Properties

Label 1440.2.h.d.1151.3
Level $1440$
Weight $2$
Character 1440.1151
Analytic conductor $11.498$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1151.3
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1440.1151
Dual form 1440.2.h.d.1151.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{5} -3.41421i q^{7} +O(q^{10})\) \(q+1.00000i q^{5} -3.41421i q^{7} +2.58579 q^{11} -3.41421 q^{13} -1.17157i q^{17} +4.82843 q^{23} -1.00000 q^{25} -6.00000i q^{29} -6.48528i q^{31} +3.41421 q^{35} -9.07107 q^{37} -11.0711i q^{41} +6.82843i q^{43} -5.65685 q^{47} -4.65685 q^{49} -1.17157i q^{53} +2.58579i q^{55} +6.58579 q^{59} +12.8284 q^{61} -3.41421i q^{65} -8.00000i q^{67} +5.65685 q^{71} -10.4853 q^{73} -8.82843i q^{77} -14.4853i q^{79} +9.17157 q^{83} +1.17157 q^{85} +4.24264i q^{89} +11.6569i q^{91} +2.48528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q + 16 q^{11} - 8 q^{13} + 8 q^{23} - 4 q^{25} + 8 q^{35} - 8 q^{37} + 4 q^{49} + 32 q^{59} + 40 q^{61} - 8 q^{73} + 48 q^{83} + 16 q^{85} - 24 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) − 3.41421i − 1.29045i −0.763992 0.645226i \(-0.776763\pi\)
0.763992 0.645226i \(-0.223237\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.58579 0.779644 0.389822 0.920890i \(-0.372537\pi\)
0.389822 + 0.920890i \(0.372537\pi\)
\(12\) 0 0
\(13\) −3.41421 −0.946932 −0.473466 0.880812i \(-0.656997\pi\)
−0.473466 + 0.880812i \(0.656997\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.17157i − 0.284148i −0.989856 0.142074i \(-0.954623\pi\)
0.989856 0.142074i \(-0.0453771\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.82843 1.00680 0.503398 0.864054i \(-0.332083\pi\)
0.503398 + 0.864054i \(0.332083\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 6.00000i − 1.11417i −0.830455 0.557086i \(-0.811919\pi\)
0.830455 0.557086i \(-0.188081\pi\)
\(30\) 0 0
\(31\) − 6.48528i − 1.16479i −0.812906 0.582395i \(-0.802116\pi\)
0.812906 0.582395i \(-0.197884\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.41421 0.577107
\(36\) 0 0
\(37\) −9.07107 −1.49127 −0.745637 0.666352i \(-0.767855\pi\)
−0.745637 + 0.666352i \(0.767855\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 11.0711i − 1.72901i −0.502624 0.864505i \(-0.667632\pi\)
0.502624 0.864505i \(-0.332368\pi\)
\(42\) 0 0
\(43\) 6.82843i 1.04133i 0.853762 + 0.520663i \(0.174315\pi\)
−0.853762 + 0.520663i \(0.825685\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.65685 −0.825137 −0.412568 0.910927i \(-0.635368\pi\)
−0.412568 + 0.910927i \(0.635368\pi\)
\(48\) 0 0
\(49\) −4.65685 −0.665265
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 1.17157i − 0.160928i −0.996758 0.0804640i \(-0.974360\pi\)
0.996758 0.0804640i \(-0.0256402\pi\)
\(54\) 0 0
\(55\) 2.58579i 0.348667i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.58579 0.857396 0.428698 0.903448i \(-0.358972\pi\)
0.428698 + 0.903448i \(0.358972\pi\)
\(60\) 0 0
\(61\) 12.8284 1.64251 0.821256 0.570560i \(-0.193274\pi\)
0.821256 + 0.570560i \(0.193274\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 3.41421i − 0.423481i
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.65685 0.671345 0.335673 0.941979i \(-0.391036\pi\)
0.335673 + 0.941979i \(0.391036\pi\)
\(72\) 0 0
\(73\) −10.4853 −1.22721 −0.613605 0.789613i \(-0.710281\pi\)
−0.613605 + 0.789613i \(0.710281\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 8.82843i − 1.00609i
\(78\) 0 0
\(79\) − 14.4853i − 1.62972i −0.579657 0.814861i \(-0.696813\pi\)
0.579657 0.814861i \(-0.303187\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.17157 1.00671 0.503355 0.864079i \(-0.332099\pi\)
0.503355 + 0.864079i \(0.332099\pi\)
\(84\) 0 0
\(85\) 1.17157 0.127075
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.24264i 0.449719i 0.974391 + 0.224860i \(0.0721923\pi\)
−0.974391 + 0.224860i \(0.927808\pi\)
\(90\) 0 0
\(91\) 11.6569i 1.22197i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.48528 0.252342 0.126171 0.992009i \(-0.459731\pi\)
0.126171 + 0.992009i \(0.459731\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 4.82843i − 0.480446i −0.970718 0.240223i \(-0.922779\pi\)
0.970718 0.240223i \(-0.0772206\pi\)
\(102\) 0 0
\(103\) − 17.5563i − 1.72988i −0.501877 0.864939i \(-0.667357\pi\)
0.501877 0.864939i \(-0.332643\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.8284 1.43352 0.716759 0.697321i \(-0.245625\pi\)
0.716759 + 0.697321i \(0.245625\pi\)
\(108\) 0 0
\(109\) −7.17157 −0.686912 −0.343456 0.939169i \(-0.611598\pi\)
−0.343456 + 0.939169i \(0.611598\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.65685i 0.155864i 0.996959 + 0.0779319i \(0.0248317\pi\)
−0.996959 + 0.0779319i \(0.975168\pi\)
\(114\) 0 0
\(115\) 4.82843i 0.450253i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −4.31371 −0.392155
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 1.00000i − 0.0894427i
\(126\) 0 0
\(127\) − 10.7279i − 0.951949i −0.879459 0.475975i \(-0.842095\pi\)
0.879459 0.475975i \(-0.157905\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 12.7279 1.11204 0.556022 0.831168i \(-0.312327\pi\)
0.556022 + 0.831168i \(0.312327\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 19.3137i 1.65008i 0.565073 + 0.825041i \(0.308848\pi\)
−0.565073 + 0.825041i \(0.691152\pi\)
\(138\) 0 0
\(139\) 3.65685i 0.310170i 0.987901 + 0.155085i \(0.0495652\pi\)
−0.987901 + 0.155085i \(0.950435\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.82843 −0.738270
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.82843i 0.723253i 0.932323 + 0.361626i \(0.117778\pi\)
−0.932323 + 0.361626i \(0.882222\pi\)
\(150\) 0 0
\(151\) 15.6569i 1.27414i 0.770807 + 0.637068i \(0.219853\pi\)
−0.770807 + 0.637068i \(0.780147\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.48528 0.520910
\(156\) 0 0
\(157\) −9.75736 −0.778722 −0.389361 0.921085i \(-0.627304\pi\)
−0.389361 + 0.921085i \(0.627304\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 16.4853i − 1.29922i
\(162\) 0 0
\(163\) 0.485281i 0.0380102i 0.999819 + 0.0190051i \(0.00604987\pi\)
−0.999819 + 0.0190051i \(0.993950\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −21.7990 −1.68686 −0.843428 0.537242i \(-0.819466\pi\)
−0.843428 + 0.537242i \(0.819466\pi\)
\(168\) 0 0
\(169\) −1.34315 −0.103319
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 8.48528i − 0.645124i −0.946548 0.322562i \(-0.895456\pi\)
0.946548 0.322562i \(-0.104544\pi\)
\(174\) 0 0
\(175\) 3.41421i 0.258090i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −24.0416 −1.79696 −0.898478 0.439019i \(-0.855326\pi\)
−0.898478 + 0.439019i \(0.855326\pi\)
\(180\) 0 0
\(181\) 1.51472 0.112588 0.0562941 0.998414i \(-0.482072\pi\)
0.0562941 + 0.998414i \(0.482072\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 9.07107i − 0.666918i
\(186\) 0 0
\(187\) − 3.02944i − 0.221534i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3.51472 −0.254316 −0.127158 0.991882i \(-0.540586\pi\)
−0.127158 + 0.991882i \(0.540586\pi\)
\(192\) 0 0
\(193\) 18.9706 1.36553 0.682765 0.730638i \(-0.260777\pi\)
0.682765 + 0.730638i \(0.260777\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 21.3137i 1.51854i 0.650776 + 0.759269i \(0.274444\pi\)
−0.650776 + 0.759269i \(0.725556\pi\)
\(198\) 0 0
\(199\) − 15.6569i − 1.10988i −0.831889 0.554942i \(-0.812740\pi\)
0.831889 0.554942i \(-0.187260\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −20.4853 −1.43778
\(204\) 0 0
\(205\) 11.0711 0.773237
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 26.0000i 1.78991i 0.446153 + 0.894957i \(0.352794\pi\)
−0.446153 + 0.894957i \(0.647206\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.82843 −0.465695
\(216\) 0 0
\(217\) −22.1421 −1.50311
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.00000i 0.269069i
\(222\) 0 0
\(223\) 2.72792i 0.182675i 0.995820 + 0.0913376i \(0.0291142\pi\)
−0.995820 + 0.0913376i \(0.970886\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.65685 0.375459 0.187729 0.982221i \(-0.439887\pi\)
0.187729 + 0.982221i \(0.439887\pi\)
\(228\) 0 0
\(229\) −14.9706 −0.989283 −0.494641 0.869097i \(-0.664701\pi\)
−0.494641 + 0.869097i \(0.664701\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 2.14214i − 0.140336i −0.997535 0.0701680i \(-0.977646\pi\)
0.997535 0.0701680i \(-0.0223535\pi\)
\(234\) 0 0
\(235\) − 5.65685i − 0.369012i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 25.4558 1.64660 0.823301 0.567605i \(-0.192130\pi\)
0.823301 + 0.567605i \(0.192130\pi\)
\(240\) 0 0
\(241\) 28.9706 1.86616 0.933079 0.359671i \(-0.117111\pi\)
0.933079 + 0.359671i \(0.117111\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 4.65685i − 0.297516i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 13.8995 0.877328 0.438664 0.898651i \(-0.355452\pi\)
0.438664 + 0.898651i \(0.355452\pi\)
\(252\) 0 0
\(253\) 12.4853 0.784943
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 30.9706i 1.92442i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −9.51472 −0.586703 −0.293351 0.956005i \(-0.594771\pi\)
−0.293351 + 0.956005i \(0.594771\pi\)
\(264\) 0 0
\(265\) 1.17157 0.0719691
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7.65685i 0.466847i 0.972375 + 0.233423i \(0.0749928\pi\)
−0.972375 + 0.233423i \(0.925007\pi\)
\(270\) 0 0
\(271\) 22.0000i 1.33640i 0.743980 + 0.668202i \(0.232936\pi\)
−0.743980 + 0.668202i \(0.767064\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.58579 −0.155929
\(276\) 0 0
\(277\) −5.27208 −0.316768 −0.158384 0.987378i \(-0.550628\pi\)
−0.158384 + 0.987378i \(0.550628\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.5858i 1.34736i 0.739025 + 0.673678i \(0.235286\pi\)
−0.739025 + 0.673678i \(0.764714\pi\)
\(282\) 0 0
\(283\) − 7.31371i − 0.434755i −0.976088 0.217377i \(-0.930250\pi\)
0.976088 0.217377i \(-0.0697502\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −37.7990 −2.23120
\(288\) 0 0
\(289\) 15.6274 0.919260
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 29.3137i 1.71253i 0.516541 + 0.856263i \(0.327219\pi\)
−0.516541 + 0.856263i \(0.672781\pi\)
\(294\) 0 0
\(295\) 6.58579i 0.383439i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −16.4853 −0.953368
\(300\) 0 0
\(301\) 23.3137 1.34378
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 12.8284i 0.734554i
\(306\) 0 0
\(307\) − 24.0000i − 1.36975i −0.728659 0.684876i \(-0.759856\pi\)
0.728659 0.684876i \(-0.240144\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 26.1421 1.48238 0.741192 0.671293i \(-0.234261\pi\)
0.741192 + 0.671293i \(0.234261\pi\)
\(312\) 0 0
\(313\) 7.65685 0.432791 0.216395 0.976306i \(-0.430570\pi\)
0.216395 + 0.976306i \(0.430570\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 16.6274i 0.933889i 0.884287 + 0.466944i \(0.154645\pi\)
−0.884287 + 0.466944i \(0.845355\pi\)
\(318\) 0 0
\(319\) − 15.5147i − 0.868657i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 3.41421 0.189386
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 19.3137i 1.06480i
\(330\) 0 0
\(331\) 28.9706i 1.59237i 0.605056 + 0.796183i \(0.293151\pi\)
−0.605056 + 0.796183i \(0.706849\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −26.4853 −1.44275 −0.721373 0.692547i \(-0.756488\pi\)
−0.721373 + 0.692547i \(0.756488\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 16.7696i − 0.908122i
\(342\) 0 0
\(343\) − 8.00000i − 0.431959i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −22.3431 −1.19944 −0.599721 0.800209i \(-0.704722\pi\)
−0.599721 + 0.800209i \(0.704722\pi\)
\(348\) 0 0
\(349\) 31.4558 1.68379 0.841896 0.539639i \(-0.181439\pi\)
0.841896 + 0.539639i \(0.181439\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 1.85786i − 0.0988841i −0.998777 0.0494421i \(-0.984256\pi\)
0.998777 0.0494421i \(-0.0157443\pi\)
\(354\) 0 0
\(355\) 5.65685i 0.300235i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −18.8284 −0.993726 −0.496863 0.867829i \(-0.665515\pi\)
−0.496863 + 0.867829i \(0.665515\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 10.4853i − 0.548825i
\(366\) 0 0
\(367\) 24.8701i 1.29821i 0.760700 + 0.649103i \(0.224856\pi\)
−0.760700 + 0.649103i \(0.775144\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) −10.7279 −0.555471 −0.277735 0.960658i \(-0.589584\pi\)
−0.277735 + 0.960658i \(0.589584\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 20.4853i 1.05505i
\(378\) 0 0
\(379\) − 9.31371i − 0.478413i −0.970969 0.239207i \(-0.923113\pi\)
0.970969 0.239207i \(-0.0768873\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 10.3431 0.528510 0.264255 0.964453i \(-0.414874\pi\)
0.264255 + 0.964453i \(0.414874\pi\)
\(384\) 0 0
\(385\) 8.82843 0.449938
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 12.3431i 0.625822i 0.949782 + 0.312911i \(0.101304\pi\)
−0.949782 + 0.312911i \(0.898696\pi\)
\(390\) 0 0
\(391\) − 5.65685i − 0.286079i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 14.4853 0.728834
\(396\) 0 0
\(397\) 14.7279 0.739173 0.369587 0.929196i \(-0.379499\pi\)
0.369587 + 0.929196i \(0.379499\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 0.727922i − 0.0363507i −0.999835 0.0181753i \(-0.994214\pi\)
0.999835 0.0181753i \(-0.00578571\pi\)
\(402\) 0 0
\(403\) 22.1421i 1.10298i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −23.4558 −1.16266
\(408\) 0 0
\(409\) −38.2843 −1.89304 −0.946518 0.322652i \(-0.895426\pi\)
−0.946518 + 0.322652i \(0.895426\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 22.4853i − 1.10643i
\(414\) 0 0
\(415\) 9.17157i 0.450215i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −38.3848 −1.87522 −0.937610 0.347690i \(-0.886966\pi\)
−0.937610 + 0.347690i \(0.886966\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.17157i 0.0568296i
\(426\) 0 0
\(427\) − 43.7990i − 2.11958i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.14214 0.103183 0.0515915 0.998668i \(-0.483571\pi\)
0.0515915 + 0.998668i \(0.483571\pi\)
\(432\) 0 0
\(433\) −14.4853 −0.696118 −0.348059 0.937473i \(-0.613159\pi\)
−0.348059 + 0.937473i \(0.613159\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 18.2843i 0.872661i 0.899787 + 0.436330i \(0.143722\pi\)
−0.899787 + 0.436330i \(0.856278\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −26.6274 −1.26511 −0.632553 0.774517i \(-0.717993\pi\)
−0.632553 + 0.774517i \(0.717993\pi\)
\(444\) 0 0
\(445\) −4.24264 −0.201120
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 18.5858i − 0.877117i −0.898703 0.438559i \(-0.855489\pi\)
0.898703 0.438559i \(-0.144511\pi\)
\(450\) 0 0
\(451\) − 28.6274i − 1.34801i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −11.6569 −0.546482
\(456\) 0 0
\(457\) 2.48528 0.116257 0.0581283 0.998309i \(-0.481487\pi\)
0.0581283 + 0.998309i \(0.481487\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 24.8284i − 1.15638i −0.815904 0.578188i \(-0.803760\pi\)
0.815904 0.578188i \(-0.196240\pi\)
\(462\) 0 0
\(463\) 17.0711i 0.793360i 0.917957 + 0.396680i \(0.129838\pi\)
−0.917957 + 0.396680i \(0.870162\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 31.1127 1.43972 0.719862 0.694117i \(-0.244205\pi\)
0.719862 + 0.694117i \(0.244205\pi\)
\(468\) 0 0
\(469\) −27.3137 −1.26123
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 17.6569i 0.811863i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 27.3137 1.24800 0.623998 0.781426i \(-0.285507\pi\)
0.623998 + 0.781426i \(0.285507\pi\)
\(480\) 0 0
\(481\) 30.9706 1.41214
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.48528i 0.112851i
\(486\) 0 0
\(487\) 10.7279i 0.486129i 0.970010 + 0.243064i \(0.0781526\pi\)
−0.970010 + 0.243064i \(0.921847\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −31.0711 −1.40222 −0.701109 0.713054i \(-0.747311\pi\)
−0.701109 + 0.713054i \(0.747311\pi\)
\(492\) 0 0
\(493\) −7.02944 −0.316590
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 19.3137i − 0.866338i
\(498\) 0 0
\(499\) − 18.6274i − 0.833878i −0.908935 0.416939i \(-0.863103\pi\)
0.908935 0.416939i \(-0.136897\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 36.2843 1.61784 0.808918 0.587922i \(-0.200054\pi\)
0.808918 + 0.587922i \(0.200054\pi\)
\(504\) 0 0
\(505\) 4.82843 0.214862
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8.14214i 0.360894i 0.983585 + 0.180447i \(0.0577544\pi\)
−0.983585 + 0.180447i \(0.942246\pi\)
\(510\) 0 0
\(511\) 35.7990i 1.58365i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 17.5563 0.773625
\(516\) 0 0
\(517\) −14.6274 −0.643313
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0.727922i 0.0318908i 0.999873 + 0.0159454i \(0.00507580\pi\)
−0.999873 + 0.0159454i \(0.994924\pi\)
\(522\) 0 0
\(523\) 7.51472i 0.328596i 0.986411 + 0.164298i \(0.0525358\pi\)
−0.986411 + 0.164298i \(0.947464\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7.59798 −0.330973
\(528\) 0 0
\(529\) 0.313708 0.0136395
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 37.7990i 1.63726i
\(534\) 0 0
\(535\) 14.8284i 0.641089i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −12.0416 −0.518670
\(540\) 0 0
\(541\) 28.8284 1.23943 0.619715 0.784827i \(-0.287248\pi\)
0.619715 + 0.784827i \(0.287248\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 7.17157i − 0.307196i
\(546\) 0 0
\(547\) − 16.4853i − 0.704860i −0.935838 0.352430i \(-0.885356\pi\)
0.935838 0.352430i \(-0.114644\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −49.4558 −2.10308
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 20.4853i 0.867989i 0.900915 + 0.433995i \(0.142896\pi\)
−0.900915 + 0.433995i \(0.857104\pi\)
\(558\) 0 0
\(559\) − 23.3137i − 0.986065i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −6.34315 −0.267332 −0.133666 0.991026i \(-0.542675\pi\)
−0.133666 + 0.991026i \(0.542675\pi\)
\(564\) 0 0
\(565\) −1.65685 −0.0697044
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 9.21320i − 0.386238i −0.981175 0.193119i \(-0.938140\pi\)
0.981175 0.193119i \(-0.0618603\pi\)
\(570\) 0 0
\(571\) − 4.97056i − 0.208012i −0.994577 0.104006i \(-0.966834\pi\)
0.994577 0.104006i \(-0.0331660\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.82843 −0.201359
\(576\) 0 0
\(577\) 15.4558 0.643435 0.321718 0.946836i \(-0.395740\pi\)
0.321718 + 0.946836i \(0.395740\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 31.3137i − 1.29911i
\(582\) 0 0
\(583\) − 3.02944i − 0.125466i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 26.1421 1.07900 0.539501 0.841985i \(-0.318613\pi\)
0.539501 + 0.841985i \(0.318613\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 10.3431i 0.424742i 0.977189 + 0.212371i \(0.0681185\pi\)
−0.977189 + 0.212371i \(0.931881\pi\)
\(594\) 0 0
\(595\) − 4.00000i − 0.163984i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 42.1421 1.72188 0.860940 0.508706i \(-0.169876\pi\)
0.860940 + 0.508706i \(0.169876\pi\)
\(600\) 0 0
\(601\) 16.6863 0.680648 0.340324 0.940308i \(-0.389463\pi\)
0.340324 + 0.940308i \(0.389463\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 4.31371i − 0.175377i
\(606\) 0 0
\(607\) − 3.21320i − 0.130420i −0.997872 0.0652100i \(-0.979228\pi\)
0.997872 0.0652100i \(-0.0207717\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 19.3137 0.781349
\(612\) 0 0
\(613\) −3.21320 −0.129780 −0.0648900 0.997892i \(-0.520670\pi\)
−0.0648900 + 0.997892i \(0.520670\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 26.8284i − 1.08007i −0.841642 0.540036i \(-0.818411\pi\)
0.841642 0.540036i \(-0.181589\pi\)
\(618\) 0 0
\(619\) − 22.9706i − 0.923265i −0.887071 0.461632i \(-0.847264\pi\)
0.887071 0.461632i \(-0.152736\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 14.4853 0.580341
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 10.6274i 0.423743i
\(630\) 0 0
\(631\) − 7.17157i − 0.285496i −0.989759 0.142748i \(-0.954406\pi\)
0.989759 0.142748i \(-0.0455938\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 10.7279 0.425725
\(636\) 0 0
\(637\) 15.8995 0.629961
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 43.5563i − 1.72037i −0.509980 0.860186i \(-0.670347\pi\)
0.509980 0.860186i \(-0.329653\pi\)
\(642\) 0 0
\(643\) − 35.1127i − 1.38471i −0.721557 0.692355i \(-0.756573\pi\)
0.721557 0.692355i \(-0.243427\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.6863 0.498750 0.249375 0.968407i \(-0.419775\pi\)
0.249375 + 0.968407i \(0.419775\pi\)
\(648\) 0 0
\(649\) 17.0294 0.668464
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 19.6569i − 0.769232i −0.923077 0.384616i \(-0.874334\pi\)
0.923077 0.384616i \(-0.125666\pi\)
\(654\) 0 0
\(655\) 12.7279i 0.497321i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6.58579 0.256546 0.128273 0.991739i \(-0.459057\pi\)
0.128273 + 0.991739i \(0.459057\pi\)
\(660\) 0 0
\(661\) −40.4264 −1.57240 −0.786202 0.617969i \(-0.787956\pi\)
−0.786202 + 0.617969i \(0.787956\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 28.9706i − 1.12174i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 33.1716 1.28057
\(672\) 0 0
\(673\) 51.4558 1.98348 0.991739 0.128276i \(-0.0409443\pi\)
0.991739 + 0.128276i \(0.0409443\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.68629i 0.256975i 0.991711 + 0.128488i \(0.0410122\pi\)
−0.991711 + 0.128488i \(0.958988\pi\)
\(678\) 0 0
\(679\) − 8.48528i − 0.325635i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 10.8284 0.414338 0.207169 0.978305i \(-0.433575\pi\)
0.207169 + 0.978305i \(0.433575\pi\)
\(684\) 0 0
\(685\) −19.3137 −0.737939
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.00000i 0.152388i
\(690\) 0 0
\(691\) − 8.00000i − 0.304334i −0.988355 0.152167i \(-0.951375\pi\)
0.988355 0.152167i \(-0.0486252\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.65685 −0.138712
\(696\) 0 0
\(697\) −12.9706 −0.491295
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 9.51472i − 0.359366i −0.983725 0.179683i \(-0.942493\pi\)
0.983725 0.179683i \(-0.0575072\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −16.4853 −0.619993
\(708\) 0 0
\(709\) 6.97056 0.261785 0.130892 0.991397i \(-0.458216\pi\)
0.130892 + 0.991397i \(0.458216\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 31.3137i − 1.17271i
\(714\) 0 0
\(715\) − 8.82843i − 0.330164i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −10.8284 −0.403832 −0.201916 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(720\) 0 0
\(721\) −59.9411 −2.23232
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6.00000i 0.222834i
\(726\) 0 0
\(727\) 26.9289i 0.998739i 0.866389 + 0.499369i \(0.166435\pi\)
−0.866389 + 0.499369i \(0.833565\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) −12.1005 −0.446942 −0.223471 0.974711i \(-0.571739\pi\)
−0.223471 + 0.974711i \(0.571739\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 20.6863i − 0.761989i
\(738\) 0 0
\(739\) 22.3431i 0.821906i 0.911657 + 0.410953i \(0.134804\pi\)
−0.911657 + 0.410953i \(0.865196\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 0 0
\(745\) −8.82843 −0.323449
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 50.6274i − 1.84989i
\(750\) 0 0
\(751\) − 28.8284i − 1.05196i −0.850496 0.525982i \(-0.823698\pi\)
0.850496 0.525982i \(-0.176302\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −15.6569 −0.569811
\(756\) 0 0
\(757\) −52.8701 −1.92159 −0.960797 0.277251i \(-0.910577\pi\)
−0.960797 + 0.277251i \(0.910577\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 30.8701i 1.11904i 0.828817 + 0.559519i \(0.189014\pi\)
−0.828817 + 0.559519i \(0.810986\pi\)
\(762\) 0 0
\(763\) 24.4853i 0.886427i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −22.4853 −0.811896
\(768\) 0 0
\(769\) −2.34315 −0.0844960 −0.0422480 0.999107i \(-0.513452\pi\)
−0.0422480 + 0.999107i \(0.513452\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 41.3137i − 1.48595i −0.669319 0.742975i \(-0.733414\pi\)
0.669319 0.742975i \(-0.266586\pi\)
\(774\) 0 0
\(775\) 6.48528i 0.232958i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 14.6274 0.523410
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 9.75736i − 0.348255i
\(786\) 0 0
\(787\) − 42.1421i − 1.50220i −0.660186 0.751102i \(-0.729522\pi\)
0.660186 0.751102i \(-0.270478\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.65685 0.201135
\(792\) 0 0
\(793\) −43.7990 −1.55535
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 18.8284i − 0.666937i −0.942761 0.333469i \(-0.891781\pi\)
0.942761 0.333469i \(-0.108219\pi\)
\(798\) 0 0
\(799\) 6.62742i 0.234461i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −27.1127 −0.956786
\(804\) 0 0
\(805\) 16.4853 0.581030
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 36.0416i 1.26716i 0.773679 + 0.633578i \(0.218414\pi\)
−0.773679 + 0.633578i \(0.781586\pi\)
\(810\) 0 0
\(811\) 3.37258i 0.118427i 0.998245 + 0.0592137i \(0.0188593\pi\)
−0.998245 + 0.0592137i \(0.981141\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −0.485281 −0.0169987
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 29.3137i 1.02306i 0.859267 + 0.511528i \(0.170920\pi\)
−0.859267 + 0.511528i \(0.829080\pi\)
\(822\) 0 0
\(823\) 6.24264i 0.217605i 0.994063 + 0.108802i \(0.0347016\pi\)
−0.994063 + 0.108802i \(0.965298\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 42.6274 1.48230 0.741150 0.671339i \(-0.234281\pi\)
0.741150 + 0.671339i \(0.234281\pi\)
\(828\) 0 0
\(829\) −19.4558 −0.675729 −0.337865 0.941195i \(-0.609705\pi\)
−0.337865 + 0.941195i \(0.609705\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.45584i 0.189034i
\(834\) 0 0
\(835\) − 21.7990i − 0.754385i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 42.4264 1.46472 0.732361 0.680916i \(-0.238418\pi\)
0.732361 + 0.680916i \(0.238418\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 1.34315i − 0.0462056i
\(846\) 0 0
\(847\) 14.7279i 0.506057i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −43.7990 −1.50141
\(852\) 0 0
\(853\) 21.7574 0.744958 0.372479 0.928041i \(-0.378508\pi\)
0.372479 + 0.928041i \(0.378508\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 35.7990i 1.22287i 0.791295 + 0.611435i \(0.209407\pi\)
−0.791295 + 0.611435i \(0.790593\pi\)
\(858\) 0 0
\(859\) 26.0000i 0.887109i 0.896248 + 0.443554i \(0.146283\pi\)
−0.896248 + 0.443554i \(0.853717\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.20101 −0.211085 −0.105542 0.994415i \(-0.533658\pi\)
−0.105542 + 0.994415i \(0.533658\pi\)
\(864\) 0 0
\(865\) 8.48528 0.288508
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 37.4558i − 1.27060i
\(870\) 0 0
\(871\) 27.3137i 0.925490i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.41421 −0.115421
\(876\) 0 0
\(877\) −1.55635 −0.0525542 −0.0262771 0.999655i \(-0.508365\pi\)
−0.0262771 + 0.999655i \(0.508365\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 35.3553i − 1.19115i −0.803299 0.595576i \(-0.796924\pi\)
0.803299 0.595576i \(-0.203076\pi\)
\(882\) 0 0
\(883\) − 42.4264i − 1.42776i −0.700267 0.713881i \(-0.746936\pi\)
0.700267 0.713881i \(-0.253064\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −12.8284 −0.430736 −0.215368 0.976533i \(-0.569095\pi\)
−0.215368 + 0.976533i \(0.569095\pi\)
\(888\) 0 0
\(889\) −36.6274 −1.22844
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) − 24.0416i − 0.803623i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −38.9117 −1.29778
\(900\) 0 0
\(901\) −1.37258 −0.0457274
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.51472i 0.0503510i
\(906\) 0 0
\(907\) − 1.85786i − 0.0616894i −0.999524 0.0308447i \(-0.990180\pi\)
0.999524 0.0308447i \(-0.00981973\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 18.3431 0.607736 0.303868 0.952714i \(-0.401722\pi\)
0.303868 + 0.952714i \(0.401722\pi\)
\(912\) 0 0
\(913\) 23.7157 0.784876
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 43.4558i − 1.43504i
\(918\) 0 0
\(919\) − 12.8284i − 0.423171i −0.977360 0.211585i \(-0.932137\pi\)
0.977360 0.211585i \(-0.0678626\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −19.3137 −0.635718
\(924\) 0 0
\(925\) 9.07107 0.298255
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 45.8995i − 1.50591i −0.658070 0.752957i \(-0.728627\pi\)
0.658070 0.752957i \(-0.271373\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3.02944 0.0990732
\(936\) 0 0
\(937\) −14.9706 −0.489067 −0.244533 0.969641i \(-0.578635\pi\)
−0.244533 + 0.969641i \(0.578635\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 42.2843i 1.37843i 0.724558 + 0.689214i \(0.242044\pi\)
−0.724558 + 0.689214i \(0.757956\pi\)
\(942\) 0 0
\(943\) − 53.4558i − 1.74076i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 51.5980 1.67671 0.838355 0.545125i \(-0.183518\pi\)
0.838355 + 0.545125i \(0.183518\pi\)
\(948\) 0 0
\(949\) 35.7990 1.16208
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 45.2548i − 1.46595i −0.680257 0.732974i \(-0.738132\pi\)
0.680257 0.732974i \(-0.261868\pi\)
\(954\) 0 0
\(955\) − 3.51472i − 0.113734i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 65.9411 2.12935
\(960\) 0 0
\(961\) −11.0589 −0.356738
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 18.9706i 0.610684i
\(966\) 0 0
\(967\) 26.7279i 0.859512i 0.902945 + 0.429756i \(0.141400\pi\)
−0.902945 + 0.429756i \(0.858600\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 27.0711 0.868752 0.434376 0.900732i \(-0.356969\pi\)
0.434376 + 0.900732i \(0.356969\pi\)
\(972\) 0 0
\(973\) 12.4853 0.400260
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 37.1716i 1.18922i 0.804013 + 0.594612i \(0.202694\pi\)
−0.804013 + 0.594612i \(0.797306\pi\)
\(978\) 0 0
\(979\) 10.9706i 0.350621i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −39.5980 −1.26298 −0.631490 0.775384i \(-0.717556\pi\)
−0.631490 + 0.775384i \(0.717556\pi\)
\(984\) 0 0
\(985\) −21.3137 −0.679111
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 32.9706i 1.04840i
\(990\) 0 0
\(991\) − 16.3431i − 0.519157i −0.965722 0.259579i \(-0.916416\pi\)
0.965722 0.259579i \(-0.0835837\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 15.6569 0.496356
\(996\) 0 0
\(997\) 5.27208 0.166968 0.0834842 0.996509i \(-0.473395\pi\)
0.0834842 + 0.996509i \(0.473395\pi\)
\(998\) 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 1440.2.h.d.1151.3 yes 4
3.2 odd 2 1440.2.h.a.1151.1 4
4.3 odd 2 1440.2.h.a.1151.4 yes 4
5.2 odd 4 7200.2.o.m.7199.4 4
5.3 odd 4 7200.2.o.e.7199.1 4
5.4 even 2 7200.2.h.i.1151.4 4
8.3 odd 2 2880.2.h.d.1151.2 4
8.5 even 2 2880.2.h.a.1151.1 4
12.11 even 2 inner 1440.2.h.d.1151.2 yes 4
15.2 even 4 7200.2.o.j.7199.4 4
15.8 even 4 7200.2.o.b.7199.1 4
15.14 odd 2 7200.2.h.c.1151.4 4
20.3 even 4 7200.2.o.j.7199.3 4
20.7 even 4 7200.2.o.b.7199.2 4
20.19 odd 2 7200.2.h.c.1151.1 4
24.5 odd 2 2880.2.h.d.1151.3 4
24.11 even 2 2880.2.h.a.1151.4 4
60.23 odd 4 7200.2.o.m.7199.3 4
60.47 odd 4 7200.2.o.e.7199.2 4
60.59 even 2 7200.2.h.i.1151.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1440.2.h.a.1151.1 4 3.2 odd 2
1440.2.h.a.1151.4 yes 4 4.3 odd 2
1440.2.h.d.1151.2 yes 4 12.11 even 2 inner
1440.2.h.d.1151.3 yes 4 1.1 even 1 trivial
2880.2.h.a.1151.1 4 8.5 even 2
2880.2.h.a.1151.4 4 24.11 even 2
2880.2.h.d.1151.2 4 8.3 odd 2
2880.2.h.d.1151.3 4 24.5 odd 2
7200.2.h.c.1151.1 4 20.19 odd 2
7200.2.h.c.1151.4 4 15.14 odd 2
7200.2.h.i.1151.1 4 60.59 even 2
7200.2.h.i.1151.4 4 5.4 even 2
7200.2.o.b.7199.1 4 15.8 even 4
7200.2.o.b.7199.2 4 20.7 even 4
7200.2.o.e.7199.1 4 5.3 odd 4
7200.2.o.e.7199.2 4 60.47 odd 4
7200.2.o.j.7199.3 4 20.3 even 4
7200.2.o.j.7199.4 4 15.2 even 4
7200.2.o.m.7199.3 4 60.23 odd 4
7200.2.o.m.7199.4 4 5.2 odd 4