Properties

Label 315.4.a.k.1.2
Level $315$
Weight $4$
Character 315.1
Self dual yes
Analytic conductor $18.586$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.5856016518\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 315.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.82843 q^{2} +6.65685 q^{4} -5.00000 q^{5} -7.00000 q^{7} -5.14214 q^{8} +O(q^{10})\) \(q+3.82843 q^{2} +6.65685 q^{4} -5.00000 q^{5} -7.00000 q^{7} -5.14214 q^{8} -19.1421 q^{10} -48.5685 q^{11} -43.6569 q^{13} -26.7990 q^{14} -72.9411 q^{16} +67.6569 q^{17} -93.2548 q^{19} -33.2843 q^{20} -185.941 q^{22} +104.167 q^{23} +25.0000 q^{25} -167.137 q^{26} -46.5980 q^{28} +58.7351 q^{29} -9.08831 q^{31} -238.113 q^{32} +259.019 q^{34} +35.0000 q^{35} -252.676 q^{37} -357.019 q^{38} +25.7107 q^{40} -276.274 q^{41} -92.6375 q^{43} -323.314 q^{44} +398.794 q^{46} +582.794 q^{47} +49.0000 q^{49} +95.7107 q^{50} -290.617 q^{52} -623.019 q^{53} +242.843 q^{55} +35.9949 q^{56} +224.863 q^{58} +524.999 q^{59} -352.794 q^{61} -34.7939 q^{62} -328.068 q^{64} +218.284 q^{65} -736.520 q^{67} +450.382 q^{68} +133.995 q^{70} +492.264 q^{71} +1164.75 q^{73} -967.352 q^{74} -620.784 q^{76} +339.980 q^{77} -872.195 q^{79} +364.706 q^{80} -1057.70 q^{82} +529.588 q^{83} -338.284 q^{85} -354.656 q^{86} +249.746 q^{88} +385.216 q^{89} +305.598 q^{91} +693.421 q^{92} +2231.18 q^{94} +466.274 q^{95} -463.892 q^{97} +187.593 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - 10q^{5} - 14q^{7} + 18q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - 10q^{5} - 14q^{7} + 18q^{8} - 10q^{10} + 16q^{11} - 76q^{13} - 14q^{14} - 78q^{16} + 124q^{17} - 96q^{19} - 10q^{20} - 304q^{22} + 16q^{23} + 50q^{25} - 108q^{26} - 14q^{28} - 188q^{29} - 120q^{31} - 414q^{32} + 156q^{34} + 70q^{35} - 132q^{37} - 352q^{38} - 90q^{40} - 100q^{41} - 536q^{43} - 624q^{44} + 560q^{46} + 928q^{47} + 98q^{49} + 50q^{50} - 140q^{52} - 884q^{53} - 80q^{55} - 126q^{56} + 676q^{58} - 104q^{59} - 468q^{61} + 168q^{62} + 34q^{64} + 380q^{65} - 1688q^{67} + 188q^{68} + 70q^{70} + 136q^{71} + 508q^{73} - 1188q^{74} - 608q^{76} - 112q^{77} - 432q^{79} + 390q^{80} - 1380q^{82} + 584q^{83} - 620q^{85} + 456q^{86} + 1744q^{88} + 1404q^{89} + 532q^{91} + 1104q^{92} + 1600q^{94} + 480q^{95} - 1188q^{97} + 98q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.82843 1.35355 0.676777 0.736188i \(-0.263376\pi\)
0.676777 + 0.736188i \(0.263376\pi\)
\(3\) 0 0
\(4\) 6.65685 0.832107
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) −5.14214 −0.227252
\(9\) 0 0
\(10\) −19.1421 −0.605327
\(11\) −48.5685 −1.33127 −0.665635 0.746278i \(-0.731839\pi\)
−0.665635 + 0.746278i \(0.731839\pi\)
\(12\) 0 0
\(13\) −43.6569 −0.931403 −0.465701 0.884942i \(-0.654198\pi\)
−0.465701 + 0.884942i \(0.654198\pi\)
\(14\) −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 \(0.339684\pi\)
\(18\) 0 0
\(19\) −93.2548 −1.12601 −0.563003 0.826455i \(-0.690354\pi\)
−0.563003 + 0.826455i \(0.690354\pi\)
\(20\) −33.2843 −0.372129
\(21\) 0 0
\(22\) −185.941 −1.80194
\(23\) 104.167 0.944357 0.472179 0.881503i \(-0.343468\pi\)
0.472179 + 0.881503i \(0.343468\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −167.137 −1.26070
\(27\) 0 0
\(28\) −46.5980 −0.314507
\(29\) 58.7351 0.376098 0.188049 0.982160i \(-0.439784\pi\)
0.188049 + 0.982160i \(0.439784\pi\)
\(30\) 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
\(33\) 0 0
\(34\) 259.019 1.30651
\(35\) 35.0000 0.169031
\(36\) 0 0
\(37\) −252.676 −1.12269 −0.561347 0.827580i \(-0.689717\pi\)
−0.561347 + 0.827580i \(0.689717\pi\)
\(38\) −357.019 −1.52411
\(39\) 0 0
\(40\) 25.7107 0.101630
\(41\) −276.274 −1.05236 −0.526180 0.850373i \(-0.676376\pi\)
−0.526180 + 0.850373i \(0.676376\pi\)
\(42\) 0 0
\(43\) −92.6375 −0.328537 −0.164268 0.986416i \(-0.552526\pi\)
−0.164268 + 0.986416i \(0.552526\pi\)
\(44\) −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 \(0.140354\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 95.7107 0.270711
\(51\) 0 0
\(52\) −290.617 −0.775026
\(53\) −623.019 −1.61468 −0.807342 0.590083i \(-0.799095\pi\)
−0.807342 + 0.590083i \(0.799095\pi\)
\(54\) 0 0
\(55\) 242.843 0.595362
\(56\) 35.9949 0.0858933
\(57\) 0 0
\(58\) 224.863 0.509068
\(59\) 524.999 1.15846 0.579229 0.815165i \(-0.303354\pi\)
0.579229 + 0.815165i \(0.303354\pi\)
\(60\) 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
\(63\) 0 0
\(64\) −328.068 −0.640758
\(65\) 218.284 0.416536
\(66\) 0 0
\(67\) −736.520 −1.34299 −0.671494 0.741010i \(-0.734347\pi\)
−0.671494 + 0.741010i \(0.734347\pi\)
\(68\) 450.382 0.803188
\(69\) 0 0
\(70\) 133.995 0.228792
\(71\) 492.264 0.822831 0.411415 0.911448i \(-0.365035\pi\)
0.411415 + 0.911448i \(0.365035\pi\)
\(72\) 0 0
\(73\) 1164.75 1.86745 0.933727 0.357987i \(-0.116537\pi\)
0.933727 + 0.357987i \(0.116537\pi\)
\(74\) −967.352 −1.51963
\(75\) 0 0
\(76\) −620.784 −0.936958
\(77\) 339.980 0.503173
\(78\) 0 0
\(79\) −872.195 −1.24215 −0.621074 0.783752i \(-0.713303\pi\)
−0.621074 + 0.783752i \(0.713303\pi\)
\(80\) 364.706 0.509692
\(81\) 0 0
\(82\) −1057.70 −1.42443
\(83\) 529.588 0.700359 0.350180 0.936683i \(-0.386121\pi\)
0.350180 + 0.936683i \(0.386121\pi\)
\(84\) 0 0
\(85\) −338.284 −0.431672
\(86\) −354.656 −0.444692
\(87\) 0 0
\(88\) 249.746 0.302534
\(89\) 385.216 0.458796 0.229398 0.973333i \(-0.426324\pi\)
0.229398 + 0.973333i \(0.426324\pi\)
\(90\) 0 0
\(91\) 305.598 0.352037
\(92\) 693.421 0.785806
\(93\) 0 0
\(94\) 2231.18 2.44818
\(95\) 466.274 0.503565
\(96\) 0 0
\(97\) −463.892 −0.485579 −0.242789 0.970079i \(-0.578062\pi\)
−0.242789 + 0.970079i \(0.578062\pi\)
\(98\) 187.593 0.193365
\(99\) 0 0
\(100\) 166.421 0.166421
\(101\) 432.725 0.426314 0.213157 0.977018i \(-0.431625\pi\)
0.213157 + 0.977018i \(0.431625\pi\)
\(102\) 0 0
\(103\) 512.626 0.490393 0.245197 0.969473i \(-0.421147\pi\)
0.245197 + 0.969473i \(0.421147\pi\)
\(104\) 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 \(0.847088\pi\)
\(108\) 0 0
\(109\) 545.176 0.479068 0.239534 0.970888i \(-0.423005\pi\)
0.239534 + 0.970888i \(0.423005\pi\)
\(110\) 929.706 0.805854
\(111\) 0 0
\(112\) 510.588 0.430768
\(113\) 231.823 0.192992 0.0964961 0.995333i \(-0.469236\pi\)
0.0964961 + 0.995333i \(0.469236\pi\)
\(114\) 0 0
\(115\) −520.833 −0.422329
\(116\) 390.991 0.312953
\(117\) 0 0
\(118\) 2009.92 1.56804
\(119\) −473.598 −0.364829
\(120\) 0 0
\(121\) 1027.90 0.772279
\(122\) −1350.65 −1.00231
\(123\) 0 0
\(124\) −60.4996 −0.0438147
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 2372.90 1.65796 0.828979 0.559280i \(-0.188922\pi\)
0.828979 + 0.559280i \(0.188922\pi\)
\(128\) 648.917 0.448099
\(129\) 0 0
\(130\) 835.685 0.563804
\(131\) −1200.04 −0.800364 −0.400182 0.916436i \(-0.631053\pi\)
−0.400182 + 0.916436i \(0.631053\pi\)
\(132\) 0 0
\(133\) 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 \(0.834096\pi\)
\(138\) 0 0
\(139\) −1245.60 −0.760078 −0.380039 0.924971i \(-0.624089\pi\)
−0.380039 + 0.924971i \(0.624089\pi\)
\(140\) 232.990 0.140652
\(141\) 0 0
\(142\) 1884.60 1.11375
\(143\) 2120.35 1.23995
\(144\) 0 0
\(145\) −293.675 −0.168196
\(146\) 4459.17 2.52770
\(147\) 0 0
\(148\) −1682.03 −0.934202
\(149\) −19.4046 −0.0106690 −0.00533452 0.999986i \(-0.501698\pi\)
−0.00533452 + 0.999986i \(0.501698\pi\)
\(150\) 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
\(153\) 0 0
\(154\) 1301.59 0.681071
\(155\) 45.4416 0.0235481
\(156\) 0 0
\(157\) −3898.46 −1.98172 −0.990862 0.134880i \(-0.956935\pi\)
−0.990862 + 0.134880i \(0.956935\pi\)
\(158\) −3339.14 −1.68131
\(159\) 0 0
\(160\) 1190.56 0.588264
\(161\) −729.166 −0.356934
\(162\) 0 0
\(163\) 1527.54 0.734024 0.367012 0.930216i \(-0.380381\pi\)
0.367012 + 0.930216i \(0.380381\pi\)
\(164\) −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 \(0.425689\pi\)
\(168\) 0 0
\(169\) −291.079 −0.132489
\(170\) −1295.10 −0.584290
\(171\) 0 0
\(172\) −616.674 −0.273378
\(173\) −685.253 −0.301149 −0.150575 0.988599i \(-0.548112\pi\)
−0.150575 + 0.988599i \(0.548112\pi\)
\(174\) 0 0
\(175\) −175.000 −0.0755929
\(176\) 3542.64 1.51725
\(177\) 0 0
\(178\) 1474.77 0.621005
\(179\) −1025.58 −0.428245 −0.214122 0.976807i \(-0.568689\pi\)
−0.214122 + 0.976807i \(0.568689\pi\)
\(180\) 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
\(183\) 0 0
\(184\) −535.638 −0.214608
\(185\) 1263.38 0.502084
\(186\) 0 0
\(187\) −3285.99 −1.28500
\(188\) 3879.57 1.50504
\(189\) 0 0
\(190\) 1785.10 0.681603
\(191\) −1074.18 −0.406939 −0.203469 0.979081i \(-0.565222\pi\)
−0.203469 + 0.979081i \(0.565222\pi\)
\(192\) 0 0
\(193\) 898.999 0.335292 0.167646 0.985847i \(-0.446383\pi\)
0.167646 + 0.985847i \(0.446383\pi\)
\(194\) −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 \(0.686898\pi\)
\(198\) 0 0
\(199\) −949.522 −0.338240 −0.169120 0.985595i \(-0.554093\pi\)
−0.169120 + 0.985595i \(0.554093\pi\)
\(200\) −128.553 −0.0454505
\(201\) 0 0
\(202\) 1656.66 0.577039
\(203\) −411.145 −0.142151
\(204\) 0 0
\(205\) 1381.37 0.470630
\(206\) 1962.55 0.663774
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) −7515.53 −2.40071
\(215\) 463.188 0.146926
\(216\) 0 0
\(217\) 63.6182 0.0199018
\(218\) 2087.17 0.648444
\(219\) 0 0
\(220\) 1616.57 0.495405
\(221\) −2953.69 −0.899033
\(222\) 0 0
\(223\) −3227.61 −0.969222 −0.484611 0.874730i \(-0.661039\pi\)
−0.484611 + 0.874730i \(0.661039\pi\)
\(224\) 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 \(0.470276\pi\)
\(228\) 0 0
\(229\) 544.774 0.157204 0.0786019 0.996906i \(-0.474954\pi\)
0.0786019 + 0.996906i \(0.474954\pi\)
\(230\) −1993.97 −0.571646
\(231\) 0 0
\(232\) −302.024 −0.0854691
\(233\) 5748.54 1.61631 0.808154 0.588972i \(-0.200467\pi\)
0.808154 + 0.588972i \(0.200467\pi\)
\(234\) 0 0
\(235\) −2913.97 −0.808878
\(236\) 3494.84 0.963961
\(237\) 0 0
\(238\) −1813.14 −0.493816
\(239\) 2678.10 0.724820 0.362410 0.932019i \(-0.381954\pi\)
0.362410 + 0.932019i \(0.381954\pi\)
\(240\) 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
\(243\) 0 0
\(244\) −2348.50 −0.616177
\(245\) −245.000 −0.0638877
\(246\) 0 0
\(247\) 4071.21 1.04877
\(248\) 46.7333 0.0119660
\(249\) 0 0
\(250\) −478.553 −0.121065
\(251\) 5716.90 1.43764 0.718820 0.695196i \(-0.244683\pi\)
0.718820 + 0.695196i \(0.244683\pi\)
\(252\) 0 0
\(253\) −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 \(0.694350\pi\)
\(258\) 0 0
\(259\) 1768.73 0.424339
\(260\) 1453.09 0.346602
\(261\) 0 0
\(262\) −4594.25 −1.08334
\(263\) −5975.36 −1.40097 −0.700487 0.713665i \(-0.747034\pi\)
−0.700487 + 0.713665i \(0.747034\pi\)
\(264\) 0 0
\(265\) 3115.10 0.722109
\(266\) 2499.14 0.576059
\(267\) 0 0
\(268\) −4902.90 −1.11751
\(269\) −4486.11 −1.01681 −0.508407 0.861117i \(-0.669766\pi\)
−0.508407 + 0.861117i \(0.669766\pi\)
\(270\) 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
\(273\) 0 0
\(274\) −10647.8 −2.34766
\(275\) −1214.21 −0.266254
\(276\) 0 0
\(277\) −3420.54 −0.741950 −0.370975 0.928643i \(-0.620976\pi\)
−0.370975 + 0.928643i \(0.620976\pi\)
\(278\) −4768.71 −1.02881
\(279\) 0 0
\(280\) −179.975 −0.0384127
\(281\) −5235.92 −1.11156 −0.555781 0.831329i \(-0.687581\pi\)
−0.555781 + 0.831329i \(0.687581\pi\)
\(282\) 0 0
\(283\) −6985.88 −1.46738 −0.733688 0.679486i \(-0.762203\pi\)
−0.733688 + 0.679486i \(0.762203\pi\)
\(284\) 3276.93 0.684683
\(285\) 0 0
\(286\) 8117.60 1.67834
\(287\) 1933.92 0.397755
\(288\) 0 0
\(289\) −335.550 −0.0682984
\(290\) −1124.31 −0.227662
\(291\) 0 0
\(292\) 7753.59 1.55392
\(293\) −7399.70 −1.47541 −0.737705 0.675123i \(-0.764090\pi\)
−0.737705 + 0.675123i \(0.764090\pi\)
\(294\) 0 0
\(295\) −2625.00 −0.518078
\(296\) 1299.30 0.255135
\(297\) 0 0
\(298\) −74.2892 −0.0144411
\(299\) −4547.58 −0.879577
\(300\) 0 0
\(301\) 648.463 0.124175
\(302\) −8996.04 −1.71412
\(303\) 0 0
\(304\) 6802.11 1.28332
\(305\) 1763.97 0.331163
\(306\) 0 0
\(307\) 2668.64 0.496116 0.248058 0.968745i \(-0.420208\pi\)
0.248058 + 0.968745i \(0.420208\pi\)
\(308\) 2263.20 0.418693
\(309\) 0 0
\(310\) 173.970 0.0318736
\(311\) −6189.25 −1.12849 −0.564244 0.825608i \(-0.690832\pi\)
−0.564244 + 0.825608i \(0.690832\pi\)
\(312\) 0 0
\(313\) −2921.59 −0.527598 −0.263799 0.964578i \(-0.584976\pi\)
−0.263799 + 0.964578i \(0.584976\pi\)
\(314\) −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 \(0.836165\pi\)
\(318\) 0 0
\(319\) −2852.68 −0.500687
\(320\) 1640.34 0.286556
\(321\) 0 0
\(322\) −2791.56 −0.483129
\(323\) −6309.33 −1.08687
\(324\) 0 0
\(325\) −1091.42 −0.186281
\(326\) 5848.07 0.993541
\(327\) 0 0
\(328\) 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
\(333\) 0 0
\(334\) 3822.75 0.626263
\(335\) 3682.60 0.600603
\(336\) 0 0
\(337\) −3693.98 −0.597103 −0.298552 0.954394i \(-0.596503\pi\)
−0.298552 + 0.954394i \(0.596503\pi\)
\(338\) −1114.38 −0.179331
\(339\) 0 0
\(340\) −2251.91 −0.359197
\(341\) 441.406 0.0700982
\(342\) 0 0
\(343\) −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 \(0.595813\pi\)
\(348\) 0 0
\(349\) 8325.22 1.27690 0.638451 0.769662i \(-0.279575\pi\)
0.638451 + 0.769662i \(0.279575\pi\)
\(350\) −669.975 −0.102319
\(351\) 0 0
\(352\) 11564.8 1.75115
\(353\) 8991.52 1.35572 0.677862 0.735189i \(-0.262907\pi\)
0.677862 + 0.735189i \(0.262907\pi\)
\(354\) 0 0
\(355\) −2461.32 −0.367981
\(356\) 2564.33 0.381767
\(357\) 0 0
\(358\) −3926.38 −0.579652
\(359\) 12893.8 1.89557 0.947783 0.318917i \(-0.103319\pi\)
0.947783 + 0.318917i \(0.103319\pi\)
\(360\) 0 0
\(361\) 1837.46 0.267891
\(362\) 11100.1 1.61163
\(363\) 0 0
\(364\) 2034.32 0.292932
\(365\) −5823.77 −0.835150
\(366\) 0 0
\(367\) −7480.17 −1.06393 −0.531964 0.846767i \(-0.678546\pi\)
−0.531964 + 0.846767i \(0.678546\pi\)
\(368\) −7598.02 −1.07629
\(369\) 0 0
\(370\) 4836.76 0.679598
\(371\) 4361.14 0.610293
\(372\) 0 0
\(373\) −3523.32 −0.489090 −0.244545 0.969638i \(-0.578639\pi\)
−0.244545 + 0.969638i \(0.578639\pi\)
\(374\) −12580.2 −1.73932
\(375\) 0 0
\(376\) −2996.81 −0.411033
\(377\) −2564.19 −0.350298
\(378\) 0 0
\(379\) 13515.4 1.83177 0.915886 0.401438i \(-0.131490\pi\)
0.915886 + 0.401438i \(0.131490\pi\)
\(380\) 3103.92 0.419020
\(381\) 0 0
\(382\) −4112.44 −0.550813
\(383\) 657.182 0.0876774 0.0438387 0.999039i \(-0.486041\pi\)
0.0438387 + 0.999039i \(0.486041\pi\)
\(384\) 0 0
\(385\) −1699.90 −0.225026
\(386\) 3441.75 0.453836
\(387\) 0 0
\(388\) −3088.06 −0.404053
\(389\) 9741.87 1.26975 0.634875 0.772615i \(-0.281052\pi\)
0.634875 + 0.772615i \(0.281052\pi\)
\(390\) 0 0
\(391\) 7047.58 0.911538
\(392\) −251.965 −0.0324646
\(393\) 0 0
\(394\) −11728.9 −1.49973
\(395\) 4360.98 0.555505
\(396\) 0 0
\(397\) 4407.42 0.557184 0.278592 0.960410i \(-0.410132\pi\)
0.278592 + 0.960410i \(0.410132\pi\)
\(398\) −3635.18 −0.457827
\(399\) 0 0
\(400\) −1823.53 −0.227941
\(401\) 11569.5 1.44078 0.720391 0.693568i \(-0.243963\pi\)
0.720391 + 0.693568i \(0.243963\pi\)
\(402\) 0 0
\(403\) 396.767 0.0490431
\(404\) 2880.59 0.354739
\(405\) 0 0
\(406\) −1574.04 −0.192410
\(407\) 12272.1 1.49461
\(408\) 0 0
\(409\) −3083.03 −0.372729 −0.186364 0.982481i \(-0.559670\pi\)
−0.186364 + 0.982481i \(0.559670\pi\)
\(410\) 5288.48 0.637023
\(411\) 0 0
\(412\) 3412.47 0.408060
\(413\) −3674.99 −0.437856
\(414\) 0 0
\(415\) −2647.94 −0.313210
\(416\) 10395.3 1.22517
\(417\) 0 0
\(418\) 17339.9 2.02900
\(419\) 5415.21 0.631385 0.315692 0.948862i \(-0.397763\pi\)
0.315692 + 0.948862i \(0.397763\pi\)
\(420\) 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
\(423\) 0 0
\(424\) 3203.65 0.366941
\(425\) 1691.42 0.193049
\(426\) 0 0
\(427\) 2469.56 0.279884
\(428\) −13068.0 −1.47585
\(429\) 0 0
\(430\) 1773.28 0.198872
\(431\) 9108.41 1.01795 0.508975 0.860781i \(-0.330024\pi\)
0.508975 + 0.860781i \(0.330024\pi\)
\(432\) 0 0
\(433\) −16847.0 −1.86979 −0.934893 0.354930i \(-0.884505\pi\)
−0.934893 + 0.354930i \(0.884505\pi\)
\(434\) 243.558 0.0269381
\(435\) 0 0
\(436\) 3629.16 0.398635
\(437\) −9714.03 −1.06335
\(438\) 0 0
\(439\) 8434.14 0.916946 0.458473 0.888708i \(-0.348397\pi\)
0.458473 + 0.888708i \(0.348397\pi\)
\(440\) −1248.73 −0.135297
\(441\) 0 0
\(442\) −11308.0 −1.21689
\(443\) 4298.49 0.461010 0.230505 0.973071i \(-0.425962\pi\)
0.230505 + 0.973071i \(0.425962\pi\)
\(444\) 0 0
\(445\) −1926.08 −0.205180
\(446\) −12356.7 −1.31189
\(447\) 0 0
\(448\) 2296.48 0.242184
\(449\) −10545.0 −1.10835 −0.554173 0.832402i \(-0.686965\pi\)
−0.554173 + 0.832402i \(0.686965\pi\)
\(450\) 0 0
\(451\) 13418.2 1.40098
\(452\) 1543.21 0.160590
\(453\) 0 0
\(454\) 2441.85 0.252426
\(455\) −1527.99 −0.157436
\(456\) 0 0
\(457\) 11952.4 1.22344 0.611719 0.791075i \(-0.290478\pi\)
0.611719 + 0.791075i \(0.290478\pi\)
\(458\) 2085.63 0.212784
\(459\) 0 0
\(460\) −3467.11 −0.351423
\(461\) 17200.9 1.73780 0.868900 0.494988i \(-0.164827\pi\)
0.868900 + 0.494988i \(0.164827\pi\)
\(462\) 0 0
\(463\) −10368.7 −1.04076 −0.520381 0.853934i \(-0.674210\pi\)
−0.520381 + 0.853934i \(0.674210\pi\)
\(464\) −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 \(0.184723\pi\)
\(468\) 0 0
\(469\) 5155.64 0.507602
\(470\) −11155.9 −1.09486
\(471\) 0 0
\(472\) −2699.62 −0.263263
\(473\) 4499.27 0.437371
\(474\) 0 0
\(475\) −2331.37 −0.225201
\(476\) −3152.67 −0.303577
\(477\) 0 0
\(478\) 10252.9 0.981082
\(479\) −7329.12 −0.699115 −0.349558 0.936915i \(-0.613668\pi\)
−0.349558 + 0.936915i \(0.613668\pi\)
\(480\) 0 0
\(481\) 11031.0 1.04568
\(482\) −8430.80 −0.796706
\(483\) 0 0
\(484\) 6842.60 0.642619
\(485\) 2319.46 0.217157
\(486\) 0 0
\(487\) −17209.5 −1.60131 −0.800655 0.599125i \(-0.795515\pi\)
−0.800655 + 0.599125i \(0.795515\pi\)
\(488\) 1814.11 0.168281
\(489\) 0 0
\(490\) −937.965 −0.0864754
\(491\) −11392.5 −1.04712 −0.523560 0.851989i \(-0.675396\pi\)
−0.523560 + 0.851989i \(0.675396\pi\)
\(492\) 0 0
\(493\) 3973.83 0.363027
\(494\) 15586.3 1.41956
\(495\) 0 0
\(496\) 662.912 0.0600113
\(497\) −3445.85 −0.311001
\(498\) 0 0
\(499\) 19079.4 1.71164 0.855822 0.517271i \(-0.173052\pi\)
0.855822 + 0.517271i \(0.173052\pi\)
\(500\) −832.107 −0.0744259
\(501\) 0 0
\(502\) 21886.7 1.94592
\(503\) 13499.4 1.19663 0.598317 0.801259i \(-0.295836\pi\)
0.598317 + 0.801259i \(0.295836\pi\)
\(504\) 0 0
\(505\) −2163.62 −0.190654
\(506\) −19368.8 −1.70168
\(507\) 0 0
\(508\) 15796.0 1.37960
\(509\) 4328.85 0.376960 0.188480 0.982077i \(-0.439644\pi\)
0.188480 + 0.982077i \(0.439644\pi\)
\(510\) 0 0
\(511\) −8153.27 −0.705831
\(512\) 14367.6 1.24017
\(513\) 0 0
\(514\) −18086.6 −1.55207
\(515\) −2563.13 −0.219311
\(516\) 0 0
\(517\) −28305.5 −2.40788
\(518\) 6771.47 0.574365
\(519\) 0 0
\(520\) −1122.45 −0.0946588
\(521\) −19395.7 −1.63098 −0.815490 0.578771i \(-0.803532\pi\)
−0.815490 + 0.578771i \(0.803532\pi\)
\(522\) 0 0
\(523\) 20413.8 1.70675 0.853377 0.521294i \(-0.174550\pi\)
0.853377 + 0.521294i \(0.174550\pi\)
\(524\) −7988.47 −0.665989
\(525\) 0 0
\(526\) −22876.2 −1.89629
\(527\) −614.887 −0.0508252
\(528\) 0 0
\(529\) −1316.34 −0.108189
\(530\) 11925.9 0.977413
\(531\) 0 0
\(532\) 4345.49 0.354137
\(533\) 12061.3 0.980171
\(534\) 0 0
\(535\) 9815.43 0.793193
\(536\) 3787.28 0.305197
\(537\) 0 0
\(538\) −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
\(543\) 0 0
\(544\) −16110.0 −1.26969
\(545\) −2725.88 −0.214246
\(546\) 0 0
\(547\) 15334.2 1.19862 0.599308 0.800518i \(-0.295442\pi\)
0.599308 + 0.800518i \(0.295442\pi\)
\(548\) −18514.4 −1.44324
\(549\) 0 0
\(550\) −4648.53 −0.360389
\(551\) −5477.33 −0.423488
\(552\) 0 0
\(553\) 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 \(0.393751\pi\)
\(558\) 0 0
\(559\) 4044.26 0.306000
\(560\) −2552.94 −0.192645
\(561\) 0 0
\(562\) −20045.3 −1.50456
\(563\) 2320.81 0.173731 0.0868654 0.996220i \(-0.472315\pi\)
0.0868654 + 0.996220i \(0.472315\pi\)
\(564\) 0 0
\(565\) −1159.12 −0.0863087
\(566\) −26744.9 −1.98617
\(567\) 0 0
\(568\) −2531.29 −0.186990
\(569\) 1736.04 0.127906 0.0639529 0.997953i \(-0.479629\pi\)
0.0639529 + 0.997953i \(0.479629\pi\)
\(570\) 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
\(573\) 0 0
\(574\) 7403.87 0.538382
\(575\) 2604.16 0.188871
\(576\) 0 0
\(577\) 8029.26 0.579311 0.289655 0.957131i \(-0.406459\pi\)
0.289655 + 0.957131i \(0.406459\pi\)
\(578\) −1284.63 −0.0924455
\(579\) 0 0
\(580\) −1954.95 −0.139957
\(581\) −3707.12 −0.264711
\(582\) 0 0
\(583\) 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 \(0.409072\pi\)
\(588\) 0 0
\(589\) 847.529 0.0592900
\(590\) −10049.6 −0.701247
\(591\) 0 0
\(592\) 18430.5 1.27954
\(593\) −12820.3 −0.887801 −0.443901 0.896076i \(-0.646406\pi\)
−0.443901 + 0.896076i \(0.646406\pi\)
\(594\) 0 0
\(595\) 2367.99 0.163157
\(596\) −129.174 −0.00887779
\(597\) 0 0
\(598\) −17410.1 −1.19055
\(599\) 15330.5 1.04572 0.522860 0.852418i \(-0.324865\pi\)
0.522860 + 0.852418i \(0.324865\pi\)
\(600\) 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
\(603\) 0 0
\(604\) −15642.3 −1.05377
\(605\) −5139.52 −0.345374
\(606\) 0 0
\(607\) −8213.06 −0.549189 −0.274595 0.961560i \(-0.588544\pi\)
−0.274595 + 0.961560i \(0.588544\pi\)
\(608\) 22205.2 1.48115
\(609\) 0 0
\(610\) 6753.23 0.448246
\(611\) −25443.0 −1.68463
\(612\) 0 0
\(613\) 1242.60 0.0818732 0.0409366 0.999162i \(-0.486966\pi\)
0.0409366 + 0.999162i \(0.486966\pi\)
\(614\) 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 \(0.358625\pi\)
\(618\) 0 0
\(619\) 19774.1 1.28399 0.641993 0.766711i \(-0.278108\pi\)
0.641993 + 0.766711i \(0.278108\pi\)
\(620\) 302.498 0.0195945
\(621\) 0 0
\(622\) −23695.1 −1.52747
\(623\) −2696.51 −0.173409
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −11185.1 −0.714132
\(627\) 0 0
\(628\) −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
\(633\) 0 0
\(634\) −37616.4 −2.35637
\(635\) −11864.5 −0.741461
\(636\) 0 0
\(637\) −2139.19 −0.133058
\(638\) −10921.3 −0.677707
\(639\) 0 0
\(640\) −3244.58 −0.200396
\(641\) −11537.5 −0.710925 −0.355463 0.934691i \(-0.615677\pi\)
−0.355463 + 0.934691i \(0.615677\pi\)
\(642\) 0 0
\(643\) 19603.0 1.20228 0.601139 0.799144i \(-0.294714\pi\)
0.601139 + 0.799144i \(0.294714\pi\)
\(644\) −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 \(0.728498\pi\)
\(648\) 0 0
\(649\) −25498.4 −1.54222
\(650\) −4178.43 −0.252141
\(651\) 0 0
\(652\) 10168.6 0.610787
\(653\) 2927.33 0.175429 0.0877145 0.996146i \(-0.472044\pi\)
0.0877145 + 0.996146i \(0.472044\pi\)
\(654\) 0 0
\(655\) 6000.18 0.357934
\(656\) 20151.7 1.19938
\(657\) 0 0
\(658\) −15618.3 −0.925326
\(659\) 4778.76 0.282480 0.141240 0.989975i \(-0.454891\pi\)
0.141240 + 0.989975i \(0.454891\pi\)
\(660\) 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
\(663\) 0 0
\(664\) −2723.21 −0.159158
\(665\) −3263.92 −0.190330
\(666\) 0 0
\(667\) 6118.23 0.355170
\(668\) 6646.99 0.385000
\(669\) 0 0
\(670\) 14098.6 0.812948
\(671\) 17134.7 0.985808
\(672\) 0 0
\(673\) −8992.15 −0.515040 −0.257520 0.966273i \(-0.582905\pi\)
−0.257520 + 0.966273i \(0.582905\pi\)
\(674\) −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 \(0.684986\pi\)
\(678\) 0 0
\(679\) 3247.25 0.183531
\(680\) 1739.50 0.0980984
\(681\) 0 0
\(682\) 1689.89 0.0948816
\(683\) 4255.14 0.238387 0.119194 0.992871i \(-0.461969\pi\)
0.119194 + 0.992871i \(0.461969\pi\)
\(684\) 0 0
\(685\) 13906.3 0.775666
\(686\) −1313.15 −0.0730850
\(687\) 0 0
\(688\) 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
\(693\) 0 0
\(694\) −14673.7 −0.802603
\(695\) 6228.02 0.339917
\(696\) 0 0
\(697\) −18691.8 −1.01579
\(698\) 31872.5 1.72836
\(699\) 0 0
\(700\) −1164.95 −0.0629014
\(701\) 3240.77 0.174611 0.0873054 0.996182i \(-0.472174\pi\)
0.0873054 + 0.996182i \(0.472174\pi\)
\(702\) 0 0
\(703\) 23563.3 1.26416
\(704\) 15933.8 0.853022
\(705\) 0 0
\(706\) 34423.4 1.83504
\(707\) −3029.07 −0.161132
\(708\) 0 0
\(709\) 19949.3 1.05672 0.528358 0.849022i \(-0.322808\pi\)
0.528358 + 0.849022i \(0.322808\pi\)
\(710\) −9422.99 −0.498082
\(711\) 0 0
\(712\) −1980.83 −0.104262
\(713\) −946.698 −0.0497253
\(714\) 0 0
\(715\) −10601.7 −0.554522
\(716\) −6827.17 −0.356345
\(717\) 0 0
\(718\) 49362.9 2.56575
\(719\) 11259.4 0.584011 0.292006 0.956417i \(-0.405677\pi\)
0.292006 + 0.956417i \(0.405677\pi\)
\(720\) 0 0
\(721\) −3588.38 −0.185351
\(722\) 7034.60 0.362605
\(723\) 0 0
\(724\) 19300.9 0.990761
\(725\) 1468.38 0.0752195
\(726\) 0 0
\(727\) −12228.5 −0.623840 −0.311920 0.950108i \(-0.600972\pi\)
−0.311920 + 0.950108i \(0.600972\pi\)
\(728\) −1571.43 −0.0800013
\(729\) 0 0
\(730\) −22295.9 −1.13042
\(731\) −6267.56 −0.317119
\(732\) 0 0
\(733\) −26635.1 −1.34214 −0.671072 0.741392i \(-0.734166\pi\)
−0.671072 + 0.741392i \(0.734166\pi\)
\(734\) −28637.3 −1.44008
\(735\) 0 0
\(736\) −24803.4 −1.24221
\(737\) 35771.7 1.78788
\(738\) 0 0
\(739\) −6074.00 −0.302349 −0.151174 0.988507i \(-0.548306\pi\)
−0.151174 + 0.988507i \(0.548306\pi\)
\(740\) 8410.14 0.417788
\(741\) 0 0
\(742\) 16696.3 0.826065
\(743\) 4016.87 0.198337 0.0991686 0.995071i \(-0.468382\pi\)
0.0991686 + 0.995071i \(0.468382\pi\)
\(744\) 0 0
\(745\) 97.0231 0.00477134
\(746\) −13488.8 −0.662010
\(747\) 0 0
\(748\) −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
\(753\) 0 0
\(754\) −9816.81 −0.474147
\(755\) 11749.0 0.566344
\(756\) 0 0
\(757\) 31044.9 1.49055 0.745275 0.666758i \(-0.232318\pi\)
0.745275 + 0.666758i \(0.232318\pi\)
\(758\) 51742.9 2.47940
\(759\) 0 0
\(760\) −2397.65 −0.114436
\(761\) −14011.8 −0.667446 −0.333723 0.942671i \(-0.608305\pi\)
−0.333723 + 0.942671i \(0.608305\pi\)
\(762\) 0 0
\(763\) −3816.23 −0.181071
\(764\) −7150.69 −0.338616
\(765\) 0 0
\(766\) 2515.97 0.118676
\(767\) −22919.8 −1.07899
\(768\) 0 0
\(769\) 3342.49 0.156740 0.0783701 0.996924i \(-0.475028\pi\)
0.0783701 + 0.996924i \(0.475028\pi\)
\(770\) −6507.94 −0.304584
\(771\) 0 0
\(772\) 5984.51 0.278999
\(773\) 21074.6 0.980594 0.490297 0.871555i \(-0.336888\pi\)
0.490297 + 0.871555i \(0.336888\pi\)
\(774\) 0 0
\(775\) −227.208 −0.0105310
\(776\) 2385.40 0.110349
\(777\) 0 0
\(778\) 37296.0 1.71867
\(779\) 25763.9 1.18496
\(780\) 0 0
\(781\) −23908.5 −1.09541
\(782\) 26981.1 1.23382
\(783\) 0 0
\(784\) −3574.12 −0.162815
\(785\) 19492.3 0.886254
\(786\) 0 0
\(787\) 21394.8 0.969048 0.484524 0.874778i \(-0.338993\pi\)
0.484524 + 0.874778i \(0.338993\pi\)
\(788\) −20394.1 −0.921968
\(789\) 0 0
\(790\) 16695.7 0.751906
\(791\) −1622.76 −0.0729442
\(792\) 0 0
\(793\) 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 \(0.651711\pi\)
\(798\) 0 0
\(799\) 39430.0 1.74585
\(800\) −5952.82 −0.263080
\(801\) 0 0
\(802\) 44293.0 1.95017
\(803\) −56570.4 −2.48608
\(804\) 0 0
\(805\) 3645.83 0.159626
\(806\) 1518.99 0.0663825
\(807\) 0 0
\(808\) −2225.13 −0.0968810
\(809\) 15939.0 0.692688 0.346344 0.938108i \(-0.387423\pi\)
0.346344 + 0.938108i \(0.387423\pi\)
\(810\) 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
\(813\) 0 0
\(814\) 46982.9 2.02303
\(815\) −7637.69 −0.328266
\(816\) 0 0
\(817\) 8638.90 0.369935
\(818\) −11803.2 −0.504508
\(819\) 0 0
\(820\) 9195.58 0.391614
\(821\) 5700.22 0.242313 0.121157 0.992633i \(-0.461340\pi\)
0.121157 + 0.992633i \(0.461340\pi\)
\(822\) 0 0
\(823\) −32438.0 −1.37390 −0.686948 0.726707i \(-0.741050\pi\)
−0.686948 + 0.726707i \(0.741050\pi\)
\(824\) −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 \(0.413532\pi\)
\(828\) 0 0
\(829\) −30766.7 −1.28899 −0.644494 0.764609i \(-0.722932\pi\)
−0.644494 + 0.764609i \(0.722932\pi\)
\(830\) −10137.4 −0.423947
\(831\) 0 0
\(832\) 14322.4 0.596804
\(833\) 3315.19 0.137892
\(834\) 0 0
\(835\) −4992.59 −0.206917
\(836\) 30150.6 1.24734
\(837\) 0 0
\(838\) 20731.7 0.854613
\(839\) 9779.71 0.402423 0.201212 0.979548i \(-0.435512\pi\)
0.201212 + 0.979548i \(0.435512\pi\)
\(840\) 0 0
\(841\) −20939.2 −0.858551
\(842\) 16034.8 0.656288
\(843\) 0 0
\(844\) 15355.0 0.626233
\(845\) 1455.40 0.0592510
\(846\) 0 0
\(847\) −7195.32 −0.291894
\(848\) 45443.7 1.84026
\(849\) 0 0
\(850\) 6475.48 0.261303
\(851\) −26320.4 −1.06023
\(852\) 0 0
\(853\) 24201.5 0.971445 0.485723 0.874113i \(-0.338556\pi\)
0.485723 + 0.874113i \(0.338556\pi\)
\(854\) 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 \(0.637708\pi\)
\(858\) 0 0
\(859\) −6179.19 −0.245438 −0.122719 0.992441i \(-0.539161\pi\)
−0.122719 + 0.992441i \(0.539161\pi\)
\(860\) 3083.37 0.122258
\(861\) 0 0
\(862\) 34870.9 1.37785
\(863\) −50256.2 −1.98232 −0.991160 0.132671i \(-0.957644\pi\)
−0.991160 + 0.132671i \(0.957644\pi\)
\(864\) 0 0
\(865\) 3426.27 0.134678
\(866\) −64497.7 −2.53085
\(867\) 0 0
\(868\) 423.497 0.0165604
\(869\) 42361.2 1.65363
\(870\) 0 0
\(871\) 32154.1 1.25086
\(872\) −2803.37 −0.108869
\(873\) 0 0
\(874\) −37189.5 −1.43930
\(875\) 875.000 0.0338062
\(876\) 0 0
\(877\) −9175.95 −0.353306 −0.176653 0.984273i \(-0.556527\pi\)
−0.176653 + 0.984273i \(0.556527\pi\)
\(878\) 32289.5 1.24114
\(879\) 0 0
\(880\) −17713.2 −0.678537
\(881\) 26172.7 1.00089 0.500444 0.865769i \(-0.333170\pi\)
0.500444 + 0.865769i \(0.333170\pi\)
\(882\) 0 0
\(883\) 18615.5 0.709471 0.354736 0.934967i \(-0.384571\pi\)
0.354736 + 0.934967i \(0.384571\pi\)
\(884\) −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 \(0.421874\pi\)
\(888\) 0 0
\(889\) −16610.3 −0.626649
\(890\) −7373.86 −0.277722
\(891\) 0 0
\(892\) −21485.7 −0.806496
\(893\) −54348.4 −2.03662
\(894\) 0 0
\(895\) 5127.92 0.191517
\(896\) −4542.42 −0.169366
\(897\) 0 0
\(898\) −40370.6 −1.50020
\(899\) −533.803 −0.0198035
\(900\) 0 0
\(901\) −42151.5 −1.55857
\(902\) 51370.7 1.89630
\(903\) 0 0
\(904\) −1192.07 −0.0438579
\(905\) −14497.0 −0.532482
\(906\) 0 0
\(907\) −26766.1 −0.979885 −0.489942 0.871755i \(-0.662982\pi\)
−0.489942 + 0.871755i \(0.662982\pi\)
\(908\) 4245.87 0.155181
\(909\) 0 0
\(910\) −5849.80 −0.213098
\(911\) −5022.67 −0.182666 −0.0913328 0.995820i \(-0.529113\pi\)
−0.0913328 + 0.995820i \(0.529113\pi\)
\(912\) 0 0
\(913\) −25721.3 −0.932367
\(914\) 45759.0 1.65599
\(915\) 0 0
\(916\) 3626.48 0.130810
\(917\) 8400.26 0.302509
\(918\) 0 0
\(919\) 9541.79 0.342497 0.171248 0.985228i \(-0.445220\pi\)
0.171248 + 0.985228i \(0.445220\pi\)
\(920\) 2678.19 0.0959754
\(921\) 0 0
\(922\) 65852.4 2.35220
\(923\) −21490.7 −0.766387
\(924\) 0 0
\(925\) −6316.90 −0.224539
\(926\) −39695.7 −1.40873
\(927\) 0 0
\(928\) −13985.6 −0.494718
\(929\) 25479.6 0.899846 0.449923 0.893067i \(-0.351451\pi\)
0.449923 + 0.893067i \(0.351451\pi\)
\(930\) 0 0
\(931\) −4569.49 −0.160858
\(932\) 38267.2 1.34494
\(933\) 0 0
\(934\) 64621.9 2.26391
\(935\) 16430.0 0.574671
\(936\) 0 0
\(937\) −33608.3 −1.17176 −0.585878 0.810399i \(-0.699250\pi\)
−0.585878 + 0.810399i \(0.699250\pi\)
\(938\) 19738.0 0.687066
\(939\) 0 0
\(940\) −19397.9 −0.673073
\(941\) 19173.6 0.664232 0.332116 0.943239i \(-0.392238\pi\)
0.332116 + 0.943239i \(0.392238\pi\)
\(942\) 0 0
\(943\) −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 \(0.494651\pi\)
\(948\) 0 0
\(949\) −50849.5 −1.73935
\(950\) −8925.48 −0.304822
\(951\) 0 0
\(952\) 2435.31 0.0829083
\(953\) −3048.61 −0.103624 −0.0518122 0.998657i \(-0.516500\pi\)
−0.0518122 + 0.998657i \(0.516500\pi\)
\(954\) 0 0
\(955\) 5370.92 0.181989
\(956\) 17827.7 0.603127
\(957\) 0 0
\(958\) −28059.0 −0.946290
\(959\) 19468.8 0.655557
\(960\) 0 0
\(961\) −29708.4 −0.997227
\(962\) 42231.6 1.41538
\(963\) 0 0
\(964\) −14659.4 −0.489781
\(965\) −4495.00 −0.149947
\(966\) 0 0
\(967\) −14467.9 −0.481133 −0.240567 0.970633i \(-0.577333\pi\)
−0.240567 + 0.970633i \(0.577333\pi\)
\(968\) −5285.62 −0.175502
\(969\) 0 0
\(970\) 8879.89 0.293934
\(971\) −12952.8 −0.428090 −0.214045 0.976824i \(-0.568664\pi\)
−0.214045 + 0.976824i \(0.568664\pi\)
\(972\) 0 0
\(973\) 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 \(0.781511\pi\)
\(978\) 0 0
\(979\) −18709.4 −0.610781
\(980\) −1630.93 −0.0531614
\(981\) 0 0
\(982\) −43615.3 −1.41733
\(983\) 1536.84 0.0498651 0.0249326 0.999689i \(-0.492063\pi\)
0.0249326 + 0.999689i \(0.492063\pi\)
\(984\) 0 0
\(985\) 15318.1 0.495509
\(986\) 15213.5 0.491376
\(987\) 0 0
\(988\) 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
\(993\) 0 0
\(994\) −13192.2 −0.420956
\(995\) 4747.61 0.151266
\(996\) 0 0
\(997\) −25894.9 −0.822566 −0.411283 0.911508i \(-0.634919\pi\)
−0.411283 + 0.911508i \(0.634919\pi\)
\(998\) 73044.0 2.31680
\(999\) 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 315.4.a.k.1.2 2
3.2 odd 2 105.4.a.e.1.1 2
5.4 even 2 1575.4.a.q.1.1 2
7.6 odd 2 2205.4.a.bb.1.2 2
12.11 even 2 1680.4.a.bo.1.1 2
15.2 even 4 525.4.d.l.274.1 4
15.8 even 4 525.4.d.l.274.4 4
15.14 odd 2 525.4.a.l.1.2 2
21.20 even 2 735.4.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.e.1.1 2 3.2 odd 2
315.4.a.k.1.2 2 1.1 even 1 trivial
525.4.a.l.1.2 2 15.14 odd 2
525.4.d.l.274.1 4 15.2 even 4
525.4.d.l.274.4 4 15.8 even 4
735.4.a.o.1.1 2 21.20 even 2
1575.4.a.q.1.1 2 5.4 even 2
1680.4.a.bo.1.1 2 12.11 even 2
2205.4.a.bb.1.2 2 7.6 odd 2