Properties

Label 7920.2.a.bo.1.1
Level $7920$
Weight $2$
Character 7920.1
Self dual yes
Analytic conductor $63.242$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.2415184009\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 660)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 7920.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} -4.60555 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -4.60555 q^{7} +1.00000 q^{11} -4.60555 q^{13} -6.60555 q^{17} +7.21110 q^{19} +1.00000 q^{25} -8.00000 q^{29} -9.21110 q^{31} +4.60555 q^{35} -3.21110 q^{37} -8.00000 q^{41} +3.39445 q^{43} -5.21110 q^{47} +14.2111 q^{49} -2.00000 q^{53} -1.00000 q^{55} +8.00000 q^{59} +7.21110 q^{61} +4.60555 q^{65} +4.00000 q^{67} -14.4222 q^{71} +0.605551 q^{73} -4.60555 q^{77} +11.2111 q^{79} -10.6056 q^{83} +6.60555 q^{85} -6.00000 q^{89} +21.2111 q^{91} -7.21110 q^{95} -12.4222 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{7} + O(q^{10}) \) \( 2 q - 2 q^{5} - 2 q^{7} + 2 q^{11} - 2 q^{13} - 6 q^{17} + 2 q^{25} - 16 q^{29} - 4 q^{31} + 2 q^{35} + 8 q^{37} - 16 q^{41} + 14 q^{43} + 4 q^{47} + 14 q^{49} - 4 q^{53} - 2 q^{55} + 16 q^{59} + 2 q^{65} + 8 q^{67} - 6 q^{73} - 2 q^{77} + 8 q^{79} - 14 q^{83} + 6 q^{85} - 12 q^{89} + 28 q^{91} + 4 q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.60555 −1.74073 −0.870367 0.492403i \(-0.836119\pi\)
−0.870367 + 0.492403i \(0.836119\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −4.60555 −1.27735 −0.638675 0.769477i \(-0.720517\pi\)
−0.638675 + 0.769477i \(0.720517\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.60555 −1.60208 −0.801041 0.598610i \(-0.795720\pi\)
−0.801041 + 0.598610i \(0.795720\pi\)
\(18\) 0 0
\(19\) 7.21110 1.65434 0.827170 0.561951i \(-0.189949\pi\)
0.827170 + 0.561951i \(0.189949\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) −9.21110 −1.65436 −0.827181 0.561935i \(-0.810057\pi\)
−0.827181 + 0.561935i \(0.810057\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.60555 0.778480
\(36\) 0 0
\(37\) −3.21110 −0.527902 −0.263951 0.964536i \(-0.585026\pi\)
−0.263951 + 0.964536i \(0.585026\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 3.39445 0.517649 0.258824 0.965924i \(-0.416665\pi\)
0.258824 + 0.965924i \(0.416665\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.21110 −0.760117 −0.380059 0.924962i \(-0.624096\pi\)
−0.380059 + 0.924962i \(0.624096\pi\)
\(48\) 0 0
\(49\) 14.2111 2.03016
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 7.21110 0.923287 0.461644 0.887066i \(-0.347260\pi\)
0.461644 + 0.887066i \(0.347260\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.60555 0.571248
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.4222 −1.71160 −0.855800 0.517306i \(-0.826935\pi\)
−0.855800 + 0.517306i \(0.826935\pi\)
\(72\) 0 0
\(73\) 0.605551 0.0708744 0.0354372 0.999372i \(-0.488718\pi\)
0.0354372 + 0.999372i \(0.488718\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.60555 −0.524851
\(78\) 0 0
\(79\) 11.2111 1.26135 0.630674 0.776048i \(-0.282779\pi\)
0.630674 + 0.776048i \(0.282779\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −10.6056 −1.16411 −0.582055 0.813149i \(-0.697751\pi\)
−0.582055 + 0.813149i \(0.697751\pi\)
\(84\) 0 0
\(85\) 6.60555 0.716473
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 21.2111 2.22353
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −7.21110 −0.739844
\(96\) 0 0
\(97\) −12.4222 −1.26128 −0.630642 0.776074i \(-0.717208\pi\)
−0.630642 + 0.776074i \(0.717208\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.21110 −0.518524 −0.259262 0.965807i \(-0.583479\pi\)
−0.259262 + 0.965807i \(0.583479\pi\)
\(102\) 0 0
\(103\) −1.21110 −0.119333 −0.0596667 0.998218i \(-0.519004\pi\)
−0.0596667 + 0.998218i \(0.519004\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.8167 1.52905 0.764527 0.644592i \(-0.222973\pi\)
0.764527 + 0.644592i \(0.222973\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.788897 0.0742132 0.0371066 0.999311i \(-0.488186\pi\)
0.0371066 + 0.999311i \(0.488186\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 30.4222 2.78880
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.60555 0.763619 0.381810 0.924241i \(-0.375301\pi\)
0.381810 + 0.924241i \(0.375301\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.78890 −0.593149 −0.296574 0.955010i \(-0.595844\pi\)
−0.296574 + 0.955010i \(0.595844\pi\)
\(132\) 0 0
\(133\) −33.2111 −2.87977
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −16.4222 −1.39291 −0.696457 0.717599i \(-0.745241\pi\)
−0.696457 + 0.717599i \(0.745241\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.60555 −0.385136
\(144\) 0 0
\(145\) 8.00000 0.664364
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.21110 0.754603 0.377301 0.926090i \(-0.376852\pi\)
0.377301 + 0.926090i \(0.376852\pi\)
\(150\) 0 0
\(151\) 19.2111 1.56338 0.781689 0.623669i \(-0.214359\pi\)
0.781689 + 0.623669i \(0.214359\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 9.21110 0.739854
\(156\) 0 0
\(157\) 3.21110 0.256274 0.128137 0.991756i \(-0.459100\pi\)
0.128137 + 0.991756i \(0.459100\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 5.21110 0.408165 0.204083 0.978954i \(-0.434579\pi\)
0.204083 + 0.978954i \(0.434579\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −19.8167 −1.53346 −0.766729 0.641970i \(-0.778117\pi\)
−0.766729 + 0.641970i \(0.778117\pi\)
\(168\) 0 0
\(169\) 8.21110 0.631623
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −0.183346 −0.0139396 −0.00696978 0.999976i \(-0.502219\pi\)
−0.00696978 + 0.999976i \(0.502219\pi\)
\(174\) 0 0
\(175\) −4.60555 −0.348147
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 23.2111 1.72527 0.862634 0.505829i \(-0.168813\pi\)
0.862634 + 0.505829i \(0.168813\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.21110 0.236085
\(186\) 0 0
\(187\) −6.60555 −0.483046
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 18.4222 1.33298 0.666492 0.745512i \(-0.267795\pi\)
0.666492 + 0.745512i \(0.267795\pi\)
\(192\) 0 0
\(193\) −21.8167 −1.57040 −0.785199 0.619244i \(-0.787439\pi\)
−0.785199 + 0.619244i \(0.787439\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.3944 0.954315 0.477157 0.878818i \(-0.341667\pi\)
0.477157 + 0.878818i \(0.341667\pi\)
\(198\) 0 0
\(199\) 10.4222 0.738811 0.369405 0.929268i \(-0.379561\pi\)
0.369405 + 0.929268i \(0.379561\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 36.8444 2.58597
\(204\) 0 0
\(205\) 8.00000 0.558744
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 7.21110 0.498802
\(210\) 0 0
\(211\) 23.2111 1.59792 0.798959 0.601385i \(-0.205384\pi\)
0.798959 + 0.601385i \(0.205384\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.39445 −0.231499
\(216\) 0 0
\(217\) 42.4222 2.87981
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 30.4222 2.04642
\(222\) 0 0
\(223\) −9.21110 −0.616821 −0.308411 0.951253i \(-0.599797\pi\)
−0.308411 + 0.951253i \(0.599797\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −15.8167 −1.04979 −0.524894 0.851168i \(-0.675895\pi\)
−0.524894 + 0.851168i \(0.675895\pi\)
\(228\) 0 0
\(229\) −8.42221 −0.556555 −0.278277 0.960501i \(-0.589763\pi\)
−0.278277 + 0.960501i \(0.589763\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.6056 0.694793 0.347396 0.937718i \(-0.387066\pi\)
0.347396 + 0.937718i \(0.387066\pi\)
\(234\) 0 0
\(235\) 5.21110 0.339935
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.78890 0.439137 0.219569 0.975597i \(-0.429535\pi\)
0.219569 + 0.975597i \(0.429535\pi\)
\(240\) 0 0
\(241\) −15.2111 −0.979833 −0.489917 0.871769i \(-0.662973\pi\)
−0.489917 + 0.871769i \(0.662973\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −14.2111 −0.907914
\(246\) 0 0
\(247\) −33.2111 −2.11317
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −10.4222 −0.657844 −0.328922 0.944357i \(-0.606685\pi\)
−0.328922 + 0.944357i \(0.606685\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 23.2111 1.44787 0.723934 0.689869i \(-0.242332\pi\)
0.723934 + 0.689869i \(0.242332\pi\)
\(258\) 0 0
\(259\) 14.7889 0.918937
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −17.0278 −1.04998 −0.524988 0.851109i \(-0.675930\pi\)
−0.524988 + 0.851109i \(0.675930\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 20.4222 1.24516 0.622582 0.782555i \(-0.286084\pi\)
0.622582 + 0.782555i \(0.286084\pi\)
\(270\) 0 0
\(271\) 4.78890 0.290905 0.145452 0.989365i \(-0.453536\pi\)
0.145452 + 0.989365i \(0.453536\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 32.2389 1.93705 0.968523 0.248925i \(-0.0800774\pi\)
0.968523 + 0.248925i \(0.0800774\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −4.60555 −0.273772 −0.136886 0.990587i \(-0.543709\pi\)
−0.136886 + 0.990587i \(0.543709\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 36.8444 2.17486
\(288\) 0 0
\(289\) 26.6333 1.56667
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −19.8167 −1.15770 −0.578851 0.815434i \(-0.696499\pi\)
−0.578851 + 0.815434i \(0.696499\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −15.6333 −0.901089
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7.21110 −0.412907
\(306\) 0 0
\(307\) −8.60555 −0.491145 −0.245572 0.969378i \(-0.578976\pi\)
−0.245572 + 0.969378i \(0.578976\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 30.4222 1.72508 0.862542 0.505985i \(-0.168871\pi\)
0.862542 + 0.505985i \(0.168871\pi\)
\(312\) 0 0
\(313\) 8.78890 0.496778 0.248389 0.968660i \(-0.420099\pi\)
0.248389 + 0.968660i \(0.420099\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −11.2111 −0.629678 −0.314839 0.949145i \(-0.601951\pi\)
−0.314839 + 0.949145i \(0.601951\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −47.6333 −2.65039
\(324\) 0 0
\(325\) −4.60555 −0.255470
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 6.78890 0.373152 0.186576 0.982441i \(-0.440261\pi\)
0.186576 + 0.982441i \(0.440261\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −28.2389 −1.53827 −0.769134 0.639087i \(-0.779312\pi\)
−0.769134 + 0.639087i \(0.779312\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −9.21110 −0.498809
\(342\) 0 0
\(343\) −33.2111 −1.79323
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −29.0278 −1.55829 −0.779146 0.626843i \(-0.784347\pi\)
−0.779146 + 0.626843i \(0.784347\pi\)
\(348\) 0 0
\(349\) −8.78890 −0.470459 −0.235229 0.971940i \(-0.575584\pi\)
−0.235229 + 0.971940i \(0.575584\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) 14.4222 0.765451
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18.7889 0.991640 0.495820 0.868425i \(-0.334868\pi\)
0.495820 + 0.868425i \(0.334868\pi\)
\(360\) 0 0
\(361\) 33.0000 1.73684
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −0.605551 −0.0316960
\(366\) 0 0
\(367\) −6.78890 −0.354378 −0.177189 0.984177i \(-0.556700\pi\)
−0.177189 + 0.984177i \(0.556700\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 9.21110 0.478217
\(372\) 0 0
\(373\) 4.60555 0.238466 0.119233 0.992866i \(-0.461956\pi\)
0.119233 + 0.992866i \(0.461956\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 36.8444 1.89758
\(378\) 0 0
\(379\) 18.4222 0.946285 0.473143 0.880986i \(-0.343120\pi\)
0.473143 + 0.880986i \(0.343120\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 18.7889 0.960068 0.480034 0.877250i \(-0.340624\pi\)
0.480034 + 0.877250i \(0.340624\pi\)
\(384\) 0 0
\(385\) 4.60555 0.234721
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −30.8444 −1.56387 −0.781937 0.623358i \(-0.785768\pi\)
−0.781937 + 0.623358i \(0.785768\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −11.2111 −0.564092
\(396\) 0 0
\(397\) 28.4222 1.42647 0.713235 0.700925i \(-0.247229\pi\)
0.713235 + 0.700925i \(0.247229\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −20.4222 −1.01984 −0.509918 0.860223i \(-0.670324\pi\)
−0.509918 + 0.860223i \(0.670324\pi\)
\(402\) 0 0
\(403\) 42.4222 2.11320
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.21110 −0.159168
\(408\) 0 0
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −36.8444 −1.81299
\(414\) 0 0
\(415\) 10.6056 0.520606
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −19.2111 −0.936292 −0.468146 0.883651i \(-0.655078\pi\)
−0.468146 + 0.883651i \(0.655078\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.60555 −0.320416
\(426\) 0 0
\(427\) −33.2111 −1.60720
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.57779 −0.0759997 −0.0379999 0.999278i \(-0.512099\pi\)
−0.0379999 + 0.999278i \(0.512099\pi\)
\(432\) 0 0
\(433\) 8.42221 0.404745 0.202373 0.979309i \(-0.435135\pi\)
0.202373 + 0.979309i \(0.435135\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 25.2111 1.19782 0.598908 0.800818i \(-0.295602\pi\)
0.598908 + 0.800818i \(0.295602\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −18.8444 −0.889323 −0.444661 0.895699i \(-0.646676\pi\)
−0.444661 + 0.895699i \(0.646676\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −21.2111 −0.994392
\(456\) 0 0
\(457\) −4.60555 −0.215439 −0.107719 0.994181i \(-0.534355\pi\)
−0.107719 + 0.994181i \(0.534355\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.21110 0.429004 0.214502 0.976724i \(-0.431187\pi\)
0.214502 + 0.976724i \(0.431187\pi\)
\(462\) 0 0
\(463\) −1.21110 −0.0562847 −0.0281424 0.999604i \(-0.508959\pi\)
−0.0281424 + 0.999604i \(0.508959\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 0 0
\(469\) −18.4222 −0.850658
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.39445 0.156077
\(474\) 0 0
\(475\) 7.21110 0.330868
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −14.7889 −0.675722 −0.337861 0.941196i \(-0.609703\pi\)
−0.337861 + 0.941196i \(0.609703\pi\)
\(480\) 0 0
\(481\) 14.7889 0.674316
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.4222 0.564063
\(486\) 0 0
\(487\) −5.57779 −0.252754 −0.126377 0.991982i \(-0.540335\pi\)
−0.126377 + 0.991982i \(0.540335\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −5.57779 −0.251722 −0.125861 0.992048i \(-0.540169\pi\)
−0.125861 + 0.992048i \(0.540169\pi\)
\(492\) 0 0
\(493\) 52.8444 2.37999
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 66.4222 2.97944
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4.18335 0.186526 0.0932631 0.995641i \(-0.470270\pi\)
0.0932631 + 0.995641i \(0.470270\pi\)
\(504\) 0 0
\(505\) 5.21110 0.231891
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 28.4222 1.25979 0.629896 0.776679i \(-0.283097\pi\)
0.629896 + 0.776679i \(0.283097\pi\)
\(510\) 0 0
\(511\) −2.78890 −0.123374
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.21110 0.0533676
\(516\) 0 0
\(517\) −5.21110 −0.229184
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 8.42221 0.368984 0.184492 0.982834i \(-0.440936\pi\)
0.184492 + 0.982834i \(0.440936\pi\)
\(522\) 0 0
\(523\) −28.2389 −1.23480 −0.617400 0.786650i \(-0.711814\pi\)
−0.617400 + 0.786650i \(0.711814\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 60.8444 2.65042
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 36.8444 1.59591
\(534\) 0 0
\(535\) −15.8167 −0.683814
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 14.2111 0.612116
\(540\) 0 0
\(541\) 3.57779 0.153821 0.0769107 0.997038i \(-0.475494\pi\)
0.0769107 + 0.997038i \(0.475494\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) 14.1833 0.606436 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −57.6888 −2.45763
\(552\) 0 0
\(553\) −51.6333 −2.19567
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.97224 0.295423 0.147712 0.989030i \(-0.452809\pi\)
0.147712 + 0.989030i \(0.452809\pi\)
\(558\) 0 0
\(559\) −15.6333 −0.661218
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −19.8167 −0.835172 −0.417586 0.908637i \(-0.637124\pi\)
−0.417586 + 0.908637i \(0.637124\pi\)
\(564\) 0 0
\(565\) −0.788897 −0.0331892
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 27.6333 1.15845 0.579224 0.815168i \(-0.303356\pi\)
0.579224 + 0.815168i \(0.303356\pi\)
\(570\) 0 0
\(571\) −12.4222 −0.519853 −0.259927 0.965628i \(-0.583698\pi\)
−0.259927 + 0.965628i \(0.583698\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 28.7889 1.19850 0.599249 0.800563i \(-0.295466\pi\)
0.599249 + 0.800563i \(0.295466\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 48.8444 2.02641
\(582\) 0 0
\(583\) −2.00000 −0.0828315
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.42221 0.265073 0.132536 0.991178i \(-0.457688\pi\)
0.132536 + 0.991178i \(0.457688\pi\)
\(588\) 0 0
\(589\) −66.4222 −2.73688
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −13.0278 −0.534986 −0.267493 0.963560i \(-0.586195\pi\)
−0.267493 + 0.963560i \(0.586195\pi\)
\(594\) 0 0
\(595\) −30.4222 −1.24719
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −8.78890 −0.358507 −0.179253 0.983803i \(-0.557368\pi\)
−0.179253 + 0.983803i \(0.557368\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −4.97224 −0.201817 −0.100909 0.994896i \(-0.532175\pi\)
−0.100909 + 0.994896i \(0.532175\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) −21.8167 −0.881166 −0.440583 0.897712i \(-0.645228\pi\)
−0.440583 + 0.897712i \(0.645228\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 24.7889 0.997963 0.498982 0.866613i \(-0.333707\pi\)
0.498982 + 0.866613i \(0.333707\pi\)
\(618\) 0 0
\(619\) 19.6333 0.789129 0.394565 0.918868i \(-0.370895\pi\)
0.394565 + 0.918868i \(0.370895\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 27.6333 1.10711
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 21.2111 0.845742
\(630\) 0 0
\(631\) −36.8444 −1.46675 −0.733376 0.679823i \(-0.762057\pi\)
−0.733376 + 0.679823i \(0.762057\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.60555 −0.341501
\(636\) 0 0
\(637\) −65.4500 −2.59322
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.8444 −0.744309 −0.372155 0.928171i \(-0.621381\pi\)
−0.372155 + 0.928171i \(0.621381\pi\)
\(642\) 0 0
\(643\) −0.366692 −0.0144609 −0.00723047 0.999974i \(-0.502302\pi\)
−0.00723047 + 0.999974i \(0.502302\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 45.2111 1.77743 0.888716 0.458458i \(-0.151598\pi\)
0.888716 + 0.458458i \(0.151598\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −4.78890 −0.187404 −0.0937020 0.995600i \(-0.529870\pi\)
−0.0937020 + 0.995600i \(0.529870\pi\)
\(654\) 0 0
\(655\) 6.78890 0.265264
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 45.2666 1.76067 0.880334 0.474355i \(-0.157319\pi\)
0.880334 + 0.474355i \(0.157319\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 33.2111 1.28787
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 7.21110 0.278382
\(672\) 0 0
\(673\) −11.3944 −0.439224 −0.219612 0.975587i \(-0.570479\pi\)
−0.219612 + 0.975587i \(0.570479\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −5.39445 −0.207326 −0.103663 0.994613i \(-0.533056\pi\)
−0.103663 + 0.994613i \(0.533056\pi\)
\(678\) 0 0
\(679\) 57.2111 2.19556
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −30.7889 −1.17810 −0.589052 0.808095i \(-0.700499\pi\)
−0.589052 + 0.808095i \(0.700499\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9.21110 0.350915
\(690\) 0 0
\(691\) 13.5778 0.516524 0.258262 0.966075i \(-0.416850\pi\)
0.258262 + 0.966075i \(0.416850\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.4222 0.622930
\(696\) 0 0
\(697\) 52.8444 2.00162
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −46.4222 −1.75334 −0.876671 0.481090i \(-0.840241\pi\)
−0.876671 + 0.481090i \(0.840241\pi\)
\(702\) 0 0
\(703\) −23.1556 −0.873330
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 24.0000 0.902613
\(708\) 0 0
\(709\) 5.63331 0.211563 0.105782 0.994389i \(-0.466266\pi\)
0.105782 + 0.994389i \(0.466266\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 4.60555 0.172238
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −26.4222 −0.985382 −0.492691 0.870204i \(-0.663987\pi\)
−0.492691 + 0.870204i \(0.663987\pi\)
\(720\) 0 0
\(721\) 5.57779 0.207728
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −8.00000 −0.297113
\(726\) 0 0
\(727\) 28.8444 1.06978 0.534890 0.844922i \(-0.320353\pi\)
0.534890 + 0.844922i \(0.320353\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −22.4222 −0.829315
\(732\) 0 0
\(733\) −7.39445 −0.273120 −0.136560 0.990632i \(-0.543605\pi\)
−0.136560 + 0.990632i \(0.543605\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) −41.2666 −1.51802 −0.759008 0.651081i \(-0.774316\pi\)
−0.759008 + 0.651081i \(0.774316\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 37.3944 1.37187 0.685935 0.727663i \(-0.259394\pi\)
0.685935 + 0.727663i \(0.259394\pi\)
\(744\) 0 0
\(745\) −9.21110 −0.337469
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −72.8444 −2.66168
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −19.2111 −0.699164
\(756\) 0 0
\(757\) −12.7889 −0.464820 −0.232410 0.972618i \(-0.574661\pi\)
−0.232410 + 0.972618i \(0.574661\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −27.6333 −1.00171 −0.500853 0.865532i \(-0.666980\pi\)
−0.500853 + 0.865532i \(0.666980\pi\)
\(762\) 0 0
\(763\) 46.0555 1.66732
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −36.8444 −1.33037
\(768\) 0 0
\(769\) −43.2111 −1.55823 −0.779116 0.626880i \(-0.784332\pi\)
−0.779116 + 0.626880i \(0.784332\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 48.0555 1.72844 0.864218 0.503117i \(-0.167814\pi\)
0.864218 + 0.503117i \(0.167814\pi\)
\(774\) 0 0
\(775\) −9.21110 −0.330873
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −57.6888 −2.06692
\(780\) 0 0
\(781\) −14.4222 −0.516067
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.21110 −0.114609
\(786\) 0 0
\(787\) −39.0278 −1.39119 −0.695595 0.718434i \(-0.744859\pi\)
−0.695595 + 0.718434i \(0.744859\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.63331 −0.129186
\(792\) 0 0
\(793\) −33.2111 −1.17936
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −40.0555 −1.41884 −0.709420 0.704786i \(-0.751043\pi\)
−0.709420 + 0.704786i \(0.751043\pi\)
\(798\) 0 0
\(799\) 34.4222 1.21777
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0.605551 0.0213694
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) 7.57779 0.266092 0.133046 0.991110i \(-0.457524\pi\)
0.133046 + 0.991110i \(0.457524\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.21110 −0.182537
\(816\) 0 0
\(817\) 24.4777 0.856367
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.8444 0.448273 0.224137 0.974558i \(-0.428044\pi\)
0.224137 + 0.974558i \(0.428044\pi\)
\(822\) 0 0
\(823\) −41.2111 −1.43653 −0.718264 0.695770i \(-0.755063\pi\)
−0.718264 + 0.695770i \(0.755063\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.60555 −0.0906039 −0.0453019 0.998973i \(-0.514425\pi\)
−0.0453019 + 0.998973i \(0.514425\pi\)
\(828\) 0 0
\(829\) 6.36669 0.221124 0.110562 0.993869i \(-0.464735\pi\)
0.110562 + 0.993869i \(0.464735\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −93.8722 −3.25248
\(834\) 0 0
\(835\) 19.8167 0.685784
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 30.4222 1.05029 0.525146 0.851012i \(-0.324011\pi\)
0.525146 + 0.851012i \(0.324011\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −8.21110 −0.282471
\(846\) 0 0
\(847\) −4.60555 −0.158249
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −40.2389 −1.37775 −0.688876 0.724879i \(-0.741896\pi\)
−0.688876 + 0.724879i \(0.741896\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −25.0278 −0.854932 −0.427466 0.904031i \(-0.640594\pi\)
−0.427466 + 0.904031i \(0.640594\pi\)
\(858\) 0 0
\(859\) −57.2111 −1.95202 −0.976009 0.217731i \(-0.930134\pi\)
−0.976009 + 0.217731i \(0.930134\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 39.6333 1.34913 0.674567 0.738214i \(-0.264330\pi\)
0.674567 + 0.738214i \(0.264330\pi\)
\(864\) 0 0
\(865\) 0.183346 0.00623396
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 11.2111 0.380311
\(870\) 0 0
\(871\) −18.4222 −0.624213
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4.60555 0.155696
\(876\) 0 0
\(877\) −24.2389 −0.818488 −0.409244 0.912425i \(-0.634208\pi\)
−0.409244 + 0.912425i \(0.634208\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −26.8444 −0.904411 −0.452206 0.891914i \(-0.649363\pi\)
−0.452206 + 0.891914i \(0.649363\pi\)
\(882\) 0 0
\(883\) −6.42221 −0.216124 −0.108062 0.994144i \(-0.534465\pi\)
−0.108062 + 0.994144i \(0.534465\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 13.0278 0.437429 0.218715 0.975789i \(-0.429814\pi\)
0.218715 + 0.975789i \(0.429814\pi\)
\(888\) 0 0
\(889\) −39.6333 −1.32926
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −37.5778 −1.25749
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 73.6888 2.45766
\(900\) 0 0
\(901\) 13.2111 0.440126
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −23.2111 −0.771563
\(906\) 0 0
\(907\) 34.0555 1.13079 0.565397 0.824819i \(-0.308723\pi\)
0.565397 + 0.824819i \(0.308723\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 18.4222 0.610355 0.305177 0.952296i \(-0.401284\pi\)
0.305177 + 0.952296i \(0.401284\pi\)
\(912\) 0 0
\(913\) −10.6056 −0.350993
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 31.2666 1.03251
\(918\) 0 0
\(919\) 26.0000 0.857661 0.428830 0.903385i \(-0.358926\pi\)
0.428830 + 0.903385i \(0.358926\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 66.4222 2.18631
\(924\) 0 0
\(925\) −3.21110 −0.105580
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −26.8444 −0.880737 −0.440368 0.897817i \(-0.645152\pi\)
−0.440368 + 0.897817i \(0.645152\pi\)
\(930\) 0 0
\(931\) 102.478 3.35857
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 6.60555 0.216025
\(936\) 0 0
\(937\) 20.9722 0.685133 0.342567 0.939494i \(-0.388704\pi\)
0.342567 + 0.939494i \(0.388704\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −53.2111 −1.73463 −0.867316 0.497758i \(-0.834157\pi\)
−0.867316 + 0.497758i \(0.834157\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −3.63331 −0.118067 −0.0590333 0.998256i \(-0.518802\pi\)
−0.0590333 + 0.998256i \(0.518802\pi\)
\(948\) 0 0
\(949\) −2.78890 −0.0905314
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 35.4500 1.14834 0.574168 0.818737i \(-0.305325\pi\)
0.574168 + 0.818737i \(0.305325\pi\)
\(954\) 0 0
\(955\) −18.4222 −0.596129
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −27.6333 −0.892326
\(960\) 0 0
\(961\) 53.8444 1.73692
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 21.8167 0.702303
\(966\) 0 0
\(967\) −25.8167 −0.830208 −0.415104 0.909774i \(-0.636255\pi\)
−0.415104 + 0.909774i \(0.636255\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) 75.6333 2.42469
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 52.4777 1.67378 0.836890 0.547372i \(-0.184372\pi\)
0.836890 + 0.547372i \(0.184372\pi\)
\(984\) 0 0
\(985\) −13.3944 −0.426783
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 35.6333 1.13193 0.565965 0.824430i \(-0.308504\pi\)
0.565965 + 0.824430i \(0.308504\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −10.4222 −0.330406
\(996\) 0 0
\(997\) 15.0278 0.475934 0.237967 0.971273i \(-0.423519\pi\)
0.237967 + 0.971273i \(0.423519\pi\)
\(998\) 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 7920.2.a.bo.1.1 2
3.2 odd 2 2640.2.a.bc.1.1 2
4.3 odd 2 1980.2.a.h.1.2 2
12.11 even 2 660.2.a.e.1.2 2
20.3 even 4 9900.2.c.q.5149.1 4
20.7 even 4 9900.2.c.q.5149.4 4
20.19 odd 2 9900.2.a.bl.1.1 2
60.23 odd 4 3300.2.c.l.1849.1 4
60.47 odd 4 3300.2.c.l.1849.4 4
60.59 even 2 3300.2.a.w.1.1 2
132.131 odd 2 7260.2.a.w.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
660.2.a.e.1.2 2 12.11 even 2
1980.2.a.h.1.2 2 4.3 odd 2
2640.2.a.bc.1.1 2 3.2 odd 2
3300.2.a.w.1.1 2 60.59 even 2
3300.2.c.l.1849.1 4 60.23 odd 4
3300.2.c.l.1849.4 4 60.47 odd 4
7260.2.a.w.1.1 2 132.131 odd 2
7920.2.a.bo.1.1 2 1.1 even 1 trivial
9900.2.a.bl.1.1 2 20.19 odd 2
9900.2.c.q.5149.1 4 20.3 even 4
9900.2.c.q.5149.4 4 20.7 even 4