Properties

Label 525.4.d.l.274.4
Level $525$
Weight $4$
Character 525.274
Analytic conductor $30.976$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.4
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.4.d.l.274.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.82843i q^{2} -3.00000i q^{3} -6.65685 q^{4} +11.4853 q^{6} +7.00000i q^{7} +5.14214i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q+3.82843i q^{2} -3.00000i q^{3} -6.65685 q^{4} +11.4853 q^{6} +7.00000i q^{7} +5.14214i q^{8} -9.00000 q^{9} +48.5685 q^{11} +19.9706i q^{12} -43.6569i q^{13} -26.7990 q^{14} -72.9411 q^{16} +67.6569i q^{17} -34.4558i q^{18} +93.2548 q^{19} +21.0000 q^{21} +185.941i q^{22} -104.167i q^{23} +15.4264 q^{24} +167.137 q^{26} +27.0000i q^{27} -46.5980i q^{28} +58.7351 q^{29} -9.08831 q^{31} -238.113i q^{32} -145.706i q^{33} -259.019 q^{34} +59.9117 q^{36} +252.676i q^{37} +357.019i q^{38} -130.971 q^{39} +276.274 q^{41} +80.3970i q^{42} -92.6375i q^{43} -323.314 q^{44} +398.794 q^{46} +582.794i q^{47} +218.823i q^{48} -49.0000 q^{49} +202.971 q^{51} +290.617i q^{52} +623.019i q^{53} -103.368 q^{54} -35.9949 q^{56} -279.765i q^{57} +224.863i q^{58} +524.999 q^{59} -352.794 q^{61} -34.7939i q^{62} -63.0000i q^{63} +328.068 q^{64} +557.823 q^{66} +736.520i q^{67} -450.382i q^{68} -312.500 q^{69} -492.264 q^{71} -46.2792i q^{72} +1164.75i q^{73} -967.352 q^{74} -620.784 q^{76} +339.980i q^{77} -501.411i q^{78} +872.195 q^{79} +81.0000 q^{81} +1057.70i q^{82} -529.588i q^{83} -139.794 q^{84} +354.656 q^{86} -176.205i q^{87} +249.746i q^{88} +385.216 q^{89} +305.598 q^{91} +693.421i q^{92} +27.2649i q^{93} -2231.18 q^{94} -714.338 q^{96} +463.892i q^{97} -187.593i q^{98} -437.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 12q^{6} - 36q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 12q^{6} - 36q^{9} - 32q^{11} - 28q^{14} - 156q^{16} + 192q^{19} + 84q^{21} - 108q^{24} + 216q^{26} - 376q^{29} - 240q^{31} - 312q^{34} + 36q^{36} - 456q^{39} + 200q^{41} - 1248q^{44} + 1120q^{46} - 196q^{49} + 744q^{51} - 108q^{54} + 252q^{56} - 208q^{59} - 936q^{61} - 68q^{64} + 1824q^{66} - 96q^{69} - 272q^{71} - 2376q^{74} - 1216q^{76} + 864q^{79} + 324q^{81} - 84q^{84} - 912q^{86} + 2808q^{89} + 1064q^{91} - 3200q^{94} - 2484q^{96} + 288q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.82843i 1.35355i 0.736188 + 0.676777i \(0.236624\pi\)
−0.736188 + 0.676777i \(0.763376\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) −6.65685 −0.832107
\(5\) 0 0
\(6\) 11.4853 0.781474
\(7\) 7.00000i 0.377964i
\(8\) 5.14214i 0.227252i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 48.5685 1.33127 0.665635 0.746278i \(-0.268161\pi\)
0.665635 + 0.746278i \(0.268161\pi\)
\(12\) 19.9706i 0.480417i
\(13\) − 43.6569i − 0.931403i −0.884942 0.465701i \(-0.845802\pi\)
0.884942 0.465701i \(-0.154198\pi\)
\(14\) −26.7990 −0.511595
\(15\) 0 0
\(16\) −72.9411 −1.13971
\(17\) 67.6569i 0.965247i 0.875828 + 0.482623i \(0.160316\pi\)
−0.875828 + 0.482623i \(0.839684\pi\)
\(18\) − 34.4558i − 0.451184i
\(19\) 93.2548 1.12601 0.563003 0.826455i \(-0.309646\pi\)
0.563003 + 0.826455i \(0.309646\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) 185.941i 1.80194i
\(23\) − 104.167i − 0.944357i −0.881503 0.472179i \(-0.843468\pi\)
0.881503 0.472179i \(-0.156532\pi\)
\(24\) 15.4264 0.131204
\(25\) 0 0
\(26\) 167.137 1.26070
\(27\) 27.0000i 0.192450i
\(28\) − 46.5980i − 0.314507i
\(29\) 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 \(0.508381\pi\)
\(32\) − 238.113i − 1.31540i
\(33\) − 145.706i − 0.768609i
\(34\) −259.019 −1.30651
\(35\) 0 0
\(36\) 59.9117 0.277369
\(37\) 252.676i 1.12269i 0.827580 + 0.561347i \(0.189717\pi\)
−0.827580 + 0.561347i \(0.810283\pi\)
\(38\) 357.019i 1.52411i
\(39\) −130.971 −0.537745
\(40\) 0 0
\(41\) 276.274 1.05236 0.526180 0.850373i \(-0.323624\pi\)
0.526180 + 0.850373i \(0.323624\pi\)
\(42\) 80.3970i 0.295370i
\(43\) − 92.6375i − 0.328537i −0.986416 0.164268i \(-0.947474\pi\)
0.986416 0.164268i \(-0.0525263\pi\)
\(44\) −323.314 −1.10776
\(45\) 0 0
\(46\) 398.794 1.27824
\(47\) 582.794i 1.80871i 0.426784 + 0.904354i \(0.359646\pi\)
−0.426784 + 0.904354i \(0.640354\pi\)
\(48\) 218.823i 0.658009i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) 202.971 0.557286
\(52\) 290.617i 0.775026i
\(53\) 623.019i 1.61468i 0.590083 + 0.807342i \(0.299095\pi\)
−0.590083 + 0.807342i \(0.700905\pi\)
\(54\) −103.368 −0.260491
\(55\) 0 0
\(56\) −35.9949 −0.0858933
\(57\) − 279.765i − 0.650100i
\(58\) 224.863i 0.509068i
\(59\) 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 \(0.620728\pi\)
\(62\) − 34.7939i − 0.0712715i
\(63\) − 63.0000i − 0.125988i
\(64\) 328.068 0.640758
\(65\) 0 0
\(66\) 557.823 1.04035
\(67\) 736.520i 1.34299i 0.741010 + 0.671494i \(0.234347\pi\)
−0.741010 + 0.671494i \(0.765653\pi\)
\(68\) − 450.382i − 0.803188i
\(69\) −312.500 −0.545225
\(70\) 0 0
\(71\) −492.264 −0.822831 −0.411415 0.911448i \(-0.634965\pi\)
−0.411415 + 0.911448i \(0.634965\pi\)
\(72\) − 46.2792i − 0.0757508i
\(73\) 1164.75i 1.86745i 0.357987 + 0.933727i \(0.383463\pi\)
−0.357987 + 0.933727i \(0.616537\pi\)
\(74\) −967.352 −1.51963
\(75\) 0 0
\(76\) −620.784 −0.936958
\(77\) 339.980i 0.503173i
\(78\) − 501.411i − 0.727867i
\(79\) 872.195 1.24215 0.621074 0.783752i \(-0.286697\pi\)
0.621074 + 0.783752i \(0.286697\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 1057.70i 1.42443i
\(83\) − 529.588i − 0.700359i −0.936683 0.350180i \(-0.886121\pi\)
0.936683 0.350180i \(-0.113879\pi\)
\(84\) −139.794 −0.181581
\(85\) 0 0
\(86\) 354.656 0.444692
\(87\) − 176.205i − 0.217140i
\(88\) 249.746i 0.302534i
\(89\) 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
\(92\) 693.421i 0.785806i
\(93\) 27.2649i 0.0304005i
\(94\) −2231.18 −2.44818
\(95\) 0 0
\(96\) −714.338 −0.759446
\(97\) 463.892i 0.485579i 0.970079 + 0.242789i \(0.0780624\pi\)
−0.970079 + 0.242789i \(0.921938\pi\)
\(98\) − 187.593i − 0.193365i
\(99\) −437.117 −0.443757
\(100\) 0 0
\(101\) −432.725 −0.426314 −0.213157 0.977018i \(-0.568375\pi\)
−0.213157 + 0.977018i \(0.568375\pi\)
\(102\) 777.058i 0.754316i
\(103\) 512.626i 0.490393i 0.969473 + 0.245197i \(0.0788525\pi\)
−0.969473 + 0.245197i \(0.921147\pi\)
\(104\) 224.489 0.211663
\(105\) 0 0
\(106\) −2385.18 −2.18556
\(107\) − 1963.09i − 1.77363i −0.462122 0.886817i \(-0.652912\pi\)
0.462122 0.886817i \(-0.347088\pi\)
\(108\) − 179.735i − 0.160139i
\(109\) −545.176 −0.479068 −0.239534 0.970888i \(-0.576995\pi\)
−0.239534 + 0.970888i \(0.576995\pi\)
\(110\) 0 0
\(111\) 758.029 0.648188
\(112\) − 510.588i − 0.430768i
\(113\) − 231.823i − 0.192992i −0.995333 0.0964961i \(-0.969236\pi\)
0.995333 0.0964961i \(-0.0307635\pi\)
\(114\) 1071.06 0.879945
\(115\) 0 0
\(116\) −390.991 −0.312953
\(117\) 392.912i 0.310468i
\(118\) 2009.92i 1.56804i
\(119\) −473.598 −0.364829
\(120\) 0 0
\(121\) 1027.90 0.772279
\(122\) − 1350.65i − 1.00231i
\(123\) − 828.823i − 0.607581i
\(124\) 60.4996 0.0438147
\(125\) 0 0
\(126\) 241.191 0.170532
\(127\) − 2372.90i − 1.65796i −0.559280 0.828979i \(-0.688922\pi\)
0.559280 0.828979i \(-0.311078\pi\)
\(128\) − 648.917i − 0.448099i
\(129\) −277.913 −0.189681
\(130\) 0 0
\(131\) 1200.04 0.800364 0.400182 0.916436i \(-0.368947\pi\)
0.400182 + 0.916436i \(0.368947\pi\)
\(132\) 969.941i 0.639565i
\(133\) 652.784i 0.425591i
\(134\) −2819.71 −1.81781
\(135\) 0 0
\(136\) −347.901 −0.219355
\(137\) − 2781.25i − 1.73444i −0.497924 0.867221i \(-0.665904\pi\)
0.497924 0.867221i \(-0.334096\pi\)
\(138\) − 1196.38i − 0.737991i
\(139\) 1245.60 0.760078 0.380039 0.924971i \(-0.375911\pi\)
0.380039 + 0.924971i \(0.375911\pi\)
\(140\) 0 0
\(141\) 1748.38 1.04426
\(142\) − 1884.60i − 1.11375i
\(143\) − 2120.35i − 1.23995i
\(144\) 656.470 0.379902
\(145\) 0 0
\(146\) −4459.17 −2.52770
\(147\) 147.000i 0.0824786i
\(148\) − 1682.03i − 0.934202i
\(149\) −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 \(0.718256\pi\)
\(152\) 479.529i 0.255888i
\(153\) − 608.912i − 0.321749i
\(154\) −1301.59 −0.681071
\(155\) 0 0
\(156\) 871.852 0.447462
\(157\) 3898.46i 1.98172i 0.134880 + 0.990862i \(0.456935\pi\)
−0.134880 + 0.990862i \(0.543065\pi\)
\(158\) 3339.14i 1.68131i
\(159\) 1869.06 0.932239
\(160\) 0 0
\(161\) 729.166 0.356934
\(162\) 310.103i 0.150395i
\(163\) 1527.54i 0.734024i 0.930216 + 0.367012i \(0.119619\pi\)
−0.930216 + 0.367012i \(0.880381\pi\)
\(164\) −1839.12 −0.875676
\(165\) 0 0
\(166\) 2027.49 0.947974
\(167\) 998.518i 0.462681i 0.972873 + 0.231340i \(0.0743111\pi\)
−0.972873 + 0.231340i \(0.925689\pi\)
\(168\) 107.985i 0.0495905i
\(169\) 291.079 0.132489
\(170\) 0 0
\(171\) −839.294 −0.375336
\(172\) 616.674i 0.273378i
\(173\) 685.253i 0.301149i 0.988599 + 0.150575i \(0.0481124\pi\)
−0.988599 + 0.150575i \(0.951888\pi\)
\(174\) 674.589 0.293911
\(175\) 0 0
\(176\) −3542.64 −1.51725
\(177\) − 1575.00i − 0.668836i
\(178\) 1474.77i 0.621005i
\(179\) −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 \(0.297020\pi\)
\(182\) 1169.96i 0.476501i
\(183\) 1058.38i 0.427529i
\(184\) 535.638 0.214608
\(185\) 0 0
\(186\) −104.382 −0.0411486
\(187\) 3285.99i 1.28500i
\(188\) − 3879.57i − 1.50504i
\(189\) −189.000 −0.0727393
\(190\) 0 0
\(191\) 1074.18 0.406939 0.203469 0.979081i \(-0.434778\pi\)
0.203469 + 0.979081i \(0.434778\pi\)
\(192\) − 984.204i − 0.369942i
\(193\) 898.999i 0.335292i 0.985847 + 0.167646i \(0.0536166\pi\)
−0.985847 + 0.167646i \(0.946383\pi\)
\(194\) −1775.98 −0.657257
\(195\) 0 0
\(196\) 326.186 0.118872
\(197\) − 3063.63i − 1.10799i −0.832519 0.553996i \(-0.813102\pi\)
0.832519 0.553996i \(-0.186898\pi\)
\(198\) − 1673.47i − 0.600648i
\(199\) 949.522 0.338240 0.169120 0.985595i \(-0.445907\pi\)
0.169120 + 0.985595i \(0.445907\pi\)
\(200\) 0 0
\(201\) 2209.56 0.775375
\(202\) − 1656.66i − 0.577039i
\(203\) 411.145i 0.142151i
\(204\) −1351.15 −0.463721
\(205\) 0 0
\(206\) −1962.55 −0.663774
\(207\) 937.499i 0.314786i
\(208\) 3184.38i 1.06152i
\(209\) 4529.25 1.49902
\(210\) 0 0
\(211\) 2306.64 0.752587 0.376294 0.926500i \(-0.377198\pi\)
0.376294 + 0.926500i \(0.377198\pi\)
\(212\) − 4147.35i − 1.34359i
\(213\) 1476.79i 0.475062i
\(214\) 7515.53 2.40071
\(215\) 0 0
\(216\) −138.838 −0.0437348
\(217\) − 63.6182i − 0.0199018i
\(218\) − 2087.17i − 0.648444i
\(219\) 3494.26 1.07817
\(220\) 0 0
\(221\) 2953.69 0.899033
\(222\) 2902.06i 0.877357i
\(223\) − 3227.61i − 0.969222i −0.874730 0.484611i \(-0.838961\pi\)
0.874730 0.484611i \(-0.161039\pi\)
\(224\) 1666.79 0.497174
\(225\) 0 0
\(226\) 887.519 0.261225
\(227\) 637.820i 0.186492i 0.995643 + 0.0932458i \(0.0297242\pi\)
−0.995643 + 0.0932458i \(0.970276\pi\)
\(228\) 1862.35i 0.540953i
\(229\) −544.774 −0.157204 −0.0786019 0.996906i \(-0.525046\pi\)
−0.0786019 + 0.996906i \(0.525046\pi\)
\(230\) 0 0
\(231\) 1019.94 0.290507
\(232\) 302.024i 0.0854691i
\(233\) − 5748.54i − 1.61631i −0.588972 0.808154i \(-0.700467\pi\)
0.588972 0.808154i \(-0.299533\pi\)
\(234\) −1504.23 −0.420234
\(235\) 0 0
\(236\) −3494.84 −0.963961
\(237\) − 2616.59i − 0.717154i
\(238\) − 1813.14i − 0.493816i
\(239\) 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 \(0.595087\pi\)
\(242\) 3935.25i 1.04532i
\(243\) − 243.000i − 0.0641500i
\(244\) 2348.50 0.616177
\(245\) 0 0
\(246\) 3173.09 0.822393
\(247\) − 4071.21i − 1.04877i
\(248\) − 46.7333i − 0.0119660i
\(249\) −1588.76 −0.404353
\(250\) 0 0
\(251\) −5716.90 −1.43764 −0.718820 0.695196i \(-0.755317\pi\)
−0.718820 + 0.695196i \(0.755317\pi\)
\(252\) 419.382i 0.104836i
\(253\) − 5059.22i − 1.25719i
\(254\) 9084.47 2.24413
\(255\) 0 0
\(256\) 5108.88 1.24728
\(257\) − 4724.29i − 1.14666i −0.819323 0.573332i \(-0.805650\pi\)
0.819323 0.573332i \(-0.194350\pi\)
\(258\) − 1063.97i − 0.256743i
\(259\) −1768.73 −0.424339
\(260\) 0 0
\(261\) −528.616 −0.125366
\(262\) 4594.25i 1.08334i
\(263\) 5975.36i 1.40097i 0.713665 + 0.700487i \(0.247034\pi\)
−0.713665 + 0.700487i \(0.752966\pi\)
\(264\) 749.238 0.174668
\(265\) 0 0
\(266\) −2499.14 −0.576059
\(267\) − 1155.65i − 0.264886i
\(268\) − 4902.90i − 1.11751i
\(269\) −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 \(0.358868\pi\)
\(272\) − 4934.97i − 1.10010i
\(273\) − 916.794i − 0.203249i
\(274\) 10647.8 2.34766
\(275\) 0 0
\(276\) 2080.26 0.453685
\(277\) 3420.54i 0.741950i 0.928643 + 0.370975i \(0.120976\pi\)
−0.928643 + 0.370975i \(0.879024\pi\)
\(278\) 4768.71i 1.02881i
\(279\) 81.7948 0.0175517
\(280\) 0 0
\(281\) 5235.92 1.11156 0.555781 0.831329i \(-0.312419\pi\)
0.555781 + 0.831329i \(0.312419\pi\)
\(282\) 6693.55i 1.41346i
\(283\) − 6985.88i − 1.46738i −0.679486 0.733688i \(-0.737797\pi\)
0.679486 0.733688i \(-0.262203\pi\)
\(284\) 3276.93 0.684683
\(285\) 0 0
\(286\) 8117.60 1.67834
\(287\) 1933.92i 0.397755i
\(288\) 2143.01i 0.438466i
\(289\) 335.550 0.0682984
\(290\) 0 0
\(291\) 1391.68 0.280349
\(292\) − 7753.59i − 1.55392i
\(293\) 7399.70i 1.47541i 0.675123 + 0.737705i \(0.264090\pi\)
−0.675123 + 0.737705i \(0.735910\pi\)
\(294\) −562.779 −0.111639
\(295\) 0 0
\(296\) −1299.30 −0.255135
\(297\) 1311.35i 0.256203i
\(298\) − 74.2892i − 0.0144411i
\(299\) −4547.58 −0.879577
\(300\) 0 0
\(301\) 648.463 0.124175
\(302\) − 8996.04i − 1.71412i
\(303\) 1298.17i 0.246133i
\(304\) −6802.11 −1.28332
\(305\) 0 0
\(306\) 2331.17 0.435504
\(307\) − 2668.64i − 0.496116i −0.968745 0.248058i \(-0.920208\pi\)
0.968745 0.248058i \(-0.0797923\pi\)
\(308\) − 2263.20i − 0.418693i
\(309\) 1537.88 0.283129
\(310\) 0 0
\(311\) 6189.25 1.12849 0.564244 0.825608i \(-0.309168\pi\)
0.564244 + 0.825608i \(0.309168\pi\)
\(312\) − 673.468i − 0.122204i
\(313\) − 2921.59i − 0.527598i −0.964578 0.263799i \(-0.915024\pi\)
0.964578 0.263799i \(-0.0849755\pi\)
\(314\) −14925.0 −2.68237
\(315\) 0 0
\(316\) −5806.08 −1.03360
\(317\) − 9825.56i − 1.74088i −0.492276 0.870439i \(-0.663835\pi\)
0.492276 0.870439i \(-0.336165\pi\)
\(318\) 7155.55i 1.26183i
\(319\) 2852.68 0.500687
\(320\) 0 0
\(321\) −5889.26 −1.02401
\(322\) 2791.56i 0.483129i
\(323\) 6309.33i 1.08687i
\(324\) −539.205 −0.0924563
\(325\) 0 0
\(326\) −5848.07 −0.993541
\(327\) 1635.53i 0.276590i
\(328\) 1420.64i 0.239151i
\(329\) −4079.56 −0.683627
\(330\) 0 0
\(331\) −9258.17 −1.53739 −0.768693 0.639618i \(-0.779093\pi\)
−0.768693 + 0.639618i \(0.779093\pi\)
\(332\) 3525.39i 0.582774i
\(333\) − 2274.09i − 0.374232i
\(334\) −3822.75 −0.626263
\(335\) 0 0
\(336\) −1531.76 −0.248704
\(337\) 3693.98i 0.597103i 0.954394 + 0.298552i \(0.0965035\pi\)
−0.954394 + 0.298552i \(0.903497\pi\)
\(338\) 1114.38i 0.179331i
\(339\) −695.470 −0.111424
\(340\) 0 0
\(341\) −441.406 −0.0700982
\(342\) − 3213.17i − 0.508037i
\(343\) − 343.000i − 0.0539949i
\(344\) 476.355 0.0746608
\(345\) 0 0
\(346\) −2623.44 −0.407622
\(347\) − 3832.83i − 0.592960i −0.955039 0.296480i \(-0.904187\pi\)
0.955039 0.296480i \(-0.0958128\pi\)
\(348\) 1172.97i 0.180684i
\(349\) −8325.22 −1.27690 −0.638451 0.769662i \(-0.720425\pi\)
−0.638451 + 0.769662i \(0.720425\pi\)
\(350\) 0 0
\(351\) 1178.74 0.179248
\(352\) − 11564.8i − 1.75115i
\(353\) − 8991.52i − 1.35572i −0.735189 0.677862i \(-0.762907\pi\)
0.735189 0.677862i \(-0.237093\pi\)
\(354\) 6029.76 0.905306
\(355\) 0 0
\(356\) −2564.33 −0.381767
\(357\) 1420.79i 0.210634i
\(358\) − 3926.38i − 0.579652i
\(359\) 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
\(362\) 11100.1i 1.61163i
\(363\) − 3083.71i − 0.445875i
\(364\) −2034.32 −0.292932
\(365\) 0 0
\(366\) −4051.94 −0.578684
\(367\) 7480.17i 1.06393i 0.846767 + 0.531964i \(0.178546\pi\)
−0.846767 + 0.531964i \(0.821454\pi\)
\(368\) 7598.02i 1.07629i
\(369\) −2486.47 −0.350787
\(370\) 0 0
\(371\) −4361.14 −0.610293
\(372\) − 181.499i − 0.0252964i
\(373\) − 3523.32i − 0.489090i −0.969638 0.244545i \(-0.921361\pi\)
0.969638 0.244545i \(-0.0786386\pi\)
\(374\) −12580.2 −1.73932
\(375\) 0 0
\(376\) −2996.81 −0.411033
\(377\) − 2564.19i − 0.350298i
\(378\) − 723.573i − 0.0984565i
\(379\) −13515.4 −1.83177 −0.915886 0.401438i \(-0.868510\pi\)
−0.915886 + 0.401438i \(0.868510\pi\)
\(380\) 0 0
\(381\) −7118.69 −0.957222
\(382\) 4112.44i 0.550813i
\(383\) − 657.182i − 0.0876774i −0.999039 0.0438387i \(-0.986041\pi\)
0.999039 0.0438387i \(-0.0139588\pi\)
\(384\) −1946.75 −0.258710
\(385\) 0 0
\(386\) −3441.75 −0.453836
\(387\) 833.738i 0.109512i
\(388\) − 3088.06i − 0.404053i
\(389\) 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
\(392\) − 251.965i − 0.0324646i
\(393\) − 3600.11i − 0.462091i
\(394\) 11728.9 1.49973
\(395\) 0 0
\(396\) 2909.82 0.369253
\(397\) − 4407.42i − 0.557184i −0.960410 0.278592i \(-0.910132\pi\)
0.960410 0.278592i \(-0.0898678\pi\)
\(398\) 3635.18i 0.457827i
\(399\) 1958.35 0.245715
\(400\) 0 0
\(401\) −11569.5 −1.44078 −0.720391 0.693568i \(-0.756037\pi\)
−0.720391 + 0.693568i \(0.756037\pi\)
\(402\) 8459.14i 1.04951i
\(403\) 396.767i 0.0490431i
\(404\) 2880.59 0.354739
\(405\) 0 0
\(406\) −1574.04 −0.192410
\(407\) 12272.1i 1.49461i
\(408\) 1043.70i 0.126645i
\(409\) 3083.03 0.372729 0.186364 0.982481i \(-0.440330\pi\)
0.186364 + 0.982481i \(0.440330\pi\)
\(410\) 0 0
\(411\) −8343.76 −1.00138
\(412\) − 3412.47i − 0.408060i
\(413\) 3674.99i 0.437856i
\(414\) −3589.15 −0.426079
\(415\) 0 0
\(416\) −10395.3 −1.22517
\(417\) − 3736.81i − 0.438831i
\(418\) 17339.9i 2.02900i
\(419\) 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 \(0.422055\pi\)
\(422\) 8830.82i 1.01867i
\(423\) − 5245.15i − 0.602902i
\(424\) −3203.65 −0.366941
\(425\) 0 0
\(426\) −5653.79 −0.643021
\(427\) − 2469.56i − 0.279884i
\(428\) 13068.0i 1.47585i
\(429\) −6361.05 −0.715884
\(430\) 0 0
\(431\) −9108.41 −1.01795 −0.508975 0.860781i \(-0.669976\pi\)
−0.508975 + 0.860781i \(0.669976\pi\)
\(432\) − 1969.41i − 0.219336i
\(433\) − 16847.0i − 1.86979i −0.354930 0.934893i \(-0.615495\pi\)
0.354930 0.934893i \(-0.384505\pi\)
\(434\) 243.558 0.0269381
\(435\) 0 0
\(436\) 3629.16 0.398635
\(437\) − 9714.03i − 1.06335i
\(438\) 13377.5i 1.45937i
\(439\) −8434.14 −0.916946 −0.458473 0.888708i \(-0.651603\pi\)
−0.458473 + 0.888708i \(0.651603\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 11308.0i 1.21689i
\(443\) − 4298.49i − 0.461010i −0.973071 0.230505i \(-0.925962\pi\)
0.973071 0.230505i \(-0.0740378\pi\)
\(444\) −5046.09 −0.539362
\(445\) 0 0
\(446\) 12356.7 1.31189
\(447\) 58.2139i 0.00615978i
\(448\) 2296.48i 0.242184i
\(449\) −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
\(452\) 1543.21i 0.160590i
\(453\) 7049.40i 0.731147i
\(454\) −2441.85 −0.252426
\(455\) 0 0
\(456\) 1438.59 0.147737
\(457\) − 11952.4i − 1.22344i −0.791075 0.611719i \(-0.790478\pi\)
0.791075 0.611719i \(-0.209522\pi\)
\(458\) − 2085.63i − 0.212784i
\(459\) −1826.74 −0.185762
\(460\) 0 0
\(461\) −17200.9 −1.73780 −0.868900 0.494988i \(-0.835173\pi\)
−0.868900 + 0.494988i \(0.835173\pi\)
\(462\) 3904.76i 0.393217i
\(463\) − 10368.7i − 1.04076i −0.853934 0.520381i \(-0.825790\pi\)
0.853934 0.520381i \(-0.174210\pi\)
\(464\) −4284.20 −0.428640
\(465\) 0 0
\(466\) 22007.9 2.18776
\(467\) 16879.5i 1.67257i 0.548296 + 0.836284i \(0.315277\pi\)
−0.548296 + 0.836284i \(0.684723\pi\)
\(468\) − 2615.56i − 0.258342i
\(469\) −5155.64 −0.507602
\(470\) 0 0
\(471\) 11695.4 1.14415
\(472\) 2699.62i 0.263263i
\(473\) − 4499.27i − 0.437371i
\(474\) 10017.4 0.970706
\(475\) 0 0
\(476\) 3152.67 0.303577
\(477\) − 5607.17i − 0.538228i
\(478\) 10252.9i 0.981082i
\(479\) −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
\(482\) − 8430.80i − 0.796706i
\(483\) − 2187.50i − 0.206076i
\(484\) −6842.60 −0.642619
\(485\) 0 0
\(486\) 930.308 0.0868305
\(487\) 17209.5i 1.60131i 0.599125 + 0.800655i \(0.295515\pi\)
−0.599125 + 0.800655i \(0.704485\pi\)
\(488\) − 1814.11i − 0.168281i
\(489\) 4582.61 0.423789
\(490\) 0 0
\(491\) 11392.5 1.04712 0.523560 0.851989i \(-0.324604\pi\)
0.523560 + 0.851989i \(0.324604\pi\)
\(492\) 5517.35i 0.505572i
\(493\) 3973.83i 0.363027i
\(494\) 15586.3 1.41956
\(495\) 0 0
\(496\) 662.912 0.0600113
\(497\) − 3445.85i − 0.311001i
\(498\) − 6082.47i − 0.547313i
\(499\) −19079.4 −1.71164 −0.855822 0.517271i \(-0.826948\pi\)
−0.855822 + 0.517271i \(0.826948\pi\)
\(500\) 0 0
\(501\) 2995.55 0.267129
\(502\) − 21886.7i − 1.94592i
\(503\) − 13499.4i − 1.19663i −0.801259 0.598317i \(-0.795836\pi\)
0.801259 0.598317i \(-0.204164\pi\)
\(504\) 323.955 0.0286311
\(505\) 0 0
\(506\) 19368.8 1.70168
\(507\) − 873.237i − 0.0764928i
\(508\) 15796.0i 1.37960i
\(509\) 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
\(512\) 14367.6i 1.24017i
\(513\) 2517.88i 0.216700i
\(514\) 18086.6 1.55207
\(515\) 0 0
\(516\) 1850.02 0.157835
\(517\) 28305.5i 2.40788i
\(518\) − 6771.47i − 0.574365i
\(519\) 2055.76 0.173869
\(520\) 0 0
\(521\) 19395.7 1.63098 0.815490 0.578771i \(-0.196468\pi\)
0.815490 + 0.578771i \(0.196468\pi\)
\(522\) − 2023.77i − 0.169689i
\(523\) 20413.8i 1.70675i 0.521294 + 0.853377i \(0.325450\pi\)
−0.521294 + 0.853377i \(0.674550\pi\)
\(524\) −7988.47 −0.665989
\(525\) 0 0
\(526\) −22876.2 −1.89629
\(527\) − 614.887i − 0.0508252i
\(528\) 10627.9i 0.875987i
\(529\) 1316.34 0.108189
\(530\) 0 0
\(531\) −4724.99 −0.386153
\(532\) − 4345.49i − 0.354137i
\(533\) − 12061.3i − 0.980171i
\(534\) 4424.32 0.358537
\(535\) 0 0
\(536\) −3787.28 −0.305197
\(537\) 3076.75i 0.247247i
\(538\) − 17174.8i − 1.37631i
\(539\) −2379.86 −0.190181
\(540\) 0 0
\(541\) −4919.18 −0.390928 −0.195464 0.980711i \(-0.562621\pi\)
−0.195464 + 0.980711i \(0.562621\pi\)
\(542\) 14654.0i 1.16133i
\(543\) − 8698.19i − 0.687431i
\(544\) 16110.0 1.26969
\(545\) 0 0
\(546\) 3509.88 0.275108
\(547\) − 15334.2i − 1.19862i −0.800518 0.599308i \(-0.795442\pi\)
0.800518 0.599308i \(-0.204558\pi\)
\(548\) 18514.4i 1.44324i
\(549\) 3175.15 0.246834
\(550\) 0 0
\(551\) 5477.33 0.423488
\(552\) − 1606.92i − 0.123904i
\(553\) 6105.37i 0.469487i
\(554\) −13095.3 −1.00427
\(555\) 0 0
\(556\) −8291.81 −0.632466
\(557\) 8613.78i 0.655256i 0.944807 + 0.327628i \(0.106249\pi\)
−0.944807 + 0.327628i \(0.893751\pi\)
\(558\) 313.145i 0.0237572i
\(559\) −4044.26 −0.306000
\(560\) 0 0
\(561\) 9857.98 0.741897
\(562\) 20045.3i 1.50456i
\(563\) − 2320.81i − 0.173731i −0.996220 0.0868654i \(-0.972315\pi\)
0.996220 0.0868654i \(-0.0276850\pi\)
\(564\) −11638.7 −0.868934
\(565\) 0 0
\(566\) 26744.9 1.98617
\(567\) 567.000i 0.0419961i
\(568\) − 2531.29i − 0.186990i
\(569\) 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 \(0.160375\pi\)
\(572\) 14114.9i 1.03177i
\(573\) − 3222.55i − 0.234946i
\(574\) −7403.87 −0.538382
\(575\) 0 0
\(576\) −2952.61 −0.213586
\(577\) − 8029.26i − 0.579311i −0.957131 0.289655i \(-0.906459\pi\)
0.957131 0.289655i \(-0.0935407\pi\)
\(578\) 1284.63i 0.0924455i
\(579\) 2697.00 0.193581
\(580\) 0 0
\(581\) 3707.12 0.264711
\(582\) 5327.93i 0.379467i
\(583\) 30259.1i 2.14958i
\(584\) −5989.32 −0.424383
\(585\) 0 0
\(586\) −28329.2 −1.99705
\(587\) 8015.14i 0.563578i 0.959476 + 0.281789i \(0.0909278\pi\)
−0.959476 + 0.281789i \(0.909072\pi\)
\(588\) − 978.558i − 0.0686310i
\(589\) −847.529 −0.0592900
\(590\) 0 0
\(591\) −9190.88 −0.639700
\(592\) − 18430.5i − 1.27954i
\(593\) 12820.3i 0.887801i 0.896076 + 0.443901i \(0.146406\pi\)
−0.896076 + 0.443901i \(0.853594\pi\)
\(594\) −5020.41 −0.346784
\(595\) 0 0
\(596\) 129.174 0.00887779
\(597\) − 2848.57i − 0.195283i
\(598\) − 17410.1i − 1.19055i
\(599\) 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 \(0.498837\pi\)
\(602\) 2482.59i 0.168078i
\(603\) − 6628.68i − 0.447663i
\(604\) 15642.3 1.05377
\(605\) 0 0
\(606\) −4969.97 −0.333154
\(607\) 8213.06i 0.549189i 0.961560 + 0.274595i \(0.0885436\pi\)
−0.961560 + 0.274595i \(0.911456\pi\)
\(608\) − 22205.2i − 1.48115i
\(609\) 1233.44 0.0820712
\(610\) 0 0
\(611\) 25443.0 1.68463
\(612\) 4053.44i 0.267729i
\(613\) 1242.60i 0.0818732i 0.999162 + 0.0409366i \(0.0130342\pi\)
−0.999162 + 0.0409366i \(0.986966\pi\)
\(614\) 10216.7 0.671519
\(615\) 0 0
\(616\) −1748.22 −0.114347
\(617\) 13170.6i 0.859367i 0.902980 + 0.429683i \(0.141375\pi\)
−0.902980 + 0.429683i \(0.858625\pi\)
\(618\) 5887.65i 0.383230i
\(619\) −19774.1 −1.28399 −0.641993 0.766711i \(-0.721892\pi\)
−0.641993 + 0.766711i \(0.721892\pi\)
\(620\) 0 0
\(621\) 2812.50 0.181742
\(622\) 23695.1i 1.52747i
\(623\) 2696.51i 0.173409i
\(624\) 9553.14 0.612871
\(625\) 0 0
\(626\) 11185.1 0.714132
\(627\) − 13587.8i − 0.865459i
\(628\) − 25951.5i − 1.64901i
\(629\) −17095.3 −1.08368
\(630\) 0 0
\(631\) −14308.8 −0.902735 −0.451367 0.892338i \(-0.649064\pi\)
−0.451367 + 0.892338i \(0.649064\pi\)
\(632\) 4484.95i 0.282281i
\(633\) − 6919.93i − 0.434507i
\(634\) 37616.4 2.35637
\(635\) 0 0
\(636\) −12442.0 −0.775722
\(637\) 2139.19i 0.133058i
\(638\) 10921.3i 0.677707i
\(639\) 4430.38 0.274277
\(640\) 0 0
\(641\) 11537.5 0.710925 0.355463 0.934691i \(-0.384323\pi\)
0.355463 + 0.934691i \(0.384323\pi\)
\(642\) − 22546.6i − 1.38605i
\(643\) 19603.0i 1.20228i 0.799144 + 0.601139i \(0.205286\pi\)
−0.799144 + 0.601139i \(0.794714\pi\)
\(644\) −4853.95 −0.297007
\(645\) 0 0
\(646\) −24154.8 −1.47114
\(647\) − 21650.0i − 1.31553i −0.753223 0.657765i \(-0.771502\pi\)
0.753223 0.657765i \(-0.228498\pi\)
\(648\) 416.513i 0.0252503i
\(649\) 25498.4 1.54222
\(650\) 0 0
\(651\) −190.855 −0.0114903
\(652\) − 10168.6i − 0.610787i
\(653\) − 2927.33i − 0.175429i −0.996146 0.0877145i \(-0.972044\pi\)
0.996146 0.0877145i \(-0.0279563\pi\)
\(654\) −6261.50 −0.374379
\(655\) 0 0
\(656\) −20151.7 −1.19938
\(657\) − 10482.8i − 0.622484i
\(658\) − 15618.3i − 0.925326i
\(659\) 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 \(0.877695\pi\)
\(662\) − 35444.2i − 2.08093i
\(663\) − 8861.06i − 0.519057i
\(664\) 2723.21 0.159158
\(665\) 0 0
\(666\) 8706.17 0.506542
\(667\) − 6118.23i − 0.355170i
\(668\) − 6646.99i − 0.385000i
\(669\) −9682.82 −0.559581
\(670\) 0 0
\(671\) −17134.7 −0.985808
\(672\) − 5000.37i − 0.287044i
\(673\) − 8992.15i − 0.515040i −0.966273 0.257520i \(-0.917095\pi\)
0.966273 0.257520i \(-0.0829054\pi\)
\(674\) −14142.1 −0.808211
\(675\) 0 0
\(676\) −1937.67 −0.110245
\(677\) − 19340.8i − 1.09797i −0.835832 0.548985i \(-0.815014\pi\)
0.835832 0.548985i \(-0.184986\pi\)
\(678\) − 2662.56i − 0.150818i
\(679\) −3247.25 −0.183531
\(680\) 0 0
\(681\) 1913.46 0.107671
\(682\) − 1689.89i − 0.0948816i
\(683\) − 4255.14i − 0.238387i −0.992871 0.119194i \(-0.961969\pi\)
0.992871 0.119194i \(-0.0380309\pi\)
\(684\) 5587.05 0.312319
\(685\) 0 0
\(686\) 1313.15 0.0730850
\(687\) 1634.32i 0.0907616i
\(688\) 6757.08i 0.374435i
\(689\) 27199.1 1.50392
\(690\) 0 0
\(691\) −17505.5 −0.963733 −0.481867 0.876245i \(-0.660041\pi\)
−0.481867 + 0.876245i \(0.660041\pi\)
\(692\) − 4561.63i − 0.250588i
\(693\) − 3059.82i − 0.167724i
\(694\) 14673.7 0.802603
\(695\) 0 0
\(696\) 906.071 0.0493456
\(697\) 18691.8i 1.01579i
\(698\) − 31872.5i − 1.72836i
\(699\) −17245.6 −0.933175
\(700\) 0 0
\(701\) −3240.77 −0.174611 −0.0873054 0.996182i \(-0.527826\pi\)
−0.0873054 + 0.996182i \(0.527826\pi\)
\(702\) 4512.70i 0.242622i
\(703\) 23563.3i 1.26416i
\(704\) 15933.8 0.853022
\(705\) 0 0
\(706\) 34423.4 1.83504
\(707\) − 3029.07i − 0.161132i
\(708\) 10484.5i 0.556543i
\(709\) −19949.3 −1.05672 −0.528358 0.849022i \(-0.677192\pi\)
−0.528358 + 0.849022i \(0.677192\pi\)
\(710\) 0 0
\(711\) −7849.76 −0.414049
\(712\) 1980.83i 0.104262i
\(713\) 946.698i 0.0497253i
\(714\) −5439.41 −0.285105
\(715\) 0 0
\(716\) 6827.17 0.356345
\(717\) − 8034.30i − 0.418475i
\(718\) 49362.9i 2.56575i
\(719\) 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
\(722\) 7034.60i 0.362605i
\(723\) 6606.47i 0.339830i
\(724\) −19300.9 −0.990761
\(725\) 0 0
\(726\) 11805.8 0.603516
\(727\) 12228.5i 0.623840i 0.950108 + 0.311920i \(0.100972\pi\)
−0.950108 + 0.311920i \(0.899028\pi\)
\(728\) 1571.43i 0.0800013i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 6267.56 0.317119
\(732\) − 7045.49i − 0.355750i
\(733\) − 26635.1i − 1.34214i −0.741392 0.671072i \(-0.765834\pi\)
0.741392 0.671072i \(-0.234166\pi\)
\(734\) −28637.3 −1.44008
\(735\) 0 0
\(736\) −24803.4 −1.24221
\(737\) 35771.7i 1.78788i
\(738\) − 9519.26i − 0.474809i
\(739\) 6074.00 0.302349 0.151174 0.988507i \(-0.451694\pi\)
0.151174 + 0.988507i \(0.451694\pi\)
\(740\) 0 0
\(741\) −12213.6 −0.605505
\(742\) − 16696.3i − 0.826065i
\(743\) − 4016.87i − 0.198337i −0.995071 0.0991686i \(-0.968382\pi\)
0.995071 0.0991686i \(-0.0316183\pi\)
\(744\) −140.200 −0.00690858
\(745\) 0 0
\(746\) 13488.8 0.662010
\(747\) 4766.29i 0.233453i
\(748\) − 21874.4i − 1.06926i
\(749\) 13741.6 0.670370
\(750\) 0 0
\(751\) −23913.2 −1.16192 −0.580962 0.813931i \(-0.697324\pi\)
−0.580962 + 0.813931i \(0.697324\pi\)
\(752\) − 42509.6i − 2.06139i
\(753\) 17150.7i 0.830022i
\(754\) 9816.81 0.474147
\(755\) 0 0
\(756\) 1258.15 0.0605269
\(757\) − 31044.9i − 1.49055i −0.666758 0.745275i \(-0.732318\pi\)
0.666758 0.745275i \(-0.267682\pi\)
\(758\) − 51742.9i − 2.47940i
\(759\) −15177.6 −0.725842
\(760\) 0 0
\(761\) 14011.8 0.667446 0.333723 0.942671i \(-0.391695\pi\)
0.333723 + 0.942671i \(0.391695\pi\)
\(762\) − 27253.4i − 1.29565i
\(763\) − 3816.23i − 0.181071i
\(764\) −7150.69 −0.338616
\(765\) 0 0
\(766\) 2515.97 0.118676
\(767\) − 22919.8i − 1.07899i
\(768\) − 15326.6i − 0.720120i
\(769\) −3342.49 −0.156740 −0.0783701 0.996924i \(-0.524972\pi\)
−0.0783701 + 0.996924i \(0.524972\pi\)
\(770\) 0 0
\(771\) −14172.9 −0.662027
\(772\) − 5984.51i − 0.278999i
\(773\) − 21074.6i − 0.980594i −0.871555 0.490297i \(-0.836888\pi\)
0.871555 0.490297i \(-0.163112\pi\)
\(774\) −3191.90 −0.148231
\(775\) 0 0
\(776\) −2385.40 −0.110349
\(777\) 5306.20i 0.244992i
\(778\) 37296.0i 1.71867i
\(779\) 25763.9 1.18496
\(780\) 0 0
\(781\) −23908.5 −1.09541
\(782\) 26981.1i 1.23382i
\(783\) 1585.85i 0.0723800i
\(784\) 3574.12 0.162815
\(785\) 0 0
\(786\) 13782.8 0.625464
\(787\) − 21394.8i − 0.969048i −0.874778 0.484524i \(-0.838993\pi\)
0.874778 0.484524i \(-0.161007\pi\)
\(788\) 20394.1i 0.921968i
\(789\) 17926.1 0.808853
\(790\) 0 0
\(791\) 1622.76 0.0729442
\(792\) − 2247.71i − 0.100845i
\(793\) 15401.9i 0.689706i
\(794\) 16873.5 0.754179
\(795\) 0 0
\(796\) −6320.83 −0.281452
\(797\) − 20645.0i − 0.917547i −0.888553 0.458773i \(-0.848289\pi\)
0.888553 0.458773i \(-0.151711\pi\)
\(798\) 7497.41i 0.332588i
\(799\) −39430.0 −1.74585
\(800\) 0 0
\(801\) −3466.95 −0.152932
\(802\) − 44293.0i − 1.95017i
\(803\) 56570.4i 2.48608i
\(804\) −14708.7 −0.645194
\(805\) 0 0
\(806\) −1518.99 −0.0663825
\(807\) 13458.3i 0.587058i
\(808\) − 2225.13i − 0.0968810i
\(809\) 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 \(0.335450\pi\)
\(812\) − 2736.94i − 0.118285i
\(813\) − 11483.0i − 0.495360i
\(814\) −46982.9 −2.02303
\(815\) 0 0
\(816\) −14804.9 −0.635141
\(817\) − 8638.90i − 0.369935i
\(818\) 11803.2i 0.504508i
\(819\) −2750.38 −0.117346
\(820\) 0 0
\(821\) −5700.22 −0.242313 −0.121157 0.992633i \(-0.538660\pi\)
−0.121157 + 0.992633i \(0.538660\pi\)
\(822\) − 31943.5i − 1.35542i
\(823\) − 32438.0i − 1.37390i −0.726707 0.686948i \(-0.758950\pi\)
0.726707 0.686948i \(-0.241050\pi\)
\(824\) −2635.99 −0.111443
\(825\) 0 0
\(826\) −14069.4 −0.592662
\(827\) 12762.6i 0.536638i 0.963330 + 0.268319i \(0.0864681\pi\)
−0.963330 + 0.268319i \(0.913532\pi\)
\(828\) − 6240.79i − 0.261935i
\(829\) 30766.7 1.28899 0.644494 0.764609i \(-0.277068\pi\)
0.644494 + 0.764609i \(0.277068\pi\)
\(830\) 0 0
\(831\) 10261.6 0.428365
\(832\) − 14322.4i − 0.596804i
\(833\) − 3315.19i − 0.137892i
\(834\) 14306.1 0.593981
\(835\) 0 0
\(836\) −30150.6 −1.24734
\(837\) − 245.384i − 0.0101335i
\(838\) 20731.7i 0.854613i
\(839\) 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
\(842\) 16034.8i 0.656288i
\(843\) − 15707.8i − 0.641760i
\(844\) −15355.0 −0.626233
\(845\) 0 0
\(846\) 20080.7 0.816061
\(847\) 7195.32i 0.291894i
\(848\) − 45443.7i − 1.84026i
\(849\) −20957.6 −0.847190
\(850\) 0 0
\(851\) 26320.4 1.06023
\(852\) − 9830.79i − 0.395302i
\(853\) 24201.5i 0.971445i 0.874113 + 0.485723i \(0.161444\pi\)
−0.874113 + 0.485723i \(0.838556\pi\)
\(854\) 9454.52 0.378837
\(855\) 0 0
\(856\) 10094.5 0.403062
\(857\) − 21036.7i − 0.838507i −0.907869 0.419254i \(-0.862292\pi\)
0.907869 0.419254i \(-0.137708\pi\)
\(858\) − 24352.8i − 0.968988i
\(859\) 6179.19 0.245438 0.122719 0.992441i \(-0.460839\pi\)
0.122719 + 0.992441i \(0.460839\pi\)
\(860\) 0 0
\(861\) 5801.76 0.229644
\(862\) − 34870.9i − 1.37785i
\(863\) 50256.2i 1.98232i 0.132671 + 0.991160i \(0.457644\pi\)
−0.132671 + 0.991160i \(0.542356\pi\)
\(864\) 6429.04 0.253149
\(865\) 0 0
\(866\) 64497.7 2.53085
\(867\) − 1006.65i − 0.0394321i
\(868\) 423.497i 0.0165604i
\(869\) 42361.2 1.65363
\(870\) 0 0
\(871\) 32154.1 1.25086
\(872\) − 2803.37i − 0.108869i
\(873\) − 4175.03i − 0.161860i
\(874\) 37189.5 1.43930
\(875\) 0 0
\(876\) −23260.8 −0.897156
\(877\) 9175.95i 0.353306i 0.984273 + 0.176653i \(0.0565271\pi\)
−0.984273 + 0.176653i \(0.943473\pi\)
\(878\) − 32289.5i − 1.24114i
\(879\) 22199.1 0.851828
\(880\) 0 0
\(881\) −26172.7 −1.00089 −0.500444 0.865769i \(-0.666830\pi\)
−0.500444 + 0.865769i \(0.666830\pi\)
\(882\) 1688.34i 0.0644549i
\(883\) 18615.5i 0.709471i 0.934967 + 0.354736i \(0.115429\pi\)
−0.934967 + 0.354736i \(0.884571\pi\)
\(884\) −19662.3 −0.748092
\(885\) 0 0
\(886\) 16456.4 0.624001
\(887\) 12837.8i 0.485964i 0.970031 + 0.242982i \(0.0781256\pi\)
−0.970031 + 0.242982i \(0.921874\pi\)
\(888\) 3897.89i 0.147302i
\(889\) 16610.3 0.626649
\(890\) 0 0
\(891\) 3934.05 0.147919
\(892\) 21485.7i 0.806496i
\(893\) 54348.4i 2.03662i
\(894\) −222.868 −0.00833759
\(895\) 0 0
\(896\) 4542.42 0.169366
\(897\) 13642.7i 0.507824i
\(898\) − 40370.6i − 1.50020i
\(899\) −533.803 −0.0198035
\(900\) 0 0
\(901\) −42151.5 −1.55857
\(902\) 51370.7i 1.89630i
\(903\) − 1945.39i − 0.0716926i
\(904\) 1192.07 0.0438579
\(905\) 0 0
\(906\) −26988.1 −0.989647
\(907\) 26766.1i 0.979885i 0.871755 + 0.489942i \(0.162982\pi\)
−0.871755 + 0.489942i \(0.837018\pi\)
\(908\) − 4245.87i − 0.155181i
\(909\) 3894.52 0.142105
\(910\) 0 0
\(911\) 5022.67 0.182666 0.0913328 0.995820i \(-0.470887\pi\)
0.0913328 + 0.995820i \(0.470887\pi\)
\(912\) 20406.3i 0.740923i
\(913\) − 25721.3i − 0.932367i
\(914\) 45759.0 1.65599
\(915\) 0 0
\(916\) 3626.48 0.130810
\(917\) 8400.26i 0.302509i
\(918\) − 6993.52i − 0.251439i
\(919\) −9541.79 −0.342497 −0.171248 0.985228i \(-0.554780\pi\)
−0.171248 + 0.985228i \(0.554780\pi\)
\(920\) 0 0
\(921\) −8005.93 −0.286432
\(922\) − 65852.4i − 2.35220i
\(923\) 21490.7i 0.766387i
\(924\) −6789.59 −0.241733
\(925\) 0 0
\(926\) 39695.7 1.40873
\(927\) − 4613.63i − 0.163464i
\(928\) − 13985.6i − 0.494718i
\(929\) 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
\(932\) 38267.2i 1.34494i
\(933\) − 18567.7i − 0.651533i
\(934\) −64621.9 −2.26391
\(935\) 0 0
\(936\) −2020.41 −0.0705545
\(937\) 33608.3i 1.17176i 0.810399 + 0.585878i \(0.199250\pi\)
−0.810399 + 0.585878i \(0.800750\pi\)
\(938\) − 19738.0i − 0.687066i
\(939\) −8764.77 −0.304609
\(940\) 0 0
\(941\) −19173.6 −0.664232 −0.332116 0.943239i \(-0.607762\pi\)
−0.332116 + 0.943239i \(0.607762\pi\)
\(942\) 44774.9i 1.54867i
\(943\) − 28778.5i − 0.993804i
\(944\) −38294.0 −1.32030
\(945\) 0 0
\(946\) 17225.1 0.592005
\(947\) 979.315i 0.0336045i 0.999859 + 0.0168023i \(0.00534857\pi\)
−0.999859 + 0.0168023i \(0.994651\pi\)
\(948\) 17418.2i 0.596749i
\(949\) 50849.5 1.73935
\(950\) 0 0
\(951\) −29476.7 −1.00510
\(952\) − 2435.31i − 0.0829083i
\(953\) 3048.61i 0.103624i 0.998657 + 0.0518122i \(0.0164997\pi\)
−0.998657 + 0.0518122i \(0.983500\pi\)
\(954\) 21466.7 0.728521
\(955\) 0 0
\(956\) −17827.7 −0.603127
\(957\) − 8558.03i − 0.289072i
\(958\) − 28059.0i − 0.946290i
\(959\) 19468.8 0.655557
\(960\) 0 0
\(961\) −29708.4 −0.997227
\(962\) 42231.6i 1.41538i
\(963\) 17667.8i 0.591211i
\(964\) 14659.4 0.489781
\(965\) 0 0
\(966\) 8374.67 0.278934
\(967\) 14467.9i 0.481133i 0.970633 + 0.240567i \(0.0773332\pi\)
−0.970633 + 0.240567i \(0.922667\pi\)
\(968\) 5285.62i 0.175502i
\(969\) 18928.0 0.627507
\(970\) 0 0
\(971\) 12952.8 0.428090 0.214045 0.976824i \(-0.431336\pi\)
0.214045 + 0.976824i \(0.431336\pi\)
\(972\) 1617.62i 0.0533797i
\(973\) 8719.23i 0.287282i
\(974\) −65885.4 −2.16746
\(975\) 0 0
\(976\) 25733.2 0.843954
\(977\) − 47244.2i − 1.54706i −0.633760 0.773529i \(-0.718489\pi\)
0.633760 0.773529i \(-0.281511\pi\)
\(978\) 17544.2i 0.573621i
\(979\) 18709.4 0.610781
\(980\) 0 0
\(981\) 4906.58 0.159689
\(982\) 43615.3i 1.41733i
\(983\) − 1536.84i − 0.0498651i −0.999689 0.0249326i \(-0.992063\pi\)
0.999689 0.0249326i \(-0.00793711\pi\)
\(984\) 4261.92 0.138074
\(985\) 0 0
\(986\) −15213.5 −0.491376
\(987\) 12238.7i 0.394692i
\(988\) 27101.5i 0.872685i
\(989\) −9649.73 −0.310256
\(990\) 0 0
\(991\) 3785.22 0.121334 0.0606668 0.998158i \(-0.480677\pi\)
0.0606668 + 0.998158i \(0.480677\pi\)
\(992\) 2164.04i 0.0692625i
\(993\) 27774.5i 0.887610i
\(994\) 13192.2 0.420956
\(995\) 0 0
\(996\) 10576.2 0.336465
\(997\) 25894.9i 0.822566i 0.911508 + 0.411283i \(0.134919\pi\)
−0.911508 + 0.411283i \(0.865081\pi\)
\(998\) − 73044.0i − 2.31680i
\(999\) −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.d.l.274.4 4
5.2 odd 4 105.4.a.e.1.1 2
5.3 odd 4 525.4.a.l.1.2 2
5.4 even 2 inner 525.4.d.l.274.1 4
15.2 even 4 315.4.a.k.1.2 2
15.8 even 4 1575.4.a.q.1.1 2
20.7 even 4 1680.4.a.bo.1.1 2
35.27 even 4 735.4.a.o.1.1 2
105.62 odd 4 2205.4.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.e.1.1 2 5.2 odd 4
315.4.a.k.1.2 2 15.2 even 4
525.4.a.l.1.2 2 5.3 odd 4
525.4.d.l.274.1 4 5.4 even 2 inner
525.4.d.l.274.4 4 1.1 even 1 trivial
735.4.a.o.1.1 2 35.27 even 4
1575.4.a.q.1.1 2 15.8 even 4
1680.4.a.bo.1.1 2 20.7 even 4
2205.4.a.bb.1.2 2 105.62 odd 4