0000 −1.57719 −0.788594 0.614914i \(-0.789191\pi\)
−0.788594 + 0.614914i \(0.789191\pi\)
\(522\) 0 0
\(523\) − 7.00000i − 0.306089i −0.988219 0.153044i \(-0.951092\pi\)
0.988219 0.153044i \(-0.0489077\pi\)
\(524\) 0 0
\(525\) − 12.1244i − 0.529150i
\(526\) 0 0
\(527\) 10.3923 0.452696
\(528\) 0 0
\(529\) −20.0000 −0.869565
\(530\) 0 0
\(531\) 18.0000i 0.781133i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 10.3923 0.449299
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −13.8564 −0.594635
\(544\) 0 0
\(545\) −18.0000 −0.771035
\(546\) 0 0
\(547\) − 20.0000i − 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) 0 0
\(549\) − 20.7846i − 0.887066i
\(550\) 0 0
\(551\) −1.73205 −0.0737878
\(552\) 0 0
\(553\) 18.0000 0.765438
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 24.2487i − 1.02745i −0.857955 0.513725i \(-0.828265\pi\)
0.857955 0.513725i \(-0.171735\pi\)
\(558\) 0 0
\(559\) −51.9615 −2.19774
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 12.0000i − 0.505740i −0.967500 0.252870i \(-0.918626\pi\)
0.967500 0.252870i \(-0.0813744\pi\)
\(564\) 0 0
\(565\) − 20.7846i − 0.874415i
\(566\) 0 0
\(567\) 1.73205 0.0727393
\(568\) 0 0
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) − 8.00000i − 0.334790i −0.985890 0.167395i \(-0.946465\pi\)
0.985890 0.167395i \(-0.0535355\pi\)
\(572\) 0 0
\(573\) − 1.73205i − 0.0723575i
\(574\) 0 0
\(575\) 12.1244 0.505621
\(576\) 0 0
\(577\) −35.0000 −1.45707 −0.728535 0.685009i \(-0.759798\pi\)
−0.728535 + 0.685009i \(0.759798\pi\)
\(578\) 0 0
\(579\) − 16.0000i − 0.664937i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −36.0000 −1.48842
\(586\) 0 0
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 0 0
\(589\) 3.46410i 0.142736i
\(590\) 0 0
\(591\) 13.8564 0.569976
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 18.0000i 0.737928i
\(596\) 0 0
\(597\) − 22.5167i − 0.921546i
\(598\) 0 0
\(599\) 27.7128 1.13231 0.566157 0.824297i \(-0.308429\pi\)
0.566157 + 0.824297i \(0.308429\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) − 26.0000i − 1.05880i
\(604\) 0 0
\(605\) − 38.1051i − 1.54919i
\(606\) 0 0
\(607\) −3.46410 −0.140604 −0.0703018 0.997526i \(-0.522396\pi\)
−0.0703018 + 0.997526i \(0.522396\pi\)
\(608\) 0 0
\(609\) 3.00000 0.121566
\(610\) 0 0
\(611\) − 54.0000i − 2.18461i
\(612\) 0 0
\(613\) − 6.92820i − 0.279827i −0.990164 0.139914i \(-0.955317\pi\)
0.990164 0.139914i \(-0.0446825\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) − 2.00000i − 0.0803868i −0.999192 0.0401934i \(-0.987203\pi\)
0.999192 0.0401934i \(-0.0127974\pi\)
\(620\) 0 0
\(621\) − 8.66025i − 0.347524i
\(622\) 0 0
\(623\) 31.1769 1.24908
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 31.1769 1.24113 0.620567 0.784154i \(-0.286903\pi\)
0.620567 + 0.784154i \(0.286903\pi\)
\(632\) 0 0
\(633\) −5.00000 −0.198732
\(634\) 0 0
\(635\) 36.0000i 1.42862i
\(636\) 0 0
\(637\) 20.7846i 0.823516i
\(638\) 0 0
\(639\) −6.92820 −0.274075
\(640\) 0 0
\(641\) −48.0000 −1.89589 −0.947943 0.318440i \(-0.896841\pi\)
−0.947943 + 0.318440i \(0.896841\pi\)
\(642\) 0 0
\(643\) 22.0000i 0.867595i 0.901010 + 0.433798i \(0.142827\pi\)
−0.901010 + 0.433798i \(0.857173\pi\)
\(644\) 0 0
\(645\) − 34.6410i − 1.36399i
\(646\) 0 0
\(647\) −22.5167 −0.885221 −0.442611 0.896714i \(-0.645948\pi\)
−0.442611 + 0.896714i \(0.645948\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) − 6.00000i − 0.235159i
\(652\) 0 0
\(653\) 3.46410i 0.135561i 0.997700 + 0.0677804i \(0.0215917\pi\)
−0.997700 + 0.0677804i \(0.978408\pi\)
\(654\) 0 0
\(655\) 62.3538 2.43637
\(656\) 0 0
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) 51.0000i 1.98668i 0.115229 + 0.993339i \(0.463240\pi\)
−0.115229 + 0.993339i \(0.536760\pi\)
\(660\) 0 0
\(661\) 19.0526i 0.741059i 0.928821 + 0.370529i \(0.120824\pi\)
−0.928821 + 0.370529i \(0.879176\pi\)
\(662\) 0 0
\(663\) −15.5885 −0.605406
\(664\) 0 0
\(665\) −6.00000 −0.232670
\(666\) 0 0
\(667\) 3.00000i 0.116160i
\(668\) 0 0
\(669\) − 6.92820i − 0.267860i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 28.0000 1.07932 0.539660 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) 0 0
\(675\) − 35.0000i − 1.34715i
\(676\) 0 0
\(677\) − 36.3731i − 1.39793i −0.715156 0.698965i \(-0.753644\pi\)
0.715156 0.698965i \(-0.246356\pi\)
\(678\) 0 0
\(679\) −24.2487 −0.930580
\(680\) 0 0
\(681\) −15.0000 −0.574801
\(682\) 0 0
\(683\) 36.0000i 1.37750i 0.724998 + 0.688751i \(0.241841\pi\)
−0.724998 + 0.688751i \(0.758159\pi\)
\(684\) 0 0
\(685\) − 72.7461i − 2.77949i
\(686\) 0 0
\(687\) 3.46410 0.132164
\(688\) 0 0
\(689\) 27.0000 1.02862
\(690\) 0 0
\(691\) 34.0000i 1.29342i 0.762736 + 0.646710i \(0.223856\pi\)
−0.762736 + 0.646710i \(0.776144\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 76.2102 2.89082
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 6.00000i 0.226941i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 36.0000 1.35584
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 24.2487i 0.910679i 0.890318 + 0.455340i \(0.150482\pi\)
−0.890318 + 0.455340i \(0.849518\pi\)
\(710\) 0 0
\(711\) 20.7846 0.779484
\(712\) 0 0
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 19.0526i − 0.711531i
\(718\) 0 0
\(719\) −8.66025 −0.322973 −0.161486 0.986875i \(-0.551629\pi\)
−0.161486 + 0.986875i \(0.551629\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) 0 0
\(723\) 8.00000i 0.297523i
\(724\) 0 0
\(725\) 12.1244i 0.450287i
\(726\) 0 0
\(727\) 19.0526 0.706620 0.353310 0.935506i \(-0.385056\pi\)
0.353310 + 0.935506i \(0.385056\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 30.0000i 1.10959i
\(732\) 0 0
\(733\) 31.1769i 1.15155i 0.817610 + 0.575773i \(0.195299\pi\)
−0.817610 + 0.575773i \(0.804701\pi\)
\(734\) 0 0
\(735\) −13.8564 −0.511101
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 28.0000i − 1.03000i −0.857191 0.514998i \(-0.827793\pi\)
0.857191 0.514998i \(-0.172207\pi\)
\(740\) 0 0
\(741\) − 5.19615i − 0.190885i
\(742\) 0 0
\(743\) −17.3205 −0.635428 −0.317714 0.948187i \(-0.602915\pi\)
−0.317714 + 0.948187i \(0.602915\pi\)
\(744\) 0 0
\(745\) −72.0000 −2.63788
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 5.19615i 0.189863i
\(750\) 0 0
\(751\) 20.7846 0.758441 0.379221 0.925306i \(-0.376192\pi\)
0.379221 + 0.925306i \(0.376192\pi\)
\(752\) 0 0
\(753\) 18.0000 0.655956
\(754\) 0 0
\(755\) 48.0000i 1.74690i
\(756\) 0 0
\(757\) − 51.9615i − 1.88857i −0.329124 0.944287i \(-0.606753\pi\)
0.329124 0.944287i \(-0.393247\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 21.0000 0.761249 0.380625 0.924730i \(-0.375709\pi\)
0.380625 + 0.924730i \(0.375709\pi\)
\(762\) 0 0
\(763\) − 9.00000i − 0.325822i
\(764\) 0 0
\(765\) 20.7846i 0.751469i
\(766\) 0 0
\(767\) 46.7654 1.68860
\(768\) 0 0
\(769\) 41.0000 1.47850 0.739249 0.673432i \(-0.235181\pi\)
0.739249 + 0.673432i \(0.235181\pi\)
\(770\) 0 0
\(771\) 6.00000i 0.216085i
\(772\) 0 0
\(773\) 32.9090i 1.18365i 0.806065 + 0.591827i \(0.201593\pi\)
−0.806065 + 0.591827i \(0.798407\pi\)
\(774\) 0 0
\(775\) 24.2487 0.871039
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 8.66025 0.309492
\(784\) 0 0
\(785\) 12.0000 0.428298
\(786\) 0 0
\(787\) − 53.0000i − 1.88925i −0.328158 0.944623i \(-0.606428\pi\)
0.328158 0.944623i \(-0.393572\pi\)
\(788\) 0 0
\(789\) − 24.2487i − 0.863277i
\(790\) 0 0
\(791\) 10.3923 0.369508
\(792\) 0 0
\(793\) −54.0000 −1.91760
\(794\) 0 0
\(795\) 18.0000i 0.638394i
\(796\) 0 0
\(797\) − 29.4449i − 1.04299i −0.853254 0.521495i \(-0.825374\pi\)
0.853254 0.521495i \(-0.174626\pi\)
\(798\) 0 0
\(799\) −31.1769 −1.10296
\(800\) 0 0
\(801\) 36.0000 1.27200
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 10.3923i 0.366281i
\(806\) 0 0
\(807\) −27.7128 −0.975537
\(808\) 0 0
\(809\) −27.0000 −0.949269 −0.474635 0.880183i \(-0.657420\pi\)
−0.474635 + 0.880183i \(0.657420\pi\)
\(810\) 0 0
\(811\) − 31.0000i − 1.08856i −0.838905 0.544279i \(-0.816803\pi\)
0.838905 0.544279i \(-0.183197\pi\)
\(812\) 0 0
\(813\) 22.5167i 0.789694i
\(814\) 0 0
\(815\) −48.4974 −1.69879
\(816\) 0 0
\(817\) −10.0000 −0.349856
\(818\) 0 0
\(819\) − 18.0000i − 0.628971i
\(820\) 0 0
\(821\) − 10.3923i − 0.362694i −0.983419 0.181347i \(-0.941954\pi\)
0.983419 0.181347i \(-0.0580457\pi\)
\(822\) 0 0
\(823\) −8.66025 −0.301877 −0.150939 0.988543i \(-0.548230\pi\)
−0.150939 + 0.988543i \(0.548230\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 45.0000i − 1.56480i −0.622774 0.782402i \(-0.713994\pi\)
0.622774 0.782402i \(-0.286006\pi\)
\(828\) 0 0
\(829\) 22.5167i 0.782036i 0.920383 + 0.391018i \(0.127877\pi\)
−0.920383 + 0.391018i \(0.872123\pi\)
\(830\) 0 0
\(831\) 10.3923 0.360505
\(832\) 0 0
\(833\) 12.0000 0.415775
\(834\) 0 0
\(835\) − 48.0000i − 1.66111i
\(836\) 0 0
\(837\) − 17.3205i − 0.598684i
\(838\) 0 0
\(839\) −38.1051 −1.31553 −0.657767 0.753221i \(-0.728499\pi\)
−0.657767 + 0.753221i \(0.728499\pi\)
\(840\) 0 0
\(841\) 26.0000 0.896552
\(842\) 0 0
\(843\) 24.0000i 0.826604i
\(844\) 0 0
\(845\) 48.4974i 1.66836i
\(846\) 0 0
\(847\) 19.0526 0.654654
\(848\) 0 0
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 13.8564i 0.474434i 0.971457 + 0.237217i \(0.0762353\pi\)
−0.971457 + 0.237217i \(0.923765\pi\)
\(854\) 0 0
\(855\) −6.92820 −0.236940
\(856\) 0 0
\(857\) 24.0000 0.819824 0.409912 0.912125i \(-0.365559\pi\)
0.409912 + 0.912125i \(0.365559\pi\)
\(858\) 0 0
\(859\) 8.00000i 0.272956i 0.990643 + 0.136478i \(0.0435784\pi\)
−0.990643 + 0.136478i \(0.956422\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −45.0333 −1.53295 −0.766476 0.642273i \(-0.777992\pi\)
−0.766476 + 0.642273i \(0.777992\pi\)
\(864\) 0 0
\(865\) 72.0000 2.44807
\(866\) 0 0
\(867\) − 8.00000i − 0.271694i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −67.5500 −2.28884
\(872\) 0 0
\(873\) −28.0000 −0.947656
\(874\) 0 0
\(875\) 12.0000i 0.405674i
\(876\) 0 0
\(877\) 5.19615i 0.175462i 0.996144 + 0.0877308i \(0.0279615\pi\)
−0.996144 + 0.0877308i \(0.972038\pi\)
\(878\) 0 0
\(879\) 1.73205 0.0584206
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) − 14.0000i − 0.471138i −0.971858 0.235569i \(-0.924305\pi\)
0.971858 0.235569i \(-0.0756953\pi\)
\(884\) 0 0
\(885\) 31.1769i 1.04800i
\(886\) 0 0
\(887\) 51.9615 1.74470 0.872349 0.488884i \(-0.162596\pi\)
0.872349 + 0.488884i \(0.162596\pi\)
\(888\) 0 0
\(889\) −18.0000 −0.603701
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 10.3923i − 0.347765i
\(894\) 0 0
\(895\) 41.5692 1.38951
\(896\) 0 0
\(897\) −9.00000 −0.300501
\(898\) 0 0
\(899\) 6.00000i 0.200111i
\(900\) 0 0
\(901\) − 15.5885i − 0.519327i
\(902\) 0 0
\(903\) 17.3205 0.576390
\(904\) 0 0
\(905\) 48.0000 1.59557
\(906\) 0 0
\(907\) − 19.0000i − 0.630885i −0.948945 0.315442i \(-0.897847\pi\)
0.948945 0.315442i \(-0.102153\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −31.1769 −1.03294 −0.516469 0.856306i \(-0.672754\pi\)
−0.516469 + 0.856306i \(0.672754\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) − 36.0000i − 1.19012i
\(916\) 0 0
\(917\) 31.1769i 1.02955i
\(918\) 0 0
\(919\) 32.9090 1.08557 0.542783 0.839873i \(-0.317370\pi\)
0.542783 + 0.839873i \(0.317370\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 0 0
\(923\) 18.0000i 0.592477i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 27.7128 0.910208
\(928\) 0 0
\(929\) 21.0000 0.688988 0.344494 0.938789i \(-0.388051\pi\)
0.344494 + 0.938789i \(0.388051\pi\)
\(930\) 0 0
\(931\) 4.00000i 0.131095i
\(932\) 0 0
\(933\) − 1.73205i − 0.0567048i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −59.0000 −1.92745 −0.963723 0.266904i \(-0.913999\pi\)
−0.963723 + 0.266904i \(0.913999\pi\)
\(938\) 0 0
\(939\) 19.0000i 0.620042i
\(940\) 0 0
\(941\) 43.3013i 1.41158i 0.708421 + 0.705791i \(0.249408\pi\)
−0.708421 + 0.705791i \(0.750592\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 30.0000 0.975900
\(946\) 0 0
\(947\) 18.0000i 0.584921i 0.956278 + 0.292461i \(0.0944741\pi\)
−0.956278 + 0.292461i \(0.905526\pi\)
\(948\) 0 0
\(949\) 5.19615i 0.168674i
\(950\) 0 0
\(951\) −8.66025 −0.280828
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) 6.00000i 0.194155i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 36.3731 1.17455
\(960\) 0 0
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) 6.00000i 0.193347i
\(964\) 0 0
\(965\) 55.4256i 1.78421i
\(966\) 0 0
\(967\) −24.2487 −0.779786 −0.389893 0.920860i \(-0.627488\pi\)
−0.389893 + 0.920860i \(0.627488\pi\)
\(968\) 0 0
\(969\) −3.00000 −0.0963739
\(970\) 0 0
\(971\) − 60.0000i − 1.92549i −0.270408 0.962746i \(-0.587159\pi\)
0.270408 0.962746i \(-0.412841\pi\)
\(972\) 0 0
\(973\) 38.1051i 1.22159i
\(974\) 0 0
\(975\) −36.3731 −1.16487
\(976\) 0 0
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) − 10.3923i − 0.331801i
\(982\) 0 0
\(983\) −45.0333 −1.43634 −0.718170 0.695868i \(-0.755020\pi\)
−0.718170 + 0.695868i \(0.755020\pi\)
\(984\) 0 0
\(985\) −48.0000 −1.52941
\(986\) 0 0
\(987\) 18.0000i 0.572946i
\(988\) 0 0
\(989\) 17.3205i 0.550760i
\(990\) 0 0
\(991\) −31.1769 −0.990367 −0.495184 0.868788i \(-0.664899\pi\)
−0.495184 + 0.868788i \(0.664899\pi\)
\(992\) 0 0
\(993\) −7.00000 −0.222138
\(994\) 0 0
\(995\) 78.0000i 2.47277i
\(996\) 0 0
\(997\) − 38.1051i − 1.20680i −0.797438 0.603401i \(-0.793812\pi\)
0.797438 0.603401i \(-0.206188\pi\)
\(998\) 0 0
\(999\) 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 1216.2.c.g.609.3 yes 4
4.3 odd 2 inner 1216.2.c.g.609.1 4
8.3 odd 2 inner 1216.2.c.g.609.4 yes 4
8.5 even 2 inner 1216.2.c.g.609.2 yes 4
16.3 odd 4 4864.2.a.w.1.1 2
16.5 even 4 4864.2.a.w.1.2 2
16.11 odd 4 4864.2.a.t.1.2 2
16.13 even 4 4864.2.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.g.609.1 4 4.3 odd 2 inner
1216.2.c.g.609.2 yes 4 8.5 even 2 inner
1216.2.c.g.609.3 yes 4 1.1 even 1 trivial
1216.2.c.g.609.4 yes 4 8.3 odd 2 inner
4864.2.a.t.1.1 2 16.13 even 4
4864.2.a.t.1.2 2 16.11 odd 4
4864.2.a.w.1.1 2 16.3 odd 4
4864.2.a.w.1.2 2 16.5 even 4