Properties

Label 1815.4.a.l.1.2
Level $1815$
Weight $4$
Character 1815.1
Self dual yes
Analytic conductor $107.088$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(107.088466660\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.46410 q^{2} +3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +10.3923 q^{6} +31.1769 q^{7} -13.8564 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+3.46410 q^{2} +3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +10.3923 q^{6} +31.1769 q^{7} -13.8564 q^{8} +9.00000 q^{9} +17.3205 q^{10} +12.0000 q^{12} -76.2102 q^{13} +108.000 q^{14} +15.0000 q^{15} -80.0000 q^{16} +46.7654 q^{17} +31.1769 q^{18} -31.1769 q^{19} +20.0000 q^{20} +93.5307 q^{21} +195.000 q^{23} -41.5692 q^{24} +25.0000 q^{25} -264.000 q^{26} +27.0000 q^{27} +124.708 q^{28} +34.6410 q^{29} +51.9615 q^{30} +97.0000 q^{31} -166.277 q^{32} +162.000 q^{34} +155.885 q^{35} +36.0000 q^{36} +274.000 q^{37} -108.000 q^{38} -228.631 q^{39} -69.2820 q^{40} +384.515 q^{41} +324.000 q^{42} +415.692 q^{43} +45.0000 q^{45} +675.500 q^{46} +3.00000 q^{47} -240.000 q^{48} +629.000 q^{49} +86.6025 q^{50} +140.296 q^{51} -304.841 q^{52} +3.00000 q^{53} +93.5307 q^{54} -432.000 q^{56} -93.5307 q^{57} +120.000 q^{58} -648.000 q^{59} +60.0000 q^{60} -458.993 q^{61} +336.018 q^{62} +280.592 q^{63} +64.0000 q^{64} -381.051 q^{65} +70.0000 q^{67} +187.061 q^{68} +585.000 q^{69} +540.000 q^{70} -372.000 q^{71} -124.708 q^{72} +935.307 q^{73} +949.164 q^{74} +75.0000 q^{75} -124.708 q^{76} -792.000 q^{78} +521.347 q^{79} -400.000 q^{80} +81.0000 q^{81} +1332.00 q^{82} +45.0333 q^{83} +374.123 q^{84} +233.827 q^{85} +1440.00 q^{86} +103.923 q^{87} +1122.00 q^{89} +155.885 q^{90} -2376.00 q^{91} +780.000 q^{92} +291.000 q^{93} +10.3923 q^{94} -155.885 q^{95} -498.831 q^{96} +470.000 q^{97} +2178.92 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} + 8q^{4} + 10q^{5} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} + 8q^{4} + 10q^{5} + 18q^{9} + 24q^{12} + 216q^{14} + 30q^{15} - 160q^{16} + 40q^{20} + 390q^{23} + 50q^{25} - 528q^{26} + 54q^{27} + 194q^{31} + 324q^{34} + 72q^{36} + 548q^{37} - 216q^{38} + 648q^{42} + 90q^{45} + 6q^{47} - 480q^{48} + 1258q^{49} + 6q^{53} - 864q^{56} + 240q^{58} - 1296q^{59} + 120q^{60} + 128q^{64} + 140q^{67} + 1170q^{69} + 1080q^{70} - 744q^{71} + 150q^{75} - 1584q^{78} - 800q^{80} + 162q^{81} + 2664q^{82} + 2880q^{86} + 2244q^{89} - 4752q^{91} + 1560q^{92} + 582q^{93} + 940q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.46410 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) 3.00000 0.577350
\(4\) 4.00000 0.500000
\(5\) 5.00000 0.447214
\(6\) 10.3923 0.707107
\(7\) 31.1769 1.68340 0.841698 0.539949i \(-0.181557\pi\)
0.841698 + 0.539949i \(0.181557\pi\)
\(8\) −13.8564 −0.612372
\(9\) 9.00000 0.333333
\(10\) 17.3205 0.547723
\(11\) 0 0
\(12\) 12.0000 0.288675
\(13\) −76.2102 −1.62592 −0.812958 0.582322i \(-0.802144\pi\)
−0.812958 + 0.582322i \(0.802144\pi\)
\(14\) 108.000 2.06173
\(15\) 15.0000 0.258199
\(16\) −80.0000 −1.25000
\(17\) 46.7654 0.667192 0.333596 0.942716i \(-0.391738\pi\)
0.333596 + 0.942716i \(0.391738\pi\)
\(18\) 31.1769 0.408248
\(19\) −31.1769 −0.376446 −0.188223 0.982126i \(-0.560273\pi\)
−0.188223 + 0.982126i \(0.560273\pi\)
\(20\) 20.0000 0.223607
\(21\) 93.5307 0.971909
\(22\) 0 0
\(23\) 195.000 1.76784 0.883920 0.467639i \(-0.154895\pi\)
0.883920 + 0.467639i \(0.154895\pi\)
\(24\) −41.5692 −0.353553
\(25\) 25.0000 0.200000
\(26\) −264.000 −1.99133
\(27\) 27.0000 0.192450
\(28\) 124.708 0.841698
\(29\) 34.6410 0.221816 0.110908 0.993831i \(-0.464624\pi\)
0.110908 + 0.993831i \(0.464624\pi\)
\(30\) 51.9615 0.316228
\(31\) 97.0000 0.561991 0.280995 0.959709i \(-0.409335\pi\)
0.280995 + 0.959709i \(0.409335\pi\)
\(32\) −166.277 −0.918559
\(33\) 0 0
\(34\) 162.000 0.817140
\(35\) 155.885 0.752837
\(36\) 36.0000 0.166667
\(37\) 274.000 1.21744 0.608721 0.793385i \(-0.291683\pi\)
0.608721 + 0.793385i \(0.291683\pi\)
\(38\) −108.000 −0.461050
\(39\) −228.631 −0.938723
\(40\) −69.2820 −0.273861
\(41\) 384.515 1.46466 0.732332 0.680948i \(-0.238432\pi\)
0.732332 + 0.680948i \(0.238432\pi\)
\(42\) 324.000 1.19034
\(43\) 415.692 1.47424 0.737122 0.675760i \(-0.236184\pi\)
0.737122 + 0.675760i \(0.236184\pi\)
\(44\) 0 0
\(45\) 45.0000 0.149071
\(46\) 675.500 2.16515
\(47\) 3.00000 0.00931053 0.00465527 0.999989i \(-0.498518\pi\)
0.00465527 + 0.999989i \(0.498518\pi\)
\(48\) −240.000 −0.721688
\(49\) 629.000 1.83382
\(50\) 86.6025 0.244949
\(51\) 140.296 0.385204
\(52\) −304.841 −0.812958
\(53\) 3.00000 0.00777513 0.00388756 0.999992i \(-0.498763\pi\)
0.00388756 + 0.999992i \(0.498763\pi\)
\(54\) 93.5307 0.235702
\(55\) 0 0
\(56\) −432.000 −1.03086
\(57\) −93.5307 −0.217341
\(58\) 120.000 0.271668
\(59\) −648.000 −1.42987 −0.714936 0.699190i \(-0.753544\pi\)
−0.714936 + 0.699190i \(0.753544\pi\)
\(60\) 60.0000 0.129099
\(61\) −458.993 −0.963411 −0.481706 0.876333i \(-0.659983\pi\)
−0.481706 + 0.876333i \(0.659983\pi\)
\(62\) 336.018 0.688295
\(63\) 280.592 0.561132
\(64\) 64.0000 0.125000
\(65\) −381.051 −0.727132
\(66\) 0 0
\(67\) 70.0000 0.127640 0.0638199 0.997961i \(-0.479672\pi\)
0.0638199 + 0.997961i \(0.479672\pi\)
\(68\) 187.061 0.333596
\(69\) 585.000 1.02066
\(70\) 540.000 0.922033
\(71\) −372.000 −0.621807 −0.310903 0.950442i \(-0.600632\pi\)
−0.310903 + 0.950442i \(0.600632\pi\)
\(72\) −124.708 −0.204124
\(73\) 935.307 1.49958 0.749791 0.661675i \(-0.230154\pi\)
0.749791 + 0.661675i \(0.230154\pi\)
\(74\) 949.164 1.49105
\(75\) 75.0000 0.115470
\(76\) −124.708 −0.188223
\(77\) 0 0
\(78\) −792.000 −1.14970
\(79\) 521.347 0.742483 0.371241 0.928536i \(-0.378932\pi\)
0.371241 + 0.928536i \(0.378932\pi\)
\(80\) −400.000 −0.559017
\(81\) 81.0000 0.111111
\(82\) 1332.00 1.79384
\(83\) 45.0333 0.0595548 0.0297774 0.999557i \(-0.490520\pi\)
0.0297774 + 0.999557i \(0.490520\pi\)
\(84\) 374.123 0.485954
\(85\) 233.827 0.298377
\(86\) 1440.00 1.80557
\(87\) 103.923 0.128066
\(88\) 0 0
\(89\) 1122.00 1.33631 0.668156 0.744021i \(-0.267084\pi\)
0.668156 + 0.744021i \(0.267084\pi\)
\(90\) 155.885 0.182574
\(91\) −2376.00 −2.73706
\(92\) 780.000 0.883920
\(93\) 291.000 0.324466
\(94\) 10.3923 0.0114030
\(95\) −155.885 −0.168352
\(96\) −498.831 −0.530330
\(97\) 470.000 0.491972 0.245986 0.969273i \(-0.420888\pi\)
0.245986 + 0.969273i \(0.420888\pi\)
\(98\) 2178.92 2.24596
\(99\) 0 0
\(100\) 100.000 0.100000
\(101\) −1932.97 −1.90433 −0.952166 0.305581i \(-0.901149\pi\)
−0.952166 + 0.305581i \(0.901149\pi\)
\(102\) 486.000 0.471776
\(103\) 968.000 0.926018 0.463009 0.886354i \(-0.346770\pi\)
0.463009 + 0.886354i \(0.346770\pi\)
\(104\) 1056.00 0.995667
\(105\) 467.654 0.434651
\(106\) 10.3923 0.00952255
\(107\) −161.081 −0.145535 −0.0727676 0.997349i \(-0.523183\pi\)
−0.0727676 + 0.997349i \(0.523183\pi\)
\(108\) 108.000 0.0962250
\(109\) −235.559 −0.206995 −0.103497 0.994630i \(-0.533003\pi\)
−0.103497 + 0.994630i \(0.533003\pi\)
\(110\) 0 0
\(111\) 822.000 0.702890
\(112\) −2494.15 −2.10424
\(113\) −1653.00 −1.37612 −0.688058 0.725655i \(-0.741537\pi\)
−0.688058 + 0.725655i \(0.741537\pi\)
\(114\) −324.000 −0.266188
\(115\) 975.000 0.790602
\(116\) 138.564 0.110908
\(117\) −685.892 −0.541972
\(118\) −2244.74 −1.75123
\(119\) 1458.00 1.12315
\(120\) −207.846 −0.158114
\(121\) 0 0
\(122\) −1590.00 −1.17993
\(123\) 1153.55 0.845624
\(124\) 388.000 0.280995
\(125\) 125.000 0.0894427
\(126\) 972.000 0.687243
\(127\) −2057.68 −1.43771 −0.718855 0.695160i \(-0.755334\pi\)
−0.718855 + 0.695160i \(0.755334\pi\)
\(128\) 1551.92 1.07165
\(129\) 1247.08 0.851155
\(130\) −1320.00 −0.890551
\(131\) −2151.21 −1.43475 −0.717373 0.696689i \(-0.754656\pi\)
−0.717373 + 0.696689i \(0.754656\pi\)
\(132\) 0 0
\(133\) −972.000 −0.633707
\(134\) 242.487 0.156326
\(135\) 135.000 0.0860663
\(136\) −648.000 −0.408570
\(137\) 921.000 0.574353 0.287176 0.957878i \(-0.407283\pi\)
0.287176 + 0.957878i \(0.407283\pi\)
\(138\) 2026.50 1.25005
\(139\) 1726.85 1.05374 0.526870 0.849946i \(-0.323365\pi\)
0.526870 + 0.849946i \(0.323365\pi\)
\(140\) 623.538 0.376419
\(141\) 9.00000 0.00537544
\(142\) −1288.65 −0.761555
\(143\) 0 0
\(144\) −720.000 −0.416667
\(145\) 173.205 0.0991993
\(146\) 3240.00 1.83660
\(147\) 1887.00 1.05876
\(148\) 1096.00 0.608721
\(149\) −2016.11 −1.10850 −0.554248 0.832352i \(-0.686994\pi\)
−0.554248 + 0.832352i \(0.686994\pi\)
\(150\) 259.808 0.141421
\(151\) −774.227 −0.417256 −0.208628 0.977995i \(-0.566900\pi\)
−0.208628 + 0.977995i \(0.566900\pi\)
\(152\) 432.000 0.230525
\(153\) 420.888 0.222397
\(154\) 0 0
\(155\) 485.000 0.251330
\(156\) −914.523 −0.469362
\(157\) −3602.00 −1.83102 −0.915512 0.402290i \(-0.868214\pi\)
−0.915512 + 0.402290i \(0.868214\pi\)
\(158\) 1806.00 0.909352
\(159\) 9.00000 0.00448897
\(160\) −831.384 −0.410792
\(161\) 6079.50 2.97597
\(162\) 280.592 0.136083
\(163\) −1672.00 −0.803443 −0.401721 0.915762i \(-0.631588\pi\)
−0.401721 + 0.915762i \(0.631588\pi\)
\(164\) 1538.06 0.732332
\(165\) 0 0
\(166\) 156.000 0.0729394
\(167\) 2527.06 1.17096 0.585479 0.810688i \(-0.300907\pi\)
0.585479 + 0.810688i \(0.300907\pi\)
\(168\) −1296.00 −0.595170
\(169\) 3611.00 1.64360
\(170\) 810.000 0.365436
\(171\) −280.592 −0.125482
\(172\) 1662.77 0.737122
\(173\) 1745.91 0.767277 0.383638 0.923483i \(-0.374671\pi\)
0.383638 + 0.923483i \(0.374671\pi\)
\(174\) 360.000 0.156848
\(175\) 779.423 0.336679
\(176\) 0 0
\(177\) −1944.00 −0.825537
\(178\) 3886.72 1.63664
\(179\) 1998.00 0.834288 0.417144 0.908840i \(-0.363031\pi\)
0.417144 + 0.908840i \(0.363031\pi\)
\(180\) 180.000 0.0745356
\(181\) −4534.00 −1.86193 −0.930966 0.365107i \(-0.881032\pi\)
−0.930966 + 0.365107i \(0.881032\pi\)
\(182\) −8230.71 −3.35220
\(183\) −1376.98 −0.556226
\(184\) −2702.00 −1.08258
\(185\) 1370.00 0.544456
\(186\) 1008.05 0.397387
\(187\) 0 0
\(188\) 12.0000 0.00465527
\(189\) 841.777 0.323970
\(190\) −540.000 −0.206188
\(191\) 2754.00 1.04331 0.521656 0.853156i \(-0.325315\pi\)
0.521656 + 0.853156i \(0.325315\pi\)
\(192\) 192.000 0.0721688
\(193\) −772.495 −0.288111 −0.144055 0.989570i \(-0.546014\pi\)
−0.144055 + 0.989570i \(0.546014\pi\)
\(194\) 1628.13 0.602540
\(195\) −1143.15 −0.419810
\(196\) 2516.00 0.916910
\(197\) −2757.42 −0.997251 −0.498625 0.866818i \(-0.666162\pi\)
−0.498625 + 0.866818i \(0.666162\pi\)
\(198\) 0 0
\(199\) −731.000 −0.260398 −0.130199 0.991488i \(-0.541562\pi\)
−0.130199 + 0.991488i \(0.541562\pi\)
\(200\) −346.410 −0.122474
\(201\) 210.000 0.0736928
\(202\) −6696.00 −2.33232
\(203\) 1080.00 0.373405
\(204\) 561.184 0.192602
\(205\) 1922.58 0.655017
\(206\) 3353.25 1.13414
\(207\) 1755.00 0.589280
\(208\) 6096.82 2.03240
\(209\) 0 0
\(210\) 1620.00 0.532336
\(211\) 2069.80 0.675313 0.337656 0.941269i \(-0.390366\pi\)
0.337656 + 0.941269i \(0.390366\pi\)
\(212\) 12.0000 0.00388756
\(213\) −1116.00 −0.359000
\(214\) −558.000 −0.178243
\(215\) 2078.46 0.659302
\(216\) −374.123 −0.117851
\(217\) 3024.16 0.946053
\(218\) −816.000 −0.253516
\(219\) 2805.92 0.865784
\(220\) 0 0
\(221\) −3564.00 −1.08480
\(222\) 2847.49 0.860861
\(223\) 5438.00 1.63298 0.816492 0.577357i \(-0.195916\pi\)
0.816492 + 0.577357i \(0.195916\pi\)
\(224\) −5184.00 −1.54630
\(225\) 225.000 0.0666667
\(226\) −5726.16 −1.68539
\(227\) 2177.19 0.636586 0.318293 0.947992i \(-0.396890\pi\)
0.318293 + 0.947992i \(0.396890\pi\)
\(228\) −374.123 −0.108671
\(229\) 6073.00 1.75247 0.876234 0.481886i \(-0.160048\pi\)
0.876234 + 0.481886i \(0.160048\pi\)
\(230\) 3377.50 0.968286
\(231\) 0 0
\(232\) −480.000 −0.135834
\(233\) 1574.43 0.442681 0.221340 0.975197i \(-0.428957\pi\)
0.221340 + 0.975197i \(0.428957\pi\)
\(234\) −2376.00 −0.663778
\(235\) 15.0000 0.00416380
\(236\) −2592.00 −0.714936
\(237\) 1564.04 0.428673
\(238\) 5050.66 1.37557
\(239\) −6290.81 −1.70259 −0.851294 0.524689i \(-0.824182\pi\)
−0.851294 + 0.524689i \(0.824182\pi\)
\(240\) −1200.00 −0.322749
\(241\) −3257.99 −0.870811 −0.435405 0.900234i \(-0.643395\pi\)
−0.435405 + 0.900234i \(0.643395\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) −1835.97 −0.481706
\(245\) 3145.00 0.820109
\(246\) 3996.00 1.03567
\(247\) 2376.00 0.612070
\(248\) −1344.07 −0.344148
\(249\) 135.100 0.0343840
\(250\) 433.013 0.109545
\(251\) 660.000 0.165971 0.0829857 0.996551i \(-0.473554\pi\)
0.0829857 + 0.996551i \(0.473554\pi\)
\(252\) 1122.37 0.280566
\(253\) 0 0
\(254\) −7128.00 −1.76083
\(255\) 701.481 0.172268
\(256\) 4864.00 1.18750
\(257\) −1641.00 −0.398299 −0.199149 0.979969i \(-0.563818\pi\)
−0.199149 + 0.979969i \(0.563818\pi\)
\(258\) 4320.00 1.04245
\(259\) 8542.47 2.04943
\(260\) −1524.20 −0.363566
\(261\) 311.769 0.0739388
\(262\) −7452.00 −1.75720
\(263\) 2558.24 0.599801 0.299901 0.953970i \(-0.403046\pi\)
0.299901 + 0.953970i \(0.403046\pi\)
\(264\) 0 0
\(265\) 15.0000 0.00347714
\(266\) −3367.11 −0.776130
\(267\) 3366.00 0.771520
\(268\) 280.000 0.0638199
\(269\) 60.0000 0.0135995 0.00679975 0.999977i \(-0.497836\pi\)
0.00679975 + 0.999977i \(0.497836\pi\)
\(270\) 467.654 0.105409
\(271\) 2818.05 0.631676 0.315838 0.948813i \(-0.397714\pi\)
0.315838 + 0.948813i \(0.397714\pi\)
\(272\) −3741.23 −0.833990
\(273\) −7128.00 −1.58024
\(274\) 3190.44 0.703436
\(275\) 0 0
\(276\) 2340.00 0.510331
\(277\) −4205.42 −0.912199 −0.456100 0.889929i \(-0.650754\pi\)
−0.456100 + 0.889929i \(0.650754\pi\)
\(278\) 5982.00 1.29056
\(279\) 873.000 0.187330
\(280\) −2160.00 −0.461017
\(281\) −5220.40 −1.10827 −0.554133 0.832428i \(-0.686950\pi\)
−0.554133 + 0.832428i \(0.686950\pi\)
\(282\) 31.1769 0.00658354
\(283\) −4077.25 −0.856421 −0.428211 0.903679i \(-0.640856\pi\)
−0.428211 + 0.903679i \(0.640856\pi\)
\(284\) −1488.00 −0.310903
\(285\) −467.654 −0.0971979
\(286\) 0 0
\(287\) 11988.0 2.46561
\(288\) −1496.49 −0.306186
\(289\) −2726.00 −0.554854
\(290\) 600.000 0.121494
\(291\) 1410.00 0.284040
\(292\) 3741.23 0.749791
\(293\) −6576.60 −1.31129 −0.655647 0.755068i \(-0.727604\pi\)
−0.655647 + 0.755068i \(0.727604\pi\)
\(294\) 6536.76 1.29671
\(295\) −3240.00 −0.639458
\(296\) −3796.66 −0.745527
\(297\) 0 0
\(298\) −6984.00 −1.35763
\(299\) −14861.0 −2.87436
\(300\) 300.000 0.0577350
\(301\) 12960.0 2.48173
\(302\) −2682.00 −0.511032
\(303\) −5798.91 −1.09947
\(304\) 2494.15 0.470558
\(305\) −2294.97 −0.430851
\(306\) 1458.00 0.272380
\(307\) 6786.18 1.26159 0.630794 0.775950i \(-0.282729\pi\)
0.630794 + 0.775950i \(0.282729\pi\)
\(308\) 0 0
\(309\) 2904.00 0.534637
\(310\) 1680.09 0.307815
\(311\) 2934.00 0.534958 0.267479 0.963564i \(-0.413809\pi\)
0.267479 + 0.963564i \(0.413809\pi\)
\(312\) 3168.00 0.574848
\(313\) 7180.00 1.29661 0.648303 0.761383i \(-0.275479\pi\)
0.648303 + 0.761383i \(0.275479\pi\)
\(314\) −12477.7 −2.24254
\(315\) 1402.96 0.250946
\(316\) 2085.39 0.371241
\(317\) 7713.00 1.36658 0.683289 0.730148i \(-0.260549\pi\)
0.683289 + 0.730148i \(0.260549\pi\)
\(318\) 31.1769 0.00549784
\(319\) 0 0
\(320\) 320.000 0.0559017
\(321\) −483.242 −0.0840248
\(322\) 21060.0 3.64481
\(323\) −1458.00 −0.251162
\(324\) 324.000 0.0555556
\(325\) −1905.26 −0.325183
\(326\) −5791.98 −0.984012
\(327\) −706.677 −0.119509
\(328\) −5328.00 −0.896919
\(329\) 93.5307 0.0156733
\(330\) 0 0
\(331\) −9349.00 −1.55247 −0.776235 0.630444i \(-0.782873\pi\)
−0.776235 + 0.630444i \(0.782873\pi\)
\(332\) 180.133 0.0297774
\(333\) 2466.00 0.405814
\(334\) 8754.00 1.43412
\(335\) 350.000 0.0570822
\(336\) −7482.46 −1.21489
\(337\) −3183.51 −0.514590 −0.257295 0.966333i \(-0.582831\pi\)
−0.257295 + 0.966333i \(0.582831\pi\)
\(338\) 12508.9 2.01300
\(339\) −4959.00 −0.794501
\(340\) 935.307 0.149189
\(341\) 0 0
\(342\) −972.000 −0.153683
\(343\) 8916.60 1.40365
\(344\) −5760.00 −0.902786
\(345\) 2925.00 0.456454
\(346\) 6048.00 0.939718
\(347\) −1470.51 −0.227496 −0.113748 0.993510i \(-0.536286\pi\)
−0.113748 + 0.993510i \(0.536286\pi\)
\(348\) 415.692 0.0640329
\(349\) −5326.06 −0.816898 −0.408449 0.912781i \(-0.633930\pi\)
−0.408449 + 0.912781i \(0.633930\pi\)
\(350\) 2700.00 0.412346
\(351\) −2057.68 −0.312908
\(352\) 0 0
\(353\) 11547.0 1.74103 0.870517 0.492139i \(-0.163785\pi\)
0.870517 + 0.492139i \(0.163785\pi\)
\(354\) −6734.21 −1.01107
\(355\) −1860.00 −0.278080
\(356\) 4488.00 0.668156
\(357\) 4374.00 0.648450
\(358\) 6921.28 1.02179
\(359\) −3547.24 −0.521494 −0.260747 0.965407i \(-0.583969\pi\)
−0.260747 + 0.965407i \(0.583969\pi\)
\(360\) −623.538 −0.0912871
\(361\) −5887.00 −0.858288
\(362\) −15706.2 −2.28039
\(363\) 0 0
\(364\) −9504.00 −1.36853
\(365\) 4676.54 0.670633
\(366\) −4770.00 −0.681235
\(367\) 5894.00 0.838322 0.419161 0.907912i \(-0.362324\pi\)
0.419161 + 0.907912i \(0.362324\pi\)
\(368\) −15600.0 −2.20980
\(369\) 3460.64 0.488221
\(370\) 4745.82 0.666820
\(371\) 93.5307 0.0130886
\(372\) 1164.00 0.162233
\(373\) 3838.22 0.532804 0.266402 0.963862i \(-0.414165\pi\)
0.266402 + 0.963862i \(0.414165\pi\)
\(374\) 0 0
\(375\) 375.000 0.0516398
\(376\) −41.5692 −0.00570151
\(377\) −2640.00 −0.360655
\(378\) 2916.00 0.396780
\(379\) −5431.00 −0.736073 −0.368037 0.929811i \(-0.619970\pi\)
−0.368037 + 0.929811i \(0.619970\pi\)
\(380\) −623.538 −0.0841759
\(381\) −6173.03 −0.830063
\(382\) 9540.14 1.27779
\(383\) 312.000 0.0416252 0.0208126 0.999783i \(-0.493375\pi\)
0.0208126 + 0.999783i \(0.493375\pi\)
\(384\) 4655.75 0.618718
\(385\) 0 0
\(386\) −2676.00 −0.352862
\(387\) 3741.23 0.491414
\(388\) 1880.00 0.245986
\(389\) 1512.00 0.197073 0.0985366 0.995133i \(-0.468584\pi\)
0.0985366 + 0.995133i \(0.468584\pi\)
\(390\) −3960.00 −0.514160
\(391\) 9119.25 1.17949
\(392\) −8715.68 −1.12298
\(393\) −6453.62 −0.828351
\(394\) −9552.00 −1.22138
\(395\) 2606.74 0.332048
\(396\) 0 0
\(397\) −10210.0 −1.29074 −0.645372 0.763869i \(-0.723298\pi\)
−0.645372 + 0.763869i \(0.723298\pi\)
\(398\) −2532.26 −0.318921
\(399\) −2916.00 −0.365871
\(400\) −2000.00 −0.250000
\(401\) −10368.0 −1.29116 −0.645578 0.763695i \(-0.723383\pi\)
−0.645578 + 0.763695i \(0.723383\pi\)
\(402\) 727.461 0.0902549
\(403\) −7392.39 −0.913750
\(404\) −7731.87 −0.952166
\(405\) 405.000 0.0496904
\(406\) 3741.23 0.457325
\(407\) 0 0
\(408\) −1944.00 −0.235888
\(409\) −15611.0 −1.88732 −0.943659 0.330920i \(-0.892641\pi\)
−0.943659 + 0.330920i \(0.892641\pi\)
\(410\) 6660.00 0.802229
\(411\) 2763.00 0.331603
\(412\) 3872.00 0.463009
\(413\) −20202.6 −2.40704
\(414\) 6079.50 0.721717
\(415\) 225.167 0.0266337
\(416\) 12672.0 1.49350
\(417\) 5180.56 0.608377
\(418\) 0 0
\(419\) 8802.00 1.02627 0.513133 0.858309i \(-0.328485\pi\)
0.513133 + 0.858309i \(0.328485\pi\)
\(420\) 1870.61 0.217325
\(421\) −2537.00 −0.293696 −0.146848 0.989159i \(-0.546913\pi\)
−0.146848 + 0.989159i \(0.546913\pi\)
\(422\) 7170.00 0.827086
\(423\) 27.0000 0.00310351
\(424\) −41.5692 −0.00476127
\(425\) 1169.13 0.133438
\(426\) −3865.94 −0.439684
\(427\) −14310.0 −1.62180
\(428\) −644.323 −0.0727676
\(429\) 0 0
\(430\) 7200.00 0.807476
\(431\) −3824.37 −0.427409 −0.213705 0.976898i \(-0.568553\pi\)
−0.213705 + 0.976898i \(0.568553\pi\)
\(432\) −2160.00 −0.240563
\(433\) −12004.0 −1.33228 −0.666138 0.745829i \(-0.732054\pi\)
−0.666138 + 0.745829i \(0.732054\pi\)
\(434\) 10476.0 1.15867
\(435\) 519.615 0.0572727
\(436\) −942.236 −0.103497
\(437\) −6079.50 −0.665496
\(438\) 9720.00 1.06036
\(439\) −13127.2 −1.42717 −0.713585 0.700569i \(-0.752930\pi\)
−0.713585 + 0.700569i \(0.752930\pi\)
\(440\) 0 0
\(441\) 5661.00 0.611273
\(442\) −12346.1 −1.32860
\(443\) 1560.00 0.167309 0.0836544 0.996495i \(-0.473341\pi\)
0.0836544 + 0.996495i \(0.473341\pi\)
\(444\) 3288.00 0.351445
\(445\) 5610.00 0.597617
\(446\) 18837.8 1.99999
\(447\) −6048.32 −0.639991
\(448\) 1995.32 0.210424
\(449\) 10890.0 1.14461 0.572306 0.820040i \(-0.306049\pi\)
0.572306 + 0.820040i \(0.306049\pi\)
\(450\) 779.423 0.0816497
\(451\) 0 0
\(452\) −6612.00 −0.688058
\(453\) −2322.68 −0.240903
\(454\) 7542.00 0.779656
\(455\) −11880.0 −1.22405
\(456\) 1296.00 0.133094
\(457\) −17746.6 −1.81652 −0.908261 0.418404i \(-0.862590\pi\)
−0.908261 + 0.418404i \(0.862590\pi\)
\(458\) 21037.5 2.14633
\(459\) 1262.67 0.128401
\(460\) 3900.00 0.395301
\(461\) −10274.5 −1.03803 −0.519016 0.854765i \(-0.673701\pi\)
−0.519016 + 0.854765i \(0.673701\pi\)
\(462\) 0 0
\(463\) 8978.00 0.901173 0.450586 0.892733i \(-0.351215\pi\)
0.450586 + 0.892733i \(0.351215\pi\)
\(464\) −2771.28 −0.277270
\(465\) 1455.00 0.145105
\(466\) 5454.00 0.542171
\(467\) 9111.00 0.902798 0.451399 0.892322i \(-0.350925\pi\)
0.451399 + 0.892322i \(0.350925\pi\)
\(468\) −2743.57 −0.270986
\(469\) 2182.38 0.214868
\(470\) 51.9615 0.00509959
\(471\) −10806.0 −1.05714
\(472\) 8978.95 0.875614
\(473\) 0 0
\(474\) 5418.00 0.525015
\(475\) −779.423 −0.0752892
\(476\) 5832.00 0.561574
\(477\) 27.0000 0.00259171
\(478\) −21792.0 −2.08524
\(479\) −13225.9 −1.26160 −0.630802 0.775944i \(-0.717274\pi\)
−0.630802 + 0.775944i \(0.717274\pi\)
\(480\) −2494.15 −0.237171
\(481\) −20881.6 −1.97946
\(482\) −11286.0 −1.06652
\(483\) 18238.5 1.71818
\(484\) 0 0
\(485\) 2350.00 0.220017
\(486\) 841.777 0.0785674
\(487\) −9200.00 −0.856041 −0.428020 0.903769i \(-0.640789\pi\)
−0.428020 + 0.903769i \(0.640789\pi\)
\(488\) 6360.00 0.589967
\(489\) −5016.00 −0.463868
\(490\) 10894.6 1.00442
\(491\) −1170.87 −0.107618 −0.0538090 0.998551i \(-0.517136\pi\)
−0.0538090 + 0.998551i \(0.517136\pi\)
\(492\) 4614.18 0.422812
\(493\) 1620.00 0.147994
\(494\) 8230.71 0.749629
\(495\) 0 0
\(496\) −7760.00 −0.702488
\(497\) −11597.8 −1.04675
\(498\) 468.000 0.0421116
\(499\) −3872.00 −0.347364 −0.173682 0.984802i \(-0.555566\pi\)
−0.173682 + 0.984802i \(0.555566\pi\)
\(500\) 500.000 0.0447214
\(501\) 7581.19 0.676053
\(502\) 2286.31 0.203273
\(503\) −19476.9 −1.72651 −0.863253 0.504771i \(-0.831577\pi\)
−0.863253 + 0.504771i \(0.831577\pi\)
\(504\) −3888.00 −0.343622
\(505\) −9664.84 −0.851643
\(506\) 0 0
\(507\) 10833.0 0.948936
\(508\) −8230.71 −0.718855
\(509\) 14010.0 1.22000 0.610002 0.792400i \(-0.291169\pi\)
0.610002 + 0.792400i \(0.291169\pi\)
\(510\) 2430.00 0.210985
\(511\) 29160.0 2.52439
\(512\) 4434.05 0.382733
\(513\) −841.777 −0.0724471
\(514\) −5684.59 −0.487814
\(515\) 4840.00 0.414128
\(516\) 4988.31 0.425577
\(517\) 0 0
\(518\) 29592.0 2.51003
\(519\) 5237.72 0.442987
\(520\) 5280.00 0.445276
\(521\) 9900.00 0.832489 0.416245 0.909253i \(-0.363346\pi\)
0.416245 + 0.909253i \(0.363346\pi\)
\(522\) 1080.00 0.0905562
\(523\) 16090.8 1.34531 0.672657 0.739954i \(-0.265153\pi\)
0.672657 + 0.739954i \(0.265153\pi\)
\(524\) −8604.83 −0.717373
\(525\) 2338.27 0.194382
\(526\) 8862.00 0.734604
\(527\) 4536.24 0.374956
\(528\) 0 0
\(529\) 25858.0 2.12526
\(530\) 51.9615 0.00425861
\(531\) −5832.00 −0.476624
\(532\) −3888.00 −0.316854
\(533\) −29304.0 −2.38142
\(534\) 11660.2 0.944915
\(535\) −805.404 −0.0650853
\(536\) −969.948 −0.0781630
\(537\) 5994.00 0.481676
\(538\) 207.846 0.0166559
\(539\) 0 0
\(540\) 540.000 0.0430331
\(541\) −15013.4 −1.19312 −0.596559 0.802569i \(-0.703466\pi\)
−0.596559 + 0.802569i \(0.703466\pi\)
\(542\) 9762.00 0.773642
\(543\) −13602.0 −1.07499
\(544\) −7776.00 −0.612855
\(545\) −1177.79 −0.0925710
\(546\) −24692.1 −1.93539
\(547\) 15429.1 1.20603 0.603017 0.797728i \(-0.293965\pi\)
0.603017 + 0.797728i \(0.293965\pi\)
\(548\) 3684.00 0.287176
\(549\) −4130.94 −0.321137
\(550\) 0 0
\(551\) −1080.00 −0.0835019
\(552\) −8106.00 −0.625026
\(553\) 16254.0 1.24989
\(554\) −14568.0 −1.11721
\(555\) 4110.00 0.314342
\(556\) 6907.42 0.526870
\(557\) 8395.25 0.638632 0.319316 0.947648i \(-0.396547\pi\)
0.319316 + 0.947648i \(0.396547\pi\)
\(558\) 3024.16 0.229432
\(559\) −31680.0 −2.39700
\(560\) −12470.8 −0.941046
\(561\) 0 0
\(562\) −18084.0 −1.35734
\(563\) 7271.15 0.544303 0.272151 0.962254i \(-0.412265\pi\)
0.272151 + 0.962254i \(0.412265\pi\)
\(564\) 36.0000 0.00268772
\(565\) −8265.00 −0.615418
\(566\) −14124.0 −1.04890
\(567\) 2525.33 0.187044
\(568\) 5154.58 0.380777
\(569\) −8182.21 −0.602840 −0.301420 0.953491i \(-0.597461\pi\)
−0.301420 + 0.953491i \(0.597461\pi\)
\(570\) −1620.00 −0.119043
\(571\) −16414.6 −1.20303 −0.601516 0.798861i \(-0.705436\pi\)
−0.601516 + 0.798861i \(0.705436\pi\)
\(572\) 0 0
\(573\) 8262.00 0.602356
\(574\) 41527.7 3.01974
\(575\) 4875.00 0.353568
\(576\) 576.000 0.0416667
\(577\) 7300.00 0.526695 0.263347 0.964701i \(-0.415173\pi\)
0.263347 + 0.964701i \(0.415173\pi\)
\(578\) −9443.14 −0.679555
\(579\) −2317.48 −0.166341
\(580\) 692.820 0.0495997
\(581\) 1404.00 0.100254
\(582\) 4884.38 0.347877
\(583\) 0 0
\(584\) −12960.0 −0.918302
\(585\) −3429.46 −0.242377
\(586\) −22782.0 −1.60600
\(587\) −5703.00 −0.401002 −0.200501 0.979694i \(-0.564257\pi\)
−0.200501 + 0.979694i \(0.564257\pi\)
\(588\) 7548.00 0.529378
\(589\) −3024.16 −0.211559
\(590\) −11223.7 −0.783173
\(591\) −8272.27 −0.575763
\(592\) −21920.0 −1.52180
\(593\) 16717.8 1.15770 0.578850 0.815434i \(-0.303502\pi\)
0.578850 + 0.815434i \(0.303502\pi\)
\(594\) 0 0
\(595\) 7290.00 0.502287
\(596\) −8064.43 −0.554248
\(597\) −2193.00 −0.150341
\(598\) −51480.0 −3.52036
\(599\) 12630.0 0.861516 0.430758 0.902468i \(-0.358246\pi\)
0.430758 + 0.902468i \(0.358246\pi\)
\(600\) −1039.23 −0.0707107
\(601\) 2757.42 0.187151 0.0935755 0.995612i \(-0.470170\pi\)
0.0935755 + 0.995612i \(0.470170\pi\)
\(602\) 44894.8 3.03949
\(603\) 630.000 0.0425466
\(604\) −3096.91 −0.208628
\(605\) 0 0
\(606\) −20088.0 −1.34657
\(607\) 13035.4 0.871649 0.435825 0.900032i \(-0.356457\pi\)
0.435825 + 0.900032i \(0.356457\pi\)
\(608\) 5184.00 0.345788
\(609\) 3240.00 0.215585
\(610\) −7950.00 −0.527682
\(611\) −228.631 −0.0151381
\(612\) 1683.55 0.111199
\(613\) −28142.4 −1.85426 −0.927128 0.374744i \(-0.877731\pi\)
−0.927128 + 0.374744i \(0.877731\pi\)
\(614\) 23508.0 1.54512
\(615\) 5767.73 0.378174
\(616\) 0 0
\(617\) −6942.00 −0.452957 −0.226478 0.974016i \(-0.572721\pi\)
−0.226478 + 0.974016i \(0.572721\pi\)
\(618\) 10059.8 0.654794
\(619\) −9220.00 −0.598680 −0.299340 0.954146i \(-0.596767\pi\)
−0.299340 + 0.954146i \(0.596767\pi\)
\(620\) 1940.00 0.125665
\(621\) 5265.00 0.340221
\(622\) 10163.7 0.655187
\(623\) 34980.5 2.24954
\(624\) 18290.5 1.17340
\(625\) 625.000 0.0400000
\(626\) 24872.2 1.58801
\(627\) 0 0
\(628\) −14408.0 −0.915512
\(629\) 12813.7 0.812267
\(630\) 4860.00 0.307344
\(631\) 7393.00 0.466419 0.233210 0.972426i \(-0.425077\pi\)
0.233210 + 0.972426i \(0.425077\pi\)
\(632\) −7224.00 −0.454676
\(633\) 6209.40 0.389892
\(634\) 26718.6 1.67371
\(635\) −10288.4 −0.642964
\(636\) 36.0000 0.00224449
\(637\) −47936.2 −2.98164
\(638\) 0 0
\(639\) −3348.00 −0.207269
\(640\) 7759.59 0.479257
\(641\) −15390.0 −0.948313 −0.474156 0.880441i \(-0.657247\pi\)
−0.474156 + 0.880441i \(0.657247\pi\)
\(642\) −1674.00 −0.102909
\(643\) −12598.0 −0.772654 −0.386327 0.922362i \(-0.626256\pi\)
−0.386327 + 0.922362i \(0.626256\pi\)
\(644\) 24318.0 1.48799
\(645\) 6235.38 0.380648
\(646\) −5050.66 −0.307609
\(647\) 7131.00 0.433305 0.216653 0.976249i \(-0.430486\pi\)
0.216653 + 0.976249i \(0.430486\pi\)
\(648\) −1122.37 −0.0680414
\(649\) 0 0
\(650\) −6600.00 −0.398267
\(651\) 9072.48 0.546204
\(652\) −6688.00 −0.401721
\(653\) −4890.00 −0.293048 −0.146524 0.989207i \(-0.546809\pi\)
−0.146524 + 0.989207i \(0.546809\pi\)
\(654\) −2448.00 −0.146368
\(655\) −10756.0 −0.641638
\(656\) −30761.2 −1.83083
\(657\) 8417.77 0.499861
\(658\) 324.000 0.0191958
\(659\) 4569.15 0.270089 0.135045 0.990840i \(-0.456882\pi\)
0.135045 + 0.990840i \(0.456882\pi\)
\(660\) 0 0
\(661\) −14438.0 −0.849581 −0.424791 0.905292i \(-0.639652\pi\)
−0.424791 + 0.905292i \(0.639652\pi\)
\(662\) −32385.9 −1.90138
\(663\) −10692.0 −0.626309
\(664\) −624.000 −0.0364697
\(665\) −4860.00 −0.283403
\(666\) 8542.47 0.497018
\(667\) 6755.00 0.392136
\(668\) 10108.2 0.585479
\(669\) 16314.0 0.942804
\(670\) 1212.44 0.0699112
\(671\) 0 0
\(672\) −15552.0 −0.892755
\(673\) 10728.3 0.614482 0.307241 0.951632i \(-0.400594\pi\)
0.307241 + 0.951632i \(0.400594\pi\)
\(674\) −11028.0 −0.630241
\(675\) 675.000 0.0384900
\(676\) 14444.0 0.821802
\(677\) 13461.5 0.764206 0.382103 0.924120i \(-0.375200\pi\)
0.382103 + 0.924120i \(0.375200\pi\)
\(678\) −17178.5 −0.973061
\(679\) 14653.1 0.828183
\(680\) −3240.00 −0.182718
\(681\) 6531.56 0.367533
\(682\) 0 0
\(683\) −4068.00 −0.227903 −0.113951 0.993486i \(-0.536351\pi\)
−0.113951 + 0.993486i \(0.536351\pi\)
\(684\) −1122.37 −0.0627410
\(685\) 4605.00 0.256858
\(686\) 30888.0 1.71911
\(687\) 18219.0 1.01179
\(688\) −33255.4 −1.84280
\(689\) −228.631 −0.0126417
\(690\) 10132.5 0.559040
\(691\) 3065.00 0.168738 0.0843691 0.996435i \(-0.473113\pi\)
0.0843691 + 0.996435i \(0.473113\pi\)
\(692\) 6983.63 0.383638
\(693\) 0 0
\(694\) −5094.00 −0.278625
\(695\) 8634.27 0.471247
\(696\) −1440.00 −0.0784239
\(697\) 17982.0 0.977212
\(698\) −18450.0 −1.00049
\(699\) 4723.30 0.255582
\(700\) 3117.69 0.168340
\(701\) 2303.63 0.124118 0.0620591 0.998072i \(-0.480233\pi\)
0.0620591 + 0.998072i \(0.480233\pi\)
\(702\) −7128.00 −0.383232
\(703\) −8542.47 −0.458301
\(704\) 0 0
\(705\) 45.0000 0.00240397
\(706\) 40000.0 2.13232
\(707\) −60264.0 −3.20574
\(708\) −7776.00 −0.412768
\(709\) 13795.0 0.730722 0.365361 0.930866i \(-0.380946\pi\)
0.365361 + 0.930866i \(0.380946\pi\)
\(710\) −6443.23 −0.340578
\(711\) 4692.13 0.247494
\(712\) −15546.9 −0.818321
\(713\) 18915.0 0.993510
\(714\) 15152.0 0.794186
\(715\) 0 0
\(716\) 7992.00 0.417144
\(717\) −18872.4 −0.982990
\(718\) −12288.0 −0.638697
\(719\) −23406.0 −1.21404 −0.607021 0.794686i \(-0.707636\pi\)
−0.607021 + 0.794686i \(0.707636\pi\)
\(720\) −3600.00 −0.186339
\(721\) 30179.3 1.55885
\(722\) −20393.2 −1.05118
\(723\) −9773.96 −0.502763
\(724\) −18136.0 −0.930966
\(725\) 866.025 0.0443633
\(726\) 0 0
\(727\) −6518.00 −0.332516 −0.166258 0.986082i \(-0.553168\pi\)
−0.166258 + 0.986082i \(0.553168\pi\)
\(728\) 32922.8 1.67610
\(729\) 729.000 0.0370370
\(730\) 16200.0 0.821355
\(731\) 19440.0 0.983604
\(732\) −5507.92 −0.278113
\(733\) 266.736 0.0134408 0.00672040 0.999977i \(-0.497861\pi\)
0.00672040 + 0.999977i \(0.497861\pi\)
\(734\) 20417.4 1.02673
\(735\) 9435.00 0.473490
\(736\) −32424.0 −1.62386
\(737\) 0 0
\(738\) 11988.0 0.597946
\(739\) −33388.7 −1.66201 −0.831005 0.556266i \(-0.812234\pi\)
−0.831005 + 0.556266i \(0.812234\pi\)
\(740\) 5480.00 0.272228
\(741\) 7128.00 0.353379
\(742\) 324.000 0.0160302
\(743\) 34362.2 1.69667 0.848335 0.529460i \(-0.177606\pi\)
0.848335 + 0.529460i \(0.177606\pi\)
\(744\) −4032.21 −0.198694
\(745\) −10080.5 −0.495735
\(746\) 13296.0 0.652548
\(747\) 405.300 0.0198516
\(748\) 0 0
\(749\) −5022.00 −0.244993
\(750\) 1299.04 0.0632456
\(751\) 6301.00 0.306161 0.153080 0.988214i \(-0.451081\pi\)
0.153080 + 0.988214i \(0.451081\pi\)
\(752\) −240.000 −0.0116382
\(753\) 1980.00 0.0958237
\(754\) −9145.23 −0.441710
\(755\) −3871.13 −0.186603
\(756\) 3367.11 0.161985
\(757\) 17026.0 0.817464 0.408732 0.912654i \(-0.365971\pi\)
0.408732 + 0.912654i \(0.365971\pi\)
\(758\) −18813.5 −0.901502
\(759\) 0 0
\(760\) 2160.00 0.103094
\(761\) 12574.7 0.598991 0.299495 0.954098i \(-0.403182\pi\)
0.299495 + 0.954098i \(0.403182\pi\)
\(762\) −21384.0 −1.01661
\(763\) −7344.00 −0.348454
\(764\) 11016.0 0.521656
\(765\) 2104.44 0.0994592
\(766\) 1080.80 0.0509803
\(767\) 49384.2 2.32485
\(768\) 14592.0 0.685603
\(769\) −13702.3 −0.642543 −0.321272 0.946987i \(-0.604110\pi\)
−0.321272 + 0.946987i \(0.604110\pi\)
\(770\) 0 0
\(771\) −4923.00 −0.229958
\(772\) −3089.98 −0.144055
\(773\) −10155.0 −0.472510 −0.236255 0.971691i \(-0.575920\pi\)
−0.236255 + 0.971691i \(0.575920\pi\)
\(774\) 12960.0 0.601857
\(775\) 2425.00 0.112398
\(776\) −6512.51 −0.301270
\(777\) 25627.4 1.18324
\(778\) 5237.72 0.241364
\(779\) −11988.0 −0.551367
\(780\) −4572.61 −0.209905
\(781\) 0 0
\(782\) 31590.0 1.44457
\(783\) 935.307 0.0426886
\(784\) −50320.0 −2.29227
\(785\) −18010.0 −0.818859
\(786\) −22356.0 −1.01452
\(787\) 33671.1 1.52509 0.762544 0.646937i \(-0.223950\pi\)
0.762544 + 0.646937i \(0.223950\pi\)
\(788\) −11029.7 −0.498625
\(789\) 7674.72 0.346296
\(790\) 9030.00 0.406675
\(791\) −51535.4 −2.31655
\(792\) 0 0
\(793\) 34980.0 1.56643
\(794\) −35368.5 −1.58083
\(795\) 45.0000 0.00200753
\(796\) −2924.00 −0.130199
\(797\) 14106.0 0.626926 0.313463 0.949600i \(-0.398511\pi\)
0.313463 + 0.949600i \(0.398511\pi\)
\(798\) −10101.3 −0.448099
\(799\) 140.296 0.00621191
\(800\) −4156.92 −0.183712
\(801\) 10098.0 0.445437
\(802\) −35915.8 −1.58134
\(803\) 0 0
\(804\) 840.000 0.0368464
\(805\) 30397.5 1.33090
\(806\) −25608.0 −1.11911
\(807\) 180.000 0.00785167
\(808\) 26784.0 1.16616
\(809\) 533.472 0.0231840 0.0115920 0.999933i \(-0.496310\pi\)
0.0115920 + 0.999933i \(0.496310\pi\)
\(810\) 1402.96 0.0608581
\(811\) −12870.9 −0.557284 −0.278642 0.960395i \(-0.589884\pi\)
−0.278642 + 0.960395i \(0.589884\pi\)
\(812\) 4320.00 0.186702
\(813\) 8454.14 0.364698
\(814\) 0 0
\(815\) −8360.00 −0.359310
\(816\) −11223.7 −0.481505
\(817\) −12960.0 −0.554973
\(818\) −54078.0 −2.31148
\(819\) −21384.0 −0.912353
\(820\) 7690.31 0.327509
\(821\) −2975.66 −0.126494 −0.0632469 0.997998i \(-0.520146\pi\)
−0.0632469 + 0.997998i \(0.520146\pi\)
\(822\) 9571.31 0.406129
\(823\) −3664.00 −0.155187 −0.0775936 0.996985i \(-0.524724\pi\)
−0.0775936 + 0.996985i \(0.524724\pi\)
\(824\) −13413.0 −0.567068
\(825\) 0 0
\(826\) −69984.0 −2.94801
\(827\) 31547.6 1.32650 0.663251 0.748397i \(-0.269176\pi\)
0.663251 + 0.748397i \(0.269176\pi\)
\(828\) 7020.00 0.294640
\(829\) −24829.0 −1.04023 −0.520113 0.854098i \(-0.674110\pi\)
−0.520113 + 0.854098i \(0.674110\pi\)
\(830\) 780.000 0.0326195
\(831\) −12616.3 −0.526658
\(832\) −4877.46 −0.203240
\(833\) 29415.4 1.22351
\(834\) 17946.0 0.745107
\(835\) 12635.3 0.523668
\(836\) 0 0
\(837\) 2619.00 0.108155
\(838\) 30491.0 1.25692
\(839\) −6048.00 −0.248868 −0.124434 0.992228i \(-0.539711\pi\)
−0.124434 + 0.992228i \(0.539711\pi\)
\(840\) −6480.00 −0.266168
\(841\) −23189.0 −0.950797
\(842\) −8788.43 −0.359702
\(843\) −15661.2 −0.639858
\(844\) 8279.20 0.337656
\(845\) 18055.0 0.735042
\(846\) 93.5307 0.00380101
\(847\) 0 0
\(848\) −240.000 −0.00971891
\(849\) −12231.7 −0.494455
\(850\) 4050.00 0.163428
\(851\) 53430.0 2.15224
\(852\) −4464.00 −0.179500
\(853\) 11656.7 0.467899 0.233950 0.972249i \(-0.424835\pi\)
0.233950 + 0.972249i \(0.424835\pi\)
\(854\) −49571.3 −1.98629
\(855\) −1402.96 −0.0561173
\(856\) 2232.00 0.0891217
\(857\) −33198.2 −1.32325 −0.661627 0.749833i \(-0.730134\pi\)
−0.661627 + 0.749833i \(0.730134\pi\)
\(858\) 0 0
\(859\) 16196.0 0.643307 0.321653 0.946858i \(-0.395761\pi\)
0.321653 + 0.946858i \(0.395761\pi\)
\(860\) 8313.84 0.329651
\(861\) 35964.0 1.42352
\(862\) −13248.0 −0.523467
\(863\) 10728.0 0.423158 0.211579 0.977361i \(-0.432139\pi\)
0.211579 + 0.977361i \(0.432139\pi\)
\(864\) −4489.48 −0.176777
\(865\) 8729.54 0.343137
\(866\) −41583.1 −1.63170
\(867\) −8178.00 −0.320345
\(868\) 12096.6 0.473026
\(869\) 0 0
\(870\) 1800.00 0.0701445
\(871\) −5334.72 −0.207532
\(872\) 3264.00 0.126758
\(873\) 4230.00 0.163991
\(874\) −21060.0 −0.815063
\(875\) 3897.11 0.150567
\(876\) 11223.7 0.432892
\(877\) −47392.4 −1.82477 −0.912387 0.409329i \(-0.865763\pi\)
−0.912387 + 0.409329i \(0.865763\pi\)
\(878\) −45474.0 −1.74792
\(879\) −19729.8 −0.757075
\(880\) 0 0
\(881\) 20664.0 0.790225 0.395112 0.918633i \(-0.370706\pi\)
0.395112 + 0.918633i \(0.370706\pi\)
\(882\) 19610.3 0.748654
\(883\) 27532.0 1.04929 0.524646 0.851320i \(-0.324198\pi\)
0.524646 + 0.851320i \(0.324198\pi\)
\(884\) −14256.0 −0.542400
\(885\) −9720.00 −0.369191
\(886\) 5404.00 0.204911
\(887\) −29822.5 −1.12891 −0.564453 0.825465i \(-0.690913\pi\)
−0.564453 + 0.825465i \(0.690913\pi\)
\(888\) −11390.0 −0.430430
\(889\) −64152.0 −2.42024
\(890\) 19433.6 0.731928
\(891\) 0 0
\(892\) 21752.0 0.816492
\(893\) −93.5307 −0.00350491
\(894\) −20952.0 −0.783825
\(895\) 9990.00 0.373105
\(896\) 48384.0 1.80401
\(897\) −44583.0 −1.65951
\(898\) 37724.1 1.40186
\(899\) 3360.18 0.124659
\(900\) 900.000 0.0333333
\(901\) 140.296 0.00518750
\(902\) 0 0
\(903\) 38880.0 1.43283
\(904\) 22904.6 0.842696
\(905\) −22670.0 −0.832681
\(906\) −8046.00 −0.295045
\(907\) −6406.00 −0.234518 −0.117259 0.993101i \(-0.537411\pi\)
−0.117259 + 0.993101i \(0.537411\pi\)
\(908\) 8708.75 0.318293
\(909\) −17396.7 −0.634777
\(910\) −41153.5 −1.49915
\(911\) 35046.0 1.27456 0.637281 0.770632i \(-0.280059\pi\)
0.637281 + 0.770632i \(0.280059\pi\)
\(912\) 7482.46 0.271677
\(913\) 0 0
\(914\) −61476.0 −2.22478
\(915\) −6884.90 −0.248752
\(916\) 24292.0 0.876234
\(917\) −67068.0 −2.41525
\(918\) 4374.00 0.157259
\(919\) −23801.8 −0.854353 −0.427176 0.904168i \(-0.640492\pi\)
−0.427176 + 0.904168i \(0.640492\pi\)
\(920\) −13510.0 −0.484143
\(921\) 20358.5 0.728378
\(922\) −35592.0 −1.27132
\(923\) 28350.2 1.01101
\(924\) 0 0
\(925\) 6850.00 0.243488
\(926\) 31100.7 1.10371
\(927\) 8712.00 0.308673
\(928\) −5760.00 −0.203751
\(929\) −42558.0 −1.50300 −0.751498 0.659736i \(-0.770668\pi\)
−0.751498 + 0.659736i \(0.770668\pi\)
\(930\) 5040.27 0.177717
\(931\) −19610.3 −0.690334
\(932\) 6297.74 0.221340
\(933\) 8802.00 0.308858
\(934\) 31561.4 1.10570
\(935\) 0 0
\(936\) 9504.00 0.331889
\(937\) 10049.4 0.350372 0.175186 0.984535i \(-0.443947\pi\)
0.175186 + 0.984535i \(0.443947\pi\)
\(938\) 7560.00 0.263159
\(939\) 21540.0 0.748596
\(940\) 60.0000 0.00208190
\(941\) −13343.7 −0.462267 −0.231133 0.972922i \(-0.574243\pi\)
−0.231133 + 0.972922i \(0.574243\pi\)
\(942\) −37433.1 −1.29473
\(943\) 74980.5 2.58929
\(944\) 51840.0 1.78734
\(945\) 4208.88 0.144884
\(946\) 0 0
\(947\) −1899.00 −0.0651628 −0.0325814 0.999469i \(-0.510373\pi\)
−0.0325814 + 0.999469i \(0.510373\pi\)
\(948\) 6256.17 0.214336
\(949\) −71280.0 −2.43819
\(950\) −2700.00 −0.0922101
\(951\) 23139.0 0.788994
\(952\) −20202.6 −0.687785
\(953\) 25766.0 0.875805 0.437903 0.899022i \(-0.355721\pi\)
0.437903 + 0.899022i \(0.355721\pi\)
\(954\) 93.5307 0.00317418
\(955\) 13770.0 0.466583
\(956\) −25163.2 −0.851294
\(957\) 0 0
\(958\) −45816.0 −1.54514
\(959\) 28713.9 0.966863
\(960\) 960.000 0.0322749
\(961\) −20382.0 −0.684166
\(962\) −72336.0 −2.42433
\(963\) −1449.73 −0.0485117
\(964\) −13032.0 −0.435405
\(965\) −3862.47 −0.128847
\(966\) 63180.0 2.10433
\(967\) 45005.6 1.49667 0.748336 0.663319i \(-0.230853\pi\)
0.748336 + 0.663319i \(0.230853\pi\)
\(968\) 0 0
\(969\) −4374.00 −0.145008
\(970\) 8140.64 0.269464
\(971\) −51840.0 −1.71331 −0.856655 0.515889i \(-0.827462\pi\)
−0.856655 + 0.515889i \(0.827462\pi\)
\(972\) 972.000 0.0320750
\(973\) 53838.0 1.77386
\(974\) −31869.7 −1.04843
\(975\) −5715.77 −0.187745
\(976\) 36719.5 1.20426
\(977\) −17133.0 −0.561037 −0.280519 0.959849i \(-0.590506\pi\)
−0.280519 + 0.959849i \(0.590506\pi\)
\(978\) −17375.9 −0.568120
\(979\) 0 0
\(980\) 12580.0 0.410054
\(981\) −2120.03 −0.0689983
\(982\) −4056.00 −0.131805
\(983\) 11241.0 0.364733 0.182366 0.983231i \(-0.441624\pi\)
0.182366 + 0.983231i \(0.441624\pi\)
\(984\) −15984.0 −0.517837
\(985\) −13787.1 −0.445984
\(986\) 5611.84 0.181255
\(987\) 280.592 0.00904899
\(988\) 9504.00 0.306035
\(989\) 81060.0 2.60623
\(990\) 0 0
\(991\) −14567.0 −0.466938 −0.233469 0.972364i \(-0.575008\pi\)
−0.233469 + 0.972364i \(0.575008\pi\)
\(992\) −16128.9 −0.516221
\(993\) −28047.0 −0.896319
\(994\) −40176.0 −1.28200
\(995\) −3655.00 −0.116454
\(996\) 540.400 0.0171920
\(997\) 29874.4 0.948979 0.474490 0.880261i \(-0.342633\pi\)
0.474490 + 0.880261i \(0.342633\pi\)
\(998\) −13413.0 −0.425432
\(999\) 7398.00 0.234297
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 1815.4.a.l.1.2 yes 2
11.10 odd 2 inner 1815.4.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.4.a.l.1.1 2 11.10 odd 2 inner
1815.4.a.l.1.2 yes 2 1.1 even 1 trivial