Properties

Label 6724.2.a.c.1.1
Level $6724$
Weight $2$
Character 6724.1
Self dual yes
Analytic conductor $53.691$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6724 = 2^{2} \cdot 41^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6724.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.6914103191\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.25808.1
Defining polynomial: \(x^{4} - 10 x^{2} - 6 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 164)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.55466\) of defining polynomial
Character \(\chi\) \(=\) 6724.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.92968 q^{3} -3.33307 q^{5} -2.77840 q^{7} +5.58303 q^{9} +O(q^{10})\) \(q-2.92968 q^{3} -3.33307 q^{5} -2.77840 q^{7} +5.58303 q^{9} -2.02837 q^{11} +3.10932 q^{13} +9.76482 q^{15} +7.91609 q^{17} -2.17964 q^{19} +8.13983 q^{21} -4.75004 q^{23} +6.10932 q^{25} -7.56744 q^{27} -3.85936 q^{29} -10.6661 q^{31} +5.94246 q^{33} +9.26060 q^{35} +6.58303 q^{37} -9.10932 q^{39} +4.80677 q^{43} -18.6086 q^{45} -4.03900 q^{47} +0.719526 q^{49} -23.1916 q^{51} +1.94327 q^{53} +6.76068 q^{55} +6.38566 q^{57} +4.75004 q^{59} +2.89068 q^{61} -15.5119 q^{63} -10.3636 q^{65} -5.97163 q^{67} +13.9161 q^{69} -14.4026 q^{71} -9.24916 q^{73} -17.8984 q^{75} +5.63562 q^{77} +0.320282 q^{79} +5.42111 q^{81} -3.69745 q^{83} -26.3849 q^{85} +11.3067 q^{87} -8.96868 q^{89} -8.63895 q^{91} +31.2484 q^{93} +7.26489 q^{95} -17.8322 q^{97} -11.3244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + 4q^{5} + 12q^{9} + O(q^{10}) \) \( 4q - 2q^{3} + 4q^{5} + 12q^{9} - 4q^{11} + 10q^{15} + 4q^{17} - 6q^{19} - 12q^{23} + 12q^{25} + 10q^{27} + 4q^{29} - 8q^{31} - 20q^{33} + 26q^{35} + 16q^{37} - 24q^{39} + 4q^{43} + 4q^{45} + 6q^{47} + 16q^{49} - 4q^{51} + 16q^{53} + 2q^{55} + 4q^{57} + 12q^{59} + 24q^{61} + 10q^{63} - 4q^{65} - 28q^{67} + 28q^{69} + 2q^{71} + 8q^{73} - 30q^{75} + 8q^{77} + 18q^{79} + 28q^{81} - 12q^{83} - 32q^{85} + 44q^{87} - 4q^{89} + 36q^{91} + 28q^{93} - 14q^{95} - 16q^{97} - 58q^{99} + O(q^{100}) \)

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
\(3\) −2.92968 −1.69145 −0.845726 0.533618i \(-0.820832\pi\)
−0.845726 + 0.533618i \(0.820832\pi\)
\(4\) 0 0
\(5\) −3.33307 −1.49059 −0.745296 0.666734i \(-0.767692\pi\)
−0.745296 + 0.666734i \(0.767692\pi\)
\(6\) 0 0
\(7\) −2.77840 −1.05014 −0.525069 0.851060i \(-0.675960\pi\)
−0.525069 + 0.851060i \(0.675960\pi\)
\(8\) 0 0
\(9\) 5.58303 1.86101
\(10\) 0 0
\(11\) −2.02837 −0.611575 −0.305788 0.952100i \(-0.598920\pi\)
−0.305788 + 0.952100i \(0.598920\pi\)
\(12\) 0 0
\(13\) 3.10932 0.862371 0.431186 0.902263i \(-0.358095\pi\)
0.431186 + 0.902263i \(0.358095\pi\)
\(14\) 0 0
\(15\) 9.76482 2.52126
\(16\) 0 0
\(17\) 7.91609 1.91993 0.959967 0.280112i \(-0.0903717\pi\)
0.959967 + 0.280112i \(0.0903717\pi\)
\(18\) 0 0
\(19\) −2.17964 −0.500044 −0.250022 0.968240i \(-0.580438\pi\)
−0.250022 + 0.968240i \(0.580438\pi\)
\(20\) 0 0
\(21\) 8.13983 1.77626
\(22\) 0 0
\(23\) −4.75004 −0.990451 −0.495226 0.868764i \(-0.664915\pi\)
−0.495226 + 0.868764i \(0.664915\pi\)
\(24\) 0 0
\(25\) 6.10932 1.22186
\(26\) 0 0
\(27\) −7.56744 −1.45636
\(28\) 0 0
\(29\) −3.85936 −0.716665 −0.358333 0.933594i \(-0.616655\pi\)
−0.358333 + 0.933594i \(0.616655\pi\)
\(30\) 0 0
\(31\) −10.6661 −1.91569 −0.957847 0.287280i \(-0.907249\pi\)
−0.957847 + 0.287280i \(0.907249\pi\)
\(32\) 0 0
\(33\) 5.94246 1.03445
\(34\) 0 0
\(35\) 9.26060 1.56533
\(36\) 0 0
\(37\) 6.58303 1.08224 0.541122 0.840944i \(-0.318000\pi\)
0.541122 + 0.840944i \(0.318000\pi\)
\(38\) 0 0
\(39\) −9.10932 −1.45866
\(40\) 0 0
\(41\) 0 0
\(42\) 0 0
\(43\) 4.80677 0.733025 0.366513 0.930413i \(-0.380552\pi\)
0.366513 + 0.930413i \(0.380552\pi\)
\(44\) 0 0
\(45\) −18.6086 −2.77401
\(46\) 0 0
\(47\) −4.03900 −0.589149 −0.294575 0.955628i \(-0.595178\pi\)
−0.294575 + 0.955628i \(0.595178\pi\)
\(48\) 0 0
\(49\) 0.719526 0.102789
\(50\) 0 0
\(51\) −23.1916 −3.24748
\(52\) 0 0
\(53\) 1.94327 0.266928 0.133464 0.991054i \(-0.457390\pi\)
0.133464 + 0.991054i \(0.457390\pi\)
\(54\) 0 0
\(55\) 6.76068 0.911609
\(56\) 0 0
\(57\) 6.38566 0.845801
\(58\) 0 0
\(59\) 4.75004 0.618402 0.309201 0.950997i \(-0.399938\pi\)
0.309201 + 0.950997i \(0.399938\pi\)
\(60\) 0 0
\(61\) 2.89068 0.370113 0.185057 0.982728i \(-0.440753\pi\)
0.185057 + 0.982728i \(0.440753\pi\)
\(62\) 0 0
\(63\) −15.5119 −1.95432
\(64\) 0 0
\(65\) −10.3636 −1.28544
\(66\) 0 0
\(67\) −5.97163 −0.729551 −0.364776 0.931095i \(-0.618854\pi\)
−0.364776 + 0.931095i \(0.618854\pi\)
\(68\) 0 0
\(69\) 13.9161 1.67530
\(70\) 0 0
\(71\) −14.4026 −1.70927 −0.854636 0.519228i \(-0.826220\pi\)
−0.854636 + 0.519228i \(0.826220\pi\)
\(72\) 0 0
\(73\) −9.24916 −1.08253 −0.541266 0.840851i \(-0.682055\pi\)
−0.541266 + 0.840851i \(0.682055\pi\)
\(74\) 0 0
\(75\) −17.8984 −2.06673
\(76\) 0 0
\(77\) 5.63562 0.642238
\(78\) 0 0
\(79\) 0.320282 0.0360345 0.0180173 0.999838i \(-0.494265\pi\)
0.0180173 + 0.999838i \(0.494265\pi\)
\(80\) 0 0
\(81\) 5.42111 0.602346
\(82\) 0 0
\(83\) −3.69745 −0.405847 −0.202924 0.979195i \(-0.565044\pi\)
−0.202924 + 0.979195i \(0.565044\pi\)
\(84\) 0 0
\(85\) −26.3849 −2.86184
\(86\) 0 0
\(87\) 11.3067 1.21220
\(88\) 0 0
\(89\) −8.96868 −0.950679 −0.475339 0.879803i \(-0.657675\pi\)
−0.475339 + 0.879803i \(0.657675\pi\)
\(90\) 0 0
\(91\) −8.63895 −0.905608
\(92\) 0 0
\(93\) 31.2484 3.24030
\(94\) 0 0
\(95\) 7.26489 0.745362
\(96\) 0 0
\(97\) −17.8322 −1.81058 −0.905292 0.424790i \(-0.860348\pi\)
−0.905292 + 0.424790i \(0.860348\pi\)
\(98\) 0 0
\(99\) −11.3244 −1.13815
\(100\) 0 0
\(101\) 13.0254 1.29608 0.648039 0.761607i \(-0.275590\pi\)
0.648039 + 0.761607i \(0.275590\pi\)
\(102\) 0 0
\(103\) −8.05673 −0.793853 −0.396927 0.917850i \(-0.629923\pi\)
−0.396927 + 0.917850i \(0.629923\pi\)
\(104\) 0 0
\(105\) −27.1306 −2.64768
\(106\) 0 0
\(107\) −4.41602 −0.426912 −0.213456 0.976953i \(-0.568472\pi\)
−0.213456 + 0.976953i \(0.568472\pi\)
\(108\) 0 0
\(109\) −18.5255 −1.77442 −0.887210 0.461366i \(-0.847360\pi\)
−0.887210 + 0.461366i \(0.847360\pi\)
\(110\) 0 0
\(111\) −19.2862 −1.83056
\(112\) 0 0
\(113\) −8.08310 −0.760394 −0.380197 0.924905i \(-0.624144\pi\)
−0.380197 + 0.924905i \(0.624144\pi\)
\(114\) 0 0
\(115\) 15.8322 1.47636
\(116\) 0 0
\(117\) 17.3594 1.60488
\(118\) 0 0
\(119\) −21.9941 −2.01620
\(120\) 0 0
\(121\) −6.88573 −0.625976
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −3.69745 −0.330710
\(126\) 0 0
\(127\) 4.44748 0.394650 0.197325 0.980338i \(-0.436775\pi\)
0.197325 + 0.980338i \(0.436775\pi\)
\(128\) 0 0
\(129\) −14.0823 −1.23988
\(130\) 0 0
\(131\) 9.85936 0.861416 0.430708 0.902491i \(-0.358264\pi\)
0.430708 + 0.902491i \(0.358264\pi\)
\(132\) 0 0
\(133\) 6.05593 0.525115
\(134\) 0 0
\(135\) 25.2228 2.17083
\(136\) 0 0
\(137\) 2.30255 0.196720 0.0983602 0.995151i \(-0.468640\pi\)
0.0983602 + 0.995151i \(0.468640\pi\)
\(138\) 0 0
\(139\) 5.16605 0.438179 0.219090 0.975705i \(-0.429691\pi\)
0.219090 + 0.975705i \(0.429691\pi\)
\(140\) 0 0
\(141\) 11.8330 0.996517
\(142\) 0 0
\(143\) −6.30684 −0.527405
\(144\) 0 0
\(145\) 12.8635 1.06826
\(146\) 0 0
\(147\) −2.10798 −0.173863
\(148\) 0 0
\(149\) −3.85936 −0.316171 −0.158086 0.987425i \(-0.550532\pi\)
−0.158086 + 0.987425i \(0.550532\pi\)
\(150\) 0 0
\(151\) 4.71103 0.383379 0.191689 0.981456i \(-0.438603\pi\)
0.191689 + 0.981456i \(0.438603\pi\)
\(152\) 0 0
\(153\) 44.1958 3.57302
\(154\) 0 0
\(155\) 35.5509 2.85552
\(156\) 0 0
\(157\) −8.30684 −0.662958 −0.331479 0.943463i \(-0.607548\pi\)
−0.331479 + 0.943463i \(0.607548\pi\)
\(158\) 0 0
\(159\) −5.69316 −0.451497
\(160\) 0 0
\(161\) 13.1975 1.04011
\(162\) 0 0
\(163\) −13.6348 −1.06796 −0.533981 0.845497i \(-0.679304\pi\)
−0.533981 + 0.845497i \(0.679304\pi\)
\(164\) 0 0
\(165\) −19.8066 −1.54194
\(166\) 0 0
\(167\) −9.97163 −0.771628 −0.385814 0.922577i \(-0.626079\pi\)
−0.385814 + 0.922577i \(0.626079\pi\)
\(168\) 0 0
\(169\) −3.33211 −0.256316
\(170\) 0 0
\(171\) −12.1690 −0.930587
\(172\) 0 0
\(173\) 9.11361 0.692895 0.346448 0.938069i \(-0.387388\pi\)
0.346448 + 0.938069i \(0.387388\pi\)
\(174\) 0 0
\(175\) −16.9742 −1.28313
\(176\) 0 0
\(177\) −13.9161 −1.04600
\(178\) 0 0
\(179\) 12.9403 0.967205 0.483602 0.875288i \(-0.339328\pi\)
0.483602 + 0.875288i \(0.339328\pi\)
\(180\) 0 0
\(181\) −8.16191 −0.606670 −0.303335 0.952884i \(-0.598100\pi\)
−0.303335 + 0.952884i \(0.598100\pi\)
\(182\) 0 0
\(183\) −8.46876 −0.626029
\(184\) 0 0
\(185\) −21.9417 −1.61318
\(186\) 0 0
\(187\) −16.0567 −1.17418
\(188\) 0 0
\(189\) 21.0254 1.52937
\(190\) 0 0
\(191\) −4.52829 −0.327656 −0.163828 0.986489i \(-0.552384\pi\)
−0.163828 + 0.986489i \(0.552384\pi\)
\(192\) 0 0
\(193\) 14.1662 1.01971 0.509853 0.860262i \(-0.329700\pi\)
0.509853 + 0.860262i \(0.329700\pi\)
\(194\) 0 0
\(195\) 30.3620 2.17427
\(196\) 0 0
\(197\) −12.5042 −0.890888 −0.445444 0.895310i \(-0.646954\pi\)
−0.445444 + 0.895310i \(0.646954\pi\)
\(198\) 0 0
\(199\) 14.5853 1.03393 0.516963 0.856008i \(-0.327062\pi\)
0.516963 + 0.856008i \(0.327062\pi\)
\(200\) 0 0
\(201\) 17.4950 1.23400
\(202\) 0 0
\(203\) 10.7229 0.752597
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −26.5196 −1.84324
\(208\) 0 0
\(209\) 4.42111 0.305815
\(210\) 0 0
\(211\) 26.3860 1.81649 0.908245 0.418439i \(-0.137423\pi\)
0.908245 + 0.418439i \(0.137423\pi\)
\(212\) 0 0
\(213\) 42.1950 2.89115
\(214\) 0 0
\(215\) −16.0213 −1.09264
\(216\) 0 0
\(217\) 29.6348 2.01174
\(218\) 0 0
\(219\) 27.0971 1.83105
\(220\) 0 0
\(221\) 24.6137 1.65570
\(222\) 0 0
\(223\) −1.89482 −0.126886 −0.0634432 0.997985i \(-0.520208\pi\)
−0.0634432 + 0.997985i \(0.520208\pi\)
\(224\) 0 0
\(225\) 34.1085 2.27390
\(226\) 0 0
\(227\) 24.9805 1.65801 0.829007 0.559238i \(-0.188906\pi\)
0.829007 + 0.559238i \(0.188906\pi\)
\(228\) 0 0
\(229\) 2.52549 0.166889 0.0834446 0.996512i \(-0.473408\pi\)
0.0834446 + 0.996512i \(0.473408\pi\)
\(230\) 0 0
\(231\) −16.5106 −1.08632
\(232\) 0 0
\(233\) −9.24835 −0.605880 −0.302940 0.953010i \(-0.597968\pi\)
−0.302940 + 0.953010i \(0.597968\pi\)
\(234\) 0 0
\(235\) 13.4623 0.878181
\(236\) 0 0
\(237\) −0.938323 −0.0609507
\(238\) 0 0
\(239\) −2.34746 −0.151844 −0.0759222 0.997114i \(-0.524190\pi\)
−0.0759222 + 0.997114i \(0.524190\pi\)
\(240\) 0 0
\(241\) 2.89068 0.186205 0.0931024 0.995657i \(-0.470322\pi\)
0.0931024 + 0.995657i \(0.470322\pi\)
\(242\) 0 0
\(243\) 6.82021 0.437516
\(244\) 0 0
\(245\) −2.39823 −0.153217
\(246\) 0 0
\(247\) −6.77721 −0.431224
\(248\) 0 0
\(249\) 10.8323 0.686471
\(250\) 0 0
\(251\) 24.9730 1.57628 0.788140 0.615496i \(-0.211044\pi\)
0.788140 + 0.615496i \(0.211044\pi\)
\(252\) 0 0
\(253\) 9.63481 0.605736
\(254\) 0 0
\(255\) 77.2992 4.84066
\(256\) 0 0
\(257\) −10.7500 −0.670569 −0.335284 0.942117i \(-0.608832\pi\)
−0.335284 + 0.942117i \(0.608832\pi\)
\(258\) 0 0
\(259\) −18.2903 −1.13650
\(260\) 0 0
\(261\) −21.5469 −1.33372
\(262\) 0 0
\(263\) 4.90250 0.302301 0.151151 0.988511i \(-0.451702\pi\)
0.151151 + 0.988511i \(0.451702\pi\)
\(264\) 0 0
\(265\) −6.47704 −0.397881
\(266\) 0 0
\(267\) 26.2754 1.60803
\(268\) 0 0
\(269\) 3.32797 0.202910 0.101455 0.994840i \(-0.467650\pi\)
0.101455 + 0.994840i \(0.467650\pi\)
\(270\) 0 0
\(271\) 4.58222 0.278350 0.139175 0.990268i \(-0.455555\pi\)
0.139175 + 0.990268i \(0.455555\pi\)
\(272\) 0 0
\(273\) 25.3094 1.53179
\(274\) 0 0
\(275\) −12.3919 −0.747262
\(276\) 0 0
\(277\) −24.2178 −1.45511 −0.727555 0.686050i \(-0.759343\pi\)
−0.727555 + 0.686050i \(0.759343\pi\)
\(278\) 0 0
\(279\) −59.5493 −3.56512
\(280\) 0 0
\(281\) −6.61369 −0.394540 −0.197270 0.980349i \(-0.563208\pi\)
−0.197270 + 0.980349i \(0.563208\pi\)
\(282\) 0 0
\(283\) 5.16605 0.307090 0.153545 0.988142i \(-0.450931\pi\)
0.153545 + 0.988142i \(0.450931\pi\)
\(284\) 0 0
\(285\) −21.2838 −1.26074
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 45.6645 2.68615
\(290\) 0 0
\(291\) 52.2426 3.06252
\(292\) 0 0
\(293\) 10.2458 0.598567 0.299284 0.954164i \(-0.403252\pi\)
0.299284 + 0.954164i \(0.403252\pi\)
\(294\) 0 0
\(295\) −15.8322 −0.921785
\(296\) 0 0
\(297\) 15.3495 0.890671
\(298\) 0 0
\(299\) −14.7694 −0.854137
\(300\) 0 0
\(301\) −13.3551 −0.769778
\(302\) 0 0
\(303\) −38.1603 −2.19225
\(304\) 0 0
\(305\) −9.63481 −0.551688
\(306\) 0 0
\(307\) −18.2754 −1.04303 −0.521515 0.853242i \(-0.674633\pi\)
−0.521515 + 0.853242i \(0.674633\pi\)
\(308\) 0 0
\(309\) 23.6036 1.34276
\(310\) 0 0
\(311\) −4.50303 −0.255343 −0.127672 0.991816i \(-0.540750\pi\)
−0.127672 + 0.991816i \(0.540750\pi\)
\(312\) 0 0
\(313\) −6.56095 −0.370847 −0.185423 0.982659i \(-0.559366\pi\)
−0.185423 + 0.982659i \(0.559366\pi\)
\(314\) 0 0
\(315\) 51.7022 2.91309
\(316\) 0 0
\(317\) −12.8363 −0.720960 −0.360480 0.932767i \(-0.617387\pi\)
−0.360480 + 0.932767i \(0.617387\pi\)
\(318\) 0 0
\(319\) 7.82820 0.438295
\(320\) 0 0
\(321\) 12.9375 0.722102
\(322\) 0 0
\(323\) −17.2543 −0.960052
\(324\) 0 0
\(325\) 18.9959 1.05370
\(326\) 0 0
\(327\) 54.2738 3.00135
\(328\) 0 0
\(329\) 11.2220 0.618688
\(330\) 0 0
\(331\) 8.99705 0.494523 0.247261 0.968949i \(-0.420469\pi\)
0.247261 + 0.968949i \(0.420469\pi\)
\(332\) 0 0
\(333\) 36.7532 2.01406
\(334\) 0 0
\(335\) 19.9038 1.08746
\(336\) 0 0
\(337\) −35.0502 −1.90930 −0.954652 0.297723i \(-0.903773\pi\)
−0.954652 + 0.297723i \(0.903773\pi\)
\(338\) 0 0
\(339\) 23.6809 1.28617
\(340\) 0 0
\(341\) 21.6348 1.17159
\(342\) 0 0
\(343\) 17.4497 0.942195
\(344\) 0 0
\(345\) −46.3832 −2.49719
\(346\) 0 0
\(347\) −18.8982 −1.01451 −0.507255 0.861796i \(-0.669340\pi\)
−0.507255 + 0.861796i \(0.669340\pi\)
\(348\) 0 0
\(349\) −23.1653 −1.24001 −0.620004 0.784599i \(-0.712869\pi\)
−0.620004 + 0.784599i \(0.712869\pi\)
\(350\) 0 0
\(351\) −23.5296 −1.25592
\(352\) 0 0
\(353\) −1.55171 −0.0825893 −0.0412946 0.999147i \(-0.513148\pi\)
−0.0412946 + 0.999147i \(0.513148\pi\)
\(354\) 0 0
\(355\) 48.0047 2.54783
\(356\) 0 0
\(357\) 64.4357 3.41030
\(358\) 0 0
\(359\) −4.28128 −0.225957 −0.112979 0.993597i \(-0.536039\pi\)
−0.112979 + 0.993597i \(0.536039\pi\)
\(360\) 0 0
\(361\) −14.2492 −0.749956
\(362\) 0 0
\(363\) 20.1730 1.05881
\(364\) 0 0
\(365\) 30.8280 1.61361
\(366\) 0 0
\(367\) −15.3067 −0.799003 −0.399501 0.916733i \(-0.630817\pi\)
−0.399501 + 0.916733i \(0.630817\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5.39918 −0.280312
\(372\) 0 0
\(373\) −5.39489 −0.279337 −0.139668 0.990198i \(-0.544604\pi\)
−0.139668 + 0.990198i \(0.544604\pi\)
\(374\) 0 0
\(375\) 10.8323 0.559379
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 26.8009 1.37667 0.688334 0.725394i \(-0.258342\pi\)
0.688334 + 0.725394i \(0.258342\pi\)
\(380\) 0 0
\(381\) −13.0297 −0.667532
\(382\) 0 0
\(383\) −21.6675 −1.10716 −0.553578 0.832797i \(-0.686738\pi\)
−0.553578 + 0.832797i \(0.686738\pi\)
\(384\) 0 0
\(385\) −18.7839 −0.957315
\(386\) 0 0
\(387\) 26.8363 1.36417
\(388\) 0 0
\(389\) 25.7187 1.30399 0.651995 0.758223i \(-0.273932\pi\)
0.651995 + 0.758223i \(0.273932\pi\)
\(390\) 0 0
\(391\) −37.6017 −1.90160
\(392\) 0 0
\(393\) −28.8848 −1.45704
\(394\) 0 0
\(395\) −1.06752 −0.0537128
\(396\) 0 0
\(397\) −28.1603 −1.41333 −0.706663 0.707551i \(-0.749800\pi\)
−0.706663 + 0.707551i \(0.749800\pi\)
\(398\) 0 0
\(399\) −17.7419 −0.888208
\(400\) 0 0
\(401\) −10.7178 −0.535220 −0.267610 0.963527i \(-0.586234\pi\)
−0.267610 + 0.963527i \(0.586234\pi\)
\(402\) 0 0
\(403\) −33.1644 −1.65204
\(404\) 0 0
\(405\) −18.0689 −0.897852
\(406\) 0 0
\(407\) −13.3528 −0.661873
\(408\) 0 0
\(409\) 22.7492 1.12488 0.562439 0.826839i \(-0.309863\pi\)
0.562439 + 0.826839i \(0.309863\pi\)
\(410\) 0 0
\(411\) −6.74575 −0.332743
\(412\) 0 0
\(413\) −13.1975 −0.649408
\(414\) 0 0
\(415\) 12.3238 0.604953
\(416\) 0 0
\(417\) −15.1349 −0.741159
\(418\) 0 0
\(419\) −7.02542 −0.343214 −0.171607 0.985165i \(-0.554896\pi\)
−0.171607 + 0.985165i \(0.554896\pi\)
\(420\) 0 0
\(421\) −16.5992 −0.808996 −0.404498 0.914539i \(-0.632554\pi\)
−0.404498 + 0.914539i \(0.632554\pi\)
\(422\) 0 0
\(423\) −22.5499 −1.09641
\(424\) 0 0
\(425\) 48.3620 2.34590
\(426\) 0 0
\(427\) −8.03147 −0.388670
\(428\) 0 0
\(429\) 18.4770 0.892080
\(430\) 0 0
\(431\) −10.7229 −0.516502 −0.258251 0.966078i \(-0.583146\pi\)
−0.258251 + 0.966078i \(0.583146\pi\)
\(432\) 0 0
\(433\) 31.1644 1.49767 0.748834 0.662758i \(-0.230614\pi\)
0.748834 + 0.662758i \(0.230614\pi\)
\(434\) 0 0
\(435\) −37.6859 −1.80690
\(436\) 0 0
\(437\) 10.3534 0.495270
\(438\) 0 0
\(439\) 34.7981 1.66082 0.830411 0.557152i \(-0.188106\pi\)
0.830411 + 0.557152i \(0.188106\pi\)
\(440\) 0 0
\(441\) 4.01714 0.191292
\(442\) 0 0
\(443\) −25.8806 −1.22963 −0.614813 0.788673i \(-0.710769\pi\)
−0.614813 + 0.788673i \(0.710769\pi\)
\(444\) 0 0
\(445\) 29.8932 1.41707
\(446\) 0 0
\(447\) 11.3067 0.534788
\(448\) 0 0
\(449\) 27.2231 1.28474 0.642368 0.766396i \(-0.277952\pi\)
0.642368 + 0.766396i \(0.277952\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −13.8018 −0.648466
\(454\) 0 0
\(455\) 28.7942 1.34989
\(456\) 0 0
\(457\) 6.19767 0.289915 0.144957 0.989438i \(-0.453695\pi\)
0.144957 + 0.989438i \(0.453695\pi\)
\(458\) 0 0
\(459\) −59.9046 −2.79611
\(460\) 0 0
\(461\) −10.9170 −0.508458 −0.254229 0.967144i \(-0.581822\pi\)
−0.254229 + 0.967144i \(0.581822\pi\)
\(462\) 0 0
\(463\) −4.07047 −0.189171 −0.0945854 0.995517i \(-0.530153\pi\)
−0.0945854 + 0.995517i \(0.530153\pi\)
\(464\) 0 0
\(465\) −104.153 −4.82997
\(466\) 0 0
\(467\) −8.63657 −0.399653 −0.199826 0.979831i \(-0.564038\pi\)
−0.199826 + 0.979831i \(0.564038\pi\)
\(468\) 0 0
\(469\) 16.5916 0.766129
\(470\) 0 0
\(471\) 24.3364 1.12136
\(472\) 0 0
\(473\) −9.74989 −0.448300
\(474\) 0 0
\(475\) −13.3161 −0.610986
\(476\) 0 0
\(477\) 10.8493 0.496756
\(478\) 0 0
\(479\) 23.5537 1.07620 0.538098 0.842882i \(-0.319143\pi\)
0.538098 + 0.842882i \(0.319143\pi\)
\(480\) 0 0
\(481\) 20.4688 0.933295
\(482\) 0 0
\(483\) −38.6645 −1.75930
\(484\) 0 0
\(485\) 59.4358 2.69884
\(486\) 0 0
\(487\) 19.9102 0.902217 0.451108 0.892469i \(-0.351029\pi\)
0.451108 + 0.892469i \(0.351029\pi\)
\(488\) 0 0
\(489\) 39.9456 1.80640
\(490\) 0 0
\(491\) 29.7585 1.34298 0.671490 0.741013i \(-0.265654\pi\)
0.671490 + 0.741013i \(0.265654\pi\)
\(492\) 0 0
\(493\) −30.5511 −1.37595
\(494\) 0 0
\(495\) 37.7450 1.69651
\(496\) 0 0
\(497\) 40.0162 1.79497
\(498\) 0 0
\(499\) −18.9297 −0.847409 −0.423704 0.905801i \(-0.639270\pi\)
−0.423704 + 0.905801i \(0.639270\pi\)
\(500\) 0 0
\(501\) 29.2137 1.30517
\(502\) 0 0
\(503\) 1.93382 0.0862248 0.0431124 0.999070i \(-0.486273\pi\)
0.0431124 + 0.999070i \(0.486273\pi\)
\(504\) 0 0
\(505\) −43.4146 −1.93192
\(506\) 0 0
\(507\) 9.76202 0.433546
\(508\) 0 0
\(509\) −34.6602 −1.53629 −0.768144 0.640277i \(-0.778819\pi\)
−0.768144 + 0.640277i \(0.778819\pi\)
\(510\) 0 0
\(511\) 25.6979 1.13681
\(512\) 0 0
\(513\) 16.4943 0.728242
\(514\) 0 0
\(515\) 26.8536 1.18331
\(516\) 0 0
\(517\) 8.19258 0.360309
\(518\) 0 0
\(519\) −26.7000 −1.17200
\(520\) 0 0
\(521\) 6.33402 0.277498 0.138749 0.990328i \(-0.455692\pi\)
0.138749 + 0.990328i \(0.455692\pi\)
\(522\) 0 0
\(523\) −29.9460 −1.30944 −0.654722 0.755869i \(-0.727215\pi\)
−0.654722 + 0.755869i \(0.727215\pi\)
\(524\) 0 0
\(525\) 49.7289 2.17035
\(526\) 0 0
\(527\) −84.4341 −3.67801
\(528\) 0 0
\(529\) −0.437142 −0.0190062
\(530\) 0 0
\(531\) 26.5196 1.15085
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 14.7189 0.636352
\(536\) 0 0
\(537\) −37.9110 −1.63598
\(538\) 0 0
\(539\) −1.45946 −0.0628635
\(540\) 0 0
\(541\) 0.834750 0.0358887 0.0179443 0.999839i \(-0.494288\pi\)
0.0179443 + 0.999839i \(0.494288\pi\)
\(542\) 0 0
\(543\) 23.9118 1.02615
\(544\) 0 0
\(545\) 61.7467 2.64494
\(546\) 0 0
\(547\) 20.6527 0.883045 0.441523 0.897250i \(-0.354438\pi\)
0.441523 + 0.897250i \(0.354438\pi\)
\(548\) 0 0
\(549\) 16.1387 0.688784
\(550\) 0 0
\(551\) 8.41203 0.358364
\(552\) 0 0
\(553\) −0.889872 −0.0378412
\(554\) 0 0
\(555\) 64.2820 2.72862
\(556\) 0 0
\(557\) 8.72447 0.369668 0.184834 0.982770i \(-0.440825\pi\)
0.184834 + 0.982770i \(0.440825\pi\)
\(558\) 0 0
\(559\) 14.9458 0.632140
\(560\) 0 0
\(561\) 47.0411 1.98608
\(562\) 0 0
\(563\) −9.80144 −0.413081 −0.206541 0.978438i \(-0.566221\pi\)
−0.206541 + 0.978438i \(0.566221\pi\)
\(564\) 0 0
\(565\) 26.9415 1.13344
\(566\) 0 0
\(567\) −15.0620 −0.632546
\(568\) 0 0
\(569\) 25.1570 1.05464 0.527318 0.849668i \(-0.323198\pi\)
0.527318 + 0.849668i \(0.323198\pi\)
\(570\) 0 0
\(571\) −27.1928 −1.13798 −0.568992 0.822343i \(-0.692666\pi\)
−0.568992 + 0.822343i \(0.692666\pi\)
\(572\) 0 0
\(573\) 13.2664 0.554214
\(574\) 0 0
\(575\) −29.0195 −1.21020
\(576\) 0 0
\(577\) 0.365186 0.0152029 0.00760144 0.999971i \(-0.497580\pi\)
0.00760144 + 0.999971i \(0.497580\pi\)
\(578\) 0 0
\(579\) −41.5025 −1.72478
\(580\) 0 0
\(581\) 10.2730 0.426196
\(582\) 0 0
\(583\) −3.94166 −0.163247
\(584\) 0 0
\(585\) −57.8601 −2.39222
\(586\) 0 0
\(587\) 37.7242 1.55704 0.778522 0.627617i \(-0.215970\pi\)
0.778522 + 0.627617i \(0.215970\pi\)
\(588\) 0 0
\(589\) 23.2484 0.957932
\(590\) 0 0
\(591\) 36.6334 1.50689
\(592\) 0 0
\(593\) 45.1333 1.85340 0.926701 0.375800i \(-0.122632\pi\)
0.926701 + 0.375800i \(0.122632\pi\)
\(594\) 0 0
\(595\) 73.3078 3.00533
\(596\) 0 0
\(597\) −42.7303 −1.74884
\(598\) 0 0
\(599\) 15.4982 0.633238 0.316619 0.948553i \(-0.397452\pi\)
0.316619 + 0.948553i \(0.397452\pi\)
\(600\) 0 0
\(601\) 5.18924 0.211674 0.105837 0.994384i \(-0.466248\pi\)
0.105837 + 0.994384i \(0.466248\pi\)
\(602\) 0 0
\(603\) −33.3398 −1.35770
\(604\) 0 0
\(605\) 22.9506 0.933074
\(606\) 0 0
\(607\) −16.8280 −0.683029 −0.341515 0.939876i \(-0.610940\pi\)
−0.341515 + 0.939876i \(0.610940\pi\)
\(608\) 0 0
\(609\) −31.4146 −1.27298
\(610\) 0 0
\(611\) −12.5586 −0.508065
\(612\) 0 0
\(613\) 25.1042 1.01395 0.506975 0.861961i \(-0.330764\pi\)
0.506975 + 0.861961i \(0.330764\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 14.3324 0.577001 0.288501 0.957480i \(-0.406843\pi\)
0.288501 + 0.957480i \(0.406843\pi\)
\(618\) 0 0
\(619\) 19.2712 0.774576 0.387288 0.921959i \(-0.373412\pi\)
0.387288 + 0.921959i \(0.373412\pi\)
\(620\) 0 0
\(621\) 35.9456 1.44245
\(622\) 0 0
\(623\) 24.9186 0.998344
\(624\) 0 0
\(625\) −18.2228 −0.728911
\(626\) 0 0
\(627\) −12.9524 −0.517271
\(628\) 0 0
\(629\) 52.1119 2.07784
\(630\) 0 0
\(631\) −30.1856 −1.20167 −0.600834 0.799374i \(-0.705165\pi\)
−0.600834 + 0.799374i \(0.705165\pi\)
\(632\) 0 0
\(633\) −77.3027 −3.07251
\(634\) 0 0
\(635\) −14.8238 −0.588263
\(636\) 0 0
\(637\) 2.23724 0.0886427
\(638\) 0 0
\(639\) −80.4100 −3.18097
\(640\) 0 0
\(641\) 25.9984 1.02687 0.513437 0.858127i \(-0.328372\pi\)
0.513437 + 0.858127i \(0.328372\pi\)
\(642\) 0 0
\(643\) −11.4402 −0.451159 −0.225580 0.974225i \(-0.572428\pi\)
−0.225580 + 0.974225i \(0.572428\pi\)
\(644\) 0 0
\(645\) 46.9372 1.84815
\(646\) 0 0
\(647\) 37.6915 1.48181 0.740904 0.671611i \(-0.234397\pi\)
0.740904 + 0.671611i \(0.234397\pi\)
\(648\) 0 0
\(649\) −9.63481 −0.378200
\(650\) 0 0
\(651\) −86.8205 −3.40277
\(652\) 0 0
\(653\) 43.6814 1.70938 0.854692 0.519136i \(-0.173746\pi\)
0.854692 + 0.519136i \(0.173746\pi\)
\(654\) 0 0
\(655\) −32.8619 −1.28402
\(656\) 0 0
\(657\) −51.6383 −2.01460
\(658\) 0 0
\(659\) −25.0900 −0.977367 −0.488684 0.872461i \(-0.662523\pi\)
−0.488684 + 0.872461i \(0.662523\pi\)
\(660\) 0 0
\(661\) −37.7392 −1.46788 −0.733942 0.679212i \(-0.762322\pi\)
−0.733942 + 0.679212i \(0.762322\pi\)
\(662\) 0 0
\(663\) −72.1102 −2.80053
\(664\) 0 0
\(665\) −20.1848 −0.782733
\(666\) 0 0
\(667\) 18.3321 0.709822
\(668\) 0 0
\(669\) 5.55121 0.214622
\(670\) 0 0
\(671\) −5.86335 −0.226352
\(672\) 0 0
\(673\) −3.63642 −0.140174 −0.0700869 0.997541i \(-0.522328\pi\)
−0.0700869 + 0.997541i \(0.522328\pi\)
\(674\) 0 0
\(675\) −46.2320 −1.77947
\(676\) 0 0
\(677\) 29.1948 1.12205 0.561024 0.827800i \(-0.310408\pi\)
0.561024 + 0.827800i \(0.310408\pi\)
\(678\) 0 0
\(679\) 49.5450 1.90136
\(680\) 0 0
\(681\) −73.1849 −2.80445
\(682\) 0 0
\(683\) 28.6527 1.09636 0.548182 0.836359i \(-0.315320\pi\)
0.548182 + 0.836359i \(0.315320\pi\)
\(684\) 0 0
\(685\) −7.67456 −0.293230
\(686\) 0 0
\(687\) −7.39888 −0.282285
\(688\) 0 0
\(689\) 6.04225 0.230191
\(690\) 0 0
\(691\) −17.6880 −0.672883 −0.336442 0.941704i \(-0.609223\pi\)
−0.336442 + 0.941704i \(0.609223\pi\)
\(692\) 0 0
\(693\) 31.4638 1.19521
\(694\) 0 0
\(695\) −17.2188 −0.653146
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 27.0947 1.02482
\(700\) 0 0
\(701\) 2.79339 0.105505 0.0527525 0.998608i \(-0.483201\pi\)
0.0527525 + 0.998608i \(0.483201\pi\)
\(702\) 0 0
\(703\) −14.3486 −0.541169
\(704\) 0 0
\(705\) −39.4401 −1.48540
\(706\) 0 0
\(707\) −36.1899 −1.36106
\(708\) 0 0
\(709\) −20.4117 −0.766578 −0.383289 0.923628i \(-0.625209\pi\)
−0.383289 + 0.923628i \(0.625209\pi\)
\(710\) 0 0
\(711\) 1.78814 0.0670606
\(712\) 0 0
\(713\) 50.6645 1.89740
\(714\) 0 0
\(715\) 21.0211 0.786145
\(716\) 0 0
\(717\) 6.87730 0.256838
\(718\) 0 0
\(719\) −34.9848 −1.30471 −0.652356 0.757912i \(-0.726219\pi\)
−0.652356 + 0.757912i \(0.726219\pi\)
\(720\) 0 0
\(721\) 22.3849 0.833655
\(722\) 0 0
\(723\) −8.46876 −0.314957
\(724\) 0 0
\(725\) −23.5781 −0.875668
\(726\) 0 0
\(727\) 24.8796 0.922734 0.461367 0.887209i \(-0.347359\pi\)
0.461367 + 0.887209i \(0.347359\pi\)
\(728\) 0 0
\(729\) −36.2444 −1.34238
\(730\) 0 0
\(731\) 38.0508 1.40736
\(732\) 0 0
\(733\) −28.7737 −1.06278 −0.531390 0.847127i \(-0.678330\pi\)
−0.531390 + 0.847127i \(0.678330\pi\)
\(734\) 0 0
\(735\) 7.02604 0.259159
\(736\) 0 0
\(737\) 12.1127 0.446176
\(738\) 0 0
\(739\) −30.1856 −1.11039 −0.555197 0.831719i \(-0.687357\pi\)
−0.555197 + 0.831719i \(0.687357\pi\)
\(740\) 0 0
\(741\) 19.8551 0.729394
\(742\) 0 0
\(743\) −22.7127 −0.833247 −0.416624 0.909079i \(-0.636787\pi\)
−0.416624 + 0.909079i \(0.636787\pi\)
\(744\) 0 0
\(745\) 12.8635 0.471282
\(746\) 0 0
\(747\) −20.6429 −0.755286
\(748\) 0 0
\(749\) 12.2695 0.448317
\(750\) 0 0
\(751\) 0.0200854 0.000732927 0 0.000366463 1.00000i \(-0.499883\pi\)
0.000366463 1.00000i \(0.499883\pi\)
\(752\) 0 0
\(753\) −73.1628 −2.66620
\(754\) 0 0
\(755\) −15.7022 −0.571461
\(756\) 0 0
\(757\) −44.9002 −1.63192 −0.815962 0.578106i \(-0.803792\pi\)
−0.815962 + 0.578106i \(0.803792\pi\)
\(758\) 0 0
\(759\) −28.2269 −1.02457
\(760\) 0 0
\(761\) −19.0203 −0.689486 −0.344743 0.938697i \(-0.612034\pi\)
−0.344743 + 0.938697i \(0.612034\pi\)
\(762\) 0 0
\(763\) 51.4713 1.86339
\(764\) 0 0
\(765\) −147.307 −5.32591
\(766\) 0 0
\(767\) 14.7694 0.533292
\(768\) 0 0
\(769\) 42.1220 1.51896 0.759480 0.650531i \(-0.225454\pi\)
0.759480 + 0.650531i \(0.225454\pi\)
\(770\) 0 0
\(771\) 31.4942 1.13424
\(772\) 0 0
\(773\) −10.7753 −0.387561 −0.193780 0.981045i \(-0.562075\pi\)
−0.193780 + 0.981045i \(0.562075\pi\)
\(774\) 0 0
\(775\) −65.1628 −2.34072
\(776\) 0 0
\(777\) 53.5848 1.92234
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 29.2137 1.04535
\(782\) 0 0
\(783\) 29.2055 1.04372
\(784\) 0 0
\(785\) 27.6873 0.988201
\(786\) 0 0
\(787\) 22.8592 0.814843 0.407421 0.913240i \(-0.366428\pi\)
0.407421 + 0.913240i \(0.366428\pi\)
\(788\) 0 0
\(789\) −14.3628 −0.511328
\(790\) 0 0
\(791\) 22.4581 0.798519
\(792\) 0 0
\(793\) 8.98805 0.319175
\(794\) 0 0
\(795\) 18.9757 0.672997
\(796\) 0 0
\(797\) −5.00444 −0.177266 −0.0886332 0.996064i \(-0.528250\pi\)
−0.0886332 + 0.996064i \(0.528250\pi\)
\(798\) 0 0
\(799\) −31.9731 −1.13113
\(800\) 0 0
\(801\) −50.0724 −1.76922
\(802\) 0 0
\(803\) 18.7607 0.662050
\(804\) 0 0
\(805\) −43.9882 −1.55038
\(806\) 0 0
\(807\) −9.74989 −0.343212
\(808\) 0 0
\(809\) −7.52135 −0.264437 −0.132218 0.991221i \(-0.542210\pi\)
−0.132218 + 0.991221i \(0.542210\pi\)
\(810\) 0 0
\(811\) 29.4103 1.03273 0.516367 0.856367i \(-0.327284\pi\)
0.516367 + 0.856367i \(0.327284\pi\)
\(812\) 0 0
\(813\) −13.4244 −0.470816
\(814\) 0 0
\(815\) 45.4457 1.59189
\(816\) 0 0
\(817\) −10.4770 −0.366545
\(818\) 0 0
\(819\) −48.2315 −1.68535
\(820\) 0 0
\(821\) 38.0205 1.32692 0.663462 0.748210i \(-0.269087\pi\)
0.663462 + 0.748210i \(0.269087\pi\)
\(822\) 0 0
\(823\) −25.7849 −0.898805 −0.449403 0.893329i \(-0.648363\pi\)
−0.449403 + 0.893329i \(0.648363\pi\)
\(824\) 0 0
\(825\) 36.3044 1.26396
\(826\) 0 0
\(827\) −7.00295 −0.243516 −0.121758 0.992560i \(-0.538853\pi\)
−0.121758 + 0.992560i \(0.538853\pi\)
\(828\) 0 0
\(829\) 0.166709 0.00579003 0.00289501 0.999996i \(-0.499078\pi\)
0.00289501 + 0.999996i \(0.499078\pi\)
\(830\) 0 0
\(831\) 70.9505 2.46125
\(832\) 0 0
\(833\) 5.69584 0.197349
\(834\) 0 0
\(835\) 33.2361 1.15018
\(836\) 0 0
\(837\) 80.7154 2.78993
\(838\) 0 0
\(839\) 13.1105 0.452625 0.226313 0.974055i \(-0.427333\pi\)
0.226313 + 0.974055i \(0.427333\pi\)
\(840\) 0 0
\(841\) −14.1053 −0.486391
\(842\) 0 0
\(843\) 19.3760 0.667345
\(844\) 0 0
\(845\) 11.1061 0.382063
\(846\) 0 0
\(847\) 19.1313 0.657361
\(848\) 0 0
\(849\) −15.1349 −0.519428
\(850\) 0 0
\(851\) −31.2696 −1.07191
\(852\) 0 0
\(853\) −0.445573 −0.0152561 −0.00762806 0.999971i \(-0.502428\pi\)
−0.00762806 + 0.999971i \(0.502428\pi\)
\(854\) 0 0
\(855\) 40.5601 1.38713
\(856\) 0 0
\(857\) −20.5519 −0.702038 −0.351019 0.936368i \(-0.614165\pi\)
−0.351019 + 0.936368i \(0.614165\pi\)
\(858\) 0 0
\(859\) −18.8891 −0.644487 −0.322243 0.946657i \(-0.604437\pi\)
−0.322243 + 0.946657i \(0.604437\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −47.9204 −1.63123 −0.815614 0.578596i \(-0.803601\pi\)
−0.815614 + 0.578596i \(0.803601\pi\)
\(864\) 0 0
\(865\) −30.3763 −1.03282
\(866\) 0 0
\(867\) −133.782 −4.54349
\(868\) 0 0
\(869\) −0.649649 −0.0220378
\(870\) 0 0
\(871\) −18.5677 −0.629144
\(872\) 0 0
\(873\) −99.5576 −3.36951
\(874\) 0 0
\(875\) 10.2730 0.347291
\(876\) 0 0
\(877\) −50.2738 −1.69762 −0.848812 0.528694i \(-0.822682\pi\)
−0.848812 + 0.528694i \(0.822682\pi\)
\(878\) 0 0
\(879\) −30.0170 −1.01245
\(880\) 0 0
\(881\) −38.6019 −1.30053 −0.650265 0.759707i \(-0.725342\pi\)
−0.650265 + 0.759707i \(0.725342\pi\)
\(882\) 0 0
\(883\) −3.07860 −0.103603 −0.0518016 0.998657i \(-0.516496\pi\)
−0.0518016 + 0.998657i \(0.516496\pi\)
\(884\) 0 0
\(885\) 46.3832 1.55916
\(886\) 0 0
\(887\) 31.0770 1.04346 0.521731 0.853110i \(-0.325286\pi\)
0.521731 + 0.853110i \(0.325286\pi\)
\(888\) 0 0
\(889\) −12.3569 −0.414437
\(890\) 0 0
\(891\) −10.9960 −0.368380
\(892\) 0 0
\(893\) 8.80358 0.294601
\(894\) 0 0
\(895\) −43.1309 −1.44171
\(896\) 0 0
\(897\) 43.2696 1.44473
\(898\) 0 0
\(899\) 41.1644 1.37291
\(900\) 0 0
\(901\) 15.3831 0.512485
\(902\) 0 0
\(903\) 39.1263 1.30204
\(904\) 0 0
\(905\) 27.2042 0.904298
\(906\) 0 0
\(907\) 42.5534 1.41296 0.706482 0.707731i \(-0.250281\pi\)
0.706482 + 0.707731i \(0.250281\pi\)
\(908\) 0 0
\(909\) 72.7213 2.41201
\(910\) 0 0
\(911\) −14.8694 −0.492645 −0.246323 0.969188i \(-0.579222\pi\)
−0.246323 + 0.969188i \(0.579222\pi\)
\(912\) 0 0
\(913\) 7.49977 0.248206
\(914\) 0 0
\(915\) 28.2269 0.933153
\(916\) 0 0
\(917\) −27.3933 −0.904606
\(918\) 0 0
\(919\) −26.5539 −0.875931 −0.437965 0.898992i \(-0.644301\pi\)
−0.437965 + 0.898992i \(0.644301\pi\)
\(920\) 0 0
\(921\) 53.5410 1.76424
\(922\) 0 0
\(923\) −44.7823 −1.47403
\(924\) 0 0
\(925\) 40.2178 1.32235
\(926\) 0 0
\(927\) −44.9810 −1.47737
\(928\) 0 0
\(929\) −2.70749 −0.0888298 −0.0444149 0.999013i \(-0.514142\pi\)
−0.0444149 + 0.999013i \(0.514142\pi\)
\(930\) 0 0
\(931\) −1.56831 −0.0513993
\(932\) 0 0
\(933\) 13.1924 0.431901
\(934\) 0 0
\(935\) 53.5181 1.75023
\(936\) 0 0
\(937\) 36.0067 1.17629 0.588143 0.808757i \(-0.299859\pi\)
0.588143 + 0.808757i \(0.299859\pi\)
\(938\) 0 0
\(939\) 19.2215 0.627269
\(940\) 0 0
\(941\) 11.6135 0.378591 0.189295 0.981920i \(-0.439380\pi\)
0.189295 + 0.981920i \(0.439380\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −70.0791 −2.27967
\(946\) 0 0
\(947\) −20.9061 −0.679355 −0.339678 0.940542i \(-0.610318\pi\)
−0.339678 + 0.940542i \(0.610318\pi\)
\(948\) 0 0
\(949\) −28.7586 −0.933544
\(950\) 0 0
\(951\) 37.6063 1.21947
\(952\) 0 0
\(953\) −13.7510 −0.445438 −0.222719 0.974883i \(-0.571493\pi\)
−0.222719 + 0.974883i \(0.571493\pi\)
\(954\) 0 0
\(955\) 15.0931 0.488401
\(956\) 0 0
\(957\) −22.9341 −0.741355
\(958\) 0 0
\(959\) −6.39742 −0.206584
\(960\) 0 0
\(961\) 82.7663 2.66988
\(962\) 0 0
\(963\) −24.6547 −0.794488
\(964\) 0 0
\(965\) −47.2169 −1.51997
\(966\) 0 0
\(967\) −7.07935 −0.227656 −0.113828 0.993500i \(-0.536311\pi\)
−0.113828 + 0.993500i \(0.536311\pi\)
\(968\) 0 0
\(969\) 50.5494 1.62388
\(970\) 0 0
\(971\) 18.7382 0.601338 0.300669 0.953729i \(-0.402790\pi\)
0.300669 + 0.953729i \(0.402790\pi\)
\(972\) 0 0
\(973\) −14.3534 −0.460148
\(974\) 0 0
\(975\) −55.6518 −1.78228
\(976\) 0 0
\(977\) −60.6756 −1.94118 −0.970592 0.240729i \(-0.922613\pi\)
−0.970592 + 0.240729i \(0.922613\pi\)
\(978\) 0 0
\(979\) 18.1918 0.581412
\(980\) 0 0
\(981\) −103.428 −3.30221
\(982\) 0 0
\(983\) 47.3720 1.51093 0.755466 0.655188i \(-0.227410\pi\)
0.755466 + 0.655188i \(0.227410\pi\)
\(984\) 0 0
\(985\) 41.6774 1.32795
\(986\) 0 0
\(987\) −32.8768 −1.04648
\(988\) 0 0
\(989\) −22.8323 −0.726026
\(990\) 0 0
\(991\) 19.2110 0.610256 0.305128 0.952311i \(-0.401301\pi\)
0.305128 + 0.952311i \(0.401301\pi\)
\(992\) 0 0
\(993\) −26.3585 −0.836461
\(994\) 0 0
\(995\) −48.6138 −1.54116
\(996\) 0 0
\(997\) 59.1077 1.87196 0.935980 0.352053i \(-0.114516\pi\)
0.935980 + 0.352053i \(0.114516\pi\)
\(998\) 0 0
\(999\) −49.8167 −1.57613
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 6724.2.a.c.1.1 4
41.40 even 2 164.2.a.a.1.4 4
123.122 odd 2 1476.2.a.g.1.4 4
164.163 odd 2 656.2.a.i.1.1 4
205.122 odd 4 4100.2.d.c.1149.2 8
205.163 odd 4 4100.2.d.c.1149.7 8
205.204 even 2 4100.2.a.c.1.1 4
287.286 odd 2 8036.2.a.i.1.1 4
328.163 odd 2 2624.2.a.y.1.4 4
328.245 even 2 2624.2.a.v.1.1 4
492.491 even 2 5904.2.a.bp.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.2.a.a.1.4 4 41.40 even 2
656.2.a.i.1.1 4 164.163 odd 2
1476.2.a.g.1.4 4 123.122 odd 2
2624.2.a.v.1.1 4 328.245 even 2
2624.2.a.y.1.4 4 328.163 odd 2
4100.2.a.c.1.1 4 205.204 even 2
4100.2.d.c.1149.2 8 205.122 odd 4
4100.2.d.c.1149.7 8 205.163 odd 4
5904.2.a.bp.1.4 4 492.491 even 2
6724.2.a.c.1.1 4 1.1 even 1 trivial
8036.2.a.i.1.1 4 287.286 odd 2