Properties

Label 1856.4.a.z.1.1
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 232)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.30242\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q-7.82921 q^{3} +7.04208 q^{5} +14.1473 q^{7} +34.2965 q^{9} +O(q^{10})\) \(q-7.82921 q^{3} +7.04208 q^{5} +14.1473 q^{7} +34.2965 q^{9} +16.9982 q^{11} -62.0351 q^{13} -55.1339 q^{15} +104.842 q^{17} +9.38323 q^{19} -110.762 q^{21} +173.620 q^{23} -75.4091 q^{25} -57.1255 q^{27} -29.0000 q^{29} -28.2215 q^{31} -133.082 q^{33} +99.6261 q^{35} +171.527 q^{37} +485.685 q^{39} +92.0027 q^{41} +519.014 q^{43} +241.518 q^{45} +109.612 q^{47} -142.855 q^{49} -820.830 q^{51} -140.553 q^{53} +119.703 q^{55} -73.4632 q^{57} -353.254 q^{59} +185.355 q^{61} +485.201 q^{63} -436.856 q^{65} -574.840 q^{67} -1359.30 q^{69} +535.054 q^{71} -508.723 q^{73} +590.393 q^{75} +240.478 q^{77} -61.0156 q^{79} -478.757 q^{81} +826.753 q^{83} +738.306 q^{85} +227.047 q^{87} -938.445 q^{89} -877.626 q^{91} +220.952 q^{93} +66.0775 q^{95} -888.766 q^{97} +582.978 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9} - 36 q^{11} - 26 q^{13} + 88 q^{15} + 82 q^{17} - 156 q^{19} - 72 q^{21} + 336 q^{23} + 151 q^{25} - 352 q^{27} - 145 q^{29} + 432 q^{31} + 108 q^{33} - 600 q^{35} + 18 q^{37} + 688 q^{39} + 82 q^{41} - 340 q^{43} + 146 q^{45} + 680 q^{47} - 115 q^{49} - 608 q^{51} + 102 q^{53} + 736 q^{55} - 576 q^{57} - 924 q^{59} + 618 q^{61} + 584 q^{63} - 704 q^{65} - 44 q^{67} + 1056 q^{69} + 1032 q^{71} - 1078 q^{73} + 468 q^{75} + 888 q^{77} + 200 q^{79} - 1843 q^{81} - 452 q^{83} + 1700 q^{85} + 116 q^{87} - 1790 q^{89} + 1128 q^{91} + 1884 q^{93} + 1024 q^{95} - 2518 q^{97} + 1500 q^{99}+O(q^{100}) \) 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
\(3\) −7.82921 −1.50673 −0.753366 0.657602i \(-0.771571\pi\)
−0.753366 + 0.657602i \(0.771571\pi\)
\(4\) 0 0
\(5\) 7.04208 0.629863 0.314931 0.949114i \(-0.398018\pi\)
0.314931 + 0.949114i \(0.398018\pi\)
\(6\) 0 0
\(7\) 14.1473 0.763880 0.381940 0.924187i \(-0.375256\pi\)
0.381940 + 0.924187i \(0.375256\pi\)
\(8\) 0 0
\(9\) 34.2965 1.27024
\(10\) 0 0
\(11\) 16.9982 0.465922 0.232961 0.972486i \(-0.425158\pi\)
0.232961 + 0.972486i \(0.425158\pi\)
\(12\) 0 0
\(13\) −62.0351 −1.32349 −0.661747 0.749727i \(-0.730185\pi\)
−0.661747 + 0.749727i \(0.730185\pi\)
\(14\) 0 0
\(15\) −55.1339 −0.949034
\(16\) 0 0
\(17\) 104.842 1.49576 0.747881 0.663833i \(-0.231071\pi\)
0.747881 + 0.663833i \(0.231071\pi\)
\(18\) 0 0
\(19\) 9.38323 0.113298 0.0566490 0.998394i \(-0.481958\pi\)
0.0566490 + 0.998394i \(0.481958\pi\)
\(20\) 0 0
\(21\) −110.762 −1.15096
\(22\) 0 0
\(23\) 173.620 1.57401 0.787005 0.616947i \(-0.211631\pi\)
0.787005 + 0.616947i \(0.211631\pi\)
\(24\) 0 0
\(25\) −75.4091 −0.603273
\(26\) 0 0
\(27\) −57.1255 −0.407178
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) −28.2215 −0.163507 −0.0817536 0.996653i \(-0.526052\pi\)
−0.0817536 + 0.996653i \(0.526052\pi\)
\(32\) 0 0
\(33\) −133.082 −0.702020
\(34\) 0 0
\(35\) 99.6261 0.481140
\(36\) 0 0
\(37\) 171.527 0.762130 0.381065 0.924548i \(-0.375557\pi\)
0.381065 + 0.924548i \(0.375557\pi\)
\(38\) 0 0
\(39\) 485.685 1.99415
\(40\) 0 0
\(41\) 92.0027 0.350449 0.175225 0.984529i \(-0.443935\pi\)
0.175225 + 0.984529i \(0.443935\pi\)
\(42\) 0 0
\(43\) 519.014 1.84067 0.920335 0.391131i \(-0.127916\pi\)
0.920335 + 0.391131i \(0.127916\pi\)
\(44\) 0 0
\(45\) 241.518 0.800077
\(46\) 0 0
\(47\) 109.612 0.340182 0.170091 0.985428i \(-0.445594\pi\)
0.170091 + 0.985428i \(0.445594\pi\)
\(48\) 0 0
\(49\) −142.855 −0.416487
\(50\) 0 0
\(51\) −820.830 −2.25371
\(52\) 0 0
\(53\) −140.553 −0.364273 −0.182136 0.983273i \(-0.558301\pi\)
−0.182136 + 0.983273i \(0.558301\pi\)
\(54\) 0 0
\(55\) 119.703 0.293467
\(56\) 0 0
\(57\) −73.4632 −0.170710
\(58\) 0 0
\(59\) −353.254 −0.779487 −0.389744 0.920923i \(-0.627436\pi\)
−0.389744 + 0.920923i \(0.627436\pi\)
\(60\) 0 0
\(61\) 185.355 0.389053 0.194526 0.980897i \(-0.437683\pi\)
0.194526 + 0.980897i \(0.437683\pi\)
\(62\) 0 0
\(63\) 485.201 0.970311
\(64\) 0 0
\(65\) −436.856 −0.833620
\(66\) 0 0
\(67\) −574.840 −1.04818 −0.524088 0.851664i \(-0.675594\pi\)
−0.524088 + 0.851664i \(0.675594\pi\)
\(68\) 0 0
\(69\) −1359.30 −2.37161
\(70\) 0 0
\(71\) 535.054 0.894355 0.447177 0.894445i \(-0.352429\pi\)
0.447177 + 0.894445i \(0.352429\pi\)
\(72\) 0 0
\(73\) −508.723 −0.815637 −0.407818 0.913063i \(-0.633710\pi\)
−0.407818 + 0.913063i \(0.633710\pi\)
\(74\) 0 0
\(75\) 590.393 0.908970
\(76\) 0 0
\(77\) 240.478 0.355909
\(78\) 0 0
\(79\) −61.0156 −0.0868961 −0.0434481 0.999056i \(-0.513834\pi\)
−0.0434481 + 0.999056i \(0.513834\pi\)
\(80\) 0 0
\(81\) −478.757 −0.656731
\(82\) 0 0
\(83\) 826.753 1.09335 0.546674 0.837345i \(-0.315894\pi\)
0.546674 + 0.837345i \(0.315894\pi\)
\(84\) 0 0
\(85\) 738.306 0.942124
\(86\) 0 0
\(87\) 227.047 0.279793
\(88\) 0 0
\(89\) −938.445 −1.11770 −0.558848 0.829270i \(-0.688757\pi\)
−0.558848 + 0.829270i \(0.688757\pi\)
\(90\) 0 0
\(91\) −877.626 −1.01099
\(92\) 0 0
\(93\) 220.952 0.246361
\(94\) 0 0
\(95\) 66.0775 0.0713621
\(96\) 0 0
\(97\) −888.766 −0.930315 −0.465158 0.885228i \(-0.654002\pi\)
−0.465158 + 0.885228i \(0.654002\pi\)
\(98\) 0 0
\(99\) 582.978 0.591833
\(100\) 0 0
\(101\) 1551.83 1.52884 0.764420 0.644719i \(-0.223026\pi\)
0.764420 + 0.644719i \(0.223026\pi\)
\(102\) 0 0
\(103\) −402.438 −0.384984 −0.192492 0.981299i \(-0.561657\pi\)
−0.192492 + 0.981299i \(0.561657\pi\)
\(104\) 0 0
\(105\) −779.994 −0.724948
\(106\) 0 0
\(107\) 821.553 0.742267 0.371133 0.928580i \(-0.378969\pi\)
0.371133 + 0.928580i \(0.378969\pi\)
\(108\) 0 0
\(109\) −803.052 −0.705674 −0.352837 0.935685i \(-0.614783\pi\)
−0.352837 + 0.935685i \(0.614783\pi\)
\(110\) 0 0
\(111\) −1342.92 −1.14832
\(112\) 0 0
\(113\) −735.506 −0.612306 −0.306153 0.951982i \(-0.599042\pi\)
−0.306153 + 0.951982i \(0.599042\pi\)
\(114\) 0 0
\(115\) 1222.64 0.991410
\(116\) 0 0
\(117\) −2127.58 −1.68116
\(118\) 0 0
\(119\) 1483.23 1.14258
\(120\) 0 0
\(121\) −1042.06 −0.782916
\(122\) 0 0
\(123\) −720.308 −0.528033
\(124\) 0 0
\(125\) −1411.30 −1.00984
\(126\) 0 0
\(127\) −1895.16 −1.32416 −0.662078 0.749435i \(-0.730325\pi\)
−0.662078 + 0.749435i \(0.730325\pi\)
\(128\) 0 0
\(129\) −4063.46 −2.77340
\(130\) 0 0
\(131\) 1145.54 0.764020 0.382010 0.924158i \(-0.375232\pi\)
0.382010 + 0.924158i \(0.375232\pi\)
\(132\) 0 0
\(133\) 132.747 0.0865461
\(134\) 0 0
\(135\) −402.283 −0.256466
\(136\) 0 0
\(137\) 2871.41 1.79067 0.895334 0.445396i \(-0.146937\pi\)
0.895334 + 0.445396i \(0.146937\pi\)
\(138\) 0 0
\(139\) −777.734 −0.474579 −0.237290 0.971439i \(-0.576259\pi\)
−0.237290 + 0.971439i \(0.576259\pi\)
\(140\) 0 0
\(141\) −858.174 −0.512562
\(142\) 0 0
\(143\) −1054.48 −0.616646
\(144\) 0 0
\(145\) −204.220 −0.116963
\(146\) 0 0
\(147\) 1118.44 0.627534
\(148\) 0 0
\(149\) −2373.29 −1.30488 −0.652440 0.757840i \(-0.726255\pi\)
−0.652440 + 0.757840i \(0.726255\pi\)
\(150\) 0 0
\(151\) 102.190 0.0550733 0.0275366 0.999621i \(-0.491234\pi\)
0.0275366 + 0.999621i \(0.491234\pi\)
\(152\) 0 0
\(153\) 3595.71 1.89998
\(154\) 0 0
\(155\) −198.738 −0.102987
\(156\) 0 0
\(157\) −585.462 −0.297611 −0.148805 0.988866i \(-0.547543\pi\)
−0.148805 + 0.988866i \(0.547543\pi\)
\(158\) 0 0
\(159\) 1100.42 0.548861
\(160\) 0 0
\(161\) 2456.24 1.20235
\(162\) 0 0
\(163\) −1440.36 −0.692131 −0.346066 0.938210i \(-0.612483\pi\)
−0.346066 + 0.938210i \(0.612483\pi\)
\(164\) 0 0
\(165\) −937.176 −0.442176
\(166\) 0 0
\(167\) 3338.00 1.54672 0.773359 0.633968i \(-0.218575\pi\)
0.773359 + 0.633968i \(0.218575\pi\)
\(168\) 0 0
\(169\) 1651.35 0.751638
\(170\) 0 0
\(171\) 321.812 0.143916
\(172\) 0 0
\(173\) 3612.07 1.58740 0.793701 0.608308i \(-0.208151\pi\)
0.793701 + 0.608308i \(0.208151\pi\)
\(174\) 0 0
\(175\) −1066.83 −0.460828
\(176\) 0 0
\(177\) 2765.70 1.17448
\(178\) 0 0
\(179\) 2415.54 1.00864 0.504319 0.863517i \(-0.331744\pi\)
0.504319 + 0.863517i \(0.331744\pi\)
\(180\) 0 0
\(181\) 1755.89 0.721073 0.360537 0.932745i \(-0.382594\pi\)
0.360537 + 0.932745i \(0.382594\pi\)
\(182\) 0 0
\(183\) −1451.18 −0.586198
\(184\) 0 0
\(185\) 1207.90 0.480037
\(186\) 0 0
\(187\) 1782.13 0.696909
\(188\) 0 0
\(189\) −808.170 −0.311035
\(190\) 0 0
\(191\) 1156.10 0.437972 0.218986 0.975728i \(-0.429725\pi\)
0.218986 + 0.975728i \(0.429725\pi\)
\(192\) 0 0
\(193\) −3214.73 −1.19897 −0.599485 0.800386i \(-0.704628\pi\)
−0.599485 + 0.800386i \(0.704628\pi\)
\(194\) 0 0
\(195\) 3420.23 1.25604
\(196\) 0 0
\(197\) 3924.03 1.41916 0.709582 0.704623i \(-0.248884\pi\)
0.709582 + 0.704623i \(0.248884\pi\)
\(198\) 0 0
\(199\) 4463.80 1.59010 0.795051 0.606543i \(-0.207444\pi\)
0.795051 + 0.606543i \(0.207444\pi\)
\(200\) 0 0
\(201\) 4500.54 1.57932
\(202\) 0 0
\(203\) −410.271 −0.141849
\(204\) 0 0
\(205\) 647.890 0.220735
\(206\) 0 0
\(207\) 5954.54 1.99937
\(208\) 0 0
\(209\) 159.498 0.0527881
\(210\) 0 0
\(211\) −4881.35 −1.59263 −0.796317 0.604880i \(-0.793221\pi\)
−0.796317 + 0.604880i \(0.793221\pi\)
\(212\) 0 0
\(213\) −4189.05 −1.34755
\(214\) 0 0
\(215\) 3654.93 1.15937
\(216\) 0 0
\(217\) −399.256 −0.124900
\(218\) 0 0
\(219\) 3982.90 1.22895
\(220\) 0 0
\(221\) −6503.89 −1.97963
\(222\) 0 0
\(223\) −1533.22 −0.460414 −0.230207 0.973142i \(-0.573940\pi\)
−0.230207 + 0.973142i \(0.573940\pi\)
\(224\) 0 0
\(225\) −2586.27 −0.766301
\(226\) 0 0
\(227\) 3110.06 0.909348 0.454674 0.890658i \(-0.349756\pi\)
0.454674 + 0.890658i \(0.349756\pi\)
\(228\) 0 0
\(229\) 4686.03 1.35223 0.676117 0.736794i \(-0.263661\pi\)
0.676117 + 0.736794i \(0.263661\pi\)
\(230\) 0 0
\(231\) −1882.75 −0.536259
\(232\) 0 0
\(233\) −908.748 −0.255511 −0.127756 0.991806i \(-0.540777\pi\)
−0.127756 + 0.991806i \(0.540777\pi\)
\(234\) 0 0
\(235\) 771.896 0.214268
\(236\) 0 0
\(237\) 477.704 0.130929
\(238\) 0 0
\(239\) −18.3950 −0.00497854 −0.00248927 0.999997i \(-0.500792\pi\)
−0.00248927 + 0.999997i \(0.500792\pi\)
\(240\) 0 0
\(241\) 6199.96 1.65716 0.828578 0.559874i \(-0.189151\pi\)
0.828578 + 0.559874i \(0.189151\pi\)
\(242\) 0 0
\(243\) 5290.68 1.39670
\(244\) 0 0
\(245\) −1006.00 −0.262330
\(246\) 0 0
\(247\) −582.089 −0.149949
\(248\) 0 0
\(249\) −6472.82 −1.64738
\(250\) 0 0
\(251\) −3325.41 −0.836248 −0.418124 0.908390i \(-0.637312\pi\)
−0.418124 + 0.908390i \(0.637312\pi\)
\(252\) 0 0
\(253\) 2951.22 0.733366
\(254\) 0 0
\(255\) −5780.35 −1.41953
\(256\) 0 0
\(257\) 2713.20 0.658541 0.329271 0.944236i \(-0.393197\pi\)
0.329271 + 0.944236i \(0.393197\pi\)
\(258\) 0 0
\(259\) 2426.63 0.582176
\(260\) 0 0
\(261\) −994.597 −0.235878
\(262\) 0 0
\(263\) 6234.94 1.46184 0.730918 0.682465i \(-0.239092\pi\)
0.730918 + 0.682465i \(0.239092\pi\)
\(264\) 0 0
\(265\) −989.787 −0.229442
\(266\) 0 0
\(267\) 7347.28 1.68407
\(268\) 0 0
\(269\) 5745.23 1.30220 0.651102 0.758990i \(-0.274307\pi\)
0.651102 + 0.758990i \(0.274307\pi\)
\(270\) 0 0
\(271\) 8855.53 1.98500 0.992501 0.122238i \(-0.0390072\pi\)
0.992501 + 0.122238i \(0.0390072\pi\)
\(272\) 0 0
\(273\) 6871.12 1.52329
\(274\) 0 0
\(275\) −1281.82 −0.281078
\(276\) 0 0
\(277\) 2805.71 0.608588 0.304294 0.952578i \(-0.401580\pi\)
0.304294 + 0.952578i \(0.401580\pi\)
\(278\) 0 0
\(279\) −967.896 −0.207693
\(280\) 0 0
\(281\) 5133.23 1.08976 0.544881 0.838514i \(-0.316575\pi\)
0.544881 + 0.838514i \(0.316575\pi\)
\(282\) 0 0
\(283\) −2928.86 −0.615204 −0.307602 0.951515i \(-0.599527\pi\)
−0.307602 + 0.951515i \(0.599527\pi\)
\(284\) 0 0
\(285\) −517.334 −0.107524
\(286\) 0 0
\(287\) 1301.59 0.267701
\(288\) 0 0
\(289\) 6078.87 1.23730
\(290\) 0 0
\(291\) 6958.34 1.40173
\(292\) 0 0
\(293\) −5685.37 −1.13359 −0.566796 0.823858i \(-0.691817\pi\)
−0.566796 + 0.823858i \(0.691817\pi\)
\(294\) 0 0
\(295\) −2487.64 −0.490970
\(296\) 0 0
\(297\) −971.031 −0.189713
\(298\) 0 0
\(299\) −10770.5 −2.08319
\(300\) 0 0
\(301\) 7342.62 1.40605
\(302\) 0 0
\(303\) −12149.6 −2.30355
\(304\) 0 0
\(305\) 1305.28 0.245050
\(306\) 0 0
\(307\) −2879.21 −0.535262 −0.267631 0.963522i \(-0.586241\pi\)
−0.267631 + 0.963522i \(0.586241\pi\)
\(308\) 0 0
\(309\) 3150.77 0.580068
\(310\) 0 0
\(311\) 513.825 0.0936860 0.0468430 0.998902i \(-0.485084\pi\)
0.0468430 + 0.998902i \(0.485084\pi\)
\(312\) 0 0
\(313\) −1508.76 −0.272460 −0.136230 0.990677i \(-0.543499\pi\)
−0.136230 + 0.990677i \(0.543499\pi\)
\(314\) 0 0
\(315\) 3416.82 0.611163
\(316\) 0 0
\(317\) −5411.89 −0.958872 −0.479436 0.877577i \(-0.659159\pi\)
−0.479436 + 0.877577i \(0.659159\pi\)
\(318\) 0 0
\(319\) −492.947 −0.0865196
\(320\) 0 0
\(321\) −6432.11 −1.11840
\(322\) 0 0
\(323\) 983.758 0.169467
\(324\) 0 0
\(325\) 4678.01 0.798429
\(326\) 0 0
\(327\) 6287.26 1.06326
\(328\) 0 0
\(329\) 1550.71 0.259858
\(330\) 0 0
\(331\) −10354.1 −1.71938 −0.859688 0.510820i \(-0.829342\pi\)
−0.859688 + 0.510820i \(0.829342\pi\)
\(332\) 0 0
\(333\) 5882.75 0.968087
\(334\) 0 0
\(335\) −4048.07 −0.660208
\(336\) 0 0
\(337\) 4707.20 0.760882 0.380441 0.924805i \(-0.375772\pi\)
0.380441 + 0.924805i \(0.375772\pi\)
\(338\) 0 0
\(339\) 5758.43 0.922581
\(340\) 0 0
\(341\) −479.714 −0.0761817
\(342\) 0 0
\(343\) −6873.52 −1.08203
\(344\) 0 0
\(345\) −9572.33 −1.49379
\(346\) 0 0
\(347\) 6733.27 1.04167 0.520837 0.853656i \(-0.325620\pi\)
0.520837 + 0.853656i \(0.325620\pi\)
\(348\) 0 0
\(349\) 9272.50 1.42219 0.711097 0.703094i \(-0.248199\pi\)
0.711097 + 0.703094i \(0.248199\pi\)
\(350\) 0 0
\(351\) 3543.79 0.538898
\(352\) 0 0
\(353\) 5776.97 0.871041 0.435520 0.900179i \(-0.356564\pi\)
0.435520 + 0.900179i \(0.356564\pi\)
\(354\) 0 0
\(355\) 3767.89 0.563321
\(356\) 0 0
\(357\) −11612.5 −1.72157
\(358\) 0 0
\(359\) 3821.93 0.561877 0.280938 0.959726i \(-0.409354\pi\)
0.280938 + 0.959726i \(0.409354\pi\)
\(360\) 0 0
\(361\) −6770.95 −0.987164
\(362\) 0 0
\(363\) 8158.51 1.17964
\(364\) 0 0
\(365\) −3582.47 −0.513739
\(366\) 0 0
\(367\) −9481.16 −1.34854 −0.674268 0.738487i \(-0.735541\pi\)
−0.674268 + 0.738487i \(0.735541\pi\)
\(368\) 0 0
\(369\) 3155.37 0.445154
\(370\) 0 0
\(371\) −1988.44 −0.278261
\(372\) 0 0
\(373\) 11849.6 1.64490 0.822450 0.568837i \(-0.192606\pi\)
0.822450 + 0.568837i \(0.192606\pi\)
\(374\) 0 0
\(375\) 11049.3 1.52156
\(376\) 0 0
\(377\) 1799.02 0.245767
\(378\) 0 0
\(379\) 14650.4 1.98560 0.992799 0.119794i \(-0.0382235\pi\)
0.992799 + 0.119794i \(0.0382235\pi\)
\(380\) 0 0
\(381\) 14837.6 1.99515
\(382\) 0 0
\(383\) −2202.45 −0.293837 −0.146919 0.989149i \(-0.546936\pi\)
−0.146919 + 0.989149i \(0.546936\pi\)
\(384\) 0 0
\(385\) 1693.46 0.224174
\(386\) 0 0
\(387\) 17800.3 2.33809
\(388\) 0 0
\(389\) −5614.61 −0.731804 −0.365902 0.930653i \(-0.619240\pi\)
−0.365902 + 0.930653i \(0.619240\pi\)
\(390\) 0 0
\(391\) 18202.7 2.35434
\(392\) 0 0
\(393\) −8968.70 −1.15117
\(394\) 0 0
\(395\) −429.677 −0.0547326
\(396\) 0 0
\(397\) −1672.32 −0.211414 −0.105707 0.994397i \(-0.533711\pi\)
−0.105707 + 0.994397i \(0.533711\pi\)
\(398\) 0 0
\(399\) −1039.30 −0.130402
\(400\) 0 0
\(401\) −1183.01 −0.147324 −0.0736619 0.997283i \(-0.523469\pi\)
−0.0736619 + 0.997283i \(0.523469\pi\)
\(402\) 0 0
\(403\) 1750.72 0.216401
\(404\) 0 0
\(405\) −3371.45 −0.413651
\(406\) 0 0
\(407\) 2915.64 0.355093
\(408\) 0 0
\(409\) −6264.74 −0.757388 −0.378694 0.925522i \(-0.623627\pi\)
−0.378694 + 0.925522i \(0.623627\pi\)
\(410\) 0 0
\(411\) −22480.9 −2.69805
\(412\) 0 0
\(413\) −4997.58 −0.595435
\(414\) 0 0
\(415\) 5822.06 0.688659
\(416\) 0 0
\(417\) 6089.04 0.715063
\(418\) 0 0
\(419\) 4224.75 0.492584 0.246292 0.969196i \(-0.420788\pi\)
0.246292 + 0.969196i \(0.420788\pi\)
\(420\) 0 0
\(421\) −10801.7 −1.25046 −0.625231 0.780440i \(-0.714995\pi\)
−0.625231 + 0.780440i \(0.714995\pi\)
\(422\) 0 0
\(423\) 3759.30 0.432112
\(424\) 0 0
\(425\) −7906.05 −0.902352
\(426\) 0 0
\(427\) 2622.26 0.297190
\(428\) 0 0
\(429\) 8255.77 0.929120
\(430\) 0 0
\(431\) 13127.7 1.46715 0.733574 0.679610i \(-0.237851\pi\)
0.733574 + 0.679610i \(0.237851\pi\)
\(432\) 0 0
\(433\) −11403.2 −1.26559 −0.632797 0.774318i \(-0.718093\pi\)
−0.632797 + 0.774318i \(0.718093\pi\)
\(434\) 0 0
\(435\) 1598.88 0.176231
\(436\) 0 0
\(437\) 1629.11 0.178332
\(438\) 0 0
\(439\) 14656.2 1.59339 0.796697 0.604378i \(-0.206578\pi\)
0.796697 + 0.604378i \(0.206578\pi\)
\(440\) 0 0
\(441\) −4899.42 −0.529038
\(442\) 0 0
\(443\) −6136.95 −0.658184 −0.329092 0.944298i \(-0.606743\pi\)
−0.329092 + 0.944298i \(0.606743\pi\)
\(444\) 0 0
\(445\) −6608.61 −0.703995
\(446\) 0 0
\(447\) 18580.9 1.96610
\(448\) 0 0
\(449\) −18012.1 −1.89319 −0.946594 0.322428i \(-0.895501\pi\)
−0.946594 + 0.322428i \(0.895501\pi\)
\(450\) 0 0
\(451\) 1563.88 0.163282
\(452\) 0 0
\(453\) −800.063 −0.0829807
\(454\) 0 0
\(455\) −6180.31 −0.636786
\(456\) 0 0
\(457\) 715.440 0.0732316 0.0366158 0.999329i \(-0.488342\pi\)
0.0366158 + 0.999329i \(0.488342\pi\)
\(458\) 0 0
\(459\) −5989.16 −0.609042
\(460\) 0 0
\(461\) 16471.2 1.66408 0.832038 0.554719i \(-0.187174\pi\)
0.832038 + 0.554719i \(0.187174\pi\)
\(462\) 0 0
\(463\) 3844.07 0.385851 0.192925 0.981213i \(-0.438202\pi\)
0.192925 + 0.981213i \(0.438202\pi\)
\(464\) 0 0
\(465\) 1555.96 0.155174
\(466\) 0 0
\(467\) −15072.5 −1.49352 −0.746759 0.665094i \(-0.768391\pi\)
−0.746759 + 0.665094i \(0.768391\pi\)
\(468\) 0 0
\(469\) −8132.41 −0.800682
\(470\) 0 0
\(471\) 4583.70 0.448420
\(472\) 0 0
\(473\) 8822.29 0.857610
\(474\) 0 0
\(475\) −707.581 −0.0683496
\(476\) 0 0
\(477\) −4820.48 −0.462714
\(478\) 0 0
\(479\) 8856.14 0.844776 0.422388 0.906415i \(-0.361192\pi\)
0.422388 + 0.906415i \(0.361192\pi\)
\(480\) 0 0
\(481\) −10640.7 −1.00867
\(482\) 0 0
\(483\) −19230.4 −1.81163
\(484\) 0 0
\(485\) −6258.76 −0.585971
\(486\) 0 0
\(487\) 6998.94 0.651237 0.325618 0.945501i \(-0.394428\pi\)
0.325618 + 0.945501i \(0.394428\pi\)
\(488\) 0 0
\(489\) 11276.8 1.04286
\(490\) 0 0
\(491\) 2169.60 0.199415 0.0997076 0.995017i \(-0.468209\pi\)
0.0997076 + 0.995017i \(0.468209\pi\)
\(492\) 0 0
\(493\) −3040.42 −0.277756
\(494\) 0 0
\(495\) 4105.38 0.372774
\(496\) 0 0
\(497\) 7569.55 0.683180
\(498\) 0 0
\(499\) 17541.6 1.57368 0.786842 0.617155i \(-0.211715\pi\)
0.786842 + 0.617155i \(0.211715\pi\)
\(500\) 0 0
\(501\) −26133.9 −2.33049
\(502\) 0 0
\(503\) −16190.4 −1.43517 −0.717587 0.696469i \(-0.754753\pi\)
−0.717587 + 0.696469i \(0.754753\pi\)
\(504\) 0 0
\(505\) 10928.1 0.962959
\(506\) 0 0
\(507\) −12928.8 −1.13252
\(508\) 0 0
\(509\) −1493.11 −0.130021 −0.0650106 0.997885i \(-0.520708\pi\)
−0.0650106 + 0.997885i \(0.520708\pi\)
\(510\) 0 0
\(511\) −7197.03 −0.623049
\(512\) 0 0
\(513\) −536.022 −0.0461325
\(514\) 0 0
\(515\) −2834.00 −0.242487
\(516\) 0 0
\(517\) 1863.20 0.158498
\(518\) 0 0
\(519\) −28279.6 −2.39179
\(520\) 0 0
\(521\) −7026.84 −0.590886 −0.295443 0.955360i \(-0.595467\pi\)
−0.295443 + 0.955360i \(0.595467\pi\)
\(522\) 0 0
\(523\) −21924.1 −1.83303 −0.916514 0.400003i \(-0.869009\pi\)
−0.916514 + 0.400003i \(0.869009\pi\)
\(524\) 0 0
\(525\) 8352.45 0.694345
\(526\) 0 0
\(527\) −2958.80 −0.244568
\(528\) 0 0
\(529\) 17976.8 1.47750
\(530\) 0 0
\(531\) −12115.4 −0.990135
\(532\) 0 0
\(533\) −5707.39 −0.463817
\(534\) 0 0
\(535\) 5785.44 0.467526
\(536\) 0 0
\(537\) −18911.8 −1.51975
\(538\) 0 0
\(539\) −2428.28 −0.194051
\(540\) 0 0
\(541\) 983.325 0.0781450 0.0390725 0.999236i \(-0.487560\pi\)
0.0390725 + 0.999236i \(0.487560\pi\)
\(542\) 0 0
\(543\) −13747.2 −1.08646
\(544\) 0 0
\(545\) −5655.16 −0.444478
\(546\) 0 0
\(547\) −6100.54 −0.476856 −0.238428 0.971160i \(-0.576632\pi\)
−0.238428 + 0.971160i \(0.576632\pi\)
\(548\) 0 0
\(549\) 6357.01 0.494190
\(550\) 0 0
\(551\) −272.114 −0.0210389
\(552\) 0 0
\(553\) −863.204 −0.0663782
\(554\) 0 0
\(555\) −9456.93 −0.723287
\(556\) 0 0
\(557\) −7531.50 −0.572926 −0.286463 0.958091i \(-0.592480\pi\)
−0.286463 + 0.958091i \(0.592480\pi\)
\(558\) 0 0
\(559\) −32197.0 −2.43612
\(560\) 0 0
\(561\) −13952.6 −1.05005
\(562\) 0 0
\(563\) 19553.5 1.46373 0.731866 0.681449i \(-0.238650\pi\)
0.731866 + 0.681449i \(0.238650\pi\)
\(564\) 0 0
\(565\) −5179.49 −0.385669
\(566\) 0 0
\(567\) −6773.10 −0.501664
\(568\) 0 0
\(569\) 23656.9 1.74297 0.871483 0.490426i \(-0.163159\pi\)
0.871483 + 0.490426i \(0.163159\pi\)
\(570\) 0 0
\(571\) 5466.08 0.400610 0.200305 0.979734i \(-0.435807\pi\)
0.200305 + 0.979734i \(0.435807\pi\)
\(572\) 0 0
\(573\) −9051.37 −0.659907
\(574\) 0 0
\(575\) −13092.5 −0.949557
\(576\) 0 0
\(577\) −653.065 −0.0471187 −0.0235593 0.999722i \(-0.507500\pi\)
−0.0235593 + 0.999722i \(0.507500\pi\)
\(578\) 0 0
\(579\) 25168.8 1.80653
\(580\) 0 0
\(581\) 11696.3 0.835187
\(582\) 0 0
\(583\) −2389.15 −0.169723
\(584\) 0 0
\(585\) −14982.6 −1.05890
\(586\) 0 0
\(587\) 19414.4 1.36511 0.682554 0.730835i \(-0.260869\pi\)
0.682554 + 0.730835i \(0.260869\pi\)
\(588\) 0 0
\(589\) −264.809 −0.0185250
\(590\) 0 0
\(591\) −30722.0 −2.13830
\(592\) 0 0
\(593\) 18910.6 1.30955 0.654777 0.755822i \(-0.272763\pi\)
0.654777 + 0.755822i \(0.272763\pi\)
\(594\) 0 0
\(595\) 10445.0 0.719670
\(596\) 0 0
\(597\) −34948.0 −2.39586
\(598\) 0 0
\(599\) −12128.7 −0.827322 −0.413661 0.910431i \(-0.635750\pi\)
−0.413661 + 0.910431i \(0.635750\pi\)
\(600\) 0 0
\(601\) −25803.3 −1.75131 −0.875656 0.482936i \(-0.839570\pi\)
−0.875656 + 0.482936i \(0.839570\pi\)
\(602\) 0 0
\(603\) −19715.0 −1.33144
\(604\) 0 0
\(605\) −7338.28 −0.493130
\(606\) 0 0
\(607\) 12183.6 0.814690 0.407345 0.913274i \(-0.366455\pi\)
0.407345 + 0.913274i \(0.366455\pi\)
\(608\) 0 0
\(609\) 3212.09 0.213728
\(610\) 0 0
\(611\) −6799.78 −0.450229
\(612\) 0 0
\(613\) 16519.2 1.08843 0.544213 0.838947i \(-0.316829\pi\)
0.544213 + 0.838947i \(0.316829\pi\)
\(614\) 0 0
\(615\) −5072.47 −0.332588
\(616\) 0 0
\(617\) −6518.09 −0.425297 −0.212649 0.977129i \(-0.568209\pi\)
−0.212649 + 0.977129i \(0.568209\pi\)
\(618\) 0 0
\(619\) 4344.67 0.282111 0.141056 0.990002i \(-0.454950\pi\)
0.141056 + 0.990002i \(0.454950\pi\)
\(620\) 0 0
\(621\) −9918.12 −0.640902
\(622\) 0 0
\(623\) −13276.4 −0.853787
\(624\) 0 0
\(625\) −512.326 −0.0327889
\(626\) 0 0
\(627\) −1248.74 −0.0795374
\(628\) 0 0
\(629\) 17983.2 1.13996
\(630\) 0 0
\(631\) 2989.38 0.188598 0.0942989 0.995544i \(-0.469939\pi\)
0.0942989 + 0.995544i \(0.469939\pi\)
\(632\) 0 0
\(633\) 38217.1 2.39967
\(634\) 0 0
\(635\) −13345.8 −0.834037
\(636\) 0 0
\(637\) 8862.02 0.551218
\(638\) 0 0
\(639\) 18350.5 1.13604
\(640\) 0 0
\(641\) 15375.4 0.947411 0.473706 0.880683i \(-0.342916\pi\)
0.473706 + 0.880683i \(0.342916\pi\)
\(642\) 0 0
\(643\) 16966.5 1.04058 0.520291 0.853989i \(-0.325823\pi\)
0.520291 + 0.853989i \(0.325823\pi\)
\(644\) 0 0
\(645\) −28615.2 −1.74686
\(646\) 0 0
\(647\) −12379.7 −0.752234 −0.376117 0.926572i \(-0.622741\pi\)
−0.376117 + 0.926572i \(0.622741\pi\)
\(648\) 0 0
\(649\) −6004.68 −0.363181
\(650\) 0 0
\(651\) 3125.86 0.188191
\(652\) 0 0
\(653\) −30418.3 −1.82291 −0.911453 0.411403i \(-0.865039\pi\)
−0.911453 + 0.411403i \(0.865039\pi\)
\(654\) 0 0
\(655\) 8067.01 0.481228
\(656\) 0 0
\(657\) −17447.4 −1.03605
\(658\) 0 0
\(659\) 6069.11 0.358754 0.179377 0.983780i \(-0.442592\pi\)
0.179377 + 0.983780i \(0.442592\pi\)
\(660\) 0 0
\(661\) 24894.6 1.46489 0.732443 0.680829i \(-0.238380\pi\)
0.732443 + 0.680829i \(0.238380\pi\)
\(662\) 0 0
\(663\) 50920.3 2.98277
\(664\) 0 0
\(665\) 934.815 0.0545121
\(666\) 0 0
\(667\) −5034.97 −0.292286
\(668\) 0 0
\(669\) 12003.9 0.693720
\(670\) 0 0
\(671\) 3150.69 0.181268
\(672\) 0 0
\(673\) 21677.5 1.24162 0.620808 0.783963i \(-0.286805\pi\)
0.620808 + 0.783963i \(0.286805\pi\)
\(674\) 0 0
\(675\) 4307.79 0.245640
\(676\) 0 0
\(677\) 21322.2 1.21046 0.605228 0.796052i \(-0.293082\pi\)
0.605228 + 0.796052i \(0.293082\pi\)
\(678\) 0 0
\(679\) −12573.6 −0.710649
\(680\) 0 0
\(681\) −24349.3 −1.37014
\(682\) 0 0
\(683\) −38.7459 −0.00217068 −0.00108534 0.999999i \(-0.500345\pi\)
−0.00108534 + 0.999999i \(0.500345\pi\)
\(684\) 0 0
\(685\) 20220.7 1.12787
\(686\) 0 0
\(687\) −36687.9 −2.03745
\(688\) 0 0
\(689\) 8719.23 0.482113
\(690\) 0 0
\(691\) −27812.0 −1.53114 −0.765571 0.643351i \(-0.777544\pi\)
−0.765571 + 0.643351i \(0.777544\pi\)
\(692\) 0 0
\(693\) 8247.54 0.452090
\(694\) 0 0
\(695\) −5476.86 −0.298920
\(696\) 0 0
\(697\) 9645.76 0.524188
\(698\) 0 0
\(699\) 7114.78 0.384987
\(700\) 0 0
\(701\) −23404.8 −1.26104 −0.630519 0.776174i \(-0.717158\pi\)
−0.630519 + 0.776174i \(0.717158\pi\)
\(702\) 0 0
\(703\) 1609.47 0.0863477
\(704\) 0 0
\(705\) −6043.33 −0.322844
\(706\) 0 0
\(707\) 21954.1 1.16785
\(708\) 0 0
\(709\) −8525.03 −0.451572 −0.225786 0.974177i \(-0.572495\pi\)
−0.225786 + 0.974177i \(0.572495\pi\)
\(710\) 0 0
\(711\) −2092.62 −0.110379
\(712\) 0 0
\(713\) −4899.80 −0.257362
\(714\) 0 0
\(715\) −7425.76 −0.388402
\(716\) 0 0
\(717\) 144.018 0.00750132
\(718\) 0 0
\(719\) −13242.3 −0.686862 −0.343431 0.939178i \(-0.611589\pi\)
−0.343431 + 0.939178i \(0.611589\pi\)
\(720\) 0 0
\(721\) −5693.39 −0.294082
\(722\) 0 0
\(723\) −48540.8 −2.49689
\(724\) 0 0
\(725\) 2186.86 0.112025
\(726\) 0 0
\(727\) 5299.22 0.270340 0.135170 0.990822i \(-0.456842\pi\)
0.135170 + 0.990822i \(0.456842\pi\)
\(728\) 0 0
\(729\) −28495.4 −1.44771
\(730\) 0 0
\(731\) 54414.5 2.75320
\(732\) 0 0
\(733\) −28433.7 −1.43278 −0.716388 0.697702i \(-0.754206\pi\)
−0.716388 + 0.697702i \(0.754206\pi\)
\(734\) 0 0
\(735\) 7876.15 0.395260
\(736\) 0 0
\(737\) −9771.23 −0.488369
\(738\) 0 0
\(739\) 19450.7 0.968209 0.484105 0.875010i \(-0.339145\pi\)
0.484105 + 0.875010i \(0.339145\pi\)
\(740\) 0 0
\(741\) 4557.30 0.225933
\(742\) 0 0
\(743\) −15260.1 −0.753484 −0.376742 0.926318i \(-0.622956\pi\)
−0.376742 + 0.926318i \(0.622956\pi\)
\(744\) 0 0
\(745\) −16712.9 −0.821896
\(746\) 0 0
\(747\) 28354.7 1.38881
\(748\) 0 0
\(749\) 11622.7 0.567003
\(750\) 0 0
\(751\) 23290.8 1.13168 0.565841 0.824514i \(-0.308551\pi\)
0.565841 + 0.824514i \(0.308551\pi\)
\(752\) 0 0
\(753\) 26035.3 1.26000
\(754\) 0 0
\(755\) 719.627 0.0346886
\(756\) 0 0
\(757\) 22951.5 1.10196 0.550981 0.834517i \(-0.314253\pi\)
0.550981 + 0.834517i \(0.314253\pi\)
\(758\) 0 0
\(759\) −23105.7 −1.10499
\(760\) 0 0
\(761\) 14975.4 0.713349 0.356675 0.934229i \(-0.383910\pi\)
0.356675 + 0.934229i \(0.383910\pi\)
\(762\) 0 0
\(763\) −11361.0 −0.539050
\(764\) 0 0
\(765\) 25321.3 1.19672
\(766\) 0 0
\(767\) 21914.1 1.03165
\(768\) 0 0
\(769\) 30907.7 1.44936 0.724682 0.689084i \(-0.241987\pi\)
0.724682 + 0.689084i \(0.241987\pi\)
\(770\) 0 0
\(771\) −21242.2 −0.992244
\(772\) 0 0
\(773\) 3728.71 0.173496 0.0867479 0.996230i \(-0.472353\pi\)
0.0867479 + 0.996230i \(0.472353\pi\)
\(774\) 0 0
\(775\) 2128.16 0.0986395
\(776\) 0 0
\(777\) −18998.6 −0.877182
\(778\) 0 0
\(779\) 863.283 0.0397051
\(780\) 0 0
\(781\) 9094.94 0.416700
\(782\) 0 0
\(783\) 1656.64 0.0756111
\(784\) 0 0
\(785\) −4122.87 −0.187454
\(786\) 0 0
\(787\) −33291.9 −1.50791 −0.753957 0.656923i \(-0.771858\pi\)
−0.753957 + 0.656923i \(0.771858\pi\)
\(788\) 0 0
\(789\) −48814.7 −2.20260
\(790\) 0 0
\(791\) −10405.4 −0.467729
\(792\) 0 0
\(793\) −11498.5 −0.514909
\(794\) 0 0
\(795\) 7749.24 0.345707
\(796\) 0 0
\(797\) 36436.3 1.61937 0.809687 0.586862i \(-0.199637\pi\)
0.809687 + 0.586862i \(0.199637\pi\)
\(798\) 0 0
\(799\) 11491.9 0.508831
\(800\) 0 0
\(801\) −32185.4 −1.41974
\(802\) 0 0
\(803\) −8647.37 −0.380024
\(804\) 0 0
\(805\) 17297.1 0.757318
\(806\) 0 0
\(807\) −44980.6 −1.96207
\(808\) 0 0
\(809\) 9776.41 0.424871 0.212435 0.977175i \(-0.431861\pi\)
0.212435 + 0.977175i \(0.431861\pi\)
\(810\) 0 0
\(811\) −30799.9 −1.33358 −0.666788 0.745248i \(-0.732331\pi\)
−0.666788 + 0.745248i \(0.732331\pi\)
\(812\) 0 0
\(813\) −69331.8 −2.99086
\(814\) 0 0
\(815\) −10143.1 −0.435948
\(816\) 0 0
\(817\) 4870.02 0.208544
\(818\) 0 0
\(819\) −30099.5 −1.28420
\(820\) 0 0
\(821\) 3080.42 0.130947 0.0654735 0.997854i \(-0.479144\pi\)
0.0654735 + 0.997854i \(0.479144\pi\)
\(822\) 0 0
\(823\) 23048.4 0.976204 0.488102 0.872787i \(-0.337689\pi\)
0.488102 + 0.872787i \(0.337689\pi\)
\(824\) 0 0
\(825\) 10035.6 0.423510
\(826\) 0 0
\(827\) 13260.3 0.557565 0.278783 0.960354i \(-0.410069\pi\)
0.278783 + 0.960354i \(0.410069\pi\)
\(828\) 0 0
\(829\) 34413.4 1.44177 0.720885 0.693055i \(-0.243736\pi\)
0.720885 + 0.693055i \(0.243736\pi\)
\(830\) 0 0
\(831\) −21966.5 −0.916978
\(832\) 0 0
\(833\) −14977.2 −0.622965
\(834\) 0 0
\(835\) 23506.5 0.974221
\(836\) 0 0
\(837\) 1612.17 0.0665766
\(838\) 0 0
\(839\) 5440.08 0.223853 0.111926 0.993717i \(-0.464298\pi\)
0.111926 + 0.993717i \(0.464298\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) −40189.1 −1.64198
\(844\) 0 0
\(845\) 11628.9 0.473429
\(846\) 0 0
\(847\) −14742.3 −0.598054
\(848\) 0 0
\(849\) 22930.7 0.926947
\(850\) 0 0
\(851\) 29780.4 1.19960
\(852\) 0 0
\(853\) −22303.5 −0.895261 −0.447631 0.894219i \(-0.647732\pi\)
−0.447631 + 0.894219i \(0.647732\pi\)
\(854\) 0 0
\(855\) 2266.22 0.0906470
\(856\) 0 0
\(857\) 13859.0 0.552411 0.276205 0.961099i \(-0.410923\pi\)
0.276205 + 0.961099i \(0.410923\pi\)
\(858\) 0 0
\(859\) 24629.1 0.978270 0.489135 0.872208i \(-0.337313\pi\)
0.489135 + 0.872208i \(0.337313\pi\)
\(860\) 0 0
\(861\) −10190.4 −0.403354
\(862\) 0 0
\(863\) −33140.6 −1.30721 −0.653604 0.756837i \(-0.726744\pi\)
−0.653604 + 0.756837i \(0.726744\pi\)
\(864\) 0 0
\(865\) 25436.5 0.999846
\(866\) 0 0
\(867\) −47592.7 −1.86428
\(868\) 0 0
\(869\) −1037.16 −0.0404869
\(870\) 0 0
\(871\) 35660.2 1.38726
\(872\) 0 0
\(873\) −30481.5 −1.18172
\(874\) 0 0
\(875\) −19966.0 −0.771398
\(876\) 0 0
\(877\) −2224.74 −0.0856604 −0.0428302 0.999082i \(-0.513637\pi\)
−0.0428302 + 0.999082i \(0.513637\pi\)
\(878\) 0 0
\(879\) 44511.9 1.70802
\(880\) 0 0
\(881\) 24242.9 0.927087 0.463544 0.886074i \(-0.346578\pi\)
0.463544 + 0.886074i \(0.346578\pi\)
\(882\) 0 0
\(883\) −24060.4 −0.916984 −0.458492 0.888698i \(-0.651610\pi\)
−0.458492 + 0.888698i \(0.651610\pi\)
\(884\) 0 0
\(885\) 19476.3 0.739760
\(886\) 0 0
\(887\) 9727.38 0.368223 0.184111 0.982905i \(-0.441059\pi\)
0.184111 + 0.982905i \(0.441059\pi\)
\(888\) 0 0
\(889\) −26811.3 −1.01150
\(890\) 0 0
\(891\) −8138.00 −0.305986
\(892\) 0 0
\(893\) 1028.51 0.0385419
\(894\) 0 0
\(895\) 17010.5 0.635304
\(896\) 0 0
\(897\) 84324.5 3.13881
\(898\) 0 0
\(899\) 818.422 0.0303625
\(900\) 0 0
\(901\) −14735.9 −0.544865
\(902\) 0 0
\(903\) −57486.9 −2.11854
\(904\) 0 0
\(905\) 12365.1 0.454177
\(906\) 0 0
\(907\) −4523.40 −0.165598 −0.0827988 0.996566i \(-0.526386\pi\)
−0.0827988 + 0.996566i \(0.526386\pi\)
\(908\) 0 0
\(909\) 53222.2 1.94199
\(910\) 0 0
\(911\) 24396.1 0.887245 0.443623 0.896214i \(-0.353693\pi\)
0.443623 + 0.896214i \(0.353693\pi\)
\(912\) 0 0
\(913\) 14053.3 0.509415
\(914\) 0 0
\(915\) −10219.3 −0.369224
\(916\) 0 0
\(917\) 16206.3 0.583620
\(918\) 0 0
\(919\) −51701.0 −1.85578 −0.927888 0.372858i \(-0.878378\pi\)
−0.927888 + 0.372858i \(0.878378\pi\)
\(920\) 0 0
\(921\) 22541.9 0.806496
\(922\) 0 0
\(923\) −33192.1 −1.18367
\(924\) 0 0
\(925\) −12934.7 −0.459772
\(926\) 0 0
\(927\) −13802.2 −0.489022
\(928\) 0 0
\(929\) −8172.60 −0.288627 −0.144313 0.989532i \(-0.546097\pi\)
−0.144313 + 0.989532i \(0.546097\pi\)
\(930\) 0 0
\(931\) −1340.44 −0.0471871
\(932\) 0 0
\(933\) −4022.84 −0.141160
\(934\) 0 0
\(935\) 12549.9 0.438957
\(936\) 0 0
\(937\) 40385.8 1.40806 0.704028 0.710173i \(-0.251383\pi\)
0.704028 + 0.710173i \(0.251383\pi\)
\(938\) 0 0
\(939\) 11812.4 0.410525
\(940\) 0 0
\(941\) −3540.29 −0.122646 −0.0613232 0.998118i \(-0.519532\pi\)
−0.0613232 + 0.998118i \(0.519532\pi\)
\(942\) 0 0
\(943\) 15973.5 0.551610
\(944\) 0 0
\(945\) −5691.20 −0.195910
\(946\) 0 0
\(947\) −53975.8 −1.85214 −0.926071 0.377349i \(-0.876836\pi\)
−0.926071 + 0.377349i \(0.876836\pi\)
\(948\) 0 0
\(949\) 31558.7 1.07949
\(950\) 0 0
\(951\) 42370.8 1.44476
\(952\) 0 0
\(953\) −19500.7 −0.662843 −0.331421 0.943483i \(-0.607528\pi\)
−0.331421 + 0.943483i \(0.607528\pi\)
\(954\) 0 0
\(955\) 8141.37 0.275862
\(956\) 0 0
\(957\) 3859.39 0.130362
\(958\) 0 0
\(959\) 40622.6 1.36786
\(960\) 0 0
\(961\) −28994.5 −0.973265
\(962\) 0 0
\(963\) 28176.4 0.942857
\(964\) 0 0
\(965\) −22638.4 −0.755186
\(966\) 0 0
\(967\) −18203.2 −0.605354 −0.302677 0.953093i \(-0.597880\pi\)
−0.302677 + 0.953093i \(0.597880\pi\)
\(968\) 0 0
\(969\) −7702.04 −0.255341
\(970\) 0 0
\(971\) −10588.8 −0.349961 −0.174981 0.984572i \(-0.555986\pi\)
−0.174981 + 0.984572i \(0.555986\pi\)
\(972\) 0 0
\(973\) −11002.8 −0.362522
\(974\) 0 0
\(975\) −36625.1 −1.20302
\(976\) 0 0
\(977\) 24805.8 0.812290 0.406145 0.913809i \(-0.366873\pi\)
0.406145 + 0.913809i \(0.366873\pi\)
\(978\) 0 0
\(979\) −15951.9 −0.520760
\(980\) 0 0
\(981\) −27541.8 −0.896375
\(982\) 0 0
\(983\) −2760.27 −0.0895613 −0.0447807 0.998997i \(-0.514259\pi\)
−0.0447807 + 0.998997i \(0.514259\pi\)
\(984\) 0 0
\(985\) 27633.3 0.893878
\(986\) 0 0
\(987\) −12140.8 −0.391536
\(988\) 0 0
\(989\) 90111.0 2.89723
\(990\) 0 0
\(991\) −5609.17 −0.179799 −0.0898997 0.995951i \(-0.528655\pi\)
−0.0898997 + 0.995951i \(0.528655\pi\)
\(992\) 0 0
\(993\) 81064.5 2.59064
\(994\) 0 0
\(995\) 31434.4 1.00155
\(996\) 0 0
\(997\) 3629.31 0.115287 0.0576436 0.998337i \(-0.481641\pi\)
0.0576436 + 0.998337i \(0.481641\pi\)
\(998\) 0 0
\(999\) −9798.55 −0.310323
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 1856.4.a.z.1.1 5
4.3 odd 2 1856.4.a.ba.1.5 5
8.3 odd 2 464.4.a.m.1.1 5
8.5 even 2 232.4.a.d.1.5 5
24.5 odd 2 2088.4.a.f.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
232.4.a.d.1.5 5 8.5 even 2
464.4.a.m.1.1 5 8.3 odd 2
1856.4.a.z.1.1 5 1.1 even 1 trivial
1856.4.a.ba.1.5 5 4.3 odd 2
2088.4.a.f.1.4 5 24.5 odd 2