Properties

Label 1400.2.g.c.449.2
Level $1400$
Weight $2$
Character 1400.449
Analytic conductor $11.179$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1400.449
Dual form 1400.2.g.c.449.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} +1.00000 q^{11} +4.00000i q^{13} -6.00000 q^{19} -2.00000 q^{21} +3.00000i q^{23} +4.00000i q^{27} +3.00000 q^{29} +2.00000i q^{33} -9.00000i q^{37} -8.00000 q^{39} +2.00000 q^{41} +9.00000i q^{43} +6.00000i q^{47} -1.00000 q^{49} +6.00000i q^{53} -12.0000i q^{57} -8.00000 q^{59} -10.0000 q^{61} -1.00000i q^{63} +1.00000i q^{67} -6.00000 q^{69} -7.00000 q^{71} -2.00000i q^{73} +1.00000i q^{77} +9.00000 q^{79} -11.0000 q^{81} +12.0000i q^{83} +6.00000i q^{87} +4.00000 q^{89} -4.00000 q^{91} -16.0000i q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 2q^{11} - 12q^{19} - 4q^{21} + 6q^{29} - 16q^{39} + 4q^{41} - 2q^{49} - 16q^{59} - 20q^{61} - 12q^{69} - 14q^{71} + 18q^{79} - 22q^{81} + 8q^{89} - 8q^{91} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 2.00000i 1.15470i 0.816497 + 0.577350i \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) 3.00000i 0.625543i 0.949828 + 0.312772i \(0.101257\pi\)
−0.949828 + 0.312772i \(0.898743\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 2.00000i 0.348155i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 9.00000i − 1.47959i −0.672832 0.739795i \(-0.734922\pi\)
0.672832 0.739795i \(-0.265078\pi\)
\(38\) 0 0
\(39\) −8.00000 −1.28103
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 9.00000i 1.37249i 0.727372 + 0.686244i \(0.240742\pi\)
−0.727372 + 0.686244i \(0.759258\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 12.0000i − 1.58944i
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) − 1.00000i − 0.125988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.00000i 0.122169i 0.998133 + 0.0610847i \(0.0194560\pi\)
−0.998133 + 0.0610847i \(0.980544\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −7.00000 −0.830747 −0.415374 0.909651i \(-0.636349\pi\)
−0.415374 + 0.909651i \(0.636349\pi\)
\(72\) 0 0
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.00000i 0.113961i
\(78\) 0 0
\(79\) 9.00000 1.01258 0.506290 0.862364i \(-0.331017\pi\)
0.506290 + 0.862364i \(0.331017\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000i 0.643268i
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 16.0000i − 1.62455i −0.583272 0.812277i \(-0.698228\pi\)
0.583272 0.812277i \(-0.301772\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) − 6.00000i − 0.591198i −0.955312 0.295599i \(-0.904481\pi\)
0.955312 0.295599i \(-0.0955191\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 20.0000i − 1.93347i −0.255774 0.966736i \(-0.582330\pi\)
0.255774 0.966736i \(-0.417670\pi\)
\(108\) 0 0
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) 18.0000 1.70848
\(112\) 0 0
\(113\) − 9.00000i − 0.846649i −0.905978 0.423324i \(-0.860863\pi\)
0.905978 0.423324i \(-0.139137\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 4.00000i − 0.369800i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 4.00000i 0.360668i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1.00000i 0.0887357i 0.999015 + 0.0443678i \(0.0141274\pi\)
−0.999015 + 0.0443678i \(0.985873\pi\)
\(128\) 0 0
\(129\) −18.0000 −1.58481
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) − 6.00000i − 0.520266i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 22.0000i 1.87959i 0.341743 + 0.939793i \(0.388983\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) 0 0
\(139\) 6.00000 0.508913 0.254457 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 0 0
\(143\) 4.00000i 0.334497i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 2.00000i − 0.164957i
\(148\) 0 0
\(149\) 17.0000 1.39269 0.696347 0.717705i \(-0.254807\pi\)
0.696347 + 0.717705i \(0.254807\pi\)
\(150\) 0 0
\(151\) 17.0000 1.38344 0.691720 0.722166i \(-0.256853\pi\)
0.691720 + 0.722166i \(0.256853\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) 12.0000i 0.939913i 0.882690 + 0.469956i \(0.155730\pi\)
−0.882690 + 0.469956i \(0.844270\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 0 0
\(173\) − 2.00000i − 0.152057i −0.997106 0.0760286i \(-0.975776\pi\)
0.997106 0.0760286i \(-0.0242240\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 16.0000i − 1.20263i
\(178\) 0 0
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) − 20.0000i − 1.47844i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) − 1.00000i − 0.0719816i −0.999352 0.0359908i \(-0.988541\pi\)
0.999352 0.0359908i \(-0.0114587\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 3.00000i − 0.213741i −0.994273 0.106871i \(-0.965917\pi\)
0.994273 0.106871i \(-0.0340831\pi\)
\(198\) 0 0
\(199\) 18.0000 1.27599 0.637993 0.770042i \(-0.279765\pi\)
0.637993 + 0.770042i \(0.279765\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) 0 0
\(203\) 3.00000i 0.210559i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 3.00000i − 0.208514i
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) − 14.0000i − 0.959264i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 26.0000i 1.74109i 0.492090 + 0.870544i \(0.336233\pi\)
−0.492090 + 0.870544i \(0.663767\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 22.0000i 1.46019i 0.683345 + 0.730096i \(0.260525\pi\)
−0.683345 + 0.730096i \(0.739475\pi\)
\(228\) 0 0
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 0 0
\(233\) − 19.0000i − 1.24473i −0.782727 0.622366i \(-0.786172\pi\)
0.782727 0.622366i \(-0.213828\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 18.0000i 1.16923i
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 0 0
\(243\) − 10.0000i − 0.641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 24.0000i − 1.52708i
\(248\) 0 0
\(249\) −24.0000 −1.52094
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 3.00000i 0.188608i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 9.00000 0.559233
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) 0 0
\(263\) − 15.0000i − 0.924940i −0.886635 0.462470i \(-0.846963\pi\)
0.886635 0.462470i \(-0.153037\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 8.00000i 0.489592i
\(268\) 0 0
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 0 0
\(271\) 4.00000 0.242983 0.121491 0.992592i \(-0.461232\pi\)
0.121491 + 0.992592i \(0.461232\pi\)
\(272\) 0 0
\(273\) − 8.00000i − 0.484182i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000i 1.32185i 0.750451 + 0.660926i \(0.229836\pi\)
−0.750451 + 0.660926i \(0.770164\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −7.00000 −0.417585 −0.208792 0.977960i \(-0.566953\pi\)
−0.208792 + 0.977960i \(0.566953\pi\)
\(282\) 0 0
\(283\) 14.0000i 0.832214i 0.909316 + 0.416107i \(0.136606\pi\)
−0.909316 + 0.416107i \(0.863394\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.00000i 0.118056i
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 32.0000 1.87587
\(292\) 0 0
\(293\) 8.00000i 0.467365i 0.972313 + 0.233682i \(0.0750776\pi\)
−0.972313 + 0.233682i \(0.924922\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.00000i 0.232104i
\(298\) 0 0
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) −9.00000 −0.518751
\(302\) 0 0
\(303\) − 28.0000i − 1.60856i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) 0 0
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) − 22.0000i − 1.24351i −0.783210 0.621757i \(-0.786419\pi\)
0.783210 0.621757i \(-0.213581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 33.0000i 1.85346i 0.375722 + 0.926732i \(0.377395\pi\)
−0.375722 + 0.926732i \(0.622605\pi\)
\(318\) 0 0
\(319\) 3.00000 0.167968
\(320\) 0 0
\(321\) 40.0000 2.23258
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 22.0000i 1.21660i
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −17.0000 −0.934405 −0.467202 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) 0 0
\(333\) 9.00000i 0.493197i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 34.0000i − 1.85210i −0.377403 0.926049i \(-0.623183\pi\)
0.377403 0.926049i \(-0.376817\pi\)
\(338\) 0 0
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 17.0000i − 0.912608i −0.889824 0.456304i \(-0.849173\pi\)
0.889824 0.456304i \(-0.150827\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) −16.0000 −0.854017
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.00000 0.158334 0.0791670 0.996861i \(-0.474774\pi\)
0.0791670 + 0.996861i \(0.474774\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) − 20.0000i − 1.04973i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 10.0000i − 0.521996i −0.965339 0.260998i \(-0.915948\pi\)
0.965339 0.260998i \(-0.0840516\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 15.0000i 0.776671i 0.921518 + 0.388335i \(0.126950\pi\)
−0.921518 + 0.388335i \(0.873050\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) 5.00000 0.256833 0.128416 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) 0 0
\(381\) −2.00000 −0.102463
\(382\) 0 0
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 9.00000i − 0.457496i
\(388\) 0 0
\(389\) 35.0000 1.77457 0.887285 0.461221i \(-0.152589\pi\)
0.887285 + 0.461221i \(0.152589\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) − 24.0000i − 1.21064i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 34.0000i − 1.70641i −0.521575 0.853206i \(-0.674655\pi\)
0.521575 0.853206i \(-0.325345\pi\)
\(398\) 0 0
\(399\) 12.0000 0.600751
\(400\) 0 0
\(401\) −39.0000 −1.94757 −0.973784 0.227477i \(-0.926952\pi\)
−0.973784 + 0.227477i \(0.926952\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 9.00000i − 0.446113i
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −44.0000 −2.17036
\(412\) 0 0
\(413\) − 8.00000i − 0.393654i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 12.0000i 0.587643i
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 23.0000 1.12095 0.560476 0.828171i \(-0.310618\pi\)
0.560476 + 0.828171i \(0.310618\pi\)
\(422\) 0 0
\(423\) − 6.00000i − 0.291730i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 10.0000i − 0.483934i
\(428\) 0 0
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) −4.00000 −0.192673 −0.0963366 0.995349i \(-0.530713\pi\)
−0.0963366 + 0.995349i \(0.530713\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 18.0000i − 0.861057i
\(438\) 0 0
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) − 36.0000i − 1.71041i −0.518289 0.855206i \(-0.673431\pi\)
0.518289 0.855206i \(-0.326569\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 34.0000i 1.60814i
\(448\) 0 0
\(449\) 9.00000 0.424736 0.212368 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) 0 0
\(453\) 34.0000i 1.59746i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 11.0000i − 0.514558i −0.966337 0.257279i \(-0.917174\pi\)
0.966337 0.257279i \(-0.0828260\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) − 16.0000i − 0.743583i −0.928316 0.371792i \(-0.878744\pi\)
0.928316 0.371792i \(-0.121256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.0000i 0.832941i 0.909149 + 0.416470i \(0.136733\pi\)
−0.909149 + 0.416470i \(0.863267\pi\)
\(468\) 0 0
\(469\) −1.00000 −0.0461757
\(470\) 0 0
\(471\) −28.0000 −1.29017
\(472\) 0 0
\(473\) 9.00000i 0.413820i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 6.00000i − 0.274721i
\(478\) 0 0
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) 36.0000 1.64146
\(482\) 0 0
\(483\) − 6.00000i − 0.273009i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 17.0000i − 0.770344i −0.922845 0.385172i \(-0.874142\pi\)
0.922845 0.385172i \(-0.125858\pi\)
\(488\) 0 0
\(489\) −24.0000 −1.08532
\(490\) 0 0
\(491\) −39.0000 −1.76005 −0.880023 0.474932i \(-0.842473\pi\)
−0.880023 + 0.474932i \(0.842473\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 7.00000i − 0.313993i
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) −24.0000 −1.07224
\(502\) 0 0
\(503\) 26.0000i 1.15928i 0.814872 + 0.579641i \(0.196807\pi\)
−0.814872 + 0.579641i \(0.803193\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 6.00000i − 0.266469i
\(508\) 0 0
\(509\) 36.0000 1.59567 0.797836 0.602875i \(-0.205978\pi\)
0.797836 + 0.602875i \(0.205978\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 0 0
\(513\) − 24.0000i − 1.05963i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 6.00000i 0.263880i
\(518\) 0 0
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 8.00000i 0.346518i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 32.0000i − 1.38090i
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 3.00000 0.128980 0.0644900 0.997918i \(-0.479458\pi\)
0.0644900 + 0.997918i \(0.479458\pi\)
\(542\) 0 0
\(543\) 44.0000i 1.88822i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 11.0000i − 0.470326i −0.971956 0.235163i \(-0.924438\pi\)
0.971956 0.235163i \(-0.0755624\pi\)
\(548\) 0 0
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) 9.00000i 0.382719i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 33.0000i 1.39825i 0.714997 + 0.699127i \(0.246428\pi\)
−0.714997 + 0.699127i \(0.753572\pi\)
\(558\) 0 0
\(559\) −36.0000 −1.52264
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 2.00000i − 0.0842900i −0.999112 0.0421450i \(-0.986581\pi\)
0.999112 0.0421450i \(-0.0134191\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 11.0000i − 0.461957i
\(568\) 0 0
\(569\) −37.0000 −1.55112 −0.775560 0.631273i \(-0.782533\pi\)
−0.775560 + 0.631273i \(0.782533\pi\)
\(570\) 0 0
\(571\) 15.0000 0.627730 0.313865 0.949468i \(-0.398376\pi\)
0.313865 + 0.949468i \(0.398376\pi\)
\(572\) 0 0
\(573\) 40.0000i 1.67102i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 0 0
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 6.00000i 0.248495i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.0000i 0.908037i 0.890992 + 0.454019i \(0.150010\pi\)
−0.890992 + 0.454019i \(0.849990\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) 44.0000i 1.80686i 0.428732 + 0.903432i \(0.358960\pi\)
−0.428732 + 0.903432i \(0.641040\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 36.0000i 1.47338i
\(598\) 0 0
\(599\) −15.0000 −0.612883 −0.306442 0.951889i \(-0.599138\pi\)
−0.306442 + 0.951889i \(0.599138\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 0 0
\(603\) − 1.00000i − 0.0407231i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 20.0000i − 0.811775i −0.913923 0.405887i \(-0.866962\pi\)
0.913923 0.405887i \(-0.133038\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) 5.00000i 0.201948i 0.994889 + 0.100974i \(0.0321959\pi\)
−0.994889 + 0.100974i \(0.967804\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23.0000i 0.925945i 0.886373 + 0.462973i \(0.153217\pi\)
−0.886373 + 0.462973i \(0.846783\pi\)
\(618\) 0 0
\(619\) 16.0000 0.643094 0.321547 0.946894i \(-0.395797\pi\)
0.321547 + 0.946894i \(0.395797\pi\)
\(620\) 0 0
\(621\) −12.0000 −0.481543
\(622\) 0 0
\(623\) 4.00000i 0.160257i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 12.0000i − 0.479234i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −45.0000 −1.79142 −0.895711 0.444637i \(-0.853333\pi\)
−0.895711 + 0.444637i \(0.853333\pi\)
\(632\) 0 0
\(633\) 8.00000i 0.317971i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 4.00000i − 0.158486i
\(638\) 0 0
\(639\) 7.00000 0.276916
\(640\) 0 0
\(641\) 49.0000 1.93538 0.967692 0.252136i \(-0.0811330\pi\)
0.967692 + 0.252136i \(0.0811330\pi\)
\(642\) 0 0
\(643\) 8.00000i 0.315489i 0.987480 + 0.157745i \(0.0504223\pi\)
−0.987480 + 0.157745i \(0.949578\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 42.0000i − 1.65119i −0.564263 0.825595i \(-0.690840\pi\)
0.564263 0.825595i \(-0.309160\pi\)
\(648\) 0 0
\(649\) −8.00000 −0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) 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
\(657\) 2.00000i 0.0780274i
\(658\) 0 0
\(659\) −44.0000 −1.71400 −0.856998 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9.00000i 0.348481i
\(668\) 0 0
\(669\) −52.0000 −2.01044
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) 14.0000i 0.539660i 0.962908 + 0.269830i \(0.0869676\pi\)
−0.962908 + 0.269830i \(0.913032\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 18.0000i − 0.691796i −0.938272 0.345898i \(-0.887574\pi\)
0.938272 0.345898i \(-0.112426\pi\)
\(678\) 0 0
\(679\) 16.0000 0.614024
\(680\) 0 0
\(681\) −44.0000 −1.68608
\(682\) 0 0
\(683\) − 23.0000i − 0.880071i −0.897980 0.440035i \(-0.854966\pi\)
0.897980 0.440035i \(-0.145034\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 52.0000i 1.98392i
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 26.0000 0.989087 0.494543 0.869153i \(-0.335335\pi\)
0.494543 + 0.869153i \(0.335335\pi\)
\(692\) 0 0
\(693\) − 1.00000i − 0.0379869i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 38.0000 1.43729
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 54.0000i 2.03665i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 14.0000i − 0.526524i
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) −9.00000 −0.337526
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 32.0000i 1.19506i
\(718\) 0 0
\(719\) −26.0000 −0.969636 −0.484818 0.874615i \(-0.661114\pi\)
−0.484818 + 0.874615i \(0.661114\pi\)
\(720\) 0 0
\(721\) 6.00000 0.223452
\(722\) 0 0
\(723\) 36.0000i 1.33885i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 8.00000i − 0.296704i −0.988935 0.148352i \(-0.952603\pi\)
0.988935 0.148352i \(-0.0473968\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) − 4.00000i − 0.147743i −0.997268 0.0738717i \(-0.976464\pi\)
0.997268 0.0738717i \(-0.0235355\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.00000i 0.0368355i
\(738\) 0 0
\(739\) 43.0000 1.58178 0.790890 0.611958i \(-0.209618\pi\)
0.790890 + 0.611958i \(0.209618\pi\)
\(740\) 0 0
\(741\) 48.0000 1.76332
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 12.0000i − 0.439057i
\(748\) 0 0
\(749\) 20.0000 0.730784
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 0 0
\(753\) 40.0000i 1.45768i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 17.0000i − 0.617876i −0.951082 0.308938i \(-0.900027\pi\)
0.951082 0.308938i \(-0.0999735\pi\)
\(758\) 0 0
\(759\) −6.00000 −0.217786
\(760\) 0 0
\(761\) 32.0000 1.16000 0.580000 0.814617i \(-0.303053\pi\)
0.580000 + 0.814617i \(0.303053\pi\)
\(762\) 0 0
\(763\) 11.0000i 0.398227i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 32.0000i − 1.15545i
\(768\) 0 0
\(769\) −38.0000 −1.37032 −0.685158 0.728395i \(-0.740267\pi\)
−0.685158 + 0.728395i \(0.740267\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 40.0000i 1.43870i 0.694648 + 0.719350i \(0.255560\pi\)
−0.694648 + 0.719350i \(0.744440\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 18.0000i 0.645746i
\(778\) 0 0
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) −7.00000 −0.250480
\(782\) 0 0
\(783\) 12.0000i 0.428845i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 38.0000i − 1.35455i −0.735728 0.677277i \(-0.763160\pi\)
0.735728 0.677277i \(-0.236840\pi\)
\(788\) 0 0
\(789\) 30.0000 1.06803
\(790\) 0 0
\(791\) 9.00000 0.320003
\(792\) 0 0
\(793\) − 40.0000i − 1.42044i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 28.0000i 0.991811i 0.868377 + 0.495905i \(0.165164\pi\)
−0.868377 + 0.495905i \(0.834836\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −4.00000 −0.141333
\(802\) 0 0
\(803\) − 2.00000i − 0.0705785i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 8.00000i − 0.281613i
\(808\) 0 0
\(809\) −43.0000 −1.51180 −0.755900 0.654687i \(-0.772800\pi\)
−0.755900 + 0.654687i \(0.772800\pi\)
\(810\) 0 0
\(811\) 14.0000 0.491606 0.245803 0.969320i \(-0.420948\pi\)
0.245803 + 0.969320i \(0.420948\pi\)
\(812\) 0 0
\(813\) 8.00000i 0.280572i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 54.0000i − 1.88922i
\(818\) 0 0
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 2.00000 0.0698005 0.0349002 0.999391i \(-0.488889\pi\)
0.0349002 + 0.999391i \(0.488889\pi\)
\(822\) 0 0
\(823\) 23.0000i 0.801730i 0.916137 + 0.400865i \(0.131290\pi\)
−0.916137 + 0.400865i \(0.868710\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.00000i 0.243414i 0.992566 + 0.121707i \(0.0388368\pi\)
−0.992566 + 0.121707i \(0.961163\pi\)
\(828\) 0 0
\(829\) 20.0000 0.694629 0.347314 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 0 0
\(831\) −44.0000 −1.52634
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −34.0000 −1.17381 −0.586905 0.809656i \(-0.699654\pi\)
−0.586905 + 0.809656i \(0.699654\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) − 14.0000i − 0.482186i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 10.0000i − 0.343604i
\(848\) 0 0
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) 27.0000 0.925548
\(852\) 0 0
\(853\) 28.0000i 0.958702i 0.877623 + 0.479351i \(0.159128\pi\)
−0.877623 + 0.479351i \(0.840872\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 38.0000i 1.29806i 0.760765 + 0.649028i \(0.224824\pi\)
−0.760765 + 0.649028i \(0.775176\pi\)
\(858\) 0 0
\(859\) −26.0000 −0.887109 −0.443554 0.896248i \(-0.646283\pi\)
−0.443554 + 0.896248i \(0.646283\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) 0 0
\(863\) − 7.00000i − 0.238283i −0.992877 0.119141i \(-0.961986\pi\)
0.992877 0.119141i \(-0.0380142\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 34.0000i 1.15470i
\(868\) 0 0
\(869\) 9.00000 0.305304
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 0 0
\(873\) 16.0000i 0.541518i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 46.0000i 1.55331i 0.629926 + 0.776655i \(0.283085\pi\)
−0.629926 + 0.776655i \(0.716915\pi\)
\(878\) 0 0
\(879\) −16.0000 −0.539667
\(880\) 0 0
\(881\) −40.0000 −1.34763 −0.673817 0.738898i \(-0.735346\pi\)
−0.673817 + 0.738898i \(0.735346\pi\)
\(882\) 0 0
\(883\) − 25.0000i − 0.841317i −0.907219 0.420658i \(-0.861799\pi\)
0.907219 0.420658i \(-0.138201\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.00000i 0.0671534i 0.999436 + 0.0335767i \(0.0106898\pi\)
−0.999436 + 0.0335767i \(0.989310\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) 0 0
\(891\) −11.0000 −0.368514
\(892\) 0 0
\(893\) − 36.0000i − 1.20469i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 24.0000i − 0.801337i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) − 18.0000i − 0.599002i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 12.0000i − 0.398453i −0.979953 0.199227i \(-0.936157\pi\)
0.979953 0.199227i \(-0.0638430\pi\)
\(908\) 0 0
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) 23.0000 0.762024 0.381012 0.924570i \(-0.375576\pi\)
0.381012 + 0.924570i \(0.375576\pi\)
\(912\) 0 0
\(913\) 12.0000i 0.397142i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 12.0000i − 0.396275i
\(918\) 0 0
\(919\) 29.0000 0.956622 0.478311 0.878191i \(-0.341249\pi\)
0.478311 + 0.878191i \(0.341249\pi\)
\(920\) 0 0
\(921\) −24.0000 −0.790827
\(922\) 0 0
\(923\) − 28.0000i − 0.921631i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 6.00000i 0.197066i
\(928\) 0 0
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) 8.00000i 0.261908i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 10.0000i − 0.326686i −0.986569 0.163343i \(-0.947772\pi\)
0.986569 0.163343i \(-0.0522277\pi\)
\(938\) 0 0
\(939\) 44.0000 1.43589
\(940\) 0 0
\(941\) −56.0000 −1.82555 −0.912774 0.408465i \(-0.866064\pi\)
−0.912774 + 0.408465i \(0.866064\pi\)
\(942\) 0 0
\(943\) 6.00000i 0.195387i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 20.0000i − 0.649913i −0.945729 0.324956i \(-0.894650\pi\)
0.945729 0.324956i \(-0.105350\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) −66.0000 −2.14020
\(952\) 0 0
\(953\) 11.0000i 0.356325i 0.984001 + 0.178162i \(0.0570153\pi\)
−0.984001 + 0.178162i \(0.942985\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 6.00000i 0.193952i
\(958\) 0 0
\(959\) −22.0000 −0.710417
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 20.0000i 0.644491i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 40.0000i − 1.28631i −0.765735 0.643157i \(-0.777624\pi\)
0.765735 0.643157i \(-0.222376\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −26.0000 −0.834380 −0.417190 0.908819i \(-0.636985\pi\)
−0.417190 + 0.908819i \(0.636985\pi\)
\(972\) 0 0
\(973\) 6.00000i 0.192351i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.00000i 0.0959785i 0.998848 + 0.0479893i \(0.0152813\pi\)
−0.998848 + 0.0479893i \(0.984719\pi\)
\(978\) 0 0
\(979\) 4.00000 0.127841
\(980\) 0 0
\(981\) −11.0000 −0.351203
\(982\) 0 0
\(983\) − 24.0000i − 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 12.0000i − 0.381964i
\(988\) 0 0
\(989\) −27.0000 −0.858550
\(990\) 0 0
\(991\) 23.0000 0.730619 0.365310 0.930886i \(-0.380963\pi\)
0.365310 + 0.930886i \(0.380963\pi\)
\(992\) 0 0
\(993\) − 34.0000i − 1.07896i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 38.0000i − 1.20347i −0.798695 0.601736i \(-0.794476\pi\)
0.798695 0.601736i \(-0.205524\pi\)
\(998\) 0 0
\(999\) 36.0000 1.13899
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 1400.2.g.c.449.2 2
4.3 odd 2 2800.2.g.f.449.1 2
5.2 odd 4 1400.2.a.l.1.1 yes 1
5.3 odd 4 1400.2.a.c.1.1 1
5.4 even 2 inner 1400.2.g.c.449.1 2
20.3 even 4 2800.2.a.bb.1.1 1
20.7 even 4 2800.2.a.f.1.1 1
20.19 odd 2 2800.2.g.f.449.2 2
35.13 even 4 9800.2.a.bk.1.1 1
35.27 even 4 9800.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1400.2.a.c.1.1 1 5.3 odd 4
1400.2.a.l.1.1 yes 1 5.2 odd 4
1400.2.g.c.449.1 2 5.4 even 2 inner
1400.2.g.c.449.2 2 1.1 even 1 trivial
2800.2.a.f.1.1 1 20.7 even 4
2800.2.a.bb.1.1 1 20.3 even 4
2800.2.g.f.449.1 2 4.3 odd 2
2800.2.g.f.449.2 2 20.19 odd 2
9800.2.a.h.1.1 1 35.27 even 4
9800.2.a.bk.1.1 1 35.13 even 4