Properties

Label 768.4.a.m.1.2
Level $768$
Weight $4$
Character 768.1
Self dual yes
Analytic conductor $45.313$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(45.3134668844\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 384)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 768.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +2.82843 q^{5} -14.1421 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +2.82843 q^{5} -14.1421 q^{7} +9.00000 q^{9} -20.0000 q^{11} +39.5980 q^{13} +8.48528 q^{15} -34.0000 q^{17} -52.0000 q^{19} -42.4264 q^{21} -62.2254 q^{23} -117.000 q^{25} +27.0000 q^{27} +200.818 q^{29} +110.309 q^{31} -60.0000 q^{33} -40.0000 q^{35} -271.529 q^{37} +118.794 q^{39} -26.0000 q^{41} -252.000 q^{43} +25.4558 q^{45} -345.068 q^{47} -143.000 q^{49} -102.000 q^{51} -681.651 q^{53} -56.5685 q^{55} -156.000 q^{57} -364.000 q^{59} +735.391 q^{61} -127.279 q^{63} +112.000 q^{65} -628.000 q^{67} -186.676 q^{69} +333.754 q^{71} +338.000 q^{73} -351.000 q^{75} +282.843 q^{77} +789.131 q^{79} +81.0000 q^{81} -1036.00 q^{83} -96.1665 q^{85} +602.455 q^{87} +234.000 q^{89} -560.000 q^{91} +330.926 q^{93} -147.078 q^{95} -178.000 q^{97} -180.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} + 18 q^{9} - 40 q^{11} - 68 q^{17} - 104 q^{19} - 234 q^{25} + 54 q^{27} - 120 q^{33} - 80 q^{35} - 52 q^{41} - 504 q^{43} - 286 q^{49} - 204 q^{51} - 312 q^{57} - 728 q^{59} + 224 q^{65} - 1256 q^{67} + 676 q^{73} - 702 q^{75} + 162 q^{81} - 2072 q^{83} + 468 q^{89} - 1120 q^{91} - 356 q^{97} - 360 q^{99}+O(q^{100}) \) 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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 2.82843 0.252982 0.126491 0.991968i \(-0.459628\pi\)
0.126491 + 0.991968i \(0.459628\pi\)
\(6\) 0 0
\(7\) −14.1421 −0.763604 −0.381802 0.924244i \(-0.624696\pi\)
−0.381802 + 0.924244i \(0.624696\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −20.0000 −0.548202 −0.274101 0.961701i \(-0.588380\pi\)
−0.274101 + 0.961701i \(0.588380\pi\)
\(12\) 0 0
\(13\) 39.5980 0.844808 0.422404 0.906408i \(-0.361186\pi\)
0.422404 + 0.906408i \(0.361186\pi\)
\(14\) 0 0
\(15\) 8.48528 0.146059
\(16\) 0 0
\(17\) −34.0000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −52.0000 −0.627875 −0.313937 0.949444i \(-0.601648\pi\)
−0.313937 + 0.949444i \(0.601648\pi\)
\(20\) 0 0
\(21\) −42.4264 −0.440867
\(22\) 0 0
\(23\) −62.2254 −0.564126 −0.282063 0.959396i \(-0.591019\pi\)
−0.282063 + 0.959396i \(0.591019\pi\)
\(24\) 0 0
\(25\) −117.000 −0.936000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 200.818 1.28590 0.642949 0.765909i \(-0.277711\pi\)
0.642949 + 0.765909i \(0.277711\pi\)
\(30\) 0 0
\(31\) 110.309 0.639097 0.319549 0.947570i \(-0.396469\pi\)
0.319549 + 0.947570i \(0.396469\pi\)
\(32\) 0 0
\(33\) −60.0000 −0.316505
\(34\) 0 0
\(35\) −40.0000 −0.193178
\(36\) 0 0
\(37\) −271.529 −1.20646 −0.603231 0.797567i \(-0.706120\pi\)
−0.603231 + 0.797567i \(0.706120\pi\)
\(38\) 0 0
\(39\) 118.794 0.487750
\(40\) 0 0
\(41\) −26.0000 −0.0990370 −0.0495185 0.998773i \(-0.515769\pi\)
−0.0495185 + 0.998773i \(0.515769\pi\)
\(42\) 0 0
\(43\) −252.000 −0.893713 −0.446856 0.894606i \(-0.647456\pi\)
−0.446856 + 0.894606i \(0.647456\pi\)
\(44\) 0 0
\(45\) 25.4558 0.0843274
\(46\) 0 0
\(47\) −345.068 −1.07092 −0.535461 0.844560i \(-0.679862\pi\)
−0.535461 + 0.844560i \(0.679862\pi\)
\(48\) 0 0
\(49\) −143.000 −0.416910
\(50\) 0 0
\(51\) −102.000 −0.280056
\(52\) 0 0
\(53\) −681.651 −1.76664 −0.883320 0.468770i \(-0.844697\pi\)
−0.883320 + 0.468770i \(0.844697\pi\)
\(54\) 0 0
\(55\) −56.5685 −0.138685
\(56\) 0 0
\(57\) −156.000 −0.362504
\(58\) 0 0
\(59\) −364.000 −0.803199 −0.401600 0.915815i \(-0.631546\pi\)
−0.401600 + 0.915815i \(0.631546\pi\)
\(60\) 0 0
\(61\) 735.391 1.54356 0.771780 0.635889i \(-0.219367\pi\)
0.771780 + 0.635889i \(0.219367\pi\)
\(62\) 0 0
\(63\) −127.279 −0.254535
\(64\) 0 0
\(65\) 112.000 0.213721
\(66\) 0 0
\(67\) −628.000 −1.14511 −0.572555 0.819866i \(-0.694048\pi\)
−0.572555 + 0.819866i \(0.694048\pi\)
\(68\) 0 0
\(69\) −186.676 −0.325698
\(70\) 0 0
\(71\) 333.754 0.557878 0.278939 0.960309i \(-0.410017\pi\)
0.278939 + 0.960309i \(0.410017\pi\)
\(72\) 0 0
\(73\) 338.000 0.541917 0.270958 0.962591i \(-0.412659\pi\)
0.270958 + 0.962591i \(0.412659\pi\)
\(74\) 0 0
\(75\) −351.000 −0.540400
\(76\) 0 0
\(77\) 282.843 0.418609
\(78\) 0 0
\(79\) 789.131 1.12385 0.561925 0.827188i \(-0.310061\pi\)
0.561925 + 0.827188i \(0.310061\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −1036.00 −1.37007 −0.685035 0.728510i \(-0.740213\pi\)
−0.685035 + 0.728510i \(0.740213\pi\)
\(84\) 0 0
\(85\) −96.1665 −0.122714
\(86\) 0 0
\(87\) 602.455 0.742413
\(88\) 0 0
\(89\) 234.000 0.278696 0.139348 0.990243i \(-0.455499\pi\)
0.139348 + 0.990243i \(0.455499\pi\)
\(90\) 0 0
\(91\) −560.000 −0.645098
\(92\) 0 0
\(93\) 330.926 0.368983
\(94\) 0 0
\(95\) −147.078 −0.158841
\(96\) 0 0
\(97\) −178.000 −0.186321 −0.0931606 0.995651i \(-0.529697\pi\)
−0.0931606 + 0.995651i \(0.529697\pi\)
\(98\) 0 0
\(99\) −180.000 −0.182734
\(100\) 0 0
\(101\) 257.387 0.253574 0.126787 0.991930i \(-0.459534\pi\)
0.126787 + 0.991930i \(0.459534\pi\)
\(102\) 0 0
\(103\) −1886.56 −1.80474 −0.902371 0.430961i \(-0.858175\pi\)
−0.902371 + 0.430961i \(0.858175\pi\)
\(104\) 0 0
\(105\) −120.000 −0.111531
\(106\) 0 0
\(107\) −1404.00 −1.26850 −0.634251 0.773127i \(-0.718692\pi\)
−0.634251 + 0.773127i \(0.718692\pi\)
\(108\) 0 0
\(109\) −39.5980 −0.0347963 −0.0173982 0.999849i \(-0.505538\pi\)
−0.0173982 + 0.999849i \(0.505538\pi\)
\(110\) 0 0
\(111\) −814.587 −0.696551
\(112\) 0 0
\(113\) 1378.00 1.14718 0.573590 0.819143i \(-0.305550\pi\)
0.573590 + 0.819143i \(0.305550\pi\)
\(114\) 0 0
\(115\) −176.000 −0.142714
\(116\) 0 0
\(117\) 356.382 0.281603
\(118\) 0 0
\(119\) 480.833 0.370402
\(120\) 0 0
\(121\) −931.000 −0.699474
\(122\) 0 0
\(123\) −78.0000 −0.0571791
\(124\) 0 0
\(125\) −684.479 −0.489774
\(126\) 0 0
\(127\) 1790.39 1.25096 0.625480 0.780241i \(-0.284903\pi\)
0.625480 + 0.780241i \(0.284903\pi\)
\(128\) 0 0
\(129\) −756.000 −0.515985
\(130\) 0 0
\(131\) −1572.00 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 735.391 0.479447
\(134\) 0 0
\(135\) 76.3675 0.0486864
\(136\) 0 0
\(137\) 2854.00 1.77981 0.889904 0.456148i \(-0.150771\pi\)
0.889904 + 0.456148i \(0.150771\pi\)
\(138\) 0 0
\(139\) −1964.00 −1.19845 −0.599224 0.800581i \(-0.704524\pi\)
−0.599224 + 0.800581i \(0.704524\pi\)
\(140\) 0 0
\(141\) −1035.20 −0.618297
\(142\) 0 0
\(143\) −791.960 −0.463126
\(144\) 0 0
\(145\) 568.000 0.325309
\(146\) 0 0
\(147\) −429.000 −0.240703
\(148\) 0 0
\(149\) 1507.55 0.828882 0.414441 0.910076i \(-0.363977\pi\)
0.414441 + 0.910076i \(0.363977\pi\)
\(150\) 0 0
\(151\) −2265.57 −1.22099 −0.610495 0.792020i \(-0.709029\pi\)
−0.610495 + 0.792020i \(0.709029\pi\)
\(152\) 0 0
\(153\) −306.000 −0.161690
\(154\) 0 0
\(155\) 312.000 0.161680
\(156\) 0 0
\(157\) −3529.88 −1.79436 −0.897181 0.441663i \(-0.854389\pi\)
−0.897181 + 0.441663i \(0.854389\pi\)
\(158\) 0 0
\(159\) −2044.95 −1.01997
\(160\) 0 0
\(161\) 880.000 0.430768
\(162\) 0 0
\(163\) −2932.00 −1.40891 −0.704454 0.709750i \(-0.748808\pi\)
−0.704454 + 0.709750i \(0.748808\pi\)
\(164\) 0 0
\(165\) −169.706 −0.0800701
\(166\) 0 0
\(167\) 3676.96 1.70378 0.851890 0.523720i \(-0.175456\pi\)
0.851890 + 0.523720i \(0.175456\pi\)
\(168\) 0 0
\(169\) −629.000 −0.286299
\(170\) 0 0
\(171\) −468.000 −0.209292
\(172\) 0 0
\(173\) −1445.33 −0.635180 −0.317590 0.948228i \(-0.602874\pi\)
−0.317590 + 0.948228i \(0.602874\pi\)
\(174\) 0 0
\(175\) 1654.63 0.714733
\(176\) 0 0
\(177\) −1092.00 −0.463727
\(178\) 0 0
\(179\) 1308.00 0.546170 0.273085 0.961990i \(-0.411956\pi\)
0.273085 + 0.961990i \(0.411956\pi\)
\(180\) 0 0
\(181\) 1996.87 0.820034 0.410017 0.912078i \(-0.365523\pi\)
0.410017 + 0.912078i \(0.365523\pi\)
\(182\) 0 0
\(183\) 2206.17 0.891175
\(184\) 0 0
\(185\) −768.000 −0.305213
\(186\) 0 0
\(187\) 680.000 0.265917
\(188\) 0 0
\(189\) −381.838 −0.146956
\(190\) 0 0
\(191\) −939.038 −0.355740 −0.177870 0.984054i \(-0.556921\pi\)
−0.177870 + 0.984054i \(0.556921\pi\)
\(192\) 0 0
\(193\) −2490.00 −0.928674 −0.464337 0.885659i \(-0.653707\pi\)
−0.464337 + 0.885659i \(0.653707\pi\)
\(194\) 0 0
\(195\) 336.000 0.123392
\(196\) 0 0
\(197\) 2723.78 0.985081 0.492540 0.870290i \(-0.336068\pi\)
0.492540 + 0.870290i \(0.336068\pi\)
\(198\) 0 0
\(199\) −2158.09 −0.768758 −0.384379 0.923175i \(-0.625584\pi\)
−0.384379 + 0.923175i \(0.625584\pi\)
\(200\) 0 0
\(201\) −1884.00 −0.661130
\(202\) 0 0
\(203\) −2840.00 −0.981916
\(204\) 0 0
\(205\) −73.5391 −0.0250546
\(206\) 0 0
\(207\) −560.029 −0.188042
\(208\) 0 0
\(209\) 1040.00 0.344202
\(210\) 0 0
\(211\) 924.000 0.301473 0.150736 0.988574i \(-0.451836\pi\)
0.150736 + 0.988574i \(0.451836\pi\)
\(212\) 0 0
\(213\) 1001.26 0.322091
\(214\) 0 0
\(215\) −712.764 −0.226093
\(216\) 0 0
\(217\) −1560.00 −0.488017
\(218\) 0 0
\(219\) 1014.00 0.312876
\(220\) 0 0
\(221\) −1346.33 −0.409792
\(222\) 0 0
\(223\) 2276.88 0.683728 0.341864 0.939749i \(-0.388942\pi\)
0.341864 + 0.939749i \(0.388942\pi\)
\(224\) 0 0
\(225\) −1053.00 −0.312000
\(226\) 0 0
\(227\) −156.000 −0.0456127 −0.0228064 0.999740i \(-0.507260\pi\)
−0.0228064 + 0.999740i \(0.507260\pi\)
\(228\) 0 0
\(229\) −639.225 −0.184459 −0.0922296 0.995738i \(-0.529399\pi\)
−0.0922296 + 0.995738i \(0.529399\pi\)
\(230\) 0 0
\(231\) 848.528 0.241684
\(232\) 0 0
\(233\) 2826.00 0.794581 0.397291 0.917693i \(-0.369951\pi\)
0.397291 + 0.917693i \(0.369951\pi\)
\(234\) 0 0
\(235\) −976.000 −0.270924
\(236\) 0 0
\(237\) 2367.39 0.648855
\(238\) 0 0
\(239\) −2466.39 −0.667521 −0.333760 0.942658i \(-0.608318\pi\)
−0.333760 + 0.942658i \(0.608318\pi\)
\(240\) 0 0
\(241\) −3354.00 −0.896474 −0.448237 0.893915i \(-0.647948\pi\)
−0.448237 + 0.893915i \(0.647948\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −404.465 −0.105471
\(246\) 0 0
\(247\) −2059.09 −0.530433
\(248\) 0 0
\(249\) −3108.00 −0.791010
\(250\) 0 0
\(251\) 6396.00 1.60841 0.804207 0.594349i \(-0.202590\pi\)
0.804207 + 0.594349i \(0.202590\pi\)
\(252\) 0 0
\(253\) 1244.51 0.309255
\(254\) 0 0
\(255\) −288.500 −0.0708492
\(256\) 0 0
\(257\) 6882.00 1.67038 0.835189 0.549962i \(-0.185358\pi\)
0.835189 + 0.549962i \(0.185358\pi\)
\(258\) 0 0
\(259\) 3840.00 0.921259
\(260\) 0 0
\(261\) 1807.36 0.428632
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −1928.00 −0.446929
\(266\) 0 0
\(267\) 702.000 0.160905
\(268\) 0 0
\(269\) 1434.01 0.325031 0.162515 0.986706i \(-0.448039\pi\)
0.162515 + 0.986706i \(0.448039\pi\)
\(270\) 0 0
\(271\) 5942.53 1.33204 0.666020 0.745934i \(-0.267997\pi\)
0.666020 + 0.745934i \(0.267997\pi\)
\(272\) 0 0
\(273\) −1680.00 −0.372448
\(274\) 0 0
\(275\) 2340.00 0.513117
\(276\) 0 0
\(277\) −1103.09 −0.239271 −0.119635 0.992818i \(-0.538173\pi\)
−0.119635 + 0.992818i \(0.538173\pi\)
\(278\) 0 0
\(279\) 992.778 0.213032
\(280\) 0 0
\(281\) 6266.00 1.33024 0.665121 0.746735i \(-0.268380\pi\)
0.665121 + 0.746735i \(0.268380\pi\)
\(282\) 0 0
\(283\) 8596.00 1.80558 0.902790 0.430082i \(-0.141515\pi\)
0.902790 + 0.430082i \(0.141515\pi\)
\(284\) 0 0
\(285\) −441.235 −0.0917070
\(286\) 0 0
\(287\) 367.696 0.0756250
\(288\) 0 0
\(289\) −3757.00 −0.764706
\(290\) 0 0
\(291\) −534.000 −0.107573
\(292\) 0 0
\(293\) −8397.60 −1.67438 −0.837189 0.546913i \(-0.815803\pi\)
−0.837189 + 0.546913i \(0.815803\pi\)
\(294\) 0 0
\(295\) −1029.55 −0.203195
\(296\) 0 0
\(297\) −540.000 −0.105502
\(298\) 0 0
\(299\) −2464.00 −0.476578
\(300\) 0 0
\(301\) 3563.82 0.682442
\(302\) 0 0
\(303\) 772.161 0.146401
\(304\) 0 0
\(305\) 2080.00 0.390493
\(306\) 0 0
\(307\) 4940.00 0.918374 0.459187 0.888340i \(-0.348141\pi\)
0.459187 + 0.888340i \(0.348141\pi\)
\(308\) 0 0
\(309\) −5659.68 −1.04197
\(310\) 0 0
\(311\) −3382.80 −0.616788 −0.308394 0.951259i \(-0.599791\pi\)
−0.308394 + 0.951259i \(0.599791\pi\)
\(312\) 0 0
\(313\) −3106.00 −0.560899 −0.280450 0.959869i \(-0.590484\pi\)
−0.280450 + 0.959869i \(0.590484\pi\)
\(314\) 0 0
\(315\) −360.000 −0.0643927
\(316\) 0 0
\(317\) 6728.83 1.19220 0.596102 0.802909i \(-0.296715\pi\)
0.596102 + 0.802909i \(0.296715\pi\)
\(318\) 0 0
\(319\) −4016.37 −0.704932
\(320\) 0 0
\(321\) −4212.00 −0.732370
\(322\) 0 0
\(323\) 1768.00 0.304564
\(324\) 0 0
\(325\) −4632.96 −0.790740
\(326\) 0 0
\(327\) −118.794 −0.0200897
\(328\) 0 0
\(329\) 4880.00 0.817760
\(330\) 0 0
\(331\) −2908.00 −0.482895 −0.241447 0.970414i \(-0.577622\pi\)
−0.241447 + 0.970414i \(0.577622\pi\)
\(332\) 0 0
\(333\) −2443.76 −0.402154
\(334\) 0 0
\(335\) −1776.25 −0.289693
\(336\) 0 0
\(337\) 4298.00 0.694739 0.347369 0.937728i \(-0.387075\pi\)
0.347369 + 0.937728i \(0.387075\pi\)
\(338\) 0 0
\(339\) 4134.00 0.662325
\(340\) 0 0
\(341\) −2206.17 −0.350355
\(342\) 0 0
\(343\) 6873.08 1.08196
\(344\) 0 0
\(345\) −528.000 −0.0823958
\(346\) 0 0
\(347\) 9996.00 1.54644 0.773218 0.634140i \(-0.218646\pi\)
0.773218 + 0.634140i \(0.218646\pi\)
\(348\) 0 0
\(349\) 3993.74 0.612550 0.306275 0.951943i \(-0.400917\pi\)
0.306275 + 0.951943i \(0.400917\pi\)
\(350\) 0 0
\(351\) 1069.15 0.162583
\(352\) 0 0
\(353\) 6738.00 1.01594 0.507971 0.861374i \(-0.330396\pi\)
0.507971 + 0.861374i \(0.330396\pi\)
\(354\) 0 0
\(355\) 944.000 0.141133
\(356\) 0 0
\(357\) 1442.50 0.213852
\(358\) 0 0
\(359\) −2132.63 −0.313527 −0.156763 0.987636i \(-0.550106\pi\)
−0.156763 + 0.987636i \(0.550106\pi\)
\(360\) 0 0
\(361\) −4155.00 −0.605773
\(362\) 0 0
\(363\) −2793.00 −0.403842
\(364\) 0 0
\(365\) 956.008 0.137095
\(366\) 0 0
\(367\) −7628.27 −1.08499 −0.542496 0.840058i \(-0.682521\pi\)
−0.542496 + 0.840058i \(0.682521\pi\)
\(368\) 0 0
\(369\) −234.000 −0.0330123
\(370\) 0 0
\(371\) 9640.00 1.34901
\(372\) 0 0
\(373\) −8383.46 −1.16375 −0.581875 0.813278i \(-0.697681\pi\)
−0.581875 + 0.813278i \(0.697681\pi\)
\(374\) 0 0
\(375\) −2053.44 −0.282771
\(376\) 0 0
\(377\) 7952.00 1.08634
\(378\) 0 0
\(379\) 12788.0 1.73318 0.866590 0.499020i \(-0.166307\pi\)
0.866590 + 0.499020i \(0.166307\pi\)
\(380\) 0 0
\(381\) 5371.18 0.722242
\(382\) 0 0
\(383\) −2319.31 −0.309429 −0.154714 0.987959i \(-0.549446\pi\)
−0.154714 + 0.987959i \(0.549446\pi\)
\(384\) 0 0
\(385\) 800.000 0.105901
\(386\) 0 0
\(387\) −2268.00 −0.297904
\(388\) 0 0
\(389\) 2684.18 0.349854 0.174927 0.984581i \(-0.444031\pi\)
0.174927 + 0.984581i \(0.444031\pi\)
\(390\) 0 0
\(391\) 2115.66 0.273641
\(392\) 0 0
\(393\) −4716.00 −0.605320
\(394\) 0 0
\(395\) 2232.00 0.284314
\(396\) 0 0
\(397\) 2206.17 0.278903 0.139452 0.990229i \(-0.455466\pi\)
0.139452 + 0.990229i \(0.455466\pi\)
\(398\) 0 0
\(399\) 2206.17 0.276809
\(400\) 0 0
\(401\) 3582.00 0.446076 0.223038 0.974810i \(-0.428403\pi\)
0.223038 + 0.974810i \(0.428403\pi\)
\(402\) 0 0
\(403\) 4368.00 0.539915
\(404\) 0 0
\(405\) 229.103 0.0281091
\(406\) 0 0
\(407\) 5430.58 0.661385
\(408\) 0 0
\(409\) 5126.00 0.619717 0.309859 0.950783i \(-0.399718\pi\)
0.309859 + 0.950783i \(0.399718\pi\)
\(410\) 0 0
\(411\) 8562.00 1.02757
\(412\) 0 0
\(413\) 5147.74 0.613326
\(414\) 0 0
\(415\) −2930.25 −0.346603
\(416\) 0 0
\(417\) −5892.00 −0.691924
\(418\) 0 0
\(419\) −2924.00 −0.340923 −0.170462 0.985364i \(-0.554526\pi\)
−0.170462 + 0.985364i \(0.554526\pi\)
\(420\) 0 0
\(421\) 7314.31 0.846741 0.423370 0.905957i \(-0.360847\pi\)
0.423370 + 0.905957i \(0.360847\pi\)
\(422\) 0 0
\(423\) −3105.61 −0.356974
\(424\) 0 0
\(425\) 3978.00 0.454027
\(426\) 0 0
\(427\) −10400.0 −1.17867
\(428\) 0 0
\(429\) −2375.88 −0.267386
\(430\) 0 0
\(431\) 15844.8 1.77081 0.885405 0.464819i \(-0.153881\pi\)
0.885405 + 0.464819i \(0.153881\pi\)
\(432\) 0 0
\(433\) −6274.00 −0.696326 −0.348163 0.937434i \(-0.613194\pi\)
−0.348163 + 0.937434i \(0.613194\pi\)
\(434\) 0 0
\(435\) 1704.00 0.187817
\(436\) 0 0
\(437\) 3235.72 0.354200
\(438\) 0 0
\(439\) 4596.19 0.499691 0.249846 0.968286i \(-0.419620\pi\)
0.249846 + 0.968286i \(0.419620\pi\)
\(440\) 0 0
\(441\) −1287.00 −0.138970
\(442\) 0 0
\(443\) 5084.00 0.545255 0.272628 0.962120i \(-0.412107\pi\)
0.272628 + 0.962120i \(0.412107\pi\)
\(444\) 0 0
\(445\) 661.852 0.0705051
\(446\) 0 0
\(447\) 4522.65 0.478555
\(448\) 0 0
\(449\) 14190.0 1.49146 0.745732 0.666246i \(-0.232100\pi\)
0.745732 + 0.666246i \(0.232100\pi\)
\(450\) 0 0
\(451\) 520.000 0.0542923
\(452\) 0 0
\(453\) −6796.71 −0.704939
\(454\) 0 0
\(455\) −1583.92 −0.163198
\(456\) 0 0
\(457\) −6474.00 −0.662672 −0.331336 0.943513i \(-0.607499\pi\)
−0.331336 + 0.943513i \(0.607499\pi\)
\(458\) 0 0
\(459\) −918.000 −0.0933520
\(460\) 0 0
\(461\) −6321.53 −0.638662 −0.319331 0.947643i \(-0.603458\pi\)
−0.319331 + 0.947643i \(0.603458\pi\)
\(462\) 0 0
\(463\) −11435.3 −1.14783 −0.573915 0.818915i \(-0.694576\pi\)
−0.573915 + 0.818915i \(0.694576\pi\)
\(464\) 0 0
\(465\) 936.000 0.0933462
\(466\) 0 0
\(467\) 3796.00 0.376141 0.188071 0.982156i \(-0.439777\pi\)
0.188071 + 0.982156i \(0.439777\pi\)
\(468\) 0 0
\(469\) 8881.26 0.874411
\(470\) 0 0
\(471\) −10589.6 −1.03598
\(472\) 0 0
\(473\) 5040.00 0.489935
\(474\) 0 0
\(475\) 6084.00 0.587691
\(476\) 0 0
\(477\) −6134.86 −0.588880
\(478\) 0 0
\(479\) 10493.5 1.00096 0.500479 0.865749i \(-0.333157\pi\)
0.500479 + 0.865749i \(0.333157\pi\)
\(480\) 0 0
\(481\) −10752.0 −1.01923
\(482\) 0 0
\(483\) 2640.00 0.248704
\(484\) 0 0
\(485\) −503.460 −0.0471360
\(486\) 0 0
\(487\) 15406.4 1.43354 0.716769 0.697311i \(-0.245620\pi\)
0.716769 + 0.697311i \(0.245620\pi\)
\(488\) 0 0
\(489\) −8796.00 −0.813433
\(490\) 0 0
\(491\) −15452.0 −1.42024 −0.710121 0.704079i \(-0.751360\pi\)
−0.710121 + 0.704079i \(0.751360\pi\)
\(492\) 0 0
\(493\) −6827.82 −0.623752
\(494\) 0 0
\(495\) −509.117 −0.0462285
\(496\) 0 0
\(497\) −4720.00 −0.425998
\(498\) 0 0
\(499\) −52.0000 −0.00466501 −0.00233250 0.999997i \(-0.500742\pi\)
−0.00233250 + 0.999997i \(0.500742\pi\)
\(500\) 0 0
\(501\) 11030.9 0.983678
\(502\) 0 0
\(503\) −12428.1 −1.10167 −0.550837 0.834613i \(-0.685691\pi\)
−0.550837 + 0.834613i \(0.685691\pi\)
\(504\) 0 0
\(505\) 728.000 0.0641497
\(506\) 0 0
\(507\) −1887.00 −0.165295
\(508\) 0 0
\(509\) −16362.5 −1.42486 −0.712429 0.701744i \(-0.752405\pi\)
−0.712429 + 0.701744i \(0.752405\pi\)
\(510\) 0 0
\(511\) −4780.04 −0.413809
\(512\) 0 0
\(513\) −1404.00 −0.120835
\(514\) 0 0
\(515\) −5336.00 −0.456567
\(516\) 0 0
\(517\) 6901.36 0.587082
\(518\) 0 0
\(519\) −4335.98 −0.366721
\(520\) 0 0
\(521\) −714.000 −0.0600401 −0.0300201 0.999549i \(-0.509557\pi\)
−0.0300201 + 0.999549i \(0.509557\pi\)
\(522\) 0 0
\(523\) −5980.00 −0.499975 −0.249988 0.968249i \(-0.580427\pi\)
−0.249988 + 0.968249i \(0.580427\pi\)
\(524\) 0 0
\(525\) 4963.89 0.412651
\(526\) 0 0
\(527\) −3750.49 −0.310008
\(528\) 0 0
\(529\) −8295.00 −0.681762
\(530\) 0 0
\(531\) −3276.00 −0.267733
\(532\) 0 0
\(533\) −1029.55 −0.0836673
\(534\) 0 0
\(535\) −3971.11 −0.320909
\(536\) 0 0
\(537\) 3924.00 0.315332
\(538\) 0 0
\(539\) 2860.00 0.228551
\(540\) 0 0
\(541\) −13729.2 −1.09106 −0.545530 0.838091i \(-0.683672\pi\)
−0.545530 + 0.838091i \(0.683672\pi\)
\(542\) 0 0
\(543\) 5990.61 0.473447
\(544\) 0 0
\(545\) −112.000 −0.00880285
\(546\) 0 0
\(547\) −18500.0 −1.44607 −0.723037 0.690809i \(-0.757255\pi\)
−0.723037 + 0.690809i \(0.757255\pi\)
\(548\) 0 0
\(549\) 6618.52 0.514520
\(550\) 0 0
\(551\) −10442.6 −0.807382
\(552\) 0 0
\(553\) −11160.0 −0.858176
\(554\) 0 0
\(555\) −2304.00 −0.176215
\(556\) 0 0
\(557\) −8765.30 −0.666782 −0.333391 0.942789i \(-0.608193\pi\)
−0.333391 + 0.942789i \(0.608193\pi\)
\(558\) 0 0
\(559\) −9978.69 −0.755015
\(560\) 0 0
\(561\) 2040.00 0.153527
\(562\) 0 0
\(563\) −268.000 −0.0200619 −0.0100310 0.999950i \(-0.503193\pi\)
−0.0100310 + 0.999950i \(0.503193\pi\)
\(564\) 0 0
\(565\) 3897.57 0.290216
\(566\) 0 0
\(567\) −1145.51 −0.0848448
\(568\) 0 0
\(569\) −13866.0 −1.02160 −0.510802 0.859698i \(-0.670652\pi\)
−0.510802 + 0.859698i \(0.670652\pi\)
\(570\) 0 0
\(571\) 5140.00 0.376712 0.188356 0.982101i \(-0.439684\pi\)
0.188356 + 0.982101i \(0.439684\pi\)
\(572\) 0 0
\(573\) −2817.11 −0.205387
\(574\) 0 0
\(575\) 7280.37 0.528022
\(576\) 0 0
\(577\) 9386.00 0.677200 0.338600 0.940930i \(-0.390047\pi\)
0.338600 + 0.940930i \(0.390047\pi\)
\(578\) 0 0
\(579\) −7470.00 −0.536170
\(580\) 0 0
\(581\) 14651.3 1.04619
\(582\) 0 0
\(583\) 13633.0 0.968477
\(584\) 0 0
\(585\) 1008.00 0.0712405
\(586\) 0 0
\(587\) −8844.00 −0.621859 −0.310929 0.950433i \(-0.600640\pi\)
−0.310929 + 0.950433i \(0.600640\pi\)
\(588\) 0 0
\(589\) −5736.05 −0.401273
\(590\) 0 0
\(591\) 8171.33 0.568737
\(592\) 0 0
\(593\) −9406.00 −0.651363 −0.325681 0.945480i \(-0.605594\pi\)
−0.325681 + 0.945480i \(0.605594\pi\)
\(594\) 0 0
\(595\) 1360.00 0.0937051
\(596\) 0 0
\(597\) −6474.27 −0.443843
\(598\) 0 0
\(599\) 23459.0 1.60018 0.800090 0.599880i \(-0.204785\pi\)
0.800090 + 0.599880i \(0.204785\pi\)
\(600\) 0 0
\(601\) −1262.00 −0.0856540 −0.0428270 0.999083i \(-0.513636\pi\)
−0.0428270 + 0.999083i \(0.513636\pi\)
\(602\) 0 0
\(603\) −5652.00 −0.381704
\(604\) 0 0
\(605\) −2633.27 −0.176955
\(606\) 0 0
\(607\) −16288.9 −1.08920 −0.544602 0.838695i \(-0.683319\pi\)
−0.544602 + 0.838695i \(0.683319\pi\)
\(608\) 0 0
\(609\) −8520.00 −0.566909
\(610\) 0 0
\(611\) −13664.0 −0.904724
\(612\) 0 0
\(613\) −7138.95 −0.470374 −0.235187 0.971950i \(-0.575570\pi\)
−0.235187 + 0.971950i \(0.575570\pi\)
\(614\) 0 0
\(615\) −220.617 −0.0144653
\(616\) 0 0
\(617\) 16874.0 1.10101 0.550504 0.834833i \(-0.314436\pi\)
0.550504 + 0.834833i \(0.314436\pi\)
\(618\) 0 0
\(619\) −20748.0 −1.34723 −0.673613 0.739085i \(-0.735258\pi\)
−0.673613 + 0.739085i \(0.735258\pi\)
\(620\) 0 0
\(621\) −1680.09 −0.108566
\(622\) 0 0
\(623\) −3309.26 −0.212813
\(624\) 0 0
\(625\) 12689.0 0.812096
\(626\) 0 0
\(627\) 3120.00 0.198725
\(628\) 0 0
\(629\) 9231.99 0.585220
\(630\) 0 0
\(631\) 14840.8 0.936294 0.468147 0.883651i \(-0.344922\pi\)
0.468147 + 0.883651i \(0.344922\pi\)
\(632\) 0 0
\(633\) 2772.00 0.174055
\(634\) 0 0
\(635\) 5064.00 0.316470
\(636\) 0 0
\(637\) −5662.51 −0.352209
\(638\) 0 0
\(639\) 3003.79 0.185959
\(640\) 0 0
\(641\) 17758.0 1.09423 0.547113 0.837059i \(-0.315727\pi\)
0.547113 + 0.837059i \(0.315727\pi\)
\(642\) 0 0
\(643\) 1148.00 0.0704086 0.0352043 0.999380i \(-0.488792\pi\)
0.0352043 + 0.999380i \(0.488792\pi\)
\(644\) 0 0
\(645\) −2138.29 −0.130535
\(646\) 0 0
\(647\) −26988.9 −1.63994 −0.819970 0.572406i \(-0.806010\pi\)
−0.819970 + 0.572406i \(0.806010\pi\)
\(648\) 0 0
\(649\) 7280.00 0.440316
\(650\) 0 0
\(651\) −4680.00 −0.281757
\(652\) 0 0
\(653\) 21069.0 1.26262 0.631311 0.775530i \(-0.282517\pi\)
0.631311 + 0.775530i \(0.282517\pi\)
\(654\) 0 0
\(655\) −4446.29 −0.265238
\(656\) 0 0
\(657\) 3042.00 0.180639
\(658\) 0 0
\(659\) −18356.0 −1.08505 −0.542525 0.840040i \(-0.682532\pi\)
−0.542525 + 0.840040i \(0.682532\pi\)
\(660\) 0 0
\(661\) 15250.9 0.897414 0.448707 0.893679i \(-0.351885\pi\)
0.448707 + 0.893679i \(0.351885\pi\)
\(662\) 0 0
\(663\) −4038.99 −0.236594
\(664\) 0 0
\(665\) 2080.00 0.121292
\(666\) 0 0
\(667\) −12496.0 −0.725408
\(668\) 0 0
\(669\) 6830.65 0.394751
\(670\) 0 0
\(671\) −14707.8 −0.846184
\(672\) 0 0
\(673\) −12082.0 −0.692016 −0.346008 0.938232i \(-0.612463\pi\)
−0.346008 + 0.938232i \(0.612463\pi\)
\(674\) 0 0
\(675\) −3159.00 −0.180133
\(676\) 0 0
\(677\) 12742.1 0.723364 0.361682 0.932302i \(-0.382203\pi\)
0.361682 + 0.932302i \(0.382203\pi\)
\(678\) 0 0
\(679\) 2517.30 0.142276
\(680\) 0 0
\(681\) −468.000 −0.0263345
\(682\) 0 0
\(683\) −33508.0 −1.87723 −0.938615 0.344967i \(-0.887890\pi\)
−0.938615 + 0.344967i \(0.887890\pi\)
\(684\) 0 0
\(685\) 8072.33 0.450260
\(686\) 0 0
\(687\) −1917.67 −0.106498
\(688\) 0 0
\(689\) −26992.0 −1.49247
\(690\) 0 0
\(691\) 364.000 0.0200394 0.0100197 0.999950i \(-0.496811\pi\)
0.0100197 + 0.999950i \(0.496811\pi\)
\(692\) 0 0
\(693\) 2545.58 0.139536
\(694\) 0 0
\(695\) −5555.03 −0.303186
\(696\) 0 0
\(697\) 884.000 0.0480400
\(698\) 0 0
\(699\) 8478.00 0.458752
\(700\) 0 0
\(701\) −3849.49 −0.207408 −0.103704 0.994608i \(-0.533070\pi\)
−0.103704 + 0.994608i \(0.533070\pi\)
\(702\) 0 0
\(703\) 14119.5 0.757507
\(704\) 0 0
\(705\) −2928.00 −0.156418
\(706\) 0 0
\(707\) −3640.00 −0.193630
\(708\) 0 0
\(709\) −23606.1 −1.25041 −0.625207 0.780459i \(-0.714986\pi\)
−0.625207 + 0.780459i \(0.714986\pi\)
\(710\) 0 0
\(711\) 7102.18 0.374617
\(712\) 0 0
\(713\) −6864.00 −0.360531
\(714\) 0 0
\(715\) −2240.00 −0.117163
\(716\) 0 0
\(717\) −7399.17 −0.385393
\(718\) 0 0
\(719\) 15799.6 0.819507 0.409753 0.912196i \(-0.365615\pi\)
0.409753 + 0.912196i \(0.365615\pi\)
\(720\) 0 0
\(721\) 26680.0 1.37811
\(722\) 0 0
\(723\) −10062.0 −0.517579
\(724\) 0 0
\(725\) −23495.7 −1.20360
\(726\) 0 0
\(727\) −4607.51 −0.235052 −0.117526 0.993070i \(-0.537496\pi\)
−0.117526 + 0.993070i \(0.537496\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 8568.00 0.433514
\(732\) 0 0
\(733\) 26219.5 1.32120 0.660600 0.750738i \(-0.270302\pi\)
0.660600 + 0.750738i \(0.270302\pi\)
\(734\) 0 0
\(735\) −1213.40 −0.0608935
\(736\) 0 0
\(737\) 12560.0 0.627752
\(738\) 0 0
\(739\) −27924.0 −1.38999 −0.694994 0.719016i \(-0.744593\pi\)
−0.694994 + 0.719016i \(0.744593\pi\)
\(740\) 0 0
\(741\) −6177.28 −0.306246
\(742\) 0 0
\(743\) 8937.83 0.441315 0.220658 0.975351i \(-0.429180\pi\)
0.220658 + 0.975351i \(0.429180\pi\)
\(744\) 0 0
\(745\) 4264.00 0.209692
\(746\) 0 0
\(747\) −9324.00 −0.456690
\(748\) 0 0
\(749\) 19855.6 0.968633
\(750\) 0 0
\(751\) 14082.7 0.684270 0.342135 0.939651i \(-0.388850\pi\)
0.342135 + 0.939651i \(0.388850\pi\)
\(752\) 0 0
\(753\) 19188.0 0.928618
\(754\) 0 0
\(755\) −6408.00 −0.308889
\(756\) 0 0
\(757\) −14871.9 −0.714039 −0.357019 0.934097i \(-0.616207\pi\)
−0.357019 + 0.934097i \(0.616207\pi\)
\(758\) 0 0
\(759\) 3733.52 0.178549
\(760\) 0 0
\(761\) −15834.0 −0.754247 −0.377124 0.926163i \(-0.623087\pi\)
−0.377124 + 0.926163i \(0.623087\pi\)
\(762\) 0 0
\(763\) 560.000 0.0265706
\(764\) 0 0
\(765\) −865.499 −0.0409048
\(766\) 0 0
\(767\) −14413.7 −0.678549
\(768\) 0 0
\(769\) −16666.0 −0.781523 −0.390762 0.920492i \(-0.627788\pi\)
−0.390762 + 0.920492i \(0.627788\pi\)
\(770\) 0 0
\(771\) 20646.0 0.964394
\(772\) 0 0
\(773\) −30957.1 −1.44043 −0.720214 0.693752i \(-0.755956\pi\)
−0.720214 + 0.693752i \(0.755956\pi\)
\(774\) 0 0
\(775\) −12906.1 −0.598195
\(776\) 0 0
\(777\) 11520.0 0.531889
\(778\) 0 0
\(779\) 1352.00 0.0621828
\(780\) 0 0
\(781\) −6675.09 −0.305830
\(782\) 0 0
\(783\) 5422.09 0.247471
\(784\) 0 0
\(785\) −9984.00 −0.453942
\(786\) 0 0
\(787\) −20228.0 −0.916201 −0.458101 0.888900i \(-0.651470\pi\)
−0.458101 + 0.888900i \(0.651470\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −19487.9 −0.875991
\(792\) 0 0
\(793\) 29120.0 1.30401
\(794\) 0 0
\(795\) −5784.00 −0.258034
\(796\) 0 0
\(797\) 9008.54 0.400375 0.200187 0.979758i \(-0.435845\pi\)
0.200187 + 0.979758i \(0.435845\pi\)
\(798\) 0 0
\(799\) 11732.3 0.519474
\(800\) 0 0
\(801\) 2106.00 0.0928987
\(802\) 0 0
\(803\) −6760.00 −0.297080
\(804\) 0 0
\(805\) 2489.02 0.108977
\(806\) 0 0
\(807\) 4302.04 0.187657
\(808\) 0 0
\(809\) −9242.00 −0.401646 −0.200823 0.979628i \(-0.564362\pi\)
−0.200823 + 0.979628i \(0.564362\pi\)
\(810\) 0 0
\(811\) −10972.0 −0.475067 −0.237533 0.971379i \(-0.576339\pi\)
−0.237533 + 0.971379i \(0.576339\pi\)
\(812\) 0 0
\(813\) 17827.6 0.769053
\(814\) 0 0
\(815\) −8292.95 −0.356429
\(816\) 0 0
\(817\) 13104.0 0.561139
\(818\) 0 0
\(819\) −5040.00 −0.215033
\(820\) 0 0
\(821\) −9336.64 −0.396895 −0.198448 0.980112i \(-0.563590\pi\)
−0.198448 + 0.980112i \(0.563590\pi\)
\(822\) 0 0
\(823\) −3566.65 −0.151064 −0.0755319 0.997143i \(-0.524065\pi\)
−0.0755319 + 0.997143i \(0.524065\pi\)
\(824\) 0 0
\(825\) 7020.00 0.296249
\(826\) 0 0
\(827\) −18876.0 −0.793691 −0.396846 0.917885i \(-0.629895\pi\)
−0.396846 + 0.917885i \(0.629895\pi\)
\(828\) 0 0
\(829\) 6974.90 0.292218 0.146109 0.989269i \(-0.453325\pi\)
0.146109 + 0.989269i \(0.453325\pi\)
\(830\) 0 0
\(831\) −3309.26 −0.138143
\(832\) 0 0
\(833\) 4862.00 0.202231
\(834\) 0 0
\(835\) 10400.0 0.431026
\(836\) 0 0
\(837\) 2978.33 0.122994
\(838\) 0 0
\(839\) 30077.5 1.23765 0.618826 0.785528i \(-0.287608\pi\)
0.618826 + 0.785528i \(0.287608\pi\)
\(840\) 0 0
\(841\) 15939.0 0.653532
\(842\) 0 0
\(843\) 18798.0 0.768016
\(844\) 0 0
\(845\) −1779.08 −0.0724287
\(846\) 0 0
\(847\) 13166.3 0.534121
\(848\) 0 0
\(849\) 25788.0 1.04245
\(850\) 0 0
\(851\) 16896.0 0.680596
\(852\) 0 0
\(853\) −41159.3 −1.65213 −0.826065 0.563575i \(-0.809426\pi\)
−0.826065 + 0.563575i \(0.809426\pi\)
\(854\) 0 0
\(855\) −1323.70 −0.0529470
\(856\) 0 0
\(857\) −25194.0 −1.00421 −0.502107 0.864806i \(-0.667441\pi\)
−0.502107 + 0.864806i \(0.667441\pi\)
\(858\) 0 0
\(859\) −9308.00 −0.369715 −0.184857 0.982765i \(-0.559182\pi\)
−0.184857 + 0.982765i \(0.559182\pi\)
\(860\) 0 0
\(861\) 1103.09 0.0436621
\(862\) 0 0
\(863\) 26802.2 1.05719 0.528596 0.848874i \(-0.322719\pi\)
0.528596 + 0.848874i \(0.322719\pi\)
\(864\) 0 0
\(865\) −4088.00 −0.160689
\(866\) 0 0
\(867\) −11271.0 −0.441503
\(868\) 0 0
\(869\) −15782.6 −0.616098
\(870\) 0 0
\(871\) −24867.5 −0.967399
\(872\) 0 0
\(873\) −1602.00 −0.0621071
\(874\) 0 0
\(875\) 9680.00 0.373993
\(876\) 0 0
\(877\) 1436.84 0.0553235 0.0276617 0.999617i \(-0.491194\pi\)
0.0276617 + 0.999617i \(0.491194\pi\)
\(878\) 0 0
\(879\) −25192.8 −0.966703
\(880\) 0 0
\(881\) −42830.0 −1.63789 −0.818944 0.573873i \(-0.805440\pi\)
−0.818944 + 0.573873i \(0.805440\pi\)
\(882\) 0 0
\(883\) 23964.0 0.913310 0.456655 0.889644i \(-0.349047\pi\)
0.456655 + 0.889644i \(0.349047\pi\)
\(884\) 0 0
\(885\) −3088.64 −0.117315
\(886\) 0 0
\(887\) 28239.0 1.06897 0.534483 0.845179i \(-0.320506\pi\)
0.534483 + 0.845179i \(0.320506\pi\)
\(888\) 0 0
\(889\) −25320.0 −0.955237
\(890\) 0 0
\(891\) −1620.00 −0.0609114
\(892\) 0 0
\(893\) 17943.5 0.672405
\(894\) 0 0
\(895\) 3699.58 0.138171
\(896\) 0 0
\(897\) −7392.00 −0.275152
\(898\) 0 0
\(899\) 22152.0 0.821814
\(900\) 0 0
\(901\) 23176.1 0.856947
\(902\) 0 0
\(903\) 10691.5 0.394008
\(904\) 0 0
\(905\) 5648.00 0.207454
\(906\) 0 0
\(907\) 31972.0 1.17047 0.585233 0.810865i \(-0.301003\pi\)
0.585233 + 0.810865i \(0.301003\pi\)
\(908\) 0 0
\(909\) 2316.48 0.0845246
\(910\) 0 0
\(911\) −26858.7 −0.976806 −0.488403 0.872618i \(-0.662420\pi\)
−0.488403 + 0.872618i \(0.662420\pi\)
\(912\) 0 0
\(913\) 20720.0 0.751075
\(914\) 0 0
\(915\) 6240.00 0.225451
\(916\) 0 0
\(917\) 22231.4 0.800596
\(918\) 0 0
\(919\) −40336.2 −1.44784 −0.723922 0.689882i \(-0.757662\pi\)
−0.723922 + 0.689882i \(0.757662\pi\)
\(920\) 0 0
\(921\) 14820.0 0.530223
\(922\) 0 0
\(923\) 13216.0 0.471300
\(924\) 0 0
\(925\) 31768.9 1.12925
\(926\) 0 0
\(927\) −16979.0 −0.601580
\(928\) 0 0
\(929\) −13650.0 −0.482069 −0.241034 0.970517i \(-0.577487\pi\)
−0.241034 + 0.970517i \(0.577487\pi\)
\(930\) 0 0
\(931\) 7436.00 0.261767
\(932\) 0 0
\(933\) −10148.4 −0.356102
\(934\) 0 0
\(935\) 1923.33 0.0672723
\(936\) 0 0
\(937\) 7098.00 0.247472 0.123736 0.992315i \(-0.460512\pi\)
0.123736 + 0.992315i \(0.460512\pi\)
\(938\) 0 0
\(939\) −9318.00 −0.323835
\(940\) 0 0
\(941\) −41326.1 −1.43166 −0.715831 0.698274i \(-0.753952\pi\)
−0.715831 + 0.698274i \(0.753952\pi\)
\(942\) 0 0
\(943\) 1617.86 0.0558693
\(944\) 0 0
\(945\) −1080.00 −0.0371771
\(946\) 0 0
\(947\) 9900.00 0.339711 0.169856 0.985469i \(-0.445670\pi\)
0.169856 + 0.985469i \(0.445670\pi\)
\(948\) 0 0
\(949\) 13384.1 0.457815
\(950\) 0 0
\(951\) 20186.5 0.688319
\(952\) 0 0
\(953\) −46938.0 −1.59546 −0.797729 0.603016i \(-0.793965\pi\)
−0.797729 + 0.603016i \(0.793965\pi\)
\(954\) 0 0
\(955\) −2656.00 −0.0899960
\(956\) 0 0
\(957\) −12049.1 −0.406993
\(958\) 0 0
\(959\) −40361.7 −1.35907
\(960\) 0 0
\(961\) −17623.0 −0.591554
\(962\) 0 0
\(963\) −12636.0 −0.422834
\(964\) 0 0
\(965\) −7042.78 −0.234938
\(966\) 0 0
\(967\) −6989.04 −0.232422 −0.116211 0.993225i \(-0.537075\pi\)
−0.116211 + 0.993225i \(0.537075\pi\)
\(968\) 0 0
\(969\) 5304.00 0.175840
\(970\) 0 0
\(971\) 53052.0 1.75337 0.876684 0.481067i \(-0.159751\pi\)
0.876684 + 0.481067i \(0.159751\pi\)
\(972\) 0 0
\(973\) 27775.2 0.915139
\(974\) 0 0
\(975\) −13898.9 −0.456534
\(976\) 0 0
\(977\) −41890.0 −1.37173 −0.685865 0.727729i \(-0.740576\pi\)
−0.685865 + 0.727729i \(0.740576\pi\)
\(978\) 0 0
\(979\) −4680.00 −0.152782
\(980\) 0 0
\(981\) −356.382 −0.0115988
\(982\) 0 0
\(983\) 10861.2 0.352408 0.176204 0.984354i \(-0.443618\pi\)
0.176204 + 0.984354i \(0.443618\pi\)
\(984\) 0 0
\(985\) 7704.00 0.249208
\(986\) 0 0
\(987\) 14640.0 0.472134
\(988\) 0 0
\(989\) 15680.8 0.504166
\(990\) 0 0
\(991\) −330.926 −0.0106077 −0.00530384 0.999986i \(-0.501688\pi\)
−0.00530384 + 0.999986i \(0.501688\pi\)
\(992\) 0 0
\(993\) −8724.00 −0.278799
\(994\) 0 0
\(995\) −6104.00 −0.194482
\(996\) 0 0
\(997\) 39948.7 1.26900 0.634498 0.772925i \(-0.281207\pi\)
0.634498 + 0.772925i \(0.281207\pi\)
\(998\) 0 0
\(999\) −7331.28 −0.232184
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 768.4.a.m.1.2 2
3.2 odd 2 2304.4.a.bh.1.1 2
4.3 odd 2 768.4.a.h.1.2 2
8.3 odd 2 inner 768.4.a.m.1.1 2
8.5 even 2 768.4.a.h.1.1 2
12.11 even 2 2304.4.a.bb.1.1 2
16.3 odd 4 384.4.d.d.193.1 4
16.5 even 4 384.4.d.d.193.2 yes 4
16.11 odd 4 384.4.d.d.193.4 yes 4
16.13 even 4 384.4.d.d.193.3 yes 4
24.5 odd 2 2304.4.a.bb.1.2 2
24.11 even 2 2304.4.a.bh.1.2 2
48.5 odd 4 1152.4.d.n.577.2 4
48.11 even 4 1152.4.d.n.577.1 4
48.29 odd 4 1152.4.d.n.577.4 4
48.35 even 4 1152.4.d.n.577.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.d.193.1 4 16.3 odd 4
384.4.d.d.193.2 yes 4 16.5 even 4
384.4.d.d.193.3 yes 4 16.13 even 4
384.4.d.d.193.4 yes 4 16.11 odd 4
768.4.a.h.1.1 2 8.5 even 2
768.4.a.h.1.2 2 4.3 odd 2
768.4.a.m.1.1 2 8.3 odd 2 inner
768.4.a.m.1.2 2 1.1 even 1 trivial
1152.4.d.n.577.1 4 48.11 even 4
1152.4.d.n.577.2 4 48.5 odd 4
1152.4.d.n.577.3 4 48.35 even 4
1152.4.d.n.577.4 4 48.29 odd 4
2304.4.a.bb.1.1 2 12.11 even 2
2304.4.a.bb.1.2 2 24.5 odd 2
2304.4.a.bh.1.1 2 3.2 odd 2
2304.4.a.bh.1.2 2 24.11 even 2