Properties

Label 1932.2.a.e.1.2
Level $1932$
Weight $2$
Character 1932.1
Self dual yes
Analytic conductor $15.427$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1932.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.4270976705\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 1932.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.30278 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.30278 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.60555 q^{11} -0.697224 q^{13} -2.30278 q^{15} -3.00000 q^{17} -7.60555 q^{19} -1.00000 q^{21} -1.00000 q^{23} +0.302776 q^{25} -1.00000 q^{27} +1.00000 q^{29} +1.00000 q^{31} +3.60555 q^{33} +2.30278 q^{35} -8.21110 q^{37} +0.697224 q^{39} -5.60555 q^{41} -7.90833 q^{43} +2.30278 q^{45} -10.6056 q^{47} +1.00000 q^{49} +3.00000 q^{51} +12.9083 q^{53} -8.30278 q^{55} +7.60555 q^{57} +10.9083 q^{59} +8.51388 q^{61} +1.00000 q^{63} -1.60555 q^{65} +10.1194 q^{67} +1.00000 q^{69} -11.9083 q^{71} +10.8167 q^{73} -0.302776 q^{75} -3.60555 q^{77} -6.39445 q^{79} +1.00000 q^{81} -0.394449 q^{83} -6.90833 q^{85} -1.00000 q^{87} +3.30278 q^{89} -0.697224 q^{91} -1.00000 q^{93} -17.5139 q^{95} -11.0000 q^{97} -3.60555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} - 5 q^{13} - q^{15} - 6 q^{17} - 8 q^{19} - 2 q^{21} - 2 q^{23} - 3 q^{25} - 2 q^{27} + 2 q^{29} + 2 q^{31} + q^{35} - 2 q^{37} + 5 q^{39} - 4 q^{41} - 5 q^{43} + q^{45} - 14 q^{47} + 2 q^{49} + 6 q^{51} + 15 q^{53} - 13 q^{55} + 8 q^{57} + 11 q^{59} - q^{61} + 2 q^{63} + 4 q^{65} - 5 q^{67} + 2 q^{69} - 13 q^{71} + 3 q^{75} - 20 q^{79} + 2 q^{81} - 8 q^{83} - 3 q^{85} - 2 q^{87} + 3 q^{89} - 5 q^{91} - 2 q^{93} - 17 q^{95} - 22 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 2.30278 1.02983 0.514916 0.857240i \(-0.327823\pi\)
0.514916 + 0.857240i \(0.327823\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.60555 −1.08711 −0.543557 0.839372i \(-0.682923\pi\)
−0.543557 + 0.839372i \(0.682923\pi\)
\(12\) 0 0
\(13\) −0.697224 −0.193375 −0.0966876 0.995315i \(-0.530825\pi\)
−0.0966876 + 0.995315i \(0.530825\pi\)
\(14\) 0 0
\(15\) −2.30278 −0.594574
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −7.60555 −1.74483 −0.872417 0.488763i \(-0.837448\pi\)
−0.872417 + 0.488763i \(0.837448\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0.302776 0.0605551
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) 3.60555 0.627646
\(34\) 0 0
\(35\) 2.30278 0.389240
\(36\) 0 0
\(37\) −8.21110 −1.34990 −0.674948 0.737865i \(-0.735834\pi\)
−0.674948 + 0.737865i \(0.735834\pi\)
\(38\) 0 0
\(39\) 0.697224 0.111645
\(40\) 0 0
\(41\) −5.60555 −0.875440 −0.437720 0.899111i \(-0.644214\pi\)
−0.437720 + 0.899111i \(0.644214\pi\)
\(42\) 0 0
\(43\) −7.90833 −1.20601 −0.603004 0.797738i \(-0.706030\pi\)
−0.603004 + 0.797738i \(0.706030\pi\)
\(44\) 0 0
\(45\) 2.30278 0.343278
\(46\) 0 0
\(47\) −10.6056 −1.54698 −0.773489 0.633809i \(-0.781490\pi\)
−0.773489 + 0.633809i \(0.781490\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) 12.9083 1.77310 0.886548 0.462638i \(-0.153097\pi\)
0.886548 + 0.462638i \(0.153097\pi\)
\(54\) 0 0
\(55\) −8.30278 −1.11955
\(56\) 0 0
\(57\) 7.60555 1.00738
\(58\) 0 0
\(59\) 10.9083 1.42014 0.710072 0.704129i \(-0.248663\pi\)
0.710072 + 0.704129i \(0.248663\pi\)
\(60\) 0 0
\(61\) 8.51388 1.09009 0.545045 0.838407i \(-0.316512\pi\)
0.545045 + 0.838407i \(0.316512\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −1.60555 −0.199144
\(66\) 0 0
\(67\) 10.1194 1.23629 0.618143 0.786066i \(-0.287885\pi\)
0.618143 + 0.786066i \(0.287885\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −11.9083 −1.41326 −0.706629 0.707584i \(-0.749785\pi\)
−0.706629 + 0.707584i \(0.749785\pi\)
\(72\) 0 0
\(73\) 10.8167 1.26599 0.632997 0.774154i \(-0.281825\pi\)
0.632997 + 0.774154i \(0.281825\pi\)
\(74\) 0 0
\(75\) −0.302776 −0.0349615
\(76\) 0 0
\(77\) −3.60555 −0.410891
\(78\) 0 0
\(79\) −6.39445 −0.719432 −0.359716 0.933062i \(-0.617126\pi\)
−0.359716 + 0.933062i \(0.617126\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −0.394449 −0.0432964 −0.0216482 0.999766i \(-0.506891\pi\)
−0.0216482 + 0.999766i \(0.506891\pi\)
\(84\) 0 0
\(85\) −6.90833 −0.749313
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 3.30278 0.350094 0.175047 0.984560i \(-0.443992\pi\)
0.175047 + 0.984560i \(0.443992\pi\)
\(90\) 0 0
\(91\) −0.697224 −0.0730890
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −17.5139 −1.79689
\(96\) 0 0
\(97\) −11.0000 −1.11688 −0.558440 0.829545i \(-0.688600\pi\)
−0.558440 + 0.829545i \(0.688600\pi\)
\(98\) 0 0
\(99\) −3.60555 −0.362372
\(100\) 0 0
\(101\) −9.51388 −0.946666 −0.473333 0.880884i \(-0.656949\pi\)
−0.473333 + 0.880884i \(0.656949\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) −2.30278 −0.224728
\(106\) 0 0
\(107\) 12.5139 1.20976 0.604881 0.796316i \(-0.293221\pi\)
0.604881 + 0.796316i \(0.293221\pi\)
\(108\) 0 0
\(109\) −10.9083 −1.04483 −0.522414 0.852692i \(-0.674968\pi\)
−0.522414 + 0.852692i \(0.674968\pi\)
\(110\) 0 0
\(111\) 8.21110 0.779363
\(112\) 0 0
\(113\) 3.51388 0.330558 0.165279 0.986247i \(-0.447148\pi\)
0.165279 + 0.986247i \(0.447148\pi\)
\(114\) 0 0
\(115\) −2.30278 −0.214735
\(116\) 0 0
\(117\) −0.697224 −0.0644584
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 2.00000 0.181818
\(122\) 0 0
\(123\) 5.60555 0.505436
\(124\) 0 0
\(125\) −10.8167 −0.967471
\(126\) 0 0
\(127\) 3.30278 0.293074 0.146537 0.989205i \(-0.453187\pi\)
0.146537 + 0.989205i \(0.453187\pi\)
\(128\) 0 0
\(129\) 7.90833 0.696289
\(130\) 0 0
\(131\) 4.81665 0.420833 0.210416 0.977612i \(-0.432518\pi\)
0.210416 + 0.977612i \(0.432518\pi\)
\(132\) 0 0
\(133\) −7.60555 −0.659485
\(134\) 0 0
\(135\) −2.30278 −0.198191
\(136\) 0 0
\(137\) −16.0278 −1.36934 −0.684672 0.728851i \(-0.740054\pi\)
−0.684672 + 0.728851i \(0.740054\pi\)
\(138\) 0 0
\(139\) −10.3028 −0.873870 −0.436935 0.899493i \(-0.643936\pi\)
−0.436935 + 0.899493i \(0.643936\pi\)
\(140\) 0 0
\(141\) 10.6056 0.893149
\(142\) 0 0
\(143\) 2.51388 0.210221
\(144\) 0 0
\(145\) 2.30278 0.191235
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 19.8167 1.62344 0.811722 0.584044i \(-0.198531\pi\)
0.811722 + 0.584044i \(0.198531\pi\)
\(150\) 0 0
\(151\) −6.60555 −0.537552 −0.268776 0.963203i \(-0.586619\pi\)
−0.268776 + 0.963203i \(0.586619\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 2.30278 0.184963
\(156\) 0 0
\(157\) 23.0278 1.83782 0.918908 0.394473i \(-0.129073\pi\)
0.918908 + 0.394473i \(0.129073\pi\)
\(158\) 0 0
\(159\) −12.9083 −1.02370
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −23.5139 −1.84175 −0.920875 0.389859i \(-0.872524\pi\)
−0.920875 + 0.389859i \(0.872524\pi\)
\(164\) 0 0
\(165\) 8.30278 0.646370
\(166\) 0 0
\(167\) −16.8167 −1.30131 −0.650656 0.759373i \(-0.725506\pi\)
−0.650656 + 0.759373i \(0.725506\pi\)
\(168\) 0 0
\(169\) −12.5139 −0.962606
\(170\) 0 0
\(171\) −7.60555 −0.581611
\(172\) 0 0
\(173\) −18.0278 −1.37062 −0.685312 0.728249i \(-0.740334\pi\)
−0.685312 + 0.728249i \(0.740334\pi\)
\(174\) 0 0
\(175\) 0.302776 0.0228877
\(176\) 0 0
\(177\) −10.9083 −0.819920
\(178\) 0 0
\(179\) −0.908327 −0.0678915 −0.0339458 0.999424i \(-0.510807\pi\)
−0.0339458 + 0.999424i \(0.510807\pi\)
\(180\) 0 0
\(181\) −9.42221 −0.700347 −0.350173 0.936685i \(-0.613877\pi\)
−0.350173 + 0.936685i \(0.613877\pi\)
\(182\) 0 0
\(183\) −8.51388 −0.629364
\(184\) 0 0
\(185\) −18.9083 −1.39017
\(186\) 0 0
\(187\) 10.8167 0.790992
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 21.0278 1.52152 0.760758 0.649036i \(-0.224828\pi\)
0.760758 + 0.649036i \(0.224828\pi\)
\(192\) 0 0
\(193\) 9.81665 0.706618 0.353309 0.935507i \(-0.385056\pi\)
0.353309 + 0.935507i \(0.385056\pi\)
\(194\) 0 0
\(195\) 1.60555 0.114976
\(196\) 0 0
\(197\) 23.3028 1.66025 0.830127 0.557574i \(-0.188268\pi\)
0.830127 + 0.557574i \(0.188268\pi\)
\(198\) 0 0
\(199\) 19.6972 1.39630 0.698150 0.715952i \(-0.254007\pi\)
0.698150 + 0.715952i \(0.254007\pi\)
\(200\) 0 0
\(201\) −10.1194 −0.713770
\(202\) 0 0
\(203\) 1.00000 0.0701862
\(204\) 0 0
\(205\) −12.9083 −0.901557
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 27.4222 1.89683
\(210\) 0 0
\(211\) −13.6056 −0.936645 −0.468322 0.883558i \(-0.655141\pi\)
−0.468322 + 0.883558i \(0.655141\pi\)
\(212\) 0 0
\(213\) 11.9083 0.815945
\(214\) 0 0
\(215\) −18.2111 −1.24199
\(216\) 0 0
\(217\) 1.00000 0.0678844
\(218\) 0 0
\(219\) −10.8167 −0.730922
\(220\) 0 0
\(221\) 2.09167 0.140701
\(222\) 0 0
\(223\) −27.5139 −1.84247 −0.921233 0.389012i \(-0.872817\pi\)
−0.921233 + 0.389012i \(0.872817\pi\)
\(224\) 0 0
\(225\) 0.302776 0.0201850
\(226\) 0 0
\(227\) 19.6972 1.30735 0.653675 0.756775i \(-0.273226\pi\)
0.653675 + 0.756775i \(0.273226\pi\)
\(228\) 0 0
\(229\) −11.6972 −0.772974 −0.386487 0.922295i \(-0.626312\pi\)
−0.386487 + 0.922295i \(0.626312\pi\)
\(230\) 0 0
\(231\) 3.60555 0.237228
\(232\) 0 0
\(233\) 21.3305 1.39741 0.698705 0.715410i \(-0.253760\pi\)
0.698705 + 0.715410i \(0.253760\pi\)
\(234\) 0 0
\(235\) −24.4222 −1.59313
\(236\) 0 0
\(237\) 6.39445 0.415364
\(238\) 0 0
\(239\) 9.69722 0.627261 0.313631 0.949545i \(-0.398455\pi\)
0.313631 + 0.949545i \(0.398455\pi\)
\(240\) 0 0
\(241\) 5.78890 0.372896 0.186448 0.982465i \(-0.440302\pi\)
0.186448 + 0.982465i \(0.440302\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.30278 0.147119
\(246\) 0 0
\(247\) 5.30278 0.337408
\(248\) 0 0
\(249\) 0.394449 0.0249972
\(250\) 0 0
\(251\) −3.39445 −0.214256 −0.107128 0.994245i \(-0.534165\pi\)
−0.107128 + 0.994245i \(0.534165\pi\)
\(252\) 0 0
\(253\) 3.60555 0.226679
\(254\) 0 0
\(255\) 6.90833 0.432616
\(256\) 0 0
\(257\) 13.0278 0.812649 0.406325 0.913729i \(-0.366810\pi\)
0.406325 + 0.913729i \(0.366810\pi\)
\(258\) 0 0
\(259\) −8.21110 −0.510213
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 18.6333 1.14898 0.574489 0.818512i \(-0.305201\pi\)
0.574489 + 0.818512i \(0.305201\pi\)
\(264\) 0 0
\(265\) 29.7250 1.82599
\(266\) 0 0
\(267\) −3.30278 −0.202127
\(268\) 0 0
\(269\) −7.72498 −0.471000 −0.235500 0.971874i \(-0.575673\pi\)
−0.235500 + 0.971874i \(0.575673\pi\)
\(270\) 0 0
\(271\) −10.8167 −0.657065 −0.328532 0.944493i \(-0.606554\pi\)
−0.328532 + 0.944493i \(0.606554\pi\)
\(272\) 0 0
\(273\) 0.697224 0.0421979
\(274\) 0 0
\(275\) −1.09167 −0.0658304
\(276\) 0 0
\(277\) −16.3305 −0.981207 −0.490603 0.871383i \(-0.663224\pi\)
−0.490603 + 0.871383i \(0.663224\pi\)
\(278\) 0 0
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 17.8167 1.06285 0.531426 0.847105i \(-0.321656\pi\)
0.531426 + 0.847105i \(0.321656\pi\)
\(282\) 0 0
\(283\) 5.69722 0.338665 0.169332 0.985559i \(-0.445839\pi\)
0.169332 + 0.985559i \(0.445839\pi\)
\(284\) 0 0
\(285\) 17.5139 1.03743
\(286\) 0 0
\(287\) −5.60555 −0.330885
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 11.0000 0.644831
\(292\) 0 0
\(293\) −13.2111 −0.771801 −0.385900 0.922540i \(-0.626109\pi\)
−0.385900 + 0.922540i \(0.626109\pi\)
\(294\) 0 0
\(295\) 25.1194 1.46251
\(296\) 0 0
\(297\) 3.60555 0.209215
\(298\) 0 0
\(299\) 0.697224 0.0403215
\(300\) 0 0
\(301\) −7.90833 −0.455828
\(302\) 0 0
\(303\) 9.51388 0.546558
\(304\) 0 0
\(305\) 19.6056 1.12261
\(306\) 0 0
\(307\) 13.4222 0.766046 0.383023 0.923739i \(-0.374883\pi\)
0.383023 + 0.923739i \(0.374883\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) 5.90833 0.335030 0.167515 0.985869i \(-0.446426\pi\)
0.167515 + 0.985869i \(0.446426\pi\)
\(312\) 0 0
\(313\) −25.2111 −1.42502 −0.712508 0.701664i \(-0.752441\pi\)
−0.712508 + 0.701664i \(0.752441\pi\)
\(314\) 0 0
\(315\) 2.30278 0.129747
\(316\) 0 0
\(317\) 2.30278 0.129337 0.0646684 0.997907i \(-0.479401\pi\)
0.0646684 + 0.997907i \(0.479401\pi\)
\(318\) 0 0
\(319\) −3.60555 −0.201872
\(320\) 0 0
\(321\) −12.5139 −0.698457
\(322\) 0 0
\(323\) 22.8167 1.26955
\(324\) 0 0
\(325\) −0.211103 −0.0117099
\(326\) 0 0
\(327\) 10.9083 0.603232
\(328\) 0 0
\(329\) −10.6056 −0.584703
\(330\) 0 0
\(331\) 19.2111 1.05594 0.527969 0.849264i \(-0.322954\pi\)
0.527969 + 0.849264i \(0.322954\pi\)
\(332\) 0 0
\(333\) −8.21110 −0.449966
\(334\) 0 0
\(335\) 23.3028 1.27317
\(336\) 0 0
\(337\) 12.5139 0.681674 0.340837 0.940122i \(-0.389290\pi\)
0.340837 + 0.940122i \(0.389290\pi\)
\(338\) 0 0
\(339\) −3.51388 −0.190848
\(340\) 0 0
\(341\) −3.60555 −0.195252
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 2.30278 0.123977
\(346\) 0 0
\(347\) 28.8167 1.54696 0.773480 0.633821i \(-0.218515\pi\)
0.773480 + 0.633821i \(0.218515\pi\)
\(348\) 0 0
\(349\) −4.48612 −0.240137 −0.120068 0.992766i \(-0.538311\pi\)
−0.120068 + 0.992766i \(0.538311\pi\)
\(350\) 0 0
\(351\) 0.697224 0.0372151
\(352\) 0 0
\(353\) 11.6056 0.617701 0.308851 0.951111i \(-0.400056\pi\)
0.308851 + 0.951111i \(0.400056\pi\)
\(354\) 0 0
\(355\) −27.4222 −1.45542
\(356\) 0 0
\(357\) 3.00000 0.158777
\(358\) 0 0
\(359\) −11.0917 −0.585396 −0.292698 0.956205i \(-0.594553\pi\)
−0.292698 + 0.956205i \(0.594553\pi\)
\(360\) 0 0
\(361\) 38.8444 2.04444
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 24.9083 1.30376
\(366\) 0 0
\(367\) −29.7250 −1.55163 −0.775816 0.630960i \(-0.782661\pi\)
−0.775816 + 0.630960i \(0.782661\pi\)
\(368\) 0 0
\(369\) −5.60555 −0.291813
\(370\) 0 0
\(371\) 12.9083 0.670167
\(372\) 0 0
\(373\) 31.4222 1.62698 0.813490 0.581579i \(-0.197565\pi\)
0.813490 + 0.581579i \(0.197565\pi\)
\(374\) 0 0
\(375\) 10.8167 0.558570
\(376\) 0 0
\(377\) −0.697224 −0.0359089
\(378\) 0 0
\(379\) −5.21110 −0.267676 −0.133838 0.991003i \(-0.542730\pi\)
−0.133838 + 0.991003i \(0.542730\pi\)
\(380\) 0 0
\(381\) −3.30278 −0.169206
\(382\) 0 0
\(383\) −30.0278 −1.53435 −0.767173 0.641440i \(-0.778337\pi\)
−0.767173 + 0.641440i \(0.778337\pi\)
\(384\) 0 0
\(385\) −8.30278 −0.423149
\(386\) 0 0
\(387\) −7.90833 −0.402003
\(388\) 0 0
\(389\) 15.7889 0.800529 0.400264 0.916400i \(-0.368918\pi\)
0.400264 + 0.916400i \(0.368918\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) −4.81665 −0.242968
\(394\) 0 0
\(395\) −14.7250 −0.740894
\(396\) 0 0
\(397\) −10.6056 −0.532277 −0.266139 0.963935i \(-0.585748\pi\)
−0.266139 + 0.963935i \(0.585748\pi\)
\(398\) 0 0
\(399\) 7.60555 0.380754
\(400\) 0 0
\(401\) −15.4222 −0.770148 −0.385074 0.922886i \(-0.625824\pi\)
−0.385074 + 0.922886i \(0.625824\pi\)
\(402\) 0 0
\(403\) −0.697224 −0.0347312
\(404\) 0 0
\(405\) 2.30278 0.114426
\(406\) 0 0
\(407\) 29.6056 1.46749
\(408\) 0 0
\(409\) −21.4222 −1.05926 −0.529630 0.848229i \(-0.677669\pi\)
−0.529630 + 0.848229i \(0.677669\pi\)
\(410\) 0 0
\(411\) 16.0278 0.790591
\(412\) 0 0
\(413\) 10.9083 0.536764
\(414\) 0 0
\(415\) −0.908327 −0.0445880
\(416\) 0 0
\(417\) 10.3028 0.504529
\(418\) 0 0
\(419\) 9.51388 0.464783 0.232392 0.972622i \(-0.425345\pi\)
0.232392 + 0.972622i \(0.425345\pi\)
\(420\) 0 0
\(421\) 11.9083 0.580376 0.290188 0.956970i \(-0.406282\pi\)
0.290188 + 0.956970i \(0.406282\pi\)
\(422\) 0 0
\(423\) −10.6056 −0.515660
\(424\) 0 0
\(425\) −0.908327 −0.0440603
\(426\) 0 0
\(427\) 8.51388 0.412015
\(428\) 0 0
\(429\) −2.51388 −0.121371
\(430\) 0 0
\(431\) −2.09167 −0.100752 −0.0503762 0.998730i \(-0.516042\pi\)
−0.0503762 + 0.998730i \(0.516042\pi\)
\(432\) 0 0
\(433\) −35.0278 −1.68333 −0.841663 0.540003i \(-0.818423\pi\)
−0.841663 + 0.540003i \(0.818423\pi\)
\(434\) 0 0
\(435\) −2.30278 −0.110410
\(436\) 0 0
\(437\) 7.60555 0.363823
\(438\) 0 0
\(439\) 32.6333 1.55750 0.778751 0.627333i \(-0.215853\pi\)
0.778751 + 0.627333i \(0.215853\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −23.6333 −1.12285 −0.561426 0.827527i \(-0.689747\pi\)
−0.561426 + 0.827527i \(0.689747\pi\)
\(444\) 0 0
\(445\) 7.60555 0.360538
\(446\) 0 0
\(447\) −19.8167 −0.937296
\(448\) 0 0
\(449\) 7.88057 0.371907 0.185954 0.982559i \(-0.440463\pi\)
0.185954 + 0.982559i \(0.440463\pi\)
\(450\) 0 0
\(451\) 20.2111 0.951704
\(452\) 0 0
\(453\) 6.60555 0.310356
\(454\) 0 0
\(455\) −1.60555 −0.0752694
\(456\) 0 0
\(457\) −0.724981 −0.0339132 −0.0169566 0.999856i \(-0.505398\pi\)
−0.0169566 + 0.999856i \(0.505398\pi\)
\(458\) 0 0
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) 28.5416 1.32932 0.664658 0.747148i \(-0.268577\pi\)
0.664658 + 0.747148i \(0.268577\pi\)
\(462\) 0 0
\(463\) 26.2111 1.21813 0.609067 0.793119i \(-0.291544\pi\)
0.609067 + 0.793119i \(0.291544\pi\)
\(464\) 0 0
\(465\) −2.30278 −0.106789
\(466\) 0 0
\(467\) −19.6056 −0.907237 −0.453618 0.891196i \(-0.649867\pi\)
−0.453618 + 0.891196i \(0.649867\pi\)
\(468\) 0 0
\(469\) 10.1194 0.467272
\(470\) 0 0
\(471\) −23.0278 −1.06106
\(472\) 0 0
\(473\) 28.5139 1.31107
\(474\) 0 0
\(475\) −2.30278 −0.105659
\(476\) 0 0
\(477\) 12.9083 0.591032
\(478\) 0 0
\(479\) −15.4222 −0.704659 −0.352329 0.935876i \(-0.614610\pi\)
−0.352329 + 0.935876i \(0.614610\pi\)
\(480\) 0 0
\(481\) 5.72498 0.261037
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −25.3305 −1.15020
\(486\) 0 0
\(487\) −19.7889 −0.896721 −0.448360 0.893853i \(-0.647992\pi\)
−0.448360 + 0.893853i \(0.647992\pi\)
\(488\) 0 0
\(489\) 23.5139 1.06333
\(490\) 0 0
\(491\) −12.3305 −0.556469 −0.278235 0.960513i \(-0.589749\pi\)
−0.278235 + 0.960513i \(0.589749\pi\)
\(492\) 0 0
\(493\) −3.00000 −0.135113
\(494\) 0 0
\(495\) −8.30278 −0.373182
\(496\) 0 0
\(497\) −11.9083 −0.534161
\(498\) 0 0
\(499\) 36.5416 1.63583 0.817914 0.575340i \(-0.195130\pi\)
0.817914 + 0.575340i \(0.195130\pi\)
\(500\) 0 0
\(501\) 16.8167 0.751313
\(502\) 0 0
\(503\) −16.3028 −0.726905 −0.363452 0.931613i \(-0.618402\pi\)
−0.363452 + 0.931613i \(0.618402\pi\)
\(504\) 0 0
\(505\) −21.9083 −0.974908
\(506\) 0 0
\(507\) 12.5139 0.555761
\(508\) 0 0
\(509\) −1.39445 −0.0618079 −0.0309039 0.999522i \(-0.509839\pi\)
−0.0309039 + 0.999522i \(0.509839\pi\)
\(510\) 0 0
\(511\) 10.8167 0.478501
\(512\) 0 0
\(513\) 7.60555 0.335793
\(514\) 0 0
\(515\) −36.8444 −1.62356
\(516\) 0 0
\(517\) 38.2389 1.68174
\(518\) 0 0
\(519\) 18.0278 0.791331
\(520\) 0 0
\(521\) 24.7889 1.08602 0.543011 0.839726i \(-0.317284\pi\)
0.543011 + 0.839726i \(0.317284\pi\)
\(522\) 0 0
\(523\) 2.00000 0.0874539 0.0437269 0.999044i \(-0.486077\pi\)
0.0437269 + 0.999044i \(0.486077\pi\)
\(524\) 0 0
\(525\) −0.302776 −0.0132142
\(526\) 0 0
\(527\) −3.00000 −0.130682
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 10.9083 0.473381
\(532\) 0 0
\(533\) 3.90833 0.169288
\(534\) 0 0
\(535\) 28.8167 1.24585
\(536\) 0 0
\(537\) 0.908327 0.0391972
\(538\) 0 0
\(539\) −3.60555 −0.155302
\(540\) 0 0
\(541\) −1.39445 −0.0599520 −0.0299760 0.999551i \(-0.509543\pi\)
−0.0299760 + 0.999551i \(0.509543\pi\)
\(542\) 0 0
\(543\) 9.42221 0.404346
\(544\) 0 0
\(545\) −25.1194 −1.07600
\(546\) 0 0
\(547\) −1.51388 −0.0647288 −0.0323644 0.999476i \(-0.510304\pi\)
−0.0323644 + 0.999476i \(0.510304\pi\)
\(548\) 0 0
\(549\) 8.51388 0.363363
\(550\) 0 0
\(551\) −7.60555 −0.324007
\(552\) 0 0
\(553\) −6.39445 −0.271920
\(554\) 0 0
\(555\) 18.9083 0.802614
\(556\) 0 0
\(557\) −29.4500 −1.24783 −0.623917 0.781490i \(-0.714460\pi\)
−0.623917 + 0.781490i \(0.714460\pi\)
\(558\) 0 0
\(559\) 5.51388 0.233212
\(560\) 0 0
\(561\) −10.8167 −0.456679
\(562\) 0 0
\(563\) 7.11943 0.300048 0.150024 0.988682i \(-0.452065\pi\)
0.150024 + 0.988682i \(0.452065\pi\)
\(564\) 0 0
\(565\) 8.09167 0.340419
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −4.00000 −0.167689 −0.0838444 0.996479i \(-0.526720\pi\)
−0.0838444 + 0.996479i \(0.526720\pi\)
\(570\) 0 0
\(571\) 20.8167 0.871150 0.435575 0.900152i \(-0.356545\pi\)
0.435575 + 0.900152i \(0.356545\pi\)
\(572\) 0 0
\(573\) −21.0278 −0.878447
\(574\) 0 0
\(575\) −0.302776 −0.0126266
\(576\) 0 0
\(577\) −17.6333 −0.734084 −0.367042 0.930204i \(-0.619630\pi\)
−0.367042 + 0.930204i \(0.619630\pi\)
\(578\) 0 0
\(579\) −9.81665 −0.407966
\(580\) 0 0
\(581\) −0.394449 −0.0163645
\(582\) 0 0
\(583\) −46.5416 −1.92756
\(584\) 0 0
\(585\) −1.60555 −0.0663814
\(586\) 0 0
\(587\) −1.30278 −0.0537713 −0.0268857 0.999639i \(-0.508559\pi\)
−0.0268857 + 0.999639i \(0.508559\pi\)
\(588\) 0 0
\(589\) −7.60555 −0.313381
\(590\) 0 0
\(591\) −23.3028 −0.958548
\(592\) 0 0
\(593\) 11.0278 0.452856 0.226428 0.974028i \(-0.427295\pi\)
0.226428 + 0.974028i \(0.427295\pi\)
\(594\) 0 0
\(595\) −6.90833 −0.283214
\(596\) 0 0
\(597\) −19.6972 −0.806154
\(598\) 0 0
\(599\) 17.7250 0.724223 0.362112 0.932135i \(-0.382056\pi\)
0.362112 + 0.932135i \(0.382056\pi\)
\(600\) 0 0
\(601\) −35.9361 −1.46586 −0.732932 0.680302i \(-0.761849\pi\)
−0.732932 + 0.680302i \(0.761849\pi\)
\(602\) 0 0
\(603\) 10.1194 0.412095
\(604\) 0 0
\(605\) 4.60555 0.187242
\(606\) 0 0
\(607\) 2.88057 0.116919 0.0584594 0.998290i \(-0.481381\pi\)
0.0584594 + 0.998290i \(0.481381\pi\)
\(608\) 0 0
\(609\) −1.00000 −0.0405220
\(610\) 0 0
\(611\) 7.39445 0.299147
\(612\) 0 0
\(613\) −44.4500 −1.79532 −0.897659 0.440692i \(-0.854733\pi\)
−0.897659 + 0.440692i \(0.854733\pi\)
\(614\) 0 0
\(615\) 12.9083 0.520514
\(616\) 0 0
\(617\) −7.11943 −0.286617 −0.143309 0.989678i \(-0.545774\pi\)
−0.143309 + 0.989678i \(0.545774\pi\)
\(618\) 0 0
\(619\) 32.3305 1.29947 0.649737 0.760159i \(-0.274879\pi\)
0.649737 + 0.760159i \(0.274879\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 3.30278 0.132323
\(624\) 0 0
\(625\) −26.4222 −1.05689
\(626\) 0 0
\(627\) −27.4222 −1.09514
\(628\) 0 0
\(629\) 24.6333 0.982194
\(630\) 0 0
\(631\) 25.0000 0.995234 0.497617 0.867397i \(-0.334208\pi\)
0.497617 + 0.867397i \(0.334208\pi\)
\(632\) 0 0
\(633\) 13.6056 0.540772
\(634\) 0 0
\(635\) 7.60555 0.301817
\(636\) 0 0
\(637\) −0.697224 −0.0276250
\(638\) 0 0
\(639\) −11.9083 −0.471086
\(640\) 0 0
\(641\) 22.1472 0.874761 0.437381 0.899276i \(-0.355906\pi\)
0.437381 + 0.899276i \(0.355906\pi\)
\(642\) 0 0
\(643\) −49.7250 −1.96096 −0.980481 0.196614i \(-0.937005\pi\)
−0.980481 + 0.196614i \(0.937005\pi\)
\(644\) 0 0
\(645\) 18.2111 0.717061
\(646\) 0 0
\(647\) −23.0917 −0.907827 −0.453914 0.891046i \(-0.649973\pi\)
−0.453914 + 0.891046i \(0.649973\pi\)
\(648\) 0 0
\(649\) −39.3305 −1.54386
\(650\) 0 0
\(651\) −1.00000 −0.0391931
\(652\) 0 0
\(653\) −7.51388 −0.294041 −0.147020 0.989133i \(-0.546968\pi\)
−0.147020 + 0.989133i \(0.546968\pi\)
\(654\) 0 0
\(655\) 11.0917 0.433388
\(656\) 0 0
\(657\) 10.8167 0.421998
\(658\) 0 0
\(659\) 20.3944 0.794455 0.397227 0.917720i \(-0.369972\pi\)
0.397227 + 0.917720i \(0.369972\pi\)
\(660\) 0 0
\(661\) −8.21110 −0.319375 −0.159687 0.987168i \(-0.551049\pi\)
−0.159687 + 0.987168i \(0.551049\pi\)
\(662\) 0 0
\(663\) −2.09167 −0.0812339
\(664\) 0 0
\(665\) −17.5139 −0.679159
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) 27.5139 1.06375
\(670\) 0 0
\(671\) −30.6972 −1.18505
\(672\) 0 0
\(673\) −4.57779 −0.176461 −0.0882305 0.996100i \(-0.528121\pi\)
−0.0882305 + 0.996100i \(0.528121\pi\)
\(674\) 0 0
\(675\) −0.302776 −0.0116538
\(676\) 0 0
\(677\) 30.6972 1.17979 0.589895 0.807480i \(-0.299169\pi\)
0.589895 + 0.807480i \(0.299169\pi\)
\(678\) 0 0
\(679\) −11.0000 −0.422141
\(680\) 0 0
\(681\) −19.6972 −0.754799
\(682\) 0 0
\(683\) −47.6056 −1.82158 −0.910788 0.412875i \(-0.864525\pi\)
−0.910788 + 0.412875i \(0.864525\pi\)
\(684\) 0 0
\(685\) −36.9083 −1.41019
\(686\) 0 0
\(687\) 11.6972 0.446277
\(688\) 0 0
\(689\) −9.00000 −0.342873
\(690\) 0 0
\(691\) −8.93608 −0.339945 −0.169972 0.985449i \(-0.554368\pi\)
−0.169972 + 0.985449i \(0.554368\pi\)
\(692\) 0 0
\(693\) −3.60555 −0.136964
\(694\) 0 0
\(695\) −23.7250 −0.899940
\(696\) 0 0
\(697\) 16.8167 0.636976
\(698\) 0 0
\(699\) −21.3305 −0.806795
\(700\) 0 0
\(701\) 1.88057 0.0710282 0.0355141 0.999369i \(-0.488693\pi\)
0.0355141 + 0.999369i \(0.488693\pi\)
\(702\) 0 0
\(703\) 62.4500 2.35534
\(704\) 0 0
\(705\) 24.4222 0.919793
\(706\) 0 0
\(707\) −9.51388 −0.357806
\(708\) 0 0
\(709\) 6.09167 0.228778 0.114389 0.993436i \(-0.463509\pi\)
0.114389 + 0.993436i \(0.463509\pi\)
\(710\) 0 0
\(711\) −6.39445 −0.239811
\(712\) 0 0
\(713\) −1.00000 −0.0374503
\(714\) 0 0
\(715\) 5.78890 0.216492
\(716\) 0 0
\(717\) −9.69722 −0.362149
\(718\) 0 0
\(719\) −25.0000 −0.932343 −0.466171 0.884694i \(-0.654367\pi\)
−0.466171 + 0.884694i \(0.654367\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) −5.78890 −0.215291
\(724\) 0 0
\(725\) 0.302776 0.0112448
\(726\) 0 0
\(727\) 3.60555 0.133722 0.0668612 0.997762i \(-0.478702\pi\)
0.0668612 + 0.997762i \(0.478702\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 23.7250 0.877500
\(732\) 0 0
\(733\) 23.0278 0.850550 0.425275 0.905064i \(-0.360177\pi\)
0.425275 + 0.905064i \(0.360177\pi\)
\(734\) 0 0
\(735\) −2.30278 −0.0849392
\(736\) 0 0
\(737\) −36.4861 −1.34398
\(738\) 0 0
\(739\) 50.0278 1.84030 0.920150 0.391565i \(-0.128066\pi\)
0.920150 + 0.391565i \(0.128066\pi\)
\(740\) 0 0
\(741\) −5.30278 −0.194802
\(742\) 0 0
\(743\) −6.72498 −0.246716 −0.123358 0.992362i \(-0.539366\pi\)
−0.123358 + 0.992362i \(0.539366\pi\)
\(744\) 0 0
\(745\) 45.6333 1.67188
\(746\) 0 0
\(747\) −0.394449 −0.0144321
\(748\) 0 0
\(749\) 12.5139 0.457247
\(750\) 0 0
\(751\) 35.5139 1.29592 0.647960 0.761674i \(-0.275622\pi\)
0.647960 + 0.761674i \(0.275622\pi\)
\(752\) 0 0
\(753\) 3.39445 0.123701
\(754\) 0 0
\(755\) −15.2111 −0.553589
\(756\) 0 0
\(757\) 17.4222 0.633221 0.316610 0.948556i \(-0.397455\pi\)
0.316610 + 0.948556i \(0.397455\pi\)
\(758\) 0 0
\(759\) −3.60555 −0.130873
\(760\) 0 0
\(761\) −13.2111 −0.478902 −0.239451 0.970908i \(-0.576967\pi\)
−0.239451 + 0.970908i \(0.576967\pi\)
\(762\) 0 0
\(763\) −10.9083 −0.394908
\(764\) 0 0
\(765\) −6.90833 −0.249771
\(766\) 0 0
\(767\) −7.60555 −0.274621
\(768\) 0 0
\(769\) −42.6611 −1.53840 −0.769199 0.639010i \(-0.779344\pi\)
−0.769199 + 0.639010i \(0.779344\pi\)
\(770\) 0 0
\(771\) −13.0278 −0.469183
\(772\) 0 0
\(773\) −24.0278 −0.864218 −0.432109 0.901821i \(-0.642230\pi\)
−0.432109 + 0.901821i \(0.642230\pi\)
\(774\) 0 0
\(775\) 0.302776 0.0108760
\(776\) 0 0
\(777\) 8.21110 0.294572
\(778\) 0 0
\(779\) 42.6333 1.52750
\(780\) 0 0
\(781\) 42.9361 1.53637
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 53.0278 1.89264
\(786\) 0 0
\(787\) −34.3583 −1.22474 −0.612370 0.790571i \(-0.709784\pi\)
−0.612370 + 0.790571i \(0.709784\pi\)
\(788\) 0 0
\(789\) −18.6333 −0.663363
\(790\) 0 0
\(791\) 3.51388 0.124939
\(792\) 0 0
\(793\) −5.93608 −0.210796
\(794\) 0 0
\(795\) −29.7250 −1.05424
\(796\) 0 0
\(797\) −14.6056 −0.517355 −0.258678 0.965964i \(-0.583287\pi\)
−0.258678 + 0.965964i \(0.583287\pi\)
\(798\) 0 0
\(799\) 31.8167 1.12559
\(800\) 0 0
\(801\) 3.30278 0.116698
\(802\) 0 0
\(803\) −39.0000 −1.37628
\(804\) 0 0
\(805\) −2.30278 −0.0811622
\(806\) 0 0
\(807\) 7.72498 0.271932
\(808\) 0 0
\(809\) 7.54163 0.265150 0.132575 0.991173i \(-0.457676\pi\)
0.132575 + 0.991173i \(0.457676\pi\)
\(810\) 0 0
\(811\) 25.2389 0.886256 0.443128 0.896458i \(-0.353869\pi\)
0.443128 + 0.896458i \(0.353869\pi\)
\(812\) 0 0
\(813\) 10.8167 0.379357
\(814\) 0 0
\(815\) −54.1472 −1.89669
\(816\) 0 0
\(817\) 60.1472 2.10428
\(818\) 0 0
\(819\) −0.697224 −0.0243630
\(820\) 0 0
\(821\) 7.81665 0.272803 0.136402 0.990654i \(-0.456446\pi\)
0.136402 + 0.990654i \(0.456446\pi\)
\(822\) 0 0
\(823\) −51.3305 −1.78927 −0.894635 0.446798i \(-0.852564\pi\)
−0.894635 + 0.446798i \(0.852564\pi\)
\(824\) 0 0
\(825\) 1.09167 0.0380072
\(826\) 0 0
\(827\) 34.5416 1.20113 0.600565 0.799576i \(-0.294942\pi\)
0.600565 + 0.799576i \(0.294942\pi\)
\(828\) 0 0
\(829\) 19.8444 0.689225 0.344612 0.938745i \(-0.388010\pi\)
0.344612 + 0.938745i \(0.388010\pi\)
\(830\) 0 0
\(831\) 16.3305 0.566500
\(832\) 0 0
\(833\) −3.00000 −0.103944
\(834\) 0 0
\(835\) −38.7250 −1.34013
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 0 0
\(839\) 30.6972 1.05979 0.529893 0.848065i \(-0.322232\pi\)
0.529893 + 0.848065i \(0.322232\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) −17.8167 −0.613638
\(844\) 0 0
\(845\) −28.8167 −0.991323
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 0 0
\(849\) −5.69722 −0.195528
\(850\) 0 0
\(851\) 8.21110 0.281473
\(852\) 0 0
\(853\) 21.8167 0.746988 0.373494 0.927633i \(-0.378160\pi\)
0.373494 + 0.927633i \(0.378160\pi\)
\(854\) 0 0
\(855\) −17.5139 −0.598962
\(856\) 0 0
\(857\) −25.8167 −0.881880 −0.440940 0.897537i \(-0.645355\pi\)
−0.440940 + 0.897537i \(0.645355\pi\)
\(858\) 0 0
\(859\) 19.2111 0.655474 0.327737 0.944769i \(-0.393714\pi\)
0.327737 + 0.944769i \(0.393714\pi\)
\(860\) 0 0
\(861\) 5.60555 0.191037
\(862\) 0 0
\(863\) −19.3944 −0.660195 −0.330097 0.943947i \(-0.607082\pi\)
−0.330097 + 0.943947i \(0.607082\pi\)
\(864\) 0 0
\(865\) −41.5139 −1.41151
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) 23.0555 0.782105
\(870\) 0 0
\(871\) −7.05551 −0.239067
\(872\) 0 0
\(873\) −11.0000 −0.372294
\(874\) 0 0
\(875\) −10.8167 −0.365670
\(876\) 0 0
\(877\) −17.2111 −0.581178 −0.290589 0.956848i \(-0.593851\pi\)
−0.290589 + 0.956848i \(0.593851\pi\)
\(878\) 0 0
\(879\) 13.2111 0.445599
\(880\) 0 0
\(881\) 46.6611 1.57205 0.786026 0.618194i \(-0.212135\pi\)
0.786026 + 0.618194i \(0.212135\pi\)
\(882\) 0 0
\(883\) 13.1194 0.441504 0.220752 0.975330i \(-0.429149\pi\)
0.220752 + 0.975330i \(0.429149\pi\)
\(884\) 0 0
\(885\) −25.1194 −0.844380
\(886\) 0 0
\(887\) 35.4861 1.19151 0.595754 0.803167i \(-0.296853\pi\)
0.595754 + 0.803167i \(0.296853\pi\)
\(888\) 0 0
\(889\) 3.30278 0.110772
\(890\) 0 0
\(891\) −3.60555 −0.120791
\(892\) 0 0
\(893\) 80.6611 2.69922
\(894\) 0 0
\(895\) −2.09167 −0.0699169
\(896\) 0 0
\(897\) −0.697224 −0.0232796
\(898\) 0 0
\(899\) 1.00000 0.0333519
\(900\) 0 0
\(901\) −38.7250 −1.29012
\(902\) 0 0
\(903\) 7.90833 0.263173
\(904\) 0 0
\(905\) −21.6972 −0.721240
\(906\) 0 0
\(907\) −7.27502 −0.241563 −0.120782 0.992679i \(-0.538540\pi\)
−0.120782 + 0.992679i \(0.538540\pi\)
\(908\) 0 0
\(909\) −9.51388 −0.315555
\(910\) 0 0
\(911\) −14.6056 −0.483904 −0.241952 0.970288i \(-0.577788\pi\)
−0.241952 + 0.970288i \(0.577788\pi\)
\(912\) 0 0
\(913\) 1.42221 0.0470681
\(914\) 0 0
\(915\) −19.6056 −0.648140
\(916\) 0 0
\(917\) 4.81665 0.159060
\(918\) 0 0
\(919\) 52.2666 1.72412 0.862058 0.506809i \(-0.169175\pi\)
0.862058 + 0.506809i \(0.169175\pi\)
\(920\) 0 0
\(921\) −13.4222 −0.442277
\(922\) 0 0
\(923\) 8.30278 0.273289
\(924\) 0 0
\(925\) −2.48612 −0.0817432
\(926\) 0 0
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) −6.90833 −0.226655 −0.113327 0.993558i \(-0.536151\pi\)
−0.113327 + 0.993558i \(0.536151\pi\)
\(930\) 0 0
\(931\) −7.60555 −0.249262
\(932\) 0 0
\(933\) −5.90833 −0.193430
\(934\) 0 0
\(935\) 24.9083 0.814589
\(936\) 0 0
\(937\) 48.8722 1.59658 0.798292 0.602271i \(-0.205737\pi\)
0.798292 + 0.602271i \(0.205737\pi\)
\(938\) 0 0
\(939\) 25.2111 0.822733
\(940\) 0 0
\(941\) −48.8444 −1.59228 −0.796141 0.605111i \(-0.793129\pi\)
−0.796141 + 0.605111i \(0.793129\pi\)
\(942\) 0 0
\(943\) 5.60555 0.182542
\(944\) 0 0
\(945\) −2.30278 −0.0749093
\(946\) 0 0
\(947\) −48.4222 −1.57351 −0.786755 0.617265i \(-0.788241\pi\)
−0.786755 + 0.617265i \(0.788241\pi\)
\(948\) 0 0
\(949\) −7.54163 −0.244812
\(950\) 0 0
\(951\) −2.30278 −0.0746726
\(952\) 0 0
\(953\) −32.3305 −1.04729 −0.523644 0.851937i \(-0.675428\pi\)
−0.523644 + 0.851937i \(0.675428\pi\)
\(954\) 0 0
\(955\) 48.4222 1.56691
\(956\) 0 0
\(957\) 3.60555 0.116551
\(958\) 0 0
\(959\) −16.0278 −0.517563
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 12.5139 0.403254
\(964\) 0 0
\(965\) 22.6056 0.727698
\(966\) 0 0
\(967\) 6.36669 0.204739 0.102370 0.994746i \(-0.467358\pi\)
0.102370 + 0.994746i \(0.467358\pi\)
\(968\) 0 0
\(969\) −22.8167 −0.732977
\(970\) 0 0
\(971\) −22.2750 −0.714839 −0.357420 0.933944i \(-0.616343\pi\)
−0.357420 + 0.933944i \(0.616343\pi\)
\(972\) 0 0
\(973\) −10.3028 −0.330292
\(974\) 0 0
\(975\) 0.211103 0.00676069
\(976\) 0 0
\(977\) 46.5139 1.48811 0.744055 0.668118i \(-0.232900\pi\)
0.744055 + 0.668118i \(0.232900\pi\)
\(978\) 0 0
\(979\) −11.9083 −0.380592
\(980\) 0 0
\(981\) −10.9083 −0.348276
\(982\) 0 0
\(983\) 1.23886 0.0395135 0.0197567 0.999805i \(-0.493711\pi\)
0.0197567 + 0.999805i \(0.493711\pi\)
\(984\) 0 0
\(985\) 53.6611 1.70978
\(986\) 0 0
\(987\) 10.6056 0.337578
\(988\) 0 0
\(989\) 7.90833 0.251470
\(990\) 0 0
\(991\) 8.69722 0.276276 0.138138 0.990413i \(-0.455888\pi\)
0.138138 + 0.990413i \(0.455888\pi\)
\(992\) 0 0
\(993\) −19.2111 −0.609646
\(994\) 0 0
\(995\) 45.3583 1.43795
\(996\) 0 0
\(997\) 12.5778 0.398343 0.199171 0.979965i \(-0.436175\pi\)
0.199171 + 0.979965i \(0.436175\pi\)
\(998\) 0 0
\(999\) 8.21110 0.259788
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 1932.2.a.e.1.2 2
3.2 odd 2 5796.2.a.k.1.1 2
4.3 odd 2 7728.2.a.bk.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1932.2.a.e.1.2 2 1.1 even 1 trivial
5796.2.a.k.1.1 2 3.2 odd 2
7728.2.a.bk.1.2 2 4.3 odd 2