Properties

Label 8550.2.a.co.1.1
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.2720937282\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 950)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.91223\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -4.22547 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -4.22547 q^{7} +1.00000 q^{8} +5.13770 q^{11} -3.16784 q^{13} -4.22547 q^{14} +1.00000 q^{16} -6.48108 q^{17} -1.00000 q^{19} +5.13770 q^{22} +7.56885 q^{23} -3.16784 q^{26} -4.22547 q^{28} -0.832162 q^{29} -4.51122 q^{31} +1.00000 q^{32} -6.48108 q^{34} -0.137699 q^{37} -1.00000 q^{38} +11.6489 q^{41} +2.51122 q^{43} +5.13770 q^{44} +7.56885 q^{46} -5.96216 q^{47} +10.8546 q^{49} -3.16784 q^{52} +0.225470 q^{53} -4.22547 q^{56} -0.832162 q^{58} -5.39331 q^{59} +14.4509 q^{61} -4.51122 q^{62} +1.00000 q^{64} +4.11021 q^{67} -6.48108 q^{68} -3.82446 q^{71} -4.70655 q^{73} -0.137699 q^{74} -1.00000 q^{76} -21.7092 q^{77} +10.6265 q^{79} +11.6489 q^{82} -12.0999 q^{83} +2.51122 q^{86} +5.13770 q^{88} +10.0000 q^{89} +13.3856 q^{91} +7.56885 q^{92} -5.96216 q^{94} -3.93972 q^{97} +10.8546 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} - 2q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} - 2q^{7} + 3q^{8} - 2q^{11} + 2q^{13} - 2q^{14} + 3q^{16} - 4q^{17} - 3q^{19} - 2q^{22} + 14q^{23} + 2q^{26} - 2q^{28} - 14q^{29} - 4q^{31} + 3q^{32} - 4q^{34} + 17q^{37} - 3q^{38} + 8q^{41} - 2q^{43} - 2q^{44} + 14q^{46} + 13q^{47} + 25q^{49} + 2q^{52} - 10q^{53} - 2q^{56} - 14q^{58} + 6q^{59} + 22q^{61} - 4q^{62} + 3q^{64} - 4q^{68} + 2q^{71} + 12q^{73} + 17q^{74} - 3q^{76} - 50q^{77} + 24q^{79} + 8q^{82} + 12q^{83} - 2q^{86} - 2q^{88} + 30q^{89} - 7q^{91} + 14q^{92} + 13q^{94} + 25q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −4.22547 −1.59708 −0.798539 0.601943i \(-0.794393\pi\)
−0.798539 + 0.601943i \(0.794393\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 5.13770 1.54907 0.774537 0.632528i \(-0.217983\pi\)
0.774537 + 0.632528i \(0.217983\pi\)
\(12\) 0 0
\(13\) −3.16784 −0.878600 −0.439300 0.898340i \(-0.644774\pi\)
−0.439300 + 0.898340i \(0.644774\pi\)
\(14\) −4.22547 −1.12930
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.48108 −1.57189 −0.785946 0.618295i \(-0.787824\pi\)
−0.785946 + 0.618295i \(0.787824\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 5.13770 1.09536
\(23\) 7.56885 1.57821 0.789107 0.614256i \(-0.210544\pi\)
0.789107 + 0.614256i \(0.210544\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −3.16784 −0.621264
\(27\) 0 0
\(28\) −4.22547 −0.798539
\(29\) −0.832162 −0.154529 −0.0772643 0.997011i \(-0.524619\pi\)
−0.0772643 + 0.997011i \(0.524619\pi\)
\(30\) 0 0
\(31\) −4.51122 −0.810239 −0.405119 0.914264i \(-0.632770\pi\)
−0.405119 + 0.914264i \(0.632770\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −6.48108 −1.11150
\(35\) 0 0
\(36\) 0 0
\(37\) −0.137699 −0.0226376 −0.0113188 0.999936i \(-0.503603\pi\)
−0.0113188 + 0.999936i \(0.503603\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) 11.6489 1.81926 0.909628 0.415425i \(-0.136367\pi\)
0.909628 + 0.415425i \(0.136367\pi\)
\(42\) 0 0
\(43\) 2.51122 0.382957 0.191479 0.981497i \(-0.438672\pi\)
0.191479 + 0.981497i \(0.438672\pi\)
\(44\) 5.13770 0.774537
\(45\) 0 0
\(46\) 7.56885 1.11597
\(47\) −5.96216 −0.869670 −0.434835 0.900510i \(-0.643193\pi\)
−0.434835 + 0.900510i \(0.643193\pi\)
\(48\) 0 0
\(49\) 10.8546 1.55066
\(50\) 0 0
\(51\) 0 0
\(52\) −3.16784 −0.439300
\(53\) 0.225470 0.0309707 0.0154853 0.999880i \(-0.495071\pi\)
0.0154853 + 0.999880i \(0.495071\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.22547 −0.564652
\(57\) 0 0
\(58\) −0.832162 −0.109268
\(59\) −5.39331 −0.702149 −0.351074 0.936348i \(-0.614184\pi\)
−0.351074 + 0.936348i \(0.614184\pi\)
\(60\) 0 0
\(61\) 14.4509 1.85025 0.925127 0.379659i \(-0.123959\pi\)
0.925127 + 0.379659i \(0.123959\pi\)
\(62\) −4.51122 −0.572925
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.11021 0.502142 0.251071 0.967969i \(-0.419217\pi\)
0.251071 + 0.967969i \(0.419217\pi\)
\(68\) −6.48108 −0.785946
\(69\) 0 0
\(70\) 0 0
\(71\) −3.82446 −0.453880 −0.226940 0.973909i \(-0.572872\pi\)
−0.226940 + 0.973909i \(0.572872\pi\)
\(72\) 0 0
\(73\) −4.70655 −0.550860 −0.275430 0.961321i \(-0.588820\pi\)
−0.275430 + 0.961321i \(0.588820\pi\)
\(74\) −0.137699 −0.0160072
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −21.7092 −2.47399
\(78\) 0 0
\(79\) 10.6265 1.19557 0.597786 0.801655i \(-0.296047\pi\)
0.597786 + 0.801655i \(0.296047\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 11.6489 1.28641
\(83\) −12.0999 −1.32813 −0.664066 0.747674i \(-0.731171\pi\)
−0.664066 + 0.747674i \(0.731171\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.51122 0.270792
\(87\) 0 0
\(88\) 5.13770 0.547681
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 13.3856 1.40319
\(92\) 7.56885 0.789107
\(93\) 0 0
\(94\) −5.96216 −0.614950
\(95\) 0 0
\(96\) 0 0
\(97\) −3.93972 −0.400018 −0.200009 0.979794i \(-0.564097\pi\)
−0.200009 + 0.979794i \(0.564097\pi\)
\(98\) 10.8546 1.09648
\(99\) 0 0
\(100\) 0 0
\(101\) −3.19798 −0.318211 −0.159105 0.987262i \(-0.550861\pi\)
−0.159105 + 0.987262i \(0.550861\pi\)
\(102\) 0 0
\(103\) 10.6868 1.05300 0.526499 0.850176i \(-0.323504\pi\)
0.526499 + 0.850176i \(0.323504\pi\)
\(104\) −3.16784 −0.310632
\(105\) 0 0
\(106\) 0.225470 0.0218996
\(107\) 10.8168 1.04570 0.522848 0.852426i \(-0.324870\pi\)
0.522848 + 0.852426i \(0.324870\pi\)
\(108\) 0 0
\(109\) 9.24791 0.885789 0.442894 0.896574i \(-0.353952\pi\)
0.442894 + 0.896574i \(0.353952\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.22547 −0.399269
\(113\) 17.6489 1.66027 0.830135 0.557562i \(-0.188263\pi\)
0.830135 + 0.557562i \(0.188263\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.832162 −0.0772643
\(117\) 0 0
\(118\) −5.39331 −0.496494
\(119\) 27.3856 2.51043
\(120\) 0 0
\(121\) 15.3960 1.39963
\(122\) 14.4509 1.30833
\(123\) 0 0
\(124\) −4.51122 −0.405119
\(125\) 0 0
\(126\) 0 0
\(127\) 12.7866 1.13463 0.567314 0.823501i \(-0.307982\pi\)
0.567314 + 0.823501i \(0.307982\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 2.11526 0.184811 0.0924057 0.995721i \(-0.470544\pi\)
0.0924057 + 0.995721i \(0.470544\pi\)
\(132\) 0 0
\(133\) 4.22547 0.366395
\(134\) 4.11021 0.355068
\(135\) 0 0
\(136\) −6.48108 −0.555748
\(137\) −5.36317 −0.458206 −0.229103 0.973402i \(-0.573579\pi\)
−0.229103 + 0.973402i \(0.573579\pi\)
\(138\) 0 0
\(139\) −10.1601 −0.861771 −0.430886 0.902407i \(-0.641799\pi\)
−0.430886 + 0.902407i \(0.641799\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −3.82446 −0.320941
\(143\) −16.2754 −1.36102
\(144\) 0 0
\(145\) 0 0
\(146\) −4.70655 −0.389517
\(147\) 0 0
\(148\) −0.137699 −0.0113188
\(149\) 5.93972 0.486601 0.243301 0.969951i \(-0.421770\pi\)
0.243301 + 0.969951i \(0.421770\pi\)
\(150\) 0 0
\(151\) −15.2978 −1.24492 −0.622460 0.782652i \(-0.713867\pi\)
−0.622460 + 0.782652i \(0.713867\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) −21.7092 −1.74938
\(155\) 0 0
\(156\) 0 0
\(157\) 12.7866 1.02048 0.510242 0.860031i \(-0.329556\pi\)
0.510242 + 0.860031i \(0.329556\pi\)
\(158\) 10.6265 0.845397
\(159\) 0 0
\(160\) 0 0
\(161\) −31.9819 −2.52053
\(162\) 0 0
\(163\) 11.4734 0.898664 0.449332 0.893365i \(-0.351662\pi\)
0.449332 + 0.893365i \(0.351662\pi\)
\(164\) 11.6489 0.909628
\(165\) 0 0
\(166\) −12.0999 −0.939131
\(167\) 19.4131 1.50223 0.751115 0.660171i \(-0.229516\pi\)
0.751115 + 0.660171i \(0.229516\pi\)
\(168\) 0 0
\(169\) −2.96480 −0.228062
\(170\) 0 0
\(171\) 0 0
\(172\) 2.51122 0.191479
\(173\) 9.78662 0.744063 0.372031 0.928220i \(-0.378661\pi\)
0.372031 + 0.928220i \(0.378661\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.13770 0.387269
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) 6.82446 0.510084 0.255042 0.966930i \(-0.417911\pi\)
0.255042 + 0.966930i \(0.417911\pi\)
\(180\) 0 0
\(181\) −0.137699 −0.0102351 −0.00511755 0.999987i \(-0.501629\pi\)
−0.00511755 + 0.999987i \(0.501629\pi\)
\(182\) 13.3856 0.992207
\(183\) 0 0
\(184\) 7.56885 0.557983
\(185\) 0 0
\(186\) 0 0
\(187\) −33.2978 −2.43498
\(188\) −5.96216 −0.434835
\(189\) 0 0
\(190\) 0 0
\(191\) −19.3779 −1.40214 −0.701068 0.713095i \(-0.747293\pi\)
−0.701068 + 0.713095i \(0.747293\pi\)
\(192\) 0 0
\(193\) 5.42851 0.390752 0.195376 0.980728i \(-0.437407\pi\)
0.195376 + 0.980728i \(0.437407\pi\)
\(194\) −3.93972 −0.282856
\(195\) 0 0
\(196\) 10.8546 0.775328
\(197\) −15.6489 −1.11494 −0.557470 0.830197i \(-0.688228\pi\)
−0.557470 + 0.830197i \(0.688228\pi\)
\(198\) 0 0
\(199\) 18.0499 1.27953 0.639763 0.768572i \(-0.279033\pi\)
0.639763 + 0.768572i \(0.279033\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −3.19798 −0.225009
\(203\) 3.51628 0.246794
\(204\) 0 0
\(205\) 0 0
\(206\) 10.6868 0.744582
\(207\) 0 0
\(208\) −3.16784 −0.219650
\(209\) −5.13770 −0.355382
\(210\) 0 0
\(211\) −7.50857 −0.516911 −0.258456 0.966023i \(-0.583214\pi\)
−0.258456 + 0.966023i \(0.583214\pi\)
\(212\) 0.225470 0.0154853
\(213\) 0 0
\(214\) 10.8168 0.739418
\(215\) 0 0
\(216\) 0 0
\(217\) 19.0620 1.29401
\(218\) 9.24791 0.626347
\(219\) 0 0
\(220\) 0 0
\(221\) 20.5310 1.38106
\(222\) 0 0
\(223\) −27.8091 −1.86223 −0.931116 0.364723i \(-0.881164\pi\)
−0.931116 + 0.364723i \(0.881164\pi\)
\(224\) −4.22547 −0.282326
\(225\) 0 0
\(226\) 17.6489 1.17399
\(227\) 8.91223 0.591525 0.295763 0.955261i \(-0.404426\pi\)
0.295763 + 0.955261i \(0.404426\pi\)
\(228\) 0 0
\(229\) −13.6489 −0.901946 −0.450973 0.892538i \(-0.648923\pi\)
−0.450973 + 0.892538i \(0.648923\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −0.832162 −0.0546341
\(233\) 23.2754 1.52482 0.762411 0.647093i \(-0.224015\pi\)
0.762411 + 0.647093i \(0.224015\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −5.39331 −0.351074
\(237\) 0 0
\(238\) 27.3856 1.77515
\(239\) −3.72898 −0.241208 −0.120604 0.992701i \(-0.538483\pi\)
−0.120604 + 0.992701i \(0.538483\pi\)
\(240\) 0 0
\(241\) −1.48878 −0.0959009 −0.0479505 0.998850i \(-0.515269\pi\)
−0.0479505 + 0.998850i \(0.515269\pi\)
\(242\) 15.3960 0.989689
\(243\) 0 0
\(244\) 14.4509 0.925127
\(245\) 0 0
\(246\) 0 0
\(247\) 3.16784 0.201565
\(248\) −4.51122 −0.286463
\(249\) 0 0
\(250\) 0 0
\(251\) 8.78662 0.554606 0.277303 0.960782i \(-0.410559\pi\)
0.277303 + 0.960782i \(0.410559\pi\)
\(252\) 0 0
\(253\) 38.8865 2.44477
\(254\) 12.7866 0.802304
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.74175 0.171025 0.0855127 0.996337i \(-0.472747\pi\)
0.0855127 + 0.996337i \(0.472747\pi\)
\(258\) 0 0
\(259\) 0.581844 0.0361540
\(260\) 0 0
\(261\) 0 0
\(262\) 2.11526 0.130681
\(263\) 2.74704 0.169390 0.0846948 0.996407i \(-0.473008\pi\)
0.0846948 + 0.996407i \(0.473008\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.22547 0.259080
\(267\) 0 0
\(268\) 4.11021 0.251071
\(269\) 9.74175 0.593965 0.296982 0.954883i \(-0.404020\pi\)
0.296982 + 0.954883i \(0.404020\pi\)
\(270\) 0 0
\(271\) 26.3555 1.60098 0.800490 0.599346i \(-0.204573\pi\)
0.800490 + 0.599346i \(0.204573\pi\)
\(272\) −6.48108 −0.392973
\(273\) 0 0
\(274\) −5.36317 −0.324001
\(275\) 0 0
\(276\) 0 0
\(277\) 0.962158 0.0578104 0.0289052 0.999582i \(-0.490798\pi\)
0.0289052 + 0.999582i \(0.490798\pi\)
\(278\) −10.1601 −0.609364
\(279\) 0 0
\(280\) 0 0
\(281\) 14.6714 0.875219 0.437610 0.899165i \(-0.355825\pi\)
0.437610 + 0.899165i \(0.355825\pi\)
\(282\) 0 0
\(283\) 26.1601 1.55506 0.777529 0.628847i \(-0.216473\pi\)
0.777529 + 0.628847i \(0.216473\pi\)
\(284\) −3.82446 −0.226940
\(285\) 0 0
\(286\) −16.2754 −0.962384
\(287\) −49.2221 −2.90549
\(288\) 0 0
\(289\) 25.0044 1.47085
\(290\) 0 0
\(291\) 0 0
\(292\) −4.70655 −0.275430
\(293\) −22.4657 −1.31246 −0.656229 0.754562i \(-0.727850\pi\)
−0.656229 + 0.754562i \(0.727850\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −0.137699 −0.00800360
\(297\) 0 0
\(298\) 5.93972 0.344079
\(299\) −23.9769 −1.38662
\(300\) 0 0
\(301\) −10.6111 −0.611612
\(302\) −15.2978 −0.880291
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −17.2204 −0.982821 −0.491410 0.870928i \(-0.663518\pi\)
−0.491410 + 0.870928i \(0.663518\pi\)
\(308\) −21.7092 −1.23700
\(309\) 0 0
\(310\) 0 0
\(311\) 7.87439 0.446516 0.223258 0.974759i \(-0.428331\pi\)
0.223258 + 0.974759i \(0.428331\pi\)
\(312\) 0 0
\(313\) 25.1678 1.42257 0.711285 0.702904i \(-0.248113\pi\)
0.711285 + 0.702904i \(0.248113\pi\)
\(314\) 12.7866 0.721590
\(315\) 0 0
\(316\) 10.6265 0.597786
\(317\) −23.9045 −1.34261 −0.671306 0.741180i \(-0.734266\pi\)
−0.671306 + 0.741180i \(0.734266\pi\)
\(318\) 0 0
\(319\) −4.27540 −0.239376
\(320\) 0 0
\(321\) 0 0
\(322\) −31.9819 −1.78228
\(323\) 6.48108 0.360617
\(324\) 0 0
\(325\) 0 0
\(326\) 11.4734 0.635451
\(327\) 0 0
\(328\) 11.6489 0.643204
\(329\) 25.1929 1.38893
\(330\) 0 0
\(331\) 30.7565 1.69053 0.845264 0.534348i \(-0.179443\pi\)
0.845264 + 0.534348i \(0.179443\pi\)
\(332\) −12.0999 −0.664066
\(333\) 0 0
\(334\) 19.4131 1.06224
\(335\) 0 0
\(336\) 0 0
\(337\) −13.4734 −0.733942 −0.366971 0.930232i \(-0.619605\pi\)
−0.366971 + 0.930232i \(0.619605\pi\)
\(338\) −2.96480 −0.161264
\(339\) 0 0
\(340\) 0 0
\(341\) −23.1773 −1.25512
\(342\) 0 0
\(343\) −16.2875 −0.879441
\(344\) 2.51122 0.135396
\(345\) 0 0
\(346\) 9.78662 0.526132
\(347\) 28.9468 1.55394 0.776971 0.629536i \(-0.216755\pi\)
0.776971 + 0.629536i \(0.216755\pi\)
\(348\) 0 0
\(349\) 27.9243 1.49475 0.747377 0.664400i \(-0.231313\pi\)
0.747377 + 0.664400i \(0.231313\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.13770 0.273840
\(353\) 28.3099 1.50678 0.753392 0.657571i \(-0.228416\pi\)
0.753392 + 0.657571i \(0.228416\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 6.82446 0.360684
\(359\) −2.60163 −0.137309 −0.0686545 0.997640i \(-0.521871\pi\)
−0.0686545 + 0.997640i \(0.521871\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −0.137699 −0.00723731
\(363\) 0 0
\(364\) 13.3856 0.701596
\(365\) 0 0
\(366\) 0 0
\(367\) −11.6489 −0.608069 −0.304034 0.952661i \(-0.598334\pi\)
−0.304034 + 0.952661i \(0.598334\pi\)
\(368\) 7.56885 0.394554
\(369\) 0 0
\(370\) 0 0
\(371\) −0.952717 −0.0494626
\(372\) 0 0
\(373\) 12.8064 0.663091 0.331545 0.943439i \(-0.392430\pi\)
0.331545 + 0.943439i \(0.392430\pi\)
\(374\) −33.2978 −1.72179
\(375\) 0 0
\(376\) −5.96216 −0.307475
\(377\) 2.63615 0.135769
\(378\) 0 0
\(379\) −20.9122 −1.07419 −0.537095 0.843522i \(-0.680478\pi\)
−0.537095 + 0.843522i \(0.680478\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −19.3779 −0.991460
\(383\) 10.3511 0.528916 0.264458 0.964397i \(-0.414807\pi\)
0.264458 + 0.964397i \(0.414807\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5.42851 0.276304
\(387\) 0 0
\(388\) −3.93972 −0.200009
\(389\) −6.56620 −0.332920 −0.166460 0.986048i \(-0.553234\pi\)
−0.166460 + 0.986048i \(0.553234\pi\)
\(390\) 0 0
\(391\) −49.0543 −2.48078
\(392\) 10.8546 0.548240
\(393\) 0 0
\(394\) −15.6489 −0.788381
\(395\) 0 0
\(396\) 0 0
\(397\) −23.5284 −1.18085 −0.590427 0.807091i \(-0.701041\pi\)
−0.590427 + 0.807091i \(0.701041\pi\)
\(398\) 18.0499 0.904761
\(399\) 0 0
\(400\) 0 0
\(401\) −6.22041 −0.310633 −0.155316 0.987865i \(-0.549640\pi\)
−0.155316 + 0.987865i \(0.549640\pi\)
\(402\) 0 0
\(403\) 14.2908 0.711876
\(404\) −3.19798 −0.159105
\(405\) 0 0
\(406\) 3.51628 0.174510
\(407\) −0.707457 −0.0350674
\(408\) 0 0
\(409\) −34.4905 −1.70545 −0.852723 0.522363i \(-0.825051\pi\)
−0.852723 + 0.522363i \(0.825051\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 10.6868 0.526499
\(413\) 22.7893 1.12139
\(414\) 0 0
\(415\) 0 0
\(416\) −3.16784 −0.155316
\(417\) 0 0
\(418\) −5.13770 −0.251293
\(419\) 10.6265 0.519138 0.259569 0.965725i \(-0.416420\pi\)
0.259569 + 0.965725i \(0.416420\pi\)
\(420\) 0 0
\(421\) −1.98021 −0.0965095 −0.0482548 0.998835i \(-0.515366\pi\)
−0.0482548 + 0.998835i \(0.515366\pi\)
\(422\) −7.50857 −0.365512
\(423\) 0 0
\(424\) 0.225470 0.0109498
\(425\) 0 0
\(426\) 0 0
\(427\) −61.0620 −2.95500
\(428\) 10.8168 0.522848
\(429\) 0 0
\(430\) 0 0
\(431\) −15.0774 −0.726254 −0.363127 0.931740i \(-0.618291\pi\)
−0.363127 + 0.931740i \(0.618291\pi\)
\(432\) 0 0
\(433\) −13.5337 −0.650386 −0.325193 0.945648i \(-0.605429\pi\)
−0.325193 + 0.945648i \(0.605429\pi\)
\(434\) 19.0620 0.915006
\(435\) 0 0
\(436\) 9.24791 0.442894
\(437\) −7.56885 −0.362067
\(438\) 0 0
\(439\) 39.1773 1.86983 0.934915 0.354872i \(-0.115476\pi\)
0.934915 + 0.354872i \(0.115476\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 20.5310 0.976560
\(443\) 19.3132 0.917600 0.458800 0.888540i \(-0.348279\pi\)
0.458800 + 0.888540i \(0.348279\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −27.8091 −1.31680
\(447\) 0 0
\(448\) −4.22547 −0.199635
\(449\) −41.4131 −1.95440 −0.977202 0.212310i \(-0.931901\pi\)
−0.977202 + 0.212310i \(0.931901\pi\)
\(450\) 0 0
\(451\) 59.8486 2.81816
\(452\) 17.6489 0.830135
\(453\) 0 0
\(454\) 8.91223 0.418272
\(455\) 0 0
\(456\) 0 0
\(457\) 37.8392 1.77004 0.885021 0.465551i \(-0.154144\pi\)
0.885021 + 0.465551i \(0.154144\pi\)
\(458\) −13.6489 −0.637772
\(459\) 0 0
\(460\) 0 0
\(461\) −1.58864 −0.0739903 −0.0369952 0.999315i \(-0.511779\pi\)
−0.0369952 + 0.999315i \(0.511779\pi\)
\(462\) 0 0
\(463\) −40.3581 −1.87560 −0.937800 0.347175i \(-0.887141\pi\)
−0.937800 + 0.347175i \(0.887141\pi\)
\(464\) −0.832162 −0.0386322
\(465\) 0 0
\(466\) 23.2754 1.07821
\(467\) −39.5130 −1.82844 −0.914221 0.405217i \(-0.867196\pi\)
−0.914221 + 0.405217i \(0.867196\pi\)
\(468\) 0 0
\(469\) −17.3676 −0.801959
\(470\) 0 0
\(471\) 0 0
\(472\) −5.39331 −0.248247
\(473\) 12.9019 0.593229
\(474\) 0 0
\(475\) 0 0
\(476\) 27.3856 1.25522
\(477\) 0 0
\(478\) −3.72898 −0.170560
\(479\) −7.61107 −0.347759 −0.173879 0.984767i \(-0.555630\pi\)
−0.173879 + 0.984767i \(0.555630\pi\)
\(480\) 0 0
\(481\) 0.436209 0.0198894
\(482\) −1.48878 −0.0678122
\(483\) 0 0
\(484\) 15.3960 0.699816
\(485\) 0 0
\(486\) 0 0
\(487\) 20.3907 0.923989 0.461995 0.886883i \(-0.347134\pi\)
0.461995 + 0.886883i \(0.347134\pi\)
\(488\) 14.4509 0.654163
\(489\) 0 0
\(490\) 0 0
\(491\) −25.0224 −1.12925 −0.564623 0.825349i \(-0.690979\pi\)
−0.564623 + 0.825349i \(0.690979\pi\)
\(492\) 0 0
\(493\) 5.39331 0.242902
\(494\) 3.16784 0.142528
\(495\) 0 0
\(496\) −4.51122 −0.202560
\(497\) 16.1601 0.724881
\(498\) 0 0
\(499\) 22.6111 1.01221 0.506105 0.862472i \(-0.331085\pi\)
0.506105 + 0.862472i \(0.331085\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 8.78662 0.392166
\(503\) 11.5035 0.512916 0.256458 0.966555i \(-0.417444\pi\)
0.256458 + 0.966555i \(0.417444\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 38.8865 1.72871
\(507\) 0 0
\(508\) 12.7866 0.567314
\(509\) −9.11526 −0.404027 −0.202013 0.979383i \(-0.564748\pi\)
−0.202013 + 0.979383i \(0.564748\pi\)
\(510\) 0 0
\(511\) 19.8874 0.879766
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 2.74175 0.120933
\(515\) 0 0
\(516\) 0 0
\(517\) −30.6318 −1.34718
\(518\) 0.581844 0.0255648
\(519\) 0 0
\(520\) 0 0
\(521\) −15.4888 −0.678576 −0.339288 0.940683i \(-0.610186\pi\)
−0.339288 + 0.940683i \(0.610186\pi\)
\(522\) 0 0
\(523\) −4.34073 −0.189807 −0.0949035 0.995486i \(-0.530254\pi\)
−0.0949035 + 0.995486i \(0.530254\pi\)
\(524\) 2.11526 0.0924057
\(525\) 0 0
\(526\) 2.74704 0.119776
\(527\) 29.2376 1.27361
\(528\) 0 0
\(529\) 34.2875 1.49076
\(530\) 0 0
\(531\) 0 0
\(532\) 4.22547 0.183197
\(533\) −36.9019 −1.59840
\(534\) 0 0
\(535\) 0 0
\(536\) 4.11021 0.177534
\(537\) 0 0
\(538\) 9.74175 0.419996
\(539\) 55.7677 2.40208
\(540\) 0 0
\(541\) 19.5491 0.840480 0.420240 0.907413i \(-0.361946\pi\)
0.420240 + 0.907413i \(0.361946\pi\)
\(542\) 26.3555 1.13206
\(543\) 0 0
\(544\) −6.48108 −0.277874
\(545\) 0 0
\(546\) 0 0
\(547\) 41.8038 1.78740 0.893700 0.448665i \(-0.148100\pi\)
0.893700 + 0.448665i \(0.148100\pi\)
\(548\) −5.36317 −0.229103
\(549\) 0 0
\(550\) 0 0
\(551\) 0.832162 0.0354513
\(552\) 0 0
\(553\) −44.9019 −1.90942
\(554\) 0.962158 0.0408782
\(555\) 0 0
\(556\) −10.1601 −0.430886
\(557\) −9.76418 −0.413722 −0.206861 0.978370i \(-0.566325\pi\)
−0.206861 + 0.978370i \(0.566325\pi\)
\(558\) 0 0
\(559\) −7.95513 −0.336466
\(560\) 0 0
\(561\) 0 0
\(562\) 14.6714 0.618874
\(563\) 11.4509 0.482600 0.241300 0.970451i \(-0.422426\pi\)
0.241300 + 0.970451i \(0.422426\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 26.1601 1.09959
\(567\) 0 0
\(568\) −3.82446 −0.160471
\(569\) 13.4338 0.563174 0.281587 0.959536i \(-0.409139\pi\)
0.281587 + 0.959536i \(0.409139\pi\)
\(570\) 0 0
\(571\) 37.6335 1.57491 0.787457 0.616370i \(-0.211397\pi\)
0.787457 + 0.616370i \(0.211397\pi\)
\(572\) −16.2754 −0.680509
\(573\) 0 0
\(574\) −49.2221 −2.05449
\(575\) 0 0
\(576\) 0 0
\(577\) −1.56885 −0.0653121 −0.0326560 0.999467i \(-0.510397\pi\)
−0.0326560 + 0.999467i \(0.510397\pi\)
\(578\) 25.0044 1.04005
\(579\) 0 0
\(580\) 0 0
\(581\) 51.1276 2.12113
\(582\) 0 0
\(583\) 1.15840 0.0479759
\(584\) −4.70655 −0.194758
\(585\) 0 0
\(586\) −22.4657 −0.928048
\(587\) 25.8693 1.06774 0.533871 0.845566i \(-0.320737\pi\)
0.533871 + 0.845566i \(0.320737\pi\)
\(588\) 0 0
\(589\) 4.51122 0.185881
\(590\) 0 0
\(591\) 0 0
\(592\) −0.137699 −0.00565940
\(593\) −28.9243 −1.18778 −0.593890 0.804547i \(-0.702408\pi\)
−0.593890 + 0.804547i \(0.702408\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 5.93972 0.243301
\(597\) 0 0
\(598\) −23.9769 −0.980488
\(599\) −41.3581 −1.68985 −0.844923 0.534887i \(-0.820354\pi\)
−0.844923 + 0.534887i \(0.820354\pi\)
\(600\) 0 0
\(601\) 2.68147 0.109379 0.0546897 0.998503i \(-0.482583\pi\)
0.0546897 + 0.998503i \(0.482583\pi\)
\(602\) −10.6111 −0.432475
\(603\) 0 0
\(604\) −15.2978 −0.622460
\(605\) 0 0
\(606\) 0 0
\(607\) −3.31324 −0.134480 −0.0672401 0.997737i \(-0.521419\pi\)
−0.0672401 + 0.997737i \(0.521419\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) 18.8871 0.764092
\(612\) 0 0
\(613\) 10.0603 0.406331 0.203165 0.979144i \(-0.434877\pi\)
0.203165 + 0.979144i \(0.434877\pi\)
\(614\) −17.2204 −0.694959
\(615\) 0 0
\(616\) −21.7092 −0.874688
\(617\) −27.6265 −1.11220 −0.556100 0.831115i \(-0.687703\pi\)
−0.556100 + 0.831115i \(0.687703\pi\)
\(618\) 0 0
\(619\) 16.6714 0.670078 0.335039 0.942204i \(-0.391250\pi\)
0.335039 + 0.942204i \(0.391250\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 7.87439 0.315734
\(623\) −42.2547 −1.69290
\(624\) 0 0
\(625\) 0 0
\(626\) 25.1678 1.00591
\(627\) 0 0
\(628\) 12.7866 0.510242
\(629\) 0.892439 0.0355839
\(630\) 0 0
\(631\) −0.709194 −0.0282326 −0.0141163 0.999900i \(-0.504494\pi\)
−0.0141163 + 0.999900i \(0.504494\pi\)
\(632\) 10.6265 0.422699
\(633\) 0 0
\(634\) −23.9045 −0.949370
\(635\) 0 0
\(636\) 0 0
\(637\) −34.3856 −1.36241
\(638\) −4.27540 −0.169265
\(639\) 0 0
\(640\) 0 0
\(641\) 2.43553 0.0961978 0.0480989 0.998843i \(-0.484684\pi\)
0.0480989 + 0.998843i \(0.484684\pi\)
\(642\) 0 0
\(643\) −7.70390 −0.303812 −0.151906 0.988395i \(-0.548541\pi\)
−0.151906 + 0.988395i \(0.548541\pi\)
\(644\) −31.9819 −1.26027
\(645\) 0 0
\(646\) 6.48108 0.254995
\(647\) −10.0499 −0.395103 −0.197552 0.980292i \(-0.563299\pi\)
−0.197552 + 0.980292i \(0.563299\pi\)
\(648\) 0 0
\(649\) −27.7092 −1.08768
\(650\) 0 0
\(651\) 0 0
\(652\) 11.4734 0.449332
\(653\) −7.09283 −0.277564 −0.138782 0.990323i \(-0.544319\pi\)
−0.138782 + 0.990323i \(0.544319\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 11.6489 0.454814
\(657\) 0 0
\(658\) 25.1929 0.982122
\(659\) 1.16346 0.0453218 0.0226609 0.999743i \(-0.492786\pi\)
0.0226609 + 0.999743i \(0.492786\pi\)
\(660\) 0 0
\(661\) 1.79432 0.0697909 0.0348955 0.999391i \(-0.488890\pi\)
0.0348955 + 0.999391i \(0.488890\pi\)
\(662\) 30.7565 1.19538
\(663\) 0 0
\(664\) −12.0999 −0.469566
\(665\) 0 0
\(666\) 0 0
\(667\) −6.29851 −0.243879
\(668\) 19.4131 0.751115
\(669\) 0 0
\(670\) 0 0
\(671\) 74.2446 2.86618
\(672\) 0 0
\(673\) 25.2824 0.974566 0.487283 0.873244i \(-0.337988\pi\)
0.487283 + 0.873244i \(0.337988\pi\)
\(674\) −13.4734 −0.518975
\(675\) 0 0
\(676\) −2.96480 −0.114031
\(677\) −4.89682 −0.188200 −0.0941001 0.995563i \(-0.529997\pi\)
−0.0941001 + 0.995563i \(0.529997\pi\)
\(678\) 0 0
\(679\) 16.6472 0.638860
\(680\) 0 0
\(681\) 0 0
\(682\) −23.1773 −0.887504
\(683\) 10.1980 0.390215 0.195107 0.980782i \(-0.437494\pi\)
0.195107 + 0.980782i \(0.437494\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −16.2875 −0.621859
\(687\) 0 0
\(688\) 2.51122 0.0957393
\(689\) −0.714253 −0.0272109
\(690\) 0 0
\(691\) −39.9846 −1.52109 −0.760543 0.649288i \(-0.775067\pi\)
−0.760543 + 0.649288i \(0.775067\pi\)
\(692\) 9.78662 0.372031
\(693\) 0 0
\(694\) 28.9468 1.09880
\(695\) 0 0
\(696\) 0 0
\(697\) −75.4975 −2.85967
\(698\) 27.9243 1.05695
\(699\) 0 0
\(700\) 0 0
\(701\) −4.91729 −0.185723 −0.0928617 0.995679i \(-0.529601\pi\)
−0.0928617 + 0.995679i \(0.529601\pi\)
\(702\) 0 0
\(703\) 0.137699 0.00519342
\(704\) 5.13770 0.193634
\(705\) 0 0
\(706\) 28.3099 1.06546
\(707\) 13.5130 0.508207
\(708\) 0 0
\(709\) −36.8865 −1.38530 −0.692650 0.721274i \(-0.743557\pi\)
−0.692650 + 0.721274i \(0.743557\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 10.0000 0.374766
\(713\) −34.1447 −1.27873
\(714\) 0 0
\(715\) 0 0
\(716\) 6.82446 0.255042
\(717\) 0 0
\(718\) −2.60163 −0.0970921
\(719\) 23.8891 0.890914 0.445457 0.895303i \(-0.353041\pi\)
0.445457 + 0.895303i \(0.353041\pi\)
\(720\) 0 0
\(721\) −45.1566 −1.68172
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) −0.137699 −0.00511755
\(725\) 0 0
\(726\) 0 0
\(727\) −19.6894 −0.730240 −0.365120 0.930961i \(-0.618972\pi\)
−0.365120 + 0.930961i \(0.618972\pi\)
\(728\) 13.3856 0.496104
\(729\) 0 0
\(730\) 0 0
\(731\) −16.2754 −0.601967
\(732\) 0 0
\(733\) −1.76418 −0.0651615 −0.0325808 0.999469i \(-0.510373\pi\)
−0.0325808 + 0.999469i \(0.510373\pi\)
\(734\) −11.6489 −0.429969
\(735\) 0 0
\(736\) 7.56885 0.278991
\(737\) 21.1170 0.777855
\(738\) 0 0
\(739\) 28.1755 1.03645 0.518227 0.855243i \(-0.326592\pi\)
0.518227 + 0.855243i \(0.326592\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.952717 −0.0349753
\(743\) −5.42851 −0.199153 −0.0995763 0.995030i \(-0.531749\pi\)
−0.0995763 + 0.995030i \(0.531749\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 12.8064 0.468876
\(747\) 0 0
\(748\) −33.2978 −1.21749
\(749\) −45.7059 −1.67006
\(750\) 0 0
\(751\) 25.8640 0.943792 0.471896 0.881654i \(-0.343570\pi\)
0.471896 + 0.881654i \(0.343570\pi\)
\(752\) −5.96216 −0.217418
\(753\) 0 0
\(754\) 2.63615 0.0960031
\(755\) 0 0
\(756\) 0 0
\(757\) 3.76947 0.137004 0.0685019 0.997651i \(-0.478178\pi\)
0.0685019 + 0.997651i \(0.478178\pi\)
\(758\) −20.9122 −0.759566
\(759\) 0 0
\(760\) 0 0
\(761\) −18.3605 −0.665568 −0.332784 0.943003i \(-0.607988\pi\)
−0.332784 + 0.943003i \(0.607988\pi\)
\(762\) 0 0
\(763\) −39.0767 −1.41467
\(764\) −19.3779 −0.701068
\(765\) 0 0
\(766\) 10.3511 0.374000
\(767\) 17.0851 0.616908
\(768\) 0 0
\(769\) −38.1300 −1.37500 −0.687501 0.726183i \(-0.741292\pi\)
−0.687501 + 0.726183i \(0.741292\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5.42851 0.195376
\(773\) −29.1575 −1.04872 −0.524361 0.851496i \(-0.675696\pi\)
−0.524361 + 0.851496i \(0.675696\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −3.93972 −0.141428
\(777\) 0 0
\(778\) −6.56620 −0.235410
\(779\) −11.6489 −0.417366
\(780\) 0 0
\(781\) −19.6489 −0.703094
\(782\) −49.0543 −1.75418
\(783\) 0 0
\(784\) 10.8546 0.387664
\(785\) 0 0
\(786\) 0 0
\(787\) 47.0165 1.67596 0.837978 0.545704i \(-0.183738\pi\)
0.837978 + 0.545704i \(0.183738\pi\)
\(788\) −15.6489 −0.557470
\(789\) 0 0
\(790\) 0 0
\(791\) −74.5750 −2.65158
\(792\) 0 0
\(793\) −45.7782 −1.62563
\(794\) −23.5284 −0.834990
\(795\) 0 0
\(796\) 18.0499 0.639763
\(797\) −42.6359 −1.51024 −0.755121 0.655586i \(-0.772422\pi\)
−0.755121 + 0.655586i \(0.772422\pi\)
\(798\) 0 0
\(799\) 38.6412 1.36703
\(800\) 0 0
\(801\) 0 0
\(802\) −6.22041 −0.219650
\(803\) −24.1808 −0.853323
\(804\) 0 0
\(805\) 0 0
\(806\) 14.2908 0.503372
\(807\) 0 0
\(808\) −3.19798 −0.112504
\(809\) −49.4630 −1.73903 −0.869514 0.493909i \(-0.835568\pi\)
−0.869514 + 0.493909i \(0.835568\pi\)
\(810\) 0 0
\(811\) −16.7816 −0.589280 −0.294640 0.955608i \(-0.595200\pi\)
−0.294640 + 0.955608i \(0.595200\pi\)
\(812\) 3.51628 0.123397
\(813\) 0 0
\(814\) −0.707457 −0.0247964
\(815\) 0 0
\(816\) 0 0
\(817\) −2.51122 −0.0878564
\(818\) −34.4905 −1.20593
\(819\) 0 0
\(820\) 0 0
\(821\) −11.5337 −0.402527 −0.201264 0.979537i \(-0.564505\pi\)
−0.201264 + 0.979537i \(0.564505\pi\)
\(822\) 0 0
\(823\) −32.6309 −1.13744 −0.568720 0.822531i \(-0.692561\pi\)
−0.568720 + 0.822531i \(0.692561\pi\)
\(824\) 10.6868 0.372291
\(825\) 0 0
\(826\) 22.7893 0.792940
\(827\) 6.64650 0.231122 0.115561 0.993300i \(-0.463133\pi\)
0.115561 + 0.993300i \(0.463133\pi\)
\(828\) 0 0
\(829\) −30.9217 −1.07395 −0.536977 0.843597i \(-0.680434\pi\)
−0.536977 + 0.843597i \(0.680434\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −3.16784 −0.109825
\(833\) −70.3495 −2.43747
\(834\) 0 0
\(835\) 0 0
\(836\) −5.13770 −0.177691
\(837\) 0 0
\(838\) 10.6265 0.367086
\(839\) 48.7109 1.68169 0.840844 0.541277i \(-0.182059\pi\)
0.840844 + 0.541277i \(0.182059\pi\)
\(840\) 0 0
\(841\) −28.3075 −0.976121
\(842\) −1.98021 −0.0682426
\(843\) 0 0
\(844\) −7.50857 −0.258456
\(845\) 0 0
\(846\) 0 0
\(847\) −65.0551 −2.23532
\(848\) 0.225470 0.00774267
\(849\) 0 0
\(850\) 0 0
\(851\) −1.04222 −0.0357270
\(852\) 0 0
\(853\) −46.1447 −1.57997 −0.789983 0.613129i \(-0.789911\pi\)
−0.789983 + 0.613129i \(0.789911\pi\)
\(854\) −61.0620 −2.08950
\(855\) 0 0
\(856\) 10.8168 0.369709
\(857\) −8.40607 −0.287146 −0.143573 0.989640i \(-0.545859\pi\)
−0.143573 + 0.989640i \(0.545859\pi\)
\(858\) 0 0
\(859\) −8.49581 −0.289873 −0.144937 0.989441i \(-0.546298\pi\)
−0.144937 + 0.989441i \(0.546298\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −15.0774 −0.513539
\(863\) 3.37352 0.114836 0.0574179 0.998350i \(-0.481713\pi\)
0.0574179 + 0.998350i \(0.481713\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −13.5337 −0.459892
\(867\) 0 0
\(868\) 19.0620 0.647007
\(869\) 54.5957 1.85203
\(870\) 0 0
\(871\) −13.0205 −0.441182
\(872\) 9.24791 0.313174
\(873\) 0 0
\(874\) −7.56885 −0.256020
\(875\) 0 0
\(876\) 0 0
\(877\) 13.2780 0.448368 0.224184 0.974547i \(-0.428028\pi\)
0.224184 + 0.974547i \(0.428028\pi\)
\(878\) 39.1773 1.32217
\(879\) 0 0
\(880\) 0 0
\(881\) 26.6489 0.897825 0.448912 0.893576i \(-0.351812\pi\)
0.448912 + 0.893576i \(0.351812\pi\)
\(882\) 0 0
\(883\) 47.7884 1.60821 0.804103 0.594490i \(-0.202646\pi\)
0.804103 + 0.594490i \(0.202646\pi\)
\(884\) 20.5310 0.690532
\(885\) 0 0
\(886\) 19.3132 0.648841
\(887\) −36.3304 −1.21985 −0.609927 0.792457i \(-0.708801\pi\)
−0.609927 + 0.792457i \(0.708801\pi\)
\(888\) 0 0
\(889\) −54.0295 −1.81209
\(890\) 0 0
\(891\) 0 0
\(892\) −27.8091 −0.931116
\(893\) 5.96216 0.199516
\(894\) 0 0
\(895\) 0 0
\(896\) −4.22547 −0.141163
\(897\) 0 0
\(898\) −41.4131 −1.38197
\(899\) 3.75406 0.125205
\(900\) 0 0
\(901\) −1.46129 −0.0486826
\(902\) 59.8486 1.99274
\(903\) 0 0
\(904\) 17.6489 0.586994
\(905\) 0 0
\(906\) 0 0
\(907\) 39.3873 1.30784 0.653918 0.756566i \(-0.273124\pi\)
0.653918 + 0.756566i \(0.273124\pi\)
\(908\) 8.91223 0.295763
\(909\) 0 0
\(910\) 0 0
\(911\) 20.5561 0.681054 0.340527 0.940235i \(-0.389395\pi\)
0.340527 + 0.940235i \(0.389395\pi\)
\(912\) 0 0
\(913\) −62.1654 −2.05738
\(914\) 37.8392 1.25161
\(915\) 0 0
\(916\) −13.6489 −0.450973
\(917\) −8.93799 −0.295158
\(918\) 0 0
\(919\) 12.4054 0.409216 0.204608 0.978844i \(-0.434408\pi\)
0.204608 + 0.978844i \(0.434408\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1.58864 −0.0523190
\(923\) 12.1153 0.398779
\(924\) 0 0
\(925\) 0 0
\(926\) −40.3581 −1.32625
\(927\) 0 0
\(928\) −0.832162 −0.0273171
\(929\) 31.3330 1.02800 0.514002 0.857789i \(-0.328162\pi\)
0.514002 + 0.857789i \(0.328162\pi\)
\(930\) 0 0
\(931\) −10.8546 −0.355745
\(932\) 23.2754 0.762411
\(933\) 0 0
\(934\) −39.5130 −1.29290
\(935\) 0 0
\(936\) 0 0
\(937\) −7.27299 −0.237598 −0.118799 0.992918i \(-0.537904\pi\)
−0.118799 + 0.992918i \(0.537904\pi\)
\(938\) −17.3676 −0.567071
\(939\) 0 0
\(940\) 0 0
\(941\) −6.72128 −0.219107 −0.109554 0.993981i \(-0.534942\pi\)
−0.109554 + 0.993981i \(0.534942\pi\)
\(942\) 0 0
\(943\) 88.1689 2.87117
\(944\) −5.39331 −0.175537
\(945\) 0 0
\(946\) 12.9019 0.419476
\(947\) 10.0757 0.327416 0.163708 0.986509i \(-0.447655\pi\)
0.163708 + 0.986509i \(0.447655\pi\)
\(948\) 0 0
\(949\) 14.9096 0.483986
\(950\) 0 0
\(951\) 0 0
\(952\) 27.3856 0.887573
\(953\) −8.14473 −0.263834 −0.131917 0.991261i \(-0.542113\pi\)
−0.131917 + 0.991261i \(0.542113\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −3.72898 −0.120604
\(957\) 0 0
\(958\) −7.61107 −0.245903
\(959\) 22.6619 0.731791
\(960\) 0 0
\(961\) −10.6489 −0.343513
\(962\) 0.436209 0.0140639
\(963\) 0 0
\(964\) −1.48878 −0.0479505
\(965\) 0 0
\(966\) 0 0
\(967\) −47.1773 −1.51712 −0.758560 0.651604i \(-0.774097\pi\)
−0.758560 + 0.651604i \(0.774097\pi\)
\(968\) 15.3960 0.494845
\(969\) 0 0
\(970\) 0 0
\(971\) −0.230528 −0.00739801 −0.00369901 0.999993i \(-0.501177\pi\)
−0.00369901 + 0.999993i \(0.501177\pi\)
\(972\) 0 0
\(973\) 42.9313 1.37632
\(974\) 20.3907 0.653359
\(975\) 0 0
\(976\) 14.4509 0.462563
\(977\) −5.80905 −0.185848 −0.0929240 0.995673i \(-0.529621\pi\)
−0.0929240 + 0.995673i \(0.529621\pi\)
\(978\) 0 0
\(979\) 51.3770 1.64202
\(980\) 0 0
\(981\) 0 0
\(982\) −25.0224 −0.798498
\(983\) −22.3511 −0.712889 −0.356444 0.934317i \(-0.616011\pi\)
−0.356444 + 0.934317i \(0.616011\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 5.39331 0.171758
\(987\) 0 0
\(988\) 3.16784 0.100782
\(989\) 19.0070 0.604388
\(990\) 0 0
\(991\) −34.9415 −1.10995 −0.554976 0.831866i \(-0.687273\pi\)
−0.554976 + 0.831866i \(0.687273\pi\)
\(992\) −4.51122 −0.143231
\(993\) 0 0
\(994\) 16.1601 0.512568
\(995\) 0 0
\(996\) 0 0
\(997\) 9.35282 0.296207 0.148103 0.988972i \(-0.452683\pi\)
0.148103 + 0.988972i \(0.452683\pi\)
\(998\) 22.6111 0.715741
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8550.2.a.co.1.1 3
3.2 odd 2 950.2.a.k.1.2 3
5.4 even 2 8550.2.a.cj.1.3 3
12.11 even 2 7600.2.a.cb.1.2 3
15.2 even 4 950.2.b.g.799.2 6
15.8 even 4 950.2.b.g.799.5 6
15.14 odd 2 950.2.a.m.1.2 yes 3
60.59 even 2 7600.2.a.bm.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.k.1.2 3 3.2 odd 2
950.2.a.m.1.2 yes 3 15.14 odd 2
950.2.b.g.799.2 6 15.2 even 4
950.2.b.g.799.5 6 15.8 even 4
7600.2.a.bm.1.2 3 60.59 even 2
7600.2.a.cb.1.2 3 12.11 even 2
8550.2.a.cj.1.3 3 5.4 even 2
8550.2.a.co.1.1 3 1.1 even 1 trivial