Properties

Label 525.4.a.l.1.2
Level $525$
Weight $4$
Character 525.1
Self dual yes
Analytic conductor $30.976$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 525.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.82843 q^{2} +3.00000 q^{3} +6.65685 q^{4} +11.4853 q^{6} +7.00000 q^{7} -5.14214 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+3.82843 q^{2} +3.00000 q^{3} +6.65685 q^{4} +11.4853 q^{6} +7.00000 q^{7} -5.14214 q^{8} +9.00000 q^{9} +48.5685 q^{11} +19.9706 q^{12} +43.6569 q^{13} +26.7990 q^{14} -72.9411 q^{16} +67.6569 q^{17} +34.4558 q^{18} -93.2548 q^{19} +21.0000 q^{21} +185.941 q^{22} +104.167 q^{23} -15.4264 q^{24} +167.137 q^{26} +27.0000 q^{27} +46.5980 q^{28} -58.7351 q^{29} -9.08831 q^{31} -238.113 q^{32} +145.706 q^{33} +259.019 q^{34} +59.9117 q^{36} +252.676 q^{37} -357.019 q^{38} +130.971 q^{39} +276.274 q^{41} +80.3970 q^{42} +92.6375 q^{43} +323.314 q^{44} +398.794 q^{46} +582.794 q^{47} -218.823 q^{48} +49.0000 q^{49} +202.971 q^{51} +290.617 q^{52} -623.019 q^{53} +103.368 q^{54} -35.9949 q^{56} -279.765 q^{57} -224.863 q^{58} -524.999 q^{59} -352.794 q^{61} -34.7939 q^{62} +63.0000 q^{63} -328.068 q^{64} +557.823 q^{66} +736.520 q^{67} +450.382 q^{68} +312.500 q^{69} -492.264 q^{71} -46.2792 q^{72} -1164.75 q^{73} +967.352 q^{74} -620.784 q^{76} +339.980 q^{77} +501.411 q^{78} -872.195 q^{79} +81.0000 q^{81} +1057.70 q^{82} +529.588 q^{83} +139.794 q^{84} +354.656 q^{86} -176.205 q^{87} -249.746 q^{88} -385.216 q^{89} +305.598 q^{91} +693.421 q^{92} -27.2649 q^{93} +2231.18 q^{94} -714.338 q^{96} +463.892 q^{97} +187.593 q^{98} +437.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 6q^{3} + 2q^{4} + 6q^{6} + 14q^{7} + 18q^{8} + 18q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 6q^{3} + 2q^{4} + 6q^{6} + 14q^{7} + 18q^{8} + 18q^{9} - 16q^{11} + 6q^{12} + 76q^{13} + 14q^{14} - 78q^{16} + 124q^{17} + 18q^{18} - 96q^{19} + 42q^{21} + 304q^{22} + 16q^{23} + 54q^{24} + 108q^{26} + 54q^{27} + 14q^{28} + 188q^{29} - 120q^{31} - 414q^{32} - 48q^{33} + 156q^{34} + 18q^{36} + 132q^{37} - 352q^{38} + 228q^{39} + 100q^{41} + 42q^{42} + 536q^{43} + 624q^{44} + 560q^{46} + 928q^{47} - 234q^{48} + 98q^{49} + 372q^{51} + 140q^{52} - 884q^{53} + 54q^{54} + 126q^{56} - 288q^{57} - 676q^{58} + 104q^{59} - 468q^{61} + 168q^{62} + 126q^{63} + 34q^{64} + 912q^{66} + 1688q^{67} + 188q^{68} + 48q^{69} - 136q^{71} + 162q^{72} - 508q^{73} + 1188q^{74} - 608q^{76} - 112q^{77} + 324q^{78} - 432q^{79} + 162q^{81} + 1380q^{82} + 584q^{83} + 42q^{84} - 456q^{86} + 564q^{87} - 1744q^{88} - 1404q^{89} + 532q^{91} + 1104q^{92} - 360q^{93} + 1600q^{94} - 1242q^{96} + 1188q^{97} + 98q^{98} - 144q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.82843 1.35355 0.676777 0.736188i \(-0.263376\pi\)
0.676777 + 0.736188i \(0.263376\pi\)
\(3\) 3.00000 0.577350
\(4\) 6.65685 0.832107
\(5\) 0 0
\(6\) 11.4853 0.781474
\(7\) 7.00000 0.377964
\(8\) −5.14214 −0.227252
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 48.5685 1.33127 0.665635 0.746278i \(-0.268161\pi\)
0.665635 + 0.746278i \(0.268161\pi\)
\(12\) 19.9706 0.480417
\(13\) 43.6569 0.931403 0.465701 0.884942i \(-0.345802\pi\)
0.465701 + 0.884942i \(0.345802\pi\)
\(14\) 26.7990 0.511595
\(15\) 0 0
\(16\) −72.9411 −1.13971
\(17\) 67.6569 0.965247 0.482623 0.875828i \(-0.339684\pi\)
0.482623 + 0.875828i \(0.339684\pi\)
\(18\) 34.4558 0.451184
\(19\) −93.2548 −1.12601 −0.563003 0.826455i \(-0.690354\pi\)
−0.563003 + 0.826455i \(0.690354\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) 185.941 1.80194
\(23\) 104.167 0.944357 0.472179 0.881503i \(-0.343468\pi\)
0.472179 + 0.881503i \(0.343468\pi\)
\(24\) −15.4264 −0.131204
\(25\) 0 0
\(26\) 167.137 1.26070
\(27\) 27.0000 0.192450
\(28\) 46.5980 0.314507
\(29\) −58.7351 −0.376098 −0.188049 0.982160i \(-0.560216\pi\)
−0.188049 + 0.982160i \(0.560216\pi\)
\(30\) 0 0
\(31\) −9.08831 −0.0526551 −0.0263276 0.999653i \(-0.508381\pi\)
−0.0263276 + 0.999653i \(0.508381\pi\)
\(32\) −238.113 −1.31540
\(33\) 145.706 0.768609
\(34\) 259.019 1.30651
\(35\) 0 0
\(36\) 59.9117 0.277369
\(37\) 252.676 1.12269 0.561347 0.827580i \(-0.310283\pi\)
0.561347 + 0.827580i \(0.310283\pi\)
\(38\) −357.019 −1.52411
\(39\) 130.971 0.537745
\(40\) 0 0
\(41\) 276.274 1.05236 0.526180 0.850373i \(-0.323624\pi\)
0.526180 + 0.850373i \(0.323624\pi\)
\(42\) 80.3970 0.295370
\(43\) 92.6375 0.328537 0.164268 0.986416i \(-0.447474\pi\)
0.164268 + 0.986416i \(0.447474\pi\)
\(44\) 323.314 1.10776
\(45\) 0 0
\(46\) 398.794 1.27824
\(47\) 582.794 1.80871 0.904354 0.426784i \(-0.140354\pi\)
0.904354 + 0.426784i \(0.140354\pi\)
\(48\) −218.823 −0.658009
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 202.971 0.557286
\(52\) 290.617 0.775026
\(53\) −623.019 −1.61468 −0.807342 0.590083i \(-0.799095\pi\)
−0.807342 + 0.590083i \(0.799095\pi\)
\(54\) 103.368 0.260491
\(55\) 0 0
\(56\) −35.9949 −0.0858933
\(57\) −279.765 −0.650100
\(58\) −224.863 −0.509068
\(59\) −524.999 −1.15846 −0.579229 0.815165i \(-0.696646\pi\)
−0.579229 + 0.815165i \(0.696646\pi\)
\(60\) 0 0
\(61\) −352.794 −0.740502 −0.370251 0.928932i \(-0.620728\pi\)
−0.370251 + 0.928932i \(0.620728\pi\)
\(62\) −34.7939 −0.0712715
\(63\) 63.0000 0.125988
\(64\) −328.068 −0.640758
\(65\) 0 0
\(66\) 557.823 1.04035
\(67\) 736.520 1.34299 0.671494 0.741010i \(-0.265653\pi\)
0.671494 + 0.741010i \(0.265653\pi\)
\(68\) 450.382 0.803188
\(69\) 312.500 0.545225
\(70\) 0 0
\(71\) −492.264 −0.822831 −0.411415 0.911448i \(-0.634965\pi\)
−0.411415 + 0.911448i \(0.634965\pi\)
\(72\) −46.2792 −0.0757508
\(73\) −1164.75 −1.86745 −0.933727 0.357987i \(-0.883463\pi\)
−0.933727 + 0.357987i \(0.883463\pi\)
\(74\) 967.352 1.51963
\(75\) 0 0
\(76\) −620.784 −0.936958
\(77\) 339.980 0.503173
\(78\) 501.411 0.727867
\(79\) −872.195 −1.24215 −0.621074 0.783752i \(-0.713303\pi\)
−0.621074 + 0.783752i \(0.713303\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 1057.70 1.42443
\(83\) 529.588 0.700359 0.350180 0.936683i \(-0.386121\pi\)
0.350180 + 0.936683i \(0.386121\pi\)
\(84\) 139.794 0.181581
\(85\) 0 0
\(86\) 354.656 0.444692
\(87\) −176.205 −0.217140
\(88\) −249.746 −0.302534
\(89\) −385.216 −0.458796 −0.229398 0.973333i \(-0.573676\pi\)
−0.229398 + 0.973333i \(0.573676\pi\)
\(90\) 0 0
\(91\) 305.598 0.352037
\(92\) 693.421 0.785806
\(93\) −27.2649 −0.0304005
\(94\) 2231.18 2.44818
\(95\) 0 0
\(96\) −714.338 −0.759446
\(97\) 463.892 0.485579 0.242789 0.970079i \(-0.421938\pi\)
0.242789 + 0.970079i \(0.421938\pi\)
\(98\) 187.593 0.193365
\(99\) 437.117 0.443757
\(100\) 0 0
\(101\) −432.725 −0.426314 −0.213157 0.977018i \(-0.568375\pi\)
−0.213157 + 0.977018i \(0.568375\pi\)
\(102\) 777.058 0.754316
\(103\) −512.626 −0.490393 −0.245197 0.969473i \(-0.578853\pi\)
−0.245197 + 0.969473i \(0.578853\pi\)
\(104\) −224.489 −0.211663
\(105\) 0 0
\(106\) −2385.18 −2.18556
\(107\) −1963.09 −1.77363 −0.886817 0.462122i \(-0.847088\pi\)
−0.886817 + 0.462122i \(0.847088\pi\)
\(108\) 179.735 0.160139
\(109\) 545.176 0.479068 0.239534 0.970888i \(-0.423005\pi\)
0.239534 + 0.970888i \(0.423005\pi\)
\(110\) 0 0
\(111\) 758.029 0.648188
\(112\) −510.588 −0.430768
\(113\) 231.823 0.192992 0.0964961 0.995333i \(-0.469236\pi\)
0.0964961 + 0.995333i \(0.469236\pi\)
\(114\) −1071.06 −0.879945
\(115\) 0 0
\(116\) −390.991 −0.312953
\(117\) 392.912 0.310468
\(118\) −2009.92 −1.56804
\(119\) 473.598 0.364829
\(120\) 0 0
\(121\) 1027.90 0.772279
\(122\) −1350.65 −1.00231
\(123\) 828.823 0.607581
\(124\) −60.4996 −0.0438147
\(125\) 0 0
\(126\) 241.191 0.170532
\(127\) −2372.90 −1.65796 −0.828979 0.559280i \(-0.811078\pi\)
−0.828979 + 0.559280i \(0.811078\pi\)
\(128\) 648.917 0.448099
\(129\) 277.913 0.189681
\(130\) 0 0
\(131\) 1200.04 0.800364 0.400182 0.916436i \(-0.368947\pi\)
0.400182 + 0.916436i \(0.368947\pi\)
\(132\) 969.941 0.639565
\(133\) −652.784 −0.425591
\(134\) 2819.71 1.81781
\(135\) 0 0
\(136\) −347.901 −0.219355
\(137\) −2781.25 −1.73444 −0.867221 0.497924i \(-0.834096\pi\)
−0.867221 + 0.497924i \(0.834096\pi\)
\(138\) 1196.38 0.737991
\(139\) −1245.60 −0.760078 −0.380039 0.924971i \(-0.624089\pi\)
−0.380039 + 0.924971i \(0.624089\pi\)
\(140\) 0 0
\(141\) 1748.38 1.04426
\(142\) −1884.60 −1.11375
\(143\) 2120.35 1.23995
\(144\) −656.470 −0.379902
\(145\) 0 0
\(146\) −4459.17 −2.52770
\(147\) 147.000 0.0824786
\(148\) 1682.03 0.934202
\(149\) 19.4046 0.0106690 0.00533452 0.999986i \(-0.498302\pi\)
0.00533452 + 0.999986i \(0.498302\pi\)
\(150\) 0 0
\(151\) −2349.80 −1.26638 −0.633192 0.773995i \(-0.718256\pi\)
−0.633192 + 0.773995i \(0.718256\pi\)
\(152\) 479.529 0.255888
\(153\) 608.912 0.321749
\(154\) 1301.59 0.681071
\(155\) 0 0
\(156\) 871.852 0.447462
\(157\) 3898.46 1.98172 0.990862 0.134880i \(-0.0430650\pi\)
0.990862 + 0.134880i \(0.0430650\pi\)
\(158\) −3339.14 −1.68131
\(159\) −1869.06 −0.932239
\(160\) 0 0
\(161\) 729.166 0.356934
\(162\) 310.103 0.150395
\(163\) −1527.54 −0.734024 −0.367012 0.930216i \(-0.619619\pi\)
−0.367012 + 0.930216i \(0.619619\pi\)
\(164\) 1839.12 0.875676
\(165\) 0 0
\(166\) 2027.49 0.947974
\(167\) 998.518 0.462681 0.231340 0.972873i \(-0.425689\pi\)
0.231340 + 0.972873i \(0.425689\pi\)
\(168\) −107.985 −0.0495905
\(169\) −291.079 −0.132489
\(170\) 0 0
\(171\) −839.294 −0.375336
\(172\) 616.674 0.273378
\(173\) −685.253 −0.301149 −0.150575 0.988599i \(-0.548112\pi\)
−0.150575 + 0.988599i \(0.548112\pi\)
\(174\) −674.589 −0.293911
\(175\) 0 0
\(176\) −3542.64 −1.51725
\(177\) −1575.00 −0.668836
\(178\) −1474.77 −0.621005
\(179\) 1025.58 0.428245 0.214122 0.976807i \(-0.431311\pi\)
0.214122 + 0.976807i \(0.431311\pi\)
\(180\) 0 0
\(181\) 2899.40 1.19067 0.595333 0.803479i \(-0.297020\pi\)
0.595333 + 0.803479i \(0.297020\pi\)
\(182\) 1169.96 0.476501
\(183\) −1058.38 −0.427529
\(184\) −535.638 −0.214608
\(185\) 0 0
\(186\) −104.382 −0.0411486
\(187\) 3285.99 1.28500
\(188\) 3879.57 1.50504
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) 1074.18 0.406939 0.203469 0.979081i \(-0.434778\pi\)
0.203469 + 0.979081i \(0.434778\pi\)
\(192\) −984.204 −0.369942
\(193\) −898.999 −0.335292 −0.167646 0.985847i \(-0.553617\pi\)
−0.167646 + 0.985847i \(0.553617\pi\)
\(194\) 1775.98 0.657257
\(195\) 0 0
\(196\) 326.186 0.118872
\(197\) −3063.63 −1.10799 −0.553996 0.832519i \(-0.686898\pi\)
−0.553996 + 0.832519i \(0.686898\pi\)
\(198\) 1673.47 0.600648
\(199\) −949.522 −0.338240 −0.169120 0.985595i \(-0.554093\pi\)
−0.169120 + 0.985595i \(0.554093\pi\)
\(200\) 0 0
\(201\) 2209.56 0.775375
\(202\) −1656.66 −0.577039
\(203\) −411.145 −0.142151
\(204\) 1351.15 0.463721
\(205\) 0 0
\(206\) −1962.55 −0.663774
\(207\) 937.499 0.314786
\(208\) −3184.38 −1.06152
\(209\) −4529.25 −1.49902
\(210\) 0 0
\(211\) 2306.64 0.752587 0.376294 0.926500i \(-0.377198\pi\)
0.376294 + 0.926500i \(0.377198\pi\)
\(212\) −4147.35 −1.34359
\(213\) −1476.79 −0.475062
\(214\) −7515.53 −2.40071
\(215\) 0 0
\(216\) −138.838 −0.0437348
\(217\) −63.6182 −0.0199018
\(218\) 2087.17 0.648444
\(219\) −3494.26 −1.07817
\(220\) 0 0
\(221\) 2953.69 0.899033
\(222\) 2902.06 0.877357
\(223\) 3227.61 0.969222 0.484611 0.874730i \(-0.338961\pi\)
0.484611 + 0.874730i \(0.338961\pi\)
\(224\) −1666.79 −0.497174
\(225\) 0 0
\(226\) 887.519 0.261225
\(227\) 637.820 0.186492 0.0932458 0.995643i \(-0.470276\pi\)
0.0932458 + 0.995643i \(0.470276\pi\)
\(228\) −1862.35 −0.540953
\(229\) 544.774 0.157204 0.0786019 0.996906i \(-0.474954\pi\)
0.0786019 + 0.996906i \(0.474954\pi\)
\(230\) 0 0
\(231\) 1019.94 0.290507
\(232\) 302.024 0.0854691
\(233\) 5748.54 1.61631 0.808154 0.588972i \(-0.200467\pi\)
0.808154 + 0.588972i \(0.200467\pi\)
\(234\) 1504.23 0.420234
\(235\) 0 0
\(236\) −3494.84 −0.963961
\(237\) −2616.59 −0.717154
\(238\) 1813.14 0.493816
\(239\) −2678.10 −0.724820 −0.362410 0.932019i \(-0.618046\pi\)
−0.362410 + 0.932019i \(0.618046\pi\)
\(240\) 0 0
\(241\) −2202.16 −0.588604 −0.294302 0.955713i \(-0.595087\pi\)
−0.294302 + 0.955713i \(0.595087\pi\)
\(242\) 3935.25 1.04532
\(243\) 243.000 0.0641500
\(244\) −2348.50 −0.616177
\(245\) 0 0
\(246\) 3173.09 0.822393
\(247\) −4071.21 −1.04877
\(248\) 46.7333 0.0119660
\(249\) 1588.76 0.404353
\(250\) 0 0
\(251\) −5716.90 −1.43764 −0.718820 0.695196i \(-0.755317\pi\)
−0.718820 + 0.695196i \(0.755317\pi\)
\(252\) 419.382 0.104836
\(253\) 5059.22 1.25719
\(254\) −9084.47 −2.24413
\(255\) 0 0
\(256\) 5108.88 1.24728
\(257\) −4724.29 −1.14666 −0.573332 0.819323i \(-0.694350\pi\)
−0.573332 + 0.819323i \(0.694350\pi\)
\(258\) 1063.97 0.256743
\(259\) 1768.73 0.424339
\(260\) 0 0
\(261\) −528.616 −0.125366
\(262\) 4594.25 1.08334
\(263\) −5975.36 −1.40097 −0.700487 0.713665i \(-0.747034\pi\)
−0.700487 + 0.713665i \(0.747034\pi\)
\(264\) −749.238 −0.174668
\(265\) 0 0
\(266\) −2499.14 −0.576059
\(267\) −1155.65 −0.264886
\(268\) 4902.90 1.11751
\(269\) 4486.11 1.01681 0.508407 0.861117i \(-0.330234\pi\)
0.508407 + 0.861117i \(0.330234\pi\)
\(270\) 0 0
\(271\) 3827.68 0.857989 0.428994 0.903307i \(-0.358868\pi\)
0.428994 + 0.903307i \(0.358868\pi\)
\(272\) −4934.97 −1.10010
\(273\) 916.794 0.203249
\(274\) −10647.8 −2.34766
\(275\) 0 0
\(276\) 2080.26 0.453685
\(277\) 3420.54 0.741950 0.370975 0.928643i \(-0.379024\pi\)
0.370975 + 0.928643i \(0.379024\pi\)
\(278\) −4768.71 −1.02881
\(279\) −81.7948 −0.0175517
\(280\) 0 0
\(281\) 5235.92 1.11156 0.555781 0.831329i \(-0.312419\pi\)
0.555781 + 0.831329i \(0.312419\pi\)
\(282\) 6693.55 1.41346
\(283\) 6985.88 1.46738 0.733688 0.679486i \(-0.237797\pi\)
0.733688 + 0.679486i \(0.237797\pi\)
\(284\) −3276.93 −0.684683
\(285\) 0 0
\(286\) 8117.60 1.67834
\(287\) 1933.92 0.397755
\(288\) −2143.01 −0.438466
\(289\) −335.550 −0.0682984
\(290\) 0 0
\(291\) 1391.68 0.280349
\(292\) −7753.59 −1.55392
\(293\) −7399.70 −1.47541 −0.737705 0.675123i \(-0.764090\pi\)
−0.737705 + 0.675123i \(0.764090\pi\)
\(294\) 562.779 0.111639
\(295\) 0 0
\(296\) −1299.30 −0.255135
\(297\) 1311.35 0.256203
\(298\) 74.2892 0.0144411
\(299\) 4547.58 0.879577
\(300\) 0 0
\(301\) 648.463 0.124175
\(302\) −8996.04 −1.71412
\(303\) −1298.17 −0.246133
\(304\) 6802.11 1.28332
\(305\) 0 0
\(306\) 2331.17 0.435504
\(307\) −2668.64 −0.496116 −0.248058 0.968745i \(-0.579792\pi\)
−0.248058 + 0.968745i \(0.579792\pi\)
\(308\) 2263.20 0.418693
\(309\) −1537.88 −0.283129
\(310\) 0 0
\(311\) 6189.25 1.12849 0.564244 0.825608i \(-0.309168\pi\)
0.564244 + 0.825608i \(0.309168\pi\)
\(312\) −673.468 −0.122204
\(313\) 2921.59 0.527598 0.263799 0.964578i \(-0.415024\pi\)
0.263799 + 0.964578i \(0.415024\pi\)
\(314\) 14925.0 2.68237
\(315\) 0 0
\(316\) −5806.08 −1.03360
\(317\) −9825.56 −1.74088 −0.870439 0.492276i \(-0.836165\pi\)
−0.870439 + 0.492276i \(0.836165\pi\)
\(318\) −7155.55 −1.26183
\(319\) −2852.68 −0.500687
\(320\) 0 0
\(321\) −5889.26 −1.02401
\(322\) 2791.56 0.483129
\(323\) −6309.33 −1.08687
\(324\) 539.205 0.0924563
\(325\) 0 0
\(326\) −5848.07 −0.993541
\(327\) 1635.53 0.276590
\(328\) −1420.64 −0.239151
\(329\) 4079.56 0.683627
\(330\) 0 0
\(331\) −9258.17 −1.53739 −0.768693 0.639618i \(-0.779093\pi\)
−0.768693 + 0.639618i \(0.779093\pi\)
\(332\) 3525.39 0.582774
\(333\) 2274.09 0.374232
\(334\) 3822.75 0.626263
\(335\) 0 0
\(336\) −1531.76 −0.248704
\(337\) 3693.98 0.597103 0.298552 0.954394i \(-0.403497\pi\)
0.298552 + 0.954394i \(0.403497\pi\)
\(338\) −1114.38 −0.179331
\(339\) 695.470 0.111424
\(340\) 0 0
\(341\) −441.406 −0.0700982
\(342\) −3213.17 −0.508037
\(343\) 343.000 0.0539949
\(344\) −476.355 −0.0746608
\(345\) 0 0
\(346\) −2623.44 −0.407622
\(347\) −3832.83 −0.592960 −0.296480 0.955039i \(-0.595813\pi\)
−0.296480 + 0.955039i \(0.595813\pi\)
\(348\) −1172.97 −0.180684
\(349\) 8325.22 1.27690 0.638451 0.769662i \(-0.279575\pi\)
0.638451 + 0.769662i \(0.279575\pi\)
\(350\) 0 0
\(351\) 1178.74 0.179248
\(352\) −11564.8 −1.75115
\(353\) 8991.52 1.35572 0.677862 0.735189i \(-0.262907\pi\)
0.677862 + 0.735189i \(0.262907\pi\)
\(354\) −6029.76 −0.905306
\(355\) 0 0
\(356\) −2564.33 −0.381767
\(357\) 1420.79 0.210634
\(358\) 3926.38 0.579652
\(359\) −12893.8 −1.89557 −0.947783 0.318917i \(-0.896681\pi\)
−0.947783 + 0.318917i \(0.896681\pi\)
\(360\) 0 0
\(361\) 1837.46 0.267891
\(362\) 11100.1 1.61163
\(363\) 3083.71 0.445875
\(364\) 2034.32 0.292932
\(365\) 0 0
\(366\) −4051.94 −0.578684
\(367\) 7480.17 1.06393 0.531964 0.846767i \(-0.321454\pi\)
0.531964 + 0.846767i \(0.321454\pi\)
\(368\) −7598.02 −1.07629
\(369\) 2486.47 0.350787
\(370\) 0 0
\(371\) −4361.14 −0.610293
\(372\) −181.499 −0.0252964
\(373\) 3523.32 0.489090 0.244545 0.969638i \(-0.421361\pi\)
0.244545 + 0.969638i \(0.421361\pi\)
\(374\) 12580.2 1.73932
\(375\) 0 0
\(376\) −2996.81 −0.411033
\(377\) −2564.19 −0.350298
\(378\) 723.573 0.0984565
\(379\) 13515.4 1.83177 0.915886 0.401438i \(-0.131490\pi\)
0.915886 + 0.401438i \(0.131490\pi\)
\(380\) 0 0
\(381\) −7118.69 −0.957222
\(382\) 4112.44 0.550813
\(383\) 657.182 0.0876774 0.0438387 0.999039i \(-0.486041\pi\)
0.0438387 + 0.999039i \(0.486041\pi\)
\(384\) 1946.75 0.258710
\(385\) 0 0
\(386\) −3441.75 −0.453836
\(387\) 833.738 0.109512
\(388\) 3088.06 0.404053
\(389\) −9741.87 −1.26975 −0.634875 0.772615i \(-0.718948\pi\)
−0.634875 + 0.772615i \(0.718948\pi\)
\(390\) 0 0
\(391\) 7047.58 0.911538
\(392\) −251.965 −0.0324646
\(393\) 3600.11 0.462091
\(394\) −11728.9 −1.49973
\(395\) 0 0
\(396\) 2909.82 0.369253
\(397\) −4407.42 −0.557184 −0.278592 0.960410i \(-0.589868\pi\)
−0.278592 + 0.960410i \(0.589868\pi\)
\(398\) −3635.18 −0.457827
\(399\) −1958.35 −0.245715
\(400\) 0 0
\(401\) −11569.5 −1.44078 −0.720391 0.693568i \(-0.756037\pi\)
−0.720391 + 0.693568i \(0.756037\pi\)
\(402\) 8459.14 1.04951
\(403\) −396.767 −0.0490431
\(404\) −2880.59 −0.354739
\(405\) 0 0
\(406\) −1574.04 −0.192410
\(407\) 12272.1 1.49461
\(408\) −1043.70 −0.126645
\(409\) −3083.03 −0.372729 −0.186364 0.982481i \(-0.559670\pi\)
−0.186364 + 0.982481i \(0.559670\pi\)
\(410\) 0 0
\(411\) −8343.76 −1.00138
\(412\) −3412.47 −0.408060
\(413\) −3674.99 −0.437856
\(414\) 3589.15 0.426079
\(415\) 0 0
\(416\) −10395.3 −1.22517
\(417\) −3736.81 −0.438831
\(418\) −17339.9 −2.02900
\(419\) −5415.21 −0.631385 −0.315692 0.948862i \(-0.602237\pi\)
−0.315692 + 0.948862i \(0.602237\pi\)
\(420\) 0 0
\(421\) 4188.34 0.484863 0.242432 0.970168i \(-0.422055\pi\)
0.242432 + 0.970168i \(0.422055\pi\)
\(422\) 8830.82 1.01867
\(423\) 5245.15 0.602902
\(424\) 3203.65 0.366941
\(425\) 0 0
\(426\) −5653.79 −0.643021
\(427\) −2469.56 −0.279884
\(428\) −13068.0 −1.47585
\(429\) 6361.05 0.715884
\(430\) 0 0
\(431\) −9108.41 −1.01795 −0.508975 0.860781i \(-0.669976\pi\)
−0.508975 + 0.860781i \(0.669976\pi\)
\(432\) −1969.41 −0.219336
\(433\) 16847.0 1.86979 0.934893 0.354930i \(-0.115495\pi\)
0.934893 + 0.354930i \(0.115495\pi\)
\(434\) −243.558 −0.0269381
\(435\) 0 0
\(436\) 3629.16 0.398635
\(437\) −9714.03 −1.06335
\(438\) −13377.5 −1.45937
\(439\) 8434.14 0.916946 0.458473 0.888708i \(-0.348397\pi\)
0.458473 + 0.888708i \(0.348397\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 11308.0 1.21689
\(443\) 4298.49 0.461010 0.230505 0.973071i \(-0.425962\pi\)
0.230505 + 0.973071i \(0.425962\pi\)
\(444\) 5046.09 0.539362
\(445\) 0 0
\(446\) 12356.7 1.31189
\(447\) 58.2139 0.00615978
\(448\) −2296.48 −0.242184
\(449\) 10545.0 1.10835 0.554173 0.832402i \(-0.313035\pi\)
0.554173 + 0.832402i \(0.313035\pi\)
\(450\) 0 0
\(451\) 13418.2 1.40098
\(452\) 1543.21 0.160590
\(453\) −7049.40 −0.731147
\(454\) 2441.85 0.252426
\(455\) 0 0
\(456\) 1438.59 0.147737
\(457\) −11952.4 −1.22344 −0.611719 0.791075i \(-0.709522\pi\)
−0.611719 + 0.791075i \(0.709522\pi\)
\(458\) 2085.63 0.212784
\(459\) 1826.74 0.185762
\(460\) 0 0
\(461\) −17200.9 −1.73780 −0.868900 0.494988i \(-0.835173\pi\)
−0.868900 + 0.494988i \(0.835173\pi\)
\(462\) 3904.76 0.393217
\(463\) 10368.7 1.04076 0.520381 0.853934i \(-0.325790\pi\)
0.520381 + 0.853934i \(0.325790\pi\)
\(464\) 4284.20 0.428640
\(465\) 0 0
\(466\) 22007.9 2.18776
\(467\) 16879.5 1.67257 0.836284 0.548296i \(-0.184723\pi\)
0.836284 + 0.548296i \(0.184723\pi\)
\(468\) 2615.56 0.258342
\(469\) 5155.64 0.507602
\(470\) 0 0
\(471\) 11695.4 1.14415
\(472\) 2699.62 0.263263
\(473\) 4499.27 0.437371
\(474\) −10017.4 −0.970706
\(475\) 0 0
\(476\) 3152.67 0.303577
\(477\) −5607.17 −0.538228
\(478\) −10252.9 −0.981082
\(479\) 7329.12 0.699115 0.349558 0.936915i \(-0.386332\pi\)
0.349558 + 0.936915i \(0.386332\pi\)
\(480\) 0 0
\(481\) 11031.0 1.04568
\(482\) −8430.80 −0.796706
\(483\) 2187.50 0.206076
\(484\) 6842.60 0.642619
\(485\) 0 0
\(486\) 930.308 0.0868305
\(487\) 17209.5 1.60131 0.800655 0.599125i \(-0.204485\pi\)
0.800655 + 0.599125i \(0.204485\pi\)
\(488\) 1814.11 0.168281
\(489\) −4582.61 −0.423789
\(490\) 0 0
\(491\) 11392.5 1.04712 0.523560 0.851989i \(-0.324604\pi\)
0.523560 + 0.851989i \(0.324604\pi\)
\(492\) 5517.35 0.505572
\(493\) −3973.83 −0.363027
\(494\) −15586.3 −1.41956
\(495\) 0 0
\(496\) 662.912 0.0600113
\(497\) −3445.85 −0.311001
\(498\) 6082.47 0.547313
\(499\) 19079.4 1.71164 0.855822 0.517271i \(-0.173052\pi\)
0.855822 + 0.517271i \(0.173052\pi\)
\(500\) 0 0
\(501\) 2995.55 0.267129
\(502\) −21886.7 −1.94592
\(503\) 13499.4 1.19663 0.598317 0.801259i \(-0.295836\pi\)
0.598317 + 0.801259i \(0.295836\pi\)
\(504\) −323.955 −0.0286311
\(505\) 0 0
\(506\) 19368.8 1.70168
\(507\) −873.237 −0.0764928
\(508\) −15796.0 −1.37960
\(509\) −4328.85 −0.376960 −0.188480 0.982077i \(-0.560356\pi\)
−0.188480 + 0.982077i \(0.560356\pi\)
\(510\) 0 0
\(511\) −8153.27 −0.705831
\(512\) 14367.6 1.24017
\(513\) −2517.88 −0.216700
\(514\) −18086.6 −1.55207
\(515\) 0 0
\(516\) 1850.02 0.157835
\(517\) 28305.5 2.40788
\(518\) 6771.47 0.574365
\(519\) −2055.76 −0.173869
\(520\) 0 0
\(521\) 19395.7 1.63098 0.815490 0.578771i \(-0.196468\pi\)
0.815490 + 0.578771i \(0.196468\pi\)
\(522\) −2023.77 −0.169689
\(523\) −20413.8 −1.70675 −0.853377 0.521294i \(-0.825450\pi\)
−0.853377 + 0.521294i \(0.825450\pi\)
\(524\) 7988.47 0.665989
\(525\) 0 0
\(526\) −22876.2 −1.89629
\(527\) −614.887 −0.0508252
\(528\) −10627.9 −0.875987
\(529\) −1316.34 −0.108189
\(530\) 0 0
\(531\) −4724.99 −0.386153
\(532\) −4345.49 −0.354137
\(533\) 12061.3 0.980171
\(534\) −4424.32 −0.358537
\(535\) 0 0
\(536\) −3787.28 −0.305197
\(537\) 3076.75 0.247247
\(538\) 17174.8 1.37631
\(539\) 2379.86 0.190181
\(540\) 0 0
\(541\) −4919.18 −0.390928 −0.195464 0.980711i \(-0.562621\pi\)
−0.195464 + 0.980711i \(0.562621\pi\)
\(542\) 14654.0 1.16133
\(543\) 8698.19 0.687431
\(544\) −16110.0 −1.26969
\(545\) 0 0
\(546\) 3509.88 0.275108
\(547\) −15334.2 −1.19862 −0.599308 0.800518i \(-0.704558\pi\)
−0.599308 + 0.800518i \(0.704558\pi\)
\(548\) −18514.4 −1.44324
\(549\) −3175.15 −0.246834
\(550\) 0 0
\(551\) 5477.33 0.423488
\(552\) −1606.92 −0.123904
\(553\) −6105.37 −0.469487
\(554\) 13095.3 1.00427
\(555\) 0 0
\(556\) −8291.81 −0.632466
\(557\) 8613.78 0.655256 0.327628 0.944807i \(-0.393751\pi\)
0.327628 + 0.944807i \(0.393751\pi\)
\(558\) −313.145 −0.0237572
\(559\) 4044.26 0.306000
\(560\) 0 0
\(561\) 9857.98 0.741897
\(562\) 20045.3 1.50456
\(563\) 2320.81 0.173731 0.0868654 0.996220i \(-0.472315\pi\)
0.0868654 + 0.996220i \(0.472315\pi\)
\(564\) 11638.7 0.868934
\(565\) 0 0
\(566\) 26744.9 1.98617
\(567\) 567.000 0.0419961
\(568\) 2531.29 0.186990
\(569\) −1736.04 −0.127906 −0.0639529 0.997953i \(-0.520371\pi\)
−0.0639529 + 0.997953i \(0.520371\pi\)
\(570\) 0 0
\(571\) 23897.8 1.75148 0.875738 0.482786i \(-0.160375\pi\)
0.875738 + 0.482786i \(0.160375\pi\)
\(572\) 14114.9 1.03177
\(573\) 3222.55 0.234946
\(574\) 7403.87 0.538382
\(575\) 0 0
\(576\) −2952.61 −0.213586
\(577\) −8029.26 −0.579311 −0.289655 0.957131i \(-0.593541\pi\)
−0.289655 + 0.957131i \(0.593541\pi\)
\(578\) −1284.63 −0.0924455
\(579\) −2697.00 −0.193581
\(580\) 0 0
\(581\) 3707.12 0.264711
\(582\) 5327.93 0.379467
\(583\) −30259.1 −2.14958
\(584\) 5989.32 0.424383
\(585\) 0 0
\(586\) −28329.2 −1.99705
\(587\) 8015.14 0.563578 0.281789 0.959476i \(-0.409072\pi\)
0.281789 + 0.959476i \(0.409072\pi\)
\(588\) 978.558 0.0686310
\(589\) 847.529 0.0592900
\(590\) 0 0
\(591\) −9190.88 −0.639700
\(592\) −18430.5 −1.27954
\(593\) −12820.3 −0.887801 −0.443901 0.896076i \(-0.646406\pi\)
−0.443901 + 0.896076i \(0.646406\pi\)
\(594\) 5020.41 0.346784
\(595\) 0 0
\(596\) 129.174 0.00887779
\(597\) −2848.57 −0.195283
\(598\) 17410.1 1.19055
\(599\) −15330.5 −1.04572 −0.522860 0.852418i \(-0.675135\pi\)
−0.522860 + 0.852418i \(0.675135\pi\)
\(600\) 0 0
\(601\) 107.658 0.00730694 0.00365347 0.999993i \(-0.498837\pi\)
0.00365347 + 0.999993i \(0.498837\pi\)
\(602\) 2482.59 0.168078
\(603\) 6628.68 0.447663
\(604\) −15642.3 −1.05377
\(605\) 0 0
\(606\) −4969.97 −0.333154
\(607\) 8213.06 0.549189 0.274595 0.961560i \(-0.411456\pi\)
0.274595 + 0.961560i \(0.411456\pi\)
\(608\) 22205.2 1.48115
\(609\) −1233.44 −0.0820712
\(610\) 0 0
\(611\) 25443.0 1.68463
\(612\) 4053.44 0.267729
\(613\) −1242.60 −0.0818732 −0.0409366 0.999162i \(-0.513034\pi\)
−0.0409366 + 0.999162i \(0.513034\pi\)
\(614\) −10216.7 −0.671519
\(615\) 0 0
\(616\) −1748.22 −0.114347
\(617\) 13170.6 0.859367 0.429683 0.902980i \(-0.358625\pi\)
0.429683 + 0.902980i \(0.358625\pi\)
\(618\) −5887.65 −0.383230
\(619\) 19774.1 1.28399 0.641993 0.766711i \(-0.278108\pi\)
0.641993 + 0.766711i \(0.278108\pi\)
\(620\) 0 0
\(621\) 2812.50 0.181742
\(622\) 23695.1 1.52747
\(623\) −2696.51 −0.173409
\(624\) −9553.14 −0.612871
\(625\) 0 0
\(626\) 11185.1 0.714132
\(627\) −13587.8 −0.865459
\(628\) 25951.5 1.64901
\(629\) 17095.3 1.08368
\(630\) 0 0
\(631\) −14308.8 −0.902735 −0.451367 0.892338i \(-0.649064\pi\)
−0.451367 + 0.892338i \(0.649064\pi\)
\(632\) 4484.95 0.282281
\(633\) 6919.93 0.434507
\(634\) −37616.4 −2.35637
\(635\) 0 0
\(636\) −12442.0 −0.775722
\(637\) 2139.19 0.133058
\(638\) −10921.3 −0.677707
\(639\) −4430.38 −0.274277
\(640\) 0 0
\(641\) 11537.5 0.710925 0.355463 0.934691i \(-0.384323\pi\)
0.355463 + 0.934691i \(0.384323\pi\)
\(642\) −22546.6 −1.38605
\(643\) −19603.0 −1.20228 −0.601139 0.799144i \(-0.705286\pi\)
−0.601139 + 0.799144i \(0.705286\pi\)
\(644\) 4853.95 0.297007
\(645\) 0 0
\(646\) −24154.8 −1.47114
\(647\) −21650.0 −1.31553 −0.657765 0.753223i \(-0.728498\pi\)
−0.657765 + 0.753223i \(0.728498\pi\)
\(648\) −416.513 −0.0252503
\(649\) −25498.4 −1.54222
\(650\) 0 0
\(651\) −190.855 −0.0114903
\(652\) −10168.6 −0.610787
\(653\) 2927.33 0.175429 0.0877145 0.996146i \(-0.472044\pi\)
0.0877145 + 0.996146i \(0.472044\pi\)
\(654\) 6261.50 0.374379
\(655\) 0 0
\(656\) −20151.7 −1.19938
\(657\) −10482.8 −0.622484
\(658\) 15618.3 0.925326
\(659\) −4778.76 −0.282480 −0.141240 0.989975i \(-0.545109\pi\)
−0.141240 + 0.989975i \(0.545109\pi\)
\(660\) 0 0
\(661\) −31510.3 −1.85417 −0.927086 0.374849i \(-0.877695\pi\)
−0.927086 + 0.374849i \(0.877695\pi\)
\(662\) −35444.2 −2.08093
\(663\) 8861.06 0.519057
\(664\) −2723.21 −0.159158
\(665\) 0 0
\(666\) 8706.17 0.506542
\(667\) −6118.23 −0.355170
\(668\) 6646.99 0.385000
\(669\) 9682.82 0.559581
\(670\) 0 0
\(671\) −17134.7 −0.985808
\(672\) −5000.37 −0.287044
\(673\) 8992.15 0.515040 0.257520 0.966273i \(-0.417095\pi\)
0.257520 + 0.966273i \(0.417095\pi\)
\(674\) 14142.1 0.808211
\(675\) 0 0
\(676\) −1937.67 −0.110245
\(677\) −19340.8 −1.09797 −0.548985 0.835832i \(-0.684986\pi\)
−0.548985 + 0.835832i \(0.684986\pi\)
\(678\) 2662.56 0.150818
\(679\) 3247.25 0.183531
\(680\) 0 0
\(681\) 1913.46 0.107671
\(682\) −1689.89 −0.0948816
\(683\) 4255.14 0.238387 0.119194 0.992871i \(-0.461969\pi\)
0.119194 + 0.992871i \(0.461969\pi\)
\(684\) −5587.05 −0.312319
\(685\) 0 0
\(686\) 1313.15 0.0730850
\(687\) 1634.32 0.0907616
\(688\) −6757.08 −0.374435
\(689\) −27199.1 −1.50392
\(690\) 0 0
\(691\) −17505.5 −0.963733 −0.481867 0.876245i \(-0.660041\pi\)
−0.481867 + 0.876245i \(0.660041\pi\)
\(692\) −4561.63 −0.250588
\(693\) 3059.82 0.167724
\(694\) −14673.7 −0.802603
\(695\) 0 0
\(696\) 906.071 0.0493456
\(697\) 18691.8 1.01579
\(698\) 31872.5 1.72836
\(699\) 17245.6 0.933175
\(700\) 0 0
\(701\) −3240.77 −0.174611 −0.0873054 0.996182i \(-0.527826\pi\)
−0.0873054 + 0.996182i \(0.527826\pi\)
\(702\) 4512.70 0.242622
\(703\) −23563.3 −1.26416
\(704\) −15933.8 −0.853022
\(705\) 0 0
\(706\) 34423.4 1.83504
\(707\) −3029.07 −0.161132
\(708\) −10484.5 −0.556543
\(709\) 19949.3 1.05672 0.528358 0.849022i \(-0.322808\pi\)
0.528358 + 0.849022i \(0.322808\pi\)
\(710\) 0 0
\(711\) −7849.76 −0.414049
\(712\) 1980.83 0.104262
\(713\) −946.698 −0.0497253
\(714\) 5439.41 0.285105
\(715\) 0 0
\(716\) 6827.17 0.356345
\(717\) −8034.30 −0.418475
\(718\) −49362.9 −2.56575
\(719\) −11259.4 −0.584011 −0.292006 0.956417i \(-0.594323\pi\)
−0.292006 + 0.956417i \(0.594323\pi\)
\(720\) 0 0
\(721\) −3588.38 −0.185351
\(722\) 7034.60 0.362605
\(723\) −6606.47 −0.339830
\(724\) 19300.9 0.990761
\(725\) 0 0
\(726\) 11805.8 0.603516
\(727\) 12228.5 0.623840 0.311920 0.950108i \(-0.399028\pi\)
0.311920 + 0.950108i \(0.399028\pi\)
\(728\) −1571.43 −0.0800013
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 6267.56 0.317119
\(732\) −7045.49 −0.355750
\(733\) 26635.1 1.34214 0.671072 0.741392i \(-0.265834\pi\)
0.671072 + 0.741392i \(0.265834\pi\)
\(734\) 28637.3 1.44008
\(735\) 0 0
\(736\) −24803.4 −1.24221
\(737\) 35771.7 1.78788
\(738\) 9519.26 0.474809
\(739\) −6074.00 −0.302349 −0.151174 0.988507i \(-0.548306\pi\)
−0.151174 + 0.988507i \(0.548306\pi\)
\(740\) 0 0
\(741\) −12213.6 −0.605505
\(742\) −16696.3 −0.826065
\(743\) 4016.87 0.198337 0.0991686 0.995071i \(-0.468382\pi\)
0.0991686 + 0.995071i \(0.468382\pi\)
\(744\) 140.200 0.00690858
\(745\) 0 0
\(746\) 13488.8 0.662010
\(747\) 4766.29 0.233453
\(748\) 21874.4 1.06926
\(749\) −13741.6 −0.670370
\(750\) 0 0
\(751\) −23913.2 −1.16192 −0.580962 0.813931i \(-0.697324\pi\)
−0.580962 + 0.813931i \(0.697324\pi\)
\(752\) −42509.6 −2.06139
\(753\) −17150.7 −0.830022
\(754\) −9816.81 −0.474147
\(755\) 0 0
\(756\) 1258.15 0.0605269
\(757\) −31044.9 −1.49055 −0.745275 0.666758i \(-0.767682\pi\)
−0.745275 + 0.666758i \(0.767682\pi\)
\(758\) 51742.9 2.47940
\(759\) 15177.6 0.725842
\(760\) 0 0
\(761\) 14011.8 0.667446 0.333723 0.942671i \(-0.391695\pi\)
0.333723 + 0.942671i \(0.391695\pi\)
\(762\) −27253.4 −1.29565
\(763\) 3816.23 0.181071
\(764\) 7150.69 0.338616
\(765\) 0 0
\(766\) 2515.97 0.118676
\(767\) −22919.8 −1.07899
\(768\) 15326.6 0.720120
\(769\) 3342.49 0.156740 0.0783701 0.996924i \(-0.475028\pi\)
0.0783701 + 0.996924i \(0.475028\pi\)
\(770\) 0 0
\(771\) −14172.9 −0.662027
\(772\) −5984.51 −0.278999
\(773\) 21074.6 0.980594 0.490297 0.871555i \(-0.336888\pi\)
0.490297 + 0.871555i \(0.336888\pi\)
\(774\) 3191.90 0.148231
\(775\) 0 0
\(776\) −2385.40 −0.110349
\(777\) 5306.20 0.244992
\(778\) −37296.0 −1.71867
\(779\) −25763.9 −1.18496
\(780\) 0 0
\(781\) −23908.5 −1.09541
\(782\) 26981.1 1.23382
\(783\) −1585.85 −0.0723800
\(784\) −3574.12 −0.162815
\(785\) 0 0
\(786\) 13782.8 0.625464
\(787\) −21394.8 −0.969048 −0.484524 0.874778i \(-0.661007\pi\)
−0.484524 + 0.874778i \(0.661007\pi\)
\(788\) −20394.1 −0.921968
\(789\) −17926.1 −0.808853
\(790\) 0 0
\(791\) 1622.76 0.0729442
\(792\) −2247.71 −0.100845
\(793\) −15401.9 −0.689706
\(794\) −16873.5 −0.754179
\(795\) 0 0
\(796\) −6320.83 −0.281452
\(797\) −20645.0 −0.917547 −0.458773 0.888553i \(-0.651711\pi\)
−0.458773 + 0.888553i \(0.651711\pi\)
\(798\) −7497.41 −0.332588
\(799\) 39430.0 1.74585
\(800\) 0 0
\(801\) −3466.95 −0.152932
\(802\) −44293.0 −1.95017
\(803\) −56570.4 −2.48608
\(804\) 14708.7 0.645194
\(805\) 0 0
\(806\) −1518.99 −0.0663825
\(807\) 13458.3 0.587058
\(808\) 2225.13 0.0968810
\(809\) −15939.0 −0.692688 −0.346344 0.938108i \(-0.612577\pi\)
−0.346344 + 0.938108i \(0.612577\pi\)
\(810\) 0 0
\(811\) 22829.2 0.988460 0.494230 0.869331i \(-0.335450\pi\)
0.494230 + 0.869331i \(0.335450\pi\)
\(812\) −2736.94 −0.118285
\(813\) 11483.0 0.495360
\(814\) 46982.9 2.02303
\(815\) 0 0
\(816\) −14804.9 −0.635141
\(817\) −8638.90 −0.369935
\(818\) −11803.2 −0.504508
\(819\) 2750.38 0.117346
\(820\) 0 0
\(821\) −5700.22 −0.242313 −0.121157 0.992633i \(-0.538660\pi\)
−0.121157 + 0.992633i \(0.538660\pi\)
\(822\) −31943.5 −1.35542
\(823\) 32438.0 1.37390 0.686948 0.726707i \(-0.258950\pi\)
0.686948 + 0.726707i \(0.258950\pi\)
\(824\) 2635.99 0.111443
\(825\) 0 0
\(826\) −14069.4 −0.592662
\(827\) 12762.6 0.536638 0.268319 0.963330i \(-0.413532\pi\)
0.268319 + 0.963330i \(0.413532\pi\)
\(828\) 6240.79 0.261935
\(829\) −30766.7 −1.28899 −0.644494 0.764609i \(-0.722932\pi\)
−0.644494 + 0.764609i \(0.722932\pi\)
\(830\) 0 0
\(831\) 10261.6 0.428365
\(832\) −14322.4 −0.596804
\(833\) 3315.19 0.137892
\(834\) −14306.1 −0.593981
\(835\) 0 0
\(836\) −30150.6 −1.24734
\(837\) −245.384 −0.0101335
\(838\) −20731.7 −0.854613
\(839\) −9779.71 −0.402423 −0.201212 0.979548i \(-0.564488\pi\)
−0.201212 + 0.979548i \(0.564488\pi\)
\(840\) 0 0
\(841\) −20939.2 −0.858551
\(842\) 16034.8 0.656288
\(843\) 15707.8 0.641760
\(844\) 15355.0 0.626233
\(845\) 0 0
\(846\) 20080.7 0.816061
\(847\) 7195.32 0.291894
\(848\) 45443.7 1.84026
\(849\) 20957.6 0.847190
\(850\) 0 0
\(851\) 26320.4 1.06023
\(852\) −9830.79 −0.395302
\(853\) −24201.5 −0.971445 −0.485723 0.874113i \(-0.661444\pi\)
−0.485723 + 0.874113i \(0.661444\pi\)
\(854\) −9454.52 −0.378837
\(855\) 0 0
\(856\) 10094.5 0.403062
\(857\) −21036.7 −0.838507 −0.419254 0.907869i \(-0.637708\pi\)
−0.419254 + 0.907869i \(0.637708\pi\)
\(858\) 24352.8 0.968988
\(859\) −6179.19 −0.245438 −0.122719 0.992441i \(-0.539161\pi\)
−0.122719 + 0.992441i \(0.539161\pi\)
\(860\) 0 0
\(861\) 5801.76 0.229644
\(862\) −34870.9 −1.37785
\(863\) −50256.2 −1.98232 −0.991160 0.132671i \(-0.957644\pi\)
−0.991160 + 0.132671i \(0.957644\pi\)
\(864\) −6429.04 −0.253149
\(865\) 0 0
\(866\) 64497.7 2.53085
\(867\) −1006.65 −0.0394321
\(868\) −423.497 −0.0165604
\(869\) −42361.2 −1.65363
\(870\) 0 0
\(871\) 32154.1 1.25086
\(872\) −2803.37 −0.108869
\(873\) 4175.03 0.161860
\(874\) −37189.5 −1.43930
\(875\) 0 0
\(876\) −23260.8 −0.897156
\(877\) 9175.95 0.353306 0.176653 0.984273i \(-0.443473\pi\)
0.176653 + 0.984273i \(0.443473\pi\)
\(878\) 32289.5 1.24114
\(879\) −22199.1 −0.851828
\(880\) 0 0
\(881\) −26172.7 −1.00089 −0.500444 0.865769i \(-0.666830\pi\)
−0.500444 + 0.865769i \(0.666830\pi\)
\(882\) 1688.34 0.0644549
\(883\) −18615.5 −0.709471 −0.354736 0.934967i \(-0.615429\pi\)
−0.354736 + 0.934967i \(0.615429\pi\)
\(884\) 19662.3 0.748092
\(885\) 0 0
\(886\) 16456.4 0.624001
\(887\) 12837.8 0.485964 0.242982 0.970031i \(-0.421874\pi\)
0.242982 + 0.970031i \(0.421874\pi\)
\(888\) −3897.89 −0.147302
\(889\) −16610.3 −0.626649
\(890\) 0 0
\(891\) 3934.05 0.147919
\(892\) 21485.7 0.806496
\(893\) −54348.4 −2.03662
\(894\) 222.868 0.00833759
\(895\) 0 0
\(896\) 4542.42 0.169366
\(897\) 13642.7 0.507824
\(898\) 40370.6 1.50020
\(899\) 533.803 0.0198035
\(900\) 0 0
\(901\) −42151.5 −1.55857
\(902\) 51370.7 1.89630
\(903\) 1945.39 0.0716926
\(904\) −1192.07 −0.0438579
\(905\) 0 0
\(906\) −26988.1 −0.989647
\(907\) 26766.1 0.979885 0.489942 0.871755i \(-0.337018\pi\)
0.489942 + 0.871755i \(0.337018\pi\)
\(908\) 4245.87 0.155181
\(909\) −3894.52 −0.142105
\(910\) 0 0
\(911\) 5022.67 0.182666 0.0913328 0.995820i \(-0.470887\pi\)
0.0913328 + 0.995820i \(0.470887\pi\)
\(912\) 20406.3 0.740923
\(913\) 25721.3 0.932367
\(914\) −45759.0 −1.65599
\(915\) 0 0
\(916\) 3626.48 0.130810
\(917\) 8400.26 0.302509
\(918\) 6993.52 0.251439
\(919\) 9541.79 0.342497 0.171248 0.985228i \(-0.445220\pi\)
0.171248 + 0.985228i \(0.445220\pi\)
\(920\) 0 0
\(921\) −8005.93 −0.286432
\(922\) −65852.4 −2.35220
\(923\) −21490.7 −0.766387
\(924\) 6789.59 0.241733
\(925\) 0 0
\(926\) 39695.7 1.40873
\(927\) −4613.63 −0.163464
\(928\) 13985.6 0.494718
\(929\) −25479.6 −0.899846 −0.449923 0.893067i \(-0.648549\pi\)
−0.449923 + 0.893067i \(0.648549\pi\)
\(930\) 0 0
\(931\) −4569.49 −0.160858
\(932\) 38267.2 1.34494
\(933\) 18567.7 0.651533
\(934\) 64621.9 2.26391
\(935\) 0 0
\(936\) −2020.41 −0.0705545
\(937\) 33608.3 1.17176 0.585878 0.810399i \(-0.300750\pi\)
0.585878 + 0.810399i \(0.300750\pi\)
\(938\) 19738.0 0.687066
\(939\) 8764.77 0.304609
\(940\) 0 0
\(941\) −19173.6 −0.664232 −0.332116 0.943239i \(-0.607762\pi\)
−0.332116 + 0.943239i \(0.607762\pi\)
\(942\) 44774.9 1.54867
\(943\) 28778.5 0.993804
\(944\) 38294.0 1.32030
\(945\) 0 0
\(946\) 17225.1 0.592005
\(947\) 979.315 0.0336045 0.0168023 0.999859i \(-0.494651\pi\)
0.0168023 + 0.999859i \(0.494651\pi\)
\(948\) −17418.2 −0.596749
\(949\) −50849.5 −1.73935
\(950\) 0 0
\(951\) −29476.7 −1.00510
\(952\) −2435.31 −0.0829083
\(953\) −3048.61 −0.103624 −0.0518122 0.998657i \(-0.516500\pi\)
−0.0518122 + 0.998657i \(0.516500\pi\)
\(954\) −21466.7 −0.728521
\(955\) 0 0
\(956\) −17827.7 −0.603127
\(957\) −8558.03 −0.289072
\(958\) 28059.0 0.946290
\(959\) −19468.8 −0.655557
\(960\) 0 0
\(961\) −29708.4 −0.997227
\(962\) 42231.6 1.41538
\(963\) −17667.8 −0.591211
\(964\) −14659.4 −0.489781
\(965\) 0 0
\(966\) 8374.67 0.278934
\(967\) 14467.9 0.481133 0.240567 0.970633i \(-0.422667\pi\)
0.240567 + 0.970633i \(0.422667\pi\)
\(968\) −5285.62 −0.175502
\(969\) −18928.0 −0.627507
\(970\) 0 0
\(971\) 12952.8 0.428090 0.214045 0.976824i \(-0.431336\pi\)
0.214045 + 0.976824i \(0.431336\pi\)
\(972\) 1617.62 0.0533797
\(973\) −8719.23 −0.287282
\(974\) 65885.4 2.16746
\(975\) 0 0
\(976\) 25733.2 0.843954
\(977\) −47244.2 −1.54706 −0.773529 0.633760i \(-0.781511\pi\)
−0.773529 + 0.633760i \(0.781511\pi\)
\(978\) −17544.2 −0.573621
\(979\) −18709.4 −0.610781
\(980\) 0 0
\(981\) 4906.58 0.159689
\(982\) 43615.3 1.41733
\(983\) 1536.84 0.0498651 0.0249326 0.999689i \(-0.492063\pi\)
0.0249326 + 0.999689i \(0.492063\pi\)
\(984\) −4261.92 −0.138074
\(985\) 0 0
\(986\) −15213.5 −0.491376
\(987\) 12238.7 0.394692
\(988\) −27101.5 −0.872685
\(989\) 9649.73 0.310256
\(990\) 0 0
\(991\) 3785.22 0.121334 0.0606668 0.998158i \(-0.480677\pi\)
0.0606668 + 0.998158i \(0.480677\pi\)
\(992\) 2164.04 0.0692625
\(993\) −27774.5 −0.887610
\(994\) −13192.2 −0.420956
\(995\) 0 0
\(996\) 10576.2 0.336465
\(997\) 25894.9 0.822566 0.411283 0.911508i \(-0.365081\pi\)
0.411283 + 0.911508i \(0.365081\pi\)
\(998\) 73044.0 2.31680
\(999\) 6822.26 0.216063
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 525.4.a.l.1.2 2
3.2 odd 2 1575.4.a.q.1.1 2
5.2 odd 4 525.4.d.l.274.4 4
5.3 odd 4 525.4.d.l.274.1 4
5.4 even 2 105.4.a.e.1.1 2
15.14 odd 2 315.4.a.k.1.2 2
20.19 odd 2 1680.4.a.bo.1.1 2
35.34 odd 2 735.4.a.o.1.1 2
105.104 even 2 2205.4.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.e.1.1 2 5.4 even 2
315.4.a.k.1.2 2 15.14 odd 2
525.4.a.l.1.2 2 1.1 even 1 trivial
525.4.d.l.274.1 4 5.3 odd 4
525.4.d.l.274.4 4 5.2 odd 4
735.4.a.o.1.1 2 35.34 odd 2
1575.4.a.q.1.1 2 3.2 odd 2
1680.4.a.bo.1.1 2 20.19 odd 2
2205.4.a.bb.1.2 2 105.104 even 2