Properties

Label 2151.4.a.h.1.20
Level $2151$
Weight $4$
Character 2151.1
Self dual yes
Analytic conductor $126.913$
Analytic rank $0$
Dimension $59$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(126.913108422\)
Analytic rank: \(0\)
Dimension: \(59\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.92673 q^{2} -4.28770 q^{4} -4.54299 q^{5} -14.1149 q^{7} +23.6751 q^{8} +O(q^{10})\) \(q-1.92673 q^{2} -4.28770 q^{4} -4.54299 q^{5} -14.1149 q^{7} +23.6751 q^{8} +8.75313 q^{10} -28.0304 q^{11} -31.9364 q^{13} +27.1957 q^{14} -11.3141 q^{16} -71.9548 q^{17} +110.437 q^{19} +19.4790 q^{20} +54.0072 q^{22} +55.0861 q^{23} -104.361 q^{25} +61.5328 q^{26} +60.5204 q^{28} +12.1306 q^{29} +19.8759 q^{31} -167.602 q^{32} +138.638 q^{34} +64.1239 q^{35} -243.779 q^{37} -212.783 q^{38} -107.556 q^{40} -349.249 q^{41} -243.760 q^{43} +120.186 q^{44} -106.136 q^{46} -96.5623 q^{47} -143.769 q^{49} +201.076 q^{50} +136.933 q^{52} -728.153 q^{53} +127.342 q^{55} -334.172 q^{56} -23.3724 q^{58} +259.682 q^{59} +616.386 q^{61} -38.2956 q^{62} +413.436 q^{64} +145.087 q^{65} -34.5550 q^{67} +308.520 q^{68} -123.550 q^{70} +110.119 q^{71} -908.850 q^{73} +469.697 q^{74} -473.522 q^{76} +395.647 q^{77} +82.5884 q^{79} +51.3996 q^{80} +672.910 q^{82} -616.389 q^{83} +326.890 q^{85} +469.661 q^{86} -663.624 q^{88} +700.181 q^{89} +450.779 q^{91} -236.193 q^{92} +186.050 q^{94} -501.716 q^{95} -1734.66 q^{97} +277.005 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 59q + 8q^{2} + 238q^{4} + 80q^{5} - 10q^{7} + 96q^{8} + O(q^{10}) \) \( 59q + 8q^{2} + 238q^{4} + 80q^{5} - 10q^{7} + 96q^{8} - 36q^{10} + 132q^{11} + 104q^{13} + 280q^{14} + 822q^{16} + 408q^{17} + 20q^{19} + 800q^{20} - 2q^{22} + 276q^{23} + 1477q^{25} + 780q^{26} + 224q^{28} + 696q^{29} - 380q^{31} + 896q^{32} - 72q^{34} + 700q^{35} + 224q^{37} + 988q^{38} - 258q^{40} + 2706q^{41} - 156q^{43} + 1584q^{44} + 428q^{46} + 1316q^{47} + 2135q^{49} + 1400q^{50} + 1092q^{52} + 1484q^{53} - 992q^{55} + 3360q^{56} - 120q^{58} + 3186q^{59} - 254q^{61} + 1240q^{62} + 3054q^{64} + 5120q^{65} + 288q^{67} + 9420q^{68} + 1108q^{70} + 4468q^{71} - 1770q^{73} + 6214q^{74} + 720q^{76} + 6352q^{77} - 746q^{79} + 7040q^{80} + 276q^{82} + 5484q^{83} + 588q^{85} + 10152q^{86} + 1186q^{88} + 11570q^{89} + 1768q^{91} + 15366q^{92} - 2142q^{94} + 5736q^{95} + 2390q^{97} + 6912q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.92673 −0.681203 −0.340602 0.940208i \(-0.610631\pi\)
−0.340602 + 0.940208i \(0.610631\pi\)
\(3\) 0 0
\(4\) −4.28770 −0.535962
\(5\) −4.54299 −0.406337 −0.203169 0.979144i \(-0.565124\pi\)
−0.203169 + 0.979144i \(0.565124\pi\)
\(6\) 0 0
\(7\) −14.1149 −0.762133 −0.381067 0.924548i \(-0.624443\pi\)
−0.381067 + 0.924548i \(0.624443\pi\)
\(8\) 23.6751 1.04630
\(9\) 0 0
\(10\) 8.75313 0.276798
\(11\) −28.0304 −0.768318 −0.384159 0.923267i \(-0.625509\pi\)
−0.384159 + 0.923267i \(0.625509\pi\)
\(12\) 0 0
\(13\) −31.9364 −0.681350 −0.340675 0.940181i \(-0.610656\pi\)
−0.340675 + 0.940181i \(0.610656\pi\)
\(14\) 27.1957 0.519167
\(15\) 0 0
\(16\) −11.3141 −0.176782
\(17\) −71.9548 −1.02656 −0.513282 0.858220i \(-0.671571\pi\)
−0.513282 + 0.858220i \(0.671571\pi\)
\(18\) 0 0
\(19\) 110.437 1.33348 0.666739 0.745292i \(-0.267690\pi\)
0.666739 + 0.745292i \(0.267690\pi\)
\(20\) 19.4790 0.217781
\(21\) 0 0
\(22\) 54.0072 0.523381
\(23\) 55.0861 0.499402 0.249701 0.968323i \(-0.419668\pi\)
0.249701 + 0.968323i \(0.419668\pi\)
\(24\) 0 0
\(25\) −104.361 −0.834890
\(26\) 61.5328 0.464138
\(27\) 0 0
\(28\) 60.5204 0.408475
\(29\) 12.1306 0.0776755 0.0388378 0.999246i \(-0.487634\pi\)
0.0388378 + 0.999246i \(0.487634\pi\)
\(30\) 0 0
\(31\) 19.8759 0.115156 0.0575778 0.998341i \(-0.481662\pi\)
0.0575778 + 0.998341i \(0.481662\pi\)
\(32\) −167.602 −0.925878
\(33\) 0 0
\(34\) 138.638 0.699299
\(35\) 64.1239 0.309683
\(36\) 0 0
\(37\) −243.779 −1.08316 −0.541581 0.840649i \(-0.682174\pi\)
−0.541581 + 0.840649i \(0.682174\pi\)
\(38\) −212.783 −0.908369
\(39\) 0 0
\(40\) −107.556 −0.425152
\(41\) −349.249 −1.33033 −0.665165 0.746697i \(-0.731639\pi\)
−0.665165 + 0.746697i \(0.731639\pi\)
\(42\) 0 0
\(43\) −243.760 −0.864490 −0.432245 0.901756i \(-0.642278\pi\)
−0.432245 + 0.901756i \(0.642278\pi\)
\(44\) 120.186 0.411789
\(45\) 0 0
\(46\) −106.136 −0.340194
\(47\) −96.5623 −0.299682 −0.149841 0.988710i \(-0.547876\pi\)
−0.149841 + 0.988710i \(0.547876\pi\)
\(48\) 0 0
\(49\) −143.769 −0.419153
\(50\) 201.076 0.568730
\(51\) 0 0
\(52\) 136.933 0.365178
\(53\) −728.153 −1.88716 −0.943580 0.331145i \(-0.892565\pi\)
−0.943580 + 0.331145i \(0.892565\pi\)
\(54\) 0 0
\(55\) 127.342 0.312196
\(56\) −334.172 −0.797422
\(57\) 0 0
\(58\) −23.3724 −0.0529128
\(59\) 259.682 0.573013 0.286506 0.958078i \(-0.407506\pi\)
0.286506 + 0.958078i \(0.407506\pi\)
\(60\) 0 0
\(61\) 616.386 1.29377 0.646886 0.762586i \(-0.276071\pi\)
0.646886 + 0.762586i \(0.276071\pi\)
\(62\) −38.2956 −0.0784444
\(63\) 0 0
\(64\) 413.436 0.807493
\(65\) 145.087 0.276858
\(66\) 0 0
\(67\) −34.5550 −0.0630084 −0.0315042 0.999504i \(-0.510030\pi\)
−0.0315042 + 0.999504i \(0.510030\pi\)
\(68\) 308.520 0.550200
\(69\) 0 0
\(70\) −123.550 −0.210957
\(71\) 110.119 0.184067 0.0920335 0.995756i \(-0.470663\pi\)
0.0920335 + 0.995756i \(0.470663\pi\)
\(72\) 0 0
\(73\) −908.850 −1.45716 −0.728581 0.684959i \(-0.759820\pi\)
−0.728581 + 0.684959i \(0.759820\pi\)
\(74\) 469.697 0.737853
\(75\) 0 0
\(76\) −473.522 −0.714694
\(77\) 395.647 0.585561
\(78\) 0 0
\(79\) 82.5884 0.117619 0.0588097 0.998269i \(-0.481269\pi\)
0.0588097 + 0.998269i \(0.481269\pi\)
\(80\) 51.3996 0.0718332
\(81\) 0 0
\(82\) 672.910 0.906225
\(83\) −616.389 −0.815150 −0.407575 0.913172i \(-0.633625\pi\)
−0.407575 + 0.913172i \(0.633625\pi\)
\(84\) 0 0
\(85\) 326.890 0.417131
\(86\) 469.661 0.588893
\(87\) 0 0
\(88\) −663.624 −0.803893
\(89\) 700.181 0.833922 0.416961 0.908924i \(-0.363095\pi\)
0.416961 + 0.908924i \(0.363095\pi\)
\(90\) 0 0
\(91\) 450.779 0.519279
\(92\) −236.193 −0.267661
\(93\) 0 0
\(94\) 186.050 0.204144
\(95\) −501.716 −0.541842
\(96\) 0 0
\(97\) −1734.66 −1.81576 −0.907879 0.419233i \(-0.862299\pi\)
−0.907879 + 0.419233i \(0.862299\pi\)
\(98\) 277.005 0.285528
\(99\) 0 0
\(100\) 447.470 0.447470
\(101\) 124.059 0.122221 0.0611104 0.998131i \(-0.480536\pi\)
0.0611104 + 0.998131i \(0.480536\pi\)
\(102\) 0 0
\(103\) −1711.43 −1.63721 −0.818604 0.574358i \(-0.805252\pi\)
−0.818604 + 0.574358i \(0.805252\pi\)
\(104\) −756.097 −0.712898
\(105\) 0 0
\(106\) 1402.96 1.28554
\(107\) 1312.09 1.18547 0.592733 0.805399i \(-0.298049\pi\)
0.592733 + 0.805399i \(0.298049\pi\)
\(108\) 0 0
\(109\) −1547.57 −1.35991 −0.679955 0.733254i \(-0.738001\pi\)
−0.679955 + 0.733254i \(0.738001\pi\)
\(110\) −245.354 −0.212669
\(111\) 0 0
\(112\) 159.697 0.134732
\(113\) −1304.61 −1.08608 −0.543040 0.839707i \(-0.682727\pi\)
−0.543040 + 0.839707i \(0.682727\pi\)
\(114\) 0 0
\(115\) −250.256 −0.202926
\(116\) −52.0122 −0.0416312
\(117\) 0 0
\(118\) −500.339 −0.390338
\(119\) 1015.63 0.782379
\(120\) 0 0
\(121\) −545.294 −0.409688
\(122\) −1187.61 −0.881322
\(123\) 0 0
\(124\) −85.2221 −0.0617191
\(125\) 1041.99 0.745584
\(126\) 0 0
\(127\) 479.592 0.335094 0.167547 0.985864i \(-0.446415\pi\)
0.167547 + 0.985864i \(0.446415\pi\)
\(128\) 544.233 0.375811
\(129\) 0 0
\(130\) −279.543 −0.188596
\(131\) 1823.07 1.21590 0.607949 0.793976i \(-0.291992\pi\)
0.607949 + 0.793976i \(0.291992\pi\)
\(132\) 0 0
\(133\) −1558.81 −1.01629
\(134\) 66.5782 0.0429215
\(135\) 0 0
\(136\) −1703.54 −1.07410
\(137\) −1887.11 −1.17684 −0.588419 0.808556i \(-0.700249\pi\)
−0.588419 + 0.808556i \(0.700249\pi\)
\(138\) 0 0
\(139\) 465.291 0.283924 0.141962 0.989872i \(-0.454659\pi\)
0.141962 + 0.989872i \(0.454659\pi\)
\(140\) −274.944 −0.165978
\(141\) 0 0
\(142\) −212.170 −0.125387
\(143\) 895.190 0.523493
\(144\) 0 0
\(145\) −55.1091 −0.0315625
\(146\) 1751.11 0.992624
\(147\) 0 0
\(148\) 1045.25 0.580534
\(149\) 2487.89 1.36789 0.683947 0.729532i \(-0.260262\pi\)
0.683947 + 0.729532i \(0.260262\pi\)
\(150\) 0 0
\(151\) −3323.76 −1.79129 −0.895643 0.444774i \(-0.853284\pi\)
−0.895643 + 0.444774i \(0.853284\pi\)
\(152\) 2614.62 1.39522
\(153\) 0 0
\(154\) −762.306 −0.398886
\(155\) −90.2962 −0.0467920
\(156\) 0 0
\(157\) −720.418 −0.366214 −0.183107 0.983093i \(-0.558615\pi\)
−0.183107 + 0.983093i \(0.558615\pi\)
\(158\) −159.126 −0.0801226
\(159\) 0 0
\(160\) 761.413 0.376219
\(161\) −777.535 −0.380611
\(162\) 0 0
\(163\) 1108.25 0.532545 0.266272 0.963898i \(-0.414208\pi\)
0.266272 + 0.963898i \(0.414208\pi\)
\(164\) 1497.47 0.713007
\(165\) 0 0
\(166\) 1187.62 0.555283
\(167\) −1676.79 −0.776968 −0.388484 0.921456i \(-0.627001\pi\)
−0.388484 + 0.921456i \(0.627001\pi\)
\(168\) 0 0
\(169\) −1177.07 −0.535762
\(170\) −629.829 −0.284151
\(171\) 0 0
\(172\) 1045.17 0.463334
\(173\) −3900.26 −1.71405 −0.857026 0.515274i \(-0.827690\pi\)
−0.857026 + 0.515274i \(0.827690\pi\)
\(174\) 0 0
\(175\) 1473.05 0.636297
\(176\) 317.138 0.135825
\(177\) 0 0
\(178\) −1349.06 −0.568070
\(179\) 663.778 0.277168 0.138584 0.990351i \(-0.455745\pi\)
0.138584 + 0.990351i \(0.455745\pi\)
\(180\) 0 0
\(181\) 2098.72 0.861859 0.430929 0.902386i \(-0.358186\pi\)
0.430929 + 0.902386i \(0.358186\pi\)
\(182\) −868.530 −0.353735
\(183\) 0 0
\(184\) 1304.17 0.522526
\(185\) 1107.48 0.440129
\(186\) 0 0
\(187\) 2016.92 0.788728
\(188\) 414.030 0.160618
\(189\) 0 0
\(190\) 966.673 0.369104
\(191\) 302.401 0.114560 0.0572801 0.998358i \(-0.481757\pi\)
0.0572801 + 0.998358i \(0.481757\pi\)
\(192\) 0 0
\(193\) −3757.87 −1.40154 −0.700770 0.713387i \(-0.747160\pi\)
−0.700770 + 0.713387i \(0.747160\pi\)
\(194\) 3342.24 1.23690
\(195\) 0 0
\(196\) 616.440 0.224650
\(197\) 2992.66 1.08233 0.541163 0.840918i \(-0.317984\pi\)
0.541163 + 0.840918i \(0.317984\pi\)
\(198\) 0 0
\(199\) −574.375 −0.204605 −0.102302 0.994753i \(-0.532621\pi\)
−0.102302 + 0.994753i \(0.532621\pi\)
\(200\) −2470.76 −0.873547
\(201\) 0 0
\(202\) −239.028 −0.0832572
\(203\) −171.222 −0.0591991
\(204\) 0 0
\(205\) 1586.63 0.540563
\(206\) 3297.47 1.11527
\(207\) 0 0
\(208\) 361.330 0.120451
\(209\) −3095.61 −1.02453
\(210\) 0 0
\(211\) −1130.97 −0.369002 −0.184501 0.982832i \(-0.559067\pi\)
−0.184501 + 0.982832i \(0.559067\pi\)
\(212\) 3122.10 1.01145
\(213\) 0 0
\(214\) −2528.05 −0.807543
\(215\) 1107.40 0.351274
\(216\) 0 0
\(217\) −280.547 −0.0877639
\(218\) 2981.75 0.926375
\(219\) 0 0
\(220\) −546.004 −0.167325
\(221\) 2297.97 0.699450
\(222\) 0 0
\(223\) 4758.26 1.42886 0.714432 0.699705i \(-0.246685\pi\)
0.714432 + 0.699705i \(0.246685\pi\)
\(224\) 2365.68 0.705642
\(225\) 0 0
\(226\) 2513.63 0.739841
\(227\) 4921.32 1.43894 0.719470 0.694523i \(-0.244385\pi\)
0.719470 + 0.694523i \(0.244385\pi\)
\(228\) 0 0
\(229\) −5415.04 −1.56260 −0.781301 0.624154i \(-0.785444\pi\)
−0.781301 + 0.624154i \(0.785444\pi\)
\(230\) 482.176 0.138234
\(231\) 0 0
\(232\) 287.193 0.0812721
\(233\) 4829.08 1.35778 0.678892 0.734238i \(-0.262460\pi\)
0.678892 + 0.734238i \(0.262460\pi\)
\(234\) 0 0
\(235\) 438.682 0.121772
\(236\) −1113.44 −0.307113
\(237\) 0 0
\(238\) −1956.86 −0.532959
\(239\) −239.000 −0.0646846
\(240\) 0 0
\(241\) 5048.84 1.34948 0.674739 0.738056i \(-0.264256\pi\)
0.674739 + 0.738056i \(0.264256\pi\)
\(242\) 1050.64 0.279080
\(243\) 0 0
\(244\) −2642.88 −0.693413
\(245\) 653.143 0.170318
\(246\) 0 0
\(247\) −3526.97 −0.908565
\(248\) 470.565 0.120488
\(249\) 0 0
\(250\) −2007.63 −0.507894
\(251\) 6977.00 1.75452 0.877259 0.480017i \(-0.159370\pi\)
0.877259 + 0.480017i \(0.159370\pi\)
\(252\) 0 0
\(253\) −1544.09 −0.383700
\(254\) −924.046 −0.228267
\(255\) 0 0
\(256\) −4356.08 −1.06350
\(257\) −7599.68 −1.84457 −0.922286 0.386507i \(-0.873681\pi\)
−0.922286 + 0.386507i \(0.873681\pi\)
\(258\) 0 0
\(259\) 3440.91 0.825514
\(260\) −622.087 −0.148385
\(261\) 0 0
\(262\) −3512.58 −0.828274
\(263\) 1590.80 0.372977 0.186489 0.982457i \(-0.440289\pi\)
0.186489 + 0.982457i \(0.440289\pi\)
\(264\) 0 0
\(265\) 3307.99 0.766823
\(266\) 3003.42 0.692298
\(267\) 0 0
\(268\) 148.161 0.0337701
\(269\) −4255.95 −0.964645 −0.482323 0.875994i \(-0.660207\pi\)
−0.482323 + 0.875994i \(0.660207\pi\)
\(270\) 0 0
\(271\) 8311.36 1.86302 0.931511 0.363713i \(-0.118491\pi\)
0.931511 + 0.363713i \(0.118491\pi\)
\(272\) 814.100 0.181478
\(273\) 0 0
\(274\) 3635.96 0.801665
\(275\) 2925.29 0.641461
\(276\) 0 0
\(277\) 3888.34 0.843421 0.421710 0.906731i \(-0.361430\pi\)
0.421710 + 0.906731i \(0.361430\pi\)
\(278\) −896.492 −0.193410
\(279\) 0 0
\(280\) 1518.14 0.324022
\(281\) 2351.52 0.499217 0.249609 0.968347i \(-0.419698\pi\)
0.249609 + 0.968347i \(0.419698\pi\)
\(282\) 0 0
\(283\) −1830.05 −0.384400 −0.192200 0.981356i \(-0.561562\pi\)
−0.192200 + 0.981356i \(0.561562\pi\)
\(284\) −472.158 −0.0986529
\(285\) 0 0
\(286\) −1724.79 −0.356605
\(287\) 4929.61 1.01389
\(288\) 0 0
\(289\) 264.488 0.0538344
\(290\) 106.180 0.0215005
\(291\) 0 0
\(292\) 3896.88 0.780984
\(293\) 4939.16 0.984808 0.492404 0.870367i \(-0.336118\pi\)
0.492404 + 0.870367i \(0.336118\pi\)
\(294\) 0 0
\(295\) −1179.73 −0.232837
\(296\) −5771.49 −1.13331
\(297\) 0 0
\(298\) −4793.51 −0.931814
\(299\) −1759.25 −0.340268
\(300\) 0 0
\(301\) 3440.65 0.658856
\(302\) 6404.01 1.22023
\(303\) 0 0
\(304\) −1249.49 −0.235735
\(305\) −2800.24 −0.525708
\(306\) 0 0
\(307\) −5082.19 −0.944808 −0.472404 0.881382i \(-0.656614\pi\)
−0.472404 + 0.881382i \(0.656614\pi\)
\(308\) −1696.42 −0.313838
\(309\) 0 0
\(310\) 173.977 0.0318749
\(311\) −249.779 −0.0455424 −0.0227712 0.999741i \(-0.507249\pi\)
−0.0227712 + 0.999741i \(0.507249\pi\)
\(312\) 0 0
\(313\) 1335.72 0.241213 0.120606 0.992700i \(-0.461516\pi\)
0.120606 + 0.992700i \(0.461516\pi\)
\(314\) 1388.05 0.249466
\(315\) 0 0
\(316\) −354.114 −0.0630395
\(317\) 4099.50 0.726344 0.363172 0.931722i \(-0.381694\pi\)
0.363172 + 0.931722i \(0.381694\pi\)
\(318\) 0 0
\(319\) −340.025 −0.0596795
\(320\) −1878.24 −0.328115
\(321\) 0 0
\(322\) 1498.10 0.259273
\(323\) −7946.50 −1.36890
\(324\) 0 0
\(325\) 3332.92 0.568852
\(326\) −2135.30 −0.362771
\(327\) 0 0
\(328\) −8268.51 −1.39193
\(329\) 1362.97 0.228398
\(330\) 0 0
\(331\) −704.646 −0.117012 −0.0585058 0.998287i \(-0.518634\pi\)
−0.0585058 + 0.998287i \(0.518634\pi\)
\(332\) 2642.89 0.436890
\(333\) 0 0
\(334\) 3230.72 0.529273
\(335\) 156.983 0.0256027
\(336\) 0 0
\(337\) −3937.55 −0.636474 −0.318237 0.948011i \(-0.603091\pi\)
−0.318237 + 0.948011i \(0.603091\pi\)
\(338\) 2267.90 0.364963
\(339\) 0 0
\(340\) −1401.60 −0.223567
\(341\) −557.132 −0.0884761
\(342\) 0 0
\(343\) 6870.70 1.08158
\(344\) −5771.05 −0.904518
\(345\) 0 0
\(346\) 7514.75 1.16762
\(347\) 4188.64 0.648006 0.324003 0.946056i \(-0.394971\pi\)
0.324003 + 0.946056i \(0.394971\pi\)
\(348\) 0 0
\(349\) −5622.54 −0.862372 −0.431186 0.902263i \(-0.641905\pi\)
−0.431186 + 0.902263i \(0.641905\pi\)
\(350\) −2838.17 −0.433448
\(351\) 0 0
\(352\) 4697.95 0.711368
\(353\) −10770.7 −1.62399 −0.811994 0.583665i \(-0.801618\pi\)
−0.811994 + 0.583665i \(0.801618\pi\)
\(354\) 0 0
\(355\) −500.271 −0.0747933
\(356\) −3002.17 −0.446951
\(357\) 0 0
\(358\) −1278.92 −0.188808
\(359\) 4653.73 0.684163 0.342081 0.939670i \(-0.388868\pi\)
0.342081 + 0.939670i \(0.388868\pi\)
\(360\) 0 0
\(361\) 5337.41 0.778162
\(362\) −4043.67 −0.587101
\(363\) 0 0
\(364\) −1932.80 −0.278314
\(365\) 4128.90 0.592099
\(366\) 0 0
\(367\) −12201.2 −1.73541 −0.867706 0.497077i \(-0.834407\pi\)
−0.867706 + 0.497077i \(0.834407\pi\)
\(368\) −623.247 −0.0882854
\(369\) 0 0
\(370\) −2133.83 −0.299817
\(371\) 10277.8 1.43827
\(372\) 0 0
\(373\) 5045.08 0.700333 0.350167 0.936687i \(-0.386125\pi\)
0.350167 + 0.936687i \(0.386125\pi\)
\(374\) −3886.07 −0.537284
\(375\) 0 0
\(376\) −2286.12 −0.313558
\(377\) −387.406 −0.0529242
\(378\) 0 0
\(379\) −4999.72 −0.677621 −0.338810 0.940855i \(-0.610024\pi\)
−0.338810 + 0.940855i \(0.610024\pi\)
\(380\) 2151.21 0.290407
\(381\) 0 0
\(382\) −582.647 −0.0780387
\(383\) 10546.4 1.40704 0.703521 0.710674i \(-0.251610\pi\)
0.703521 + 0.710674i \(0.251610\pi\)
\(384\) 0 0
\(385\) −1797.42 −0.237935
\(386\) 7240.41 0.954734
\(387\) 0 0
\(388\) 7437.72 0.973177
\(389\) 11923.9 1.55415 0.777075 0.629408i \(-0.216702\pi\)
0.777075 + 0.629408i \(0.216702\pi\)
\(390\) 0 0
\(391\) −3963.71 −0.512668
\(392\) −3403.76 −0.438561
\(393\) 0 0
\(394\) −5766.06 −0.737284
\(395\) −375.198 −0.0477931
\(396\) 0 0
\(397\) 4067.23 0.514178 0.257089 0.966388i \(-0.417237\pi\)
0.257089 + 0.966388i \(0.417237\pi\)
\(398\) 1106.67 0.139377
\(399\) 0 0
\(400\) 1180.75 0.147594
\(401\) 14266.9 1.77670 0.888350 0.459168i \(-0.151852\pi\)
0.888350 + 0.459168i \(0.151852\pi\)
\(402\) 0 0
\(403\) −634.765 −0.0784613
\(404\) −531.926 −0.0655057
\(405\) 0 0
\(406\) 329.899 0.0403266
\(407\) 6833.23 0.832213
\(408\) 0 0
\(409\) 8198.77 0.991205 0.495603 0.868549i \(-0.334947\pi\)
0.495603 + 0.868549i \(0.334947\pi\)
\(410\) −3057.02 −0.368233
\(411\) 0 0
\(412\) 7338.11 0.877482
\(413\) −3665.39 −0.436712
\(414\) 0 0
\(415\) 2800.25 0.331226
\(416\) 5352.59 0.630847
\(417\) 0 0
\(418\) 5964.41 0.697916
\(419\) 3509.69 0.409212 0.204606 0.978844i \(-0.434409\pi\)
0.204606 + 0.978844i \(0.434409\pi\)
\(420\) 0 0
\(421\) −1228.24 −0.142188 −0.0710938 0.997470i \(-0.522649\pi\)
−0.0710938 + 0.997470i \(0.522649\pi\)
\(422\) 2179.09 0.251365
\(423\) 0 0
\(424\) −17239.1 −1.97454
\(425\) 7509.29 0.857068
\(426\) 0 0
\(427\) −8700.23 −0.986027
\(428\) −5625.86 −0.635365
\(429\) 0 0
\(430\) −2133.66 −0.239289
\(431\) 15746.8 1.75985 0.879926 0.475111i \(-0.157592\pi\)
0.879926 + 0.475111i \(0.157592\pi\)
\(432\) 0 0
\(433\) 15416.5 1.71101 0.855507 0.517792i \(-0.173246\pi\)
0.855507 + 0.517792i \(0.173246\pi\)
\(434\) 540.539 0.0597851
\(435\) 0 0
\(436\) 6635.51 0.728860
\(437\) 6083.57 0.665941
\(438\) 0 0
\(439\) −13029.2 −1.41651 −0.708256 0.705956i \(-0.750518\pi\)
−0.708256 + 0.705956i \(0.750518\pi\)
\(440\) 3014.84 0.326652
\(441\) 0 0
\(442\) −4427.58 −0.476467
\(443\) −8891.43 −0.953599 −0.476800 0.879012i \(-0.658203\pi\)
−0.476800 + 0.879012i \(0.658203\pi\)
\(444\) 0 0
\(445\) −3180.92 −0.338854
\(446\) −9167.90 −0.973346
\(447\) 0 0
\(448\) −5835.61 −0.615417
\(449\) 12338.8 1.29689 0.648447 0.761260i \(-0.275419\pi\)
0.648447 + 0.761260i \(0.275419\pi\)
\(450\) 0 0
\(451\) 9789.60 1.02212
\(452\) 5593.76 0.582098
\(453\) 0 0
\(454\) −9482.07 −0.980211
\(455\) −2047.88 −0.211003
\(456\) 0 0
\(457\) −14357.2 −1.46958 −0.734791 0.678293i \(-0.762720\pi\)
−0.734791 + 0.678293i \(0.762720\pi\)
\(458\) 10433.3 1.06445
\(459\) 0 0
\(460\) 1073.02 0.108761
\(461\) −13873.5 −1.40164 −0.700818 0.713341i \(-0.747181\pi\)
−0.700818 + 0.713341i \(0.747181\pi\)
\(462\) 0 0
\(463\) 6720.52 0.674577 0.337288 0.941401i \(-0.390490\pi\)
0.337288 + 0.941401i \(0.390490\pi\)
\(464\) −137.246 −0.0137316
\(465\) 0 0
\(466\) −9304.36 −0.924927
\(467\) −13853.0 −1.37267 −0.686337 0.727284i \(-0.740782\pi\)
−0.686337 + 0.727284i \(0.740782\pi\)
\(468\) 0 0
\(469\) 487.740 0.0480208
\(470\) −845.222 −0.0829515
\(471\) 0 0
\(472\) 6148.01 0.599545
\(473\) 6832.70 0.664203
\(474\) 0 0
\(475\) −11525.4 −1.11331
\(476\) −4354.73 −0.419325
\(477\) 0 0
\(478\) 460.489 0.0440634
\(479\) −6813.41 −0.649922 −0.324961 0.945727i \(-0.605351\pi\)
−0.324961 + 0.945727i \(0.605351\pi\)
\(480\) 0 0
\(481\) 7785.41 0.738012
\(482\) −9727.76 −0.919269
\(483\) 0 0
\(484\) 2338.06 0.219577
\(485\) 7880.56 0.737810
\(486\) 0 0
\(487\) 6464.04 0.601465 0.300733 0.953708i \(-0.402769\pi\)
0.300733 + 0.953708i \(0.402769\pi\)
\(488\) 14593.0 1.35368
\(489\) 0 0
\(490\) −1258.43 −0.116021
\(491\) −4063.25 −0.373467 −0.186733 0.982411i \(-0.559790\pi\)
−0.186733 + 0.982411i \(0.559790\pi\)
\(492\) 0 0
\(493\) −872.852 −0.0797389
\(494\) 6795.53 0.618917
\(495\) 0 0
\(496\) −224.878 −0.0203575
\(497\) −1554.32 −0.140284
\(498\) 0 0
\(499\) 15417.9 1.38317 0.691583 0.722297i \(-0.256913\pi\)
0.691583 + 0.722297i \(0.256913\pi\)
\(500\) −4467.72 −0.399605
\(501\) 0 0
\(502\) −13442.8 −1.19518
\(503\) −10985.5 −0.973792 −0.486896 0.873460i \(-0.661871\pi\)
−0.486896 + 0.873460i \(0.661871\pi\)
\(504\) 0 0
\(505\) −563.597 −0.0496629
\(506\) 2975.05 0.261377
\(507\) 0 0
\(508\) −2056.35 −0.179598
\(509\) 12010.2 1.04586 0.522929 0.852376i \(-0.324839\pi\)
0.522929 + 0.852376i \(0.324839\pi\)
\(510\) 0 0
\(511\) 12828.3 1.11055
\(512\) 4039.15 0.348646
\(513\) 0 0
\(514\) 14642.6 1.25653
\(515\) 7775.02 0.665259
\(516\) 0 0
\(517\) 2706.68 0.230251
\(518\) −6629.73 −0.562343
\(519\) 0 0
\(520\) 3434.94 0.289677
\(521\) −6895.43 −0.579836 −0.289918 0.957052i \(-0.593628\pi\)
−0.289918 + 0.957052i \(0.593628\pi\)
\(522\) 0 0
\(523\) 16667.0 1.39349 0.696747 0.717317i \(-0.254630\pi\)
0.696747 + 0.717317i \(0.254630\pi\)
\(524\) −7816.79 −0.651676
\(525\) 0 0
\(526\) −3065.05 −0.254073
\(527\) −1430.17 −0.118215
\(528\) 0 0
\(529\) −9132.52 −0.750598
\(530\) −6373.62 −0.522363
\(531\) 0 0
\(532\) 6683.72 0.544692
\(533\) 11153.7 0.906420
\(534\) 0 0
\(535\) −5960.83 −0.481699
\(536\) −818.093 −0.0659258
\(537\) 0 0
\(538\) 8200.07 0.657119
\(539\) 4029.92 0.322043
\(540\) 0 0
\(541\) 5011.21 0.398242 0.199121 0.979975i \(-0.436191\pi\)
0.199121 + 0.979975i \(0.436191\pi\)
\(542\) −16013.8 −1.26910
\(543\) 0 0
\(544\) 12059.7 0.950473
\(545\) 7030.59 0.552582
\(546\) 0 0
\(547\) 22306.8 1.74364 0.871819 0.489828i \(-0.162940\pi\)
0.871819 + 0.489828i \(0.162940\pi\)
\(548\) 8091.36 0.630740
\(549\) 0 0
\(550\) −5636.26 −0.436965
\(551\) 1339.67 0.103579
\(552\) 0 0
\(553\) −1165.73 −0.0896416
\(554\) −7491.79 −0.574541
\(555\) 0 0
\(556\) −1995.03 −0.152173
\(557\) −8349.63 −0.635162 −0.317581 0.948231i \(-0.602871\pi\)
−0.317581 + 0.948231i \(0.602871\pi\)
\(558\) 0 0
\(559\) 7784.81 0.589020
\(560\) −725.501 −0.0547464
\(561\) 0 0
\(562\) −4530.75 −0.340068
\(563\) −10064.6 −0.753417 −0.376708 0.926332i \(-0.622944\pi\)
−0.376708 + 0.926332i \(0.622944\pi\)
\(564\) 0 0
\(565\) 5926.81 0.441315
\(566\) 3526.02 0.261854
\(567\) 0 0
\(568\) 2607.09 0.192590
\(569\) 1651.40 0.121670 0.0608350 0.998148i \(-0.480624\pi\)
0.0608350 + 0.998148i \(0.480624\pi\)
\(570\) 0 0
\(571\) 23898.3 1.75151 0.875757 0.482752i \(-0.160363\pi\)
0.875757 + 0.482752i \(0.160363\pi\)
\(572\) −3838.31 −0.280573
\(573\) 0 0
\(574\) −9498.05 −0.690664
\(575\) −5748.86 −0.416946
\(576\) 0 0
\(577\) 10181.4 0.734586 0.367293 0.930105i \(-0.380285\pi\)
0.367293 + 0.930105i \(0.380285\pi\)
\(578\) −509.598 −0.0366721
\(579\) 0 0
\(580\) 236.291 0.0169163
\(581\) 8700.27 0.621253
\(582\) 0 0
\(583\) 20410.4 1.44994
\(584\) −21517.1 −1.52463
\(585\) 0 0
\(586\) −9516.44 −0.670854
\(587\) 9971.33 0.701126 0.350563 0.936539i \(-0.385990\pi\)
0.350563 + 0.936539i \(0.385990\pi\)
\(588\) 0 0
\(589\) 2195.05 0.153557
\(590\) 2273.03 0.158609
\(591\) 0 0
\(592\) 2758.13 0.191484
\(593\) −16799.7 −1.16337 −0.581686 0.813414i \(-0.697607\pi\)
−0.581686 + 0.813414i \(0.697607\pi\)
\(594\) 0 0
\(595\) −4614.02 −0.317910
\(596\) −10667.3 −0.733140
\(597\) 0 0
\(598\) 3389.60 0.231791
\(599\) 4059.29 0.276892 0.138446 0.990370i \(-0.455789\pi\)
0.138446 + 0.990370i \(0.455789\pi\)
\(600\) 0 0
\(601\) −2424.31 −0.164542 −0.0822711 0.996610i \(-0.526217\pi\)
−0.0822711 + 0.996610i \(0.526217\pi\)
\(602\) −6629.21 −0.448815
\(603\) 0 0
\(604\) 14251.3 0.960062
\(605\) 2477.27 0.166471
\(606\) 0 0
\(607\) 17901.3 1.19702 0.598510 0.801115i \(-0.295760\pi\)
0.598510 + 0.801115i \(0.295760\pi\)
\(608\) −18509.5 −1.23464
\(609\) 0 0
\(610\) 5395.31 0.358114
\(611\) 3083.85 0.204188
\(612\) 0 0
\(613\) 3978.84 0.262160 0.131080 0.991372i \(-0.458156\pi\)
0.131080 + 0.991372i \(0.458156\pi\)
\(614\) 9792.03 0.643606
\(615\) 0 0
\(616\) 9366.99 0.612673
\(617\) 14969.5 0.976741 0.488371 0.872636i \(-0.337591\pi\)
0.488371 + 0.872636i \(0.337591\pi\)
\(618\) 0 0
\(619\) −19925.6 −1.29383 −0.646913 0.762564i \(-0.723940\pi\)
−0.646913 + 0.762564i \(0.723940\pi\)
\(620\) 387.163 0.0250788
\(621\) 0 0
\(622\) 481.258 0.0310236
\(623\) −9882.99 −0.635560
\(624\) 0 0
\(625\) 8311.43 0.531931
\(626\) −2573.58 −0.164315
\(627\) 0 0
\(628\) 3088.93 0.196277
\(629\) 17541.0 1.11194
\(630\) 0 0
\(631\) −14340.9 −0.904757 −0.452379 0.891826i \(-0.649424\pi\)
−0.452379 + 0.891826i \(0.649424\pi\)
\(632\) 1955.29 0.123065
\(633\) 0 0
\(634\) −7898.65 −0.494788
\(635\) −2178.78 −0.136161
\(636\) 0 0
\(637\) 4591.47 0.285590
\(638\) 655.138 0.0406539
\(639\) 0 0
\(640\) −2472.44 −0.152706
\(641\) −3101.58 −0.191116 −0.0955578 0.995424i \(-0.530463\pi\)
−0.0955578 + 0.995424i \(0.530463\pi\)
\(642\) 0 0
\(643\) −24212.2 −1.48497 −0.742486 0.669861i \(-0.766354\pi\)
−0.742486 + 0.669861i \(0.766354\pi\)
\(644\) 3333.84 0.203993
\(645\) 0 0
\(646\) 15310.8 0.932499
\(647\) 14136.6 0.858992 0.429496 0.903069i \(-0.358691\pi\)
0.429496 + 0.903069i \(0.358691\pi\)
\(648\) 0 0
\(649\) −7279.01 −0.440256
\(650\) −6421.64 −0.387504
\(651\) 0 0
\(652\) −4751.84 −0.285424
\(653\) −8407.94 −0.503872 −0.251936 0.967744i \(-0.581067\pi\)
−0.251936 + 0.967744i \(0.581067\pi\)
\(654\) 0 0
\(655\) −8282.21 −0.494065
\(656\) 3951.42 0.235178
\(657\) 0 0
\(658\) −2626.08 −0.155585
\(659\) 406.791 0.0240460 0.0120230 0.999928i \(-0.496173\pi\)
0.0120230 + 0.999928i \(0.496173\pi\)
\(660\) 0 0
\(661\) −22192.4 −1.30588 −0.652939 0.757410i \(-0.726464\pi\)
−0.652939 + 0.757410i \(0.726464\pi\)
\(662\) 1357.66 0.0797086
\(663\) 0 0
\(664\) −14593.1 −0.852893
\(665\) 7081.67 0.412955
\(666\) 0 0
\(667\) 668.226 0.0387913
\(668\) 7189.55 0.416425
\(669\) 0 0
\(670\) −302.464 −0.0174406
\(671\) −17277.6 −0.994029
\(672\) 0 0
\(673\) −17824.1 −1.02090 −0.510452 0.859906i \(-0.670522\pi\)
−0.510452 + 0.859906i \(0.670522\pi\)
\(674\) 7586.60 0.433568
\(675\) 0 0
\(676\) 5046.92 0.287148
\(677\) −27332.0 −1.55163 −0.775815 0.630961i \(-0.782661\pi\)
−0.775815 + 0.630961i \(0.782661\pi\)
\(678\) 0 0
\(679\) 24484.6 1.38385
\(680\) 7739.15 0.436446
\(681\) 0 0
\(682\) 1073.44 0.0602702
\(683\) −31853.8 −1.78456 −0.892278 0.451486i \(-0.850894\pi\)
−0.892278 + 0.451486i \(0.850894\pi\)
\(684\) 0 0
\(685\) 8573.12 0.478193
\(686\) −13238.0 −0.736778
\(687\) 0 0
\(688\) 2757.91 0.152826
\(689\) 23254.5 1.28582
\(690\) 0 0
\(691\) 24069.9 1.32513 0.662564 0.749006i \(-0.269468\pi\)
0.662564 + 0.749006i \(0.269468\pi\)
\(692\) 16723.1 0.918667
\(693\) 0 0
\(694\) −8070.40 −0.441424
\(695\) −2113.81 −0.115369
\(696\) 0 0
\(697\) 25130.1 1.36567
\(698\) 10833.1 0.587450
\(699\) 0 0
\(700\) −6315.99 −0.341031
\(701\) 610.991 0.0329199 0.0164599 0.999865i \(-0.494760\pi\)
0.0164599 + 0.999865i \(0.494760\pi\)
\(702\) 0 0
\(703\) −26922.3 −1.44437
\(704\) −11588.8 −0.620411
\(705\) 0 0
\(706\) 20752.3 1.10627
\(707\) −1751.08 −0.0931485
\(708\) 0 0
\(709\) −15273.0 −0.809014 −0.404507 0.914535i \(-0.632557\pi\)
−0.404507 + 0.914535i \(0.632557\pi\)
\(710\) 963.888 0.0509494
\(711\) 0 0
\(712\) 16576.9 0.872535
\(713\) 1094.89 0.0575090
\(714\) 0 0
\(715\) −4066.84 −0.212715
\(716\) −2846.08 −0.148552
\(717\) 0 0
\(718\) −8966.49 −0.466054
\(719\) −27319.3 −1.41702 −0.708510 0.705700i \(-0.750632\pi\)
−0.708510 + 0.705700i \(0.750632\pi\)
\(720\) 0 0
\(721\) 24156.7 1.24777
\(722\) −10283.8 −0.530087
\(723\) 0 0
\(724\) −8998.67 −0.461924
\(725\) −1265.96 −0.0648505
\(726\) 0 0
\(727\) 15530.0 0.792262 0.396131 0.918194i \(-0.370353\pi\)
0.396131 + 0.918194i \(0.370353\pi\)
\(728\) 10672.2 0.543323
\(729\) 0 0
\(730\) −7955.28 −0.403340
\(731\) 17539.7 0.887454
\(732\) 0 0
\(733\) 18585.2 0.936507 0.468254 0.883594i \(-0.344883\pi\)
0.468254 + 0.883594i \(0.344883\pi\)
\(734\) 23508.4 1.18217
\(735\) 0 0
\(736\) −9232.53 −0.462385
\(737\) 968.591 0.0484105
\(738\) 0 0
\(739\) −27867.1 −1.38715 −0.693577 0.720382i \(-0.743966\pi\)
−0.693577 + 0.720382i \(0.743966\pi\)
\(740\) −4748.56 −0.235893
\(741\) 0 0
\(742\) −19802.6 −0.979752
\(743\) −26465.7 −1.30677 −0.653387 0.757024i \(-0.726653\pi\)
−0.653387 + 0.757024i \(0.726653\pi\)
\(744\) 0 0
\(745\) −11302.5 −0.555826
\(746\) −9720.52 −0.477069
\(747\) 0 0
\(748\) −8647.96 −0.422728
\(749\) −18520.1 −0.903483
\(750\) 0 0
\(751\) −17010.6 −0.826533 −0.413267 0.910610i \(-0.635612\pi\)
−0.413267 + 0.910610i \(0.635612\pi\)
\(752\) 1092.51 0.0529784
\(753\) 0 0
\(754\) 746.428 0.0360522
\(755\) 15099.8 0.727866
\(756\) 0 0
\(757\) −16202.7 −0.777934 −0.388967 0.921252i \(-0.627168\pi\)
−0.388967 + 0.921252i \(0.627168\pi\)
\(758\) 9633.12 0.461597
\(759\) 0 0
\(760\) −11878.2 −0.566930
\(761\) 13718.1 0.653457 0.326729 0.945118i \(-0.394054\pi\)
0.326729 + 0.945118i \(0.394054\pi\)
\(762\) 0 0
\(763\) 21843.8 1.03643
\(764\) −1296.61 −0.0613999
\(765\) 0 0
\(766\) −20320.2 −0.958482
\(767\) −8293.31 −0.390422
\(768\) 0 0
\(769\) −19139.3 −0.897505 −0.448752 0.893656i \(-0.648131\pi\)
−0.448752 + 0.893656i \(0.648131\pi\)
\(770\) 3463.15 0.162082
\(771\) 0 0
\(772\) 16112.6 0.751173
\(773\) 1454.06 0.0676572 0.0338286 0.999428i \(-0.489230\pi\)
0.0338286 + 0.999428i \(0.489230\pi\)
\(774\) 0 0
\(775\) −2074.28 −0.0961423
\(776\) −41068.4 −1.89983
\(777\) 0 0
\(778\) −22974.1 −1.05869
\(779\) −38570.1 −1.77396
\(780\) 0 0
\(781\) −3086.69 −0.141422
\(782\) 7637.01 0.349231
\(783\) 0 0
\(784\) 1626.62 0.0740988
\(785\) 3272.85 0.148806
\(786\) 0 0
\(787\) −34703.9 −1.57187 −0.785934 0.618311i \(-0.787817\pi\)
−0.785934 + 0.618311i \(0.787817\pi\)
\(788\) −12831.6 −0.580086
\(789\) 0 0
\(790\) 722.907 0.0325568
\(791\) 18414.4 0.827738
\(792\) 0 0
\(793\) −19685.1 −0.881512
\(794\) −7836.47 −0.350260
\(795\) 0 0
\(796\) 2462.75 0.109660
\(797\) −5327.34 −0.236768 −0.118384 0.992968i \(-0.537771\pi\)
−0.118384 + 0.992968i \(0.537771\pi\)
\(798\) 0 0
\(799\) 6948.12 0.307643
\(800\) 17491.1 0.773006
\(801\) 0 0
\(802\) −27488.6 −1.21029
\(803\) 25475.5 1.11956
\(804\) 0 0
\(805\) 3532.33 0.154656
\(806\) 1223.02 0.0534481
\(807\) 0 0
\(808\) 2937.10 0.127880
\(809\) 23677.3 1.02899 0.514494 0.857494i \(-0.327980\pi\)
0.514494 + 0.857494i \(0.327980\pi\)
\(810\) 0 0
\(811\) −24380.5 −1.05563 −0.527814 0.849360i \(-0.676988\pi\)
−0.527814 + 0.849360i \(0.676988\pi\)
\(812\) 734.148 0.0317285
\(813\) 0 0
\(814\) −13165.8 −0.566906
\(815\) −5034.76 −0.216393
\(816\) 0 0
\(817\) −26920.2 −1.15278
\(818\) −15796.8 −0.675212
\(819\) 0 0
\(820\) −6803.01 −0.289721
\(821\) 21296.0 0.905283 0.452641 0.891693i \(-0.350482\pi\)
0.452641 + 0.891693i \(0.350482\pi\)
\(822\) 0 0
\(823\) 4908.88 0.207913 0.103957 0.994582i \(-0.466850\pi\)
0.103957 + 0.994582i \(0.466850\pi\)
\(824\) −40518.4 −1.71301
\(825\) 0 0
\(826\) 7062.23 0.297490
\(827\) 38572.6 1.62189 0.810944 0.585124i \(-0.198954\pi\)
0.810944 + 0.585124i \(0.198954\pi\)
\(828\) 0 0
\(829\) 30118.0 1.26181 0.630905 0.775860i \(-0.282684\pi\)
0.630905 + 0.775860i \(0.282684\pi\)
\(830\) −5395.33 −0.225632
\(831\) 0 0
\(832\) −13203.7 −0.550185
\(833\) 10344.9 0.430288
\(834\) 0 0
\(835\) 7617.62 0.315711
\(836\) 13273.0 0.549112
\(837\) 0 0
\(838\) −6762.25 −0.278756
\(839\) −24615.7 −1.01291 −0.506454 0.862267i \(-0.669044\pi\)
−0.506454 + 0.862267i \(0.669044\pi\)
\(840\) 0 0
\(841\) −24241.8 −0.993967
\(842\) 2366.50 0.0968586
\(843\) 0 0
\(844\) 4849.28 0.197771
\(845\) 5347.41 0.217700
\(846\) 0 0
\(847\) 7696.78 0.312237
\(848\) 8238.36 0.333616
\(849\) 0 0
\(850\) −14468.4 −0.583838
\(851\) −13428.8 −0.540933
\(852\) 0 0
\(853\) 24728.8 0.992614 0.496307 0.868147i \(-0.334689\pi\)
0.496307 + 0.868147i \(0.334689\pi\)
\(854\) 16763.0 0.671685
\(855\) 0 0
\(856\) 31064.0 1.24036
\(857\) 1605.12 0.0639788 0.0319894 0.999488i \(-0.489816\pi\)
0.0319894 + 0.999488i \(0.489816\pi\)
\(858\) 0 0
\(859\) −43424.7 −1.72483 −0.862417 0.506199i \(-0.831050\pi\)
−0.862417 + 0.506199i \(0.831050\pi\)
\(860\) −4748.19 −0.188270
\(861\) 0 0
\(862\) −30339.9 −1.19882
\(863\) 28312.7 1.11678 0.558388 0.829580i \(-0.311420\pi\)
0.558388 + 0.829580i \(0.311420\pi\)
\(864\) 0 0
\(865\) 17718.8 0.696483
\(866\) −29703.4 −1.16555
\(867\) 0 0
\(868\) 1202.90 0.0470382
\(869\) −2314.99 −0.0903690
\(870\) 0 0
\(871\) 1103.56 0.0429308
\(872\) −36638.9 −1.42288
\(873\) 0 0
\(874\) −11721.4 −0.453641
\(875\) −14707.5 −0.568234
\(876\) 0 0
\(877\) 16723.1 0.643897 0.321948 0.946757i \(-0.395662\pi\)
0.321948 + 0.946757i \(0.395662\pi\)
\(878\) 25103.7 0.964932
\(879\) 0 0
\(880\) −1440.75 −0.0551907
\(881\) −986.914 −0.0377412 −0.0188706 0.999822i \(-0.506007\pi\)
−0.0188706 + 0.999822i \(0.506007\pi\)
\(882\) 0 0
\(883\) 21840.2 0.832368 0.416184 0.909280i \(-0.363367\pi\)
0.416184 + 0.909280i \(0.363367\pi\)
\(884\) −9853.01 −0.374879
\(885\) 0 0
\(886\) 17131.4 0.649595
\(887\) −629.867 −0.0238431 −0.0119216 0.999929i \(-0.503795\pi\)
−0.0119216 + 0.999929i \(0.503795\pi\)
\(888\) 0 0
\(889\) −6769.40 −0.255386
\(890\) 6128.78 0.230828
\(891\) 0 0
\(892\) −20402.0 −0.765817
\(893\) −10664.1 −0.399619
\(894\) 0 0
\(895\) −3015.54 −0.112624
\(896\) −7681.79 −0.286418
\(897\) 0 0
\(898\) −23773.6 −0.883449
\(899\) 241.107 0.00894478
\(900\) 0 0
\(901\) 52394.1 1.93729
\(902\) −18862.0 −0.696269
\(903\) 0 0
\(904\) −30886.7 −1.13637
\(905\) −9534.45 −0.350205
\(906\) 0 0
\(907\) 34384.5 1.25879 0.629393 0.777088i \(-0.283304\pi\)
0.629393 + 0.777088i \(0.283304\pi\)
\(908\) −21101.1 −0.771218
\(909\) 0 0
\(910\) 3945.72 0.143736
\(911\) −33862.9 −1.23153 −0.615767 0.787928i \(-0.711154\pi\)
−0.615767 + 0.787928i \(0.711154\pi\)
\(912\) 0 0
\(913\) 17277.6 0.626294
\(914\) 27662.4 1.00108
\(915\) 0 0
\(916\) 23218.1 0.837496
\(917\) −25732.5 −0.926677
\(918\) 0 0
\(919\) −26656.8 −0.956830 −0.478415 0.878134i \(-0.658788\pi\)
−0.478415 + 0.878134i \(0.658788\pi\)
\(920\) −5924.83 −0.212322
\(921\) 0 0
\(922\) 26730.6 0.954798
\(923\) −3516.81 −0.125414
\(924\) 0 0
\(925\) 25441.1 0.904321
\(926\) −12948.7 −0.459524
\(927\) 0 0
\(928\) −2033.11 −0.0719181
\(929\) 12246.8 0.432514 0.216257 0.976336i \(-0.430615\pi\)
0.216257 + 0.976336i \(0.430615\pi\)
\(930\) 0 0
\(931\) −15877.5 −0.558931
\(932\) −20705.7 −0.727721
\(933\) 0 0
\(934\) 26691.0 0.935070
\(935\) −9162.86 −0.320490
\(936\) 0 0
\(937\) −11620.7 −0.405155 −0.202577 0.979266i \(-0.564932\pi\)
−0.202577 + 0.979266i \(0.564932\pi\)
\(938\) −939.745 −0.0327119
\(939\) 0 0
\(940\) −1880.93 −0.0652652
\(941\) 50209.2 1.73940 0.869698 0.493584i \(-0.164313\pi\)
0.869698 + 0.493584i \(0.164313\pi\)
\(942\) 0 0
\(943\) −19238.8 −0.664369
\(944\) −2938.06 −0.101298
\(945\) 0 0
\(946\) −13164.8 −0.452457
\(947\) 28445.8 0.976097 0.488049 0.872816i \(-0.337709\pi\)
0.488049 + 0.872816i \(0.337709\pi\)
\(948\) 0 0
\(949\) 29025.4 0.992838
\(950\) 22206.3 0.758388
\(951\) 0 0
\(952\) 24045.3 0.818605
\(953\) 38782.3 1.31824 0.659119 0.752038i \(-0.270929\pi\)
0.659119 + 0.752038i \(0.270929\pi\)
\(954\) 0 0
\(955\) −1373.81 −0.0465501
\(956\) 1024.76 0.0346685
\(957\) 0 0
\(958\) 13127.6 0.442729
\(959\) 26636.4 0.896907
\(960\) 0 0
\(961\) −29395.9 −0.986739
\(962\) −15000.4 −0.502736
\(963\) 0 0
\(964\) −21647.9 −0.723269
\(965\) 17072.0 0.569498
\(966\) 0 0
\(967\) 54398.6 1.80904 0.904520 0.426432i \(-0.140230\pi\)
0.904520 + 0.426432i \(0.140230\pi\)
\(968\) −12909.9 −0.428657
\(969\) 0 0
\(970\) −15183.7 −0.502598
\(971\) 2476.35 0.0818432 0.0409216 0.999162i \(-0.486971\pi\)
0.0409216 + 0.999162i \(0.486971\pi\)
\(972\) 0 0
\(973\) −6567.54 −0.216388
\(974\) −12454.5 −0.409720
\(975\) 0 0
\(976\) −6973.83 −0.228716
\(977\) −48041.8 −1.57318 −0.786588 0.617478i \(-0.788154\pi\)
−0.786588 + 0.617478i \(0.788154\pi\)
\(978\) 0 0
\(979\) −19626.4 −0.640717
\(980\) −2800.48 −0.0912838
\(981\) 0 0
\(982\) 7828.81 0.254407
\(983\) −3188.75 −0.103464 −0.0517322 0.998661i \(-0.516474\pi\)
−0.0517322 + 0.998661i \(0.516474\pi\)
\(984\) 0 0
\(985\) −13595.6 −0.439789
\(986\) 1681.75 0.0543184
\(987\) 0 0
\(988\) 15122.6 0.486957
\(989\) −13427.8 −0.431728
\(990\) 0 0
\(991\) −48105.8 −1.54201 −0.771005 0.636829i \(-0.780246\pi\)
−0.771005 + 0.636829i \(0.780246\pi\)
\(992\) −3331.24 −0.106620
\(993\) 0 0
\(994\) 2994.77 0.0955616
\(995\) 2609.38 0.0831385
\(996\) 0 0
\(997\) 54934.0 1.74501 0.872507 0.488602i \(-0.162493\pi\)
0.872507 + 0.488602i \(0.162493\pi\)
\(998\) −29706.2 −0.942218
\(999\) 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 2151.4.a.h.1.20 yes 59
3.2 odd 2 2151.4.a.g.1.40 59
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.4.a.g.1.40 59 3.2 odd 2
2151.4.a.h.1.20 yes 59 1.1 even 1 trivial