Properties

Label 1216.2.c.b.609.2
Level $1216$
Weight $2$
Character 1216.609
Analytic conductor $9.710$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 609.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1216.609
Dual form 1216.2.c.b.609.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000i q^{3} -4.00000i q^{5} -1.00000 q^{7} -6.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} -4.00000i q^{5} -1.00000 q^{7} -6.00000 q^{9} +5.00000i q^{13} +12.0000 q^{15} -5.00000 q^{17} +1.00000i q^{19} -3.00000i q^{21} +3.00000 q^{23} -11.0000 q^{25} -9.00000i q^{27} -7.00000i q^{29} -10.0000 q^{31} +4.00000i q^{35} -2.00000i q^{37} -15.0000 q^{39} -6.00000 q^{41} -4.00000i q^{43} +24.0000i q^{45} -8.00000 q^{47} -6.00000 q^{49} -15.0000i q^{51} -9.00000i q^{53} -3.00000 q^{57} +1.00000i q^{59} +2.00000i q^{61} +6.00000 q^{63} +20.0000 q^{65} +7.00000i q^{67} +9.00000i q^{69} +12.0000 q^{71} +11.0000 q^{73} -33.0000i q^{75} -16.0000 q^{79} +9.00000 q^{81} +14.0000i q^{83} +20.0000i q^{85} +21.0000 q^{87} -4.00000 q^{89} -5.00000i q^{91} -30.0000i q^{93} +4.00000 q^{95} -12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{7} - 12q^{9} + O(q^{10}) \) \( 2q - 2q^{7} - 12q^{9} + 24q^{15} - 10q^{17} + 6q^{23} - 22q^{25} - 20q^{31} - 30q^{39} - 12q^{41} - 16q^{47} - 12q^{49} - 6q^{57} + 12q^{63} + 40q^{65} + 24q^{71} + 22q^{73} - 32q^{79} + 18q^{81} + 42q^{87} - 8q^{89} + 8q^{95} - 24q^{97} + O(q^{100}) \)

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
\(3\) 3.00000i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) 0 0
\(5\) − 4.00000i − 1.78885i −0.447214 0.894427i \(-0.647584\pi\)
0.447214 0.894427i \(-0.352416\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 5.00000i 1.38675i 0.720577 + 0.693375i \(0.243877\pi\)
−0.720577 + 0.693375i \(0.756123\pi\)
\(14\) 0 0
\(15\) 12.0000 3.09839
\(16\) 0 0
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i
\(20\) 0 0
\(21\) − 3.00000i − 0.654654i
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) −11.0000 −2.20000
\(26\) 0 0
\(27\) − 9.00000i − 1.73205i
\(28\) 0 0
\(29\) − 7.00000i − 1.29987i −0.759991 0.649934i \(-0.774797\pi\)
0.759991 0.649934i \(-0.225203\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.00000i 0.676123i
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) −15.0000 −2.40192
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 24.0000i 3.57771i
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) − 15.0000i − 2.10042i
\(52\) 0 0
\(53\) − 9.00000i − 1.23625i −0.786082 0.618123i \(-0.787894\pi\)
0.786082 0.618123i \(-0.212106\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.00000 −0.397360
\(58\) 0 0
\(59\) 1.00000i 0.130189i 0.997879 + 0.0650945i \(0.0207349\pi\)
−0.997879 + 0.0650945i \(0.979265\pi\)
\(60\) 0 0
\(61\) 2.00000i 0.256074i 0.991769 + 0.128037i \(0.0408676\pi\)
−0.991769 + 0.128037i \(0.959132\pi\)
\(62\) 0 0
\(63\) 6.00000 0.755929
\(64\) 0 0
\(65\) 20.0000 2.48069
\(66\) 0 0
\(67\) 7.00000i 0.855186i 0.903971 + 0.427593i \(0.140638\pi\)
−0.903971 + 0.427593i \(0.859362\pi\)
\(68\) 0 0
\(69\) 9.00000i 1.08347i
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) − 33.0000i − 3.81051i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 14.0000i 1.53670i 0.640030 + 0.768350i \(0.278922\pi\)
−0.640030 + 0.768350i \(0.721078\pi\)
\(84\) 0 0
\(85\) 20.0000i 2.16930i
\(86\) 0 0
\(87\) 21.0000 2.25144
\(88\) 0 0
\(89\) −4.00000 −0.423999 −0.212000 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 0 0
\(91\) − 5.00000i − 0.524142i
\(92\) 0 0
\(93\) − 30.0000i − 3.11086i
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 0 0
\(105\) −12.0000 −1.17108
\(106\) 0 0
\(107\) − 9.00000i − 0.870063i −0.900415 0.435031i \(-0.856737\pi\)
0.900415 0.435031i \(-0.143263\pi\)
\(108\) 0 0
\(109\) − 9.00000i − 0.862044i −0.902342 0.431022i \(-0.858153\pi\)
0.902342 0.431022i \(-0.141847\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 8.00000 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 0 0
\(115\) − 12.0000i − 1.11901i
\(116\) 0 0
\(117\) − 30.0000i − 2.77350i
\(118\) 0 0
\(119\) 5.00000 0.458349
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) − 18.0000i − 1.62301i
\(124\) 0 0
\(125\) 24.0000i 2.14663i
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) − 1.00000i − 0.0867110i
\(134\) 0 0
\(135\) −36.0000 −3.09839
\(136\) 0 0
\(137\) −15.0000 −1.28154 −0.640768 0.767734i \(-0.721384\pi\)
−0.640768 + 0.767734i \(0.721384\pi\)
\(138\) 0 0
\(139\) 14.0000i 1.18746i 0.804663 + 0.593732i \(0.202346\pi\)
−0.804663 + 0.593732i \(0.797654\pi\)
\(140\) 0 0
\(141\) − 24.0000i − 2.02116i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −28.0000 −2.32527
\(146\) 0 0
\(147\) − 18.0000i − 1.48461i
\(148\) 0 0
\(149\) 6.00000i 0.491539i 0.969328 + 0.245770i \(0.0790407\pi\)
−0.969328 + 0.245770i \(0.920959\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 30.0000 2.42536
\(154\) 0 0
\(155\) 40.0000i 3.21288i
\(156\) 0 0
\(157\) 20.0000i 1.59617i 0.602542 + 0.798087i \(0.294154\pi\)
−0.602542 + 0.798087i \(0.705846\pi\)
\(158\) 0 0
\(159\) 27.0000 2.14124
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) 10.0000i 0.783260i 0.920123 + 0.391630i \(0.128089\pi\)
−0.920123 + 0.391630i \(0.871911\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) − 6.00000i − 0.458831i
\(172\) 0 0
\(173\) − 14.0000i − 1.06440i −0.846619 0.532200i \(-0.821365\pi\)
0.846619 0.532200i \(-0.178635\pi\)
\(174\) 0 0
\(175\) 11.0000 0.831522
\(176\) 0 0
\(177\) −3.00000 −0.225494
\(178\) 0 0
\(179\) 12.0000i 0.896922i 0.893802 + 0.448461i \(0.148028\pi\)
−0.893802 + 0.448461i \(0.851972\pi\)
\(180\) 0 0
\(181\) 6.00000i 0.445976i 0.974821 + 0.222988i \(0.0715812\pi\)
−0.974821 + 0.222988i \(0.928419\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 9.00000i 0.654654i
\(190\) 0 0
\(191\) 5.00000 0.361787 0.180894 0.983503i \(-0.442101\pi\)
0.180894 + 0.983503i \(0.442101\pi\)
\(192\) 0 0
\(193\) 20.0000 1.43963 0.719816 0.694165i \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) 0 0
\(195\) 60.0000i 4.29669i
\(196\) 0 0
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) 0 0
\(199\) −5.00000 −0.354441 −0.177220 0.984171i \(-0.556711\pi\)
−0.177220 + 0.984171i \(0.556711\pi\)
\(200\) 0 0
\(201\) −21.0000 −1.48123
\(202\) 0 0
\(203\) 7.00000i 0.491304i
\(204\) 0 0
\(205\) 24.0000i 1.67623i
\(206\) 0 0
\(207\) −18.0000 −1.25109
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) − 1.00000i − 0.0688428i −0.999407 0.0344214i \(-0.989041\pi\)
0.999407 0.0344214i \(-0.0109588\pi\)
\(212\) 0 0
\(213\) 36.0000i 2.46668i
\(214\) 0 0
\(215\) −16.0000 −1.09119
\(216\) 0 0
\(217\) 10.0000 0.678844
\(218\) 0 0
\(219\) 33.0000i 2.22993i
\(220\) 0 0
\(221\) − 25.0000i − 1.68168i
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) 66.0000 4.40000
\(226\) 0 0
\(227\) − 17.0000i − 1.12833i −0.825662 0.564165i \(-0.809198\pi\)
0.825662 0.564165i \(-0.190802\pi\)
\(228\) 0 0
\(229\) − 22.0000i − 1.45380i −0.686743 0.726900i \(-0.740960\pi\)
0.686743 0.726900i \(-0.259040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 32.0000i 2.08745i
\(236\) 0 0
\(237\) − 48.0000i − 3.11794i
\(238\) 0 0
\(239\) −11.0000 −0.711531 −0.355765 0.934575i \(-0.615780\pi\)
−0.355765 + 0.934575i \(0.615780\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 24.0000i 1.53330i
\(246\) 0 0
\(247\) −5.00000 −0.318142
\(248\) 0 0
\(249\) −42.0000 −2.66164
\(250\) 0 0
\(251\) 20.0000i 1.26239i 0.775625 + 0.631194i \(0.217435\pi\)
−0.775625 + 0.631194i \(0.782565\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −60.0000 −3.75735
\(256\) 0 0
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) 2.00000i 0.124274i
\(260\) 0 0
\(261\) 42.0000i 2.59973i
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) −36.0000 −2.21146
\(266\) 0 0
\(267\) − 12.0000i − 0.734388i
\(268\) 0 0
\(269\) 14.0000i 0.853595i 0.904347 + 0.426798i \(0.140358\pi\)
−0.904347 + 0.426798i \(0.859642\pi\)
\(270\) 0 0
\(271\) −11.0000 −0.668202 −0.334101 0.942537i \(-0.608433\pi\)
−0.334101 + 0.942537i \(0.608433\pi\)
\(272\) 0 0
\(273\) 15.0000 0.907841
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 6.00000i − 0.360505i −0.983620 0.180253i \(-0.942309\pi\)
0.983620 0.180253i \(-0.0576915\pi\)
\(278\) 0 0
\(279\) 60.0000 3.59211
\(280\) 0 0
\(281\) −16.0000 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 0 0
\(283\) − 6.00000i − 0.356663i −0.983970 0.178331i \(-0.942930\pi\)
0.983970 0.178331i \(-0.0570699\pi\)
\(284\) 0 0
\(285\) 12.0000i 0.710819i
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) − 36.0000i − 2.11036i
\(292\) 0 0
\(293\) − 7.00000i − 0.408944i −0.978872 0.204472i \(-0.934452\pi\)
0.978872 0.204472i \(-0.0655478\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 15.0000i 0.867472i
\(300\) 0 0
\(301\) 4.00000i 0.230556i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) 6.00000i 0.341328i
\(310\) 0 0
\(311\) 35.0000 1.98467 0.992334 0.123585i \(-0.0394392\pi\)
0.992334 + 0.123585i \(0.0394392\pi\)
\(312\) 0 0
\(313\) −9.00000 −0.508710 −0.254355 0.967111i \(-0.581863\pi\)
−0.254355 + 0.967111i \(0.581863\pi\)
\(314\) 0 0
\(315\) − 24.0000i − 1.35225i
\(316\) 0 0
\(317\) 23.0000i 1.29181i 0.763418 + 0.645904i \(0.223520\pi\)
−0.763418 + 0.645904i \(0.776480\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 27.0000 1.50699
\(322\) 0 0
\(323\) − 5.00000i − 0.278207i
\(324\) 0 0
\(325\) − 55.0000i − 3.05085i
\(326\) 0 0
\(327\) 27.0000 1.49310
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) 11.0000i 0.604615i 0.953211 + 0.302307i \(0.0977569\pi\)
−0.953211 + 0.302307i \(0.902243\pi\)
\(332\) 0 0
\(333\) 12.0000i 0.657596i
\(334\) 0 0
\(335\) 28.0000 1.52980
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 0 0
\(339\) 24.0000i 1.30350i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) 36.0000 1.93817
\(346\) 0 0
\(347\) − 2.00000i − 0.107366i −0.998558 0.0536828i \(-0.982904\pi\)
0.998558 0.0536828i \(-0.0170960\pi\)
\(348\) 0 0
\(349\) − 28.0000i − 1.49881i −0.662114 0.749403i \(-0.730341\pi\)
0.662114 0.749403i \(-0.269659\pi\)
\(350\) 0 0
\(351\) 45.0000 2.40192
\(352\) 0 0
\(353\) 29.0000 1.54351 0.771757 0.635917i \(-0.219378\pi\)
0.771757 + 0.635917i \(0.219378\pi\)
\(354\) 0 0
\(355\) − 48.0000i − 2.54758i
\(356\) 0 0
\(357\) 15.0000i 0.793884i
\(358\) 0 0
\(359\) −27.0000 −1.42501 −0.712503 0.701669i \(-0.752438\pi\)
−0.712503 + 0.701669i \(0.752438\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 33.0000i 1.73205i
\(364\) 0 0
\(365\) − 44.0000i − 2.30307i
\(366\) 0 0
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 0 0
\(369\) 36.0000 1.87409
\(370\) 0 0
\(371\) 9.00000i 0.467257i
\(372\) 0 0
\(373\) − 15.0000i − 0.776671i −0.921518 0.388335i \(-0.873050\pi\)
0.921518 0.388335i \(-0.126950\pi\)
\(374\) 0 0
\(375\) −72.0000 −3.71806
\(376\) 0 0
\(377\) 35.0000 1.80259
\(378\) 0 0
\(379\) − 11.0000i − 0.565032i −0.959263 0.282516i \(-0.908831\pi\)
0.959263 0.282516i \(-0.0911690\pi\)
\(380\) 0 0
\(381\) − 6.00000i − 0.307389i
\(382\) 0 0
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 24.0000i 1.21999i
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) −15.0000 −0.758583
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 64.0000i 3.22019i
\(396\) 0 0
\(397\) − 2.00000i − 0.100377i −0.998740 0.0501886i \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) 0 0
\(399\) 3.00000 0.150188
\(400\) 0 0
\(401\) −4.00000 −0.199750 −0.0998752 0.995000i \(-0.531844\pi\)
−0.0998752 + 0.995000i \(0.531844\pi\)
\(402\) 0 0
\(403\) − 50.0000i − 2.49068i
\(404\) 0 0
\(405\) − 36.0000i − 1.78885i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 0 0
\(411\) − 45.0000i − 2.21969i
\(412\) 0 0
\(413\) − 1.00000i − 0.0492068i
\(414\) 0 0
\(415\) 56.0000 2.74893
\(416\) 0 0
\(417\) −42.0000 −2.05675
\(418\) 0 0
\(419\) 24.0000i 1.17248i 0.810139 + 0.586238i \(0.199392\pi\)
−0.810139 + 0.586238i \(0.800608\pi\)
\(420\) 0 0
\(421\) − 21.0000i − 1.02348i −0.859141 0.511739i \(-0.829002\pi\)
0.859141 0.511739i \(-0.170998\pi\)
\(422\) 0 0
\(423\) 48.0000 2.33384
\(424\) 0 0
\(425\) 55.0000 2.66789
\(426\) 0 0
\(427\) − 2.00000i − 0.0967868i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −22.0000 −1.05970 −0.529851 0.848091i \(-0.677752\pi\)
−0.529851 + 0.848091i \(0.677752\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) − 84.0000i − 4.02749i
\(436\) 0 0
\(437\) 3.00000i 0.143509i
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) 36.0000 1.71429
\(442\) 0 0
\(443\) − 32.0000i − 1.52037i −0.649709 0.760183i \(-0.725109\pi\)
0.649709 0.760183i \(-0.274891\pi\)
\(444\) 0 0
\(445\) 16.0000i 0.758473i
\(446\) 0 0
\(447\) −18.0000 −0.851371
\(448\) 0 0
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 24.0000i 1.12762i
\(454\) 0 0
\(455\) −20.0000 −0.937614
\(456\) 0 0
\(457\) 11.0000 0.514558 0.257279 0.966337i \(-0.417174\pi\)
0.257279 + 0.966337i \(0.417174\pi\)
\(458\) 0 0
\(459\) 45.0000i 2.10042i
\(460\) 0 0
\(461\) 20.0000i 0.931493i 0.884918 + 0.465746i \(0.154214\pi\)
−0.884918 + 0.465746i \(0.845786\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) −120.000 −5.56487
\(466\) 0 0
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 0 0
\(469\) − 7.00000i − 0.323230i
\(470\) 0 0
\(471\) −60.0000 −2.76465
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) − 11.0000i − 0.504715i
\(476\) 0 0
\(477\) 54.0000i 2.47249i
\(478\) 0 0
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) − 9.00000i − 0.409514i
\(484\) 0 0
\(485\) 48.0000i 2.17957i
\(486\) 0 0
\(487\) 30.0000 1.35943 0.679715 0.733476i \(-0.262104\pi\)
0.679715 + 0.733476i \(0.262104\pi\)
\(488\) 0 0
\(489\) −30.0000 −1.35665
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 35.0000i 1.57632i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 28.0000i 1.25345i 0.779240 + 0.626726i \(0.215605\pi\)
−0.779240 + 0.626726i \(0.784395\pi\)
\(500\) 0 0
\(501\) 6.00000i 0.268060i
\(502\) 0 0
\(503\) −29.0000 −1.29305 −0.646523 0.762894i \(-0.723778\pi\)
−0.646523 + 0.762894i \(0.723778\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 36.0000i − 1.59882i
\(508\) 0 0
\(509\) − 22.0000i − 0.975133i −0.873086 0.487566i \(-0.837885\pi\)
0.873086 0.487566i \(-0.162115\pi\)
\(510\) 0 0
\(511\) −11.0000 −0.486611
\(512\) 0 0
\(513\) 9.00000 0.397360
\(514\) 0 0
\(515\) − 8.00000i − 0.352522i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 42.0000 1.84360
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) − 1.00000i − 0.0437269i −0.999761 0.0218635i \(-0.993040\pi\)
0.999761 0.0218635i \(-0.00695991\pi\)
\(524\) 0 0
\(525\) 33.0000i 1.44024i
\(526\) 0 0
\(527\) 50.0000 2.17803
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) − 6.00000i − 0.260378i
\(532\) 0 0
\(533\) − 30.0000i − 1.29944i
\(534\) 0 0
\(535\) −36.0000 −1.55642
\(536\) 0 0
\(537\) −36.0000 −1.55351
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 22.0000i 0.945854i 0.881102 + 0.472927i \(0.156803\pi\)
−0.881102 + 0.472927i \(0.843197\pi\)
\(542\) 0 0
\(543\) −18.0000 −0.772454
\(544\) 0 0
\(545\) −36.0000 −1.54207
\(546\) 0 0
\(547\) 20.0000i 0.855138i 0.903983 + 0.427569i \(0.140630\pi\)
−0.903983 + 0.427569i \(0.859370\pi\)
\(548\) 0 0
\(549\) − 12.0000i − 0.512148i
\(550\) 0 0
\(551\) 7.00000 0.298210
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) 0 0
\(555\) − 24.0000i − 1.01874i
\(556\) 0 0
\(557\) 24.0000i 1.01691i 0.861088 + 0.508456i \(0.169784\pi\)
−0.861088 + 0.508456i \(0.830216\pi\)
\(558\) 0 0
\(559\) 20.0000 0.845910
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 24.0000i − 1.01148i −0.862686 0.505740i \(-0.831220\pi\)
0.862686 0.505740i \(-0.168780\pi\)
\(564\) 0 0
\(565\) − 32.0000i − 1.34625i
\(566\) 0 0
\(567\) −9.00000 −0.377964
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 26.0000i 1.08807i 0.839064 + 0.544033i \(0.183103\pi\)
−0.839064 + 0.544033i \(0.816897\pi\)
\(572\) 0 0
\(573\) 15.0000i 0.626634i
\(574\) 0 0
\(575\) −33.0000 −1.37620
\(576\) 0 0
\(577\) −33.0000 −1.37381 −0.686904 0.726748i \(-0.741031\pi\)
−0.686904 + 0.726748i \(0.741031\pi\)
\(578\) 0 0
\(579\) 60.0000i 2.49351i
\(580\) 0 0
\(581\) − 14.0000i − 0.580818i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −120.000 −4.96139
\(586\) 0 0
\(587\) 20.0000i 0.825488i 0.910847 + 0.412744i \(0.135430\pi\)
−0.910847 + 0.412744i \(0.864570\pi\)
\(588\) 0 0
\(589\) − 10.0000i − 0.412043i
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) −2.00000 −0.0821302 −0.0410651 0.999156i \(-0.513075\pi\)
−0.0410651 + 0.999156i \(0.513075\pi\)
\(594\) 0 0
\(595\) − 20.0000i − 0.819920i
\(596\) 0 0
\(597\) − 15.0000i − 0.613909i
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) − 42.0000i − 1.71037i
\(604\) 0 0
\(605\) − 44.0000i − 1.78885i
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 0 0
\(609\) −21.0000 −0.850963
\(610\) 0 0
\(611\) − 40.0000i − 1.61823i
\(612\) 0 0
\(613\) − 6.00000i − 0.242338i −0.992632 0.121169i \(-0.961336\pi\)
0.992632 0.121169i \(-0.0386643\pi\)
\(614\) 0 0
\(615\) −72.0000 −2.90332
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) − 34.0000i − 1.36658i −0.730149 0.683288i \(-0.760549\pi\)
0.730149 0.683288i \(-0.239451\pi\)
\(620\) 0 0
\(621\) − 27.0000i − 1.08347i
\(622\) 0 0
\(623\) 4.00000 0.160257
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 10.0000i 0.398726i
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) 3.00000 0.119239
\(634\) 0 0
\(635\) 8.00000i 0.317470i
\(636\) 0 0
\(637\) − 30.0000i − 1.18864i
\(638\) 0 0
\(639\) −72.0000 −2.84828
\(640\) 0 0
\(641\) 46.0000 1.81689 0.908445 0.418004i \(-0.137270\pi\)
0.908445 + 0.418004i \(0.137270\pi\)
\(642\) 0 0
\(643\) 20.0000i 0.788723i 0.918955 + 0.394362i \(0.129034\pi\)
−0.918955 + 0.394362i \(0.870966\pi\)
\(644\) 0 0
\(645\) − 48.0000i − 1.89000i
\(646\) 0 0
\(647\) −25.0000 −0.982851 −0.491426 0.870919i \(-0.663524\pi\)
−0.491426 + 0.870919i \(0.663524\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 30.0000i 1.17579i
\(652\) 0 0
\(653\) 6.00000i 0.234798i 0.993085 + 0.117399i \(0.0374557\pi\)
−0.993085 + 0.117399i \(0.962544\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −66.0000 −2.57491
\(658\) 0 0
\(659\) 21.0000i 0.818044i 0.912525 + 0.409022i \(0.134130\pi\)
−0.912525 + 0.409022i \(0.865870\pi\)
\(660\) 0 0
\(661\) 13.0000i 0.505641i 0.967513 + 0.252821i \(0.0813583\pi\)
−0.967513 + 0.252821i \(0.918642\pi\)
\(662\) 0 0
\(663\) 75.0000 2.91276
\(664\) 0 0
\(665\) −4.00000 −0.155113
\(666\) 0 0
\(667\) − 21.0000i − 0.813123i
\(668\) 0 0
\(669\) 42.0000i 1.62381i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 99.0000i 3.81051i
\(676\) 0 0
\(677\) 1.00000i 0.0384331i 0.999815 + 0.0192166i \(0.00611720\pi\)
−0.999815 + 0.0192166i \(0.993883\pi\)
\(678\) 0 0
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) 51.0000 1.95432
\(682\) 0 0
\(683\) − 16.0000i − 0.612223i −0.951996 0.306111i \(-0.900972\pi\)
0.951996 0.306111i \(-0.0990280\pi\)
\(684\) 0 0
\(685\) 60.0000i 2.29248i
\(686\) 0 0
\(687\) 66.0000 2.51806
\(688\) 0 0
\(689\) 45.0000 1.71436
\(690\) 0 0
\(691\) − 6.00000i − 0.228251i −0.993466 0.114125i \(-0.963593\pi\)
0.993466 0.114125i \(-0.0364066\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 56.0000 2.12420
\(696\) 0 0
\(697\) 30.0000 1.13633
\(698\) 0 0
\(699\) 54.0000i 2.04247i
\(700\) 0 0
\(701\) 20.0000i 0.755390i 0.925930 + 0.377695i \(0.123283\pi\)
−0.925930 + 0.377695i \(0.876717\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) 0 0
\(705\) −96.0000 −3.61557
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 14.0000i 0.525781i 0.964826 + 0.262891i \(0.0846758\pi\)
−0.964826 + 0.262891i \(0.915324\pi\)
\(710\) 0 0
\(711\) 96.0000 3.60028
\(712\) 0 0
\(713\) −30.0000 −1.12351
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 33.0000i − 1.23241i
\(718\) 0 0
\(719\) −39.0000 −1.45445 −0.727227 0.686397i \(-0.759191\pi\)
−0.727227 + 0.686397i \(0.759191\pi\)
\(720\) 0 0
\(721\) −2.00000 −0.0744839
\(722\) 0 0
\(723\) − 42.0000i − 1.56200i
\(724\) 0 0
\(725\) 77.0000i 2.85971i
\(726\) 0 0
\(727\) −25.0000 −0.927199 −0.463599 0.886045i \(-0.653442\pi\)
−0.463599 + 0.886045i \(0.653442\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 20.0000i 0.739727i
\(732\) 0 0
\(733\) − 40.0000i − 1.47743i −0.674016 0.738717i \(-0.735432\pi\)
0.674016 0.738717i \(-0.264568\pi\)
\(734\) 0 0
\(735\) −72.0000 −2.65576
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 40.0000i − 1.47142i −0.677295 0.735712i \(-0.736848\pi\)
0.677295 0.735712i \(-0.263152\pi\)
\(740\) 0 0
\(741\) − 15.0000i − 0.551039i
\(742\) 0 0
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) 24.0000 0.879292
\(746\) 0 0
\(747\) − 84.0000i − 3.07340i
\(748\) 0 0
\(749\) 9.00000i 0.328853i
\(750\) 0 0
\(751\) 22.0000 0.802791 0.401396 0.915905i \(-0.368525\pi\)
0.401396 + 0.915905i \(0.368525\pi\)
\(752\) 0 0
\(753\) −60.0000 −2.18652
\(754\) 0 0
\(755\) − 32.0000i − 1.16460i
\(756\) 0 0
\(757\) − 2.00000i − 0.0726912i −0.999339 0.0363456i \(-0.988428\pi\)
0.999339 0.0363456i \(-0.0115717\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.00000 0.0362500 0.0181250 0.999836i \(-0.494230\pi\)
0.0181250 + 0.999836i \(0.494230\pi\)
\(762\) 0 0
\(763\) 9.00000i 0.325822i
\(764\) 0 0
\(765\) − 120.000i − 4.33861i
\(766\) 0 0
\(767\) −5.00000 −0.180540
\(768\) 0 0
\(769\) −7.00000 −0.252426 −0.126213 0.992003i \(-0.540282\pi\)
−0.126213 + 0.992003i \(0.540282\pi\)
\(770\) 0 0
\(771\) 36.0000i 1.29651i
\(772\) 0 0
\(773\) 17.0000i 0.611448i 0.952120 + 0.305724i \(0.0988984\pi\)
−0.952120 + 0.305724i \(0.901102\pi\)
\(774\) 0 0
\(775\) 110.000 3.95132
\(776\) 0 0
\(777\) −6.00000 −0.215249
\(778\) 0 0
\(779\) − 6.00000i − 0.214972i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −63.0000 −2.25144
\(784\) 0 0
\(785\) 80.0000 2.85532
\(786\) 0 0
\(787\) − 13.0000i − 0.463400i −0.972787 0.231700i \(-0.925571\pi\)
0.972787 0.231700i \(-0.0744288\pi\)
\(788\) 0 0
\(789\) − 72.0000i − 2.56327i
\(790\) 0 0
\(791\) −8.00000 −0.284447
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) 0 0
\(795\) − 108.000i − 3.83037i
\(796\) 0 0
\(797\) − 27.0000i − 0.956389i −0.878254 0.478195i \(-0.841291\pi\)
0.878254 0.478195i \(-0.158709\pi\)
\(798\) 0 0
\(799\) 40.0000 1.41510
\(800\) 0 0
\(801\) 24.0000 0.847998
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 12.0000i 0.422944i
\(806\) 0 0
\(807\) −42.0000 −1.47847
\(808\) 0 0
\(809\) −29.0000 −1.01959 −0.509793 0.860297i \(-0.670278\pi\)
−0.509793 + 0.860297i \(0.670278\pi\)
\(810\) 0 0
\(811\) 7.00000i 0.245803i 0.992419 + 0.122902i \(0.0392200\pi\)
−0.992419 + 0.122902i \(0.960780\pi\)
\(812\) 0 0
\(813\) − 33.0000i − 1.15736i
\(814\) 0 0
\(815\) 40.0000 1.40114
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 0 0
\(819\) 30.0000i 1.04828i
\(820\) 0 0
\(821\) 2.00000i 0.0698005i 0.999391 + 0.0349002i \(0.0111113\pi\)
−0.999391 + 0.0349002i \(0.988889\pi\)
\(822\) 0 0
\(823\) −51.0000 −1.77775 −0.888874 0.458151i \(-0.848512\pi\)
−0.888874 + 0.458151i \(0.848512\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 43.0000i − 1.49526i −0.664117 0.747628i \(-0.731193\pi\)
0.664117 0.747628i \(-0.268807\pi\)
\(828\) 0 0
\(829\) − 5.00000i − 0.173657i −0.996223 0.0868286i \(-0.972327\pi\)
0.996223 0.0868286i \(-0.0276732\pi\)
\(830\) 0 0
\(831\) 18.0000 0.624413
\(832\) 0 0
\(833\) 30.0000 1.03944
\(834\) 0 0
\(835\) − 8.00000i − 0.276851i
\(836\) 0 0
\(837\) 90.0000i 3.11086i
\(838\) 0 0
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) − 48.0000i − 1.65321i
\(844\) 0 0
\(845\) 48.0000i 1.65125i
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) 0 0
\(849\) 18.0000 0.617758
\(850\) 0 0
\(851\) − 6.00000i − 0.205677i
\(852\) 0 0
\(853\) − 26.0000i − 0.890223i −0.895475 0.445112i \(-0.853164\pi\)
0.895475 0.445112i \(-0.146836\pi\)
\(854\) 0 0
\(855\) −24.0000 −0.820783
\(856\) 0 0
\(857\) 20.0000 0.683187 0.341593 0.939848i \(-0.389033\pi\)
0.341593 + 0.939848i \(0.389033\pi\)
\(858\) 0 0
\(859\) − 14.0000i − 0.477674i −0.971060 0.238837i \(-0.923234\pi\)
0.971060 0.238837i \(-0.0767661\pi\)
\(860\) 0 0
\(861\) 18.0000i 0.613438i
\(862\) 0 0
\(863\) −10.0000 −0.340404 −0.170202 0.985409i \(-0.554442\pi\)
−0.170202 + 0.985409i \(0.554442\pi\)
\(864\) 0 0
\(865\) −56.0000 −1.90406
\(866\) 0 0
\(867\) 24.0000i 0.815083i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −35.0000 −1.18593
\(872\) 0 0
\(873\) 72.0000 2.43683
\(874\) 0 0
\(875\) − 24.0000i − 0.811348i
\(876\) 0 0
\(877\) − 37.0000i − 1.24940i −0.780864 0.624701i \(-0.785221\pi\)
0.780864 0.624701i \(-0.214779\pi\)
\(878\) 0 0
\(879\) 21.0000 0.708312
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 0 0
\(883\) − 34.0000i − 1.14419i −0.820187 0.572096i \(-0.806131\pi\)
0.820187 0.572096i \(-0.193869\pi\)
\(884\) 0 0
\(885\) 12.0000i 0.403376i
\(886\) 0 0
\(887\) −26.0000 −0.872995 −0.436497 0.899706i \(-0.643781\pi\)
−0.436497 + 0.899706i \(0.643781\pi\)
\(888\) 0 0
\(889\) 2.00000 0.0670778
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 8.00000i − 0.267710i
\(894\) 0 0
\(895\) 48.0000 1.60446
\(896\) 0 0
\(897\) −45.0000 −1.50251
\(898\) 0 0
\(899\) 70.0000i 2.33463i
\(900\) 0 0
\(901\) 45.0000i 1.49917i
\(902\) 0 0
\(903\) −12.0000 −0.399335
\(904\) 0 0
\(905\) 24.0000 0.797787
\(906\) 0 0
\(907\) − 37.0000i − 1.22856i −0.789086 0.614282i \(-0.789446\pi\)
0.789086 0.614282i \(-0.210554\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 24.0000i 0.793416i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −55.0000 −1.81428 −0.907141 0.420826i \(-0.861740\pi\)
−0.907141 + 0.420826i \(0.861740\pi\)
\(920\) 0 0
\(921\) 36.0000 1.18624
\(922\) 0 0
\(923\) 60.0000i 1.97492i
\(924\) 0 0
\(925\) 22.0000i 0.723356i
\(926\) 0 0
\(927\) −12.0000 −0.394132
\(928\) 0 0
\(929\) 29.0000 0.951459 0.475730 0.879592i \(-0.342184\pi\)
0.475730 + 0.879592i \(0.342184\pi\)
\(930\) 0 0
\(931\) − 6.00000i − 0.196642i
\(932\) 0 0
\(933\) 105.000i 3.43755i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 31.0000 1.01273 0.506363 0.862320i \(-0.330990\pi\)
0.506363 + 0.862320i \(0.330990\pi\)
\(938\) 0 0
\(939\) − 27.0000i − 0.881112i
\(940\) 0 0
\(941\) − 51.0000i − 1.66255i −0.555860 0.831276i \(-0.687611\pi\)
0.555860 0.831276i \(-0.312389\pi\)
\(942\) 0 0
\(943\) −18.0000 −0.586161
\(944\) 0 0
\(945\) 36.0000 1.17108
\(946\) 0 0
\(947\) − 36.0000i − 1.16984i −0.811090 0.584921i \(-0.801125\pi\)
0.811090 0.584921i \(-0.198875\pi\)
\(948\) 0 0
\(949\) 55.0000i 1.78538i
\(950\) 0 0
\(951\) −69.0000 −2.23748
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) − 20.0000i − 0.647185i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 15.0000 0.484375
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 54.0000i 1.74013i
\(964\) 0 0
\(965\) − 80.0000i − 2.57529i
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 0 0
\(969\) 15.0000 0.481869
\(970\) 0 0
\(971\) 32.0000i 1.02693i 0.858111 + 0.513464i \(0.171638\pi\)
−0.858111 + 0.513464i \(0.828362\pi\)
\(972\) 0 0
\(973\) − 14.0000i − 0.448819i
\(974\) 0 0
\(975\) 165.000 5.28423
\(976\) 0 0
\(977\) 26.0000 0.831814 0.415907 0.909407i \(-0.363464\pi\)
0.415907 + 0.909407i \(0.363464\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 54.0000i 1.72409i
\(982\) 0 0
\(983\) −6.00000 −0.191370 −0.0956851 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 0 0
\(985\) 8.00000 0.254901
\(986\) 0 0
\(987\) 24.0000i 0.763928i
\(988\) 0 0
\(989\) − 12.0000i − 0.381578i
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) −33.0000 −1.04722
\(994\) 0 0
\(995\) 20.0000i 0.634043i
\(996\) 0 0
\(997\) 2.00000i 0.0633406i 0.999498 + 0.0316703i \(0.0100827\pi\)
−0.999498 + 0.0316703i \(0.989917\pi\)
\(998\) 0 0
\(999\) −18.0000 −0.569495
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.b.609.2 yes 2
4.3 odd 2 1216.2.c.c.609.1 yes 2
8.3 odd 2 1216.2.c.c.609.2 yes 2
8.5 even 2 inner 1216.2.c.b.609.1 2
16.3 odd 4 4864.2.a.o.1.1 1
16.5 even 4 4864.2.a.p.1.1 1
16.11 odd 4 4864.2.a.b.1.1 1
16.13 even 4 4864.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.b.609.1 2 8.5 even 2 inner
1216.2.c.b.609.2 yes 2 1.1 even 1 trivial
1216.2.c.c.609.1 yes 2 4.3 odd 2
1216.2.c.c.609.2 yes 2 8.3 odd 2
4864.2.a.a.1.1 1 16.13 even 4
4864.2.a.b.1.1 1 16.11 odd 4
4864.2.a.o.1.1 1 16.3 odd 4
4864.2.a.p.1.1 1 16.5 even 4