Properties

Label 2352.4.a.ck.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.145408.2
Defining polynomial: \(x^{4} - 24 x^{2} + 142\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1176)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.66254\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -16.6019 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -16.6019 q^{5} +9.00000 q^{9} +49.2481 q^{11} +64.4774 q^{13} +49.8056 q^{15} +132.429 q^{17} +82.0605 q^{19} -82.0466 q^{23} +150.622 q^{25} -27.0000 q^{27} +157.717 q^{29} +185.291 q^{31} -147.744 q^{33} -51.9002 q^{37} -193.432 q^{39} -49.4929 q^{41} -313.788 q^{43} -149.417 q^{45} +553.259 q^{47} -397.287 q^{51} -619.868 q^{53} -817.611 q^{55} -246.181 q^{57} +712.989 q^{59} -287.421 q^{61} -1070.45 q^{65} -226.636 q^{67} +246.140 q^{69} -55.3834 q^{71} +799.071 q^{73} -451.867 q^{75} +120.313 q^{79} +81.0000 q^{81} +857.862 q^{83} -2198.57 q^{85} -473.151 q^{87} -377.167 q^{89} -555.873 q^{93} -1362.36 q^{95} -1265.20 q^{97} +443.233 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{3} - 8q^{5} + 36q^{9} + O(q^{10}) \) \( 4q - 12q^{3} - 8q^{5} + 36q^{9} + 40q^{11} - 48q^{13} + 24q^{15} + 152q^{17} + 224q^{19} + 8q^{23} - 28q^{25} - 108q^{27} - 144q^{29} + 400q^{31} - 120q^{33} - 304q^{37} + 144q^{39} + 152q^{41} - 160q^{43} - 72q^{45} + 544q^{47} - 456q^{51} - 1320q^{53} + 16q^{55} - 672q^{57} + 1040q^{59} - 896q^{61} - 648q^{65} + 416q^{67} - 24q^{69} - 248q^{71} + 752q^{73} + 84q^{75} - 864q^{79} + 324q^{81} + 1456q^{83} - 1608q^{85} + 432q^{87} - 2936q^{89} - 1200q^{93} + 80q^{95} - 144q^{97} + 360q^{99} + O(q^{100}) \)

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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −16.6019 −1.48492 −0.742458 0.669892i \(-0.766340\pi\)
−0.742458 + 0.669892i \(0.766340\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 49.2481 1.34990 0.674949 0.737865i \(-0.264166\pi\)
0.674949 + 0.737865i \(0.264166\pi\)
\(12\) 0 0
\(13\) 64.4774 1.37560 0.687801 0.725900i \(-0.258576\pi\)
0.687801 + 0.725900i \(0.258576\pi\)
\(14\) 0 0
\(15\) 49.8056 0.857317
\(16\) 0 0
\(17\) 132.429 1.88934 0.944669 0.328024i \(-0.106383\pi\)
0.944669 + 0.328024i \(0.106383\pi\)
\(18\) 0 0
\(19\) 82.0605 0.990840 0.495420 0.868654i \(-0.335014\pi\)
0.495420 + 0.868654i \(0.335014\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −82.0466 −0.743822 −0.371911 0.928268i \(-0.621297\pi\)
−0.371911 + 0.928268i \(0.621297\pi\)
\(24\) 0 0
\(25\) 150.622 1.20498
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 157.717 1.00991 0.504954 0.863146i \(-0.331509\pi\)
0.504954 + 0.863146i \(0.331509\pi\)
\(30\) 0 0
\(31\) 185.291 1.07352 0.536762 0.843734i \(-0.319647\pi\)
0.536762 + 0.843734i \(0.319647\pi\)
\(32\) 0 0
\(33\) −147.744 −0.779363
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −51.9002 −0.230604 −0.115302 0.993331i \(-0.536784\pi\)
−0.115302 + 0.993331i \(0.536784\pi\)
\(38\) 0 0
\(39\) −193.432 −0.794204
\(40\) 0 0
\(41\) −49.4929 −0.188524 −0.0942622 0.995547i \(-0.530049\pi\)
−0.0942622 + 0.995547i \(0.530049\pi\)
\(42\) 0 0
\(43\) −313.788 −1.11284 −0.556421 0.830901i \(-0.687826\pi\)
−0.556421 + 0.830901i \(0.687826\pi\)
\(44\) 0 0
\(45\) −149.417 −0.494972
\(46\) 0 0
\(47\) 553.259 1.71705 0.858523 0.512775i \(-0.171382\pi\)
0.858523 + 0.512775i \(0.171382\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −397.287 −1.09081
\(52\) 0 0
\(53\) −619.868 −1.60652 −0.803259 0.595630i \(-0.796903\pi\)
−0.803259 + 0.595630i \(0.796903\pi\)
\(54\) 0 0
\(55\) −817.611 −2.00448
\(56\) 0 0
\(57\) −246.181 −0.572062
\(58\) 0 0
\(59\) 712.989 1.57328 0.786638 0.617414i \(-0.211820\pi\)
0.786638 + 0.617414i \(0.211820\pi\)
\(60\) 0 0
\(61\) −287.421 −0.603287 −0.301644 0.953421i \(-0.597535\pi\)
−0.301644 + 0.953421i \(0.597535\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1070.45 −2.04265
\(66\) 0 0
\(67\) −226.636 −0.413254 −0.206627 0.978420i \(-0.566249\pi\)
−0.206627 + 0.978420i \(0.566249\pi\)
\(68\) 0 0
\(69\) 246.140 0.429446
\(70\) 0 0
\(71\) −55.3834 −0.0925746 −0.0462873 0.998928i \(-0.514739\pi\)
−0.0462873 + 0.998928i \(0.514739\pi\)
\(72\) 0 0
\(73\) 799.071 1.28115 0.640577 0.767894i \(-0.278695\pi\)
0.640577 + 0.767894i \(0.278695\pi\)
\(74\) 0 0
\(75\) −451.867 −0.695694
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 120.313 0.171345 0.0856725 0.996323i \(-0.472696\pi\)
0.0856725 + 0.996323i \(0.472696\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 857.862 1.13449 0.567244 0.823550i \(-0.308009\pi\)
0.567244 + 0.823550i \(0.308009\pi\)
\(84\) 0 0
\(85\) −2198.57 −2.80551
\(86\) 0 0
\(87\) −473.151 −0.583071
\(88\) 0 0
\(89\) −377.167 −0.449209 −0.224604 0.974450i \(-0.572109\pi\)
−0.224604 + 0.974450i \(0.572109\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −555.873 −0.619799
\(94\) 0 0
\(95\) −1362.36 −1.47132
\(96\) 0 0
\(97\) −1265.20 −1.32435 −0.662174 0.749351i \(-0.730366\pi\)
−0.662174 + 0.749351i \(0.730366\pi\)
\(98\) 0 0
\(99\) 443.233 0.449966
\(100\) 0 0
\(101\) −2011.24 −1.98144 −0.990722 0.135906i \(-0.956605\pi\)
−0.990722 + 0.135906i \(0.956605\pi\)
\(102\) 0 0
\(103\) 38.9204 0.0372325 0.0186162 0.999827i \(-0.494074\pi\)
0.0186162 + 0.999827i \(0.494074\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1293.26 1.16845 0.584223 0.811593i \(-0.301399\pi\)
0.584223 + 0.811593i \(0.301399\pi\)
\(108\) 0 0
\(109\) 1403.08 1.23294 0.616470 0.787378i \(-0.288562\pi\)
0.616470 + 0.787378i \(0.288562\pi\)
\(110\) 0 0
\(111\) 155.701 0.133139
\(112\) 0 0
\(113\) −448.950 −0.373749 −0.186875 0.982384i \(-0.559836\pi\)
−0.186875 + 0.982384i \(0.559836\pi\)
\(114\) 0 0
\(115\) 1362.13 1.10451
\(116\) 0 0
\(117\) 580.297 0.458534
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1094.38 0.822222
\(122\) 0 0
\(123\) 148.479 0.108845
\(124\) 0 0
\(125\) −425.377 −0.304375
\(126\) 0 0
\(127\) 634.373 0.443240 0.221620 0.975133i \(-0.428866\pi\)
0.221620 + 0.975133i \(0.428866\pi\)
\(128\) 0 0
\(129\) 941.363 0.642499
\(130\) 0 0
\(131\) 627.531 0.418532 0.209266 0.977859i \(-0.432893\pi\)
0.209266 + 0.977859i \(0.432893\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 448.251 0.285772
\(136\) 0 0
\(137\) 1795.53 1.11973 0.559863 0.828585i \(-0.310854\pi\)
0.559863 + 0.828585i \(0.310854\pi\)
\(138\) 0 0
\(139\) −565.754 −0.345228 −0.172614 0.984990i \(-0.555221\pi\)
−0.172614 + 0.984990i \(0.555221\pi\)
\(140\) 0 0
\(141\) −1659.78 −0.991337
\(142\) 0 0
\(143\) 3175.39 1.85692
\(144\) 0 0
\(145\) −2618.40 −1.49963
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −583.578 −0.320863 −0.160432 0.987047i \(-0.551289\pi\)
−0.160432 + 0.987047i \(0.551289\pi\)
\(150\) 0 0
\(151\) −874.848 −0.471484 −0.235742 0.971816i \(-0.575752\pi\)
−0.235742 + 0.971816i \(0.575752\pi\)
\(152\) 0 0
\(153\) 1191.86 0.629780
\(154\) 0 0
\(155\) −3076.18 −1.59409
\(156\) 0 0
\(157\) −203.881 −0.103640 −0.0518201 0.998656i \(-0.516502\pi\)
−0.0518201 + 0.998656i \(0.516502\pi\)
\(158\) 0 0
\(159\) 1859.61 0.927524
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2329.09 1.11919 0.559596 0.828765i \(-0.310956\pi\)
0.559596 + 0.828765i \(0.310956\pi\)
\(164\) 0 0
\(165\) 2452.83 1.15729
\(166\) 0 0
\(167\) 2742.47 1.27077 0.635385 0.772196i \(-0.280841\pi\)
0.635385 + 0.772196i \(0.280841\pi\)
\(168\) 0 0
\(169\) 1960.34 0.892279
\(170\) 0 0
\(171\) 738.544 0.330280
\(172\) 0 0
\(173\) 2797.24 1.22931 0.614654 0.788796i \(-0.289295\pi\)
0.614654 + 0.788796i \(0.289295\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −2138.97 −0.908332
\(178\) 0 0
\(179\) 1363.49 0.569340 0.284670 0.958626i \(-0.408116\pi\)
0.284670 + 0.958626i \(0.408116\pi\)
\(180\) 0 0
\(181\) −1845.23 −0.757763 −0.378882 0.925445i \(-0.623691\pi\)
−0.378882 + 0.925445i \(0.623691\pi\)
\(182\) 0 0
\(183\) 862.264 0.348308
\(184\) 0 0
\(185\) 861.640 0.342427
\(186\) 0 0
\(187\) 6521.88 2.55041
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −496.967 −0.188268 −0.0941342 0.995560i \(-0.530008\pi\)
−0.0941342 + 0.995560i \(0.530008\pi\)
\(192\) 0 0
\(193\) 3290.12 1.22709 0.613544 0.789660i \(-0.289743\pi\)
0.613544 + 0.789660i \(0.289743\pi\)
\(194\) 0 0
\(195\) 3211.34 1.17933
\(196\) 0 0
\(197\) −2989.83 −1.08130 −0.540651 0.841247i \(-0.681822\pi\)
−0.540651 + 0.841247i \(0.681822\pi\)
\(198\) 0 0
\(199\) −4746.57 −1.69083 −0.845417 0.534108i \(-0.820648\pi\)
−0.845417 + 0.534108i \(0.820648\pi\)
\(200\) 0 0
\(201\) 679.909 0.238592
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 821.676 0.279943
\(206\) 0 0
\(207\) −738.419 −0.247941
\(208\) 0 0
\(209\) 4041.32 1.33753
\(210\) 0 0
\(211\) −5034.43 −1.64258 −0.821289 0.570512i \(-0.806745\pi\)
−0.821289 + 0.570512i \(0.806745\pi\)
\(212\) 0 0
\(213\) 166.150 0.0534480
\(214\) 0 0
\(215\) 5209.46 1.65248
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −2397.21 −0.739674
\(220\) 0 0
\(221\) 8538.68 2.59898
\(222\) 0 0
\(223\) −2448.32 −0.735208 −0.367604 0.929982i \(-0.619822\pi\)
−0.367604 + 0.929982i \(0.619822\pi\)
\(224\) 0 0
\(225\) 1355.60 0.401659
\(226\) 0 0
\(227\) −3573.33 −1.04480 −0.522401 0.852700i \(-0.674964\pi\)
−0.522401 + 0.852700i \(0.674964\pi\)
\(228\) 0 0
\(229\) −3378.69 −0.974978 −0.487489 0.873129i \(-0.662087\pi\)
−0.487489 + 0.873129i \(0.662087\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3130.51 0.880199 0.440100 0.897949i \(-0.354943\pi\)
0.440100 + 0.897949i \(0.354943\pi\)
\(234\) 0 0
\(235\) −9185.14 −2.54967
\(236\) 0 0
\(237\) −360.939 −0.0989261
\(238\) 0 0
\(239\) 3964.01 1.07285 0.536424 0.843949i \(-0.319775\pi\)
0.536424 + 0.843949i \(0.319775\pi\)
\(240\) 0 0
\(241\) −3173.03 −0.848102 −0.424051 0.905638i \(-0.639392\pi\)
−0.424051 + 0.905638i \(0.639392\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5291.05 1.36300
\(248\) 0 0
\(249\) −2573.59 −0.654997
\(250\) 0 0
\(251\) 2509.92 0.631174 0.315587 0.948897i \(-0.397799\pi\)
0.315587 + 0.948897i \(0.397799\pi\)
\(252\) 0 0
\(253\) −4040.64 −1.00408
\(254\) 0 0
\(255\) 6595.71 1.61976
\(256\) 0 0
\(257\) −5695.39 −1.38237 −0.691184 0.722679i \(-0.742911\pi\)
−0.691184 + 0.722679i \(0.742911\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1419.45 0.336636
\(262\) 0 0
\(263\) 6420.02 1.50523 0.752614 0.658462i \(-0.228792\pi\)
0.752614 + 0.658462i \(0.228792\pi\)
\(264\) 0 0
\(265\) 10291.0 2.38555
\(266\) 0 0
\(267\) 1131.50 0.259351
\(268\) 0 0
\(269\) −290.460 −0.0658351 −0.0329175 0.999458i \(-0.510480\pi\)
−0.0329175 + 0.999458i \(0.510480\pi\)
\(270\) 0 0
\(271\) −3681.21 −0.825157 −0.412578 0.910922i \(-0.635372\pi\)
−0.412578 + 0.910922i \(0.635372\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7417.86 1.62660
\(276\) 0 0
\(277\) −5427.03 −1.17718 −0.588590 0.808432i \(-0.700317\pi\)
−0.588590 + 0.808432i \(0.700317\pi\)
\(278\) 0 0
\(279\) 1667.62 0.357841
\(280\) 0 0
\(281\) −5237.25 −1.11184 −0.555922 0.831235i \(-0.687634\pi\)
−0.555922 + 0.831235i \(0.687634\pi\)
\(282\) 0 0
\(283\) 8302.83 1.74400 0.872000 0.489506i \(-0.162823\pi\)
0.872000 + 0.489506i \(0.162823\pi\)
\(284\) 0 0
\(285\) 4087.07 0.849464
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 12624.5 2.56960
\(290\) 0 0
\(291\) 3795.60 0.764612
\(292\) 0 0
\(293\) −2773.58 −0.553018 −0.276509 0.961011i \(-0.589178\pi\)
−0.276509 + 0.961011i \(0.589178\pi\)
\(294\) 0 0
\(295\) −11837.0 −2.33618
\(296\) 0 0
\(297\) −1329.70 −0.259788
\(298\) 0 0
\(299\) −5290.15 −1.02320
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 6033.72 1.14399
\(304\) 0 0
\(305\) 4771.73 0.895832
\(306\) 0 0
\(307\) 6528.52 1.21369 0.606844 0.794821i \(-0.292435\pi\)
0.606844 + 0.794821i \(0.292435\pi\)
\(308\) 0 0
\(309\) −116.761 −0.0214962
\(310\) 0 0
\(311\) 4350.89 0.793300 0.396650 0.917970i \(-0.370173\pi\)
0.396650 + 0.917970i \(0.370173\pi\)
\(312\) 0 0
\(313\) −5411.10 −0.977168 −0.488584 0.872517i \(-0.662486\pi\)
−0.488584 + 0.872517i \(0.662486\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7013.79 −1.24269 −0.621346 0.783536i \(-0.713414\pi\)
−0.621346 + 0.783536i \(0.713414\pi\)
\(318\) 0 0
\(319\) 7767.27 1.36327
\(320\) 0 0
\(321\) −3879.77 −0.674603
\(322\) 0 0
\(323\) 10867.2 1.87203
\(324\) 0 0
\(325\) 9711.73 1.65757
\(326\) 0 0
\(327\) −4209.23 −0.711839
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −142.799 −0.0237128 −0.0118564 0.999930i \(-0.503774\pi\)
−0.0118564 + 0.999930i \(0.503774\pi\)
\(332\) 0 0
\(333\) −467.102 −0.0768679
\(334\) 0 0
\(335\) 3762.59 0.613648
\(336\) 0 0
\(337\) −1489.10 −0.240702 −0.120351 0.992731i \(-0.538402\pi\)
−0.120351 + 0.992731i \(0.538402\pi\)
\(338\) 0 0
\(339\) 1346.85 0.215784
\(340\) 0 0
\(341\) 9125.23 1.44915
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −4086.38 −0.637691
\(346\) 0 0
\(347\) −4017.06 −0.621461 −0.310730 0.950498i \(-0.600574\pi\)
−0.310730 + 0.950498i \(0.600574\pi\)
\(348\) 0 0
\(349\) −4235.87 −0.649687 −0.324844 0.945768i \(-0.605312\pi\)
−0.324844 + 0.945768i \(0.605312\pi\)
\(350\) 0 0
\(351\) −1740.89 −0.264735
\(352\) 0 0
\(353\) 11399.1 1.71873 0.859366 0.511361i \(-0.170858\pi\)
0.859366 + 0.511361i \(0.170858\pi\)
\(354\) 0 0
\(355\) 919.468 0.137466
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 713.980 0.104965 0.0524825 0.998622i \(-0.483287\pi\)
0.0524825 + 0.998622i \(0.483287\pi\)
\(360\) 0 0
\(361\) −125.080 −0.0182359
\(362\) 0 0
\(363\) −3283.13 −0.474710
\(364\) 0 0
\(365\) −13266.1 −1.90241
\(366\) 0 0
\(367\) 12235.8 1.74033 0.870167 0.492757i \(-0.164011\pi\)
0.870167 + 0.492757i \(0.164011\pi\)
\(368\) 0 0
\(369\) −445.437 −0.0628415
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −9060.32 −1.25771 −0.628854 0.777523i \(-0.716476\pi\)
−0.628854 + 0.777523i \(0.716476\pi\)
\(374\) 0 0
\(375\) 1276.13 0.175731
\(376\) 0 0
\(377\) 10169.2 1.38923
\(378\) 0 0
\(379\) −10607.9 −1.43770 −0.718852 0.695163i \(-0.755332\pi\)
−0.718852 + 0.695163i \(0.755332\pi\)
\(380\) 0 0
\(381\) −1903.12 −0.255905
\(382\) 0 0
\(383\) −4281.14 −0.571164 −0.285582 0.958354i \(-0.592187\pi\)
−0.285582 + 0.958354i \(0.592187\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2824.09 −0.370947
\(388\) 0 0
\(389\) −4334.27 −0.564926 −0.282463 0.959278i \(-0.591151\pi\)
−0.282463 + 0.959278i \(0.591151\pi\)
\(390\) 0 0
\(391\) −10865.4 −1.40533
\(392\) 0 0
\(393\) −1882.59 −0.241639
\(394\) 0 0
\(395\) −1997.42 −0.254433
\(396\) 0 0
\(397\) −12347.5 −1.56097 −0.780485 0.625175i \(-0.785028\pi\)
−0.780485 + 0.625175i \(0.785028\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 8321.66 1.03632 0.518159 0.855284i \(-0.326617\pi\)
0.518159 + 0.855284i \(0.326617\pi\)
\(402\) 0 0
\(403\) 11947.1 1.47674
\(404\) 0 0
\(405\) −1344.75 −0.164991
\(406\) 0 0
\(407\) −2555.99 −0.311291
\(408\) 0 0
\(409\) 15242.4 1.84276 0.921382 0.388659i \(-0.127062\pi\)
0.921382 + 0.388659i \(0.127062\pi\)
\(410\) 0 0
\(411\) −5386.59 −0.646474
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −14242.1 −1.68462
\(416\) 0 0
\(417\) 1697.26 0.199317
\(418\) 0 0
\(419\) −8251.84 −0.962121 −0.481061 0.876687i \(-0.659748\pi\)
−0.481061 + 0.876687i \(0.659748\pi\)
\(420\) 0 0
\(421\) −1825.97 −0.211383 −0.105691 0.994399i \(-0.533706\pi\)
−0.105691 + 0.994399i \(0.533706\pi\)
\(422\) 0 0
\(423\) 4979.34 0.572349
\(424\) 0 0
\(425\) 19946.8 2.27661
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −9526.18 −1.07209
\(430\) 0 0
\(431\) 11399.2 1.27397 0.636986 0.770876i \(-0.280181\pi\)
0.636986 + 0.770876i \(0.280181\pi\)
\(432\) 0 0
\(433\) 2484.03 0.275692 0.137846 0.990454i \(-0.455982\pi\)
0.137846 + 0.990454i \(0.455982\pi\)
\(434\) 0 0
\(435\) 7855.20 0.865812
\(436\) 0 0
\(437\) −6732.78 −0.737008
\(438\) 0 0
\(439\) 11572.5 1.25814 0.629071 0.777348i \(-0.283435\pi\)
0.629071 + 0.777348i \(0.283435\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −11072.6 −1.18753 −0.593766 0.804638i \(-0.702360\pi\)
−0.593766 + 0.804638i \(0.702360\pi\)
\(444\) 0 0
\(445\) 6261.67 0.667038
\(446\) 0 0
\(447\) 1750.73 0.185250
\(448\) 0 0
\(449\) 12248.0 1.28735 0.643673 0.765300i \(-0.277410\pi\)
0.643673 + 0.765300i \(0.277410\pi\)
\(450\) 0 0
\(451\) −2437.43 −0.254489
\(452\) 0 0
\(453\) 2624.54 0.272211
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 12667.5 1.29663 0.648315 0.761372i \(-0.275474\pi\)
0.648315 + 0.761372i \(0.275474\pi\)
\(458\) 0 0
\(459\) −3575.58 −0.363603
\(460\) 0 0
\(461\) 11444.0 1.15618 0.578089 0.815973i \(-0.303798\pi\)
0.578089 + 0.815973i \(0.303798\pi\)
\(462\) 0 0
\(463\) −5577.93 −0.559888 −0.279944 0.960016i \(-0.590316\pi\)
−0.279944 + 0.960016i \(0.590316\pi\)
\(464\) 0 0
\(465\) 9228.53 0.920350
\(466\) 0 0
\(467\) −13494.5 −1.33716 −0.668578 0.743642i \(-0.733097\pi\)
−0.668578 + 0.743642i \(0.733097\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 611.644 0.0598366
\(472\) 0 0
\(473\) −15453.5 −1.50222
\(474\) 0 0
\(475\) 12360.1 1.19394
\(476\) 0 0
\(477\) −5578.82 −0.535506
\(478\) 0 0
\(479\) 5321.13 0.507576 0.253788 0.967260i \(-0.418323\pi\)
0.253788 + 0.967260i \(0.418323\pi\)
\(480\) 0 0
\(481\) −3346.39 −0.317219
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 21004.7 1.96655
\(486\) 0 0
\(487\) −5915.38 −0.550414 −0.275207 0.961385i \(-0.588746\pi\)
−0.275207 + 0.961385i \(0.588746\pi\)
\(488\) 0 0
\(489\) −6987.27 −0.646166
\(490\) 0 0
\(491\) −20748.9 −1.90710 −0.953548 0.301240i \(-0.902600\pi\)
−0.953548 + 0.301240i \(0.902600\pi\)
\(492\) 0 0
\(493\) 20886.3 1.90806
\(494\) 0 0
\(495\) −7358.50 −0.668162
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −10673.6 −0.957546 −0.478773 0.877939i \(-0.658918\pi\)
−0.478773 + 0.877939i \(0.658918\pi\)
\(500\) 0 0
\(501\) −8227.41 −0.733679
\(502\) 0 0
\(503\) 18494.4 1.63941 0.819706 0.572784i \(-0.194137\pi\)
0.819706 + 0.572784i \(0.194137\pi\)
\(504\) 0 0
\(505\) 33390.3 2.94228
\(506\) 0 0
\(507\) −5881.01 −0.515158
\(508\) 0 0
\(509\) −5766.36 −0.502140 −0.251070 0.967969i \(-0.580782\pi\)
−0.251070 + 0.967969i \(0.580782\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −2215.63 −0.190687
\(514\) 0 0
\(515\) −646.152 −0.0552871
\(516\) 0 0
\(517\) 27247.0 2.31784
\(518\) 0 0
\(519\) −8391.73 −0.709742
\(520\) 0 0
\(521\) 4394.77 0.369555 0.184778 0.982780i \(-0.440844\pi\)
0.184778 + 0.982780i \(0.440844\pi\)
\(522\) 0 0
\(523\) −1063.00 −0.0888752 −0.0444376 0.999012i \(-0.514150\pi\)
−0.0444376 + 0.999012i \(0.514150\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 24537.9 2.02825
\(528\) 0 0
\(529\) −5435.35 −0.446729
\(530\) 0 0
\(531\) 6416.90 0.524425
\(532\) 0 0
\(533\) −3191.18 −0.259334
\(534\) 0 0
\(535\) −21470.5 −1.73505
\(536\) 0 0
\(537\) −4090.47 −0.328709
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −4002.20 −0.318056 −0.159028 0.987274i \(-0.550836\pi\)
−0.159028 + 0.987274i \(0.550836\pi\)
\(542\) 0 0
\(543\) 5535.70 0.437495
\(544\) 0 0
\(545\) −23293.7 −1.83081
\(546\) 0 0
\(547\) −13365.7 −1.04475 −0.522373 0.852717i \(-0.674953\pi\)
−0.522373 + 0.852717i \(0.674953\pi\)
\(548\) 0 0
\(549\) −2586.79 −0.201096
\(550\) 0 0
\(551\) 12942.3 1.00066
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −2584.92 −0.197700
\(556\) 0 0
\(557\) 5675.70 0.431754 0.215877 0.976421i \(-0.430739\pi\)
0.215877 + 0.976421i \(0.430739\pi\)
\(558\) 0 0
\(559\) −20232.2 −1.53083
\(560\) 0 0
\(561\) −19565.6 −1.47248
\(562\) 0 0
\(563\) 281.060 0.0210396 0.0105198 0.999945i \(-0.496651\pi\)
0.0105198 + 0.999945i \(0.496651\pi\)
\(564\) 0 0
\(565\) 7453.41 0.554987
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6736.73 0.496342 0.248171 0.968716i \(-0.420171\pi\)
0.248171 + 0.968716i \(0.420171\pi\)
\(570\) 0 0
\(571\) 16445.8 1.20532 0.602659 0.797999i \(-0.294108\pi\)
0.602659 + 0.797999i \(0.294108\pi\)
\(572\) 0 0
\(573\) 1490.90 0.108697
\(574\) 0 0
\(575\) −12358.0 −0.896289
\(576\) 0 0
\(577\) 6097.01 0.439900 0.219950 0.975511i \(-0.429411\pi\)
0.219950 + 0.975511i \(0.429411\pi\)
\(578\) 0 0
\(579\) −9870.36 −0.708460
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −30527.4 −2.16863
\(584\) 0 0
\(585\) −9634.01 −0.680884
\(586\) 0 0
\(587\) 19531.2 1.37332 0.686659 0.726980i \(-0.259077\pi\)
0.686659 + 0.726980i \(0.259077\pi\)
\(588\) 0 0
\(589\) 15205.1 1.06369
\(590\) 0 0
\(591\) 8969.48 0.624290
\(592\) 0 0
\(593\) 17701.7 1.22583 0.612917 0.790147i \(-0.289996\pi\)
0.612917 + 0.790147i \(0.289996\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 14239.7 0.976203
\(598\) 0 0
\(599\) 16786.0 1.14500 0.572501 0.819904i \(-0.305973\pi\)
0.572501 + 0.819904i \(0.305973\pi\)
\(600\) 0 0
\(601\) 6281.32 0.426324 0.213162 0.977017i \(-0.431624\pi\)
0.213162 + 0.977017i \(0.431624\pi\)
\(602\) 0 0
\(603\) −2039.73 −0.137751
\(604\) 0 0
\(605\) −18168.7 −1.22093
\(606\) 0 0
\(607\) 2452.50 0.163993 0.0819967 0.996633i \(-0.473870\pi\)
0.0819967 + 0.996633i \(0.473870\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 35672.7 2.36197
\(612\) 0 0
\(613\) 14792.7 0.974670 0.487335 0.873215i \(-0.337969\pi\)
0.487335 + 0.873215i \(0.337969\pi\)
\(614\) 0 0
\(615\) −2465.03 −0.161625
\(616\) 0 0
\(617\) −15044.2 −0.981614 −0.490807 0.871268i \(-0.663298\pi\)
−0.490807 + 0.871268i \(0.663298\pi\)
\(618\) 0 0
\(619\) −8948.32 −0.581039 −0.290520 0.956869i \(-0.593828\pi\)
−0.290520 + 0.956869i \(0.593828\pi\)
\(620\) 0 0
\(621\) 2215.26 0.143149
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11765.7 −0.753006
\(626\) 0 0
\(627\) −12124.0 −0.772225
\(628\) 0 0
\(629\) −6873.09 −0.435688
\(630\) 0 0
\(631\) 27769.9 1.75198 0.875991 0.482327i \(-0.160208\pi\)
0.875991 + 0.482327i \(0.160208\pi\)
\(632\) 0 0
\(633\) 15103.3 0.948343
\(634\) 0 0
\(635\) −10531.8 −0.658174
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −498.451 −0.0308582
\(640\) 0 0
\(641\) −15807.9 −0.974063 −0.487031 0.873384i \(-0.661920\pi\)
−0.487031 + 0.873384i \(0.661920\pi\)
\(642\) 0 0
\(643\) −21911.4 −1.34386 −0.671929 0.740616i \(-0.734534\pi\)
−0.671929 + 0.740616i \(0.734534\pi\)
\(644\) 0 0
\(645\) −15628.4 −0.954058
\(646\) 0 0
\(647\) −11245.6 −0.683323 −0.341662 0.939823i \(-0.610990\pi\)
−0.341662 + 0.939823i \(0.610990\pi\)
\(648\) 0 0
\(649\) 35113.4 2.12376
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 8668.43 0.519482 0.259741 0.965678i \(-0.416363\pi\)
0.259741 + 0.965678i \(0.416363\pi\)
\(654\) 0 0
\(655\) −10418.2 −0.621484
\(656\) 0 0
\(657\) 7191.64 0.427051
\(658\) 0 0
\(659\) 12625.9 0.746334 0.373167 0.927764i \(-0.378272\pi\)
0.373167 + 0.927764i \(0.378272\pi\)
\(660\) 0 0
\(661\) −1884.10 −0.110867 −0.0554334 0.998462i \(-0.517654\pi\)
−0.0554334 + 0.998462i \(0.517654\pi\)
\(662\) 0 0
\(663\) −25616.0 −1.50052
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12940.2 −0.751192
\(668\) 0 0
\(669\) 7344.95 0.424473
\(670\) 0 0
\(671\) −14155.0 −0.814376
\(672\) 0 0
\(673\) −25932.5 −1.48533 −0.742663 0.669665i \(-0.766438\pi\)
−0.742663 + 0.669665i \(0.766438\pi\)
\(674\) 0 0
\(675\) −4066.80 −0.231898
\(676\) 0 0
\(677\) 18226.4 1.03471 0.517354 0.855772i \(-0.326917\pi\)
0.517354 + 0.855772i \(0.326917\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 10720.0 0.603217
\(682\) 0 0
\(683\) 10680.6 0.598362 0.299181 0.954196i \(-0.403287\pi\)
0.299181 + 0.954196i \(0.403287\pi\)
\(684\) 0 0
\(685\) −29809.2 −1.66270
\(686\) 0 0
\(687\) 10136.1 0.562904
\(688\) 0 0
\(689\) −39967.5 −2.20993
\(690\) 0 0
\(691\) 20500.4 1.12862 0.564308 0.825565i \(-0.309143\pi\)
0.564308 + 0.825565i \(0.309143\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9392.58 0.512634
\(696\) 0 0
\(697\) −6554.30 −0.356186
\(698\) 0 0
\(699\) −9391.53 −0.508183
\(700\) 0 0
\(701\) 13903.9 0.749137 0.374568 0.927199i \(-0.377791\pi\)
0.374568 + 0.927199i \(0.377791\pi\)
\(702\) 0 0
\(703\) −4258.95 −0.228491
\(704\) 0 0
\(705\) 27555.4 1.47205
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 18055.3 0.956391 0.478195 0.878253i \(-0.341291\pi\)
0.478195 + 0.878253i \(0.341291\pi\)
\(710\) 0 0
\(711\) 1082.82 0.0571150
\(712\) 0 0
\(713\) −15202.5 −0.798510
\(714\) 0 0
\(715\) −52717.5 −2.75737
\(716\) 0 0
\(717\) −11892.0 −0.619409
\(718\) 0 0
\(719\) −9649.29 −0.500498 −0.250249 0.968182i \(-0.580512\pi\)
−0.250249 + 0.968182i \(0.580512\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 9519.08 0.489652
\(724\) 0 0
\(725\) 23755.7 1.21692
\(726\) 0 0
\(727\) 12451.5 0.635214 0.317607 0.948222i \(-0.397121\pi\)
0.317607 + 0.948222i \(0.397121\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −41554.6 −2.10253
\(732\) 0 0
\(733\) −21799.1 −1.09845 −0.549227 0.835673i \(-0.685078\pi\)
−0.549227 + 0.835673i \(0.685078\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11161.4 −0.557850
\(738\) 0 0
\(739\) 15158.8 0.754568 0.377284 0.926098i \(-0.376858\pi\)
0.377284 + 0.926098i \(0.376858\pi\)
\(740\) 0 0
\(741\) −15873.1 −0.786929
\(742\) 0 0
\(743\) −5094.25 −0.251534 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(744\) 0 0
\(745\) 9688.49 0.476455
\(746\) 0 0
\(747\) 7720.76 0.378163
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 13213.6 0.642038 0.321019 0.947073i \(-0.395975\pi\)
0.321019 + 0.947073i \(0.395975\pi\)
\(752\) 0 0
\(753\) −7529.76 −0.364408
\(754\) 0 0
\(755\) 14524.1 0.700114
\(756\) 0 0
\(757\) 32248.9 1.54836 0.774180 0.632966i \(-0.218163\pi\)
0.774180 + 0.632966i \(0.218163\pi\)
\(758\) 0 0
\(759\) 12121.9 0.579707
\(760\) 0 0
\(761\) 15715.4 0.748596 0.374298 0.927308i \(-0.377884\pi\)
0.374298 + 0.927308i \(0.377884\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −19787.1 −0.935170
\(766\) 0 0
\(767\) 45971.7 2.16420
\(768\) 0 0
\(769\) 24907.4 1.16799 0.583994 0.811758i \(-0.301489\pi\)
0.583994 + 0.811758i \(0.301489\pi\)
\(770\) 0 0
\(771\) 17086.2 0.798111
\(772\) 0 0
\(773\) −22027.2 −1.02492 −0.512461 0.858710i \(-0.671266\pi\)
−0.512461 + 0.858710i \(0.671266\pi\)
\(774\) 0 0
\(775\) 27908.9 1.29357
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4061.41 −0.186798
\(780\) 0 0
\(781\) −2727.53 −0.124966
\(782\) 0 0
\(783\) −4258.36 −0.194357
\(784\) 0 0
\(785\) 3384.81 0.153897
\(786\) 0 0
\(787\) 38388.7 1.73877 0.869384 0.494137i \(-0.164516\pi\)
0.869384 + 0.494137i \(0.164516\pi\)
\(788\) 0 0
\(789\) −19260.0 −0.869044
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −18532.2 −0.829883
\(794\) 0 0
\(795\) −30872.9 −1.37730
\(796\) 0 0
\(797\) 28628.1 1.27235 0.636174 0.771546i \(-0.280516\pi\)
0.636174 + 0.771546i \(0.280516\pi\)
\(798\) 0 0
\(799\) 73267.6 3.24408
\(800\) 0 0
\(801\) −3394.50 −0.149736
\(802\) 0 0
\(803\) 39352.8 1.72943
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 871.379 0.0380099
\(808\) 0 0
\(809\) −11542.2 −0.501608 −0.250804 0.968038i \(-0.580695\pi\)
−0.250804 + 0.968038i \(0.580695\pi\)
\(810\) 0 0
\(811\) 29374.0 1.27184 0.635919 0.771756i \(-0.280621\pi\)
0.635919 + 0.771756i \(0.280621\pi\)
\(812\) 0 0
\(813\) 11043.6 0.476404
\(814\) 0 0
\(815\) −38667.2 −1.66191
\(816\) 0 0
\(817\) −25749.6 −1.10265
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −17945.2 −0.762839 −0.381419 0.924402i \(-0.624565\pi\)
−0.381419 + 0.924402i \(0.624565\pi\)
\(822\) 0 0
\(823\) −42524.3 −1.80110 −0.900549 0.434755i \(-0.856835\pi\)
−0.900549 + 0.434755i \(0.856835\pi\)
\(824\) 0 0
\(825\) −22253.6 −0.939116
\(826\) 0 0
\(827\) 43291.0 1.82029 0.910143 0.414293i \(-0.135971\pi\)
0.910143 + 0.414293i \(0.135971\pi\)
\(828\) 0 0
\(829\) −28580.9 −1.19741 −0.598707 0.800968i \(-0.704319\pi\)
−0.598707 + 0.800968i \(0.704319\pi\)
\(830\) 0 0
\(831\) 16281.1 0.679645
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −45530.1 −1.88699
\(836\) 0 0
\(837\) −5002.85 −0.206600
\(838\) 0 0
\(839\) −11526.0 −0.474281 −0.237141 0.971475i \(-0.576210\pi\)
−0.237141 + 0.971475i \(0.576210\pi\)
\(840\) 0 0
\(841\) 485.697 0.0199146
\(842\) 0 0
\(843\) 15711.7 0.641923
\(844\) 0 0
\(845\) −32545.3 −1.32496
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −24908.5 −1.00690
\(850\) 0 0
\(851\) 4258.23 0.171528
\(852\) 0 0
\(853\) 25692.6 1.03130 0.515650 0.856799i \(-0.327551\pi\)
0.515650 + 0.856799i \(0.327551\pi\)
\(854\) 0 0
\(855\) −12261.2 −0.490438
\(856\) 0 0
\(857\) −33824.0 −1.34820 −0.674100 0.738640i \(-0.735468\pi\)
−0.674100 + 0.738640i \(0.735468\pi\)
\(858\) 0 0
\(859\) −13254.7 −0.526479 −0.263240 0.964730i \(-0.584791\pi\)
−0.263240 + 0.964730i \(0.584791\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −5166.82 −0.203801 −0.101901 0.994795i \(-0.532492\pi\)
−0.101901 + 0.994795i \(0.532492\pi\)
\(864\) 0 0
\(865\) −46439.5 −1.82542
\(866\) 0 0
\(867\) −37873.4 −1.48356
\(868\) 0 0
\(869\) 5925.18 0.231298
\(870\) 0 0
\(871\) −14612.9 −0.568473
\(872\) 0 0
\(873\) −11386.8 −0.441449
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −25268.0 −0.972907 −0.486453 0.873707i \(-0.661710\pi\)
−0.486453 + 0.873707i \(0.661710\pi\)
\(878\) 0 0
\(879\) 8320.74 0.319285
\(880\) 0 0
\(881\) 42652.8 1.63111 0.815555 0.578680i \(-0.196432\pi\)
0.815555 + 0.578680i \(0.196432\pi\)
\(882\) 0 0
\(883\) −31070.9 −1.18416 −0.592082 0.805877i \(-0.701694\pi\)
−0.592082 + 0.805877i \(0.701694\pi\)
\(884\) 0 0
\(885\) 35510.9 1.34880
\(886\) 0 0
\(887\) −36035.6 −1.36410 −0.682050 0.731306i \(-0.738911\pi\)
−0.682050 + 0.731306i \(0.738911\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 3989.10 0.149989
\(892\) 0 0
\(893\) 45400.7 1.70132
\(894\) 0 0
\(895\) −22636.5 −0.845423
\(896\) 0 0
\(897\) 15870.5 0.590746
\(898\) 0 0
\(899\) 29223.5 1.08416
\(900\) 0 0
\(901\) −82088.6 −3.03526
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 30634.3 1.12522
\(906\) 0 0
\(907\) −17945.5 −0.656967 −0.328484 0.944510i \(-0.606538\pi\)
−0.328484 + 0.944510i \(0.606538\pi\)
\(908\) 0 0
\(909\) −18101.2 −0.660481
\(910\) 0 0
\(911\) −17672.5 −0.642717 −0.321359 0.946958i \(-0.604139\pi\)
−0.321359 + 0.946958i \(0.604139\pi\)
\(912\) 0 0
\(913\) 42248.1 1.53144
\(914\) 0 0
\(915\) −14315.2 −0.517209
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −30422.3 −1.09199 −0.545995 0.837788i \(-0.683848\pi\)
−0.545995 + 0.837788i \(0.683848\pi\)
\(920\) 0 0
\(921\) −19585.6 −0.700723
\(922\) 0 0
\(923\) −3570.98 −0.127346
\(924\) 0 0
\(925\) −7817.32 −0.277872
\(926\) 0 0
\(927\) 350.284 0.0124108
\(928\) 0 0
\(929\) 16017.6 0.565683 0.282841 0.959167i \(-0.408723\pi\)
0.282841 + 0.959167i \(0.408723\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −13052.7 −0.458012
\(934\) 0 0
\(935\) −108275. −3.78715
\(936\) 0 0
\(937\) −7370.43 −0.256970 −0.128485 0.991711i \(-0.541011\pi\)
−0.128485 + 0.991711i \(0.541011\pi\)
\(938\) 0 0
\(939\) 16233.3 0.564168
\(940\) 0 0
\(941\) 14999.7 0.519634 0.259817 0.965658i \(-0.416338\pi\)
0.259817 + 0.965658i \(0.416338\pi\)
\(942\) 0 0
\(943\) 4060.73 0.140229
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −12699.6 −0.435776 −0.217888 0.975974i \(-0.569917\pi\)
−0.217888 + 0.975974i \(0.569917\pi\)
\(948\) 0 0
\(949\) 51522.0 1.76236
\(950\) 0 0
\(951\) 21041.4 0.717469
\(952\) 0 0
\(953\) 5347.41 0.181762 0.0908811 0.995862i \(-0.471032\pi\)
0.0908811 + 0.995862i \(0.471032\pi\)
\(954\) 0 0
\(955\) 8250.59 0.279563
\(956\) 0 0
\(957\) −23301.8 −0.787086
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 4541.70 0.152452
\(962\) 0 0
\(963\) 11639.3 0.389482
\(964\) 0 0
\(965\) −54622.2 −1.82212
\(966\) 0 0
\(967\) −38787.2 −1.28988 −0.644939 0.764234i \(-0.723117\pi\)
−0.644939 + 0.764234i \(0.723117\pi\)
\(968\) 0 0
\(969\) −32601.6 −1.08082
\(970\) 0 0
\(971\) 28064.9 0.927545 0.463772 0.885954i \(-0.346496\pi\)
0.463772 + 0.885954i \(0.346496\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −29135.2 −0.956998
\(976\) 0 0
\(977\) 20896.8 0.684287 0.342144 0.939648i \(-0.388847\pi\)
0.342144 + 0.939648i \(0.388847\pi\)
\(978\) 0 0
\(979\) −18574.8 −0.606386
\(980\) 0 0
\(981\) 12627.7 0.410980
\(982\) 0 0
\(983\) 1791.08 0.0581147 0.0290573 0.999578i \(-0.490749\pi\)
0.0290573 + 0.999578i \(0.490749\pi\)
\(984\) 0 0
\(985\) 49636.7 1.60564
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 25745.2 0.827755
\(990\) 0 0
\(991\) −22843.9 −0.732251 −0.366126 0.930565i \(-0.619316\pi\)
−0.366126 + 0.930565i \(0.619316\pi\)
\(992\) 0 0
\(993\) 428.396 0.0136906
\(994\) 0 0
\(995\) 78802.0 2.51075
\(996\) 0 0
\(997\) −12875.8 −0.409007 −0.204504 0.978866i \(-0.565558\pi\)
−0.204504 + 0.978866i \(0.565558\pi\)
\(998\) 0 0
\(999\) 1401.30 0.0443797
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 2352.4.a.ck.1.1 4
4.3 odd 2 1176.4.a.bc.1.1 yes 4
7.6 odd 2 2352.4.a.cr.1.4 4
28.27 even 2 1176.4.a.bb.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.4.a.bb.1.4 4 28.27 even 2
1176.4.a.bc.1.1 yes 4 4.3 odd 2
2352.4.a.ck.1.1 4 1.1 even 1 trivial
2352.4.a.cr.1.4 4 7.6 odd 2